text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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{"url":"https:\/\/www.physicsforums.com\/threads\/canonical-transformation-for-harmonic-oscillator.643173\/","text":"# Canonical transformation for Harmonic oscillator\n\n1. Oct 11, 2012\n\n### aaaa202\n\nFind under what conditions the transformation from (x,p) to (Q,P) is canonical when the transformation equations are:\nQ = ap\/x , P=bx2\nAnd apply the transformation to the harmonic oscillator.\nI did the first part and found a = -1\/2b\nI am unsure about the next part tho:\nWe have the hamiltonian in (x,p):\nH = 1\/2m(p2+(m$\\omega$x)2 , $\\omega$2=k\/m\nSo transforming to (Q,P) whilst setting am=-1 you get the nice equation:\nH = P(Q2+$\\omega$2)\nShould I now use hamiltons equation in the new coordinate basis, i.e.;\ndH\/dP = Q2+$\\omega$2 = dQ\/dt\ndH\/dQ = 2QP = -dP\/dt\nAnd solve these differential equations for Q,P and transform back? I think so but these equations are just not very easu. I mean the second one is easy to do by means of substitution but really the first one is just a mess. What is the smartest way to do this? I can't really get the first one solved.\n\n2. Oct 14, 2012\n\n### gabbagabbahey\n\nDo you mean $a=-\\frac{1}{2b}$, or $a=-\\frac{1}{2}b$? Brackets are important!\n\nAgain, brackets are needed if you meant $H=\\frac{1}{2m}\\left(p^2+(m\\omega x)^2\\right)$\n\n(1) Why are you setting am=-1?\n(2) You don't get that when you set am=-1\n\nWithout setting am=-1, you should get $H=-\\frac{1}{4m}PQ^2 -am\\omega^2P$\n\nThe second DE is only easy to solve once you know Q. The first one is seperable, so al that is needed is some striaghtforward integration... try a trig substitution if you are stuck on the integration part","date":"2018-02-20 12:01:43","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\": 0.8953652381896973, \"perplexity\": 1410.8916077012195}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-09\/segments\/1518891812938.85\/warc\/CC-MAIN-20180220110011-20180220130011-00330.warc.gz\"}"} | null | null |
Церковь Сошествия Святого Духа (Духосошественская церковь) — приходская церковь Русской православной церкви в посёлке Лесные Поляны в Подольском районе (с 2015 года — городской округ Подольск) Московской области.
Храм расположен на территории усадьбы Воробьёво, являющейся памятником культурного наследия Российской Федерации.
История
В 1845 году княгиня Елизавета Ростиславовна Вяземская — мать княжны Варвары Вяземской (Ершовой), проживавшая в те годы у своей дочери, обратилась к митрополиту Московскому Филарету (Дроздову) с просьбой:
…беру смелость прибегнуть к Вашему Высокопреосвященству и просить архипастырского Вашего соизволения и благословения дозволить мне устроить при доме дочери моей в упомянутом сельце Воробьеве домовую церковь во имя Сошествия Св. Духа. При сем нужным признаю изъяснить, что хотя дом, в котором предполагаю устроить церковь, находится в имении, принадлежащем дочери моей, однако по совершенному семейному согласию устроение для меня в нём церкви и жительство мое в оном не подлежит ни малейшему препятствию или сомнению….
Указом от 22 июля 1845 года Московская духовная консистория разрешила устройство этого домового храма. Домовая Духосошественская церковь была построена в августе 1848 года и освящена 5 сентября 1848 года протоиереем Подольского Троицкого собора Василием Берёзкиным. С весны 1860 года церковь вместо домовой стала именоваться приписной к Успенской церкви.
Авторство проекта храма в русском стиле с колокольней приписывают архитектору Михаилу Быковскому. К квадратному в плане зданию храма примыкают пониженные притвор и трёхчастная апсида. В центре его боковых фасадов — трёхчастные окна в архивольтах. Двери на входе в церковь обрамляет широкая профилированная арка, по бокам которой, в углах притвора расположены массивные кувшинообразные столбы.
После революции 1917 года здание церкви использовалось в качестве водонапорной башни, для чего была разрушена её верхняя часть. Затем здесь были организованы склад, лыжная база и развлекательное заведение. После распада СССР частично разрушенное здание храма было передано верующим. С осени 1993 года в храме возобновилась богослужебная и приходская жизнь; начались восстановительные работы, продолжающиеся по настоящее время под руководством архитекторов Андрея Анисимова и Татьяны Беляевой.
Сейчас богослужения в храме совершаются еженедельно по субботам. Настоятель Духосошественской церкви — протоиерей Сергий Кожемяк.
Источники
Семёнов К. А. Святыни Подмосковья. Подольское благочиние. — Подольск: Артист-Медиа, 2013. — 368 с. — ISBN 978-5-9904473-1-8.
Примечания
Ссылки
Сайт храма
Храм Сошествия Святого Духа пос. Лесные Поляны
Лесные Поляны
Лесные Поляны | {
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Well thanks for making a private message sent respect tfully your way a public one ...have a great day!
OK, so being new to this site and working on multiple issues I am trying to get pictures from a Discovery , and PDF fi!es to download through the messenger service here withing FindLaw ... Basically , I am trying to receive pictures and discovery files from membes of our FindLaw community . He has info and pictures of a research we are doing together and is trying to send them to me .... My internet server is blocked with restrictions due to my situation , however I am able to research and do work on this site (ThanxFindLaw!).... If anyone can help with this I would appreciate the knowledge and information! Chris Nashville , Tn.
I was recently informed about a program being given to offenders as part of sentencing for drug offenders / addicts . The program is in Nashville Tn and the Judge over it is Honorable Norman out of Davidson County. Does anyone have information about this program? The rules , requirements and the impatient/out patient terms to be completed? I don't have access to actually Google or search it online , so if anyone has researched information or online information they can post here that would be great and very helpful. I would also like to know of anyone who may be personally aware of the program , by experience or by working / being a counselor within it. ( so please tell me about it!) Again , it is called DC4 , Judge Norman is over it in Davidson County / Nashville . Thank You!
Send message to cell from Email?
I have a friend who is going through pre trial phases and is trying to present the evidence of a falsified police report that resulted in him being charged. The situation is as follows.... He was invited into a home ,where he knew the residents, ultimately a po!ice report was made which listed items being stolen. However it is on video within the home that items that were taken was not listed and items that were shown taken was not on the report.... This is on camera however and the report and insurance clearly do not match up . The alleged victim has in fact falsified a police report to prevent from certain items being found stolen that were in his possession. Basically , we need to know if a police report is falsified and or misleading is it admissible in court and what should he do?
Waiver of Miranda Rights while intoxicated or under the influence?
I appreciate your opinion on this issue , I guess just because Its not what I wanted to hear I'm not going to give up on the fight of this matter . I have to firmly believe that it was my being intoxicated and under the influence of multiple narcotics is the result of this whole issue of whether my Rights were violated . Don't you think its impossible for someone not OF SOUND MIND to waive anything legally ? Especially your right to self incriminate?
Can you legally waive your Miranda rights if you are under the influence or intoxicated? ......... I am currently in a criminal issue with the court....on the day of arrest the detectives took me to an interview room where multiple questions were asked and vague answers were given , however the DA now is trying to use this interview as a confession and of course I'm trying to fight it . I am represented by a public defender , and doing some of my own research to make sure proceedings are proper and fair toward my defence. I DID NOT sign a waiver of the Miranda rights , however the lead detective has audio of my saying I would talk with him from the scene of arrest .Then about an hour later , when I was taken and placed in the interview room ( where full recordings and video are available) ... The question I was asked was " do you remember your Miranda rights" and I stated" I knew them and was aware of them ".... And at that point questions proceeded and most answers given were very vague and a lot of " I don't remember or i dont knows" but there is some possible incriminating answers that are trying to be used against me. *** yes , on the day of the arrest , right after I left the interview room and was placed in the county jail .... drug and intoxication screens were given for detox issues with all incoming inmates due to medical policy. So there is a record of my being intoxicated or under the influence. Can anyone help me with this ? I need any case sitations or legal insight to present at the argument in a future suppression hearing because obviously I need to suppress the evidence of the alladged confession. Just to answer the obvious questions.... The crime(s) I am charged with are NON VIOLENT PROPERTY OFFENCES.... Thank you for your help and guidance ! | {
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Occurring after the transfigurative rains, the Rainbow represents hope and reward for prolonged sacrifice. Accordingly, the rainbow is the bridge to personal illumination. Consequently, since the rainbow contains all the primary colors, our unconscious may be illustrating the paramount gift available to humanity is the embodiment of the full spectrum of our emotions. Furthermore, since a rainbow is a direct refraction of sunlight, we may interpret the dream rainbow as pertaining to the arrangement of wisdom, guidance and perceptive clarity in our waking life. | {
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India, All India
Alwar lynching: Rajasthan govt sets up probe team
Published : Apr 7, 2017, 6:22 pm IST
Updated : Apr 9, 2017, 7:59 pm IST
The Rajasthan government gave a preliminary report about the incident in which a 55-year-old man, Pehlu Khan, was beaten to death in Alwar.
A mob of cow vigilantes associated with the Bajrang Dal and Vishwa Hindu Parishad (VHP) beat up Khan and 4 others badly. (Photo: Screengrab)
New Delhi: The Union Home Ministry on Friday received a report from the Rajasthan government which said a special police team has been constituted to arrest all accused allegedly involved in the lynching of a Muslim man in Alwar.
The Rajasthan government gave a preliminary report about the incident in which a 55-year-old man, Pehlu Khan, was beaten to death in Alwar district allegedly by a group of cow vigilantes on April 1.
The local police reached the spot soon after receiving the report about the incident, rescued four of Khan's colleagues and took him to a hospital, the report to the Union Home Ministry said.
The Rajasthan government said three people have been arrested so far and a special police team has been constituted to probe the incident and find out the circumstances leading to it.
The police team will also try to arrest the remaining accused, sources said quoting the report.
The incident took place when as many as 16 people were allegedly transporting 36 bovines in six pickup vans.
The deceased, Khan, and four others, including his two sons, were beaten brutally by some locals at Behror in Alwar suspecting they were smuggling cows, police said.
The incident rocked both Houses of Parliament where the Congress attacked the BJP saying the Constitution was being violated in the name of cow protection in the states ruled by the saffron party.
Tags: alwar lynching, rajasthan police, gau rakhshak, pehlu khan, union home ministry
Latest From India
Team of top ministers start J&K outreach tour
VHP to reach out to kar sevaks to begin temple work
NIA takes over case involving DSP Davinder Singh | {
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Q: Screenshot from video at different time I have 3 canvas on a line, and in each I want to put an image (screenshot form a video, at different time). The problem is that all 3 screenshots are at the same time (the last time specified).
Bellow is my JavaScript code.
function getVideoScreenShot(videoFile, currentIncidentTime, idx) {
var images = [];
for(var canvas=1; canvas<=3; canvas++) {
var canvasId = "#canvas"+ idx + canvas;
var milisec = parseInt(currentIncidentTime)%1000;
var secFromMilisec = milisec/1000;
var sec = (parseInt(currentIncidentTime)/1000)%60;
sec = "" + sec + "";
sec = parseInt(sec.substring(0,2));
var min = ((parseInt(currentIncidentTime)/1000)/60)%60;
min = "" + min + "";
min = parseInt(min.substring(0,2));
var secFromMin = min * 60;
var seconds = secFromMilisec+sec+secFromMin;
if (canvas == 1) {
seconds = seconds - 1;
} else if (canvas == 3) {
seconds = seconds + 1;
} else {
seconds = seconds;
}
images.push({canvas: canvasId, time: seconds});
}
console.log(images);
var video = document.createElement('video');
video.addEventListener('loadedmetadata', function ()
{
var ratio = video.videoWidth / video.videoHeight;
var w = video.videoWidth;
var h = w / ratio;
for (var i = 0; i < images.length; i++) {
images[i].canvas = document.querySelector(images[i].canvas);
images[i].canvas.width = w;
images[i].canvas.height = h;
}
draw(video, images);
});
video.src = videoFile;
}
function drawFrame(video, time, canvas) {
var context = canvas.getContext("2d");
video.addEventListener("seeked", function (e) {
e.target.removeEventListener(e.type, arguments.callee);
context.fillRect(0, 0, canvas.width, canvas.height);
context.drawImage(video, 0, 0, canvas.width, canvas.height);
});
console.log(time, "time to draw frame");
video.currentTime = time;
}
function draw(video, images) {
for (var i = 0; i < images.length; i++) {
drawFrame(video, images[i].time, images[i].canvas);
}
}
A: Here's an example letting the user click to select when a video frame is grabbed to the canvas.
Up to 4 frames can be grabbed from the video and displayed on a single canvas.
If you need a timed frame capture you can create a requestAnimationFrame loop and trigger the #snap.click code when your desired elapsed time(s) occur.
JavaScript:
var canvas=document.getElementById("canvas");
var ctx=canvas.getContext("2d");
var cw=canvas.width;
var ch=canvas.height;
var position=0;
var v = document.getElementById('v');
v.addEventListener( "loadedmetadata", function(e){
// resize canvas
cw=canvas.width=v.videoWidth;
ch=canvas.height=v.videoHeight;
// play the video
v.play();
}, false );
$('#snap').click(function(){
var x,y;
if(position==0){ x=0;y=0; }
if(position==1){ x=cw/2;y=0; }
if(position==2){ x=0;y=ch/2; }
if(position==3){ x=cw/2;y=ch/2; }
ctx.clearRect(v,x,y,cw/2,ch/2);
ctx.drawImage(v,x,y,cw/2,ch/2);
position++;
if(position>3){ position=0; }
});
HTML -- you must set the video source to your own .mp4
<button id=snap>Capture</button>
<br>
<h4>Captured Snapshots -- 4 snapshots per canvas</h4>
<canvas id="canvas" width=300 height=300></canvas>
<h4>Video Element below</h4>
<video id=v controls loop>
<source src=yourClip.mp4 type=video/mp4>
</video>
| {
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} | 1,372 |
China: The Roots of Madness is a 1967 Cold War era, made-for-TV documentary film produced by David L. Wolper, written by Pulitzer Prize winning journalist Theodore H. White with production cost funded by a donation from John and Paige Curran. The film has been released under Creative Commons license. It won an Emmy Award in the documentary category.
The film attempts to analyze the anti-Western sentiment in China from the official American perspective, covering of China's political history, from Boxer Rebellion of the Qing Dynasty to Red Guards of Cultural Revolution. The film focuses on the power struggle between the Kuomintang and the Communist Party of China, amid heavy political intervention from Moscow, with Sun Yat-sen, Chiang Kai-shek and Mao Zedong playing the pivotal role at the center stage.
Introduction
The documentary film was made for television in 1967, during the Cold War. It was written by Pulitzer Prize-winning journalist Theodore H. White, directed by Mel Stuart, edited by William T. Cartwright, and produced by David L. Wolper. Production costs were funded by a donation from John and Paige Curran. The film has been released under Creative Commons license.
White's access to important political figures of the time allowed him to create some rare footage, which included the wedding of Chang and the funeral of Sun. The film won an Emmy Award in the documentary category.
As evidenced by his commentary throughout the films, White, Time magazine's China correspondent during World War II, was scathing about the People's Republic of China. Remarking that Chinese had been suffering in a 100-year tragedy, he added:
Synopsis
Episode one
White referred to Empress Dowager Cixi as "China's evil spirit... a Manchu concubine... said to have poisoned her own son upon his throne, install her infant nephew as the emperor, killed his mother, and then imprisoned him in 1898."
Episode two
White's impression on the downfall of Qing Dynasty: "...and then it vanished, simply vanished, the Manchu Dynasty disappeared overnight, nothing like that had ever happen in all the history, 2000 years of tradition, the whole structure of the imperial confucianism, political thought, dissolving to dust...."
On post-Qing China, "out of this turbulence, there appeared two types of Asian leaders, arch symbols, the man of gun, and the man of idea, and these two types of gunman and the dreamer, have perplexed all our efforts in Asia for 50 years since, and they still perplexed and haunted all our policy, even today..."
Sun "was a man of dream, the dream of China, powerful, free of emperors and foreigners, made him from his youth a revolutionary.... Slowly from the early 1920, Sun Yatsen had somehow built a government, a tiny southern foothold at Canton, ringed by hostile warlords. By 1924 the ageing revolutionary had learned, idea and gun must go together... in 1923 he tells the New York Times: We have lost hope of help from America, England, France, the only country that show any sign of helping us in the south is the Soviet government of Russia...."
Episode three
The Kuomintang left wing "no longer trust their army leader at the front. Borodin is urging: 'Get rid of Chiang Kai-shek.' In four short years, the communist had grown 60,000 members. To hear the left wing Nationalist: 'No revolution is completed, until peasants own their land, and workers their factories.' Chiang disagreed."
Episode four
"When Mao left at 1927 with a dozen of his ally, it was as if a ghost had risen from the dead, he was disgusted with Borodin and his Russian documents. He felt the key of revolution in Asia lied in the countryside, not the big city proletariat. Mao's idea was simple, turn the hidden peasant's anger towards the local gentry, the local rich.... Mao transformed the countryside into a total environment of hate, women, children, everyone, not to be afraid to die... by 1932, he controlled a good chunk of Hunan, Jiangxi, claimed the loyalty of 9 million people."
Episode five
"Day and night the bombs continue, yet Chiang persists. Powerless to strike back, Chiang knows, only the Americans can help.... It is about this time 1941, at the height of the bombing, I had my first talk with Chiang Kai-shek, about the war with Japan, and strategy. At the end, almost just an afterthought, he said, "remember, Japanese is a disease of the skin, the communists are the disease of the heart." It seems odd to me, because at that time the Japanese were bombing the daylight out of both Chiang Kai-shek and the communist, both of them are ally against the Japanese. And now in retrospect, almost a vision of the apocalypse to come."
Episode six
On Manchuria after the Japanese surrender, "the Russian hed temporarily occupied Manchuria by the surrender term of Japan. Communist expected to get from Russian surrender Japanese equipment and guns, and hold the countryside before Chiang arrived.... Manchuria is the topic of the struggle. Industries Japan had built and left is the greatest pride in China. Chiang's American equipped troops seize all major cities, to find a hollow triumph. The Russian occupiers had rooted every factories before withdrawn, rip out shops and stores where great machines once stood."
Episode seven
"In American custody in Hong Kong there are cascades and piles of translation coming from the Chinese, these are sands, gritty, gravels, little bit of information, meaningless, because we don't know who does what to who in Peking, we don't know how they think, or how they make up their mind, because no matter how hard we study China, we cannot predict such thing as the Great Leap Forward in 1958, we can't predict such thing as Red Guard purge in 1966, as if there was struggle of sea monsters going on, deep deep beneath the surface of our vision, only bubbles come to the surface, to tell us terrible struggle, we don't know what the struggle is about."
Critical reception
While the film won an Emmy Award in the documentary category soon after its release, contemporary critics have criticised his "callous and condescending" portrayal of Chinese. Film Threat remarked that White never attempted to take on board the Chinese viewpoint, and points out there were unconfirmed rumours that the CIA was involved in the film's making.
See also
List of American films of 1967
Notes
References
Theodore H. White Mel Stuart, China: The Roots of Madness; a Documentary (New York,: Norton, 1968).
American documentary films
Emmy Award-winning programs
Documentary films about China
Documentary films about the Republic of China (1912–1949)
Black-and-white documentary films
1967 documentary films
1967 films
Cold War films
Articles containing video clips
American black-and-white films
1960s English-language films
1960s American films | {
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What is Wiring Diagram? A wiring diagram is a schematic that uses abstract reflective symbols to exhibit all the interconnections of parts in a very system. Wiring diagrams are made up of a couple of items: symbols that signify the ingredients within circuit, and traces that represent the relations between them. Therefore, from wiring diagrams, you understand the relative location of the ingredients and the way they may be linked. It's a terminology engineers need to learn whenever they operate on electronic equipment projects.
How to Read Wiring Diagram? To see a wiring diagram, is in fact a program have to know exactly what basic components are included in an exceedingly wiring diagram, and which pictorial symbols are utilized to reflect them. The typical elements in a wiring diagram include floor, energy, cable and link, output devicesand switches, resistors, logic gate, lights, etc.. A list of symbols and descriptions is available about the"electrical symbol" page.
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Do you have corrections or additions for this page? All are welcomed! Comment below or email [email protected] – Thank you!!
While much has been written and even speculated about Elizabeth Fones Winthrop Feake Hallett, this much is certain – she was a pioneer and survivor, and her courage and indominitable spirit make her amongst the most fascinating and important individuals of the early American landscape. She is also the subject of the recently published historical biography, 'Insubordinate Spirit: A true story of life and loss in earliest America, 1610-1655" by Missy Wolfe. Ms. Wolfe's incredible text is exhaustively researched, and serves not only to document the history of those times, but to correct facts – or establish as unsubstantiated and at times untrue – items from Anya Seton's historical fiction novel, "The Winthrop Woman" that have seeped into the fabric of history and now appear as "facts" on genealogy websites far and wide.
Born: January 21, 1610 Groton Manor, Suffolk England, to Thomas Fones (A London Apothecarist) and Anne Winthrop, the sister of John Winthrop. John Winthrop would go on to become Governor of the Massachusetts Bay Colony, and Elizabeth's connection to him would become an important part of her life – and survival – in America.
Died: Widely published as February 1, 1673, however a reliable citation for this date of death has not been established to my knowledge. | {
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\section{Introduction}\label{sec:introduction}
This work is based on results from the author's undergraduate thesis
\cite{KobayashiThesis}.
The goals of this document are to understand when one can apply
countable sequences of Reidemeister moves and preserve ambient isotopy
in the limit. The motivating examples are the following two curves.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{.49\linewidth}
\centering
\includegraphics[width=.7\linewidth]{figures/countable-r1.pdf}
\caption{A wild-looking unknot.}
\label{subfig:wild-looking-unknot}
\end{subfigure}
\begin{subfigure}[t]{.49\linewidth}
\centering
\includegraphics[width=.84\linewidth]{figures/smooth-fox-artin-labeled.pdf}
\caption{An unknotted-looking wild knot.}
\label{subfig:remarkable-curve}
\end{subfigure}
\caption{Two scintillating curves.}
\label{fig:scintillating-curves}
\end{figure}
\textbf{Tameness of \cref{subfig:wild-looking-unknot}.} As indicated
by the caption, the curve in \cref{subfig:wild-looking-unknot} is
tame. In fact, it is the unknot. An explicit ambient isotopy taking it
to the unknot can be constructed as follows. From time $t=0$ to
$t=\frac{1}{2}$, use a Reidemeister~I move to remove loop $\ms L_1$.
Then, from time $t=\frac{1}{2}$ to $t=\frac{3}{4}$, use a Reidemeister
I move to remove loop $\ms L_2$. So on and so forth, removing loop
$\ms L_n$ between time $t=1 - \frac{1}{2^{n-1}}$ and $t= 1 -
\frac{1}{2^n}$.
If one is careful about exactly how the Reidemeister I moves are
performed, then the result will be an ambient isotopy. We provide the
details in \cref{prop:wild-looking-unknot-tame}.
\textbf{Wildness of \cref{subfig:remarkable-curve}.} By contrast, the
curve in \cref{subfig:remarkable-curve} is wild, a result established
by Ralph Fox in \cite{Fox}. His argument uses techniques co-developed
with Emil Artin
in \cite{Fox-Artin}, namely, a sort of ``invariant'' for tameness of
arcs. We summarize the relevant result in
\cref{sec:tameness-invariant}.
\textbf{The Problem.} At first, the wildness of the curve in
\cref{subfig:remarkable-curve} can appear very counterintuitive. As
with the loops in \cref{subfig:wild-looking-unknot}, any \emph{finite}
number of the ``stitches'' in \cref{subfig:remarkable-curve} can be
safely removed using Reidemeister II moves. The procedure is as
follows. First, use a Reidemeister II move to move $\color{red} \ms
L_{-1}$ into $\color{red} \ms L_1$. Next, use another Reidemeister II
move to remove both $\color{red} \ms L_{-1}$ and $\color{blue} \ms
L_2$.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{.490\linewidth}
\centering
\includegraphics[scale=1.75]{figures/smooth-fox-artin-unstitching-1-labeled.pdf}
\caption{Moving $\color{red} \ms L_{-1}$ into $\color{red} \ms L_1$.}
\end{subfigure}
\begin{subfigure}[t]{.490\linewidth}
\centering
\includegraphics[scale=1.75]{figures/smooth-fox-artin-unstitching-2-labeled.pdf}
\caption{Removing $\color{red} \ms L_{-1}$ and $\color{blue} \ms
L_2$.}
\end{subfigure}
\caption{Removing one stitch.}
\end{figure}
Note, this leaves us with a shrunk copy of the same diagram we started
with. Hence we can repeat the process any finite number of times. In
general, to perform step $n$ we use a Reidemeister II move to slide
loop $\color{red} \ms L_{2n-3}$ into loop $\color{red} \ms L_{2n-1}$,
then we remove $\color{red} \ms L_{2n - 3}$ and $\color{blue} \ms
L_{2n}$ by using another Reidemeister II move.
What's to stop us from using the same approach as in
\cref{subfig:wild-looking-unknot}, where we just performed step $n$
between $t=1-\frac{1}{2^{n-1}}$ and $t=1 - \frac{1}{2^n}$? Indeed, if
we choose our Reidemeister II moves carefully, it's possible to
guarantee continuity of the resulting limit function.
The issue is with bijectivity on the ambient space. As we show in
\cref{sec:does-not-apply}, there is no way to choose our sequence of
Reidemeister II moves without accidentally dragging points from the
ambient space down to the wild point in the limit. The proof is a
simple geometric argument but it can be easy to overlook.
\textbf{The Upshot.} To clarify the situation, in
\cref{thm:vks-ambient-homeomorphism,thm:vks-ambient-isotopy} we
generalize the strategy we described for
\cref{subfig:wild-looking-unknot} and make it rigorous.
\cref{thm:vks-ambient-homeomorphism} uses the language of
\emph{ambient homeomorphisms} while \cref{thm:vks-ambient-isotopy}
uses the language of \emph{ambient isotopies}; other than that, they
are equivalent.
The layout of the rest of the paper is as follows:
\begin{enumerate}[label=(\S\arabic*)] \setcounter{enumi}{1}
\item In \cref{sec:background}, we go over the definitions of knots,
ambient homeomorphisms, ambient isotopies, PL-ness, and
tameness/wildness. It might be helpful to review these concepts
since we will be working directly with equivalence in this paper
instead of by proxy through Reidemeister's Theorem. We also give
the definition of uniform convergence and recall two elementary
results from first courses in Analysis and Topology, respectively;
the main theorems will follow from these.
\item In \cref{sec:uniform-convergence}, we give the definition of
uniform convergence and use it to build up our main results
(\cref{thm:vks-ambient-homeomorphism,thm:vks-ambient-isotopy}). In
most cases \cref{thm:vks-ambient-homeomorphism} tends to be more
ergonomic than \cref{thm:vks-ambient-isotopy}, but we will
generally employ \cref{thm:vks-ambient-isotopy} because it seems
the language of ambient isotopy is more ubiquitous than that of
ambient homeomorphism.
\item In \cref{sec:various-applications}, we apply
\cref{thm:vks-ambient-isotopy} to various curves with
countably-many crossings.
\item In \cref{sec:does-not-apply}, we show what goes wrong if we
try to apply \cref{thm:vks-ambient-isotopy} to two noteworthy
examples; the second is the wild curve from
\cref{subfig:remarkable-curve}.
\end{enumerate}
Finally, some notation.
\begin{table}[H]
\centering
\footnotesize
\begin{tabular}{ccl}
\toprule
Symbol && Interpretation \\ \midrule
$f$ && Embedding and/or knot (usually). \\
\midrule
$h$ && Homeomorphism. \\
$\msf{h}$ && PL Homeomorphism. \\
$\ms h$ && Homeomorphism constructed by composing other
homeomorphisms. \\ \midrule
$H$ && Ambient isotopy. \\
$\msf{H}$ && PL Ambient isotopy. \\
$\ms H$ && Ambient isotopy constructed by iteratively gluing other
ambient isotopies. \\\midrule
$\metspace{X}$ && Metric space $X$ with metric $\metric{X}$. \\
$\topspace{X}$ && Topological space $X$ with topology $\ms T_{X}$.
\\\midrule
$\comp_{k=1}^n f_k$ && The composite function $f_n \circ f_{n-1} \circ
\cdots \circ f_2 \circ f_1$. \\
$k$ && Generally reserved for an ``free'' index (e.g., $k
\in \set{1, \ldots, n}$ above).\\
$n$ && Generally reserved for a ``particular'' index (e.g., $n$ is
the ``stop'' index above).\\\midrule
$\pn{f_k}_{k=1}^\infty$ && A sequence $f_1, f_2, \ldots,
f_k, \ldots$. \\
$f_k \to f$ && The $f_k$ converge to $f$ pointwise. \\
$f_k \uconv f$ && The $f_k$ converge to $f$ uniformly. Note, some
authors use $\rightrightarrows$ instead of
$\uconv$. \\\midrule
$A^\circ$ && Interior of $A$. \\
$\ol{A}$ && Closure of $A$. \\
$A^c$ && $X \setminus A$ (whenever $X$ is understood) \\
$A_1 \setminus A_2$ && Set difference of $A_1$ and $A_2$.\\
$A_1 \sqcup A_2$ && Disjoint union of $A_1$ and $A_2$.\\\midrule
\np{\ldots} && Used to indicate parsing order in
grammatically-ambiguous sentences. \\
$\cmark$ && Indicates the completion of a case in a proof by
casework.\\\bottomrule
\end{tabular}
\end{table}
\section{Background}\label{sec:background}
Today we will be working with knot equivalence directly instead of
making appeals to \emph{Reidemeister's theorem}. This is because we're
interested in knots that might be \emph{wild}
(\cref{def:tame-and-wild}), but Reidemeister's theorem assumes
\emph{tameness} (also \cref{def:tame-and-wild}). We begin with a
reminder of some fundamental definitions.
\subsection{Fundamental Definitions}\label{subsec:fundamental-definitions}
Let $\topspace{X}$, $\topspace{Y}$ be topological spaces. An
\emph{embedding} of $X$ into $Y$ is a function $f \colon X \to Y$ such
that restricting the codomain of $f$ to $f(X)$ gives us a
homeomorphism $\widetilde{f} \colon X \to f(X)$.
\begin{figure}[H]
\centering
\includegraphics[scale=.25]{figures/including-x.pdf}
\caption{Example of embedding an $X$ into $\RR^2$.}
\end{figure}
Since embeddings must be injective, some authors choose to denote them
by $f \colon X \into Y$. Here we call $Y$ the \emph{ambient space} and
refer to $f(X)$ as \emph{$X$ embedded by $f$ in $Y$}.
Two embeddings $f_1$, $f_2 \colon X \to Y$ are said to be \emph{equivalent}
(denoted $f_1 \cong f_2$) if there exists a homeomorphism $h \colon Y \to
Y$ such that for all $x \in X$,
\[
(h \circ f_1)(x) = f_2(x).
\]
Since $h$ is a homeomorphism on the \emph{ambient} space, we refer to
it as an \emph{ambient homeomorphism}.
\begin{remark}
This definition requires pointwise equality for $(h \circ f_1)$,
$f_2$. In general, this is stronger than requiring $h(f_1(X)) =
f_2(X)$ as sets.\footnote{The correspondence holds for \emph{tame}
knots and certain {everywhere-wild} knots. An example of a knot
for which it fails is \cref{subfig:remarkable-curve}. The idea is
that the ambient homeomorphism can only send wild points to other
wild points, and this makes it impossible to pull the strand
``through'' the wild point.} The interested reader should see
\cite{Bothe1981, Shilepsky} for more.
\end{remark}
Geometrically, we think of $h$ as ``deforming'' the ambient space to
take $f_1(X)$ to $f_2(X)$.
\begin{figure}[H]
\centering
\includegraphics[scale=.25]{figures/deformed-x-graphic.pdf}
\caption{An example $h$ taking $f_1(X)$ to a distorted version
representing $f_2(X)$.}
\label{fig:example-homeomorphism}
\end{figure}
A \emph{knot} is an embedding $f \colon S^1 \into \RR^3$. One should note
that some authors take the codomain to be $S^3$ instead of $\RR^3$
because $S^3$ is compact. Our proofs today only require that $Y$ be a
metric space, hence we are free to choose either option. We could even
choose a thickened orientable surface in order to work with virtual
knots. However, we will do neither of these things and instead choose
$Y = \RR^3$ because it is easier to represent graphically.
For embeddings in $\RR^3$, defining equivalence through ambient
homeomorphisms is equivalent to defining equivalence with
\emph{ambient isotopy} (defined below). We refer to this fact as the
\emph{equivalence of equivalences}. For further discussion (as well as
a list of references about the correspondence in each of the
\textbf{PL}, $\mb{C^\infty}$, and \textbf{Topological} categories),
see \cite{KobayashiThesis}, particularly \S 6.3.
\begin{definition}[Ambient Isotopy]
Let $\topspace{X}$, $\topspace{Y}$ be topological spaces. Let $f_1$,
${f_2} \colon X \into Y$ be embeddings. Then a function $H \colon
[0,1] \times Y \to Y$ is called an \emph{ambient isotopy} iff
\begin{enumerate}
\item $H$ is continuous,
\item $H(0, \cdot)$ is the identity on $Y$,
\item For all $t \in [0,1]$, the function $H(t, \cdot) \colon Y
\to Y$ is a homeomorphism, and
\item For all $x \in X$, we have
\[
(H(1, \cdot) \circ {f_1})(x) = {f_2}(x).
\]
\end{enumerate}
We often refer to $H$ as an \emph{ambient isotopy from
${f_1}$ to ${f_2}$}.
\end{definition}
Note that $H(1,\cdot)$ is an ambient homeomorphism from
$
f_1}$ to $
f_2}$. We can think of the $t$ variable as describing a ``time''
parameter in a movie connecting $H(0,\cdot)$ to $H(1,\cdot)$.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/deformed-x-movie.pdf}
\caption{5 freeze-frames from an ambient isotopy where $H(1,\cdot)$
is the $h$ in \cref{fig:example-homeomorphism}}
\end{figure}
We will prove our results today for both ambient homeomorphisms and
ambient isotopies. Although this is not strictly necessary in light of
the equivalence of equivalences, we have chosen to include both
arguments to illustrate the additional step required for working with
ambient isotopy.
\subsection{Tameness and Wildness}
Oftentimes, knot theory is restricted to the study of \emph{tame}
knots, which we define in a moment. We think of tame knots as being
well-behaved because they belong to equivalence classes of knots that
have representative elements that can described with finite
information, namely \emph{polygonal} or \emph{PL knots}.\footnote{PL
stands for Piecewise Linear. For more on PL topology, the reader
might look at J.L.\ Bryant's \emph{Piecewise Linear Topology}. An
online version can be found at
\url{https://www.maths.ed.ac.uk/~v1ranick/papers/pltop.pdf}} This is
the loose intuition underpinning Reidemeister's theorem.
\begin{definition}[Polygonal Knot]
A \emph{polygonal knot} is a knot that is comprised of a finite
union of straight line segments.
\end{definition}
\begin{figure}[H]
\centering
\includegraphics[scale=.4]{figures/polygonal-knots/ex-polyknots.pdf}
\caption{Examples of some polygonal knots}
\end{figure}
\begin{definition}[Tame \& Wild Knots]\label{def:tame-and-wild}
A \emph{tame knot} is a knot that is ambient isotopic to a polygonal
knot. A \emph{wild knot} is a knot that is not tame.
\end{definition}
\begin{figure}[H]
\centering
\includegraphics{figures/trefoil-to-polygonal-trefoil.pdf}
\caption{Example of a tame knot}
\end{figure}
\begin{remark}
There are many other common definitions for tameness and wildness. A
discussion of these definitions (and some of the equivalences) can
be found in \cite{KobayashiThesis}.
\end{remark}
\subsection{Uniform Convergence}
Finally, we recall the definition of \emph{uniform convergence}.
\begin{definition}[Uniform Convergence]
Let $\metspace{X}$, $\metspace{Y}$ be metric spaces, and consider a
sequence of functions $f_n \colon X \to Y$. Suppose that the $f_n$
converge pointwise to some $f \colon X \to Y$. Then we say the $f_n$
\emph{converge to $f$ uniformly} iff for all $\varepsilon > 0$,
there exists $n_0 \in \NN$ such that for all $x \in X$, $n > n_0$
implies
\[
\metric{Y}(f_n(x), f(x)) < \varepsilon.
\]
\end{definition}
We typically denote uniform convergence by $f_n \uconv f$.
Geometrically, we think of this in terms of pictures like the
following:
\begin{figure}[H]
\centering
\includegraphics[scale=.5]{figures/function-ball.pdf}
\caption{Example of some $f_n \colon \RR \to \RR$ satisfying $\norm{f_n -
f}_{\infty} < \varepsilon$. Dashed lines indicate $f(x) \pm
\varepsilon$.}
\end{figure}
\begin{adjustwidth}{1em}{}
\begin{leftbar}\vspace{-.5em}
\begin{remark}
It's worth noting that, as with the definition of ambient
homeomorphism, this definition is generally stronger than
requiring convergence of the images as sets. In fact, we can
construct simple examples of $f_n \colon [0,1] \into \RR^3$ such
that for all $n$, $m\in\NN$, $f_n([0,1]) = f_m([0,1])$, but the
$f_n$ do not even converge pointwise. One way to achieve this is
to alternate between two different parameterizations of the same
curve.
\begin{figure}[H]
\centering
\begin{subfigure}[t]{.49\linewidth}
\centering
\includegraphics[scale=.4]{figures/3d-uconv-1-balls.pdf}
\caption{One parameterization\ldots}
\end{subfigure}
\begin{subfigure}[t]{.49\linewidth}
\centering
\includegraphics[scale=.4]{figures/3d-uconv-2-balls.pdf}
\caption{\ldots And another.}
\end{subfigure}
\caption{Two parameterizations of the same curve.}
\end{figure}
In the figures above, the colors correspond to value of $t \in
[0,1]$ that yields the given point under $f_n$. In order for
$f_n \uconv f$, not only do the curves have to take on the same
``shape,'' but also every point of a given color in $f_n([0,1])$
has to be less than $\varepsilon$ away from the corresponding
point in $f([0,1])$.
\end{remark}
\end{leftbar}
\end{adjustwidth}
We recall two results from first courses in Analysis and Topology,
respectively.
\begin{proposition}\label{prop:uniform-convergence-continuity}
Let $(X, \metric{X})$, $(Y, \metric{Y})$ be metric spaces. For each
$k \in \NN$, let $f_k \colon X \to Y$ be continuous. Suppose that
there exists $f \colon X \to Y$ such that $f_k \uconv f$. Then $f$
is continuous.
\end{proposition}
\begin{proposition}\label{prop:compact-hausdorff-homeomorphism}
Let $\topspace{X}$, $\topspace{Y}$ be topological spaces. Suppose
that $X$ is compact and $Y$ is Hausdorff. Now suppose that $f \colon X
\to Y$ is bijective and continuous. Then $f$ is a homeomorphism.
\end{proposition}
We are now ready to begin our discussion of these results in the
context of knots.
\section{Uniform Convergence \& Knots}\label{sec:uniform-convergence}
These two propositions give us the following simple result.
\begin{corollary}\label{cor:uniformly-convergent-homeomorphism}
Let $(X, \metric{X})$, $(Y, \metric{Y})$ be metric spaces, and
suppose that $X$ is compact. For each $k \in \NN$, let $f_k \colon X
\into Y$ be an embedding. Suppose that the $f_k$ converge uniformly
to some $f \colon X \to Y$. Then if $f$ is injective, it follows
that $f$ is an embedding.
\end{corollary}
\begin{proof}
By hypothesis, for all $k \in \NN$ we have $f_k$ is an embedding and
hence continuous. Since we also have $f_k \uconv f$,
\cref{prop:uniform-convergence-continuity} implies $f$ is
continuous.
$\metspace{Y}$ is a metric space and thus Hausdorff. So $f(X)$ with
the subspace topology is Hausdorff. Now, $f$ is injective and thus
$f$ is a bijection between $X$ and $f(X)$. By
\cref{prop:compact-hausdorff-homeomorphism} it follows that $f$ is a
homeomorphism between $X$ and $f(X)$. Thus $f$ is an embedding.
\end{proof}
Among other things, \cref{cor:uniformly-convergent-homeomorphism} can
be used to construct fractal-like knot diagrams.
\begin{example} \label{ex:koch-knot}
Consider the knot constructed by the following iterative procedure,
loosely inspired by the Koch Snowflake:
\begin{figure}[H]
\centering
\begin{subfigure}{.32\linewidth}
\centering
\includegraphics[scale=.75]{figures/koch-knot/koch-knot-0.pdf}
\caption{$f_1$}
\end{subfigure}
\begin{subfigure}{.32\linewidth}
\centering
\includegraphics[scale=.75]{figures/koch-knot/koch-knot-1.pdf}
\caption{$f_2$}
\end{subfigure}
\begin{subfigure}{.32\linewidth}
\centering
\includegraphics[scale=.75]{figures/koch-knot/koch-knot-2.pdf}
\caption{$f_3$}
\end{subfigure}
\end{figure}
\begin{figure}[H]\ContinuedFloat
\centering
\begin{subfigure}{.32\linewidth}
\centering
\includegraphics[scale=.75]{figures/koch-knot/koch-knot-3.pdf}
\caption{$f_4$}
\end{subfigure}
\begin{subfigure}{.32\linewidth}
\centering
\includegraphics[scale=.75]{figures/koch-knot/koch-knot-4.pdf}
\caption{$f_5$}
\end{subfigure}
\begin{subfigure}{.32\linewidth}
\centering
\includegraphics[scale=.75]{figures/koch-knot/koch-knot-5.pdf}
\caption{$f_6$}
\end{subfigure}
\caption{A ``snowflake'' knot.}
\label{fig:snowflake-knot}
\end{figure}
One can show that with the proper choice of parameterizations for
the $f_k$, \cref{cor:uniformly-convergent-homeomorphism} guarantees
the limit function $f_\infty = \lim_{k\to\infty} f_k$ is an
embedding. The main challenge is explicitly proving injectivity;
this can be done as long as the ``shrink'' factor for the twists is
sufficiently small.
\end{example}
\subsection{Iteratively Constructing (Ambient) Homeomorphisms}
For the rest of this document we will be primarily interested in a
special case of \cref{cor:uniformly-convergent-homeomorphism}. Namely,
when $X = Y$, the $f_k$'s become homeomorphisms from $X$ to itself.
When $X$ is a compact subset of $\RR^3$ this will give us a way to
construct ambient homeomorphisms (and later, ambient isotopies) by
composing countably-many Reidemeister moves.
To that end we repackage \cref{cor:uniformly-convergent-homeomorphism}
into a form that is more ergonomic when working with this special
case. In particular, we'll write the limit function in terms of a
\emph{composition} of homeomorphisms, which is more in line with our
intuition of applying multiple Reidemeister moves in succession. This
will also allow us to offload the uniform convergence requirement to
one of acting on a shrinking collection of neighborhoods, which is
easier to interpret in terms of knot diagrams. Note, the following
theorem is true for homeomorphisms in general, but we are only
interested in applying it to ambient homeomorphisms.
\begin{theorem}\label{thm:vks-ambient-homeomorphism}
$(Y, \metric{Y})$ be a metric space. For all $k \in \NN$, let $h_k
\colon Y \to Y$ be a homeomorphism, and for all $n \in \NN$, define
\[
\ms h_n = \comp_{k=1}^n h_k = (h_n \circ h_{n-1} \circ \cdots
\circ h_{2} \circ h_1).
\]
For each $k$ let $V_k \subseteq Y$ such that $h_k$ is identity on
$V_k^c$. Then provided
\begin{enumerate}
\item The $V_k$ satisfy
\[
\lim_{n\to\infty} \diam\pn[bigg]{\bigcup_{k=n}^\infty V_k} = 0,
\]
\label{cond:vk-vanish}
\item There exists a compact $A \subseteq Y$ such that
\[
\pn[bigg]{\bigcup_{k=1}^\infty V_k} \subseteq A^\circ ,
\]
and \label{cond:vk-compactly-contained}
\item $\ms h_\infty$ defined by
\begin{align*}
\ms h_\infty
&= \lim_{n\to\infty} \ms h_n
\end{align*}
exists and is bijective,
\end{enumerate}
then $\ms h_\infty$ is a homeomorphism.
\end{theorem}
Before continuing to the proof, we make some remarks about the
statement.
\begin{remark}
The hypothesis that the limit $\ms h_\infty$ exists is superfluous
as it is implied by conditions (\ref{cond:vk-vanish}) and
(\ref{cond:vk-compactly-contained}).
\end{remark}
\begin{remark}
One can replace the somewhat-technical conditions on the $V_k$ with
simpler ones. E.g., conditions (\ref{cond:vk-vanish}) and
(\ref{cond:vk-compactly-contained}) could be substituted with
\begin{enumerate}
\item Requiring
\[
\cdots \subseteq V_{k+1} \subseteq V_k \subseteq \cdots
\subseteq V_1 \subseteq A^\circ
\]
and
\item $\lim_{k\to\infty} \diam(V_k) = 0$.
\end{enumerate}
Another option would be to replace condition (\ref{cond:vk-vanish})
with something like ``$\diam(\limsup V_n) = 0$.'' In any case, we
avoided simplifications like these in an effort to make
correspondence with the hypotheses of
\cref{cor:uniformly-convergent-homeomorphism} more direct.
\end{remark}
\begin{remark}\label{rem:harmonic-series-BAD}
We require $\lim_{n\to\infty} \diam\pn{\bigcup_{k=n}^\infty
V_k} = 0$ instead of $\lim_{n\to\infty} \diam\pn{V_k} = 0$ to
avoid situations where the $V_k$ have diameters like $\frac{1}{k}$.
If we were to allow cases like these the divergence of the harmonic
series would cause problems.
\end{remark}
\begin{remark}
Though perhaps tempting, it is not sufficient to do away with the
conditions on the $V_k$ by requiring something like ``$h_k \uconv
1_Y$.'' As a counterexample, consider $Y = [0,1]$ with the standard
metric on $\RR$. For all $k \in \NN$, define
\[
h_k = x^{(k+1)/k}.
\]
Then $h_k \uconv 1_{\bk{0,1}}$. But note, $\ms h_k = x^k$, and thus
\[
\ms h_\infty(x) =
\begin{cases}
0 & x \in [0, 1) \\
1 & x = 1
\end{cases}
\]
which is not a homeomorphism.
\end{remark}
\begin{proof}
We will employ the gluing lemma. To that end, we need to partition
$Y$ into two closed sets and show that $\ms h_\infty$ is a
homeomorphism on both. A natural choice is to consider $\ol{A^c}$
and $A$. Note, the compactness of the latter will allow us to appeal
to \cref{cor:uniformly-convergent-homeomorphism}.
\begin{figure}[H]
\centering
\includegraphics[width=.7\linewidth]{figures/a-aprime-vks-gluing.pdf}
\caption{Example $A$ shown shaded on the left; example $\ol{A^c}$
shown shaded on the right.}
\end{figure}
We examine these two sets separately.
\begin{enumerate}
\item (On $\ol{A^c}$): By construction, each $h_k$ is
identity on $V_k^c$. Since each $V_k \subseteq A^\circ$, it follows
$\ms h_\infty$ is identity (and hence a homeomorphism) on
$(A^\circ)^c = \ol{A^c}$.
\item (On $A$): Now, we show that $\ms h_\infty$ is a
homeomorphism on $A$. By
\cref{cor:uniformly-convergent-homeomorphism}, because $\ms
h_\infty$ was assumed to be bijective it suffices to
show that the restrictions $\ms h_k |_{A}$ converge uniformly to
$\ms h_\infty |_{A}$. We will suppress writing the $|_{A}$
for now because it clutters the notation too much.
Let $\varepsilon > 0$ be given. Recall that by hypothesis, we
have
\[
\lim_{n\to\infty} \diam\pn[bigg]{\bigcup_{k=n}^\infty V_k} = 0,
\]
hence there exists $n_0 \in \NN$ such that
\[
\diam\pn[bigg]{\bigcup_{k> n_0}^\infty V_k} < \varepsilon.
\]
We have the following claim.
\textbf{Claim:} For all $n > n_0$, for all $y \in A$, we have
\[
d(\ms h_n (y), \ms h_\infty (y)) < \varepsilon.
\]
\textbf{Proof of Claim:} Fix an $n > n_0$ and let $y \in A$ be
arbitrarily chosen. We have two subcases.
\begin{enumerate}
\item First, suppose $\ms h_{n_0}(y) \not\in
\bigcup_{k>n_0}^\infty V_k$.
Recall that we defined the $h_k$ such that each $h_k$ is
identity outside $V_k$. It follows that for all $k > n_0$ we
have $h_k(y) = y$. Hence
\begin{align*}
\ms h_n (y)
&= \ms h_\infty (y),
\end{align*}
so
\begin{align*}
d(\ms h_n(y), \ms h_{\infty}(y))
&= 0 \\
&< \varepsilon,
\end{align*}
as desired. \cmark
\begin{figure}[H]
\centering
\includegraphics[width=.5\linewidth]{figures/a-2-a.pdf}
\caption{An example of this case with $n_0 = 4$. The
shaded portions represent $\bigcup_{k>n_0}^\infty V_k$.
Here, $y$ starts in $V_1$, is mapped into $V_2$ by
$h_1$, into $V_4$ by $h_2$, skipped by $h_3$, then
finally mapped to another point of $V_4$ by $h_4$,
before remaining fixed for all $k > 4$. }
\end{figure}
\item Now, suppose $\ms h_{n_0} (y) \in
\bigcup_{k>n_0}^\infty V_k$. Note, since \np{for all $k >
n_0$, $h_k$ is bijective and is identity outside
$\bigcup_{k>n_0}^\infty V_k$}, it follows that for all $n >
n_0$,
\[
\ms h_{n}(y) \in\ \ \bigcup_{\mathclap{k>n_0}}^\infty\,
V_k
\]
and hence
\[
\ms h_{\infty} (y) \in \ol{\ \
\bigcup_{\mathclap{k>n_0}}^\infty\, V_k}.
\]
Note, the set in the second is just the closure of the set
in the first; thus they have the same diameter. By
definition of $n_0$, we have $\diam
\pn{\bigcup_{k>n_0}^\infty V_k} < \varepsilon$, hence
\begin{align*}
d(\ms h_n(y), \ms h_\infty (y)) < \varepsilon
\end{align*}
as desired. \cmark
\end{enumerate}
In either case, we get that $d(\ms h_n(y),\ \ms h_\infty (y)) <
\varepsilon$. Now (writing the restrictions explicitly again) it
follows that $\ms h_n|_{A}$ is a sequence of homeomorphisms with
$\pn{\ms h_n|_{A}} \uconv \pn{\ms h_\infty|_{A}}$.
Finally, recall that by hypothesis, $\ms h_\infty$ is a
bijection. This implies $\ms h_\infty|_{A}$ is too, and since
$A$ is compact, \cref{cor:uniformly-convergent-homeomorphism}
now guarantees $\ms h_\infty|_{A}$ is a homeomorphism on $A$.
\end{enumerate}
Now, applying the gluing lemma to $(\ms h_\infty)|_{A}$ and $(\ms
h_\infty)|_{A^c}$ we conclude that $\ms h_\infty$ is continuous.
An identical argument shows $\ms h_\infty^{-1}$ is continuous. It
follows that $\ms h_\infty$ is a homeomorphism, as desired.
\end{proof}
\subsection{Iteratively Constructing (Ambient) Isotopies}
We now state the analogous result for isotopies. As with
\cref{thm:vks-ambient-homeomorphism}, the theorem below is valid for
isotopies in general but we are only interested in applying it to
ambient isotopies. It might be easy to get bogged down by the
additional details so we summarize the main ideas.
Given a sequence of isotopies $H_k$, if the associated homeomorphisms
$h_k(\cdot) \coloneqq H_k(1, \cdot)$ satisfy the hypotheses of
\cref{thm:vks-ambient-homeomorphism}, then we can stitch the $H_k$'s
together into an isotopy $\ms H_\infty$ as follows. First, define $t_0
= 0$ and let $\pn{t_k}_{k=1}^\infty$ be a strictly increasing sequence
in $\pn{0,1}$. Define $\ms H_\infty$ to apply the effects of $H_1$
over the compressed time interval $\bk{t_0, t_1}$. Then, do the same
to apply $H_2$ over $\bk{t_1, t_2}$. Continue this process, in general
applying $H_k$ over the interval $\bk{t_{k-1}, t_k}$.
Stopping the construction above after $n$ steps will give us an
isotopy $\ms H_n$. Taking $n \to \infty$ we will get a function $\ms
H_\infty$ with $\ms H_\infty(1, \cdot) = \ms h_\infty$. By
\cref{thm:vks-ambient-homeomorphism}, $\ms h_\infty$ will be a
homeomorphism. And since the $\ms H_n$ are all isotopies, we'll see
that $\ms H_\infty(t, \cdot)$ will be a homeomorphism for all $t \in
\bp{0,1}$. Applying a uniform convergence argument to the $\ms H_n$
will then show $\ms H_\infty$ is continuous overall and thus an
isotopy!
\begin{theorem}\label{thm:vks-ambient-isotopy}
Let $\metspace{Y}$ be a metric space. For all $k \in \NN$, let $H_k
\colon [0,1] \times Y \to Y$ be an isotopy, and let $V_k \subseteq
Y$ such that $H_k$ is identity on $[0,1] \times (V_k^c)$. For each
$k$ define $h_k \colon Y \to Y$ by $h_k(y) = H_k(1, y)$; note that
by definition of an isotopy, $h_k$ is a homeomorphism.
Suppose that the $h_k$'s and $V_k$'s satisfy the hypotheses of
\cref{thm:vks-ambient-homeomorphism}, and for all $n \in \NN$ define
$\ms h_n = \comp_{k=1}^n h_k$. Let $t_0 = 0$ and let
$\pn{t_k}_{k=1}^\infty$ be a strictly increasing sequence in
$\pn{0,1}$ converging to $1$. Then $\ms H_\infty \colon [0,1] \times
Y \to Y$ defined by
\[
\ms H_\infty(t,y) =
\begin{cases}
H_1\pn{\frac{t - t_0}{t_1 - t_0},\ y} & \text{ if } t
\in \bk{t_0, t_1} \\
H_2\pn{\frac{t - t_1}{t_2 - t_1},\ \ms h_1(y)} & \text{ if } t \in
\pb{t_1, t_2} \\
H_3\pn{\frac{t - t_2}{t_3 - t_2},\ \ms h_2(y)} & \text{ if } t \in
\pb{t_2, t_3} \\
\,\,\,\vdots & \\
H_{k}\pn{\frac{t - t_{k-1}}{t_{k} - t_{k-1}},\ \ms h_{k-1}(y)} & \text{ if } t \in
\pb{t_{k-1}, t_{k}} \\
\,\,\,\vdots & \\
\ms h_\infty(y) & \text{ if } t = 1,
\end{cases}
\]
is an isotopy.
\end{theorem}
\begin{proof}
To show $\ms H_\infty$ is an isotopy we must show
\begin{enumerate}
\item $\ms H_\infty(0, \cdot)$ is
identity, \label{cond:is-identity-at-0}
\item For each $t \in [0,1]$, $\ms H_\infty(t, \cdot)$ is a
homeomorphism, and \label{cond:is-homeomorphism-at-each-t}
\item $\ms H_\infty$ is
continuous. \label{cond:is-continuous-overall}
\end{enumerate}
We prove these in the order above.
\begin{enumerate}
\item For all $y \in Y$, $\ms H_\infty(0, \cdot) = H_1(0, \cdot)$.
Since $H_1$ is an isotopy, $H_1(0, \cdot)$ is identity (by
definition) and this proves (\ref{cond:is-identity-at-0}).
\item To prove (\ref{cond:is-homeomorphism-at-each-t}) we break
things up into three subcases.
\begin{enumerate}
\item Suppose $t = 0$. Then $\ms H_\infty(t, \cdot) = \ms
H_\infty(0, \cdot)$ which is the identity and thus a
homeomorphism. \cmark
\item Suppose $t \in \pn{0,1}$. Then there exists $k \in \NN$
such that $t \in \pb{t_{k-1}, t_{k}}$. Recall that by
construction,
\begin{equation}
\ms H_\infty(t,\cdot) = H_{k}\pn{\frac{t - t_{k-1}}{t_{k} -
t_{k-1}},\ \ms h_{k-1}(\cdot)}. \label{eq:h-inf-h-k}
\end{equation}
Define $\ms g(\cdot) \coloneqq H_k\pn{\frac{t - t_{k-1}}{t_k
- t_{k-1}},\ \cdot}$. By definition of an isotopy, $\ms g$
is a homeomorphism. Also, $\ms h_{k-1}$ is a finite
composition of homeomorphisms and thus a homeomorphism.
Rewriting \cref{eq:h-inf-h-k} gives us that
\[
\ms H_\infty(t, \cdot) = \ms
(g \circ \ms h_{k-1})(\cdot)
\]
which is a finite composition of homeomorphisms and hence
itself a homeomorphism, as desired. \cmark
\item Now suppose $t = 1$. Then $\ms H_\infty(t, \cdot) = \ms
h_\infty(\cdot)$. Since the $h_k$'s, $V_k$'s were assumed
to satisfy the hypotheses of
\cref{thm:vks-ambient-homeomorphism} we get that $\ms
h_\infty$ is a homeomorphism as desired. \cmark
\end{enumerate}
In either case, we see $\ms H_\infty(t, \cdot)$ is a
homeomorphism.
\item Finally, it remains to show $\ms H_\infty$ is continuous. We
employ uniform convergence.
Define a sequence of isotopies $\ms H_n$ as follows: For each $n
\in \NN$, let $\ms H_n \colon [0,1] \times Y \to Y$ be given by
\[
\ms H_n(t,y) =
\begin{cases}
H_1\pn{\frac{t - t_0}{t_1 - t_0},\ y} & \text{ if } t
\in \bk{t_0, t_1} \\
H_2\pn{\frac{t - t_1}{t_2 - t_1},\ \ms h_1(y)} & \text{ if } t \in
\pb{t_1, t_2} \\
\,\,\,\vdots & \\
H_{n}\pn{\frac{t - t_{n-1}}{t_{n} - t_{n-1}},\ \ms h_{n-1}(y)}
& \text{ if } t \in \pb{t_{n-1}, t_{n}} \\
\ms h_n(y) & \text{ if } t \in \pb{t_{n}, 1}.
\end{cases}
\]
That is, we follow $\ms H_\infty(t, y)$ until we reach
$t=t_{n}$ and then we freeze. One can verify that the $\ms
H_n(t, y)$ are indeed isotopies; of particular note, they are
continuous. We now show $\ms H_n \uconv \ms H_\infty$.
Let $\varepsilon > 0$ be given. Then by the hypotheses on the
$V_k$ there exists $n_0 \in \NN$ such that
\[
\diam\pn[bigg]{\bigcup_{k>n_0}^\infty V_k} < \varepsilon.
\]
Let $n > n_0$ be arbitrarily chosen, and similarly let $(t, y)
\in [0,1] \times Y$. We show $d(\ms H_n(t,y), \ms H_\infty(t,
y)) < \varepsilon$. We have two subcases.
\begin{enumerate}
\item First, suppose $t \in \bk{0, t_{n}}$. Then
$\ms H_n(t, y) = \ms H_\infty(t,y)$ and so we have $d(\ms
H_n(t,y), \ms H_\infty(t,y)) = 0$ and the bound holds.
\item Now, suppose $t \in \pb{t_{n}, 1}$. If $y
\in \pn{\bigcup_{k>n_0}^\infty V_k}^c$, then $\ms
H_n(t, y) = \ms H_\infty(t, y)$, so we have $d(\ms H_n(t,y),
\ms H_\infty(t,y)) = 0$ and the bound holds. Else, note that
both of $\ms H_n(t,y)$, $\ms H_\infty(t,y) \in
\ol{\bigcup_{k>n_0}^\infty V_k}$, hence
\[
d(\ms H_n(t,y), \ms H_\infty(t,y)) < \varepsilon
\]
as desired.
\end{enumerate}
In either case, we have $d(\ms H_n(t,y), \ms H_\infty(t,y)) <
\varepsilon$. As $(t,y)$ were arbitrarily chosen, this implies
$\ms H_n \uconv \ms H_\infty$. By
\cref{prop:uniform-convergence-continuity}, $\ms H_\infty$ is
continuous.
\end{enumerate}
It follows that $\ms H_\infty$ is an isotopy as desired.
\end{proof}
In the next section, we apply this result to a variety of curves,
beginning with the example from \cref{fig:scintillating-curves}.
\section{Various Applications of
\cref{thm:vks-ambient-isotopy}}\label{sec:various-applications} The
first few examples will all make use of the following lemma, which
allows us to remove the bijectivity hypothesis from
\cref{thm:vks-ambient-isotopy}.
\begin{lemma}\label{lem:disjoint-vks}
Let all variables be quantified as in
\cref{thm:vks-ambient-isotopy}. Then if the $V_k$'s are all
disjoint, $\ms H_\infty(1, \cdot)$ is guaranteed to be a bijection.
\end{lemma}
\begin{proof}
If the $V_k$ are all disjoint, then defining $U =
\bigcup_{k=1}^\infty V_k$ we can write $Y$ as the disjoint union
\[
Y = (U^c) \sqcup \pn[bigg]{\bigsqcup_{k=1}^\infty V_k}.
\]
Note, $\ms H_\infty(1, \cdot)$ is identity on $U^c$ and hence a
bijection. Since the $V_k$ are all disjoint, $\ms H_\infty(1,
\cdot)|_{V_k} = \ms H_k(1, \cdot)|_{V_k}$, the latter of which is a
homeomorphism and thus bijective, so $\ms H_\infty(1, \cdot)$ is a
bijection on each of the $V_k$.
Thus $\ms H_\infty(1, \cdot)$ is a bijection overall.
\end{proof}
\begin{proposition}\label{prop:wild-looking-unknot-tame}
The curve shown in \cref{fig:wild-looking-unknot-redux} below is
tame.\footnote{Strictly speaking we have not defined tameness for
curves, only for knots. A \emph{tame curve} is a curve that's
ambient homeomorphic (equivalently, ambient isotopic) to a
polygonal curve.} In particular, it is an unknot.
\begin{figure}[H]
\centering
\centering
\includegraphics[width=.4\linewidth]{figures/countable-r1.pdf}
\caption{A wild-looking unknot, redux.}
\label{fig:wild-looking-unknot-redux}
\end{figure}
\end{proposition}
\begin{proof}
Let $f_0 \colon S^1 \into \RR^3$ be the standard unknot, and let
$f_1 : S^1 \into \RR^3$ be an embedding yielding a diagram like
\cref{fig:wild-looking-unknot-redux}.\footnote{A parametrization can
be found in \cite{KobayashiThesis}, although it is given in the
context of a ``theorem'' about tameness and parametrizations that
turns out to be incorrect.} We apply
\cref{thm:vks-ambient-isotopy} to construct an ambient isotopy $\ms
H_\infty \colon [0,1] \times \RR^3 \to \RR^3$ taking $f_0$ to $f_1$.
Consider the sequence of $V_k$'s chosen as follows.
\begin{figure}[H]
\centering
\includegraphics[scale=.508]{figures/countable-r1-vks.pdf}
\caption{$V_k$'s.}
\end{figure}
One can verify that $\lim_{n\to\infty} \diam\pn{\bigcup_{k=n}^\infty
V_k} = 0$ and that there exists a compact set $A \subseteq \RR^3$
such that $\bigcup_{k=1}^\infty V_k \subseteq A$.
For all $k \in \NN$, let $H_k \colon [0,1] \times \RR^3 \to \RR^3$
be an ambient isotopy inserting a Reidemeister I into the arc bound
in $V_k$. Use these $H_k$ to define $\ms H_\infty$ as in
\cref{thm:vks-ambient-isotopy}. By \cref{lem:disjoint-vks}, $\ms
H_\infty(1, \cdot)$ is a bijection. Thus by
\cref{thm:vks-ambient-isotopy}, $\ms H_\infty$ is an ambient isotopy
from $f_0$ to $f_1$.
\end{proof}
\begin{proposition}\label{prop:countable-r2}
The following curve is tame.
\begin{figure}[H]
\centering
\includegraphics[angle=-90]{figures/countable-r2-v2.pdf}
\end{figure}
\end{proposition}
\begin{proof}[Proof (Sketch)]
We apply \cref{thm:vks-ambient-isotopy} twice. This two-step method
is not strictly necessary, but the diagram is a bit less cluttered
this way. Consider the following starting curve:
\begin{figure}[H]
\centering
\includegraphics[angle=-90]{figures/countable-r2-two-step-0.pdf}
\end{figure}
Apply Reidemeister II moves within the dotted regions below:
\begin{figure}[H]
\centering
\includegraphics{figures/countable-r2-two-step-0-vs.pdf}
\end{figure}
Since these $V_k$ are disjoint, we can again apply
\cref{lem:disjoint-vks} to obtain an ambient isotopy. The result
looks something like the following:
\begin{figure}[H]
\centering
\includegraphics[angle=-90]{figures/countable-r2-two-step-1.pdf}
\end{figure}
Now, perform Reidemeister II moves in the following regions:
\begin{figure}[H]
\centering
\includegraphics[angle=-90]{figures/countable-r2-two-step-1-vs-2.pdf}
\end{figure}
Again, the $V_k$ here are all disjoint, hence one can apply
\cref{lem:disjoint-vks} to show that this is an ambient isotopy. The
end result is
\begin{figure}[H]
\centering
\includegraphics[angle=-90]{figures/countable-r2-v2.pdf}
\end{figure}
which is the desired diagram.
\end{proof}
We now consider a similar curve, this time constructed using
Reidemeister I moves. This will be the most technical argument of the
paper. We advise the reader to read through
\cref{ex:remarkable-curve-redux} in the next section before
continuing. This is because \cref{ex:remarkable-curve-redux} shows how
we can lose bijectivity in the limit, and the bulk of the challenge in
\cref{prop:countable-r1-2} is addressing similar concerns. We have to
address bijectivity in a manner like this whenever we have points that
are moved by infinitely many of the $H_k$'s (whereas in
\cref{prop:countable-r2}, each point is moved by only finitely many
$H_k$).
\begin{proposition}\label{prop:countable-r1-2}
Let $f_0 \colon [0,1] \into \RR^3$ be
\begin{figure}[H]
\centering
\includegraphics[scale=.625]{figures/recursive-countable-r1/initial-bend.pdf}
\caption{Our starting arc.}
\label{fig:recursive-r1-starting-arc}
\end{figure}
and let $f_1 \colon [0,1] \into \RR^3$ be
\begin{figure}[H]
\centering
\includegraphics[angle=-90]{figures/countable-r1-v2.pdf}
\caption{A different countable sequence of Reidemeister I moves.}
\label{fig:another-countable-r1}
\end{figure}
Then $f_0 \cong f_1$.
\end{proposition}
\begin{proof}[Proof (Sketch)]
We will construct the ambient isotopy from $f_0$ to $f_1$ by a
recursive process. We will repeatedly insert Reidemeister I moves
like the following:
\begin{figure}[H]
\centering
\includegraphics{figures/recursive-countable-r1/initial-to-final.pdf}
\caption{The general procedure.}
\label{fig:general-procedure}
\end{figure}
We must do so in such a way that we can still argue bijectivity of
$\ms H_\infty(1, \cdot)$. The key idea is to choose our $H_k$'s so
that only one point (denoted $y_\infty$) gets moved infinitely-many
times.\footnote{$y_\infty$ will be the point that gets sent to the
limit of the twists in \cref{fig:another-countable-r1}. In our
construction, $y_\infty$ will be the vertex in
\cref{fig:recursive-r1-starting-arc}, but one can create other
constructions where it is a different point.} We explicitly
guarantee this by constructing our $H_k$'s so that for all $y \in
\RR^3 \setminus \set{y_\infty}$, there exists $n_0$ such that for $n
> n_0$, $\ms h_{n_0}(y)$ is unmoved by $H_n(t, \cdot)$.
To that end, define $\ell$ as shown in \cref{fig:with-l}, and let
$\varepsilon > 0$ with $\varepsilon \ll 1$.\footnote{Actually,
$\varepsilon > 0$ can be arbitrarily chosen so long as for all $k
\in \NN$ we have $\ol{V_{k+1}} \subseteq V_{k}^\circ$. We just
choose $\varepsilon \ll 1$ because it makes for cleaner-looking
diagrams.} Define $V_1$ to be a closed rectangular prism of
dimensions $(6 + \varepsilon)\ell \times (2+\varepsilon)\ell \times
(2+\varepsilon)\ell$, and $\msf{H_1}$ to be a PL ambient isotopy
inserting the first loop such that $\msf{H}_1$ is identity off
$V_1$.\footnote{We can assume PL-ness because the modifications can
be realized by \emph{elementary
moves}.}\textsuperscript{,}\footnote{The $6$ in our prism
dimensions comes from the fact $\ell$ is defined to be
$1/3$\textsuperscript{rd} of the length of the twist inserted in
\cref{fig:with-l}, and the moves halve in size at each iteration.}
Note, even though we define $V_1$ to be closed, we'll draw it with
dotted lines in the below.
\begin{figure}[H]
\centering
\includegraphics{figures/recursive-countable-r1/initial-to-final-l.pdf}
\caption{The same figure, now showing $V_1$.}
\label{fig:with-l}
\end{figure}
Now, we describe the general strategy for inserting the
$k+1$\textsuperscript{st} loop given the first $k$ loops. We want
$V_{k+1}$, $\msf{H}_{k+1}$ to be half-scale versions of $V_k$,
$\msf{H}_k$. The figure below shows this for $k = 1$.
\begin{figure}[H]
\centering
\includegraphics{figures/recursive-countable-r1/next-to-next-next.pdf}
\caption{$V_1$ and $V_2$, with application of $\msf H_2$ shown.}
\label{fig:v2}
\end{figure}
But, to make things work, it will be important to first apply some
\emph{intermediate} ambient isotopy $\msf{H}_{k+.5}$ in between
$\msf{H}_k$ and $\msf{H}_{k+1}$ such that $\msf H_{k+.5}$ does not
change the diagram, but {does} help ensure that points in the
ambient space aren't lost in the limit.
\textbf{Desired Properties of $\bm{\msf H_{k+.5}}$:} We want $\msf
H_{k+.5}$ to preemptively ``unsquish'' points that might be
compressed together by $\msf H_{k+1}$. To determine exactly how much
unsquishing we have to do, we look at a sort of inverse Lipschitz
condition.
Let $\msf{h}_{k+1} = \msf{H}_{k+1}(1, \cdot)$. Since $\msf{H}_{k+1}$
is a PL ambient isotopy, $\msf{h}_{k+1}$ is a PL ambient
homeomorphism, and thus there exists $c \in (0,1)$ such that for all
$x_1, x_2 \in \RR^3$,
\begin{equation}
c \cdot d(x_1,x_2) \leq d(\msf{h_{k+1}}(x_1),
\msf{h_{k+1}}(x_2)).\footnote{This essentially pops out of the
finiteness condition on our simplicial complexes for PL
maps.} \label{eq:bound-on-squishing}
\end{equation}
Note that because the $\msf H_{k}$ are all identical up to scaling,
$c$ is independent of $k$.\footnote{One might ask why we can't have
$c \geq 1$. Note that $V_{k+1}$ being bounded precludes $c > 1$.
For $c = 1$, note that $\msf{h}_{k+1}$ is not a vector space
isomorphism of $\RR^3$, and hence not an isometry on $\RR^3$;
since $\msf h_{k+1}$ is identity outside $V_{k+1}$, isometry must
fail on $V_{k+1}$.} We interpret \cref{eq:bound-on-squishing} as
giving us a bound on how much $\msf{h}_{k+1}$ can ``squish'' points
in the space together. Let $q_k$ be the tip of the twist before
applying $\msf H_{k+1}$:
\begin{figure}[H]
\centering
\includegraphics[scale=.75]{figures/recursive-countable-r1/what-is-qk.pdf}
\caption{$q_k$ labeled.}
\end{figure}
For all $p \in V_{k+1}$, we want $\msf H_{k+.5}$ to be constructed
to guarantee that either
\begin{enumerate}
\item $\msf{h}_{k+1}(\msf h_{k+.5}(p)) \in V_k \setminus V_{k+1}$
(i.e.\ $p$ gets moved to the outer box), or
\item $p$ gets ``moved farther from $q_k$ than it can be squished
in later'':
\begin{equation}
\frac{1}{c} \cdot d(p, q_k) \leq d(\msf h_{k +.5}(p),
\msf h_{k+.5}(q_k)). \label{eq:h1.5-bound}
\end{equation}
\end{enumerate}
\textbf{Constructing $\bm{\msf H_{k+.5}}$:} Let $V_{k+.5}$ be a
slightly-scaled-up version of $V_{k+1}$ such that $V_{k+1}
\subsetneq V_{k+.5} \subsetneq V_k$. To make things easier, we will
require that $V_{k+.5}$ also only intersects with $(\msf{h}_{k}
\circ \msf h_{k-1} \circ \cdots \circ \msf h_{1} \circ f_0)([0,1])$
in a wedge-shape and that $V_{k+.5}$ and $V_{k+1}$ share the same
center of mass and have all sides parallel (see
\cref{fig:v1.5-and-v2}).
\begin{figure}[H]
\centering
\includegraphics[scale=.75]{figures/recursive-countable-r1/3d-triangulated-boxes.pdf}
\caption{$V_{k+.5}$ and $V_{k+1}$}
\label{fig:v1.5-and-v2}
\end{figure}
We can parameterize every point $p \in V_{k+.5}$ in terms of a
piecewise linear function as detailed below. The construction is a
bit unergonomic to formalize explicitly, but it is meant to capture
the ideas of \cref{fig:vs-with-lines}.
\begin{figure}[H]
\centering
\includegraphics[scale=1.25]{figures/recursive-countable-r1/3d-triangulation-lines-top.pdf}
\caption{The piecewise-linear parameterization, with some of the
$v_{k+.5}$'s (outer points; defined below) and $v_{k+1}$'s
(inner points; also defined below) shown in white.}
\label{fig:vs-with-lines}
\end{figure}
We construct it in two parts and then scale them by half and glue
them together.
\begin{enumerate}
\item If $p \in V_{k+.5} \setminus V_{k+1}$, there exist unique
points $v_{k+.5} \in \partial {V_{k+.5}}$ and $v_{k+1} \in
\partial {V_{k+1}}$ such that $v_{k+.5}$ is the point in
$V_{k+.5}$ ``corresponding'' to $v_{k+1}$ in $V_{k+1}$, and $p$
is on the line segment $\ol{v_{k+.5} v_{k+1}}$.\footnote{By
``corresponds,'' we mean that given a linear function that
scales up $V_{k+1}$ to yield $V_{k+.5}$, $v_{k+1}$ gets mapped
to $v_{k+.5}$.} Thus there exists a unique $s \in [0,1]$ such
that we can write $p$ as a convex combination
\[
p = s \cdot v_{k+.5} + (1-s) \cdot v_{k+1}.
\]
\item If $p \in V_{k+1}$, there exists a unique point $v_{k+1}
\in\partial {V_{k+1}}$ such that $p$ is on the line segment
$\ol{q_k v_{k+1}}$. Analogously to the above, there exists a
unique $s \in [0,1]$ such that we can write $p$ as
\[
p = s \cdot v_{k+1} + (1-s) \cdot q_k.
\]
\end{enumerate}
We re-scale $s$ to glue these two parameterizations into a single
function which we call $\msf H'_{k+.5}$:
\[
\msf H'_{k+.5}(s, v_{k+.5}, v_{k+1}) =
\begin{cases}
\qquad \ \, 2s \cdot v_{k+1} \, + (1-2s) \cdot q_k & s \in
\bk{0, \frac{1}{2}} \\
(2s-1) \cdot v_{k+.5} + (2-2s) \cdot v_{k+1} & s \in
\pb{\frac{1}{2}, 1}.
\end{cases}
\]
We'll now do something a bit bizarre and rewrite the parameters in
$\msf H'_{k+.5}$ as functions of $p$. Note that with the re-scaling
of $s$, we now have $s$ uniquely determined by $p$. Also recall that
by construction, $v_{k+.5}$ and $v_{k+1}$ are each uniquely
determined by $p$. Hence, we can think of $s$, $v_{k+.5}$, $v_{k+1}$
as being functions of $p$. One can show that these are all
continuous. As such, we can indeed think of $\msf H'_{k+.5}$ as just
being a complicated way of writing the identity function on
$V_{k+.5}$.
To turn $\msf H'_{k+.5}$ into our ambient isotopy $\msf H_{k+.5}$,
we now introduce time dependence in $s$. Define $s_{c_0} =
\frac{c}{2}$
and $s_{c_1} = \frac{1}{2}$, and observe $s_{c_0} <
s_{c_1}$.\footnote{The reason that $c$ appears in this expression is
because we're trying to get \cref{eq:h1.5-bound} out in the end.}
Define $s_c \colon [0,1] \to [s_{c_0}, s_{c_1}]$ by
\[
s_c(t) = t \cdot s_{c_1} + (1-t) \cdot s_{c_0},
\]
and use this to define
\[
s'(t, p) =
\begin{cases}
\pn{\frac{s(p)}{s_{c_0}}} \cdot s_c(t) & \text{if } s(p) \in
\bk{0, s_{c_0}} \\
\pn{\frac{s(p) - s_{c_0}}{1 - s_{c_0}}} \cdot 1 + \pn{1 -
\pn{\frac{s(p) - s_{c_0}}{1 - s_{c_0}}}} \cdot s_c(t) &
\text{if } s(p) \in \pb{s_{c_0}, 1}
\end{cases}
\]
This looks unpleasant but the idea is simple. First, recall that
$s(p)$ represents a parameter in $[0,1]$ that tells us how to write
$p$ as a convex combination of other points. One can verify that
when $t = 0$, $s'(t,p)$ reduces to $s(p)$. Then, as $t$ increases to
$1$, $s'(t,p)$ distorts the interval represented by $s(p)$ until we
end up with something like the following, in which $s_{c_0}$ gets
mapped to where $s_{c_1}$ was initially:
\begin{figure}[H]
\centering
\includegraphics[scale=1.5]{figures/recursive-countable-r1/interval-start.pdf}
\caption{The interval $[0,1]$ represented by $s'(t,p)$ as $t$ goes
from $0$ to $1$. The light portion represents the values where
$s(p) \in [0, s_{c_0}]$ and the dark portion represents $s(p)
\in [s_{c_0}, 1]$.}
\end{figure}
The net effect of $\msf H_{k+.5}$ is to take a diagram like the
following
\begin{figure}[H]
\centering
\includegraphics{figures/recursive-countable-r1/3d-triangulation-lines-top-colored.pdf}
\end{figure}
and turn it into
\begin{figure}[H]
\centering
\includegraphics{figures/recursive-countable-r1/3d-triangulation-lines-top-colored-unsquished.pdf}
\caption{The effects of $\msf H_{k+.5}$.}
\end{figure}
Finally, we have the following:
\begin{adjustwidth}{1em}{}\vspace{.5em}
\begin{leftbar}
\textbf{Claim:} $\msf H_{k+.5} \colon [0,1] \times \RR^3 \to \RR^3$
given by
\[
\msf H_{k+.5}(t, p) =
\begin{cases}
\msf H'_{k+.5}(s'(t, p), p) & p \in V_{k+.5} \\
p & p \not \in V_{k+.5}
\end{cases}
\]
satisfies the desired properties of $\msf H_{k+.5}$.
\textbf{Proof of Claim:} One can verify that $\msf H_{k+.5}$
satisfies all the properties of an ambient
isotopy.\footnote{Intuitively, all it is doing is sliding all the
points in $V_{k+1}$ along the lines in \cref{fig:vs-with-lines}
until they are \emph{either} in $V_{k+.5} \setminus V_{k+1}$
\emph{or} $\frac{1}{c}$ times as far from $q_k$ as they were at
the start.} It remains to show that $\msf h_{k+.5} = \msf
H_{k+.5}(1, \cdot)$ satisfies the conditions stipulated near
\cref{eq:h1.5-bound}.
Let $p \in V_{k+.5}$ be arbitrary. We have two cases.
\begin{enumerate}
\item Suppose $s(p) \in \pb{s_{c_0}, 1}$. Then $\msf h_{k+.5}(p)
\in V_{k+.5} \setminus V_{k+1}$, and hence $\msf h_{k+1}(\msf
h_{k+.5}(p)) \in V_{k} \setminus V_{k+1}$. \cmark
\item Suppose $s(p) \in \bk{0, s_{c_0}}$. One can verify that in
this case, $\msf H_{k+.5}(t, p)$ only slides $p$ along a ray
segment originating from $q_k$, with the sliding dictated by
$s'(t,p)$. Hence
\begin{align*}
\frac{d(p, q_k)}{d(h_{k+.5}(p),\ \msf h_{k+.5}(q_k))}
&= \frac{d(\msf H_{k+.5}(0, p), H_{k+.5}(0,
q_k))}{d(H_{k+.5}(1, p), H_{k+.5}(1, q_k))} \\
&= \frac{s'(0, p)}{s'(1, p)} \\
&= \frac{\cancel{\frac{s(p)}{s_{c_0}}}
s_{c_0}}{\cancel{\frac{s(p)}{s_{c_0}}}
s_{c_1}} \\
&= c.
\end{align*}
Simplifying gives us
\[
\frac{1}{c} \cdot d(p, q_k) = d(H_{k+.5}(1, p),\ H_{k+.5}(1,
q_k)),
\]
as desired.
\end{enumerate}
\end{leftbar}
\end{adjustwidth}
\textbf{Guaranteeing Bijectivity:} Observe that for all $k\in\NN$,
for all $p \in V_{k+1}$, we have
\begin{align}
d(\msf{h}_{k+1}(\msf h_{k+.5}(p)),\ \msf{h}_{k+1}(\msf
h_{k+.5}(q_{k})))
&\geq c \cdot d(\msf h_{k+.5} (p),\ \msf h_{k+.5}(q_{k}))
\nonumber\\
&\geq c \cdot \frac{1}{c} \cdot d(p, q_{k}) \nonumber\\
&\geq d(p, q_{k}). \label{eq:final-bound}
\end{align}
For each $n \in \NN$, let $\ms h_n$ denote the composition of all
these homeomorphisms:
\begin{align*}
\ms h_n
&= \pn{\comp_{k=1}^{n-1} \pn{\msf{h}_{k+1} \circ \msf h_{k+.5}}} \circ
\msf h_1 \\
&= (\msf h_n \circ \msf h_{n-.5} \circ \cdots \circ \msf h_2 \circ
\msf h_{1.5} \circ \msf h_1).
\end{align*}
Note that the sequence of points $\ms h_n^{-1}(q_n)$ is constant,
hence the limit $\lim_{n\to\infty} \ms h_n^{-1}(q_{n})$ exists; in
particular, it is $y_\infty$. For all $y \in \RR^3 \setminus
\set{y_\infty}$, \cref{eq:final-bound} shows that at each step, $y$
is sent no closer to $q_{k+1}$ than it was to $q_k$. Since the boxes
are shrinking it follows that each such $y$ will eventually leave
the boxes and thus remain fixed at subsequent steps. Explicitly: If
$n_0$ satisfies
\[
\frac{(6+2\varepsilon)\ell}{2^{n_0}} < d(y, y_\infty),
\]
Then for all $n > n_0$ we have $\ms h_{n}(y) \not \in V_n$, and
hence
\[
\ms h_n(y) = \ms h_{n_0}(y).
\]
This implies $\ms h_\infty$ is a bijection between $\RR^3 \setminus
\set{y_\infty}$ and $\RR^3 \setminus \set{\ms h_\infty(y_\infty)}$.
So \cref{thm:vks-ambient-homeomorphism} implies $\ms h_\infty$ is a
homeomorphism between $\RR^3 \setminus \set{y_\infty}$ and $\RR^3
\setminus \set{\ms h_\infty(y_\infty)}$. Thus $\ms h_\infty$ is
bijective on $\RR^3$, and \cref{thm:vks-ambient-isotopy} implies
that $\ms H_\infty(1, \cdot)$ is an ambient isotopy.
\end{proof}
Finally, we have the following famous example.
\begin{proposition}\label{prop:countable-csum-of-trefoils}
The following curve is a tame arc.
\begin{figure}[H]
\centering
\includegraphics[scale=.5]{figures/countable-trefoil/countable-trefoil-arc.pdf}
\end{figure}
\end{proposition}
\begin{proof}[Sketch]
We apply \cref{lem:disjoint-vks}. Consider a sequence of properly
nested boxes $V_1$, $V_2$, $\ldots$, as follows:
\begin{figure}[H]
\centering
\includegraphics[scale=.5]{figures/countable-trefoil/countable-trefoil-arc-boxed.pdf}
\caption{A countable connected sum of trefoils.}
\label{fig:countable-csum-of-trefoils}
\end{figure}
For all $k \in \NN$, define $V'_k$ by $V'_k = V_k \setminus
V_{k+1}$. Note the $V'_k$ are disjoint. We can define the ambient
isotopies $H_k$ such that $H_k$ performs the following modification
on $V'_k$:
\begin{figure}[H]
\centering
\includegraphics[scale=.5]{figures/countable-trefoil/countable-trefoil-arc-boxed-removing-1.pdf}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=.5]{figures/countable-trefoil/countable-trefoil-arc-boxed-removing-2.pdf}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=.5]{figures/countable-trefoil/countable-trefoil-arc-boxed-removing-3.pdf}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=.5]{figures/countable-trefoil/countable-trefoil-arc-boxed-removing-35.pdf}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=.5]{figures/countable-trefoil/countable-trefoil-arc-boxed-removing-4.pdf}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=.5]{figures/countable-trefoil/countable-trefoil-arc-boxed-removing-5-1.pdf}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=.5]{figures/countable-trefoil/countable-trefoil-arc-boxed-removing-6.pdf}
\end{figure}
Taking the limit, we obtain an ambient isotopy unknotting the arc.
\end{proof}
Having established the versatility of \cref{thm:vks-ambient-isotopy},
we now discuss situations in which it cannot be applied. In a sense,
we will see that each of the hypotheses of the theorem are sharp.
\section{Cases Where \cref{thm:vks-ambient-isotopy} Does Not
Apply}\label{sec:does-not-apply}
\begin{example}
If we extend the right hand side of
\cref{fig:countable-csum-of-trefoils} with a straight line segment,
then we cannot apply \cref{thm:vks-ambient-isotopy}. In fact, the
curve is wild --- see \cite{Daverman}, Exercise 2.8.4.
\begin{figure}[H]
\centering
\includegraphics[scale=.5]{figures/countable-trefoil/countable-trefoil-arc-extended.pdf}
\end{figure}
What breaks here is that if we try to apply the same argument as we
did with the non-extended version in
\cref{prop:countable-csum-of-trefoils}, we can't force
$\lim_{n\to\infty} \diam\pn{\bigcup_{k=n}^\infty V_k} = 0$. In
particular, $\diam\pn{\bigcup_{k=n}^\infty V_k}$ is bounded below by
the length of the straight line segment.
\end{example}
And now, as promised, we discuss the curve from
\cref{subfig:remarkable-curve}.
\begin{example}\label{ex:remarkable-curve-redux}
One cannot apply \cref{thm:vks-ambient-isotopy} to the following
curve:
\begin{figure}[H]
\centering
\includegraphics[scale=1.5]{figures/smooth-fox-artin.pdf}
\caption{Fox's ``Remarkable Knotted Curve.''}
\label{fig:remarkable-curve-redux}
\end{figure}
Here, the $V_k$'s are not the problem; rather it is bijectivity on
the ambient space. Consider a ``line'' of points in the ambient
space passing through the first loop:
\newcommand{8}{8}
\begin{figure}[H]
\centering
\includegraphics[scale=8]{figures/smooth-fox-artin-pole-1.pdf}
\caption{The curve, now with an imaginary ``line'' of points from
the ambient space.}
\label{fig:remarkable-curve-with-line}
\end{figure}
As we remove the first loop, some points on the ``line'' get pulled
through:
\begin{figure}[H]
\centering
\includegraphics[scale=8]{figures/smooth-fox-artin-pole-2.pdf}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=8]{figures/smooth-fox-artin-pole-3.pdf}
\caption{Removing the first loop. Note, by continuity, the
``line'' must remain unbroken.}
\end{figure}
As we remove the second loop, a similar process occurs:
\begin{figure}[H]
\centering
\includegraphics[scale=8]{figures/smooth-fox-artin-pole-4.pdf}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=8]{figures/smooth-fox-artin-pole-5.pdf}
\caption{Removing the second loop.}
\end{figure}
As $n \to \infty$, the stitching process continues, with the number
of lines doubling at each iteration. No matter what we try, in the
limit, a countable subset of the original line gets mapped to the
wild point. In fact, if we thicken the line in
\cref{fig:remarkable-curve-with-line} to a cylinder, we see that
\emph{uncountably}-many points are lost in the limit!
This is reflected in the fact that if we were to try something like
the approach taken in \cref{prop:countable-r1-2}, we would not be
able to define ambient isotopies that do the jobs of the $\msf H_{k
+ .5}$'s.
\end{example}
\begin{remark}\label{rem:cant-tie-it}
Another perspective one might consider here is that there is no
obvious way to ``tie'' the curve in
\cref{fig:remarkable-curve-redux} if starting with just an unknotted
line. Indeed, there's no ``first loop'' to insert --- to tie one, we
require infinitely-many of the others to be present already!
\end{remark}
\section{Discussion}
We conclude with a discussion of directions for future work.
\cref{thm:vks-ambient-homeomorphism,thm:vks-ambient-isotopy} give us
one direction to a loose ``countable analogue'' of Reidemeister's
theorem. The restrictions on the $V_k$'s have a nice diagrammatic
interpretation --- ``the total region we're going modify has to shrink
in the limit'' --- but so far, a similarly-concise description of the
bijectivity requirement has eluded the author.
Qualitatively, it seems that problems tend to occur when the set of
{points that get moved infinitely-many times} is not topologically
discrete; however, it's been difficult to find the right language to
distinguish between cases when this gives rise to \emph{legitimate}
problems versus ones where the issue is superficial. As an example of
the former, consider \cref{ex:remarkable-curve-redux}, and as an
example of the latter, consider a sequence of homeomorphisms $h_k :
\RR^3 \to \RR^3$ where each $h_k$ is defined by
\[
h_k(y) = y +
\begin{bmatrix}
\frac{1}{2^k} \\[.5em]
0 \\
0
\end{bmatrix}.
\]
Then
\[
h_\infty(y) = y +
\begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix},
\]
and so \emph{all} points in $\RR^3$ are moved infinitely-many times by
$h_\infty$, yet we have no problems.
Thus, we have the following question:
\begin{adjustwidth}{1em}{}\vspace{.5em}
\begin{leftbar}\vspace{-.5em}
\begin{question}
Is there a simpler way to guarantee bijectivity of the limit
function in \cref{thm:vks-ambient-isotopy}? In particular, is there
a purely diagrammatic condition?
\end{question}
\end{leftbar}
\end{adjustwidth}
One of the things that makes the problem in
\cref{ex:remarkable-curve-redux} tricky to spot at first is that the
limiting process yields an \emph{isotopy}, just not an \emph{ambient
isotopy} (this is reminiscent of \emph{Bachelor's unknotting}). We
wonder whether a similar effect can be obtained using only
Reidemeister I or Reidemeister III moves, as detailed in the following
questions:
\begin{adjustwidth}{1em}{}\vspace{.5em}
\begin{leftbar}\vspace{-.5em}
\begin{question}\label{q:what-about-r1}
Does there exist a knot $f \colon S^1 \into \RR^3$ such that applying a
countable sequence of Reidemeister I moves to $f$ yields an isotopy,
but not an ambient isotopy?
\end{question}
\begin{question}
Same as \cref{q:what-about-r1}, but with Reidemeister III moves
instead of Reidemeister I moves.
\end{question}
\end{leftbar}
\end{adjustwidth}
Of course, we want to avoid trivial examples like taking $H_1$ to be
Bachelor's unknotting and then taking the remaining $H_k$'s to be
identity.
Now, returning to the question of a countable analogue for
Reidemeister's theorem:
\begin{question}
If we restrict ourselves to Reidemeister moves, when do the
converses of
\cref{thm:vks-ambient-homeomorphism,thm:vks-ambient-isotopy} hold?
That is, given an ambient isotopy between two embeddings $f_0, f_1 :
S^1 \into \RR^3$, when can we guarantee the existence of the desired
$V_k$'s and $h_k$'s, where each of the $h_k$'s represent single
Reidemeister moves?
\end{question}
We have a conjecture in this direction. In \cite{KobayashiThesis} the
author employed \cref{thm:vks-ambient-isotopy} to argue the following
result:
\begin{theorem}
Call a knot diagram a \textbf{discrete diagram} if it satisfies all
the axioms of a regular diagram except perhaps having finitely-many
crossings.\footnote{The ``discrete'' in ``discrete diagram'' comes
from the fact that the set of crossing points only needs to be
topologically discrete rather than finite.}
Now, let $f \colon S^1 \into \RR^3$ be an arbitrary knot. Then if $f$
admits a discrete diagram, $f$ is ambient isotopic to a
representative comprised of countably-many line segments.
\end{theorem}
This gave rise to the following conjecture (we thank Kye Shi for the
insight of adding the fourth move):
\begin{conjecture}
Define the \emph{extended} Reidemeister moves to be the standard
move set together with a fourth move
\begin{figure}[H]
\centering
\includegraphics{figures/extended-move.pdf}
\caption{Fourth move}
\end{figure}
where in the above, $A$ is a compact set whose interior remains
fixed relative to its boundary. Note, $A$ can contain wild points.
Let $f_0, f_1 \colon S^1 \into \RR^3$ be knots that admit discrete
diagrams. Suppose $f_0 \cong f_1$. Then there exists a countable
sequence of extended Reidemeister moves satisfying the hypotheses of
\cref{thm:vks-ambient-isotopy} and taking $f_0$ to $f_1$.
\end{conjecture}
For more details on the proposed approach, see \cite{KobayashiThesis},
\S 9.3.1. We have some partial results in this direction but there
remain important gaps.
\section{Acknowledgements}
We thank Kye Shi for proofreading and for offering many helpful
suggestions on how to improve the exposition. We also thank Francis Su
and Sam Nelson for the many insights they provided when advising the
author's undergraduate thesis \cite{KobayashiThesis}, on which this
work is based.
\bibliographystyle{amsplain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,223 |
Q: Failing bluetooth sequence I'm trying to make a simple connection sequence to a serial bluetooth device at the beginning of my app. Right now, all of this is inside onCreate ():
BT = BluetoothAdapter.getDefaultAdapter();
BT.enable();
if(!BT.isEnabled()) {
Intent enabler = new Intent(BluetoothAdapter.ACTION_REQUEST_ENABLE);
startActivityForResult(enabler, REQUEST_ENABLE);
BT.enable();
Toast.makeText(getApplicationContext(), "Bluetooth On.", Toast.LENGTH_LONG).show();
//finish apk
finish();
}
else {
Toast.makeText(getApplicationContext(), "Bluetooth On.", Toast.LENGTH_LONG).show();
}
pairedDevices = BT.getBondedDevices();
pDevices = new ArrayList<BluetoothDevice>();
if (pairedDevices.size()>0) {
for(BluetoothDevice bt : pairedDevices)
{
pDevices.add(bt); //Get the device's name and the address
}
}
else {
Toast.makeText(getApplicationContext(), "Nothing paired.", Toast.LENGTH_LONG).show();
}
try {
BluetoothDevice dispositivo = BT.getRemoteDevice(pDevices.get(0).getAddress());
btSocket = dispositivo.createInsecureRfcommSocketToServiceRecord(myUUID);
btSocket.connect();
}
catch (IOException e){
Toast.makeText(getApplicationContext(), "Failed.", Toast.LENGTH_LONG).show();
}
The goal is to connect to the first available paired device. So far it always displays "Failed." even when I have an unconnected paired device sitting next to the phone.
Should I be doing this somewhere else in the app? I'm not really concerned with delaying the main activity since this is for a personal project.
Edit: spelling
A: All BT 2.1 devices and newer ones require the secure connection, so use the createRfcommSocketToServiceRecord instead of the createInsecureRfcommSocketToServiceRecord.
A: I actually figured it out... It was trying to connect to only the first paired connection. I put the try catch on a for loop, now it connects fine :)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,149 |
<?php
namespace Meling\Document;
/**
* Ссылки страницы
* Class Links
*
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$href .= '?' . htmlspecialchars($version);
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$this->links[$href] = $this->builder->buildLink($rel, $href, $type, $attributes);
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if ($sizes) {
$attributes = array_merge(array('sizes' => $sizes), $attributes);
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$this->links[$href] = $this->builder->buildLink('shortcut icon', $href, $type, $attributes);
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/**
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$this->links[$href] = $this->builder->buildLink('stylesheet', $href, $type, $attributes);
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/**
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$result = '';
foreach ($this->links as $link) {
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"redpajama_set_name": "RedPajamaGithub"
} | 6,073 |
{"url":"https:\/\/www.physicsforums.com\/threads\/projectile-motion-problem-involving-air-resistance.901556\/","text":"# Homework Help: Projectile Motion problem involving air resistance\n\nTags:\n1. Jan 25, 2017\n\n1. The problem statement, all variables and given\/known data\nOk, so I am attempting to solve a projectile motion problem involving air resistance that requires me to find the total x-distance the projectile traverses before landing again.\n\nGiven:\n$$\\\\ m=0.7\\text{kg} \\\\ k=0.01 \\frac{\\text{kg}}{\\text{m}} \\\\ \\theta=30 \\degree$$\n\n2. Relevant equations\n$$F_{air}=kv^2$$\n\n3. The attempt at a solution\nI divided the dynamics of the problem into x-component and y-component equations:\nFrom $$F_{x}=-kv_x^2$$, I got:\n$$x\\left(t\\right)=\\frac{m}{k}\\ln \\left(kv_it-m\\sec \\theta \\right)$$\n\nI divided the y-component of the motion into two parts--going up and going down:\nI used $$F_{y1}=-mg-kv_y^2$$ for the going up part, and I used $$F_{y2}=-mg+kv_y^2$$\nsolving the differential equation for the going up part, I got:\n$$v_y\\left(t\\right)=\\sqrt{\\frac{mg}{k}}\\tan \\left(\\arctan \\left(v_{yi}\\sqrt{\\frac{k}{mg}}\\right)-t\\sqrt{\\frac{g}{m}}\\right)$$ -- which is where I am stuck on... because when I plugged in my given values, the graph doesn't look right as its t-intercept is greater than one which is found from solving this kinematically without air resistance.\nsolving the differential equation for $$v_y(t)$$ of the going down part and integrating it, I got:\n$$y\\left(t\\right)=-\\frac{m}{k}\\ln \\left(\\cosh \\left(t\\sqrt{\\frac{gk}{m}}\\right)\\right)+y_i$$\n\nCan someone help me solve this problem?\n\n2. Jan 25, 2017\n\n### ehild\n\nNo, this is not correct. The force of air resistance is opposite to the velocity vector and its magnitude is proportional to v2. v2 is a scalar, it does not have components.\n\n3. Jan 26, 2017\n\n### Ray Vickson\n\nYou have not given an initial speed, so your problem is not completely specified.\n\nFor any given initial speed you can set up and solve the DEs numerically, and I think that is about the best you can do (in view of the criticism of \"ehild\" in #2).\n\nPerhaps, though, you can get usable approximations by attempting something like a perturbation theory approach in which you essentially expand in powers of the small parameter $k$.\n\n4. Jan 26, 2017\n\nSorry, I forgot to mention that-- the initial velocity is 9 m\/s\n\n5. Jan 26, 2017\n\nWhat do you mean? Can't any force in vectorspace be divided into\n\nLast edited by a moderator: May 8, 2017\n6. Jan 26, 2017\n\n### Ray Vickson\n\nThe friction force acts along the negative of the tangent to the trajectory. See, eg.,\nhttp:\/\/wps.aw.com\/wps\/media\/objects\/877\/898586\/topics\/topic01.pdf\nor\nhttp:\/\/young.physics.ucsc.edu\/115\/range.pdf\n\nThe case of $\\vec{f}_{\\text{friction}} = - k \\vec{v}$ is tractable, but not your case of $\\vec{f}_{\\text{friction}} = - k |v|^2\\, \\vec{v}\/|v| = -k |v| \\vec{v}$.\n\nLast edited by a moderator: May 8, 2017\n7. Jan 28, 2017","date":"2018-05-27 21:52:51","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\": 0.6997114419937134, \"perplexity\": 809.6293791454535}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-22\/segments\/1526794870470.67\/warc\/CC-MAIN-20180527205925-20180527225925-00517.warc.gz\"}"} | null | null |
\section{Introduction}
\noindent The first direct observation of gravitational waves in
2015~\cite{LIGOScientific:2016aoc} started a new revolutionary era in
gravitational wave astronomy. For the first time, it was possible to make
observations without the limitations brought by detecting electromagnetic
radiation or particles. Gravitational waves travel at the speed of light, and
unlike electromagnetic radiation, interact extremely weakly with matter,
travelling mostly undisturbed through the universe, carrying with them
unfiltered information of their origins. These properties make them an
outstanding probe of the pre-recombination era
universe~\cite{Ricciardone:2016ddg}. Sources of gravitational waves in the very
early universe produce a stochastic gravitational wave
background~\cite{Christensen:2018iqi, Caprini:2018mtu} that could be detectable
with future gravitational wave detectors~\cite{Allen:1996vm, Maggiore:1999vm},
like the upcoming Laser Interferometer Space Antenna
(LISA)~\cite{LISA:2017pwj}.
One potential source of contributions to the stochastic gravitational
background of the very early universe is a first order cosmological phase
transition~\cite{Witten:1984rs,Hogan:1986qda,Kamionkowski:1993fg,
Caprini:2019egz}. Such transitions proceed via the nucleation, expansion, and
merger of bubbles containing the new low temperature phase~\cite{Guth:1981uk,
Steinhardt:1981ct,Ignatius:1993qn, Espinosa:2010hh, Megevand:2017vtb}. The
phase transition comes to an end when all bubbles have merged with the
neighbouring bubbles so that the old phase has been replaced by the new one
everywhere in the fluid, leaving behind a characteristic spectrum of sound
waves~\cite{Hindmarsh:2013xza,Hindmarsh:2015qta,Hindmarsh:2017gnf,
Hindmarsh:2019phv} and if the transition is strong enough, significant
vorticity~\cite{Cutting:2019zws}. The sound waves are an important source of
gravitational waves. They persist in the fluid long after the phase transition
has completed, until dissipated away by the viscosity. Over time, these sound
waves can steepen into shock waves. Such a statistically random field of shocks
moving in various directions is known as acoustic turbulence~\cite{L_vov_1997,
Lvov:2000bdb}, and is the focus of this article.
Over the years the shock-containing compressional modes have received some
study but to a much lesser degree when compared to vortical turbulence. Perhaps
the most famous of such studies is that of the relatively simple Burgers'
equation~\cite{Frisch:2000td, BEC_2007}, which shares many of the properties
seen in the Navier-Stokes equations, apart from the chaotic behaviour and
randomness rising from small perturbations in the initial conditions. This is
because it is possible to integrate Burgers' equation explicitly. Burgers'
equation appears in the asymptotic limit in many physical situations, and has
been extensively studied due to its simplicity.
As for the Navier-Stokes equations, there have been some earlier studies that
deal with the compressional modes in fluids with a non-relativistic equation of
state. Numerical simulations of the two- and three-dimensional Navier-Stokes
equations with a longitudinal velocity component were performed by Porter,
Pouquet and Woodward in Ref.~\cite{Porter1992ANS} in the supersonic limit. They
pay attention to the power laws seen in the energy spectra and the kinetic
energy fractions between the longitudinal and transverse modes. The
interactions between the compressible and rotational modes in a
three-dimensional case were studied in 1990's in
Refs.~\cite{Kida1992EnergyAS,DUCROS1999517} with resolutions up to $1024^3$. Of
the more cosmologically oriented papers using relativistic fluid equations, one
worth highlighting is a paper by Pen and Turok~\cite{Pen:2015qta} that contains
one-, two- and three-dimensional simulations of shock formation in primordial
acoustic oscillations.
In this paper we study two-dimensional decaying acoustic turbulence using
numerical simulations with relativistic fluid equations and random initial
conditions. The emphasis is on the profile of the generated shock waves, their
effect on the shape of the energy spectrum, and the decay properties of the
kinetic energy and the integral length scale. Using the obtained results, we
also make an estimate for the gravitational wave power spectrum resulting from
shocks in a three-dimensional fluid flow.
We have chosen to conduct the simulations in two dimensions for several
reasons. The most important of these is that based on the existing literature,
the shocks -- amongst other phenomena -- have the same properties, like the
inertial range power laws in the energy spectrum, in two and three dimensions.
However, the two-dimensional case is simpler to analyse: for example, it is
easier to locate the shocks in two dimensions; some quantities, like the
vorticity, are simpler (it being a scalar in 2D); and there are additional
conserved quantities compared to 3D. In addition, there is the clear advantage
of 2D being more computationally efficient, allowing for the use of larger grid
sizes, increasing the dynamic range of the simulations. This makes it easier to
study non-linear phenomena like turbulence and shocks. In 3D, the largest
simulations to date have lattice sizes of $4200^3$~\cite{Hindmarsh:2017gnf},
and have not yet simulated sufficiently fast fluid flows for long enough to
show the development of turbulence after a cosmological phase transition.
Here, we simulate on grid sizes up to $10000^2$, for long enough to easily see
the development and decay of shocks.
We also save some compute time by starting simulations with the velocity and
density perturbations as a random field with given power spectra, rather than
simulating the whole phase transition. This allows us to conclude that the
effects we observe are not special to phase transitions. This is similar to the
approach of Ref.~\cite{RoperPol:2019wvy}, studying gravitational wave
production by vortical turbulence, which starts its simulations with the
Kolmogorov spectrum.
The contents of this article are as follows: Section~\ref{Methods} contains
information about the fluid equations, details of the numerical simulations and
the initial conditions, and lists some useful quantities used in characterising
the state of the fluid. Section~\ref{Results} concerns the results of our
numerical simulations and is divided into several subsections. In
Section~\ref{shockshape}, an analytical form for the shock shape is derived
using the fluid equations. Section~\ref{EnSpec} focuses on the energy spectrum
and its evolution over time, and in~\ref{kinendec} the decay of kinetic energy
and the integral length scale is studied. The last
subsection,~\ref{transenerg}, takes a closer look at the transverse kinetic
energy that arises from the longitudinal only initial conditions under these
fluid equations. In Section~\ref{GravSpec} an estimate is built for the
gravitational wave power spectrum resulting from a collection of sound waves
seen in our simulations. Two appendices are also included, Appendix~\ref{AppA}
being about testing the results obtained in Section~\ref{shockshape} by
conducting runs in a shock tube. Appendix~\ref{AppB} provides a more in depth
look at the initial conditions and some more technical aspects of the
simulations. Also listed are the runs used to obtain the tables and figures
presented in this paper, and the initial conditions for each of the runs. In
this paper we take the speed of light $c=1$.
\section{Methods} \label{Methods}
\noindent In this paper we study the evolution and properties of
two-dimensional decaying acoustic turbulence using numerical simulations of a
relativistic fluid. The equations we have employed are obtained from the
relativistic fluid equations by expanding them to second order in first
order small quantities that we have taken to be the non-relativistic bulk
velocity $\textbf{v}$, and the bulk and shear viscosities. We also relate the
pressure $p$ and energy density $\rho$ via the ultra-relativistic
equation of state $p = c_s^2 \rho$, where $c_s$ is the speed of sound
parameter, which has the value $1/\sqrt{3}$ in the case of a radiation fluid.
The derivation of the inviscid part of these equations is discussed in
more detail in~\cite{Brandenburg:1996fc}. They can be written as
\begin{widetext}
\begin{align}
\frac{\partial \rho}{\partial t} + (1+c_s^2) \nabla \cdot (\rho \textbf{v}) &
= 0 \label{cont} \\
\frac{\partial \textbf{v}}{\partial t} + \textbf{v} \cdot \nabla \textbf{v} - c_s^2 \textbf{v}
(\nabla \cdot \textbf{v}) + \frac{c_s^2}{\rho(1+c_s^2)} \nabla \rho & =
\frac{1}{1+c_s^2} \left[ \eta \nabla^2\textbf{v} +
\left(\frac{1}{3} \eta + \nu \right) \nabla (\nabla \cdot \textbf{v}) \right] \, ,
\label{EulerEq}
\end{align}
\end{widetext}
where $\eta$ and $\nu$ are the kinematic shear and bulk viscosity respectively.
They enter the equations via the additions of the anisotropic stress tensor and
the viscous bulk pressure to the energy momentum tensor. In the very early
universe the Reynolds number is expected to be large and the shear viscosity to
be dominant; its magnitude can be expressed in terms of the temperature and the
electromagnetic gauge coupling parameter~\cite{Arnold:2006fz}. With our choice
of scheme, viscosity is required to keep the numerical solution stable in cases
where there is significant power in the longitudinal modes. In the longitudinal
case shear and bulk viscosities also act in effectively the same way. The lack
of an external forcing term means that the kinetic energy in the system decays
over time, as it is being dissipated into internal energy by the viscosity at
small length scales.
We define the spectral density $P(k)$ through the two-point correlation
function of a homogenous and isotropic velocity field as
\begin{equation}
\left\langle v_i (\mathbf{k}) v_i (\mathbf{k}^\prime) \right\rangle = (2 \pi)^2
P(k) \delta(\mathbf{k} - \mathbf{k}^\prime)
\end{equation}
with $v_i(\mathbf{k})$ being the Fourier components of the velocity, related
through the Fourier transform pair
\begin{align}
v_i(\mathbf{k}) &= \int v_i(\mathbf{r}) e^{-i \mathbf{r} \cdot \mathbf{k}} \,
d^2r \\
v_i(\mathbf{r}) &= \frac{1}{(2 \pi)^2}\int v_i(\mathbf{k}) e^{i \mathbf{r}
\cdot \mathbf{k}} \, d^2k \, ,
\end{align}
where $\mathbf{k}$ is the wave vector. We define a quantity $E(k)$ through
\begin{equation} \label{PowSpecEnSpecRel}
E(k) = \frac{k}{4 \pi} P(k) \, ,
\end{equation}
where $P(k)$ is the spectral density, such that
\begin{equation} \label{En_spec_def}
\frac12 \left\langle \textbf{v}^2 \right\rangle = \int\limits_0^{\infty} E(k) \, dk \, ,
\end{equation}
from which we directly obtain the root mean square (rms) value of the
velocity vector field.
In a system with a non-relativistic equation of
state, $E(k)$ is also the linear spectrum of the kinetic energy per unit
mass, so we will refer to it as the energy spectrum. The true specific kinetic
energy in our system is
$(1 + c_\text{s}^2) \left\langle \textbf{v}^2 \right\rangle$.
The velocity field is decomposed into longitudinal and transverse components so
that
\begin{equation}
\textbf{v} = \textbf{v}_\parallel + \textbf{v}_\perp \, ,
\end{equation}
where the components fulfil the properties
\begin{equation}
\nabla \cdot \textbf{v}_\perp = 0 \, , \qquad \nabla \times \textbf{v}_\parallel = 0 \, .
\end{equation}
This decomposition also splits the energy spectrum into two parts,
$E(k) = E_\parallel(k) + E_\perp (k)$, where the longitudinal spectrum contains
the contribution from acoustic turbulence, and the transverse part the vortical
contributions associated with traditional fluid turbulence that consists of
vortices of various sizes.
The fluid equations are integrated numerically using finite difference methods.
Time integration is performed using the fourth order Runge-Kutta scheme, and
spatial derivatives are evaluated with a second order central difference
scheme.\footnote{We have checked that no significant changes to our results
would be introduced by a fourth order scheme. For more information about
the scheme and its viability, see Appendix~\ref{AppA}.} The
spatial grid is a square with $N^2$ points and unit spacing. The time step size
is chosen as $\Delta t = 0.2 \Delta x$, providing a stable numerical solution.
Periodic boundary conditions are enforced on all edges of the grid. The
corresponding reciprocal lattice is spanned by the wave vectors $\mathbf{k}_i$,
whose elements obtain values in the range $[- \pi, \pi)$ with a spacing of
$\Delta k = 2 \pi /(N \Delta x)$.
The initial conditions are given for the longitudinal and transverse components
using an initial spectral density of the form
\begin{equation} \label{initPowSpec}
P(k) = A \frac{
(k/k_p)^{\beta_0}}{\left[ 1 + (k/k_p)^{\alpha_0/ \gamma} \right]^\gamma}
e^{-(k/k_d)^2} \, .
\end{equation}
The parameters $\alpha_0$ and $\beta_0$ set the initial values for the inertial
range and low-$k$ power laws, $\gamma$ affects the shape around the peak of the
spectrum, and $k_p$ is the initial wavenumber around which the peak is located.
The inverse of $k_p$ deterimines the integral length scale $L$ characterizing
the length scale at which most of the energy is located. We have also
introduced an exponential suppression factor in order to reduce discretization
effects at high wavenumbers. This suppression is controlled by the parameter
$k_d$. Since we are interested in acoustic turbulence only, we set
$P_\perp(k)=0$ initially, i.e.~the initial velocity field is purely
compressible. The Fourier components obtained from $P_\parallel(k)$ are given
random phases in such a way that the resulting initial velocity field is real
and statistically random. The energy density is initialized by writing it out
as
\begin{equation}
\rho (\mathbf{r}, t) = \rho_0 + \delta \rho (\mathbf{r}, t) \,
\end{equation}
where the density perturbation $\delta \rho$ is initialized in the same way as
the velocity components. More information about the execution of the
simulations and their initial conditions can be found in Appendix~\ref{AppB}.
Next, we define some quantities that are useful in analysing the flows. We
write the rms velocity of the longitudinal component as
$\bar{v}=\sqrt{\left\langle \textbf{v}_\parallel^2 \right\rangle}$, and define the
integral length scale of the longitudinal component as
\begin{equation} \label{int_len_scale}
L = \frac{2}{\bar{v}^2} \int\limits_0^\infty \frac{1}{k} E_\parallel(k) \, dk
\, .
\end{equation}
From the initial values of these two quantities we can define a time scale,
\begin{equation}
t_s=L_0/\bar{v}_0
\end{equation}
where $\bar{v}_0$ is the initial value of the rms longitudinal velocity, and
$L_0$ is the initial value of the integral scale. In addition to the integral
scale, there are other relevant length scales constructed from the effective
viscosity $\mu = 4 \eta/3 + \nu$, the rms velocity $\bar{v}$, and the quantity
\begin{equation}
\label{eq:dilatationsquared}
\mathcal{D} = \left\langle (\nabla \cdot \textbf{v})^2 \right\rangle,
\end{equation}
the compressional part of the enstrophy, which can be used to quantify
``shockiness'' in the system. First, we have
\begin{equation} \label{swidth}
\delta_s = \mu/\bar{v} \, ,
\end{equation}
which we shall see characterises the shock width. We also have the longitudinal
counterparts of the Kolmogorov and Taylor microscales $L_K$ and $L_T$. We
define the Kolmogorov microscale as
\begin{equation}
L_K = \left( \frac{\mu^3}{\epsilon} \right)^{1/6} \, ,
\end{equation}
where $ \epsilon = - \dot{\mathcal{D}}$ is the dissipation rate of
$\mathcal{D}$. From the equations of motion it follows that
\begin{equation}
\epsilon
= \frac{4 \mu}{1+c_s^2} \int\limits_0^\infty k^4 E_\parallel (k) \, dk \, .
\end{equation}
As in the case of vortical fluid turbulence, the Kolmogorov microscale
specifies the length scale at which viscosity is dominant and dissipates
kinetic energy into internal energy. The Taylor microscale is an intermediate
length scale located between the integral and Kolmogorov length scales at which
viscous effects become significant, and is defined by
\begin{equation}
L_\text{T} = \sqrt{ \frac{ \bar{v}^2 } {\mathcal{D}} } \, .
\end{equation}
The Taylor and Kolmogorov wavenumbers are defined as inverses of the
corresponding length scales. We also define the longitudinal counterpart of the
Reynolds number
\begin{equation} \label{ReynoldsNum}
\text{Re} = \frac{\bar{v} L}{\mu} \, ,
\end{equation}
that, as in the vortical only case, characterises the strength of non-linear
effects in the flow, which in the longitudinal case means shocks. In other
words, large values of the longitudinal Reynolds number lead to a formation of
very strong and sharp shocks.
\section{Results} \label{Results}
\noindent We have performed numerical simulations of acoustic turbulence with
grid sizes of $N=4080$ and $N=10080$ with various initial power spectra
(\ref{initPowSpec}) for the longitudinal component, leading to various shock
formation times, and initial longitudinal Reynolds numbers in the range 16-223.
We call runs with an initial Reynolds number that lies at the end of this range
high Reynolds number runs. These kind of runs are obtained by increasing the
initial rms velocity and also by increasing the initial integral length scale,
which moves the top of the energy spectrum to lower wavenumbers. We have run
for about 60 shock formation times in all of our runs to give the system enough
time to show sufficient decay characteristics. A table listing each run and
their initial conditions is found in Appendix~\ref{AppB}. In this section we
shall present our findings from these runs focusing on the shape of the shocks,
their impact on the energy spectrum, the decay of the longitudinal kinetic
energy and the integral length scale, and the generation of transverse kinetic
energy under these equations from the longitudinal only initial conditions.
\subsection{Shock shape} \label{shockshape}
\noindent In order to study the shape of the shock waves, we solve equations
(\ref{cont}) and (\ref{EulerEq}) for a single shock moving towards the positive
x-axis using the ansatz
\begin{equation}
\rho (\mathbf{r}, t) = L(k_s(x-ut)), \quad \vec{v} (\mathbf{r}, t) =
V(k_s(x-ut)) \vec{e}_x \, .
\end{equation}
Here $u$ denotes the shock velocity. The resulting differential equation is
then written in terms of $\chi=k_s(x-ut)$ for $V(\chi)$ and is simplified by
assuming that $V(\chi) \ll u$. Its solution is
\begin{equation} \label{ShockProf}
V(x, t) = \frac{\sqrt{a^2+2bC}}{b} \tanh \left[ k_s (x-x_0-ut) \right]
- \frac{a}{b} \, ,
\end{equation}
where the parameters $a$ and $b$ can be written as:
\begin{gather}
a = u \left(1 - \frac{c_s^2}{u^2} \right) \\
b = (1+c_s^2) \left( \frac{c_s^2-1}{c_s^2+1} - \frac{c_s^2}{u^2} \right) \, .
\end{gather}
The integration constant $C$ is fixed using the conditions that on the right
side of the shock $V$ approaches the value $V_+$, and on the left side the
value $V_-$ while the derivative of $V$ approaches zero on both sides. For a
right-moving shock we also have $V_- > V_+$. These conditions fix $C$ as
\begin{equation}
C = a V_+ + \frac{b}{2} V_+^2 = a V_- + \frac{b}{2} V_-^2 \, .
\end{equation}
The shock velocity is solved from this equation and can be written in the form
\begin{equation} \label{ShockVel}
u = c_s \left( 1 - \frac{2}{1+c_s^2} \xi \right)^{- \frac{1}{2}} \, , \quad
\xi = \left( 1 + \frac{2}{\widetilde{\delta \rho}_+
+ \widetilde{\delta \rho}_-} \right)^{-1}
\end{equation}
where $\widetilde{\delta \rho}_+$ and $\widetilde{\delta \rho}_-$ are the
values of the fractional density perturbation on the left and right sides of
the shock, obtained by replacing $V_+$ and $V_-$ using the relation between $V$
and the fractional density perturbation
\begin{equation} \label{VelDenRel}
V(x,t) = \frac{u}{1+c_\text{s}^2} \widetilde{\delta \rho} (x, t) \, , \quad
\widetilde{\delta \rho} (x, t) = \frac{\delta \rho(x,t)}{\rho_0}.
\end{equation}
Equation (\ref{ShockVel}) gives us the expected result of the shock velocity
always being larger than the speed of sound for a right-moving shock with
$\widetilde{\delta \rho}_+=0$, since the smallest value $\xi$ can obtain is
zero in the limit of the shock wave amplitude going to zero. It is also evident
that the shock velocity increases with increasing amplitude.
The width of the shock is controlled by the parameter $k_s$ that can be written
in terms of the above quantities as
\begin{equation}
k_s = \frac{3 (1+c_s^2) \sqrt{a^2+2bC}}{8 \mu}
\end{equation}
and whose inverse is of the same order of magnitude as the shock width
$\delta_s$.
The parameters $a$, $b$ and $C$ all increase with increasing shock velocity,
which indicates that steep shocks are obtained when the amplitude of the shocks
is large. Here the effects of the viscosities are clearly seen, with small
viscosity values leading to steep shocks. We have conducted shock tube runs to
study and verify the results obtained here by investigating shocks in a very
narrow and long grid. These are discussed in Appendix~\ref{AppA}.
In our 2D simulations the initially smooth density and velocity fields
generate multiple shock waves moving in various directions, after a time of
order $t_s$. This can be seen in Figure~\ref{fig:rhodiv}, which on the left
shows a contour plot of the density perturbation shortly after the shocks have
formed. In the second plot on the right, the divergence of the velocity field
has been plotted to highlight the shocks. Figure~\ref{fig:shockslices} shows
zoomed in slices of the fractional density perturbation both in the high and
low Reynolds number cases. In the case of the former, oscillations can be seen
near the top of the shock, similar to the Gibbs phenomenon~\cite{Gibbs}. This
limits the obtainable Reynolds numbers, as reducing the viscosity too much
causes these oscillations to grow, eventually ruining the solution. This also
has an effect on the shape of the energy spectrum around the Kolmogorov
microscale, creating a bump in the spectrum at this wavenumber range.
\begin{figure*}
\centering
\subfloat[]{
\includegraphics[width=0.5\textwidth]{Figure1a.pdf}
\label{fig:fig1a}
}
\subfloat[]{
\includegraphics[width=0.5\textwidth]{Figure1b.pdf}
\label{fig:fig1b}
}
\caption{The density perturbation $\delta \rho$ of a $4080^2$
resolution run that has the same initial conditions as run 9 (a),
and the divergence of the corresponding velocity field $\nabla \cdot
\textbf{v}$ (b), showing the locations of the shocks at $t \approx 6.5 t_s$.}
\label{fig:rhodiv}
\end{figure*}
\begin{figure*}
\centering
\subfloat[]{
\includegraphics[width=0.5\textwidth]{Figure2a.pdf}
\label{fig:fig2a}
}
\subfloat[]{
\includegraphics[width=0.5\textwidth]{Figure2b.pdf}
\label{fig:fig2b}
}
\caption{Zooms of the fractional density perturbation slices of the same
$4080^2$ resolution run as in Figure~\ref{fig:rhodiv} (a), and a similar
moderate Reynolds number run (b) that has identical initial conditions to those
of run 2. Both figures show the shocks after about $6.5$ shock formation
times.}
\label{fig:shockslices}
\end{figure*}
\subsection{Shocks and the energy spectrum} \label{EnSpec}
\noindent Based on our simulations that use various different initial spectral
densities (\ref{initPowSpec}), we find that as the initial conditions steepen
into shocks, the features induced by the initial conditions near the peak of
the energy spectrum are erased, and that the energy spectrum obtains a
universal broken power law form whose power law values differ from those of
the initial conditions. After a single shock formation time, the inertial range
power law located between the integral length scale and the Taylor microscale
settles into the well-known value of -2, first proposed and obtained by
Burgers~\cite{Burgers_1948} for the one-dimensional Burgers equation, and later
generalised to multiple dimensions in the case of the Euler and the continuity
equation by Kadomtsev and Petviashvili~\cite{KP}.
The evolution of the inertial range power law in one of our runs has been
plotted in Figure~\ref{fig:inerpowlaw} as a function of the number of shock
formation times. Due to strong oscillations in the spectrum at early times,
the data for the plot has been obtained by fitting a power law $k^{- \varphi}$
to two intervals; between the Taylor and half the Kolmogorov wavenumber at
early times when $t/t_s < 0.6$, and between the integral wavenumber and the
Taylor wavenumber otherwise. Early on, obtaining decent fits of the inertial
range is obstructed by these oscillations, so we have tracked the evolution of
the power law range of the initial conditions instead, which initially develops
towards a similar power law value but at a higher wavenumber range. At
$t/t_s = 0.6$ the oscillations have weakened and the two ranges coincide, to a
reasonable accuracy, so we have opted to change the limits of the fit at this
particular time.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Figure3.pdf}
\end{center}
\caption{\label{fig:inerpowlaw} Evolution of the inertial range power law
index obtained by fitting a power law $k^{- \varphi}$ to the data of run 9. The
bounds of the fit are the Taylor and half times the Kolmogorov wavenumbers when
$t/t_s<0.6$ (dashed curve) and the integral wavenumber and the Taylor
wavenumber otherwise.}
\end{figure}
In order to study and determine the universal shape of the spectrum, we extract
the time dependence from the spectrum. Figure~\ref{fig:energyspecs} shows the
time evolution of the spectrum with dark lines corresponding to late times.
Over time the integral length scale $L$, introduced in section~\ref{Methods},
increases as evidenced by the shift of the peak of the spectrum towards small
wavenumbers. Thus we fix the location of the peak by scaling the wavenumber by
$L$, so that the spectrum becomes a function of $\kappa = L(t)k$. The other
time dependent feature of the spectrum is the decay of the compressional
kinetic energy $\mathcal{E}_\parallel$, which causes the magnitude of the
spectrum to decrease. Thus, we write the spectrum in the form
\begin{equation} \label{EnPsi}
E_{\parallel} (\kappa, t) = L(t) \mathcal{E}_{\parallel}(t) \Psi(\kappa) \, ,
\quad \kappa = L(t)k \, .
\end{equation}
The function $\Psi(\kappa)$ is plotted in Figure~\ref{fig:collapseplots} at
even time intervals after the shocks have formed, and we see the spectra
collapsing onto a single function on all but the very smallest length scales.
\begin{figure*}
\centering
\begin{minipage}[t]{1.0\columnwidth}
\centering
\includegraphics[width=\columnwidth]{Figure4.pdf}
\caption{\label{fig:energyspecs} The scaled energy spectrum of run 2 plotted
every $10 t_s$ from $3 t_s$ onwards. Dark colours correspond to late times.}
\end{minipage} \hfill
\begin{minipage}[t]{1.0\columnwidth}
\centering
\includegraphics[width=\columnwidth]{Figure5.pdf}
\caption{\label{fig:collapseplots} The energy spectra of
Figure~\ref{fig:energyspecs} collapsing into the function $\Psi(Lk)$. The black
line above the inertial range describes the $k^{-2}$ power law, while the black
line above the low-$k$ range goes like $k^{2.4}$.}
\end{minipage}
\end{figure*}
In the range where the spectra collapse well, we model the function $\Psi$ by
a broken power law form. We assume that this form holds at wavenumbers
corresponding to length scales larger than the Taylor microscale, so that
\begin{equation} \label{BrokPowLaw}
\Psi(\kappa) = \Psi_0 \frac{(\kappa/ \kappa_p)^{\beta}}{
1+(\kappa/ \kappa_p)^\alpha} \, , \quad \kappa \ll L/L_T \, ,
\end{equation}
where $\beta$ is the low-$k$ power law index, and the inertial range
power law is given by $\beta-\alpha$. From equation (\ref{EnPsi}) it follows
that the integral of $\Psi$ over all values of $\kappa$ must equal unity, and
another condition follows from substituting (\ref{EnPsi}) into the definition
of the integral length scale in Equation (\ref{int_len_scale}). For both of
these conditions to be satisfied simultaneously, the parameters $\Psi_0$ and
$\kappa_p$ must fulfil
\begin{equation} \label{Afix}
\Psi_0 = \frac{\alpha}{\pi} \sin \left(\frac{\pi \beta}{\alpha}\right)
\end{equation}
and
\begin{equation} \label{KappaFix}
\kappa_p = \frac{ \sin \left(\frac{\pi (\beta+1)}{\alpha}\right)}{ \sin
\left(\frac{\pi \beta}{\alpha}\right)} \, ,
\end{equation}
when $\beta - \alpha < -1$, meaning that these parameters get fixed by the
normalisation condition and the choice of $L$ as the integral scale. In the
high-$\kappa$ region where the collapse is not as good, and the function still
changes a little over time. We ascribe this temporal behaviour to the changing
shape of the shocks caused by the viscous dissipation. Thus, we expect the
function $\Psi$ to be a broken power law modulated by a function that depends
on the width of the shocks. To quantify the dilatation of the shocks, we use
the dimensionless quantity $\mathcal{D} L^2 / \mathcal{E}_\parallel$ to measure
shockiness in the system, where $\mathcal{D}$ is defined through
Eq.~(\ref{eq:dilatationsquared}). The quantity $\nabla \cdot \mathbf{v}$, often
called the dilatation, obtains large values at the locations of the shock waves
briefly after shock formation in comparison to the values seen in the initial
conditions, which leads to an increase in its rms value $\sqrt{\mathcal{D}}$.
The dimensionless quantity is plotted in Figure~\ref{fig:avg_div_sqrd}, and a
sharp increase in its value can be seen around one shock formation time, after
which the quantity decreases, as the shocks deteriorate.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Figure6.pdf}
\end{center}
\caption{\label{fig:avg_div_sqrd} Plot of the dimensionless quantity
$\mathcal{D} L^2 / \mathcal{E}_\parallel$ that is used to measure shockiness of
the system as a function of shock formation times. The data for the plot has
been obtained from run 9. The black line indicates a $t^{-0.4}$ power law.}
\end{figure}
In order to determine what impact the shocks have on the energy spectrum, we
follow the method presented in Ref.~\cite{Kuznetsov} to find the form of the
two-dimensional energy spectrum using the one-dimensional spectrum. The
one-dimensional energy spectrum of a tanh shock is obtained using the Fourier
transform and has the form
\begin{equation} \label{1dspec}
E_1(k) = |\mathcal{F}(\tanh(k_s x))|^2 = \frac{\pi^2}{ k_s^2} \text{csch}^2
\left( \frac{\pi k}{2 k_s} \right) \, ,
\end{equation}
which can be related to the $D$-dimensional spectrum by separating the
wavevector $\mathbf{k}$ into two parts; $\mathbf{k}_1$ and its transverse
projection $\mathbf{k}_\perp$ and then by integrating over the latter
\begin{align}
E_1(k_1) &= \int E_D (|\mathbf{k}|) d \mathbf{k}_\perp \\
&= \frac12 \Omega_{D-1} \int\limits_{k_1^2}^\infty E_D(s) (s^2 - k_1^2)^{
\frac{D-3}{2}} \, ds^2 \, .
\end{align}
where $\mathbf{k} = \mathbf{k}_\perp + \mathbf{k}_1$ and $\Omega_{D-1}$ is the
solid angle of the ($D-2$)-sphere. For $D=2$ the equation can be written as
\begin{equation}
E_1(k_1) = \int\limits_{k_1^2}^\infty \frac{E_2(\sqrt{u})}{\sqrt{u-k_1^2}} \,
du \, .
\end{equation}
Now we can use the property, that $E_1$ is a first order Liouville-Weyl
fractional integral~\cite{Erdelyi} of $E_2$ to solve for the two-dimensional
spectrum, yielding
\begin{equation} \label{E2LW}
E_2(k) = \frac{1}{\pi} \int\limits_{k^2}^\infty \frac{1}{\sqrt{u-k^2}}
\frac{d}{du} E_1(\sqrt{u}) \, du \, .
\end{equation}
Substituting Equation (\ref{1dspec}), changing the variable, and defining
\begin{equation}
E_2 = \frac{\pi^2}{k_s^3} \mathcal{I} \, , \quad P= \frac{\pi k}{2 k_s}
\end{equation}
allows us to write (\ref{E2LW}) as
\begin{equation} \label{IP}
\mathcal{I}(P) = \int\limits_1^\infty \frac{ds}{\sqrt{s^2-1}}
\frac{\cosh(P s)}{\sinh^3(P s)} \, .
\end{equation}
The integral in $\mathcal{I}$ does not have a closed form solution, but its
asymptotic behaviour at small and large values of the argument can be found to
be
\begin{equation}
\mathcal{I} (P) \sim
\begin{cases}
\dfrac{\pi}{4 P^3} \, , \quad P \ll 1 \\
\\
2 \sqrt{\dfrac{\pi}{P}} e^{-2P} \, , \quad P \gg 1
\end{cases} \, .
\end{equation}
We now propose the function $\Psi(\kappa)$ to have the form
\begin{equation}
\Psi(\kappa) = \widetilde{\Psi}_0 \frac{(\kappa/ \kappa_p)^{\beta+3}}{
1+(\kappa/ \kappa_p)^\alpha} \mathcal{I}
\left( \frac{\pi \kappa}{2 \kappa_s} \right) \, , \label{CollapsePsiEq}
\end{equation}
where $\kappa_s=k_s L$. Note that the parameter $\beta$ still denotes the
low-$k$ power law. Figure~\ref{fig:Psifit} shows $\Psi(\kappa)$ obtained from
simulation data in comparison to the fit resulting from using the equation
above. It is seen that the fit is very good in the high-$\kappa$ region. At the
wavenumber range between the Taylor and the Kolmogorov wavenumbers, the fit
deviates a bit from the simulation data, leading to slightly steeper values for
the inertial range power than $k^{-2}$. The fit in this range can be improved
by increasing the complexity of the fitting function, for example, by using a
double broken power law instead, but for our purposes we deem
Equation~(\ref{CollapsePsiEq}) to be a good enough estimate for the spectral
collapse function $\Psi$.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Figure7.pdf}
\end{center}
\caption{\label{fig:Psifit} The function $\Psi(\kappa)$, where the blue line
is the curve obtained from simulation data of run 2 at $t \approx 20 t_s$, and
the dashed red line is a fit using Equation~(\ref{CollapsePsiEq}). The obtained
values for the fit parameters are $\widetilde{\Psi}_0=0.0034$, $\alpha \approx
4.801$, $\beta \approx 2.013$. $\kappa_p \approx 0.976$, and
$\kappa_s=12.541$.}
\end{figure}
\subsection{Decay of longitudinal kinetic energy} \label{kinendec}
\noindent Energy is dissipated into heat by the viscosity at small length
scales, and since our fluid equations do not contain a forcing term, the total
kinetic energy decreases over time. Figure~\ref{fig:kin_en} plots the kinetic
energy normalised by its initial value for several runs as a function of the
number of shock formation times. It is seen that after about 10 shock formation
times the kinetic energy decays following a power law form.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Figure8.pdf}
\end{center}
\caption{\label{fig:kin_en} The kinetic energy normalised by the
initial value plotted for some the runs listed in
Table~\ref{tab:table3} as a function of the shock formation time.}
\end{figure}
In order to find an analytical function that models the kinetic energy
behaviour seen in the figure, we have applied the analysis made by Saffman in
Ref.~\cite{saffman1971}, but to the longitudinal-only case instead. The
starting point is the relation describing the kinetic energy decay due to
viscous dissipation, which for the fluid equations (\ref{cont}) and
(\ref{EulerEq}) can be shown to be
\begin{equation} \label{viscousdiss}
\frac12 \frac{d \left\langle \textbf{v}^2_\parallel \right\rangle}{dt} = - \frac{2 \mu}{1+c_\text{s}^2} \int\limits_0^\infty k^2 E_\parallel(k) \, dk \, ,
\end{equation}
when the vorticity $\nabla \times \mathbf{v}$ is zero. There is some
vorticity generated from longitudinal only initial conditions under
these fluid equations, as discussed in the next section, but the
transverse kinetic energy is still small enough in comparison to the
longitudinal kinetic energy for the above equation to be
approximately valid. After the shocks have formed, the energy
spectrum has the familiar behaviour of $k^{-2}$ in the inertial range.
According to Saffman, in the case of Burger's equation, the spectrum in the
inertial range has the form
\begin{equation} \label{saffman_spec}
E_\parallel(k) = \frac{\mathcal{L} \overline{J}^2}{4 \pi k^2} \, ,
\end{equation}
which we assume to also hold for the fluid equations employed in this
paper by applying the physical interpretations of $\mathcal{L}$ and
$\overline{J}^2$ to the longitudinal case. Here $\mathcal{L}$ is the mean
length of shocks per unit area, and $\overline{J}^2$ is the mean square jump
in velocity across the shock. In analogy to~\cite{saffman1971}, we cut off the
integral at the wavenumber corresponding to the length scale of the shock width
$\delta_s$ and substitute (\ref{saffman_spec}) into (\ref{viscousdiss}), which
after integration gives
\begin{equation} \label{KinEnDifEq}
\frac12 \frac{d \left\langle \textbf{v}^2_\parallel \right\rangle}{dt} = - \frac{\mu}{2
\pi (1+c_s^2)} \frac{\mathcal{L} \overline{J}^2}{\delta_s} \propto
\frac{\bar{v}^3(t)}{L(t)} \, ,
\end{equation}
where to obtain the latter expression we have used the proportionality
relations
\begin{equation} \label{PropRels}
\quad \mathcal{L} \propto L^{-1} \, , \quad \overline{J}^2 \propto \bar{v}^2 \,
,
\end{equation}
and the definition for the shock width $\delta_s$ in equation (\ref{swidth}).
Now in order to make progress, we need to find a relation between the time
behaviour of the integral length scale and the rms velocity. To this end, we
write the spectrum in the form
\begin{equation} \label{BrokPowLawk}
E_\parallel (k, t) = D(t) \frac{ \left[ k/k_p(t) \right]^\beta}{
1+ \left[ k/k_p(t) \right]^\alpha} \, ,
\end{equation}
where the prefactor $D(t)$ contains the time dependence of the spectral
magnitude. Here we have ignored the high-$k$ behaviour of the spectrum found in
the previous section. Now in the low-$k$ power law range, when $k \ll k_p$, the
spectrum becomes
\begin{equation}
E_\parallel (k, t) \approx D(t) k_p(t)^{- \beta} k^\beta \, .
\end{equation}
The very low-$k$ end of the spectrum stays mostly unchanged, maintaining its
magnitude and power law index, as seen in Figure~\ref{fig:energyspecs}. Hence,
it can be approximated that
\begin{equation} \label{IRpowlaw}
D(t) k_p(t)^{- \beta} = \text{const.}
\end{equation}
Substituting this spectrum into Equation~(\ref{En_spec_def}) gives
\begin{equation}
\frac12 \left\langle \textbf{v}_\parallel^2 \right\rangle = D(t) \int\limits_0^{\infty}
\frac{ \left[ k/k_p(t) \right]^\beta}{1+ \left[ k/k_p(t) \right]^\alpha} \,
dk \, .
\end{equation}
Using this form for the spectrum leads to an overestimation of the integral,
since we have ignored the high-$k$ behaviour, but we argue that this does not
affect the value of the energy significantly, since the largest contribution to
the integral comes from the energy containing scales around the peak of the
spectrum, and the contributions from scales smaller than the Taylor microscale
are small in comparison. After a change of variables $s = k/k_p$ the integral
becomes
\begin{equation}
\frac12 \left\langle \textbf{v}_\parallel^2 \right\rangle = D(t) k_p(t)
\int\limits_0^\infty \frac{s^\beta}{1+s^\alpha} \, ds \, .
\end{equation}
Since the power laws stay the same after the shocks have formed, the parameters
$\alpha$ and $\beta$ are mostly constant. Thus, the integral gives
approximately a constant value, and by using the definition of the rms velocity
and relation (\ref{IRpowlaw}) with $k_p(t)^{-1}=L(t)$, we get
\begin{equation} \label{CondForL}
\bar{v}^2 (t) L(t)^{\beta + 1} = \text{const.} \equiv \xi^{-(\beta + 1)} \, .
\end{equation}
Using this, we can now solve Equation~(\ref{KinEnDifEq}) for the energy
$\mathcal{E} = \left\langle \textbf{v}_\parallel^2 \right\rangle/2=\bar{v}^2/2$ with
the initial condition ${\mathcal{E}(t=0)=\mathcal{E}_0}$, yielding
\begin{equation} \label{KinEnAnalytical}
\mathcal{E}(t) = \frac{\mathcal{E}_0}{
\left( 1 + C \frac{t}{t_s} \right)^{\zeta}} \, , \quad \zeta =
\frac{2 (\beta + 1)}{\beta + 3} \, ,
\end{equation}
where in the denominator we have used (\ref{CondForL}) to write the constant
$\xi$ in terms of the initial value of the integral length scale $L_0$ and the
initial energy $\mathcal{E}_0$, resulting in $\xi (2 \mathcal{E}_0)^{(\beta +
3)/(2(\beta + 1))}=L_0^{-1} \bar{v}_0 = t_s^{-1}$, which is used as an estimate
for the shock formation time. We have also absorbed all constants into the
parameter $C$, whose value depends on the values of the prefactors of the
relations listed in Equation~(\ref{PropRels}). Without knowing the numerical
values of the prefactors, the value of $C$ can be obtained by fitting.
Based on our fits detailed later in this section, its typical value lies
in the range between 0.29 and 0.47. With the help of the above result,
Equation~(\ref{CondForL}) can be used to find $L(t)$, which reads as
\begin{equation} \label{IntLenAnalytical}
L(t) = L_0 \left( 1 + C \frac{t}{t_s} \right)^\lambda \, , \quad \lambda =
\frac{2}{\beta + 3} \, .
\end{equation}
The integral length scales of some of the runs featured in
table~\ref{tab:table3} of Appendix~\ref{AppB} are plotted in
Figure~\ref{fig:l_int_lon} against time in units of $t_s$ the shock formation
time.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Figure9.pdf}
\end{center}
\caption{\label{fig:l_int_lon} The longitudinal integral length scale
normalised by the initial value plotted for multiple runs as a function of the
number of shock formation times.}
\end{figure}
The results obtained for the power law values as functions of the
low-$k$ power law $\beta$ in equations (\ref{KinEnAnalytical}) and
(\ref{IntLenAnalytical}) coincide with those found by similar
methods for three-dimensional classical vortical
turbulence~\cite{lesieur2008turbulence}.
Equations (\ref{KinEnAnalytical}) and (\ref{IntLenAnalytical}) can now be used
as fitting functions to the curves seen in Figures~\ref{fig:kin_en}
and~\ref{fig:l_int_lon}, and the obtained power law indices can then be
compared to the analytical ones by measuring the low-$k$ power law index
$\beta$ of each run. The fits to the kinetic energy and the integral length
scale are constrained to the shock containing phase by using fitting ranges
whose lower boundary lies in the range $t/t_s \geq 1$. In these ranges the
inertial range has a power law of $k^{-2}$ and the fitting equations are valid.
Figure~\ref{fig:kin_en_L_fits} shows a pair of such fits for a single run. We
have varied the lower bound of the fit to all data points in the range
$1 \leq t/t_s \leq 3$ and averaged over the results to obtain the averaged
power law indices $\hat{\zeta}$ and $\hat{\lambda}$.
\begin{figure*}
\centering
\subfloat[]{
\includegraphics[width=0.5\textwidth]{Figure10a.pdf}
\label{fig:kin_en_fit}
}
\subfloat[]{
\includegraphics[width=0.5\textwidth]{Figure10b.pdf}
\label{fig:l_int_fit}
}
\caption{Fits of the functions (\ref{KinEnAnalytical}) and
(\ref{IntLenAnalytical}) (red dashed curves) to the time evolution of the
kinetic energy (a) and the integral length scale (b) of run 2 (blue curves).
The fitting range is $t/t_s \geq 1$ and the parameter values obtained are
$\mathcal{E}_0 \approx 0.00263$, $\zeta \approx 1.207$, and $C \approx 0.391$
for the kinetic energy, and $L_0 \approx 20.8$, $\lambda \approx 0.346$ and $C
\approx 0.455$ for the integral length scale.}
\label{fig:kin_en_L_fits}
\end{figure*}
The low-$k$ power law indices are measured by fitting a broken power law, akin
to that in Equation~(\ref{BrokPowLawk}), on a suitable wavenumber range and
averaging the obtained values for the fit parameters over times $8 \leq t/t_s
\leq 12$, which in our simulations results to around 120 data points on
average. The suitable range in question has been chosen to be
$k \in [1/6L, 1/L_T]$, which contains the inertial range and a sufficient
amount of the low-$k$ power law. Time averaging like this is necessary because
there are oscillations in the spectrum that the fitting algorithm is sensitive
to. We have used the standard deviations of the time averaging to quantify the
strength of these oscillations.
It is also possible to derive relations between $\beta$, $\zeta$, and $\lambda$
that can be used to test the robustness of the theory by comparing to the
values obtained from the simulations by fitting. Such relations have been
obtained in Refs.~\cite{Olesen:1996ts, Brandenburg:2016odr} by considering
appropriate scaling of the energy spectrum and by making use of the rescaling
invariance of the hydrodynamic equations. Here, one relation follows
immediately from Equation~(\ref{CondForL}), which requires
\begin{equation} \label{ParamCond1}
\lambda (\beta + 1) - \zeta = 0
\end{equation}
for it to be valid. This relation can also be obtained directly from the power
laws in equations (\ref{KinEnAnalytical}) and (\ref{IntLenAnalytical}). A
relation containing only $\zeta$ and $\lambda$ can also be derived by replacing
$\beta$ in the equation above by using either of these two equations, giving
\begin{equation} \label{ParamCond2}
\zeta - 2(1-\lambda) = 0 \, .
\end{equation}
Table~\ref{tab:table1} lists the averaged power law indices and the standard
deviations obtained from fits to the time evolutions of the kinetic energy and
the integral length scale. Power laws obtained from time averaging are denoted
by hats, and alongside them are the power laws obtained from equations
(\ref{KinEnAnalytical}) and (\ref{IntLenAnalytical}) using the values obtained
for the time averaged low-$k$ power law $\hat{\beta}$. These values are listed
in Table~\ref{tab:table2} alongside $\hat{\alpha}$ and the averaged value of
the inertial range power law $\widehat{\beta - \alpha}$. Also listed are the
standard deviations of these averages, denoted by sigmas, the initial low-$k$
power law index of the energy spectrum $\beta_0$ and the initial high-$k$ power
law $\beta_0 - \alpha_0$. The errors obtained from the fitting covariances are
negligible in comparison to the standard deviations of the time fluctuations in
all of these cases. We have also measured the magnitude of the statistical
fluctuations resulting from different initial random phases given to the
Fourier velocity components by making runs with the same initial conditions but
with different random seeds. Based on these runs, the fluctuations are found to
be either smaller or at the largest comparable in magnitude to the standard
deviations in Tables~\ref{tab:table1} and~\ref{tab:table2}. The values in these
two tables are used to test the relations (\ref{ParamCond1}) and
(\ref{ParamCond2}), which are listed in Table~\ref{tab:table2b} along with
their standard deviations obtained from the error propagation formula. These
are denoted as $\Delta C_i$ where the index $i$ marks the column of the table
(the run ID column being column 0). These relations are also plotted in a
$\zeta \lambda$-coordinate system in Figure~\ref{fig:ZLplot} where different
low-$k$ power law values correspond to lines with different slopes converging
at the origin~\cite{Brandenburg:2016odr}. The diagonal solid black line is the
curve $\zeta = 2(1-\lambda)$ of Equation~(\ref{ParamCond2}). The error bars for
the data points obtained from Table~\ref{tab:table1} are smaller than the data
point markers and are thus not drawn in the figure. The scaling law following
from the self-similarity is fulfilled well, with the value of zero lying within
the margin of error, whereas the one using the scaling invariance is not as
good due to the small standard deviations in the values of $\zeta$ and
$\lambda$.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Figure11.pdf}
\end{center}
\caption{\label{fig:ZLplot} A $\zeta \lambda$-plot that illustrates the
relations in equations (\ref{ParamCond1}) and (\ref{ParamCond2}). The diagonal
solid black line is the curve $\zeta = 2(1-\lambda)$. The data points for each
run have been obtained using the values of $\hat{\zeta}$ and $\hat{\lambda}$
from Table~\ref{tab:table1}.}
\end{figure}
\begin{table}
\begin{ruledtabular}
\begin{tabular}{D{.}{.}{1.0} D{.}{.}{1.3} D{.}{.}{1.3} D{.}{.}{1.3}
D{.}{.}{1.3} D{.}{.}{10.0} D{.}{.}{10.0}}
\multicolumn{1}{c}{ID} &\multicolumn{1}{c}{$\hat{\zeta}$} &\multicolumn{1}{c}{
$\zeta$} &\multicolumn{1}{c}{$\hat{\lambda}$} &\multicolumn{1}{c}{$\lambda$}
&\multicolumn{1}{c}{$\sigma_\zeta$} &\multicolumn{1}{c}{$\sigma_\lambda$} \\
\hline
\rule{0pt}{3ex}
\textcolor{col1}{1} & 1.521 & 1.294 & 0.417 & 0.353 & $\num{1.09E-02}$
& $\num{7.48E-03}$ \\
\textcolor{col2}{2} & 1.201 & 1.252 & 0.339 & 0.374 & $\num{2.58E-03}$
& $\num{1.43E-03}$ \\
\textcolor{col3}{3} & 1.200 & 1.252 & 0.339 & 0.374 & $\num{2.56E-03}$
& $\num{1.42E-03}$ \\
\textcolor{col4}{4} & 1.333 & 1.284 & 0.350 & 0.358 & $\num{1.08E-02}$
& $\num{5.07E-03}$ \\
\textcolor{col5}{5} & 1.443 & 1.246 & 0.454 & 0.377 & $\num{1.85E-02}$
& $\num{1.48E-02}$ \\
\textcolor{col6}{6} & 1.377 & 1.359 & 0.374 & 0.320 & $\num{1.53E-02}$
& $\num{7.96E-03}$ \\
\textcolor{col7}{7} & 1.403 & 1.352 & 0.426 & 0.324 & $\num{1.83E-02}$
& $\num{1.32E-02}$ \\
\textcolor{col8}{8} & 1.307 & 1.330 & 0.265 & 0.335 & $\num{4.54E-03}$
& $\num{6.60E-03}$ \\
\textcolor{col9}{9} & 1.164 & 1.296 & 0.294 & 0.352 & $\num{3.07E-03}$
& $\num{2.07E-03}$ \\
\textcolor{col10}{10} & 1.160 & 1.237 & 0.294 & 0.381 & $\num{3.38E-03}$
& $\num{1.88E-03}$ \\
\textcolor{col11}{11} & 1.314 & 1.422 & 0.265 & 0.289 & $\num{7.46E-03}$
& $\num{4.15E-03}$
\end{tabular}
\end{ruledtabular}
\caption{\label{tab:table1}
Time averaged fit parameters for the kinetic energy and integral length scale
power laws $\hat{\zeta}$ and $\hat{\lambda}$, obtained by fitting the curves
seen in Figure~\ref{fig:kin_en_L_fits} so that the lower boundary of the
fitting range uses all data points in the range $1 \leq t/t_s \leq 3$, and by
averaging over the results. Also listed are the standard deviations, and the
predicted values for the power laws given by equations (\ref{KinEnAnalytical})
and (\ref{IntLenAnalytical}) by using the values for the time averaged low-$k$
power law $\hat{\beta}$ listed in Table~\ref{tab:table2}.}
\end{table}
\begin{table}
\begin{ruledtabular}
\begin{tabular}{D{.}{.}{1.0} D{.}{.}{1.0} D{.}{.}{3.0} D{.}{.}{1.3}
D{.}{.}{1.3} D{.}{.}{2.3} D{.}{.}{1.3} D{.}{.}{1.3} D{.}{.}{1.3}}
\multicolumn{1}{c}{ID} &\multicolumn{1}{c}{$\beta_0$} &\multicolumn{1}{c}{
$\beta_0 - \alpha_0$} &\multicolumn{1}{c}{$\hat{\alpha}$}
&\multicolumn{1}{c}{$\hat{\beta}$}
&\multicolumn{1}{c}{$\widehat{\beta-\alpha}$}
&\multicolumn{1}{c}{$\sigma_\alpha$} &\multicolumn{1}{c}{$\sigma_\beta$}
&\multicolumn{1}{c}{$\sigma_{\beta-\alpha}$} \\
\hline
\rule{0pt}{3ex}
\textcolor{col1}{1} & 4 & -8 & 3.576 & 2.669 & -0.907 & 0.368 & 0.167
& 0.363 \\
\textcolor{col2}{2} & 3 & -3 & 4.464 & 2.348 & -2.116 & 0.072 & 0.113
& 0.069 \\
\textcolor{col3}{3} & 3 & -3 & 4.464 & 2.349 & -2.115 & 0.072 & 0.113
& 0.068 \\
\textcolor{col4}{4} & 4 & -5 & 4.182 & 2.586 & -1.596 & 0.522 & 0.440
& 0.189 \\
\textcolor{col5}{5} & 5 & -15 & 4.238 & 2.305 & -1.933 & 0.426 & 0.299
& 0.182 \\
\textcolor{col6}{6} & 5 & -5 & 4.986 & 3.240 & -1.745 & 0.381 & 0.453
& 0.139 \\
\textcolor{col7}{7} & 9 & -6 & 5.258 & 3.176 & -2.082 & 0.261 & 0.264
& 0.027 \\
\textcolor{col8}{8} & 5 & -2 & 4.891 & 2.967 & -1.924 & 0.443 & 0.464
& 0.050 \\
\textcolor{col9}{9} & 3 & -3 & 4.693 & 2.683 & -2.010 & 0.710 & 0.739
& 0.052 \\
\textcolor{col10}{10} & 3 & -3 & 4.374 & 2.243 & -2.131 & 0.669 & 0.681
& 0.017 \\
\textcolor{col11}{11} & 7 & -4 & 5.970 & 3.916 & -2.055 & 0.514 & 0.530
& 0.026
\end{tabular}
\end{ruledtabular}
\caption{\label{tab:table2}
The initial low-$k$ power law of energy spectrum $\beta_0$ and the initial
inertial range power law $\beta_0 - \alpha_0$, and the same parameters after
the shocks have formed obtained by time averaging the results obtained from
broken power law fits of Equation~(\ref{BrokPowLawk}) over the interval
$8 \leq t/t_s \leq 12$, denoted by hats. The last three columns list the
standard deviations for the time fluctuations of the parameters $\alpha$ and
$\beta$, and the inertial range power law.}
\end{table}
\begin{table}
\begin{ruledtabular}
\begin{tabular}{D{.}{.}{1.0} D{.}{.}{2.3} D{.}{.}{2.3} D{.}{.}{1.3}
D{.}{.}{10.0}}
\multicolumn{1}{c}{ID} &\multicolumn{1}{c}{$\hat{\lambda} (\hat{\beta} + 1)
- \hat{\zeta}$} &\multicolumn{1}{c}{$\hat{\zeta} - 2(1-\hat{\lambda})$}
&\multicolumn{1}{c}{$\Delta C_1$} &\multicolumn{1}{c}{$\Delta C_2$} \\
\hline
\rule{0pt}{3ex}
\textcolor{col1}{1} & 0.008 & 0.354 & 0.076 & $\num{1.85E-02}$ \\
\textcolor{col2}{2} & -0.066 & -0.122 & 0.039 & $\num{3.85E-03}$ \\
\textcolor{col3}{3} & -0.066 & -0.122 & 0.039 & $\num{3.82E-03}$ \\
\textcolor{col4}{4} & -0.078 & 0.033 & 0.155 & $\num{1.48E-02}$ \\
\textcolor{col5}{5} & 0.056 & 0.351 & 0.145 & $\num{3.49E-02}$ \\
\textcolor{col6}{6} & 0.209 & 0.125 & 0.173 & $\num{2.21E-02}$ \\
\textcolor{col7}{7} & 0.376 & 0.255 & 0.126 & $\num{3.21E-02}$ \\
\textcolor{col8}{8} & -0.256 & -0.163 & 0.126 & $\num{1.40E-02}$ \\
\textcolor{col9}{9} & -0.080 & -0.247 & 0.218 & $\num{5.15E-03}$ \\
\textcolor{col10}{10} & -0.206 & -0.252 & 0.201 & $\num{5.06E-03}$ \\
\textcolor{col11}{11} & -0.013 & -0.157 & 0.142 & $\num{1.12E-02}$
\end{tabular}
\end{ruledtabular}
\caption{\label{tab:table2b} Numerical values for the relations of equations
(\ref{ParamCond1}) and (\ref{ParamCond2}) that are obtained using the fit
parameters in Tables~\ref{tab:table1} and~\ref{tab:table2}. The last two
columns contain the standard deviations of the relations in columns 1 and 2
obtained using the standard deviations of the fit parameters with the error
propagation formula.}
\end{table}
\subsection{Generation of transverse kinetic energy} \label{transenerg}
\noindent In our simulations we see an emergence of small amounts of transverse
kinetic energy from longitudinal-only initial conditions. In order to study the
vorticity generation more closely, we can take a look at the vorticity
equation, obtained by taking a curl of Equation~(\ref{EulerEq}). The equation
can be written for the vorticity $\omega = \nabla \times \textbf{v}$, which in
two-dimensional case can be treated as a scalar, giving
\begin{align}
\frac{\partial \omega}{\partial t} + (1-2c_s^2) \omega (\nabla \cdot \textbf{v})
&+ (1-c_s^2) (\textbf{v} \cdot \nabla) \omega \\ \nonumber
&- c_s^2 \textbf{v} \times \nabla^2 \textbf{v} = \frac{\eta}{1+c_s^2} \nabla^2 \omega \, .
\end{align}
From this it follows that if initially $\omega = 0$
\begin{equation}
\frac{\partial \omega}{\partial t} = c_s^2 \textbf{v} \times
\nabla ( \nabla \cdot \textbf{v}) \, ,
\end{equation}
meaning that there is a vorticity generating term resulting from the last term
on the left hand side of (\ref{EulerEq}), giving rise to some transverse
kinetic energy even when the initial conditions contain only longitudinal
modes.
\begin{figure*}
\centering
\subfloat[$v(x, y) = |\mathbf{v}|$]{
\includegraphics[width=0.5\linewidth]{Figure12a}
\label{fig:vmag1}
}
\subfloat[$L \omega (x, y)$]{
\includegraphics[width=0.5\linewidth]{Figure12b}
\label{fig:vor1}
} \\
\subfloat[$v(x, y) = |\mathbf{v}|$]{
\includegraphics[width=0.5\linewidth]{Figure12c}
\label{fig:vmag2}
}
\subfloat[$L \omega (x, y)$]{
\includegraphics[width=0.5\linewidth]{Figure12d}
\label{fig:vor2}
}
\caption{The magnitude of the velocity field $|\textbf{v}|$ (a) and the corresponding
vorticity field $\omega = \nabla \times \mathbf{v}$ scaled by the integral
length scale (b) of a moderate Reynolds number $4080^2$ resolution run after
about 13 shock formation times. Figures (c) and (d) show the same quantities
for a high Reynolds number run at the end of the run at about $t = 67 t_s$. The
runs have the same initial conditions as runs 2 and 9. In Figures (b) and (d)
vortex-like structures can be seen, appearing in pairs of different signs.}
\label{fig:vmagvor}
\end{figure*}
In the simulations we see that early on the vorticity field attains its largest
values in the regions containing overlapping or colliding shocks. This is
illustrated in Figures~\ref{fig:vmag1} and~\ref{fig:vor1} that show the
magnitude of the velocity field and the corresponding vorticity that has been
scaled by the integral length scale to obtain a dimensionless quantity. As the
shocks overlap with each other, their amplitude increases, and regions with
much higher amplitudes than seen in the initial conditions are formed, shown in
the figure in yellow. The largest values of vorticity right after the shocks
are formed are obtained around these regions, shown as thin short dark red
lines in the contour plot. These features are short-lived and change location
as the shocks travel. The other part of the vorticity field after shock
formation is the background vorticity that changes slowly in comparison to the
vorticity from shock collisions, and contains vortex-like structures that often
appear in pairs of different signs. Over time as the shocks get dissipated, the
background vorticity becomes dominant, with the shocks being only faintly
visible in comparison, as seen in Figure~\ref{fig:vor2}, which plots the
dimensionless vorticity field at the very end of a run. The higher the Reynolds
number of the run is, the higher the generated transverse kinetic energy is
relative to the longitudinal kinetic energy. In Figure~\ref{fig:lontranen} the
energy fraction $\mathcal{E}_\perp/ \mathcal{E}_\parallel$ has been plotted for
several runs, and the group of curves with the highest values corresponds to
the high Reynolds number runs. It shows that the transverse kinetic energy is
still small compared to the longitudinal kinetic energy, even after 60 shock
formation times.
Runs 3 and 10 contain only bulk viscosity. We find that in the longitudinal
case, both the bulk and the shear viscosity affect the fluid almost in an
identical way. Runs 2 and 3, and 9 and 10 have the same initial conditions and
random phases, with the only difference being the viscosity type. The values
for the viscosities in these runs are chosen so that the value of the effective
viscosity is the same. In the longitudinal case, these pairs of runs produce
results that are very close to each other, which can also be seen from the
plots of longitudinal quantities, such as in
Figures~\ref{fig:kin_enA},~\ref{fig:l_int_lonA} of Appendix~\ref{AppC},
and~\ref{fig:ZLplot}, where the runs overlap, or from the tables of the
previous section.
The same is not true in the transverse case, as is evident by
Figures~\ref{fig:lontranen} and \ref{fig:lontranenA}, where the curves of the
previously mentioned run pairs clearly separate from each other some time after
the start of the run, with the bulk viscosity only run having a larger
transverse kinetic energy at the end in both cases (see Table \ref{tab:table3}
in appendix \ref{AppB} for the colour and viscosity type of each run in the
case of Figure~\ref{fig:lontranenA}). This is because in the shear viscosity
only case the dissipation of energy is larger, as under these fluid equations
the dissipation due to viscosity can be shown to be
\begin{equation}
\frac12 \frac{d \left\langle \textbf{v}^2 \right\rangle}{dt} =
\begin{cases}
- \frac{2 \mu}{1+c_\text{s}^2} \int\limits_0^\infty k^2 E(k) \, dk \, ,
\quad \text{when } \nabla \times \textbf{v}=0 \\
- \frac{2 \eta}{1+c_\text{s}^2} \int\limits_0^\infty k^2 E(k) \, dk \, ,
\quad \text{when } \nabla \cdot \textbf{v}=0
\end{cases}
\end{equation}
meaning that for the transverse component the viscous dissipation is caused
only by the shear viscosity. This also strongly affects the shape of the
transverse energy spectrum at large-$k$ between the bulk and shear-viscosity
only runs.
The focus of this paper is the study of the longitudinal case, and thus there
is more potential work to be done in studying the transverse case under these
fluid equations. The transverse only case (incompressible flow) has been
extensively studied and forms part of the standard understanding of turbulence
presented in textbooks (see e.g. Ref.~\cite{lesieur2008turbulence}).
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Figure13.pdf}
\end{center}
\caption{\label{fig:lontranen}
Development of the ratio of the transverse to longitudinal kinetic energy of
several runs with time, in units of the shock formation time $t_s$. The high
Reynolds number runs 7 and 9, and the low Reynolds number runs 2
and 4 are clearly separated into two groups by at least an order of
magnitude, with the high Re runs having higher transverse kinetic energy. The
exception is the low Re bulk viscosity only run (ID 2, green), which joins the
high Re curves at the end of the run.}
\end{figure}
\section{Estimate for the gravitational wave power spectrum} \label{GravSpec}
\noindent While there are no gravitational waves in two dimensions, we can
estimate the gravitational wave power spectrum generated by shocks in three
dimensions by using the results in Ref.~\cite{KP}, according to which the
energy spectrum maintains the $k^{-2}$ inertial range power law in any number
of space dimensions. By assuming the energy spectrum to have a simple broken
power law form, the GW power spectrum can be obtained by adapting standard
methods~\cite{Kosowsky:2001xp,Gogoberidze:2007an,Caprini:2007xq,Caprini:2009yp,
Caprini:2009fx,Hindmarsh:2019phv}.
The source of the gravitational waves is taken to be the shear stresses
resulting from a velocity field consisting of randomly distributed sound waves,
generated on a timescale long compared the light-crossing time of any important
scales in the velocity field. The resulting GW power spectrum can be calculated
from the unequal time velocity field correlators for the system. Our
calculation assumes that the shock lifetime $t_s$ is much less than a Hubble
time, meaning that the expansion of the universe can be approximated by
setting the velocities to zero after a Hubble time~\cite{Hindmarsh:2015qta}.
It is also assumed that the fluid velocities are non-relativistic, and that the
velocity can be treated as a Gaussian random field, with any non-Gaussianity
leading to negligible contributions to the connected four-point correlator. As
the initial velocity field steepens into shocks, the velocity field loses its
Gaussianity but we assume the deviation from Gaussianity to be small, so that
the correlator can still be approximately treated as Gaussian. Measuring the
unequal time correlators for a collection of shock waves to test the validity
of this assumption stands as possible future work.
We begin by citing Equation (3.46) of Ref.~\cite{Hindmarsh:2019phv}, which
gives the growth rate of the gravitational wave power spectrum
$ \mathcal{P}_\text{gw}$ as
\begin{equation} \label{dPgwdt}
\frac{1}{H_*} \frac{d}{dt} \mathcal{P}_{\text{gw}} = 3
\left( \Gamma \bar{v}^2 \right)^2 (H_* L) \frac{(kL)^3}{2 \pi^2}
\tilde{P}_{\text{gw}} (kL) \, ,
\end{equation}
where $H_*$ is the Hubble rate at the time of the transition and $\Gamma$ is
the mean adiabatic index of the fluid. We take $\Gamma = (1+c_s^2) = 4/3$, as
appropriate for an ultrarelativistic fluid. The final factor in the expression
is a dimensionless spectral density function, defined as
\begin{align} \label{DSpecDen}
\tilde{P}_{\text{gw}} (y) = \frac{1}{4 \pi y c_s} &
\left( \frac{1-c_s^2}{c_s^2} \right)^2 \int\limits_{z_-}^{z_+} \frac{dz}{z}
\frac{(z-z_+)^2 (z-z_-)^2}{z_+ + z_- - z} \\ \nonumber
& \times \tilde{P}_v (z) \tilde{P}_v (z_+ + z_- - z) \, ,
\end{align}
where $z_\pm = y(1 \pm c_s)/(2 c_s)$, $z=qL$, and $\tilde{P}_v (z)$ is the
scaled velocity spectral density, which is related to the actual spectral
density as
\begin{equation} \label{scaledPv}
P_v(qL) = L^3 \bar{v}^2 \tilde{P}_{v} (qL)
\end{equation}
with $q$ being the wavenumber. The relation between the energy spectrum and
the spectral density in 3D is
\begin{equation} \label{3DPowSpecEnSpecRel}
E(k) = \frac{k^2}{2 \pi^2} P_v (k) \, .
\end{equation}
On the other hand, the energy spectrum can also be written in terms of the
collapse function $\Psi$ as seen in Equation~(\ref{EnPsi}), from which it
follows that
\begin{equation}
P_v (z) = \pi^2 L^3 \bar{v}^2 \frac{\Psi(z)}{z^2} \, .
\end{equation}
Now for the function $\Psi(z)$ we use the broken power law form of
Equation~(\ref{BrokPowLaw}) that by using Equation~(\ref{scaledPv}) gives
\begin{equation}
\tilde{P}_v (z) = \frac{\Psi_0 \pi^2}{z_p^2}
\frac{(z/z_p)^{\beta-2}}{1+(z/z_p)^\alpha} \, .
\end{equation}
Here the parameter $\kappa_p$ of Equation~(\ref{BrokPowLaw}) has been denoted
with $z_p$ to coincide notationally with $z$ and is fixed in terms of $\alpha$
and $\beta$ along with $\Psi_0$ through Equations (\ref{Afix}) and
(\ref{KappaFix}). Using this and Equation~(\ref{DSpecDen}), and integrating
Equation~(\ref{dPgwdt}) with respect to time with a change of variable
${z=kLs}$ gives the following expression for the gravitational wave power
spectrum
\begin{align} \label{PgwFin}
\frac{1}{(H_* L_0)^2} & \mathcal{P}_{\text{gw}} (k, t_{H_\star}) = \frac{3 \pi
\Psi_0^2 \Gamma^2 (1-c_s^2)^2}{8 L_0^2 z_p^4 c_s^5} k^{5} \\ \nonumber
& \times \int\limits_{0}^{t_{H_\star}} dt \, \bar{v}^4(t) L^{6}(t)
\int\limits_{s_-}^{s_+} ds \, I(s, t)
\end{align}
where $t_{H_\star}$ is the lifetime of the GW source, which we recall is
taken to be the Hubble time at the time of the
phase transition,~\cite{Hindmarsh:2015qta}. The integrand $I$ has the form
\begin{equation}
I(s, t) = \frac{(s-s_+)^2 (s-s_-)^2 [s(s_+ + s_- - s)/s_p^2(t)]^{\beta - 3}}{
s_p^2(t)[1+[s/s_p(t)]^\alpha][1+[(s_+ + s_- - s)/s_p(t)]^\alpha]} \, ,
\end{equation}
with $s_p(t) = z_p/kL(t)$, and $s_\pm = (1 \pm c_s)/(2 c_s)$\
Now we write the time integral only in terms of the integral scale $L(t)$ by
relating it to $\bar{v}(t)$ using equation (\ref{CondForL}) and
substituting the time development Equations (\ref{KinEnAnalytical}) and
(\ref{IntLenAnalytical}) into it (while keeping in mind that
$\mathcal{E} = \bar{v}^2/2$). Because of the relation between the power
law indices in Equation~(\ref{ParamCond1}) the time dependence vanishes and the
equation can be written in the form
\begin{equation}
\bar{v}^2 (t) = \bar{v}_0^2 \left( \frac{L(t)}{L_0} \right)^{-(\beta + 1)} \, .
\end{equation}
Here $\bar{v}_0$ denotes the initial value of the rms velocity. Using
this, the time integral in Equation~(\ref{PgwFin}) can be written as
\begin{equation} \label{TimeInt}
\int\limits_{0}^{t_{H_\star}} dt \, \bar{v}^4(t) L^{6}(t) =
\bar{v}_0^4 L_0^6 \int\limits_{0}^{t_{H_\star}} dt \,
\left( \frac{L(t)}{L_0} \right)^{2(2-\beta)} \, .
\end{equation}
Next we make a change of variables
$\tau = k L(t) / z_p = s_p^{-1}$ in the time integral. This is tantamount to
integrating over the integral length scale, which in the scenario considered
here is a monotonically increasing quantity with time.
The time differential can be related to the differential of this new
variable by using Equation~(\ref{IntLenAnalytical}), which yields
\begin{equation}
dt = \frac{z_p}{\lambda C k \bar{v}_0}
\left( \frac{\tau}{\tau_0} \right)^{1 / \lambda-1} \, d \tau \, ,
\end{equation}
where $\lambda$ is the decay power law of the integral length scale, $C$ is a
decay parameter whose inverse gives the number of shock formation times that it
takes for the flow to start decaying, and $\tau_0 = k L_0 / z_p$. Now, the $s$
integrand can also be written as
\begin{equation}
I(s, \tau) = \tau^{2(\beta - 2)} I_\tau (s, \tau) \, ,
\end{equation}
where
\begin{equation}
I_\tau (s, \tau) = \frac{(s-s_+)^2 (s-s_-)^2 \left[ s (s_+ + s_- - s)
\right]^{\beta-3}}{\left[ 1 + ( \tau s)^\alpha \right] \left[ 1 +
\tau^\alpha (s_+ + s_- - s)^\alpha \right]} \, ,
\end{equation}
meaning that the factor of $\tau$ resulting from Equation~(\ref{TimeInt}) ends
up cancelling with that coming from the $s$ integrand. The
Equation~(\ref{PgwFin}) now becomes
\begin{align}
&\frac{1}{(H_* L_0)^2} \mathcal{P}_{\text{gw}} (k) = \frac{3 \pi
\Psi_0^2 z_p \Gamma^2 (1-c_s^2)^2}{8 c_s^5} \\ \nonumber
& \times \left( \frac{k L_0}{z_p} \right)^4 \tau_0^{2 \beta - 1/\lambda - 3}
\int\limits_{\tau_0}^{\tau_{H_\star}} d \tau \, \tau^{1/\lambda - 1}
\int\limits_{s_-}^{s_+} ds \, I_\tau(s, \tau) \, ,
\end{align}
where $\tau_{H_\star} = k L(t_{H_\star}) / z_p$, and in the prefactor the
powers of $k$ and $L_0$ are equal, so that they can be written in terms of
$\tau_0 = k L_0 / z_p$. It is worth noting that with this formulation the
integration limits also depend on the wavenumber $k$. We can now write the
power law index $\lambda$ in terms of the low-$k$ power law index of the energy
spectrum $\beta$ using the relation between them in
Equation~(\ref{IntLenAnalytical}). We then factorise the result and write it in
the form
\begin{equation} \label{GWspecTau}
\frac{1}{(H_* L_0)^2} \mathcal{P}_{\text{gw}} (k L_0, \tau_{H_\star})
= \frac{\bar{v}_0^3}{C} \mathcal{N} S(k L_0, \tau_{H_\star}) \, ,
\end{equation}
where the numerical factor $\mathcal{N}$ is determined by the speed of sound in
the fluid and the power law parameters $\alpha$ and $\beta$ appearing in the
energy spectrum and has the form
\begin{equation}
\mathcal{N} = \frac{3 \pi}{8} \frac{(\beta + 3) \Psi_0^2 z_p}{2}
\frac{\Gamma^2 (1-c_s^2)^2}{c_s^5} \, .
\end{equation}
The function $S(k L_0, \tau_{H_\star})$ determines the shape of the
spectrum, and can be written as
\begin{equation} \label{GWSpecShape}
S(k L_0, \tau_{H_\star}) = \tau_0^{(3 \beta -1)/2}
\int\limits_{\tau_0}^{\tau_{H_\star}} d \tau \,
\tau^{(\beta + 1)/2} \int\limits_{s_-}^{s_+} ds \, I_\tau (s, \tau) \, .
\end{equation}
Using Equation (\ref{GWSpecShape}), we have plotted the shape of the GW power
spectrum numerically. In the three-dimensional case the low-$k$ power law
index of the energy spectrum $\beta$ is not expected to be the same as in 2D,
and should be determined by numerical simulations. After the
phase transition has completed, the fluid contains shocks and has the $k^{-2}$
power law at the inertial range. For this estimate, we have assumed a value of
$\beta = 4$, and taken $\alpha = 6$ to obtain the correct value for the
high-$k$ power law. The spectrum is then obtained by numerically integrating
the two integrals that appear in (\ref{GWSpecShape}) for a given ratio
$L(t_{H_\star})/L_0$, which we have taken to be 6.1 for
illustrative purposes when plotting the spectrum in Figure~\ref{fig:GWspec}.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Figure14.pdf}
\end{center}
\caption{\label{fig:GWspec} The function $S(k L_0)$ obtained numerically from
Equation (\ref{GWSpecShape}) with parameter values $\alpha = 6$, $\beta = 4$,
and $L(t_{H_\star})/L_0 = 6.1$. A bend in the spectrum at the low-$k$ end is
seen at $k L_0 \simeq z_p L_0/L(t_{H_\star}) \simeq 0.1$. The black line
demonstrates a power law of $k^{5.5}$.}
\end{figure}
The figure highlights an
interesting aspect in the low-$k$ end of the spectrum, in that there is a
change in the low-$k$ power law index around
$k L_0 \simeq z_p L_0/L(t_{H_\star}) \simeq 0.095$, after which the power law
changes from a steeper $k^{9}$ power law to a shallower power law of $k^{5.5}$.
The location where this change occurs is determined by the lifetime of the
source $t_{H_\star}$ through the ratio $L(t_{H_\star})/L_0$, so that the
shallower power law appears in the range
\begin{equation}
z_p L_0/L(t_{H_\star}) \lesssim kL_0 \lesssim z_p \, .
\end{equation}
Therefore, for short enough lifetimes, where the integral scale does not have
enough time to grow significantly compared to its initial value, the range is
short and close to the peak, meaning that effectively only the steeper slope is
obtained, and for long lifetimes, where $L(t_{H_\star}) >> L_0$, the bend
occurs at very small wavenumbers close to the origin, so that the spectrum
effectively only possesses the shallower slope. In the first case, the shock
formation time $t_s$ is close in magnitude to the duration of the GW source
$t_{H_\star}$, which is the Hubble time. Hence only short-lived source
$t_s \ll t_{H_\star}$, as assumed here, will show the intermediate power law.
The power law behaviour of the GW power spectrum can be inspected by extracting
the wavenumber behaviour of Equation (\ref{GWspecTau}) in different limits. At
very small wavenumbers fulfilling the condition $\tau \ll 1$ for any $\tau \in
[\tau_0, \tau_{H_\star}]$ the integral over $s$ yields essentially a constant,
from which it follows that
\begin{equation}
\frac{1}{(H_* L_0)^2} \mathcal{P}_{\text{gw}} (k) \propto k^{2 \beta + 1} \, ,
\end{equation}
which for $\beta = 4$ gives the value of the power law index seen in
Figure~\ref{fig:GWspec}. At large wavenumbers, so that $\tau \gg 1$ for any
$\tau \in [\tau_0, \tau_{H_\star}]$, it can be approximated
\begin{equation}
I_\tau (s, \tau) \approx \frac{(s-s_+)^2 (s-s_-)^2 \left[ s (s_+ + s_- - s)
\right]^{\beta-\alpha-3}}{\tau^{2 \alpha}} \, ,
\end{equation}
which means that the $s$-integral yields a constant once more, and after
integrating over $\tau$, the $k$-dependence is found to be
\begin{equation}
\frac{1}{(H_* L_0)^2} \mathcal{P}_{\text{gw}} (k) \propto k^{2(\beta-\alpha)+1}
\, ,
\end{equation}
which for acoustic turbulence gives the power law of $k^{-3}$ at high
wavenumbers. To touch on the intermediate power law seen in
Figure~\ref{fig:GWspec}, we need to understand the behaviour of the
$\tau$-integrand in the regime where the $s$-integral does not yield a
constant. To this end, we rewrite the integrals of Equation~(\ref{GWSpecShape})
in the form
\begin{equation}
\frac{1}{(H_* L_0)^2} \mathcal{P}_{\text{gw}} (k)
\propto k^{\frac{3 \beta - 1}{2}} \int\limits_{\tau_0}^{\tau_{H_\star}}
f(\tau) \, ,
\end{equation}
where the function $f(\tau)$ denotes the integrand
\begin{equation} \label{taufunc}
f(\tau) = \tau^{\frac{\beta+1}{2}} \int\limits_{s_-}^{s_+} ds \,
I_\tau (s, \tau) \, .
\end{equation}
This function has been obtained numerically by using the same parameter values
as in Figure~\ref{fig:GWspec}, and is plotted in Figure~\ref{fig:ftau}.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{Figure15.pdf}
\end{center}
\caption{\label{fig:ftau} The integrand function $f(\tau)$
(see Eq.~\ref{taufunc}) plotted using the same parameter values as in
Figure~\ref{fig:GWspec}. The highlighted area shows the part of the curve
contributing to the GW power spectrum at $k L_0=0.24$, which lies roughly in
the middle of the intermediate power law range.}
\end{figure}
Since the integration limits depend on the wavenumber $k$, a different part of
this curve is integrated for each value of $k$. It turns out that the
intermediate power law is obtained at wavenumbers for which the integration
range spans the peak of the function $f(\tau)$, that is, when the separation
between $\tau_{H_\star}$ and $\tau_0$ is larger than the width of the peak in
the integrand $f(\tau)$, which is located approximatively in the range
$0.5 \lesssim \tau \lesssim 2$. The width of the integration range for a given
$k$ is determined by the the ratio $L(t_{H_\star})/L_0$. When it is large, the
peak is panned even for small wavenumbers, resulting in the narrower power law
at low-$k$, and when it is small, the integration range is narrow and does not
span the peak entirely for any $k$ so that only the steeper power law is
obtained. For the wavenumbers in the intermediate power law range, the integral
over $f(\tau)$ is effectively a constant, since the largest contribution to
the integral is obtained around the peak, which is spanned for all such
wavenumbers, and since the contributions from the edges of the integration
range are small in comparison. Therefore, it follows that in the intermediate
power law range the GW power spectrum goes as
\begin{equation}
\frac{1}{(H_* L_0)^2} \mathcal{P}_{\text{gw}} (k)
\propto k^{\frac{3 \beta - 1}{2}} \, ,
\end{equation}
giving the power law seen in Figure~\ref{fig:GWspec} when $\beta = 4$.
To conclude, apart from giving a power law of $k^{-3}$ in the high-$k$ range,
the decay of the shocks also induces a change in the low-$k$ power law, going
from $k^{2 \beta + 1}$ to a shallower $k^{(3 \beta - 1)/2}$ one, over a range
depending on the integral scale of the fluid flow after a Hubble time. Note
that the rate at which the flow was originally generated may also appear as a
scale in the gravitational wave power spectrum, below which another power law
may apply~\cite{Caprini:2009fx}. We have assumed that this happens at a lower
wavenumber than any considered here.
\section{Conclusions}
\noindent We have studied decaying acoustic turbulence using two-dimensional
numerical simulations with the emphasis being on the impact of the shocks upon
the energy spectrum, and on the decay of the kinetic energy. Conducting the
simulations in two dimensions allows for better computational efficiency and
the use of larger grid sizes in comparison to 3D, which leads to there being
more dynamic range in the wavenumber space. Two-dimensional systems are also
simpler to analyse and in the case of shocks share some properties with
three-dimensional systems. By making use of the universality of the power
spectra, the obtained two-dimensional decay properties and power laws of the
system have been applied in three dimensions to calculate an estimate for the
gravitational wave power spectrum resulting from a collection of shock waves.
The longitudinal energy spectrum of the fluid can be written in terms of the
longitudinal kinetic energy, integral scale, and the dimensionless function
$\Psi(k L)$ as seen in Equation~(\ref{EnPsi}). The function $\Psi(kL)$ has the
property that it maintains its shape over time at length scales above the
dissipation range. Using the tanh shock profile obtained from the fluid
equations, we have presented an analytical universal form for this function,
which is found to be a broken power law modulated by an integral function
$\mathcal{I}$ that is shown in Equation~(\ref{IP}). This function depends on
the steepness of the shocks via the wavenumber parameter $k_s$ appearing in the
argument of the tanh shocks. Between the wavenumbers corresponding to the
integral scale and the Taylor microscale, the power law is found to be
$k^{-2.08 \pm 0.08}$, which agrees very well with the $k^{-2}$ power law
associated with acoustic turbulence~\cite{KP}, obtained as an inverse-variance
weighted average of the measurements in Table \ref{tab:table2}. At lower
wavenumbers, using the same method, the power law is $k^\beta$, with
$\beta = 2.50 \pm 0.31$.
In order to find the time evolution of the longitudinal kinetic energy, we have
used the $k^{-2}$ inertial range power law, and the self-similarity of the
spectrum at low-$k$ to find equations (\ref{KinEnAnalytical}) and
(\ref{IntLenAnalytical}), the latter of which describes the decay of the
longitudinal integral length scale. At times much larger than the shock
formation time, these produce power law forms, where the values of the power
law indices depend on the low-$k$ power law index of the energy spectrum. From
the simulations using the earlier averaging technique with the means and
standard deviations listed in Table \ref{tab:table1}, we find the kinetic
energy to decay as $t^{-1.21 \pm 0.06}$, and the integral scale to increase as
$t^{0.32 \pm 0.03}$. To test the validity of our results, we have used the
analytical results and the scaling relations between the power law parameters,
and compared the results from those to the independent data obtained from the
simulations by fitting. In general, we find these to be in good agreement.
Lastly, we have produced an estimate for the shape of the gravitational wave
power spectrum in three dimensions, using the universality of the $k^{-2}$
spectrum for a shocked fluid, and the evolution laws for the kinetic energy and
the integral scale. The power spectrum is peaked at a wavenumber set by the
initial integral scale. At higher wavenumbers the GW power spectrum is found to
go as $k^{-3}$, which is the same as the power law predicted from linear
evolution of acoustic waves produced by first order phase transitions~\cite{
Hindmarsh:2017gnf,Hindmarsh:2019phv}.
At wavenumbers lower than the peak of the spectrum, there is a change in the
power law from $k^{2 \beta + 1}$ to a less steep $k^{(3 \beta - 1)/2}$. This
power law is maintained down to values of $k$ of order the inverse integral
scale at the end of the effective sourcing of GWs, expected to be about a
Hubble time.
Our work is of direct relevance for calculations of the gravitational wave
power spectrum produced by first order thermal phase transitions in the early
Universe, in cases where the shock formation and decay time $t_s$ is shorter
than the Hubble time, often the case for phase transitions strong enough to be
observed. The acoustic turbulence simulated here in two dimensions will also
develop in three dimensions, with the same $k^{-2}$ power law in the energy
spectrum at high $k$. This is also the same power law as found in the linear
approximation to the evolution of the sound waves following the phase
transition, and so we do not expect qualitative changes to the GW power
spectrum as a result of the appearance and decay of shocks. However, we do
expect the acoustic turbulence to significantly affect the power-law behaviour
of the gravitational wave power spectrum at wavenumbers lower than the peak,
where a non-trivial power law may develop in the energy spectrum. The index of
this power law cannot, however, be predicted from two-dimensional numerical
simulations. In any case, the low-$k$ power law in the gravitational wave
spectrum will be different from that from the linear evolution of acoustic
waves and from vortical turbulence. Finding this characteristic power law is
clearly a high priority for reliable predictions for the gravitational wave
power spectrum following a phase transition.
\begin{acknowledgments}
\noindent
We acknowledge useful discussions with Carl Bender. J.D was supported by the
Magnus Ehrnrooth Foundation. D.J.W. (ORCID ID 0000-0001-6986-0517) was
supported by Academy of Finland grant nos. 324882 and 328958, J.D. and K.R.
(ORCID ID 0000-0003-2266-4716) by Academy of Finland grant nos. 319066 and
320123, and M.~H. (ORCID ID 0000-0002-9307-437X) by Academy of Finland grant
no.~333609. The authors would also like to thank Finnish Grid and Cloud
Infrastructure at the University of Helsinki
(urn:nbn:fi:research-infras-2016072533) and CSC – IT Center for Science,
Finland, for computational resources.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,199 |
Mobile Platforms: Sharing Data between Truck Cab and Back Office
April 12, 2019 • by Jim Beach
Mobile communications systems have been used by fleets for many years, and all those systems provide a gateway from the vehicle to the back office.
HDT File Image
The mobile revolution in trucking, or any other industry, doesn't work without reliable, secure transfer of data from the mobile device to the trucking company's back office. That's where a mobile platform comes in.
In the most general sense, a mobile platform means the operating system and hardware used in a smartphone, tablet or laptop – the piece that enables your device to do the things it does, from making calls to making videos and everything else in between. More specifically, in trucking's case, a mobile platform means the software and hardware that connects mobile devices with fleet managers.
"For me, a mobile platform is an enterprise solution that has an end-to-end integration," from the trucking company's back office to the driver, explains Salem Elnahwy, chief technology officer, Transflo. "It allows drivers to do their job, and it does a lot of things in the background," whether the device is in the cab or in "the palm of their hand."
A mobile platform should not be confused with the mobile devices, themselves, says Eric Witty, vice president of product management at Trimble Transportation Mobility. The devices may be loaded with apps for specific tasks, such as performing driver vehicle inspection reports, scanning documents, or recording vehicle location and performance. But for a fleet to make use of that data, it needs a platform, or backbone. "There's a difference between a product or software and a backbone," he says. "A product is meant for that need; but it is dependent on the mobile platform to serve other purposes."
To put it another way, mobile devices, such as smartphones or tablets, are data collectors via various, specific applications. Jason Penkethman, chief product and strategy officer, Spireon, notes these applications are used in a variety of tasks, "such as driver safety, vehicle health and status, operational efficiency, communications, job management, routing." And the list goes on.
Mobile communications systems have been used by fleets for many years, and all those systems provide a gateway from the vehicle to the back office. Traditionally, those were devices hard-wired into the vehicle's on-board computer. Many fleets now use mobile devices to extend driver connectivity outside the truck for signature capture or other tasks. Those may extend the capability of cab-mounted units or perform the same tasks of traditional in-cab devices. The electronic logging device mandate prompted many trucking companies and owner-operators to adopt some sort of mobile technology to meet the rule's requirements.
For all of these operators, the important ingredients of a trucking-centered mobile platform are its ability to integrate with a company's existing back-office and/or other technologies currently in use, and its ability to generate data that can be useful beyond its original purpose.
In-cab devices, such as this Omnitracs unit, provide the gateway between the driver and the back office while easing the driver's workload each day.
Photo Jim Beach
Longtime mobile communications providers have been integrating their products with trucking management software providers for years, providing GPS data for dispatch systems to use when assigning vehicles to loads. That hasn't changed, but the kinds of data generated by the vehicle and the driver now go way beyond location data – and the flow now goes both ways.
"I think integration has been key for years," Witty says. "What we think these kinds of systems provide to companies is the value to the driver," in interactions with customers and with dispatch. "Capturing the data is certainly one of the things," he adds, but there also are ways to push data back to drivers by automating some functions such as data entry. The driver has some of his or her workload eased. Instead of entering data into a device, drivers simply validate what has been automatically filled in.
The data coming in from the vehicle is important beyond the telematics information. Fleets can use it in a number of ways if that data is readily available to other parts of their systems.
"There's a lot of information coming in from various platforms and devices, and it all needs to be connected to the back office," says Adam Bruttell, vice president of sales and marketing North America for Mix Telematics. Companies can put this data to work to do things such as optimize routes, calculate cost per mile, generate payroll data for employees, track maintenance, and evaluate vehicle utilization.
A mobile platform allows fleets to put the data generated by the vehicle and driver to work in monitoring efficiency, navigation and other details.
Photo: Jim Beach
Kevin Aries, global product success, Verizon Connect, says integrating the mobile systems with the back office helps companies with driver retention by "improving work experience and communication" for drivers. In his view, an integrated mobile platform should help in a number of areas, from dispatch to collecting form data in the field.
Not all data has to be integrated into the back office, or not in real-time at least. Mobile services that do need such integration include ELD data, vehicle/driver routing applications, mobile forms, and vehicle/trailer location and status information.
But fleets might consider other app data optional, according to Roni Taylor, senior vice president, FAI strategy and business development, at Spireon. For example, there are mobile apps that can provide direct feedback to drivers on their speed. "The same information can be sent to a field supervisor in the form of an alert or report, but it would be optional to capture that speeding data and integrate it with a back-office system." But some fleets might want that data in their system to be used as part of a company safety initiative.
In the most general sense, a mobile platform means the operating system and hardware used in a smartphone, tablet or laptop.
Photo: Verizon Connect
How mobile platforms augment your current system
Once mobile data has been integrated into a company's transportation management system, it can be put to work in a number of ways. TMS providers generally offer modules for integrating data for specific purposes. Mobile communication and telematics vendors also offer services for analyzing raw data collected from vehicles. Aries says mobile platforms make a company's enterprise system "more robust," by providing ways to tackle new problems as businesses grow. They also help by providing greater levels of automation and efficiency.
The mobile data can give company managers new ways of looking at things, Penkethman says. "They add additional context or dimension" that helps managers find quality and process issues
While much of the mobile data fleets collect involve location and status, Taylor says such data can be "integrated into enterprise systems for non-telematics needs. For example, second-order analytics can provide insights into risk, or correlation of events between telematics data and data from other systems." That analysis can be used to establish vehicle maintenance schedules or transportation flow analysis.
Devices plugged into the vehicle's onboard computer can serve as the gateway between the truck and the back office.
Witty notes his company has always worked with trucking companies and third-party providers to meld the data from their mobile platforms with their existing systems and applications. The next step is to open up the platform for fleets that have developed their own applications. "The element of our platform that will become valuable is letting fleets and third-parties put their apps on there; to provide the platform that allows everything to communicate."
Related: ZF on How Advanced Technologies Can Improve Trucking Efficiency and Safety
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Dupré Logistics and international logistics provider Hoyer Group announced a joint venture, Hoyer Bulk LLC, to help meet bulk logistics demands and international and domestic supply chain needs in North America. | {
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I'll come back to that in a bit. Why don't we start by establishing my bona fides. After spending about the first ten years of my working career in electrical engineering for medium and larger companies, I had what Michael Gerber's book "The E-Myth" calls an "entrepreneurial seizure". As a result, I ended up co-founding a Gigabit Ethernet start-up in the late '90's.
Although we fortunately avoided what Gerber calls "the most disastrous assumption anyone can make about going into business." ("That Fatal Assumption is: if you understand the technical work of a business, you understand a business that does technical work."), we did manage to find yet another of the thousands of ways a business fails. After that, I founded a consulting company. Being a good acquaintance with a fellow who did contracting work for my first start-up, we started talking about working together so we could get contracts that required more resources than we had individually. In the end we merged companies, and I became a "partner".
Ah! So there it is. Partner. Partnership. Mine is just one of the motivations for finding a partner, or going into a partnership with someone else. There are lots of other reasons. For those of you who read the Business of Software forum, this might sound familiar. There are a fair number of people asking about how they should find a partner to make up for this or that shortcoming that they have seen in themselves.
After spending a little over 5 years with the two other partners, I made the decision in August 2005 to leave the company. I have heard this called a "business divorce". Although, I felt strongly that leaving was the best decision for me personally – I had hoped the process would not be like a "business divorce". In fact, I had made it clear that I wished for my exit to be as friendly and painless as possible. That it wasn't "their fault" or something they "did". I thought that the fact we had put a partnership agreement in place would prevent this wish from being simple naivet??. In the end, it was not to be. Although the partnership agreement largely accomplished it's purpose with respect to valuation, many feelings were hurt all around, and emotions laid bare. At this point, I can't even tell you for sure whether it was more their fault or mine. More to the point, it probably doesn't matter.
However, all of this does give me a certain perspective. In a sense, it is perhaps a bit unique because I left a company that was viable and successful. It is more common to hear cautionary tales coming from folks whose partnerships came apart because of business failure. In my experience an interesting, yet sad, note is that a partner leaving even a functioning business can be very difficult for both sides. I am sure that I can't give you all, or even most of the reasons for that – they will surely be unique to every circumstance. However, maybe I can talk a little about some of the issues that you will should confront before taking on, or becoming, a partner.
Okay. This is a big one. Perhaps the biggest. You might think I am getting ready to tell you to never enter a partnership without a written agreement. I will – later. This issue even surpasses that one. Here it is: Partnerships are not equal. Although there are a small handful of counterexamples (i.e. two close friends as partners in a small company), equal partnerships are problematic. Someone needs to be the "Panjandrum". This needs to be reflected in the partnership arrangement somehow. If you have equal financial stakes, you can solve this by having someone with additional voting rights. This is not to say that your other partners might not be able to collectively out-vote the Panjandrum, but organizations abhor a power-vacuum. If the organization is not clear on who the Panjandrum is, they will pick one (or more than one, thus creating fiefdoms). In my estimation, this causes many problems. Not the least is that there may be more that one person who thinks that he should be the Panjandrum. In our case, we never directly addressed this before I became a partner. In retrospect, had we tried – I would probably not have become a partner. As it was, this one issue caused frequent problems.
Related to this, is that everyone needs a clear idea of what their roles are, and their boundaries. There are plenty of leadership positions to go around, but partners need to be clear what the roles are, and what the other partners expect from them in those roles. Again, in some small two-person partnerships there may not be a need to be explicit, simply because both people know what is going on, and feel alright with some small amount of duplicated effort from time-to-time.
You may have heard the old saw about never having your best friend as a roommate or business partner. Having had both circumstances, I think it is dead wrong. I have found that if you really know someone, and are honest about your own strengths and weaknesses, as well as theirs, then good friends are better roommates and business partners. Maybe it's just me, but it's the things that you don't know until much later that start really tearing up partnerships. I realize that very few people out there have really close friends that have the right technical skills to be ready made partners. However, the more you can get to know someone before forming a partnership the better. This might mean having to work with someone a while before discussing a partnership.
A great example of this is my first major start-up. The founders of the Gigabit Ethernet start-up consisted of myself, my wife and a best friend. Although the business failure was a difficult time financially and emotionally for all of us (and our investors), I am still married and my best friend is still a best friend. I think that knowing each other so well allowed us to define roles very accurately, and to have the healthy and necessary disagreements without suspicions of ulterior motives.
With apologies to Billy Joel, this is really the fundamental place that I think a lot of folks go wrong. As you are talking with your partners to be, there are a lot things you need to agree on. Are you building the company to sell, or is it a lifestyle play? Product or Service? Who do you want as customers? Who is in charge of what (business, sales, production, etc.)?
The biggest danger here is not that others will be dishonest with you (although that is a danger). The biggest danger is that you are not honest with yourself. Brutally honest. What do you want? Really? What are you great at doing? Really? Write it down. Concentrate on it. This is lonely work. You can't even really consult with other people about it – they certainly can't tell you what you want. This is big-time grown-up stuff.
If you take the time to do this, and it could take quite a bit of time, you will have a firm foundation to really engage in the tough questions. More importantly, perhaps, is that you might find issues that you need to address, that might never have even been discussed. The big problem with a written partnership agreement is that it can never address things that you don't know need to be addressed. Honest self-examination can help to uncover issues that if otherwise overlooked can cause problems within a partnership.
This is another one of those topics, that my view is maybe a little different than some others. I am not talking about how much each person gets paid (important though it may be). I am talking about the fundamental way all the partners view Money with a capital 'M'. Sooner or later, you are probably going to know more about your partner's personal finances than you thought you wanted to. It's better to talk about it before you become partners. Are any of the prospective partners in deep trouble if they miss a paycheck? Does anyone have poor credit? Deeply in credit card debt? Too much house? Greedy spouse? Do they balance their checkbook every month? Do they have any cash reserves?
I am not saying to ditch someone as a partner because they don't have perfect finances (who does?). However, what I can tell you for sure is that the way your partners view their personal finances will impact they way they view their business finances. Further, depending on the partnership structure – the tax consequences can affect everyone involved. Be open about this, and demand openness in return. Remember, in the early days any financing or leases will probably involve personal guarantees from the partners. Don't get too caught up in particular numbers (e.g. don't get too worked up if someone doesn't want to tell you exactly how much they have in retirement savings), but the more you know about how every partner views money and deals with it – the better.
Finally, in a foreshadowing to the next section, you should also probably discuss your prospective partner's spouses view of money, and their outlook of the business partnership from a financial perspective. Specifically, when it comes time to draft those magical partnership documents, you need to make sure that everyone does their homework along with your corporate attorney as to how to protect the company financially in the case a partner divorces or is subject to a "palimony" type suit.
Very true about the "Panjandrum" even if you are childhood friends. Case in point: Richard Branson (of Virgin and British billionaire fame) and Nik Powell had a 60/40 stake in Virgin Records when Nik left around 1980-ish. Their "who is boss" relationship had sorted itself in an earlier venture, the "Student" magazine when Richard sent Nik packing after Nik had tried to assert a kind of re-organization. Later Richard invited Nik back to help run things because of their strong relationship but certainly the power issue was clearer at that point (although of course you can never avoid all conflict). Anyway, I would hold that it needs to be clear who is running the show even when (especially when?) you are childhood friends. | {
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<!-- 链接文件,用于被其它md文件包含 -->
<!-- 人名 -->
[AlanDonovan]: https://github.com/adonovan
[BrianKernighan]: http://www.cs.princeton.edu/~bwk/
[DennisRitchie]: http://genius.cat-v.org/dennis-ritchie/
[RobertGriesemer]: http://research.google.com/pubs/author96.html
[RobPike]: http://genius.cat-v.org/rob-pike/",
[KenThompson]: http://genius.cat-v.org/ken-thompson/
[RussCox]: http://research.swtch.com/
[NiklausWirth]: https://en.wikipedia.org/wiki/Niklaus_Wirth
[TonyHoare]: https://en.wikipedia.org/wiki/Tony_Hoare
[FredBrooks]: http://www.cs.unc.edu/~brooks/
<!-- 图书 -->
[gopl]: http://gopl.io
[tcpl]: http://s3-us-west-2.amazonaws.com/belllabs-microsite-dritchie/cbook/index.html
[TheCProgrammingLanguage]: http://s3-us-west-2.amazonaws.com/belllabs-microsite-dritchie/cbook/index.html
[ThePracticeOfProgramming]: https://en.wikipedia.org/wiki/The_Practice_of_Programming
<!-- Go语言 -->
[Golang]: https://golang.org/
[Golang-oracle]: https://godoc.org/golang.org/x/tools/oracle
[Golang-godoc-analysis]: https://godoc.org/golang.org/x/tools/cmd/godoc
[Golang-gorename]: https://godoc.org/golang.org/x/tools/cmd/gorename
<!-- 其他语言 -->
[Alef]: http://doc.cat-v.org/plan_9/2nd_edition/papers/alef/
[APL]: https://en.wikipedia.org/wiki/APL_(programming_language)
[Limbo]: http://doc.cat-v.org/inferno/4th_edition/limbo_language/
[Modula-2]: https://en.wikipedia.org/wiki/Modula-2
[Newsqueak]: http://doc.cat-v.org/bell_labs/squeak/
[Oberon]: https://en.wikipedia.org/wiki/Oberon_(programming_language)
[Oberon-2]: https://en.wikipedia.org/wiki/Oberon-2_(programming_language)
[Pascal]: https://en.wikipedia.org/wiki/Pascal_(programming_language)
[Scheme]: https://en.wikipedia.org/wiki/Scheme_(programming_language)
[Squeak]: http://doc.cat-v.org/bell_labs/squeak/
<!-- 系统 -->
[Unix]: http://doc.cat-v.org/unix/
[UNIX]: http://doc.cat-v.org/unix/
[Linux]: http://www.linux.org/
[FreeBSD]: https://www.freebsd.org/
[OpenBSD]: http://www.openbsd.org/
[MacOSX]: http://www.apple.com/cn/osx/
[Plan9]: http://plan9.bell-labs.com/plan9/
[Windows]: https://www.microsoft.com/zh-cn/windows/
<!-- 其他 -->
[BellLabs]: http://www.cs.bell-labs.com/
[CSP]: https://en.wikipedia.org/wiki/Communicating_sequential_processes
[KR]: https://en.wikipedia.org/wiki/K%26R
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,598 |
1 Thursday, 20 June 2013
4 [The witness takes the stand]
6 JUDGE KWON: Good morning, everyone. Please continue,
7 Mr. Tieger.
8 MR. TIEGER: Thank you, Mr. President. Good morning to all in
9 the courtroom.
10 WITNESS: BOGDAN SUBOTIC [Resumed]
12 Cross-examination by Mr. Tieger: [Continued]
13 Q. Good morning, Mr. Subotic. When we adjourned --
14 A. Thank you, good morning.
15 Q. When we adjourned yesterday, we were discussing aspects of
16 Srebrenica in 1995. I had -- you had mentioned or the issue of
17 directives came up in the context of your statement. I mentioned that we
18 would be turning to the question of directives, and so I wanted to turn
19 to paragraphs 231 through 233 of your statement, and those paragraphs
20 reflect a discussion about directive for further operations number 7
21 signed by the Supreme Commander in March of 1995 as we see in
22 paragraph 231, and you explain in your statement that you learned about
23 directive number 7 after it had been drafted. That we see at the
24 beginning of paragraph 232. You describe that Dr. Karadzic went to the
25 Main Staff, and as you say in paragraph 233, a few days after his return
1 from the Main Staff I asked the president what was discussed. He told me
2 it was about drafting and signing a directive and a large number of other
3 documents. The paragraph goes on to describe how you reminded him
4 that -- of your understanding or agreement that you would be present and
5 review contents to which he replied, well, General, who am I supposed to
6 trust if not the generals at the Main Staff, and then you state that the
7 president would never knowingly put his signature under that disputable
8 text of directive number 7. That's found at the last sentence of
9 paragraph 233.
10 First, General Subotic, what were you referring to when you
11 indicated the "disputable text of directive number 7"?
12 A. I can tell you that I personally did not see the directive at
13 all. I did not have it in my hands ever. However, later on before I
14 testified in 2006, there was a dispute, as far as I understood it, and as
15 I say, I have never seen the directive personally. As far as I know,
16 there was a dispute about the sentence which allowed some use of force or
17 something of that kind. To be honest, I would like you to quote from
18 that directive, because I don't know. I have never learnt about that,
19 which is why I claimed that. Because we had a system, President Karadzic
20 and myself, according to which all the documents that were of some
21 significance had to be reviewed. Otherwise, what would have been my role
22 in the office? I was the expert, especially in terms of the military.
23 Q. You described that in your statement, and I referred to it just a
24 few minutes ago. In fact, your statement contains an excerpt or two from
25 the document itself at paragraph 231, there's a reference to the sentence
1 as follows with reference to the Srebrenica and Zepa enclaves "by planned
2 and well thought-out combat operations create an unbearable situation of
3 total insecurity with no hope of further survival or life for the
4 inhabitants of Srebrenica and Zepa."
5 So as a matter of fact, you yourself refer to portions of that,
6 and is that one -- is that part of the disputable text that you were
7 referring to? In other words, were you -- were you saying in your
8 statement that this is the kind of thing that you claim Dr. Karadzic
9 would not knowingly have signed because of the nature of what that says,
10 or are you referring to another part of the document?
11 A. No, no. I had in mind that part. I have not got the translation
12 of paragraph 231 into Serbian, but now you have jogged my memory.
13 Q. Okay, good. I would also call up 838, the document itself, and
14 turn to B/C/S page 15 and you can see it. And that's also found on
15 English page 10 in 838.
16 A. I still don't have it on the screen.
17 Q. It's on the screen, sir. You see under the Drina Corps in the
18 middle of the page?
20 Q. You see the end of that first paragraph?
21 "By planned and well thought-out combat operations --"
22 A. I've found it.
23 Q. "-- create an unbearable situation of total insecurity with no
24 hope of further survival --"
25 A. [Overlapping speakers]
1 Q. "-- or life for the inhabitants of Srebrenica and Zepa."
2 And, sir, when you were calling it disputable --
4 Q. -- isn't it the case that what you meant was this is a patently
5 illegal order to create no hope of further survival or life for the
6 inhabitants of those enclaves?
7 A. Yes, yes, absolutely. Absolutely. I accept what you are saying.
8 This is absolutely what was disputable for me. Knowing
9 President Karadzic and what had been done by the two of us up to then, I
10 have a serious doubt that President Karadzic ever read this. I don't
11 know how things unfolded in the Main Staff. However, I doubt that he
12 would have accepted this if he had read it. Moreover, now that I look at
13 it and when I reflect on this, I may not be mistaken if I said that I
14 believed that President Karadzic never read this, and on the other hand,
15 I would not exclude the possibility that that page had been subsequently
16 either altered or tempered with in another way. Because you see, you
17 legal professionals, all the lawyers in the world initial all the pages
18 of any serious document. I personally believe, or, rather, I doubt that
19 President Karadzic ever read this. I don't know whether he read this,
20 whether that was how it was written. I would not be able to say. So
21 these are my serious doubts. However, knowing him --
22 Q. General, you don't have to make the same point repeatedly. I
23 understand your position on that. So it's your position -- let's set
24 aside for the moment your speculation that somehow this was substituted
25 afterwards, it was your position in your statement and as you just stated
1 here that the Supreme Commander signed a directive that laid out the
2 course of operations for his military forces after participating in the
3 drafting of it without bothering to read it. That's your position. And
4 that explains why Dr. Karadzic's signature is on this document that
5 contains this order; correct? That's what you're saying?
6 A. What confuses me is this: The principle concerning strategic
7 documents pertaining to the army was such that a document was prepared by
8 the Main Staff. That was the first draft. The first draft was then
9 discussed, and then possible corrections were made to that draft. If I
10 had participated in the drafting of this document, I am giving you a
11 serious guarantee that I would not have allowed this be written in this
12 document.
13 Q. Right. And apparently your testimony is that Dr. Karadzic is the
14 kind of guy who when he gets a document that dictates what his forces
15 will be doing doesn't bother to read it but says, Great, here's my
16 signature. That's it.
17 A. I cannot say that he did not bother. I cannot say that. I
18 wasn't there. If I had been there, I would have sat down together with
19 Radovan Karadzic, I would have looked at the document together with him,
20 I would have perused it together with him, and then I would alert his
21 attention to a possible irregularities. I'm telling you this very
22 honestly and I adhere that.
23 Q. And what prevented you from looking at the document after it had
24 been signed? Did you consider it to be the equivalent as a nonrefundable
25 plane ticket?
1 A. He was not in the office -- or, rather, that document was not in
2 my office. I did not have an occasion to see it. It was signed in
3 Han Pijesak and that's where he stayed as far as I know. I personally
4 never saw it.
5 Q. I'm asking you why you didn't ask to see it. I didn't have to --
6 why didn't you say to Dr. Karadzic, You know, I didn't a chance to review
7 it, as you say you had agreed, so let me take a look at it and make sure
8 there's nothing problematic, disputable, or more realistically illegal
9 about it?
10 A. I gave a statement, and I said that I should have seen it.
11 However, I don't know whether he was convinced that that's not how it was
12 written. I don't know. What I stated in my statement is correct. I
13 can't change it now. I can't speculate. However, I have my doubts about
14 President Karadzic having allowed this to be said or written. These are
15 my doubts, and I have to share them with you. And there's nothing else
16 to it.
17 Q. Let's move on to a somewhat different aspect of the operation and
18 consequences of the operation --
19 JUDGE KWON: Just one question from me for the witness. I'm not
20 sure whether the witness could assist us. Could we see the first page of
21 this document in B/C/S. And the second page in English.
22 General Subotic, this directive 7 is dated as 8th of March, 1995.
23 Do you see that?
24 THE WITNESS: [Interpretation] Yes, I do.
25 JUDGE KWON: Can we see the second last page in the B/C/S and the
1 first page in the English. But the date on which General Milovanovic
2 sent out this directive to the unit was 17th of March, almost ten days
3 after the signature date. Can you tell us why it was so, if you could?
4 THE WITNESS: [Interpretation] I can see here that the Main Staff
5 of the VRS - and this is first time I see the document - sent this to the
6 command of the 1st Krajina Corps, and I don't know why it was sent to the
7 1st Krajina Corps. I wouldn't know that. Let me just read the two
8 sentences here. I don't understand anything. I don't know why this was
9 sent to the command of the 1st Krajina Corps in Banja Luka in the first
10 place. I wouldn't know.
11 JUDGE KWON: It's so natural that if a directive, some document
12 like directive has been signed it should be circulated to the subordinate
13 units, but is it usual to send or circulate this kind of document after
14 almost ten days after the signature? Would it be usual?
15 THE WITNESS: [Interpretation] As far as I know, in general terms,
16 in principle I would not find it commonplace, but I don't know why they
17 did that. To put it simply, I never saw this document, and I can't say
18 anything concrete about it.
20 Back to you, Mr. Tieger.
22 Q. In paragraph 185 of your statement, General Subotic --
23 A. 185?
24 Q. 185; right.
25 A. Just bear with me, please. Very well.
1 Q. You describe some issues raised around the beginning of
2 October 1995 concerning the exchange of prisoners, first talking -- that
3 is you and Dr. Karadzic, talking to Serbs who were interested in having
4 prisoners exchanged and presumably therefore having Serbian prisoners
5 come home, and then a conversation or -- or a discussion that
6 Dr. Karadzic had apparently following that discussion with the
7 representatives of the people who had been killed in Vijenac about
8 exchange, Dr. Karadzic talking to the commission for exchange to try to
9 arrange an exchange. So if I understand that correctly, after the
10 conversation with these Serbian civilians who were concerned about the
11 fate of their family members, Dr. Karadzic then contacted the commission
12 for exchange and said get in touch with the Muslims and arrange for
13 exchange. That's an accurate understanding of what you've -- the
14 information you provided? That's just the backdrop to the question I
15 want to ask you. So is that -- do we have an accurate grasp of the
16 situation at that time? Talk to these people, they wanted an exchange,
17 Dr. Karadzic then talked to the commission to assist those -- those
18 concerns.
19 A. I can only confirm very precisely and say that things happened as
20 they are described in paragraph 185. I can't change any of that. I can
21 read the paragraph for the benefit of the Trial Chamber, but I wouldn't
22 change a single word in this paragraph. The conversation that happened
23 that night had a huge impact on me. I'm very familiar with the
24 situation, and I can only confirm that the veracity of what is written in
25 paragraph 185.
1 Q. All right. And then the people he was talking to at the
2 Commission for Exchange said where are the prisoners from Zepa and
3 Srebrenica? And Dr. Karadzic said well, as far as he knew, they had been
4 exchanged. That's what you say here; correct?
6 Q. Now, the fact is, General Subotic, that there had been enormous
7 pressure that began very shortly after the fall of the enclave for the
8 Bosnian Serb authorities to identify where the missing Muslim men and
9 boys were and to -- where they were held or what had happened to them.
10 Isn't that right? Do you acknowledge that? The enormous pressure from
11 the international community, and indeed even ultimately pressure from the
12 Serbian community because exchanges weren't -- were -- of Serbs were
13 being held up because of the question of where the men and boys from
14 Srebrenica were?
15 A. I accept just the last part that you said, and the other thing
16 remains as I said it. Because what is stated here by those people, by
17 the parents, they are angry at Mladic and at others about why there was
18 no exchange carried out between Zepa, meaning the Muslim prisoners and
19 the Serb prisoners. What I wrote I cannot change, and I do not accept
20 any other variations.
21 Q. First of all, is it your testimony that you were unaware of any
22 pressure from the international community, either the international
23 media, or international representatives on the Bosnian Serb authorities,
24 and particularly Dr. Karadzic, to identify where the thousands of missing
25 Muslim men and boys were?
1 A. I didn't know that. No, not at all. I did not have any
2 information, nor did I know, and it wasn't my area of responsibility
3 either. There are other commissions. Others were entrusted with that.
4 That was not something that I was entrusted with. I stand by this
5 completely, behind everything that is written in paragraph
6 170 [as interpreted].
7 Q. Well, I thought I understood you in the course of your statement
8 to make a point about your concerns about the -- what the international
9 media was saying; in fact, there's a portion in your statement devoted to
10 propaganda and what the international media was saying about the
11 Bosnian Serbs that you considered unfair. Let me give -- let me just
12 recite for you quickly a little bit of the evidence in this Court about
13 what was being written in the international media in the aftermath of
14 Srebrenica. So at P4397, on the 17th of July, there was an article in
15 the British publication "The Independent."
16 THE INTERPRETER: Interpreter's correction as to the previous
17 answer of the witness: It's paragraph 185.
19 Q. That discusses a film that was made two days after Srebrenica and
20 what it shows, a reference by a UN official to General Mladic reportedly
21 saying that Serbs had been forced to kill lots of people because they
22 were trying to break out of Srebrenica. The article says the fate of the
23 menfolk of Srebrenica has been a concern to their families and
24 international human rights since the enclave fell last Tuesday, and first
25 a delegation of the ICRC being prohibited from visiting Bratunac where
1 many of the captured Muslim men were believed to be held. Just a few
2 days later P4398, referring to possibly the biggest mass execution of
3 Muslim prisoners by the Bosnian Serbs in more than three years of war.
4 According to residents of the Bosnian Serb controlled town of Bratunac
5 and Serbs from Serbia who had visited the area, as many as 4.000 captured
6 Muslim men from Srebrenica have been killed by the Bosnian Serbs. Two
7 days after that, 4400, 23 July another article even if -- saying even if
8 a fraction of the stories emerging from Srebrenica are true, the men of
9 Zepa have every reason to be afraid of becoming POWs of General Mladic.
10 And again referring to information from residents of the Serb controlled
11 town of Bratunac and Serbs from Serbia about thousands of captured men
12 from Srebrenica who had been summarily executed. And stories of mass
13 executions of prisoners have started to cross the Drina River, irritating
14 Bosnian Serb authorities because this time the tales are recounted by
15 Serbs. Or just one more, P4401, two days after that reporting that in
16 the days after Srebrenica fell, residents reported seeing truckloads of
17 men being brought to shallow pits dug on the other side of the river bank
18 and shot by Bosnian Serb soldiers, and stating -- and reporting that
19 thousands of men from Srebrenica were taken prisoner by the Bosnian Serbs
20 after the "liberation" of the Bosnian enclave on 11 July. Some estimates
21 of prisoners executed are as high as 4.000.
22 Now, is it your testimony, General, that none of this information
23 and none of these allegations and none the concern of the international
24 community reflected in these articles was known to you at the time, that
25 you had no awareness of this whatsoever?
1 A. I assert that all of this is a lie. I would not assert that it
2 was a lie had these gentlemen from England called Serbian journalists so
3 that together at the actual location they could establish what the
4 situation was, and then they could write the article together in English
5 and in Serbian. Then I would believe them. I read six books about
6 Srebrenica which were written after the war with all the details.
7 Q. You've mentioned that [overlapping speakers]
8 A. And that is why I do not believe, I do not believe at all,
9 Mr. Tieger, this. I didn't even know about this, this -- this title,
10 nothing like that. I don't believe that. I just simply don't believe
11 it.
12 Q. I wasn't asking you --
13 A. Had it been realistic.
14 Q. I wasn't asking for your opinion about the accuracy of these
15 articles. We have a lot of information about that, General. I was
16 asking you if you claim to have been totally unaware that these reports
17 were being made and disseminated internationally.
18 A. I state that I did not know that at all. I state that
19 absolutely. Only after the end of the war I learned some other things.
20 At that time, I didn't know anything.
21 Q. Were you unaware that on July 24th, 1995, the Special Rapporteur
22 of the commission on human rights sent Dr. Karadzic a letter expressing
23 his deepest - and I'm referring now to P6396 - expressing his deepest
24 concern regarding the recent events in the Srebrenica area which resulted
25 in the forced displacement of some 40.000 individuals and referring to
1 reports that as a result of these events several thousand individuals are
2 unaccounted for and there is fear that many have been killed or detained
3 and calling for a proper investigation and evaluation, and in particular,
4 calling for access to those who had been detained during the recent
5 events? Were you unaware of that?
6 A. I did not know. I absolutely did not know.
7 Q. What about a letter sent to Dr. Karadzic by the
8 Special Representative for the Secretary-General on the 12th of August
9 also requesting access to investigate these allegations and what happened
10 to the men and boys of Srebrenica, to the thousands of men and boys who
11 were the subject of the international community's concern. Were you
12 unaware of that too? Is that your testimony?
13 A. I did not have the opportunity to see the letter or to know about
14 it. I didn't have the opportunity. I don't know.
15 Q. And you claim this is something Dr. Karadzic didn't mention to
16 his special advisor. He kept this from you.
17 A. I would not say that he did not mention it or that he concealed
18 it from me. I don't know. Perhaps I wasn't there. I have no idea. I
19 am simply not aware of it. I simply don't know about it. If there is a
20 letter, this letter in Serbian, perhaps I could refresh my memory, but
21 like this, no, I -- I am not aware of it.
22 JUDGE KWON: Yes, Mr. Robinson.
23 MR. ROBINSON: Yes, Mr. -- could we have the exhibit number for
24 that letter? I couldn't find it in e-court [overlapping speakers] by the
25 date.
1 JUDGE KWON: It's in front of us.
2 MR. ROBINSON: I'm speaking of the 12th of August, letter.
3 MR. TIEGER: 2288.
4 MR. ROBINSON: Thank you.
5 THE ACCUSED: [Interpretation] Line 8 of the transcript, the
6 witness did not say the -- the -- I would not say that he did not mention
7 it or conceal it. This "did not mention it" is superfluous.
8 JUDGE KWON: Let's continue. Thank you.
10 Q. Okay. This letter has not been translated. I can read it to
11 you. Dear -- it's dated the 12th of August, 1995. It's from the
12 Special Representative the Secretary-General for the former Yugoslavia,
13 Mr. Akashi. It states: Dear, Dr. Karadzic, it refers to the
14 Resolution 1010 of 10 August in which the Security Council expressed its
15 deep concern and reports of grave violations of international
16 humanitarian law in and around Srebrenica and at the fact that many of
17 the former inhabitants of Srebrenica are not accounted for. I am equally
18 concerned at these reports, especially at the allegation of the existence
19 of a mass grave identified by the government of the
20 United States of America.
21 And he requests as a matter of urgency and in accordance with the
22 directives of the resolution he mentioned, that Dr. Karadzic allow
23 UNPROFOR, and co-operate with UNPROFOR, to investigate the report of the
24 existence of a mass grave and requests immediate access for
25 representatives of UNHCR, the ICRC, and other international agencies to
1 persons displaced from Srebrenica and Zepa in areas of
2 Bosnia and Herzegovina under Bosnian Serb control. Then he asks further
3 that ICRC representatives be permitted to visit and register any persons
4 detained against their will, including any members of the forces of the
5 Republic of Bosnia and Herzegovina.
6 So I mention that to you as another reflection of the concern by
7 the international community and pressures placed on the Bosnian Serb
8 authorities. Is that a document again that you profess to be -- to have
9 been completely unaware of reflecting concerns by the international
10 community and demands by the international community that you also
11 profess to be unaware of at the time?
12 A. I attended practically all the meetings between Akashi and
13 Mr. Karadzic, and I took notes, kept the minutes, and if any of this was
14 referred to, then I am aware of it. And I knew that Mr. Akashi and
15 Mr. Karadzic would always and frequently speak in Pale or in some other
16 places, and I was always present, and I knew that. And this was talked
17 about. All the questions were talked about. There was an understanding.
18 There were agreements about these things that were happening, but this
19 specific something that you offered here I did not take part in that, but
20 I know that the president always allowed, as far as the Red Cross is
21 concerned and UNPROFOR and as far as all the international organisations
22 are concerned. He actually had problem with the army and the commanders,
23 actually, because he always permitted everything that the international
24 community asked to be done.
25 Q. All right. So I'm a little confused now. Are you saying you
1 were aware of these pressures or you were not aware of these pressures by
2 the international community to allow to identify where the men and boys
3 of Srebrenica were and to allow access to them if they were still alive?
4 A. Mr. Tieger, the international community was pressuring
5 Republika Srpska every minute, every moment, and you are asking me now to
6 split hairs. I always was under pressure, not just myself but everybody
7 in Republika Srpska. How could I have known about every instance of
8 pressure by the international community? These pressures were there
9 every day. So I will tell you what I know sincerely. I'm not afraid of
10 anything. But don't impose any mention of any pressure on me that I'm
11 not aware of it. There's no reason for that.
12 I am not aware of this. Therefore, I cannot discuss it.
13 Q. Let's turn back --
14 JUDGE KWON: Just a second. Mr. Subotic, you said you are not
15 aware of this. You were not aware of this at the time. But do you agree
16 that Mr. Karadzic received this letter from Mr. Akashi?
17 THE WITNESS: [Interpretation] I believe that he did. If you say
18 so, then I believe that he did receive it, because I know that he had
19 contacts very, very frequently and very solidly with Akashi. I don't
20 have any doubts about it.
21 JUDGE KWON: If Mr. Karadzic had received this, you would also
22 have received this as well. So is the thing that you now don't remember
23 this?
24 THE WITNESS: [Interpretation] No, I did not receive the letter.
25 I did not have the letter in my hands in the Serbian language.
1 Therefore, had I had it, I don't know if he had received the letter, I
2 would believe that he would have taken measures to do what Akashi was
3 requesting in the letter. But this was not in my remit to deal with
4 this. This was in the remit of other structures in Republika Srpska.
5 JUDGE KWON: I think you -- this is one part I don't follow among
6 your answer. You said:
7 "I did not receive the letter. I did not have the letter in my
8 hands in the Serbian language."
9 When an international actor like Mr. Akashi, who was in
10 particular the Special Representative for the Secretary-General, when
11 they -- when such international actors sent that letter to Mr. Karadzic,
12 did they send also in Serbian language?
13 THE WITNESS: [Interpretation] That I don't know. I know mostly
14 whenever I was at the talks with Mr. Akashi there was interpretation into
15 Serbian. I would take notes. Ninety per cent of the conversations then
16 I would note down. As for this specific case, we did not have, as far as
17 I can remember, talks, because then I would remember it. I would have it
18 written down somewhere.
19 JUDGE KWON: You answered that you didn't have the letter in the
20 Serbian language. It sounded to me that you received the letter in
21 English.
22 THE WITNESS: [Interpretation] No. What I wanted to say was I
23 would have known about it had the letter been in Serb. I would have
24 known. Perhaps I didn't express myself properly. It's logical if I had
25 seen the letter I would not have know what it was about because Karadzic
1 did not give me the letter or speak to me about the letter. I don't know
2 if I was at the office at that time. I don't recall simply.
3 JUDGE KWON: Thank you. Mr. Tieger.
5 Q. Getting back to paragraph 18 --
6 JUDGE KWON: Just a second. There seems to have been an omission
7 on the part of French translation. I take it it will be supplemented
8 later on. Shall we continue. Yes.
9 MR. TIEGER: Thank you.
10 Q. So in light of this backdrop, General Subotic, getting back to
11 paragraph 185 where you state that Dr. Karadzic said that as far as he
12 knew the prisoners from Zepa and Srebrenica had been exchanged, I want to
13 ask you what you meant by -- what you understood he meant by "as far as
14 he knew." So did -- how many prisoners did you understand had been
15 exchanged, prisoners from Srebrenica and Zepa?
16 A. I don't know. The number of prisoners was not -- actually, you
17 saw here that these parents were angry. They were asking where Mladic
18 was and where Tolimir was and why were the prisoners from Zepa not
19 exchanged for their prisoners. That's how I understood it in that
20 conversation and that is what I noted down. Therefore, that is
21 disputable. We didn't know Radovan or I when we spoke with them whether
22 they were actually exchanged or not, but he said, As far as I know, the
23 prisoners were exchanged. But we were not there. Other teams were there
24 for the exchange of prisoners, but at that meeting there were just the
25 two of us because the parents came to address us.
1 Q. And according to you, what steps did you and Dr. Karadzic take in
2 the wake of this meeting to find out whether or not prisoners from
3 Srebrenica and Zepa had been exchanged, how many had been exchanged, and
4 when that exchange had taken place?
5 A. We did not have any information prior to the meeting nor did we
6 know that they would come. They came at midnight to the office in
7 Banja Luka. At midnight, Mr. Tieger. We were just caught up, caught out
8 in the situation, both Radovan and I, and then we asked for the people
9 who were deal with that, who knew what was happening with the prisoners
10 to come. The two of us didn't know anything about it. I mean, simply
11 what I wrote down is what I stand by. I have nothing else to say
12 regarding that.
13 Q. Okay. And your testimony is that by October of 1995, there was
14 no particular reason for you or Dr. Karadzic to have any awareness of
15 where Srebrenica prisoners were or how many there were or whether they
16 had been exchanged; correct? That's your testimony. So you were just
17 caught out when this issue came up.
18 A. Correct. Correct.
19 Q. All right. I want to turn to another subject, and that is
20 something you raised in paragraphs 240 and 241 of your statement, and
21 that's about Blagaj Japra, the Japra Valley, and in those paragraphs --
22 okay.
23 A. Could I please see the text in the Serbian.
24 Q. General Subotic, this statement that I'm referring to was
25 prepared by you and the Defence, or at least by -- I mean, I don't know
1 the extent of your participation in this firsthand, but this is a
2 document that has your signature on it and to which you attest -- yes.
3 A. Yes. Can I please see the document so I can just remind myself.
4 I cannot memorise so many pages.
5 Q. What's in front of you right now? I see there's a document in
6 front of you. What is it?
7 A. What I see before me is in English. It states "Blagaj Japra,"
9 Q. And what's the document --
10 A. But it's in English.
11 Q. Mr. Subotic, what is the document that you have on the table in
12 front of you if it's not your statement?
13 A. I have a document on the table. Paragraph 240, page 91,
14 Blagaj Japra, but it's in English.
15 Q. So --
16 A. 241 is also in English, on page 92.
17 Q. General Subotic, there's only -- there's only --
18 JUDGE KWON: Let me ask the witness. If you did not understand
19 your statement, how could you sign the document as your statement? You
20 confirmed that you signed the document.
21 THE WITNESS: [Interpretation] I received a statement in Serbian,
22 but it statement I didn't bring my own. I was told that I could not
23 bring anything into the courtroom, but this is something that I found on
24 the desk here, the statement, and it's in English. You have Serbian and
25 English sporadically. I don't know what that is.
1 JUDGE KWON: Could you see the last page. Last page of the
3 THE WITNESS: [Interpretation] I signed. I signed the document,
4 but I didn't know that in my document -- well, in my document everything
5 is in the Serbian language, the document that I received from the
6 Defence, but this one is partially in English and partially in Serbian.
7 The paragraphs and the pagination are the same. I don't know why this
8 happened. Perhaps it's because of something on your ...
9 JUDGE KWON: So I'll read out the passage where Mr. Karadzic
10 asked you questions. It's yesterday's transcript, page 39984.
11 Dr. Karadzic showed you the statement that you have with you now:
12 "Do you see that statement of yours on the screen?
13 "Yes.
14 "Have you read that statement and have you signed it?
15 "Yes."
16 THE WITNESS: [Interpretation] I read it in the Serbian language
17 and I signed it, but this, this, theirs, I mean mine, the one I have in
18 the hotel, all of it is in Serbian. I didn't even know that there is
19 this mixed thing, Serbian-English. I don't know why.
20 JUDGE KWON: Shall we strike out the witness statement? I think
21 probably Mr. Karadzic or Mr. Robinson need to clarify.
22 MR. ROBINSON: Yes, Mr. President. I believe that the witness is
23 mistaken. What is in front of him is what he signed, and what he signed
24 is a compilation of his statement as he gave it to Mr. Sladojevic and
25 also the excerpts from his verbatim testimony in the Krajisnik case. So
1 that's what the statement consists of.
2 JUDGE KWON: What is contained in paragraph 24 is part of his
3 statement which he gave on 20th of May, 2006.
4 THE ACCUSED: [Interpretation] Prosecution. Prosecution.
5 JUDGE KWON: Mr. Subotic, having heard this clarification, are
6 you minded to change your answer with regard to paragraph 240?
7 THE WITNESS: [Interpretation] What do you mean change? I don't
8 understand the question now. If I knew how to read this answer or if
9 someone would read it out to me, then I could recognise it or not
10 recognise it. But I don't know how to read it, so I don't know whether I
11 can change it. I believe that that's it, but it's just written in the
12 English language.
14 JUDGE KWON: Let me ask one question to Mr. Robinson for
15 clarification. Take paragraph 240. That's a part of witness's
16 statement. I take it there's a Serbian version of his witness statement,
17 isn't it? Why do we have it English here?
19 MR. ROBINSON: Well, Mr. President, Mr. Sladojevic has told me
20 he's not sure actually what the status is of the portions of the
21 statement that relate to interview by OTP of Mr. Subotic, so he believes
22 that they may also have been in English like the Krajisnik testimony, but
23 he's not sure.
24 JUDGE KWON: Speaking for myself, I'm concerned whether we need
25 to strike out all the English part from his statement. If you could
1 assist us in that regard.
2 MR. ROBINSON: Yes. First, definitely with respect to the
3 Krajisnik testimony, this is no different than other 92 ter witnesses who
4 have listened to their testimony before --
5 JUDGE KWON: Just a second. But we need to satisfy the
6 requirement that if asked, witness would have answered the same question
7 here. That's not satisfied here in this case.
8 MR. ROBINSON: I believe he would say that if -- that his
9 testimony he gave in the Krajisnik case was the same.
10 JUDGE KWON: But he wasn't offered that opportunity, because he
11 can't read English at all.
12 MR. ROBINSON: That's true, but he gave the testimony, and I also
13 believe he listened to the recordings of the testimony from the Krajisnik
14 case. So he can say that the answers that he gave would be the same,
15 which is what I believe he has said when he was first questioned by
16 Dr. Karadzic. So those portions that are in English from the Krajisnik
17 case I think fall with the same parameters that we have for other
18 witnesses who have testified in this Tribunal and whose testimony have
19 come under 92 ter.
20 With respect to the statement from the interview, we would have
21 to check on that.
22 JUDGE KWON: Any observation to make, Mr. Tieger?
23 MR. TIEGER: Well, just a couple. I mean, first of all, it is
24 the case that witnesses in the past who cannot read English, instead of
25 being shown a transcript had reviewed the previous testimony or the
1 previous statement in another form so that they were in a position to
2 affirm that that was accurate. So I don't have a problem with the
3 modality, and I'm not trying to suggest that. And the -- and I'm not
4 trying to make this particularly a precise memory game for the witness so
5 that he has to be able to correlate the paragraph number with information
6 he previously reviewed for the purpose of making the statement, but
7 that's -- and, in fact, the -- the -- my point in raising this was not to
8 challenge that the witness had said what he said - it is reflected in
9 paragraphs 240 and 241 - but instead to point out what was omitted that
10 he also said but didn't make it into the statement that appears in front
11 of Your Honours.
12 So I wasn't actually challenging the position that he didn't
13 stand by 240 and 241. He's repeated several times, and I think he
14 would -- whether he recalled what he said in Krajisnik or not, that he
15 doesn't back down from that. But in this particular instance, it's a
16 question of selective identification of what he previously said.
17 JUDGE KWON: Whether or not the Prosecution is challenging, I'm
18 concerned about the requirements of Rule 92 ter have been met in this
19 case.
20 Did you try to say something, Mr. Subotic?
21 THE WITNESS: [Interpretation] Mr. President, we have two
22 variants. The first variant is that you allow me within 10 minutes of
23 time to bring my own statement that I have at the hotel and everything
24 will be fine.
25 The second variant is that this be read out to me from English
1 into Serbian, and I will immediately confirm whether that's it or whether
2 it's not, and I will say what I know, but precisely --
3 JUDGE KWON: So by way of -- by way of example, let me turn to
5 THE WITNESS: [Interpretation] If somebody could read that out to
6 me in Serbian, then I will answer Mr. Tieger.
7 JUDGE KWON: I will do that. This is pat of your testimony in
8 Krajisnik case. So Judge Orie asked you:
9 "Judge Orie: This Chamber received evidence which suggests that
10 such an event did take place, and that is not just stories, but that
11 includes documentary evidence, that includes some pictures. So,
12 therefore, if you say, well, hardly could have taken place since we
13 didn't know about such a massive event, I just inform you that of course
14 there was no finding from the Chamber in that respect, but there is quite
15 a bit of evidence."
16 And you answered like this:
17 "I have no reason to doubt what you're saying, but I do have
18 reason to doubt that anybody in Pale was informed. I can simply not
19 fathom that somebody could have been informed at Pale without raising a
20 campaign on that issue. That's the only thing I'm saying."
21 And this is my question for you: Before you signed the statement
22 which is in front of you, did anybody of the Defence team read out that
23 paragraph to you?
24 THE WITNESS: [Interpretation] They read that part out to me, and
25 they gave it to me in the Serbian language in my copy of the statement.
1 I don't know. I didn't go into this.
2 JUDGE KWON: Just a second, Mr. Subotic. I would like you to be
3 precise. If you don't remember, please say you don't remember. You said
4 that the Defence team gave this part in the Serbian language. Is it
5 correct?
6 THE WITNESS: [Interpretation] Correct. It's not that it was
7 given to me. It was read out to me. But I was told that this would
8 be -- I mean, in my copy in the Serbian language, all of it, not only
9 this, but also those other ones.
10 JUDGE KWON: Be precise, please. Did you receive this part in
11 Serbian language, or the Defence team only read out the part to you?
12 Which is correct?
13 THE WITNESS: [Interpretation] I -- well, they had two copies. I
14 didn't know that in their copy this was in the English language. I
15 received a copy in the Serbian language. All the items, all the
16 paragraphs, all the pages. I have that copy, but since I was told that
17 I'm not allowed to bring anything into the courtroom, I didn't bring
18 that, but I have a copy of the statement.
19 JUDGE KWON: Thank you, Mr. Subotic.
20 Yes, Mr. Tieger, please continue.
22 Q. Well, His Honour just kindly read out to you what 241 says, and
23 that's the gist of what you convey in your -- the information you provide
24 about events in the Japra Valley and the awareness in Pale and among the
25 Bosnian Serb leadership about what happened; that is, that you have
1 reason to doubt that anybody in Pale was informed because nobody raised a
2 campaign on that issue.
3 Now, that did not reflect the entirety of the discussion that
4 took place during your testimony in the Krajisnik case, and if we turn to
5 pages 26470 --
6 A. What -- I mean, I don't understand this question sufficiently.
7 What I said in the Krajisnik case, well, I wasn't asked for anything
8 precise. I wasn't asked for anything precise. I was just -- I mean,
9 well, the statement I gave is what I knew then. It's not that I had any
10 precise information and so on, and no one asked me for precise
11 information. But later when the statement was taken from me, I mean in
12 2012, that was something different because there was information that was
13 accessible to me and that was generally accessible, a lot of information.
14 Q. And what you said in the Krajisnik case to Judge Orie was that:
15 "You see, Mr. President, this involves," and again this is
16 referring to the Japra Valley operations, "... this involves a huge
17 amount of people, 4.000 people. It's not a small group. It's neither 4
18 nor 40. I believe that there is nor municipal authority or police force
19 in that area that could have kept this from the government or Pale.
20 Since that was never a subject of discussion ever, I simply cannot
21 conceive that something like that could have happened without anybody
22 having been informed whereas the communication lines were working."
23 So what's not contained in paragraphs 240 and 241 is the fact --
24 is your position that there is no way that this event, that any municipal
25 authority or police force from that area could have kept this information
1 from the republic-level authorities or from Pale; correct?
2 A. Correct. Correct. I stand by that and that is what is written
3 in my statement, the one in the Serbian language, likewise what
4 Mr. Tieger read out just now. And I stand by that and I cannot claim
5 anything else.
6 Q. Let me turn to another event not involving 4.000 people but
7 involving a large number of people that was known -- also known to the
8 authorities in Pale, and that's the events at Koricanske Stijene. And
9 that's referred to in your statement at paragraphs 242 through 260.
10 And in those paragraphs, Mr. Subotic, you relate the awareness of
11 Dr. Karadzic about the event, refer to a session of the Presidency or at
12 least the attendance of Presidency members and others at an ad hoc
13 meeting or session, Dr. Karadzic giving you the order - and at this time
14 you were the minister of defence - giving you the order to go to
15 Banja Luka and hold a meeting. Your participation in a meeting involving
16 prosecutors, judges, chiefs of police stations in the area and others,
17 and passing on verbally the order of the president and the Presidency to
18 take immediate investigative actions and all actions necessary,
19 et cetera. And then it also provides information in the subsequent
20 paragraphs about some of the -- about what happened afterwards and what
21 transpired. For example, you were asked the question, it appears in
22 paragraph 244, how many of those policemen who escorted or guarded the
23 convoy were interviewed about what happened to what, as it turned out,
24 were more than 200 Muslim civilians who were massacred. And you
25 indicated you didn't know, you were not informed about that.
1 So that's the event in the portion of your testimony that I'm
2 referring to, Mr. Subotic.
3 Now, as part of your statement, you indicate that one -- you
4 suggest there that there were three people responsible for the killings
5 at Koricanske Stijene, that is three policemen, one of whom was
6 prosecuted, two of whom were killed, presumably meaning in combat after
7 they fled, after they went to the army.
8 First of all, when you refer to one person prosecuted, you're
9 referring to one person prosecuted many years later by the ICTY and
10 convicted of that crime here; right? Or do you claim to be aware of
11 somebody else who was prosecuted by Republika Srpska authorities for the
12 murder at Koricanske Stijene?
13 A. Mr. Tieger, I gave that statement in the Krajisnik case very
14 precisely and in great detail, and in that statement I said all the
15 things that I learned at that meeting in Banja Luka, when this entire
16 team went to the site, to the scene of this crime, where we visually saw
17 this area. So I gave this statement only on the basis of the notes I
18 took at that meeting. What I was told by the police, the judges, in
19 order to be able to convey all of that to the state Presidency. That was
20 my task. Everything that you repeated just now about those three
21 persons, I noted all of that down at that meeting in Banja Luka. That is
22 to say I -- actually, that is what I was told. I was not an
23 investigator. I was not aware of the case. Quite simply, I conveyed the
24 information to the Presidency. I know, I do know, later when the trial
25 took place and so on, that it was, after all, the way you are saying. It
1 wasn't only these three, and so on and so forth, but I could not judge
2 anything, I could not investigate, and that is why my statement -- I
3 mean, I claim that it's only from that meeting and that event where we
4 were on that day, nothing else. And I personally condemned that at that
5 meeting, personally, as a human being, and I asked absolutely for very
6 precise investigations, and so on and so forth. So I have nothing else
7 to say with regard to that case.
8 Q. Mr. Subotic, this is not exactly a major whodunit. Everybody
9 there knew that the people who had been killed were part of a convoy that
10 was escorted by a policeman from the Prijedor Police Station; right?
11 THE INTERPRETER: The interpreters note: We cannot hear the
12 witness.
13 JUDGE KWON: Mr. Subotic, probably you turned off your
14 microphone. The interpreters did not hear you. Could you repeat your
15 answer.
16 THE WITNESS: [Interpretation] I'm saying -- I mean, I don't know
17 which part I should repeat.
18 JUDGE KWON: Could you repeat all over.
19 THE WITNESS: [Interpretation] Well, my task was in Banja Luka at
20 the Ministry of the Interior to bring together all relevant persons and
21 institutions that should explain to me what it was that had happened.
22 The president, who entrusted me with this task, or, rather, the member of
23 the Presidency were primarily interested in the following: Whether the
24 military, the army took part in this. Immediately at the beginning of
25 the meeting that was eliminated. The army had nothing to do with that
1 event. So it was the police from Prijedor, and all of it was explained
2 to me. I wrote all of that down precisely, who said what. The head of
3 one centre, the other centre, and everybody else who took part, I
4 explained all of that when I testified at Mr. Orie's. I did not know
5 whether it was true or whether it was not true, whether that is exactly
6 the way things happened or not.
7 Later on in the later period I found out because that was made
8 public. There were trials, arrests, and so on. What Mr. Tieger said to
9 me just now, that was confirmed exactly, but I didn't know about that. I
10 could not inform the Presidency about that because the information I
11 provided was what I wrote down specifically. That's it.
13 Q. General, you said just said that according to you you eliminated
14 from the outset that it was the army and knew that it was the police from
15 Prijedor. That was the question I had asked you. So policemen had been
16 sent to escort the convoy, which means that -- that since you had the
17 police chief -- the relevant police chiefs there, everybody knew who
18 those police member were; right? Right there on the spot, their
19 identities were easy to identify.
20 A. Mr. Tieger, they just announced three persons to me, I mean as an
21 answer that I could convey to the Presidency. I don't know anything
22 else. Whether they did know or whether they did not know, that was their
23 affair.
24 Q. You're suggesting that this matter was considered so important by
25 the leadership that it sent you out personally to make sure to order that
1 all appropriate measures were undertaken to solve it. I'm saying to you
2 it was crystal clear who was involved. Do you even know if any of those
3 people were interviewed, interviewed, by Republika Srpska authorities in
4 connection with this crime in 1992 or 1993 or 1994?
5 A. I don't know about later when I left. I know that the ministry
6 took part, the MUP, and the Ministry of Justice. As I said, there were
7 judges, and I don't know all the relevant ones, but I didn't take part in
8 that. Did I not have an opportunity to learn about this.
9 Q. This Trial Chamber has received evidence that the people involved
10 and responsible, directly responsible, hands on, for these killings were
11 not in hiding and had anyone been looking for them they would have been
12 found. And you can turn to P4257 for that evidence. And that makes
13 sense, doesn't it, I mean, since their identities were known as
14 policemen, it wouldn't have been so difficult to get ahold of if anyone
15 had wanted to find them; right?
16 A. If you have that, I believe that's the way it is, but I do not
17 know about that, but, say, logically I can infer that these persons
18 should know that if, as I said, they said that later. Well -- but I was
19 not in the know. Had I known about that, I would have conveyed that
20 immediately, and I would have included it in my note.
21 Q. And so that would further suggest that people involved didn't
22 want to find them. That also logically follows, doesn't it?
23 A. According to what you are claiming, this would be a logical
24 conclusion. However, I cannot claim that because I didn't participate in
25 any of that. That was not my task. It was a task that was supposed to
1 be carried out by investigators, judges, prosecutors. I don't know who
2 else, but in any case, professionals. I was not at the receiving end of
3 any further information. I didn't receive any information, and I didn't
4 really express any interest in receiving further information.
5 Q. So according to you, these most elementary aspects, the most
6 rudimentary aspects of a serious investigation were not undertaken, but
7 you didn't care, and as far as you know, I take it, Dr. Karadzic didn't
8 do anything about it or care about it either?
9 A. Mr. Tieger, we had zillions of other problems. We had
10 institutions that were in charge of those matters. It was not up to the
11 two of us to monitor the work of the police, the formers [as interpreted]
12 of everybody else and their uncle. You have to understand that we are
13 not computers. Competent people were tasked with doing that. Whether
14 they did it or not, I don't know. I know practically nothing about what
15 they did. However, that was our system. So the system had to function,
16 and things had to be done according to that system.
17 I cannot invent things in order to satisfy somebody's needs or
18 requirements. I don't know. How was I supposed to know what the police
19 did? I was not the chief of the police or the minister of police, for
20 that matter. I knew for a fact that the military did not take part, and
21 that was what mattered to me. And the rest I did according to the
22 Presidency's instructions. I submitted my report. The ministry did
23 their own job. The prosecutors, the judges did what they were supposed
24 to do, and I don't know what they did. I did not get involved, therefore
25 I cannot discuss that matter.
1 Q. Last question before the break. Slobodan Avlijas, the deputy
2 minister of justice or at least a high ranking official in the
3 Ministry of Justice at that time, participated in the meetings concerning
4 what to do about Koricanske Stijene. That is the meeting in Banja Luka
5 attended by senior officials, including you. That's T35186. And he
6 testified and told this Court that everybody knew what had happened and
7 that in any properly functioning state, Simo Drljaca would have been
8 arrested right there. Do you dispute that?
9 A. This is not what I noted. What I noted I conveyed at the trial
10 of with Mr. Orie. And as for that word, Simo Drljaca was removed from
11 his position, so I don't know. I suppose that what you're saying is
12 correct.
13 Q. Simo Drljaca became an assistant minister in the MUP, a higher
14 ranking position, and in fact was given a major award by Dr. Karadzic
15 after this event, isn't that right?
16 A. I don't know whether he was decorated because of that.
17 Dr. Karadzic did not explain why people were decorated. It was the
18 chiefs and commanders who were supposed to provide their input as to
19 whether somebody deserved to be decorated or not. This is how things are
20 done all over the world, in all the states. That's how people are
21 decorated. And people can deceive a president, people can deceive a
22 king, because things are not double checked; because if things were
23 checked then no decoration would serve any purpose. I'm telling you this
24 as an expert as the chief of a body in the Republika Srpska government.
25 But look here, there is a law on decorations, and in the law on
1 decorations it says that anybody who commits a crime or a big
2 transgression, any decoration is taken away from them. I am the author
3 of that law. I don't know whether this was really followed through if --
4 whether Simo Drljaca was decorated and whether the decoration was taken
5 away from him. If he was decorated, then the chief of police, his boss,
6 had to take the decoration from him because the law on decorations is a
7 public instrument, and I don't know whether that was ever done.
8 Q. For the record, decoration is found at P4261. I see we're
9 overdue for the break, so that I will resume afterwards.
10 MR. ROBINSON: Mr. President, I would request that the witness be
11 allowed during the break to go to his hotel which is nearby to bring the
12 version that he has of the statement so he can look at it and we could
13 verify what it is that he has. I think he could do that within the
14 30 minutes of our break.
15 JUDGE KWON: I have no difficulty with it.
16 We'll have a break for half an hour and resume at 10 past 11.00.
17 I'm sorry. We'll resume at quarter past 11.00.
20 JUDGE KWON: Yes. Please continue, Mr. Tieger.
21 MR. TIEGER: Just a note, Mr. President, before I resume that I
22 understand the witness brought the statement back from the hotel room. I
23 considered that in the interests of time I might ask to take a look at it
24 before the Court resumed the Bench, but when we began to do so, the
25 witness was very interested in explaining something about it, so we just
1 said, Stop, wait until the court resumes and anything that needs to be
2 explained can be handled then. Mr. Robinson was with me, and I believe a
3 representative of the Registrar as well.
4 So that's where we are now. The witness has that document. I --
5 I haven't seen it. I know the Court inquired about it. Mr. Robinson
6 raised it. It's difficult to know exactly in whose court this
7 immediately falls, figuratively, but I presume the Bench wants to either
8 ask about it or see it, and I certainly would like to see it at some
9 point.
10 JUDGE KWON: I leave it to the parties. Let's continue.
12 Q. Okay. Mr. Subotic, I understand that you went back to your hotel
13 room, came back with the document, the Serbian version of the statement
14 that you referred to earlier and you have it with you now. I would ask
15 if with the assistance of the Registrar if I could briefly examine that
17 A. Can I provide a certain explanation for the benefit of the
18 Presiding Judge? There was some confusion, indeed --
19 JUDGE KWON: You will have an opportunity to explain that, but at
20 this time could you kindly provide that to Mr. Tieger.
22 Q. Mr. Subotic, I understood you to stay, and I'm referring back to
23 page 25 or 26 of the -- of today's -- of the transcript of today's
24 testimony that you had two -- there were two copies. You didn't know
25 theirs was in the English language. You received a copy in the Serbian
1 language, all the items, all the paragraphs, all the pages. I have that
2 copy, but since you were told not to bring anything to the courtroom you
3 didn't bring it but you have a copy. Is this document that was just
4 handed to me the document you were referring to?
5 A. Yes. Yes. It was my mistake. I was mistaken. If I may be
6 allowed to explain in very simple terms. Can I explain things to you,
7 sir?
8 JUDGE KWON: Yes. Yes, please go ahead.
9 THE WITNESS: [Interpretation] Mr. President, so far I have
10 provided three statements to The Hague Tribunal, in 2007, 2008, and in
11 2006. All the statements have been translated into Serbian. I have
12 received them, and I have them in Banja Luka, and I have initialed all
13 the pages in those statements, and those statements are in Serbian,
14 exclusively in Serbian. That's why I was sure that I have a statement in
15 Serbian. However, when Messers. from the Defence took a statement from
16 me, they were reading sections in English, and they asked me -- at the
17 same time I had my own copies which were translated, and that's why I
18 thought that those were valid statements, and I adhere by that all the
19 time, because I initialed every page, and now that I have arrived here to
20 testify, they showed me all that. I initialed everything, and this is
21 theirs, their signature as well as mine. But I believed -- or, rather, I
22 simply forgot. Otherwise, I would have told you that I had that
23 translation. In other words, in that sense I suggest that when you're
24 putting questions to me to read everything in Serbian and then I will
25 confirm what I stated. I can confirm, because everything has been
1 initialed. Well, this was my mistake. I apologise to the Trial Chamber
2 and to you for having been convinced that I had that in my hotel. I
3 don't. I did not bring of that. I did not dare take those statements
4 with me on my travel. This is what this is all about.
6 Q. Okay. And just to make this clear, because I'm not sure that
7 your explanation was completely clear to the Court, when you said you
8 were mistaken, this document I have in front of me now that you brought
9 back from the hotel room is a mixture of English and Serbian; right? So
10 some paragraphs in English, some paragraphs in Serbian.
11 A. Only those paragraphs which were taken from my previous evidence
12 given before The Hague Tribunal, only those paragraphs. I suppose that
13 the Defence did not have them translated into translation, whereas I do
14 have all that translated into Serbian, because after each time I provided
15 evidence I received the translation of the evidence into Serbian.
16 Q. Mr. Subotic, I am not aware that your -- that the testimony you
17 gave in the Krajisnik case was transcribed into Serbian.
18 A. Yes. I have it in Banja Luka. I received that.
19 Q. And who do you say transcribed that and provided it to you?
20 A. I believe that it was the office in Banja Luka. This was done
21 subsequently. And as for what happened in Pale, I received that
22 personally in Pale, those two days in Pale. Since I was in Banja Luka on
23 several occasions in that office, I believe that I received it from them.
24 In any case, I got it from Banja Luka.
25 Q. And you say you have a copy of your Krajisnik testimony in the
1 Serbian language in Banja Luka that you can provide us.
2 A. Yes. I can send it to you from Banja Luka. I don't know how,
3 but I can.
4 Q. We can make arrangements for that. Meanwhile let me return this
5 document to you for such use as you may make of it with the -- with any
6 portions that are in Serbian that may be referred to.
7 MR. ROBINSON: Mr. President, just to round out this topic, I
8 would like to just inform the Chamber that we -- I have an e-mail here
9 where we requested the transcript of -- the portions of the statement
10 that were in English be transcribed -- be translated into Serbian but the
11 language section declined to do that. So that's why, at least for our
12 side of it, the statement was in both languages.
13 JUDGE KWON: The version of statement the witness signed was in
14 the B/C/S in most part.
15 MR. ROBINSON: Yes. He signed the version that's uploaded into
16 e-court as the B/C/S original, which contains B/C/S and English.
17 JUDGE KWON: But his original statement was translated into
18 B/C/S.
19 MR. ROBINSON: No. His original statement was taken -- was --
20 his original statement is the one that he has in front of him. It was
21 then translated into English by the CLSS.
22 JUDGE KWON: No, no. I mean his statement of, for example,
23 20th of May, 2006, was it not translated?
24 MR. ROBINSON: It was not translated. It was in English that we
25 had, as far as we were working with the English, and so it was contained
1 in English in the version that we considered to be the original version,
2 along with the Krajisnik transcript references.
3 JUDGE KWON: Just a second. Do you have his statement with you?
4 Paragraph 240, footnote 250, it refers to 65 ter 20639, witness statement
5 dated 20th of May, 2006.
6 MR. ROBINSON: Yes, that's correct.
7 JUDGE KWON: Which is in B/C/S.
8 MR. ROBINSON: As I understand it, we were working with English
9 version at the time.
10 JUDGE KWON: So you could have inserted this part into B/C/S,
11 which was sent to the CLSS.
12 MR. ROBINSON: Yes, apparently that's correct. There are four
13 statements as Mr. Sladojevic has explained to me. That's the only one
14 that's in Serbian. And that could have been done but it wasn't.
15 JUDGE KWON: So when witness said that he had all the statements
16 in B/C/S version which he initialed every page, I understood in that way
17 that -- so 240 could have been put in B/C/S, but I will leave it at that.
18 Shall we continue.
19 MR. TIEGER: Just on the heels of Mr. Robinson's comment, just to
20 note that the information that CLSS declined to translate, the Krajisnik
21 testimony is consistent with my understanding that there is no B/C/S
22 version of that testimony, but I'll move on as the Court has suggested.
23 Q. Mr. Subotic, at paragraphs 219 through 226, you deal with the
24 subject as the header indicates, "Prisoners of war and collection
25 centres." And on the heels of our discussion before the adjournment
1 about matters about which the Bosnian Serb authorities in Pale were
2 informed, I want to talk about the information you provided in the
3 statement and the information that was available at the time.
4 Now, first of all, included in that portion of your statement is
5 a colloquy between Judge Orie and yourself during the Krajisnik testimony
6 when Judge Orie quoted to you from a record of the government session
7 held on the 15th of June, 1992, which stated:
8 "'The government has considered the proposed report. It has been
9 concluded that the issue of prisoner exchange is extremely important,
10 complex, and delicate, and that if sufficient attention is not paid to
11 it, it can cause a number of negative consequences for the whole
12 republic. It has been agreed that a working group consisting of
13 Professor Branko Djeric,'" who was, as we know, the president of the
14 government at the time, the prime minister, "'Milan Trbojevic,'" who was
15 the vice-prime minister, "'Dr. Dragan Kalanic,'" minister of health,
16 "'Mico Stanisic,'" minister of the interior, "'Bogdan Subotic,'" you, the
17 minister of defence, "'and Momcilo Mandic,'" the minister of justice, and
18 it provided they "'should consider all the aspects of the prisoner
19 exchange problem and they should propose systematic and other solutions,
20 taking into account our international regulations. It is obvious that
21 solving this problem is urgent and that the regulations and concrete
22 measures for solving this issue should be proposed as soon as possible.'"
23 And you indicated to Judge Orie you -- that this was a reference
24 to prisoners who had been taken, you said, on both sides of the
25 municipality. Prior to that -- although you emphasised that these were
1 prisoners taken on a local scale and not major battles.
2 Now, Mr. Subotic, the language that accompanied the establishment
3 of this working group, referring to the problem as important, complex and
4 delicate, that's a reflection, is it not, of the awareness at that time
5 that thousands of Muslims and Muslim civilians in particular had been
6 taken under detention and if that problem wasn't resolved there would be
7 significant -- could be or would be significant negative consequences for
8 the Bosnian Serb Republic?
9 A. I did not hear at that session that thousands were mentioned. I
10 didn't hear anybody mentioning any figure. And as for the rest, I accept
11 that, but no figures were mentioned at all.
12 Q. Well, did you -- when -- when you came to understand, as
13 reflected in the report, that the problem was extremely important and
14 complex and required the involvement of the highest officials of the
15 government, including, as mentioned, the prime minister himself, the
16 vice-prime minister, the minister of the interior, and so on, did you
17 make any effort to find out the extent and scope of the problem?
18 A. For us and for me personally it didn't make any difference
19 whether we were talking about three, five, a hundred, or a thousand. It
20 was all the same to me.
21 As far as I can remember, and I didn't note any such thing, that
22 any figures were mentioned. It's different if different places. This
23 was based on different information. In any case, that problem was very
24 seriously taken into account, and this is how things were done at the
25 time as far as I know during the first government.
1 Q. Now, you had an opportunity during the course of your Krajisnik
2 case to see a photograph of at least some of the people who had been held
3 in custody, and you acknowledged, and it's there in your statement in
4 paragraph 226, that it was "... illegal, to hold people in detention and
5 treat them in a fashion that resulted in a condition like that" depicted
6 in the photograph; correct?
7 A. It's not quite like that. You remember very well if you read
8 everything what I -- you were also present in the courtroom. You
9 remember very well that I said things were staged, and I showed how some
10 images, some photographs, were not adequate. I explained a lot of that
11 to you. You are aware of that, so I don't see any need to discuss that.
12 I entirely stand by my statement of that day. I have nothing new to say.
13 Q. So is that your testimony to this Trial Chamber, that you are not
14 aware of -- or you did not acknowledge in any way that prisoners were
15 held in Bosnian Serb detention facilities in condition that reduced them
16 to an emaciated condition, that were not held in unhygienic [sic]
17 conditions, under -- with -- with inadequate food? You're denying all
18 that and saying it was staged, is that it?
19 A. Mr. Tieger, I stand by my statement absolutely, and if you wish,
20 because the Trial Chamber and all the people present here would need to
21 hear my statement and then decide on it. I stand by my statement, and I
22 have nothing further to add, the one that I provided to Orie. I cannot,
23 otherwise that is that. I have no other statement. But -- and I cannot
24 give you a different answer, because you did not read to the
25 Trial Chamber all that I said in the statement. I would like the Defence
1 and the others to find out.
2 THE INTERPRETER: The interpreter did not hear the last part of
3 that sentence.
4 THE WITNESS: [Interpretation] I know what it is. I could read it
5 out. If I had a Serbian translation here, I would read it out. I would
6 ask the Trial Chamber to allow me to read it out. I have no other answer
7 for you other than that.
9 Q. No, sir, I'm asking you a question in front of this
10 Trial Chamber, in front of these Judges right here, and I'm asking you if
11 your position is that Muslim -- Muslims were held in Bosnian Serb
12 detention facilities and treated in a fashion that resulted in their near
13 starvation, that -- that they were held in conditions where hygienic
14 facilities were totally lacking. Basically they were held in brutal and
15 inhumane conditions. Do you -- do you accept that that was the case, do
16 you deny that that was the case, or do you say that you don't know? What
17 is it as you're testifying before these Judges now?
18 A. I am saying that I did not give such a statement the way you are
19 telling it now. I provided a different statement, and I stand by that
20 statement that I provided to Mr. Orie. And you are asking me in a
21 suggestive way for me to accuse somebody or defend somebody in a way, and
22 that's something that I cannot do. What I said in that statement stands
23 before this Court and the court of God, and I have nothing more to add to
25 Q. Well, one of the things you acknowledge in that statement,
1 Mr. Subotic, is that you were the person responsible for providing -- for
2 ensuring that soldiers had sufficient food and supplies - that's at
3 page 26533 - correct?
4 A. Which soldiers? Whose soldiers?
5 Q. As minister of defence, were you or were you not responsible for
6 ensuring that the VRS had adequate supplies, ammunition, food, and so on,
7 or was that not part of your portfolio?
8 A. I was, yes. I was. That's true, but it's not the way you are
9 telling it. It's different. Read exactly what I said, exactly what I
10 said, please. Please do not impose things in a leading way. I do not
11 accept that. I did not say it the way you are telling it.
12 Q. Question at page 26533:
13 "Q. What was in your domain, Mr. Subotic, according to what you
14 told us, was logistics including such things was food; right?
15 "A. Yes."
16 A. No, no. Just specifically -- all right. Yes. Yes. And it is
17 still a yes.
18 Q. And who do you claim was responsible for providing food to the
19 prisoners taken and held by the members of the army for whose supply of
20 food you were responsible for?
21 A. I don't know. I don't know. I don't know who was responsible.
22 I don't know who was responsible. I don't know who that was. You did
23 not ask me that there, and I did not give an answer to that then. You're
24 asking me now that now, and I don't know to tell you who it was. I was
25 not the head of any camp or a quartermaster in order to be able to know
1 that.
2 Q. Well, in fact -- in fact, you said in your testimony it was the
3 responsibility of whoever set up the camp to ensure that the prisoners
4 were fed, and that's at page 26523.
5 A. Mr. Tieger. Mr. Tieger, read to me -- read to me word-for-word
6 what I said. Then I will give you an answer yes or no.
7 Q. Mr. Subotic, when this working group was set up about the complex
8 and delicate and extremely important problem of prisoners, did you or
9 anybody else make -- on that working group make an effort to find out how
10 many prisoners there were and how they were being treated?
11 A. Yes, that was done, but I don't have any indicators now, because
12 I took the least part in that, so I didn't have any numbers or anything.
13 THE INTERPRETER: Could the witness please be asked to speak into
14 the microphone.
15 THE WITNESS: [Interpretation] I did not see a camp, actually, for
16 myself, with my very own eyes. I just saw it on television or something
17 like that. I did not see it in real life.
19 Q. Are you denying that there were thousands of Bosnian Muslim and
20 Bosnian Croat prisoners being held at that time, civilian prisoners being
21 held at that time, or do you say that you don't know or that it was true?
22 A. I don't have any figures at my disposal. I personally have no
23 figures available, and I'm not able to say anything with any certainty,
24 because that's not something that I knew, and I didn't have any
25 possibility of knowing it.
1 Q. You had no possibility of knowing it, Mr. Subotic? The -- on the
2 1st of June, 1992, the 1 KK sent a report to the Main Staff, that's
3 P5398, about the 7.000 prisoners they captured during events around
4 Prijedor. On the 17th of July, 1992, a report was sent to Dr. Karadzic
5 and to Professor Djeric from the Ministry of the Interior stating:
6 "The army, Crisis Staffs and War Presidencies have requested that
7 the army round up or capture as many Muslim civilians as possible, and
8 they leave them -- leave such undefined camps to internal affairs organs.
9 The conditions in some of these camps are poor. There is to food.
10 Individuals sometimes do not observe international norms, et cetera."
11 At the 17th Assembly session -- and that by the way was P1096.
12 At the 17th Assembly session held on July 24th to 26th, 1992,
13 Representative Milanovic stood up and said - and that's at page 30
14 through 31 of the English and page 30 of the B/C/s:
15 "We have a huge problem with captured people of other
16 nationalities. We have hundreds and thousands of these prisoners."
17 On the 2nd of June, 1992, a day after the VRS -- the -- the
18 report from the 1 KK to the Main Staff I referred to earlier,
19 Dr. Karadzic met with Radislav Brdjanin - and that's according to as we
20 see in P1478, at pages 55 through 61 - and Brdjanin talked about the
21 problem of the Krajina with 14.500 Muslims and wanted a position about
22 prisoners from the highest level.
23 Now, in light of that information plus more, do you seriously
24 assert to this Court that you had no possibility of knowing that
25 thousands of Bosnian Muslim and Croat civilians were being held by the
1 Bosnian Serb authorities?
2 A. No. No, I did not.
3 Q. You don't deny that was the case. You just say you weren't aware
4 of it; is that it?
5 A. I don't know. I'm not denying anything, not do I know that it
6 was like that, and I don't -- this report was not something that I had in
7 front of me. I'm seeing this for the first time.
8 THE INTERPRETER: The interpreter did not hear the end of that
9 sentence.
10 JUDGE KWON: Could you repeat your last sentence, Mr. Subotic.
11 THE WITNESS: [Interpretation] Should I repeat the last sentence?
12 Is that right?
14 THE WITNESS: [Interpretation] I said that I did not know and that
15 I don't know and that I was not aware either of this report or of the
16 figures, and I have nothing else to say.
18 Q. I've got a limited amount of time left, so I want to move on to
19 just a couple of more topics, Mr. Subotic. At paragraph 26 of your
20 statement, you say the Serbs made no -- "there was no plan. Serbs did
21 not make any plans before the war." Then you go on to say that only the
22 Muslims made plans.
23 Now, this Court has received evidence that Dr. Karadzic talked
24 both before and after about the steps that he had formulated and was
25 prepared to take. For example, P1387, English pages 74 through 75, B/C/S
1 pages 57 through 58, at the 38th session of the Bosnian Serb Assembly,
2 Dr. Karadzic recalled the earlier days, and he explained how all the
3 steps were carefully planned to be executed once at a time, and he said:
4 "Let us use Alija's mistake to increase the price of wine.
5 Remember how all the SAOs," that's Serbian autonomous regions, "and all
6 those measures before the war always took place following Alija's
7 mistakes. There were nine to ten actions that we carried out. We
8 brainstormed them all together. However, we did not pull all nine moves
9 straight away, but we carried them out after Alija made a mistake. It is
10 then we'd make a move and the Muslims would curse his mother afterwards
11 and not ours."
12 And similarly in P2554 at a SFRY Presidency meeting in
13 December 1991, well before the 38th session, Dr. Karadzic again
14 explained:
15 "We made a list of moves, ten moves in the direction that we want
16 that the result be. But we do not put them into action until
17 Alija Izetbegovic messes something up. When Alija messes something up,
18 we make move number five and then we wait, and when he messes something
19 else up we make move number six."
20 And said the same thing in P953 --
21 JUDGE KWON: Let's break it down.
22 MR. TIEGER: Okay.
23 Q. So is it your position that Dr. Karadzic never told you, despite
24 the fact that he told everybody in the Bosnian Serb Assembly and
25 everybody present at the SFRY meeting in December 1991 that he had plans
1 for what was going to happen in Bosnia?
2 A. I did not have such contacts with Mr. Karadzic in 1991. I didn't
3 know him, and I didn't even know he existed. Specifically, in this case,
4 I have no ideas about that. This is something that I never -- I'm
5 hearing of this from you for the first time, so I cannot give any
6 comments on this. We can move on. Let's go to the next question.
7 Q. Well, let's talk about one of those very concrete steps that were
8 taken. You referred in paragraphs 52 through 55 to Crisis Staffs, and
9 tried to assert to the Court, did assert to the Court that they were all
10 self-organised and had nothing to do with the Bosnian Serb leadership;
11 right? That was your position?
12 A. Yes. Yes, that is correct.
13 Q. Well, this Court has heard a lot of evidence about the document
14 commonly known as Variant A and B, which called for the establishment of
15 Crisis Staffs. For example, P2568. We heard from a member of the
16 Main Board and Executive Board about his presence at a meeting of SDS
17 functionaries at the Holiday Inn during which Mr. Karadzic distributed
18 numbered copies of the Variant A and B document to municipal
19 representatives and gave a speech about the threat that Serbs were under,
20 and their municipal leaders were called up to take the Variant A and B
21 document. That's something you didn't know about, Mr. Subotic?
22 A. Absolutely not. In 1991, I did not have any information along
23 those lines, and I did not act along those lines. I was doing some other
24 things in 1991 to prevent the attack of Croatia on Bosnia-Herzegovina,
25 and I have a lot of statements about that. As for this, I don't know
1 anything about this. I'm not aware of it.
2 Q. Well, let's move to 1992. For example, on January 26th 1992, at
3 the 6th Assembly session of the -- of Bosnian Serb Republic, in front of
4 the Assembly, during which there was a discussion about the further steps
5 that the Bosnian Muslims and Bosnian Croats had taken toward a sovereign
6 and independent Bosnia, Mr. Cizmovic stood up and said: To resolve this
7 problem I propose that we begin with an urgent operationalisation and a
8 declaration on the establishment and promulgation of the Serbian Republic
9 of Bosnia-Herzegovina. Tasks set out in the instructions of
10 19 December 1991 should be carried out.
11 And then this Trial Chamber has heard repeated evidence that on
12 the 14th of February, at an extended session of the Main and
13 Executive Boards, including municipal presidents and so on, Dr. Karadzic
14 activated the second level of variant A and B in a speech that referred
15 to that second level four separate times, including saying that's why we
16 called you here today. You didn't know about that either?
17 A. I didn't know any of that. I began to acquire knowledge on the
18 8th of April, 1992, when I came to Pale. Before that, I never went to
19 Pale, and I did not contact anyone. So I have no information or any
20 other evidence that I know anything about it.
21 Q. Well, let me just tell you about -- you refer to one -- two
22 Assembly sessions you were present at where this information was brought
23 to your attention in a very explicit manner. First the 46th Assembly
24 session, that's P1403, and then we're talk about the 50th which we talked
25 about before. So at the 46th Assembly section at English pages 347
1 through 348 and B/C/S 304 --
2 A. What is the date of that, please? Could you please say it?
3 Q. I'll get that for you in a second. I thought I had it. That was
4 November 1994, sir.
5 A. Could you please tell me which paragraph, what the number is.
6 Q. It's not in your statement?
7 A. All right. Very well.
8 Q. And I'm also telling you that the transcript of that session,
9 which is in evidence, also reflects your presence at the session. But
10 more importantly, here's what Dr. Karadzic said at that session and then
11 said at the 50th:
12 Everything was as clear as day in the municipalities. He said,
13 Please, remember how we used to work before the war. Everything was as
14 clear as day in the municipalities where we were majority and in those
15 where we were minority. Do you remember the instruction A and
16 instruction B? We had Crisis Staffs, and it was clear they were the
17 authority. They could make mistakes, but they were still the authority.
18 The people were not left without the authority because they were the
19 Crisis Staffs.
20 And then at the 50th Assembly session to which you've explicitly
21 referred in your statement, Dr. Karadzic said the following - that's P970
22 at English pages 316 to through 317, B/C/S page 278:
23 "At the moment the war began in the municipalities where we were
24 in the majority we had municipal power, held it firmly, controlled
25 everything. In the municipalities where we were in the minority, we set
1 up secret government, Municipal Boards, Municipal Assemblies, presidents
2 of Executive Boards. You will remember the Variant A and B variants. In
3 the B variant where we were in the minority, 20 per cent, 15 per cent, we
4 had set up a government and a brigade, a unit, no matter what size, but
5 there was a detachment with a commander. Distribution of weapons was
6 carried out thanks to the JNA. What could be withdrawn was withdrawn and
7 distributed to the people in the Serbian areas but it was the SDS which
8 organised the people and created the army. It was an army. Together
9 with the people, those were the armed forces of the Serbian Republic of
10 Bosnia and Herzegovina. They created the space, liberated and created
11 the space."
12 Mr. Subotic, those are words by Dr. Karadzic spoken in your
13 presence that clearly indicated the plans that were set in place and
14 implemented by the Bosnian Serb authorities before the war contrary to
15 your assertion in your statement that there were no plans by the
16 Bosnian Serbs; isn't that correct?
17 A. Do you have those plans? Could you provide those plans so that I
18 can see them? I cannot understand that you can say this from I don't
19 know where. Do you have the plans of the Bosnian Muslims? Do you have
20 the plan of the Serbs in order to be able to assert this before the
21 Trial Chamber? You cannot just be talking off the top of your head.
22 Crisis Staffs existed. All of that existed, but these were necessary, a
23 necessary evil. And of course we organised ourselves in some fashion,
24 but you don't have any documents that were not legal. Show any illegal
25 document on the basis of which Republika Srpska was established and then
1 we can talk and allow anyone, let anyone, anyone from Bosnia and
2 Herzegovina, give you such a plan, a copy. We cannot just be talking off
3 the top of our heads, Mr. Tieger. These are just leading things.
4 Q. I suggest we not talk off the top of our heads. Look at
5 paragraph 26 of your statement, please, and tell me if you didn't testify
6 before this Court in the form of this statement that, "Serbs did not make
7 any plans before the war"? Is that your testimony or not?
8 A. Let's see. Let's look at paragraph 26. It's in the English
9 language. Please read it out for me, and then I'll see when I did or did
10 not. Read it in B/C/S.
11 Q. "There was no plan. Serbs did not make any plans before the
12 war."
13 That's a direct quote from the signed statement.
14 A. Yes. They did not make any plans.
15 Q. Okay. Thank you. I just have one more topic to cover with you,
16 sir, in the limited time I have, and that's in relation to the document
17 that Dr. Karadzic showed you yesterday. That's 11274. During which
18 he -- this is a session of the Bosnian Presidency during which he took
19 your attention -- drew your attention to a portion of that document that
20 referred to the declaration of a state of war, and he asked you:
21 "Did we declare a war on the Muslim-Croat coalition? Did they
22 declare a war on us?"
23 And then you were shown that document. Now, I'd like to show you
24 other parts of that document just to identify the context in which that
25 declaration of war was discussed, and if we can call up 11274, please.
1 If we can look at page 1 of the English translation and page 24
2 of the B/C/S moving on to 25.
3 That's a reflection of Mr. Halilovic talking, and he's talking
4 about indications about agreement of exchange and resettlement, and then
5 he talks about playing directly into the hands of those whose intention
6 it is to create some ethnic territories, to move the population, and to
7 create national units.
8 Turning to page 2 of the English and page 25 of the Serbian -- or
9 B/C/S, we see Mr. Abdic talking, who expresses concern about the need to
10 talk more about the fact that UNPROFOR has been put into a situation
11 where ethnic cleansing is being carried out under its supervision.
12 Page 3 of the English, page 28 of the B/C/S, Mr. Izetbegovic
13 says: If we don't accept the ultimatum, these people really could be
14 hurt. If we accept it, we're legalising ethnic division, that is the
15 alteration of the demographic picture of Bosnia, the creation of
16 ethnically clean territories, like some precondition for the creation of
17 some kind of a Serbian state in BH.
18 Page 4 of the English and page 28 through 29 of the Serbian,
19 Mr. Izetbegovic continues noting that: Wherever they surrendered their
20 weapons, et cetera, they ended up getting hurt. He's talking about, I
21 think, they later simply point their cannons and kill all those people.
22 That's the context in which the Bosnian Muslim authorities
23 declared or discussed the declaration of a state of war, isn't it?
24 A. Bosnian Muslims declared war from our side. Everything was done
25 to prevent a war, everything. And the only guilty party for starting the
1 war is the Bosnian Muslims or, rather, Alija Izetbegovic and the
2 international community. Had that -- had they not done what they had
3 done, there would have been no war. We Serbs asked for just one thing,
4 to have the right in Bosnia-Herzegovina for the Serb community to have
5 its rights like in other countries all over the world that have mixed
6 communities. So my conscience is clear there, and I have no dilemma in
7 terms of who is guilty of starting the war. So the international
8 community assisted Alija Izetbegovic and had that not happened, there
9 would have been no war in Bosnia-Herzegovina. As regards this part. If
10 you have some more, because you intimated that there would be another
11 question in this regard, but I did not hear that right.
12 Q. There will be.
13 MR. TIEGER: But let me tender those -- tender those pages along
14 with what was tendered during the examination-in-chief.
15 JUDGE KWON: Yes, we'll receive them.
17 JUDGE KWON: Microphone.
18 THE REGISTRAR: That will be added to Exhibit D3716.
20 Q. And, Mr. Subotic, the reason -- first of all, Dr. Karadzic asked
21 you whether or not the Bosnian Serbs had declared a state of war, and the
22 reason that the Bosnian Serb authorities refrained from introducing a
23 state of war is because of their position that if something is done by
24 the civilian authorities without a state of war, such as settlement and
25 resettlement, but that is much better and it is not discarded after the
1 war. In other words, that all the demographic redistribution that was
2 taking place at the time that was referred to in the excerpts I talked to
3 you about earlier, that was reflected in the Bosnian Muslims and
4 Bosnia Croats by the thousands in the camps, that that democratic
5 redistribution wouldn't be discarded after the war or was less likely to
6 be discarded if a state of war was not declared. And that was the
7 reason, wasn't it?
8 A. Where is that written?
9 Q. Well, I'm glad you asked that question because apparently you
10 demand that. Let's turn to D00456 --
11 A. Where is that written?
12 Q. -- D00456, English page 57, B/C/S page 61 through 62. That's the
13 20th session of the Bosnian Serb Assembly, and we'll see Radovan Karadzic
14 speaking.
15 A. Please read it out to me so that I can hear what it is. I cannot
16 remember that.
17 Q. In English it states the following: "We have refrained --
18 JUDGE KWON: Just a second. Let's find the passage.
20 Q. "We --"
21 JUDGE KWON: Where is it?
22 MR. TIEGER: It's about the -- in English it's about the fourth
23 line down, and --
24 Q. Are you able to find it, sir, "We have refrained from introducing
25 a state of war"?
1 A. I don't have it in Serbian on my screen. Could somebody please
2 react?
3 MR. TIEGER: Can we scroll down until we can see -- we can see
4 more. Or scroll up until we can find a point of reference. Another
5 point of reference would be the reference to Presidency and government.
6 THE WITNESS: [Interpretation] No. It's not on this page. I do
7 not see it on this page.
8 MR. TIEGER: All right. Let's turn to -- is this -- B/C/S
9 page -- it should be the Serbian page 61, going on into 62. I -- I can't
10 see if this is the --
11 THE WITNESS: [Interpretation] I have page 59 here on the screen.
12 MR. TIEGER: Sometimes the electronic versions don't conform to
13 the hard copy notations.
14 JUDGE KWON: Yes, Mr. Karadzic. Microphone, please.
15 THE ACCUSED: [Microphone not activated]
16 MR. TIEGER: I'm sorry?
17 JUDGE KWON: Mr. Karadzic, turn on your microphone.
18 THE ACCUSED: I couldn't before Mr. Tieger turned off.
19 JUDGE KWON: Okay.
20 THE ACCUSED: Now I see, but it is not at beginning of sentence.
21 [Interpretation] Until now, "we have refrained," [In English] This is
22 sort of 15th line from the bottom. [Interpretation] It's actually around
23 the 15th line from the bottom of the page and the sentence does not start
24 with those words, "We have refrained." I'll tell you how the sentence
25 actually starts, "I don't know what else we'll be doing tonight." So
1 that is the lower third of the page, "I don't know what we will be doing
2 tonight. Please, if we are to continue tonight, could somebody else us
3 from a legal point of view whether we have to change it."
4 THE WITNESS: [Interpretation] I have it.
5 JUDGE KWON: Okay. It starts on the second line in English.
6 Shall we start from there, "I don't know what else we will do tonight."
8 Q. Okay. And the quote is as follows in the translation:
9 "We have refrained from introducing a state of war because it is
10 much more important, more founded and feasible if something is done by
11 the civilian authorities without a state of war under regular
12 circumstances such as settlement and resettlement. When the civilian
13 authorities do this, it is much better. It is not discarded after the
15 And that's the reason or at least one of the reasons,
16 Mr. Subotic, why a state of war was not declared by the Bosnian Serb
17 authorities; correct? At least according to Dr. Karadzic.
18 A. Yes, but what is negative there? I would like to know in terms
19 of your question what is negative here.
20 Q. I have one more matter to -- I ask you the questions, sir. I
21 have one more matter to raise with you and then we'll adjourn and I'll be
22 finished, and I'd like to call up P1483, page 155 in both versions. And
23 it's in May of 1993, Mr. Subotic -- excuse me a second.
25 MR. TIEGER: Sorry. And I'd like both the English and the B/C/S
1 called up of 1483, at page 155.
2 On the 27th of May, 1993, Colonel Bogojevic met with
3 General Mladic, as reflected in his notebook, and he related that four to
4 five days ago Simo Drljaca had arrived, sent by the minister of the
5 interior, and Drljaca came about the Tomasica mine, the mine near
6 Prijedor where earlier they had buried around 5.000 Muslim bodies.
7 MR. ROBINSON: Excuse me one second, Mr. Tieger. Mr. President,
8 this appears not to have been notified to us as being an exhibit which
9 will be used in cross-examination. If I'm mistaken about that. I'm
10 looking at the two e-mails, and I don't see this one, but -- so I would
11 object to it being brought to the witness unless there's some explanation
12 for why we weren't notified.
13 MR. TIEGER: Well, I regret the omission. If so, that would be
14 the first time that I recall that ever happening in a long period of
15 time. I'm not quite sure what difference it would make, if they need
16 in -- in those terms. I'm not quite sure why the Defence doesn't want
17 the witness confronted with this, but I -- I -- I -- given the kinds of
18 notification we normally receive from the Defence on matters of this sort
19 and the accommodation we have provided, I find this to be a fairly
20 extraordinary objection, and I'm not -- I don't think it's well founded.
21 JUDGE KWON: Shall I take it as a word of apology for your
22 omission?
23 MR. TIEGER: I regret the omission, but I --
24 JUDGE KWON: All right. Let's continue.
25 MR. TIEGER: All right. Thank you, Mr. President. I'll
1 continue.
2 JUDGE KWON: But let us find the passage. Probably in English it
3 starts from the previous page.
4 MR. TIEGER: It does but where I was reading from is -- we are
5 now on -- on the correct page. We can go to the previous page to see
6 the -- but this is -- this is the passage I was reading from.
7 JUDGE KWON: Yes. Probably. We need to start from the previous
8 page in both versions so that witness can follow. Yes. In B/C/S as
9 well, shall we go back to the previous page?
11 Q. All right. This page reflects what I had just mentioned to you
12 earlier, the meeting of Colonel Bogojevic who related Drljaca's arrival,
13 having been sent by the minister of the interior about the Tomasica mine.
14 And if we continue to the next page in both languages, please. That's
15 the mine near Prijedor where earlier they had buried around 5.000 Muslim
16 bodies. And then General Bogojevic said:
17 "I'm sure the world knows about this from the released prisoners.
18 Drljaca came to leave this with us and they want to get rid of it by
19 burning, grinding, or some other way. There are all kinds of bodies, and
20 they have involved Subotic in this. The team includes Drljaca. He was
21 in charge even while this was being done. At the meeting where General
22 Subotic, Arsic, Drljaca, me, and Mile Matijevic from the Banja Luka SUP."
23 He asked for the position. The position was "They killed them,
24 so they should get rid of them. And an investigation must be launched in
25 connection with this case and information retained to prevent it from
1 getting into the hands of unauthorised people."
2 A. What it's date? I cannot see the date here.
3 Q. May 27th 1993.
4 A. The 27th of May, 1993. This is the first time I see this and
5 hear of it, but I know what this is about. So it is well known that the
6 first conflict in Bosnia-Herzegovina occurred in Prijedor, the first one
7 as far as the Bosnian Krajina is concerned. I wouldn't dare speak about
8 Herzegovina and the rest down there, because I'm not familiar with that.
9 This conflict occurred already in the beginning of May in
10 Prijedor, and during that time I was minister in the Government of
11 Republika Srpska and Bosnia-Herzegovina. That is to say I came on the
12 5th -- on the 8th of April. After that I assumed my duty, and, I don't
13 know, for more than a month, I think, I did not move from Pale at all.
14 Since communications were very bad at the time, we had not even
15 established a government yet in full. There were five or six of us. So
16 that is to say that the formation of the government was underway.
17 Some information was coming in about clashes between Serbs and
18 Muslims, ethnic groups in the town of Prijedor itself. Now, what
19 happened there and is to what extent, we did not know. We were not in a
20 position, we from the government, not at all. We hadn't even been
21 established yet as such for -- for us to get any meaningful information.
22 Time elapsed. Sometime in 1993, in the month of May, Haris Silajdzic, in
23 a public statement on the radio said that in Prijedor, in the beginning
24 of May 1992, 5- to 7.000 Muslims were killed. Around the 12th of May -
25 I'm not sure. I don't remember correctly - in 2003, among other things,
1 I became the chief inspector of the army. I personally heard this
2 information in Pale. And that was the first time I heard of anything
3 like that. Then I decided, because that was my task, to check because I
4 knew it wasn't the army. In Prijedor, in May, there was the JNA, a
5 military unit, but it was in Slavonia. As far as I remember, it was
6 precisely in Slavonia.
7 So in this conflict with the Muslim population or fighters or
8 whatever, I don't know how it was that they clashed, it was the
9 Territorial Defence that took part. It only could have been the
10 Territorial Defence, possibly the police. That is why I, around the
11 22nd or 23rd of May, I cannot remember exactly, I went to Banja Luka. I
12 asked for information as to what it was that happened in Prijedor in May,
13 what Haris Silajdzic had stated in the media. It turned out that no one
14 could tell me anything specific, and I addressed precisely this
15 Colonel Bogojevic who was head of security in the 1st Krajina Corps, and
16 I asked him and he said, We don't know all of it either, but the Prijedor
17 police would have to know and the Territorial Defence. They were the
18 ones who took part in this. I asked the colonel that we go there and
19 see. We found Colonel Arsic there who was commander of the JNA unit who
20 maybe could have or should have had some information about that. I,
21 Bogojevic, and Arsic went together to the police to see Drljaca, who was
22 chief of police at the time. I don't know exactly what his exact title
23 was. I first put this question: Did the army take part in this? The
24 explanation I was given was that this military unit, I think it was
25 called the 27th or some brigade, that was a JNA unit, that it was
1 somewhere in Okucani or towards Papuk, but that that was this clash
2 between the opposing parties, that is to say the Muslims and the
3 Territorial Defence in Prijedor.
4 Simo Drljaca, well, I asked where were these people who were
5 killed buried, and he said at the mine of Tomasica. I asked that we all
6 go there together to see whether anything can be seen there or something
7 like that, because according to military regulations in a state of war,
8 and so on, which had already been regulated, well, not exactly in May but
9 by the end of June I as minister of defence issued these regulations.
10 This was regulating how this should be done, how people should be buried,
11 one's own soldiers, enemy soldiers, and so on and so forth. I asked that
12 we take a look, and when we arrived there, so that is about a year after
13 this massacre, I asked Mr. Drljaca. He said that it was the materials
14 that took part in this and part of the police. I asked how many were
15 there, because I said Haris Silajdzic -- well, this is what initiated my
16 interest in it. When I asked the question how many people, fighters, or
17 whoever they were, civilians, were buried there, he said up to 500, up to
19 I could not make any decision then. For me it was important that
20 the army didn't do it, and that was confirmed by Colonel Arsic and
21 Drljaca. When actually after that Krajisnik and I went to see the
22 president of the municipality we talked a bit about that, but he said
23 that's how they were buried. That's how it was. No one touched
24 anything. The location and time are well known. When I returned to
25 Pale, I notified the minister of justice, it was Mr. Jovo Rosic, and I
1 asked him, I mean I explained this to him, what it was that I found out,
2 and I asked him to start an investigation about this case.
3 What happened after that I don't know. That is what I know about
4 this case.
5 Q. One follow-up question. So your testimony is that before the
6 attack on Kozarac on May 24th, 1992, before the confinement of thousands
7 of Muslims and Croats in Omarska and Keraterm, before the room 3 massacre
8 where 150 people were killed, before the attack on the Brdo in
9 July of 1992, before the killing of over 150 people from the Brdo and
10 Omarska and so on, that already at least 500 and according to Bogojevic,
11 5.000 Muslims had been killed before that in Prijedor? That's your
12 testimony?
13 A. That's what we were told by Drljaca to me, Bogojevic and Arsic.
14 I didn't know anything about that, to be honest, although there were
15 rumours. There was information that we received in Pale that there had
16 been conflicts between Muslims and Serbs in the territory of Prijedor.
17 There was nothing strange about that. The war had already started,
18 therefore there was nothing specific. And as for Kozarac and the rest, I
19 really at that time -- I did not have a possibility or time to deal with
20 that problem at all throughout the entire first war -- year of the war I
21 didn't have the time, or at least up to June I didn't. Perhaps two or
22 three times I turned up in Banja Luka and once I was there with
23 Krajisnik. When we went to visit the president of the municipality, I
24 know that Krajisnik asked him about the situation with the population in
25 Prijedor, and that was sometime around the 12th or the 13th of May. He
1 said we have about 5.000 Muslims, about 5- to 6.000 Croats in terms of
2 the other population. That's what I know. I remember that very well.
3 After that case, in 1993 I remember that, and then I wondered how
4 come that there were 5.000 Muslims and that they were killed by the
5 Territorial Defence at the beginning of the war since the
6 Territorial Defence perhaps had one company. I don't know how many. I
7 didn't know how many men they had at their disposal. So who was it who
8 was capable of killing that many people? I didn't find any of that
9 logical. But listen, see, according to Haris Silajdzic there were 5- to
10 7.000 and so on and so forth. In any war the figures get exaggerated,
11 and that was also true of Srebrenica and so on and so forth. This is
12 what I know about that case.
13 I don't know where this has come from, where this information has
14 come from. From Bogojevic? I haven't a clue really. I really don't --
15 am surprised. What did he say? Let me see. Just bear with me for a
16 moment. That Subotic got involved. What does that mean? What does it
17 mean when he says that I got involved? What was I involved in? Maybe he
18 thought that I as the main inspector of the army and I came to check the
19 situation. I don't know what I was involved in. As a minister, what was
20 I supposed to be involved in during the first days of the war. I was in
21 Pale. I didn't know whether I was coming or going, what was I supposed
22 to do, how I was supposed to set up my ministry from the 12th of May
23 onwards? I really don't understand. He should explain his own words to
24 you, I guess.
25 MR. TIEGER: Thank you, Mr. Subotic.
1 Thank you, Mr. President.
2 JUDGE KWON: Yes, Mr. Robinson.
3 MR. ROBINSON: Mr. President, in light of the time for redirect
4 which will probably be somewhat extensive, I'm proposing that we excuse
5 Colonel Salapura for today so that he can start his testimony in the
6 morning.
8 Mr. Subotic, before Mr. Karadzic starts his re-examination, I
9 will put some questions to you, and I ask you to be very precise, because
10 depending upon your answer, the Chamber may order redaction of some part
11 of your statement.
12 So you have your statement, signed statement with you right now.
13 THE WITNESS: [Interpretation] Yes.
14 JUDGE KWON: And some parts of the paragraphs are written in
15 English as we saw before.
16 THE WITNESS: [Interpretation] That's correct.
17 JUDGE KWON: It is my understanding that there are three kinds of
18 paragraphs that were written in English, so shall we go to paragraph 236.
19 Or we shall go to paragraph 240 first which we saw earlier today. Could
20 we upload it. Next page. Yes, 240. Under the subtitle of Blagaj Japra.
21 This is written in English, but this is a quotation from your
22 witness statement you gave on the 20th of May, 2006, which --
24 JUDGE KWON: Which was also written in the B/C/S on which you
25 signed and initialed every page. Do you remember?
1 THE WITNESS: [Interpretation] I remember that, yes.
2 JUDGE KWON: So you were confident this is true because you
3 signed the original B/C/S version at the time when the witness statement
4 was made.
5 THE WITNESS: [Interpretation] That's correct.
6 JUDGE KWON: All right. Then let us move to next paragraph,
7 paragraph 241. It's a question from Judge Orie and your answer, and
8 there's another question from Judge Hanoteau and your answer. This is a
9 transcript from the Krajisnik case. When you gave testimony at the time,
10 the transcript was made only in English and no -- and it is my
11 understanding and I am confident --
13 JUDGE KWON: -- that no Serbian translation was made at the time.
14 THE WITNESS: [Interpretation] No, there wasn't, but I did provide
15 the statement. I signed it, and only later did I receive its
16 translation.
17 JUDGE KWON: I don't understand your answer, Mr. Subotic. What
18 we are talking about is your testimony in the Krajisnik case.
19 THE WITNESS: [Interpretation] Yes, yes. Yes, yes, yes. I did
20 provide that statement. I remember that very well.
21 JUDGE KWON: What do you mean by the statement, providing that
22 statement?
23 THE WITNESS: [Interpretation] I uttered those words. When
24 Mr. Orie asked me questions, I answered, yes.
25 JUDGE KWON: Yes, I'm coming to that. So when Defence produced
1 this part of statement, i.e., that is paragraph 241, how did you find
2 that the paragraph 241 was correct? Did the Defence read out this
3 paragraph to you?
4 THE WITNESS: [Interpretation] Yes, yes, yes. The Defence read
5 out that paragraph to me when they took a statement from me. Whenever
6 they took statements from me, they read those statements to me first from
7 English, and the statements that I provided in the Krajisnik case, and I
8 confirmed their correctness because I knew that there were no alterations
9 to them made.
10 JUDGE KWON: And shall we go back to paragraph 236.
11 Mr. Tieger, who was JR?
12 THE ACCUSED: Maybe Jean-Rene Ruez.
13 JUDGE KWON: So --
14 MR. TIEGER: Sorry, Mr. President. Which footnote are we
15 referring to?
16 JUDGE KWON: 236.
17 MR. TIEGER: That's the problem. I believe that was
18 John Ralston.
19 JUDGE KWON: Yes. Do you see paragraph 236? This is an
20 interview you gave at the -- at the Prosecution --
21 THE WITNESS: [Interpretation] I can see that, yes.
22 JUDGE KWON: So you were asked by the representative of the
23 Office of the Prosecutor, John Ralston, and you gave your answer. You
24 see the initial BS?
25 THE WITNESS: [Interpretation] Yes, yes.
1 JUDGE KWON: How did you confirm that this part was also correct?
2 THE WITNESS: [Interpretation] Well, Mr. Sladojevic interpreted
3 that to me when he was taking my statement.
4 JUDGE KWON: Thank you.
6 JUDGE KWON: I'm not sure the Chamber understand you when you
7 said Mr. Sladojevic interpreted that to you. What did exactly he do?
8 THE WITNESS: [Interpretation] He told me that he would also
9 include some statements from my previous testimony which in essence
10 correspond with what I know. He -- he told me that those statements
11 would be included verbatim from your previous evidence because in essence
12 they correspond with my entire statement; i.e., I repeat them in one way
13 or another almost verbatim in the same vein, and I accepted that that
14 could be exclusively done in the way I testified before Mr. Orie. I
15 accepted that voluntarily. Therefore, there's no reason for things to be
16 repeated.
17 JUDGE KWON: Yes. I understand that, Mr. Sladojevic told you
18 that those parts would be included -- included verbatim from your
19 previous evidence, but my question is whether Mr. Sladojevic or any other
20 person, including interpreter, read out your statement in its entirety
21 that is to be included in your statement.
22 THE WITNESS: [Interpretation] Yes. What is now before me was
23 read out to me, both the parts in English as well as the parts in
24 Serbian. We spent an entire day during proofing on that.
1 JUDGE KWON: Very well. We'll an a break for 45 minutes and
2 resume at 1.45.
3 --- Recess taken at 1.01 p.m.
4 --- On resuming at 1.51 p.m.
5 JUDGE KWON: Yes, Mr. Tieger.
6 MR. TIEGER: If I could just quickly note one matter unrelated to
7 this witness that I've already mentioned to Mr. Robinson, that is that in
8 light of the timing and nature of Mr. Zametica's proposed 92 ter
9 statement, the Prosecution will be asking at an appropriate moment that
10 his testimony be led live, so I wanted to let the Chamber know in case it
11 had anything in mind by way of arguments for timing just before the
12 witness testifies or any time before.
13 JUDGE KWON: To be led live in its entirety.
14 MR. TIEGER: Correct. Well, in light of the fact that there is
15 apparently no way of segregating the new portions from the previously --
16 the previous draft, that is correct.
18 MR. ROBINSON: Mr. President, we're against that but if the
19 Chamber were minded to do that we would prefer to postpone his testimony
20 until the 3rd of July, which would be after the 14 days would have
21 expired from the receipt of the more recent statement, so if you could
22 let us know, but we are unlikely to make much progress with his testimony
23 tomorrow in any event, so if it's determined that there was not enough
24 notice to allow him to testify under Rule 92 ter at this time, we would
25 simply prefer that he be recalled at a time when there is sufficient
1 notice.
2 JUDGE KWON: We can't decide in vacuum. We'll hear from the
3 parties sometime today or tomorrow.
4 Yes, Mr. Karadzic, please proceed.
5 THE ACCUSED: [Interpretation] Thank you, Excellencies; good
6 afternoon to everybody.
8 Q. [Interpretation] Good afternoon, General, sir.
9 A. Good afternoon.
10 Q. I'll start with last things first, and I will base my questions
11 on what we heard from you yesterday. Could you please tell us, General,
12 sir, how you understood the situation which resulted in casualties that
13 were buried in the mine in Prijedor.
14 A. Those casualties were result of a conflict as Drljaca confirmed,
15 and that conflict involved the Territorial Defence and police forces
16 during May in Prijedor. I have nothing else to say because I didn't hear
17 anything different.
18 Q. Thank you. Were there any signs, were there any rumours that
19 they were the result of a crime, that they were killed? Did Bogojevic
20 specify the way casualties came by?
21 A. To be honest, I didn't ask any questions since he said that they
22 were killed during clashes in the town of Prijedor, I did not ask any
23 questions. I didn't have any reason to doubt those words because it was
24 publicly known that there was fighting going on in Prijedor. I was in
25 Pale, so I did not have precise possibilities to know more than I did.
1 Q. When it comes to that burial and sanitization, did that happen
2 before or after you issued your instructions?
3 A. They were buried immediately after.
5 MR. TIEGER: All right. This is the second time Dr. Karadzic is
6 introducing terms of art that have particular connotations that were not
7 either -- they're not contained in the document, for example. Now we
8 have sanitization. Previously we had casualties. It's clear what he's
9 intending to convey with his use of the words, and I want to urge
10 open-ended questions that don't direct the witness to particular
11 formulations of events.
12 JUDGE KWON: Thank you, Mr. Tieger.
13 MR. KARADZIC: [Interpretation].
14 Q. General, sir, what instructions did you give in the month of June
15 after the army started functioning properly?
16 A. I prescribed everything in terms of documents concerning warfare,
17 the burial of casualties, and so on and so forth. Therefore, I didn't do
18 anything else.
19 Q. Thank you. That provision concerning burials, what does it set
20 out? Which casualties is the army supposed to bury?
21 A. I know from some parts, for example, in Zvornik where I was
22 present when General Stankovic, who is a pathologist, dealt with some
23 casualties in Zvornik that were buried at the cemetery, and they were
24 Muslims. Pathologists did their bit that they were supposed to do. They
25 had documents. The graves were marked with numbers. I saw that, and
1 that was in accordance with the military regulations that prevailed in
2 the JNA and in all the other militaries in the world. I had an occasion
3 to see that. That was prescribed by our rules and regulations. People
4 could not be buried in any way. Their graves had to be marked. Their
5 names had to be known, and so on and so forth. And later on the mortal
6 remains could be transferred to other places depending on requirements
7 and some other thing. I don't know what.
8 Q. Thank you. Was every killed enemy soldier buried in the place
9 where they were found? What procedure was followed?
10 A. Listen, it depended on the conditions. For example; I don't know
11 why those people in Prijedor did it the way they did. In Prijedor there
12 are Muslim cemeteries and so on and so forth, so they could bury them in
13 Muslim cemeteries, which would have only been fair, and so on and so
14 forth. They should have marked those graves. However, maybe the
15 conditions were not in place. Maybe it was not possible due to fighting
16 and conflicts. I don't know. I never investigated the circumstances.
17 My information dated from one year after the conflict and after the
18 burial.
19 Q. Thank you. Tell me, after the war or at any time, were those
20 bodies exhumed when that burial site was discovered? Did anybody mention
21 the number of bodies that were found there?
22 A. I know from the media, from the press, and from TV that that
23 burial site in the Omarska mine was exhumed. To be honest, I was never
24 in those mines, so I don't know what the name refers to. However, that
25 information did not contain the number of casualties, but that was done
1 after the war. I know that. In peacetime I don't know when, in what
2 year, but that can be checked in the territory of Banja Luka.
3 Q. Thank you. You were asked whether you denied that Omarska,
4 Keraterm were camps where the civilians and fighters were detained and
5 other people. Can you tell us how many people were released from Omarska
6 and Keraterm and how many were found as guilty of having participated in
7 fighting?
8 A. According to the information that I had even before the end of
9 the war, 60 per cent of the people who were detained there were released.
10 I don't know what happened to the remaining 40 per cent. I don't know.
11 I'm not privy to that information, but I know for a fact that I came
12 across information according to which 60 per cent were released.
13 Q. Who was it who was in charge of the investigations and triages
14 that happened there? Who was it who decided that some should be released
15 and that some should be kept? Do you know that?
16 A. Yes, I did know that, because --
17 JUDGE KWON: Yes, Mr. Tieger.
18 MR. TIEGER: More of the same and I'm going to keep rising when
19 this keeps happening. You can scour the transcript. There is no mention
20 of triage up to now. This is part of a continuing pattern by
21 Dr. Karadzic trying to indicate to witnesses where he wants them to go
22 and what assumptions he wants them to build into their answers.
24 MR. ROBINSON: Actually, Mr. President, I don't agree with that.
25 This is not a leading question and the witness is free to give any answer
1 he wants. I don't think that Dr. Karadzic is asking an improper question
2 at all in this way. I don't really understand the basis for an objection
3 that Dr. Karadzic is creating some kind of suggestion to the witness.
4 The question is open-ended. He can answer it in any way he wishes.
5 JUDGE KWON: How about the word "triage"?
6 MR. ROBINSON: Nevertheless, that's not suggested. It's up to --
7 if that's how Dr. Karadzic characterises and the witness understands what
8 the import of the question is that's not suggestive simply charactering
9 of something one way or the other.
10 JUDGE KWON: Just a second. Yes, Mr. Tieger.
11 MR. TIEGER: In my system, as Mr. Robinson well knows -- first of
12 all, we should be arguing this outside the presence of the witness, but
13 never mind. This assumes facts not in evidence. So Dr. Karadzic wants
14 to ask the witness about something that the witness has not testified to,
15 that was not raised in the examination, that is now presented as a fact
16 and ask him who was in charge of that fact that the witness hasn't
17 testified about and that wasn't part of any earlier answer. That is the
18 existence of triage. Now, and -- and as I said before, this is another
19 reflection of the same pattern that exists of providing the witness with
20 something as a fact and then having the witness automatically build it
21 into his answers.
22 THE ACCUSED: [Interpretation] I will rephrase.
24 THE ACCUSED: [Interpretation] I will rephrase the question.
1 Q. General, sir, these 60 per cent, were they released --
2 JUDGE KWON: Just a second. Bear that in mind. The Chamber
3 agrees with Mr. Tieger's observation. Please continue.
5 Q. General, sir, according to what you heard and found out these
6 60 per cent, were they just released indiscriminately, and what about
7 these other per cent and what is this the result of? Did others decide
8 about that?
9 A. Others were aware of that. I didn't know about it, but nobody
10 asked me about it. Mr. Orie didn't ask me about that. I was responding
11 only to Mr. Orie's questions. I didn't know who would be asking me what.
12 So this means that the investigative organs interviewed individuals.
13 This was publicly shown by the organs of power. It's not a secret. All
14 those who were not -- who were not -- who did not participate in the
15 fighting, and so on and so forth, who did not, they were released. I
16 know that for sure. There's nothing else to it.
17 Q. Thank you. Now I'm going to move to page 54 of today's
18 transcript where the declaration of war was discussed. First, I'm asking
19 you this: It was read how I said that the municipalities had their own
20 units. Are you able to tell us whether the municipality has the right to
21 any armed formations and to which ones?
22 A. According to the law on the national defence of the
23 Socialist Republic of Yugoslavia or the Federal Republic of Yugoslavia,
24 the municipality had all the powers in that sense at the municipal level.
25 This is particularly interesting until the formation of
1 Republika Srpska -- or, rather, for the beginning of the war, until we
2 formed our own organs, laws, and so on and so forth. So we copied
3 certain things from those laws. We did not change the laws of
4 Yugoslavia. Therefore, the municipalities did have such powers at the
5 municipal level. They had their own Municipal Staffs. They had their
6 own weaponry. They had -- all of this existed on one side and the other
7 side, the Muslim side, the Serb side, and the Croat side.
8 Q. Thank you. You were asked or actually what was disputed was your
9 assertion in the testimony in the Krajisnik case that Serbs did not have
10 any preparations. What sort of preparations did you mean when you said
11 that you did not have any preparations?
12 A. Serbs had no preparations at all for armed fighting. They simply
13 had -- Serbs had political preparations, preparations through which they
14 attempted, and it wasn't just the Serbs in Bosnia and Herzegovina but in
15 Croatia, Slovenia, and so on and so forth. They were trying to convince
16 people not to go into war, not to embark on war, that if we needed to
17 split up then we should split up. So that was it. No Serb was willing
18 to participate in the war. Not a single Serb. Believe me, we loved
19 Yugoslavia. Yugoslavia was a wonderful country. Therefore there were no
20 military preparations that were carried out until the beginning of the
21 war. As one says only with the entrance of the army to Sarajevo, they
22 were escaping in front of the enemy, they were getting killed in
23 Sarajevo. It was then that people understood what it was all about, how
24 things were proceeding and then they began something at their own level,
25 they made some attempts.
1 Q. Thank you. And then my learned friend Mr. Tieger said as I --
2 that I said that we had a series of pre-planned moves. When
3 Mr. Izetbegovic made one move, we made another move. So this political
4 counter game, did that lead to war? Did it have a military nature?
5 A. No, no. It was just precisely that political game that tried by
6 all means. This is generally known. Everybody knows this. Muslims in
7 Sarajevo know this, and many Muslims would confirm that, and -- and they
8 could testify to it. Thus no one was in favour of war. Quite the
9 contrary. We accepted that Bosnia and Herzegovina should be Bosnia and
10 Herzegovina. It should be one state, because Yugoslavia was dismembered.
11 MR. TIEGER: Excuse me.
13 MR. TIEGER: I'm sorry, but I'm going to keep objecting to this.
14 So this is another example so Dr. Karadzic says to the witness about this
15 matter. So did these political counter moves and so on. What the
16 witness actually said when he was presented with that information was, I
17 didn't have any contacts with Mr. Karadzic in 1991. I didn't know him.
18 I didn't know he existed, specifically in this case. I have no ideas
19 about that. Well, now Mr. Karadzic gave him some ideas and it comes out
20 in his testimony and that's what's happening over and over.
21 JUDGE KWON: I also agree, and also it reduces the probative
22 value of the witness's answer. While the Chamber is able to recognise
23 such -- such the probative value of such questioning, but please refrain
24 from putting a leading nature in your -- in your questions. Please
25 proceed.
1 THE ACCUSED: [Interpretation] Yes, I just wanted to respond
2 briefly to Mr. Tieger. This was a consequence of the fact that
3 Mr. Tieger presented our political measures to the witness as military,
4 war measures, and that brought him into the situation of having to say
5 that he didn't know anything about it. Had he read which measures we had
6 taken, then ...
7 JUDGE KWON: It is you that inserted the term political
8 counter-gain and it's not appropriate when you conduct re-examination.
9 Please proceed.
10 THE ACCUSED: [Interpretation] Very well, your Excellency, but in
11 that case I would ask that the witness be shown the measures to which
12 Mr. Tieger alluded so that we can see. He presented them as part of
13 preparations for war, but they were not of a military nature, so he
14 confused the witness.
15 JUDGE KWON: Please proceed. It's up to you how to conduct your
16 re-examination, Mr. Karadzic.
17 THE ACCUSED: [Interpretation] Well, just give me a little time to
18 collect myself and I will come back to that.
20 Q. General, sir, you were shown the transcript of the
21 20th of June, 1992, where the Muslim side -- or, rather, the Presidency
22 declares war and proclaims a state of war. Was this the first time that
23 they undertook and adopted some aggressive warmongering actions in
24 relation to the Serbs?
25 A. No, no. They already from the attack by Croatia to -- and
1 preparations for the attack on.
2 JUDGE KWON: Just a second. Is this the first time that they
3 adopted some aggressive warmongering actions in relation to the Serbs?
4 Please reformulate your question, Mr. Karadzic.
5 THE ACCUSED: [Interpretation] All right, very well. I apologise.
7 Q. Was that the first time that any law was being referred to or
8 mentioned which would direct towards fighting against the Serbs?
9 A. No, it was not. Already from the beginning, after the formation
10 of Bosnia and Herzegovina, after the referendum, this already started to
11 be prepared and so on and so forth.
12 Q. Thank you. Can we look at D332 in e-court, please. General,
13 sir, please, could you read the date and the first sentence and then
14 could you read paragraph 4.
15 A. The Republic of Bosnia and Herzegovina, Ministry of
16 National Defence, Territorial Defence Staff, Sarajevo, number such and
17 such. The date, 20 -- I don't know if this is 20 -- 20th or the 27th -
18 I'm not sure - of April, 1992. Order on the implementation of decision
19 number such and such of the RBiH Presidency pursuant to the decision of
20 the Presidency of the Republic of Bosnia-Herzegovina.
21 Q. Thank you. Could you read item four now?
22 A. Four: Hurriedly plan and begin combat operations in the whole
23 territory of the Republic of Bosnia and Herzegovina and co-ordinate them
24 with the Territorial Defence staffs of regions, districts, and the
25 Republic of Bosnia and Herzegovina. In planning combat operations, plan
1 extensive measures of protection of the civilian population and property
2 of the citizens of the Republic of Bosnia and Herzegovina.
3 Commander Colonel Efendic. Hasan. I'm missing some letters, but last
4 name is Efendic. This is the person who replaced Vukosavljevic,
5 General Vukosavljevic.
6 Q. General, sir, against whom should combat actions be planned and
7 commenced on the territory of the entire Bosnia and Herzegovina?
8 A. Well, it's not against angels or anyone. It's against Serbs and
9 probably Croats at that time. Possibly.
10 Q. Thank you. And --
11 JUDGE KWON: Please remember, Mr. Subotic, to put a pause before
12 you start answering the question.
14 THE WITNESS: [Interpretation] Very well. Very well.
16 Q. The Serb side at that time or at any point later did it make a
17 plan and issue a similar directive for offensives on the entire territory
18 of Bosnia and Herzegovina or in territories which were majority Muslim or
19 Croat?
20 A. Absolutely not ever until perhaps the end of 1992 when there were
21 orders by the army for certain operations like that, but otherwise before
22 that this was not definitely the case, no.
23 Q. Thank you. Can we now see 1D44050 in e-court, please. There is
24 no translation, but I'm briefly just going to read five or six lines.
25 First of all, let's identify the document though.
1 JUDGE KWON: Mr. Tieger.
2 MR. TIEGER: We're about to see a document that doesn't have an
3 English translation, so I won't be able to tell why it's being raised at
4 this point to this witness, but -- so I'm a little bit lost as to the
5 manner in which this arises from the cross-examination and what is
6 intended by putting this particular document in front of the witness
7 rather than seeking whatever information he might have that's relevant to
8 redirect examination.
9 JUDGE KWON: Yes, Mr. Karadzic.
10 THE ACCUSED: [Interpretation] Your Excellencies, there was a
11 question as to the declaration of war and the conduct and actions of the
12 Muslim army and possible preparations on the Serb side. The topic is the
13 same as the one dealt with with the previous document, and the author is
14 the same, General Hasan Efendic.
16 MR. TIEGER: That's not justification for leading the witness,
17 which is apparently what this is an effort to do. Meanwhile this
18 document is in front of the witness and he's reading it. So
19 Dr. Karadzic, as has been his custom, one way or another continues to try
20 to lead witnesses.
21 JUDGE KWON: He put a question, albeit very brief, on the
22 previous page. Very well. Now that the witness has read the document,
23 what is your question?
24 THE ACCUSED: [Interpretation] Well, he didn't read the part that
25 I'm interested in. It's on a different ...
2 Q. Is this directive correct, General, sir, is this how they acted
3 based on this directive from the 29th of April?
4 A. I am familiar with that situation. First of all, I would like to
5 tell the Trial Chamber that in Bosnia and Herzegovina this post was
6 health by General Vukosavljevic, and he was replaced because he is a Serb
7 by ethnicity even though he was married to a Muslim woman from Sarajevo.
8 Therefore, he was not suitable for the preparations, because he wouldn't
9 do that. He was lieutenant-general by rank. He was a very serious man.
10 I knew him personally. We all knew him, because he was the commander of
11 the Territorial Defence for the whole of Bosnia and Herzegovina for a
12 number of years. So he was deliberately replaced so that Hasan Efendic
13 could subordinate himself to Alija Izetbegovic and then --
15 THE WITNESS: [Interpretation] -- he could be acting according to
16 the wishes of Alija Izetbegovic.
17 JUDGE KWON: In order to put a document to the witness, first you
18 should put a foundational question and lead the witness to tell us what
19 this document is about, and you may then proceed to ask some content of
20 the document. I don't see from the document what this directive is about
21 at all, and I don't know what this document is about.
22 THE ACCUSED: [Interpretation] Excellencies, the directive has
23 been taken down.
24 THE WITNESS: [Interpretation] May I say something?
25 THE ACCUSED: [Interpretation] No. Just wait. The directive has
1 been taken down. I already put questions on it, then it was adopted.
2 This is D332 it was earlier on the screen. This is a review by the same
3 man about the development of the military --
4 JUDGE KWON: You're not giving evidence. Put a question to the
7 Q. All right. General, do you see this document? Can you tell us
8 what the document deals with?
9 A. He is justifying the declaration of war in Bosnia-Herzegovina,
10 rather, Alija's declaration of war, and he is trying in some way to
11 explain that, the reasons for declaring war. That is the essence.
12 Q. Can we see page 4. What is written there? Can you read out the
13 heading for the Trial Chamber.
14 A. Well, now I have page 3.
15 Q. Can we please have the first page yet again and could you read it
16 out.
17 A. "Development and growth of defensive liberation forces." Please,
18 this is clear. Defensive liberation forces. These are forces that were
19 attacked, not those that are defending themselves.
20 JUDGE KWON: Mr. Subotic, what you were asked about is to read
21 out the title. Please concentrate on answering the question.
22 THE WITNESS: [Interpretation] All right.
24 MR. TIEGER: Well, I agree with the Court's guidance, and I also
25 think it's clear that what's happening is that in lieu of asking this
1 witness focused questions that might elicit information that he has in a
2 non-leading way that arise from the cross-examination, instead he's being
3 turned to documents that Dr. Karadzic wants information to be elicited
4 about and being asked to read it out and affirm -- it's classical
5 leading. It's also very peripheral to the cross-examination which dealt
6 with the matter that Dr. Karadzic raised in a very specific way and now
7 he's returning using that subject as a broad platform from which to
8 discuss everything about Muslim preparations for war.
9 THE ACCUSED: [Interpretation] I'll withdraw that, Excellency.
10 I'm going to abandon this topic altogether and I'm going to go back to --
11 JUDGE KWON: Speaking for myself, and I take it my colleagues
12 would agree with me but I tend to agree with Mr. Tieger's observation.
13 So, Mr. Robinson, if you could have a word with Mr. Karadzic how to
14 conduct his re-examination in the future.
15 Shall we adjourn for today with the witness -- with this witness
16 and I will hear from the parties about Zametica's evidence.
17 MR. ROBINSON: Yes, Mr. President.
18 JUDGE KWON: Yes. I was informed that we can -- Mr. Salapura's
19 evidence has not been concluded which he will continue tomorrow for about
20 half an hour or an hour.
21 Yes. Mr. Subotic, you may be excused. We'll continue tomorrow
22 morning at 9.00.
23 [The witness stands down]
24 [Trial Chamber and legal officer confer]
25 MR. ROBINSON: Excuse me, Mr. President, I think I can shortcut
1 this discussion just by doing some math here. I don't think we'll reach
2 Mr. Zametica tomorrow, so instead I think we'll reschedule his testimony
3 for the 3rd of July and that will obviate any problems.
5 MR. TIEGER: I certainly haven't thought through the impact --
6 what I understand Mr. Robinson to be saying is that the circumstances are
7 such that he's in a position to reformulate the schedule in a manner that
8 doesn't implicate any loss of time to the Court or any impact on the
9 Prosecution. I actually haven't thought it through from that respect,
10 but if his point is he can do something unilaterally that doesn't have
11 any implications to the Prosecution's schedule, I suppose that's a way of
12 saying it doesn't really matter what the Prosecution says in response. I
13 haven't thought it through fully from that point of view and to see what
14 impact it has, so my position was predicated upon the impact of dealing
15 with that statement under the circumstances that we expected to occur,
16 and you'd have to give me just about at least two minutes to consider the
17 implications of what Mr. Robinson just suggested.
18 JUDGE KWON: Very well, Mr. Tieger. So we'll deal with it when
19 it will arise as a matter of fact. So we'll deal with it immediately
20 before Mr. Zametica is due to testify.
21 There's one matter.
23 JUDGE KWON: So I'd like to ask the parties whether -- or how
24 they view this: Mr. Salapura is going to testify in the Mladic case
25 tomorrow morning for some time, after which he's due to come to this
1 courtroom and to give testimony, this time as a Defence witness. In
2 terms of perception or appearance, whether there's any problem or
3 whatever, can I hear from the parties.
4 MR. ROBINSON: We don't see any problem. Maybe I'm missing
5 something that is not blatantly obvious to me, but in reality, the
6 Prosecution is interested in proving the events in Srebrenica and some of
7 the things that occurred there, and they're using Colonel Salapura to do
8 that in the Mladic case. We're interested in showing that Dr. Karadzic
9 didn't have any knowledge of those events, and we're using
10 Colonel Salapura in our case for that purpose. So I don't see any
11 inconsistency, but if I'm missing something about the perception, maybe
12 you can let me know.
13 JUDGE KWON: Mr. Tieger.
14 MR. TIEGER: Yeah, I'm also happy to focus on particular concerns
15 the Court might have, but my initial reaction is that this is a
16 reflection of an aspect of witness testimony that we see reflected in
17 various ways when insider witnesses are called, and that is in part that
18 parties may call witnesses in respect of particular information and ask
19 the Court to consider the witness's testimony in light of the totality of
20 evidence and the extent to which that witness's testimony about certain
21 matters is convincing and credible and corroborated by the totality of
22 the evidence and disregard other aspects of the witness's testimony that
23 may be, for whatever reason, whether it's -- well, for various reasons,
24 not considered reliable by the Court. So I -- given the position that
25 certainly has been taken in respect of insider testimony in other
1 instances, I think it would not be necessarily surprising that a
2 witness's evidence is of interest in some respects to either party,
3 although all of the evidence that that witness may provide may not be
4 accepted or indeed may be challenged by the party that calls the witness,
5 and I say that completely in the abstract, because I haven't conducted
6 myself the particular analysis of what evidence I think the Defence
7 specifically intends to rely on, what evidence I think is likely to be
8 adduced from this witness in the Mladic case, but I think that broad
9 principle is -- is one that we're familiar with in the institution.
10 Given the fact that people close to the events are also people
11 close to the -- very often to the accused and are other parties of
12 interest, there is often a viewed need to elicit information from those
13 people, understanding the possibility that not all information elicited
14 will -- can be assessed in and given precisely the same weight.
15 JUDGE KWON: In conclusion, you do not see it problematic?
16 MR. TIEGER: I can say this, Mr. President, when you raise it, it
17 makes me rethink it and -- and consider the possibility of -- of giving
18 it further thought, but the -- the answer to your question at this moment
19 is no, it is not something that we had intended to raise with the Court
20 in the manner you just asked about.
21 MR. ROBINSON: Maybe there's some English custom that Mr. Tieger
22 and I are missing, but we're going under the principle that the witness
23 is not the property of either party and therefore it could be okay.
24 JUDGE MORRISON: I don't think you're missing any English
25 customs.
1 JUDGE BAIRD: You're not, you're not.
2 JUDGE MORRISON: I think the Chamber's just acting out of an
3 abundance of caution. But I speak purely for myself, if neither party
4 sees any difficulty, then so be it.
5 JUDGE BAIRD: This is the point. I mean, if both sides agree,
6 then the matter comes to an end, but from the standpoint of perception, I
7 thought that instinctively it might not have been the best course of
8 action and possibly a day to interpose himself between the two bodies of
9 testimony, but if both sides agree, then, you know, so be it.
10 MR. ROBINSON: Yes. I think especially under the circumstances
11 where we wouldn't have any other witnesses tomorrow if we did have this
12 problem, a gap, so I think it's better to go ahead.
13 JUDGE KWON: Thank you. We'll do so.
14 THE ACCUSED: [Interpretation] May I?
15 JUDGE KWON: Yes?
16 THE ACCUSED: [Interpretation] Indeed, I have exercised restraint,
17 but this is not the first time that Mr. Tieger is mentioning that
18 witnesses are close to me. Mr. Tieger called all the witnesses who are
19 not close to me. And none of these witnesses that are called by the
20 Defence are particularly close to me. Simply they are not Prosecution
21 witnesses.
22 JUDGE KWON: Very well. The hearing is adjourned.
23 --- Whereupon the hearing adjourned at 2.40 p.m.,
24 to be reconvened on Friday, the 21st day
25 of June, 2013, at 9.00 a.m. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,651 |
Q: Update model with WTForms form data I have some Flask-SQLAlchemy models and Flask-WTF forms generated with wtforms_alchemy to represent them. I implemented a method on each model to update its attributes from a form's data. For each new model and field I have to update these methods, which is annoying. Is there a way to make this more automatic, or a a feature in the libraries I'm using that I'm missing?
def edit_car(car_id):
form = CarForm(request.form)
if form.is_valid():
car = Car.query.get_or_404(car_id)
car.from_form(form) # Update car fields
...
# save car in database ...
class Car(db.Model):
color = db.Column(db.String(10))
...
def from_form(self, form):
self.color = form.color.data
... # all other fields
A: Use the form's populate_obj method to fill in the model. It sets an attribute of the same name as each field.
form.populate_obj(car)
db.session.commit()
If the simple "set attribute by field name" behavior isn't appropriate for a given model/form pair (although it should be in your case), you can override the method.
class SpecialCarForm(FlaskForm):
...
def populate_obj(obj):
# mess with data, set extra fields, etc.
# potentially call super after
super().populate_obj(obj)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,085 |
Medical-Objects would like to thank our loyal customers for all of your support over the last year and give our best wishes to you all over the holiday period. We look forward to continuing to provide you with Australia's Fastest Secure Messaging as well as a range of exciting new products throughout 2017.
During the holiday period, Medical Objects will operate under our normal trading hours, see below for exclusions. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,378 |
Dead??? I didn't even know Kim Jong was il
The years most hilarious post-heading aside, it seems that Christmas has come early for the people of North Korea as the evil dictator and long-suspected extra terrestrial Kim Jong-il [pictured left] died earlier in the week.
Despite this great news, we must err on the side of caution. Jong-il's death may not plunge the country into chaos as he had begun grooming his son, Kim Jong-un to succeed him and placed him in a high-ranking position. It remains to be seen however if the young whelp has had sufficient time to consolidate a power base and form the necessary alliances required to rule the troubled country.
Jong-il's death could set back efforts by the west to get Pyongyang to abandon its nuclear weapons ambitions. Concerns are also high that Kim Jong-un may feel he needs to prove himself by precipitating a crisis or displaying his swagger on the international stage.
The "mourners" in N.Korea must pretend to wail at the loss of their "beloved" leader or they'll be shot.
Whatever the future; this was a great time to vacation in North Korea. Pure coincidence I swear...
Mercury Invaded!
An Extra Terrestrial spacecraft is currently in orbit of the planet Mercury possibly sending in ground forces to establish a permanent staging area from which to launch an assault on Earth.
That was the chilling revelation from Rear Admiral "Smokestack" Henderson Acting Director of UNETIDA speaking from the UN in New York this afternoon. Henderson said he had received actionable intelligence that there could be an imminent threat in our own solar system.
Colonel "Whopper" Creedon Acting Director of Intelligence, UNETIDA revealed images of a gigantic object almost the size of Mercury itself which has appeared on both astronomers and UNETIDA extra-planetary surveillance officers' screens lurking near the planet."The only explanation is that it's a cloaked Extra Terrestrial craft the size of the Death Star from Star Wars" said Colonel Creedon. "I mean what else could it be?"
The Colonel theorizes that the craft secretly is offloading troops and war machines including inter-planetary missile launchers which may deliver payloads loaded with personnel "like in Quake 2" or warheads capable of completely obliterating our planet."If they have the technology to get here and cloak themselves, we have to assume that the −183 °C to 427 °C temperatures won't be much of an issue for them," he added.
Admiral Henderson was at the UN to ask for increased funding to the tune of US$5.5 Trillion for UNETIDA space defence projects including a number of first strike initiatives designed to prevent attacks on the Earth by eliminating the enemy first. It is assumed Colonel Creedon's trademark "theatrical scaremongering" will secure the much needed budget.
Colonel "Whopper" Creedon reveals a cloaked alien craft near Mercury.
Source: Civilian Overseer / The Daily Mail
December 7th 1941, "a date which will live in infamy."
Today marks the 70th Anniversary of the cowardly attack on Pearl Harbor by the Empire of Japan "which awakened a skleeping giant" and broughtthe United States into World War II.
Seven decades later President Barack Obama called for the Stars and Stripes to be flown at half mast on federal buildings across the country, to mark National Pearl Harbor Remembrance Day.
"On a serene Sunday morning 70 years ago, the skies above Pearl Harbor were darkened by the bombs of Japanese forces in a surprise attack that tested the resilience of our armed forces and the will of our nation," Obama said in an address.
"In the wake of the bombing of our harbor and the crippling of our Pacific Fleet, there were those who declared the United States had been reduced to a third-class power. But rather than break the spirit of our Nation, the attack brought Americans together and fortified our resolve. Patriots across our country answered the call to defend our way of life at home and abroad."
Obama paid tribute to "the more than 3,500 Americans killed or wounded during that deadly attack and ... to the heroes whose courage ensured our nation would recover from this vicious blow. "As a nation, we look to December 7, 1941, to draw strength from the example set by these patriots and to honor all who have sacrificed for our freedoms."
U.S. Navy vetran David Callahan, one of the few living surviors of the attack will attend a memorial ceremony today. It will be his fist time back at Pearl since then.
Sources AFP / Foz News
EARTH 2 Located! Threat Imminent!
NASA concept artist's impression of Keplar 22b
Defying UNETIDA suggestions, NASA have revealed the discovery of the first ever planet in a habitable zone outside our solar system.
While earlier this year astronomers "confirmed" the first rocky exoplanet to meet key requirements for sustaining life was Kepler-22b, initially glimpsed in 2009, their findings were much dismissed as those astronomers were actually french, and therefore cretins. However now that the US space agency has confirmed the planet, it can be taken as being fact and is emerging in news stories across the globe.
Confirmation means that astronomers have seen it crossing in front of its star three times and that the conditions are right for our form of life and not that they know life actually exists there. It has the right distance from its star to support water, plus a suitable temperature and atmosphere to support life as we know it. "We have now got good planet confirmation with Kepler-22b," said Bill Borucki, Kepler principal investigator at NASA Ames Research Center. "We are certain that it is in the habitable zone and if it has a surface, it ought to have a nice temperature," he told reporters.
"If we can see their planet, then we have to assume they can see ours" declared Colonel "Whopper" Creedon, Acting Director of Intelligence for UNETIDA. "It's naive to believe that they have the same level of technology we have and that somehow being 600 light years away is the same kind of insurmountable obstacle that it is to us presently."
Colonel "Whopper" Creedon during his briefing to the UNSC
The Colonel outlined his "first strike" strategy at a meeting of the UN Security Council this afternoon. Present were representatives of world military research agencies who are now drawing up plans to both conceive of a payload effective against a rocky, gaseous or liquid planet [as Kepler-22b's composition is unknown] and a method of delivering it as quickly as possible.
In the meantime NASA also announced that Kepler has uncovered more than a thousand other potential target... er planets, twice the number it previously had been tracking, according to research being presented at a conference in California this week.
4 comments Labels: NASA, Science and Technology, UNETIDA, Whopper
Movie Mini-Reviews 30-11-11
In this crime drama, Ryan Gosling portrays a nameless Hollywood Stuntman who moonlights as a Driver for the criminal underworld. It is similar in many respects to the 1978 Walter Hill movie The Driver but is a very different beast. Some may even call Drive a modern Western in it's execution.
Drive is directed by the little-known Danish director Nicolas Winding Refn and features excellent performances from Brian Cranston and Carey Public Enemies Mulligan with cameos from Ron [I'm in a dozen movies this year] Perlman and Christina Mad Men Hendricks. Cliff Martinez takes us back to the '80s with his vibe for the score.
I don't normally appreciate crime movies where somehow lawless nefarious actions are glorified on screen but I was intrigued by this movie as grotesque violence was promised after I heard that several reviewers had walked out of a press screening in disgust. This "extreme gore" was certainly delivered and it features some truly shocking moments not seen since Scorcese's The Departed. Sadly however these moments of violence are too few and far between to recommend the movie outside of its niche audience. Exciting at times but the time waiting between its violent and exciting moments can drag a bit.
Colonel Creedon Rating: **1/2
Johnny English Reborn
Johnny English, British comedy's answer to James Bond was expertly brought to life by Rowan Mr. Bean Atkinson in a 2004 movie with Natalie Imbruglia and John Malkovich. It apparently made a mint but it still took 6 years for a sequel to emerge, an eternity in movie making terms for an industry which seems intent on turning out sequels in the fastest humanly possible time. Johnny English's producers however bided their time with Johnny English Reborn and despite bringing nothing relatively new to the formula it was good enough to equal it's progenitor.
Rubber-faced Atkinson naturally returns as the elite secret agent, elite that is on being almost oblivious to what's going on around him. The movie explains in-universe why it's been so long since the sequel – a protection detail in Mozambique [cue nervous twitch] some years ago wiped the shine off Johnny's star and he fell out of favour with MI7. Since then he's been holed up in a Tibetan monastery learning how to use his wits to solve problems and to ignore pain by dragging boulders with his genitals. Needless to say a situation arises that needs his "unique" talents and he returns to civilisation and the agency where he puts his training to work. He still remains oblivious to the larger picture and retainis his sense of superior arrogance which almost constantly backfires with much hilarity.
There are some great action scenes and almost Bond-class set pieces. While far from the class of Edgar Wright or Will Ferrel, William Davies has woven an expectantly implausible plot that a blind man could see the climax of before the end of the first act, this of course is to cement the idea that everyone watching is more observant that Johnny himself. Atkinson is supported on screen by Gillian X-Files Anderson as Pegasus the head of MI7, Dominic Centurion West as Agent Ambrose, Rosamund Die Another Day Pike as Kate and Daniel Kaluuya as Tucker. A worthy sequel in the same vein as the original. Fans will be satisfied.
We all know our disaster movies, earthquakes, volcanoes, meteor strikes, modern ice-ages, the sun being blocked out and even some Zombie movies can be shoe-horned into that category. But perhaps the most chilling, because it's more believable, is the global pandemic. SARS, H1N1, Bird-Flu have been recent buzz words to signal the end or humanity but were dealt with before annihilating all life on the planet. When it was announced that "realistic" director Steven Soderbergh would be bringing us Contagion, one knew that while you'd be fed a jarringly grim tale, humanity would win out in the end.
Despite it's well researched viral science - praised by science writers and vaccination experts [as well as an "official" nod of approval from the CDC], this is still high drama as opposed to a documentary and thusly entertaining. The movie has some outstanding performances from a plethora of famous faces; Matt Damon, Gwyneth Paltrow, Lawrence Fishburne, Jude Law, Marion Collitard and Kate Winslet lend their acting chops to this well directed and hauntingly scored [by Cliff Martinez] movie. They are supported by Brian Cranston who made an excellent Admiral with the Public Health Commissioned Corps and Elliot Gould who had been long absent from the silver screen.
I know there was plenty of laughing and sneezing after watching Outbreak in 1995, but this was so evocative of reality that there was no such nonsense here. Gripping stuff.
Machine Gun Preacher is unusually entertaining for a true story that wouldn't be really out of place in the realm of Kris Kristofferson/Sally Field Lifetime movies – albeit late at night when intense violence could be permitted. Gerard 300 Butler plays Sam Childers an ex-con who swiftly returns to his biker-gang drug-using ways upon his release from prison. In his absence his stripper trailer-trash wife Lynn [2-time Whopper Award Winner Michelle Monaghan] finds Jesus and helps Sam on his road to redemption. Not only does Sam himself find Jesus, work, start his own construction company, build a non-denominational church in which he becomes a preacher but he also travels to The Sudan during the height of civil war to build an orphanage and rescue young South-Sudanese children from the hands of those of the North who kidnap them to be child-soldiers and sex-slaves.
It's this "rescuing" that provides the true meat of the movie. Sam is not military trained – he just "likes his guns" and he seems to relish being in an environment where he can shoot people and not be frowned upon by either the law or The Lord. His journey is even more fascinating when you know it's true although Christian groups have slammed the movie for it's unnecessary focus on Childers' most violent actions as opposed to his deeply spiritual journey – a fact that I insist on rewarding with stars!
Hypersonic Weapon tests successful
The Pentagon announced this week that on Nov 17th it successfully test fired an Advanced Hypersonic Weapon [above], understood to be one of a number of alternatives [including the Falcon programme] under consideration as part of it's "prompt global strike" system that can deliver long-range weapons anywhere in the world while avoiding flying over third-party nations as revealed through a congressional report earlier in the year.
The missile was launched from Hawaii and reached its target on Kwajalein atoll, part of the Marshall Islands in the Pacific 3700km away in less than 30 minutes by exceeding Mach 5. Such a hypersonic weapon concept flies at a relatively flat trajectory within the atmosphere, rather than soaring up toward space like a ballistic missile and eventually coming back down.
The Department of Defence said "The objective of the test is to collect data on hypersonic boost-glide technologies and test range performance for long-range atmospheric flight." Nonetheless defence analysts Global Security.org say the aim of the programme is to be able to strike a target 6000km away in 35 minutes, with an accuracy of 10m. They claim the weapon is part of a programme to allow a conventional weapon to strike "fleeting targets around the globe faster than today's munitions".
The X-51 Missile is another Hypersonic Weapon in development
The Tinfoil Helmet Brigade [THB] a group of UFOlogists and conspiracy theorists believe that the AHW is part of a UNETIDA project to quickly destroy Extra Terrestrial assets that successfully evade the orbital defence grid and make planetfall. There was no statement from UNETIDA in response as their Office of Public Affairs is currently being audited by the Special Investigations Committee.
6 comments Labels: Science and Technology, UNETIDA, US Army, Weapons
So Long Afghanistan
Sadly the work of Task Force Razor is at an end, but and it had a hell of a ride over the past 10 years. A super-secret multi-service Tier-1 Special Operations unit commanded directly from "someone" in the NMCC. It began with 48 operators in 2001 who rotated in and out every few months to form four six-man standard teams. It'll make a great TV show one day if only they could ever declassify anything we did, but I think it's somewhat unlikely.
The Colonel [right] with a Name/Rank-classified operator of TF Razor in late 2011
With the death of 42 of the operators over the course of OEF and with no orders to replace the fallen; it was certain that the days of TF Razor were coming to an end. I rotated back in earlier in the month to lead a series of suitably suicidal but successful missions in a classified urban area but before extraction after the final mission, a building collapsed, killing 3 of the remaining operators and seriously injuring another. The only operator who would have been ever able to walk again 'Patches' told me "It's time to go sir!" I knew what he meant, our time here was done, his even more so because a sniper nailed him as he carried our wounded comrade back to the helo. Sadly he too died before we landed leaving me to do all the debriefings alone.
I am on my way next month to assume a permanent staff position with UNETIDA [though appointments are currently being held up by the Special Investigation Committee], a position which would have conflicted with the types of operations TF Razor handled with exceptional valor and intrepidity above and beyond the call of duty. At least that's what the citations for the medals given to the 47 members' families say. As always, I'll wear my own new medals to honour their sacrifice and dream of past glory as I watch everything with more eyes then ever before.
2 comments Labels: GWOT, UNETIDA, Whopper
Happy Birthday USMC
A special message from General James F. Amos, Commandant of the Marine Corps and Sergeant Major Micheal P. Barrett, Sergeant Major of The Marine Corps to mark the 236th Birthday.
4 comments Labels: General Amos, USMC
Movie Mini Reviews 8-11-11
The last several movies I've seen since Captain America haven't compelled me to write full reviews for them - they literally weren't anything to write home about, which is a bit sad considering the performance of summer movies in recent years. I managed to avoid both Rise of the Planet of the Apes and Cowboys And Aliens in the last batch despite enjoying the previous Planet of the Apes remake with Mark Wahlberg and wanting to see if Jon Favreau could do something as impressive as Iron Man with something else. Sadly I was warned away by several sources who claimed that neither movie was in any way up to the hype that fuelled them.
Hype however, was one thing that J.J.Abrams is able to generate if Cloverfield was anything to go by [where the hype was actually better than the movie itself], but in his sci-fi monster movie Super 8 he paid tribute to someone he obviously has the deepest respect for: Steven Spielberg. The entire movie struck me as a deeply personal homage on J.J.'s behalf to Spielberg who was obviously a great influence on his career choice and it was certainly a fitting tribute [perhaps even more than Pegg/Frost's attempt earlier in the year with Paul]. The problem is however that I dislike most of the peacenik crap Spielberg put out earlier in his career like E.T. which I have always despised and Close Encounters, I mean seriously; giant mile-long ships come that close to the planet and we don't fire off volleys of ICBMs at them? Nonsense!
Nonetheless Super 8 is a well paced movie that sadly has more to offer 10-14 year olds then a nostalgic trip down memory lane remembering at all those kid-friendly sci-fi movies you grew up with.
The reboot of Conan The Barbarian was sadly only a middling attempt to revitalise an extraordinary character for the silver screen. Jason Game of Thrones Mamoa was cast as the titular hero and got all oiled up for a chance at movie stardom. Sadly while they got the look for Robert E. Howard's most famous character down perfectly, the true spirit of Conan wasn't bottled in this movie and the Millius/Schwarzenegger version written by Oliver Stone can lord that over this pale imitation. Don't get me wrong, as an adventure tale itself, it wasn't too shabby. The set pieces, architecture, locations and action elevated this from the mediocre direct-to-DVD bin but this was Conan! So more was expected. I will point the finger at Lionsgate because promotion for this movie in this part of the world anyway was anaemic at best.
Stephen Avatar Lang nailed the overacting required to be a truly reprehensible villain and a terribly disfigured Rose Charmed McGowan [girls, this is what happens when you try to change your appearance unnecessarily] under layers of make up makes us glad that Rodriguez never got his vision for Red Sonja off the ground. Ron [I'm in about 5 movies this year] Perlman is well suited as Conan's aged barbarian father and his death is one of the most unique I've ever witnessed on screen. Rachel G.I. Joe Nichols is suitably cast as the eye candy and sets back female role-models about twenty years with her screaming damsel in distress once her pitiful fighting skills fail her. As good as they were, James Earl Jones subdued Thulsa Doom, Sandahl Bergman's feisty Valeria and Mako's hilarious Wizard are still worth ten times the characters in this movie.
Colonel Creedon Rating: ***1/2
Columbiana starts off as an exciting thriller as the titular character as a young schoolgirl is chased through the favalas of Columbia by her parent's killers as she races to the U.S. embassy with highly sensitive intelligence data stored on a flash drive which she has swallowed. Had the movie been just about that, I'd probably be writing a great 5-star review now. Sadly no such thing happened as the movie soon transformed into a predictable revenge thriller as the little schoolgirl grew up into a frighteningly thin Zoe Star Trek Saldana [Lucas woman, eat a ham sandwich or three will you?] who proceeded to get her well deserved retribution.
Bland performances all round from Saldana who really can't carry a movie, Michael Alias Vartan, Lenny Human Target James and Cliff Die Hard 4.0 Curtis were sadly not much better with the material here. I was expecting more from the director of The Transporter 3, Oliver Megaton. It ended up being a hideously misjudged train wreck, punctuated with some satisfying Besson-like action scenes.
Colonel Creedon Rating: ***
The stylish if uneventful thriller Tinker Tailor Soldier Spy may be a tad over long and light on action but its heavy on intrigue, nail biting suspense and brings seriously big acting guns to the game. Featuring extraordinary performances from Gary Oldman, John Hurt, Colin Firth, Mark Strong, Toby Jones and Tom Hardy. A welcome change of pace that is well worth watching especially if classic spy movies are your bag.
Drive, Johnny English Reborn, Contagion and Machine Gun Preacher to follow
Kill All Kittens!
Lance Corporal Spazweki once asked me: "How come you always shoot every animal we encounter Sir?"
Always one to impart knowledge, my response was exceedingly simple, "Because you never know son, you just never know..."
Army Public Health Command poster
From Bruce via Wired.
He vowed to die a Martyr but instead he died a coward
Moammar Gadhafi was born in 1942 and abandoned Geography studies to pursue a military career. He rose to power at only 27 after a bloodless coup against King Idris in 1969. In the 70's he laid out his Third Universal Theory which walked a line between communism and capitalism and oversaw the rapid development of Libya with regards to her oil industry and military. He improved living standards making him popular with the poor at least initially.
As other Gulf nations such as Saudi Arabia also improved oil production to the same degree, Libyans noticed that the standard of living in these places was becoming much higher whereas in Libya; while the economy and infrastructure was improved - freedom and liberty became severely restrained. All political parties were banned and any attempts at creating one were met with execution. Intrusive surveillance was set up in government, education and industrial sectors. 10% of the Libyan people became informants. Freedom of expression was outlawed and any objections to the regime were met with public mutilations and dismemberment. Adultery was punished by flogging, homosexuals were jailed and thieves were amputated. The list goes on, but I'll stop it there...
Castro and Gadhafi BFF
Growing increasingly Islamist, Gadhafi supported militant groups in the Philippines and Indonesia. In '79 he supported the Iranian revolution and later in the '80's he supported the Red Brigades in Italy and the IRA in Ireland. However not content with supporting others, he got his own hands dirty by having his diplomats murder a British policewoman in London, his agents bombed a nightclub in West Berlin but perhaps his most heinous act of all was personally ordering the bombing of Pan Am Flight 103 in 1988 which killed all on board and more in Lockerbie, Scotland the town on which the aircraft debris fell. These events saw many UN countries impose trade sanctions against Libya.
President Reagan set Libya as a high priority for his administration in '81 due to the madman's support for extremist groups. SecDef Haig wanted pro-active measures against him and after year of occasional skirmishes over Libya's claim over the Gulf of Sidra in the Mediterranean, Operation El Dorado Canyon was launched. 18 F-111 bombers dropped 60 tons of munitions on military targets in Tripoli but alas, Gadhafi survived.
In the mid 1990's Gadhafi seemed to reassess his image and began a process of atonement by extraditing the "Lockerbie bombers" to Scotland. Due to a fall in oil prices in the early 2000's, Gadhafi publicly renounced terrorism and once Saddam Hussein was captured in Iraq in 2006 he renounced his own WMD programme. It seemed Libya was finally coming in from the cold as I noted then.
In February 2011, with vigour infused by the revolutions in Tunisia and Egypt, there were major political protests against Gadhafi. The protesters were met with stiff opposition as the tyrant claimed all who opposed him were drunk and had consumed drugged Nescafé. When he ordered that his troops begin gunning down the protesters "like dogs and rats", that was the final straw for the Libyan people and a full-scale civil war began, and the world took note.
While President Obama promised "no boots on the ground" the U.S. war machine joined it's NATO allies including france and the United Kingdom as they began an intense bombing campaign against Gadhafi-controlled military targets. They systematically whittled down the numbers of his forces with precision strikes and over several months, city by city, allowed the Libyan people to retake control of their country, a privilege denied them for over 4 decades.
Sirte, the dictator's birthplace, became the last bastion of Gadhafi's regime after the fall of Tripoli to the rebels - who had since become The National Transitional Council [NTC]. There on Wednesday evening, RAF Tornados flew surveillance missions which cleared the way for french fighter jets to fly a sortie. The same frenchies that grew some balls and actually started NATO's campaign in Libya earlier in the year, on Thursday morning together with U.S. Predator drones, attacked a convoy at a roundabout two miles west of the city. Over a dozen armed trucks were destroyed killing 50 or so but some of the personnel escaped from their wreckage - among them, unbeknownst to NATO was Colonel Gadhafi himself!
The tyrant fled with a handful of loyal men into a storm drain but the forces of the NTC were in hot pursuit. They fired anti-aircraft guns into the drains before going in on foot. After an intense firefight, the loyalists were dead and a snivelling whimpering Moammar Gadhafi was dragged out from from the sewer pipe in which he was trapped like a rat.
The 69 year old 'Mad Dog' begged for his life, a far cry from his trademark defiant posturing while in power. He was beaten with shoes as a sign of great disrespect, but just in case he didn't get the message he was summarily executed like "a dog in the street" by the very people he had treated worse than animals. His body was hauled onto a truck and dragged around Sirte before being brought to Misrata where it was driven about the streets as the jubilant inhabitants rejoiced shouting "God is great".
"He called us rats, but look where we found him," said Ahmed Al Sahati, standing next to two stinking drainage pipes under a six-lane highway.
The entrance to this now-famous storm drain in Sirte is already a monument. The graffiti says "The hiding place of the vile rat Gadhafi."
The man has no basis in any event of reality we exist in. He is, in my opinion quite mad. It's obvious he's playing out some sort of insane plot in his head and the Libyan people are simply playthings in his sick game.
Upon taking power a Lieutenant simply promotes himself to Colonel and starts dressing in completely over the top elaborate military uniforms - I mean what's up with that? Today, if someone just promoted himself to Colonel and started wearing dozens of medals and badges on an array of colourful uniforms and declares that he's in charge - I hope that this would be a sign to people that this bozo needs serious professional help... or a bullet - to prevent a madman from taking power again.
-Excerpt from a declassified psychological report on Colonel Moammar Gadhafi prepared by Lieutenant Colonel C. Creedon, USMC in 1998 for the Joint Chiefs of Staff.
May the victims of the Lockerbie bombing and all those even among his own people finally rest in peace now that this tyrant has been sent to rot in hell.
14 comments Labels: Obituary, President Obama, World Events
Steven Seagal: Border Sheriff
There was some unfounded speculations that after he left the governor's position that Arnold Schwarzenegger would pick up a weapon and stand on the U.S. / Mexican border to prevent illegal aliens from crossing - but he didn't, he's going back to acting. So you therefore have to hand it to fellow 80's action star Steven Seagal though, because next year he's grabbing a gun and going to do it instead!
Give a man a badge. [via Splash]
That's right, this is not, I repeat not Creedon's insane ramblings for the week this is real! In January 2012, Seagal will be swapping Louisiana for Texas where he has just been sworn in as Sheriff's Deputy this week to go out on patrol in Hudspeth County, which runs along the Rio Grande east of El Paso. Yes Seagal will be on the front line in the border wars against drug lords and to stop illegals from crossing the border, or at least a 98 mile long stretch of it.
Seagal, 59, a 7th dan black belt in Aikido, will bring his martial arts prowess to bear in protecting the border, as well as offering up his expertise to train the other sheriff's deputies, working full-time to help secure the border Texas shares with Mexico.
As you know from reading here that for the past couple of years cameras have been following him around Jefferson Parish, Louisiana where Seagal is a fully-commissioned Reserve Deputy Sheriff. What's the bet that those camera's will be heading Southwest now too?
"Holy Shit! They're coming through the fence - don't just stand there slackjawed! You gotta badge now! Just point an shoot!" [picture via Splash]
4 comments Labels: Celebrities, Wacky News Stories
Special Investigation Committee formed
On Friday the U.N. Office of Internal Oversight Services at the request of the Security Council announced the appointment of a Special Investigation Committee to examine the manpower, assets, finances and procedures of UNPASID after a number of high profile debacles put the directorate under scrutiny. The committee will also jointly assess UNETIDA with regard to manpower and budgetary issues being faced by many countries who contribute personnel and assets to the organisations.
The five permanent member countries of the Security Council will be represented on the committee by: Professor Wai Chen, Chinese Academy of Sciences [Peoples Republic of China], Lt. General Andrew G. Kelly [Ret.], NASA [United States], Deputy Director Sergei Sitnikov, FSB [Russian Federation], Rear Admiral Oliver Braithwaite III CBE, Defence Intelligence [United Kingdom] and Cardinal Antoine Pascal, Roman Catholic Church [France].
The committee will be chaired by Mrs. Anna Scherzer, a senior economist with the Swiss Financial Markets Authority and will make recommendations to the UNSC together with a report of the committee's findings in due course.
All UNPASID and UNETIDA directors, officers, personnel and staff have received instructions through the organisations respective Inspector Generals offices to comply with the wishes of the committee and to make themselves available for questioning at their behest.
1 comments Labels: UNETIDA, UNPASID
World War Z without guns?
The filming of the movie adaptation of Max Brooks' World War Z starring Brad Pitt hit a snag this week as Hungarian police confiscated some 100 weapons at Budapest's Ferenc Liszt Airport destined for use on the movie.
Apparently Hungaran police were dissatisfied with the methods that the London armourers used to deactivate the real weapons from being able to fire live ammunition. Janos Hajdu, head of Hungary's Counterterrorism Center, explained that in Hungary; weapons were considered to be deactivated only if the process "was irreversible," while the weapons seized could still be fired even though screws had been used to fill the end of the barrels.
Bela Gajdos, a weapons supervisor for World War Z, said Mafilm, a Hungarian film company based near Budapest which had the guns brought to Hungary, had the necessary permits, including a detailed list of the weapons in question, issued by local police authorities. Gajdos was questioned by investigators who thoroughly searched his Budapest home before dawn Monday and confiscated the permits.
In an unrelated story, Lt.Colonel "Spinner" Kilar, Chief Logistics Officer for the United Nations Paranormal and Supernatural Interdiction Directorate [UNPASID], was arrested along with several other personnel today under charges of conspiracy and misappropriation of UNPASID assets. The charges were as a result of some 100 UNPASID weapons not being reported as missing over a number of weeks. The weapons whereabouts are unknown at this time.
1 comments Labels: Movies, UNPASID
OEF 10 years on
The factory-applied layer of silver polish had just about worn off my first set of eagles, when I was given command of Task Force Razor - not two weeks after the twin towers fell. Rummy gave me my orders in the executive bathroom of the White House of all places. There would be no records - I had to flush the orders after reading. The man stood there and watched as I did so. "We're not declaring war so anything goes" I was told. "Don't let torture, or the Geneva Convention stand in your way, but keep the civilian kill-count on the minimum because you never know who will be watching you, or if you will know who will be watching them watch you." Classic Rummy.
We deployed 'in country' on Sept 23rd 2001 and from then until mid '02 we did some crazy shit over there let me tell you. We took our commands from a company spook called "Smith" from the Special Activities Division. We marked the targets for the October 7th airstrikes, helped coordinate anti-Taliban forces on the ground with U.S. firepower from the air and generally eliminated anyone we found with a 'banana clip' at least until we found one that talked, or persuaded to do so later.
Some "influential" donkey senator got word that there was a rogue CIA op causing havoc in the Kandahar province and that their leader was called "د هغه سړي سره کوم روح" [the man with no soul]. We got shut down pretty quickly after that. Smith vanished and my team were absorbed into regular SpecOps forces under Col. Mulholland and I was sent stateside to assist the selection process fot MAR DET ONE. At that point the U.S. was trying to win the hearts and minds of the local populace to thwart the efforts of the insurgents and to be fair it wasn't helpful that we were accidentally shooting women and children every couple of months.
It's because I can remember it as yesterday is what makes it so hard to believe that President Obama marked the 10th anniversary of Operation Enduring Freedom on Friday, honouring those who have served and noting their efforts toward bringing the war to a responsible end from a position of strength. Despite the tremendous losses of almost 1,800 American patriots, he noted progress in taking the fight against violence extremism to the source. "In delivering justice to Osama bin Laden and many other al-Qaida leaders, we are closer than ever to defeating al-Qaida and its murderous network," he said in an address.
Personally I think enormous challenges remain it's certainly worth noting that the Taliban have been pushed out of their key strongholds, Afghan security forces are growing stronger and the Afghan people are in a better position to craft their own destiny if they can manage to hold onto the freedom and hope that the U.S. and coalition allies have brought them over the subsequent years.
Mulholland the aforementioned colonel is now a Lieutenant General and ending a tour as commander of U.S. Army Special Operations Command. "We're moving toward an increased special operations role," together with U.S. intelligence, he said, "whether it's counterterrorism-centric, or counterterrorism blended with counterinsurgency."
So basically while most American troops prepare to withdraw from Afghanistan in 2014, the CIA and military special operations forces who were first to fight will also be the last to leave as they begin girding for the next great pivot of the campaign, as they continue to train and support the Afghans and closely observe if they have what it takes to keep their country above the mire of terrorism. That in itself is no small task and one that could stretch their war up to another decade.
Sources: U.S. Department of Defense / Marine Corps Times
1 comments Labels: CIA, GWOT, President Obama, Secretary Rumsfeld, US Army, Whopper, World Events
Aw crap! "That guy!" died! RIP Charles Napier
You probably won't recognise the name Charles Napier too quickly as he wasn't particularly famous for it, but because of his voice, his face and the square-jawed tough guys and military types he played in many movies and TV shows, he will be remembered forever - even if only as "that guy!"
Gen. Gilmore [Austin Powers: International Man of Mystery]
Napier was born in Scottsville, Kentucky, served in 11th Airborne Division and studied teaching before becoming an actor. He made his film debut in Cherry, Harry & Raquel! in 1970. Among his more memorable movie roles were Murdock, the intelligence officer in Rambo: First Blood Part II and Lt. Boyle in Silence of the Lambs. Less serious roles included Loaded Weapon 1 and as generals in the first two Austin Powers movies. His role as Judge Garnett in Philadelphia was described [by those who saw his whole performance] as "career defining" but sadly much if it was cut for running time.
Col. Briggs [The A-Team]
Napier was just as recognised for his numerous television appearances. He guest starred dozens of TV shows like Mission: Impossible, The Incredible Hulk, The Rockford Files and Knight Rider. He also did two episodes of The A-Team. His most memorable TV work is of course the role of Adam the "space hippie" in the Star Trek: The Original Series episode "The Way to Eden". 25 years later he returned to Star Trek to portray Lt. General Rex Denning in the Star Trek: Deep Space Nine episode "Little Green Men".
On Star Trek - Adam [L] and Gen. Rex Denning [R]
The voice of Duke Phillips [The Critic]
Napier's voice is instantly recognisable. He voiced many of The Incredible Hulk's growls in the '70's has also provided several guest voices for episodes of The Simpsons. Fans of the animated Superman series in the late 90's and later The Justice League will know his voice as General Hardcastle. His greatest voice work in my opinion however is as the Ted Turner-esque media mogul Duke Phillips in the 1990's animated series The Critic.
Napier was married twice and had three children. He collapsed in his home on Monday and was found the following morning and taken to Memorial Hospital in Bakersfield, California. He was taken off life support just before 1pm EST on Wednesday and died shortly thereafter. He was 75.
1 comments Labels: Movies, Obituary, Star Trek, TV
Dempsey replaces Mullen as CJCS
Admiral Michael Glenn "Mike" Mullen, retired from the United States Navy at Joint Base Myer-Henderson Hall, Va. on Friday Sept. 30. He passed the reigns as Chairman of the Joint Chiefs of Staff to Army General Martin E. Dempsey during a change of responsibility ceremony there before an audience that included President Obama, Vice President Biden, Secretary of Defense Panetta, senior military officers and other distinguished officials.
Gen. Dempsey [L] and SECDEF Panetta [R]
In his final speech as Chairman, Adm. Mullen said he reminded Gen. Dempsey that he will not only be the POTUS' advisor, but also the representative of 2.2 million members of the armed forces. He said that his successor's biggest challenge will be in Afghanistan "making sure that the security gains we have made are not squandered by the scourge of corruption or the lack of good governance."
"Time is both his best friend and his worst enemy. I never seemed to have enough of it to do the things I wanted, and it's hard to believe it's over," Mullen said as he finished a 43 year career.
Gen. Dempsey was sworn in at the 18th Chairman and he thanked his predecessor for his patriotism and friendship. He vowed to maintain and strengthen the military during his term in office. In his speech, Dempsey said U.S. armed forces "are powerful, responsive, resilient, versatile and admired. [They] provide leaders with a wide range of options to counter threats and crises. And when sent to do the nation's bidding, we are an unambiguous signal of our nation's resolve."
Adm. Mullen [L] shakes the hand of his sucessor
"Our people -- America's sons and daughters -- are our decisive edge ... We'll change and we'll be challenged, but when I complete my tenure as the chairman of the Joint Chiefs of Staff, I intend to be able to say exactly the same thing: We will be the joint force the nation needs us to be, so help me, God."
"I am supremely confident of the future because we have the strongest military force in our history and in the history of the world," SECDEF Panetta said. "And it is strong exactly because we can replace one great warrior with another."
On Monday, General Dempsey briefly spoke with reporters in his office at The Pentagon. While the session was considered off the record, Dempsey's staff allowed some details about the general's office to be revealed: He sits behind a 4 by 6 foot desk, the same one Gen. Douglas MacArthur used in the Philippines during World War II. He said his desk at home is one used by Gen. Omar Bradley, the first chairman. A larger than life painting of Gen. George C. Marshall hangs in the office in such a way that the chairman can see it clearly from his desk. On the desk however is something uniquely Dempsey's: a box that holds cards he had made when he commanded 1st Armored Division during its '03-'04 deployment to Iraq. The cards each bear the photo, personal and family information for a division solider killed in action there.
Credits: Jim Garamone & Karen Parrish, American Forces Press Service
2 comments Labels: Admiral Mullen, President Obama, Secretary Panetta
You wait for ages and all of a sudden: 7 of them show up at once!
A Supernova, the death cry of a star. One supernova is a rare occurrence but in the galaxy designated Arp 220, a mere 250 million light-years from our Solar System, not only one supernova is occurring - but SEVEN!
According to IO9, a team of astronomers at Chalmers and Onsala Space Observatory found that all seven supernovas went off in the last sixty years (allowing for the 250 million years for the light to reach Earth). In cosmic terms, that's the blink of an eye. European Southern Observatory astronomer Rodrigo Parra explains: "In Arp 220, we see far more supernovae than in our galaxy. We estimate that a star explodes in Arp 220 once every quarter [century]. In the Milky Way, there is only one supernova per century."
MEHAU KULYK/SCIENCE PHOTO LIBRARY
Astronomers believe it's proof that Arp 220 is one of the most efficient galaxies in the universe when it comes to making stars...and then destroying them. There's so many stars that have formed in that galaxy that there's always plenty of potential supernovae, even allowing for the fact that exactly when each star explodes in its own lifetime can vary by a whole lot more than sixty years. Team member John Conway adds: "Arp 220 is well-known as a place where star formation is very efficient" as well as being a supernova factory.
Astronomers however are not privy to the information gathered by UNETIDA. Acting Intelligence director for UNETIDA, Colonel "Whopper" Creedon, said that the multiple supernovae are consistent with information that he acquired recently regarding an ancient conflict. A conflict between two immensely powerful empires a quarter of a billion years ago in Arp 220 or Sangesh'ka as it was known locally. "Basically you had two immense empires, the Jurathi and the Wevnarr," explained Creedon. "Both were highly technically advanced and quite belligerent. We understand that each expanded their rule from a single world at either side of the Sangesh'ka galaxy conquering all they encountered - until of course they encountered each other. We are now witnessing the final moments of their eons-long war as they wipe each other out – totally, in a way we can barely fathom - by destroying their suns."
Advertisment leaves three dead!
A poorly conceived advertisement for a Cork city music event at a nightclub has left three dead in the wake of a UNETIDA raid on the premises on Friday morning.
The advertisement for a music event sponsored by a prominent European beer company was presented in a "wrap around" format over the real cover of the local newspaper. It presented the ad in such a way that it posed as the newspaper's headline "MIDDAY INVASION" [in the Impact font as 44mm high lettering]. It also featured a green "tractor beam" over the nightclub venue, doctored photographs of extra-terrestrial spacecraft over the prominent Shandon landmark and testimony from a non-existent professor from a society that was invented for the purposes of the advertisement. Only a tiny message at the bottom identifies the page as an advertisement [4mm high non-bold Times New Roman lettering].
Friday morning shoppers in Cork city were horrified to see a group of helicopters deposit two dozen armed men on the roof and an armoured car simultaneously plough through the front doors of the ground floor of the nightclub; which by day is a small shopping arcade. Witnesses saw flashes from windows and heard small explosions and automatic weapons fire.
A UNETIDA statement later in the afternoon revealed that a Special Operations strike team was promptly dispatched to raid the venue described in the newspaper. A ground unit entered from street level while helicopters deposited a second unit the roof of the establishment where they blew in skylights and windows to gain access. The team reported that they "encountered resistance" once inside. They terminated three targets who were reported as "babbling and screaming alien language" and were disguised as cleaning staff. A fourth target was wounded and apprehended.
On Saturday, Colonel "Whopper" Creedon, Acting Director of UNETIDA Intelligence revealed that personnel, acting on the information presented in the newspaper, did perform an assault on the premises and shot and killed one Lithuanian and two Polish nationals employed by a local cleaning firm to service the establishment. He also slammed the newspaper and the European beer company for presenting the information in the manner that they did so, which he described as "grossly irresponsible". The only good thing about the ad according to Creedon "is that it covers George Hook's exceptionally ugly mug from being seen".
"UNETIDA is a fast-reaction force to this type of threat" added Creedon, "that sometimes means we don't have time to read all the way down to the samll print at the bottom of a page."
9 comments Labels: UNETIDA, Whopper
UNETIDA's Week In The News 27-09-11
UNETIDA has appeared not once, not twice but three times in the press in the past few days...
The Commander, UNETIDA Naval Tactical Support Capt. "Harpoon" Dutton KBE, speaking in the absence of Contraalmirante "Tridente" Carlos UNETIDA Naval Operations Commander, said "UNETIDA is currently not in command of either the USS Jefferson City nor any other U.S. naval asset afloat or submerged in the Pacific Ocean at this time. The NASA UARS satellite that crashed to Earth did not contain any UNETIDA controlled "black box" or other equipment on board. We are not involved in any search or recovery operation with the U.S. Space Agencies."
UNETIDA's Director of Research and Development Dr. "Quantum" Pataal praised the work of Scientists at CERN near Geneva, Switzerland who have reportedly debunked half of "known physics" and rendered Einstein "the intellectual equivalent of a public school science teacher." Pataal said that science fiction writers will have to put on their creative caps now and come up with even more fantastical ideas than those of Asimov, Speilberg, Bradbury, Roddenberry, Dick and even Lucas himself now that space and time travel will be "reality in a few years".
Source: Anorak
Colonel "Whopper" Creedon, UNETIDA's Acting Director of Intelligence gave evidence at an inquest into the death of one Micheal Faherty, 76, of Ballybane, Galway on December 22nd 2010. Col. Creedon proved that UNETIDA's prototype "Microwave Gun" could not have been responsible for the man's death as it produced far too much collateral damage. West Galway coroner, C. McLoughlin and Assistant chief fire officer, G. O'Malley said there was no other adequate explanation for the death of Mr. Faherty, other than "Spontaneous Combustion" - now the first recorded case in the country.
Captain America: The First Avenger rules the summer
I like to convince myself that the Matt Salinger Captain America movie never even happened. So for me, this was my first time seeing the star spangled avenger come to life in a live action motion picture. Marvel Studios and Paramount have collaborated on several movies now and each entry in their catalogue has been continuously impressive. Captain America: The First Avenger is inexorably linked to the The Avengers movie next year. The Incredible Hulk, Iron Man [1&2] and Thor all worked pretty flawlessly as movies but should Captain America fail, it could endanger what is going to be one of the most incredible cinematic events in history as these heroes finally come together to battle evil in the one movie.
Captain America: The First Avenger serves a dual purpose both as an introduction to the character who will lead The Avengers super-hero team and also brings the final pieces of the grand mystery that has been glimpsed in post-credits stingers for more than 3 years now. Like the other movies in the franchised, Captain America stands firmly on it's own two feet and never reduces itself to being simply the piece of a larger whole.
When I saw Chris Evans was picked to portray Steve Rogers I was unsure. His performance in the Fantastic Four series was hardly stellar, but to be honest, he didn't have a lot to work with those movies. Come to think of them in fact I believe Evans now puts a nail in the coffin of that franchise which isn't exactly bringing a tear to my eye. Then I saw him in the even-more-awful Scott Pilgrim vs. The World as Lucas Lee and changed my mind – he'd certainly work as Cap and I was right. His performance as a weedy but ambitiously patriotic young Rogers is flawless but he really comes into his own when dosed with the Super Soldier Serum so he can become a living weapon, the first in an army of super-soldiers with which the U.S. can better combat the Nazi menace sweeping across Europe.
That's right, while the tale is framed by present day scenes, the majority of the movie takes place during WWII just as a Captain America origin tale should [the character first appeared in 1941]. While a comic book delivers it's story through illustrations with speech bubbles, it's up to a solid array of actors to make those drawings flesh and voice, and The First Avenger's cast is stellar. Hugo The Matrix Weaving was a deliciously OTT Johann Schmidt AKA Red Skull and single-minded in his quest for the Cosmic Cube, witnessed in the stinger at the end of Thor. I'm wondering however which of Weaving or Mark Strong gets the call "I want you to be the villain in my next movie" first nowadays as they seem to have cornered that market in recent years. Kings actor Sebastian Stan gets to be Captain America's buddy Bucky Barnes, Tommy Lee Jones is excellent as Col. Chester Phillips. Hayley The Pillars of the Earth Atwell is Strategic Scientific Reserve officer Peggy Carter, Stanley The Core Tucci is Dr. Abraham Erskine, creator the Super Soldier Serum, British theatrical actor Dominic Cooper as Howard Stark – Tony's father and not forgetting Neil Star Trek: First Contact McDonagh as "Dum-Dum" Dugan of The Howling Commandos.
Director Joe The Rocketeer Johnston is know for his effects-laden movies and he doesn't disappoint with the cinematic spectacle he delivers here. Among the 1600 effects shots were a multitude of scenes with Steve Rogers before he was "bulked up" by the Super Soldier Serum using complex green screen and body double technology. There was one or two hokey effects shots [one scene where Cap was running after a bomber inside a massive hangar was especially crud] but they weren't enough to snap you away from your immersion in this incredible stylised wartime world that Johnston and his team created for us. Veteran composer Alan Back To The Future Silvestri composed an instantly classic score, evoking the great epic cinematic qualities of old WWII adventures.
I'm pleased to announce that there is nothing to worry about, the world has accepted The First Avenger as a hero to admire [he did surprising well overseas for someone with "America" in his name and literally dressed in an American flag suit and has made over $350m]. The hype for The Avengers is already assured some measure of success provided Joss Serenity Whedon doesn't screw it up, but the Marvel Super-Hero train is going so fast now, I doubt even he could derail it.
Final Verdict: An old-fashined adventure tale, another outstanding work of Marvel cinematic gold and another kick in the teeth for DC/Warner's epic fails at everything except Batman.
Colonel Creedon Rating: *****+
Dragon*Con 2011
On September 2nd 1864 the forces of Union General William T. Sherman marched into Atlanta, Georgia, one day after the Confederates evacuate the city. Exactly 147 years later, the people of Atlanta stood aside as a 40,000+ force of fans of Science Fiction, Fantasy, TV shows, Games, Comic Books and Movies descended on downtown Atlanta and literally became an occupying force of five of the largest hotel complexes I've ever been in.
The crowds before rush hour. Photo by Mark Twomey, used with permission.
It was Bruce Russell who demanded that Mark and I fly to Atlanta to join him and his buddies, Jer and Sean, at the 25th annual Dragon*Con. We were joined by Constance who helped us navigate the colossal space invaded by thousands of people just like ourselves. The experience was an assault on the senses, hundreds of attendees, many dressed in sometimes elaborate costumes from all aspects of popular culture from fully armoured Imperial Stormtroopers to scantily clad Anime girls, moved as a crowd like a river of colour through the hotel lobbies, skywalks and convention centres on their way to get their own experiences.
In the Navy! Yes this bloke likes G.I.Joe even more than I. Photo by Mark Twomey
Many celebrities wandered amongst us, themselves seeking hero worship on their way to a panel or a fast buck from autographing pictures. We spotted the muscular Lou The Incredible Hulk Ferrigno riding an escalator, Laura Smallville Vandervoort, who really is as pretty as she is on screen, was walking about with V co-star Joel Taken Gretsch [who is also William Shatner's son-in-law]. A wide-eyed Robert Nightmare on Elm Street Englund and a haggard-looking Lance Aliens Henrickson took a wrong turning out a door into a sea of people and were promptly ushered through another door by quick thinking staff before the stars were too mobbed.
Some less-than-iconic stars like flame haired Felicia The Guild Day and a newly lean David Stargate: Universe Blue mingled after hours in the party-like atmosphere that formed each evening in all locations. There was as much to see at night as there was by day. Drink flowed from many locations catering for thousands, it was even served from an automated BAR-2-D-2 droid. Goth and Steampunk themed bands put on concerts for those who wished to dance the night away - lightsabers in the air. Some retired early however, regenerating themselves for the next hectic day of adventure.
Well holding up cigarette lighters would be a fire hazard.
Several dealers rooms were established where companies and individuals gathered and set up stalls to hawk their much-sought wares. Literally everything imaginable was available; from Superhero belt-buckles to Star Trek costumes, real metal fantasy and anime-themed swords and blades, valuable comic books and graphic novels, DVDs of almost forgotten yet beloved cartoons, trading cards, patches, badges, action-figures, posters, t-shirts and games. There was also an abundance of hand-made unique goods such as corsets, leather-ware, steampunk devices, jewelery and even kilts!
One room, "The Hall of Fame" was set up where you could go to meet the stars present and get a personalised autograph. I met Sean Patrick Young Indiana Jones Flanery, Eddie Warehouse 13 McClintock and the fascinating Richard Battlestar Galactica Hatch. I myself sought only two autographs - those of Michael Battlestar Galactia Hogan who scribbled "Frak Em All" on a Colonel Tigh picture for me and of course Erin Buck Rogers Grey who still looks stunning at 61 and whom I hope I didn't scare with my nervous dribbling. One true gentleman whom I must mention is Tahmoh Battlestar Galactica Penikett who waived his autograph fee on a picture for a friend of Constance who survived cancer.
Bill Shatner having fun. Photo via Dragon*Con
Finally it was the panels themselves that were of most interest. One may have to queue for an hour in the blistering 36°C heat to get a good seat, or what we did was to wait in relative comfort until the queue had dissipated and one could simply sit in the back and listen to the fascinating tales and memories recounted by the likes of William Star Trek Shatner and Carrie Star Wars Fisher. Shatner is doing well for a man of 80 and had some fascinating insight into his work. He also told us that he has discovered that most people in attendance had not come to the con to see him, but to see each other and that he was glad to be a part of that. Carrie Fisher was equally fascinating and a lot more R-rated in her speech than Shatner. She is conscious that her battles with her weight and drug abuse have taken their toll on her star, but she has made it her mission to educate people on such pitfalls.
G.I.Joe / Iron Man artist Tom Feister's panel got way out of hand.
All in all it was a fascinating worthwhile experience and an excellent primer before I dare brave the San Diego Comic Con at some point in the future. I will however look forward to returning to Dragon*Con and to Bruce and the boys in a few years for even more adventures.
6 comments Labels: Battlestar Galactica, Celebrities, Comics, G.I.Joe, Mark, Movies, Star Trek, Star Wars, Stargate, Television, TV
Next Year... STALLONE.... will be.... VAN DAMAGED!!!!
That's just one of the cheesy promo lines that I wish Don LaFontaine was still with us to deliver during an Expendables 2 teaser trailer. And it's true, Sylvester Stallone himself has confirmed that he will be battling "The Muscles from Brussels" himself - Jean-Claude Van Damme on screen in the sequel to one of the greatest action movies in history, The Expendables.
JCVD turned down Sly's offer to appear in the original because Stallone could not tell Van Damme who his character was. Obviously this time there was more fleshed out before offers were made because we understand from Stallone that: "We'll have a big showdown between me and Jean-Claude Van Damme, which has been anticipated for a long time, so it should be a good one."
This is almost the greatest news ever until we realise that not only is JCVD on board but Stallone has also apparently secured the legendary CHUCK NORRIS to appear. The first thought you have may be forgiven because, as you know, if Norris is the bad guy, then he could never be beaten and the villains would win but if he's a good guy, then the bad guys will just give up as soon as they see him and that'd be a boring movie. However I think at 71, 'ol Chuck should be used for dispensing on screen sage advice and instruction like Yoda as opposed to actually fighting as it's possible he could, albeit accidentally, destroy the universe.
Sources have said that Stallone had been talking with Hulk Hogan, Mr. T, Carl Weathers, Vin Diesel, The Rock, Charlie Sheen and Lorenzo Lamas about the sequel, which he intends to be "a love letter to martial arts" and that Nicholas Cage will soon be confirmed to star and perhaps even John Travolta. "I like using people that had a moment and then maybe have fallen on some hard times and give them another shot," Stallone told Entertainment Weekly.
Earlier this month Stallone confirmed that both his Planet Hollywood partners, Arnold Schwarzenegger and Bruce Willis would have more extended roles in the sequel than their brief-but-awesome cameo appearances in the original, joining the returning cast of Dolph Lundgren, Jason Statham, Mickey Rourke and others. The Expendables 2 will be directed by Simon Con Air West.
To be honest, I don't think G.I.Joe 2 out around the same time will get much of a look in with this up against it.
Source: Vaughan / Screenrant / Entertainment Weekly / AICN
December 7th 1941, "a date which will live in infa...
He vowed to die a Martyr but instead he died a cow...
You wait for ages and all of a sudden: 7 of them s...
Captain America: The First Avenger rules the summe...
Next Year... STALLONE.... will be.... VAN DAMAGED!... | {
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{"url":"https:\/\/math.stackexchange.com\/questions\/1898594\/norm-of-self-adjoint-member-of-c-algebra","text":"# Norm of self-adjoint member of $C^*$-algebra\n\nThis question arose from the proof of proposition $1.11(e)$ in chapter $8$ of John B. Conway's A Course in Functional Analysis. This portion of the proposition can be stated:\n\nLet $\\mathscr{A}$ be a $C^*$-algebra, and let $a\\in\\mathscr{A}$ be given. If $a=a^*$, then $\\|a\\|=r(a)$.\n\n(Here, $r(a)$ denotes the spectral radius of $a$.)\n\nThe proof, as stated in the book, proceeds as follows:\n\nSince $a^*=a$, $\\|a^2\\|=\\|a^*a\\|=\\|a\\|^2$; by induction, $\\|a^{2n}\\|=\\|a\\|^{2n}$ for $n\\geq1$ That is, $\\|a^{2n}\\|^{1\/2n}=\\|a\\|$ for $n\\geq1$. Hence $r(a)=\\lim\\|a^{2n}\\|^{1\/2n}=\\|a\\|$.\n\nNow I was able to show by induction that $$\\|a^{2^n}\\|=\\|a\\|^{2^n} \\qquad (n\\geq1),$$ from which the result follows, but I could not prove it as it is stated in the book.\n\nSo my question is: How can we prove (presumably by induction) that $\\|a^{2n}\\|^{1\/2n}=\\|a\\|$ for $n\\geq1$? Is this simply an error in the book, or can it be done?\n\n\u2022 I suppose if someone can point me to an errata for the book with this being mentioned, that would be an acceptable answer. Aug 21 '16 at 2:25\n\u2022 Seems likely to me this is just an error. Aug 21 '16 at 2:27\n\u2022 @EricWofsey That's what I was thinking, as it would be a small error in typesetting. But still I am curious. Aug 21 '16 at 2:31\n\nIt is definitely a typo. The proof of $\\|a^{2n}\\|=\\|a\\|^{2n}$ cannot be trivial, since for instance it implies that $\\|a^3\\|=\\|a\\|^3$ (which I don't think can be easily obtained from the axioms): $$\\|a^3\\|^2=\\|a^{2\\times3}\\|=\\|a\\|^{2\\times3}=(\\|a\\|^3)^2.$$ So, even if one can find an argument for the formula for $2n$, it is not worth it for Conway's proof, as any subsequence is good enough.","date":"2021-09-28 08:22:43","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\": 0.9523380398750305, \"perplexity\": 95.25738266055887}, \"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-2021-39\/segments\/1631780060538.11\/warc\/CC-MAIN-20210928062408-20210928092408-00663.warc.gz\"}"} | null | null |
Il ghiacciaio Gillock (in inglese: "Gillock glacier") è un ghiacciaio lungo circa 9 km situato sulla costa della Principessa Ragnhild, nella Terra della Regina Maud, in Antartide. Il ghiacciaio, il cui punto più alto si trova a circa 1.451 m s.l.m., si trova in particolare nelle montagne Sør Rondane, dove fluisce verso nord a partire dal monte Walnum fino ad arrivare al versante ovest della dorsale Smalegga.
Storia
Il ghiacciaio Gillock è stato mappato nel 1957 da cartografi norvegesi grazie a fotografie aeree scattate nel corso dell'operazione Highjump, 1946-1947, ed è stato in seguito così battezzato in onore del tenente della marina militare statunitense Robert A. Gillock, che prese parte alla suddetta operazione come navigatore durante diversi voli effettuati sulle zone comprese tra 14°E e 164°E.
Note
Voci correlate
Ghiacciai dell'Antartide
Collegamenti esterni
Gillock, Ghiacciaio
Gillock, Ghiacciaio | {
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La Sirène des bas-fonds () est un film américain réalisé par Richard L. Bare, sorti en 1949.
Synopsis
Fiche technique
Titre : La Sirène des bas-fonds
Titre original :
Réalisation : Richard L. Bare
Scénario : David Lang
Direction artistique : Ted Smith
Décors : Lyle B. Reifsnider
Costumes : Doris Stutz (non créditée)
Photographie : Carl E. Guthrie
Effets visuels : Edwin B. DuPar
Montage : Frank Magee
Musique : William Lava
Production : Saul Elkins
Société de production et de distribution : Warner Bros. Pictures
Pays d'origine :
Langue : anglais
Format : Noir et blanc — 35 mm — 1,37:1 - Son : Mono (RCA Sound System)
Genre : Film noir
Durée : 86 minutes
Date de sortie :
Distribution
Virginia Mayo : Flaxy Martin
Zachary Scott : Walter "Walt" Colby
Dorothy Malone : Nora Carson
Tom D'Andrea : Sam Malko
Helen Westcott : Peggy Farrar
Douglas Kennedy : Hap Richie
Elisha Cook Jr. : Roper
Douglas Fowley : Max, détective
Monte Blue : Joe, détective
Jack Overman : Caesar
Rory Mallinson : l'assistant du coroner
Liens externes
Film américain sorti en 1949
Film noir américain
Film de Warner Bros
Film américain en noir et blanc | {
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\section{Introduction}
In the last decades, it has become clear that energetic feedback processes are the key ingredients in shaping
the star formation history of the Universe and regulating the formation and evolution of galaxies.
In particular, supernovae (SNe) and black holes (BHs) are considered the principal sources of such feedback \citep[see][for a review]{Silk2007}.
Cosmological simulations have been one of the most powerful tool for investigating the formation and evolution of galaxies \citep[see review from][]{Bertschinger1998,SpringelRev2010}.
However, current cosmological simulations lack both the resolution and the physics needed to model small scale feedback phenomena like the explosion of a single SN or the accretion into a BH.
Several numerical, sub-grid recipes have been employed to mimic the large scale effect of feedback by injecting the energy either in thermal or kinetic form.
Thermal feedback has the advantage of having an isotropic effect on the closest surrounding medium.
However, in the presence of radiative cooling and poor resolution, a large part of the energy is radiated away before any hydrodynamical response of the medium.
Several studies concentrated their efforts in numerically solving the over-cooling problem and the consequent dissipation of thermal feedback energy
\citep[e.g.][]{Gerritsen1997,Mori1997,Thacker2000,Kay2002,Sommer-Larsen2003,Brook2004,Stinson2006,Booth2009}.
In order to avoid the over-cooling problem, other studies favoured the kinetic feedback approach, in which all or part of the energy is given to the surrounding medium as momentum
\citep[e.g.][]{Navarro1993,Mihos1994,Kawata2001,Kay2002,Springel2003,Oppenheimer2006,DallaVecchia2008,Dubois2008}.
However, over-cooling can still play an important role in quickly dissipating the heat produced by the outflowing gas shocks, and underestimating the increase of the temperature of the medium.
Until now, despite the large variety of implementation of feedback models, very little work has been done to compare
the two feedback approaches in the context of galaxy formation.
\cite{Kay2002} compared the results of cosmological simulations with either thermal or kinetic SN feedback.
They showed that even when artificially preventing cooling of heated particles the results of the two schemes are considerably different.
Nevertheless, before tackling the over-cooling problem, one should first make sure that the input energy is accurately conserved.
In a recent study of the modelling of SN feedback in SPH simulations, \cite{Saitoh2009} \citep[hereafter SM09, see
also][]{Merlin2010} noted that strong perturbations of the internal
energy of gas particles lead to the violation of energy conservation
when an individual particle time-step scheme is used (see Fig.~\ref{fig:sedovindiv} for an illustration).
That can be summarised as follows. If a source of
energy alters the internal energy of a (active) gas particle, the
particle will eventually decrease its time-step according to the Courant criterion. However, if its
neighbouring particles are inactive,\footnote{Their hydrodynamical
state is not updated at the same time.} they will not react
immediately to the change of the thermal state of the
region, and the integration accuracy may be strongly compromised. The lack of a
prompt response to the nearby energy injection can lead to effects
like inter-particle crossing (due to particles missing the deceleration given by
viscous forces) and non-conservation of energy.
SM09 proved that, when using an appropriate time-step limiter, accurate energy
conservation is achieved for strong internal energy perturbations.
They proposed a scheme in which the ratio of the time-steps (longer over shorter) of
neighbouring gas particles cannot be larger than a given factor
$f_{\rm step}$. A value of $f_{\rm step}=4$ is enough to ensure
energy conservation to similar level of accuracy given by a global
time-step integration scheme (\cite{Merlin2010} suggested the values $f_{\rm
step}=[4,8]$).
In this work and for the first time, we investigate under what conditions thermal and kinetic feedback schemes lead to qualitatively identical results.
We first show that the feedback prescriptions are equivalent if the integration of the hydrodynamical equations is done over global, system time-steps.
In order to do that, we compare test simulations of Sedov's blast wave problem \citep{Sedov1959,Landau1959} to its analytic solution, finding very good agreement and energy conservation to within 1\%.
With the aim of generalising the results, we investigate whether one can achieve similar agreement and accuracy
with the more computationally efficient individual time-step integration scheme.
We implement a modified version of the SM09 time-step
limiter in the public version of \textsc{gadget}-2{} \citep{Springel2005} consistently taking into account the time-step synchronisation and the underlying leapfrog integrator.
We test the robustness and speed of the limiter with Sedov's explosion
problem injecting either thermal or kinetic energy at a random time.
We demonstrate that thermal and kinetic feedback methods are equivalent
and energy is accurately conserved, provided that
the system promptly responds to the energy perturbation.
With the application of the time-step scheme to galaxy formation and cosmological simulations in mind, we study a more realistic problem: the release of energy
in the presence of density and pressure gradients. We apply the scheme to an off-centre
explosion in a self-gravitating gas halo, and show that the
concordance of feedback methods and accurate energy conservation are still achieved even for the
extreme case where all available energy (either thermal or kinetic) is injected into only one particle.
\begin{figure}
\begin{center}
\hspace{-2mm}\includegraphics[trim = 0mm 2mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig01a.ps}
\hspace{-1mm}\includegraphics[trim = 0mm 2mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig01b.ps}\\
\hspace{-2mm}\includegraphics[trim = 0mm 4mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig01c.ps}
\hspace{-1mm}\includegraphics[trim = 0mm 4mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig01d.ps}
\end{center}
\caption{Sedov's explosion test for thermal (left column) and kinetic (right column) energy injection,
when the individual time-step integration is used. Results are given in the natural system of units soon after the explosion at time $t=0.004$.
We show in the upper panels the projected gas density in a slice of thickness $\Delta z=0.1$ centred on the box origin.
The dashed circle corresponds to the position of the spherical shock-front given by the similarity solution in Eq.~\ref{eq:sedovblast} at that time.
The white dots mark the position of all particles that initially received the energy.
In the lower panels, the average radial density profiles are compared to the profiles given by Sedov's analytic solution (red lines).
Given the large energy jump introduced by the explosion, it is clear from these plots that none of the feedback scheme are able to describe properly Sedov's test.
\label{fig:sedovindiv}}
\end{figure}
In standard cosmological simulation practice the new integration approach presented in this paper
could, in principle, require more simulation steps, but will mainly populate the intermediate time-bin levels with more particles.
Therefore, we could expect a slight increase of the computational time for galaxy formation simulations,
a price to pay for a better handling of feedback events.
However, we still want to emphasise that, in idealised simulations like the ones presented in this work,
the proposed scheme enables much faster integration than the standard, individual time-step scheme
since it preserve a high energy conservation accuracy.
The paper is organised as follows. We first show in Sec.~\ref{sec:concordance} that concordance
of feedback methods is achieved with a global time-step scheme.
We then apply our individual time-step algorithm to Sedov's blast wave
test and to the off-centre explosion in a self-gravitating gas halo in Sec.~\ref{sec:individual}.
We conclude in Sec.~\ref{sec:discuss}.
In all the simulations presented in this paper, we use the so-called \textit{natural} system of units in which the units of velocity, length and mass are unity.
\section{Concordance under accurate time integration}
\label{sec:concordance}
\bigskip
\begin{table*}
\begin{center}
\begin{tabular}{ l | l | c | c | cc | cc }
\hline
\hline
\multirow{2}{*}{Name} & \multirow{2}{*}{Scheme} & \multirow{2}{*}{$E_{\star}$} & \multirow{2}{*}{$\alpha$} &
\multicolumn{2}{c}{ Energy (\%) } & \multicolumn{2}{c}{ Time (h) } \\
& & & & Thermal & Kinetic & Thermal & Kinetic \\
\hline
\hline
{\bf SEDOV-G} & {\bf Global} & $\mathbf{1}$ & {\bf 2} & {\bf 0.81} & {\bf 0.78} & {\bf 3.71} & {\bf 2.50} \\
\hline
SEDOV-G-$E01$ & Global & $0.1$ & 2 & 0.83 & 0.82 & 1.97 & 1.46 \\
SEDOV-G-$E001$ & Global & $0.01$ & 2 & 0.84 & 0.74 & 1.09 & 0.96 \\
\hline
SEDOV-G-$\alpha\, 1$ & Global & $1$ & 1 & 1.99 & 2.30 & 3.87 & 2.73 \\
SEDOV-G-$\alpha\, 4$ & Global & $1$ & 4 & 0.73 & 0.61 & 3.58 & 2.38 \\
\hline
\end{tabular}
\end{center}
\caption{Numerical parameters and energy conservation estimate and computational time for Sedov's blast wave tests with a global time-step scheme.
From left to right: names of the runs specifying the integration scheme and the change of numerical parameters, time integration scheme;
injected energy, $E_{\star}$; artificial viscosity, $\alpha$; energy conservation estimates in percent (from Eq.~\ref{eq:energyconservation}); and computational time in hours.
Statistics are given at the end of the runs, at time $t=0.1$. The wall-clock time is given for runs on 8 cores. The reference simulation is highlighted in bold.}
\label{table:concordstat}
\end{table*}
In this section we demonstrate the concordance of thermal and kinetic feedback schemes.
We run several simulation tests of Sedov's explosion problem varying the numerical parameters to show the robustness of our findings.
Here, we assume the most favourable case in term of integration accuracy.
The integration is done over global time-steps in order to concentrate only on how and why concordance is achieved.
We recall that Sedov's problem with thermal energy input has already been numerically solved with SPH \citep[e.g.][]{Springel2002,Tasker2008},
reaching the desired integration accuracy by limiting the particle maximum time-step or evolving the system on small, constant time-steps.
\subsection{Simulation setup}
\label{subsec:globalsedovsetup}
\bigskip
We generate glass-like initial conditions \citep{White1996} with $N=128^3$ gas particles by evolving a random distribution of particles with inverted gravitational force sign.
We further relax the glass particle distribution by running it with hydro forces only, in order to smooth the pressure gradients that may remain due to the small density fluctuations.
In the \textit{natural} system of units the size of the box is $L=1$. In order to obtain a uniform background density, $\rho_0=1$, the gas particle mass is set to $m_g=1/N$.
We assign to each particle the internal energy $u_0=10^{-3}$ so that the late evolution of the explosion is not affected by the energy of the medium
that the expanding shell accumulates with time.
The total energy $E_{\star}$ is injected in a small region in the centre of the volume either in thermal or kinetic form.
In the simulations presented in this section, we heat/kick the $n_{\rm [h,k]}=32$ particles within a sphere of radius $r_0$
(we refer to heated and kicked particles with the subscripts `h' and `k', respectively).
The energy is distributed using the SPH kernel weight normalised to unity, and with kernel size $r_0$
Particle $i$ at distance $r_i$ from the centre of injection receives the energy per unit mass,
\begin{equation}
u_i = \frac{w(r_i,r_0)}{\sum_{j=1}^{n_{\rm [h,k]}} w(r_j,r_0)}\,\frac{E_\star}{m_{\rm g}}\,,
\end{equation}
where $m_{\rm g}$ is the mass of the particle, and $w(r_i,r_0)$ is the kernel weight at distance $r_i$.
To conserve energy, the kernel weight is normalised to the sum of the weights over all heated/kicked particles.
If the particle is heated, its internal energy increases by $u_i$. If
the particle is kicked, its velocity is increased by $v_i=\sqrt{2
u_i}$, and the change of kinetic energy in this case equals the
change of thermal energy in the other. The momentum kick is radial,
along the direction from the centre of the explosion to the particle.
For a given number of heated/kicked
particles, because the injected energy per unit mass is
$u_\star\propto E_\star/m_g\propto N$, the mass resolution
defines the energy jump. Our choice of
initial conditions leads to an energy jump of the order of
$\sim10^9$ with respect to the medium initial energy and for the reference value $E_\star=1$.
For given values of $E_\star$ and
$\rho_0$, and assuming monatomic gas with a ratio of specific heats $\gamma=5/3$, the analytic solution for the time evolution
of the shock-front radius is given by \citep{Sedov1959}:
\begin{equation}
R(t) \simeq 1.1527\left(\frac{E_\star}{\rho_0}\right)^{1/5}t^{2/5}\,.
\label{eq:sedovblast}
\end{equation}
Regarding the time integration, we employ in this section global, adaptive time-steps.
At each simulation step, the whole system is advanced in time by integrating over the minimum time-step of all particles.
To define the time-steps, we use the standard Courant parameter, $C=0.15$, and the value of the time accuracy coefficient $\eta=0.0025$
in equations \ref{eq:courdt} and \ref{eq:accdtnew} presented in Appendix~\ref{app:dtcriteria}.
Regarding the accuracy parameter $\eta$, we performed accuracy and efficiency tests with different values,
and chose $\eta=0.0025$ as fiducial parameter for this study. For a comparison of those tests,
the reader should refer to Sec.~\ref{sec:sedov} and \ref{sec:evrard}.
Finally, in order to define the SPH kernel, we set the number of neighbours to $32 \pm 2$ particles.
The list of runs is given in Table~\ref{table:concordstat}.
In the reference model (highlighted in bold) we adopt the values $E_{\star}=1$ and $\alpha=2$ for the injected energy and the artificial viscosity coefficient, respectively.
We also run a sample of simulations varying $E_{\star}$ and $\alpha$.
Beside the numerical parameters of the tests, we also list in Table~\ref{table:concordstat} a measure of the conservation of the injected energy at the end of the runs:
\begin{equation}
\frac{\Delta E_{\star}}{E_{\star}} = \frac{(E_{+}-E_{0})-E_{\star}}{E_{\star}}\,,
\label{eq:energyconservation}
\end{equation}
where $E_{+} = E_{\rm t} + E_{\rm k}$ is the time evolution of the sum of the thermal and kinetic energies in the simulated volume
and $E_0$ is the initial, total energy of the system (excluding the input energy).
We also list the total simulation running time on 8 cores.
\subsection{Simulation results}
\label{subsec:globalsedovresults}
\bigskip
\begin{figure}
\begin{center}
\hspace{-2mm}\includegraphics[trim = 0mm 2mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig02a.ps}
\hspace{-1mm}\includegraphics[trim = 0mm 2mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig02b.ps} \\
\hspace{-2mm}\includegraphics[trim = 0mm 4mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig02c.ps}
\hspace{-1mm}\includegraphics[trim = 0mm 4mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig02d.ps}
\end{center}
\caption{Sedov's explosion test for the reference simulations, where the system is evolved on global time-steps:
energy is injected either in thermal (left column) or kinetic (right column) form.
Results are given at $t=0.02$ after the explosion.
We see that both methods agree remarkably well in reproducing the similarity solution for both the location and the width of the blast shell.
\label{fig:concordprofile}}
\end{figure}
We first describe the reference model by showing in Fig.~\ref{fig:concordprofile} the blast wave behaviour at time $t=0.02$.
In the upper row, the projected gas density of a slice of thickness $\Delta z=0.1$ centred on the box
origin is plotted for the thermal and kinetic feedback runs.
The white dots mark the position of all particles that initially received the energy.
The dashed circle corresponds to the position of the spherical shock-front given by the analytic solution in Eq.~\ref{eq:sedovblast} at that time.
In the lower row, the simulated, average radial density profiles (black lines) are compared against Sedov's similarity solution (red lines).
The agreement with the analytic solution is very good for the two feedback schemes, for both the location and the width of the blast shell.
As shown by other authors, the lack of resolution and the SPH smoothing do give a density contrast at the shell radius that is smaller than the theoretical prediction
\citep[see for instance a comparative study for AMR and SPH codes by][]{Tasker2008}.
Nevertheless, we see on the radial profiles that our resolution is sufficient enough to describe correctly the asymmetry of the shock front in both cases.
As expected from a global time-step integration, the computation time is large, ranging from $\sim3$ to $\sim4$ hours.
With both feedback schemes, the accuracy of energy conservation arises because at each simulation step the entire system is integrated.
Therefore, all particles are aware of the hydrodynamical state of their neighbours.
We will develop this argument in more detail in Sec.~\ref{sec:individual}.
Here, we focus on how and why there is agreement between the two schemes.
It is somehow expected that both methods give the same results.
Indeed, Sedov's initial conditions are the total energy of the explosion and the medium density.
The only requirement is that a large amount of energy is instantaneously injected in a small volume,%
\footnote{It has been proved that injecting energy in a region of the size of the spatial resolution gives an accurate treatment of the problem
\protect\citep[e.g.][]{Tasker2008}.} but its form is not specified \citep{Landau1959}.
One can input either all thermal, all kinetic or a combination of both forms and obtain the same similarity solution at any given time.
Since the hydrodynamics conservation laws are used to derive the solution, a property of the similarity solution is that
the fractions of thermal and kinetic energies of the blast are constant in time.
However, in the numerical integration, a finite time is required to convert one form of energy to the other and reach the energy budget
given by the analytic solution (see illustration in Fig.~\ref{fig:concordratio}).
As long as momentum can be converted into thermal energy by physical processes like e.g. viscosity,%
\footnote{In SPH simulations kinetic energy is converted to thermal energy by the numerical, artificial viscosity scheme \citep{Monaghan1983,Balsara1995}.}
the numerical integration of the conservation laws should reach the similarity solution.
We show in Fig.~\ref{fig:concordenergy} the time evolution of the energy conservation relative error
(given by Eq.~\ref{eq:energyconservation}) for all test simulations,
where solid and dashed lines refer to the thermal and kinetic energy injection methods, respectively.
Lines of the same colour are for simulations with the same numerical parameters. Energy injection happens at time $t=0$.
We first concentrate on the reference simulation (black lines).
We show in the plot that the violation of energy conservation happens in the early stage of the explosion ($t\le10^{-2}$), when the energy contrast is the largest.
In the thermal case (solid line), there is initially a jump of $\sim0.15$\% around $t=10^{-5}$.
The conversion of energy into momentum happens very quickly, and after $t=4\times10^{-4}$ the evolution follows the kinetic case with an offset slowly decreasing.
At later times, both curves flatten to roughly $0.8$\%.
The same behaviour can be seen in the energy variation tests (blue and green lines), where the input energy is decreased by a factor of 10 and 100, respectively.
Decreasing the input energy slows the blast evolution and gives the time offset seen the plot. With constant $\alpha$, varying the input energy gives
similar relative errors, providing an estimate that is independent of the energy value.
Moreover, energy conservation is achieved at a comparable level for both thermal and kinetic feedback.
We show in Fig.~\ref{fig:concordenergy} that the choice of the artificial viscosity parameter plays an important role in the input energy conservation.
We compare three models with $\alpha=1$ (red), $2$ (black) and $4$ (orange).
The simulations with $\alpha=1$ keep an excess of momentum for longer time
(see discussion of Fig.~\ref{fig:concordratio} below), which gives a larger energy conservation error.
On the other hand, simulations with higher $\alpha$ flatten to a smaller energy violation value (below 1\%) at earlier times.
We argue that the error accumulates during the conversion from kinetic to thermal energy in the shock front.
This indicates that a higher artificial viscosity preserves a better accuracy in shocks.
\begin{figure}
\begin{center}
\hspace{-4mm}\includegraphics[trim = 0mm 4mm 0mm 4mm, clip, width=0.46\textwidth]{./Fig03.ps}
\end{center}
\caption{Time evolution of the input energy conservation for the Sedov's blast wave tests with global time-step integration.
Solid lines show the results for the thermal injection of energy. Dashed lines represent the evolution of the kinetic feedback runs.
The colour coding is defined by the properties of the runs.
\label{fig:concordenergy}}
\end{figure}
While the evolution of the input energy relative error informs us on the accuracy of the simulations,
additional information is needed to understand the behaviour of the blast.
The time evolution of the conversion of one form of energy to the other following the explosion is described in Fig.~\ref{fig:concordratio}.
In this figure, we show an estimate of the energy partition as $(E_{-}-E_{0-})/(E_{+}-E_{0+})$, where
$E_{-}=E_{\rm t}-E_{\rm k}$ and $E_{+}=E_{\rm t}+E_{\rm k}$
are the time evolution of the difference and the sum of the thermal and kinetic budget of the whole system, respectively.
In order to follow only the conversion of the input energy $E_{\star}$, we thus remove the initial (before injection) values $E_{0-}$ and $E_{0+}$.
If $E_{\star}$ is in thermal form (solid lines), the ratio is unity at injection time $t=0$;
if $E_{\star}$ is in kinetic form (dashed lines) the ratio is initially $-1$.
We compare the results with the energy partition expected value given by Sedov's similarity solution.
At a given time, we calculate the radial profiles of density, velocity and pressure,
and integrate them numerically to obtain the total thermal and kinetic energy within the blast radius.%
\footnote{Note that in Sedov's solution the total injected energy is within the blast radius.}
The integration gives 71.7\% of the blast energy in thermal and 28.3\% in kinetic form \citep[see][for the same result]{Chevalier1974}.
We plot the analytic ratio (dotted line) and find agreement with our results.
\begin{figure}
\begin{center}
\hspace{-4mm}\includegraphics[trim = 0mm 4mm 0mm 4mm, clip, width=0.46\textwidth]{./Fig04.ps}
\end{center}
\caption{Time evolution of the energy partition for the Sedov's blast wave tests with global time-step integration.
$E_{-}$ and $E_{+}$ are the time evolution of the difference and the sum of
the thermal and kinetic budget of the whole system, respectively.
Quantities with subscript `0' correspond to the initial values, before energy injection.
Solid lines show the results for the thermal injection of energy.
Dashed lines represent the evolution of the kinetic feedback runs.
The colour coding is as in Fig.~{\protect\ref{fig:concordenergy}}.
\label{fig:concordratio}}
\end{figure}
Indeed, the evolution shows that energy of one form is quickly converted into the other, and that the ratio tends to the same energy partition limit value.
However, as already mentioned, Fig.~\ref{fig:concordratio} shows clearly that a finite time is needed
in order for the simulations to converge to the similarity solution.
We see from the results of the different setups that this convergence time depends on both
the physical properties of the explosion and the numerical parameters of the simulation.
This property will be studied in more detail in an upcoming work.
We will especially investigate the effect of radiative cooling processes and compare the numerical convergence to the timescale
at which the adiabatic Sedov's blast evolves to the radiative snowplow phase.
Nevertheless, since decreasing the input energy makes the pressure gradient shallower (thermal feedback) and the viscosity term smaller (kinetic feedback)
for a given artificial viscosity parameter, the offset seen in the time evolution can already be explained by the lower conversion rates.
However, we see from Fig.~\ref{fig:concordratio} that changing the conversion rate clearly modifies the convergence process.
With smaller $\alpha$, viscous forces are reduced and more momentum is created in the thermal feedback case.
Conversely, less momentum is dissipated in the kinetic case, even though this excess of momentum from both feedback methods tends,
slowly with time, to the same expected limit.
On the opposite, a higher conversion rate ($\alpha=4$) produces an excess of thermal energy, creating too few (thermal case)
or converting too much (kinetic case) momentum. With this higher $\alpha$ the convergence to Sedov's energy partition happens again slowly with time.
We confirm here that the stability of the numerical scheme depends on the artificial viscosity normalisation.%
\footnote{This supports our claim that the error builds up mostly in the viscous term.}
We also demonstrate that the value of $\alpha=2$ reproduces best Sedov's test through a faster numerical convergence.
This is in agreement with the result of \cite{Monaghan1992} for the choice of $\alpha$ when considering shock fronts.
However, we cannot yet advise for this somewhat higher value than typically used in cosmological simulations since,
in the artificial viscosity formalism of \textsc{gadget}-2{}, the $\alpha$ parameter does not differentiate shocks from shear flows.
In the latter, a too high conversion rate would affect dramatically the velocity field at the contact surface.
In this specific matter of the artificial viscosity in SPH simulations, we assess that more work is required before obtaining a more general criterion.
\section{Concordance under efficient time integration}
\label{sec:individual}
\bigskip
\begin{table*}
\begin{center}
\begin{tabular}{ l | l | c | c | c | cc | cc }
\hline
\hline
\multirow{2}{*}{Name} & \multirow{2}{*}{Scheme} & \multirow{2}{*}{$f_{\rm step}$} & \multirow{2}{*}{$\eta$} & \multirow{2}{*}{$n_{\rm [h,k]}$} &
\multicolumn{2}{c}{ Energy (\%) } & \multicolumn{2}{c}{ Time (min) } \\
& & & & & Thermal & Kinetic & Thermal & Kinetic \\
\hline\hline
{\bf SEDOV-G-D} & {\bf Global - Update} & {\bf --} & {\bf 0.0025} & {\bf 32} & {\bf 0.93} & {\bf 1.35} & {\bf 139} & {\bf 101} \\
\hline
SEDOV-L2-NU & Limiter - No Update & 2 & 0.0025 & 32 & $4.0\times 10^4$ & $43.46$ & 119 & 35.4 \\
\hline
SEDOV-L2-U & Limiter - Update & 2 & 0.0025 & 32 & 2.37 & 2.19 & 24.6 & 20.8 \\
{\bf SEDOV-L4-U} & {\bf Limiter - Update} & {\bf 4} & {\bf 0.0025} & {\bf 32} & {\bf 3.03} & {\bf 2.86} & {\bf 21.8} & {\bf 18.2} \\
SEDOV-L8-U & Limiter - Update & 8 & 0.0025 & 32 & 4.85 & 5.28 & 21.0 & 17.7 \\
\hline
SEDOV-L4-U-$\eta$ & Limiter - Update & 4 & 0.025 & 32 & 3.90 & 4.28 & 21.0 & 16.8 \\
SEDOV-L4-U-N$4$ & Limiter - Update & 4 & 0.0025 & 4 & 4.12 & 4.33 & 26.4 & 25.7 \\
\hline
\end{tabular}
\end{center}
\caption{Energy conservation estimate and computational time for Sedov's blast wave tests with a limiter time-step scheme
are compared to a reference simulation using global time-steps, when the injection of energy is delayed (D).
The reference simulations, including our recommended set of parameters when using the limiter, are highlighted in bold characters.
Names are given to each simulations according to the integration scheme used and when numerical parameters differ from the run of reference.
Energy conservation estimates (in percent) are given at the end of the runs, at time $t=0.04$ after energy injection. The wall-clock time (in minutes) is for runs on 8 processors.
\label{table:sedovstat}}
\end{table*}
In simulations with a large dynamical range (e.g. astrophysical problems of galaxy and structure formation),
the required time scales can span several orders of magnitude.
In state-of-the-art cosmological simulations, up to several billions of particles are integrated over time.
The use of constant and global time integration steps is then prohibitive in terms of computational costs.
The individual time-step integration scheme \citep[first introduced by][]{Aarseth1963,Makino1991}
allows particles to be integrated on time-steps which are functions of the local state.
SM09 demonstrated that an accurate description of feedback processes which involve strong energy perturbations
can only be achieved by ensuring fast information transfer when using individual time-steps.
They proposed an innovative scheme in which the time-step length of neighbouring gas particles is constrained by a limiter.
SM09 validated their time-step limiter algorithm with Sedov's blast wave test.
They also tested the effect of the time-step limiter with a SN-like explosion in a self-gravitating halo of cold gas.
They showed that the conservation of energy and linear momentum agree remarkably well
with those obtained with a more conservative (but computationally more expensive) global time-step scheme.
However, SM09 used initial conditions that included the explosion energy.
This leads to setting the time-step of heated/kicked particles and limiting the time-step of their neighbours at the start of the simulation.
Therefore, the integration was correctly performed over the appropriate time-step from the beginning.
In the most general case, e.g. in cosmological simulations where the feedback events from SN or AGN activity do not affect only active particles,
this would not happen.
We specifically design our tests to consider the energy input after the simulation has been running for several steps.
This prevents any time-step adjustment before the integration of the dynamical and hydrodynamical equations is performed.
We also state here that we do not limit the maximum size of the time-step.
This setup is justified by the aim of describing the more general case of energetic feedback,
where individual time-steps reflect only the hydrodynamical state of particles.
In order to study an asymmetric problem, we modify the case of the self-gravitating gas halo by off-setting the explosion.
In the following sections, we briefly present our method to maintain a high accuracy
when considering strong feedback events in simulations using an individual time-step scheme.
The reader interested in the implementation of this method should refer to the appendices for a detailed description.
The results of the two sets of test simulations are then presented, focusing on the conditions needed to achieve the concordance between feedback methods.
\subsection{Individual time-stepping for feedback}
\label{subsec:individualscheme}
\bigskip
In the context of the hierarchical time-stepping scheme presented in Appendix~\ref{app:leapfrog},
it is important to note that if the energy content of a gas particle is suddenly increased, the particle itself will be informed as soon as it becomes active.
Within a few active time-steps, a fraction of the energy is efficiently converted from one form to the other (thermal to kinetic or \textit{vice versa}).
If thermal energy is injected, it will be converted into momentum through strong hydro accelerations due to the large pressure gradient.
If heated particles and their neighbours do not adjust their time-steps, the integration would lead to very large velocities and an artificial excess of kinetic energy.
In the other case, the injected kinetic energy is converted into thermal energy through shocks.
Eventually, the integration over a long time-step would lead to an excess of thermal energy, again violating energy conservation.
In both scenarios, the violation of energy conservation is due to the fact that the particles do not react soon enough to the sudden change of their hydrodynamical state.
Even if the particle becomes active at the proper time, if the new energy state is not taken into account in defining the length of the next step,
the integration following the energy increase will also be done over a too long time-step, leading again to non-conservation of energy.
In any case, it is therefore crucial to capture the initial stage of energy injection.
We emphasise here the problems that one may encounter when implementing feedback modules in SPH codes.
We show in Appendix~\ref{app:dtcriteria} that the signal velocity and the acceleration are the key information needed to define the time-step.
In the public release of \textsc{gadget}-2{} both quantities are calculated during the computation of the local hydrodynamical forces.
Consequently, any later change of the energetics of an active particle would not be taken into account during the calculation of the next time-step.
On the other hand, injecting the energy before the hydro calculation would make the current time-step inconsistent with the new hydrodynamical state of the particle.
If the current time-step is overestimated, the integration could lead to unphysically too strong feedback effects.
We also want to warn the reader against any feedback implementation that would inject the energy as a rate to be applied over a given length of time.
Modifying either the rate of entropy change \text{(thermal fedback)} or the hydrodynamical acceleration \text{(kinetic fedback)} would not conserve the input energy
if the kick operator (from the leap-frog integration scheme) is applied on two successive steps of different length.
Considering the problems mentioned above, we recommend injecting instantaneously the feedback energy just before the computation of the next time-step,
whilst, taking into account the following two actions to preserve the accuracy of the time integration.
Firstly, to ensure the fast information transfer of the change of energy in the medium, we use the time-step limiter proposed by SM09.
In Appendix~\ref{app:timelimiter} we describe an implementation of this limiter in \textsc{gadget}-2{} which preserve the time-step synchronisation.
Secondly, to ensure that the computation of the time-step following the explosion takes properly into account the local hydrodynamical change,
we impose that the particles which receive the energy update their signal velocity and hydrodynamical acceleration.
Doing so, we impose for these particles an update of the computation of their next time-step
that will lead to an update of their hydrodynamical properties right after the explosion time
(see Appendix~\ref{app:timeupdate} for the detail about this time-step update).
We will now show that these actions are essential to be considered altogether in order to preserve the concordance of feedback methods when using individual time-steps.
\subsection{Sedov's blast wave test}
\label{sec:sedov}
In this section we describe the Sedov's blast wave tests performed with an individual time-step scheme.
Before showing the results of our simulations, we mention the differences with the setup used in Sec.~\ref{subsec:globalsedovsetup}.
\subsubsection{Simulation setup}
\label{subsec:individualsedovsetup}
\smallskip
Starting with the same glass-like uniform conditions presented in Sec.~\ref{subsec:globalsedovsetup},
all particles are evolved over global, background time-steps that are of the order of $\Delta t_{\rm back} \sim 10^{-3}$.
Since all particles are synchronised from the beginning of the simulation, the issues about an explosion occurring in the middle of active steps
or about neighbouring particles being initially on different time-bin levels, will not be addressed in this section.
We refer to Sec.~\ref{sec:evrard} for an analysis of these effects on the long-term evolution of the medium.
To avoid a pre-defined population of the time-bin levels in these Sedov's test simulations,
the injection of a total energy of $E_\star=1$ (either in thermal or kinetic form), is delayed by a few of the background steps.
In order to focus on feedback processes similar to SN explosions or BH activity,
we have set the total initial internal energy of the system to $E_0=1$. Given the resolution $N=128^3$ of the simulations,
the initial energy contrast between the heated/kicked particles and the background ones is of the order of $\sim 10^6$.
All tests presented in this section have been run until $t=0.04$ after the explosion time.
It is interesting to remind here that the amount of injected energy constrains the time-steps that follow the explosion.
Since the signal velocity \citep[see][]{Monaghan1997} at the explosion location is related to the energy injection as $v_{\rm sig} \propto \sqrt{u_\star}$,
we can already anticipate through the Courant criterion that, for the energy contrast considered here,
the step of the heated/kicked particles will be $\sim 10^3$ times smaller than the background step.
This will correspond to an abrupt drop of about 10 levels in the hierarchy of time-bins.
To analyse the behaviour of our time-step scheme, we use a set of simulations (see Table~\ref{table:sedovstat})
where combinations of the integration techniques and numerical parameters are investigated.
Here we use again a simulation with a global time-stepping scheme as reference.
Then, we test the limiter technique without the time-step update.
Finally, we enforce the time-step update, as describe in Appendix~\ref{app:timeupdate}.
With the latter setup, we estimate the impact of the limiter parameter $f_{\rm step}$,
as well as the time integration efficiency parameter $\eta$ (from Eq.~\ref{eq:accdtnew}), for both the thermal and kinetic feedback methods.
With the aim of studying two different initial energy distributions, we also compare simulations
where the energy is injected over a different number of particles.
In order to quantify the accuracy of the different methods,
we estimate the input energy conservation error at $t=0.04$ for each simulation using Eq.~\ref{eq:energyconservation}.
To estimate the performances of the different time-step schemes, we also list the total running time of each test in Table~\ref{table:sedovstat}.
\begin{figure}
\begin{center}
\hspace{-2mm}\includegraphics[trim = 0mm 2mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig05a.ps}
\hspace{-1mm}\includegraphics[trim = 0mm 2mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig05b.ps} \\
\hspace{-2mm}\includegraphics[trim = 0mm 4mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig05c.ps}
\hspace{-1mm}\includegraphics[trim = 0mm 4mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig05d.ps}
\end{center}
\caption{Sedov's explosion test for thermal (left column) and kinetic (right column) energy injection,
when the limiter is applied without the time-step update. Results are given soon after the explosion at time $t=0.004$.
We see that for the thermal implementation of feedback a stable shell has developed far ahead from the similarity solution
because the large violation of energy conservation largely increases the system total energy (see Table~\ref{table:sedovstat}).
In the kinetic feedback case inter-particle crossing is evident.
No density contrast has yet developed given that kicked particles have traveled far from the explosion site.
These results illustrate how the lack of a prompt response of the medium leads to two distinct effects from the two feedback schemes.
\textit{Time animation available as supporting online material.}
\label{fig:sedovnocorrect}}
\end{figure}
\begin{figure}
\begin{center}
\hspace{-2mm}\includegraphics[trim = 0mm 2mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig06a.ps}
\hspace{-1mm}\includegraphics[trim = 0mm 2mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig06b.ps} \\
\hspace{-2mm}\includegraphics[trim = 0mm 4mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig06c.ps}
\hspace{-1mm}\includegraphics[trim = 0mm 4mm 2mm 4mm, clip, width=0.24\textwidth]{./Fig06d.ps}
\end{center}
\caption{Sedov's explosion test for thermal (left column) and kinetic (right column) energy injection,
when both the limiter and the time-step update are applied. Compared with the previous figure, results are shown at later time $t=0.02$.
We show here that, for our recommended set of parameters, the two feedback schemes are able to reproduce precisely the similarity solution
and hence provide again concordant results as in Fig.~\ref{fig:concordprofile}.
This demonstrate that both a fast information transport and a prompt response to the explosion are needed in order to resolve strong feedback events
with individual time-step integration.
\textit{Time animation available as supporting online material.}
\label{fig:sedovcorrect}}
\end{figure}
\subsubsection{Simulation results}
\label{subsec:sedovresults}
\smallskip
We present in Fig.~\ref{fig:sedovnocorrect} the early state of the medium at $t=0.004$, for both thermal (left column) and kinetic (right column) energy injection,
when using the time-step limiter without any update at the explosion time.
This corresponds to the time integration that has been illustrated in the appendices in Fig.~\ref{fig:timelines}-b).
For those runs, we have chosen the conservative value of $f_{\rm step}=2$ in order to ensure the fastest information propagation between neighbouring particles.
It is striking to see how the two feedback methods give extremely deviant but very different results.
In the thermal case, heated particles are found close to the analytical blast radius but since the first step after the explosion is much too long,
the acceleration given to the surrounding medium produces an important excess of kinetic energy.
Eventually, the propagation of this excess of energy will lead to a violation of energy conservation to a level of $\sim 40,000\:\%$ at the end of the run.
For the kinetic injection of energy, we see that the isotropic nature of the blast is destroyed.
Indeed, the lack of an immediate response of the medium makes kicked-particles cross their neighbours
before they partially thermalise their momentum. Later on, they will eventually start to expand individual bubbles around them.
Even with the use of the limiter, some particles are able to travel up to the border of the volume before interacting with the medium.
Here, the energy creation reaches a level of $\sim 43\:\%$.
This two pictures clearly show the importance of properly computing the time-step following the energy injection event.
In Fig.~\ref{fig:sedovcorrect}, we show the results for the simulations in which we additionally enforce the time-step update.
Here we use the fiducial value of $f_{\rm step}=4$. The shell position and the radial density profile are in extremely good agreement with the analytic solution.
Sedov's solution is remarkably well represented for both feedback methods, reproducing again the concordance
with a conservation of input energy close to $3\,\%$ at the end of the runs.
\begin{table*}
\begin{center}
\begin{tabular}{ l | l | c | c | c | cc | cc }
\hline\hline
\multirow{2}{*}{Name} & \multirow{2}{*}{Scheme} & \multirow{2}{*}{$f_{\rm step}$} & \multirow{2}{*}{$\eta$} & \multirow{2}{*}{$n_{\rm [h,k]}$} &
\multicolumn{2}{c}{ Energy (\%) } & \multicolumn{2}{c}{ Time (h) } \\
& & & & & Thermal & Kinetic & Thermal & Kinetic \\
\hline
\hline
{\bf HALO-G-D} & {\bf Global - Update} & {\bf --} & {\bf 0.0025} & {\bf 32} & {\bf 1.62} & {\bf 1.84} & {\bf 25.70} & {\bf 25.87} \\
\hline
HALO-I-U & Individual - Update & -- & 0.0025 & 32 & $5.7\times 10^4$ & $1.7\times 10^5$ & 14.20 & 19.73 \\
HALO-L2-NU & Limiter - No Update & 2 & 0.0025 & 32 & $3.0\times 10^3$ & 19.42 & 6.07 & 3.73 \\
\hline
HALO-L2-U & Limiter - Update & 2 & 0.0025 & 32 & 2.01 & 1.27 & 3.63 & 3.60 \\
{\bf HALO-L4-U} & {\bf Limiter - Update} & {\bf 4} & {\bf 0.0025} & {\bf 32} & {\bf 2.16} & {\bf 2.60} & {\bf 3.41} & {\bf 3.39} \\
HALO-L8-U & Limiter - Update & 8 & 0.0025 & 32 & 2.27 & 3.26 & 3.35 & 3.34 \\
\hline
HALO-L4-U-$\eta$ & Limiter - Update & 4 & 0.025 & 32 & 4.05 & 5.51 & 2.40 & 2.38 \\
HALO-L4-U-N$1$ & Limiter - Update & 4 & 0.0025 & 1 & 2.60 & 0.95 & 3.44 & 3.47 \\
\hline
\end{tabular}
\end{center}
\caption{Energy conservation estimate and computational time for the off-centre explosion tests, where the energy injection is delayed (D) after the start of the simulation.
The reference simulations, including our recommended set of parameters when using the limiter, are highlighted in bold characters.
Names are given to each simulations according to the integration scheme used and when numerical parameters differ from the run of reference.
Energy conservation estimates (in percent) are given at the end of the runs, at time $t=0.04$ after energy injection. The wall-clock time (in hours) is for runs on 8 processors.
\label{table:evrardstat}}
\end{table*}
Regarding the parameter study shown in Table~\ref{table:sedovstat}, we can see that increasing the value of $f_{\rm step}$ slowly enhances the energy violation,
while keeping the computational time nearly constant. Using a less conservative accuracy parameter $\eta=0.025$ increases also the energy creation
but both thermal and kinetic simulations are completed faster.
We note that even distributing the energy over a smaller number of neighbours, which provides a larger energy contrast, gives an acceptable energy conservation.
\begin{figure}
\begin{center}
\hspace{-4mm}\includegraphics[trim = 0mm 4mm 0mm 4mm, clip, width=0.46\textwidth]{./Fig07.ps}
\end{center}
\caption{Time evolution of the energy conservation for the most relevant Sedov's blast wave tests.
Solid lines show the results for the thermal injection of energy.
Dashed lines represent the evolution of the kinetic feedback runs.
\label{fig:sedovenergy}}
\end{figure}
In Fig.~\ref{fig:sedovenergy}, the time evolution of the energy
conservation for the runs using the fiducial value of $f_{\rm step}=4$
is compared with the reference simulations using the global time-step
scheme. We notice that the ratio of energy increase between matching
models is fairly constant. This behaviour tells us that the
convergence between the two feedback methods is achieved since
the very beginning of the expansion of the blast. Moreover, we see in
this figure that all simulations experience first a sharp increase
of the energy creation before reaching a stable level of energy
conservation. Indeed, the rate of energy violation is high right
after the explosion (when the energy needs to be absorbed by the
surrounding of the injection region) but as soon as the blast becomes
stable and starts to expand, the propagation of the error becomes very
small. This happens approximately for $t>0.004$. We also confirm from
this analysis that the time accuracy parameter $\eta$ needs to be
chosen small enough in order to get the best convergence with the
feedback methods. Finally, regarding the spatial energy distribution,
it is clear that concentrating the injection over a smaller number of
particles gives a higher energy jump (meaning a larger time-level
drop), and hence produces a slightly worse energy conservation.
We conclude that, when applying the proposed integration scheme,
there are no dramatic differences either in concordance, or in energy conservation accuracy,
or in performances for the set of parameters we explored.
\subsection{A more realistic test}
\label{sec:evrard}
\bigskip
In this section we simulate an explosion event in a self-gravitating
gas halo. The initial conditions are similar to the ones in SM09.
However, the injected energy is placed off-centre to follow the blast
evolution in the presence of pressure and density gradients. In
cosmological simulations of structure formation, the sources of
mechanical and thermal feedback (SNe and BHs) are generally situated
in a similar environment. Though idealised, the setup gives a
qualitative and quantitative view of the above scenario.
\subsubsection{Simulation setup}
\label{subsec:evrardsetup}
\smallskip
We create a spherical particle distribution of density profile
$\rho\propto r^{-2}$ by spatially remapping a uniform, spherical
distribution of particles through the radial transformation
\begin{equation}
r_{{\rm old},i}\rightarrow r_{{\rm new},i}=\left(\frac{r_{{\rm old},i}}{R}\right)^3 R\,,
\end{equation}
where $R=1$ in the same system of units used for Sedov's blast wave
problem. The sphere is cut out from a uniform distribution of
$N=128^3$ particles in a cubic volume of side $L=2$. The total mass
of the gas sphere is unity. We assign an internal energy of 0.05. The
system is evolved including self-gravity up to time $t=3$ when relaxation has already
taken place \citep{Evrard1988}. We chose the gravitational softening
$\epsilon=3.125\times 10^{-3}$ and the artificial viscosity parameter
$\alpha=1$.
We start the simulation tests using the relaxed gas halo as initial conditions, and inject either thermal or kinetic energy in a region centred on $[0.05,0,0]$.
The off-centre explosion is generated by enhancing the energy of particles by $u_{\rm [h,k]}$ as described in Sec.~\ref{subsec:globalsedovsetup}.
We avoid the first simulation step in which all particles are active, and delay the energy injection to the time $t_{\star}=10^{-4}$.
For the sake of simplicity, in the whole analysis, we refer to the explosion time as the initial time $t_0=t_{\star}$.
Any other time is relative to $t_0$. To remove another bias, we also ensure that at injection time all particles receiving energy are inactive.
Doing so, we can apply our time-stepping scheme to the most generic situation, but more importantly, put it also under the least favourable conditions.
\begin{figure}
\begin{center}
\hspace{-.35cm} \includegraphics[trim = 0mm 8mm 4mm 8mm, clip, width=0.245\textwidth]{./Fig08a.ps}
\includegraphics[trim = 8mm 8mm 4mm 8mm, clip, width=0.23\textwidth]{./Fig08b.ps}\\
\hspace{-.35cm} \includegraphics[trim = 0mm 0mm 4mm 8mm, clip, width=0.245\textwidth]{./Fig08c.ps}
\includegraphics[trim = 8mm 0mm 4mm 8mm, clip, width=0.23\textwidth]{./Fig08d.ps}
\end{center}
\caption{The off-centre explosion in a self-gravitating gas sphere for thermal (left column) and kinetic (right column) energy injection.
In all pictures we show the gas density at time $t=0.02$ in a slice of thickness $\Delta z=0.1$ centred on the box origin.
The white dots mark the position (at the same time) of all particles that received thermal or kinetic energy.
\textit{Top row:} only the time-step limiter is applied using the conservative value of $f{\rm step}=2$.
The energy violation is severe in the thermal case and the bubble have blown away a large fraction of the gas halo.
\textit{Bottom row:} only the time-step update is enforced. The behaviour is similarly wrong in both cases.
The halo atmosphere is disrupted and no expanding bubble forms.
\textit{Time animation available as supporting online material.}
\label{fig:evrwrong}}
\end{figure}
\begin{figure}
\begin{center}
\hspace{-.3cm} \includegraphics[trim = 0mm 8mm 4mm 8mm, clip, width=0.245\textwidth]{./Fig09a.ps}
\includegraphics[trim = 8mm 8mm 4mm 8mm, clip, width=0.23\textwidth]{./Fig09b.ps}\\
\hspace{-.3cm} \includegraphics[trim = 0mm 0mm 4mm 8mm, clip, width=0.245\textwidth]{./Fig09c.ps}
\includegraphics[trim = 8mm 0mm 4mm 8mm, clip, width=0.23\textwidth]{./Fig09d.ps}
\end{center}
\caption{The off-centre explosion in a self-gravitating gas sphere for thermal and kinetic energy injection when time-step limiter
and update are applied. The plots are as in Fig.~\protect\ref{fig:evrwrong}.
\textit{Top row:} the energy is injected on 32 particles in thermal (left panel) and kinetic (right panel) form.
Impacted particles are made active and the time-step of surrounding particles has been corrected.
The results are qualitatively identical, showing concordance of the two feedback methods.
\textit{Bottom row:} all available energy is injected into one particle.
The results excellently agree with each other and with the cases in which the energy is injected into 32 particles.
\textit{Time animation available as supporting online material.}
\label{fig:evrcorrect}}
\end{figure}
The mass-weighted background internal energy at the position of the
explosion is $u=1.75$, whereas the mass-weighted velocity is
$v=4.43\times 10^{-2}$. The average internal energy and velocity are
measured in a spherical shell of thickness 0.004 and average radius 0.05. In
the case of a single heated particle, the jump in energy would be
$\simeq 5\times 10^5$ times its initial value. If only one particle
is kicked, its momentum change would be $\simeq 3\times 10^4$ times
the background initial value.
We compute the energy conservation as the relative variation of the total injected energy as in Sec.~\ref{sec:concordance}, and including the potential energy.
The simulation parameters together with the energy conservation estimate and the running time are listed in Table~\ref{table:evrardstat}
where the reference simulations are highlighted in bold.
\subsubsection{Simulation results}
\label{subsec:evrardresults}
\smallskip
We show in Fig.~\ref{fig:evrwrong} the off-centre explosion in a self-gravitating gas sphere
for thermal (left column) and kinetic (right column) energy injection.
In all panels we show the projected gas density at time $t=0.02$ in a slice of thickness $\Delta z=0.1$ centred on the box centre.
The white dots mark the position (at the same time) of all particles that initially received the energy.
We first note the effect of the pressure and density gradients.
In all pictures, the heated/kicked particles have moved from the location of energy injection along the direction of decreasing gradients (x-axis).
In the upper panels, we see the result of correcting the feedback scheme by making impacted particles active.
Although heated/kicked particles are integrated accurately, their neighbours do not react soon enough.
Indeed, before neighbours are active again, heated particles accelerate along the decreasing pressure gradients,
and kicked particles build thermal energy through shocks.
Once the neighbours feel the energy perturbation, they receive large accelerations.
Integrating over a too long time-step gives them unrealistic momentum values,
and in a few steps they are ejected from the vicinity of the explosion.
Without the use of the limiter, the propagation of the effect from neighbour to neighbour
builds up energy to such levels that its conservation is violated by
$\sim 57,000\:\%$ and $\sim 170,000\:\%$ for thermal and kinetic energy injection, respectively.
It is interesting to see that both feedback implementations lead to the complete disruption of the halo atmosphere, which lost its initial spherical shape.
This confirms the results of SM09 about the problems encounter with the standard individual time-stepping scheme and shows the importance of ensuring a smooth transport of the information from particle to particle.
In the lower panels of Fig.~\ref{fig:evrwrong}, we show the tests using the time-step limiter but without the time-step update.
Though the limiter factor is set to the conservative value of two, the energy conservation is still largely violated.%
With thermal feedback (bottom-left panel), we recover the behaviour discussed in Sec.~\ref{subsec:individualsedovsetup}.
The blast wave expands to large radii because thermal energy is overproduced within the hot bubble.
Indeed, energy conservation is violated by $\sim3,000$\%, and the total thermal energy of the system is boosted to a factor of 20 or more the input one.
At the final time, half of the thermal energy has been converted into momentum, and the total kinetic energy is two thirds of the total.
Simulations of strong BH and SN feedback should be carefully checked against this behaviour.
If kinetic energy is injected (bottom-right panel), the picture is similar to that discussed in Sec.~\ref{subsec:individualsedovsetup}.
Inter-particle crossing is clearly recognisable in the picture.
The particles with highest velocities (the closest to the injection position) have crossed several smoothing lengths and shock-heated the gas farther away.
We note that within the blast wave several small bubbles are created, and dense gas is entrained, possibly decreasing the shell mass.
Though less severe than the thermal case, energy conservation is violated by $\simeq20\%$, which,
in cosmological simulations may lead to overestimating the effect of SN winds.
In summary, we showed that neither the update nor the limiter of impacted particles time-step are able by themselves
to conserve energy within a reasonable accuracy level.
We show in Fig.~\ref{fig:evrcorrect} the off-centre explosion test when both the time-step limiter and update are applied (all plots are as in Fig.~\ref{fig:evrwrong}).
We show in the upper panels the tests with $n_{\rm [h,k]}=32$, and in the lower ones the test with $n_{\rm [h,k]}=1$.
The agreement between the same model with different $n_{\rm [h,k]}$, and between different models, is striking.
The evolution of the hot bubble is very similar in all cases, expanding and buoyantly moving from the place of energy injection to larger radii.
The reference simulations with the global time-stepping integration show a conservation error $\lesssim2\%$,
whereas the reference simulations with individual time-steps ($f_{\rm step}=4$, $n_{\rm [h,k]}=32$) have conservation error $\lesssim3\%$.
We warn the reader that the gravitational force calculation is accurate at $\sim1\%$ level and could contribute to the errors listed in Table~\ref{table:evrardstat}.
Nonetheless, the lower panels of Fig.~\ref{fig:evrcorrect} show that concordance of thermal and kinetic feedback
is achieved even in the case of heating/kicking just one particle with
all the available energy.
The time evolution of the energy conservation relative error is plotted in Fig.~\ref{fig:evrardenergy}.
We plot some of the most relevant thermal (solid lines) and kinetic (dashed lines) feedback tests.
Each line colour refers to simulations that were run with the same numerical parameters.
The agreement between kinetic and thermal feedback is excellent for the global time-stepping scheme (black lines),
proving that a prompt response of the system to the energy perturbation is necessary to achieve concordance of the two methods.
Applying the proposed time-step scheme (red lines, reference model), the integration slightly looses accuracy,
but simulations get a considerable speed up by a factor of $\sim8$ (see Table~\ref{table:evrardstat}).
The agreement between the two feedback schemes is again excellent.
We show in blue the tests with $\eta=0.025$.
Given the lower time integration accuracy, energy violation
naturally reaches higher values, but the ratio of relative errors is similar to that of the reference tests.
We see that the dashed blue curve goes to higher values right after the explosion.
That shows that it is crucial to properly capture the conversion of momentum through
viscous forces at the very earliest time. At the final time, the relative error is 4 and $5.5\:\%$ for thermal and kinetic schemes, respectively.
Finally, simulations where only one particle is heated/kicked (green
lines) also show a good concordance of feedback methods. However,
injecting kinetic energy leads this time to a loss of total energy.
This can be explained because all particles close to the explosion are
inactive when energy is injected. Since all the energy is
concentrated, the kicked particle has the time to move for a few steps
while decelerated by viscous forces and before the surrounding medium
reacts. During this time, there is no energy transfer to neighbours which lead to the loss of energy.
This problem could be solved by making all neighbours active at the
same time of the impacted particles. We have tested this hypothesis,
and obtained again the concordance. In any case, energy is still
conserved with good accuracy.
\begin{figure}
\begin{center}
\hspace{-4mm}\includegraphics[trim = 0mm 4mm 0mm 4mm, clip, width=0.46\textwidth]{./Fig10.ps}
\end{center}
\caption{Time evolution of the energy conservation for the most relevant tests of the explosion in a gas halo.
Solid lines show the results for the thermal injection of energy.
Dashed lines represent the evolution of the kinetic feedback runs.
\label{fig:evrardenergy}}
\end{figure}
\bigskip
\section{Conclusions}
\label{sec:discuss}
\bigskip
In this work we investigated the concordance of thermal and kinetic feedback methods in the absence of cooling.
We numerically demonstrated that the two methods are equivalent using simulations of Sedov's explosion with a global time-step integration scheme.
We argued that this result is expected from the hypothesis of Sedov's problem: a large amount of energy is instantaneously injected in a small volume.
The blast physical properties are given by the integration of hydrodynamics conservation laws which also give the fraction of energy in thermal
(72\%) and kinetic (28\%) form.
We focused on the conservation of input energy which we achieved within 1\% in the reference simulation.
In test simulations with varying input energy, the energy conservation relative error flattens with time to roughly the same value,
and is independent of the energy for a constant artificial viscosity parameter.
Decreasing the artificial viscosity parameter results in a larger conservation error.
We also investigated the evolution of the thermal and kinetic fractions of the input energy.
We found excellent agreement with the expected values derived from Sedov's analytic solution.
Using the global time-step scheme in cosmological simulations is computationally expensive.
To reduce the computational costs most of the numerical codes use an adaptive, individual time-step scheme.
SM09 showed that to achieve good energy conservation of feedback energy a time-step limiter is necessary.
We implemented a similar limiter in \textsc{gadget}-2{} taking into account time-step synchronisation and hierarchy, and performed simulations of the Sedov's blast wave.
We showed that the limiter does not give accurate energy conservation if the explosion energy is injected at random time
and a maximum value of the time-step for ambient gas particles is not enforced.
We proposed a solution to the problem: the system must not only propagate the information rapidly but should also promptly react to the energy input.
That is done by updating the time-step of particles receiving energy accordingly to their new hydrodynamical state.
We tested the modified time integration scheme on the Sedov's blast wave, and obtained good energy conservation accuracy and concordance of methods.
We applied both the limiter and the time-step update to a more realistic case of an explosion in a self-gravitating gas halo,
and showed that concordance and accuracy are achieved also in the extreme case where all the available energy injected into one particle.
In the test simulations presented here, we did not consider radiative cooling,
but we are aware of the potential breaking of concordance when such processes become important.
Cooling being a non-linear physical process,
the amount of radiated energy before reaching energy partition will depend on the specific feedback scheme and the consequent gas temperature evolution.
In the thermal feedback case, the bubble temperature decreases by adiabatic expansion and conversion of thermal energy into momentum.
In the kinetic case, the temperature increases through shocks. It is thus important to reach the same energy configuration before any significant cooling loss.
As mentioned by \cite{DallaVecchia2008}, if the gas is heated to a temperature where the cooling time becomes longer than the local dynamical time,
radiative over-cooling may be prevented. These considerations will be addressed in more detail in an upcoming work.
\bigskip
We summarize this work as follows:
\smallskip
\begin{itemize}
\item concordance of thermal and feedback methods arises by accurate time integration and is expected from theoretical arguments;
\smallskip
\item keeping all other numerical parameters fixed, the maximum energy conservation relative error is roughly constant when varying the input energy, and at the same level with both feedback schemes;
\smallskip
\item high artificial viscosity coefficient enables a fast conversion of thermal energy to momentum,
and hence permit to numerically converge to a stable solution in a shorter time;
\smallskip
\item concordance and energy conservation can be achieved with a time-step limiter, provided that the system promptly reacts to the input of energy;
\smallskip
\item in cosmological simulations with strong kinetic and/or thermal feedback from SNe and BHs, one should take into account accurate energy conservation before inferring any results dependent on feedback processes.
\end{itemize}
\section*{Acknowledgements}
We thank the anonymous referee for his useful comments that helped clarifying technical parts of this paper.
We are grateful to Volker Springel for making \textsc{gadget}-2{} publicly available.
We acknowledge Takayuki Saitoh for the detailed description on his simulation setup.
The authors are grateful to S.~Khochfar, E.~Neistein and J.~Johnson for helping improving the writing of the paper.
We also thanks Tom Theuns and Joop Schaye for enlightening comments and discussions.
CDV is supported by the Marie Curie Reintegration Grant FP7-RG-256573.
All simulations were performed on the SFC cluster of the TMoX group at the Max Planck Rechenzentrum Garching.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,410 |
Q: Unicode character rendered at a different size in IE6 In a web application, I have to display a special unicode character, know as BLACK DIAMOND (U+25C6) (see here for more details). Here is a sample : ◆
The font defined for the page is Arial, with size 13px.
Surprisingly, the character is rendered with a bigger size in IE6 vs other browsers (FF, Chrome, ...).
Is there any reason of this weird behavior and what is the solution to avoid this ?
A: This is because the specified character is missing from the font you specified. So the browser looks for a suitable font to use for that character so it can still display it. Different browsers pick different fonts, so you'll see it a little bit differently on each.
There isn't much in general you can do to avoid this because missing fonts are very common on the web and thus you cannot really rely on any font to be present on the user's machine. You can try mitigating it, though:
*
*You can put the U+25C6 in a span that's styled to use a different but specific font that has the character (and which works well with your main font).
*Same as above, but distribute a web font (WOFF seems to be a reasonable choice nowadays) that contains the glyph. That way you have more control about what is displayed.
*Stay far, far away from specifying fallback fonts like Arial Unicode MS. Just don't use them at all.
*If you're just after the looks of U+25C6 and don't care about having it actually in text form you can use an image or a CSS hack.
A: After being puzzled for a while, I realized that browsers render Unicode characters with different fonts depending on the order of the characters. Here's an example using the N-Ary Union (U+22C3), (U+1D54A), and ℝ (U+211D):
<p>⋃</p>
<p>⋃</p>
<p>ℝ⋃</p>
<p>⋃ℝ</p>
On my Mac, Chrome renders the first paragraph with STIXGeneral, the second paragraph with Apple Symbols and STIXGeneral, the third paragraph with Menlo and Apple Symbols, and the fourth paragraph only with Apple Symbols. Firefox renders everything with STIXGeneral except the ℝ in the third paragraph, which it renders with Geneva.
(Chrome shows the rendered fonts at the bottom of the Computed tab when inspecting an element with the developer tools. Firefox has a Fonts tab when inspecting an element. I couldn't find anything similar for Safari, which is confirmed by this answer.)
As far as I can tell, this is a simple optimization: If a glyph exists in a font already loaded for a particular "text node", use this font. Otherwise, search for another font which can render this glyph. Interestingly, I observed the same behavior (large ⋃ after and small ⋃ before ) also in Visual Studio Code and Apple Pages.
This optimization has a subtle security implication: If you print out a document and black out some text, the rendering of the later characters can reveal information about the blacked-out text.
(I put "text node" in quotation marks because <span></span>⋃ leads to the same behavior while <span style="font-weight: bold;"></span>⋃ does not.)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,271 |
One day a little man was walking on the road when suddenly he saw a very pretty woman in a house having a bath, and his eyes litted up. The woman then saw him and pulled her curtains from unwards every day he went at the doorway and peeped on her untill the woman started noticing so she got an bottle urinated in it and threw it on him and from that day the little man learned a valuable lesson.
This story has been read 966 times. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,920 |
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using Accord.Statistics.Distributions.Fitting;
using Accord.Statistics.Distributions.Multivariate;
using Accord.Statistics.Models.Fields;
using Accord.Statistics.Models.Fields.Functions;
using Accord.Statistics.Models.Fields.Learning;
using Accord.Statistics.Models.Markov;
using Accord.Statistics.Models.Markov.Learning;
using Accord.Statistics.Models.Markov.Topology;
namespace ConsoleApplication1
{
class CTrainer
{
private HiddenMarkovClassifierLearning _Teacher;
public void func_train(int[][] InputSequences, int[] OutputClass, HiddenMarkovClassifier Classifier)
{
_Teacher = new HiddenMarkovClassifierLearning(Classifier,
modelIndex => new BaumWelchLearning(Classifier.Models[modelIndex])
{
Tolerance = 0.001,
Iterations = 0
});
double error = _Teacher.Run(InputSequences, OutputClass);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,377 |
A very friendly and scenic full hookup campground just ~6 miles from Monument Valley visitor center.
Very decent sites with just a few dings. There is a selection of full hookup back-in, pull-through and dry-camping/tent sites all basic, cleared, flat dirt with small "sitting area" containing picnic table and grill. Not a lot of separation or privacy, plus some sites are somewhat difficult to get into (our first site had a tree that made it impossible to get our rig in). On positive side you have very nice all-round views of red rocks plus there is a "slot" view of Monument Valley especially from the upper sites.
Decent facilities. Clean toilets and bathrooms with good water-pressure and temperature. Slight ding is that showers were a tad older and somewhat small/tight leaving not a lot of room for changing.
Good amenities. There is decent on-site store, pool (although it was closed while we were there) and laundry which was modern, but somewhat pricey ($3 per wash). On-site dump station and workable WiFi. The Campground also offers tours of the Valley and nearby Lodge has additional amenities incl, car-wash, gas station, propane and market.
Location is what you come here for. Gouldings is the only hookup campground next to Monument Valley, and for visiting the area it's a perfect place to stay. As an additional bonus there's an on-site short hike to a lovely arch (nice surprise) and lovely views all-around.
Great area for doggie. There's limited space at camp, but good space to walk right next to the campground and on the trail to the nearby arch.
BONUS ALERT: Red Rocks and distant views of Monument Valley from your campsite!
Summary: If you're coming to Monument Valley and you want full hookups Gouldings is the only deal in town. Considering it's a monopoly we were pleasantly surprised. The sites are basic dirt with not a lot of privacy or separation, but the views are lovely (red rocks all around and slot view of Monument Valley in the back-ground) and the owners super-friendly and responsive. As an additional bonus there's a lovely little hike to an arch right by camp, perfect for both humans and paws, on-site WiFi (a bonus where there's almost zero Verizon signal), plus the nearby Lodge has lots of additional amenities incl. movies of the area. The campground also offers tours of the Valley and is conveniently located only ~6 miles from the visitor center. Given our preferences we'd prefer to stay at the tribal campground which is dry-camping, cheaper ($10/night) and closer to the park (unfortunately it was closed while we were there…it's opening up again Dec 2013), but if you want full hookups I'd certainly recommend this park.
Extra Info: Very, very poor Verizon signal (1X only). On-site WiFi is workable. Dry-camping sites cost $26/night. Full hookups cost $45/night. Good Sam's and AAA discounts. On-site dump station.
View down the row from top. Our RV on right. Slot view of Monument Valley in background.
View from bottom of row towards top of campground.
"Aerial" view of main campground from arch hike. Our RV is on right side.
Great review, thanks. We have not yet ventured into Utah but really want to. That looks like a nice place but we better start saving up to stay there. Since I like hook-ups, I'd probably bite the bullet and go for it.
Just curious – do you not use the shower in your RV? Or are you just looking at the showers to do your reviews?
It's both. If we're in a spot without sewer hookups we'll always use the campground showers, but even when we have full hookups I'll go ahead and try them out for the campground review.
Elaine and I depart April 25 for a month in Costa Rica's rain forests.
I've got a Nepal trek planned for Spring 2014 should you be tempted.
Oh wow…I can't recall if I saw that video or I just relived it again. Wonderful! ENJOY your time in Costa Rica. Can't wait to see your shots of the place.
And…I'll have to think about that Nepal trip. It's so very tempting.
Thanks for the review. We are thinking of visiting Utah come fall.
Cool! Fall is a great time to visit, and you can't go wrong staying here. I recommend staying a few days so you can see sunrise/sunset and do one or more of the Valley Tours.
Well that's where we are now and it sure is nice. Goosenecks is a little far from Monument Valley for our tastes (it would be a long day to leave the dog in the RV), but for folks without pets it's great spot to park. Of course once the Tribal Campground opens up again I suspect that would become our first choice.
I could gladly spend a fair amount of time in this area. It's been on my list for some time. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,249 |
\section{Introduction}
\label{Intro}
Many recent works in relation to the weak gravity conjecture \cite{ArkaniHamed:2006dz} and the swampland conjecture \cite{Ooguri:2006in,Baume:2016psm,Klaewer:2016kiy,Blumenhagen:2017cxt} have investigated the question of both the size of moduli spaces as well as the range of validity of an effective field theory (EFT) description of trans-Planckian excursions of scalar fields in the context of the string landscape. While so far the focus has primarily been on the case of axions \cite{Rudelius:2015xta, Montero:2015ofa, Heidenreich:2015wga, Brown:2015iha, Brown:2015lia}, recently there has been interest in the axionic superpartners and, hence, in the total moduli space \cite{Palti:2017elp,Hebecker:2017lxm}. Generically, the geometric moduli space of an underlying compact geometry features at least one universal flat direction given by the modulus describing the total volume of the internal space. Without further assumptions regarding specific details of the string vacuum, this direction is limited only by EFT restrictions. Here we would like to take a different approach to this question by eliminating at least this universal flat direction via a leading order mechanism of partial moduli stabilisation which leaves at least one remaining flat direction. Thus, instead of focusing on the `full' moduli space, as it would arise in the 4-dimensional effective supergravity description of a specific string compactification, we study the `reduced' moduli space of the remaining flat directions of a string vacuum with a trustworthy EFT description. Notice that in general this `reduced' moduli space could still be non-compact with geodesic trajectories of infinite size.
We address this question in a particular class of vacua, namely for the moduli space of K\"ahler deformations in the context of type IIB Calabi-Yau (CY) orientifold compactifications with background fluxes. This arena is well suited for this purpose since it allows for rather robust moduli stabilisation proposals. A popular mechanism for moduli stabilisation in this class of models is given by the Large Volume Scenario (LVS) which leads to phenomenologically viable vacua where the EFT approximation is expected to hold \cite{Balasubramanian:2005zx}. Moreover, for $h^{1,1}>2$, LVS models can feature a `reduced' moduli space of flat directions since leading order $\alpha'$ and non-perturbative effects can lift only the overall volume modulus and any blow-up mode resolving a point-like singularity \cite{Cicoli:2008va}. This key-property of LVS vacua has allowed very promising cosmological applications \cite{Cicoli:2011zz}. In fact, these remaining flat directions are natural inflaton candidates associated with effective shift symmetries \cite{Burgess:2014tja} which can be used to construct explicit models of large and small field inflation \cite{Conlon:2005jm, Cicoli:2008gp, Broy:2015zba, Burgess:2016owb, Cicoli:2016chb, Cicoli:2011ct, Cicoli:2015wja}. In order to trust these inflationary models, it is therefore crucial to study the size of the `reduced' LVS moduli space of inflationary flat directions.
In this paper, we shall perform this analysis by scanning for LVS vacua through the Kreuzer-Skarke list of CY threefolds with $h^{1,1}=2$, $3$ and $4$ obtained from 4-dimensional reflexive polytopes \cite{Kreuzer:2000xy}. Using a general scan based on the topology of the toric divisors, we find several LVS geometries which were so far unknown. We then focus on the $h^{1,1}=3$ cases and study their reduced moduli space $\mathcal{M}_r$ which turns out to be 1-dimensional. The volume of $\mathcal{M}_r$ can be parameterised in terms of the canonically normalised inflaton $\phi$ and depends on the moduli space metric as well as the CY K\"ahler cone. By computing different approximations to the K\"ahler cones, we show that for all $h^{1,1}=3$ cases $\mathcal{M}_r$ is compact and its volume is bounded from above:
\begin{equation}
\frac{\Delta \phi}{M_p} \leq c\, \ln\mathcal{V}\,,
\label{Delta_bound}
\end{equation}
where $c$ is an $\mathcal{O}(1)$ numerical coefficient and $\mathcal{V}$ is the CY volume in string units, i.e.\ ${\rm Vol}({\rm CY}) = \mathcal{V}\, \ell_s^6$ with $\ell_s=2\pi\sqrt{\alpha'}$.\footnote{The fact that the volume of the reduced moduli space is logarithmically bounded should not come as a surprise since it was shown that proper distances in moduli space grow at best logarithmically \cite{Palti:2017elp}.} This result has important implications for cosmological applications of LVS vacua since it sets a geometrical upper bound on the allowed inflaton range. In particular, this upper bound sets strong constraints on inflationary models which require a trans-Planckian field range to obtain enough e-foldings of inflationary expansion. In the LVS framework, large field inflationary models with this feature involve CY geometries with a K3 fibration \cite{Cicoli:2008gp, Broy:2015zba, Burgess:2016owb, Cicoli:2016chb}. These models are particularly interesting, not just because they can predict a large tensor-to-scalar ratio of order $r\sim 0.005 - 0.01$, but also because they can be successfully embedded into global CY orientifold compactifications with explicit moduli stabilisation and chiral brane setup \cite{Cicoli:2016xae, Cicoli:2017axo}. Interestingly, we find that the size of the reduced moduli space can generically be trans-Planckian only for this kind of examples which feature a K3 fibration.
Based on our systematic analysis for $h^{1,1}=3$ and the characteristic properties of any LVS vacuum combined with the CY K\"ahler cone conditions, we expect our results to hold more generally for $h^{1,1}>3$ too. This leads us to formulate the following conjecture: \\[2mm]
\noindent\textbf{LVS moduli space conjecture:}\\
\textit{The reduced moduli space $\mathcal{M}_r$ of LVS vacua is compact and its volume is bounded by:
\begin{equation}
{\rm Vol}(\mathcal{M}_r) \lesssim \, \left[ \ln \left(\frac{M_p}{\Lambda}\right) \right]^{\rm{dim}(\mathcal{M}_r)}\,,
\label{Delta_bound2}
\end{equation}
where $\Lambda$ is the cut-off of the EFT.}\\[2mm]
In general the EFT cut-off $\Lambda$ is given by the Kaluza-Klein (KK) scale associated with the bulk or with some internal 4-cycle wrapped by a stack of D-branes. Expressing $\Lambda$ in terms of the CY volume (for example $\Lambda \sim M_p / \mathcal{V}^{2/3}$ if we take the cut-off to be the bulk KK scale), it is immediate to realise that the more general bound (\ref{Delta_bound2}) reduces to the simpler result (\ref{Delta_bound}) valid for the 1-dimensional case. Moreover, the general bound (\ref{Delta_bound2}) is very similar to the constraints coming from the weak gravity conjecture and the swampland conjecture \cite{Baume:2016psm,Klaewer:2016kiy,Blumenhagen:2017cxt}. However, it is important to stress that, even if the bound (\ref{Delta_bound2}) is expressed in terms of the EFT cut-off, our results are more explicit since they are purely based on the geometrical features of the underlying compactification threefold. Notice however that KK and $\alpha'$ corrections modify the background geometry away from CY but we expect our main result, i.e. the compactness of the LVS reduced moduli space, to remain qualitatively correct since the exponentially large volume should help to control these corrections. This implies that the inflaton field range should be upper bounded even in the presence of a UV complete theory that would include the effect of heavy KK or stringy modes.
A simplified picture which gives a good intuition in support of our LVS moduli space conjecture is the following. In LVS vacua the leading order moduli stabilisation effects fix only the total volume modulus of the CY threefold together with the volume of a local blow-up divisor. Hence any remaining flat directions parameterising the reduced moduli space $\mathcal{M}_r$ correspond to divisors which can typically no longer shrink to zero or grow to infinite volume as this process would be obstructed by the presence of a blow-up divisor with fixed size inside a stabilised overall volume.
This paper is organised as follows. In Sec.~\ref{sec:setup} we explain the structure of LVS vacua which allow for a reduced moduli space whereas in Sec.~\ref{ComputeCone} we describe our computation of the K\"ahler cone of all LVS vacua with $h^{1,1}=3$ obtained from 4-dimensional reflexive polytopes. In Sec.~\ref{Results} we then show that the reduced moduli space of these string vacua is compact. This result is based both on a systematic scan through the existing list of toric CY threefolds and on an analytic proof. We finally discuss the implication of our results in Sec. \ref{Concl} and present our conclusions in Sec.~\ref{Concl2}.
\section{LVS vacua with flat directions}
\label{sec:setup}
In this section we first review the necessary ingredients for the realisation of LVS vacua which feature a reduced moduli space after leading order moduli stabilisation. We then focus on the $h^{1,1}=3$ case and describe different classes of LVS vacua depending on the geometry and topology of the underlying CY threefold.
\subsection{General conditions for LVS vacua}
\label{GenLVS}
The effective 4-dimensional supergravity obtained from type IIB CY orientifold compactifications with D3/D7-branes, O3/O7-planes and background fluxes is characterised by the following K\"ahler potential and superpotential \cite{Grimm:2004uq}:
\begin{equation}
K = - 2 \, \ln\left(\mathcal{V} + \frac{\hat\xi}{2} \right) \qquad \qquad W = W_0 + \sum_{i \,\in\, I} A_i \,e^{-a_i T_i} \,,
\label{KW}
\end{equation}
where we included only the leading order $\alpha'$ correction to $K$ controlled by the parameter $\hat\xi = - \frac{\chi(X) \zeta(3)}{2 (2\pi)^3 g_s^{3/2}}$ \cite{Becker:2002nn} with $\chi(X)$ denoting the Euler number of the CY threefold $X$ and $g_s$ the string coupling. On the other hand, $W_0$ is the vacuum expectation value (VEV) of the tree-level flux superpotential \cite{Gukov:1999ya} which fixes the dilaton and the complex structure moduli \cite{Giddings:2001yu}. The superpotential includes also non-perturbative corrections which depend only on the moduli receiving instanton contributions (with index running over the subset $I$) together with model-dependent constants $A_i$ and $a_i$. The dependence of $K$ on the K\"ahler moduli is hidden inside the CY volume $\mathcal{V}$. In fact, if the K\"ahler form of $X$ is expanded as $J = t^i \hat{D}_i$ in a basis of harmonic $(1,1)$-forms $\hat{D}_i$ with $i=1,...,h^{1,1}$ (we assume $h^{1,1}_+ = h^{1,1}$), the internal volume becomes:
\begin{equation}
\mathcal{V} = \frac{1}{3!} \int_X J \wedge J \wedge J = \frac16 \,k_{ijk}\, t^i t^j t^k \qquad \text{where} \qquad
k_{ijk} = \int_X \hat{D}_i \wedge \hat{D}_j \wedge \hat{D}_k \,.
\label{volForm}
\end{equation}
The corrected K\"ahler coordinates are given by $T_i = \tau_i + {\rm i}\, \int_{D_i} C_4$ and $\tau_i$ is the volume of the 4-cycle $D_i$ dual to the $(1,1)$-form $\hat{D}_i$:
\begin{equation}
\tau_i = \frac{1}{2!} \int_X \hat{D}_i \wedge J \wedge J = \frac12\, k_{ijk}\, t^j t^k\,.
\label{taui}
\end{equation}
After inverting this relation and using the volume form (\ref{volForm}), the K\"ahler potential in (\ref{KW}) can finally be expressed in terms of the K\"ahler moduli $\tau_i$.
Focusing on the large volume regime $\mathcal{V} \gg \hat\xi$ where the EFT is under control and on natural $\mathcal{O}(1-10)$ values of $W_0$, the generic F-term scalar potential arising from the expressions of $K$ and $W$ in (\ref{KW}) takes the form:
\begin{equation}
V = \sum_{i,j \,\in\, I} a_i a_j A_i A_j \,K^{i\bar{j}}\,\frac{e^{-\left(a_i \tau_i + a_j \tau_j\right)}}{\mathcal{V}^2}
-\sum_{i \,\in\, I} 4 A_i W_0 \,a_i\tau_i\,\frac{e^{-a_i \tau_i}}{\mathcal{V}^2}
+ \frac{3\hat\xi W_0^2}{4\mathcal{V}^3}\,,
\label{VF}
\end{equation}
where, without loss of generality, we consider $W_0$ and all $A_i$ to be real. Furthermore, the $C_4$-axions have been fixed already and the inverse K\"ahler metric has the general expression \cite{Bobkov:2004cy}:
\begin{equation}
K^{i\bar{j}} = - \frac49\, \left(\mathcal{V} + \frac{\hat\xi}{2} \right) \, k_{ijk}\, t^k + \frac{4 \,\mathcal{V} - \hat\xi}{\mathcal{V} - \hat\xi} \, \tau_i \tau_j
\stackrel{\mathcal{V}\gg \hat\xi}{\simeq} - \frac49\,\mathcal{V} \,k_{ijk}\, t^k + 4 \, \tau_i \tau_j\,.
\label{eq:Kinv}
\end{equation}
The scalar potential (\ref{VF}) admits an LVS minimum at $\mathcal{V} \sim W_0 \,e^{a_i \tau_i}$ only if each of the three contributions in (\ref{VF}) has the same volume scaling. As pointed out in \cite{Cicoli:2008va}, this is guaranteed if:\footnote{As explained in \cite{Cicoli:2008va} there may exist further LVS-type vacua where the third requirement is relaxed.}
\begin{enumerate}
\item $X$ has negative Euler number $\chi(X) < 0$.\footnote{Note however that LVS vacua with $\chi(X)>0$ can exist if string loop corrections are included \cite{Cicoli:2012fh}.}
\item $X$ features at least one divisor $D_s$ which supports non-perturbative effects and can be made `small', i.e.\ the CY volume $\mathcal{V}$ does not become zero or negative when $\tau_s\to 0$.
\item The element $K^{s\bar{s}}$ of the inverse K\"ahler metric scales as (for $\mathcal{V} \gg \tau_s^{3/2} \sim \hat\xi$):
\begin{equation}
K^{s\bar{s}} \simeq \lambda \mathcal{V} \sqrt{\tau_s} \,,
\label{Kinv}
\end{equation}
where $\lambda$ is an $\mathcal{O}(1)$ coefficient.
\end{enumerate}
Combining (\ref{taui}) and (\ref{eq:Kinv}), it is easy to realise that the third condition above is equivalent to:
\begin{equation}
k_{ssi}\,k_{ssj} = k_{sss}\, k_{sij} \qquad \forall \,i,j\,,
\label{kRel}
\end{equation}
if $k_{ssi}\neq 0$ for some $i$, otherwise we can clearly see from (\ref{eq:Kinv}) that $K^{s\bar{s}} \simeq 4\tau_s^2$. Moreover, the relation (\ref{kRel}) is equivalent also to the fact that $\tau_s$ is a perfect square since:
\begin{equation}
\tau_s = \frac12\, k_{sij}\, t^i t^j = \frac{1}{2 k_{sss}}\, k_{ssi} t^i \,k_{ssj} t^j = \frac{1}{2 k_{sss}}\left( k_{ssi} t^i \right)^2\,.
\label{square}
\end{equation}
Notice also that (\ref{kRel}) implies necessarily that $k_{sss}\neq 0$ since otherwise $k_{ssi}=0$ $\forall\,i$. The last condition in eq.~\eqref{square} is algorithmically simple to implement and will be the relevant one in the next sections. The second requirement above can be satisfied if $D_s$ is a del Pezzo (dP) divisor. Such divisors are key-ingredients for the realisation of LVS vacua \cite{Cicoli:2011it}. In fact, a dP divisor $D_s$ is rigid, i.e.\ $h^{2,0}(D_s)=0$, and without Wilson lines, i.e.\ $h^{1,0}(D_s)=0$. These are sufficient conditions to guarantee a non-vanishing $T_s$-dependent non-perturbative contribution to $W$. If in addition a del Pezzo satisfies the square relation in eq.~\eqref{square}, then it turns out that for an appropriate choice of basis this del Pezzo can 'diagonalise' the volume form in the sense that $k_{sss}\neq 0$ and $k_{ssi}=0$ for all $i \neq s$. Therefore, from now on we refer to them as 'diagonal' del Pezzos (ddP).\footnote{So far all the LVS examples found in the literature feature a diagonal dP divisor.} Moreover note that, being diagonal in the volume, $\tau_s$ is a natural `small' modulus since it can be shrunk to zero size without affecting the overall volume. This is related to the fact that diagonal dP divisors are blow-ups of point-like singularities \cite{Cicoli:2008va}. Notice however that not all dP divisors are diagonal, since for some of them it is never possible to find a basis of smooth divisors where the only non-zero intersection number is $k_{sss}$. These divisors are called `non-diagonal' dP and can be seen as resolutions of line-like singularities \cite{Cicoli:2011it}.
To summarise we identify LVS vacua as CY geometries such that:
\begin{enumerate}
\item $X$ is favourable and has negative Euler number $\chi(X) < 0$.
\item $X$ features at least one del Pezzo divisor that satisfies the square relation in eq.~\eqref{square}.
\end{enumerate}
Under the above conditions, the general scalar potential (\ref{VF}) depends only on the overall volume and $n_s$ small K\"ahler moduli which control the volume of a diagonal dP divisor and support non-perturbative effects. Hence the leading order $\alpha'$-correction to $K$ and non-perturbative corrections to $W$ stabilise $n_s$ $C_4$-axions and $(n_s+1)$ 4-cycle volume moduli at \cite{Balasubramanian:2005zx}:
\begin{equation}
\frac{3\,\hat\xi}{2} \simeq \sum_{i=1}^{n_s}\sqrt{\frac{2}{d_i}}\,\tau_{{\scriptscriptstyle 0},i}^{3/2}\qquad\text{and}\qquad
\mathcal{V}_{\scriptscriptstyle 0} \simeq \sqrt{\frac{2}{d_i}} \, \frac{W_0\sqrt{\tau_{{\scriptscriptstyle 0},i}}}{4\, a_i A_i}\,e^{a_i \tau_{{\scriptscriptstyle 0},i}} \quad \forall\, i=1,...,n_s\,,
\label{eq:LVS_minimum}
\end{equation}
with $d_i = (9-p_i)$ where $p_i$ is the degree of the $i$-th diagonal dP divisor (e.g.\ $d_i=9$ for a dP$_0\equiv {\mathbb P}^2$ divisor). This leaves a reduced moduli space $\mathcal{M}_r$ characterised by $(h^{1,1}-n_s-1)$ 4-cycle volume flat directions and $(h^{1,1}-n_s)$ axionic flat directions. Given that the axionic directions are periodic, they cannot give rise to a non-compact moduli space. We shall therefore neglect them and focus only on the remaining $(h^{1,1}-n_s-1)$ flat directions for the 4-cycle volume moduli.
These directions can be lifted by the inclusion of additional contributions to the effective 4-dimensional scalar potential like D-terms from magnetised D-branes \cite{Dine:1987xk, Dine:1987gj}, string loop corrections \cite{Berg:2005ja, Berg:2007wt, Cicoli:2007xp, Berg:2014ama, Haack:2015pbv}, higher derivative $\alpha'$ effects \cite{Ciupke:2015msa, Grimm:2017okk}, poly-instantons \cite{Blumenhagen:2008ji, Blumenhagen:2012kz, Blumenhagen:2012ue} or non-perturbative effects for divisors rigidified by fluxes \cite{Bianchi:2011qh, Bianchi:2012pn, Louis:2012nb}. These corrections have indeed been used to achieve full closed string moduli stabilisation and to generate the inflationary potential in several LVS models. In what follows, we shall however neglect these additional effects in order to study the size of the $(h^{1,1}-n_s-1)$-dimensional reduced moduli space $\mathcal{M}_r$. Clearly the simplest situation is for $n_s=1$, and so $\mathcal{M}_r$ is non-trivial only for $h^{1,1}\geq 3$.
\subsection{Classes of LVS vacua with \texorpdfstring{$h^{1,1}=3$}{h11=3}}
\label{sec:class_LVS}
The simplest LVS vacua with $h^{1,1} \geq 3$ and only $n_s=1$ feature an $(h^{1,1}-2)$-dimensional reduced moduli space $\mathcal{M}_r$ parameterised by the unfixed 4-cycle volume moduli. $\mathcal{M}_r$ can be thought of as a subspace of the full K\"ahler moduli space $\mathcal{M}$ defined by the hypersurface equations:
\begin{equation}
\mathcal{V}(\tau_i) = \mathcal{V}_{\scriptscriptstyle 0} \qquad\text{and}\qquad\tau_s = \tau_{\scriptscriptstyle 0} \,,
\label{eq:hypersurface}
\end{equation}
where $\mathcal{V}_0$ and $\tau_0$ are the fixed values given in~\eqref{eq:LVS_minimum}. The geometry of the full K\"ahler moduli space $\mathcal{M}$ of the CY threefold $X$ is derived from the metric descending from the K\"ahler potential in (\ref{KW}) and the K\"ahler cone conditions defined as:
\begin{equation}
\int_{C_i} J > 0 \,,
\end{equation}
for all curves $C_i$ in the Mori cone of $X$, that is the cone of curves. As explained above, from now on we understand $\mathcal{M}$ as the $h^{1,1}$-dimensional real manifold excluding the axionic directions. A pictorial visualisation of the $\mathcal{M}_r$ hypersurface inside the K\"ahler cone for $h^{1,1}=3$ is given in Fig.~\ref{cone}.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.3\textwidth]{kahlercone.pdf}
\caption{Pictorial representation of a K\"ahler cone with $h^{1,1}=3$ parametrised by the 2-cycle volumes $t^i$. The hypersurfaces of~\eqref{eq:hypersurface} are represented respectively in blue and red. The intersection between these two hypersurfaces inside the cone represents the reduced moduli space $\mathcal{M}_r$.}
\label{cone}
\end{center}
\end{figure}
Using the induced metric coming from $\mathcal{M}$, the volume of the $\mathcal{M}_r$ hypersurface can be defined as:
\begin{equation}
{\rm Vol}\,(\mathcal{M}_r) = \int_{\mathcal{M}_r} * \mathbf{1}_{r} \,.
\label{eq:M_vol}
\end{equation}
Due to the technical difficulty to compute $\mathcal{M}_r$, in what follows we shall perform this task only for $h^{1,1}=3$. In this case the reduced moduli space $\mathcal{M}_r$ turns out to be 1-dimensional. To simplify the discussion, we subdivide the LVS vacua with $h^{1,1}=3$ into different classes depending on the number $1 \leq n_{\scriptscriptstyle {\rm ddP}} \leq 2$ of diagonal dP divisors and the number $0 \leq n_{\scriptscriptstyle {\rm K3f}} \leq 1$ of K3 fibrations. Each of these classes of LVS vacua is characterised by a different volume form (\ref{volForm}) expressed in terms of the 4-cycle moduli $\tau_i$ as follows:
\begin{itemize}
\item $n_{\scriptscriptstyle {\rm ddP}}=2$ and $n_{\scriptscriptstyle {\rm K3f}}=0$: \\
These cases feature 2 diagonal dP moduli, and no fibred K3 divisor. The volume is completely diagonal and takes the so-called \textbf{strong Swiss cheese} form:
\begin{equation}
\mathcal{V} = \alpha \, \tau_b^{3/2} - \beta_1 \, \tau_{s_1}^{3/2}- \beta_2 \, \tau_{s_2}^{3/2} \,,
\label{vol:2dp}
\end{equation}
where $\alpha$, $\beta_1$ and $\beta_2$ are positive and depend on the CY intersection numbers with $\beta_i$ expressed in terms of the degree of the corresponding dP divisor as $\beta_i=\frac13 \,\sqrt{\frac{2}{d_{s_i}}}$. According to the discussion in Sec.~\ref{GenLVS}, if both dP divisors support non-perturbative effects, there is no flat direction left over. We shall therefore consider only cases where at non-perturbative level $W$ depends just on a single dP modulus. This situation reproduces the case of K\"ahler moduli inflation \cite{Conlon:2005jm} which is a small field model that has been recently embedded in a chiral global construction \cite{Cicoli:2017shd} since in the inflationary regime inflaton-dependent non-perturbative effects become negligible.
\item $n_{\scriptscriptstyle {\rm ddP}}=1$ and $n_{\scriptscriptstyle {\rm K3f}}=1$: \\
In these cases the CY threefold $X$ is a \textbf{K3 fibration} over a ${\mathbb P}^1$ base together with a diagonal dP divisor \cite{Cicoli:2011it}. The volume form simplifies to:
\begin{equation}
\mathcal{V} = \alpha \sqrt{\tau_f} \, \tau_b - \beta \, \tau_s^{3/2} \,,
\label{vol:K3}
\end{equation}
where $\tau_f$ controls the volume of the K3 fibre and $\beta = \frac13 \,\sqrt{\frac{2}{d_s}}$ as before. This CY geometry allows the construction of several fibre inflation models depending on the different microscopic origin of the effects which lift the inflationary direction. These constructions include both small \cite{Cicoli:2011ct, Lust:2013kt} and large field models \cite{Cicoli:2008gp, Broy:2015zba, Burgess:2016owb, Cicoli:2016chb} which can be successfully embedded in global CY compactifications \cite{Cicoli:2016xae} with chiral matter \cite{Cicoli:2017axo}.
\item $n_{\scriptscriptstyle {\rm ddP}}=1$ and $n_{\scriptscriptstyle {\rm K3f}}=0$: \\
These cases admit only 1 diagonal dP divisor without any K3 fibration. They can be subdivided further into two subclasses depending on the structure of the volume form in the limit $\tau_s\to 0$:
\begin{enumerate}
\item For $\tau_s\to 0$, the intersection polynomial cannot be simplified by any choice of cohomology basis, and so it appears \textbf{structureless}. Hence the CY volume is:
\begin{equation}
\mathcal{V} = f_{\scriptscriptstyle 3/2}(\tau_1,\tau_2) - \beta \, \tau_s^{3/2}\,,
\end{equation}
where $f_{\scriptscriptstyle 3/2}(\tau_1,\tau_2)$ is a homogeneous function of degree $3/2$ in $\tau_1$ and $\tau_2$.
\item In a proper basis of smooth divisors, when the diagonal dP divisor $\tau_s$ shrinks to zero size, the CY volume takes a simple diagonal form:
\begin{equation}
\mathcal{V} = \alpha \tau_b^{3/2} - \beta_1 \, \tau_s^{3/2}- \beta_2 \left(\gamma_1\, \tau_s + \gamma_2\, \tau_*\right)^{3/2} \,,
\label{vol:ssc}
\end{equation}
with $\gamma_1$ and $\gamma_2$ positive coefficients which depend on the intersection numbers. In this case, similarly to $\tau_s$, also $\tau_b$ in (\ref{vol:ssc}) satisfies a perfect square relation. We shall call these examples as \textbf{strong Swiss cheese-like}. Notice however that the combination $\gamma_1 D_s+ \gamma_2 D_*$ does not correspond to a smooth divisor, and so there is no choice of basis of smooth divisors where $\mathcal{V}$ takes the standard strong Swiss cheese (SSC) form. Moreover, even if $\mathcal{V}$ looks diagonal when $\tau_s\to 0$, the divisor $D_*$ might not be shrinkable. We found by direct search that $D_*$ is rigid, i.e.\ $h^{2,0}(D_*)=0$, but it can be either a `non-diagonal' dP, a `Wilson' divisor with $h^{1,0}(D_*)\geq 1$ \cite{Blumenhagen:2012kz} or a `rigid but non-dP' divisor with Hodge diamond as a standard dP divisor but with $h^{1,1}(D_*) > 9$ \cite{Cicoli:2016xae}. Notice that cases where $D_*$ is a Wilson divisor with $h^{1,0}(D_*)= 1$ have been used to construct small field inflationary models where the inflaton potential is generated by poly-instanton effects \cite{Blumenhagen:2012ue}.
\end{enumerate}
\end{itemize}
\section{The reduced moduli space}
\label{ComputeCone}
In this section we describe how to compute the reduced moduli space for all LVS vacua with $h^{1,1}=3$.
\subsection{Computation of the K\"ahler cone}
In the following we will be interested in CY threefolds which arise as hypersurfaces in toric varieties obtained from 4-dimensional reflexive lattice polytopes which have been classified by Kreuzer and Skarke \cite{Kreuzer:2000xy}. The geometry of the full K\"ahler moduli space $\mathcal{M}$, and so also of the reduced moduli space $\mathcal{M}_r$ and the associated volume ${\rm Vol}\,(\mathcal{M}_r)$, depend both on the metric as well as on the K\"ahler cone conditions. While the metric on the moduli space is readily obtained from the intersection tensor, unfortunately, there is no known algorithmic procedure to determine the exact K\"ahler cone $\mathcal{K}_X$ for the hypersurface CY threefolds. However, we will provide two approximate expressions for $\mathcal{K}$ using the following strategy: the CY Mori cone $M_X$ is contained inside a larger Mori cone $M_A$ and contains a smaller Mori cone $M_\cap$:
\begin{equation}
M_A \supseteq M_X \supseteq M_\cap \,.
\end{equation}
Conversely, the dual K\"ahler cones satisfy:
\begin{equation}
\mathcal{K}_A \subseteq \mathcal{K}_X \subseteq \mathcal{K}_\cap \,.
\end{equation}
If we then impose the hypersurface equations (\ref{eq:hypersurface}) which define the reduced moduli space $\mathcal{M}_r$, its volume turns out to have a lower and a upper bound:
\begin{equation}
{\rm Vol}\,(\mathcal{M}_{A,r}) \leq {\rm Vol}\,(\mathcal{M}_r) \leq {\rm Vol}\,(\mathcal{M}_{\cap,r}) \,,
\end{equation}
where $\mathcal{M}_{A,r}$ and $\mathcal{M}_{\cap,r}$ denote the reduced moduli spaces obtained respectively from the approximated Mori cones $M_A$ and $M_\cap$. The label $A$ of the larger Mori cone $M_A$ (or the smaller K\"ahler cone $\mathcal{K}_A$) stays for `ambient' since $M_A$ is obtained by intersecting all the Mori cones of the ambient varieties which are connected via `irrelevant' flop transitions. These are transitions that, though changing the intersection ring of the ambient space, do not have any effect on the CY hypersurface. Therefore, we can omit curves which are flopped in these `irrelevant' transitions, when looking for the constraints on a form in the ambient variety whose pullback to the hypersurface gives a valid K\"ahler form. These irrelevant curves are discarded when we take the intersection of the Mori cones.
On the other hand, the label $\cap$ of the smaller Mori cone $M_\cap$ (or the larger K\"ahler cone $\mathcal{K}_\cap$) stays for `intersection' since $M_\cap$ is obtained by intersecting all toric surfaces of the ambient variety with the CY hypersurface. These toric surfaces are given by the set of the 2-dimensional cones in the fan of the toric variety. Hence, in order to obtain the curves in $M_\cap$, we have to intersect the hypersurface divisor with the two toric divisors corresponding to the two extremal rays of the 2-dimensional cones.
In the fortunate case that the larger and smaller Mori cones coincide, i.e.\ $M_A=M_\cap$, we know the actual CY K\"ahler cone by this rather simple algorithmic calculation. However, for most of the geometries this is not the case and we have to work a bit harder to obtain the actual CY K\"ahler cone (or a better approximation thereof). Given that our goal is to show that the reduced moduli space is compact, a better determination of the CY K\"ahler cone is crucial in particular for those cases where ${\rm Vol}\,(\mathcal{M}_{\cap,r})$ fails to be finite. The next section deals with such examples.
\subsection{Improved computation of the K\"ahler cone}
\label{sec:improv_KC}
Unfortunately, there are CY geometries for which $M_A \supsetneq M_\cap$ and ${\rm Vol}\,(\mathcal{M}_{\cap,r})$ is infinite and, hence, the smaller Mori cone $M_\cap$ is not sufficient to show that $\mathcal{M}_r$ is compact. For $h^{1,1}=3$ there are 32 toric CY hypersurfaces with $1 \leq n_{\scriptscriptstyle {\rm ddP}} \leq 2$ for which this is the case. These CY threefolds, which we have to study in more detail, fall into 3 different categories: they can be either torus fibred, K3 fibred,\footnote{In 4 cases they are both K3 and $T^2$ fibred.} or a double cover of a 3-dimensional toric variety branched over a divisor different from the anti-canonical bundle of the respective toric variety.
Since in the recent years, in the course of the F-theory phenomenology hype, all toric hypersurface $T^2$ fibrations have been worked out in detail \cite{Morrison:2012ei,Morrison:2014era,Mayrhofer:2014opa,Klevers:2014bqa}, it is rather straightforward to obtain all the curves in the fibre. The curves from the base can be pulled back via sections or multi-sections. In this way we obtain all the generating curves of the Mori cone of the CY hypersurface.
In the case of K3 fibrations, the situation is a bit more complicated because the details of all possible degenerations have not been worked out yet to the same extent as for the $T^2$ fibred examples. However, for the K3 fibrations at hand, it is not too difficult to figure out all the surfaces into which the K3 factorises at degeneration points of the fibration. With the curves from these surfaces and the pullback of the $\mathbb P^1$ of the base, we are able to obtain a much better approximation to the actual CY Mori cone than $M_\cap$.
For the CY threefolds which are double covers of toric varieties we can, first of all, employ all the effective curves of the underlying 3-dimensional toric space. To really use them for the construction of the Mori cone, we have to pull them back to the hypersurface which is a quadratic in the $\mathbb P^1$ bundle over the toric threefold, i.e.\ a double cover. In addition to these curves, there is the $\mathbb P^1$ fibre itself which is realised over the points of the base where the 3 sections which are in front of the 3 quadratic monomials vanish simultaneously.
The exact CY Mori cones for the list of these 32 CY threefolds are added as a text file to version of this paper uploaded on the \href{https://arxiv.org/}{arXiv} website.
\section{Compactness of the reduced moduli space}
\label{Results}
In this section, we present the results of our scan for toric CY threefolds with diagonal del Pezzos for $h^{1,1}=2$, $3$ and $4$. For $h^{1,1}=3$ we determine the subclasses of LVS vacua and give reference values as well as histograms for the distribution of the size of $\mathcal{M}_r$. In the second part, we explicitly prove that the moduli space $\mathcal{M}_r$ is compact for these vacua for all values of $\mathcal{V}_0, \tau_0 > 0$.
\subsection{Scanning results}
\label{scan}
We now turn to the explicit scan for LVS vacua which, according to the discussion in Sec.~\ref{sec:setup}, we identify with CY geometries with diagonal dP divisors. Here we use the database of \cite{Altman:2014bfa} which includes CY threefolds with $h^{1,1}=2$, $3$ and $4$ built as hypersurfaces in toric varieties. For each CY threefold $X$ this database gives its topological data together with the Mori cone of the ambient space $M_A$. Since there are in general several triangulations giving rise to the same CY geometry, we do not distinguish them here and, hence, only count the number of different CY geometries $n_{\scriptscriptstyle {\rm CY}}$. We focus only on the favorable geometries as otherwise $h^{1,1}>3$. In order to compute the divisor topologies from the topological data of $X$, we use the tool \texttt{cohomCalg} \cite{Blumenhagen:2010pv, Blumenhagen:2011xn}. When searching for diagonal dP divisors, it suffices to compute the topology of toric divisors since a dP 4-cycle cannot come from a non-toric ambient space divisor. In fact, if this were the case, the dP divisor would have to be a formal sum of coordinate divisors which would allow for deformations, in clear contradiction with its rigidity. On the other hand, given that a K3 divisor can be deformed, we need to scan also for linear combinations of coordinate divisors in order to identify all possible K3 fibrations.
The general result of the scan for the total number of LVS vacua $n_{\scriptscriptstyle {\rm LVS}}$ for $h^{1,1}=2$, $3$ and $4$ is displayed in Tab.~\ref{tab:gen_res}. For the case with $h^{1,1}=3$, the number of LVS vacua for the different subclasses described in Sec.~\ref{sec:class_LVS} is given in Tab.~\ref{tab:h113}.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c||c|c|c|c|}
\hline
$h^{1,1}$ & $n_{\scriptscriptstyle {\rm CY}}$ & $n_{\scriptscriptstyle {\rm LVS}}$ & \% & $n_{\scriptscriptstyle {\rm ddP}}=1$ & $n_{\scriptscriptstyle {\rm ddP}}=2$ & $n_{\scriptscriptstyle {\rm ddP}}=3$ \\\hline
$2$ & $39$ & $22$ & 56.4\% & $22$ & $-$ & $-$ \\
$3$ & $305$ & $132$ & 43.3\% & $93$ & $39$ & $-$ \\
$4$ & $1997$ & $749$ & 37.5\% & $464$ & $261$ & $24$ \\
\hline
\end{tabular}
\caption{Number $n_{\scriptscriptstyle {\rm CY}}$ of favorable CY geometries in the database of \cite{Altman:2014bfa} for $h^{1,1}=2$, $3$, $4$ and number $n_{\scriptscriptstyle {\rm LVS}}$ of LVS vacua classified in terms of the number $n_{\scriptscriptstyle {\rm ddP}}$ of diagonal dP divisors.}
\label{tab:gen_res}
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c||c|c|c|c|}
\hline
$h^{1,1}$ & $n_{\scriptscriptstyle {\rm CY}}$ & $n_{\scriptscriptstyle {\rm LVS}}$ & SSC & K3 fibred & SSC-like & structureless \\
\hline
$3$ & $305$ & $132$ & $39$ & $43$ & $36$ & $14$ \\
\hline
\end{tabular}
\caption{Different LVS subclasses for the cases in Tab.~\ref{tab:gen_res} with $h^{1,1}=3$.}
\label{tab:h113}
\end{table}
The next step consists in computing the reduced moduli space $\mathcal{M}_r$ for all LVS cases with $h^{1,1}$=3. Following the procedure described in Sec.~\ref{sec:improv_KC}, we start by computing the Mori cone $M_\cap$ given by the intersections of all toric divisors on the CY hypersurface $X$. The fixed values of $\mathcal{V}_{\scriptscriptstyle 0}$ and $\tau_{\scriptscriptstyle 0}$ in \eqref{eq:LVS_minimum} determine then the two hypersurface equations \eqref{eq:hypersurface} which identify the 1-dimensional reduced moduli spaces $\mathcal{M}_{\cap,r}$ for $M_\cap$ and $\mathcal{M}_{A,r}$ for $M_A$. The volumes ${\rm Vol}\,(\mathcal{M}_{A,r})$ and ${\rm Vol}\,(\mathcal{M}_{\cap,r})$ of these two approximations of the CY reduced moduli space can finally be computed using (\ref{eq:M_vol}).
Since the size of the reduced moduli space depends on the fixed value of the overall volume (and, in a much milder way, also on $\tau_{\scriptscriptstyle 0}$), in Fig.~\ref{histolb} we show the distribution of ${\rm Vol}\,(\mathcal{M}_{A,r})$ and ${\rm Vol}\,(\mathcal{M}_{\cap,r})$ for $g_s=0.1$, which fixes $\tau_{\scriptscriptstyle 0}$ from (\ref{eq:LVS_minimum}), and different values of the CY volume $\mathcal{V}_{\scriptscriptstyle 0}=10^3$, $10^4$ and $10^5$. For the 32 cases where ${\rm Vol}\,(\mathcal{M}_{\cap,r})$ is infinite, we replace $M_\cap$ with the exact CY Mori cone obtained by direct computation as explained in Sec.~\ref{sec:improv_KC}. For the geometries with $n_{\scriptscriptstyle {\rm ddP}}=2$ we double-count the number of vacua since, as explained in Sec.~\ref{sec:class_LVS}, the remaining flat direction can be either of the two diagonal dP blow-ups. According to Tab.~\ref{tab:h113}, this gives a total of 171 distinct LVS vacua with $h^{1,1}=3$ and 1 flat direction.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1\textwidth]{histogramsNew2.pdf}
\caption{Distribution of volumes in units of $M_p$ of reduced moduli spaces for different subclasses of LVS vacua with $h^{1,1}=3$ for $g_s=0.1$ and $\mathcal{V}_{\scriptscriptstyle 0}=10^3$, $10^4$ and $10^5$. The histograms on the left display the distribution of ${\rm Vol}\,(\mathcal{M}_{A,r})$ whereas the ones on the right correspond to ${\rm Vol}\,(\mathcal{M}_{\cap,r})$.}
\label{histolb}
\end{center}
\end{figure}
In Tab.~\ref{tab:mean_delta} we also display the average values of ${\rm Vol}\,(\mathcal{M}_{A,r})$ and ${\rm Vol}\,(\mathcal{M}_{\cap,r})$ together with the maximal value of ${\rm Vol}\,(\mathcal{M}_{A,r})$ for all different LVS subclasses. Notice that we do not quote the maximal value of ${\rm Vol}\,(\mathcal{M}_{\cap,r})$ since it could overestimate the size of the actual CY reduced moduli space when $M_A \supsetneq M_\cap$. However, except for the $n_{\scriptscriptstyle {\rm ddP}}=2$ SSC case, for all CY geometries which have ${\rm Vol}\,(\mathcal{M}_{A,r})$ maximal, the Mori cone is exact since $M_A=M_\cap$.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
& $\mathcal{V}_{\scriptscriptstyle 0}$ & SSC & K3 fibred & SSC-like & structureless \\ \hline
& $10^3$ & $0.58$ & $2.27$ & $0.43$ & $0.57$ \\
$\langle{\rm Vol}\,(\mathcal{M}_{A,r})\rangle$ & $10^4$ & $0.67$ & $3.62$ & $0.55$ & $0.80$ \\
& $10^5$ & $0.76$ & $4.98$ & $0.62$ & $0.97$ \\ \hline
& $10^3$ & $0.71$ & $2.47$ & $0.70$ & $0.57$ \\
$\langle{\rm Vol}\,(\mathcal{M}_{\cap,r})\rangle$ & $10^4$ & $0.81$ & $3.81$ & $0.82$ & $0.80$ \\
& $10^5$ & $0.91$ & $5.17$ & $0.89$ & $0.97$ \\\hline
& $10^3$ & $1.44$ & $3.31$ & $0.87$ & $1.48$ \\
${\rm max}({\rm Vol}\,(\mathcal{M}_{A,r}))$ & $10^4$ & $1.91$ & $5.29$ & $1.38$ & $2.41$ \\
& $10^5$ & $2.38$ & $7.29$ & $1.87$ & $2.79$ \\\hline
\end{tabular}
\caption{Average volume in units of $M_p$ of the two different approximations $\mathcal{M}_{A,r}$ and $\mathcal{M}_{\cap,r}$ of the reduced moduli space of LVS vacua with $h^{1,1}=3$. We also show the maximal value of the size of the reduced moduli space $\mathcal{M}_{A,r}$ obtained from the ambient Mori cone $M_A$.}
\label{tab:mean_delta}
\end{table}
Tab.~\ref{tab:mean_delta} shows also that the size of the reduced moduli space varies significantly with $\mathcal{V}_{\scriptscriptstyle 0}$ only for K3 fibred geometries. In this cases it is therefore important to include also the dependence of the field range $\Delta\phi$ on the fixed value of the blow-up mode $\tau_{\scriptscriptstyle 0}$. This effect can be taken into account simply considering different values of $g_s$ since, as can be seen from (\ref{eq:LVS_minimum}), $\tau_{\scriptscriptstyle 0}$ is determined just by the string coupling and the underlying CY topology. Fig.~\ref{histogs} shows the distribution of ${\rm Vol}\,(\mathcal{M}_{A,r})$ for all 43 K3 fibred LVS geometries with $h^{1,1}=3$ for $g_s=0.1$, $0.2$ and $0.3$, and $\mathcal{V}_{\scriptscriptstyle 0}=10^3$, $10^4$ and $10^5$. The field range $\Delta\phi$ takes the largest maximal value for $g_s=0.3$ and $\mathcal{V}_{\scriptscriptstyle 0}=10^5$, signaling the presence of a lower bound proportional to $\tau_{\scriptscriptstyle 0}$ (which decreases when $g_s$ increases) and an upper bound proportional to $\ln\mathcal{V}_{\scriptscriptstyle 0}$. We shall confirm this result with an analytical calculation in the next section.
\begin{figure}[htb!]
\begin{center}
\includegraphics[width=1\textwidth]{histograms-gs.pdf}
\caption{Distribution of volumes in units of $M_p$ of the reduced moduli space $\mathcal{M}_{A,r}$ for all 43 K3 fibred LVS vacua for $g_s=0.1$, $0.2$ and $0.3$, and $\mathcal{V}_{\scriptscriptstyle 0}=10^3$, $10^4$ and $10^5$.}
\label{histogs}
\end{center}
\end{figure}
\subsection{Analytic proof}
\label{sec:proof_comp}
Let us now provide an analytical proof of the compactness of the reduced moduli space $\mathcal{M}_r$ for all subclasses of LVS vacua with $h^{1,1}=3$. Since $\mathcal{M}_r$ depends through the hypersurface equation on the values $\mathcal{V}_0, \tau_0$ it is important to demonstrate this for all values $\mathcal{V}_0 , \tau_0 > 0$. For SSC, K3 fibred and SSC-like LVS geometries, ${\rm Vol}(\mathcal{M}_r)$ can be shown to be finite by directly computing the metric on the reduced moduli space and the associated field range $\Delta\phi$. We achieve this result starting from the line element of the full moduli space obtained from the K\"ahler potential in (\ref{KW}) together with the relevant volume form as follows:
\begin{equation}
{\rm d}s^2 = g_{ij} \,{\rm d}\tau_i {\rm d}\tau_j\qquad\text{with}\qquad g_{ij} = 2 \,\frac{\partial^2 K}{\partial T_i\partial \bar{T}_j}=\frac12 \frac{\partial^2 K}{\partial \tau_i\partial \tau_j}\,,
\label{ds}
\end{equation}
where the factor $2$ in front of the metric is needed to match the standard definition of the kinetic Lagrangian $\mathcal{L}_{\rm kin}= \frac{\partial^2 K}{\partial T_i\partial \bar{T}_j}\,\partial_\mu T_i\partial^\mu \bar{T}_j=\frac12 \,g_{ij}\partial_\mu \tau_i \partial^\mu \tau_j$ (neglecting the axionic terms). Given that the overall volume and the diagonal dP modulus are fixed respectively at $\mathcal{V}=\mathcal{V}_{\scriptscriptstyle 0}$ and $\tau_s=\tau_{\scriptscriptstyle 0}$, it is useful to trade ${\rm d}\tau_b$ for ${\rm d}\mathcal{V}$ via ${\rm d}\mathcal{V} = \frac{\partial \mathcal{V}}{\partial \tau_i}\,{\rm d}\tau_i$ and finally set ${\rm d}\mathcal{V}= {\rm d}\tau_s=0$. Then (\ref{ds}) reduces to the line element ${\rm d}s_r^2$ of the 1-dimensional reduced moduli space parametrised by the remaining flat direction $\tau$ which, depending on the particular LVS subclass, can be either another diagonal dP blow-up, a K3 fibre or a modulus with a more complicated topology. The field range in units of $M_p$ can then be computed as:
\begin{equation}\label{field-range}
\Delta\phi = \int_{\tau_{\rm min}}^{\tau_{\rm max}} {\rm d}s_r\,,
\end{equation}
where the minimum and maximal values of $\tau$ are derived from the K\"ahler cone conditions of either $M_A$ or $M_\cap$ as discussed in Sec.~\ref{scan}. Note that in general we need to choose several charts and corresponding transition functions to rewrite eq.~\eqref{eq:M_vol} in terms of local coordinates on $\mathcal{M}_r$ and, thus, eq.~\eqref{field-range} holds only if the moduli space can be covered by a single chart. For the vacua we are considering this approach is justified for all SSC, K3 fibred and SSC-like cases, but not for the structureless geometries. For all SSC, K3 fibred and SSC-like cases we also find that:
\begin{equation}
\tau_{\rm min} = a \,\tau_{\scriptscriptstyle 0} \qquad\text{and}\qquad \tau_{\rm max} = f_{\scriptscriptstyle 1}(\mathcal{V}_{\scriptscriptstyle 0},\tau_{\scriptscriptstyle 0}) \,,
\label{eq:field_range}
\end{equation}
where $a\geq 0$ and $f_{\scriptscriptstyle 1}$ is a homogeneous function of degree 1 in the 4-cycle moduli that is strictly positive for all $\mathcal{V}_{\scriptscriptstyle 0}>0$ and $\tau_{\scriptscriptstyle 0} >0$.
Let us now present the metric on $\mathcal{M}_r$ and the associated field range for these 3 subclasses of LVS vacua with $h^{1,1}=3$:
\newpage
\begin{itemize}
\item \textbf{SSC vacua} with $n_{\scriptscriptstyle {\rm ddP}}=2$ and $n_{\scriptscriptstyle {\rm K3f}}=0$:
Using the volume form in eq.~\eqref{vol:2dp}, the line element of the reduced moduli space parametrised by $\tau = \tau_{s_2}$ is given by:
\begin{equation}
{\rm d}s_r^2 = \frac{3 \beta_2}{4 \mathcal{V}_{\scriptscriptstyle 0} \sqrt{\tau}}\left(\frac{1 + \beta_1 \, \epsilon}{1 + \beta_1\,\epsilon + \beta_2\, \frac{\tau^{3/2}}{\mathcal{V}_{\scriptscriptstyle 0}}}\right) \,{\rm d} \tau^2\,\qquad\text{with}\qquad \epsilon \equiv \frac{\tau_{\scriptscriptstyle 0}^{3/2}}{\mathcal{V}_{\scriptscriptstyle 0}} \ll 1\,.
\label{dsSSC}
\end{equation}
In this case $a=0$ for each toric CY we have analysed, and so at one of the boundaries of $\mathcal{M}_r$ we find a singularity. In the full moduli space this corresponds to the face of the cone where the respective dP divisor shrinks to zero size. Despite that, the volume of the reduced moduli space becomes:
\begin{equation}
\Delta \phi = \frac{2}{\sqrt{3}} \sqrt{1 +\beta_1 \,\epsilon } \,{\rm arcsinh} \left( f_{\scriptscriptstyle 1}^{3/4} \sqrt{\frac{\beta_2}{\mathcal{V}_{\scriptscriptstyle 0} \left(1+ \beta_1 \,\epsilon\right)}} \right),
\label{DphiSSC}
\end{equation}
which, in turn, implies that $\Delta \phi$ is finite for all $\mathcal{V}_0,\tau_0>0$. Hence, in these cases $\mathcal{M}_r$ is compact due to the $1/\sqrt{\tau}$ behaviour of the metric near the singularity. Notice also that for $f_{\scriptscriptstyle 1} \rightarrow \infty$ the volume of $\mathcal{M}_r$ behaves logarithmically and, thus, satisfies the general bound~\eqref{Delta_bound}.
\item \textbf{K3 fibred vacua} with $n_{\scriptscriptstyle {\rm ddP}}=1$ and $n_{\scriptscriptstyle {\rm K3f}}=1$:
From the volume form (\ref{vol:K3}) and parameterising the flat direction with the K3 modulus, i.e.\ $\tau = \tau_f$, we find that:
\begin{equation}
{\rm d}s_r^2 = \frac{3 }{4 \tau^2} \left( 1 + \beta \,\epsilon \right) {\rm d} \tau^2 \,.
\label{K3_mod_spm}
\end{equation}
In this case $a>0$ for each toric CY we have analysed, and so the metric is regular at the boundaries of $\mathcal{M}_r$. This implies that $\mathcal{M}_r$ is compact. In fact, its volume looks like:
\begin{equation}
\Delta \phi = \frac{\sqrt{3}}{2} \sqrt{1 +\beta \,\epsilon}\ln\left( \frac{f_{\scriptscriptstyle 1}(\mathcal{V}_{\scriptscriptstyle 0},\tau_{\scriptscriptstyle 0})}{a\,\tau_{\scriptscriptstyle 0}}\right) \,,
\label{DphiK3}
\end{equation}
where the homogeneous function $f_{\scriptscriptstyle 1}$ can take two different forms:
\begin{enumerate}
\item[($i$)] $f_{\scriptscriptstyle 1}(\mathcal{V}_{\scriptscriptstyle 0},\tau_{\scriptscriptstyle 0}) = b \, \mathcal{V}_{\scriptscriptstyle 0}^{2/3} + \mathcal{O}(\epsilon)$
\item[($ii$)] $f_{\scriptscriptstyle 1}(\mathcal{V}_{\scriptscriptstyle 0},\tau_{\scriptscriptstyle 0}) = b \, \mathcal{V}_{\scriptscriptstyle 0}/\sqrt{\tau_{\scriptscriptstyle 0}}$
\end{enumerate}
where $b>0$. Clearly (\ref{DphiK3}) satisfies the general distance bound (\ref{Delta_bound}).
\item \textbf{SSC-like vacua} with $n_{\scriptscriptstyle {\rm ddP}}=1$ and $n_{\scriptscriptstyle {\rm K3f}}=0$:
Starting from the overall volume (\ref{vol:ssc}), if the flat direction of the reduced moduli space is parameterised by $\tau=\gamma_1\tau_s +\gamma_2\tau_*$,
the line element of $\mathcal{M}_r$ takes the same form as in (\ref{dsSSC}). However this case is different since $a>0$ for each toric CY we have analysed. Thus the singularity of the metric at $\tau=0$ is outside the reduced moduli space, implying that $\mathcal{M}_r$ is compact. The field range $\Delta\phi$ here takes the same form as in (\ref{DphiSSC}) but with an additional subleading negative contribution coming from the lower boundary of $\mathcal{M}_r$.
\end{itemize}
For the remaining \textbf{structureless} LVS threefolds, it is in general not possible to derive explicit formulae for the metric on $\mathcal{M}_r$ and the associated field range since the hypersurface eq.~\eqref{eq:hypersurface} is equivalent to an irreducible polynomial of degree 6 in the divisor volumes, and so it cannot be solved explicitly for one of these volumes. However, we can still claim that ${\rm Vol}(\mathcal{M}_r)$ has to be finite via the following argument. For each structureless case, we checked that ($i$) the hypersurface (\ref{eq:hypersurface}) intersects precisely two faces of the K\"ahler cone for all values $\mathcal{V}_{\scriptscriptstyle 0}>0$ and $\tau_{\scriptscriptstyle 0} > 0$, and ($ii$) the full metric on the interior of two faces of the cone is non-singular.\footnote{With interior of the faces we mean the faces without the edges defined by the intersections of two faces.} This implies that also the induced metric on the two boundary points of $\mathcal{M}_r$ has to be non-singular, and so $\mathcal{M}_r$ has to be compact since it can be parametrised by a finite interval.
\section{Implications}
\label{Concl}
In this section we discuss phenomenological and conceptual implications of our result.
\subsection{Phenomenological implications}
Our result has several implications for inflationary cosmology and moduli stabilisation:
\begin{itemize}
\item \textbf{Detectable tensor modes}
The number of e-foldings of inflation between horizon exit $\phi_*$ and the end of inflation $\phi_{\rm end}$ is given by:
\begin{equation}
N_e = \int_{\phi_{\rm end}}^{\phi_*} \sqrt{\frac{8}{r(\phi)}} \,{\rm d}\phi\,,
\label{Ne}
\end{equation}
where $r$ is the tensor-to-scalar ratio. Present CMB data constrain $r(\phi_*) \lesssim 0.1$ \cite{Ade:2015xua} while future cosmological observations should be able to probe values of $r(\phi_*)$ of order $0.05$ \cite{Cabass:2015jwe}. We stress that a detection of primordial gravity waves requires, strictly speaking, a large tensor-to-scalar ratio just at horizon exit. Hence, in models where $r(\phi)$ varies significantly during inflation, $r$ could be of order $0.01$ at horizon exit and then quickly decrease during the final $40$-$50$ e-foldings. As can be easily seen from (\ref{Ne}), if $r$ is very small, $N_e$ becomes quickly very large even for a small, i.e.\ a sub-Planckian, field range. Despite this observation, let us mention that so far no sub-Planckian model with detectable $r$ has been found in the context of LVS inflationary models \cite{Conlon:2005jm, Cicoli:2008gp, Broy:2015zba, Burgess:2016owb, Cicoli:2016chb, Cicoli:2011ct, Cicoli:2015wja}.
Thus, our K\"ahler cone bound (\ref{Delta_bound}) does not necessarily imply a model-independent upper bound on the prediction of the tensor modes. However it can lead to very interesting constraints on large classes of inflationary models. If, for example, $r(\phi)$ is roughly constant during the whole inflationary evolution, (\ref{Ne}) gives (restoring the dependence on the Planck mass):
\begin{equation}
\frac{\Delta\phi}{M_p} \simeq \frac{N_e}{2}\, \sqrt{\frac{r(\phi_*)}{2}}\qquad\text{with}\qquad \Delta\phi\equiv \phi_* - \phi_{\rm end}\,.
\label{rconst}
\end{equation}
If we now use our geometrical constraint (\ref{Delta_bound}) on $\Delta\phi$, we can obtain a theoretical upper bound on the observed tensor-to-scalar ratio.
As shown in Fig.~\ref{histogs}, the LVS vacua which allow for the largest inflaton range are K3 fibred CY threefolds with $g_s=0.3$. In order to be very conservative, we therefore focus on this LVS subclass and use the maximal values of $\Delta\phi$ for $\mathcal{V}_{\scriptscriptstyle 0}=10^3$, $10^4$ and $10^5$ which are in the right ballpark to match the observed amplitude of the density perturbations \cite{Cicoli:2008gp, Broy:2015zba, Cicoli:2016chb, Cicoli:2011ct}. The results for $N_e=50$ are presented in Fig.~\ref{FigBound0} which implies $r(\phi_*) \lesssim 0.1$-$0.2$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.8\textwidth]{Bound0.pdf}
\caption{Inflaton range $\Delta\phi$ in units of $M_p$ as a function of the tensor-to-scalar ratio $r$ for constant $r$, and corresponding upper bounds for $g_s=0.3$, $N_e=50$ and $\mathcal{V}_{\scriptscriptstyle 0}=10^3$, $10^4$ and $10^5$.}
\label{FigBound0}
\end{center}
\end{figure}
This upper bound on the value of $r$ at horizon exit is approximately of the same order of magnitude as the one derived from present CMB data which give the
constraint $r(\phi_*) \lesssim 0.1$ \cite{Ade:2015xua}. Hence our geometrical upper bound on the inflaton range might not seem to rule out primordial gravity waves at the edge of detectability. However the results shown in Fig.~\ref{FigBound0} might very well be too naive for two main reasons: ($i$) the field range $\Delta\phi$ considered in (\ref{rconst}) does not include the post-inflationary region where the potential should develop a minimum, and so the additional requirement that also this region should be inside the CY K\"ahler cone, would necessarily make the upper bound on $r$ stronger; ($ii$) for most of the inflationary models, the approximation of constant tensor-to-scalar ratio during inflation is not very reliable.
For example, most of the string inflation models derived in the LVS framework feature a potential which in the inflationary region can very well be approximated as \cite{Conlon:2005jm, Cicoli:2008gp, Broy:2015zba, Burgess:2016owb, Cicoli:2016chb, Cicoli:2011ct, Cicoli:2015wja}:
\begin{equation}
V \simeq V_0\left(1-c_1\,e^{-c_2\phi}\right)\,,
\label{Vstring}
\end{equation}
which gives:
\begin{equation}
\epsilon = \frac12\left(\frac{V'}{V}\right)^2 \simeq \frac12\,c_1^2\,c_2^2\,e^{-2c_2\phi}\,.
\end{equation}
Defining the endpoint of inflation as $\epsilon(\phi_{\rm end})\simeq 1$ and the value of the tensor-to-scalar ratio at horizon exit as $r(\phi_*)=16\,\epsilon(\phi_*)$, (\ref{Ne}) can be solved explicitly to obtain (restoring again the dependence on $M_p$):
\begin{equation}
\frac{\Delta\phi}{M_p} \simeq \frac{N_e}{2}\, \sqrt{\frac{r(\phi_*)}{2}}\,\ln\left(\frac{4}{\sqrt{r(\phi_*)}}\right)\,.
\label{rStaro}
\end{equation}
Due to the extra logarithmic dependence compared with the relation (\ref{rconst}) for the case with $r$ constant, we now obtain a stronger upper bound on the predicted tensor-to-scalar ratio when we combine (\ref{rStaro}) with our geometrical constraint (\ref{Delta_bound}). Focusing again on K3 fibred LVS vacua with $g_s=0.3$ and considering the maximal values of $\Delta\phi$ for $\mathcal{V}_{\scriptscriptstyle 0}=10^3$, $10^4$ and $10^5$, we get the results shown in Fig.~\ref{FigBound} for $N_e=50$. Interestingly, we now find $r(\phi_*) \lesssim 0.02$ for $\mathcal{V}_{\scriptscriptstyle 0}=10^5$, $r(\phi_*) \lesssim 0.01$ for $\mathcal{V}_{\scriptscriptstyle 0}=10^4$ and $r(\phi_*) \lesssim 0.005$ for $\mathcal{V}_{\scriptscriptstyle 0}=10^3$. This result implies that the next generation of cosmological observations, which should be sensitive to values of $r(\phi_*)$ of order $0.05$ \cite{Cabass:2015jwe}, should not detect primordial gravity waves.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.8\textwidth]{Bound.pdf}
\caption{Inflaton range $\Delta\phi$ in units of $M_p$ as a function of the tensor-to-scalar ratio $r$ for inflationary potentials of the form (\ref{Vstring}), and corresponding upper bounds for $g_s=0.3$, $N_e=50$ and $\mathcal{V}_{\scriptscriptstyle 0}=10^3$, $10^4$ and $10^5$.}
\label{FigBound}
\end{center}
\end{figure}
\item \textbf{Large and small field models}
As shown in Fig.~\ref{histolb}, LVS single-field inflationary models built from CY threefolds with $h^{1,1}=3$ split into large and small field models characterised respectively by a trans- and a sub-Planckian inflaton field range. Models with $\Delta\phi \gtrsim M_p$ feature a K3 fibration and are about $25\%$ of the total, while models with $\Delta \phi\lesssim M_p$ are based on CY threefolds without a K3 fibred structure which constitute about $75\%$ of the total number of LVS vacua with $h^{1,1}$=3 and 1 flat direction. The reason why K3 fibrations feature a much larger moduli space is that they saturate the inequality (\ref{Delta_bound}) which, in turn, follows from the form (\ref{K3_mod_spm}) of their moduli space metric.
These purely geometric considerations, combined with the particular shape of the inflationary potential, determine the inflaton range needed to obtain enough e-foldings of inflation to solve the flatness and horizon problems. In models where the inflaton is a K3 fibre modulus, $\Delta\phi$ can be either trans-Planckian, if the inflationary potential is generated by perturbative corrections \cite{Cicoli:2008gp, Broy:2015zba, Cicoli:2016chb, Burgess:2016owb, Cicoli:2016xae, Cicoli:2017axo}, or sub-Planckian, if $V(\phi)$ is developed by non-perturbative effects \cite{Cicoli:2011ct}. On the other hand, scenarios where the inflaton is the volume of either a blow-up mode \cite{Conlon:2005jm,Cicoli:2017shd} or a Wilson divisor \cite{Blumenhagen:2012ue} are necessarily small field models where the inflationary potential can be generated only at non-perturbative level since in these cases string loop or higher derivative effects would require a trans-Planckian $\Delta\phi$ in contradiction with our analysis.
Notice that the allowed inflaton field range depends on the stabilised values of the volume of the CY threefold $\mathcal{V}_{\scriptscriptstyle 0}$ and of a diagonal dP divisor $\tau_{\scriptscriptstyle 0}$ which, in turn, depends on the string coupling $g_s$. The upper bounds presented in Fig.~\ref{histolb} are referred to the values $10^3 \leq \mathcal{V}_{\scriptscriptstyle 0} \leq 10^5$ which are in general required to reproduce the observed amplitude of the scalar modes, and $g_s=0.1$ which is a standard reference value to trust the perturbative expansion. The distribution of moduli space sizes for SSC cases do not change significantly for different values of $g_s$ since in these cases the lower bound on the inflaton range is $\tau_{\scriptscriptstyle 0}$-independent, as can be seen from the fact that $a=0$ in (\ref{eq:field_range}). Also SSC-like geometries are not very sensitive to the value of $g_s$ since their moduli space is on average smaller than the one of SSC LVS vacua. The situation is different for K3 fibred LVS vacua which, as illustrated in Fig.~\ref{histogs}, can slightly increase their allowed $\Delta\phi$ for larger values of the string coupling even if the inflaton field range can never be larger than $10$ Planck units for values of $g_s$ which are still compatible with perturbation theory.
\item \textbf{Consistency of inflationary model building}
If we restrict our analysis to large field models with an inflationary potential of the form (\ref{Vstring}) which emerge naturally in LVS compactifications, our geometrical upper bound (\ref{Delta_bound}) does not just affect the prediction of primordial tensor modes but it sets also strong constraints on the allowed underlying parameter space. In fact, a fully consistent string model should not just be able to drive enough e-foldings of inflation but should also satisfy several conditions like: ($i$) a viable global embedding into a concrete CY orientifold compactification with explicit brane set-up and fluxes compatible with tadpole cancellation; ($ii$) the presence of a chiral visible sector with a GUT or an MSSM-like gauge group whose degrees of freedom can be successfully excited at reheating after the end of inflation; ($iii$) the generation of the correct amplitude of the density perturbations by the inflaton fluctuations; ($iv$) a safe decoupling of all the modes orthogonal to the inflaton direction so that the inflationary dynamics is stable; ($v$) the minimum of the potential, and not just the inflationary plateau, should be inside the CY K\"ahler cone; ($vi$) the EFT should be fully under control throughout the whole inflationary dynamics, i.e.\ no KK mode should become light during inflation.
Explicit chiral CY embeddings of fibre inflation models have been recently derived in \cite{Cicoli:2017axo} but not all the models are able to satisfy simultaneously all these constraints. Using the inflaton field range (\ref{DphiK3}) for a given value of $\mathcal{V}_{\scriptscriptstyle 0}$, models where the form of $f_{\scriptscriptstyle 1}$ satisfy case ($ii$) seem more promising than case ($i$) for cosmological applications since they feature a larger range for the inflaton field which can be used to achieve enough e-foldings together with a large value of the tensor-to-scalar ratio of order $r\sim 0.005 - 0.01$.
\item \textbf{Curvaton-like models}
The main condition which becomes very constraining when combined with our upper bound (\ref{Delta_bound}), is the requirement to match the COBE normalisation for the amplitude of the density perturbations. In fact, if we focus again on the inflationary models described by the potential (\ref{Vstring}), the COBE normalisation condition fixes $V_0$ which is a function of the overall volume $\mathcal{V}_{\scriptscriptstyle 0}$. In turn, given that the upper bound $\Delta \phi \lesssim \ln\mathcal{V}$ depends on the CY volume $\mathcal{V}$, the maximal size of the reduced moduli space is fixed by the observation of the amplitude of the density perturbations. This typically requires values of the internal volume of order $\mathcal{V}_{\scriptscriptstyle 0} \sim 10^3$-$10^5$, depending on the exact value of the VEV of the flux superpotential.
However multi-field models with more than one field lighter than the Hubble constant during inflation, might be characterised by non-standard mechanisms for the generation of the density perturbations. In these curvaton-like models, the inflaton fluctuations give a negligible contribution to the amplitude of the density perturbations which are instead generated by another light field \cite{Burgess:2010bz,Cicoli:2012cy}. In LVS models natural candidates for curvaton-like fields are very light axions associated with large 4-cycles. These scenarios might open up the possibility to match the COBE normalisation condition for larger values of $\mathcal{V}_{\scriptscriptstyle 0}$, leading to a larger allowed field range for the inflaton field.
\item \textbf{$\alpha$-attractors}
As pointed out in \cite{Burgess:2016owb} (see in particular Fig. 3 on page 23), LVS inflationary models characterised by the scalar potential (\ref{Vstring}) represent stringy realisations of $\alpha$-attractor and Starobinsky-like models \cite{Kallosh:2013maa, Kallosh:2015zsa, Carrasco:2015pla} which seem to describe present CMB data rather well. A peculiar feature of the scalar potential (\ref{Vstring}) is the presence of an inflationary plateau which in standard $\alpha$-attractor models, derived just within the supergravity framework, is assumed to be of arbitrary length in field space. However we have shown that, when these promising inflationary models are embedded in concrete string compactifications, an additional consistency constraint should be taken into account. In fact, the K\"ahler cone of the underlying CY threefold forces the inflationary plateau to be of finite size with important implications for the prediction of crucial cosmological observables like $N_e$ or $r$.
\item \textbf{Moduli stabilisation}
Besides inflation, our results constrain also any moduli stabilisation scenario which exploits subleading perturbative or non-perturbative effects to lift the flat directions which parametrise the reduced moduli space $\mathcal{M}_r$. One such a scenario was discussed in \cite{Cicoli:2016chb} where these moduli were lifted in a model-independent way for any LVS vacuum by subleading $F^4$-type corrections \cite{Ciupke:2015msa}. In this case the moduli VEVs depend purely on the topological data of $X$ as well as the values of $\mathcal{V}_{\scriptscriptstyle 0}$ and $\tau_{\scriptscriptstyle 0}$. Indeed, direct computation for some examples where the exact Mori cone is available shows that these vacua still lie within $\mathcal{M}_r$. It would also be interested to extend our results to heterotic vacua which feature an LVS-like moduli stabilisation procedure \cite{Cicoli:2013rwa}.
\end{itemize}
\subsection{Conceptual implications}
Let us now discuss some more theoretical implications of our result:
\begin{itemize}
\item \textbf{LVS moduli space conjecture}
The main result of this paper is the proof of the compactness of the 1-dimensional reduced moduli space $\mathcal{M}_r$ of LVS CY threefolds with $h^{1,1}=3$. Following the intuitive picture presented in the introduction, $\mathcal{M}_r$ can be seen to be parameterised by a divisor which cannot collapse or grow to infinite volume since it is obstructed by the fact that the sizes of both the overall volume and a diagonal dP divisor are kept fixed. This simple understanding has led us to conjecture the validity of this result also for LVS vacua with arbitrary $h^{1,1}$ where the reduced the moduli space $\mathcal{M}_r$ becomes higher-dimensional. Similarly to the $h^{1,1}=3$ case, we expect CY threefolds with several K3 fibrations to feature the largest reduced moduli space also for $h^{1,1}>3$.
\item \textbf{Comparison with weak gravity and swampland conjectures}
The formulation (\ref{Delta_bound2}) of the LVS moduli space conjecture in terms of the cut-off $\Lambda$ of the EFT treatment, allows us to compare our results with other bounds on field range excursions in moduli space which have been recently invoked by using the weak gravity and swampland conjectures \cite{Baume:2016psm,Klaewer:2016kiy,Blumenhagen:2017cxt,Palti:2017elp,Hebecker:2017lxm}.
Our upper bound (\ref{Delta_bound2}) looks very similar to the ones derived in \cite{Baume:2016psm,Klaewer:2016kiy,Blumenhagen:2017cxt,Palti:2017elp,Hebecker:2017lxm}. However our constraint comes from purely geometric considerations associated with the size of the K\"ahler cone of the underlying CY compactification manifold, while the other bounds are basically derived by requiring a trustable EFT approach. Due to this different origin, our geometrical upper bound on the inflaton field range can be considered to be both weaker and stronger than the ones coming from the the weak gravity and swampland conjectures, depending on point of view. In fact, it can be thought to be weaker in the sense that, as the inflaton travels in field space, it would reach the point where some KK or winding modes become light before hitting the walls of the K\"ahler cone. On the contrary, our result (\ref{Delta_bound2}) can also be thought to be stronger, i.e.\ more constraining, since it sets an upper bound for $\Delta\phi$ which might hold even if the effect of heavy KK or stringy modes would be taken into account. These corrections would definitely modify both the background geometry and the definition of the correct K\"ahler coordinates, and so our upper bound (\ref{Delta_bound2}) would certainly be quantitatively modified. However we believe that our main result, i.e. the fact that the volume of the reduced moduli space is finite, would still be qualitatively correct since the exponentially large volume characteristic of LVS models should provide a powerful tool to control the effect of $\alpha'$ and KK corrections.
\item \textbf{Systematic search of new LVS vacua}
In order to demonstrate the compactness of the LVS reduced moduli space in the 1-dimensional case, we had to determine the complete ensemble of LVS vacua with 3 K\"ahler moduli. Hence we performed a systematic search through the existing list of CY threefolds realised as hypersurfaces embedded in toric varieties \cite{Kreuzer:2000xy}. This analysis led to the discovery of several new LVS vacua compared with similar searches performed in the past \cite{Gray:2012jy,Altman:2017vzk}. However, these searches were looking for the desired structure in the intersection polynomial, and so they can be considered as complementary to our approach which is instead based on scans for divisor topologies. As the authors of \cite{Gray:2012jy, Altman:2017vzk} did not claim generality, it is not surprising that our results differ from theirs. In particular, we find 25 new LVS geometries for $h^{1,1}=3$ and 561 new LVS geometries for $h^{1,1}=4$ compared to \cite{Altman:2017vzk}. For $h^{1,1}=2$ our results precisely match with those of \cite{Altman:2017vzk}, most likely due to the simplicity of this subclass of LVS vacua. Moreover, our results are also compatible with the search for K3 fibred LVS geometries performed in \cite{Cicoli:2011it}.
Notice also that Tab.~\ref{tab:h113} shows the presence of 43 CY geometries which have all the required properties to realise fibre inflation \cite{Cicoli:2008gp} since they feature a K3 fibred structure together with a diagonal dP divisor. This is in agreement the scanning results of \cite{Cicoli:2016xae} where a non-chiral global embedding of fibre inflation models has been presented.\footnote{Unlike the present case, the earlier scan in \cite{Cicoli:2016xae} used 526 examples corresponding to different triangulations, some of which may however correspond to the same CY geometry.}
Let us finally point out that the scanning results presented in Tab.~\ref{tab:gen_res} show that LVS vacua are highly generic in this corner of the type IIB landscape since a very large percentage of toric CY hypersurface threefolds allow for the presence of an LVS minimum, namely 56.4$\%$ for $h^{1,1}=2$, 43.4$\%$ for $h^{1,1}=3$ and 37.5$\%$ for $h^{1,1}=4$.
\end{itemize}
\section{Conclusions}
\label{Concl2}
In this paper we investigated the space of flat directions of IIB Calabi-Yau orientifold models after partial moduli stabilization in an LVS vacuum.\footnote{Note that the stabilization in an LVS vacuum is crucial to derive the field-range bounds, for other moduli stabilization proposals such as KKLT \cite{Kachru:2003aw} our argument would not imply any field-range bound.} Our main result is captured by an LVS moduli space conjecture which states that due to the CY K\"ahler cone this moduli space is compact and that its volume respects the bound in eq.~\eqref{Delta_bound2}. This bound takes a form similar to recent results in relation to the Swampland conjecture \cite{Baume:2016psm,Klaewer:2016kiy,Blumenhagen:2017cxt,Palti:2017elp,Hebecker:2017lxm}. We supported our claim by proving it for a complete subset of CY geometries, namely the $h^{1,1}=3$-Kreuzer-Skarke list giving rise to one-dimensional moduli spaces. We demonstrated how the compactness arises as a result of the CY K\"ahler cone conditions and their determination posed the main computational challenge in our analysis. To this end it was necessary to determine the ensemble of LVS vacua. By searching for diagonal del Pezzo divisors for $h^{1,1}=3,4$ we found 586 previously unknown CY LVS-geometries. In total roughly every second CY geometry in this corner of the landscape of toric hypersurface threefolds admits at least one LVS-vacuum.
The bound in eq.~\eqref{Delta_bound2} gives rise to a new strong constraint on inflationary model building in LVS, constraining any model where the inflaton is described by a K\"ahler modulus \cite{Conlon:2005jm, Cicoli:2008gp, Broy:2015zba, Burgess:2016owb, Cicoli:2016chb, Cicoli:2011ct, Cicoli:2015wja}. The distribution of the field-ranges displayed in fig.~\eqref{histolb} and fig.~\eqref{histogs} splits into two practically disjoint pieces which correspond to CYs featuring a K3-fibration and those without such a fibration-structure. The former class induce super-Planckian field-ranges and are, therefore, more likely to lead to an observable tensor-to-scalar ratio, while the remaining geometries feature mostly sub-Planckian field-spaces.
In the future it would be desirable to explicitly prove the conjectured compactness of the reduced moduli space of LVS vacua with $h^{1,1}>3$.
\acknowledgments
We would like to thank Roberto Valandro and Ross Altman for useful discussions. The work of PS has been supported by the ERC Advanced Grant ``String Phenomenology in the LHC Era" (SPLE) under contract ERC-2012-ADG-20120216-320421. PS is also grateful to INFN-Bologna for hospitality during the initial stage of this work.
\bibliographystyle{JHEP}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,913 |
Xterra World Championships; Super League Malta; Air Relax recovery boots.
The Xterra World Championships on Sunday was one of the hardest in recent memory, with a week of rain prior to the race turning the bike and run course into thick, energy sapping mud. Rom Akerson, a relatively unknown triathlete from Costa Rica took the men's title, while 2x champion Lesley Paterson took the women's.
Super League Malta took place this weekend, and as always the racing was intense and perfect to watch for winter trainer workouts. Frenchmen Vincent Luis took the men's title, while the women's podium was dominated by Americans with Katie Zaferes taking the top spot.
The swim was cancelled at 70.3 Waco in Texas this weekend. First pack swimmer Andrew Starykowicz didn't let that hold him back, as he averaged an amazing 28.5 mph average on the 56mile bike (1hr 58min bike time) then held on during the run to win over a competitive field. Haley Chura took the women's title with a dominating run split.
The aftermath of Hurricane Michael has required the Florida Ironman to move the race date from this Saturday to Sunday, and the race location from Panama City to Haines City (about 6hrs drive away).
Great insight from Daniela Ryf's coach on the challenges Daniela has had to overcome over the last two seasons, and why she may be the greatest Ironwoman of all time.
Progress is coming along well on the new shop, and thank you to everyone who has stopped by to tour the building. Our time frame for the move is still relatively open, as we are waiting on city inspections and permits. We'll post pictures of our progress soon.
Pro triathletes Guy Crawford and Kate Bevilaqua let me take some of their recovery tools for a 'test ride' yesterday. I was impressed with the Compex Muscle Stimulator and Air Relax Recovery boots, and even more impressed with how much the price point has dropped on these type of recovery tools over the years.
-Rom Akerson after his win in Maui at the Xterra World Champs. | {
"redpajama_set_name": "RedPajamaC4"
} | 624 |
Nobleton kan syfta på följande platser:
Kanada
Nobleton, Ontario, ort,
USA
Nobleton (ort i USA), Florida, Hernando County,
Robotskapade Kanadaförgreningar
Robotskapade USAförgreningar | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,013 |
Nexus S and Nexus S 4G get early Ice Cream Sandwich ROMs
Michael Crider
Those lucky Nexus owners, they get everything sooner. In this case, it's a relatively stable AOSP version of Ice Cream Sandwich for the previous generation of Google's developer phone. The ROM is available for eager Android fans right now – the only thing that appears to be missing is a reliable video record function.
The work comes from XDA member "kwiboo", and he's been hard at work on the software for the last few days; the two separate ROMs are already on their second version. Functions are mostly complete, though the WiFi drivers need a quick patch. Since Android's source code doesn't include Gmail, YouTube, the Android Market and similar Google-branded apps, those packages are being directly loaded from the Galaxy Nexus.
Check out the ROM in action below:
https://www.youtube.com/watch?v=8h0lIajxzoM
On the example phone at least, Ice Cream Sandwich is running admirably fast. Take particular note of the ICS software buttons – or rather don't, because they aren't there. ICS is able to forgo its software buttons if the hardware it's running on already has hardware-based navigation buttons. The Ice Cream Sandwich ROMs should be coming fast and furious for the next few weeks as more and more modders get a hang of the source code, and the uber-popular CyanogenMod should release its version of ICS early next year.
[via PD]
Nexus S 4G | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,184 |
Metro Manila, Filippseyjar
Menning og saga
Mannkynssaga
Travel Back In Time At Intramuros
4,98 (51)Einkunn 4,98 af 5 í 51 umsögn.
·Manila, Filippseyjar
Upplifun sem Omar býður upp á
Innifalið: miðar
We will meet at the Plaza San Luis Complex within the historic walled city of Intramuros.
We will travel back in time, to the Old Manila, as occupied by the Spaniards by taking a walking tour of a typical Filipino Spanish House called Casa Manila.
Cross the street to the Oldest Stone Church of the country and explore the Museum that is used as a monastery and marvel at the illusion paintings that adorns the church walls. The church is one of the 4 Baroque Churches that is a Unesco World Heritage Site for the Philippines.
Walk further to see the imposing Manila Cathedral before entering Fort Santiago. The old gateway to the military fort that has become a witness to many historical events.
Not many people know that Intramuros is the only place to try Sampaguita (Jasmine) Ice Cream. Trying a scoop is a unique sensory experience! Before we end the day, will take you to the only restaurant that serves them for an optional sampling!
Taxi, is the most convenient transportation from any hotels, but Grab (an app) is the most reliable where the fare is listed prior to departure.
Cross the street to the Oldest Stone Church of the country and explore the Museum that is used as a monastery and marvel at the illusion paintings that adorns the church walls.…
Omar lofar öryggi
Hvað er innifalið
Miðar
All museum entrance tickets and guide fee
Þetta er gestgjafi þinn, Omar
51 umsögn
I'm Omar, an Architect.
2 years ago I retired from my day job to pursue a startup featured on CNN Philippines, The Final Pitch (PH version of Shark Tank).
Initially, I wanted to establish my own business to have control of my time & finance my love for travels documented on my blog (www.moleinthefoot.blogspot.com)
The travel bug bit me first, and is now a Department of Tourism Accredited Tour Guide.
Together with my batch mates in the tourism guiding course, we're taking this opportunity to improve and show guests the beauty of our city. I have 10 experiences to offer.
I am now creating a tour agency with my batch mates as my core team.
On a personal note, I was raised in the historical part of Manila, & my university within the walls of Intramuros.
Check out my IG: @omar.palero, @travellingphilippine, & @rainbowasiatravel
Together with my batch mates in the tourism guiding course, we're takin…
Casa Manila Complex
A replica of an Old Filipino Spanish House within a complex that includes souvenir shops, restaurants and hotels
San Agustin Complex
The Oldest Church in the Philippines that also has a Monastery that is converted into a museum, that showcase religious arts
Manila Cathedral
The seat of the Archbishop of Manila
A park complex rich in history, that features a replica of the National Heroes residence, Jose Rizal
4,98 (51 umsögn)
Einkunn 4,98 af 5 í 51 umsögn.
James & Geertje
We had a great time exploring Intramuros with Omar. We arrived in Manila just before the covid-19 lockdown, so a lot of the sights were actually closed, but Omar showed us an adjusted walk, and drive. He picked us up at the hotel, and during the drive to Intramuros talked about the things we were seeing. It was a great experience and Omar is a funny, knowledgeable and kind person.
We had a great time exploring Intramuros with Omar. We arrived in Manila just before the covid-19 lockdown, so a lot of the sights were actually closed, but Omar showed us an adju…
Omar's "Travel Back In Time At Intramuros" experience was the type of walking tour I was looking for. Being a Filipino-American born in the US, while I could visit Makati, Poblacion, BGC and all of its nightlife and expansive shopping malls, I sought to learn and experience more of Manila's rich history and heritage...and what no better way than inside the start of it all—Intramuros. I joined a small group and Omar took us through churches, museums, and Fort Santiago—it really did feel like I time traveled hundreds of years ago. During the entirety Omar provided excellent supplemental commentary. It's very evident Omar is very passionate about the Philippines' history and no doubt is dedicated to share his in-depth knowledge of it with others. He is very warm and friendly, even showing a funny side delivering some witty jokes. I felt Omar's extensive knowledge of the areas and history made this experience very personal and authentic. It ended with me knowing more about the Manila's history and also making a new friend. Highly recommended.
Omar's "Travel Back In Time At Intramuros" experience was the type of walking tour I was looking for. Being a Filipino-American born in the US, while I could visit Makati, Poblacio…
What a wonderful tour. It was interesting to learn more about the impact of the Spaniards on the Philippines. Omar really captured what it was like then. He was very patient with all of our questions. Omar was very flexible with the start/finish time of the trip.
What a wonderful tour. It was interesting to learn more about the impact of the Spaniards on the Philippines. Omar really captured what it was like then. He was very patient with a…
Omar was an amazing tour guide. He provided Filipino history, architectural information, personal insights, jokes, and more. He was a docent that skipped the things we didn't care about and went into more detail when we expressed interest in something else. I did a self-guided tour of the area 8 years ago with my family but with a knowledgeable guide like Omar, this was 1000x better and more interesting! If we had more time, my husband and I would definitely take one of Omar's other tours. 10/10 I'd recommend his services!
Omar was an amazing tour guide. He provided Filipino history, architectural information, personal insights, jokes, and more. He was a docent that skipped the things we didn't care…
Omar was amazing, met us near a beautiful open fountain, and gave us a wonderful tour with witty jokes and thoughtful interpretations of everything we say. I learned a lot about the history of the Philippines from him. At the end of the tour, we ended at beautiful place to take in the sunset. Special note: wasn't use to the heat and a jet lagged from flying, had a really bad headache, Omar drove me to a chemist and then back to my hotel. Great guy.
Omar was amazing, met us near a beautiful open fountain, and gave us a wonderful tour with witty jokes and thoughtful interpretations of everything we say. I learned a lot about th…
I showed up late, but Omar was very accommodating and ensured that I caught up with the group. The tour was very comprehensive, taking us through a number of churches and cathedrals to a discovery of the old Spanish citadel, including a museum dedicated to Filipino independence and its national hero, Jose Rizal. Omar was very knowledgeable about the historic, political and cultural significance of the locations we visited. He dedicated additional time to explore the main cathedral and graciously offered to give me and other guests a ride back to our hotels in Makati despite having a personal appointment to make. Great experience with Omar!
I showed up late, but Omar was very accommodating and ensured that I caught up with the group. The tour was very comprehensive, taking us through a number of churches and cathedra…
Sýna 51 umsögn
Extra cash should you want to indulge! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,384 |
{"url":"https:\/\/en.wikipedia.org\/wiki\/Forecasting","text":"# Forecasting\n\nForecasting is the process of making statements about events whose actual outcomes (typically) have not yet been observed. A commonplace example might be estimation of some variable of interest at some specified future date. Prediction is a similar, but more general term. Both might refer to formal statistical methods employing time series, cross-sectional or longitudinal data, or alternatively to less formal judgemental methods. Usage can differ between areas of application: for example, in hydrology, the terms \"forecast\" and \"forecasting\" are sometimes reserved for estimates of values at certain specific future times, while the term \"prediction\" is used for more general estimates, such as the number of times floods will occur over a long period.\n\nRisk and uncertainty are central to forecasting and prediction; it is generally considered good practice to indicate the degree of uncertainty attaching to forecasts. In any case, the data must be up to date in order for the forecast to be as accurate as possible.[1]\n\n## Categories of forecasting methods\n\n### Qualitative vs. quantitative methods\n\nQualitative forecasting techniques are subjective, based on the opinion and judgment of consumers, experts; they are appropriate when past data are not available. They are usually applied to intermediate- or long-range decisions. Examples of qualitative forecasting methods are[citation needed] informed opinion and judgment, the Delphi method, market research, and historical life-cycle analogy.\n\nQuantitative forecasting models are used to forecast future data as a function of past data; they are appropriate when past data are available. These methods are usually applied to short- or intermediate-range decisions. Examples of quantitative forecasting methods are[citation needed] last period demand, simple and weighted N-Period moving averages, simple exponential smoothing, and multiplicative seasonal indexes.\n\n### Na\u00efve approach\n\nNa\u00efve forecasts are the most cost-effective objective forecasting model, and provide a benchmark against which more sophisticated models can be compared. For stationary time series data, this approach says that the forecast for any period equals the historical average. For time series data that are stationary in terms of first differences, the na\u00efve forecast equals the previous period's actual value.\n\n### Time series methods\n\nTime series methods use historical data as the basis of estimating future outcomes.\n\ne.g. Box-Jenkins\n\n### Causal \/ econometric forecasting methods\n\nSome forecasting methods try to identify the underlying factors that might influence the variable that is being forecast. For example, including information about climate patterns might improve the ability of a model to predict umbrella sales. Forecasting models often take account of regular seasonal variations. In addition to climate, such variations can also be due to holidays and customs: for example, one might predict that sales of college football apparel will be higher during the football season than during the off season.[2]\n\nSeveral informal methods used in causal forecasting do not employ strict algorithms[clarification needed], but instead use the judgment of the forecaster. Some forecasts take account of past relationships between variables: if one variable has, for example, been approximately linearly related to another for a long period of time, it may be appropriate to extrapolate such a relationship into the future, without necessarily understanding the reasons for the relationship.\n\nCausal methods include:\n\n\u2022 Regression analysis includes a large group of methods for predicting future values of a variable using information about other variables. These methods include both parametric (linear or non-linear) and non-parametric techniques.\n\nQuantitative forecasting models are often judged against each other by comparing their in-sample or out-of-sample mean square error, although some researchers have advised against this.[4]\n\n### Judgmental methods\n\nJudgmental forecasting methods incorporate intuitive judgements, opinions and subjective probability estimates.\n\n### Artificial intelligence methods\n\nOften these are done today by specialized programs loosely labeled\n\n## Forecasting accuracy\n\nThe forecast error is the difference between the actual value and the forecast value for the corresponding period.\n\n$\\ E_t = Y_t - F_t$\n\nwhere E is the forecast error at period t, Y is the actual value at period t, and F is the forecast for period t.\n\nMeasures of aggregate error:\n\n Mean absolute error (MAE) $\\ MAE = \\frac{\\sum_{t=1}^{N} |E_t|}{N}$ Mean Absolute Percentage Error (MAPE) $\\ MAPE = \\frac{\\sum_{t=1}^N |\\frac{E_t}{Y_t}|}{N}$ Mean Absolute Deviation (MAD) $\\ MAD = \\frac{\\sum_{t=1}^{N} |E_t|}{N}$ Percent Mean Absolute Deviation (PMAD) $\\ PMAD = \\frac{\\sum_{t=1}^{N} |E_t|}{\\sum_{t=1}^{N} |Y_t|}$ Mean squared error (MSE) or Mean squared prediction error (MSPE) $\\ MSE = \\frac{\\sum_{t=1}^N {E_t^2}}{N}$ Root Mean squared error (RMSE) $\\ RMSE = \\sqrt{\\frac{\\sum_{t=1}^N {E_t^2}}{N}}$ Forecast skill (SS) $\\ SS = 1- \\frac{MSE_{forecast}}{MSE_{ref}}$ Average of Errors (E) $\\ \\bar{E}= \\frac{\\sum_{i=1}^N {E_i}}{N}$\n\nBusiness forecasters and practitioners sometimes use different terminology in the industry. They refer to the PMAD as the MAPE, although they compute this as a volume weighted MAPE.[citation needed] For more information see Calculating demand forecast accuracy.\n\n## Applications of forecasting\n\nClimate change and increasing energy prices have led to the use of Egain Forecasting for buildings. This attempts to reduce the energy needed to heat the building, thus reducing the emission of greenhouse gases. Forecasting is used in Customer Demand Planning in everyday business for manufacturing and distribution companies.\n\nForecasting has also been used to predict the development of conflict situations. Forecasters perform research that uses empirical results to gauge the effectiveness of certain forecasting models.[5] However research has shown that there is little difference between the accuracy of the forecasts of experts knowledgeable in the conflict situation and those by individuals who knew much less.[6]\n\nSimilarly, experts in some studies argue that role thinking[clarification needed] does not contribute to the accuracy of the forecast.[7] The discipline of demand planning, also sometimes referred to as supply chain forecasting, embraces both statistical forecasting and a consensus process. An important, albeit often ignored aspect of forecasting, is the relationship it holds with planning. Forecasting can be described as predicting what the future will look like, whereas planning predicts what the future should look like.[8][9] There is no single right forecasting method to use. Selection of a method should be based on your objectives and your conditions (data etc.).[10] A good place to find a method, is by visiting a selection tree. An example of a selection tree can be found here.[11] Forecasting has application in many situations:\n\n## Limitations\n\nLimitations pose barriers beyond which forecasting methods cannot reliably predict.\n\n### Performance limits of fluid dynamics equations\n\nAs proposed by Edward Lorenz in 1963, long range weather forecasts, those made at a range of two weeks or more, are impossible to definitively predict the state of the atmosphere, owing to the chaotic nature of the fluid dynamics equations involved. Extremely small errors in the initial input, such as temperatures and winds, within numerical models double every five days.[13]\n\n### Complexity introduced by the technological singularity\n\nThe technological singularity is the theoretical emergence of superintelligence through technological means.[14] Since the capabilities of such intelligence would be difficult for an unaided human mind to comprehend, the technological singularity is seen as an occurrence beyond which events cannot be predicted.\n\nRay Kurzweil predicts the singularity will occur around 2045 while Vernor Vinge predicts it will happen some time before 2030.\n\n## References\n\n1. ^ Scott Armstrong, Fred Collopy, Andreas Graefe and Kesten C. Green. \"Answers to Frequently Asked Questions\". Retrieved May 15, 2013.\n2. ^ Nahmias, Steven (2009). Production and Operations Analysis.\n3. ^ Ellis, Kimberly (2008). Production Planning and Inventory Control Virginia Tech. McGraw Hill. ISBN\u00a0978-0-390-87106-0.\n4. ^ J. Scott Armstrong and Fred Collopy (1992). \"Error Measures For Generalizing About Forecasting Methods: Empirical Comparisons\". International Journal of Forecasting 8: 69\u201380.\n5. ^ J. Scott Armstrong, Kesten C. Green and Andreas Graefe (2010). \"Answers to Frequently Asked Questions\".\n6. ^ Kesten C. Greene and J. Scott Armstrong (2007). \"The Ombudsman: Value of Expertise for Forecasting Decisions in Conflicts\". Interfaces (INFORMS) 0: 1\u201312.\n7. ^ Kesten C. Green and J. Scott Armstrong (1975). \"Role thinking: Standing in other people\u2019s shoes to forecast decisions in conflicts\". Role thinking: Standing in other people\u2019s shoes to forecast decisions in conflicts 39: 111\u2013116.\n8. ^ \"FAQ\". Forecastingprinciples.com. 1998-02-14. Retrieved 2012-08-28.\n9. ^ Kesten C. Greene and J. Scott Armstrong. [http:\/\/www.qbox.wharton.upenn.edu\/documents\/mktg\/research\/INTFOR3581%20-%20Publication% 2015.pdf \"Structured analogies for forecasting\"] (PDF). qbox.wharton.upenn.edu.\n10. ^ \"FAQ\". Forecastingprinciples.com. 1998-02-14. Retrieved 2012-08-28.\n11. ^ \"Selection Tree\". Forecastingprinciples.com. 1998-02-14. Retrieved 2012-08-28.\n12. ^ J. Scott Armstrong (1983). \"Relative Accuracy of Judgmental and Extrapolative Methods in Forecasting Annual Earnings\". Journal of Forecasting 2: 437\u2013447.\n13. ^ Cox, John D. (2002). Storm Watchers. John Wiley & Sons, Inc. pp.\u00a0222\u2013224. ISBN\u00a00-471-38108-X.\n14. ^ Superintelligence. Answer to the 2009 EDGE QUESTION: \"WHAT WILL CHANGE EVERYTHING?\": http:\/\/www.nickbostrom.com\/views\/superintelligence.pdf","date":"2014-03-09 15:35:20","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\": 0, \"img_math\": 9, \"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\": 0.5802057981491089, \"perplexity\": 2812.426625342646}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-10\/segments\/1393999679512\/warc\/CC-MAIN-20140305060759-00083-ip-10-183-142-35.ec2.internal.warc.gz\"}"} | null | null |
Succeeding daily through Apps, Books, and Coaching.
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A little about the creator of My Productive leadership.
Hello, my name is Todd Mckeever, I have had the privilege to be able to purposefully pursue life and leadership journey since 1992 covering states like Iowa, Florida, the greater DC area, Arkansas, and Missouri. Some of my God-given opportunities include or have included: Serving as Open Bible Central KidsMin Director, Executive Operations Coordinator for Kidology, Central Region Open Bible Coach, National and International Kidology Coach, National Trainer for Kids Quest USA and Character Connex along with several opportunities to speak at camps, conferences plus numerous seminars.
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"redpajama_set_name": "RedPajamaC4"
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What is the UK's intention with regard to its allocation?
What is the discussion from other countries?
UK and others must consider what is the best avenue of delivering resources – reallocating SDRs or delivering other kinds of bi-lateral aid.
How is this going forward at IMF? What is the IMF relationship to be with unions?
IMF conditionality habit is not kicked, we fear new money will be applied in old way. Rich countries are using counter-cyclical spending.
How can the IMF go from austerity conditions to enabling fiscal expansion?
Now is not the time to be forcing pro-cyclical spending cuts, this should wait until after the downturn is over.
The shift in assessment of conditionality is important – programmatic rather than on individual line items.
IMF programmes attempt to ensure a responsible fiscal adjustment, where necessary, including protecting the most vulnerable groups.
Composition of public spending is a matter for country authorities. The IMF has no mandate to discuss composition of public spending.
Do not believe that IMF is a significant driver of donor provision of aid through vertical funds.
A question of distribution of open chairs if consolidation – who to give them to. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,593 |
Q: Improve performance of inserting data for one to many relationship in EF I am getting really poor performance in EF because of a particular design structure for my database. Here are the relevant relationships:
I have the following data model:
public class Sensor
{
[Key]
public int Id { get; set; }
[Required, MaxLength(64)]
public string Name { get; set; }
[Required, ForeignKey("Type")]
public int SensorTypeId { get; set; }
public virtual SensorType Type { get; set; }
public virtual ICollection<SensorSample> SensorSamples { get; set; }
}
public class SensorSample
{
[Key]
public int Id { get; set; }
[Required, ForeignKey("Sensor")]
public int SensorId { get; set; }
public virtual Sensor Sensor { get; set; }
[Required]
public DateTime SampleTime { get; set; }
[Required]
public virtual ICollection<SampleData> SampleData { get; set; }
}
public class SampleData
{
[Key]
public int Id { get; set; }
[Required, ForeignKey("DataType")]
public int SampleDataTypeId { get; set; }
public virtual SampleDataType DataType { get; set; }
[Required, ForeignKey("Unit")]
public int SampleUnitId { get; set; }
public virtual SampleUnit Unit { get; set; }
[Required, ForeignKey("Sample")]
public int SensorSampleId { get; set; }
public virtual SensorSample Sample { get; set; }
[MaxLength(128)]
public string Value { get; set; }
}
Because a SensorSample can have multiple data sample types (i.e. temperature, pressure, etc), an INSERT must query for existing samples to make the appropriate association with the correct SampleTime. This is done using the following code:
SensorSample sample = null;
foreach (var d in input)
{
SampleData data = new SampleData();
data.SampleDataTypeId = dataTypeId;
data.SampleUnitId = unitId;
data.Value = d.Value;
// check for existing sample for this sensor and timestamp
sample = SensorSamples.FirstOrDefault(s => s.SensorId == sensor.Id && s.SampleTime == d.Timestamp);
if (sample == null)
{
// sample doesn't exist, create a new one
sample = new SensorSample();
sample.SampleTime = d.Timestamp;
sample.SensorId = sensor.Id;
sensor.SensorSamples.Add(sample);
}
// add the data to the sample
sample.SampleData.Add(data);
}
I have tried optimizing the inserting of sample data by doing it in batches (i.e. 1000 records at a time). This does help, but even though there is an index on the SampleTime field, the lookup query seems to take longer as more records are added.
So, my question is, how do I improve the design and/or performance of adding sample data to the database? Is there a better database structure for handling the one-to-many relationship? I am willing to make some compromises on database design if I can get an appropriate offset in performance, but I still need to be able to handle different data associated with a given SampleTime.
A: Entity Framework maintains a local cache of all the local entities, and tracks any changes are made in those entities. As the number of entities grows, the checking gets more expensive.
Here is a very interesting post series on how does DetectChanges work and what can you do about it. Look especially in part 3.
When I need to bulk load a lot of data I disable DetectChanges and also clear the local cache after saving, so that memory can be freed:
public static void ClearDbSet<T>(this DbContext context) where T : class {
var entries = context.ChangeTracker.Entries<T>().Where(e => e.State == EntityState.Unchanged);
foreach (DbEntityEntry<T> entry in entries.ToList()) {
entry.State = EntityState.Detached;
}
}
The ToList call is necessary otherwise the iterator will throw an exception.
A: to maximize LOAD performance for test data
DONT run project in Debug mode (multiple factor slower for EF)
use these settings:
Context.Configuration.LazyLoadingEnabled = false;
Context.Configuration.ProxyCreationEnabled = false;
Context.Configuration.AutoDetectChangesEnabled = false;
Context.Configuration.ValidateOnSaveEnabled = false;
every 100 entries or fewer, discard Context.
Using( new context)
try
Context.Set<TPoco>().AddOrUpdate(poco);
Instead of
Context.Set<TPoco>().firstorDefault(lamba);
Context.Set<TPoco>().Add(poco);
A: EF6 beta 1 has an AddRange function that may suit your purpose:
INSERTing many rows with Entity Framework 6 beta 1
Note that the article I link to refers to the technique of setting AutoDetectChangesEnabled to false in EF5 that @felipe refers to
| {
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} | 7,818 |
{"url":"https:\/\/chemistry.tutorvista.com\/organic-chemistry\/test-for-carboxylic-acid.html","text":"Top\n\n# Test for Carboxylic Acid\n\nThe organic acid group contains carboxyl group $\u2013COOH$ bonded to either an alkyl group $(RCOOH)$ or an aryl group $(ArCOOH)$. To understand how the test for this functional group works we need to get the facts about the various forms of acids that this group can result in.\n\nIn aliphatic series, we have formic acid or $HCOOH$, acetic acid or $CH_3COOH$ and $Ar-COOH$ as the benzoic acid. The longest chain having the functional group carboxyl is considered as the parent structure and is suitably named by removing the \u2018-e\u2019 suffix of the corresponding alkane with a suffix of \u2018oic\u2019 acid.\n\nThe name of salt of a carboxylic acid consists of the cation followed by the name of the acid with the ending \u2018-ic\u2019 acid changing to \u2018-ate\u2019.\u00a0There are many tests which help in identifying these functional groups and their respective acids, like Tollens\u2019s Fehling\u2019s and Benedict\u2019s tests etc.\n\n Related Calculators 1 Sample T Test Anova Test Calculator Calculate F Test Chi Square Test Calculator\n\n## Test for Carboxylic Acid Functional Group\n\nThe acids do not show the typical reaction of carbonyl groups because of the following resonance, however they have the tendency to donate protons and act as acids. This property of donating protons is helpful in the identification of a -COOH group.\n\nLitmus test:\nThe carboxyl group because of their acidic nature, it turns blue litmus into red.\n\nThe blue litmus solution of around one drop is added to an aqueous solution of acid of around 1 mL. The appearance of a red colour indicates the presence of a carboxylic acid group. Blue litmus paper may be used as well in place of a blue litmus solution.\n\nSodium bicarbonate test:\nCarboxylic acid group reacts with a solution of sodium bicarbonate with effervescence which shows the evolution of $CO_{2}$.\n\n$RCOOH + NaHCO_{3} \\rightarrow RCOONa + H_{2}O + CO_{2}$\n\nTo a saturated solution of sodium bicarbonate in water of around 1 mL we need to add the given compound (a pinch of solid sample or 2 -3 drops of the liquid, or even aqueous or alcoholic solution of the compound. The quick effervescence indicates the presence of carboxylic acid. The effervescence of CO2 can be tested with lime water which turn milky when this gas is passed through it.\n\n$CO_{2} + Ca (OH)_{2} \\rightarrow CaCO_{3} + H_{2}O$\n\nThe carbon di oxide in aqueous form acts as dilute acid turns the mild alkali to react and produce salt and water. The cloudy appearance of $CaCO_{3}$ is basically the presence of salt, white in colour which remains suspended.\n\n## Ferric Chloride Test for Carboxylic Acid\n\nIn order to complete this test, the ethanoic acid is first reacted with ammonium hydroxide and the resultant ammonium acetate is then further reacted with ferric chloride.\n\n$CH_{3}COOH + NH_{4}OH \\rightarrow CH_{3}COONH_{4} + H_{2}$O\n\nThe ammonium acetate so formed is put in warm bath where the ammonia vapor is lost to an extent. Once the vapor is lost it is then reacted with ferric chloride. The resultant ferric acetate formed is confirmed with the appearance of red hue colour.\n\n$CH_{3}COONH_{4} + FeCl_{3} \\rightarrow (CH_{3}COO)_{3} Fe + NH_{4}Cl$\n\n## Test for Carboxylic Acid and Alcohol\n\nThe test for carboxylic acid and alcohol is better known as esterification. The esterification reactions are very slow as homogeneous and heterogeneous acids act catalytically in the esterification because the initiating step in the reaction mechanism is protonation of the carboxylic used. The presence of efficient homogeneous mineral acid H2SO4 helps in the process of protonation.\n\nThe heterogeneous catalyst containing sulphonic acid groups are active in esterification.\u00a0The process of esterification is basically the reaction between alcohol and carboxylic acid in presence of sulphuric acid at about 170oC temperature. The formation of ester is important as most of the synthetic flavoured edibles are based on ester and their various derivatives.\u00a0The alcohol reacts with carboxylic acid in presence of catalyst H2SO4 acid and produce ester along with water as only other product.\n\nCarboxylic acid caries a proton or hydrogen ion from the catalyst sulphuric acid which finally attaches to one of the oxygen\u2019s lone pairs. This oxygen is bonded with carbon with a double bond and this transfer of proton eventually gives the positive charge to oxygen.\n\nThe delocalisation of positive charge results in the shifting of electron pair resulting in canonical structures.\n\nThe carbon atoms charge attracts the ethanol molecule\u2019s lone pair which finally results in the formation of a water molecule. The remaining resultant is the ester that we have as the other product.\u00a0Test ends in the formation of a sweet smelling compound ester.\n\n$CH_{3}COOH + C_{2}H_{5}OH \\rightarrow H_{2}SO_{4} \/ 170 C \\rightarrow CH_{3}COOC_{2}H_{5} + H_{2} O$\n\n## Test for Carboxylic Acid Using Sodium Carbonate\n\nThe test involving sodium carbonate with carboxylic acid results in the evolution of carbon di oxide gas which turns lime water milky.\nEthanoic acid when poured into sodium carbonate solution, there is an immediate effervescence of carbon di oxide which is then confirmed by passing it through lime water.\n\nThe evolution of carbon di oxide is confirmed once the lime water turns milky which also confirms the presence of carboxylic acid.\n\n$2 CH_{3}COOH + Na_{2}CO_{3} \\rightarrow 2 CH_{3}COONa + CO_{2} + H_{2}O$\n\n$CO_{2} + Ca (OH)_{2} \\rightarrow CaCO_{3} + H_{2}O$","date":"2019-07-23 00:51:53","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\": 0.4828183352947235, \"perplexity\": 4402.194186515383}, \"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-2019-30\/segments\/1563195528635.94\/warc\/CC-MAIN-20190723002417-20190723024417-00138.warc.gz\"}"} | null | null |
Christina Applegate Celebrity desktop wallpaper, Christina Applegate wallpaper, Celebrity wallpaper - Celebrities no. 12893. Download this Christina Applegate Christina Applegate desktop wallpaper in multiple resolutions for free. | {
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\section{Introduction} One of the major challenges
in group cohomology is the computation of the
cohomology of nilpotent groups. Such computations
are important because general questions about modular group
cohomology can often be reduced to questions that
concern only the cohomology of its
Sylow $p$-subgroup. The structure of the cohomology of $p$-groups can be quite
complicated, but in the case when the cohomology ring is Cohen-Macaulay
(i.e, when the depth equals the Krull dimension), the homological algebra
of the representation theory has more orderly structural features.
In the realm of Lie theory and modular representations
of algebraic groups the nilpotent restricted $p$-Lie algebras
play a similar role to that of the $p$-groups in group
representations. For example, an element in the restricted
cohomology ring of a restricted Lie algebra is nilpotent if and only if
its restriction to every nilpotent subalgebra is nilpotent.
In this paper we address some
basic questions on the structure of the cohomology
rings for these algebras.
Suppose that $({\mathfrak n},[p])$ is a nilpotent restricted
$p$-Lie algebra and that $k$ is an algebraically closed field of
characteristic $p>0$. The spectrum of the cohomology
ring identifies with the restricted nullcone
${\mathcal N}_{1}({\mathfrak n})=\{x\in {\mathfrak n}:\ x^{[p]}=0\}$.
If we assume that $p > 2$, then there
is a spectral sequence \cite{FP}
\[
E_2^{2i,j} = S^{2i}(\mathfrak n^*)^{(1)}\otimes \operatorname{H}\nolimits^{j}(\mathfrak n,k) \Rightarrow
\operatorname{H}\nolimits^{2i+j}(u(\mathfrak n), k)
\]
where $S^{*}({\mathfrak n}^{*})^{(1)}$ is the Frobenius twist of the
symmetric algebra on the dual of the underlying vector space of $\mathfrak n$,
$\operatorname{H}\nolimits^{*}(\mathfrak n,k)$ is the ordinary Lie algebra of $\mathfrak n$, and
$\operatorname{H}\nolimits^{*}(u(\mathfrak n), k)$, is the cohomology ring of the restricted enveloping algebra
$u(\mathfrak n)$ of $\mathfrak n$.
There are many cases in which it is known that
the spectral sequence collapses at the $E_2$ page, so that
$E_2^{*,*}$ is isomorphic to the associated graded
ring of $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$. In this situation, the
cohomology ring $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$ is a free module
over the symmetric algebra $S^{*}({\mathfrak n}^{*})^{(1)}$,
and the cohomology ring is Cohen-Macaulay. In 1986,
Friedlander and Parshall \cite{FP} showed this happens
when $\mathfrak n$ is the nilpotent
radical of the Borel subalgebra of the restricted
Lie algebra of an algebraic group, provided $p>h$
where $h$ is the Coxeter number of the associated
root system. Twenty five years later, Drupieski, Ngo
and the second author \cite{DNN}
showed that a stronger result holds when $p\geq 2(h-1)$,
that is, there is a ring isomorphism $\operatorname{H}\nolimits^*(u(\mathfrak n), k)
\cong S^{*}({\mathfrak n}^{*})^{(1)}\otimes \operatorname{H}\nolimits^{*}(\mathfrak n,k)$.
The investigations of this paper were originally inspired
by a computer calculation by the first author
which demonstrated that if $\mathfrak n$ is the Lie
algebra of the nilpotent radical of a Borel
subalgebra of a group of type $B_2$ and if $p=5$ (which is
larger than $h$ but {\em not} larger than $2(h-1)$), then the
isomorphism $\operatorname{H}\nolimits^*(u(\mathfrak n), k) \cong S^{*}({\mathfrak n}^{*})^{(1)}\otimes
\operatorname{H}\nolimits^{*}(\mathfrak n,k)$ holds a modules
over the symmetric algebra $S^{*}({\mathfrak n}^{*})^{(1)}$ (so that the
cohomology ring is Cohen-Macaulay), but it does {\em not}
hold as rings. A search for the reason for this
phenomenon led to the discovery of a one-dimensional central
extension of $\mathfrak n$ (with trivial $p$th power)
whose cohomology ring is easily proved to
be not Cohen-Macaulay. Indeed, this illustrates
a general situation. One of the theorems in the paper is that if
$\operatorname{H}\nolimits^*(u(\mathfrak n), k) \cong S^{*}({\mathfrak n}^{*})^{(1)}\otimes \operatorname{H}\nolimits^{*}(\mathfrak n,k)$
holds as an isomorphism of rings, then when $\mathfrak n$ is
replaced by a one-dimensional extension, there is the
same isomorphism, though perhaps only as an isomorphism of
$S^{*}({\mathfrak n}^{*})^{(1)}$-modules. In the paper, we present numerous examples of this
phenomenon.
Cohen-Macaulay rings are of general interest, in part, because they have
very nice structural properties. For example, it can
be shown that for any restricted Lie algebra, if its cohomology ring is
Cohen-Macaulay, then the cohomology ring admits a formal
Poincar\'e duality and its Poincar\'e series, as a rational
polynomial, satisfies a functional equation \cite{BC1, BC2}. In the
case of a $p$-nilpotent Lie algebra with vanishing $p$-power
operation, we show that the cohomology ring is Cohen-Macaulay
if and only if the spectral sequence
given above collapses at the $E_2$-page.
The paper is organized as follows. In the following
section of the paper, we present
preliminaries about cohomology rings and the
definitions from commutative algebra.
A proof that the cohomology ring is Cohen-Macaulay
if and only if the spectral sequence collapses is given
in Section 3. We also prove results which
describe how the spectral sequence behaves under
central extensions in that section. The next
four sections are concerned with specific examples. In the case that if $\mathfrak n$
is the nilpotent radical of a Borel
subalgebra of $\mathfrak{sl}_3$ (i.e., type $A_2$), and the
field has characteristic 3, then the isomorphism
$\operatorname{H}\nolimits^*(u(\mathfrak n), k) \cong
S^{*}({\mathfrak n})^{(1)}\otimes \operatorname{H}\nolimits^{*}(\mathfrak n,k)$
holds as modules over the symmetric algebra, but not as ring.
Similar examples are given in type $B_2$ in characteristic 5 and in
type $G_2$ in characteristic 7. In each example, there is a one-dimensional
extension of the Lie algebra whose cohomology ring is not Cohen-Macaulay.
Examples of Lie algebra whose cohomology rings are not Cohen-Macaulay
are given in all characteristics. In Section 8, we
outline which Lie algebra of dimension 5 have
cohomology rings that are Cohen-Macaulay. In Section 9, we look at the
special case of the nilpotent radical of the Borel
subalgebra of $\mathfrak{sl}_4$ when $h<p<2(h-1)$ and
show that the cohomology ring can be identified the
symmetric algebra tensored with the ordinary Lie algebra cohomology as rings.
\section{Preliminaries}
Let $({\mathfrak g},[p])$ be a restricted Lie algebra
over an algebraically closed field of characteristic
$p>0$. Throughout this paper we will work with
the added assumption that $p\geq 3$.
The restricted representations for ${\mathfrak g}$
correspond to modules for the restricted enveloping
algebra $u({\mathfrak g})$. Since $u({\mathfrak g})$
is a finite-dimensional cocommutative Hopf
algebra the cohomology ring
$\text{H}^{*}(u({\mathfrak g}),k)$ is a finitely
generated graded-commutative $k$-algebra.
For these types of rings, notions like
Krull dimension and spectrum are well defined.
The spectrum of the cohomology ring
${\mathcal V}_{\mathfrak g}$ is homeomorphic to
the restricted nullcone:
\[
{\mathcal N}_{1}({\mathfrak g}):=
\{x\in {\mathfrak g}:\ x^{[p]}=0\}.
\]
\vskip.1in
When $p\geq 3$ there exists a the spectral
sequence
\begin{equation}\label{specseq1}
E_2^{2i,j} \ \ = \ \ S^i({\mathfrak g}^*)^{(1)}
\otimes \operatorname{H}\nolimits^i({\mathfrak g},k) \Rightarrow
\operatorname{H}\nolimits^{2i+j}(u({\mathfrak g}), k)
\end{equation}
where $\text{H}^{*}({\mathfrak g},k)$ is the
ordinary Lie algebra cohomology.
In particular, the map of the universal
enveloping algebra of $\mathfrak g$ to the restricted
enveloping algebra of $\mathfrak g$ induces an edge
homomorphism from the restricted cohomology
to the ordinary Lie
algebra cohomology. On the other hand,
there is another edge
homomorphism $\Phi: S^{*}({\mathfrak g}^{*})^{(1)}\rightarrow
\operatorname{H}\nolimits^{2*}(u({\mathfrak g}),k)$ which induces
an inclusion of
$R=k[{\mathcal N}_{1}({\mathfrak g})]\hookrightarrow
\operatorname{H}\nolimits^{*}(u({\mathfrak g}),k)$. Furthermore,
$\operatorname{H}\nolimits^{*}(u({\mathfrak g}),k)$ is an integral
extension of $R$.
The following observation is a consequence of these facts in the
event that the $p$-power operation vanishes on a
nilpotent restricted $p$-Lie algebra $\mathfrak n$. In this situation, ${\mathcal{N}}_1(\mathfrak n)
= \mathfrak n$ and its coordinate ring is $S^*(\mathfrak n^*)^{(1)}$. Indeed, any
restricted Lie algebra with vanishing $p$-power operation is nilpotent.
\begin{prop} \label{prop:symmetric}
Suppose that $\mathfrak n$ is a restricted Lie algebra such
that $x^{[p]} = 0$ for all $x \in \mathfrak n$. Then the edge
homomorphism $S^*(\mathfrak n^*)^{(1)} \to \operatorname{H}\nolimits^{2*}(u(\mathfrak n),k)$
is an injection.
\end{prop}
We require several ring theoretic notions which, though usually
defined for commutative rings or commutative local rings,
apply also to graded-commutative $k$-algebras.
Suppose that
$A = \sum_{i \geq 0} A_i$ is a graded-commutative $k$-algebra and that
$M = \sum_{i \geq 0} M_i$ is a graded $A$-module. A sequence
$x_1, \dots, x_r$ of homogeneous elements of $A$ is said to be
a {\it regular sequence} for $M$ if for every $i = 1, \dots, r$, we
have that multiplication by $x_i$ is an injective map from
$M/(x_1, \dots, x_{i-1})M$ to itself. The {\it depth} of $M$ is
the length of the longest regular sequence for $M$, and the
depth of $A$ is the depth of $A$ as a module over itself.
A sequence of homogeneous elements $x_1, \dots, x_r$ is a homogeneous
set of parameters for $A$, if $x_1, \dots, x_r$
generate a polynomial subring $R = k[x_1, \dots, x_r] \subseteq A$
and that $A$ is finitely generated as a module over $R$. In this case,
the number $r$ must be the Krull dimension of $A$. The module
$M$ is {\it Cohen-Macaulay} if its depth is equal to the Krull
dimension of $A$. The algebra $A$ is Cohen-Macaulay if it is
Cohen-Macaulay as a module over itself. This is equivalent to
the condition that there be a homogeneous set of parameters
$x_1, \dots, x_r$ for $A$ such that $A$ is a finitely generated
free module over the polynomial subring $k[x_1, \dots, x_r]$.
It is a theorem that if there is a homogeneous set of parameters
$x_1, \dots, x_r$ such that $A$ is a finitely generated free
module over $k[x_1, \dots, x_r]$, then $A$ is a finitely generated
free module over $k[y_1, \dots, y_r]$ for any homogeneous
set of parameters $y_1, \dots, y_r$. For a reference,
see Proposition 3.1 of \cite{St} or Theorem 2, page IV-20 of
\cite{Se}. Note that in the book of Serre, the proof is given
only for commutative local rings, but can be easily adapted to
the graded-commutative case.
With these preliminaries, we can extract the results
that we need for this paper.
\begin{thm}\label{thm:CM-prelim}
Suppose that $\mathfrak n$ is a $p$-nilpotent Lie algebra with trivial
$p$th-power operation (i.e., $x^{[p]} = 0$ for all $x \in \mathfrak n$).
Then the cohomology ring $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$
is Cohen-Macaulay if and only if it is a free module over
the polynomial subring $S^{*}(\mathfrak n^*)^{(1)}$. In particular,
no nonzero element of $S^{*}(\mathfrak n^*)^{(1)}$ can be a divisor of
zero if the cohomology ring is Cohen-Macaulay.
\end{thm}
\begin{proof} The last statement clearly follows from the first part of the
theorem. Suppose that $x_1, \dots, x_r$ is a basis for $\mathfrak n^*$.
By Proposition~\ref{prop:symmetric},
$$S^{*}(\mathfrak n^*)^{(1)}= k[x_1, \dots, x_r] \subseteq \operatorname{H}\nolimits^*(u(\mathfrak n),k).$$
Because the ordinary Lie algebra cohomology $\operatorname{H}\nolimits^*(\mathfrak n,k)$
is finite dimensional, we have that $x_1, \dots, x_r$
is a homogeneous set of parameters for $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$.
If $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$ is a finitely generated
free module over $S(\mathfrak n^*)^{(1)}$, then
it is Cohen-Macaulay. On the other hand, if $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$
is Cohen-Macaulay, then it must be free as a
module over $S(\mathfrak n^*)^{(1)}$, by the results
cited above.
\end{proof}
\section{Consequences of the Lie cohomology spectral sequence}
Let $\mathfrak n$ be a restricted Lie algebra with trivial $p$-restriction.
We consider the first quadrant spectral sequence given as
\begin{equation} \label{eq:specseq}
E_2^{2i,j} \ \ = \ \ S^i(\mathfrak n^*)^{(1)}
\otimes \operatorname{H}\nolimits^i(\mathfrak n,k) \Rightarrow
\operatorname{H}\nolimits^{2i+j}(u(\mathfrak n), j)
\end{equation}
One of important fact to note about the spectral sequence
is that $E_2^{i,j} = \{0\}$ if $j > \operatorname{Dim}\nolimits(\mathfrak n)$. This is simply because
the ordinary Lie algebra cohomology of $\mathfrak n$ has the property that
$\operatorname{H}\nolimits^j(\mathfrak n, k) = \{0\}$ for $j > \operatorname{Dim}\nolimits(\mathfrak n)$.
This spectral sequence can be used to show that
the cohomology of $u({\mathfrak n})$ is Cohen-Macaulay.
\begin{prop}\label{prop:specseq}
If the spectral sequence $E_*^{*,*}$ collapses at the $E_2$ page,
then $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$ is free as module over the polynomial
subalgebra $S= S^{*}(\mathfrak n^*)^{(1)}.$ In particular, it is
Cohen-Macaulay.
\end{prop}
\begin{proof}
The spectral sequence is a filtered version of the cohomology
$\operatorname{H}\nolimits^*(u({\mathfrak n}),k)$. That is, if $\zeta \in E_2^{i,j}$ and $\eta
\in E_2^{r,s}$ then $\zeta\eta \in \sum_{\ell \geq 0} E_2^{i+r+\ell,
j+s-\ell}$. For this reason, for any $m$ the collection of
the lowest $m$ rows, ($U_m = \sum E_2^{i,j}$ with $i \geq 0$ and
$0 \leq j \leq m$) is a module over $S$, which lies in the
bottom row. Because the spectral sequence collapses,
$E_2 = E_{\infty}$ and the
quotients $U_m/U_{m-1}$ are free $S$-modules. Hence,
the proposition follows from the fact that every one of the
quotient maps $U_m \to U_m/U_{m-1}$ must split as a map of
$S$-modules.
\end{proof}
Many of the results of
\cite{BC1} apply in the case that the polynomial ring is
Cohen-Macaulay. In particular, we have the following
adaptation of \cite[Theorem 1.1]{BC1}. We refer the reader
to that paper for the proof which carries over from group
cohomology to restricted Lie algebra cohomology with only
minimal changes.
\begin{thm} \label{thm:bc}
Suppose that $\mathfrak n$ is a restricted $p$-Lie algebra with trivial $p$th-power
operation. Let $d$ denote the dimension of $\mathfrak u$.
If the cohomology ring $\operatorname{H}\nolimits^*(u(\mathfrak n), k)$
is Cohen-Macaulay, then any basis for $\mathfrak n^*$, meaning
any complete linearly independent set of
degree-two generators $X_1, \dots, X_d$ for the symmetric algebra
$S^{*}(\mathfrak n^*)^{(1)}$, is a homogeneous set of parameters for
$\operatorname{H}\nolimits^*(u(\mathfrak n),k)$ and the quotient
\[
\operatorname{H}\nolimits^*(u(\mathfrak n),k)/(X_1, \dots, X_d)
\]
satisfies Poincar\'e duality in formal dimension $d$. Moreover,
the Poincar\'e series $P_k(t) = \sum_{i \geq 0}
\operatorname{Dim}\nolimits \operatorname{H}\nolimits^i(u(\mathfrak u),k)\ t^{i}$, regarded as a rational function of $t$
satisfies the functional equation
\[
P_k(1/t) = (-t)^d P_k(t).
\]
\end{thm}
A consequence of the preceding result is the following.
\begin{cor}\label{cor:bc}
Suppose that $\mathfrak n$ is a restricted $p$-Lie algebra of dimension $d$
whose $p$-power operation is trivial. Then the
spectral sequence (\ref{eq:specseq}) collapses at the $E_2$ page if and only if
the cohomology ring $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$ is Cohen-Macaulay.
In this case, the edge homomorphism $\operatorname{H}\nolimits^*(u(\mathfrak n),k) \to \operatorname{H}\nolimits^*(\mathfrak n,k)$
is surjective and the Poincar\'e series, as a rational
function has the form
\[
P_k(t) = \frac{f_k(t)}{(1-t^2)^d},
\]
where $f_k(t)$ is the Poincar\'e polynomial for the ordinary
Lie algebra cohomology $\operatorname{H}\nolimits^*(\mathfrak n,k)$.
\end{cor}
\begin{proof}
For convenience of notation let $S = S^{*}(\mathfrak n^*)^{(1)}$ and let
$\operatorname{H}\nolimits^* = \operatorname{H}\nolimits^*(u(\mathfrak n).k)$. By Proposition \ref{prop:specseq},
if the spectral sequence (\ref{eq:specseq}) collapses at the $E_2$
page then $\operatorname{H}\nolimits^*$ is Cohen-Macaulay. We need to prove the
converse. So assume that $\operatorname{H}\nolimits^*$ is Cohen-Macaulay. For $t \geq 0$,
let $M_t = S \cdot \sum_{j=0}^t \operatorname{H}\nolimits^j$, the $S$-submodule of
$\operatorname{H}\nolimits^*$ generated by elements of degree at most $t$. By Theorem
\ref{thm:bc}, this is a free $S$-module. That is, let $I = (X_1,
\dots, X_d)$ (in the notation of the theorem). Then $\operatorname{H}\nolimits^*$
is a free $S$-module on a set of homogeneous elements
$\zeta_1, \dots, \zeta_m$ whose classes form a basis of $\operatorname{H}\nolimits^*/I$.
If $\zeta_1, \dots, \zeta_\ell$ are all of those element of
degree at most $t$, then it is easily seen that $M_t$ is a
free module on these elements.
Now we proceed by induction on $t$ to prove that
\[
M_t/I \cdot M_t \ \cong \ \sum_{i = 0}^t E_2^{0,i},
\]
as vector spaces, and that no differential $d_r$ on the
$r^{th}$ page of the spectral
sequence has a nonzero image on any of the lines $E_r^{*,0}, \dots,
E_r^{*,t}$. This is true if $t = 0$, by Proposition \ref{prop:symmetric}.
That is, the differential $d_r$ on the $r^{th}$-page cannot have a
nonzero image $d_r:E_r^{j,r-1} \to E_r^{j+r,0}$ as otherwise the
edge homomorphism onto the bottom row would not be injective.
So assume that the statement is true for a certain value of $t$.
Then the differential $d_r$ on the $r^{th}$ page of the spectral
sequence must vanish on $E_r^{*,t+1}$, as otherwise there would be
a nonzero image on one of the lower lines. Therefore, $E_2^{0,j}
= E_\infty^{0,j}$ for all $j$ with $0 \leq j \leq t+1$. As a consequence,
$M_{t+1}$ is generated as an $S$-module by elements representing a
$k$-basis for $\sum_{i=0}^{t+1} E_\infty^{0,i} \cong \sum_{i=0}^{t+1}
E_2^{0,i}$. That is, $M_{t+1} \cong \sum_{i=1}^{t+1} E_\infty^{*,i}$.
Because, this is a free $S$-module, it must be that
\[
\sum_{i=1}^{t+1} E_\infty^{*,i} \ \cong \ \sum_{i=1}^{t+1} E_2^{*,i}
\]
since these are both free modules on the same number of generators.
It follows that no differential of the spectral sequence can have a
nonzero image on row $t+1$. The induction proves the corollary.
\end{proof}
We record the following lemma which will be later
useful in comparing spectral sequences.
\begin{lemma} \label{lem:eqdim}
Suppose that $\operatorname{Dim}\nolimits \operatorname{H}\nolimits^m(u(\mathfrak n),k) = \sum_{2i+j = m}
\operatorname{Dim}\nolimits(S^{2i}(\mathfrak n^*) \otimes \operatorname{H}\nolimits^j(\mathfrak n,k))$ for
all $m \geq 0$. Then the spectral sequence (\ref{eq:specseq})
collapses at the $E_2$ page and the cohomology ring $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$
is Cohen-Macaulay.
\end{lemma}
\begin{proof}
The hypotheses of the lemma asserts that $\operatorname{Dim}\nolimits \operatorname{H}\nolimits^m(u({\mathfrak n}),k) =
\sum_{m = 2i+j} \operatorname{Dim}\nolimits E_2^{2i+j}$. The condition forces the
spectral sequence to collapse at the $E_2$ page, because
otherwise there would be some further nontrivial differential
that would reduce the dimension.
\end{proof}
Nilpotent Lie algebras can be built up from central extensions. The
next theorem provides conditions
on when the spectral sequence will
collapse at $E_{2}$ under a central extension and
yield an isomorphism of $S^{*}({\mathfrak n}^{*})^{(1)}$-modules.
\begin{thm}\label{thm:tensoralg}
Let $\mathfrak n$ be a nilpotent restricted $p$-Lie algebra with
trivial $p$-power operation. Assume that $\mathfrak z$ is a central ideal
of dimension one. Suppose that we have
an isomorphism of rings
\[
\operatorname{H}\nolimits^*(u(\mathfrak n/\mathfrak z), k) \cong S^{*}((\mathfrak n/\mathfrak z)^*)^{(1)}
\otimes \operatorname{H}\nolimits^*(\mathfrak n/\mathfrak z,k).
\]
Then
\[
\operatorname{H}\nolimits^*(u(\mathfrak n), k) \cong S^{*}(\mathfrak n^*)^{(1)}
\otimes \operatorname{H}\nolimits^*(\mathfrak n,k).
\]
as modules over $S^{*}(\mathfrak n^*)^{(1)}$. In particular,
$\operatorname{H}\nolimits^*(u(\mathfrak n),k)$ is Cohen-Macaulay.
\end{thm}
\begin{proof}
We use the Lyndon-Hochschild-Serre (LHS) spectral sequence
\[
E_2^{i,j} = \operatorname{H}\nolimits^i(u(\mathfrak n/\mathfrak z), \operatorname{H}\nolimits^j(u(\mathfrak z),k))
\Rightarrow \operatorname{H}\nolimits^{i+j}(u(\mathfrak n),k).
\]
Since ${\mathfrak z}$ is central, $\mathfrak n$ acts trivially on $\mathfrak z$
and hence also trivially on its cohomology. Thus we have
$E_{2}^{i,j}\cong \operatorname{H}\nolimits^i(u(\mathfrak n/\mathfrak z),k)\otimes \operatorname{H}\nolimits^j(u(\mathfrak z),k)$.
Next note that $E_2^{0,1}$ has dimension one, and is spanned by an
element $x$. Then $d_2(x) = s + v$, where $s$ is in
$S^1((\mathfrak n/\mathfrak z)^*)^{(1)} \otimes 1$ and $v$ is in
$1 \otimes \operatorname{H}\nolimits^2(\mathfrak n/\mathfrak z,k)$ by the hypothesis. Also
by the hypothesis, we have that $v^n = 0$ for $n$ sufficiently
large, since $\operatorname{H}\nolimits^*(\mathfrak n/\mathfrak z,k)$ has finite dimension.
Consequently, for $r$ sufficiently large we have that
$0 = (s+v)^{p^r} = s^{p^r} + v^{p^r}= s^{p^r}$ on the
$E_3$ page of the spectral sequence. However, if $s$ is
not zero, then we have a contradiction to the fact that
$S^{*}(\mathfrak n^*)^{(1)}$ injects into the cohomology ring
$\operatorname{H}\nolimits^{2*}(u(\mathfrak n),k)$. So $d_2(x) \in 1 \otimes
\operatorname{H}\nolimits^2(\mathfrak n/\mathfrak z,k)$.
Next we see that the rows $E_2^{*,0}$ and $E_2^{*,1}$ are
both isomorphic to $\operatorname{H}\nolimits^*(u(\mathfrak n/\mathfrak z), k)$ as modules
over the symmetric algebra $S^{*}((\mathfrak n/\mathfrak z)^*)^{(1)}$. Moreover,
the differential $d_2: E_2^{*,1} \to E_2^{*+2,0}$
is a homomorphism of $S^{*}((\mathfrak n/\mathfrak z)^*)^{(1)}$-modules. More
specifically, we have that the differential
\[
d_2 : S^{*}(\mathfrak n^*)^{(1)} \otimes \operatorname{H}\nolimits^{*}(\mathfrak n,k) \cong E_2^{*,1}
\longrightarrow E_2^{*,0} \cong S^{*}(\mathfrak n^*)^{(1)} \otimes \operatorname{H}\nolimits^{*}(\mathfrak n,k)
\]
is multiplication by $d_2(x)$ which has the form
$d_2(x) = 1 \otimes w \in 1 \otimes
\operatorname{H}\nolimits^2(\mathfrak n/\mathfrak z,k)$. Hence, by the hypothesis, we have
that $E_3^{*,1} \cong S(\mathfrak n^*)^{(1)} \otimes K$ and
$E_3^{*,1} \cong S(\mathfrak n^*)^{(1)} \otimes C$, where
$K$ and $C$ are
respectively the kernel and cokernel of multiplication
by $w$ on $\operatorname{H}\nolimits^*(\mathfrak n/\mathfrak z,k)$.
We now observe that the element $w \in \operatorname{H}\nolimits^2(\mathfrak n/\mathfrak z, k)$
is the extension class associated to the extension
$\mathfrak z \hookrightarrow \mathfrak n \rightarrow \mathfrak n/\mathfrak z$. In the LHS
spectral sequence $\hat{E}_2^{i,j} = \operatorname{H}\nolimits^i(\mathfrak n/\mathfrak z,
\operatorname{H}\nolimits^j(\mathfrak z,k)) \Rightarrow \operatorname{H}\nolimits^{i+j}(\mathfrak n,k)$,
of ordinary Lie algebra cohomology associated to that
sequence, the differential $d_2: \hat{E}_2^{*,1}
\to \hat{E}_2^{*,0}$ is multiplication by $w$. In
addition, there can be no further differentials
in that spectral sequence, since the sequence has
only two nonzero rows. It
follows that $E_2^{*,0} \oplus E_2^{*,1} \cong
S^{*}((\mathfrak n/\mathfrak z)^*)^{(1)} \otimes \operatorname{H}\nolimits^*(\mathfrak n, k)$ as
(free) modules over $S^{*}((\mathfrak n/\mathfrak z)^*)^{(1)}$.
Finally, we note that $E_3 = E_\infty$. The reason is that
the $E_3$ page is generated, as a ring by the elements on
the bottom two rows ($E_3^{*,0}$ and $E_3^{*,1}$) and
an element $X$ in $E_3^{0,2}$ that represents the class
of a generator in $S^{*}(\mathfrak z^*)^{(1)} \subseteq S^{*}(\mathfrak n^*)^{(1)}$.
The class $X$ must survive until the $E_\infty$ page
of the spectral sequence. So $d_2(X) = 0$, and we must have
that $XE_3^{i,j} = E_3^{i, j+2}$ for all $i, j \geq 0$.
Consequently, the differential
$d_3$ must vanish, because it vanishes on a collection of
ring generators. The same holds for all further
differentials in the spectral sequence.
We have verified the hypothesis of Lemma \ref{lem:eqdim}
and the theorem is proved.
\end{proof}
As an initial application we offer the following. Notice that the
algebra $\mathfrak n$ in the corollary must be the direct sum of a commutative
Lie algebra and a Heisenberg Lie algebra. Assuming $p \geq 3$, any such
ordinar Lie algebra can be made into a restricted Lie algebra by
assuming a vanishing $p$-power operation.
\begin{cor} \label{cor:dim1comm}
Suppose that $\operatorname{Dim}\nolimits([\mathfrak n, \mathfrak n]) = 1$.
Then $\operatorname{H}\nolimits^*(u({\mathfrak n}), k) \cong S^{*}(\mathfrak n^*)^{(1)} \otimes
\operatorname{H}\nolimits^*(\mathfrak n,k)$, as $S^{*}(\mathfrak n^*)^{(1)}$-modules,
and $\operatorname{H}\nolimits^*(u({\mathfrak n}),k)$ is Cohen-Macaulay.
\end{cor}
\begin{proof} Observe that the quotient algebra $\mathfrak v = \mathfrak n/[\mathfrak n,\mathfrak n]$
is commutative and hence we have that
$\operatorname{H}\nolimits^*(u(\mathfrak v), k) \cong S^{*}(\mathfrak v^*)^{(1)}
\otimes \operatorname{H}\nolimits^*(\mathfrak v,k)$ as rings. Thus, Theorem~\ref{thm:tensoralg}
implies the corollary.
\end{proof}
The next theorem provides
stronger conditions which will show when one can identify
$\operatorname{H}\nolimits^{*}(u({\mathfrak n}),k)$ with $S^{*}({\mathfrak n}^{*})^{(1)}\otimes
\operatorname{H}\nolimits^{*}({\mathfrak n},k)$ as rings. Together this theorem in conjunction with Theorem~\ref{thm:tensoralg}
can be applied to inductively compute cohomology rings.
\begin{thm}\label{th:splitting} Let ${\mathfrak n}$ be a
$p$-nilpotent Lie algebra and suppose that
there is an isomorphism of $S^{*}({\mathfrak n}^{*})^{(1)}$-modules,
$$
\operatorname{H}\nolimits^{*}(u({\mathfrak n}),k)\cong S^{*}({\mathfrak n}^{*})^{(1)}
\otimes \operatorname{H}\nolimits^{*}({\mathfrak n},k).
$$
Moreover, assume that there exists a subalgebra $B$ in
$\operatorname{H}\nolimits^{*}(u({\mathfrak n}),k)$ such that
$B\cong \operatorname{H}\nolimits^{*}({\mathfrak n},k)$ under the map
$\phi:\operatorname{H}\nolimits^{*} (u({\mathfrak n}),k) \rightarrow
\operatorname{H}\nolimits^{*}({\mathfrak n},k)$. Then
$\operatorname{H}\nolimits^{*}(u({\mathfrak n}),k)\cong S^{*}({\mathfrak n}^{*})^{[1]}\otimes
\operatorname{H}\nolimits^{*}({\mathfrak n},k)$ as rings.
\end{thm}
\begin{proof} Let $A$ be the subalgebra in
$\operatorname{H}\nolimits^{*}(u({\mathfrak n}),k)$ isomorphic to $S^{*}({\mathfrak n}^{*})^{(1)}$.
We have an algebra homomorphism $\Gamma$ defined by
\[
S^{*}({\mathfrak n}^{*})^{(1)}\otimes \operatorname{H}\nolimits^{*}({\mathfrak n},k)
\rightarrow A\otimes B \rightarrow \operatorname{H}\nolimits^{*}(u({\mathfrak n}),k)\otimes
\operatorname{H}\nolimits^{*}(u({\mathfrak n}),k) \rightarrow \operatorname{H}\nolimits^{*}(u({\mathfrak n}),k).
\]
The last map is given by the cup product. This map is bijective because
\[
\operatorname{H}\nolimits^{*}(u({\mathfrak n}),k)\cong S^{*}({\mathfrak n}^{*})^{(1)}\otimes
\operatorname{H}\nolimits^{*}({\mathfrak n},k).
\]
as $S^{*}({\mathfrak n}^{*})^{(1)}$-modules.
\end{proof}
\section{Some examples of type A}
We begin with the example of the nilpotent radical $\mathfrak n$ of a Borel
subalgebra of $\mathfrak{sl}_3$. The relations for the cohomology in
characteristic 3 were
calculated by computer using the system Magma \cite{BoCa}.
Specifically, we use the package for basic algebras written by
the first author. Two of the three unusual relation can be
derived from the second example in this section.
Note that $\mathfrak n$ has an action of a two dimensional torus.
Let $\alpha$ and $\beta$ be the simple roots. By convention
the weights of $\mathfrak n$ consist of sums of negative roots so that its
cohomology has weights in the positive cone of roots. For convenience, we
subscript elements by their weights whenever this causes no
problems.
\begin{lemma}\label{sl3-ordin}
Let $\mathfrak n$ be the nilpotent radical of a Borel subalgebra of
$\mathfrak{sl}_3$ over a field of characteristic at least three.
Then the ordinary Lie algebra cohomology of $\mathfrak n$ is
given as
\[
\operatorname{H}\nolimits^*(\mathfrak n, k) = k[\eta_\alpha, \eta_\beta, \eta_{2\alpha+\beta},
\eta_{\alpha+2\beta}]/I
\]
where $I$ is the ideal generated by
\[
\eta_\alpha^2, \ \eta_\alpha\eta_\beta, \
\eta_\beta^2, \ \eta_\alpha\eta_{2\alpha+\beta}, \
\eta_\beta\eta_{\alpha+2\beta}, \
\eta_\beta\eta_{2\alpha+\beta} +\eta_{\alpha}\eta_{\alpha+2\beta}, \
\eta_{2\alpha+\beta}^2, \
\eta_{\alpha+2\beta}^2, \
\eta_{2\alpha+\beta}\eta_{\alpha+2\beta} \
\]
\end{lemma}
\begin{proof}
The result follows easily from the LHS spectral sequence
$E_2^{i,j} = \operatorname{H}\nolimits^i(\mathfrak n/\mathfrak z, \operatorname{H}\nolimits^j(\mathfrak z,k))
\Rightarrow \operatorname{H}\nolimits^{i+j}(\mathfrak n,k)$. Note that the
torus $T$ acts on the spectral sequence. We let
$\eta_\alpha$ and $\eta_\beta$ be the generators of $E_2^{1,0}$
having weights $\alpha$ and $\beta$ and let
$\eta_{\alpha+\beta}$ be the generator of $E_2^{0,1}$ with weight
$\alpha+\beta$. It is easy to check
that $d_2(\eta_{\alpha+\beta}) =
\eta_\alpha\eta_\beta$, which is the extension class. The elements
$\eta_\alpha\eta_{\alpha+\beta}$ and
$\eta_{\beta}\eta_{\alpha+\beta}$ survive to
the $E_3 = E_{\infty}$ page, and are ring generators \--
which we call $\eta_{2\alpha+\beta}$ and
$\eta_{\alpha+2\beta}$, respectively. The
relations follow easily from the relations
in the exterior algebra of ${\mathfrak n}^{*}$.
\end{proof}
With this lemma, we can compute the cohomology of the
restricted Lie algebra. The following calculation is
computer generated in part. However,
as we see in the next example all but one of the
relations can be derived by hand.
\begin{prop}\label{sl3-rest}
Let $k$ be a field of characteristic 3.
Let $u(\mathfrak n)$ be the restricted enveloping algebra
of the Lie algebra $\mathfrak n$ which is the nilpotent
radical of the Borel subalgebra of $\mathfrak{sl}_3$. Then the
cohomology ring $\operatorname{H}\nolimits^{*}(u(\mathfrak n),k) \cong S^{*}(\mathfrak n^*)^{(1)}
\otimes \operatorname{H}\nolimits^*(\mathfrak n, k)$ is a free
$S^{*}(\mathfrak n^*)^{(1)}$-module with basis consisting
of the images of a basis of the ordinary Lie algebra
cohomology of $\mathfrak n$ as in Lemma \ref{sl3-ordin}.
The ring $S^{*}(\mathfrak n^*)^{(1)}$ is a polynomial ring
in variables $X_\alpha, X_\beta$ and
$X_{\alpha+\beta}$ having weights
$3\alpha, 3\beta$ and $3(\alpha+\beta)$ under
the action of the torus.
The multiplicative relations are given by
\[
\eta_\alpha^2, \
\eta_\alpha\eta_\beta, \
\eta_\beta^2, \
\eta_\beta\eta_{2\alpha+\beta} +\eta_{\alpha}\eta_{\alpha+2\beta}, \
\eta_{2\alpha+\beta}^2, \
\eta_{\alpha+2\beta}^2, \
\]
\[
\eta_\alpha\eta_{2\alpha+\beta}- \eta_\beta X_\alpha, \
\eta_\beta\eta_{\alpha+2\beta} - \eta_\alpha X_\beta, \
\eta_{2\alpha+\beta}\eta_{\alpha+2\beta} - X_\alpha X_\beta
\]
\end{prop}
\begin{proof}
First note that the isomorphism
$\operatorname{H}\nolimits^{*}(u(\mathfrak n),k) \cong S^{*}(\mathfrak n^*)^{(1)}
\otimes \operatorname{H}\nolimits^*(\mathfrak n, k)$ is a consequence of
Corollary \ref{cor:dim1comm}.
Consider that LHS spectral sequence
\[
E_2^{i,j} = \operatorname{H}\nolimits^i(u(\mathfrak n/\mathfrak z), \operatorname{H}\nolimits^j(u(\mathfrak z),k))
\Rightarrow \operatorname{H}\nolimits^{i+j}(u(\mathfrak n),k)
\]
We have elements $a,b$ in $E_2^{1,0}$ and $u$ in $E_2^{0,1}$.
The differential on the $E_2$ page has the form $d_2(u) = ab$
as in the proof of Lemma~\ref{sl3-ordin}. A representative of
$X_{\alpha+\beta}$ is in $E_2^{0,2}$, and this element must live until
the $E_{\infty}$ page. Because the resulting ring at the
$E_3$ page is generated in degrees one and two, we conclude
that all further differentials must vanish and
$E_3 = E_\infty$. The problem is to ungrade the spectral
sequence. The first six relations are forced by the
grading on the spectral sequence and by the action of
the torus. For example,
the relation
$\eta_\beta\eta_{2\alpha+\beta} +
\eta_{\alpha}\eta_{\alpha+2\beta}$ holds on the
(graded) $E_3$ page and must hold in the ungrading
because there is no nonzero element of that
weight ($2\alpha+2\beta$) in $E_3^{3,0}$.
For the last three relations, we rely on the computer.
For example, $\eta_\alpha\eta_{2\alpha+\beta} = 0$ in the graded
$E_3$ page. This element lies in $E_3^{2,1}$ in the
ungrading it is equal to $\eta_\beta X_\alpha \in E_3^{4,0}$.
Note that both elements
have weight $3\alpha+\beta$. The other two relations
are similar.
\end{proof}
Next we extend the example slightly to get a
nilpotent Lie algebra (with trivial $p$th power)
where the restricted cohomology fails to
be Cohen-Macaulay. We consider the algebra $\mathfrak n$, labeled
as $L_{5,9}$ in the list of de Graaf \cite{deG}.
This algebra is isomorphic to the quotient of the
Lie algebra of all upper triangular $4 \times 4$
matrices by its center. The algebra can also be represented
as the algebra of all strictly upper triangular matrices
such that the all entries in the third (and fourth) row
are zero. Notice that in this representation, the algebra has trivial
$p$-power operation. The algebra has a
basis consisting of $u_1, \dots, u_5$, where
$u_4$ and $u_5$ are central, and with
the additional relations:
\[
[u_1, u_2] = u_4, [u_2, u_3] = u_5,
[u_1, u_3] = 0.
\]
The reader should be aware that this is a minor change
from the presentation in \cite{deG}.
\begin{prop}\label{prop:notcm1}
Suppose that the characteristic of $k$ is 3,
and let $\mathfrak n$ be as above. The cohomology ring
$\operatorname{H}\nolimits^*(u(\mathfrak n),k)$ is not Cohen-Macaulay. In
particular, it has an associated prime $\mathfrak P$
such that $\operatorname{H}\nolimits^*(u(\mathfrak n),k)/\mathfrak P$ has Krull
dimension four.
\end{prop}
\begin{proof}
First we observe that the algebra has an action
of a three dimensional torus, $T$ (in the representation
as upper triangular $4 \times 4$ matrices modulo the
center of that algebra). With this action,
the basis elements $u_1, \dots, u_5$ have weights
$-\alpha, \ -\beta, \ -\gamma, \ -\alpha-\beta$ and $-\beta-
\gamma$, respectively. The torus also acts on the
cohomology.
Let $\mathfrak v$ be the commutative subalgebra of $\mathfrak n$ spanned
by $u_1, u_3, u_4, u_5$. Its Lie algebra cohomology
is an exterior algebra generated by elements
$\eta_\alpha, \eta_{\alpha+\beta},
\eta_{\beta+\gamma}, \eta_\gamma$
all in degree one and having weights
as indicated by the subscripts. The element $u_2$ acts
on $\mathfrak v$ and on its cohomology. After possible
rescaling, we have that $u_2\eta_{\alpha+\beta} =
\eta_\alpha$ and $u_2\eta_{\beta+\gamma} =
\eta_\gamma.$ Recall that the action on the
cohomology is dual to the action on the algebra,
so multiplying by $u_2$ on an element of
cohomology subtracts the root $\beta$.
Now we consider the spectral sequence
\[
E_2^{i,j} = \operatorname{H}\nolimits^i(u(\mathfrak n/\mathfrak v), \operatorname{H}\nolimits^j(u(\mathfrak v),k)
\Rightarrow \operatorname{H}\nolimits^{i+j}(u(\mathfrak n), k).
\]
As a module over $u(\mathfrak n/\mathfrak v)$,
\[
\operatorname{H}\nolimits^1(u(\mathfrak v)) \ \cong \ M_1 \oplus M_2
\]
where $M_1$ is the span of
$\{\eta_\alpha, \eta_{\alpha+\beta} \}$,
$M_2$ is the span of
$\{ \eta_\gamma, \eta_{\beta+\gamma} \}$ and the
action of the class of $u_2$ is given as above.
In degree 2, we have that
\[
\operatorname{H}\nolimits^2(u(\mathfrak n), k) \ \cong \Lambda^{2}(M_1) \ \
\oplus \ \ M_1 \otimes M_2 \ \ \oplus \ \
\Lambda^{2}(M_2) \ \ \oplus \ \ k^4
\]
where the last factor is spanned by the
generators $X_\alpha, X_\gamma, X_{\alpha+\beta}, X_{\beta+\gamma}$
(each having weight equal to three times its index) of
$S^{*}(\mathfrak v^*)^{(1)}$ that are fixed by the action
of $u_2$. The interesting part of this is
the tensor product $M_1 \otimes M_2$ which
is the direct sum of a trivial module
spanned by $\eta_\alpha \wedge \eta_{\beta+\gamma} -
\eta_{\alpha+\beta} \wedge \eta_\gamma$
and an indecomposable three dimensional
module generated by $\eta_{\alpha+\beta} \wedge
\eta_{\beta+\gamma}$ and
with socle spanned by $w = \eta_\alpha \wedge \eta_\gamma$.
Because the characteristic is 3, this is a
free module over $u(\mathfrak n/\mathfrak v)$. Thus there
is an element $E_2^{0,2} \cong
\operatorname{H}\nolimits^2(u(\mathfrak v),k)^{\mathfrak n/\mathfrak v}$ that is
determined by $w$. Write $M_1 \otimes M_2
\cong k \oplus N$, where $N$ is the free
submodule with socle spanned by $w$.
Let $X_\beta \in E_2^{2,0}$
be the generator of $S^{1}((\mathfrak n/\mathfrak v)^{*})^{(1)}$. This
elements survives to the $E_\infty$ page.
Moreover, every element in $E_2^{i,j}$
for $j \geq 2$ is a multiple of this element.
We claim that this element must be contained
in an associated prime of $\operatorname{H}\nolimits^*(u(\mathfrak n), k)$.
Specifically, the element $\hat{w} \in
\operatorname{H}\nolimits^0(u(\mathfrak n/\mathfrak v),N)$
determined by $w$ in $E_2^{2, 0}$
has the property that $X_\beta\hat{w} = 0$
on the $E_2$ page because $N$ is free
over $u(\mathfrak n/\mathfrak v)$. So we only need to
show that $\hat{w}$ survives to the $E_\infty$
page and that the product $X_\beta\hat{w}$
does not ungrade to something that is nonzero. Both
of these statements can be deduced from
looking at the action of the torus.
That is, $w$ has weight $\alpha + \gamma$
and there is no element of that weight in
either $E_2^{2,1}$ or $E_2^{0,3}$. Hence
both $d_2$ and $d_3$ must both vanish on
the class of $w$. Likewise, $X_\beta\hat{w}$
has weight $\alpha +3\beta +\gamma$ and
there is no element of that weight in
$E_\infty^{3,1}$ or $E_\infty^{4,0}$.
So we must have that $X_\beta$ annihilates
the class of $w$ in $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$
\end{proof}
\begin{rem} \label{rem:derive-rel}
As mentioned earlier in the paper, two of the
computer generated relations in Proposition
\ref{sl3-rest} are indicated by the calculation
above. For this we require the spectral
sequence
\[
E_2^{i,j} \ = \ \operatorname{H}\nolimits^i(u(\mathfrak n/\mathfrak z),
\operatorname{H}\nolimits^j(u(\mathfrak z), k))
\Rightarrow \operatorname{H}\nolimits^{i+j}(u(\mathfrak n),k)
\]
where $\mathfrak n$ is the 5-dimensional Lie algebra as
above and $\mathfrak z$ is the one-dimensional subalgebra
spanned by $u_5$. Note that $\mathfrak n/\mathfrak z \cong
\mathfrak a \oplus \mathfrak b$ where $\mathfrak a$ (generated by the
classes of $u_1, u_2, u_4$) is the nilpotent
radical of a Borel subalgebra of $\mathfrak{sl}_3$, and
$\mathfrak b$ (generated by $u_3$) is a one dimensional
Lie algebra. The bottom row of the spectral
sequence is generated by elements $\eta_\alpha, \eta_\beta,
\eta_{2\alpha+\beta}, \eta_{\alpha+2\beta},
X_\alpha, X_\beta,X_{\alpha+\beta}$, generating
$\operatorname{H}\nolimits^*(u(\mathfrak a),k)$, and $\eta_\gamma, X_\gamma$, generating
$\operatorname{H}\nolimits^*(u(\mathfrak b),k)$. The weights are as indicated
by the subscripts.
The term $E_2^{1,0}$ is spanned by an element
$\eta_{\beta+\gamma}$. Its image
under the differential $d_2$ is $\eta_\beta \eta_\gamma$.
What we know from the proof of Proposition~\ref{prop:notcm1} is
that $\eta_\alpha \eta_\gamma X_\beta = 0$.
This element can only be zero if it is in the
image of $d_2$. Then by an examination of
weights we see that (up to some nonzero
scalar multiple) $d_2(\eta_{\beta+\gamma}\eta_{\alpha+2\beta}) =
\eta_\beta \eta_\gamma \eta_{\alpha+2\beta} =
\eta_\alpha \eta_\gamma X_\beta$. Note that this relation
occurs in the ring $E_2^{*,0} \cong
\operatorname{H}\nolimits^*(u(\mathfrak a),k) \otimes \operatorname{H}\nolimits^*(u(\mathfrak b),k)$.
Consequently, we must have that
$\eta_\beta \eta_{\alpha+2\beta} =
\eta_\alpha X_\beta$ in $\operatorname{H}\nolimits^*(u(\mathfrak a),k)$, as asserted.
The relation $\eta_\alpha \eta_{2\alpha+\beta}
= \eta_\beta X_\alpha$ follows by
symmetry (interchanging $u_1$ with $u_3$
and $u_4$ with $u_5$).
\end{rem}
We shall see in Section \ref{sec:metabelian} that the
example in Proposition \ref{prop:notcm1}
can be generalized, giving a metabelian Lie algebra
whose cohomology ring is not Cohen-Macaulay
for any prime $p$.
\section{Some examples of type B}
In this section, we consider the nilpotent radical $\mathfrak n$
of the Borel subalgebra of a Lie algebra of type
$B_2$ and some extensions thereof. We show that in
characteristic ~5, the cohomology of $\mathfrak n$ has
the form $\operatorname{H}\nolimits^*(u(\mathfrak n),k) \cong
S(\mathfrak n^*)^{(1)} \otimes \operatorname{H}\nolimits^*(\mathfrak n,k)$
as a module over $S(\mathfrak n^*)^{(1)}$, but not as a ring.
Moreover, there is a one-dimensional extension of
this Lie algebra whose cohomology is not Cohen-Macaulay.
The basic idea of the construction applies other
non-simply laced cases in other characteristics.
Note that if the characteristic of $k$ is greater than
$6 = 2(h-1)$ then the isomorphism $\operatorname{H}\nolimits^*(u(\mathfrak n),k) \cong
S(\mathfrak n^*)^{(1)} \otimes \operatorname{H}\nolimits^*(\mathfrak n,k)$ is an
isomorphism of rings by \cite[Theorem 3.1.1]{DNN}.
The Lie algebra $\mathfrak n$ has a basis $u_1, u_2, u_3, u_4$
and the Lie bracket is given by $[u_1,u_2] = u_3$,
$[u_1, u_3] = u_4$, $[u_2,u_3] = 0$ with $u_4$ being
central. There is an action of a two-dimensional
torus, $T$, relative to which the basis elements
have weights $-\alpha, -\beta, -\alpha-\beta, -2\alpha-\beta$
respectively ($\alpha$ being the short simple root).
We begin with a calculation of the ordinary
Lie algebra cohomology of $\mathfrak n$. As in the last section, we
let the subscripts of the elements denote their weights.
\begin{lemma}\label{lem:ordcohob2}
The cohomology ring $\operatorname{H}\nolimits^*(\mathfrak n,k)$ is generated
by elements which we denote $\eta_\alpha, \eta_\beta,
\eta_{\alpha+2\beta}, \eta_{3\alpha+\beta},
\eta_{4\alpha+2\beta}, \eta_{3\alpha+3\beta}$
in degrees 1,1,2,2,3,3. All products
of two of the given generators are zero except
for the products
\[
\eta_{\alpha} \eta_{3\alpha+3\beta} =
-\eta_\beta\eta_{4\alpha+2\beta}
= \eta_{3\alpha+\beta} \eta_{\alpha+2\beta}
\]
\end{lemma}
\begin{proof}
Let $\mathfrak z$ be the subalgebra spanned by $u_4$.
Then we have a spectral sequence given as
\[
E_2^{i,j} = \operatorname{H}\nolimits^i(\mathfrak n/\mathfrak z,
\operatorname{H}\nolimits^j(\mathfrak z,k)) \Rightarrow
\operatorname{H}\nolimits^{i+j}(\mathfrak n,k).
\]
The bottom row is the cohomology of the nilpotent
radical of a Borel subalgebra of $\mathfrak{sl}_3$, which
is provided in Lemma \ref{sl3-ordin}. We adopt the
notation for the elements as given in that lemma.
The $E_2$ term of the spectral sequence is generated
as a ring by one additional element $\zeta = \zeta_{2\alpha+\beta}
\in E_2^{0,1}$. Its image
under the $d_2$ differential is the extension class
$\eta_{2\alpha+\beta}$ (having the same weight).
The differential vanishes on every
product of $\zeta$ with an element on the bottom row
except $\eta_\beta \zeta$, and there $d_2(\eta_\alpha
\zeta) = \eta_\beta \eta_{2\alpha+\beta}
= -\eta_\alpha \eta_{\alpha+2\beta}$.
The product structure is derived
from the product on the $E_2$ page as well as
weight considerations.
\end{proof}
Now we extend this to the restricted Lie algebra
cohomology. Some of the relations in the proposition
given below were calculated using the basic
algebra package in Magma.
\begin{prop} \label{prop:b2rest}
Suppose that $k$ is a field of characteristic 5.
Let $\mathfrak n$ be the nilpotent
radical of the Borel subalgebra of a Lie
algebra of type $B_2$. The
cohomology ring $\operatorname{H}\nolimits^{*}(u(\mathfrak n),k) \cong S^{*}(\mathfrak n^*)^{(1)}
\otimes \operatorname{H}\nolimits^*(\mathfrak n, k)$ is a free
$S^{*}(\mathfrak n^*)^{(1)}$-module with basis consisting
of the images of a basis of the ordinary Lie algebra
cohomology of $\mathfrak n$ as in Lemma \ref{sl3-ordin}.
The ring $S^{*}(\mathfrak n^*)^{(1)}$ is a polynomial ring
in variables $X_\alpha, X_\beta, X_{\alpha+\beta}$
and $X_{2\alpha+\beta}$ having weights
$5\alpha, 5\beta, 5(\alpha+\beta)$ and
$5(2\alpha+\beta)$ under
the action of the torus. The multiplicative relations
among the generators of $1 \otimes \operatorname{H}\nolimits^*(\mathfrak n,k)
\subseteq \operatorname{H}\nolimits^*(u(\mathfrak n),k)$ are exactly as given
in Lemma \ref{lem:ordcohob2}, except that
$\eta_{3\alpha+\beta}^2 =
\eta_{\alpha+2\beta}X_\alpha$. In particular,
the isomorphism $\operatorname{H}\nolimits^{*}(u(\mathfrak n),k) \cong S^{*}(\mathfrak n^*)^{(1)}
\otimes \operatorname{H}\nolimits^*(\mathfrak n, k)$ does not hold as rings.
\end{prop}
\begin{proof}
The first statement follows from \cite[Theorem 3.1.1]{DNN}. A large
part of the remainder of the proof can be derived
from the LHS spectral sequence and weight
considerations. The unusual relation was verified by the computer.
\end{proof}
Next we consider an extension of the algebra, whose
cohomology ring is not Cohen-Macaulay. As in the
case of Remark \ref{rem:derive-rel}, we can derive
the unusual relation in the above proposition from
the calculations of the cohomology of the extension.
Let $\mathfrak n$ be the restricted Lie algebra of dimension
~5, with basis $u_1, \dots, u_5$ and Lie bracket
given by $[u_1,u_2] = u_3$,
$[u_1, u_3] = u_4$, $[u_1,u_4] = u_5$,
with $u_2, \dots, u_5$ forming a commutative subalgebra
that we denote $\mathfrak v$. We continue to assume that the
characteristic of $k$ is ~5. The algebra $\mathfrak n$ has
an action of a 2-dimensional torus, so that the elements
$u_1, \dots, u_5$ have weights $-\alpha$, $-\beta$,
$-\alpha-\beta$, $-2\alpha-\beta$ and $-3\alpha-\beta$.
\begin{prop}\label{prop:b2notcm}
The cohomology ring $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$ is not
Cohen-Macaulay. In particular, there is an element
in $\operatorname{H}\nolimits^2(u(\mathfrak n),k)$ whose annihilator $\mathfrak P$ has
the property that
$\operatorname{H}\nolimits^*(u(\mathfrak n),k)/\mathfrak P$ has Krull dimension
four.
\end{prop}
\begin{proof}
We consider the spectral sequence:
$E_2^{i,j} = \operatorname{H}\nolimits^i(u(\mathfrak n/\mathfrak v), \operatorname{H}\nolimits^j(u(\mathfrak v),k))\Rightarrow
\operatorname{H}\nolimits^{i+j}(u(\mathfrak n),k)$. As a module over $u(\mathfrak n/\mathfrak v)$
$M = \operatorname{H}\nolimits^1(u(\mathfrak v),k)$ is uniserial of length ~4,
generated by an element $\eta_{3\alpha + \beta}$.
Then $\operatorname{H}\nolimits^2(u(\mathfrak v),k)$ is the exterior square of
$M$ and it has a free $u(\mathfrak n/\mathfrak v)$-summand
generated by $\eta_{3\alpha+\beta} \wedge
u_{\alpha}\eta_{3\alpha+\beta}$
and having socle generated by $\eta_{\alpha+\beta}
\wedge \eta_\beta$. Hence there is
a class $\zeta$ in $E_2^{0,2}$ represented by a
$u(\mathfrak n/\mathfrak v)$-homomorphism of $k$ onto the socle of this
summand. This class is annihilated on the $E_2$ page
of the spectral sequence by the class $X_\alpha$
in $E_2^{2,0}$. As in the case of Proposition
\ref{prop:notcm1}, we can argue by weights that $\zeta$
is represented by a nonzero class $\hat{\zeta} \in
\operatorname{H}\nolimits^2(u(\mathfrak n),k)$ such that $X_\alpha \hat{\zeta} =0$,
and the annihilator of $\hat{w}$ is contained in a prime
$\mathfrak P$ having the asserted properties.
\end{proof}
\begin{rem}\label{rem:notorus}
We should note that the action of the torus is not
required to show that the example is not Cohen-Macaulay.
If it were the case that one of the differentials
$d_2$ or $d_3$ failed to vanish on the class $\zeta$,
then we would have that $d_2(\zeta) = X_\alpha \mu$ for
$\mu$ in $E_2^{0,1}$ or that $d_3(\zeta) = X_\alpha \mu$
for $\mu \in E_3^{1,0}$. In either case we would have
a class in $\operatorname{H}\nolimits^1(u(\mathfrak n),k)$ that is annihilated
by $X_\alpha$. Similarly, there is no way to ungrade
the spectral sequence to avoid having a large associate
prime in the cohomology ring.
\end{rem}
\begin{rem}\label{rem:derive-rel2}
We remark, as in \ref{rem:derive-rel}, that the
unusual relation $\eta_{3\alpha+\beta}^2 =
\eta_{\alpha+2\beta} X_\alpha$
in Proposition \ref{prop:b2rest} can be derived
from the last Proposition. The proof is almost
exactly the same as in Remark~\ref{rem:derive-rel} and
we leave the details to the interested reader.
\end{rem}
\begin{rem}\label{rem:b2other-primes}
The situation in Proposition \ref{prop:b2notcm} can
be extended to give examples for other primes. For
suppose that $p > 5$, and let $\mathfrak n$ be the restricted
$p$-Lie algebra of dimension $n+1$ for some $n$
with $(p+3)/2 \leq n < p$, defined as follows.
A basis for $\mathfrak n$ consists of the elements $v,
u_1, \dots, u_n$, where $u_1, \dots, u_n$ span a
commutative subalgebra, which we denote $\mathfrak v$.
Then the product is given
by $[v,u_i] = u_{i+1}$ for $i = 1, \dots, n-1$,
and $[v,u_n]= 0$. The $p$th-power operation is zero
on $\mathfrak n$. The algebra $\mathfrak n$ has an action of a
two-dimensional torus such that the basis elements
$v, u_1, \dots, u_n$ have weights $-\alpha, -\beta,
-\alpha-\beta, \dots, -(n-1)\alpha-\beta$, respectively.
Then $M = \operatorname{H}\nolimits^1(u(\mathfrak v), k)$ is an indecomposable
uniserial module of dimension $n$ over the algebra
$u(\mathfrak n/\mathfrak v)$. Because $n \geq (p+3)/2$, its
exterior square $\Lambda^{2}(M)$ has a free summand.
Hence, considering the spectral sequence
with $E_2$ term $E_2^{i,j} = \operatorname{H}\nolimits^i(u(\mathfrak n/\mathfrak v),
\operatorname{H}\nolimits^j(u(\mathfrak v),k)) \Rightarrow \operatorname{H}\nolimits^{i+j}(u(\mathfrak n),k)$
and arguing exactly as in the proof of
Proposition \ref{prop:b2notcm}, we get that
$\operatorname{H}\nolimits^*(u(\mathfrak n),k)$ has an associated prime $\mathfrak P$
such that $\operatorname{H}\nolimits^*(u(\mathfrak n),k)/\mathfrak P$ has Krull
dimension at most $n$.
\end{rem}
\section{An example of type $G_2$ in characteristic 7}
In this section we consider the nilpotent radical of a
Borel subalgebra of the restricted Lie algebra of type $G_2$.
We obtain a similar result to that in Proposition
\ref{sl3-rest} and Proposition \ref{prop:b2rest}.
Because the methods are also very similar to those in the
aforementioned propositions, we give only a sketch.
\begin{prop}\label{prop:g2rest}
Suppose that $\mathfrak n$ is the nilpotent radical of a
Borel subalgebra of the restricted Lie algebra of type $G_2$.
Then as a module over the symmetric algebra $S^{*}(\mathfrak n^*)^{(1)}$,
we have that $\operatorname{H}\nolimits^*(u(\mathfrak n),k) \cong S^{*}(\mathfrak n^*)^{(1)}
\otimes \operatorname{H}\nolimits^*(\mathfrak n,k)$. However, this is not an isomorphism
as rings.
\end{prop}
\begin{proof}
The characteristic of the field $k$ is larger than the
Coxeter number and hence the first statement is a consequence of
\cite[(3.5) Prop.]{FP}. Our task is to prove the second statement.
Suppose that $\alpha$ and $\beta$ are the simple roots for
the root system of type $G_2$. Assume that $\alpha$ is the
short root. The other positive roots are $\alpha+\beta, 2\alpha+\beta,
3\alpha+\beta, 3\alpha+2\beta$. We construct the central extension $\mathfrak g$
\[
\xymatrix{
0 \ar[r] & \mathfrak a \ar[r] & \mathfrak g \ar[r] & \mathfrak n \ar[r] & 0
}
\]
where $\mathfrak a$ has dimension one, and $\mathfrak g$ has an action by
the two dimensional torus so that an element of $\mathfrak a$ has
weight $-4\alpha-\beta$. Thus $\mathfrak g$ has basis $u_\alpha,
u_\beta, u_{\alpha+\beta}, u_{2\alpha+\beta}, u_{3\alpha+\beta},
u_{4\alpha+\beta}, u_{3\alpha+2\beta}$ where the subscript on
each element indicates the negative of its weight.
Let $\mathfrak v$ be the subalgebra generated by
$u_\beta, u_{\alpha+\beta}, u_{2\alpha+\beta}, u_{3\alpha+\beta},
u_{4\alpha+\beta}, u_{3\alpha+2\beta}$, and let $\mathfrak z$ be the
subalgebra generated by $u_{3\alpha+2\beta}$. The cohomology
of $\mathfrak v$ can be computed from the spectral sequence
$E_2^{i,j} = \operatorname{H}\nolimits^i(\mathfrak v/\mathfrak z, \operatorname{H}\nolimits^j(\mathfrak z,k)) \Rightarrow
\operatorname{H}\nolimits^{i+j}(\mathfrak v,k)$. Let $\eta_\gamma$ denote an element of
weight $\gamma$ on this $E_2$ page. The differential must
take the element $\eta_{3\alpha+2\beta} \in E_2^{0,1}$
to the extension class
$d_2(\eta_{3\alpha+2\beta}) = \eta_\beta \wedge
\eta_{3\alpha+\beta} - \eta_{\alpha+\beta} \wedge
\eta_{2\alpha+\beta} \in E_2^{2,0}$.
Note that this element is annihilated
by the action of $u_\alpha$, as is $\eta_{2\alpha+\beta}$.
The point of this calculation is that $\operatorname{H}\nolimits^2(\mathfrak v,k)$ contains
a free module under the action of $u(\mathfrak g/\mathfrak v)$. That is the
element $\eta_{4\alpha+\beta} \wedge \eta_{3\alpha+\beta}$
generates a uniserial module of dimension $7$ over
$u(\mathfrak g/\mathfrak v)$, whose socle (the submodule annihilated by
the action of $u_\alpha$) is spanned by $\eta_{\alpha+\beta}
\wedge \eta_{\beta}$, a class that survives to the $E_\infty$
page of the spectral sequence. This implies that $\operatorname{H}\nolimits^{*}(u({\mathfrak g}),k)$ is not Cohen-Macaulay.
That is, we see from the spectral sequence $E_2^{i,j} = \operatorname{H}\nolimits^i(u(\mathfrak g/\mathfrak v),
\operatorname{H}\nolimits^j(u(\mathfrak v),k))\Rightarrow \operatorname{H}\nolimits^{i+j}(u({\mathfrak g}),k)$, that the element $X_\alpha \in E_2^{0,2}$ in the
symmetric algebra annihilates the element corresponding
to $\eta_{\alpha+\beta} \wedge \eta_{\beta} \in E_2^{0,2}$.
This spectral sequence collapses at the $E_2$ page, and hence
the relation exists in $\operatorname{H}\nolimits^*(u(\mathfrak g),k)$.
The proposition is a consequence of Theorem \ref{thm:tensoralg}.
More specifically, by following the arguments in Proposition
\ref{sl3-rest} and using the weight information, we see that
in $\operatorname{H}\nolimits^4(u(\mathfrak n),k)$ there must be a relation having
roughly the form $(\eta_\alpha \wedge \eta_{3\alpha+\beta})^2
= (\eta_{\alpha+\beta} \wedge \eta_{\beta}) X_\alpha$. Note that
both are elements of weight $8\alpha+2\beta$.
\end{proof}
\section{A metabelian example} \label{sec:metabelian}
In this section we present an example of a metabelian restricted
Lie algebra with the property that its cohomology ring is not
Cohen-Macaulay. Such an example can be constructed for any value
of $p\geq 3$, except that the dimension of the example depends on
the prime $p$.
Let $\mathfrak n$ be the nilpotent restricted Lie algebra with basis
consisting of the elements $u, v_i, w_i$ for $i = 1, \dots,
n$, and Lie bracket defined
by the rule
\[
[u,v_i] = w_i, \quad [u,w_i] = 0 = [v_i, v_j] = [v_i, w_j] =
[w_i, w_j]
\]
for all $i,j$ such that $1 \leq i, j, \leq n$. Note that $\mathfrak n$
is isomorphic to a subalgebra of $\mathfrak{sl}_{n+2}$. That is, we can
define a homomorphism $\varphi: \mathfrak n \to \mathfrak{sl}_{n+2}$ as follows.
Let $E_{i,j}$ be the matrix with $1 \in k$ in the $(i,j)$
position and 0 elsewhere. Then define $\varphi$ by $\varphi(u)
= E_{1,2}$, $\varphi(v_i) = E_{2,i+2}$ and $\varphi(w_i) =
E_{1,i+2}$. The image of $\varphi$ has the property that the
$p^{th}$-power of any element in the algebra is zero, since
$p \geq 3$. Also, we note that the algebra has an action of
the diagonal torus of $\mathfrak{sl}_{n+2}$ of dimension $n+1$.
Let $\mathfrak v$ be the subalgebra with basis consisting of all of the
elements $v_i, w_i$ for $i = 1, \dots, n$. This subalgebra is
commutative. We consider the
spectral sequence $E_2^{r,s} =
\operatorname{H}\nolimits^r(u(\mathfrak n/\mathfrak v), \operatorname{H}\nolimits^s(u(\mathfrak v), k) \Rightarrow
\operatorname{H}\nolimits^{r+s}(u(\mathfrak n),k)$. The cohomology group $\operatorname{H}\nolimits^1(u(\mathfrak v),k) =
\operatorname{H}\nolimits^1(\mathfrak v,k)$ has dimension $2n$ and is spanned by
elements $\gamma_i$ (of weight $\alpha_2 +\dots + \alpha_{i+1}$)
and $\eta_i$ (of weight $\alpha_1 + \dots + \alpha_{i+1}$,
for $i = 1, \dots, n$. The action of the element $u \in \mathfrak n/\mathfrak v$ on
$\operatorname{H}\nolimits^1(u(\mathfrak v),k)$ is given by $u \cdot \eta_i = \gamma_i$ and
$u \cdot \gamma_i = 0$. Thus, $\operatorname{H}\nolimits^1(u(\mathfrak v),k)$ is a direct sum
of $n$ uniserial $u(\mathfrak n/\mathfrak v)$-modules of dimension 2.
With this information, we can prove the following.
\begin{prop}\label{prop:metabelian}
Let $n = p-1$. Then the cohomology ring $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$ is
not Cohen-Macaulay.
\end{prop}
\begin{proof}
Let $M$ denote the uniserial $u(\mathfrak n/\mathfrak v)$-module of dimension 2. As noted
$E_2^{0,1} = \operatorname{H}\nolimits^1(u(\mathfrak v),k)$ is a directs sum of $p-1$ copies of
$M$. Hence, $E_2^{0,p-1}$ contains the $p-1$ exterior power of
$\operatorname{H}\nolimits^1(u(\mathfrak v),k)$ which includes the $p-1$ tensor power of $M$.
The $p-1$ tensor power of $M$ has a projective $u(\mathfrak n/\mathfrak v)$-module,
the uniserial module of dimension $p$ generated by
$\eta_1 \wedge \dots \wedge \eta_{p-1}$, and having socle
spanned by $\gamma_1 \wedge \dots \wedge \gamma_{p-1}$. Consequently,
$E_2^{0,p-1}$ has an element $y$ of weight
$\alpha_2 + \dots + \alpha_{p}$.
This element must survive to the $E_{\infty}$ page of the spectral
sequence, because there is no element of the same
weight in $E_r^{p-1-r,r+2}$
for any value of $r>0$. On the other hand, because
$\gamma_1 \wedge \dots \wedge \gamma_{p-1}$ is in the socle of a
projective $u(\mathfrak n/\mathfrak v)$-module, we must have that $X.y = 0$
for $X$ the generator of the symmetric algebra in $E_{2.0}$.
Again, by a weight argument we see that the product $X.y$ cannot
ungrade to an element that is not zero. Consequently, this
relation must exist in $\operatorname{H}\nolimits^*(u(\mathfrak n).k)$. This implies that $X$
is not a regular element and that the depth of $\operatorname{H}\nolimits^*(u(\mathfrak n),k)$
is less than the Krull dimension.
\end{proof}
\section{Nilpotent Lie algebras of dimension $\leq 5$}
In this section we will use deGraaf's classification
of indecomposable nilpotent Lie algebras over
fields of characteristic not equal to 2. Again our interest is in
whether there is an isomorphism
\[
\operatorname{H}\nolimits^*(u({\mathfrak n}),k)\cong
S^{*}({\mathfrak n}^{*})^{(1)}\otimes
\Lambda^{*}({\mathfrak n}^{*})
\cong S^{*}({\mathfrak n}^{*})^{(1)} \otimes
\operatorname{H}\nolimits^{*}({\mathfrak n},k)
\]
as rings or as modules over the symmetric algebra. If the algebra
is commutative, then the isomorphism holds as rings. On the other
hand if the $p$-power operation on the Lie algebra fails to vanish
($x^{[p]} \neq 0$ for some $x$ in $\mathfrak n$), then the above isomorphism
can not hold.
Given a finitel dimensional nilpotent Lie algebra, it is not always
possible to make it a restricted Lie algebra with trivial $p$-power
operation. For one thing, the adjoint action of any element on the
algebra would have to be nilpotent of degree less than $p$.
We have listed below the indecomposable
non-abelian nilpotent Lie algebras of
dimension less than or equal to 5 along with the restrictions
on the prime $p$ that are necessary to impose a trivial $p$-power
operation. In this case, our standing assumption
that ${\mathcal N}_{1}({\mathfrak n})\cong {\mathfrak n}$ holds. The notation for
the Lie algebras is taken from DeGraaf's list \cite{deG}.
\vskip .25cm
\noindent
Dimension 3: \ $L_{3,2}$ ($p\geq 3$)
\vskip .5cm
\noindent
Dimension 4: \ $L_{4,3}$ ($p \geq 5$)
\vskip .5cm
The Lie algebras $L_{3,2}$ (reps. $L_{4,3}$) arise naturally
as the unipotent radicals of the Borel subalgebras of
simple Lie algebras of type $A_{2}$ (resp. $B_{2}$).
We have $L_{3,2}=\langle x_{-\alpha}, x_{-\beta}, x_{-\alpha-\beta}
\rangle$ and $L_{4,3}=\langle x_{-\alpha}, x_{-\beta},
x_{-\alpha-\beta}, x_{-2\alpha-\beta} \rangle$. The restricted cohomology rings
of these algebras are given in Propositions \ref{sl3-rest} and
\ref{prop:b2rest}.
\vskip .5cm
\noindent
Dimension 5: $L_{5,4}$ ($p\geq 3$), $L_{5,5}$ ($p\geq 5$),
$L_{5,6}$ ($p\geq 5$), $L_{5,7}$ $(p\geq 5$),
$L_{5,8}$ ($p\geq 3$), $L_{5,9}$ ($p\geq 5$).
\vskip .5cm
We now use deGraaf's description of the five dimensional
nilpotent Lie algebra (cf. \cite[Section 4]{deG}) and
describe natural gradings on these
Lie algebras. The natural gradings are induced by toral
actions given by outer automorphisms. When we compute
the cohomology of the algebras, differentials in the spectral sequences
respect the actions of these tori.
\vskip .25cm
\noindent
$L_{5,4}$: This nilpotent Lie algebra arises as a
subalgebra of the nilpotent radical of a simple Lie
algebra of type $A_{3}$. Let
$\alpha_{1},\alpha_{2},\alpha_{3}$ denote the
simple roots. The $L_{5,4}$ consists of the span of
the root vectors $\{x_{-\alpha_{1}},x_{-\alpha_{3}},
x_{-\alpha_{1}-\alpha_{2}},x_{-\alpha_{2}-\alpha_{3}},
x_{-\alpha_{1}-\alpha_{2}-\alpha_{3}}\}$.
\vskip .25cm
\noindent
$L_{5,5}$: This Lie algebra has a double grading with
basis $\langle x_{-\alpha},x_{-\beta},x_{-\alpha-\beta},
x_{-2\alpha-\beta},x_{-2\alpha} \rangle$.
\vskip .25cm
\noindent
$L_{5,6}$: Let $W(1)=\langle e_{i}:\ i\in {\mathbb Z}\}$
be the Witt algebra defined over ${\mathbb Z}$ with
Lie bracket $[e_{i},e_{j}]=(i+j)e_{i+j}$. One can
consider the subalgebra ${\mathfrak a}=\langle e_{i}:\ i< 0\}$
and factor this out by the ideal
${\mathfrak z}=\{e_{i}:\ i\leq -6\}$. The Lie algebra
$L_{5,6}$ is the Lie algebra ${\mathfrak a}/{\mathfrak z}$
tensored by $k$. This has a natural ${\mathbb Z}$ grading,
thus an action of a one-dimensional torus
on $L_{5,6}$. This Lie algebra can also be viewed as a non-graded
central extension of the unipotent radical of type $B_{2}$.
\vskip .25cm
\noindent
$L_{5,7}$: The Lie algebra $L_{5,7}$ is a graded central
extension of $L_{4,3}$. The Lie algebra can be graded by a two-dimensional torus and has
basis given by $\langle x_{-\alpha},x_{-\beta},x_{\alpha-\beta},
x_{-2\alpha-\beta},x_{-3\alpha-\beta} \rangle$.
\vskip .25cm
\noindent
$L_{5,8}$: Let ${\mathfrak a}$ be the nilpotent radical for the Borel subalgebra of
a Lie algebra of type $A_{3}$, and ${\mathfrak z}$ be the
center of this Lie algebra. The Lie algebra
$L_{5,8}$ can be realized as ${\mathfrak a}/{\mathfrak z}$
and has basis (with an action of a three dimensional torus) given by
$\langle x_{-\alpha_{1}},x_{-\alpha_{2}},x_{-\alpha_{3}},
x_{-\alpha_{1}-\alpha_{2}},x_{-\alpha_{2}-\alpha_{3}} \rangle$.
\vskip .25cm
\noindent
$L_{5,9}$: One can realize this Lie algebra as another
graded central extension of $L_{4,3}$. This Lie
algebra has a double grading with basis
$\langle x_{-\alpha},x_{-\beta},x_{-\alpha-\beta},
x_{-2\alpha-\beta}, x_{-\alpha-2\beta} \rangle$.
\vskip .5cm
The ordinary Lie algebra cohomology can be computed recursively
using central extensions and the LHS spectral sequence. For example,
if ${\mathfrak a}$ is a nilpotent Lie algebra and
${\mathfrak z}$ is a one-dimensional central
subalgebra then the LHS spectral sequence:
$$E_{2}^{i,j}=\operatorname{H}^{i}({\mathfrak a}/{\mathfrak z},k)
\otimes \operatorname{H}^{j}({\mathfrak z},k)
\Rightarrow \operatorname{H}^{i+j}({\mathfrak a},k)$$
will converge after the second page
(i.e., $E_{3}\cong E_{\infty}$). We have
$$\operatorname{H}^{2}({\mathfrak a},k)\cong
\operatorname{H}^{2}({\mathfrak a}/{\mathfrak z},k)/\langle
\text{Im }\delta_{2} \rangle \oplus \langle
\text{Ker }\hat{\delta}_{2} \rangle.$$
where $\delta_{2}:E_{2}^{0,1}\rightarrow E_{2}^{2,0}$
and $\hat{\delta_{2}}:E_{2}^{1,1}\rightarrow E_{2}^{3,0}$.
With appropriate choices of central subalgebras one
can guarantee that the differentials respect the
gradings above. This allows us to compute the
differentials $\delta_{2}$ and $\hat{\delta}_{2}$ inductively.
The weight spaces for the ordinary Lie algebra
cohomology for these Lie algebras are multiplicity
free (i.e., one-dimensional) and given in the
following tables.
To aid in the computation we use some facts
about the ordinary Lie algebra
cohomology. For example, that if $\mathfrak g$ has dimension
$d$, then the ordinary Lie algebra cohomology vanishes
in degrees greater than $d$. In addition there is a
Poincar\'e duality that is also respected by the action of
the tori. So for example, if $d$ is the dimension of the algebra
$\mathfrak n$ and if the element in $\operatorname{H}\nolimits^d(\mathfrak n,k)$ has weight $\gamma$,
then the weights of the cohomology element in degrees $d-1$ will be
$\gamma-\zeta_1, \gamma-\zeta_2, \dots $ where $\zeta_1,
\zeta_2, \dots$ are the weights of the cohomology elements in
degree 1.
\vskip.3in
\begin{table}[htbp]
\begin{tabular}{||r|r| |r|r||}
\hline
$L_{3,2}$ & & $L_{4,3}$ & \\
\hline
\hline
degree & weights & degree & weights \\
\hline
0 & $0$ & 0 & $0$ \\
1 & $\alpha$, $\beta$ & 1 & $\alpha$, $\beta$ \\
2 & $\alpha+2\beta$, $2\alpha+\beta$ & 2 & $\alpha+2\beta$, $3\alpha+\beta$ \\
3 & $2\alpha+2\beta$ & 3 & $3\alpha+3\beta$, $4\alpha+2\beta$ \\
& & 4 & $4\alpha+3\beta$ \\
\hline
\end{tabular}
\end{table}
\begin{table}[htbp]
\begin{tabular}{||r|r||}
\hline
$L_{5,4}$ & \\
\hline
\hline
degree & weights \\
\hline
0 & $0$ \\
1 & $\alpha_{1}$, $\alpha_{3}$, $\alpha_{1}+\alpha_{2}$, $\alpha_{2}+\alpha_{3}$ \\
2 & $\alpha_{1}+\alpha_{3}$, $2\alpha_{1}+\alpha_{2}$, $\alpha_{1}+\alpha_{2}+\alpha_{3}$, \\
& $\alpha_{2}+2\alpha_{3}$, $\alpha_{1}+2\alpha_{2}+\alpha_{3}$ \\
3 & $2\alpha_{1}+3\alpha_{2}+2\alpha_{3}$, $\alpha_{1}+2\alpha_{2}+3\alpha_{3}$, $2\alpha_{1}+2\alpha_{2}+2\alpha_{3}$, \\
& $3\alpha_{1}+2\alpha_{2}+\alpha_{3}$, $2\alpha_{1}+\alpha_{2}+2\alpha_{3}$ \\
4 & $2\alpha_{1}+3\alpha_{2}+3\alpha_{3}$, $3\alpha_{1}+3\alpha_{2}+2\alpha_{3}$, $2\alpha_{1}+2\alpha_{2}+3\alpha_{3}$, \\
& $3\alpha_{1}+2\alpha_{2}+2\alpha_{3}$ \\
5 & $3\alpha_{1}+3\alpha_{2}+3\alpha_{3}$ \\
\hline
\end{tabular}
\end{table}
\begin{table}[htbp]
\begin{tabular}{||r|r||}
\hline
$L_{5,8}$ & \\
\hline
\hline
degree & weights \\
\hline
0 & $0$ \\
1 & $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ \\
2 & $\alpha_{1}+2\alpha_{2}$, $2\alpha_{1}+\alpha_{2}$, $\alpha_{1}+\alpha_{3}$, \\
& $2\alpha_{2}+\alpha_{3}$, $\alpha_{2}+2\alpha_{3}$ \\
3 & $\alpha_{1}+\alpha_{2}+2\alpha_{3}$, $2\alpha_{2}+2\alpha_{3}$, $\alpha_{1}+3\alpha_{3}+\alpha_{3}$, \\
& $2\alpha_{1}+\alpha_{2}+\alpha_{3}$, $2\alpha_{1}+2\alpha_{2}$ \\
4 & $\alpha_{1}+3\alpha_{2}+2\alpha_{3}$, $2\alpha_{1}+2\alpha_{2}+2\alpha_{3}$, $2\alpha_{1}+3\alpha_{2}+\alpha_{3}$ \\
5 & $2\alpha_{1}+3\alpha_{2}+2\alpha_{3}$ \\
\hline
\end{tabular}
\end{table}
\begin{table}[htbr]
\begin{tabular}{||r|r| |r|r||}
\hline
$L_{5,5}$ & & $L_{5,6}$ & \\
\hline
\hline
degree & weights & degree & weights \\
\hline
0 & $0$ & 0 & $0$ \\
1 & $\alpha$, $\beta$, $2\alpha$ & 1 & $1$, $2$ \\
2 & $2\alpha+\beta$, $\alpha+2\beta$, $3\alpha$ & 2 & $3$ \\
3 & $4\alpha+2\beta$, $5\alpha+\beta$, $3\alpha+3\beta$ & 3 & $12$ \\
4 & $5\alpha+3\beta$, $6\alpha+2\beta$, $4\alpha+3\beta$ & 4 & $13$, $14$ \\
5 & $6\alpha+3\beta$ & 5 & $15$ \\
\hline
\end{tabular}
\end{table}
\begin{table}[htbr]
\begin{tabular}{||r|r| |r|r||}
\hline
$L_{5,7}$ & & $L_{5,9}$ & \\
\hline
\hline
degree & weights & degree & weights \\
\hline
0 & $0$ & 0 & $0$ \\
1 & $\alpha$, $\beta$ & 1 & $\alpha$, $\beta$ \\
2 & $\alpha+2\beta$, $4\alpha+\beta$ & 2 & $3\alpha+\beta$, $\alpha+3\beta$ \\
3 & $6\alpha+2\beta$, $3\alpha+3\beta$ & 3 & $2\alpha+4\beta$, $4\alpha+2\beta$ \\
4 & $6\alpha+4\beta$, $7\alpha+3\beta$ & 4 & $4\alpha+5\beta$, $5\alpha+4\beta$ \\
5 & $7\alpha+4\beta$ & 5 & $5\alpha+5\beta$ \\
\hline
\end{tabular}
\end{table}
\vfill\eject
With this information about the ordinary Lie algebra
cohomology $\text{H}^{*}({\mathfrak n},k)$ we
can deduce structural properties about the
restricted Lie algebra cohomology.
\begin{thm} Let ${\mathfrak n}$ be a five-dimensional
nilpotent Lie algebra. Then
\begin{itemize}
\item[(a)] $\operatorname{H}^{*}(u({\mathfrak n}),k)
\cong S^{*}({\mathfrak n}^{*})^{(1)}\otimes
\operatorname{H}^{*}({\mathfrak n},k)$ as
$S^{*}({\mathfrak n}^{*})^{(1)}$ module for $L_{5,4}$
($p\geq 3$), $L_{5,5}$ ($p\geq 5$), $L_{5,6}$
($p\geq 7$), $L_{5,7}$ $(p\geq 7$), $L_{5,8}$
($p\geq 5$), and $L_{5,9}$ ($p\geq 5$).
In these cases the cohomology ring
$\operatorname{H}\nolimits^{*}(u({\mathfrak n}),k)$ is Cohen-Macaulay.
\item[(b)] In addition, the above isomorphism holds
as rings for the algebras
$L_{5,4}$ ($p\geq 5$), $L_{5,5}$ ($p\geq 7$), $L_{5,6}$
($p\geq 13$), $L_{5,7}$ $(p\geq 11$), $L_{5,8}$
($p\geq 5$), and $L_{5,9}$ ($p\geq 5$).
\end{itemize}
\end{thm}
\begin{proof} (a) There exist a one-dimensional central ideal ${\mathfrak z}$ such that ${\mathfrak n}/{\mathfrak z}$
is isomorphic to (i) the nilpotent radical for a simple Lie algebra of Type $B_{2}$ for $L_{5,5}$, $L_{5,6}$, $L_{5,7}$ and $L_{5,9}$, (ii) an abelian Lie algebra
for $L_{5,4}$ and (iii) the nilpotent radical of type $A_{2}\times A_{1}$ for $L_{5,8}$.
Now assume that $p\geq 7$ in case (i), $p\geq 3$ in case (ii), and $p\geq 5$ in case (iii). Then
$\operatorname{H}^{*}(u({\mathfrak n}/{\mathfrak z}),k)\cong
S^{*}(({\mathfrak n}/{\mathfrak z})^{*})^{(1)}\otimes \operatorname{H}^{*}({\mathfrak n}/{\mathfrak z},k)$ as rings. Hence, by
Theorem~\ref{thm:tensoralg}, $\operatorname{H}^{*}(u({\mathfrak n}),k)\cong
S^{*}({\mathfrak n}^{*})^{(1)}\otimes
\operatorname{H}^{\bullet}({\mathfrak n},k)$ as
$S^{*}({\mathfrak n}^{*})^{(1)}$ module.
The remaining cases when $L_{5,8}$ ($p=3$), $L_{5,5}$ ($p=5$) and $L_{5,9}$ ($p=5$) can be verified directly by showing that the
spectral sequence (\ref{eq:specseq}) collapses.
(b) We construct a subalgebra $B$ in
$\text{H}^{*}(u({\mathfrak n}),k)$ which is
isomorphic to $\text{H}^{*}({\mathfrak n},k)$.
This is accomplished using the gradings to show that the
following conditions cannot simultaneously hold. First,
\[
\gamma_{1}+\gamma_{2}=\gamma_{3}+p\sigma
\]
where $\gamma_{j}$ is a weight of
$\text{H}^{a_{j}}({\mathfrak n},k)$ for $j=1,2,3$ and $\sigma\neq 0$
is a weight of $S^{*}({\mathfrak n}^{*})^{(1)}$. Second,
$$a_{1}+a_{2}=a_{3}+\text{deg}(\sigma)$$
were $\text{deg}(\sigma)$ is the cohomological degree of the element that $\sigma$ represents.
In all the cases listed in the statement this was verified, and proves there do not exist
two elements of weights $\gamma_1$ and $\gamma_2$ whose product is the product
of an element of the symmetric algebra with an element of weight
$\gamma_3$. Furthermore, this shows that a basis of weight vectors in
$\text{H}^{*}({\mathfrak n},k)$ form
a subalgebra of $\text{H}^{*}(u({\mathfrak n}),k)$.
\end{proof}
\begin{thm} Let ${\mathfrak n}$ be a five-dimensional
nilpotent Lie algebra. Then
$\operatorname{H}^{*}(u({\mathfrak n}),k)$
is not Cohen-Macaulay in the cases that $\mathfrak n$ is
$L_{5,7}$ for $p=5$ and $L_{5, 8}$ for $p=3$. In these cases, the depth of the
cohomology ring is one less than the dimension.
\end{thm}
\begin{proof}
The results are proved in Propositions \ref{prop:notcm1}
and \ref{prop:b2notcm}.
\end{proof}
We remark that the work in \cite{BC1, BC2} can be adapted for restricted Lie algebra cohomology to show that
in the case when the cohomology ring has depth at least one less than the Krull dimension, the cohomology
ring is Cohen-Macaulay if and only if it satisfies the functional equation in Theorem~\ref{thm:bc}. We can
conclude that the cohomology rings for $L_{5,7}$ ($p=5$) and $L_{5, 8}$ ($p=3$) do not satisfy the
functional equation.
\section{Type $A_{3}$, $p>h$}
From the Examples \ref{sl3-rest}, \ref{prop:b2rest}, \ref{prop:g2rest},
one might get the impression that if ${\mathfrak n}$ is a
unipotent radical of a Borel subalgebra of a simple Lie algebra
and $h<p <2(h-1)$, then the restricted cohomology is not isomorphic
as rings to the tensor of the symmetric algebra and the ordinary Lie
algebra cohomology. However, this is not true. In this section
we analyze the smallest example of this sort where the ring isomorphism
exists. In this case, the root system $\Phi$ is of type $A_{3}$
and ${\mathfrak n}$ is the six dimensional Lie algebra of
strictly upper triangular $4\times 4$ matrices over
a field of characteristic $5$. In the DeGraaf
notation this is the Lie algebra $L_{6,19}(\epsilon)$.
Since $p>h$, the ordinary Lie algebra cohomology is given by Kostant's
theorem. A proof in characteristic $p$ with $p>h$
is found in \cite[Theorem 4.1.1]{UGA}. As a module for $T$ we have
\begin{equation} \label{eq:Kostant}
\operatorname{H}^{n}({\mathfrak n},k)\cong \bigoplus_{w\in W,\ l(w)=n} -w\cdot 0
\end{equation}
As before we are using the convention that ${\mathfrak n}$
consists of negative root vectors. Let
$\Delta=\{\alpha_{1},\alpha_{2},\alpha_{3}\}$
denote the simple root vectors. Then (\ref{eq:Kostant})
can be used to produce the following table which describes the weights in
the cohomology groups $\operatorname{H}^{n}({\mathfrak n},k)$.
\begin{table}[htbp]
\begin{tabular}{||r|r||}
\hline
$L_{6,19}(\epsilon)$ & \\
\hline
\hline
degree & weights \\
\hline
0 & $0$ \\
1 & $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ \\
2 & $\alpha_{1}+2\alpha_{2}$, $\alpha_{1}+\alpha_{3}$, $2\alpha_{1}+\alpha_{2}$, \\
& $\alpha_{2}+2\alpha_{3}$, $2\alpha_{2}+\alpha_{3}$ \\
3 & $2\alpha_{1}+2\alpha_{2}$, $\alpha_{1}+2\alpha_{2}+3\alpha_{3}$, $\alpha_{1}+3\alpha_{2}+\alpha_{3}$, \\
& $2\alpha_{1}+\alpha_{2}+2\alpha_{3}$, $2\alpha_{2}+2\alpha_{3}$, $3\alpha_{1}+2\alpha_{2}+\alpha_{3}$ \\
4 & $2\alpha_{1}+2\alpha_{2}+3\alpha_{3}$, $2\alpha_{1}+4\alpha_{2}+2\alpha_{3}$, $\alpha_{1}+3\alpha_{2}+3\alpha_{3}$ \\
& $3\alpha_{1}+3\alpha_{2}+\alpha_{3}$, $3\alpha_{1}+2\alpha_{2}+2\alpha_{3}$ \\
5 & $2\alpha_{1}+4\alpha_{2}+3\alpha_{3}$, $3\alpha_{1}+3\alpha_{2}+3\alpha_{3}$, $3\alpha_{1}+4\alpha_{2}+2\alpha_{3}$ \\
6 & $3\alpha_{1}+4\alpha_{2}+3\alpha_{3}$ \\
\hline
\end{tabular}
\end{table}
\begin{thm} Let ${\mathfrak n}$ be the unipotent radical
corresponding to the simple group with root system $A_{3}$
(i.e., $4\times 4$ upper triangular matrices) with $p>h$. Then
$$\operatorname{H}^{*}(u({\mathfrak n}),k)\cong
S^{*}({\mathfrak n}^{*})^{(1)}\otimes
\operatorname{H}^{*}({\mathfrak n},k)$$
as rings.
\end{thm}
\begin{proof} For $p>2(h-1)$ we can apply \cite[Theorem 3.1.1]{DNN}.
This leaves us with the case when $p=5$.
In order to invoke Theorem~\ref{th:splitting}, we need
to analyze the equation
\begin{equation} \label{eq:condition1}
-w_{1}\cdot 0-w_{2}\cdot 0=p\sigma-w_{3}\cdot 0
\end{equation}
where $-w_{j}\cdot 0$ is a weight of
$\text{H}^{a_{j}}({\mathfrak n},k)$ for $j=1,2,3$ and $\sigma$
is a weight of $S^{*}({\mathfrak n}^{*})^{(1)}$.
Furthermore, by consideration
of cohomological degrees we must have that
\begin{equation} \label{eq:condition2}
l(w_{1})+l(w_{2})-l(w_{3})=2\text{deg}(\sigma)
\end{equation}
where $\text{deg}(\sigma)$ is the cohomological degree
corresponding to the element of weight $\sigma$.
The first equation (\ref{eq:condition1}) does have solutions. For example,
$$(2\alpha_{1}+4\alpha_{2}+3\alpha_{3})+(3\alpha_{1}+3\alpha_{2}+3\alpha_{3})
=(2\alpha_{2}+\alpha_{3})+5(\alpha_{1}+\alpha_{2}+\alpha_{3}).$$
Here $l(w_{1})=l(w_{2})=5$ and $l(w_{3})=2$. However,
$\text{deg}(\sigma)\leq 6$. Thus, the second
equation (\ref{eq:condition2}) cannot hold. A
careful, case by case analysis rules out the possibility
that both equations could simultaneously be satisfied.
\end{proof}
We end the paper with the following intriguing question.
\begin{quest}
Suppose that $\mathfrak n$ is the nilpotent radical of a Borel
subalgebra of a Lie algebra arising from a reductive algebraic group.
In the case that the root system $\Phi$ is simply laced
and that $p >h$, is there always a ring isomorphism
$$\operatorname{H}^{*}(u({\mathfrak n}),k)\cong
S^{*}({\mathfrak n}^{*})^{(1)}\otimes
\operatorname{H}^{*}({\mathfrak n},k)?$$
\end{quest}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 673 |
Q: Transaction issue in ASP Classic I am facing HTTP 500 error with classic asp application hosted in IIS 7.5 server and Windows server 2008.
I have done the below steps.
*
*32 Bit Enabled in App Pool
*Enable Parent Paths = true
*MSDTC Security settings updated
Still that is not working with the above changes. But on uninstalling and installing MSDTC and doing a IISRESET works fine.
Note: ASP page enabled with Transaction=required and removing this also works fine.
A: *
*Open IIS manager and go to ASP section
*set send error to browser = true
*remove the default script error message
*save and restart IIS
*Uncheck the "Show friendly error message" from browser. If you have performed the above steps you will be able to see the correct error in you browser.
Check the screenshot below
IIS settings
What is the error you are seeing in the browser??
Check these below links to configure MSDTC and your DB for transaction and authentication (inbound,outbound). For you description it looks like a authentication and transaction related issue in ASP.
Where MSDTC needs to be installed in Distributed transaction case
https://msdn.microsoft.com/en-us/library/dd327979.aspx
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,045 |
Sarah is our insurance specialist, reporting the latest important developments in the industry.
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Despite this lack of transparency, a recent study by the Norwich and Peterborough Building Society (N&P) revealed the ****dreadful reality of the state of household finances for many British families. Amongst other data, research discovered that the Welsh population are the worst savers within the UK while Scottish people are the best, and that one in four Brits over 55 has no available savings at all. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,453 |
\section{Introduction}
\label{SecIntro}
A central question of much current interest is whether an extended symmetry algebra
${\cal W}$~\cite{Zam85,BS9210} exists for logarithmic conformal
field theories~\cite{Gur9303,Flo0111,Gab0111,Kaw0204} like the logarithmic minimal models
${\cal LM}(p,p')$~\cite{PRZ0607,RP0706,RP0707}. These models contain a countably
{\em infinite} number of inequivalent Virasoro modules which the extended symmetry should
reorganize into a {\em finite} number of ${\cal W}$-extended modules closing under fusion.
In the case of the logarithmic minimal models ${\cal LM}(1,p')$, the existence and properties of
such an extended ${\cal W}$-symmetry, including the associated fusion rules, are by now largely
understood~\cite{GK9606,FHST0306,FGST0504,CF0508,GR0707,AM0707,GT0711,PRR0803}.
The works~\cite{FGST0606a,FGST0606b} strongly indicate the existence of a
${\cal W}_{p,p'}$ symmetry algebra for general augmented minimal models, but offer only very
limited insight into the associated fusion algebras. Recently, a detailed description of these fusion
algebras has been provided in~\cite{RP0804,Ras0805,Ras0812} generalizing the approach
of~\cite{PRR0803}. Extending ideas originating with
Cardy~\cite{Cardy86,Cardy89}, this approach uses a strip-lattice implementation of fusion to
obtain the fusion rules of the entire series of logarithmic minimal models ${\cal LM}(p,p')$ in the
${\cal W}$-extended picture where they are denoted by ${\cal WLM}(p,p')$. It is stressed,
that the extended picture is described by the {\em same} lattice model as the Virasoro picture.
Contrary to the situation in the Virasoro picture, for $p>1$, there is no identity
nor a pair of so-called fundamental modules in the lattice approach to ${\cal WLM}(p,p')$.
In~\cite{Ras0812}, we found that one can supplement the set of indecomposable modules
associated with boundary conditions by a set of reducible yet indecomposable rank-1 modules.
This algebraically enlarged set was shown to yield a well-defined fusion algebra called the
{\em fundamental fusion algebra}. This algebra is so named since it is
generated from repeated fusions of the two {\em fundamental modules}
$\ketw{2,1}$ and $\ketw{1,2}$ in addition to the identity $\ketw{1,1}$ which is now
present for all $p$. It was also found
that the fusion algebra generated by modules associated with boundary conditions
is an {\em ideal} of the fundamental fusion algebra. Further algebraic extensions exist.
In particular, for $p>1$, there are additional {\em irreducible} modules not associated with
boundary conditions. Their fusion properties have been systematically examined only very
recently~\cite{GRW0905,Ras0906,Wood0907}. Here we restrict ourselves to the
modules generating the fundamental fusion algebra.
The fusion matrices of a standard rational conformal field theory are diagonalizable.
This is made manifest by the Verlinde formula~\cite{Ver88} where the diagonalizing
similarity matrix is the modular $S$-matrix of the characters in the theory.
In a logarithmic conformal field theory, on the other hand, there are typically
more linearly independent representations
than linearly independent characters due to the presence of indecomposable modules
of rank greater than 1. Consequently, there is no Verlinde formula in the usual sense
and the fusion matrices may not all be diagonalizable. This is indeed the situation for the
${\cal W}$-extended logarithmic minimal models ${\cal WLM}(p,p')$ analyzed in the present work.
In the regular representation of a fusion algebra, the fusion matrices are mutually commuting.
Viewing the fusion matrices as adjacency matrices of graphs, the fusion rules are succinctly
encoded in these fusion graphs. In this context, the regular representation of a fusion algebra is
referred to as the {\em graph fusion algebra}. Fusion graphs have been the key to the
classification of rational conformal field theories on the cylinder~\cite{BPPZ9809,BPPZ9908}
and on the torus~\cite{Ocn99,Ocn00,PZ0011,PZ0101}.
In the rational $A$-type theories, the Verlinde algebra yields a diagonal modular
invariant, while $D$- and $E$-type theories are related to non-diagonal
modular invariants. The Ocneanu algebras arise when considering fusion on the
torus, with left and right chiral halves of the theory, and involve Ocneanu graphs.
We refer to~\cite{Kos88,DiFZ90,DiF92,PZ9510} for earlier results on the interrelation
between fusion algebras, graphs and modular invariants.
It is our hope that the present work will be a step in the direction of
extending these fundamental insights to the logarithmic conformal field theories.
As already indicated, the fusion matrices in the regular representation of the fundamental
fusion algebra are mutually commuting, but in general not diagonalizable.
Nevertheless, we show that they can be brought simultaneously to
block-diagonal forms whose blocks are upper-triangular matrices of dimension 1, 3, 5 or 9.
The directed graphs associated with the two fundamental modules are described in detail.
They consist of a number of connected components of which there are two prototypes.
The adjacency matrices of these {\em tadpole} graphs and {\em eye-patch} graphs
are Jordan decomposed explicitly.
Combining them, the adjacency matrices $X$ and $Y$ of the two fundamental graphs
are found to share a complete set of common generalized
eigenvectors organized as a web constructed by interlacing the Jordan chains of $X$ and $Y$.
This web is here called a {\em Jordan web} and it consists of connected
subwebs with 1, 3, 5 or 9 generalized eigenvectors.
The similarity matrix, formed by concatenating these vectors, {\em simultaneously} brings
$X$ and $Y$ to Jordan canonical form {\em modulo} permutation similarity.
For $p>1$, it is simply not possible to properly Jordan decompose them simultaneously.
The ranks of the participating Jordan blocks are 1 or 3, and the corresponding eigenvalues
are given by $2\cos \frac{j\pi}{\rho}$ where $j=0,\ldots,\rho$ and $\rho=p,p'$.
For $p=1$, the fundamental fusion graph with adjacency matrix $Y$ is given by a {\em single}
eye-patch graph and is thus connected. The fundamental fusion matrix $X$ acts
as a conjugation on this eye-patch graph. In contrast to the situation for $p>1$,
as demonstrated in~\cite{Ras0908}, these simple properties allow for the existence of
a similarity matrix which simultaneously brings all fusion matrices of the fundamental
fusion algebra of ${\cal WLM}(1,p')$ to Jordan form.
The two fundamental fusion matrices, in particular, are both brought to Jordan {\em canonical}
form by this similarity transformation. The present work is an extension of the
paper~\cite{Ras0908} on ${\cal WLM}(1,p')$ to the general series of ${\cal W}$-extended
logarithmic minimal models ${\cal WLM}(p,p')$.
For $p>1$, only some of the modules in the fundamental fusion algebra of ${\cal WLM}(p,p')$
are associated with boundary conditions within our lattice approach. The fusion matrices in
the regular representation of the corresponding fusion subalgebra
have features similar to the ones for the larger fundamental fusion algebra. From~\cite{Ras0812},
we know that the modules associated with boundary conditions form an ideal of the
fundamental fusion algebra. Their matrix realizations $\Nh_\mu$ therefore follow from the
realizations
$N_\mu$ of the generators of the fundamental fusion algebra by elimination of the rows
and columns corresponding to the modules not associated with boundary conditions.
According to~\cite{Ras0812}, every fusion matrix $N_\mu$
can be written as a polynomial in the fundamental fusion matrices $X$ and $Y$.
Likewise, every fusion matrix $\Nh_\mu$
can be written as a polynomial in the {\em auxiliary fusion matrices} $\Xh$ and $\Yh$
obtained from $X$ and $Y$ by the aforementioned elimination procedure.
For $p>1$, the matrix $\Yh$ does {\em not} correspond to
a module associated with a boundary condition and is, in this sense, auxiliary. For $p>2$,
this applies to both $\Xh$ and $\Yh$.
Despite their auxiliary status, the matrices $\Xh$ and $\Yh$ are very useful
in the description of the spectral decomposition of the fusion matrices $\Nh_\mu$.
We refer to the corresponding directed graphs as {\em auxiliary fusion graphs}.
As in the case of the fundamental fusion graphs, the auxiliary fusion graphs consist of a
certain number of connected components of which there are two prototypes:
{\em cycle} graphs and the eye-patch graphs above.
We show that the auxiliary adjacency matrices $\Xh$ and $\Yh$ share a complete set of
common generalized eigenvectors, and that the corresponding Jordan web consists of connected
subwebs with 1, 2, 3 or 8 generalized eigenvectors.
We subsequently show that the fusion matrices $\Nh_\mu$ can be brought simultaneously to
block-diagonal forms whose blocks are upper-triangular matrices of dimension 1, 2, 3 or 8.
The remaining part of this paper is organized as follows.
Section~\ref{SecWLM} briefly reviews some basics of ${\cal WLM}(p,p')$ and its fundamental
fusion algebra. The associated graph fusion algebras are formally introduced and the fusion rules
involving the fundamental modules are summarized.
The cycle, tadpole and eye-patch graphs are defined in Section~\ref{SecAdjacency},
and the spectral decompositions of their adjacency matrices are worked out in detail.
These results are conveniently expressed in terms of Chebyshev polynomials.
Using the summarized fusion rules just mentioned, in Section~\ref{SecFundAux},
we determine the fundamental and auxiliary fusion graphs as well as their adjacency matrices.
We recall that the connected components of these graphs are of the form discussed
in Section~\ref{SecAdjacency}.
In Section~\ref{SecSpectral}, we work out the spectral decompositions of the fundamental
and auxiliary fusion matrices. In both cases, we determine a complete set of common
generalized eigenvectors and describe the corresponding Jordan web and its connected
components. The Jordan canonical forms of the fundamental and auxiliary fusion matrices
follow readily. Arising as the result of the
simultaneous similarity transformation of the general fusion matrices,
we also present explicit expressions for the block-diagonal forms of these matrices.
Section~\ref{SecConclusion} contains some concluding remarks and indications of future
work, while Appendix~\ref{AppJordan} provides elementary examples demonstrating that two
commuting matrices may not share a complete set of common generalized eigenvectors nor
necessarily be brought simultaneously to Jordan form. In Appendix~\ref{AppJordanWeb},
the Jordan subwebs formed by the common
generalized eigenvectors of $X$ and $Y$ are collected in table form with respect to
the corresponding eigenvalues. At various places in the paper, some of the key results
are illustrated for ${\cal W}$-extended critical percolation ${\cal WLM}(2,3)$.
\section{${\cal W}$-extended logarithmic minimal models}
\label{SecWLM}
A logarithmic minimal model ${\cal LM}(p,p')$ is defined~\cite{PRZ0607,RP0707}
for every coprime pair of positive integers $p<p'$. The model has central charge
\be
c\;=\;1-6\frac{(p'-p)^2}{pp'}
\label{c}
\ee
and conformal weights
\be
\D_{\rho,\sigma}\;=\;\frac{(\rho p'-\sigma p)^{2}-(p'-p)^2}{4pp'},\hspace{1.2cm}
\rho,\sigma\in\mathbb{N}
\label{D}
\ee
Its ${\cal W}$-extension ${\cal WLM}(p,p')$ is discussed
in~\cite{PRR0803,RP0804,Ras0805,Ras0812} and briefly reviewed in the following.
Throughout, we are using the following notation and conventions
\be
\mathbb{Z}_{n,m}\;=\;\mathbb{Z}\cap[n,m],\qquad
\eps(n)\;=\;\frac{1-(-1)^n}{2},\qquad
n\cdot m\;=\;1+\eps(n+m),\qquad n,m\in\mathbb{Z}
\label{eps}
\ee
and
\be
\kappa,\kappa'\in\mathbb{Z}_{1,2},\qquad
a\in\mathbb{Z}_{1,p-1},\qquad
b\in\mathbb{Z}_{1,p'-1},\qquad
r\in\mathbb{Z}_{1,p},\qquad
s\in\mathbb{Z}_{1,p'}
\label{kkabrs}
\ee
\subsection{Modules associated with boundary conditions}
The indecomposable modules in ${\cal WLM}(p,p')$, which can be associated with
Yang-Baxter integrable boundary conditions on the strip lattice and ${\cal W}$-invariant
boundary conditions in the continuum scaling limit,
were identified in~\cite{RP0804,Ras0805} by extending constructions in~\cite{PRR0803}
pertaining to the case $p=1$. The set of these modules is given by
\be
\big\{ \Wc(\D_{\kappa p,b}),\Wc(\D_{a,\kappa p'}),\Wc(\D_{\kappa p,p'}),
\ketw{\R_{\kappa p,s}^{a,0}},\ketw{\R_{r,\kappa p'}^{0,b}},\ketw{\R_{\kappa p,p'}^{a,b}}\big\}
\label{JWout}
\ee
and is of cardinality
\be
6pp'-2p-2p'
\label{6pp}
\ee
Here we have adopted the notation of~\cite{GRW0905} denoting a
${\cal W}$-irreducible module of conformal weight $\D$ by $\Wc(\D)$.
Thus, there are $2p+2p'-2$ irreducible (hence indecomposable rank-1) modules
\be
\big\{\Wc(\D_{\kappa p,s}),\Wc(\D_{r,\kappa p'})\big\}
\label{r1}
\ee
where the two modules $\Wc(\D_{\kappa p,p})=\Wc(\D_{p,\kappa p'})$ are listed twice,
in addition to $4pp'-2p-2p'$ indecomposable rank-2 modules
\be
\big\{\ketw{\R_{\kappa p,s}^{a,0}}, \ketw{\R_{r,\kappa p'}^{0,b}}\big\}
\label{r2}
\ee
and $2(p-1)(p'-1)$ indecomposable rank-3 modules
\be
\big\{\ketw{\R_{\kappa p,\kappa' p'}^{a,b}}\big\}
\hspace{1.2cm}\mathrm{subject\ to}\ \ \
\ketw{\R_{p,2p'}^{a,b}}\equiv\ketw{\R_{2p,p'}^{a,b}}\quad
\mathrm{and}\quad \ketw{\R_{2p,2p'}^{a,b}}\equiv\ketw{\R_{p,p'}^{a,b}}
\label{r3}
\ee
The associative and commutative fusion algebra of the modules (\ref{JWout}) was determined
in~\cite{Ras0805,Ras0812}. There is no algebra unit or identity for $p>1$, while, for
$p=1$, the irreducible module $\Wc(\D_{1,1})$ is the identity.
\subsection{Fundamental fusion algebra}
\label{SecFundFus}
In~\cite{Ras0812}, we found that one can supplement the set of indecomposable
modules (\ref{JWout}) by the $(p-1)(p'-1)$ reducible yet indecomposable rank-1 modules
\be
\big\{\ketw{a,b}\big\},\qquad\quad \D(\ketw{a,b})\;=\;\D_{a,b}
\label{ab}
\ee
with conjectured embedding patterns given by
\be
\mbox{
\begin{picture}(100,60)(-30,0)
\unitlength=0.8cm
\thinlines
\put(-3.3,1){$\ketw{a,b}:$}
\put(-0.7,1.9){$\ketw{\D_{2p-a,b}}$}
\put(2.7,0){$\ketw{\D_{a,b}}$}
\put(2.6,0.6){\vector(-4,3){1.2}}
\put(5,1){$=$}
\end{picture}
}
\hspace{2cm}
\mbox{
\begin{picture}(100,60)(-30,0)
\unitlength=0.8cm
\thinlines
\put(-0.7,1.9){$\ketw{\D_{a,2p'-b}}$}
\put(2.7,0){$\ketw{\D_{a,b}}$}
\put(2.6,0.6){\vector(-4,3){1.2}}
\end{picture}
}
\label{abemb}
\ee
Their characters read
\be
\chit[\ketw{a,b}](q)\;=\;\frac{1}{\eta(q)}\sum_{k\in\mathbb{Z}}(k^2-1)
\Big(q^{(ap'+bp+2kpp')^2/4pp'}-q^{(ap'-bp+2kpp')^2/4pp'}\Big)
\label{abchar}
\ee
where $\eta(z)$ is the Dedekind eta function
\be
\eta(q)\;=\;q^{\frac{1}{24}} \prod_{n=1}^\infty (1-q^n)
\label{eta}
\ee
The cardinality of the enlarged set of indecomposable modules is readily seen to be given by
\be
7pp'-3p-3p'+1
\label{CardJFund}
\ee
and this set was shown in~\cite{Ras0812} to yield a well-defined fusion algebra called the
{\em fundamental fusion algebra}
\be
\mathrm{Fund}[{\cal WLM}(p,p')]\;=\;\big\langle\ketw{1,1},\ketw{2,1},\ketw{1,2}\big\rangle
\ee
This algebra is so named since it is
generated from repeated fusions of the two {\em fundamental modules}
$\ketw{2,1}$ and $\ketw{1,2}$ in addition to the identity $\ketw{1,1}$ which is now
present for all $p$.
The module $\ketw{1,1}$ is irreducible for $p=1$ in which case $\ketw{1,1}=\Wc(\D_{1,1})$.
The module $\ketw{2,1}$ is irreducible for $p=1,2$ in which case $\ketw{2,1}=\Wc(\D_{2,1})$.
The module $\ketw{1,2}$ is irreducible for $p'=2$ in which case $\ketw{1,2}=\Wc(\D_{1,2})$.
{}From~\cite{Ras0812},
we know that the fusion algebra generated by the modules (\ref{JWout}) is an {\em ideal}
of the fundamental fusion algebra.
To simplify the notation, we sometimes write $\ketw{\R_{r,s}^{0,0}}=\ketw{r,s}$, or
$\ketw{r,s}=\Wc(\D_{r,s})$ if $\ketw{r,s}$ happens to be irreducible.
Further algebraic extensions exist.
In particular, for $p>1$, there are irreducible modules not associated with boundary conditions
as the ones in (\ref{JWout}). Their
fusion properties have been systematically examined only very
recently~\cite{GRW0905,Ras0906,Wood0907}. Here we restrict ourselves to the
modules generating the fundamental fusion algebra.
\subsection{Fusion products of fundamental modules}
\label{SecFusProducts}
Since the associative and commutative
fundamental fusion algebra is generated from repeated fusions of the two
fundamental modules $\ketw{2,1}$ and $\ketw{1,2}$, the complete set of fusion rules can
be reconstructed from knowledge of the basic fusion products involving these two modules.
Here we list all such fusion products. For $p=1$, we have
\bea
\ketw{2,1}\otimes\ketw{\kappa,s}&=&\ketw{2\cdot\kappa,s}\nn
\ketw{2,1}\otimes\ketw{\R_{1,\kappa p'}^{0,b}}&=&\ketw{\R_{1,(2\cdot\kappa)p'}^{0,b}}
\label{21a}
\eea
while for $p>1$, we have
\bea
\ketw{2,1}\otimes\ketw{a,b}&=&\big(1-\delta_{a,1}\big)\ketw{a-1,b}\oplus\ketw{a+1,b}\nn
\ketw{2,1}\otimes\ketw{\kappa p,s}&=&\ketw{\R_{\kappa p,s}^{1,0}}\nn
\ketw{2,1}\otimes\ketw{a,\kappa p'}
&=&\big(1-\delta_{a,1}\big)\ketw{a-1,\kappa p'}\oplus\ketw{a+1,\kappa p'} \nn
\ketw{2,1}\otimes\ketw{\R_{\kappa p,s}^{a,0}}&=&2\delta_{a,1}\ketw{\kappa p,s}
\oplus2\delta_{a,p-1}\ketw{(2\cdot\kappa)p,s} \nn
&&\oplus \big(1-\delta_{a,1}\big)\ketw{\R_{\kappa p,s}^{a-1,0}}
\oplus\big(1-\delta_{a,p-1}\big)\ketw{\R_{\kappa p,s}^{a+1,0}}
\nn
\ketw{2,1}\otimes\ketw{\R_{r,\kappa p'}^{0,b}}&=&\delta_{r,1}\ketw{\R_{2,\kappa p'}^{0,b}}
\oplus\delta_{r,p}\ketw{\R_{\kappa p,p'}^{1,b}}\nn
&&\oplus\big(1-\delta_{r,1}\big)\big(1-\delta_{r,p}\big)\big(\ketw{\R_{r-1,\kappa p'}^{0,b}}
\oplus\ketw{\R_{r+1,\kappa p'}^{0,b}}\big)\nn
\ketw{2,1}\otimes\ketw{\R_{\kappa p,p'}^{a,b}}&=&2\delta_{a,1}\ketw{\R_{p,\kappa p'}^{0,b}}
\oplus2\delta_{a,p-1}\ketw{\R_{p,(2\cdot\kappa)p'}^{0,b}}\nn
&&
\oplus \big(1-\delta_{a,1}\big)\ketw{\R_{\kappa p,p'}^{a-1,b}}
\oplus\big(1-\delta_{a,p-1}\big)\ketw{\R_{\kappa p,p'}^{a+1,b}}
\label{21}
\eea
Since $p'>p\geq1$, we simply have
\bea
\ketw{1,2}\otimes\ketw{a,b}&=&\big(1-\delta_{b,1}\big)\ketw{a,b-1}\oplus\ketw{a,b+1}\nn
\ketw{1,2}\otimes\ketw{\kappa p,b}
&=&\big(1-\delta_{b,1}\big)\ketw{\kappa p,b-1}\oplus\ketw{\kappa p,b+1}\nn
\ketw{1,2}\otimes\ketw{r,\kappa p'}&=&\ketw{\R_{r,\kappa p'}^{0,1}}\nn
\ketw{1,2}\otimes\ketw{\R_{\kappa p,s}^{a,0}}&=&\delta_{s,1}\ketw{\R_{\kappa p,2}^{a,0}}
\oplus\delta_{s,p'}\ketw{\R_{\kappa p,p'}^{a,1}}\nn
&&\oplus\big(1-\delta_{s,1}\big)\big(1-\delta_{s,p'}\big)\big(\ketw{\R_{\kappa p,s-1}^{a,0}}
\oplus\ketw{\R_{\kappa p,s+1}^{a,0}}\big)\nn
\ketw{1,2}\otimes\ketw{\R_{r,\kappa p'}^{0,b}}&=&2\delta_{b,1}\ketw{r,\kappa p'}
\oplus2\delta_{b,p'-1}\ketw{r,(2\cdot\kappa)p'}\nn
&&\oplus\big(1-\delta_{b,1}\big)\ketw{\R_{r,\kappa p'}^{0,b-1}}\oplus
\big(1-\delta_{b,p'-1}\big)\ketw{\R_{r,\kappa p'}^{0,b+1}}\nn
\ketw{1,2}\otimes\ketw{\R_{\kappa p,p'}^{a,b}}&=&2\delta_{b,1}\ketw{\R_{\kappa p,p'}^{a,0}}
\oplus2\delta_{b,p'-1}\ketw{\R_{(2\cdot\kappa)p,p'}^{a,0}}\nn
&&\oplus\big(1-\delta_{b,1}\big)\ketw{\R_{\kappa p,p'}^{a,b-1}}\oplus\big(1-\delta_{b,p'-1}\big)
\ketw{\R_{\kappa p,p'}^{a,b+1}}
\label{12}
\eea
for all $p\in\mathbb{N}$.
\subsection{Graph fusion algebras}
Let $\Ic_f$ denote the set of indecomposable modules appearing in the fundamental fusion
algebra of ${\cal WLM}(p,p')$. In the regular representation
\be
N_\mu N_\nu\;=\;\sum_{\la\in\Ic_f} N_{\mu,\nu}{}^\la N_\la,\qquad\quad
\mu,\nu\in\Ic_f
\label{NNN}
\ee
of this fusion algebra, the fusion matrices $N_\mu$ are mutually commuting, but in general
not diagonalizable. Viewing the fusion matrices as
adjacency matrices of graphs, the fusion rules are neatly encoded in the graphs.
In this context, (\ref{NNN}) is referred to as the {\em graph fusion algebra} of ${\cal WLM}(p,p')$,
in this case corresponding to the fundamental fusion algebra.
As demonstrated in Section~\ref{SecFundFusGra}, the {\em fundamental fusion graphs},
the ones associated to the two fundamental modules, have two
particular types of connected and directed components.
In Section~\ref{SecAdjacency}, we discuss the
spectral decomposition of the adjacency matrices of these subgraphs.
The adjacency matrices of the fundamental fusion graphs themselves are given by the
matrix realizations of the two fundamental modules. We use
\be
X\;=\;(1+\delta_{p,1})N_{\ketw{2,1}},\qquad\quad
Y\;=\;N_{\ketw{1,2}}
\label{XNYN}
\ee
as a convenient shorthand for these matrices. The normalization of $X$ is
chosen to ensure universality of notation in the following. In Section~\ref{SecSpectral},
we will demonstrate that $X$ and $Y$ can be {\em simultaneously} brought
to Jordan form, {\em modulo} permutation similarity, by a common similarity transformation.
It is recalled that two matrices $A$ and $B$ are {\em permutation similar} if for some
permutation matrix $P$,
\be
A\;=\;P^{-1}BP
\ee
In~\cite{Ras0812}, we found that the fundamental fusion algebra is isomorphic to
the polynomial ring
\be
\mathbb{C}[X,Y]\big/\big(P_{p}(X),P_{p'}(Y),P_{p,p'}(X,Y)\big)
\label{FundPol}
\ee
where
\be
P_n(x)\;=\;2\big(T_{2n}(\tfrac{x}{2})-1\big)U_{n-1}(\tfrac{x}{2}),\qquad
P_{n,n'}(x,y)\;=\;\big(T_n(\tfrac{x}{2})-T_{n'}(\tfrac{y}{2})\big)
U_{n-1}(\tfrac{x}{2})U_{n'-1}(\tfrac{y}{2})
\label{Pn}
\ee
Here $T_n(z)$ and $U_n(z)$ denote the Chebyshev polynomials of the first and second kind,
respectively. The isomorphism is given by
\bea
\ketw{a,b}&\leftrightarrow&
U_{a-1}\big(\tfrac{X}{2}\big)U_{b-1}\big(\tfrac{Y}{2}\big)\nn
\Wc(\D_{\kappa p,s})&\leftrightarrow&
\tfrac{1}{\kappa}U_{\kappa p-1}\big(\tfrac{X}{2}\big)U_{s-1}\big(\tfrac{Y}{2}\big)\nn
\Wc(\D_{a,\kappa p'})&\leftrightarrow&
\tfrac{1}{\kappa}U_{a-1}\big(\tfrac{X}{2}\big)U_{\kappa p'-1}\big(\tfrac{Y}{2}\big)\nn
\ketw{\R_{\kappa p,s}^{a,0}}&\leftrightarrow&
\tfrac{2}{\kappa}T_a\big(\tfrac{X}{2}\big)U_{\kappa p-1}\big(\tfrac{X}{2}\big)
U_{s-1}\big(\tfrac{Y}{2}\big)\nn
\ketw{\R_{r,\kappa p'}^{0,b}}&\leftrightarrow&
\tfrac{2}{\kappa}U_{r-1}\big(\tfrac{X}{2}\big)
T_b\big(\tfrac{Y}{2}\big)U_{\kappa p'-1}\big(\tfrac{Y}{2}\big)\nn
\ketw{\R_{\kappa p,p'}^{a,b}}&\leftrightarrow&
\tfrac{4}{\kappa}T_a\big(\tfrac{X}{2}\big)
U_{\kappa p-1}\big(\tfrac{X}{2}\big)
T_b\big(\tfrac{Y}{2}\big)U_{p'-1}\big(\tfrac{Y}{2}\big)
\label{abR4}
\eea
where it is noted that
\be
U_{\kappa p-1}\big(\tfrac{X}{2}\big)U_{p'-1}\big(\tfrac{Y}{2}\big)
\;\equiv\; U_{p-1}\big(\tfrac{X}{2}\big)U_{\kappa p'-1}\big(\tfrac{Y}{2}\big)\qquad
(\mathrm{mod}\ P_{p,p'}(X,Y))
\ee
Identifying the formal entities $X$ and $Y$ appearing in (\ref{FundPol}) with the
two fundamental matrices (\ref{XNYN}) of matching notation, we obtain the regular
representation (\ref{NNN}) of the fundamental fusion algebra.
Letting $\Ic_b$ denote the set of indecomposable modules (\ref{JWout}) associated with
boundary conditions, the regular representation of the corresponding fusion algebra is given by
\be
\Nh_\mu \Nh_\nu\;=\;\sum_{\la\in\Ic_b} \Nh_{\mu,\nu}{}^\la \Nh_\la,\qquad\quad
\mu,\nu\in\Ic_b
\label{NNNb}
\ee
Since this fusion algebra is an ideal of the fundamental fusion algebra, $\Nh_\mu$, for every
$\mu\in\Ic_b$, is obtained from $N_\mu$ by elimination of the rows and columns corresponding
to the $(p-1)(p'-1)$ modules (\ref{ab}) {\em not} associated with boundary conditions.
Indeed, ordering the elements of $\Ic_f$ according to $\Ic_f=(\Ic_f\setminus\Ic_b)\cup\Ic_b$
yields
\be
N_\mu\;=\;\left(\!\!
\begin{array}{c|c}
\ast&\ast \\
\hline
0&\ast
\end{array}
\!\!\right),\quad \mu\in\Ic_f\setminus\Ic_b;\qquad\quad
N_\mu\;=\;\left(\!\!
\begin{array}{c|c}
0&\ast \\
\hline
\\[-.4cm]
0&\Nh_\mu
\end{array}
\!\!\right),\quad \mu\in\Ic_b
\ee
Utilizing this block-triangular structure for $X$ and $Y$
\be
X\;=\;\left(\!\!
\begin{array}{c|c}
\ast&\ast \\
\hline
\\[-.4cm]
0&\Xh
\end{array}
\!\!\right),\qquad\quad
Y\;=\;\left(\!\!
\begin{array}{c|c}
\ast&\ast \\
\hline
\\[-.4cm]
0&\Yh
\end{array}
\!\!\right)
\ee
we have
\be
N_\mu\;=\;\mathrm{pol}_\mu(X,Y)\;=\;\left(\!\!
\begin{array}{c|c}
\ast&\ast \\
\hline
\\[-.4cm]
0&\mathrm{pol}_\mu(\Xh,\Yh)
\end{array}
\!\!\right)
\label{NpolXY}
\ee
where $\mathrm{pol}_\mu(X,Y)$ is the polynomial appearing in (\ref{abR4}) for $\mu\in\Ic_f$.
It follows that we can express the fusion matrices $\Nh_\mu$ in terms of the
matrices $\Xh$ and $\Yh$
\be
\Nh_\mu\;=\;\mathrm{pol}_\mu(\Xh,\Yh),\qquad \mu\in\Ic_b
\ee
using the {\em same} polynomial as in the description of $N_\mu$ in terms of $X$ and $Y$
(\ref{NpolXY}).
For $p>2$, we have $\ketw{2,1},\ketw{1,2}\in\Ic_f\setminus\Ic_b$, in which case
the matrices $\Xh$ and $\Yh$ should be thought of as {\em auxiliary} matrices.
Similarly, for $p=2$, we have $\ketw{1,2}\in\Ic_f\setminus\Ic_b$.
Despite their auxiliary status, the matrices $\Xh$ and $\Yh$ are very useful
in the description of the spectral decomposition of the fusion matrices $\Nh_\mu$.
We refer to the corresponding directed graphs as {\em auxiliary fusion graphs}.
As in the case of the fundamental fusion graphs, the auxiliary ones consist of
two particular types of connected and directed components,
one of which also appears as subgraphs of the fundamental fusion graphs.
\section{Spectral decomposition of adjacency matrices}
\label{SecAdjacency}
\subsection{Cycle, tadpole and eye-patch graphs}
As already mentioned, a fundamental (or auxiliary) fusion graph consists of a number of
connected components.
There are three prototypes: {\em cycle} graphs, {\em tadpole} graphs and {\em eye-patch} graphs,
and they are the topic of the present section.
The connected subgraphs of the fundamental fusion graphs are all tadpole
or eye-patch graphs, while the connected subgraphs of the auxiliary fusion graphs are
all cycle or eye-patch graphs.
All of these connected graphs depend on a single integer {\em order parameter} $\rho\geq2$.
We refer to a connected and directed graph of the type
\be
\mbox{
\begin{picture}(100,100)(160,-45)
\unitlength=0.75cm
\thinlines
\put(6.3,-0.1){$L$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.65,1.25){$U_1$}
\put(8.8,1.75){\vector(2,1){0.6}}
\put(9,1.85){\vector(-2,-1){0.5}}
\put(9.55,2){$\ldots$}
\put(10.7,1.85){\vector(2,-1){0.55}}
\put(10.9,1.75){\vector(-2,1){0.55}}
\put(11.45,1.25){$U_{\rho-1}$}
\put(12.2,1.1){\vector(1,-1){0.6}}
\put(12.35,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.65,-1.25){$D_{\rho-1}$}
\put(8.8,-1.55){\vector(2,-1){0.6}}
\put(9,-1.65){\vector(-2,1){0.5}}
\put(9.6,-1.9){$\ldots$}
\put(10.95,-1.6){\vector(-2,-1){0.55}}
\put(10.75,-1.7){\vector(2,1){0.55}}
\put(11.45,-1.25){$D_1$}
\put(12.3,-0.85){\vector(1,1){0.5}}
\put(12.45,-0.7){\vector(1,1){0.5}}
\put(12.7,-0.45){\vector(-1,-1){0.5}}
\put(13,-0.1){$R$}
\end{picture}
}
\label{GCp}
\ee
as a {\em cycle graph} with order parameter $\rho$.
Its order is $2\rho$ and the labeling of the $2\rho$ vertices has been chosen to reflect their
position in the graph. The cycle graph with order parameter $\rho=2$ is given by
\be
\mbox{
\begin{picture}(100,80)(120,-30)
\unitlength=0.75cm
\thinlines
\put(6.3,-0.1){$L$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.7,1.25){$U_1$}
\put(8.4,1.1){\vector(1,-1){0.6}}
\put(8.55,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.7,-1.25){$D_1$}
\put(8.5,-0.85){\vector(1,1){0.5}}
\put(8.65,-0.7){\vector(1,1){0.5}}
\put(8.9,-0.45){\vector(-1,-1){0.5}}
\put(9.2,-0.1){$R$}
\end{picture}
}
\label{GC2}
\ee
In the ordered basis
\be
\big\{ L,U_1,\ldots,U_{\rho-1},R,D_1,\ldots ,D_{\rho-1}\big\}
\ee
the adjacency matrix associated to the cycle graph (\ref{GCp}) is given by
\be
\Cc_\rho\;=\;
\left(\!\!
\begin{array}{c|ccccc|c|ccccc}
0&1&&&&&&&&&&\\
\hline
2&0&1&&&&&&&&&\\
&1&0&&&&&&&&&\\
&&&\ddots&&&&&&&&\\
&&&&0&1&&&&&&\\
&&&&1&0&2&&&&&\\
\hline
&&&&&0&0&1&&&&\\
\hline
&&&&&&2&0&1&&&\\
&&&&&&&1&0&&&\\
&&&&&&&&&\ddots&&\\
&&&&&&&&&&0&1\\
2&&&&&&&&&&1&0
\end{array}
\!\!\right)
\label{MCp}
\ee
The first and $(\rho+1)$'th rows and columns (corresponding to $L$ and $R$)
are emphasized to signal their special status. For $\rho=2$, the adjacency matrix is
\be
\Cc_2\;=\;
\left(\!\!
\begin{array}{cccc}
0&1&0&0\\
2&0&2&0\\
0&0&0&1\\
2&0&2&0
\end{array}
\!\!\right)
\label{MC2}
\ee
We refer to a connected and directed graph of the type
\be
\mbox{
\begin{picture}(100,100)(100,-45)
\unitlength=0.75cm
\thinlines
\put(-0.5,0){$L_{1}$}
\put(0.8,0.15){\vector(-1,0){0.4}}
\put(0.8,0.15){\vector(1,0){0.35}}
\put(1.6,0){$\ldots$}
\put(3,0.15){\vector(-1,0){0.4}}
\put(3,0.15){\vector(1,0){0.35}}
\put(3.5,0){$L_{\rho-1}$}
\put(4.85,0.15){\vector(1,0){0.85}}
\put(6,0){$L_{\rho}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.65,1.25){$U_1$}
\put(8.8,1.75){\vector(2,1){0.6}}
\put(9,1.85){\vector(-2,-1){0.5}}
\put(9.55,2){$\ldots$}
\put(10.7,1.85){\vector(2,-1){0.55}}
\put(10.9,1.75){\vector(-2,1){0.55}}
\put(11.45,1.25){$U_{\rho-1}$}
\put(12.2,1.1){\vector(1,-1){0.6}}
\put(12.35,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.65,-1.25){$D_{\rho-1}$}
\put(8.8,-1.55){\vector(2,-1){0.6}}
\put(9,-1.65){\vector(-2,1){0.5}}
\put(9.6,-1.9){$\ldots$}
\put(10.95,-1.6){\vector(-2,-1){0.55}}
\put(10.75,-1.7){\vector(2,1){0.55}}
\put(11.45,-1.25){$D_1$}
\put(12.3,-0.85){\vector(1,1){0.5}}
\put(12.45,-0.7){\vector(1,1){0.5}}
\put(12.7,-0.45){\vector(-1,-1){0.5}}
\put(13,-0.1){$R$}
\end{picture}
}
\label{GTp}
\ee
as a {\em tadpole graph} with order parameter $\rho$.
Its order is $3\rho-1$ and the labeling of the $3\rho-1$ vertices has been chosen to reflect their
position in the graph. The tadpole graph with order parameter $\rho=2$ is given by
\be
\mbox{
\begin{picture}(100,80)(100,-30)
\unitlength=0.75cm
\thinlines
\put(4,0){$L_{1}$}
\put(4.95,0.15){\vector(1,0){0.75}}
\put(6,0){$L_{2}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.7,1.25){$U_1$}
\put(8.4,1.1){\vector(1,-1){0.6}}
\put(8.55,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.7,-1.25){$D_1$}
\put(8.5,-0.85){\vector(1,1){0.5}}
\put(8.65,-0.7){\vector(1,1){0.5}}
\put(8.9,-0.45){\vector(-1,-1){0.5}}
\put(9.2,-0.1){$R$}
\end{picture}
}
\label{GT2}
\ee
In the ordered basis
\be
\big\{ L_1,\ldots, L_{\rho-1},L_\rho,U_1,\ldots,U_{\rho-1},R,D_1,\ldots ,D_{\rho-1}\big\}
\ee
the adjacency matrix associated to the tadpole graph (\ref{GTp}) is given by
\be
\Tc_\rho\;=\;
\left(\!\!
\begin{array}{ccccc|c|ccccc|c|ccccc}
0&1&&&&&&&&&&&&&&&\\
1&0&&&&&&&&&&&&&&&\\
&&\ddots&&&&&&&&&&&&&&\\
&&&0&1&&&&&&&&&&&&\\
&&&1&0&1&&&&&&&&&&&\\
\hline
&&&&0&0&1&&&&&&&&&&\\
\hline
&&&&&2&0&1&&&&&&&&&\\
&&&&&&1&0&&&&&&&&&\\
&&&&&&&&\ddots&&&&&&&&\\
&&&&&&&&&0&1&&&&&&\\
&&&&&&&&&1&0&2&&&&&\\
\hline
&&&&&&&&&&0&0&1&&&&\\
\hline
&&&&&&&&&&&2&0&1&&&\\
&&&&&&&&&&&&1&0&&&\\
&&&&&&&&&&&&&&\ddots&&\\
&&&&&&&&&&&&&&&0&1\\
&&&&&2&&&&&&&&&&1&0
\end{array}
\!\!\right)
\label{MTp}
\ee
The $\rho$'th and $(2\rho)$'th rows and columns (corresponding to $L_\rho$ and $R$)
are emphasized to signal their special status. For $\rho=2$, the adjacency matrix is
\be
\Tc_2\;=\;
\left(\!\!
\begin{array}{ccccc}
0&1&0&0&0\\
0&0&1&0&0\\
0&2&0&2&0\\
0&0&0&0&1\\
0&2&0&2&0
\end{array}
\!\!\right)
\label{MT2}
\ee
We also introduce what we call an {\em eye-patch graph} with order parameter $\rho$
\be
\mbox{
\begin{picture}(100,100)(160,-45)
\unitlength=0.75cm
\thinlines
\put(-0.5,0){$L_{1}$}
\put(0.8,0.15){\vector(-1,0){0.4}}
\put(0.8,0.15){\vector(1,0){0.35}}
\put(1.6,0){$\ldots$}
\put(3,0.15){\vector(-1,0){0.4}}
\put(3,0.15){\vector(1,0){0.35}}
\put(3.5,0){$L_{\rho-1}$}
\put(4.85,0.15){\vector(1,0){0.85}}
\put(6,0){$L_{\rho}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.65,1.25){$U_1$}
\put(8.8,1.75){\vector(2,1){0.6}}
\put(9,1.85){\vector(-2,-1){0.5}}
\put(9.55,2){$\ldots$}
\put(10.7,1.85){\vector(2,-1){0.55}}
\put(10.9,1.75){\vector(-2,1){0.55}}
\put(11.45,1.25){$U_{\rho-1}$}
\put(12.2,1.1){\vector(1,-1){0.6}}
\put(12.35,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.65,-1.25){$D_{\rho-1}$}
\put(8.8,-1.55){\vector(2,-1){0.6}}
\put(9,-1.65){\vector(-2,1){0.5}}
\put(9.6,-1.9){$\ldots$}
\put(10.95,-1.6){\vector(-2,-1){0.55}}
\put(10.75,-1.7){\vector(2,1){0.55}}
\put(11.45,-1.25){$D_1$}
\put(12.3,-0.85){\vector(1,1){0.5}}
\put(12.45,-0.7){\vector(1,1){0.5}}
\put(12.7,-0.45){\vector(-1,-1){0.5}}
\put(13,0){$R_{\rho}$}
\put(14.6,0.15){\vector(-1,0){0.75}}
\put(15,0){$R_{\rho-1}$}
\put(16.8,0.15){\vector(-1,0){0.4}}
\put(16.8,0.15){\vector(1,0){0.35}}
\put(17.55,0){$\ldots$}
\put(18.9,0.15){\vector(-1,0){0.4}}
\put(18.9,0.15){\vector(1,0){0.35}}
\put(19.5,0){$R_{1}$}
\end{picture}
}
\label{GPp}
\ee
which, for $\rho=2$, reduces to
\be
\mbox{
\begin{picture}(100,80)(120,-30)
\unitlength=0.75cm
\thinlines
\put(4,0){$L_{1}$}
\put(4.95,0.15){\vector(1,0){0.75}}
\put(6,0){$L_{2}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.7,1.25){$U_1$}
\put(8.4,1.1){\vector(1,-1){0.6}}
\put(8.55,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.7,-1.25){$D_1$}
\put(8.5,-0.85){\vector(1,1){0.5}}
\put(8.65,-0.7){\vector(1,1){0.5}}
\put(8.9,-0.45){\vector(-1,-1){0.5}}
\put(9.2,0){$R_{2}$}
\put(10.9,0.15){\vector(-1,0){0.75}}
\put(11.2,0){$R_{1}$}
\end{picture}
}
\label{GP2}
\ee
The order of the graph (\ref{GPp}) is $4\rho-2$, and the labeling of the $4\rho-2$ vertices has
been chosen to reflect their position in the graph.
In the ordered basis
\be
\big\{ L_1,\ldots, L_{\rho-1},L_\rho,U_1,\ldots,U_{\rho-1},
R_1,\ldots, R_{\rho-1},R_\rho,D_1,\ldots ,D_{\rho-1}\big\}
\ee
the adjacency matrix associated to the graph (\ref{GPp}) is given by
\be
\Ec_{\rho}\;=\;
\left(\!\!
\begin{array}{ccccc|c|ccccc|ccccc|c|ccccc}
0&1&&&&&&&&&&&&&&&&&&&&\\
1&0&&&&&&&&&&&&&&&&&&&&\\
&&\ddots&&&&&&&&&&&&&&&&&&&\\
&&&0&1&&&&&&&&&&&&&&&&&\\
&&&1&0&1&&&&&&&&&&&&&&&&\\
\hline
&&&&0&0&1&&&&&&&&&&&&&&&\\
\hline
&&&&&2&0&1&&&&&&&&&&&&&&\\
&&&&&&1&0&&&&&&&&&&&&&&\\
&&&&&&&&\ddots&&&&&&&&&&&&&\\
&&&&&&&&&0&1&&&&&&&&&&&\\
&&&&&&&&&1&0&0&&&&&2&&&&&\\
\hline
&&&&&&&&&&0&0&1&&&&&&&&&\\
&&&&&&&&&&&1&0&&&&&&&&&\\
&&&&&&&&&&&&&\ddots&&&&&&&&\\
&&&&&&&&&&&&&&0&1&&&&&&\\
&&&&&&&&&&&&&&1&0&1&&&&&\\
\hline
&&&&&&&&&&&&&&&0&0&1&&&&\\
\hline
&&&&&&&&&&&&&&&&2&0&1&&&\\
&&&&&&&&&&&&&&&&&1&0&&&\\
&&&&&&&&&&&&&&&&&&&\ddots&&\\
&&&&&&&&&&&&&&&&&&&&0&1\\
&&&&&2&&&&&&&&&&&&&&&1&0
\end{array}
\!\!\right)
\label{MPp}
\ee
The $\rho$'th and $(3\rho-1)$'th rows and columns (corresponding to $L_\rho$ and $R_\rho$)
are emphasized to signal their special status.
For $\rho=2$, the adjacency matrix (\ref{MPp}) is meant to reduce to
\be
\Ec_2\;=\;
\left(\!\!
\begin{array}{cccccc}
0&1&0&0&0&0\\
0&0&1&0&0&0\\
0&2&0&0&2&0\\
0&0&0&0&1&0\\
0&0&0&0&0&1\\
0&2&0&0&2&0
\end{array}
\!\!\right)
\label{MP2}
\ee
Extending the definition of the order parameter to $\rho=1$, we let a cycle,
tadpole or eye-patch graph collapse to the following directed order-2 graph
\be
\mbox{
\begin{picture}(60,20)(0,0)
\unitlength=0.75cm
\thinlines
\put(0,0){$L$}
\put(1.3,.15){\vector(1,0){0.35}}
\put(1.3,.15){\vector(1,0){0.6}}
\put(1.3,.15){\vector(-1,0){0.45}}
\put(1.3,.15){\vector(-1,0){0.7}}
\put(2.05,0){$R$}
\end{picture}
}
\label{Gp1}
\ee
This type of graph is relevant only when considering the series ${\cal WLM}(1,p')$.
The corresponding adjacency matrix is
\be
\Cc_1\;=\;\Tc_1\;=\;\Ec_1\;=\;
\left(\!\!
\begin{array}{cc}
0&2\\
2&0
\end{array}
\!\!\right)
\label{MT1P1}
\ee
\subsection{Spectral decompositions}
In preparation for the spectral decomposition of the adjacency matrices (\ref{MCp}),
(\ref{MTp}) and (\ref{MPp}), we recall that canonical Jordan blocks of rank-2 or -3
associated to the eigenvalue $\la$ of a matrix $A$ are given by
\be
\Jc_{\la,2}\;=\;\left(\!\!\begin{array}{cc} \la&1 \\ 0&\la \end{array}\!\!\right),\qquad\quad
\Jc_{\la,3}\;=\;\left(\!\!\begin{array}{ccc} \la&1&0 \\ 0&\la&1 \\ 0&0&\la \end{array}\!\!\right)
\label{Jb}
\ee
They appear in the Jordan decomposition of $A$ if the eigenvalue $\la$ gives rise to a
Jordan chain of length 2 or 3, where a Jordan chain of length 3, in particular, is given by
\be
Av^{(2)}\;=\;\la v^{(2)}+v^{(1)},\qquad \quad
Av^{(1)}\;=\;\la v^{(1)}+v^{(0)},\qquad \quad
Av^{(0)}\;=\;\la v^{(0)}
\label{Av}
\ee
This chain of relations implies that
\be
\big(A-\la I\big) v^{(2)}\;=\;v^{(1)},\qquad\quad
\big(A-\la I\big) v^{(1)}\;=\;v^{(0)},\qquad\quad
\big(A-\la I\big)^{\ell+1} v^{(\ell)}\;=\;0,\quad \ell\in\mathbb{Z}_{0,2}
\label{Alv}
\ee
indicating that the vectors are {\em generalized eigenvectors}.
A proper eigenvector is merely a special type of generalized eigenvector.
We say that an upper-triangular (square) matrix with identical entries
$\la$ on the diagonal is a Jordan block if the geometric multiplicity of (the single eigenvalue)
$\la$ is 1. It is a Jordan {\em canonical} block (as in (\ref{Jb}))
if the entries on the super-diagonal are 1 while all entries above the super-diagonal are 0.
A block-diagonal matrix is of Jordan (canonical) form if every block is a Jordan
(canonical) block.
Following~\cite{Ras0908}, we introduce the $2\rho-1$ functions
$f_h(x)$, $h\in\mathbb{Z}_{1,2\rho-1}$, defined by
\be
f_k(x)\;=\;U_{k-1}(\tfrac{x}{2}),\qquad\quad
f_\rho(x)\;=\;U_{\rho-1}(\tfrac{x}{2}),\qquad \quad
f_{\rho+k}(x)\;=\;2T_k(\tfrac{x}{2})U_{\rho-1}(\tfrac{x}{2})
\label{f}
\ee
Here, and in the following, we are using the convention
\be
k\in\mathbb{Z}_{1,\rho-1}
\ee
Certain useful properties of $f_h(x)$ are listed here, while further details can be
found in~\cite{Ras0908}.
For $\rho>2$, the functions satisfy recursive relations allowing us to express
$x f_h(x)$, for $h\in\mathbb{Z}_{1,2\rho-2}$, as
\bea
f_2(x)&=&x f_1(x) \nn
f_{h-1}(x)+f_{h+1}(x)&=&x f_h(x),\qquad\quad h\in\mathbb{Z}_{2,\rho-1}\nn
f_{\rho+1}(x)&=&x f_\rho(x)\nn
2f_\rho(x)+f_{\rho+2}(x)&=&x f_{\rho+1}(x)\nn
f_{h-1}(x)+f_{h+1}(x)&=&x f_h(x),\qquad\quad h\in\mathbb{Z}_{\rho+2,2\rho-2}
\label{ff}
\eea
It follows that
\be
\begin{array}{rll}
f_2'(x)&=\quad\!\!x f_1'(x)+f_1(x),\qquad\qquad\qquad\qquad\quad\ \
f_2''(x)&=\quad\!\! xf_1''(x)+2f_1'(x) \\[3pt]
f_{h-1}'(x)+f_{h+1}'(x)&=\quad\!\! x f_h'(x)+f_h(x),\qquad\qquad\ \!
f_{h-1}''(x)+f_{h+1}''(x)&=\quad\!\!xf_h''(x)+2f_h'(x)\\[3pt]
f_{\rho+1}'(x)&=\quad\!\!x f_\rho'(x)+f_\rho(x),\qquad\qquad\qquad\qquad\ \
f_{\rho+1}''(x)&=\quad\!\!x f_\rho''(x)+2f_\rho'(x)\\[3pt]
2f_\rho'(x)+f_{\rho+2}'(x)&=\quad\!\!x f_{\rho+1}'(x)+f_{\rho+1}(x),\qquad\ \ \!
2f_\rho''(x)+f_{\rho+2}''(x)&=\quad\!\!x f_{\rho+1}''(x)+2f_{\rho+1}'(x)\\[3pt]
f_{h-1}'(x)+f_{h+1}'(x)&=\quad\!\!x f_h'(x)+f_h(x),\qquad\qquad
f_{h-1}''(x)+f_{h+1}''(x)&=\quad\!\!x f_h''(x)+2f_h'(x)
\end{array}
\label{fdiff}
\ee
with the conditions on $h$ adopted from (\ref{ff}).
It is noted that we have not included any relations involving $xf_{2\rho-1}(x)$ for general $x$.
Instead, we focus on evaluations at $x=\la_j$, for $j\in\mathbb{Z}_{0,\rho}$, where
\bea
2(-1)^if_\rho(\la_j)+f_{2\rho-2}(\la_j)&=&\la_j f_{2\rho-1}(\la_j),
\qquad\qquad\qquad j\in\mathbb{Z}_{1,\rho-1} \ \mathrm{or}\ i=j\in\{0,\rho\}\nn
2(-1)^kf_\rho'(\la_k)+f_{2\rho-2}'(\la_k)&=&\la_k f_{2\rho-1}'(\la_k)+f_{2\rho-1}(\la_k)\nn
2(-1)^kf_\rho''(\la_k)+f_{2\rho-2}''(\la_k)&=&\la_k f_{2\rho-1}''(\la_k)+2f_{2\rho-1}'(\la_k)
\label{f2p}
\eea
We also note that
\be
f_h(\la_k)\;=\;0,\qquad h\in\mathbb{Z}_{\rho,2\rho-1}
\label{fhlk}
\ee
A convenient notation to be used below is
\be
\tfrac{1}{\ell!}f_h^{(\ell)}(x)\;=\;\tfrac{1}{\ell!}\pa_x^\ell f_h(x),
\qquad \ell\in\mathbb{Z}_{0,2},
\quad h\in\mathbb{Z}_{1,2\rho-1}
\ee
where $0!=1$.
Since the spectral decompositions of $\Cc_\rho$, $\Tc_\rho$ and $\Ec_{\rho}$ are worked out
in Section~\ref{SecCycle}, \ref{SecTad} and \ref{SecEye} for $\rho\geq2$, we here discuss
the rather trivial spectral decomposition of $\Cc_1=\Tc_1=\Ec_1$ (\ref{MT1P1}) in the
framework employed in those sections.
With $\rho=1$, the eigenvalues are
\be
\la_j\;=\;2\cos\frac{j\pi}{\rho},\qquad j\in\mathbb{Z}_{0,\rho}
\label{laj}
\ee
with corresponding eigenvectors given by
\be
V_0\;=\;\left(\!\!\!\begin{array}{c}
f_{\rho}(\la_0)
\\[4pt]
(-1)^0 f_{\rho}(\la_0)
\end{array}\!\!\!\right)\;=\;
\left(\!\!\begin{array}{c}
2 \\[4pt] 2
\end{array}\!\!\right),\qquad
V_\rho\;=\;\left(\!\!\!\begin{array}{c}
f_{\rho}(\la_\rho)
\\[4pt]
(-1)^\rho f_{\rho}(\la_\rho)
\end{array}\!\!\!\right)\;=\;
\left(\!\!\!\begin{array}{c}
-2 \\[4pt] 2
\end{array}\!\!\!\right)
\label{V}
\ee
The minimal and characteristic polynomials of $\Cc_1$ are given by
\be
m(\Cc_1)\;=\;(\Cc_1-\la_0I)(\Cc_1-\la_\rho I)\;=\;\Cc_1^2-4I,\qquad
\det(\la I-\Cc_1)\;=\;(\la-\la_0)(\la-\la_\rho)\;=\;\la^2-4
\label{charC1}
\ee
while the similarity matrix constructed by concatenating the two eigenvectors
\be
Q_{\Cc_1}\;=\;\big(V_0\ V_\rho\big)
\ee
diagonalizes $\Cc_1$
\be
Q_{\Cc_1}^{-1}\Cc_1 Q_{\Cc_1}
\;=\;\mathrm{diag}\big(\la_0,\la_\rho\big)\;=\;\mathrm{diag}\big(2,-2\big)
\label{JC1}
\ee
\subsubsection{Cycle graphs}
\label{SecCycle}
It follows from the explicit construction of generalized eigenvectors below that
the eigenvalues of $\Cc=\Cc_\rho$, $\rho>1$, are given by (\ref{laj}),
while the minimal and characteristic polynomials of $\Cc$ are given by
\bea
&m(\Cc)\;=\;(\Cc-\la_0I)(\Cc-\la_\rho I)\displaystyle{\prod_{k=1}^{\rho-1}(\Cc-\la_k I)^2}
\;=\;(\Cc^2-4I)U_{\rho-1}^2(\tfrac{\Cc}{2})\nn
&\det(\la I-\Cc)\;=\;(\la-\la_0)(\la-\la_\rho)\displaystyle{\prod_{k=1}^{\rho-1}(\la-\la_k)^2}
\;=\; (\la^2-4)U_{\rho-1}^2(\tfrac{\la}{2})
\label{charC}
\eea
This implies that the Jordan canonical form of $\Cc$ consists of $\rho-1$ rank-2 blocks
associated to the eigenvalues $\la_k$,
and two rank-1 blocks associated to the eigenvalues $\la_0=2$, $\la_\rho=-2$.
The number of linearly independent eigenvectors of $\Cc$ is thus $\rho+1$. Since
the null-space of $\Cc$ is empty for $\rho$ odd but one-dimensional for $\rho$ even,
the rank of $\Cc$ is
\be
\mathrm{rank}(\Cc)\;=\;2\rho-\eps(\rho-1)\;=\;2\rho-1+\eps(\rho)
\label{rankC}
\ee
To establish these results on $\Cc$, we now discuss the two eigenvectors
corresponding to $\la_0=2$, $\la_\rho=-2$, and the $\rho-1$ Jordan chains of length 2
associated to $\la_k$. Using the various properties of the functions $f_h(x)$ discussed
above, it is straightforward to verify that
\be
C_0\;=\;\left(\!\!\!\begin{array}{c}
f_{\rho}(\la_0)\\ \vdots\\ f_{2\rho-1}(\la_0)
\\[.15cm] \hline\\[-.3cm]
(-1)^0 f_{\rho}(\la_0)\\ \vdots\\ (-1)^0 f_{2\rho-1}(\la_0)
\end{array}\!\!\!\right),\qquad
C_\rho\;=\;\left(\!\!\!\begin{array}{c}
f_{\rho}(\la_\rho)\\ \vdots\\ f_{2\rho-1}(\la_\rho)
\\[.15cm] \hline\\[-.3cm]
(-1)^\rho f_{\rho}(\la_\rho)\\ \vdots\\ (-1)^\rho f_{2\rho-1}(\la_\rho)
\end{array}\!\!\!\right)
\label{Cj}
\ee
are eigenvectors of $\Cc$
\be
\Cc C_0\;=\;\la_0 C_0,\qquad
\Cc C_\rho\;=\;\la_\rho C_\rho
\ee
For every $k\in\mathbb{Z}_{1,\rho-1}$, it is likewise verified that
\be
C_k^{(0)}\;=\;\left(\!\!\!\begin{array}{c}
f'_{\rho}(\la_k)\\ \vdots\\ f'_{2\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
(-1)^k f'_{\rho}(\la_k)\\ \vdots\\ (-1)^k f'_{2\rho-1}(\la_k)
\end{array}\!\!\!\right),\qquad
C_k^{(1)}\;=\;\left(\!\!\!\begin{array}{c}
\tfrac{1}{2}f''_{\rho}(\la_k)\\ \vdots\\ \tfrac{1}{2}f''_{2\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
\tfrac{1}{2}(-1)^k f''_{\rho}(\la_k)\\ \vdots\\ \tfrac{1}{2}(-1)^k f''_{2\rho-1}(\la_k)
\end{array}\!\!\!\right)
\label{Ck12}
\ee
form the Jordan chain
\be
\Cc C_k^{(0)}\;=\;\la_k C_k^{(0)},\qquad
\Cc C_k^{(1)}\;=\;\la_k C_k^{(1)}+C_k^{(0)}
\ee
Finally, the $2\rho$-dimensional matrix $Q_\Cc$ is constructed by concatenating
the generalized eigenvectors (\ref{Cj}) and (\ref{Ck12})
\be
Q_\Cc\;=\;\left(\!\!\begin{array}{c|cc|c|cc|c} C_0&C_1^{(0)}&C_1^{(1)}&\ldots&
C_{\rho-1}^{(0)}&C_{\rho-1}^{(1)}&C_\rho \end{array}\!\!\right)
\label{QC}
\ee
By a similarity transformation, this matrix converts $\Cc$ into its Jordan canonical form
\be
J_\Cc\;=\;Q_\Cc^{-1}\Cc Q_\Cc
\;=\;\mathrm{diag}\big(\la_0,\Jc_{\la_1,2},\ldots,\Jc_{\la_{\rho-1},2},\la_\rho\big)
\label{JC}
\ee
\subsubsection{Tadpole graphs}
\label{SecTad}
It follows from the explicit construction of generalized eigenvectors below that
the eigenvalues of $\Tc=\Tc_\rho$, $\rho>1$, are given by (\ref{laj}),
while the minimal and characteristic polynomials of $\Tc$ are given by
\bea
&m(\Tc)\;=\;(\Tc-\la_0I)(\Tc-\la_\rho I)\displaystyle{\prod_{k=1}^{\rho-1}(\Tc-\la_k I)^3}
\;=\;(\Tc^2-4I)U_{\rho-1}^3(\tfrac{\Tc}{2})\nn
&\det(\la I-\Tc)\;=\;(\la-\la_0)(\la-\la_\rho)\displaystyle{\prod_{k=1}^{\rho-1}(\la-\la_k)^3}
\;=\; (\la^2-4)U_{\rho-1}^3(\tfrac{\la}{2})
\label{charT}
\eea
This implies that the Jordan canonical form of $\Tc$ consists of $\rho-1$ rank-3 blocks
associated to the eigenvalues $\la_k$,
and two rank-1 blocks associated to the eigenvalues $\la_0=2$, $\la_\rho=-2$.
The number of linearly independent eigenvectors of $\Tc$ is thus $\rho+1$. Since
the null-space of $\Tc$ is empty for $\rho$ odd but one-dimensional for $\rho$ even,
the rank of $\Tc$ is
\be
\mathrm{rank}(\Tc)\;=\;3\rho-1-\eps(\rho-1)\;=\;3\rho-2+\eps(\rho)
\label{rankT}
\ee
To establish these results on $\Tc$, we now discuss the two eigenvectors
corresponding to $\la_0=2$, $\la_\rho=-2$, and the $\rho-1$ Jordan chains of length
3 associated to $\la_k$. Using the various properties of the functions $f_h(x)$ discussed
above, it is straightforward to verify that
\be
T_j\;=\;\left(\!\!\!\begin{array}{c}
f_1(\la_j)\\ \vdots\\ f_{\rho-1}(\la_j)
\\[.15cm] \hline\\[-.3cm]
f_{\rho}(\la_j)\\ \vdots\\ f_{2\rho-1}(\la_j)
\\[.15cm] \hline\\[-.3cm]
(-1)^j f_{\rho}(\la_j)\\ \vdots\\ (-1)^j f_{2\rho-1}(\la_j)
\end{array}\!\!\!\right),\quad j=0,\rho;\qquad
T_k^{(0)}\;=\;\left(\!\!\!\begin{array}{c}
f_1(\la_k)\\ \vdots\\ f_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
f_{\rho}(\la_k)\\ \vdots\\ f_{2\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
(-1)^k f_{\rho}(\la_k)\\ \vdots\\ (-1)^k f_{2\rho-1}(\la_k)
\end{array}\!\!\!\right)
\;=\;\left(\!\!\!\begin{array}{c}
f_1(\la_k)\\ \vdots\\ f_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
0\\ \vdots\\ 0
\\[.15cm] \hline\\[-.3cm]
0\\ \vdots\\ 0
\end{array}\!\!\!\right)
\label{Tj}
\ee
are eigenvectors of $\Tc$
\be
\Tc T_0\;=\;\la_0 T_0,\qquad
\Tc T_k^{(0)}\;=\;\la_k T_k^{(0)},\qquad
\Tc T_\rho\;=\;\la_\rho T_\rho
\ee
For every $k\in\mathbb{Z}_{1,\rho-1}$, it is likewise verified that $T_k^{(0)}$ together with
\be
T_k^{(1)}\;=\;\left(\!\!\!\begin{array}{c}
f'_1(\la_k)\\ \vdots\\ f'_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
f'_{\rho}(\la_k)\\ \vdots\\ f'_{2\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
(-1)^k f'_{\rho}(\la_k)\\ \vdots\\ (-1)^k f'_{2\rho-1}(\la_k)
\end{array}\!\!\!\right),\qquad
T_k^{(2)}\;=\;\left(\!\!\!\begin{array}{c}
\tfrac{1}{2}f''_1(\la_k)\\ \vdots\\ \tfrac{1}{2}f''_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
\tfrac{1}{2}f''_{\rho}(\la_k)\\ \vdots\\ \tfrac{1}{2}f''_{2\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
\tfrac{1}{2}(-1)^k f''_{\rho}(\la_k)\\ \vdots\\ \tfrac{1}{2}(-1)^k f''_{2\rho-1}(\la_k)
\end{array}\!\!\!\right)
\label{Tk12}
\ee
form the Jordan chain
\be
\Tc T_k^{(0)}\;=\;\la_k T_k^{(0)},\qquad
\Tc T_k^{(1)}\;=\;\la_k T_k^{(1)}+T_k^{(0)},\qquad
\Tc T_k^{(2)}\;=\;\la_k T_k^{(2)}+T_k^{(1)}
\ee
Finally, the $(3\rho-1)$-dimensional matrix $Q_\Tc$ is constructed by concatenating
the generalized eigenvectors (\ref{Tj}) and (\ref{Tk12})
\be
Q_\Tc\;=\;\left(\!\!\begin{array}{c|ccc|c|ccc|c} T_0&T_1^{(0)}&T_1^{(1)}&T_1^{(2)}&\ldots&
T_{\rho-1}^{(0)}&T_{\rho-1}^{(1)}&T_{\rho-1}^{(2)}&T_\rho \end{array}\!\!\right)
\label{QT}
\ee
By a similarity transformation, this matrix converts $\Tc$ into its Jordan canonical form
\be
J_\Tc\;=\;Q_\Tc^{-1}\Tc Q_\Tc
\;=\;\mathrm{diag}\big(\la_0,\Jc_{\la_1,3},\ldots,\Jc_{\la_{\rho-1},3},\la_\rho\big)
\label{JT}
\ee
\subsubsection{Eye-patch graphs}
\label{SecEye}
It follows from the explicit construction of generalized eigenvectors below that
the eigenvalues of $\Ec=\Ec_{\rho}$, $\rho>1$, are given by (\ref{laj}),
while the minimal and characteristic polynomials of $\Ec$ are given by
\bea
&m(\Ec)\;=\;(\Ec-\la_0I)(\Ec-\la_\rho I)
\displaystyle{\prod_{k=1}^{\rho-1}(\Ec-\la_k I)^3}
\;=\;(\Ec^2-4I)U_{\rho-1}^3(\tfrac{\Ec}{2})\nn
&\det(\la I-\Ec)\;=\;(\la-\la_0)(\la-\la_\rho)\displaystyle{\prod_{k=1}^{\rho-1}(\la-\la_k)^4}
\;=\; (\la^2-4)U_{\rho-1}^4(\tfrac{\la}{2})
\label{charP}
\eea
This implies that the Jordan canonical form of $\Ec$ consists of $\rho-1$ rank-3 blocks
associated to the eigenvalues $\la_k$,
and $\rho+1$ rank-1 blocks associated to the eigenvalues $\la_j$.
The number of linearly independent eigenvectors of $\Ec$ is thus $2\rho$. Since
the null-space of $\Ec$ is empty for $\rho$ odd but two-dimensional for $\rho$ even,
the rank of $\Ec$ is
\be
\mathrm{rank}(\Ec)\;=\;4\rho-2-2\eps(\rho-1)\;=\;4(\rho-1)+2\eps(\rho)
\label{rankP}
\ee
To establish these results on $\Ec$, we now discuss the $\rho-1$ Jordan chains of length
3 associated to $\la_k$, and the additional $\rho+1$ eigenvectors
corresponding to $\la_j$. Using the various properties of the functions $f_h(x)$ discussed
above, it is straightforward to verify that
\be
E_j\;=\;\left(\!\!\!\begin{array}{c}
f_1(\la_j)\\ \vdots\\ f_{\rho-1}(\la_j)
\\[.15cm] \hline\\[-.3cm]
f_{\rho}(\la_j)\\ \vdots\\ f_{2\rho-1}(\la_j)
\\[.15cm] \hline\\[-.3cm]
(-1)^j f_1(\la_j)\\ \vdots\\ (-1)^j f_{\rho-1}(\la_j)
\\[.15cm] \hline\\[-.3cm]
(-1)^j f_{\rho}(\la_j)\\ \vdots\\ (-1)^j f_{2\rho-1}(\la_j)
\end{array}\!\!\!\right),\quad j=0,\rho;\qquad
E_k\;=\;\left(\!\!\!\begin{array}{c}
f_1(\la_k)\\ \vdots\\ f_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
f_{\rho}(\la_k)\\ \vdots\\ f_{2\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
(-1)^{k-1} f_{1}(\la_k)\\ \vdots\\ (-1)^{k-1} f_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
(-1)^{k-1} f_{\rho}(\la_k)\\ \vdots\\ (-1)^{k-1} f_{2\rho-1}(\la_k)
\end{array}\!\!\!\right)
\;=\;\left(\!\!\!\begin{array}{c}
f_1(\la_k)\\ \vdots\\ f_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
0\\ \vdots\\ 0
\\[.15cm] \hline\\[-.3cm]
(-1)^{k-1}f_1(\la_k)\\ \vdots\\ (-1)^{k-1}f_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
0\\ \vdots\\ 0
\end{array}\!\!\!\right)
\label{Ej}
\ee
and
\be
E_k^{(0)}\;=\;\left(\!\!\!\begin{array}{c}
f_1(\la_k)\\ \vdots\\ f_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
f_{\rho}(\la_k)\\ \vdots\\ f_{2\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
(-1)^{k} f_{1}(\la_k)\\ \vdots\\ (-1)^{k} f_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
(-1)^{k} f_{\rho}(\la_k)\\ \vdots\\ (-1)^{k} f_{2\rho-1}(\la_k)
\end{array}\!\!\!\right)
\;=\;\left(\!\!\!\begin{array}{c}
f_1(\la_k)\\ \vdots\\ f_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
0\\ \vdots\\ 0
\\[.15cm] \hline\\[-.3cm]
(-1)^{k}f_1(\la_k)\\ \vdots\\ (-1)^{k}f_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
0\\ \vdots\\ 0
\end{array}\!\!\!\right)
\label{Ek0}
\ee
are eigenvectors of $\Ec$
\be
\Ec E_0\;=\;\la_0 E_0,\qquad
\Ec E_k\;=\;\la_k E_k,\qquad
\Ec E_k^{(0)}\;=\;\la_k E_k^{(0)},\qquad
\Ec E_\rho\;=\;\la_\rho E_\rho
\ee
The vectors $E_k$ and $E_k^{(0)}$ are readily seen to be linearly independent.
For every $k\in\mathbb{Z}_{1,\rho-1}$, it is likewise verified that $E_k^{(0)}$ together with
\be
E_k^{(1)}\;=\;\left(\!\!\!\begin{array}{c}
f'_1(\la_k)\\ \vdots\\ f'_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
f'_{\rho}(\la_k)\\ \vdots\\ f'_{2\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
(-1)^k f'_1(\la_k)\\ \vdots\\ (-1)^k f'_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
(-1)^k f'_{\rho}(\la_k)\\ \vdots\\ (-1)^k f'_{2\rho-1}(\la_k)
\end{array}\!\!\!\right),\qquad
E_k^{(2)}\;=\;\left(\!\!\!\begin{array}{c}
\tfrac{1}{2}f''_1(\la_k)\\ \vdots\\ \tfrac{1}{2}f''_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
\tfrac{1}{2}f''_{\rho}(\la_k)\\ \vdots\\ \tfrac{1}{2}f''_{2\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
\tfrac{1}{2}(-1)^k f''_1(\la_k)\\ \vdots\\ \tfrac{1}{2}(-1)^k f''_{\rho-1}(\la_k)
\\[.15cm] \hline\\[-.3cm]
\tfrac{1}{2}(-1)^k f''_{\rho}(\la_k)\\ \vdots\\ \tfrac{1}{2}(-1)^k f''_{2\rho-1}(\la_k)
\end{array}\!\!\!\right)
\label{Ek12}
\ee
form the Jordan chain
\be
\Ec E_k^{(0)}\;=\;\la_k E_k^{(0)},\qquad
\Ec E_k^{(1)}\;=\;\la_k E_k^{(1)}+E_k^{(0)},\qquad
\Ec E_k^{(2)}\;=\;\la_k E_k^{(2)}+E_k^{(1)}
\ee
Finally, the $(4\rho-2)$-dimensional matrix $Q_\Ec$ is constructed by concatenating
the generalized eigenvectors (\ref{Ej}), (\ref{Ek0}) and (\ref{Ek12})
\be
Q_\Ec\;=\;\left(\!\!\begin{array}{c|cccc|c|cccc|c} E_0&E_1&E_1^{(0)}&E_1^{(1)}&E_1^{(2)}&\ldots&
E_{\rho-1}&E_{\rho-1}^{(0)}&E_{\rho-1}^{(1)}&E_{\rho-1}^{(2)}&E_\rho \end{array}\!\!\right)
\label{QE}
\ee
By a similarity transformation, this matrix converts $\Ec$ into its Jordan canonical form
\be
J_\Ec\;=\;Q_\Ec^{-1}\Ec Q_\Ec
\;=\;\mathrm{diag}\big(\la_0;\la_1,\Jc_{\la_1,3};\ldots;\la_{\rho-1},\Jc_{\la_{\rho-1},3};
\la_\rho\big)
\label{JE}
\ee
\section{Fundamental and auxiliary fusion graphs}
\label{SecFundAux}
\subsection{Fundamental fusion graphs}
\label{SecFundFusGra}
The graphs associated to the two fundamental modules are called
{\em fundamental fusion graphs} and consist of certain connected components.
Every such subgraph is a tadpole graph or an eye-patch graph.
The fundamental fusion graph, whose adjacency matrix is
given by $X$, consists of $p'-1$ tadpole graphs and $p'$
eye-patch graphs, all with order parameter $\rho=p$. For every $b\in\mathbb{Z}_{1,p'-1}$,
the $3p-1$ vertices of the tadpole graphs are given by
\be
\Tc_p:\quad
\big(L_r,U_a,R,D_a\big)\;=\;\big(\ketw{r,b},\ketw{\R_{p,b}^{a,0}},\ketw{2p,b},
\ketw{\R_{2p,b}^{a,0}}\big),
\qquad r\in\mathbb{Z}_{1,p},\quad a\in\mathbb{Z}_{1,p-1}
\ee
while for every $\beta\in\mathbb{Z}_{0,p'-1}$, the $4p-2$ vertices of the eye-patch graphs
are given by
\be
\Ec_p:\quad
\big(L_r,U_a,R_r,D_a\big)\;=\;\big(\ketw{\R_{r,p'}^{0,\beta}},\ketw{\R_{p,p'}^{a,\beta}},
\ketw{\R_{r,2p'}^{0,\beta}},\ketw{\R_{2p,p'}^{a,\beta}}\big),
\qquad r\in\mathbb{Z}_{1,p},\quad a\in\mathbb{Z}_{1,p-1}
\ee
Likewise, the fundamental fusion graph, whose adjacency matrix is
given by $Y$, consists of $p-1$ tadpole graphs and $p$
eye-patch graphs, all with order parameter $\rho=p'$. For every $a\in\mathbb{Z}_{1,p-1}$,
the $3p'-1$ vertices of the tadpole graphs are given by
\be
\Tc_{p'}:\quad
\big(L_s,U_b,R,D_b\big)\;=\;\big(\ketw{a,s},\ketw{\R_{a,p'}^{0,b}},\ketw{a,2p'},
\ketw{\R_{a,2p'}^{0,b}}\big),
\qquad s\in\mathbb{Z}_{1,p'},\quad b\in\mathbb{Z}_{1,p'-1}
\ee
while for every $\al\in\mathbb{Z}_{0,p-1}$, the $4p'-2$ vertices of the eye-patch graphs
are given by
\be
\Ec_{p'}:\quad
\big(L_s,U_b,R_s,D_b\big)\;=\;\big(\ketw{\R_{p,s}^{\al,0}},\ketw{\R_{p,p'}^{\al,b}},
\ketw{\R_{2p,s}^{\al,0}},\ketw{\R_{p,2p'}^{\al,b}}\big),
\qquad s\in\mathbb{Z}_{1,p'},\quad b\in\mathbb{Z}_{1,p'-1}
\ee
In accord with the results obtained in~\cite{Ras0908} on ${\cal WLM}(1,p')$, the graph
corresponding to $X$ consists of $2p'-1$ order-2 graphs (\ref{Gp1})
\be
\big(L,R\big)\;\in\;\Big\{\big(\ketw{1,b},\ketw{2,b}\big),
\big(\Wc(\D_{1,p'}),\Wc(\D_{2,p'})\big),
\big(\ketw{\R_{1,p'}^{0,b}},\ketw{\R_{1,2p'}^{0,b}}\big)\Big\},\qquad
b\in\mathbb{Z}_{1,p'-1}
\ee
where $\D_{2,p'}=\D_{1,2p'}$, while there is a single eye-patch graph associated to $Y$
\be
\big(L_s,U_b,R_s,D_b\big)\;=\;\big(\Wc(\D_{1,s}),\ketw{\R_{1,p'}^{0,b}},
\Wc(\D_{2,s}),\ketw{\R_{1,2p'}^{0,b}}\big),
\qquad s\in\mathbb{Z}_{1,p'},\quad b\in\mathbb{Z}_{1,p'-1}
\ee
\subsubsection{${\cal W}$-extended critical percolation ${\cal WLM}(2,3)$}
${\cal W}$-extended critical percolation is described by ${\cal WLM}(2,3)$ where
$p=2$ and $p'=3$. In this case,
the fundamental fusion graph, whose adjacency matrix is given by $X$, consists
of five connected components, all with order parameter 2. The order of the graph is 28.
The connected components are the two tadpole graphs
\be
\mbox{
\begin{picture}(100,80)(100,-35)
\unitlength=0.75cm
\thinlines
\put(2.8,0){$\ketw{1,1}$}
\put(4.55,0.15){\vector(1,0){0.75}}
\put(5.45,0){$\ketw{2,1}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.25,1.4){$\ketw{\R_{2,1}^{1,0}}$}
\put(8.4,1.1){\vector(1,-1){0.6}}
\put(8.55,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.25,-1.6){$\ketw{\R_{4,1}^{1,0}}$}
\put(8.5,-0.85){\vector(1,1){0.5}}
\put(8.65,-0.7){\vector(1,1){0.5}}
\put(8.9,-0.45){\vector(-1,-1){0.5}}
\put(9.25,-0.05){$\ketw{4,1}$}
\end{picture}
}
\hspace{4cm}
\mbox{
\begin{picture}(100,80)(100,-35)
\unitlength=0.75cm
\thinlines
\put(2.8,0){$\ketw{1,2}$}
\put(4.55,0.15){\vector(1,0){0.75}}
\put(5.45,0){$\ketw{2,2}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.25,1.4){$\ketw{\R_{2,2}^{1,0}}$}
\put(8.4,1.1){\vector(1,-1){0.6}}
\put(8.55,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.25,-1.6){$\ketw{\R_{4,2}^{1,0}}$}
\put(8.5,-0.85){\vector(1,1){0.5}}
\put(8.65,-0.7){\vector(1,1){0.5}}
\put(8.9,-0.45){\vector(-1,-1){0.5}}
\put(9.25,-0.05){$\ketw{4,2}$}
\end{picture}
}
\ee
and the three eye-patch graphs
\be
\mbox{
\begin{picture}(100,80)(120,-35)
\unitlength=0.75cm
\thinlines
\put(2.8,0){$\ketw{1,3}$}
\put(4.55,0.15){\vector(1,0){0.75}}
\put(5.45,0){$\ketw{2,3}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.25,1.4){$\ketw{\R_{2,3}^{1,0}}$}
\put(8.4,1.1){\vector(1,-1){0.6}}
\put(8.55,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.25,-1.6){$\ketw{\R_{4,3}^{1,0}}$}
\put(8.5,-0.85){\vector(1,1){0.5}}
\put(8.65,-0.7){\vector(1,1){0.5}}
\put(8.9,-0.45){\vector(-1,-1){0.5}}
\put(9.25,-0.05){$\ketw{2,6}$}
\put(11.7,0.1){\vector(-1,0){0.75}}
\put(11.85,0){$\ketw{1,6}$}
\end{picture}
}
\label{X23E1}
\ee
and
\be
\mbox{
\begin{picture}(100,80)(120,-35)
\unitlength=0.75cm
\thinlines
\put(2.4,0){$\ketw{\R_{1,3}^{0,1}}$}
\put(4.35,0.15){\vector(1,0){0.75}}
\put(5.25,0){$\ketw{\R_{2,3}^{0,1}}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.25,1.4){$\ketw{\R_{2,3}^{1,1}}$}
\put(8.4,1.1){\vector(1,-1){0.6}}
\put(8.55,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.25,-1.6){$\ketw{\R_{4,3}^{1,1}}$}
\put(8.5,-0.85){\vector(1,1){0.5}}
\put(8.65,-0.7){\vector(1,1){0.5}}
\put(8.9,-0.45){\vector(-1,-1){0.5}}
\put(9.25,-0.05){$\ketw{\R_{2,6}^{0,1}}$}
\put(11.9,0.1){\vector(-1,0){0.75}}
\put(12.05,0){$\ketw{\R_{1,6}^{0,1}}$}
\end{picture}
}
\label{X23E2}
\ee
and
\be
\mbox{
\begin{picture}(100,80)(120,-35)
\unitlength=0.75cm
\thinlines
\put(2.4,0){$\ketw{\R_{1,3}^{0,2}}$}
\put(4.35,0.15){\vector(1,0){0.75}}
\put(5.25,0){$\ketw{\R_{2,3}^{0,2}}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.25,1.4){$\ketw{\R_{2,3}^{1,2}}$}
\put(8.4,1.1){\vector(1,-1){0.6}}
\put(8.55,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.25,-1.6){$\ketw{\R_{4,3}^{1,2}}$}
\put(8.5,-0.85){\vector(1,1){0.5}}
\put(8.65,-0.7){\vector(1,1){0.5}}
\put(8.9,-0.45){\vector(-1,-1){0.5}}
\put(9.25,-0.05){$\ketw{\R_{2,6}^{0,2}}$}
\put(11.9,0.1){\vector(-1,0){0.75}}
\put(12.05,0){$\ketw{\R_{1,6}^{0,2}}$}
\end{picture}
}
\label{X23E3}
\ee
Likewise, the fundamental fusion graph, whose adjacency matrix is given by $Y$, consists
of three connected components, all with order parameter 3. The order of the graph is 28.
The connected components are the single tadpole graph
\be
\mbox{
\begin{picture}(100,80)(100,-35)
\unitlength=0.75cm
\thinlines
\put(0,0){$\ketw{1,1}$}
\put(2.1,0.15){\vector(-1,0){0.4}}
\put(2.1,0.15){\vector(1,0){0.45}}
\put(2.75,0){$\ketw{1,2}$}
\put(4.45,0.15){\vector(1,0){0.85}}
\put(5.4,0){$\ketw{1,3}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.55,1.4){$\ketw{\R_{1,3}^{0,1}}$}
\put(9.9,1.55){\vector(-1,0){0.45}}
\put(9.9,1.55){\vector(1,0){0.45}}
\put(10.5,1.4){$\ketw{\R_{1,3}^{0,2}}$}
\put(12,1.1){\vector(1,-1){0.6}}
\put(12.15,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.55,-1.4){$\ketw{\R_{1,6}^{0,2}}$}
\put(9.9,-1.25){\vector(-1,0){0.45}}
\put(9.9,-1.25){\vector(1,0){0.45}}
\put(10.5,-1.4){$\ketw{\R_{1,6}^{0,1}}$}
\put(12.1,-0.85){\vector(1,1){0.5}}
\put(12.25,-0.7){\vector(1,1){0.5}}
\put(12.5,-0.45){\vector(-1,-1){0.5}}
\put(12.8,-0.05){$\ketw{1,6}$}
\end{picture}
}
\ee
and the two eye-patch graphs
\be
\mbox{
\begin{picture}(100,80)(160,-35)
\unitlength=0.75cm
\thinlines
\put(0,0){$\ketw{2,1}$}
\put(2.1,0.15){\vector(-1,0){0.4}}
\put(2.1,0.15){\vector(1,0){0.45}}
\put(2.75,0){$\ketw{2,2}$}
\put(4.45,0.15){\vector(1,0){0.85}}
\put(5.4,0){$\ketw{2,3}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.55,1.4){$\ketw{\R_{2,3}^{0,1}}$}
\put(9.9,1.55){\vector(-1,0){0.45}}
\put(9.9,1.55){\vector(1,0){0.45}}
\put(10.5,1.4){$\ketw{\R_{2,3}^{0,2}}$}
\put(12,1.1){\vector(1,-1){0.6}}
\put(12.15,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.55,-1.4){$\ketw{\R_{2,6}^{0,2}}$}
\put(9.9,-1.25){\vector(-1,0){0.45}}
\put(9.9,-1.25){\vector(1,0){0.45}}
\put(10.5,-1.4){$\ketw{\R_{2,6}^{0,1}}$}
\put(12.1,-0.85){\vector(1,1){0.5}}
\put(12.25,-0.7){\vector(1,1){0.5}}
\put(12.5,-0.45){\vector(-1,-1){0.5}}
\put(12.8,-0.05){$\ketw{4,3}$}
\put(15.3,0.1){\vector(-1,0){0.85}}
\put(15.45,-0.05){$\ketw{4,2}$}
\put(17.55,0.1){\vector(-1,0){0.4}}
\put(17.55,0.1){\vector(1,0){0.45}}
\put(18.15,-0.05){$\ketw{4,1}$}
\end{picture}
}
\label{Y23E1}
\ee
and
\be
\mbox{
\begin{picture}(100,80)(160,-35)
\unitlength=0.75cm
\thinlines
\put(-0.4,0){$\ketw{\R_{2,1}^{1,0}}$}
\put(1.85,0.15){\vector(-1,0){0.4}}
\put(1.85,0.15){\vector(1,0){0.45}}
\put(2.4,0){$\ketw{\R_{2,2}^{1,0}}$}
\put(4.3,0.15){\vector(1,0){0.85}}
\put(5.25,0){$\ketw{\R_{2,3}^{1,0}}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.55,1.4){$\ketw{\R_{2,3}^{1,1}}$}
\put(9.9,1.55){\vector(-1,0){0.45}}
\put(9.9,1.55){\vector(1,0){0.45}}
\put(10.5,1.4){$\ketw{\R_{2,3}^{1,2}}$}
\put(12,1.1){\vector(1,-1){0.6}}
\put(12.15,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.55,-1.4){$\ketw{\R_{2,6}^{1,2}}$}
\put(9.9,-1.25){\vector(-1,0){0.45}}
\put(9.9,-1.25){\vector(1,0){0.45}}
\put(10.5,-1.4){$\ketw{\R_{2,6}^{1,1}}$}
\put(12.1,-0.85){\vector(1,1){0.5}}
\put(12.25,-0.7){\vector(1,1){0.5}}
\put(12.5,-0.45){\vector(-1,-1){0.5}}
\put(12.8,-0.05){$\ketw{\R_{4,3}^{1,0}}$}
\put(15.5,0.1){\vector(-1,0){0.85}}
\put(15.65,-0.05){$\ketw{\R_{4,2}^{1,0}}$}
\put(17.9,0.1){\vector(-1,0){0.4}}
\put(17.9,0.1){\vector(1,0){0.45}}
\put(18.5,-0.05){$\ketw{\R_{4,1}^{1,0}}$}
\end{picture}
}
\label{Y23E2}
\ee
We recall that
\be
\ketw{4,3}\;=\;\Wc(\D_{4,3})\;=\;\Wc(\D_{2,6})\;=\;\ketw{2,6},\quad
\ketw{\R_{4,3}^{1,1}}\;=\;\ketw{\R_{2,6}^{1,1}},\quad
\ketw{\R_{4,3}^{1,2}}\;=\;\ketw{\R_{2,6}^{1,2}}
\ee
\subsection{Fundamental fusion matrices}
We choose to work with the basis
\bea
&&\Big\{\ketw{1,1},\ldots,\ketw{1,p'},\ketw{\R_{1,p'}^{0,1}},\ldots,\ketw{\R_{1,p'}^{0,p'-1}},\nn
&&\hspace{4.5cm}\ketw{1,2p'},\ketw{\R_{1,2p'}^{0,1}},\ldots,\ketw{\R_{1,2p'}^{0,p'-1}},\nn
&&\vdots\nn
&& \ketw{p-1,1},\ldots,\ketw{p-1,p'},\ketw{\R_{p-1,p'}^{0,1}},\ldots,\ketw{\R_{p-1,p'}^{0,p'-1}},\nn
&&\hspace{4.5cm}\ketw{p-1,2p'},\ketw{\R_{p-1,2p'}^{0,1}},\ldots,\ketw{\R_{p-1,2p'}^{0,p'-1}},\nn
&&\ketw{p,1},\ldots,\ketw{p,p'},\ketw{\R_{p,p'}^{0,1}},\ldots,\ketw{\R_{p,p'}^{0,p'-1}},\nn
&&\hspace{4.5cm}\ketw{2p,1},\ldots,\ketw{2p,p'},
\ketw{\R_{p,2p'}^{0,1}},\ldots,\ketw{\R_{p,2p'}^{0,p'-1}},\nn
&&\ketw{\R_{p,1}^{1,0}},\ldots,\ketw{\R_{p,p'}^{1,0}},\ketw{\R_{p,p'}^{1,1}},\ldots,
\ketw{\R_{p,p'}^{1,p'-1}},\nn
&&\hspace{4.5cm}\ketw{\R_{2p,1}^{1,0}},\ldots,\ketw{\R_{2p,p'}^{1,0}},
\ketw{\R_{p,2p'}^{1,1}},\ldots,\ketw{\R_{p,2p'}^{1,p'-1}},\nn
&&\vdots\nn
&&\ketw{\R_{p,1}^{p-1,0}},\ldots,\ketw{\R_{p,p'}^{p-1,0}},\ketw{\R_{p,p'}^{p-1,1}},\ldots,
\ketw{\R_{p,p'}^{p-1,p'-1}},\nn
&&\hspace{4.5cm}\ketw{\R_{2p,1}^{p-1,0}},\ldots,\ketw{\R_{2p,p'}^{p-1,0}},
\ketw{\R_{p,2p'}^{p-1,1}},\ldots,\ketw{\R_{p,2p'}^{p-1,p'-1}}\Big\}
\label{Ybasis}
\eea
in which $Y$ has the simple form
\be
Y\;=\;\mathrm{diag}\big(
\underbrace{\Tc_{p'},\ldots,\Tc_{p'}}_{p-1},\,\underbrace{\Ec_{p'},\ldots,\Ec_{p'}}_{p}\big)
\label{Ydiag}
\ee
while $X$ is the matrix
\be
X\;=\;\Pc^{-1}\mathrm{diag}\big(
\underbrace{\Tc_{p},\ldots,\Tc_{p}}_{p'-1},\,\underbrace{\Ec_{p},\ldots,\Ec_{p}}_{p'}\big)\Pc\;=\;
\left(\!\!
\begin{array}{ccccc|c|cccccc}
0&I&&&&&&&&& \\
I&0&&&&&&&&& \\
&&\ddots&&&&&&&& \\
&&&0&I&&&&&& \\
&&&I&0&\It&&&&& \\
\hline
&&&&0&0&I&&&& \\
\hline
&&&&&2I&0&I&&& \\
&&&&&&I&0&&& \\
&&&&&&&&\ddots& \\
&&&&&&&&&0&I \\
&&&&&2C&&&&I&0
\end{array}
\!\!\right)
\label{X}
\ee
Here $\Pc$ is a permutation matrix, while the $(4p'-2)\times(4p'-2)$-dimensional matrix $C$
and the $(3p'-1)\times(4p'-2)$-dimensional matrix $\It$ are given by
\be
C\;=\;\left(\!\!
\begin{array}{cc}
0&I \\
I&0 \end{array} \!\!\right),\qquad
\It\;=\;\left(\!\!
\begin{array}{ccc}
I_{(2p'-1)\times(2p'-1)}&0_{(2p'-1)\times(p'-1)}&0_{(2p'-1)\times(p')} \\
0_{(p')\times(2p'-1)}&0_{(p')\times(p'-1)}&I_{(p')\times(p')}
\end{array}
\!\!\right)
\ee
In (\ref{X}), $X$ is written as a $(2p-1)\times(2p-1)$-dimensional matrix whose entries
are blocks. Every block (indicated by $0$ or $I$) to the left of the leftmost vertical delimiter
has $3p'-1$ columns, while every block
(indicated by $0$, $I$, $\It$, $2I$ or $2C$) to the right of this delimiter has $4p'-2$ columns.
Likewise, every block (indicated by $0$, $I$ or $\It$) above the upper vertical delimiter has
$3p'-1$ rows, while every block (indicated by $0$, $I$, $2I$ or $2C$)
below this delimiter has $4p'-2$ rows.
For small values of $p$, the matrix $X$ in (\ref{X}) is meant to reduce to
\be
X\big|_{p=2}\;=\;
\left(\!\!
\begin{array}{c|c|c}
0&\It&0 \\
\hline
0&0&I \\
\hline
0&2I+2C&0
\end{array}
\!\!\right),\qquad
X\big|_{p=1}\;=\;2C
\label{X2X1}
\ee
with $p'$-dependent dimensions of the blocks given as above.
For later convenience, it is noted that
\be
\It E_0\;=\;T_0,\qquad \It E_k\;=\;T_k^{(0)},
\qquad \It E_k^{(\ell)}\;=\;T_k^{(\ell)},\qquad \It E_{p'}\;=\;T_{p'}
\label{ItE}
\ee
and
\be
C E_0\;=\;E_0,\qquad C E_k\;=\;(-1)^{k-1}C_k,
\qquad C E_k^{(\ell)}\;=\;(-1)^k E_k^{(\ell)},\qquad C E_{p'}\;=\;(-1)^{p'}E_{p'}
\label{CE}
\ee
where $k\in\mathbb{Z}_{1,p'-1}$ and $\ell\in\mathbb{Z}_{0,2}$, while the order parameter
appearing in the entries of the vectors is $\rho=p'$. We also note that the basis used
in~\cite{Ras0908} on ${\cal WLM}(1,p')$ is different from (\ref{Ybasis}) for $p=1$
\bea
\!\!\ketw{p,1},\ldots,\ketw{p,p'},\ketw{\R_{p,p'}^{0,1}},\ldots,\ketw{\R_{p,p'}^{0,p'-1}},
\ketw{2p,1},\ldots,\ketw{2p,p'},
\ketw{\R_{p,2p'}^{0,1}},\ldots,\ketw{\R_{p,2p'}^{0,p'-1}}
\eea
The two bases are related by a permutation.
\subsection{Auxiliary fusion graphs}
The two auxiliary fusion graphs consist of certain connected components.
Every such subgraph is a cycle graph or an eye-patch graph.
The auxiliary fusion graph, whose adjacency matrix is
given by $\Xh$, consists of $p'-1$ cycle graphs and $p'$
eye-patch graphs, all with order parameter $\rho=p$. For every $b\in\mathbb{Z}_{1,p'-1}$,
the $2p$ vertices of the cycle graphs are given by
\be
\Cc_p:\quad
\big(L,U_a,R,D_a\big)\;=\;\big(\ketw{p,b},\ketw{\R_{p,b}^{a,0}},\ketw{2p,b},
\ketw{\R_{2p,b}^{a,0}}\big),
\qquad a\in\mathbb{Z}_{1,p-1}
\ee
while for every $\beta\in\mathbb{Z}_{0,p'-1}$, the $4p-2$ vertices of the eye-patch graphs
are given by
\be
\Ec_p:\quad
\big(L_r,U_a,R_r,D_a\big)\;=\;\big(\ketw{\R_{r,p'}^{0,\beta}},\ketw{\R_{p,p'}^{a,\beta}},
\ketw{\R_{r,2p'}^{0,\beta}},\ketw{\R_{2p,p'}^{a,\beta}}\big),
\qquad r\in\mathbb{Z}_{1,p},\quad a\in\mathbb{Z}_{1,p-1}
\ee
Likewise,the auxiliary fusion graph, whose adjacency matrix is
given by $\Yh$, consists of $p-1$ cycle graphs and $p$
eye-patch graphs, all with order parameter $\rho=p'$. For every $a\in\mathbb{Z}_{1,p-1}$,
the $2p'$ vertices of the cycle graphs are given by
\be
\Cc_{p'}:\quad
\big(L,U_b,R,D_b\big)\;=\;\big(\ketw{a,p'},\ketw{\R_{a,p'}^{0,b}},\ketw{a,2p'},
\ketw{\R_{a,2p'}^{0,b}}\big),
\qquad b\in\mathbb{Z}_{1,p'-1}
\ee
while for every $\al\in\mathbb{Z}_{0,p-1}$, the $4p'-2$ vertices of the eye-patch graphs
are given by
\be
\Ec_{p'}:\quad
\big(L_s,U_b,R_s,D_b\big)\;=\;\big(\ketw{\R_{p,s}^{\al,0}},\ketw{\R_{p,p'}^{\al,b}},
\ketw{\R_{2p,s}^{\al,0}},\ketw{\R_{p,2p'}^{\al,b}}\big),
\qquad s\in\mathbb{Z}_{1,p'},\quad b\in\mathbb{Z}_{1,p'-1}
\ee
\subsubsection{${\cal W}$-extended critical percolation ${\cal WLM}(2,3)$}
In the case of ${\cal W}$-extended critical percolation ${\cal WLM}(2,3)$, there are exactly
two indecomposable modules (\ref{ab}) in the fundamental fusion algebra {\em not}
associated with boundary conditions, namely the identity $\ketw{1,1}$ and
the fundamental module $\ketw{1,2}$.
The auxiliary fusion graph, whose adjacency matrix is given by $\Xh$, consists
of five connected components, all with order parameter 2. The order of the graph is 26.
The connected components are the two cycle graphs
\be
\mbox{
\begin{picture}(100,80)(120,-35)
\unitlength=0.75cm
\thinlines
\put(5.45,0){$\ketw{2,1}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.25,1.4){$\ketw{\R_{2,1}^{1,0}}$}
\put(8.4,1.1){\vector(1,-1){0.6}}
\put(8.55,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.25,-1.6){$\ketw{\R_{4,1}^{1,0}}$}
\put(8.5,-0.85){\vector(1,1){0.5}}
\put(8.65,-0.7){\vector(1,1){0.5}}
\put(8.9,-0.45){\vector(-1,-1){0.5}}
\put(9.25,-0.05){$\ketw{4,1}$}
\end{picture}
}
\hspace{3cm}
\mbox{
\begin{picture}(100,80)(120,-35)
\unitlength=0.75cm
\thinlines
\put(5.45,0){$\ketw{2,2}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.25,1.4){$\ketw{\R_{2,2}^{1,0}}$}
\put(8.4,1.1){\vector(1,-1){0.6}}
\put(8.55,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.25,-1.6){$\ketw{\R_{4,2}^{1,0}}$}
\put(8.5,-0.85){\vector(1,1){0.5}}
\put(8.65,-0.7){\vector(1,1){0.5}}
\put(8.9,-0.45){\vector(-1,-1){0.5}}
\put(9.25,-0.05){$\ketw{4,2}$}
\end{picture}
}
\ee
and the three eye-patch graphs (\ref{X23E1}), (\ref{X23E2}) and (\ref{X23E3}).
Likewise, the auxiliary fusion graph, whose adjacency matrix is given by $\Yh$, consists
of three connected components, all with order parameter 3. The order of the graph is 26.
The connected components are the single cycle graph
\be
\mbox{
\begin{picture}(100,80)(160,-35)
\unitlength=0.75cm
\thinlines
\put(5.4,0){$\ketw{1,3}$}
\put(7.45,1){\vector(-1,-1){0.5}}
\put(7.3,0.85){\vector(-1,-1){0.5}}
\put(7.05,0.6){\vector(1,1){0.5}}
\put(7.55,1.4){$\ketw{\R_{1,3}^{0,1}}$}
\put(9.9,1.55){\vector(-1,0){0.45}}
\put(9.9,1.55){\vector(1,0){0.45}}
\put(10.5,1.4){$\ketw{\R_{1,3}^{0,2}}$}
\put(12,1.1){\vector(1,-1){0.6}}
\put(12.15,0.95){\vector(1,-1){0.6}}
\put(7.55,-1){\vector(-1,1){0.6}}
\put(7.4,-0.85){\vector(-1,1){0.6}}
\put(7.55,-1.4){$\ketw{\R_{1,6}^{0,2}}$}
\put(9.9,-1.25){\vector(-1,0){0.45}}
\put(9.9,-1.25){\vector(1,0){0.45}}
\put(10.5,-1.4){$\ketw{\R_{1,6}^{0,1}}$}
\put(12.1,-0.85){\vector(1,1){0.5}}
\put(12.25,-0.7){\vector(1,1){0.5}}
\put(12.5,-0.45){\vector(-1,-1){0.5}}
\put(12.8,-0.05){$\ketw{1,6}$}
\end{picture}
}
\ee
and the two eye-patch graphs (\ref{Y23E1}) and (\ref{Y23E2}).
\subsection{Auxiliary fusion matrices}
We choose to work with the basis
\bea
&&\Big\{\ketw{1,p'},\ketw{\R_{1,p'}^{0,1}},\ldots,\ketw{\R_{1,p'}^{0,p'-1}},
\ketw{1,2p'},\ketw{\R_{1,2p'}^{0,1}},\ldots,\ketw{\R_{1,2p'}^{0,p'-1}},\nn
&&\vdots\nn
&& \ketw{p-1,p'},\ketw{\R_{p-1,p'}^{0,1}},\ldots,\ketw{\R_{p-1,p'}^{0,p'-1}},
\ketw{p-1,2p'},\ketw{\R_{p-1,2p'}^{0,1}},\ldots,\ketw{\R_{p-1,2p'}^{0,p'-1}},\nn
&&\ketw{p,1},\ldots,\ketw{p,p'},\ketw{\R_{p,p'}^{0,1}},\ldots,\ketw{\R_{p,p'}^{0,p'-1}},\nn
&&\hspace{4.5cm}\ketw{2p,1},\ldots,\ketw{2p,p'},
\ketw{\R_{p,2p'}^{0,1}},\ldots,\ketw{\R_{p,2p'}^{0,p'-1}},\nn
&&\ketw{\R_{p,1}^{1,0}},\ldots,\ketw{\R_{p,p'}^{1,0}},\ketw{\R_{p,p'}^{1,1}},\ldots,
\ketw{\R_{p,p'}^{1,p'-1}},\nn
&&\hspace{4.5cm}\ketw{\R_{2p,1}^{1,0}},\ldots,\ketw{\R_{2p,p'}^{1,0}},
\ketw{\R_{p,2p'}^{1,1}},\ldots,\ketw{\R_{p,2p'}^{1,p'-1}},\nn
&&\vdots\nn
&&\ketw{\R_{p,1}^{p-1,0}},\ldots,\ketw{\R_{p,p'}^{p-1,0}},\ketw{\R_{p,p'}^{p-1,1}},\ldots,
\ketw{\R_{p,p'}^{p-1,p'-1}},\nn
&&\hspace{4.5cm}\ketw{\R_{2p,1}^{p-1,0}},\ldots,\ketw{\R_{2p,p'}^{p-1,0}},
\ketw{\R_{p,2p'}^{p-1,1}},\ldots,\ketw{\R_{p,2p'}^{p-1,p'-1}}\Big\}
\label{Yhbasis}
\eea
in which $\Yh$ has the simple form
\be
\Yh\;=\;\mathrm{diag}\big(
\underbrace{\Cc_{p'},\ldots,\Cc_{p'}}_{p-1},\,\underbrace{\Ec_{p'},\ldots,\Ec_{p'}}_{p}\big)
\label{Yhdiag}
\ee
while $\Xh$ is the matrix
\be
\Xh\;=\;\Pch^{-1}\mathrm{diag}\big(
\underbrace{\Cc_{p},\ldots,\Cc_{p}}_{p'-1},\,\underbrace{\Ec_{p},\ldots,\Ec_{p}}_{p'}\big)\Pch\;=\;
\left(\!\!
\begin{array}{ccccc|c|cccccc}
0&I&&&&&&&&& \\
I&0&&&&&&&&& \\
&&\ddots&&&&&&&& \\
&&&0&I&&&&&& \\
&&&I&0&\Ih&&&&& \\
\hline
&&&&0&0&I&&&& \\
\hline
&&&&&2I&0&I&&& \\
&&&&&&I&0&&& \\
&&&&&&&&\ddots& \\
&&&&&&&&&0&I \\
&&&&&2C&&&&I&0
\end{array}
\!\!\right)
\label{Xh}
\ee
Here $\Pch$ is a permutation matrix, while the $(4p'-2)\times(4p'-2)$-dimensional matrix $C$ and
the $(2p')\times(4p'-2)$-dimensional matrix $\Ih$ are given by
\be
C\;=\;\left(\!\!
\begin{array}{cc}
0&I \\
I&0 \end{array} \!\!\right),\qquad
\Ih\;=\;\left(\!\!
\begin{array}{cccc}
0_{(p')\times(p'-1)}&I_{(p')\times(p')}&0_{(p')\times(p'-1)}&0_{(p')\times(p')} \\
0_{(p')\times(p'-1)}&0_{(p')\times(p')}&0_{(p')\times(p'-1)}&I_{(p')\times(p')}
\end{array}
\!\!\right)
\ee
In (\ref{Xh}), $\Xh$ is written as a $(2p-1)\times(2p-1)$-dimensional matrix whose entries
are blocks. Every block (indicated by $0$ or $I$) to the left of the leftmost vertical delimiter
has $2p'$ columns, while every block
(indicated by $0$, $I$, $\Ih$, $2I$ or $2C$) to the right of this delimiter has $4p'-2$ columns.
Likewise, every block (indicated by $0$, $I$ or $\Ih$) above the upper vertical delimiter has
$2p'$ rows, while every block (indicated by $0$, $I$, $2I$ or $2C$)
below this delimiter has $4p'-2$ rows.
For small values of $p$, the matrix $\Xh$ in (\ref{Xh}) is meant to reduce to
\be
\Xh\big|_{p=2}\;=\;
\left(\!\!
\begin{array}{c|c|c}
0&\Ih&0 \\
\hline
0&0&I \\
\hline
0&2I+2C&0
\end{array}
\!\!\right),\qquad
\Xh\big|_{p=1}\;=\;2C
\label{Xh2Xh1}
\ee
with $p'$-dependent dimensions of the blocks given as above.
For later convenience, it is noted that
\be
\Ih E_0\;=\;C_0,\qquad \Ih E_k\;=\;\Ih E_k^{(0)}\;=\;0,
\qquad \Ih E_k^{(1)}\;=\;C_k^{(0)},\qquad \Ih E_k^{(2)}\;=\;C_k^{(1)},\qquad \Ih E_{p'}\;=\;C_{p'}
\label{IhE}
\ee
where $k\in\mathbb{Z}_{1,p'-1}$, while the order parameter appearing in the entries of
the vectors is $\rho=p'$.
\section{Spectral decomposition of fusion matrices}
\label{SecSpectral}
The objective here is to examine to what extent the fusion matrices $N_\mu$ (or $\Nh_\mu$)
can be simultaneously brought to Jordan form. Our first goal is thus to
devise a similarity transformation in the form of a
matrix $Q$ ($\Qh$) which simultaneously brings the fundamental (auxiliary) fusion matrices
$X$ and $Y$ ($\Xh$ and $\Yh$) to Jordan form. For $p>1$, this is only
possible modulo permutation similarity.
With the Jordan decompositions of $Y$ and $\Yh$ implemented, the best we can do is therefore
\be
Q^{-1}XQ\;=\;P^{-1}J_XP,\qquad\quad Q^{-1}YQ\;=\; J_Y
\label{QXQ}
\ee
and
\be
\Qh^{-1}\Xh\Qh\;=\;\Ph^{-1}J_{\Xh}\Ph,\qquad\quad \Qh^{-1}\Yh\Qh\;=\; J_{\Yh}
\label{QhXhQh}
\ee
where $J_X$ and $J_Y$ ($J_{\Xh}$ and $J_{\Yh}$) are {\em Jordan canonical forms} of
$X$ and $Y$ ($\Xh$ and $\Yh$), while $P$ ($\Ph$) is a permutation matrix.
For every fusion matrix $N_\mu$ in (\ref{NNN}), it then follows that
\bea
Q^{-1}N_\mu Q&=&Q^{-1}\mathrm{pol}_{\mu}(X,Y)Q
\;=\;\mathrm{pol}_{\mu}(Q^{-1}XQ,Q^{-1}YQ)
\;=\;\mathrm{pol}_{\mu}(P^{-1}J_XP,J_Y)\nn
&=&P^{-1}\mathrm{pol}_{\mu}^{(x)}(J_X)P\,\mathrm{pol}_{\mu}^{(y)}(J_Y)
\label{QNQ}
\eea
where we have used that the polynomials $\mathrm{pol}_{\mu}(X,Y)$ (\ref{abR4})
factorize and thus can be written as
\be
\mathrm{pol}_{\mu}(X,Y)\;=\;\mathrm{pol}_{\mu}^{(x)}(X)\,\mathrm{pol}_{\mu}^{(y)}(Y)
\label{polXY}
\ee
As we will demonstrate, $Q^{-1}N_\mu Q$ is a {\em block-diagonal matrix whose blocks
are upper-triangular matrices of dimension 1, 3, 5 or 9}, while $P$ is a {\em symmetric}
permutation matrix.
Likewise, for every $\Nh_\mu$ in (\ref{NNNb}), it follows that
\bea
\Qh^{-1}\Nh_\mu\Qh&=&\Qh^{-1}\mathrm{pol}_{\mu}(\Xh,\Yh)\Qh
\;=\;\mathrm{pol}_{\mu}(\Qh^{-1}\Xh\Qh,\Qh^{-1}\Yh\Qh)
\;=\;\mathrm{pol}_{\mu}(\Ph^{-1}J_{\Xh}\Ph,J_{\Yh})\nn
&=&\Ph^{-1}\mathrm{pol}_{\mu}^{(x)}(J_{\Xh})\Ph\,\mathrm{pol}_{\mu}^{(y)}(J_{\Yh})
\label{QhNhQh}
\eea
where $\Qh^{-1}\Nh_\mu \Qh$ turns out to be a {\em block-diagonal matrix whose blocks
are upper-triangular matrices of dimension 1, 2, 3 or 8}, while $\Ph$ is a {\em symmetric}
permutation matrix.
By reversing the conjugation in (\ref{QNQ}) or (\ref{QhNhQh}), one obtains an explicit
expression for the given fusion matrix.
We will describe the relations (\ref{QNQ}) and (\ref{QhNhQh}) in detail in
Section~\ref{SecGeneral} and Section~\ref{SecGeneralHat}.
\subsection{Jordan webs}
As discussed in Appendix~\ref{AppJordan}, two commuting matrices need not share
a complete set of common generalized eigenvectors. However, we
will demonstrate that the two fundamental (or the two auxiliary) adjacency matrices $X$ and $Y$
($\Xh$ and $\Yh$) {\em do} have a common complete set of generalized eigenvectors.
These generalized eigenvectors
are organized as a web constructed by interlacing the Jordan chains of the two matrices.
We refer to such a web as a {\em Jordan web}. It consists of a number of
connected components or subwebs which we will characterize in the following.
\subsubsection{Fundamental fusion matrices}
With respect to $X$ or $Y$ separately, we only encounter Jordan chains of length 1 or 3.
As we will demonstrate in Section~\ref{SecFundWebs}, five different types of connected
Jordan webs arise in the description of the common generalized eigenvectors
$G_{\la,\la'}^{(\ell,\ell')}$ of $X$ and $Y$
\bea
XG_{\la,\la'}^{(0,\ell')}\;=\;\la G_{\la,\la'}^{(0,\ell')},\qquad
& XG_{\la,\la'}^{(1,\ell')}\;=\;\la G_{\la,\la'}^{(1,\ell')}+G_{\la,\la'}^{(0,\ell')},&\qquad
XG_{\la,\la'}^{(2,\ell')}\;=\;\la G_{\la,\la'}^{(2,\ell')}+G_{\la,\la'}^{(1,\ell')}\nn
YG_{\la,\la'}^{(\ell,0)}\;=\;\la' G_{\la,\la'}^{(\ell,0)},\qquad
& YG_{\la,\la'}^{(\ell,1)}\;=\;\la' G_{\la,\la'}^{(\ell,1)}+G_{\la,\la'}^{(\ell,0)},&\qquad
YG_{\la,\la'}^{(\ell,2)}\;=\;\la' G_{\la,\la'}^{(\ell,2)}+G_{\la,\la'}^{(\ell,1)}
\label{XYG}
\eea
These Jordan webs are
\be
\mbox{
\begin{picture}(100,100)(40,0)
\unitlength=0.75cm
\thinlines
\put(-0.2,2){$W_{\la,\la'}^{(1,1)}:$}
\put(2,2){$G_{\la,\la'}^{(0,0)}$}
\end{picture}
}
\hspace{.4cm}
\mbox{
\begin{picture}(100,100)(40,0)
\unitlength=0.75cm
\thinlines
\put(-0.2,2){$W_{\la,\la'}^{(1,3)}:$}
\put(2,0){$G_{\la,\la'}^{(0,0)}$}
\put(2,2){$G_{\la,\la'}^{(0,1)}$}
\put(2,4){$G_{\la,\la'}^{(0,2)}$}
\put(2.4,1.6){\vector(0,-1){0.8}}
\put(2.4,3.6){\vector(0,-1){0.8}}
\end{picture}
}
\hspace{.4cm}
\mbox{
\begin{picture}(100,100)(40,0)
\unitlength=0.75cm
\thinlines
\put(-0.2,2){$W_{\la,\la'}^{(3,1)}:$}
\put(2,2){$G_{\la,\la'}^{(0,0)}$}
\put(5,2){$G_{\la,\la'}^{(1,0)}$}
\put(8,2){$G_{\la,\la'}^{(2,0)}$}
\put(4.5,2.1){\vector(-1,0){1}}
\put(7.5,2.1){\vector(-1,0){1}}
\end{picture}
}
\label{web13}
\ee
and
\be
\mbox{
\begin{picture}(100,100)(60,0)
\unitlength=0.75cm
\thinlines
\put(-0.2,2){$W_{\la,\la'}^{(3,3)^\dagger}:$}
\put(2,0){$G_{\la,\la'}^{(0,0)}$}
\put(5,0){$G_{\la,\la'}^{(1,0)}$}
\put(8,0){$G_{\la,\la'}^{(2,0)}$}
\put(2,2){$G_{\la,\la'}^{(0,1)}$}
\put(2,4){$G_{\la,\la'}^{(0,2)}$}
\put(4.5,0.1){\vector(-1,0){1}}
\put(7.5,0.1){\vector(-1,0){1}}
\put(2.4,1.6){\vector(0,-1){0.8}}
\put(2.4,3.6){\vector(0,-1){0.8}}
\end{picture}
}
\hspace{3.5cm}
\mbox{
\begin{picture}(100,100)(40,0)
\unitlength=0.75cm
\thinlines
\put(-0.2,2){$W_{\la,\la'}^{(3,3)}:$}
\put(2,0){$G_{\la,\la'}^{(0,0)}$}
\put(5,0){$G_{\la,\la'}^{(1,0)}$}
\put(8,0){$G_{\la,\la'}^{(2,0)}$}
\put(2,2){$G_{\la,\la'}^{(0,1)}$}
\put(5,2){$G_{\la,\la'}^{(1,1)}$}
\put(8,2){$G_{\la,\la'}^{(2,1)}$}
\put(2,4){$G_{\la,\la'}^{(0,2)}$}
\put(5,4){$G_{\la,\la'}^{(1,2)}$}
\put(8,4){$G_{\la,\la'}^{(2,2)}$}
\put(4.5,0.1){\vector(-1,0){1}}
\put(7.5,0.1){\vector(-1,0){1}}
\put(4.5,2.1){\vector(-1,0){1}}
\put(7.5,2.1){\vector(-1,0){1}}
\put(4.5,4.1){\vector(-1,0){1}}
\put(7.5,4.1){\vector(-1,0){1}}
\put(2.4,1.6){\vector(0,-1){0.8}}
\put(5.4,1.6){\vector(0,-1){0.8}}
\put(8.4,1.6){\vector(0,-1){0.8}}
\put(2.4,3.6){\vector(0,-1){0.8}}
\put(5.4,3.6){\vector(0,-1){0.8}}
\put(8.4,3.6){\vector(0,-1){0.8}}
\end{picture}
}
\label{web59}
\ee
\\[-.35cm]
where a horizontal arrow from $G_{\la,\la'}$ to $G'_{\la,\la'}$ indicates that
$XG_{\la,\la'}=\la G_{\la,\la'}+G'_{\la,\la'}$, while a vertical arrow from
$G_{\la,\la'}$ to $G'_{\la,\la'}$ indicates that $YG_{\la,\la'}=\la' G_{\la,\la'}+G'_{\la,\la'}$.
We note that these five connected Jordan webs are all subwebs of $W_{\la,\la'}^{(3,3)}$.
A web of the type $W_{\la,\la'}^{(1,1)}$ is merely a common eigenvector of
$X$ and $Y$ not appearing in any non-trivial Jordan chain.
To describe the matrix realizations of the restrictions of $X$ and $Y$ to these connected
Jordan webs, we introduce an ordered basis $B_{\la,\la'}^{(\ell,\ell')}$ of generalized eigenvectors
associated to $W_{\la,\la'}^{(\ell,\ell')}$. As always, we favour $Y$ and thus introduce
\bea
&B_{\la,\la'}^{(1,1)}\;=\;\big\{G_{\la,\la'}^{(0,0)}\big\},\qquad
B_{\la,\la'}^{(1,3)}\;=\;\big\{G_{\la,\la'}^{(0,0)},G_{\la,\la'}^{(0,1)},G_{\la,\la'}^{(0,2)}\big\},\qquad
B_{\la,\la'}^{(3,1)}\;=\;\big\{G_{\la,\la'}^{(0,0)},G_{\la,\la'}^{(1,0)},G_{\la,\la'}^{(2,0)}\big\}\nn
&B_{\la,\la'}^{(3,3)^\dagger}\;=\;\big\{G_{\la,\la'}^{(0,0)},G_{\la,\la'}^{(0,1)},G_{\la,\la'}^{(0,2)},
G_{\la,\la'}^{(1,0)},G_{\la,\la'}^{(2,0)}\big\}\nn
&B_{\la,\la'}^{(3,3)}\;=\;\big\{G_{\la,\la'}^{(0,0)},G_{\la,\la'}^{(0,1)},G_{\la,\la'}^{(0,2)},
G_{\la,\la'}^{(1,0)},G_{\la,\la'}^{(1,1)},G_{\la,\la'}^{(1,2)},
G_{\la,\la'}^{(2,0)},G_{\la,\la'}^{(2,1)},G_{\la,\la'}^{(2,2)}\big\}
\label{B}
\eea
The corresponding matrix realizations are denoted by $X_{\la,\la'}^{(\ell,\ell')}$
and $Y_{\la,\la'}^{(\ell,\ell')}$ and are given by
\bea
&X_{\la,\la'}^{(1,1)}\;=\;\la I_{1\times1},\qquad
X_{\la,\la'}^{(1,3)}\;=\;\la I_{3\times3},\qquad
X_{\la,\la'}^{(3,1)}\;=\; \Jc_{\la,3}\nn
&X_{\la,\la'}^{(3,3)^\dagger}\;=\;\begin{pmatrix}
\la&0&0&1&0\\ &\la&0&0&0\\ &&\la&0&0\\ &&&\la&1\\ &&&&\la\end{pmatrix},\qquad
X_{\la,\la'}^{(3,3)}
\;=\;\begin{pmatrix} \la I&I&0\\ 0&\la I&I\\ 0&0&\la I \end{pmatrix}
\label{Xweb}
\eea
and
\bea
&Y_{\la,\la'}^{(1,1)}\;=\;\la' I_{1\times1},\qquad
Y_{\la,\la'}^{(1,3)}\;=\; \Jc_{\la',3},\qquad
Y_{\la,\la'}^{(3,1)}\;=\;\la' I_{3\times3} \nn
&Y_{\la,\la'}^{(3,3)^\dagger}\;=\;\mathrm{diag}\big(\Jc_{\la',3},\la',\la'\big),\qquad
Y_{\la,\la'}^{(3,3)} \;=\;\mathrm{diag}\big(\Jc_{\la',3},\Jc_{\la',3},\Jc_{\la',3}\big)
\label{Yweb}
\eea
In (\ref{Xweb}), the nine-dimensional matrix $X_{\la,\la'}^{(3,3)}$ is written as a three-dimensional
matrix whose entries are three-dimensional matrices.
\subsubsection{Auxiliary fusion matrices}
\label{SecAuxFusMat}
With respect to $\Xh$ or $\Yh$ separately, we only encounter Jordan chains of length 1, 2 or 3.
As we will demonstrate in Section~\ref{SecAuxWebs}, six different types of
connected Jordan webs
arise in the description of the common generalized eigenvectors of $\Xh$ and $\Yh$
\bea
\Xh\Gh_{\la,\la'}^{(0,\ell')}\;=\;\la \Gh_{\la,\la'}^{(0,\ell')},\qquad
& \Xh\Gh_{\la,\la'}^{(1,\ell')}\;=\;\la \Gh_{\la,\la'}^{(1,\ell')}+\Gh_{\la,\la'}^{(0,\ell')},&\qquad
\Xh\Gh_{\la,\la'}^{(2,\ell')}\;=\;\la \Gh_{\la,\la'}^{(2,\ell')}+\Gh_{\la,\la'}^{(1,\ell')}\nn
\Yh\Gh_{\la,\la'}^{(\ell,0)}\;=\;\la' \Gh_{\la,\la'}^{(\ell,0)},\qquad
& \Yh\Gh_{\la,\la'}^{(\ell,1)}\;=\;\la' \Gh_{\la,\la'}^{(\ell,1)}+\Gh_{\la,\la'}^{(\ell,0)},&\qquad
\Yh\Gh_{\la,\la'}^{(\ell,2)}\;=\;\la' \Gh_{\la,\la'}^{(\ell,2)}+\Gh_{\la,\la'}^{(\ell,1)}
\label{XYGh}
\eea
Three of these Jordan webs are inherited from $X$ and $Y$ as
$W_{\la,\la'}^{(1,1)}\to\Wh_{\la,\la'}^{(1,1)}$,
$W_{\la,\la'}^{(3,1)}\to\Wh_{\la,\la'}^{(3,1)}$ and
$W_{\la,\la'}^{(1,3)}\to\Wh_{\la,\la'}^{(1,3)}$, that is,
\be
\mbox{
\begin{picture}(100,100)(40,0)
\unitlength=0.75cm
\thinlines
\put(-0.2,2){$\Wh_{\la,\la'}^{(1,1)}:$}
\put(2,2){$\Gh_{\la,\la'}^{(0,0)}$}
\end{picture}
}
\hspace{.4cm}
\mbox{
\begin{picture}(100,100)(40,0)
\unitlength=0.75cm
\thinlines
\put(-0.2,2){$\Wh_{\la,\la'}^{(1,3)}:$}
\put(2,0){$\Gh_{\la,\la'}^{(0,0)}$}
\put(2,2){$\Gh_{\la,\la'}^{(0,1)}$}
\put(2,4){$\Gh_{\la,\la'}^{(0,2)}$}
\put(2.4,1.6){\vector(0,-1){0.8}}
\put(2.4,3.6){\vector(0,-1){0.8}}
\end{picture}
}
\hspace{.4cm}
\mbox{
\begin{picture}(100,100)(40,0)
\unitlength=0.75cm
\thinlines
\put(-0.2,2){$\Wh_{\la,\la'}^{(3,1)}:$}
\put(2,2){$\Gh_{\la,\la'}^{(0,0)}$}
\put(5,2){$\Gh_{\la,\la'}^{(1,0)}$}
\put(8,2){$\Gh_{\la,\la'}^{(2,0)}$}
\put(4.5,2.1){\vector(-1,0){1}}
\put(7.5,2.1){\vector(-1,0){1}}
\end{picture}
}
\label{webh13}
\ee
The Jordan web $W_{\la,\la'}^{(3,3)^\dagger}$\!, on the other hand, breaks down as only
the quotient $W_{\la,\la'}^{(3,3)^\dagger}\!/G_{\la,\la'}^{(0,0)}$ survives the reduction to
$\Xh$ and $\Yh$ in the sense that
$W_{\la,\la'}^{(3,3)^\dagger}\to\Wh_{\la,\la'}^{(2,1)}\cup\Wh_{\la,\la'}^{(1,2)}$, where
\be
\mbox{
\begin{picture}(100,60)(10,-5)
\unitlength=0.75cm
\thinlines
\put(-0.2,1.15){$\Wh_{\la,\la'}^{(1,2)}:$}
\put(2,0){$\Gh_{\la,\la'}^{(0,0)}$}
\put(2,2){$\Gh_{\la,\la'}^{(0,1)}$}
\put(2.4,1.6){\vector(0,-1){0.8}}
\end{picture}
}
\hspace{.6cm}
\mbox{
\begin{picture}(100,60)(10,-5)
\unitlength=0.75cm
\thinlines
\put(-0.2,1.15){$\Wh_{\la,\la'}^{(2,1)}:$}
\put(2,1.15){$\Gh_{\la,\la'}^{(0,0)}$}
\put(5,1.15){$\Gh_{\la,\la'}^{(1,0)}$}
\put(4.5,1.25){\vector(-1,0){1}}
\end{picture}
}
\label{webh1221}
\ee
Finally, only the quotient $W_{\la,\la'}^{(3,3)}/G_{\la,\la'}^{(0,0)}$ survives
the reduction of the Jordan web $W_{\la,\la'}^{(3,3)}$ to $\Xh$ and $\Yh$.
This eight-dimensional connected Jordan web $\Wh_{\la,\la'}^{(3,3)}$ is given by
\be
\mbox{
\begin{picture}(100,100)(40,0)
\unitlength=0.75cm
\thinlines
\put(-0.2,2){$\Wh_{\la,\la'}^{(3,3)}:$}
\put(5,0){$\Gh_{\la,\la'}^{(1,0)}$}
\put(8,0){$\Gh_{\la,\la'}^{(2,0)}$}
\put(2,2){$\Gh_{\la,\la'}^{(0,1)}$}
\put(5,2){$\Gh_{\la,\la'}^{(1,1)}$}
\put(8,2){$\Gh_{\la,\la'}^{(2,1)}$}
\put(2,4){$\Gh_{\la,\la'}^{(0,2)}$}
\put(5,4){$\Gh_{\la,\la'}^{(1,2)}$}
\put(8,4){$\Gh_{\la,\la'}^{(2,2)}$}
\put(7.5,0.1){\vector(-1,0){1}}
\put(4.5,2.1){\vector(-1,0){1}}
\put(7.5,2.1){\vector(-1,0){1}}
\put(4.5,4.1){\vector(-1,0){1}}
\put(7.5,4.1){\vector(-1,0){1}}
\put(5.4,1.6){\vector(0,-1){0.8}}
\put(8.4,1.6){\vector(0,-1){0.8}}
\put(2.4,3.6){\vector(0,-1){0.8}}
\put(5.4,3.6){\vector(0,-1){0.8}}
\put(8.4,3.6){\vector(0,-1){0.8}}
\end{picture}
}
\label{webh8}
\ee
where, with reference to (\ref{XYGh}), $\Gh_{\la,\la'}^{(0,0)}\equiv0$.
To describe the matrix realizations of the restrictions of $\Xh$ and $\Yh$ to these connected
Jordan webs, we introduce the $\Yh$-favouring ordered bases $\Bh_{\la,\la'}^{(\ell,\ell')}$
of generalized eigenvectors associated to $\Wh_{\la,\la'}^{(\ell,\ell')}$ by
\bea
&\Bh_{\la,\la'}^{(1,1)}\;=\;\big\{\Gh_{\la,\la'}^{(0,0)}\big\},\qquad
\Bh_{\la,\la'}^{(1,3)}\;=\;\big\{\Gh_{\la,\la'}^{(0,0)},\Gh_{\la,\la'}^{(0,1)},
\Gh_{\la,\la'}^{(0,2)}\big\},\qquad
\Bh_{\la,\la'}^{(3,1)}\;=\;\big\{\Gh_{\la,\la'}^{(0,0)},\Gh_{\la,\la'}^{(1,0)},\Gh_{\la,\la'}^{(2,0)}\big\}\nn
&\Bh_{\la,\la'}^{(1,2)}\;=\;\big\{\Gh_{\la,\la'}^{(0,0)},\Gh_{\la,\la'}^{(0,1)}\big\},\qquad
\Bh_{\la,\la'}^{(2,1)}\;=\;\big\{\Gh_{\la,\la'}^{(0,0)},\Gh_{\la,\la'}^{(1,0)}\big\}\nn
&\Bh_{\la,\la'}^{(3,3)}\;=\;\big\{\Gh_{\la,\la'}^{(0,1)},\Gh_{\la,\la'}^{(0,2)},
\Gh_{\la,\la'}^{(1,0)},\Gh_{\la,\la'}^{(1,1)},\Gh_{\la,\la'}^{(1,2)},
\Gh_{\la,\la'}^{(2,0)},\Gh_{\la,\la'}^{(2,1)},\Gh_{\la,\la'}^{(2,2)}\big\}
\label{Bh}
\eea
The corresponding matrix realizations are denoted by $\Xh_{\la,\la'}^{(\ell,\ell')}$
and $\Yh_{\la,\la'}^{(\ell,\ell')}$ and are given by
\bea
&\Xh_{\la,\la'}^{(1,1)}\;=\;\la I_{1\times1},\qquad
\Xh_{\la,\la'}^{(1,3)}\;=\;\la I_{3\times3},\qquad
\Xh_{\la,\la'}^{(3,1)}\;=\; \Jc_{\la,3}\nn
&\Xh_{\la,\la'}^{(1,2)}\;=\;\la I_{2\times2},\qquad
\Xh_{\la,\la'}^{(2,1)}\;=\; \Jc_{\la,2},\qquad
\Xh_{\la,\la'}^{(3,3)}
\;=\;\left(\!\!
\begin{array}{cc|ccc|ccc}
\la&0&0&1&0&0&0&0 \\
&\la&0&0&1&0&0&0 \\
\hline
&&\la&0&0&1&0&0 \\
&&&\la&0&0&1&0 \\
&&&&\la&0&0&1 \\
\hline
&&&&&\la&0&0 \\
&&&&&&\la&0 \\
&&&&&&&\la
\end{array}
\!\!\right)
\label{Xhweb}
\eea
and
\bea
&\Yh_{\la,\la'}^{(1,1)}\;=\;\la' I_{1\times1},\qquad
\Yh_{\la,\la'}^{(1,3)}\;=\; \Jc_{\la',3},\qquad
\Yh_{\la,\la'}^{(3,1)}\;=\;\la' I_{3\times3}\nn
&\Yh_{\la,\la'}^{(1,2)}\;=\; \Jc_{\la',2},\qquad
\Yh_{\la,\la'}^{(2,1)}\;=\;\la' I_{2\times2},\qquad
\Yh_{\la,\la'}^{(3,3)} \;=\;\mathrm{diag}\big(\Jc_{\la',2},\Jc_{\la',3},\Jc_{\la',3}\big)
\label{Yhweb}
\eea
The eight-dimensional matrix $\Xh_{\la,\la'}^{(3,3)}$ in (\ref{Xhweb}) is obtained from the
nine-dimensional matrix $X_{\la,\la'}^{(3,3)}$ in (\ref{Xweb}) by elimination of the first row
and column. By similar eliminations, the eight-dimensional matrix $\Yh_{\la,\la'}^{(3,3)}$
in (\ref{Yhweb}) is obtained from the nine-dimensional matrix $Y_{\la,\la'}^{(3,3)}$ in (\ref{Yweb}),
while the four-dimensional matrices $\mathrm{diag}(\Xh_{\la,\la'}^{(1,2)},\Xh_{\la,\la'}^{(2,1)})$
and $\mathrm{diag}(\Yh_{\la,\la'}^{(1,2)},\Yh_{\la,\la'}^{(2,1)})$
follow from the five-dimensional matrices $X_{\la,\la'}^{(3,3)^\dagger}$
and $Y_{\la,\la'}^{(3,3)^\dagger}$\!, respectively.
\subsection{Fundamental fusion algebra}
In the following, we write
\be
\beta_i\;=\;2\cos\frac{i\pi}{p},\quad i\in\mathbb{Z}_{0,p},\qquad\quad
\beta_j'\;=\;2\cos\frac{j\pi}{p'},\quad j\in\mathbb{Z}_{0,p'}
\label{betabeta}
\ee
We also recall our label conventions $a\in\mathbb{Z}_{1,p-1}$ and
$b\in\mathbb{Z}_{1,p'-1}$ introduced in (\ref{kkabrs}).
\subsubsection{Fundamental fusion matrices}
\label{SecFundWebs}
Due to the block-diagonal structure (\ref{Ydiag}) of the fundamental fusion matrix $Y$,
its spectral decomposition
follows readily from the spectral decompositions of $\Tc_{p'}$ and $\Ec_{p'}$ discussed
in Section~\ref{SecTad} and Section~\ref{SecEye}, respectively. The Jordan canonical form of
$Y$ thus consists of $2p-1$ rank-1 blocks of eigenvalue $\beta'_j$ for every $j\in\{0,p'\}$,
$p$ rank-1 blocks of eigenvalue $\beta'_b$ for every $b\in\mathbb{Z}_{1,p'-1}$,
and $2p-1$ rank-3 blocks of eigenvalue $\beta'_b$ for every $b\in\mathbb{Z}_{1,p'-1}$.
Likewise, the Jordan canonical form of
$X$ consists of $2p'-1$ rank-1 blocks of eigenvalue $\beta_i$ for every $i\in\{0,p\}$,
$p'$ rank-1 blocks of eigenvalue $\beta_a$ for every $a\in\mathbb{Z}_{1,p-1}$,
and $2p'-1$ rank-3 blocks of eigenvalue $\beta_a$ for every $a\in\mathbb{Z}_{1,p-1}$.
To characterize the connected components of the Jordan web of the complete set of
common generalized eigenvectors of $X$ and $Y$, we choose to work in the
$Y$-favouring basis (\ref{Ybasis}). A generalized vector $G_{\la,\la'}^{(\ell,\ell')}$ can thus
be written as a $(2p-1)$-dimensional vector whose $p-1$ upper entries are
$(3p'-1)$-dimensional vectors of the type $T$ appearing in Section~\ref{SecTad},
while the $p$ lower entries are $(4p'-2)$-dimensional vectors of the type $E$
appearing in Section~\ref{SecEye}.
The connected subwebs of the type $W_{\la,\la'}^{(1,1)}$ are given by the following eigenvectors
\be
W_{\beta_i,\beta'_j}^{(1,1)}:\quad G_{\beta_i,\beta'_j}^{(0,0)}\;=\;\left(\!\!\!\begin{array}{c}
f_{1}(\beta_i)T_j\\ \vdots\\ f_{p-1}(\beta_i)T_j
\\[.15cm] \hline\\[-.3cm]
f_{p}(\beta_i)E_j\\ \vdots\\ f_{2p-1}(\beta_i)E_j
\end{array}\!\!\!\right),\qquad
i+j\ \mathrm{even},\quad i\in\{0,p\},\ j\in\{0,p'\}
\label{Wij11}
\ee
or
\be
W_{\beta_a,\beta'_j}^{(1,1)}:\quad G_{\beta_a,\beta'_j}^{(0,0)}\;=\;\left(\!\!\!\begin{array}{c}
f_{1}(\beta_a)T_j\\ \vdots\\ f_{p-1}(\beta_a)T_j
\\[.15cm] \hline\\[-.3cm]
f_{p}(\beta_a)E_j\\ \vdots\\ f_{2p-1}(\beta_a)E_j
\end{array}\!\!\!\right)\;=\;
\left(\!\!\!\begin{array}{c}
f_{1}(\beta_a)T_j\\ \vdots\\ f_{p-1}(\beta_a)T_j
\\[.15cm] \hline\\[-.3cm]
0\\ \vdots\\ 0
\end{array}\!\!\!\right),\qquad
a+j\ \mathrm{odd},\quad j\in\{0,p'\}
\ee
or
\be
W_{\beta_i,\beta'_b}^{(1,1)}:\quad G_{\beta_i,\beta'_b}^{(0,0)}\;=\;\left(\!\!\!\begin{array}{c}
f_{1}(\beta_i)T_b^{(0)}\\ \vdots\\ f_{p-1}(\beta_i)T_b^{(0)}
\\[.15cm] \hline\\[-.3cm]
f_{p}(\beta_i)E_b\\ \vdots\\ f_{2p-1}(\beta_i)E_b
\end{array}\!\!\!\right),\qquad
i+b\ \mathrm{odd},\quad i\in\{0,p\}
\ee
The connected subwebs of the type $W_{\la,\la'}^{(1,3)}$ consist of the following
generalized eigenvectors
\be
W_{\beta_i,\beta'_b}^{(1,3)}:\quad G_{\beta_i,\beta'_b}^{(0,\ell')}\;=\;\left(\!\!\!\begin{array}{c}
f_{1}(\beta_i)T_b^{(\ell')}\\ \vdots\\ f_{p-1}(\beta_i)T_b^{(\ell')}
\\[.15cm] \hline\\[-.3cm]
f_{p}(\beta_i)E_b^{(\ell')}\\ \vdots\\ f_{2p-1}(\beta_i)E_b^{(\ell')}
\end{array}\!\!\!\right),\qquad
i+b\ \mathrm{even},\quad i\in\{0,p\},\quad \ell'\in\mathbb{Z}_{0,2}
\ee
The connected subwebs of the type $W_{\la,\la'}^{(3,1)}$ consist of the following
generalized eigenvectors
\be
W_{\beta_a,\beta'_j}^{(3,1)}:\quad G_{\beta_a,\beta'_j}^{(\ell,0)}\;=\;\left(\!\!\!\begin{array}{c}
\tfrac{1}{\ell!}f_{1}^{(\ell)}(\beta_a)T_j\\ \vdots\\ \tfrac{1}{\ell!}f_{p-1}^{(\ell)}(\beta_a)T_j
\\[.15cm] \hline\\[-.3cm]
\tfrac{1}{\ell!}f_{p}^{(\ell)}(\beta_a)E_j\\ \vdots\\ \tfrac{1}{\ell!}f_{2p-1}^{(\ell)}(\beta_a)E_j
\end{array}\!\!\!\right),\qquad
a+j\ \mathrm{even},\quad j\in\{0,p'\},\quad \ell\in\mathbb{Z}_{0,2}
\ee
The connected subwebs of the type $W_{\la,\la'}^{(3,3)^\dagger}$ consist of the following
generalized eigenvectors
\be
W_{\beta_a,\beta'_b}^{(3,3)^\dagger}:\quad
G_{\beta_a,\beta'_b}^{(\ell,0)}\;=\;\left(\!\!\!\begin{array}{c}
\tfrac{1}{\ell!}f_{1}^{(\ell)}(\beta_a)T_b^{(0)}\\ \vdots\\ \tfrac{1}{\ell!}f_{p-1}^{(\ell)}(\beta_a)T_b^{(0)}
\\[.15cm] \hline\\[-.3cm]
\tfrac{1}{\ell!}f_{p}^{(\ell)}(\beta_a)E_b\\ \vdots\\ \tfrac{1}{\ell!}f_{2p-1}^{(\ell)}(\beta_a)E_b
\end{array}\!\!\!\right),\quad
G_{\beta_a,\beta'_b}^{(0,\ell')}\;=\;
\left(\!\!\!\begin{array}{c}
f_{1}(\beta_a)T_b^{(\ell')}\\ \vdots\\ f_{p-1}(\beta_a)T_b^{(\ell')}
\\[.15cm] \hline\\[-.3cm]
f_{p}(\beta_a)E_b^{(\ell')}\\ \vdots\\ f_{2p-1}(\beta_a)E_b^{(\ell')}
\end{array}\!\!\!\right),\qquad
a+b\ \mathrm{odd}, \quad \ell,\ell'\in\mathbb{Z}_{0,2}
\ee
where (\ref{fhlk}) ensures consistency of the two expressions for $G_{\beta_a,\beta'_b}^{(0,0)}$.
Finally, the connected subwebs of the type $W_{\la,\la'}^{(3,3)}$ consist of the following
generalized eigenvectors
\be
W_{\beta_a,\beta'_b}^{(3,3)}:\quad G_{\beta_a,\beta'_b}^{(\ell,\ell')}\;=\;\left(\!\!\!\begin{array}{c}
\tfrac{1}{\ell!}f_{1}^{(\ell)}(\beta_a)T_b^{(\ell')}\\ \vdots\\
\tfrac{1}{\ell!}f_{p-1}^{(\ell)}(\beta_a)T_b^{(\ell')}
\\[.15cm] \hline\\[-.3cm]
\tfrac{1}{\ell!}f_{p}^{(\ell)}(\beta_a)E_b^{(\ell')}\\ \vdots\\
\tfrac{1}{\ell!}f_{2p-1}^{(\ell)}(\beta_a)E_b^{(\ell')}
\end{array}\!\!\!\right),\qquad
a+b\ \mathrm{even},\quad \ell,\ell'\in\mathbb{Z}_{0,2}
\label{Wab33}
\ee
Using properties of the $T$ and $E$ vectors as generalized eigenvectors of
$\Tc_{p'}$ and $\Ec_{p'}$, together with (\ref{ItE}) and (\ref{CE}), in particular,
it is straightforward to prove that the vectors given in (\ref{Wij11}) through (\ref{Wab33})
indeed correspond to the Jordan webs (\ref{web13}) and (\ref{web59})
consistent with (\ref{XYG}).
We also note that the number $\Nc^{(\ell,\ell')}$ of connected Jordan webs of the type
$W^{(\ell,\ell')}$ is given by
\be
\Nc^{(1,1)}=\;p+p',\quad
\Nc^{(1,3)}=\;p'-1,\quad
\Nc^{(3,1)}=\;p-1,\quad
\Nc^{(3,3)^\dagger}=\;\Nc^{(3,3)}=\;\tfrac{1}{2}(p-1)(p'-1)
\ee
consistent with the total number (\ref{CardJFund}) of generalized eigenvectors.
In Appendix~\ref{AppJordanWeb},
we list the connected Jordan subwebs $W_{\beta_i,\beta'_j}^{(\ell,\ell')}$ with respect
to the labeling $i,j$ of the corresponding eigenvalues.
The similarity matrix $Q$ appearing in (\ref{QXQ}) is constructed by concatenating
the common generalized eigenvectors of $X$ and $Y$ according to any ordering of the
ordered bases (\ref{B}). The permutation matrix $P$
depends on this choice of ordering, and the degree of
convenience of such a choice depends on the intended application. Here we consider
a general ordering reflecting the partitioning
\be
\big\{B_{\la,\la'}^{(1,1)}\big\}\cup
\big\{B_{\la,\la'}^{(1,3)}\big\}\cup
\big\{B_{\la,\la'}^{(3,1)}\big\}\cup
\big\{B_{\la,\la'}^{(3,3)^\dagger}\big\}\cup
\big\{B_{\la,\la'}^{(3,3)}\big\}
\label{BBBBB}
\ee
such that every set (of generalized eigenvectors) of the type $B_{\la,\la'}^{(1,1)}$
comes before every set (of generalized eigenvectors) of the type $B_{\la,\la'}^{(1,3)}$,
and so on.
The Jordan canonical forms $J_X$ and $J_Y$ in (\ref{QXQ}) are then of the form
\bea
J_X\!\!&=&\!\!\mathrm{diag}\Big(
\underbrace{\la,\ldots}_{p+p'},\,
\underbrace{\la,\ldots}_{3p'-3},\,
\underbrace{\Jc_{\la,3},\ldots}_{p-1},\,
\underbrace{\mathrm{diag}\big(\Jc_{\la,3},\la,\la\big),\ldots}_{\tfrac{1}{2}(p-1)(p'-1)},\,
\underbrace{\mathrm{diag}\big(\Jc_{\la,3},\Jc_{\la,3},\Jc_{\la,3}\big),
\ldots}_{\tfrac{1}{2}(p-1)(p'-1)}\Big)
\label{JX}
\\
J_Y\!\!&=&\!\!\mathrm{diag}\Big(
\underbrace{\la',\ldots}_{p+p'},\,
\underbrace{\Jc_{\la',3},\ldots}_{p'-1},\,
\underbrace{\la',\ldots}_{3p-3},\,
\underbrace{\mathrm{diag}\big(\Jc_{\la',3},\la',\la'\big),\ldots}_{\tfrac{1}{2}(p-1)(p'-1)},\,
\underbrace{\mathrm{diag}\big(\Jc_{\la',3},\Jc_{\la',3},\Jc_{\la',3}\big),
\ldots}_{\tfrac{1}{2}(p-1)(p'-1)}\Big)
\eea
We stress that the eigenvalues $\la$ and $\la'$ vary in these expressions but are always of the
form (\ref{betabeta}).
The corresponding permutation matrix $P$ is a block-diagonal matrix whose blocks are
of dimension 1, 3, 3, 5 or 9, corresponding to the dimensions of the sets
$B_{\la,\la'}^{(\ell,\ell')}$. By a similarity transformation (\ref{QXQ}), these $P$-blocks convert the
blocks in $J_X$ into the corresponding upper-triangular matrices $X_{\la,\la'}^{(\ell,\ell')}$
in (\ref{Xweb}). The $P$-blocks of dimension 1 or 3 are identity matrices,
while the $P$-blocks of dimension 5 or 9 are the symmetric permutation matrices
\be
P_5\;=\;\begin{pmatrix} 1&0&0&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\\ 0&1&0&0&0\\ 0&0&1&0&0
\end{pmatrix},\qquad
P_9\;=\;\begin{pmatrix}
1&0&0&0&0&0&0&0&0\\
0&0&0&1&0&0&0&0&0\\
0&0&0&0&0&0&1&0&0\\
0&1&0&0&0&0&0&0&0\\
0&0&0&0&1&0&0&0&0\\
0&0&0&0&0&0&0&1&0\\
0&0&1&0&0&0&0&0&0\\
0&0&0&0&0&1&0&0&0\\
0&0&0&0&0&0&0&0&1 \end{pmatrix}
\label{P5P9}
\ee
where it is recalled that a symmetric permutation matrix equals its inverse.
The symmetric permutation matrix $P$ is thus given by
\be
P\;=\;\mathrm{diag}\Big(\underbrace{1,\ldots,1}_{4p+4p'-6},\,
\underbrace{P_5,\ldots,P_5}_{\tfrac{1}{2}(p-1)(p'-1)},\,
\underbrace{P_9,\ldots,P_9}_{\tfrac{1}{2}(p-1)(p'-1)}\Big)
\ee
As actions on the connected Jordan webs $W_{\la,\la'}^{(3,3)^\dagger}$ and $W_{\la,\la'}^{(3,3)}$,
these permutations reflect the vertices (generalized eigenvectors) with respect to the line from
south-west to north-east through $G_{\la,\la'}^{(0,0)}$.
$P_5$ and $P_9$ thus have one and three fix-points, respectively, in accord with the numbers of
units on their diagonals.
\subsubsection{General fusion matrices}
\label{SecGeneral}
Here we determine the upper-triangular block-diagonal
matrix $Q^{-1}N_\mu Q$ obtained from the general fusion matrix $N_\mu$ by
a similarity transformation with respect to $Q$ defined according to (\ref{BBBBB}).
{}From (\ref{QNQ}) and Section~\ref{SecFundWebs}, we have that
\bea
Q^{-1}N_\mu Q\!\!&=&\!\!\mathrm{diag}\Big(
\underbrace{g(X_{\la,\la'}^{(1,1)})h(Y_{\la,\la'}^{(1,1)}),\ldots}_{p+p'},\,
\underbrace{g(X_{\la,\la'}^{(1,3)})h(Y_{\la,\la'}^{(1,3)}),\ldots}_{p'-1},\,
\underbrace{g(X_{\la,\la'}^{(3,1)})h(Y_{\la,\la'}^{(3,1)}),\ldots}_{p-1},\nn
&&\qquad\qquad\qquad
\underbrace{g(X_{\la,\la'}^{(3,3)^\dagger})h(Y_{\la,\la'}^{(3,3)^\dagger}),
\ldots}_{\tfrac{1}{2}(p-1)(p'-1)},\,
\underbrace{g(X_{\la,\la'}^{(3,3)})h(Y_{\la,\la'}^{(3,3)}),\ldots}_{\tfrac{1}{2}(p-1)(p'-1)}\Big)
\eea
where, for simplicity, $g(z)=\mathrm{pol}_{\mu}^{(x)}(z)$ and
$h(z)=\mathrm{pol}_{\mu}^{(y)}(z)$.
In a given block $g(X_{\la,\la'}^{(\ell,\ell')})h(Y_{\la,\la'}^{(\ell,\ell')})$, the pairs of labels $\la,\la'$
(eigenvalues (\ref{betabeta}) of $X$ and $Y$) of $X_{\la,\la'}^{(\ell,\ell')}$ and
$Y_{\la,\la'}^{(\ell,\ell')}$ are the same, while they generally vary from block to block.
For the five types of blocks, we have
\bea
g(X_{\la,\la'}^{(1,1)})h(Y_{\la,\la'}^{(1,1)})\!\!&=&\!\! g(\la)h(\la')\nn
g(X_{\la,\la'}^{(1,3)})h(Y_{\la,\la'}^{(1,3)})\!\!&=&\!\! g(\la)h(\Jc_{\la',3}),\qquad
g(X_{\la,\la'}^{(3,1)})h(Y_{\la,\la'}^{(3,1)})\;=\;g(\Jc_{\la,3})h(\la')\nn
g(X_{\la,\la'}^{(3,3)^\dagger})h(Y_{\la,\la'}^{(3,3)^\dagger})\!\!&=&\!\!\begin{pmatrix}
g(\la)h(\la')&g(\la)h'(\la')&\tfrac{1}{2}g(\la)h''(\la')&g'(\la)h(\la')&\tfrac{1}{2}g''(\la)h(\la') \\
0&g(\la)h(\la')&g(\la)h'(\la')&0&0 \\
0&0&g(\la)h(\la')&0&0 \\
0&0&0&g(\la)h(\la')&g'(\la)h(\la') \\
0&0&0&0&g(\la)h(\la')
\end{pmatrix}\nn
g(X_{\la,\la'}^{(3,3)})h(Y_{\la,\la'}^{(3,3)})\!\!&=&\!\! g(\Jc_{\la,3})\times h(\Jc_{\la',3})
\label{gh}
\eea
where $g(\Jc_{\la,3})\times h(\Jc_{\la',3})$ denotes the nine-dimensional Kronecker product of
the two three-dimensional matrices $g(\Jc_{\la,3})$ and $h(\Jc_{\la',3})$.
It is recalled that, for a function $f$ expandable as a power series in its argument, we have
\be
f(\Jc_{\la,2})
\;=\;\begin{pmatrix} f(\la)&f'(\la)
\\ 0&f(\la)\end{pmatrix} ,\qquad\quad
f(\Jc_{\la,3})
\;=\;\begin{pmatrix} f(\la)&f'(\la)&\tfrac{1}{2}f''(\la)
\\ 0&f(\la)&f'(\la) \\ 0&0&f(\la) \end{pmatrix}
\ee
whose ranks depend on $f'(\la)$ and $f''(\la)$.
The first of these matrix expressions will be relevant in (\ref{ghhh}) below.
This completes the description of the upper-triangular block-diagonal matrix
$Q^{-1}N_\mu Q$.
\subsubsection{${\cal W}$-extended critical percolation ${\cal WLM}(2,3)$}
In the case of ${\cal WLM}(2,3)$, the eigenvalues of $X$ and $Y$ are
\be
\beta_i\;=\;2\cos\frac{i\pi}{2},\quad i\in\{0,1,2\},\qquad\qquad
\beta'_j\;=\;2\cos\frac{j\pi}{3},\quad j\in\{0,1,2,3\}
\label{beta23}
\ee
respectively. As displayed in Figure~\ref{web23}, the connected components
$W_{\beta_i,\beta'_j}^{(\ell,\ell')}$ of the Jordan web associated to the fundamental
fusion algebra are neatly organized with respect to the labels $i$ and $j$.
\psset{unit=1cm}
\begin{figure}
$$
\renewcommand{\arraystretch}{1.5}
\begin{array}{c|cccc}
\mbox{}_i\diagdown\mbox{}^j&0&1&2&3\\[4pt]
\hline
\rule{0pt}{16pt}
0&W_{2,2}^{(1,1)}&W_{2,1}^{(1,1)}&W_{2,-1}^{(1,3)}&$\O$
\\[4pt]
1&W_{0,2}^{(1,1)}&W_{0,1}^{(3,3)}&W_{0,-1}^{(3,3)^\dagger}&W_{0,-2}^{(3,1)}
\\[4pt]
2&W_{-2,2}^{(1,1)}&W_{-2,1}^{(1,1)}&W_{-2,-1}^{(1,3)}&$\O$
\end{array}
$$
\caption{The connected components $W_{\beta_i,\beta'_j}^{(\ell,\ell')}$ of the Jordan web
associated to the fundamental fusion algebra of ${\cal W}$-extended critical percolation
${\cal WLM}(2,3)$. The two \O's
indicate that there are no common generalized eigenvectors corresponding to the pair
$(\beta_0,\beta'_3)=(2,-2)$ or to the pair $(\beta_2,\beta'_3)=(-2,-2)$ of eigenvalues of the
fundamental fusion matrices $X$ and $Y$.}
\label{web23}
\end{figure}
An example of an ordering of the common generalized eigenvectors of $X$ and $Y$
respecting (\ref{BBBBB}) is
\bea
&\!\!G_{2,2}^{(0,0)}; G_{0,2}^{(0,0)}; G_{-2,2}^{(0,0)}; G_{2,1}^{(0,0)}; G_{-2,1}^{(0,0)};
G_{2,-1}^{(0,0)}, G_{2,-1}^{(0,1)}, G_{2,-1}^{(0,2)};
G_{-2,-1}^{(0,0)}, G_{-2,-1}^{(0,1)}, G_{-2,-1}^{(0,2)};
G_{0,-2}^{(0,0)}, G_{0,-2}^{(1,0)}, G_{0,-2}^{(2,0)};\nn
&\!\!G_{0,-1}^{(0,0)}, G_{0,-1}^{(0,1)}, G_{0,-1}^{(0,2)}, G_{0,-1}^{(1,0)}, G_{0,-1}^{(2,0)};
G_{0,1}^{(0,0)}, G_{0,1}^{(0,1)}, G_{0,1}^{(0,2)}, G_{0,1}^{(1,0)}, G_{0,1}^{(1,1)}, G_{0,1}^{(1,2)},
G_{0,1}^{(2,0)}, G_{0,1}^{(2,1)}, G_{0,1}^{(2,2)}
\eea
We define the similarity matrix $Q$ by concatenating these vectors in the order given.
Modulo a similarity transformation, $Q$ converts $X$ and $Y$ into the Jordan canonical forms
\bea
&J_X\;=\;P^{-1}Q^{-1}XQP\;=\;\mathrm{diag}\big(
2,0,-2,2,-2,2,2,2,-2,-2,-2,\Jc_{0,3},\Jc_{0,3},0,0,\Jc_{0,3},\Jc_{0,3},\Jc_{0,3}\big) \nn
&J_Y\;=\;Q^{-1}YQ\;=\;\mathrm{diag}\big(
2,2,2,1,1,\Jc_{-1,3},\Jc_{-1,3},-2,-2,-2,\Jc_{-1,3},-1,-1,\Jc_{1,3},\Jc_{1,3},\Jc_{1,3}\big)
\eea
where $P$ is the symmetric permutation matrix
\be
P\;=\;\mathrm{diag}\big(I_{14\times14},P_5,P_9\big)
\ee
The fusion matrix $N_\mu$ associated to the general module $\mu\in\Ic_f$
is polynomial in $X$ and $Y$
\bea
&N_{1,1}\;=\;I,\quad N_{1,2}\;=\;Y,\quad N_{1,3}\;=\;Y^2-I,\quad
N_{2,1}\;=\;X,\quad N_{2,2}\;=\;XY,\quad N_{2,3}\;=\;X(Y^2-I)\nn
&N_{1,6}\;=\;\tfrac{1}{2}Y(Y^2-I)(Y^2-3I),\quad
N_{2,6}\;=\;\tfrac{1}{2}XY(Y^2-I)(Y^2-3I)\nn
&N_{4,1}\;=\;\tfrac{1}{2}X(X^2-2I),\quad
N_{4,2}\;=\;\tfrac{1}{2}X(X^2-2I)Y \nn
&N_{1,3}^{0,1}\;=\;Y(Y^2-I),\quad
N_{1,3}^{0,2}\;=\;(Y^2-I)(Y^2-2I)\nn
&N_{1,6}^{0,1}\;=\;\tfrac{1}{2}Y^2(Y^2-I)(Y^2-3I),\quad
N_{1,6}^{0,2}\;=\;\tfrac{1}{2}Y(Y^2-I)(Y^2-2I)(Y^2-3I)\nn
&N_{2,3}^{0,1}\;=\;XY(Y^2-I),\quad
N_{2,3}^{0,2}\;=\;X(Y^2-I)(Y^2-2I)\nn
&N_{2,6}^{0,1}\;=\;\tfrac{1}{2}XY^2(Y^2-I)(Y^2-3I),\quad
N_{2,6}^{0,2}\;=\;\tfrac{1}{2}XY(Y^2-I)(Y^2-2I)(Y^2-3I)\nn
&N_{2,1}^{1,0}\;=\;X^2,\quad
N_{2,2}^{1,0}\;=\;X^2Y,\quad
N_{2,3}^{1,0}\;=\;X^2(Y^2-I) \nn
&N_{4,1}^{1,0}\;=\;\tfrac{1}{2}X^2(X^2-2I),\quad
N_{4,2}^{1,0}\;=\;\tfrac{1}{2}X^2(X^2-2I)Y,\quad
N_{4,3}^{1,0}\;=\;\tfrac{1}{2}X^2(X^2-2I)(Y^2-I)\nn
&N_{2,3}^{1,1}\;=\;X^2Y(Y^2-I),\quad
N_{2,3}^{1,2}\;=\;X^2(Y^2-I)(Y^2-2I)\nn
&N_{4,3}^{1,1}\;=\;\tfrac{1}{2}X^2(X^2-2I)Y(Y^2-I),\quad
N_{4,3}^{1,2}\;=\;\tfrac{1}{2}X^2(X^2-2I)(Y^2-I)(Y^2-2I)
\label{N23}
\eea
where we have introduced the abbreviations $N_{r,s}=N_{\ketw{r,s}}$,
$N_{r,s}=N_{\Wc(\D_{r,s})}$ and
$N_{r,s}^{\al,\beta}=N_{\ketw{\R_{r,s}^{\al,\beta}}}$\!.
The similarity transformation of $N_\mu$ is the block-diagonal matrix $Q^{-1}N_\mu Q$ whose
blocks are upper-triangular matrices. As illustrations of such block-diagonal matrices,
we here consider
\bea
Q^{-1}N_{4,2}Q\!\!&=&\!\!\mathrm{diag}\Big(
4,0,-4,2,-2,\begin{pmatrix} -2&2&0\\ 0&-2&2\\ 0&0&-2\end{pmatrix},
\begin{pmatrix} 2&-2&0\\ 0&2&-2\\ 0&0&2\end{pmatrix},
\begin{pmatrix} 0&2&0\\ 0&0&2\\ 0&0&0\end{pmatrix},\nn
&&\hspace{3cm} \begin{pmatrix} 0&0&0&1&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\
0&0&0&0&1\\ 0&0&0&0&0\end{pmatrix},
\begin{pmatrix} 0&-1&0\\ 0&0&-1\\ 0&0&0\end{pmatrix}\times
\begin{pmatrix} 1&1&0\\ 0&1&1\\ 0&0&1\end{pmatrix}\Big)\nn
Q^{-1}N_{2,3}^{1,1}Q\!\!&=&\!\!\mathrm{diag}\Big(
24,0,24,0,0,\begin{pmatrix} 0&8&-12\\ 0&0&8\\ 0&0&0\end{pmatrix},
\begin{pmatrix} 0&8&-12\\ 0&0&8\\ 0&0&0\end{pmatrix},
\begin{pmatrix} 0&0&-6\\ 0&0&0\\ 0&0&0\end{pmatrix},\nn
&&\hspace{5cm}0,0,0,0,0,
\begin{pmatrix} 0&0&1\\ 0&0&0\\ 0&0&0\end{pmatrix}\times
\begin{pmatrix} 0&2&3\\ 0&0&2\\ 0&0&0\end{pmatrix}\Big)
\eea
\subsection{Fusion algebra associated with boundary conditions}
\subsubsection{Auxiliary fusion matrices}
\label{SecAuxWebs}
Due to the the block-diagonal structure (\ref{Yhdiag}) of the auxiliary fusion matrix
$\Yh$, its spectral decomposition
follows readily from the spectral decompositions of $\Cc_{p'}$ and $\Ec_{p'}$ discussed
in Section~\ref{SecCycle} and \ref{SecEye}, respectively. The Jordan canonical form of
$\Yh$ thus consists of $2p-1$ rank-1 blocks of eigenvalue $\beta'_j$ for every $j\in\{0,p'\}$,
$p$ rank-1 blocks of eigenvalue $\beta'_b$ for every $b\in\mathbb{Z}_{1,p'-1}$,
$p-1$ rank-2 blocks of eigenvalue $\beta'_b$ for every $b\in\mathbb{Z}_{1,p'-1}$,
and $p$ rank-3 blocks of eigenvalue $\beta'_b$ for every $b\in\mathbb{Z}_{1,p'-1}$.
Likewise, the Jordan canonical form of
$\Xh$ consists of $2p'-1$ rank-1 blocks of eigenvalue $\beta_i$ for every $i\in\{0,p\}$,
$p'$ rank-1 blocks of eigenvalue $\beta_a$ for every $a\in\mathbb{Z}_{1,p-1}$,
$p'-1$ rank-2 blocks of eigenvalue $\beta_a$ for every $a\in\mathbb{Z}_{1,p-1}$,
and $p'$ rank-3 blocks of eigenvalue $\beta_a$ for every $a\in\mathbb{Z}_{1,p-1}$.
To characterize the connected components of the Jordan web of the complete set of
common generalized eigenvectors of $\Xh$ and $\Yh$, we choose to work in the
$\Yh$-favouring basis (\ref{Yhbasis}). A generalized vector $\Gh_{\la,\la'}^{(\ell,\ell')}$ can thus
be written as a $(2p-1)$-dimensional vector whose $p-1$ upper entries are
$2p'$-dimensional vectors of the type $C$ appearing in Section~\ref{SecCycle},
while the $p$ lower entries are $(4p'-2)$-dimensional vectors of the type $E$
appearing in Section~\ref{SecEye}.
The connected subwebs of the type $\Wh_{\la,\la'}^{(1,1)}$ are given by the following eigenvectors
\be
\Wh_{\beta_i,\beta'_j}^{(1,1)}:\quad \Gh_{\beta_i,\beta'_j}^{(0,0)}\;=\;\left(\!\!\!\begin{array}{c}
f_{1}(\beta_i)C_j\\ \vdots\\ f_{p-1}(\beta_i)C_j
\\[.15cm] \hline\\[-.3cm]
f_{p}(\beta_i)E_j\\ \vdots\\ f_{2p-1}(\beta_i)E_j
\end{array}\!\!\!\right),\qquad
i+j\ \mathrm{even},\quad i\in\{0,p\},\ j\in\{0,p'\}
\label{Whij11}
\ee
or
\be
\Wh_{\beta_a,\beta'_j}^{(1,1)}:\quad \Gh_{\beta_a,\beta'_j}^{(0,0)}\;=\;\left(\!\!\!\begin{array}{c}
f_{1}(\beta_a)C_j\\ \vdots\\ f_{p-1}(\beta_a)C_j
\\[.15cm] \hline\\[-.3cm]
f_{p}(\beta_a)E_j\\ \vdots\\ f_{2p-1}(\beta_a)E_j
\end{array}\!\!\!\right)\;=\;
\left(\!\!\!\begin{array}{c}
f_{1}(\beta_a)C_j\\ \vdots\\ f_{p-1}(\beta_a)C_j
\\[.15cm] \hline\\[-.3cm]
0\\ \vdots\\ 0
\end{array}\!\!\!\right),\qquad
a+j\ \mathrm{odd},\quad j\in\{0,p'\}
\ee
or
\be
\Wh_{\beta_i,\beta'_b}^{(1,1)}:\quad \Gh_{\beta_i,\beta'_b}^{(0,0)}\;=\;\left(\!\!\!\begin{array}{c}
0\\ \vdots\\ 0
\\[.15cm] \hline\\[-.3cm]
f_{p}(\beta_i)E_b\\ \vdots\\ f_{2p-1}(\beta_i)E_b
\end{array}\!\!\!\right),\qquad
i+b\ \mathrm{odd},\quad i\in\{0,p\}
\ee
The connected subwebs of the type $\Wh_{\la,\la'}^{(1,3)}$ consist of the following
generalized eigenvectors
\be
\Wh_{\beta_i,\beta'_b}^{(1,3)}:\quad \Gh_{\beta_i,\beta'_b}^{(0,\ell')}\;=\;\left(\!\!\!\begin{array}{c}
f_{1}(\beta_i)C_b^{(\ell'-1)}\\ \vdots\\ f_{p-1}(\beta_i)C_b^{(\ell'-1)}
\\[.15cm] \hline\\[-.3cm]
f_{p}(\beta_i)E_b^{(\ell')}\\ \vdots\\ f_{2p-1}(\beta_i)E_b^{(\ell')}
\end{array}\!\!\!\right),\qquad
i+b\ \mathrm{even},\quad i\in\{0,p\},\quad \ell'\in\mathbb{Z}_{0,2}
\ee
where $C_b^{(-1)}\equiv0$.
The connected subwebs of the type $\Wh_{\la,\la'}^{(3,1)}$ consist of the following
generalized eigenvectors
\be
\Wh_{\beta_a,\beta'_j}^{(3,1)}:\quad \Gh_{\beta_a,\beta'_j}^{(\ell,0)}\;=\;\left(\!\!\!\begin{array}{c}
\tfrac{1}{\ell!}f_{1}^{(\ell)}(\beta_a)C_j\\ \vdots\\ \tfrac{1}{\ell!}f_{p-1}^{(\ell)}(\beta_a)C_j
\\[.15cm] \hline\\[-.3cm]
\tfrac{1}{\ell!}f_{p}^{(\ell)}(\beta_a)E_j\\ \vdots\\ \tfrac{1}{\ell!}f_{2p-1}^{(\ell)}(\beta_a)E_j
\end{array}\!\!\!\right),\qquad
a+j\ \mathrm{even},\quad j\in\{0,p'\},\quad \ell\in\mathbb{Z}_{0,2}
\ee
The connected subwebs of the type $\Wh_{\la,\la'}^{(1,2)}$ consist of the following
generalized eigenvectors
\be
\Wh_{\beta_a,\beta'_b}^{(1,2)}:\quad \Gh_{\beta_a,\beta'_b}^{(0,\ell')}\;=\;
\left(\!\!\!\begin{array}{c}
f_{1}(\beta_a)C_b^{(\ell')}\\ \vdots\\ f_{p-1}(\beta_a)C_b^{(\ell')}
\\[.15cm] \hline\\[-.3cm]
0\\ \vdots\\ 0
\end{array}\!\!\!\right),\qquad
a+b\ \mathrm{odd},\quad \ell'\in\mathbb{Z}_{0,1}
\ee
The connected subwebs of the type $\Wh_{\la,\la'}^{(2,1)}$ consist of the following
generalized eigenvectors
\be
\Wh_{\beta_a,\beta'_b}^{(2,1)}:\quad \Gh_{\beta_a,\beta'_b}^{(\ell,0)}\;=\;
\left(\!\!\!\begin{array}{c}
0\\ \vdots\\ 0
\\[.15cm] \hline\\[-.3cm]
\tfrac{1}{(\ell+1)!}f_{p}^{(\ell+1)}(\beta_a)E_b\\ \vdots\\
\tfrac{1}{(\ell+1)!}f_{2p-1}^{(\ell+1)}(\beta_a)E_b
\end{array}\!\!\!\right),\qquad
a+b\ \mathrm{odd},\quad \ell\in\mathbb{Z}_{0,1}
\ee
Finally, the connected subwebs of the type $\Wh_{\la,\la'}^{(3,3)}$ consist of the following
generalized eigenvectors
\be
\Wh_{\beta_a,\beta'_b}^{(3,3)}:\quad \Gh_{\beta_a,\beta'_b}^{(\ell,\ell')}\;=\;\left(\!\!\!\begin{array}{c}
\tfrac{1}{\ell!}f_{1}^{(\ell)}(\beta_a)C_b^{(\ell'-1)}\\ \vdots\\
\tfrac{1}{\ell!}f_{p-1}^{(\ell)}(\beta_a)C_b^{(\ell'-1)}
\\[.15cm] \hline\\[-.3cm]
\tfrac{1}{\ell!}f_{p}^{(\ell)}(\beta_a)E_b^{(\ell')}\\ \vdots\\
\tfrac{1}{\ell!}f_{2p-1}^{(\ell)}(\beta_a)E_b^{(\ell')}
\end{array}\!\!\!\right),\qquad
a+b\ \mathrm{even},\quad \ell,\ell'\in\mathbb{Z}_{0,2},\quad (\ell,\ell')\neq(0,0)
\label{Whab33}
\ee
where $C_b^{(-1)}\equiv0$ as above.
Using properties of the $C$ and $E$ vectors as generalized eigenvectors of
$\Cc_{p'}$ and $\Ec_{p'}$, together with (\ref{IhE}) and (\ref{CE}), in particular,
it is straightforward to prove that the vectors given in (\ref{Whij11}) through (\ref{Whab33})
indeed correspond to the Jordan webs (\ref{webh13}), (\ref{webh1221}) and (\ref{webh8})
consistent with (\ref{XYGh}).
We also note that the number $\Nch^{(\ell,\ell')}$ of connected Jordan webs of the type
$\Wh^{(\ell,\ell')}$ is given by
\be
\Nch^{(1,1)}=\;p+p',\quad
\Nch^{(3,1)}=\;p-1,\quad
\Nch^{(1,3)}=\;p'-1,\quad
\Nch^{(2,1)}=\;\Nch^{(1,2)}=\;\Nch^{(3,3)}=\;\tfrac{1}{2}(p-1)(p'-1)
\ee
consistent with the total number (\ref{6pp}) of generalized eigenvectors.
The similarity matrix $\Qh$ appearing in (\ref{QhXhQh}) is constructed by concatenating
the common generalized eigenvectors of $\Xh$ and $\Yh$ according to any ordering of the
ordered bases (\ref{Bh}). The permutation matrix $\Ph$
depends on this choice of ordering, and the degree of
convenience of such a choice depends on the intended application. Here we consider
a general ordering reflecting the partitioning
\be
\big\{\Bh_{\la,\la'}^{(1,1)}\big\}\cup
\big\{\Bh_{\la,\la'}^{(1,3)}\big\}\cup
\big\{\Bh_{\la,\la'}^{(3,1)}\big\}\cup
\Big(\big\{\Bh_{\la,\la'}^{(1,2)}\big\}\cup\big\{\Bh_{\la,\la'}^{(2,1)}\big\}\Big)\cup
\big\{\Bh_{\la,\la'}^{(3,3)}\big\}
\label{BBBBBh}
\ee
such that every set (of generalized eigenvectors) of the type $\Bh_{\la,\la'}^{(1,1)}$
comes before every set (of generalized eigenvectors) of the type $\Bh_{\la,\la'}^{(1,3)}$,
and so on. In addition, for every pair $\la,\la'$ in $\{\Bh_{\la,\la'}^{(1,2)}\}$
(or equivalently in $\{\Bh_{\la,\la'}^{(2,1)}\}$), the two vectors $\Gh_{\la,\la'}^{(0,0)}$
and $\Gh_{\la,\la'}^{(0,1)}$ in $\Bh_{\la,\la'}^{(1,2)}$ are followed immediately by the two
vectors $\Gh_{\la,\la'}^{(0,0)}$ and $\Gh_{\la,\la'}^{(1,0)}$ in $\Bh_{\la,\la'}^{(2,1)}$.
The Jordan canonical forms $J_{\Xh}$ and $J_{\Yh}$ in (\ref{QhXhQh}) are then of the form
\bea
J_{\Xh}\!\!&=&\!\!\mathrm{diag}\Big(
\underbrace{\la,\ldots}_{p+p'},\,
\underbrace{\la,\ldots}_{3p'-3},\,
\underbrace{\Jc_{\la,3},\ldots}_{p-1},\,
\underbrace{\mathrm{diag}\big(\Jc_{\la,2},\la,\la\big),\ldots}_{\tfrac{1}{2}(p-1)(p'-1)},\,
\underbrace{\mathrm{diag}\big(\Jc_{\la,2},\Jc_{\la,3},\Jc_{\la,3}\big)
\ldots}_{\tfrac{1}{2}(p-1)(p'-1)}\Big)
\label{JXh}
\\
J_{\Yh}\!\!&=&\!\!\mathrm{diag}\Big(
\underbrace{\la',\ldots}_{p+p'},\,
\underbrace{\Jc_{\la',3},\ldots}_{p'-1},\,
\underbrace{\la',\ldots}_{3p-3},\,
\underbrace{\mathrm{diag}\big(\Jc_{\la',2},\la',\la'\big),\ldots}_{\tfrac{1}{2}(p-1)(p'-1)},\,
\underbrace{\mathrm{diag}\big(\Jc_{\la',2},\Jc_{\la',3},\Jc_{\la',3}\big)
\ldots}_{\tfrac{1}{2}(p-1)(p'-1)}\Big)
\eea
We stress that the eigenvalues $\la$ and $\la'$ vary in these expressions but are always of the
form (\ref{betabeta}).
The corresponding permutation matrix $\Ph$ is a block-diagonal matrix whose blocks are
of dimension 1, 3, 3, 4 or 8, corresponding to the dimensions of the sets
$\Bh_{\la,\la'}^{(1,1)}$, $\Bh_{\la,\la'}^{(1,3)}$, $\Bh_{\la,\la'}^{(3,1)}$,
$\Bh_{\la,\la'}^{(1,2)}\cup\Bh_{\la,\la'}^{(2,1)}$ and $\Bh_{\la,\la'}^{(3,3)}$, respectively.
By a similarity transformation (\ref{QhXhQh}), these $\Ph$-blocks convert the
blocks in $J_{\Xh}$ into the corresponding upper-triangular matrices $\Xh_{\la,\la'}^{(\ell,\ell')}$
in (\ref{Xhweb}) (where $\Xh_{\la,\la'}^{(1,2)}$ and $\Xh_{\la,\la'}^{(2,1)}$ are viewed as the
single four-dimensional matrix $\mathrm{diag}(\la,\la,\Jc_{\la,2})$).
The $\Ph$-blocks of dimension 1 or 3 are identity matrices,
while the $\Ph$-blocks of dimension 4 or 8 are the symmetric permutation matrices
\be
\Ph_4\;=\;\begin{pmatrix} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0
\end{pmatrix},\qquad
\Ph_8\;=\;\begin{pmatrix}
0&0&1&0&0&0&0&0\\
0&0&0&0&0&1&0&0\\
1&0&0&0&0&0&0&0\\
0&0&0&1&0&0&0&0\\
0&0&0&0&0&0&1&0\\
0&1&0&0&0&0&0&0\\
0&0&0&0&1&0&0&0\\
0&0&0&0&0&0&0&1 \end{pmatrix}
\label{Ph4Ph8}
\ee
The symmetric permutation matrix $\Ph$ is thus given by
\be
\Ph\;=\;\mathrm{diag}\Big(\underbrace{1,\ldots,1}_{4p+4p'-6},\,
\underbrace{\Ph_4,\ldots,\Ph_4}_{\tfrac{1}{2}(p-1)(p'-1)},\,
\underbrace{\Ph_8,\ldots,\Ph_8}_{\tfrac{1}{2}(p-1)(p'-1)}\Big)
\ee
Acting on the non-connected Jordan web $\Wh_{\la,\la'}^{(1,2)}\cup\Wh_{\la,\la'}^{(2,1)}$,
the permutation matrix $\Ph_4$ interchanges the two connected components.
As an action on the connected Jordan webs $\Wh_{\la,\la'}^{(3,3)}$, $\Ph_8$
reflects the vertices (generalized eigenvectors) with respect to the line from
south-west to north-east through $\Gh_{\la,\la'}^{(1,1)}$ and $\Gh_{\la,\la'}^{(2,2)}$.
$\Ph_8$ thus has two fix-points in accord with the two units on the diagonal.
By eliminating the first row and column of the permutation matrices $P_5$ and $P_9$
in (\ref{P5P9}), one obtains the permutation matrices $\Ph_4$ and $\Ph_8$, respectively.
Likewise, the Jordan canonical forms $J_{\Xh}$ and $J_{\Yh}$ follow from the Jordan
canonical forms $J_X$ and $J_Y$ by elimination of the corresponding rows and columns.
Instead of preserving this elimination property, $\Ph_4$ could have been chosen as the
four-dimensional identity matrix in which case the blocks $\mathrm{diag}(\Jc_{\la,2},\la,\la)$
in (\ref{JXh}) are replaced by $\mathrm{diag}(\la,\la,\Jc_{\la,2})$.
\subsubsection{General fusion matrices}
\label{SecGeneralHat}
Here we determine the upper-triangular block-diagonal
matrix $\Qh^{-1}\Nh_\mu \Qh$ obtained from the general fusion matrix $\Nh_\mu$ by
a similarity transformation with respect to $\Qh$ defined according to (\ref{BBBBBh}).
{}From (\ref{QhNhQh}) and Section~\ref{SecAuxWebs}, we have that
\bea
\Qh^{-1}\Nh_\mu \Qh\!\!&=&\!\!\mathrm{diag}\Big(
\underbrace{g(\Xh_{\la,\la'}^{(1,1)})h(\Yh_{\la,\la'}^{(1,1)}),\ldots}_{p+p'},\,
\underbrace{g(\Xh_{\la,\la'}^{(1,3)})h(\Yh_{\la,\la'}^{(1,3)}),\ldots}_{p'-1},\,
\underbrace{g(\Xh_{\la,\la'}^{(3,1)})h(\Yh_{\la,\la'}^{(3,1)}),\ldots}_{p-1},\nn
&&\qquad
\underbrace{\mathrm{diag}\big(g(\Xh_{\la,\la'}^{(1,2)})h(\Yh_{\la,\la'}^{(1,2)}),
g(\Xh_{\la,\la'}^{(2,1)})h(\Yh_{\la,\la'}^{(2,1)})\big),
\ldots}_{\tfrac{1}{2}(p-1)(p'-1)},\,
\underbrace{g(\Xh_{\la,\la'}^{(3,3)})h(\Yh_{\la,\la'}^{(3,3)}),\ldots}_{\tfrac{1}{2}(p-1)(p'-1)}\Big)
\eea
where, as before, $g(z)=\mathrm{pol}_{\mu}^{(x)}(z)$ and
$h(z)=\mathrm{pol}_{\mu}^{(y)}(z)$, and where
\bea
g(\Xh_{\la,\la'}^{(1,1)})h(\Yh_{\la,\la'}^{(1,1)})\!\!&=&\!\! g(\la)h(\la')\nn
g(\Xh_{\la,\la'}^{(1,3)})h(\Yh_{\la,\la'}^{(1,3)})\!\!&=&\!\! g(\la)h(\Jc_{\la',3}),\qquad
g(\Xh_{\la,\la'}^{(3,1)})h(\Yh_{\la,\la'}^{(3,1)})\;=\;g(\Jc_{\la,3})h(\la')\nn
g(\Xh_{\la,\la'}^{(1,2)})h(\Yh_{\la,\la'}^{(1,2)})\!\!&=&\!\! g(\la)h(\Jc_{\la',2}),\qquad
g(\Xh_{\la,\la'}^{(2,1)})h(\Yh_{\la,\la'}^{(2,1)})\;=\;g(\Jc_{\la,2})h(\la')\nn
g(\Xh_{\la,\la'}^{(3,3)})h(\Yh_{\la,\la'}^{(3,3)})\!\!&=&\!\! \left(\!\!\begin{array}{cccccccc}
gh&gh'&0&g'h&g'h'&0&\tfrac{1}{2}g''h&\tfrac{1}{2}g''h'\\[3pt]
0&gh&0&0&g'h&0&0&\tfrac{1}{2}g''h\\[3pt]
0&0&gh&gh'&\tfrac{1}{2}gh''&g'h&g'h'&\tfrac{1}{2}g'h''\\[3pt]
0&0&0&gh&gh'&0&g'h&g'h'\\[3pt]
0&0&0&0&gh&0&0&g'h\\[3pt]
0&0&0&0&0&gh&gh'&\tfrac{1}{2}gh''\\[3pt]
0&0&0&0&0&0&gh&gh'\\[3pt]
0&0&0&0&0&0&0&gh
\end{array}\!\!\right)
\label{ghhh}
\eea
To simplify the notation, we have used the abbreviations $g=g(\la)$ and $h=h(\la')$.
The eight-dimensional matrix $g(\Xh_{\la,\la'}^{(3,3)})h(\Yh_{\la,\la'}^{(3,3)})$ in (\ref{ghhh})
is obtained from the nine-dimensional matrix $g(X_{\la,\la'}^{(3,3)})h(Y_{\la,\la'}^{(3,3)})$
given in (\ref{gh}) by elimination of the first row and column.
This completes the description of the upper-triangular block-diagonal matrix
$\Qh^{-1}\Nh_\mu \Qh$.
\subsubsection{${\cal W}$-extended critical percolation ${\cal WLM}(2,3)$}
As for $X$ and $Y$, the eigenvalues of $\Xh$ and $\Yh$ are given in (\ref{beta23})
in the case of ${\cal WLM}(2,3)$.
As displayed in Figure~\ref{web23h}, the connected components
$\Wh_{\beta_i,\beta'_j}^{(\ell,\ell')}$ of the Jordan web
associated to the fusion algebra of modules associated with boundary conditions
are neatly organized with respect to the labels $i$ and $j$.
\psset{unit=1cm}
\begin{figure}
$$
\renewcommand{\arraystretch}{1.5}
\begin{array}{c|cccc}
\mbox{}_i\diagdown\mbox{}^j&0&1&2&3\\[4pt]
\hline
\rule{0pt}{16pt}
0&\Wh_{2,2}^{(1,1)}&\Wh_{2,1}^{(1,1)}&\Wh_{2,-1}^{(1,3)}&$\O$
\\[4pt]
1&\Wh_{0,2}^{(1,1)}&\Wh_{0,1}^{(3,3)}&\Wh_{0,-1}^{(1,2)}\cup\Wh_{0,-1}^{(2,1)}
&\Wh_{0,-2}^{(3,1)}
\\[4pt]
2&\Wh_{-2,2}^{(1,1)}&\Wh_{-2,1}^{(1,1)}&\Wh_{-2,-1}^{(1,3)}&$\O$
\end{array}
$$
\caption{The connected components $\Wh_{\beta_i,\beta'_j}^{(\ell,\ell')}$ of the Jordan web
associated to the fusion algebra of modules associated with boundary conditions in
${\cal W}$-extended critical percolation ${\cal WLM}(2,3)$. The two \O's
indicate that there are no common generalized eigenvectors corresponding to the pair
$(\beta_0,\beta'_3)=(2,-2)$ or to the pair $(\beta_2,\beta'_3)=(-2,-2)$ of eigenvalues of the
auxiliary fusion matrices $\Xh$ and $\Yh$.}
\label{web23h}
\end{figure}
An example of an ordering of the common generalized eigenvectors of $\Xh$ and $\Yh$
respecting (\ref{BBBBBh}) is
\bea
&\Gh_{2,2}^{(0,0)}; \Gh_{0,2}^{(0,0)}; \Gh_{-2,2}^{(0,0)}; \Gh_{2,1}^{(0,0)}; \Gh_{-2,1}^{(0,0)};
\Gh_{2,-1}^{(0,0)}, \Gh_{2,-1}^{(0,1)}, \Gh_{2,-1}^{(0,2)};
\Gh_{-2,-1}^{(0,0)}, \Gh_{-2,-1}^{(0,1)}, \Gh_{-2,-1}^{(0,2)};
\Gh_{0,-2}^{(0,0)}, \Gh_{0,-2}^{(1,0)}, \Gh_{0,-2}^{(2,0)};\nn
&\Gh_{0,-1}^{(0,1)}, \Gh_{0,-1}^{(0,2)}, \Gh_{0,-1}^{(1,0)}, \Gh_{0,-1}^{(2,0)};
\Gh_{0,1}^{(0,1)}, \Gh_{0,1}^{(0,2)}, \Gh_{0,1}^{(1,0)}, \Gh_{0,1}^{(1,1)}, \Gh_{0,1}^{(1,2)},
\Gh_{0,1}^{(2,0)}, \Gh_{0,1}^{(2,1)}, \Gh_{0,1}^{(2,2)}
\eea
We define the similarity matrix $\Qh$ by concatenating these vectors in the order given. Modulo
a similarity transformation, $\Qh$ converts $\Xh$ and $\Yh$ into the Jordan canonical forms
\bea
&J_{\Xh}\;=\;\Ph^{-1}\Qh^{-1}\Xh\Qh\Ph\;=\;\mathrm{diag}\big(
2,0,-2,2,-2,2,2,2,-2,-2,-2,\Jc_{0,3},\Jc_{0,2},0,0,\Jc_{0,2},\Jc_{0,3},\Jc_{0,3}\big) \nn
&J_{\Yh}\;=\;\Qh^{-1}\Yh\Qh\;=\;\mathrm{diag}\big(
2,2,2,1,1,\Jc_{-1,3},\Jc_{-1,3},-2,-2,-2,\Jc_{-1,2},-1,-1,\Jc_{1,2},\Jc_{1,3},\Jc_{1,3}\big)
\eea
where $\Ph$ is the symmetric permutation matrix
\be
\Ph\;=\;\mathrm{diag}\big(I_{14\times14},\Ph_4,\Ph_8\big)
\ee
The fusion matrix $\Nh_\mu$ associated to the general module $\mu\in\Ic_b$
is polynomial in $\Xh$ and $\Yh$. It is given by the same polynomial as in (\ref{N23})
but as a function of $\Xh,\Yh$ instead of $X,Y$. We recall that the only two modules
in the fundamental fusion algebra
{\em not} associated with boundary conditions are $\ketw{1,1}$ and $\ketw{1,2}$, that is,
\be
\Ic_f\setminus\Ic_b\;=\;\{\ketw{1,1},\ketw{1,2}\}
\ee
The similarity transformation of $\Nh_\mu$ is the block-diagonal matrix $\Qh^{-1}\Nh_\mu \Qh$
whose blocks are upper-triangular matrices. As illustrations of such block-diagonal matrices,
we here consider
\bea
\Qh^{-1}\Nh_{4,2}\Qh\!\!&=&\!\!\mathrm{diag}\Big(
4,0,-4,2,-2,\begin{pmatrix} -2&2&0\\ 0&-2&2\\ 0&0&-2\end{pmatrix},
\begin{pmatrix} 2&-2&0\\ 0&2&-2\\ 0&0&2\end{pmatrix},
\begin{pmatrix} 0&2&0\\ 0&0&2\\ 0&0&0\end{pmatrix},\nn
&&\hspace{4.5cm} 0,0,\Jc_{0,2},
\begin{pmatrix}
0&0&0&-1&-1&0&0&0\\
0&0&0&0&-1&0&0&0\\
0&0&0&0&0&-1&-1&0\\
0&0&0&0&0&0&-1&-1\\
0&0&0&0&0&0&0&-1\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0
\end{pmatrix}\Big)\nn
\Qh^{-1}\Nh_{2,3}^{1,1}\Qh\!\!&=&\!\!\mathrm{diag}\Big(
24,0,24,0,0,\begin{pmatrix} 0&8&-12\\ 0&0&8\\ 0&0&0\end{pmatrix},
\begin{pmatrix} 0&8&-12\\ 0&0&8\\ 0&0&0\end{pmatrix},
\begin{pmatrix} 0&0&-6\\ 0&0&0\\ 0&0&0\end{pmatrix},\nn
&&\hspace{4.5cm}0,0,0,0,
\begin{pmatrix}
0&0&0&-1&-1&0&0&0\\
0&0&0&0&-1&0&0&0\\
0&0&0&0&0&-1&-1&0\\
0&0&0&0&0&0&-1&-1\\
0&0&0&0&0&0&0&-1\\
0&0&0&0&0&0&0&0\\
0&0&0&0&0&0&0&0
\end{pmatrix}\Big)
\eea
\section{Conclusion}
\label{SecConclusion}
We have extended the work~\cite{Ras0908} on ${\cal WLM}(1,p')$ by considering the spectral
decompositions of the regular representations of the graph fusion algebras of the general
${\cal W}$-extended logarithmic minimal model ${\cal WLM}(p,p')$.
In preparation therefore, we first defined and examined three types of directed and
connected graphs, here called cycle, tadpole and eye-patch graphs.
As in the rational minimal models, the fundamental fusion algebra of ${\cal WLM}(p,p')$
is described by a simple graph fusion algebra.
The graphs associated with the two fundamental modules consist of a number of
tadpole and eye-patch graphs.
The corresponding adjacency matrices share a complete set of common generalized
eigenvectors organized as a web. This Jordan web is constructed by interlacing the
Jordan chains of the two matrices and consists of connected
subwebs with 1, 3, 5 or 9 generalized eigenvectors.
The similarity matrix, formed by concatenating these vectors, simultaneously brings
the two fundamental adjacency matrices to Jordan canonical form modulo permutation similarity.
By the same similarity transformation, the general fusion matrices are brought simultaneously to
block-diagonal forms whose blocks are upper-triangular matrices of dimension 1, 3, 5 or 9.
For $p>1$, only some of the modules in the fundamental fusion algebra of ${\cal WLM}(p,p')$
are associated with boundary conditions within our lattice approach.
The regular representation of the corresponding fusion subalgebra has features similar to the
ones in the regular representation of the fundamental fusion algebra, but with dimensions
of the connected Jordan-web components and upper-triangular blocks given by 1, 2, 3 or 8.
In addition to eye-patch graphs, cycle graphs appear as connected components of the two
auxiliary fusion matrices obtained from the fundamental fusion matrices by elimination of certain
rows and columns. The general fusion matrices associated with boundary conditions
are conveniently described in terms of the two auxiliary fusion matrices.
Some of the key results have been illustrated for ${\cal W}$-extended critical percolation
${\cal WLM}(2,3)$.
There are several natural continuations of this work, all of which we hope to discuss elsewhere.
The first one concerns an algebraic extension of the fundamental fusion algebra of
${\cal WLM}(p,p')$ for $p>1$. It amounts to including all modules arising from fusions
of the complete set of irreducible modules in the model as discussed
in~\cite{GRW0905,Ras0906,Wood0907}.
The second continuation concerns the derivation of a generalized Verlinde formula from the
spectral decomposition of the various fusion matrices of ${\cal WLM}(p,p')$.
This problem was solved in~\cite{Ras0908} for $p=1$.
Other approaches to a Verlinde-like formula for ${\cal WLM}(1,p')$ have been proposed
in~\cite{FHST0306,FK0705,GR0707,GT0711,PRR0907}.
In the case of the so-called {\em projective modules} in ${\cal WLM}(p,p')$
\be
\big\{\ketw{\kappa p,p'},\ketw{\R_{\kappa p,p'}^{a,0}},\ketw{\R_{p,\kappa p'}^{0,b}},
\ketw{\R_{\kappa p,p'}^{a,b}};\ \kappa\in\mathbb{Z}_{1,2},\ a\in\mathbb{Z}_{1,p-1},\
b\in\mathbb{Z}_{1,p'-1}
\big\}
\ee
of which there are $2pp'$~\cite{Ras0805},
the structure of the corresponding Verlinde-like formula~\cite{PR0912} resembles the ordinary
Verlinde formulas. This is intimately related to the observation that the
auxiliary fusion graphs underlying the restrictions of the fundamental matrices $X$ and $Y$
to the projective modules are simply given by $p'$ cycle graphs $\Cc_p$ and $p$ cycle
graphs $\Cc_{p'}$, respcetively. Their spectral decompositions are much simpler than the ones
considered here as they only involve rank-1 and rank-2 blocks. The two matrices share a
complete set of ($2pp'$) common generalized eigenvectors with the numbers of connected
Jordan webs given by
\be
\Nc^{(1,1)}_{\mathrm{proj}}\,=\;2,\qquad
\Nc^{(1,2)}_{\mathrm{proj}}\,=\;p'-1,\qquad
\Nc^{(2,1)}_{\mathrm{proj}}\,=\;p-1,\qquad
\Nc^{(2,2)}_{\mathrm{proj}}\,=\;\tfrac{1}{2}(p-1)(p'-1)
\label{Nproj}
\ee
As in the case of the connected Jordan webs associated with the auxiliary (boundary) fusion
matrices $\Xh$ and $\Yh$, cf. Section~\ref{SecAuxFusMat},
the connected Jordan webs (\ref{Nproj}) associated with the
auxiliary (projective) fusion matrices can be viewed as quotients of the connected Jordan
webs associated with the fundamental fusion matrices $X$ and $Y$.
The third continuation concerns the spectral decomposition of the (matrix) generators
of the Grothendieck ring associated to ${\cal WLM}(p,p')$. For $p=1$, this ring is obtained
by elevating the various character identities to equivalence relations between the corresponding
generators (modules) of the fusion algebra. For $p>1$, on the other hand, the
situation is more complicated as also pointed out in~\cite{GRW0905,Wood0907}.
Partition functions only concern characters, not the full-fledged fusion algebra. It thus suffices
to consider the Grothendieck ring instead of the fusion algebra when discussing partition
functions. In such circumstances, one is simply not concerned with the reducible yet
indecomposable module structures, only in their characters.
Based on spectral decompositions of the regular representation of the Grothendieck ring of
${\cal WLM}(1,p')$, a Verlinde-like formula was derived in~\cite{PRR0907}.
In~\cite{Ras0908}, a general framework is outlined within which it makes sense to discuss rings
of equivalence classes of fusion-algebra generators. Together
with the insight we have just gained by studying the graph fusion
algebras and fusion graphs, this may provide the means to classify
Grothendieck-like rings associated to ${\cal WLM}(p,p')$.
\subsection*{Acknowledgments}
\vskip.1cm
\noindent
This work is supported by the Australian Research Council.
The author thanks Paul A. Pearce for helpful discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,887 |
Q: Is the wildcard character allowed on PHP prepared statements? I'm using AJAX to send the user's input to a PHP script. The PHP script shows records from a MySQL table.
If the user enter the wildcard character (%) then all the records from the database are listed.
Is this OK from a coding perspective? Should be the % char allowed? Or could my code be wrong?
I really don't care from the user experience, actually this could be a good feature, but I want to know if this is a normal behavior for prepared statements.
...
$stmt = $dbh->prepare("SELECT * FROM tblStudent where lastNameStudent LIKE ?");
$stmt->bindParam(1, $lastname, PDO::PARAM_STR);
$stmt->execute();
...
When I set $lastname='%' I get all the records in the tblStudent table.
A:
Is this fine from the code perspective?
Yes.
Should be the % char allowed?
Nothing wrong with it. At least from security point of view.
Speaking of the application logic - it depends. You may escape this character if you want it be searched literally, or not escape if you want to use it as a wildcard.
This behavior has nothing to do with prepared statements.
A prepared statement is just a way to put a variable into query. The content of this variable is none of its business.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,552 |
Le Jewish Renewal ou judaïsme du renouveau est un mouvement contemporain du judaïsme né aux États-Unis avec des pratiques mystiques et méditatives.
À la fin des années 1960 et début des années 1970, de jeunes rabbins constituent en marge des yeshivot et autres académies talmudiques le mouvement des Havourot (pluriel de Havourah) ou "confrérie" pour la prière et l'étude en réaction à la saturation institutionnelle mais aussi à la perte spirituelle qui règneraient en Amérique du Nord. Ces juifs observants s'inspirent au départ des pratiques cultuelles des pharisiens et autres sectes antiques.
Bien vite ce mouvement se diversifie. Dans la région de Boston la Havourat Shalom pratique une vie communautaire rurale en autarcie. C'est surtout l'arrivée d'une population urbaine qui rapprochera le judaïsme du renouveau de la pensée du judaïsme libéral. Notamment par l'action d'Arthur Waskow (ancien rabbin du mouvement reconstructionniste, tandis que la teinte mystique provient de l'influence du rabbin Zalman Schachter-Shalomi.
Après avoir quitté le mouvement loubavitch, ce dernier fonda dans les années 1960 : The B'nai Or Religious Fellowship La Confrérie Religieuse B'nai Or. B'nai Or signifiant en hébreu les "fis de la lumière", Schachter-Shalomi fonde son mouvement en référence et aux Esséniens (et aux Manuscrits de la mer Morte popularisés depuis peu). Cependant la trame mystique ne que peu les matériaux théologiques, doctrinaires et eschatologiques de Qumran, étant influencée par la méditation bouddhiste et le soufisme.
En 1985, une première conférence nationale change le nom des B'nai Or en P'nai Or ou "visages de lumière", nouvelle nomination motivée par le tournant libéral et féministe auquel aspire le mouvement de Schachter-Shalomi. En 1993, la famille libérale de Waskow - née des premières havourot - finit par s'unir à la confrérie de Schachter-Shalomi, fondant l'association ALEPH sous l'intitulé Alliance for Jewish Renewal.
Liens externes
ALEPH : Alliance for Jewish Renewal
Courant du judaïsme | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,748 |
Home » Maryland News » Hogan calls for new…
Hogan calls for new Chesapeake Bay crossing
Amanda Iacone
Gov. Larry Hogan announced Tuesday afternoon that the state would begin the long process to study where to put another bay bridge in order to accommodate future traffic demand.
In this file photo, traffic waits to cross the Chesapeake Bay Bridge on a Memorial Day weekend. On Tuesday, Maryland Gov. Larry Hogan announced a $5 million, four-year study to look into a location for a new bay crossing in order to address congestion on the existing Bay Bridge. (WTOP File Photo/Dave Dildine) (WTOP/Dave Dildine)
Gov. Larry Hogan speaks at a news conference on Tuesday, Aug. 30, 2016 near Annapolis, Md., with the Chesapeake Bay Bridge in the backdrop. Hogan announced a $5 million study to explore a potential new Chesapeake Bay crossing. (AP Photo/Brian Witte) (AP)
WASHINGTON — Maryland Gov. Larry Hogan wants to build another Chesapeake Bay crossing.
Hogan announced Tuesday afternoon that the state would begin the long process to study where to put another bay bridge span in order to accommodate future traffic demand.
The Chesapeake Bay Bridge can be safely maintained for another 50 years. But estimates show that by 2040, traffic delays could back up as much as 14 miles for drivers trying to cross between Kent Island and Annapolis.
14-mile backups on Bay Bridge could become the new normal
Traffic deaths at 6-year low on dangerous Indian Head Highway
Drivers today already face backups especially on summer weekends but the bridge also snarls commuter traffic.
"Hours that could be spent with your family or at work or doing things you enjoy are instead spent sitting in bumper to bumper traffic," Hogan said. "As governor, I believe that working to keep traffic flowing safely and efficiently into the future is an important priority."
Hogan said he directed the Maryland Transportation Authority to begin the study, which will narrow down possible locations for a new span and explore financing, engineering, land use and economic impacts.
He promised that Marylanders would have plenty of opportunities to weigh in during the study.
The Maryland Transportation Authority approved the $5 million study last week. Work will begin this fall and take four years to complete.
In December, the state released an analysis of the long-term viability of the dual curved spans, which carry five lanes of traffic along U.S. 50/301. The study recommended adding three more travel lanes to mitigate future congestion.
Demolishing the two spans and replacing them with an eight-lane mega bridge — the most expensive of several options suggested in the report — would cost an estimated $7 billion.
Local News Maryland News Transportation News Washington, DC Traffic
chesapeake bay chesapeake bay bridge larry hogan traffic congestion | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,467 |
The former Country home of Shell oil baron Henri Deterding
Property in Norfolk: Very flat indeed – and rising all the time
Norfolk has never seen a county pursuit quite like it.
By Clive Aslet
Last Updated: 1:30PM GMT 26 Nov 2008
Graze and favoured: Easton Lodge, outside Norwich, forms part of the 2,417-acre estate sold by Savills to a European buyer
What is it about Norfolk? At a time when the rest of the property market in the UK is in the doldrums, this county, in the bulge of East Anglia, has had three top-quality estates on the market. Two of them – guide price of £25 million each – have either sold or gone under offer.
Excuse me, but aren't we meant to be tightening belts? Prices are falling here, yet this extraordinary performance says something about Norfolk, farmland and the upper echelons of the country-house market, which for the time being is sailing on as though there isn't an iceberg in sight.
James Laing, of Strutt & Parker, is flabbergasted. "I haven't seen three Norfolk estates like this on the market at the same time in all of my 40-year career," he says.
Coincidence has something to do with it, but so does the fact that indigenous Norfolk people are enormously proud that no motorway goes through the county; and there is no doubt that north Norfolk seems cut off from the rest of the world.
En route, you pass through what the Hitchhiker's Guide to the Galaxy would probably call planet Mangelwurzel, otherwise known as Swaffham. The writer Raffaella Barker, who was brought up in Norfolk and still lives there, is a typical native.
"You have to insulate yourself with goose fat during the winter," she warns. "It's very cold, only great grey slabs of cloud, and all your neighbours are fish."
When she was growing up in the 1960s and 1970s, Norfolk was still a feudal land of great estates, leavened by a smattering of artists. "It has changed enormously. It's richer and more London-centred now."
Visit Holt and you can see what she means. The translocated urbanite will find every means necessary to support life: smart restaurants, bookshops and a retro outfitters called Old Town, selling tank-tops and knitted ties.
So while the number of people wanting to acquire property in Norfolk may be limited – Crispin Holborow, of Savills, who has negotiated the sale of the 2,417-acre Easton estate, outside Norwich, speaks of "having to share buyers around" – the gene pool is growing. According to reports, Easton has gone to a European with Anglophile tastes. In other words, he likes shooting – pretty well de rigueur in these parts.
"Very flat, Norfolk," sighs Amanda in Noël Coward's Private Lives. But not everywhere is, and the right kind of curves fetch a premium. The pleasingly contoured Kelling Hall estate has gone under offer at about £25 million, having caught the eye of a British entrepreneur. Even the house is curved.
It is one of a group of Arts and Crafts country houses on the coast, built to what was known in the early 20th century as a butterfly plan. This was the era when sunlight and fresh air had been identified as an antidote to T B, and butterfly-plan houses – in this dry, breezy location – were constructed to allow the maximum degree of sunshine into their rooms. Like Kelling, whose architect was Edward Maufe, they made much of flints and other local materials. But if the traditional construction was backward-looking, the client was a man of his time: H W Deterding, director general of Royal Dutch Petroleum, soon to be known as Shell.
According to Mark Lawson, of the Buying Solution, who acted for Kelling Hall's purchaser, it is a sporty sort of place, with an "incredible" wild bird shoot. Two tennis courts are set at right angles to each other, so that you will never have to serve into the sun. There are, in addition, 30 cottages, a self-catering development known as The Lowes and a 1,000-acre arable farm, all producing an annual income of nearly £600,000. And it goes right down to the sea.
The late-Victorian author Clement Scott (drama critic of The Daily Telegraph) popularised this part of the coast as Poppyland, a place of restorative ozone and wholesome walks; around Kelling it seems completely unchanged.
If you have missed Easton and Kelling, don't despair. Strutts still has High House, at Castle Acre, on the market. This handsome late-Georgian house – formed of five perfect 18?ft cubes, according to the vendor, Henry Birkbeck – comes with 1,000 acres and a price tag of £9.5 million. Near Sandringham, it would be a passport to the pleasures of country life, particularly shooting.
As Laing observes, the place "sits in its own 300-acre grassland park, so it is not surrounded by potatoes, sugar beet or people". It will attract someone who loves architecture and everything that goes with it; in Norfolk, 700 acres of farmland – all that's left, after you have subtracted the park – is more than a sneeze, but not much. Still, a new owner could work on it.
At Great Massingham, Strutt & Parker could fit you up with West Heath Farm, 225 acres with a barn that has planning permission to be converted to a five-bedroom house. They are asking £1.6 million for that, and £800,000 for 150 acres at Swaffham.
As with other global commodities, wheat prices have fallen away from their peak earlier this year. Land prices, similarly, have gone soggy after the dreadful summer. But at £6,000 an acre, they are still double four years ago, and farmers – if not City boys – are in the market. In a world that hasn't a clue how it will feed itself when the population reaches 9?billion around the middle of this century, the long-term trend can only be up.
The bijou Godfrey's Hall near Hindringham, with 19 acres of parkland and woodland, would suit someone who doesn't relish the whole farming malarkey. It was built of mellow Holkham brick in 1868 for the Waters family in place of an earlier Jacobean hall, and features a sweeping cantilevered staircase and oak panelling salvaged from the old house. It has been sensitively and comprehensively restored by the current owners.
Big estates don't come up often, particularly in counties such as Norfolk, still dominated by traditional landowners such as the Earl of Leicester (Holkham) and the Marquess of Cholmondeley (Houghton), who will never sell. If you want one, you have to move quickly, even in a recession. A man, let's say, in his fifties, doesn't want to wait another five or 10 years.
Not everyone who made money over the past decade put it in Icelandic banks. Last year seemed to be an all-time boom at the top end of the country market. Incredibly, given the financial apocalypse that is upon us, 2008 looks as though it will do even better.
Clive Aslet is Editor at Large of Country Life
High House Estate, Castle Acre (£9.5m): Strutt & Parker 020 7629 7282www.struttandparker.com
West Heath Farm, Massingham (£1.6m): Strutt & Parker 01603 617431www.struttandparker.com
Godfrey's Hall, Hindringham (£2.65m): Knight Frank 020 7629 8171www.knightfrank.co.uk
Telegraph Article
Posted in: Daily Telegraph, Royal Dutch Shell Plc, Shell.
Tagged: Henri Deterding · Royal Dutch Petroleum Co. · Shell
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0 Comments on "The former Country home of Shell oil baron Henri Deterding" | {
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Charges were stayed earlier this week against Jamie Bacon in relation to the Surrey Six killings in October 2009. —Image: Abbotsford News
Kelowna girlfriend of notorious B.C. gang member dies
Madison Fine overdoses as Jamie Bacon has murder charges stayed
Dec. 8, 2017 10:05 a.m.
The Kelowna girlfriend of Red Scorpion gang member Jamie Bacon has passed away.
According to her obituary, Madison Fine, 25, died of an accidental drug overdose Dec. 1.
News reports in Vancouver, say Fine had gone to the Lower Mainland to be with Bacon following the staying of murder charges against him in relation to the Surrey Six massacre case. She was reportedly found unresponsive in a Richmond hotel by Bacon's mother
Fine's obituary describes Jamie Bacon as "her true love."
"A young woman with a big boisterous laugh and a loud talker, she could make you laugh so hard you would cry with her stories," says Fine's obituary. "Maddie spent her last few years, loving Jamie, sharing the laughs and travelling the world, making sure she did as much as she could while she was here. Generous to a fault, she was the best, most thoughtful gift-giver.
"We will miss her spirit, her stubbornness and her energy, and even, in a small way, the constant chaos she brought with her wherever she went."
A celebration of Fine's life will be held in the new year.
In lieu of flowers, the Fine Family plans to identify a charity to which donations can be in Madison's name.
To report a typo, email: edit@kelownacapnews.com.
Vancouver's latest tool for housing the homeless getting rough reception
Human foot found on Vancouver Island beach | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,617 |
{"url":"https:\/\/forum.azimuthproject.org\/plugin\/ViewComment\/16798","text":"\\$$X = { a b c } \\$$\n\n\\$$Y = { \u25cf \u25b2 \u2610 } \\$$\n\n\\$$f(a) = \u25b2 \\$$\n\n\\$$f(b) = \u25b2 \\text{or} \u2610 \\$$\n\n\\$$f(c) = \u25cf \\$$\n\nthen\n\n\\$$P := { { \u25b2 \u2610 } { \u25cf } } \\$$\n\n\\$$f*({\u25b2 \u25fb\ufe0e}) = { a b } \\$$\n\n\\$$f*({ \u25cf }) = { c } \\$$\n\nand\n\n\\$$Q = { { \u25b2 } { \u2610 } { \u25cf } } \\$$\n\n\\$$f*({ \u25b2 }) = { a } \\$$\n\n\\$$f*({ \u2610 }) = { b } \\$$\n\n\\$$f*({ \u25cf }) = { c } \\$$\n\nBeyond the construction of the examples above, I'm tempted to think of f* as a function, but that feels incorrect and I can't quite say why...","date":"2022-05-25 03:31:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"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\": 0.9340261220932007, \"perplexity\": 451.02044141981406}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662578939.73\/warc\/CC-MAIN-20220525023952-20220525053952-00015.warc.gz\"}"} | null | null |
{"url":"https:\/\/2022.help.altair.com\/2022\/hwsolvers\/os\/topics\/solvers\/os\/param_tolrsc_bulk_r.htm","text":"# PARAM, TOLRSC\n\nBulk Data Entry Connecting grid points of the shell element are moved onto the solid face.\n\nParameter Values Description\nTOLRSC <Real>\n\nDefault = 0.05\n\nWhen the RSSCON shell-to-solid element connector is used, the connecting grid points of the shell element are moved onto the solid face if the grid points are close enough. The tolerable distance of the shell grid point to the solid edge or face is $\\epsilon \\cdot h$ where $h$ is the height of the solid edge. Rigid body invariance is satisfied with double-precision accuracy if the shell grid points are adjusted.","date":"2022-12-05 13:59:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 2, \"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\": 0.5655853152275085, \"perplexity\": 1740.2258530392546}, \"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-2022-49\/segments\/1669446711017.45\/warc\/CC-MAIN-20221205132617-20221205162617-00633.warc.gz\"}"} | null | null |
{"url":"https:\/\/neatpour.com\/employer-address-ufkgmi\/viewtopic.php?tag=b7ba9e-all-electrical-formulas","text":"# all electrical formulas\n\n|\n\nElectric power is a widely used term in physics as you must be aware of. question_answer Answers(2) edit Answer . The various electrical formulas and their derivations are as follows: The relationship between volt, watt, and ohms is below Volts = square root of (watts \u00d7 ohms) The relationship between volts, watts, and amperes is as follows This theorem is applicable to all types of source of e.m.f. Every student begins at a different level of understanding, and you may find this unit an easy review, or you may find it requires a high level of concentration. Cookies are only used in the browser to improve user experience. Every student begins at a different level of understanding, and you may find this unit an easy review, or you may find it requires a high level of concentration. Academia.edu is a platform for academics to share research papers. Electrical Engineering Formulas Ohms Law Rectier Eciency. Press the Ohm's Law button after you have made your entries: VOLTS: AMPS: OHMS: So, if you change one side, you must do the same thing to the other side. Electrical Formulas Here i discuss some of important electrical formulas.All this formulas are useful for basic calculation in Electrical Engineering including Voltage,Ampere,Power, efficiency,power factor and many more .I hope it can make your basic understanding about electrical calculation is clear. Above all, these four physical quantities in electricity are \u00e2\u0080\u00a6 Single Phase: So, the formula V+1=IR+1 is valid, but not very useful. Now you don't need to remember all those complex formulas. Unknown May 21, 2015 at 8:59 AM Utter Bullshit.. You can target the Engineering ToolBox by using AdWords Managed Placements. In the above formulas 1 is the angle of lead or lag between current and voltage and cos 1 = P\/EI = power factor or pf. Determine the magnitude of the current flowing through it. Electrical Formulas: Horse Power HP, Watts, KiloWatts, KW, KVA, Kilo Volt Amps, Volts, Amps, Amperes, Power Factor, Pump HP, Fan HP E x I x EFF x PF x \u2026 Solution: Unknown May 21, 2015 at 8:59 AM Utter Bullshit.. I want all the formulas used in chapter electricity. Example 1: A wire carrying a current of 4 Amperes is having a resistance of 5 $\\omega$. Note: This information is provided as a quick reference resource and is not intended to serve as a substitute for qualified engineering assistance. o 40293 10895 PRINTED IN U.S.A. A Division of Tandy Corporation Fort Worth, TX 76102 Rad.e 'haek . Torque to Horsepower (hp) Horsepower (hp) to Torque. Voltage applied V = 25 V, Your email address will not be published. I x E x 2 x EFF x PF. Ohm's Law Formulas for A-C Circuits and Power Factor. Solution: Electrical motors are the machines which convert the electrical input to the mechanical energy. Please read AddThis Privacy for more information. OHM\u00e2\u0080\u0099S LAW CALCULATOR VOLTS=Voltage, AMPS=Current, OHMS=Resistance, WATTS=Power Give me any TWO numeric values and I'll give you all FOUR. Equivalent Resistance - Series & Parallel Circuit. Google use cookies for serving our ads and handling visitor statistics. The Ohm Law formulas above show the relationship between Voltage, Amps, Watts, and Ohms. In these calculations:V = voltage (in volts)I = current (in amps)R = resistance (in ohms)P = power (in watts) In any case, be certain that you fully understand the concepts of \u2026 In these calculations:V = voltage (in volts)I = current (in amps)R = resistance (in ohms)P = power (in watts) Your email address will not be published. Also, register to \u00e2\u0080\u009cBYJU\u00e2\u0080\u0099S \u2013 The Learning App\u00e2\u0080\u009d for loads of interactive, engaging Physics-related videos and an unlimited academic assist. Electrical Formulas helps us to calculate the parameters related to electricals in any electrical components. If the calculated power is positive, (+P) in value for any formula the component absorbs the power, that is it is consuming or using power. . Ohm\u2019s Law. Glossary :-I = Amperes. Electricity Formulas are applied in calculating the unknown electrical parameters from the known in electric circuits.. Electrical & Electronics, Ohm's Law, Formulas & Equations. We\u00e2\u0080\u0099ve seen the formula for determining the power in an electric circuit: by multiplying the voltage in \u00e2\u0080\u009cvolts\u00e2\u0080\u009d by the current in \u00e2\u0080\u009camps\u00e2\u0080\u009d we arrive at an answer in \u00e2\u0080\u009cwatts.\u00e2\u0080\u009d Let\u00e2\u0080\u0099s apply this to a circuit example: How to Use Ohm\u00e2\u0080\u0099s Law to Determine Current. this pdf sheet of chapter Electricity is useful for last minutes revision before exam . Most commonly used electrical formulas are formulas related to voltage, current, power, resistance etc. Amps = square root of (watts \/ ohms) AC Motor Formulas: E = voltage \/ I = amps \/ W = watts \/ PF = power factor \/ Eff = efficiency \/ HP = horsepower. He was a Journeyman Electrician, Master Electrician, and Electrical Contractor. What are the basic Laws of Electrical Engineering? Learn the Power Formula. Similarly, you will learn about the electric power formula in detail over here. Ohm's Law Formulas for A-C Circuits and Power Factor. Ohm\u00e2\u0080\u0099s Law and Joule\u00e2\u0080\u0099s Law are commonly used in calculations dealing with electronic circuits. Electrical - Electrical units, amps and electrical wiring, wire gauge and AWG, electrical formulas and motors; Related Documents . all electrical formula free download - Electrical Engineer Formula, Electrical formula and calculation, Electrical Formulator, and many more programs Given: Current I = 4 A, If resistances (i.e. Watts = volts x amperes. The power rating - energy per unit time - of the stove can be calculated as, P = (5 MJ) (106 J\/MJ) \/ ((60 min) (60 s\/min)), \u03bc = 746 Php \/ Pinput_w\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 (6), Pinput_w = input electrical power (watts), \u03bc = 746 Php \/ (1.732 V I PF)\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (6b), P3-phase = (U I PF 1.732) \/ 1,000\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0 (7), P3-phase = electrical power 3-phase motor (kW), I3-phase = (746 Php) \/ (1.732 V \u03bc PF)\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 (8), I3-phase = electrical current 3-phase motor (amps). All electrical formulas are available in the electricity calculation application. All Electrical Engineering Formulas 1. ELECTRONIC FORMULAS Ohm's Law Formulas for D-C Circuits. Avoid All Electrical Formula hack cheats for your own safety, choose our tips and advices confirmed by pro players, testers and users like you. These formulas for your electrical exam are your core formulas. Electrical is the branch of Physics dealing with electricity, electronics and electromagnetism. Also, the test will ask you to determine conductor resistance or even voltage drop (and more). how_to_reg Follow . With just a handful of basic mathematical formulas, you can get pretty far in analyzing the goings-on in electronic circuits and in choosing values for electronic components in circuits you design. = 125 $\\times$ 10-3 C Ohm\u2019s law and the electrical formulas related to it are the foundation of all electrical circuits. Moreover, it is a very important thing that allows us to see the rate at which electrical energy transfers from an electric circuit. Academia.edu is a platform for academics to share research papers. Mike Holt worked his way up through the electrical trade from apprentice electrician through electrical contractor, to become one of the most recognized experts in the world as it relates to electrical power installations. Most of the time all you need is a good calculator, but these days many electricians use software and even apps to help with electrical formulas. 746. Electricity is a part of science which come under physics our expert uploaded all required notes for Electricity in few pages . Now you don't need to remember all those complex formulas. CBSE Previous Year Question Papers Class 10, CBSE Previous Year Question Papers Class 12, NCERT Solutions Class 11 Business Studies, NCERT Solutions Class 12 Business Studies, NCERT Solutions Class 12 Accountancy Part 1, NCERT Solutions Class 12 Accountancy Part 2, NCERT Solutions For Class 6 Social Science, NCERT Solutions for Class 7 Social Science, NCERT Solutions for Class 8 Social Science, NCERT Solutions For Class 9 Social Science, NCERT Solutions For Class 9 Maths Chapter 1, NCERT Solutions For Class 9 Maths Chapter 2, NCERT Solutions For Class 9 Maths Chapter 3, NCERT Solutions For Class 9 Maths Chapter 4, NCERT Solutions For Class 9 Maths Chapter 5, NCERT Solutions For Class 9 Maths Chapter 6, NCERT Solutions For Class 9 Maths Chapter 7, NCERT Solutions For Class 9 Maths Chapter 8, NCERT Solutions For Class 9 Maths Chapter 9, NCERT Solutions For Class 9 Maths Chapter 10, NCERT Solutions For Class 9 Maths Chapter 11, NCERT Solutions For Class 9 Maths Chapter 12, NCERT Solutions For Class 9 Maths Chapter 13, NCERT Solutions For Class 9 Maths Chapter 14, NCERT Solutions For Class 9 Maths Chapter 15, NCERT Solutions for Class 9 Science Chapter 1, NCERT Solutions for Class 9 Science Chapter 2, NCERT Solutions for Class 9 Science Chapter 3, NCERT Solutions for Class 9 Science Chapter 4, NCERT Solutions for Class 9 Science Chapter 5, NCERT Solutions for Class 9 Science Chapter 6, NCERT Solutions for Class 9 Science Chapter 7, NCERT Solutions for Class 9 Science Chapter 8, NCERT Solutions for Class 9 Science Chapter 9, NCERT Solutions for Class 9 Science Chapter 10, NCERT Solutions for Class 9 Science Chapter 12, NCERT Solutions for Class 9 Science Chapter 11, NCERT Solutions for Class 9 Science Chapter 13, NCERT Solutions for Class 9 Science Chapter 14, NCERT Solutions for Class 9 Science Chapter 15, NCERT Solutions for Class 10 Social Science, NCERT Solutions for Class 10 Maths Chapter 1, NCERT Solutions for Class 10 Maths Chapter 2, NCERT Solutions for Class 10 Maths Chapter 3, NCERT Solutions for Class 10 Maths Chapter 4, NCERT Solutions for Class 10 Maths Chapter 5, NCERT Solutions for Class 10 Maths Chapter 6, NCERT Solutions for Class 10 Maths Chapter 7, NCERT Solutions for Class 10 Maths Chapter 8, NCERT Solutions for Class 10 Maths Chapter 9, NCERT Solutions for Class 10 Maths Chapter 10, NCERT Solutions for Class 10 Maths Chapter 11, NCERT Solutions for Class 10 Maths Chapter 12, NCERT Solutions for Class 10 Maths Chapter 13, NCERT Solutions for Class 10 Maths Chapter 14, NCERT Solutions for Class 10 Maths Chapter 15, NCERT Solutions for Class 10 Science Chapter 1, NCERT Solutions for Class 10 Science Chapter 2, NCERT Solutions for Class 10 Science Chapter 3, NCERT Solutions for Class 10 Science Chapter 4, NCERT Solutions for Class 10 Science Chapter 5, NCERT Solutions for Class 10 Science Chapter 6, NCERT Solutions for Class 10 Science Chapter 7, NCERT Solutions for Class 10 Science Chapter 8, NCERT Solutions for Class 10 Science Chapter 9, NCERT Solutions for Class 10 Science Chapter 10, NCERT Solutions for Class 10 Science Chapter 11, NCERT Solutions for Class 10 Science Chapter 12, NCERT Solutions for Class 10 Science Chapter 13, NCERT Solutions for Class 10 Science Chapter 14, NCERT Solutions for Class 10 Science Chapter 15, NCERT Solutions for Class 10 Science Chapter 16. Solved Examples. Such properties include a number of poles, speed, frequency, slip, starting current and rated horsepower. This spreadsheet calculates the most common and basic electrical engineering formulas: Single phase and three-phase power in kVA, current in Amps Improve Formulas with Cell References . The Potential difference is given by V = IR Download All Electrical Formula apk 1.9 for Android. All Electrical Formula tricks hints guides reviews promo codes easter eggs and more for android application. The formulas for your electrical exam should remain the same but always make sure. = 4 A $\\times$ 5 $\\omega$ Electric Power Formulas & Equations in DC and AC 1-\u00ce\u00a6 & 3-\u00ce\u00a6 Circuits. Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with the amazing, fun and free SketchUp Make and SketchUp Pro .Add the Engineering ToolBox extension to your SketchUp from the SketchUp Pro Sketchup Extension Warehouse! Electrical Calculator This app is a gift for all Electrical \/ Electronics Engineering graduates and students. 12 Volt Current and Maximum Wire Length - Maximum copper wire length with 2% voltage drop; Abbreviations According the International Electrotechnical Commission - Conforming abbreviations according IEC; All-Aluminum Conductor \u00e2\u0080\u00a6\n\nShare","date":"2021-04-15 08:29:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.4950076937675476, \"perplexity\": 3716.318658044706}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038084601.32\/warc\/CC-MAIN-20210415065312-20210415095312-00392.warc.gz\"}"} | null | null |
Celebrities Support the #FreeBritney Movement: See Snooki, Paris Hilton and More Speak Out
Jacklyn Krol Published: July 19, 2020
Twitter Video
Celebrities are voicing their support of #FreeBritney, the viral movement surrounding Britney Spears' controversial conservatorship which was first put in place in 2008.
Bella Thorne posted resources for fans to sign petitions to let the "Toxic" hit-maker have more rights, while Nicole "Snooki" Polizzi has voiced her concern for Spears in several Instagram Stories and social media posts over the last few months.
Modern Family star Ariel Winter shared a link to an article that allegedly revealed what Spears can and cannot do in her day to day life. "What her 'father' and team is doing to her is absolutely disgusting and devastating," she claimed in an Instagram Story. "Please read and share."
The hosts of CBS' The Talk also spoke out against Spears' management team and father's alleged controlling ways.
On Instagram, Spears' Crossroads co-star, Taryn Manning, also shared her support for the pop star.
"Britney I know you're strong [as f--k] and have your own brain and thought process," she wrote. "To me you look happy and like you're having a blast! Keep up the happy posts and your faith in Jesus Christ. How about instead of #freebritney we say #GodIsWatchingOverBritney."
"How about let's give this wonderful woman the dignity she deserves and earned," she added, addressing recent conspiracies that have gained steam within the broader movement. "Until you know the facts stop speculating and perpetuating the father of lies. We all know she's under a type of control that's unfair and things will be fixed. Have faith. Wish her well! Send good vibes of no fear! Please!"
Meanwhile, last year, on May 5, 2019, Miley Cyrus screamed "free Britney" during a concert. Cyrus voiced her support during a performance of her hit "Party in the U.S.A.," in which she name-drops the Princess of Pop in the lyrics.
See their posts and more, below.
https://twitter.com/bellaffupdates/status/1283904566023479298?s=19
https://www.instagram.com/p/CCwepr7hx0l/
https://twitter.com/BritsBitch19/status/1194836689765580800?s=19
https://www.instagram.com/p/CCt4ybjFZ1r/
https://twitter.com/tanamongeau/status/1284307158851444736?s=20
https://twitter.com/Tinashe/status/1119259622710267904?s=20
https://twitter.com/cher/status/1220146545636429825?s=20
https://twitter.com/hxrryspears/status/1217525880916004867?s=20
https://twitter.com/MissyElliott/status/1279923773098151937?s=20
https://twitter.com/ObjetivoBArmy/status/1127036499663437825?s=20
https://twitter.com/FemmeFatales/status/1284658685608951815?s=20
https://twitter.com/raeraeverret/status/1130227503816159237?s=20
https://twitter.com/JeffreeStar/status/1174449155084886016?s=20
Screenshot_20200718-014339_Instagram
Britney Spears Through the Years
Source: Celebrities Support the #FreeBritney Movement: See Snooki, Paris Hilton and More Speak Out
Filed Under: Ariel Winter, Bella Thorne, britney spears, cher, Courtney Love, Kim Petras, miley cyrus, paris hilton, snooki, Tinashe
Categories: Celebrity News | {
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Q: Find the minimum value of the Hermitian polynomial Let $a\in\mathbb{R}$, $b\in\mathbb{C}$, and $c>0$. Find the minimum of the Hermitian polynomial $R$:
$R(t,\bar t)=a+bt+\bar b \bar t+c|t|^2$.
The only progress I have made is rewriting the function as : $(a+c|t|^2)+ 2Re(bt)$. My problem is that, from what I remember about complex differentiation, is that $f(z)=|z|^2$ is only differentiable at the origin and $g(z)=Re(z)$ is not differentiable.
There must be a way to work with the the differential equation involving $R$.
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{"url":"https:\/\/www.physicsforums.com\/threads\/definite-integration-of-f-ma.668195\/","text":"Definite Integration of F=ma\n\n1. Jan 30, 2013\n\nAstrum\n\nThis isn't a HM question, and I'm asking for an explanation.\n\nThis is \"The effect of a Radio Wave on an Ionospheric Electron\"\n\nThe integration is weird, I don't follow what is being done.\n\n$$a=\\frac{-eE}{m}$$ - reworking of F=ma\n\n$$\\frac{-eE}{m}sin(\\omega t$$\n\nonly interested in the x axis.\n\n$$\\int\\frac{dv}{dt}=\\int^{t}_{0}a_{0}sin(\\omega t) dt$$\n\nThis becomes: $$v(t)=v_{0}-\\frac{a_{0}}{\\omega}cos(\\omega t-1)$$\n- I don't get where this came from, I understand the indefinite integration, but not where the \"\u03c9t-1\" came from.\n\nAnd the last step:\n\n$$\\int\\frac{dx}{dt}=\\int^{t}_{0}[v_{0}-\\frac{a_{0}}{\\omega}cost(\\omega t-1)]dt$$\n\n= $$x_{0} + (v_{0}+\\frac{a_{0}}{\\omega})t-\\frac{a_{0}}{\\omega^{2}}sin(\\omega t)$$\n\nNot sure where the final answer comes from. Could't you just integrate it twice, then tack on the definite integral?\n\n2. Jan 30, 2013\n\nAlephZero\n\nI think some parentheses are in the wrong place.\n$$\\int_0^t a_0 \\sin(\\omega t)\\, dt = \\left[ -\\frac{a_0}{\\omega} \\cos(\\omega t)\\right]_0^t$$\n$$= - \\frac{a_0}{\\omega}(\\cos(\\omega t) - \\cos 0)$$\n$$= - \\frac{a_0}{\\omega}(\\cos(\\omega t) - 1)$$","date":"2018-02-21 08:02:07","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\": 2, \"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\": 0.9382447004318237, \"perplexity\": 1569.139868338029}, \"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-2018-09\/segments\/1518891813571.24\/warc\/CC-MAIN-20180221063956-20180221083956-00759.warc.gz\"}"} | null | null |
Imports System
Imports OpenCvSharp
' Namespace OpenCvSharpSamplesVB
Imports SampleBase
''' <summary>
''' samples/c/polar_transform.c
''' </summary>
Friend Module PolarTransform
Public Sub Start()
Using imgSrc As IplImage = Cv.LoadImage(FilePath.Image.Fruits, LoadMode.Color)
Using imgLog As IplImage = Cv.CreateImage(Cv.Size(256, 256), BitDepth.U8, 3)
Using imgLinear As IplImage = Cv.CreateImage(Cv.Size(256, 256), BitDepth.U8, 3)
Using imgRecovered1 As IplImage = Cv.CreateImage(Cv.GetSize(imgSrc), BitDepth.U8, 3)
Using imgRecovered2 As IplImage = Cv.CreateImage(Cv.GetSize(imgSrc), BitDepth.U8, 3)
Cv.LogPolar(imgSrc, imgLog, Cv.Point2D32f(imgSrc.Width / 2.0F, imgSrc.Height / 2.0F), 40, Interpolation.Linear Or Interpolation.FillOutliers)
Cv.LogPolar(imgLog, imgRecovered1, Cv.Point2D32f(imgSrc.Width / 2.0F, imgSrc.Height / 2.0F), 40, Interpolation.Linear Or Interpolation.FillOutliers Or Interpolation.InverseMap)
Cv.LinearPolar(imgSrc, imgLinear, Cv.Point2D32f(imgSrc.Width / 2.0F, imgSrc.Height / 2.0F), imgLinear.Width, Interpolation.Linear Or Interpolation.FillOutliers)
Cv.LinearPolar(imgLinear, imgRecovered2, Cv.Point2D32f(imgSrc.Width / 2.0F, imgSrc.Height / 2.0F), imgLinear.Width, Interpolation.InverseMap Or Interpolation.Linear Or Interpolation.FillOutliers)
Cv.NamedWindow("log-polar")
Cv.ShowImage("log-polar", imgLog)
Cv.NamedWindow("inverse log-polar")
Cv.ShowImage("inverse log-polar", imgRecovered1)
Cv.NamedWindow("linear-polar")
Cv.ShowImage("linear-polar", imgLinear)
Cv.NamedWindow("inverse linear-polar")
Cv.ShowImage("inverse linear-polar", imgRecovered2)
Cv.WaitKey()
Cv.DestroyAllWindows()
End Using
End Using
End Using
End Using
End Using
End Sub
End Module
' End Namespace
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Sacramento won federal funds for a new federal building at Seventh and K Streets that included a postal facility to handle the increasing volume of mail coming into the city (shown here in 1894).
# Sacramento:
# Indomitable City
# Steven M. Avella
Copyright © 2003 by Steven M. Avella
9781439630587
Published by Arcadia Publishing
Charleston SC, Chicago IL, Portsmouth NH, San Francisco CA
Library of Congress control number: 2003108859
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Front cover: Pedestrians cross the street at busy Eighth and K.
# Table of Contents
Title Page
Copyright Page
ACKNOWLEDGMENTS
INTRODUCTION
1. PREHISTORY TO THE GOLD RUSH
2. GOLD, NATURE, AND CITY BUILDING
3. A MATURING CITY
4. RETOOLING FOR MODERN TIMES
5. THE NEW DEAL AND THE WAR
6. THE EMERGENCE OF A METROPOLIS
7. U RBAN RENAISSANCE
BIBLIOGRAPHY
INDEX
# ACKNOWLEDGMENTS
I am deeply indebted to the authors whose works I have used in putting together this history of Sacramento. Their work is found on every page. Any error is my own. Special thanks go to James Henley, Joseph Pitti, Mead Kibbey, and William Mahan, who read the text and scanned it for errors of omission and commission. My dear, patient friend, Susan Silva, read the various drafts and took time off work to help me with picture selection. I am grateful to Ruth Ellis and her staff at the Sacramento Room at the City Library for their excellent resources and service. Likewise, Gary Kurutz and the staff of the California Room of the State Library have also provided a rich array of resources. A brief stint as an adjunct at California State University Sacramento allowed me to sample a variety of master's theses and student papers related to Sacramento topics. The staff at Arcadia Publishing, especially editors Jim Kempert, Rob Kangas, and Barbie Langston, have been marvelous. My deepest thanks go to James Henley, Pat Johnson, and the entire "family" of researchers, volunteers, and scholars that gather at the Sacramento Archives and Museum Collection Center (SAMCC). For years these archivists, curators, and historians have been my colleagues and friends as I have pursued any number of Sacramento themes. I could not begin to reciprocate all the favors they have done for me. To Jim Henley and the SAMCC staff and volunteers do I gratefully dedicate this book.
—Steven M. Avella
Associate Professor of History, Marquette University, Milwaukee, Wisconsin.
Photo credits: SAMCC (pp. 12, 13 courtesy California State Library Collection; p. 25 courtesy Edwin Beach Collection; p. 110 courtesy Antonia Castenada Collection; pp. 75, 102, 125 courtesy Frank Christy Collection; p. 136 courtesy Franz Dicks Collection; p. 105 courtesy Mather Field Collection; pp. 2, 8, 10, 14, 18, 19, 21, 22, 29, 30, 34, 37, 39, 40, 42, 43, 44, 46, 49, 52–54, 57, 59, 63, 64, 67, 70, 76, 79, 83, 84, 91, 147 courtesy Eleanor McClatchy Collection; p. 89 courtesy Arthur H. McCurdy Collection; back cover and pp. 33, 50, 69, 96, 98, 106, 109, 112, 115, 116, 118, 121, 123, 128, 135, 136, 138, 141, 144, 155 courtesy Sacramento Bee Collection; p. 151 courtesy Sacramento Chapter of the American Society of Civil Engineers Collection; pp. 131, 142, 145 courtesy Sacramento Ethnic Survey Collection; front cover and pp. 60, 92, 126 courtesy Sacramento Metropolitan Chamber of Commerce Collection; p. 27 courtesy Sacramento Society of California Pioneers Collection; p. 81 and p. 86 [Silver Wings Collection] courtesy Sacramento Valley Photography Survey Collection; pp. 149, 152 courtesy Suttertown News Collection; p. 134 Unknown Collection; p. 95 courtesy Joyce Vernon Collection; p. 73 courtesy Weinstock-Lubin Collection)
# INTRODUCTION
Indomitable is an old-fashioned word. "Unconquered" and "not easily subdued" are terms usually applied to individuals, nations, or causes. But when city leaders in Sacramento crafted a new city seal in the early 1860s, they chose the classical Latin "Urbs Indomita"—Indomitable City—to characterize the California state capital. Indeed, the first years of Sacramento's existence did not bode well for the future. By the time the city seal was adopted, the city had already survived all-consuming fires, disastrous floods, and catastrophic epidemics. It had an unstable population, hot summers, and damp winters. Some believed that the site, picked largely for the convenience of gold miners who only needed a place to get off the boat, would fade into oblivion as did other gold rush towns. But something different happened here. By an act of its collective will, Sacramento decided to fight back. Instead of surrendering to the raging waters of the two rivers that embraced the city (the American and the Sacramento), they built levees. To combat the scourge of fire, they mandated brick construction for the commercial district. Sacramento even managed to snag the prize of the state capital, which had bounced around several Northern California towns. Sacramento's first miracle was that it survived the gold rush and its aftermath; other California communities didn't. One effusive publicist, writing boilerplate for a historical celebration commemorating the "Days of '49" in the 1920s, made this observation: "Columbia, Mokelumne Hill, Aurora ... Seven Forks, Jackass Hill, Angel's Camp, Poker Flat—all famous in the days of old and the days of gold—live now for the most part in the memories of other and better times, while Sacramento fulfilled its destiny." To be a part of Sacramento was to participate in an ethic of survival and to continually seek new ways to adapt the city and its development to ever changing situations. Sacramento exists today because it was indomitable. No one easily subdued it.
The following pages provide a sweeping historical overview of the city of Sacramento, California. Home to native peoples like the Nisenan and Miwok, explored and examined by Spanish soldiers and padres, settled by an ambitious Swiss adventurer, Sacramento burst into urban life during one of America's periods of mass hysteria, the gold rush. Here, local merchants and entrepreneurs decided to open their shops along the banks of the Sacramento River and to survey and plat the lands running east and south of the two rivers.
Sacramento had the good fortune of being the western origin hub of the transcontinental railroad. Its ability to process and market the produce of California's rich Central Valley assured its economic viability. But if some believed the city's survival to be the working out of some inexorable "destiny" or the fulfillment of some unchangeable purpose, they couldn't be more wrong. For those who fought, planned, and executed the schemes and developments that kept the city alive, the city's future unfurled as the result of much courage, vision, and hard work, and also because of Sacramentans' conscious desire to keep the city vital no matter what the challenge. Indomitability is in Sacramento's collective DNA. Like all American cities, Sacramento is the product of purposeful human planning that at times cooperates with and, more often than not, overcomes the liabilities of its environment.
Today the city is at the center of a rapidly expanding metropolitan area and is the capital of a state with the fifth largest economy in the world. As yet, Sacramento's history has not been studied as thoroughly nor has the city received as much attention as its two coastal competitors, San Francisco and Los Angeles. But that is changing. Professional historians like Mark Eifler, Albert Hurtado, and Kenneth Owens have brought new perspectives to selected eras of the city's early years and to important figures like John Sutter. Sociologists and others note the ways in which the city has accommodated a remarkable diversity throughout its history as Sacramento has become a crossroads of sorts of the larger American reality. Journalists on the Lehrer News Hours, in search of a diverse American community, have interviewed Sacramentans as representative American citizens. Literary figures like Joan Didion and Richard Rodriguez reflect on their formative years in Sacramento and bring to it a new appreciation of ways in which it influenced their respective imaginations. To travel down its tree-lined streets, to sample life in its ever-renewing downtown, and to share the advantages of Sacramento's natural environment, its purposeful building of city institutions, and its distinct urban culture is to step into the living reality of the indomitable city.
The early seal for Sacramento City boasts the motto Urbs Indomita, Indomitable City.
# 1. PREHISTORY TO THE GOLD RUSH
Sacramento is the capital of the most populous state of the American Union. From its beginnings as one of many small gold rush era towns, it became a political, social, and economic hub of California's interior. Today it stands at the center of a five county metropolitan area that embraces over a million inhabitants and continues to grow.
# THE PHYSICAL ENVIRONMENT
Sacramento is located in the "other California," the vast Central Valley that runs 450 miles through the heart of the Golden State. The valley was once part of the ocean floor, and in its prehistoric period, four great mountain ranges emerged—the Sierra Nevada to the east, the Klamath and the Cascades to the north, and the Coastal Range to the west. These mountains surrounded a huge depression into which they poured waters, sand, gravel, and other sediment. Eventually this "inland sea" receded, some say by bursting through the Coastal Range at Carquinez, leaving behind a valley containing volcanic rock and alluvial fans, the latter from the washed rock of the Coastal Range. But the valley's key characteristic is its flatness.
Sacramento is located in the northern part of this valley, an area watered by the Sacramento River. The Sacramento Valley represents a distinct region of the larger Central Valley. As historian Joseph McGowan has noted, "The physical geography of the valley has been a continuous factor in valley history. Transportation, settlement, irrigation, reclamation, floods and agriculture have all reflected this physical environment, especially the presence of the rivers." The Sacramento River begins on the southern slopes of the Klamath and provides the central waterway for the valley. Into it flow tributary streams fed from snow-capped mountains to the east. To the south, the waters of the Cosumnes and American Rivers also run into the Sacramento. Dozens of smaller streams with names like Antelope, Deer, Mill, and Butte enter the Sacramento as well. Three creeks—Stony Creek, Cache Creek and Putah Creek—add to the river's flow. These waterways bring a rich diversity of soils and dump them on the ground in alluvial fans, providing the basis for the rich agriculture of the valley, an important ingredient in Sacramento's economic stability.
Like many American cities, Sacramento's destiny was shaped by its strategic location. Situated on low lands at the confluence of the Sacramento and American Rivers, Sacramento was initially a gateway for the legendary Argonauts of '49, a convenient drop-off point for miners and a place where they returned for supplies and recreation. Later, the agricultural riches of the valley were "mined" and processed by enterprising Sacramentans. Venture capitalists underwrote these endeavors and evolving transportation systems conveyed them to markets all over the nation and the world.
Sacramento stands out, as one popular history called it, a "City of the Plain." Although the valley slopes steadily from Red Bluff to Sacramento, the perception of the land around Sacramento is flatness. Indeed, visitors seeing its skyline today from the east or west can see the stark lines of its nest of tall buildings from a great distance. The city can be oppressively hot during summer but, thanks to the oceanic breezes that come up through the Sacramento Delta, often pleasantly cool enough in the evenings. Winters are often damp and rainy with daytime temperatures hovering in the 50s and 60s, while nights sometime plunge to the 40s and 30s. Snow and freezing rain are unusual, but on rare occasions Sacramento has been blanketed in winter white. Precipitation varies from year to year. Some years, the rains barely soak the soil. Other years, the heavens open in such a deluge that fear of flooding is real.
Albert Bierstadt portrays a romantic, pastoral Sacramento Valley.
Archaeologist Norman Wilson relates that the valley was inhabited by ancient prehistoric creatures: mastodons, horses, camels, ferocious saber-toothed tigers, huge bears, and fearless wolves. Flocks of waterfowl clouded the skies. Fish like sturgeon and salmon were found in abundance while large herds of elk, deer, and antelope roamed at will. California's symbolic grizzly also gamboled through the region. Giant oaks, sycamores, cottonwoods, willows, and ash once dominated the land near Sacramento's rivers, creeks, and streams. Wild oats grew in abundance, tules choked the riverbanks, and open patches of heavy grass flourished in flat areas. Human settlement, however, permanently altered the environment. Trees were cut down, wetlands filled in, and ecosystems that once sustained prehistoric life were disrupted. The mountains surrounding the valley meant that it was isolated from the rest of the world. Indeed, because it was so remote, population grew slowly in Sacramento and in California's interior in general for many years.
# HUMAN DEVELOPMENT IN PREHISTORIC SACRAMENTO
The lure of the valley was its abundant natural resources. No one is quite sure if the first inhabitants came from the north and east, working their way down the coast, or came up from the south, but no doubt they were nomadic food-gatherers who discovered a rich treasure trove in the valley. The first humans arrived more than 12,000 years ago. The simple law of inertia may have contributed to the first settlements in the valley; people simply found everything they needed and did not want to scale any of the mountain chains to get out.
Permanent villages appeared as people hunted and adapted their food-gathering techniques to accommodate their locality. Native peoples became skilled anglers and hunters. They discovered in the seeds and nuts of the area a new and nutritious food source. Indeed, the millions of acorns that fell from the abundant oaks provided an important and preservable staple for Indian peoples who settled in the area.
The Indians of the Central Valley were a varied and variegated lot with different languages, cultures, and ways of life. Early images of them were largely derived from the observations of white explorers like Jedediah Smith, who characterized the Sacramento Nisenan as "degraded" and "miserable." Later, literary figures such as Gertrude Atherton described the tribes as "lethargic." Hinton R. Helper, a notorious apologist for southern slavery, also weighed in with negative observations on California natives, characterizing them as "filthy and abominable." The offensive ethnocentricity of these comments stand in sharp contrast to the recent scholarship of archaeologists, historians, and ethnographers who have evaluated these tribal peoples on their own terms and provided a new framework for understanding them. The Native Americans skillfully adapted to their environment. Among the tribes of California existed a diversity of languages, housing and clothing styles, religions, and life-passage customs. In Sacramento County, the two major Indian groups that dominated the region were the Nisenan and Miwok. Both of these spoke a variety of the Penutian phylum of languages (other tribes that shared them were the Coastanoan, Wintun, and Yokuts).
An Indian woman whose clothing reflects European influence grinds acorns and seeds.
The Nisenan, the group that occupied most of the area later encompassed by the city of Sacramento, was a branch of the Maidu (sometimes referred to as the Southern Maidu). This group occupied strategic areas along the rivers. One tribal center at the mouth of the American River was called Pusune. Villages between the Cosumnes River and the south fork of the American River, near Placerville, formed another important center. East along the American River were the villages of Sek and Kadema. Between the mouth of the American River and Folsom there were estimated to be ten village sites—four of them in present-day Sacramento alone: Momel at present-day Fifth and Richards, Samor at Fifth and J, Yalis at 30th and B, and another at today's city plaza. Unfortunately, native names were ignored by European settlers, thereby erasing memory of their presence in the area for many years.
Isolation was a defining feature of Nisenan life. Historians suggest that they had few contacts outside their tribelet. They existed in an uneasy relationship with other tribal groups in the area, particularly the Plains-Miwok who lived south of the Cosumnes River. Nisenan people lived in a variety of settings. Some existed in small, extended family groupings of 15–25 people. Others lived in fairly large villages numbering over 500. The central village often acted as the pole of a more extended settlement pattern, and the headman of the village drew these tributary villages together for religious and social gatherings and for hunting. Valley Nisenan lived in dome-shaped houses 10–15 feet across made from cut saplings placed in holes at the perimeter and bent inward toward the center. These were covered with earth, tules, and grasses around an excavated hole in the earth that sank the structure anywhere from 12 to 24 inches. In larger villages, tribes had a similar but larger semi-subterranean structure called a "kum" or dance house with a smoke hole in the center. Sweat houses (one was on present-day J Street) and acorn granaries were also common in village life. As with every tribelet, Nisenan peoples laid out cemeteries, trading sites, ceremonial grounds, and sacred spaces for shrines.
Nisenan people were food-gatherers, taking advantage of the abundance around them. Hunting, gathering, and fishing went on year-round, but in the late summer and early fall, extended families and villages gathered the precious acorn. Drained of its bitter tannic acid, the acorn provided sustenance in times of food scarcity as it was eaten as mush or bread. Nisenan, however, were omniverous. Deer drives were a common collective endeavor, while black bears, wild cats, mountain lions, and rabbits provided pelts and occasionally food. Nisenan people ate birds and fish of various kinds and also insects such as grasshoppers that they drove out of the bushes and roasted. To the horror of visiting Europeans, Nisenan people often wore few clothes (a trait adopted by later inhabitants of Sacramento, especially during the torrid summer months). Men often went naked or wore only a breech cloth. Women were bare breasted, wearing only short skirts made of local materials.
An Indian spears salmon.
Social structure among the Nisenan focused on the headman, a chief or captain who received his authority from the leaders of each extended family and from the village shamans. The headman arbitrated disputes, convened the people to discuss major problems, and oversaw the gathering activities. Property was both communal and personal. Families staked out fishing sites and oak groves while each person kept his or her own personal hunting accouterments. Women owned and could inherit cooking utensils.
The Nisenan found the meaning of their world through religious symbols and myths. Mountains like Mount Diablo and the Sutter Buttes were invested with religious significance. Like every society, the Nisenan had myths to explain the reasons for existence and the purpose of life. Creation stories include the activity of a trinity of beings: a huge turtle that brought the earth up from the bottom of the sea, a world creator who fashioned the land, and the coyote, a human spirit who did both good and bad in bringing about the human race. Religious expression took form in various cultic rituals. The most elaborate form of Nisenan ritual, the Kuksu ceremony, consisted of a dance, done by initiates with costumed feather headbands who represented the spirits of gods. The ceremonies were held in winter inside a large, earth-covered dance house. Another religious observance ritualized mourning with an annual celebration in late September or early October. This involved a ritual marking of days and the construction of a central pyre around which mourners and dancers performed as the belongings of the deceased were burned. Rituals indicating the change of seasons included the Kamin Dance, which celebrated the beginning of spring, and the Lole Dance honoring the first fruit. The Nisenan had doctors or shamans, whose ceremonies also took place in the dance house. The religious doctor or ocpe gained control over spirits through dreams or other specialized experiences. He was the chief bridge between the human and the supernatural and figured prominently in the rituals of the dance.
Nisenan villages consisted of semi-subterranean, dome-shaped houses.
While the Nisenan were the primary people living on the lands later to be encompassed by Sacramento, nearby were the Miwok, another Penutian-speaking people with a different culture. Miwoks did concentrate south of the Cosumnes River, but through intermarriage and sometimes-permeable boundaries, they also figured into the ethnographic realities of the region. The Valley Miwok (one of three groups including the Coast and Lake Miwok) were also food-gatherers like the nearby Nisenan, subsisting on acorns, small game, and fish. One distinction of Miwok culture was that some of its tribal members had actually been sent to the missions and inducted into the religio-cultural world of the Spanish settlers. These interior converts—called tularenos by the mission padres because they lived in the tule-choked areas along the rivers—were to be a beachhead for subsequent missionization of the California interior. Miwok people in general, however, were resistant to Spanish influence.
# CONTACT AND ENCOUNTER WITH EUROPEAN CULTURE
The eighteenth century, a great epoch of European contact with California, brought some of the first Europeans to the Central Valley. The Spanish were the first to explore the valley as part of their larger policy of extending influence and settlement into Alta California. From the beachhead of presidios, pueblos, and missions along the coast, forays into the Central Valley took place. However, the valley held no particular allure and early encounters were from a distance. Spanish explorer (and later governor) Pedro Fages recorded his vista of the Sacramento Valley in 1772 as he looked over the broad area from a spur of Mount Diablo around present-day Antioch. On this same expedition, Franciscan Friar Juan Crespi looked out over the Sacramento Delta and beheld a land "as level as the palm of the hand." In 1776, Juan Bautista de Anza headed an expedition that slightly probed into the delta.
In the nineteenth century, the pace of exploration and engagement picked up. In September 1808, Ensign Gabriel Moraga left with 13 soldiers from Mission San Jose to explore the rivers, locate Indian settlements, and scout for possible mission sites. He was also charged (an order he wisely ignored) with recapturing tularenos who had abandoned the missions. He reached the American River between present-day Rancho Cordova and Folsom in October, calling the American Las Llagas (literally "the wounds" to commemorate the suffering of Christ). His most significant move, in terms of the future state capital of California, was his naming of the Feather River, the Rio del Santissimo Sacramento—the River of the Most Blessed Sacrament (a reference to the Catholic doctrine of the real presence of Christ in the Eucharist). He then moved west to the present Sacramento River and named it the Jesus Maria. He followed the east bank of the river upstream as far north as Butte City before he turned back. Moraga had mixed encounters with the Indians and reported negatively to his superiors on the possibility of a mission in the Central Valley. His advice would be followed.
However, others still believed that California's mission chain should extend into the Central Valley. A visit up the Sacramento River in October 1811, spearheaded by two Franciscan priests, Ramon Abella and Buenaventura Fortuini, suggested again that a Catholic outpost was needed in the area. Still, the Spanish were reluctant. In 1817, Abella returned with Friar Narciso Duran and Captain Luis Arguello, still looking for a good mission site. However, by this time Spain's grip on its American colonies was weakening. In 1821, Mexico declared its independence and California became a Mexican province.
European penetration continued. In 1823, a vessel of the Russian Imperial Navy led by Estonian Otto von Kotzebue is thought to have ascended the Sacramento as far as the American River. In this party were 20 Aleuts who camped near present-day Freeport. The American presence began in 1827 when the famous Jedediah "Bible Totin' " Smith explored the valley from the south and camped along the American River near the present-day site of California State University, Sacramento. Traveling east over the snowy Sierras, Smith related to his trapper friends and to others in the East the abundant resources in the region. The Hudson Bay Company headquartered at Fort Vancouver on the Columbia River took note of Smith's descriptions. Members of the Hudson Bay Company's fur brigades began to visit the valley every year from 1830 to 1844. Michel Laframboise was perhaps the best known of the Hudson Bay Company trappers. Other explorers noted the large size of the various Indian encampments they encountered. By the middle of the 1830s, as McGowan notes, the Sacramento Valley was increasingly well known, but visited only by transients looking for pelts. In May 1833, Captain Juan Bautista (John) Cooper, an American who became a naturalized Mexican citizen, petitioned for a land grant along the American River south of present-day Power Inn Road and Fruitridge. Cooper, however, never followed through on the application and allowed the petition to lapse.
Conflict with Native Americans was inevitable as more whites came into the region. Moraga was hit with flying debris during his trip in 1817. Armed clashes between trappers and Indians took place near the Cosumnes River near Galt in 1828. Some fighting was spurred by the growing practice of horse-stealing by the Miwok and other valley Indians. (Native peoples either ate or mounted the horses to improve their chances against Europeans.) But more fatal than lead or steel, the European encounter with the Indians of the Sacramento region brought deadly disease. In 1833, tribes' numbers were decimated by a devastating epidemic of malaria, brought by Hudson Bay Company trappers. This disease obliterated whole villages of the Nisenan and forced survivors into the hills. One estimate suggests that nearly 75 percent of the native population died during this plague. Those who survived were too weakened to resist white settlement. Scholars disagree over the number of Native Americans who lived in the Central Valley before and after the malaria epidemic. It is not known how many lived in the region of Sacramento. By 1839, however, the year John Augustus Sutter appeared to claim the lands that would make up the future city, the numbers were smaller than they had been before.
# JOHN SUTTER AND HIS LEGACY
Sutter was the first European to establish a permanent settlement on the lands that would encompass the city. Different viewpoints exist on whether to accord the title "Founder of Sacramento" to John Sutter. Indeed, the first efforts at city-building in the region, the ill-fated town of Sutterville, were done at his direction. His role as a civic official was evidenced by his participation as judge for the election for magistrate of the Sacramento District in September 1846. But others note that the present city of Sacramento was born after he had sold the fort and repaired to his farm along the Yuba River. Like every other figure in the American West, Sutter has been the subject of revisionist scholarship. These scholars have noted that his life has been more defined by myth (some of it of his own making) than hard historical fact. Throughout most of Sacramento's history, he has been memorialized by a myriad of romantic accounts reprinted in tour guides, promotional materials, and even a host of books. He has even been the subject of a Hollywood movie. Even more, school children of the region were taught over the years to hail him as the founder of the state capital. Distilling the factual from the fanciful reveals a core of important facts about Sutter.
Johann Augustus Sutter was born February 15, 1803 in Kandern in the German margrave of Baden, 13 miles north of Basel, Switzerland. He was the son of Johann Jakob Sutter, a foreman in the paper mill, and Christina Wilhelmina Strober, a clergyman's daughter from Grenzach, located up the Rhine River. Sutter grew up in Kandern, a town that witnessed the movements of Napoleon's army, and one historian suggests that Sutter's love for the military, for uniforms and military pomp stemmed from his boyhood experiences of witnessing armies coming and going through his native town. He left Kandern at age 15 and attended school at Neuchatel, Switzerland, and he began an apprenticeship with Emmanuel Thurneysen, a publisher, bookseller, and printer in Basel. Sutter did not take to the publishing and bookselling trade and quit Basel for a job as a draper's clerk in nearby Aarburg. There he met Anna or Annette Dubeld and impregnated her, marrying her one day before the birth of his first child, a son they named Johann Augustus Sutter, Jr. Four more children followed. Sutter had by then moved to the Canton of Bern, where Annette's mother, a comfortable widow in Burgdorf, financed him in a dry goods business. When this failed, he entered the reserve corps at Bern in 1828, rising to the rank of first under-lieutenant. In later years he embellished this slim military record with stories of participation in campaigns in Spain in 1823–1824 and service under the French King Charles X at Grenoble in 1830. Sutter never lost his boyhood love of military titles and regalia. For whomever he worked, Mexican or American governments, he always managed to wangle titles such as "Captain" or "General."
Adventurer and settler John (Johann) Augustus Sutter embellished his brief military career and fostered a boyhood love for military titles and regalia.
Sutter's love for titles and his own exaggerated accounts of his past life contributed to the historic romanticization of him by successive generations of Sacramentans. However, modern historians are not so slow to ignore his short-comings and quick to pierce the bubble of his sometimes egregious self-promotion. Among his many character defects, Sutter had an unfortunate propensity to pile up debts and compounded his problems by continuing to borrow. This provided the context for his hasty departure from Europe. In mid-May 1834, he secretly liquidated his assets and abandoned his struggling family, taking passage for America and landing in New York. He not only left behind his family but also a host of debts.
Sutter became a trader and set out for Missouri, the gateway to the American West, with two German and two French companions. He arrived first in St. Louis where he met a Westphalian, Johann August Laufkotter, with whom he traveled for two years. Here, his biographers note, Sutter began to embellish his previous life (free from worry that anyone would check the veracity of his stories). In Missouri, Sutter witnessed the adventure and profit potential in the commercial exploitation of the American West. In 1835 and 1836, Sutter joined two caravans headed for the lucrative trading center of Santa Fe. On the way he learned the importance of Bent's Fort, a military/commercial outpost on the upper Arkansas River in present-day Colorado that functioned as a magnet for commercial interaction between whites and Indians. Some have suggested that this was the model for Sutter's Fort in Sacramento. (Others claim it was Fort Vancouver or Sitka.) Through a fellow traveler on the Santa Fe expedition, fur-trader Charles Beaubien, he also heard stories of far-off California with its natural wealth and soft climate.
At Santa Fe, Sutter's less than honest side emerged when he became involved in a scheme that defrauded some of his German clients. His companion Laufkotter soon came to mistrust his voluble and glad-handing friend. When the situation became untenable in 1837, Sutter simply moved further west and spent a year in Westport, Missouri (today Kansas City). Here too, in pursuit of commercial success, Sutter skirted the boundaries of the law and of propriety by selling illegal whiskey and enjoying sexual favors from Shawnee women.
A year later, when his presence in Westport became compromised, Sutter gathered a small band (including a young Indian boy he had "purchased"), linked up with a group representing the American Fur Company, and accompanied them west along the Oregon Trail, reaching western Oregon in early November. He tarried for a time in the Willamette Valley and then moved north to the headquarters of the Hudson Bay Company at Fort Vancouver. Here he was anxious to head directly south to California, but officials of the company dissuaded him from the winter-time trek and urged instead that he go first to Hawaii. Sutter concurred and with eight companions boarded the Columbia, making for Honolulu in mid-December. In the Hawaiian kingdom, he claimed that he had so impressed King Kameahamea III that the monarch wanted him to stay and command his native army. But Sutter was restless to move on, so the Polynesian king let him go with the gift of ten indentured servants—two women and eight men for three years. These kanakas as they became known were invaluable to Sutter in his projects.
Prior to the discovery of gold, Sutter's Fort was the first stop for settlers coming into what would become Sacramento.
Sutter left Hawaii and visited the Russian colony at New Archangel (Sitka), Alaska. Here he learned of yet another commercial opportunity as he heard of the fur trade—pelts claimed from sea otters and seals—and of the Russian commercial outpost near Bodega Bay, Fort Ross. He then set sail on the brig Clementine for Mexican California and on July 4, 1839 was deposited in Monterey, the capital of the province. He arrived at a time when the Mexican government was concerned about the frequency of American incursions into the Central Valley. Of the valley Sutter had no doubt learned much from the Hudson Bay officials at Fort Vancouver and the Russians at Sitka—both of whom had given him coveted letters of recommendation to Alta California Governor Juan Bautista Alvarado. He presented himself to Alvarado and requested the opportunity to establish a rancho in the valley. The Mexican governor surely believed that the eager stranger could serve his goal of pacifying and securing California's interior and also curtail the activities of Indians who used inland bases to stage horse-stealing raids on coastal ranches. He freely gave Sutter permission to pick a site, settle on it, and then return a year hence to file for Mexican citizenship and apply for the land grant he desired.
Before he debarked, Sutter consulted with the Russians at Fort Ross. He also stopped to visit with General Mariano Vallejo at Sonoma. Sutter then proceeded up the Sacramento River on a small schooner, the Isabella, along with a smaller boat, both of which he had chartered in Yerba Buena (San Francisco). His crew contained a medley of nationalities, both German carpenters and kanakas. Using Miwok guides (who deftly directed him away from their territories), Sutter meandered along the riverbanks until he finally traveled upstream on the American River. Near the Nisenan Pusune Rancheria, a short distance from the mouth of the river, he unloaded his vessel and pitched his tents. The local Indians may have feared that Sutter was hunting for escaped mission Indians, but he made gestures of friendship and conciliation. Nonetheless, Sutter still felt compelled to remind the native inhabitants of the deadly firepower of Europeans. Shortly after his landing, he symbolically positioned the three cannons he had brought with him from Hawaii and fired a nine-gun salute when the Isabella (with several who bailed out of the project) sailed back to San Francisco. The roar of the guns frightened elk and deer and put thousands of waterfowl to flight, and it reminded the native inhabitants that Sutter meant to defend himself.
The kanakas made temporary grass huts near the river. Sutter moved a bit south of the landing to higher ground and there began his fort, which grew slowly but surely on a small knoll well back from the rivers.
Sutter's land grant was redrawn by Jean Jacques Vioget, who made the original in 1841.
# CREATING THE NEW OUTPOST
On August 29, 1840, Sutter petitioned Governor Alvarado for Mexican citizenship and then made application for his land grant. This claim would later become the subject of a great deal of controversy. During the winter of 1840–1841, Sutter engaged engineer and navigator Jean Jacques Vioget, a former Swiss drummer in Napoleon's army, to make a survey and prepare the maps required by the Mexican officials to validate his land grant. Sutter believed that his grant lay between the present Sutterville Road and a line drawn east to west through the northern edge of the Sutter Buttes and from the Sacramento River to a line a few miles east of the Feather River. However, Vioget, who used homemade equipment to plot the location, made an error of 14 miles on the southern boundary, drawing a latitude 11-and-a-half miles north of the American River. Ironically, not even Sutter's Fort was included in the map, and since the grant specified that Sutter was not to include in his possessions the overflow lands of the river, the site of modern Sacramento was also excluded. The ensuing confusion of land titles was further complicated when the two copies of Vioget's map were lost. McGowan notes that although there would be controversy, the basic outline of Sutter's grant included lands along the south bank of the American River from the big bend near Brighton (present site of California State University at Sacramento) to the fort, the area around Sutterville, and both banks of the Feather River from a point west of present-day Nicolaus to a few miles north of Marysville. Sutter's grant was not an unbroken area, but rather a few choice regions along the river where the soil was deep and the site accessible by water.
This has been called the "earliest known" painting of Sutter's Fort.
In 1841, Sutter was awarded 11 leagues or 44,000 acres of an "empresario" grant that envisioned a colony of Swiss families. Thus, he named the territory Nueva Helvetia (New Switzerland). The main requirement of an empresario grant was that the grantee had to settle 12 families on the land. Sutter began to organize the area according to his plans to establish an inland empire. Using the clay-like soils found around the area, Indian laborers constructed modest adobe buildings and finally a rectangular fortress measuring 300 by 160 feet with high walls. Bastions crowned the northwest and southwest corners. Within the fort, Sutter constructed buildings to be used for residences and shops. New Helvetia was from the start intended to make money for Sutter and to establish him as a benign grandee of the area. This first outpost of European civilization was and continues to be historically significant. The fort, allowed to fall into ruins after Sutter quit the area in 1850, was nearly totally destroyed until the romantic 1890s when the Native Sons of the Golden West began its restoration, a project that the State of California finished. The fort is today one of Sacramento's most important tourist attractions.
From the fort, commercial and development activities were launched. Imitating what he had seen of Anglo-American, British, and Russian traders, Franciscan missionaries, and Californios, Sutter made his profits through Indian labor. Sutter's relationships with the Indians were often rocky, and despite their growing dependence on white goods, the Nisenans actually kept their distance from Sutter. Eventually, however, a combination of bribes and force drew them into his service. He learned the social system of the Nisenan and Miwoks and appealed to the local headmen for help in securing workers. Soon a respectable Indian workforce, watched over by white overseers, was in place. During the all-important wheat harvest, close to 600 Indians worked in Sutter's fields. Sutter's wealth, as Albert Hurtado notes, was built almost totally on Indian labor.
Indeed, their hard work was the economic lifeline of the fort. They hunted for pelts and game; they processed animal hides and rendered precious tallow for candles. They made the adobe bricks that were used for the fort, they planted and harvested the grain crops, and they tended herds of cattle, sheep, and pigs. Sutter even drafted the Indians into a makeshift army (replete with uniforms purchased from the departing Russians at Fort Ross). With these "troops" he warded off Miwok raids on his fields and herds and launched a punitive expedition against the Mokelumne Miwok, who were stealing horses and selling them to traders. Nonetheless, Indians were not passive recipients of Sutter's direction. Some rebelled and fled his service. Sutter employed corporal punishment to keep Indians in line and violently punished two "rebellious" Miwoks, Rufino and Raphero, placing the head of Raphero on a grisly pike outside the fort. Sutter also used Indian labor to pay off debts he owed to creditors like Antonio Sunol and William Leidesdorff.
Acts of violence notwithstanding, Sutter's conquest of the Indians came primarily through his attaching them to a European system of peonage and commerce. He also sold them liquor. Indians were paid in European goods that they craved. Sutter issued metal disks to the Indians that were punched with a distinctive hole for a day's labor. These disks, worn like a necklace, were redeemable only in Sutter's store, increasing their growing dependence on white accouterments. Two weeks' work, for example, secured a muslin shirt or cotton for trousers. Most Indians who worked for Sutter were not slaves but peonized labor, as was true throughout the Mexican empire. In this first set of economic activities, the city's modern commercial life was born.
Sutter also altered the patterns of native life by injecting new work demands into the traditional cycle of food-gathering that had marked the life of the Central Valley tribes. The Native Americans learned to adapt to European work schedules and notions of time. He altered traditional Miwok and Nisenan marriage customs such as polygamy, forbidding chiefs from having more than one or two wives. To prevent the domination of women by the chiefs, Sutter matched eligible Indian women with available males. Sutter himself, however, continued to cohabit with a number of women. Before his abandoned wife, Annette Dubeld, caught up with him in 1850, Sutter lived with a kanaka woman named Manuiki and also consorted with Indian women, according to the negative recollections of one of his overseers, Heinrich Lienhard.
Sutter's Fort was also a workshop, providing small manufactured goods and supplies for those coming into the province and anxious to settle in California. Sutter himself served as a generous grandee for those who subsequently received large grants to other parts of the valley. The New Helvetia grant was but the first viable land concession made by the Mexican government in the Sacramento Valley, and imposed a new arrangement of property holding, development, and future historical development on the northern valley. Sutter's Fort was also the first stop for other grantees in what would become Sacramento County. John Sinclair and his wife, Mary, were the first settlers on the Del Paso land grant, north of the American River, one of the last to be broken up. To the east of Del Paso were Rancho San Juan and Rancho Rio de los Americanos. To the south of New Helvetia, the Mexican government awarded the Omochumnes or Sheldon grant to Jared Sheldon. Peter Lassen and William Ide received land from Mexico farther north in the valley. Sutter helped Pierson B. Reading receive lands from Mexico. John Bidwell, an overland immigrant who was a major player in the founding of the city of Chico and one of Sutter's first faithful retainers, arrived with the Bidwell-Bartelson party in 1841 and worked alongside his patron, taking over his Hock Farm operation in 1844. Bidwell did not receive lands from Mexico but from a former landholder near Chico.
# THE DYNAMIC OF MANIFEST DESTINY
Ultimately, Sutter's New Helvetia was drawn into the vortex of larger events taking place within the Mexican empire and in the rapidly expanding United States. In 1844, Sutter became enmeshed in a complicated internal struggle for dominance in California that pitted Governor Manuel Micheltorena against a cabal led by General Jose Castro. Loyal to Micheltorena, Sutter assembled a military force and marched south to fight the governor's foes. A series of shifting alliances left Sutter out on a limb, and he was captured at the Battle of Cahuenga. Micheltorena was deposed, but his successor Pio Pico and his allies did not punish the Sacramento adventurer. However, when he returned to his fort in 1845, he found things unraveling. Many of his native workers had fled, and Sutter responded with violence, executing the Miwok Raphero. Continual difficulties with the Native Americans ensued.
Even more importantly, in the 1840s, a burst of ideologically inspired expansionism gripped the popular imagination. Manifest Destiny, as it was termed, summoned American economic and cultural energies to press the new nation westward to the Pacific Coast. Thousands upon thousands began the trek west, hoping to find land, security, and fortune in what Anglos perceived was an "undeveloped wilderness." Sutter soon began to catch the wave of westward migrants, and his fort became an important stopping-off point for visitors who continued to make their way into the valley. Hunters, trappers, and British sailors (who had deserted in San Francisco) were welcomed at the fort—all receiving warm and gracious hospitality from Sutter, whose reputation as a generous and congenial host left many warm memories, even among his critics. The overland emigrants, tired and worn out from the strenuous journey and in particular the arduous trek over the Sierras, found Sutter's Fort an oasis. In 1844, the Townsend-Stevens-Murphy Party brought wagons over the top of the Sierras. Two years later, the ill-fated Donner Party unsuccessfully tried to do the same. It was Sutter who attempted to bring succor to the trapped and starving party by donating flour, meat, and some horses.
U.S. interest in California began to peak as well, especially with the expansionist politics of the Democratic Party in the 1840s. By 1841, the mapping expedition of Captain Charles Wilkes had sent a group of scientists to Sutter's Fort to examine the territory. One of the West's most intrepid explorers, John Charles Fremont, then a lieutenant in the U.S. Corps of Topographical Engineers, crossed the Sierra in the winter of 1844 in company with scout Kit Carson. Coming down to Sacramento, he encouraged Anglo settlers in the valley to rise up against Mexico. He too was graciously received by Sutter, who fed and supplied him. This growing American presence presaged the disruption of relations with Mexico that began to intensify after the election of expansionist James Knox Polk in 1844. By mid-1846, the United States and Mexico were at war. A local revolutionary movement, begun in Sacramento County, raised the standard of revolt and hoisted up the Bear Flag as a symbol of secession from Mexico. With the onset of hostilities, Colonel Fremont captured Sutter's Fort and temporarily renamed it "Fort Sacramento." An incurable opportunist, Sutter acquiesced in the American take-over and even accepted an appointment as a U.S. Federal Indian Subagent in 1847. He even permitted, after months of negotiation, the interment of General Mariano Vallejo at the fort. By 1848, the Treaty of Guadalupe Hidalgo ended the war and allowed the purchase of California by the United States.
Explorer John Charles Fremont encouraged Anglo settlers to rise up against Mexico.
While Sutter managed to tailor his loyalties appropriately during these days of transition, the dynamics of change affected his dreams for New Helvetia. The shifting politics and the disruptions of his labor supply placed his plans for an inland empire in jeopardy. Sutter owed money to virtually everyone. He gradually realized that he had to market the lands of his grant.
# CITY BUILDING
Interestingly, few of the many immigrants who availed themselves of Sutter's legendary hospitality remained in the area, in part because Sutter himself seemed disinclined to create a city or a town around his fort that could sustain a stable population. However, by 1845, this changed, and Sutter began to plan a city along the river. His first effort at city building took place on a plot of land 3 miles downstream on the Sacramento River, a location with banks high enough to keep back the waters that occasionally flooded the areas around the fort. "Tomorrow we are surveying at least the town or city: but not close by the fort," Sutter wrote to Pierson B. Reading on January 29, 1845. His friend John Bidwell surveyed the site working together with pioneer immigrant Lansford B. Hastings. The new city was near the intersection of present-day Sutterville Road and the Sacramento River. The grid planned by the surveyor contained 200 lots. Later the town was named "Suttersville" and finally "Sutterville." However, as with many of Sutter's moneymaking schemes, he soon lost control of the project to his creditors. Hastings managed to gain title to many of the lots, along with Bidwell and another early Sacramentan, George McKinstry. Sutterville might have proven a more popular site than Sacramento, but a number of events intervened that shifted the foundation of the city to the north along the riverfront.
Debt had stalked Sutter throughout his life and, lacking a benefactor or a company or a royal patron, he had been compelled to finance the fort himself. In the end, the money he borrowed and promised to repay eventually cut off his expansionist dreams. One of his more pressing notes was for $30,000, which he owed the Russians for the purchase of the contents of Fort Ross in 1841. Sutter had dismantled the building, the fort walls, and some of the redwood planking and moved it to New Helvetia. He also had acquired a schooner that he renamed the Sacramento and additional livestock, including 2,000 head of cattle, horses, and sheep driven overland to Sutter's Fort. The beginnings of American rule over California were the beginning of the end for Sutter and New Helvetia. Pressed by the debts he owed to so many, Sutter characteristically began to expand his operations.
With the help of a detachment of Mormons, who had arrived after the Mexican War, Sutter began to build a gristmill on the American River. Mormon leader Samuel Brannan also arrived at the fort, recognized the location as a good spot for retailing, and built a store outside the fort. In 1845, James Wilson Marshall arrived from Oregon. Hoping once again to provide needed commodities for the steady flow of people coming into the Central Valley, Sutter negotiated a treaty with the Koloma Indians to build a lumber sawmill on the banks of the American River. With Marshall's skills as craftsman and mechanic, Sutter put him to work in the Sierras, near Coloma, to build the lumber mill, unwittingly setting in motion a new epoch of western and even world history when Marshall's workers discovered gold in the traces of the mill.
Sacramento had not yet been born in 1848, but some of its elements were already in place. Sutter had built a network of commercial activity that drew its wealth from the rich natural environment. Other settlers began to manufacture things that people needed: hides, tallow, food, and grain. Wealth was generated by the labor of the native peoples. The location was enhanced by its proximity to rivers and waterways, which could become vehicles of commerce. When gold was discovered, the world was turned upside down. By this time, however, Sutter had done as much as he could. The task of city formation and the creation of a community would be left to his son.
This 1848 view of the Sacramento landing, painted by C.A. Tabor, shows the foot of I Street. The vessel moored by the shanty is the Provinence, which was dismantled and reconfigured into a store owned by George McDougal and William Blackburn.
# 2. GOLD, NATURE, AND CITY BUILDING
The past was prelude in the formation of Sacramento. Indian inhabitants shaped the human ecology. Sutter and the people he attracted produced commercial possibilities, but it was the gold rush that created Sacramento. This massive movement of people and capital literally brought the world to the shores of the Sacramento River and created a center for social, commercial, and cultural exchange that exists to this day.
# THE GOLD MINING BOOM
Sacramento did not just happen; it was planned. Like so many cities in America, a handful of enterprising and even avaricious developers saw advantages to developing the site and took financial risks to do so. In Sacramento, this drive is embodied in the career of Samuel Brannan, who played a major role in the development of the city. Brannan was a native of Maine. Fleeing an abusive father, he made a picaresque journey across America, causing historian Kevin Starr to compare him to Mark Twain's Huck Finn. Brannan early on became interested in land speculation and lost money in his first efforts. To recoup his fortunes, he learned printing and became a writer and an editor. In the 1830s, he converted to Mormonism and turned his energies to the propagation of this expansive "westering" denomination, which was in its formative years. When the persecuted Mormons searched for a secluded site on which to construct their new Zion, Brannan laid his skills for writing and publishing at the service of the growing church. He combined his religious zeal with the expansionist doctrines of the Democratic Party of the 1840s and played a role in a scheme to settle Mormons in the Mexican province of California and then turn over their land to the U.S. government. These plans coincided with President James K. Polk's efforts to provoke war with Mexico in order to seize Texas and California. In February 1846, Brannan and a party of more than 200 left New York and arrived in Yerba Buena in July. By the time he arrived, the Mexican War had begun and California would soon be permanently detached from Mexico. Nonetheless, Brannan set to work to develop a Mormon enclave in California and also to create a profitable climate for land speculators. He and his followers built homes, shops, and food processing facilities. Brannan also began a newspaper, the California Star. As his new enterprise boomed, Brannan traveled back to Salt Lake in order to convince Brigham Young to relocate Mormon headquarters to California. Young balked and Brannan moved back west.
In 1847, Brannan formed a new partnership with fellow merchant C.C. Smith and opened a store at Sutter's Fort. Brannan had long seen the agriculture of the valley as an important component of future growth and wealth. His store traded groceries and other dry goods for raw grains, animals, and foodstuffs. But profits were slow at New Helvetia—until early 1848 when sawmill builder Marshall discovered gold. News of this was communicated to Sutter, who traveled to the site and determined to keep the matter secret. However, when a teamster took a nugget found along the mill trace to the C.C. Smith store and tried to purchase a bottle of brandy with it, the cat was out of the bag. Smith informed Brannan. Quick to seize the opportunity, Brannan and Smith stocked their store with everything necessary for a stay in the mountains. Then on May 12, 1848, Brannan literally shouted the news of the discovery on the streets of San Francisco. The gold rush was launched and so was Brannan's fortune and the city of Sacramento.
Land speculator, merchant, and newspaper publisher Samuel Brannan played a major role in the city's development.
As the news spread, Brannan purchased more goods in order to outfit the steady flow of miners coming up the Sacramento River. The shop at New Helvetia was too small and too removed from the patterns of movement up the river, so Brannan and another associate, P.B. Cornwall, sought to relocate to a more strategic location and appealed to developer Lansford Hastings for free land in Sutterville. In exchange, Brannan held out the tantalizing prospect of future riches for all involved when his warehouse and shop would become the economic hub of the new community. But, because of previous dealings, Hastings was mistrustful of Brannan and refused the offer. Brannan then moved up the Sacramento River to the foot of modern K Street and claimed a small clearing cleaned out by Sutter's men for use as a boat landing, called Sutter's Embarcadero. On this spot, he simply overrode the claims of previous occupant George McDougal, who there ran a ferry service across the Sacramento. Brannan erected a few tents and set up shop alongside the river. By this simple act, Brannan established the location of the present-day city of Sacramento, for around this wilderness embarcadero, a new community would arise. The site, however, would have its problems. As historian Mark Eifler noted, Sutterville enjoyed a geographically better position, safely secure from periodic flooding. Nonetheless, Brannan's command of goods and capital exercised a defining role. Despite its being in a flood plain, the strategically located embarcadero trumped any fears generated by Mother Nature; it had a better claim by virtue of what Eifler called the "geography of trade."
Sam Brannan relocated his store to Sacramento's early embarcadero, quickly putting up this frame storehouse at Front and J Streets.
Hundreds and then thousands—nearly 6,000 alone in the summer and fall of 1848—came to the embarcadero, tethered their schooners in the still-deep waters of the river, and jumped off to begin the trek into the gold country. With them came more merchants, entertainers, gamblers, and people who, like Brannan, were looking to capitalize on this mighty rush. Brannan made as much as $5,000 a day, almost all of which was sheer profit. In the midst of it all, Sutter, who actually owned the land on the embarcadero, left the burgeoning community just as it was beginning to make money. In fact he was in debt to Brannan (as he was to many people in the valley), who seized a precious opportunity to solidify his claims to this lucrative location. In the fall of 1848, Sutter's long-abandoned son, John Sutter, Jr., arrived at the fort, anxious to meet up with his father. The solemn younger Sutter soon discovered that his father had many debts and little liquid capital. When the elder Sutter delegated to his son the "honor" of straightening out his tangled finances (and promptly left for the Coloma gold site with a gang of Indian peons), Brannan was ready to pounce. With the assistance of another fort merchant, Samuel Hensley, he pressed young Sutter to hire engineers to survey and plat Sutter's holdings and to plan a new city that would begin at the embarcadero. City lots could be sold to retire the debts.
Brannan's suggestions were really the best since the land stretching east from the river was now in high demand. Young Sutter gave his approval and agreed to name the new community "Sacramento City." In late December 1848, Hensley and Sutter Jr. secured the services of Peter H. Burnett, a lawyer who had migrated from Missouri to Oregon and taken an active hand in the formation of territorial government. Burnett had come to California in September 1848 but, like many, found mining distasteful. He arrived at Sutter's Fort just when he was most needed. When they offered the attorney one-fourth of the gross proceeds from the sale of the lots, he snapped at it.
In December 1848, Sutter Jr. commissioned Captain William H. Warner, an army engineer, and his assistants, Lieutenants William Tecumseh Sherman and Edward O.C. Ord, to survey and plat the new city. A traditional American grid pattern was superimposed on the lands from the river east to just beyond Sutter's Fort and south from Sutter's Slough—a finger of the Sacramento River extending inland to Sixth Street between I Street and the American River. East-west streets bore the letters of the alphabet while north-south streets had numbers. The street along the river was Front Street. Young Sutter donated alleys and streets as well as space for ten public parks. The city was now "packaged" for sale, and a land rush began in earnest.
# CREATING SACRAMENTO CITY
Even before Warner's survey of the city, the embarcadero exploded with life as miners turned it into the starting point for their march to the gold fields. Sacramento's first business establishments were ships tied up along the riverside. These vessels, along with the makeshift tents of Brannan and others, served nascent Sacramento as stores, freight warehouses, jails, and even water purification centers. Samuel Hensley and Pierson B. Reading put up the first frame building on the corner of Front and I Streets. Brannan quickly put up a frame storehouse at Front and J. By early April 1848, 12 simple buildings rose near the busy embarcadero and hundreds of eager immigrants bought their supplies as they moved through. The sale of the lots began in January 1849, just as the pace of gold fever was accelerating.
Optimistic miners and merchants who hoped to "mine the miners" descended on Sacramento from every direction. Overland emigrant groups, weary and weather-beaten, tramped over the Sierras and camped in Sacramento. Ships that had come past Cape Horn disgorged their passengers in San Francisco who then transferred to sloops or schooners that moved up the river to Sacramento. Brannan and his competitors Hensley and Reading purchased sailing vessels to transport goods and people to Sacramento. Steamboats began chugging up the river. By 1850, two sailing ships unloaded their passengers every day. Mules and wagons hauled people and goods to the gold fields. The paths of those early gold rush trails brought miners down a primitive J Street where they turned left on 12th Street and then forded the American River by Lisle's Ferry, later Lisle's Bridge, the site of the present 16th Street Bridge. From there they headed for the American, Bear, Yuba, and Feather Rivers. Other roads led to the mines, including present-day Stockton, Folsom, and Auburn Boulevards and Marysville Road. Before they went, they stopped and shopped at the busy emporia along the river. J and K Streets became an important commercial area. At Sixth and K, a Horse Market also thrived. Brannan and his associates went forward and created the three-story City Hotel out of an old flour mill they had disassembled from an inland Sutter-held site and rebuilt on the riverfront.
Many of the early businesses were housed in large canvas tents, some salvaged from the sails of abandoned vessels. The first church building in the city was the Baltimore Chapel of the Methodist Episcopal Church, made of pre-fabricated materials brought around the Horn. James Lee pitched an early tent structure that provided space for gambling and other entertainment on J Street. The smell of urine and feces from the drinkers and gamblers in the large, damp canvas led to the legendary sobriquet "The Stinking Tent." Gambling halls, bars, restaurants, and houses of prostitution were important pieces of Sacramento's early social life.
Gold rush era Sacramento was a challenging place to live. Forty-niner James Winchester wrote to his relatives in mid-1849:
Sacramento City contains more than ten thousand inhabitants. Most of the stores and houses are without floors, with canvas roofs and walls. No building is enclosed by a fence; but all are, as it were, in one immense open lot, one great cesspool of mud, offal, garbage, dead animals and that worst of nuisances consequent upon the entire absence of outhouses. I can't describe it as it is, but it is desolate beyond description.
Since living conditions could get no worse, things did improve. The number of residences grew steadily, first simple frame structures and later more sturdy brick houses. Society and local culture became increasingly more "respectable" as one of Sacramento's first civic events, a large Fourth of July ball held in 1849, took place in the partially completed city hotel.
Sacramento's emergence on the embarcadero was not assured without some struggle. Stung by Brannan's arrogant disregard of his own prior rights on the riverfront, rival George McDougal attempted to halt the development of the new community. He rousted Sutter from his Coloma hideaway and brought him back to take control of the enterprise. Sutter was powerless to do much except fire Burnett, who in return for his fees demanded and received a large number of city lots. McDougal claimed that he had leasing rights to a large portion of the embarcadero. Hence, efforts to build on this leased land would have to be halted. Sutter Jr., at Brannan's urging, contested this "claim" in court and McDougal's lease was thrown out. McDougal also attempted to shift commercial dominance to Sutterville by offering to sell his supply of goods at cost in order to undercut the embarcadero merchants. Further, he induced Lansford Hastings and Sutter to offer 80 free lots to lure merchants to Sutterville. Brannan and his cronies replied with violence, destroying McDougal's goods and then turning to Sutter Jr. to up the ante on free land. With more land to give, Sutter Jr. offered the merchants 500 lots. Sutterville slumped into oblivion, except for a brewery and a brick works.
Artist J. Cameron created this view of an encampment in early Sacramento (lithographed by G.V. Cooper in 1849).
John Sutter Sr. retired to his property at Hock Farm on the Yuba River. When an arsonist destroyed his property, he spent the remainder of his life alternately reminiscing about early days in Sacramento and petitioning the U.S. government for recompense for his lost lands. He moved east to Lititz, Pennsylvania and, after the failure of his last petition to Congress, died on June 18, 1880.
Sacramento's early economy and its future were fueled by capital investment. People put money in and did what they could to make sure the money earned a profit. Brannan typified this spirit of entrepreneurial self-interest that helped to shape the city. He became the leading merchant, buying land for speculation and erecting buildings for his own use and to lease out to others. Brannan and other capitalists like him invested in the future of Sacramento's embarcadero location and the projection of a commercial district along J and K Streets. These men were the first wave of successful businessmen who made their money by "mining the miners." The firms of Hensley, Reading & Company and Priest, Lee & Company had also begun at Sutter's Fort. For a time both equipped individual miners, but eventually Barton Lee (of Priest, Lee), one of Sacramento's wealthiest men, edged out Hensley by concentrating on the wholesale trade and leaving retailing to smaller operators. Any speculator with capital could inflate land prices. Sutter Jr. insisted on a $250 cap on lot sales, but once the lots were sold at that minimal price, their value increased considerably.
John Sutter Jr. hired engineers to survey and plan a new city that would begin at the embarcadero.
# FORGING A GOVERNMENT, SETTLING COMPETING CLAIMS
People with an active stake in Sacramento's economic future also played a key role in devising and planning stable city institutions. Assuring urban order was another way of protecting investments. Although population patterns followed no predictable formula, Sacramento's numbers generally ebbed and flowed with the rhythms of the mining seasons. During the dry months, the city emptied as its inhabitants trudged the mountains looking for gold. In the wet winter months, Sacramento filled with people waiting for the rains to cease and the streams to become workable again. More inhabitants meant more merchants who settled in the city to provide more products and services for the miners. More investment in the city meant more interest in providing a stable government and assuring order in the sometimes disorderly community.
Inherent class tensions increased among early Sacramentans as patterns of land ownership emerged. Most immigrants arrived with hope but little cash. With the great demand for land, real estate prices began to command higher prices. Lots that had initially sold for $250 were now marketed for $8,000. Consequently, those who owned more land soon rose to the top of an economic aristocracy. One group of men, defined by historian Mark Eifler as the "great speculators," acquired the resources to hold substantial real estate. According to his research, about one-fourth of the residents of the city held property, but the bulk of those holdings consisted of only one lot. Most Sacramentans were renters or transients who owned nothing. Of the others, 100 owned at least two lots, 42 owned at least $20,000 in property, and 21 held more than $50,000. This loose coalition of merchants, traders, and speculators were the movers and shakers and the most insistent about creating government entities to protect their rights.
The origins of Sacramento's first city government are not altogether clear. Under Mexican law, Sutter had held the position of alcalde (a term with no precise English equivalent to describe its duties but it combined executive, legislative, and judicial functions in one person). In September 1846, John Sinclair was elected magistrate of the Sacramento District. The situation grew muddled after Mexican law and military rule formally ended in May 1848 when Congress approved the Treaty of Guadalupe Hidalgo. The U.S. Congress had anticipated organizing a California territory shortly after the Mexican War, but the issue of California's admission to the Union got snared in the escalating debate over the extension of slavery. The matter would not be resolved until late summer of 1850 when the last of the sectional compromises that kept the Union together would be forged in Congress, and California was admitted as a free state. But between the start of the gold rush and California's admission to the Union in September 1850, the pressing needs for social order and stability grew more intense as the gold rush telescoped the "normal" time line of growth and social development into a relatively short space. Sacramento needed a stable government but lurched toward it with uncertainty and some conflict.
In early 1849, a group of merchants laid out a simple government structure for Sacramento County consisting of a sheriff and an alcalde. John Frederick Morse, a physician who wrote one of the first histories of the city, noted that elections had already taken place in late 1848 for two alcaldes. In a more romantic version, Morse describes the first laws of Sacramento being crafted at a meeting under a spreading oak tree at the foot of I Street in the spring of 1849. A territory-wide election in August 1849 to select delegates to a state constitutional convention was called, which gave the umbrella for the election of Sacramento's first city council. William Stout became the first mayor and when Stout left town, General A.M. Winn, a land agent, replaced him. The council consisted largely of the wealthiest men in the city, including Sam Brannan, Barton Lee, John McDougal, and Samuel Hensley.
These councilmen set up a series of committees to deal with specific areas of urban need: finances, the waterfront, roads, and a city center. They also set to drafting a charter, which initially voters rejected, in part because of proposed prohibitions against gambling. In October, a second charter election was held, and this time the document succeeded. The newly formed California Legislature accepted the charter in February 1850, the first in California, and the city officially commenced on March 18, 1850. A new city council was elected on April 1 and Hardin Bigelow, a local businessman, became the first mayor under this new charter.
# THE PERILOUS YEAR: 1850
Sacramento's first official year of existence was its most perilous. On the front burner was the growing dispute over property rights. The clash between land developers—who had indeed played an important role in creating the new city despite natural disasters—and the landless (and those who held little land) may not have been inevitable, but it did come. The city's real estate was falling under the control of a core of well-endowed capitalists with the funds to purchase large numbers of lots, parcels that were often subdivided and sold at higher prices or used as collateral to purchase needed inventory for shops and stores. However, questions surfaced about the legality of Sutter's land grant. The federal government empowered a land commission to begin the arduous process of evaluating the validity of all the Mexican grants, so Sutter's New Helvetia grant, plagued by the imperfect lines drawn by Vioget, was one of the first to come under scrutiny.
As the city flooded with miners, especially overland immigrants who needed a place to stay, extensive landowners began charging the weary newcomers nightly fees to camp on their lands. Some of these came with sharply etched ideas about their rights under government pre-emption policies and began to deeply resent the monopolization of land in and around Sacramento. In the middle of October 1850, one Z.M. Chapman built a log dwelling on a vacant lot outside the city limits near Sutter's Fort. Representatives of Priest, Lee & Company visited the enterprising Chapman and informed him that he had to stop building since their company owned the property. When Chapman refused, a court case was launched and the doughty woodsman insisted that the company show evidence of ownership. When they were unable to do so to his satisfaction, Chapman challenged the Sutter claim on which the company based its ownership and then called into question all city land titles. Emboldened by this, a wave of squatters began to sink roots on city lands. The squatters themselves formed an organization called the Sacramento City Settlers Association. A Massachusetts physician, Dr. Charles Robinson, became its president and the intellectual wellspring of the movement. Robinson not only vigorously endorsed Chapman's claim, but he also built his own squatter's cabin near the levee.
Naturally, landowners, led by Brannan, upheld the legality of Sutter's claim and mobilized against the squatters. Strongly represented on the city council, the landowners decried Robinson's provocative action and had the shanty torn down. The destruction of the Robinson structure might have set off tensions right away, but floods drove the transients out of Sacramento in the winter of 1849–1850 and this brought a temporary end to the problems. Moreover, the opening of the placer mines in the spring further emptied the city. However, slowly but surely the transients made their way back—many of them weary of the slim pickings in the gold fields and others looking for a place to settle, dry out, and relax. The settlers' organization did its best to keep alive the question of the validity of Sutter's grant. A controversial and widely disseminated pamphlet by John Plumbe argued vigorously that Sutter's claim, which now encompassed the city limits, was questionable. Sutter was himself, in the logic of the pamphlet, a squatter. With renewed vigor, squatters again began to settle on undeveloped Sacramento land, and the landowners hit back. Aided by an April 1850 law passed by the transitional state legislature, "An Act concerning forcible entry and Unlawful Detainer," they now had the legal authority to eject squatters. Their next move was to have a city ordinance passed forbidding the building of tents, shanties, or houses on any vacant lot belonging to a private person. The landowners also formed a Law and Order Association.
Land agent and brigadier general, A.M. Winn was one of Sacramento's early mayors.
Anxious to test the new laws, city council member John P. Rogers brought suit against squatter John T. Madden, who had settled on a vacant lot on the corner of Second and N Streets. Madden found himself hauled before the newly organized recorder's court, which had jurisdiction over city ordinances. Judge B.F. Washington ruled against Madden, who promptly appealed the decision to a county court. Squatters and landowners then squared off in their respective organizations. One of the loudest voices in defense of the squatters was a young Irish immigrant, James McClatchy, whose militant rhetoric urged squatters to active resistance. When County Judge E.J. Willis ruled against Madden's appeal on August 10, 1850, the stage was set for violence. McClatchy and fellow squatter supporter Michael Moran were arrested by city authorities and charged with resisting or attempting to resist the sheriff who was trying to get Madden off the contested lot. Arrest warrants for other squatter sympathizers, including the militant Robinson, also went forth.
Four days later, fighting erupted when a band of settlers (perhaps 40 or 50) led by Robinson and James Maloney attempted to re-take Madden's claim. As their rag-tag band marched through the heart of the city, local officials thought the group intended to spring McClatchy and Moran. When this did not happen, Mayor Hardin Bigelow and Sheriff Joseph McKinney ordered them to disperse. At the corner of Fourth and J, the long simmering hostility exploded as shots were fired. Bigelow was wounded four times and city assessor J.W. Woodland, standing near Bigelow, was killed. Other casualties included one of the settlers, a small boy standing nearby, and one of the citizens. Maloney's horse was shot out from under him, and he was pursued and shot to death in an alley. Robinson, wounded in the leg, was arrested. General A.M. Winn, head of the city council, invoked martial law and recruited 500 volunteers to keep order. The following day, Sheriff McKinney and a posse tracked down a dozen settlers at the Five Mile House in Brighton. In another gunfight, McKinney and three settlers were killed.
The episode, so traumatic and dramatic, has often been told as a tale of good versus evil—land speculators versus the democratic land equalizers. In part this is true, but in reality the issues are far more complex and reveal important class tensions that marked the city during its formative era. In the end, the bitter standoff redoubled the efforts of civic leaders to create more stability. Eventually the federal government upheld the Sutter grant and the speculators won. In some respects, the victory of the landholders was temporary. Their efforts to use Sacramento as a speculative enterprise eventually gave way to a more welcoming attitude toward newcomers. Land and settlement in Sacramento needed to be opened up, not restricted to the profiteering motives of a relative few. The bitter experiences with land monopoly in Sacramento framed the life and career of James McClatchy, destined to be a major force in city journalism.
Squatters riot in Sacramento City.
# FLOODS, FIRES, AND EPIDEMICS
The decision to locate Sacramento on the embarcadero may have been an economically sound one, but the location had some serious problems. Anxious to secure their economic foothold, Brannan and others did not give much attention to the fact that the land was very low. They may even have ignored the bits and pieces of driftwood around the site and even in the trees—signs that the area was sometimes under water. Indeed, Sacramento was prone to flooding, and a primitive levee hastily erected along the embarcadero was inadequate. Heavy rains in late 1849 extended into January 1850. On January 8, the American River burst into the city, sweeping away virtually everything in its path. Future mayor Hardin Bigelow urged the building of higher levees, advice that was only heeded in the spring when the Sierra snowpack melted and the rivers rose again. Bigelow's makeshift levee held back the next torrent. On April 29, Sacramentans voted to tax themselves $250,000 to build a levee to encircle the city. Only 3 feet high, 6 feet wide at the top, and 12 feet wide at the base, the earthwork was a first effort to alter the liabilities entailed in Sacramento's location. But in 1852, these modest levees failed, and floods again inundated the city. In response, local landowner Samuel Norris laid out a nice, neat little town called Hoboken (near what is today the J Street entrance to California State University), and for six weeks about 1,000 people found refuge there. Prospects looked bright for Norris's town and steamers plied their way to the new site with supplies. However, once the waters receded, Hoboken slipped into oblivion. Sacramentans gave no thought to moving the city to higher ground but set to work building higher levees. Before it was over, Sacramentans spent nearly $600,000 protecting their city from floods.
Early Sacramento City was prone to flooding.
Water would be an enemy in other ways as well. On October 18, 1850, a steamship bringing news of California statehood to Sacramento also carried passengers with the dreaded cholera. Before the epidemic ran its course, nearly 600 died and many more fled. Dr. John Frederick Morse, one of the city's first physicians and chroniclers, referred to Sacramento during those months as "a veritable lazar house." Hundreds died and were buried in the city cemetery, donated by John Sutter on the southern fringe of the city, now Tenth and Broadway.
Although floods and disease were serious threats to Sacramento's well-being, its counterpart, fire, did just as much damage until it too was controlled. Early Sacramento was a tinder box, with many of its structures made of canvas tent material or framed with dried-out pine lumber. Long dry summers raised the potential for fire, and in September 1849 the city was wiped out by a blaze—a harbinger of future conflagrations. In mid-1850, the annus horribilis of Sacramento's existence, another roaring blaze destroyed again the canvas and flimsy wooden structures. On November 4, 1852, by the time Sacramento had begun to develop a more settled "urban look" with large buildings, streets, and a distinct commercial district, fire broke out in Madame Lanos's millinery shop near Fourth and J Streets. Fanned by a strong north wind, the blaze destroyed 55 blocks. Two years later, another destructive fire hit in the scalding month of July. Between 1855 and 1870, fire ravaged the city periodically. This impelled a steady improvement in the quality of its fire services, creating two new volunteer companies, but the city did not have a full-time, paid fire department until 1872.
As with the water crisis, the collective forces of urban regeneration invoked the spirit if not the letter of Urbs Indomita and rebuilt the city soon after the ashes cooled. Brick structures soon began to dot the streets (in 1855, the city passed an ordinance mandating them in the business district). Indeed, by 1856, Sacramento had 500 brick (built with materials manufactured in one of the 30 brick yards around the city) and 2,000 frame buildings. Major hotels and gathering places included the Golden Eagle at Seventh and K, the Clarendon on Second, the Dawson House, later known as the St. George, at Fourth and J, and the What Cheer House at Front and K Streets. Other industries included breweries, soda water manufacturers, and two iron and brass foundries.
# BUILDING A STATE CAPITOL
The early process of city building reached a pivotal moment when the state government decided to locate California's wandering state capital in Sacramento in 1854. California's constitutional convention met at Colton Hall in the old Mexican capital of Monterey. The subject of the permanent seat of California's capital had been taken up and for a time it was decided to locate it in San Jose. However, when San Jose proved unready for the new government, a bidding war was set off among a number of California cities—including San Luis Obispo and Santa Barbara in the south and Benicia, Stockton, San Francisco, and Vallejo in the northern part of the state. Sacramento had also placed a bid and, in fact, had served as temporary quarters in 1852 while Vallejo, the choice for a time, readied itself. The first Sacramento courthouse hosted the legislature in January 1852, but the city's chances to hold the temporary capital evaporated when flooding hit, and the legislators migrated to Benicia. Although the legislature had capacious quarters there, the town proved too small for state government. Sacramento then made another bid. In January 1854, the mayor and common council offered the legislature and state officers free use of their handsome new courthouse, fireproof vaults for public money and records, and removal from Benicia to Sacramento without charge. The city further sweetened the pot by granting the state a public square between I and J and Ninth and Tenth Streets for a permanent capitol.
Legislators may have worried about Sacramento's climate and propensity to be wiped off the map, but they also admired the city's willingness to rebuild. They were also probably attracted by the city's various entertainments and abundant hotels. Sacramento boasted 55 hotels, planked streets, 14 stages, and 28 river steamers. In February 1854, Senator Amos Parnall Catlin, a longtime advocate of Sacramento, introduced a bill to fix the permanent location of the government in Sacramento. The bill passed both houses, and on February 25, Governor John Bigler signed it into law. When the courthouse was destroyed in the fire of 1854, a newer and even more capacious building was erected in anticipation of the legislature's relocation in January 1855. The state paid $12,000 a year for the Sacramento courthouse, and it served as the hub of California government until 1869.
On March 15, 1856, Senator William J. Ferguson introduced a bill to issue bonds to construct a new state capitol. Work actually began on a Greek-style building, but stopped after two weeks when state officers decided the bond issue violated the state constitution. The project sat for a while, and San Francisco, Oakland, and San Jose pitched hard to secure the capital for their cities. Nonetheless, in 1860, Sacramento trumped its rivals by offering the state new land between L and N, Tenth and Twelfth Streets. Governor John Downey signed a bill appropriating $500,000 for a capitol building project, and the design submitted by Miner Frederick Butler was accepted. However, before it could get underway, Sacramento was hit by serious flooding in 1861–1862, raising again the prospect of capital removal. The legislature adjourned temporarily to San Francisco but was back in January 1863. The construction of the capitol resumed in June, and although it would not be entirely completed until 1874, it was ready for occupancy in November 1869.
Constructed of brick with facades, outer steps, and columns, of granite from California quarries, the capitol covered an area of 52,480 square feet, cost nearly $2.6 million, and resembled the U.S. capitol. The structure was shaped like an "E" with the arms facing east and was capped by a 125-foot dome, the most important feature of Sacramento's skyline for many years. All the offices of state government and the legislature were contained within the new building. The legislature formally took possession of its respective chambers on December 6, 1869. It is still the most elegant building in the city and a natural benchmark for urban excellence. In 1866, a writer for the Evening Bee expressed in poetic fashion the hope that the new building would bring a defining presence to Sacramento: "It is a masterful piece of architecture.... The effect it will have on the taste and manners and morality of our city will be almost incredible. No man will be able to look upon it and not feel its refining impulses."
A new county court house served as the state capitol from 1855 to 1869.
Construction of the California state capitol building was interrupted by serious flooding in 1861–1862.
Around the grounds of the capitol, a beautiful public park was laid out with handsome granite pillars marking the access paths to the statehouse. The business of state government was initially slow, with the legislature meeting biennially and for only short periods. Some state offices, like the Supreme Court, eventually moved to San Francisco, but the capital remained firmly ensconced in Sacramento. For Sacramento, the state proved to be an extremely stable employer. The number of jobs has grown steadily and has only slightly been offset by boom and bust cycles. The presence of government employees in Sacramento has historically provided it with a cadre of well-educated and cultured men and women who have supported good education and cultural development.
By constitutional amendment in 1861, Sacramento became the permanent home of the annual state fair. Housed for many years in the State Agricultural Pavilion on Sixth and M Streets and on 43 acres of prime city property called Union Park (today the Boulevard Park neighborhood), it also included a popular race track. The state fair brought an annual stream of visitors to the state capital and provided a substantial boost to the local economy. In 1906, it was moved to an 80-acre site on Stockton Boulevard.
# THE CREATION OF URBAN COMMUNITY
Sacramento continued the process of community building throughout the 1850s. As the highly-individualistic "easy pickings" of the placer fields gave way to the more labor intensive and expensive hard rock and hydraulic mining, more people moved to Sacramento and devoted themselves to urban life.
Demographically, Sacramento had always been a diverse place. Most gold rush denizens hailed from eastern states and often gathered together in state groupings to maintain contact with one another. Irish settlers were quite prominent in early Sacramento, many of them emigrating from Irish enclaves in the eastern states. Irish denizens brought with them a love for the old land and a hearty Irish patriotism, as well as an attachment to the Catholic Church. The Irish in Sacramento found their natural home in St. Rose of Lima Church at Seventh and K Streets, which they dominated for many years. Irish priests and nuns provided an important core of literate and competent leaders. Irish culture was perpetuated in Sacramento through public celebration, especially of Ireland's patron St. Patrick, and fund-raising and agitation for Irish nationalism were continuous. A number of leading Irish nationalists of the nineteenth century visited Sacramento enroute to San Francisco. The two Catholic schools conducted by Irish nuns and Christian Brothers were bastions of Irish nationalism as students learned Irish history, recited Irish poetry in closing exercises, and celebrated Irish holidays.
Other ethnic groups included German-speaking Swiss, like Sutter, and Germans. Small numbers of Hispanic Californians were also present in early Sacramento. Likewise, a contingent of Native Americans lived in the city. African Americans made their presence felt in the burgeoning city as well. Some made for the gold fields, but like others many remained in Sacramento and offered services to transient miners. Sacramento's African-American population, like the Irish, tended to coalesce around its church, the African Methodist Episcopal (AME) Church, the oldest independent African-American denomination. In 1850, St. Andrew's African Methodist Episcopal Church was organized after an earlier attempt had failed. Services were held in the home of Daniel Blue under the leadership of Barney and Daniel Fletcher. Shortly after, a Baptist church, Siloam (later Shiloh), began under the direction of Charles Satchell, holding services in a Chinese Baptist Church on Sixth and H Streets. In May 1854, Elizabeth Thorne Scott opened her home as a school to 14 African-American children. The next year, Reverend Jeremiah Sanderson, a black educator and abolitionist, came from San Francisco and moved the school to facilities offered by the AME Church.
St. Rose of Lima Church, shown here c. 1868, provided a natural home for Sacramento's Irish immigrants.
Chinese residents also came in quest of gold. Targeted for discrimination from the first, Chinese miners lost a share of their hard-earned diggings to a noxious foreign miners tax that was passed to dissuade the Chinese from competing with American-born miners. As a result, the Chinese immigrants learned to provide services to urban Sacramento and were ghettoized. Chinese Sacramentans settled along I Street, near "China Slough" (also known as Sutter Slough), forming one of the first distinct ethnic enclaves in the city. Chinese restaurants, gambling establishments, and businesses (especially laundries) popped up in this section of town. Almost exclusively male, Chinese residents slowly welcomed a growing female population. The number of Chinese steadily grew in the city, numbering nearly 1,200 by 1860.
Although issues of class and economic difference continued to surface throughout the city's history, the so-called "civic middlemen" described by historian Mark Eifler stepped forward to even out sharp class differences and provide a solid middle class. These merchants included those who made a living carting goods—teamsters and river line operators. Likewise, physicians were part of this guild of solid citizens, having burnished their credentials in the disastrous cholera epidemic of 1850 and occupying important roles in the city's growing concern for public health. The city's religious leaders, members of the press, and teachers were vital parts of that "civic middle" in that they had significant influence on public opinion. Invested in the city and anxious to provide for its greater advancement and development, these middlemen bridged the gap between the urban working class and the great speculators, providing a solid community with leadership in civic office and social affairs and also making accessible the resources of the city to those who aspired to their status. These groups centered their activities symbolically in the bustling trade along the embarcadero and also near the Horse Market.
# WOMEN'S ROLES
The forging of deeper community ties was also facilitated by the increasing presence of women. With its raucous gaming tables and hoop-ti-do gold rush ribaldry, Sacramento seemed an uncongenial place for many nineteenth-century women whose social training excluded such "unrespectable" entertainment. The 1850 census noted that only 6.96 percent of the 6,820 inhabitants of Sacramento were women (475 women to 6,345 men). The gendered world of Sacramento grew steadily, bringing with it a stability that moved the city beyond its hurly-burly origins. Women in early Sacramento did indeed speak of their efforts to tame the wild miners, but they also set themselves to earn a living. Women ran hotels and restaurants, but according to historians Dian Self and Elaine Connolly, the majority of women employed outside the home earned their livelihood in the clothing trade. Other jobs like housekeeping, millinery, and midwifery were also available to Sacramento women. One brave woman, Mrs. John Zwicker, operated a saloon and shooting gallery at Third and J.
Chinese Sacramentans settled along I Street, forming one of the first distinct ethnic enclaves in the city.
Two important aspects of women's life represented opposite ends of the moral spectrum: prostitution and the convent. Although viewed as a moral vice (and one that city leaders periodically railed against but never seriously tried to eliminate), prostitution provided another source of income for enterprising women. The "world's oldest profession" apparently enjoyed an open field for many years, and ladies of the night were seldom arrested. One prominent madam, Johanna Hiegel, ran a cigar business and a house of prostitution on Second between I and J for nearly 30 years beginning in 1854.
Another breed of dedicated and enterprising women, the Roman Catholic Sisters of Mercy, appeared in Sacramento in October 1857 and set to work educating the young and caring for the homebound. Later, they would become prominent providers of health care. This community exercised their religious life in a distinctive way—wearing special garb, living celibate lives in community, and having enormous influence in business and educational affairs, perhaps more than any other group of Sacramento women. In 1860, they opened St. Joseph's Academy for girls, a school that took in grade school to young adult women of all religious denominations and trained them in classical subjects as well as in more practical domestic duties. Their first convent was planned for the same site that was later claimed by the state capitol. Forced to move when the city gave the land to the state, the sisters opened their large convent at Eighth and G Streets and remained there with their school until the late 1960s. The sisters exercised an important role in the distribution of charity. Visiting the homes of the poor and destitute, educating the young, and providing relief in disasters were just a few of the ways in which the Sisters of Mercy endeared themselves to the wider population of Sacramento.
# RELIGIOUS INSTITUTIONS
The presence of the Sisters of Mercy highlights the role of religion in shaping Sacramento's early culture. As in many areas overwhelmed by a flood of footloose young men in search of gold, organized religion of any sort had a hard time gaining traction in Sacramento. Optimism did burn bright, however. Since the area around Sacramento had not been missionized by Roman Catholics, Protestant evangelicals anxious to win "God's gold" moved quickly to claim the area. Reverend Sylvester Woolbridge, a representative of the old school Presbyterians, preached the first "sermon" on the Sacramento riverbank in the spring of 1848. Other ministers began in temporary quarters and some took to preaching on the levee. Methodist-Episcopalian pastor W. Grove Deal began the first regular religious services in May 1849. Sacramento's most famous minister was Joseph Augustine Benton, a Congregational minister who did not come originally to preach the gospel but to mine gold. When this failed in early 1849, he took up residence in Sacramento, founding the First Church of Christ in September 1849 and remained for nearly ten years, longer than any other minister in the city during the gold rush era. Notably eloquent, Benton played an important role as a religious leader and a public citizen. Benton's castle-like church, first erected on Sixth Street, was designed by local architect Albion Sweetser and became a popular lecture and music hall for Sacramento. Sacramento's Republican Party held its first convention in the church.
However, despite their best efforts, the promotion of Sabbath observance in early Sacramento was frustrating. John Morse's history notes, "Sunday was almost a forgotten day. ..." But gradually more ministers came and with them more stable congregations. Methodists had established themselves early on in Sacramento. Indeed, Sacramento's first church building was put up in 1849 by Methodist preacher Isaac Owens, the Baltimore Chapel located at Seventh and L Streets. Methodists later put up the city's first brick church on Seventh Street. Baptist and Episcopal churches followed. In August 1850, Dominican Father Peter Anderson founded St. Rose of Lima Church on lots donated by Catholic convert and first governor Peter Burnett. Chinese inhabitants worshiped at the Chinese Baptist Church on Sixth and H, but most retained their own Buddhist rituals. Sacramento had its own Joss House in the Chinese district. Religion, it was felt, tended to tame "a most vitiated and chaotic community." The common struggles of religious leaders in early Sacramento produced a kind of practical ecumenism. Church members of various denominations contributed to the building funds of others. Non-Catholics sent their sons and daughters to Catholic schools. Sometimes inter-denominational kindness had a hook. For example, the Baltimore Chapel was loaned to Sacramento's fledgling Jewish community for its High Holy days "so that the spirit of the building would bring them back to the true religion." Jewish Sacramentans did not become Christians, but they did purchase the chapel to house Congregation B'Nai Israel in June 1852.
# THE PRESS
As religious institutions helped create a more stable community, so also did the rise of a popular press. Samuel Brannan backed editor Edward C. Kemble in issuing the city's first paper, Placer Times, on April 28, 1849. Located at the fort, it moved to the Front Street location on July 9, 1849 and covered some of the first events of city life. In April 1850, a second newspaper, the Sacramento Transcript commenced. In all, 44 newspapers came and went between 1849 and 1858, but only two remained: the Sacramento Union and the Sacramento Daily Bee. The Union, the eldest of the two papers, had begun in 1851 when a rate war between the Placer Times and the Transcript resulted in a pay cut for the printers. The Union continued until January 1994. It would change ownership and editorial direction many times.
The evening Sacramento Daily Bee began when the California American went out of existence. Owned by a consortium of businessmen, Levi C. Chandler, William H. Tobey, John Church, and Lyman P. Davis, the Bee's first editor was Cherokee poet John Rollin Ridge. Ridge lasted only a few months and was replaced by another prominent local journalist, James McClatchy. A native of Ireland's County Antrim, McClatchy had come to the United States in 1840, settling first in New York, where he became associated with Horace Greeley's New York Tribune. Perhaps hearkening to Greeley's famed injunction to "Go west young man and grow up with the country," McClatchy migrated to California via a harrowing trip through Mexico and up the California coast. After a brief stint in the mining fields, he settled in Sacramento in the late summer of 1849. From the beginning, McClatchy interested himself in the welfare and future of Sacramento and became deeply involved in virtually every aspect of its development. He joined in a number of journalistic endeavors and also in local politics, writing for the Placer Times, Democratic State Journal, and the Californian. Once he replaced Ridge, McClatchy remained linked with the Bee, only taking off time to become Sacramento sheriff between 1864 and 1866 and for a brief stint working at a San Francisco paper. His interest in the Bee gradually evolved into a partnership with others and eventually, sole ownership of the paper. By the time of his death in 1883, he was able to turn the paper over to his two sons, Charles Kenny and Valentine, who secured family dominance over Sacramento journalism down to the present.
The Sacramento Bee was located on Third between J and K Streets.
# SCHOOLS AND OTHER ELEMENTS OF COMMUNITY
Schools also contributed to the development of community. It took time for public schools to emerge since private institutions moved first to meet the growing demands for education. Already in 1849, Congregationalist minister Joseph Augustine Benton had begun a school under the auspices of his church. Sacramento Methodists formed a successful school as well. By 1852, there were nine private schools. The Sacramento Academy and the Young Ladies Seminary were two of the best known. A Roman Catholic school later associated with the Sisters of Mercy was formed at St. Rose of Lima Church.
On April 26, 1853, the state legislature authorized the Sacramento City Council to establish free common schools, and on February 20, 1854, the first public school (Franklin School) opened at Fifth and K. By July nearly 500 children were attending public and private schools in the Sacramento area. Between 1854 and 1893, the city built and maintained 13 schools. In 1856, Sacramento opened the second high school in California. Sacramento schools were segregated until 1885. In May 1856, a small public school for African-American children was built on Fifth and O Streets.
To accentuate the growing civilization of the city, a public library opened in November 1857, with 800 volumes secured from gifts and an energetic subscription campaign. Likewise, theater was present from the beginning, aligned perhaps to the somewhat bawdy acts in the gambling tents. Sacramento's first theater was a frame and canvas structure crowned by a tin roof called the Eagle Theater. Rowdy miners sat on benches facing the stage, while a modest balcony accessed by ladder was arranged for the ladies. Wiped out in the floods of 1850, the site was sold to new owners, who moved to Second Street and renamed it the Tehama. By the time the Tehama succumbed to the 1851 fire, small theaters and entertainment halls proliferated to meet the needs of lonely and bored transients. One of Sacramento's grandest theaters, the Clunie, built by future Congressman Thomas Jefferson Clunie in 1870 as a hotel, was later enlarged with part of it being converted into a playhouse.
For the hundreds of transients whose life ended here, Sacramento prepared "hallowed ground." The earliest cemetery was started by Sutter near his fort. The land was eventually sold to private developers and is today the site of the Sutter Middle School. The public City Cemetery at Y (Broadway) and Tenth was given to the city by Sutter in 1850 and became Sacramento's prime burial ground. Roman Catholics opened their cemetery south of Y Street in 1860 and Jewish Sacramentans buried their dead at the New Helvetia cemetery at Alhambra and J. Later burial grounds were secured on Stockton Boulevard.
Sacramento had experienced its first tumultuous generation and survived its tenuous location through the sheer determination and strength of its settlers. As a new era dawned Sacramento left behind the early problems of civic disorder, fire and flood, and epidemics.
The first Sacramento High School was located at Ninth and M Streets.
# 3. A MATURING CITY
The decade of the 1860s that found most of America caught up in the Civil War presented Sacramento with a reprise of the vulnerabilities inherent in its location. Already in 1853, city leaders had raised the level of I, J, and K Streets, lifting them to the level of the city plaza. They erected new sidewalks and buildings to render them less prone to flooding. However, tremendous flooding in the winter of 1861–1862 created havoc and spurred even greater efforts to protect the city. In 1862, determined city leaders redirected the American River, removing a troublesome curve. They also relieved pressure on the mouth of the American River, bringing it into the Sacramento River at a point about one-fourth of a mile north of the original confluence. Using wheelbarrows and shovels, Sacramentans hauled in dirt from the new channel of the American River to fill in 10-foot high bulkheads along the streets. New wooden sidewalks crowned the streets. Buildings were lifted up by jackscrews and given new foundations. Sometimes first floors of buildings were simply abandoned and second stories became the new entrance. Not all parts of the city had to be raised to the same heights, but by 1873, 12 blocks of I, J, and K Streets were raised as much as 15 feet. In 1877, the first cement sidewalks in Sacramento were laid in Capitol Park under the direction of Adolph Teichert.
Connecting Sacramento to the outside world constituted one of its first important industries. Steamboats coming up from San Francisco plied the waters of the Sacramento. At first a myriad of independent contractors handled this burgeoning business, but eventually the California Steam Navigation Company came to monopolize 75 percent of the river traffic. Ultimately, however, the use of the Sacramento River as an artery of trade and commerce began to falter. Riverboats and other vessels found it difficult to move up the channel due to the large amount of silt and "slickens" that were washed from the mountains into the river by hydraulic miners. Sacramento denizens, led by James McClatchy of the Bee, waged a relentless war against the environmental damage done by the "hydraulickers." Legal action came to a head in 1884 when a ruling of the California Supreme Court, the Sawyer Decision, ended the practice.
Teamster companies hauled supplies and mining equipment over the mountains to miners. The Horse Market provided individual riders with animals, saddles, and harnesses, as well as carriages. Stagecoach travel began in 1849. New Englander James Birch and a partner initiated stage service between Sacramento and Mormon Island. Birch's success opened the field for others. In 1854, Birch and a friend, Frank Stevens, consolidated smaller stage lines into the California Stage Company. At its peak, this firm generated huge profits as it commanded nearly 80 percent of the stage traffic over 3,000 miles of routes connecting major parts of the American West. In 1856, Birch and others pressed Congress to create three wagon roads to the Pacific Coast and approve overland mail from St. Louis to San Francisco. Likewise, the state legislature appropriated funds for the expansion of roads from Sacramento to the state line via Placerville. With roads came telegraph lines that connected Sacramento with Marysville and San Francisco. In 1861, these lines went all the way to Salt Lake City and eventually to the East Coast. Carrying mail via the fabled Pony Express also had a Sacramento connection. The dispatch of these swift couriers lasted only 18 months from 1860 to early 1861, but their romantic appeal of relaying mail across the country from St. Joseph, Missouri to the B.F. Hastings building in Sacramento left a tale repeated from generation to generation and memorialized by a statue in the state historic park in Old Sacramento. However, the Pony Express was quickly replaced by the telegraph.
Riverboats plied the waters of the Sacramento River, docking at Sacramento's riverfront, shown here in 1910.
# A DEFINITIVE IMPACT: THE RAILROAD
Sacramento was born just when the railroad was beginning to crisscross America and Congress was debating routes for a major transcontinental system intended to tie the East to the West Coast. The first plans for a railroad in Sacramento date to 1852 when Peter Burnett and others plotted a line that skirted the Sierra foothills to Marysville. Like many infrastructure projects of this magnitude, the project fell apart due to lack of funding. In 1854, another group of entrepreneurs organized the Sacramento Valley Railroad Company (SVRR). While searching for funding sources in the east, the company managed to interest a young engineer, Theodore Judah, who had only recently built a railroad over the Niagara Gorge in New York. From a perch at 31st and M Streets (today Folsom and Alhambra Boulevards), Judah took careful note of the number of mule-driven teams passing from Sacramento into the mountains and realized that a railroad could make a good profit.
Judah plotted a path for a railroad that would plunge into the Sierra Nevada foothills and branch north and south along the base of the mountains. The SVRR track-laying commenced on August 9, 1855, and seven months later, in February 1856, the SVRR officially opened. The rail line began at Third and R and turned east at the R Street levee moving toward Folsom. The 22 miles of tracks cost over $1.3 million but proved their worth by transporting tons of cobblestone and granite for buildings and streets in Sacramento and San Francisco. The discovery of the Comstock Lode in Nevada in 1859 assured the railroad's profitability. Soon a tandem arrangement with the river lines allowed river steamers to haul goods from San Francisco, load them on rail cars, and transfer them to Folsom for distribution to points east in the Sierra Nevada. Reverse shipments brought Comstock silver to Folsom, which was then unloaded in Sacramento and San Francisco. According to historian Walter Gray, the chief significance of the SVRR was to demonstrate to Californians the profitability and efficiency of rail travel.
Engineer Theodore Judah lobbied for support of a railroad through the towering peaks of the Sierra Nevada.
The first locomotive to operate in California was brought around Cape Horn and put into use by the Sacramento Valley Railroad (photo taken in 1855).
The problem was the Sierra Nevada, which loomed nearly 7,000 feet high at its crest. Judah was fascinated by the challenge of the towering peaks and explored potential routes through them. He even wrote a pamphlet, A Practical Plan for Building the Pacific Railroad, and lobbied far and wide for support. Seeking financing, he was dismissed by San Francisco capitalists and others as "Crazy Judah," but their snickering did not deter the young engineer. Turning to Sacramento's increasingly wealthy merchants, he approached four entrepreneurs who had made their livelihood in the state capital—wholesale grocer Leland Stanford (destined to become governor of the state in 1862); his friend, dry goods merchant Charles Crocker; and two hardware retailers, Collis P. Huntington and Mark Hopkins. These four, later known as the "Big Four," met in the upper room of the Stanford Brothers Store, heard Judah, and agreed to underwrite him. Thus, the Central Pacific Railroad was incorporated on June 28, 1861.
Throughout the tumultuous 1850s, Congress had debated the route of a future Pacific railroad and actually pondered several routes across the vast "American Desert." However, the final decision was held hostage to the sectional politics of the 1850s. Once the South withdrew from the Union during the Civil War, Congress approved and President Lincoln signed a bill sanctioning the route cutting through the heart of the country with two railroad companies building from opposite ends. In July 1862, Congress granted the franchise for building the western end to the Central Pacific. Leland Stanford, railroad owner and by then governor of California, provided additional support with state subsidies. (Conflict of interest was not then the issue that it is today.)
Groundbreaking ceremonies for the Central Pacific took place at Front and K Streets on January 8, 1863. With a prayer by Reverend Joseph Augustine Benton and predictions of future prosperity from Governor Stanford and others, the tracks laid in Sacramento moved steadily eastward over the Sierras. Judah never saw the completion of what his imagination helped create. He had broken with the Big Four and died in New England in October 1863.
# THE RAILROAD AND SACRAMENTO
It is nearly impossible to exaggerate the importance of the railroad to Sacramento. Railroad building invigorated the economy and increased demand for unskilled labor. Initially, a considerable amount of labor was needed for construction. The rail yards required men and shop workers for their various operations, bringing a host of newcomers to Sacramento. Later, the railroad recruited large gangs of maintenance workers who were needed to keep rail lines open.
Urban demography started to change. Into the older parts of the city came a flood of new residents, mostly immigrants, who worked in the rail yards, in rail-connected industries, and provided services for the growing city: barbers, painters, restaurateurs, and retailers of various sorts. Ethnic groups began to come and arrange themselves mostly in the close-knit neighborhoods on the West End of the city. Although the comparative spatial smallness of Sacramento precluded the development of sharply etched ethnic enclaves such as were found in midwestern and eastern industrial cities, nonetheless, various ethnic groups staked out social and cultural space, contributing to the city's cosmopolitan character.
Chinese men and women had lived in Sacramento since the gold rush days, the 1852 census showing 814 males and 10 females. Chinese workers increased in numbers when they were recruited to help build the Central Pacific. These early settlers were merchants, restaurateurs, laundrymen, peddlers, and providers of other services. Settling in on I Street and also near Sutter Slough (rechristened China Slough and China Lake), the Chinese enclave contained theaters, gambling houses, a local newspaper, and religious houses including not only family altars, but also a joss house and a Christian chapel run by the Congregationalists. Other Christian denominations also opened missions in the sometimes labeled Chinatown area and assisted the newcomers in learning English.
Distaste for the Chinese had always been in evidence in Sacramento, but it grew during the 1870s after the railroad work was completed. Popular presentations of the Chinese by the press derided the prevalence of Asian prostitution, the high incidence of gambling, and the existence of lotteries and opium dens. Legal efforts to exclude and restrict the Chinese began in the 1870s. In 1876, 4,000 Sacramentans, mostly members of the Sacramento Order of Caucasians, promoted white labor at an anti-Chinese meeting. In 1878, the Sacramento City Council sent a telegram to President Rutherford B. Hayes, urging him to sign legislation to limit Chinese immigration. The famous Workingman's Party, an offshoot of the anti-Chinese movement in San Francisco, led by Irish immigrant Denis Kearney, petitioned to have the Chinese excluded from municipal employment in Sacramento. In July 1879, Kearney himself appeared in Sacramento and, at a rally that drew nearly 3,000 people, raged against land monopoly and "consumptive politicians," specifically calling on them to "drive out the Chinese who are swarming all over the state of California." Although efforts by the Board of Trustees failed to exclude Chinese from the city, the anti-Chinese movement essentially achieved its goal when an 1882 Federal Exclusion Law, endorsed by California congressmen, barred future Chinese immigration. This law was extended for another ten years in 1892.
German-speaking Sacramentans, like the Chinese, had been present from the beginning of the city. The 1852 census identified 730 Germans in the county. By 1860, the number climbed to 1,634 and the number of German speakers (Germans, Swiss, and Austrians) peaked at 2,200 in the 1890s. German-speaking Sacramentans, although diverse in background, came to positions of prominence. One such figure was August Coolot, a local tobacconist (and early investor in the Central Pacific) whose reclusive ways belied a vast fortune. Others included his son-in-law Melchior Diepenbrock, the patriarch of a large and professionally prominent Sacramento family. Prominent Sacramentans of German background included Charles Matthias Goethe, a real estate speculator and eugenicist. Goethe had married into a large fortune and bestowed money liberally on pet causes, especially Sacramento State University. The Breuner family dominated the furniture industry; Joseph Hobrecht was well-known for his lighting fixtures, while Adolph Teichert raked in millions as an engineering contractor and aggregate tycoon. The community had its own newspaper, the Nord California Herold and maintained a spirit of cohesiveness focused around social organizations like the Turnverein (Turners), an athletic-social club that erected a large social hall. German life also focused around the German Evangelical Church, begun in 1867. In 1919, the congregation renamed itself St. John's Lutheran. St. John's welcomed a substantial number of well-to-do German-speaking families, and under the 40-year pastorate of Reverend Charles Oehler continued German language services until World War I. St. Francis of Assisi Catholic Church, although officially a multiethnic parish, also catered to German-speaking Catholics through the work of the Franciscan Friars of Teutopolis, Illinois.
Italians also joined Sacramento's working class in the nineteenth and twentieth centuries. Though only 41 Italians were registered in the 1852 census, the Italian contingent had risen to nearly 2,700 by 1910. According to historian Bruce Pierini, many Italians were recruited from eastern states and directly from Italy for the Southern Pacific Railyards. Census figures placed Italians as the third largest unskilled group working for the railroads in 1900. By 1910, they were the largest and held that place until 1924. Most of Sacramento's Italian colony were northern Italians, but the city also had a small cadre of Sicilians. Early Sacramento Italians also worked as cooks or in the business of lodge keeping. David DeBernardi ran the popular Sacramento Market. Italian Sacramentans enjoyed informal leadership from men like Luigi Caffaro, the owner of the Commercial Hotel, and had their own newspaper (La Capitale) under the editorship of Adriano Mazzini from 1907 until the 1940s. Most Sacramento Italians were Catholics, and in 1906 the Diocese of Sacramento permitted the establishment of St. Mary's at Eighth and N, a national church. The church moved to Seventh and T Streets in 1914 and later relocated to the eastern end of the city at 58th and M after World War II. St. Mary's Italian priests played an important role in community formation through religious services, newspaper articles, and later radio broadcasts. One of the pastors of St. Mary's, Father Dominic Taverna, helped found the social/ cultural Dante Club.
The first Breuner's store (established in 1856) was located at Sixth and K Streets.
Sacramento was also a popular magnet for various groups of Portuguese. Several distinct groups of Portuguese-speaking immigrants made their way to Sacramento and its environs in the latter part of the nineteenth century. These included Portuguese from the continent as well as the more numerous Azoreans (others came from the Madeiras and Cape Verde Islands). As with most groups, the first Portuguese came with the gold rush. Many who came later first devoted themselves to agricultural pursuits across the river in Bryte and Broderick, on the south side of Sacramento, and in the Freeport/Clarksburg area. Others established successful businesses. Sacramento Portuguese arrayed themselves in a neighborhood south of R Street called "Arizona" (a corruption of the term Azores), and there built a small community that had as its center a Portuguese national church named for St. Elizabeth at 12th and S Streets (founded in 1912).
# THE RAILROAD YARDS
The medley of ethnic groups that took up residence in Sacramento did so for railroad related work. The Central Pacific reached Promontory Summit in Utah on April 30, 1869 and was joined to the Union Pacific. The driving of the golden spike and the completion of the line on May 10, 1869 put Sacramento into the national transportation network and made the city a terminus for people, freight, mail, and locally grown produce headed for the tables of the East. The Central Pacific became the most important business in the state and absorbed other railroads, including the SVRR, as well as other transportation enterprises—river lines, stagecoaches, and cartage companies. It also absorbed a potential rival, the Southern Pacific (SP), a company whose name became the umbrella for the Big Four's vast transportation monopoly. Under SP were river boats, bay ferries, ocean liners, cable car lines, extensive real estate holdings, and timber and oil companies. SP became Sacramento's largest employer. Eventually the Big Four took up residence in more opulent San Francisco, but they left a growing workforce of engineers, laborers, shop workers, and administrators in Sacramento, most of whom worked in the repair and construction shops. The Goss & Lambard machine shop and foundry were the nucleus of the later sprawling railroad yards, which became one of the largest industrial sites in the western United States.
The shops grew steadily, with the city graciously ceding land the company required for expansion. In the shops, workers repaired rolling stock and produced an amazing array of products needed by the company: from office equipment to boilers and engines. From 1872 to 1937, some 272 engines rolled off the production lines. Shop craftsmen also made elaborate silver palace sleepers, water pumps, and even 100 of San Francisco's fabled cable cars. One of the first electric light plants on the Pacific Coast was designed and built in the shops. The presence of the yards brought stable work at good wages for the denizens of Sacramento. As the shops dominated the city's economy, so also did the railroad exercise significant influence over class status and politics in the city.
# AGRICULTURE
The railroad also stimulated the development of valley agriculture, a mainstay of California's economy for many years. The rich soils of the Sacramento and San Joaquin Valleys produced needed grains and fruits that could now be marketed beyond the relatively small California market. The railroad gave impetus to the production of wheat, which could be easily shipped. The city had a number of mills. One of the most prominent was the Phoenix Mills, established in 1881. By 1894, the mill was running night and day, churning out 200 barrels of flour per day. The Pioneer Mills belonged to the milling giant, Sperry Flour Company, and had begun in 1856 with old-fashioned grinding stones activated by water from the American River. Indeed, Sacramento mills churned out thousands of barrels of flour until eventually business slowed in the wake of declining international prices and stiff competition from other parts of the country.
Crop lands in the county produced grapes near Rancho Cordova, pears, asparagus, and berries in the Delta, and oranges and other citrus fruits in Citrus Heights, Orangevale, and Fair Oaks. The invention of workable refrigerator cars in the 1880s made it possible to ship Central Valley fruits and vegetables long distances. Many of these fruits were shipped east in refrigerated cars cooled by huge blocks of ice from the high Sierras. By 1901, a huge ice house had been built in nearby Roseville that chilled fruits for shipments going east. Canning became another important Sacramento industry stimulated by the transcontinental railroad. Canning technology did not progress substantially until the time of the Civil War. However, by 1864, Sacramento had its first salmon cannery. J. Routier began the first fruit cannery in the Sacramento Valley in 1876. In 1882, the Capitol Packing Company on Front and K Streets was opened and by 1888 employed 450 people.
With the opening of the Panama Canal in 1914, the market for California canned goods increased dramatically and other regional canneries opened. Hunt Brothers began canning operations in Davis and Marysville. By 1912, Libby, McNeill & Libby had already opened a Sacramento facility that became one of California's largest. The California Packing Company and Bercut Richards also built huge facilities. Some growers and shippers reacted to the increasing consolidation of the packing industry by forming cooperatives, associations, and exchanges. Sacramento's most famous example of this is the California Almond Growers Exchange, which still ships tons of locally produced almonds around the world. The seasonal work patterns of the canneries also set a tone for the working-class culture of Sacramento. Canneries provided an opportunity for immigrant women to develop work skills and add to their family's income. The net result of this increasingly diverse workforce was a slow but steady increase in Sacramento's population. In 1860, 13,875 people resided in the city. By 1900, nearly 30,000 called Sacramento home.
The Goss & Lambard Sacramento Iron Works assembled the first locomotives until the Central Pacific Railroad built their own shops, which became the nucleus of the expansive Southern Pacific rail yards.
By 1894, the Phoenix Milling Company ran day and night, churning out 200 barrels of flour per day.
# RAILROAD DOMINANCE OF SACRAMENTO POLITICS
The economic dominance of the railroad extended to other aspects of the city's development. Sacramento was a microcosm of the advantages and liabilities of American industrialization. Industrial work and wages brought wealth, stability, and the prospect of serious urban building. It also brought pollution, periodic depression, the absorption of prime city lands, and the undemocratic manipulation of city politics.
The railroad hooked Sacramento on its jobs and occasionally played the city like a flute when it needed additional land and space for shop expansion. One such incident occurred in 1875 when company officials wanted to take over the old city waterworks property at I and Sixth Streets and demanded a portion of the riverfront to transform it into a rolling mill. Summoning city officials to their offices in San Francisco in February, the company promised to build the mill and assured the city fathers that this would bring even greater stability to the city. Recalling a similar incident two years earlier when railroad officials had insisted on representation on the city board of trustees in exchange for keeping the shops in the city, the editors of the Sacramento Union denounced the land grab and the company's efforts to subvert democracy. Union opposition displeased the railroad leadership, who began dropping dark hints about relocating the railroad yards to Oakland. Local citizens, fearful of angering the city's chief employer, reacted immediately and organized a mass meeting of nearly 600 people at Turner Hall. Those attending heard the report of a meeting with Leland Stanford and another committee at which the company had assured them of its willingness to stay in Sacramento. But the railroad made no bones about its desire to control the city: they asked that in local elections no one be nominated to an elected city office who was hostile to the railroad. Those assembled at Turner Hall capitulated to railroad demands and, through a series of propositions, declared "that it is manifestly for the best interests of the city to cultivate friendly relations with the Directors of the Central Pacific Company."
When the headstrong Union kept up a sarcastic barrage against the company, decrying those who were "bringing Sacramento in chains before Leland Stanford," its days were numbered. By the end of February 1875, the offending Union had been purchased by the railroad's organ, the rival Sacramento Record. The Record-Union soon squashed any critical spirit and lauded the economic development of Sacramento, counting the railroad yards as the city's main hope for a bright future.
From the 1870s on, the railroad exercised considerable control over local politics, to the extent that their political operatives directed the voting patterns of the yard workers. Railroad agents became increasingly adept at working with local politicians of both parties. Company "bosses," like gambler Frank Rhodes and hops-grower Bartley Cavanaugh (father of a later city manager), picked candidates for city trustee as well as state and federal office. William F. Herrin, head of the Southern Pacific political bureau, conveyed railroad wishes to local politicians and plied officials with free passes on the rail lines. Though technically illegal under the State Constitution of 1879, this standard bribe continued for many years. Herrin resided part-time in Sacramento and gave Bartley Cavanaugh's address as his residence.
# CLASS STRUGGLE
Once the basic parameters of railroad control were established, the relationship between city and railroad was usually peaceful. A series of competent superintendents oversaw the needs of the railroad workers, perhaps the most legendary being Andrew Jackson Stevens, memorialized by a bronze statue in present-day Cesar Chavez Park. Stevens was master mechanic of the shops and the one responsible for introducing the construction of locomotives to the Sacramento yards. Stevens practiced a form of paternalism, refusing to lay off workers during tough times, choosing instead to reduce hours and declining to keep a "black list" of those fired. When he died in 1888, the workers collected nearly $5,300 for the statue.
Even though men like Stevens demonstrated a benign paternalism, others did not feel positively inclined toward the railroad. There were those who felt shortchanged by the inability of the industrial world to distribute wealth equitably or to offer respite from monopolistic or heavy-handed actions by industrial leaders who manipulated government and other public institutions to serve their own ends. In California, farmers disgruntled with the practices and rates of Southern Pacific spearheaded the first reactions against big business. In 1878, a branch of the national farmer's organization, the Grange, was organized in Sacramento. As farmer discontent spread to more organizations, Sacramento found itself receptive to an early coherent voice of disgruntlement: the Populist Party.
The emergence of Populism in the 1890s reflected a widespread discontent in the nation's agricultural areas and drew strong support in Sacramento. In fact, the reformist agenda of the Populists in 1896 caused the Daily Bee, by then edited by the son of James McClatchy, Charles Kenny McClatchy, to turn from its traditional backing of Republican candidates and to throw support behind the Democratic/Populist nominee, William Jennings Bryan. Charles Kenny McClatchy, known as C.K., had taken over the editorial writing at the paper after the death of his father in 1883. C.K., like his father, would become a powerful voice for reform in the city. Ruggedly independent, he withdrew somewhat from his strong support of local Populist politicians when many of them became involved in the anti-Catholic American Protective Association (APA), which attempted to exclude Catholics from public office. The association of reformers with the APA undercut their agenda for a time.
# THE PULLMAN STRIKE: LABOR VIOLENCE
The presence of the railroad also played a critical role in one of the major labor disruptions of nineteenth-century America, the Pullman Strike of 1894. The rise of organized labor was slow in Sacramento. Historian Carson Sheetz noted that from the 1850s through the 1880s, assorted strike actions punctuated various labor venues (cartage, teamsters, and construction) and that various groups of workers pushed for the eight-hour workday. However, Sacramento's only recognized labor organization was the Typographical Union, a fairly moribund association chartered in November 1859. Working-class issues quickened with the activity of the Workingman's Party in the 1870s. Labor issues usually centered on the desire to drive out the "cheap labor" of the Chinese. One of Sacramento's early labor leaders was a young carpenter, J.D. Jost, who was at the center of whatever labor organizing activities took place from the 1870s through the 1890s. Jost was a highly respected figure and won general acceptance by avoiding inflammatory and class-based rhetoric. The advent of the national Knights of Labor in the last decades of the nineteenth century stirred some interest in Sacramento, but there was no enthusiasm for an organization that combined skilled and unskilled laborers, the hallmark of the Knight's membership.
Young Sacramento Bee editor Charles Kenny (C.K.) McClatchy became a powerful voice for reform in the city.
The turning point for local labor came when Sacramento experienced a population and housing boom in the 1880s. Spurred partly by competition with rapidly growing Los Angeles, Sacramento began to lay out more areas for development, building more homes, more public transportation, and other urban infrastructure. The increasing demands for labor created a temporary labor shortage in Sacramento and skilled workers from a variety of trades picked up the pace of labor association. In August 1889, a Sacramento Federated Trades Council was formed, comprised of 11 skilled local trade unions. The numbers soon grew.
Sacramento's reputation for fairly amicable labor relations hit bumps throughout its history. A strike (endorsed by the Trades Council), by the typographers at the Bee in late 1890 and early 1891, nearly brought the paper to its knees. But the worst flare-up of labor unrest took place in the midst of a national depression and the 1894 Pullman Strike.
On May 1, 1894, workers at the Pullman Yards, a railroad car manufacturing site near Chicago, went on strike, protesting reduced wages and benefits. They refused to couple Pullman cars, and they blocked trains that carried the offensive cars. The Pullman workers were not unionized at the time, but socialist leader Eugene V. Debs, head of the United Railway Workers, helped the workers press their cause. Around the nation, the prospect of rail turmoil sent shock waves through a country now heavily dependent on regular rail traffic. The standoff between workers and the company resulted in a federal injunction against the workers, who responded with acts of violence and destruction of cars. The strike found sympathetic supporters in the American West, especially in rail centers like San Francisco, Oakland, and Sacramento. In June 1894, workers at the Southern Pacific yards declared solidarity with their fellow laborers in Illinois and declared a secondary boycott, refusing to move any train with a Pullman car. Southern Pacific retaliated by closing the freight operations, and eventually all work in the yards stopped.
Strikers during the Pullman Strike sabotage a train on the Yolo trestle, 3 miles west of Sacramento, July 11, 1894.
Workers then took over the yards and stations. Despite Southern Pacific demands that strikers be arrested, city officials resisted. Efforts to calm the situation were ignored and eventually federal marshals were sent in to end the boycott. When an impasse was reached, the U.S. marshal called on the governor for three regiments of the local National Guard. In a tense standoff on the Fourth of July 1894, nearly 1,000 guardsmen confronted strikers. When they were ordered to fix bayonets and march on the strikers, many of whom were friends and neighbors, the Guard refused and withdrew. Federal intervention came next. On July 10, President Cleveland ordered the strikers to stand down or be arrested. When shots were exchanged, federal troops arrived on July 11 from the San Francisco Presidio, and U.S. Marines and Army cavalry secured the Sacramento depot and rail yards. In retaliation, local radicals wrecked a train on the Yolo Bypass, killing several soldiers and rail workers. Labor peace eventually returned, but a number of the strikers were blackballed by a vengeful company.
# BUILDING THE CITY: ELECTRICITY AND TRANSPORTATION
The development of urban infrastructure continued apace. By 1854, a city hall had been constructed. The creation of a stable city water supply had been an item on the public agenda since the disastrous fires of the early 1850s. In 1852, city residents approved a tax increase to pay for a new waterworks and it was completed in April 1854. However, the production of a clean, reliable water supply would continue to vex Sacramento for many years. City drinking water contained visible sediment derisively called "Sacramento Straight," and disagreements over the best method of providing clean water prevented a comprehensive city-wide approach until voters approved bonds for a new water filtration plant in the 1920s.
The city's sewer system also required continual updating. The first Sacramento sewers were put in place in 1853. This primitive water disposal system was only designed to handle the excess of rain water and not the increasing amount of human waste deposited by the growing population. Citizens dumped their personal waste and their used bath and laundry water into backyard privies and the China Slough for many years. Outdoor privies were the rule in most homes—literally a trench behind the home. Water closets made their first appearance in Sacramento in 1870 and were soon integrated into the design of public buildings like the County Hospital and large city hotels and eventually private homes. Garbage collection was improved in 1895 when the city hired a private scavenger. In 1922, the city assumed the responsibility. For many years, city garbage was incinerated.
In 1855, gas mains began to be laid down the main streets, illuminating Sacramento's nighttime darkness. Telegraph transmission had been the city's first form of instant communication, and after Alexander Graham Bell had successfully demonstrated the telephone at the 1876 Centennial Exposition in Philadelphia, Sacramentans, like other Americans, soon developed a fascination with the talking box. In 1878, the first two phones in Sacramento connected the Carriage Manufacture Company and the Music Parlor. Later, the phone was hooked up to the telegraph lines and communication with San Francisco commenced. Sacramento's first continuous phone line was begun in 1879—a 5-mile long circuit that connected 29 businesses with a central exchange. The Western Union office served as the first informal telephone exchange, with lines strung from homes and businesses to the telegraph office. In 1880, the Sacramento Telephonic Exchange began and was then purchased by the Bell Company. A rival company originated in 1895 that for a time created confusion since calls could not be patched through from one company to another. These two companies merged in 1902, and in 1906 renamed themselves the Pacific Telephone & Telegraph Company.
Sacramentans gradually embraced electricity as their main power source. Privately owned steam generators had powered the first electric lights in Sacramento as early as 1879, when the Sacramento Union and the Weinstock, Lubin & Co. department store co-sponsored a magnificent display of electric lighting during State Fair week. The newspaper generated the energy from its steam presses, and it was carried over wires on rooftops to a set of arc lights in store display windows.
In the 1880s, a number of hydraulic mining companies tested the possibility of hydroelectric power in the Mother Lode. The leaders of the Natoma Water and Mining Company—Horatio Gates Livermore along with his sons and Albert Gallatin, a manager of the Huntington/Hopkins Hardware Store—secured enough capital to build a small dam and powerhouse on the American River at Folsom. On July 13, 1895, Sacramentans were awakened at 4 a.m. to the sound of a 100-gun salute announcing that electrical power was pulsing 22 miles from Folsom into the state capital, at the time the longest electrical transmission in the world and the second in the United States. On September 9, Admission Day, Sacramento hosted a spectacular electrical parade to celebrate the coming of electricity to the capital city. Over 60,000 witnessed the spectacle, which included 25,000 colored lights wrapped around poles and the illumination of the state capitol building. By 1899, the demand for power had escalated to the point that a second power station was created on the North Fork of the Yuba River. Sacramento soon became fully electric, with street lights, neon signs, and streetcars powered by the alternating current generated by the same mountain streams that had once threatened doom for the city.
Electricity also provided a power source for urban transportation and for the spatial expansion of the city. The city's road paving technology evolved from dirt paths to the simple plank roads that had paved the way to the gold fields to later roads paved with macadamized stone. With the advent of public transportation by streetcars, roads that had long been dirt and dust now needed hard surfaces to accommodate the heavy streetcars zigzagging down the main business districts and out to the residential areas. Beginning in 1858, horse drawn carriages conveyed passengers from Third and R to the commercial district at Second and K. In 1861, car tracks were laid on city streets to facilitate transport systems. In 1890, the Livermores incorporated the Sacramento Electric Power & Light Company and obtained a franchise from the city to build an electric streetcar system, and by 1895, this system was tied to the hydroelectric power that flowed in Sacramento. In 1906, Pacific Gas & Electric (PG & E) took over the system and controlled the supply of electricity and gas for years. From the company's car barns at 28th and N Streets ran 11 lines that crisscrossed the city. The shops of the company also built a number of streetcars. This public transit system dominated Sacramento until the 1940s when it was replaced by buses and automobiles.
Streetcars meant spatial expansion and a greater distinction between residential and working centers. As the old horse-drawn cars were transformed to the new electrical cars, a sorting out of urban space soon segregated the "homes" (east of 16th Street) from the business and commercial district—and also from the older housing stock of the West End that would soon be predominately occupied by the foreign-born and the working class.
Hydroelectric power came to Sacramento, lighting up the capitol building in September 1895.
# PROMOTING THE CITY: CREATING URBAN LEGENDS
A new generation of Sacramentans emerged to lead the city into the new century. Earlier community leaders like Hopkins, Huntington, Stanford, and Crocker had stepped forward to meet the challenges of a changing city life. But in 1873, they moved the headquarters of the Central Pacific out of Sacramento (and themselves with it) and took up residence in San Francisco. The departure of this wealthy cadre (a not uncommon occurrence with Sacramentans who had made their fortunes) gave the city something of a collective inferiority complex. However, a new generation of leaders surfaced, led by Albert Gallatin and local businessman Joseph Steffens. In 1873, a Board of Trade was formed to promote business interests. Steffens, Gallatin, and Sacramento mayor Christopher Green formed part of this group dedicated to creating a better climate for urban development.
The 1870s were a time of financial stress and difficulty for Sacramento as promised railroad wealth did not immediately materialize. However, in the 1880s, a land boom, spurred in part by aggressive railroad advertising in the East, began to bring more people to California. When the majority of these newcomers moved to Southern California, Sacramento developers scrambled to seek their share of the settlers. To compete with the allure of citrus fruits promoted by Southern California advertisers, Sacramento merchants began their own fruit colonies in the eastern part of the county in communities like Orangevale and Fair Oaks. These two small villages were actively marketed as places of health and sunshine, able to produce a superior breed of orange, grapefruit, and lemon. The McClatchy brothers at the Bee joined in the effort to move Sacramento in a new direction. Both brothers railed incessantly against anything or anybody that stood in the way. Relentlessly they pressed Sacramento to improve its water supply, pave its roads, clean up its politics, eliminate its vice and crime, and beautify its streets by planting trees. In 1894, the Bee issued a huge promotional tract, Where California Fruits Grow: A Resource of Sacramento County. Written in the "high booster" style of nineteenth-century city "boomers" and lavishly illustrated, the publication was distributed far and wide to lure people to the city. Goaded by the increasing demand for more organization and planning in city development, in 1895, a new Chamber of Commerce succeeded the somewhat informal Board of Trade and became even more focused on marketing Sacramento.
Historical memory was also drafted in the quest to create a new, improved Sacramento. One group that contributed mightily to the creation of a historic past for Sacramento was the Sacramento Pioneers. This organization of citizens who had been residents of California before 1850 was founded in 1854. The initial group consisted of 70 male citizens who still had first-hand memories of early Sacramento. The group devoted itself to collecting books and memorabilia of the city's past and eventually built a hall to store them. Pioneer Hall at 1009 Seventh Street hosted meetings and public events. Pioneer Society members, especially James McClatchy, spoke often at public gatherings, evoking the spirit of Sacramento's past. Annual celebrations of the Fourth of July became an important moment for Pioneer speakers to recall their memories for a new generation as they were often asked to give the formal address for the day. As time went on, these memories took on a warmer and more romantic glow—and in some cases were so embellished as to bear little resemblance to the truth. Nonetheless, like most cities, Sacramento developed its core of urban myths that helped define its identity and could also be used to attract others to the city. Books, such as Thompson and West's 1888 compendium of Sacramento life and history, included lithographs of city residences and also biographies of leading citizens wealthy enough to purchase a space in the tome.
The Pioneers left their mark on Sacramento through their efforts to reclaim a decrepit Sutter's Fort. By 1895, the fort was nearly a complete ruin. When it was proposed to run the city streetcar system through the property, the Sacramento Pioneers mobilized to save this remnant of Sacramento's past. They forged an alliance with a new group, the Native Sons and Daughters of the Golden West, purchased the property, and began to rehabilitate it. Later, the State of California finished the job and it became a state park. These nostalgic groups did much to market the legend of John Sutter as the "founder" of Sacramento—a myth that passed into the textbooks printed by the State of California and was mandated as part of the statewide curriculum for students in elementary schools.
Historical memories played an important role in the creation of promotional materials for the city. Pamphlets, brochures, posters, and other materials constantly stressed Sacramento's gold rush past and down-played memories of the early struggles of the city: fires, floods, and epidemics (except to note the heroism of the people in combating them). Sacramento's benign climate, its economic vitality, its cultural institutions, schools, health care facilities, the elegance of its buildings (especially the state capitol and the nearby Catholic Cathedral of the Blessed Sacrament), as well as its entrepreneurial drive were promoted in order to bring more people to live in Sacramento.
# ARCHITECTURAL AND CULTURAL ENHANCEMENTS
With stability and wealth enough, Sacramento began to look more and more like a city. In 1872, a retrospective by the Sacramento Union lauded the city's growth made possible by "her mechanics" and the "new industries." Public schools were enlarged, the seat of government had been finished, the Odd Fellows had erected a new temple, and Sacramentans looked forward to a permanent governor's mansion. Local developers like Joseph Steffens pressed hard for river improvements and encouraged vigorous city promotion. At Steffens's urging, Sacramento won federal funds in 1887 for a new federal building at Seventh and K that included a postal facility to handle the increasing volume of mail and business coming into Sacramento. This handsome sandstone structure resembled a castle and remained on the site until it was torn down in the 1950s. Churches also played an important role, not only in the maturation of city life but also in providing buildings of exceptional elegance.
Pioneer Hall, here decorated for a parade to honor President Benjamin Harrison's visit on May 2, 1891, hosted meetings and events.
Roman Catholics were the first to make Sacramento a regional church headquarters. Irish immigrant Bishop Patrick Manogue transferred the center of operations of the Catholic Church of Northern California from Grass Valley to Sacramento in 1886. Soon after, he commenced the building of a new Roman Catholic cathedral based on the design of the Parisian church of the Holy Trinity. Set deliberately at 11th and K, one block north of the capitol, it was intended to complement the state house. In May 1886, workmen began digging the foundation of the Cathedral of the Blessed Sacrament. The mammoth new church was dedicated in solemn rites held at the end of June 1889.
In 1899, the Episcopal Church made Sacramento the headquarters of a Missionary District under the leadership of Bishop William Hall Moreland. Between 1903 and 1908, the Episcopalians built the handsome neo-Gothic St. Paul Church on the corner of 15th and K. A small chapel named for St. Andrew was moved from 23rd and K to 26th and M Streets. Later, a bishop's residence and the Episcopal Trinity Cathedral were erected on this site.
Sacramento in this period began to develop as a mature city. In the 1880s and 1890s, developing city neighborhoods began to showcase elegant Victorian homes, especially along H Street. Perhaps the most elegant was hardware merchant Albert Gallatin's magnificent residence on the corner of 16th and H. Later sold to Joseph Steffens (father of future muckraker Lincoln Steffens), the home embodied many of the finest features of Sacramento's Victorians with gilded ceilings, hardwood floors, and odd shaped rooms. In 1903, it became the governor's mansion (previously California governors had lived in hotels or private homes). Governor George Pardee became its first full-time resident and subsequent governors lived there until 1967 when California First Lady Nancy Reagan refused to live in the "old fire trap." It is now a state museum.
The Cathedral of the Blessed Sacrament, based on the design of the Parisian Church of the Holy Trinity, was the second highest building in Sacramento, second only to the state capitol, in 1889.
Social and cultural life quickened as well. In 1884, a coterie of Sacramentans led by merchant David Lubin formed the California Museum Association "to foster art, science, mechanics and literature" in Sacramento. The association brokered the transfer to the city of a private gallery attached to the estate of E.B. and Margaret Crocker on Third and O Streets. The residence had been the former home of merchant B.F. Hastings and was sold to Judge E.B. Crocker (brother of Big Four magnate Charles Crocker) and his wife Margaret. Local architect Seth Babson renovated, expanded, embellished, and remade it into an Italianate mansion for the Crockers. Behind the home, Babson constructed a gallery that incorporated bowling alleys, a skating rink, and a billiards room. The gallery was intended to house an eclectic art collection that the Crockers had accumulated in their various travels abroad. This collection included 1,000 drawings and other works by contemporary California artists. In 1885, Margaret Crocker deeded the gallery to the city and the Museum Association organized displays of its various objects. Thousands of grateful Sacramentans attended a Flower Festival held in honor of "Lady Bountiful" in May 1885. Her later benefactions included the donation of imported windows to the Catholic Cathedral of the Blessed Sacrament and to St. Paul's Episcopal Church. Sadly, her affection for the city evaporated in 1891 when a lawsuit against a German maid accused of stealing from Crocker's daughter Aimee declared the defendant innocent—despite Margaret's corroborating testimony. Angered that her word in favor of her daughter had not been accepted by a Sacramento court, she quit Sacramento in 1892 and moved to New York, where she died in 1901.
Her late pique at the city notwithstanding, she was one of its greatest benefactors. Even before she donated the gallery, in 1882 she had purchased and expanded a large house, known as the Marguerite Home, at Seventh and Q Streets as a shelter for older women lacking visible means of support. In 1884, she purchased two square blocks across from the City Cemetery on Broadway and built a $40,000 botanical garden. In 1901, Margaret Crocker signed over her mansion to the Peniel Rescue Mission for the care of "erring young women." Margaret's daughter Jennie later repurchased the former family home and it, along with the attached gallery, became the West's first public art museum. Crocker's generosity was matched by Jane Stanford, wife of the governor, who donated art works to local churches and her own home to the Sisters of Mercy for an orphanage.
Since few others in Sacramento had the wealth of the Crockers or the Stanfords, charitable individuals formed private associations, mostly comprised of Sacramento women, who cared for the city's poor. For example, an association of women who founded the Protestant Orphan Asylum took up the care of orphan children.
# HEALTH CARE
Sacramento's early health care history paralleled the development of the field of medicine. From gold rush days on, Sacramento had a cadre of doctors who helped to care for the needs of the sick. Hospital care, which in the early nineteenth century did not have a strong therapeutic thrust, also existed in the city. One of the first city hospitals was in an adobe building at Sutter's Fort. A medley of private hospitals were created around the city. Sacramento County opened its public hospital in 1853 at the corner of I and Seventh Streets, moving it in 1867 to L Street between Tenth and 11th (the capitol grounds today). In 1870, the county purchased 60 acres on Upper Stockton Road where over the years they have enlarged and modernized the public hospital. Important doctors in Sacramento's health care history include Dr. John Frederick Morse, also noted as one of the city's first historians. Dr. J.D.T. Stillman worked with Morse to create a respectable health care presence in Sacramento, and Dr. Gregory Phelan, a New York doctor, played an important role in shaping policies related to public health as well as public education. The advent of the railroad shops and the significant number of job-related injuries caused the railroad to open The Central Pacific Hospital in November 1868. Providing an employer-subsidized medical service for injured workers, it collected dues from the workers and employed a number of city physicians who tended to injured or ill workers at the hospital and at home.
Early Sacramento doctors were a mix of professionally trained and self-proclaimed physicians. After several false starts, the city formed the Sacramento Society for Medical Improvement in 1868. Leading this association of physicians was Dr. Gustavus Simmons, who attempted to professionalize the health care field and opened the Ridge Home Sanitarium in the 1880s. The home was a combination of nursing home and operating hospital, but clearly not the kind of specialized institution that would be recognized as a hospital today. Eventually, Simmons and other city physicians cajoled the Sisters of Mercy to take over the project, which they did in 1895, creating Mater Misericordiae Hospital, known popularly as Sisters Hospital. The sisters re-did the hodge-podge of buildings and wards and created a unified institution that resembled more clearly a hospital. However, the city was slow to respond to public health issues, and some believe that the prevalence of contagious illnesses in Sacramento slowed its growth.
# THE LEGACY OF DAVID LUBIN
Sacramento's adoption of consumerism as a way of life took hold gradually as the downtown began to arrange itself into a new commercial district. Popular emporia or department stores led the way. In 1874, David Lubin opened the Mechanics Store, a dry goods business on the southeast corner of K and Fourth Streets. The store introduced an innovation in Sacramento retailing by operating on a cash-only basis and insisting on fixed prices for its goods. He was joined by his half-brother, Harris Weinstock, and the store was renamed Weinstock, Lubin & Company in imitation of eastern emporia that used family names. As business picked up, Weinstock and Lubin added on to their store, which eventually engulfed most of the block. Once his fortune was made, the intense Lubin turned to public affairs and played a key role in the development of Sacramento cultural life. In 1901, he wrote a philosophical treatise entitled Let There Be Light and devoted the rest of his life to the issue of balancing agricultural supply with world demand for food. In 1905, with the blessing of the Italian royal family, he opened the International Institute of Agriculture in Rome, with 46 nations participating. Lubin died in Rome on New Year's Day 1919 and is memorialized in the name of a Sacramento grade school and a street in the Eternal City, Viale Lubin at the edge of the Villa Borghese. In 1946, the United Nations Food and Agriculture Organization absorbed the work he began.
The railroad helped define a new era in Sacramento and the forces of development helped the city create a more visible and settled atmosphere. Sacramento did not grow rapidly during this time, hampered in part by disease, slow development of its roads, transportation system, and other appurtenances of urban living. However, the twinkling lights of the electric parade, the successful effort to stave off capital removal, and the formation of a new and even more aggressive development association, the Chamber of Commerce, provided yet another moment when Urbs Indomita asserted itself.
Merchant and philanthropist David Lubin, shown here c. 1913, devoted the latter part of his life to agricultural issues and the world demand for food.
# 4. RETOOLING FOR MODERN TIMES
Igniting the fires of reform in late nineteenth-century Sacramento was not easy. For nearly ten years, C.K. McClatchy of the Bee had derided the city administration and the pace of urban improvement as "Silurian"—a typical McClatchy neologism for backwards. Others shared the editor's dim view of Sacramento's poor streets, brackish water, and undeveloped cultural and social life. State legislators in particular bridled against staying too long in the heat and smell and disease of Sacramento for the infrequent sessions of the state legislature. State offices scattered to more pleasant locations. Legislators particularly resented the hypercritical tone of the Bee. Smouldering discontent among the solons over sessions in "dull" Sacramento was ignited by McClatchy's needling in March 1893 when the Bee sarcastically "celebrated" the end of a legislative session with a bold headline begun "Thank God." Angry legislators denounced the newspaper and in retaliation introduced a bill to remove the capital from Sacramento. Fortunately, the efforts faltered (as did another attempt spearheaded by a vengeful Southern Pacific early in the twentieth century). But the very possibility that state government might pull out of Sacramento galvanized city leaders to take action and mobilize a cadre of middle-class reformers who insisted on a cleaner and more efficient Sacramento. McClatchy later claimed credit for bringing Sacramento out of its "Silurian" period. But the editor's self-promotion notwithstanding, Sacramento did embark on an important period of physical and social transformation in the late nineteenth and early twentieth centuries. At the same time the restoration of national prosperity in the latter half of the 1890s and energetic local leadership stimulated by the national and state Progressive movements gave Sacramento a new lease on life.
# GOVERNMENT REFORM
The cumulative impact of industrial expansion and population growth presented new challenges. The development of electrical power, the internal combustion engine, the increasing popularity of secondary education, and the respect tendered to "scientific thinking" encouraged a local movement for efficiency and democracy. City government felt the first pressures of the new civic mood. The first city charter, forged in the 1850s, had provided for an elected mayor and a common council derived from the geographical boundaries of city wards. In 1857, the city and county governments merged for a time, an arrangement that lasted only until 1863. Urban-rural clashes doomed the alliance, and the city and county governments uncoupled when a new charter was drafted. The charter of 1863, which would be in place for more than 30 years, provided for three trustees elected by the voters at large. Each trustee took care of an aspect of city life: roads, public health, fire and police, etc.
As the city grew in size and complexity, the charter of 1863 became inadequate. Historian William Mahan has described the growing discontent with the existing system, which resulted in the 1893 revision. The process began when the city elected 15 freeholders, charged with the task of writing the new charter, in December 1891. The freeholders were a mix of some of the city's pioneers, like Judge J. Wesley Armstrong (who had come during the gold rush), along with new city leaders, including department store mogul Harris Weinstock, attorneys Robert Devlin and Grove Johnson, and future mayor Clinton White. The group completed their work by March 1892. The new charter created an independent mayor who ran the city like the executive of a company. An expanded board of trustees, which functioned as a legislative branch, complemented the newly empowered mayor. The city was divided into nine wards, with each ward electing its own trustee and its own representative on the board of education. The new document gave the mayor the right to appoint a number of officers with the consent of the Board of Trustees, including the chief of police, all 15 police officers, the city surveyor, superintendent of streets, fire chief and firemen, directors of cemeteries, and employees of the waterworks.
Sacramento lawyer and legislator Grove L. Johnson, shown here in 1906, was one of 15 freeholders elected in 1891 to write a new city charter.
# THE NEW URBAN POLITICS: RAILROAD CONTROL
The charter of 1893 created a more effective government to deal with issues plaguing Sacramento. Anxious city development types were eager to improve Sacramento's image and enhance its suitability for newcomers.
Public health issues continued to dog the city. Ineffective means of rodent extermination and foul-tasting drinking water continued to be problems, and city denizens refused to pass the tax bonds necessary to remedy these problems. Street cleaning, especially for the animal ordure left daily, was limited. Sacramento stunk, especially in the hot summer months. In addition to the waste left by animals on the streets, the city had seven dairies and several breweries within its limits, a foul smelling glue factory at 30th and U, as well as hog and cattle yards. Arguments over the surfacing of city streets raged continually. Some roads had been surfaced; others were still gravel and hardened soil, dusty in summer and mud pots in the rainy season. City vice, long a bête noire of respectable citizens and the object of erstwhile crusades by the newspapers, flourished. Prostitution was carried on in houses with the distinctive "1/2" on their addresses. Saloons, gambling, illegal lotteries, opium dens, and bars that stayed open all night were as hard to erase as original sin—and all of this vice appeared to go on under the noses (and perhaps with the tacit support) of the city administration and the police force.
Sacramento's poor streets were a source of complaint, as in this view of yet to be paved J Street looking west from about Seventh Street, 1868.
For some city officials, the time was ripe for a renegotiation of city dealings with the powerful railroad. The railroad's political control was obvious, and wily railroad political agents were still able to work with the new city administrations and especially the city council. However, city reformers soon began to identify the railroad with everything that kept Sacramento from urban greatness. The railroad, once hailed as the city's salvation, was increasingly viewed as a detriment to urban improvement. Donning the mantle of urban reformers, local progressives in Sacramento framed virtually every public issue as a struggle between the forces of urban advance and a reactionary cabal consisting of the overly powerful railroad, its puppet politicians, and immigrants. In fact, the division of the city into geographic wards by the charter made it easier to pit one group against another and accentuated class and socioeconomic differences as never before. Naturally, the poorer and more immigrant-heavy wards were targeted by reformers as sources of urban corruption while the more prosperous, upscale, Americanized wards were the sources of civic renewal. William Mahan relates that the "worst case" wards (as far as reformers were concerned) were the four in the city's West End, an area that took in the now decaying waterfront and the rail yards. The city's fifth ward was a swing ward, while wards six through nine were in newer residential districts—an area referred to as the "homes." Representing the poorer, working-class wards was the local Democratic Party. In this broad Democratic coalition were also the city's immigrant communities. These too were viewed suspiciously by reformers.
# A MORE DIVERSE CITY
Among the growing number of nationalities clustered in the West End were Italians, Portuguese, Japanese, Croatians, and Filipinos. Japanese immigration to America began in the late nineteenth century. Most Japanese migrated to the West Coast, and thanks largely to the dynamic of chain-migration, Sacramento became one of the major cities for Japanese settlement. By 1900, there were 336 in the city and 1,209 in the county. By 1910, the numbers had escalated to nearly 1,500 and 3,874 respectively. Sacramentans awoke to the visibility of Japanese residents when nearly 700 gathered in McKinley Park in 1895 to celebrate the victory of their nation over China. Scattered Japanese agricultural settlements dotted the county, and in the city the Japanese Quarter (Japantown) was one of the larger ethnic enclaves stretching east from Second to Fifth Streets and south from L to O (the heart was between Third and Fourth and L and M). By 1911, historian Wayne Maeda notes that Sacramento's Japantown was self-contained and self-sufficient, with over 200 Japanese-owned businesses. The community had its own barber shops, pool halls, banks, hospitals, mutual aid societies, and an array of churches—Buddhist, Methodist, Presbyterian, and Baptist.
Joining the ethnic medley were also immigrants from the Balkans, largely Croats but also some Serbs. By 1930, there were nearly 1,000 of the first generation, many of whom worked in mining, railroading, farming, and canning. The center of Croatian culture was the Rosemont Grill, where newly arrived Croatians found assistance and an informal labor exchange. Filipinos also joined the mix, and though few initially, large numbers began coming through Hawaii in 1898. The mix also included some Koreans and Punjabis as well.
Although the railroad worked well with both parties, Sacramento immigrants gravitated to the Democrats. The diverse Democratic coalition was held together by one of Sacramento's most skilled politicians, Thomas Fox, a native of Oak Park who was a career executive with the Pacific Mutual Life Insurance Company. Fox, twice postmaster of Sacramento under Democratic administrations, was a popular and effective politician. Edward J. Carraghar, who owned the popular Saddle Rock Restaurant on Second Street, assisted him. Southern Pacific political bureau director William F. Herrin used the time-honored "boss" system to continue to bend public policy to support railroad interests. Sacramento reformers took a dim view of railroad-oriented politics with its hint of corruption, based on transient and immigrant voters whom they believed were too easily manipulated by politicians and railroad company masters. City reformers were determined to end the dominance of what novelist Frank Norris referred to as the "Octopus."
# PROGRESSIVE HIGH TIDE: THE WESTERN PACIFIC AND CLINTON F. WHITE
Sacramento's progressive reformers built on old Populist ideas and energy. New political leaders who disregarded and rejected old parties came on the scene and began to press a reform agenda. Interestingly, reform began with old-line politicians. Even though he was a "tool" of the Southern Pacific, Mayor William Land (1898–1899) and his successor George Clark (1900–1903), a local mortician, fought a successful battle to end illegal gambling at Sacramento's poolrooms. Victories piled up. In 1903, initiative and referendum were added to the city charter. Using the initiative helped end ward politics by mandating the election of each trustee at-large rather than by ward, making it possible for middle-class Sacramentans to outvote the working-class immigrant wards.
Sacramento women also contributed to the climate of reform by mobilizing to eliminate the influence of urban vice in the form of gambling halls and taverns. They provided critical support to the efforts of Mayors Land and Clark to end poolroom gambling. Linked to these efforts to eliminate sin was the cause of women's suffrage. Early efforts promoting women's suffrage began in 1871 when Mrs. L.G. Waterhouse, a midwife, "hydropathic" physician, and leader of the Sacramento Women's Suffrage Association, hosted veteran women's rights activist Susan B. Anthony for a rousing talk. Nonetheless, efforts by local women to move the state legislature to debate the matter waxed and waned until 1911 when it was put before the entire state in a referendum. By that time, the support of both papers, Bee and Union, was strong, and after a tense election day the state voted to grant women the vote. In August 1920, the rest of the country followed California's example and the 19th Amendment guaranteed women's suffrage. The vote of women allowed middle-class voters to outnumber the transient-immigrant votes, which were exclusively male.
The Western Pacific passenger depot in Sacramento was approved in spite of interference by Southern Pacific operatives.
As time went on, a more sharply defined reform agenda emerged that linked change with the efforts of city boosters to promote Sacramento as a good place to live. Reformers insisted on honesty in government, a more stable local economy, and greater efficiency in public and private ventures. McClatchy was the voice of this movement, endlessly lampooning and haranguing city leaders and policies that gave the city a bad reputation. One of McClatchy's allies was Hiram F. Johnson, the son of longtime Sacramento lawyer and legislator Grove Johnson. Hiram Johnson, who had supported Clark against Land in mayoral politics, wedded himself to the Progressive cause. His public profile as a reformer was enhanced through his work on the prosecution team of the San Francisco Graft Trials of the early 1900s. The successful prosecution of political malefactor "Boss" Abe Ruef gave birth to the California Progressive movement. Hiram Johnson would eventually be elected governor by those same forces in 1910.
Sacramento's Progressive revolt was sparked when Southern Pacific hegemony in the city was challenged by the rival Western Pacific Railroad (WP). When WP sought a route into the city and a place to build its own yards, Southern Pacific henchmen on the city council did not openly oppose the project, but did everything they could to slow it down. Encouraged by events in San Francisco and angered at blatant railroad interference in the gubernatorial election of 1906, Sacramento reformers struck back and showed their strength in October 1907 by pushing through a right-of-way for the Western Pacific Railroad. A month later, reform forces rode to a second victory when they narrowly elected Progressive Clinton F. White as mayor. White had won by denouncing Southern Pacific as a monopoly that spoiled the prospects of Sacramento's future advance. "Why not cast out the evil influences that have so long kept the city down?" White asked in his campaign speeches. C.K. McClatchy lauded the election of White as an apocalyptic event in the history of Sacramento: "This election is a triumph for right against wrong; for law against lawlessness; for the interest of the People against the grasping clutch of corporate monopoly: for better civic government in every way."
White disappointed McClatchy's hopes by lasting only two years before he returned to his private law practice. He was replaced by the man he defeated in 1907, Marshall R. Beard. Despite McClatchy's editorial groaning that Beard represented a return to "boss rule," the mayor continued the process of city improvement begun under White. Meanwhile, McClatchy turned his interests to state politics, where the Progressive movement was led by the Lincoln-Roosevelt League, a cadre of Progressive reformers that propelled Sacramentan Hiram Johnson to the governor's seat. Under Johnson, California became a showcase of Progressive reform and Sacramento whirred with new legislative initiatives, realizing many of the hopes and dreams of reformers.
# A GROWING AND CHANGING CITY
Progressive era Sacramento was not only bursting with reformist hopes, but it also began to expand spatially. In 1911, Sacramento added substantial new lands to its 1849 boundaries. The chief focus of annexationist efforts was the streetcar suburb of Oak Park on the southeastern fringe of the city. Oak Park had been laid out in the 1880s by developer Edward Alsip. A neat community of small homes, businesses, and churches, it attracted a large number of Sacramentans employed in service trades, such as telephone operators and railroad workers. Other Sacramentans had been attracted to Oak Park largely because of a popular amusement park, Joyland, which included a roller coaster, a tunnel of love, and swimming pool. In 1909, long discussed annexation plans were pushed by the Oak Park Improvement Club. Likewise lands east of 31st Street, the city's back door, also became ripe for the picking of a growing Sacramento. East Sacramento, as it was known, was a mixture of homes, a school, a dairy, and a privately owned park, as well as a race track. Although less populated than Oak Park, East Sacramento stood right in the path of the city's projected eastward expansion. The campaign for annexation began in March 1910 and succeeded in convincing both Oak Park and East Sacramento residents that joining Sacramento proper would be in their best interests. In September 1911, they approved the annexation, thereby adding 6,000 acres to the city.
The heart of Sacramento's subsequent suburban growth was north of Sacramento across the American River. The sale of the old Rancho del Paso Grant in 1910 released over 44,000 acres for county development. These lands fell into the hands of O.A. Robertson, a St. Paul capitalist working with the Sacramento Valley Colonization Company, a consortium of ten investors including former Sacramento Mayor Daniel Webster Carmichael, who established a rural idyll bearing his own name some 12 miles east of Sacramento. Another consortium of developers planned Citrus Heights also on the Del Paso grant. Earlier in the 1880s, investors had planned Orangevale and Fair Oaks, small citrus growing communities that were intended to burnish the northern state's reputation as a sunshine capital. The North Sacramento Land Company's D.W. Johnston and his son Carl E. Johnston secured a huge 4,000-acre tract that had once been grain and grazing fields. Subdividing about 2,500 acres of this property, the Johnstons laid out a small city called North Sacramento. Cut through by 100-foot wide Del Paso Boulevard, the city thrived as Sacramento's first automobile suburb. Grammar schools, businesses, and other utilities began to give the community shape and identity. By 1929, the population was estimated at 10,000. North Sacramento cherished a strong community identity for many years, until it eventually merged with a burgeoning Sacramento in 1964. Sacramento also purchased 828 acres of the Del Paso grant along Arcade Creek to be preserved as a city park. Two 18-hole golf courses now dominate the park.
# GOVERNMENTAL REFORM AGAIN
In 1910, Sacramento's newly rejuvenated city government erected a handsome new city hall on I Street. Designed by architect Rudolph Herold, the brick/terracotta structure with a modified onion dome presided over the city plaza and formed a link in the chain of public buildings that included the old post office/ court house and later a Carnegie-funded public library.
The streetcar suburb of Oak Park, shown here c. 1910, was annexed to Sacramento in 1911.
In June 1911, as Sacramento voters pondered the fate of Oak Park, they selected 15 freeholders to rewrite the city charter yet again. The freeholders hammered out a document that called for five non-partisan, elected commissioners who would possess equal power—thereby further undercutting bossism. In a November election that re-elected Mayor Marshall Beard, voters approved the new charter. A reform-minded Municipal Voter's League pressed hard for four reform candidates with a battle ensuing as Beard and Carraghar, old city politicians, scored well in the primaries. But the Municipal League fielded an effective counter-weight with Catholics and Democrats by endorsing Michael Burke, a prominent Catholic and labor union politician. Likewise, they took advantage of voting women by endorsing Laura Johnston, who came in fifth and became the first woman elected to municipal office in Sacramento. Beard and Carraghar continued to serve, but represented the end of the old working-class ward politics. After the commission system was up and running, Sacramento adopted a civil service system in 1913, thereby ending the practice of ward bosses handing out public jobs and of appointed officials "feeding the kitty" of political associations. Gradually, a professionalized bureaucracy emerged.
The commission system, however, lasted only a short time, unable to cope with the rapid acceleration of the city's growing needs. In 1921, Sacramento again revised its charter and created a nine-member city council. The most important innovation in the new charter was the creation of a city manager's position, which installed an urban professional to oversee city life. Sacramento's first city manager was Illinois-born Clyde L. Seavey. A former journalist and member of the Sierra Club, Seavey had been strongly endorsed by the Chamber of Commerce. Seavey's reforms trimmed the budget, consolidated governmental departments, and insisted on licensing new businesses (putting fortune-tellers, hypnotists, and mesmerists out of business). Seavey also tackled the problem of urban vice by denying the Japanese licenses to operate poolrooms, closing 30 saloons, and reviving the chain gang to scare off transients. The city council was chosen by at-large voting and the mayor, no longer the strong figure of the 1893 charter, was the councilman with the highest number of votes. The end of ward elections cost the working class its power over city government. Power passed to the educated and established and those endorsed by the Chamber of Commerce.
Progressive era emphasis on city planning reflected the increased value of city property and a desire to manage city life more efficiently. To this end, city leaders engaged the services of professional city planners. In 1908, Charles Mulford Robinson, a nationally-known urban planner, visited Sacramento and proposed a city design that included streets radiating from the state capitol. In 1916, another national leader, John Nolen, provided plans for the use of Sacramento's increasingly more valuable urban space, including a re-working of the capital's grid pattern and also an ambitious program for Del Paso Park. In 1923, the city adopted its first zoning ordinance and in 1924 created the first City Planning Commission. Although the commission was quickly disbanded after only a few months of service, in June 1926 it was revived and became integral to rational city development.
Former journalist Clyde L. Seavey became Sacramento's first city manager.
# WORLD WAR I: DEMANDS FOR LOYALTY AND A NEW MOOD
Sacramento entered World War I. By the time Woodrow Wilson delivered his dramatic request for a war declaration against the Central Powers in April 1917, most Sacramentans agreed that the time had come to teach the German Kaiser and his allies a lesson. A huge turnout at the Empress Theater fanned the flames of patriotism and urged enlistment in the armed forces. In all, about 4,000 Sacramento County youth responded to their country's call, with nearly 100 dying in the conflict. Hundreds of Sacramentans on the home front offered their services to the Red Cross and sold and bought Liberty Bonds. A local Council of Defense kept a close eye for any sign of disloyalty. In the anti-German hysteria of the era, Sacramento's Lutheran church abandoned its German language services and local Irish nationalists were barred from welcoming the widow of one of the Easter Rebellion martyrs for fear of giving offense to the American alliance with Great Britain.
During World War I, Sacramento's fears of radicalism flowered once again. The city occasionally had brushes with radical groups. Bands of unemployed workers periodically descended on the city and created jitters in the press and in city government. In 1913, for example, one of these groups, Kelley's Army, marched into Sacramento and set up shop in one of the city parks. Eventually, city officials were able to bribe them to move, but not until C.K. McClatchy sufficiently raised public fears that these "bummers" threatened the city by their demands for food and shelter. In 1917, a violent explosion damaged a portion of the back porch and kitchen of the governor's mansion. City officials blamed the blast on violent elements of the International Workers of the World (IWW), a militant labor organization that had begun to attract followers on the West Coast. A number were rounded up and jailed by the federal government, but were never convicted of this bombing. Several of the detainees died of influenza in the city jail.
When the war ended in November 1918, the city erupted with joyful demonstrations. In April 1919, Sacramentans strewed the streets with golden poppies to welcome home "the returning heroes" of the 363rd Infantry of the "Fighting 91st." Hoping to win public support for a League of Nations to prevent future conflicts, President Woodrow Wilson visited Sacramento in September 1919. Taking this internationalist message into Sacramento was an act of political courage since both McClatchy and Hiram Johnson bitterly opposed American entrance into the League. Nonetheless, nearly 10,000 Sacramentans lined the tracks on R Street shouting their approval and throwing roses at Wilson and his wife Edith. "I feel you voice approval of the great covenant signed to make peace permanent in the world," the President shouted in gratitude from the back of his rail car.
Patriotism became increasingly visible, as evidenced by this Portuguese float built by the Nunes Brothers shipbuilders for the Fourth of July parade, 1916.
The war brought a new element to Sacramento's economy: military spending. The U.S. Army's growing interest in air power sent the War Department scurrying for factories to build the biplanes that would challenge German power. One such plant, the Liberty Works near present-day North Sacramento, made the Curtiss JN-4 (Jenny) airplane for the Army Air Corps, and in 1918 Sacramento made a strong pitch to the War Department to locate a flight school in or near the Sacramento area. Military commanders agreed, and in June, Mather Field at Mills Station became the home of an airfield and training school. City leaders welcomed the new facility, not only because it brought airplanes and military personnel (both considered an interesting novelty) but also because Mather's construction and needs pumped money into the local economy. Mather Field had a relatively brief existence because the war ended that November, and the nation demobilized rapidly. Mather Field went into dormancy in 1923, with periodic use as the site of air shows, the delivery of airmail, and even as a stop-off point for Charles "Lucky Lindy" Lindbergh, who traversed the nation after his showstopping transatlantic flight in 1927. Mather Field, later Mather Air Force Base, reopened as a military facility just prior to World War II.
# SACRAMENTO AND PROHIBITION
One of the things that War Department officers demanded of Sacramento as a "price" for locating a flight school near the city was the clean up of its West End. Drinking, gambling, and prostitution raged unabated, despite periodic clean-up attempts. A high profile commission led by Simon Lubin (son of David Lubin) scored the moral squalor of the city's "dives" in 1919 and blamed their continued existence on lax police enforcement. Not much came of the furor, but moral reforms were also high on the Progressive agenda.
Sacramento had always supported the brewing industry. Henry Grau founded the Buffalo Brewing Company in 1889 and built a huge brewery on 21st Street between Q and R—once the largest brewery west of the Mississippi. Swiss brewer Frank Ruhstaller had purchased a Sacramento brewery in 1881 and produced a popular steam beer. Ruhstaller merged his holdings with Buffalo in 1897 and eventually came to dominate the new company. The popularity of these beers and the host of taverns and saloons in the city suggested that the atmosphere of the state capital was never sympathetic to either temperance or prohibition.
Since the 1850s, organizations like the Women's Christian Temperance Union and the Sons of Temperance had pressed for saloon restrictions, but saloons were popular with the mostly male working class, who treated them as social clubs. Support for prohibition of any kind from local breweries and wineries was weak. Saloons proliferated. By 1884, there were almost 400 drinking establishments, nearly 130 of them never bothering to pay for a liquor license. (Mark Twain called Sacramento "The City of Saloons.") Groups that favored prohibition (Grangers, the American Protective Association, and social moralists in the Progressive movement) attracted supporters but were never able to rise above the opposition. Rules banning the sale and consumption of alcohol never passed muster with city officials. In 1907, an effort to ban saloons from the "Homes" section of the city failed by a wide margin in a popular vote. Anti-Saloon League activity was clearly visible, fueled by reformers and the Ministerial Association (an alliance of local Protestant clergy), but efforts spearheaded in 1914 and in 1916 by the group were turned back. As a result, when the 18th Amendment was passed in 1919, Sacramento's embrace of the dry cause was tenuous. New Year's Eve 1919 produced its share of blowout parties—before the dread hand of Prohibition formally gripped on January 17, 1920. But despite the federal directive, Sacramentans flouted the regulation. City historians note that Sacramento had a revolving door of police chiefs during the 1920s, none of whom were able to adequately handle the problem. Bootlegging seemed to go unchallenged in both city and county. Despite periodic raids and revocation of business licenses, the less than enthusiastic or consistent enforcement of the 18th Amendment allowed the secret liquor trade to flourish. It took until 1922 for Sacramento to enforce the new law. By then speakeasies and other private drinking establishments had been established and had gained general acceptance. "Virtuecrats" in Sacramento were dismayed by the city's resistance to prohibition—their organized displeasure playing a role in the popularity of the Ku Klux Klan in the California state capital.
Mather Field's engineering officers, shown here in 1918, were among those stationed at the facility, instituted during World War I.
# SACRAMENTO AND THE KU KLUX KLAN
The Klan had experienced a national revival even before the United States entered the war and had taken as its pet themes Prohibition and the disparagement of Catholics and immigrants. Klan organizers had come to Sacramento in 1921, making a dramatic appearance at the city's Westminster Presbyterian Church where they approved of the minister's preaching. From his quarters in the Traveler's Hotel, Klan organizer (Kleagle) Edgar Fuller began to tap into a wide stream of pro-Prohibition and anti-Catholic sentiment that reflected itself in the numbers willing to take out Klan membership. Sacramento Klansmen attempted to portray themselves as upstanding American patriots, devoid of prejudice (they even painted a local African-American church), but clearly were suspicious of Catholic "interests" in the city and worried about the potential for radicalism.
City Manager Clyde Seavey, with the public support of McClatchy and the Bee, waged war on the Klan. The paper attempted to embarrass participants in the huge induction ceremonies that took place in Oak Park and in Folsom's dredgings by sending reporters to copy down license plate numbers and then revealing the owners' names in the newspaper. A raid on Southern California Klan membership headquarters also revealed the names of several Sacramento city officials: seven policemen, three fire captains, and the city harbor master. When Seavey attempted to fire them, their appealed case was brought before the city council, which refused to support the city manager. Seavey was pilloried by some as a "red" and resigned from the manager's post. He then served in state government, and in 1934 President Franklin D. Roosevelt appointed him to the Federal Power Commission.
# SACRAMENTO ASSOCIATIONALISM
The Klan's appeal to many Sacramentans (hundreds showed up for the induction ceremonies) certainly was related to the fears and reaction of the postwar period. Progressivism had spent its energies, and people were anxious for the comfort of the "old times" promised by Klan rhetoric. But the Klan's popularity was also related to the organization's fraternal appeal. Sacramentans were joiners and had a strong network of fraternal organizations that bonded businessmen and fellow citizens in collective endeavors.
Close associations often began in the city high school where students formed sororities and fraternities. Fraternal groups like the Howards and the Odd Fellows made a significant contribution to the emergence of city benevolence to the poor and to professional health care. Strong Masonic lodges developed and banded together for charitable as well as political reasons. Likewise groups like the Elks, the Improved Order of Redmen, and the Knights of Pythias built substantial meeting halls. Irish citizens could join the Ancient Order of Hibernians, and young Catholic men and women could participate in the Young Men's and Young Women's Institutes. Singers could warble with the McNeill Club. Ethnic communities like the Germans formed the Turnverein, while the Portuguese formed devotional societies like the Irmandade do Divino Espirito Santo (IDES), which underwrote annual festivals and celebrations.
One Sacramento association that has stood the test of time is the Sutter Club, founded in 1889 with rules drawn up by Valentine McClatchy and Judge William Cary Van Fleet. The founders were up and coming (and in some cases quite accomplished) professionals in business, law, medicine, and agriculture. Desiring a better place to meet, socialize, and discuss affairs than the rough and tumble haunts on Front Street, the new club began its operation in the upper room of the California State Bank on Fourth and J Streets. Later, the association built a clubhouse on Ninth and L, once the site of a Baptist church. Their annual Christmas ball was one of the social highlights of Sacramento's year. Nearly all the city's civic leaders were members and the roster read like a Who's Who of Sacramento elite.
Another popular gathering spot was the country club, which meant the Del Paso Country Club. Originally called the Sacramento Golf Club, in 1916 it moved from its headquarters to a location on the old Rancho del Paso. Here city residents met for golf, tennis, trapshooting, and dances. For Sacramento women, the major social gathering was the Tuesday Club, founded December 1, 1896. A second group dedicated to cultural advancement among Sacramento women was the Saturday Club.
# TOWARD A GREATER SACRAMENTO
The fears of Sacramentans, reflected in their embrace of harsh measures against radicals and membership in the Klan, was counterbalanced by their wholesale embrace of modernity. Leading the charge to modernize Sacramento life were the collective forces of the Chamber of Commerce. Under executive secretary Harry Muddox, the chamber made great strides in improving the city and in attracting federal dollars. In 1922, Muddox stepped down and was replaced by one of the most indefatigable of Sacramento's boosters, Arthur Serviss Dudley.
Dudley was a native of West Salem, Wisconsin and a graduate of the Illinois College of Photography. Attracted by a career in boosterism, he became secretary of the Riverside Chamber of Commerce and later took a similar position in Los Angeles. He arrived in Sacramento in 1920 as secretary to the chamber and helped pull the city's development energies together. To unite the city, Dudley drew memories of the past into the service of a new modernized Sacramento, which he believed needed civic unity. He hosted a huge historical festival called the "Days of '49" adjacent to the rail yards on the site of the present-day railroad depot in May 1922. It featured a "whiskerino" contest for which some Sacramento men had grown beards for six months. For the occasion, event planners constructed a mining town, an Indian village, and a wooden mountain that offered donkey rides. Fireworks, parades, rodeos, and music also punctuated the event. "Sacramento has reincarnated the spirit of the Days of '49," crowed an account of the celebration. "The march of life, the very atmosphere of the romantic days of the Argonauts has been brought back." The message was nostalgic but the goals thoroughly modern: Sacramento had to pull itself together for the task of urban improvement and civic growth. Dudley later played a pivotal role in the economic development of Sacramento, recruiting and helping initiate the development of infrastructure, buildings, and greater employment opportunities.
# TRANSPORTATION AND THE RISE OF AUTOMOBILE CULTURE
As people-moving technology improved, Sacramento became increasingly connected with a wider area. A series of inter-urban lines ran back and forth from the state capital, connecting the city with Chico, San Francisco, Woodland, and other cities. Sacramento gradually eased into one of its most significant social changes with its embrace of the automobile. In 1900, the first automobile was exhibited at one of the city's periodic street fairs. In 1902, car races were held for the first time in the city, and in 1903, merchant Joseph Schnerr leased a shop on Tenth and J for the sale of automobiles and bicycles. By the next year, 27 automobiles were registered in Sacramento. By 1910, the number had leapt to 700, and in 1912 there were 16 automobile agencies, 1 assembly plant, and 11 garages. By 1929, one in three Sacramentans owned a car. Trucks and buses made their appearance in 1910. A bus route between Sacramento and Folsom opened in 1910; in 1914, buses connected the city with Stockton. Trucking firms also began to move in on the short-haul traffic generally carried by the railroads. The advent of the automobile ended the work of horseshoers, blacksmiths, wagon makers, and feed and harness stores. Ultimately, the vehicles undermined the powerful railroad and ended the steamboat era.
The creation of a network of passable and interconnected roads became a state priority in 1895 when a committee was established to draw up a plan. In 1895 as well, the state purchased the Lake Tahoe Wagon Toll Road, which became the skeleton of Highway 50, connecting Sacramento with the popular resort area. The state created the California Highway Commission in 1909, which secured a bond issue financing the construction of the Yolo Causeway, an elevated two-lane roadway propped up by a wooden infrastructure. This dependable land link with the Bay Area opened in 1916.
The Days of '49 celebration in 1922 featured a constructed mining town, an Indian village, and a wooden mountain that offered donkey rides.
A group of local bicycle aficionados formed the Capital City Wheelmen in 1886, and during their races became aware of the perilous conditions around the state capital. After taking inventory, the Wheelmen made common cause with local merchants and held a Good Roads Convention in 1903. But funding new roads posed problems. An early effort to vote bond money for a road project ran into Supreme Court opposition; however, in 1907 a successful bond issue was passed that began a network of paved roads, including the major laterals out of the city: Folsom Boulevard, Stockton and Franklin Boulevards, Auburn Boulevard, part of Riverside Boulevard, and Jackson Road to Plymouth. In 1911–1912, new bridges spanned the American River: the 12th Street Bridge and the H Street Bridge. In 1916, additional bond money was made available for further improvements and bridge construction.
# THE BUILDING BOOM OF THE 1920S
Civic pride and World War I patriotism combined to re-work the built-space of Sacramento in the early decades of the twentieth century. According to historians Paula Boghosian and Bonnie Snyder, nearly 30 new buildings were erected to beautify the downtown. In 1918, a new Carnegie library was built on I Street next to the City Plaza. On or near J Street alone the following new buildings went up: the D.O. Mills Bank, the Capital National Bank, the Masonic Temple at 12th and J with its glowering sculpted Knights Templar, the Public Market at 13th and J, the California Western Life Insurance building at Ninth and J, and the Elks Building on 11th and J. The crowning achievement of the decade was the Memorial Auditorium completed in 1927 on 16th and J. Designed by architects Dean & Dean, the Memorial Auditorium honored the heroes of World War I and was the city's public center and the site of entertainments as diverse as operatic performances, boxing matches, and big band orchestras. Added to these new structures were the Southern Pacific Depot at Third and I and the graceful granite Capitol Extension buildings that faced each other west of the state house.
A raft of church buildings also began to grace the state capital. Many of them had originally been on the city's deteriorating West End but relocated east of the riverfront. Sacramento's first church, the First Congregational, had been torn down in the early 1920s and a new building, resembling the old, was built near Sutter's Fort. The members of the First Methodist Episcopal built an elegant church on the corner of 21st and J, but most dramatic was the Westminster Presbyterian Church with its Byzantine-like dome, which rose south of Capitol Park. Likewise, the beautiful First Baptist Church on 24th and L soon dominated its city space. In 1922, Sacramento's Catholic bishop, Patrick Keane, urged the Christian Brothers to sell their decaying property on the corner of 12th and K where it had been since 1876. The brothers eventually relocated their popular school to 21st and Broadway, and on the old Christian Brothers site, the Weinstock, Lubin & Company store relocated from its historical Fourth and J location into an elegant new store modeled after the Parisian department store Le Printemps.
Health care professionals also sought to reinvigorate the city's antiquated health care system. Even before World War I and the deadly Spanish Influenza epidemic that hit the city with gale force in 1918, city physicians had been pressing for a new hospital. An elaborate plan was worked out with the Sisters of Mercy and the local physician's association to build such a structure (to be named Memorial Hospital in honor of the city's World War I veterans), but these negotiations fell apart. In the aftermath of the collapsed deal, a consortium of physicians erected Sutter Hospital in 1923. Undaunted by the increased financial risks, the Sisters of Mercy followed through on their plans and transferred their health care facility to former dairy lands on 40th and J Streets. Designed by city hall architect Rudolph Herold, Sisters Hospital (Mater Misericordiae) opened in 1925. In the 1930s, it changed its name to Mercy Hospital.
Electric street railways made suburban development possible. Handsome residential areas developed in East Sacramento as developers Charles Wright and Howard Kimbrough, in business since 1892, moved into the "back door" of the growing city. By this time the city's fascination with Victorian homes was done. A brief infatuation with craftsmen style homes and various "revival" styles of housing architecture reflected itself in the new urban housing stock. Wright and Kimbrough developed Tract 24, an expansive housing subdivision later known as the Fabulous Forties. Historian Kerry Philips has observed the new technology of home building during this time with better plumbing, utilities, and spatial efficiency—as well as more developed notions of health, family size, and privacy inherent in the new homes. Tract 24 sported Southern style mansions, villas, and mission-style homes set in from the tree-lined streets that became the residence of choice for Sacramento's upwardly mobile elites. Meanwhile, Christopher Jones, Wilbur Brand, and Frank Williams formed the Sierra Oaks Development Corporation and along H Street transformed the formerly bucolic area into a tract of beautiful homes.
The Elks Temple was one of the city's first skyscrapers.
Sutter Hospital was erected by a consortium of physicians after World War I.
Likewise, evolution of the city's water plans led to the removal of the old Y Street Levee. Y Street was renamed Broadway in 1938 and south of it William Land Park, a plot of land purchased with a handsome bequest from a former mayor and hotelier, took shape. Around the beautiful park, a series of small homes and comfortable residences took shape. Likewise the area around Sacramento City College on Freeport Boulevard blossomed with new developments. Next door to Land Park was Curtis Park, a neighborhood that grew in the shadow of the Western Pacific yards.
Schools too were making their appearance in greater number. A successful bond issue in 1921 resulted in a new Sacramento High School completed three years later. This state of the art school was the single most important educational entity in the city. Additional spending approved in 1921 resulted in six new elementary schools. The capstone, though, was the establishment of a junior college. The new college originally met in the upper floor of the old Sacramento High School until a bond issue in the 1920s made possible the construction of new buildings on Freeport Boulevard. Public subscriptions erected Hughes Stadium, home of the City College Panthers, and contributed to Sacramento's growing love for spectator sports. Events at Hughes Field helped unite the burgeoning city. In later years, the annual football match between the rival city high schools—Sacramento High and McClatchy High—would be waged on Thanksgiving Day.
In 1924, the McGeorge School of Law opened in Sacramento, providing legal training for local citizens. In 1966, the school affiliated itself with the Stockton-based University of the Pacific. Dean Gordon Schaber joined the faculty in 1953 and eventually became a Sacramento Superior Court judge and dean of the growing school. Well known in Sacramento for his civic activism, he became a respected figure in the community. McGeorge faculty member Anthony M. Kennedy, son of a respected legal family, reached the heights of the American judiciary when President Gerald Ford named him to the Ninth Circuit in 1975. In 1988, President Ronald Reagan appointed him to the Supreme Court.
# NEW INFRASTRUCTURE
"Sacramento's impure water supply [is] one of the greatest obstacles to growth of the city," noted a Chamber of Commerce report in 1895. City water was still undrinkable; as one resident recalled, "The water came out of the faucet almost a brick red." Individual Sacramentans attempted to purify it with home devices such as Pasteur filters and holding tanks that allowed the sediment to sink to the bottom. The city improved the water considerably when they agreed to chlorinate it. Yet it was not until 1915 that a report presented to the Board of Trustees urged that Sacramento use river water and that proper filtration be implemented to make the water "brilliantly clear, colorless, satisfactorily soft, and hygienically most gratifying and safe." In June 1919, voters approved a bond issue of $1.8 million for the new filtration plant. Situated near the Jibboom Street Bridge, the handsome plant had etched on its cornice the words of the Prophet Ezechiel, "And everything shall live whithersoever the river cometh." On December 31, 1923, President Calvin Coolidge pressed a button in Washington, D.C., that electrified Sacramento's city plaza and caused clear water to gush forth from the fountain, heralding the completion of the city's new filtration plant. Sacramento had always had legal rights to the abundant waters of the Sacramento and American Rivers and their tributaries in mountain streams. Now, as mermaids frolicked in the city plaza fountain, Sacramento finally had a supply of clean water.
Aligned with the need for more water was the need for more electricity. In 1921, the state legislature approved the creation of municipal utility districts with the power to raise money. Voters in 1923 approved the formation of a publicly owned utility company, the Sacramento Municipal Utility District (SMUD) to implement the city's claim to water and power rights on Silver Creek in El Dorado County. However, privately-owned PG&E and Great Western utility companies refused to give up their hold on water and hydroelectric generation in Sacramento. SMUD tried in vain to purchase facilities for generation, and three separate bond issues fell short of the necessary two-thirds majority needed for the construction of a dam. However, the building of the Central Valley Project in the 1930s contributed to sentiment in favor of publicly owned power. SMUD also continued to add various localities to its service area, expanding its ability to pass bond issues. After years of litigation and negotiation, SMUD took over the operation of Sacramento's electric distribution system from PG&E on December 31, 1946. The utility providers arrived just as a new wave of growth would place tremendous demands on Sacramento's power supply.
# CITY JOURNALISM, ENTERTAINMENT, AND LEISURE
During this time, the Sacramento Bee became the dominant daily, outdistancing its longtime rival, the Union. The paper accomplished its goals by responding to a variety of different constituencies in the Sacramento Valley, including calling it "Superior California." Under the solid financial leadership of Valentine McClatchy, the Bee substantially increased its readership by an innovative "block-subscription" marketing strategy and attracted more advertising accounts. The popular evening daily managed to walk a fine line between its crusading editorializing and the sensibilities of local panjandrums who might be offended by C.K.'s sometimes trenchant rhetoric. The paper took over the rival Scripps-owned Sacramento Star in 1924, and for the first time the Bee had popular funny pages.
The rival Sacramento Union maintained its niche as the morning daily of the city and the chief rival to the Bee's dominance. As the city's communication giant, the Bee was the natural source for the introduction of radio to the city. Two stations shared the same radio frequency; KVQ and the Bee-operated KFBK began broadcasting from the Kimball-Upson Building on K Street on September 23, 1922.
Only a handful of theaters existed in Sacramento as the new century dawned: the Clunie, the Acme Theater, the Grand, and the Alisky Theater. By the early 1900s, Sacramento theaters provided a mix of popular vaudeville and stock dramatic productions. By 1913, ten theaters were to be found on K Street. However, even though vaudeville would continue to play in Sacramento until the 1930s, these venues began to lose their appeal as talking movies became popular. When the majestic Clunie was torn down in 1923, Guy B. Post, an actor who had played on its stage in the early twentieth century, viewed the ruins and remarked, "What a desecration." Ironically, as historian Andrew Flink notes, he was in town to promote his movie, "Gold Madness."
Movies supplanted stage and vaudeville productions as one of Sacramento's most popular pastimes. Sacramento's first movie was George Melies "Trip to the Moon," which debuted March 9, 1903 in Grauman's Vaudeville Theater. A local entrepreneur, Charles W. Godard, came to control a number of city theaters and took a major hand in retooling them to show motion pictures. In 1915, he opened his own theater (modestly named the Godard), which became the first Sacramento theater devoted to the full-time screening of motion pictures. Fox West Coast Theaters began to dominate the city's movie theater market, taking over the Hippodrome on K Street (today the Crest) in 1928 along with several other downtown theaters. Ethnic theaters included the Mexico, which operated from 1930 to 1933, and the Nippon, a Japanese movie house.
Movie theaters had their historic moments. The controversial film, Birth of a Nation (billed as The Clansman), appeared at the Clunie. In March 1928, the Capitol Theater gave Sacramento its first talking picture, The Jazz Singer with Al Jolson. Sacramento theaters also became more comfortable and patron-friendly. One of the most elegant was the Alhambra, designed by architects Leonard Stark and Edward Flanders to resemble a Moorish castle. The exotic architecture, the gardens, the courtyards, and the small oranges hanging from the trees, as well as quotations from The Rubaiyat of Omar Khayyam, made movie-going a total sensory experience. In 1938, the "thoroughly modern" Tower Theater was built on the fringe of the Land Park district.
Sacramento also became a popular site for Hollywood filmmaking. Between 1914 and 1935, Sacramento area locales provided the backdrop for 45 feature length films such as Huckleberry Finn, Show Boat, and Steamboat 'Round the Bend. Famous directors who visited Sacramento include King Vidor, Cecile B. DeMille, and John Ford. While Hollywood directors favored the Sacramento River for its cinematographic verisimilitude, in fact the use of the river for pleasure craft was not just a celluloid dream. Well into the 1930s, two paddle wheel steamers, the Delta Queen and Delta King, plied the waters of the Sacramento offering food, lodging, entertainment, and gambling. Although eventually taken out of service for lack of passengers, for many years the vessels provided a tangible reminder of Sacramento's early transportation history.
The elegant Alhambra Theatre resembled a Moorish castle.
Public entertainment also included taking advantage of the little slices of nature within the built environment of Sacramento—city parks. In 1911, a $250,000 legacy from former mayor and hotelier William Land mandated the selection of a park site. The city purchased 236 acres of the Swanston-McDevitt Tract between Riverside and Freeport Boulevards in 1918. Land's heirs balked at the purchase of "remote" swampland and a strong campaign sought to reverse the acquisition and transfer the Land bequest to rural Del Paso Park. (There was even a citywide vote in its favor). However, a court battle resolved only in 1922 insisted that the purchase was permanent, and the park began to take shape. The city built a new levee, drained wetlands, planted 4,000 trees and 6,000 shrubs, laid out a nine hole golf course, and opened William Land Park. Sculptures and fountains were donated for its beautification. As a suburban neighborhood developed around the beautiful park, it quickly became one of Sacramento's finest and soon added a city zoo and a children's recreation park called Fairy Tale Town.
By the late 1920s, Sacramento's population edged closer to the 100,000 mark hoped for by developers. In 1928, the annual meeting of the state Association of Realtors was held in Sacramento, and visiting agents noted, "Sacramento [had] struck her real stride in the development of a modern metropolis and the building of a real skyline." In a myriad of statistics, from water connections to school enrollments, the future seemed bright and beautiful for California's state capital.
The movie Steamboat 'Round the Bend with Will Rogers was filmed in Sacramento in 1935. Insert shows Rogers chatting with Buck McKee of Roseville.
# 5. THE NEW DEAL AND THE WAR
Nearly 15,000 Sacramentans crowded the Southern Pacific Depot in September 1932 to catch a glimpse of Franklin Delano Roosevelt, the Democratic nominee for President. As New York governor, Roosevelt motored down K Street on a whirlwind tour of the business district, the state capitol, and Sutter's Fort, thousands thronged his route. A local band played his campaign theme song, "Happy Days are Here Again," but social and economic conditions in California's capital were anything but happy.
# THE GRIP OF THE DEPRESSION
If the crowds had permitted him, FDR, soon to be elected to the presidency, may have seen some of the shacks of the unemployed near the rail yards. The economic and social devastation of the Great Depression was painfully evident in Sacramento. Unemployment first hit the seasonal canning industry. Already in September 1930, 153 employees of the California Cooperative Producers Canning Company had been turned out of their jobs without pay as demand plummeted for canned goods. Ultimately, the once prosperous company found itself $25,000 in arrears to 600 seasonal employees. Angry workers filed petitions at the State Labor Bureau but to no avail. Added to the misery caused by market downturns, nature itself waged war on the local fruit and canning industries when a terrible freeze hit valley citrus crops in early December 1932, destroying at least half of the citrus trees—especially oranges. Cannery officials struggled to keep operations going and promised city officials that they would give work only to residents. In 1933, local residents were heartened when the canneries began an aggressive hiring program. However, such optimism was short-lived when cannery officials decided to take advantage of the glutted and desperate job market by paying workers as little as 20¢ per hour; less than $10 a week for six eight-hour days.
Cannery workers were not alone. By December 22, 1931, local banks were promising financial assistance to school teachers and all county employees whose salary warrants had been postponed until January. Employment problems multiplied. Workers at the two rail yards experienced layoffs and reduced workweeks. In 1931, the Western Pacific shops laid off two-thirds of its workers. Later in the year, the shops reopened with a skeleton crew of 200, working a four-day week. In early February 1932, the desperation of the increasing number of unemployed burst on the public scene when local newspapers recorded a mini-riot of 200 men all scrambling for 20 temporary jobs unloading heavy granite from a flatcar to river barges. "The twenty best fighters completed the work and drew their pay early in the afternoon," observed the local press. Unemployment skyrocketed. Adding to the distress, two Sacramento banks—the California National Bank and the California Trust and Savings Bank—failed to open on January 21, 1933. Heavy withdrawals had begun, culminating in a bank run on the Friday before the closing, drawing the banks's reserves below their legal limit. By 1932, there were 27,000 unemployed in Sacramento.
As the Democratic nominee for President, Franklin Delano Roosevelt visited Sacramento in September 1932.
Adding to the mix of unemployed residents of Sacramento was the presence of a large number of transients who set up "Hoovervilles" or transient camps. In order to discourage them from going to the city's shelter, which had begun feeding the homeless, City Manager Dean banned all feeding, even when donors appeared with tubs of beans and loaves of bread. Transients, nonetheless, only increased in numbers as the desperation spread across the nation. In 1935, a state survey taken of California's transients noted that nearly 3,000 lived in shanty villages "almost within the shadow of the capitol dome." Researchers identified several different areas, mostly along the route of the railroad tracks. In the area around the Jibboom Street Bridge many homeless camped and built shelters. Near the city incinerator with its piles of rotting garbage was a district called Shooksville, where nearly 1,000 people of all races, age groups, and nationalities lived. Named for Samuel Shooks, an African American who was hailed as "mayor" of the area, Shooksville contended constantly with rodents, flies, and mosquitoes drawn by the rotting garbage (from which transients also picked meals and salvaged junk) as well as stagnant pools of water. Along Y Street, several groups of seasonal workers—Finns, Mexicans, and Russians—lived a quasi-communal life when times were slow in the fields or the mines. On 20th Street was an area known as the "Rattlesnake District." Huddled here were mostly young, single men, many of them ex-convicts who lived in dampness and mud from the overflow of the spring rains. On 25th Street, Sacramento's traditional "hobo jungle" was another small colony.
As the number of unemployed grew, city and county government could or would do little. For many years, city council leaders had taken a laissez faire approach to many city problems. As noted earlier, when they came under pressure from local newspapers, churches, and women's groups to clean up vice, they did little. The office of mayor, filled by the council person with the most votes in an election year, was largely a ceremonial task. Real power resided with the city manager who at the time of the Great Depression was James S. Dean, a local architect who remained at the post until 1943. But even he was unable to confront this crisis head-on. The city had no agency responsible for relief, but instead relied on the network of private charitable organizations that had traditionally been the city's social safety net: the Salvation Army, the Ann Land Memorial Fund, the Isador Cohen Fund (which provided shoes for needy children), the Catholic Ladies Relief Society, and other sectarian organizations. Twenty-two of these organizations received substantial donations from the highly popular Community Chest, which raised thousands during the prosperous 1920s.
Faith in private sector solutions was also propagated by local business leaders who sought to be cheerleaders for the ailing local economy and formed a committee on unemployment. Private unemployment exchanges, "outsourcing" the care of the indigent to the agencies of the Community Chest, and above all helping to curb the "psychology" of depression were some of the ideas put forward by the committee in November 1930. City leaders picked up on this. City Manager Dean expressed their spirit when he urged a better psychological outlook, because after all, "it [the Depression] would not last." City council members suggested hiring the unemployed for neglected home repairs and even recommended that unemployed Sacramentans consider the example of New Yorkers who sold apples on the street.
But the inadequacy of private initiatives was painfully evident as the Depression only got worse. Serious problems at the city's homeless shelter at Front and I Streets accentuated the problems with privatizing charity. Located in the old city waterworks, the shelter was operated by the city recreation department with funds from the Community Chest. However, once the demands on the shelter grew, officials turned its direction over to the Salvation Army. Angry homeless men protested the decision, rejecting the Salvation Army's liberal doses of religion served with the soup they dished out.
Faith in private sector initiatives to combat poverty were shaken even more in November 1930, when the Community Chest announced it would be $32,000 short of its goal. Community Chest collections plummeted even further in 1932, when the appeal fell short by $100,000. Chest officers cut the budget 35 percent and jettisoned their "quality of life" donations. By early January 1932, the number of family relief cases handled by Community Chest agencies had increased from 496 in December 1931 to 1,011. In May, the Chest simply ran out of money and turned all their cases over to the county.
# COUNTY RESPONSES
City officials must have been puzzled by the stubborness of the Depression, but they could take some solace in the fact that under California law, relief of the poor was not a city but a county responsibility. Here too, however, the structure of relief efforts was inadequate for the demands of the time. Public relief was administered by a small county office that dispensed a limited number of goods and also relied on the county poor farm and poor house to pick up the slack. Sacramento's registrar of charities for years was the doughty Mary Judge, a single woman who administered the tiny welfare office. Under her wing were not only care for the indigent but also relief for senior citizens and aid for widows and the blind. Judge was a gruff and forbidding woman who nonetheless had a heart of gold. She often helped indigents out of her own pocket after unceremoniously ejecting them from the county office.
Judge's handle on local welfare needs was marked by the old-fashioned informality that reigned in Sacramento for many years. A longtime resident of Sacramento, she knew most of the indigents in the county and believed that she could separate the worthy from the unworthy. Transients of any kind, a serious problem because of the sheer numbers who came into the city "riding the rails," were given the back of her hand and perhaps a boot out of the locality. Judge lamented the advent of the automobile, which only increased the problem of transiency. By rail, car, or foot, the transients kept coming, and there was no money to help them. Not only had the Community Chest dried up, but tax revenue had withered as well. County assessments also showed a decline, as did city revenues. Tax delinquencies skyrocketed. Judge seized the moment with characteristic toughness and declared that only Sacramento residents would be given relief—all others would have to depart. To save money, she abandoned monetary relief and began to dispense rations, expending $5 per week for a family of five.
But Judge's "get tough" tactics reached their limits. Angry groups of unemployed banded together to form an Unemployed Council to pressure city and county officials for more food, lodging, and bathing facilities. Dissatisfaction with the amount of city and county relief led to angry gatherings, and in early 1933, a group of 100 men descended on Judge's offices. Judge faced the mob down and then heard them as they begged for additional help, the dispensing of the "work test" as a way of getting food, and freedom from police harassment that prevented them from even discussing politics. In July 1933, a near riot erupted when a number of the hungry and homeless, many of them transients, again stormed Judge's office. This time she called for law enforcement.
The militancy of the unemployed created a backlash among the city's leaders, especially in the police department, led by Chief William A. Hallanan. A hunger march held in April 1934 brought accusations of communist influence. In the late summer of that year, a raid on communist headquarters in Sacramento led to the arrest of 24 persons. Seventeen were charged and fourteen actually went to trial, accused of vagrancy and criminal syndicalism. Sacramento District Attorney J.W. Babcock built strong public opinion against the defendants before they even set foot in the courtroom. These trials became a showcase for political statements from both sides as widely irrelevant issues were injected. The Sacramento trials resulted in the conviction of eight of the fourteen. Nevertheless, Sacramento's economic problems did not go away by simply tagging protestors "communist" or "socialist." Even before the trials, Sacramento leaders had begun to realize that relief could not be dispensed as it had been in the past. Grudgingly, in 1933, Judge allowed Sacramento Board of Supervisors clerk Earl Desmond to move into her office and begin a new filing system. New measures and new income were needed.
Although Sacramento had been the temporary beneficiary of federal largesse in Mather Field and road construction, for the first time, the city had to turn to Washington, D.C. for help in dealing with what had always been a local responsibility: social welfare. The city applied for $23,000 in government assistance with the Reconstruction Finance Corporation (RFC), a Hoover-era program devised to help reignite business. The RFC loan along with other government programs began to direct money back into Sacramento. With the election of Franklin Delano Roosevelt in 1932 and the commencement of the myriad of New Deal programs, the direct impact of the federal government on the city and people of Sacramento increased substantially.
# GOVERNMENT PROGRAMS TO THE RESCUE
Several New Deal programs poured money into Sacramento. Some federal funds, especially for transients, came through the State Employment Relief Administration (SERA). SERA's help with the transients enabled the cash-strapped city to feed and clothe them. SERA also helped to coordinate the flow of federal dollars that came from other New Deal agencies. The Civil Works Administration, begun in the first months of the Roosevelt Administration, provided Sacramento with a few needed infrastructure projects. County executive Charles W. Deterding proposed a loan of $130,000 to add bridges on Lower Stockton Road (Franklin Boulevard) and a clinic building at the County Hospital. Federal aid also came to the state for a variety of local projects in late 1934, including road grading for city streets, improvement of hospital grounds, tree removal, golf course cleanup, and the installation of street signs and traffic signals. Likewise, the construction of the landmark Tower Bridge in 1935 was made possible by a transfer from the Civil Works Administration. In January 1934, the Public Works Administration (PWA) approved a loan of $340,000 to build a new Home for Aged Women on the grounds of the County Hospital. C.K. McClatchy High School on Freeport Boulevard and the auditorium of the City College were also built with PWA funds. In all the PWA pumped in $3 million to Sacramento.
But the biggest donor to the city's infrastructure was the Works Progress Administration (WPA), created by the Roosevelt Administration in 1935 and overseen by social worker Harry Hopkins. WPA projects benefited the entire county and included 46 new public buildings, over 220 miles of new highway and streets, improvements in William Land and Del Paso Parks, 14 new exhibition halls and administration buildings at the state fairgrounds, and additional runways at McClellan Field and the Municipal Airport. The beautification and upgrading of Sacramento gave the aging city a new look. Symbolic of the changes was the renaming of M Street into the grandiloquent Capitol Avenue in 1940. WPA workers provided hours of cataloguing and indexing of state records, books and newspapers, and vital statistics—including old city records dating from 1849. Historical materials for various California counties were inventoried along with materials in the state Indian exhibit.
The construction of the landmark Tower Bridge, shown here when it opened in 1935, was made possible by the Civil Works Administration.
Federal money enhanced the cultural infrastructure of Sacramento as well. Under the auspices of the Federal Art Project, directed by Joseph Briton Matthew of the City College's art department, some of the famed New Deal art was created for public places. The newly built auditorium of the City College was adorned with a mural painted by Bay Area artist Ralph Stackpole. Stackpole, heavily influenced by Mexican muralist Diego Rivera, had also sculpted fountains for William Land Park and the city plaza. Artist Kathryn Uhl Ball's drawings and watercolors of Sacramento buildings were also a legacy of this era. A short-lived Sacramento Art Center offered classes and exhibits at various locations. In addition a Sacramento Federal Orchestra offered free concerts to area residents while orchestral training was provided at the Oak Park Library. By the time the WPA office closed in June 1942, over $4 million had been pumped into Sacramento County.
Equally important, the approval of the massive Central Valley Project in 1937, and its extension to the American River watershed in 1944, began a long-awaited systematization of the river, water, and energy systems of the Sacramento Valley. This included the building of levees and plans for a dam at Folsom, making possible a safe system of flood control. The surge of federal funding was only the beginning of a long-term relationship with the federal government, as the Roosevelt era transformed the relationship between the federal government and American cities. An ongoing array of federally mandated and funded programs soon became an important part of urban life in the California capital. The energetic Chamber of Commerce, under the leadership of Arthur S. Dudley, usually simply pro-business, aggressively lobbied for federal funds.
# DUDLEY: PROMOTER OF AIR POWER
Although Dudley had left Sacramento for short stints as an urban booster in Stockton and in Oregon, he returned to the city permanently in 1927 and resumed his work with the Chamber of Commerce. It was Dudley's lobbying that brought the single most important change to Sacramento's economy: military installations. City developers like Dudley and others looked enviously on the impact of military installations on locales like San Diego, San Francisco, and Los Angeles. Sacramento had had its brief flirtation with a military base when Mather Field opened in 1918. However, the base was almost immediately downsized once World War I was over. Although it lingered through the 1920s as a base for airmail and was resuscitated in the early 1930s when the government staged war games over Sacramento, the federal government permanently closed it down in 1932 as a budget saving measure.
Dudley knew how to lobby. He had helped keep March Field alive in Los Angeles and determined to do the same for Mather. Even before Mather closed, Dudley had begun shuttling regularly to Washington to lobby for commercial air routes. On one of his trips he met Sacramento Bee correspondent Gladstone Williams. Williams put him in touch with General Henry "Hap" Arnold, who was waging a strong campaign to enhance the role of the air force. Through Arnold he met maverick military men like Generals Billy Mitchell and Carl Spatz, who also urged the resistant military establishment to build a credible American air force. Joining private business support to their military perspective, Dudley argued that a larger air force would require logistical centers located at strategic places scattered throughout the country—especially along the coastal areas—locations that were more vulnerable to attack.
In November 1934, Dudley joined forces with fellow Chamber of Commerce secretary Reginald Waters of Miami to create the National Air Defense Frontier Association, a lobbying firm that mobilized chambers of commerce around the nation to press for a series of supply and logistical centers for the Army Air Corps on the coasts of the United States. Bombarding legislators, public officials, and military men with studies and requests for more military air preparedness, Dudley and his associates achieved success in the early 1930s with the approval of a plan to create a ring of 20 air bases as a foundation for continental defense. Making league with Florida Democratic Congressman J. Mark Wilcox, they aided in drafting a bill to create new air bases at strategic coastal locations. Senator Hiram Johnson of California pushed the bill through the Senate, and it became law in 1935. Appropriating the money took time, but the matter was pressed effectively by local Congressman Frank Buck. Before the money came through, Dudley urged Alden Anderson, president of the Capital National Bank and a Chamber of Commerce official, to snap up lands that might be available for an air depot. Anderson dispatched an agent from the firm of Artz and Cook to obtain options on 1,200 acres on the old Rancho del Paso at the low price of $111,855. When the money came through and the selection panel chose Sacramento as the site for one of the new air depots, the community was ready to move.
On September 8, 1936, at ceremonies presided over by Governor Frank Merriam, the army began construction of the base, soon to be named for Hezekiah McClellan, an Indiana army pilot who died in a test crash in 1936. To the new site were transferred men, supplies, and materiel from the Rockwell Air Depot in San Diego. The new arrivals flowed in between December 1938 and January 1939. In April 1939, the base was dedicated. Conceived as a supply and aircraft maintenance base, it was one of the best equipped in the nation. Completed just as World War II was beginning, the base soon grew rapidly. The workweek was expanded to round-the-clock operations, and the base became one of Sacramento's major employers of women. By 1943, women had been promoted to the all-male preserve of production inspectors. Its employment force zoomed to over 22,000 by 1943 but declined predictably after the end of World War II.
An unexpected bonus was the re-opening of Mather Field, which a generous congress re-established in May 1936. Mather was chosen as a training facility for navigators and new buildings were constructed north of its original site. In 1941, a class of 46 navigators (including two Sacramentans) reported for school at the base. In 1958, it was selected as a base for the Strategic Air Command. Employment numbers hovered between 6,000 and 7,000, with more military than civilian wage earners.
Sacramento booster Arthur S. Dudley greets "Hap Arnold," 1930.
In 1942 the army spun off a Sacramento-based sub-depot of the Quartermaster Corps to reduce congestion at the San Francisco–Oakland Port of Embarkation. Later the Signal Corps took charge of the depot, and the site became an important storage and repair location for army communications equipment. Eventually the depot moved from its initial location at the State Fairgrounds on Broadway and Stockton Boulevard to temporary quarters at the Bercut-Richards packing plant in the northwest part of the city. In 1945, the army constructed permanent facilities for the depot on a 485-acre site 8 miles southeast of the capital. Its workforce peaked at 4,000 mostly civilian employees in 1968. McClellan, Mather, and the Signal Depot soon became major economic players in Sacramento County, and their locations outside of the city limits stimulated even more suburban development.
# THE WAR
Sacramentans learned of the air raid on Pearl Harbor late in the morning of Sunday, December 7, 1941. By 2 p.m. hundreds of workers were ordered to McClellan Field to begin outfitting B-26s and P-40s for shipment to Alaska. City government coordinated emergency plans. Mayor Thomas Monk organized civil defense procedures and beefed up security around public buildings, keeping a vigilant eye on Delta levees for signs of sabotage. The first blackout mandated by civil defense authorities took place one day after Pearl Harbor Day at 7:23 in the evening. Nevertheless, not everyone heard the order; city display windows and even the water tower lights remained lit. Subsequent efforts resulted in similar patterns of erratic non-compliance—including the continued lighting of Sacramento's 14-story Elks Building. Eventually, a county ordinance was passed establishing an adequate system of signaling blackouts and informing people when they were over. By 1943, over 14,000 Sacramentans had signed on as Civil Defense volunteers.
Sacramentans once again became used to uniformed men walking their downtown streets. Local residents were urged to welcome the lads. The popular United Service Organization set up headquarters on Eighth and K Streets and made rooms available for social gatherings, dances, and places to write letters. In the basement of the Cathedral of the Blessed Sacrament, Fathers Richard Dwyer and Vito Mistretta ran a drop-in center in the cathedral basement for Catholic military personnel. There too young people in uniform could relax, borrow from the local library, develop photographs, and write letters home. Beds were set up in the Memorial Auditorium for service people who could not find a vacancy in Sacramento hotels.
Sacramento's retailers pledge not to gouge buyers during World War II.
Wartime privations hit Sacramento as rationing and conservation became a part of home-front mobilization. Sacramento retailers gathered en masse and with hands uplifted swore not to use the war emergency as an excuse to gouge customers. Increased wartime taxes stymied spending anyway. "We couldn't sell anything during the war," complained one businessman. "The government took 85 percent of our profits for the war effort." Rubber and gasoline shortages meant fewer automobiles. Restrictions on sugar meant the substitution of the gooey thick Karo syrup in candy and other sweets. Shoes were limited to three pairs per year per person. Nylon stockings, the most famous black-market item of the war years, were also in short supply in Sacramento. Sacramentans were urged to grow "victory gardens," and an annual contest sponsored by the Sacramento Bee awarded a prize to the best garden.
# JAPANESE RELOCATION
In 1941, nearly 5,000 Japanese-Americans resided in the greater Sacramento area. The FBI had undertaken surveillance of potential German, Italian, and Japanese "fifth columnists," and although the agency reported that the Japanese were not a serious threat, anti-Japanese sentiment fostered a climate of suspicion. Sacramento's Japanese had been subjected to increasing mistrust since the beginning of the twentieth century. The Gentlemen's Agreement of 1907 began to restrict immigration from abroad, and in Sacramento anti-Japanese predilection was stirred up by Valentine McClatchy, who after a trip to the Far East early in the century returned with a deep conviction that Asians, especially Japanese, could not be assimilated. On the pages of the Sacramento Bee, McClatchy preached a virulent anti-Japanese message. In 1913, the California legislature passed the Alien Land Act, forbidding Japanese from owning land. Schools were also segregated.
Once war began, officials of the Western Defense Command under General John L. DeWitt pressed for evacuation and internment. On February 19, 1942, President Roosevelt signed Executive Order 9066 giving the secretary of war the authority to designate "military areas" from which "any and all persons may be excluded." General DeWitt then designated the western half of the Pacific Coast states and Arizona as military areas. These became the prohibitive zones for the Japanese. Subsequent proclamations imposed curfews and initiated the process of evacuation.
Crackdowns on Japanese Sacramentans began almost immediately. Three prominent Sacramento businessmen—Rikitaro Sato, F.J. Miyagawa, and Gichi Aoki—were arrested as enemy aliens. FBI officers swept down on the Sumitomo Bank in Sacramento, freezing its assets. Agents seized and closed Japanese-owned companies like the New Eagle Drug Company, the Highland Investment Company, and the Pacific Trading Company. Local Japanese tried to assure city leaders of their loyalty and support for the American war effort; however, in March 1942, orders came that Sacramento was in the restricted zone and that mass evacuation must begin. The evacuation started in May. The Bee sported banner headlines on May 7: "All Japanese Must Get Out of Sacramento." By May 13, 3,800 Japanese were bussed to an assembly point northeast of Sacramento called Walerga Alien Induction Center from buses departing from the Memorial Auditorium. Housed in primitive barracks, the Japanese waited for transportation to "secure" locations. By May 16 the evacuation to Walerga was completed. By mid-June the large group at Walerga was transported to Tule Lake, just a few miles from the Oregon border. Walerga was then transformed into a new army base named Camp Kohler.
Although internees were permitted to return after 1945, of the 6,764 Japanese citizens of the county exiled by internment, only 4,000 returned. A number simply turned their backs on California and Sacramento and relocated to other parts of the country. Historian Cheryl Cole notes, "The reception given Japanese returning to their hometown was cool. There was no welcome, no apologies, no sympathy expressed for their suffering." For those who did come back to Sacramento's Japantown, they found their homes occupied by others. Many found temporary quarters in churches until they got on their feet. Historian Wayne Maeda observes that the returning Japanese expanded the boundaries of Japantown, moving east from the original Second and Fifth, L and O Street boundary to Seventh and south to S Street. But in searching for housing, they ran into legal road blocks set up by restrictive covenants—clauses written into property transactions that barred homeowners from selling to African Americans or people of the Mongolian race. Such restrictions were ruled unconstitutional after the war. Other Japanese returnees came back to agricultural pursuits in Elk Grove, Florin, and in the Pocket area of Sacramento.
# NEW DEMOGRAPHICS: THE LATINO PRESENCE
In 1940, only 2,196 Mexicans lived in Sacramento County, roughly one percent of the total population. Initially attracted by agricultural work, the first "Great Mexican Harvest" had been in 1920. Mexicans in Sacramento found employment at Southern Pacific and local canneries, especially after Congress imposed immigration restrictions on Southern and Eastern Europeans in the 1920s. Early Mexican barrios included Alkali Flat near the rail yards and across the river in Broderick and Gardenland. Clearly, the highest concentration of Mexican residents was in the West End, a section of the city richly described by Ernesto Galarza in his famous autobiographical account Barrio Boy. With the increase in Mexican population during and after the Second World War, Mexican residential areas began to spread south on Franklin Boulevard when immigrants from Chihuahua and Durango in Northern Mexico arrived to work in the canneries. Through the 1940s Mexicans made up 40-50 percent of all employment in Sacramento canneries. Agricultural workers strung hops vines, cut asparagus, and thinned sugar beet plants.
The growing visibility of Mexicans was perhaps first noticed by local Catholic churches, especially the cathedral where Mexican Catholics came for baptism and weekly Mass. Likewise, the records of St. Mary's Church, then on Seventh and T, noted larger numbers of Hispanic baptisms. Already in 1919, the Ortiz family of Broderick had hosted a celebration in honor of the Mexican national icon, Our Lady of Guadalupe, on December 12 with local Catholic bishops participating in the event. In the late 1920s, Father Stephen Keating, an assistant priest at the cathedral with a background in social work, began to seek out Mexican families. Two church workers, Frederico Falcon, a native of Fresnillo, and former nun Magdalena Martinez, assisted in rounding up the scattered flock. Keating held religious services and provided instructions for Hispanics. He also collected food and clothing for them, helped them with legal problems, and sponsored social gatherings. Keating and Falcon (an accomplished musician) formed the popular Santo Nombre marching band, a Hispanic troupe that marched and played in civic parades and on different occasions. Keating's departure from the priesthood in the late 1930s left Falcon with the task of keeping the flock together.
Japanese man is interrogated by police prior to internment.
In 1942, wartime shortages of labor compelled Congress to approve the importation of Mexican labor under the Bracero program, and Mexicans were brought north to work at the Southern Pacific yards. Hence the number of Mexicans living in the Sacramento area surged dramatically. During and after the war, Mexican Sacramentans developed clusters of community. Alkali Flat continued to be "Barrio Centro," and along 12th Street a Mexican commercial district flourished with shops and various clubs like Los Reyes, La Mexicali, El San Diego, and El Xochimilco. Other cultural institutions included a Mexican Baptist church and a Methodist church. By 1944, Sacramento's Catholic Mexicans managed to raise enough money to purchase a former Catholic church on Third and O Streets that had become a Japanese theater house. Falcon and the cathedral priests opened the new church in December, naming it Our Lady of Guadalupe. This religious center provided yet another important gathering point for the Mexican citizens of Sacramento and a place to crystallize their social, religious, and cultural life. Countless fairs, dances, and religious festivals emanated from the church. In 1956, when the community had outgrown the church, the pastor petitioned for a new site at Seventh and T on property that had once housed St. Mary's Italian Church (it had moved east in 1948). The huge new church, designed by Sacramento architect Harry Devine, was dedicated in 1959. Facing Southside Park, it sported a huge mosaic of Our Lady of Guadalupe that dominated the small stretch of T Street.
Mexican workers, like these at Southern Pacific, came in greater numbers during and after the Second World War (c. 1940s).
# THE AFRICAN-AMERICAN PRESENCE
The center of African-American cultural life was to be found at the city's two historic black churches, St. Andrew's African Methodist Episcopal Church (AME) and Siloam (after 1891 Shiloh) Baptist Church. A new church, Kyle Baptist, founded by minister Louis Harvey, added yet another center of African-American visibility. Black businesses maintained a sustaining trade in Sacramento. In 1917, George Dunlap opened the Dunlap Dining Room at 612 J Street. Its home-style cuisine attracted a diverse crowd. The restaurant later moved to Oak Park. Black journalist William Collins founded a small black newspaper, the Western Observer, broadcasting news of the black community. Then in 1942, a successor founded the Sacramento Outlook, followed by the Sacramento Observer, which began in 1962. Harvey and Collins also helped to found a chapter of the NAACP in 1923, and the two men took a role in blunting the influence of the Ku Klux Klan (although Klan members in a public relations stunt had ostentatiously repainted Harvey's first West End church).
Black Sacramentans fared badly in the Depression. But with the advent of the airbases, their numbers grew, reaching 1,500 by 1940. Wartime also brought additional black residents, especially military men. Racial discrimination was a fact of life in Sacramento. African Americans were not permitted to use the facilities of the YMCA or to join either the Girl or Boy Scouts. They were barred from service in other public facilities. In June 1943, 31-year-old Private Nathan Randall of McClellan was refused a drink in a bar at the Bank Café on Fourth Street. When Randall demanded service, other African Americans witnessing the event joined him and a brief scuffle with police ensued. Later, housing would continue to be an important part of the struggle for civil rights in Sacramento.
# ENTERTAINMENT
Sacramento life was not always consumed during the 1930s and 1940s with the grim realities of Depression and war. Movies were still popular during the economic downturn. Feature length films, together with newsreels, shorts, and cartoons, drew Sacramentans downtown to popular movie theaters. Sacramentans could dance to the sound of the big bands at the Trianon, located above the Senator Theater, and listen to jazz at the Zanzibar Club, an African-American club that featured musical greats Duke Ellington, Count Basie, and Dinah Washington. Big band sounds also pulsed from the Memorial Auditorium as Tommy Dorsey, Benny Goodman, Glenn Miller, and Les Brown ("and his band of renown") played the sometimes mellow and sometimes animated syncopations of Swing Jazz.
The military also contributed to Sacramento's cultural life. The McClellan Band played at public concerts at Land Park. Camp Kohler hosted a popular beauty pageant and dance. But as they had in World War I, local military officials tried to curtail the activities of soldiers on liberty by urging them to stay away from gambling and prostitution. Sacramento's vice spots flourished nonetheless as illegal lotteries, illicit gambling, prostitution, and after-hours drinking took place both on the West End and across the river in nearby Broderick. Likewise, despite a presidential order outlawing prostitution near military bases, Sacramento's houses of ill repute flourished until 1943 when military commanders threatened to put the entire city off-limits unless the brothels were closed down.
Sports-minded Sacramentans got the thrill of their lives in the summer of 1942 when the local Sacramento Solons won the Pacific Coast League pennant. Professional baseball had been a popular Sacramento pastime since 1886 when the California League invited the Altas, a local team, to join the new professional circuit. The Altas, later renamed the Senators, were a popular sensation, packing in crowds at a stadium at Snowflake Park. A new team, the Gilt Edge, succeeded the Senators in 1898. In 1905, the recently formed Pacific Coast League franchise from Tacoma moved to Sacramento. In 1910, the team moved to a new ballpark at Riverside and Broadway, known over the years as Buffalo Park. In 1922, the 10,000 seat Moreing Field (named for owner Lew Moreing) replaced Buffalo Park's wooden grandstand and bleachers. The field, once acclaimed as one of the finest in the West, was subsequently renamed Cardinal Field, Doubleday Park, and finally, to honor a local sportswriter, Edmonds Field. The team was acquired in 1935 by businessman Branch Rickey, general manager of the St. Louis Cardinals who made Sacramento the Cardinal's tenth farm club. He renamed the park Cardinal Field and the team became known as the Solons. The Solons had a difficult time securing the pennant—at one point forcing a change in league rules that awarded the prize only to those who had won the most games instead of the all important play-offs. In September 1942, over 11,000 crammed into the newly-renamed Edmonds Field to watch pitcher Tony Freitas hurl a four hitter to defeat the Los Angeles Angels and clinch the first and only pennant the Solons won. Later, however, they lost the league play-offs to Seattle and the next year fell into a disappointing last place. In July 1948, the old wooden Edmonds Field burned to the ground and a new (and aesthetically less satisfying) stadium took its place. When the team departed for Hawaii in 1960, the field was demolished. A brief revival of the Solons took place between 1974 and 1976, but for a time the city appeared to have lost interest.
Edmonds Field was home to the Sacramento Solons, who won the Pacific Coast pennant in 1942.
# THE END OF THE WAR
Few Sacramentans took to the streets when Hitler's Germany finally surrendered in May 1945. Stores remained open, workers at their desks, and children in school. At McClellan Field, Mather, the Army Signal Depot, and Camp Kohler it was "just another day." Ongoing coverage of the bloody Pacific theater vied with local news. Indeed, the headline story of the atomic bombing of Hiroshima shared the spotlight with the death of longtime U.S. Senator and Sacramento native Hiram Johnson. Editorial writer Walter Jones of the Sacramento Bee warned, "Atomic power must not become a Frankenstein monster to America whose scientists played such a major role in the production of this terrifying instrument of war." But few appeared to be worried about the implications of the giant mushroom cloud. Sacramento was still insulated from most international trends.
But eventually Sacramento burst the bonds of its public reserve. On the afternoon of August 14, 1945, the wait ended. Anxious but subdued crowds began to congregate on K Street in anticipation of President Truman's 4 p.m. address formally announcing the end of the conflict. (Worried police Chief Alec McAllister had banned liquor sales—a fruitless effort as subsequent events revealed.) As soon as Truman's nasally Missouri twang announced the end of the Japanese war, the city erupted in celebratory chaos. The main thoroughfare of K Street became a parking lot as horns blared, whistles shrieked, firecrackers popped, and crowds surged into the streets. At Tenth and K, sailors tried to direct snarled traffic, contributing to the confusion. Overhead, a huge four-motored bomber twisted and dipped as it buzzed over the city again and again. "Even the roar of its great motors were drowned by the tumult," reported a jubilant Sacramento Bee. Less obtrusively but still near the center of action, worshippers, some of them dabbing tears from their eyes, went in and out of the Cathedral of the Blessed Sacrament at 11th and K, "giving thanks to Him that the war was at an end." An old era closed for Sacramento.
# 6. THE EMERGENCE OF A METROPOLIS
In early 1942, the Chamber of Commerce, encouraged by the economic vitality pumped into Sacramento by the military installations, laid out an ambitious program to enhance Sacramento as a commercial and population center for Northern California. City futurists envisioned Sacramento in the far-off year of 2000 as a city with anywhere between 400,000 to 800,000 people and of thousands of square acres coming from fresh annexations. The Chamber prophesied a city skyline in which "a half dozen more office buildings from fourteen to twenty stories" would dwarf the stately Elks Temple, the Cathedral of the Blessed Sacrament, the capitol building, and the California Western Life Insurance Company building. The Sacramento of 2000 would have solved its growing traffic problems and increased its economic prosperity. And above all she would keep her trees. The report predicted: "For Sacramento, in her march to metropolitan greatness, has not sacrificed the shade trees for which she was renowned so widely back in 1942."
Amazingly, many of the prognostications proved true. The Sacramento of 2000 in fact has surpassed the 400,000 population mark and annexations have augmented the city. New buildings indeed dwarf the cathedral and the Elks Temple. Through reliance on federal and state dollars and employment, the city did grow significantly, especially in undeveloped hinterlands of Sacramento County. As in most communities in the nation, Sacramento coped with the reality of a deteriorating downtown and also with the demands of a mass consumption culture that transformed it dramatically.
# NEW URBAN LEADERS
The reign of longtime Mayor Thomas Monk (city leader since 1938) came to an end in 1945. Among those who later held the office were funeral home director George L. Klumpp and Sacramento City College dean Belle Cooledge, who was the city's first female mayor. The more powerful position, however, continued to be the city manager. Since 1930, architect James S. Dean had held the post. When Dean departed for a state job in 1943, Elton Sherwin briefly held the post until his death in 1946 brought about the appointment of 42-year-old Bartley Cavanaugh. The namesake and son of a famous Sacramento city politician of earlier years, Cavanaugh remained in the office until 1964. Besides having a common touch and unusual political skills, Cavanaugh was an important force in responding to the challenges of the postwar era.
Cavanaugh shook up the lethargic city government, firing the chief of police and several police captains. He aggressively pursued new business and infrastructure improvements. Between 1946 and 1955, the city annexed 27 districts and increased its size by nearly 10 square miles. Unoccupied areas, like River Park, came into the city limits without too many difficulties. Annexation strategies took organizers first to the less-affluent south area suburbs. In early 1948, elections were scheduled for Colonial Heights, Fruitridge, and Coloma Heights. However, in these populated areas, the annexation movement took place more slowly. Many residents of Sacramento's borderlands who feared higher taxes and less control were lukewarm at best to the prospect of annexation.
City manager Bartley Cavanaugh Jr. aggressively pursued new business and infrastructure improvements.
McClellan Air Force Base was once one of the best-equipped supply and aircraft maintenance bases in the nation, shown here in 1948.
The Colonial Heights-Fruitridge opponents filed court appeals to block the elections, but elections moved forward. Coloma Heights only agreed to join the city after a deal was struck over who would assume the community's heavy bonded indebtedness. Annexation was slowed further when the vote for Sutterville Heights failed narrowly in late 1947. Despite the fact that more south area communities were added in the early 1950s, annexation proponents were dealt a big blow when affluent Arden Arcade refused to join the city in a decisive vote in 1959. The last big annexation battle was over North Sacramento, a town that once dreamed of being a city in its own right. After earlier failed votes, North Sacramento voted narrowly to join Sacramento in 1964. By this time, the once strong proponents of annexation began to change their minds as the merits of taking in less developed and poorer districts became problematic.
# ECONOMIC EXPANSION: GOVERNMENT AND PRIVATE SECTORS
Cavanaugh and others were caught in a whirlwind of growth and change after World War II. Important modifications in government policies delivered the national economy from revisiting the economic morass of the Great Depression. In particular, government sponsorship of higher education for returning G.I.s and substantial subsidies and incentives for independent home-ownership helped to transform the social landscape of America. Sacramento in particular benefited from the massive expenditures for military and defense related products. Increases in the defense budget was the wind in Sacramento's economic sails during the postwar era. Expanded employment opportunities generated by federal military installations, burgeoning state and local governments, and private enterprise created a new economic climate in Sacramento. The jobs generated by these not only produced new wealth, but also rearranged population centers.
The three military installations—McClellan, Mather, and the Army Signal Depot—provided the majority of employment. McClellan was the flagship. When Cold War tensions rose after World War II, the base buzzed with life. By 1952, it employed more than 17,000 (most of them civilians and 20 percent of them women), and the numbers grew steadily, hitting a peak of 25,900 in 1968. The government poured in hundreds of millions of dollars to expand and upgrade the base over the years. By 1955, its $72 million payroll was the largest in Sacramento County. Two years earlier the Sacramento Bee noted that base employees made local purchases of around $14 million.
Similar increases took place in the local, state, and federal government workforce. Since the 1920s, the presence of state government had been growing. Sacramento legislators had made repeated efforts to end the evil of "scatteration" (i.e., the diffusion of state offices to different locations, especially San Francisco) and by the end of World War II had had some success. Sacramento made important improvements for the legislators, such as the introduction of air conditioning and successful mosquito abatement programs. But more than anything else, the rapid postwar growth in California required a growing, full-time government. Governor Edmund G. Brown reorganized the executive branch in the early 1960s, creating centralized departments. Jess Unruh, a Los Angeles assemblyman, helped expand and modernize the state legislature. In 1966, voters approved Proposition 1A, which created a full-time legislature. This retooled government required more workers and specialists, and they usually lived in Sacramento.
Even before Proposition 1A, city life quickened as more bureaucrats were hired to staff the increasing number of state agencies. Lobbyists, legislators, and other hangers-on frequented the fading Senator Hotel. Here in the 1940s and early 1950s reigned the "uncrowned king" of the legislature, Artie Samish, who was the lobbyist for the truck and bus lines, the liquor companies, and the race tracks. An expose by Collier's magazine spelled the decline of his influence, and in 1953 he was "dethroned" by a conviction for federal income tax evasion. Sacramento became more of a government town than it had ever been. Between the air bases and government employment, Sacramento's traditional reliance on the rail yards, retailing, and agricultural production faded. By 1956, state Director of Industrial Relations Ernest B. Webb noted that government units employed 40 percent of all non-farm workers in Sacramento, far outstripping the rest of the state where government workers made up only 17 percent of the non-farm total.
City and county governments worked closely with the Chamber of Commerce to attract new businesses. One of the biggest coups was snaring a share of the growing aerospace industry that came to California after the war. A Christmastime 1950 headline in the Sacramento Bee, "Rocket Plants to be Erected East of Capital," announced the arrival of Aerojet General Corporation, a subsidiary of General Tire and Rubber. The aerospace giant that made rocket engines for the federal government secured a 7,200-acre tract 16 miles east of the city. Paying a whopping $6.6 million for the new plant, Aerojet peaked in 1963, giving paychecks to 19,792 employees and accounting for more than 60 percent of the region's manufacturing employment. Aerojet was but one of a number of private companies that discovered the cheap land, abundant power sources, and friendly business climate of Sacramento County. In 1956, Aerojet sold 2,000 acres of its vast holdings to the Douglas Aircraft Company, which also began missile testing.
While the aerospace acquisitions were the big prize, other private firms came as well. By 1948, the city had an "industrial park" on Richards Boulevard. Its occupants included paper giant Crown Zellerbach, which opened in July 1949. In May 1951, Procter & Gamble, the Cincinnati-based soap king, opened a plant on Power Inn Road. Other firms like Federal Mogul, a ball-bearing manufacturer, Firestone Tire, and Campbell Soup all relocated to prime sites, providing hundreds of jobs and diversifying the economic profile of the community.
Aerojet General, a subsidiary of General Tire and Rubber, sets off a couple of its rockets.
# POPULATION GROWTH
The lure of jobs caused Sacramento's population to skyrocket. The location of job centers outside the city shifted the demographic center to the heretofore-undeveloped lands of the county. In 1940, 62 percent of county residents lived in the city and 38 percent in the county. In 1950, the scales balanced evenly with 50 percent in the city and 50 percent in the county. However, by 1960 the percentage of city to county dwellers flipped exactly from the numbers of 1940 with 62 percent residing in the county and 38 percent in the city.
Source: Sacramento Archives and Museum Collection Center
Much of this growth continued to be focused on the northeast corridor extending out along Highway 40, Auburn Boulevard, and Fair Oaks Boulevard. However, in the south area of Stockton Boulevard, Franklin, and Freeport, expansion continued as well, stretching until it touched the tip of rural Elk Grove and began to draw rural Galt into its swirl.
# SACRAMENTO'S NEW SUBURBS
Before the war, few Sacramentans lived more than three or four blocks from the streetcar tracks, but new industries, new freeways, and automobility allowed Sacramentans to empty out into the suburbs. In some instances, growth occurred in existing settlements. The area immediately around McClellan Field moved swiftly to take advantage of the opening of the base. "Come to North Sacramento, Hagginwood and Del Paso Heights," ran one newspaper ad. "The Giant $7,000,000 Army Air Depot Insures Rapid But Solid Development of the District!" In fact, Del Paso Heights grew so rapidly that by 1955 local citizens were pushing for incorporation as a sixth class city. Military bases themselves built government housing for base residents. McClellan's Capeheart community and the Wherry Homes of Mather provided neat residential enclaves for on-base personnel. Likewise, older communities like Carmichael, Fair Oaks, Orangevale, and Florin, which had enjoyed a rather sleepy semi-rural existence, burst into life as developers turned over fields, knocked down orchards, and built new homes.
Whole new colonies sprang up as well. McClellan Air Force Base stimulated the growth northeast of the city and north of the American River (one subdivision was named "Aerohaven"). Flagship developments of the suburban frontier included River Park and Arden Park that filled with ranch-style homes and upwardly mobile, white middle-class inhabitants. Developer Jere Strizek would capitalize on the rapid growth of McClellan by creating a whole new suburb: North Highlands, a concentrated image of Sacramento's suburban growth. With the encouragement of base commanders, Strizek purchased a 2,000-acre tract near McClellan in 1950 and began building. At its peak, Strizek's workers were churning out a new home every 12 hours. North Highlands skyrocketed from about 150 people in 1951 to over 22,000 eight years later.
To the south of the city, the construction of the U.S. Army Signal Depot and the opening of the Campbell Soup Plant brought working-class subdivisions called Hollywood Park, Sutterville Heights, and Freeport Village. Along Florin Road, housing tracts, service stations, and supermarkets began to pop up like dandelions. Parkway Estates began in the 1950s, advertising their tract homes with promises of freeway proximity and assuring slogans, "Quality is not expensive." Virtually all of these new homes were eligible for FHA loans.
# NEW SCHOOLS
Into these new colonies were also inserted schools. The growth in Sacramento's young reflected national baby-boom trends and consequently an increasing demand for schools. Since the late 1940s, voters had passed bond issues to improve the aging grade schools and the two city high schools. Burgeoning baby boomers taxed existing school facilities. By 1949–1950, the enrollment of Sacramento's schools spiked over 30,000. In May 1951, voters approved a $6.5 million bond issue and a 50¢ increase in taxes to provide funds for new schools and additions to others. By the 100th anniversary of the Sacramento City Unified School District in 1954, there were twenty-four elementary schools, five junior highs, two senior high schools, a junior college, and a highly successful adult education program. New Sacramento high schools opened, including Hiram Johnson in 1959, Luther Burbank in 1962, and John F. Kennedy High in 1967.
Sacramento also finally received a four-year institution of higher learning in 1947. Postwar needs for teachers and an expanding educational system provided an opening for State Senator Earl Desmond to propose the formation of a new branch of the network of state teacher's colleges that already dotted the state. After much debate, the legislature approved the establishment of the new college and appointed former Chico professor Guy L. West as the first president of what would become known as "Sac State."
The college began in the buildings of the City College while the campus was being built on former farming lands near the growing River Park subdivision. This small "academic village" grew rapidly and became an important asset as its programs began to enrich cultural life. In addition to teaching duties, the faculty provided a cadre of experts for local media and other state and city needs. The schools of engineering and education poured their graduates into local businesses and schools. The university's trained historians provided some of the resources used to write this history. Sac State professors V. Aubrey Neasham and Joseph McGowan spearheaded the research that undergirded the restoration of "Old Sacramento" and explored other areas of local development. An array of local politicians came from Sacramento State, including Mayor Joe Serna Jr. and Grantland Johnson, a city councilman and state official. Concerts, dance recitals, and art exhibits found a showcase at the college. Public radio broadcasting from the campus began in 1964 with a 10-watt transmitter under the title KERS. Gradually expanding its wattage and scope, the station eventually became KXPR, a classical music station with programming in public affairs and a late-night jazz program.
California State University, Sacramento, shown here in March 1957, became an important cultural asset as its programs began to enrich cultural life.
With the formation of American River Junior College in 1955, yet another outlet for college-bound Sacramentans was available. Setting up its location in suburban Carmichael, the junior college became a magnet for suburban youth who took their first two years of general education before moving on to the state college or university system. The Los Rios Community College District took Sacramento City College out of the city school district and began a network of two year colleges that absorbed American River College. In 1961, a master plan for higher education was devised by the education-friendly administration of Governor Edmund G. "Pat" Brown (1958–1966) that brought additional resources and upgrading to Sacramento's college system. In 1972, Sacramento State was incorporated into the larger state university system and became known as California State University at Sacramento.
# INFRASTRUCTURE: A NEW SYSTEM OF ROADS AND FREEWAYS
Sacramento's love affair with the automobile became permanent in the postwar era, and the proliferation of cars soon affected the quality of life downtown. Public transit began to evaporate. The venerable streetcar made its exit in the late 1940s when PG&E sold its long-held franchise to the National City Lines. This large company, a front for auto, bus, and tire companies, specialized in replacing streetcars with more efficient buses, and in 1947 Sacramento's last streetcar service shut down. Eventually, the bus lines would be transferred to city ownership since private businesses could no longer make a profit on a service that was largely replaced by independent automobile ownership. With the proliferation of cars, city officials worried openly that Sacramento would suffer if there were not adequate provisions, so they quickly approved the construction of a 1,400-space parking garage on an entire city block north of I Street in the 1950s.
Dependence on the automobile was further enhanced by the development of the freeway system. The route of U.S. 40, the main continental freeway, ran through the state capital over the Tower Bridge and out 16th Street. In the early 1940s, the North Sacramento freeway was opened, replacing an older road that was subject to periodic water inundation. In 1947, the California Division of Highways and the U.S. Bureau of Public Roads undertook an "origin and destination" survey of the greater Sacramento area. They chronicled the increase in the volume of traffic, and by 1949, city officials tagged the traffic situation as the "number 1 problem" affecting the area's future. An extensive survey revealed that nearly 350,000 automobile trips were generated daily, 84 percent of which were within the county. Plotting a new freeway plan in 1950, engineers went to work designing a new road system around Sacramento that accommodated (and encouraged) the new automobile culture. In 1955, the construction of the "Elvas Freeway" improved traffic flow to the northeast corner of the city. The plan expanded in 1955 to include the southeast corner of the city in a section known as the 29th and 30th Street Freeway, which also included a new span across the American River.
The enactment of the 1957 National Defense Highway Act provided additional funds to connect California with the nationwide interstate system. Preparations for the Winter Olympics at Squaw Valley in 1960 hastened the development of Interstate 80, which followed the old Southern Pacific route eastward over Donner Pass, overtaking the old Highway 40. Sacramentans now had free access to Reno and Carson City. Interstate 80 continued with a spur across the old city limits and East Sacramento along 30th Street. Ribbons of concrete went down across the city along 29th and 30th Streets and another at W and X Streets. A new bridge spanned the Sacramento River joining the W-X Street Freeway with Yolo County.
Modernization of Highway 50 extended Sacramento's reach beyond the confines of Folsom Road and Fair Oaks Boulevard and proceeded outward toward Folsom and into the foothills. To the south the widening and expansion of and rerouting of Highway 99 from Stockton Boulevard knitted the rapidly developing south area even more firmly to the wider metropolitan area. The extension of Interstate 5, a roadway that runs the full length of the state, was delayed in Sacramento due to routing difficulties. But once it was complete, all roads converged on the state capital as a spaghetti bowl of freeway interchanges and off-ramps reconfigured venerable neighborhoods, created and destroyed businesses, and reinforced Sacramento's dependence on the automobile.
# AIR TRANSPORT AND A DEEP-WATER PORT
Adding to the transportation revolution in the city were two major developments: a new airport and a deep water port. The need for improved air transportation systems also pressed hard. Sacramento's municipal airport (Metropolitan Field) came into being in 1931 off Freeport Boulevard. It had three runways, and its main commercial carrier was Boeing Airways, later consolidated into United Airlines. The airport grew steadily throughout the 1930s and 1940s, fortified by government airmail service.
In 1958, a joint city-county committee contracted with the airport consultation firm of Leigh Fisher & Associates. A year later, Fisher submitted a report calling for a new airport to serve the growing area. The most controversial aspect of the report called for relocation. In spite of criticism, the board went forward and selected a site in the North Natomas area 12 miles northwest of the capital along I-5. With the help of effective lobbying by Congressman John E. Moss and city officials, ground was broken in 1964. By late fall of 1967, a proud Moss appeared at the dedication of the new $22 million Sacramento Metropolitan Field (later renamed Sacramento International Airport). Under the direction of James Carr, the airport expanded to receive jumbo jets.
The expansion of Interstate 5, shown here near Second and Third Streets in May 1968, was delayed because of routing difficulties but once completed conributed to the reconfiguration of the city.
The desire to make Sacramento a deep-water port went back to the Progressive era. The prospect of using the river as a commercial artery again became possible after hydraulic mining ended and the possibility of keeping the channel clear enough to receive sailing vessels re-emerged. Federal dollars pursued by area congressmen brought about navigation improvements, and the City built new wharves and terminals. Over 5,000 tons of freight and 840 tons of rice grown in the region's wetlands were shipped through Sacramento in the 1930s. Legislators managed to create a joint Sacramento/Yolo Port District in the 1930s, but it was not until mid-1945 that the Army Corps of Engineers signed off on the construction of a deep channel to connect Sacramento with the Bay Area. In July 1946, President Harry Truman approved the plan, and on August 7, 1949, groundbreaking took place.
Although delayed by the Korean War and problems with funding, the project went slowly forward, pushed every step of the way by Congressman Moss. Despite further delays due to Sacramento's silty river bottom, the port welcomed its first visitor in June 1963, the 8,000 ton Taiwanese freighter S.S. Taipei Victory, whose cargo hull was filled with Sacramento area rice. The port never materialized as its advocates predicted. Strong competition from the nearby Port of Stockton along with continued problems keeping the channel of the Sacramento River clean have hampered its potential.
# WATER PROJECTS AND NEW POWER
A long delayed Folsom Dam was completed in 1955 and with it the accompanying Nimbus spill-off. Like the port, efforts for damming the American River had been in the works since after World War I. In 1946, Folsom incorporated as a city and plans were laid for the construction of a new dam and additional hydroelectric generation. By the early 1950s, in the wake of the need to prevent floods, promote recreation, and meet the electrical needs of Sacramento businesses, as well as the growing number of "all electric" homes, federal monies flowed into Folsom. In 1956, the new structure was dedicated after having already proven its worth in the wet winter of 1955 when it held back the waters of a swollen American River in one of the worst flooding seasons the region would see until the inundation of 1985–1986. In 1955, SMUD was approved for a generation plant on the American River, and in 1961 kilowatt hours began pumping from the new facility. The dam also created Folsom Lake, a popular recreation spot, and Lake Natoma, which sits below the bluffs of suburban Orangevale. In another development, the long-stalled plans to transform the banks of the American River into a jogging and recreational trail materialized in the 1970s. Stretching 23 miles from Hazel Avenue and the Nimbus Fish Hatchery, west to Discovery Park at the confluence of the American and Sacramento Rivers, joggers, naturalists, and rafters flocked to the trail, making it one of the most popular recreation sites in the area.
The Taipei Victory arrived at the Port of Sacramento on June 29, 1963.
# HEALTH CARE
Sacramento desperately needed more modern health care facilities. Area hospitals, benefiting from federal loans and grants under the Hill-Burton Act of 1949, applied for and received substantial aid to expand their facilities and improve their technology. A massive building project at the County Hospital in the early 1950s added over 400 beds. Likewise, private health care centers like Sutter and Mercy General Hospitals also expanded. Maternity wards in both were booming during these years, and the general increase in children pressed the Sisters of Mercy to convert their old nursing school (it had closed in 1950) into a children's hospital. In addition, the Episcopal Church erected the million-dollar St. Luke Medical Center on Capitol Avenue. Scores of new office buildings housed clinics for a variety of health care professionals, including the growing number of pediatricians, orthodontists, and optometrists. In response to federal initiatives, the University of California at Davis Medical School made the Sacramento County Hospital its teaching hospital in 1966. In 1973, the school purchased the hospital outright. Renamed the U.C. Davis Medical Center in 1978, the hospital serves as a teaching facility for physicians in training as well as expanding the scope of services and therapies for the region. In 1965, Kaiser-Permanente health care came to Sacramento and opened a facility on Morse Avenue. Secured by lucrative contracts with local federal employees, Kaiser provides pre-paid health care benefits for scores of Sacramentans.
Efforts to dam the American River had been in the works since after World War I.
# REDEVELOPMENT
New residents, as well as city families, poured into the suburbs, thereby decreasing the city's tax base and leaving behind the maintenance and upgrading of its infrastructure. Urban churches lost congregations, stores closed, and even the lure of movies and other cultural events at downtown theaters and the Memorial Auditorium became less and less attractive. Many of the questions about the downtown's future focused on the deteriorating buildings of the West End, whose boundaries though somewhat fluid stretched from the river east to Seventh Street and south of the Southern Pacific Depot to R or S Street.
Sacramentans held two different images of the West End. Suburban developers, politicians, and newcomers accustomed to small-town life viewed it as a cauldron of urban pathologies. It had lots of bars, fast food joints, flop houses, employment agencies, and houses of ill-repute. It also had a very low tax base, yet consumed the lion's share of expenditures for fire and police calls. The residential structures of the district were not only a public relations liability, but also a public health hazard. Absentee landlords owned over 80 percent of its buildings. In a series of articles in 1949, Sacramento Bee reporter Hale Champion painted the area in the darkest possible hues. Another writer depicted the area as "the worst skid row west of Chicago."
Other Sacramentans had a different perspective. No one denied the physical deterioration of the buildings and the prevalence of crime; however, they also noted that despite its rough character, the West End provided affordable housing for nearly 10,000 transients—mostly men over the age of 55, pensioners, people of color, and other minorities. There were also businesses and entertainment venues that catered to city clientele. The West End functioned as a labor market for those who needed cheap workers for the canneries, the fields, and other assorted jobs. Some saw it as one of the last bastions of the vibrant ethnic cultures that co-existed for generations within blocks of each other in Sacramento. Within its boundaries were ethnic clubs, social halls, and remnants of Sacramento's past.
Already during the Great Depression, the federal government had begun taking a more active role in urban affairs. After the war, the commitment of the central and state governments to urban rejuvenation grew stronger. The first steps in doing something about Sacramento's rapidly decaying West End were modest. When the California Redevelopment Act, which provided state funds for local improvement projects, passed in 1945, the Sacramento City Council studied possible options for redevelopment. In October 1947, it voted to widen the parkway between the state capitol and the Tower Bridge to remove unsightly buildings and give those entering the city an uninhibited view of the seat of California's government.
Little changed until 1949 when Congress passed the Federal Housing Act. This sweeping legislation promised government subsidies not only for slum clearance but also for the construction of public housing for the displaced. Sacramento already had two public housing projects, New Helvetia on Broadway and Dos Rios off North B Street.
The nexus of slum clearance and relocation needs would complicate redevelopment issues for years. In December of 1950, the city released its first detailed redevelopment plan. Crafted by architects Richard Neutra and Robert Alexander, it proposed an extensive program of slum removal and the construction of high rise public housing facilities interspersed among new buildings in the redevelopment zones. The Sacramento Redevelopment Agency was formed to implement this plan under the leadership of Joseph T. Bill. Studies progressed throughout the 1950s with developers identifying 233 blocks in need of transformation. However, opposition from businessmen and others to the public housing aspects of the plan mounted throughout 1951 and 1952. One way that Bill and others found to evade the requirement to relocate those who had lived in a redevelopment area for at least a year was to insist that the law's provisions only applied to families. This immediately reduced the number eligible for relocation significantly by leaving out the largest single group of denizens of the area: single men. Meanwhile, the plan stalled and was tabled in May 1953.
In the meantime, a new Republican administration had taken over in Washington, D.C. that backed away from earlier plans linking redevelopment with public housing. In July 1954, San Francisco developer Ben Swig offered a fresh plan that focused almost entirely on new shopping and business buildings in the redevelopment area. At the heart of the Swig Plan was a new K Street pedestrian mall, with moving sidewalks, that spanned from Second Street to 12th. Opponents of Swig's plan noted that it made little mention of public housing and relocation. Anger at the destruction of their property prompted West End residents to resist change and stall a ballot measure (Proposition B) in 1956 that would have enabled the city to borrow money to proceed with demolition. The Proposition B defeat, however, was only temporary. Undeterred, city leaders found other sources of financing and demolition plans proceeded, targeting the area directly in front of the state capitol as the first site for rehabilitation.
In 1957, Governor Goodwin Knight presided at the symbolic wrecking of a dilapidated Victorian on Sixth and Capitol, commencing a process that continued into the 1960s. Demolition proceeded rapidly and before long a new Capitol Mall emerged, providing a sweeping open space between the Tower Bridge and the statehouse. As one Sacramento Union columnist put it: "Where honky-tonks and gas stations once stood, a thoroughly businesslike Capitol Mall sprouted, an appropriate gateway to the Capital." In 1960, yet another plan laid out an ambitious agenda for the next phase of urban renewal—a pedestrian mall for K Street, a convention center across from the aging Memorial Auditorium, and a new historic park with a "gold rush" motif called "Old Sacramento."
Sacramento redevelopment left large vacant city blocks for years until development projects became final.
Redevelopment did hit bumps. Ambitious plans for ethnic "blocks" (e.g., the Hispanic Plaza de los Flores) in redevelopment failed as money and support were not sufficient. Another major controversy erupted over the placement of Interstate 5, the north/south freeway that cuts through the heart of California. The State Department of Highways floated several plans for the new link. One proposal was to locate the freeway along the river in Old Sacramento, thereby demolishing the city's historic district. Arguing forcefully in favor of this was the Macy's Company of New York, which contemplated an anchor store for a proposed new shopping mall near the river. Arguing against the riverfront freeway was a coalition of preservationists, including Eleanor McClatchy of the Bee. William W. Winster, dean of the School of Architecture at the University of California at Berkeley, best articulated the concerns of this group:
It is beyond me how a city can think of routing a freeway through so important a historic quarter. A freeway is a freeway and it can be routed anywhere. Why, this is where the Argonauts landed and this was where your merchant princes planned and built the first transcontinental railroad, and this was where California was planned and governed in those first early years.
Ultimately, a compromise was reached that preserved the first three blocks of the old city and ran the freeway between Second and Third Streets with an off-ramp close enough to the proposed new Macy's.
Redevelopment again slowed to a halt in the mid-1960s as an unsteady American economy caused many early backers to withdraw. Things looked bleak until 1967 when developers formed the Downtown Plaza Properties Group, consisting of developer George McKeon, aggregate millionaire Henry Teichert, and construction mogul William Campbell who reinvigorated the plan. "Those guys were saviors," Mayor Burnett Miller later recalled. Dividing their efforts between office and retail operations, McKeon and his associates began to fill in the empty spaces of the nearly abandoned redevelopment projects.
In the mid-1960s, Mayor Walter Christiansen made plans to develop the K Street Mall. Envisioning a broad pedestrian walkway, he and city designers attempted to recreate the feel of the Sacramento River running from the mountains (the large concrete blocks in the eastern mall) to the sea (the flatter pools and grass near Seventh Street). Likewise it was hoped that retail shopping would continue. Later, these concrete piles would be derided as "tank traps."
The long-stalled Old Sacramento project got underway in the late 1960s. City officials began to look seriously at the rehabilitation of nearly 100 historic buildings. A new wave of entrepreneurship in the area began as early as 1960 when restaurateur Newton Cope restored an 1853 firehouse and opened the posh Firehouse Restaurant in the middle of the blight. The city began to clear out the last of the residents of the run-down buildings, make assessments about salvageability, and dispatch historians and others to assist in the historic reconstruction of Sacramento's past. In 1966, the National Register for Historic Places designated this district a national landmark and the state legislature declared it a state park. In all, $33 million was poured into the Old Sacramento project.
The neglected side of the story was the plight of those displaced by the demolition of the housing in the area. The City Council empaneled a Community Welfare Council consisting of local clergy and others who studied the conditions created by redevelopment. This group issued a report in 1964 noting that from 1957 to 1963, the population of single men dropped from 5,500 to 1,400. Nearly 4,000 of these men left and the redevelopment agency could account for only 555 of them. Of this group, a little over half found housing within or adjacent to the redevelopment area while the rest either moved out of town or to other places in the metropolitan area. Sacramento simply ejected these men from their midst. Some settled in other areas of the city, but many quit Sacramento, never to return.
# THE CHANGING CULTURE OF SACRAMENTO
One group displaced from the West End were the nearly 2,000 African Americans living in the redevelopment area. They saw their historic churches demolished (St. Andrew's and Shiloh) and their housing and community institutions removed. The search for affordable housing was difficult. Black Sacramentans had already defined a zone for themselves near McClellan Field, where a bus shuttled them to their jobs. After redevelopment, others were scattered all over the city with relocation points in the Oak Park area (where they already had a foothold), and south in the Glen Elder and Meadowview areas. Later, clusters at Freeport Manor and in smaller residential areas near the Western Pacific rail yards and Hughes Stadium created urban space for the African-American community. Indeed, the black population was growing at this time. By the end of the 1960s, upwardly mobile African Americans were moving into suburban areas like North Highlands, Rancho Cordova, and to the Parkway-South area. In 1960, Sacramento's black population stood at 19,805 or 3.9 percent of the total county population.
Sacramento's history of racial exclusion began to be tackled in a more systematic fashion as the national civil rights movement picked up steam, and Sacramento citizens stepped forward to end long-standing practices of racial discrimination. African Americans like Frank Canson were inducted into the police force in 1947, and his wife Fannie Canson in 1948 was the first African-American teacher in Sacramento public schools since Sarah Mildred Jones, a black principal of Fremont School at 24th and J (1894–1914).
In early 1950, black attorney Nathaniel Colley represented the growing black professional class. Colley was one of the first, in league with the revived NAACP, to challenge cases of racial discrimination in Sacramento. In 1949, he exposed police brutality against two African-American suspects who had been beaten into confessing to the robbery of a bus passenger. In one of his first cases argued for the NAACP, Colley challenged racial exclusion at the Land Park swimming pool (Hazel Jackson v. Land Park Plunge, 1950)—a case he won only to see the pool closed. Later, he effectively challenged racial discrimination in the River Oaks Public Housing Project (1951–1953).
Clarence Canson and Douglas Greer, African-American attorneys, ran for city council in the 1950s. Although they lost their bids, African-American political strength mustered to elect Milton McGhee to the City Council in 1967—the first black elected official in municipal office in Sacramento. A year earlier, Ebony Magazine's Negro Handbook had cited Sacramento as one of the "ten best cities for Negroes." The increasing prominence of statewide black office holders like Wilson Riles and Byron Rumford also enhanced the visibility of Sacramento's growing black community.
Attorneys Douglas Greer and Nathaniel Colley shake hands at the NAACP life membership banquet.
Sacramento caught the crest of the national civil rights movement, and African-American leaders participated in marches and pressured city leaders on such issues as access to housing and job opportunities. The issue of segregation in Sacramento schools was highlighted in August 1963 when the racially-mixed Stanford Junior High School burned to the ground. Minority students were then sent to previously all-white schools to create racial balance. School desegregation in the Sacramento City Unified District went forward and influenced similar policies in neighboring districts. The federal government's war on poverty provided funds and a framework for community organization among Sacramento's poor, many of whom were black. The Sacramento Area Economic Opportunity Council (SAEOC) was the recipient of funds from the federal Office of Economic Opportunity. African Americans played an important role in this organization. Through the annexation of areas like Del Paso Heights and the growing popularity of the Sacramento Observer, African-American visibility increased.
Toward the end of the 1960s, civil rights militancy eclipsed the legal strategies of Colley and others. Sacramento had chapters of most national civil rights organizations, such as Congress of Racial Equality (CORE), Student Non-violent Coordinating Committee (SNCC), and the NAACP. It also had branches of more militant groups such as the Black Panther Party, the Black Student Union, and the Nation of Islam. Racial tensions flared when police arrested three heavily armed members of the militant Black Panthers who had stalked into the state capitol to protest a gun control measure the legislature had approved. An October 1968 symposium at Sac State on "Racism in America" drew nearly 10,000 participants. Black Panther leader Eldridge Cleaver delivered an impassioned 75-minute speech at Hornet Stadium denouncing capitalism and oppression and tapping into the widespread anger felt by many over the death of Martin Luther King Jr. the previous April. Simmering tensions were given a voice when federal anti-poverty legislation provided for the creation of neighborhood councils and organizations. The Oak Park Neighborhood Council reflected one such umbrella group dedicated to soothing tensions, improving services, and keeping open channels of communication.
The national difficulties of the year 1968 found their way to Sacramento. For a time, trouble had been brewing in the Oak Park district and in the area around James McClatchy Park. Police and local youths had engaged in several stand-offs, but tensions had cooled. Then a fight broke out in July between a large group of black youth and an all-white baseball team playing in McClatchy Park. A second day of unrest followed. Nervously convened meetings between the Oak Park Neighborhood Organization and city officials allowed residents to air grievances. The meetings accentuated the need for activities to undercut further violence. Recreational and neighborhood revitalization projects were launched; educational workshops, meal programs, and musical groups helped to siphon off ill will. However, efforts fell short when in the summer of 1969 gunfire erupted near the park between residents and police. The night of June 16 saw nearly 100 shots fired between police and snipers. Injuries to police and rioters resulted in 38 arrests and calls for an investigation of conditions in the deteriorating Oak Park area—as well as into the behavior of the police.
While conflict between African Americans and police continued into the 1970s, historian Clarence Cesar notes the rising economic and political clout of Sacramento's African-American community. Community leaders Rosenwald "Robbie" Robinson and Callie Carney served terms on the City Council. African Americans took prominent roles in various school boards of the area. Representation in the judiciary was increased as well. In 1975, Sacramento native William K. Morgan (owner of the Morgan-Jones Funeral Home) was appointed to the Superior Court.
# MASS CONSUMPTION CULTURE
Although Sacramentans had long been a community of shoppers and frequented the well developed commercial districts along J and K Streets, in the postwar era, the culture of consumerism surged more powerfully than ever. Sacramentans joined their fellow Americans in enshrining mass consumption as a defining characteristic of U.S. culture. Historian Lizabeth Cohen has described the rise of this phenomenon in the postwar era. Government programs encouraged better education for higher paying jobs, underwrote home ownership, and expanded automobile ownership. Shopping, buying, and consuming defined local culture as never before. The shopping center emerged as the most powerful visible symbol of this new way of life.
Smaller retailing outlets like the Hollywood Mall and Arden Town led the way. In 1946, developer Jere Strizek built the first major shopping mall in Sacramento County: Town & Country. This simple shopping center was constructed with surplus materials and affected an "Old West" mystique with its use of wagon wheels and hitching rails. Town & Country's 46 stores and nearby theater were a fabulous success. In 1951, developer James Cordano worked with the Blumenfeld family to open Sacramento's first regional shopping center called Country Club Center. Anchored by J.C. Penney and Rhodes (another department store), it took root in an already rapidly growing area. The remaining lands soon filled in with tract homes, stores, and churches. Set off by an enormous parking lot, ringed with palm trees, Country Club was a smashing success. By December 1955, nearly 2,000 businesses did a healthy trade in the new shopping areas. Another rival, Arden Fair, joined the fray in 1957 and all continued to flourish in the 1960s. In the south area, Cordano developed Southgate Shopping Center in 1960, and later Sacramento's first enclosed and air-conditioned mall, Florin Center, opened in 1967. Subsequent mall development in Citrus Heights and Roseville further decentralized Sacramento's shopping experiences.
The modern supermarket made its appearance in Sacramento, displacing small retail grocery operations like those of the Arata brothers or butchers like Moehring and Yorke. Grocery chains became popular after the war, bringing all of these various food buying operations under one roof and surrounding them with capacious parking lots. Some of Sacramento's supermarkets evolved from small "mom and pop" operations. Such was the case with the Kassis brothers, a Lebanese family who came to Sacramento in the 1920s. Under the leadership of the family patriarch, A.G. Kassis, the sons began their successful business with a fruit stand on 28th and Broadway. Later, their enterprise expanded into a network of 12 stores that they christened "Stop 'n Shop," and they effectively marketed themselves with the help of a popular radio jingle.
The Kassis brothers' competitors included another family chain, lead by the Inks brothers—Charles, Russell, and Dick—who began a grocery business in 1925. Adapting new supermarket retailing techniques, the brothers forged a grocery empire of 33 supermarkets, 29 of them in the Sacramento area. Contributing to their success was their ability to move into the rapidly growing suburbs.
The popular Raley's Markets were the brainchild of Thomas Raley, who had come to California from Arkansas in 1925. He took his apprenticeship in the grocery business in the Bay Area, and in 1935 opened a small store in Placerville. In 1938, he opened his first Sacramento store on Stockton Boulevard. By 1953, he owned seven stores. Raley's eventually became the largest locally owned grocery operation in the Sacramento area. Other chains came. Lucky's opened its first market on 30th and Broadway in March 1948. The next year, florist Bert Geisreiter sold Margaret Crocker's Conservatory across from the city cemetery to the Safeway chain.
Independent grocery retailing persisted in Sacramento largely through Chinese family markets. Historian Alfred Yee has chronicled the rise and decline of locally run Chinese markets like Giant Foods and Farmer's Market, which opened in the 1930s. Belair, begun by the Wong Family as a fruit stand in rural Penryn in 1956, grew dramatically as well. Ultimately these family-owned markets were gobbled up by chains such as Raley's.
Town & Country Shopping Center opened in 1946.
The Kassis brothers owned the Stop 'n Shop grocery chain.
Popular restaurants contributed to Sacramento's culture of mass consumption. As prosperity spread, Sacramentans began to eat meals away from home and an active restaurant industry scrambled to accommodate the new tastes. Sacramento had its revered old restaurants, like the Rosemont Grill at Ninth and J, and Posey's Cottage (a favorite spot for Governor Ronald Reagan) on Tenth and O. Frank Fat's popular Chinese eatery on L Street was also a hang-out of the political classes. More upscale dining was available at the Firehouse Restaurant in Old Sacramento. Suburban dwellers found a good venue at Aldo's Restaurant located in the Town & Country shopping center.
Sacramento also had it share of the old fashioned diners and quick lunch places typical of many cities. Stan's Drive-Ins, run by proprietor Stanley Burke, began in Sacramento in 1933. These first drive-in restaurants sported a modern design and service to the car. The huge sloppy hamburgers made at Jim Denny's on Terminal Way and 12th Street still attracts a non-stop lunch crowd. Likewise the popular Sam's Hof Brau on J Street featured piled high sandwiches served in a traditional "gasthaus" atmosphere. Vic's ice cream parlor, established in 1949 by Vic Vito and Ash Rutledge at Riverside and Eighth Avenue, perpetuated an old fashioned soda parlor ambience. Merlino's orange freeze, hawked in an orange-shaped kiosk, was a popular summer beverage.
But the real growth was in the mass produced fast-food industry that took off in Sacramento like a rocket. Fosters Freeze, a popular fast-food chain, came to Sacramento after the war and featured soft-served ice milk, fried hamburgers, and French fries. Hart's Hamburgers on Freeport Boulevard featured a neon Highlander doing a flashing light "fling." The most successful innovation in fast-food came when the McDonald's hamburger kingdom came to Sacramento in February 1954 and opened a stand on Fruitridge Boulevard. Sherwood "Shakey" Johnson and his associates are credited with popularizing pizza around the Sacramento area. The popular Shakey's Pizza shops included low-cost food, beer, and occasional entertainment by local bands.
The fast-food industry took off quickly in Sacramento with places like Shakey's Pizza Parlor doing a brisk business.
# THE NEW STATE FAIRGROUNDS
Everything changed in Sacramento in the postwar era—even the location of the state fair. Since 1906, the fairgrounds had been located on Stockton Boulevard. With the proliferation of automobile ownership, the area around the fairgrounds grew cramped and congested as local residents rented their driveways and front lawns to desperate motorists. Likewise, the aging buildings of the site began to feel their limitations. State Senator Earl Desmond proposed relocation to a new site on former grazing lands of the Swanston meat-packing company (closed since 1949) near Arden Fair. Claimed under the power of eminent domain by the State of California, the site was soon being studied by planners recruited from the Disney Corporation in Southern California to design the new exposition park. The result was a huge new concrete center surrounded by a mammoth parking lot and named the modernistic-sounding "Cal Expo." In 1967, Governor Ronald Reagan formally dedicated the site, and the old fairgrounds passed into mythic memory. The huge Golden Bears that had greeted fair-goers at the old site were transferred to Cal Expo, but seemed dwarfed by its concrete structures. It would take time for Sacramentans to warm up to the new site.
# ENTERTAINMENT
If Sacramentans found common ground in shopping malls and supermarket chains, movies and television provided even more common culture. Virtually everyone in Sacramento (and in America) went to the movies. Nearly everyone owned a television set.
Sacramentans still frequented movie palaces in the late 1940s. Filmmakers who still owned the theaters experimented with new screen technology (Cinerama and Cinemascope), and Sacramento theaters like the Senator adapted. On the site of the old Hippodrome, the new Crest Theater had a gala opening in October 1949, hosting stars William Demarest, Kathryn Grayson, and tenor Mario Lanza for the premier of The Midnight Kiss. However, movie studios began to unload their branch theaters. Moreover, the popularity of television and the decline in the central business district meant the end of downtown movie palaces. By 1969, historian Andrew Flink notes that only nine theaters still operated in the downtown. Suburban theaters, however, prospered. The Village Theater opened at Fulton and El Camino in late 1949, the Rainbow Theater in Carmichael in the same year. Drive-in theaters too began to dot the landscape. In 1946, the El Rancho Drive-In Theater commenced operation in West Sacramento. In June 1950, the Fruitridge opened its ports to motorists as did the Bell Drive-In on Marysville Road. Others followed.
One development that slowed and even arrested the entertainment exodus from downtown was the Music Circus, a summer stock musical theater providing first-rate entertainment. The prototype premiered in Lambertville, New Jersey where musicals were performed in the round for appreciative audiences in a Chatauqua-like tent. This format came to the attention of two Beverly Hills producers, Russell Lewis and Howard Young, both veterans of Broadway productions. Eleanor McClatchy, Sacramento's chief arts patron, invited the two producers to come to Sacramento and attempt a summer musical theater in the state capital. McClatchy's generous financial backing and Young's and Lewis's expertise made a powerful match, and in 1951 the new Sacramento Light Opera Association was formed and space rented at 15th and H from the Sacramento Civic Repertory. The Music Circus opened its first season with a performance of Jerome Kern's Showboat. The audiences, packed into the large flowing tent, grew and grew through the summer, and the Music Circus became a popular summer pastime. Although summer heat and uncomfortable seating sometimes marred the evening's pleasure, steady upgrading of shows, music, and the physical structure assured continued popularity for the summer theater.
More "highbrow" entertainment was provided by the Sacramento Philharmonic Orchestra, conducted for years by Fritz Berens. The Saturday Club arranged a cultural season that brought soloists, the San Francisco symphony, and ballet troupes to Sacramento. Annual visits of the San Francisco Opera company brought divas like soprano Leontyne Price to perform in the Memorial Auditorium.
As the 1950s opened, radio was still the predominant medium. Sacramento had five major stations (KFBK, KXOA, KGMS, KCRA, and KROY) with national hookups. Public broadcasting took an important turn in 1944 when advertising executive Ewing C. Kelly formed the Central Valley Broadcasting Company. Working with partners David R. McKinley and C. Vernon Hansen, Kelly made application to the Federal Communications Commission (FCC) for a local 250-watt AM radio station. KCRA began broadcasting in April 1945 as an NBC affiliate, and in 1951 expanded its power. In August 1949, Kelly and Hansen (McKinley was bought out) began an FM station.
However, radio was soon eclipsed by the rise of television. Television had first been demonstrated in Sacramento at the 1947 State Fair when students of the Grant Technical Schools built the first viewer and transmitter in Sacramento County. Sacramentans with high aerials and favorable weather could pick up television broadcasts from San Francisco. Sacramento's first television station, KCCC, opened to great fanfare in 1953, but its signal was weak. In September 1954, KOVR began broadcasting. In March 1955, KBET began broadcasting, and in 1959 changed its title to KXTV. In August 1948, KCRA applied to the FCC for a television station, but was compelled to await approval until 1955.
Sacramento Bee artist Jack Oglesby sketches a happy Russell Lewis and Howard Young, co-creators of the Music Circus.
# YOUTH CULTURE
In 1946, Sacramento's baby boom began when county births soared from 4,236 in 1945 to 5,354. This group and subsequent large "classes" of newborns began to work their way through the Sacramento school system, creating huge changes. A new cohort of young people compelled businesses and public facilities to cater to the youngsters in a variety of ways. William Land Park added Fairy Tale Town, a youngster's attraction in 1959. Entertainment venues for youngsters were added to local theaters, which ran popular matinees of Disney cartoon movies like Snow White, Sleeping Beauty, and Song of the South. Television broadcasting carved out its own children's hours in the early morning and with after school cartoon shows like KXTV's Diver Dan and his puppet pal O-U Squid, and Captain Delta on KOVR. The proximity of the airbases made the air force fighter Captain Sacto (played by the popular Harry Martin for many years) a natural Sacramento child entertainer. Children's clothing and toy stores like Bob's Toyland appeared with great success.
Sacramento's pop music culture, fed by local youth, evolved with the rest of the state and nation. Locally, teen dancers rocked to the tunes of Bill Rase and others. Local radio station KROY began to spin pop tunes and later KXOA became the button most pushed on car radios by Sacramento teens. In February 1963, Sacramento native Fred Vail brought the newly formed Beach Boys to the Memorial Auditorium. The following year the popular group recorded a live album from the Memorial. A year later, the Rolling Stones rolled into Sacramento to play at the Memorial Auditorium.
Sacramento had its collective anxieties about "wild youth." A particular source of concern was cruising down K Street in a souped-up car or "hot rod." Although discouraged by local police and parent groups, the K Street drag became a popular rite-of-passage. Ministers, PTAs, and other groups worried about the effects of rock music, "suggestive" attire, and movies with the surly images of teen idols such as Elvis Presley, James Dean, and others. Historian William Mahan recalled Marian Stebbins, Dean of Students at Sacramento High School in 1960, expelling every girl who came to school wearing jeans.
The years after World War II completed Sacramento's transformation to a metropolitan center. The city's boundaries were now more fluid than ever as the automobile and freeways created new configurations of population and commercial life. The net effect of the changes seemed to leave the old city behind and transfer energy and vitality to the suburbs. The downtown faded as the shopping centers and theaters of new areas buzzed with life. But the pace of change was only beginning. Sacramento would burst to life again in the final decades of the twentieth century.
# 7. U RBAN RENAISSANCE
In the latter years of the twentieth century, Sacramento emerged even more sharply as the hub of a thriving five county metropolitan area. By the end of the century, a new skyscape had emerged dramatically from the flat floor of the Central Valley. The city had weathered significant challenges, especially the loss of federal dollars that had poured into the region through the military installations and the aerospace industry. It had met the sometimes tense social challenges brought about by shifting demographics and was acknowledged by some as a showcase of diversity. Thanks to new leadership, the aging downtown was recreated and the city and its neighborhoods became a sought-after alternative to suburban life. The city itself began to think in regional rather than local concerns. New forms of transportation and communication knit the Sacramento metropolitan area together in a manner vaguely reminiscent of the city's earlier days when Sacramento was a relatively self-contained urban center.
# THE CHALLENGES OF THE 1970S
The bright promises of redevelopment and road construction had left a mixed legacy. The new superhighways had united the metropolis and improved transportation significantly. Its aggressive urban redevelopment program had eliminated the worst slums, and high-rise buildings and town houses began to be built where taverns and flophouses once stood. However, all of this had come at a price. Urban neighborhoods in the path of the freeways had been destroyed, and property values of the remaining structures in the vicinity had dropped. Ambitious plans to restore K Street to its role as a shopping center by making it a pedestrian mall had also failed. People had abandoned the downtown, and one wag quipped that one could "roll a bowling ball" down the nearly vacant K Street Mall without hitting anything. Meanwhile, pranksters had taken to filling the fountain at 11th and K with laundry detergent and occasionally someone could be found taking a bath in the suds.
A series of unfortunate incidents in the 1970s further diminished Sacramento's self-esteem. The most traumatic was a tragic air accident in 1972 that sent a Korean War–era aircraft careening across Freeport Boulevard and smashing into the popular Farrell's Ice Cream Parlour, killing 22 people, many of them youngsters. The searing memory of this event—memorialized by a plaque in 2003—cast a pall over the city. In April 1975, members of the violent Symbionese Liberation Army, with heiress Patty Hearst participating, robbed the Crocker Bank in Carmichael and killed Myrna Lee Opsahl, a 42-year-old church worker depositing her congregation's Sunday collection. The perpetrators of these crimes were not brought to justice until 2003. In September of 1975 a visit by President Gerald Ford nearly turned into disaster when Lynne "Squeaky" Fromme, an acolyte of mass murderer Charles Manson, tried to fire a shot at the chief executive. The jarring nature of these events and the discomfort of the future of the downtown laid bare a collective anxiety in Sacramento's citizenry. At the same time, the city's demographics began to change as well.
# A CHANGING POPULATION
Sacramento had always hosted an array of ethnic groups and communities. However, community leaders had continually pressed newcomers to "Americanize." This emphasis, combined with the relative proximity of ethnic neighborhoods (Sacramento never had the self-contained ethnic enclaves typical of many other American cities), had tended to homogenize the city.
Filipinos constituted one of the first waves of Sacramento's ethnic identity. Young, single Filipino men had been coming to Sacramento since the 1920s. Attracted by the prospect of jobs as farm laborers, they lived in the West End where employment agencies recruited them. After World War II, more Filipinos moved to the state capital—so much so that in 1961, Sacramento and the Philippine capital of Manila became sister cities. In December of that year, Sacramento mayor James B. McKinney declared December 28 "Jose Rizal Day" in honor of the Filipino nationalist hero who had been executed by the Spanish 65 years earlier. (Rizal had actually visited Sacramento in 1888 and expressed admiration for the harmony that existed among various ethnic groups.) The locus of the Filipino community was the Catholic Church. A group of Filipino Sisters had come to Sacramento in the 1950s and had begun to organize cultural and religious celebrations. After 1965, there was a huge increase in Sacramento's Filipino community.
The Farrell's Ice Cream Parlour tragedy in September 1972 spawned new rules for pilots and airports.
Filipinos came because immigration laws were changed. In 1965, Congress revised the immigration legislation of the 1920s and removed nationality and race as a basis for inclusion or exclusion from the United States. They also eliminated ethnic quotas. The new law placed a priority on the family as the unit of immigration. At the same time, American society experienced a revival of interest in the preservation of ethnicity. Sacramento had to learn a new way of accommodating immigrants. Among the newcomers were thousands of southeast Asian families dislocated by the Vietnam War. After the collapse of South Vietnam in 1975, Sacramento welcomed thousands of Vietnamese, Laotians (Mien), and Hmong to the city. By 1995, there were over 40,000 Vietnamese living in Sacramento. By 2000, from 18,000 to 25,000 Hmong resided in the county, and by 2002, over 12,000 Mien also lived in the region. Jobs and educational opportunities, especially in the high-tech and medical fields, attracted scores of Indians and Pakistanis as well. Stockton Boulevard, a declining commercial corridor leading south from the city, sprouted into new life as Chinese, Vietnamese, Filipino, and Hmong Sacramentans bought homes and opened businesses along a 24-mile stretch from 14th Avenue to Morrison Creek.
Mary Gorre is crowned queen of the Filipino community during the Philippine Independence Day celebration, 1946.
The new cultural diversity reflected itself in places of worship. Sikhs and Hindus came from the Indian subcontinent. Muslims had been in Sacramento since the 1890s, and with an influx of Pakistani immigrants, Sacramento had one of the first mosques on the West Coast by the 1940s. Korean and Vietnamese Catholics built their own churches in the 1980s. Russian families arrived in Sacramento, building on communities across the river in Bryte. Pentecostal and other evangelical churches flourished as radio evangelists played a role in bringing thousands of Ukrainians. By 1995, over 20,000 Ukrainians lived in the Sacramento area.
# GOVERNMENTAL CHANGE
The growing ethnic diversity of the city began to reflect itself in politics. Sacramento city government reached a crossroads of sorts by the end of the 1960s. The general political ferment of the period had brought to the fore a new group of leaders, many of them representatives of the city's diverse ethnic pool. In 1965, Sun Wong was elected to office, the first Chinese American in the city's history. Two years later, attorney Milton McGhee, the first African American took his seat on the board, and in 1969 the rising tide of Latino voters elected Manuel Ferrales to office. In this social cauldron, citizens mobilized to create a more responsive and updated city government. One such group, the "Citizens for Better Government," took aim at the 1921 City Charter and proposed a series of reforms that they thought would make the city more responsive to the shifting needs and changing demographics of Sacramento. They urged, among other things, the creation of voting districts for city council members (replacing the at-large election that had kept the City Council largely a bastion of white, middle-class males) and the at-large election of an independent mayor. Carefully saved, however, was the heart of the 1921 reform: the post of city manager. Whatever the other changes of city government, its city managers, including Richard Rathfon, Walter Slipe, William Edgar, and Robert Thomas, continued to exercise a powerful influence on the shape of Sacramento public life.
These reform proposals worked their way through supporters on the existing city council and voters approved them in a November 1970 election. Eight new council posts were created, each elected by separate districts, and an at-large mayor was approved. The new city council included one incumbent, Manuel Ferrales, but also welcomed a host of newcomers such as Anne Rudin, Philip Isenberg, Robert Matsui, Rosenwald Robinson, R. Burnett Miller, and Ritz Naygrow. Richard Marriott, a pro-labor activist, was elected mayor.
Attorney Milton McGhee, shown here in October 1974, became the first African American to serve on the city council.
The mayor's position evolved dramatically during this time. While mayors of the 1960s were highly visible figures popular with the city's business elite, the mayors of the closing decades of the twentieth century cut more dynamic figures as movers and shakers. Mayor Philip Isenberg, who took over after Marriott resigned in 1975, helped shape the mayor's position into a major political force. But the two dueling politicos that most strongly affected the tide of city government were Anne Rudin and Joe Serna Jr. Rudin was elected mayor in 1983 and served until 1992—not the first woman to hold that office but the first woman elected directly to the mayor's job by Sacramento voters. When Isenberg departed city government for the legislature, Rudin and Serna competed to dominate city government. Both were ambitious politicians, eager to play a role in developing Sacramento life. Rudin was more of a political and social liberal, opposed to war, openly supportive of socially liberal causes such as gay and lesbian rights. To the consternation of many, she openly speculated that Sacramento might be able to survive without the presence of its three major military installations. Serna was less easy to characterize. His base of support came from the growing number of Latino Sacramentans and organized labor. He cut his political teeth on his work with farm labor organizer Cesar Chavez. But Serna was above all a pragmatist, willing to make deals with ideological opponents to advance the cause of Sacramento's civic and economic health.
Sacramento's public officials faced a lot of issues from the 1970s on. Perhaps the single most important was the pace and scope of urban growth. This pitted developers, anxious to take advantage of new land and new markets and expand aggressively, against those who proposed a more orderly phased-in approach. The latter group continually warned against the "Los Angelization" of Sacramento—meaning ill-planned urban sprawl. This faction had Rudin's support. Serna by contrast, a protege of former Mayor Isenberg, who had appointed him to the City Redevelopment Board, favored a more aggressive approach to growth and change. A careful politician, Serna was not willing to align himself too openly with the pro-development group, but clearly believed that the future of Sacramento required more boldness and risk-taking than he thought Rudin was willing to provide.
In the 1980s, the signal issue focused on the section of land called North Natomas, a 9,000-acre strip of farmland between Sacramento and the Metropolitan Airport. Since it was located within a natural flood plain, Sacramento's master plan had insisted that this area remain agricultural and not be subdivided for other uses. Developers wanted to open it for growth, and they had secured important backing from the county. In order to get the city to agree to re-zone the land for development, developers dangled the bait of a major sports stadium—something that appealed to a wide spectrum of Sacramentans. The issue of North Natomas figured largely in the 1983 mayoral campaign between Anne Rudin and florist Ross Relles. Rudin, who urged slow growth in the area, barely won against the less-experienced Relles. In 1985, developers Greg Lukenbill and Joseph and Richard Benvenuti forced the city's hand when they purchased a National Basketball Association team, the Kansas City Kings, and completed a 10,000 seat arena just east of the North Natomas land. Swept away by the public support for the Sacramento Kings and the prospect of professional sports in Sacramento, the city council ignored Rudin's objections and approved the re-zoning. A second arena holding over 18,000 fans was then built to accommodate the surging fans in the area.
In 1969, the rising tide of Latino voters elected Manuel Ferrales, shown here in October 1974, to the city council. (Photo by Rudolfo Cueller.)
Serna's turn for city governance came when Rudin turned down a third term in 1992. Easily besting a host of competitors, Serna, long a veteran of city politics, plunged into the task of governing the city, putting his brand of aggressive leadership to work. Although the mayor had no power over local schools, he nonetheless waded into the middle of contentious school district politics by pressing for improvements in the public schools, which had some of the worst drop-out rates in the state. Attempting to build on the success of the Kings, he worked aggressively to court other sports teams, seeking professional football and baseball franchises for the city. To attract businesses to relocate to Sacramento, he helped put together an expensive incentive package for the Packard Bell Company to take over the old Army Depot. Serna celebrated the increased cultural diversity of the city not only by his own ethnicity but by creating the popular Festival of the Family, replacing the more staid Camellia Festival that was to his mind a relic of the old Sacramento elite. Serna died in November 1999 and was succeeded by interim mayor Jimmy Yee. A spirited campaign waged in 2000 brought the election of Heather Fargo, Sacramento's third woman mayor. At the same time, Sacramento approved the creation of a full-time mayor's position.
The election of Anne Rudin and Heather Fargo also placed a spotlight on the evolving role of women in Sacramento life. In 1960, thanks largely to the air bases and the expansion of state government, Sacramento had one of the highest percentages of women in the workforce in California. A variety of women's clubs and organizations had always created a sphere of influence for Sacramento women, but in the latter twentieth century, the somewhat generic women's clubs gave way to professional organizations such as the League of Women Voters, the American Association of University Women, and various political clubs. Three very powerful women helped shape Sacramento culture in the 1950s and 1960s: Eleanor McClatchy, head of the Bee, Marion Armstrong, president of Weinstock's, and Lucy Ritter, vice president of Cal-Western Life Insurance Company. As the general consciousness of women's issues and status increased in the 1960s and 1970s, Sacramento women became more visible in "traditional" male bastions such as state and local government, fire fighting, and law enforcement, as well as in sports and the legal profession. In 2001, Sacramento ranked high in women-owned businesses. Political agitation by such groups as Grandmothers for Peace highlighted the role of women in a military city. Barriers to women's advance such as the "males only" policy of the Sutter Club came under fire. Politicians like Sandra Smoley, Muriel Johnson, Illa Collins, and Deborah Ortiz assumed prominent positions in local and state government, and gender-equity issues moved more prominently into Sacramento public life.
Eleanor McClatchy, president of McClatchy newspapers for more than 40 years, helped shape Sacramento culture in the 1950s and 1960s.
# THE DECLINE OF OLD INDUSTRIES AND BASE CLOSURES
Toward the end of the twentieth century, Sacramento's economy changed decisively. Earlier economic mainstays, the railroad and the canneries, went the way of all flesh. The rail yards had been waning since the 1930s. In 1937, the last locomotive rolled off the lines, and the rail shops turned to the maintenance of large diesel engines. The decline of the railroads had begun. Fewer and fewer people took the train, opting for airplane or automobile travel. In addition, the re-shifting of management and the relocation of facilities eventually spelled an end to the longtime service of the shops. The Western Pacific closed its shops on Sutterville Road about 1960. In 1979, Southern Pacific car repairs were transferred to the rail yards in Roseville. By 1981, the Sacramento Locomotive Works, as the shops had been retitled, began to shrink in size and labor force. The acquisition of the Southern Pacific by the Union Pacific temporarily staved off closure, but by 1999, the Sacramento yards were too old and the demands of centralized corporate management required change; the yards closed, leaving behind acres of highly toxic soil. Clean-up progresses today, and eventually the yards and the adjoining depot and storage areas may be the site of major new urban development.
Sacramento's canning industry, once an economic mainstay, began to decline as well. Sacramento's four major packing plants were hit by the twin blows of improved technology and decreasing consumer demand. Canning technology became more and more mechanized, requiring fewer workers. Likewise, the development of frozen foods seriously undercut the longtime popularity of canned goods. Consumer demand further declined when health-conscious consumers began to prefer fresh fruits and vegetables and shied away from preservation processes that added calories and robbed fruits and vegetables of their vitamins. Libby, McNeil & Libby closed their plant on C Street in 1980, followed by Del Monte in 1981. Even the huge Bercut-Richards plant on Seventh and Richards folded in the early 1980s. The only remnants of the once thriving food processing industry were the Campbell Soup plant on Franklin Boulevard and the Blue Diamond Almond Growers Cooperative off 16th Street.
But the biggest change in Sacramento's economy came with the closure of its three military bases that had been in the region since the 1930s and 1940s. The unsteady reliance on federal dollars had first been felt when Aerojet General, a private company depending heavily on federal contracts, began to experience substantial declines in the sale of propulsion systems for the Titan, Minuteman, and Polaris missiles in 1965. From its peak employment of 22,000, the aerospace giant began laying off employees until by 1972 only 9,539 worked at the plant. Military installations seemed safe during the Vietnam conflict. However, national pressures brought on by the growing peace movement and the wind-down of the Vietnam War during the Nixon administration brought the three Sacramento facilities under the microscope of federal budget-cutters. Already in the 1970s rumors had been floating that McClellan Air Force Base was slated to close. Mather Field appeared on a "hit list" of possible closures in 1987, but vigorous efforts by Congressmen Vic Fazio and Robert Matsui stalled the decision. But in 1988, Congress created the Base Realignment and Closure Commission (BRAC), an independent agency that evaluated the need for military bases and made recommendations to the secretary of defense. By late 1988, the commission had placed Mather on its list, and in 1993 the base closed permanently.
The end of the Cold War in 1989 created additional pressure for a "peace dividend" that included reduced defense expenditures. Secretary of Defense Richard Cheney proposed a series of base closures and consolidations, many of them targeting California bases and among them the Sacramento Army Depot with its 3,498 jobs. In June 1991, the commission voted to shutter the depot. Some of its jobs were transferred to other places. Some were permanently eliminated. With extensive city help, the buildings were made available to Packard Bell.
McClellan Field braced for the worst, and in 1993 its name appeared on the dreaded list. However, here the fight to save the field was more intense. Sacramentans attempted to work creatively with BRAC to build a case for McClellan's ongoing existence. Mayor Serna and Chamber of Commerce President Thomas Eres lobbied incessantly to keep the base open. But their efforts and those of area congressmen were in vain. On June 22, 1995, BRAC voted to close McClellan, and the base slowly faded, closing permanently in April 2001.
Government efforts to help during the transition were considerable. In addition to the standard option of job relocation, workers from McClellan and elsewhere were given training for new positions. The lands and buildings of the military bases were turned over to local government and to private industry for development. Mather retained the Veteran's Hospital and McClellan the base exchange (BX) for the benefit of the numerous military retirees in the area. Two thousand acres of Mather became a county-run airport used by air-cargo firms. Another 2,000 acres were transformed into a regional park. McClellan also transitioned much of its 8.5 million square feet of space to private sector purposes. County officials worked closely with private development and created McClellan Business Park, a mixed-use complex of office space and industrial facilities.
# HIGH TECH AND STATE GOVERNMENT
The departure of these three bases and the precipitous decline in the once strong aerospace industry dealt Sacramento a blow in her solar plexus. The ripple effect of the departures and job losses on schools, businesses, churches, and the security of community life created apprehension and tension. Fortunately, state government jobs remained stable and a new economic mainstay, the computer industry, appeared, helping to replace the bread and butter jobs that created Sacramento and continued the city's reputation for economic solidity.
Mayor Joseph A. Serna Jr. pressed hard to improve the city's public schools and worked aggresssively to attract new businesses to Sacramento.
The computer chip technology that had transformed the Silicon Valley of California in the 1970s found its way to the Sacramento metropolitan area in the spring of 1978 when Alan L. Seely, an executive for Hewlett Packard, opened a plant in Roseville. Hewlett Packard began operation in 1980 and soon became one of the top ten employers in the Sacramento region. Other computer-manufacturing firms came to Roseville, including Shugart Electronics, a Sunnyvale-based firm, and Japanese-owned NEC. Intel built a plant in Folsom, Apple in Laguna, and Oracle in Rocklin. The hi-tech revolution of the 1980s and 1990s brought scores of highly-trained and well-paid employees to the region as the demand for personal computers skyrocketed.
State government continued to grow. Even with conservative Republican governors at the helm, the number of people required to keep the state government functional escalated. Bright young professionals continued to move to the capital in the 1970s and 1980s—some of them attracted by the quirky liberalism of Governor Jerry Brown (1975–1983). These new state workers and reorganized agencies soon began to demand new office space and more up-to-date communications technology. New state office buildings for the Environmental Protection Agency, the Justice Department, and the Departments of Education and Health Services were built in the downtown, causing the town to buzz with life during the day. State workers also played a role in urging a revivification of the city's night life, dining options, and public entertainment. Likewise, scores of lobbyists anxious to press their causes set up shop in the growing number of capital office buildings. They too resided in city neighborhoods and pressed city leaders to make Sacramento a first-class capital city through the provision of social amenities such as theater, restaurants, and other forms of recreation.
Among the evolving venues of city life was the city's media establishment. The long domination of the Sacramento Bee continued even after the death of Eleanor McClatchy in 1980. When the Bee switched from its long-standing evening format to a morning edition, the rival Union's days were numbered. In 1994, the Union finally gave up the ghost. Already in 1978 Eleanor McClatchy's nephew Charles K. (also known as C.K. like his grandfather) had taken over as president of the family company. Young McClatchy made significant changes in the business, pulling the company out of its television and radio ventures and buying smaller newspapers in growing cities like Tacoma, Anchorage, and Minneapolis. In 1988, the McClatchy company went public and issued stocks. Young C.K. died unexpectedly in 1989 and Erwin Potts became the first non-family member to head McClatchy Company. In 1999, the company's revenues exceeded $1 billion.
# BRINGING THE SUBURBS TO THE CITY
New state workers and lobbyists anxious to live near their jobs and suburbanites anxious for a break from their cul-de-sacs and malls looked for ways to make Sacramento more socially appealing. With public and private initiatives, Sacramento took on a new look. In effect, as one developer noted, Sacramento transferred many of the appealing features of the suburbs to the city. Likewise, a renewed appreciation for the charm of the old city's architecture, its tree-shaded streets and its convenience helped to re-shape the city's appeal. Sacramentans rediscovered the charm of some its old buildings. Old Sacramento's transformation from skid row to a tourist area was sealed in 1976 with the building of the California State Railroad Museum in the old town. As well, popular sites like the docked Delta King riverboat provided dining and light theater entertainment.
Sacramento's most visible public building, the state capitol, also came in for major renovations. A 1972 seismic study revealed that the venerable old structure was unsafe. In 1976, a major renovation was undertaken, which rebuilt and retrofitted the capitol from the basement up, reusing original materials or carefully reproducing historical replicas. While the legislature met in temporary buildings to the rear of the capitol, the $67.7 million project proceeded ahead, finally opening to the public in 1982.
The destruction of the 1927 movie palace, the Alhambra Theater, and its replacement with a Safeway market, roused Sacramentans from their torpor. The theater tumbled to the wrecker's ball in 1972 after the public voted against a bond measure to preserve it. In reaction to that failure, many Sacramentans mobilized to preserve some of their past. In 1973, the Sacramento branch of the American Association of University Women issued Vanishing Victorians: A Guide to the Historic Homes of Sacramento. This popular history of Sacramento's elegant old Victorian homes played a role in awakening Sacramentans to their architectural heritage and arresting the destruction of historic city buildings. The Sacramento Old City Association was then formed to preserve the city's architectural heritage. This association and young government staffers who came to Sacramento during the term of Governor Jerry Brown took a new interest in rehabbing old homes and in snapping up inexpensive properties in midtown—an area that had been virtually redlined by insurance companies. Thanks in part to the increasing congestion on Sacramento freeways, the efforts of vocal neighborhood associations, and successful efforts to alter busy traffic patterns in residential neighborhoods all over Sacramento, a new interest in gentrification developed that revived the downtown area. Midtown properties, along with those in Land Park and Curtis Park, soon became premium locations.
A major renovation of the state capitol was undertaken in 1976.
In 1976, the downtown received a boost when a long-awaited Sacramento Community Center opened its doors. The huge complex sprawled south from J to L Street and included a major meeting hall, breakout rooms, and other amenities. Later additions included a theater and symphony hall. The community center created renewed interest in the renovation of K Street that had been discussed during the early phases of urban renewal. Despite the creation of a pedestrian mall, the once thriving commercial street and adjoining J Street had little foot traffic and businesses attracted little more than the noontime luncheon crowds. The offending "tank traps" were torn out and a new mall emerged. The centerpiece was an 18-mile modern light rail system completed in 1987, shuttling commuters from Sacramento's northern suburbs to the downtown. Subsequent stops moved southward with proposed plans to link the entire region by this highly popular form of mass transit. Around the downtown, two new hotels (a modern Hyatt Regency on the corner of 12th and L and a new Sheraton, using the facade of the Public Market on J Street and 12th, designed by architect Julia Morgan in the 1920s, as its lobby entrance) set the stage for making Sacramento a convention center.
The Downtown Plaza opened in October 1993, adding to the revitalization efforts.
Malls had become the defining points of suburban Sacramento and city developers sought to bring the more attractive aspects of the suburban malls downtown. In 1989, the Downtown Plaza Associates took a fresh look at the plight of the lonely Macy's and Weinstock stores that had been planted at the opposite end of K Street in 1960. Ernest Hahn of the Hahn Company of San Diego floated a bold plan by architect Louis Jerde to expand the shopping area at the foot of K Street by 250,000 square feet and to build an outdoor plaza with ample parking, food and theater concessions, and a medley of specialty shops designed to bring people in from the suburbs. The $157 million plan went forward and in October 1993 the Downtown Plaza opened with a gala bash. Predicted Sacramento councilwoman Deborah Ortiz, "If anything is going to turn around the Sacramento revenue base, this is it." Old Sacramento was connected to the Plaza Mall by a tunnel, which included a history wall that displayed the names and faces of many of Sacramento's movers and shakers from various eras.
Sacramento sprouted a skyline in the 1980s and 1990s. In addition to a host of new state government buildings, city developers helped to build an array of new office space for the growing downtown workforce and for cultural needs. The 28-story Renaissance Tower was built in 1989. Behind the city library an expansive Galleria named for local developer Angelo Tsakapoulous provided a venue for library-sponsored lectures and social events. An addition to the library included the establishment of a room devoted solely to Sacramento history. A much-needed refurbishing of the old city hall and a major extension of its facilities was commenced in 2003. Across from the city hall, Plaza Park was renamed in honor of labor activist Cesar Chavez and decorative banners, new sidewalks, and refurbished fountains made it a favored spot for transients, residents, and downtown employees.
Older buildings in the downtown underwent extensive renovation. In 1992, joint public and private efforts spared the Memorial Auditorium from demolition and underwrote a major restoration of the "Old Barn." Re-opened to great acclaim in November 1996, the modernized Memorial Auditorium continued to be a popular site for concerts, some indoor athletic events, graduations, and public ceremonies. In 2003, California governor Gray Davis held his inauguration ceremonies in the auditorium. Major renovations of the city's stately Cathedral of the Blessed Sacrament began in the summer of 2003. Around the cathedral, Sacramentans enjoyed a popular weekly street fair on Thursday nights on K Street from Memorial Day through the summer. Later, the venue was reduced to a more manageable outdoor market.
The beginning of the summer holidays were marked by another popular Sacramento event, the Jazz Jubilee. This important festival was spearheaded by a group of local musicians who formed the New Sacramento Traditional Jazz Society in 1968. The group won public visibility when they organized a jazz concert to support the rehabilitation of the Delta King, anchored along Front Street in Old Sacramento, in 1969. Nearly 4,000 turned out for the concert, and with good organization the group managed to put on a jazz festival in 1974, modeled after successful ones in Monterey and Newport, Rhode Island. Almost 3,000 showed up for this event held on Memorial Day weekend, and in every subsequent year the numbers attending grew, as did the number of bands. Originally the festival highlighted Dixieland, but by 1992 a wide array of musical styles (Gospel, Latin, Zydeco, Blues, and even Barbershop) were added to the popular medley of performers. Nearly 100,000 attended the festival in 1992.
Movie going picked up again in the 1970s and 1980s, spearheaded by a number of exciting feature films that found new fans. The new customers found that films in a theater had significant advantage over television or even the later popular video rental stores. In 1967, the dome-shaped Century Theaters opened near Arden Fair Mall. The comfortable seating and the tiered rows provided comfort and an unobstructed view for moviegoers. A new downtown theater attracting much attention was the big-screen technology called IMAX that became the chief venue of the newly reconditioned Esquire Theater on K Street. Using the marquee of the old theater that had long since closed, the Esquire combined the nostalgic memory of old Sacramento with the technology of the new.
One development that kept Sacramentans coming downtown was the popular Music Circus. The familiar Music Circus tent had undergone a major expansion and improvement for the 1968–1969 season. However, it could still be a hot and sticky experience, especially for the entertainers who worked under the lights, as well as for unlucky patrons who occasionally found themselves sitting near the huge tent poles that obstructed their view of the stage. After battles with its neighbors, Music Circus leaders managed to secure permission for a new parking structure and tent facility. Sponsored by Wells Fargo, the Music Circus's new pavilion premiered in the summer of 2003 and included comfortable seating, better air conditioning, and more rest rooms to remove whatever discomfort may have been a part of this now familiar Sacramento institution. The demand for Broadway productions apart from the summer season was met with a popular Broadway Series begun by the Light Opera Association at the Community Center theater in 1989.
# SPORTS
Nothing brought people back to Sacramento more successfully than professional sports. Far and away the most popular cultural expansion in Sacramento during this period was the addition of professional sports in the form of a National Basketball Association team.
Although interconnected with the larger issue of the development of North Natomas, the advent of the popular Sacramento Kings is a story in its own right. In 1979, a young 24-year-old local contractor, Greg Lukenbill, teamed with developers Joseph and Richard Benvenuti to propose the building of a 32,500-seat multipurpose stadium and 20,000-seat sports complex in North Natomas. Thwarted for a time by the city's unwillingness to re-zone the area, Luckenbill persevered, formed the Sacramento Sports Association, and in April 1985 purchased the Kansas City Kings. The advent of the Kings launched the building of a new sports arena underwritten by the Atlantic Richfield Company and called ARCO Arena. Lukenbill eventually sold the franchise to Jim Thomas in 1992, and then in January of 1998, the Las Vegas–based Maloof family acquired partial ownership of the team, which then lead to their controlling interest in the team and ARCO Arena the following year. Under the ownership of the Maloof family, the leadership of General Manager Geoff Petrie, and the coaching of Rick Adelman, who joined the club in the 1998–1999 season, the Kings soon emerged as a major NBA play-off team, moving forward to the Conference finals in 2002. Sacramentans thronged Kings games. Kings sports apparel, the hero worship of Kings players, and the economic boost and prestige of the popular team made the Kings one of the defining elements of urban life.
In April 1996, a women's team, the Sacramento Monarchs, premiered at the ARCO as part of the Women's National Basketball Association. Other efforts to recruit professional teams to Sacramento were endorsed by Sacramento mayor Joe Serna. Serna hoped to attract the NFL Oakland Raiders to Sacramento. There was also some interest expressed in relocating the Pittsburgh Pirates, owned by Kevin McClatchy, a member of the prominent Sacramento Bee family. More realistic voices, however, urged the return of Triple A baseball to Sacramento and the construction of a new park to host games. Arguments between Mayor Serna and local developers over the location of the park ensued, and the city pulled out of the process when a group of local business leaders decided to build the park in West Sacramento rather than on the old rail yard grounds that Serna had pushed. With a generous $8 million from the Raley Company, a new field (Raley Field) opened in April 2000, and a new team (the former Vancouver Canadians), purchased from Japan Sports Systems and renamed the Sacramento River Cats, stepped up to the plate.
Sacramento's Jazz Jubilee, shown here in May 1975, takes place on Memorial Day weekend and includes a wide array of musical styles.
# THE POOR IN A CHANGING SACRAMENTO
Amid the glitter and glamour of the new service and entertainment industries in the downtown, the reality of urban poverty also began to register an imprint on Sacramento. First the gold rush and later the presence of the railroad and the soft winter climate of California brought unemployed transients to the city. However, the demolition of the West End with its cheap flophouses and its day labor market as well as the cutbacks in care for the mentally ill during the administration of Governor Ronald Reagan turned loose a new crop of homeless men, women, and families that roamed the streets of the city, sleeping in alleys, in front of churches, under bridges, and along the riverbank. Efforts to feed the hungry have been a special mission of the religious communities of the area. Private charities such as the Union Rescue Mission and the Salvation Army took the lead in feeding the homeless for many years. In more recent times, the Episcopal and Roman Catholic Churches have taken a more active role in providing food, shelter, and care for the poor. Along 12th Street, a homeless feeding and care center begun by Dan and Chris Delany known as Loaves & Fishes was established by a number of religious groups and has expanded substantially since its fairly modest beginnings in the late 1970s.
Sacramento, like any other American city, is still a work in progress. The city continues to improve and expand its urban space; its neighborhoods and still manageable public ambience attract many who are weary of life in the overdeveloped Bay Area and Silicon Valley. Sacramento's reputation for diversity is its proudest boast to this day, as it continues to encourage its multicultural population to work together for the common good. The city above all has never lost any of its "indomitable" spirit. The same forces that allowed the city to defy nature and rise as a major social, economic, and cultural center near two fast rivers still live to define another generation.
# BIBLIOGRAPHY
Avella, Steven M. A History of the Diocese of Sacramento. Private publication, 2002.
B'Nai Israel, 150. Sacramento: The Congregation, 1999.
Burns, John, et al. Sacramento: Gold Rush Legacy, Metropolitan Destiny. Carlsbad: Heritage Media Corporation, 1999.
Bruno, Lloyd. Old River Town: A Personal History of Sacramento. Dunsmuir: Suttertown Publishing, 1996.
Cole, Cheryl. A History of the Japanese Community in Sacramento, 1993–1972. San Francisco: R and E Research Associates, 1974.
Connolly, Elaine and Dian Self. Capital Women: An Interpretative History of Women in Sacramento, 1850–1920. Sacramento: Capital Women's History Project, 1995.
Comstock, Timothy F. The Sutter Club: One Hundred Years. Sacramento: Sutter Club, 1989.
Craft, George S. California State University Sacramento: The First Forty Years: 1947–1987. Sacramento: The Hornet Foundation, 1987.
Davis, Winfield J. An Illustrated History of Sacramento County, California. Chicago: Lewis Publishing, 1890.
Dillon, Richard. Captain John Sutter: Sacramento Valley's Sainted Sinner. Santa Cruz: Western Tanager Press, 1967 and 1981.
Eifler, Mark A. Gold Rush Captialists: Greed and Growth in Sacramento. Albuquerque: University of New Mexico Press, 2002.
Flink, Andrew. A Century of Cinema in Sacramento, 1900–2000. Rancho Cordova, 1999.
Hurtado, Albert. Indian Survival on the California Frontier. New Haven: Yale University Press, 1988.
Kelley, Robert. Battling the Inland Sea: Floods, Public Policy and the Sacramento Valley. Berkeley: University of California Press, 1989.
Kibbey, Mead, ed. 1853–1854 Sacramento Directory. Sacramento: California State Library Foundation, 1997.
———, ed. J. Horace Culver's Sacramento City Directory for the Year 1851. Sacramento: California State Library Foundation, 2000.
Kleinschmidt, Bruce, et al. 150 Years of Faith: The Stories of Three Trailblazing Sacramento Churches, 1849–1999. Sacramento: Tri-Church Sesquicentennial Committee, 1999.
Leland, Dorothy Kupcha. A Short History of Sacramento. San Francisco: Lexikon, 1989.
Lord, Myrtle Shaw. A Sacramento Saga. Sacramento: Sacramento Chamber of Commerce, 1946.
Maeda, Wayne. Changing Dreams and Treasured Memories: A Story of Japanese Americans in the Sacramento Region. Sacramento: Sacramento Japanese Citizens League, 2000.
McGowan Joseph A. and Terry Willis. Sacramento: Heart of the Golden State. Woodland Hills: Windsor Publications, 1983.
McGowan Joseph, A. History of the Sacramento Valley. 3 vols. New York and West Palm Beach: Lewis Historical Publishing, 1961.
Mims, Julie Elizabeth and Kevin Michael Mims. Sacramento: A Pictorial History of California's Capital. Virginia Beach: The Donning Company, 1981.
Morse, John F. The First History of Sacramento City. Sacramento: Sacramento Book Collectors Club, 1945.
Neasham, V. Aubrey and James E. Henley. The City of the Plain. Sacramento: Sacramento Pioneer Foundation/Sacramento Historic Landmarks Commission, 1969.
Ottley, Alan, ed. The Sutter Family and the Origins of Gold Rush Sacramento. Norman: University of Oklahoma Press, 2002.
Owens, Kenneth N., ed. John Sutter and a Wider West. Lincoln and London: University of Nebraska Press, 1994.
Pierini, Bruce. The Italians of Sacramento. Sacramento: Sacramento County Historical Society, 1997.
Reed, Walter G., ed. History of Sacramento County. Los Angeles: Historic Record Company, 1923.
Rogers, Richard C. The First 100 Years of Sacramento City Schools, 1854–1954. Sacramento: California Retired Teachers Association/Sacramento Branch, 1991.
Rodgerson, Eleanor. Adobe, Brick and Steel: A History of Hospitals and Shelters for the Sick in Sacramento and El Dorado Counties. Sacramento: Sacramento–El Dorado County Medical Society, 1993.
Sacramento Guide Book. Sacramento Bee, 1939.
Severson, Thor. Sacramento: An Illustrated History: 1839–1874. Fifth ed. San Francisco: California Historical Society, 1973.
Smith, Jesse M., ed. Sketches of Old Sacramento. Sacramento: Sacramento County Historical Society, 1976.
Sutherland, Ruth Ward. "... for the people": The Story of SMUD. Sacramento: Sacramento Municipal Utilities District, 1973.
Willis, William L. History of Sacramento County. Los Angeles: Historic Record Company, 1913.
Wilson, Burt. A History of Sacramento Jazz, 1948–1966. Canoga Park: Burt Wilson, 1986.
Woolridge, Jesse W. History of the Sacramento Valley. Chicago: Pioneer Historical Publishing, 1931.
Wright, George, ed. History of Sacramento County. Berkeley: Howell-North, 1960.
150 Years of the Chinese Presence in California. Sacramento: Sacramento Chinese Culture Foundation, 2001.
# INDEX
Aerojet
African Americans
Alvarado, Juan Bautista
American River
ARCO Arena
Army Signal Depot
Benton, Joseph Augustine
Big Four
Bigelow, Hardin
Brannan, Samuel
Burnett, Peter H.
canneries
Carmichael
Cathedral of the Blessed Sacrament
Cavanaugh, Bartley
Central Pacific
Central Valley
Chinese
Clunie Theater
Colley, Nathaniel
Coloma
Community Chest
Cosumnes River
County Hospital
Croatians
Crocker, Charles
Crocker, Margaret
dance house
Dean, James S.
Depression
Dudley, Arthur Serviss
electricity
Elks Temple
Fair Oaks
Feather River
floods
Florin
Folsom
food-gatherers
Fort Ross
Fort Vancouver
Gallatin, Albert
gold rush
Hastings, Lansford B.
healthcare
Hensley, Samuel
Hopkins, Mark
Horse Market
Hudson Bay Company
Huntington, Collis P.
Irish
Italians
Japanese
Johnson, Hiram F.
Judah, Theodore
Judge, Mary
Ku Klux Klan
Land Park
Lubin, David
Lukenbill, Greg
Manogue, Patrick
Marshall, James Wilson
Marysville
Mather Field
McClatchy, C.K.
McClatchy, James
McClellan Field
Memorial Auditorium
Mexicans
miners
Miwok
Moraga, Gabriel
Moreland, William Hall
Morse, John
Moss, John
Music Circus
Muslims
New Helvetia
Nisenan
Oak Park
Old Sacramento
Orangevale
Portuguese
Progressive movement
Prohibition
Public Market
Pullman Strike
radio
Rancho Cordova
Rancho del Paso
River Park
Robinson, Charles
Roosevelt, Franklin D.
Roseville
Rudin, Anne
Sacramento Bee (and Evening Bee)
Sacramento River
Sacramento Union
schools
Seavey, Clyde L.
Serna, Joseph A., Jr.
Sisters of Mercy
Smith, Jedidiah
Southern Pacific
squatters
Stanford, Leland
streetcars
Sutter, John (Johann) Augustus
Sutter, John, Jr.
Sutter's Fort
Sutterville
telegraph
television
temperance
Town & Country Shopping Center
Urbs Indomita
Vietnamese
Vioget, Jean Jacques
waterworks
Weinstock, Harris
Weinstock, Lubin & Co.
West End
Western Pacific
White, Clinton
women's suffrage
World War I,
World War II,
Yuba River
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Со́сенка — топоним и фамилия:
Сосенка, Ондржей (род. 1975) — чешский велогонщик.
Населённые пункты
Белоруссия
Сосенка — деревня в Костеневичском сельсовете Вилейского района Минской области.
Сосенка — деревня в Засульском сельсовете Столбцовского района Минской области.
Россия
Сосенка — деревня в Козельском районе Калужской области.
Сосенка — деревня в Шимском районе Новгородской области.
Сосенка — деревня в Борском районе Самарской области.
Водные объекты
Реки
Сосенка (приток Десны) — в Москве.
Сосенка (приток Мечи) — в Рязанской Московской областях.
Сосенка (приток Хапиловки) — в Москве и Московской области.
Сосенка (приток Шелони) — в Шимском районе Новгородской области.
Озёра
Сосенка (озеро) — озеро в Ушачском районе Витебской области Белоруссии.
См. также
Сосенки (значения)
Сосна (значения) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,914 |
\section{Design Tasks}
\label{sec:design-tasks}
We formulated a set of design tasks across three different physical domains with high-dimensional state spaces and complex dynamics. Each domain uses a different ground-truth simulator $f_S$, which is used to evaluate designs and pre-train the learned simulator model $f_M$.
\subsection{\twodfluids{}}
\label{sec:2d-fluid-design}
Inspired by existing 2D physical reasoning benchmarks \citep{allen2019tools,bakhtin2019phyre}, these tasks involve creating one or more 2D ``tool'' shapes to direct fluid into a particular goal region (\autoref{fig:2dfluids_results}, \autoref{app:2dfluids}).
We consider three tasks with 4--48 design parameters where the goal is to guide the fluid such that each particle comes as close as possible to the center of a randomly-sampled yellow reward region.
In \contain{}, the joint angles $\phi_\mathrm{joints}$ of a multi-segment tool must be optimized to catch the fluid by creating cup- or spoon-like shapes.
In \ramp{}, the joint angles $\phi_\mathrm{joints}$ of a multi-segment tool must be optimized to guide the fluid to a distant location.
In \maze{}, the rotation $\phi_\mathrm{rot}$ of multiple tools must be optimized to funnel the fluid to the target location.
Fluid dynamics are represented using $10^2$--$10^3$ particles, and unrolled for up to 300 time steps.
The learned model $f_M$ is trained on the 2D WaterRamps dataset released by \citet{sanchez2020learning} which uses the solver in \citet{hu2018moving} to generate trajectories.
The dataset contains scenes with 1--4 straight line segments; no curved lines or large numbers of obstacles are shown.
Therefore, designs which solve \twodfluids{} tasks are necessarily far out of distribution for the learned model.
\subsection{\threedfluids{}}
\label{sec:3d-fluid-design}
To evaluate higher-dimensional inverse design, we created a ``landscaping'' task that requires optimizing a 3D surface to guide fluids into different areas of an environment (\autoref{fig:3dfluids}, \autoref{app:3dfluids}).
Specifically, water flows out of a pipe and onto an obstacle parameterized by $\phi_\mathrm{map}$, a $25 \times 25$ heightmap (625 design parameters).
In \direction{}, $\phi_\mathrm{map}$ is optimized to redirect the fluid stream towards a specified direction.
In \twopools{} and \threepools{}, $\phi_\mathrm{map}$ is optimized to split the fluid stream such that particles hitting the floor land as close as possible to one of two or three specified pools.
Each task has multiple variants, such as different target directions in \direction{}.
The simulation is unrolled for 50 time steps, and contains up to 2048 particles.
The learned model $f_M$ is trained on data generated from the simulator in \citet{bender2015divergence}.
\begin{figure*}[h!]
\centering
\includegraphics[width=0.24\textwidth]{figures/init_airfoil_mesh.png}%
\includegraphics[width=0.24\textwidth]{figures/airfoil_stack.png}%
\includegraphics[width=0.24\textwidth]{figures/airfoil_stats_drag.pdf}%
\includegraphics[width=0.24\textwidth]{figures/airfoil_stats_reward.pdf}
\vspace{-0.25em}
\caption{\airfoil{} results. (a) An initial airfoil design is warped by moving 10 control points (orange dots). The physics model simulates the resulting aerodynamics on a 4158 node mesh, based on which lift and drag are computed. (b) For the task of finding a minimum-drag configuration under constant lift constraint, gradient-based learned design is able to find similar designs to specialized solver \dafoam{}, both using single models and ensembles. (c-d) Larger ensemble sizes of 3--5 achieve a close quantitative match to \dafoam{} for both drag and overall reward.
Shown are means over 10 randomized initial designs, with bootstrapped 95\% confidence intervals.}
\vspace{-0.5em}
\label{fig:airfoil}
\end{figure*}
\subsection{\airfoil{}}
\label{sec:airfoil}
Shape optimization in aerodynamics is one area where gradient-based optimization is routinely applied using traditional simulators \citep{buckley2010airfoil}.
Here, we consider the well-studied task of drag optimization of a 2D airfoil profile (\autoref{fig:airfoil}, \autoref{app:airfoil}).
In this task, a wing is defined using a curve on a 2D mesh, which can be deformed using a set of 10 control points $\phi_\textrm{ctrl}$.
The reward function is formulated as optimizing the wing shape to minimize drag under certain constraints such as constant lift and bounds on the shape to prevent degenerate (e.g. infinitely thin) configurations.
Lift and drag coefficients are computed by running an aerodynamics simulation on a 4158 node mesh.
The learned model $f_M$ is trained on data generated from the ground-truth simulator $f_S$, for which we use the OpenFOAM solver \cite{openfoam}.
\subsection{Model learning and optimization}
While each domain has a different state space structure and uses a different ground-truth simulator $f_S$, the learned simulators $f_M$ all share the same architecture (with identical hyperparameters in \twodfluids{} and \threedfluids{}, and only minor variations of the hyperparameters for \airfoil{} due to it being a steady state simulation; see \autoref{app:model}).
The models are trained for next-step prediction on task-independent datasets (random perturbations of the design space for \airfoil{} and \threedfluids{}, and an open-source, qualitatively distinct dataset for \twodfluids{}), and are unrolled for up to 300 time steps during design optimization without further fine-tuning (see \autoref{sec:optimization} for details).
\section{Background}
\label{sec:background}
Solving inverse problems with physical simulators has a long history in science and engineering, spanning data assimilation for weather modeling \citep{navon2009data}, system identification in robotics \citep{guevara2017adaptable, seita_fabrics_2020}, and tomographic and geophysical imaging \citep{cui2016review, pascual1999review}.
Inverse design can be framed as an inverse problem in which the objective is to optimize design parameters to produce some desired target property.
Simulation-based inverse design has been studied in a variety of disciplines, including nanophotonics \citep{molesky2018inverse}, material science \citep{Dijkstra2021predictive}, mechanical design \cite{coros2013computational}, and aerodynamics \cite{anderson1999aerodynamic, Rhie_1983}.
Classical numerical solvers used for inverse design can be highly accurate, but are often inefficient (making sampling-based inference methods infeasible for high-dimensional designs) and domain-specialized ( prohibiting general-purpose inverse design across domains \citep{choi2021use}).
Differentiable simulators \citep{freeman2021brax,hu2019difftaichi,Schenck2018SPNetsDF} have recently garnered attention, as they allow for more sample-efficient gradient-based optimization.
However, like classical solvers, they are still typically narrow in application scope, as many simulation techniques are hard to express as a differentiable program (e.g. constraint dynamics and multiphysics coupling).
The last few years have seen increased interest in using machine learning to accelerate inverse design across a variety of applications \citep[e.g.][]{Challapalli2021inverse,Christensen2020predictive,Bombarelli2018Automatic,Forte2022Inverse,
Hoyer2019neural,Kumar2020Inverse,Li2020efficient,Liu2018Generative,Sha2021Machine,ZHENG2021113894}.
These methods can provide impressive speedups over classical approaches by using learned generative models to propose designs (thus calling an expensive simulator fewer times), or by learning a scoring function which maps designs to target values (replacing the simulator entirely). However, they are based on components with limited out-of-domain generalization,
restricting new designs to configurations near the training data or requiring model refinement.
We instead propose to replace classical simulators with learned simulators.
Like learned scoring functions, learned simulators can be faster than classical simulators \citep{kochkov2021machine,stachenfeld2021learned} and are differentiable when parameterized as a neural network.
Furthermore, learned simulators mimic the underlying physical dynamics independent of the design task and are therefore more likely to generalize.
Physics simulators have been successfully implemented as learned, differentiable models of complex dynamics such as fluids, rigid-body interactions, and soft-body systems \citep{bhatnagar2019prediction,li2020fourier,mrowca2018,Rudy2017data,thuerey2020deep,ummenhofer2020lagrangian,wang2020physicsinformed}.
Graph-based models in particular are promising candidates for design problems, having demonstrated high accuracy, stability, efficiency, and generalization performance \citep{belbute2020combining, pfaff2021learning,sanchez2020learning}.
However, high-quality forward models do not necessarily translate into better downstream task performance \cite{hamrick2020role,lutter2021learning}. While learned simulators have been used successfully for planning and control in low-dimensional state spaces \citep[up to 21 degrees of freedom (DOF), e.g.][]{bharadhwaj2020model,sanchez2018graph,wang2019benchmarking} or more complex domains with small action spaces \citep[6 DOF, e.g.][]{li2018learning},
they require replanning at every timestep to avoid error accumulation. It is not known whether learned simulators can support gradient-based, high-dimensional inverse design which demands high accuracy, well-behaved gradients, long-term rollout stability, and generalization beyond the training data.
Here we study inverse design in non-rigid, graph-based physical systems with up to 625 design dimensions and 2000 state dimensions, over 50--300 timesteps, without replanning after the design period. We show that using gradient-based optimization with learned, general-purpose simulators is an effective choice for inverse design.
\section{Discussion}
We used state-of-the-art learned, differentiable physics simulators with gradient-based optimization to solve challenging inverse design problems.
Across three domains and seven tasks, which involved designing landscapes and tools to control water flows or optimizing the shape of an airfoil, we demonstrated that gradient descent with pre-trained simulators can discover high-quality designs that match or exceed the quality of those found using alternative methods.
This approach succeeds in a variety of interesting and surprising ways:
it permits gradient backpropagation through complex physical trajectories for hundreds of steps; scales to tasks with large design and state spaces (100s and 1000s of dimensions, respectively); and successfully generates designs which require the learned simulator to generalize far beyond its training data.
In the classic aerodynamics problem of airfoil shape optimization, our approach produces a design comparable to that of a specialized solver using only simple, general-purpose strategies like model ensembling.
While our results have exciting implications for inverse design, they also open up possibilities for explaining everyday human behavior like tool invention---a longstanding puzzle in cognitive science \citep{allen2019tools,osiurak2016,Shumaker2011}.
With general-purpose learned simulators, we have the potential not just to create highly specialized tools in engineering domains, but also to model everyday tool creation, such as creating a hook from a pipe cleaner, building a blanket fort, or folding a paper boat.
Our approach has limitations that should be visited in future work.
Gradient descent is inappropriate for design spaces with regions of zero gradients, such as in \twodfluids{} tasks where the fluid may not always make contact with the tool (\autoref{fig:mpm_zero_grad}).
Many interesting design tasks also have variably-sized or combinatorial design spaces that cannot easily be optimized with gradient descent, such as computer-aided design (CAD) approaches to 3D modeling.
An exciting future direction will be to integrate general-purpose learned simulators with hybrid optimization techniques such as those used in material science and robotics \citep{chen2020generative,toussaint2018differentiable}.
As learned simulators continue to improve, we could also use them to do even broader cross-domain, multi-physics design.
While challenges remain, our results represent a promising step towards faster and more general-purpose inverse design.
\section{Results}
\label{sec:results}
Our results show that learned simulators can be used to effectively optimize various designs despite significant domain shift and long rollout lengths.
The same underlying model architecture is used for each domain, highlighting the generality of learned simulators for design.
Examples of designs found with our approach are available at: \url{https://sites.google.com/view/optimizing-designs}.
Performance is always evaluated using the ground-truth simulator $f_S$ (see \autoref{sec:optimization}).
Here we discuss these results, and compare the capabilities of gradient descent with learned simulators over classical simulators and sampling-based optimization techniques.
\subsection{Overall results}
\label{sec:overall-results}
We first asked whether a learned simulator combined with gradient descent (GD-M) could produce good-quality designs at all.
This approach might fail in various ways: accumulating model error, vanishing or exploding gradients \citep{bengio1994learning}, or domain shift \citep{hamrick2020role}.
However, as the following results show, GD-M produced high-quality designs across all three domains.
\autoref{fig:2dfluids_results}a shows qualitative results for \twodfluids{} (\autoref{sec:2d-fluid-design}), where our approach (GD-M) produces intuitive, functional designs to contain (\contain{}), transport (\ramp{}), or funnel (\maze{}) the fluid to a target location.
On average, GD-M outperforms CEM-M by 16.1--118.9\%, indicating a substantial benefit of gradient-based optimization.
GD-M also outperforms CEM-S by 3.9--37.5\%, despite using a learned simulator rather than the ground-truth.
However, these design spaces are still relatively small (between 16 and 36 dimensions).
In \threedfluids{} (\autoref{sec:3d-fluid-design}), we substantially increase the dimensionality to a 625-dimensional landscape.
Here, GD-M produces robust designs, creating ridges to re-route water in particular directions or valleys to direct water into pools (\autoref{fig:3dfluids}a-c).
In comparison, CEM-M cannot solve any of these tasks, with performance 30--85$\times$ worse than GD-M.
In \airfoil{} (\autoref{fig:airfoil}b), GD-M recovers the characteristic S-curve shape for a low-drag airfoil under a small angle of attack, and matches the design obtained with \dafoam{}, an adjoint aerodynamics solver which computes close-to-optimal designs for this task.
Specifically, the design obtained with \dafoam{} yields a drag coefficient of 0.01902, while GD-M finds designs with drag between 0.01898--0.01919 depending on ensemble size (see \autoref{sec:ensembles}).
Importantly, \dafoam{}'s solver and optimizer are highly specialized for the particular task of airfoil design, while our approach is more general-purpose in that it requires only trajectory data for training and a generic gradient-based optimizer.
\begin{figure*}[!t]
\centering
\includegraphics[width=\textwidth]{figures/episode_joints_ablations_wmaze_norm_v2.pdf}
\vspace{-1.6em}
\caption{
Ablation experiments on the \contain{} and \maze{} tasks.
(a) Performance of all optimizers increase with rollout length; GD-M performance starts to deteriorate around step 225. (b) In \contain{}, CEM performance drops when increasing the number of joints above 24, while GD-M remains stable. (c) We observe a similar trend with the number of tools in \maze{}. (d) CEM often gets stuck in sub-optimal solutions early in optimization, while GD-M performance continues to increase.}
\vspace{-0.5em}
\label{fig:2dfluids_ablations}
\end{figure*}
\subsection{Model stability \& gradient quality}
We investigated accuracy over long timescales by measuring the effect of rollout length on design quality in \twodfluids{} (\autoref{fig:2dfluids_ablations}a and \ref{fig:hg_episode_sweep_demos}).
Longer rollouts can in principle allow for higher reward in this task as they give the fluid time to settle; however, with learned models, they can also be unstable due to error accumulation \citep{talvitie2014model,venkatraman2015improving}.
Nevertheless, we find that the learned simulator does not seem to be severely impacted by this problem.
Specifically, the quality of designs found by GD-M increases up to 225 steps (\autoref{fig:2dfluids_ablations}a), indicating that the learned simulator's accuracy and gradients remain stable for a surprisingly long time.
Across the episode lengths evaluated, we find that GD-M outperforms not only CEM-M (by 18.1\% on average) but also CEM-S (by 4.4\% on average).
This indicates that the benefits of a having a learned model that supports better optimization techniques can outweigh the error incurred by long rollouts.
The strong performance of GD-M on longer rollout lengths is noteworthy.
Gradients tend to degrade when passed through chains of many model evaluations, and as a result, previous work generally only optimizes gradients in small action spaces over just a few time-steps \cite{li2018learning}.
We speculate that one reason for the success of GD-M is the addition of noise in training $f_M$, which promotes stability on the forward pass and may also force smoother gradients.
\subsection{Generalization}
Deep networks often struggle to generalize far from their training data \cite{geirhos2018generalisation}.
This poses a problem for design: to produce in-distribution training data, we would already need to know what good designs look like, thus defeating the aim of wanting to find \emph{new} designs.
However, we find that the GNN-based simulators studied here overcome this issue.
As noted in \autoref{sec:2d-fluid-design}, the learned simulator for \twodfluids{} was trained on a pre-existing, highly simplified dataset where only one to four straight line segments interact with a fluid
(\autoref{app:domains}).
In contrast, the design tasks studied here involve highly articulated, curved obstacles (\contain{}, \ramp{}) or a larger number of obstacles (\maze{}); yet, GD-M still discovers effective designs without requiring any finetuning (\autoref{fig:2dfluids_results}).
We suspect this is because the model is trained to learn local collision rules, making it more robust to global distribution shift.
Using a learned simulator trained in a relatively simple environment has another unexpected advantage.
In rare cases, classical simulators suffer from degenerate behavior around certain edge cases.
For example, with the classical simulator for \twodfluids{}, particles can get stuck in between joint segments, especially when there are a large number of parts or joints (\autoref{fig:mpm_instability}).
However, since the learned simulator was trained on simpler data where these effects are unobserved, it picks up only on the appropriate collision performance and not the unrealistic edge cases.
Thus, the learned simulator produces more plausible rollouts than the classical simulator in these cases, and might therefore be a better candidate for producing designs that would transfer to the real world.
\subsection{Improving accuracy with ensembles}
\label{sec:ensembles}
In engineering tasks like \airfoil{}, simulators must be especially accurate, as small differences in the predicted pressure field can cause large errors in lift and drag coefficients.
While GD-M (without ensembling) can produce designs close to \dafoam{}'s, we notice a slightly rounder wing front (\autoref{fig:airfoil}b, bottom) causing a small increase in drag (0.01919 versus 0.01902 in \dafoam{}).
To further improve performance, we implemented an ensemble of
learned simulators trained on separate splits of the training set.
Ensembles are a popular choice for training transition models for use in control \citep{chua2018deep}, as they can provide higher quality predictions and are more resistant to delusions---a particularly problematic issue for accuracy-sensitive domains such as airfoil design.
During optimization, we make predictions with all models in the ensemble, each trained on a different data split, and average the gradients.
As shown in \autoref{fig:airfoil}b-c, larger ensembles yield designs with significantly lower drag ($\beta = -5.3\times 10^{-5}$, $p = 0.0003$, where $\beta$ is a linear regression coefficient) and higher overall reward ($\beta = 5.6\times 10^{-5}$, $p = 0.0001$), and are able to produce designs very close to the solution found by \dafoam{}, with a drag coefficient of 0.01898 (size-5 ensemble).
Thus, with ensembles, we are able to achieve performant designs with a general-purpose learned simulator, indicating that we can use learned models for design optimization in spaces traditionally reserved for specialized solvers like \dafoam{}.
\subsection{Scalability to larger design spaces}
\label{sec:design-dimensionality}
In larger design spaces, sampling-based optimization procedures quickly become intractable, especially with relatively slow simulators.
We hypothesized that gradient descent with fast, learned simulators could overcome this issue, especially as the size of the design space is increased.
We therefore compared different optimizers on \twodfluids{} as a function of the dimensionality of the design space (the number of tool joints in \contain{} or the number of tools in \maze{}) and on the higher-dimensional \threedfluids{}.
For \contain{} (\autoref{fig:2dfluids_ablations}b), the performance of GD-M increases with the number of joints as increasingly fine grained solutions are made possible.
In contrast, for both CEM-M and CEM-S, design quality deteriorates with more joints as high quality solutions become harder to find with random sampling. In the highest dimensional \contain{} task with 48 tool joints, GD-M outperforms CEM-M by $154.9\%$ and CEM-S by $126.5\%$.
When CEM does find solutions (\autoref{fig:2dfluids_results}b), they lack global coherence and appear more jagged than solutions found with GD-M.
Similarly, for \maze{} (\autoref{fig:2dfluids_ablations}c), the performance of GD-M is largely unaffected by the number of tools, while the performance of CEM-M and CEM-S both degrade as the design space grows. For the highest dimensional \maze{} problem with 36 joints, GD-M outperforms CEM-M by $207.4\%$ and CEM-S by $133.7\%$.
In the 625-dimensional \threedfluids{} task, CEM-M performs 30--85$\times$ worse than GD-M (\autoref{fig:3dfluids}) despite extensive hyperparameter tuning. This trend held across all tasks (\autoref{fig:accuracy_comparison}a), and even persisted when using fewer control points in the design space (\autoref{fig:design_space_3d}).
This is due not only to \threedfluids{}'s larger design space, but also because this problem requires a globally coherent solution: modifying small areas independently is unlikely to have much effect on the global movement of the fluid.
\subsection{Model speed and sample efficiency}
Learned simulators can provide large speedups over traditional simulators in certain domains by learning to compensate for coarser sub-stepping and making optimal use of hardware acceleration.
In \airfoil{}, although we use a very simple GD setup, our approach is able to find very similar designs as \dafoam{}'s specialized optimizer.
Moreover, our approach requires only 21s (single model) to 62s (size-5 ensemble) on a single A100 GPU, compared to 1021s for \dafoam{} run on an 8-core workstation, despite requiring $10\times$ more optimization steps.
In \twodfluids{}, the ground-truth simulator runs at a similar speed to the learned model \citep[see][]{sanchez2020learning}, but is non-differentiable and therefore depends on more expensive gradient-free optimization techniques which require more function evaluations.
We use 20--40 function evaluations per optimization step of CEM, compared to a single evaluation with GD (which is about $3\times$ more costly, due to the gradient computation). Thus, GD with a differentiable learned model can be much more efficient than using the ground truth simulator with a sampling-based method.
\section{Introduction}
Humans are creators.
Our ancestors created stone tools which led to innovations in hunting and food consumption, aqueducts and irrigation systems which revolutionized farming and urban habitation, and more recently, airplanes which let us cross the globe in hours.
Automatically designing objects to exhibit a desired property---often referred to as \emph{inverse design}---promises to transform science and engineering, including aerodynamics \cite{eppler2012airfoil}, material design \cite{butler2016computational}, optics \citep{colburn2021inverse}, and robotics \cite{Gupta2021, xu2021diffsim}.
Despite its promise, widespread practice of inverse design has been limited by the availability of fast, general-purpose simulators.
In science and engineering, many methods rely on specialized ``classical'' solvers, which are handcrafted to simulate a particular physical process.
While accurate and reliable, these solvers can be quite slow, may not provide gradients, and are narrow in their applicability \citep{cranmer2020frontier}.
In robotics and reinforcement learning, simulators are often learned, but accumulate errors over long time horizons and often struggle to generalize beyond their training data \citep{janner2019mbpo,talvitie2014model,venkatraman2015improving}, making them unsuitable for design optimization without further finetuning.
Recently, a class of learned physics simulators based on graph neural networks (GNNs) has been proposed \citep{pfaff2021learning,sanchez2020learning}.
These models have shown success in general-purpose physical prediction, exhibiting high accuracy and generalization ability.
However, this may still be insufficient for inverse design problems, as optimizers can exploit regions in the state space where model predictions are unreliable \citep{lutter2021learning}. Models must therefore be more than just accurate overall: they must also be robust and smooth.
It is unknown whether GNN-based physics simulators exhibit these properties.
\begin{figure*}[!t]
\centering
\includegraphics[width=0.96\textwidth]{figures/design_schematic.pdf}
\vspace{-0.25em}
\caption{Optimizing a physical design. Here, the goal is to direct a stream of water (shown in blue) into two ``pools'' (shown in purple) by designing a ``landscape'' (shown in green) parameterized as a 2D height field. (a) The simulation pipeline takes in a design $\phi$ and initial conditions $\alpha$ and uses the design function $f_D$ to produce an initial state. The simulation is rolled out with a pre-trained learned simulator $f_M$ for $K$ steps, at which point the final state is passed (along with reward parameters $\theta_R$) to the reward function $f_R$, which computes the quality of the design. (b) Each step of optimization involves rolling out the simulation and then adjusting the design ($\phi$) accordingly using an optimizer such as gradient descent or CEM. Shown are selected frames from gradient-based optimization in the \twopools{} task of the \threedfluids{} domain.}
\vspace{-0.5em}
\label{fig:schematic}
\end{figure*}
In this paper, we optimize physical designs by performing gradient descent through pretrained, GNN-based, state-of-the-art learned simulators.
We use this approach to perform successful inverse design without requiring further finetuning of the simulator.
Across two high-dimensional fluid manipulation tasks (\twodfluids{} and \threedfluids{}) and a design task from aerodynamics (\airfoil{}), we show that learned simulators:
(1) produce high-quality designs across diverse physical tasks with complex particle- or mesh-based physics, while using the same underlying GNN architecture;
(2) generalize sufficiently to permit designs far outside their training data;
(3) support gradient-based optimization over hundreds of time steps, through states with thousands of particles, in tasks with up to 625 design parameters (and as a result, produce better designs than sampling-based optimization using a classical simulator); and
(4) can be much faster than specialized simulators used in engineering, while generating designs of similar quality.
Overall, our results are a proof-of-concept for how state-of-the-art learned simulators can be used at scale to optimize designs for different physical tasks.
\section{Problem Formulation}
\begin{figure*}[h!]
\centering
\includegraphics[width=0.48\textwidth]{figures/optimization_trajectories_relative.png}%
\hspace{1em}\includegraphics[width=0.16\textwidth]{figures/cem_designs_relative.png}%
\hspace{1em}\includegraphics[width=0.16\textwidth]{figures/happy_glass_barplots_norm_gt_v2.pdf}
\vspace{-0.25em}
\caption{\twodfluids{} results. The state spaces consist of $10^2$--$10^3$ particles and the design spaces of 16--36 parameters. (a) Evolution of designs found by GD-M during optimization for each \twodfluids{} task. Visualizations correspond to simulations of the designs under $f_S$. The design is shown in black, fluid particles in blue, and Gaussian reward in yellow. The transparent particles show the location of fluid for $t<t_K$, and the solid particles show the location of fluid at the final frame $(K=150)$. $r$ denotes reward for the current design.
(b) Final designs found by CEM-M.
(c) Mean reward over 50 reward locations (with bootstrapped $95\%$ confidence intervals) obtained by each optimizer across the \twodfluids{} tasks. For \contain{} and \ramp{}, results are shown for 16 joints; for \maze{}, results are shown for a $6\times6$ grid of 36 rotors. Across these tasks, GD-M outperforms both CEM-M and CEM-S.}
\vspace{-0.5em}
\label{fig:2dfluids_results}
\end{figure*}
Consider the design task depicted in \autoref{fig:schematic}, in which the goal is to direct a stream of water (shown in blue) into two ``pools'' (shown in purple) by designing a ``landscape'' (shown in green) parameterized as a 2D height field.
Here, an ideal design will create ridges and valleys that direct fluid into the two targets.
In the next sections, we formalize what it means to find and evaluate such a design and discuss our choices for simulator and optimizer.
\subsection{Learned simulators}
To demonstrate the utility of learned simulators for finding physical designs, we rely on the recently developed \mgn{} model \citep{pfaff2021learning}, which is an extension of the GNS model for particle simulation \citep{sanchez2020learning}.
\mgn{} is a type of message-passing graph neural network (GNN) that performs both edge and node updates \citep{battaglia2018relational,gilmer2017neural}, and which was designed specifically for physics simulation.
Here, we briefly summarize how the learned simulator works, and refer interested readers to the original papers for details.
We consider simulations over physical states represented as graphs $G\in\mathcal{G}$.
The state $G = (V, E)$ has nodes $V$ connected by edges $E$, where each node $v\in V$ is associated with a position $\mathbf{u}_v$ and additional dynamical quantities $\mathbf{q}_v$.
These graphs may be either meshes (as in \mgn{}) or particle systems (as in GNS).
In a mesh-based system (such as \airfoil{}), $V$ and $E$ correspond to vertices and edges in the mesh, respectively.
In a particle system (such as \twodfluids{}), each node corresponds to a particle and edges are computed dynamically based on proximity.
Under this framework, we can also consider hybrid mesh-particle systems (such as \threedfluids{}).
See \autoref{app:model} for model implementation details, and \autoref{app:domains} for details on the representation used for each domain.
The simulation dynamics are given by a ``ground-truth'' simulator $f_S:\mathcal{G}\rightarrow\mathcal{G}$ which maps the state at time $t$ to that at time $t+\Delta t$.
The simulator $f_S$ can be applied iteratively over $K$ time steps to yield a trajectory of states, or a ``rollout,'' which we denote $(G^{t_0}, ..., G^{t_K})$.
Using \mgn{}, we learn an approximation $f_M$ of the ground-truth simulator $f_S$.
The learned simulator $f_M$ can be similarly applied to produce rollouts $(\tilde{G}^{t_0}, \tilde{G}^{t_1}, ..., \tilde{G}^{t_K})$, where $\tilde{G}^{t_0} = G^{t_0}$ represents initial conditions given as input.
We note that a learned simulator allows us to take much larger time-steps than $f_S$, permitting shorter rollout lengths: in our running example, one model step corresponds to 200 internal steps of the classical simulator.
See \autoref{fig:schematic}a for an illustration of simulation using a learned model.
\subsection{Optimizing design parameters}
\label{sec:optimization}
To optimize a physical design, we leverage the pipeline shown in \autoref{fig:schematic}: (1) transform design parameters into an initial scene, (2) simulate the scene using $f_M$ or $f_S$, (3) evaluate how well the simulation achieves the desired behavior, and (4) adjust the design parameters accordingly.
\vspace{-0.5em}
\paragraph{Design parameters}
To produce the initial state $G^{t_0}$, we introduce a differentiable design function $f_D: \Phi\times \mathcal{A} \rightarrow \mathcal{G}$ which maps design parameters $\phi\in\Phi$ and other initial conditions $\alpha\in \mathcal{A}$ to an initial state: $G^{t_0}=f_D(\phi, \alpha)$.
In our landscape design task (\autoref{fig:schematic}), $\phi$ is the 2D height field of the mesh, while $\alpha$ is the non-controllable objects in the scene like the initial position of the fluid.
\vspace{-0.5em}
\paragraph{Maximizing reward}
The reward function $f_R: \mathcal{G}\times\Theta_R\rightarrow\mathbb{R}$ maps the final state of a length-$K$ trajectory ($G^{t_K}$ or $\tilde{G}^{t_K}$) and parameters $\theta_R\in\Theta_R$ to a scalar value.
In our running example, the reward function is defined as the Gaussian likelihood of each fluid particle under the closest ``pool'', averaged across particles (\autoref{fig:schematic}a).
We define the full objective under the ground-truth simulator $f_S$ as $J_S(\phi):=f_R(f_S^{(K)}(f_D(\phi, \alpha)); \theta_R)$, where $f_S^{(K)}$ indicates $K$ applications of the simulator.
We want to find the design parameters that maximize $J_S$, i.e. $\phi^* = \argmax_\phi J_S(\phi)$.
We can approximate this optimization using a learned simulation model instead by maximizing $J_M(\phi):=f_R(f_M^{(K)}(f_D(\phi, \alpha)); \theta_R)$.
\looseness=-1
\paragraph{Optimizers}
Optimal design parameters $\phi^*$ can be found using any generic optimization technique.
Given the differentiability of the learned simulator $f_M$, we are particularly interested in evaluating gradient-based optimization, which requires fewer function evaluations and scales better to large design spaces than sampling-based techniques \citep{bharadhwaj2020model}.
We focus on the Adam optimizer \citep{kingma:adam}, which we use to find $\phi^*$ by computing the gradient $\nabla_\phi J_M(\phi)$.
This involves backpropagating gradients through the reward function $f_R$, length-$K$ rollout produced by $f_M^{(K)}$, and design function $f_D$.
As a baseline, we consider the cross-entropy method (CEM) \citep{rubinstein2004cem}, a gradient-free sampling-based technique that is popular in model-based control \cite{chua2018deep,wang2019benchmarking}.
CEM can be used with any simulator and works by sampling a population of candidates for $\phi$ and evolving them to maximize the reward.
However, CEM requires multiple evaluations of $f_M$ or $f_S$ per optimizer step (depending on the population size, which is 20--40), whereas Adam considers only a single candidate $\phi$ and thus only a single evaluation per step.
Across our design tasks (\autoref{sec:design-tasks}) we compared: gradient descent with the learned simulator (\textbf{GD-M}), CEM with the learned simulator (\textbf{CEM-M}), and CEM with the ground-truth simulator (\textbf{CEM-S}).
In all tasks, $f_S$ is non-differentiable, preventing a comparison to GD-S.
However, in the special case of \airfoil{}, we compare to \textbf{\dafoam{}} \cite{he2020dafoam}, a specialized solver which computes gradients with the adjoint method.
\begin{figure*}[!h]
\centering
\includegraphics[width=0.96\textwidth]{figures/waterworks.pdf}
\vspace{-0.25em}
\caption{\threedfluids{} results found with GD-M or CEM-M and evaluated with $f_S$. Simulations use up to 2000 particles and 625 design parameters. The heightmap of a 2D landscape is optimized to redirect the fluid towards the purple targets; birds eye views of heightmaps are shown in the upper right corner of each subplot. In this high-dimensional task domain, GD finds designs with high reward (a-c), while CEM fails to find meaningful solutions for \twopools{} (d) as well as the other tasks (\autoref{fig:accuracy_comparison}a).
Each subplot reports the mean reward (r) and bootstrapped [lower, upper] 95\% confidence intervals for the corresponding optimizer and task (averaged over 10 randomized initial designs for each task variation, see \autoref{app:3dfluids}).
}
\vspace{-0.5em}
\label{fig:3dfluids}
\end{figure*}
\looseness=-1
\paragraph{Evaluation}
Unless otherwise noted, we always evaluate the quality of an optimized design $\phi^*$ using the ground-truth objective $J_S(\phi^*)$, regardless of whether $\phi^*$ was found using the learned model $f_M$ (as in CEM-M and GD-M) or the ground-truth simulator $f_S$ (as in CEM-S).
We also use rollouts from $f_S$ to produce visualizations in the figures.
\section{Optimizer hyperparameters}
\label{app:hyperparameters}
Optimization hyperparameters were chosen to reflect good performance for each optimizer in each domain.
We therefore performed sweeps for major hyperparameters of each optimizer for each domain, with those used for experiments in the paper shown in the table below.
CEM maintains a population of samples and uses these to estimate the mean $\mu$ and standard deviation $\sigma$ of a Gaussian distribution over design parameters. To optimize $\mu$ and $\sigma$, it takes the top performing fraction, deemed the ``elite portion,'' from the current step. The initial standard deviation of this distribution is given by ``Initial $\sigma$'', and the initial mean is set to 0.
We found that for CEM, the population sample size had a significant effect on overall optimization quality (\autoref{fig:cem_samples}). Due to computational considerations, we picked the smallest value for this hyperparameter that performed within 1 standard deviation of the optimal sample size. Both the elite portion and initial $\sigma$ parameters were chosen as the best performing values on a set of held-out random tasks for each domain.
For GD, we only performed a hyperparameter sweep over the learning rate, which was the only parameter to significantly affect performance. For the \twodfluids{} tasks, we introduced gradient clipping to eliminate the effect of rare gradient spikes over the course of optimization. However, not using gradient clipping still produced qualitatively and quantitatively similar results. For additional Adam parameters, we used the default values for the exponential decay rates that track the first and second moment of past gradients of $b_1=0.9$ and $b_2=0.999$ \cite{optax}.
\begin{center}
\begin{tabular}{ r | c c c c c c}
\toprule
& \multicolumn{3}{c}{\twodfluids{}} & \multicolumn{2}{c}{\threedfluids{}} & \airfoil{} \\
GD & \contain{} & \ramp{} & \maze{} & \direction{} & \emph{Pools} & \\
\midrule
Learning rate & 0.005 & 0.005 & 0.01 & 0.01 & 0.01 & 0.01 \\
Momentum term $b_1$ & 0.9 & 0.9 & 0.9 & 0.9 & 0.9 & 0.9 \\
Momentum term $b_2$ & 0.999 & 0.999 & 0.999 & 0.999 & 0.999 & 0.999 \\
Gradient clip & 10 & 10 & 10 & --- & --- & --- \\
& & & & & & \\
CEM & & & & & & \\
\midrule
Sampling size & 20 & 20 & 20 & 40 & 40 & --- \\
Elite portion & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & --- \\
Initial $\mu$ & 0 & 0 & 0 & 0 & 0 & --- \\
Initial $\sigma$ & 0.5 & 0.5 & 1.5 & 0.1 & 0.1 & --- \\
Evolution smoothing & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & --- \\
\midrule
Optimization steps & 1000 & 1000 & 1000 & 200 & 200 & 200 \\
\bottomrule
\end{tabular}
\label{tab:optimizer_hypers}
\end{center}
\begin{figure}[H]
\centering
\includegraphics[width=0.7\textwidth]{figures/CEM_sample_size_combined.png}
\caption{For CEM, increasing population size, while more computationally expensive, can lead to improvements in performance. (a) In the \threedfluids{} domain, CEM benefits from large sample sizes, although returns are diminishing for sizes beyond 40 (\direction{} task, 36 design parameters). (b) In the \twodfluids{} domain, CEM benefits from larger sample sizes, although returns are diminishing for sizes beyond 20 (\contain{} task, 40 design parameters).}
\label{fig:cem_samples}
\end{figure}
\section{Model architecture and training}
\label{app:model}
For each task domain, we train a GNN for next-step prediction of the system state.
For the domains considered in this paper, we unify the approaches of GNS \citep{sanchez2020learning} and \mgn{} \citep{pfaff2021learning}:
In the \airfoil{} domain, we encode/decode mesh nodes and mesh edges as a graph as described in the aerodynamics examples of \mgn{}, while for particle-based fluids, edges are generated based on proximity as in GNS. In the case of \threedfluids{}, both particles (fluid) and a mesh (the designed obstacle) are present; hence, edges are generated based on proximity (for fluid-fluid and fluid-obstacle interaction) or from the landscape mesh. As the landscape does not have any internal dynamics, we did not find it necessary to distinguish between world- and mesh edges, and use a single edge type.
Once encoded as a graph, the core model and training procedure is largely identical between GNS and \mgn{}, and we refer to the above papers for full details on architecture and model training.
Briefly, we use an Encode-Process-Decode GNN with 10 processor blocks. All edge and node functions are 2-layer MLPs of width 128, with ReLu activation and LayerNorm after each MLP block. The model is trained with Adam and a mini-batch size of 2, with training noise, for up to 10M steps. We implemented this model in JAX \citep{jax2018github}.
In addition to the different encoding procedures for mesh vs. particle systems, the parameters for training noise and connectivity radius have to be set per-domain, to account for differences in particle size/mesh spacing. These details are described in \autoref{app:domains}.
\paragraph{Gradient computation}
\label{app:gradients}
In our experiments, we pass gradients through long model rollouts of up to 300 steps. As it is prohibitive to store all forward activations for the backwards pass, we use gradient checkpointing \citep{chen2016training} to store activations only at the beginning of each step of the trajectory during the forward pass, and recompute the intermediate activations for each step as needed when the backwards pass walks the trajectory in reverse. Gradient calculation using this method has roughly 3 times the time cost of a pure forward simulation: forward dynamics have to be computed twice for each step, in addition to the computation of the backwards pass itself.
\section{Task domains}
\label{app:domains}
\subsection{\twodfluids{}}
\label{app:2dfluids}
Tasks in \twodfluids{} are procedurally generated from templates specified in \autoref{tab:2dfluids_tasks}.
The simulation domain is a 2D box, with the lower left corner specified as $[0, 0]$, and upper right corner specified as $[1,1]$. Fluid particles are initialized as a box of size \emph{Initial fluid box} with bounding boxes given in format $[x_\text{min}, y_\text{min}, x_\text{max}, y_\text{max}]$. Certain task parameters were varied for ablation experiments in \autoref{fig:2dfluids_ablations} (rollout length, \# joints (\contain{}), \# tools (\maze{})); \autoref{tab:2dfluids_tasks} contains default values used unless otherwise specified.
\begin{table}[t]
\begin{center}
\begin{tabular}{ r | c c c }
\toprule
& Contain & Ramp & Maze (nxn) \\
\midrule
Environment size & 1x1 & 1x1 & 1x1 \\
Rollout length & 150 & 150 & 150 \\
\\
Initial fluid box & [0.2, 0.5, 0.3, 0.6] & [0.2, 0.5, 0.3, 0.6] & [0.2, 0.75, 0.8, 0.8] \\
\\
Reward sampling box & [0.4, 0.1, 0.6, 0.3] & [0.8, 0, 1, 0.2] & [0.1, 0.1, 0.9, 0.2] \\
Reward $\sigma$ & 0.1 & 0.1 & 0.1 \\
\\
Design parameter & joint angles & joint angles & rotation \\
\# tools & 1 & 1 & $n^2$ \\
\# joint angles & 16 & 16 & 1 \\
\\
Tool position (left) & [0.15, 0.35] & [0.15, 0.35] & --- \\
Tool domain box (3x3) & --- & --- & [0.14, 0.3, 0.65, 0.6] \\
Tool domain box (4x4) & --- & --- & [0.14, 0.3, 0.71, 0.6] \\
Tool domain box (5x5) & --- & --- & [0.14, 0.3, 0.75, 0.6] \\
Tool domain box (6x6) & --- & --- & [0.14, 0.25, 0.77, 0.65] \\
\\
Tool Length & 0.8 & 0.8 & --- \\
Tool length (3x3) & --- & --- & 0.72 \\
Tool length (4x4) & --- & --- & 0.64 \\
Tool length (5x5) & --- & --- & 0.65 \\
Tool length (6x6) & --- & --- & 0.63 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Task Parameters for \twodfluids{} tasks. Boxes are described as $[x_\text{min}, y_\text{min}, x_\text{max}, y_\text{max}]$.}
\label{tab:2dfluids_tasks}
\end{table}
\paragraph{Design space}
A ``tool`` in this task domain is a 2D curve composed of several line segments connected by joints. For a large number of joints, a tool can thus approximate a smooth curve (\autoref{fig:hg_designspace_appendix}).
Each task's design space consists of the relative joint angles controlling the tool's shape. We consider tasks with a single, multi-segment tool (\contain{}, \ramp{}) and a task with multiple, single-segment tools (\maze{}).
For each tool, relative angles are calculated by moving from the anchor point on the left, along the tool segments to right, such that $\text{angle}_i = \text{angle}_{i-1} + {\phi_\mathrm{joints}}_i$ for the $i^\text{th}$ joint from the anchor.
We also experimented with two additional design space parameterizations: (1) jointly optimizing the joint angles and a global position offset $[x, y]$ for each tool, and (2) changing the parameterization of angles to be absolute (such that $\text{angle}_i = {\phi_\mathrm{joints}}_i$ directly).
We discuss the effects of these alternate parameterizations in \autoref{app:design-param-results}.
\begin{figure}[t!]
\centering
\includegraphics[width=0.8\textwidth]{figures/pivots.png}
\caption{Visualization of the design space parameterization for the \twodfluids{} task. Each red dot corresponds to the anchor points (\contain{} and \ramp{}) and center of rotation (\maze{}) being optimized.}
\label{fig:hg_designspace_appendix}
\end{figure}
\paragraph{Simulation and objective}
Both fluids and tools are represented as particles with different types, and simulated with the learned model for 150 steps (with the exception of the ablation experiment on rollout length). Scenes consist of $N=100 \ldots 1000$ fluid particles.
For ground-truth evaluation of the designs, we simulate particle dynamics with an MPM solver~\cite{hu2018moving}.
Task reward is calculated using the Gaussian likelihood of the final particle positions after rollout ($\mathbf{u_v}$ from $\tilde{G}^{t_K}$). That is, for a task with reward parameterized with mean $\mu$ and spherical covariance $\sigma$ ($\theta_R = [\mu, \sigma]$), the reward is calculated as
\begin{equation*}
f_R := \mathrm{mean}_v]\, \mathcal{N}(\mathbf{u_v}; \mu, \sigma)\,.
\end{equation*}
\paragraph{\contain{}}
For this task, the center of the goal region $\mu$ is sampled uniformly from a rectangular reward region in the lower-middle section of the $1\times1$ simulation domain ($[0.4, 0.6]\times[0.2, 0.4]$).
A tool protruding to the right is initially placed below the fluid rectangle.
By optimizing a single tool's relative joint angles, successful solutions must ``contain'' the fluid in the region by creating a cup or spoon.
\paragraph{\ramp{}}
The fluid and tool are initialized as in \contain{}, and $\mu$ is sampled from a region lower and further to the right than in \contain{} ($[0.8, 1]\times[0, 0.2]$).
By again optimizing a single tool's relative joint angles, successful solutions will create a ``ramp'' from the initial fluid position to the goal location in the bottom right.
\paragraph{\maze{}}
The goal is sampled from a long region near the bottom of the domain ($[0.1, 0.9]\times[0.1, 0.2]$).
By optimizing the rotation angles of a grid of rigid, linear tools, successful solutions will create a directed path from the top of the screen to the goal location at the bottom.
\paragraph{Model training}
We trained the learned simulator on the \textsc{WaterRamps} datasets released by \citet{sanchez2020learning}. This dataset consists of 1000 trajectories featuring a single large block of water falling on one to four randomized straight line segments (see image below for examples). Model architecture and hyperparameters are described in \autoref{app:model}, with a training noise scale of $6.7\,10^{-4}$ and connectivity radius of $0.015$.
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.7\textwidth]{figures/hg_trainig_data.pdf}
\caption{Four examples of trajectories from the \textsc{WaterRamps} dataset released by \citep{sanchez2020learning} used as training data for the supervised prediction model.}
\end{center}
\end{figure}
\subsection{\threedfluids{}}
\label{app:3dfluids}
\paragraph{Design space}
This domain has a design space $\phi_\mathrm{map}$ of 625 parameters, which determine the y coordinate offset to nodes of a $25 \times 25$ square mesh centered at $\mathbf{c}=(0.5, 0.5, 0.5)$ in the simulation domain.
While we could directly mapping the parameters to coordinates, we use the design function $y_i = \gamma_H \mathrm{tanh}(\phi_{\mathrm{map}_i})$ ($\gamma_H = 0.3$ for all tasks) to prevent trivial task solutions (i.e. obstacles which touch the floor).
\paragraph{Simulation}
The simulation consists of an inflow pipe located at (-0.5, 1.0, 0.5) above the landscape which continually emits a stream of liquid, represented as particles. These particles are then redirected by the designed landscape, and finally removed once they hit the floor at $y=0$.
In our experiments we observed up to 2084 particles present in the scene at one time. We unroll the learned simulation model for trajectories of 50 time steps, and store the final particle positions $\mathbf{u_v}$, as well as the positions of removed particles that touched the floor at any point $\mathbf{u_v^D}$ to be passed to the reward function.
Ground truth simulations for evaluation are performed by running the same setup with an SPH solver. We note that SPH requires very small simulation time steps, and performs $\approx 10^4$ internal steps for a trajectory of the same length.
\paragraph{\direction{}}
In this task, we want to align the water stream with a given direction vector $\mathbf{d}$. We can formalize this using the reward function
\begin{equation*}
f_{R_\mathrm{dir}} := \mathrm{mean}_v \left( (\mathbf{u_v} - \mathbf{c})\cdot \mathbf{d}\right) - \mathrm{std}_v \left( (\mathbf{u_v} - \mathbf{c})\cdot\mathbf{d_\perp}\right) - \gamma_R\, \mathrm{mean}(\nabla \phi_\mathrm{map})
\end{equation*}
where $\mathbf{d}_\perp$ is orthogonal to $\mathbf{d}$. The first term aligns the direction of the particle relative to the domain center, and the second term concentrates the stream. The last term is a smoothness regularizer on the design landscape, which prefers smooth solutions ($\gamma_R=300$ for both tasks). Absolute reward numbers for this task can be positive or negative, hence we report the normalized reward $f_{R_\mathrm{dir}} - f_{R_\mathrm{dir}}^\mathrm{initial}$, i.e. an unchanged initial design corresponds to a zero reward, to make the scores easier to interpret.
Rewards can be in 8 different directions, spaced between $0$ and $180\deg$. We collapse across directions for reporting reward means and confidence intervals for each optimizer.
\paragraph{\twopools{} and \threepools{}}
In these tasks, we define two and three pools, respectively, with center $\mathbf{\mu_p}$ on the floor. For each particle which has hit the floor, we assign it to its closest pool $\mathbf{\hat{\mu_p}}$, and define the reward as the Gaussian probability under $\mathbf{\hat{\mu_p}}$, i.e.
\begin{equation*}
f_{R_\mathrm{pools}} := \mathrm{mean}_v (\mathcal{N}(\mathbf{u_v^D};\mathbf{\hat{\mu_p}, \sigma})) - \gamma_R\,\mathrm{mean}(\nabla \phi_\mathrm{map})
\end{equation*} with $\sigma=0.4$ and a regularization term as above.
To showcase different ways of splitting the water stream, we consider one positioning of the pools for the two pool case, and two for the three pool case. In the two pool case, pools are placed at $[1.49, -0.35]$ and $[1.49, 1.35]$. In the three pool case, pools are placed either at $[1.6, -0.45]$, $[1.85, 0.5]$, and $[1.6, 1.45]$, or at $[0.5, -0.5]$, $[1.7, 0.5]$, and $[0.55, 1.5]$. These were selected to ensure the task was solveable -- pools directly beneath the landscape, or too far away from the landscape, would not be reachable even with dramatically warped surfaces.
\paragraph{Model training}
We trained a model on next-step prediction of particle positions, on a dataset of 1000 trajectories of water particles interacting with a randomized obstacle plane (random rotations and sine-wave deformations of the planar obstacle surface). The data was generated using the SPH simulator SPlisHSPlasH \citep{bender2015divergence}.
The noise scale is set to $0.003$ and a connectivity radius of $0.01$ to account for the different particle radius of the 3D SPH simulation compared to 2D MPM. All other architectural and hyperparameters are as described in \autoref{app:model}.
\subsection{\airfoil{}}
\label{app:airfoil}
The airfoil optimization task is modeled similarly to the NACA0012 aerodynamic shape optimization configuration for incompressible flow for the DAFoam solver (see details \href{https://dafoam.readthedocs.io/en/latest/Tutorial_Aerodynamics_NACA0012_Incompressible.html}{here}), to make it easier to compare design solutions to this solver.
\paragraph{Design space}
The design space consists of the $y$-coordinate of 10 control points (see \autoref{fig:airfoil}a). Moving these control points deforms both the airfoil, and the simulation mesh surrounding it. The airfoil shape is deformed using B-spline interpolation as described by \citet{ffd}, and the mesh is deformed using IDWarp~\citep{Secco2021}.
We thus define a design function $G^{t0} = f_D(\phi_\mathrm{ctrl}, G_{\alpha})$ which takes an initial, undeformed airfoil mesh (we use the standard NACA0012 airfoil), encoded as a graph $G_{\alpha}$, as well as the control point position $\phi_\mathrm{ctrl}$ as input. It returns the graph of the deformed airfoil mesh $G^{t_0}$ to be passed to the simulator.
We note that the coefficients for spline interpolation and mesh warping can be precomputed for a given initial mesh, making it easy to define a differentiable function to use for design optimization.
\paragraph{Simulation}
Given the initial mesh, as well as simulation parameters, the simulator or learned model predict the steady-state incompressible airflow around the wing, sampled on each of the 4158 nodes on the simulation mesh. The entire simulation domain and an example prediction of the pressure field are shown in \autoref{fig:airfoil_appendix}a,b.
For drag minimization we require predictions of the pressure field $p$, as well as the effective Reynolds stress $\rho_\mathrm{eff}$ at each mesh node, i.e. $\mathbf{q}_v=(p,\rho_{eff})$. Unlike the other domains in this paper, this is a single-step prediction task, and model rollouts are of length one.
For this task, we consider an inflow speed of 0.1 mach, under an $5.1^{\circ}$ angle of attack.
\paragraph{Task objective}
The task reward is defined as $f_R := -C_D - \gamma_L ||C_L - C_{L0}||^2 - \gamma_A\, a(\phi_\mathrm{ctrl})$, i.e. we minimize the drag coefficient $C_D$ under soft constraints of unchanged lift $C_L$ and a wing area $a$ of 1-3 times the initial area. We use $\gamma_L = 10, \gamma_A = 1$, and a tanh nonlinearity to enforce the volume inequality. Lift and drag can be computed from the simulation output $p, \rho_{eff}$ by integration around the airfoil, see e.g. \citep{ladson1988effects}. We report the normalized reward $f_R - f_R^\mathrm{initial}$ such that the initial, undeformed wing design corresponds to a zero reward.
\paragraph{Model training}
We trained a model to predict $p, \rho_{eff}$ on a dataset of 10000 randomized airfoil meshes, simulated with OpenFoam \citep{openfoam}. For training ensemble models, this dataset is split into 5 non-overlapping blocks, and a separate model is trained on each section. Since this is a steady-state prediction task, information needs to propagate further at each model evaluation. We therefore use twice-repeated processor blocks with shared parameters, i.e. the model performs 20 message passing steps, with 10 blocks of learnable parameters. We found that this increases accuracy in the one-step setup by being able to pass messages further across the mesh.
Training noise is often cited for stability over long rollouts, but even in this one-step setting, training noise and data variation can be useful. To increase robustness to unseen wing configurations, we varied the grid resolution between 1000-10000 nodes for each sample in the training set, and added training noise to the input mesh coordinates. We use a normal noise distribution with the scale of $1\%$ of the average edge lengths surrounding the node noise is applied to.
All other aspects of model architecture and training procedure are as described in \autoref{app:model}.
\begin{figure}[t!]
\centering
\includegraphics[width=0.6\textwidth]{figures/wing_pred.png}
\caption{(a) Aerodynamics are computed on a large 4158 node mesh centered around the airfoil, with the closeup regions around the airfoil in (b, c) marked as a white square in the center. (b, c) Pressure predictions and control points (orange) for the initial and final optimized wing design (Ensemble-5 model).}
\label{fig:airfoil_appendix}
\end{figure}
\section{Further results}
\subsection{Model accuracy}
\label{app:model-accuracy}
In order for a learned simulator to be useful for design, it must be sufficiently accurate in the forward direction. We study this question directly for each of the domains (\threedfluids{}, \twodfluids{} and \airfoil{}) by examining the magnitude of the error between the model predictions of reward for the discovered designs, and the ground truth reward for those designs. The results for each domain are shown in \autoref{fig:accuracy_comparison}.
Broadly, the learned model very successfully mimics the ground truth simulator in reward prediction across all three domains. The accuracy for \airfoil{} is within a single standard deviation across all ensemble sizes, while the predictions in \threedfluids{} match very closely for both the high performing designs (GD-M) and low performing designs (CEM-M).
However, we do notice some discrepancies in the predicted and ground truth reward for the \twodfluids{} domain, particularly the \maze{} task. As mentioned in the main text, the ground truth solver sometimes produces unrealistic rollouts for this domain (see \autoref{fig:mpm_instability}), with fluid particles becoming stuck between the different tools. Despite this issue, we find that the model is sufficiently similar to the ground truth to produce designs that still achieve high reward overall.
\begin{figure}[H]
\centering
\includegraphics[width=0.96\textwidth]{figures/appendix_performance_comparisons_combined.png}
\vspace{-1em}
\caption{(a) For the \threedfluids{} domain, reward predicted by the learned model (GD-M eval w/ M, CEM-M eval w/ M) is very close to the ground-truth simulator evaluation (GD-M, CEM-M) for all tasks. (b) This is also true for the \twodfluids{} domain, though the reward is slightly overestimated when using the model. This effect is amplified in \maze{}, where the ground-truth dynamics sometimes struggles to correctly simulate ``sticky'' bottlenecks (see \autoref{fig:mpm_instability}). (c) Model predictions of drag (GD-M eval w/ M) are relatively close to the ground-truth simulator evaluation (GD-M), particularly for larger ensemble sizes.}
\label{fig:accuracy_comparison}
\end{figure}
\subsection{Effects of design parameterization}
\label{app:design-param-results}
In this section, we study how different parameterization choices for the design space affect both gradient-based and sampling-based optimizers.
First, in the \threedfluids{} domain, we investigate a parameterization of the design space that uses interpolation to minimize the number of control points on the 2D heightfield (\autoref{fig:design_space_3d}). Control points are placed evenly across the grid, and bi-linearly interpolated onto the $25 \times 25$ mesh. We vary the number of control points from $2\times2$ up to $14\times14$, and find that while CEM-M performs similarly to GD-M when very few control points are allowed, its performance quickly drops as more control points are added.
\begin{figure}[H]
\centering
\hspace{-4mm}\includegraphics[width=0.8\textwidth]{figures/waterworks_design_space.png}
\caption{Performance on the \threedfluids{} \direction{} task with variable design space resolution: GD performs well for large design spaces, while CEM performance quickly drops with increased number of design parameters. (b) Examples of CEM designs at two different design space resolutions. (C) Examples of GD designs at two different design space resolutions.}
\label{fig:design_space_3d}
\end{figure}
Second, in the \twodfluids{} domain, we investigate what happens when we change the design space to use \emph{absolute} joint angles rather than relative ones. When using relative joint angles, changes to joints near the tool's pivot (left side) affect the global properties of the tool. We hypothesized that this could be selectively benefiting the sampling-based approaches, as this makes the effective design space much lower dimensional. We therefore change the design space to be absolute, with a tool's joint angles calculated directly: $angle_i = \phi_i$.
As hypothesized, this change does dramatically decrease the performance of the sampling-based technique (see \autoref{fig:absolute_vs_relative}). Perhaps more surprisingly, the gradient-based optimizer is almost completely unaffected by this reparameterization. While the qualitative solutions it finds differ (with tools now containing ``kinks'' to prevent the motion of the fluid rather than curves, \autoref{fig:absolute_vs_relative} top), the overall reward achieved is similar.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{figures/absolute_vs_relative.png}
\vspace{-2em}
\caption{\textbf{(a)} Example solutions for each optimizer across the \contain{} and \ramp{} tasks when optimizing over \textit{Relative} vs \textit{Absolute} angles. \textbf{(b)} Mean reward with $95\%$ confidence intervals obtained by each optimizer across the \contain{} and \ramp{} tasks when optimizing over \textit{Relative} vs \textit{Absolute} angles.}
\label{fig:absolute_vs_relative}
\end{figure}
\subsubsection{Failure modes of gradient descent}
\label{app:twodfluids-zero-gradients}
Other parameterizations of the design space can badly affect the performance of gradient-based optimizers. In particular, gradient-based optimizers suffer when there are regions of zero gradients. In the \airfoil{} and \threedfluids{} domains, this is not normally a problem, as the design always interacts with the physical system on which reward is being measured. But in the \twodfluids{} domain, we can manipulate this.
In particular, for this experiment we changed the design space for \twodfluids{} to include a global position offset $[x,y]$ for the tool. Making this simple change often has no effect on the discovered designs, but occasionally the gradient-based optimization procedure can move the tool such that it no longer interacts with the fluid (\autoref{fig:mpm_zero_grad}). Once the tool has been moved out of the range of the fluid, there is no longer any way to affect the reward, and therefore there is no gradient signal to recover. To overcome this problem, future work would need to consider more sophisticated hybrid optimization techniques \cite{toussaint2018differentiable}.
\begin{figure}[H]
\centering
\includegraphics[width=0.74\textwidth]{figures/zero_grad.png}
\includegraphics[width=0.24\textwidth]{figures/zero_grad_shift2.png}
\caption{Failure mode of the GD optimizer: in some instances where the translation of the tool is included in the design, the tool may end up outside of the scope of the fluid. In this cases, the optimization can no longer recover as it will get zero gradients from there on.}
\label{fig:mpm_zero_grad}
\end{figure}
\subsection{Failure modes of the MPM solver}
\label{app:twodfluids-mpm-instability}
One of the advantages of using a learned simulator over a classic simulator is learned simulators can be trained in regions of the state and action space that are known to exhibit regularized, smooth behaviors. For example, as mentioned in \autoref{app:model-accuracy}, the MPM solver \cite{hu2018moving} we use for evaluation in \twodfluids{} shows surprising irregularities with ``sticking'' behavior when there are a large number of different tools. In the \maze{} task, this is particularly prevalent, as fluids often become stuck stochastically in some funnels but not others of similar sizes (\autoref{fig:mpm_instability}).
Since the learned simulator was trained on much simpler scenarios where this effect is not observed, it only learns the smooth behavior of the fluid's movement, which makes the resulting trajectories look more realistic. This may enable better generalization to real world scenarios.
\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{figures/taichi_sadness.png}
\caption{Left: MPM simulation \citep{hu2018moving} of a problem with many separate solid objects. As highlighted in the red circle, the MPM solver struggles with water movement between obstacles, often creating artificially sticky bottlenecks. Right: Learned model rollout for the same setup. The model rollout looks significantly more plausible, without any ``stickiness'' artifacts. Please see \url{https://sites.google.com/view/optimizing-designs} for videos demonstrating this effect clearly.}
\label{fig:mpm_instability}
\end{figure}
\subsubsection{Further designs found in \twodfluids{}}
\label{app:twodfluids-more-designs}
In the figures below, we demonstrate the range of found solutions for different solvers in tasks in the \twodfluids{} domain.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/hg_episode_sweep_demo_relative_seed2_v2.png}
\caption{Example solutions for each optimizer across the range of rollout lengths sampled for \contain{} in \autoref{fig:2dfluids_ablations}.}
\label{fig:hg_episode_sweep_demos}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/hg_n_joints_sweep_demo_relative_seed2_v2.png}
\caption{Example solutions for each optimizer across the range of joint angle numbers sampled for \contain{} in \autoref{fig:2dfluids_ablations}.}
\label{fig:hg_n_joints_sweep_demos}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=.7\textwidth]{figures/hg_maze_num_tools_demo_seed1_v2.png}
\caption{Example solutions for each optimizer across the range of grid sizes sampled for \maze{} in \autoref{fig:2dfluids_ablations}.}
\label{fig:hg_maze_num_tools_demos}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/hg_seed_sweep_demo_relative_seed2_v2.png}
\caption{Example solutions on \contain{} task for each optimizer across 6 random seeds.}
\label{fig:hg_contain_seeds_demos}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/hg_ramp_seeds_sweep_demo_relative_seed2_v2.png}
\caption{Example solutions on \ramp{} task for each optimizer across 6 random seeds.}
\label{fig:hg_ramp_seeds_demos}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/hg_maze_seeds_sweep_demo_relative_v2.png}
\caption{Example solutions on \maze{} for each optimizer across 6 random seeds.}
\label{fig:hg_maze_seeds_demos}
\end{figure}
\section{Design Tasks}
\label{sec:design-tasks}
We formulated a set of design tasks across three different physical domains with high-dimensional state spaces and complex dynamics. Each domain uses a different ground-truth simulator $f_S$, which is used to evaluate designs and pre-train the learned simulator model $f_M$.
\subsection{\twodfluids{}}
\label{sec:2d-fluid-design}
Inspired by existing 2D physical reasoning benchmarks \citep{allen2019tools,bakhtin2019phyre}, these tasks involve creating one or more 2D ``tool'' shapes to direct fluid into a particular goal region (\autoref{fig:2dfluids_results}, \autoref{app:2dfluids}).
We consider three tasks with 4--48 design parameters where the goal is to guide the fluid such that each particle comes as close as possible to the center of a randomly-sampled yellow reward region.
In \contain{}, the joint angles $\phi_\mathrm{joints}$ of a multi-segment tool must be optimized to catch the fluid by creating cup- or spoon-like shapes.
In \ramp{}, the joint angles $\phi_\mathrm{joints}$ of a multi-segment tool must be optimized to guide the fluid to a distant location.
In \maze{}, the rotation $\phi_\mathrm{rot}$ of multiple tools must be optimized to funnel the fluid to the target location.
Fluid dynamics are represented using $10^2$--$10^3$ particles, and unrolled for up to 300 time steps.
The learned model $f_M$ is trained on the 2D WaterRamps dataset released by \citet{sanchez2020learning} which uses the solver in \citet{hu2018moving} to generate trajectories.
The dataset contains scenes with 1--4 straight line segments; no curved lines or large numbers of obstacles are shown.
Therefore, designs which solve \twodfluids{} tasks are necessarily far out of distribution for the learned model.
\subsection{\threedfluids{}}
\label{sec:3d-fluid-design}
To evaluate higher-dimensional inverse design, we created a ``landscaping'' task that requires optimizing a 3D surface to guide fluids into different areas of an environment (\autoref{fig:3dfluids}, \autoref{app:3dfluids}).
Specifically, water flows out of a pipe and onto an obstacle parameterized by $\phi_\mathrm{map}$, a $25 \times 25$ heightmap (625 design parameters).
In \direction{}, $\phi_\mathrm{map}$ is optimized to redirect the fluid stream towards a specified direction.
In \twopools{} and \threepools{}, $\phi_\mathrm{map}$ is optimized to split the fluid stream such that particles hitting the floor land as close as possible to one of two or three specified pools.
Each task has multiple variants, such as different target directions in \direction{}.
The simulation is unrolled for 50 time steps, and contains up to 2048 particles.
The learned model $f_M$ is trained on data generated from the simulator in \citet{bender2015divergence}.
\begin{figure*}[h!]
\centering
\includegraphics[width=0.24\textwidth]{figures/init_airfoil_mesh.png}%
\includegraphics[width=0.24\textwidth]{figures/airfoil_stack.png}%
\includegraphics[width=0.24\textwidth]{figures/airfoil_stats_drag.pdf}%
\includegraphics[width=0.24\textwidth]{figures/airfoil_stats_reward.pdf}
\vspace{-0.25em}
\caption{\airfoil{} results. (a) An initial airfoil design is warped by moving 10 control points (orange dots). The physics model simulates the resulting aerodynamics on a 4158 node mesh, based on which lift and drag are computed. (b) For the task of finding a minimum-drag configuration under constant lift constraint, gradient-based learned design is able to find similar designs to specialized solver \dafoam{}, both using single models and ensembles. (c-d) Larger ensemble sizes of 3--5 achieve a close quantitative match to \dafoam{} for both drag and overall reward.
Shown are means over 10 randomized initial designs, with bootstrapped 95\% confidence intervals.}
\vspace{-0.5em}
\label{fig:airfoil}
\end{figure*}
\subsection{\airfoil{}}
\label{sec:airfoil}
Shape optimization in aerodynamics is one area where gradient-based optimization is routinely applied using traditional simulators \citep{buckley2010airfoil}.
Here, we consider the well-studied task of drag optimization of a 2D airfoil profile (\autoref{fig:airfoil}, \autoref{app:airfoil}).
In this task, a wing is defined using a curve on a 2D mesh, which can be deformed using a set of 10 control points $\phi_\textrm{ctrl}$.
The reward function is formulated as optimizing the wing shape to minimize drag under certain constraints such as constant lift and bounds on the shape to prevent degenerate (e.g. infinitely thin) configurations.
Lift and drag coefficients are computed by running an aerodynamics simulation on a 4158 node mesh.
The learned model $f_M$ is trained on data generated from the ground-truth simulator $f_S$, for which we use the OpenFOAM solver \cite{openfoam}.
\subsection{Model learning and optimization}
While each domain has a different state space structure and uses a different ground-truth simulator $f_S$, the learned simulators $f_M$ all share the same architecture (with identical hyperparameters in \twodfluids{} and \threedfluids{}, and only minor variations of the hyperparameters for \airfoil{} due to it being a steady state simulation; see \autoref{app:model}).
The models are trained for next-step prediction on task-independent datasets (random perturbations of the design space for \airfoil{} and \threedfluids{}, and an open-source, qualitatively distinct dataset for \twodfluids{}), and are unrolled for up to 300 time steps during design optimization without further fine-tuning (see \autoref{sec:optimization} for details).
\section{Background}
\label{sec:background}
Solving inverse problems with physical simulators has a long history in science and engineering, spanning data assimilation for weather modeling \citep{navon2009data}, system identification in robotics \citep{guevara2017adaptable, seita_fabrics_2020}, and tomographic and geophysical imaging \citep{cui2016review, pascual1999review}.
Inverse design can be framed as an inverse problem in which the objective is to optimize design parameters to produce some desired target property.
Simulation-based inverse design has been studied in a variety of disciplines, including nanophotonics \citep{molesky2018inverse}, material science \citep{Dijkstra2021predictive}, mechanical design \cite{coros2013computational}, and aerodynamics \cite{anderson1999aerodynamic, Rhie_1983}.
Classical numerical solvers used for inverse design can be highly accurate, but are often inefficient (making sampling-based inference methods infeasible for high-dimensional designs) and domain-specialized ( prohibiting general-purpose inverse design across domains \citep{choi2021use}).
Differentiable simulators \citep{freeman2021brax,hu2019difftaichi,Schenck2018SPNetsDF} have recently garnered attention, as they allow for more sample-efficient gradient-based optimization.
However, like classical solvers, they are still typically narrow in application scope, as many simulation techniques are hard to express as a differentiable program (e.g. constraint dynamics and multiphysics coupling).
The last few years have seen increased interest in using machine learning to accelerate inverse design across a variety of applications \citep[e.g.][]{Challapalli2021inverse,Christensen2020predictive,Bombarelli2018Automatic,Forte2022Inverse,
Hoyer2019neural,Kumar2020Inverse,Li2020efficient,Liu2018Generative,Sha2021Machine,ZHENG2021113894}.
These methods can provide impressive speedups over classical approaches by using learned generative models to propose designs (thus calling an expensive simulator fewer times), or by learning a scoring function which maps designs to target values (replacing the simulator entirely). However, they are based on components with limited out-of-domain generalization,
restricting new designs to configurations near the training data or requiring model refinement.
We instead propose to replace classical simulators with learned simulators.
Like learned scoring functions, learned simulators can be faster than classical simulators \citep{kochkov2021machine,stachenfeld2021learned} and are differentiable when parameterized as a neural network.
Furthermore, learned simulators mimic the underlying physical dynamics independent of the design task and are therefore more likely to generalize.
Physics simulators have been successfully implemented as learned, differentiable models of complex dynamics such as fluids, rigid-body interactions, and soft-body systems \citep{bhatnagar2019prediction,li2020fourier,mrowca2018,Rudy2017data,thuerey2020deep,ummenhofer2020lagrangian,wang2020physicsinformed}.
Graph-based models in particular are promising candidates for design problems, having demonstrated high accuracy, stability, efficiency, and generalization performance \citep{belbute2020combining, pfaff2021learning,sanchez2020learning}.
However, high-quality forward models do not necessarily translate into better downstream task performance \cite{hamrick2020role,lutter2021learning}. While learned simulators have been used successfully for planning and control in low-dimensional state spaces \citep[up to 21 degrees of freedom (DOF), e.g.][]{bharadhwaj2020model,sanchez2018graph,wang2019benchmarking} or more complex domains with small action spaces \citep[6 DOF, e.g.][]{li2018learning},
they require replanning at every timestep to avoid error accumulation. It is not known whether learned simulators can support gradient-based, high-dimensional inverse design which demands high accuracy, well-behaved gradients, long-term rollout stability, and generalization beyond the training data.
Here we study inverse design in non-rigid, graph-based physical systems with up to 625 design dimensions and 2000 state dimensions, over 50--300 timesteps, without replanning after the design period. We show that using gradient-based optimization with learned, general-purpose simulators is an effective choice for inverse design.
\section{Discussion}
We used state-of-the-art learned, differentiable physics simulators with gradient-based optimization to solve challenging inverse design problems.
Across three domains and seven tasks, which involved designing landscapes and tools to control water flows or optimizing the shape of an airfoil, we demonstrated that gradient descent with pre-trained simulators can discover high-quality designs that match or exceed the quality of those found using alternative methods.
This approach succeeds in a variety of interesting and surprising ways:
it permits gradient backpropagation through complex physical trajectories for hundreds of steps; scales to tasks with large design and state spaces (100s and 1000s of dimensions, respectively); and successfully generates designs which require the learned simulator to generalize far beyond its training data.
In the classic aerodynamics problem of airfoil shape optimization, our approach produces a design comparable to that of a specialized solver using only simple, general-purpose strategies like model ensembling.
While our results have exciting implications for inverse design, they also open up possibilities for explaining everyday human behavior like tool invention---a longstanding puzzle in cognitive science \citep{allen2019tools,osiurak2016,Shumaker2011}.
With general-purpose learned simulators, we have the potential not just to create highly specialized tools in engineering domains, but also to model everyday tool creation, such as creating a hook from a pipe cleaner, building a blanket fort, or folding a paper boat.
Our approach has limitations that should be visited in future work.
Gradient descent is inappropriate for design spaces with regions of zero gradients, such as in \twodfluids{} tasks where the fluid may not always make contact with the tool (\autoref{fig:mpm_zero_grad}).
Many interesting design tasks also have variably-sized or combinatorial design spaces that cannot easily be optimized with gradient descent, such as computer-aided design (CAD) approaches to 3D modeling.
An exciting future direction will be to integrate general-purpose learned simulators with hybrid optimization techniques such as those used in material science and robotics \citep{chen2020generative,toussaint2018differentiable}.
As learned simulators continue to improve, we could also use them to do even broader cross-domain, multi-physics design.
While challenges remain, our results represent a promising step towards faster and more general-purpose inverse design.
\section{Results}
\label{sec:results}
Our results show that learned simulators can be used to effectively optimize various designs despite significant domain shift and long rollout lengths.
The same underlying model architecture is used for each domain, highlighting the generality of learned simulators for design.
Examples of designs found with our approach are available at: \url{https://sites.google.com/view/optimizing-designs}.
Performance is always evaluated using the ground-truth simulator $f_S$ (see \autoref{sec:optimization}).
Here we discuss these results, and compare the capabilities of gradient descent with learned simulators over classical simulators and sampling-based optimization techniques.
\subsection{Overall results}
\label{sec:overall-results}
We first asked whether a learned simulator combined with gradient descent (GD-M) could produce good-quality designs at all.
This approach might fail in various ways: accumulating model error, vanishing or exploding gradients \citep{bengio1994learning}, or domain shift \citep{hamrick2020role}.
However, as the following results show, GD-M produced high-quality designs across all three domains.
\autoref{fig:2dfluids_results}a shows qualitative results for \twodfluids{} (\autoref{sec:2d-fluid-design}), where our approach (GD-M) produces intuitive, functional designs to contain (\contain{}), transport (\ramp{}), or funnel (\maze{}) the fluid to a target location.
On average, GD-M outperforms CEM-M by 16.1--118.9\%, indicating a substantial benefit of gradient-based optimization.
GD-M also outperforms CEM-S by 3.9--37.5\%, despite using a learned simulator rather than the ground-truth.
However, these design spaces are still relatively small (between 16 and 36 dimensions).
In \threedfluids{} (\autoref{sec:3d-fluid-design}), we substantially increase the dimensionality to a 625-dimensional landscape.
Here, GD-M produces robust designs, creating ridges to re-route water in particular directions or valleys to direct water into pools (\autoref{fig:3dfluids}a-c).
In comparison, CEM-M cannot solve any of these tasks, with performance 30--85$\times$ worse than GD-M.
In \airfoil{} (\autoref{fig:airfoil}b), GD-M recovers the characteristic S-curve shape for a low-drag airfoil under a small angle of attack, and matches the design obtained with \dafoam{}, an adjoint aerodynamics solver which computes close-to-optimal designs for this task.
Specifically, the design obtained with \dafoam{} yields a drag coefficient of 0.01902, while GD-M finds designs with drag between 0.01898--0.01919 depending on ensemble size (see \autoref{sec:ensembles}).
Importantly, \dafoam{}'s solver and optimizer are highly specialized for the particular task of airfoil design, while our approach is more general-purpose in that it requires only trajectory data for training and a generic gradient-based optimizer.
\begin{figure*}[!t]
\centering
\includegraphics[width=\textwidth]{figures/episode_joints_ablations_wmaze_norm_v2.pdf}
\vspace{-1.6em}
\caption{
Ablation experiments on the \contain{} and \maze{} tasks.
(a) Performance of all optimizers increase with rollout length; GD-M performance starts to deteriorate around step 225. (b) In \contain{}, CEM performance drops when increasing the number of joints above 24, while GD-M remains stable. (c) We observe a similar trend with the number of tools in \maze{}. (d) CEM often gets stuck in sub-optimal solutions early in optimization, while GD-M performance continues to increase.}
\vspace{-0.5em}
\label{fig:2dfluids_ablations}
\end{figure*}
\subsection{Model stability \& gradient quality}
We investigated accuracy over long timescales by measuring the effect of rollout length on design quality in \twodfluids{} (\autoref{fig:2dfluids_ablations}a and \ref{fig:hg_episode_sweep_demos}).
Longer rollouts can in principle allow for higher reward in this task as they give the fluid time to settle; however, with learned models, they can also be unstable due to error accumulation \citep{talvitie2014model,venkatraman2015improving}.
Nevertheless, we find that the learned simulator does not seem to be severely impacted by this problem.
Specifically, the quality of designs found by GD-M increases up to 225 steps (\autoref{fig:2dfluids_ablations}a), indicating that the learned simulator's accuracy and gradients remain stable for a surprisingly long time.
Across the episode lengths evaluated, we find that GD-M outperforms not only CEM-M (by 18.1\% on average) but also CEM-S (by 4.4\% on average).
This indicates that the benefits of a having a learned model that supports better optimization techniques can outweigh the error incurred by long rollouts.
The strong performance of GD-M on longer rollout lengths is noteworthy.
Gradients tend to degrade when passed through chains of many model evaluations, and as a result, previous work generally only optimizes gradients in small action spaces over just a few time-steps \cite{li2018learning}.
We speculate that one reason for the success of GD-M is the addition of noise in training $f_M$, which promotes stability on the forward pass and may also force smoother gradients.
\subsection{Generalization}
Deep networks often struggle to generalize far from their training data \cite{geirhos2018generalisation}.
This poses a problem for design: to produce in-distribution training data, we would already need to know what good designs look like, thus defeating the aim of wanting to find \emph{new} designs.
However, we find that the GNN-based simulators studied here overcome this issue.
As noted in \autoref{sec:2d-fluid-design}, the learned simulator for \twodfluids{} was trained on a pre-existing, highly simplified dataset where only one to four straight line segments interact with a fluid
(\autoref{app:domains}).
In contrast, the design tasks studied here involve highly articulated, curved obstacles (\contain{}, \ramp{}) or a larger number of obstacles (\maze{}); yet, GD-M still discovers effective designs without requiring any finetuning (\autoref{fig:2dfluids_results}).
We suspect this is because the model is trained to learn local collision rules, making it more robust to global distribution shift.
Using a learned simulator trained in a relatively simple environment has another unexpected advantage.
In rare cases, classical simulators suffer from degenerate behavior around certain edge cases.
For example, with the classical simulator for \twodfluids{}, particles can get stuck in between joint segments, especially when there are a large number of parts or joints (\autoref{fig:mpm_instability}).
However, since the learned simulator was trained on simpler data where these effects are unobserved, it picks up only on the appropriate collision performance and not the unrealistic edge cases.
Thus, the learned simulator produces more plausible rollouts than the classical simulator in these cases, and might therefore be a better candidate for producing designs that would transfer to the real world.
\subsection{Improving accuracy with ensembles}
\label{sec:ensembles}
In engineering tasks like \airfoil{}, simulators must be especially accurate, as small differences in the predicted pressure field can cause large errors in lift and drag coefficients.
While GD-M (without ensembling) can produce designs close to \dafoam{}'s, we notice a slightly rounder wing front (\autoref{fig:airfoil}b, bottom) causing a small increase in drag (0.01919 versus 0.01902 in \dafoam{}).
To further improve performance, we implemented an ensemble of
learned simulators trained on separate splits of the training set.
Ensembles are a popular choice for training transition models for use in control \citep{chua2018deep}, as they can provide higher quality predictions and are more resistant to delusions---a particularly problematic issue for accuracy-sensitive domains such as airfoil design.
During optimization, we make predictions with all models in the ensemble, each trained on a different data split, and average the gradients.
As shown in \autoref{fig:airfoil}b-c, larger ensembles yield designs with significantly lower drag ($\beta = -5.3\times 10^{-5}$, $p = 0.0003$, where $\beta$ is a linear regression coefficient) and higher overall reward ($\beta = 5.6\times 10^{-5}$, $p = 0.0001$), and are able to produce designs very close to the solution found by \dafoam{}, with a drag coefficient of 0.01898 (size-5 ensemble).
Thus, with ensembles, we are able to achieve performant designs with a general-purpose learned simulator, indicating that we can use learned models for design optimization in spaces traditionally reserved for specialized solvers like \dafoam{}.
\subsection{Scalability to larger design spaces}
\label{sec:design-dimensionality}
In larger design spaces, sampling-based optimization procedures quickly become intractable, especially with relatively slow simulators.
We hypothesized that gradient descent with fast, learned simulators could overcome this issue, especially as the size of the design space is increased.
We therefore compared different optimizers on \twodfluids{} as a function of the dimensionality of the design space (the number of tool joints in \contain{} or the number of tools in \maze{}) and on the higher-dimensional \threedfluids{}.
For \contain{} (\autoref{fig:2dfluids_ablations}b), the performance of GD-M increases with the number of joints as increasingly fine grained solutions are made possible.
In contrast, for both CEM-M and CEM-S, design quality deteriorates with more joints as high quality solutions become harder to find with random sampling. In the highest dimensional \contain{} task with 48 tool joints, GD-M outperforms CEM-M by $154.9\%$ and CEM-S by $126.5\%$.
When CEM does find solutions (\autoref{fig:2dfluids_results}b), they lack global coherence and appear more jagged than solutions found with GD-M.
Similarly, for \maze{} (\autoref{fig:2dfluids_ablations}c), the performance of GD-M is largely unaffected by the number of tools, while the performance of CEM-M and CEM-S both degrade as the design space grows. For the highest dimensional \maze{} problem with 36 joints, GD-M outperforms CEM-M by $207.4\%$ and CEM-S by $133.7\%$.
In the 625-dimensional \threedfluids{} task, CEM-M performs 30--85$\times$ worse than GD-M (\autoref{fig:3dfluids}) despite extensive hyperparameter tuning. This trend held across all tasks (\autoref{fig:accuracy_comparison}a), and even persisted when using fewer control points in the design space (\autoref{fig:design_space_3d}).
This is due not only to \threedfluids{}'s larger design space, but also because this problem requires a globally coherent solution: modifying small areas independently is unlikely to have much effect on the global movement of the fluid.
\subsection{Model speed and sample efficiency}
Learned simulators can provide large speedups over traditional simulators in certain domains by learning to compensate for coarser sub-stepping and making optimal use of hardware acceleration.
In \airfoil{}, although we use a very simple GD setup, our approach is able to find very similar designs as \dafoam{}'s specialized optimizer.
Moreover, our approach requires only 21s (single model) to 62s (size-5 ensemble) on a single A100 GPU, compared to 1021s for \dafoam{} run on an 8-core workstation, despite requiring $10\times$ more optimization steps.
In \twodfluids{}, the ground-truth simulator runs at a similar speed to the learned model \citep[see][]{sanchez2020learning}, but is non-differentiable and therefore depends on more expensive gradient-free optimization techniques which require more function evaluations.
We use 20--40 function evaluations per optimization step of CEM, compared to a single evaluation with GD (which is about $3\times$ more costly, due to the gradient computation). Thus, GD with a differentiable learned model can be much more efficient than using the ground truth simulator with a sampling-based method.
\section{Introduction}
Humans are creators.
Our ancestors created stone tools which led to innovations in hunting and food consumption, aqueducts and irrigation systems which revolutionized farming and urban habitation, and more recently, airplanes which let us cross the globe in hours.
Automatically designing objects to exhibit a desired property---often referred to as \emph{inverse design}---promises to transform science and engineering, including aerodynamics \cite{eppler2012airfoil}, material design \cite{butler2016computational}, optics \citep{colburn2021inverse}, and robotics \cite{Gupta2021, xu2021diffsim}.
Despite its promise, widespread practice of inverse design has been limited by the availability of fast, general-purpose simulators.
In science and engineering, many methods rely on specialized ``classical'' solvers, which are handcrafted to simulate a particular physical process.
While accurate and reliable, these solvers can be quite slow, may not provide gradients, and are narrow in their applicability \citep{cranmer2020frontier}.
In robotics and reinforcement learning, simulators are often learned, but accumulate errors over long time horizons and often struggle to generalize beyond their training data \citep{janner2019mbpo,talvitie2014model,venkatraman2015improving}, making them unsuitable for design optimization without further finetuning.
Recently, a class of learned physics simulators based on graph neural networks (GNNs) has been proposed \citep{pfaff2021learning,sanchez2020learning}.
These models have shown success in general-purpose physical prediction, exhibiting high accuracy and generalization ability.
However, this may still be insufficient for inverse design problems, as optimizers can exploit regions in the state space where model predictions are unreliable \citep{lutter2021learning}. Models must therefore be more than just accurate overall: they must also be robust and smooth.
It is unknown whether GNN-based physics simulators exhibit these properties.
\begin{figure*}[!t]
\centering
\includegraphics[width=0.96\textwidth]{figures/design_schematic.pdf}
\vspace{-0.25em}
\caption{Optimizing a physical design. Here, the goal is to direct a stream of water (shown in blue) into two ``pools'' (shown in purple) by designing a ``landscape'' (shown in green) parameterized as a 2D height field. (a) The simulation pipeline takes in a design $\phi$ and initial conditions $\alpha$ and uses the design function $f_D$ to produce an initial state. The simulation is rolled out with a pre-trained learned simulator $f_M$ for $K$ steps, at which point the final state is passed (along with reward parameters $\theta_R$) to the reward function $f_R$, which computes the quality of the design. (b) Each step of optimization involves rolling out the simulation and then adjusting the design ($\phi$) accordingly using an optimizer such as gradient descent or CEM. Shown are selected frames from gradient-based optimization in the \twopools{} task of the \threedfluids{} domain.}
\vspace{-0.5em}
\label{fig:schematic}
\end{figure*}
In this paper, we optimize physical designs by performing gradient descent through pretrained, GNN-based, state-of-the-art learned simulators.
We use this approach to perform successful inverse design without requiring further finetuning of the simulator.
Across two high-dimensional fluid manipulation tasks (\twodfluids{} and \threedfluids{}) and a design task from aerodynamics (\airfoil{}), we show that learned simulators:
(1) produce high-quality designs across diverse physical tasks with complex particle- or mesh-based physics, while using the same underlying GNN architecture;
(2) generalize sufficiently to permit designs far outside their training data;
(3) support gradient-based optimization over hundreds of time steps, through states with thousands of particles, in tasks with up to 625 design parameters (and as a result, produce better designs than sampling-based optimization using a classical simulator); and
(4) can be much faster than specialized simulators used in engineering, while generating designs of similar quality.
Overall, our results are a proof-of-concept for how state-of-the-art learned simulators can be used at scale to optimize designs for different physical tasks.
\section{Problem Formulation}
\begin{figure*}[h!]
\centering
\includegraphics[width=0.48\textwidth]{figures/optimization_trajectories_relative.png}%
\hspace{1em}\includegraphics[width=0.16\textwidth]{figures/cem_designs_relative.png}%
\hspace{1em}\includegraphics[width=0.16\textwidth]{figures/happy_glass_barplots_norm_gt_v2.pdf}
\vspace{-0.25em}
\caption{\twodfluids{} results. The state spaces consist of $10^2$--$10^3$ particles and the design spaces of 16--36 parameters. (a) Evolution of designs found by GD-M during optimization for each \twodfluids{} task. Visualizations correspond to simulations of the designs under $f_S$. The design is shown in black, fluid particles in blue, and Gaussian reward in yellow. The transparent particles show the location of fluid for $t<t_K$, and the solid particles show the location of fluid at the final frame $(K=150)$. $r$ denotes reward for the current design.
(b) Final designs found by CEM-M.
(c) Mean reward over 50 reward locations (with bootstrapped $95\%$ confidence intervals) obtained by each optimizer across the \twodfluids{} tasks. For \contain{} and \ramp{}, results are shown for 16 joints; for \maze{}, results are shown for a $6\times6$ grid of 36 rotors. Across these tasks, GD-M outperforms both CEM-M and CEM-S.}
\vspace{-0.5em}
\label{fig:2dfluids_results}
\end{figure*}
Consider the design task depicted in \autoref{fig:schematic}, in which the goal is to direct a stream of water (shown in blue) into two ``pools'' (shown in purple) by designing a ``landscape'' (shown in green) parameterized as a 2D height field.
Here, an ideal design will create ridges and valleys that direct fluid into the two targets.
In the next sections, we formalize what it means to find and evaluate such a design and discuss our choices for simulator and optimizer.
\subsection{Learned simulators}
To demonstrate the utility of learned simulators for finding physical designs, we rely on the recently developed \mgn{} model \citep{pfaff2021learning}, which is an extension of the GNS model for particle simulation \citep{sanchez2020learning}.
\mgn{} is a type of message-passing graph neural network (GNN) that performs both edge and node updates \citep{battaglia2018relational,gilmer2017neural}, and which was designed specifically for physics simulation.
Here, we briefly summarize how the learned simulator works, and refer interested readers to the original papers for details.
We consider simulations over physical states represented as graphs $G\in\mathcal{G}$.
The state $G = (V, E)$ has nodes $V$ connected by edges $E$, where each node $v\in V$ is associated with a position $\mathbf{u}_v$ and additional dynamical quantities $\mathbf{q}_v$.
These graphs may be either meshes (as in \mgn{}) or particle systems (as in GNS).
In a mesh-based system (such as \airfoil{}), $V$ and $E$ correspond to vertices and edges in the mesh, respectively.
In a particle system (such as \twodfluids{}), each node corresponds to a particle and edges are computed dynamically based on proximity.
Under this framework, we can also consider hybrid mesh-particle systems (such as \threedfluids{}).
See \autoref{app:model} for model implementation details, and \autoref{app:domains} for details on the representation used for each domain.
The simulation dynamics are given by a ``ground-truth'' simulator $f_S:\mathcal{G}\rightarrow\mathcal{G}$ which maps the state at time $t$ to that at time $t+\Delta t$.
The simulator $f_S$ can be applied iteratively over $K$ time steps to yield a trajectory of states, or a ``rollout,'' which we denote $(G^{t_0}, ..., G^{t_K})$.
Using \mgn{}, we learn an approximation $f_M$ of the ground-truth simulator $f_S$.
The learned simulator $f_M$ can be similarly applied to produce rollouts $(\tilde{G}^{t_0}, \tilde{G}^{t_1}, ..., \tilde{G}^{t_K})$, where $\tilde{G}^{t_0} = G^{t_0}$ represents initial conditions given as input.
We note that a learned simulator allows us to take much larger time-steps than $f_S$, permitting shorter rollout lengths: in our running example, one model step corresponds to 200 internal steps of the classical simulator.
See \autoref{fig:schematic}a for an illustration of simulation using a learned model.
\subsection{Optimizing design parameters}
\label{sec:optimization}
To optimize a physical design, we leverage the pipeline shown in \autoref{fig:schematic}: (1) transform design parameters into an initial scene, (2) simulate the scene using $f_M$ or $f_S$, (3) evaluate how well the simulation achieves the desired behavior, and (4) adjust the design parameters accordingly.
\vspace{-0.5em}
\paragraph{Design parameters}
To produce the initial state $G^{t_0}$, we introduce a differentiable design function $f_D: \Phi\times \mathcal{A} \rightarrow \mathcal{G}$ which maps design parameters $\phi\in\Phi$ and other initial conditions $\alpha\in \mathcal{A}$ to an initial state: $G^{t_0}=f_D(\phi, \alpha)$.
In our landscape design task (\autoref{fig:schematic}), $\phi$ is the 2D height field of the mesh, while $\alpha$ is the non-controllable objects in the scene like the initial position of the fluid.
\vspace{-0.5em}
\paragraph{Maximizing reward}
The reward function $f_R: \mathcal{G}\times\Theta_R\rightarrow\mathbb{R}$ maps the final state of a length-$K$ trajectory ($G^{t_K}$ or $\tilde{G}^{t_K}$) and parameters $\theta_R\in\Theta_R$ to a scalar value.
In our running example, the reward function is defined as the Gaussian likelihood of each fluid particle under the closest ``pool'', averaged across particles (\autoref{fig:schematic}a).
We define the full objective under the ground-truth simulator $f_S$ as $J_S(\phi):=f_R(f_S^{(K)}(f_D(\phi, \alpha)); \theta_R)$, where $f_S^{(K)}$ indicates $K$ applications of the simulator.
We want to find the design parameters that maximize $J_S$, i.e. $\phi^* = \argmax_\phi J_S(\phi)$.
We can approximate this optimization using a learned simulation model instead by maximizing $J_M(\phi):=f_R(f_M^{(K)}(f_D(\phi, \alpha)); \theta_R)$.
\looseness=-1
\paragraph{Optimizers}
Optimal design parameters $\phi^*$ can be found using any generic optimization technique.
Given the differentiability of the learned simulator $f_M$, we are particularly interested in evaluating gradient-based optimization, which requires fewer function evaluations and scales better to large design spaces than sampling-based techniques \citep{bharadhwaj2020model}.
We focus on the Adam optimizer \citep{kingma:adam}, which we use to find $\phi^*$ by computing the gradient $\nabla_\phi J_M(\phi)$.
This involves backpropagating gradients through the reward function $f_R$, length-$K$ rollout produced by $f_M^{(K)}$, and design function $f_D$.
As a baseline, we consider the cross-entropy method (CEM) \citep{rubinstein2004cem}, a gradient-free sampling-based technique that is popular in model-based control \cite{chua2018deep,wang2019benchmarking}.
CEM can be used with any simulator and works by sampling a population of candidates for $\phi$ and evolving them to maximize the reward.
However, CEM requires multiple evaluations of $f_M$ or $f_S$ per optimizer step (depending on the population size, which is 20--40), whereas Adam considers only a single candidate $\phi$ and thus only a single evaluation per step.
Across our design tasks (\autoref{sec:design-tasks}) we compared: gradient descent with the learned simulator (\textbf{GD-M}), CEM with the learned simulator (\textbf{CEM-M}), and CEM with the ground-truth simulator (\textbf{CEM-S}).
In all tasks, $f_S$ is non-differentiable, preventing a comparison to GD-S.
However, in the special case of \airfoil{}, we compare to \textbf{\dafoam{}} \cite{he2020dafoam}, a specialized solver which computes gradients with the adjoint method.
\begin{figure*}[!h]
\centering
\includegraphics[width=0.96\textwidth]{figures/waterworks.pdf}
\vspace{-0.25em}
\caption{\threedfluids{} results found with GD-M or CEM-M and evaluated with $f_S$. Simulations use up to 2000 particles and 625 design parameters. The heightmap of a 2D landscape is optimized to redirect the fluid towards the purple targets; birds eye views of heightmaps are shown in the upper right corner of each subplot. In this high-dimensional task domain, GD finds designs with high reward (a-c), while CEM fails to find meaningful solutions for \twopools{} (d) as well as the other tasks (\autoref{fig:accuracy_comparison}a).
Each subplot reports the mean reward (r) and bootstrapped [lower, upper] 95\% confidence intervals for the corresponding optimizer and task (averaged over 10 randomized initial designs for each task variation, see \autoref{app:3dfluids}).
}
\vspace{-0.5em}
\label{fig:3dfluids}
\end{figure*}
\looseness=-1
\paragraph{Evaluation}
Unless otherwise noted, we always evaluate the quality of an optimized design $\phi^*$ using the ground-truth objective $J_S(\phi^*)$, regardless of whether $\phi^*$ was found using the learned model $f_M$ (as in CEM-M and GD-M) or the ground-truth simulator $f_S$ (as in CEM-S).
We also use rollouts from $f_S$ to produce visualizations in the figures.
\section{Optimizer hyperparameters}
\label{app:hyperparameters}
Optimization hyperparameters were chosen to reflect good performance for each optimizer in each domain.
We therefore performed sweeps for major hyperparameters of each optimizer for each domain, with those used for experiments in the paper shown in the table below.
CEM maintains a population of samples and uses these to estimate the mean $\mu$ and standard deviation $\sigma$ of a Gaussian distribution over design parameters. To optimize $\mu$ and $\sigma$, it takes the top performing fraction, deemed the ``elite portion,'' from the current step. The initial standard deviation of this distribution is given by ``Initial $\sigma$'', and the initial mean is set to 0.
We found that for CEM, the population sample size had a significant effect on overall optimization quality (\autoref{fig:cem_samples}). Due to computational considerations, we picked the smallest value for this hyperparameter that performed within 1 standard deviation of the optimal sample size. Both the elite portion and initial $\sigma$ parameters were chosen as the best performing values on a set of held-out random tasks for each domain.
For GD, we only performed a hyperparameter sweep over the learning rate, which was the only parameter to significantly affect performance. For the \twodfluids{} tasks, we introduced gradient clipping to eliminate the effect of rare gradient spikes over the course of optimization. However, not using gradient clipping still produced qualitatively and quantitatively similar results. For additional Adam parameters, we used the default values for the exponential decay rates that track the first and second moment of past gradients of $b_1=0.9$ and $b_2=0.999$ \cite{optax}.
\begin{center}
\begin{tabular}{ r | c c c c c c}
\toprule
& \multicolumn{3}{c}{\twodfluids{}} & \multicolumn{2}{c}{\threedfluids{}} & \airfoil{} \\
GD & \contain{} & \ramp{} & \maze{} & \direction{} & \emph{Pools} & \\
\midrule
Learning rate & 0.005 & 0.005 & 0.01 & 0.01 & 0.01 & 0.01 \\
Momentum term $b_1$ & 0.9 & 0.9 & 0.9 & 0.9 & 0.9 & 0.9 \\
Momentum term $b_2$ & 0.999 & 0.999 & 0.999 & 0.999 & 0.999 & 0.999 \\
Gradient clip & 10 & 10 & 10 & --- & --- & --- \\
& & & & & & \\
CEM & & & & & & \\
\midrule
Sampling size & 20 & 20 & 20 & 40 & 40 & --- \\
Elite portion & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & --- \\
Initial $\mu$ & 0 & 0 & 0 & 0 & 0 & --- \\
Initial $\sigma$ & 0.5 & 0.5 & 1.5 & 0.1 & 0.1 & --- \\
Evolution smoothing & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & --- \\
\midrule
Optimization steps & 1000 & 1000 & 1000 & 200 & 200 & 200 \\
\bottomrule
\end{tabular}
\label{tab:optimizer_hypers}
\end{center}
\begin{figure}[H]
\centering
\includegraphics[width=0.7\textwidth]{figures/CEM_sample_size_combined.png}
\caption{For CEM, increasing population size, while more computationally expensive, can lead to improvements in performance. (a) In the \threedfluids{} domain, CEM benefits from large sample sizes, although returns are diminishing for sizes beyond 40 (\direction{} task, 36 design parameters). (b) In the \twodfluids{} domain, CEM benefits from larger sample sizes, although returns are diminishing for sizes beyond 20 (\contain{} task, 40 design parameters).}
\label{fig:cem_samples}
\end{figure}
\section{Model architecture and training}
\label{app:model}
For each task domain, we train a GNN for next-step prediction of the system state.
For the domains considered in this paper, we unify the approaches of GNS \citep{sanchez2020learning} and \mgn{} \citep{pfaff2021learning}:
In the \airfoil{} domain, we encode/decode mesh nodes and mesh edges as a graph as described in the aerodynamics examples of \mgn{}, while for particle-based fluids, edges are generated based on proximity as in GNS. In the case of \threedfluids{}, both particles (fluid) and a mesh (the designed obstacle) are present; hence, edges are generated based on proximity (for fluid-fluid and fluid-obstacle interaction) or from the landscape mesh. As the landscape does not have any internal dynamics, we did not find it necessary to distinguish between world- and mesh edges, and use a single edge type.
Once encoded as a graph, the core model and training procedure is largely identical between GNS and \mgn{}, and we refer to the above papers for full details on architecture and model training.
Briefly, we use an Encode-Process-Decode GNN with 10 processor blocks. All edge and node functions are 2-layer MLPs of width 128, with ReLu activation and LayerNorm after each MLP block. The model is trained with Adam and a mini-batch size of 2, with training noise, for up to 10M steps. We implemented this model in JAX \citep{jax2018github}.
In addition to the different encoding procedures for mesh vs. particle systems, the parameters for training noise and connectivity radius have to be set per-domain, to account for differences in particle size/mesh spacing. These details are described in \autoref{app:domains}.
\paragraph{Gradient computation}
\label{app:gradients}
In our experiments, we pass gradients through long model rollouts of up to 300 steps. As it is prohibitive to store all forward activations for the backwards pass, we use gradient checkpointing \citep{chen2016training} to store activations only at the beginning of each step of the trajectory during the forward pass, and recompute the intermediate activations for each step as needed when the backwards pass walks the trajectory in reverse. Gradient calculation using this method has roughly 3 times the time cost of a pure forward simulation: forward dynamics have to be computed twice for each step, in addition to the computation of the backwards pass itself.
\section{Task domains}
\label{app:domains}
\subsection{\twodfluids{}}
\label{app:2dfluids}
Tasks in \twodfluids{} are procedurally generated from templates specified in \autoref{tab:2dfluids_tasks}.
The simulation domain is a 2D box, with the lower left corner specified as $[0, 0]$, and upper right corner specified as $[1,1]$. Fluid particles are initialized as a box of size \emph{Initial fluid box} with bounding boxes given in format $[x_\text{min}, y_\text{min}, x_\text{max}, y_\text{max}]$. Certain task parameters were varied for ablation experiments in \autoref{fig:2dfluids_ablations} (rollout length, \# joints (\contain{}), \# tools (\maze{})); \autoref{tab:2dfluids_tasks} contains default values used unless otherwise specified.
\begin{table}[t]
\begin{center}
\begin{tabular}{ r | c c c }
\toprule
& Contain & Ramp & Maze (nxn) \\
\midrule
Environment size & 1x1 & 1x1 & 1x1 \\
Rollout length & 150 & 150 & 150 \\
\\
Initial fluid box & [0.2, 0.5, 0.3, 0.6] & [0.2, 0.5, 0.3, 0.6] & [0.2, 0.75, 0.8, 0.8] \\
\\
Reward sampling box & [0.4, 0.1, 0.6, 0.3] & [0.8, 0, 1, 0.2] & [0.1, 0.1, 0.9, 0.2] \\
Reward $\sigma$ & 0.1 & 0.1 & 0.1 \\
\\
Design parameter & joint angles & joint angles & rotation \\
\# tools & 1 & 1 & $n^2$ \\
\# joint angles & 16 & 16 & 1 \\
\\
Tool position (left) & [0.15, 0.35] & [0.15, 0.35] & --- \\
Tool domain box (3x3) & --- & --- & [0.14, 0.3, 0.65, 0.6] \\
Tool domain box (4x4) & --- & --- & [0.14, 0.3, 0.71, 0.6] \\
Tool domain box (5x5) & --- & --- & [0.14, 0.3, 0.75, 0.6] \\
Tool domain box (6x6) & --- & --- & [0.14, 0.25, 0.77, 0.65] \\
\\
Tool Length & 0.8 & 0.8 & --- \\
Tool length (3x3) & --- & --- & 0.72 \\
Tool length (4x4) & --- & --- & 0.64 \\
Tool length (5x5) & --- & --- & 0.65 \\
Tool length (6x6) & --- & --- & 0.63 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Task Parameters for \twodfluids{} tasks. Boxes are described as $[x_\text{min}, y_\text{min}, x_\text{max}, y_\text{max}]$.}
\label{tab:2dfluids_tasks}
\end{table}
\paragraph{Design space}
A ``tool`` in this task domain is a 2D curve composed of several line segments connected by joints. For a large number of joints, a tool can thus approximate a smooth curve (\autoref{fig:hg_designspace_appendix}).
Each task's design space consists of the relative joint angles controlling the tool's shape. We consider tasks with a single, multi-segment tool (\contain{}, \ramp{}) and a task with multiple, single-segment tools (\maze{}).
For each tool, relative angles are calculated by moving from the anchor point on the left, along the tool segments to right, such that $\text{angle}_i = \text{angle}_{i-1} + {\phi_\mathrm{joints}}_i$ for the $i^\text{th}$ joint from the anchor.
We also experimented with two additional design space parameterizations: (1) jointly optimizing the joint angles and a global position offset $[x, y]$ for each tool, and (2) changing the parameterization of angles to be absolute (such that $\text{angle}_i = {\phi_\mathrm{joints}}_i$ directly).
We discuss the effects of these alternate parameterizations in \autoref{app:design-param-results}.
\begin{figure}[t!]
\centering
\includegraphics[width=0.8\textwidth]{figures/pivots.png}
\caption{Visualization of the design space parameterization for the \twodfluids{} task. Each red dot corresponds to the anchor points (\contain{} and \ramp{}) and center of rotation (\maze{}) being optimized.}
\label{fig:hg_designspace_appendix}
\end{figure}
\paragraph{Simulation and objective}
Both fluids and tools are represented as particles with different types, and simulated with the learned model for 150 steps (with the exception of the ablation experiment on rollout length). Scenes consist of $N=100 \ldots 1000$ fluid particles.
For ground-truth evaluation of the designs, we simulate particle dynamics with an MPM solver~\cite{hu2018moving}.
Task reward is calculated using the Gaussian likelihood of the final particle positions after rollout ($\mathbf{u_v}$ from $\tilde{G}^{t_K}$). That is, for a task with reward parameterized with mean $\mu$ and spherical covariance $\sigma$ ($\theta_R = [\mu, \sigma]$), the reward is calculated as
\begin{equation*}
f_R := \mathrm{mean}_v]\, \mathcal{N}(\mathbf{u_v}; \mu, \sigma)\,.
\end{equation*}
\paragraph{\contain{}}
For this task, the center of the goal region $\mu$ is sampled uniformly from a rectangular reward region in the lower-middle section of the $1\times1$ simulation domain ($[0.4, 0.6]\times[0.2, 0.4]$).
A tool protruding to the right is initially placed below the fluid rectangle.
By optimizing a single tool's relative joint angles, successful solutions must ``contain'' the fluid in the region by creating a cup or spoon.
\paragraph{\ramp{}}
The fluid and tool are initialized as in \contain{}, and $\mu$ is sampled from a region lower and further to the right than in \contain{} ($[0.8, 1]\times[0, 0.2]$).
By again optimizing a single tool's relative joint angles, successful solutions will create a ``ramp'' from the initial fluid position to the goal location in the bottom right.
\paragraph{\maze{}}
The goal is sampled from a long region near the bottom of the domain ($[0.1, 0.9]\times[0.1, 0.2]$).
By optimizing the rotation angles of a grid of rigid, linear tools, successful solutions will create a directed path from the top of the screen to the goal location at the bottom.
\paragraph{Model training}
We trained the learned simulator on the \textsc{WaterRamps} datasets released by \citet{sanchez2020learning}. This dataset consists of 1000 trajectories featuring a single large block of water falling on one to four randomized straight line segments (see image below for examples). Model architecture and hyperparameters are described in \autoref{app:model}, with a training noise scale of $6.7\,10^{-4}$ and connectivity radius of $0.015$.
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.7\textwidth]{figures/hg_trainig_data.pdf}
\caption{Four examples of trajectories from the \textsc{WaterRamps} dataset released by \citep{sanchez2020learning} used as training data for the supervised prediction model.}
\end{center}
\end{figure}
\subsection{\threedfluids{}}
\label{app:3dfluids}
\paragraph{Design space}
This domain has a design space $\phi_\mathrm{map}$ of 625 parameters, which determine the y coordinate offset to nodes of a $25 \times 25$ square mesh centered at $\mathbf{c}=(0.5, 0.5, 0.5)$ in the simulation domain.
While we could directly mapping the parameters to coordinates, we use the design function $y_i = \gamma_H \mathrm{tanh}(\phi_{\mathrm{map}_i})$ ($\gamma_H = 0.3$ for all tasks) to prevent trivial task solutions (i.e. obstacles which touch the floor).
\paragraph{Simulation}
The simulation consists of an inflow pipe located at (-0.5, 1.0, 0.5) above the landscape which continually emits a stream of liquid, represented as particles. These particles are then redirected by the designed landscape, and finally removed once they hit the floor at $y=0$.
In our experiments we observed up to 2084 particles present in the scene at one time. We unroll the learned simulation model for trajectories of 50 time steps, and store the final particle positions $\mathbf{u_v}$, as well as the positions of removed particles that touched the floor at any point $\mathbf{u_v^D}$ to be passed to the reward function.
Ground truth simulations for evaluation are performed by running the same setup with an SPH solver. We note that SPH requires very small simulation time steps, and performs $\approx 10^4$ internal steps for a trajectory of the same length.
\paragraph{\direction{}}
In this task, we want to align the water stream with a given direction vector $\mathbf{d}$. We can formalize this using the reward function
\begin{equation*}
f_{R_\mathrm{dir}} := \mathrm{mean}_v \left( (\mathbf{u_v} - \mathbf{c})\cdot \mathbf{d}\right) - \mathrm{std}_v \left( (\mathbf{u_v} - \mathbf{c})\cdot\mathbf{d_\perp}\right) - \gamma_R\, \mathrm{mean}(\nabla \phi_\mathrm{map})
\end{equation*}
where $\mathbf{d}_\perp$ is orthogonal to $\mathbf{d}$. The first term aligns the direction of the particle relative to the domain center, and the second term concentrates the stream. The last term is a smoothness regularizer on the design landscape, which prefers smooth solutions ($\gamma_R=300$ for both tasks). Absolute reward numbers for this task can be positive or negative, hence we report the normalized reward $f_{R_\mathrm{dir}} - f_{R_\mathrm{dir}}^\mathrm{initial}$, i.e. an unchanged initial design corresponds to a zero reward, to make the scores easier to interpret.
Rewards can be in 8 different directions, spaced between $0$ and $180\deg$. We collapse across directions for reporting reward means and confidence intervals for each optimizer.
\paragraph{\twopools{} and \threepools{}}
In these tasks, we define two and three pools, respectively, with center $\mathbf{\mu_p}$ on the floor. For each particle which has hit the floor, we assign it to its closest pool $\mathbf{\hat{\mu_p}}$, and define the reward as the Gaussian probability under $\mathbf{\hat{\mu_p}}$, i.e.
\begin{equation*}
f_{R_\mathrm{pools}} := \mathrm{mean}_v (\mathcal{N}(\mathbf{u_v^D};\mathbf{\hat{\mu_p}, \sigma})) - \gamma_R\,\mathrm{mean}(\nabla \phi_\mathrm{map})
\end{equation*} with $\sigma=0.4$ and a regularization term as above.
To showcase different ways of splitting the water stream, we consider one positioning of the pools for the two pool case, and two for the three pool case. In the two pool case, pools are placed at $[1.49, -0.35]$ and $[1.49, 1.35]$. In the three pool case, pools are placed either at $[1.6, -0.45]$, $[1.85, 0.5]$, and $[1.6, 1.45]$, or at $[0.5, -0.5]$, $[1.7, 0.5]$, and $[0.55, 1.5]$. These were selected to ensure the task was solveable -- pools directly beneath the landscape, or too far away from the landscape, would not be reachable even with dramatically warped surfaces.
\paragraph{Model training}
We trained a model on next-step prediction of particle positions, on a dataset of 1000 trajectories of water particles interacting with a randomized obstacle plane (random rotations and sine-wave deformations of the planar obstacle surface). The data was generated using the SPH simulator SPlisHSPlasH \citep{bender2015divergence}.
The noise scale is set to $0.003$ and a connectivity radius of $0.01$ to account for the different particle radius of the 3D SPH simulation compared to 2D MPM. All other architectural and hyperparameters are as described in \autoref{app:model}.
\subsection{\airfoil{}}
\label{app:airfoil}
The airfoil optimization task is modeled similarly to the NACA0012 aerodynamic shape optimization configuration for incompressible flow for the DAFoam solver (see details \href{https://dafoam.readthedocs.io/en/latest/Tutorial_Aerodynamics_NACA0012_Incompressible.html}{here}), to make it easier to compare design solutions to this solver.
\paragraph{Design space}
The design space consists of the $y$-coordinate of 10 control points (see \autoref{fig:airfoil}a). Moving these control points deforms both the airfoil, and the simulation mesh surrounding it. The airfoil shape is deformed using B-spline interpolation as described by \citet{ffd}, and the mesh is deformed using IDWarp~\citep{Secco2021}.
We thus define a design function $G^{t0} = f_D(\phi_\mathrm{ctrl}, G_{\alpha})$ which takes an initial, undeformed airfoil mesh (we use the standard NACA0012 airfoil), encoded as a graph $G_{\alpha}$, as well as the control point position $\phi_\mathrm{ctrl}$ as input. It returns the graph of the deformed airfoil mesh $G^{t_0}$ to be passed to the simulator.
We note that the coefficients for spline interpolation and mesh warping can be precomputed for a given initial mesh, making it easy to define a differentiable function to use for design optimization.
\paragraph{Simulation}
Given the initial mesh, as well as simulation parameters, the simulator or learned model predict the steady-state incompressible airflow around the wing, sampled on each of the 4158 nodes on the simulation mesh. The entire simulation domain and an example prediction of the pressure field are shown in \autoref{fig:airfoil_appendix}a,b.
For drag minimization we require predictions of the pressure field $p$, as well as the effective Reynolds stress $\rho_\mathrm{eff}$ at each mesh node, i.e. $\mathbf{q}_v=(p,\rho_{eff})$. Unlike the other domains in this paper, this is a single-step prediction task, and model rollouts are of length one.
For this task, we consider an inflow speed of 0.1 mach, under an $5.1^{\circ}$ angle of attack.
\paragraph{Task objective}
The task reward is defined as $f_R := -C_D - \gamma_L ||C_L - C_{L0}||^2 - \gamma_A\, a(\phi_\mathrm{ctrl})$, i.e. we minimize the drag coefficient $C_D$ under soft constraints of unchanged lift $C_L$ and a wing area $a$ of 1-3 times the initial area. We use $\gamma_L = 10, \gamma_A = 1$, and a tanh nonlinearity to enforce the volume inequality. Lift and drag can be computed from the simulation output $p, \rho_{eff}$ by integration around the airfoil, see e.g. \citep{ladson1988effects}. We report the normalized reward $f_R - f_R^\mathrm{initial}$ such that the initial, undeformed wing design corresponds to a zero reward.
\paragraph{Model training}
We trained a model to predict $p, \rho_{eff}$ on a dataset of 10000 randomized airfoil meshes, simulated with OpenFoam \citep{openfoam}. For training ensemble models, this dataset is split into 5 non-overlapping blocks, and a separate model is trained on each section. Since this is a steady-state prediction task, information needs to propagate further at each model evaluation. We therefore use twice-repeated processor blocks with shared parameters, i.e. the model performs 20 message passing steps, with 10 blocks of learnable parameters. We found that this increases accuracy in the one-step setup by being able to pass messages further across the mesh.
Training noise is often cited for stability over long rollouts, but even in this one-step setting, training noise and data variation can be useful. To increase robustness to unseen wing configurations, we varied the grid resolution between 1000-10000 nodes for each sample in the training set, and added training noise to the input mesh coordinates. We use a normal noise distribution with the scale of $1\%$ of the average edge lengths surrounding the node noise is applied to.
All other aspects of model architecture and training procedure are as described in \autoref{app:model}.
\begin{figure}[t!]
\centering
\includegraphics[width=0.6\textwidth]{figures/wing_pred.png}
\caption{(a) Aerodynamics are computed on a large 4158 node mesh centered around the airfoil, with the closeup regions around the airfoil in (b, c) marked as a white square in the center. (b, c) Pressure predictions and control points (orange) for the initial and final optimized wing design (Ensemble-5 model).}
\label{fig:airfoil_appendix}
\end{figure}
\section{Further results}
\subsection{Model accuracy}
\label{app:model-accuracy}
In order for a learned simulator to be useful for design, it must be sufficiently accurate in the forward direction. We study this question directly for each of the domains (\threedfluids{}, \twodfluids{} and \airfoil{}) by examining the magnitude of the error between the model predictions of reward for the discovered designs, and the ground truth reward for those designs. The results for each domain are shown in \autoref{fig:accuracy_comparison}.
Broadly, the learned model very successfully mimics the ground truth simulator in reward prediction across all three domains. The accuracy for \airfoil{} is within a single standard deviation across all ensemble sizes, while the predictions in \threedfluids{} match very closely for both the high performing designs (GD-M) and low performing designs (CEM-M).
However, we do notice some discrepancies in the predicted and ground truth reward for the \twodfluids{} domain, particularly the \maze{} task. As mentioned in the main text, the ground truth solver sometimes produces unrealistic rollouts for this domain (see \autoref{fig:mpm_instability}), with fluid particles becoming stuck between the different tools. Despite this issue, we find that the model is sufficiently similar to the ground truth to produce designs that still achieve high reward overall.
\begin{figure}[H]
\centering
\includegraphics[width=0.96\textwidth]{figures/appendix_performance_comparisons_combined.png}
\vspace{-1em}
\caption{(a) For the \threedfluids{} domain, reward predicted by the learned model (GD-M eval w/ M, CEM-M eval w/ M) is very close to the ground-truth simulator evaluation (GD-M, CEM-M) for all tasks. (b) This is also true for the \twodfluids{} domain, though the reward is slightly overestimated when using the model. This effect is amplified in \maze{}, where the ground-truth dynamics sometimes struggles to correctly simulate ``sticky'' bottlenecks (see \autoref{fig:mpm_instability}). (c) Model predictions of drag (GD-M eval w/ M) are relatively close to the ground-truth simulator evaluation (GD-M), particularly for larger ensemble sizes.}
\label{fig:accuracy_comparison}
\end{figure}
\subsection{Effects of design parameterization}
\label{app:design-param-results}
In this section, we study how different parameterization choices for the design space affect both gradient-based and sampling-based optimizers.
First, in the \threedfluids{} domain, we investigate a parameterization of the design space that uses interpolation to minimize the number of control points on the 2D heightfield (\autoref{fig:design_space_3d}). Control points are placed evenly across the grid, and bi-linearly interpolated onto the $25 \times 25$ mesh. We vary the number of control points from $2\times2$ up to $14\times14$, and find that while CEM-M performs similarly to GD-M when very few control points are allowed, its performance quickly drops as more control points are added.
\begin{figure}[H]
\centering
\hspace{-4mm}\includegraphics[width=0.8\textwidth]{figures/waterworks_design_space.png}
\caption{Performance on the \threedfluids{} \direction{} task with variable design space resolution: GD performs well for large design spaces, while CEM performance quickly drops with increased number of design parameters. (b) Examples of CEM designs at two different design space resolutions. (C) Examples of GD designs at two different design space resolutions.}
\label{fig:design_space_3d}
\end{figure}
Second, in the \twodfluids{} domain, we investigate what happens when we change the design space to use \emph{absolute} joint angles rather than relative ones. When using relative joint angles, changes to joints near the tool's pivot (left side) affect the global properties of the tool. We hypothesized that this could be selectively benefiting the sampling-based approaches, as this makes the effective design space much lower dimensional. We therefore change the design space to be absolute, with a tool's joint angles calculated directly: $angle_i = \phi_i$.
As hypothesized, this change does dramatically decrease the performance of the sampling-based technique (see \autoref{fig:absolute_vs_relative}). Perhaps more surprisingly, the gradient-based optimizer is almost completely unaffected by this reparameterization. While the qualitative solutions it finds differ (with tools now containing ``kinks'' to prevent the motion of the fluid rather than curves, \autoref{fig:absolute_vs_relative} top), the overall reward achieved is similar.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{figures/absolute_vs_relative.png}
\vspace{-2em}
\caption{\textbf{(a)} Example solutions for each optimizer across the \contain{} and \ramp{} tasks when optimizing over \textit{Relative} vs \textit{Absolute} angles. \textbf{(b)} Mean reward with $95\%$ confidence intervals obtained by each optimizer across the \contain{} and \ramp{} tasks when optimizing over \textit{Relative} vs \textit{Absolute} angles.}
\label{fig:absolute_vs_relative}
\end{figure}
\subsubsection{Failure modes of gradient descent}
\label{app:twodfluids-zero-gradients}
Other parameterizations of the design space can badly affect the performance of gradient-based optimizers. In particular, gradient-based optimizers suffer when there are regions of zero gradients. In the \airfoil{} and \threedfluids{} domains, this is not normally a problem, as the design always interacts with the physical system on which reward is being measured. But in the \twodfluids{} domain, we can manipulate this.
In particular, for this experiment we changed the design space for \twodfluids{} to include a global position offset $[x,y]$ for the tool. Making this simple change often has no effect on the discovered designs, but occasionally the gradient-based optimization procedure can move the tool such that it no longer interacts with the fluid (\autoref{fig:mpm_zero_grad}). Once the tool has been moved out of the range of the fluid, there is no longer any way to affect the reward, and therefore there is no gradient signal to recover. To overcome this problem, future work would need to consider more sophisticated hybrid optimization techniques \cite{toussaint2018differentiable}.
\begin{figure}[H]
\centering
\includegraphics[width=0.74\textwidth]{figures/zero_grad.png}
\includegraphics[width=0.24\textwidth]{figures/zero_grad_shift2.png}
\caption{Failure mode of the GD optimizer: in some instances where the translation of the tool is included in the design, the tool may end up outside of the scope of the fluid. In this cases, the optimization can no longer recover as it will get zero gradients from there on.}
\label{fig:mpm_zero_grad}
\end{figure}
\subsection{Failure modes of the MPM solver}
\label{app:twodfluids-mpm-instability}
One of the advantages of using a learned simulator over a classic simulator is learned simulators can be trained in regions of the state and action space that are known to exhibit regularized, smooth behaviors. For example, as mentioned in \autoref{app:model-accuracy}, the MPM solver \cite{hu2018moving} we use for evaluation in \twodfluids{} shows surprising irregularities with ``sticking'' behavior when there are a large number of different tools. In the \maze{} task, this is particularly prevalent, as fluids often become stuck stochastically in some funnels but not others of similar sizes (\autoref{fig:mpm_instability}).
Since the learned simulator was trained on much simpler scenarios where this effect is not observed, it only learns the smooth behavior of the fluid's movement, which makes the resulting trajectories look more realistic. This may enable better generalization to real world scenarios.
\begin{figure}[H]
\centering
\includegraphics[width=0.5\textwidth]{figures/taichi_sadness.png}
\caption{Left: MPM simulation \citep{hu2018moving} of a problem with many separate solid objects. As highlighted in the red circle, the MPM solver struggles with water movement between obstacles, often creating artificially sticky bottlenecks. Right: Learned model rollout for the same setup. The model rollout looks significantly more plausible, without any ``stickiness'' artifacts. Please see \url{https://sites.google.com/view/optimizing-designs} for videos demonstrating this effect clearly.}
\label{fig:mpm_instability}
\end{figure}
\subsubsection{Further designs found in \twodfluids{}}
\label{app:twodfluids-more-designs}
In the figures below, we demonstrate the range of found solutions for different solvers in tasks in the \twodfluids{} domain.
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/hg_episode_sweep_demo_relative_seed2_v2.png}
\caption{Example solutions for each optimizer across the range of rollout lengths sampled for \contain{} in \autoref{fig:2dfluids_ablations}.}
\label{fig:hg_episode_sweep_demos}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/hg_n_joints_sweep_demo_relative_seed2_v2.png}
\caption{Example solutions for each optimizer across the range of joint angle numbers sampled for \contain{} in \autoref{fig:2dfluids_ablations}.}
\label{fig:hg_n_joints_sweep_demos}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=.7\textwidth]{figures/hg_maze_num_tools_demo_seed1_v2.png}
\caption{Example solutions for each optimizer across the range of grid sizes sampled for \maze{} in \autoref{fig:2dfluids_ablations}.}
\label{fig:hg_maze_num_tools_demos}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/hg_seed_sweep_demo_relative_seed2_v2.png}
\caption{Example solutions on \contain{} task for each optimizer across 6 random seeds.}
\label{fig:hg_contain_seeds_demos}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/hg_ramp_seeds_sweep_demo_relative_seed2_v2.png}
\caption{Example solutions on \ramp{} task for each optimizer across 6 random seeds.}
\label{fig:hg_ramp_seeds_demos}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=\textwidth]{figures/hg_maze_seeds_sweep_demo_relative_v2.png}
\caption{Example solutions on \maze{} for each optimizer across 6 random seeds.}
\label{fig:hg_maze_seeds_demos}
\end{figure} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,946 |
package MakeMaker::Test::Setup::Problem;
@ISA = qw(Exporter);
require Exporter;
@EXPORT = qw(setup_recurs teardown_recurs);
use strict;
use File::Path;
use File::Basename;
use MakeMaker::Test::Utils;
my %Files = (
'Problem-Module/Makefile.PL' => <<'END',
use ExtUtils::MakeMaker;
WriteMakefile(
NAME => 'Problem::Module',
);
END
'Problem-Module/subdir/Makefile.PL' => <<'END',
printf "\@INC %s .\n", (grep { $_ eq '.' } @INC) ? "has" : "doesn't have";
warn "I think I'm going to be sick\n";
die "YYYAaaaakkk\n";
END
);
sub setup_recurs {
while(my($file, $text) = each %Files) {
# Convert to a relative, native file path.
$file = File::Spec->catfile(File::Spec->curdir, split m{\/}, $file);
my $dir = dirname($file);
mkpath $dir;
open(FILE, ">$file") || die "Can't create $file: $!";
print FILE $text;
close FILE;
# ensure file at least 1 second old for makes that assume
# files with the same time are out of date.
my $time = calibrate_mtime();
utime $time, $time - 1, $file;
}
return 1;
}
sub teardown_recurs {
foreach my $file (keys %Files) {
my $dir = dirname($file);
if( -e $dir ) {
rmtree($dir) || return;
}
}
return 1;
}
1;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,090 |
Hełm – rodzaj uzbrojenia ochronnego osłaniającego głowę. Znany i użytkowany od starożytności po współczesność. Dawniej stanowiący element zbroi chroniący przed ciosami zadawanymi bronią białą (np. mieczem, włócznią) lub przed pociskami broni neurobalistycznej (np. łuku, kuszy). Współcześnie funkcjonujący jako standardowy element wyposażenia żołnierzy chroniący przed odłamkami i szrapnelami.
Historia
Hełmy znane były już w starożytności (m. in. w Asyrii, Babilonii, Egipcie, Grecji czy Rzymie). Pierwotnie wykonywano je ze skóry lub płytek rogowych, następnie zaczęto stosować metal (początkowo brąz a następnie żelazo i stal). Najstarsze egzemplarze hełmów zachowanych do czasów współczesnych pochodzą z terenów Mezopotoamii z okresu ok. 3000 lat p. n. e. Przyjmuje się jednak, że wytwarzano je już wcześniej i wykształciły się niezależnie w różnych kręgach kulturowych na świecie. Hełmy w ciągu swojej historii przybierały bardzo zróżnicowane formy, w zależności od kręgu kulturowego, dostępnych technologii i przeznaczenia. Szczególnie intensywnym etapem ich rozwoju był okres średniowiecza i renesansu, kiedy to pojawiały się ich najbardziej złożone konstrukcje (np. przyłbica, armet, salada), co miało również związek z intensywnym rozwojem zbroi. Oprócz pełnienia funkcji bojowej, hełmy zyskiwały niekiedy również ważne znaczenie symboliczne jako oznaka władzy lub statusu (zob. hełm heraldyczny). Stosowano je również w charakterze sportowym na turniejach rycerskich (zob. hełm turniejowy).
Wraz z rozwojem broni palnej zarówno hełm jak i cała zbroja zaczęły stopniowo tracić na znaczeniu, ponieważ nie były w stanie zapewnić należytej ochrony przed coraz powszechniej stosowaną bronią strzelecką. Od ok. XVIII w. ze względu na niską przydatność, w wojsku powszechnie zaczęto zastępować je płóciennymi czapkami (np. trikorn, czako, rogatywka), co doprowadziło niemal do zaniku ich stosowania. Sam hełm zaczął zaś pełnić bardziej funkcję ozdobną niż praktyczną, pozostając niekiedy na wyposażeniu nielicznych formacji (jak np. ciężkiej jazdy – kirasjerów).
Hełm na nowo pojawił się w powszechnym użyciu wojskowym dopiero w czasie I wojny światowej. Miało to związek z masowym wykorzystaniem nowoczesnej artylerii, generującej olbrzymie straty wśród żołnierzy, których płócienne czapki nie zapewniały żadnej ochrony przed odłamkami i szrapnelami. Ze względu na fakt, iż gwałtownie zaczęła rosnąć liczba ofiar z ranami głowy (ciężkimi w leczeniu i często śmiertelnymi), zwrócono uwagę na konieczność zapewniania żołnierzom należytej ochrony. Pierwsi na ten problem zareagowali Francuzi, wprowadzając już w 1914 r. "stalowe ochraniacze" zakładane pod czapki, a następnie wprowadzając w 1915 r. stalowy hełm Adriana, który stał się pierwszym nowoczesnym hełmem wojskowym. Niemal od razu w ich ślady poszli Brytyjczycy, wprowadzając hełm Brodiego. Niedługo potem Inni członkowie Ententy, rozpoczęli przyjmowanie hełmów francuskich lub brytyjskich, bądź tworzyli ich lokalne modyfikacje. Zapotrzebowanie na stalowe hełmy nie pozostało również niezauważone w armiach państw centralnych, gdzie początkowo wprowadzono prowizoryczne hełmy Gaede, a następnie opracowano nowoczesny stahlhelm M1916.
Od tego czasu hełmy stały się standardowym wyposażeniem żołnierzy we wszystkich armiach i utrzymują się w tej roli do dziś. Współcześnie w ich produkcji zamiast stali stosuje się najczęściej włókna syntetyczne, zapewniające większą wytrzymałość (np. kevlar czy nylon balistyczny).
Budowa
Konstrukcję wszystkich hełmów stosowanych w historii łączył zasadniczo ten sam punkt odniesienia, determinowany przez kształt ludzkiej głowy i ułożenia na niej narządów (oczu, uszu, ust). W związku z czym, mimo występowania zróżnicowanych form konstrukcyjnych łączyła je konieczność osłony przede wszystkim mózgoczaszki, umożliwienia obserwacji, oddychania i słyszenia. Główną częścią hełmu jest jego dzwon, osłaniający największą część głowy i stanowiący podstawę do montażu ewentualnych elementów dodatkowych (w zależności od potrzeb).
W zależności od konstrukcji możemy wyróżnić 3 podstawowe typy hełmów:
Hełmy otwarte – osłaniające głowę, ale nie zakrywające twarzy (np. łebka, morion, a także wszystkie hełmy współczesne).
Hełmy półotwarte – osłaniające głowę i częściowo zakrywające twarz (np. niektóre typy kapalinów i salad).
Hełmy zamknięte – osłaniające głowę i zakrywające twarz (np. hełm garnczkowy, przyłbica).
Elementy składowe
1 dzwon
2 nakarczek
3 grzebień
4 daszek
5 nosal
6 policzek
7 zasłona
8 wizura
9 czepiec/kołnierz kolczy
10 fasunek
11 podpinka
Przykłady hełmów na przestrzeni wieków
Zobacz też
kask
hełmofon
hauba lotnicza
Przypisy
Bibliografia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 567 |
Q: Basic SSH port change not working on EC2 instance I am trying to change the port of SSH on an EC2 instance, but I am not getting this to work.
What I am doing is the following:
Open the file /etc/ssh/ssh_config
Uncomment and change the line Port 2345
Save and close
run the command service sshd restart
Now I can still connect to port 22. And when I run the command ssh -p 2345 localhost i get:
ssh: connect to host localhost port 2345: Connection refused
A: It may be a typo, but if you want to change the port that sshd listens on, you need to edit the Port setting in /etc/ssh/sshd_config, not ssh_config.
Also, you're using two different port numbers above (2345 and 2232), but I'm sure that's a typo.
If it's not that, could you edit into your question the output of iptables -L -n -v so we can see your firewall rules?
A: ssh_config is the SSH client configuration file.
sshd_config is the SSH daemon (server) configuration file.
So, if you want to change the port for the ssh server You have to edit the sshd_config file,
Uncomment the line
Port 22
and change port 22 to 2345.
After editing the file, restart the ssh service.
/etc/init.d/sshd restart
And also allow ssh port in the iptables. and restart the iptables.
A: Maybe there is something simple but you have Port 2345 in your config and then try to connect to 2232.
2345 != 2232
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,075 |
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"redpajama_set_name": "RedPajamaC4"
} | 4,598 |
When rolling percentages, do 1d100 and two d10s (percentiles) share the same probabilities?
I know d100s are shunned because they take too much time to roll, and 1d10+1percentile is really fun, but do they share the same probabilities? Is it better to use a (digital) d100?
Also, do d100s have 0? You can't roll a 0 with percentiles, I think.
dice statistics
GolokopitenkoGolokopitenko
Yes, a d100 is the same as 2d10 with one as the percentile.
A d100 goes 1–100, a d10 goes 0-9. Neither allows you to roll a 0, because of the way you count a percentile dice. (00 on the percentile and a 6 on the other dice forms 6, 00 on one and 0 on the other is 100, no option will result in 0.)
Do remember to use different colors of dice, else you will find things getting confusing quickly.
As an extra to the answer, it just occured to my why you asked the question. I guess you were wondering because 2d10 is not the same as a d20?
This is because (asides from one being a range of 2–20), there are multiple ways of getting the same result. If you roll 2d10, you can get a result of 7 by having: a 5 and a 2; a 6 and a 1, a 3 and a 4, etc. Because there is more than one way to get a '7' result, you are more likely to get a 7 than a 2, which you can only get with 2× 1.
In a situation with a percentile dice and a 'normal' one, there is only one way to get every result (1 in a 100, to get 54 you need 5 on the percentile and 4 on the normal, no other result will work).
LegendaryDude
TheikTheik
27.5k104104 silver badges145145 bronze badges
\$\begingroup\$ It's probably worth noting that the reckoning is different depending on whether or not you're using special percentile dice, where the first one is labeled 00-90 and thus 10 and 6 would be 16, not 6. \$\endgroup\$ – SirTechSpec Jun 10 '16 at 1:04
\$\begingroup\$ By the way: You can substitute a D20 with a D10 and a coin (aka "D2"). When the coin lands heads-up, add 10 to the D10. \$\endgroup\$ – Philipp Jul 28 '16 at 15:18
D100 and d%+d10 have exactly the same probabilities. If all 3 dice involved are fair, then they should come up with very similar distributions when rolled repeatedly. Obviously this isn't always the case as dice aren't consistent and there is a lot of randomness unless you roll a lot of times.
It seems there might be some confusion as to why d% doesn't have a bell shaped curve since it's two dice being rolled. It looks a lot like rolling 2d10 which is not the same as rolling d20 (or d19+1). 2d10 has a bell shaped curve because the results are added. there are the same number of permutations of 2d10 as there are of d%, but the number of possible results is 19 instead of 100.
To show this lets take 10 possible results from 2d10 and d%
\begin{array}{r|cc} & \rlap{\text{meaning as a...}}\\ \text{dice faces} & \text{2d10 result} & \text{d% result}\\ \hline 1,1 & 2 & 11\%\\ 1,2 & 3 & 12\%\\ 2,1 & 3 & 21\%\\ 2,2 & 4 & 22\%\\ 3,1 & 4 & 31\%\\ 3,2 & 5 & 32\%\\ 4,1 & 5 & 41\%\\ 4,2 & 6 & 42\%\\ 5,1 & 6 & 51\%\\ 5,2 & 7 & 52\%\\ 1,5 & 6 & 15\%\\ 2,5 & 7 & 25\%\\ \end{array}
As you can see there is no duplication of results between 2d10 and the d%. You can also observe that ordering of the dice is significant, this is why when we roll percentage dice we roll either two different colors, specifying which die is the 10s place and which is the 1s, or we use specially marked dice with 1-0 and 10-00.
As far as rolling 0%, no, neither a d100 nor a d% can roll a 0. They range from 1-100%.
wax eaglewax eagle
The standard method for 2d10-as-1d100 is to designate ahead of time which die will serve as the tens digit and which die will serve as the ones digit, treating a roll of 00 as equal to 100.
From a probabilistic standpoint (assuming fair dice), this is equal to 1d100. The key to understanding this is that you don't add the numbers up: each number is its own digit, independent of the others. This is what makes it work. There are two factors at play in that:
There are no outcomes on 2d10 which produce a result that couldn't be rolled on 1d100. Likewise, there are no result on a d100 that can't be produced on 2d10.
For every result that could be rolled on 1d100, there is exactly one 2d10 outcome which produces that result.
Let's compare this to, say, a claim that rolling 2d10 and summing the numbers (treating a 0 on each die as 10) is equivalent to rolling 1d20. The dice involved in the two claims are the same, but your claim is equivalent and mine is not.
Your claim fits the first point: I could name any result that a d100 could roll, and show how 2d10 could roll it. I could also do this backwards: if I name any result that 2d10 could roll, I can show how 1d100 could roll it. My claim fails this test, because I can roll a 1 on 1d20, but I can't roll it on 2d10.
Your claim also fits the second point: for any number I named that 2d10 could roll, I can show only one way to roll it. The real key here is that each result has the exactly the same number of ways to roll it: that number happens to be one, but that doesn't really matter here (though it makes the math easier). Because of this, all numbers on 2d10 are equally likely, just like on 1d100.
My claim is different. There's only one way to roll a 2 on 2d10-and-sum roll (1+1), and there's also only one way to roll a 20 (0+0): these two numbers are equally likely. But, for example, there are three ways to roll a 4 (1+3, 3+1, 2+2). This means that a 4 is more likely to come up than a 2. That's not like 1d20, where all results are equally likely, so they're not equivalent.
This is why 2d10-as-digits is equivalent to 1d100, even though 2d10-and-sum isn't equivalent to 1d20.
The SpooniestThe Spooniest
No, they do not yield the same probabilities. A d100, trademarked as a Zocchihedron, is not perfectly symmetrical, and it is not a fair die. Some numbers turn up with a higher frequency than others. The first version of the die turned up numbers lower than 8 and higher than 92 much less frequently than other numbers. These were slightly addressed in later versions, where the high and low values were painted on different faces, but that only adjusted the statistical distribution of the bad rolls; it couldn't fix the underlying problem.
The real problem is they are not evenly round. The ends are slightly pointed, so numbers painted on those faces rarely turn up. They are not fair, they cannot be made fair, and they do not produce the same probabilities as rolling 2d10.
A d10 is a pentagonal trapezohedron, which is a symmetrical fair die. Even an imperfect d10 is far more fair than a d100 can be, due to the limits of design. And 2d10 can be rolled as the two digits representing the 10's and 1's digits to yield a perfectly symmetrical distribution of values from 00-99, giving a nice flat probability curve.
A digital d100 dice app will also yield a symmetrical distribution of values from 0-99%, but it won't inspire as much excitement in your players as when you trot out the giant golf-ball of randomness.
John DetersJohn Deters
\$\begingroup\$ Note that the question asks for a comparison with a digital d100, putting the focus on ideal probabilities rather than dice imperfections. \$\endgroup\$ – SevenSidedDie Sep 16 '15 at 4:33
\$\begingroup\$ @SevenSidedDie, the questioner explicitly mentioned both the "slow" and "fun" of rolling d100s and d10s, implying comparisons of the physical objects. I answered the question asked, which was if the physical d100 shares the same probabilities as rolling two physical d10 dice, which it does not. In his second question, "is it better to use a (digital) d100?" he used the word digital only in parenthesis, de-emphasizing it. And it only asks about the subjective "better", without saying what outcome he wants: a "fun" experience, or a "statistically perfect outcome". \$\endgroup\$ – John Deters Sep 16 '15 at 4:54
\$\begingroup\$ Ah, I have finally hit upon why this answer seems so off-target. We all know that regular dice shapes are theoretically fair, but not usually actually fair. I'm not a fan of the golfball and I do think it's actually less fair, but comparing the actual-fairness of a d100 to the theoretical-fairness of d% is cherry-picking the evidence, by making an apples to oranges comparison, and such a maneuver actually invalidates the assertion it's meant to support. \$\endgroup\$ – SevenSidedDie Feb 10 '17 at 20:41
\$\begingroup\$ Yes, the golfball isn't redeemable with better tolerances. So, a simple improvement to the answer would be something like "A d10 is a pentagonal trapezohedron, which is a theoretically symmetrical fair die, and even imperfect real ones are more fair than the d100". That would eliminate the apples to oranges issue. And it would probably end up damning the the d100 more thoroughly, for pointing out its unfairness is worse even than the unfairness in common drum-rolled dice sets. \$\endgroup\$ – SevenSidedDie Feb 10 '17 at 21:07
\$\begingroup\$ Cool! Now I can feel good about giving an upvote. Thanks! \$\endgroup\$ – SevenSidedDie Feb 11 '17 at 22:18
Rolling 2 d10s is the same as 1d100 for one simple fact:
They are rolling the digits independently as opposed to rolling 3d6 or 1d18 in which the results are added together and the lower cap for each set can be different (3 or 1 respectively).
When you roll 2 d10s for the purpose of replacing 1d100 the lowest result you can have is 1 (created by rolling 10 on tens digit and a 1 on the ones digit) and the highest you can roll is 100 (10 on the tens digit and 10 on the ones digit).
TechImpTechImp
Yes. A 1d100 and a 2d10 share the same probability chances.
A 1d100 die, and 2d10 both have equal opportunities to represent numbers from 1-100. The 1d100 can display numbers 1-100, and the 2d10 can display the numbers 00-99. (In some games, a 0 on both dice is interpreted as "100" instead of "0". In this interpretation, the 2d10 displays 1-100, just like the d100.)
Both dice combinations have 100 possible results, and each of the 100 results are unique, which results an an equal probability that either has a chance to land on any specific number.
SandwichSandwich
the following only applies when dice rolls are treated as unique. In most cases, we don't treat dice like we do below. Normally, 1d6 + 1d8 would usually have a range of \$[2\mathrel{{.}\,{.}}14]\$, not \$[2\mathrel{{.}\,{.}}48]\$ because in most cases we sum dice instead of counting each individual combination as unique. Below, I use the notation 1d6 + 1d8 to mean a single roll of these and we are counting the unique possibilities the dice could provide. So 2d6 treats \$[1,3]\$ as a different roll than \$[3,1]\$ even though when we roll for damage, they're both just 4 (a pretty crappy roll, honestly...). So keep that in mind when reading this post!!
Mathematically, two rolls (\$A\text{d}B\$ and \$X\text{d}Y\$) have an exact mapping if and only if \$max(B^A,Y^X)\mod min(B^A,Y^X) = 0\$. That is to say if you take for each roll the number of sides raised to a power equal to the number of dice, and the lower value divides into the higher value, there will exist an exact mapping.
In your case, not only do they divide each other but they're equal (\$100^1 = 10^2)\$ so they have a 1:1 mapping. This means you can apply any 1:1 mapping you want. You could have \$[5,2]\$ on your d10s map to 59 on the d% if you really felt like it and had a methodology for tracking what results map to each other.
You can try this with any combination: 5d2 ≉ 2d6 because \$2^5 = 32\$ and \$6^2 = 36\$. However, you could come very close to approximating your 5 coins with a 2d6 roll and just mapping 4 values to reroll.
You can even combine dice, like so:
1d4, 1d2 ≈ 1d8 because \$4✕2 = 8\$
2d5, 1d4 ≈ 2d10 because \$5^2 ✕ 4^1 = 25 ✕ 4 = 10^2 = 100\$
2d8, 1d10 ≈ 3d6, 1d12, 1d20 because \$8^2 = 64 ✕ 10 = 640\$ and \$6^3 = 216 ✕ 12 ✕ 20 = 51840\$ and \$51840 \mod 640 = 0\$
When you add dice rolls, you multiply their power results together. So that last roll means that if you had needed to simulate [2d8, 1d10] but you only had [3d6, 1d12, 1d20], you could still do it with a proper mapping. (For those keeping track, it's an 81:1 mapping because \$51840 / 640 = 81\$. This means that you can do anything you want to eliminate 81 (\$3^4\$) from the dice you do have to make it work. There's only four 3s in the dice you have, so you have to map your 3d6 to 3d2 (using 1,2,3 vs 4,5,6 or odd vs even) and then mapping your 1d12 to 1d4. Now you have mapped both sides to 640 combinations, from which you can apply your garden variety 1:1 mapping.)
corsiKacorsiKa
\$\begingroup\$ What is an exact mapping? I know what injections, surjections and bijections are. \$\endgroup\$ – Tommi May 10 '17 at 8:19
\$\begingroup\$ @Thanuir For example, you can make an exact mapping of a regular six sided die to a coin (a two sided die) by mapping 1,2,3 to heads and 4,5,6 to tails. You cannot make an exact mapping of a six sided die to a four sided die because there would be leftover elements. So while bijective is one-to-one, the exact mapping is X-to-one where no element in set A points to multiple elements in B. If you grouped the the elements in A into sets, those sets would be bijective onto B. \$\endgroup\$ – corsiKa May 10 '17 at 16:15
\$\begingroup\$ Okay, so a mapping where the pre-image of every element has the same cardinality (and so a surjection, in particular, unless one considers the map from the empty set to a non-empty set). Is this standard terminology in number theory, algebra, or in some other field? Anyway, you might want to clarify this in the post itself, as I highly doubt I am the only reader who did not immediately understand the terminology. \$\endgroup\$ – Tommi May 11 '17 at 6:30
\$\begingroup\$ As I researched the topic I was unable to find a standard term to describe the phenomenon. Perhaps a question for math stack? \$\endgroup\$ – corsiKa May 14 '17 at 17:09
\$\begingroup\$ I am reasonably familiar with mathematics terminology, which is why I asked - this would have been something I should know if it was established terminology. \$\endgroup\$ – Tommi May 15 '17 at 10:44
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Lawrence Solomon continues to vocalize on his views that smart grids are an uneconomic, politically motived fantasy. Instead of seeing the potential for Smart Grids to usher in competitive, market-based pricing for electricity, he appears to believe that we are best served with the current rigid system where it is quite difficult to balance supply and demand.
Perhaps clarifying some of the assertions from his most recent post can help people understand the situation more clearly.
Correction #1 – Like the Internet, the electrical grid is a system that transports a commodity from where it is produced to where it is consumed. The Internet transportation network itself is not any more inherently profitable than the power network. Both systems can be subject to congestion and both operate as a cost center, not a profit center.
Correction #2 – Assigning profitability to customer equipment is a somewhat misleading concept. Are laptops and other capable electronic devices inherently any more profitable than electrical appliances such as water heaters, lighting fixtures, fridges, stoves, dishwashers and air conditioners?
Correction #3 – Value can and should be defined by the interplay of supply and demand. In the media world, a proxy for that interplay of supply and demand is advertising spot rates that sell for premium prices during prime time hours. Why would it not make sense for the spot price of electricity to move up and down in response to the interaction of the forces of supply and demand in a comparable manner?
Faulty Assertion #2 – Unlike most conventional power plants, solar and wind technologies can't be powered on and off as needed to meet the varying demands of customers.
Correction #4 – In Ontario, 46% of power capacity is supplied by the combination of coal and nuclear thermal electric plants. These plants required several hours to several days to ramp up from a cold start. They operate most effectively when they operate at full capacity all the time. As a result, coal and nuclear plants tend to be operated more constantly and contributed 63% of power produced in 2010. The ideal customer for these plants is one that has a steady demand around the clock and from day-to-day.
Unfortunately, customers are not ideal. Demand can and does cycle up and down by over 35% during the course of the day. Peak demand on hot or cold days can be even more extreme. If the increase of demand is predictable, a gas fired plant can be fired up to produce power, but that comes at the cost of consuming fuel. Hydro is much more flexible, but there is a limited amount of hydro power that can be generated at any one time or can be ultimately developed.
Another issue that arises is capacity utilization. Both hydro dams and natural gas plants are expensive capital infrastructure assets. Those assets may operate 50% or less of the time to meet peak demand. The rest of the time they are idle. In 2006, for example, peak power consumption reached 25,000 MW for only 32 hours of the year or the equivalent of a 0.3% capacity utilization rate for the last 1 MW of production capacity. At a capital cost of at least $1 million per MW of production capacity, some might consider that peak power quite expensive.
Hydro power has one very appealing characteristic, most dams can store water so that power can be produced on demand. Sometimes, the call for that power may come from peak demand by customers, at other times, it may arise because alternative sources of production are not available. They are the closest thing to an energy storage "battery" in the system.
Coupling wind and solar systems with hydro assets can be an elegant solution because the combination enables hydro operators to conserve water to be used to meet peak demand and hopefully capture premium prices for that energy. The hydro capacity that needs to be in place to meet peak demand, can also be used to shape and firm power from intermittent sources of production. In a sense, the coupling of solar and wind with hydro is somewhat equivalent to expanding the storage capacity of the hydro reservoir to store more water and thus sell more premium priced electricity when prices are at their highest.
Faulty Assertion #3 – The smart grid would solve the problem of instability by controlling the customers instead of the technologies. To protect the grid from sudden drops in the power being produced, for example, the smart grid engineers would reach into our homes and businesses to instantly turn off our refrigerators, freezers, washing machines, air conditioners, and other smart appliances as needed to match the sudden power losses.
Correction #5 – The scenario presented is more characteristic of a dumb grid instead of a smart grid. A smart grid should include "smart appliances" that can be preset to consume power in response to changes in pricing. Some uses of power, such as heating water for the house, a pool or a hot tub can be deferred to a period when energy is inexpensive. The same goes for energy consumed by cooling storage systems like fridges or freezers. Why not set the dishwasher to run at night when prices are lower?
At any time, the customer would still have the ability to over-ride the programming if they wanted power on demand. The only difference would be those "energy hogs" would be paying a premium to satisfy their desire to consumer energy during periods of peak demand. How is this situation any different than a home owner who uses a programmable thermostat today? Does the consumer really need to keep the temperature of the hot tub at 104 degrees around the clock when it is not being used? Does it matter if the freezer varies in temperature a bit as long as the food remains frozen? If the consumer had a price signal that turning on the air condition during peak hours will require buying premium priced power, might they opt to turn on a fan instead? Do the dinner dishes really need to be washed right now?
Smart Grids are not about enabling "the grid engineer to reach into our homes." No, a true Smart Grid will present visible, transparent, and competitively-determined price signals to consumers so they can see the cost associated with individual actions. That would be a substantial improvement over today's situation where price signals are not getting through to end users in on a timely basis. To parrot the CRTC and Bell UBB line, why should I pay for the slothful, anti-social habits of energy hogs who persist on loading the network with demand during peak periods when supplying energy is quite expensive? Smart Networks enables the system to make those energy hogs pay for their behavior.
Faulty Assertion #4 – …smart grid engineers would reach into our pocketbooks, by pricing power cheaper in the middle of the night, on political criteria, to encourage us to soak up an excess of power that their anti-fossil fuel scheme has produced.
Correction #6 – If Smart Grids present real-time pricing to the consumer and those prices are determined by competitive markets that balance supply and demand, how can it be a bad thing to price energy at a cheaper price when it is less expensive to produce and deliver it? It will be the market sending the price signal not some bogeyman "network engineer." Where is there "political criteria" embedded in this type of market-based system?
Correction #7 – That excess power that needs to be soaked up will primarily come from big coal and nuclear thermal plants that need operate 24-7 to be most efficient. If anything, the operation of wind or solar during off-peak hours enables hydro plants to store water so it can be used when energy prices produce peak revenues. Referring to the pattern of demand going up during the day, is it not apparent that solar energy will tend to generate power when it is needed during the day instead of the middle of the night?
Faulty Assertion #5 – Most of that excess power, they fantasize, will recharge the batteries of our electric vehicles as we sleep. But electric cars are going nowhere, the marketplace has made clear — they remain unaffordable even with big rebates on the vehicle's purchase price.
Correction #8 – As indicated previously, there are uses of energy in the home, such as hot water heating, operating fridges and freezers, washing dishes, and space heating/cooling that can behave like a battery in that the benefit can be stored or deferred to a less expensive time to operate. Why won't a consumer want to heat their pools and hot tubs or wash their dishes at night when power prices are at a minimum?
Second, I wonder how demand for plug-in electric cars or even fully electric commuter cars would fare if they could be "filled up" over night with low cost electricity instead of consuming expensive gasoline? I might purchase one as a second commuter vehicle if I could operate it at a per-kilometer cost that is lower than the gasoline powered alternative. Why not let the Smart Grid reflect market pricing and let consumers make an informed decision?
Faulty Assertion #6 – The world's electricity systems will remain predominantly fossil fuelled, and because fossil fuels are both flexible and cheap, they won't require a smart grid to manage them.
Correction #9 – As indicated previously, coal fired thermal plants are not very flexible. Neither are nuclear thermal electric plants. Presenting real-time price signals to consumers can enable those components of the production mix to operate a peak efficiency and rates of profitability.
Natural gas itself is plentiful and is currently reasonably inexpensive. However, if natural gas was used to both replace legacy coal capacity and meet new demand, is there sufficient capacity in pipeline networks to meet that demand? How many billions of dollars would need to be spent to expand that capacity? Even if that pipeline capacity is expanded, will the increased demand for natural gas result in escalating natural gas prices that could be double or triple current prices? How inexpensive will electricity from natural gas plants be under that situation?
Replacing legacy power plants with new capacity and expanding capacity to meet future growth of demand is not going to be inexpensive. Hydro capacity is limited and recent experience indicates their costs and the cost of their long transmission lines can be subject to considerable inflation over time. Besides, there is not enough potential hydro resources that can be developed in North America to fully satisfy the market.
Assertion #7 – The one smart grid that was completed — a small smart grid in Boulder, Colo., called Smart Grid City — came in at $100-million, three times the original cost estimates, and at a cost of $2,000 per billpayer, it has little value to show for itself.
Explanation #10 – This assertion could benefit from having Paul Harvey around to tell "the rest of the story." The Boulder Smart Grid system included a build out of a broadband fibre communications network into the neighborhood level of distribution. They essentially built out a broadband delivery network that could compete with cable or telco broadband services. To suggest that the full $2,000 cost per subscriber should be allocated to the Smart Grid purpose is akin to suggesting that the full cost of cellular telephone service should be allocated and paid for by people's texting use of the network.
If Lawrence dug deeper, he would find that several Canadian utilities, including Toronto Hydro, have built out fibre communications networks in the past to support both their internal needs and the delivery of commercial broadband services. Some of the municipal utilities in Ontario went on to sell a portion of those networks for a considerable profit. For example, Ottawa Hydro booked a $20 million profit on their network to Atria Networks. Rogers in turn bought that company for $425 million. It is premature to write the Boulder network off as a loss or a boondoggle.
What should concern Lawrence is the situation where a power utility deploys fibre optic infrastructure to satisfy its internal needs and then enters into a non-compete agreement with incumbent telco's to never make that broadband capacity available to third parties. If the utility is going to string fibre lines to support internal communications with their substations and other equipment in the distribution network, surely it must make sense to use those assets to support competition and the delivery of competitive broadband services.
Smart Grids can offer a means to send price signals to consumers that enables competition and competitive supply markets to work effectively. What we need today is informed discussion about how the system can be used to create open and dynamic competition throughout the system instead of perpetuating the past model of dumb networks, dumb pricing, and dumb regulation by government. The technology offer the possibility but steps need to be taken in corporate and public policy to ensure the benefits are realized. | {
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Alice Stokes Paul (January 11, 1885 – July 9, 1977) was an American Quaker, suffragist, feminist, and women's rights activist, and one of the foremost leaders and strategists of the campaign for the Nineteenth Amendment to the U.S. Constitution, which prohibits sex discrimination in the right to vote. Paul initiated, and along with Lucy Burns and others, strategized events such as the Woman Suffrage Procession and the Silent Sentinels, which were part of the successful campaign that resulted in the amendment's passage in August 1920.
Paul often suffered police brutality and other physical abuse for her activism, always responding with nonviolence and courage. She was jailed under terrible conditions in 1917 for participating in a Silent Sentinels protest in front of the White House, as she had been several times during earlier efforts to secure the vote for women in England.
After 1920, Paul spent a half-century as leader of the National Woman's Party, which fought for the Equal Rights Amendment, written by Paul and Crystal Eastman, to secure constitutional equality for women. She won a major permanent success with the inclusion of women as a group protected against discrimination by the Civil Rights Act of 1964.
Early life and education
Alice Stokes Paul was born on January 11, 1885, to William Mickle Paul I (1850–1902) and Tacie Parry Paul (1859–1930) at Paulsdale in Mount Laurel Township, New Jersey. She was a namesake for Alice Stokes (1821–1889), her maternal grandmother and the wife of William Parry (1817–1888). Her siblings were Willam Mickle Paul II (1886–1958), Helen Paul Shearer (1889–1971), and Parry Haines Paul (1895–1956). She was a descendant of William Penn, the Quaker founder of Pennsylvania. Her ancestors included participants in the New Jersey Committee of Correspondence in the Revolutionary era and a state legislative leader in the 19th century. She grew up in the Quaker tradition of public service. Alice Paul first learned about women's suffrage from her mother, a member of the National American Woman Suffrage Association (NAWSA), and would sometimes join her mother in attending suffragist meetings.
Paul attended Moorestown Friends School, where she graduated at the top of her class. In 1901, she entered Swarthmore College, which had been co-founded in 1864 by her grandfather and other Hicksite Friends. While at Swarthmore, Paul served on the Executive Board of Student Government, an experience which may have sparked her excitement for political activism. She graduated from Swarthmore with a bachelor's degree in biology in 1905.
After graduation, partly to avoid going into teaching, Paul pursued a fellowship year in New York City, living on the Lower East Side at the Rivington Street Settlement House. Working in the settlement movement reinforced her determination to right perceived injustices in America, but Paul soon realized that social work was not the way she was to achieve this goal: "I knew in a very short time I was never going to be a social worker, because I could see that social workers were not doing much good in the world... you couldn't change the situation by social work."
In 1907, after completing coursework in political science, sociology, and economics, Paul earned a Master of Arts degree from the University of Pennsylvania. She continued her studies at the Woodbrooke Quaker Study Centre in Birmingham, England. Paul also took economics classes from the University of Birmingham while continuing to earn money doing social work. It was at Birmingham that she first heard Christabel Pankhurst speak. When Paul later moved to London to study sociology and economics at the London School of Economics, she joined the militant suffrage group the Women's Social and Political Union (WSPU) led by Christabel and her mother, Emmeline Pankhurst. Paul was arrested repeatedly in London during suffrage demonstrations and served three jail terms. After returning from England in 1910, she attended the University of Pennsylvania, earning a Ph.D. in sociology. Her dissertation was entitled "The Legal Position of Women in Pennsylvania"; it addressed the history of the women's movement in Pennsylvania and the rest of the U.S. and urged woman suffrage as the key issue of the day.
After the ratification of the 19th Amendment, Paul enrolled at two law schools, taking day and evening classes to finish more quickly. In 1922, Paul received her law degree (LL.B) from the Washington College of Law at American University. In 1927, she earned a master of laws degree, and in 1928, a doctorate in civil law from American University.
Career
Britain
Early work in British woman suffrage
In 1907, after completing her master's degree at the University of Pennsylvania, Paul moved to England, where she eventually became deeply involved with the British women's suffrage movement, regularly participating in demonstrations and marches of the Women's Social and Political Union (WSPU). After a "conversion experience" seeing Christabel Pankhurst speak at the University of Birmingham, Paul became enamored of the movement. She first became involved by selling a Suffragist magazine on street corners. Considering the animosity towards the Suffragists, this was an arduous task and opened her eyes to the abuse women involved in the movement faced. These experiences, combined with the teachings of Professor Beatrice Webb, convinced Paul that social work and charity could not bring about the needed social changes in society: this could only be accomplished through equal legal status for women.
While in London, Paul also met Lucy Burns, a fellow American activist, while arrested in a British police station, who would become an essential ally for the duration of the suffrage fight, first in England, then in the United States. The two women quickly gained the trust of prominent WSPU members and began organizing events and campaign offices. When Emmeline Pankhurst attempted to spread the movement to Scotland, Paul and Burns accompanied her as assistants.
Paul quickly gained the trust of fellow WSPU members through her talent with visual rhetoric and her willingness to put herself in physical danger to increase the visibility of the suffrage movement. While at the WSPU's headquarters in Edinburgh, Paul and local suffragists made plans to protest a speech by the Minister of Foreign Affairs, Sir Edward Grey. For a week prior, they spoke with people on the streets to promote knowledge about why they were protesting against the Cabinet member. After Grey discussed proposed legislation he claimed would lead to prosperity at the meeting, Paul stood up and exclaimed: "Well, these are very wonderful ideals, but couldn't you extend them to women?" Police responded by dragging her out of the meeting and through the streets to the police station, where she was arrested. As planned, this act was viewed by many as a public silencing of legitimate protest and increased press coverage and public sympathy.
Later events involved even more risk of bodily harm. Before a political meeting at St. Andrew's Hall in Glasgow in August 1909, Paul camped out on the hall's roof so that she could address the crowd below. When police forced her to descend, crowds cheered her effort. Later, when Paul, Burns, and fellow suffragettes attempted to enter the event, they were beaten by police as sympathetic bystanders attempted to protect them. After Paul and her fellow protesters were taken into custody, crowds gathered outside the police station demanding the women's release.
On November 9, 1909, in honor of Lord Mayor's Day, the Lord Mayor of London hosted a banquet for cabinet ministers in the city's Guild Hall. Paul planned the WSPU's response; she and Amelia Brown disguised themselves as cleaning women and entered the building with the normal staff at 9:00 am. Once in the building, the women hid until the event started that evening. Then they came out of hiding and "took their stand". When Prime Minister H. H. Asquith stood to speak, Brown threw her shoe through a pane of stained glass, and both women yelled, "Votes for women!" Following this event, both women were arrested and sentenced to one-month hard labor after refusing to pay fines and damages for the window damage. She was imprisoned at Holloway Prison in London.
Civil disobedience and hunger strikes
While associated with the Women's Social and Political Union, Paul was arrested a total of seven times and imprisoned three times. It was during her time in prison that she learned the tactics of civil disobedience from Emmeline Pankhurst. Chief among these tactics was demanding to be treated as a political prisoner upon arrest. This not only sent a message about the legitimacy of the suffragists to the public but also had the potential to provide tangible benefits. In many European countries, including England, political prisoners were given a special status: "[T]hey were not searched upon arrest, not housed with the rest of the prisoner population, not required to wear prison garb, and not force-fed if they engaged in hunger strikes." Though arrested suffragettes often were not afforded the status of political prisoners, this form of civil disobedience provided much press for the WSPU. For example, during a London arrest (after being denied political prisoner status), Paul refused to put on prisoner's clothing. After the prison matrons could not undress her forcibly, they requested assistance from male guards. This act, considered shockingly improper by Victorian era standards, provided extensive press coverage for the suffrage movement.
Another popular civil disobedience tactic used by the Suffragettes was hunger striking. The first WSPU-related hunger strike was conducted by sculptor Marion Wallace Dunlop in June 1909. By that fall, it was being widely used by WSPU members because of its effectiveness in publicizing their mistreatment and gaining quick release from prison wardens. Refusing food worked in securing an early release for Paul during her first two arrests. However, during her third prison stint, the warden ordered twice daily force-feeding to keep Paul strong enough to finish her month-long sentence.
Though the prisons staunchly maintained that the force-feeding of prisoners was for their own benefit, Paul and other women described the process as torturous. Paul had developed severe gastritis at the end of her month in prison. She was carried out of prison and immediately tended to by a doctor. However, after this event, her health was permanently scarred; she often developed colds and flu, which would sometimes require hospitalization.
Paul had been given a Hunger Strike Medal 'for Valour' by WSPU.
United States
After the ordeal of her final London imprisonment, Paul returned to the United States in January 1910 to continue her recovery and to develop a plan for suffrage work back home. Paul's experiences in England were well-publicized, and the American news media quickly began following her actions upon her return home. She drew upon the teachings of Woodbrooke and her religion and quickly decided that she wanted to embrace a single goal as a testimony. The single goal she chose was the recognition of women as equal citizens.
Paul reenrolled at the University of Pennsylvania, pursuing her Ph.D. while speaking about her experiences in the British suffrage movement to Quaker audiences and starting to work towards United States suffrage on the local level. After completing her dissertation, a comprehensive overview of the history of the legal status of United States women, she began participating in National American Woman Suffrage Association (NAWSA) rallies, and in April 1910, was asked to speak at NAWSA's annual convention. After this significant opportunity, Paul and Burns proposed to NAWSA leadership a campaign to gain a federal amendment guaranteeing the vote for women. This was wholly contrary to NAWSA's state-by-state strategy. Paul and Burns were laughed at by NAWSA leadership; the only exception was Jane Addams, who suggested that the women tone down their plan. As a response, Paul asked to be placed on the organization's Congressional Committee.
1913 Woman Suffrage Procession
One of Paul's first big projects was initiating and organizing the 1913 Woman Suffrage Procession in Washington, D.C., the day before President Wilson's inauguration. Paul was determined to pressure Wilson because the office of the President would be able to influence Congress the most. She assigned volunteers to contact suffragists nationwide and recruit supporters to march in the parade. In a matter of weeks, Paul succeeded in gathering roughly eight thousand marchers representing most of the country. However, she had much more trouble gaining institutional support for the protest parade. Paul insisted the parade route go through Pennsylvania Avenue where President Wilson would be. Her goal was to send the message that the push for women's suffrage existed before Wilson and would outlast him if need be. Washington, D.C. officials originally resisted this route, and according to biographer Christine Lunardini, Paul was the only one who truly believed the parade would take place on that route. Eventually, the city ceded the route to NAWSA. However, the city supervisor claimed that the women would not be safe marching along the Pennsylvania Avenue route and strongly suggested the group move the parade. Paul responded by demanding the city supervisor provide more police, which was not done. On March 3, 1913, the parade gained legitimacy with Congress passing a special resolution ordering the city supervisor to prohibit all ordinary traffic along the parade route and prevent any interference with the suffrage marchers.
On the event day, the procession proceeded along Paul's desired route. The event, which was led by notable labor lawyer Inez Milholland dressed in white and riding a horse, was described by the New York Times as "one of the most impressively beautiful spectacles ever staged in this country". Multiple bands, banners, squadrons, chariots, and floats were also displayed in the parade representing all women's lives. One of the most notable sights was the lead banner in the parade which declared, "We Demand an Amendment to the United States Constitution Enfranchising the Women of the Country." Some participating groups and leaders, however, wanted black and white women's organizations and state delegations to be segregated; after much discussion, NAWSA decided black women could march where they wished. Still, Ida B. Wells was asked not to march with the Illinois delegation; ultimately, she joined the Chicago group and continued the march with the state delegation.
Over half a million people came to view the parade. With insufficient police protection, the situation soon devolved into a near-riot, with onlookers pressing so close to the women that they could not proceed. Police largely did nothing to protect the women from rioters. A senator who participated in the march later testified that he personally took the badge numbers of 22 officers who had stood idle, including 2 sergeants. Eventually, the Massachusetts and Pennsylvania national guards stepped in, and students from the Maryland Agricultural College provided a human barrier to help the women pass. Some accounts even describe Boy Scouts as stepping in and providing first aid to the injured. The incident mobilized public dialogue about the police response to the women's demonstration, producing greater awareness and sympathy for NAWSA.
After the parade, the NAWSA's next focus was lobbying for a constitutional amendment to secure the right to vote for women. Such an amendment had been initially sought by suffragists Susan B. Anthony and Elizabeth Cady Stanton who, as leaders of the NWSA, fought for a federal amendment to the constitution securing women's suffrage until the 1890 formation of NAWSA, which campaigned for the vote on a state-by-state basis.
National Woman's Party
Paul's militant methods started to create tension between her and the leaders of NAWSA, who thought she was moving too aggressively in Washington. Eventually, disagreements about strategy and tactics led to a break with NAWSA. Paul formed the Congressional Union for Woman Suffrage and, later, the National Woman's Party (NWP) in 1916.
The NWP began introducing some of the methods used by the suffrage movement in Britain and focused entirely on achieving a constitutional amendment for woman suffrage. Alva Belmont, a multi-millionaire socialite at the time, was the largest donor to Paul's efforts. The NWP was accompanied by press coverage and the publication of the weekly newspaper, The Suffragist.
Silent Sentinels
In the U.S. presidential election of 1916, Paul and the National Woman's Party (NWP) campaigned in western states where women could already vote against the continuing refusal of President Woodrow Wilson and other incumbent Democrats to actively support the Suffrage Amendment. Paul went to Mabel Vernon to help her organize a picketing campaign. In January 1917, the NWP staged the first political protest and picketing at the White House. Picketing had been legalized by the 1914 Clayton Antitrust Act, so the women were not doing anything illegal. The pickets, participating in a nonviolent civil disobedience campaign known as the "Silent Sentinels", dressed in white, silent and with 2,000 taking part over two years, maintained a presence six days a week, holding banners demanding the right to vote. Paul knew the only way they could accomplish their goal was by displaying the President's attitude toward suffrage, so picketing would achieve this in the best manner. Each day Paul would issue "General Orders", selecting women to be in charge and who would speak for the day. She was the "Commandant", and Mabel Vernon was the "Officer of the Day". Paul created state days to get volunteers for the pickets, such as Pennsylvania Day, Maryland Day, and Virginia Day. She also made special days for professional women, such as doctors, nurses, and lawyers.
After the United States entered World War I in April 1917, many people viewed the picketing Silent Sentinels as disloyal. Paul made sure the picketing would continue. In June 1917, picketers were arrested for "obstructing traffic". Over the next six months, many, including Paul, were convicted and incarcerated at the Occoquan Workhouse in Virginia (which later became the Lorton Correctional Complex) and the District of Columbia Jail.
When the public heard the news of the first arrests, some were surprised that leading suffragists and very well-connected women were going to prison for peacefully protesting. President Wilson received bad publicity from this event and was livid with the position he was forced into. He quickly pardoned the first women arrested on July 19, two days after they had been sentenced, but reporting on the arrests and abuses continued. For example, the Boston Journal stated, "The little band representing the NWP has been abused and bruised by government clerks, soldiers, and sailors until its efforts to attract the President's attention has sunk into the conscience of the whole nation."
Suffragists continued picketing outside the White House after the Wilson pardon and throughout World War I. Their banners contained such slogans as "Mr. President, How Long Must Women Wait For Liberty?" and "We Shall Fight for the Things Which We Have Always Held Nearest Our Hearts—For Democracy, For The Right of Those Who Submit To Authority To Have A Voice in Their Own Governments." The capitalization of each word emphasized the gravity of the situation. With the hope of embarrassing Wilson, some of the banners quoted Wilson's own words against him. Wilson ignored these women, but his daughter Margaret waved in acknowledgment, a major victory for the protesters. Although the suffragists protested peacefully, their protests were sometimes violently opposed. While protesting, young men would harass and beat the women, with the police never intervening on behalf of the protesters. Police would even arrest other men who tried to help the women who were getting beaten. Even though they protested during wartime, they maintained public support by agitating peacefully. More protesters were arrested and sent to Occoquan or the District Jail throughout this time. Pardons were no longer given.
Prison, hunger strikes, and passage of 19th Amendment
In solidarity with other activists in her organization, Paul purposefully strove to receive the seven-month jail sentence that started on October 20, 1917. She began serving her time in the District Jail.
Whether sent to Occoquan or the District Jail, the women were given no special treatment as political prisoners. They had to live in harsh conditions with poor sanitation, infested food, no chocolate, and dreadful facilities. In protest of the conditions at the District Jail, Paul began a hunger strike. This led to her being moved to the prison's psychiatric ward and being force-fed raw eggs through a feeding tube. "Seems almost unthinkable now, doesn't it?" Paul told an interviewer from American Heritage when asked about forced feeding, "It was shocking that a government of men could look with such extreme contempt on a movement that was asking nothing except such a simple little thing as the right to vote."
On November 14, 1917, the suffragists who were imprisoned at Occoquan endured brutality allegedly endorsed by prison authorities which became known as the "Night of Terror". The National Woman's Party (NWP) went to court to protest the treatment of the women such as Lucy Burns, Dora Lewis, and Alice Cosu, her cellmate in Occoquan Prison, who suffered a heart attack at seeing Dora's condition. The women were later moved to the District Jail where Paul languished. Despite the brutality that she experienced and witnessed, Paul remained undaunted. On November 27 and 28, all the suffragists were released from prison. Within two months, Wilson announced a bill on women's right to vote.
Post-Suffrage
After Suffrage, the National Women's Party (NWP) continued to lobby in Congress and abroad, advocating for legal equality for women. Alice Paul and NWP members successfully lobbied to include equality provisions into the United Nation's charter, such as the phrase "the equal rights of men and women and of nations large and small." NWP is credited with drafting over 300 pieces of legislation that became law. Paul remained in leadership positions, officially and unofficially, until she moved to Connecticut in 1974.
Equal Rights Amendment
Once suffrage was achieved in 1920, Paul and some members of the National Woman's Party shifted attention to constitutional guarantees of equality through the Equal Rights Amendment (ERA), which was written by Paul and Crystal Eastman. Drafted and delivered to Congress in 1923, the original text of the Equal Rights Amendment—which Paul and the National Woman's Party dubbed the "Lucretia Mott Amendment" in honor of this antislavery and suffrage activist of an earlier generation—read, "Men and women shall have equal rights throughout the United States and every place subject to its jurisdiction." In 1943, the amendment was renamed the "Alice Paul Amendment," and contained wording was changed to the version that still exists today: "Equality of rights under the law shall not be denied or abridged by the United States or by any state on account of sex." For Paul, the ERA had the same appeal as suffrage in that it was a constitutional amendment and a single-issue campaign that she believed could and should unite women around a common core goal. Paul understood the value of single-issue politics for building coalitions and securing success.
Not everyone agreed about next steps or the ERA; from the start, the amendment had its critics. While Paul's activism in the years after suffrage centered on securing legal protections for women's equality in the U.S. and abroad, other activists and some members of the NWP focused on a wide range of issues from birth control and air conditioning to educating newly enfranchised women voters. Some of Paul's earlier allies in suffrage found the ERA troubling, especially since they believed it would erode protective legislation—laws about working conditions or maximum hours that protected women in the workplace. If the ERA guaranteed equality, opponents argued, protective legislation for women would be null and void. The rival League of Women Voters (LWV), which championed workplace legislation for women, opposed the Equal Rights Amendment. Paul and her cohorts, including a small group from the NWP, thought that sex-based workplace legislation restricted women's ability to compete for jobs with men and earn good wages. In fact, Paul believed that protective legislation hurt women wage earners because some employers simply fired them rather than implement protections on working conditions that safeguarded women. Women were paid less than men, lost jobs requiring them to work late nights—often a prohibition under protective legislation—and had long been blocked from joining labor unions on par with men. She also believed that women should be treated under the law like men were and not as a class that required protection. To Paul, such protections were merely a form of entrenched "legalized inequality," a position shared by suffragist Harriot Stanton Blatch. To Paul, the ERA was the most efficient way to ensure legal equality. Paul expected women workers to rally behind the ERA; some did, many did not. While early on, there was hope among NWP members that they could craft a bill that would promote equality while also guaranteeing labor protection for women, to Paul, that was a contradiction. What's more, she was surprised when Florence Kelley, Ethel Smith, Jane Addams, and other suffragists parted with her and aligned with protective legislation.
While Paul continued to work with the NWP and even served as president again in the 1940s, she remained steadfastly committed to women's equality as her singular mission. Along with the ERA, Paul worked on behalf of similar efforts in state legislation and international contexts. She helped ensure that the United Nations proclamations include equality for women. She hoped that this would encourage the United States to follow suit. Paul worked to change laws that had altered the status of a woman's citizenship based on that of her husband's. In the U.S., women who married men from foreign countries lost their U.S. citizenship and were considered by the U.S. to be citizens of whatever country their husbands were from. To Paul, this was a violation of equal rights. As such, she successfully worked on behalf of the international Equal Nationality Treaty in 1933 and in the U.S. for the successful passage of the Equal Nationality Act in 1934, which let women retain their citizenship upon marriage. Just after the founding of the United Nations in 1945, Paul wanted to ensure that women's equality was a part of the organization's charter and that its Commission on Human Rights included a focus on women's equality in its Universal Declaration of Human Rights. She prevailed: the final version of the Declaration in 1948 opened with a reference to "equal rights of men and women".
The ERA was introduced in Congress in 1923 and had various peaks and valleys of support in the following years as Paul continued to push for its passage. There were favorable committee reports in Congress in the late 1930s, and with more women working in men's jobs during the war, public support for the ERA also increased. In 1946, the ERA passed by three votes in the Senate, not the majority needed for it to advance. Four years later, it would garner the Senate votes but fail in the House, thereby halting it from moving forward.
Paul was encouraged when women's movement activism gained steam in the 1960s and 1970s, which she hoped would spell victory for the ERA. When the bill finally passed Congress in 1972, Paul was unhappy about the changes in the wording of the ERA that now included time limits for securing its passage. Advocates argued that this compromise—the newly added seven-year deadline for ratification in the states—enabled the ERA's passage in Congress, but Paul accurately predicted that the inclusion of a time limit would ensure its defeat. In addition, this version put enforcement power in the hands of the federal government only; Paul's original and 1943 reworded versions required both states and the federal government to oversee its provisions. Paul's version was politically insightful and strategic: politicians who believed in states' rights, including many Southern states, were more likely to support an ERA that gave states some discretion of enforcement authority than a version that did not. Paul was proved correct: while the ERA did receive a three-year extension from Congress, it remained three states short of those needed for ratification.
States continued to attempt to ratify the ERA long after the deadline passed, including Nevada in 2017 and Illinois in 2018. In 2017 and again in 2019, the Senate and House introduced resolutions to remove the deadline from the ERA. These or similar measures, if passed, according to some experts, would make the amendment viable again, although other experts disagree.
1964 Civil Rights Act
Paul played a significant role in adding protection for women in Title VII of the Civil Rights Act of 1964, despite the opposition of liberals who feared it would end protective labor laws for women. The prohibition on sex discrimination was added to the Civil Rights Act by Howard W. Smith, a powerful Virginia Democrat who chaired the House Rules Committee. Smith's amendment was passed by a teller vote of 168 to 133. For twenty years, Smith had sponsored the Equal Rights Amendment in the House because he believed in equal rights for women, even though he opposed equal rights for blacks. For decades, he had been close to the National Woman's Party, especially to Paul. She and other feminists had worked with Smith since 1945, trying to find a way to include sex as a protected civil rights category.
Personal life and death
Paul had an active social life until she moved to Washington, D.C. in late 1912. She enjoyed close relationships with women and befriended and occasionally dated men. Paul did not preserve private correspondence for the most part, so few details about her personal life are available. Once Paul devoted herself to winning the vote for women, she placed the suffrage effort first in her life. Nevertheless, Elsie Hill and Dora Kelly Lewis, two women whom she met early in her work for NAWSA, remained close to her all their lives. She knew William Parker, a scholar she met at the University of Pennsylvania, for several years; he may have tendered a marriage proposal in 1917.
Paul became a vegetarian around the time of the suffrage campaign.
In 1974, Paul suffered a stroke and was placed in a nursing home under the guardianship of her nephew, who depleted her estate. News of her penniless state reached friends, and a fund for indigent Quakers quickly aided Paul. Paul died at the age of 92 on July 9, 1977, at the Greenleaf Extension Home a Quaker facility in Moorestown, New Jersey, less than a mile from her birthplace and childhood home]. She is buried at Westfield Friends Burial Ground in Cinnaminson, New Jersey. Visitors fequently leave notes at her tombstone to thank her for her lifelong work on behalf of women's rights.
Legacy
Paul was posthumously inducted into the National Women's Hall of Fame in 1979, and into the New Jersey Hall of Fame in 2010.
Her alma mater, Swarthmore College, named a dormitory Alice Paul Residence Hall in her honor. Montclair State University in New Jersey has also named a dormitory (Alice Paul Hall) in her honor. On April 12, 2016, President Barack Obama designated Sewall-Belmont House as the Belmont–Paul Women's Equality National Monument, named for Alice Paul and Alva Belmont. The University of Pennsylvania, her doctoral alma mater, maintains the Alice Paul Center for Research on Gender, Sexuality, and Women.
Two countries have honored her by issuing a postage stamp: Great Britain in 1981 and the United States in 1995. The U.S. stamp was the $0.78 Great Americans series.
Paul appeared on a United States half-ounce $10 gold coin in 2012 as part of the First Spouse Gold Coin Series. A provision in the Presidential $1 Coin Program directs that Presidential spouses be honored. As President Chester A. Arthur was a widower, Paul is shown representing "Arthur's era". The U.S. Treasury Department announced in 2016 that an image of Paul will appear on the back of a newly designed $10 bill along with Lucretia Mott, Sojourner Truth, Susan B. Anthony, Elizabeth Cady Stanton, and the 1913 Woman Suffrage Procession that Paul initiated and organized. Designs for the new $5, $10, and $20 bills will be unveiled in 2020 in conjunction with the 100th anniversary of American women winning the right to vote via the 19th Amendment.
In 1987, a group of New Jersey women raised the money to purchase Paul's papers when they came up for auction so that an archive could be established. Her papers and memorabilia are now held by the Schlesinger Library at Harvard University, and the Smithsonian Institution in Washington D.C. In 1990, the same group, now the Alice Paul Institute, purchased the brick farmhouse, Paulsdale, in Mount Laurel, New Jersey, where Paul was born. Paulsdale is a National Historic Landmark and is on the New Jersey and National Registers of Historic Places. The Alice Paul Institute keeps her legacy alive with educational exhibits about her life, accomplishments, and advocacy for gender equality.
Hilary Swank played Paul in the 2004 film Iron Jawed Angels, which portrayed the 1910s women's suffrage movement for passage of the 19th Amendment. In 2018, Alice Paul was a central character in an episode of Timeless (Season 2, Episode 7) which alludes to Paul giving an impassioned speech to President Woodrow Wilson during a march that ends in police violence upon the suffragist marchers. According to history, Paul was at the event and was arrested, but there is no evidence that she spoke to Wilson on that day. In 2022, SUFFS, a musical written by Shaina Taub, premiered at the Public Theater with Alice Paul as a main character.
On January 11, 2016, Google Doodle commemorated her 131st birthday.
See also
Iron Jawed Angels, 2004 film about Alice Paul and Lucy Burns and their movement, which resulted in the passage of the 19th Amendment.
List of civil rights leaders
List of suffragists and suffragettes
List of women's rights activists
Timeline of women's suffrage
Timeline of women's suffrage in the United States
Women's suffrage organizations
References
Further reading
Baker, Jean H. Sisters: The Lives of American Suffragists. New York: Hill and Wang, 2005.
_. Votes for Women: The Struggle for Suffrage Revisited. New York: Oxford University Press, 2002.
Butler, Amy E. Two Paths to Equality: Alice Paul and Ethel M. Smith in the ERA Debate, 1921–1929. Albany: State University of New York Press, 2002.
Cahill, Bernadette. Alice Paul, the National Woman's Party and the Vote: The First Civil Rights Struggle of the 20th Century. Jefferson: McFarland & Company, Inc., Publishers, 2005.
Cassidy, Tina. Mr. President, How Long Must We Wait?: Alice Paul, Woodrow Wilson, and the Fight for the Right to Vote (2019).
Cullen-Dupont, Kathryn. American Women Activists' Writings: An Anthology, 1637–2002. New York: Cooper Square Press, 2002.
Dodd, Lynda G. "Parades, Pickets, and Prison: Alice Paul and the Virtues of Unruly Constitutional Citizenship." Journal of Law and Politics 24 (2008): 339–433. online
Evans, Sara M. Born for Liberty: A History of Women in America. New York: The Free Press, 1989.
Hartmann, Susan M. "Paul, Alice"; American National Biography Online Feb. 2000 Access June 5, 2014
Hill, Jeff. Defining Moments: Women's Suffrage. Detroit: Omnigraphics, Inc., 2006.
Irwin, Inez Haynes. The Story of Alice Paul and the National Woman's Party. Fairfax: Denlinger's Publishers, LTD, 1964.
Leleux, Robert. "Suffragettes March on Washington." The American Prospect 24 (2013): 81.
Lunardini, Christine. Alice Paul: Equality for Women. Boulder: Westview Press, 2013.
___. From Equal Suffrage to Equal Rights: Alice Paul and the National Woman's Party, 1910–1928. New York: New York University Press, 1986.
Olson, Tod. "One Person, One Vote." Scholastic Update 127 (1994): 15
* Piott, Steven L. American Reformers, 1870-1920: Progressives in Word and Deed (2006); chapter 12 is on Paul.
Stevens, Doris. Jailed for Freedom. New York: Liverwright Publishing Corporation, 1920.
Stillion Southard, Belinda Ann. "The National Woman's Party's Militant Campaign for Woman Suffrage: Asserting Citizenship Rights through Political Mimesis." (2008). PhD thesis, U of Maryland online
Willis, Jean L. "Alice Paul: The Quintessential Feminist," in Feminist Theorists, ed. Dale Spender (1983).
External links
The Alice Paul Institute
Alice Paul at Lakewood Public Library: Women In History
The Sewall-Belmont House & Museum – Home of the historic National Woman's Party
Biographical sketch at the University of Pennsylvania
Manuscript version of Paul's PhD dissertation, "The Legal Position of Women in Pennsylvania" at the University of Pennsylvania
Papers, 1785–1985. Schlesinger Library, Radcliffe Institute, Harvard University.
"I Was Arrested, Of Course…", American Heritage, February 1974, Volume 25, Issue 2. Interview of Alice Paul by Robert S. Gallagher.
Conversations with Alice Paul: Woman Suffrage and the Equal Rights Amendment, An Interview Conducted by Amelia R. Fry, 1979, The Bancroft Library
Michals, Debra. "Alice Paul". National Women's History Museum. 2015.
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Bezděčí u Trnávky är en ort i Tjeckien. Den ligger i regionen Pardubice, i den östra delen av landet, km öster om huvudstaden Prag. Bezděčí u Trnávky ligger meter över havet och antalet invånare är .
Terrängen runt Bezděčí u Trnávky är huvudsakligen kuperad, men åt sydost är den platt. Bezděčí u Trnávky ligger nere i en dal. Den högsta punkten i närheten är meter över havet, km nordost om Bezděčí u Trnávky. Runt Bezděčí u Trnávky är det ganska glesbefolkat, med invånare per kvadratkilometer. Närmaste större samhälle är Moravská Třebová, km nordväst om Bezděčí u Trnávky. I omgivningarna runt Bezděčí u Trnávky växer i huvudsak blandskog.
Trakten ingår i den hemiboreala klimatzonen. Årsmedeltemperaturen i trakten är °C. Den varmaste månaden är juli, då medeltemperaturen är °C, och den kallaste är januari, med °C. Genomsnittlig årsnederbörd är millimeter. Den regnigaste månaden är juni, med i genomsnitt mm nederbörd, och den torraste är mars, med mm nederbörd.
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"redpajama_set_name": "RedPajamaWikipedia"
} | 7,752 |
{"url":"https:\/\/math.stackexchange.com\/questions\/2186984\/how-can-i-solve-this-weird-integral","text":"# How can I solve this weird integral?\n\na friend of mine asked me this question, given by a book with many difficult math problems. And then, I saw that is quite interesting to start by trying to evaluate the problem by the substitution of $x^4+1$, so we can call it u. But, unfortunately, this couldn't help me, since the integral just got harder, as I found a new integral in terms of u, that has the annoying $4x$ messing up with this possible solution.\n\nSo, finally, after failing with Wolfram Alpha, Symbolab and others, I came here to ask you guys, how can I solve it? $$\\int _1^3\\:\\sqrt{x^4+1} \\ \\ dx$$\n\nAlso, it even gets more difficult when he asks: What is the value of x, since $$\\int _1^3\\:\\sqrt{x^4+1} \\ \\ dx\\ge \\frac{2x}{3}$$\n\nThank you all for spending some time analysing my question!\n\n\u2022 Be careful with your second integral: the $x$ on the right-hand side is completely unrelated to the $x$s on the left-hand side! (I'm not sure your second expression is meaningful, actually.) \u2013\u00a0Patrick Stevens Mar 14 '17 at 22:17\n\u2022 I sense a strog smell of elliptic integral. \u2013\u00a0Sangchul Lee Mar 14 '17 at 22:20\n\u2022 Mathematica says the numerical answer is about 8.98005623469252857877121527304; it can't find a nice approximation to this number. \u2013\u00a0Patrick Stevens Mar 14 '17 at 22:24\n\u2022 One usually tries $u$-substitution when the integrand contains a factor to correspond with $\\mathrm{d}u$. It is possible a trignometric or other \"special function\" substitution will be required. \u2013\u00a0hardmath Mar 14 '17 at 22:25\n\u2022 And the exact answer is, as simply as Mathematica can make it, $\\frac{1}{3} \\sqrt{2} \\left(-1+3 \\sqrt{41}+(1+i) F\\left(\\left.i \\sinh ^{-1}\\left(\\sqrt[4]{-1}\\right)\\right|-1\\right)-(1+i) F\\left(\\left.i \\sinh ^{-1}\\left(3 \\sqrt[4]{-1}\\right)\\right|-1\\right)\\right)$, where $F$ is the elliptic integral of the first kind. \u2013\u00a0Patrick Stevens Mar 14 '17 at 22:26\n\n$\\int_1^3\\sqrt{x^4+1}~dx$\n$=\\int_1^3x^2\\sqrt{1+\\dfrac{1}{x^4}}~dx$\n$=\\int_1^3x^2\\sum\\limits_{n=0}^\\infty\\dfrac{(-1)^n(2n)!}{4^n(n!)^2(1-2n)x^{4n}}~dx$\n$=\\int_1^3\\sum\\limits_{n=0}^\\infty\\dfrac{(-1)^n(2n)!x^{2-4n}}{4^n(n!)^2(1-2n)}~dx$\n$=\\left[\\sum\\limits_{n=0}^\\infty\\dfrac{(-1)^n(2n)!x^{3-4n}}{4^n(n!)^2(1-2n)(3-4n)}\\right]_1^3$\n$=\\sum\\limits_{n=0}^\\infty\\dfrac{27(-1)^n(2n)!}{324^n(n!)^2(2n-1)(4n-3)}-\\sum\\limits_{n=0}^\\infty\\dfrac{(-1)^n(2n)!}{4^n(n!)^2(2n-1)(4n-3)}$","date":"2019-07-23 08:55:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 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\": 0.7791768312454224, \"perplexity\": 477.2713207721198}, \"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-2019-30\/segments\/1563195529175.83\/warc\/CC-MAIN-20190723085031-20190723111031-00015.warc.gz\"}"} | null | null |
<?php
namespace CodeRefactor\Mixin;
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return true;
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,614 |
\section{Introduction}\label{intro.sec}
Recent years, exactly calculable quantities
in gauge theories play important roles
in study of gauge theories themselves and their relation to string/M theory.
In this paper we discuss a relation between two of such quantities.
One is
the ${\bf S}^3$ partition function\cite{Kapustin:2009kz,Jafferis:2010un,Hama:2010av}.
It is used to confirm dualities among 3d theories
\cite{Kapustin:2010xq,Jafferis:2011ns,Kapustin:2011gh,Willett:2011gp}
and predictions of AdS$_4$/CFT$_3$
\cite{Drukker:2010nc,Herzog:2010hf,Martelli:2011qj,Cheon:2011vi,Jafferis:2011zi}.
Furthermore, this function provides a simple way to determine
the R-charge at IR fixed points\cite{Jafferis:2010un}.
The partition function is evaluated exactly
by localization.
We choose a nilpotent supercharge ${\cal Q}$ and
deform the action by ${\cal Q}$-exact terms.
The partition function is given by
\begin{equation}
Z=\int {\cal D}\Phi \exp\left(
-S_0^{(3d)}-u\int_{{\bf S}^3}\sqrt{g}{\cal L}^{(3d)} d^3x
\right).
\label{zs3}
\end{equation}
$S^{(3d)}_0$ is the original action of the 3d theory and
the second term in the exponent is the ${\cal Q}$ exact action.
This path integral does not depend on $u$,
and is evaluated exactly in the weak coupling limit $u\rightarrow\infty$.
The other exactly calculable quantity we consider is
the ${\cal N}=1$ superconformal index
for 4d theories\cite{Kinney:2005ej,Romelsberger:2005eg}.
The index is defined by
\begin{equation}
I(t,x,h_i)=\tr\left[(-1)^F
q^{D-\frac{3}{2}R-2J_L}
t^{R+2J_L}
x^{2J_R}
\prod_ih_i^{{\cal F}_i}\right],
\label{indexdef}
\end{equation}
where $D$ (the dilatation), $R$ (the R-charge),
$J_L$ and $J_R$ ($SU(2)_L\times SU(2)_R$ spins),
are Cartan generators of the ${\cal N}=1$ superconformal algebra
$PSU(2,2|1)$,
and ${\cal F}_i$ are Cartan generators of the flavor symmetry.
Only operators saturating the BPS bound
\begin{equation}
D-\frac{3}{2}R-2J_L\geq 0
\end{equation}
contribute to the index, and
(\ref{indexdef}) is independent of the
variable $q$.
This quantity is exactly calculable, and is conveniently used as a tool to check
Seiberg-duality\cite{Romelsberger:2007ec,Dolan:2008qi,Spiridonov:2008zr,Spiridonov:2009za}
and AdS$_5$/CFT$_4$\cite{Kinney:2005ej,Nakayama:2005mf,Nakayama:2006ur,Benvenuti:2006qr,Gadde:2010en}.
One way to compute the index is to use localization.
We choose a nilpotent supercharge ${\cal Q}$ and
deform the action by $\cal Q$-exact terms.
The index can be expressed in the path integral form
\begin{equation}
I(t,x,h_i)=\int {\cal D}\Phi \exp\left(
-S_0^{(4d)}-u\int_{{\bf S}^3\times{\bf S}^1}\sqrt{g}{\cal L}^{(4d)} d^4x
\right),
\label{ideform}
\end{equation}
where $S_0^{(4d)}$ is the original action of the theory
defined in ${\bf S}^3\times{\bf S}^1$,
and the second term in the exponent is the ${\cal Q}$-exact deformation action.
The chemical potentials are introduced as non-trivial Wilson lines around
${\bf S}^1$.
Let $r$ and $\beta r$ be the ${\bf S}^3$ radius and the ${\bf S}^1$ period,
respectively.
The ratio $\beta$ is related to the parameter $q$ by
$q=e^{-\beta}$.
In the case of the index, the deformation term does not necessarily
have to be ${\cal Q}$-exact
because the index does not depend on continuous coupling constants;
even so, we adopt a ${\cal Q}$-exact deformation action in this paper
for the reason which will become clear shortly.
The similarity between (\ref{zs3}) and (\ref{ideform})
strongly suggests that there exists some relation between
the index and the partition function.
If we consider 4d and 3d theories with the same gauge group $G$
and the same matter contents,
we naturally expect
that the partition function is obtained by taking a small ${\bf S}^1$ limit
of the index.
Such a relation was recently studied in \cite{Dolan:2011rp,Gadde:2011ia}.
In \cite{Dolan:2011rp} it is shown for particular examples of gauge theories
that a relation between the 3d partition function and the 4d index follows
from certain mathematical properties of special functions appearing in the
index and partition function.
A similar relation is also studied in
\cite{Gadde:2011ia}, and a limiting procedure which reduces
the superconformal index of 4d ${\cal N}=2$ theories
to the ${\bf S}^3$ partition function of corresponding
3d ${\cal N}=4$ theories is proposed.
In these works, only the final expressions for
the partition function and the index are studied,
and physical origin of the relation is not so obvious.
The purpose of this paper is to extend the relation obtained
in \cite{Dolan:2011rp,Gadde:2011ia} to
general 3d ${\cal N}=2$ and 4d ${\cal N}=1$ theories,
and to establish the relation
at more fundamental level
by comparing 3d and 4d actions.
For this purpose, it is convenient to use as similar deformation terms
as possible in two computations.
We use ${\cal Q}$-exact deformation terms in both cases with
closely related supercharges ${\cal Q}$ in 3d and 4d theories.
In both (\ref{zs3}) and (\ref{ideform}),
the deformation terms dominate the actions
in the weak coupling limit $u\rightarrow\infty$,
and only few terms in the original actions are relevant to the partition function and the index.
Let $S^{(3d)}_{\rm rel}$ and $S^{(3d)}_{\rm rel}$
be the relevant terms including the deformation terms.
$S^{(3d)}_{\rm rel}$ consists of
(supersymmetric completion of) Chern-Simons and FI terms
in the original action $S^{(3d)}_0$ and
the ${\cal Q}$-exact terms
\begin{equation}
S^{(3d)}_{\rm rel}=S^{(3d)}_{\rm CS}+S_{\rm FI}^{(3d)}+
u\int_{{\bf S}^3}\sqrt{g}{\cal L}^{(3d)}d^3x,
\label{relevant}
\end{equation}
while $S_{\rm rel}^{(4d)}$ consists of
the (supersymmetric completion of) FI terms
and the deformation terms
\begin{equation}
S^{(4d)}_{\rm rel}=S_{\rm FI}^{(4d)}+
u\int_{{\bf S}^3\times{\bf S}^1}\sqrt{g}{\cal L}^{(4d)}d^4x.
\label{relevant4d}
\end{equation}
We consider 3d and 4d theories with the same gauge group $G$ and
chiral multiplets $\Phi_I$
belonging to the same $G$-representations $R_I$.
We assume that the Weyl weight $\Delta_I$\footnote
The deformation terms are not invariant under the
dilatation, and the dilatation is broken in the deformed theories.
For this reason, the parameters $\Delta_I$ in the deformed theories
should be regarded
not as the weyl weights
but as parameters appearing in the ${\cal Q}$ transformation
laws for chiral multiplets.
The absence of the dilatation symmetry in the deformed theories
does not cause any problem because
we need only the fermionic symmetry ${\cal Q}$
for the computation of the exact results.
} of each chiral multiplet
is the same in 3d and 4d.
We explicitly show
for a 3d theory without Chern-Simons terms on round ${\bf S}^3$
that $S_{\rm rel}^{(3d)}$ is obtained
by dimensional reduction
of $S_{\rm rel}^{(4d)}$
provided that an appropriate Wilson line is turned on.
The symmetry associated with the Wilson line may not be a
symmetry of the original action $S_0^{(4d)}$,
but is a symmetry of $S^{(4d)}_{\rm rel}$.
(The symmetry may be anomalous.
We discuss the treatment of anomalous symmetries
at the end of \S\ref{comparison.sec}.)
$S^{(4d)}_{\rm rel}$ has the symmetry rotating each chiral multiplet
independently.
For each chiral multiplet $\Phi_I$
we define the charge ${\cal F}_I$ rotating only $\Phi_I$
and the corresponding chemical potential $h_I$.
By comparing $S_{\rm rel}^{(3d)}$
and $S_{\rm rel}^{(4d)}$,
we obtain a formula which
gives the partition function $Z$
as a small radius limit of the index
$I(t,x,h_I)$.
We further generalize the relation
by using the recently proposed\cite{Gadde:2011ia} connection
between squashing parameter $s$ of ${\bf S}^3$\cite{Hama:2011ea}
and $SU(2)_R$ Wilson line.
The most general formula we propose in this paper is
\begin{equation}
Z=\lim_{q\rightarrow 1}I(t=q,x=q^s,h_I=q^{-ir\mu_I+\frac{1}{3}\Delta_I})
|_{\zeta_A^{(4d)}=\frac{1}{\beta r}\zeta_A^{(3d)}},
\label{mainresult}
\end{equation}
where $\mu_I$ are real mass parameters, and
$\zeta_A^{(4d)}$ and $\zeta_A^{(3d)}$ are 4d and 3d FI parameters, respectively.
Unfortunately, when Chern-Simons levels $k_a$ of 3d theory
are non-vanishing,
we could not reproduce the ${\bf S}^3$
partition function from the index due to the difficulty in obtaining
Chern-Simons terms
by dimensional reduction.
The paper is organized as follows.
After explaining our notation for spinors in the next section,
we summarize the superconformal algebra and the supersymmetry
transformation laws in \S\ref{algebra.sec} and \S\ref{susy.sec}.
Exact computations of the ${\bf S}^3$ partition function and
the 4d superconformal index are briefly reviewed in
\S\ref{partition.sec} and \S\ref{index.sec},
respectively.
In \S\ref{comparison.sec} we compare the 3d and 4d actions,
and find the relation
between the partition function and the index in the case
of $\mu_I=k_a=\zeta^{(3d)}_A=s=0$.
Generalization to non-vanishing parameters is discussed in \S\ref{generalization.sec}.
Conclusions are presented in \S\ref{conc.sec}.
\section{Notation for spinors}\label{notation.sec}
Because we consider both 3d and 4d theories,
we use notation for spinors such that the expression of 3d
and 4d theories look as similar as possible.
For 3d spacetime,
we use coordinates $x^m$ ($m=1,2,3$).
Although we can define Majorana spinors in
3d Minkowski spacetime, all spinors we use
are complex spinors.
For a complex spinor $\psi$, we denote its Majorana conjugate by $\ol\psi$.
In Euclidean spacetime $\psi$ and $\ol\psi$ should be treated as
independent spinors.
For 4d spacetime, we use coordinates
$x^\mu$ ($\mu=1,2,3,4$).
When we consider ${\bf S}^3\times{\bf S}^1$ background,
we use $x^m$ for ${\bf S}^3$ and $x^4$ for ${\bf S}^1$.
The 4d Dirac's matrices are expressed
in terms of the 3d Dirac's matrices by
\begin{equation}
\gamma^m=\left(\begin{array}{cc} 0 & \gamma^m \\
\gamma^m & 0 \end{array}\right)\quad m=1,2,3,\quad
\gamma^4=\left(\begin{array}{cc}
0 & -i \\
i & 0
\end{array}\right).
\end{equation}
We use the same symbol $\gamma^m$ for 3d and 4d Dirac's matrices.
The charge conjugation and the chirality in 4d are
\begin{equation}
C_{ab}=\left(\begin{array}{cc} \epsilon_{ab} & 0 \\
0 & \epsilon_{ab} \end{array}\right),\quad
\gamma^5=\left(\begin{array}{cc} {\bf 1}_2 & 0 \\
0 & -{\bf 1}_2 \end{array}\right).
\end{equation}
We call the upper (lower) half of a four-component spinor
left-handed (right-handed).
Namely, a left-handed (right-handed) spinor has positive (negative) chirality.
3d and 4d completely anti-symmetric tensors $\epsilon_{mnp}$ and $\epsilon_{\mu\nu\rho\sigma}$ are defined by
\begin{equation}
\gamma_{mnp}=i\epsilon_{mnp}{\bf1}_2,\quad
\gamma^5\gamma_{\mu\nu\rho\sigma}=-\epsilon_{\mu\nu\rho\sigma}{\bf1}_4.
\end{equation}
We raise and lower spinor indices by
the relation $\psi_a=\psi^b\epsilon_{ba}$.
Spinor indices are contracted by the
NW-SE rule.
For example, for spinors $\psi$ and $\chi$,
$\psi\chi\equiv\psi^a\chi_a=\psi^a\chi^b\epsilon_{ba}$.
In 4d we use two-component representation.
We use a symbol without and with bar for a left-handed and right-handed
spinor, respectively.
For example, when we use symbol $\psi$ and $\ol\psi$ as 4d
two-component spinors,
their four-component representations are
\begin{equation}
\left(\begin{array}{c}\psi \\ 0\end{array}\right),\quad
\left(\begin{array}{c}0 \\ \ol\psi\end{array}\right).
\end{equation}
Note that $\ol\psi$ is not the Dirac's conjugate of $\psi$.
We will never use Dirac's conjugate in this paper.
We use indices $\mu,\nu,\ldots$ not only in 4d but also in 3d.
In that case
we assume that all fields do not depend on $x^4$,
and the $4$-th component of a gauge field $A_\mu$ is regarded as a
Hermitian scalar field $\sigma$.
For example, if the gauge covariant derivative is given by
$D_\mu=\partial_\mu-iA_\mu$,
the fermion kinetic term $-(\ol\psi\gamma^\mu D_\mu\psi)$
represents in 3d the sum of two terms
$-(\ol\psi\gamma^m D_m\psi)$ and
$-(\ol\psi\sigma\psi)$.
\section{Superconformal algebra}\label{algebra.sec}
Before considering actions and transformation laws,
let us compare the 4d ${\cal N}=1$ superconformal algebra
and 3d ${\cal N}=2$ superconformal algebra.
The 4d algebra contains the generators
\begin{equation}
M_{\mu\nu},\quad
P_\mu,\quad
K_\mu,\quad
D,\quad
R,\quad
Q,\quad
\ol Q,\quad
S,\quad
\ol S,
\end{equation}
while the 3d algebra contains
the same generators with vector indices $\mu$ and $\nu$ running over
$1,2,3$ only.
For later use we define Cartan generators of the rotation groups,
\begin{equation}
M_{12}=iJ_3\quad(3d),\quad
M_{12}=i(J_L+J_R),\quad
M_{34}=i(J_L-J_R)\quad(4d).
\end{equation}
Almost all (anti-)commutation relations are the same in 3d and 4d.
\begin{align}
&[M_{\mu\nu},M_{\rho\sigma}]=
\eta_{\mu\rho}M_{\nu\sigma}
-\eta_{\mu\sigma}M_{\nu\rho}
-\eta_{\nu\rho}M_{\mu\sigma}
+\eta_{\nu\sigma}M_{\mu\rho},\nonumber\\
&[M_{\mu\nu},P_\rho]=\eta_{\mu\rho}P_\nu-\eta_{\nu\rho}P_\mu,\quad
[M_{\mu\nu},K_\rho]=\eta_{\mu\rho}K_\nu-\eta_{\nu\rho}P_\mu,\nonumber\\
&[D,P_\mu]=P_\mu,\quad
[D,K_\mu]=-K_\mu,\quad
[P_\mu,K_\nu]=-2M_{\mu\nu}+2\eta_{\mu\nu}D,
\nonumber\\
&[M_{\mu\nu},{\cal S}]=\frac{1}{2}\gamma_{\mu\nu}{\cal S},\quad
({\cal S}=Q,\ol Q,S,\ol S),\nonumber\\
&[R,Q]=-Q,\quad
[R,\ol Q]=\ol Q, \quad
[R,S]=S,\quad
[R,\ol S]=-\ol S,
\nonumber\\&
[D,Q]=\frac{1}{2}Q,\quad
[D,\ol Q]=\frac{1}{2}\ol Q, \quad
[D,S]=-\frac{1}{2}S, \quad
[D,\ol S]=-\frac{1}{2}\ol S,
\nonumber\\&
[S,P_\mu]=\gamma_\mu\ol Q, \quad
[\ol S,P_\mu]=\gamma_\mu Q, \quad
[Q,K_\mu]=\gamma_\mu\ol S, \quad
[\ol Q,K_\mu]=\gamma_\mu S,
\nonumber\\&
\{Q_a,\ol Q_b\}=2(\gamma^\mu)_{ab}P_\mu, \quad
\{S_a,\ol S_b\}=2(\gamma^\mu)_{ab}K_\mu.
\label{common}
\end{align}
Differences between 3d and 4d arise only in
$\{S,Q\}$ and $\{\ol S,\ol Q\}$.
In the 3d algebra, they are
\begin{align}
\{S_a,Q_b\}=&(\gamma^{mn})_{ab}M_{mn}+2\epsilon_{ab}D+2\epsilon_{ab}R,
\nonumber\\
\{\ol S_a,\ol Q_b\}=&(\gamma^{mn})_{ab}M_{mn}+2\epsilon_{ab}D-2\epsilon_{ab}R,
\label{qs3d}
\end{align}
while in the 4d algebra,
the coefficients of the $R$-charge terms are different.
\begin{align}
\{S_a,Q_b\}=&(\gamma^{\mu\nu})_{ab}M_{\mu\nu}+2\epsilon_{ab}D+3\epsilon_{ab}R,
\nonumber\\
\{\ol S_a,\ol Q_b\}=&(\gamma^{\mu\nu})_{ab}M_{\mu\nu}+2\epsilon_{ab}D-3\epsilon_{ab}R.
\label{qs4d}
\end{align}
In radial quantization, the dilatation $D$ is regarded as Hamiltonian,
and $\ol Q^a$ and $\ol S_a$ are treated to be Hermitian conjugate to each other.
From (\ref{qs3d}) and (\ref{qs4d}) we can derive BPS bounds.
In particular, the bound obtained from $\{\ol S_1,\ol Q^1\}$
is important in the following computations.
In 3d, it is
\begin{equation}
\{\ol S_1,\ol Q^1\}=2D-2R-2J_3\geq 0.
\end{equation}
In 4d, we obtain the bound with different coefficients
\begin{equation}
\{\ol S_1,\ol Q^1\}=2D-3R-4J_L\geq 0.
\end{equation}
\section{Supersymmetry transformations}\label{susy.sec}
Because the Poincare subalgebra
in (\ref{common})
generated by
$M_{\mu\nu}$, $P_\mu$, $Q$ and $\ol Q$ is the same in 3d and 4d,
(up to the absence of $P_4$ and $M_{m4}$ in 3d,)
$Q$ and $\ol Q$-transformation laws in the flat background
take the same form in 3d and 4d.
For a vector multiplet
$(A_\mu,\lambda,\ol\lambda,D)$,
the $Q$ and $\ol Q$-transformations are
\begin{align}
\delta^0 A_\mu=&i(\epsilon\gamma_\mu\ol\lambda)-i(\ol\epsilon\gamma_\mu\lambda),\nonumber\\
\delta^0\lambda=&
\frac{i}{2}\gamma^{\mu\nu}\epsilon F_{\mu\nu}
+D\epsilon,\nonumber\\
\delta^0\ol\lambda=&
-\frac{i}{2}\gamma^{\mu\nu}\ol\epsilon F_{\mu\nu}
+D\ol\epsilon,\nonumber\\
\delta^0D=&
-(\epsilon\gamma^\mu D_\mu\ol\lambda)-(\ol\epsilon\gamma^\mu D_\mu\lambda).
\label{d0vector}
\end{align}
We use the symbol $\delta^0$ rather than $\delta$ to
emphasize that these are rules for the flat background.
When we regard these as rules for 3d theory,
all fields are assumed to be independent of $x^4$, and
$A_4$ should be regarded as a Hermitian scalar field $\sigma$.
For a chiral multiplet $(\phi,\psi,F)$,
the transformation laws are
\begin{align}
\delta^0 \phi=&\sqrt2(\epsilon\psi),\nonumber\\
\delta^0 \phi^\dagger=&\sqrt2(\ol\epsilon\ol\psi),\nonumber\\
\delta^0\psi=&
-\sqrt{2}\gamma^\mu\ol\epsilon D_\mu\phi
+\sqrt{2}\epsilon F,\nonumber\\
\delta^0\ol\psi=&
-\sqrt{2}\gamma^\mu\epsilon D_\mu\phi^\dagger
+\sqrt{2}\ol\epsilon F^\dagger,\nonumber\\
\delta^0 F=&
-\sqrt{2}(\ol\epsilon\gamma^\mu D_\mu\psi)
-2(\ol\epsilon\ol\lambda)\phi,\nonumber\\
\delta^0 F^\dagger=&
-\sqrt{2}(\epsilon\gamma^\mu D_\mu\ol\psi)
-2\phi^\dagger(\epsilon\lambda).
\label{d0chiral}
\end{align}
We can construct supersymmetry transformation laws for
an arbitrary conformally flat background
from (\ref{d0vector}) and (\ref{d0chiral})
by Weyl-covariantization.
By a Weyl transformation
\begin{equation}
e_\mu^a=e^{-\alpha}e_\mu'^a,
\label{weyle}
\end{equation}
a field $\varphi$ with Weyl weight $\Delta_\varphi$ is transformed by
\begin{equation}
\varphi=e^{\Delta_\varphi\alpha}\varphi'.
\label{weylphi}
\end{equation}
Even if a field $\varphi$ has definite Weyl weight,
its derivative is not transformed covariantly as (\ref{weylphi})
and terms containing $\partial_\mu\alpha$ arise.
There are such non-covariant terms in the transformation laws
(\ref{d0vector}) and (\ref{d0chiral}).
To extend them to a general conformally flat background,
we should covariantize them with respect to Weyl transformation
by adding terms containing
derivatives of parameters $\epsilon$ and $\ol\epsilon$.
$\delta^0\lambda$ in 3d and $\delta^0\psi$
contain terms
proportional to $(D_\mu\varphi)\gamma^\mu\ol\epsilon$
with $\varphi=\sigma$ and $\phi$, respectively.
$\delta^0\ol\lambda$ in 3d and $\delta^0\ol\psi$ also contain
similar scalar derivative terms.
In $d$-dimensional spacetime,
we can covariantize terms of this form by the replacement
\begin{align}
(D_\mu\varphi)
\gamma^\mu\ol\epsilon
\rightarrow
&(D_\mu\varphi)
\gamma^\mu\ol\epsilon
+\frac{2\Delta_\varphi}{d}\varphi\gamma^\mu D_\mu\ol\epsilon,
\nonumber\\
(D_\mu\varphi)
\gamma^\mu\epsilon
\rightarrow
&(D_\mu\varphi)
\gamma^\mu\epsilon
+\frac{2\Delta_\varphi}{d}\varphi\gamma^\mu D_\mu\epsilon.
\label{covdf}
\end{align}
The fermion derivative terms
in $\delta^0 D$ $\delta^0F$, and $\delta^0F^\dagger$
are covariantized by the replacement
\begin{align}
(\ol\epsilon\gamma^\mu D_\mu\chi)\rightarrow
&(\ol\epsilon\gamma^\mu D_\mu\chi)
+\frac{2\Delta_\chi+1-d}{d}(D_\mu\ol\epsilon\gamma^\mu\chi),
\nonumber\\
(\epsilon\gamma^\mu D_\mu\ol\chi)\rightarrow
&(\epsilon\gamma^\mu D_\mu\ol\chi)
+\frac{2\Delta_\chi+1-d}{d}(D_\mu\epsilon\gamma^\mu\ol\chi).
\label{covariantization}
\end{align}
We can easily confirm that (\ref{covdf}) and (\ref{covariantization})
are transformed covariantly by the Weyl transformation
(\ref{weyle}) and
(\ref{weylphi}) as fields with weight
$\Delta_\varphi+1/2$ and
$\Delta_\chi+1/2$, respectively.
\section{${\bf S}^3$ partition function}\label{partition.sec}
In this section we briefly review the computation
of the ${\bf S}^3$ partition function.
We here only consider the case with $\mu_I=\zeta_A=k_a=s=0$.
Both a 3d ${\cal N}=2$ theory
and a 4d ${\cal N}=1$ theory have
eight supercharges.
Four of them ($Q$ and $\ol S$)
correspond to the parameter $\epsilon$
and the other four ($\ol Q$ and $S$) to $\ol\epsilon$.
When we use localization, we choose
a nilpotent supercharge ${\cal Q}$,
and add ${\cal Q}$-exact terms
to the action.
Because we should use a linear combination
of $\ol Q$ and $S$ for computation of the index
(\ref{indexdef}),
we consider only transformations by $\ol\epsilon$
in the following.
On a conformally flat 3d background
the parameter $\ol\epsilon$ must satisfy the Killing equation\cite{Kapustin:2009kz}
\begin{equation}
D_m\ol\epsilon=\gamma_m\kappa,
\label{killing}
\end{equation}
where $\kappa$ is an arbitrary spinor.
Corresponding to four supercharges $\ol Q_a$ and $S_a$,
there are four linearly independent solutions to (\ref{killing}).
In the case of ${\bf S}^3$,
two of them are right-invariant, and belong to the $({\bf 2},{\bf 1})$ representation
of the $SO(4)=SU(2)_L\times SU(2)_R$ isometry group.
Let us denote spinors with $J_L=+1/2$ and $J_L=-1/2$ by
$\ol\epsilon_1$ and $\ol\epsilon_2$, respectively.
We adopt $\delta(\ol\epsilon_1)$ as ${\cal Q}$.
Both $\ol\epsilon_1$ and $\ol\epsilon_2$ satisfy
\begin{equation}
D_m\ol\epsilon
=-\frac{i}{2r}\gamma_m\ol\epsilon,
\label{leftkilling}
\end{equation}
and $(\ol\epsilon_1\ol\epsilon_2)$ is constant on ${\bf S}^3$.
The other two solutions of
(\ref{killing}), which we will not use in this paper,
are left-invariant, and
satisfy a similar equation to
(\ref{leftkilling})
with opposite sign on the right hand side.
In the following the parameter $\ol\epsilon$ is always
assumed to satisfy (\ref{leftkilling}).
The $\delta(\ol\epsilon)$ transformation laws for fields
on ${\bf S}^3$ are
obtained from (\ref{d0vector}) and (\ref{d0chiral}) by the Weyl-covariantization.
The vector multiplet transformation laws are
\begin{align}
&\delta(\ol\epsilon) A_m=-i(\ol\epsilon\gamma_m\lambda),\quad
\delta(\ol\epsilon) \sigma=(\ol\epsilon\lambda),\quad
\delta(\ol\epsilon) \lambda=0,\nonumber\\
&\delta(\ol\epsilon) \ol\lambda=
-\frac{i}{2}\gamma^{mn}\ol\epsilon F_{mn}
-\gamma^m\ol\epsilon D_m\sigma+D\ol\epsilon+\frac{i}{r}\ol\epsilon \sigma
,\nonumber\\
&\delta(\ol\epsilon)D=
-(\ol\epsilon\gamma^mD_m\lambda)
-(\ol\epsilon[\sigma,\lambda])
-\frac{i}{2r}(\lambda\ol\epsilon).
\label{d3vecsusy}
\end{align}
The chiral multiplet transformation laws are
\begin{align}
&\delta(\ol\epsilon) \phi^\dagger=\sqrt2(\ol\epsilon\ol\psi),\quad
\delta(\ol\epsilon) \phi=0,\quad
\delta(\ol\epsilon)\ol\psi=\sqrt{2}\ol\epsilon F^\dagger,\quad
\delta(\ol\epsilon) F^\dagger=0,\nonumber\\
&\delta(\ol\epsilon)\psi=
-\sqrt{2}\gamma^m\ol\epsilon D_m\phi
+\sqrt{2}\ol\epsilon\sigma\phi
+\frac{\sqrt2i}{r}\Delta_\Phi\ol\epsilon\phi
,\nonumber\\
&\delta(\ol\epsilon) F=
-\sqrt{2}(\ol\epsilon\gamma^mD_m\psi)
-\sqrt{2}\sigma(\ol\epsilon\psi)
-2(\ol\epsilon\ol\lambda)\phi
-\frac{\sqrt2i}{r}\left(\Delta_\Phi-\frac{1}{2}\right)(\ol\epsilon\psi),
\label{d3chisusy}
\end{align}
where $\Delta_\Phi$ is the Weyl weight of the chiral multiplet,
which is defined as the Weyl weight of the dynamical scalar component field.
There is an ambiguity in the choice of
the ${\cal Q}$-exact deformation Lagrangian density ${\cal L}$.
We adopt the following one obtained by applying
$\delta(\ol\epsilon_1)$ and
$\delta(\ol\epsilon_2)$ to an anti-chiral operator,
\begin{equation}
(\ol\epsilon_1\ol\epsilon_2){\cal L}=
\delta(\ol\epsilon_1)\delta(\ol\epsilon_2)
\left(-\frac{1}{4}\tr(\ol\lambda\ol\lambda)
-\frac{1}{2}\sum_I \phi_I^\dagger F_I\right),
\label{antichiral}
\end{equation}
where $\tr$ represents
a gauge invariant positive definite inner product.
(\ref{antichiral}) can be used both in 3d and 4d.
In 3d, by using the 3d transformation laws
(\ref{d3vecsusy}) and (\ref{d3chisusy}),
we obtain
\begin{align}
{\cal L}^{(3d)}
=&
\tr\bigg[
\frac{1}{4}F_{mn}F^{mn}
+\frac{1}{2}D_m \sigma D^m\sigma
-\frac{1}{2}\epsilon^{mnp}F_{mn}D_p\sigma
+\frac{1}{2}\left(\frac{1}{r}\sigma-iD\right)^2
\nonumber\\&
-(\ol\lambda\gamma^mD_m\lambda)
-(\ol\lambda[\sigma,\lambda])
+\frac{i}{2r}(\ol\lambda\lambda)\bigg]
\nonumber\\
&+
\sum_I\bigg[
-\phi_I^\dagger D_m D^m\phi_I
+\phi_I^\dagger\sigma\sigma\phi_I
+\phi_I^\dagger D\phi_I
\nonumber\\&
-\frac{i(1-2\Delta_I)}{r}\phi_I^\dagger\sigma\phi_I
-\frac{\Delta_I(\Delta_I-2)}{r^2}\phi_I^\dagger\phi_I
-F_I^\dagger F_I
\nonumber\\&
-(\ol\psi_I\gamma^mD_m\psi_I)
-(\ol\psi_I\sigma\psi_I)
-\frac{i(\Delta_I-\frac{1}{2})}{r}(\ol\psi_I\psi_I)
-\sqrt2\phi_I^\dagger(\lambda\psi_I)
-\sqrt2(\ol\psi_I\ol\lambda)\phi_I
\bigg].
\label{3dchirallag}
\end{align}
(We use notation that in Euclidean signature the Hermitian conjugate
of the auxiliary fields $D$ and $F_I$ are $-D$ and $-F_I^\dagger$, respectively.)
In the large $u$ limit,
we can perform the path integral (\ref{zs3}),
and obtain
the matrix model integral\cite{Kapustin:2009kz,Jafferis:2010un,Hama:2010av}
\begin{align}
Z=\int_{\RR^{\rank G}} d\sigma J^{(3d)}(\sigma)
Z^{\rm vector}(\sigma)\prod_IZ_{\Phi_I}^{\rm chiral}(\sigma).
\label{zresult}
\end{align}
Integration variable $\sigma$ in (\ref{zresult}) is
an element of the Cartan subalgebra of the gauge group $G$.
The Jacobian factor $J^{(3d)}(\sigma)$ is
\begin{equation}
J^{(3d)}(\sigma)=\prod_{\alpha\in \Delta}\pi\alpha(r\sigma).
\label{j3d}
\end{equation}
$Z^{\rm vector}(\sigma)$ and $Z^{\rm chiral}_{\Phi_I}(\sigma)$
are $1$-loop partition function of
vector and chiral multiplets.
They are given by
\begin{align}
Z^{\rm vector}(\sigma)
=&
\prod_{\alpha\in\Delta}\frac{\sinh(\pi\alpha(r \sigma))}{\pi\alpha(r \sigma)},
\nonumber\\
Z_{\Phi_I}^{\rm chiral}(\sigma)
=&
\prod_{\rho\in R_I}
\prod_{k=1}^\infty
\left(
\frac{k+1-\Delta_I-i\rho(r\sigma)}
{k-1+\Delta_I+i\rho(r\sigma)}
\right)^k.
\end{align}
\section{Superconformal index}\label{index.sec}
Let us consider a 4d ${\cal N}=1$ theory in ${\bf S}^3\times{\bf S}^1$.
The background is conformally flat, and the parameter $\ol\epsilon$ must satisfy
the Killing equation
\begin{equation}
D_\mu \ol\epsilon=\gamma_\mu\kappa.
\label{4dlikking}
\end{equation}
To relate 3d spinor $\ol\epsilon(x^m)$ satisfying
(\ref{leftkilling})
and 4d spinor $\ol\epsilon(x^\mu)$, we take the anzats
\begin{equation}
\ol\epsilon(x^\mu)=f(x^4)\ol\epsilon(x^m).
\end{equation}
From (\ref{leftkilling})
the 4d spinor $\ol\epsilon(x^\mu)$ satisfies
\begin{equation}
D_\mu\ol\epsilon(x^\mu)=\frac{1}{2r}\gamma_\mu\gamma_4\ol\epsilon(x^\mu)
\label{killingin4}
\end{equation}
for $\mu=1,2,3$.
For $\ol\epsilon$ to be a Killing spinor
in 4d, this must hold for $\mu=4$, too.
This determines the function $f(x^4)$ up to normalization as
\begin{equation}
f(x^4)= e^{\frac{x^4}{2r}}.
\end{equation}
Corresponding to the Killing spinors $\ol\epsilon_1(x^m)$ and $\ol\epsilon_2(x^m)$
in 3d, we define two Killing spinors in 4d, which
are denote by the same symbols $\ol\epsilon_1$ and $\ol\epsilon_2$.
We adopt $\delta(\ol\epsilon_1(x^\mu))$ as ${\cal Q}$ in the same way as in 3d.
We want to compute a quantity in the form
\begin{equation}
I=\tr[(-1)^F {\cal O}q^D],
\label{igen}
\end{equation}
where ${\cal O}$ is an operator constructed from the
Cartan generators of the superconformal and flavor symmetries.
The most general form of ${\cal O}$ is
\begin{equation}
{\cal O}=y^{-\frac{3}{2}R-2J_L}
t^{R+2J_L}
x^{2J_R}
\prod_i
h_i^{{\cal F}_i}.
\end{equation}
This is equivalent to
imposing the boundary condition
\begin{equation}
\Phi(x^m,x^4)={\cal O}
\Phi(x^m,x^4+\beta r),
\label{phibc}
\end{equation}
on an arbitrary field $\Phi$.
For localization to be applicable the supercharge ${\cal Q}$
must commute with the operator ${\cal O}$.
Equivalently, the Killing spinor $\ol\epsilon_1$ must
satisfy the boundary condition
(\ref{phibc}).
This requires
$y=q$, and in this case
(\ref{igen}) becomes the index (\ref{indexdef}).
The 4d supersymmetry transformation laws
are obtained from (\ref{d0vector}) and (\ref{d0chiral})
by using (\ref{covdf}), (\ref{covariantization}),
and (\ref{killingin4}).
The transformation laws for a vector multiplet are
\begin{align}
&\delta(\ol\epsilon) A_\mu=-i(\ol\epsilon\gamma_\mu\lambda),\quad
\delta(\ol\epsilon)\lambda=0,\nonumber\\
&\delta(\ol\epsilon)\ol\lambda=-\frac{i}{2}\gamma^{\mu\nu}\ol\epsilon F_{\mu\nu}+D\ol\epsilon,\nonumber\\
&\delta(\ol\epsilon) D=-(\ol\epsilon\gamma^\mu D_\mu\lambda).
\label{4dvector}
\end{align}
The chiral multiplet transformation laws are
\begin{align}
&\delta(\ol\epsilon) \phi^\dagger=\sqrt{2}(\ol\epsilon\ol\psi),\quad
\delta(\ol\epsilon) \phi=0,\quad
\delta(\ol\epsilon)\ol\psi=\sqrt{2}\ol\epsilon F^\dagger,\quad
\delta(\ol\epsilon) F^\dagger=0,\nonumber\\
&\delta(\ol\epsilon)\psi=
-\sqrt{2}\gamma^\mu\ol\epsilon D_\mu\phi
-\frac{\sqrt{2}\Delta_\Phi}{r}\gamma^4\ol\epsilon\phi,\nonumber\\
&\delta(\ol\epsilon) F
=-\sqrt{2}(\ol\epsilon\gamma^\mu D_\mu\psi)
-2(\ol\epsilon\ol\lambda)\phi
-\frac{\sqrt{2}(\Delta_\Phi-1)}{r}(\ol\epsilon\gamma^4\psi).
\label{4dchitr}
\end{align}
The 4d deformation Lagrangian density ${\cal L}^{(4d)}$
is given by (\ref{antichiral}) with
the 4d transformation laws (\ref{4dvector}) and
(\ref{4dchitr}),
\begin{align}
{\cal L}^{(4d)}=&\tr\bigg[
\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
-\frac{1}{8}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}
-\frac{1}{2}D^2
-(\ol\lambda\gamma^\mu D_\mu\lambda)\bigg]
\nonumber\\
&+\sum_I\bigg[-F_I^\dagger F_I
-\phi_I^\dagger D_\mu D^\mu \phi_I
+\phi_I^\dagger D\phi_I
-\frac{\Delta_I^2-2\Delta_I}{r^2}\phi_I^\dagger\phi_I
-\frac{2(\Delta_I-1)}{r}\phi_I^\dagger D_4\phi_I
\nonumber\\&
-(\ol\psi_I\gamma^\mu D_\mu\psi_I)
-\frac{\Delta_I-1}{r}(\ol\psi_I\gamma^4\psi_I)
-\sqrt2\phi_I^\dagger(\lambda\psi_I)
-\sqrt2(\ol\psi_I\ol\lambda)\phi_I\bigg].
\label{4dlag}
\end{align}
Note that this action contains only the anti-self-dual part of $F_{\mu\nu}$,
and we need to change the coefficient of the topological term
$\propto\tr(F\wedge F)$ to localize the path integral to flat connections.
This is possible
because the index does not depend on the coefficient of this term
as well as other coupling constants consistent with the symmetry of the system.
This Lagrangian density is essentially the same as
what is
derived in \cite{Romelsberger:2005eg}.
The index can be computed exactly
by performing the path integral (\ref{ideform}) in the large $u$ limit.
The result is\cite{Romelsberger:2007ec}
\begin{equation}
I(t,x,h_i)=\int_{\TT^{\rank G}} dA_4 J^{(4d)}(A_4) \pexp f(q^{i r A_4},t,x,h_i).
\label{iresult}
\end{equation}
The $A_4$ integral is taken over the maximal torus of the gauge group $G$.
$\pexp$ is the plethystic exponential
\begin{equation}
\pexp f(g,t,x,h_i)=
\exp\left(\sum_{m=1}^\infty\frac{1}{m}f(g^m,t^m,x^m,h_i^m)\right).
\end{equation}
$J^{(4d)}(A_4)$ is the Jacobian factor associated with the gauge fixing,
\begin{equation}
J^{(4d)}(A_4)=\prod_{\alpha\in\Delta}\frac{\sin(\pi\beta\alpha( r A_4))}{\beta}.
\label{j4d}
\end{equation}
$f(g,t,x,h_i)$ is the letter index.
The contribution of vector multiplets
is
\begin{align}
&f^{\rm vector}(q^{ir A_4},t,x,h_i)
\nonumber\\
&=
\sum_{\alpha\in G}q^{i\alpha(r A_4)}\left(
\sum_{l=0}^\infty\sum_{k=-l/2}^{l/2}
t^{l+2}x^{2k}
-
\sum_{l=1}^\infty\sum_{k=-l/2}^{l/2}
t^lx^{-2k}
\right)
\nonumber\\
&=
\frac{2t^2-t(x+x^{-1})}
{(1-tx)(1-tx^{-1})}
\sum_{\alpha\in G}q^{i\alpha(r A_4)}.
\label{vectorletter}
\end{align}
The contribution of a chiral multiplet $\Phi_I$
belonging to a gauge representation $R_I$ is
\begin{align}
&f^{\rm chiral}_{\Phi_I}(q^{ir A_4},t,x,h_i)
\nonumber\\
&=
\sum_{\rho\in R_I}\sum_{l=0}^\infty
\sum_{k=-l/2}^{l/2}
\left(
q^{i\rho(r A_4)}t^{l+\frac{2}{3}\Delta_I}x^{-2k}\prod_ih_i^{{\cal F}_i(\Phi_I)}
-q^{-i\rho(r A_4)}t^{l-\frac{2}{3}\Delta_I+2}x^{2k}\prod_ih_i^{-{\cal F}_i(\Phi_I)}
\right)
\nonumber\\
&=
\sum_{\rho\in R_I}\frac{
q^{i\rho(r A_4)}
t^{\frac{2}{3}\Delta_I}
\prod_ih_i^{{\cal F}_i(\Phi_I)}
-
q^{-i\rho(r A_4)}
t^{2-\frac{2}{3}\Delta_I}
\prod_ih_i^{-{\cal F}_i(\Phi_I)}
}
{(1-tx)(1-tx^{-1})}.
\label{chiralletter}
\end{align}
\section{Comparison of the deformation actions}\label{comparison.sec}
In order to relate the ${\bf S}^3$ partition function and the index,
let us compare the Lagrangian densities ${\cal L}^{(3d)}$
in (\ref{3dchirallag}) and ${\cal L}^{(4d)}$ in (\ref{4dlag}).
They
look similar, but not the same.
The difference is partially absorbed by shifting the auxiliary $D$-field.
\begin{equation}
D^{(4d)}= D^{(3d)}+\frac{i}{r}\sigma.
\label{dshift}
\end{equation}
Even after this shift the actions are still different.
If we assume there are no non-trivial background Wilson lines
around ${\bf S}^1$
and the covariant derivative
$D_4$ reduces to $-iA_4=-i\sigma$ in dimensional reduction,
the difference is
\begin{equation}
{\cal L}^{(3d)}
-{\cal L}^{(4d)}
=\frac{1}{2r}\left[
(\ol\lambda\gamma^4\lambda)
-\sum_I(\ol\psi_I\gamma^4\psi_I)\right].
\label{additional}
\end{equation}
This difference can be removed by introducing
a suitable Wilson line
if the theory has the symmetry $R_0$ with the charge assignments
\begin{equation}
R_0(\lambda)=+1,\quad
R_0(\psi_I)=-1,\quad
R_0(A_\mu)=R_0(\phi)=0.
\end{equation}
We weakly gauge this symmetry and introduce
the gauge field $V_\mu$ for this symmetry.
If we turn on the Wilson line
\begin{equation}
\langle V_4\rangle=-\frac{i}{2r},
\end{equation}
the difference (\ref{additional})
is canceled by the terms arising from
the 4d fermion kinetic terms in (\ref{4dlag}).
This is equivalent to the insertion of the operator
\begin{equation}
{\cal O}=q^{-\frac{1}{2}R_0}
\end{equation}
in (\ref{igen}),
and thus we expect that the partition function $Z$
is given by
\begin{equation}
Z=\lim_{q\rightarrow 1}\tr[(-1)^Fq^{-\frac{1}{2}R_0}q^{D}].
\label{zlimtr}
\end{equation}
If the 4d parent theory
has non-vanishing superpotential,
the symmetry $R_0$ is in general broken.
However, the superpotential does not affect the index.
The relevant part of the deformed action $S_{\rm rel}^{(4d)}$
has the large symmetry
rotating chiral multiplets independently.
Let ${\cal F}_I$ denote the generator rotating only a chiral multiplet $\Phi_I$
by charge $1$.
Correspondingly, we introduce chemical potentials $h_I$.
The symmetry $R_0$ is related to $R$,
the $R$-symmetry in the superconformal algebra, by
\begin{equation}
R_0=R-\frac{2}{3}\sum_I\Delta_I{\cal F}_I,
\end{equation}
and we can express (\ref{zlimtr}) as a special limit of
the index,
\begin{equation}
Z=\lim_{q\rightarrow 1} I(t=q,x=1,h_I=q^{\frac{1}{3}\Delta_I}).
\label{relation0}
\end{equation}
It is easily checked that
this relation indeed holds for
(\ref{zresult}) and (\ref{iresult}) as follows.
Because the radius of the maximal torus $\TT^{\rank G}$ is inversely proportional
to the ${\bf S}^1$ period $\beta r$, it becomes
$\RR^{\rank G}$ in the limit $\beta\rightarrow 0$.
We also see that the Jacobian factor (\ref{j4d}) reduces to (\ref{j3d})
when $\beta\rightarrow 0$.
We obtain
\begin{equation}
\lim_{q\rightarrow 1}\int_{\TT^{\rank G}} dA_4J^{(4d)}(A_4)
=\int_{\RR^{\rank G}} d\sigma J^{(3d)}(A_4).
\label{intrel}
\end{equation}
For the letter indices,
we first express the plethystic exponential
of (\ref{vectorletter}) and (\ref{chiralletter})
as the infinite products,
\begin{align}
\pexp f^{\rm vector}(q^{ir A_4},t,x,h_I)
=&
\prod_{\alpha\in G}\frac{\prod_{l=1}^\infty\prod_{k=-l/2}^{l/2}
(1-q^{i\alpha(r A_4)}t^lx^{-2k})}
{
\prod_{l=0}^\infty\prod_{k=-l/2}^{l/2}
(1-q^{i\alpha(r A_4)}t^{l+2}x^{2k})},
\nonumber\\
\pexp f^{\rm chiral}_{\Phi_I}(q^{ir A_4},t,x,h_I)
=&
\prod_{\rho\in R_I}
\prod_{l=0}^\infty
\prod_{k=-l/2}^{l/2}
\left(
\frac{1-q^{-i\rho(r A_4)}t^{l-\frac{2}{3}\Delta_I+2}x^{2k}h_I^{-{\cal F}_I}}
{1-q^{i\rho(r A_4)}t^{l+\frac{2}{3}\Delta_I}x^{-2k}h_I^{{\cal F}_I}}
\right).
\label{pexpform}
\end{align}
Once we obtain these infinite products, it is straightforward to
confirm
the following relations.
\begin{align}
\lim_{q\rightarrow1}\pexp f^{\rm vector}(q^{ir A_4},q,1,q^{\frac{1}{3}\Delta_I})=&Z^{\rm vector}(A_4),
\nonumber\\
\lim_{q\rightarrow 1}\pexp f_{\Phi_I}^{\rm chiral}(q^{ir A_4},q,1,q^{\frac{1}{3}\Delta_I})=&Z^{\rm chiral}_{\Phi_I}(A_4).
\label{letterlimit}
\end{align}
Combining (\ref{intrel}) and (\ref{letterlimit}),
we obtain the relation
(\ref{relation0}).
Before ending this section,
let us argue the anomaly associated with the
inserted operator ${\cal O}$.
We consider the quantity
\begin{equation}
I(t,x,h_I)=\tr[(-1)^F q^X q^D],
\label{xinserted}
\end{equation}
where we denote the inserted operator ${\cal O}$ by $q^X$.
As we mentioned in \S\ref{intro.sec} the symmetry
generated by $X$
may be anomalous,
and then
the quantity
(\ref{xinserted})
is not well defined.
This can be regarded as inconsistency in the ${\bf S}^1$ compactification.
If $X$ is anomalous, the rotation by $q^X$ does not keep
the effective action $\Gamma$ invariant but changes it by
\begin{align}
\Gamma\rightarrow \Gamma'
=\Gamma+\int_{{\bf S}^3\times{\bf S}^1}\frac{\beta}{8\pi^2}\tr_F(XF\wedge F)
\label{gammashify}
\end{align}
where $\tr_F$ is the trace over Weyl fermions of positive chirality, which
contribute to the anomaly.
This change of the effective action obstacles the compactification
$x^4+\beta r\sim x^4$.
We can remove this obstruction by adding the following term
to the tree-level action.
\begin{align}
S'
&=-\int_{{\bf S}^3\times{\bf S}^1}\frac{x^4}{8\pi^2 r} \tr_F(XF\wedge F)
\nonumber\\
&=\int_{{\bf S}^1}dx^4\int_{{\bf S}^3}\frac{1}{8\pi^2 r} \tr_F\left[X\left(A\wedge F-\frac{2i}{3}A\wedge A\wedge A\right)\right].
\label{improve}
\end{align}
Due to the $x^4$ dependence of the $\theta$ angle,
the change of $S'$ under the shift $x^4\rightarrow x^4+\beta r$
cancels the anomalous change (\ref{gammashify}).
With the inclusion of the term
(\ref{improve}) in the action,
we can consistently compactify the $x^4$ direction
with the twist by ${\cal O}=q^X$.
When $X$ is anomalous, we define the quantity
(\ref{xinserted}) by the path integral (\ref{indexdef})
with the action improved by (\ref{improve}).
Let us consider whether it is possible to extend the additional term $S'$
in a supersymmetric way.
(\ref{improve}) is a three-dimensional Chern-Simons term except that
fields depend on the fourth coordinate $x^4$ along ${\bf S}^1$.
If all fields were $x^4$-independent,
we could actually construct the supersymmetric completion
\begin{align}
S'_{\rm SUSY}
=&
\int_{{\bf S}^1\times{\bf S}^3}d^4x\frac{\sqrt{g}}{8\pi^2 r}
\nonumber\\&\tr_F
\left[X\left\{\frac{i}{2}\epsilon^{mnp}\left(A_m\partial_n A_p-\frac{2i}{3}A_mA_nA_p\right)
+(\lambda\ol\lambda)
-D^{(4d)}A_4
+\frac{i}{r}A_4^2\right\}
\right].
\label{lpsusy}
\end{align}
For fields depending on $x^4$, however,
this action is not supersymmetry invariant.
We have non-vanishing supersymmetry transformation of the action
\begin{equation}
\delta
S'_{\rm SUSY}=
\int_{{\bf S}^1\times{\bf S}^3}d^4x\frac{\sqrt{g}}{8\pi^2 r}
\tr_F[X(\lambda\gamma^\mu\ol\epsilon)\partial_4 A_\mu].
\end{equation}
It is even worse that
(\ref{lpsusy}) is not even gauge invariant
due to terms containing $A_4$.
Unfortunately, we could not remedy these defects in
(\ref{lpsusy}),
and we use the non-supersymmetric term (\ref{improve}) to
turn on non-trivial Wilson lines for
anomalous symmetries.
In the large $u$ limit, the term
(\ref{improve}) is irrelevant, and
$I(t,x,h_I)$ is still given by the formula (\ref{iresult}).
However, the absence of the supersymmetry spoils the
$u$-independence of the path integrals, and
we can no longer regard $I(t,x,h_I)$ computed by the formula (\ref{iresult})
as the index of the original theory.
In the small ${\bf S}^1$ limit $\beta\rightarrow 0$, the term (\ref{improve}) vanishes
and the relation (\ref{relation0}) still holds.
\section{Generalization}\label{generalization.sec}
Up to here we have been assuming that
parameters of the 3d theory, $\mu_I$, $\zeta_A$, $k_a$, and $s$ all vanish.
Let us consider how we can obtain partition function
for a theory with these parameters turned on.
If the 3d theory has a flavor $U(1)$ symmetry,
we can introduce a real mass proportional to
the flavor charge for each chiral multiplet.
We here focus only on the relevant part $S_{\rm rel}^{(3d)}$,
and we can introduce real mass $\mu_I$ for each chiral multiplet $\Phi_I$
by weakly gauging ${\cal F}_I$ and turning on the
scalar component $\sigma_I$ of the corresponding vector multiplet $(\sigma_I,A_{I,m},\lambda_I,\ol\lambda_I,D_I)$.
(If some of ${\cal F}_I$ are anomalous, we need to introduce
the term (\ref{improve}) in the definition of the index.)
Note that we should turn on the auxiliary field $D_I$, too, to preserve
the supersymmetry (\ref{d3vecsusy}).
\begin{equation}
\langle \sigma_I\rangle=\mu_I,\quad
\langle D^{(3d)}_I\rangle=-\frac{i}{r}\mu_I,\quad
\langle A_{I,m}\rangle=
\langle\lambda_I\rangle=
\langle\ol\lambda_I\rangle=0.
\end{equation}
From the viewpoint of 4d theory, this is realized by
turning on the Wilson line for the flavor symmetry ${\cal F}_I$,
\begin{equation}
\langle A_{I,4}\rangle=\mu_I,\quad
\langle D^{(4d)}_I\rangle=
\langle A_{I,m}\rangle=
\langle\lambda_I\rangle=
\langle\ol\lambda_I\rangle=0.
\end{equation}
This is equivalent to the insertion of the operator
\begin{equation}
q^{-ir\sum_I\mu_I{\cal F}_I}
\label{fiinsertion}
\end{equation}
in (\ref{zlimtr}).
The next parameter we consider is a squashing parameter $s$.
The partition function of a theory on squashed ${\bf S}^3$ is
investigated in \cite{Hama:2011ea},
and it is found that
the partition function is changed
when both the isometries $SU(2)_L$ and $SU(2)_R$
are broken to $U(1)$.
It is proposed recently in \cite{Gadde:2011ia}
that the partition function depending on the squashing parameter
is reproduced from the index by turning on
$SU(2)_R$ Wilson line
in the case of 4d ${\cal N}=2$ theories.
We consider the insertion of the operator
\begin{equation}
q^{2sJ_R},
\label{jrinsertion}
\end{equation}
in a general 4d ${\cal N}=1$ theory.
By inserting (\ref{fiinsertion}) and (\ref{jrinsertion}) into
(\ref{zlimtr}),
we obtain
\begin{align}
Z=&\lim_{q\rightarrow 1}\tr[(-1)^Fq^{-\frac{1}{2}R_0}q^{-ir\mu_I{\cal F}_I}q^{2sJ_R}q^D]
\nonumber\\
=&\lim_{q\rightarrow 1}
I(t=q,x=q^s,h_I=q^{-ir\mu_I+\frac{1}{3}\Delta_I}).
\label{final}
\end{align}
This is the relation (\ref{mainresult})
with vanishing FI parameters.
Let us confirm that
(\ref{final}) reproduces
the partition function
of a 3d theory
with non-vanishing real masses and squashing parameter.
From the infinite product representation (\ref{pexpform})
we easily obtain
\begin{align}
&
\lim_{q\rightarrow 1}\pexp
f_{\Phi_I}^{\rm chiral}(q^{ir A_4},q,q^s,q^{r\mu_I+\frac{1}{3}\Delta_I})
\nonumber\\
&=
\prod_{\rho\in R}
\prod_{m,n\geq 0}
\left(
\frac{m(1+s)+n(1-s)-\Delta_I+2-i\rho(r A_4)-ir\mu_I}
{m(1-s)+n(1+s)+\Delta_I+i\rho(r A_4)+ir\mu_I}
\right),\nonumber\\
&\lim_{q\rightarrow 1}\pexp
f^{\rm vector}(q^{ir A_4},q,q^s,q^{r\mu_I+\frac{1}{3}\Delta_I})
\nonumber\\
&=
\prod_{\alpha\in G}\frac{\prod_{m,n\geq 0,(m,n)\neq(0,0)}
(m(1-s)+n(1+s)+i\alpha(r A_4))}
{
\prod_{m,n\geq0}
(m(1+s)+n(1-s)+2+i\alpha(r A_4)}.
\label{dblsine}
\end{align}
These are consistent with known results.
When $\mu_I=0$, these agree with the results
in \cite{Hama:2011ea} by the identification of parameters
\begin{equation}
\frac{\wt\ell}{\ell}=\frac{1+s}{1-s}.
\end{equation}
The $\mu_I$ dependence of
(\ref{dblsine})
is consistent with
the holomorphic dependence of the
partition function on $\Delta_I+ir\mu_I$\cite{Jafferis:2010un}.
One may think that this result is inconsistent with the result
in \cite{Hama:2011ea} because
the expression (\ref{final}) for the partition function
does not break the $SU(2)_L$ symmetry.
Ref \cite{Hama:2011ea} shows that an $SU(2)\times U(1)$ invariant
squashing does not change the partition function.
The reason for these different results is as follows.
The squashing considered in \cite{Hama:2011ea} is a left-invariant
squashing which preserves $SU(2)_L$ isometry,
and a Wilson line is turned on so that a half of
left-invariant Killing spinors is preserved.
There is in fact another essentially inequivalent possibility.
We can realize a left-invariant squashing with right-invariant Killing spinors
by taking a different
graviphoton background
from \cite{Hama:2011ea}.
In the above we use right-invariant Killing spinors,
and the $SU(2)_R$ Wilson line (\ref{jrinsertion})
preserves $SU(2)_L$ isometry.
This is a different situation from \cite{Hama:2011ea}.
In our case, the squashed metric is obtained from
the 4d background metric corresponding to
the insertion (\ref{jrinsertion})
\begin{align}
ds^2
=&
r^2\left[
(\mu^1)^2+(\mu^2)^2+(\mu^3+is dx^4)^2
+(dx^4)^2
\right]
\nonumber\\
=&r^2\left[
(\mu^1)^2+(\mu^2)^2
+\frac{1}{1-s^2}(\mu^3)^2
+(1-s^2)\left(dx^4
+\frac{is}{1-s^2}\mu^3\right)^2
\right],
\end{align}
where $\mu^a$ are left-invariant one-forms used in \cite{Hama:2011ea}.
We can read off the squashed metric of the base manifold,
\begin{equation}
ds^2=r^2[(\mu^1)^2+(\mu^2)^2]
+\frac{r^2}{1-s^2}(\mu^3)^2.
\end{equation}
It is interesting problem to confirm directly in 3d
that the partition function
for this squashed manifold with right-invariant Killing spinors
agree with (\ref{dblsine}).
As the last extension, let us introduce FI parameters.
Let $(A_{A,m},\sigma_A,\lambda_A,\ol\lambda_A,D_A^{(3d)})$
be $U(1)$ vector multiplets
for which we want to turn on the FI parameters.
If the 3d original action
contains the supersymmetry completion of FI terms
\begin{equation}
S^{(3d)}_{\rm FI}=-\sum_A\zeta_A^{(3d)}\int_{{\bf S}^3}\sqrt{g}\left(D^{(3d)}_A-\frac{i}{r}\sigma_A\right)d^3x,
\label{fi3d}
\end{equation}
the additional factor
\begin{equation}
\exp\left(-4\pi^2i r^2\sum_A\zeta^{(3d)}_A\sigma_A\right)
\label{fifactor}
\end{equation}
should be included
in the integrand in (\ref{zresult}).
$S_{\rm FI}^{(3d)}$ in (\ref{fi3d})
is obtained by dimensional reduction of 4d FI terms.
Note that the 4d FI term must be accompanied by smeared Wilson line
to preserve the supersymmetry,
\begin{equation}
S^{(4d)}_{\rm FI}=-\sum_A\zeta_A^{(4d)}\int_{{\bf S}^3\times{\bf S}^1}
\sqrt{g}\left(D^{(4d)}_A-\frac{2i}{r}A_{A,4}\right)d^4x.
\label{4dfi}
\end{equation}
Due to the coupling to the gauge fields, the 4d FI parameters
must be quantized, and thus the index can depend on them.
If we keep the relation $\beta r\zeta_A^{(4d)}=\zeta_A^{(3d)}$ in the small radius limit,
we reproduce
(\ref{fi3d}) from
(\ref{4dfi}) and the factor
corresponding to (\ref{fifactor}) arises in
the index formula (\ref{iresult}).
Taking account of this relation,
we obtain the most general relation (\ref{mainresult}).
Finally we comment on Chern-Simons terms.
The supersymmetric completion of Chern-Simons term is
\begin{equation}
S_{\rm CS}^{(3d)}
=\int_{{\bf S}^3}\sqrt{g}\tr'\left[\frac{i}{2}\epsilon^{mnp}\left(A_m\partial_n A_p-\frac{2i}{3}A_mA_nA_p\right)
+(\lambda\ol\lambda)
-D\sigma\right]d^3x,
\label{ch3d}
\end{equation}
where $\tr'$ is a gauge invariant inner product containing Chern-Simons levels.
If these terms exist in the original action in (\ref{zs3}),
the extra factor
\begin{equation}
e^{-2\pi^2i\tr'(r^2\sigma^2)}
\end{equation}
arises in the integrand in (\ref{zresult}).
Unfortunately, we cannot reproduce this contribution from the
index due to the difficulty in constructing
4d action which gives Chern-Simons terms through dimensional reduction.
\section{Conclusions}\label{conc.sec}
In this paper we investigated
a relation between 3d and 4d actions used for computation of two
exactly calculable quantities,
the ${\bf S}^3$ partition function and
the 4d superconformal index.
When the 3d theory does not have Chern-Simons terms,
the relevant part of the action, which affects the ${\bf S}^3$
partition function,
consists of ${\cal Q}$-exact deformation terms
and the supersymmetric completion of FI terms.
In the case of round ${\bf S}^3$, we showed that this relevant part of the
3d action is obtained by dimensional reduction
from the corresponding terms in 4d action used for the computation
of the 4d superconformal index.
From this fact, we obtained a relation
which gives the ${\bf S}^3$ partition function as a
small radius limit of the 4d superconformal index
suitably generalized so that we can introduce chemical potentials
to anomalous symmetries.
To obtain the most general relation (\ref{mainresult}),
we used a connection between a squashing of ${\bf S}^3$
and $SU(2)_R$ Wilson line.
Although the squashing we considered in this paper,
the left-invariant squashing with right-invariant Killing spinors,
is different from squashings studied in \cite{Hama:2011ea},
our result agree with the partition function
for the $U(1)\times U(1)$ symmetric squashed ${\bf S}^3$ derived in \cite{Hama:2011ea}.
For 3d theory with Chern-Simons terms,
we could not give a 4d action reproducing the ${\bf S}^3$
partition function.
\section*{Acknowledgments}
I would like to thank Daisuke Yokoyama and Shuichi Yokoyama
for daily discussions.
I would also like to thank Kazuo Hosomichi and Yu Nakayama for
valuable comments.
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\section{Introduction}
During their growth via accretion, supermassive black holes become visible as active galactic nuclei \citep[AGN;][]{Rees82, LyndenBell69, Soltan82,Merloni04}. The feedback from AGN is a key ingredient in cosmological simulations, as it injects energy into the host galaxies and circumgalactic medium (CGM). This is required in order to reproduce key galaxy properties including the M-sigma relationship, colour bi-modality, enrichment of the of the IGM by metals, galaxy sizes and broader range of specific star formation rates \citep[e.g.,][]{Silk98,DiMatteo05,Alexander12,Dubois13,Dubois13b,Vogelsberger14,Hirschmann14, Crain15,Segers16,Beckmann17,Harrison17,Choi18,Scholtz18}.
The CGM surrounding a galaxy is spatially extended material (on scales larger than the galaxy's stellar size; $>8$\,kpc at $z\sim2$; \citealt{ForsterSch18a}) that serves as a fuel reservoir for future star formation. This reservoir contains both the processed gas ejected from the galaxy via star-formation feedback \citep[e.g.][]{Ginolfi17,Spilker18,Gallerani18,Jones19} or AGN feedback \citep[e.g.][]{Bischetti19, Vayner21,Travascio20,Cicone21} as well as streams of gas from the large scale filaments \citep[e.g.][]{ArrigoniBattaia18, Umehata19}. Therefore, it can be used to study past star formation and AGN activity as well as future fuelling of galaxy growth.
The most reliable efforts to map the CGM come from integral-field-spectroscopy observations of Ly$\alpha$ emission, which traces ionised gas \citep[e.g.,][]{drak20,sand21}. These observations constrained the size of these ionised gas halos up to $\sim170$\,kpc. However, Ly$\alpha$ observations are tracing only a single phase of the gas in the CGM, omitting the cold phase.
Cold gas halos have been detected in both stacked [\ion{C}{ii}] 158 $\mu$m emission and individual star-forming main sequence galaxies \citep[SFMS; ][]{Schreiber15} at $z>4$ \citep[][]{Fujimoto19,Fujimoto20, Herrera-Camus21, Lambert22} that extend on scales larger than both the beam of the observations and UV emission tracing the young stars. These extended halos are taken as evidence of enrichment of the CGM by starbursts driven outflows \citep[see][for more details]{Ginolfi20}. However, in these observations, AGN were purposefully discarded to study the effect of starburst-driven outflows on the CGM.
Although we have a wealth of observations of cold gas halos, the picture around AGN host galaxies is still emerging. More recently, two studies have detected CO(3-2) halos around X-ray AGN at $z\sim$ 2.4 (\citealt{Cicone21} in a single AGN host galaxy, \citealt{Jones22} in stacked emission). In radio loud AGN, cold gas molecular halos have been detected in the Spiderweb galaxy, a galaxy protocluster at $z\sim$2.2, with a size of up to 70 kpc using CO(4-3), [\ion{C}{i}](1-0), [\ion{C}{ii}] emission \citep[][]{Emonts+16, Emonts+18, DeBreuck+22}, and in a radio loud quasar at $z\sim$2.2, where the molecular gas reservoir is aligned with a radio jet on scales of 100 kpc \citep[][]{Li+21}. However, we still lack a systematic study of cold gas halos around quasar host galaxies.
Extremely red quasars (ERQs), a population of unique obscured quasars, are often luminous dust-reddened sources that are believed to be at a different evolutionary stage compared to blue quasars and moderate luminosity AGN \citep[][]{Ross15, Hamann17, Klindt+19}.
ERQs were initially discovered through a selection based on high rest-frame ultraviolet to infrared colours (i-W3$>$ 4.6 mag) from the Wide-Field Infrared Survey Explorer \citep[WISE; ][]{Wright10} and Sloan Digital Sky survey (SDSS; \citealt{Blanton+17}). This selection results in high nuclear obscuration with column densities up to 10$^{24}$ cm$^{-2}$ \citep[][]{Goulding18, Ishikawa21}.
Previous detections of cold gas halos have used [\ion{C}{ii}] emission, which is tracing multiple phases of gas, or CO(3-2), which requires multiple conversion factors to estimate molecular gas mass (e.g., $r_{J1}$ and $\alpha_{CO}$; \citealt{Bolatto13}). In this work we focus on analysing ALMA observations of three emission lines: CO(7-6), [\ion{C}{i}]\,($^{3} \rm P_{2} \rightarrow {\rm ^3 P}_{1}$) and H$_2$O 2$_{11}$--2$_{02}$, as well as the dust continuum (rest-frame $\sim 350 \mu$m). The CO(7-6) and [\ion{C}{i}](2-1) are among the most luminous emission lines, and therefore important lines to cool the ISM and CGM, each tracing a different cold gas phase. Similarly to CO(1-0), the [\ion{C}{i}] emission line is tracing primarily molecular gas from the outer layers of molecular clouds \citep[see][]{Walter11,AlaghbandZadeh16,Glover16,Papadopoulos18,Valentino18,Bisbas19} as well as the diffuse molecular gas in the interstellar medium, such as photon dominated regions \citep[PDRs;][]{Tielens85} with relatively low critical density ($n_{\rm crit}$) of $\sim 10^{3}$ cm$^{-3}$. Hence [\ion{C}{i}] is an excellent trace of the cold molecular gas \citep[][]{Bothwell17, Jiao17}. Furthermore, the conversion of [\ion{C}{i}] integrated line luminosity to H$_2$ mass has smaller uncertainties compared to that of CO, and has a lesser dependency on the metallicity of the gas. On the other hand, the CO(7-6) is tracing warmer and denser gas ($T_{\rm ex}\sim$150 K and $n_{\rm crit}\sim 10^{5}$ cm$^{-3}$). This emission line is an excellent tracer of current or recent star formation \citep[see][]{Lu18,Zhao20} in star-forming galaxies, but the exact source of excitation in quasar host galaxies is yet unknown.
In \S~\ref{sec:data} we describe our targets, and the observations used in our study, while \S~\ref{sec:analyses} describes the data analyses of the ALMA observations, including spectral fitting, stacking and modelling the extracted radial surface brightness profile.
In \S~\ref{sec:results} we present our results and discuss the physical properties and origins of the discovered cold gas and dust halos. In this work, we consider any additional component large than the galaxy as a halo.
In all of our analyses, we adopt the cosmological parameters of
$H_{\rm0} = 67.3$\,km\,s$^{-1}$, $\rm \Omega_M = 0.3$, $\rm
\Omega_\Lambda = 0.7$ \citep{Planck13} and assume a \citet{Chabrier03} initial mass
function (IMF).
\section{Data and observations}\label{sec:data}
\subsection{ALMA Observations and data reduction}
In this project we investigate the molecular gas emission of fifteen extremely red quasars (ERQs) at $z\sim$ 2.4, observed in CO(7-6), [\ion{C}{i}]\,($^{3} \rm P_{2} \rightarrow {\rm ^3 P}_{1}$) and H$_2$O 2$_{11}$--2$_{02}$ emission and 1.2 mm dust continuum in ALMA project 2017.1.00478.S (PI: Fred Hamann) in Cycle 5. These ERQs were identified in \citet{Hamann17} with the following selection criteria: 1) QSOs with red colours (i-W3 $>$ 4.6); 2) large equivalent with of the CIV emission line (RW(CIV) $>$ 100 \AA ). The IDs, coordinates, redshifts, black hole masses and bolometric luminosities of the sample are summarised in Table \ref{Table:Sample}.
The raw science data models were processed by the ALMA staff through their calibration pipeline, which resulted in the calibrated measurement sets for each observation. As two sources were observed over two separate executions, we joined these executions using CASA's \texttt{concat} task. The measurement sets included also the calibrator sources, so we created measurement sets containing only calibrated visibilities of the target source with CASA's \texttt{split} task.
With the measurement sets for each source, we imaged the sources using a uniform imaging pipeline. As the optical and sub-mm line emission can have a significant velocity offsets, we first imaged the cubes using the natural weighting (to maximise the sensitivity) to find the redshift of the CO(7-6), [\ion{C}{i}]\,($^{3} \rm P_{2} \rightarrow {\rm ^3 P}_{1}$) (hereafter [\ion{C}{i}](2-1)) and H$_2$O 2$_{11}$--2$_{02}$ (752.033 GHz; hereafter H$_2$O) emission lines. Indeed we find that the velocity offset between the optical and CO redshifts is up to 2000\, km\,s$^{-1}$. To identify the ``line-fre'' channels we picked channels outside of the $\pm600$\, km\,s$^{-1}$ of the centre of the detected emission lines. We then used CASA's \texttt{uvcontsub} task to subtract the continuum emission in the \textit{uv}-plane.
The continuum-free visibilities were imaged using the \texttt{tclean} task in the ``cube'' mode using 0.1 arcsecond cells and natural weighting to create the ``dirt'' line cubes. Given that we are ultimately searching for faint emission on large scales, natural weighting is the ideal weighting scheme for us. The channel width is kept to the intrinsic value of $\sim$20\,km\,s$^{-1}$. Once we estimate the RMS of the ``dirt'' cubes, we repeat the \texttt{tclean} task, this time cleaning the emission down to $3\times$RMS. The final beam size of the natural weighted line cubes are 0.7-1.0 arcsecond. We imaged the continuum using the \texttt{tclean} task in the continuum (mfs) mode, using only the line-free channels (i.e. channels $>\pm2\times$ FWHM of the systematic redshift; which corresponds to our original assumption of line-free channels of $\pm 600$ km s$^{-1}$).
\begin{table}
\caption{Table of basic properties of our sample. (1) Object ID; (2,3) Coordinates; (4) optical redshift ; (5,6) Black hole masses and bolometric luminosities from \citealt{Hamann17}.
}
\input{./Tables/Sample.tex}
\label{Table:Sample}
\end{table}
\subsection{Extraction of the integrated line emission}\label{sec:spectra}
In this section, we describe the extraction and analyses of the total integrated emission lines. To get the emission-line profiles of the ERQs, we extract the spectra from the naturally weighted cubes using apertures with a radius of 0.7--0.9 arcsecond, corresponding to the beam size of these observations, centred on the peak of the emission, which we found by fitting a two-dimensional Gaussian to the emission.
To model the extracted emission-line profiles, each emission line was fitted with one or two Gaussian components with the centroid, line width and normalisation (flux) as free parameters. We note that we do not give a physical meaning to either of these components and we use these to characterise the total emission line profile. To distinguish between individual models, several different statistics can be used. One of the most basic models is the reduced $\chi^{2}_{\rm red}$ parameter:
\begin{equation}
\chi^{2}_{\rm red} = \frac{\chi^{2}}{N_{\rm data}- N_{\rm var}}
\end{equation}
where N$_{\rm data}$ is the number of data points and N$_{\rm var}$ is the number of variables that are fitted (e.g. three for a single gaussian profile). Alternatively, we can use the Bayesian Information Criterion (BIC; \citealt{Schwarz78}, \citealt{Liddle07}, \citealt{Concas19}), which further penalises the $\chi^{2}$ for more variables:
\begin{equation}
\rm BIC = \chi^{2} + N_{\rm var}\log(N_{\rm data})
\end{equation}
Similarly to the $\chi^{2}_{\rm red}$, the model with lower BIC is a statistically better fit to the data. The following criteria are used: <2 - no difference; 2--6 - slight evidence; 6--10 significant evidence; >10 a better fit. However, it is worth noting that using BIC or reduced $\chi^{2}$ metrics to distinguish between models does not change the conclusions of this work.
The resulting extracted spectra with best fits are presented in Figures \ref{fig:CO_data}, \ref{fig:CI_data} \& \ref{fig:H2O_data} and we present the results of the best-fits in Table 2. We define the FWHM of the line as the velocity width containing 68\% of the flux of the line, similar to previous works in the literature \citep[][]{Bothwell13, Wardlow18} to describe the emission linewidth in a way that is independent of the fitted model. We report SNR, integrated line flux and the velocity line width for all lines detected at SNR>5 in Table \ref{Table:Lines}.
\begin{landscape}
\begin{table}
\caption{Table of basic properties of our sample. (1) Object ID; (2) SNR of the CO(7-6) line; (3) FWHM of the CO(7-6) line; (4) redshift of the CO(7-6) line; (5) integrated flux of the CO(7-6) line; (6) SNR of the [\ion{C}{i}] line; (7) FWHM of the [\ion{C}{i}] line; (8) redshift of the [\ion{C}{i}] line; (9) integrated flux of the [\ion{C}{i}] line; (10) SNR of the H$_2$O line; (11) FWHM of the H$_2$O line; (12) redshift of the H$_2$O line; (13) integrated flux of the H$_2$O line; (14) Continuum flux at 1200 $\mu$m (observed frame).
}
\input{./Tables/Line_emission.tex}
\label{Table:Lines}
\end{table}
\end{landscape}
\begin{landscape}
\begin{figure}
\includegraphics[width=1.1\paperwidth]{Graphs/CO_data_summary.pdf}
\caption{ Summary of the CO(7-6) data. Left column: Moment-0 map of the emission line. The red solid contours show 1, 2, 3, 4, 5 $\sigma$ and the red dashed contours -1, -2, -3, -4, -5 $\sigma$. Middle panel: Spectrum extracted from the region corresponding to the beam size. The orange line indicates the best fit to the data according to the BIC statistics. Right panel: Plot of the \textit{uv}-visibilities (flux (real component) vs the \textit{uv}-distance). The red solid line shows the best fit to the data. We have also included the single resolved source fit (blue dashed line), unresolved source model (green dashed line) and a combination of resolved and a point source (magenta dashed line). We only show the \textit{uv}-visibilities for targets detected over 10 $\sigma$.
}
\label{fig:CO_data}
\end{figure}
\end{landscape}
\begin{landscape}
\begin{figure}
\includegraphics[width=1.1\paperwidth]{Graphs/CI_data_summary.pdf}
\caption{ Summary of the [\ion{C}{i}] data. Left column: Moment-0 map of the emission line. The red solid contours show 1, 2, 3, 4, 5 $\sigma$ and the red dashed contours -1, -2, -3, -4, -5 $\sigma$. Middle panel: Spectrum extracted from the region corresponding to the beam size. The orange line indicates the best fit to the data according to the BIC statistics. Right panel: Plot of the \textit{uv}-visibilities (flux (real component) vs the \textit{uv}-distance). The red solid line shows the best fit to the data. We have also included the single resolved source fit (blue dashed line), unresolved source model (green dashed line) and a combination of resolved and a point source (magenta dashed line). We only show the \textit{uv}-visibilities for targets detected over 10 $\sigma$.
}
\label{fig:CI_data}
\end{figure}
\end{landscape}
\begin{landscape}
\begin{figure}
\includegraphics[width=1.1\paperwidth]{Graphs/H2O_data_summary.pdf}
\caption{ Summary of the H$_2$O data.Left column: Moment-0 map of the emission line. The red solid contours show 1, 2, 3, 4, 5 $\sigma$ and the red dashed contours -1, -2, -3, -4, -5 $\sigma$. Middle panel: Spectrum extracted from the region corresponding to the beam size. The orange line indicates the best fit to the data according to the BIC statistics. Right panel: Plot of the \textit{uv}-visibilities (flux (real component) vs the \textit{uv}-distance). The red solid line shows the best fit to the data. We have also included the single resolved source fit (blue dashed line), unresolved source model (green dashed line) and a combination of resolved and a point source (magenta dashed line). We only show the \textit{uv}-visibilities for targets detected over 10 $\sigma$.
}
\label{fig:H2O_data}
\end{figure}
\end{landscape}
\section{Analysis}\label{sec:analyses}
In this section, we present the analysis used in this work to achieve our goal of tracing cold gas halos around these quasars. In \S \ref{sec:uv-analyses}, we describe our analyses of the ALMA data in the \textit{uv}-plane, and in \S \ref{sec:RP_extractio} we present the extraction and modelling of the radial brightness profiles. Finally, in \S \ref{sec:image_stack} \& \ref{sec:uv_stack} we present our method for stacking in both image and \textit{uv}-plane stacking, respectively.
\subsection{Investigating \textit{uv}-visibilities}\label{sec:uv-analyses}
In order to measure the flux and the sizes of the emission, we have extracted and collapsed the \textit{uv}-visibilities. We first split the calibrated measurement sets to include only channels with the emission line. We selected these line emission channels based on the extracted spectra in the image plane, by selecting channels with at least 10\% of the peak flux of the combined spectra. We extracted the \textit{uv}-visibilities using \texttt{uvplot} Python library \citep[][]{uvplot_tazzari} and binned them in 30\,k$\lambda$-wide bins. We show these visibilities for each emission line in Figures \ref{fig:CO_data}, \ref{fig:CI_data} \& \ref{fig:H2O_data} for sources with SNR>10.
We fitted these amplitudes as a function of $uv$ distance with four different functions, a constant, to represent an unresolved source \citep[e.g.,][]{Rohlfs96}, a half Gaussian model centred on 0, to represent a resolved source with a Gaussian morphology, a combination of the previous two models representing both a central unresolved source and resolved source and two half Gaussians representing two resolved sources. Similarly to the spectral line fitting, we used BIC to distinguish between the models. The final uncertainties on the sizes comes as 68\% confidence interval uncertainties from the MCMC fitting. We summarise the estimated sizes in Table \ref{Table:Sizes}.
\subsection{Extraction and modelling of radial profiles}\label{sec:RP_extractio}
One of the goals of this work is to search for extended molecular gas emission (in CO(7-6), [\ion{C}{i}], H$_2$O and dust continuum). To achieve this, we construct the radial brightness profiles of the moment-0 map, i.e. the flux map. Compared to curve-of-growth, radial brightness profiles are more sensitive to faint emission at larger radii - i.e. extended halo emission. Although our sources were the primary targets of our ALMA observations, the emission line regions are not in the exact center of the field (1-2 pixels offset). Hence, we fitted the objects with a 2D Gaussian profile to find the centre of the emission. We then calculated the median flux in annuli with a width of 0.2 arcsecond (2 pixels), with the annuli centred on the coordinates determined from the 2D Gaussian fitting. The uncertainty on the flux is calculated as the RMS of the moment-0 map divided by the square root of the number of beams covered by the annuli (the number of pixels in the annuli divided by the number of pixels in the beam). We estimated the RMS of the noise of the moment-0 map using the \texttt{astropy}'s sigma-clipping function (SNR=3). We have repeated this procedure on an image of the clean elliptical beam to compare the radial profiles of the object to the radial profiles of the beam.
We investigated the effect of the width of the annuli versus the number of the annuli on the radial profile modelling. As we increase the width of the annuli the uncertainty on the median flux decreases. However, the increased size of annuli results in fewer annuli, to avoid overlap of the bins. Comparing the $\Delta$BIC between the models (see below) showed that the decrease of the flux uncertainties with increased annulus width does not out-weight the downside of more coarse sampling of the radial surface brightness profile. That is, in this case, we found that having more data points with large uncertainties is better than having a smaller number of data points with smaller uncertainties.
\subsubsection{Modelling of the radial brightness profiles}\label{sec:RP_analyses}
In order to investigate whether the data contains a single galaxy component (i.e. the gas in the galaxy) or whether it also contains a diffuse outer larger-scale component (i.e. a halo), we modelled radial surface density profiles of a single symmetrical 2D Gaussian or two symmetrical 2D Gaussian components.
We start by creating a 2D circular Gaussian model with arbitrary amplitude (=1) and set the intrinsic size of the object (FWHM). We convolved the model with the beam and extracted the radial brightness profile of this mock convolved image as described in \S~\ref{sec:RP_extractio}. Finally, we compared the modelled and observed radial brightness profiles and used \texttt{Python}'s \texttt{emcee} library to find the best value of FWHM using the MCMC ensemble sampler algorithm.
We set a flat prior on the FWHM to be between 0 and 3 arcseconds. For the double 2D Gaussian model, we fit three free parameters: FWHM$_{\rm galaxy}$, FWHM$_{\rm halo}$ and the log$_{10}$ ratio between the peak flux of the galaxy and outer components log$_{10}\frac{\rm peak_{\rm halo}}{\rm peak_{\rm galaxy}}$. For these three parameters we set the priors to [0, 1 arcsec], [1, 4 arcsec] and [-1, -5] for FWHM$_{\rm galaxy}$, FWHM$_{\rm halo}$ and log$_{10}\frac{\rm peak_{\rm halo}}{\rm peak_{\rm galaxy}}$, respectively. We adopt the likelihood function of \citet{Nikolic09} as:
\begin{equation}
\log L(\sigma) = \sum_{i} \left[ \left[ \frac{D_i- M_i}{\delta D_i}\right]^2 +\log(2\pi\delta D_i^2)\right]
\end{equation}
where the D$_{i}$ and $\delta$D$_{i}$ are the data and uncertainties on the data, respectively, M$_{i}$ is the model. The final results quoted in this work are the 50th, 16th and 84th percentiles of the posterior distribution.
\subsection{Stacking in the image plane}\label{sec:image_stack}
Although we detect a majority of the sources in each observed emission line, we also stack the emission line cubes to search for the faint emission on large scales. We stacked the individual cubes rather than moment-0 maps of the emission lines. There is no reason to assume that the line widths of the extended and galaxy component are the same, which would result in removing some extended component signal. Stacking the cube rather than moment-0 maps allows us to search for the extended halo emission across multiple velocity ranges later on. Furthermore, we only stacked cubes which contain a detected emission line. As we described above, there can be a significant offset between the optical and submm lines and which would potentially result in including noise only.
To stack the emission cubes, we used the same code from \citet{Jones22}, a similar technique to \citet{Delhaize13,Bischetti19, Jolly20}. Here we briefly outline the method. To stack the emission line cubes, we first start with an empty cube with a spatial dimension of 128$\times$128 pixels (corresponding to 12.8$\times$ 12.8 arcseconds) with a spectral axis of -2000--2000 km s$^{-1}$ in 200 channels, resulting in channels width of 20 km s$^{-1}$. This stacked cube setup was chosen to match the individual cubes from the imaging pipeline. For each of the emission line cubes, we created a cutout corresponding to the size of the new empty cube. However, the spectral scale is different for each cube as it is tuned to the different central frequencies and hence velocity width. As a result, we distribute the flux to individual velocity bins of the stacked cube as described by equation 1 in \citet{Jones22}.
We stacked the cubes based on four different weighting schemes:
\begin{enumerate}
\item \textit{Uniform:} The cubes are stacked without any weighting schemes.
\item \textit{Normalisation (Normed):} We estimated the maximum value of each cube and calculated the weighting as 1/(maximum value). This weighting scheme effectively results in normalising the data cubes by their maximum values.
\item \textit{Inverse Variance (InvV):} Normalising by the noise levels in the cube. We first estimated the RMS of each cube using sigma clipping of SNR=3. We then calculated the weights as 1/(RMS$^{2}$), which penalises data cubes with a larger noise. However, given the uniformity of the depth of our observations, the weights are very similar resulting to similar results to the uniform weighting.
\item \textit{Inverse Variance \& Normalization (InNo):} Combination of the previous two weighting schemes, with the final weights being 1/(RMS$^{2}\times$ max value).
\end{enumerate}
To accurately describe the radial emission line profiles, we need to also stack the beams of the individual cubes to find the final beam of the stacked cube. As each spectral window has a narrow frequency range, the beam size is changing only by maximum of $\sim 0.5$\%. As this difference is negligible, we use only a single common beam for the cube to simplify the procedure and the analysis of the stacked cubes. Using a single common beam per data cube simplifies the beam stacking and data analysis from three dimensions to two dimensions.
We extracted the clean beam information from the header of the individual cube and created an image of the beam for each cube. We stacked these beam images on the same grid as the stacked cube (128$\times$128 pixels corresponding to 12.8$\times$ 12.8 arcsec), giving the beam the same weight as to the individual cube in the cube stacking. We then fit this stacked beam image with a 2D Gaussian, similarly to the method used by the \texttt{tclean} to find the size and shape of the clean beam. This then became the new clean beam for the stacked data.
Using the method described above, we stacked the data cubes of CO(7-6), [\ion{C}{i}], H$_2$O and continuum emission. We constructed moment-0 maps of stacked cubes for each of the weightings in the range of $\pm$100 km s$^{-1}$, $\pm$200 km s$^{-1}$, $\pm$300 km s$^{-1}$, $\pm$400 km s$^{-1}$ and $\pm$500 km s$^{-1}$. For CO(7-6), the emission line is at the edge of the band in seven objects, with five objects with CO(7-6) less than 500 km s$^{-1}$ from the edge of the band. However, as we are not measuring the emission line profiles of the stacked data this does not influence our results; although it will decrease the SNR of our stacked data. We showed these moment-0 maps and their extracted brightness profiles in Figures A\ref{fig:CO_data}, A\ref{fig:CI_data}, A\ref{fig:H2O_data}.
\subsection{Stacking in \textit{uv}-plane}\label{sec:uv_stack}
Since our data was taken with an interferometer, it is important to verify any morphology results by investigating the data in the \textit{uv}-plane. The image analysis from interferometric observations can be sensitive to the exact cleaning procedure, which can accidentally introduce faint artefacts into the data. Furthermore, any extended emission detected in the image stacks can be caused by stacking the residuals of the dirty beam. Fortunately, working in the \textit{uv}-plane circumvents all of these downsides of image-plane stacking.
For each object, we first split the visibility data into separate measurement sets, containing: CO(7-6), [\ion{C}{i}], H$_2$O and continuum emission only. For the emission line visibilities, we extracted channels in the velocity range as the stacked cube with strongest halo emission from the image-based stacking: $\pm$300, $\pm$200 and $\pm$200 kms$^{-1}$ for CO(7-6), [\ion{C}{i}] and H$_2$O, respectively (see \S \ref{sec:image_stack_res}). We centred velocity ranges on the frequency peak (the 3D position in our ALMA cubes. For the continuum, we selected line-free channels (defined as 2$\times$ FWHM from the spectral fitting). During the \texttt{split} task, we also binned the data in the time domain with 30s bins, to make the data sizes more manageable. We verified that the time-binning does not affect our final results, by stacking a subset of the [\ion{C}{i}] data set.
In order to stack the visibilities, we used the \texttt{STACKER} \citep[][]{Lindroos2015}. We first concatenate (using the CASA \texttt{concat} task) the individual emission line and continuum measurement sets to create a single measurement set per emission tracer. We shifted the coordinate of the visibility data sets by rewriting the source centre determined from the moment-0 maps as "00:00:00.00, 00:00:00.0" using the \texttt{uv.stack} task in \texttt{STACKER}. Finally, we recalculated the data weights for the combined visibility data sets with the \texttt{statwt} task, based on the scatter of visibilities, which includes the effects of integration time, channel width, and system temperature. This is comparable to using the inverse rms weighting for the image-based stacking.
We have adapted our \textit{uv}-stacking procedure for the continuum stacking. Both \texttt{stacker} and \texttt{uvplot} require for all the spectral windows to have the same number of channels. This is not a problem for the emission line stacking, as we always stack the same number of channels (i.e. same line width). However, for the continuum, each spw has a different number of line-free channels. The above-outlined process would result in a significant number of continuum channels being excluded as we would have to settle for the lowest common number of channels across thirteen objects with four spw each. Instead, for the continuum emission, we shifted the coordinate of each MS containing the continuum emission only using the \texttt{uv.stacker}, extracted the visibilities using \texttt{uvplot} and then concatenated the text files containing each of the objects \textit{uv}-visibilities. This is the equivalent of stacking the objects in a uniform scheme, as we cannot use the \texttt{statwt} on the already extracted $uv$-visibilities.
We analyse the stacked visibility datasets in the same way as visibilities of the individual objects.
We extracted the visibilities and collapsed the visibilities as described in \S \ref{sec:uv-analyses}, with bin sizes of 20 k$\lambda$. We present the stacked visibilities in Figure \ref{fig:uv_stacking}.
\section{Results \& Discussion}\label{sec:results}
In this section, we present the results of our data analyses described above. We show the overview of the emission lines in \S \ref{sec:em-lines}, sizes of the different cold gas tracers in \S~\ref{sec:galaxy-sizes} and investigate the stacked data for presence of cold gas halos in \S~\ref{sec:image_stack_res} \& \ref{sec:uv_stack_res}. In \S~\ref{sec:mass_estimates} we derived the physical properties of these halos and in \S~\ref{sec:origins} we discuss their origins.
\subsection{Individual quasar host galaxies}\label{sec:em-lines}
First, we give a brief overview of the detected emission line properties of our sample. We detect twelve objects in CO(7-6), eleven in [\ion{C}{i}] emission, ten objects in H$_2$O emission and fourteen objects in continuum, across a range of SNR (3--67). The measured FWHM of the emission lines ranged 220--1060 km s$^{-1}$. Although some objects require two Gaussian components to accurately describe the emission line profiles, we do not detect any evidence for broad wings indicating a large scale outflow. For objects which were detected in multiple emission lines, the line widths of different emission lines agree within the 1 sigma errors. We summarise the redshifts, SNRs, FWHM and integrated flux in Table \ref{Table:Lines}.
\subsubsection{Galaxy component sizes}\label{sec:galaxy-sizes}
We used the collapsed \textit{uv}-visibilities to measure the sizes of the CO(7-6), [\ion{C}{i}], H$_2$O and continuum in \S \ref{sec:uv-analyses} and we present the collapsed visibilities in the right panels of Figures \ref{fig:CO_data}, \ref{fig:CI_data}, \ref{fig:H2O_data}. We investigate the \textit{uv}-visibilities of all emissions above SNR=10, as this is considered a minimal SNR limit to reliably measure sizes for interferometric data \citep[see][for details]{Simpson15,Harrison16Alm, Scholtz20}. We resolve eleven out of twelve sources in CO(7-6), nine out of ten in [\ion{C}{i}] and seven out of eight sources in H$_2$O emission. For objects J1348-0250 and J2323-0100, we are able to decompose the emission into a point source and a resolved component in CO(7-6) and [\ion{C}{i}] emission respectively. In these cases, we report the values of the resolved components and mark each source with an asterisk in Table~\ref{Table:Sizes}.
\begin{figure}
\includegraphics[width=0.9\columnwidth]{Graphs/Sizes_comparison.pdf}
\caption{ Comparison of sizes from different cold gas tracers (CO(7-6), [\ion{C}{i}](2-1), H$_2$O) and dust continuum. In each panel, the dashed black line shows the 1:1 ratio between the sizes. If a source is not detected in one tracer, we only show the size for the other tracer. Points with a size of -0.5 kpc indicate an upper limit on the size in that tracer. For comparison of dust continuum vs [\ion{C}{i}](2-1), we compared our sample to other studies which measured CO(3-2) or lower, as they trace molecular gas of same temperature and density.
}
\label{fig:Size_comp}
\end{figure}
We present the comparison of the CO(7-6), [\ion{C}{i}], H$_2$O and dust continuum sizes in Figure \ref{fig:Size_comp}. We measured the sizes of the four emission tracers to be in the range of 3.1--8.6 kpc (median of $3.8\pm 2.0$ kpc) for CO(7-6), 2.9--11.2 kpc (median of $5.3\pm2.5$ kpc) for [\ion{C}{i}], 2.7--6.0 kpc (median of $4.3\pm 1.5$ kpc) for H$_2$O and 2.1-9.5 kpc (median of $5.1\pm2.6$) for FIR emission. The error on the median was estimated as a standard deviation. We summarise the galaxy sizes in each tracer in Table \ref{Table:Sizes}. There is no evidence for the evolution of either dust continuum or [\ion{C}{i}] sizes as a function of QSO bolometric luminosity, however, this can be due to covering a very narrow range of bolometric luminosities in this sample.
In the top panel of Figure \ref{fig:Size_comp}, we compare the sizes of the dust emission with [\ion{C}{i}] for our objects (red points), SMGs \citep[blue points;][]{Chen17, Tadaki17}, star-forming galaxies \citep[green points;][]{Kaasinen20} and results from the FIRE-2 simulations \citep[grey points;][]{Cochrane19}. For the SMGs and star-forming galaxies, we use the observations of CO(3-2) emission, as [\ion{C}{i}](2-1) is rarely detected let alone resolved in high redshift galaxies. The cold molecular gas and dust sizes lie very close to the dashed 1:1 line, indicating very similar sizes.
The CO(7-6), [\ion{C}{i}] and H$_2$O sizes are consistent with CO sizes measured by \citet{Chen17, Callistro-Rivera18} and [\ion{C}{ii}] sizes measured by ALPINE survey \citep[][]{Fujimoto20} and Hot Dust Obscured Galaxies (Hot DOGs; Scholtz et al. in prep). The median FIR sizes of our sample agree with those found in AGN host galaxies \citep[][]{Harrison16Alm, Scholtz20, Lamperti21, Scholtz21}, Hot DOGs (Scholtz et al., in prep) and sub--mm and star-forming galaxies \citep[e.g.][]{Ikarashi15, Simpson15,Hodge16, Spilker16, Tadaki17,Fujimoto18, Lang19, Chen20}, however, the range of the values is a factor of $\sim$1.5 higher than those found in submm and AGN host galaxies. This can support the hypothesis that these objects are in a blowout phase of galaxy evolution.
\begin{table}
\caption{Sizes of the different emission region from the \textit{uv}-visibilities (see \S \ref{sec:uv-analyses}). (1) Object ID; (2) Size of the CO(7-6) emission; (3) Size of the [\ion{C}{i}] emission; (4) Size of the H$_2$O emission; (5) Size of the dust continuum.
}
\input{./Tables/Sizes_emission.tex}
\par $^*$ Object has two components: a point source and a resolved component.
\label{Table:Sizes}
\end{table}
\subsubsection{Detection of cold gas halos in individual sources}\label{sec:halos_ind}
We extracted the radial brightness profiles for our sources in CO(7-6), [\ion{C}{i}] and H$_2$O emission in \S~\ref{sec:RP_extractio} and we modelled these radial brightness profiles as described in \S~\ref{sec:RP_analyses}. Overall we detect an additional large-scale extended (halo) emission in two sources: J1232+0912 (CO 7-6) and J2323-0100 ([\ion{C}{i}]) and we show these in Figures \ref{fig:123241_CO_Aperture_modelling} \& \ref{fig:232326_CI_Aperture_modelling}. We show the results of the fitting of a single resolved component in the middle panels of Figures \ref{fig:123241_CO_Aperture_modelling} \& \ref{fig:232326_CI_Aperture_modelling}, the residuals (green lines) showing a clear need for an additional large scale halo component. We present the results of fitting two resolved components - galaxy component and a large scale halo component - in the bottom panels of Figures \ref{fig:123241_CO_Aperture_modelling} \& \ref{fig:232326_CI_Aperture_modelling}. Based on the BIC and reduced $\chi^2$, the radial brightness profiles require both components to be fitted. The FWHM size of the large-scale extended halo emission extended emission is $ 22.38 ^{+ 5.26 }_{- 4.09 }$ and $ 20.5 ^{+ 7.53 }_{- 5.32 }$\,kpc for J1232+0912 (CO 7-6) and J2323-0100 ([\ion{C}{i}]), respectively. The $\Delta$BIC for J1232+0912 (CO 7-6) and J2323-0100 are -24.0 and -4.0 in favour of a double component fit, respectively. We present the sizes of the individual components, BIC values and ratio between the peaks of the components in Table: \ref{Table:halo_solo}. This is a first detection of large-scale gas reservoirs around individual quasar host galaxies using the [CI] \& CO(7-6) emission line at high redshift.
\begin{table*}
\caption{Radial surface brightness profile fitting results for objects with detected extended emission: J1232+0912 and J2323-0100. (1) Object ID; (2) Tracer in which we detect the extended emission; (3) Model fitted to the data (best fit model is in bold); (4) Size of the galaxy component; (5) Size of the outer component; (6) log$_{10}$ ratio between the peaks of the two-component; (7) BIC of the fit; (8) $\chi^2$ of the fit. }
\input{./Tables/Extended_emission_solo.tex}
\label{Table:halo_solo}
\end{table*}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{Graphs/SDSS_J123241.73+091209.3CO_aperture_modelling.pdf}
\caption{ Modelling of the aperture profiles of the CO(7-6) emission from J1232+0912. Top panel: The red solid contours show 1, 2, 3, 4, 5 $\sigma$ and the red dashed contours -1, -2, -3, -4, -5 $\sigma$. Middle and bottom panels: Modelling of the radial brightness profiles using a single resolved source model (middle panel) and two resolved sources model. The blue points show the extracted radial brightness profile and their uncertainties. The orange and green lines show 100 randomly drawn solutions from the MCMC chain for the fit and residuals, respectively. The black line shows the intrinsic model before the convolution. The shaded region shows the 0.5$\times$ RMS of the moment-0 maps.
}
\label{fig:123241_CO_Aperture_modelling}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{Graphs/SDSS_J232326.17-010033.1CI_aperture_modelling.pdf}
\caption{ Modelling of the aperture profiles of the [\ion{C}{i}] emission from J2323-0100. Top panel: The red solid contours show 1, 2, 3, 4, 5 $\sigma$ and the red dashed contours -1, -2, -3, -4, -5 $\sigma$. Middle and bottom panels: Modelling of the radial brightness profiles using a single resolved source model (middle panel) and two resolved sources model. The blue points show the extracted radial brightness profile and their uncertainties. The orange and green lines show 100 randomly drawn solutions from the MCMC chain for the fit and residuals, respectively. The black line shows the intrinsic model before the convolution. The shaded region shows the 0.5$\times$ RMS of the moment-0 maps.
}
\label{fig:232326_CI_Aperture_modelling}
\end{figure}
\subsection{Stacking results}
In this section we describe the results obtained from stacking the data in both image-plane and the \textit{uv}-plane. We described the stacking methods in \S \ref{sec:image_stack} \& \ref{sec:uv_stack}.
\subsubsection{Image-plane stacking results}\label{sec:image_stack_res}
We show the full results of the image-stacked data for CO(7-6), [\ion{C}{i}], H$_2$O and continuum in Figures \ref{fig:Stacking_CO}, \ref{fig:Stacking_[CI]}, \ref{fig:Stacking_H2O]}, \ref{fig:Stacking_cont]}, showing both the moment-0 map and extracted radial surface brightness profiles for each of the velocity ranges ($\pm100$, $\pm200$, $\pm 300$,$\pm 400$ and $\pm500$ km s$^{-1}$) and stacking weighting schemes. For further analysis, we use the inverse RMS weighting scheme, as it allows more direct comparison with \textit{uv}-plane stacking. Since we do not expect any \textit{a priori} correlation between the galaxy and halo emission, we do not bias the stacks based on the galaxy gas brightness such as in the Normed and InNo weighting schemes.
We further investigate and model the image-plane stacking in Figures: \ref{fig:CO_Aperture_modelling}, \ref{fig:CI_Aperture_modelling}, \ref{fig:H2O_Aperture_modelling}, \ref{fig:Cont_Aperture_modelling}. In the top subplot of each Figure, we show moment-0 map of the stack from the velocity range which gives the most robust evidence for extended halo emission ($\pm 200$ km s$^{-1}$ for [\ion{C}{i}](2-1) and H$_2$O emission and $\pm 300$ km s$^{-1}$ for CO(7-6) emission). The extracted radial surface brightness profiles (second and third subplots) show emission on scales larger than the beam for CO(7-6), [\ion{C}{i}](2-1) and dust continuum.
Using the methods described in \S~\ref{sec:RP_analyses}, we fitted the extracted radial brightness profiles with a single resolved galaxy component in the middle panels of Figures \ref{fig:CO_Aperture_modelling}, \ref{fig:CI_Aperture_modelling}, \ref{fig:H2O_Aperture_modelling}, \ref{fig:Cont_Aperture_modelling}. The residual of the fits (green solid line) shows significant emission on scale larger than one arcsecond for CO(7-6), [\ion{C}{i}] and dust continuum stacked data. We show the image of the model galaxy component and the moment-0 residual in the second and third row of Figure \ref{fig:Stack_resid}, respectively, further showing the residual emission on scales of $>1$ arcsecond, that are not accounted for by the fitting a resolved galaxy component only.
We fitted the galaxy resolved galaxy component and an extended halo in the bottom panels of \ref{fig:CO_Aperture_modelling}, \ref{fig:CI_Aperture_modelling}, \ref{fig:H2O_Aperture_modelling}, \ref{fig:Cont_Aperture_modelling}. The BIC and reduced $\chi^{2}$ indicates that the two-component model is a better fit for the stacked CO(7-6), [\ion{C}{i}] and dust continuum data, while the single galaxy component is a better fit for the stacked H$_2$O data. We show the moment-0 residual from fitting the two-component model in the bottom row of Figure \ref{fig:Stack_resid}. We see small residual emission in the CO(7-6) and dust emissions, suggesting that the emission halo is not strictly symmetrical as we model. Overall, the residual images confirm the presence of large extended halos in the image-stacked data as shown in the aperture growth analysis. This is the first detection of a cold molecular gas emission in the CGM of ERQs at high redshift using emission lines.
The estimated deconvolved halo emission FWHM sizes are $13.5\pm0.66$, $12.6\pm1.24$ and $14.6\pm2.7$ kpc for CO(7-6), [\ion{C}{i}](2-1) and dust emission, respectively. These sizes are smaller than the previous detection of [\ion{C}{ii}] emission halos at $z>5$ of $\sim 22$ kpc \citep[][]{Fujimoto19,Fujimoto20} and CO(3-2) emission of $z\sim2.5$ AGN from the SUPER survey ($\sim27$\,kpc, \citealt{Jones22}). We do not detect any extended halo emission in H$_2$O, however, this is expected as H$_2$O emission is mostly tracing dense warm gas. We confirmed that the stacked emission is not dominated by the objects with individually detected extended emission (J1232+0912 and J2323-0100), by repeating the stacking procedure excluding these sources from the stacking. After removing the objects with individually detected halos from the stacks we measured the sizes of the halo component as $13.9\pm0.8$ and $12.2\pm1.78$ kpc for CO(7-6) and [\ion{C}{i}](2-1), respectively. Therefore, this detected extended emission is not from a single source but is present in all of the sources.
\begin{table*}
\caption{Radial surface brightness profile fitting results for stacked data and results of the modelling the \textit{uv}-visibilities of the stacked data. (1) Tracer in which we detect the extended emission; (2) Model fitted to the data (best fit model is in bold); (3) Size of the galaxy component from radial surface brightness profiles; (4) Size of the outer component from radial surface brightness profiles; (5) log$_{10}$ ratio between the peaks of the two-component from radial surface brightness profiles; (6) BIC; (7) Size of the galaxy component from \textit{uv}-visibilities fit; (8) Size of the outer component from \textit{uv}-visibilities fit; (9) log$_{10}$ ratio between the flux of the two-component from \textit{uv}-visibilities fit; (10) BIC
}
\input{./Tables/Stacking_Res.tex}
\label{Table:stack_res}
\end{table*}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{Graphs/CO_aperture_modelling.pdf}
\caption{ Modelling of the aperture profiles of the CO(7-6) stacked emission. Top panel: The red solid contours show 1, 2, 3, 4, 5 $\sigma$ and the red dashed contours -1, -2, -3, -4, -5 $\sigma$. Middle and bottom panels: Modelling of the radial brightness profiles using a single resolved source model (middle panel) and two resolved sources model. The blue points show the extracted radial brightness profile and their uncertainties. The orange and green lines show 100 randomly drawn solutions from the MCMC chain for the fit and residuals, respectively. The black line shows the intrinsic model before the convolution.
}
\label{fig:CO_Aperture_modelling}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{Graphs/CI_aperture_modelling.pdf}
\caption{ Modelling of the aperture profiles of the [\ion{C}{i}](2-1) stacked emission. Top panel: The red solid contours show 1, 2, 3, 4, 5 $\sigma$ and the red dashed contours -1, -2, -3, -4, -5 $\sigma$. Middle and bottom panels: Modelling of the radial brightness profiles using a single resolved source model (middle panel) and two resolved sources model. The blue points show the extracted radial brightness profile and their uncertainties. The orange and green lines show 100 randomly drawn solutions from the MCMC chain for the fit and residuals, respectively. The black line shows the intrinsic model before the convolution. The shaded region shows the 0.5$\times$ RMS of the moment-0 maps.
}
\label{fig:CI_Aperture_modelling}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{Graphs/H2O_aperture_modelling.pdf}
\caption{ Modelling of the aperture profiles of the H$_{2}$O stacked emission. Top panel: The red solid contours show 1, 2, 3, 4, 5 $\sigma$ and the red dashed contours -1, -2, -3, -4, -5 $\sigma$. Middle and bottom panels: Modelling of the radial brightness profiles using a single resolved source model (middle panel) and two resolved sources model. The blue points show the extracted radial brightness profile and their uncertainties. The orange and green lines show 100 randomly drawn solutions from the MCMC chain for the fit and residuals, respectively. The black line shows the intrinsic model before the convolution. The shaded region shows the 0.5$\times$ RMS of the moment-0 maps.
}
\label{fig:H2O_Aperture_modelling}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\columnwidth]{Graphs/Continuum_aperture_modelling.pdf}
\caption{ Modelling of the aperture profiles of the dust continuum stacked emission. Top panel: The red solid contours show 1, 2, 3, 4, 5 $\sigma$ and the red dashed contours -1, -2, -3, -4, -5 $\sigma$. Middle and bottom panels: Modelling of the radial brightness profiles using a single resolved source model (middle panel) and two resolved sources model. The blue points show the extracted radial brightness profile and their uncertainties. The orange and green lines show 100 randomly drawn solutions from the MCMC chain for the fit and residuals, respectively. The black line shows the intrinsic model before the convolution. The shaded region shows the 0.5$\times$ RMS of the moment-0 maps.
}
\label{fig:Cont_Aperture_modelling}
\end{figure}
\begin{figure*}
\includegraphics[width=0.8\paperwidth]{Graphs/Stacking_residual.pdf}
\caption{ Image visualisation of the aperture growth modelling. Each column represents different emission tracers. From left to right: CO(7-6), [\ion{C}{i}](2-1), H$_2$O an dust continuum. From top to bottom: First row: Data from the image-based stacking. The red dot indicates the centre of the image. Second row: Model image constructed from the best-fit to the radial brightness profiles convolved with the beam. The model is dominated by the galaxy component. Third row: Residual image after subtracting the galaxy component model. Fourth row: Residual image after subtracting the total model image (i.e. both galaxy and outer components). In each residual images, the solid contours show SNR levels of 2 and increasing by one and dashed contours show negative contours starting at -2.
}
\label{fig:Stack_resid}
\end{figure*}
\subsubsection{\textit{uv}-plane stacking results}\label{sec:uv_stack_res}
We stacked the targets in \textit{uv}-plane for all four emission tracers. We extracted the visibilities from the stacked measurement sets and we show these in Figure \ref{fig:uv_stacking}. Visually, all of these stacked \textit{uv}-visibilities show a Gaussian-like profile, indicating at least a single resolved source in the stacked data. However, in the CO(7-6), [\ion{C}{i}](2-1) and dust continuum \textit{uv}-stacked data, we also see a sudden upturn in flux density at low \textit{uv}-distances, indicating an additional large scale component.
In order to confirm the presence of extended emission, we fitted the collapsed stacked \textit{uv}-visibilities using three separate models: a single resolved source (half-Gaussian model), resolved source and a point source (half-Gaussian model + a constant), and two separate resolved sources (two half-Gaussian models). Based on the BIC and $\chi^{2}$, our modelling of the \textit{uv}-visibilities favoured two resolved components in CO(7-6), [\ion{C}{i}](2-1) and dust continuum stacks, while for the H$_2$O data, the BIC favours a single resolved model, in agreement with the image-based stacking.
We measured the sizes of the extended (halo) component to be $18.4\pm2.4$, $13.3\pm3.2$ and $12.3\pm1.6$ kpc for CO(7-6), [\ion{C}{i}](2-1) and dust emission, respectively. These values agree with the results from the image-stacking within 1$\sigma$ error. Overall, the results of the \textit{uv}-stacking confirm our previous results of the existence of extended cold gas halos around these quasars. We summarise the \textit{uv}-based stacking results in Table \ref{Table:stack_res}.
\begin{figure}
\includegraphics[width=0.99\columnwidth]{Graphs/UV_stacking_summary.pdf}
\caption{Results of the \textit{uv}-based stacking. Panels from top: CO(7-6), [\ion{C}{i}](2-1), H$_2$O and dust continuum. The stacked and binned \textit{uv}-visibilities are shown as blue points, while the best fit is shown as a red dashed line. When two separate components are required in the fit we show the individual components as a green and blue dashed line. The stacked \textit{uv}-visibilities show a presence of additional extended emission in CO(7-6), [\ion{C}{i}](2-1) and dust continuum data.
}
\label{fig:uv_stacking}
\end{figure}
\subsection{Estimating dust and molecular gas masses in the halos}\label{sec:mass_estimates}
The \textit{uv}-stacking allowed us to reliably estimate the fluxes of the halo components and as a result, we can estimate the total dust and molecular gas mass. We derive the dust masses using a single modified black body (MBB) curve, (e.g., \citealt{Jones20}) as:
\begin{equation}
S_{\rm obs}(\nu_{\rm obs})=\frac{(1+z)\pi R^2}{D_{\rm L}^2}B'(\nu,T_{\rm dust})\left( 1-e^{\frac{-M_{\rm dust}\kappa_o(\nu/\nu_o)^{\beta}}{\pi R^2}} \right)
\end{equation}
where $S_{\rm obs}(\nu_{\rm obs})$ is the observed band 6 flux, $R$ is the size of the galaxy or halo, $\kappa_o$ = 4 cm$^2$ g$^{-1}$ with $\nu_0$ = 1.2 THz \citep[see][]{Bianchi13} and $\beta$ = 1.8. As the source is at high redshift, it is necessary to also include the effect of the dust heating by the CMB. Removing this contribution results in a modified black body function ($B'(\nu,T_{\rm dust})$):
\begin{multline}
B'(\nu,T_{\rm dust})=B(\nu,T_{\rm dust}')-B(\nu,T_{\rm CMB})\\
= \frac{2h\nu^3}{c^2}\left[\frac{1}{e^{h\nu/k_BT_{\rm dust}'}-1}-\frac{1}{e^{h\nu/k_BT_{\rm CMB}}-1} \right]
\end{multline}
where $T_{\rm dust}$ is the true dust temperature, $T_{\rm dust}'$ is an effective dust temperature, $T_{\rm CMB}=(1+z) T_o$, with $T_{\rm o}=2.73$\,K is the CMB temperature at $z=0$. We assume dust temperature of 30 K and we discuss this value below and beta value of 1.8
For estimating the cold gas mass in the host galaxies and in the halo we use the [\ion{C}{i}](2-1) emission line. The CO(7-6) traces more excited gas (150 K ) and the conversion to CO(1-0) necessary to estimate cold gas mass is very uncertain, even in star-forming galaxies, let alone in uncertain conditions of these extended halos. To calculate the cold gas mass ($M$(H$_2$)) from [\ion{C}{i}](2-1), we use the method described in \citet{Bothwell17}. We calculate $M$(H$_2$) as:
\begin{equation}
\begin{aligned}
M(H_{2}) = 1375.8 \times D^2_L (1+z)^{-1} \left(\frac{X_{[\ion{C}{i}]}}{10^{-5}}\right)^{-1} \left(\frac{A_{10}}{10^{-7} s^{-1}}\right)^{-1}\\
\times Q_{10}^{-1} I_{\rm [\ion{C}{i}]_{(1-0)}}
\end{aligned}
\end{equation}
where $X_{\rm[CI_{1-0}]}$ is the [\ion{C}{i}]/H$_2$ abundance ratio, we adopt a literature-standard [\ion{C}{i}]/H2 abundance ratio of $3\times 10^{-5}$ and the Einstein A coefficient ($A_{10}$) of $7.93 \times 10^{-8}$s$^{-1}$ with excitation factor ($Q_{10}$) of 0.6. The value of $Q_{10}$ is dependent on the specific conditions within the gas \citep[][]{Papadopoulos04}. As we observed the [\ion{C}{i}](2-1) line rather than the [\ion{C}{i}](1-0) and hence we need to convert these, using the [\ion{C}{i}](1-0)/[\ion{C}{i}](2-1) conversion factor of 3 \citep[][]{Jiao17}
Using the method and assumptions above, we estimated average dust mass of $10^{8.3 \pm 0.16}$ and $10^{7.6 \pm 0.12}$ M$_{\odot}$ for the galaxy and halo components, respectively, and average molecular gas masses of $10^{10.8\pm 0.14}$ and $10^{10.2\pm 0.16}$ M$_{\odot}$ for the galaxy and halo components, respectively. We note that the quoted uncertainties are estimated from the random flux uncertainties. We discuss the systematic uncertainties below. The average host galaxy masses are in agreement with molecular gas and dust masses measured in high-z quasars \citep[][]{Bischetti21, Decarli22}. These halo molecular gas masses indicate a massive cold gas reservoir around these luminous quasars. Previous studies estimating these molecular gas halos around star-forming galaxies estimated $10^{11.3-11.77}$M$\odot$ \citep[with a range of CO(4-3)/CO(1-0) and $\alpha_{\rm CO}=10$;][]{Ginolfi17} and $1.3\times 10^{11}$M$\odot$ \citep[using CO(4-3)/CO(1-0) = 0.45--1 and $\alpha_{\rm CO}$=3.6;][]{Li21}.
Here, we discuss the uncertainties in estimating the dust and molecular gas masses, which in both cases, are dominated by systematic rather than random uncertainties. The primary source of uncertainties in estimating the dust masses is the assumed dust temperature of 30 K, as this is a typical value for $z\sim$ 2.5 star-forming galaxies \citep[30-40 K; ][]{Schreiber18,Liang19, Reuter20}. However, the dust in the halo is located away from any of the heating sources such as star-formation and quasar, it can be significantly cooler than dust in the galaxy. Recalculating the dust mass in the halos for a temperature of 20 K (lowest measured temperature for SF galaxy at $z\sim$2.4; \citealt{Reuter20}) yields $10^{8.3}$ M$_{\odot}$, a factor of 5 higher than the original estimate for the dust mass in the halo.
There have been numerous studies focusing on the [\ion{C}{i}] abundance and its effect on estimates of the molecular gas mass. Although [\ion{C}{i}] is certainly more stable and more reliable than the CO for determining the molecular gas mass, the [\ion{C}{i}]-to-H$_2$ conversion factor is also associated with uncertainties. Both \citet{Offner14} and \citet{Glover16} show results from post-processing of hydrodynamical simulations of star-forming clouds, claiming that the [\ion{C}{i}] abundance varies as a function of interstellar radiation field (ISRF), metallicity and H$_2$ column density. At high A$_{\rm v}$, the [\ion{C}{i}] abundance is raised by the presence of cosmic rays, while at low A$_{\rm v}$, increasing the ISFR by a factor of 100--1000 can increase the [\ion{C}{i}] abundance by 30--50\%, a similar effect to $\alpha_{\rm CO}$. Furthermore, \citet{Glover16} have showed evidence for evolving [\ion{C}{i}] abundance as a function of metallicity given as: $\sim \rm Z^{-1}$. The metallicity of CGM material can vastly vary. Indeed, \citet{Pointon19} found that the metallicity of cooler CGM ($10^{4}$ K) can vary by a factor 10--100, and there are no measurements of the metallicity of $<100$\,K CGM. However, if we consider a range of metallicity values for our molecular gas measurement, the average molecular gas mass in these cold gas halos can be up to $10^{12.1}$ M$\odot$ (for log$_{10}$ OH = -1.5). However, given that the origin of this cold is most likely AGN or star formation driven outflows, the metallicity of the gas is going to be closer to that of a galaxy.
An alternative approach to calculating molecular gas masses is to use the dust mass as a proxy tracer of the molecular gas \citep[e.g.][]{Leroy11, Magdis11, Magnelli12, Scoville14, Genzel15}. Therefore, assuming a gas-to-dust ratio ($\delta$GDR), molecular gas masses can be estimated as M($ H_{2}$ = $\delta$GDR M$_{\rm dust}$. As mentioned above, the lack of metallicity estimates of the cold CGM, we adopt a fixed value of $\delta$GDR = 100-1000 \citep[for solar metallicity star-forming galaxies ][]{Sandstrom13, Remy-Ruyer14}. We estimate molecular gas masses from dust measurement of $10^{9.6-10.6}$ M$_{\odot}$, within the estimated values using the [\ion{C}{i}](2-1) emission.
Overall, we estimated a range of average cold gas masses inside these halos to be in the range of $10^{10.2}-10^{12.1}$ M$\odot$. This indicates that these quasars have significant gas reservoirs surrounding their host galaxies. In the next section, we discuss the origin of these cold gas halos.
\subsection{Origins of the halo emission}\label{sec:origins}
In this section we discuss the origin of these extended cold gas halos. The potential origins of these halos are: 1) current merger events; 2) companions; 3) cold gas halos from AGN or SF driven outflows. To distinguish between these scenarios we investigated the emission line kinematics and HST archival i-band and 1.4 $\mu$m imaging of our targets. Given the double peak nature of the emission lines (see Figures \ref{fig:CO_data}, \ref{fig:CI_data} \& \ref{fig:H2O_data}) and smooth velocity gradient of moment-1 maps, we see no evidence of disturbed kinematics suggesting a recent merger.
Overall, six targets have deep HST i-band and $1.4 \mu$m imaging, with five of six targets showing no sign of additional objects within 1.5 arcsecond. However, in Figure \ref{fig:HST_merger} we present HST 1.4$\mu$m map ($\lambda_{\rm rest=}$400 \AA) imaging of J2323-0100 with ALMA [\ion{C}{i}] contours overlaid in red. The presence of an additional bright object 1.2 arcsecond away from the main quasar with additional faint emission possibly around the bright point source may indicate a galaxy merger with tidal stellar streams between them \citep[see also][ for merger discussion]{Vayner21}. As a result, we cannot confirm that the individually detected large-scale cold gas emission in J2323-0100 is a cold gas halo. For the rest of the targets, we see no evidence of mergers in either the emission line kinematics or HST imaging. Furthermore, given that we do not see any evidence for companion galaxies in the HST imaging and we do not detect the extended emission in H$_2$O emission, we conclude that our extended emission are indeed cold gas halos, rather than contamination from mergers or companions.
We investigated the ionised outflow velocities of the two ERQs with individually detected large scale cold gas halos. \citet{Perrotta19} have observed the ionised outflow velocities in [\ion{O}{iii}] of 4800 and 3000 km s$^{-1}$. Although these are extreme velocities, they are in a common range for high luminosity ERQs such as our sample \citep[][]{Perrotta19, Vayner21}.
\begin{figure}
\includegraphics[width=0.99\columnwidth]{Graphs/SDSS_J232326_merger.pdf}
\caption{ HST $1.4\mu$m image of the J2323-0100 with ALMA [\ion{C}{i}] map overlayed as red solid (2,3, 5 and 10 $\sigma$ levels) contours. The image is centred on the location of the quasar. The HST imaging show clear signs of a merger, suggesting that the detection of large scale cold gas emission in this particular object is caused by the merger, rather than a cold gas halo (see Figure \ref{fig:232326_CI_Aperture_modelling}).
}
\label{fig:HST_merger}
\end{figure}
It is now important to discuss the origin of the cold gas halo. Previous studies \citep[][]{Fujimoto19, Fujimoto20, Ginolfi20}) showed evidence of cold gas halos detected in [\ion{C}{ii}] emission in star-forming galaxies, suggesting that the origin of the halos can be starburst driven winds. A possibility, cosmological simulations showed that bright galaxies at high redshift were likely encased in diffuse filaments of gas (e.g., \citealt{Pallotini17}, \citealt{Kohandel19}). However, the presence of dust and [\ion{C}{i}] emission in these halos indicates significant metal enrichment of these halos, pointing towards its origin in the galaxy.
Assuming molecular gas outflow velocities of 700--2000 km s$^{-1}$ \citep[][]{Bischetti19, Stanley19}, we calculated the travel times of 7-35 Myr for the gas to reach such distances from the host galaxy. As quasar and AGN can switch can vary drastically on scales of 1-10 Myr \citep[e.g.][]{Hickox14, Schawinski15, King15, McAlpine17}, it is less likely that the current quasar episode is responsible for the creation of these cold gas halos, but rather than they are relics of previous AGN or starbursts episodes. However, assuming the velocity gas similar to that of the detected ionised outflows in these objects \citep[$\sim$ 3000 km s$^{-1}$;][]{Vayner21}, the time taken for the gas to reach 13 kpc would only be 4.2 Myr. However, this would require this ionised gas to cool down to 40 K, traced by [\ion{C}{i}](2-1) emission, in the same time.
As the [\ion{C}{i}](2-1) line is tracing the cold phase of the gas (T$\sim$30 K) and the CO(7-6) is tracing excited dense gas (T$\sim$150 K), we speculate that the cold gas halos contain a large amount of both cold and warm excited gas, as both [\ion{C}{i}] and CO(7-6) have similar halo extension. Finally, these metal-enriched cold molecular gas halos can be either created by AGN or star-formation driven outflows \citep[][]{Maiolino12, Cicone15, Fiore17, Spilker18}. The most likely scenario is that these halos were created by a combination of past AGN and starburst activity. Overall, this is the first detection of cold molecular gas halos traced in [\ion{C}{i}], CO(7-6) and dust continuum emission in high redshift galaxies. Our result confirms the predictions of the cosmological simulations that the baryon cycle the enriched gas exchanges with the CGM are at work \citep[see e.g., ][]{Hopkins14, Somerville15, Hayward17}.
\section{Conclusions}
We present the results of ALMA Band 6 observations of 15 extremely red quasars. We detect the 13, 11, 10 and 13 objects in CO(7-6), [\ion{C}{i}](2-1), H$_2$O and continuum emission, with SNR ranging from 5 to 64 $\sigma$. We constructed a radial brightness profile for both individual sources and for stacked data (in both image and \textit{uv}-plane) to search for extended emission around these objects. Based on our analyses we find:
\begin{enumerate}
\item We measured the sizes of the four emission tracers to be in the range of 3.1--8.6 kpc (median of 4.0 kpc) for CO(7-6), 2.9--11.2 kpc (median of 5.3 kpc) for [\ion{C}{i}](2-1), 2.7--6.0 kpc (median of 4.3 kpc) for H$_2$O and 2.0--8.0 kpc (median of 5.1 kpc) for continuum emission. These value are consistent with those found in the literature for sub-mm galaxies and other AGN host galaxies (see Figure \ref{fig:Size_comp}).
\item Modelling the observed radial surface brightness profiles, we found extended emission in two objects in either CO(7-6) or [\ion{C}{i}](2-1) emission (see Figures \ref{fig:123241_CO_Aperture_modelling} \& \ref{fig:232326_CI_Aperture_modelling}). We measured the FWHM sizes of these extended halos to be $21.6^{+5.0}_{-4.0}$ kpc for the CO(7-6) halo in J1232+0912 and $19.8^{+7.2}_{-5.0}$ kpc for the [\ion{C}{i}](2-1) halo in J2323+0100.
\item We stacked our sample in CO(7-6), [\ion{C}{i}](2-1), H$_2$O and dust continuum emission in the image plane and extracted the the radial surface brightness profiles from the moment-0 maps of the stacked emission cubes (see Figures \ref{fig:CO_Aperture_modelling}, \ref{fig:CI_Aperture_modelling}, \ref{fig:H2O_Aperture_modelling} \& \ref{fig:Cont_Aperture_modelling}). Modelling the profiles showed evidence of large scales cold gas halos in CO(7-6), [\ion{C}{i}](2-1) and dust continuum with size of $13.5\pm0.66$, $12.6\pm1.24$ and $14.6\pm2.7$ kpc for CO(7-6), [\ion{C}{i}](2-1) and dust emission, respectively. Investigating the residual stacked images after subtracting a central galaxy source (see Figure \ref{fig:Stack_resid}) confirms the radial surface brightness profile modelling.
\item Stacking our data in the \textit{uv}-plane across the four emission tracers and extracting the \textit{uv}-visibilities confirms the result of the image stacking extended cold gas halos around these quasar host galaxies (see Figure \ref{fig:uv_stacking}). We measure the sizes of the halo component to be $18.4\pm2.4$, $13.3\pm3.2$ and $12.3\pm1.6$ kpc for CO(7-6), [\ion{C}{i}](2-1) and dust emission, respectively. These cold gas halo sizes agree within 1$\sigma$ with the sizes measured in the image-plane.
\item Using the measured fluxes of the dust continuum and the [\ion{C}{i}](2-1) emission line from the \textit{uv}-plane stacking we derived the average dust and molecular gas mass inside the halo of $10^{7.6}$ and $10^{10.2}$ M$_{\odot}$, respectively. These dust and molecular gas masses indicate substantial dust and gas reservoirs around these quasar host galaxies and evidence of enrichment of CGM from the past AGN or starbursts activity.
\end{enumerate}
Overall, our analysis of this deep ALMA band 6 data shows evidence for a central host galaxy source surrounded by an extended cold gas and dust halo. Assuming typical molecular and ionised gas outflow velocities implies long travel times for the gas to reach such distances (5-30 Myr), and hence suggesting these halos are relics of past AGN or star-formation activity.
\section*{Data Availability}
The datasets were derived from sources in the public domain: ALMA data from \url{https://almascience.nrao.edu/aq/
?result_view=observation}. The images, spectra and stacked data in this article will be shared on reasonable request to the corresponding author.
\section*{Acknowledgements}
J.S. and R.M. acknowledge ERC Advanced Grant 695671 "QUENCH" and support by the Science and Technology Facilities Council (STFC).
G.C.J. acknowledges funding from ERC Advanced Grants 789056 ``FirstGalaxies'' under the European Union's Horizon 2020 research and innovation programme.
S.C is supported by European Union's
HE ERC Starting Grant No. 101040227 - WINGS.
This paper makes use of ALMA data: 2017.1.00478.S.
ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ.
\bibliographystyle{mnras}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,364 |
Route numbers to identify SBB's long-distance services
SWITZERLAND: Swiss Federal Railways will introduce route numbers for its long-distance services from the start of the winter timetable on December 10. Intended to offer passengers 'simple and reliable' identification of their services, the numbering is based on that used for the national road network.
The main east-west route from Genève Aéroport to Bern, Zürich and St Gallen will be known as IC1 and the Zürich – Lugano route via the Gotthard Base Tunnel as IC2. Basel – Luzern – Lugano services will be identified as IC21 and Basel – Zürich – Chur services will be shown as IC3. The Genève – St Gallen route via Biel will be designated IC5.
The ICN designation for services operated by tilting EMUs will be dropped, with all long-distance trains designated as Intercity (IC) or Interregio (IR).
Twindexx fleet to enter service in 2018
SWITZERLAND: Swiss Federal Railways confirmed on November 30 that it will launch the first Twindexx double-deck electric multiple-units in commercial service during 2018. The decision follows an interim ruling by the Federal Office for Transport authorising the trains to carry passengers after supplier Bombardier Transportation provided all the necessary safety ...
SBB targets 15 min frequencies in 'next quantum leap'
SWITZERLAND: Swiss Federal Railways is drawing up long-term proposals to run its long-distance services at 15 min intervals. Describing this as 'the next quantum leap in Swiss public transport', SBB was responding on November 9 to a government consultation document on long-term investment in the national rail network.
Swiss long-distance concession delayed
SWITZERLAND: The Federal Transport Office, BAV, has decided to postpone the award of new concessions to operate long-distance passenger services on the national network, and to directly award SBB a two-year extension to its existing contract until the December 2019 timetable change. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,322 |
Anglicanos de São Tomás (muitas vezes chamados de cristãos sírios anglicanos ou cristãos sírios da Igreja do Sul da Índia) são os membros Cristãos de São Tomé da Igreja do Sul da Índia, a província autônoma do sul da Índia da Comunhão Anglicana. Eles estão entre as várias comunidades eclesiásticas diferentes que se separaram dos outrora indivisos Cristãos de São Tomé, uma antiga comunidade cristã cujas origens remontam às atividades missionárias do apóstolo São Tomé no primeiro século, no atual Estado de Kerala, no sul da Índia. O Apóstolo, segundo a lenda, chegou a Malankara (derivado de Maliankara perto de Muziris) em 52 dC.
A comunidade começou como uma facção de Cristãos Sírios Malankara, que optaram por se juntar à Igreja Anglicana, principalmente entre 1836 e 1840. Isso aconteceu devido à influência dos missionários da Church Mission Society, que trabalhavam entre os cristãos ortodoxos orientais de Travancore. Em 1879, essas congregações anglicanas de São Tomé foram organizadas como a Diocese de Travancore e Cochin da Igreja da Inglaterra. Outros cristãos de São Tomé influenciados pela prática e crença anglicanas continuariam a fundar a Igreja Síria Mar Thoma, uma Igreja em plena comunhão com a Comunhão Anglicana.
Em 1930, uma província eclesiástica anglicana separada foi fundada a partir das dioceses da Igreja da Inglaterra no Império Indiano Britânico, estabelecendo a Igreja da Índia, Birmânia e Ceilão. Em 1947, logo após a independência da Índia, as dioceses anglicanas do sul da Índia fundiram-se com outras Igrejas protestantes da região, com base no Quadrilátero de Lambeth, formando a Igreja do Sul da Índia. Os cristãos sírios anglicanos são membros da Igreja do Sul da Índia desde então.
Galeria
Ver também
Igreja do Sul da Índia
Igreja Mar Thoma
Referências
Cristianismo na Índia
Cristãos de São Tomé
Anglicanismo na Índia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 624 |
{"url":"https:\/\/programmer.group\/5e3562d220b63.html","text":"# [machine learning note day22] 4.2. Linear regression case analysis + Boston house price forecast\n\nKeywords: network\n\n# Linear regression case analysis\n\n## Boston house price forecast\n\nThe regression model built in scikit learn is used to predict the \"Boston house price\" data. Some competition data can be obtained from the official website of kaggle, https:\/\/www.kaggle.com\/datasets\n\n1. Data description of housing prices in Boston\n\nfrom sklearn.datasets import load_boston\n\nprint boston.DESCR\n\n\n2. Boston area housing price data segmentation\n\nfrom sklearn.cross_validation import train_test_split\nimport numpy as np\nX = boston.data\ny = boston.target\n\nX_train,X_test,y_train,y_test = train_test_split(X,y,random_state=33,test_size = 0.25)\n\n\n3. Standardized processing of training and test data\n\nfrom sklearn.preprocessing import StandardScaler\nss_X = StandardScaler()\nss_y = StandardScaler()\n\nX_train = ss_X.fit_transform(X_train)\nX_test = ss_X.transform(X_test)\ny_train = ss_X.fit_transform(y_train)\nX_train = ss_X.transform(y_test)\n\n\n4. Using the simplest linear regression model, linear regression and gradient decline to estimate sgdregger to predict house prices\n\nfrom sklearn.linear_model import LinearRegression\nlr = LinearRegression()\nlr.fit(X_train,y_train)\nlr_y_predict = lr.predict(X_test)\n\nfrom sklearn.linear_model import SGDRegressor\nsgdr = SGDRegressor()\nsgdr.fit(X_train,y_train)\nsgdr_y_predict = sgdr.predict(X_test)\n\n\n5. Performance evaluation\n\nFor different types of prediction, we can not strictly require that the numerical results of regression prediction should be strictly the same as the real values. In general, we want to measure the difference between the predicted value and the real value. Therefore, the evaluation function can be used for evaluation. Among them, the most intuitive evaluation index, mean squared error (MSE), is the objective of linear regression model optimization.\n\nThe calculation method of MSE is as follows:\n\n{MSE=}\\frac{1}{m}\\sum_{i=1}{m}\\left({y{i}-\\bar{y}}\\right)^{2}MSE=m1\u2211i=1m(y**i\u2212y\u00af)2\n\nUsing MSE evaluation mechanism to evaluate the regression performance of the two models\n\nfrom sklearn.metrics import mean_squared_error\n\nprint 'The mean square error of linear regression model is:',mean_squared_error(ss_y.inverse_transform(y_test),ss_y.inverse_tranform(lr_y_predict))\nprint 'The mean square error of gradient descent model is:',mean_squared_error(ss_y.inverse_transform(y_test),ss_y.inverse_tranform(sgdr_y_predict))\n\n\nThrough this comparison, it is found that the performance of the gradient descent estimation method is not as good as that of the linear regression method, but if the training data scale is very large, then the gradient method is very efficient in both classification and regression problems, which can save a lot of computing time without losing too much performance. According to the proposal of scikit learn optical network, if the data scale is more than 100000, the random gradient method is recommended to estimate the parameter model.\n\nNote: linear regression is the simplest and most easy to use regression model. Because of the linear hypothesis between the feature and the regression target, it also limits its application scope to some extent. In particular, the vast majority of real-life case data can not guarantee strict linear relationship between various characteristics and regression objectives. However, we can still use linear regression model as the baseline system for most data analysis without knowing the relationship between features.\n\nThe complete code is as follows:\n\nfrom sklearn.linear_model import LinearRegression, SGDRegressor, Ridge\nfrom sklearn.preprocessing import StandardScaler\nfrom sklearn.cross_validation import train_test_split\nfrom sklearn.metrics import mean_squared_error,classification_report\nfrom sklearn.cluster import KMeans\n\ndef linearmodel():\n\"\"\"\n\/\/Linear regression for Boston dataset processing\n:return: None\n\"\"\"\n\nx_train,x_test,y_train,y_test = train_test_split(ld.data,ld.target,test_size=0.25)\n\n# 2. Standardized treatment\n\n# Eigenvalue processing\nstd_x = StandardScaler()\nx_train = std_x.fit_transform(x_train)\nx_test = std_x.transform(x_test)\n\n# Target value processing\n\nstd_y = StandardScaler()\ny_train = std_y.fit_transform(y_train)\ny_test = std_y.transform(y_test)\n\n# 3. Estimator process\n\n# LinearRegression\nlr = LinearRegression()\n\nlr.fit(x_train,y_train)\n\n# print(lr.coef_)\n\ny_lr_predict = lr.predict(x_test)\n\ny_lr_predict = std_y.inverse_transform(y_lr_predict)\n\nprint(\"Lr Predicted value:\",y_lr_predict)\n\n# SGDRegressor\nsgd = SGDRegressor()\n\nsgd.fit(x_train,y_train)\n\n# print(sgd.coef_)\n\ny_sgd_predict = sgd.predict(x_test)\n\ny_sgd_predict = std_y.inverse_transform(y_sgd_predict)\n\nprint(\"SGD Predicted value:\",y_sgd_predict)\n\n# Ridge regression with regularization\n\nrd = Ridge(alpha=0.01)\n\nrd.fit(x_train,y_train)\n\ny_rd_predict = rd.predict(x_test)\n\ny_rd_predict = std_y.inverse_transform(y_rd_predict)\n\nprint(rd.coef_)\n\n# Evaluation results of two models\n\nprint(\"lr The mean square error of is:\",mean_squared_error(std_y.inverse_transform(y_test),y_lr_predict))\n\nprint(\"SGD The mean square error of is:\",mean_squared_error(std_y.inverse_transform(y_test),y_sgd_predict))\n\nprint(\"Ridge The mean square error of is:\",mean_squared_error(std_y.inverse_transform(y_test),y_rd_predict))\n\nreturn None\n\n432 original articles published, praised 196, 20000 visitors+\n\nPosted by pinxue on Sat, 01 Feb 2020 03:36:31 -0800","date":"2020-07-04 21:47:39","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\": 0.3360118865966797, \"perplexity\": 3456.5077210745517}, \"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\/1593655886706.29\/warc\/CC-MAIN-20200704201650-20200704231650-00084.warc.gz\"}"} | null | null |
Q: How to pass an object's variable stored as java's Object to another object's Function which is also stored as java's Object? I am trying to make collision detector in Libgdx using Bullet. Here I want pass one colliding object's power variable as parameter to another object's onCollision() function. Here Ball and Brick extend AbstractObject. Also power and onCollision() is declared in AbstractObject but initialized in Brick and Ball. I have set btCollisionObject.userData=this in each class.What is the most efficient way to do this?
Here is my current contactListener:
package com.anutrix.brickbreaker3d.Helpers;
import com.anutrix.brickbreaker3d.gameObjects.AbstractObject;
import com.anutrix.brickbreaker3d.gameObjects.Ball;
import com.anutrix.brickbreaker3d.gameObjects.Brick;
import com.badlogic.gdx.Gdx;
import com.badlogic.gdx.graphics.g3d.ModelInstance;
import com.badlogic.gdx.physics.bullet.collision.ContactListener;
import com.badlogic.gdx.physics.bullet.collision.btCollisionObject;
import com.badlogic.gdx.utils.Array;
public class CollisionListener extends ContactListener {
@Override
public boolean onContactAdded(btCollisionObject ob0, int partId0, int index0, btCollisionObject ob1, int partId1, int index1) {
Gdx.app.log("sdkjg", "fsfgsdg");
Ball bl = null;
Brick br = null;
AbstractObject aO0 = (AbstractObject) ob0.userData;
AbstractObject aO1 = (AbstractObject) ob1.userData;
if (aO0 instanceof Ball) {
bl = (Ball) aO0;
} else if (aO1 instanceof Ball) {
bl = (Ball) aO1;
}
if (aO0 instanceof Brick) {
br = (Brick) aO0;
} else if (aO1 instanceof Brick) {
br = (Brick) aO1;
}
bl.onCollision(br.power);
br.onCollision(bl.power);
return true;
}
}
Here is the Ball Class:
public class Ball extends AbstractObject {
public Integer power;
public Ball(Integer id, Integer type, Vector3 position) {
super(id, type, position);
modelInstance = new ModelInstance(Assets.instance.ball.get(type));
shape = new btSphereShape(0.2f);
body = new btCollisionObject();
body.setCollisionShape(shape);
super.setPosition(position);
this.power = type + 1;
this.body.setCollisionFlags(this.body.getCollisionFlags() | btCollisionObject.CollisionFlags.CF_CUSTOM_MATERIAL_CALLBACK);
active=true;
body.userData=this;
}
public Integer getPower() {
return power;
}
public void resetPower() {
this.power = this.getType()+1;
}
public void onCollision(Integer power) {
this.collided=true;
}
@Override
public void getDetails(){
Gdx.app.log("Life", power.toString());
Gdx.app.log("Active", Boolean.toString(this.active));
super.getDetails();
}
}
Here is Brick class:
package com.anutrix.brickbreaker3d.gameObjects;
import com.anutrix.brickbreaker3d.Helpers.Assets;
import com.badlogic.gdx.Gdx;
import com.badlogic.gdx.graphics.g3d.ModelInstance;
import com.badlogic.gdx.math.Vector3;
import com.badlogic.gdx.physics.bullet.collision.btBoxShape;
import com.badlogic.gdx.physics.bullet.collision.btCollisionObject;
/**
*
* @author Anutrix
*/
public class Brick extends AbstractObject {
public Integer power;
public Brick(Integer id, Integer type, Vector3 position) {
super(id, type, position);
modelInstance = new ModelInstance(Assets.instance.brick.get(type));
shape = new btBoxShape(new Vector3(1f, 0.5f, 1f));
body = new btCollisionObject();
body.setCollisionShape(shape);
super.setPosition(position);
this.power = type + 1;
this.body.setCollisionFlags(this.body.getCollisionFlags() | btCollisionObject.CollisionFlags.CF_CUSTOM_MATERIAL_CALLBACK);
active=true;
body.userData=this;
}
public Integer getPower() {
return power;
}
public void onCollision(Integer power) {
this.power = this.power-power;
if(this.power<=0){
this.active=false;
}
this.collided=false;//reset
}
@Override
public void getDetails(){
Gdx.app.log("Life", power.toString());
Gdx.app.log("Active", Boolean.toString(this.active));
super.getDetails();
}
}
Here is AbstractObject Class:
package com.anutrix.brickbreaker3d.gameObjects;
import com.anutrix.brickbreaker3d.Helpers.Assets;
import com.badlogic.gdx.Gdx;
import com.badlogic.gdx.graphics.g3d.ModelInstance;
import com.badlogic.gdx.math.Vector3;
import com.badlogic.gdx.physics.bullet.collision.btCollisionObject;
import com.badlogic.gdx.physics.bullet.collision.btCollisionShape;
public class AbstractObject {
private Integer id;
private Integer type;
private Vector3 position;
public ModelInstance modelInstance;
public btCollisionShape shape;
public btCollisionObject body;
public Integer power;
public boolean collided;
public boolean active;
public AbstractObject(Integer id, Integer type, Vector3 position) {
this.id = id;
this.type = type;
this.position = position;
this.collided = false;
}
public void setPosition(float x, float y, float z) {
this.setPosition(new Vector3(x, y, z));
}
public Integer getId() {
return id;
}
public void setId(Integer id) {
this.id = id;
}
public Integer getType() {
return type;
}
public void setType(Integer type) {
this.type = type;
this.setModelInstance(new ModelInstance(Assets.instance.brick.get(type)));
}
public Vector3 getPosition() {
return position;
}
public void setPosition(Vector3 position) {
this.position = position;
this.modelInstance.transform.translate(position);
this.body.setWorldTransform(modelInstance.transform);
}
public ModelInstance getModelInstance() {
return modelInstance;
}
public void setModelInstance(ModelInstance modelInstance) {
this.modelInstance = modelInstance;
}
public btCollisionObject getObject() {
return body;
}
public void onCollision(Integer power){
}
public void getDetails() {
Gdx.app.log("ID", id.toString());
Gdx.app.log("Type", type.toString());
Gdx.app.log("Position", position.toString());
Gdx.app.log("Collision", Boolean.toString(collided));
Gdx.app.log("---------------", "---------------");
}
public void dispose() {
shape.dispose();
body.dispose();
Gdx.app.log(this.toString(), "dispose");
}
}
Is there an alternative to all that casting? Casting decreases performance right?
A: I think what you hit is kind of a classical problem in OOP design in mainstream modern languages i.e. lack of multiple dispatch or multimethods. There are a few typical ways to fight it and the most traditional one uses double dispatch and optionally a visitor pattern.
General idea looks something like this
public abstract class AbstractObject {
...
public final void dispatchCollision(AbstractObject other) {
other.dispatchCollisionImpl(this);
}
protected abstract void dispatchCollisionImpl(AbstractObject other);
protected abstract void onCollisionWithBall(Ball ball);
protected abstract void onCollisionWithBrick(Brick ball);
}
public class Ball extends AbstractObject {
...
@Override
protected void dispatchCollisionImpl(AbstractObject other) {
other.onCollisionWithBall(this); // this is where main "magic" happens
}
@Override
protected void onCollisionWithBall(Ball ball) {
throw new UnsupportedOperationException("Ball-ball collision should never happen");
}
@Override
protected void onCollisionWithBrick(Brick ball) {
// your actual brick-ball collision logic
}
}
Code in the Brick class is pretty symmetrical to the code in the Ball.
And then in your CollisionListener you can simply do something like:
public class CollisionListener extends ContactListener {
@Override
public boolean onContactAdded(btCollisionObject ob0, int partId0, int index0, btCollisionObject ob1, int partId1, int index1) {
AbstractObject aO0 = (AbstractObject) ob0.userData;
AbstractObject aO1 = (AbstractObject) ob1.userData;
aO0.dispatchCollision(aO1);
//aO1.dispatchCollision(aO0); // if you want to do both
return true;
}
}
The main drawback of this approach is that if you have many subclasses of your AbstractObject, you need to add methods for each of them in each subclass. On the other hand you can put some default generic logic to such methods in some base classes.
If you have many subclasses or need some plugin-like support you should probably go to more advanced techniques for multi-methods simulation such as for example having explicit global Map<Tuple<Class,Class>, Handler> for dispatch.
Explicit Multi-methods
Here is one idea on how to create something similar to multi-methods more explicitly:
public class ClassesPair {
public final Class<? extends AbstractObject> targetClass;
public final Class<? extends AbstractObject> objectClass;
public ClassesPair(Class<? extends AbstractObject> targetClass, Class<? extends AbstractObject> objectClass) {
this.targetClass = targetClass;
this.objectClass = objectClass;
}
@Override
public boolean equals(Object o) {
if (this == o) return true;
if (o == null || getClass() != o.getClass()) return false;
ClassesPair that = (ClassesPair) o;
if (!targetClass.equals(that.targetClass)) return false;
return objectClass.equals(that.objectClass);
}
@Override
public int hashCode() {
int result = targetClass.hashCode();
result = 31 * result + objectClass.hashCode();
return result;
}
}
public interface CollisionHandler<T extends AbstractObject, O extends AbstractObject> {
void handleCollision(T target, O object);
}
public class CollisionsDispatcher {
private final Map<ClassesPair, CollisionHandler> originalDispatchMap = new HashMap<>();
private Map<ClassesPair, CollisionHandler> extendedDispatchMap = new HashMap<>();
private CollisionHandler getHandlerOrParent(Class<? extends AbstractObject> targetClass, Class<? extends AbstractObject> objectClass) {
//Need to decide on the rules, for now target is more important
Class stopClass = AbstractObject.class.getSuperclass();
for (Class tmpTarget = targetClass; tmpTarget != stopClass; tmpTarget = tmpTarget.getSuperclass()) {
for (Class tmpObject = objectClass; tmpObject != stopClass; tmpObject = tmpObject.getSuperclass()) {
CollisionHandler collisionHandler = originalDispatchMap.get(new ClassesPair(tmpTarget, tmpObject));
if (collisionHandler != null)
return collisionHandler;
}
}
return null;
}
public CollisionHandler getHandler(Class<? extends AbstractObject> targetClass, Class<? extends AbstractObject> objectClass) {
ClassesPair key = new ClassesPair(targetClass, objectClass);
CollisionHandler collisionHandler = extendedDispatchMap.get(key);
if (collisionHandler == null) {
// choice #1
// Just fail every time nothing was found
//throw new UnsupportedOperationException("Collision of " + targetClass.getName() + " with " + objectClass.getName() + "' is not supported");
// choice #2 go through handlers for parents.
// It provides ability to put some generic logic only once
// Need to decide on the rules, for now target is more important
collisionHandler = getHandlerOrParent(targetClass, objectClass);
if (collisionHandler != null) {
extendedDispatchMap.put(key, collisionHandler); // put it back for faster future usages
} else {
throw new UnsupportedOperationException("Collision of " + targetClass.getName() + " with " + objectClass.getName() + "' is not supported");
}
// choice #3
// Just do nothing. Everything that has no explicit handler is not affected by collision
// return null;
}
return collisionHandler; // God save Java with its type erasure for generics!
}
public void handleCollision(AbstractObject target, AbstractObject object) {
CollisionHandler handler = getHandler(target.getClass(), object.getClass());
if (handler != null) { // this check only for choice #3
handler.handleCollision(target, object); // God save Java with its type erasure for generics!
}
}
public <T extends AbstractObject, O extends AbstractObject> void registerHandler(Class<T> targetClass, Class<O> objectClass, CollisionHandler<? super T, ? super O> handler) {
ClassesPair key = new ClassesPair(targetClass, objectClass);
originalDispatchMap.put(key, handler);
// just clear extended cache. It is much easier than to track all possible propagated values
// and handle them properly. On the other hand registerHandler should be called only a few
// time during set up so it shouldn't be real penalty in performance
extendedDispatchMap = new HashMap<>();
}
}
So now for an usage example assume that you want to create some Arkanoid-like game with 3 kinds of bricks:
*
*one-hit brick which is always blue
*two-hits brick that changes color from red to pink after first hit
*super-brick that is black and can not be destroyed at all
public abstract class AbstractBrick extends AbstractObject {
protected int hitCount;
public AbstractBrick(int hitCount) {
this.hitCount = hitCount;
}
public int getHitCount() {
return hitCount;
}
public void setHitCount(int hitCount) {
this.hitCount = hitCount;
}
public abstract Color getColor();
@Override
protected void dispatchCollisionImpl(AbstractObject other) {
other.onCollisionWithBrick(this);
}
@Override
protected void onCollisionWithBall(Ball ball) {
}
@Override
protected void onCollisionWithBrick(AbstractBrick ball) {
}
}
// takes one hit to break
public class SimpleBrick extends AbstractBrick {
public SimpleBrick() {
super(1);
}
@Override
public Color getColor() {
return Color.BLUE;
}
}
// takes two hits to break
public class DoubleBrick extends AbstractBrick {
public DoubleBrick() {
super(2);
}
@Override
public Color getColor() {
if (hitCount == 2)
return Color.RED;
else
return Color.PINK;
}
}
// never breaks
public class SuperBrick extends AbstractBrick {
public SuperBrick() {
super(-1);
}
@Override
public Color getColor() {
return Color.BLACK;
}
}
So now you create specific instance of CollisionsDispatcher with all the necessary handlers registered in it
public class MyCollisionsDispatcher extends CollisionsDispatcher {
public MyCollisionsDispatcher() {
// Pre-register all required handlers
// using Java-8 syntax for "::" instead of anonymous classes
registerHandler(Ball.class, AbstractBrick.class, this::handleBallBrick);
registerHandler(AbstractBrick.class, Ball.class, this::handleUsualBrickBall);
registerHandler(SuperBrick.class, Ball.class, this::handleSuperBrickBall);
}
void handleBallBrick(Ball ball, AbstractBrick brick) {
// bounce of the ball
// in this case it is not important which brick we hit
System.out.println("Ball hit some brick");
}
void handleUsualBrickBall(AbstractBrick brick, Ball ball) {
int newCount = brick.getHitCount() - 1;
if (newCount != 0) {
brick.setHitCount(newCount);
} else {
// remove brick
}
System.out.println("Usual brick was hit by a ball. newCount = " + newCount);
}
void handleSuperBrickBall(SuperBrick brick, Ball ball) {
// do nothing. Super brick is so super!
System.out.println("Super brick was hit by a ball but nothing happened");
}
}
and with that you can do something like this:
public void test() {
AbstractObject simpleBrick = new SimpleBrick();
AbstractObject doubleBrick = new DoubleBrick();
AbstractObject superBrick = new SuperBrick();
AbstractObject ball = new Ball();
CollisionsDispatcher dispatcher = new MyCollisionsDispatcher();
dispatcher.handleCollision(ball, simpleBrick);
dispatcher.handleCollision(simpleBrick, ball);
dispatcher.handleCollision(ball, doubleBrick);
dispatcher.handleCollision(doubleBrick, ball);
dispatcher.handleCollision(doubleBrick, ball);
dispatcher.handleCollision(ball, superBrick);
dispatcher.handleCollision(superBrick, ball);
dispatcher.handleCollision(superBrick, ball);
}
and the output is exactly like one would expect:
Ball hit some brick
Usual brick was hit by a ball. newCount = 0
Ball hit some brick
Usual brick was hit by a ball. newCount = 1
Usual brick was hit by a ball. newCount = 0
Ball hit some brick
Super brick was hit by a ball but nothing happened
Super brick was hit by a ball but nothing happened
So in your CollisionListener you can call just
@Override
public boolean onContactAdded(btCollisionObject ob0, int partId0, int index0, btCollisionObject ob1, int partId1, int index1) {
AbstractObject aO0 = (AbstractObject) ob0.userData;
AbstractObject aO1 = (AbstractObject) ob1.userData;
dispatcher.handleCollision(aO0, aO1);
// dispatcher.handleCollision(aO1, aO0); // if you want to do both
return true;
}
The main drawbacks here are the other sides of the main advantages:
*
*you can put all your collision-related code in a single place MyCollisionsDispatcher but that "single place" might get pretty big.
*Another bonus is that with such approach you might have a "plug-in" system i.e. someone can add a new AbstractObject subclass without touching anything in the existing code by just registering proper handlers in the dispatcher. The disadvantage of this is that in any mainstream language I know you loose your compile time checks that every necessary handler is actually implemented as it is in double-dispatch.
Summary (and a bit of comparison)
In terms of long term management and code clarity it is in my opinion a matter of taste which solution to prefer unless you have other limitations which make some of them not applicable. Every suitable technique is relatively advanced and might stumble a developer how is not aware of it.
In terms of performance the first rules is: Measure it!. Still I'll break it and do my forecast that double dispatch is faster then explicit Map which is faster than a bunch of instanceof if there are many subclasses (still, YMMV) As for memory consumption I don't see any significant differences.
As someone said Software Engineering Is Art Of Compromise so finally it is up to you to make right trade-offs.
| {
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} | 4,661 |
Bazylika Eufrazjusza (chorw. Eufrazijeva bazilika) – bazylika w Poreču w Chorwacji, w 1997 roku wpisana na listę światowego dziedzictwa UNESCO.
Obecny budynek wzniesiono w VI wieku n.e. na miejscu kościoła z V wieku i wcześniejszej, mniejszej bazyliki z IV stulecia. Mozaiki pierwotnych budowli wciąż są widoczne w północnej nawie. Bazylika stanowi jeden z najlepiej zachowanych przykładów sztuki bizantyńskiej w Chorwacji. We wnętrzu, oprócz wczesnobizantyjskich mozaik przedstawiających sceny biblijne, znajdują się także dekoracje z XVI wieku. Cyborium nad głównym ołtarzem pochodzi natomiast z końca XIII wieku.
Przypisy
Bibliografia
Oliver, Jeanne, Croatia, LonelyPlanet 2005, str. 145-146.
Poreč
Poreč
Obiekty z listy dziedzictwa UNESCO w Chorwacji
Poreč
Żupania istryjska
Religia w Poreču
Zabytkowe kościoły w Chorwacji | {
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} | 5,379 |
The Australian Financial Security Authority (AFSA) is responsible for the administration and regulation of the personal insolvency system, proceeds of crime, trustee services and the administration of the Personal Property Securities Register (PPSR).
AFSA has produced a number of useful publications on personal insolvency, which are listed below.
Do you need general information about personal insolvency?
Do you need information about bankruptcy?
Do you need general information about bankruptcy?
Do you need information about voluntary bankruptcy?
Do you need information about how creditors can make you bankrupt?
Do you need information about debt?
Do you need information about borrowing and credit?
Do you need information about mortgages? | {
"redpajama_set_name": "RedPajamaC4"
} | 3,160 |
The German Army (, "army") is the land component of the armed forces of Germany. The present-day German Army was founded in 1955 as part of the newly formed West German Bundeswehr together with the Marine (German Navy) and the Luftwaffe (German Air Force). , the German Army had a strength of 62,766 soldiers.
History
Overview
A German army equipped, organized, and trained following a single doctrine and permanently unified under one command was created in 1871 during the unification of Germany under the leadership of Prussia. From 1871 to 1919, the title Deutsches Heer (German Army) was the official name of the German land forces. Following the German defeat in World War I and the end of the German Empire, the main army was dissolved. From 1921 to 1935 the name of the German land forces was the Reichsheer (Army of the Empire) and from 1935 to 1945 the name Heer was used. The Heer was one of two ground forces of the Third Reich during World War II but, unlike the Heer, the Waffen-SS was not a branch of the Wehrmacht but was a combat force under the Nazi Party's own Schutzstaffel forces. The Heer was formally disbanded in August 1946.
After World War II, Germany was divided into the Federal Republic of Germany (West Germany) and the German Democratic Republic (East Germany), which both formed their own armed forces: on 12 November 1955 the first recruits began their service in the West German Heer, while on 1 March 1956 the East German Landstreitkräfte der NVA (Land Forces of the National People's Army) were founded. During the Cold War, the West German Army was fully integrated into NATO's command structure while the Landstreitkräfte were part of the Warsaw Pact. Following the process of German reunification in 1990, the Landstreitkräfte were partially integrated into the German Army. Since then, the German Army has been employed in peacekeeping operations worldwide and since 2002 also in combat operations in Afghanistan as part of NATO's International Security Assistance Force.
Founding of the Army
Following World War II the Allies dissolved the Wehrmacht with all its branches on 20 August 1946. However already one year after the founding of the Federal Republic of Germany in May 1949 and because of its increasing links with the West under German chancellor Konrad Adenauer, the Consultative Assembly of Europe began to consider the formation of a European Defence Community with German participation on 11 August 1950. Former high-ranking Wehrmacht officers outlined in the Himmeroder memorandum a plan for a "German contingent in an international force for the defense of Western Europe." For the German land forces the memorandum envisioned the formation of a 250,000 strong army. The officers saw the need for the formation of twelve Panzer divisions and six corps staffs with accompanying Corps troops, as only armoured divisions could muster a fighting force to throw back the numerically far superior forces of the Warsaw Pact.
Theodor Blank was appointed "officer of the Federal Chancellor for the Strengthening of Allied Troops questions". This Defence Ministry forerunner was known somewhat euphemistically as the Blank Office (Amt Blank), but explicitly used to prepare for the rearmament of West Germany (Wiederbewaffnung). By March 1954 the Blank Office had finished plans for a new German army. Plans foresaw the formation of six infantry, four armoured, and two mechanised infantry divisions, as the German contribution to the defense of Western Europe in the framework of a European Defence Community. On 8 February 1952 the Bundestag approved a German contribution to the defense of Western Europe and on 26 February 1954 the Basic Law of the Republic was amended with the insertion of an article regarding the defence of the sovereignty of the federal government. Following a decision at the London Nine Power Conference of 28 September to 3 October 1954, Germany's entry into NATO effective from 9 May 1955 was accepted as a replacement for the failed European Defence Community plan. Afterwards the Blank Office was converted to the Defence Ministry and Theodor Blank became the first Defence Minister. The nucleus of army was the so-called V Branch of the Department of Defence. Subdivisions included were VA Leadership and Training, VB Organisation and VC Logistics.
The army saw itself explicitly not as a successor to the defeated Wehrmacht, but as in the traditions of the Prussian military reformers of 1807 to 1814 and the members of the military resistance during National Socialism, such as the officers which undertook the failed 20 July plot to assassinate Adolf Hitler in 1944. Nevertheless, for lack of alternatives the officer corps was made up largely of former Wehrmacht officers. The first Chief of the Army was the former Wehrmacht General der Panzertruppe Hans Rottiger, who had been involved in the drafting of the Himmeroder memorandum.
The official date of the founding of the army is 12 November 1955 when the first soldiers began their service in Andernach. In 1956 the first troops set up seven training companies in Andernach and began the formation of schools and training centers. On 1 April 1957, the first conscripts arrived for service in the army. The first military organisations created were instructional battalions, officer schools, and the Army Academy, the forerunner to the Führungsakademie der Bundeswehr in Hamburg. In total of twelve armoured and infantry divisions were to be established by 1959, as planned in Army Structure I. To achieve this goal existing units were split approximately every six months. However the creation of all twelve divisions did not take place until 1965. At the end of 1958 the strength of the army was about 20,200 men. The army was equipped at first with American material, such as the M-47 Patton main battle tank. Three corps commands were formed beginning in 1957: the I Corps, II Corps, and the III Corps.
Also in 1957 the "Office for Territorial Defence" was established as the highest Territorial Army authority. The Office for Territorial Defence was under the direct command of the Federal Ministry of Defence and commanded the Territorial Army (Germany) (Territorialheer), a reserve formation. While the Heer along with the Marine and Luftwaffe were firmly integrated into the NATO Military Command Structure, the Territorialheer remained under national command. The main function of the Territorialheer was to maintain the operational freedom of NATO forces through providing rear area defence against saboteurs, enemy special forces, and the like. There were three Territorial Commands (Territorialkommandos), including North, South, and Schleswig-Holstein, and up to six Wehrbereichskommandos (WBKs), military regional commands. By 1985 each of the WBKs had two Heimatschutzbrigades (HSBs, home defence brigades).
The development of Soviet tactical nuclear weapons required the development of a new Army structure even before Army Structure 1 was fully achieved. To minimize the effects of attacks with tactical nuclear weapons on massed forces, the 28,000 strong divisions of the Heer were broken up into smaller and more mobile brigades under Army Structure 2. These smaller units were also to be capable of self-sustainment on a nuclear battlefield for several days, and to be capable of moving quickly from defense and to attack. The new armoured and mechanised brigades were capable of combined arms combat. Each division was composed of three brigades. The armoured brigades consisted of an armoured infantry battalion, two armoured battalions, a self-propelled artillery battalion and a supply battalion. The mechanised brigades consisted of a motorised infantry battalion, two mechanised infantry battalions, an armoured battalion, a field artillery battalion and a supply battalion. The motorised brigades consisted of three motorised infantry battalions, an anti-tank battalion, a field artillery battalion and a supply battalion. The alpine brigades consisted of three alpine battalions, a mountain artillery battalion and a supply battalion. By 1959 the Heer consisted of 11 divisions of 27 brigades, four Panzer (armoured), four Panzergrenadier (mechanised), two Jäger (motorised), and one Gebirgsjäger (alpine).
From roughly 1970 onward, Army Structure 3 saw the targeted number of 36 active brigades raised by 1975 while the 2nd and 4th Panzergrenadier divisions were reorganised into Jäger formations. The armies Fallschirmjäger (paratrooper) brigades were renamed into Luftlande (airborne) brigades and a third brigade (Luftlandebrigade 27) was formed.
Under Army Structure 4 from 1980/81 on, the German Army fielded 12 divisions (with 38 active brigades): six Panzer (armoured), four Panzergrenadier (mechanised), one Luftlande (airborne), and one Gebirgs (alpine) divisions. Ten active divisions were grouped into three corps: I German Corps as part of NATO's Northern Army Group, II German Corps and III German Corps as part of Central Army Group. The remaining heavy division (6th Panzergrenadier Division) was part of Allied Forces Baltic Approaches. In peacetime the 1st Airborne Division was assigned to II German Corps with its three brigades to be distributed among the three Corps respectively in wartime, forming a quick reaction reserve.
The number of active brigades rose compared to Army Structure 3 due to two Heimatschutz territorial defense brigades (51 and 56) being assigned to the field army as part of a mechanised and mountain division respectively. The non-NATO assigned territorial army formed 10 further territorial defense brigades for rear area security at varying readiness levels, with most units being partially manned in peacetime and others being entirely non-active units with equipment in storage.
Brigades in the field army grew to four combat battalions instead of three. Mechanised brigades typically consisted of one Panzer and three Panzergrenadier battalions, of which one was a partially active and mixed formation, containing a tank company and two mechanised companies. Armoured brigades similarly consisted of one Panzergrenadier and three Panzer battalions, with one armoured battalion being mixed and partially active (containing one mechanised and two tank companies).
Mechanised infantry battalions in mechanised brigades typically had one of three companies equipped as motorised infantry with M113 APCs instead of Marder IFVs.
Post Cold War
After 1990, the Heer absorbed the Nationale Volksarmee, the armed forces of East Germany. The former East German forces were initially controlled by the Bundeswehr Command East under the command of Lieutenant General Jörg Schönbohm and disbanded on 30 June 1991. In the aftermath of the merger, the German Army consisted of four Corps (including IV Corps at Potsdam in the former DDR) with a manpower of 360,000 men. It was continuously downsized from this point. In 1994 III Corps was reorganised as the German Army Forces Command. In 1996, the 25th Airborne Brigade was converted into a new command leading the Army's special forces, known as the Kommando Spezialkräfte.
Logistics, CBRN defense, territorial defense and military police units were split off into the newly formed Joint Support Service and medical units into the Joint Medical Service in 2000. The transferred units continue to wear army uniforms.
The 2001 onwards restructuring of the German Army saw it move to a seven division structure – five mechanised (each with two mechanised brigades), one special forces, and one air assault.
In 2003, three Corps still existed, each including various combat formations and a maintenance brigade, as well as the I. German/Dutch Corps, a joint German-Netherlands organization, used to control in peacetime the 1st Panzer and 7th Panzer Divisions as well as Dutch formations. The 1st Panzer would have reported to the corps in wartime while the 7th would be posted to the Allied Rapid Reaction Corps. II Corps was German in peacetime but would have exchanged a division with the V U.S. Corps in time of war (the 5th Panzer). The 5th Panzer Division was formally disbanded as of 30 June 2001. In peacetime it also commanded the 10th Panzer Division, which was allocated to Eurocorps and which parents the German half of the Franco-German Brigade. The 1st Mountain Division at Munich was also subordinate to this headquarters.
The IV Corps was headquartered at Potsdam in eastern Germany and controlled two Panzergrenadier Divisions, the 13th and 14th. The 14th Panzergrenadier Division also took control of units in Western Germany re-subordinated from the 6th Panzergrenadier Division when it lost its command function. It would have made up the German contribution to the Multinational Corps Northeast in time of war. IV Corps also used to have under its command the Military District Command I, the 1st Airmobile Brigade, and the Berlin Command (:de:Standortkommando Berlin).
The current structure was assumed with the most recent German Army reform which also suspended conscription by 1 July 2011 and saw the army move to a purely professional three division structure with a view on creating smaller, more flexible and more deployable units, emphasising global employment against non-state threats such as international terrorism or as part of UN and EU missions.
, the German Army had a strength of 62,766 soldiers.
Structure and organisation
The German Army is commanded by the Inspector of the Army (Inspekteur des Heeres) based at the Army Command (Kommando Heer) in Strausberg near Berlin. The training centers are supervised by the Army Training Command in Leipzig.
The combat units of the army now include two armoured divisions and the lighter rapid forces division. Unlike other European armies such as neighbouring France, regiments are not a common form of organization and are thus rare in the German army. Battalions and regiments are directly subordinate to brigades or to divisions as divisional troops. German infantry battalions field 1,000 men, considerably larger than most NATO armies, (e.g. twice the size of a US Army battalion). While some brigades are still designated as either Panzer (armour) or Panzergrenadier (mechanised infantry) formations, these names are by now traditional and no longer imply a different organisation, for example an armoured brigade would not be expected to contain more tanks than a mechanised one.
1. Panzerdivision in Oldenburg
Panzerlehrbrigade 9, in Munster
Panzer Brigade 21, in Augustdorf
Panzergrenadier Brigade 41, in Neubrandenburg
43 Mechanized Brigade (Royal Netherlands Army), in Havelte (Netherlands)
Divisional troops
10. Panzerdivision, in Veitshöchheim
Panzer Brigade 12, in Cham
Gebirgsjäger Brigade 23, in Bad Reichenhall
Panzergrenadier Brigade 37, in Frankenberg
Franco-German Brigade, in Müllheim
Divisional troops
Rapid Forces Division, in Stadtallendorf
Airborne Brigade 1, in Saarlouis
11 Airmobile Brigade (Royal Netherlands Army), in Schaarsbergen (Netherlands)
Special Forces Command (KSK), in Calw
Helicopter Command, in Bückeburg
Divisional troops
Eurocorps (German elements), in Strasbourg (France)
Command Support Brigade
German elements in two permanent battalions and one staff company
1 (German/Netherlands) Corps, in Münster
German elements in two permanent battalions and one staff company
Multinational Corps North East, in Szczecin (Poland)
610th Signal Battalion, in Prenzlau
Central Army Depot, in Herongen
Central Army Depot, in Pirmasens
Central Mobilisation Base, in Brück
Equipment
Further vehicles include:
Armoured personnel carrier and fighting vehicles:
Puma (IFV) infantry fighting vehicle
Boxer (armoured fighting vehicle)
TPz Fuchs as armoured personnel carrier
ATF Dingo as armoured infantry mobility vehicle
Trucks:
Mercedes-Benz Zetros off-road transport truck
MAN KAT1 high-mobility off-road truck
Unimog all-wheel drive army personnel or equipment carriers
Truppengattungen
The German Army has eleven different branches of troops, designated as Truppengattungen. Each Truppengattung is responsible for training and readiness of its units and disposes of its own schools and centres of excellence for doing so. Optically this distinction can be made by the branch colour, called Waffenfarbe which is displayed by a cord attached to the rank insignia, and the colour of their beret with a specific badge attached to it.
Beret Colour (Army only and Security Units of Navy and Air Force)
Black: Armoured Corps, Reconnaissance Corps
Green: Mechanised and Light Infantry Corps
Dark Red: Aviation Corps, Airborne Corps, Special Forces, formations assigned to airborne division
Light Red: Combat Support Corps and Military Police
Dark Blue: Medical Corps
Navy Blue: Multinational Units, Officer Cadet Battalions, Navy and Air Force Security Units
Bright Blue: Troops with United Nations Missions
Grey mountain cap (Bergmütze): Mountain Troops Gebirgsjäger
Waffenfarbe (Army and army support branch only)
Bright Red: General ranks (only Kragenspiegel, not Litze),
Crimson: General Staff
Rank structure
The rank structure of the German army is adjusted to the rank structure of NATO. Unlike its predecessors, the modern German Army does not use the rank of Colonel General. The highest rank for an army officer is Lieutenant General, as the rank of Full General is reserved for the Armed Forces chief of staff or officers serving as NATO officers.
Officers
NCOs and enlisted
See also
Bavarian Army
History of Germany during World War II
Imperial Army (German Empire) (to 1806)
Kaiserliche Armee (1870–1918):
Imperial German Army
, the Airforce
, the Navy
Prussian Army
Tank battalions of the German Army 1956–2008
List of military weapons of Germany
References
Further reading
Addington, Larry H. The Blitzkrieg Era and the German General Staff, 1865–1941 (1971).
Bartov, Omer. Hitler's army: Soldiers, Nazis, and war in the Third Reich (1992).
Bull, Stephen. German Assault Troops of the First World War: Stosstrupptaktik—The First Stormtroopers (History Press, 2014).
Citino, Robert M. The Path to Blitzkrieg: Doctrine and Training in the German Army, 1920–39 (2007).
Citino, Robert M. Quest for Decisive Victory: From Stalemate to Blitzkrieg in Europe, 1899–1940 (2002).
Dupuy, Trevor Nevitt. A Genius for War: The German Army and General Staff, 1807–1945 (1977).
Gross, Gerhard P. The Myth and Reality of German Warfare: Operational Thinking From Moltke the Elder to Heusinger (2016).
Deist, Wilhelm, ed. The German Military in the Age of Total War (Berg, 1985).
Hughes, Daniel J., and Richard L. DiNardo, eds. Imperial Germany and War, 1871–1918 (University Press of Kansas, 2018).
Karau, Mark D. Germany's Defeat in the First World War: The Lost Battles and Reckless Gambles That Brought Down the Second Reich (ABC-CLIO, 2015).
Kelleher, Catherine M. "Fundamentals of German Security: The Creation of the Bundeswehr: Continuity and Change", in Stephen F. Szabo (ed.), The Bundeswehr and Western Security, (St. Martin's Press, New York, 1990).
Lummel, Peter. "Food Provisioning in the German Army of the First World War." in Food and War in Twentieth Century Europe (Routledge, 2016) pp. 31-44.
Seaton, Albert. The German Army: 1933-45 (1982).
Showalter, Dennis (2016). Instrument of War: The German army 1914–18
Showalter, Dennis (2015). The Wars of German Unification
Online free
External links
Official Homepage of the German Army (Heer)
Historical links
German Army pre-1914
German Army 1914–1918
German Army Organization 1914
The Nazi German Army 1935-1945 (Heer)
German Infantry Photographs from World War II—Colour photographs
Gebirgsjaeger—German Mountain Troops
Axis History—Axis History site including German troops
Bundeswehr | {
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Acteon tornatilis é uma espécie de molusco pertencente à família Acteonidae.
A autoridade científica da espécie é Linnaeus, tendo sido descrita no ano de 1758.
Trata-se de uma espécie presente no território português, incluindo a zona económica exclusiva.
Referências
Acteon tornatilis - World Register of Marine Species (consultado em 29 de dezembro de 2013).
Ligações externas
Acteon tornatilis - Biodiversity Heritage Library - Bibliografia
Acteon tornatilis - NCBI Taxonomy Database
Acteon tornatilis - Global Biodiversity Information Facility
Acteon tornatilis - Encyclopedia of Life
Moluscos de Portugal
Acteon
Moluscos descritos em 1758 | {
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Kulki is een plaats in het Poolse district Miński, woiwodschap Mazovië. De plaats maakt deel uit van de gemeente Siennica en telt 70 inwoners.
Plaats in Mazovië | {
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} | 1,825 |
\section{Introduction} \label{sec:introduction}
Ever since Mitchell (1767) pointed out that the observed frequency of visual
double stars was too high to be due to random chance, the study of binary stars
has occupied an important place in astrophysics. William Herschel (1802)
discovered and cataloged hundreds of visual pairs and produced the first
observations of a rudimentary binary orbit. In doing so he established that the
double stars were indeed physical pairs and that Newtonian physics operated
nicely in the distant sidereal universe. By the beginning of the twentieth
century tens of thousands of binary stars were known and cataloged (e.g., Burnham
1906). By the middle to late twentieth century the first systematic attempts to
establish the binary frequency of main sequence F and G stars suggested that a
very high fraction (70 - 80\%) of all such stellar systems consist of binary or
multiple stars (Heintz 1969; Abt \& Levy 1976; Abt 1983). The most comprehensive
and complete study of the multiplicity of G stars was performed by Duquennoy \&
Mayor (1991) who argued that two-thirds of all such stellar systems are
multiple.
It has often been assumed but never clearly demonstrated that similar statistics
applied to stars of all spectral types. This assumption has led to the commonly held
opinion that most all stars form in binary or multiple systems with the Sun (and its
system of planets) being atypical as a single star.
But how robust is the assumption
that the binary statistics for G stars is representative of all stars?
Over the last decade two important developments have occurred in stellar research which
directly bear on this question. First, the functional form of the stellar initial mass
function (IMF) has been better constrained by observations of both field stars (e.g.,
Kroupa, 2002) and young embedded clusters (e.g., Muench et al. 2002). The IMF has been
found to peak broadly between 0.1 - 0.5 M$_\odot$, indicating that most stars formed in the
Galactic disk are M stars. Second, surveys for binary stars have suggested that the binary
star frequency may be a function of spectral type (e.g., Fischer \& Marcy 1992). In
particular, there have been a number of attempts to ascertain the binary frequency of M type
stars and even for L and T dwarfs, objectss near and below the hydrogen burning limit. These
studies suggest that the binary frequency declines from the G star value, being only around
30\% for M stars (e.g., Leinert et al. 1997; Reid \& Gizis 1997; Delfosse et al. 2004;
Siegler et al. 2005) and as much as a factor of 2 lower for L and T dwarfs (e.g., Gizis et
al. 2003). I argue in this communication that these two facts together suggest that most
stellar systems in the Galaxy consist of single rather than binary or multiple stars.
\section{The Single Star Fraction and Spectral Type}
\label{sec:observations}
In this section I use data compiled from the literature to examine the single
star fraction as a function of stellar spectral type, in particular for the range
spanning G to M stars. I consider the single star fraction (SSF) to be the
fraction of stellar systems without a {\it stellar} companion, that is, primary
stars without a companion whose mass exceeds 0.08 M$_\odot$. Figure 1 displays the
single star fraction as a function of spectral type for G and later type stars.
This plot suggests that the SSF is significantly greater for M stars than for G
stars. Indeed the SSF for M stars appears to be at least 70\%. It is difficult
to evaluate the significance of this difference at face value given that the
differing binary surveys suffer from differing biases and varying degrees of
incompleteness. The systematic differences that can arise between the surveys
mostly derive from varying sensitivities to primary/secondary separations and
mass ratios. Below I attempt to evaluate the results from the surveys used to
construct Figure 1.
\begin{figure}
\centering
\includegraphics[scale=0.40]{binaries2.EPS}
\caption{The single star fraction vs spectral type. The single star
fraction increases significantly with spectral type reaching values
of $\sim$ 75\% for M stars, the most populous stars in the IMF and
the field. Vertical error bars represent statistical uncertainties in
the SSF. The horizontal error bars indicate the approximate extent in
spectral type covered by the individual surveys and do not represent
an uncertainty in this coordinate. Data taken from
Duquennoy \& Mayor (1991), Reid \& Gizis (1997), Fischer \& Marcy (1992),
Delfosse et al. (2004), Leinert et al. 1997, and Siegler et al. (2005).
}
\end{figure}
In their seminal study, Duquennoy \& Mayor (1991) obtained a spectroscopic survey
of a distance-limited complete sample of F7-G9 stars in the Northern Hemisphere
and within 22 pc of the Sun. They examined radial velocities obtained for these
stars over a 13 year period. They combined their detections of spectroscopic
binaries with known visual binaries and common proper motion pairs to examine 164
primaries for evidence of multiplicity. They derive multiplicity ratios of
57:38:4:1 for single:double:triple:quadruple systems, respectively. They
considered all the various detection biases to estimate the incompleteness of
their study and concluded that there was a slight bias against detecting low mass
companions, this resulted in a 14\% upward correction to the multiplicity
fraction such that 57\% of systems were estimated to be multiple for a
primary/companion mass ratio, q $>$ 0.1. They further extrapolated this
incompleteness correction to include substellar secondaries and estimated a
multiplicity fraction of 2/3 and a single star fraction of 1/3 for their sample.
However, in recent years sensitive and precise radial velocity surveys of 1330
single FGKM stars have indicated a paucity of substellar companions within 5 AU
of the primary stars (Marcy \& Butler 2000; Marcy et al. 2005). In addition
coronographic imaging surveys have found a similar dearth of substellar
companions around GK and M stars over separations between 75 and 300 AU (McCarthy
\& Zuckerman 2004). The existence of this so-called ``brown dwarf desert''
indicates that Duquennoy \& Mayor may have overestimated the multiplicity
fraction of G stars and the true value is likely 57\% or even somewhat smaller.
For the purposes of this paper I adopt 57\% as the multiplicity fraction of G
type stars and thus 43\% for the SSF.
The first extensive examination of the multiplicity of M stars was performed by Fischer \&
Marcy (1992) who studied radial velocity, speckle and visual binary data for a sample of
stars within 20 pc. The full range of separations, $a$ $<$ 10$^4$ AU, was examined, similar
to the G star study. These authors pointed out that M star surveys suffer less from the
effects of incompleteness than G star surveys because the M star sample is on the whole a
factor of 2 closer in distance and M star primaries are sufficiently faint to enable
detection of very faint companions more readily. They derived a SSF of 58\% which is higher
than the G star value.
Reid \& Gizis (1997) determined the SSF for a volume complete sample of 79
M2-M4.5 primary stars within 8 pc of the Sun and derived a SSF of 70 $\pm$ 12\%
for this sample. The range of binary separations they were able to probe was 0.1
- 10$^4$ AU. A similar volume complete search for M dwarf binaries within 5 pc
of the Sun was performed by Leinert et al. (1997) who reported a SSF of 74 $\pm$
19\%. However, their sample of 29 stars is smaller than the Reid \& Gizis (1997)
and Fischer \& Marcy (1997) samples accounting for the larger uncertainty. More
recently Delfosse et al. (2004) presented statistics for a much larger sample of
100 M dwarfs which they estimated was 100\% complete for stellar mass companions
over the entire separation range and out to 9 pc from the Sun. Delfosse et al.
(2004) derive a multiple star fraction of 26 $\pm$ 3 \% which corresponds to a
SSF of 74 $\pm$ 6\%. This may represent the most accurate determination for the
M star SSF yet made. I note here that even if one considers substellar
companions this estimate for the SSF will not likely alter significantly since as
mentioned earlier, surveys have revealed a dearth of substellar companions to G,
K {\it and} M stars (Marcy \& Butler 2000; McCarthy and Zuckerman 2004).
Surveys for multiplicity among very late M stars and even L and T dwarfs have
also been recently reported. These studies typically explore more limited
separation ranges and somewhat smaller samples of stars. The multiplicity
fractions they find are however all lower than that reported for the earlier type
M stars. For example, Siegler et al. (2005) examined a magnitude-limited survey
of 36 M6 - M 7.5 stars and derived a binary fraction of 9 $\pm$ 4\% corresponding
to a SSF of 91 $\pm$ 5\%. However this sample is not volume limited and may be
incomplete. Thus the inferred SSF is likely an upper limit. Despite this
limitation Siegler et al. were able to conclude that wide (a$>$20 AU) binaries
are very rare among these stars. Although not considered for inclusion in Figure
1 because of the large fraction of brown dwarfs in their samples, surveys by
Gizis et al. (2003) and Bouy et al. (2003) find similarly small binary
fractions for ultra low mass objects. For example, Gizis et al. examined 82
nearby late M and L dwarfs and derived a (incompleteness corrected) binary
fraction of 15 $\pm$ 5\% (corresponding to a SSF of 85 $\pm$ 14\%) for
separations, $a > $ 1.6 AU. Estimating the possible contribution of
companions at smaller separations they suggest a binary star fraction (BSF)
of 15 $\leq$ BSF $\leq$ 25 \% corresponding to 75 $\leq$ SSF $\leq$ 85 \%
for these
objects near and just below the hydrogen burning limit. Bouy et al. (2003)
examined the binary statistics for a sample of 134 late M and L field dwarfs and
estimated a binary fraction for a separation range of about 2 - 140 AU of only
10\% corresponding to a SSF of 90\% for these objects. They also noted a dearth
of companions with wide (i.e., $a >$ 15 AU) separations. Although these surveys
of very low mass and substellar objects suffer from some degree of incompleteness
it is quite unlikely that sensible corrections for such effects would decrease
the estimated single star fraction to a value similar to that of G stars or even
typical M stars.
The observations discussed above lead to the conclusion that the single
star fraction is a function of spectral type and increases from about
43\% for G stars to $\sim$ 85\% for brown dwarfs. The most secure estimate
for M stars appears to be about 74\% based on the complete volume-limited
sample of Delfosse et al. (2004) for M stars with stellar companions.
\section{M Stars and the IMF} \label{sec:results}
The stellar IMF is one of the most fundamental
distribution functions in astrophysics. A great deal of effort has been expended
in determining its form since the first attempt to measure its shape by Salpeter
(1954). He found that the IMF is a power-law which decreases with stellar mass
for field stars with masses in the range between 1-10 M$_\odot$. More recent
determinations of the IMF for field stars and young embedded clusters have
expanded the mass range covered by Salpeter. These studies have found the IMF
to break from a single power-law shape near 0.5 M$_\odot$\ and to have a broad peak
between $\sim$ 0.1 - 0.5 M$_\odot$. On either side of this peak the IMF falls off
rapidly (e.g., Miller \& Scalo 1979; Kroupa 2002; Muench et al. 2002; Chabrier
2003; Luhman et al. 2006).
\begin{figure}
\centering
\includegraphics[scale=0.36, angle=90.0]{cimf2.eps}
\caption{The cumulative frequency distributions for all hydrogen burning
stars in two versions of the primary star IMF and the
PDMF of main sequence field stars. The two IMFs correspond to the
Miller-Scalo field star IMF and the IMF derived for the young embedded
Trapezium cluster by Muench et al. (2002). The vertical line marks the location
of the M star boundary (Torres \& Ribas 2002). The fraction of M stars is high for all these
mass functions ranging between 73 and 84\%. The latter value representing
fraction of all main sequence field stars that are M stars currently
residing in the Galactic disk. Based on data from Miller \& Scalo 1989 and
Muench et al. 2002.}
\end{figure}
The broad peak of the IMF encompasses the M stars and indicates that these stars
are the most numerous objects created in the star formation process. This is
illustrated in Figure 2 which shows the cumulative fraction of all stars above
the hydrogen burning limit given by the IMF. Two different IMFs are plotted
which span the range of modern day determinations of this function. One is the
log-normal field star IMF derived by Miller \& Scalo (1979) and the other
represents a determination of the IMF for the embedded Trapezium cluster in Orion
in which the IMF is characterized by a series of broken power-laws (Muench et al.
2002). This latter IMF is very similar to that determined for the field by
Kroupa (2002) but is more sensitive to substellar masses (not plotted). The
vertical dashed line shows the boundary for the M star population. The fraction
of all stars {\it above the hydrogen burning limit (HBL)} that are M stars is
73\% for the Muench et al. IMF and 78\% for the Miller-Scalo IMF. (It is
important to note here that these two IMFs are essentially primary star IMFs,
that is, IMFs that do not include companion star masses.) This analysis
indicates that roughly 3/4 of all stars formed are M stars.
The IMF represents the frequency distribution of stars at birth and differs from
the present day mass function (PDMF) which represents the frequency distribution
of all stars currently living within the Galactic disk. Stellar evolution has
significantly depleted the high mass end of the PDMF relative to the IMF.
Therefore, the fraction M stars in the PDMF is somewhat higher than the fraction
in the IMF. Indeed, for the PDMF derived by Miller \& Scalo (1979) we find from
Figure 2 that 84\% of all stars in the Galactic disk are M stars.
\vskip 0.2in
\section{The Total Single Star Fraction} \label{sec:SSF}
To estimate the total fraction of single stars, I assume that all stars earlier
than M are characterized by the single star fraction for G stars determined by
Duquennoy \& Mayor (1991), that is, $SSF_{<M} =$ 43\%. The single star
fraction for M-type stars (i.e., $SSF_{M}$) is assumed to be that (74\%)
determined by Delfosse et al. (2004) for a complete, volume limited sample. The
total SSF is then simply given by:
\small
$$ {\rm SSF(total)} = SSF_{<M} \times ETF + SSF_{M} \times MTF $$
\normalsize
\noindent
Here $MTF$ is the M-type fraction, that is, the fraction of all stars that are M-type stars
and $ETF = 1 - MTF$ is the early-type fraction, that is the fraction of all stars that have
spectral types earlier than M. To determine the SSF for all stars produced at any one time
by the star formation process I adopt the Muench et al. and Miller-Scalo IMFs,
specifically, MTF = 0.73 and 0.78, respectively. The total SSF is found to be 66\% and 67\%
for these two IMFs, respectively. Therefore, single stars must ultimately account for as
many as two-thirds of all stellar systems that formed at any one time in the Galaxy.
Similarly, if we consider the MTF (0.84) for the Miller-Scalo PDMF we find the total SSF to
be 69\%. Thus, {\it two thirds of all (main sequence) primary stars currently residing in
the Galactic disk are single stars}.
\section{Discussion and Conclusions}
\label{sec:discussion}
The primary result of this paper is the recognition that most stellar systems in
the Galaxy consist of single rather than binary stars. This fact has important
consequences for star and planet formation theory. For example, contrary to the
current accepted paradigm that most, if not all, stars form in binary or
multiple systems (e.g., Larson 1972, 2001; Mathieu 1994), this result could
indicate that the theoretical frameworks developed to explain the formation of
single, sunlike stars (e.g., Shu, Adams \& Lizano 1987) have wide applicability.
Indeed, when appropriately modified for a cluster-forming environment (e.g.,
Myers 1998; Shu, Li \& Allen 2004), they may even describe most star forming
events in the Galaxy. On the other hand, most stars could still initially form
in binary or multiple systems provided that most such systems promptly
disintegrate via dynamical interactions or decay in an early, perhaps even
protostellar, stage of evolution (e.g., Kroupa 1995; Sterzik \& Durisen 1998,
Reipurth 2000).
The current paradigm that most, if not all stars, form in binaries was
strengthened by early multiplicity surveys of pre-main sequence (PMS) stars. In
particular, surveys of the PMS population of the Taurus cloud indicated a binary
fraction that was twice that of field G stars (Ghez et al. 1993; Leinert et al.
1993; Reipurth \& Zinnecker 1993). However, most field stars are now known to
have formed in embedded clusters, environments quite different than represented
by the Taurus PMS population (e.g., Lada \& Lada 2003). Binary surveys of both
young embedded and Galactic clusters have revealed binary fractions
indistinguishable from that of the field (e.g., Petr et al. 1998; Duch\^ene,
Bouvier \& Simon 1999; Patience \& Duch\^ene 2001). The most simple and
straightforward hypothesis to explain these two facts and the finding of
a high SSF in this paper is that the most common outcome of the star formation
process is a single rather than multiple star.
Observations of dust emission and extinction of molecular cloud cores have found
that the shape of the primordial or dense core mass function is very similar to
that of the stellar IMF except that the core mass function is offset to higher
mass by a factor of 2-3 (e.g., Stanke et al. 2005, Alves, Lombardi \& Lada
2005). These observations indicate that a 1-to-1 mapping of core mass to
stellar mass, modified by a more or less constant star formation efficiency of
30-50\%, is possible, if not likely. This idea is consistent with single
star systems being most often produced once the cores undergo collapse.
The fact that stellar multiplicity is a function of stellar mass, however, may provide
important clues to the nature of the physical process of star formation. For example,
Durisen, Sterzik \& Pickett (2001) have shown that if individual protostellar cores can
further fragment and produce small N clusters, the dynamical decay of these clusters into
binary and single stars can in certain circumstances produce a binary star fraction that
declines with decreasing primary mass, similar to what is observed. However, to be
consistent with the SSF derived here and to simultaneously produce reasonable binary
component separations, such models would require N $\geq$ 5, within a region $\sim$ 300
AU in size (Sterzik \& Durisen 1998). This would correspond to a stellar surface density
($\sim$ 7.5 $\times$ $10^5$ stars pc$^{-2}$) about two orders of magnitude higher than the
peak density (7.2 $\times$ 10$^3$ stars pc$^{-2}$) measured for the rich Trapezium cluster
(Lada et al. 2004). Such ultra-dense protostellar groups have not yet been identified, but
could be revealed with high resolution infrared imaging surveys of deeply embedded
candidates. A related possibility, proposed by Kroupa (1995) and collaborators, posits that
all stars are formed in binaries in modestly dense embedded clusters. Dynamical
interactions between these systems can disrupt some binaries and modify the separations of
others. These models can produce the observed dependance of binary frequency with mass, but
at the expense of a SSF (50\%) that is too low to be consistent with that derived here.
These models could be made consistent with the high Galactic SSF by assuming more compact
configurations for the birth clusters, however it is unclear whether
the required higher cluster densities would remain consistent with observed values.
Another possibility is that binary star formation is related to the initial
angular momentum content of the primordial cores. In this case the
initial angular momentum of a protostellar core would be expected to be a
function of core mass, with low mass cores being endowed with considerably less
angular momentum than high mass cores. A systematic molecular-line survey of
cores of varying mass within a molecular cloud could test this idea. A related
possibility is that turbulence may play a role in the propensity for a core to
fragment. For example, Shu, Li \& Allen (2004) posit that the break in the
stellar IMF at 0.5 M$_\odot$\ is a result of the transition from turbulent to
thermal support of the envelopes of dense pre-collapse cloud cores. The more
massive the core, the more turbulence is required to insure its support.
Ammonia observations of dense cores in fact do suggest that massive cores are
more turbulent than low mass cores (Jijina, Myers \& Adams 1999). Perhaps
increased cloud turbulence in the more massive dense cores can also promote, in
some fashion, more efficient core fragmentation and a higher incidence of binary
star formation. In this context it would be interesting to know if the trend of
increasing stellar multiplicity with stellar mass continues to the more massive
A, B and O stars, as has been suggested in some studies (e.g., Preibisch,
Weigelt, \& Zinnecker 2001, Shatsky \& Tokovinin 2002).
Finally I note that the large fraction of single star systems in the field is
consistent with the idea that most stars could harbor planetary systems
unperturbed by binary companions and thus extra-solar planetary systems
that are characterized by architectures and stabilities similar
to that of the solar system could be quite common around M stars, provided
planetary systems can form around M stars in the first place.
\vskip -0.1in
\acknowledgments
I am indebted to August Muench for constructing the cumulative IMFs presented in
Figure 2 and many useful discussions. I thank David Latham and Bo Reipurth for
their careful reading of the paper and detailed suggestions and Kevin Luhman,
Geoff Marcy, Frank Shu and Pavel Kroupa for useful comments which
improved the paper.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,165 |
Absolute Fanny Of The Week
Posted on January 20, 2018 by Rev. Stuart Campbell
We've got a shock result for the inaugural winner of this exciting new award, readers. Remarkably it's NOT David Torrance – who wrote an arch review of a chapter of a book about Donald Trump that didn't actually exist. Good effort, Davey, but no cigar.
And it's not even notorious Lib Dem atom-wit Alex Cole-Hamilton MSP, who bleated to Ofcom when an SNP party political broadcast unrelatedly took the mickey out of a character who may (or may not) have been the selfsame Torrance.
No, the first ever Wings Absolute Fanny Of The Week is… Anas Sarwar!
In common with half of Scottish Labour, Sarwar has been raging at Paisley MP Mhairi Black for her "silence" over the closure of a 16-bed children's ward in the town, which has been replaced by an entire new state-of-the-art 256-bed children's hospital 10 minutes down the road, right next to the sparkly new Southern General.
But only Sarwar did it by enclosing a picture of the local paper's front-page splash on the story. In which there was one teeny-weeny slight problem for him.
Now, we've got no idea as to whether the Greater Glasgow And Clyde Health Board (GGCHB) made the right decision about the ward or not. Trading a dilapidated 16-bed facility for a brand spanking new 256-bed one with fantastic amenities just six miles away seems a pretty good deal to us, and the decision was made by an almost 3:1 majority of the GGCHB committee, not by the SNP or the Scottish Government. But we don't know enough about the specifics to judge.
(As with any change of location, some people will lose out because they live closer to the old one than the new one, and others will gain for the opposite reason.)
What we CAN judge, however, is that anyone who gets themselves caught indignantly shrieking about Mhairi Black's "silence" on the subject while actually including a picture of the newspaper front page in which she's anything BUT silent about it makes you the Absolute Fanny Of The Week.
Congratulations, Anas! It's not a secret any more!
Category comment, idiots, scottish politics
Never be a truth teller. | Chic Gibson – A blog of sorts
424 to "Absolute Fanny Of The Week"
I know you have limited time, Rev., but I'm sure there would be scope for:
Monday's Moron.
Tuesday's Twat.
Wednesday's Wanker.
Thursday's Tosspot
Friday's Fuckwit
And Saturday we could just make do with Chris's Cartoon.
And even God needed a rest on the seventh day.
Anas sarwar should win the "Fanny of the Week" award every week.
fillofficer says:
heh, hilarious, he must have been off school the day they got taught reading, ffs
And that little list of suggestions is before you even get past the Slab mob and onto the Effin' Tories.
Ah Anas, I forgot he was still fannying around loking for a real job.
There is the most apt quote from Dorothy L Sayers the brilliant author
" But it is the mark of all movements, however well intentioned, that their pioneers, tend, by much lashing of themselves into excitement, to lose sight of the obvious"
Just about sums it up.
CmonIndy says:
James Kelly is the clear winner.
Jeez, after Sarwar's role in the last indyref, and ever since, I've had him down as Fanny of the Decade.
But it does prove to everyone that the London branch office of Labour in Scotland, is nothing more than a unionist party harping from the sidelines.
A thoroughly deserved accolade.
Stoker says:
Another to add to the 'Anas Sarwar is a liar' file.
Not Good says:
I have some ideas, should you decide to have a 14 day Fud cycle? I can think of at least 13 to start with….
mark johnston says:
another one that only pretends to read stuff
Abulhaq says:
Mr Sarwar do at least get the grammar right, it's 'standing up for WHOM' not who.
Oh boy … Fanny of the week award. 😀
I wonder if either Scruffy Fluffy or Scruffy Fluffy Junior would like to try for next week's award. I have the perfect cause for them … the closure of D.G.R.I. (Dumfries and Galloway Royal Infirmary) in Dumfries.
Admittedly there is a NEW D.G.R.I. on the outskirts of Dumfries on the A75 but, just like the 256-bed children's hospital in Paisley, a change in physical location and a brand new building (financed by the E.U. by the way) will surely NOT be a barrier to such "esteemed" politicians as this pair of TWATS!
Gordon Hutchison says:
Well played & well deserved Mr Sarwar.
Hamish100 says:
RAH used to be under NHS Argyll and Clyde with the then Chairman Mullin (labour) overseeing the closing of the maternity hospital at Inverclyde and services at Dunoon, RAH and Vale of Leven all under labour. Labour then allowed NHS Glasgow to take over. The Board was full of labour place men with the odd tories and UNITE and UNISON place men in senior management places as they proposed to close Vale of Leven and the other hospitals services— but not with a new hospital to replace them only 6 miles away
How much Sarwar has managed to forget about labour and tories in charge of the NHS. Thankfully many of us have not.
If truth be told, he was a fanny before this week. In fact he's one of many who has made a career of being a fanny.
Graeme McCormick says:
Fanny might have asked Jackie Baillie for her thoughts on the RAH. She told the non alligned North of the River Campaign Group that if there was an A & E at Clydebank then she wasn't bothered if the RAH and the Vale of Leven closed.
She stopped convening the group when the Holyrood Election was on the horizon and announced a "Save the Vale" campaign when Nicola saved it years ago.
shiregirl says:
Yoghurt for brains. And he is an MSP…
It is beyond my comprehension that a fud like this is allowed to sit in the Scottish Parliament.
256 beds or 16…
As someone who has lived in a number of rural areas from the highlands to the borders, travel to hospital has been a given over the years. If there's a state of the art unit ten minutes away I'd say that's a pretty fair trade.
So betrayal? Hardly. Mind you, it is Mr Sarwar. (shrugs)
This is what happens when you rely too much on biased reporting silencing your opponents.
That said the vulnerability of mega hospitals to a flu outbreak does bring in to question the ideal size of a facility.
Shouldn't there be a vocative comma after 'Mr Sarwar'? 🙂
Michael Cavanagh says:
I feel it is unfair to have a fanny of the week award that offers no chance to the random, amateur fanny. Seasoned professional fannies like 'the humungous Anas' have an unfair advantage and should have a sponsored league that can allow for a Fanny league; perhaps organized through the Professional Fanny Federation (the Pee Fi Fi). This would enable people like Klever Vague, a dog walking economist, to engage in a junior league along with retired hysterical leprechauns and other such innocents.
Sarwar should check into the nearest Britnat desperation ward. He's probably BUPA on the quiet.
shiregirl @ 15:58,
Alas, there isn't any political equivalent of the driving licence, alas! It's the dodgems there in Holyrood.
Perhaps what is really beyond your – and my – comprehension is that there are still plenty of deluded folk around who can bring themselves to vote for utter dolts like Anas and Murdo and the rubbish parties which sustain and promote them.
I've never heard a "Unionist" campaign or demand "equal partner" status or even equality in the Union for Scotland. They only ever denigrate Scotland and campaign for greater Colonialist control over Scotland. They're not Unionists only colonialist and colonialist enablers. Everyone of them are fannies. Never an easy choice picking the biggest fanny from that lot.
Using their term "Unionists" gives them more credit and credibility than they deserve. They're hoose Tams and colonialist enablers, subjugating Scotland under Westminster's rule.
"Trading a dilapidated 16-bed facility"
You need to changed that. First the links says it's 40 beds anyway, second there's no picture of this "dilapidated" ward, and thirdly having been to the RAH every day for 10 days while my wife's lkife was saved and then recovered, the hospiatly is NOT dilapidated, it's a lovely hospital, though it wasn't the children's ward I was at it was Ward 32, outside the operating theatre and in HDU – none of them were in the slightest fucking dilapidated.
'Bites lip'
And nor was A&E and triage, or even the tunnel.
"You need to changed that. First the links says it's 40 beds anyway"
No, the link says there are 40 beds free on any given day at the new hospital. Be better at reading.
"thirdly having been to the RAH every day for 10 days while my wife's lkife was saved and then recovered, the hospiatly is NOT dilapidated"
Nobody said the entire hospital was dilapidated.
@yesindyref2 4:28pm
"However, bosses at NHSGGC have defended their proposals and said the 16-bed children's ward at the RAH is usually only half full.
Jamie Redfern, general manager of paediatric services, said: "On any given day, there are eight patients in the RAH ward and, at the [new] children's hospital, there are 40 free beds."
Good to hear your wife is doing well.
And Sarwar's big mistake?
He didn't bother to read beyond the headline.
He should listen to The Rev more often!
Well PROVE the children's ward is dilpaidated. Get a photo and a link thaty proves it.
And yes my apologies about HALF of my posting, yes it is a 16 bed ward in the article, the 40 beds are the free ones at the children's ward.
It really is time that lying politicians were held to account,legally. Caught 3 times and your out. That would empty the Unionist benches.
@Andy-B says: 20 January, 2018 at 2:57 pm:
" … it does prove to everyone that the London branch office of Labour in Scotland, is nothing more than a unionist party harping from the sidelines."
Harping? That lot couldn't even twang a Jaws Harp.
Wullie B says:
There will be more of this soon Rev, Portree A&E is closing at the behest of NHS Highland and a new hospital is being built in Broadford, yet the SNP are getting slated for it, both Ian Blackford and Kate Forbes are fighting hard to stop the closure to the dept, and Councillor Ronald MacDonald of Bitter Together fameas a economist who was brought out of some dark cupboard every time the BEEB needed a soundbite in 2014, feck knows how he conned his way into being elected up here, but that will def be coming out soon I would say, especially with most services being performed at Raigmore 120 miles away.
He already has previous in the locally published Labour News, I mean West Highland Free Press, seems to be the new Brian WIlson up here in Wilsons old paper
@Spock says: 20 January, 2018 at 3:50 pm:
As the old advert on STV used to say, "He comes from a long line of Fannies"
More as above cant edit post I dont think but you will remember this fud
https://www.theguardian.com/politics/2014/aug/14/independent-scotland-economy-crash-sterling-ronald-macdonald
https://www.facebook.com/CllrRonaldMacDonald/
But he will def push the SNPbad for folks who stay in North Skye, doesnt seem to realise that Broadford, the ideal choice covers quite a bit of the mainland to
"10 minutes down the road"
See discussion on the Herald from a solid pro-indy poster, and I can back that up too. Even at the best of times you'd have to have wings to get there in 10 minutes from RAH, and at rush hour it could take up to an hour from Paisley and around. Traffic can be crawling – I've driven it enough times in both directions.
Don't slip your standards to those of the unionist media.
Facts facts facts.
It has been a remarkable week of idiocy by the Better Together Yoons, even by their standards.
Next week Johann Lamont shares tattie neeps and haggis with Liar Carmichael and Murdo The Queen's Eleven 'protecting children is an SNP Vanity Project' Fraser. A 'Robbie' Burns fund raiser for the Filthy Rich who are determined to crush Scotland and starve the people into submission.
Well done, you rabid socialist, JoLo.
I pre nomionate her for Idiot of the Week.
Is it 'something for nothing' gravy train beano, or is it a millionaire's do, £1000 a plate?
They KNOW in their heart of hearts that their days, and their corrupt, immoral illegal Union days, are numbered.
Boy will we party like there's no tomorrow when we kick this mob of freeloading buffoons out of office.
Eh don't know the place personally , but apparently it was built 1988 – so 30 years from first build possibly improvements and extensions since then , I don't know.
I don't doubt the comment of hardly delapitated from someone who has actually been on the premises , but 30 years however it's presented and however well it's been maintained is not a short time and possibly nearing the end of its useful life , just a thought .
I suppose that leaves the airhead of the week award to a Mr Cole-Hamilton. And like the Jules Rimet Trophy winning it three times allows him to keep the award in perpetuity.
Eh as I said last week.
Are S I U operating today , under another name ?
In fact some pretty dubious comments recently from apparently avid Independence supporters , Either that or strong liquor has been partaken of , and sense has gone out the window .
wull2 says:
I think you should have a Wings Poll every week on a Saturday for … Fanny of the week award.
Nominations to be in for Friday evening, only one entry per person, there is not enough Days in the week.
The article in the Gazette (links in text) states that the Royal Sick Children's 256 bed children's facility would struggle to cope with the estimated 8,000 extra patients that would be using it each year if the 16 bed children's ward at RAH closes.
8,000 patients per year is 22 patients per day every day. How does a 16 bed ward cope with that given that the average length of stay in hospital for elective procedures is just under 4 days and for emergency admissions just under 7 days?
@Robert Graham
I don't know the children's ward there. If it's in the main building it'll be fine, but if there's some annexe I don't know about it MIGHT be delapidated. It would need a photo or some sort of proof about that. It's Inverclyde our kids were in, a few times 🙂 Fine hospital too.
It's not neccessary to the article ("dilapidated"), and I've been caught out on that (I posted a commment here about one recently where I misread something – like I did that 40 bed thing). It can destroy the whole argument, even if it shoudln't.
We need to be hard with each other to get accuracy, but with the other side, patient, nice, friendly, determined, – and correct. So that lurkers know where the right of it is.
This is just the latest in a process which is going to go on and on for some time. The medics are persuaded and I tend to agree with them. Excellence is limited, experience is invaluable and having lots of little specialist services dotted about is not good for patients.
Some of those 8 kids in Paisley might well benefit from the experienced specialists in the larger hospital as well as services like scanners etc.
The stats don't lie, you are more likely to be properly diagnosed and properly and appropriately treated at a large teaching hospital than a small local one.
There is also the inevitable problem with recruiting enough specialist staff to staff all these multiple centres. So you get non specialists and inexperienced juniors and lots of turnover in smaller centres.
In almost all these cases the decision will be made on the recommendation of the medics who have all the relevant evidence on their side.
Do you want your kid treated by the best in the best facilities or just hopefully in an alright facility?
John Alexander Ferguson says:
I think it should be a rule, that the Presiding Officer insists that all information given by MSPs be factually accurate and true. Any MSP found wanting, be removed from the chamber and disciplined. Sick of the weak efforts of our present PO.
Any body having problems with the site today ie posting comments
@yesindyref2 says:20 January, 2018 at 5:02 pm:
"at rush hour it could take up to an hour from Paisley and around. Traffic can be crawling – I've driven it enough times in both directions."
Yeah, yesindyref2, but do you have a siren and lots of flashing lights?
Is anus on the wordcatcher list 4 times ave posted today and not one post appeared an two test test posts .
Dear ronnie Anderson,
I have had problems for months, perhaps what you are posting is spoiling there game. So they just remove it.
The staff want it closed for better facilities. Tgere must Dr surgery in the area. An ambulance could get there in 10mins. Others might take longer. Just as in every other healthcare facilities. People have to travel, especially if it is not an urgency. Everybody can't have facilities across the road. How many people would use a 16 bed facilities instead of a 256 facility. With a better waiting areas and better cafe facility etc.
OT … An interesting opinion piece in the Guardian.
" Britain's tired old economy isn't strong enough for Brexit "
http://archive.is/l5wAV
The message I took from it is that the UK economy has never really moved from it's post imperial state into a modern balanced and stable economy.
Accordingly, Scotland needs to disentangle itself from UKOK and move to an economy with a better mix, especially more manufacturing.
Anas is weird. Why would a millionaire dentist want to get into faux lefty SLab politics anyway. Its been a decade now of trying to clear up the SLab era of unspeakable wastes of money for starters.
The English are currently going nuts right the noo over their £200_bn PFI bill, and they will never actually own the hospitals etc that they're paying £200+bn PFI business/golden goose.
Who was the great PFI king, Crash Gordon Brown.
Every time I see Anas there, I always hear, "Who wants to be a slave wage paying millionaire, Anas!"
http://www.youtube.com/watch?v=YG6UllZwj9c
The RAH may be fine for many, but children come from farther afield and they will be perfectly happy with the new hospital.
People think a ward looks fine for their needs but looked at by staff as past its prime!
Because we live in a world of fast paced change and equipment is being updated sometimes faster than you can learn how to use it, space for patients changes too!
Modern hospitals are more geared up to prevent the mass infections of old…small cottage hospitals for eg can be closed because of flu whereas larger ones can isolate easily.
We all have to understand that no hospital is closed these days by a whim of a politician but by careful medical advice.
Sure, but it's not 10 minutes down the road for parents taking their kids, going home, and visiting a couple of times a day for a few days or even weeks. And going through the St James even an ambulance would be slowed down.
Yes, in the highlands it could be hours travel, from the islands where Glasgow is used for specialist stuff it can be a day each way.
Still doesn't mean the "10 minutes down the road" is accurate, it's not. From a poster in the H: "having worked in Paisley for 35 years I can also tell you without fear of contradiction that while St James' Interchange is not the middle of Glasgow, at rush hour, or if there are roadworks, it can be chaotic"
From another fierce Indy supporter: "According to Google Maps the driving time between RAH and RHC is 17 mins., alternative route 18 mins, half for what is being claimed. By bus it is 46 min".
Google doesn't do rush hour at the St James, or anywhere come to that.
They do not tell us on the TV, the papers, it might be their next move, night be to remove certain posts that inform us.
@Yesindyref2,
Are you making the assumption that most patients live closer to the old hospital than the new. The time from the old hospital to the new may therefore be irrelevant.
The moanin faced cunts want to come down to Ayrshire and get a bus from Dalmellington or Drongan to Crosshouse hospital to see their sick bairns let's just say it isn't ten minutes down the road.
Really makes me sick politicising the NHS over a decision taken by the NH fucking S board. total and utter fanny
It seems a reasonable assumption. If they were nearer the new one I guess they'd go there anyway. If referred via a GP they seem mostly to have a reasonable choice where they refer people, and an ambluance would presumably go to the nearest or most approptiate (or both).
RAH would have quite a large catchment area to its west and south, and a bit to the east and north compared with RHC.
But my point is about accuracy, not whether it's the right decision or not. I've mixed feelings, but hey, budgets are budgets. It's what got me thinking: "How could we get £69 million a year to save RAH Children's ward (and presumably other cuts)", and that's how I got into that palace in Paris that houses one British ambassador – and the extravagance of the FCO which iScotland could do without.
Do you trust BBC Scotland?
I see a poll on this as I don't do twitter cant vote would be a resounding NO from me.
Doug McGregor says:
Look on the bright side , at least he's not inside your mouth trying to mend your teeth.
Thanks for that, just saw your post.
Yes, she's doing fine, just arranging the 6 week clinic which could be at RAH or ironically VOL, same gynaecologist who with the senior consultant / surgeon, saved her life.
@Scott 6:01pm
Looks like over 4000 folk have voted. 96% say No.
https://twitter.com/neil1pat/status/953406765151981569
colin alexander says:
I believe it's been the policy to centralise NHS services to save money.
In the West End of Glasgow, Blawarthill shut, Knightswood hospital shut, Drumchapel H shut, Yorkhill children's shut, the Western Infirmary shut etc.
Gartnavel is still open.
Most services are now transferred to "Betty's" or the Queen Elizabeth University Hospital in Govan. Not the easiest place to get to for much of the North of Glasgow.
Many Westenders continue to complain about the lack of an A&E Unit since the one at the Western has closed. I think they have a good point.
If there is a problem with the Clyde Tunnel, it could add quite a bit to journey times, especially in an emergency. The difference could be life or death.
Also, I would just like to praise Mhairi Black for speaking out and criticising decisions, not just toeing the party line. We need more politicians like her who will say what they really think.
Hud the bus this surely qualifies fur ah fanny .
https://www.facebook.com/NowThisPolitics/videos/1881091321922349/?hc_ref=ARRKbixyJIK7LR_0Ao9XhOMX_XbiLTpTyjvp2PRTX9izjJ_k6-BcsGfWReZ5Bj2_Yt8
Thats a change to far lol.
Hospital closures are always a tricky subject at the best of times. Some people will inevitably be disadvantaged by any move, and those are the ones from whom you will hear loud protests. (As is their right.) You won't hear a toot though from or about those whose situation has improved.
There has also been a latterday policy of centralisation that is entirely medical-driven, from a well-intentioned wish to maintain a sufficient depth of expertise in any one location.
This does somewhat ignore the wider social context, though. For example, the new Southern here in Glasgow is a fine institution, but after withdrawal of the excellent but short-lived direct bus services from the North side, it is rather expensive now (or alternatively bothersome and time-consuming) to make a visit from large parts of the city. That needs to be changed.
Furthermore, on one occasion I was attempting a visit on a very rainy late afternoon during rush hour, and the Govan Road outside was completely chock-a-bloc with traffic, and nothing much was moving anywhere. None of which was the medics' fault, but there you are. (Just don't have a heart attack in Maryhill at 5:30pm on a wet Friday night, peeps!)
However, none of that has anything directly to do with the politics of independence. It is the "weaponisation" of such issues by the BritNat Party, the BBC and the hyenas of the dead tree scrolls that has made any kind of rational analysis and response to such matters nigh-on impossible.
Damn them. The quicker independence is settled so we can all get back to a normal existence the better.
yesindy2ref i have close relatives that stay close to RAH , I have travelled that route countless times and would say it takes me on average ten minutes . Someone posted a photo of the ward on twitter last night, if its indeed the ward in question it looked as if it could do with a lick of paint.
I think the Rev's description of the journey the journey time is correct as I have done it hundreds of times.
When you talk to somebody and they ask you how long a journey takes you dont say well it can take over an hour because a car may break down at the interchange etc etc do you?
Mare shite from Colin A.
Colin A, public health care in the UK is actually a terrific way to compare and contrast how each major UK party is doing running the NHS, SNP Scotland, tories England, Labour Wales.
As we all know, beeb Scotland gimps rage at us all the time that the Scottish NHS is truly terrible and we would all be so wise and clever to let Anas and chums run it all, again.
Would we though, even starting at free prescriptions for Scots.
http://www.independent.co.uk/news/health/christmas-ambulance-delays-dead-serious-harm-east-of-england-winter-nhs-leak-a8168001.html
This could get exciting, a close run thing, with so many contenders.
Who's going to get the keep the trophy after winning it three times?
One_Scot says:
Said this on the last post, but I think it is worth saying twice,
'You know what, I have all the time in the world for unionist supporters making valid points and engaging in debate, but unionists pretending to be Independence supporters, well they are just the scummiest lowest of the low.
The quicker independence is settled so we can all get back to a normal existence the better.
YES …. normal [ like every other small country in Northern Europe ] existence ASAP!
heedtracker
I refer you to the post by Robert J. Sutherland.
He clearly knows what he's talking about, as do I. I'm sure anyone from the North or Westend of Glasgow will agree with our posts.
Whereas, where are you Heedtracker? North or West End of Glasgow? No.
You just argue the toss for the sake of it.
colin alexander @ 18:38,
Here you are at your divisive little game again, CoCo. Heedy and I don't disagree as you assert. As with the rest of us, we're not clones and may well differ on something or other, but not on anything significant.
Which is more than I can say about me and you, wormtongue.
Precisely!
Southside Glasgow, your arch enemy Nic Sturgeon's my MSP. She was here this morning but I was too hungover.
The local old Victoria's gone, demolished almost completely for flats, high density quick buck flats.
This was built lately behind, beeb gimp network have still to drive by smear it though. Soon no doubt.
http://www.nhsggc.org.uk/patients-and-visitors/main-hospital-sites/new-victoria-hospital/
Precisely, you don't live in the North side of Glasgow.
As you mention Nicola Sturgeon: http://archive.is/2KZ6d
It's the Express' take on the ward closure. I've archived it, as I wouldn't want people clicking on the Express website and giving them ad revenue.
Sorry it's not the Guardian.
@Heed
In a previous life I had a contract with the GGHB and regularly visited them all (not medical staff mostly), very nice people, I loved the work. The people at the Vic were very relaxed 🙂
But the squinty bridge is never shut, so if the tunnel is, that's what say 5 minute divert past the Hydro delay Colin A?
There's a pretty nifty helipad on the roof of the new QE too.
The people at the Vic were very relaxed
Its almost all gone now though. Really miss it too. We do not appreciate our heritage in Scotland today, sometimes. Its all about the bucks.
Robert J. Sutherland
Wormtongue, haha. Cheers Robert, I like it. You know how to flatter me.
Privatisation came in and I need say no more.
Robert J. Sutherland re policy of centralisation to maintain a level of expertise, i am living proof that the policy works , during June and July 2015 I was in the new southern general, i spent over a week in critical care several days of which was in an induced coma. I have been told by a relative who is a senior healthcare professional that if I was anywhere else I would not have made it.
frogesque says:
It's called a splat. It's what a fly does as its head goes up its airse when it hits the windscreen.
Too much whataboutery. Local mat flats are ideal when everything goes smoothly but when it goes wrong, and can go wrong spectacularly and fast, you want the all singing all dancing sooper duper kit and caboodle with highly trained staff to use it.!
GGS is 9 months old, brain hemorrhage and heart stopped twice on the day he was born 10 weeks premium. Wasn't expected to last the night.
Several infections, major bowel ops and he's a bright wee button, thriving, trying to walk and making all the proper goo gah noises he should. Still has to have a heart valve repaired when he's a bit bigger.
Don't anyone ever dare run down our SNHS or the good folk who work in it. Hospitals get old, priorities and procedures change. Specialities change all the time and hard decisions have to be made.
I know absolutely that our latest GGS would not be alive if he hadn't received the best of care.
End of *
Now fuck off!
@Lenny Hartley / @RJS
Centralisation or not is an issue that shouldn't be party political. Sadly it is.
Premium should have been prem as in premature, bloody autocorrect!
@frogesque
Before that contract at the GGHB I'd been all private sector, and the accepted attitude was private sector good people, public sector just in it for the benefits and the pension, lazy, incompetent, you name it. I took it because I was in Edinburgh long long commute back home, hardly got to see the baby / weans (can't remember the year) and it was a lot nearer. What an eye-opener it was working there and I worked with a lot of people.
It only needs one word to describe them: dedicated.
http://www.isdscotland.org/Health-Topics/Emergency-Care/
For those who want to check waiting time performance etc
See also: http://www.nhsperforms.scot
Talking of respected Scottish journalists, remind which ones were tied up with the obscenity of the Brian Spanner scandal?
yesindyref2 @ 19:43,
My view also, as already indicated.
In fact I got an inside view on this some years back (centralisation is by no means a new issue) from someone I knew from when we were both on the local school board, who was both a medical professional and also a LibDem party activist, and very much in support of the clinical advantages.
Like other issues we have seen, it's only post-IR1 that the Red/Blue/Orange Tories and their faithful friends in the media have poisoned all rational debate on this. Largely due to their manifest inadequacies as politicians, it must be added.
And in so doing they have also attempted to airbrush away all their inconvenient previous views and public statements.
A blag which they can't get away with, thanks to sites like this.
shug says:
I have a sense that unionists are becoming more polarized in their view. They agree with the propaganda they are being served up by the BBC and it reinforces their belief hence this type of headline and fake news will be very effective.
There is part of me thinks the SNP should serve up Westminster cuts to the full
The yes side need to break the media impasse and this means making Wings mainstream
@yesindyref2 7.59
Actually the wee fella was born at the new Simpsons, Little France Edinburgh, and the bowel operated on at The Sick Kids Edn, he will get his heart op in Glasgow though.
It's a tribute to the organisation and dedication of health care teams. They are bloody fantastic!
Here's the performance figures for the A&E at Betty's in Govan:
http://www.nhsperforms.scot/hospital-data/indicator-hospital?hospitalid=59&indicatorid=2
The latest figure for meeting the A&E 4 hour target:
7 January 2018 – 67.6%,
the previous figure for 31 December 2017 was
I wish the SNP would just stop building hospitals and clinics and updating services all the time all this modernisation's too much we need to get back to basics when we just went to the phone box put our four pennies in dialled and waited to be connected then pressed button A to talk then went to the bus stop to wait for the bus, one every half hour before you changed buses to go where you wanted to go then went into the freshly updated Labour refurbished hospital with the new smell of refurbishment paint because that's how Labour used to refurbish hospitals, you knew where you were with Labour, by the time you'd done all that you were satisfied you'd done all you could then you died aged 57 and had had a good life everybody said at your funeral, and your kids had died from diptheria anyway so what was the point of your living anyway
Going to be some tough competition for this new award.
There are a lot if practical and financial reasons that this decision makes sense, whether parents think/feel hospitals should be near to them.
By the accounts I read on FB, the children's ward at RAH was very outdated/lacking. Children were exposed to adult patients etc. First hand accounts of the new facility include individual bedrooms, pull out beds for parents, TVs, then its all modern medical equipment. Of course the big plus is that world renowned paediatric staff from old Yorkhill are on hand. Call me daft, but that last one is pretty key.
We are a small country. We inevitably work around the centres of population that attract the brightest and best doctors to their hospitals, that coincidentally have accommodation to buy by those doctors.
Travel to a centre of medical excellence happens in heart, cancer etc. and as far as im aware, if you are asked to attend elsewhere, you get travel expenses.
In terms of travel time, half of Paisley has no need to get through the St James interchange, they go on at the airport, and it would take you 15-20 mins in a car. Certainly an ambulance or transport will have no real issue. Public transport is a bit more hassle, but it is for most places.
Until this country has an oil wealth fund of £1 trillion like Norway, this is the best decision for our pocket money.
I grew up in Paisley during the 70s/80s, and it was known as The Butcher's shop. A bit cruel, but it is certainly not at the cutting edge. I was visiting a sick female adult there about 3 months ago, and the best thing I can say about it is it looks 'tired'. These comments are not in any way reflective of the staff, because I know all staff do their best under very difficult circs.
Dr Jim I was just thinking the SNP could build more hospitals & schools & housing if they went back to gerry building Nissen Huts .
Four Pennies tae phone lol , did ye no learn tae tap ( morse code ) worked every time .
Dr Jim says
your kids had died from diptheria anyway so what was the point of your living anyway
My great grandparents lost four of their five children in the 1874 scarlet fever epidemic. Aged 0-7. They went on to have ten in total, including my grandfather. They lived in crap accommodation around where the M8/M77 now split.
Apparently this epidemic raged globally, yet history books never mention such event. Child death was a normal occurrence than.
Dr Jim ah forgot tae mention the Nissen huts wie nae windies would be Sanatoriums fresh air for the Tuberculosis sufferers .
That's what Law hospital was originally, old barracks! It's where I trained and really loved it but, it wouldnae be suitable in these high tech days. 😀
I myself had scarlet fever, measles and chicken pox like most of the kids then, and polio was still a thing when they had to get around wearing leg braces
Rickets was not unknown, pneumonia was every winter for many
and yet here we are with state of the art hospitals, paramedics, helicopters, scanning machines that can see amazing amounts of stuff never seen before allowing for medical interventions and treatments so quick people don't even realise their lives are being saved on a daily basis they just expect star trek medicine because they've seen it on the telly
And all of it free for an extra ten minutes on a bus to visit a facility that has Television, reading, computers, games for the kids and the confidence that somebody can help you with whatever your problem might be
OK I might sound like and old duffer to the youngsters but I remember and was there when Labour used any extra cash to line their own pockets with wee holidays to Cuba (for cultural exchanges) while Scotlands homes were damp dark and and full of closes with lead paint but as they like to keep reminding us
What a sense of humour and community we all had
And Billy Connolly was funny when he made his living from it and good luck to him but he was and is also a total dickhead because he supports the regime that made it that way
End of History reminder for the young who might think they've got troubles
Cag-does-thinking says:
Actually accolades for the campaign to save children's services should go to the hilarious independent councillor who sticks it to Labour on a regular basis in Paisley. He may be an outsider but he is no dummy when it comes to looking at how Labour works.
It is however a sore point about the children's ward at RAH. Labour conjoined Inverclyde Dunbartonshire and Renfrewshire into Greater Glasgow which they are most definately not.
The new "state of the art|" hospital has struggled from the beginning to accomodate both patients and visitors. It's essentially too big. If you have a child with major illness it's a nightmare to park at the new hospital. You have to go there anyway if the illness is serious enough.
What we are losing is a local facility accessible by transport links which are local. The important thing is that people in Paisley, Mhairi Black and all, recognise this. We are being steamrollered into being part of Glasgow for health services and that is always a bad idea.
Ha ha. What a fanny.
mogabee The Law/ Stonehouse/ Belvidere /Hairmiers if the Scottish Gov did return to Nissen Hospitals it would give these moaners something to moan about . In the last 5 years I've been in more Glasgow Hospitals than my local Monklands & more recently the Jubilee in Clydebank if that where the Specialist treatment is the onus is on the Patient to ether go or suffer .
In respect of RAH children's ward I would rather my child was in the best place of care under full Pediatric staff .
I had two of my sons in Yorkhill when younger ( dif times ) I traveled by train from Airdrie to Partick every night after work with no thought of the time or journey , that's what Parents do . My Wife & myself were grateful to the Staff for their dedication .
Mmm, Yorkhill, parties, nurses, Overflow and the Stirling Castle was it? I think I was born in nissen huts which was Ayr Hospital (Heathfield) back then but not sure if my leg was being pulled. Strangely enough I don't remember.
….not fair. Unionists have an unfair advantage in this contest!
I guess anyway we're pretty spoilt in the Central Belt or even on the finges of it. There must be people reading these comments shaking their heads in wonder about discussions about a few minutes either way. Everybody should have an A&E to go to, and they do use phones to get opinions.
yesindyref2,
How much do you spend on The National every year?
How do you compare the standards there with the standards of articles on this site?
How many comments do you post on The National every week?
yesindyref2 (3rd March 2017 – "Pushing the accelerator"):
"To pick up what Rev says, all I gave was £5, and posted as much on the first thread.
Am I ashamed it's so little? Why should I be? It's all I can afford at the moment."
I spent almost a decade working on the Paisley Daily Express, back when it was actually based in New Street, Paisley.
We had a great team then, under a charismatic Editor, who was a born and bred Buddie, who loved the town and the paper.
Today, the PDE sells less than half the copies it did then, oh, an the main editorial office is in GLASGOW. So, the PDE really has a nerve getting aerated about the closure of Ward 15 – the new unit is a damned sight closer to Paisley than the PDE is.
Cag-does-thinking @ 21:18,
You touch on the heart of it here. It isn't really a medical issue pure-and-simple at all. If anything the medical case supports centralisation as the medics prefer, and people like Lenny Hendry upthread can personally attest.
Rather it's an issue of adequate resourcing for ancillary support, not least accessible and affordable public transport. Which is a much wider issue than a purely medical one, granted, but it has a particular sensitivity for this issue.
In my case, for example, I don't have a private car, so travel to that big central facility by taxi takes something like 30mins at best, and costs £20 round trip (which isn't refundable for a visitor). The only alternative is the bus, which is cheaper, but takes far onger and involves changeovers.
Initially there was a great and very convenient direct bus service, but the private operator discontinued it after a year because it was severely underutilised. (All those people taking to their cars instead.) It ought to have been maintained by subsidy as a public good, but that's austerity for you. (During the time of the previous Labour administration, as it happens, but I'm not trying to political point-score here.)
The conclusion: public support for clinically-necessary improvements in medical facilities may depend critically on apparently tangential social issues such as ease of access.
And unfortunately for us all, the continued high dependence on private car ownership and the absolutely antiquated bus system operating in much of Scotland (not helped by the idiotic privatisations of the Thatcher/Major years) is having an impact on otherwise unrelated matters.
Meanwhile, the best that Anas+NorthBritLab can muster is inane sound-biting like this.
We desperately need a medical treatment for parasites. The human kind.
Sarwar deserves this title permanently because he makes his millions partly by not paying the living wage 'because he does not have to' that's socialism in action for you – labour style. And he thought he could be leader of even a red tory party.
A longish post but covers some of the points raised by other posters above
1. Hospital provision in Glasgow:
http://www.nhsggc.org.uk/patients-and-visitors/main-hospital-sites/new-victoria-hospital/new-victoria-hospital/
Click on the Servicesand Outpatients link to see what is offered at the hospital. There are 21 Services including Minor Injuries Unit, Day hospital, Day Surgery and out of hours GP service.
All in a brand new facility.
New Stobhill Hospital
http://www.nhsggc.org.uk/patients-and-visitors/main-hospital-sites/stobhill-campus/new-stobhill-hospital/#
Again click on the Services and outpatients link to see what is offered at the hospital which serves the North of Glasgow. There are 28 services in all including Minor injuries unit etc same as Victoria.
Again in a brand new facility.
So provision of local services particularly some of the most commonly used diagnostic and outpatient services as well as Day Surgery which means people do not have to travel far to access them. The provision of modern facilities also helps to cut infections and improve the throughput of patients in an efficient manner. Access to a minor injuries facilities helps to take the strain off larger A&E departments.
2. Provision of Services in general
Last year Shona Robison outlined the future direction of service provision in the NHS. This will be based on centres of excellence dealing with what are described as life changing procedures – heart OPS, orthopedics (hip replacement etc) and probably key hole surgery. The model for this type of centre would be the likes of the Golden Jubilee Hospital. Local hospitals would provide day surgery etc similar in fact to the new Victoria and Stobhill facilities.
This is driven by the recognition that the changes in medical delivery and the need for medical professionals -doctors, nurses and midwives – to maintain their skills requires a certain level of throughput of patients in order to maintain their skill levels.
Do you want your key-hole surgery carried out by someone who does 2-3 such procedures per month in your local hospital or someone who does 2 per day in a centre of excellence? That is what it boils down to across so many disciplines in medicine.
There is also staffing issues. A growing problem as we lose EU doctors and nurses. Of relevance to Paediatrics in RAH and in Livingston is a workforce report published in 2017 by the Royal College of Pediatrics and Child Health. You can read it here by following the links in the page. It is short – 6 pages -but worth reading
https://www.rcpch.ac.uk/workforce
@ Dr Jim
I always feel like some kind of cultural sell out when I confess I NEVER found Bill Connelly funny at any point in his career. The acting role I like him best is the Uncle who kept poisonous creatures, but the kids like him. The Lemony Snicket movie.
The rest is, imo, guff. As a young man, doing his big banana feet etc., talking about Govan, I just didnt find it funny, and I wanted to.
When I was about 11, we lived on 3rd floor of a tenement in West end of Paisley. About 1am one night, I felt sick, and my face felt numb. I felt I had no choice but to go wake my mother, so shuffled through to her bedroom, followed by my young sister, saying I felt funny.
My mother was raging at being woken, and was cursing as she fumbled for the switch. When the big light was turned on, my mother and sister started screaming – at me.
My face had blown up to more than twice its size, and my eyes were almost shut. There was a lot of omg, and my mother put on her coat over her nightdress, to run up to a phone box that was two streets away.
An indian doctor came, who was very kind to me, because my mother apologised to him for me being a nuisance.
The doctor snapped at her that I could have died if the jag and pill he gave me hadnt slowed the swelling. He was prepared to do a tracheotomy, and cut a hole into my windpipe.
We never found out what caused it, but everyone told me I looked like a hideous doll for a week. It was a severe allergic reaction to something.
Ah, those were the days. Im glad it wasnt raining that night, or you might be a cybernat short today :).
Dundee built a new hospital in 1974, Ninewells and it is about as far West as you can go before leaving the city.
The main hospital until then had been Dundee Royal Infirmary which first opened in 1798. A big grand imposing building that continued to be used after Ninewells opened for a further 24 years before finally closing in 1998.
I've had reason to use Ninewells on more than one occasion since 1998 and have always been impressed with the treatment I received. In fact when I last looked Ninewells has consistently been very close to the top of the A&E figures and is one of Scotland's best performing hospitals.
OK it is now over 40 years old but I doubt you would notice, I'd much rather be there than the old DRI.
Things change and often for the better, it's the making of difficult decisions that are not so easy but I'll tell you what even if initially you might disagree with a decision made on your behalf. Often it can turn out to be the right one.
Me personally? I'm all for modernising Scotland in all ways possible, out with the decrepit and in with the new if it will make peoples lives better.
By the way can anyone tell me where that £69 million figure came from.
Aye yer right Ronnie.
Mind you, Monklands frae Wishaw was incredibly hard to get to without a car, as I had one of my boys in the isolation unit. And I recently had an op in the Jubilee!!
Just shows how things have changed and some haven't changed much at all. 😀
Slab probably dodged a bullet when they didn't elect Anas as leader.
They may have sat on a hand grenade minus pin with the one they got but they probably dodged a bullet. 🙂
@yesindyref2
Yes, some folks live a few hours drive from a full A&E. That's why so much of Scotland is covered by planes and helicopters, both Scottish Government and chastity. Hospitals in Scotland with full A&E 24/7 facilities ….
(I think this is the current situation)
University Hospital Ayr
University Hospital Crosshouse
Borders General Hospital
Dumfries & Galloway Royal Infirmary
Galloway Community Hospital
Forth Valley
Forth Valley Royal Hospital
Aberdeen Royal Infirmary
Dr Gray's Hospital
Royal Aberdeen Children's Hospital
Greater Glasgow & Clyde
Glasgow Royal Infirmary
Inverclyde Royal Hospital
Royal Alexandra Hospital
RHSC Glasgow
Queen Elizabeth University Hospital
Belford Hospital
Caithness General Hospital
Lorn & Islands District General Hospital
Raigmore Hospital
Hairmyres Hospital
Monklands Hospital
Wishaw General Hospital
RHSC Edinburgh
Royal Infirmary of Edinburgh
St John's Hospital at Howden
Balfour Hospital
Gilbert Bain Hospital
Ninewells Hospital
Perth Royal Infirmary
Western Isles Hospital
Chastity!? Charity of course.
mark robertson says:
fanny of the week more like fud of the century
Ah well, you know what they say: "abstention is better than cure".
Or something…?! =grin=
some folks live a few hours drive from a full A&E
Of course I'm thinking of the mainland only here. All across the Hebrides any serious emergency and it has to be an air ambulance!
@mark roberston –
Fud of the Century is different league altogether, surely!? It's Mundell-level fanniness, rarely reached by mere mortals.
@olin alexander says: 20 January, 2018 at 7:14 pm:
Such illogical thinking there Colin. Perhaps we should situate the Hospital equidistant from north and south Glasgow, just to be fair to both area covered by the new hospital, and a site in east or west Glasgow may well fit that bill.
Oh! Wait! Then we will have the voters of either east or west Glasgow complaining unless it is also equidistant for them as well.
The point is that no matter where you site a new hospital in any given area, unless that area boundaries are absolutely circular, there will be people less conveniently situated than others.
Oh! Wait up! Some will still be closer to the hospital than others and complaining.
Did you understand the logic and the point of this comment, Colin? It really isn't difficult to understand.
As to bringing Nicola Sturgeon into the matter – any excuse will do – would that also be the case Colin?
@Colin alexander says: 20 January, 2018 at 7:14 pm:
Totally OT, but showing what small independent countries can achieve.
New Zealand planning a space launch tomorrow!
https://www.rocketlabusa.com
Hi Thepnr at 10:04 pm.
You typed,
"In fact when I last looked Ninewells has consistently been very close to the top of the A&E figures and is one of Scotland's best performing hospitals.
I've been working at Ninewells (in the Medical School) since 2001. What struck me, in my first few months there, was that there is always maintenance and refurbishment work going on – and new builds, like the Clinical Research Centre, the Jackie Wood Cancer Centre, the (newish) Maternal & Child Health Sciences building, at the far west of the campus, which physically connects with the Childrens' Hospital, based around Ward 29 and children's outpatients areas. I don't think a week goes past without work going on somwhere on the Ninewells Campus.
Apart from maintenance work going on within the actual hospital, at the moment a new building is nearing completion, next to the Jackie Wood Cancer Centre. This new building will house a manufacturing facility producing generic drugs for all NHS Scotland.
Yes, the original (expansive) building is over 40 years old but it has been expanded over the years and the original building is still in good shape.
@Brian Doonthetoon
I have a certain bias towards Ninewells, I too worked there for 6 months back in 1987 in their engineering department while a student at what is now Abertay University.
Most people would not believe the amount of work that goes on behind the scenes to keep a hospital running. Even then planned maintenance cards printed out everyday for all the essential equipment that few ever see.
The maintenance for every pump, generator, boiler (and they were big) etc. has it's own maintenance scheme. Then the small things like the radiators, leaky taps, kitchen equipment. All have to be maintained and it's not a small job to keep everything running smoothly.
They manage it though and the proof is the comparison with Scotland and the other NHS's elsewhere. Well done to all NHS Scotland staff and not just the front-line doctors and nurses.
New modern hospitals are much cleaner too. Mind the outbreaks of the MRSA superbugs. No idea how much of that was down to our noble UKOK hackdom bullshit but I know they're designed now for much more effectively cleaning.
Interesting list, was looking for Islay and that's what it is at Bowmore "Community Casualty Unit". I went up and had a look one time I was over there just for interest. I'm guessing they take people there before ferry / air ambulance.
Yes, the air ambulances. See them sometimes where I am on the mainland, or at least I guess that's what they are.
There's the hand dispensers too, we used them all the time at RAH. I did manage to use the wrong one by a sink, thought it was a bit sticky all the same. Most visitors seemed to use them, it's education too so in that way the media done good for a change.
Talking about air ambulances, I often amn't able to book a ferry not knowing if I'll get that far sometimes, or what time, so end up in the wait queue – every time at Mallaig for Armadale. They really work hard to get you on, one time I was 8th there and just got on, a couple of times diagaonally in the ramp and one time they had to get an officer down to check it was safe.
Anyway to get to the point of a shaggy dog story, I was told a long time ago they keep spaces for an ambulance and they are rare, and I think it was about 3 car or small van spaces worth, a reason you usually get on though they can't properly tell you that just in case you get angry you don't get on if you don't, though a little blether don't go amiss.
You do see the ambulance transfer sometimes, the mainland ambulance transferring patient to the island one and going back on the ferry.
Not a lot of people know that innit!
Robert Peffers
Good evening Robert, how's it going?
The best thing about Scotland is the NHS.
When I was campaigning for YES, I highlighted the work of Professor Allyson Pollock:
https://www.allysonpollock.com/?p=1872
One of my themes was that the Scottish NHS is a health care system the world envies. That remaining part of this UK Union poses a threat to our NHS.
In my opinion, the best way of protecting the Scottish NHS is by Scotland becoming an independent country.
That's what I said in 2014 and I still believe that in 2018.
So, I laugh when people like yourself imagine I'm a Unionist just because I criticise the SNP on occasion.
"I criticise the SNP on occasion." Zoomer LOL
Its all you ever do Colin A, you and Rock. You are Nic Sturgeon nemesis aren't you. I think most WoS btl commenters understand how annoying it must be for yoons to read btl comment after btl comment, monstering the tories red and blue.
So hopefully its good old NHS level therapy for you Colin A, to do your Nic Sturgeon ate my British hamster stuff on here.
@Colin A –
It's okay to be critical of the SNP and still be an indy supporter.
Look at Loki!
Why don't you collaborate with him and do something productive instead of annoying folk here – he's already done loads of stuff about 'pollock'.
Like Stu I don't know enough about the ins and outs of hospital management to determine what the capacities and requirements are but a couple of things strike me from the story.
1) The rather bizarre claim that children will die seems a tad intemperate. Do all children who live more than a couple of miles from a dedicated ward run this risk or only children in Paisley who have the misfortune to live six miles from the best facility in Scotland? (Carolann Davidson is a Labour Councillor btw).
2) RAH A&E is still there and would be first port of call for an emergency.
3) Getting to the Southern from Paisley is really not that hard and if you are from outside Paisley it is easier getting to the Southern than trying to negotiate Paisley's insane road system.
galamcennalath to your list add Arran War Memorial Hospital which has 24/7 A+E, I suspect that many Island hospitals are the same. Following job advert I saw today gives details, I can confirm that I have used the A+E facilities many times over the years and the standard of Care is second to none.
YesIndyref2 you are correct about a space being left for an Ambulance and also fire Appliances . However the main reason that Cal-Mac have space when the ship is fully booked is because their booking system is not fit for purpose. Part of the recent contract they were awarded specified a "Smart Ticketing System" We will see, Im not holding my breath.
Location: Remote & Rural North, Arran War Memorial Hospital, Grade: Band 5
Category: Nursing & Midwifery Salary: £22440.00 to £29034.00
Contract: Permanent Duration: undefined
Region: Ayrshire_Arran Hours: 34.50
Job type: PartTime Date posted: 19 Jan 2018
Ref: N/671/17/SHOW Closing date: 03 Feb 2018
The Isle of Arran often described as 'Scotland in Miniature' is an island some 20 miles long by 10 across separated from the mainland Ayrshire by an hour long ferry crossing from Ardrossan to Brodick and in summer by a shorter ferry from Kintyre to Lochranza. Arran's stable population if just 5250 increases approximately 3 fold during the summer months. In support of our core purpose of Working together to achieve the healthiest life possible for everyone in Ayrshire and Arran we are committed to a culture that is Caring Safe and Respectful. The post holder is required to work collaboratively in a safe caring and respectful way. Dynamic and forward thinking Arran War Memorial Hospital is a 17 bedded hospital providing care to a diverse patient population on an inpatient and outpatient basis including Accident and Emergency services day therapies and outpatient clinics. Patient care services are provided by a wide range of clinicians from both Primary and Secondary care setting and are expanding as the Integration of Health and Social Care progresses locally. We are seeking an enthusiastic and flexible nurse to join our team of staff providing care to patients within all of the hospitals settings. The successful candidate will be a first level Registered Nurse who will be able to provide evidence of work experience in a variety of acute clinical care settings. Staff will be required to work on all shift patterns and provide care in all general areas within the hospital and also the opportunity of working in the community. Hours: 34.5 hours per week.
Corbyn bounce is bouncing less and less. Anas knows why. Who wants to be a millionaire socialist worker/slave wager, Anas!
https://www.politicalite.com/tories/tories-take-hulton-tories-shock-win-local-bolton-election/
Nicola Sturgeon and I have much in common: we want the best for Scotland and her people.
We also believe independence is the best way of achieving that.
Of course, I have differences of opinion on best strategies and some policies etc but, that doesn't mean we are enemies or opponents when it comes to many core issues such as independence and the NHS.
SNP PBB being critisised now for it's production values and acting
Yoonworld now being sucked up into it's own bowels
What do they expect for a PBB anyway *Braveheart* I know that's one of their all time favourites
All the LOLZ
@heedy
I know you like playing with the guy but he's a mental, careful he's not infectious, don't tell him your name Pike
Damn! fell for it again
Nearly but not quite, last king of Scothland likes it.
@afneil
6h6 hours ago
I don't retweet PPBs of any party but the new one from the SNP which is causing such controversy I thought was very effective in getting its message across. Big cut above most PPBs.
Kerly says:
He,he, he, Fanny
@Lenny Hartley
Yes, that too. I think there was something about the mezzanine deck not being bookable on for instance the Islay crossings, even though they were actually working.
The other thing is block freight. For Coll for instance which I did on Calmac 8 day tickets with other islands (carefully planned trips) I could never get a booking back to Oban, as they were filled from Tiree (the Barra ferry). But I managed to get on OK. It's annoying, but on the other hand particualrly for the Outer Hebrides freight is the lifeline. Different problems these days of course with RET which takes away the benefit of the 8 day ticket for me, though making it cheaper even than a 6 ticket book for the likes of Arran.
Anyway zzzzzs for me, I see oor Colin and Rock still haven't come up with their factoids, though some other types of oids seem to be oozing out of them.
Yes dear. There's a bloke at my work like you too, he voted YES but NEVER again, we can never make it alone etc.
Whatever floats your boat Colin A. You'll get Nicola in the end though, we all know how political careers end in failure right. Its democracy.
Even the great UKOK newsrooms are looking at the end of everything, sales, influence, advertisers don't want to be seen in their shitrags much now. BBC gimps will also be wondering how on earth they can get millenials to watch their relentless vote tory, Scotland's a shithole garbage.
Nothing's forever Colin A, except you ofcourse:D
The MRSA infection rate in hospitals and its sometimes tragic consequences for patients was very real. The media, for once, were not over-hyping the story. If I remember correctly Scottish hospitals had some of the worst rates in Europe for hospital acquired infections at that time – early noughties.
Labour-LibDem coalition were in office in Holyrood at the time and seemed to be less than pro-active in dealing with the problem.
When the SNP came to power, M&s Sturgeon as Health Secretary, they made a huge effort to tackle the problem.
Cleaning services brought back in house and more cleaners employed. Very pro-active campaign to ensure hand cleaning. Not wearing long sleeves – nothing below the elbow. Pre-testing of patients coming into hospital for elective treatments to ensure they were not carriers. And, as you said, improving facilities which are easier to clean.
Introducing unannounced inspections of hospitals with follow-up inspections to ensure recommendations have been acted upon if problems with cleanliness had been identified.
These and other steps have helped to bring infection rates in hospitals under control and reduce them.
Dug's in good form.
https://weegingerdug.wordpress.com/2018/01/20/being-deafened-by-the-silenced/
It's like, unionist politicians keep getting more like scary Zombies.
http://www.youtube.com/watch?v=qVYE8vG3wqg
Just for laughs, mind.
AFoTW 🙂
AFOTW.
As someone who actually knows about the logistics of looking after sick kids, I can absolutely confirm that it is much, much better to look after paediatric patients in a dedicated facility where the entire staff are skilled at delivering safe care to small people. Having paediatric resuscitation teams, critical care services and surgeons on call 24/7 365 is well worth the 6 mile trip down the road. Children's Hospitals are much more attractive to elite specialist staff so recruitment is easier than at a District General. Centralising services is also more efficient than having wards scattered around the region so will give better value for money. It's called the supercentre model.
The whole hospital closure meme is a political football which ultimately distracts from the central aim of health boards to provide high quality safe care at reasonable cost.
@me at 10:59pm
New Zealand launch successful. Wee countries can kick ass too 🙂
http://www.nzherald.co.nz/business/news/article.cfm?c_id=3&objectid=11979201
RE: SNP PPB
With the yoons and their media chums in meltdown over this, it is becoming obvious just how effective it has been.
Well, SNP folk, you have stumbled on a brilliant formula.
I hope to see more like it. Maybe our hipster could return with one of his yoon pals. More of the same please. 🙂
Luigi at 0711am,
Exactly. What stands out however, from all of this, is just how totally uptight the unionists are. Absolutely zero sense of humour – and let's face it, I and many others did not when watching the PPB think immediately of that Torrance bloke, since I'd never seen him very much, and the last time I checked he wore dark glasses.
The mind boggling sense of entitlement amongst unionists and their 'scottishy' media puppets, is truly staggering. What a right greetin faced bunch.
Let's have more of this from the SNP. Truly superb.
The curious case of the disappearing parliament;
https://itisintruthnotforglory.wordpress.com/2018/01/20/the-curious-case-of-the-disappearing-parliament/
The Scottish Press;
https://grousebeater.wordpress.com/2018/01/20/the-scottish-press/
'Unfair' delivery charge dossier sent to watchdog;
http://archive.is/MivV4
How Brexiteers want to reassert British power in Asia;
http://www.youtube.com/watch?v=SfplaX4UpaQ&feature=em-subs_digest
Being deafened by the "silenced"; 'First posted by Macart 2:12am'
Sinn Féin to be led by a woman for the first time in its modern history;
http://archive.is/Rkw5f
People are crowdfunding to buy Katie Hopkins' house to turn it into a Refugee Shelter;
https://evolvepolitics.com/people-are-crowdfunding-to-buy-katie-hopkins-house-and-make-it-a-refugee-shelter/
No Brexit deal on financial services if UK diverges from EU, says Merkel ally;
https://www.politico.eu/article/no-brexit-deal-on-financial-services-if-uk-diverges-from-eu-says-merkel-ally/
Exclusive: leaked files reveal Carillion's payments to blacklisting agency. And far more.;
https://www.thecanary.co/uk/2018/01/20/exclusive-leaked-files-reveal-carillions-payments-blacklisting-agency-far/
VIRGIN CARE 'ASKED STAFF NOT TO REPORT SAFETY CONCERNS';
https://skwawkbox.org/2018/01/20/virgin-care-asked-staff-not-to-report-safety-concerns/
Carillion's missing millions;
https://www.redpepper.org.uk/carillions-missing-millions/
How not to be ignorant about the world; (Ted Talk)
https://www.gapminder.org/videos/how-not-to-be-ignorant-about-the-world/
A BBC Scotland complaint – Coverage of EU Withdrawal Bill;
http://indyref2.scot/a-bbc-scotland-complaint-coverage-of-eu-withdrawal-bill
That's All for now, Wingers. Will post any late links as and when.
Have a peaceful and profitable Sunday.
As ever, top marks to Stu, our Champion who steps up every day to take on the BritNat forces of propaganda in hand-to-hand 'media combat'- and wins every time.
While detail will always remain very important, everything has context and should never be looked at as separate from the big picture. When the electorate are distracted they are effectively blinded. This is a vital part of this Tory Government's propaganda strategy. It helps win elections after all.
So it is that the UK is filled with media noise every single day from morning till night in an attempt to drown out the reporting of the truth.
Currently, the British Establishment, operating through the Government, is seeking to strengthen its power over Scotland and Northern Ireland as part of the UK leaving the EU . In Scotland, by the repatriation of devolved powers, and in NI by redrawing the political boundaries in order to ensure the DUP dominate, a practice known as gerrymandering and not new to Unionist politics in the Province.
The Government's big picture is framed by the desire of the English ruling classes to grab as much political power as possible within the UK and then, free from EU regulations, to create a UK which will be effectively democracy-free and controlled by the wealthy elite via its political wing, the Tory Party and with only two over-riding policies – to continually transfer the wealth of the UK to the rich, and to cement Austerity into the lives of the masses on a permanent basis.
People like Anas Sarwar and David Torrance will always be promoted and protected by and in the Media because they are part and parcel of that strategy. The British Establishment always looks after its own.
Our big picture is of course Scottish Independence, and it's becoming clearer by the day why Scotland needs it.
Over in NI, the same basic problem of British Colonial power continues to usurp the sovereignty and integrity of the island of Ireland and disturb the peace of the people. This is the same colonial legacy which the English Upper class has left scattered across half the world. And still they cling to their delusions of superiority in their insistence on different classes of people, only now there are it seems only two classes, the very rich and the rest.
Everyone ready, eyes open, and…. say 'cheese'.
Stephen Paterson says:
Is Mhairi Black going to apologise for her plastic paddy remark? Is she another big jock knew headbanger who can't accept an Irish identity?
Is it still Labour – unionists plants and placepeople running (ruining) the Health boards or inciting malicious protests. That new hospital looks really adequate for the many and not the few.
Sarwar has a habit of asking ridiculous questions in Holyrood which should have been for the Labour run Health Board. With manipulated facts and figures. They can't count or read a balance sheet. Killing and maiming people the world over. £Trns of debt. Milking Scotland of £Billions which would provide even better essential services.
Most people would be delighted with a brand new up to the minute facility. Instead of complaining.
I have been told many times, keep putting the drops in the eyes, although you can see better, you will see even more and tell others.
You know those 'Jimmy' hats etc much loved by footy supporters?
Anyone know where I can get a hipster beard tie and glasses from?
Listening to President Macron is like listening to Nicola Sturgeon
Spookily promising for Scotland I think
Did he demand Scotland's Remain ( in the EU) vote be respected based on the Claim of Right and the sovereignty of the people of Scotland, then do the opposite and support Soft Brexit?
findlay farquaharson says:
The Scottish Fanny Society is holding its Anas Sarwar of the week competition this evening (allegedly)
@Stephen Paterson,
I will deal with your points. She is a Partick Thistle supporter and has been since she was a little girl She went to a Catholic girls school and attended Mass every week.
The issue you are referring to about "plastic Irish" is an interview she gave to Holyrood magazine. She was not decrying Irish folk in any way, but was alluding to those native Scots on both sides of the religious divide here in Scotland, who utilise Irish culture and history to their own ends,
She is perfectly correct in that view in my opinion
26 January 2012, c5805
Nicola Sturgeon: "Presiding officer, the motion for this afternoon's debate is deliberately simple. It states that
"This Parliament acknowledges the sovereign right of the Scottish people to determine the form of Government best suited to their needs, and declares and pledges that in all its actions and deliberations their interests shall be paramount."
Nicola, the people voted to REMAIN part of the EU. That is a form of government. So, you have no right to support Soft, medium, or hard-bolied Brexit.
The FM or Scot Govt or Holyrood is not sovereign; it's the people of Scotland, as you have accepted.
Frogesque https://www.tamshepherdstrickshop.co.uk/ Hope that helps
Phydaux says:
Another fanny exposed and ridiculed, ignorance and lies oozing out of them, like a poisonous pus.
The SNP PPB was perfectly pitched, in my view, by eliminating the need for any argie-bargies and allowed a narrative of a real and rightful national pride in Scotland's achievements, full of facts and information. Hence the faux outrage from the Colonialists/Unionists who can't stand it when they are denied argie-bargies to sow their division and hatred and lies.The PPB exemplified ,for me, a line I read in a fictional book by the great James Lee Burke, that the only argument you ever win is the one you don't have.
@Stephen Paterson says: 21 January, 2018 at 8:57 am:
Isn't that comment from you in the wrong place, Stephen?
What exactly did you intend to achieve by asking your rather idiotic question here on Wings?
Seems to me that Wings has, of late, acquired a small, but overly vociferous, faction of totally illogical thinkers who ask, or state, absolutely ridiculous questions.
However, as I seem to be making a reply to one such illogical thinker, I had best explain the illogical thinking I refer to.
It really is, as all logical things are, quite simple.
You enquire, "Is Mhairi Black going to apologise for her plastic paddy remark?"
Logically there is only one person in the entire World who can answer your question – that one person is none other than Ms Black herself and to the best of my knowledge, (and I stand ready to be corrected), Ms Black may or may not read Wings but, sure as hell, is not well known as a Wings over Scotland commenter.
So may I ask you the obvious question – What exactly is your motive for questioning Wingers if the SNP MP will answer your question?
Which question incidentally refers to some alleged remark quoted completely without context. Without which context, (or having personally heard/read or viewed the remark elsewhere), the average Winger will have no idea what the hell you are mumping about.
By the way, your own denigration, that you apply to Scots in general instead of to Ms Black in particular, is hardly likely to encourage Wingers to answer your, somewhat overblown and much exaggerated, feelings of insult.
What was it again?
Oh! Yes! Here it is:-
"Is she another big jock knew headbanger who can't accept an Irish identity?"
BTW: It makes little sense – what exactly is a, "knew headbanger"?
Incidentally, are you aware that most Scots will likely have a fair proportion of Irish blood. For Myself I had two Irish Grandmothers. One born in what was to later become a republic and one from what was and still is, from 1542, the English province.
Glamaig says:
O/T completely random:
My road map shows total border crossings between Scotland and England:
2 railway lines
1 motorway
4 A-roads
17 minor roads
Doesn't seem very much
The SNP PPB has opened a door to more exposure of the ways that
corrupt media reporters follow the Westminster directive for Scotland.Their squealing over the Torrence issue shows how they could be hurt, annoyed and outraged by similar kinds of exposure.
A phantom Films type on the same lines of "No to Yes" could be deployed, deliberately targeted on social media and with a themed end punchline making fools of them.
That would outrage them further, and the more rabid they become, the more mistakes they make, which again may trend on social media, which the PPB shows is very effective. Just my way of thinking. However opportunities are endless with idea.
What thoughts do you have on this?
That made you jump huh Colin A.
Yoons are a hoot sometimes:D
If you ever dare to mention online anywhere that Scotland could be an EU member state and there might be a Scottish EU president like,
http://www.euronews.com/2018/01/11/bulgaria-renews-calls-for-euro-as-it-takes-on-eu-presidency
Yoon and English fury and mockery melts the line. You're an odd lot Colin A but who's the odder, yoons or Scots that voted to stay under yoon rule?
Another though re my last posting, imagine what could be done with this idea in exposing the unionist parties in Scotland!
Yesindy2ref, the mezz deck would probably not be bookable as if more than a couple of trailers booked then they cant use them. The freight block bookings are an issue, however if
I have seen one photo on fb of car decks half empty when folk have been told the sailing is full, ive seen a hundred and that cannot be down entirely to block bookings. Mind you cannot keep everybody happy, the number of folk complaining that they cant get their car on to attend a medical appointment on the mainland or have a short notice medical appointment seems to have risen considerably since RET has been introduced.
When I was a lad you accepted the good and bad of living on an Island. Now folk expect no downsides.
So it turns out that Keith Cochrane, who is the chief executive of Carillion, is also the lead non-executive at the Scotland Office.
Surely Fluffy Mundell must remove this incompetent aid as soon as possible.
Cochrane backed no in the 2014 independence referendum, he's also an advisor to the pro-unionist group These Islands.
LATE LINKS!
http://www.pilaraymara.com/2018/01/watching-news-reporting-scotland-from.html?spref=tw
https://www.salon.com/2018/01/20/signaling-more-independence-from-the-us-the-world-bank-phases-out-its-support-for-fossil-fuels_partner/
https://www.theregister.co.uk/2018/01/20/twitter_russia_bots/
They Were Bad. He May Be Worse.
http://archive.is/QZ7Lg
Well Sunday Politics show is at it again, showing a clip of the new SNP PPB, speaking about new schools. The programmes shows Edinburgh's Oxgang primary school, which was in the news due its shoddy building and the unfortunate death of a young pupil.
Of course there's no mention of the Labour/Libdem admin, that the PFI school was built under. The show even rolled out Tom Harris, to say Labour's PFI wasn't that bad.
Heedtracker said:
"yoons or Scots that voted to stay under yoon rule"
You mean, 55% agreed with a central tenet of the BT NO campaign that said:
"vote NO to guarantee Scotland's continued membership of the EU"
Then 62% in Scotland voted Remain in the EU.
Thus, TWO referendums backed Scotland remaining part of the EU.
Now, since the Turkeys voted for Christmas (the NO vote), the status quo that they were supposed to be voting for will no longer exists with Brexit.
However, Alex Salmond warned of this very situation in 2013: "real uncertainty on Scotland's future in Europe is coming from Westminster".
He urged people to vote YES or: "The alternative is Scotland being dragged to the EU exit door against our will as Westminster descends into a right-wing debating society that threatens jobs and prosperity in the real world."
Brexit is against the will of the clear majority of the sovereign people of Scotland. It is a betrayal of TWO clear referendum decisions, so the Scot Govt should not touch it with a barge pole.
Soft Brexit is Brexit. It is being dragged out the EU against our will. Thus, it is unacceptable for the FM, Scottish Govt or Scottish Parliament to aid or facilitate that in any way.
It's good that the SNP warn of the very serious damage of leaving the EU, highlight the betrayal of promises and argue that Hard Brexit is the most damaging of all.
But that does not give them a mandate to try to achieve ANY form of Brexit when it is clearly contrary to the will of the people of Scotland.
'So it turns out that Keith Cochrane, who is the chief executive of Carillion, is also the lead non-executive at the Scotland Office.'
As reported in the Herald. Good grief this sounds corrupt as f*ck
http://archive.is/ynqbW
Robert Kerr says:
@Glamaig
Thanks for the border crossing count. Saved me the effort. Was on my wish-list/todo-list.
You are correct Scotland is isolated pretty much from our neighbour.
Most of the heavy stuff goes by rail via Carlisle. Xustoms checks at Mossend in N Lanarkshire would be easy to implement. Then we would be able to quantify actual Scottish exports by rail. All that whisky exported via Felixstowe would not be English anymore.
Two Off Topic links,the first being David Mundell MP "under pressure".
The Sunday Herald:
"SCOTLAND Secretary David Mundell is under pressure after it emerged that the interim chief executive of failed construction giant Carillion is one of his top advisers.
Keith Cochrane CBE, who sat on Carillion's "business integrity" and audit committees before taking temporary charge of the the firm last year, is also the lead non-executive director at the Scotland Office……" [!!!] 🙁
https://tinyurl.com/y82jdq4v
The second link is the Guardian:
"Scotland's new social security system will include an unprecedented degree of independent scrutiny – with the express intention of future-proofing the powers against the kinds of austerity measures that have devastated vulnerable groups in the rest of the UK………"
By Libby Brooks.Scotland correspondent.
http://archive.is/6SxX7
@colin alexander says:21 January, 2018 at 10:20 am:
Very telling that you, and your yoon pals, need to ask that little question.
There is one stand out EU/EC view that has been held and stated but goes mainly unreported by the Westminster Establishment. It is precisely worded, "The United Kingdom cannot cherry pick which of the four freedoms it chooses".
That simple means that the freedoms that are the entire essence of the EU are sacrosanct.
"sacrosanct:-
adjective – (especially of a principle, place, or routine) regarded as too important or valuable to be interfered with."
Got it now, Colin?
The EU and EC have said since the United Kingdom declared BRUKEXIT that in no way will the EU allow any dilution by the United Kingdom of any of the four core freedoms of the EU.
The 'four core freedoms' of the European Union are the freedom of movement of goods, people, services and capital, over borders.
Is that David Torrence on Politics Scotland a look alike how many are there.
Where is Davidson sent Mary Scanlon on Police Scotland now Brewer the same as Gary.
PictAtRandom says:
And I can only think that's 'lead' as in 'balloon'.
Totally bizarre that Unionism can have an unchallenged monopoly on broadcasting, a virtual monopoly on press media, can see the Labour Party liaising with the BBC to write the news agenda two weeks in advance, and see the BBC's flagrant violation's of Purdah rules all but ignored by spineless, dare I say, complicit Electoral commission, and yet, look at the farcical hysterical reaction to what, a minute and a half of what might at a push be described as abstract ridicule.
You can smell their terror as Scotland wakes up to their widespread complicity in keeping Scotland subjugated and downtrodden for generations.
Their reaction is like a pompous BritNat drill instructor screaming and busting a blood vessel trying to keep conventional order in the ranks, but floundering and impotent because it's his own idiotic caricature the troops are laughing at.
It is real "Emperors new clothes" stuff. The SNP'S broadcast has pointed a finger at the buck-naked Imperial media …and giggled.
Andy-B
Tom Harris, it seems, appears on Scottish political shows more than any elected politician.
Whether it's BBC or STV, there he is again, Tom Harris. Political reject. Hasbeen. A nobody. An EX-Red Tory Labour MP.
WHY ??!!
Breeks
I printed the Electoral Commission rules.
They said the Electoral Commission does not have any rules on impartiality for the BBC or ITV in General Elections or referendums ( well, that's what it said for indyref)
These organisations have their own codes of conduct and are regulated by OFCOM.
And this is just a glimpse of what direct rule from Westminster would look like.
BBC POLITICS Scotland Peter Geaghan? is Torrance's doppelgänger or is it the other way around?
Leonard (LABOUR NORTH) will side with the tories in any possible vote over our independence.
There hasn't been a material change since 2014!!!! Brexit in his view is a wee side show obviously. Tell that to the workers who will be losing their jobs in the next few years. What a total f******- my nomination for next week. Or maybe ex MP Harris or Jackie Baillie or Davidson, Torrance Mundell…. so many to choose from.
Ok Colin A, so what do you actually want to happen next for Brexit Scotland?
You're on here all the time ragin away, SNP bad, Nic Sturgeon ate my hamster etc, but you never say what you want to be done about it all.
Constructive critics are rather rare in yoon culture to be fair Colin A, smash, wreck, lie, frighten, UK first, Scotland is our region of greater England, Holyrood's just a shitty town council for a shithole region of the UK etc
So what to do next Colin A, other than stop voting SNP?
Why Heed why…y'know he's just gonnae launch intae mair fud fanny waffle? 😉
Why Heed why…y'know he's just gonnae launch intae mair fud fanny waffle
He is. But Nic Sturgeon ate his hamster, that's always interesting to hear about:D
@colin alexander says: 21 January, 2018 at 12:15 pm:
colin must be the only Wings commenter who does not already know why.
Looks like I've upset @heedys resident wee playmate by saying Scotlands beloved leader has similar views to an actual President of another country
He'll be at it all night now @heedy and you know it's you he comes here to play with
Nae pals the wee soul
I want the SNP to 100% oppose Brexit, as that's the will of the people (62%).
If the UK Govt ignore that, they ignore that. The SNP stick to their guns. The SNP do NOTHING to facilitate any type of Brexit. Instead they fight to keep Scotland in the EU.
Re devolution. The SNP sticks to: devolved = devolved. EU powers are devolved so if Scotland is dragged out the EU against her will, those EU powers go to Scotland.
It would then be for Scotland to decide if they want to give WM any say in administering them – or not. After the Scottish election (seeking a mandate for independence) or an indyref.
That's what I want, because that's what the people voted for.
As I am a democrat and defender of the sovereignty of the people of Scotland.
Fud*
*Fear Uncertainty Doubt
He'll be at it all night now @heedy and you know it's you he comes here to play with,
Indeed, but more generally as I mentioned, Colin A's jump into UKOK action at the site of your Macron/Sturgeon point, is much like the yoon rage, you really only ever see in BBC Scotland telly, at Scots, with the damn cheek to even suggest that there may well be a Scottish EU president one day, can melt the internet.
England first, never forget:D
Dr Jim
What same views? You never explained.
Do you mean France is pro-nuclear power, pro-nuclear weapons?
President Macron is the directly Elected head of state of France.
No disrespect to Ms Sturgeon but, the FM is only elected as an MSP and only the appointed head of the North UK Holyrood branch office of devolved Westminster Govt.
The SNP prefer the unelected Royals to be head of state of Scotland.
Claim of Right: Kings can only reign by the will of the people. But the SNP intended to deny the people a say about who would be head of state of indy Scotland.
I'm not saying the Royals cannot rule, but that should be only following a decision by the people, not decided by the SNP's White Paper.
Let's hope that's another lesson learned for next time.
@heedtracker says: 21 January, 2018 at 12:38 pm:
I'd hazard a guess that there are far darker psychological reasons than well loved hamster eating at work there, heedtracker.
I think colin/Colin really, really, fancies Nicola but she ignored his advances.
So it may well be a case of, (the oft misquote William Cosgrove's), quote is most likely Colin/colin's motivation:-
"Heav'n has no rage, like love to hatred turn'd
Nor hell a fury, like a woman scorn'd."
Either that or colin/Colin just has a rather poor understanding of the meanings of words coupled with a poor knowledge of politics and diplomacy.
A well written and to the point article by Iain MacWhiter in the SH.
http://archive.is/5kNBo
Watched a few minutes of Politics Scotland as they inferred that collapsed schools was an SNP initiative funded under PFI also the SNP system was basically PFI under another name. Thoroughly unscrupulous reporting, switched off after that. They will of course make some pathetic excuse that it was an overview, but most of us can spot manipulation of facts.
As you are so knowledgeable:
How many more Tom Harris TV appearances are needed before he overtakes the all time record set by that other Red Tory reject, Brian Wilson?
Nice. But what does all that hoohaa mean Colin A, in the real world, of elections and referendums.
You wrapping yourself up into all kinds of contortions here Colin A, at the simplest of questions too,
"I want the SNP to 100% oppose Brexit, as that's the will of the people (62%)"
That's nonsense too. Scots voted to Remain in the EU. We did not vote for the SNP to oppose Brexit, in any shape or form. And thusly, your arch enemy Nic Sturgeon is not opposing Brexit.
That's the kind of gibberjabber we Scots have to face all the time from idiots like you Colin A, in all beeb Scotland gimp tv and radio and every newspaper.
Did Nic Sturgeon really eat your hamster Colin A? sorry your British hamster, great British hamster!
jfngw
Do you mean SNP PFI like Wheatley Group / Glasgow Housing Association taking out huge private loans then paid for by above inflation rent rises while the fat cat boss is on a mega salary?
"Wheatley Group has secured £100 million of new private investment as it drives forward the country's largest house-building programme.
The debt funding deal with investment firm BlackRock Real Assets follows days after Wheatley's financial outlook had been revised upward. In its annual review, S&P Global Ratings revised its forecast to "stable" from "negative", while retaining Wheatley's A+ credit rating for its £300 million public bond, issued in November, 2014."
http://www.scottishfinancialnews.com/13256/glasgow-housing-giant-to-deliver-thousands-more-affordable-homes-following-100m-debt-funding-deal/
Oh dear…i think coco is making a desperate late bid to wrestle away Sarwar's "fanny of the week" award…!!
My name is coco…and I weely weely support independence…it's just I hate Nicol Sturgeon, the SNP, the Yes campaign, and everything positive connected with Independence".
"I also never criticise ANY unionists in any way, nor the msm, but only because I am such an 'uber yesser' you understand…and hopefully you forgot everything i posted over the christmas/new year period…that was coco with a 'C'…honest guv".
Bolt, ya diddy.
Scot Finlayson says:
Keith "I`M VOTING NO" Cochrane CBE,
Fellow of the Royal Society of Edinburgh,
ex of pro Better Together Weir Group,
UK Government's Lead Non-Executive Director for the Scotland Office and Office of the Advocate General,
is Chief Executive of the failing Carillion construction empire.
A Lead Non-Executive Director is the way Big Business has been allowed to infiltrate our Government,
there are 80 unelected Lead Non-Executive Directors infesting 17 depts of Government,
their chief is Sir Ian Michael Cheshire who worked for Ernest Saunders during the illegal Guinness takeover of Scottish company Distillers,
Saunders was given a 3 year prison sentence but let out early because, according to bent doctors ,he had alzheimer's,
he later recovered from alzheimer's,the only human being to have recovered from alzheimer's ever,
Sir Ian Michael Cheshire disappeared to later emerge as controller of the 80 business leaders infesting all Government Departments,
worth a read if you are interested in democracy
https://tinyurl.com/ydg4xv6y
His name was Alexander, though Colin was his first
Only SNPbad mantra could quench his toxic thirst
He talked a load of shite from mornings through to nights
But Heed was here to goad him on with phoney little fights
And so we all watched passively to all the pish he spouted
But Alexander thought 'I'm great!' though he had been pure routed
So on it went relentlessly, on Wings below the line
The needless pointless crap from Colin, with not one person chimed
What will we all do now, that we know the score?
We cannot rid ourselves of Colin's obvious ordure
The only course of action, it is plain to see
Is simply to ignore the prick that needles you and me
Colin A's currently trying to think of a way of explaining what this means, Nicola Sturgeon at my hamster style, because as we all know, yoons of the great UKOK zone, always know what the "will of the Scottish people" is.
Actually to be fair, future Sir Gordon Brewer told that Leonard nobody this morning on his beeb Scotland gimp network that the English will not permit Scots indyref2, at least until they decide when the time is right, probably after dudes like future Lord Gordon Brewer have actually taken out SNP Scots gov.
See Colin A, even though Nic Sturgeon ate your hamster, the great British UKOK card deck, is always stacked in your favour.
Re the SNP's versions of PFI.
I have a modicum of sympathy. With Austerity, it's difficult or nigh on impossible for Govt to fund lots of public building projects.
So, these Quangos and Aleos etc , eg Wheatley Group turn to the likes of BlackRock to borrow money privately.
Ultimately that money will be paid back WITH INTEREST FOR PROFIT to the private investors.
That money will come from taxpayers as Wheatley Group is supported by the Scot Govt and from tenants / factored home owners.
So, while we shouldn't pretend private money is not happening under the SNP, We must also ask WHY? Are the SNP following the same ideology as New Labour?
The Why, I'd argue, may be because of UK Govt Austerity supported by Tories and Labour, so that the amount of public – not for profit – money that can be invested in house building and public infrastructure projects is severely limited.
Cochrane and his position as a Non-Executive Director in the Scottish Office
I love the acronym for that position: NED so appropriate. And collectively when you include those in other departments: NEDS.
Noticed the Sunday Herald used it quite a bit in the article.
@K1 –
Re your reply @Dr Jim:-
What a right old load of utter infantile pish, colin.
You have no actual grasp on reality and live in a childlike fantasy world.
In the first place, (whether you or they like it or not), the actuality is that Holyrood, and hence the Scottish Government, no matter of which political spectrum colour(s), are legally, under Westminster's, (Kingdom of England law), – thus an actual legally created part of the Westminster Establishment's, de facto parliament of the country of England.
Note those very carefully chosen words Colin/colin. They mean exactly what the describe.
The consequences of any SG not abiding by the Westminster, (de facto Parliament of the country of England), rules would give Westminster the instant legal right, (under English law), to dissolve the Scottish Parliament.
The ensuing battles would be long and deliberately drawn out by Westminster and probably conducted in the Westminster created United Kingdom Supreme Court by mainly English/UK created judges on the bench.
You fail to differentiate between the SNP as a purely Scottish political party and the SNP/SG who have no option but be a part of the Westminster Establishment. For example, Nicola Sturgeon is, through no choice of her own, a Privy Councillor.
Now pause a moment before jumping to your feet yelling, "I told you she was one of them", and consider this –
"What is the Privy Council?
The Privy Council is an advisory body to the Monarch; its members are known as Privy Counsellors. It is one of the oldest parts of the UK's constitutional arrangements, with its origins dating back to at least the thirteenth century.
The Privy Council advises the Queen on the carrying out of her duties, including the exercise of the Royal Prerogative and other functions assigned to the Sovereign by Acts of Parliament. Although some of the Privy Council's powers are ceremonial in nature, many relate to matters of constitutional importance. In almost every instance, however, the advisory role of the Privy Council is a fiction, and the body is effectively a vehicle for executive decisions made by the Government which are then formally issued in the Queen's name."
http://researchbriefings.parliament.uk/ResearchBriefing/Summary/CBP-7460
Now ask yourself – is it not better to have an FM on the inside to know what the Privy Council are advising/demanding what the Queen should do?
Whether any of us like it or not, for at least the time being, we must play within the rules as laid down by Westminster even when they are using Kingdom of England legalities that dates from long before there even was a United Kingdom.
We must thole such laws until the SNP and SG command a clear majority of the Electorate. The clear fact is that although the Rule of Law in Scotland categorically states that the people of Scotland are sovereign there must be a clear majority of the electorate backing the SG and/or the SNP before the matter can be legally taken before the High Court of Scotland.
For only then can we be sure that the High Court of Scotland has no other option than accept that the legally sovereign people have spoken and demand they accept the people's sovereign decision(s).
In Westminster, under the English rule of law, Her Majesty remains legally sovereign but, (since 1688), the monarchy OF THE KINGDOM OF ENGLAND, must legally delegate their divine right to rule powers, (sovereignty), to the Parliament of England. However, there has been no legally elected as such Parliament of England since the last day of April 1707.
Westminster, though, uses that delegated sovereignty as if it applies also under Scottish law. It doesn't but, even although the High Court of Scotland has ruled otherwise, Westminster has ignored the true facts and, to date no Scottish political party, (or government organ), has had any proof that the legally sovereign people of Scotland want to end the union.
So what you, colin/Colin want is for either the SG or the SNP to take an unlawful, (under English law), action to give Westminster the legal right, (under English law), to take back all Holyrood's devolved powers. How stupid is that? We are but a hair's breadth from having a legal, under Scottish law, right to use the people of Scotland's sovereignty properly.
In your war against Nicola, the SG and the SNP you have the backing of all unionists, (including that other false flag yoon fool, Rock and other yoon sock puppets.
Nice poem.
I don't force anyone to engage in dialogue.
But when the SNP go talking ( or is that crawling) to the unelected House of Lords to try and save devolution, it seems a bit hypocritical for some Wingers to regard me as someone persona non grata for political discussion.
Dr Jim Says: Looks like I've upset @heedys resident wee playmate.
He is doing Wings people a favour, of what to skip by, and YES people from even reading his post in the first place.
Oooh I know this one Colin A. Its because Scots.gov cannot borrow, like what the great and good UK gov Treasury does, by the hundreds of billions, nae trillions, and then there's their £200+bn PFI debt mountain/golden goose bullshit.
And all of that despite their great 2014 The Vow shyste on Scotland Colin A, where we all promised devo-max, if we would just please please please, not end this shithead UKOK zone.
UK gov Treasury dingdongs have borrowed and wasted so much money Colin A, in Scotland's book too, it does all look like its their cunning UKOK plan, dump so much debt on their Scotland region, indy really may not be a viable financial option, or at the very least, giant debt does give yoons like you a massive Project Fear 2 stick, to beat the YES out of Scots again. Surely not eh Colin A.
Although, watching SLab's Leonard nobody flounder about under the very gentlest of future Sir Gordon Brewer questions this morn, it is hard to gauge just how much fiscal planning yoon culture does for its Scotland region, what dudes like Leonard nobody do rightfully own. Leave it all up to the beeb gimp network to crush the vile seps, seems the way UKOK forward.
Interesting too how you've ducked away from this nonsense too Colin A. You yoons eh:D
Scott Shaw says:
How dare you call Sarwar a fanny. Fannies are very useful things and Sarwar and his ilk certainly ain't that.
Things in Ireland, north and south, just got more interesting, and more encouraging, with Mary Lou McDonald becoming Sinn Fein's new president. The Dublin TD was quick to announce her continuing ambition of uniting the island of Ireland.
Bit by bit the tatty so-called united kingdom crumbles away.
How would I know, Colin? I don't use the Westminster Establishment, (of which the United Kingdom Media is big part), to get misinformation at subliminal, subconscious or, increasingly, at conscious level.
Some weekend reading when television is crap:
Our useless press: https://wp.me/p4fd9j-miL
A hostile western: https://wp.me/p4fd9j-mff
Colin Alexander: "Are the SNP following the same ideology as New Labour?"
No. And you state nothing to disprove that.
Remarkable how the irritants that appear on this site to remind us of our master's existence all have Scottish names, rather than say, Anthony Smith-Jones, or Aubrey Lytton, or Maid Marion of Sherwood.
@heedtracker
There's a strong similarity between your playmate and the guy in the SNP PPB, it disnae matter how many folk tell him he's a dumplin and even leave the room he still hangs out at the fridge
Robert Peffers said:
"we must play within the rules as laid down by Westminster even when they are using Kingdom of England legalities that dates from long before there even was a United Kingdom".
No we don't.
It's good you recognise the subservient, devolved WM Govt nature of Holyrood.
I infer from your suggestion that WM can just dissolve the SP despite it clearly saying in the Scotland Act 2016 about the "permanent" nature of the Scottish Parliament that shouldn't be abolished except by a vote by the people of Scotland. ( You could be right. The "permanent" may be an expression of sentiment, not a guarantee of a legally enforceable right.)
The Gina Miller case upheld the right of WM parliament over Royal Prerogative with regard to Brexit. Other case law has upheld that parliament must vote when it's constitutional change.
What "unlawful action"? There is no law that says the SG or SP MUST lick their Lordships' "noble" airses. There is no law that says the SG or SP MUST give an LCM.
The SG and SP could simply do nothing, but continue to criticise and verbally oppose Brexit and argue the Claim of Right.
There was nothing to stop them asserting the breach of Claim of Right rather than Sewell Convention during the Gina Miller case. That the democratic decision of the people of Scotland should be upheld when such major constitutional change, such as leaving the EU, is taking place.
They could or anyone could assert Claim of Right and take it to court. The finding of the court, if biased, if unjust, would be open to national and international legal scrutiny.
To summarise, I agree with some of your points such as the subservience of devolution. Other parts are sheer speculation, sheer unproven assertions and predictions without proof.
PBBs tend to pass me by but thanks to Cole-Hamilton the latest SNP is certainly getting way more attention than these things normally do, although the classic Python format helps too.
Nevertheless it just proves the old adage, if you need a tube a double barrelled tube is better.
Colin A is angry that Nic Sturgeon ate his hamster but he's quite a good yoon tool, a fanny even, at diverts.
Look at this week on planet toryboy, where the top of the toryboy catastrofuck, red and blue, Carillion, is effectively a very big chunk of our noble and honest tory UK gov.
Carillion's catastrofuck is a toryboy catastrofuck, right in to No.10, yet from massed ranks of beeb Scotland gimps, not a peep.
How it works in toryboy teamGB, current owners of the Scotland region, kept down and in its place by the BBC Scotland gimp network, only £350 million a year too.
To compensate for Colin A's rage at Nic Sturgeon eating his hamster, UKOK Prime Minister Teresa has appointed a Loneliness Minister.
Yes I know, its all SNP bad but can you really invent bigger and creepy shysters than toryboy Colin A.
TROLLISH NATIONAL ANTHEM
Some speak of Alexander (as you do on days like these)
Of Podmore, Caledonia and those who love to tease
And when the outlook's rocky we're all reduced to tears
By the trollolololololol of the British Volunteers
Accordin ti The Manual ti wreck a brand-new threid
Ye either plant a time-bomb or banter wi The Heed
Or throw things fae the margins an play on secret fears
Wi a trollolololololol for the British Volunteers
So let us fight for freedom but be aware of those
Who dress in party colours or wear their mufti clothes
May they and their commanders stay happy down the years
With their trollolololololol for the British Volunteers
Exactly. You are 100% correct.
I'm saying the SNP DO USE private finance by a different name.
BUT: I think they are probably forced to do so, due to the lack of public money, due to Austerity policy. (Whereas New Labour did it for political ideological reasons).
So, That's not an attack on the SNP, but a defence of their use of private money, as there is little other option due to UK WM Govt.
Even if WM did fund those Scottish projects via public money. Much of that "public money" is again from borrowing that is paid back with interest – or rather it's interest on interest with ever increasing debt under the UK.
Currently just under £2 Trillion and rising by £5,170 per second.
Nevertheless it just proves the old adage, if you need a tube a double barrelled tube is better
To be fair to all the wetfart toryboys like Cole-Jollyposh, that PPB they're enraged by, does kick off with the party guests arriving and the host explaining that, "Davey's bashing on about politics again."
Not sure if Mr T goes by Davey, probably David, Davey is too common by half, even so…
Much as I admire the poems, I'm being nice by warning you:
It seems Stu hates poetry and you risk getting banned by writing poems on his blog.
Grouse Beater
I noticed that too. Many of the Unionists commenters use names like Donald McDonald, Murdo MacLeod, Gregor McGregor etc.
I think they are made up or, I say old chap, it's the Eton educated SIU clan chiefs / "Scottish" aristocracy writing in..
Iain Crighton says:
Can it be just like the World Cup , if you win it three times you get to keep it ?
Until the uk government exercises the use of its asset purchase facility and the BOE and buys up the bonds.
It isnt debt at all
Much as I admire the poems,
I'm being nice by warning you:
If you're all so nasty to me,
I'll go running off to Stu
I'm weely just a fanny
And a pretendy fanny too
So now I'm rumbled as a yoon
I'm going boo hoo boo hoo.
c(C)olin a(A)lexander circa 2018
Funny how Rock also goes running off to Stu "help me Stu".
O/T .
https://www.facebook.com/soulseekers.worldwide/videos/1810889285894025/?hc_ref=ARSLXMI2lNEpEw6ePjgN6JGmzkrc3Mu-oxUrW3uED0hFO4OidYsq1i2yut-AObDSUF8
One of nature's wonders, unless your trying to sleep lol .
Cumoan Wingers get ah grip Yous ur aw playing the diversion game wie the Usual suspect Im amazed at some of the better known Wingers responding to clowns .
Apologies for long post, this is literally a copy and paste from the main BBC News Scotland web page a few minutes ago, keeping the story titles only and deleting the rest. I have left none of them out.
It is just one long litany of doom and gloom. This is the impression the BBC want you to have of Scotland. They are totally trying to mess with our heads!
The swimming hedgehog is still there though, obviously more significant than the FM's speech at the David Hume institute, or anything related to Brexit or the devolution settlement that happened this week, and may or may not be still hidden in the Politics section…
Climbers rescued from snow-covered ridge
Travel warning as heavy snow hits
Police ask drivers to avoid the A82 around Glencoe Mountain Resort where conditions are "hazardous".
Man found dead after 'altercation'
'Despicable' attacker hid weapon in bag
Man found dead in Glasgow street
Police target drivers in pensioner hunt
Small improvement for polluted streets
Staff member threatened by man with knife
Appeal after officer forced to evade van
Fire crews tackle early morning blaze
'Six mile tailback' at Scottish ski resort
Murder inquiry into 'brutal' street attack
'Deadliest winter' since 2004 The Sunday Post says freezing weather and the flu have pushed winter deaths to a 13 year high.
Hunting The Old Fox Historian and broadcaster Dan Snow on the Lord Lovat mystery
Reborn babies Making the lifelike dolls loved by some and feared by others
Mental health plea 'I'm just a mum desperate to make a change'
'I am a victim too' The wife of a man caught with child abuse images wants the same support as victims
More of your pictures of snow from around Scotland
No shortage of rhetoric at FMQs – just of novelty
Hard frost on Scotland's economy
Live Hearts 0-0 Hibernian – cup tie goalless at half time
Brora's Duff eyes Highland derby
'How I lost my legs loomed over me'
Travel warning across Scotland
Strangers at 103-year-old woman's funeral
Drivers face major road disruption
Meet Phelps, the swimming hedgehog
Runaway lorry ploughs into cars
Don't forget these new 'local democracy reporters' advertised and funded by the BBC and disproportionately more of them in Scotland.
Their job will be to "offer in depth coverage of our local authorities". In other words spy for the BBC and plague hospitals/council offices and anywhere else you can think of looking for snp baaaad stories.
There's an interesting monument in Glasgow's Sandymount Cemetery to all the nursing staff at the Belvedere Fever Hospital who died treating the sick there in the various epidemics which swept the city. Very brave lassies, & they were mostly lassies, who went to their work fully knowing the risks.
ronnie anderson @ 15:23,
We know, Ron, we know. But it's devlish hard sometimes to ignore the human wart – the unwanted excrescence that's hard to get rid of.
And the poems, for all the dire warnings, were a good laugh!
(I just wish he were honest enough to come out and say what political movement he's actually for. Dozy old Labour, I suspect, though RISE or even UKIP are good runners-up.)
@Colin Alexander
I don't object to private money, most of us have used it at some time. It's poor control of the profits that I objected to with PFI, something I believe the current Scottish system is trying to control.
Of course these companies are devious and will try and load the system, it's up to the government to be alert to this.
Winifred.
Yes I'd wager that these BBC funded reporters will be used against the SNP government, and the independence movement in general.
I'd imagine it's the first step to combat the ever decreasing unionist press demise.
Lets not forget that the BBC currently buys over a thousand copies of the Guardian newspaper every week.
"BBC spent £127,643 on 80,000 copies of Left-leaning newspaper last year
Startling figure is nearly 45 per cent higher than its bill for any other title
This despite fact Guardian accounts for tiny fraction of Britain's newspaper sales, shifting just 176,000 copies a day"
http://archive.is/FCTCh
Robert J Sutherland I have no problem with the Wingers poems taking the pish out of the Usual suspects ( their not conversing with them as such ) . How many times has it to be said the Usual suspects are on Wings to cause disruption & sow the seeds of division .
PictAtRandom @ 2.59 pm =)
Ah, that's me untelt!
Not a lot happening at the moment, just the BBC doing it's normal stuff trying to blacken the names of anyone who isn't a Tory – it's Marr giving John McDonnell the treatment now apparently for something explained 4 years ago.
OT/ Enjoying listening to Truly Scottish TV News while browsing Wings comments =)
Fasinating hearing international news from France 24, a completely different perspective to the British media. In a good way!
Looks like our newest Scottish broadcasters are going from strength to strength 🙂
The BBC provide material for 700 titles or something like that. The BBC is bound by its Charter to protect Britain, which is the establishment, which is the Tories.
So when it's Tories v Labour the BBC support the Tories.
When it's Labour versus the SNP the BBC support Labour.
The media pick up their stories from the BBC because they have no money.
That's it, basically, the end of democracy in the UK.
ronnie anderson
Open question:
What's the difference between the Non-Profit Distributing model under SNP and PFI under Labour or the Tories?
https://www.scottishfuturestrust.org.uk/page/non-profit-distributing
What's the difference in private sector profits between NPD and PFI?
Things slow in politics, so I'll have a go at the msm. That seems to be in vogue anyway!
We can be pissed of at the msm for their bias. There is much evidence of simply poor journalism as well. We complain about plagiarism and failure to check facts.
However, what is the most basic requirement of a journalist? I would say to understand the language they are writing in and have sufficient knowledge to use it accurately!
An example from the last hour, from the Daily Express. Fails in every measure of competence, not least basic English!
Headline …
"French politician admits EU is jealous of Britain's booming economy following Brexit"
"A FRENCH politician has claimed that Britain's post-Brexit future is positive – something which has made the EU jealous, as he mocked those who predicted a 'cataclysm' in the wake of Brexit."
The writer should have used envious. The text makes absolutely no sense with 'jealous'.
You are jealous when another person threatens to take something/someone from you.
You are envious when another person has something/someone you want.
Opposites, in effect. Yes, sometimes folks misuse the words in colloquial speech, but there is absolutely no excuse for someone employed to use the English language 'professionally'!
"Launched in 2017 following more than two years of talks and planning with experts from across the UK media industry, the LNPs support more than 700 print, broadcast and online news outlets by providing free access to news content generated by the BBC News Hub, the Shared Data Unit and up to 150 local democracy reporters.
Those 700 news outlets known as Section One partners – are spread across England, Wales, Scotland and Northern Ireland."
http://www.bbc.co.uk/lnp/partners
The MSM is strapped for cash and most reporters are cut and pasters. Here endeth freedom of the press and democracy in the UK.
Sensible answer.
So, I'll ask you too:
What's the difference in private sector profits between the SNP's NPD and Labour or Tory PFI schemes?
It's not a trick question, I don't know the answer. Do you or anyone else?
You should do some research then Colin A, before going Nic Sturgeon ate my hamster WoS btl.
Would you like me to enlighten you Colin A, seeing as you clearly do not have a clue, or the stones either, to tell us what this means, shock!
If we look at Wheatley Group:
"Long term debt facilities
As at 31 March 2017, Wheatley
Group had £1,183.5m of bond and
bank funding facilities in place with
total Group drawn debt balances of
£1,014.2m" .
The interest rates they are paying aren't made clear.
https://www.wheatley-group.com/__data/assets/pdf_file/0029/47549/Wheatley-Group-annual-report-and-accounts-2016-2017.pdf
"Wheatley Group was issued with a revised credit rating of A+ (negative outlook) in July 2016, following the Brexit vote."
Interesting effect of the Brexit vote.
https://www.sfha.co.uk/news/news-category/sector-news/news-article/wheatley-groups-financial-outlook-revised-to-stable-by-sp-global-ratings
The interest rates aren't made clear. Only the £50 million one is mentioned because they had a 3.542% interest rate, so presumably all the others are higher.
Speaking of Scotland's rancid media [just in case ye hivnae read it yet]
Today's Mail On Sunday contains lengthy excerpts from a book by a former Head Of The Diplomatic Service which lifts the lid on just how horrible a person Thatcher was which makes one wonder why the men in grey suits took so long to defenestrate her.
How she didn't start a major war is a mystery.
Any difference between her views and fascism is difficult to discern.
Try to imagine a world with Thatcher as PM at the same time as Trump as POTUS – makes one almost grateful that we currently have the useless idiot Maybot as PM.
If the thought of reading The Heil is off-putting hold your nose and have a look at the article – it is both scary and revolting.
The Irish perspective always makes much more sense than anything coming out of the loyal UKOK msm!
" … preparations …. all devoted to having your cake and eating it: a bespoke free-trade deal, encompassing all goods and services, plus complete freedom to negotiate third-party trade deals; no more freedom of movement; and no more European Court of Justice. It doesn't matter how many times or ways in which European politicians say no; the British just don't seem to listen."
https://www.irishtimes.com/business/economy/why-uk-s-juvenile-brexiteers-can-t-see-what-s-staring-them-in-the-face-1.3363037
colin alexander @ 5.08pm
"The key difference between PFI and NPD is that, whereas in the former, the [private structure's] capital includes a small element of private equity, in the latter its members invest only loans. In consequence, while [private company] shareholders receive returns on their capital in NPD, the level of these returns is to a large extent 'capped' at the point at which contracts are signed, and any surpluses remaining at the end of the contract are passed to a designated charity. This is distinct from the PFI model, in which surpluses are passed to [private company] members as dividends." (Hellowel and Pollock, 2009: 406)
It's good you recognise the subservient, devolved WM Govt nature of Holyrood. "
You get more lunatic and illogical with every comment you make, colin. I've wasted enough time on you but will just point out I reasoned for you exactly why Holyrood has to stick by Westminster's rules. You obviously cannot take that reason in.
The legally, under the laws of Scotland, sovereign people of Scotland must, to legally, (under anyone's laws), have a majority of the legally sovereign people of Scotland agree to anything in order to make any legal moves.
The consequence of acting without that majority would be to set Scotland up for a civil war with at least three factions.
The YES, the NO and the Dinna Ken factions. The Dinna Kens will abstain because they dinna ken but the civil war would eclipse the Northern Ireland situation in both length of time and ferocity and the YES faction would be on a hiding to nothing being backed by both Westminster and the Northern Irish Unionists.
Is that really what you are advocating for Scotland, Colin?
You have preached like a demented religious nutter against Nicola Sturgeon, the SNP and everything they stand for as long as you have commented on Wings as colin alexander and your style is rather familiar under other titles.
You want Scotland's elected representatives, and remember they are in both Holyrood and Westminster to act without the sovereign permission of the legally, (under Scots law), sovereign people of Scotland.
It will be a hard enough fight to have a Scotland with a legal mandate under Scots law recognized by the more prominent World Powers without attempting to do so without such a legal mandate.
Re Sunday Politics and The Marr Program – at least these as a rule have interviews with government and opposition.
As for the Scottish segment of Sunday Politics it seems to becoming almost the norm not to have anybody on from the SNP or supporters thereof.
I was impressed with the interview of Macron on Marr – answered all the questions without any woffling or evasion – a pleasant change from The Maybot.
This is distinct from the PFI model, in which surpluses are passed to [private company] members as dividends." (Hellowel and Pollock, 2009: 406)
The private company also owns all the schools and hospitals for example, once their PFI contracts are up, usually 30 years. That's all down to Crash Gordon, the greatest big spender/borrower ever really.
Like or not, England has been showered with PFI contracts but look at what happens to yoon rage, The Graun style, when SNP Scots.gov try to use their PFI variant and its is NOT spent on England.
Severin's one of the sneakiest yoon attack propagandists around, and this is some of his of creepiest work.
https://www.theguardian.com/uk-news/2015/jul/31/scotlands-private-finance-infrastructure-programme-turmoil-ons-ruling
Beeb Scotland gimpery just flat out refuses to even mention that its all on SLabour.
http://www.bbc.co.uk/news/uk-scotland-37135611
"PFI schools built in Scotland 'owned by offshore firms'
You are correct that I'm not a qualified accountant and can't give a breakdown of the exact differences of NPD and PFI. I need to take it as a matter of trust that the system caps the costs and profits of projects done under this system. Of course if it is proven that I have been deceived then they will have lost my trust and I would be unlikely to vote for them again, it would be risky strategy for a party wanting to achieve independence.
lengthy excerpts from a book by a former Head Of The Diplomatic Service which lifts the lid on just how horrible a person Thatcher was
Thanks for highlighting.
Worse than I imagined even. A truly dispicable woman.
Next time some old Tory nutter says something positive about Thatcher, plenty of come backs in this!
http://archive.is/V4xZB
If you do not know by now that Colin Alexander is a black ops
Then you never will.
His methodology is pick pick pick. Try and sow doubt. Appear aloof but quote finance figures and have reference sites to back it up.
I have given him the benefit of the doubt as being Labour or RISE, but I think now it is more than that. Eh Colin?
I shall not reply to him again. He wastes our time with distrsction
Legerwood
Thank you for the explanation. I hope others found it helpful too.
He uses exactly the same arguments and points as the unionist and anti-SNP posters do on the Herald, but at least they have the honesty not to pretend to be Indy supporters, unlike the cowardly faker who posts here.
WoS btl is not a quarantine zone from the yoons though, red or blue tory, like Colin A and Rock especially.
No need to guess. Happy to tell you.
Colin Alexander's voting history:
Nearly always voted SNP since I turned 18.
2011: SNP : SNP
2014 indyref: YES
2015 UK GE: SNP ;
2016 Scot Parl: constituency vote: Labour (only time I've voted Labour).
List vote: Rise (Then wished I'd voted Green)
2016 Eu-Ref: Remain.
2017 Council: 1 SNP 2 Green ( should have voted till I boaked. eg voted 1,2,3,4,5 etc.
2017 GE: SNP
CV: Never been in the armed forces.
Political affiliations:
Currently: None.
All previous affiliations:
Former SNP member.
Politically motivated donations 2017:
£'s to Stuart Campbell's legal fund
£'s to Craig Murray's legal fund
Bob, I'm criticised if I don't have figures. When I do have figures, I try to quote the website link. No mystery there: I Google and read some blogs etc.
Black Ops? For BTL comments? Robert Peffers assures me that does happen on here. But that's Robert for you. Bless him.
Personally, the only similarity I can see between many Btl commenters, and the likes of revolutionaries like Fidel Castro, is that many were born round about the same year.
Somehow, I cannae see Theresa May having many sleepless nights as a result of WoS comments.
gala—- @6.21
And the bastards gave her a State Funeral.
Last word Colin and then we're done. You may know much, but I bet you any amount you want that if the SNP had any contracts that came anywhere close to the PFI schemes used by Labour, then every Unionist politician and every media outlet would have jumped on it like fleas on a dog.
Big business has much in common with Unionists and I can tell you for certain that there are precious few secrets in business when the need arises, given that many Labour and Tory politicians have entrenched relations with big business.
So spare me the phoney figures please.
P.s. I think many Unionists ,and I believe you are one, are having many sleepless nights thanks to Wings.
Re PFI/NPD
http://archive.is/sZcvR
This appears to be the main clue to the ripoff.
However, what the Treasury missed is that the equity returns on a PFI project are commonly scheduled to accrue late in the life of the project: so that the outstanding capital on which the return is earned builds up
But I've no idea what that means in layperson's language.
Pretty good slams Colin A. At least a lot better than your yoon chum Rock.
Just you keep bashing away, Nic Sturgeon ate my hamster, stop voting SNP for Scottish independence, they're a terrible party because of Scottish sovereignty n shit.
Its clearly the way to bring it all crashing down dude:D
yesindyref2 (to the Rev. Stuart Campbell):
Robert Peffers,
You are an aggressive verbal bully here and almost certainly a nasty person in real life.
And like all bullies, you are a coward who runs away the moment he is challenged.
If you are as clever as you think, why don't you have your own site to flog your "knowledge"?
Instead of contaminating every article on someone else's website with your verbal diarrhoea.
Why haven't you ever written a book on the history of the union?
After 310 years as a colony of England, only an utter "numpty" like yourself would pretend that Scotland is "sovereign".
Of the last 80 odd posts (didn't have the time or inclination to go any further) more than half of them are from Mr Alexander (around 18) plus a number of people replying to him (around 26), some of whom think he's a troll. Just wondering why people who think he's a troll are clogging up the site by responding to him and in doing so probably encouraging him to continue to post on here? It's also extremely off-putting for people looking for some half-decent information having to trawl through this.
In other words Mr Alexander is doing a grand job, from his point of view. Renamed the Colin Alexander site. 10/10 Colin. Does this mean that you now get your bonus?
Sunday Politics: Great to see that Brewer didn't get all of his own way today. Gillian Martin SNP nipped him in the bud when he was questioning her about Carillion and PFI. She reminded him twice that PFI doesn't exist in Scotland now with, for example, "you're conflating what's going on in the rest of the UK with Scotland." Brewer then said that PFI had just been renamed under the SNP and she said no and spelt out the differences for him. She also managed to mention the Scottish Investment Bank. WELL DONE GILLIAN.
Leonard was on the show today too and the minute that Nicola Sturgeon's name was mentioned he began to behave oddly with his head going backwards and forwards constantly, like a tortoise (in and out), and his shoulders going up and down. And with his face screwed up he said, for example, "I've never witnessed from the days of Margaret Thatcher a political leader who was so divisive (forgot about Corbyn), because of that call for another Referendum." And finished up, when talking about Nicola, by saying "I can't see there being a sufficient material change to call for this second Referendum, when it was understood that it was a once in a lifetime opportunity for people to vote."
Dafty or BBC fly man Brewer left it at that. Didn't bother to ask Leonard if Brexit constitutes a material change.
Last weeks Sunday Politics:
Brewer is totally taken aback when Tomkins tells him that "if legislative consent motion is not passed by the Scottish Parliament it will be difficult to deliver Brexit at all." Around 4 minutes in.
I watched the full programme on the Iplayer, but when I tried to find it on youtube to post on here Tomkins speech had been edited out, however someone by the name of Tracy has kindly posted this on youtube.
http://www.youtube.com/watch?v=C0uq7CHV-Xc
Kevin McKenna who was also on the show brings this up again later, by saying that Tomkins statement should make front page news (or words to that effect). Did I miss something? Was it on the News / front page news?
As an aside, Kevin McKenna comes across as being anti-SNP at times, especially when he is on TV, so why does he try to make out that he's a supporter on other occasions?
Evidence? Surely you must have it with your high standards?
ScotsRenewables says:
Why is everyone wasting so many electrons, so much energy, discussing things with 'Colin Alexander'?
He may be a genuine indy supporter who has grave doubts about the path the SNP is treading. Or he may (more likely) be a black ops false flag flyer. It really doesn't matter which, IMO.
I suppose it all depends on what you see the 'below the line' discussions on Wings as being good for. For me, I come here to learn stuff I didn't know, to feel good about being in the company of like-minded people, to garner arguments to direct at persuadable 'no' voters and to have my own position confirmed and strengthened.
I am an SNP member, but I don't agree with everything the party says and does. What I do know is that solidarity and voting SNP all the time is the only way we are ever going to get past this 48% limbo we are stuck in. Plenty of time for slagging the SNP once the job is done, but what purpose it serves in open forum at this point in time escapes me.
I fail to see that 'Colin's posts or responding to him is achieving anything other than making us look divided or weak to the casual observer.
My humble suggestion is, STOP FEEDING THE TROLL and concentrate on the good stuff.
Yesindyref2 says: "He uses exactly the same arguments and points as the unionist and anti-SNP posters do on the Herald".
For example?
I don't think that's fair. I'm openly critical of the SNP's YES campaign. That's because WE LOST when we should have WON. Daily we pay the price for that. I'm blatant in criticisms of the SNP. Nothing subtle or sleekit about it.
Sometimes I praise them.
I'm openly critical about how little the SNP have done for indy between 2014 to now. However, the Scot Govt's growth commission have reported back, so let's wait and see what the SNP say about a future currency and Scottish indy economics. I hope, I'll have something good to say.
I think Unionists were always on about Gers figures. Making out Scotland is an economic basket case. I don't do that. For the average income person, I think indy Scotland could be a more prosperous place with better public services and higher quality of life.
Probably higher taxes though and more opportunities for middle / lower income people, so that's why many rich are so vehemently opposed to indy.
Unionists went on about North Sea Oil ( Brent Crude). It was at c. US 99 in 2014, down to US 43 in 2016, but is now back up to c. US69 per barrel. So, maybe they'll go back to Gers.
Or just quote the latest BBC SNP bad story.
Groue Beater says:
Colin Alexander: the Eton educated SIU clan chiefs / "Scottish" aristocracy writing in.
I know three clan chiefs personally, all Eton educated, and landowners. Guess what – they all believe Scotland should be independent.
Anyhow, you dodge the scorn in my comment. You've too much to say and next to nothing of substance worth saying. But like your predecessor on here, and the one that will follow you when you disappear, they whine the same dirge. And like you there's an unmistakeable mixture of arrogance and aggression in their posts.
heedtracker,
Guardian reader with a Slovene (ex-)girlfriend, you are living proof that not all people in Scotland are stupid.
Only the cleverest people like yourself and some of your pals here would think I am a Tory.
Rock (22nd June 2014 – "Good faith and bad practice"):
"I could perhaps agree with you about Scottish Tories but Ruth "line in the sand" Davidson IS NOT TO BE TRUSTED ONE BIT, to say the least.
She is a total bully on TV debates and her tone at FM Questions is disgraceful.
She came 4th in her constituency with less than 10% of the vote yet she gets the privilege of abusing and insulting the democratically elected First Minister.
She is worse than an English Tory and represents everything that makes the Tories so hated in Scotland.
And then there is that Tory scum Alex Johnstone —."
yesindyref2 (10th May 2017 – Slicing the shrinking pie):
"I have no idea why so much of the electorate actually respects Davidson, though in fairness I did before they became 2nd party and totally dishonest."
Rock (10th May 2017 – Slicing the shrinking pie):
Some of us can say things with foresight rather than hindsight.
He is an odd choice Petra, Leonardo nobody.
He's like some bloke that's gone up to Scotland from Yorkshire on his hols, to visit his Scottish rellies, once a decade and somehow he's landed the SLabour leader gig, a SLabour Britnat Zelig, Zleonardig, if you will.
packhorse pete says:
This. Personally I never read his stuff now. Just scroll by and ignore. Very easy.
Tinto Chiel says:
Petra, Kevin, like Mr MacWhirter, has made a nice wee living with his columns/publications in recent years talking about independence. He is boilerplate Labour, always talks about Socialism's Holy Grail and has never, to my recollection, said anything positive about the SNP or SG. I suspect the thought of a Corbyn government gives him a multiplied organism.
Ian's just a nice middle-class SDP boy who pretends federalism is the answer, while he makes snide disaster-filled comments re the SNP. Two books (and counting) about independence, though.
They might not want an independent Scotland really, but, heh, ho, it pays the bills to kick the can up and down the road.
Could we all just take Ronnie's advice and ignore the tiresome bacon rolls?
They'll go back to their mammies in their dressing-gowns for some chicken soup and a cuddle soon enough.
"Plenty of time for slagging the SNP once the job is done",
It's meant to be constructive criticisms to sort things BEFORE any indy campaign begins. Heedtracker challenged me to give my alternatives, not just criticise. I did.
We've discussed PFI / NPD .
We've established I'm not the only person who was confused by the differences.
Hopefully, thanks to helpful commenters people are learning why in theory NPD is preferable to PFI.
When the £ figures are available and compared,hopefully the differences will be very clear.
If people have a problem with the % of posts being mine in this thread, please write more comments Wingers.
Please readers, join in the debates / discussions. Don't be put off by the occasional nasty remark.
It would be good to hear from new commenters.
Awe, Rock, you actually think I give a shit what you think.
You mess with my man Rab Peffers, you mess with me, not that Rabbie needs it:D
'Last weeks Sunday Politics:
Thanks for that Petra. I don't watch the programme but I hauled it up on iplayer there and youre absolutely right there it is (I make it 55 mins in).
That sounds pretty sensational to me but I never heard a cheep about it before and I do monitor Imperial Radio and the BBC website a fair bit. Even the National didnt seem to pick it up.
I'd better withdraw the "cowardly" thing all the same while I remember, it was an old-fashioned troll (designed to elicit a response) and not actually merited.
Glamaig
The BBC's Brian Taylor's view on Brexit and LCMs:
His view, In brief: "No, Holyrood cannot 'veto' or 'block' Brexit"
It would. But if you do, can you please not endlessly repeat yourself over and over and over, like what our tory yoons Colin A and Rock here do do.
Variety is the spice etc:D
Here's a couple of good things
https://twitter.com/Ms_Longo_learns/status/955052982046900224
https://twitter.com/SNPStudents/status/955150051533377538
Petra @ 7.32 pm
I don't know about the Sunday Politics show and Prof Tomkins' position on the EU ( Withdrawal Bill) but in a report in the Herald on 9th Jan it was reported that the prof and Murdo Fraser both signed up to a unanimous cross party report savaging – Herald's word – the EU (Withdrawal) Bill.
The report in question was an interim report of Holyrood's Finance and Constitution Committee. Guess they are members of the committee. The whole committee think the Bill as drafted is incompatible with the devolution settlement.
So, one way or another his position is a matter of public rec
As is the classic Trollers way when you ignore them they resort to abuse in the end , just to incite a response .
Chin up Mr Peffers we're infuriating in our silence lol.
(spotted on Rev's twitter)
Well who knew?
https://archive.is/oxbQk
So we're all clear then? Labour, once again, reneges on home rule. The federal solution. The VOW? A complete lie. Any such pledge made in future by Labour. Any pledge at all of a devolution 'journey' whether it be at cooncil level on upward. That pledge was, is and always shall be utterly worthless.
M'kay?
"Ian's just a nice middle-class SDP boy."
Apologies: meant to say "LD boy" (age-related inaccuracy). The SDP were formed by some right-wing Labour MPs, and an early defector was George Cunningham, the Islington Massey Ferguson who thought up the 40% Rule for the 1979 Devolution Referendum.
It's hard to keep up with their infamy, these Yoons, tbh.
I know one thing for sure. Leadership is a heavy burden to bear. Nicola is a great leader.
Throughout history great leaders have overcome the odds by sometimes going against the concensus of all who surrounded them. Napoleon, Alexander, Caesar. All had one thing in common. They did it their own way.
Do not for one second believe that Nicola does not know what we all want and how desperately we want it. I am sure people close to her read all the Indy sites and know exactly how we all feel. She herself has fought for indy since childhood.
Now is when we must trust her to lead the way and plan our strategy. Yes we are outnumbered in terms of resources, but that never stopped Horatio at the bridge or the Spartans at Thermopylae .Yes we are impatient, but we have waited for hundreds of years.
A great leader always knows how to win.
@ Legerwood at 8:45pm ….. "His (Tomkins) position is a matter of public rec…"
I've come across articles, etc, to that effect Legerwood and was kind of surprised to hear that the Holyrood Tories were in agreement with the SNP (re. Withdrawal Bill) and then we found out that the Scottish Tories had voted at Westminster in support Theresa May. Hearing / reading about the first in the Media, whilst the second voting fiasco information was suppressed.
However I haven't read / heard that if "legislative consent motion is not passed by the Scottish Parliament it will (would) be difficult to deliver Brexit at all." That's quite a statement, imo, and much as I don't like Tomkins he IS a Constitutional expert.
If anyone is interested in what Tomkins has to say, check out the video at 7:32pm.
Scotsrenewables
I tend to agree. It is perfectly possible that Colin is just a grumpy indy supporter. Every workplace, organisation or club has a "glass half empty" character. Given that, there really isn't any point in threads tail spinning off topic to go over the same issues time and time again. Colin isn't going to cheer up and the SNP are the only serious game in town. We know where Colin stands, leave it at that.
If it is concern trolling then it is doubly the case that there is no point in getting bogged down in endless discussions. However, if it is concern trolling then at least it is a bit more sophisticated than that complete eedjit that used to pretend he was a Green.
Meg merrilees says:
There was a TV interview with Lord Robertson after the 13 Tory tractors had voted against Scotland and he said that now that the bill would pass to the Lords, it might be possible to get some amendments passed to alter some of it for without it, as it stood, it would need a Legislative Consent Motion and if that was not forthcoming from Holyrood then it would scupper Brexit.
I didn't hear the interview repeated, and there has certainly been almost zero mention of the 13 tractors and their dastardly deed – it was most noticeable on Brian Taylor's summing up of FMQ's last week.
He reports much of the exchanges between Nicola, (t)Ruthless and Leonard but did not mention a single thing about Nicola's brilliant silencing of the Tories when she stated that the 13 were "Lobby -fodder" and should be ashamed of themselves for their actions against the Interests of Scotland.
Come on Donalda – thought you were going to ensure there was no bias – simply not good enough!!!
That's always been the BBC's greatest power, just not making public issues that may influence.
Most people can make a decision with what info they are given but if we do not know what's happening…
Its not just politics either. How many of the BBC knew about sex crime carried about BBC stars and buried it, let alone monsters in Westminster?
I was surprised when I read the article but that together with the behaviour of their MPs when it came to voting against the amendment suggests they are trying to play both sides of the street on this. Different positions for different audiences but all the time one aim: doing down Scotland.
@Heed: "It is perfectly possible that Colin is just a grumpy indy supporter."
Oh good grief Heed I can't believe you're saying that. And yet I agree wtih you, it is possible. It's possible that, as he said before, he's pointing out all the arguments that people will use, and giving us practice in arguing against them.
I find it annoying because the forums are full of those arguments from unionists, I also look occasionally at the blogs of KH and FW, and their twitter feeds and you see that stuff there. Or on Sturgeon's twitter. But in fairness you get it amongst Indy supporters. CM for instance, whether seriously or as a devil's advocate, Peat Worrier with some, and posters at times on Wings with one or two. And then there's Bella / commonspace. Like otheres I'd prefer to get the likes of Nana's links, Petra's with info, and others with postitive useful stuff – and even "just" opinions and views.
The problem with Colin is he uses all the negative arguments!
Coco..how about turning up at the next wings meet up ?
You could be a guest speaker, or are your little balls only big on the internet you rancid fud ?
Another noteworthy thing about the Tomkins clip on Sunday Politics is that he is actually complementary about the efforts of the SNP and Scottish Government, which is not something I ever heard before.
Correction.
It was not Lord Robertson but Lord Foulkes who was talking about the bill. The clip is viewable on this link from Smallaxe this morning:
I think i slightly misunderstood as he says that without the LCM from Holyrood, Lord's would not vote for the bill and that action would scupper Brexit
slightly subtler version of my previous post but the same outcome.
I did study Heed's Slovene girlfriend a bit during the first Indy ref, specifically his blog "Notes from North Britain
On Law, Politics and the British Constitution"
He's a weird one, and to be honest, if I was the Tories I wouldn't trust him to stay on my side. He might just as easliy get something into his head and change his mind completely, even including this Independence thing.
Totally agree with you about Nicola.
She is a great leader, knows her stuff and is highly respected internationally.
She certainly has a canny head on her shoulders.
She can steer us to victory.
@ Mike Cassidy, re: NPD/PFI
Thanks for posting the link. I'm like you, hoping for a more 'lay' version, but it was good to read anyway.
Oh good grief Heed I can't believe you're saying that.
I didn't, that was HandandShrimp.
Colin A's merely a yoon with a trick or two to play and then its all SNP bad.
Sorry about that, wrong attribution. Yes it was H&S.
Good/bad one from Ireland. Our neighbours are losing interest in Brexit but they're not going to give up control of their Scotland region. Sorry this one's not about you Colin A.
https://www.irishtimes.com/business/economy/why-the-uk-s-juvenile-brexiteers-can-t-see-what-s-staring-them-in-the-face-1.3363037?mode=amp
schrodingers cat says:
yawns
@Rock says: 21 January, 2018 at 7:32 pm:
Hilarious, now Rock just you tell the Wingers any instances when I have said, wrote, commented or claimed that Scotland, (country or kingdom), was sovereign after 1320.
You cannot even read and comprehend either Scottish language or even standard English properly for I have never claimed such a thing in my life. I'll say it again just for you, Rock, The people of Scotland are legally sovereign.
What is more the High Court of Scotland has ruled that to be so.
However, here is an article from, "The National":-
http://www.thenational.scot/news/15187409._The_people_of_Scotland_are_sovereign____MSPs_give_Sturgeon_mandate_for_second_referendum_on_independence/
Now away and think about what I have actually claimed you silly unionist numptie and it most certainly is NOT what you have claimed on this thread today, (Well yesterday now, as from two three ago).
Labour to vote with the Tories to install Ruth Davidson as FM if the SNP are in a minority at the next election
I'll just leave that to allow time for folk to ponder
@ Macart at 8:48pm ……."Labour reneges."
Good one Macart. To my mind Corbyn's our greatest threat now, one of many of course, so we should focus on getting as much data on here to highlight that he's a waste of space as far as the Scots are concerned. Let the Scottish youngsters know what's gone on previously with Blair, Brown, Unionist politicians in general and what our future would be like with Mr Indecisive U-turn Corbyn of the Divided party.
@ Meg at 9:52pm and 10:51pm …..
Thanks for the info Meg. One thing for sure is that I'm feeling more relaxed about this 'power grab', since I came across Tomkin's comments yesterday, in that it looks as though the SNP Government has MUCH more control over this than I realised. 'Constitutional crisis' gets mentioned frequently, by a number of characters. I say bring it on. Bring the Constitutional crisis on. That's what we're all waiting for.
@ Legerwood at 9:59pm ……. "Different positions for different audiences but all the time one aim: doing Scotland down."
Spot on Legerwood. The MSM is going all out to show Ruth Davidson and her wee Scottish cabal in a favourable light, whilst going hell for leather to suppress information about the Scottish Tory tractors in London. Too bad for them that we're no longer reliant on the MSM for our information.
@ yesindyref2 at 10:22pm …….. "Tomkins – he's a weird one."
I get the same feeling as you do about Tomkins, yesindyref2. There's no doubt that he's highly intelligent, while he can lack in commonsense as per some of his strange ideas that I came across on Hansard. He's definitely, cognitively, head and shoulders above the Tory dross at Holyrood and probably even in relation to the inept Tories at Westminster too. Maybe he's actually scunnered with Davidson and May's, a couple of doughballs, antics? Then there's Arlene Foster!
He swung from being a fervent Independence supporter to becoming a hard-line Tory, but I wonder if he's having second thoughts now? I would imagine that he's highly ambitious and had set his sights on being Westminster PM or a lardy Lordy eventually, however maybe even he can see that England is finished now? Was he a Remainer? If so, now totally disillusioned with the Brexit mess? Add to that, he's not a supporter of the Monarchy and that his US wife seems keen on bringing their four children up in Scotland. Is it possible that he's keeping his options open or even looking forward to being PM of an Independent Scotland?
@Petra
Thanks for the support on that, sometimes I think I'm getting a bit wacky. There are two or three other Conservatives I've got my eye on, no intention of naming them as I'd be laughed off Wings.
There's 7 months of turbulence for Union supporters, a lot of mind and heart-searching, and the thing is there's maybe only one problem for Indy supporters, and that's that Brexit gets thrown out, Article 50 withdrawn and that accepted by the EU-27. Even then, Indy ref 2 could go ahead as people may well think it was so close we need to be in control rather than apparently helpless.
It means we can be fairly relaxed, whereas a lot of Unionists will be having a maelstorm of emotions.
Having said that I'm maybe not the best judge, as I was sure there'd be some Labour politicians coming over to YES in Indy Ref 1 and was savagely disappointed nobody prominent did. Maybe some were close, and who knows how they voted in the privacy of the polling booth.
@Dr Jim.
If we are not independent before the next Holyrood election, there will be no meaningful budget left by the time WM slash and burn it.
The unionists do not want control of Holyrood, they KNOW they are not up to the job.
@ yesindyref2 ….
There's a couple of others I've got my eye on too, but will we ever know how they'll actually vote when the time comes?
@ Geeo ..
We know they're not up to the job, as do they Geeo, but it would depend on who was ruling the roost at Westminster as to how well they would (seemingly) do. With their lackey's in charge at Holyrood we'd probably find that our pocket money would be raised overnight and our NHS, education etc, etc would be fantastic, as per the MSM.
The hope would be it becomes important enough for them to become openly verbal about it before the Ref. It could happen, maybe, perhaps, possibly 🙂
CameronB Brodie says:
I watched the interview with Leonard earlier, what a plank.
@AnasSarwar
I know you support the cause of the Palestinians but assume you also support the state of Israel, so why not a Scottish state? You do support the principle of universal human rights? Are you a British nationalist?
Relocating the British subject: Ethnographic encounters with identity politics and nationalism during the 2014 Scottish independence referendum
In this article, the author uses an ethnographic encounter in the aftermath of the 2014 Scottish independence referendum to explore questions of identity and nationalism in Scotland. During that encounter, he was confronted by his own, sometimes contradictory, thoughts and feelings about Britain, Britishness and Scotland. Taking inspiration from a genre of social scientific writing called 'ethnographic memoir', the article is his attempt to work through, and make sense of, those thoughts and feelings, by drawing on the work of both social anthropologists and sociologists who have written about identity, nationalism and legacies of empire in the UK.
Following particularly the work of the social anthropologist Georgie Wemyss, who argues that contemporary discourses around 'Britain' legitimate what she calls 'the invisible empire', it is suggested that affirmations of Britain during the independence referendum helped empower an insidious, but largely taken for granted, discourse of imperial nationalism. This insight allows the author to locate the source of his own disenchantment with the identity label 'British'. The article concludes with consideration of some of the wider implications this might have for a subdiscipline that calls itself the Anthropology of Britain.
http://journals.sagepub.com/doi/abs/10.1177/0081176917693538
The Invisible Empire
This book offers a significant and original contribution to critical race theory. Georgie Wemyss offers an anthropological account of the cultural hegemony of the West through investigations of the central and pivotal constituent of the dominant white discourse of Britishness – the Invisible Empire. She demonstrates how the repetitive burying of British Empire histories of violence in the retelling of Britain's past works to disguise how power operates in the present, showing how other related elements have been substantially reproduced through time to accommodate the challenges of history. The book combines ethnographic and discourse analysis with the study of connected histories to reveal how the dominant discourse maintains its dominance through its flexibility and its strategic alliances with subordinate groups.
https://www.routledge.com/The-Invisible-Empire-White-Discourse-Tolerance-and-Belonging/Wemyss/p/book/9780754673477
The Social Consequences of Brexit for the UK and Europe
This article examines the 2016 Referendum on the United Kingdom's membership of the European Union and draws on initial research into the reasons that the UK voted to leave and demographics of the leave vote. This initial analysis suggests that the Brexit (British Exit) vote reveals wider and deeper societal tensions along the lines of age, class, income, and education (Goodwin and Heath 2016). By providing an account of the background and events of the referendum, this article asserts that the vote was a case study in populist right-wing Eurosceptic discourse (Leconte 2010; Taggart 2004), but it also reveals strong elements of English nationalism (including British exceptionalism and social conservatism) in parts of British society (Henderson et al. 2016; Wellings 2010). Given this, the article begins to make sense of Brexit from a social quality perspective and outlines a possible social quality approach to the UK and Europe post-Brexit.
https://www.berghahnjournals.com/view/journals/ijsq/6/1/ijsq060102.xml
Mr Corbyn has a woeful track record on Scottish issues right enough.
https://weegingerdug.wordpress.com/2017/08/25/a-kinder-politics/
Something from August last year.
K.A.Mylchreest says:
There's a most interesting Irish analysis of the English perceptions behind Brexit here :
https://www.irishtimes.com/opinion/brexit-is-a-collective-english-mental-breakdown-1.3356258
It concludes :
"On June 23rd, 2016, the English rejected that offer and opted to continue living the fiction of splendid isolation that sustained the UK and the British empire before it, and to continue denying the Scots and the Irish a will of their own. Any recovery from this collective mental breakdown will involve treating it in the light of its deep historical causes. Not until there is a separate English parliament, giving England at last the distinctive political identity it has shunned for 300 years, will the delusions that led the country to Brexit finally be dissipated by contact with reality. Perhaps then, with their psychosis healed, the English will apply to rejoin the EU."
Well worth reading the whole short article IMO 🙂
mr thms says:
Has this been posted before?
The 'original' SNP Political Broadcast from 2011
http://www.youtube.com/watch?v=POP-DjI0wvw
"What has the Scottish Government Ever Done for Us?"
Thanks for posting mr thms.
Marco Biagi mentioned this on Rev's feed the other day, about how it was almost identical to the current one that's causing the stooshie wi the unionists. I wonder why that one didn't produce anything like the furore that the latest one has?
I can see the latest one is set in a more recognisable contemporary set up, perhaps it's this that drives them mad? That it portrays the younger generation appreciating what the SNP has done for Scotland, whereas the older ppb is in a much more 'traditional' pub setting and the casting was certainly older too?
But still doesn't explain the yoon froth on this at all…yes, they are 'latching' on to the 'character' likeness of Torrance et al, but perhaps the 'divisions' where not in such stark contrast pre 2014 referendum, from a unionist perspective?
They don't like the idea of their naysaying as represented by the bearded one in the ppb being so outnumbered by the newer generation? Very threatened they are by the prospect of a majority for independence coming up through the youth vote?
If the Tories have traditionally had the 'well off voters' and Labour all the rest (I'm sorry, just can't take the libdems seriously) and as per R Peffers reasoning in terms of the cronyism within unionist ranks in Scotland. Then it's gotta be killing them knowing that a new and more sussed out generation know the 'Union' holds nothing for them and it is the SNP who have taken care of 'everyone' in Scotland. That demographic aren't replacing the unionists ranks in Scotland as they once did?
Perhaps this does explain some of the quite bizarre over reactions that we've been seeing of late…hmmm.
Nicolas Boyle is also the author of the excellent article in The New European about the English not being team players.
http://www.theneweuropean.co.uk/top-stories/the-problem-with-the-english-england-doesn-t-want-to-be-just-another-member-of-a-team-1-4851882
Hard to believe it was written a year ago…
I'm curious Yesindyref2 and Petra about your "closet" YES sympathisers…
I still think there is a time/ opportunity coming when it might be possible to gain some purchase over unionism, not the BritNat knob ends perhaps, but the actual thinkers, via the argument that a Brexited England outside the EU and an Indy Scotland inside the EU, might actually be positive result for all, and create the firm foundation for a soft-Brexit type trading arrangement for England – with EU concessions and trading flexibility argued and derived from the former unitary trading conditions of the UK.
Scotland would of course be Sovereign and formally independent, but there is a possibility that some kind of buffer status might exist so that the EU doe's not lose England's trade, England trade does tailspin to 3rd world status, nor deregulate itself to toxicity, and that the end of the United Kingdom has a small chance of a constructive agreement rather than an acrimonious dispute.
I know the issue is heavily laden with problems, especially considering England's exports are predominantly weapons, dodgy economic skullduggery and manipulative propaganda, but from Scotland's perspective, we should not underestimate the value of cooperation in bringing the UK to its end.
Ireland has surely given Scotland an illustration of how EU membership adds clout to their bargaining position, and economic obscurity on the periphery of Europe will sooner or later focus the minds of the English.
Morning all and a big thank you to Smallaxe for posting my links while I've been out of action. I will link when I can so here we go
https://handfulofearth.scot/a-fuel-poverty-free-scotland
https://michaelgreenwell.wordpress.com/2018/01/21/not-to-be-forgotten/
https://itisintruthnotforglory.wordpress.com/2018/01/21/what-did-the-unionist-media-never-do-for-us/
https://scottovoce.wordpress.com/2018/01/21/a-dark-web/
http://newsnet.scot/news-analysis/carillion-sunny-outlook-bust-5-months/
http://www.lbc.co.uk/radio/presenters/alex-salmond/lord-kerr-predicts-second-eu-referendum-in-autumn/
http://www.stevenpaterson.scot/tory-speaks-forked-tongue/
https://skwawkbox.org/2018/01/21/mays-new-unit-fb-news-change-signal-estab-clampdown-who-decides-whats-trusted/
Top Tories boast of close links to Brexit politician in 'Chinese cash-for-access' sting
http://archive.is/BYuwo
The Telegraph reports that crime is spiraling out of control The Tories have cut 22,000 police officers and closed hundreds of police stations since 2010. Former Met Chief Inspector, @Peter_Kirkham, says Theresa May is responsible
https://twitter.com/UKDemockery/status/955222441571377153
https://www.politico.eu/article/british-businesses-call-on-may-to-stay-in-customs-union-with-eu/
Interview with financial journalist & author Ian Fraser on RBS' turnaround unit GRG. Full interview available 22nd January 2018
https://twitter.com/RealMediaGB/status/955121023665524739
@Nana
Good to see you Nana. Hope you're keeping better and kettle's on. 🙂
Herald comment:
I've said that for a while, that there's a lot less meat ( almost none) in Fray Bentos pies nowadays.
Now I know why: it's those Baxters, major donors to Better Together, that make them.
("Gordon Baxter, late President of Baxter Food group. Amount: £10,000
Audrey Baxter: "My father believed firmly all his life in the United Kingdom.")
The ones that told us: Vote NO to guarantee Scotland's continued membership of the EU. We were told we'll get Devo to the Max with a No vote. Those promises also turned out to be all gravy with no meat.
Now, Scotland is being dragged out of the EU against the will of the majority of the sovereign people of Scotland.
Also, not only did we not get Devo to the Max. We got EVEL which created second-class Scots MPs.
Now, the UK Govt is preparing to totally betray the whole devolution settlement by grabbing back DEVOLVED powers from the EU.
It was supposed to be that what is devolved is devolved.
Thus, devolved EU powers should return to the devolved parliaments.
Just to make a terrible situation worse, Brexit Bill amendments are being done by the unelected and undemocratic House of Lords.
Baxters!
John McTernan 'our political strategist' says shortbread radio just on defending PFI schemes and suggesting it is ludicrous that existing schemes should be reviewed / or bought out.
No 'Dear John' letter for him yet from Auntie! 🙂
Wings has always been a very sensible platform, so, let us show magnanimity this morning and give praise to someone we all too often denigrate.
How refreshing to see his few minutes of TV stardom last week has not changed wee Davie Torrance one iota. He's back in The Herald this morning, spouting the same Jackie Baillie as always.
What a trouper.
Well I suppose we cannot expect much less in this "democracy" which is the UK. On youtube now there are appearing videos of the SNP PPB, that are showing the debunking of the claims made, by the incorporation of lies. Black arts at work I suspect.
One particular one comes in numerous different headings in order to mislead the viewer into looking at it. There is no end to their deviousness.
Nana Smith says:
Good morning Macart 🙂
Liz Rannoch says:
Good morning Nana, lovely to have you back. Thanks to Smallaxe also, what a team! Links and porridge, yummmm. What better way to start the day?
Am I the only one who doesn't see the 'lookylikey' as Mr T?
For me, he's more like Fluffy's wean.
I really do not know why good posters on this site, who should know better, can't seem to avoid giving the "Rock" and "C,A", all the attention they do not deserve. They are a disruptive force and should not be given the oxygen that many are so are willing to give them.
No doubt with good intentions and trying to debunk what these people have said. However they only persist because of the attention given to them.
call me dave . Left left left everybuddies oota step but oor Jock . McTernan Half Mooth For Rent , he used tae be ah full mooth but who want's tae listen tae regurgitated keeeh .
Morning Liz. Hackalumpoff posted a wee song for Davey on off topic. It's a love song of sorts. Go have a listen.
Morning Nana I hope Norman has fluffed up your pillows & keeping you supplied wie tea , take it easy XX .
Marie Clark says:
Nana, great to see you back, now just don't overdue thing. Many thanks for the links, and to Smallaxe who kept them going for you.
I have to say I second all the folks who are asking for the Rock and coco show to be ignored. They seem to have taken over the thread at times, boring as hell, same old same old. No use whatsoever trying to debate points with them. Makes the thread so much shorter when you have to scroll past them, and the replies. Not good folks.
Giving Goose says:
call me dave
I heard that phrase "our political strategist" on radio Scotland this morning when introducing McTernan and I thought that I was hearing things.
Well, well…
A Freudian slip or a statement of genuine fact?
And whose "politcal strategist"? Radio Scotland's?
If so, then Radio Scotland is acting like a political party. Actually, we shouldn't be surprised, because it is a political party, with a manifesto and all the trappings.
The Anti Scotland Party AKA BBC Scotland.
Ronnie, I'm waiting………..
Morning Marie. You are right, boring as hell. It's the easiest thing just scrolling along, keeps me sane!
Oooh Nana, gid job I wisnae eating my porridge! Loved the wee quiet bit at the end – great song, thank you.
@call me dave says:
Two articles in the National today very relevant on this:
http://www.thenational.scot/news/15886943.Labour_are_still_haunted_by_the_fiasco_of_PFI_deals/
http://www.thenational.scot/news/15887130.How_the_banks_perpetrated_the_greatest_robbery_in_British_history/
(The relevance of the latter is that the robbery happened on Gordon Brown's watch and he is also the main culprit for PFI).
BTW re one of the arguments above I'm sure there was a report recently that said on average NPDs cost half as much as PFIs. I've searched but can't find it (funny how that happens a lot) but maybe someone here can recall it. NPD I believe is better but still unnecessarily expensive however that is down to Westminster's ban on us being able to raise money in the old fashioned way via government bonds or whatever. The SNP have been forced to use private money and have of course been criticised by many for "just using another form of PFI": which though true is hypocritical and dishonest in most cases as they never mention this is forced on them by the UK government.
@Petra says:21 January, 2018 at 7:32 pm:
" … some of whom think he's a troll. Just wondering why people who think he's a troll are clogging up the site by responding to him and in doing so probably encouraging him to continue to post on here?"
Probably, Petra, the answer is some of us do not think these people are trolls. We know them to be undercover agents of the Westminster Establishment. Whether by personal choice or paid for agents is neither here nor there.
These people are not just trying to disrupt the threads they are attempting to destroy the case for Scottish independence. Trolls have a quite different motivation and Trolls do not espouse any particular cause. These people most certainly are anti-SNP, anti- SG and anti-Scottish independence.
The point of replying to some of their stuff is to draw them out and expose the lack of a case for unionism and a secondary aim is to ridicule them. Note how loud they squeal when you show how stupid they are.
Wee example on this very thread. Rock claims Scotland is not sovereign and that I wrongly claim it is. I have never made such a claim and, without even being challenged by me, Rock immediately becomes abusive and attempts to bully me yet claims I'm the bully.
The actual facts are very clear. No country on Earth is sovereign for countries are just lumps of rock. Sovereignty is the God given right, (initially of monarchs, to rule a population.
In the Kingdom of England the English monarch is still legally sovereign but under only the law of the Kingdom of England the monarch of England's was forced to legally delegate their sovereignty to the parliament of the Kingdom of England. However, there has not been a legally elected parliament of the Kingdom of England since 30 April 1707.
On the 1st May 1707 the Westminster parliament became legally the parliament of a united kingdom and the United Kingdom resulted from a bipartite Treaty of Union.
In Scotland neither the monarch or the parliament of Scotland were/are legally sovereign as from at least 1320 and the Declaration of Arbroath. The people of Scotland are, under Scottish law, legally sovereign and that is still the case today. This has been ruled as such by the High court of Scotland.
The actuality is that Westminster, (always indirectly), claims it has sovereignty over Scotland. Which idea may have been acceptable if Westminster did not, illegally, assume that on the 1st May 1707 it became the Parliament of the Kingdom of England that was renamed as the United Kingdom. Westminster continued from 1st May 1707 as if it was still the parliament of the Kingdom of England but then, by devolution, it became the de facto parliament of the country of England but retained its claim to be still the Parliament of the United Kingdom and simultaneously, (without any elected as such members), the parliament of the country of England illegally devolving the sovereign powers of the country of England to Wales, N.I. and her sole kingdom partner of the Kingdom of Scotland. Ignoring the simple truth that if one of the only two kingdom partners ends the partnership there cannot remain a United Kingdom of any sort. What remains is two independent kingdoms.
The evidence for that truth is contained in a Westminster commissioned paper they produced during Indyref1 that claimed just that. Then, when challenged on this claim we had the Secretary of State foragainst Scotland, David Mundell who openly claimed, "The Treaty of Union, 'EXTINGUISHED'. the Kingdom of Scotland and renamed the Kingdom of England as the United Kingdom".
And here he is admitting that was so on Scottish National TV:-
http://www.youtube.com/watch?v=_OIRUr4R7GQ
Just to also highlight some of the other, more subtle, work of the agents of the union here on Wings who constantly claim the SNP do nothing at Westminster. Here is a recording of a Pete Wishart speech in the Westminster debating chamber that the Yoons shouted down. Yet none of the, so called, Scottish MSM even bothered to even mention it:-
http://www.youtube.com/watch?v=cDN3C76RXZM
fireproofjim says:
Welcome back Nana.
Don't know how you manage to find such great links. Just what we need.
An academic survey of MPs (as reported in the Guardian) has going that a majority of Tory MPs don't want a transition deal which gives ongoing freedom of movement and includes the EJC. They haven't been listening to the EU, again. The only likely transition deal will be full single market membership.
Does this imply the Tories also fantasise about cherry picking a transition deal?
The same survey shows the majority of Labour MPs what to stay in the single market permanently.
It would seem there is a real and wide political gulf between what Tory and Labour MPs want. Yet there is little sign of Corbyn reflecting this.
Thank you fireproofjim 🙂
But but! McTernan is the only game it town..Oh wait!
Ian Murray fears he's on Corbyn's deselection list. 🙂
Good to see you posting these links again Nana.
I hope the SNP has got a battery of PPB'S lined up and ready to go.
I hope Phantom Power films have a load more Journeys to Yes films made.
I hope Chris Cairns has all his cartoons selected for Billboards.
I hope Rev Stu has his next Wee Book ready to rumble.
I hope that YES is ready to capture the winds that the SNP can't hold in its own sails.
I hope that the SNP is properly and thoroughly conversant with Scotland's inalienable sovereignty and has a robust and detailed strategy to affect its implementation.
I hope Europe is alert to the uphill struggle that Scotland faces against Unionist propaganda, unionist placemen in strategic jobs, and the ocean of dark money that is swamping Scotland's political landscape.
I hope that all of us, are ready.
I hope that we have learned from defeat. I hope we have mechanisms in place that when somebody stands up for something, like Prof Robertson, GA Ponsonby and Craig Murray, there is coordinated support from them coming from the highest levels and the controversies which they, and others like them, expose are backed to the hilt with the full weight of a coordinated YES campaign and Scottish Government.
I hope we have our YES narrative and agenda as thoroughly planned and crystal clear in our heads as the BBC and Labour Party have their pre-scripted News agenda ready a fortnight in advance.
I hope we have enough quick thinking and able lieutenants to keep us all coordinated, focused and on target to defeat our Unionist suppressors and end the subjugation of our country.
Corbyn is aware of the position favoured by the majority of his membership and support. His problem is that it's not his favoured position. He's very much an out means out kinda guy. As hard a brexit position as any of his brexit supportive colleagues on the government benches.
Apparently his policy stance though is that he'd deliver exactly the same Brexit as Ms May, only with a sad face and by walking as slowly as possible to the cliff edge whilst wearing slippers.
He really needs to have a sit down with his membership and chat about this position, but he won't. He can't. He needs to project the whole party unity/government in waiting/new, nu, noo even cooler Britannia thing. I'd wish him luck with that, but…(shrugs)
Excellent point: "I still think there is….opportunity…it might be possible to gain some purchase over unionism….via the argument that a Brexited England outside the EU and an Indy Scotland inside the EU"
That's one reason why I thought it's a mistake for independists to campaign for Soft Brexit – after Scotland voted Remain.
The EU seems to assume rUK / England would be the continuing state ( if UK were members), so presumably would continue the Single Market / Customs Union membership ( commonly referred to as Soft Brexit).
Whereas, presumably indy-Scotland would have to start from scratch.
So, if there's an indyref campaign: Unionists would repeat Project Fear, replacing: "indy-Scotland would be out the EU" with :"indy-Scotland would be out the Single Market. So, vote NO to guarantee Scotland remains in the UK Single Market and EU's Single Market."
Whereas, if rUK/England is out( EU and Single Market) and Scotland is going in the Single Market, some in the business community, who overwhelmingly back the Union / Tories for financial reasons, may start to break ranks from the British Nationalist faction, if they see a threat to profits because of Brexit.
Some could even go from Union supporters to neutral or even support indy-Scotland in the Single Market or EU.
Breeks: well said; a plan!
Thank you. Some mornings I may be a little late with links [stiff little fingers,lol] or I might have to get up earlier 🙁
Its absolutely disgusting the lengths these Labour ambulance chasers will go to attack and attempt to discredit this SNP Scottish Government.
To throw Neil biddy's words back into his mouth they aren't fit to be politicians.
@Nana, welcome back!
One more link
https://spice-spotlight.scot/2018/01/22/guest-blog-a-continuity-bill-in-the-scottish-parliament/
Good front page in The National this morning
http://www.thenational.scot/news/15887107.Bridge_from_Scotland_to_Ireland_could_create__Celtic_powerhouse_/
If we had a truly Scottish Broadcasting Corporation this would feature prominently on our TV news channels with expert analysis.
A Road / Rail bridge / tunnel as in Malmo to Copenhagen, which is 16k long, would transform south west scotland and northern ireland's economy and Port Patrick to Bangor is only 20 miles.
You think even the Yoons would be up for that
Morning Fred 🙂
Mr Mundell didn't say that. It was John Mckay who referred to a report that McKay claims "suggests..Scotland was extinguished in international law" by the Union.
Do you know which report he's referring to? ( I don't).
He then asks Mr Mundell: " Are you comfortable with that?" He answers: "Yes, because the Act of Union is irrelevant".
Thus, he neither agrees or disagrees with that assertion. He evades answering, as politicians do all the time.
I can't speak for others, but I've praised the SNP MPs on being much more hardworking ( than Labour predecessors) and a very talented group of people.
I voted SNP at the GE because of my constituency SNP MP ( Carol Monaghan) being an excellent SNP MP. I've already said, I know constituents that were helped by her.
I voted FOR HER, not for the SNP campaign of keeping Scotland Strong at WM
( because that's an oxymoron).
What I've said is that their talents are wasted at WM. As Scotland's participation at WM is a waste of time, because Scotland does not get a fair deal.
Only today, I've pointed out Scottish MPs are now second-class MPs due to EVEL.
But,thank you for your comment and posting the links. An excellent speech by Pete Wishart surrounded by the sneering MPs mocking his every word. ( Reminds me of something).
If you know what report John McKay was referring to, could you tell me?
Also, could you please give details of the High Court case which you refer to : "The people of Scotland are, under Scottish law, legally sovereign and that is still the case today. This has been ruled as such by the High court of Scotland."
Morning Nana! great to see back with acttual news. That Facebook censorship thing in Skawkbox made RT jump too. They think they're going to be kicked off FB.
"The changes will reduce the news that people see overall by 20% – but Zuckerberg's statement that 'the community' will decide what constitutes 'trusted' are problematic – which community?
Who can be trusted to decide who can be trusted?"
I have an alternative candidate – the person who posted this on bbc:
Learn how the BBC is working to strengthen trust and transparency in online news
http://www.bbc.co.uk/news/help-41670342
SNP PPB viewing figures on Twitter almost at 250,000.
In case you missed it:
Twitter: https://twitter.com/twitter/statuses/954040730686812160
Youtube: http://www.youtube.com/watch?v=Yy8dEz-1upM&feature=youtu.be
Let's keep the momentum going. The BBC really won't like it. LOLZ!!!!
Great summary of the ammunition we need for the next Yes campaign.
If I may add a couple of items.
1. Above all. A powerful, weel kent Yes campaign leader who is combative enough to withstand the inevitable media attacks. First thoughts -Angus Robertson would be good – pro Europe, bilingual in German, ex MP. Well liked.
There are several others who would be good. But appointment must be soon.
2. A short and hard campaign with less but better quality literature than Inde1. Preferably through the spring and early summer of 2019 to allow for evening campaigning and good weather for rallies. Vote in say July/August 2019. (2018 is too soon to properly organise and Brexit result must be clear).
Morning Heedy 🙂
Who decides what the truth is or is not, why the Tory government of course. The fakers will decide.
We will soon see how far this inquiry into child abuse is allowed to go. Westminster will be watching closely, of that I am certain.
Starts today
https://www.iicsa.org.uk/live
Great timing and all. 🙂
This is the perfect cue for the SNP to take back the initiative (let's face they lost a fair bit with BREXIT). Here's yer chance.
Now commission a full-scale feasibility study (socioeconomic, environmental, cost-benefit, risk analysis – the lot). Show the people you are serious about this.
A bridge to Ireland. Get support/interest from Ireland and NI. Heck even the yoons over there might support it if they thought it would tie Ulster to the UK.
The yoons will scoff. Boy will they scoff, but their red faces and foaming mouths will give the game away. The SNP sometimes seem a bit over-cautious – playing safe to avoid ridicule. But ridicuyle from a useful yoon idiot ain't a bid thing.
The SNP should be inviting controversy. If they want the media to take notice, they have to be clever.
C'mon, Humza – commission a bridge to Ireland study. Send thousands of pounds on it. Set the agenda and let the greetin yoons try and label "SNP bad" on it. The yoons are auffering cognitive dissonance big time. That makes them vulnerable – very easily triggered. The whole point is to get people taking about what you want them to talk about.
Set the agenda, SNP – stir the pot and do something outrageous and the media lapdogs will not let you down. 🙂
75% of us don't trust Social media for news now and are returning to proper trusted news says SKY news
Only 25% of us trust the internet news and are returning to getting our news from trusted news outlets says the BBC
In other news the Russians are coming to get us and the Chinese are going to get us and of course North Korea is on the verge of getting us
If anyone's in any doubt and worried about everything the best thing to do is trust the existing establishment who got us into this mess to get us all out of it because they're smarter than us and are destined to be in charge as of birthright and position
And it must be true because they're practically screaming it out the telly at us, the newspapers tell us it every day, so does the radio
So it's vote for them and all will be well
Phew! I'm reassured, what about you
An extremely positive (not for the first time) editorial in today's Daily Record supporting Nicola Sturgeon and putting Richard Leonard down very strongly indeed.
I suspect the Record knows that a very large proportion of it readership supports the SNP and independence and it may think it wise indeed to recognise that. Or perhaps the Record understands the way the game is going and wants to be sure it is not backing a loser. Whatever.
Much of the anti SNP venom in the Record is now confined to the Readers Letters but it is the same handful of nawbag scribes much of the time – Hamish Leishman, Andy from Newarthill, Keith Howell, Allan Sutherland, Jill Stephenson and other names which pop up regularly. It would do no harm if we all wrote regularly to the paper.
Actually it is no longer the Record or the Sun (which is also regularly fairly evenhanded) that are our major enemies in the tabloids but the vicious Daily Mail and the infantile Daily Express.
Morning Heedy ?
No doubt. The web's self censoring anyway. Boring stuff gets little attention, big stuff does. Btl newspapers are all heavily moderated/censored. Its probably heat from mainstream media that's making Zuckerberg do something.
Tory THe Graun for example really hates him and that crew dine out on their faux lefty liberal progressive free speech hoohaa.
It used to be, complain all you want, as long as nothing changes, the rich get richer. They're stamping out Holyrood but change is coming.
Can they stop change in England?
From that Salmond LBC thing,
"The author of Article 50, Lord Kerr has predicted that there will be a second EU Referendum in the autumn.
Speaking to Alex Salmond Lord Kerr said: "The parliamentary row of the autumn will be when the government bring back an outline, a framework, of the terms they think they can get for a permanent settlement.
"If it doesn't look very good, quite a lot of people in the House of Commons and House of Lords will say now hang on, this isn't exactly as was promised during the Referendum in 2016.
"It turns out we can't have our cake and eat it.
"A lot of new facts have come to light".
http://archive.is/iGNSa
"But Farage's interventions over recent days are highly significant, and about far more than the rogueish populist yearning to be back in the headlines. He is nothing if not a shrewd interpreter of prevailing political and public moods. "He is dangerous – and brilliant, in equal measure," says one ardent Tory MP from the Remain side. "We know that from painful experience."
That's the Graun not the torygraph and all the usual The Graun black out of anything Scots too, with their, "The brilliant and dangerous in equal measure Farage."
Its going to be a bumpy ride this UKOK summer. So much for Vote NO Thanks for a strong, safe, secure union with the English:D
"… it is no longer the Record or the Sun (which is also regularly fairly evenhanded) that are our major enemies in the tabloids…"
'The VOW' will most assuredly be followed up with 'The VOW2'.
'Nuff said.
If the UK so much as utters a word even similar to VOW their fate is sealed coz even the ardent dense people will reject that
Phronesis says:
Isn't it time that Scotland re-affirms its relationship with WM- particularly taking back its independence of WM with coinage, taxation etc. The 'Union' is thoroughly imbalanced, built on injustice and deception- it should be re-imagined for the modern era via Scotland's autonomy (which will be good for democracy in general).
'Scotland kept its independence with respect to its legal and religious systems, but coinage, taxation, sovereignty, trade, parliament and flag became one. The red cross of St. George combined with the blue cross of St. Andrew resulting in the 'old' union flag. This is popularly called the Union Jack, although strictly speaking, this only applies when it is flown on the jackstaff of a warship'
But the 'Union' has never been about equality or fairness,WM attitudes and behaviours are deeply ingrained;
'Scotland was expressing its deepest anger about its negligent treatment in the 1690s by King William, and by the high-handed action of the English parliament in determining the succession in 1701 in the Act of Settlement without consulting the Scots…
A total of 96 petitions were presented against the union, most in November and December 1706, during the debates on the Articles. They were designed to show to undecided MPs the widespread unpopularity of the proposed terms…
It is possible that the petitions and their messages had some influence in the changes made to the Articles. But the Duke of Argyll, one of the leaders of the Scottish Court party, said that petitions were little more than "paper kites" – a revealing insight into how governments of the day regarded public opinion…'
Pomp and pageantry don't really disguise the underlying historical wrongs of the 'Union' that are still being played out today.
'At about the time the clearances (mass evictions and emigration of Highland populations) entered a new and more intense phase in the 1820s, the high point of what was known as Highlandism was reached when King George IV made a state visit to Scotland in August 1822.
This visit, the first by a British monarch since Charles II in 1650, was deliberately contrived to reaffirm the bond between the Scots and their monarch, which had been challenged so repeatedly in the past.
It could be argued that the pageantry of 1822, which at the time was called "one and twenty daft days", glossed over what really was happening in Scotland (and England) at this time – clearances, economic hardship, radical discontent'
http://www.historic-uk.com/HistoryUK/HistoryofBritain/The-Act-of-Union/
http://www.parliament.uk/about/living-heritage/evolutionofparliament/legislativescrutiny/act-of-union-1707/overview/the-scottish-parliament-in-revolt-1703/
http://www.parliament.uk/about/living-heritage/evolutionofparliament/legislativescrutiny/act-of-union-1707/overview/thanksgiving-and-lament/
Last links for now
https://www.rt.com/uk/416621-bbc-funds-bully-pay/
SMEs may be worst affected by Brexit, research suggests
http://archive.is/IQh0W
What will the EU look like after Brexit
https://www.ft.com/content/dec6968c-f6ca-11e7-8715-e94187b3017e
@colin alexander says: 22 January, 2018 at 8:25 am:
And there you go, colin, showing once more your lack of comprehension of reality.
Yes indeed EVEL did relegate Scottish MPs but that was but a tiny symptom of what was really going on right under your nose that you still do not recognise for what it is.
Heaven knows I've pointed it out often enough but it still goes way over your head.
Facts:- There was no actual legal Union of the Crowns in 1603 because the two kingdom's Rule of Law could not be compatible and if there had been then under the English law the one kingdom would have been the Kingdom of Scotland. So between 1603 and 1707 the two kingdoms remained independent.
In 1688 England had a so called, "Glorious Revolution", that was, under English law, a rebellion against their still independent monarchy and they applied their change of monarchy to the still independent Kingdom of Scotland. They were still murdering what they called rebel Jacobites in 1745, almost 40 years after the had illegally forced a Treaty of Union upon the Scots.
However, that Treaty of Union was/is an agreement between only two still independent and equally sovereign kingdoms. It thus former a claimed United Kingdom with both the former independent kingdoms ending their existing parliaments.
Westminster was legally ended as the Parliament of the Kingdom of England but the Scottish parliament was only legally prorogued. The entirely new parliament that opened on 1 May 1707 was the Parliament of two equally sovereign formerly independent kingdoms.
Then Westminster that had increasingly acted as if it were the Kingdom of England parliament forced upon the two kingdom United Kingdom what they claimed was devolved powers.
Thing is they only devolved Westminster's powers upon three of the four United Kingdom's countries but they devolved some Westminster powers to the kingdom of Scotland who were one of the two United Kingdom partners and unlike Wales and N.I. who were parts of the Kingdom of England. What was worse they devolved the powers unequally and differently.
In effect by splitting a two kingdom union as a four country union with England not devolved Westminster has assumed itself the parliament of the Country of England and simultaneously the parliament of the United Kingdom ignoring the legal fact that Scotland is their only equally sovereign kingdom partner.
Ergo, Westminster has made itself simultaneously the, unelected as such, de facto Parliament of the country of England and the Parliament of the United Kingdom but has thus made disappear the Kingdom of Scotland and put the country of England as the master country devolving its assumed English sovereignty to the so called devolved administrations and that includes a Westminster instigate supreme court it claims has sovereignty over Scottish law which two legal systems are actually stated in the Treaty of Union as incompatible.
They did not just relegate Scots MPs as second class members of the United Kingdom Parliament they wiped out the entire Kingdom of Scotland and installed the country of England as the master race who rule the country of England's Scottish region.
As per the claims of Westminster as told by David Mundell:-
@Liz Rannoch says: 22 January, 2018 at 8:57 am:
No you are not the only one who doesn't see the actor as a Torrance lookalike … but I'm not going to say on an open forum what it looks like to me.
I'll just leave it to Wingers to use their undoubted imaginations.
Re the suggested bridge to Ireland:
Where would the Customs Post and border be? Would they be needed?
Would that be a "hard border" or "soft border" ?
Re Irish border, What exactly has been agreed so far during Brexit talks? Do Ireland and the UK Govt even know?
Has anything been put in writing by the EU and UK Govt or will the UK Govt just rely on the Daily Record publish a Vow?
Heedtracker@11.01
I think that Facebook is losing customers. Especially the young and the politics. I'm in my 70s and I've really stopped doing politics on Facebook. It's really for wee photos of the family( which I would never, ever post) and we "thoughts for the day stuff".
Yes people mostly use Twitter now which is quicker easier. "They" will probably attack twitter soon.
We are learning the truth about corrupt politicians and the very corrupt and sleazy British Government and it's treacherous colonial history. We can learn and debate all round the world now and that horrifies the British State and its Allies. It's just NOT good enough that people get to see the truth, the very whole truth!
Joanna Cherry QC MP tweeted
UK govt's refusal to admit the #Article50 notice of intention to leave the EU can be unilaterally revoked is typical of the lack of candour which characterises current UK Govt policy on #Brexit. Read about the Scottish court case here
UK government questions Scottish bid to show Britain can alone revoke Brexit
http://archive.is/2BkhV
@Les Wilson says: 22 January, 2018 at 8:59 am:
"I really do not know why good posters on this site, who should know better, can't seem to avoid giving the "Rock" and "C,A", all the attention they do not deserve."
You are, of course, entitled to your opinion Les, but these people are far from being just simply Trolls who seek attention for the sake of just attention.
Their presence on Wings has a far more sinister motive and if you are naïve enough to imagine them as simply Trolls you are not seeing their real subversive and divisive intentions. What's more you have just shown they have succeeded in their intentional divisiveness.
Robert Peffers @ 12:08,
Oh, give your CoCo promos a rest, FGS!
I think that Facebook is losing customers.
Maybe. Its certainly a giant cash cow, sucking in ad money and worse, sucking it out of the newsstands.
This is probably what's freaking out our press barons. We're just not looking at their products enough anymore with the web, for all sorts of reasons. If youre selling something, you're hardly going to fork hard cash money to likes of stinky olde The Graun, if far more eyes are on facebook sites, let alone WoS.
Anyway we've all had to sit and be trolled by smirking beeb gimps for decades, we saw lately that the beeb is pumping millions in to very rich dude owned newscorps, Scotland the most for a change, but its clearly to keep them in the UKOK tory game.
The mega rich tory that owns Dundee's tory rags, is a now being handed millions from the BBC, for example. Why would a mega rich tory dude in Dundee, suddenly find a big fat juicy BBC cheque of several millions payable to him, land on his lap?
Will Wings over Scotland get some of free beeb gimp dosh:D
And all of its backed up by nonsense like,
http://www.bbc.co.uk/editorialguidelines/guidelines/impartiality
Impartiality lies at the heart of public service and is the core of the BBC's commitment to its audiences. It applies to all our output and services – television, radio, online, and in our international services and commercial magazines. We must be inclusive, considering the broad perspective and ensuring the existence of a range of views is appropriately reflected."
They do love a jolly good larf, do beeb gimps. That pile of BBC "Editorial Guidelines," is actually Pacific Quay bog roll, gets right in there, no messing.
Welcome back nana and thanks once again for all the links, and smallaxe likewise for standing-in during your absence.
Seems there is increasing resistance down south to a full-bore Brexit, maybe even any kind of Brexit, but how that all develops still remains to be seen. What an almighty guddle!So Nicola's current caution in proceeding with indy looks to be well justified.
@UKunityorg
You are deluded and quite possibly a danger to yourself and the general public.
A Brexit in the World, and How It Found Its Place
The rise of insular populism
….Relating to that history, one question regularly discussed in Finland is whether the Brexit Thing is part of a transnational political and social trend against transnational collaboration of the kind that the EU represents. For a time, it seemed as if the world was heading back into the populist orbit of nationalist socialism—a form of exclusionary and often racist nationalism built on the back of fury about ever-growing wealth gaps, austerity, and precarity….
http://www.anthropology-news.org/index.php/2017/10/16/a-brexit-in-the-world-and-how-it-found-its-place/
#Brexit, Europe and Anthropology: time to say something
Brexit means trouble, that is for certain; what is less certain is what kind of trouble. Some might sympathise with the immediate response of Chris Gregory (ANU):
"I guess like most people I am totally bewildered by it all. Is it a major political and economic upheaval or will it be a minor blip in the madness that is 'business as usual' these days? My gut feeling is that the politics of inequality is catching up with the economics of inequality and that the 99% are giving voice to grievances that the hard right are exploiting to the nth degree."
As the co-editor of Social Anthropology/Athropologie Sociale, the European Association of Social Anthropologists' journal (and coincidentally a British citizen who lives in Finland), I felt a responsibility to provide a forum for anthropologists to respond.
"Brexit concerns the state of Europe today; it concerns deeply divisive social, economic and political issues – the kinds of issues that anthropologists work hard to understand in social and cultural terms; and it is about precarity. Perhaps more than anything else, it is about precarity."
http://allegralaboratory.net/brexit-europe-and-anthropology-time-to-say-something/
Waking up to Brexit
I come from a Brexit heartland – the Medway Towns of Kent, a working-class, predominately Tory social-world of small businesses, builders, and weakly unionised and now all but defunct industries (cement and paper). I know that world well, and yet somehow I have again realised over the last few days I had forgotten or dislocated from it too. As I contemplated the seeming small-minded, racist, Imperial residues undergirding the Brexit decision I knew intrinsically and closely too of the disenfranchised anger or ambivalence of those that felt otherwise. So how could I not have taken that more into account? And if I failed in that what hope does an elite political class have of ever connecting to or realising such social-worlds?
The political and populist connections did not happen of course – another socio-economic fault-line. The politicians misjudged the nation. The nation revealed its mistrust of the politicians, in full glare. It was obviously coming: after the Scottish referendum and the SNP resurgence, the distance between the people and Westminster has never been more palpable. And yet, lead by a Prime Minister with a party-political issue to settle, we sleepwalked into it. And now we wake up, too late. Boris Johnson has gotten what he pretend to want – the unexpected lightening-rod for voters expressing dispossession. And he looks chastened.
https://cultureandcapitalismblog.wordpress.com/2016/06/27/waking-up-to-brexit/
Can you give references as requested? ( see above post)
From my own research, I understand the Claim of Right by the Scots Parliament remains constitutional law, as it has never been repealed ( it has been amended, but then I couldn't find on Parliament's website what the amendments are). I
I presume it's not repealed as it's the basis for deposing the Stewarts as Kings of the Scots and installing the Hanoverians.
That document asserts the monarch rules by the consent of the people. (Thus the people of Scotland are sovereign).
If the document has not been repealed, then that assertion of sovereignty of the people of Scotland remains valid.
Thus, whether Scotland remained as Kingdom or not is a moot point. It is largely irrelevant. As you point out the land itself is just rock. The King or Queen, thus their kingdom is not sovereign.
By the Claim of Right, it's the people themselves that are the sovereign people. That is what matters, not the structures of govt / kingdom or physical geography.
Please See my previous post.
Mr Mundell does not say the words you claim. It's John McKay who says them. Mr Mundell does not challenge them, that's true, but neither does he say them or agree with them.
We target one Scottish media, and if it is not reported by another Scottish media we target them and so on.
The people and the media will soon get the message,
If we just target the large media, which we don't appear to be winning the smaller Scottish media get off Scot free, because they never get attacked.
I sense another "SNP BAD" story shaping up:
https://www.pressandjournal.co.uk/fp/news/politics/holyrood/1397453/councils-stunned-by-accounting-blunders/
(Great to see The National featuring the Scotland – Ireland bridge which we'd been discussing, BTW.)
You are welcome Robert. Smallaxe did a great job.
For everyone reading Wings
Don't forget to support this channel. There's an interesting video from Mark Nicol on The Black Douglas, a Scottish border legend.
http://www.trulyscottishtv.com/channel-5—spotlight.html
@Breeks says: 22 January, 2018 at 10:13 am:
I hope, Breeks, that your long list of hopes is met but think many of them will be answered with this :-
https://www.snp.org/constitution
Here is another very interesting speech by the FM:-
https://www.snp.org/nicola_sturgeon_speech_on_brexit_at_the_david_hume_institute
I have no doubt that if we could encourage more people to use the SNP website we would have a much better informed electorate.
It should be the main duty of the SMSM to propagate the Scottish Government's doings for the people of Scotland but, ""The Silence of the Bams", is deafning.
Breeks @ 22 January, 2018 at 5:20 am, said :
Thanks for that. It would seem that the Irish piece I stumbled across was nothing but a greatly abridged version of this more complete article. I highly recommend it, it just makes so much sense, no really! Almost every paragraph has quotable lines, but you might as well all read it for yourselves.
Part of the reason this strikes home so effectively is, I'm sure, because I must confess to having soaked up a great deal of BritNattery in my early years. When I think rationally I (hopefully!) know better, but at a deeper irrational emotional level, the specters of 'British' (i.e. English) exceptionalism still lurk. Perhaps the best we can do is recognise them for the propaganda and conditioning they are. But honestly it's not easy.
No one likes the taste of humble pie, it 'feels' like groveling to admit that all those silly Europeans, with their funny accents and comic antics, are neither any worse nor any better than we 'Brits'. That's the nub of the psychosis — that admitting simple objective equality has to feel like some kind of self-deprecation and submission.
Hopefully this is no more than a symptom of us 'peace babies', from which younger folk are largely immune?
@AnasSarwar, Leonard, Lokie & all other "leftist" planks
What is British nationalism and Brexit, if it isn't the articulation of Imperial identity politics?
@Neil Oliver
Are you sure you have Scotland's best interests at heart and that you're the right man to represent Scotland's unique cultural heritage?
Brexit, boundaries and imperial identities: A comparative view
The year 2016 will be marked as a year in which identity politics reached new levels of significance. Among numerous dramatic events, the UK referendum on membership of the European Union has brought many issues of interest to archaeologists to the fore. These range from entirely contemporary concerns, such as the future of research funding in Britain, to topics of more longitudinal significance, including the interactions between different identity groups in particular economic and political circumstances. In this paper, I wish to explore aspects of the distinctive position of Britain as an illustration of identity dynamics in the long term, focussing on the relationship between imperialism and identities and viewed through the lens of recent work in Border Studies.
Brexit can be seen as the culmination of the collapse of the British empire, and transformation of British identity, in the post-Second World War era and the particular dynamics of this process invite comparison with Britain's earlier position as one of the frontier provinces of the Roman empire, especially in the 4th and 5th centuries AD. This comparison reveals two paradoxical dimensions of imperial identities, the first being that so-called 'peripheries' can be more important than 'cores' in the creation of imperial identities and the second that such identities can be simultaneously ideologically powerful yet practically fragile in the circumstances which follow imperial collapse. Such insights are important because, at a time of apparently resurgent nationalism in many countries, archaeologists need to work harder than ever to understand identity dynamics with the benefit of time depth.
I don't trust the daily stranger or Murdoch's sun an inch, of course they will give a balanced editorial from time to time.
But they are no friends of ours.
In my opinion the only reason we are seeing the daily stranger attacking Leonard is because he is a Corbynista and it wants to see Corbyn defeated and replaced with an establishment Labour figure.
Its for sales Peter. Never forget these rags are businesses that have to make a profit first.
Peter McCulloch @ 13:39,
Interesting thought, that. There is stuff going on in both the Labour and Tory parties over Brexit, but you hear nary a peep about such things from the yoonitariat, except very indirectly as in this case.
One wonders how long it will take for the NorthBritLab faithful to twig that their new leader is no more capable than any of his predecessors were of sorting out their communal failure to establish any kind of coherent and forward-looking policy agenda. (Maybe after everyone has had their turn as leader will they even begin to realise they have a problem!)
As for the Tories, if things get any trickier down south, as in the Brexit deliberations in the Great Peoples' Assembly HoL, I could see a real split coming in the Tory Party. The inability of a section to see anything beyond their superior Little England could easily turn to outright frustration with another lot that is business-friendly and is intent on remaining in SM/CU at the very least. This problem the Tories have had over "Europe" has festered for decades, and this year could finally erupt into open internecine warfare.
Meanwhile, we are routinely ignored and trampled over. With a constitutional crisis also likely coming to a head. (Thanks for all the interesting contributions to that topic yesterday evening, BTW.)
Time people up here realised that things just can't go on like this. We need to get out, and the sooner the better.
Yay, wife just got the letter saying tests show (huge) cyst was benign. Well done Mrs YIR2! Letter dictated 23rd December …
Delighted for you both.
Yesindyref2 @ 2.16
Great news indeed, you must both be so relived ….. my best to Mrs YIR2.
And you get off the internet and go treat your selves…. By order!!!
@Bob Mack / @Liz g
Thanks! I daresay flowers might be a good idea to start with.
Yesindyref2, great news on Mrs YIR2, what a sense of relief for you both. Delighted to hear your news.
Now I hope that you're going to something nice to celebrate.
Apparently the 'scottish' tory twats hsve been shafting Scotland again !!
Voting down a motion to refund VAT taken from the Police Service in Scotland, by 10-9 (labour ACTUALLY supported the SNP on this for a change).
https://www.facebook.com/groups/zinjanthropus/permalink/1434996996627699/
"….Thanks for that. It would seem that the Irish piece I stumbled across was nothing but a greatly abridged version of this more complete article. I highly recommend it, it just makes so much sense, no really! Almost every paragraph has quotable lines, but you might as well all read it for yourselves."
It might be a pipe dream, but the sooner England comes under new management, management that respects the likes of Professor Nicholas Boyle, then the sooner Constitutional dialogue can become a constructive and objective reality.
It is very difficult to foresee anything constructive emerging from a million years of dialogue with the current brigade of charlatans, liars and cretans inhabiting the corridors of Westminster masquerading as UK government. Unfortunately, I fear the same sophistry and mendacity we see in Westminster Government is endemic throughout the "British" Establishment, especially the media, the "Imperial" views of Eton, but unfortunately also the Courts. An "old school" Brit like Professor Boyle seems sadly out of sync with the modern English mainstream.
You get the feeling the British Establishment will attempt to lay claim to everything which "exists" in the Realm UK, and that Scotland, the "great disrupter" should be impoverished and silenced, and require to establish its credentials and right to exist from scratch.
I fear Scotland trying to negotiate the dissolution of the Union with Westminster will encounter the same problems as Michel Barnier, trying to negotiate the UK's Brexit. Before Mr Barnier gets to negotiate anything meaningful about Brexit, he has first been required to school the UK government about its own competencies and delusions as though they were a distracted brat of child forever looking out the window.
I think it would be quite the Historic spectacle to witness the dissolution of the United Kingdom, with cases presented and being discussed between learned types like Professor Boyle and a comparable champion of the Scottish Constitutional position, but sadly, I very much fear the "debate" if it happens at all, will be yet another Constitutional "whack-a-mole", with one disingenuous BritNat sock-puppet, appearing after another on the good old BBC's "Two Minute Hate" every hour rather than broadcast news or constructive informative debate.
With the likes of Professor Boyle in a position of authority however, you get the sense that the correct, righteous and honourable Constitutional judgement would still be delivered, even if the Scottish delegates had been delayed by a snow or train drivers strike, and unable to make their submissions themselves, leaving Professor Boyle to make both submissions.
Did I say that was a pipe dream? Maybe. Maybe we will see who rules England. The Eton Royal Establishment or the gutter Press mob. Something tells me this could get messy…
Been listening to the radio and seeing the news headlines this morning. Already some early contenders in for next weekends WAFOTW winner.
I foretell next weeks award will go to a 'top' tory unionist politician.
Moving along.
@ yesindyref2:
Great news for you both. Take a wee break, you both deserve it. Then come out fighting stronger than ever lol
Hi Breeks.
I replied to you yesterday.
https://wingsoverscotland.com/a-sense-of-inclusion/#comment-2333502
"Is anus on the wordcatcher list"
Yesindyref2 great news, must be a weight off your shoulder, nae wonder you blew a fusecatbthe Rev the other night 🙂
All the best to u both and remember to give me a shout next time your on Arran.
Yesindyref2
Catching up here.
Pleased to hear the excellent news for you both.
Yes I understand that, it would want to prevent its sales
from falling any further.
But how many pro indy supporters would actually buy that rag after what it did in 2014?
@Robert J. Sutherland
It will be very interesting to see what happens in Labour when momentum purges the Blairites.
Lets not forget Corbyn was lucky in 2017 to up against an incompetent like Theresa that May as prime minister,
most of the media attacks were on her and not Corbyn.
I don't see the Tories making that mistake again, they will replace her for the 2022 General election with some one younger and a lot less incompetent.
Then watch the right wing media in England get tore into Corbyn and his policies.
We've now found out that Keith Cochrane CEO of Carillion had also been appointed as the lead "non-executive director" for the Scotland Office and Office of the Advocate General by David Mundell.
Don't you think that we have a right to know who else is working for Mundell / Westminster? Following a bit of a (short) search, I can't find anything online to that effect at all.
We know who our MP's, MSP's and MEP's are, so why not those who are working at the Scotland Office, especially when their budget has risen dramatically? Is it because the Scotland Office IS the UK's propaganda outfit? Lack of transparency, secrecy and so on?
Does anyone on here have a comprehensive list of employees?
http://www.thenational.scot/news/15811316.Scotland_Office_is_just_the_UK___s_propaganda_outfit/
https://en.wikipedia.org/wiki/Scottish_Office
http://www.gov.uk/government/organisations/office-of-the-secretary-of-state-for-scotland/about
Thanks for the links Nana and great to see that you're back with us. Big thanks, too, to Smallaxe for keeping the show on the road.
Fantastic news for you and your wife, yesindyref2. A massive burden lifted from your shoulders with your wife no doubt feeling much better than she has for some time now.
Rev 3.37 nice to see you getting back to me on that question I was using the dictionary alternative for arsehole seems more politer .
Thanks all for the good wishes. Both doctors had said they thought it was non-cancerous, but one did say with a cyst that large it can come back cancerous. But after they'd saved her life somehow that seemed a relatively minor problem – one at a time! And not hearing anything kind of presumed they'd prioritise people not so lucky, and then Christmas together which might not have happened, and the flu outbreak and all.
But yes, a relief, and life goes on. Managed to get her a card saying "You're a living legend" from the card factory 🙂
"come back cancerous" sorry, that should be "borderline" cancerous.
Thanks Brian. I've skim read it, but I need to read something a few times to really get my head into it. But that seems a terrific level of interest being shown by Europe into circumstances where Scotland was being governed by an unelected Secretary of State.
My immediate thoughts are that it's dynamite, since what would Europe make of our sovereign Remain referendum result being overturned, our devolved preambles being shat upon by Westminster's power grab, and if it happens, drawing the teeth and claws from our Holyrood Parliament.
Bit dated perhaps, but the principles, and triggers which allowed Europe to get involved need much closer inspection…
Cheers Brian.
I have also noted the SNP's literature about Constitution, but frankly it seems lame and anaemic. I hope there's is a helluva lot more work being done behind the scenes than the prospectus would suggest. The constitutional issue will make THE critical difference about whether or not Europe and the wider world can and will recognise the significance of Scottish sovereignty.
Sovereignty should be forming the bedrock of our whole constitutional strategy, not merely be the rubber stamp we intend to apply to a democratic mandate we "hopefully" secure.
Unlike 2014, I hope we put a great deal more thought and preparation into what happens in Constitutional terms as a direct consequence of a Yes AND No result. Of course a YES means Independence, but let us be clear and explicit what a No vote means and doesn't mean relative to our constitutional sovereignty.
Don't get me wrong, I pray to the heavens we never have to worry about another No result, but I feel certain we need a much greater level of awareness about Scotland's inalienable and popular Sovereignty before we risk presenting ourselves with a de facto ratification of a constitutional oxymoron in the same way we did in 2014- a Sovereign people ostensibly trying to rid themselves of sovereignty.
Good luck with that. Many of the bampots think it's as simple as Yes or NO settling everything.
Sovereignty is more than a four letter word, so it's beyond their comprehension.
Glad to hear your GOOD NEWS. and the relief all round must be welcome.
Here's to the year ahead.
Ronnie and Rev
Being cheeky now but what about the 7th planet?
Uran** or do we call it Urarsehole?
O/T Ronnie
re the comment at 3.37
Think I might have just incurred the Rev's Hammers!!!
Just asked him what we should call the 7th planet…
Hey Meg merrilees ~
The seventh rock from ra sun (Ra) is Ur…
Thankx for that xx.
To Learn.
Great news about Mrs YI2. 🙂
Brek oot the bubbly.
North chiel says:
" Breeks" Briandoonthetoon" 0317pm/ " my immediate thoughts are this is dynamite since what would Europe make of our sovereign remain result being overturned" . Agree entirely with Breeks here , this is so relevant to the present situation .Can we please get this printed on the Front page of the "National" ASAP . All WOS readers should read this link !
Best OT news ever. Bet that's all been a nightmare/life affirming experience.
My Grandad likes to say, we're not promised a minute.
Robert Peffers (23rd October 2016 – "All around the houses"):
"What's more is as much an equally sovereign partner kingdom in the Union as is the three country Kingdom of England."
Robert Peffers (24th October 2016 – "A reminder of the obvious"):
"The stark truth is that there are only two equally sovereign kingdoms in the United Kingdom and there are three distinct countries in the Kingdom of England."
I won't bother to add the numerous other posts in which you have claimed exactly what you are now denying.
@Heed, Macart and all.
Slept about 12 hours or more, didn't even have my tea!
It was just great her life was saved, then Christmas and even the week or so after. But then perhaps the worry starts again, could there be cancer, need for chemotherapy, perhaps even a shortened life expectancy? Before that was all irrelevant compared to the possibility of losing her there and then. I never thought she had cancer, seemed to me the other symptoms didn't exist, but who am I to do a diagnosis?
So I guess it's a kind of hidden stress that has now got relief, and it's onwards and upwards.
tiderium says:
Massive over reaction from the BBC to the story on tonight's mis-reporting Scotland. Using a severely disabled child as sob piece to drive home how despicable the SNP are. Parents in outrage, health secretary apology etc, this is the main story on the news. tiny wee bit about 84% seen within 4 hours which appeared to be said grudgingly with a bit tagged on that the target is 95% which was said smugly.
I would rather go an extra 7 miles to get to a newer hospital with state of the art equipment. I have to sometimes travel to Dundee from Perth for treatment which is 28 miles. I have been blue-lighted from Perth to Dundee with Sepsis as there was no beds available at Perth. I Had to wait 2 hours for an Ambulance on fluids, morphine and oxygen all the while knowing sepsis can kill… rapidly. I don't think a 10 minute journey is going to make a major difference to the care of the child. I have also had to travel to Strathcathro a journey of 52 miles a round trip of 104 miles for a procedure. which can be carried out in Perth because it has been… twice. So complaining about a journey of 7 miles or 10 minutes really hacks me off.
← Nothing to see
Taking things personally → | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,872 |
\section{Introduction}
We present a methodology for approximate maximum likelihood estimation that uses approximate Bayesian computation (ABC, \citealp{tavare1997inferring}, \citealp{pritchard1999population}, \citealp{marjoram2003markov}). The method is applicable to a very general experimental setup, valid for both ``static'' and ``dynamic'' models.
Our main question is: since in ABC studies artificial datasets are produced from the data generating model and compared to the observed data according to a threshold parameter $\delta$ (the smaller the $\delta$ the better the inference), what can we do if we are unable to reduce $\delta$ below a certain level? Or alternatively, can we perform inference using a relatively large $\delta$ and fewer Markov chain Monte Carlo (MCMC) simulations, instead of progressively decrease $\delta$ at the expense of using many MCMC iterations?
Suppose we have obtained a very rough approximation to the posterior distribution for the unknowns, if we have at least located the main mode for the posterior can we conduct approximate maximum likelihood inference? Basically by accepting of having obtained a poor approximation to the posterior, except for the location of its main mode, we switch to maximum likelihood estimation by proposing draws around such approximated mode using a special ABC-MCMC sampler.
Let $Y\sim f(y|\cdot,\phi)$ denote a realization from an observable random variable, i.e. $f(\cdot)$ is the data generating mechanism. Depending on the modelling scenario, $Y$ might be observed conditionally on an unobservable random variable $X$, i.e. $Y\sim f(y|X,\phi)$, or $X$ might be fixed and known (e.g. a set of deterministic inputs or covariates). In any case, there is dependence on an unknown (vector) parameter $\phi$. The statistical methods we are going to introduce have general appeal, however in order to set a working framework for the time being we assume to deal with state-space models (also known as Hidden Markov Models, \citealp{cappe2005inference}). We remark that our methods based on ABC are \textit{not} restricted to state-space, nor dynamic models. For example, below we assume $X$ a Markov process, but this is not a requirement nor is conditional independence of measurements.
Consider an observable, discrete-time stochastic process $\{Y_t\}_{t\geq t_0}$, $Y_t\in\mathsf{Y}\subseteq \mathbb{R}^{d_y}$ and a latent and unobserved continuous-time stochastic process $\{X_t\}_{t\geq t_0}$, $X_t\in\mathsf{X}\subseteq \mathbb{R}^{d_x}$.
Process $X_t\sim g(x_t|x_{t-1},\eta)$ is assumed Markov with transition densities $g(\cdot)$ depending on another unknown (vector) parameter $\eta$.
We think at $\{Y_t\}$ as a measurement-error-corrupted version of $\{X_t\}$ and assume that observations for $\{Y_t\}$ are conditionally independent given $\{X_t\}$.
Our state-space model can be summarised as
\begin{equation}
\begin{cases}
Y_t\sim f(y_t|X_t,\phi),\qquad t\geq t_0\\
X_t\sim g(x_t|x_{t-1},\eta).
\end{cases}
\label{eq:state-space-general}
\end{equation}
Typically $f(\cdot)$ is a known function set by the modeller, whereas $g(\cdot)$ is often unknown (e.g. when $\{X_t\}$ is a diffusion process, i.e. the solution of a stochastic differential equation) except for very simple toy models.
Goal of our work is to estimate the parameters $\theta=(\eta,\phi)$ using observations $y=(y_0,y_1,...,y_n)$ from $\{Y_t\}_{t\geq t_0}$ collected at discrete times $\{t_0,t_1,...,t_n\}$.
As an example, $\{Y_t\}_{t\geq t_0}$ may be defined as
\begin{equation}
Y_{t_i}=X_{t_i} + \epsilon_{t_i}, \quad \epsilon_{t_i}\sim p_{\epsilon}(\phi), \qquad i=0,1,...,n \label{eq:measurement-model}
\end{equation}
with $\{\epsilon_t\}$ representing unobservable noise sources (e.g. measurement errors) having distribution with probability density function (pdf) $p_\epsilon(\cdot)$.
In Bayesian inference the goal is to analytically derive the posterior distribution $\pi(\theta|y)$ or, most frequently, implement an algorithm for sampling draws from the posterior distribution. Sampling procedures are often carried out using Markov chain Monte Carlo (MCMC) or Sequential Monte Carlo (SMC) methods embedded in MCMC procedures \citep{andrieu2010particle}. However the last ten years have seen an explosion of methodological advances for the so-called approximate Bayesian computational methods. We propose to use an ABC-MCMC sampler as a ``workhorse'' to obtain an approximate MLE for $\theta$. Notice that (approximate) Bayesian algorithms leading to an approximate MLE have also been considered in \cite{rubio2013simple}. See \cite{grazian-liseo(2015)} for ABC strategies for ``integrated likelihoods'', where nuisance parameters are integrated out.
The paper is organized as follows: in section \ref{sec:data-cloning} we briefly review some properties of ``data cloning'' for maximum likelihood estimation; in section \ref{sec:ABC-all} we introduce topics of approximate Bayesian computation methodology, first by considering some basics (section \ref{sec:abc-basics}) then introducing our original contribution in sections \ref{sec:abc-dc}--\ref{sec:dynamic-abcdc}. Finally in section \ref{sec:simulations} simulation studies illustrate results.
\subsection{Data cloning}\label{sec:data-cloning}
A Bayesian procedure for maximum likelihood estimation based on ``replicating'' data was first introduced in \cite{robert1993} (see also \citealp{robert-titterington}) and then studied under different flavors by e.g. \cite{doucet-godsill-robert}, \cite{jacquier-johannes-polson} and \cite{lele-dennis-lutscher}. The latter introduced the term ``data-cloning'' which we employ.
For simplicity, in the following we always consider the full vector parameter $\theta$, and it should be understood that some of its components enter $f(\cdot)$ while others enter $g(\cdot)$.
The likelihood function of $\theta$ for the state-space model \eqref{eq:state-space-general} can be written as
\begin{align}
L(\theta;y)&=p(y_0;\theta)\prod_{i=1}^np(y_i|y_1,...,y_{i-1};\theta)\nonumber\\
&=
\int f(y_0|x_0;\theta)p(x_0) \prod_{i=1}^n \bigl\{ f(y_i|x_i,\theta)g(x_i|x_{i-1},\theta)\bigr\}dx_0\cdots dx_n
\label{eq:likelihood}
\end{align}
where $x_0=X_{t_0}$ and $p(x_0)$ the corresponding unconditional density. Actually in what follows, and without loss of generality, we assume $x_0$ deterministic and known, hence we remove $p(x_0)$ from the expression of the likelihood function.
Notice the latter equality in \eqref{eq:likelihood} exploits the notion of conditional independence between observations and the Markovian nature of the latent state. We now consider ``cloning'' the data $y$, i.e. we choose a positive integer $K$, produce $K$ copies of $y$ and stack them in $y^{(K)}=(y,y,...,y)$ where $y$ is replicated $K$ times.
Now, we generate $K$ independent vectors for process $\{X_t\}_{t\geq t_0}$ from $g(\cdot)$ at times $\{t_0,...,t_n\}$, say $X^{(1)},X^{(2)},...,X^{(K)}$, all simulated conditionally on the same value of $\theta$. Therefore we set $X^{(k)}=(X_{t_0}^{(k)},...,X_{t_n}^{(k)}\}$ for a generic $k$ ($k=1,...,K$).
All vectors in $y^{(K)}$ are assumed to be conditionally independent of each other, given their individual latent state $X^{(k)}$, for example we could imagine a series of $K$ independent experiments leading to exactly the same result. Then the likelihood function for the cloned data $y^{(K)}$ results in
\begin{equation}
L(\theta;y^{(K)})=\int \prod_{k=1}^K \{f(y|X^{(k)},\theta)p(X^{(k)}|\theta)\}dX^{(1)}\cdots dX^{(K)}
\label{eq:cloned-likelihood}
\end{equation}
where we denote with $p(X^{(k)}|\cdot)$ the joint density of vector $X^{(k)}$, hence $p(X^{(k)}|\theta)=\prod_{i=1}^n g(X^{(k)}_i|X^{(k)}_{i-1},\theta)$.
Now, for each $k$ a given term in the product in \eqref{eq:cloned-likelihood} depends on $X^{(k)}$ and not on terms having different indices $k'$ ($k'\neq k$). Therefore \eqref{eq:cloned-likelihood} is a product of $K$ integrals, each returning the likelihood function based on $y$, and we can write
\begin{equation}
L(\theta;y^{(K)})=\prod_{k=1}^K \int f(y|X^{(k)},\theta)p(X^{(k)}|\theta)dX^{(k)} = (L(\theta;y))^K.
\label{eq:cloned-likelihood-bis}
\end{equation}
Therefore the likelihood function for the cloned data is the likelihood based on the actual measurements raised to the power $K$. Now, it is clear that the MLE of $\theta$ is the argmax for both $L(\theta;y^{(K)})$ and $L(\theta;y)$.
By considering a prior distribution $\pi(\theta)$, we have the posterior distribution resulting from a data-cloned likelihood
\begin{equation}
\pi(\theta|y^{(K)})\propto L(\theta;y^{(K)})\pi(\theta)=(L(\theta;y))^K\pi(\theta).
\label{eq:cloned-data-posterior}
\end{equation}
It is easy to prove that for a large enough $K$ the mean of the posterior $\pi(\theta|y^{(K)})$ approaches the MLE of $\theta$ regardless the specific choice of $\pi(\theta)$ \citep{lele-dennis-lutscher} and a central limit theorem can be derived \citep{jacquier-johannes-polson,lele2010estimability}. However, simulations experiments in \cite{lele-dennis-lutscher} show that using informative priors enable a more rapid convergence to the MLE.\\
In algorithm \ref{alg:generalized-lele-algorithm} we consider a generalization of the data-cloning Metropolis-Hastings algorithm for sampling from $\pi(\theta|y^{(K)})$ (we call it ``generalization'' simply because in \citealp{lele-dennis-lutscher} the proposal distribution for $\theta$ is $\pi(\theta)$, while we consider a general proposal $u(\cdot)$).
Consider a proposed value $\theta^{\#}$ generated from a distribution having density $u(\theta^{\#}|\theta^*)$ and a proposal $X^{\#(k)}$ generated from $v(X^{\#(k)}|\theta^{\#})$ which we set for convenience to be $v(X^{\#(k)}|\theta^{\#})\equiv p(X^{\#(k)}|\theta^{\#})$.
\begin{algorithm}
\caption{A data-cloning MCMC algorithm}
\begin{algorithmic}
\State 1. Initialization: Fix a starting value $\theta^*$ or generate it from its prior $\pi(\theta)$ and set $\theta_1=\theta^*$. Set $j=1$.
\State 2. Generate $K$ independent values of $X$, denoted $X^{*(1)},...,X^{*(K)}$ from $p(X|\theta^{*})$.
\State 3. Calculate
\begin{equation*}
q^*=\prod_{k=1}^K f(y|X^{*(k)},\theta^*).
\end{equation*}
\State 4. Generate a $\theta^{\#}\sim u(\theta^{\#}|\theta^*)$. Generate independent $X^{\#(1)},...,X^{\#(K)}$ from $p(X|\theta^{\#})$. Compute $q^{\#}=\prod_{k=1}^K f(y|X^{\#(k)},\theta^{\#})$.
\State 5. Generate a uniform random variable $\omega\sim U(0,1)$, and calculate the acceptance probability
\begin{eqnarray}
& \alpha = \min\biggl[1,\underbrace{\frac{q^{\#}\cdot p(X^{\#(1)}|\theta^{\#})\cdots p(X^{\#(K)}|\theta^{\#})}{q^* \cdot p(X^{*(1)}|\theta^{*})\cdots p(X^{*(K)|\theta^{*}}) }}_{\text{ratio of likelihoods}}\nonumber\\
&\times \underbrace{\frac{v(X^{*(1)}|\theta^{*})\cdots v(X^{*(K)}|\theta^{*})u(\theta^*|\theta^{\#})}{v(X^{\#(1)}|\theta^{\#})\cdots v(X^{\#(K)}|\theta^{\#})u(\theta^{\#}|\theta^{*})}}_{\text{ratio of proposals}} \times \underbrace{\frac{\pi(\theta^{\#})}{\pi(\theta^*)}}_{\text{ratio of priors}} \biggr]\nonumber\\
&= \min\biggl[1,\frac{q^{\#}}{q^* }
\times \frac{u(\theta^*|\theta^{\#})}{u(\theta^{\#}|\theta^{*})} \times \frac{\pi(\theta^{\#})}{\pi(\theta^*)} \biggr]. \label{eq:generalized-acceptance-prob}
\end{eqnarray}
If $\omega>\alpha$, set $\theta_{j+1}:=\theta_{j}$ otherwise set $\theta_{j+1}:=\theta^{\#}$, $\theta^*:=\theta^{\#}$ and $q^*:=q^{\#}$. Increase $j$ by 1 and go to step 6.
\State 6. Repeat steps 4--5 as long as $j\leq R$ for $R$ ``large''.
\end{algorithmic}
\label{alg:generalized-lele-algorithm}
\end{algorithm}
The notation $:=$ means ``assign the value on the right hand side to the left hand side''. For a large enough number of iterations $R$ this algorithm produces a chain having $\pi(\theta,X|y^{(K)})$ as its stationary distribution, with $X=(X^{(1)},...,X^{(K)})$. In order to obtain draws from the desired marginal distribution $\pi(\theta|y^{(K)})$ it is sufficient to discard the $\{X^{(1)},...,X^{(K)}\}_{j=1,...,R}$ obtained from the algorithm output $\{\theta,X^{(1)},...,X^{(K)}\}_{j=1,...,R}$ (after some appropriate burnin period). For $K\rightarrow\infty$ the sample mean of the $\{\theta\}_j$ is the MLE of $\theta$ and $K$ times the covariance matrix of the draws returns the covariance of the MLE, the inverse of the Fisher information based on the original data \citep{jacquier-johannes-polson,lele2010estimability}. Also, for $K\rightarrow \infty$ and independently of the chosen prior, $\pi(\theta|y^{(K)})$ is degenerate at $\theta=\hat{\theta}$, where $\hat{\theta}$ is the MLE of $\theta$.
Notice the simplification occurring in the expression for $\alpha$, due to taking $v(X|\theta)\equiv p(X|\theta)$, this resulting in \eqref{eq:generalized-acceptance-prob}. The simplification above solves the typically difficult problem of not having a ready expression for the transition densities of $\{X_t\}_{t\geq t_0}$. In fact, here all we need is the ability to (somehow) simulate the process $\{X_t\}_{t\geq t_0}$, and having access to transition densities is not strictly required. For example, in section \ref{sec:gompertz} we know the solution of the considered stochastic differential equation (SDE) model, so we can simulate from it. When the exact solution to an SDE is not available, a numerical discretization with stepsize $h$ (e.g. the Euler-Maruyama scheme) generates an approximate solution, converging to the exact one as $h\rightarrow 0$. \cite{beskos2006exact} even devised a numerical scheme resulting in exact simulation of the SDE solution (i.e. without discretization error), though this is of not so general applicability.
It is important to realize that dealing with a ``powered-up'' posterior as in \eqref{eq:cloned-data-posterior} results in a surface having increasingly peaked modes for increasing $K$ and deeper ``valleys'' in-between modes enclosing smaller and smaller probability mass (assuming the existence of multiple modes). Therefore we believe it is important not to let $K$ fixed to a large value from the start of the algorithm, but instead start with a small value for $K$ and then increase it progressively. Enabling a smooth and not too rapid increase of $K$ should help the chain from being stuck in low-probability regions. However in the examples discussed in sections \ref{sec:g-and-k}--\ref{sec:2GBM} a rapid increase in $K$ is possible.
\section{Approximate inference using ABC with data-cloning}\label{sec:ABC-all}
Acceptance of proposals in MCMC algorithms is particularly challenging when the modelled process is highly erratic, for example when the unobserved state is a diffusion process, that is a solution to a stochastic differential equation (SDE, e.g. \citealp{fuchs2013inference}). For such class of models, trajectories for $\{X_t\}$ may result quite distant from the observed data $y$, even for values of the parameters in the bulk of their posterior distributions. In such circumstance $q^{\#}$ will often be small compared to $q^{*}$, and the proposal will rarely be accepted. For example, when using an approach as the one described above, where trajectories are simulated ``blindly'' from $p(X|\cdot)$, that is unconditionally to data, then trajectories do not exploit direct knowledge of the data. This sometimes result in many rejected proposals if the sample size is large. Carefully tuned Sequential Monte Carlo (SMC) strategies can be constructed so that the best trajectories (``particles'') are selected according to their proximity to data, and this have pushed forward Bayesian inference via MCMC methods incorporating SMC \citep{andrieu2010particle}.
However for complex (ideally multidimensional) stochastic models and a large number of observations, use of SMC methods is computer intensive. Approximate Bayesian computation (ABC, see reviews by \citealp{sisson-fan(2011)} and \citealp{marin-et-al(2011)}) eases sampling from an approximation of the posterior distribution, by substituting likelihood function evaluations with simulations from the data generating model. Here follows a short discussion on ABC which will ease the introduction to our original contribution.
\subsection{Basics of ABC}\label{sec:abc-basics}
Here we summarize some minimal notions of ABC methodology, without considering for the moment the data-cloning scenario, hence in this section it can be assumed that $K=1$.
The ABC approach considers generating samples $z$ from $f(\cdot)$ in \eqref{eq:state-space-general} (i.e. $z\in \mathsf{Y}$, same as the actual data) and corresponding proposals $\theta^{\#}$ are accepted if the $z$ are ``close'' to data $y$, according to a threshold $\delta>0$. Several criterion for ``closeness'' can be postulated, as described below. In ABC we aim at simulating draws from the augmented approximated posterior
\begin{equation}
\pi_{\delta}(\theta,z|y)\propto J_{\delta}(y,z)\underbrace{L(\theta;z)\pi(\theta)}_{\propto \pi(\theta|z)}
\label{eq:abc-posterior}
\end{equation}
where $z=(z_0,...,z_n)$ and $L(\theta;z)$ is the (intractable) likelihood function for $\theta$ based on $z$. Then $\pi_{\delta}(\theta|y)\propto \int \pi_{\delta}(\theta,z|y)dz$. Here $J_{\delta}(\cdot)$ is some function that depends on $\delta$ and weights the intractable posterior for simulated data $\pi(\theta|z)\propto L(\theta;z)\pi(\theta)$ with high values in regions where $z$ and $y$ are similar. Therefore we would like (i) $J_{\delta}(\cdot)$ to give higher rewards to proposals corresponding to $z$ having values close to $y$. In addition (ii) $J_{\delta}(y,z)$ is assumed to be a constant when $z=y$ (i.e. when $\delta=0$) so that the \textit{exact} marginal posterior $\pi(\theta|y)$ is recovered.
A common choice for $J_{\delta}(y,z)$ is the uniform kernel
\[J_{\delta}(y,z)\propto \mathbb{I}_{\{\rho(z,y)\leq\delta\}}\]
where $\rho(z,y)$ is some measure of closeness between $y$ and $z$ and $\mathbb{I}$ is the indicator function. Important alternatives are the Epanechnikov and Gaussian kernels \citep{beaumont2010approximate} or sums of discrepancies \citep{toni2009approximate}.
An ABC-MCMC algorithm targeting the distribution \eqref{eq:abc-posterior} has been proposed in \cite{marjoram2003markov}. However, one of the difficulties is that, in practice, $\delta$ has to be set as a tradeoff between statistical accuracy (with a small positive $\delta$) and computational feasibility ($\delta$ not too small). Also, notice that ABC methods are most often applied to models where a set of low-dimensional summaries of the data $S(y)$ is employed rather than the full dataset $y$. That is whenever is possible (and even more so when $S(\cdot)$ is sufficient for $\theta$), it is advisable to consider $J_{\delta}(S(y),S(z))$ instead, so that for example we have
\[
\pi_{\delta}(\theta,z|\rho(S(z),S(y))\leq\delta)\propto L(\theta;z)\pi(\theta)\mathbb{I}_{\{\rho(S(z),S(y))\leq\delta\}}
\]
when $J_{\delta}(S(y),S(z))\propto \mathbb{I}_{\{\rho(S(z),S(y))\leq\delta\}}$.
The introduction of summaries $S(\cdot)$ is a double edged sword. On one hand the specification of appropriate (i.e. informative, though usually not sufficient) statistics is not trivial, especially for dynamic models, whereas it is somehow more intuitive to specify them for static models. On the other hand having an informative set of statistics implies a considerable reduction in the number of elements to be compared ($d_s$ comparisons, where $d_s:=\dim(S)$ instead of the $n$ comparisons required when simulated and observed data have to be compared, with $d_s\ll n$) and consequently a much smaller $\delta$ can be employed.
\subsection{ABC-DC: Data-cloning ABC}\label{sec:abc-dc}
In most problems $\delta$ is a strictly positive value, sometimes set ``small enough'', some other times set to a larger than desired value, depending on the complexity of the experimental scenario. In fact when using a very small $\delta$ to obtain accurate inference, this results in a high rejection rate, often too high to be computationally feasible. Therefore we propose to consider a larger $\delta$ than what would typically be considered as appropriate, coupled with data-cloning. Our idea is that if we use $K=1$ while dynamically decrease $\delta$ in an ABC-MCMC algorithm, to reach a moderately large $\delta$-value that still allows exploration of the posterior surface (producing an acceptance rate of, say, 10-15\%, and this phase could be considered as ``burn-in'') we can then start a data-cloning procedure and progressively enlarge $K$ while keeping $\delta$ constant to its last value. During the initial exploration (burn-in with $K=1$) we require a $\delta$ which is small enough to locate the approximate position of the maxima for the exact marginal posterior $\pi(\theta|y)$, not a $\delta$ producing an accurate approximation to the surface of $\pi(\theta|y)$. When we start increasing $K$ the marginal posterior $\pi_{\delta}(\theta|y^{(K)})$ will concentrate around its maxima, which for large enough $K$ should be approximately at the same location as the MLE.
More in detail we propose to consider $R$ iterations of an ABC-MCMC algorithm later denoted as ABC-DC: (i) start an ABC-MCMC algorithm without data-cloning ($K=1$) and let $\delta$ decrease during this phase; an initially large $\delta$ will enable a rapid exploration of the posterior surface at a high acceptance rate (say 30\%) to locate the bulk of the approximated posterior. (ii) while $\delta$ progressively decreases, the algorithm focus on exploring a more accurate approximation of the posterior until $\delta$ reaches say 10-20\% acceptance rate. Such acceptance rate is typically too high in ABC studies (a typical value would be 1\% or less) however we plan to increase the number of clones and focus on the peak of such posterior. (iii) At this point $\delta$ is kept fixed to its last value and data-cloning starts, by progressively increasing the value of $K$. (iv) Once $R$ iterations are completed, we collect the draws generated with the largest value of $K$ and use those for maximum likelihood inference. As such the resulting samples are not from $\pi_{\delta}(\theta,z|y)$ but from the powered-approximated posterior $\pi_{\delta}(\theta,z^{(1)},...,z^{(K)}|y^{(K)})$ for finite $K$
\begin{align*}
\pi_{\delta}(\theta,z^{(1)},...,z^{(K)}|y^{(K)}) \propto \pi(\theta)\prod_{k=1}^K J_\delta(y,z^{(k)})L(\theta;z^{(k)})
\end{align*}
where the $L(\theta;z^{(k)})$ can be simplified out in the Metropolis-Hastings acceptance probability as previously illustrated. As mentioned in section \ref{sec:abc-basics}, we assume as implicit the dependence on data via summary statistics, therefore here and in the rest of our work $J_\delta(y,z^{(k)})\equiv J_\delta(S(y),S(z^{(k)}))$.
Of course enlarging $K$ shrinks the area of the support of the cloned posterior where most of the probability mass is located, hence it becomes increasingly difficult to explore a progressively peaked surface: this is why in step (ii) of the schedule above we do not recommend to go below, say, 10-20\% acceptance as this rate will reduce drastically when $K$ increases in step (iii).
In our applications we use a Gaussian kernel, that is
\begin{equation}
J_{\delta}(y,z^{(k)}) \propto \exp\{-\bigl(S(z^{(k)})-S(y)\bigr)'\Omega^{-1}\bigl(S(z^{(k)})-S(y)\bigr)/2\delta^2\}
\label{eq:gauss-kernel}
\end{equation}
where $'$ denotes transposition.
Clearly equation \eqref{eq:gauss-kernel} respects desired criteria, i.e. (i) it is constant when $S(z^{(k)})\equiv S(y)$ and (ii) it gets larger values when $S(z^{(k)})\approx S(y)$. For a generic $z$ this implies writing $S(z)\sim N_{d_s}(S(y),\delta^2\Omega)$ with $N_{d_s}(\cdot,\cdot)$ a $d_s$-dimensional Gaussian distribution centred at $S(y)$ and $\Omega$ a positive definite matrix.
For simplicity we assume a diagonal $\Omega$ with elements $\Omega=\mathrm{diag}\{\omega^{2}_1,...,\omega^{2}_{d_s}\}$. Of course, using a diagonal $\Omega$ might have an impact on the inference as it does not take into account the correlation among summary statistics. Recall that we keep writing $J_{\delta}(y,z)$ instead of $J_{\delta}(S(y),S(z))$, the dependence on data via summary statistics being considered as implicit.
When the elements in vector $S(\cdot)$ are varying approximately on the same range of values it is possible to consider $(\omega^{2}_1,...,\omega^{2}_{d_s})=(1,...,1)$, however in general the variability of the statistics is unknown and, depending on the type of data and the underlying model, these can have very different magnitude. An inappropriate choice for the elements in $\Omega$ affects the accuracy of the ABC inference negatively, with $J_{\delta}(\cdot)$ being dominated by the most variable
statistic so that $\delta$ will bound the distance with respect to such statistic, not the remaining ones.
To our knowledge the only systematic study on the weighting of summary statistics in ABC is \cite{prangle-adaptiveABC}. When using summary statistics in our experiments, before starting the data cloning procedure we run a pilot study using ABC-MCMC (i.e. $K=1$) with $(\omega_1,...,\omega_{d_s})=(1,...,1)$ and collect the values of the accepted $S(z)$. At the end of the pilot run we compute (after some appropriate burnin) the mean or the median absolute deviation MAD for each coordinate of the accepted $S(z)$ and define $(\omega_1,...,\omega_{d_s}):=(\mathrm{MAD}_1,...,\mathrm{MAD}_{d_s})$. Then we plug the obtained $\Omega$ to weight the summary statistics into data-cloning ABC (introduced in sections \ref{sec:abc-dc}--\ref{sec:dynamic-abcdc}) or in a further run of ABC-MCMC, for comparison purposes. See \cite{prangle-adaptiveABC} for a thorough study on this approach.
For ease of reading we produce two ABC-DC algorithms: the ``static'' ABC-DC is given in algorithm \ref{alg:abc-dc}, where both $\delta$ and $K$ are assumed fixed, allowing for a more immediate understanding. However, in our applications we use algorithm \ref{alg:abc-dc-dynam} which is discussed later.
\begin{algorithm}
\caption{Static ABC-DC}
\begin{algorithmic}
\State 1. Initialization: Fix a starting value $\theta^*$ or generate it from its prior $\pi(\theta)$ and set $\theta_1=\theta^*$. Set $j=1$, fix $\delta>0$ and a positive integer $K$. A vector of statistics $S(\cdot)$ and weights $\Omega$ is available.
\State 2. Generate $K$ independent values of the latent process $X^{*(1)},...,X^{*(K)}$ from $p(X|\theta^{*})$. Conditionally on each $X^{*(k)}$ generate a corresponding $z^{*(k)}$ from equation \eqref{eq:state-space-general} for each $k=1,...,K$.
\State 3. Calculate $J_{\delta}(y,z^{*(k)})$ for every $k$ and compute
\begin{equation*}
q^*=\prod_{k=1}^K J_{\delta}(y,z^{*(k)}).
\end{equation*}
\State 4. Generate a $\theta^{\#}\sim u(\theta^{\#}|\theta^*)$. Generate $K$ independent $X^{\#(k)}$'s from $p(X|\theta^{\#})$ and corresponding $z^{\#(k)}$.
\State 5. Calculate $J_{\delta}(y,z^{\#(k)})$ for every $k$ and set $q^{\#}=\prod_{k=1}^K J_{\delta}(y,z^{\#(k)})$.
Generate $\omega\sim U(0,1)$, and calculate
\[
\alpha = \min\biggl[1,\frac{q^{\#}}{q^* }
\times \frac{u(\theta^*|\theta^{\#})}{u(\theta^{\#}|\theta^{*})} \times \frac{\pi(\theta^{\#})}{\pi(\theta^*)} \biggr].
\]
If $\omega>\alpha$, set $\theta_{j+1}:=\theta_{j}$ and, increase $j$ by 1 and go to step 6. Otherwise set $\theta_{j+1}:=\theta^{\#}$, $\theta^*:=\theta^{\#}$, $q^*:=q^{\#}$, increase $j$ by 1 and go to 6.
\State 6. Repeat steps 4--5 as long as $j\leq R$.
\end{algorithmic}
\label{alg:abc-dc}
\end{algorithm}
\subsection{Dynamic ABC-DC}\label{sec:dynamic-abcdc}
In our experiments, and unlike in \cite{lele-dennis-lutscher} and \cite{jacquier-johannes-polson} where $K$ is kept fixed during the MCMC algorithm execution, we let $K$ increase (see also \citealp{doucet-godsill-robert}). A ``dynamic'' version of ABC-DC with varying $\delta$ and $K$ is presented in algorithm \ref{alg:abc-dc-dynam}. Notice that previously cited work did not use ABC with data-cloning, so to the best of our knowledge ours is the first work proposing doing so.
As discussed by Christian P. Robert at \url{http://xianblog.wordpress.com/2010/09/22/feedback-on-data-cloning/} data-cloning share features with the simulated annealing global optimization method: keeping $K$ fixed to a high value once and for all
removes the dynamic features of a simulated annealing random walk that first explores the whole space and then progressively focus on the highest modes, achieving convergence if the cooling is slow enough. In other words, if $K$ is ``large enough'', the Metropolis algorithm will face difficulties in the exploration of the parameter space, and hence in the subsequent discovery of the global modes, while, if $K$ is ``too small'', there is no certainty that the algorithm will identify the right mode of possibly multiple modes. Here follow a ``dynamic'' ABC-DC algorithm, where schedules are defined for $\delta$ and $K$, namely $\{\delta_{r_1},\delta_{r_2},...,\delta_{r_p}\}$ and $\{ K_{s_1},K_{s_2},...,K_{s_q}\}$ with $\delta_{r_1}>...>\delta_{r_p}>0$ and $1=K_{s_1}<...<K_{s_q}$. This version of ABC-DC starts with $s_1$ iterations of ABC-MCMC algorithm with decreasing thresholds, where $s_1=\sum_{l=1}^pr_l$, $K=1$ constantly throughout the $s_1$ iterations. The threshold is $\delta:=\delta_{r_1}$ in the first $r_1$ iterations, $\delta:=\delta_{r_2}$ in the next $r_2$ iterations etc. In summary the first $r_1$ iterations use $(\delta,K):=(\delta_{r_1},1)$ and in general during iterations $(r_l:r_{l+1})$ we use $(\delta,K):=(\delta_{r_l},1)$. During the last $r_p$ iterations of ABC-MCMC we keep track of the maximum value $\mathrm{max}\{\pi_{\delta_{r_p}}(\theta|y)\}$ of the approximated posterior $\pi_{\delta_{r_p}}(\theta|y)$ and corresponding $\tilde{\theta}:=\mathrm{argmax}\pi_{\delta_{r_p}}(\theta|y)$: this is easily accomplished and cheap to implement by initializing $\mathrm{max}\{\pi_{\delta_{r_p}}(\theta|y)\}:=0$ just before setting $(\delta,K):=(\delta_{r_p},1)$. Then, whenever we have $J_{\delta}(y,z^{\#})\pi(\theta^{\#})>\mathrm{max}\pi_{\delta_{r_p}}(\theta|y)$ for the current maximised value of the posterior kernel, we set $\mathrm{max}\{\pi_{\delta_{r_p}}(\theta|y)\}:=J_{\delta}(y,z^{\#})\pi(\theta^{\#})$ and $\tilde{\theta}:=\theta^{\#}$. This search for the maximum has to be performed only when $\delta\equiv\delta_{r_p}$ as we are not interested in the maximum obtained for poorer approximations of the posterior. An alternative approach to the search for a mode $\tilde{\theta}$ is given by adjusting the output of ABC-MCMC as from \cite{beaumont2002approximate}, hence step $5'$ denoted as ``optional'' in algorithm \ref{alg:abc-dc-dynam}; this step is described in section \ref{sec:regression-adjustment}.
\begin{algorithm}
\caption{Dynamic ABC-DC}
\small
\begin{algorithmic}
\State \textbf{ABC-MCMC stage}
\State 1. Initialization: Fix a starting value $\theta^*$ or generate it from its prior $\pi(\theta)$ and set $\theta_1=\theta^*$. Set $j:=1$, $\delta:=\delta_{r_1}$ and $\mathrm{max}\pi_{\delta_{r_p}}:=0$.
\State 2. Generate $X^{*}$ from $p(X|\theta^{*})$ and a corresponding $z^{*}$ from \eqref{eq:state-space-general}. Compute
$q^*=J_{\delta}(y,z^{*})$.
\State 3. Generate $\theta^{\#}:= \texttt{AMRW}(\theta^*,\Sigma_j)$. Generate $X^{\#}$'s from $p(X|\theta^{\#})$ and corresponding $z^{\#}$. Compute $q^{\#}=J_{\delta}(y,z^{\#})$.
\State 4. Generate $\omega\sim U(0,1)$, and calculate
\[
\alpha = \min\biggl[1,\frac{q^{\#}}{q^* } \times \frac{u_1(\theta^*|\theta^{\#},\Sigma_j)}{u_1(\theta^{\#}|\theta^{*},\Sigma_j)}
\times \frac{\pi(\theta^{\#})}{\pi(\theta^*)} \biggr].
\]
If $\omega>\alpha$, set $\theta_{j+1}:=\theta_{j}$ otherwise set $\theta_{j+1}:=\theta^{\#}$, $\theta^*=\theta^{\#}$ and $q^*:=q^{\#}$. If $\delta=\delta_{r_p}$ go to 5. Otherwise increase $j$ by 1 and if $j\in \{r_2,...,r_p\}$ update $\delta:=\delta_j$. Then go to 3.
\State 5. Check current maximum (only when $\delta=\delta_{r_p}$): if $q^{\#}\pi(\theta^{\#})>\mathrm{max}\pi_{\delta_{r_p}}$ set $\mathrm{max}\pi_{\delta_{r_p}}:=q^{\#}\pi(\theta^{\#})$ and $\tilde{\theta}:=\theta^{\#}$. Increase $j$ by 1. If $j\leq s_1$ go to 3 else go to step 6.
\State $5'.$ \textit{(optional)} Apply regression adjustment on the draws obtained with $\delta_{r_p}$. Take as $\tilde{\theta}$ either the mean or the mode of the adjusted draws and call $\Sigma_{s_1}$ the covariance of the adjusted draws.
\State
\State \textbf{Data-cloning stage}
\State 6. Take the last accepted value $\theta^*$, $\tilde{\theta}$ and the current covariance $\Sigma_{s_1}$ from ABC-MCMC. Set $\hat{\Sigma}_k:=\Sigma_{s_1}$, $K:=K_{s_2}$ and $\delta:=\delta_{r_p}$.
\State 7. Generate $K$ independent vectors denoted $X^{*(1)},...,X^{*(K)}$ from $p(X|\theta^{*})$. Conditionally on each $X^{*(k)}$ generate a corresponding $z^{*(k)}$. Calculate
$q^*=\prod_{k=1}^K J_{\delta}(y,z^{*(k)}).$
\State 8. Generate $\theta^{\#}:=\texttt{MIS}(\tilde{\theta},\hat{\Sigma}_k)$. Generate $K$ independent $X^{\#(k)}$'s from $p(X|\theta^{\#})$ and corresponding $z^{\#(k)}$. Compute $q^{\#}=\prod_{k=1}^K J_{\delta}(y,z^{\#(k)})$.
\State 9.
Generate $\omega\sim U(0,1)$ and calculate
\[
\alpha = \min\biggl[1,\frac{q^{\#}}{q^* }
\times \frac{u_2(\theta^*|\tilde{\theta},\hat{\Sigma}_k)}{u_2(\theta^{\#}|\tilde{\theta},\hat{\Sigma}_k)} \times \frac{\pi(\theta^{\#})}{\pi(\theta^*)} \biggr].
\]
If $\omega>\alpha$, set $\theta_{j+1}:=\theta_{j}$ otherwise set $\theta_{j+1}:=\theta^{\#}$, $\theta^*:=\theta^{\#}$ and $q^*:=q^{\#}$.
Increase $j$ by 1. If $j\in \{s_{3},...,s_{q}\}$ increase $K$ and update $\hat{\Sigma}_k:=\hat{cov}(\theta)_{k'}$ then go to 7, otherwise go to 10. Here $\hat{cov}(\theta)_{k'}$ is the sample covariance computed on draws generated using the previous value $k'$ of $K$ in the schedule.
\State 10. If $j\leq R$ go to 8 otherwise stop.
\end{algorithmic}
\label{alg:abc-dc-dynam}
\end{algorithm}
During ABC-MCMC we generate parameter proposals using adaptive Gaussian Metropolis random walk \citep{haario-saksman-tamminen}. We write $\theta^{\#}:= \texttt{AMRW}(\theta^*,\Sigma_j)$ to denote such a proposal (and the corresponding Gaussian proposal function is denoted $u_1(\cdot)$). During the exploration of the approximate posterior surface with $K=1$ we aim at locating the principal mode $\tilde{\theta}$ of the distribution for the smallest threshold $\delta_{r_p}$ as described above. At the end of the $s_1$ iterations of ABC-MCMC we fix $\delta:=\delta_{r_p}$ for the rest of the ABC-DC execution and increase $K$ progressively. At this stage parameters are proposed using a Metropolis independent sampler $\texttt{MIS}(\tilde{\theta},\hat{\Sigma}_k)$ generating from the Gaussian distribution $N(\tilde{\theta},\hat{\Sigma}_k)$ (and the corresponding Gaussian proposal function is denoted $u_2(\cdot)$) where $\hat{\Sigma}_k$ is the sample covariance matrix obtained from draws generated using the previous value of $K$.
We switched from a Gaussian random walk to an independent Metropolis sampler when powering-up because ABC-MCMC should have located the bulk of the approximated posterior (optionally with the help of the regression adjustment in section \ref{sec:regression-adjustment}), as well as its highest mode, and therefore we use such information to propose parameters in the next stage corresponding to a larger $K$. Not following random walk dynamics helps in avoiding getting trapped in local modes that might emerge when powering-up the posterior surface for increasing $K$.
This is why, in order to accommodate the reduced support of the current targeted distribution for increasing $K$, we recompute the covariance matrix $\hat{\Sigma}_{k}$ for the independence sampler at iterations $j\in\{s_1,...,s_q\}$.
Notice that at the end of step 9 when we reach an iteration $j\in\{s_1,...,s_q\}$ and $K$ has to be enlarged, we need to ``balance'' the information contained in the numerator of $\alpha$ with the one in the denominator, and this is why we go back to step 7 instead of 8 and recompute $q^*$. Basically after increasing $K$ the number of clones in the numerator would be larger than the one in the denominator, hence the need to first go back to 7, where we recompute the denominator using the same value of $K$ employed for the numerator. This re-evaluation is performed only when $j\in \{s_1,...,s_q\}$ and we can safely interpret this step as the starting point for a new chain, targeting the corresponding powered distribution. The effect of step 7 is removed after some burnin period (and a presumably short one, as at this point the chain should already be in the bulk of the powered posterior). In the end, from the inference point of view, what matters are the draws generated at the largest $K$, which are generated with a fixed $\delta\equiv\delta_{r_p}$, hence are genuinely distributed according to the corresponding (marginal) posterior $\pi_\delta(\theta|y^{(K)})$.
For complex models a slow transition between different number of clones $K_{s_{j'}}$ might be required, meaning that differences $K_{s_{j'}}-K_{s_{j'+1}}$ should not be too large. This is because the chain needs to adapt to a narrower support for the posterior when using $K_{s_{j'+1}}$ clones, and to ease the exploration we use the covariance obtained from draws generated with $K_{s_{j'}}$ clones, which is hopefully appropriate. Otherwise if the difference above is too large the covariance used will be inappropriate (too large variances) and many proposals will be rejected.
A maximum likelihood ABC algorithm has also been proposed in \cite{rubio2013simple}, however in that work a non-parametric kernel estimator of the posterior density is constructed (using draws from the ABC rejection algorithm proposed in \citealp{pritchard1999population}), then the maximizer of the non-parametric density is found numerically. In our approach we do not require any kernel estimation procedure (which is onerous unless the dimension of $\theta$ is low), nor direct optimization procedures. An approach similar to the one by Rubio and Johansen is in \cite{grazian-liseo(2015)}.
In conclusion, we use draws produced by algorithm \ref{alg:abc-dc-dynam} under $(\delta,K)\equiv(\delta_{r_p},K_{s_q})$ and for these draws we compute their sample mean $\hat{\theta}_{\delta,K}$ to obtain an approximate maximum likelihood estimate (MLE) of $\theta$. In principle, if we were able to construct sufficient summaries $S(\cdot)$ for $\theta$, and by letting $\delta\rightarrow 0$ and then $K\rightarrow\infty$, we would have $\hat{\theta}_{MLE}\equiv \hat{\theta}_{\delta,K}$ and $\hat{\Sigma}_{MLE} \equiv K\cdot\hat{\Sigma}_{\delta,K}$, where $\hat{\theta}_{MLE}$ is the MLE of $\theta$ and $\Sigma_{MLE}$ is the covariance of the MLE (i.e. the inverse of the Fisher information based on $y$) computed using the sample covariance $\hat{\Sigma}_{\delta,K}$ derived under $(\delta,K)\equiv(\delta_{r_p},K_{s_q})$. The reasoning behind these results, when $S(\cdot)$ is sufficient, is that (i) for an ABC approximation to a posterior it holds that in distribution $\lim_{\delta\rightarrow 0}\pi_{\delta}(\theta|y)=\pi(\theta|y)$, and that (b) data-cloning implies that in distribution $\lim_{K\rightarrow\infty}\pi(\theta|y^{(K)})=N(\hat{\theta}_{MLE},\hat{\Sigma}_{MLE})$ (\citealp{jacquier-johannes-polson}, \citealp{lele2010estimability}). Therefore by first taking the limit for $\delta$ and then applying the limit for $K$ we have that
\[\hat{\theta}_{\delta,K}\sim N(\hat{\theta}_{MLE},\hat{\Sigma}_{MLE}),\qquad \delta\rightarrow 0, K\rightarrow \infty.\]
Then (when asymptotics holds) it would be easy to compute approximate standard errors and construct confidence intervals for the true value $\theta^o$ of $\theta$ (using the fact that $\hat{\theta}_{MLE}\rightarrow \theta^o$ when $n\rightarrow\infty$), by noting that $\hat{\Sigma}_{MLE}$ is provided ``for free'' as detailed above. However, in reality our reasoning is based on not using a small $\delta$, therefore the confidence bounds resulting from using asymptotics are often wide, even though in our experiments we obtain good point approximations to the MLE. This is because $\delta$ is large and an increasing $K$ does not necessarily reduce the chain variability for all parameters. Therefore multiplying $\hat{\Sigma}_{\delta,K}$ by $K$ might give too large standard errors (this fact holds regardless of whether we are able to make use of sufficient summary statistics, which is in general not the case). This is particularly true when summary statistics are not informative for a certain parameter, see the case of parameter $\log\sigma$ in Figure \ref{fig:gompertz-regularization-top}. For all these reasons, focus of this work is on parameters point estimation.
In conclusion, even if we use a $\delta$ which is not small for accurate Bayesian inference, but small enough for locating the maximum of the posterior, then we can power-up the ABC posterior and propose samples around such maximum to obtain a good (point) approximation of the MLE. A formal study on the properties of the obtained estimators for positive $\delta$ and finite $K$ is not considered here and is left for future research.
Finally note that in \cite{lele2010estimability} it is shown that a criterion to choose the number of clones is to monitor the decay (as $K$ increases) of the largest eigenvalue $\lambda_1(K)$ of the sample covariance matrix obtained from a chain using $K$ clones. They used such criterion to diagnose parameters estimability, that is if $\lambda_1(K)$ decreases to zero then the parameters are deemed estimable, with a covariance matrix approaching degeneracy. The examples treated with standard data-cloning (i.e. papers not using ABC methodology) are sometimes able to consider clones in the order of hundreds \citep{baghishani-mohammadzadeh} with a good acceptance rate, hence it makes sense to monitor the convergence of $\lambda_1(K)$. When using ABC we do not enjoy the luxury of letting an automatic criterion decide when to stop increasing $K$, as we are anyway bounded to use a much smaller number of clones with a small acceptance rate.
\subsection{Regression adjustment}\label{sec:regression-adjustment}
In section \ref{sec:g-and-k} we consider an example with a static model, where ABC inference is easily enabled thanks to the existence of intuitive and informative summary statistics (albeit not sufficient ones). However for dynamic models it is usually way more difficult to identify informative summaries. Therefore in section \ref{sec:gompertz} we automatically retrieve summaries using the method in \cite{fearnhead-prangle(2011)}. Although automatized construction of summaries is certainly a handy tool, since we plan to use a large threshold $\delta$ at the end of the burnin ($K=1$) phase it may be useful to ``adjust'' the obtained draws before starting the data-cloning phase, in order to obtain a more informative independence sampler. For the example in section \ref{sec:gompertz} we consider the regression adjustment proposed in \cite{beaumont2002approximate}. In this section for ease of writing we assume a scalar $\theta$.
Denote with $\theta_{\delta}=(\theta_1,...,\theta_{r_p})$ the sequence of draws for $\theta$ produced at the smallest threshold $\delta\equiv\delta_{r_p}$ when $K=1$ and denote with $(S_1,...,S_{r_p})$ the corresponding simulated summary statistics (each summary can be a vector). Consider the following regression model
\begin{equation}
\theta_i = \alpha + (S_i-S)'{\beta}+\xi_i,\qquad i=1,...,r_p
\label{eq:beaumont-regression}
\end{equation}
where $S$ is the summary statistic for the observed data, $\alpha$ and $\beta$ are regression parameters and $\xi_i$ is mean zero homoscedastic noise (see \citealp{blum2013comparative} for alternative approaches). Parameters $(\alpha,\beta)$ can be estimated via local linear regression by minimizing the following criterion
\[
\sum_{i=1}^{r_p}(\theta_i-\alpha-(S_i-S)'\beta)^2 J_{\delta_{r_p}}(S,S_i)
\]
for some appropriate kernel $J_{\delta}$ (e.g. uniform, Gaussian kernel). Solution to the least squares problem is given by
\[
(\hat{\alpha},\hat{\beta}) = (Z^TWZ)^{-1}Z^TW\theta_{ \delta}
\]
where $Z$ is the design matrix for model \eqref{eq:beaumont-regression} and $W$ a diagonal matrix with $i$th entry given by $J_{\delta_{r_p}}(S,S_i)$ (see \citealp{beaumont2002approximate} for details). The adjusted parameters are given by
\[
\theta_i^* = \theta_i - (S_i-S)^T\hat{\beta},\qquad i=1,...,r_p.
\]
When the employed $\delta_{r_p}$ is relatively large the posterior obtained from the adjusted draws $\theta_i^*$ is usually more concentrated than the one based on $\theta_i$, see section \ref{sec:gompertz}. This means that, based on the set of $\theta_i^*$, we are able to construct a more informative empirical covariance matrix for the independence sampler used when $K>1$. Also, we may consider taking the mean or median of the adjusted parameters and centre the independence sampler at such value ($\tilde{\theta}$ in algorithm \ref{alg:abc-dc-dynam}).
\section{Simulation studies}\label{sec:simulations}
This section considers approximate inference for three simulation studies. The first two studies deal with observations from a $g$-and-$k$ distribution and from a state space model respectively, both lacking explicit expressions for the likelihood function. The third one is based on a two-dimensional stochastic differential equation with correlated noise. For the latter it is possible to write the exact likelihood function and therefore identify by numerical optimization the maximum likelihood estimate, which we compare with the ABC-DC estimator. In all examples whenever we refer to ABC-DC we mean the ``dynamic ABC-DC'' in algorithm \ref{alg:abc-dc-dynam}.
From the computer coding point of view, the three examples are easily vectorised (our codes are written in MATLAB) and therefore simulating model realizations for a certain $K>1$, say $5\leq K \leq 15$, did not result in any serious slow-down compared to using $K< 5$. However, consider having a computationally expensive model simulator such that producing a single realizations from the model requires several seconds or minutes. Assume therefore that running many ($R$) iterations of an ABC-MCMC algorithm for Bayesian inference is impractical, whereas running $\tilde{R}\ll R$ iterations of ABC-DC over $M>1$ processors is feasible (for simplicity, assume $K$ a multiple of $M$). Here the task of computing the cloned likelihood would be performed by distributing $K/M$ model simulations to each of the $M$ processors.
In our examples timing is obtained on simulations running on a i7-4790 CPU 3.60 GHz PC desktop.
\subsection{$g$-and-$k$ distribution}\label{sec:g-and-k}
An interesting case study is given by $g$-and-$k$ distributions, first analysed via ABC methods in \cite{allingham2009bayesian}. This is a flexibly shaped distribution that is used to model non-standard data
through a small number of parameters. It is defined by its inverse distribution
function, but has no closed form density. The quantile function (inverse distribution function) is given by
\begin{equation}
F^{-1}(x;A,B,c,g,k)= A+B\biggl[1+c\frac{1-\exp(-g\cdot r(x))}{1+\exp(-g\cdot r(x))}\biggr](1+r^2(x))^kr(x)
\label{eq:g-k-inverse}
\end{equation}
where $r(x)$ is the $x$th standard normal quantile, $A$ and $B$ are location and scale parameters and $g$ and $k$ are related to skewness and kurtosis. We assume $\theta=(A,B,g,k)$ as parameter of interest, given that we keep $c$ fixed to $c=0.8$ \citep{rayner2002numerical}. Parameters restrictions are $B>0$ and $k>-0.5$.
An evaluation of \eqref{eq:g-k-inverse} returns a draw ($x$th quantile) from the $g$-and-$k$ distribution or, in other words, the $i$th sample $r_i:=r_i(x)\sim N(0,1)$ produces a draw $z_i:=F^{-1}(\cdot;A,B,c,g,k)$ from the $g$-and-$k$ distribution. Notice in this case there is no hidden/latent process, hence all simulated values $z_i$ are independent draws from said distribution.
We follow the simulation setup for data $(y_1,...,y_n)$ of size $n=10^4$ generated as in \cite{allingham2009bayesian} (see also \citealp{fearnhead-prangle(2011)}) with $\theta=(3,1,2,0.5)$. They set uniform priors $U(0,10)$ on each parameter then used a standard ABC-MCMC algorithm (\citealp{marjoram2003markov}) to estimate parameters, with summaries $S(y)=(y_{(1)},...,y_{(n)})$ (the sequence of ordered data) and $J(y,z)=(\sum_{i=1}^n[S_i(z)-S_i(y)]^2)^{1/2}$ with $S_i$ the $i$th element of $S$ and $z=(z_1,...,z_n)$ a vector of samples from the $g$-and-$k$ distribution. They obtain good inference for all parameters but $g$ which is essentially unidentified. Actually it is extremely simple to obtain accurate inference for all parameters by reducing the dimensionality of the problem using a smaller set of summaries. We set $S(y)=(P_{20},P_{40},P_{60},P_{80},\mathrm{skew}(y))$, that is the 20-40-60-80th percentiles of the data and the sample skewness. As comparison function $J_{\delta}$ we consider a different criterion, namely a Gaussian kernel
as in \eqref{eq:gauss-kernel}. With such setup, we first run a standard ABC-MCMC without data-cloning and let $\delta$ decrease with schedule $\delta\in\{5,3,1\}$. Results were not encouraging, and that's because of using a matrix of weights $\Omega$ with unit diagonal thus giving the same weight to each of the five summary statistics. Then we formed a new matrix of weights from the output of such preliminary (pilot) run, as described in section \ref{sec:abc-basics}, to obtain $[\omega_1,...,\omega_5]=[0.22, 0.19, 0.53, 2.96, 1.90]$.
With the new $\Omega$ we run ABC-MCMC once more, this time with
$\delta\in\{0.3, 0.1, 0.05, 0.015\}$ where the largest value of $\delta$ was used for the first 10,000 iterations, then decreased every 10,000 iterations and the smallest value was used for the last 20,000 iterations of overall $R=50,000$ ABC-MCMC iterations, starting from parameter values (5,5,3,2). The 50,000 iterations were completed in 103 seconds. Trace plots are in Figure \ref{fig:g-k_abcmcmc_trace}.
\begin{figure}
\centering
\includegraphics[width=17cm,height=9cm]{g-k_abcmcmc_trace}
\caption{\footnotesize{g-k distribution: trace plots for the ABC-MCMC run. A major variability reduction occurs at iteration 10,000 when reducing $\delta$ from 0.3 to 0.1. Horizontal lines give the true parameter values.} }
\label{fig:g-k_abcmcmc_trace}
\end{figure}
At the smallest $\delta=0.015$ we obtain an acceptance rate of 1-2\% and corresponding posterior means and 95\% posterior intervals: $A=2.98$ (2.97,2.99), $B=0.95$ (0.92,0.98), $g=1.95$ (1.82,2.07), $k=0.53$ (0.50,0.56).
Considering a pilot run was doubly useful, because we can now use the determined $\Omega$ into ABC-DC. We start the algorithm at the same starting parameter values used in ABC-MCMC and keep the threshold fixed to a large value, that is $\delta=0.3$ and in this case we do not let it decrease. Notice that $\delta=0.3$ is the largest value used in the previous ABC-MCMC experiment. The first 7,000 iterations are run with $K=1$, useful to identify a temporary main mode $\tilde{\theta}$ (see Figure \ref{fig:g-k_density_gparameter}), then we enlarge it to $K=15$ and during the next 20,000 iterations observe an acceptance rate of 1-2\%. Therefore we ran in total $R=27,000$ iterations which are completed in 402 seconds. Trace plots are in Figure \ref{fig:g-k_abcdc_trace}.
\begin{figure}
\centering
\includegraphics[width=17cm,height=9cm]{g-k_abcdc_trace}
\caption{\footnotesize{g-k distribution: trace plots for the ABC-DC run. Here $\delta=0.3$ constantly and the variability reduction at iteration 7,000 is only due to $K$ increasing from $K=1$ to $K=15$. Horizontal lines give the true parameter values.}}
\label{fig:g-k_abcdc_trace}
\end{figure}
Using $(\delta,K)=(0.3,15)$ we obtain the following asymptotic means and standard errors from the large samples arguments outlined in section \ref{sec:dynamic-abcdc}: $\hat{A}=2.99$ (0.07), $\hat{B}=0.98$ (0.26), $\hat{g}=1.97$ (0.77), $\hat{k}=0.48$ (0.14). As previously remarked, in the present work we focus on point estimation and in this section confidence intervals based on asymptotics are reported only to show that these are likely to be overestimated when using ABC-DC, see the discussion below where we also consider a bootstrap approach.
We considered the setup above for ABC-DC to perform a fair comparison with ABC-MCMC, namely an $R$ long enough to return 20,000 draws (when $K=15$) and a $K$ large enough to return the same acceptance rate as in ABC-MCMC. Of course the setting is time consuming, however we can show how to obtain the same results using a quicker ABC-DC. We run 7,000 iterations with $(\delta,K)=(0.3,1)$ then increase the number of clones to $K=5$ for further 5,000 iterations; overall time is 48 seconds and the acceptance rate is 10-12\%. We obtain $\hat{A}=2.98$ (0.06), $\hat{B}=0.97$ (0.23), $\hat{g}=1.99$ (0.74), $\hat{k}=0.49$ (0.16) essentially the same results obtained under the more expensive setup with computations an order of magnitude faster. For both ABC-DC simulations it is interesting to appreciate how the variability of the chain for parameter $g$ (when $K=1$), and the automatic identification of its maximum, allowed for a dramatic improvement in the mode identification for a larger $K$. See Figure \ref{fig:g-k_density_gparameter} for the approximate marginal distribution of $g$ based on $K=1$: ABC-DC automatically identifies the highest mode at about $g=2$ and focuses on such mode for an increasing $K$.
\begin{figure}
\centering
\includegraphics[scale=0.4]{g-k_density_gparameter}
\caption{\footnotesize{g-k distribution: marginal posterior for $g$ when $K=1$ and $\delta=0.3$. At this point the current main mode $\tilde{\theta}$ is automatically identified by ABC-DC and the algorithm will use it to propose samples under a larger value of $K$.}}
\label{fig:g-k_density_gparameter}
\end{figure}
We conclude that ABC-DC returns very good point estimates, however standard errors are likely to be overestimated. To verify the latter claim we run a parametric bootstrap procedure of size 100: that is we produce 100 independent datasets using the parameters obtained under the more computationally conservative approach ($\hat{A}=2.98$, $\hat{B}=0.97$, $\hat{g}=1.99$, $k=0.49$ when $K=5$ and $\delta=0.3$) then re-estimate parameters on each simulated dataset via ABC-DC. For each simulation we used, again, 7,000 iterations with $(\delta,K)=(0.3,1)$ followed by 5,000 iterations with $(\delta,K)=(0.3,5)$. Bootstrap results are in Table \ref{tab:g-and-k-bootstrap}, showing reasonable variation, whereas previously standard errors built on asymptotic arguments are overestimated. Finally, we perform a parametric bootstrap procedure employing an exact maximum likelihood estimation obtained by numerical optimization \citep{rayner2002numerical}, using the R package in \url{https://github.com/dennisprangle/gk}. Results are in Table \ref{tab:g-and-k-bootstrap}, and we can once more appreciate the good approximation provided by ABC-DC. Thanks to this further simulation, we can definitely confirm that trying to apply the asymptotic considerations from section \ref{sec:dynamic-abcdc} to compute standard errors (instead of, say, bootstrap procedures) results in inflated confidence intervals.
\begin{table}
\centering
\caption {g-k distribution: means over 100 parametric bootstrap replications and 2.5-97.5\% empirical percentiles using an exact MLE procedure and ABC-DC. For each replication using ABC-DC we considered $(\delta,K)=(0.3,5)$.}
\begin{tabular}{cccc}
\hline
\hline
& True values & MLE & ABC-DC\\
\hline
$A$ & 2.98 & 2.99 [2.97,3.01] & 2.98 [2.95,3.01]\\
$B$ & 0.97 & 0.98 [0.95,1.02] & 0.98 [0.91,1.04]\\
$g$ & 1.99 & 1.97 [1.92,2.02] & 2.15 [1.89,2.70]\\
$k$ & 0.49 & 0.49 [0.47,0.51] & 0.48 [0.41,0.58]\\
\hline
\end{tabular}
\label{tab:g-and-k-bootstrap}
\end{table}
\subsection{Stochastic Gompertz model}\label{sec:gompertz}
Here we consider a state-space model with stochastic Gompertz dynamics. \cite{donnet2010bayesian} and \cite{ditlevsen2013introduction} used a hierarchical (mixed-effects) version of this model to study chicken growth. We do not use a mixed-effects model so our results cannot be directly compared to the cited references. We have the stochastic differential equation (SDE)
\begin{equation}
dX_t = BCe^{-Ct}X_tdt + \sigma X_tdW_t,\qquad X_0=Ae^{-B}
\label{eq:gompertz-state}
\end{equation}
with $A$, $B$, $C$ and $\sigma$ unknown positive constants. It is easy to prove by It\^{o}'s formula on the transformed process $Z_t=\log(X_t)$ that \eqref{eq:gompertz-state} has explicit solution $X_t=Ae^{-(Be^{-Ct})-\frac{1}{2}\sigma^2t+\sigma W_t}$ and $X_0=Ae^{-B}$. Same as in \cite{donnet2010bayesian} and \cite{ditlevsen2013introduction} we consider data on a logarithmic scale according to
\[
(y_0,y_1,...,y_n) = (\log(A)-B,\log(X_1),...,\log(X_n))+\epsilon
\]
where $\epsilon\sim N_{n+1}(0,\sigma^2_{\epsilon}I_{n+1})$ has $(n+1)$-dimensional multivariate Gaussian distribution (here $I_{n+1}$ is the identity matrix). For simplicity we assume $X_0$ known hence an estimate of either $A$ or $B$ can be determined from an estimate of the other parameter, e.g. $B=\log(A/X_0)$. In the following we choose to determine $B$. We assume $\sigma_{\epsilon}$ known as this is a difficult parameter to identify without access to repeated measurements. Therefore unknowns are $(A,C,\sigma)$ and, in order to preserve positivity, in practice we conduct inference for $\theta=(\log A,\log C,\log\sigma)$. We set the following priors $\log A\sim U(1,15)$, $\log C \sim U(0.5,4)$, $\sigma \sim LN(0.1,0.2)$. Here $LN(a,b)$ denotes the log-Normal distribution with parameters $(a,b)$ ($a$ and $b$ being the mean and standard deviation respectively of the associated Normal distribution).
We simulate $n+1=51$ data points at equispaced observational times $(t_0,...,t_n)=(0,1,...,50)$ with parameters $(\log A,\log B,\log C,\log\sigma,\log\sigma_{\epsilon})=(8.01,1.609,2.639,0,-1.609)$. In practice we normalize times to be in [0,1] for numerical stability.
We wish to consider summary statistics to ease inference via ABC, however the determination of summaries for dynamic models is way less intuitive than for static models. We employ the regression approach suggested in \cite{fearnhead-prangle(2011)} to determine a set of three summaries $S(\cdot)=(S_1(\cdot),S_2(\cdot),S_3(\cdot))$. Essentially their ``semi-automatic ABC'' is such that the vector $S(\cdot)$ has the same dimension as $\theta$, that is each element of $S(\cdot)$ is supposed to be informative for a given component of $\theta$. We do not illustrate their method here, but the reader can refer to \cite{picchini-2013} for an exposition targeting SDE models and to our MATLAB package \citep{abc-sde} implementing the Fearnhead-Prangle method for the determination of $S(\cdot)$ (but not implementing data-cloning).
We first illustrate the results from an ABC-MCMC without data-cloning: we start at initial parameter values $(\log A_0,\log C_0,\log\sigma_0)=(11,0.6,-2.3)$ and first run a pilot study to determine weights $(\omega_1,\omega_2,\omega_3)$ weighting the three automatically obtained summaries $(S_1,S_2,S_3)$. Then we run ABC-MCMC once more using these weights and produce a total of $R=40,000$ draws. The first 8000 iterations use $\delta=20$, then we decrease it to $\delta=5$ for 7000 iterations and finally to $\delta=1.5$ for the last 25000 iterations with a 1\% acceptance rate at the smallest threshold. Automatic summaries construction together with the 40,000 ABC-MCMC iterations required about 55 seconds. Trace plots are in Figure \ref{fig:gompertz_abcmcmc_traces}. Posterior means are $\log\hat{A}=7.52$, $\log\hat{C}=2.58$, $\log\hat{\sigma}=-0.52$. As we see in a moment the identification of $\sigma$ is difficult because we can't really make use of an informative summary statistic.
\begin{figure}
\centering
\includegraphics[scale=0.7]{gompertz_abcmcmc_traces}
\caption{\footnotesize{Gompertz model: traceplots for $\log A$, $\log C$ and $\log \sigma$ when using ABC-MCMC. Horizontal lines are parameter true values.}}
\label{fig:gompertz_abcmcmc_traces}
\end{figure}
We now consider ABC-DC where $K=1$ is used for the first 10,000 iterations and we keep $\delta=18$ constant for the entire simulation, i.e. this $\delta$ is about thirteen times larger than the smallest $\delta$ used in ABC-MCMC. At the end of the 10,000 iterations we apply the regression adjustment described in section \ref{sec:regression-adjustment}. This is trivially implemented and has negligible impact on the overall computational budget, amounting to solve a system of linear equations using draws already sampled at previous iterations, i.e. it is an almost instantaneous operation. In Figure \ref{fig:gompertz-regularization-top} circles denote pairs $(\theta_i,S_i)$ where the $\theta_i$ are draws generated with $K=1$ (after burnin) and $S_i$ the corresponding three-dimensional simulated summary statistic, one for each dimension of $\theta_i$. Plusses denote the pairs $(\theta_i^*,S_i)$, i.e. regression-adjusted parameters. We deduce that regularization enables an improved identification of $A$ and $C$, which is of great help given the large value of $\delta$ we are using. However the summary statistic used for $\sigma$ seems totally uninformative for the said parameter and in fact the regularization has no effect, see also Figure \ref{fig:gompertz-regularization-densities}. As explained in section \ref{sec:regression-adjustment} we can use the sample covariance from the adjusted draws to create a more effective independence sampler when $K>1$. We employ this strategy here, and furthermore we center the independence sampler to the mean of the adjusted draws $\theta^*_i$ and execute 20,000 further iterations using $K=11$ clones. Trace plots are in Figure \ref{fig:gompertz_abcdc_traces}. Sample means computed on the last 20,000 draws returns point estimates $\log\hat{A}=7.88$, $\log\hat{C}=2.71$, $\log\hat{\sigma}=-0.288$ which are closer to the true parameter values than those obtained via ABC-MCMC.
\begin{figure}
\centering
\includegraphics[scale=0.7]{gompertz-regularization-top}
\caption{\footnotesize{Gompertz model: ABC-DC with $K=1$. Abscissas have values of simulated summary statistics and ordinates have the corresponding simulated parameters, labelled with gray circles (o). Black plusses (+) denote corresponding regularized parameters.}}
\label{fig:gompertz-regularization-top}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.7]{gompertz-regularization-densities}
\caption{\footnotesize{Gompertz model: ABC-DC with $K=1$. Kernel smoothed marginals for regression adjusted parameters (solid lines) and from the original non-adjusted draws (dashed lines). For $\log\sigma$ the two marginals superimpose.}}
\label{fig:gompertz-regularization-densities}
\end{figure}
With $K=11$ we obtain a 1\% acceptance rate, for comparison with inference via ABC-MCMC. The entire algorithm (including the calculation of summary statistics and the regression adjustment) required about 48 seconds. This was possible thanks to a carefully vectorized MATLAB code so that simulating multiple instances of our model does not have any significant impact on the computational performance. Notice that without adjustment we are not able to jump from $K=1$ straight to $K=11$, as this would result in a extremely low acceptance rate. Finally, in order to show the robustness of the method, we perturb the parameters starting values and produce three different chains given in Figure \ref{fig:gompertz-threestartingvalues}, all converging to the same values despite the large ABC tolerance and very different starting parameters. For example, in Figure \ref{fig:gompertz-logA-1clone-density} we show the marginal posteriors for $\log A$ for the three chains resulting from $K=1$ and before applying regression adjustment: as we can see the main modes are rather different. Despite this, for increasing $K$ the three ABC-DC chains converge to about the same value, as we have just discussed.
\begin{figure}
\centering
\includegraphics[scale=0.7]{gompertz_abcdc_traces}
\caption{\footnotesize{Gompertz model: traceplots for $\log A$, $\log C$ and $\log \sigma$ when using ABC-DC. Horizontal lines are parameter true values.}}
\label{fig:gompertz_abcdc_traces}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.7]{gompertz-threestartingvalues}
\caption{\footnotesize{Gompertz model: three independent chains of ABC-DC starting at different values.}}
\label{fig:gompertz-threestartingvalues}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.4]{gompertz-logA-1clone-density}
\caption{\footnotesize{Gompertz model: marginal posteriors without regression adjustment for each of three chains for $\log A$ resulting from iterations 7,000-10,000 in Figure \ref{fig:gompertz-threestartingvalues}, that is from $K=1$ and $\delta=18$. See main text for comments.}}
\label{fig:gompertz-logA-1clone-density}
\end{figure}
Our main comment is that we obtain a reasonable point estimate using ABC-DC without having to reduce $\delta$ too much, which should be particularly relevant for even more complex modelling scenarios, as commented at the beginning of section \ref{sec:simulations}.
\subsection{Two dimensional correlated Geometric Brownian motion }\label{sec:2GBM}
In this section we apply ABC-DC to estimate the parameters of a two-dimensional SDE model having a likelihood function analytically available. It is therefore possible to compare results based on our methodology with exact maximum likelihood estimation. The model considered is a two-dimensional geometric Brownian motion or Black-Scholes model, the standard asset price model in finance.
The process $\left\{X_{t},Y_{t}\right\}_{t\geq0}$ is solution of
the system of SDEs
\begin{align}
\begin{cases}
dX_{t}&=\mu_{1}X_{t}dt+\sigma_{1}X_{t}dW_{t}^1 \\
dY_{t}&=\mu_{2}Y_{t}dt+\sigma_{2}Y_{t}dW_{t}^2
\label{eq:GBM}
\end{cases}
\end{align}
with $\sigma_{1},\sigma_{2}>0$, $\{W^{1}_t\}$ and $\{W^{2}_t\}$ correlated Wiener processes such that $dW_{t}^{1}\cdot dW_{t}^{2}=\rho \cdot dt$ with a correlation coefficient $|\rho|<1$.
We can explicitly write the two components of the exact solution for
$t\in[0,\infty)$ in terms of two independent Wiener processes $\{B^{1}_t\}$
and $\{B^{2}_t\}$ as
\begin{align*}
\ X_{t}&=X_{0}\exp\left\{ \left(\mu_{1}-\frac{1}{2}\sigma_{1}^{2}\right)t+\sigma_{1}B_{t}^{1}\right\} \\
\ Y_{t}&=Y_{0}\exp\left\{ \left(\mu_{2}-\frac{1}{2}\sigma_{2}^{2}\right)t+\sigma_{2}\left(\rho B_{t}^{1}+\sqrt{1-\rho^{2}}B_{t}^{2}\right)\right\}
\end{align*}
where $ (X_{0},Y_{0}) $ is the deterministic starting state of the process.
By application of the multi-dimensional It{\^o} formula we can derive an explicit expression for the conditional transition densities. It follows that for $s\in\left[0,t\right),\, t\in[0,\infty),$
the transition density is
\[
p(s,x_{s},y_{s};t,x_{t},y_{t})=\frac{1}{2\pi(t-s)\sqrt{1-\rho^{2}}\sigma_{1}\sigma_{2}x_{t}y_{t}}\cdot
\]
\[
\exp\biggl(-\frac{(\ln(x_{t})-\ln(x_{s})-(\mu_{1}-\frac{1}{2}\sigma_{1}^{2})(t-s))^{2}}{2\sigma_{1}^{2}(t-s)(1-\rho^{2})}
\]
\[
-\frac{(\ln(y_{t})-\ln(y_{s})-(\mu_{2}-\frac{1}{2}\sigma_{2}^{2})(t-s))^{2}}{2\sigma_{2}^{2}(t-s)(1-\rho^{2})}+
\]
\[
\frac{(\ln(x_{t})-\ln(x_{s})-(\mu_{1}-\frac{1}{2}\sigma_{1}^{2})(t-s))(\ln(y_{t})-\ln(y_{s})-(\mu_{2}-\frac{1}{2}\sigma_{2}^{2})(t-s))\rho}{\sigma_{1}\sigma_{2}(t-s)(1-\rho^{2})}\biggr).
\]
We assume observations generated from model \eqref{eq:GBM}, hence due to the Markovian property of the model solution it is possible to write the exact likelihood function for a discrete sample from the process as proportional to the product of its transition densities. MLEs are therefore obtained by maximizing numerically the resulting likelihood function, see Table \ref{tab:GBM-abcdc}.
We wish to conduct inference for the set of parameters $\theta = \left(\mu_1, \ln(\sigma_1), \mu_2, \ln(\sigma_2), \rho \right)$ using $500$ equispaced observations taken on the time interval $\left[0,1\right]$. The starting state of the model is set to $({X}_{0}, {Y}_{0})=\left(1,2\right)$ and the true values of the parameters are $\mu_{1}=1.7$, $\ln\sigma_{1}=-0.8$, $\mu_{2}=1.3,$ $\ln\sigma_{2}=-1.2$ , $\rho=0.3$.
The dimension of the problem creates difficulties in comparing data and pseudo-data, since to preserve a reasonable acceptance rate we would need to fix the threshold $\delta$ to high values. However for this example we are able to identify sufficient statistics, resulting in much higher acceptance rates compared to using the entire dataset. Sufficiency implies that comparing summaries of data and pseudo-data is equivalent to comparing actual and simulated data.
We denote with $\left(M_{1},V_{1},M_{2},V_{2},R_{1},R_{2}\right)$ the vector
of sufficient statistics based on discrete observations $\{Z_i\}_{i=0,..,n}=\{X_{i},Y_{i}\}_{i=0,..,n}$. The statistics are
\begin{align*}
\ M_{1}&=\sum_{i=1}^{n}(\ln X_{i}-\ln X_{i-1})=(\ln X_{n}-\ln X_{0}) ,
\ V_{1}=\sum_{i=1}^{n}(\ln X_{i}-\ln X_{i-1})^{2} \\
\ M_{2}&=\sum_{i=1}^{n}(\ln Y_{i}-\ln Y_{i-1})=(\ln Y_{n}-\ln Y_{0}) ,
\ V_{2}=\sum_{i=1}^{n}(\ln Y_{i}-\ln Y_{i-1})^{2} \\
\ R_{1}&=\sum_{i=1}^{n}(\ln X_{i}-\ln X_{i-1})(\ln Y_{i}-\ln Y_{i-1}) ,
\ R_{2}=\sum_{i=1}^{n}\ln(X_{i}Y_{i}).
\end{align*}
Since these statistics vary on different scales, we weight them using the estimated standard deviation $\hat{\sigma}_{j}$ of each statistic, obtained using a pilot run of ABC-MCMC considering 20,000 iterations and using thresholds $\delta=\left(1,0.9,0.8\right)$, updated at iterations 7,000 and 14,000. We denote with ${S}\left(z\right)=\left(S_{j}(z)\right)_{j=1,..6}$ the vector of statistics for a dataset $z$ and with $S(z^*)$ the corresponding quantity for a simulated $z^*$. By considering for $J_{\delta}\left({S}(z),{S}(z^{*})\right)$ a Gaussian kernel
$
\prod_{k=1}^{K}\exp\left(-u/(2\delta^{2})\right)/\delta
$, we weight summary statistics by writing $u={D^{T}}{\Sigma}{D}$ where
${D^{T}}=\left({S}(z^{*(k)})-{S}(z)\right)^{T}$
and diagonal matrix ${\Sigma}=\mathrm{diag}\left(\omega_{1},...,\omega_{6}\right)$
with $\omega_{j}=1/\hat{\sigma}^2_j$.
We conduct inference on the logarithms of the two volatility parameters $\sigma_{1}$ and $\sigma_{2}$, since they are meant to be strictly positive. Moreover, the correlation parameter $\rho$ should be in $(-1,1)$, therefore we truncate its prior on this interval. We set priors ${N}\left(1.5,0.5^2\right)$ on the drift parameters $\mu_{1}$ and $\mu_{2}$, whereas the priors on $\ln\sigma_{1}$ and $\ln\sigma_{2}$ are chosen to be ${N}\left(-1,0.5^2\right)$, and as prior on
$\rho$ we set a truncated Gaussian $N_{(-1,1)}\left(0.5,0.3^2\right)$ with subscript denoting the truncation at the specified interval. The starting values for the parameters are set to $\theta_0=\left(1.5,-1,1.5,-1,0.1\right)$.
We compare the estimates obtained using ABC-MCMC and ABC-DC on a total of 100,000 iterations. In both cases, the covariance for the adaptive Metropolis algorithm is updated every 1,000 iterations.
With ABC-MCMC the threshold $\delta$ is updated every 10,000 iterations within the values $\left(0.8,0.5,0.4,0.3,0.2\right)$, then after further 20,000 iterations is decreased to $\delta=0.2$, and finally from iteration 60,000 to the end of the simulation the threshold is fixed at $\delta = 0.15$. During this last stage the acceptance ratio varies between 1 and 4\%. After a total running time of 198 seconds, we obtain the estimated posterior means $ \hat{\theta} = (1.6426 , -0.8495 ,1.2979, -1.1599 , 0.3860)$. Traceplots are given in Figure \ref{fig:GBM_ABC}.
For comparison, similar settings are applied to estimate the model parameters with ABC-DC. The first 10,000 iterations are spent in the initial ABC-MCMC stage, where the threshold $\delta$ is updated once, at iteration 5,000, from 0.8 to 0.5. At the end of the ABC-MCMC stage the acceptance rate is about 15\%. Then every 10,000 iterations the number of clones is increased progressively to 3, 5, 6, 7, and the final 40,000 iterations are run with $K=8$. In this last stage the acceptance ratio is 1\%. The simulation required 282 seconds, returning $ \hat{\theta} = (1.7065 ,-0.8695 , 1.3254 , -1.1679 , 0.4075 ) $. However, good estimates with ABC-DC can be obtained with only 40,000 iterations, passing directly from 1 to 8 clones at iteration 10,000, without updating the threshold $\delta$, with a final acceptance rate of 3.3\%. In this case estimates are $ \hat{\theta} = ( 1.7145 , -0.8739 , 1.3378 , -1.1804 , 0.4040)$, returned in only 112 seconds. Traceplots are shown in Figure \ref{fig:GBM_ABCDC_fast}. A similar cut in the number of iterations failed with ABC-MCMC, since a slow reduction of the threshold $\delta$ was required in order to preserve an acceptable mixing of the chains. Results are summarized in Table \ref{tab:GBM-abcdc}.
The presented estimates are based on a single dataset generated from the set of parameters $\theta$, but in repetitions of the experiment ABC-DC has proved to be a robust method, as shown below. We now run $B=30$ independent simulations: for each simulation a different dataset is generated using the same value for the true parameter $\theta$ as considered in previous experiments. Then, each of these datasets is fitted using several algorithms, each algorithm returning the mean value of the proposed parameters. On the set of $B$ estimates $\hat{\theta}_{b}$ we compute the mean bias $\sum_{b=1}^{B}(\hat{\theta}_{b}-\theta)/B$ and the root mean square error (RMSE) $\sqrt{\sum_{b=1}^{B}(\hat{\theta}_{b}-\theta)^2/B} $. Results obtained with ABC-MCMC and ABC-DC are comparable, see Table \ref{tab:GBM-robustness}.
This example shows that both ABC-MCMC and ABC-DC can be applied with success to multi-dimensional SDE models when the likelihood function is unavailable, if we can identify a set of sufficient (or at least informative) summary statistics. The advantage of ABC-DC in this case is that we can obtain results comparably good in almost half of the time required for ABC-MCMC.
\begin{table}
\footnotesize
\centering{}
\begin{tabular}{cccccc}
\hline
\hline
& & & ABC-MCMC & ABC-DC \footnotesize{progressive} & ABC-DC \footnotesize{fast} \\
& True values & Exact MLE & $K=1$, $\delta$=0.15 & $K=8$, $\delta = 0.5$ & $K=8$, $\delta = 0.8$\\
&& & 100,000 iterations & 100,000 iterations & 40,000 iterations\\
\hline
$\mu_{1}$ & 1.7 & 1.7359 & 1.6426 & 1.7065 & 1.7145 \\
$\ln\sigma_{1}$ & -0.8 & -0.8381 & -0.8495 & -0.8695 & -0.8739 \\
$\mu_{2}$ & 1.3 & 1.2913 & 1.2979 & 1.3254 & 1.3378 \\
$\ln\sigma_{2}$ & -1.2 & -1.1599 & -1.1599 & -1.1679 & -1.1804 \\
$\rho$ & 0.3 & 0.3742 & 0.3860 & 0.4075 & 0.4040 \\
\hline
\end{tabular}
\caption {\footnotesize{True parameters, exact MLE, ABC-MCMC and ABC-DC estimates. Results comparably good can be obtained with ABC-DC in a bit more than half of the time required to ABC-MCMC. ``ABC-DC progressive'' considers the case where $\delta$ is reduced and $K$ is progressively increased. ``ABC-DC fast'' denotes a fixed $\delta$ and a $K$ increased rapidly.}}
\label{tab:GBM-abcdc}
\end{table}
\begin{table}
\footnotesize
\centering{}
\begin{tabular}{ccccccc}
\hline
\hline
& \multicolumn{2}{c}{ABC-MCMC} & \multicolumn{2}{c}{ABC-DC \footnotesize{progressive}} & \multicolumn{2}{c}{ABC-DC \footnotesize{fast}}\\
& Mean Bias & RMSE & Mean Bias & RMSE & Mean Bias & RMSE \\
\hline
$\mu_{1}$ & -0.0911 & 0.2782 & -0.0674 & 0.3605 & -0.0114 & 0.3021 \\
$\ln\sigma_{1}$ & -0.0092 & 0.0408 & -0.0134 & 0.0435 & -0.0409 & 0.0552 \\
$\mu_{2}$ & 0.0287 & 0.2309 & -0.0244 & 0.2241 & 0.0021 & 0.1727 \\
$\ln\sigma_{2}$ & 0.0109 & 0.0269 & 0.0083 & 0.0522 & -0.0093 & 0.0317 \\
$\rho$ & -0.0039 & 0.0423 & -0.0036 & 0.0561 & 0.0321 & 0.0570 \\
\hline
\end{tabular}
\caption {\footnotesize{Mean bias and root mean square error for the parameters estimates on 30 different simulations.}}
\label{tab:GBM-robustness}
\end{table}
\begin{figure}
\centering
\includegraphics[scale=0.58]{GBM_ABC_1000}
\vspace{-1cm}
\caption{Geometric Brownian motion: (top) traceplots for $\mu_1$, $\log \sigma_1$; (middle) $\mu_2$, $\log\sigma_2$ and (bottom) $\rho$ when using ABC-MCMC and updating progressively the threshold $\delta$ to the values $\left(0.8,0.5,0.4,0.3,0.2,0.15\right)$. Horizontal lines are the true parameters.}
\label{fig:GBM_ABC}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.58]{GBM_ABCDC_fast_8}
\vspace{-1cm}
\caption{Geometric Brownian motion: (top) traceplots for $\mu_1$, $\log \sigma_1$; (middle) $\mu_2$, $\log\sigma_2$ and (bottom) $\rho$ when using ABC-DC and passing directly from 1 to 8 clones at iteration 10,000, without updating the threshold $\delta$. Horizontal lines are the exact MLEs.}
\label{fig:GBM_ABCDC_fast}
\end{figure}
\section{Summary}
We have presented a strategy to integrate approximate Bayesian computation (ABC) in the so-called ``data cloning'' framework for approximate maximum likelihood estimation. A standard ABC-MCMC algorithm is initially used as a ``workhorse'' to locate an approximate maximum for the ABC posterior, which we use to center a Metropolis independent sampler, to be employed during the data-cloning stage. We note that the accuracy of the final inference, beyond mere identification of the location of the approximate MLE, can be enhanced for small $\delta$ and large $K$. However expecting our sampler to satisfy simultaneously both requirements is unrealistic as these are competitive criteria, particularly for highly erratic stochastic processes. That is, when considering an ABC framework it becomes increasingly unlikely to accept a proposed parameter as $K$ increases.
We note that for the considered examples ABC-DC produces reasonable inferences even though it is run with a larger $\delta$ than typically desired. In previous works using data-cloning MCMC (\citealp{lele-dennis-lutscher}, \citealp{baghishani-mohammadzadeh}, \citealp{jacquier-johannes-polson}) the algorithms were started at a high value of $K$, this opening the possibility for a chain to get stuck in some local maximum, instead we start the simulation with $K=1$ and then we increase it. Furthermore, in simulations considered in \cite{baghishani-mohammadzadeh} the starting parameter for their MCMC experiments was set to the exact MLE, which will \textit{of course} produce good results for $K$ large since their MCMC procedure starts in a peaked distribution already centred at its maximum.
Besides statistical inference considerations, a scenario where we could see our method being employed is when considering a computationally expensive model simulator such that producing a single realizations from the model requires several seconds or minutes. Assume therefore that running many ($R$) iterations of ABC-MCMC is impractical, whereas running $\tilde{R}\ll R$ iterations of ABC-DC over $M>1$ processors is feasible (for simplicity, assume $K$ a multiple of $M$). Here the task of computing the cloned likelihood would be performed by distributing $K/M$ model simulations to each of the $M$ processors.
A further application of the method can be envisaged when the experimenter is in need of reasonable and rapidly available estimates to be used as starting values for expensive procedures requiring a careful initialization, such as particle MCMC (pMCMC, \cite{andrieu2010particle}). For example in \cite{owen2015likelihood} it was found that using a specific ABC strategy (ABC-SMC) prior to starting pMCMC was of benefit, instead of investing a large amount of computational budget in trying to hand-tune pMCMC.
\section*{Acknowledgements}
Research was funded by the Swedish Research Council (VR grant 2013-5167).
\bibliographystyle{elsarticle-harv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,160 |
The Adventures of Lomax est un jeu vidéo de plates-formes développé et édité par Psygnosis en 1996 sur PlayStation et Windows. Le jeu est un spin-off tiré de la série Lemmings. Il met en scène Lomax, un lemming qui doit sauver ses congénères à travers de nombreux niveaux de plates-formes en scrolling horizontal.
Le jeu a été conçu par Erwin Kloibhofer et Henk Nieborg, les créateurs de Lionheart (1993) et The Misadventures of Flink (1994).
Système de jeu
Les Lemmings, créatures aux cheveux verts et à la robe violette, ont été ensorcelés par Evil Ed. Le joueur doit alors guider Lomax, un chevalier lemming à travers plus de quarante niveaux aux thèmes variés (collines, paysages engloutis, cimetière...). En traversant ces niveaux, le joueur a l'occasion de libérer les lemmings (transformés en zombis, enfermés dans des tonneaux...) en les frappant afin de leur rendre leur forme originelle.
Au fil du jeu, Lomax obtiendra de nouvelles capacités (que l'on trouve dans les vases disséminés dans les niveaux), propres à l'espèce des lemmings et à son casque, qui l'aideront dans sa progression :
L'attaque tournoyante (attaque de base)
Le casque magique (permettant de toucher les ennemis à distance, mais également d'absorber une attaque. Dans ce cas, le casque disparaitra)
Le casque enflammé (permettant de toucher plusieurs ennemis au lieu d'un lors d'un lancer. Limité en utilisation)
Le casque-grappin (permettant de s'agripper aux plates-formes flottant en hauteur, ou bien de traverser des gouffres. Limité en utilisation)
Le casque-planeur (permettant de sauter plus haut et de planer durant un certain laps de temps. Limité en utilisation)
Construction de ponts (place une passerelle permettant d'atteindre les endroits en hauteur. Limité en utilisation)
Forage de galeries (permet de creuser dans les murs de poussière. Limité en utilisation).
Les lemmings permettent également d'ouvrir un niveau bonus : une fois un certain nombre de lemmings libérés (sans perdre de continue), le portail de fin de niveau se transforme et nous téléporte vers un monde dans lequel le but est de récolter un maximum de pièces et d'atteindre la sortie avant la fin du temps imparti.
Accueil
Dengeki PlayStation : 60 % / 55 %
GameSpot : 7,7/10
Notes et références
Liens externes
The Adventures of Lomax sur GameSpot
Jeu de plates-formes
Jeu vidéo sorti en 1996
Jeu PlayStation
Jeu Windows
Jeu vidéo développé au Royaume-Uni
Jeu Psygnosis
Lemmings | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,472 |
Jenna Elizabeth Johnson: FREE READS!
Hello readers! Here are some free reads - extra scenes, short stories and the like - from both the Oescienne and Otherworld series. Be sure to scroll down to get a look at everything. Happy reading!
RECEIVE FREE EBOOKS FROM JENNA ELIZABETH JOHNSON IN EXCHANGE FOR AN HONEST REVIEW!
1.) Read and review one of J.E. Johnson's books.
2.) Copy and paste the review OR include a screenshot of the review in an email and send it to authorjejohnson@gmail.com (put Read, Review, Receive Program OR RRR Program in the subject). Be sure to include your name and WHERE you left the review (Amazon, B&N, iTunes, Smashwords, Kobo, Goodreads and/or Google Play/Google Books).
3.) Tell me if you'd prefer a Mobi (Kindle), ePub (Nook) or PDF copy of the NEXT book in the series you are reviewing (for example, if you read and review Faelorehn, I will send you a copy of Dolmarehn).
4.) After checking the review, I will send your new ebook as an attachment, along with directions on how to upload it onto your Kindle or Nook.
5.) Thank you and as always, HAPPY READING!
START READING WITH THESE FREE BOOKS!!!
Hello Readers! A long while back, when I first started writing the Legend of Oescienne series, I thought it would be fun to keep a journal told from Jahrra's perspective. On my old blog, I used to post these when the time didn't slip away from me as much as it has of late. Anyways, I'd like to revive Jahrra's Journal here and post her journal entries from time to time. New entries will go up on my main feed as well as on this page (where you can read them all). I hope those of you who enjoy the Oescienne books also enjoy these peeks into the mind of Jahrra. Happy reading!
I thought about running. After all, Pada has always told me about that dangerous forest and how dangerous animals and monsters live in there, but I wanted to know who the green cloaked man was.
Why does he come to my dreams? Why can't I see his face? Who is he? But by the time I turned back around, he was gone and I was awake.
I learned all about dragons today! I had no idea there were so many different kinds. But I'll tell you about my favorite: the Korli. Master Hroombra is a Korli dragon. I used to think he had so many wrinkles because he was old, but he says it is a, what was the word? Characteristic of his race. That means it is something all Korli dragons share, but he did say he had more wrinkles than most.
Korli dragons come in all the colors of blue and gray, from nearly black to nearly white. Master Hroombra is a medium-dark gray blue. Some of their bottom teeth stick out of their mouths, making them look frightening. But Master Hroombra has never frightened me!
Korli dragons were in charge of watching over the humans. Master Hroombra says that long ago humans lived in Ethöes. I asked him what they were like and he said they were very similar to Nesnan elves, only they didn't live nearly as long and they weren't as strong or magical like the elves were.
I thought it would be an awful thing to be human, but he only said that sometimes being strong and living for a long time wasn't always such a good thing. I still think it wouldn't be such a good thing to be human, but he just smiled like he always does.
It is late and Nida will be wondering about the candlelight coming from my room. I'll tell you more tomorrow, dear Journal, and then I'll tell you about my second favorite dragon, the Tanaan!
Today I learned mathematics with Master Hroombra. I hate mathematics! I thought it was going to be another boring lesson, but when Master Hroombra left me to add the numbers on my own I noticed a pile of paper in the great big fireplace. I crawled closer and peeked in, hoping to see what the notes said, but the markings on the paper were so strange! I picked up the one closest to me and tried to read it, but whoever wrote it used marks not at all like the ones Master Hroombra has taught me to read. I know I should have put it back, but I kept it, tucking it away to put into my journal later. Someday I'm going to figure out what it says, but for now, I'll just look at the marks. They are pretty in an odd way.
I asked Master Hroombra today what exactly a boarlaque is. I had an idea from what Pada had told me, but I wasn't sure. Master Hroombra just smiled and pulled out a large, old book, trailing dust through the air like a comet. I think it is funny when dragon's smile, especially Master Hroombra with all his teeth sticking out. The yellowed pages crackled and stuck as I turned them, but I almost fell over when I noticed what was on them. Not words like all the other books he gives me, but pictures. Loads and loads of pictures of animals and maps and flowers, all colored in! I asked him where he got it and he told me the elves of Felldreim created it. It was the most wonderful book I've ever seen! Finally, I got to a fearsome looking beast and he told me to stop. The boarlaque, he told me. It made me shiver. It was bigger than a bear and covered in brown and gray fur that was all tangled. It had stripes on its face and giant teeth and claws. I've decided two things: one, this book of creatures is my absolute favorite book ever, and two, I'm never going near the Wreing Florenn again, Pada or no Pada.
It is hard to stay awake, I'm so tired! But Master Hroombra is teaching me about the stars tonight. Oh, I'm so excited! Right now I'm sitting by the fire in the Castle Guard Ruin (and can you believe it, Nida and Pada are letting me stay here for the night!), but soon we'll go outside to watch a meteor shower. He says that it's like watching magic sparks streak through the air, and that sometimes the sparks are blue and red and yellow. I want to go outside right now, but Master Hroombra says we must wait. I hate waiting! The sky is all dark, like the ink in my pen clouding up water, and the crickets are chirping in the wood, it should be time to go! But no, we must wait. Adults always want to wait. Until then I'll write in my journal and watch the stars come out through the doorway.
There's one right now. Master Hroombra says it is called Rhian and it is part of a constellation called Echainn and Echnia. They are the Horses he says, and that the other star I see, just to the left and above Rhian, is Epohra. He was about to tell me the story of Echainn and Echnia but a meteor just streaked across the sky. Oh! It is fiery yellow, just like he said. I'll have to ask him about the stars later, because he's grabbing the old quilt and we're going to watch the fire fall from the sky!
I tried to pretend to be sick, I even tried getting sick, but it was no use. Nida practically dragged me out to the end of the path leading to the main road this morning and lifted me into the mail cart. I think I frightened Gieaun and Scede, but when I told them I didn't want to go to school, they understood. Mr. Dharedth the mailman told us jokes the entire way (we didn't stop at Master Hroombra's today and I didn't even see him when we passed by the Castle Guard Ruin). Once we got to the schoolhouse though, I felt like my bones were melting. I could barely walk and I felt all shaky. But we got there before the mean boys and girls did, and we went and hid behind a big huge rock sitting in front of a giant oak tree. No one even noticed us until the teacher called us into class, and by then it was too late for the horrible girl and boy to do anything mean. I felt better when they ignored me and Gieaun and Scede at lunch time, and when Mr. Cohrbin dismissed us, Mr. Dharedth was already there with his mail cart. I was so relieved I almost fell out when we started moving. I'm not sure, but maybe school won't be so bad after all.
I came home from school today with my face all puffy and my eyes all red. Nida asked what had happened and I lied - 'I hit my knee against the wheel of the mail cart while getting down.' I told her. She gave me one of her looks, you know the ones where she places her hands on her hips and arches one eyebrow? But she didn't ask any more questions. And I only lied because I was too ashamed to tell the truth.
I got in trouble today. I had to sit on the fallen log on the edge of the schoolyard. The only reason I can bring myself to write this down instead of curling up in my bed and slowly dying of shame is that Ellysian, the evil girl I've told you about before, got in trouble, too. She and her nasty brother had started it, by throwing dirt clods at Gieaun, Scede, and me. Well, she's been bugging us ever since school started, and I'd had it. I picked up a dirt clod and threw it right back at her. Too bad I missed, but Mr. Cohrbin saw me. I'm sure my face turned the color of ripe autumn apples! I lowered my head in embarrassment and listened to Ellysian wail in anger and false injury.
Gieaun and Scede told Mr. Cohrbin what had happened, but since we had both been throwing things at one another, we were both in trouble. We had to sit on the fallen log all through lunch. I sat quietly, feeling very much ashamed of myself, but Ellysian only huffed and puffed and sent me nasty glances. I was glad when Mr. Cohrbin finally rang the bell for us to return to class. The good thing was, however, Ellysian and her brother didn't bother me once the rest of the day. Maybe I should get in trouble more often . . .
It's so very hard to be little, especially when you are little and unable to tell people what you are thinking. On the other hand, sometimes being stuck inside your own head can be a good thing. I can think better about stuff that way. Not the usual stuff everyone else my age thinks about, like when the newest episode of their favorite cartoon is going to air, or if their moms will let them go on a play date with their best friend, but the kind of stuff they don't notice normally. I think it's because most kids and adults are so distracted by all the noise and lights in the world that they miss the cool stuff.
Well, I guess 'cool' is the wrong word to describe it. 'Creepy' is probably a better description. Just yesterday, something creepy happened at my school. I don't go to the same kind of school my brothers go to, but I'm pretty sure weird little monsters don't just climb into their classrooms in the middle of lunch recess.
At normal schools, you have to talk to people and answer your teacher if she asks a question. Where I go to school, I don't have to do any of that. I like learning and discovering new things just fine, but at a regular school, I think the noises and the mean kids would be too much for me. Mom and Dad say that I have autism, and that's why I have trouble behaving like a normal kid. I don't know that much about autism, but from what I heard Mom and Dad saying when they thought I wasn't listening is that I have many of the symptoms.
At the time I didn't know what a symptom was, so I had to look it up in Logan's dictionary. He's my older brother. It was hard pulling the dictionary from his bookshelf and even harder to find the word.
I knew it was in the S section, but sometimes when I'm doing something, my body won't stop when I tell it to. Like looking for the word. My fingers kept flipping the pages long after I realized I had wandered in the T's. Other times, my body works faster than my mind and I'll stop in the middle of doing something even when my brain is asking me to keep going. It is very frustrating, but I have learned to live with it.
When I finally made it back to the right spot in the S's, I read that a symptom is a sign that something is wrong. That had scared me. For a whole week I moved around the house more carefully and quietly than normal. Being scared of something when you can't tell your family why is very hard.
Eventually, Mom sat down with me and gradually discovered the reason for my glum mood.
"Symptoms," was all I could say.
She must have realized I had overheard her and Dad, because she told me that my autism made me different than other kids, but not sick or bad. She said that the doctor was going to try and find some medicine to make me better, but that I didn't need to worry because she and Dad would look after me.
"Meggy?" I had asked in a small voice.
Meggy was what I called my older sister, but her real name is Meghan. Of my entire family, I love Meghan the most, maybe even more than Mom and Dad. It sounds awful, to love your sister more than your parents, but for some reason I just do. It's like my heart has special little rooms inside for all my family and friends, but Meggy's spot is bigger than all the rest.
And so it was because of my autism that I went to a different school than the other kids on our street.
I didn't mind, and most days were calm with very little excitement. Except for what happened yesterday.
Some days we stayed inside during our lunch break and played with the toys in the classroom. Tuesday was one of those days. After we were done eating, Mrs. Warren excused us from our tables to go play. I went directly to my cubby and pulled out a comic book from home, the pages worn and creased from the dozens of times I'd looked through it.
I was just sitting down on the carpet when a foul scent wafted in my direction. I wrinkled my nose and instinctively looked toward the open window, my eyes widening at what I spotted there. Our classroom was on the second floor of the building, so the only things we ever saw outside were the trees at the edge of the lawn, the road beyond that, and the birds that visited the feeders hanging in the oak tree nearby. But it wasn't a tree or a bird that I saw.
It was one of the ghoulies from my backyard. But what was it doing here? Ever since I was little, strange creatures that looked like half-rotten monsters had been showing up in my yard and the swamp behind it. At first, they had frightened me, but I remembered Meggy talking about them before I started going to school, so I figured everyone saw them. When my sister stopped talking about them, I thought they had gone away, but then they started showing up again. I waited for Meghan to say something, but she never did, and so I didn't either. I learned to watch her carefully after that, and I realized that she could still see them. She just wasn't telling anyone. I decided to do the same thing.
And now one of them was hanging onto the ledge and peering through the open window. It had shiny skin, like a frog or some other animal that lived in wet places, and it was the color of splotchy mud. Long, crooked yellow teeth jutted out from its lower jaw, and its nose was upturned like a pig's. When I squinted, I could just make out a nasty halo of blackish red smoke radiated from its dark hide, like silt unfurling and forming clouds of color in clear water. All of the nasty things, the ghoulies, always had this weird color surrounding them, and they all smelled terrible.
I would have been afraid, but like I said, I had seen them before. Not all of them looked exactly like this one, but I knew it was a ghoulie and I knew it wouldn't hurt me. The other ones had stayed away from me the way mosquitoes avoid me when they realize I'm wearing bug repellent. Anyway, I was mostly afraid it might hurt my classmates. Just like everyone else, except for Meggy and me, they couldn't see it. As I watched it, the ghoulie shoved the upper half of its body through the window and was now reaching for the ground with its short front legs.
I jerked my attention away from the window and cast a glance at the girl, Maddie, sitting closest to me. She had pulled over a barrel of wooden blocks onto the carpet and was cracking them together, her glasses making her pale blue eyes as large as an owl's. The blocks came in many different shapes, the kind you used to make wooden cities with arches and columns.
One of the blocks, a rectangular-shaped one, had toppled away from the rest of the pile. I reached out, and ignoring Maddie's screech of disapproval, grasped the wooden block in my hand, my fingers clumsy and slow to respond to my thoughts. The corners were smoothed and it was almost as heavy as a rock. I knew there was a risk of breaking the window if I threw it, but the monster now had its orange eyes fixed on Bella. She had an even harder time than I did communicating with normal people, and she couldn't move very fast. The creature would go for her first.
Sticking my tongue out in concentration, I drew my arm back and threw the wood as hard as I could, holding my breath and praying it didn't miss. To my great relief, the wood cracked against the creature's skull, causing it to squeal and jerk away from Bella, scrambling backwards through the window.
Mrs. Rodriguez turned away from one of the other boys, Jake, and glanced at the open window, then straightened and narrowed her eyes at the girl beside me.
Maddie just grinned and continued to click the wooden blocks together, making a burbling sound as she did so.
Miss Rodriguez must not have been too worried about us throwing blocks because she looked back down at Jake. Or maybe it was because Jake had started flinging paint around with his paintbrush and she needed to return his focus on keeping the paint on the paper.
Now that the scary monster was gone, I breathed a sigh of relief, one of the only sounds I could make without trying very hard. It was so frustrating. I knew how to read, and how to write, even if my letters didn't always turn out neat or facing in the proper direction. I could picture them in my mind, the words I wanted to say as well, but every time I tried to speak or print out a sentence, my throat would close up, or my mind would go suddenly blank. It was as if the string of words belonging to the sentence I wanted to write broke loose and went flying all over the place like deflating balloons.
My eyebrows drew together and I glanced up, looking around the room and taking note of all of my classmates. There were only seven of us altogether. We were in what is called a Special Needs class. None of us were like normal kids, but that was okay. My mom and my teachers reminded me that I was extra special, and that's why I couldn't go to a school like my brothers or sister.
Knowing all this didn't bother me. I didn't want to be like the normal kids. Bradley and Logan had told me stories about some of the kids at their school who were mean to them and the other students. The nice thing about my school was that all my classmates usually got along and none of them ever did anything just to be mean. Even so, something, some feeling deep inside of me, told me that I was very different from my fellow classmates as well. Seeing the ghoulies was only part of it.
Maddie made a loud noise and I jumped, breaking out of my own personal reflection. When I looked up at her, she was smiling and holding out one of the wooden arches.
I tried smiling back. She was asking me if I would help her build a city. Deciding it was best for me to forget about the monster in the window, I took the block and scooted closer to her. As I started stacking wooden shapes with as much coordination as my hands would allow me, I thought about some of the other reasons I was different. I created a bridge with one of the larger arches and glanced up at my friend.
Maddie didn't glow like me. Neither did Jake, Russell, Bella or Mira. Mrs. Warren and her assistant Miss Rodriguez didn't glow, either. In fact, most people didn't glow. Just me and my sister, Meghan. When I was little, I didn't think too much about it. But now that I was seven, I wondered about stuff all the time. Not just why we glowed, but also why we saw the ghoulies and no one else did.
I sighed and got back to building castles with Maddie. Thinking about monsters reminded me of the ghoulie from earlier. I still had no idea why it had showed up here, or why it was out during the day. Usually I only saw them before and after sunset. Trying to forget about it, I picked up a cone-shaped block and was about to add it to the top of a turret when a shadow momentarily blocked the light streaming in through the window. I glanced up, my hand freezing in mid-air like a crane jerking to a stop over a half-constructed skyscraper. The ghoulie from earlier was back, its wicked claws scratching at the corner of the ledge, trying to find purchase on the windowsill.
My mouth went dry and my skin grew hot. I risked a glance at my arm. Just as I suspected, the pale blue glow of my skin had grown a little brighter. This always scared me. It happened a lot when I was frightened or mad or sad. Each time it happened, I was worried my skin might actually melt, but I always managed to calm down before it grew worse.
I returned my attention to the window and nearly gasped. A second ghoulie had joined the other and was trying to force its way in as well. And then a third one appeared, its beady eyes fierce and glowing orange as it peered at me through the window. The fear crept over my skin like icy spiders and I tried to remain calm. Why were there so many? What were they doing out in the middle of the day? And then I remembered: Halloween was only three days away. That would explain why there were so many, but not why they were here.
Halloween night had always made me nervous, ever since before I could walk. Mom or Meghan would help me make my costume, and then I'd go out trick-or-treating with my brothers and sister, just like all the other kids. I would laugh and scream and try to look like I was having a good time. That's what grown-ups expected of kids on Halloween. Only, that was the one time of year that the ghoulies came out the most. Kind of like the way worms crawl out onto the sidewalks after a rainstorm. Even if they could never get close enough to hurt me, they still scared me and gave me bad dreams.
The slight rustle of cloth distracted me from the creatures scrabbling at the window sill. Mari, who had been completely absorbed in her picture book over in the library nook, had flung the book down and was making a bee line for the window.
For a few seconds I sat gaping at her. Could she see the monsters, too? I returned my eyes to the window. One of the ghoulies had pulled its attention away from the classroom and was busy eyeing a hummingbird buzzing around the feeder in the oak tree.
"Hummie, hummie!" Mari squealed, her hands outstretched as she moved closer to the window.
Without giving it a second thought, I dropped the block I was holding. It crashed down and crumbled the castle in progress into a pile of rubble. Maddie gasped, then made a sound of outrage, but I hardly noticed. I was too busy making my way to the window.
With my heart racing in time with my feet, I shoved Mari out of the way and reached for the window latch. The girl went down screeching, crying at the top of her lungs when her bottom hit the ground.
The monsters latched their attention onto me, their predatory eyes narrowing as they bared their teeth and hissed. One of them had slipped its arm through the open window and was trying to sink its claws into the wall.
Panicking, I spun around, trying to find something to use as a weapon against it. A shelf of books, some bean bag chairs, the table with our math activity from earlier …. A cup of pencils sat on the bookshelf closest to me. As quickly as I could, I reached out and grabbed one of the largest pencils, nearly shouting in triumph as my thoughts and actions worked in unison for once. The tip was dull from use and the eraser was worn all the way down to the metal band, but it would have to do.
Another sharp hiss, followed by a growl, caught my attention and I spun back around. The first ghoulie had wedged the window open further and had stuck its head through, its pig nose sniffing the air. It glared at me, but couldn't swipe me with its claws because both front feet were being used to hang onto the windowsill. The smell coming off of it made me gag, and I held my arm up to my nose. The monster lunged again and the others behind it forgot all about the hummingbird as they turned to regard me and the other students once again.
A fresh shriek of anger and pain from Mari had drawn the attention of our teachers. Mrs. Warren and Miss Rodriguez were now looking in our direction with worry on their faces. I continued to stare at the three ghoulies, scraping my brain for any sort of idea that might make them go away. They were still struggling, but it wouldn't be long before they made it into the classroom, especially since the first one had almost succeeded earlier. I didn't have much time. Taking a deep breath and telling myself to be brave, I turned the pencil around, pointy end out, and shoved it as hard as I could into the eye of the closest ghoulie.
The beast screeched in pain and let go of the windowsill to claw at its eye. The others, who had mostly been hanging onto the first one, wheeled back and the whole lot of them crashed to the ground below. Taking a shaky breath, I stepped forward, almost falling over my own feet, and pulled the window shut. It clicked into place and I fumbled with the latch until it locked.
By that time both Mrs. Warren and Miss Rodriguez were kneeling beside Mari to see if she had hurt herself. The dark-haired girl mumbled something and pointed in my direction. Her face was scarlet and there were streaks of tears painting her cheeks.
I stood still, my hands clamped behind my back, and hung my head. It was times like these that I really wished I could talk like normal kids.
I nodded my head. Even if I couldn't communicate like everyone else, I never lied. Mom had told me that lying made you weaker and being honest made you stronger, even if it was hard to be honest.
I was already weak in so many ways. I wanted to be strong, so I never lied.
I looked up toward the window. Blackish streaks and a few scratches showed on the wall, but like the creatures, I knew I was the only one who noticed the marks. I wanted to tell them about the ghoulies, but I knew that telling people you saw scary monsters that no one else could see was worse than lying. Instead, I told them something else that may not have been entirely truthful, but wasn't really a lie either.
There was something dangerous trying to get in through the window, a wasp I think. Mari was distracted by the hummingbird and didn't see it, so I pushed her so she wouldn't get stung, then closed the window so it wouldn't get inside.
I wanted to cry. Not because I thought I was in trouble, or even because I felt bad for knocking Mari down, but because I couldn't tell my teachers what had happened. I couldn't even apologize properly to Mari for being so rude. But I did try.
Turning toward my classmate, I mumbled, "Sworry," and offered her my hand.
She stared at it with big brown eyes and sniffed once before reaching out her own hand to accept my help. Her fingers were wet, but I knew it was because of her tears.
"Thankoo Aiden," she said, her rage and tears suddenly gone.
She smiled at me and gave me a hug, which made me blush. Mom was always teasing me by saying that Mari had a crush on me, and now I was beginning to wonder if saving her from the ghoulies had made it worse.
"Next time, you need to get one of us if you see any wasps," Mrs. Warren said, her voice more stern than before.
Both she and Miss Rodriguez got up and returned to their previous tasks. Taking a deep breath and letting it out, I shuffled back over to Maddie and helped her pick up the blocks. I expected her to be angry with me as well, but during the whole incident at the window she managed to start building the castle's foundation once again.
For five more minutes I helped with the blocks until Miss Rodriguez rang the bell announcing our lunch break was over. We all put back our toys and games and then headed to our cubbies where our pillows and blankets were stored. I recovered my comic book and put it back as well.
Jake and Mari groaned, but Maddie and Bella started rolling out their blankets. Russell refused to follow directions at first, but then Miss Rodriguez went over to encourage him to join the rest of the class. Not wanting to draw any more attention to myself, I did as I was told, smoothing out the corners of my small quilt and fluffing my pillow. I especially loved my pillow because my sister had bought it for me.
Logan and Bradley had laughed and told me it made me smell like a girl, but I didn't care. Meggy had got it for me.
As I lay still, listening to the relaxing music Miss Rodriguez had put on for us, I breathed in deeply, the faint scent of lavender tickling my nose. The smell reminded me of my big sister and thinking of Meghan made me feel much better. Soon the scary images of the ghoulies faded from my mind and I fell asleep.
Mom picked me up later that afternoon. She worked at one of the local high schools and had a different schedule than most teachers. Some days, she had to go to work early and got to come home early. Other days she started late and picked me up later. My school, like me, wasn't normal. At least not really. I didn't have to be in class at the same time every day. Mom could drop me off on her way to work and pick me up when she got off, whenever that happened to be. Today, she picked me up just before two o'clock.
Miss Rodriguez talked with Mom for a few minutes before turning toward me. "Ready to go home, Aiden?"
She smiled and held out a hand as I rubbed at my eyes, trying to wipe away the sleepiness. On most days, I didn't have any trouble sleeping during nap time. Today I could only lie there and stare at the window, my stomach tying itself into knots as I waited for the ghoulies to come back.
I nodded my head as I got closer to Mom, reaching up to take her hand. Her fingers were warm and strong, and she held onto mine as if she'd never let go.
Downstairs, her car waited in one of the slots in the parking lot. As we passed under the window to my classroom, I held my breath, expecting to see the ghoulies hiding in the bushes. Either they were gone for good, or they were staying out of sight because I didn't see them as I climbed into the backseat of Mom's car. Maybe I had imagined the whole thing, after all. Maybe I really had slept during nap time and I had only dreamed the whole thing.
"How was school today?" Mom asked, her eyes watching me in the rear-view mirror.
Feeling somewhat ashamed about the ghoulies, I lowered my eyes and studied my hands. I couldn't tell her about it even if I wanted to. Even now, as I tried to put the thoughts together in my mind, they buckled against some invisible barrier, the way a soda can crunches and folds up when one of my older brothers steps on it.
When I didn't answer her, Mom took a deep breath and released it on a sigh. "I talked to Dr. Sellers on the phone this morning. She says she has some medicine she wants to try out with you. To see if it will make you feel better when you get upset."
Mom's voice trailed off at the end and I felt my fingers curl into fists. When she said she wanted to make me feel better, she meant the times I had my fits. Even though I couldn't talk like my brothers and sister, I had no trouble letting them know I was angry, frustrated or scared about something. Usually, I only got mad or upset when the ghoulies came around because I was afraid they'd hurt Jack, Joey, Logan and Bradley, or even Meghan. And because I could do nothing, I would throw what Bradley called 'temper tantrums'. My doctor said it was common for children with autism to 'express aggression'. I didn't really know what those words meant, but I could tell Dr. Sellers had been talking about my fits.
My parents didn't talk too much about my autism with me, but they always let me know what they planned to do to help me with it. In this world, kids were expected to be able to do certain things at certain ages. Since I was seven, I should be able to read and write sentences and add and subtract numbers, as well as many other things, including getting along with other kids. I could read, and I could add and subtract just fine. But I couldn't prove it to my parents or my teachers. My fingers wouldn't hold a pencil like they should, so I couldn't write my name down or show them that eight plus two was ten. When I snuck into my older brothers' room and took their chapter books from the bookshelves, it wasn't just because I liked the sound of the paper fluttering as I flipped the pages.
Despite what my family thought, I liked the books because they were about awesome adventures with dragons and magicians and superheroes. But if I were to open my mouth to say the words out loud, nothing ever came out. Once in a while, one or maybe two words would break free, but then my throat would clog up or my tongue would get stuck to the roof of my mouth. I couldn't explain it. It was like the real me was living inside a glass ball made of a mirror that only I could see out of, but no one else could see in. I wondered if that's what it was like for my other classmates too, but I had no way of asking them.
Before I ever started school, Mom used to take me to therapy with other moms who had kids like me. We would learn what the teacher called 'social skills'. I didn't learn much there because most of the stuff they taught us I already knew from listening to Mom and Dad talk to Bradley, Logan and Meghan. Sighing, I thought about what Mom had said about taking medicine. Part of me felt nervous about it, and another part of me felt guilty that, no matter how hard they tried, Mom and Dad and the doctors couldn't seem to help me get better.
Eventually, I looked up, only to find Mom glancing at me in the rearview mirror. She looked worried and I wondered why. Mom always got this way after talking to the doctor. I turned my eyes quickly away once again and stared out the window. The trees lining the road were just starting to change color and some of them had already lost most of their leaves, their bare white branches like the hands of skeletons.
Mom flicked on her turn signal and our car rolled to a stop as she approached a traffic light. On the left, a large shopping center sat partially hidden behind a row of neatly trimmed hedges. I peeked up between the two front seats and glanced out the windshield. Traffic rushed by in a hurry, each car that flew by making ours rock a little. Mom was taking the back road home today and I smiled. I liked the fast roads, but the freeway always scared me a little. It felt like we were driving on a racetrack with all those other cars, some of them really big, but on the highways we could still go fast and there were only two lanes most of the time. No big scary trucks taking forever to pass you if you were stuck behind a slow driver.
The light turned green and Mom pulled forward, turning right after the car in front of her crossed the intersection. We quickly picked up speed and I glanced out the window once again, smiling when we passed the airport. A small plane was landing as we drove by, its wing lights blinking like little stars.
The open, rolling landscape of rural San Luis Obispo was blanketed with acres of vineyards, the golden leaves of the grapevines a cheerful, sunny yellow splashed against the blue sky. An occasional farm house, complete with a paddock for horses or goats, helped add variety to the scenery. Just as I was starting to put the bad memories of the day behind me, I noticed a pumpkin patch coming up on the left. The orange gourds, glowing like burnished beads in the afternoon sunlight, reminded me once again that Halloween was only a few days away. Normally, the sight of pumpkins wouldn't bother me, but the small, rustic shed set up nearby had one of those giant blow up jack-o-lanterns next to it, the black slash of its eyes and mouth reminding me of the ghoulies.
I jerked my head to the side and glanced out the other window, quickly trying to banish the memories. We had come to another traffic stop and as Mom waited for the light to change, a crow came to rest on a fence post on the side of the road. It turned its head, one black, glossy eye staring at me. A red sheen flashed over the bird's eye and a sharp pain pierced my head. Drawing in a quick breath, I squeezed my eyes shut. A memory, harsh as corrosive acid spilling against the backs of my eyes, flooded my mind, and a kaleidoscope of scenes played out in my head. A huge black bird with burning red eyes ... Weird, dark red smoke curling in unnatural tendrils around a tree ... A terrifying voice speaking in my head ... Where had these memories come from? Tangled up with all the images was another memory, one of my sister cutting her hand in the kitchen. Yesterday. Meghan had hurt herself yesterday. Were these memories from the same time? I felt my face pale and my heart begin to race. What was going on?
The light changed and our car quickly picked up speed, leaving the crow behind. Clenching my teeth, I squeezed my eyes shut and tried very hard to remember everything that had happened yesterday. I had a bad feeling it wouldn't be pleasant, but for some reason it was important for me to remember. I never forgot any of my weird encounters with the ghoulies, or other strange stuff in my neighborhood. Why had I let this one slip? Had something, or someone, made me forget?
As we rolled down the highway, the pumpkin field now a faint blur in the distance, I thought long and hard about the day before. Mom had picked me up, just as she had today, and then we'd gone to get Bradley and Logan at their school in Arroyo Grande. Once we got home, my older brothers had raced to the front door, pushing and shoving each other despite the many warnings from Mom. I had lingered behind. I always lingered behind because my brothers liked to torment me. Usually I didn't mind, but it got tiring after a while.
My sister Meggy had already come home and was downstairs in her room doing her homework. We weren't supposed to go into her room without her permission, but Logan and Bradley always did. I only went into her room without permission if it was a dire emergency, like on Saturday mornings when no one else was up and I needed her to turn on the TV for me.
Bradley and Logan were so preoccupied with their argument over who had scored more points in basketball during lunch recess, and Mom was too busy keeping them from getting into a physical fight, that they didn't notice my sluggishness. I was okay with that. I didn't want to get stuck in the middle of it anyway, so I had dragged my feet and stared at the path leading up to our front door.
A low, grumbling complaint from a eucalyptus tree growing beside our house grabbed my attention before I could reach the front door. I forgot the uneven path and the patches of moss growing between the cracks and cast my eyes toward the sliver, sickle-shaped leaves above.
It didn't take long to spot the bird that had made the noise. It was a crow, at least I thought it was a crow. A large, black smudge grasping one of the dead tree branches with its clawed feet, its red eyes trained on me. Suddenly, I felt very aware of my surroundings. I could smell the sweet, spongy scent of the swamp in the canyon behind our house. I could feel the chill of the fog rolling in off the ocean, its damp breath coating my skin. A car pulled into one of the driveways three houses down, and the distinctive crunch of tires crushing the gravelly asphalt grated against my ears. I shivered, but I was pretty sure it wasn't from the encroaching cold.
Shaking my head, I glanced back up at the black bird and found it still studying me. I narrowed my eyes. Was it really a crow? I mean, it looked just like the crows I saw flying around the neighborhood, but this one was so big, much bigger than the others. And it had red eyes. Crows didn't have red eyes. They had black eyes. Then something even more bizarre happened. A dense coat of smoke, deep red in color, seemed to seep through crow's feathers, like steam rising off of frost-coated roofs on cold mornings.
The giant black bird fluffed its feathers, then craned its neck forward and let out one long, low caw. If I didn't know any better, I would have sworn it said Aiden.
With my heart pounding in my throat, I ran to the front door. Well, I tried to run. My clumsy legs and feet refused to move as quickly as I wanted them to. It was like wading through thick mud. I pawed at the doorknob once I reached the door, frantic to get it open so I could hide in the safety of the house. This black bird was like those ghoulies. I knew it. It had the same dark red, murky, black light glow to it that the other monsters had.
The siren of a fire engine wailing like a banshee snapped me out of my reflection. I blinked and looked up, surprised to find that I was still in the back seat of my Mom's car and not standing on the threshold of our living room. She had pulled over onto the shoulder of the road to let the fire engine pass, its bright color painting a red streak against the grey and brown faces of the distant hills to the east. As she pulled back out onto the highway, I took a few steadying breaths. The recollection of yesterday's ordeal was now fresh and bright in my mind. Wanting to recall the rest, I let my thoughts descend back into my mind, hoping to dredge up the remainder of the memory.
I must have gotten the front door open at some point because the next thing I remembered was me standing in our living room, staring into the kitchen as Mom asked Meghan to peel the potatoes while she went to get Jack and Joey from day care. Still feeling rattled from the strange bird, I had headed toward my big sister, ignoring Bradley and Logan, who had transferred their argument about who was the better basketball player to who was better at playing video games. Sometimes, I liked to watch them beat each other up while in the guise of the colorful characters on the TV screen, but not that afternoon. I wanted the comfort only my big sister could give me.
Meghan was tall, and kind, and pretty. She was also a lot more quiet than my brothers and she knew a lot of stuff. But my favorite thing about my sister was that she had the same glowing skin as me. Crinkling my nose, I held out my arm, the loose flannel sleeve slipping from my wrist and bunching up at my elbow. Usually, the pale blue glow wasn't so bright, but today it was almost the color of the TV screen when Bradley or Logan shut off their video game console. I wondered if it had anything to do with the big crow.
Shivering a little, I padded across the cold kitchen floor toward Meggy. She must not have heard me, because I had to say something to get her attention. What I wanted to say was, Meggy, there is a weird bird outside and it's scaring me. Can you help? What came out was only, "Help?"
Meghan paused and glanced down at me. She smiled, her hazel eyes flashing to green, blue and then grey. That was another cool thing about Meggy. She had neat eyes. I wondered if my eyes changed too. Sometimes, it felt like they did, but I never looked at them in the mirror. Maybe they were like the rest of me, hidden behind that one-way mirror that surrounded me.
"Sure buddy," she answered. "Do you know where the colander is? Big yellow bowl with holes like lemons?"
Although I hadn't been asking if she needed help, I nodded anyway. I knew what she was talking about, and helping my sister might take my mind off the bird. We used the colander to drain the water from boiled noodles, or to wash off berries or salad, and it was stored below the microwave with all the other big pots and pans. The drawer was stuck a little, and it took me a while to pull it open and fish out the colander. It wasn't heavy, but my arms refused to lift it. Instead, I ended up dragging it behind me, making all kinds of noise. It was so embarrassing, never being able to do anything the way normal kids could. Meghan never seemed to notice, though. Sometimes, I imagined she could see me, the real me, but she never said so.
After taking the colander from me, Meghan got back to work washing and peeling the potatoes. My brothers continued to shoot imaginary fireballs at one another and that icy chill still clung to my skin.
I jumped and felt my eyes grow wide and the blood drain from my face. A voice, like the crackle of dry bones snapping under heavy feet, had called my name. I spun around, wondering if Logan and Bradley had heard it.
"Die! Die you orange mutant!" Logan was hissing at the TV as he pressed the button on his controller with the enthusiasm of a woodpecker.
Bradley only laughed, a maniacal cackle. "I don't think so! You can't get past my shield!"
No. As crazy as their conversation was, it hadn't been them who'd said my name, and clearly they hadn't heard it.
I glanced up at Meghan. She was still peeling the potatoes. She hadn't heard it, either. The voice was in my head.
As if it read my thoughts, the voice spoke again.
Little Fae-child ... it crooned in that eerie whisper.
This time I didn't check to see if anyone else heard. All I wanted to do was not be scared anymore. Taking three steps forward, I flung my arms around Meghan's leg and held on tight, burying my face against her jeans right above her knee. I squeezed my eyes shut and prayed. Please, please go away, whoever you are!
My plea was answered with a raspy chuckle, followed by, Hello, little Fae-child. Do you know what you are?
It was such a cold voice, with the kind of tone that witches used in movies when they were trying to make kids think they were nice, when really they just wanted to stick you in a great big cauldron and turn you into stew.
I used to think witches and goblins and monsters were all make-believe, especially since Logan and Bradley had always try to scare me and Jack and Joey with stories about them. At first, I thought they could see the monsters too, but when they started describing what they looked like I knew they were only teasing. If they really knew about them, they would be even more scared than me, I bet. The ghoulies didn't scare me so much anymore, since I had grown used to them, but this strange voice terrified me.
I clung to my big sister for a long time. When the voice didn't return, I stole a glance upward only to find Meggy standing unnaturally still, the potato peeler gripped tightly in one hand, her other palm pressed against the rim of the sink. Her eyes were fixed on something outside the window. Curious, I followed her gaze. What I saw then frightened me so much I gaped, letting my sister's leg go. It was the black bird, only it looked as if it had grown in size. The strange, dark red glow had grown, too, seeping from the bird like some demon fog. Long, wispy tendrils of it crawled down the trunk of the tree and over the branches, the pointed ends looking like earth worms seeking damp soil to burrow into. One of the tendrils had pulled away from the tree and was stretching toward the window.
My instincts told me to run, even as my conscience insisted I stay and guard my big sister. I took several steps back and opened my mouth, wanting to scream. I must have made some sound because Meghan snapped out of her daze, her hands jerking up as if returning to the chore she had forgotten about. Her hands must have moved too fast because instead of peeling the potato skin, the peeler slipped and cut her knuckle.
"Crud!" Meghan hissed, dropping the peeler and clutching her hand.
I jumped. When did Mom get home?
"Bradley! Logan! Turn those video games off and finish the potatoes. I need to see to your sister's hand."
All around me, chaos erupted. Groaning, Logan and Bradley obeyed, shuffling into the kitchen with glum looks on their faces. Mom was pulling Meghan down the hallway toward the bathroom and Jack and Joey, recently released from the captivity of the day care center, started chasing each other around the house, screeching in delight.
As my two older brothers commenced with the potato peeling, resuming their fight, this time using soggy potato peels instead of fireballs and magical shields, I quietly snuck out of the kitchen and headed for the room I shared with the twins. Jack and Joey hardly noticed as I disappeared down the hallway, having abandoned their chase game for the toy basket kept next to the couch.
I passed the bathroom on the way, peeking in to make sure Meghan was okay. Mom was clucking like a hen while my sister tried to assure her it was only a nick. Meggy didn't see me as I passed, but she looked shaken, no matter how she tried to reassure Mom. Her skin was pale and her eyes were shifting color faster than usual. Had she heard the voice as well? Had she seen what I'd seen?
Hoping that the horrible bird was gone now and the strange voice with it, I slipped into my room and climbed onto my bed. Although the window on the far wall looked out over the opposite side of our yard, I closed the blinds anyway. It took me a long time to get the plastic rod to twirl in the right direction, but once I could no longer see any daylight, I breathed a sigh of relief. Selecting one of my favorite comic books, one sporting a hero wearing green, I had curled up on my pillow and read until Mom called us all into the kitchen for dinner.
"Aiden honey, did you hear me?"
I blinked and glanced up to find my mom turned in the driver's seat and giving me a worried look. We were home and parked in our driveway. When had that happened? I shook my head. No. I hadn't heard her. I had been so preoccupied with recalling the incident in the kitchen with Meggy and the weird voice that I hadn't even noticed most of the drive home, let alone what Mom had been saying.
She smiled kindly and said, "We are going to see Doctor Sellers tomorrow morning. We are going to try this medication, and if it doesn't help, you won't have to keep taking it, okay?"
Will it make me sick? Will it make me different? Will it make the creepy voice go away for good and make it so I don't see the ghoulies anymore?
Knowing none of those words would ever come out of my mouth, I simply said, "Kay."
Mom smiled and ruffled my hair, then released her seat belt and stepped out of the car. Although it had scared me, I was glad I remembered everything. If the black bird and the ghoulies were really trying to hurt me or Meghan, knowing as much about them as I could might help someday.
Not wanting to linger in the car, especially if the crow was back, I quickly unsnapped my seatbelt and climbed out onto the concrete after my mom. The car door slammed shut a little harder than I had meant it to, but I didn't let the guilt get to me. Instead, I hurried toward Mom, grabbing her hand as we walked the short distance to the house. I didn't even look into the trees. The last thing I wanted to see was that black bird staring at me with glowing red eyes again.
Thank you! As an author it is always so wonderful to hear from a reader who has enjoyed the story so far ;).
I'm absolutely in love with the Otherworld series. Once I found the first book on iBooks, I had to download all the rest, including the novellas. I finished them all within a matter of days. I admit, I still check iBooks every few months to see if you have anything new, and if I don't find anything, I re-read my favourite excerpts from the books. It's a series that I will always come back to as the years pass. Thank you so much Jenna!
I'll always keep my fingers crossed for more tales from the Otherworld.
Hello Laila! Thank you so much for your kind words! They are the stuff that keeps us authors chugging along during the dark days of our own self-doubt ;). I WILL be adding more Otherworld books to my list - plan on using the summer to work on a sequel trilogy to the original three, this one told from Cade's perspective ;). I also need to write Robyn's sequel as well as a book for Enorah. Further down the line Aiden will have his own series, too. I also have a few more characters in mind for single novels, but they are very fresh right now. Hoping to have something out this fall, but I have SO many writing projects on my plate right now I have no idea when I'll get them all done! Until then, happy reading!
Thank you for both 'Oescienne' series and 'Otherworld' series, both amazing reads. Was so sad to finish both as now feel lost !
Cant wait for the next in either series. Which brings me to my question, where do we find any new parts these series?
FLAME AND FORM (a new novella set in the universe of the Otherworld and available in the anthology A Plague of Dragons).
Thanks again and feel free to reach out at any time! | {
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Vatnajökull National Park, established in 2008, encompasses not only all of Vatnajökull glacier but also extensive surrounding areas. These include the national parks previously existing at Skaftafell in the southwest and Jökulsárgljúfur in the north, so that today's national park covers 13% of Iceland and ranks as Europe's largest.
The objective of the Vatnajökull National Park's establishment is to conserve the area's landscape, biology, geological formations and cultural heritage and enable the public to experience and enoy its nature and history.
To reach these objectives Vatnajökull National Park is divided into four areas. Each area has a park manager whose role is to ensure that operations within the park are in accordance with the park's rules and regulations.
Vatnajökull National Park offers a unique opportunity to experience the interplay between the forces of glaciers and volcanic activities.The National Park stretches around the Vatnajökull ice cap, with visitor centres in South-East Iceland (Skaftafell), East Iceland (Skriðuklaustur) and North-East Iceland (Jökulsárgljúfur).
Vatnajökull National Park operates campsites in Skaftafell and Jökulsárgljúfur.
For the most part, the park lies in highland areas. If you are heading into the Icelandic highlands, you must be properly prepared, as the weather, river flow and driving or walking conditions may change very suddenly. Important safety equipment, such as communication and navigation devices, is essential in places with little traffic. If help is needed, phone Iceland's emergency number, 112, where your message will be passed on to suitable parties.
It must also be emphasized that off-road driving is prohibited in Iceland. What may seem like an innocent de-tour can leave a lasting scar in the landscape, something that Icelanders have learned the hard way through their own actions. The National Park authorities strongly recommend that visitors seek further information about driving in the park prior to their visit. | {
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The Bwindi Impenetrable National Park (BINP) is in southwestern Uganda. The park is part of the Bwindi Impenetrable Forest and is situated along the Democratic Republic of the Congo (DRC) border next to the Virunga National Park and on the edge of the Albertine Rift. Composed of of both montane and lowland forest, it is accessible only on foot. BINP is a United Nations Educational, Scientific and Cultural Organization-designated World Heritage Site.
Species diversity is a feature of the park. It provides habitat for 120 species of mammals, 350 species of birds, 310 species of butterflies, 27 species of frogs, chameleons, geckos, and many endangered species. Floristically, the park is among the most diverse forests in East Africa, with more than 1,000 flowering plant species, including 200 species of trees and 104 species of ferns. The northern (low elevation) sector has many species of Guineo-Congolian flora, including two endangered species, the brown mahogany and Brazzeia longipedicellata. In particular, the area shares in the high levels of endemisms of the Albertine Rift.
The park is a sanctuary for colobus monkeys, chimpanzees, and many birds such as hornbills and turacos. It is most notable for the 400 Bwindi gorillas, half of the world's population of the endangered mountain gorillas. 14 habituated mountain gorilla groups are open to tourism in four different sectors of Buhoma, Ruhijja, Rushaga and the Nkuringo in the Districts of Kanungu, Kabale and Kisoro respectively all under the management of Uganda Wildlife Authority.
History
In 1932, two blocks of the Bwindi Impenetrable Forest were designated as Crown Forest Reserves. The northern block was designated as the "Kayonza Crown Forest Reserve", and the southern block designated as the "Kasatora Crown Forest Reserve". These reserves had a combined area of . In 1942, the two reserves were combined and enlarged, then renamed the Impenetrable Central Crown Forest. This new protected area covered and was under the joint control of the Ugandan government's game and forest departments.
In 1964, the reserve was designated as an animal sanctuary to provide extra protection for its mountain gorillas and renamed the Impenetrable Central Forest Reserve. In 1966, two other forest reserves became part of the main reserve, increasing its area to almost . The park continued to be managed as both a game sanctuary and forest reserve.
In 1991, the Impenetrable Central Forest Reserve, along with the Mgahinga Gorilla Reserve and the Rwenzori Mountains Reserve, was designated as a national park and renamed the Bwindi Impenetrable National Park. It covered an area of . The national park was declared in part to protect a range of species within it, most notably the mountain gorilla. The reclassification of the park had a large impact on the Batwa pygmy people, who were evicted from the forest and no longer permitted to enter the park or access its resources. Gorilla tracking became a tourist activity in April 1993, and the park became a popular tourist destination. In 1994, a area was incorporated into the park and it was inscribed on the World Heritage List. The park's management changed: Uganda National Parks, since renamed the Uganda Wildlife Authority, became responsible for the park. In 2003 a piece of land next to the park with an area of was purchased and incorporated into the park.
In March 1999, a force of 100–150 former Rwandan Interahamwe guerrillas infiltrated across the border from the DRC and kidnapped 14 foreign tourists and their Ugandan guide from the park headquarters, eventually releasing six and murdering the remaining eight with machetes and clubs. Several victims were reportedly tortured, and at least one of the female victims was raped. The Ugandan guide was doused with gasoline and lit on fire. The Interahamwe attack was reportedly intended to "destabilize Uganda" and frighten away tourist traffic from the park, depriving the Ugandan government of income. The park was forced to close for several months, and the popularity of the gorilla tours suffered badly for several years, though attendance has since recovered due to greater stability in the area. An armed guard also now accompanies every tour group.
Geography and climate
Kabale town to the south-east is the nearest main town to the park, away by road. The park is composed of two blocks of forest that are connected by a corridor of forest. The shape of the park is a legacy of previous conservation management, when the original two forest blocks were protected in 1932. There is agricultural land where there were previously trees directly outside the park's borders. Cultivation in this area is intense.
The park's underlying geology consists of Precambrian shale phyllite, quartz, quartzite, schist, and granite. The park is at the edge of the Western Rift Valley in the highest parts of the Kigezi Highlands, which were created by up-warping of the Western Rift Valley. Its topography is very rugged, with narrow valleys intersected by rivers and steep hills. Elevations in the park range from above sea level, and 60 percent of the park has an elevation of over . The highest elevation is Rwamunyonyi Hill at the eastern edge of the park. The lowest part of the park is at its most northern tip.
The forest is an important water catchment area. With a generally impermeable underlying geology where water mostly flows through large fault structures, water infiltration and aquifers are limited. Much of the park's rainfall forms streams, and the forest has a dense network of streams. The forest is the source of many rivers that flow to the north, west, and south. Major rivers that rise in the park include the Ivi, Munyaga, Ihihizo, Ishasha, and Ntengyere rivers, which flow into Lake Edward. Other rivers flow into Lakes Mutanda and Bunyonyi. Bwindi supplies water to local agricultural areas.
Bwindi has a tropical climate. Annual mean temperature ranges from a minimum of to a maximum of . Its annual rainfall ranges from . Peak rainfall occurs from March to April and from September to November. The park's forest plays an important role in regulating the surrounding area's environment and climate. High amounts of evapotranspiration from the forest's vegetation increases the precipitation that the region outside the park receives. They also lessen soil erosion, which is a serious problem in south-western Uganda. They lessen flooding and ensure that streams continue to flow in the dry season.
Biodiversity
The Bwindi Impenetrable Forest is old, complex, and biologically rich. Diverse species are a feature of the park, and it became an UNESCO World Heritage Site because of its ecological importance. Among East African forests, Bwindi has some of the richest populations of trees, small mammals, birds, reptiles, butterflies, and moths. The park's diverse species are partly a result of the large variations of elevation and habitat types in the park, and may also be because the forest was a refuge for species during glaciations in the Pleistocene epoch.
The park's forests are afromontane, which is a rare vegetation type on the African continent. Located where plain and mountain forests meet, there is a continuum of low-altitude to high altitude primary forests in the park, one of the few large tracts of East African forest where this occurs. The park has more than 220 tree species, and more than 50% of Uganda's tree species, and more than 100 fern species. The brown mahogany is a threatened plant species in the park.
Bwindi Impenetrable National Park is important for the conservation of the afromontane fauna, especially species endemic to the Western Rift Valley's mountains. It is thought to have one of the richest faunal communities in East Africa, including more than 350 bird species and more than 200 butterfly species. There are an estimated 120 mammal species in the park, of which 10 are primates, and more than 45 are small mammals. Along with mountain gorilla, species in the park include common chimpanzee, L'Hoest's monkey, African elephant, African green broadbill, and cream-banded swallowtail, black and white colobus, red-tailed monkeys, vervets, the giant forest hog, and small antelope species. The fish species in the park's rivers and streams are not well known. There are also many carnivores, including the side-striped jackal, African golden cat, and African civet.
Mountain gorillas
The park is inhabited by about 459 individual mountain gorillas as per the last 2019 Gorilla Census (Gorilla Fund)(Gorilla beringei beringei), known as the Bwindi population, which makes up almost half of all the mountain gorillas in the world. The rest of the worldwide mountain gorilla population lives in the nearby Virunga Mountains. A 2006 census of the mountain gorilla population in the park showed that its numbers had increased modestly from an estimated 300 individuals in 1997, to 320 individuals in 2002 to 340 individuals in 2006, and 400 in 2018. Poaching, disease and habitat loss are the greatest threat to the gorillas.
Research on the Bwindi population lags behind that of the Virunga National Park population, but some preliminary research on the Bwindi gorilla population has been carried out by Craig Stanford. This research has shown that the Bwindi gorilla's diet is markedly higher in fruit than that of the Virunga population, and that the Bwindi gorillas, even silverbacks, are more likely to climb trees to feed on foliage, fruits, and epiphytes. In some months, the Bwindi gorilla diet is very similar to that of Bwindi chimpanzees. It was also found that Bwindi gorillas travel farther per day than Virunga gorillas, particularly on days when feeding primarily on fruit than when they are feeding on fibrous foods. Additionally, Bwindi gorillas are much more likely to build their nests in trees, nearly always in Alchornea floribunda (locally, "Echizogwa"), a small understory tree.
The mountain gorilla is an endangered species, with an estimated total population of about 650 individuals. There are no mountain gorillas in captivity, but during the 1960s and 1970s, some were captured to start captive breeding.
Conservation
The park is owned by the Uganda Wildlife Authority, a parastatal government body. The park has total protection, although communities adjacent to the park can access some of its resources.
The areas bordering the park have a high population density of more than 300 people per square kilometer. Some of the people who live in these areas are among the poorest people in Uganda. The high population and poor agricultural practises place a great pressure on the Bwindi forest, and are one of the biggest threats to the park. Ninety percent of the people are dependent on subsistence agriculture, as agriculture is one of the area's few ways of earning income.
Prior to Bwindi's gazetting as a national park in 1991, the park was designated as a forest reserve, and regulations about the right to access the forest were more liberal and seldom enforced. Local people hunted, mined, logged, pit sawed, and kept bees in the park. It was gazetted as a national park in 1991 because of its rich biodiversity and threats to the integrity of the forest. Its designation as a national park gave the park higher protection status. State agencies increased protection and control of the park. Adjacent communities' access to the forest immediately ended. This closing of access caused large amounts of resentment and conflict among these local communities and park authorities. The Batwa, a group that had relied on the forest, were badly affected. The Batwa fished, harvested wild yams and honey, and had ancestral sites within the park. Despite the Batwa people's historical claim to land rights and having lived in the area for generations without destroying the area's ecosystem, they did not benefit from any national compensation scheme when they were evicted. Non-Batwa farmers who had cut down the forested areas in order to cultivate them, received compensation and their land rights were recognised. People have lost livestock and crops from wildlife, and there have been some human deaths. The habituation of gorillas to humans in order to facilitate tourism may have increased the damage they do to local people's property because their fear of people has decreased.
Tourism
Tourists may visit the park any time during the year, although conditions in the park are more difficult during the rainy season. The park is in a remote location and the roads are in poor condition, the drive time from Kampala is estimated at 8 hours. The park is connected by domestic flights from Entebbe International Airport, which land at Kihihi Airstrip and Kisoro Airstrip. Tourist accommodations include a lodge, tented camps, and rooms run by the community located near the Buhoma entrance gate.
Bwindi Community Hospital provides health care to 40,000 people in the area and to visiting tourists.
Gorilla tracking is the park's main tourist attraction, and it generates much revenue for Uganda Wildlife Authority. Gorilla tracking first became available in April 1993, when tourists tracked the Mubare gorilla group. Tourists wishing to track gorillas must first obtain a permit. Selected gorillas families have been habituated to human presence, and the number of visitors is tightly controlled to prevent risks to the gorillas and degradation of the habitat. Several tour companies are able to reserve gorilla tracking permits for prospective visitors to Bwindi. The gorillas seldom react to tourists. There are strict rules for tourists to minimize the risk of disease transmission to the gorillas. Uganda, Rwanda, and the Democratic Republic of the Congo are the only countries where it is possible to visit mountain gorillas. Guided walks through the forest include a walk to a waterfall, and walks for monkey watching and birding.
See also
Institute of Tropical Forest Conservation — a forest research institute in the park.
References
External links
National parks of Uganda
Kanungu District
Kisoro District
Important Bird Areas of Uganda
World Heritage Sites in Uganda
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\section{Introduction}
Interface problems arise widely in the multi-physics and multi-material applications in the fluid mechanics and material science. The governing partial differential equations (PDEs) for interface problems are usually characterized with discontinuous coefficients that represent different material properties. The solutions to the interface problems often involve kinks, singularities, discontinuity, and other non-smooth behaviors. It is therefore challenging to obtain accurate numerical approximations for interface problems. Moreover, the complexity of the interface geometry may add an extra layer of difficulty to the numerical approximation.
In general, there are two classes of numerical methods for solving interface problems. The first class of methods use interface-fitted meshes, i.e., the meshes are tailored to fit the interface, see the left plot in Figure \ref{fig: domain}. Methods of this type include classical finite element methods \cite{ChZou:98}, discontinuous Galerkin methods \cite{ArBrCoMa:02} and recently developed weak Galerkin methods \cite{2016MuWangYeZhao}, to name a few. The second class of methods use unfitted meshes which are independent of the interface, as illustrated in the right plot in Figure \ref{fig: domain}. In the past few decades, many numerical methods based on unfitted meshes have been developed. In the finite difference framework, since the pioneering work of immersed boundary method \cite{2002Peskin} by Peskin, many numerical methods of finite difference type have been developed such as immersed interface method \cite{LeLi:94, 2006LiIto}, matched interface and boundary method \cite{2004ZhaoWei}. In finite element framework, there are quite a few numerical methods developed, for instance, the general finite element method \cite{BaOs:83}, unfitted finite element method \cite{HaHa:02}, multi-scale finite element method \cite{HouWuCai:99}, extended finite element method \cite{1999MoesDolbowBelytschko}, and immersed finite element method (IFEM) \cite{Li:98, 2003LiLinWu}. A great advantage for unfitted numerical methods is that they can circumvent (re)meshing procedure which can be very expensive especially for time dependent problems with complex interface geometry or for shape optimization processes that require repeated updates of the mesh.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.4\textwidth]{general_partition_triangulation.pdf}~~
\includegraphics[width=0.4\textwidth]{cartesian_partition.pdf}
\end{center}
\caption{An interface-fitted mesh (left), and an unfitted mesh (right).}
\label{fig: domain}
\end{figure}
The IFEM was first developed in \cite{Li:98} for a one-dimensional elliptic interface problem
and then extended to higher-order approximations \cite{AdjeridGuoLin:2017, AdjeridLin:2009, CaZhZh:17, 2017CaoZhangZhangZou} and to higher-dimensional elliptic interface problems \cite{GuSaSa:16, 2011HeLinLin, LiLinLinRo:04, 2018LinSheenZhang, 2010VallaghePapadopoulo, 2011WuLiLai}.
Recently, the partially penalized immersed finite element method was introduced in \cite{LinLinZhang:15}.
Compared to classical IFEM, the partially penalized IFEM contains normal flux jump terms on interface edges to ensure the consistency of the scheme. In addition, the new IFEM includes a stabilization term on interface edges to guarantee the stability of the scheme. The partially penalized IFEM significantly improves the numerical approximation, especially the accuracy around the interface. The optimal a priori error estimate is theoretically proved for partially penalized IFEM in the energy norm \cite{LinLinZhang:15}.
We note that the partially penalized IFEM can solve elliptic interface problems accurately on uniform Cartesian meshes provided that the exact solution of interface problems is piecewise smooth, and the contrast of the coefficient is ``moderate". However, for interface problems that also involve singularity or steep gradient,
the partially penalized IFEM alone may not be efficient to obtain an accurate approximation on uniform meshes. In such cases, it is necessary to apply certain adaptive mesh refinement (AMR) strategy to IFEM. The goal for AMR is to obtain an approximate solution within prescribed error tolerance with minimum computational cost which is particularly rewarding for interface problems with non-smooth solutions.
The key success for AMR is the a posteriori error estimation which provides both global and local information on the approximation error. Moreover, in many applications even without the intention of performing adaptive mesh refinement, the a posteriori error estimation is also important in assessing the quality of the simulation by providing an effective error control.
We note that the a priori error estimation of IFEM is getting mature in the past decade, but the a posteriori error estimation of IFEM is still in the infancy. In this paper, we will develop and analyze a residual-based a posteriori error estimate for the partially penalized IFEM for second-order elliptic interface problems in the two dimensional space.
Comparing to the residual-based error estimation for the classical finite element solution \cite{BeVe:00}, the newly introduced estimator in this work additionally
includes the jump of tangential derivative of numerical solutions on interface edges
besides the standard terms of element residual and jump of normal flux on all edges. This is necessary since the numerical solution is in general discontinuous across the interface edges due to the
construction of the IFEM basis functions.
Moreover, the new error estimator also includes the geometrical fitting error due to the polygonal approximation of the curved interface.
Theoretically, we prove that the error estimation is both globally reliable and locally efficient.
To prove reliability, we use the Helmholtz decomposition and a $L^2$ representation technique recently introduced in \cite{CaHeZh-mcom:17, CaHeZh-sinum:17}; moreover, we introduce a new type of Cl\'ement-type interpolation in the IFEM space that allows us to take advantage of the error equation of IFEM.
In the efficiency analysis for IFEM, the technique using the standard bubble functions \cite{Ve:1996} is invalid because the edge jumps of the normal flux and the tangential derivative become piecewise constant on the interface edges.
\textcolor{black}{Instead, we prove the efficiency in two different approaches that aims to provide an optimal efficiency constant for both regular and irregular interface edges.}
The rest of the article is organized as follows. In Section \ref{sec:2}, we recall the partially penalized IFEM for elliptic interface problems. In Section \ref{sec:3}, we introduce our residual-based error estimator specially designed for IFEM. Section \ref{sec:4} and Section \ref{sec:5} are dedicated to the analysis of global reliability and local efficiency, respectively.
Finally, in Section \ref{sec:6}, we present several numerical experiments to test the performance of our a posteriori error estimators.
\section{Interface Problems and Partially Penalized IFEM}\label{sec:2}
Let $\O\subset \mathbb{R}^2 $ be a polygonal domain with Lipschitz boundary
$\partial \O = \overline \Gamma_D \cup \overline \Gamma_N$, where
$ \Gamma_D \cap \Gamma_N = \emptyset$. Assume that $\mbox{meas}(\Gamma_D) > 0$.
We consider the elliptic interface problem:
\begin{equation} \label{problem}
-\nabla \cdot (\alpha \nabla u) =f \quad \mbox{ in } {\O^+\cup\O^-}
\end{equation}
with boundary conditions
\[
u= 0 ~\mbox{ on } \Gamma_D~~
\mbox{ and } ~~
-\alpha \nabla u \cdot {\bf n}=g_{_N} ~\mbox{ on } \Gamma_N.
\]
Here,
$f \in L^2(\O)$, $g_{_N} \in L^2(\Gamma_N)$, and ${\bf n}$ is the unit vector outward normal to $\partial \O$. The notations $\nabla$ and $\nabla \cdot$ are the gradient and divergence operators, respectively. Furthermore, assume that $\O$ is separated by a closed smooth interface curve $\Gamma$ into $\O^+$ and $\O^-$ such that
$\overline{\O} = \overline{\O^+\cup\G\cup\O^-}$. The diffusion coefficient $\alpha$ is assumed to be a positive piecewise constant function as follows
\[
\alpha(x,y)=\left\{ \begin{array}{lll}
\alpha^+ & \mbox{for} \,(x, y) \in \O^+,\\[2mm]
\alpha^- & \mbox{for} \,(x,y) \in \O^-.
\end{array}
\right.
\]
Denote by $\rho = \displaystyle\frac{\alpha^+}{\alpha^-}$ the ratio of the coefficient jump.
The solution is assumed to satisfy the following interface jump conditions:
\begin{equation}
\jump{u}_{\G} = 0 \quad \mbox{and} \quad \jump{\alpha \nabla u\cdot {\bf n}}_{\G} = 0,
\end{equation}
where the jump of a function $v$ across the interface $\G$ is defined by
\[
\jump{v}_{\G} = v^+|_\G - v^-|_\G.
\]
We use the standard notations for the Sobolev spaces.
Let
\[
H_D^1(\O)=\{v \in H^1(\O) : v =0 \mbox{ on } \Gamma_D\}.
\]
Then the variational problem for \eqref{problem} is to find $u \in H_D^1(\O)$ such that
\begin{equation}\label{continuous-weak-solution}
a(u,v) \triangleq (\alpha \nabla u, \nabla v) =(f,v) -(g_{_N},v)_{\Gamma_N}, \quad \forall \, v\in H_D^1(\O),
\end{equation}
where $(\cdot,\cdot)_\o$ is the $L^2$ inner product on $\o$. The subscript $\o$ is omitted when $\o=\O$.
\subsection{Triangulation}
In this paper, we only consider the triangular meshes in two dimensions.
Let ${\mathcal T} =\{K\}$ be a triangulation of $\O$ that is regular but not necessarily body-fitted.
Denote the set of all vertices of the triangulation ${\mathcal T}$ by
\[{\mathcal N}:= {\mathcal N}_I \cup {\mathcal N}_D \cup {\mathcal N}_N\]
where ${\mathcal N}_I$ is the set of all interior vertices, and ${\mathcal N}_D$ and ${\mathcal N}_N$ are the sets of vertices
on $\bar\Gamma_D$ and $\Gamma_N$, respectively.
Denote the set of all edges of the triangulation ${\mathcal T}$ by
\[
{\mathcal E} := {\mathcal E}_I \cup {\mathcal E}_D \cup {\mathcal E}_N
\]
where ${\mathcal E}_I$ is the set of all interior edges and ${\mathcal E}_D$ and ${\mathcal E}_N$ are
the sets of boundary edges on $\Gamma_D$ and $\Gamma_N$, respectively.
For each element $K \in {\mathcal T}$, denote by $h_K$ the diameter of $K$,
and by ${\mathcal N}_K$ and ${\mathcal E}_K$ the sets of
all vertices and edges on $K$, respectively.
For simplicity, we assume that the interface cuts the partition
with the following properties:
\begin{description}
\item[(I)] If $\Gamma$ meets an edge at more than one point, then this edge is part of $\Gamma$.
\item[(II)] If the case (I) does not occur, then $\Gamma$ must intersect a triangle at two points, and these two points must be on different edges of this triangle.
\end{description}
Based on the above assumption, all triangular elements in the partition can be categorized into two classes: \textit{non-interface elements} that either has no intersection with $\Gamma$ or $\Gamma \cap K \subset \partial K$, and \textit{interface elements} whose interior is cut through by $\Gamma$.
Denote the set of all interface elements by ${\mathcal T}^{int}$.
For each interface triangle $K$ we let $\Gamma_K = \Gamma \cap K$ and
$\tilde\Gamma_K$ be the line segment approximating $\Gamma_K$ by connecting two endpoints of $\Gamma_K$.
Let $K^+ = K \cap \O^+$ and $K^- = K \cap \O^-$. Also we let $\tilde K^+$ and $\tilde K^-$ be the two sub-elements of $K$ separated by $\tilde \Gamma_K$.
From the setting above, it is easy to see that $\tilde K^\pm$ is either a triangle or a quadrangle. Also we define
\begin{equation}\label{eq: SK}
S_K \triangleq(K^+ \setminus \tilde K^+) \cup (K^- \setminus \tilde K^-),
\quad \forall \,K \in {\mathcal T}^{int},
\end{equation}
which is the region enclosed by $\Gamma_K$ and $\tilde\Gamma_K$. Under the assumption that $\Gamma_K$ is $C^2$-continuous, the area of the $S_K$ is of at least $O(h^3_K)$ \cite{LiLinLinRo:04}. Figure \ref{fig: triangle} provides an illustration of a typical triangular interface element.
\begin{figure}[hbt!]
\begin{center}
\includegraphics[width=.35\textwidth]{FIG-1.pdf}
\end{center}
\caption{A triangular interface element}
\label{fig: triangle}
\end{figure}
For an edge $F \in {\mathcal E}$, if $F$ is cut through by $\Gamma$, i.e., $F \cap \Gamma \neq \emptyset$ and
$F \not \subset \Gamma$, then $F$ is called an interface edge and
denote by ${{{\mathcal E}}^{int}}$ the set of all such interface edges.
For each $F \in {\mathcal E}$, denote by $h_F$ the length of $F$.
Denote by ${\bf n}_F =(n_1, n_2)$ and ${\bf t}_F=(-n_2, n_1)$ the unit vectors
normal and tangential to $F$, respectively.
Let $K_{F,1}$ and $K_{F,2}$ be the two elements sharing the common edge $F \in {\mathcal E}_I$
such that the unit vector out normal to $K_{F,1}$ coincides with ${\bf n}_F$. When $F\subset \partial \O$,
${\bf n}_F$ is the unit outward vector normal to $\partial \O$, and denote by $K_{F,1}$ the boundary element having the
edge $F$.
For a function $v$ that is defined on $K_{F,1} \cup K_{F,2}$, denote its traces on $F$ by $v|_F^1$ and $v|_F^2$
restricted on $K_{F,1}$ and $K_{F,2}$, respectively.
Define the jump of a function $v$ on the edge $F$ by
\[
\jump{v}_F = \left\{
\begin{array}{lll}
v|_F^1 -v|_F^2,& \mbox{for }F \in {\mathcal E}_I,\\[2mm]
v|_F^1,& \mbox{for } F \in {\mathcal E}_D \cup {\mathcal E}_N
\end{array}
\right.
\]
and the average of a function $v$ on the edge $F$ by
\[
\{v\}_F = \left\{
\begin{array}{lll}
\left( v|_F^1 +v|_F^2 \right)/2, & \mbox{for }F \in {\mathcal E}_I,\\[2mm]
\,v|_F^1, & \mbox{for } F \in {\mathcal E}_D \cup {\mathcal E}_N.
\end{array}
\right.
\]
It is easy to verify that
\begin{equation}\label{jump-formula}
\jump{vw}_F= \jump{v}_F \{w\}_F+ \{v\}_F \jump{w}_F, \quad \forall \, F \in {\mathcal E}.
\end{equation}
For simplicity, we may drop the subscript $F$ in the notations $\jump{\cdot}_F$ and $\{\cdot\}_F$ if there is no confusion on where the jump and average are defined.
\subsection{IFEM approximation}
For simplicity, we assume that the interface does not intersect with the boundary, i.e., ${{{\mathcal E}}^{int}} \subset {\mathcal E}_I$. Let $\tilde \alpha$ be an approximation of $\alpha$ such that
\[
\tilde \alpha(x,y) = \left\{\begin{array}{lll}
\alpha^+ & \mbox{if}& (x,y) \in \tilde K^+,\\[2mm]
\alpha^- & \mbox{if}& (x,y) \in \tilde K^-,
\end{array}
\right.
\quad \forall \,(x,y) \in K \in {\mathcal T}^{int}.
\]
For each interface element $K \in {\mathcal T}^{int}$, define the local IFE space by
\[
\tilde P_1(K) = \left\{ v \in H^1(K): \tilde \alpha \nabla v \in H(\mbox{div},K), v|_{\tilde K^\pm} \in P_1(\tilde K^\pm)\right\}
\]
where $P_1(w)$ is the space of all polynomial functions in $w$ of degree no more than $1$.
The global IFE space $\mathcal{S}({\mathcal T})$ is then defined to include all functions such that
\begin{enumerate}
\item $v|_K \in \tilde P_1(K)$ for all $K \in {\mathcal T}^{int}$ and $v|_K \in P_1(K)$ for all $K \in {\mathcal T}/{\mathcal T}^{int}$, and
\item $v$ is continuous at every vertex $z \in {\mathcal N}$.
\end{enumerate}
Note that for each $z \in {\mathcal N}$, there exists a unique IFE nodal basis function \cite{LiLinLinRo:04,2003LiLinWu}, denoted by $\tilde\lambda_z \in \mathcal{S}({\mathcal T})$, such that
\[
\tilde\lambda_z(z') = \delta_{zz'}, \quad \forall \, z' \in {\mathcal N}
\]
where $\delta$ is the Kronecker delta function.
The partially penalized IFEM solution for the interface problem is to find \\$u_{\mathcal T} \in \mathcal{S}_D({\mathcal T}) = \left\{ v \in \mathcal{S}({\mathcal T}) \,:\, v =0 \mbox{ on } \Gamma_D \right\}$ such that
\begin{equation} \label{IFE approx}
a_h(u_{\mathcal T},v)=(f,v) - (g_{_N}, v)_{\Gamma_N}, \quad \forall \,v \in \mathcal{S}_D({\mathcal T})
\end{equation}
where the bilinear form $a_h(w,v )$ is defined by
\begin{eqnarray*} \label{FE approximation}
a_h(w,v)
&=&\sum_{K\in {\mathcal T}} \int_{K} \tilde\alpha \nabla w \cdot \nabla v \,dx
-\sum_{F \in {{{\mathcal E}}^{int}}} \int_F\{\tilde \alpha \nabla w \cdot {\bf n}_F \} \jump{v} \,ds\\[2mm]
&&+ \epsilon \sum_{F\in {{{\mathcal E}}^{int}}} \int_F \{ \tilde \alpha \nabla v \cdot {\bf n}_F\} \jump{w} \,ds
+\sum_{F\in {{{\mathcal E}}^{int}}} \gamma\int_F \displaystyle\frac{\tilde \alpha}{h_F} \jump{w}\jump{v} \,ds.
\end{eqnarray*}
Here $\epsilon$ may take the values $-1$, $0$, and $1$, corresponding to symmetric, incomplete, and non-symmetric IFEM. The constant $\gamma$ is the
stability parameter and needs to be chosen large enough for symmetric and incomplete IFEMs guarantee the coercivity. For non-symmetric IFEM, the constant $\gamma$ is only required to be positive. For more details on the partially penalized IFEM, we refer readers to \cite{LinLinZhang:15}.
\begin{remark}
By the definition of $\tilde \alpha$ and $\tilde \Gamma_K$ it is easy to see that
$\tilde \alpha = \alpha $ on all $F \in {\mathcal E}^{int}$.
\end{remark}
\subsection{Inconsistency error}
Due to the geometrical approximation of the interface curve $\Gamma$ by a polygonal interface $\tilde \Gamma = \bigcup\limits_{K\in{\mathcal T}^{int}} \tilde \Gamma_K$, the following geometrical inconsistent error exists. By \eqref{IFE approx} and integration by parts we have for any $v \in \mathcal{S}_D({\mathcal T})$
\begin{equation} \label{eq: inconsistency}
\begin{split}
a_h(u - u_{\mathcal T},v)
=&\sum_{K\in {\mathcal T}} \int_{K} \tilde\alpha \nabla u \cdot \nabla v \,dx
-\sum_{F \in {{{\mathcal E}}^{int}}} \int_F\{\tilde \alpha \nabla u \cdot {\bf n}_F \} \jump{v} \,ds - (f,v) + (g_{_N}, v)_{\Gamma_N} \\
=&\sum_{K\in {\mathcal T}} \int_{K} \alpha \nabla u \cdot \nabla v \,dx
-\sum_{F \in {{{\mathcal E}}^{int}}} \int_F\{\alpha \nabla u \cdot {\bf n}_F \} \jump{v} \,ds \\
&+ \sum_{K\in {\mathcal T}} \int_{K} (\tilde\alpha - \alpha) \nabla u \cdot \nabla v \,dx - (f,v) +(g_{_N}, v)_{\Gamma_N}
\\
=&
\sum_{K\in {\mathcal T}^{int}} \int_{K} (\tilde\alpha - \alpha) \nabla u \cdot \nabla v \,dx.
\end{split}
\end{equation}
\begin{remark}
If the interface $\Gamma$ is a polygon such that $\tilde\alpha = \alpha$ on each interface element, then the term \eqref{eq: inconsistency} vanishes. In this case, the partially penalized IFEM scheme is consistent. In general, for a curved interface $\Gamma$, the global convergence of IFEM will not be affected by such linear approximation of the interface, since the partially penalized IFEM uses piecewise linear approximation \cite{2007Braess}.
\end{remark}
\section{Residual-Based A Posteriori Error Estimation}\label{sec:3}
In this section, we introduce the residual-based
error estimator for the partially penalized IFEM.
We note that the classical residual-based a posteriori error estimation for conforming finite element methods on fitted meshes consists of element residual and the jump of the normal flux on edges. For the IFEM, it is also necessary to include the jump of the tangential
derivative on interface edges since the IFEM solution
may not be continuous across interface edges.
Define the normal flux jump of $u_{\mathcal T}$ on each edge by
\[
j_{n,F}=\left\{
\begin{array}{lll}
\jump{\alpha \nabla u_{\mathcal T} \cdot {\bf n}_F}_F, & \mbox{for }F \in {\mathcal E}_I,\\[2mm]
0,& \mbox{for }F \in {\mathcal E}_D, \\[2mm]
\alpha \nabla u_{\mathcal T} \cdot {\bf n}_F+g_{_N}|_F, & \mbox{for }F \in {\mathcal E}_N,
\end{array}
\right.
\]
and the tangential derivative jump of $u_{\mathcal T}$ on each edge by
\[
j_{t,F}=\left \{
\begin{array}{lll}
\jump{ \nabla u_{\mathcal T} \cdot {\bf t}_F}_F, &\mbox{for } F \in {\mathcal E}_I,\\[2mm]
0, & \mbox{for }F \in {\mathcal E}_D \cup {\mathcal E}_N.
\end{array}
\right.
\]
Note that on each interface edge $F \in {{{\mathcal E}}^{int}}$ both $j_{n,F}$ and $j_{t,F}$ are piecewise constant.
For all $K \in {\mathcal T}$ we define the local error indicator $\eta_K$ by
\begin{equation} \label{eta-K}
\begin{split}
\eta_K^2&=
\sum \limits_{F \in {\mathcal E}_K \cap {{{\mathcal E}}^{int}}} \left(\displaystyle\frac{h_F}{2} \|\tilde\alpha_F^{-1/2} j_{n,F}\|_{0,F}^2
+\displaystyle\frac{h_F}{2} \| \tilde\alpha_F^{1/2} j_{t,F}\|_{0,F}^2\right)
+ \|\tilde \alpha^{1/2} \nabla u_{\mathcal T} \|_{S_K}^2
\\
&+
\sum \limits_{F \in {\mathcal E}_K \cap {\mathcal E}_I \setminus {{{\mathcal E}}^{int}} }
\displaystyle\frac{h_F}{2} \| \tilde\alpha_F^{-1/2}j_{n,F}\|_{0,F}^2
+
\sum \limits_{F \in {\mathcal E}_K \cap {\mathcal E}_N} h_F \| \tilde\alpha_F^{-1/2}j_{n,F}\|_{0,F}^2
\end{split}
\end{equation}
where $\tilde\alpha_F(x)=\max\left(\tilde\alpha|_F^1(x), \tilde\alpha|_F^2(x)\right)$, and $S_K$ is defined in \eqref{eq: SK}. Note that $\tilde\alpha_F(x)$ is a constant on $F$ when $F \notin {{{\mathcal E}}^{int}}$. The global error estimator $\eta$ is then defined by
\begin{equation}\label{est}
\eta=\left( \sum_{K \in {\mathcal T}} \eta_K^2\right)^{1/2}.
\end{equation}
\begin{remark}
If $K$ is a non-interface element, i.e., ${\mathcal E}_K \cap {{{\mathcal E}}^{int}} = \emptyset$, the first and second terms in \eqref{eta-K} vanish.
In this case, the local error indicator $\eta_K$ is identical to the residual-based error indicator for
the classical body-fitting conforming finite element method \cite{BeVe:00}.
\end{remark}
\section{Global Reliability}\label{sec:4}
In this section, we establish the reliability bound of the global estimator $\eta$ given in \eqref{est}.
For each $z \in {\mathcal N}$, let $\o_z$ be the union of all elements sharing
$z$ as a common vertex.
To this end, let ${\mathcal N}_\Gamma$ be the set of all vertices $z$ such that
$\mbox{meas}_{d-1} (\o_z \cap \Gamma) >0 $, and define
\begin{eqnarray*}
H_{f}({\mathcal T}) &=&
\left( \sum_{z \in {\mathcal N} \setminus ({\mathcal N}_\Gamma \cup {\mathcal N}_D)}
\displaystyle\frac{\mbox{diam}(\o_z)^2}{\alpha_z} \|f -f_z\|^2_{0,\o_z} \right. \\[2mm]
&&
\left. \quad + \sum_{z \in {\mathcal N}_\Gamma \cup {\mathcal N}_D}
\left( \displaystyle\frac{\mbox{diam}(\o_z)^2}{\alpha^-}\|f\|_{0,\o_z^-}^2 +
\displaystyle\frac{\mbox{diam}(\o_z)^2}{\alpha^+}\|f\|_{0,\o_z^+}^2\right) \right)^{1/2}
\end{eqnarray*}
where $\alpha_z= \alpha(x)|_{\o_z}$ and
$f_z$ is the average value of $f$ on $\o_z$.
\begin{remark}
The first term in $H_f({\mathcal T})$ is a higher order term for $f \in L^2(\O)$ \cite{CaVe:99}.
It is also well known that for linear finite element methods the edge residuals are dominant. In our adaptive algorithm the element residual is also omitted.
\end{remark}
\subsection{Helmholtz decomposition}
Let
\[
H_N^1(\O)=
\left\{ v \in H^1(\O) \,: \int_\O v \,dx =0 \quad\mbox{and} \quad \displaystyle\frac{\partial v}{ \partial t} = 0
\mbox{ on } \Gamma_N
\right\}.
\]
For $\phi \in H^1(\O)$, define
the adjoint curl operator by
$\nabla^{\perp} \, \phi = \left(- \displaystyle\frac{\partial \phi}{\partial y}, \,\displaystyle\frac{\partial \phi}{\partial x} \right)$.
For each $v \in \mathcal{S}({\mathcal T})$, we define the discrete gradient operator $\nabla_h$ by
\[
(\nabla_h v)|_K = \nabla (v|_K), \quad \forall \, K \in {\mathcal T}.
\]
\begin{lemma}[Helmholtz Decomposition] \label{HD}
Let $u$ and $u_{\mathcal T}$ be the solutions of \eqref{continuous-weak-solution} and \eqref{IFE approx}, respectively.
Then there exist uniquely $\phi \in H_{D}^1(\O)$ and $\psi \in H_N^1(\O)$
such that
\begin{equation} \label{HD:1}
\alpha \nabla u - \tilde \alpha \nabla_h u_{\mathcal T}=\alpha \nabla \phi+\nabla^{\perp}\,\psi.
\end{equation}
Moreover,
\begin{equation} \label{HD:3}
( \nabla \phi, \nabla^{\perp} \psi)=0.
\end{equation}
\end{lemma}
\begin{proof}
The proof can be referred to \cite{DaDuPaVa:1996, AinsworthRankin:08}. Here we also sketch a proof for the convenience of readers. Let $\tilde {\bf e}_\sigma := \alpha \nabla u - \tilde \alpha \nabla_h u_{\mathcal T}$ and $e_{\mathcal T} = u - u_{\mathcal T}$.
Assume that $\phi \in H^1(\O)$ solves the following equation:
\[
- \nabla \cdot \alpha \nabla \phi = -\nabla_h \cdot \tilde {\bf e}_\sigma \quad \mbox{in } \O
\]
with the boundary conditions
\[
\phi = 0 \quad \mbox{on }\Gamma_D \quad \mbox{and} \quad
-\alpha \nabla \phi \cdot {\bf n} = -\tilde {\bf e}_\sigma \cdot {\bf n} \quad \mbox{on } \Gamma_N.
\]
Then there exists $\psi \in H^1(\O)$ such that
\[
\nabla^\perp \psi = \tilde {\bf e}_\sigma - \alpha \nabla \phi.
\]
Moreover, we have
$ \displaystyle\frac{\partial \psi}{\partial t}= \nabla^\perp \psi \cdot {\bf n} =(\tilde {\bf e}_\sigma - \alpha \nabla \phi) \cdot {\bf n}=0$ on $\Gamma_N$, hence, $\psi \in H_N^1(\O)$.
By integration by parts and the boundary conditions it is easy to check that
\[
(\nabla \phi, \nabla^\perp \psi) =0.
\]
This completes the proof of the lemma.
\end{proof}
\begin{lemma} \label{L2 representation with Helmholtz Decomposition}
Let $\phi$ and $\psi$ be given in \eqref{HD:1}.
Then we have the following error representations in the weighted semi-$H^1$ norm:
\begin{equation} \label{p-representation}
\begin{split}
\|\alpha^{1/2} \nabla \phi\|_{0,\O}^2&=
(f, \phi-v)
-\sum_{F\in {\mathcal E}_I \cup {\mathcal E}_N} \int_F j_{n,F} \{\phi-v\} ds
\\
&+\epsilon \sum_{F\in {{{\mathcal E}}^{int}}} \int_F \{\tilde \alpha \nabla v \cdot {\bf n}_F \} \jump{u_{\mathcal T}} \,ds
+
\sum_{F\in {{{\mathcal E}}^{int}}} \displaystyle\frac{\gamma}{h_F}\int_F\tilde \alpha\, \jump{u_{\mathcal T}}\jump{v} \,ds,
\end{split}
\end{equation}
for any $v \in \mathcal{S}_D({\mathcal T})$ and
\begin{equation}\label{q-representation}
\|\alpha^{-1/2} \nabla^{\perp} \psi \|_{0,\O}^2=
-\sum_{F\in {{{\mathcal E}}^{int}}} \int_F \jump{u_{\mathcal T}} \big( \nabla^{\perp} \psi \cdot {\bf n}_F \big) ds +
((1 - \tilde \alpha/\alpha) \nabla_h u_{\mathcal T}, \nabla^{\perp} \psi).
\end{equation}
\end{lemma}
\begin{proof}
Let $v \in \mathcal{S}_D({\mathcal T})$ be arbitrary. Applying \eqref{HD:3}, \eqref{HD:1},
and integration by parts gives
\begin{equation} \label{L20}
\begin{split}
&(\alpha \nabla \phi,\nabla \phi)
=(\tilde {\bf e}_\sigma, \nabla \phi)
=\Big(\tilde {\bf e}_\sigma,\nabla_h ( \phi- v) \Big)
+\big(\tilde {\bf e}_\sigma, \nabla_h v \big)
\\
=& \sum_{K\in {\mathcal T}} \left( \int_K \big(f,\phi-v \big) \,dx
+ \int_{\partial K} \big(\alpha \nabla u \cdot {\bf n} \big) \big(\phi-v \big) \,ds\right)
\\
&- \sum_{K \in {\mathcal T}} \left(\tilde \alpha\nabla u_{\mathcal T}, \nabla_h ( \phi- v) \right)_K
+\big(\tilde {\bf e}_\sigma, \nabla_h v \big)
\\
=&
(f, \phi -v) -\sum_{F \in {\mathcal E}^{int}} \int_F (\alpha \nabla u \cdot {\bf n}_F) \jump{v} \,ds
- \sum_{F \in {\mathcal E}_N} \int_F g_N ( \phi -v) \,ds
\\
&- \sum_{K \in {\mathcal T}} \left(\tilde \alpha\nabla u_{\mathcal T}, \nabla_h ( \phi- v) \right)_K
+\big(\tilde {\bf e}_\sigma, \nabla_h v \big).
\end{split}
\end{equation}
Applying integration by parts again gives
\begin{equation} \label{L21}
\begin{split}
&\sum_{K \in {\mathcal T}} \left(\tilde\alpha\nabla u_{\mathcal T}, \nabla_h ( \phi- v) \right)_K
=
\sum_{K \in {\mathcal T}} \int_{\partial K} (\tilde\alpha \nabla u_{\mathcal T}) \cdot {\bf n} ( \phi- v) \,ds
\\
=&
\sum_{F \in {\mathcal E}_I } \int_F \jump{ \tilde\alpha \nabla u_{\mathcal T} \cdot {\bf n}_F} \{\phi-v\} \,ds
-\sum_{F \in {\mathcal E}^{int}} \int_F \{\tilde\alpha \nabla u_{\mathcal T} \cdot {\bf n}_F \} \jump{v} \,ds
\\
&+
\sum_{F \in {\mathcal E}_N} \int_F (\tilde \alpha \nabla u_{\mathcal T} \cdot {\bf n}_F) (\phi-v) \,ds.
\end{split}
\end{equation}
The last equality used
\eqref{jump-formula}, $\phi-v =0$ on $\Gamma_D$, and the facts that
\[
\jump{v}_F=0, \quad \forall \, F \in {\mathcal E}_I \setminus {{{\mathcal E}}^{int}} \quad \mbox{and} \quad
\jump{\phi}_F =0, \quad \forall \, F \in {\mathcal E}_I.
\]
By integration by parts and \eqref{IFE approx} we also have
\begin{equation}
\begin{split}
\big(\tilde {\bf e}_{\sigma} , \nabla_h v \big)
=&
(\alpha \nabla u, \nabla_h v) - (\tilde \alpha \nabla_h u_{\mathcal T}, \nabla_h v)
\\
=&\sum_{F\in {{{\mathcal E}}^{int}}} \int_F \{ \alpha \nabla u \,\cdot\, {\bf n}_F\} \jump{v} \,ds
-
\sum_{F\in {{{\mathcal E}}^{int}}} \int_F \{ \tilde \alpha \nabla u_{\mathcal T} \,\cdot\, {\bf n}_F\} \jump{v} \,ds
\\&
+\epsilon \sum_{F\in {{{\mathcal E}}^{int}}} \int_F \{ \tilde\alpha \nabla v \,\cdot\, {\bf n}_F\} \jump{u_{\mathcal T}} \,ds
+\sum_{F\in {{{\mathcal E}}^{int}}} \displaystyle\frac{\gamma }{h_F}\int_F \tilde\alpha \, \jump{u_{\mathcal T}}\jump{v} \,ds,
\end{split}
\end{equation}
which, together with \eqref{L20} and \eqref{L21}, gives \eqref{p-representation}.
To prove \eqref{q-representation}, by \eqref{HD:3}, \eqref{HD:1}, integration by parts, and the facts that
\[
\jump{e_{\mathcal T}}_F = - \jump{u_{\mathcal T}}_F
\quad \mbox{and} \quad
\jump{ \nabla^{\perp} \psi \cdot {\bf n}_F}_F = 0, \quad \forall \,F \in {\mathcal E}_I,
\]
we have
\begin{equation*}
\begin{split}
&(\alpha^{-1} \nabla^{\perp} \psi, \nabla^{\perp} \psi)=( \nabla u - (\tilde \alpha/\alpha) \nabla_h u_{\mathcal T}, \nabla^{\perp} \psi)
\\
=& ( \nabla u - \nabla_h u_{\mathcal T}, \nabla^{\perp} \psi) + ((1 - \tilde \alpha/\alpha) \nabla_h u_{\mathcal T}, \nabla^{\perp} \psi)
\\
=&\sum_{K \in {\mathcal T}} \int_{\partial K} e_{\mathcal T} (\nabla^{\perp} \psi \cdot {\bf n}) \, ds
+ ((1 - \tilde \alpha/\alpha) \nabla_h u_{\mathcal T}, \nabla^{\perp} \psi)\\
= &- \sum_{F \in {{{\mathcal E}}^{int}}}\int_F \jump{u_{\mathcal T}} \big( \nabla^{\perp} \psi \cdot {\bf n}_F \big)\,ds
+ ((1 - \tilde \alpha/\alpha) \nabla_h u_{\mathcal T}, \nabla^{\perp} \psi).
\end{split}
\end{equation*}
Hence, we obtain \eqref{q-representation}. This completes the proof of the lemma.
\end{proof}
\subsection{Modified Cl\'ement-type interpolation}
Define a modified Cl\'ement-type interpolation operator $\mathcal{I}_h\,:\, H_{D}^1(\O) \rightarrow S_{D}({\mathcal T})$ by
\begin{equation} \label{cle:1}
\mathcal{I}_h(v)=\sum_{z \in {\mathcal N} } (\pi_z v) \tilde\lambda_z(x)
\end{equation}
where $\pi_z$ is defined by
\[
\pi_z(v)= \left\{
\begin{array}{ll}
\displaystyle\frac{\int_{\o_z} \lambda_z v \,dx}{ \int_{\o_z} \lambda_z \,dx}, & \forall \, z \in {\mathcal N} \setminus {\mathcal N}_D, \\[4mm]
0,& \forall \, z \in {\mathcal N}_D,
\end{array}
\right.
\]
where $\lambda_z $ and $\tilde \lambda_z$ are the classical barycentric hat function and the linear IFEM nodal basis function
of $\mathcal{S}({\mathcal T})$ associated to $z$, respectively. Note that
\begin{equation} \label{weighted-interpolation}
(v-\pi_z v, \lambda_z)_{\o_z}=0 \quad \forall \, z \in {\mathcal N} \setminus {\mathcal N}_D.
\end{equation}
By Lemma~6.1 in \cite{CaVe:99} there holds
\begin{equation}\label{approximation}
\| v - \pi_z v\|_{0,\o_z} \le C \mbox{diam}(\o_z) \| \nabla v\|_{0,\o_z}, \quad \forall \, z\in {\mathcal N} \quad \mbox{and} \quad
\forall \,v \in H_D^1(\O).
\end{equation}
The following lemma provides the approximation and stability properties of the modified Cl\'ement-type interpolation operator.
\begin{lemma}[Cl\'ement-type Interpolation]\label{lem:clement}
Let $v \in H_{D}^1(\O)$, and $\mathcal{I}_h v \in \mathcal{S}_D({\mathcal T})$ be the interpolation of $v$ defined in \eqref{cle:1}.
Then there exists a constant $C>0$ that is independent of the mesh size and the location of the interface such that
\begin{align}
\big\|v-\mathcal{I}_h v \big\|_{0,K} &\le C h_K \big\|\nabla v\big\|_{0,\o_K},
\quad \forall \, K \in {\mathcal T}, \label{cle:2}
\\
\big\|\nabla (v-\mathcal{I}_h v) \big\|_{0,K} &\le C \big\|\nabla v \big\|_{0,\o_K},
\quad \quad \;\forall \,K \in {\mathcal T}, \; \label{cle:3}
\\
\big\| (v-\mathcal{I}_h v)|_{K} \big\|_{0,F} &\le C h_F^{1/2} \big\|\nabla v\big\|_{0,\o_{K}},
\; \; \forall \,F \in {\mathcal E}_K, \;K \in \{K_{F,1}, K_{F,2}\} , \label{cle:4}
\end{align}
where
$\o_K$ is the union of all elements sharing at least one vertex with $K$.
\end{lemma}
\begin{proof}
Without loss of generality, assume that $K \in {\mathcal T}$ is an interior element.
To prove \eqref{cle:2}, by the partition of unity, the triangle inequality, the boundedness of IFE basis functions
$\|\tilde\lambda_z\|_{\infty,K} \le C(\rho)$ (Theorem 2.4, \cite{LiLinLinRo:04}), and \eqref{approximation},
we have
\begin{eqnarray*}\label{cle:2a}
\big\|v-\mathcal{I}_h v \big\|_{0,K} &\le& \sum_{z\in {\mathcal N}_K} \big\| (v- \pi_zv) \tilde\lambda_z \big\|_{0,K}
\le C \sum_{z\in {\mathcal N}_K} \big\| v- \pi_z v \big\|_{0,K} \\[2mm]
& \le& C \sum_{z\in {\mathcal N}_K} \big\| v- \pi_z v \big\|_{0,\o_z} \le
C \sum_{z\in {\mathcal N}_K} h_K \| \nabla v \|_{0,\o_z} \le C h_K \|\nabla v\|_{0,\o_K}.
\end{eqnarray*}
To prove \eqref{cle:3}, by the partition of unity, the triangle inequality, the fact that
$\|\nabla \tilde \lambda_z\|_{\infty,K} \le C h_K^{-1}$ (Theorem 2.4, \cite{LiLinLinRo:04}), and \eqref{approximation}, we have
\begin{eqnarray*}
\big\|\nabla (v-\mathcal{I}_h v) \big\|_{0,K} &=&
\left\| \sum_{z \in {\mathcal N}_K} \nabla \Big((v-\pi_z v)\tilde\lambda_z \Big) \right\|_{0,K}
\le \sum_{z \in {\mathcal N}_K} \Big( \big\|\tilde\lambda_z \nabla v \big\|_{0,K}
+ \big\|(v-\pi_z v) \nabla \tilde\lambda_z \big\|_{0,K} \Big)\\[2mm]
&\le& C \left( \big\|\nabla v \big\|_{0,K} + h_K^{-1}\big\|v-\pi_z v \big\|_{0,K} \right)
\le C \big\|\nabla v \big\|_{0,\o_K}.
\end{eqnarray*}
Finally, \eqref{cle:4}
follows from the partition of unity, the triangle inequality, the trace inequality, and \eqref{approximation}:
\begin{eqnarray*}
\big\|(v-\mathcal{I}_h v)|_K \big\|_{0,F} &\le& \sum_{z\in {\mathcal N}_K} \big\|(v-\pi_z v) \tilde \lambda_z|_K\big\|_{0,F}
\le C \sum_{z\in {\mathcal N}_K} \big\|(v-\pi_z v)|_K\big\|_{0,F}
\\[2mm]
&\le& C \sum_{z \in {\mathcal N}_K}
\left(h_F^{-1/2} \big\| v-\pi_z v \big\|_{0,K}
+h_F^{1/2}\big \|\nabla v \big\|_{0,K}\right)
\le C h_F^{1/2}\big \|\nabla v \big\|_{0,\o_{K}}.
\end{eqnarray*}
This completes the proof of the lemma.
\end{proof}
\begin{lemma}
There exists a constant $C>0$, independent of the mesh size
and the location of the interface, such that
\begin{equation} \label{norm-0-1}
\| \jump{u_{\mathcal T}}\|_{0,F} \le C h_F\| j_{t,F}\|_{0,F}, \quad \forall \,F \in {{{\mathcal E}}^{int}}
\end{equation}
and
\begin{equation}\label{norm-1/2-1}
\| \jump{u_{\mathcal T}}\|_{1/2,F} \le h_F^{1/2}\|j_{t,F}\|_{0,F}, \quad \forall \,F \in {{{\mathcal E}}^{int}}.
\end{equation}
\end{lemma}
\begin{proof}
We first prove the results on the reference element $K$ formed by vertices
$((0,0), (1,0), (0,1))$ and let $F$ be the edge of $K$ on the $x$-axis. Without loss of generality, let
$(a,0), 0< a < 1$, be the interface point on $F$ and $\jump{u_{\mathcal T}}_F(a,0) =b$.
Note that $\jump{u_{\mathcal T}}_F$ takes the value $0$ at both endpoints $(0,0)$ and $(1,0)$. By direct calculations, we have
\[
\|\jump{u_{\mathcal T}}\|_{0,F}^2 = \displaystyle\frac{1}{3}b^2 \quad \mbox{and} \quad
\| \jump{\nabla u_{\mathcal T} \cdot {\bf t}}\|_{0,F}^2
= \left(\displaystyle\frac{1}{a} + \displaystyle\frac{1}{1-a}\right) b^2 \ge 4b^2.
\]
Regarding the $H^{1/2}$-norm, we have
\[
\|\jump{u_{\mathcal T}}\|_{1/2,F} = \inf_{v \in \tilde H^1(K)} \|v\|_{1,K}
\]
where
$\tilde H^1(K) =
\{v \in H^1(K) : v = \jump{u_{\mathcal T}} \,~\mbox{on}\, F, ~v = 0\, ~\mbox{on } \,\partial K \setminus \{F \}$.
In particular, let
\[
v = \begin{cases}
\displaystyle\frac{b}{a}x &\mbox{ in } K_1 \\
\displaystyle\frac{b}{a-1} (y+x -1) &\mbox{ in } K_2
\end{cases}
\]
where $K_1$ is the subtriangle fromed by $((0,0),(a,0),(0,1))$ and $K_2 = K \setminus K_1$. It is easy to verify that $v \in \tilde H^1(K)$.
Another direct calculation gives
\[
\|v\|_{1,K}^2 = \displaystyle\frac{1}{12}b^2 + \left( \displaystyle\frac{1}{2a} + \displaystyle\frac{1}{1-a}\right)b^2 .
\]
It is easy to verify that
\[
\| \jump{u_{\mathcal T}}\|_{0,F} \le \| j_{t,F}\|_{0,F} \quad\mbox{ and} \quad
\| \jump{u_{\mathcal T}}\|_{1/2,F} \le \|v\|_{1,K} \le 2\|j_{t,F}\|_{0,F},
\]
which, together with the scaling argument, gives \eqref{norm-0-1} and \eqref{norm-1/2-1}. This completes the proof of the lemma.
\end{proof}
\begin{lemma} \label{lemma:con}
Let $\phi$ be given in \eqref{HD:1}. There exists a constant $C$ independent of the mesh size and the
location of the interface such that
\begin{equation}\label{rel:con}
\|\alpha^{1/2} \nabla \phi\|_{0,\O} \le
C \left(
\sum_{F\in {\mathcal E}_I \cup {\mathcal E}_N} h_F
\big\|\tilde \alpha_F^{-1/2}j_{n,F} \big\|_{0,F}^2
+
\sum_{F\in {\mathcal E}^{int}} h_F
\big\| \tilde \alpha^{1/2} j_{t,F} \big\|_{0,F}^2
+H_f({\mathcal T})^2
\right)^{1/2}.
\end{equation}
\end{lemma}
\begin{proof}
By Lemma \ref{L2 representation with Helmholtz Decomposition},
\begin{eqnarray*}
\|\alpha^{1/2} \nabla \phi\|_{0,\O}^2
&=&
(f, \phi-v)
-\sum_{F\in {\mathcal E}_I \cup {\mathcal E}_N} \int_F j_{n,F} \{\phi-v\} ds
\\[2mm]
&&~+ \epsilon \sum_{F\in {{{\mathcal E}}^{int}}} \int_F \{ \tilde \alpha \nabla v \cdot {\bf n}_F \} \jump{u_{\mathcal T}} \,ds
+
\sum_{F\in {{{\mathcal E}}^{int}}} \displaystyle\frac{\gamma}{h_F}\int_F \tilde \alpha\, \jump{u_{\mathcal T}}\jump{v} \,ds\\[2mm]
&
\triangleq& I_1+I_2+I_3+I_4.
\end{eqnarray*}
Let $v= \phi_I \in \mathcal{S}_D({\mathcal T})$ be the modified Cl\'ement-type interpolation defined in \eqref{cle:1} of $\phi$.
By the partition of unity, the fact that $\tilde \lambda_z = \lambda_z$ for $z \in {\mathcal N}\setminus {\mathcal N}_\Gamma$,
the Cauchy-Schwarz inequality, \eqref{weighted-interpolation}, and \eqref{approximation}, we have
\begin{equation}\label{I-1}
\begin{split}
I_1
& =
\sum_{z \in {\mathcal N}_\Gamma \cup {\mathcal N}_D} \bigg(f\tilde\lambda_z , \phi- \pi_z \phi\bigg)_{\o_z}
+
\sum_{z \in {\mathcal N} \setminus ({\mathcal N}_\Gamma \cup {\mathcal N}_D)} \bigg((f - f_z) \lambda_z, \phi- \pi_z \phi\bigg)_{\o_z}
\\
&\le
C\left( \sum_{z \in {\mathcal N}_\Gamma \cup {\mathcal N}_D} \mbox{diam}(\o_z) \big\| f \big\|_{0,\o_z} \big\|\nabla \phi \big\|_{0,\o_z}
+
\sum_{z \in {\mathcal N} \setminus ({\mathcal N}_\Gamma \cup {\mathcal N}_D)} \mbox{diam}(\o_z) \big\| f-f_z \big\|_{0,\o_z} \big\|\nabla \phi \big\|_{0,\o_z}
\right)
\\
&\le C H_f({\mathcal T}) \big\|\alpha^{1/2} \nabla \phi \big\|_{0,\O}.
\end{split}
\end{equation}
For any $F \in ({\mathcal E}_I \cup {\mathcal E}_N) \setminus {{{\mathcal E}}^{int}}$,
it follows from the continuity of $\phi-\phi_{I}$ on $F$, Cauchy-Schwarz inequality, and \eqref{cle:4} that
\begin{equation}\label{I_2b}
\begin{split}
\int_F j_{n,F} \{\phi- \phi_{I}\}\,ds=&\int_F j_{n,F} (\phi- \phi_{I})|_{K_{F,1}}\,ds
\le C h_F^{1/2} \big\| j_{n,F} \big\|_{0,F} \big\|\nabla \phi \big\|_{0,\o_{K_{F,1}}}
\\
\le& C h_F^{1/2} \big\| \alpha_F^{-1/2} j_{n,F} \big\|_{0,F} \big\| \alpha^{1/2} \nabla \phi \big\|_{0,\o_{K_{F,1}}}.
\end{split}
\end{equation}
For any $F \in {{{\mathcal E}}^{int}}$, it then follows from the Cauchy-Schwarz and Young's inequality and \eqref{cle:4} that
\begin{eqnarray*}\label{I_2a}
&&\int_F j_{n,F} \{ \phi-\phi_{I}\} \,ds \\\
&=&
\int_{F+} j_{n,F} \{ \phi-\phi_{I}\} \,ds +\int_{F^-} j_{n,F} \{ \phi-\phi_{I}\} \,ds\\[2mm]
&\le&
\|j_{n,F}\|_{0,F^+}
\bigg( \big\|(\phi-\phi_{I})|_{K_{F,1}} \big\|_{0,F^+}+ \big\|(\phi-\phi_{I})|_{0,K_{F,2}} \big\|_{0,F^+}\bigg)\\[2mm]
&&+
\|j_{n,F}\|_{0,F^-}
\bigg( \|(\phi-\phi_{I})|_{K_{F,1}} \big\|_{0,F^-}+ \|(\phi-\phi_{I})|_{K_{F,2}} \big\|_{0,F^-} \bigg)\\[2mm]
&\le&
C \left( \displaystyle\frac{1}{\alpha^+} \|j_{n,F}\|_{0,F^+}^2
\!+\! \displaystyle\frac{1}{\alpha^{-}} \|j_{n,F}\|_{0,F^-}^2 \right)^{1/2}
\!\! \left(
\sum_{K \in \{K_{F,1}, K_{F,2}\}} \big\|\alpha^{1/2}(\phi-\phi_{I})|_{K} \big\|^2_{0,F}
\right)^{1/2}\\[2mm]
&\le& C h_F^{1/2} \big\| \tilde\alpha_F^{-1/2} j_{n,F}\big\|_{0,F}
\| \alpha ^{1/2} \nabla \phi\|_{0,\o_F},
\end{eqnarray*}
where $\o_F = \o_{K_{F,1}} \cup \o_{K_{F,2}}$, which, together with
\eqref{I_2b} and the Cauchy-Schwarz inequality, yields
\begin{equation} \label{I-2}
I_2 \le
C\left( \sum_{F \in {\mathcal E}_I \cup {\mathcal E}_N} h_F \big\| \tilde\alpha_F^{-1/2} j_{n,F}\big\|_{0,F}^2 \right)^{1/2} \|\alpha^{1/2} \nabla \phi\|_{0,\O}.
\end{equation}
To bound $I_3$, we apply the following trace inequality
for functions in the IFEM space (see Lemma 3.2 in \cite{LinLinZhang:15}).
Let $v \in \mathcal{S}_D({\mathcal T})$ be arbitrary, then
\begin{equation}\label{trace:n}
\big\|\tilde\alpha \nabla v|_K \cdot {\bf n}_F \big\|_{0,F} \le
C h_F^{-1/2}\big\|\tilde\alpha \nabla v \big\|_{0,K},\quad \forall \,F \in {{{\mathcal E}}^{int}}.
\end{equation}
It now follows from the Cauchy-Schwarz inequality, \eqref{trace:n}, and
\eqref{cle:3} that
\begin{eqnarray*}
I_3 &\le&\sum_{F\in {{{\mathcal E}}^{int}}} \|\jump{u_{\mathcal T}}\|_{0,F}
\left( \left\| \tilde \alpha \nabla \phi_{I}|_{K_{F,1}} \cdot {\bf n}_F \right\|_{0,F}
+ \left\| \tilde \alpha \nabla \phi_{I}|_{K_{F,2}}\cdot {\bf n}_F \right\|_{0,F}\right)
\\[2mm]
&\le& \sum_{F\in {{{\mathcal E}}^{int}}} C \, h_F^{-1/2} \|\jump{u_{\mathcal T}}\|_{0,F}
\bigg( \left \| \tilde\alpha \nabla \phi_{I} \right \|_{0,K_{F,1}}
+ \left\| \tilde\alpha \nabla \phi_{I} \right\|_{0,K_{F,2}}
\bigg) \\[2mm]
&\le&C \left( \sum_{F\in {{{\mathcal E}}^{int}}} h_F^{-1}
\| \tilde \alpha_F^{1/2}\jump{u_{\mathcal T}}\|_{0,F}^2 \right)^{1/2}
\|\alpha^{1/2}\nabla \phi\|_{0, \O}.
\end{eqnarray*}
Together with \eqref{norm-0-1} we obtain
\begin{equation}\label{I-3}
I_3 \le
C \left( \sum_{F\in {{{\mathcal E}}^{int}}} h_F
\|\tilde \alpha_F^{1/2} j_{t,F}\|_{0,F}^2 \right)^{1/2}
\|\alpha^{1/2}\nabla \phi\|_{0,\O}.
\end{equation}
To bound $I_4$, we use the fact that
\[
\int_F \tilde\alpha \jump{u_{\mathcal T}} \jump{\phi_{I}} \,ds =
-\int_F \tilde \alpha \jump{u_{\mathcal T}} \jump{\phi- \phi_{I} }\,ds
\quad \forall \,F \in {{{\mathcal E}}^{int}},
\]
the Cauchy-Schwarz and triangle inequalities, and \eqref{cle:4} to obtain
\begin{eqnarray*}
I_4
&=&-\sum_{F\in {\mathcal E}^{int}}\displaystyle\frac{\gamma}{h_F} \int_F \tilde \alpha \,\jump{u_{\mathcal T}} \jump{\phi-\phi_{I}} \,ds\nonumber\\[2mm]
&\le& C \sum_{F\in {\mathcal E}^{int}} \displaystyle\frac{\gamma }{h_F} \|\tilde \alpha_F^{1/2} \jump{u_{\mathcal T}}\|_{0,F}
\left( \|\alpha^{1/2}(\phi-\phi_{I})|_{K_{F,1}} \|_{0,F}+ \|\alpha^{1/2}(\phi-\phi_{I})|_{K_{F,2}} \|_{0,F}\right)\nonumber\\[2mm]
&\le&
C \bigg( \sum_{F\in {\mathcal E}^{int}} h_F^{-1}
\|\tilde \alpha_F^{1/2} \jump{u_{\mathcal T}}\|_{0,F}^2 \bigg)^{1/2}
\|\alpha^{1/2}\nabla \phi\|_{0,\O},
\end{eqnarray*}
which, combining with \eqref{norm-0-1}, yields
\begin{equation}\label{I-4}
I_4 \le C \left( \sum_{F\in {\mathcal E}^{int}} h_F
\|\tilde \alpha_F^{1/2} j_{t,F}\|_{0,F}^2 \right)^{1/2}
\|\alpha^{1/2}\nabla \phi\|_{0,\O}.
\end{equation}
Finally, \eqref{rel:con} is a direct consequence of \eqref{I-1}, \eqref{I-2}, \eqref{I-3}, \eqref{I-4} and the
Young's inequality.
This completes the proof of the lemma.
\end{proof}
\begin{lemma} \label{lemma:noncon}
Let $\psi$ be given in \eqref{HD:1}. There exists a constant $C$ independent of the mesh size and the location of the interface such that
\begin{equation}\label{nonconformist-estimate}
\|\alpha^{-1/2} \nabla^{\perp} \psi\|_{0,\O} \le C \bigg( \sum_{F \in {\mathcal E}^{int}} h_F
\| \tilde\alpha_F^{1/2} j_{t,F}\|_{0,F}^2
+
\sum_{K \in {\mathcal T}^{int}} \| \tilde \alpha^{1/2} \nabla u_{\mathcal T}\|_{0,S_K}^2
\bigg)^{1/2}.
\end{equation}
\end{lemma}
\begin{proof}
For the first term in \eqref{q-representation} it follows from the duality inequality, \eqref{norm-1/2-1}, the trace and Young's inequalities that
\begin{equation}\label{I-21}
\begin{split}
&\sum_{F \in {\mathcal E}^{int}} \int_F \jump{u_{\mathcal T}} (\nabla^{\perp} \psi \cdot {\bf n}_F) \,ds
\le \sum_{F \in {\mathcal E}^{int}} \big\| \jump{u_{\mathcal T}} \big\|_{1/2,F}
\left\|\nabla^{\perp} \psi \cdot {\bf n}_F \right\|_{-1/2,F}
\\
\le~&C\sum_{F \in {\mathcal E}^{int}} h_F^{1/2}
\big\| j_{t,F} \big\|_{0,F} \big\|\nabla^{\perp}\psi \big\|_{0,K_{F,1}}
\\
\le~& C \bigg( \sum_{F \in {\mathcal E}^{int}} h_F
\| \tilde \alpha_F^{1/2} j_{t,F}\|_{0,F}^2 \bigg)^{1/2}
\|\alpha^{-1/2} \nabla^{\perp} \psi\|_{0,\O}.
\end{split}
\end{equation}
For the second term in \eqref{q-representation} it follows from the Cauchy-Schwarz inequality that
\begin{equation}\label{I-22}
\begin{split}
&((1 - \tilde \alpha/\alpha) \nabla u_{\mathcal T}, \nabla^{\perp} \psi) \le C
\sum_{K \in {\mathcal T}^{int}} \| \tilde\alpha^{1/2}\nabla u_{\mathcal T} \|_{0,S_K} \|\alpha^{-1/2} \nabla^{\perp} \psi\|_{0,S_K}
\\
\le~ &C \left( \sum_{K \in {\mathcal T}^{int}} \| \tilde \alpha^{1/2} \nabla u_{\mathcal T}\|_{0,S_K}^2 \right)^{1/2}
\|\alpha^{-1/2} \nabla^{\perp} \psi\|_{0,\O}.
\end{split}
\end{equation}
Combining \eqref{I-21}-\eqref{I-22} and \eqref{q-representation} gives \eqref{nonconformist-estimate}.
This completes the proof of the lemma.
\end{proof}
\begin{theorem}\label{global-reliability}{\em(Global Reliability)}
There exists a constant $C_r>0$ that is independent of the location of the interface and the mesh size, such that
\begin{equation}\label{reliability}
\|\alpha^{1/2} ( \nabla u - \nabla_hu_{\mathcal T}) \|_{0,\O} \le C_r (\eta +H_f({\mathcal T})).
\end{equation}
\end{theorem}
\begin{proof}
First by adding and subtracting proper terms we have
\begin{equation} \label{rel:1}
\begin{split}
&\| \alpha^{1/2} ( \nabla u - \nabla_hu_{\mathcal T})\|_{0,\O}^2\\
=&
(\alpha \nabla u - \tilde \alpha \nabla_h u_{\mathcal T}, \nabla u - \nabla_hu_{\mathcal T})
+( (\tilde \alpha - \alpha)\nabla_h u_{\mathcal T}, \nabla u - \nabla_hu_{\mathcal T} )
\\=&
(\alpha \nabla u - \tilde \alpha \nabla_h u_{\mathcal T}, \alpha^{-1}(\alpha\nabla u - \tilde \alpha \nabla_h u_{\mathcal T}) )
\\&+
(\alpha \nabla u - \tilde \alpha \nabla_h u_{\mathcal T},(\tilde \alpha / \alpha -1)\nabla_h u_{\mathcal T}) )+
( (\tilde \alpha - \alpha)\nabla_h u_{\mathcal T}, \nabla u - \nabla_hu_{\mathcal T} )
\\=&
(\alpha \nabla u - \tilde \alpha \nabla_h u_{\mathcal T}, \alpha^{-1}(\alpha\nabla u - \tilde \alpha \nabla_h u_{\mathcal T}) )
+
2( \nabla u - \nabla_hu_{\mathcal T},(\tilde \alpha -\alpha)\nabla_h u_{\mathcal T})
\\&
-
(\alpha^{-1}(\tilde \alpha -\alpha)\nabla_h u_{\mathcal T},(\tilde \alpha -\alpha)\nabla_h u_{\mathcal T}) )\\
\le&
(\alpha \nabla u - \tilde \alpha \nabla_h u_{\mathcal T}, \alpha^{-1}(\alpha\nabla u - \tilde \alpha \nabla_h u_{\mathcal T}) )
+
2(\nabla u - \nabla_hu_{\mathcal T},(\tilde \alpha -\alpha)\nabla_h u_{\mathcal T}).
\end{split}
\end{equation}
By \eqref{HD:1} and \eqref{HD:3} we have
\begin{equation} \label{rel:2}
\begin{split}
&(\alpha \nabla u - \tilde \alpha \nabla_h u_{\mathcal T}, \alpha^{-1}(\alpha\nabla u - \tilde \alpha \nabla_h u_{\mathcal T}) )
=(\alpha \nabla \phi + \nabla^{\perp} \psi, \nabla \phi +\alpha^{-1} \nabla^{\perp} \psi)\\
=&\|\alpha^{1/2} \nabla \phi\|_{0,\Omega}^2 + \|\alpha^{-1/2} \nabla^{\perp} \psi\|_{0,\Omega}^2.
\end{split}
\end{equation}
Applying the Cauchy-Schwarz and Young's inequalities also gives
\begin{equation}\label{rel:3}
(\nabla u - \nabla_hu_{\mathcal T},(\tilde \alpha -\alpha)\nabla_h u_{\mathcal T}) \le C
\left( \sum_{K \in {\mathcal T}^{int}} \|\tilde \alpha \nabla_h u_{\mathcal T} \|_{0,S_K}^2 \right)^{1/2}
\|\alpha^{1/2} ( \nabla u - \nabla_hu_{\mathcal T})\|_{0,\Omega}.
\end{equation}
\eqref{reliability} is then a direct result of \eqref{rel:1}--\eqref{rel:3}, Lemma \ref{lemma:con}, Lemma \ref{lemma:noncon}
and the Young's inequality. This completes the proof of theorem.
\end{proof}
\section{Local Efficiency}\label{sec:5}
In this section, we establish the efficiency bound for the error indicators $\eta_K$ defined in
\eqref{eta-K} for every element $K \in {\mathcal T}$. For non-interface elements, the efficiency bound for $\eta_K$ is well known (see \cite{BeVe:00}) and the key technique is using the local edge and element bubble functions.
However, the same technique without modification becomes invalid for interface elements because jumps of the normal flux and the tangential derivative on the interface edges become piecewise constant. To overcome this difficulty it is natural to design more localized bubble functions that allow us to derive the efficiency bounds on $F^+$ and $F^-$ separately for each $F \in {{{\mathcal E}}^{int}}$.
For each $F \in {{{\mathcal E}}^{int}}$ we first design auxiliary elements and edge bubble functions associated to $F^-$.
Choose $\tilde K_{F,1} \subset K_{F,1}$ and
$\tilde K_{F,2} \subset K_{F,2}$ to be two regular triangular sub-elements such that
$F^-$ is a common edge of both $\tilde K_{F,1}$ and $\tilde K_{F, 2}$, to be more precise,
\[
\partial \tilde K_{F,1} \cap \partial \tilde K_{F,2} = F^-
\quad \mbox{and} \quad
(\partial \tilde K_{F,1} \cup \partial \tilde K_{F,2}) \cap F^+ =\emptyset.
\]
Note that $\tilde K_{F,1}$ and $\tilde K_{F,2}$ are not necessarily in $K_{F,1}^-$ and $K_{F,2}^-$, respectively. The key requirement here is to make sub-elements $\tilde K_{F,1}, \tilde K_{F,2}$ regular while $K_{F,1}^-$ and $K_{F,2}^-$ may not be in general.
For each sub-element $\tilde K \in \{\tilde K_{F,1}, \tilde K_{F,2}\}$, define the auxiliary element bubble function $\upsilon_{\tilde K}$ such that
(i) $\upsilon_{\tilde K} \in P_3(\tilde K)$,
(ii) $\upsilon_{\tilde K}|_{\partial \tilde K} \equiv 0$ and
(iii) $\upsilon_{\tilde K}$ takes the value $1$ at the barycentric center of $\tilde K$.
We also define the auxiliary edge bubble function $\upsilon_{F^-}$ for $F^-$ such that
(i) $\upsilon_{F^-} |_{\tilde K} \in P_2(\tilde K)$ for $\tilde K \in \{ \tilde K_{F,1}, \tilde K_{F,2}\}$,
(ii) $\upsilon_{F^-}|_{\partial (\tilde K_{F,1} \cup \tilde K_{F,2})} \equiv 0$ and
(iii) $ \upsilon_{F^-} $ takes value $1$ at the middle point of $F^-$.
It is easy to verify that $\upsilon_{\tilde K}$ for $\tilde K \in \{ \tilde K_{F,1}, \tilde K_{F,2}\} $
and $\upsilon_{F^-}$ uniquely exist. Let $w_{F^-} = (j_{n,F}|_{F^-})v_{F^-} $ and
$w_{\tilde K} = v_{\tilde K}f_K$ with $f_K$ being the average of $f$ on $K$.
Applying the continuity of $\alpha \nabla u \cdot {\bf n}_F$, the divergence theorem, and the Cauchy-Schwarz inequality gives
\begin{eqnarray*}
\| j_{n,F}\|_{0,F^-}^2 &\le& C \int_{F^-} \jump{\tilde\alpha \nabla u_{\mathcal T} \cdot {\bf n}_F} w_{F^-} \,ds \\[2mm]
&=&
C\int_{F^-} \left(\jump{\tilde \alpha \nabla u_{\mathcal T} \cdot {\bf n}_F} - \jump{\alpha \nabla u \cdot {\bf n}_F}\right) w_{F^-} \,ds\\[2mm]
&=&
C\int_{\tilde K_{F,1} \cup \tilde K_{F,2}} (\tilde \alpha \nabla_h u_{\mathcal T} - \alpha \nabla u) \cdot \nabla w_{F^-} \,dx
+\int_{\tilde K_{F,1} \cup \tilde K_{F,2}} f w_{F^-}\,dx\\[2mm]
&\le& C
\| \tilde \alpha \nabla_h u_{\mathcal T} - \alpha \nabla u \|_{0,\tilde K_{F,1} \cup \tilde K_{F,2}} \| \nabla w_{F^-}\|_{0,\tilde K_{F,1} \cup \tilde K_{F,2}} +\|f\|_{0,\tilde K_{F,1} \cup \tilde K_{F,2}}\|w_{F^-}\|_{0,\tilde K_{F,1} \cup \tilde K_{F,2}},
\end{eqnarray*}
which, combining with the properties of $w_{F^-}$ that
\[
\|\nabla w_{F^-}\|_{0,\tilde K_{F,1} \cup \tilde K_{F,2}} \le C \displaystyle\frac{1}{h_F^-} \|w_{F^-}\|_{0,\tilde K_{F,1} \cup \tilde K_{F,2}} \le C\displaystyle\frac{1}{\sqrt{h_F^-}} \| j_{n,F}\|_{0,F^-},
\]
yields
\begin{equation} \label{effi-flux-jump}
\| j_{n,F}\|_{0,F^-} \le C \left( \displaystyle\frac{1}{\sqrt{h_F^-}}\| \tilde \alpha \nabla_h u_{\mathcal T} - \alpha \nabla u \|_{0,\tilde K_{F,1} \cup \tilde K_{F,2}} + \sqrt{h_F^-}\|f\|_{0,\tilde K_{F,1} \cup \tilde K_{F,2}}\right).
\end{equation}
We now establish the efficiency bound for the element residual $\|f\|_{0,\tilde K}$ for $\tilde K \in \{ \tilde K_{F,1}, \tilde K_{F,2}\}$. By the property of $w_{\tilde K}$, the triangle inequality, the
divergence theorem, and the Cauchy-Schwarz inequality, we have
\begin{eqnarray*}
\|f_{K}\|_{0,\tilde K}^2 &\le&
C \left(\int_{\tilde K} f w_{\tilde K} \,dx + \| f - f_{K}\|_{0,\tilde K} \|w_K\|_{0,\tilde K} \right) \\[2mm]
&\le& C \left( \int_{\tilde K} \nabla \cdot(\tilde \alpha \nabla u_{\mathcal T} - \alpha \nabla u )w_{\tilde K} \,dx
+ \| f - f_{K}\|_{0,\tilde K} \|w_{\tilde K}\|_{0,\tilde K} \right)\\[2mm]
&\le&C \left( \|\tilde \alpha \nabla u_{\mathcal T} - \alpha \nabla u \|_{0,\tilde K} \|\nabla w_{\tilde K}\|_{0,\tilde K}
+\| f - f_{K}\|_{0,\tilde K} \|w_{\tilde K}\|_{0,\tilde K}
\right),
\end{eqnarray*}
which, combining with similar properties for $w_{\tilde K}$:
\[
\| \nabla w_{\tilde K}\|_{0,\tilde K} \le C \displaystyle\frac{1}{h_F^-} \|w_{\tilde K}\|_{0,\tilde K}
\le C\displaystyle\frac{1}{h_F^-} \|f_{K}\|_{0,\tilde K},
\]
gives
\begin{equation} \label{effi:elemen-residual}
h_F^-\|f_{K}\|_{0,\tilde K} \le
\left( \|\tilde \alpha \nabla u_{\mathcal T} - \alpha \nabla u \|_{0,\tilde K}+ h_F^-\|f - f_K\|_{0,\tilde K}
\right).
\end{equation}
By \eqref{effi-flux-jump} and \eqref{effi:elemen-residual}, we have
\[\sqrt{h_F^-}\|j_{n,F}\|_{0, F^-}
\le C \sum_{K \in \{K_{F,1}, K_{F,2}\}}
\left(
\| \alpha \nabla u - \tilde \alpha \nabla u_{\mathcal T} \|_{0,\tilde K}
+ h_F^- \|f - f_K\|_{0,\tilde K} \right).
\]
Adding proper weights gives
\begin{equation}\label{effi:flux-jump-2}
\sqrt{h_F^-}\|\tilde \alpha_F^{-1/2}j_{n,F}\|_{0,F^-}
\le C \sum_{K \in \{K_{F,1}, K_{F,2}\}} \!\!\!\!\left(
\| \alpha^{1/2} (\nabla u -\nabla u_{\mathcal T} )\|_{0,K}
+
\|\tilde \alpha^{1/2} \nabla u_{\mathcal T}\|_{0,S_K}
+ H_{f,K} \right)
\end{equation}
where
\[
H_{f,K} = h_K \| \alpha^{-1/2}(f -f_K)\|_{0,K}, \quad \forall \, K \in {\mathcal T}
\]
and the constant $C$ is independent of the mesh size and the location of the interface but may depend on $\rho$.
Similarly, one can prove that
\[
\|j_{t,F}\|_{0,F^-} \le C \sum_{\tilde K \in \{\tilde K_{F,1}, \tilde K_{F,2}\}} \displaystyle\frac{1}{\sqrt{h_F^-}} \| \nabla u-\nabla u_{\mathcal T}\|_{0,\tilde K}
\]
and, after adding proper weights, that
\begin{equation}\label{effi:tan-jump}
\sqrt{h_F^-} \|\tilde \alpha_F^{1/2}j_{t,F}\|_{0,F^-} \le C \sum_{ K \in \{ K_{F,1}, K_{F,2}\}}
\| \alpha^{1/2} (\nabla u-\nabla u_{\mathcal T})\|_{0,K}.
\end{equation}
Similarly, by defining auxiliary bubble functions on $F^+$, we also have the following local efficiency results on $F^+$:
\begin{equation}\label{effi:flux-jump-3}
\sqrt{h_F^+}\|\tilde \alpha_F^{-1/2}j_{n,F}\|_{0,F^+} \le C \sum_{K \in \{K_{F,1}, K_{F,2}\}} \left(
\| \alpha^{1/2} \nabla (u -u_{\mathcal T} )\|_{K}
+
\| \tilde \alpha^{1/2} \nabla u_{\mathcal T}\|_{S_K}
+ H_{f,K} \right)
\end{equation}
and
\begin{equation}\label{effi:tan-jump-2}
\sqrt{h_F^+} \|\tilde \alpha_F^{1/2}j_{t,F}\|_{0,F^+} \le C \sum_{K \in \{K_{F,1}, K_{F,2}\}}
\| \alpha^{1/2} (\nabla u-\nabla u_{\mathcal T})\|_{0,K}.
\end{equation}
For the first case we assume that for each $F \in {{{\mathcal E}}^{int}}$, there exist positive constants $\lambda$ and
$\Lambda$ such that
\begin{equation}\label{localtion-assump}
\lambda \le \displaystyle\frac{h_{F}^+}{h_{F}^-} \le \Lambda.
\end{equation}
\begin{lemma}\label{lem:efficiency1}
Let $u$ and $u_{\mathcal T}$ be the solution in \eqref{continuous-weak-solution} and \eqref{IFE approx}, respectively.
Then there exists a constant $C_1$ that is independent of the mesh size but may depend on $\lambda, \Lambda,$ and
$ \rho$ such that
\begin{equation} \label{local-efficiency}
\eta_K \le C_1 \left( \| \alpha^{1/2} (\nabla u- \nabla_h u_{\mathcal T}) \|_{0,\o_K}
+ \sum_{K \subset \o_K} \| \tilde \alpha^{1/2} \nabla u_{\mathcal T}\|_{0,S_K}
+ \sum_{K \subset \o_K} H_{f,K}
\right),\; \forall \, K \in {\mathcal T}.
\end{equation}
\end{lemma}
\begin{proof}
In the case that K is not an interface element,
\eqref{local-efficiency} is a well known result (see, e.g., \cite{CaHeZh-mcom:17}).
In the case that $K$ is an interface element, by \eqref{effi:flux-jump-2} and \eqref{effi:flux-jump-3}, we have
\begin{eqnarray*}
h_F^{1/2} \|\tilde a_F^{-1/2}j_{n,F}\|_{0,F} &=&
h_F^{1/2} \left( \|\tilde \alpha_F^{-1/2}j_{n,F}\|_{0,F^+}^2 + \| \tilde \alpha_F^{-1/2}j_{n,F}\|_{0,F^-}^2\right)^{1/2} \\[2mm]
&\le&
C_1\sum_{K \in \{ K_{F,1} , K_{F,2} \}}
\left( \| \alpha^{1/2} \nabla e_{\mathcal T}\|_{0,K}+ \|\tilde \alpha^{1/2} \nabla u_{{\mathcal T}}\|_{0,K} + H_{f,K} \right),
\end{eqnarray*}
where $C_1$ is independent of the mesh size but might depend on $\rho$, $\lambda$, and $\Lambda$.
Similarly, by \eqref{effi:tan-jump} and \eqref{effi:tan-jump-2}, we have
\begin{equation}
h_F^{1/2} \|\tilde \alpha_F^{1/2}j_{t,F}\|_{0,F} \le C_1\sum_{K \in \{ K_{F,1} , K_{F,2} \}} \| \alpha^{1/2} \nabla e_{\mathcal T}\|_{0,K},
\end{equation}
where $C_1$ is independent of the mesh size but might depend on $\rho$, $\lambda$, and $\Lambda$.
This completes the proof of the lemma.
\end{proof}
In the above lemma the efficient constant $C_1$ blows up when the $\lambda$ and $\Lambda$ become extreme. However, we note that in the extreme circumstances the related basis functions for IFEM are very ``close" to classical FE nodal basis functions thanks to its consistency with standard FE basis functions (see \cite{LiLinLinRo:04} for more detail). The partially penalized IFEM solution then should be also ``close" to
the classical finite element solution on fitted meshes. In the following lemma we prove the efficiency in a different approach that yields a bounded coefficient constant when the $\lambda$ and $\Lambda$ are extreme.
\begin{lemma}\label{lem:efficiency2}
There exists a constant $C_2$ that is independent of the mesh size and the location of the interface such that
for each interface edge $F \in {\mathcal E}^{int}$ the following efficiency bound holds:
\begin{equation} \label{local-efficiency-a}
\begin{split}
&\|j_{n,F}\|_{0,F} + \|j_{t,F}\|_{0,F} \\
\le& C_2 \!\!\!\! \sum_{K \in \{K_{F,1}, K_{F,2}\}}
\!\!\!\!\left( \| \alpha^{1/2} (\nabla u- \nabla_h u_{\mathcal T}) \|_{0,K} \!+\!
\| \tilde \alpha^{1/2} \nabla u_{\mathcal T}\|_{0,S_K}
\!+\! H_{f,K} +h_F^{1/2}\min \left\{ \sqrt{h_{F}^-}, \sqrt{h_{F}^+} \right\}
\right).
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
Without loss of generality assume that $h_F^+ \ge h_F^-$. Let $K \in \{K_{F,1}, K_{F,2} \}$. Define
$\hat K \subset K$ be the triangle such that (i) $F^+$ is a complete edge of $\hat K$ and (ii) the vertex in $K$ that is opposite to $F$, denote by $z_{F,K}$, is also a vertex of $\hat K$. It is obvious that $\hat K$ is regular.
Also define $\hat u_K$ be a piecewise linear function such that $\hat u_K \equiv u_{\mathcal T}$ on $K^+$ (or $\hat K$), if $\hat K \subset K^+$ (or $K^+ \subset \hat K$) and that $\hat u_K|_F$ is linear. It is obvious that $\hat u_K$ uniquely exists.
Finally define $\hat u_{{\mathcal T}}$ such that $\hat u_{{\mathcal T}}|_K = \hat u_K$, $K \in \{K_{F,1}, K_{F,2}\}$.
By the triangle inequality, we have
\begin{equation} \label{part0}
\begin{split}
\| \jump{\tilde \alpha^{1/2} D_n u_{\mathcal T}}\|_{0,F} &\le
\| \jump{\tilde\alpha^{1/2} D_n \hat u_{\mathcal T}}\|_{0,F} + \| \jump{\tilde \alpha^{1/2} D_n (u_{\mathcal T} - \hat u_{\mathcal T})}\|_{0,F}\\
& \le
\| \jump{\tilde\alpha^{1/2} D_n \hat u_{\mathcal T}}\|_{0,F} +
\sum_{K \in \{K_{F,1}, K_{F,2}\}}\|\tilde \alpha^{1/2} \nabla (u_{\mathcal T} - \hat u_{\mathcal T})|_K\|_{0,F}.
\end{split}
\end{equation}
Then by the definition of $\hat u_{\mathcal T}$ and a direct computation, we have for each $K \in \{K_{F,1}, K_{F,2}\}$:
\begin{equation}
\begin{split}
&\|\tilde \alpha^{1/2} \nabla (u_{\mathcal T} - \hat u_{\mathcal T})|_K\|_{0,F}
\lesssim \displaystyle\frac{\sqrt{h_{F}^-}}{ \rho_{K^-}} |(u_{\mathcal T} - \hat u_{\mathcal T})(z_F^-)|,
\end{split}
\end{equation}
where $\rho_{K^-}$ denotes that radius of the ball inscribed in $K^-$. This, combining
with the fact that $|(u_{\mathcal T} - \hat u_{\mathcal T})(z_F^-)| \lesssim \rho_{K^-}$ (See \cite{2003LiLinWu,LiLinLinRo:04} for the consistence of FE and IFE functions when one piece of the interface element is small.) yields
\begin{equation}\label{part1}
\begin{split}
&\|\alpha^{1/2} \nabla (u_{\mathcal T} - \hat u_{\mathcal T})|_K\|_{0,F}
\lesssim \sqrt{h_{F}^-}.
\end{split}
\end{equation}
Now applying the fact that $\jump{D_n \hat u_{\mathcal T}}|_F$ is a constant and the standard efficiency result on $\hat K_{F,1}$ and $\hat K_{F,2}$ gives
\begin{equation}\label{part2}
\begin{split}
&h_F^{1/2}\| \jump{\tilde\alpha^{1/2} D_n \hat u_{\mathcal T}}\|_{0,F} \lesssim
h_F^{1/2}\| \jump{\tilde\alpha^{1/2} D_n u_{\mathcal T}}\|_{0,F^+} \\
\lesssim &
\| \alpha^{1/2} \nabla u - \tilde \alpha^{1/2} \nabla u_{\mathcal T})\|_{\hat K_{F,1} \cup \hat K_{F,2}}
+ h_F \|f\|_{{\hat K_{F,1} \cup \hat K_{F,2}}}\\
\lesssim & \sum_{K \in \{K_{F,1}, K_{F,2}\}}\| \alpha^{1/2} \nabla (u - u_{\mathcal T})\|_{K} + \| \tilde \alpha^{1/2} \nabla u_{\mathcal T}\|_{0,S_K}
+ H_{f,K}.
\end{split}
\end{equation}
Combing \eqref{part0}--\eqref{part2} we have
\begin{equation}\label{part3}
\begin{split}
&h_F^{1/2}\| \jump{\tilde\alpha^{1/2} D_n u_{\mathcal T}}\|_{0,F} \\
\le & C_2 \sum_{K \in \{K_{F,1}, K_{F,2}\}} \left( \| \alpha^{1/2} \nabla (u - u_{\mathcal T})\|_{ K} + h_F^{1/2}\sqrt{h_{F}^-} +H_{f,K}
+\| \tilde \alpha^{1/2} \nabla u_{\mathcal T}\|_{0,S_K}\right)
\end{split}
\end{equation}
where the constant $C_2$ is independent of the mesh size and the location of the interface.
Similarly we can prove that
\begin{equation}\label{part4}
h_F^{1/2} \|\tilde \alpha_F^{1/2}j_{t,F}\|_{0,F} \le C_2\sum_{K \in \{ K_{F,1} , K_{F,2} \}}
\left( \| \alpha^{1/2} \nabla (u-u_{\mathcal T})\|_{0,K} + h_F^{1/2}\sqrt{h_{F}^-} \right),
\end{equation}
where the constant $C_2$ is also independent of the mesh size and the location of the interface.
Finally \eqref{local-efficiency-a} is a direct result of \eqref{part3} and \eqref{part4}.
\end{proof}
\begin{remark}
\eqref{part3} indicates that for the case when $\lambda$ and $\Lambda$ are extreme the term $h_F^{1/2}\sqrt{h_{F}^-}$ becomes negligible, and, therefore, $C_2$ can be used as an effective efficiency constant.
\end{remark}
\begin{theorem}
Let $u$ and $u_{\mathcal T}$ be the solution in \eqref{continuous-weak-solution} and \eqref{IFE approx}, respectively.
The following efficiency bound holds for any $K \in {\mathcal T}^{int}$:
\begin{equation}\label{part5}
\begin{split}
&\eta_K \le \min\{ C_1, C_2\}
\left( \| \alpha^{1/2} \nabla (u - u_{\mathcal T})\|_{0,\omega_K}
+\delta_K\right).
\end{split}
\end{equation}
\end{theorem}
where
\[
\delta_K = \sum_{K \subset \o_K} \| \tilde \alpha^{1/2} \nabla u_{\mathcal T}\|_{0,S_K}
+ \chi(C_2\le C_1) h_F^{1/2}\sqrt{h_{F}^-} +\sum_{K \subset \o_K} H_{f,K},
\]
$\chi$ is the characteristic function and
$C_1$ and $C_2$ are the efficiency constants in Lemma \ref{lem:efficiency1} and Lemma \ref{lem:efficiency2}, respectively.
\section{Numerical Results}\label{sec:6}
In this section, we report some numerical results to demonstrate the performance of the residual-based error estimator for partially penalized IFEM.
For the first three examples, we consider a diffusion interface problem with a smooth elliptical interface curve which has been reported in
\cite{LinLinZhang:15, LinYangZhang:15:2}. Let $\Omega=[-1,1]^2$, and the interface $\Gamma$ is an ellipse centered at $(x_0,y_0) = (0,0)$ with horizontal semi-axis $a= \frac{\pi}{6.28}$ and the vertical semi-axis $b = \frac{3}{2}a$.
The interface separates $\Omega$ into two sub-domains, denoted by $\Omega^-$ and $\Omega^+$ such that
\begin{equation*}
\Omega^- = \{(x,y): r(x,y) < 1\} ~~~\text{and}~~~
\Omega^+ = \{(x,y): r(x,y) > 1\},
\end{equation*}
where
\begin{equation*}
r(x,y) = \sqrt{\frac{(x-x_0)^2}{a^2} + \frac{(y-y_0)^2}{b^2}}.
\end{equation*}
The exact solution to this interface problem is
\begin{equation}\label{eq: true solution ellipse}
u(x,y) =
\left\{
\begin{array}{ll}
\frac{1}{\beta^-}r^{p}, & \text{if~} (x,y) \in\Omega^-, \\
\frac{1}{\beta^+}r^{p} + \frac{1}{\beta^-} - \frac{1}{\beta^+}, &\text{if~} (x,y)\in\Omega^+.
\end{array}
\right.
\end{equation}
Here $\beta^\pm>0$ are the diffusion coefficients, and $p>0$ is the regularity parameter. In the following, we use $\rho=\frac{\beta^+}{\beta^-}$ to denote
the ratio of the coefficient jump.
Our adaptive mesh refinement
follows the standard procedure:
\begin{equation*}
\textbf{Solve} \longrightarrow ~ \textbf{Estimate} \longrightarrow~ \textbf{Mark} \longrightarrow~ \textbf{Refine}.
\end{equation*}
We solve the interface problem using IFEM in \eqref{IFE approx}, then we compute the residual-based error indicator $\eta_K$ on each element by \eqref{eta-K}. We adopt the equilibration marking strategy, i.e., construct a minimal subset $\hat{\mathcal{T}}$ of $\mathcal{T}$ such that
\begin{equation}
\sum_{K\in\hat{\mathcal{T}}} \eta_K^2 \geq \theta^2\eta^2
\end{equation}
where the threshold $\theta=0.5$. Finally we refine the marked triangles by newest vertex bisection \cite{Maubach:95}.
The initial mesh is formed by first partitioning the domain into a $4\times 4$ congruent rectangles, then cutting each rectangle into two congruent triangles by connecting its diagonal with positive slope.
\subsection*{Example 6.1 (Piecewise smooth solution with moderate jump)}
\label{ex1}
In this example, we let $\rho = 100$ and $p=5$ which represents a moderate coefficient contrast and piecewise smooth solution.
In Figure \ref{fig: mesh small}, we list, from left to right, some typical meshes of similar number of elements and degrees of freedom (DOF) generated by the uniform IFEM, the adaptive IFEM, and the adaptive FEM on unfitted mesh\cite{ChXiZh:09}. We observe that there is not much local mesh refinement around the interface for the adaptive IFEM in the middle of Figure \ref{fig: mesh small}, comparing with the uniform mesh in the left plot. This indicates that the errors of the partially penalized IFEM on uniform mesh are almost equally distributed, and IFEM itself can resolve the interface accurately for moderate coefficient jump. Whereas using the finite element method on unfitted meshes requires much more local mesh refinement around the interface (see the plot on the right of Figure \ref{fig: mesh small}) to resolve the non-smooth feature of interface problems.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.32\textwidth]{IFEmeshUnif.pdf}
\includegraphics[width=0.32\textwidth]{IFEmeshSmall.pdf}
\includegraphics[width=0.32\textwidth]{chenIFEmesh.pdf}
\end{center}
\caption{Uniform mesh for IFEM (left), adaptive mesh for IFEM (center), adaptive mesh for FEM {\em\cite{ChXiZh:09}} (right) for Example 6.1}
\label{fig: mesh small}
\end{figure}
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.6\textwidth]{CvgSmall.pdf}
\end{center}
\caption{Convergence of uniform IFEM, adaptive IFEM, and adaptive FEM for Example 6.1}
\label{fig: cvg small}
\end{figure}
In Figure \ref{fig: cvg small}, we report the convergence of these three methods
and the residual-based error estimator for the adaptive IFEM.
The slopes of the log(DOF)-log$(\| \alpha^{1/2}(\nabla u-\nabla_h u_{\mathcal T})\|_{0,\Omega} )$
and the log(DOF)-log($\eta$) for the adaptive IFEM are both very close to $-1/2$,
which indicates the optimal-order decay of errors with respect to the number of unknowns and, hence, the efficiency of our local error indicators. We use the following efficiency index,
\[
\mbox{eff-index} = \displaystyle\frac{{\eta}}{\| \alpha^{1/2}(\nabla u-\nabla_h u_{\mathcal T})\|_{0,\Omega}}
\]
to test the efficiency of our residual-based error estimator. The eff-index is very stable at every mesh level and the value is around 3. We note that the errors of uniform IFEM are very close to the errors of adaptive IFEM. This again indicates that the IFEM itself can resolve the interface accurately for moderate coefficient jumps and piecewise smooth solutions. However, using the standard FEM, the magnitudes of errors are much larger than those of IFEM, with similar degrees of freedom.
\subsection*{Example 6.2 (Piecewise smooth solution with large jump)}
\label{ex2}
In this example, we test the large jump case by choosing $\rho=10^6$. In this case, the true solution possesses a very steep gradient at the interface.
The left plot of Figure \ref{fig: IFEM large} shows a typical mesh for the adaptive IFEM, which, compared with Figure \ref{fig: mesh small}, has much denser refinement around the interface. In the right plot of Figure \ref{fig: IFEM large}, we observe the optimal-rate decay of the errors and the estimators.
Nevertheless, even if the convergence rate is optimal for the uniform IFEM, the magnitudes of errors are significantly larger than the errors of adaptive IFEM.
Hence, applying adaptive mesh refinement is computationally more efficient for interface problems with large coefficient jump even for IFEM.
The efficiency indices are between 2.5 - 3.5 on all meshes except the first few coarse ones, and they become more stable (close to 3) as the computations reach the asymptotical convergence region. This phenomenon indicates the robustness of the error estimation with respect to the ratio of coefficient jump.
Furthermore, the numerical solutions and the error surfaces on the uniform and adaptive meshes with similar DOFs are depicted in Figure \ref{fig: sol large} and Figure \ref{fig: error large}, respectively. We can observe that the error is significantly diminished for the adaptive solution.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.48\textwidth]{IFEmeshLarge.pdf}
\includegraphics[width=0.5\textwidth]{CvgLarge.pdf}
\end{center}
\caption{Mesh generated by the adaptive IFEM (left) and the convergence of
adaptive and uniform IFEM (right) for Example 6.2}
\label{fig: IFEM large}
\end{figure}
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.5\textwidth]{IFEMsolUnifLarge.pdf}~
\includegraphics[width=0.5\textwidth]{IFEMsolAdptLarge.pdf}
\end{center}
\caption{Numerical solutions of uniform (left) and adaptive (right) IFEM with similar DOFs for Example 6.2}
\label{fig: sol large}
\end{figure}
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.5\textwidth]{IFEMerrUnifLarge.pdf}~
\includegraphics[width=0.5\textwidth]{IFEMerrAdptLarge.pdf}
\end{center}
\caption{Point-wise errors of uniform (left) and adaptive (right) IFEM with similar DOFs for Example 6.2}
\label{fig: error large}
\end{figure}
\subsection*{Example 6.3 (Solution with singularity and large coefficient jump)}
\label{ex3}
In this example, we consider the interface problem with large coefficient jump and a solution singularity. We choose $\rho=10^6$ and $p=0.5$ in the exact solution Figure \ref{eq: true solution ellipse}. Note that the solution is merely in $H^{1.5-\epsilon}(\Omega)$ for any $\epsilon>0$ and the solution becomes singular at the origin.
The left plot of Figure \ref{fig: singularity} shows a typical mesh of the adaptive IFEM, and
we observe that the mesh is densely refined around the interface as well as the point of singularity. The right plot of Figure \ref{fig: singularity} shows the optimal-rate decay of the errors and the estimators of our adaptive IFEM.
Again, the averaging effectivity index is close to 3, which is similar to that in previous examples. This indicates the uniform effectivity of the error estimate with respect to the type of elements, i.e., interface and non-interface elements. The comparison of error in the right plot of Figure \ref{fig: singularity} shows a stronger superiority of the adaptive mesh refinement than the uniform mesh refinement. In fact, the convergence of IFEM with uniform mesh refinement is not optimal, due to the singularity of the solution.
The numerical solutions and error surfaces for the adaptive IFEM and uniform IFEM are depicted in Figure \ref{fig: sol singularity} and Figure \ref{fig: error singularity}, respectively. It is easy to see that the numerical solution on uniform mesh cannot resolve the behavior of exact solution accurately at the singularity point, and the error of the uniform solution possesses a very high peak around the singular point, while we
can barely observe this phenomenon from the adaptive solution.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.48\textwidth]{adpIFEmeshSingul.pdf}
\includegraphics[width=0.5\textwidth]{ifemAdaptCvgSingul.pdf}
\end{center}
\caption{Mesh generated by the adaptive IFEM (left) and the convergence of
adaptive and uniform IFEM (right) for Example 6.3}
\label{fig: singularity}
\end{figure}
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.49\textwidth]{IFEMsolUnifSingul.pdf}
\includegraphics[width=0.49\textwidth]{IFEMsolAdptSingul.pdf}
\end{center}
\caption{Numerical solutions of uniform (left) and adaptive (right) IFEM with similar DOFs for Example 6.3}
\label{fig: sol singularity}
\end{figure}
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.49\textwidth]{IFEMerrUnifSingul.pdf}
\includegraphics[width=0.49\textwidth]{IFEMerrAdptSingul.pdf}
\end{center}
\caption{Point-wise errors of uniform (left) and adaptive (right) IFEM with similar DOFs for Example 6.3}
\label{fig: error singularity}
\end{figure}
\subsection*{Example 6.4 (Solution with complicated interface shape)}
\label{ex4}
In this example, we consider an interface problem with a
more complicated interfacial shape. The exact solution has a petal-shaped interface and it is defined through the following level set function:
\begin{equation}\label{eq: true solution 2}
u(x,y) =
\left\{
\begin{array}{ll}
\frac{1}{\beta^-}\phi(x,y), & \text{if~} \phi(x,y) < 0, \\
\frac{1}{\beta^+}\phi(x,y), &\text{if~} \phi(x,y) \geq 0,
\end{array}
\right.
\quad
\mbox{in } \O = [-1\,,1]^2
\end{equation}
where
\[
\phi(x,y) = (x^2+y^2)^2\left(1+0.5\sin\left(12\tan^{-1}\left(\frac{y}{x}\right)\right)\right)-0.3 .
\]
Due to the complexity of interface shape, we start the AMR procedure with a finer initial mesh, a $16\times 16$ Cartesian triangular mesh. A typical mesh is depicted on the left plot of Figure \ref{fig: petal}.
Comparing with Example 6.1, in which the interface is an ellipse (see Figure \ref{fig: mesh small}), the refinement around interface is denser. This is because the larger curvature of the interface causes the larger value of the inconsistency term in the error indicator in \eqref{eta-K}.
The convergence plot depicted in the right plot of Figure \ref{fig: petal} indicates the optimal-rate decay for both the errors and estimators. The errors of adaptive solution are a little smaller than the errors of the uniform solution with similar degrees of freedom, although the latter also converge in optimal rate. Moreover, the efficient index for this example is close to $3$ which is similar to those in the previous examples. Numerical solutions and error surfaces of uniform and adaptive IFEMs are reported in Figure \ref{fig: solution petal} and Figure \ref{fig: err petal}, respectively. We can again see that the errors of adaptive IFEM are smaller than the errors of uniform IFEM given similar degrees of freedom.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{adpIFEpetalMesh.pdf}
\includegraphics[width=0.5\textwidth]{ifemAdaptCvgPetal.pdf}
\end{center}
\caption{Mesh generated by the adaptive IFEM (left) and the convergence of
adaptive and uniform IFEM for Example 6.4}
\label{fig: petal}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{IFEsolPetal.pdf}~
\includegraphics[width=0.48\textwidth]{adpIFEsolPetal.pdf}
\end{center}
\caption{Numerical solutions of uniform (left) and adaptive (right) IFEM with similar DOFs for Example 6.4}
\label{fig: solution petal}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.48\textwidth]{IFEerrPetal.pdf}~
\includegraphics[width=0.48\textwidth]{adpIFEerrPetal.pdf}
\end{center}
\caption{Point-wise errors of uniform (left) and adaptive (right) IFEM for Example \ref{ex4} with similar DOFs for Example 6.4}
\label{fig: err petal}
\end{figure}
\subsection*{Example 6.5 (Effect of the inconsistent error terms)}
\label{ex5}
In this example, we will explore the effect of inconsistent error term for different interface geometries by revisiting the Example 6.1 and Example 6.4 which has a simple ellipse interface and a more complicated petal interface. We consider the following error indicator
\begin{equation} \label{xi-K}
\begin{split}
\xi_K^2&=
\sum \limits_{F \in {\mathcal E}_K \cap {{{\mathcal E}}^{int}}} \left(\displaystyle\frac{h_F}{2} \|\tilde\alpha_F^{-1/2} j_{n,F}\|_{0,F}^2
+\displaystyle\frac{h_F}{2} \| \tilde\alpha_F^{1/2} j_{t,F}\|_{0,F}^2\right)
\\
&+
\sum \limits_{F \in {\mathcal E}_K \cap {\mathcal E}_I \setminus {{{\mathcal E}}^{int}} }
\displaystyle\frac{h_F}{2} \| \tilde\alpha_F^{-1/2}j_{n,F}\|_{0,F}^2
+
\sum \limits_{F \in {\mathcal E}_K \cap {\mathcal E}_N} h_F \| \tilde\alpha_F^{-1/2}j_{n,F}\|_{0,F}^2
\end{split}
\end{equation}
which is same as $\eta_K^2$ in \eqref{eta-K} but without the inconsistent error term $\|\tilde \alpha^{1/2} \nabla u_{\mathcal T} \|_{S_K}^2$. The global error estimator is defined in the standard way:
\[
\xi=\left( \sum_{K \in {\mathcal T}} \xi_K^2\right)^{1/2}.
\]
First we compare the convergences using the error estimator $\xi$ and $\eta$ for Example 6.1. In the left plot of Figure \ref{fig: effect of Sk}, the estimators $\xi$ and $\eta$ are very similar at all meshes, and the errors of the corresponding IFEM solutions guided by these two estimators are also close. We believe that due to the simple and smooth interface shape of this example (an ellipse interface), the inconsistent error term is negligible. Next, we use the new error estimator $\xi$ in Example 6.4 In which the geometry of interface is more complicated. As we can see in the right plot of Figure \ref{fig: effect of Sk}, the error estimators $\eta$ and $\xi$ and the corresponding IFEM solutions show notable differences, especially on the first few coarse meshes. In this sense, including the geometrical correction (inconsistent error) term leads to better error indication particularly on coarse meshes. We also note that as the mesh is adaptively refined, the error estimators $\eta$ and $\xi$ become closer, as well as the IFEM solutions leading by these two estimators.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.49\textwidth]{ex1EtaXi.pdf}
\includegraphics[width=0.49\textwidth]{ex4EtaXi.pdf}
\end{center}
\caption{Comparison of estimators $\eta$ and $\xi$ for Example 1 (left) and Example 4 (right)}
\label{fig: effect of Sk}
\end{figure}
\subsection*{Example 6.6 (Additional Comments on Large Jump Scenarios)}
\label{ex6}
In this test, we revisit the large jump scenario using test problem from the Example 6.2. To show that necessity of performing adaptive mesh strategy for IFEM, we use the true error in energy norm as our error indicator, i.e.,
\[\eta^*_K = \|\alpha^{1/2}(\nabla u-\nabla_h u_{\mathcal T})\|_{0,K},\]
and the global error estimator is the true error in the energy norm, i.e.,
\[
\eta^*=\left( \sum_{K \in {\mathcal T}} \eta_K^{*2}\right)^{1/2} = \|\alpha^{1/2}(\nabla u-\nabla_h u_{\mathcal T})\|_{0,\O}.
\]
An adaptive mesh using the error estimator $\eta^*$ with similar number of triangles are shown in the left plot of Figure \ref{fig: estimator eta-star}. In both cases the refinement is concentrated around the interface, which is similar to the adaptive mesh in Figure \ref{fig: IFEM large}. This again shows that the partially penalized IFEM itself may not be sufficient to obtain accurate solution for extremely large jumps of the coefficient. In this case, the adaptive mesh refinement is more advantageous. The right plot of Figure \ref{fig: estimator eta-star} shows the convergence of the errors governed by the estimator $\eta$ and the true error $\eta^*$. We can see that both converge in an optimal rate, although using the true error as estimator gives slightly more accurate solutions.
\begin{figure}[ht!]
\begin{center}
\includegraphics[width=0.43\textwidth]{MeshLv25TrueError.pdf}
\includegraphics[width=0.55\textwidth]{LargeJumpCvgTrueError.pdf}
\end{center}
\caption{Mesh generated by the adaptive IFEM (left) using true error as the error indicators
and convergence of IFEM solutions guided by $\eta$ and by the true error $\eta^*$ for Example 6.2}
\label{fig: estimator eta-star}
\end{figure}
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{"url":"https:\/\/forum.allaboutcircuits.com\/threads\/reading-frequency-with-a-pic16f877.66101\/","text":"# Reading frequency with a PIC16F877?\n\n#### GuitarMan216\n\nJoined Oct 6, 2011\n18\nHey guys, I've decided to start a little guitar tuner project, using assembly with a PIC16F877. I have the circuit set up, so my guitar output is essentially a square wave that I can go into the pic with. My DMM is reading the right frequencies and voltage there, so I know that's good.\n\nMy issue is, I'm really green to this stuff, so I'm not sure how to start with the programming and input. Is there a way I could increment a timer every time the input is high, and compare say 20 input pulses with the external oscillator cycles in the time frame it took? Or is there an easier way? Any help would be appreciated. Thanks!\n\n#### ErnieM\n\nJoined Apr 24, 2011\n8,010\nThere are several timers in there and at least one of them has a setting to use an external pin as the clock: that means it is \"increment a timer every time the input is high\" (actually it is edge triggered). You can cleat the timer register, wait a specific interval, then read the timer register and there you have counts per your specific interval, or the inverse of frequency.\n\nIf you have nothing else happening just waiting a given number of statements can give you the specific interval.\n\n#### GuitarMan216\n\nJoined Oct 6, 2011\n18\nThere are several timers in there and at least one of them has a setting to use an external pin as the clock: that means it is \"increment a timer every time the input is high\" (actually it is edge triggered). You can cleat the timer register, wait a specific interval, then read the timer register and there you have counts per your specific interval, or the inverse of frequency.\n\nIf you have nothing else happening just waiting a given number of statements can give you the specific interval.\nThanks for the quick reply! I just found this forum the other day, and it seems like a goldmine.\n\nI'm just unsure how this code would go, would I want to create some sort of loop with some set value for the TMR?:\n\nbanksel TMR1\nmovlw 0xFF00\nmovwf TMR1\nbtfsc TMR1\ngoto whatever\n\nWould that then allow the timer to increment about 30 times until it's cleared, and then somehow read the interval?\n\nSorry, I know I'm being incredibly vague.\n\n#### Markd77\n\nJoined Sep 7, 2009\n2,806\nThere is a capture mode on this PIC, which will store the value of timer1, and optionally trigger an interrupt, every 16 rising edges (or 1,2,4,8) of the CCP pin.\ntimer1 doesen't get reset so you just subtract the previous value. Ignore the first result, it won't be accurate unless you put in extra effort.\nHave a look at section 8 of the datasheet for more info.\n\n#### GuitarMan216\n\nJoined Oct 6, 2011\n18\nThere is a capture mode on this PIC, which will store the value of timer1, and optionally trigger an interrupt, every 16 rising edges (or 1,2,4,8) of the CCP pin.\ntimer1 doesen't get reset so you just subtract the previous value. Ignore the first result, it won't be accurate unless you put in extra effort.\nHave a look at section 8 of the datasheet for more info.\nThanks man. I've been reading up in section 8 for the past hour or so. This stuff is just so confusing, that I'm having a hard time getting started.\n\n#### GuitarMan216\n\nJoined Oct 6, 2011\n18\nOk, so I've spent some time in Section 8, and am starting to get an idea. I switched over to C, so I'm a little more comfortable. I'm using Hi-Tech C.\n\nHere's my code so far, but I have no clue if I'm anywhere close. Any insight?\n\nThanks!\n\nRich (BB code):\n#include <pic16f877.h> \/\/Microcontroller header file\n\n#define E_S\t\t164.81\t\/\/Low E(2nd harmonic)\n#define A_S\t\t220.00\n#define D_S\t\t293.67\n#define G_S\t\t392.00\n#define B_S\t\t493.88\n#define EH_S\t659.26\t\/\/High E(2nd harmonic)\n\nvoid main()\n{\n\nunsigned int TIM1, TIM2, capold, capnew, diff, TMRdiff;\n\nTRISC=0b00000000; \/\/Setting PORTC pins to inputs\nTRISB=0b11111111; \/\/Setting PORTB pins to outputs\nT1CON=0b00001001; \/\/Initiating TMR1(Fosc\/4)\nCCP1CON=0b00000110; \/\/Capture every 4th rising edge\nTMR1IF=0;\n\nfloat FREQ;\n\nMain:\n\nif(CCP2IF) \/\/Timer1 Register Capture\n{\ncapold=CCPR1H;\nTIM1=TMR1;\nLoop:\n\nif(CCP2IF)\n{\ncapnew=CCPR2H;\nTIM2=TMR1;\ngoto StringDetect;\n}\ngoto Loop;\n}\n\ngoto Main;\n\nStringDetect:\n\ndiff=capnew-capold;\nTMRdiff=TIM2-TIM1;\n\nFREQ=(diff\/TMRdiff)*1000000*4; \/\/Multiply by 4, since it's finding every 4th trigger\n\nif(FREQ<((A_S+E_S)\/2))\n{\nif(FREQ>70.00)\n{\nif(FREQ<(E_S-1.0)){\nRB0=1;\nRB1=0;\nRB2=0;\n}\nif(FREQ>(E_S+1.0)){\nRB2=1;\nRB1=0;\nRB0=0;\n}\nif((E_S-1.0)<FREQ<(E_S+1.0)){\nRB1=1;\nRB2=0;\nRB0=0;\n}\n}\n}\nif(FREQ<((D_S+A_S)\/2))\n{\nif(FREQ>((A_S+E_S)\/2))\n{\nif(FREQ<(A_S-1.0)){\nRB0=1;\nRB1=0;\nRB2=0;\n}\nif(FREQ>(A_S+1.0)){\nRB2=1;\nRB1=0;\nRB0=0;\n}\nif((A_S-1.0)<FREQ<(A_S+1.0)){\nRB1=1;\nRB2=0;\nRB0=0;\n}\n}\n}\nif(FREQ<((G_S+D_S)\/2))\n{\nif(FREQ>((D_S+A_S)\/2))\n{\nif(FREQ<(D_S-1.0)){\nRB0=1;\nRB1=0;\nRB2=0;\n}\nif(FREQ>(D_S+1.0)){\nRB2=1;\nRB1=0;\nRB0=0;\n}\nif((D_S-1.0)<FREQ<(D_S+1.0)){\nRB1=1;\nRB2=0;\nRB0=0;\n}\n}\n}\nif(FREQ<((B_S+G_S)\/2))\n{\nif(FREQ>((G_S+D_S)\/2))\n{\nif(FREQ<(G_S-1.0)){\nRB0=1;\nRB1=0;\nRB2=0;\n}\nif(FREQ>(G_S+1.0)){\nRB2=1;\nRB1=0;\nRB0=0;\n}\nif((G_S-1.0)<FREQ<(G_S+1.0)){\nRB1=1;\nRB2=0;\nRB0=0;\n}\n}\n}\nif(FREQ<((E_S+B_S)\/2))\n{\nif(FREQ>((B_S+G_S)\/2))\n{\nif(FREQ<(B_S-1.0)){\nRB0=1;\nRB1=0;\nRB2=0;\n}\nif(FREQ>(B_S+1.0)){\nRB2=1;\nRB1=0;\nRB0=0;\n}\nif((B_S-1.0)<FREQ<(B_S+1.0)){\nRB1=1;\nRB2=0;\nRB0=0;\n}\n}\n}\nif(FREQ>((EH_S+B_S)\/2))\n{\nif(FREQ<1000.00)\n{\nif(FREQ<(EH_S-1.0)){\nRB0=1;\nRB1=0;\nRB2=0;\n}\nif(FREQ>(EH_S+1.0)){\nRB2=1;\nRB1=0;\nRB0=0;\n}\nif((EH_S-1.0)<FREQ<(EH_S+1.0)){\nRB1=1;\nRB2=0;\nRB0=0;\n}\n}\n}\n}\n\nLast edited by a moderator:\n\n#### Markd77\n\nJoined Sep 7, 2009\n2,806\nI don't know much about C, but the way I'd do it is:\nevery CCP1IF (don't confuse CCP1 and CCP2, they are seperate counters):\nstore CCPR1H and CCPR1L as a 16 bit C variable (capold),\nsubtract that from the current value of CCPR1H and CCPR1L (I think you can use CCPR1H*256+CCPR1L),\nmake sure that the case where TMR0 has changed from FFFF to 0000 during the period doesn't cause problems,\ncompare that to precalculated periods, eg for 220Hz use (1000000*4)\/220 = 18182\nThis saves clock cycles and program memory.\n\n#### GuitarMan216\n\nJoined Oct 6, 2011\n18\nI don't know much about C, but the way I'd do it is:\nevery CCP1IF (don't confuse CCP1 and CCP2, they are seperate counters):\nstore CCPR1H and CCPR1L as a 16 bit C variable (capold),\nsubtract that from the current value of CCPR1H and CCPR1L (I think you can use CCPR1H*256+CCPR1L),\ncompare that to precalculated periods, eg for 220Hz use (1000000*4)\/220 = 18182\nThis saves clock cycles and program memory.\nThanks again Mark! The CCPR1H*256+CCPR1L helps. I'll throw it in, do a little more research and see what pops up.\n\n#### ErnieM\n\nJoined Apr 24, 2011\n8,010\nHey GuitarMan: first off, where is your jungle band?\n\nI'm trying to follow your work here but I'm confused what you are doing. You said you have a nice square wave of your guitar note, what pin are you applying that to the PIC?\n\nWhy would that square wave be the 2nd harmonic as the comments in your code suggest? I would expect it to be the fundamental but I'm no player.\n\nAlso, using HiTech C they define the symbol \"CCPR2\" for your processor that gives you the whole 16 bit value of Timer 1 in one shot.\n\nI'm still trying to figure out the settings for Timer1 to use T1CKI an the external clock If that is how you are doing it then I don't check your T1CON setting (but don't have a new value for you).\n\n#### GuitarMan216\n\nJoined Oct 6, 2011\n18\nHey GuitarMan: first off, where is your jungle band?\n\nI'm trying to follow your work here but I'm confused what you are doing. You said you have a nice square wave of your guitar note, what pin are you applying that to the PIC?\n\nWhy would that square wave be the 2nd harmonic as the comments in your code suggest? I would expect it to be the fundamental but I'm no player.\n\nAlso, using HiTech C they define the symbol \"CCPR2\" for your processor that gives you the whole 16 bit value of Timer 1 in one shot.\n\nI'm still trying to figure out the settings for Timer1 to use T1CKI an the external clock If that is how you are doing it then I don't check your T1CON setting (but don't have a new value for you).\nHey Ernie, first off, thanks for the response.\n\nOk, I'm about as new to this as can be, so I don't even know what a jungle band is. Whoops lol.\n\nMy computer died, and I left the charger, so I don't have all the info right now, but I believe I was going into RC2.\n\nAs far as the 2nd harmonic, I tune by playing a 12 fret harmonic(One octave above), which would give that frequency, with a more pure and accurate signal. The fundamental if you play it open, is less \"stable\", so you have more harmonics. Not sure if I made sense there.\n\nI don't know what you mean by: \"Also, using HiTech C they define the symbol \"CCPR2\" for your processor that gives you the whole 16 bit value of Timer 1 in one shot.\"\n\nAgain, thanks so much for your help. I can picture you guys shaking your head staring at the computer monitor, but hey, I gotta start somewhere right?\n\n#### Markd77\n\nJoined Sep 7, 2009\n2,806\nAlso, using HiTech C they define the symbol \"CCPR2\" for your processor that gives you the whole 16 bit value of Timer 1 in one shot.\nThis would mean you can use CCPR1 instead of CCPR1H*256+CCPR1L, good call.\n\nI'm still trying to figure out the settings for Timer1 to use T1CKI an the external clock If that is how you are doing it then I don't check your T1CON setting (but don't have a new value for you).\nThis would be for the original way that GuitarMan suggested, and would be a valid method.\nI think the easier method is setting Timer1 on the internal clock and using capture mode.\n\n#### GuitarMan216\n\nJoined Oct 6, 2011\n18\nThis would mean you can use CCPR1 instead of CCPR1H*256+CCPR1L, good call.\n\nThis would be for the original way that GuitarMan suggested, and would be a valid method.\nI think the easier method is setting Timer1 on the internal clock and using capture mode.\nAh, so CCPR1 stores the capture info? And can I just say: NewCap=CCPR1?\n\nSo you're saying I don't even need the external oscillator? I'm not sure how the internal clock works...\n\n#### Markd77\n\nJoined Sep 7, 2009\n2,806\nSorry for confusing you, I've been less than precise with some of my descriptions.\nThe 16F877 needs an external oscillator, usually a crystal.\nWhat I've been calling the internal clock is the instruction cycle clock, Tcy, which is 1\/4 of the external oscillator.\nThe method I've been describing is to set Timer1 to the instruction cycle clock and use capture mode to make it store the Timer1 value every (n) times that a CCP pin toggles.\nThe other option would be to set Timer1 to counter mode (so your guitar input would become an external clock on the T1CKI pin and Timer1 would run at the frequency of the guitar) and have a timer based on the instruction cycle clock.\nBoth methods would work, but I think method 1 is simpler.\nI hope I've cleared it up a bit.\n\n#### GuitarMan216\n\nJoined Oct 6, 2011\n18\nOk guys, I have the code \"almost\" working. the middle LED (RB1), is constantly on though, and the side LED's just flicker when it's out of tune. It seems to be reading the frequency right, but the output is messed up. Any suggestions?\n\nRich (BB code):\n#include <htc.h> \/\/Microcontroller header file\n\n#define E_S\t\t164.81\t\/\/Low E(2nd harmonic)\n#define A_S\t\t220.00\n#define D_S\t\t293.67\n#define G_S\t\t392.00\n#define B_S\t\t493.88\n#define EH_S\t659.26\t\/\/High E(2nd harmonic)\n#define OSC 4000000L\n\nvoid main()\n{\n\nunsigned int capold, capnew, cap, FREQ;\n\nINTCON = 0b11000000;\nPIE1 = 0b00100000;\nPIR1 = 0b00000000;\nTRISC=0b00000100; \/\/Setting RC2 to input\nTRISB=0b00000000; \/\/Setting PORTB pins to outputs\nT1CON=0b00110001; \/\/Initiating TMR1(Fosc\/4)(1:8 prescaler)\n\nTMR1L = 0b00000000;\t\t\/\/ Set Timer1 register to zero\nTMR1H = 0b00000000;\nCCPR1L = 0b00000000;\t\t\/\/ Set CCP register to 0\nCCPR1H = 0b00000000;\nCCP1CON=0b00000110; \/\/Capture every 4th rising edge\n\nTMR0IE = 0; \/\/ Disable interrupt on TMR0 overflow\nPEIE = 1; \/\/ Enable peripheral interrupts\nGIE = 1; \/\/ Global interrupt enable\nCCP1IF = 0;\t\t\/\/Clear interrupt flag\n\nMain:\n\n{\nif (CCP1IF) \/\/ A TMR1 register capture occurred\n{\ncapold = capnew;\ncapnew = 256*CCPR1H + CCPR1L;\n\nif (capnew > capold)\n{\ncap = capnew - capold;\nFREQ=(((OSC\/4)*4)\/(8*(cap)));\n\nif(FREQ<((A_S+E_S)\/2))\n{\nif(FREQ>70.00)\n{\nif(FREQ<(E_S-1)){\nRB0=1;\nRB1=0;\nRB2=0;\n}\nif(FREQ>(E_S+1)){\nRB2=1;\nRB1=0;\nRB0=0;\n}\nif((E_S-1)<FREQ<(E_S+1)){\nRB1=1;\nRB2=0;\nRB0=0;\n}\n}\n}\n}\n\ngoto Main;\n}\n}\n}\n\n#### GuitarMan216\n\nJoined Oct 6, 2011\n18\nNevermind, I got it!!! Thanks so much for the help guys, it's working great! Now I might add a servo motor for automated tuning.\n\n#### djsfantasi\n\nJoined Apr 11, 2010\n5,839\nGlad you got it, but could you share what you found for us following the thread?\n\nThanks\n\n#### GuitarMan216\n\nJoined Oct 6, 2011\n18\nGlad you got it, but could you share what you found for us following the thread?\n\nThanks\nSure, after a lot of research and reading, as well as your guys suggestions, I messed with the code a bunch and ended up with the code below. It wasn't so much one thing that I found, but a bunch. I'm trying to make it even better, but as of now, it's working pretty well.\n\nRich (BB code):\n\/*********************************************\n\/\n\/PIC based guitar tuner\n\/\n\/Ryan Monteleone\n\/\n**********************************************\/\n\n#define PIC_CLK 4000000 \/\/External Oscillator speed\n\n#define NOTE\t1.059463094\t\/\/The 12th root of 2(The difference between each note)\n\n#define E_S\t\t164.81\t\/\/Low E Standard(2nd harmonic)\n#define A_S\t\t220.00\n#define D_S\t\t293.67\n#define G_S\t\t392.00\n#define B_S\t\t493.88\n#define EH_S\t659.26\t\/\/High E Standard(2nd harmonic)\n\n#define E_F\t\tE_S\/NOTE\t\/\/Low E Flat(2nd harmonic)\n#define A_F\t\tA_S\/NOTE\n#define D_F\t\tD_S\/NOTE\n#define G_F\t\tG_S\/NOTE\n#define B_F\t\tB_S\/NOTE\n#define EH_F\tEH_S\/NOTE\t\/\/High E Flat(2nd harmonic)\n\n#define OSC 4000000L \/\/Defining a long variable for the clock speed\n\nvoid main()\n{\n\nunsigned int capold, capnew, cap, FREQ;\t\/\/Creating unsigned variables to capture the frequency\n\nint x=0, y=0;\t\/\/Creating delay variables\n\nINTCON= 0b11000000;\t\/\/Enabling all masked interrupts(including peripherals)\nPIE1= 0b00100000;\t\/\/Enabling the USART receive interrupt\nPIR1= 0b00000000;\t\/\/\nTRISC= 0b00000110; \/\/Setting RC2 to input\nTRISB= 0b00000000; \/\/Setting PORTB pins to outputs\nTRISA= 0b00000000; \/\/Setting PORTA pins to outputs for 7-segment display\nT1CON= 0b00100001; \/\/Initiating TMR1(Fosc\/4)(1:8 prescaler)\n\nTMR1L= 0b00000000;\t\/\/ Set Timer1 register to zero\nTMR1H= 0b00000000; \/\/\nCCPR1L= 0b00000000;\t\/\/ Set CCP register to 0\nCCPR1H= 0b00000000; \/\/\nCCP1CON=0b00000110; \/\/Capture every 4th rising edge\n\nTMR0IE=0; \/\/ Disable interrupt on TMR0 overflow\nPEIE=1; \/\/ Enable peripheral interrupts\nGIE=1; \/\/ Global interrupt enable\nCCP1IF=0;\t\/\/Clear interrupt flag\n\nRB0=0;\t\/\/clearing pins\nRB1=0;\t\/\/\nRB2=0;\t\/\/\n\nRA0=0;\t\/\/Turning on all 7 LED's of the 7-segment display\nRA1=0;\t\/\/\nRA2=0;\t\/\/\nRA3=0;\t\/\/\nRA4=0;\t\/\/\nRA5=0;\t\/\/\n\nMain:\n\nTRISC= 0b00000110;\n{\nif (CCP1IF) \/\/TMR1 capture\n{\ncapold=capnew;\ncapnew=256*CCPR1H+CCPR1L; \/\/Storing capture value in capnew\n\nif(capnew>capold)\t\/\/Verifying both captures are independent\n{\ncap=capnew-capold;\nFREQ=(((OSC\/4)*4)\/(4*(cap)));\t\/\/Calculating the frequency based on timer, and 4 rising edges\nif(FREQ<((A_S+E_S)\/2))\n{\nif(FREQ>100.00)\t\/\/Filtering out any low frequency noise\n{\nif(FREQ<(E_S-3.5)){\t\/\/Setting the tolerance\nRB0=0;\nRB1=0;\nRB2=1;\n\nTRISA=0b00011000;\t\/\/7-segment display\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\t\/\/Start the capture again\n}\nif(FREQ>(E_S+3.5)){\nRB2=0;\nRB1=0;\nRB0=1;\n\nTRISA=0b00011000;\n\nfor(x;x<9999;x++){\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\nif((E_S-3.5)<FREQ<(E_S+3.5)){\nRB1=1;\nRB2=0;\nRB0=0;\n\nTRISA=0b00011000;\n\nfor(x;x<9999;x++){\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\n}\n}\nif(FREQ<((D_S+A_S)\/2))\n{\nif(FREQ>((A_S+E_S)\/2))\n{\nif(FREQ<(A_S-3.5)){\nRB0=0;\nRB1=0;\nRB2=1;\n\nTRISA=0b00000100;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\nif(FREQ>(A_S+3.5)){\nRB2=0;\nRB1=0;\nRB0=1;\n\nTRISA=0b00000100;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\nif((A_S-3.5)<FREQ<(A_S+3.5)){\nRB1=1;\nRB2=0;\nRB0=0;\n\nTRISA=0b00000100;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\n}\n}\nif(FREQ<((G_S+D_S)\/2))\n{\nif(FREQ>((D_S+A_S)\/2))\n{\nif(FREQ<(D_S-3.5)){\nRB0=0;\nRB1=0;\nRB2=1;\n\nTRISA=0b00100010;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\nif(FREQ>(D_S+3.5)){\nRB2=0;\nRB1=0;\nRB0=1;\n\nTRISA=0b00100010;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\nif((D_S-3.5)<FREQ<(D_S+3.5)){\nRB1=1;\nRB2=0;\nRB0=0;\n\nTRISA=0b00100010;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\n}\n}\nif(FREQ<((B_S+G_S)\/2))\n{\nif(FREQ>((G_S+D_S)\/2))\n{\nif(FREQ<(G_S-6.5)){\nRB0=0;\nRB1=0;\nRB2=1;\n\nTRISA=0b00010001;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\nif(FREQ>(G_S+6.5)){\nRB2=0;\nRB1=0;\nRB0=1;\n\nTRISA=0b00010001;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\nif((G_S-8.5)<FREQ<(G_S+6.5)){\nRB1=1;\nRB2=0;\nRB0=0;\n\nTRISA=0b00010001;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\n}\n}\nif(FREQ<((EH_S+B_S)\/2))\n{\nif(FREQ>((B_S+G_S)\/2))\n{\nif(FREQ<(B_S-12.5)){\nRB0=0;\nRB1=0;\nRB2=1;\n\nTRISA=0b00110000;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\nif(FREQ>(B_S+12.5)){\nRB2=0;\nRB1=0;\nRB0=1;\n\nTRISA=0b00110000;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\nif((B_S-12.5)<FREQ<(B_S+12.5)){\nRB1=1;\nRB2=0;\nRB0=0;\n\nTRISA=0b00110000;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\n}\n}\nif(FREQ>((EH_S+B_S)\/2))\n{\nif(FREQ<1000.00)\n{\nif(FREQ<(EH_S-14.5)){\nRB0=0;\nRB1=0;\nRB2=1;\n\nTRISA=0b00011000;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\nif(FREQ>(EH_S+14.5)){\nRB2=0;\nRB1=0;\nRB0=1;\n\nTRISA=0b00011000;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\nif((EH_S-14.5)<FREQ<(EH_S+14.5)){\nRB1=1;\nRB2=0;\nRB0=0;\n\nTRISA=0b00011000;\n\nfor(x;x<9999;x++){ \/\/creating a delay\nfor(y;y<6;y++){\n}\n}\ngoto Main;\n}\n}\n}\n}\n\ngoto Main;\n}\n}\n}","date":"2020-01-24 04:49:02","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, 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\section{Introduction}
The Brout-Englert-Higgs mechanism \cite{Higgs} is one of the cornerstones of the Standard Model of particle physics. Part of the Higgs mechanism involves a scalar field developing a non-zero vacuum expectation value rather than having a vacuum expectation value of zero. An example of this non-zero vacuum expectation value comes from $\Lambda \Phi ^4$ theory with a complex scalar field
whose Lagrangian density is
\begin{equation}
\label{higgs}
{\cal L} = \partial_\mu \Phi ^* \partial ^\mu \Phi - \frac{1}{2} m^2 | \Phi | ^2 - \frac{1}{4} \Lambda | \Phi |^4 ~.
\end{equation}
The equation of motion from \eqref{higgs} is
\begin{equation}
\label{higgs-em}
\partial _\mu \partial ^\mu \Phi - m^2 \Phi - \Lambda \Phi^3 = 0 ~.
\end{equation}
If we look for solutions, $\Phi$, which are space and time independent ({\it i.e.} $\partial _\mu \Phi =0$) {\it and} if $m^2 >0$ then the only solution is $\Phi =0$. However for a tachyonic mass term ({\it i.e.} $m^2 <0$) \eqref{higgs-em} has a non-zero, constant solution $\Phi_0 = \langle 0 |\sqrt{\Phi ^* \Phi} | 0 \rangle = \sqrt{\frac{-m^2}{\Lambda}}$. The vacuum solution is now given by $\Phi = \sqrt{\frac{-m^2}{\Lambda}} e^{i \theta}$ with magnitude $\sqrt{\frac{-m^2}{\Lambda}}$ and a phase $e^{i \theta}$ ($0 \le \theta \le 2 \pi$). Due to the phase of $e^{i \theta}$ there are an infinite number of equivalent vacua labeled by $\theta$. Usually one takes the arbitrary choice of
$\theta =0$ as the vacuum for $\Phi$. This non-zero vacuum expectation value of the scalar field is responsible for giving masses to the $W^\pm$ and $Z^0$ gauge bosons of the $SU(2) \times U(1)$ Standard Model, while leaving the photon massless.
Aside from the Standard Model, the Higgs mechanism has found application in the theory of superconductors via the Ginzburg-Landau model
\cite{GL}. In the Ginzburg-Landau model the source of the non-zero order parameter/scalar field vacuum expectation value is due to the interaction between the electrons and the phonons of the background lattice.
Another set of phenomena where a non-trivial vacuum is important are the Casimir effect \cite{casimir} and dynamical Casimir effect \cite{d-casimir}. In the canonical Casimir effect there are two, neutral, conducting plates which are placed a fixed distance apart. This restricts the type of quantum fluctuations that can occur between the plates as compared to outside the plates leading to an attractive force between the plates. In the dynamical Casimir effect the plates are moved with respect to one another and this results in the creation of photons out of the vacuum -- a result which has been observed relatively recently \cite{d-casimir1}.
Below we will show that a massless scalar field placed in a gravitational wave background leads to the scalar
field developing a non-zero vacuum expectation value. We make a comparison of this gravitationally induced effect with
the scalar field vacuum expectation value of spontaneous symmetry breaking as found in the Higgs mechanism and the Ginzburg-Landau model.
The comparison to the Ginzburg-Landau model is especially relevant since there the symmetry breaking is driven by the interactions induced by the phonons from the background lattice, whereas in the usual Higgs mechanism the symmetry breaking comes from the scalar field's self interaction. As in the Ginzburg-Landau model, here the scalar field's vacuum value is driven by interactions with the gravitational wave background. We also make a comparison between the present gravitationally induced vacuum expectation value and the dynamical Casimir. In the dynamical Casimir effect and the present case there is the possibility of producing {\it massless} particles from the vacuum. There are earlier works \cite{gibbons} \cite{garriga} which show that a plane gravitational wave background will not produce particles from the vacuum. We show how this is avoided exactly for the case of {\it massless} (scalar) fields.
Finally, we connect the results of the present paper with other recent works that propose there is a shift of the pre-existing Higgs vacuum expectation value of the Standard Model either via stationary gravitational fields \cite{onifrio1, onifrio2} or via a gravitational wave background \cite{kurkov}. There is also very recent work \cite{caldwell} which discusses the consequences of the interaction of a gravitational wave background with a time-dependent vacuum expectation value from a (non-Abelian) gauge field.
\section{Scalar field in gravitational wave background}
\subsection{Approximate gravitational wave background}
The equation for a complex scalar field, $\varphi$, in a general gravitational background is
\begin{equation}
\label{KGvacuum1}
\frac{1}{{\sqrt { - det \left[ {g_{\mu \nu } } \right]} }}\left( {\partial _\mu g^{\mu \nu } \sqrt { - det \left[ {g_{\mu \nu } } \right]} \partial _\nu } \right)\varphi = 0.
\end{equation}
\noindent Following \cite{Jones16} we take the gravitational wave to travel in the positive $z$ direction and have the $+$ polarization. For this situation the metric \cite{Schutz} can be written as,
\begin{equation}
\label{metric}
ds^2 = -dt^2 + dz^2 + f(t-z)^2 dx^2 + g(t-z)^2 dy^2 = du dv + f(u)^2 dx^2 + g(u)^2 dy^2,
\end{equation}
where in the last step we have switched to light front coordinates $u = z - t$ and $v = z+t$ with metric components $g_{uv} = g_{vu} = \frac{1}{2}$ and $g_{xx} = f(u) ^2$ and $g_{yy} = g(u)^2$. The metric functions $f(u)$ and $g(u)$ will be taken to be oscillatory functions of $u$. The determinant term in \eqref{KGvacuum1} is
$\sqrt{-det {\left[ {g_{\mu \nu } } \right]}} = \frac{fg}{2}$. Substituting the light front version of the metric into equation \eqref{KGvacuum1} gives
\begin{equation}
\left( {4 f^2 g^2 \partial _u \partial _v + 2 fg {\partial _u (fg)} \partial _v + g^2 \partial _x^2 + f^2 \partial _y^2 } \right)\varphi = 0.
\label{KGvacuum4}
\end{equation}
\noindent We take the metric ansatz functions of the form $f\left( u \right) = 1 + \varepsilon \left( {u} \right)$, and $g\left( u \right) = 1 - \varepsilon \left( {u} \right)$ and substitute these into equation \eqref{KGvacuum4}. This form for $f\left( u \right)$ and $g\left( u \right)$ describes a wave propagating in the $z$ direction so that $x$ and $y$ directions should
be physically identical. Thus we require of the solution that $\left( {\partial _y^2 - \partial _x^2 } \right)\varphi=0$. Using this equation \eqref{KGvacuum4} becomes,
\begin{widetext}
\begin{equation}
\left[ {4 \left( {1 - 2\varepsilon ^2 + \varepsilon ^4 } \right)\partial _u \partial _v - 4 \left( {1 - \varepsilon ^2 } \right)\varepsilon \left( {\partial _u \varepsilon } \right)\,\partial _v +
(1 + \varepsilon ^2 ) (\partial _x^2 + \partial _y^2 )} \right]\varphi = 0.
\label{KGvacuum7}
\end{equation}
\end{widetext}
\noindent Finally we consider a sinusoidal, plane gravitational wave by taking $\varepsilon \left( u \right) = h_ + e^{i Ku}$, where $h_{+}$ is a dimensionless amplitude and $K$ is a wave number. The metric must be real so it is understood that the metric components are obtained by taking the real part of the ansatz functions so that $f(u), g(u) =1 \pm h_+ e^{i Ku} \rightarrow 1 \pm h_+ \cos(K u)$. This real form still satisfies the linearized general relativistic field equations to which $f(u), g(u)$ are solutions. Substituting this choice of $\varepsilon (u)$ into equation \eqref{KGvacuum7} gives
\begin{equation}
\left( {4 F(u) \partial _u \partial _v - 4iKG(u)\,\partial _v + H(u) (\partial _x^2 + \partial _y^2) }\right)\varphi = 0,
\label{KGvacuum10}
\end{equation}
\noindent where $F\left( {u} \right) \equiv \left( {1 - 2h_ + ^2 e^{2iKu} + h_ + ^4 e^{4iKu} } \right)$,
$G\left( {u} \right) \equiv \left( {h_ + ^2 e^{2iKu} - h_ + ^4 e^{4iKu} } \right)$, and $H(u) = \left( 1 + h_+^2 e^{2iKu} \right)$ . We separate equation \eqref{KGvacuum10} using
$\varphi = X\left( x \right)Y\left( y \right) U \left( u \right) V \left( v \right)$. The eigenvalue equations and associated solutions
for $X(x)$ and $Y(y)$ are
\begin{equation}
\partial _x^2 X = - p^2 X \to X(x) = e^{ip x} ~~~~,~~~~
\partial _y^2 Y = - p^2 Y \to Y(y) = e^{ip y}.
\label{XYequations}
\end{equation}
The function $X(x)$ and $Y(y)$ are simply free waves as is to be expected since the gravitational wave is traveling
in the $u=z-t$ direction, and $p$ is the common momentum in the $x, y$ directions. The common momentum in the $x$ and $y$ directions comes from the assumed symmetry in these transverse directions, and it also realizes the condition $\left( {\partial _y^2 - \partial _x^2 } \right)\varphi=0$ which we took above. Using \eqref{XYequations} we find that \eqref{KGvacuum10} becomes
\begin{equation}
F(u) \frac{{\partial _u U}}{U}\frac{{\partial _v V}}{V} - iKG(u)\frac{{\partial _v V}}{V} - H(u) \frac{p^2}{2} = 0.
\label{ASequation}
\end{equation}
\noindent Since the light front coordinate $v$ is orthogonal to $u$ and since the gravitational wave only depends
on $u$ one expects that the eigenfunction $V(v)$ also is solved by a free, plane wave, as was the case for
$X(x)$ and $Y(y)$. This is the case and we find
\begin{equation}
- i\partial _v V = p_v V \to V(v) = e^{ip_v v} .
\label{eigenvalueV}
\end{equation}
\noindent Substituting equation \eqref{eigenvalueV} into equation \eqref{ASequation} and defining $\lambda \equiv \frac{p ^2}{2 p_v}$ yields
\begin{equation}
i\frac{{\partial _u U(u)}}{U(u)} = \lambda \frac{H(u)}{F(u)} - K\frac{G(u)}{F(u)} ~.
\label{eigenvalueU3}
\end{equation}
\noindent The term $i\frac{{\partial _u U(u)}}{U(u)}$ in \eqref{eigenvalueU3} represents the kinetic energy of the scalar field; the term $\lambda \frac{H(u)}{F(u)}$ represents
the interaction of the scalar field, via $\lambda$, with the gravitational background, via $\frac{H(u)}{F(u)}$; the term $K\frac{G(u)}{F(u)}$ represents a pure gravitational
potential term. Equation \eqref{eigenvalueU3} can be integrated to give,
\begin{equation}
U(u) = A e^{\frac{\lambda }{K}} e^{ \frac{- \lambda }{{K\left( {1 - h_ + ^2 e^{2iKu} } \right)}}}
\left( {1 - h_ + ^2 e^{2iKu} } \right)^{\frac{1}{2}\left( {\frac{\lambda }{K} - 1} \right)} e^{ - i\lambda u} ~,
\label{eigenvalueU4}
\end{equation}
\noindent where $A e^{\frac{\lambda }{K}}$ is constant. The factor $e^{\frac{\lambda }{K}}$ was chosen to ensure that the eigenfunction for the $u$ direction becomes a free plane wave, $e^{- i\lambda u}$, as $h_+ \to 0$ ({\it i.e.} as the gravitational wave is turned off). Collecting together all the solutions in $x, y, v$ and $u$ directions gives the solution of the scalar field in the gravitational background,
\begin{equation}
\varphi = A e^{\frac{\lambda }{K}} e^{ - \frac{\lambda }{{K\left( {1 - h_ + ^2 e^{2iKu} } \right)}}} \left( {1 - h_ + ^2 e^{2iKu} } \right)^{\frac{1}{2}\left( {\frac{\lambda }{K} - 1} \right)}
e^{ - i\lambda u} e^{ip_v v} e^{ip x} e^{ip y} + B .
\label{Sfield}
\end{equation}
\noindent $A$ is a normalization constant and we have added a constant $B$ which is allowed by the shift symmetry of solutions to \eqref{KGvacuum4}. Below we choose $B= -A$. This choice of $B$ ensures that if one turns off the gravitational background ($h_+ \to 0$) and also takes the field momenta to zero ($\lambda, p_v, p \to 0$) then $\varphi \to 0$. This solution for the scalar field given in \eqref{Sfield} is very similar to the form of the solution found in \cite{Padmanabhan99} for the {\it static} electric field evaluated in light front coordinates. Here we have a massless scalar field in a gravitational wave background.
In the limit $h_+ \to 0$ \eqref{Sfield} goes to the expected flat space-time solution of a free wave scalar field,
$\varphi _0 \propto e^{ - i\lambda u} e^{ip_v v} e^{ip x} e^{ip y} \to \varphi_0 \propto e^{i(p_v + \lambda)t} e^{i(p_v -\lambda) z}
e^{ip x} e^{ip y}$. The second version of $\varphi _0$ is the conversion from light front coordinates back to $t ,z$ coordinates. Defining
an energy $E_0 = p_v + \lambda$ and a momentum in the z-direction $p_z = p_v -\lambda$ one sees that the usual dispersion relationship
for a massless field ({\it i.e.} $E_0 ^2 = p_x^2 + p_y^2 + p_z^2 \to 2p^2 + p_z^2$ is recovered if one recalls that $\lambda = \frac{p^2}{2 p_v}$). Next, taking the limit of \eqref{Sfield} when all the transverse momenta of the scalar field go to zero ({\it i.e.} $p_x=p_y=p \to 0$ and $\lambda \rightarrow 0$) one finds
\begin{eqnarray}
\label{phi-higgs}
\varphi (u, v) &=& \mathop {\lim }\limits_{p_v \to 0} A \left( {1 - h_ + ^2 e^{2iKu} } \right)^{-\frac{1}{2}} e^{i p_v v} - A \nonumber \\
& & \to A \left[ \left( {1 - h_ + ^2 e^{2iKu} } \right)^{-\frac{1}{2}} - 1 \right] \approx \frac{A}{2} h_+ ^2 e^{2i Ku}+ \frac{3A}{8} h_+ ^4 e^{4i Ku}~.
\end{eqnarray}
\noindent In \eqref{phi-higgs} we have taken the limit $p_v \to 0$ after the limit of the transverse momenta going to zero
$p_x=p_y=p \rightarrow 0$. Using the vacuum solution in \eqref{phi-higgs} the magnitude square of $\varphi$ is
\begin{equation}
\varphi ^* \varphi \approx \frac{A^2 h_+ ^4}{4} + \frac{3 A^2 h_+^6}{8} \cos(2 K u ).
\label{phi-higgs-2}
\end{equation}
\noindent Equations \eqref{phi-higgs} and \eqref{phi-higgs-2} show that in the presence of a gravitational wave background, that in addition to the vacuum state solution $\varphi = 0$ ($\varphi=0$ is always a solution to \eqref{KGvacuum1}), there are also the vacuum solutions given by $\varphi (u,v)$ in \eqref{phi-higgs}. A common feature shared by the present example and the usual scalar field symmetry breaking is that the scalar field magnitude only depends on the parameters of the interaction -- $\varphi^* \varphi$ depends on $h_+$ and $K$ (the amplitude and wave number of the gravitational wave background) while for the canonical Higgs field example of \eqref{higgs} $\Phi^* \Phi$ depends on $m^2$ and $\Lambda$ (the parameters of the scalar fields self interaction).
One can compare the above results with the Casimir effect \cite{casimir} and the dynamical Casimir effect \cite{d-casimir}.
Both of the Casimir and dynamical Casimir effects involve a non-trivial vacuum state due to the restriction of the quantum fluctuations of fields placed between two neutral conducting, infinite plates. And in the dynamical Casimir effect the time dependent oscillation of the plates creates photons out of the vacuum. In the present case our interpretation of the scalar field solution in \eqref{phi-higgs} is that the time dependent oscillations of the gravitational field create scalar field quanta out of the vacuum.
In support of this interpretation, that the scalar field solution from \eqref{phi-higgs} represents the creation of scalar field quanta by the gravitational wave background, we look at the current in the $u$-direction associated with $\varphi$ from \eqref{phi-higgs} which to lowest order in $h_+$ is given by
\begin{equation}
\label{current-u}
j_u = -i (\varphi ^* \partial _u \varphi - \varphi \partial _u \varphi ^*) \approx A^2 h_+^4 K + \frac{9}{4} A^2 h_+ ^6 K \cos(2 K u) ~,
\end{equation}
\noindent If we take the lowest order of the current in \eqref{current-u} (or time averaging $j_u$ in \eqref{current-u}) we find a constant scalar field current in the $u$ direction of magnitude $A^2 h_+^4 K$. Our interpretation of this result is that the incoming gravitational wave current in the $u$ direction creates an outgoing scalar field current given, to leading order, by the first term in \eqref{current-u}. This picture is further supported by looking at the tree-level, Feynman diagram processes of $graviton + graviton \to photon + photon$, where in our case $photon$ is really the massless scalar field quanta. In reference \cite{skobelev} this tree level process was calculated and found to be non-zero in general. In particular it is non-zero when the incoming gravitons and outgoing photons travel in the same direction . In this work we are looking at collections of gravitons and scalar field quanta ({\it i.e.} gravitational plane waves and scalar field plane waves). Thus, if the process $graviton + graviton \to photon + photon$ is non-zero at the individual quanta level, as shown in \cite{skobelev}, then this implies it should be non-zero for a collection of these quanta {\it i.e.} gravitational and scalar field plane waves.
To conclude this subsection we recall that there are well known restrictions against the creation of field quanta by a plane gravitational wave background \cite{gibbons} \cite{garriga}, which is what we are proposing above. However in reference \cite{gibbons} a loop hole was given -- particle creation might occur if the scalar field were massless and if the momenta of the scalar field quanta were in the same direction as the gravitational plane wave. These are exactly the conditions we have here -- the scalar field is massless and is traveling in the same direction as the gravitational wave since from \eqref{phi-higgs} $\varphi$ only depends on $u=z-t$. In more detail it was shown in \cite{garriga} that the Bogoliubov $\beta$ coefficients, which indicate particle production, were proportonal to energy-momentum conserving delta functions
\begin{equation}
\label{b-beta}
\beta_{ij} = \langle u_i ^{out} | u_j ^{in~*} \rangle \propto \delta (k_{-} + l_{-}) ~,
\end{equation}
where $k_{-} = \frac{\omega - k_z}{2}$ and $l_{-} = \frac{\omega - l_z}{2}$ are the light front momenta of the scalar field before and after \footnote{In \cite{garriga} as well as in \cite{gibbons} a sandwich gravitational wave background was considered. The plane gravitational wave background of \eqref{metric} was sandwiched, before and after, by Minkowski space-time. The functions $u_i ^{in}$ and $u_j ^{out}$ are the solutions in the asymptotic Minkowski regions that are connected to each other through the intermediate plane wave gravitational background.},
$\omega =\sqrt{{\bf k}^2 + m^2}$ or $\omega =\sqrt{{\bf l}^2 + m^2}$ respectively, and the indices $i,j$ label the momenta of the outgoing and ingoing scalar field quanta. If $m \ne 0$ it is easy to see that $k_{-} + l_{-}$ can not vanish. If however, as is true in the case consider here, $m=0$ and ${\bf k, l} \to k_z, l_z$ ({\it i.e.} the before and after momemta of the scalar field is purely along the $+z$ direction) then $k_{-} + l_{-}$ vanishes and the Bogoliubov $\beta$ coefficient is non-zero indicating particle production.
\subsection{Exact gravitational wave background}
One might ask to what extent the linear approximation for the gravitational wave -- namely that $f(u) = 1 + \epsilon (u)$ and
$g(u) = 1- \epsilon (u)$ with $\epsilon (u) = h_+ e^{iKu}$ -- is crucial in obtaining the result in \eqref{phi-higgs}. What if one
took an exact, gravitational plane wave solution instead of a linearized approximation? To this end we now repeat briefly the above analysis for an exact solution for the plane wave metric in the $+$ polarization. The ansatz functions $f(u)$ and $g(u)$ will be exact, plane wave solutions if they satisfy the general relativistic field equations in this case which are of the form ${\ddot f}/ f + {\ddot g}/ g = 0$ \cite{Schutz}. One simple exact, plane wave, solution is $f(u) = e^{-iKu} e^{ - Ku}$ and $g(u) = e^{-iKu} e^{Ku} $. These ansatz functions have plane wave parts ($e^{-iKu}$) but they also have exponentially growing or decaying amplitudes ($e^{ \pm Ku}$). Near $u=0$ one has oscillating, wave solutions due to the $e^{-iKu}$ parts of the ansatz function, but as $u$ moves away from $u=0$ the $e^{\pm Ku}$ terms act like growing/decaying amplitudes. Because of this these solutions can only be of use for a restricted range of $u$ near $u=0$. Asymptotically, as $u\rightarrow \infty$, the functions $f(u), g(u)$ are not physically acceptable. In this case we are dealing with an exact solution to the non-linear general relativistic field equations so one may ask if taking the real part of the complex form of the ansatz functions will still be a solution to ${\ddot f}/ f + {\ddot g}/ g = 0$ due to the non-linearity of general relativity. One can show that taking the real part of the ansatz functions ({\it i.e.} $f(u) = e^{-iKu} e^{ - Ku} \to \cos(Ku) e^{- Ku}$ and $g(u) = e^{-iKu} e^{Ku} \to \cos(Ku) e^{Ku}$) is still a solution to ${\ddot f}/ f + {\ddot g}/ g = 0$. However, as in the previous linearized case, it is much easier to work with the complex form of the ansatz functions when one uses the background metric in the equation for the complex scalar field.
Using the above metric background we substitute $f(u) = e^{-iKu} e^{ - Ku}$ and $g(u) = e^{-iKu} e^{Ku}$ into equation \eqref{KGvacuum4},
\begin{widetext}
\begin{equation}
\left( {4 e^{-4iKu} \partial _u \partial _v + 2 e^{-2iKu} \partial _u \left( {e^{-2iKu} } \right)\partial _v + e^{-2iKu} e^{2Ku} \partial _x^2 + e^{-2iKu} e^{ - 2Ku} \partial _y^2 } \right)\varphi = 0,
\label{Exact1}
\end{equation}
\end{widetext}
\noindent and making the substitution $\varphi = U(u) V(v) X(x) Y(y) = U(u) e^{ip_v v} e^{ip x} e^{ip y}$, we find
\begin{equation}
\left( i \frac{{\partial _u U}}{U} + K - \lambda e^{ 2iKu} \cosh(2 K u) \right) = 0 ~,
\label{Exact2}
\end{equation}
\noindent where $\lambda =\frac{p^2}{2p_v}$ as before. Equation \eqref{Exact2} can be compared to \eqref{eigenvalueU3} in the sense that $i \frac{{\partial _u U}}{U}$ is the kinetic energy term of the scalar field, $\lambda e^{ 2iKu} \cosh(2 K u)$ is an interaction between the scalar field and the gravitational background, and $K$ is a pure gravitational potential term.
In the limit when the gravitational wave is absent ({\it i.e.} $K \to 0$) the solution to \eqref{Exact2} is again given by
$\varphi _0 \propto e^{ - i\lambda u} e^{ip_v v} e^{ip x} e^{ip y}$. Restoring the gravitational background ({\it i.e.} $K \ne 0$) the solution to \eqref{Exact2} is $U (u) = A e^{\left( {\frac{{\left( {-1 - i} \right)\lambda}}{{8K }} e^{ 2iKu} e^{2Ku} + \frac{{\left( {-1 + i} \right)\lambda}}{{8K}} e^{ 2iKu} e^{ - 2Ku} } \right)} e^{iKu}$, where $A$ is a constant similar to that found in \eqref{eigenvalueU4} and the scalar field takes the form
\begin{equation}
\varphi (u, v, x, y) = A e^{\left( {\frac{{\left( {-1 - i} \right)\lambda}}{{8K }} e^{ 2iKu} e^{2Ku} + \frac{{\left( {-1 + i} \right)\lambda}}{{8K}} e^{ 2iKu} e^{ - 2Ku} } \right)} e^{iKu}
e^{ip_v v} e^{ip x} e^{ip y} + B,
\label{Exact4}
\end{equation}
\noindent where $A$ is a normalization constant. We have again added a constant $B$ via the shift symmetry of solutions to \eqref{KGvacuum4}. As before we set $B=-A$ so that $\varphi (u,v,x,y) \to 0$ when the gravitational wave background is turned off and when the scalar field momenta go to zero. Here we do not have an $h_+$ since the ``amplitude" is given by the $e^{\pm Ku}$ terms in $f(u), g(u)$. As before if we take the limit of the massless scalar field to its vacuum state by taking its energy and momenta parameters to zero ({\it i.e.} taking the limit $p_x=p_y=p \to 0$ and $\lambda \to 0$) one finds that as before $\varphi$ and $\varphi \varphi^*$ do not go to zero but rather
\begin{equation}
\label{phi-exact}
\varphi = \mathop {\lim }\limits_{p_v \to 0} A e^{iKu} e^{i p_v v} = A (e^{iKu} -1) ~~~~;~~~~ \varphi ^* \varphi = 2 A^2 (1- \cos(K u))~.
\end{equation}
In \eqref{phi-exact} we have again taken the limit that $p_v$ becomes arbitrarily small ($p_v \rightarrow 0$). With this we see that
$\varphi$ depends on the gravitational wave background through the wave number, $K$. For this exact solution metric we again find that the scalar field acquires a non-zero, space-time dependent vacuum value even when one takes the limit of all the scalar field momenta going to zero. Also as in the previous subsection we find that $\varphi ^* \varphi$ has a constant term (the $2 A^2$ term in \eqref{phi-exact} which corresponds to the $\frac{A^2 h_+ ^4}{4}$ term in \eqref{phi-higgs-2}) and a space-time dependent part (the $-2 A^2 \cos(Ku)$ term in \eqref{phi-exact} which corresponds to the $\frac{3 A^2 h_+^6}{8} \cos(2 K u )$ term in \eqref{phi-higgs-2}). As for the plane wave solution of the previous subsection we can calculate the current in the $u$-direction and find
\begin{equation}
\label{current-u-2}
j_u = -i (\varphi ^* \partial _u \varphi - \varphi \partial _u \varphi ^*) = 2 A^2 K (1 - \cos(Ku)) ~.
\end{equation}
The current above is similar to the one found in the previous subsection in \eqref{current-u} -- there is a constant term, $2 A^2 K$, and a space-time dependent term $-2 A^2 K\cos(Ku)$. These can be compared to the terms $A^2 h_+^4 K$ and $\frac{9}{4} A^2 h_+ ^6 K \cos(2 K u)$ in \eqref{current-u}. If one time averages the current in \eqref{current-u-2} one finds $\langle j_u \rangle = 2 A^2 K$ {\it i.e.} one has a constant current in the $u$ direction. We take the same interpretation of $\langle j_u \rangle$ as the leading term of $j_u$ from \eqref{current-u} -- $\langle j_u \rangle$ represents a scalar field plane wave traveling in the $u$-direction produced by the initial gravitational wave.
\section{Discussion and Conclusions}
We have shown that a massless scalar field placed in a plane, gravitational wave background will develop a space-time dependent, non-zero vacuum value given by \eqref{phi-higgs} \eqref{phi-higgs-2} even in the limit when all the momentum parameters of the scalar field are taken to zero ({\it i.e.} $p_v, p , \lambda \rightarrow 0$). This is different from what happens to the massless scalar field solution in Minkowski space-time, where when one takes the zero energy-momentum limit the scalar field vanishes. We have drawn attention to the similarity of this gravitationally induced scalar field vacuum value with the vacuum expectation value in the Higgs phenomenon and with the dynamical Casimir effect.
Three potential physical consequences of this gravitationally induced vacuum value for the scalar field are: (i) the production of massless field quanta, such as photons, from the gravitational wave background; (ii) the usual Higgs vacuum expectation value of the Standard Model may be modified or even generated by stationary and/or time dependent gravitational backgrounds; (iii) the interplay between gravitational waves and scalar and gauge fields in the early Universe may lead to observational consequences at the present time.
Point (i) was investigated in references \cite{Jones16, Jones15}. Here we calculated the currents connected with the scalar field (see equations \eqref{current-u} and \eqref{current-u-2}) that the time averaged current, $\langle j_u \rangle$ was a constant which we interpreted as the incoming gravitational wave creating an outgoing scalar field. There is a well known restriction against the creation of fields from an incident, plane gravitational plane wave \cite{gibbons}. The present work avoids this conclusion by using the loop hole mentioned in \cite{gibbons} that the prohibition only applied to massive fields. Here we considered a massless field that travels in the same direction as the initial gravitational wave. Furthermore in \eqref{b-beta} we have shown that the the Bogoliubov coefficients indicating
particle production, as calculated in \cite{garriga} for a gravitational plane wave sandwich background, are non-zero exactly in the limit $m \to 0$ and the momenta of the field quanta being in the same direction as the gravitational wave ({\it i.e.} exactly the conditions of this work). The production of fields by an incoming gravitational plane wave background is also in agreement with the non-zero, tree-level amplitudes for $graviton + graviton \to photon + photon$ coming from Feynman diagram calculations \cite{Calmet16, skobelev, bohr}. Also since these processes occur at the tree-level Feynman diagrams they are in some sense classical effects.
Point (ii) was discussed in references \cite{onifrio1, onifrio2, kurkov}. In the works \cite{onifrio1, onifrio2} the idea was considered that the usual Higgs expectation value can be shifted by the effect of a static or stationary gravitational background ({\it e.g.} the Schwarzschild or Kerr space-times). This effect requires a coupling between the scalar field and the gravitational field of the form $\xi \phi R$ or $\xi \phi K$ where $\xi$ is the coupling and $R$ is the Ricci scalar and $K=R_{\alpha \beta \gamma \delta}
R^{\alpha \beta \gamma \delta}$ is the Kretcschmann scalar. This coupling of the scalar field to the gravitational field results
in a shifting of the pre-existing Higgs vacuum expectation value with an associated shift in particle masses. This shift is potentially observable. In the work \cite{kurkov} the idea of a scalar field-gravitational background coupling of the form $\xi \phi R$ is again considered but now the gravitational background is both space and time dependent. Again it is found that the gravitational background can shift the Higgs vacuum expectation value, but now this shift is space-time dependent as is the case for the results of the present work. Further it was found in \cite{kurkov} that even when there is no vacuum expectation value of the scalar field, one can be generated from the interaction with the gravitational background as is the case for our results.
Point (iii) was very recently proposed in reference \cite{caldwell} where the interplay of a gravitational wave background with a cosmological
non-Abelian gauge field was considered. The non-Abelian gauge field was assumed to have a {\it pre-existing} vacuum expectation value of the form $A_i ^a = \phi (\tau) \delta _i ^a$ where $\phi (\tau)$ is a scalar function of the proper time $\tau$, and the indices $i$ and $a$ were space and ``color" indices respectively. This interplay between the gravitational wave and the pre-existing, non-Abelian gauge field might lead to interesting and potentially observable phenomenon such as neutrino-like oscillations between the gravitational field and the non-Abelian gauge field. In contrast, in the present work, a space-time dependent scalar field, as given in \eqref{phi-higgs}, is generated out of the vacuum by the gravitational wave background. Similarly one might conjecture that the phenomenon proposed in \cite{caldwell} could work with the gravitational wave background generating a non-zero vacuum expectation value for the gauge field from the vacuum rather than requiring a pre-existing gauge field.
Finally we want to point out that, like the standard Higgs mechanism of particle physics, the present generation of the vacuum expectation value of the scalar field by the gravitational wave background is already implied at the classical level. In the usual Higgs mechanism, as given in \eqref{higgs}, the non-zero vacuum expectation value of $\Phi_0 = \sqrt{\frac{-m^2}{\Lambda}}$ is obtained from the classical, scalar field Lagrangian \eqref{higgs}. In a similar way the non-zero vacuum value for $\varphi$ from the gravitational wave background already emerges by examining the system of a classical scalar field interacting with a classical gravitational background as given in \eqref{phi-higgs} and \eqref{phi-exact}. Also the view that the leading term in the scalar field currents, $j_u$, from \eqref{current-u} and \eqref{current-u-2} represented production of the scalar field from the incoming gravitational wave, finds support from the non-zero, tree-level Feynman diagram process $graviton + graviton \to photon + photon$. This further indicates the classical nature of the scalar field vacuum expectation values found here, since tree-level Feynman diagrams represent the classical limit of a given interaction.
{\bf Acknowledgment}
DS is supported by grant $\Phi.0755$ in fundamental research in Natural Sciences by the Ministry of Education and Science of Kazakhstan.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,453 |
Discoverer of the Titanic
Founder and President, Ocean Exploration Trust
Director, Center of Ocean Exploration
Watery images of the once-glorious Titanic were some of the most enchanting and enthralling images of recent years, something most people only dream of seeing in real life. For Robert Ballard they represented a concrete accomplishment, and just one in a series of triumphs.
World-famous for his discovery of the Titanic and the Bismark, Ballard leads audiences on a journey to plumb some of history's greatest mysteries. Ballard's adventurous spirit is matched by a practical and hugely successful approach to goal setting, team building and execution. The skills that led to his headline-grabbing accomplishments are the same skills that now lead people from all walks of life toward a realization of their dreams.
A renowned scientist and explorer, he is the author of 16 books, including Discovery of the Titanic, and Collision with History: The Search for John F. Kennedy's PT 109. After many years of exploring ancient history in the Mediterranean and Black Seas with the discovery of the largest concentration of ancient Roman ships ever found under the sea, most recently Ballard has turned his sights on the Pacific ocean and his team and ship of exploration, E/V Nautilus are working off the west coast of the US.
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Dr. Ballard's program went very well and the 350 or so people who attended really enjoyed his program. Dr. Vora and I received many favorable comments about his program and as to how personable and gracious he was in talking with our guests and picture taking. I have to say that I enjoyed his program much more than last year's speaker and can say that everything went smooth and it was a pleasure working with you.
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The Personal Account of How the Titanic Was Found and a Look into the Future
Dr. Ballard shares his incredible story of the significant discovery of the Titanic, the advanced technology he used for this world famous expedition and what's the future holds for deep sea exploration. ...
Dr. Ballard shares his incredible story of the significant discovery of the Titanic, the advanced technology he used for this world famous expedition and what's the future holds for deep sea exploration.
Deep Sea Exploration
Dr. Ballard uses the emerging science of deep water archeology to share his findings from numerous expeditions where he searched for, located and document sites of historical significance. ...
Dr. Ballard uses the emerging science of deep water archeology to share his findings from numerous expeditions where he searched for, located and document sites of historical significance.
Dr. Ballard speaks about how he has turned his dreams into reality by taking risks and learning how to face failure and turn it into success. ...
Dr. Ballard speaks about how he has turned his dreams into reality by taking risks and learning how to face failure and turn it into success.
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Exploring the Titanic: How the Great Ship Ever Lost- Was Found
Best known for his 1985 discovery of the Titanic, Dr. Ballard has succeeded in tracking down numerous other significant shipwrecks, including the German battleship Bismarck, the lost fleet of Guadalcanal, the U.S. aircraft carrier Yorktown (sunk in the World War II Battle of Midway), and John F. Kennedys boat, PT-109.
While those discoveries have captured the imagination of the public, Dr. Ballard believes his most important discoveries were of hydrothermal vents and black smokers in the Galapagos Rift and East Pacific Rise in 1977 and 1979 along with their exotic life forms living off the energy of the Earth through a process now called chemosynthesis.
In addition to being a National Geographic Society Explorer-In-Residence and a commissioner on the U.S. Commission on Ocean Policy, Dr. Ballard is the president of the Ocean Exploration Trust (OET).
Ballard was born June 30, 1942, in Wichita, KS but moved to California at a very young age and grew up exploring the shore in San Diego. Dr. Ballard has a Ph.D. in marine geology and geophysics from the University of Rhode Island. He spent 30 years at Woods Hole Oceanographic Institution, where he helped develop telecommunications technology to create tele-presence for his education initiative, which allows hundreds of thousands of schoolchildren to accompany him from afar on undersea explorations around the globe each year. In 2001, he returned to the Graduate School of Oceanography at the University of Rhode Island where he is presently a tenured Professor of Oceanography and Director of the Center for Ocean Exploration.
Dr. Ballard has 21 honorary degrees and six military awards. He was also a Commander in the U.S. Naval Reserve, serving in the Navy from 1967 to 1997. He received the National Geographic Societys prestigious Hubbard Medal in 1996 for extraordinary accomplishments in coaxing secrets from the worlds oceans and engaging students in the wonder of science. In 2014, he was elected to the American Academy of Arts and Sciences. Dr. Ballard has published numerous books, scientific papers, and a dozen articles in National Geographic magazine. Dr. Ballard also has been featured in several National Geographic television programs, including the record-breaking Secrets of the TITANIC.
His discoveries also include sunken remains of ships along ancient trade routes in the Mediterranean Sea; two ancient Phoenician ships off Israel, the oldest shipwrecks ever found in deep water; and four 1,500- year-old wooden ships, one almost perfectly preserved in the Black Sea. Dr. Ballards Black Sea project seeks evidence of a great flood that may have struck the region thousands of years ago.
His 1997 best-selling book, Lost Liners, tells the story of the great transatlantic liners through memorable wrecks he has visited. Dr. Ballard was also a special advisor on Steven Spielbergs futuristic Sea Quest, DSV television show.
An explorer, discoverer and historian, Dr. Ballards fascinating journeys can teach us a great deal about our past, and they have encouraged others to take tremendous strides in the survey of the undiscovered mysteries of the deep sea.
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<p style="font-weight: normal;">Best known for his 1985 discovery of the <span>Titanic</span>, Dr. Ballard has succeeded in tracking down numerous other significant shipwrecks, including the German battleship <span>Bismarck, </span>the lost fleet of Guadalcanal, the U.S. aircraft carrier <span>Yorktown </span>(sunk in the World War II Battle of Midway), and John F. Kennedys boat, <span>PT-109</span>.</p> <p style="font-weight: normal;">While those discoveries have captured the imagination of the public, Dr. Ballard believes his most important discoveries were of hydrothermal vents and black smokers in the Galapagos Rift and East Pacific Rise in 1977 and 1979 along with their exotic life forms living off the energy of the Earth through a process now called chemosynthesis.</p> <p style="font-weight: normal;">In addition to being a National Geographic Society Explorer-In-Residence and a commissioner on the U.S. Commission on Ocean Policy, Dr. Ballard is the president of the Ocean Exploration Trust (OET).</p> <p style="font-weight: normal;">Ballard was born June 30, 1942, in Wichita, KS but moved to California at a very young age and grew up exploring the shore in San Diego. Dr. Ballard has a Ph.D. in marine geology and geophysics from the University of Rhode Island. He spent 30 years at Woods Hole Oceanographic Institution, where he helped develop telecommunications technology to create tele-presence for his education initiative, which allows hundreds of thousands of schoolchildren to accompany him from afar on undersea explorations around the globe each year. In 2001, he returned to the Graduate School of Oceanography at the University of Rhode Island where he is presently a tenured Professor of Oceanography and Director of the Center for Ocean Exploration.</p> <p style="font-weight: normal;">Dr. Ballard has 21 honorary degrees and six military awards. He was also a Commander in the U.S. Naval Reserve, serving in the Navy from 1967 to 1997. He received the National Geographic Societys prestigious Hubbard Medal in 1996 for extraordinary accomplishments in coaxing secrets from the worlds oceans and engaging students in the wonder of science. In 2014, he was elected to the American Academy of Arts and Sciences. Dr. Ballard has published numerous books, scientific papers, and a dozen articles in National Geographic magazine. Dr. Ballard also has been featured in several National Geographic television programs, including the record-breaking Secrets of the TITANIC.</p> <p style="font-weight: normal;">His discoveries also include sunken remains of ships along ancient trade routes in the Mediterranean Sea; two ancient Phoenician ships off Israel, the oldest shipwrecks ever found in deep water; and four 1,500- year-old wooden ships, one almost perfectly preserved in the Black Sea. Dr. Ballards Black Sea project seeks evidence of a great flood that may have struck the region thousands of years ago.</p> <p style="font-weight: normal;">His 1997 best-selling book, <span><em><strong>Lost Liners</strong></em>, </span>tells the story of the great transatlantic liners through memorable wrecks he has visited. Dr. Ballard was also a special advisor on Steven Spielbergs futuristic Sea Quest, DSV television show.</p> <p style="font-weight: normal;">An explorer, discoverer and historian, Dr. Ballards fascinating journeys can teach us a great deal about our past, and they have encouraged others to take tremendous strides in the survey of the undiscovered mysteries of the deep sea.</p> | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,341 |
The ban on Luis Suarez has been slammed as "absurd, out of proportion and unfair" by Liverpool keeper Pepe Reina.
The striker was hit with a massive 10 game ban on Wednesday following the biting incident with Branislav Ivanovic, and there are now genuine fears that the frontman may decide to move on and continue his career abroad.
Reina has hit out at the ban saying the length of it is wrong and that people making the decisions have got it in for Suarez.
"I consider myself a friend of Luis. He is being treated differently, I don't know if it's because he's Uruguayan or because he's had a previous episode like this," Reina told Spanish radio station Cadena Cope.
"He knows full well that what he did was wrong but a 10-game ban seems to me absurd, out of proportion and unfair.
"It seems that the people making the decisions have got it in for Luis a little bit. That's the way I see it. I am not justifying what he did but the punishment is very disproportionate.
"He knows he was in the wrong, he knows he has made a mistake but the treatment is completely out of place. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,712 |
package org.ovirt.engine.core.common.action;
import java.util.logging.Logger;
import org.ovirt.engine.core.compat.Guid;
import com.google.gwt.user.client.rpc.SerializationException;
import com.google.gwt.user.client.rpc.SerializationStreamReader;
import com.google.gwt.user.client.rpc.SerializationStreamWriter;
public class DetachStorageDomainFromPoolParameters_CustomFieldSerializer {
private static Logger logger = Logger.getLogger(DetachStorageDomainFromPoolParameters.class
.getName());
public static void deserialize(SerializationStreamReader streamReader,
DetachStorageDomainFromPoolParameters instance) throws SerializationException {
}
public static DetachStorageDomainFromPoolParameters instantiate(
SerializationStreamReader streamReader)
throws SerializationException {
logger.severe("Instantiating DetachStorageDomainFromPoolParameters via custom serializer.");
DetachStorageDomainFromPoolParameters instance =
new DetachStorageDomainFromPoolParameters((Guid) streamReader.readObject(),
(Guid) streamReader.readObject());
instance.setRemoveLast(streamReader.readBoolean());
instance.setDestroyingPool(streamReader.readBoolean());
return instance;
}
public static void serialize(SerializationStreamWriter streamWriter,
DetachStorageDomainFromPoolParameters instance) throws SerializationException {
logger.severe("Serializing DetachStorageDomainFromPoolParameters.");
streamWriter.writeObject(instance.getStorageDomainId());
streamWriter.writeObject(instance.getStoragePoolId());
streamWriter.writeBoolean(instance.getRemoveLast());
streamWriter.writeBoolean(instance.getDestroyingPool());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,599 |
\section{Introduction}
Athermal biopolymer fibrous networks, such as collagen and fibrin networks, play crucial roles in determining the mechanical stability and functionality of individual cells and tissues~\cite{kees_nature_2005,Janmey_soft_matter_2007,RevModPhys.86.995}. This class of disordered biological materials, in which thermal fluctuations and entropic effects are negligible, is characterized by two generic features; first, the resistance of the constituent fibers to elongation (stretching) and shortening (compression) is much larger than to bending, i.e.~these networks exhibit strong rigidity/stiffness scale separation. Second, the degree of connectivity of the fibers, i.e.~the average number of fibers per network's node, is small such that the network is sub-isostatic~\cite{robbie_nature_physics_2016}. That is, such materials are underconstrained, implying that in the absence of any bending resistance of the fibers, the network would possess \emph{floppy} (\emph{zero}) \emph{modes}, which are nontrivial, collective deformation modes that do not involve any stretching or compression of fibers, and hence no energetic cost~\cite{phonon_gap_2012}. These two generic features endow athermal biopolymer fibrous networks with intriguing universal properties, which are of both practical and fundamental importance.
The two aforementioned properties of athermal biopolymer fibrous networks are quantified by the dimensionless ratio $\kappa\!\ll\!1$ between fibers' bending and stretching/compression stiffnesses, and by the average connectivity $z\!<\!z_{\rm c}$, where $z_{\rm c}\!=\!2{\,\mathchar'26\mkern-12mu d}$ (${\,\mathchar'26\mkern-12mu d}$ is the space dimension) is the Maxwell rigidity criterion for $\kappa\=0$~\cite{maxwell_1864}. A remarkable property of biopolymer fibrous networks is that they undergo dramatic macroscopic stiffening transition when subjected to external strains. In particular, an initially undeformed network that features a vanishing shear modulus $G\!=0$ for $\kappa\=0$ and $z\!<\!z_{\rm c}$ (cf.~Fig.~\ref{fig:fig1}a) undergoes a sharp stiffening transition as its shear strain $\gamma$ attains a critical value $\gamma_{\rm c}$, upon which $G$ features a jump discontinuity, attaining a finite value $G(\gamma_{\rm c},\kappa=0)$~\cite{wouter_pre_2017,robbie_thesis,merkel_pnas_2019}. This strain-stiffening transition is associated with the emergence of a special internal state, commonly termed a state-of-self-stress (SSS, cf.~Fig.~\ref{fig:fig1}b), which corresponds to a set of putative stretching or compressive forces in the fibers that exactly balance each other on the network's nodes~\cite{gustavo_pre_2014,robbie_pre_2018,merkel_pnas_2019}.
Substantial experimental, computational and theoretical effort has been made in order to characterize and understand the strain-stiffening transition of athermal biopolymer fibrous networks~\cite{kees_nature_2005,Janmey_soft_matter_2007,RevModPhys.86.995,gustavo_pre_2014,robbie_pre_2018,merkel_pnas_2019,mackintosh_pre_2016,Rens_JPCB_2016,robbie_nature_physics_2016,wouter_pre_2017,robbie_pre_2018,robbie_thesis,maha_prl_2008,mackintosh_prl_2019,fred_arXiv_2022}. Here, we first develop a complete scaling theory of the critical strain-stiffened state. This is achieved by conceptualizing the entire network as composed of two interacting sub-networks; a stiff underconstrained sub-network at its sharply defined critical state $\gamma\=\gamma_{\rm c}$ and a soft sub-network of a characteristic dimensionless stiffness $\kappa\!\ll\!1$, where the latter is treated as a perturbation on top of the former, see Fig.~\ref{fig:fig1}c. This framework allows to derive a set of nonlinear scaling relations for all basic quantities that characterize the critical state, which are verified through highly accurate numerical simulations. We then go beyond the critical state, deriving scaling predictions for $G(\gamma,\kappa)$ away from $\gamma_{\rm c}$, which reveal a previously unidentified characteristic strain scale in athermal biopolymer fibrous networks.\\
\begin{figure*}[ht!]
\includegraphics[width = 1\textwidth]{fig1_strain_stiffening_final.pdf}
\caption{\footnotesize {\bf An example of a model of a strain-stiffened athermal biopolymer network obtained in a numerical simulation.} (a) An example of the isotropic edge-diluted, disordered floppy networks used in this work, here with $N\!=\!400$ nodes and connectivity of $z\!=\!3.8$ edges per node. (b) The same network shown in (a), after being deformed to the strain-stiffening point. The thickness of the edges represents components of the state-of-self-stress that emerges at the strain-stiffening point, that endows the floppy network with a finite shear modulus~\cite{matthieu_thesis}. Blue (green) edges mark stretching (compression) forces. (c) An illustration of the two-step procedure aimed at mimicking strain-stiffened fibrous networks in the presence of bending forces. Black edges represent stiff springs of stiffness $k$ (taken to be unit), while yellow edges represent soft springs with stiffness $\kappa\!\ll\!1$ (mimicking bending). (d) Illustration of the force-balance (see text for the definition of the various symbols) between the stiff and soft sub-networks after the introduction of the soft sub-network, followed by node displacement $\mathitbf u_\star$ (dashed arrow).}
\label{fig:fig1}
\end{figure*}
\section{Results}
\!\!\!\!\!\!{\bf Scaling theory of the critical state.} Consider first the stiff, underconstrained sub-network in the absence of strain, see Fig.~\ref{fig:fig1}a. Being sub-isostatic, this sub-network features floppy (zero) modes, i.e.~modes that can be deformed without any energetic cost. As a shear strain $\gamma$ is applied at a given direction, the system self-organizes anisotropically to give rise to the SSS at $\gamma\=\gamma_{\rm c}$, cf.~Fig.~\ref{fig:fig1}b. There exist, however, an extensive number of zero modes that are not coupled to the applied strain and remain floppy at $\gamma\=\gamma_{\rm c}$. In more formal terms, one can show that floppy modes are orthogonal to the applied shear strain~\cite{footnote}. Consider then adding the soft sub-network to the stiff one, where the two sub-networks interact at the network's nodes, see Fig.~\ref{fig:fig1}c. The soft sub-network may be regarded as a perturbation applied to the stiff one, introducing forces of order $\kappa$, i.e.~the soft sub-network features a characteristic deformation that is independent of $\kappa$ (but depends on the average connectivity $z$)~\cite{robbie_pre_2018}.
How would the stiff sub-network respond to the forces introduced to it by the soft sub-network? To address this question, we first note that the soft forces are random and isotropic in nature, and hence will inevitably have projections on the floppy modes of the stiff sub-network, those that persist at $\gamma_{\rm c}$ as they are not coupled to the strain. As the floppy modes are energetically cheap to move, one expects them to control the response of the stiff sub-network, for example the node displacements (relative to the node positions prior to the addition of the soft sub-network) of characteristic magnitude $u_*$, see Fig.~\ref{fig:fig1}d.
As the response of the stiff sub-network to the random perturbations introduced by the soft sub-network is expected to be dominated by its floppy modes, its energy $U_{\mbox{\tiny stiff}}(u)$ as function of the node displacement $u$ features vanishing first (mechanical equilibrium) and second (floppy normal modes) derivatives at $u\=0$. Since $u\=0$ is a stable state, a cubic term $\sim u^3$ is also excluded. Hence, we expect $U_{\mbox{\tiny stiff}}(u)\!\sim\!u^4$, i.e.~that the energy of the stiff sub-network is dominated by quartic anharmonicity, in line with the arguments recently spelled out in~\cite{manning_rigidity1_pre_2022,manning_rigidity2_pre_2022}. Consequently, the net force applied to the network's nodes by the stiff sub-network follows $F_{\mbox{\tiny stiff}}\!\sim\!dU_{\mbox{\tiny stiff}}/du\!\sim\!u^3$. As this force balances the imposed soft forces $F_{\mbox{\tiny soft}}\!\sim\!\kappa$, i.e.~$F_{\mbox{\tiny stiff}}\=F_{\mbox{\tiny soft}}$ (cf.~Fig.~\ref{fig:fig1}d), we predict the network to feature node displacements of characteristic magnitude $u_*$ that scales as
\begin{equation}
\label{eq:displacement}
u_*(\kappa) \sim \kappa^{1/3} \ .
\end{equation}
The basic result in Eq.~\eqref{eq:displacement} has immediate implications for the energetics of the system. First, it implies that the energy of the stiff sub-network scales as $U_{\mbox{\tiny stiff}}\!\sim\!u_*^4\!\sim\!\kappa^{4/3}$. Since the soft sub-network undergoes $\kappa$-independent deformation (of exactly $\gamma_c$), its energy scales as $U_{\mbox{\tiny soft}}\!\sim\!\kappa$. Taken together, the total energy $U$ of the systems is predicted to scale as
\begin{equation}
\label{eq:energy}
U(\kappa) = U_{\mbox{\tiny stiff}}(\kappa) + U_{\mbox{\tiny soft}}(\kappa) \sim U_{\mbox{\tiny soft}}(\kappa) \sim \kappa \ ,
\end{equation}
since $U_{\mbox{\tiny stiff}} \ll U_{\mbox{\tiny soft}}$ in the limit of small $\kappa$. That is, we predict that the total energy of the network is dominated by the energy of the perturbing soft sub-network.
It is natural to consider next the forces in the problem. To that aim, we distinguish between two characteristic forces. The first one corresponds to the net force of magnitude $F$ applied to each node in the network by the two sub-networks (cf.~Fig.~\ref{fig:fig1}d), $F\=F_{\mbox{\tiny stiff}}\=F_{\mbox{\tiny soft}}$, where the latter equality follows from mechanical equilibrium, as already invoked above (the {\em total} net force at each node obviously vanishes, $F_{\mbox{\tiny total}}\=F_{\mbox{\tiny stiff}}-F_{\mbox{\tiny soft}}\=0$). The second one corresponds to the magnitude of the fiber scale force $f$ (cf.~Fig.~\ref{fig:fig1}d), which includes a contribution from the stiff sub-network $f_{\mbox{\tiny stiff}}$ (corresponding to fiber stretching/compression) and a contribution from the soft sub-network $f_{\mbox{\tiny soft}}$ (corresponding to fiber bending). Obviously, the vectorial sum of the individual fiber stretching/compression forces equals the net force applied by the stiff sub-network on the network's nodes and likewise the vectorial sum of the individual fiber bending forces equals the corresponding net force applied by the soft sub-network.
With this distinction in mind, we set out to derive a scaling estimate for the eigenvalue $\lambda$ of the geometrical operator defining the SSS~\cite{sss_epje_2018}, see also Supplementary Information. At the critical $\gamma_{\rm c}$ state in the absence of a soft sub-network perturbation, $\kappa\=0$, the geometrical operator associated with the SSS features an identically vanishing eigenvalue, $\lambda\=0$. That means that the eigenvector corresponding to $\lambda\=0$ is interpreted as composed of a set of putative stretching/compression fiber forces that exactly balance on the network's nodes. With the application of a soft sub-network perturbation, $\kappa\!>\!0$, the network node displacements $u_*$ predicted in Eq.~\eqref{eq:displacement} emerge and one expects $\lambda$ to become finite as the $\kappa\=0$ state is distorted. Previous work has shown that $\lambda$ can be expressed in terms of the ratio $F/f$ as $\lambda \!\sim\!(F/f)^2$~\cite{asm_pnas_2012}. Furthermore, in the Supplementary Information we show that $\lambda$ increases with the node displacements $u_*$ as $\lambda\!\sim (F/f)^2\!\sim u_*^2$. Using Eq.~\eqref{eq:displacement}, we then predict
\begin{equation}
\label{eq:SSS}
\lambda(\kappa) \sim \kappa^{2/3} \ .
\end{equation}
The above results can be readily used to obtain a prediction for the fiber scale forces $f$, and consequently for the stress $\sigma$ in the network, which satisfies $\sigma\!\sim\!f$. First, recall that $F\=F_{\mbox{\tiny stiff}}\=F_{\mbox{\tiny soft}}\!\sim\!\kappa$ and that $f_{\mbox{\tiny soft}}\!\sim\!\kappa$. Consequently, Eq.~\eqref{eq:SSS} --- i.e.~$\lambda\!\sim\!(F/f)^2\!\sim\! \kappa^{2/3}$ --- can be satisfied if $f$ is dominated by the fiber stretching/compression forces, i.e.~$f\!\sim\!f_{\mbox{\tiny stiff}}\!\ll\!F_{\mbox{\tiny stiff}}$. Therefore, we obtain $f_{\mbox{\tiny stiff}} \sim \kappa/\sqrt{\lambda}$, which implies that the network's stress $\sigma$ satisfies
\begin{equation}
\label{eq:stress}
\sigma(\kappa) \sim f_{\mbox{\tiny stiff}}(\kappa) \sim \kappa^{2/3}\qquad\hbox{with}\qquad f_{\mbox{\tiny stiff}}\gg f_{\mbox{\tiny soft}} \ .
\end{equation}
\begin{figure*}[ht!]
\includegraphics[width = 1\textwidth]{fig2_strain_stiffening_final.pdf}
\caption{\footnotesize {\bf High-precision numerical validation of the theoretical predictions at the critical state.} (a) The magnitude $u_\star$ of node displacement vectors $\mathitbf u_\star$ between the $\kappa\!=\!0$ strain-stiffened state, and the mechanical equilibrium state reached after introducing the soft interactions of stiffness $\kappa$ (in units of the stiffness of the stiff interactions), cf.~Fig.~\ref{fig:fig1}a. (b) Energy-per-node $U/N$ of the soft (empty symbols) and stiff (full symbols) sub-networks. (c) The minimal eigenvalue $\lambda$ of the geometric operator defining the SSS (see Supplementary Information for details), indicating the ratio between the stretching/compression forces in the floppy network's springs, and the resulting net-force on the nodes. (d) The macroscopic shear stress follows $\sigma$. All quantities are plotted against $\kappa$.}
\label{fig:fig2}
\end{figure*}
Quite counter-intuitively and surprisingly, the last relation in Eq.~\eqref{eq:stress} implies that the while the overall network's energy $U$ is dominated by the soft sub-network (corresponding to bending energy), cf.~Eq.~\eqref{eq:energy}, the overall network's stress $\sigma$ is dominated by the stiff sub-network (corresponding to stretching/compression forces). This intriguing result is a manifestation of the singular nature of the response of the critical strain-stiffened state to the perturbation introduced by the soft sub-network, to be further discussed below.
The most pronounced macroscopic effect of the strain-stiffening transition in biopolymer fibrous networks is the dramatic increase in the shear modulus $G$. This stiffening plays important roles in many cellular and tissue-level physiological processes, e.g.~it allows the transmission of cellular and tissue forces over large scales, and enables long-range mechanical communication between cells~\cite{GOREN20201152}. Our next goal is to understand the properties of $G$ at $\gamma_{\rm c}$.
In view of the finite jump discontinuity experienced by the shear modulus at $\gamma_{\rm c}$, we expect $G(\gamma_{\rm c},\kappa)$ to approach a finite value in the $\kappa\!\to\!0$ limit (see Supplementary Information for additional discussion), i.e.~that
\begin{equation}
\label{eq:G}
G(\gamma_{\rm c},\kappa\!\to\!0^+) \sim \kappa^{0} \ ,
\end{equation}
even though the $\kappa\!\to\!0$ limit turns out to be subtle in this context, as discussed below. Next, we aim at calculating the derivative $dG/d\gamma$, evaluated at $\gamma\=\gamma_{\rm c}$, which is nothing but the first nonlinear shear modulus. This observable is sensitive to the presence of low-frequency vibrational modes, and their coupling to external deformation, as pointed out in~\cite{exist} and explained in the Supplementary Information, where it is shown that
\begin{equation}
\label{eq:G_derivative}
\frac{dG(\gamma,\kappa)}{d\gamma}\biggr\rvert_{\gamma=\gamma_{\rm c}} \!\!\sim \kappa^{-2/3} \ .
\end{equation}
Equation~\eqref{eq:G_derivative} predicts that $G(\gamma)$ varies strongly with $\gamma$ near $\gamma_{\rm c}$ and that this variation becomes singular as $\kappa\!\to\!0$.\\
\!\!\!\!\!\!{\bf The singular perturbation nature of the critical state.} As mentioned above, the perturbation introduced by the soft sub-network to the critical strain-stiffened state is of a singular nature. This fundamental point is further highlighted by comparing the response forces $f_{\mbox{\tiny stiff}}$ to the imposed force perturbation $f_{\mbox{\tiny soft}}$, i.e.~the ratio
\begin{equation}
\label{eq:singular_f}
f_{\mbox{\tiny stiff}}(\kappa)/f_{\mbox{\tiny soft}}(\kappa) \sim \kappa ^{-1/3} \ ,
\end{equation}
which diverges in the $\kappa\!\to\!0$ limit. That is, the response is infinitely stronger than the perturbation.
In addition, let us consider again the shear modulus $G$ at $\gamma_{\rm c}$. A careful calculation, presented in the Supplementary Information, reveals that the value approached in the $\kappa\!\to\!0^+$ limit, differs from $G(\gamma_{\rm c},\kappa=0)$, i.e.~the value obtained at $\kappa\=0$. It is shown that in fact we have
\begin{equation}
\label{eq:singular_G}
G(\gamma_{\rm c},\kappa\!\to\!0^+) = G(\gamma_{\rm c},\kappa=0) - \Delta{G} \equiv G(\gamma_{\rm c})
\end{equation}
in the limit $\kappa\!\to\!0^+$, where $\Delta{G}\!>\!0$ contains a product of two contributions, one that scales as $\kappa^{-2/3}$ and another as $\kappa^{2/3}$ such that the limit $\kappa\!\to\!0^+$ is finite. Consequently, Eq.~\eqref{eq:singular_G} indicates that the limit $\kappa\!\to\!0^+$ is different from setting $\kappa\=0$, yet again revealing the singular nature of the $\kappa$ perturbation in the problem.\\
\section{Numerical validation.}
Our next goal is to quantitatively test the predictions of the scaling theory developed above through highly accurate numerical calculations. To this aim, we employ disordered networks of relaxed Hookean springs in two-dimensions (${\,\mathchar'26\mkern-12mu d}\=2$, 2D) with mean connectivity of $z\!=\!3.8$, see Methods.
\begin{figure*}[ht!]
\includegraphics[width = 1\textwidth]{fig3_strain_stiffening_final.pdf}
\caption{\footnotesize {\bf High-precision numerical validation of the theoretical predictions for the linear and first nonlinear shear moduli.} (a) The shear modulus $G$ of a strain-stiffened network in the presence of the soft interactions, of stiffness $\kappa$. We find that $G$ approaches a constant as $\kappa\!\to\!0^+$, as predicted (see Eq.~\eqref{eq:singular_G} and Supplementary Information). The dashed lines indicate the value of the shear modulus when $\kappa\!=\!0$ identically, underlining the singular-perturbative nature of the soft (bending) interactions. (b) The first nonlinear shear modulus $dG/d\gamma$ follows the theoretical prediction $\sim\!\kappa^{-2/3}$ in Eq.~\eqref{eq:G_derivative}.}
\label{fig:fig3}
\end{figure*}
We subject our networks to simple shear strain of magnitude $\gamma$ (see Methods for details); as long as $\gamma\!<\!\gamma_{\rm c}$, the potential energy remains zero (demonstrated in the Supplementary Information). However, at $\gamma_c$ the system strain-stiffens --- it acquires a finite shear modulus. Accurately resolving the critical strain $\gamma_{\rm c}$ is difficult from a numerical/computational perspective; the reason is that, at $\gamma_{\rm c}$, the energy in the spring network should be vanishingly small. This means that the actual length of springs is very close to their rest-length. In addition, since we aim at finding a minimum of the potential energy, at which the forces exerted by the springs on the nodes balance each other \emph{exactly}, we must resolve the \emph{differences} between the aforementioned differences between the spring lengths and their corresponding rest-lengths.
The computational problem of accurately resolving differences-of-differences of numbers that are very similar to each other --- requires employing 128-bit floating point precision numerics. The latter turns out to be essential for probing the physics of the critical strain stiffening point. By employing 128-bit numerics, we are able to generate strain-stiffened networks (i.e.~at $\gamma_{\rm c}$) with an energy-per-node of order $10^{-20}k\ell^2$ (see Supplemental Information), where $k$ denotes the spring stiffnesses (shared by all springs) and $\ell\!\equiv\!\sqrt{A/N}$ forms the microscopic units of length of our networks of area $A$.
In order to mimic strain-stiffened networks at finite-$\kappa$, we employ a `two-step procedure' as explained in Methods, which involves introducing an additional, soft interaction potential to the critical networks at $\gamma_{\rm c}$, which represents the fibers' bending energy. The addition of the soft interaction does not alleviate the numerical challenge, at small $\kappa$ (of order $10^{-10}$ or lower): as explained above, now the vanishing differences between the forces in the stiff network (which, we reiterate, themselves require resolving the vanishing difference between the springs' lengths and rest-lengths) need to be balanced by a force of order $\kappa$ --- which becomes challenging for the smallest $\kappa$ probed in our work.
These aforementioned numerical challenges that force us to resort to 128-bit numerics dramatically limit the size and number of systems one can generate. For this reason, we generated a \emph{single} realization of strain-stiffened networks for different system sizes $N$ (number of network's nodes), both at $\kappa\!=\!0$ and at various finite $\kappa$ (using the two-step procedure), rather than ensembles of systems, as conventionally done. Nevertheless, these single realizations appear sufficient to validate our theoretical predictions, as shown next.
In Fig.~\ref{fig:fig2}, we validate the predictions of Eqs.~\eqref{eq:displacement}-\eqref{eq:stress}, for our 3 networks of $N\!=\!400,1600$ and $6400$ nodes at $z\!=\!3.8$. Additional details can be found in the figure caption. We find excellent agreement with all scaling predictions. In Fig.~\ref{fig:fig3}, we validate the predictions of Eqs.~\eqref{eq:G}-\eqref{eq:G_derivative}, in addition to demonstrating the singular-perturbative nature of the soft-subsystem and illustrating the observables defined in Eq.~\eqref{eq:singular_G}. Moreover, in Fig.~\ref{fig:fig3} we use $\kappa$ values as high as $\kappa\=10^{-3}$ in order to highlight the fact that the critical scaling at $\gamma_{\rm c}$ for small $\kappa$ must break down for large enough $\kappa$. This is nicely demonstrated in Fig.~\ref{fig:fig3}a, where $G(\gamma_{\rm c},\kappa)$ reveals a clear deviation from the $G(\gamma_{\rm c},\kappa\!\to\!0^+)$ value with increasing $\kappa$.
\vspace{0.2cm}
\section{Scaling theory beyond the critical point.}
Our next goal is to go beyond the critical point, i.e.~to understand the scaling structure of $G(\gamma,\kappa)$ near $\gamma_{\rm c}$ using Eqs.~\eqref{eq:G}-\eqref{eq:G_derivative}. The first step in achieving this goal is to formulate what is the physical meaning of ``being near $\gamma_{\rm c}$''. The latter entails the existence of a characteristic $\kappa$-dependent strain scale $\delta\gamma_*(\kappa)\!>\!0$ such that $|\gamma-\gamma_{\rm c}|\!\ll\!\delta\gamma_*(\kappa)$ defines ``being near $\gamma_{\rm c}$'', both below and above the critical strain $\gamma_{\rm c}$. Consequently, we pose the simplest scaling ansatz of the form
\begin{equation}
\label{eq:G_scaling}
G(\gamma,\kappa) \sim {\cal F}\!\left(\frac{\gamma-\gamma_{\rm c}}{\delta\gamma_*(\kappa)} \right) \ ,
\end{equation}
where ${\cal F}(\cdot)$ is a dimensionless scaling function and $G$ is nondimensionalized by $k/\ell$. The challenge then is to derive both $\delta\gamma_*(\kappa)$ and ${\cal F}(\cdot)$.
Denoting $x\!\equiv\!(\gamma-\gamma_{\rm c})/\delta\gamma_*(\kappa)$ and taking the derivative of $G$ in Eq.~\eqref{eq:G_scaling} with respect to $\gamma$, evaluated at $\gamma_{\rm c}$, we obtain $dG/d\gamma\rvert_{\gamma=\gamma_{\rm c}}\=d{\cal F}/dx\rvert_{x=0}[\delta\gamma_*(\kappa)]^{-1}$. To comply with Eq.~\eqref{eq:G_derivative}, $d{\cal F}/dx\rvert_{x=0}$ has to be finite, i.e.~${\cal F}$ varies linearly with $x$ to leading order, and
\begin{equation}
\label{eq:typical_strain}
\delta\gamma_*(\kappa) \sim \kappa^{2/3} \ .
\end{equation}
Moreover, Eq.~\eqref{eq:G} implies ${\cal F}(x\=0)\=G(\gamma_{\rm c})$ (the latter is defined in Eq.~\eqref{eq:singular_G}). Consequently, we end up with
\begin{equation}
\label{eq:small_strain_scaling}
G(\gamma,\kappa)-G(\gamma_{\rm c}) \sim \kappa^{-2/3} (\gamma-\gamma_{\rm c}) \quad \hbox{for} \quad |\gamma-\gamma_{\rm c}|\!\ll\!\kappa^{2/3} \ .
\end{equation}
Equation~\eqref{eq:typical_strain} predicts that the characteristic strain scale $\delta\gamma_*(\kappa)$ vanishes in the limit $\kappa\!\to\!0$ with a nontrivial exponent. Equation~\eqref{eq:small_strain_scaling} predicts that $G(\gamma,\kappa)$ varies linearly with $\gamma$ on top of the constant $G(\gamma_{\rm c})$, both below and above $\gamma_{\rm c}$. That is, we predict that $G(\gamma,\kappa)$ is regular near $\gamma_{\rm c}$ in terms of its variation with the strain $\gamma$.
Can the scaling form $G(\gamma,\kappa)\!\sim\!{\cal F}\!\left(\frac{\gamma-\gamma_{\rm c}}{\kappa^{2/3}} \right)$ be used to obtain predictions also at smaller strains $\gamma$ below the characteristic strain scale, i.e.~for $\gamma_{\rm c}-\gamma\!\gtrsim\!\kappa^{2/3}$? Below the strain-stiffening transition, the characteristic scale of $G$ is determined by the soft sub-network, i.e.~$G$ has to be linear in $\kappa$. The above scaling form then immediately predicts the $\gamma$ dependence of $G(\gamma,\kappa)$, leading to
\begin{equation}
\label{eq:gamma_3halves}
G(\gamma,\kappa) \sim \kappa\,(\gamma_{\rm c}-\gamma)^{-3/2} \ ,
\end{equation}
i.e.~we expect ${\cal F}(x)\!\sim\!x^{-3/2}$ in this regime below the critical point, in perfect agreement with the arguments and numerical result of Ref.~\cite{robbie_pre_2018}. Equation~\eqref{eq:gamma_3halves} is expected to hold at an intermediate crossover regime below $\gamma_{\rm c}$, where $G$ significantly increases with $\gamma$. As will be shown below, this prediction is consistent with extensive numerical data available in the literature.
At the same time, it is clear that the scaling ansatz $G(\gamma,\kappa)\!\sim\! {\cal F}\!\left(\frac{\gamma-\gamma_{\rm c}}{\kappa^{2/3}} \right)$ cannot hold in other regimes further away from the critical point, as explained in the Supplementary Information. The scaling theory of Eqs.~\eqref{eq:G_scaling}-\eqref{eq:gamma_3halves} also implies that previous attempts~\cite{robbie_nature_physics_2016,mackintosh_prl_2019,fred_arXiv_2022} to describe the strain-stiffening transition using a Widom-like scaling~\cite{widom_1965} of the form $G(\gamma,\kappa)\!\sim\!|\gamma-\gamma_{\rm c}|^f{\cal G}_\pm(\kappa/|\gamma-\gamma_{\rm c}|^\phi)$, with $f,\phi\!>\!0$ (here $\pm$ corresponds to the strain regimes above and below $\gamma_{\rm c}$, respectively), cannot be strictly valid, as is further discussed in the Supplementary Information.\\
\section{Agreement with numerical results available in the literature.}
As the strain-stiffening transition has been quite extensively studied in the literature using numerical simulations, it would be interesting at this point to test some of our predictions against numerical results available in the literature. Very recently~\cite{fred_arXiv_2022}, the rheology of fluid-immersed, strain-stiffened networks near $\gamma_{\rm c}$ has been numerically studied using overdamped simulations. It has been shown (cf.~Eq.~5 and Fig.~3c therein) that the excess viscosity at $\gamma_{\rm c}$ scales as $\kappa^{-\xi}$. The excess viscosity is related to the square of the non-affine displacements~\cite{asm_pnas_2012}, which we predict to scale as $\sim\kappa^{-2/3}$ (see Supplementary Information). In~\cite{fred_arXiv_2022}, it was found that $\xi\!\approx\!1.5/2.2\!\approx\!0.68$, in great quantitative agreement with our prediction.
The scaling theory of Eqs.~\eqref{eq:G_scaling}-\eqref{eq:gamma_3halves}, and Eq.~\eqref{eq:typical_strain} in particular, predicts the existence of a characteristic strain scale $\delta\gamma_*(\kappa)$ from both sides of $\gamma_{\rm c}$, which shrinks with decreasing $\kappa$. Evidence for the existence of such a shrinking strain scale below $\gamma_{\rm c}$ is provided in Fig.~1 of~\cite{robbie_pre_2018}, where $G(\gamma,\kappa)$ is plotted as a function of $\gamma$ for various values of $\kappa$ (note that the parameter varied in that figure is $\kappa^{-1}$), and above $\gamma_{\rm c}$ in Fig.~5.3. of~\cite{robbie_thesis}, where the square of the non-affine displacements is plotted as a function of $\gamma$ for various values of $\kappa$. While these figures --- reproduced for the sake of completeness in the Supplementary Information --- do not allow to extract the functional form of $\delta\gamma_*(\kappa)$ (to be compared with Eq.~\eqref{eq:typical_strain}), they clearly indicate its existence.
The prediction in Eq.~\eqref{eq:gamma_3halves} is consistent with extensive numerical data available in the literature. In~\cite{Rens_JPCB_2016,robbie_nature_physics_2016,mackintosh_prl_2019}, it has been observed that $G(\gamma_{\rm c}-\gamma)^{-f}\!\sim\!\kappa(\gamma_{\rm c}-\gamma)^{-\phi}$ (i.e.~$G\!\sim\!\kappa(\gamma_{\rm c}-\gamma)^{f-\phi}$) with $f\!-\!\phi$ close to $-3/2$, below $\gamma_{\rm c}$. Specifically, in~\cite{Rens_JPCB_2016}, $f\!-\!\phi\!\approx\!0.53\!-\!2.0\=-1.47$ (2D undistorted honeycomb lattice, cf.~Fig.~4a therein) and $f\!-\!\phi\!\approx\!0.63\!-\!1.9\=-1.27$ (2D undistorted triangular lattice, cf.~Fig.~4d therein) have been reported; in~\cite{robbie_nature_physics_2016}, $f\!-\!\phi\!\approx\!0.75\!-\!2.1\=-1.35$ (2D triangular lattice, cf.~Fig.~2b therein), $f\!-\!\phi\!\approx\!0.84\!-\!2.2\=$ (2D Mikado network, cf.~Fig.~2b therein) and $f\!-\!\phi\!\approx\!0.8\!-\!2.2\=-1.4$ (3D fcc lattice, cf.~Fig.~2b therein) have been reported; and in~\cite{mackintosh_prl_2019}, $f\!-\!\phi\!\approx\!0.73\!-\!2.26\=-1.53$ (2D triangular network, cf.~Fig.~3a therein) and $f\!-\!\phi\!\approx\!0.68\!-\!2.05\=-1.37$ (2D packing-derived network, cf.~Fig.~3b therein) have been reported. These are all in reasonable agreement with our prediction $f\!-\!\phi\=-3/2\=-1.5$ in Eq.~\eqref{eq:gamma_3halves}.
The observations discussed above provide strong independent support to our scaling theory. Moreover, as these literature calculations involved deforming fibrous networks at finite $\kappa$ values, their consistency with our predictions also supports the developed framework in which the entire network is viewed as two interacting sub-networks, where the soft (fiber bending) sub-network is treated as a perturbation to the stiff (fiber stretching/compression) sub-network near $\gamma_{\rm c}$ (the so-called `two-step procedure'). In addition, in~\cite{merkel_pnas_2019} (cf.~Fig.~4D therein) and in~\cite{robbie_thesis} (cf.~Fig.~5.1 therein), $G-G(\gamma_{\rm c})\!\sim\!(\gamma-\gamma_{\rm c})$ has been observed above and close to $\gamma_{\rm c}$ for $\kappa\=0$, apparently in agreement with the prediction in Eq.~\eqref{eq:small_strain_scaling}. This apparent agreement, however, should be taken with caution as Eq.~\eqref{eq:small_strain_scaling} is valid in the $\kappa\!\to\!0$ limit (and note the predicted diverging $\kappa^{-2/3}$ prefactor), which may (or may not) differ from its $\kappa\=0$ counterpart in terms of the scaling with $\gamma$. Future work should further clarify this point.\\
\section{Discussion}
In this work, we developed a comprehensive scaling theory of the strain-stiffening transition of athermal biopolymer fibrous networks, at and near the critical strain $\gamma_{\rm c}$. Building on the intrinsic stiffness scale separation between fiber bending and stretching/compression, we treated the sub-network of weak bending forces as a random and isotropic perturbation of dimensionless magnitude $\kappa\!\ll\!1$ applied to the anisotropic stiff sub-network at $\gamma_{\rm c}$. With this conceptual framework in mind, we theoretically predicted the dependence of the salient physical quantities in the problem on $\kappa$ at the critical strain $\gamma_{\rm c}$.
The existence of floppy modes that are uncoupled (orthogonal) to the applied strain have been shown to dominate the stiff sub-network's response to the weak bending forces, which is characterized by quartic anharmonicity, in line with recent theoretical arguments~\cite{manning_rigidity1_pre_2022,manning_rigidity2_pre_2022}. The $\kappa$ scaling of the network's node displacements and of its energy at $\gamma_{\rm c}$ then follow, cf.~Eqs.~\eqref{eq:displacement}-\eqref{eq:energy}. Furthermore, analyzing the state-of-self-stress (SSS) and its breakdown in the presence of weak bending forces, as well as the accompanying non-affinity, allowed us to predict the $\kappa$ scaling of the forces in the problem and of the macroscopic modulus $G$ at $\gamma_{\rm c}$, cf.~Eqs.~\eqref{eq:SSS}-\eqref{eq:G_derivative}. The structure of the theory highlights the role of $\kappa$ as a singular perturbation applied to the critical strain-stiffened state, two manifestations of which are discussed in the context of Eqs.~\eqref{eq:singular_f}-\eqref{eq:singular_G}
Numerically testing the set of $\kappa$ scaling predictions for various physical quantities at $\gamma_{\rm c}$ pose a significant challenge and require high-precision numerical simulations. This challenge has been met and led to excellent quantitative agreement with the theoretical predictions. Independent support to the scaling at $\gamma_{\rm c}$ is provided by recent numerical results in the literature~\cite{fred_arXiv_2022} for the rheology of fluid-immersed, strain-stiffened networks, see discussion above.
With the complete $\kappa$ scaling theory at $\gamma_{\rm c}$ and its numerical validation at hand, we extended the theory beyond the critical state and derived scaling relations for the macroscopic modulus $G(\gamma,\kappa)$ in Eqs.~\eqref{eq:G_scaling}-\eqref{eq:gamma_3halves}. These scaling relations highlight the existence of a previously unidentified characteristic $\kappa$-dependent strain scale $\delta\gamma_*(\kappa)$ near $\gamma_{\rm c}$. Available numerical results in the literature lend independent support to some of the predictions, which should be further tested in their entirety in future work. Furthermore, the structure of the emerging scaling theory indicates that the Widom-like scaling form assumed in previous work~\cite{robbie_nature_physics_2016, mackintosh_prl_2019} cannot be strictly and self-consistently valid.
The progress made in this work also opens the way for additional future investigations. The most immediate and pressing ones are the substantiation of the predicted strain scale in Eq.~\eqref{eq:typical_strain} and of the linear $\gamma$ scaling near $\gamma_{\rm c}$ according to Eq.~\eqref{eq:small_strain_scaling}. In addition, a more thorough quantitative comparison between the two-step procedure (in which a critical $\gamma_{\rm c}$ state is supplemented with a $\kappa\!\ll\!1$ soft sub-network) and the continuous straining procedure at $\kappa\!>\!0$ should be performed. Achieving this comparison is a substantial computational challenge for the same reasons spelled out above, i.e.~each network realization requires performing thousands of minimization steps of the potential energy with 128-bit precision, which are extremely costly from a running time perspective.
In this work, we did not address spatial aspects of the strain-stiffening transition. Yet, it is conceivable that a characteristic lengthscale that increases with decreasing $\kappa$ exists in this problem~\cite{gustavo_pre_2014,robbie_pre_2018}. In this context, it would be interesting to explore finite size effects encountered in computer simulations, which have not been discussed and quantified here (according to Figs.~\ref{fig:fig2}-\ref{fig:fig3}, these appear to be rather weak). Moreover, the existence of a characteristic lengthscale is expected to be intimately related to the spatial decay of mechanical perturbation throughout the fibrous network, which may have profound implications for cell-cell communication in realistic biopolymer networks.
As discussed above and in the Supplementary
Information, the scaling form in Eq.~\eqref{eq:G_scaling} cannot remain valid away from the critical point $\gamma_{\rm c}$, i.e.~different scaling relations are expected for $|\gamma-\gamma_{\rm c}|\!\gg\!\delta\gamma_*(\kappa)$. While these scaling relations have been numerically explored~\cite{robbie_thesis,mackintosh_pre_2016,mackintosh_prl_2019,robbie_nature_physics_2016,fred_arXiv_2022}, theoretically deriving them remains a challenge for future work. Finally, we did not consider in this work possible differences between the effect of applied shear and dilatational strains, and in particular all numerical validation tests were performed under shear strains. It would be interesting to systematically explore dilatational strains in future work.\\
\acknowledgments
We thank Gustavo D\"uring for enlightening discussions that led to this work and Eric Lerner for his assistance with the graphics of Fig.~1. E.L.~acknowledges Support from the NWO (Vidi grant no.~680-47-554/3259). E.B.~acknowledges support from the Ben May Center for Chemical Theory and Computation and the Harold Perlman
Family.\\
\vspace{-2cm}
\section*{Methods}
\subsection{Preparation of initial isotropic networks.}
In order to generate disordered networks of relaxed Hookean springs, we first create packings of harmonic discs --- similar to the ones studied e.g.~in~\cite{breakdown} ---, at a packing fraction of 1.0. We then adopt the network of contacts between the harmonic discs to form an initial disordered network of nodes and edges. The edges of these initial random networks are then diluted following the algorithm described in~\cite{quantifier_PRE_2021}, which maintains low fluctuations in the local connectivity of nodes. Edge-dilution is stopped once the network reaches a connectivity (per node) of $z\!=\!3.8$, which resides \emph{below} the Maxwell threshold $z_{\rm c}\=2{\,\mathchar'26\mkern-12mu d}$~\cite{maxwell_1864} (they are thus referred to as \emph{underconstrained} or \emph{floppy} networks). Then, each edge is substituted with a fully relaxed Hookean spring of stiffness $k$ (which is used as the unit of stiffness in the problem). An example of such an edge-diluted disordered network is displayed in Fig.~\ref{fig:fig1}a. The resulting disordered floppy networks are then subjected to athermal, quasistatic deformation, which involves repeatedly applying small shear-strain increments, and following each increment with a potential energy minimization~\cite{fire}. Further technical details are provided in the Supplementary Information.\\
\subsection{Adding the soft sub-network (representing bending forces).}
In order to mimic strain-stiffened networks, whose fibers feature both stretching and bending energies, we add a highly-coordinated network of weak (soft) Hookean springs --- of spring stiffness $\kappa$ (made dimensionless by $k$) --- to the strain-stiffened spring networks, as illustrated in Fig.~\ref{fig:fig1}c, and described in detail in the Supplementary Information. Adding the weak interaction introduces net-forces of order $\kappa$ on the nodes. We then minimize again the potential energy, while including the weak interaction as well. This `two-step procedure' also requires 128-bit numerics, for the same reasons previously spelled out.\\
\subsection{Observables}
All derivatives of the potential energy with respect to strain are calculated with 128-bit precision numerics, using the exact expressions for linear and nonlinear athermal elasticity of disordered solids, see e.g.~\cite{lutsko,athermal_elasticity_pre_2010}. The exact expressions used are also provided in the Supplemental Information.
\bibliographystyle{apsrev4-2}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,492 |
<?php
namespace Magento\OfflineShipping\Model\Source\SalesRule;
use Magento\Framework\Data\OptionSourceInterface;
use Magento\OfflineShipping\Model\SalesRule\Rule;
class FreeShippingOptions implements OptionSourceInterface
{
/**
* {@inheritdoc}
* @codeCoverageIgnore
*/
public function toOptionArray()
{
return [
[
'value' => 0,
'label' => __('No')
],
[
'value' => Rule::FREE_SHIPPING_ITEM,
'label' => __('For matching items only')
],
[
'value' => Rule::FREE_SHIPPING_ADDRESS,
'label' => __('For shipment with matching items')
]
];
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,493 |
\section{Introduction}
\label{sec:intro}
Non-destructive evaluation (NDE) is an increasingly important application of CT, which motivates improved capabilities for quantitative energy-independent reconstruction and precise material characterization.
Such reconstruction techniques utilize physically accurate forward models that account for the polyenergetic nature of the X-ray source radiation and associated physical phenomena such as beam-hardening\cite{jin2015beamhardening}, rather than simpler models based on mono-energetic approximations.
However, an accurate estimate of the CT system's source-detector spectral response is a necessity for the above methods.
For example, the method for reconstructing energy-independent material properties like effective atomic-number and electron density from dual-energy CT scans \cite{busi2019method,azevedo2016system,champley2019method} and the method for tissue characterization in
\cite{mccollough2015dual, so2021spectral}
require a precise calibration of the source-detector spectral response.
Direct measurement of the spectral response of the whole X-Ray CT system is difficult since the detector response is hard to measure.
Based on Beer–Lambert's law\cite{lambert1760photometria}, we can use a linear model for transmission measurements of objects with known dimensions and composition to reconstruct the discretized spectral response of a CT scanner, but this yields a highly ill-conditioned system.
Champley et al. \cite{champley2019method} use linear least-squares to do spectral estimation (LSSE) with constraints to enforce non-negativity. However, this method requires an accurate initial guess close to the true spectrum.
Various regularization methods\cite{ruth1997estimation, yan1999modeling} are used to overcome the issues introduced by the ill-conditioned nature of the problem.
SVD-based algorithms have also been applied to the spectral estimation problem\cite{tominaga1986530, armbruster2004spectrum, leinweber2017x}.
Sidky et al.\cite{osti_20711775} represent the spectrum as a linear combination of B-splines and use EM (Expectation–maximization) to find the solution.
Zhao et al.\cite{zhao2014indirect} estimate the spectrum as a linear combination of six Monte Carlo model spectra.
Liu et al.\cite{liu20201495} introduce compressed sensing to estimate the spectrum.
The central ideas of the above methods have a common theme: try to solve the ill-conditioned inverse problem either with regularized optimization or by introducing basis spectra to perform the estimation.
In this paper, we introduce a novel dictionary-based spectral estimation (DictSE) method that can efficiently reconstruct the overall spectral response of a CT system from transmission scans of multiple known objects, without the need for accurate initialization.
We represent the unknown spectral response using an over-complete dictionary that accounts for vast combinations of different source spectra, filter attenuation characteristics, and detector energy-response models.
We formulate the reconstruction problem as a MAP estimation framework that combines a linear beam-hardening forward model along with prior constraints.
Specifically, we impose an $L_0$ sparsity constraint to limit the support for the spectrum representation and a simplex constraint to account for the bright-dark normalization of the transmission data.
We present a novel iterative optimization strategy that alternates between support selection and pairwise iterative coordinate descent (ICD) update to find the optimal sparse representation of the spectrum.
Finally, we demonstrate DictSE through a cross-validation experiment on four datasets collected at beamline 8.3.2 of ALS.
\section{Reconstruction model}
\label{sec:rec_model}
In this section, we describe the spectral estimation problem and our proposed solution in more detail. We use a physics-based model for X-ray transmission measurements and discretize the model in energy and by projection to produce a linear measurement model.
Then we introduce a dictionary-based framework and express the reconstruction as a MAP estimation problem.
Finally, we solve this dictionary-based MAP problem using support selection to enforce sparse coding and an ICD algorithm modified to enforce a simplex constraint on the coefficients of the selected dictionary elements.
\vspace{-2ex}
\subsection{Physics-based Model and Discretization}
\label{ssec:phy_model}
\begin{figure}[!t]
\centering
\centerline{\includegraphics[width=7cm]{Figs/physic_model.png}}
\caption{\textbf{Physics model of CT scanning.} A homogenous cylindrical object is scanned. The entire spectral response can be decomposed into the product of the X-ray source spectrum, filter response, and detector response, respectively.}
\label{fig:phy_model}
\end{figure}
Fig.~\ref{fig:phy_model} illustrates the setup for X-ray spectral estimation.
Using notation as in that figure, the spectral response of this CT system is the product of the X-ray source spectrum, filter response, and detector response, which yields the response function
\begin{equation}
\label{equ:spec_model}
S(E)=S_{sr}(E) S_{ft}(E) S_{dt}(E) \ ,
\end{equation}
and our goal is to estimate $S(E)$.
We relate $S(E)$ to measurements by first defining $I$ and $I_0$ to be the measured intensity of the object scan (with sample) and the blank scan (without sample), respectively.
Based on Beer–Lambert's law, we have
\begin{equation}
\label{equ:obj_scan}
I=\int_0^{E_{\max }} S(E) \cdot \exp \left\{-\int_L \mu(E, r) d r\right\} d E \ ,
\end{equation}
where $\mu(E, r)$ is linear attenuation coefficient (LAC) with units $mm^{-1}$. Likewise, $I_0$ is defined as in \eqref{equ:obj_scan} with $\mu = 0$.
The (normalized) transmission through line $L$ is then defined as
\begin{equation}
\label{equ:transm}
\begin{aligned}
y & =\frac{I}{I_0}=\int_0^{E_{\max }} \bar{S}(E) \cdot \exp \left\{-\int_L \mu(E, r) d r\right\} d E \ ,
\end{aligned}
\end{equation}
where $\bar{S}(E)=\frac{S(E)}{\int_0^{E_{\max }} S(E) d E}$ is the normalized response.
We assume each sample consists of solid rods made from known materials taken from a specified reference set $\Phi$.
For each material $s\in \Phi$ and projection $i\in\{0,...,M-1\}$, we define $L_{i, s}$ to be the path length of the $i^{th}$ projection through the $s^{th}$ material and $\mu_s(E)$ to be the LAC of the $s$ material at energy $E$.
Assuming that $I_0$ is independent of projection and noise-free, this yields the transmission for the $i^{th}$ projection as
\begin{equation}
\label{equ:scaled_transm}
\begin{aligned}
y_i & =\int_0^{E_{\max }} \bar{S}(E) \cdot \exp \left\{-\sum_{s \in \Phi} \mu_s(E) L_{i, s}\right\} d E+\tau_i \ ,
\end{aligned}
\end{equation}
where $\tau_i$ is additive noise.
We discretize in energy by subdividing into non-overlap\-ping bins $[E_j, E_{j+1}]$ and defining $x_j=\int_{E_j}^{E_{j+1}} \bar{S}\left(E_j\right) d E$.
Making the approximation that each $\mu_s$ is constant on each bin, we define the coefficient from the $j^{th}$ energy bin to the $i^{th}$ projection as $F_{i, j}=\exp \left\{-\sum_{s \in \Phi} \mu_s\left(E_j\right) L_{i, s}\right\}. $
Then \eqref{equ:scaled_transm} becomes
\begin{equation}
\label{equ:y_discrete}
\begin{aligned}
y_i & \approx \sum_{j=0}^{J-1} \int_{E_j}^{E_{j+1} } \bar{S}(E) F_{i,j} d E+\tau_i = \sum_{j=0}^{J-1} F_{i,j} x_j + \tau_i \ .
\end{aligned}
\end{equation}
Using this along with constraints to yield a normalized spectrum, the forward model is
\begin{equation}
\begin{aligned}
\label{equ:smp_model}
&Y=F x + \tau, \\
&\text{s.t. } \|x\|_1=1, x_j \geq 0 \; \forall j,
\end{aligned}
\end{equation}
where $Y\in \mathbb{R}^M$ is a vector of $M$ normalized transmission measurements, $F \in \mathbb{R}^{M \times N_e}$ is a forward matrix over $N_e$ energy bins, $\tau$ is additive noise, and $x\in \mathbb{R}^{N_e}$ is the unknown vector discretization of $\bar{S}\left(E\right)$ that we seek to estimate.
\vspace{-2ex}
\subsection{Dictionary-based Model}
\label{ssec:dict_model}
Developing a dictionary-based method instead of directly estimating the spectrum at each energy bin has several motivations.
When the discretization of $x$ is very fine, the projection model matrix $F$ has a large null space, leading to an underdetermined reconstruction, which requires a good initial estimate.
Also, with a dictionary-based method, we can apply a sparsity-promoting penalty so that the dimensionality of the optimization problem can be significantly decreased and the time for reconstruction reduced.
More specifically, as indicated in \eqref{equ:smp_model}, we model $x$ as a discrete probability distribution, so that the entries of $x$ are nonnegative and sum to 1. We call this the simplex constraint and use $\mathcal{S}$ to represent the set of all such possible vectors.
We use a fixed, over-complete dictionary $D$ to represent the unknown normalized spectrum, $x$, as
\vspace{-1ex}
\begin{equation}
x=D\omega
\end{equation}
where $D$ is an $\mathbb{R}^{N_e\times N_k}$ matrix, each column $D_{*,k}\in \mathcal{S}$ represents a normalized basic spectrum.
With this $D$, the transmission model can be rewritten as
\vspace{-1ex}
\begin{equation}
\label{equ:dict_model}
Y=F D \omega + \tau \ .
\end{equation}
\vspace{-2ex}
\subsection{MAP Estimate}
\label{ssec:map}
We use a Bayesian framework to estimate $\omega$ from transmission measurements under sparsity and simplex constraints. We define transmission weights using a diagonal matrix $\Lambda$, where we take $\Lambda_{i,i} = 1/(y_i M)$. We then define
the loss function $l(\omega) = \frac{1}{2}\|Y-F D \omega\|_{\Lambda}^2$, in which case the MAP estimate is given by
\begin{equation}
\label{equ:map_model}
\widehat{\omega}=\arg \min _{\substack{\omega \in \mathcal{S} \\\|\omega\|_0 \leq C}}l(\omega) \ ,
\end{equation}
where $\omega \in \mathcal{S}$ enforces $x \in \mathcal{S}$ since each column of $D$ is a simplex; the $L_0$ constraint ensures $\omega$ has no more than $C$ non-zero components.
\begin{algorithm}[!t]
\caption{Dictionary-based Spectrum Reconstruction Algorithm (see equation \eqref{equ:kbfw_model} for notation)}
\label{alg:mp_est}
\begin{algorithmic}[1]
\State{Initialize $k^* \gets \arg \min _{\substack{k }}l(\epsilon_{k})$, $\omega=\epsilon_{k^*}$, $\Omega=\{k^*\}$}
\While{$|\Omega| < C$} \mbox{\hspace{12pt}} // Support selection
\State{$\left(k^*, \beta^*\right) \gets \arg \min _{\substack{k \notin \Omega \\ \beta \in[0,1)}}l'(k,\beta | \widehat{\omega})$ }
\State{$\omega \gets \beta^* \omega+\left(1-\beta^*\right) \epsilon_{k^*}$}
\State{$\Omega \gets \Omega \cup \{k^*\}$}
\State{$g \gets k^*$} \mbox{\hspace{12pt}} // Select new element for next loop
\While{not converged} \mbox{\hspace{12pt}} // Pairwise ICD update
\For{$k\in \Omega$ and $k\neq g$}
\State{$\alpha^* \leftarrow \arg \min _{\alpha \in\left[-\omega_k, \omega_g\right]}l\left(\omega+\alpha (\epsilon_k -\epsilon_g)\right)$}
\State{$\omega \leftarrow \omega+\alpha^* (\epsilon_k -\epsilon_g)$}
\EndFor
\EndWhile
\EndWhile
\State{$\hat{\omega} \gets \omega$ \mbox{\hspace{12pt}} // Return the final estimate }
\end{algorithmic}
\end{algorithm}
\vspace{-2ex}
\subsection{Support Selection and Pairwise ICD update}
\label{ssec:ds_pwicd}
To minimize the MAP cost function, we alternate between greedy support selection and a pairwise ICD update.
Inspired by the Orthogonal Matching Pursuit (OMP)\cite{pati1993}, our DictSE builds support set $\Omega=\{k: \omega_k \neq 0\}$ by adding one basic spectrum from the dictionary at a time and then updating the coefficients.
However, the conventional support selection method in OMP does not account for simplex constraints on $\omega$.
Further, the OMP method assumes that the dictionary atoms are normalized, whereas, in our spectral estimation problem, the product of forward matrix $F$ and the spectrum dictionary $D$ is not.
Thus choosing a basis spectrum with conventional matching pursuit is inappropriate.
To describe our alternative method for support selection, we first define a function $l'(k,\beta | \widehat{\omega})$ that measures the fit to data obtained by scaling the existing coefficients $\hat{\omega}$ by $\beta$ and using the remaining weight on the $k^{th}$ coefficient. That is,
\begin{equation}
\label{equ:kbfw_model}
\begin{aligned}
l'(k,\beta | \widehat{\omega})&=l(\beta \widehat{\omega}+\left(1-\beta\right) \epsilon_{k})\\
&=\frac{1}{2}\left\|Y-\beta F D \widehat{\omega}-\left(1-\beta\right) F D_{*, k}\right\|_{\Lambda}^2
\end{aligned}
\end{equation}
where $\beta \in [0,1)$ enforces $\omega \in \mathcal{S}$ and $\epsilon_{k}$ is a one-hot vector for the $k^{th}$ spectrum.
Then we select a new element from the dictionary by minimizing $l'(k,\beta | \widehat{\omega})$ to obtain
\vspace{-1ex}
\begin{equation}
\begin{aligned}
\left(k^*, \beta^*\right) &\leftarrow
\arg \min _{\substack{k \notin \Omega\\ \beta \in[0,1)}}\left\{l'(k,\beta | \widehat{\omega})\right\}.
\end{aligned}
\end{equation}
This minimization can be solved easily for each $k \notin \Omega$ since equation (\ref{equ:kbfw_model}) is quadratic in the scalar $\beta$. In fact, defining $e_k = FD\widehat{\omega}- FD_{*,k}$, we have
\vspace{-1ex}
\begin{equation}
\beta_k=\frac{e_k^T \Lambda (Y-FD_{*,k})}{\| e_k\|^2_{\Lambda}} \ .
\end{equation}
Using $\beta_k$ in equation \eqref{equ:kbfw_model} allows us to find $k^*$ that minimizes equation \ref{equ:kbfw_model}; this $k^*$ is then included in $\Omega$ with $\omega_{k^*} = 1- \beta^*$ and the remaining $\omega_k$ scaled by $\beta^*$.
To rebalance the weights while enforcing the simplex constraint, we use pairwise ICD between the most recently added dictionary element with index $g$ and the remaining elements in $\Omega$, as shown in lines 6-12 of Algorithm \ref{alg:mp_est}.
More precisely, after choosing $g$, we loop repeatedly over $k \in \Omega \setminus \{g\}$, in each case finding an optimal pairwise update $\omega \leftarrow \omega+\alpha^* (\epsilon_k -\epsilon_g)$, where $\alpha^*$ is chosen by
\vspace{-1ex}
\begin{equation}
\alpha^* \leftarrow \arg \min _{\alpha \in\left[-\omega_k, \omega_g\right]}\left\{l\left(\omega+\alpha (\epsilon_k-\epsilon_g) \right)\right\} \ .
\end{equation}
The constraints on $\alpha$ in the minimization ensure each $\omega_k \geq 0$. Since $l(\omega)$ is quadratic, $\alpha^*$ can be computed as below
\vspace{-1ex}
\begin{equation}
\alpha^*=\text{Clip}\left\{\frac{(Y-F D \omega)^{\mathrm{T}} \Lambda F D (\epsilon_k -\epsilon_g)}{\left\|F D (\epsilon_k -\epsilon_g)\right\|_{\Lambda}^2}, [-\omega_k, \omega_g]\right\}
\end{equation}
The pairwise ICD update will stop when the total update is less than $10^{-6}$.
The algorithm is summarized in Algorithm \ref{alg:mp_est}.
\begin{table}[t]
\centering
\caption{Setup for X-ray Scanning}
\begin{tabular}{ll}
\hline
Projection Geometry: & Parallel beam geometry \\
Source filter: & 2 $mm$ Silicon \\
Scintillator: & 50 $\mu m$ $Lu_{3}Al_{5}O_{12}$ \\
Max Energy: & 100 KeV \\
Views Spanning: & Equi-spaced in $[0,2\pi]$ \\
Detector Pixel Size: & 0.00065 mm \\
Nviews $\times$ Nrows $\times$ Ncolumns: & $2625 \times 100 \times 2560$ \\
Sample-detector distance: & $0.3$ mm \\ \hline
\end{tabular}
\label{table:setup_dataset}
\end{table}
\label{sec:imp}
\begin{figure}[!t]
\centering
\centerline{\includegraphics[width=8.5cm]{Figs/dataset.png}}
\caption{\textbf{Real scans of 4 rods with different materials.}}
\label{fig:scans}
\end{figure}
\section{ Implementation}
\label{sec:implementation}
In this section, we describe the normalized transmission data $Y$, the projection matrix $F$, and the dictionary of spectra $D$, which are required to estimate the spectral response of an X-ray CT system using Algorithm \ref{alg:mp_est}.
\vspace{-2ex}
\subsection{ Transmission data $Y$}
\label{ssec:trans_data}
We collected four CT datasets of different metal rods, $\Phi = \{Ti, V, Al, Mg\}$, at beamline 8.3.2 of the ALS.
As shown in Fig.~\ref{fig:scans}, for each dataset, we scanned a single rod.
The same CT scanner setup was used to collect all datasets.
Table~\ref{table:setup_dataset} gives more detailed information about the X-ray CT measurements.
We scanned $15$ bright scans and $10$ dark scans and averaged them to obtain $\bar{I}_{\text {bright }}$ and $\bar{I}_{\text {dark }}$.
Then, for each view, we normalized measurement data $I_{\text {scan }}$ to obtain
$ Y=\frac{I_{\text {scan }}-\bar{I}_{\text {dark }}}{\bar{I}_{\text {bright }}-\bar{I}_{\text {dark }}}$.
\vspace{-2ex}
\subsection{ Forward Matrix $F$}
\label{ssec:fw_mat}
For each dataset, the sample is a single rod that is solid and pure.
To calculate $F_{i,j}$, we need the path length of the $i^{th}$ projection through this rod.
To obtain the path length, we used filtered back projection to reconstruct the volume of the rod and then generated a mask to represent the object area.
Using this mask, we calculated the path length for each projection and used this to calculate the forward matrix $F$.
\vspace{-2ex}
\subsection{ Dictionary Generation $D$}
\label{ssec:dict_gen}
As mentioned in equation (\ref{equ:spec_model}), the spectral response can be modeled as the product of the X-ray source spectrum, filter response, and detector response, so we can create a dictionary by varying one or more of these elements.
The X-ray source spectrum is fixed in this experiment; we used an estimate provided by the beamline scientists at the ALS Beamline 8.3.2.
Therefore, in this implementation, we generated the dictionary by varying the filter and detector responses.
Based on the spectrum models in Ref.~\cite{champley2019method}, the filter response and detector response are determined by their material properties and thicknesses, which we vary as in Table~\ref{table:dict} to generate our dictionary.
By combining two groups of responses, we obtain an over-complete dictionary $D$ containing $60\times36=2160$ normalized responses spectra.
\begin{table}[t]
\centering
\caption{Dictionary Generation List}
\begin{tabular}{|cccc|}
\hline
\multicolumn{4}{|c|}{Filter response} \\ \hline
\multicolumn{1}{|c|}{Material} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Thickness Range\\ $mm$\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Step\\ $mm$\end{tabular}} & \begin{tabular}[c]{@{}c@{}}\# of \\ responses\end{tabular} \\ \hline
\multicolumn{1}{|c|}{$Al$} & \multicolumn{1}{c|}{0.1$\sim$5.9} & \multicolumn{1}{c|}{0.2} & 30 \\ \hline
\multicolumn{1}{|c|}{$Cu$} & \multicolumn{1}{c|}{0.2$\sim$0.49} & \multicolumn{1}{c|}{0.01} & 30 \\ \hline
\multicolumn{4}{|c|}{Detector response(Scintillator)} \\ \hline
\multicolumn{1}{|c|}{$Lu_{3}Al_{5}O_{12}$} & \multicolumn{1}{c|}{0.025$\sim$0.095} & \multicolumn{1}{c|}{0.002} & 36 \\ \hline
\end{tabular}
\label{table:dict}
\end{table}
\section{ Experimental Results}
\label{sec:exp}
We compared our proposed DictSE method with a least-squares spectral estimation (LSSE) method provided by Livermore tomography tools (LTT) \cite{CHAMPLEY2022102595} on four datasets described in section \ref{ssec:trans_data}.
An initial spectrum for the LSSE method was generated by LTT using $3 mm$ silicon as a source filter and $50 \mu m$ $Lu_{3}Al_{5}O_{12}$ as a scintillator.
We then evaluated the estimated spectra of DictSE and LSSE using leave-one-out cross-validation since we do not have a ground truth response.
Table~\ref{table:res} demonstrates that DictSE's reconstructed spectra outperform the LSSE's reconstructed spectra in NRMSE for all cross-validation cases.
For each case $v\in\{1,2,3,4\}$, we computed NRMSE $= \frac{\| Y_{v}-\widehat{Y}_{v}\|_{2} }{\| Y_{v}\|_{2} }$ to compare transmission measurements $Y_{v}$ and transmission value of the forward model using estimated spectrum $\widehat{Y}_{v}=F_{v}D\widehat{\omega}_v$ on the validation rod.
Fig.~\ref{fig:est_sp} shows all four cases of cross-validation reconstructed spectra using both the DictSE and LSSE methods.
For each case, DictSE's estimated spectra are smoother than the LSSE's.
Also, from the shape of the reconstructed spectra over all cases, DictSE is less data-sensitive than LSSE.
\begin{table}[!t]
\centering
\caption{Leave-One-Out Cross-Validation NRMSE}
\begin{tabular}{c|c|c|c|c}
\hline
Case & Fit & Test & LSSE & DictSE \\
\hline
1 & $Ti, Mg, Al$ & $V$ & 0.0331 & \textbf{0.0315} \\
2 & $V, Mg, Al$ & $Ti$ & 0.0624 & \textbf{0.0242} \\
3 & $V, Ti, Al$ & $Mg$ & 0.0343 & \textbf{0.0122} \\
4 & $V, Ti, Mg$ & $Al$ & 0.0483 & \textbf{0.0093} \\
\hline
\end{tabular}
\label{table:res}
\end{table}
\begin{figure}[!t]
\centering
\centerline{\includegraphics[width=8.5cm]{Figs/est_sp.png}}
\caption{\textbf{Reconstructed Spectra with DictSE and LSSE.} }
\label{fig:est_sp}
%
\end{figure}
\section{ Conclusion}
\label{sec:conclude}
This work provides a novel application of dictionary learning to X-Ray spectral estimation, allowing an efficient spectrum reconstruction from a vast dictionary obtained from CT datasets.
Our method uses a greedy support selection method to do sparse coding followed by pairwise ICD to do minimization while enforcing a simplex constraint.
Leave-one-out cross-validation experiments on four datasets demonstrated that our DictSE method outperforms the LSSE method in NRMSE.
\section{ACKNOWLEDGMENTS}
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and LDRD project 22-ERD-011. The authors acknowledge Dula Parkinson for his support during beamtime. Charles Bouman was partially supported by the Showalter Trust, and Greg Buzzard was partially supported by NSF CCF-1763896.
\vfill\pagebreak
\bibliographystyle{IEEEbib}
\section{Introduction}
\label{sec:intro}
Non-destructive evaluation (NDE) is an increasingly important application of CT, which motivates improved capabilities for quantitative energy-independent reconstruction and precise material characterization.
Such reconstruction techniques utilize physically accurate forward models that account for the polyenergetic nature of the X-ray source radiation and associated physical phenomena such as beam-hardening\cite{jin2015beamhardening}, rather than simpler models based on mono-energetic approximations.
However, an accurate estimate of the CT system's source-detector spectral response is a necessity for the above methods.
For example, the method for reconstructing energy-independent material properties like effective atomic-number and electron density from dual-energy CT scans \cite{busi2019method,azevedo2016system,champley2019method} and the method for tissue characterization in
\cite{mccollough2015dual, so2021spectral}
require a precise calibration of the source-detector spectral response.
Direct measurement of the spectral response of the whole X-Ray CT system is difficult since the detector response is hard to measure.
Based on Beer–Lambert's law\cite{lambert1760photometria}, we can use a linear model for transmission measurements of objects with known dimensions and composition to reconstruct the discretized spectral response of a CT scanner, but this yields a highly ill-conditioned system.
Champley et al. \cite{champley2019method} use linear least-squares to do spectral estimation (LSSE) with constraints to enforce non-negativity. However, this method requires an accurate initial guess close to the true spectrum.
Various regularization methods\cite{ruth1997estimation, yan1999modeling} are used to overcome the issues introduced by the ill-conditioned nature of the problem.
SVD-based algorithms have also been applied to the spectral estimation problem\cite{tominaga1986530, armbruster2004spectrum, leinweber2017x}.
Sidky et al.\cite{osti_20711775} represent the spectrum as a linear combination of B-splines and use EM (Expectation–maximization) to find the solution.
Zhao et al.\cite{zhao2014indirect} estimate the spectrum as a linear combination of six Monte Carlo model spectra.
Liu et al.\cite{liu20201495} introduce compressed sensing to estimate the spectrum.
The central ideas of the above methods have a common theme: try to solve the ill-conditioned inverse problem either with regularized optimization or by introducing basis spectra to perform the estimation.
In this paper, we introduce a novel dictionary-based spectral estimation (DictSE) method that can efficiently reconstruct the overall spectral response of a CT system from transmission scans of multiple known objects, without the need for accurate initialization.
We represent the unknown spectral response using an over-complete dictionary that accounts for vast combinations of different source spectra, filter attenuation characteristics, and detector energy-response models.
We formulate the reconstruction problem as a MAP estimation framework that combines a linear beam-hardening forward model along with prior constraints.
Specifically, we impose an $L_0$ sparsity constraint to limit the support for the spectrum representation and a simplex constraint to account for the bright-dark normalization of the transmission data.
We present a novel iterative optimization strategy that alternates between support selection and pairwise iterative coordinate descent (ICD) update to find the optimal sparse representation of the spectrum.
Finally, we demonstrate DictSE through a cross-validation experiment on four datasets collected at beamline 8.3.2 of ALS.
\section{Reconstruction model}
\label{sec:rec_model}
In this section, we describe the spectral estimation problem and our proposed solution in more detail. We use a physics-based model for X-ray transmission measurements and discretize the model in energy and by projection to produce a linear measurement model.
Then we introduce a dictionary-based framework and express the reconstruction as a MAP estimation problem.
Finally, we solve this dictionary-based MAP problem using support selection to enforce sparse coding and an ICD algorithm modified to enforce a simplex constraint on the coefficients of the selected dictionary elements.
\vspace{-2ex}
\subsection{Physics-based Model and Discretization}
\label{ssec:phy_model}
\begin{figure}[!t]
\centering
\centerline{\includegraphics[width=7cm]{Figs/physic_model.png}}
\caption{\textbf{Physics model of CT scanning.} A homogenous cylindrical object is scanned. The entire spectral response can be decomposed into the product of the X-ray source spectrum, filter response, and detector response, respectively.}
\label{fig:phy_model}
\end{figure}
Fig.~\ref{fig:phy_model} illustrates the setup for X-ray spectral estimation.
Using notation as in that figure, the spectral response of this CT system is the product of the X-ray source spectrum, filter response, and detector response, which yields the response function
\begin{equation}
\label{equ:spec_model}
S(E)=S_{sr}(E) S_{ft}(E) S_{dt}(E) \ ,
\end{equation}
and our goal is to estimate $S(E)$.
We relate $S(E)$ to measurements by first defining $I$ and $I_0$ to be the measured intensity of the object scan (with sample) and the blank scan (without sample), respectively.
Based on Beer–Lambert's law, we have
\begin{equation}
\label{equ:obj_scan}
I=\int_0^{E_{\max }} S(E) \cdot \exp \left\{-\int_L \mu(E, r) d r\right\} d E \ ,
\end{equation}
where $\mu(E, r)$ is linear attenuation coefficient (LAC) with units $mm^{-1}$. Likewise, $I_0$ is defined as in \eqref{equ:obj_scan} with $\mu = 0$.
The (normalized) transmission through line $L$ is then defined as
\begin{equation}
\label{equ:transm}
\begin{aligned}
y & =\frac{I}{I_0}=\int_0^{E_{\max }} \bar{S}(E) \cdot \exp \left\{-\int_L \mu(E, r) d r\right\} d E \ ,
\end{aligned}
\end{equation}
where $\bar{S}(E)=\frac{S(E)}{\int_0^{E_{\max }} S(E) d E}$ is the normalized response.
We assume each sample consists of solid rods made from known materials taken from a specified reference set $\Phi$.
For each material $s\in \Phi$ and projection $i\in\{0,...,M-1\}$, we define $L_{i, s}$ to be the path length of the $i^{th}$ projection through the $s^{th}$ material and $\mu_s(E)$ to be the LAC of the $s$ material at energy $E$.
Assuming that $I_0$ is independent of projection and noise-free, this yields the transmission for the $i^{th}$ projection as
\begin{equation}
\label{equ:scaled_transm}
\begin{aligned}
y_i & =\int_0^{E_{\max }} \bar{S}(E) \cdot \exp \left\{-\sum_{s \in \Phi} \mu_s(E) L_{i, s}\right\} d E+\tau_i \ ,
\end{aligned}
\end{equation}
where $\tau_i$ is additive noise.
We discretize in energy by subdividing into non-overlap\-ping bins $[E_j, E_{j+1}]$ and defining $x_j=\int_{E_j}^{E_{j+1}} \bar{S}\left(E_j\right) d E$.
Making the approximation that each $\mu_s$ is constant on each bin, we define the coefficient from the $j^{th}$ energy bin to the $i^{th}$ projection as $F_{i, j}=\exp \left\{-\sum_{s \in \Phi} \mu_s\left(E_j\right) L_{i, s}\right\}. $
Then \eqref{equ:scaled_transm} becomes
\begin{equation}
\label{equ:y_discrete}
\begin{aligned}
y_i & \approx \sum_{j=0}^{J-1} \int_{E_j}^{E_{j+1} } \bar{S}(E) F_{i,j} d E+\tau_i = \sum_{j=0}^{J-1} F_{i,j} x_j + \tau_i \ .
\end{aligned}
\end{equation}
Using this along with constraints to yield a normalized spectrum, the forward model is
\begin{equation}
\begin{aligned}
\label{equ:smp_model}
&Y=F x + \tau, \\
&\text{s.t. } \|x\|_1=1, x_j \geq 0 \; \forall j,
\end{aligned}
\end{equation}
where $Y\in \mathbb{R}^M$ is a vector of $M$ normalized transmission measurements, $F \in \mathbb{R}^{M \times N_e}$ is a forward matrix over $N_e$ energy bins, $\tau$ is additive noise, and $x\in \mathbb{R}^{N_e}$ is the unknown vector discretization of $\bar{S}\left(E\right)$ that we seek to estimate.
\vspace{-2ex}
\subsection{Dictionary-based Model}
\label{ssec:dict_model}
Developing a dictionary-based method instead of directly estimating the spectrum at each energy bin has several motivations.
When the discretization of $x$ is very fine, the projection model matrix $F$ has a large null space, leading to an underdetermined reconstruction, which requires a good initial estimate.
Also, with a dictionary-based method, we can apply a sparsity-promoting penalty so that the dimensionality of the optimization problem can be significantly decreased and the time for reconstruction reduced.
More specifically, as indicated in \eqref{equ:smp_model}, we model $x$ as a discrete probability distribution, so that the entries of $x$ are nonnegative and sum to 1. We call this the simplex constraint and use $\mathcal{S}$ to represent the set of all such possible vectors.
We use a fixed, over-complete dictionary $D$ to represent the unknown normalized spectrum, $x$, as
\vspace{-1ex}
\begin{equation}
x=D\omega
\end{equation}
where $D$ is an $\mathbb{R}^{N_e\times N_k}$ matrix, each column $D_{*,k}\in \mathcal{S}$ represents a normalized basic spectrum.
With this $D$, the transmission model can be rewritten as
\vspace{-1ex}
\begin{equation}
\label{equ:dict_model}
Y=F D \omega + \tau \ .
\end{equation}
\vspace{-2ex}
\subsection{MAP Estimate}
\label{ssec:map}
We use a Bayesian framework to estimate $\omega$ from transmission measurements under sparsity and simplex constraints. We define transmission weights using a diagonal matrix $\Lambda$, where we take $\Lambda_{i,i} = 1/(y_i M)$. We then define
the loss function $l(\omega) = \frac{1}{2}\|Y-F D \omega\|_{\Lambda}^2$, in which case the MAP estimate is given by
\begin{equation}
\label{equ:map_model}
\widehat{\omega}=\arg \min _{\substack{\omega \in \mathcal{S} \\\|\omega\|_0 \leq C}}l(\omega) \ ,
\end{equation}
where $\omega \in \mathcal{S}$ enforces $x \in \mathcal{S}$ since each column of $D$ is a simplex; the $L_0$ constraint ensures $\omega$ has no more than $C$ non-zero components.
\begin{algorithm}[!t]
\caption{Dictionary-based Spectrum Reconstruction Algorithm (see equation \eqref{equ:kbfw_model} for notation)}
\label{alg:mp_est}
\begin{algorithmic}[1]
\State{Initialize $k^* \gets \arg \min _{\substack{k }}l(\epsilon_{k})$, $\omega=\epsilon_{k^*}$, $\Omega=\{k^*\}$}
\While{$|\Omega| < C$} \mbox{\hspace{12pt}} // Support selection
\State{$\left(k^*, \beta^*\right) \gets \arg \min _{\substack{k \notin \Omega \\ \beta \in[0,1)}}l'(k,\beta | \widehat{\omega})$ }
\State{$\omega \gets \beta^* \omega+\left(1-\beta^*\right) \epsilon_{k^*}$}
\State{$\Omega \gets \Omega \cup \{k^*\}$}
\State{$g \gets k^*$} \mbox{\hspace{12pt}} // Select new element for next loop
\While{not converged} \mbox{\hspace{12pt}} // Pairwise ICD update
\For{$k\in \Omega$ and $k\neq g$}
\State{$\alpha^* \leftarrow \arg \min _{\alpha \in\left[-\omega_k, \omega_g\right]}l\left(\omega+\alpha (\epsilon_k -\epsilon_g)\right)$}
\State{$\omega \leftarrow \omega+\alpha^* (\epsilon_k -\epsilon_g)$}
\EndFor
\EndWhile
\EndWhile
\State{$\hat{\omega} \gets \omega$ \mbox{\hspace{12pt}} // Return the final estimate }
\end{algorithmic}
\end{algorithm}
\vspace{-2ex}
\subsection{Support Selection and Pairwise ICD update}
\label{ssec:ds_pwicd}
To minimize the MAP cost function, we alternate between greedy support selection and a pairwise ICD update.
Inspired by the Orthogonal Matching Pursuit (OMP)\cite{pati1993}, our DictSE builds support set $\Omega=\{k: \omega_k \neq 0\}$ by adding one basic spectrum from the dictionary at a time and then updating the coefficients.
However, the conventional support selection method in OMP does not account for simplex constraints on $\omega$.
Further, the OMP method assumes that the dictionary atoms are normalized, whereas, in our spectral estimation problem, the product of forward matrix $F$ and the spectrum dictionary $D$ is not.
Thus choosing a basis spectrum with conventional matching pursuit is inappropriate.
To describe our alternative method for support selection, we first define a function $l'(k,\beta | \widehat{\omega})$ that measures the fit to data obtained by scaling the existing coefficients $\hat{\omega}$ by $\beta$ and using the remaining weight on the $k^{th}$ coefficient. That is,
\begin{equation}
\label{equ:kbfw_model}
\begin{aligned}
l'(k,\beta | \widehat{\omega})&=l(\beta \widehat{\omega}+\left(1-\beta\right) \epsilon_{k})\\
&=\frac{1}{2}\left\|Y-\beta F D \widehat{\omega}-\left(1-\beta\right) F D_{*, k}\right\|_{\Lambda}^2
\end{aligned}
\end{equation}
where $\beta \in [0,1)$ enforces $\omega \in \mathcal{S}$ and $\epsilon_{k}$ is a one-hot vector for the $k^{th}$ spectrum.
Then we select a new element from the dictionary by minimizing $l'(k,\beta | \widehat{\omega})$ to obtain
\vspace{-1ex}
\begin{equation}
\begin{aligned}
\left(k^*, \beta^*\right) &\leftarrow
\arg \min _{\substack{k \notin \Omega\\ \beta \in[0,1)}}\left\{l'(k,\beta | \widehat{\omega})\right\}.
\end{aligned}
\end{equation}
This minimization can be solved easily for each $k \notin \Omega$ since equation (\ref{equ:kbfw_model}) is quadratic in the scalar $\beta$. In fact, defining $e_k = FD\widehat{\omega}- FD_{*,k}$, we have
\vspace{-1ex}
\begin{equation}
\beta_k=\frac{e_k^T \Lambda (Y-FD_{*,k})}{\| e_k\|^2_{\Lambda}} \ .
\end{equation}
Using $\beta_k$ in equation \eqref{equ:kbfw_model} allows us to find $k^*$ that minimizes equation \ref{equ:kbfw_model}; this $k^*$ is then included in $\Omega$ with $\omega_{k^*} = 1- \beta^*$ and the remaining $\omega_k$ scaled by $\beta^*$.
To rebalance the weights while enforcing the simplex constraint, we use pairwise ICD between the most recently added dictionary element with index $g$ and the remaining elements in $\Omega$, as shown in lines 6-12 of Algorithm \ref{alg:mp_est}.
More precisely, after choosing $g$, we loop repeatedly over $k \in \Omega \setminus \{g\}$, in each case finding an optimal pairwise update $\omega \leftarrow \omega+\alpha^* (\epsilon_k -\epsilon_g)$, where $\alpha^*$ is chosen by
\vspace{-1ex}
\begin{equation}
\alpha^* \leftarrow \arg \min _{\alpha \in\left[-\omega_k, \omega_g\right]}\left\{l\left(\omega+\alpha (\epsilon_k-\epsilon_g) \right)\right\} \ .
\end{equation}
The constraints on $\alpha$ in the minimization ensure each $\omega_k \geq 0$. Since $l(\omega)$ is quadratic, $\alpha^*$ can be computed as below
\vspace{-1ex}
\begin{equation}
\alpha^*=\text{Clip}\left\{\frac{(Y-F D \omega)^{\mathrm{T}} \Lambda F D (\epsilon_k -\epsilon_g)}{\left\|F D (\epsilon_k -\epsilon_g)\right\|_{\Lambda}^2}, [-\omega_k, \omega_g]\right\}
\end{equation}
The pairwise ICD update will stop when the total update is less than $10^{-6}$.
The algorithm is summarized in Algorithm \ref{alg:mp_est}.
\begin{table}[t]
\centering
\caption{Setup for X-ray Scanning}
\begin{tabular}{ll}
\hline
Projection Geometry: & Parallel beam geometry \\
Source filter: & 2 $mm$ Silicon \\
Scintillator: & 50 $\mu m$ $Lu_{3}Al_{5}O_{12}$ \\
Max Energy: & 100 KeV \\
Views Spanning: & Equi-spaced in $[0,2\pi]$ \\
Detector Pixel Size: & 0.00065 mm \\
Nviews $\times$ Nrows $\times$ Ncolumns: & $2625 \times 100 \times 2560$ \\
Sample-detector distance: & $0.3$ mm \\ \hline
\end{tabular}
\label{table:setup_dataset}
\end{table}
\label{sec:imp}
\begin{figure}[!t]
\centering
\centerline{\includegraphics[width=8.5cm]{Figs/dataset.png}}
\caption{\textbf{Real scans of 4 rods with different materials.}}
\label{fig:scans}
\end{figure}
\section{ Implementation}
\label{sec:implementation}
In this section, we describe the normalized transmission data $Y$, the projection matrix $F$, and the dictionary of spectra $D$, which are required to estimate the spectral response of an X-ray CT system using Algorithm \ref{alg:mp_est}.
\vspace{-2ex}
\subsection{ Transmission data $Y$}
\label{ssec:trans_data}
We collected four CT datasets of different metal rods, $\Phi = \{Ti, V, Al, Mg\}$, at beamline 8.3.2 of the ALS.
As shown in Fig.~\ref{fig:scans}, for each dataset, we scanned a single rod.
The same CT scanner setup was used to collect all datasets.
Table~\ref{table:setup_dataset} gives more detailed information about the X-ray CT measurements.
We scanned $15$ bright scans and $10$ dark scans and averaged them to obtain $\bar{I}_{\text {bright }}$ and $\bar{I}_{\text {dark }}$.
Then, for each view, we normalized measurement data $I_{\text {scan }}$ to obtain
$ Y=\frac{I_{\text {scan }}-\bar{I}_{\text {dark }}}{\bar{I}_{\text {bright }}-\bar{I}_{\text {dark }}}$.
\vspace{-2ex}
\subsection{ Forward Matrix $F$}
\label{ssec:fw_mat}
For each dataset, the sample is a single rod that is solid and pure.
To calculate $F_{i,j}$, we need the path length of the $i^{th}$ projection through this rod.
To obtain the path length, we used filtered back projection to reconstruct the volume of the rod and then generated a mask to represent the object area.
Using this mask, we calculated the path length for each projection and used this to calculate the forward matrix $F$.
\vspace{-2ex}
\subsection{ Dictionary Generation $D$}
\label{ssec:dict_gen}
As mentioned in equation (\ref{equ:spec_model}), the spectral response can be modeled as the product of the X-ray source spectrum, filter response, and detector response, so we can create a dictionary by varying one or more of these elements.
The X-ray source spectrum is fixed in this experiment; we used an estimate provided by the beamline scientists at the ALS Beamline 8.3.2.
Therefore, in this implementation, we generated the dictionary by varying the filter and detector responses.
Based on the spectrum models in Ref.~\cite{champley2019method}, the filter response and detector response are determined by their material properties and thicknesses, which we vary as in Table~\ref{table:dict} to generate our dictionary.
By combining two groups of responses, we obtain an over-complete dictionary $D$ containing $60\times36=2160$ normalized responses spectra.
\begin{table}[t]
\centering
\caption{Dictionary Generation List}
\begin{tabular}{|cccc|}
\hline
\multicolumn{4}{|c|}{Filter response} \\ \hline
\multicolumn{1}{|c|}{Material} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Thickness Range\\ $mm$\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Step\\ $mm$\end{tabular}} & \begin{tabular}[c]{@{}c@{}}\# of \\ responses\end{tabular} \\ \hline
\multicolumn{1}{|c|}{$Al$} & \multicolumn{1}{c|}{0.1$\sim$5.9} & \multicolumn{1}{c|}{0.2} & 30 \\ \hline
\multicolumn{1}{|c|}{$Cu$} & \multicolumn{1}{c|}{0.2$\sim$0.49} & \multicolumn{1}{c|}{0.01} & 30 \\ \hline
\multicolumn{4}{|c|}{Detector response(Scintillator)} \\ \hline
\multicolumn{1}{|c|}{$Lu_{3}Al_{5}O_{12}$} & \multicolumn{1}{c|}{0.025$\sim$0.095} & \multicolumn{1}{c|}{0.002} & 36 \\ \hline
\end{tabular}
\label{table:dict}
\end{table}
\section{ Experimental Results}
\label{sec:exp}
We compared our proposed DictSE method with a least-squares spectral estimation (LSSE) method provided by Livermore tomography tools (LTT) \cite{CHAMPLEY2022102595} on four datasets described in section \ref{ssec:trans_data}.
An initial spectrum for the LSSE method was generated by LTT using $3 mm$ silicon as a source filter and $50 \mu m$ $Lu_{3}Al_{5}O_{12}$ as a scintillator.
We then evaluated the estimated spectra of DictSE and LSSE using leave-one-out cross-validation since we do not have a ground truth response.
Table~\ref{table:res} demonstrates that DictSE's reconstructed spectra outperform the LSSE's reconstructed spectra in NRMSE for all cross-validation cases.
For each case $v\in\{1,2,3,4\}$, we computed NRMSE $= \frac{\| Y_{v}-\widehat{Y}_{v}\|_{2} }{\| Y_{v}\|_{2} }$ to compare transmission measurements $Y_{v}$ and transmission value of the forward model using estimated spectrum $\widehat{Y}_{v}=F_{v}D\widehat{\omega}_v$ on the validation rod.
Fig.~\ref{fig:est_sp} shows all four cases of cross-validation reconstructed spectra using both the DictSE and LSSE methods.
For each case, DictSE's estimated spectra are smoother than the LSSE's.
Also, from the shape of the reconstructed spectra over all cases, DictSE is less data-sensitive than LSSE.
\begin{table}[!t]
\centering
\caption{Leave-One-Out Cross-Validation NRMSE}
\begin{tabular}{c|c|c|c|c}
\hline
Case & Fit & Test & LSSE & DictSE \\
\hline
1 & $Ti, Mg, Al$ & $V$ & 0.0331 & \textbf{0.0315} \\
2 & $V, Mg, Al$ & $Ti$ & 0.0624 & \textbf{0.0242} \\
3 & $V, Ti, Al$ & $Mg$ & 0.0343 & \textbf{0.0122} \\
4 & $V, Ti, Mg$ & $Al$ & 0.0483 & \textbf{0.0093} \\
\hline
\end{tabular}
\label{table:res}
\end{table}
\begin{figure}[!t]
\centering
\centerline{\includegraphics[width=8.5cm]{Figs/est_sp.png}}
\caption{\textbf{Reconstructed Spectra with DictSE and LSSE.} }
\label{fig:est_sp}
%
\end{figure}
\section{ Conclusion}
\label{sec:conclude}
This work provides a novel application of dictionary learning to X-Ray spectral estimation, allowing an efficient spectrum reconstruction from a vast dictionary obtained from CT datasets.
Our method uses a greedy support selection method to do sparse coding followed by pairwise ICD to do minimization while enforcing a simplex constraint.
Leave-one-out cross-validation experiments on four datasets demonstrated that our DictSE method outperforms the LSSE method in NRMSE.
\section{ACKNOWLEDGMENTS}
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and LDRD project 22-ERD-011. The authors acknowledge Dula Parkinson for his support during beamtime. Charles Bouman was partially supported by the Showalter Trust, and Greg Buzzard was partially supported by NSF CCF-1763896.
\vfill\pagebreak
\bibliographystyle{IEEEbib}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,129 |
1. "Sweet Caroline" by Neil Diamond 2. "Bohemian Rhapsody" by Queen 3. "Love Shack" by The B-52's song 4. "Dancing Queen" by Abba 5. "Say It Ain't So" by Weezer 6. "Stand By Me" by Ben E. King 7. "Take Me Home, Country Roads" by John Denver 8. "Build Me Up, Buttercup" by The Foundations 9. "Hotel California" by Eagles 10. Brown Eyed Girl By Van Morrison. I'm a Believer by Smashmouth. Fat Bottomed Girls by Queen. American Pie by Don Mclean. Carry On, Wayward Son by Kansas. I'm Gonna Be (500 Miles) by The Proclaimers. Fergalicious by Fergie. Stacy's Mom by Fountains of Wayne. Follow Me by Uncle Kracker. how long do spawn bags take to colonize shroomery [Top 21 Songs] List Of Best Sing Along Songs 60s And 50s. This list is purely based on the evaluation and comments of many people on social networks, famous websites as well as prestigious magazines. So, please let us know through the comments section if you have other ideas. #1. Maybellene. Single by Chuck Berry; Genre: Rock and roll, rockabilly4 thg 12, 2013 ... "I am home, I am free," sings 48-year old Trent Reznor, and he proves it by tapping his inner alt-rock manbeast for a proudly anthemic grinder ... range rover tdv8 timing chain replacement Aug 01, 2020 · The Bouncing Souls – True Believers. Few songs feel as deeply, earnestly good to sing along to as True Believers. Not only does the track epitomise the big, everybody-knows-the-words punk of New ... 26 thg 9, 2019 ... If You Can Actually Sing · "Love on the Brain," Rihanna · "The Story," Brandi Carlile · "Angel Of the Morning," Juice Newton · "I Like It," DeBarge. cornville 10 mile yard sale 2022 Find your new go-to sing-along with our list of the best karaoke songs, from easy anthems to crowd-pleasing gems. 🙌 Awesome, you're subscribed! Thanks for subscribing! Look out for your first newsletter in your inbox soon! We know this cit...The Bouncing Souls – True Believers. Few songs feel as deeply, earnestly good to sing along to as True Believers. Not only does the track epitomise the big, everybody-knows-the-words punk of New ...1. Journey – Don't Stop Believin'. Because, regardless of where you are from, everyone relates to that small town girl or city boy. The gold medal for the most likely song to initiate a spontaneous crowd sing-a-long, without a doubt, goes to Don't Stop Believin'. john deere 425 overheatingThe feeling "Dancing Queen" is known for is enough to make it one of the best sing-along songs on this list. 5. Livin' On a Prayer - Bon Jovi. A hard-hitting ham that offers unforgettable production, "Livin' On a Prayer" is easy to follow word for word, but the hook is easily the most powerful part of the song. severe pain in stomach Jun 27, 2022 · So without further ado, we loudly present the 25 best rock songs of all time. Video unavailable Watch on YouTube Watch on "I Love Rock 'N Roll" (1981) — Joan Jett So what if it's a cover – Joan... As far as sing-along choruses go, few metal songs are as awesome as Twisted Sister's We're Not Gonna Take It. Not only is it a tune everyone's familiar with, but it embodies the grand ...#1. Maybellene Single by Chuck Berry Genre: Rock and roll, rockabilly Released: July 1955 This song was the first hit by singer Chuck Berry. There's no denying that it's one of the best rock and roll songs of all time. The proof is that Rolling Stone magazine said that rock & roll guitar starts here. #2. Heartbreak Hotel Single by Elvis Presley 11. "Just a Friend" Biz Markie. The song "Just a Friend" was released in 1989, it is a love song with a twist, the artist Biz Markie sings it in an ironic fashion. The part of the song "Oh baby you have got what I need" in a sing-along with several people will make for some hilarious moments, especially while drunk.Sweet Home Chicago - Robert Johnson. This unassuming folk blues song comes to us only from field recordings, but it was incredibly influential on many British rock stars. Johnson's raw guitar style and troubled lyrics heavily influenced the Stones, Eric Clapton, Led Zeppelin, and others.2. Bohemian Rhapsody - Queen. Made more famous by Mike Meyers and his crew in Wayne's World, this song is the ultimate car sing-along, and not just because of the awesome head-banging bridge. Everyone usually loves the break into the fast-paced part of the song that starts with "I see a little silhouetto of a man…". pf940v2 safariland holster Freya Riding's simple and stripped back piano makes you truly appreciate the vocals, which should be the focus of any performance. With moving and haunting lyrics, this would be a good cover song to get creative with. Check out this version of Castles sung and played live.Besides, it should be considered a crime if you don't sing along. Also, it's perhaps one of the most effective means of transporting you back to the '70s. 2016 highlander transmission fluid change 28 thg 9, 2021 ... "O Come, O Come Emmanuel" by Sufjan Stevens · "Pretty Paper" by Dolly Parton and Willie Nelson · "Christmas (Baby Please Come Home)" by Darlene ...And we are singing with you! 1. Journey – Don't Stop Believin'. Because, regardless of where you are from, everyone relates to that small town girl or city boy. The gold medal for the most likely song to initiate a spontaneous crowd sing-a-long, without a …Top 50 Rock Songs of the '80s🔴 YouTube Playlist: https://goo.gl/f2j18x Spotify Playlist: https://goo.gl/c5Qz44 woman jumps off bridge in pennsylvania 10 thg 8, 2022 ... Instead find a song that people remember the words to and can sing along with. That said, try to pick something other than the top hit by that ...17. 'Faith' by George Michael. 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Get ready to rock out to these top 20 karaoke rock songs!Apr 15, 2019 · As you can imagine, getting the list down to just ten songs was such a struggle! Because of this, we had to sneak in a few that we just love: 'Love Shack' – The B52s. 'Sweet Home Alabama' – Lynyrd Skynyrd. 'I've Had The Time of My Life' – Bill Medley and Jennifer Warnes. 'I'm Gonna Be (500 Miles)' – The Proclaimers. 10. Chicago - "If You Leave Me Now". Even if you find this a bit cheesy, today's music still won't hold a candle to it. It's an absolute classic and that first line alone is enough to send you into a state of incredible calm. Besides, it should be considered a crime if you don't sing along. best buy screen protector installation Unlike its evil twin in 1980s rock, Billy Joel's 'We Didn't Start the Fire,' the song was not a huge pop hit; on its 1987 album, Document, R.E.M. was still emerging from the niche of ...15. "Born Under a Bad Sign" by Albert King. Song year: 1967. Albert King's single "Born Under a Bad Sign" was a crossover blues hit. The track employed R&B rhythms and the '60s growing fascination with astrology to make one of the most influential blues songs ever. Today the song is considered a blues standard.Something that always makes a song a perfect choice to play in a bar setting is a 'sing along' chorus, and few songs have choruses that resonate quite as well as 'Shake It Off' by Taylor Swift. This is the main single from her fifth studio album titled, "1989", and it even went Diamond in the USA, with 10 million copies sold.Top 50 Rock Songs of the '80s🔴 YouTube Playlist: https://goo.gl/f2j18x Spotify Playlist: https://goo.gl/c5Qz44 15. "Born Under a Bad Sign" by Albert King. Song year: 1967. Albert King's single "Born Under a Bad Sign" was a crossover blues hit. The track employed R&B rhythms and the '60s growing fascination with astrology to make one of the most influential blues songs ever. 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Listen now only on Spotify: Top pop and rock anthems from the 90's until today. Spotify · Playlist · 50 songs · 1.1M ... Aug 01, 2020 · The Bouncing Souls – True Believers. Few songs feel as deeply, earnestly good to sing along to as True Believers. Not only does the track epitomise the big, everybody-knows-the-words punk of New ... can private investigators hack social media But 'Like a Rolling Stone' changed it all. I mean it was something that I myself could dig. It's very tiring having other people tell you how much they dig you if you yourself don't dig you.". So here, have some inspiration and sing – this song won't give you a hard time. 2. The Beatles – "Yellow Submarine".Jun 03, 2022 · 1 of 22 1. Bon Jovi's "Livin' on a Prayer" Credit: Theo Wargo/Getty Images '''Livin' On a Prayer' is the so-obvious choice. Really, can there be any other choice?'' — lilbooth02 '''Livin on a... Popular songs for kids! 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But the thing that most classic rock hits have in common is that will probably never die. Similar Playlists: Best 80s Rock Songs. Best Rock Love Songs The song is best played loud, and you can scream or sing along. The song hit number 1 on the chart list in 1988. The lyrics came from a poem. Next: The best rock love songs of all time (our list of favs) 11. Tom Sawyer - Rush. Here are the best R&B karaoke songs, ranked by fans everywhere.Jan 17, 2022 · As a Heartland rock staple, Free Fallin by the Heartbreakers, the tune peaked as number 7 on the Billboard Hot 100, while being praised as an homage by countless artists, it is considered as one of the Greatest Songs of All Time by Rolling Stone Magazine. great pyrenees male vs female Discover 100 Greatest Sing-a-long Songs by Various Artists. Find album reviews, track lists, credits, ... The Best [Edit] Mike Chapman / Holly Knight. Tina Turner: 04:10 ... I Wanna Rock. …Turn on with our far-out list of the best '60s rock songs. 1. "Louie Louie" by The Kingsmen. Song year: 1963. Portland, Oregon's The Kingsmen changed the face of rock and roll with their raw, garage rock version of "Louie Louie.". The raucous and incoherent song was a runaway hit with teens, leading concerned parents to complain.Best Classic Rock Songs – Final Thoughts. We hope you enjoyed our list of the best classic rock songs! As you can see, classic rock is a genre of its own – and it's quite powerful. But the thing that most classic rock hits have in common is that will probably never die. Similar Playlists: Best 80s Rock Songs. Best Rock Love Songs kioti kl130 manual And for gigging guitarists, these rock n roll hits are an easy way to garner more attention from the kind of rowdy crowd that demands drunken sing-alongs. bank of america account locked Sweet (1973) The fast-paced tempo and skilled vocals of Brian Connolly inspire anyone to join him in screaming, "Oh yeah!" 14. "Free Fallin'". Tom Petty and the Heartbreakers (1989) The mellow tune and memorable hook make this 1989 hit a perfect sing-along song that's recognizable from the first strum of the guitar. 13.The Top Karaoke Songs: The Bottom Line. It's best to sing a song you already know. That way you won't have to stare at the machine to read the lyrics. You can relax, get into your performance, and just have fun. Choose your favorite songs that people already know and love. When everyone in the room is on the same page there's a connection.15 thg 4, 2021 ... These Are The Best Karaoke Songs Of All Time ... this melancholy song about a changing world, Styles revived the entire classic rock genre."Raise Your Glass," by Pink, is the ideal sing-along tune for any bar and individuals of all stages of inebriation. Even if you don't know the lyrics, everyone will raise their glass and sing along as the music erupts into a chorus. So, how about that, lads and gals? Raise your glass! #9. "Old Time Rock And Roll" by Bob Seger ford transit 22 mass air flow sensor fault Jan 17, 2022 · As a Heartland rock staple, Free Fallin by the Heartbreakers, the tune peaked as number 7 on the Billboard Hot 100, while being praised as an homage by countless artists, it is considered as one of the Greatest Songs of All Time by Rolling Stone Magazine. Sweet Caroline – Neil Diamond. Try to count how many times you have heard, "Sweet Caroline, bum-bum-bum!". Neil Diamond delivered a classic that everyone seems to love to sing … darvin furniture outlet Dec 10, 2021 · "Jailhouse Rock" is one of Elvis' most popular songs, and also incredibly easy to sing along to as a group. It gets the party going with its peppy lyrics and sound. "Folsom Prison Blues" by Johnny Cash Song Year: 1957 Johnny Cash's signature gruff singing voice is one that many seniors love to try to emulate. Spotify · Playlist · 50 songs · 1.1M likes. Listen now only on Spotify: Top pop and rock anthems from the 90's until today. Spotify · Playlist · 50 songs · 1.1M ... We Will Rock You by Queen Queen gets two songs on our list because they have the most popular songs that people love to sing along to. When someone starts up "We Will Rock You," a crowd of all ages is likely to start singing, clapping, and stomping along acapella-style. Stomp, stomp, clap, pause, stomp, stomp, clap, pause. Yeah. beatles bootlegs blogspot Criteria: A rock anthem is a powerful, celebratory rock song with arena-rock sound often with lyrics celebrating rock music itself and simple sing-a-long choruses, chants, or … austin face and body May 17, 2016 · 15. "Ballroom Blitz" Sweet (1973) The fast-paced tempo and skilled vocals of Brian Connolly inspire anyone to join him in screaming, "Oh yeah!" 14. "Free Fallin'" Tom Petty and the Heartbreakers (1989) The mellow tune and memorable hook make this 1989 hit a perfect sing-along song that's recognizable from the first strum of the guitar. 13. It's the chorus in a song that gravitates the song to a whole new level. Rock songs are often groove-laden, and the catchy lyrics of songs evoke a special feeling among listeners. The space below showcases a list of the best rock songs with a catchy chorus. 100 Greatest Rock Songs with a Catchy Chorus. Paradise City- Guns N' RosesElton John - Tiny Dancer. One of those rare occasions where a song gets ultra-popular more than 30 years after it's original release thanks to just one epic movie scene. Tiny Dancer is, of course, more of a piano song, but it works nicely on the guitar even though the chord progression is a bit harder to memorize.15. "Born Under a Bad Sign" by Albert King. Song year: 1967. Albert King's single "Born Under a Bad Sign" was a crossover blues hit. The track employed R&B rhythms and the '60s growing fascination with astrology to make one of the most influential blues songs ever. Today the song is considered a blues standard. ford 1900 tractor engine rebuild kit If you really want to excite the crowd, do a crazy little dance while you sing. It's also a great sing along song. Lyrics here. Green Day — Boulevard of Broken Dreams No Green Day song is all that hard to sing and most are good karaoke songs. For a lot of similar bands, check out our lists of groups like Fall Out Boy and bands like Blink 182.Aug 01, 2020 · The Bouncing Souls – True Believers. Few songs feel as deeply, earnestly good to sing along to as True Believers. Not only does the track epitomise the big, everybody-knows-the-words punk of New ... 11. "Just a Friend" Biz Markie. The song "Just a Friend" was released in 1989, it is a love song with a twist, the artist Biz Markie sings it in an ironic fashion. The part of the song "Oh baby you have got what I need" in a sing-along with several people will make for some hilarious moments, especially while drunk.Song year: 1972. "Melisa" was first written in 1967, before the Allman Brothers Band was even formed. Greg Allman is the man behind the lyrics of this standout ballad. This song is about a gypsy-man vagabond who can only think about a woman named Melissa. 36 inch styrofoam ball It is now renowned as one of the greatest rock songs of all time, and without a doubt one of the best acoustic tunes out there, so much that Chris Cornell recorded a cover version that was only released posthumously in 2020. 4. Hotel California by The Eagles. Album: Hotel California. Release Date: February 22nd, 1977.3 thg 6, 2022 ... 22 top stadium-concert sing-along songs: Your picks! ; 1 · 1. Bon Jovi's "Livin' on a Prayer" · Credit: Theo Wargo/Getty Images ; 2 · 2. Neil ...Here is a list of 10 favorites that might inspire you the next time you decide to burst into song. 01 of 10 Bon Jovi - 'Livin' On a Prayer' (1986) Mercury Bon Jovi's "Livin' On a Prayer" stands as one of the genuinely iconic songs of 80s rock. The band performed an acoustic version as part of a concert to help heal after the 9/11 terrorist attacks. kobalt 40v hedge trimmer Feb 02, 2022 · 1. "Sweet Caroline" by Neil Diamond Neil Diamond – Sweet Caroline High Quality neildiamond "Sweet Caroline" is the catchiest song by Neil Diamond. The best part of the tune is the chorus. Listeners cannot help but belt it out once that part plays. Playing this 1969 hit will encourage people to sing along with passion. May 27, 2022 · Nearly every list of the best songs ever recorded has 'Be My Baby' somewhere near the top, and deservedly so. Ronnie Spector was rock & roll's first bad girl, so pay your respects by putting this... nfc tools examples 19 thg 11, 2020 ... Nonetheless, these are examples of rock music. Each piece has lyrics you can sing along to, instrumental parts you can picture yourself ... spokane superior court filing fees 11 thg 12, 2018 ... Gather the family and get ready to rock. This is the perfect Christmas song for kids to sing along to. Double points if using the Kids ...Discover 100 Greatest Sing-a-long Songs by Various Artists. Find album reviews, track lists, credits, awards and more at AllMusic. ... The Best [Edit] Mike Chapman / Holly Knight. Tina Turner: 04:10 ... I Wanna Rock. Dee Snider / Tom Werman. Twisted Sister: 03:0529. "Shake It Off" by Taylor Swift. 30. "Hey Jude" by The Beatles. Sing-Along Songs – Final Thoughts. 1. "Sweet Caroline" by Neil Diamond. This is one of those enjoyable chanting tunes that includes "parts" that were added to the song at a later time in order to encourage more people to participate in the enjoyment.Whatever you think about singing karaoke, remember it's all in the song selection, and classic rock has the most memorable picks for your fifteen minutes. Many people in their 40s will look at this list with songs from their youth and think how are that classic rock but think about the younger up-and-coming Karaoke singers who want to sing a ... italian mobster name generator | {
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