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Q: ETIMEDOUT error when querying mysql database Using sequel pro I have created a database called test. It has one table called users. In that table there is one user -> id=1, name=Phantom.
I have installed the mysql node module
When I run the code below I get The solution is: undefined.
Can anyone advise how I can connect to database and show the users?
var mysql = require('mysql');
var connection = mysql.createConnection({
host : 'localhost:8889',
user : 'root',
password : 'root',
database : 'test'
});
connection.connect();
connection.query('SELECT * from users', function(err, rows, fields) {
if (err) throw err;
console.log('The solution is: ', rows);
});
connection.end();
It shows the following error :
Error: connect ETIMEDOUT
at Connection._handleConnectTimeout (/Users/fitz035/Desktop/sony/presave/node_modules/mysql/lib/Connection.js:425:13)
I am running the db through MAMP. These are the db settings :
Host: localhost
Port: 8889
User: root
Password: root
Socket: /Applications/MAMP/tmp/mysql/mysql.sock
A: Node is asynchronous, so connection.end() is likely to happen before your query calls back. Also, specify the port Mysql is running on when non-standard.
try this :
var mysql = require('mysql');
var connection = mysql.createConnection({
host : 'localhost',
user : 'root',
password : 'root',
database : 'test',
port: 8889
});
connection.connect();
connection.query('SELECT * from users', function(err, rows, fields) {
if(err) console.log(err);
console.log('The solution is: ', rows);
connection.end();
});
A: If it on VPS configured with Firewall, you need to whitelist your IP via SSH. Otherwise it still throws exactly above error even after adding via cPanel's RemoteMYSQl
# csf -a [RemoteIP]
# csf -r
You can do it quickly via WHM too. Just posted as it may help someone.
A: Check your current MySQL server port and change to:
DB_PORT=3304
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{"url":"https:\/\/gamedev.stackexchange.com\/questions\/125786\/texture-mapping-artifacts-on-certain-surfaces","text":"# Texture mapping artifacts on certain surfaces\n\nI am currently working on a 3d game engine, but am complete stumped on a problem involving texture rendering. Here is the rundown.\n\nMy game engine renders meshes of the OBJ wavefront type. I have created a OBJ mesh parser that simply looks through all the vertice\/normal\/texture indexes(Line by Line) and loads the corresponding data into the right places without altering it.\n\nMesh Parser Code: (NOTE pseudo codeish)\n\nMY VERTEX STRUCTURE: struct VERTEX { D3DXVECTOR3 Pos; D3DXVECTOR3 Normal; D3DXVECTOR2 TexCoords; };\n\n if (line[i]->c_str()[0] == 'v' && line[i]->c_str()[1] == ' ') {\n\nfloat tmpx = 0;\nfloat tmpy = 0;\nfloat tmpz = 0;\n\nsscanf(line[i]->c_str(), \"v %f %f %f\", &tmpx, &tmpy, &tmpz);\n\nD3DXVECTOR3 myVec = {tmpx, tmpy, tmpz };\n\nvertices.push_back(myVec);\n\n}\nelse if (line[i]->c_str()[0] == 'v' && line[i]->c_str()[1] == 't') {\n\nfloat tmpx = 0;\nfloat tmpy = 0;\n\nsscanf(line[i]->c_str(), \"vt %f %f\", &tmpx, &tmpy);\n\nD3DXVECTOR2 myTexCoords = { tmpx, tmpy };\n\ntexCoords.push_back(myTexCoords);\n}\nelse if (line[i]->c_str()[0] == 'v' && line[i]->c_str()[1] == 'n') {\n\nfloat tmpx = 0;\nfloat tmpy = 0;\nfloat tmpz = 0;\n\nsscanf(line[i]->c_str(), \"vn %f %f %f\", &tmpx, &tmpy, &tmpz);\n\nD3DXVECTOR3 myNorm = { tmpx, tmpy, tmpz };\n\nnormals.push_back(myNorm);\n\n}\n\nsscanf(line[i]->c_str(), \"f %d\/%d\/%d %d\/%d\/%d %d\/%d\/%d %d\/%d\/%d\",\n&tmpindices1, &tmpTexCoordsIndex1, &tmpNormalsIndex1,\n&tmpindices2, &tmpTexCoordsIndex2, &tmpNormalsIndex2,\n&tmpindices3, &tmpTexCoordsIndex3, &tmpNormalsIndex3,\n&tmpindices4, &tmpTexCoordsIndex4, &tmpNormalsIndex4);\n\nverticeIndex.push_back(tmpindices1 - 1);\nverticeIndex.push_back(tmpindices2 - 1);\nverticeIndex.push_back(tmpindices3 - 1);\nverticeIndex.push_back(tmpindices4 - 1);\n\ntexCoordsIndex.push_back(tmpTexCoordsIndex1 - 1);\ntexCoordsIndex.push_back(tmpTexCoordsIndex2 - 1);\ntexCoordsIndex.push_back(tmpTexCoordsIndex3 - 1);\ntexCoordsIndex.push_back(tmpTexCoordsIndex4 - 1);\n\nnormalsIndex.push_back(tmpNormalsIndex1 - 1);\nnormalsIndex.push_back(tmpNormalsIndex2 - 1);\nnormalsIndex.push_back(tmpNormalsIndex3 - 1);\nnormalsIndex.push_back(tmpNormalsIndex4 - 1);\n\nfor (int i = 0; i < texCoordsIndex.size(); i++) {\nModelToFill.Vertices[i].TexCoords = texCoords.at(texCoordsIndex.at(i));\n}\n\nfor (int i = 0; i < normalsIndex.size(); i++) {\nModelToFill.Vertices[i].Normal = normals.at(normalsIndex.at(i));\n}\n\nfor (int i = 0; i < vertices.size(); i++) {\nModelToFill.Vertices[i].Pos = vertices.at(i);\n}\n\nfor (int i = 0; i < verticeIndex.size(); i++) {\nModelToFill.Indices[i] = verticeIndex.at(i);\n}\n\n\nBut now when my renderer renders the mesh it looks like this:\n\nAs You can see the texture coordinates are all messed up, (for some surfaces). The Bottom, Leftmost and back are messed up but the front and top are fine.\n\nNow I should mention that my mesh is exported from blender with materials, and the mesh is also triangulated. Before importing I also performed the smart UV unwrap.\n\nFinally, I should mention that the parts of the mesh which are not texture mapped correctly vary depending on the mesh.\n\nMY QUESTION though quite general: What is the most probable cause of this issue, the mesh parser, form of blender mesh exporting, or renderer.\n\nYou need to be careful, wavefront files and gpu does not works the same, in the file, you can have a missmatch of indices between position, texcoord and normal because they can be reuse differently. Extreme example, a cube with a texture per face only need 8 point values, 4 texcoord values and 6 normal values.\n\nNow, on the GPU, a vertex need to be a full tuple, it means for our cube, that you need to split points for divergent texcoord and normal, and you ends with 24 vertex.\n\nEDIT: tips for building a proper vertex buffer and index buffer from a wavefront file\n\nuse std::map<std::tuple<int,int,int>,int> remap;, the key is the triplet position\/uv\/normal, the value is an increasing counter, starting at 0, and that you increment every time you that a new triplet.\n\nOnce you insert every triplet, counter is your number of unique vertex. you can now create a vertex buffer of that many values, and fill it by reversing the std::map, the value become the vertex offset, the key is the vertex attribut indices.\n\nYou do the same for the index buffer, you use the triplet as a key to retrieve the final vertex index.\n\n\u2022 Hey! Thanks for the response. Upon recent debugging using the Visual Studio graphics debugger, I did notice that my graphics card was using the same vertex indices for both texture coordinates and normal coordinates. I am currently trying to construct an algorithm to sort the texcoords and normals so that when referenced using the vertex indices, the correct texcoord and normal is in the proper indexed position relative to the vertex indice. Unfortunate I have no clue on how to make this algorithm. Do you have any idea how I can? Jul 22 '16 at 0:16\n\u2022 @NJMercaldo I had a tips on how to easy turn the multi indices logic and generate the vertex buffer in the answer. Jul 22 '16 at 0:32\n\u2022 Hey, I am making good progress at implementing your provided algorithm, but I am stuck at one place. How do I reverse the map so that the key becomes the counter number, and the data the map outputs becomes the vertice\/texcoord\/normal tuple? Is this what you mean by reverse? Thanks :) Jul 22 '16 at 14:22\n\u2022 Algorithm is now working beautifully! Thanks again for the help. Jul 22 '16 at 17:19","date":"2021-09-27 10:11:00","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\": 0.23182614147663116, \"perplexity\": 4600.865296417559}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780058415.93\/warc\/CC-MAIN-20210927090448-20210927120448-00466.warc.gz\"}"} | null | null |
Home > Bus Life > Shuttle Bus Conversion > Nomadic Bus Life Q&A
Nomadic Bus Life Q&A
I opened up to the community to ask us questions about bus life or living as nomads. Here is the conversation.
What was the most difficult part of starting bus life? What was the most rewarding?
While building the bus itself was more challenging than we thought it would be (and took much longer), the most diffcult part was probably overcoming fear. Fear of change. Fear of the unknown. Fear of challenging the status quo. Fear of not having a permanent address with a mailbox. Fear of not making enough money to support the lifestyle.
But it has been the most rewarding adventure! Not having a permanent address gives us the ultimate freedom to go anywhere, do anything. Every day is exciting. The fear melts away and then there is only the question, "Where will we go next?". We laugh like kids every day!
How much did you spend on the bus?
We spent a whopping $2,000 to buy the bus (fantastic deal for a bus with low miles), and put about $20,000 into building it out. Some of the things we spent money on were luxuries, like our wood stove and lithium batteries, but we were willing to splurge a little on things that were quality and would have longevity. We've also spent more after the build on some mechanical repair work. All told, it's a great deal to have a traveling home that goes where you go, and not have to pay rent or a mortgage!
How'd the install go on the wood stove with the fiberglass roof?? Nervous about ours!
We were nervous too. The thought of cutting a hole in the roof was a scary thought, and we put it off for a long time… But once we decided it was time there was nothing to be afraid of! We just measured carefully and made the cut. Now our wood stove is a permanent part of our bus. We love it and couldn't live without it!
Kimberly wood stove, best on the market
How do you make money on the road?
When we were ready to start the next chapter and hit the road we both quit our jobs. Surprisingly, my employer asked me to stay on and work remotely. No one else at my job was doing it, and I wouldn't have known it was an option without attempting to resign first. Now almost the entire department is working remotely due to COVID. Times change. The lesson here is you never know, it might be a possibility for you, and it doesn't hurt to ask!
(I am an IT analyst working for a health system, and work Mon-Thu with plenty of time for travel and play. All I need is wi-fi on the road, which I have and is working great. I made a post about what I use for internet on the road, check it out if you are looking.)
What kind of camera do you use?
My regular camera is a Nikon DF, but it's not an "adventure camera". It's too big and too heavy to climb mountains or ski with. So the majority of my photos here were taken with my Pixel 3 (cell phone). They aren't as good as they could be, but it's like they say, the best camera is the one you have with you!
Backpacking on the Olympic Coast
Very curious why you chose a wood stove over other options like a propane heater (propex)?
Firstly, we didn't want to rely on an electric heater because we weren't sure if our solar would be enough to sustain that. We seriously looked into propane heat as an option, but ultimately it was our lifestyle that caused us to go with wood heat. We spend a lot of time in the mountains at higher elevations. Propane heaters require the correct mix of propane to oxygen, and at higher altitudes there is less oxygen, which could cause issues with ignition or noxious gasses. There are adapters to make adjustments to the amount of propane flow, but ultimately we didn't want to have to keep making adjustments based on our elevation, so we went with a wood burner.
We love our wood stove and feel like it was the right choice! Wood heat and the ambiance of flickering flames is so special. We do also use a tiny space heater that heats really well, but only use it when we are connected to shore power.
It's warm and cozy inside, even in the snowy mountains!
What was the most difficult part of the build/ or surprisingly the easiest?
Framing out and walling in all of those odd angles corners and nooks was so frustrating. It took ages and a whole lot of patience. There are no right angles in a bus! Also, fiberglassing the shower by hand was grueling and very messy. We breathed a lot of awful fumes (even with PPE) that burned our sinuses. The result was well worth it, but it was a lot of work!
The easiest was plumbing and electrical. It's surprising because neither of us knew anything about plumbing or electrical when we started. We just researched and learned it all as we went. YouTube became our best friend.
So many curves and odd angles!
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Q: Postgresql , comparar time extraido de um timestamp Necessito extrair as horas e minutos de um timestamp no mysql, eu tentei usar o maketime();
CREATE OR REPLACE FUNCTION agendar(dia timestamp) returns bool as
$$
declare
hora time := extract(hour from dia);
minuto time := extract(minute from dia);
horaminuto := maketime(hora,minuto,0);
porém acusa de sintaxe incorreta.
Eu preciso extrair a data e a hora para comparar e ver se o horario do timestamp atingiu ou não um limite de tempo. Por exemplo, um comercio que abre as 9:30 da manhã, preciso verificar se a hora que foi passada no timestamp se passa antes ou depois desse horario
A: Não entendi muito bem sua função, onde ela retorna o valor, onde declara variável, comparação... Eu estou acostumado com procedimentos ao invés de funções, talvez por isso, mas vamos lá, talvez te ajude um pouco a resolver sem criar uma função.
Você pode extrair as horas através desta função:
SELECT DATE_FORMAT(datetimeVariable, '%H') AS 'hora' FROM tabela
Onde o %H significa o flag de horas, por exemplo, para extrair minutos seria o flag %i:
SELECT DATE_FORMAT(datetimeVariable, '%i') AS 'minuto' FROM tabela
Agora se você quer uma maneira mais simples de comparar diretamente, tenta com essa função aqui:
SELECT Count(*) AS 'valido' FROM tabelaHorarios
WHERE TIME_TO_SEC(TIME(primeiroDatetime)) > TIME_TO_SEC('09:30:00') AND
id='1'
O que acontece é que, ele vai retornar 0 se o primeiroDatetime não tiver passado das 9h30, do contrário, um valor maior do que 1, para quantos dados ele encontrar... o id você define qual valor da tabela você quer testar... Por que isso? Provavelmente seu sistema vai comparar 1 mês do comércio, para ver em que dias fechou, certo? Você pode definir um range do mês inteiro e contar quantos dias do mês o comércio abriu após as 9h30...
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White Snake Projects Presents Sing Out Strong: Incarcerated Voices, Including Writers from Virtual Opera Death by Life (Press Release)
Galvanized by the killing of George Floyd in Minneapolis in May of 2020, Cerise Jacobs and her activist opera company, White Snake Projects, developed the virtual opera Death by Life – exploring the intersection of systemic racism and mass incarceration with texts written by incarcerated writers and their families and a score by five Black composers – to stand as a monument of support for the Black Lives Matter movement. This month, Death by Life provides the thematic basis for Sing Out Strong: Incarcerated Voices, the fourth installment of White Snake's popular community song initiative. Two of Death by Life's seven writers, Joe Dole and Devon Terrell, are also contributing texts for Sing Out Strong, along with eight other incarcerated writers, paired with ten composers from across the spectrum of backgrounds and influences. Tickets are free, with a requested donation going to support the Justice Arts Coalition, which will also be curating artwork for each song. To register for Sing Out Strong: Incarcerated Voices, click here.
As an activist opera company, White Snake Projects establishes ties with community partners for each production, many of whom then become ongoing parts of the company's creative and activist ecosystem. For Death by Life, a key partner was Alice Kim, Director of the Human Rights Lab at the University of Chicago, who alerted the White Snake team to a trove of essays written by currently and formerly incarcerated individuals. Seven were purchased as the basis of the libretto, and Joe Dole and Devon Terrell were among those writers. Dole wrote "a fantastical and unexpectedly funny 'Yard Scene with the Animals' between an incarcerated man and a bird family" (I Care if You Listen), while Terrell wrote the poem "The Mourn," which concluded the first scene of the opera. Both Dole and Terrell are students in the Stateville Think Tank run by Alice Kim. Joining them are eight newcomers to the White Snake family: Sarah Allen, Hal Cobb, Shareaf Fleming, Terry Hedin, Jevon Jackson, W. M. Peeples, Dana P. R. Schultze, and James Soto. The roster includes numerous prizewinners of the PEN America Prison Writing Contest as well as members of the Northwestern Prison Education Program (NPEP), an initiative of Northwestern University to provide a high-quality liberal arts education to incarcerated students in Illinois.
Selected to be paired with these writers are ten composers representing a diverse range of ages, backgrounds and ethnicities: Adore Alexander, Andrew Amendola, Jake Berran, Andrew Conklin, Sarah Taylor Ellis, Patricio Molina, Johanny Navarro, Iván Enrique Rodríguez, Del'Shawn Taylor, and SiHyun Uhm. They include a Juilliard student (Uhm), an ASCAP Leonard Bernstein Award winner (Rodríguez), an Opera America Discovery Grant winner (Ellis), an activist for arts access in marginalized communities (Taylor), a jazz guitarist (Amendola), and a former child prodigy (Molina). Their influences range from Afro-Caribbean and early twentieth-century California immigrant music to electronic music and multi-media, and they hail from Chile, Puerto Rico, Korea and all corners of the United States.
Performers for Sing Out Strong: Incarcerated Voices include soprano Sarah Rogers, a young artist at the Boston University Opera Institute and one of eight winners of the Boston University at Carnegie Hall competition; Dominican tenor José Heredia, first prize winner in the New Jersey Association of Verismo Opera's International Vocal Competition; baritone Brandon Bell, who joins Fort Worth Opera as one of their Lesley Resident Artists this season; cellist Clare Monfredo, a Fulbright Scholarship recipient who studied in Leipzig and is now pursuing her DMA on a five-year graduate fellowship at the CUNY Graduate Center; and pianist Nathan Ben-Yehuda, winner of the 2017 Yamaha Young Performing Artists award. White Snake regular Tian Hui Ng serves as Music Director.
Early in the pandemic, when White Snake Projects pivoted to move its offerings to an online format, the Boston Musical Intelligencer declared that the company had "set a new standard for online concerts in the age of pandemic." Thanks to the technical wizardry of audio engineer Jon Robertson, live synchronous performances by singers and instrumentalists in multiple locations can now be taken for granted. Writing about Death by Life, the contemporary music online hub I Care if You Listen declared that the "singers … were impressively synced, tuned, and compelling." The same review was quick to add that much more was achieved than just technical feats: "The digital medium creates a container to elevate original stories, avoid melodrama, and advance a nuanced artistic vision. … Death by Life's success is not just as art that transcends its impressive technology, but in the company's communal infrastructure that informs thoughtful work and builds a bridge to further action." Seen and Heard International agreed: "Death by Life is great theater with an activist edge that brings to mind Larry Kramer's The Normal Heart or Jonathan Larson's Rent. … And, like the other two plays, Death by Life left me with tears in my eyes." | {
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Having searched for business suppliers in Bramhall, Stockport ( which is a town with an approximate population of 17,436 ), we have found 2 suppliers of services such as Wall Art, & Accountants in Bramhall and have listed them below, we hope to add more in the near future.
If you know of any more suppliers of business related services, either matching the list of 2 services we already have or new services, in Bramhall that you can recommend please contact us and we will look at adding them to this page.
Accountancy in Bramhall - Castletree are a firm of Chartered Certified Accountants and Business Advisers based in Bramhall.
Wall art suitable for interior design and office receptions, available online and shipped to Bramhall. Printed to order on canvas, acrylic and other surfaces with 30 day guarantee.
Not found what you are looking for in Bramhall? We have other listings in locations such as Widnes, Runcorn, Warrington, Paddington, Hale, Altrincham, Wilmslow, Cheadle, Poynton & Whaley Bridge that you may find helpful.
Bramhall is a suburb in the Metropolitan Borough of Stockport, Greater Manchester, England.Historically in Cheshire, it had a population of 17,436 at the 2011 Census.
The manor of Bramall dates from the Anglo-Saxon period, when it was held as two separate estates by two Saxon freemen, Brun and Hacun. In 1070, William the Conqueror subdued the north-west of England, and divided the land among his followers. The manor of "Bramale" was given to Hamon de Massey, who eventually became the first Baron of Dunham Massey. The earliest reference to Bramall was recorded in the Domesday Book as "Bramale", a name derived from the Old English words brom meaning broom, both indigenous to the area, and halh meaning nook or secret place, probably by water. De Masci received the manor as wasteland, since it had been devastated by William the Conqueror's subdual. By the time of the Domesday survey, the land was recovering and cultivated again.
The above introduction to Bramhall uses material from the Wikipedia article 'Bramhall' and is used under licence.
The Google map below shows a scrollable map of Bramhall and the surrounding area ( the amount of surrounding area depends on the location you are looking for). | {
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} | 6,878 |
{"url":"https:\/\/www.physicsforums.com\/threads\/fluid-flow-around-a-sphere-whats-the-net-force.723580\/","text":"# Fluid flow around a sphere, what's the net force?\n\n1. Nov 18, 2013\n\n### Shmi\n\n1. The problem statement, all variables and given\/known data\n\nA sphere of radius $a$ is submerged in a fluid which is flowing in the z-hat direction. There is some associated viscosity in the fluid which will exert a force on the sphere. Use symmetry to argue that the net force will be in the z-direction. Show that it will have the form\n\n$$F_z = \\oint \\left( -P' \\cos{\\theta} + \\sigma_{rr} \\cos{\\theta} - \\sigma_{r \\theta} \\sin{\\theta} \\right) \\; dA$$\n\n2. Relevant equations\n\nWe are told earlier that it satisfies the Navier-Stokes equation\n\n$$- \\nabla P' + \\eta \\nabla^2 v = 0$$\n\nwith the boundary conditions that $v = 0$ at $r=a$. Also, it satisfies ${\\bf v} = v_0 \\hat{z}$ very far from the sphere.\n\n$\\sigma$ is the stress tensor\n\nAlso, we earlier defined fluid force as\n\n$$F_i = \\oint \\sigma_{ij} dA_j$$\n\n3. The attempt at a solution\n\nSo, clearly as the fluid moves around the face of the sphere, the components in y-hat cancel while the z-hat components add, and this argument applies all the way around the azimuthal coordinate to x-hat and back. So, I get that the force is only in z-hat, but I don't get the form of the integral expression. Why is there a pressure term, and what are the sin and cos terms doing next to the stress tensor components? I'm not great with tensor notation, so maybe I'm missing it. Could someone clarify what that expression means so that I can better derive it?\n\n2. Nov 18, 2013\n\n### Staff: Mentor\n\nP' is the pressure distribution at the surface of the sphere and acts normal to the sphere surface, and \u03c3rr and \u03c3 are the viscous portions of the stress, normal and tangential to the surface, respectively. The sines and cosines give the components of these stresses in the flow direction.","date":"2017-08-21 02:02:53","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\": 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\": 0.7898747324943542, \"perplexity\": 391.7510535668472}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-34\/segments\/1502886107065.72\/warc\/CC-MAIN-20170821003037-20170821023037-00493.warc.gz\"}"} | null | null |
{"url":"https:\/\/brainsanswers.com\/mathematics\/question12850694","text":", 21.06.2019 14:10, hardwick744\n\n# How many real and imaginary solutions does the equation x^2-3x=-2x-7 have?\n\n### Other questions on the subject: Mathematics\n\nMathematics, 21.06.2019 15:10, jenashaqlaih\nWhat is 32+4 x (16 x 1\/2) -2 show your work}\nMathematics, 21.06.2019 18:00, drealtania21\nHi, we are learning circumscribed angles in geometry and i'm really stuck on this one! would be appreciated, all i know is that quadratic equations have to be used.\nMathematics, 21.06.2019 20:10, MikeWrice3615\nWhat additional information could be used to prove abc =mqr using sas? check all that apply.","date":"2021-03-08 11:32:17","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9204718470573425, \"perplexity\": 4166.709856339829}, \"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-2021-10\/segments\/1614178375439.77\/warc\/CC-MAIN-20210308112849-20210308142849-00306.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.shaalaa.com\/question-bank-solutions\/proportions-given-x-b-c-y-c-z-a-b-prove-that-ax-cz-0_39008","text":"Share\n\n# Given, X\/(B - C ) = Y\/(C - a ) = Z\/(A - B) , Prove that Ax+ by + Cz = 0 - Mathematics\n\nCourse\n\n#### Question\n\nGiven, x\/(b - c ) = y\/(c - a ) = z\/(a - b) , Prove that\n\nax+ by + cz = 0\n\n#### Solution\n\nGiven, x\/(b - c ) = y\/(c - a ) = z\/(a - b)= k (say)\n\nx = k (b - c ), y = k (c - a), z = k (a - b)\nax + by + cz\n= ak (b - c ) + bk (c - a) + ck (a - b)\n= abk - ack + bck - abk + ack - bck\n= 0\n\nIs there an error in this question or solution?\nGiven, X\/(B - C ) = Y\/(C - a ) = Z\/(A - B) , Prove that Ax+ by + Cz = 0 Concept: Proportions.","date":"2020-08-13 14:48:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 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\": 0.616374671459198, \"perplexity\": 4246.709949575852}, \"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-2020-34\/segments\/1596439739046.14\/warc\/CC-MAIN-20200813132415-20200813162415-00367.warc.gz\"}"} | null | null |
{"url":"http:\/\/rxiv.org\/ai\/","text":"# Artificial Intelligence\n\nPrevious months:\n2007 - 0703(1)\n2010 - 1003(33) - 1004(9) - 1005(5) - 1008(2) - 1009(1) - 1010(1) - 1012(1)\n2011 - 1101(2) - 1106(1) - 1107(1) - 1109(2)\n2012 - 1201(1) - 1204(3) - 1206(2) - 1207(6) - 1208(7) - 1209(1) - 1210(4) - 1211(2)\n2013 - 1301(5) - 1302(2) - 1303(6) - 1304(9) - 1305(1) - 1308(1) - 1309(8) - 1310(7) - 1311(1) - 1312(4)\n2014 - 1404(2) - 1405(3) - 1406(1) - 1408(5) - 1410(1) - 1411(1) - 1412(1)\n2015 - 1501(1) - 1502(3) - 1503(6) - 1504(3) - 1506(5) - 1507(4) - 1508(1) - 1509(4) - 1510(2) - 1511(4) - 1512(1)\n2016 - 1601(1) - 1602(10) - 1603(2) - 1605(4) - 1606(6) - 1607(5) - 1608(7) - 1609(5) - 1610(12) - 1611(14) - 1612(10)\n2017 - 1701(4) - 1702(9) - 1703(5) - 1704(11) - 1705(11) - 1706(14) - 1707(24) - 1708(19) - 1709(21) - 1710(15) - 1711(21) - 1712(18)\n2018 - 1801(4)\n\n## Recent submissions\n\nAny replacements are listed further down\n\n[425] viXra:1801.0192 [pdf] submitted on 2018-01-16 07:03:26\n\n### FastNet: An Efficient Architecture for Smart Devices\n\nAuthors: John Olafenwa, Moses Olafenwa\n\nInception and the Resnet family of Convolutional Neural Network architectures have broken records in the past few years, but recent state of the art models have also incurred very high computational cost in terms of training, inference and model size. Making the deployment of these models on Edge devices, impractical. In light of this, we present a new novel architecture that is designed for high computational efficiency on both GPUs and CPUs, and is highly suited for deployment on Mobile Applications, Smart Cameras, Iot devices and controllers as well as low cost drones. Our architecture boasts competitive accuracies on standard datasets even outperforming the original Resnet. We present below the motivation for this research, the architecture of the network, single test accuracies on CIFAR 10 and CIFAR 100, a detailed comparison with other well-known architectures and link to an implementation in Keras.\nCategory: Artificial Intelligence\n\n[424] viXra:1801.0102 [pdf] submitted on 2018-01-09 11:34:24\n\n### Bayesian Transfer Learning for Deep Networks\n\nAuthors: J. Wohlert, A. M. Munk, S. Sengupta, F. Laumann\n\nWe propose a method for transfer learning for deep networks through Bayesian inference, where an approximate posterior distribution q(w|\u03b8) of model parameters w is learned through variational approximation. Utilizing Bayes by Backprop we optimize the parameters \u03b8 associated with the approximate distribution. When performing transfer learning we consider two tasks; A and B. Firstly, an approximate posterior q_A(w|\u03b8) is learned from task A which is afterwards transferred as a prior p(w) \u2192 q_A(w|\u03b8) when learning the approximate posterior distribution q_B(w|\u03b8) for task B. Initially, we consider a multivariate normal distribution q(w|\u03b8) = N (\u00b5, \u03a3), with diagonal covariance matrix \u03a3. Secondly, we consider the prospects of introducing more expressive approximate distributions - specifically those known as normalizing flows. By investigating these concepts on the MNIST data set we conclude that utilizing normalizing flows does not improve Bayesian inference in the context presented here. Further, we show that transfer learning is not feasible using our proposed architecture and our definition of task A and task B, but no general conclusion regarding rejecting a Bayesian approach to transfer learning can be made.\nCategory: Artificial Intelligence\n\n[423] viXra:1801.0050 [pdf] submitted on 2018-01-06 00:20:25\n\n### Fruit Recognition from Images Using Deep Learning\n\nAuthors: Horea Muresan, Mihai Oltean\n\nIn this paper we introduce a new, high-quality, dataset of images containing fruits. We also present the results of some numerical experiment for training a neural network to detect fruits. We discuss the reason why we chose to use fruits in this project by proposing a few applications that could use this kind of neural network.\nCategory: Artificial Intelligence\n\n[422] viXra:1801.0041 [pdf] submitted on 2018-01-05 06:09:53\n\n### Taking Advantage of BiLSTM Encoding to Handle Punctuation in Dependency Parsing: A Brief Idea\n\nAuthors: Matteo Grella\n\nIn the context of the bidirectional-LSTMs neural parser (Kiperwasser and Goldberg, 2016), an idea is proposed to initialize the parsing state without punctuation-tokens but using them for the BiLSTM sentence encoding. The relevant information brought by the punctuation-tokens should be implicitly learned using the errors of the recurrent contributions only.\nCategory: Artificial Intelligence\n\n[421] viXra:1712.0659 [pdf] submitted on 2017-12-29 06:21:14\n\n### TDBF: Two Dimensional Belief Function\n\nAuthors: Yangxue Li; Yong Deng\n\nHow to ef\ufb01ciently handle uncertain information is still an open issue. Inthis paper, a new method to deal with uncertain information, named as two dimensional belief function (TDBF), is presented. A TDBF has two components, T=(mA,mB). The \ufb01rst component, mA, is a classical belief function. The second component, mB, also is a classical belief function, but it is a measure of reliability of the \ufb01rst component. The de\ufb01nition of TDBF and the discounting algorithm are proposed. Compared with the classical discounting model, the proposed TDBF is more \ufb02exible and reasonable. Numerical examples are used to show the ef\ufb01ciency of the proposed method.\nCategory: Artificial Intelligence\n\n[420] viXra:1712.0654 [pdf] submitted on 2017-12-29 11:33:39\n\n### Partitioned Singular Value Decomposition to Digital Image Encryption\n\nAuthors: Reza Nazarian\n\nImage encryption means a set of automated techniques which turn original images, known as input images, into cipher or encrypted image. A vast amount of digital images produced by social media and newspapers makes image encryption as an imperative toolset to the community. This paper implements a manifold learning-based method and examines its efficacy to digital image encryption. A list of experimental studies is performed to ensure the accuracy and general performance of the proposed method.\nCategory: Artificial Intelligence\n\n[419] viXra:1712.0647 [pdf] submitted on 2017-12-28 23:25:34\n\n### A Total Uncertainty Measure for D Numbers Based on Belief Intervals\n\nAuthors: Xinyang Deng, Wen Jiang\n\nAs a generalization of Dempster-Shafer theory, the theory of D numbers is a new theoretical framework for uncertainty reasoning. Measuring the uncertainty of knowledge or information represented by D numbers is an unsolved issue in that theory. In this paper, inspired by distance based uncertainty measures for Dempster-Shafer theory, a total uncertainty measure for a D number is proposed based on its belief intervals. The proposed total uncertainty measure can simultaneously capture the discord, and non-specificity, and non-exclusiveness involved in D numbers. And some basic properties of this total uncertainty measure, including range, monotonicity, generalized set consistency, are also presented.\nCategory: Artificial Intelligence\n\n[418] viXra:1712.0495 [pdf] submitted on 2017-12-18 08:50:22\n\n### Just Keep it in Mind: Information is a Complex Notion with Physical and Semantic Information Staying for Real and Imaginary Parts of the Expression\n\nAuthors: Emanuel Diamant\nComments: 3 Pages. Presented at the IS4SI 2017 Summit, Information Theory Section, Gothenburg, Sweden, 12\u201316 June 2017\n\nShannon\u2019s Information was devised to improve the performance of a data communication channel. Since then, the situation has changed drastically and today a more generally applicable and suitable definition of information is urgently required. To meet this demand, I have proposed a definition of my own. According to it, information is a complex notion with Physical and Semantic information staying for Real and Imaginary parts of the term. The scientific community has very unfriendly accepted this idea. But without a better solution for the problem of: 1) intron-exon partition in genes, 2) information flow in neuronal networks, 3) memory creation and potentiation in brains, 4) thoughts and thinking materialization in human heads, and 5) the undeniable shift from Computational (that is, data processing based) approach to Cognitive (that is, information processing based) approach in the field of scientific research, they would be forced to admit one day that something worthy is in this new definition.\nCategory: Artificial Intelligence\n\n[417] viXra:1712.0494 [pdf] submitted on 2017-12-18 09:05:26\n\n### Shannon's Definition of Information is Obsolete and Inadequate. it is Time to Embrace Kolmogorov\u2019s Insights on the Matter\n\nAuthors: Emanuel Diamant\nComments: 3 Pages. Presented at the 2016 ICSEE International Conference, Eilat, Israel, 16 \u2013 18 November 2016.\n\nInformation Theory, as developed by Claude Shannon in 1948, was about the communication of messages as electronic signals via a transmission channel. Only physical properties of the signal and the channel have been taken into account. While the meaning of the message has been ignored totally. Such an approach to information met very well the requirements of a data communication channel. But recent advances in almost all sciences put an urgent demand for meaningful information inclusion into the body of a communicated message. To meet this demand, I have proposed a new definition of information. In this definition, information is seen as a complex notion composed of two inseparable parts: Physical information and Semantic information. Classical informations such as Shannon, Fisher, Renyi, Kolmogorov\u2019s complexity, and Chaitin\u2019s algorithmic information \u2013 they are all physical information variants. Semantic information is a new concept and it desires to be properly studied, treated, and used.\nCategory: Artificial Intelligence\n\n[416] viXra:1712.0469 [pdf] submitted on 2017-12-15 23:33:47\n\n### Predicting Yelp Star Reviews Based on Network Structure with Deep Learning\n\nAuthors: Luis Perez\n\nIn this paper, we tackle the real-world problem of predicting Yelp star-review rating based on business features (such as images, descriptions), user features (average previous ratings), and, of particular interest, network properties (which businesses has a user rated before). We compare multiple models on different sets of features -- from simple linear regression on network features only to deep learning models on network and item features. In recent years, breakthroughs in deep learning have led to increased accuracy in common supervised learning tasks, such as image classification, captioning, and language understanding. However, the idea of combining deep learning with network feature and structure appears to be novel. While the problem of predicting future interactions in a network has been studied at length, these approaches have often ignored either node-specific data or global structure. We demonstrate that taking a mixed approach combining both node-level features and network information can effectively be used to predict Yelp-review star ratings. We evaluate on the Yelp dataset by splitting our data along the time dimension (as would naturally occur in the real-world) and comparing our model against others which do no take advantage of the network structure and\/or deep learning.\nCategory: Artificial Intelligence\n\n[415] viXra:1712.0468 [pdf] submitted on 2017-12-15 23:41:37\n\n### The Effectiveness of Data Augmentation in Image Classification using Deep Learning\n\nAuthors: Luis Perez, Jason Wang\n\nIn this paper, we explore and compare multiple solutions to the problem of data augmentation in image classification. Previous work has demonstrated the effectiveness of data augmentation through simple techniques, such as cropping, rotating, and flipping input images. We artificially constrain our access to data to a small subset of the ImageNet dataset, and compare each data augmentation technique in turn. One of the more successful data augmentations strategies is the traditional transformations mentioned above. We also experiment with GANs to generate images of different styles. Finally, we propose a method to allow a neural net to learn augmentations that best improve the classifier, which we call neural augmentation. We discuss the successes and shortcomings of this method on various datasets.\nCategory: Artificial Intelligence\n\n[414] viXra:1712.0467 [pdf] submitted on 2017-12-15 23:43:11\n\n### Gaussian Processes for Crime Prediction\n\nAuthors: Luis Perez, Alex Wang\n\nThe ability to predict crime is incredibly useful for police departments, city planners, and many other parties, but thus far current approaches have not made use of recent developments of machine learning techniques. In this paper, we present a novel approach to this task: Gaussian processes regression. Gaussian processes (GP) are a rich family of distributions that are able to learn functions. We train GPs on historic crime data to learn the underlying probability distribution of crime incidence to make predictions on future crime distributions.\nCategory: Artificial Intelligence\n\n[413] viXra:1712.0465 [pdf] submitted on 2017-12-16 00:36:46\n\n### Reinforcement Learning with Swingy Monkey\n\nAuthors: Luis Perez, Aidi Zhang, Kevin Eskici\n\nThis paper explores model-free, model-based, and mixture models for reinforcement learning under the setting of a SwingyMonkey game \\footnote{The code is hosted on a public repository \\href{https:\/\/github.com\/kandluis\/machine-learning}{here} under the prac4 directory.}. SwingyMonkey is a simple game with well-defined goals and mechanisms, with a relatively small state-space. Using Bayesian Optimization \\footnote{The optimization took place using the open-source software made available by HIPS \\href{https:\/\/github.com\/HIPS\/Spearmint}{here}.} on a simple Q-Learning algorithm, we were able to obtain high scores within just a few training epochs. However, the system failed to scale well after continued training, and optimization over hundreds of iterations proved too time-consuming to be effective. After manually exploring multiple approaches, the best results were achieved using a mixture of $\\epsilon$-greedy Q-Learning with a stable learning rate,$\\alpha$, and $\\delta \\approx 1$ discount factor. Despite the theoretical limitations of this approach, the settings, resulted in maximum scores of over 5000 points with an average score of $\\bar{x} \\approx 684$ (averaged over the final 100 testing epochs, median of $\\bar{m} = 357.5$). The results show an continuing linear log-relation capping only after 20,000 training epochs.\nCategory: Artificial Intelligence\n\n[412] viXra:1712.0464 [pdf] submitted on 2017-12-16 00:38:28\n\n### Multi-Document Text Summarization\n\nAuthors: Luis Perez, Kevin Eskici\n\nWe tackle the problem of multi-document extractive summarization by implementing two well-known algorithms for single-text summarization -- {\\sc TextRank} and {\\sc Grasshopper}. We use ROUGE-1 and ROUGE-2 precision scores with the DUC 2004 Task 2 data set to measure the performance of these two algorithms, with optimized parameters as described in their respective papers ($\\alpha =0.25$ and $\\lambda=0.5$ for Grasshopper and $d=0.85$ for TextRank). We compare these modified algorithms to common baselines as well as non-naive, novel baselines and we present the resulting ROUGE-1 and ROUGE-2 recall scores. Subsequently, we implement two novel algorithms as extensions of {\\sc GrassHopper} and {\\sc TextRank}, each termed {\\sc ModifiedGrassHopper} and {\\sc ModifiedTextRank}. The modified algorithms intuitively attempt to maximize'' diversity across the summary. We present the resulting ROUGE scores. We expect that with further optimizations, this unsupervised approach to extractive text summarization will prove useful in practice.\nCategory: Artificial Intelligence\n\n[411] viXra:1712.0446 [pdf] submitted on 2017-12-13 08:17:06\n\n### A New Divergence Measure for Basic Probability Assignment and Its Applications in Extremely Uncertain Environments\n\nAuthors: Liguo Fei, Yong Hu, Yong Deng, Sankaran Mahadevan\n\nInformation fusion under extremely uncertain environments is an important issue in pattern classification and decision-making problem. Dempster-Shafer evidence theory (D-S theory) is more and more extensively applied to information fusion for its advantage to deal with uncertain information. However, the results opposite to common sense are often obtained when combining the different evidences using Dempster\u2019s combination rules. How to measure the difference between different evidences is still an open issue. In this paper, a new divergence is proposed based on Kullback-Leibler divergence in order to measure the difference between different basic probability assignments (BPAs). Numerical examples are used to illustrate the computational process of the proposed divergence. Then the similarity for different BPAs is also defined based on the proposed divergence. The basic knowledge about pattern recognition is introduced and a new classification algorithm is presented using the proposed divergence and similarity under extremely uncertain environments, which is illustrated by a small example handling robot sensing. The method put forward is motivated by desperately in need to develop intelligent systems, such as sensor-based data fusion manipulators, which need to work in complicated, extremely uncertain environments. Sensory data satisfy the conditions 1) fragmentary and 2) collected from multiple levels of resolution.\nCategory: Artificial Intelligence\n\n[410] viXra:1712.0444 [pdf] submitted on 2017-12-13 08:59:01\n\n### Environmental Impact Assessment Using D-Vikor Approach\n\nAuthors: Liguo Fei, Yong Deng\n\nEnvironmental impact assessment (EIA) is an open and important issue depends on factors such as social, ecological, economic, etc. Due to human judgment, a variety of uncertainties are brought into the EIA process. With regard to uncertainty, many existing methods seem powerless to represent and deal with it effectively. A new theory called D numbers, because of its advantage to handle uncertain information, is widely used to uncertainty modeling and decision making. VIKOR method has its unique advantages in dealing with multiple criteria decision making problems (MCDM), especially when the criteria are non-commensurable and even conflicting, it can also obtain the compromised optimal solution. In order to solve EIA problems more effectively, in this paper, a D-VIKOR approach is proposed, which expends the VIKOR method by D numbers theory. In the proposed approach, assessment information of environmental factors is expressed and modeled by D numbers. And a new combination rule for multiple D numbers is defined. Subjective weights and objective weights are considered in VIKOR process for more reasonable ranking results. A numerical example is conducted to analyze and demonstrate the practicality and effectiveness of the proposed D-VIKOR approach.\nCategory: Artificial Intelligence\n\n[409] viXra:1712.0439 [pdf] submitted on 2017-12-13 11:45:00\n\n### Large Scale Traffic Surveillance :Vehicle Detection and Classification Using Cascade Classifier and Convolutional Neural Network\n\nAuthors: Shaif Chowdhury\n\nIn this Paper, we are presenting a traffic surveillance system for detection and classification of vehicles in large scale videos. Vehicle detection is crucial part of Road safety. There are lots of different intelligent systems proposed for traffic surveillance. The system presented here is based on two steps, a descriptor of the image type haar-like, and a classifier type convolutional neural networks. A cascade classifier is used to extract objects rapidly and a neural network is used for final classification of cars. In case of Haar Cascades, the learning of the system is performed on a set of positive images (vehicles) and negative images (non-vehicle), and the test is done on another set of scenes. For the second, we have used faster R-CNN architecture. The cascade classifier gives faster processing time and Neural Network is used to increase the detection rate.\nCategory: Artificial Intelligence\n\n[408] viXra:1712.0432 [pdf] submitted on 2017-12-13 22:28:48\n\n### DS-Vikor: a New Methodology for Supplier Selection\n\nAuthors: Liguo Fei, Yong Deng, Yong Hu\n\nHow to select the optimal supplier is an open and important issue in supply chain management (SCM), which needs to solve the problem of assessment and sorting the potential suppliers, and can be considered as a multi-criteria decision-making (MCDM) problem. Experts\u2019 assessment play a very important role in the process of supplier selection, while the subjective judgment of human beings could introduce unpredictable uncertainty. However, existing methods seem powerless to represent and deal with this uncertainty effectively. Dempster-Shafer evidence theory (D- S theory) is widely used to uncertainty modeling, decision making and conflicts management due to its advantage to handle uncertain information. The VIKOR method has a great advantage to handle MCDM problems with non-commensurable and even conflicting criteria, and to obtain the compromised optimal solution. In this paper, a DS- VIKOR method is proposed for the supplier selection problem which expends the VIKOR method by D-S theory. In this method, the basic probability assignment (BPA) is used to denote the decision makers\u2019 assessment for suppliers, Deng entropy weight-based method is defined and applied to determine the weights of multi-criteria, and VIKOR method is used for getting the final ranking results. An illustrative example under real life is conducted to analyze and demonstrate the practicality and effectiveness of the proposed DS-VIKOR method.\nCategory: Artificial Intelligence\n\n[407] viXra:1712.0400 [pdf] submitted on 2017-12-13 06:52:57\n\n### Adaptively Evidential Weighted Classifier Combination\n\nAuthors: Liguo Fei, Bingyi Kang, Van-Nam Huynh, Yong Deng\n\nClassifier combination plays an important role in classification. Due to the efficiency to handle and fuse uncertain information, Dempster-Shafer evidence theory is widely used in multi-classifiers fusion. In this paper, a method of adaptively evidential weighted classifier combination is presented. In our proposed method, the output of each classifier is modelled by basic probability assignment (BPA). Then, the weights are determined adaptively for individual classifier according to the uncertainty degree of the corresponding BPA. The uncertainty degree is measured by a belief entropy, named as Deng entropy. Discounting-and-combination scheme in D-S theory is used to calculate the weighted BPAs and combine them for the final BPA for classification. The effectiveness of the proposed weighted combination method is illustrated by numerical experimental results.\nCategory: Artificial Intelligence\n\n[406] viXra:1712.0347 [pdf] submitted on 2017-12-07 09:10:57\n\n### Finding The Next Term Of Any Time Series Type Or Non Time Series Type Sequence Using Total Similarity & Dissimilarity {Version 6} ISSN 1751-3030.\n\nIn this research investigation, the author has detailed a novel scheme of finding the next term of any given time series type or non-time series type sequence.\nCategory: Artificial Intelligence\n\n[405] viXra:1712.0138 [pdf] submitted on 2017-12-05 14:07:08\n\n### Topological Clustering as a Method of Control for Certain Critical-Point Sensitive Systems\n\nAuthors: Martin J. Dudziiak\nComments: 6 Pages. submitted to CoDIT 2018 (Thessaloniki, Greece, April 2018)\n\nNew methods can provide more sensitive modeling and more reliable control, through use of dynamically-alterable local neighborhood clusters comprised of of the state-space parameters most disposed to be influential in non-linear systemic changes. Particular attention is directed to systems with extreme non-linearity and uncertainty in measurement and in control communications (e.g., micro-scalar, remote and inaccessible to real-time control). An architecture for modeling based upon topological similarity mapping principles is introduced as an alternative to classical Turing machine models including new \u201cquantum computers.\u201d\nCategory: Artificial Intelligence\n\n[404] viXra:1712.0071 [pdf] submitted on 2017-12-03 19:12:51\n\n### The Intelligence Quotient of the Artificial Intelligence\n\nAuthors: Dimiter Dobrev\nComments: 15 Pages. Bulgarian. Serdica Journal of Computing.\n\nTo say which programs are AI, it's enough to run an exam and recognize for AI those programs that passed the exam. The exam grade will be called IQ. We cannot say just how big the IQ has to be in order one program to be AI, but we will choose one specific value. So our definition of AI will be any program whose IQ is above this specific value. This idea has already been realized in [1], but here we will repeat this construction by bringing some improvements.\nCategory: Artificial Intelligence\n\n[403] viXra:1711.0477 [pdf] submitted on 2017-11-30 18:22:54\n\n### Okay, Google: a Preliminary Evaluation of the Robustness of Scholar Metrics\n\nAuthors: H Qadrawxu-Korbau, D Smith, K Beryllium\n\nGoogle Scholar provides a number of metrics often used as proxies for scientific productivity. It is, however, possible to consciously manipulate Scholar metrics, for instance via copious self-citation or upload of fake papers to indexed websites. Here, we post a paper on vixra, a preprint forum, and arbitrarily cite a completely random study to evaluate whether Scholar will count this submission toward the overall citation count of that study. We publish no results, as the publication of the paper is, in this case, the experiment.\nCategory: Artificial Intelligence\n\n[402] viXra:1711.0470 [pdf] submitted on 2017-11-30 02:13:24\n\n### Multi-Scalar Multi-Agent Control for Optimization of Dynamic Networks Operating in Remote Environment\n\nAuthors: Martin Dudziak\n\nMulti-agent control systems have demonstrated effectiveness in a variety of physical applications including cooperative robot networks and multi-target tracking in high-noise network and group environments. We introduce the use of multi-scalar models that extend cellular automaton regional neighborhood comparisons and local voting measures based upon stochastic approximation in order to provide more efficient and time-sensitive solutions to non-deterministic problems. The scaling factors may be spatial, temporal or in other semantic values. The exercising of both cooperative and competitive functions by the devices in such networks offers a method for optimizing system parameters to reduce search, sorting, ranking and anomaly evaluation tasks. Applications are illustration for a group of robots assigned different tasks in remote operating environments with highly constrained communications and critical fail-safe conditions.\nCategory: Artificial Intelligence\n\n[401] viXra:1711.0433 [pdf] submitted on 2017-11-26 23:19:36\n\n### Finding The Next Term Of Any Time Series Type Sequence Using Total Similarity & Dissimilarity {Version 5} ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel scheme of finding the next term of any given time series type sequence.\nCategory: Artificial Intelligence\n\n[400] viXra:1711.0429 [pdf] submitted on 2017-11-27 05:14:34\n\n### Finding The Next Term Of Any Sequence Using Total Similarity & Dissimilarity {Version 5}. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel scheme of finding the next term of any given sequence.\nCategory: Artificial Intelligence\n\n[399] viXra:1711.0420 [pdf] submitted on 2017-11-26 01:39:24\n\n### Move the Tip to the Right a Language Based Computeranimation System in Box2d\n\nAuthors: Frank Schr\u00f6der\n\nNot only \u201crobots need language\u201d, but sometimes a human-operator too. To interact with complex domains, he needs a vocabulary to init the robot, let him walk and grasping objects. Natural language interfaces can support semi-autonomous and fully-autonomous systems on both sides. Instead of using neural networks, the language grounding problem can be solved with object-oriented programming. In the following paper a simulation of micro-manipulation under a microscope is given which is controlled with a C++ script. The small vocabulary consists of init, pregrasp, grasp and place.\nCategory: Artificial Intelligence\n\n[398] viXra:1711.0382 [pdf] submitted on 2017-11-22 02:30:08\n\n### A Survey on Evolutionary Computation: Methods and Their Applications in Engineering\n\nAuthors: Morteza Husainy Yar, Vahid Rahmati, Hamid Reza Dalili Oskouei\n\nEvolutionary computation is now an inseparable branch of artificial intelligence and smart methods based on evolutional algorithms aimed at solving different real world problems by natural procedures involving living creatures. It's based on random methods, regeneration of data, choosing by changing or replacing data within a system such as personal computer (PC), cloud, or any other data center. This paper briefly studies different evolutionary computation techniques used in some applications specifically image processing, cloud computing and grid computing. These methods are generally categorized as evolutionary algorithms and swarm intelligence. Each of these subfields contains a variety of algorithms and techniques which are presented with their applications. This work tries to demonstrate the benefits of the field by presenting the real world applications of these methods implemented already. Among these applications is cloud computing scheduling problem improved by genetic algorithms, ant colony optimization, and bees algorithm. Some other applications are improvement of grid load balancing, image processing, improved bi-objective dynamic cell formation problem, robust machine cells for dynamic part production, integrated mixed-integer linear programming, robotic applications, and power control in wind turbines.\nCategory: Artificial Intelligence\n\n[397] viXra:1711.0370 [pdf] submitted on 2017-11-20 22:14:32\n\n### Finding The Next Term Of Any Given Sequence Using Total Similarity & Dissimilarity {Version 3} ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel scheme of finding the next term of any given sequence.\nCategory: Artificial Intelligence\n\n[396] viXra:1711.0367 [pdf] submitted on 2017-11-21 00:18:32\n\n### One Step Evolution Of Any Real Positive Number {Version 2}\n\nIn this research investigation, the author has detailed the Theory Of One Step Evolution Of Any Real Positive Number.\nCategory: Artificial Intelligence\n\n[395] viXra:1711.0361 [pdf] submitted on 2017-11-20 02:12:39\n\n### Finding The Next Term Of Any Given Sequence Using Total Similarity & Dissimilarity. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel scheme of finding the next term of any given sequence.\nCategory: Artificial Intelligence\n\n[394] viXra:1711.0360 [pdf] submitted on 2017-11-20 02:43:10\n\n### Ontology Engineering for Robotics\n\nAuthors: Frank Schr\u00f6der\n\nOntologies are a powerfull alternative to reinforcement learning. They store knowledge in a domain-specific language. The best-practice for implementing ontologies is a distributed version control system which is filled manually by programmers.\nCategory: Artificial Intelligence\n\n[393] viXra:1711.0359 [pdf] submitted on 2017-11-20 05:21:55\n\n### Finding The Next Term Of Any Given Sequence Using Total Similarity & Dissimilarity {New} ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel scheme of finding the next term of any given sequence.\nCategory: Artificial Intelligence\n\n[392] viXra:1711.0292 [pdf] submitted on 2017-11-12 09:29:57\n\n### Strengths and Potential of the SP Theory of Intelligence in General, Human-Like Artificial Intelligence\n\nAuthors: J Gerard Wolff\n\nThis paper first defines \"general, human-like artificial intelligence\" (GHLAI) in terms of five principles. In the light of the definition, the paper summarises the strengths and potential of the \"SP theory of intelligence\" and its realisation in the \"computer model\", outlined in an appendix, in three main areas: the versatility of the SP system in aspects of intelligence; its versatility in the representation of diverse kinds of knowledge; and its potential for the seamless integration of diverse aspects of intelligence and diverse kinds of knowledge, in any combination. There are reasons to believe that a mature version of the SP system may attain full GHLAI in diverse aspects of intelligence and in the representation of diverse kinds of knowledge.\nCategory: Artificial Intelligence\n\n[391] viXra:1711.0266 [pdf] submitted on 2017-11-11 03:38:23\n\n### Revisit Fuzzy Neural Network: Demystifying Batch Normalization and ReLU with Generalized Hamming Network\n\nAuthors: Lixin Fan\nComments: 10 Pages. NIPS 2017 publication.\n\nWe revisit fuzzy neural network with a cornerstone notion of generalized hamming distance, which provides a novel and theoretically justified framework to re-interpret many useful neural network techniques in terms of fuzzy logic. In particular, we conjecture and empirically illustrate that, the celebrated batch normalization (BN) technique actually adapts the \u201cnormalized\u201d bias such that it approximates the rightful bias induced by the generalized hamming distance. Once the due bias is enforced analytically, neither the optimization of bias terms nor the sophisticated batch normalization is needed. Also in the light of generalized hamming distance, the popular rectified linear units (ReLU) can be treated as setting a minimal hamming distance threshold between network inputs and weights. This thresholding scheme, on the one hand, can be improved by introducing double-thresholding on both positive and negative extremes of neuron outputs. On the other hand, ReLUs turn out to be non-essential and can be removed from networks trained for simple tasks like MNIST classification. The proposed generalized hamming network (GHN) as such not only lends itself to rigorous analysis and interpretation within the fuzzy logic theory but also demonstrates fast learning speed, well-controlled behaviour and state-of-the-art performances on a variety of learning tasks.\nCategory: Artificial Intelligence\n\n[390] viXra:1711.0265 [pdf] submitted on 2017-11-11 04:14:07\n\n### Revisit Fuzzy Neural Network: Bridging the Gap Between Fuzzy Logic and Deep Learning\n\nAuthors: Lixin Fan\nComments: 75 Pages. bridging the gap between symbolic versus connectionist.\n\nThis article aims to establish a concrete and fundamental connection between two important fields in artificial intelligence i.e. deep learning and fuzzy logic. On the one hand, we hope this article will pave the way for fuzzy logic researchers to develop convincing applications and tackle challenging problems which are of interest to machine learning community too. On the other hand, deep learning could benefit from the comparative research by re-examining many trail-and-error heuristics in the lens of fuzzy logic, and consequently, distilling the essential ingredients with rigorous foundations. Based on the new findings reported in [38] and this article, we believe the time is ripe to revisit fuzzy neural network as a crucial bridge between two schools of AI research i.e. symbolic versus connectionist [93] and eventually open the black-box of artificial neural networks.\nCategory: Artificial Intelligence\n\n[389] viXra:1711.0250 [pdf] submitted on 2017-11-08 06:37:55\n\n### Total Intra Similarity And Dissimilarity Measure For The Values Taken By A Parameter Of Concern. {Version 1}. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of finding the \u2018Total Intra Similarity And Dissimilarity Measure For The Values Taken By A Parameter Of Concern\u2019. The advantage of such a measure is that using this measure we can clearly distinguish the contribution of Intra aspect variation and Inter aspect variation when both are bound to occur in a given phenomenon of concern. This measure provides the same advantages as that provided by the popular F-Statistic measure.\nCategory: Artificial Intelligence\n\n[388] viXra:1711.0241 [pdf] submitted on 2017-11-07 03:26:43\n\n### Dysfunktionale Methoden der Robotik\n\nAuthors: Frank Schr\u00f6der\n\nBei der Realisierung von Robotik-Projekten kann man eine ganze Menge verkehrt machen. Damit sind nicht nur kalte L\u00f6tstellen oder abst\u00fcrzende Software gemeint, sondern sehr viel grunds\u00e4tzlichere Dinge spielen eine Rolle. Um Fehler zu vermeiden, muss man sich zun\u00e4chst einmal mit den Failure-Patterns n\u00e4her auseinandersetzen, also jenen Entwicklungsmethoden, nach denen man auf gar keinen Fall einen Roboter bauen und wie die Software m\u00f6glichst nicht funktionieren sollte.\nCategory: Artificial Intelligence\n\n[387] viXra:1711.0235 [pdf] submitted on 2017-11-06 20:27:28\n\n### Not Merely Memorization in Deep Networks: Universal Fitting and Specific Generalization\n\nAuthors: Xiuyi Yang\n\nWe reinterpret the training of convolutional neural nets(CNNs) with universal classification theorem(UCT). This theory implies any disjoint datasets can be classified by two or more layers of CNNs based on ReLUs and rigid transformation switch units(RTSUs) we propose here, this explains why CNNs could memorize noise and real data. Subsequently, we present another fresh new hypothesis that CNN is insensitive to some variant from input training data example, this variant relates to original training input by generating functions. This hypothesis means CNNs can generalize well even for randomly generated training data and illuminates the paradox Why CNNs fit real and noise data and fail drastically when making predictions for noise data. Our findings suggest the study about generalization theory of CNNs should turn to generating functions instead of traditional statistics machine learning theory based on assumption that the training data and testing data are independent and identically distributed(IID), and apparently IID assumption contradicts our experiments in this paper.We experimentally verify these ideas correspondingly.\nCategory: Artificial Intelligence\n\n[386] viXra:1711.0226 [pdf] submitted on 2017-11-07 01:52:12\n\n### Theory Of Universal Evolution Along Prime Basis (Time Like) ISSN 1751-3030.\n\nIn this research investigation, the author has detailed the Theory Of Evolution.\nCategory: Artificial Intelligence\n\n[385] viXra:1711.0208 [pdf] submitted on 2017-11-07 02:22:45\n\n### Theory Of Universal Evolution Along Prime Basis (Time Like) {Version 2} ISSN 1751-3030.\n\nIn this research investigation, the author has detailed the Theory Of Evolution.\nCategory: Artificial Intelligence\n\n[384] viXra:1711.0116 [pdf] submitted on 2017-11-02 23:51:41\n\n### Dynamic Thresholding For Linear Binary Classifiers. {Version 2} ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of finding the Thresholding for Linear Binary Classifiers.\nCategory: Artificial Intelligence\n\n[383] viXra:1711.0034 [pdf] submitted on 2017-11-02 06:05:21\n\n### Dynamic Thresholding For Linear Binary Classifiers. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of finding the Thresholding for Linear Binary Classifiers.\nCategory: Artificial Intelligence\n\n[382] viXra:1710.0336 [pdf] submitted on 2017-10-31 23:50:38\n\n### Scheme For Finding The Next Term Of A Sequence Based On Evolution. {Version 7}. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of finding the next term of a sequence based on Evolution.\nCategory: Artificial Intelligence\n\n[381] viXra:1710.0299 [pdf] submitted on 2017-10-27 04:13:49\n\n### Scheme For Finding The Next Term Of A Sequence Based On Evolution. {Version 6}. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of finding the next term of a sequence based on Evolution.\nCategory: Artificial Intelligence\n\n[380] viXra:1710.0297 [pdf] submitted on 2017-10-25 03:57:32\n\n### Scheme For Finding The Next Term Of A Sequence Based On Evolution {File Closing Version 2}. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of finding the next term of a sequence based on Evolution.\nCategory: Artificial Intelligence\n\n[379] viXra:1710.0294 [pdf] submitted on 2017-10-25 23:47:37\n\n### Scheme For Finding The Next Term Of A Sequence Based On Evolution {File Closing Version 3}. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of finding the next term of a sequence based on Evolution.\nCategory: Artificial Intelligence\n\n[378] viXra:1710.0293 [pdf] submitted on 2017-10-26 01:24:46\n\n### Scheme For Finding The Next Term Of A Sequence Based On Evolution {File Closing Version 4}. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of finding the next term of a sequence based on Evolution.\nCategory: Artificial Intelligence\n\n[377] viXra:1710.0289 [pdf] submitted on 2017-10-26 03:56:28\n\n### Scheme For Finding The Next Term Of A Sequence Based On Evolution {File Closing Version 5}. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of finding the next term of a sequence based on Evolution.\nCategory: Artificial Intelligence\n\n[376] viXra:1710.0279 [pdf] submitted on 2017-10-24 04:45:19\n\n### Scheme For Finding The Next Term Of A Sequence Based On Evolution {File Closing Version 1}. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of finding the next term of a sequence based on Evolution.\nCategory: Artificial Intelligence\n\n[375] viXra:1710.0271 [pdf] submitted on 2017-10-23 23:14:04\n\n### The Average Computed In Primes Basis {File Closing Version 2}. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of finding the average of a sequence in Primes Basis.\nCategory: Artificial Intelligence\n\n[374] viXra:1710.0267 [pdf] submitted on 2017-10-23 06:21:13\n\n### The Average Computed In Primes Basis {File Closing Version 1}. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of finding the average of a sequence in Primes Basis.\nCategory: Artificial Intelligence\n\n[373] viXra:1710.0259 [pdf] submitted on 2017-10-23 00:38:01\n\n### Universe\u2019s Way Of Recursively Finding The Next Term Of Any Sequence {File Closing Version 3}. ISSN 1751-3030\n\nIn this research investigation, the author has detailed a novel method of Universe\u2019s Way Of Recursively Finding The Next Term Of Any Sequence.\nCategory: Artificial Intelligence\n\n[372] viXra:1710.0208 [pdf] submitted on 2017-10-18 23:07:44\n\n### The Recursive Future Equation Based On The Ananda-Damayanthi Normalized Similarity Measure. {File Closing Version 4}. ISSN 1751-3030\n\nIn this research Technical Note the author have presented a Recursive Future Average Of A Time Series Data Based on Cosine Similarity.\nCategory: Artificial Intelligence\n\n[371] viXra:1710.0141 [pdf] submitted on 2017-10-12 10:42:50\n\n### Miguel A. Sanchez-Rey\n\nAuthors: Advances in the Collective Interface\n\nA byproduct of 2AI.\nCategory: Artificial Intelligence\n\n[370] viXra:1710.0139 [pdf] submitted on 2017-10-12 11:12:18\n\n### Advances in the Collective Interface\n\nAuthors: Miguel A. Sanchez-Rey\n\nA byproduct of 2AI.\nCategory: Artificial Intelligence\n\n[369] viXra:1710.0003 [pdf] submitted on 2017-10-01 06:54:11\n\n### Nature-Like Technology for Communication Network Selfactualization in the Mode Advancing Real-Time\n\nAuthors: Popov Boris\n\nIn order to provide control system operation in real-time mode, communication system should operate in the mode advancing real-time that can be achieved only by means of providing the communication system with mechanism for network structure forward adaptation to the variations in the user query topics and rates as well as their self-actualization. A technique for developing such nature-like technology that is based on fundamental natural inertia phenomenon and widespread symbiotic cooperation, distinguished by building-up (developing) resources being used, is proposed.\nCategory: Artificial Intelligence\n\n[368] viXra:1709.0404 [pdf] submitted on 2017-09-26 13:26:47\n\n### A Suggestion on CLIPS\/JIProlog\/JNNS\/ImageJ\/Java Agents\/JikesRVM Based Analysis of Cryo-EM\/TEM\/SEM Images Using HDF5 Image Format \u2013 Some Interesting & Feasible Implementations of Expert Systems to Understand Nano- Bio Material Systems and EM\n\nAuthors: D.N.T.Kumar\nComments: 7 Pages. Prolog\/NN\/Expert Systems\/JikesRVM\/Informatics\/EM\/Cryo-EM\/TEM\/SEM\/Material Science\/Java Agents\/Nanotechnology.\n\nIn this short communication the importance of expert systems based imaging framework to probe Cryo-EM images is presented from a practical implementation point of view. Neural Networks or NN are an excellent tool to probe various domains of science and technology. Cryo-EM Technique holds bright future based on the application of NN.Prolog-NN based algorithms could form a powerful informatics and computational framework for researching the challenges of nano-bio Applications. Further,it is useful and important to study the behavior of NN in domains where knowledge does not exist, i.e to use the models to make bold predictions which form the basis for Cryo-EM Image Processing tasks and the discovery of new nano-bio phenomena.Indeed, the performance of NN is most useful to researchers in domains where the modeling and predicting \u201cuncertainty\u201d is known to be the greatest factor. All the methods presented here are also applicable to TEM\/SEM\/other EM Image Processing tasks as well.\nCategory: Artificial Intelligence\n\n[367] viXra:1709.0403 [pdf] submitted on 2017-09-26 13:33:02\n\n### Kernel Principal Component Analysis as Mathematical Tool In Processing Cryo- EM Images \u2013 A Suggestion Using Kernel Based Data Processing Techniques in a Java Virtual Machine(JVM) Environment.\n\nAuthors: D.N.T.Kumar\nComments: 7 Pages. A Suggestion Using Kernel Based Data Processing Techniques in a Java Virtual Machine(JVM) Environment.\n\nIn this short communication,it was proposed to highlight some novel methodologies to probe,process and compute Cryo-EM Images in a unique way by using an open source Kernel-PCA and by interfacing the KERNEL-PCA via Java Matlab Interface(JMI) \u2013 JikesRVM system or any other Java Virtual Machine(JVM).The main reason to design and develop this kind of computing approach is to utilize the features of Java based technologies for futuristic applications in the promising and demanding domains of CRYO-EM Imaging in the nano-bio domains.This is one of the pioneering research topics in this domain with a lot of promise.Image de-noising and novelty detection paves the way and holds the key for better Cryo-EM image processing.\nCategory: Artificial Intelligence\n\n[366] viXra:1709.0394 [pdf] submitted on 2017-09-26 11:50:52\n\n### How Does the ai Understand What's Going on\n\nAuthors: Dimiter Dobrev\n\nMost researchers regard AI as a static function without memory. This is one of the few articles where AI is seen as a device with memory. When we have memory, we can ask ourselves: \"Where am I?\", and \"What is going on?\" When we have no memory, we have to assume that we are always in the same place and that the world is always in the same state.\nCategory: Artificial Intelligence\n\n[365] viXra:1709.0323 [pdf] submitted on 2017-09-21 05:35:00\n\n### Recursive Future Average Of A Time Series Data Based On Cosine Similarity-RF\n\nIn this research Technical Note the author have presented a Recursive Future Average Of A Time Series Data Based on Cosine Similarity.\nCategory: Artificial Intelligence\n\n[364] viXra:1709.0322 [pdf] submitted on 2017-09-21 05:49:31\n\n### Recursive Future Average Of A Time Series Data Based On Cosine Similarity-RF {Version 2}\n\nIn this research Technical Note the author have presented a Recursive Future Average Of A Time Series Data Based on Cosine Similarity.\nCategory: Artificial Intelligence\n\n[363] viXra:1709.0313 [pdf] submitted on 2017-09-22 00:01:00\n\n### The Recursive Future Equation Based On The Ananda-Damayanthi Normalized Similarity Measure. {File Closing Version 2}. ISSN 1751-3030\n\nIn this research Technical Note the author have presented a Recursive Future Average Of A Time Series Data Based on Cosine Similarity.\nCategory: Artificial Intelligence\n\n[362] viXra:1709.0242 [pdf] submitted on 2017-09-15 20:34:58\n\n### Exact Map Inference in General Higher-Order Graphical Models Using Linear Programming\n\nAuthors: Ikhlef Bechar\n\nThis paper is concerned with the problem of exact MAP inference in general higher-order graphical models by means of a traditional linear programming relaxation approach. In fact, the proof that we have developed in this paper is a rather simple algebraic proof being made straightforward, above all, by the introduction of two novel algebraic tools. Indeed, on the one hand, we introduce the notion of delta-distribution which merely stands for the difference of two arbitrary probability distributions, and which mainly serves to alleviate the sign constraint inherent to a traditional probability distribution. On the other hand, we develop an approximation framework of general discrete functions by means of an orthogonal projection expressing in terms of linear combinations of function margins with respect to a given collection of point subsets, though, we rather exploit the latter approach for the purpose of modeling locally consistent sets of discrete functions from a global perspective. After that, as a first step, we develop from scratch the expectation optimization framework which is nothing else than a reformulation, on stochastic grounds, of the convex-hull approach, as a second step, we develop the traditional LP relaxation of such an expectation optimization approach, and we show that it enables to solve the MAP inference problem in graphical models under rather general assumptions. Last but not least, we describe an algorithm which allows to compute an exact MAP solution from a perhaps fractional optimal (probability) solution of the proposed LP relaxation.\nCategory: Artificial Intelligence\n\n[361] viXra:1709.0217 [pdf] submitted on 2017-09-14 08:11:16\n\n### Quantum Thinking Machines\n\nAuthors: George Rajna\n\nCategory: Artificial Intelligence\n\n[360] viXra:1709.0211 [pdf] submitted on 2017-09-14 06:46:27\n\n### Analyzing Huge Volumes of Data\n\nAuthors: George Rajna\n\nNeural networks learn how to carry out certain tasks by analyzing large amounts of data displayed to them. [15] Who is the better experimentalist, a human or a robot? When it comes to exploring synthetic and crystallization conditions for inorganic gigantic molecules, actively learning machines are clearly ahead, as demonstrated by British Scientists in an experiment with polyoxometalates published in the journal Angewandte Chemie. [14] Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that - surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9]\nCategory: Artificial Intelligence\n\n[359] viXra:1709.0161 [pdf] submitted on 2017-09-13 10:30:45\n\n### AI is Reinforcing Stereotypes\n\nAuthors: George Rajna\n\nFollowing the old saying that \"knowledge is power\", companies are seeking to infer increasingly intimate properties about their customers as a way to gain an edge over their competitors. [27] Researchers from Human Longevity, Inc. (HLI) have published a study in which individual faces and other physical traits were predicted using whole genome sequencing data and machine learning. [26] Artificial intelligence can improve health care by analyzing data from apps, smartphones and wearable technology. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17]\nCategory: Artificial Intelligence\n\n[358] viXra:1709.0159 [pdf] submitted on 2017-09-13 06:47:26\n\n### Mergeable Nervous Robots\n\nAuthors: George Rajna\n\nResearchers at the Universit\u00e9 libre de Bruxelles have developed self-reconfiguring modular robots that can merge, split and even self-heal while retaining full sensorimotor control. [29] A challenging brain technique called whole-cell patch clamp electrophysiology or whole-cell recording (WCR) is a procedure so delicate and complex that only a handful of humans in the whole world can do it. [28] ComText allows robots to understand contextual commands such as, \" Pick up the box I put down. \" [27] McMaster and Ryerson universities today announced the Smart Robots for Health Communication project, a joint research initiative designed to introduce social robotics and artificial intelligence into clinical health care. [26] Artificial intelligence can improve health care by analyzing data from apps, smartphones and wearable technology. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19]\nCategory: Artificial Intelligence\n\n[357] viXra:1709.0142 [pdf] submitted on 2017-09-11 20:53:40\n\n### Brain Emotional Learning Based Intelligent Controller for Velocity Control of an Electro Hydraulic Servo System\n\nAuthors: Zohreh Alzahra Sanai Dashti, Milad Gholami, Masoud Hajimani\nComments: 7 Pages. IOSR Journal of Electrical and Electronics Engineering (IOSR - JEEE) e - ISSN: 2278 - 1676,p - ISSN: 2320 - 3331, Volume 12, Issue 4 Ver. I I (Jul. \u2013 Aug. 2017), PP 29 - 35\n\nIn this paper, a biologically motivated controller based on mammalian limbic system called Brain Emotional Learning Based Intelligent Controller (BELBIC) is used for velocity control of an Electro Hydraulic Servo System (EHSS) in presence of flow nonlinearities, internal friction and noise. It is shown that this technique can be successfully used to stabilize any chosen operating point of the system with noise and without noise. All derived results are validated by computer simulation of a nonlinear mathematical model of the system. The controllers which introduced have big range for control the system. We compare BELBIC controller results with feedbacks linearization, backstepping and PID controller.\nCategory: Artificial Intelligence\n\n[356] viXra:1709.0141 [pdf] submitted on 2017-09-11 20:56:32\n\n### Design & Implementation of Fuzzy Parallel Distributed Compensation Controller for Magnetic Levitation System\n\nAuthors: Milad Gholami, Zohreh Alzahra Sanai Dashti, Masoud Hajimani\nComments: 9 Pages. IOSR Journal of Electrical and Electronics Engineering (IOSR - JEEE) e - ISSN : 2278 - 1676,p - ISSN: 2320 - 3331, Volume 12, Issue 4 Ver. I I (Jul. \u2013 Aug. 2017), PP 20 - 28\n\nThis study applies technique parallel distributed compensation (PDC) for position control of a Magnetic levitation system. PDC method is based on nonlinear Takagi-Sugeno (T-S) fuzzy model. It is shown that this technique can be successfully used to stabilize any chosen operating point of the system. All derived results are validated by experimental and computer simulation of a nonlinear mathematical model of the system. The controllers which introduced have big range for control the system.\nCategory: Artificial Intelligence\n\n[355] viXra:1709.0125 [pdf] submitted on 2017-09-11 07:51:38\n\n### Machine Learning Monitoring Air Quality\n\nAuthors: George Rajna\n\nUCLA researchers have developed a cost-effective mobile device to measure air quality. It works by detecting pollutants and determining their concentration and size using a mobile microscope connected to a smartphone and a machine-learning algorithm that automatically analyzes the images of the pollutants. [15] Who is the better experimentalist, a human or a robot? When it comes to exploring synthetic and crystallization conditions for inorganic gigantic molecules, actively learning machines are clearly ahead, as demonstrated by British Scientists in an experiment with polyoxometalates published in the journal Angewandte Chemie. [14] Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11]\nCategory: Artificial Intelligence\n\n[354] viXra:1709.0108 [pdf] submitted on 2017-09-10 06:02:53\n\n### A New Semantic Theory of Nature Language\n\nAuthors: Kun Xing\n\nFormal Semantics and Distributional Semantics are two important semantic frameworks in Natural Language Processing (NLP). Cognitive Semantics belongs to the movement of Cognitive Linguistics, which is based on contemporary cognitive science. Each framework could deal with some meaning phenomena, but none of them fulfills all requirements proposed by applications. A unified semantic theory characterizing all important language phenomena has both theoretical and practical significance; however, although many attempts have been made in recent years, no existing theory has achieved this goal yet. This article introduces a new semantic theory that has the potential to characterize most of the important meaning phenomena of natural language and to fulfill most of the necessary requirements for philosophical analysis and for NLP applications. The theory is based on a unified representation of information, and constructs a kind of mathematical model called cognitive model to interpret natural language expressions in a compositional manner. It accepts the empirical assumption of Cognitive Semantics, and overcomes most shortcomings of Formal Semantics and of Distributional Semantics. The theory, however, is not a simple combination of existing theories, but an extensive generalization of classic logic and Formal Semantics. It inherits nearly all advantages of Formal Semantics, and also provides descriptive contents for objects and events as fine-gram as possible, descriptive contents which represent the results of human cognition.\nCategory: Artificial Intelligence\n\n[353] viXra:1709.0096 [pdf] submitted on 2017-09-08 13:34:21\n\n### Robots Understand Brain Function\n\nAuthors: George Rajna\n\nA challenging brain technique called whole-cell patch clamp electrophysiology or whole-cell recording (WCR) is a procedure so delicate and complex that only a handful of humans in the whole world can do it. [28] ComText allows robots to understand contextual commands such as, \" Pick up the box I put down. \" [27] McMaster and Ryerson universities today announced the Smart Robots for Health Communication project, a joint research initiative designed to introduce social robotics and artificial intelligence into clinical health care. [26] Artificial intelligence can improve health care by analyzing data from apps, smartphones and wearable technology. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18]\nCategory: Artificial Intelligence\n\n[352] viXra:1709.0068 [pdf] submitted on 2017-09-06 07:16:28\n\n### Identification of Individuals\n\nAuthors: George Rajna\n\nResearchers from Human Longevity, Inc. (HLI) have published a study in which individual faces and other physical traits were predicted using whole genome sequencing data and machine learning. [26] Artificial intelligence can improve health care by analyzing data from apps, smartphones and wearable technology. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16]\nCategory: Artificial Intelligence\n\n[351] viXra:1709.0067 [pdf] submitted on 2017-09-06 07:32:22\n\n### Analyzing the Monotonicity of Belief Interval Based Uncertainty Measures in Belief Function Theory\n\nAuthors: Xinyang Deng, Shiyu Wang, Yong Deng\n\nMeasuring the uncertainty of pieces of evidence is an open issue in belief function theory. A rational uncertainty measure for belief functions should meet some desirable properties, where monotonicity is an very important property. Recently, measuring the total uncertainty of a belief function based on its associated belief intervals becomes a new research idea and have attracted increasing interest. Several belief interval based uncertainty measures have been proposed for belief functions. In this paper, we summarize the properties of these uncertainty measures and especially investigate whether the monotonicity is satisfied by the measures. This study provide a comprehensive comparison to these belief interval based uncertainty measures and is very useful for choosing the appropriate uncertainty measure in the practical applications.\nCategory: Artificial Intelligence\n\n[350] viXra:1709.0048 [pdf] submitted on 2017-09-05 04:26:06\n\n### On the Dual Nature of Logical Variables and Clause-Sets\n\nAuthors: Elnaserledinellah Mahmood Abdelwahab\n\nThis paper describes the conceptual approach behind the proposed solution of the 3SAT problem recently published in [Abdelwahab 2016]. It is intended for interested readers providing a step-by-step, mostly informal explanation of the new paradigm proposed there completing the picture from an epistemological point of view with the concept of duality on center-stage. After a brief introduction discussing the importance of duality in both, physics and mathematics as well as past efforts to solve the P vs. NP problem, a theorem is proven showing that true randomness of input-variables is a property of algorithms which has to be given up when discrete, finite domains are considered. This insight has an already known side effect on computation paradigms, namely: The ability to de-randomize probabilistic algorithms. The theorem uses a canonical type of de-randomization which reveals dual properties of logical variables and Clause-Sets. A distinction is made between what we call the syntactical Container Expression (CE) and the semantic Pattern Expression (PE). A single sided approach is presumed to be insufficient to solve anyone of the dual problems of efficiently finding an assignment validating a 3CNF Clause-Set and finding a 3CNF-representation for a given semantic pattern. The deeply rooted reason, hereafter referred to as The Inefficiency Principle, is conjectured to be the inherent difficulty of translating one expression into the other based on a single-sided perspective. It expresses our inability to perceive and efficiently calculate complementary properties of a logical formula applying one view only. It is proposed as an alternative to the commonly accepted P\u2260NP conjecture. On the other hand, the idea of algorithmically using information deduced from PE to guide the instantiation of variables in a resolution procedure applied on a CE is as per [Abdelwahab 2016] able to provide an efficient solution to the 3SAT-problem. Finally, linking de-randomization to this positive solution has various well-established and important consequences for probabilistic complexity classes which are shown to hold.\nCategory: Artificial Intelligence\n\n[349] viXra:1709.0019 [pdf] submitted on 2017-09-02 07:01:40\n\n### Understanding Robots\n\nAuthors: George Rajna\n\nComText allows robots to understand contextual commands such as, \" Pick up the box I put down. \" [27] McMaster and Ryerson universities today announced the Smart Robots for Health Communication project, a joint research initiative designed to introduce social robotics and artificial intelligence into clinical health care. [26] Artificial intelligence can improve health care by analyzing data from apps, smartphones and wearable technology. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17]\nCategory: Artificial Intelligence\n\n[348] viXra:1709.0007 [pdf] submitted on 2017-09-01 10:31:26\n\n### Computing, Cognition and Information Compression\n\nAuthors: J Gerard Wolff\n\nThis article develops the idea that the storage and processing of information in computers and in brains may often be understood as information compression. The article first reviews what is meant by information and, in particular, what is meant by redundancy, a concept which is fundamental in all methods for information compression. Principles of information compression are described. The major part of the article describes how these principles may be seen in a range of observations and ideas in computing and cognition: the phenomena of adaptation and inhibition in nervous systems; 'neural' computing; the creation and recognition of 'objects' and 'classes'in perception and cognition; stereoscopic vision and random-dot stereograms; the organisation of natural languages; the organisation of grammars; the organisation of functional, structured, logic and object-oriented computer programs; the application and de-referencing of identifiers in computing; retrieval of information from databases; access and retrieval of information from computer memory; logical deduction and resolution theorem proving; inductive reasoning and probabilistic inference; parsing; normalisation of databases.\nCategory: Artificial Intelligence\n\n[347] viXra:1709.0004 [pdf] submitted on 2017-09-01 06:49:15\n\n### Simple Chess Puzzle\n\nAuthors: George Rajna\n\nResearchers at the University of St Andrews have thrown down the gauntlet to computer programmers to find a solution to a \"simple\" chess puzzle which could, in fact, take thousands of years to solve and net a $1m prize. [11] It appears that we are approaching a unique time in the history of man and science where empirical measures and deductive reasoning can actually inform us spiritually. Integrated Information Theory (IIT)-put forth by neuroscientists Giulio Tononi and Christof Koch-is a new framework that describes a way to experimentally measure the extent to which a system is conscious. [10] There is also connection between statistical physics and evolutionary biology, since the arrow of time is working in the biological evolution also. From the standpoint of physics, there is one essential difference between living things and inanimate clumps of carbon atoms: The former tend to be much better at capturing energy from their environment and dissipating that energy as heat. [8] This paper contains the review of quantum entanglement investigations in living systems, and in the quantum mechanically modeled photoactive prebiotic kernel systems. [7] The human body is a constant flux of thousands of chemical\/biological interactions and processes connecting molecules, cells, organs, and fluids, throughout the brain, body, and nervous system. Up until recently it was thought that all these interactions operated in a linear sequence, passing on information much like a runner passing the baton to the next runner. However, the latest findings in quantum biology and biophysics have discovered that there is in fact a tremendous degree of coherence within all living systems. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to understand the Quantum Biology. Category: Artificial Intelligence [346] viXra:1708.0482 [pdf] submitted on 2017-08-31 14:55:22 ### AI Analyzes Gravitational Lenses Authors: George Rajna Comments: 25 Pages. Researchers from the Department of Energy's SLAC National Accelerator Laboratory and Stanford University have for the first time shown that neural networks - a form of artificial intelligence - can accurately analyze the complex distortions in spacetime known as gravitational lenses 10 million times faster than traditional methods. [16] By listening to the acoustic signal emitted by a laboratory-created earthquake, a computer science approach using machine learning can predict the time remaining before the fault fails. [15] Who is the better experimentalist, a human or a robot? When it comes to exploring synthetic and crystallization conditions for inorganic gigantic molecules, actively learning machines are clearly ahead, as demonstrated by British Scientists in an experiment with polyoxometalates published in the journal Angewandte Chemie. [14] Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that - surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] Category: Artificial Intelligence [345] viXra:1708.0471 [pdf] submitted on 2017-08-30 12:48:09 ### Earthquake Machine Learning Authors: George Rajna Comments: 23 Pages. By listening to the acoustic signal emitted by a laboratory-created earthquake, a computer science approach using machine learning can predict the time remaining before the fault fails. [15] Who is the better experimentalist, a human or a robot? When it comes to exploring synthetic and crystallization conditions for inorganic gigantic molecules, actively learning machines are clearly ahead, as demonstrated by British Scientists in an experiment with polyoxometalates published in the journal Angewandte Chemie. [14] Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] Category: Artificial Intelligence [344] viXra:1708.0414 [pdf] submitted on 2017-08-28 08:55:08 ### Artificial Intelligence Cyber Attacks Authors: George Rajna Comments: 24 Pages. The next major cyberattack could involve artificial intelligence systems. [13] Steve was a security robot employed by the Washington Harbour center in the Georgetown district of the US capital. [12] Combining the intuition of humans with the impartiality of computers could improve decision-making for organizations, eventually leading to lower costs and better profits, according to a team of researchers. [11] A team researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [343] viXra:1708.0381 [pdf] submitted on 2017-08-27 07:40:31 ### Security Robots Authors: George Rajna Comments: 22 Pages. Combining the intuition of humans with the impartiality of computers could improve decision-making for organizations, eventually leading to lower costs and better profits, according to a team of researchers. [11] A team researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [342] viXra:1708.0341 [pdf] submitted on 2017-08-24 22:13:50 ### Routing Games Over Time with Fifo Policy Authors: Anisse Ismaili Comments: 16 Pages. Submission to conference WINE 2017 on August 2nd. We study atomic routing games where every agent travels both along its decided edges and through time. The agents arriving on an edge are first lined up in a \\emph{first-in-first-out} queue and may wait: an edge is associated with a capacity, which defines how many agents-per-time-step can pop from the queue's head and enter the edge, to transit for a fixed delay. We show that the best-response optimization problem is not approximable, and that deciding the existence of a Nash equilibrium is complete for the second level of the polynomial hierarchy. Then, we drop the rationality assumption, introduce a behavioral concept based on GPS navigation, and study its worst-case efficiency ratio to coordination. Category: Artificial Intelligence [341] viXra:1708.0331 [pdf] submitted on 2017-08-24 13:23:16 ### Computers Improve Decision Making Authors: George Rajna Comments: 20 Pages. Combining the intuition of humans with the impartiality of computers could improve decision-making for organizations, eventually leading to lower costs and better profits, according to a team of researchers. [11] A team researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [340] viXra:1708.0246 [pdf] submitted on 2017-08-21 10:02:18 ### AI that can Understand Us Authors: George Rajna Comments: 47 Pages. Computing pioneer Alan Turing's most pertinent thoughts on machine intelligence come from a neglected paragraph of the same paper that first proposed his famous test for whether a computer could be considered as smart as a human. [27] Predictions for an AI-dominated future are increasingly common, but Antoine Blondeau has experience in reading, and arguably manipulating, the runes\u2014he helped develop technology that evolved into predictive texting and Apple's Siri. [26] Artificial intelligence can improve health care by analyzing data from apps, smartphones and wearable technology. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Category: Artificial Intelligence [339] viXra:1708.0239 [pdf] submitted on 2017-08-20 09:31:39 ### Artificial Intelligence Revolution Authors: George Rajna Comments: 45 Pages. Predictions for an AI-dominated future are increasingly common, but Antoine Blondeau has experience in reading, and arguably manipulating, the runes\u2014he helped develop technology that evolved into predictive texting and Apple's Siri. [26] Artificial intelligence can improve health care by analyzing data from apps, smartphones and wearable technology. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] Category: Artificial Intelligence [338] viXra:1708.0238 [pdf] submitted on 2017-08-19 14:27:54 ### Machine-Learning Device Authors: George Rajna Comments: 24 Pages. In what could be a small step for science potentially leading to a breakthrough, an engineer at Washington University in St. Louis has taken steps toward using nanocrystal networks for artificial intelligence applications. [16] Physicists have applied the ability of machine learning algorithms to learn from experience to one of the biggest challenges currently facing quantum computing: quantum error correction, which is used to design noise-tolerant quantum computing protocols. [15] Who is the better experimentalist, a human or a robot? When it comes to exploring synthetic and crystallization conditions for inorganic gigantic molecules, actively learning machines are clearly ahead, as demonstrated by British Scientists in an experiment with polyoxometalates published in the journal Angewandte Chemie. [14] Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Category: Artificial Intelligence [337] viXra:1708.0176 [pdf] submitted on 2017-08-16 01:32:34 ### Machine Learning Quantum Error Correction Authors: George Rajna Comments: 23 Pages. Physicists have applied the ability of machine learning algorithms to learn from experience to one of the biggest challenges currently facing quantum computing: quantum error correction, which is used to design noise-tolerant quantum computing protocols. [15] Who is the better experimentalist, a human or a robot? When it comes to exploring synthetic and crystallization conditions for inorganic gigantic molecules, actively learning machines are clearly ahead, as demonstrated by British Scientists in an experiment with polyoxometalates published in the journal Angewandte Chemie. [14] Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that - surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] New research demonstrates that particles at the quantum level can in fact be seen as behaving something like billiard balls rolling along a table, and not merely as the probabilistic smears that the standard interpretation of quantum mechanics suggests. But there's a catch - the tracks the particles follow do not always behave as one would expect from \"realistic\" trajectories, but often in a fashion that has been termed \"surrealistic.\" [8] Category: Artificial Intelligence [336] viXra:1708.0167 [pdf] submitted on 2017-08-15 06:17:08 ### Organismic Learning Authors: George Rajna Comments: 24 Pages. A new computing technology called \"organismoids\" mimics some aspects of human thought by learning how to forget unimportant memories while retaining more vital ones. [15] Who is the better experimentalist, a human or a robot? When it comes to exploring synthetic and crystallization conditions for inorganic gigantic molecules, actively learning machines are clearly ahead, as demonstrated by British Scientists in an experiment with polyoxometalates published in the journal Angewandte Chemie. [14] Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] Category: Artificial Intelligence [335] viXra:1708.0131 [pdf] submitted on 2017-08-11 13:16:12 ### Adaptive Plant Propagation Algorithm for Solving Economic Load Dispatch Problem Authors: Sayan Nag Comments: 11 Pages. Optimization problems in design engineering are complex by nature, often because of the involvement of critical objective functions accompanied by a number of rigid constraints associated with the products involved. One such problem is Economic Load Dispatch (ED) problem which focuses on the optimization of the fuel cost while satisfying some system constraints. Classical optimization algorithms are not sufficient and also inefficient for the ED problem involving highly nonlinear, and non-convex functions both in the objective and in the constraints. This led to the development of metaheuristic optimization approaches which can solve the ED problem almost efficiently. This paper presents a novel robust plant intelligence based Adaptive Plant Propagation Algorithm (APPA) which is used to solve the classical ED problem. The application of the proposed method to the 3-generator and 6-generator systems shows the efficiency and robustness of the proposed algorithm. A comparative study with another state-of-the-art algorithm (APSO) demonstrates the quality of the solution achieved by the proposed method along with the convergence characteristics of the proposed approach. Category: Artificial Intelligence [334] viXra:1708.0065 [pdf] submitted on 2017-08-06 17:11:22 ### Meta Mass Function Authors: Yong Deng Comments: 11 Pages. In this paper, a meta mass function (MMF) is presented. A new evidence theory with complex numbers is developed. Different with existing evidence theory, the new mass function in complex evidence theory is modelled as complex numbers and named as meta mass function. The classical evidence theory is the special case under the condition that the mass function is degenerated from complex number as real number. Category: Artificial Intelligence [333] viXra:1708.0038 [pdf] submitted on 2017-08-04 04:30:39 ### Holistic Unique Clustering. {File Clsoing Version 4} ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 3 Pages. In this research Technical Note the author has presented a novel method to find all Possible Clusters given a set of M points in N Space. Category: Artificial Intelligence [332] viXra:1708.0030 [pdf] submitted on 2017-08-03 10:30:43 ### Machine Learning for Discovery Authors: George Rajna Comments: 22 Pages. Who is the better experimentalist, a human or a robot? When it comes to exploring synthetic and crystallization conditions for inorganic gigantic molecules, actively learning machines are clearly ahead, as demonstrated by British Scientists in an experiment with polyoxometalates published in the journal Angewandte Chemie. [14] Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] New research demonstrates that particles at the quantum level can in fact be seen as behaving something like billiard balls rolling along a table, and not merely as the probabilistic smears that the standard interpretation of quantum mechanics suggests. But there's a catch-the tracks the particles follow do not always behave as one would expect from \"realistic\" trajectories, but often in a fashion that has been termed \"surrealistic.\" [8] Quantum entanglement\u2014which occurs when two or more particles are correlated in such a way that they can influence each other even across large distances\u2014is not an all-or-nothing phenomenon, but occurs in various degrees. The more a quantum state is entangled with its partner, the better the states will perform in quantum information applications. Unfortunately, quantifying entanglement is a difficult process involving complex optimization problems that give even physicists headaches. [7] A trio of physicists in Europe has come up with an idea that they believe would allow a person to actually witness entanglement. Valentina Caprara Vivoli, with the University of Geneva, Pavel Sekatski, with the University of Innsbruck and Nicolas Sangouard, with the University of Basel, have together written a paper describing a scenario where a human subject would be able to witness an instance of entanglement\u2014they have uploaded it to the arXiv server for review by others. [6] The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the relativistic quantum theory. Category: Artificial Intelligence [331] viXra:1708.0029 [pdf] submitted on 2017-08-03 10:54:39 ### Future Search Engines Authors: George Rajna Comments: 25 Pages. The outcome is the result of two powerful forces in the evolution of information retrieval: artificial intelligence\u2014especially natural language processing\u2014and crowdsourcing. [15] Who is the better experimentalist, a human or a robot? When it comes to exploring synthetic and crystallization conditions for inorganic gigantic molecules, actively learning machines are clearly ahead, as demonstrated by British Scientists in an experiment with polyoxometalates published in the journal Angewandte Chemie. [14] Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] Category: Artificial Intelligence [330] viXra:1708.0025 [pdf] submitted on 2017-08-02 23:22:10 ### Similarity Measure Of Any Two Vectors Of Same Size Authors: Ramesh Chandra Bagadi Comments: 2 Pages. In this research Technical Note the author has presented a novel method of finding a Generalized Similarity Measure between two Vectors of the same size. Category: Artificial Intelligence [329] viXra:1708.0019 [pdf] submitted on 2017-08-03 06:42:09 ### Holistic Unique Clustering. ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 2 Pages. In this research Technical Note the author has presented a novel method to find all Possible Clusters given a set of M points in N Space. Category: Artificial Intelligence [328] viXra:1708.0010 [pdf] submitted on 2017-08-02 04:36:45 ### A Generalized Similarity Measure {File Closing Version 3} ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 2 Pages. In this research Technical Note the author has presented a novel method of finding a Generalized Similarity Measure between two Vectors or Matrices or Higher Dimensional Data of different sizes. Category: Artificial Intelligence [327] viXra:1707.0394 [pdf] submitted on 2017-07-30 02:17:51 ### The Recursive Future Equation And The Recursive Past Equation Based On The Ananda-Damayanthi Normalized Similarity Measure. {File Closing Version-2} Authors: Ramesh Chandra Bagadi Comments: 3 Pages. In this research Technical Note the author have presented a Recursive Future Equation and Recursive Past Equation to find one Step Future Element or a one Step Past Element of a given Time Series data Set. Category: Artificial Intelligence [326] viXra:1707.0389 [pdf] submitted on 2017-07-29 07:23:01 ### Machine Learning and Deep Learning Authors: George Rajna Comments: 27 Pages. Deep learning and machine learning both offer ways to train models and classify data. This article compares the two and it offers ways to help you decide which one to use. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [325] viXra:1707.0372 [pdf] submitted on 2017-07-28 06:35:21 ### The Recursive Future Equation And The Recursive Past Equation Based On The Ananda-Damayanthi Normalized Similarity Measure. {Future} Authors: Ramesh Chandra Bagadi Comments: 2 Pages. In this research Technical Note the author have presented a Recursive Future Equation and Recursive Past Equation to find one Step Future Element or a one Step Past Element of a given Time Series data Set. Category: Artificial Intelligence [324] viXra:1707.0268 [pdf] submitted on 2017-07-20 02:20:32 ### Finding The Optimal Number \u2018K\u2019 In The K-Means Algorithm Authors: Ramesh Chandra Bagadi Comments: 2 Pages. In this research Technical Note the author has presented a novel method to find the Optimal Number \u2018K\u2019 in the K-Means Algorithm. Category: Artificial Intelligence [323] viXra:1707.0255 [pdf] submitted on 2017-07-19 05:05:13 ### Humanize Artificial Intelligent Authors: George Rajna Comments: 41 Pages. Google recently launched PAIR, an acronym of People + AI Research, in an attempt to increase the utility of AI and improve human to AI interaction. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] Category: Artificial Intelligence [322] viXra:1707.0254 [pdf] submitted on 2017-07-19 06:01:57 ### Using the Appropriate Norm In The K-Nearest Neighbours Analysis. ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 1 Page. In this research Technical Note, the author has detailed a novel technique of finding the distance metric to be used for any given set of points. Category: Artificial Intelligence [321] viXra:1707.0252 [pdf] submitted on 2017-07-19 06:40:54 ### A Generalized Similarity Measure {File Closing Version 2} ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 2 Pages. In this research Technical Note the author has presented a novel method of finding a Generalized Similarity Measure between two Vectors or Matrices or Higher Dimensional Data of different sizes. Category: Artificial Intelligence [320] viXra:1707.0230 [pdf] submitted on 2017-07-17 05:49:20 ### A Generalized Similarity Measure ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 2 Pages. In this research Technical Note the author has presented a novel method of finding a Generalized Similarity Measure between two Vectors or Matrices or Higher Dimensional Data of different sizes. Category: Artificial Intelligence [319] viXra:1707.0225 [pdf] submitted on 2017-07-17 01:50:21 ### Multi Class Classification Using Holistic Non-Unique Clustering {File Closing Version 8}. ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 3 Pages. In this research Technical Note the author has presented a novel method to find all Possible Clusters given a set of M points in N Space. Category: Artificial Intelligence [318] viXra:1707.0200 [pdf] submitted on 2017-07-14 04:55:42 ### Multi Class Classification Using Holistic Non-Unique Clustering ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 1 Page. In this research technical Note the author have presented a novel method to find all Possible Clusters given a set of M points in N Space. Category: Artificial Intelligence [317] viXra:1707.0198 [pdf] submitted on 2017-07-14 05:30:10 ### Multi Class Classification Using Holistic Non-Unique Clustering. {File Closing Version 7} ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 1 Page. In this research technical Note the author have presented a novel method to find all Possible Clusters given a set of M points in N Space. Category: Artificial Intelligence [316] viXra:1707.0179 [pdf] submitted on 2017-07-13 01:20:46 ### Modification To The Scaling Aspect In Gower\u2019s Scheme Of Calculating Similarity Coefficient Authors: Ramesh Chandra Bagadi Comments: 1 Page. In this research technical Note the author have presented a tiny modification to the Numeric Variables Scaling Aspect In Gower\u2019s Scheme of calculating Similarity Coefficient. Category: Artificial Intelligence [315] viXra:1707.0178 [pdf] submitted on 2017-07-13 02:34:27 ### Recursive Future Average Of A Time Series Data Based On Cosine Similarity Authors: Ramesh Chandra Bagadi Comments: 2 Pages. In this research Technical Note the author have presented a Recursive Future Average Of A Time Series Data Based on Cosine Similarity. Category: Artificial Intelligence [314] viXra:1707.0166 [pdf] submitted on 2017-07-12 01:12:07 ### Theoretical Materials Authors: George Rajna Comments: 49 Pages. University have created the first general-purpose method for using machine learning to predict the properties of new metals, ceramics and other crystalline materials and to find new uses for existing materials, a discovery that could save countless hours wasted in the trial-and-error process of creating new and better materials. [28] As machine learning breakthroughs abound, researchers look to democratize benefits. [27] Machine-learning system spontaneously reproduces aspects of human neurology. [26] Surviving breast cancer changed the course of Regina Barzilay's research. The experience showed her, in stark relief, that oncologists and their patients lack tools for data-driven decision making. [25] New research, led by the University of Southampton, has demonstrated that a nanoscale device, called a memristor, could be used to power artificial systems that can mimic the human brain. [24] Scientists at Helmholtz-Zentrum Dresden-Rossendorf conducted electricity through DNA-based nanowires by placing gold-plated nanoparticles on them. In this way it could become possible to develop circuits based on genetic material. [23] Researchers at the Nanoscale Transport Physics Laboratory from the School of Physics at the University of the Witwatersrand have found a technique to improve carbon superlattices for quantum electronic device applications. [22] The researchers have found that these previously underestimated interactions can play a significant role in preventing heat dissipation in microelectronic devices. [21] LCLS works like an extraordinary strobe light: Its ultrabright X-rays take snapshots of materials with atomic resolution and capture motions as fast as a few femtoseconds, or millionths of a billionth of a second. For comparison, one femtosecond is to a second what seven minutes is to the age of the universe. [20] A 'nonlinear' effect that seemingly turns materials transparent is seen for the first time in X-rays at SLAC's LCLS. [19] Category: Artificial Intelligence [313] viXra:1707.0165 [pdf] submitted on 2017-07-12 01:25:24 ### Multi Class Classification Using Holistic Non-Unique Clustering Authors: Ramesh Chandra Bagadi Comments: 2 Pages. In this research technical Note the author have presented a novel method to find all Possible Clusters given a set of M points in N Space. Category: Artificial Intelligence [312] viXra:1707.0145 [pdf] submitted on 2017-07-11 02:29:17 ### A Novel Type Of Time Series Type Forecasting Authors: Ramesh Chandra Bagadi Comments: 3 Pages. In this research investigation, the author has detailed a novel Time series type of forecasting. Category: Artificial Intelligence [311] viXra:1707.0142 [pdf] submitted on 2017-07-11 04:48:06 ### A Novel Type Of Time Series Type Forecasting. {File Closing Version 1} Authors: Ramesh Chandra Bagadi Comments: 3 Pages. In this research investigation, the author has detailed a novel Time series type of forecasting. Category: Artificial Intelligence [310] viXra:1707.0102 [pdf] submitted on 2017-07-07 01:23:03 ### Holistic Non-Unique Clsutering. {File Closing Version 1} ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 2 Pages. In this research technical Note the author have presented a novel method to find all Possible Clusters given a set of points in N Space. Category: Artificial Intelligence [309] viXra:1707.0098 [pdf] submitted on 2017-07-07 01:44:57 ### Holistic Non-Unique Clsutering. {File Closing Version 2} ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 2 Pages. In this research technical Note the author have presented a novel method to find all Possible Clusters given a set of M points in N Space. Category: Artificial Intelligence [308] viXra:1707.0071 [pdf] submitted on 2017-07-05 08:51:43 ### Seeing All The Clusters Authors: Ramesh Chandra Bagadi Comments: 1 Page. In this technical note the author has presented a novel method to find all the clusters (overlapping and non-unique) formed by a given set of points. Category: Artificial Intelligence [307] viXra:1707.0070 [pdf] submitted on 2017-07-05 08:58:23 ### Seeing All Clusters Formed By A Given Set Of Points (File Closing Version) ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 1 Page. In this research investigation, the author has presented a novel technique to find all Clusters that may be overlapping to some extent. Category: Artificial Intelligence [306] viXra:1707.0061 [pdf] submitted on 2017-07-05 06:54:24 ### Holistic Non-Unique Clsutering. ISSN 1751-3030 Authors: Ramesh Chandra Bagadi Comments: 1 Page. In this technical note, the author has presented a novel scheme of Holistic Non-Unique Clustering. Category: Artificial Intelligence [305] viXra:1707.0043 [pdf] submitted on 2017-07-03 22:47:02 ### Using the Appropriate Norm In The K-Nearest Neighbours Analysis Authors: Ramesh Chandra Bagadi Comments: 1 Page. In this Technical Note the author has presented and alternative to the use of L2 Norm for Nearness Analysis in K-Nearest Neighbours Algorithm. Category: Artificial Intelligence [304] viXra:1707.0002 [pdf] submitted on 2017-07-01 04:24:01 ### Inner Workings of Neural Networks Authors: George Rajna Comments: 33 Pages. Neural networks learn to perform computational tasks by analyzing large sets of training data. But once they've been trained, even their designers rarely have any idea what data elements they're processing. [20] Researchers from Disney Research, Pixar Animation Studios, and the University of California, Santa Barbara have developed a new technology based on artificial intelligence (AI) and deep learning that eliminates this noise and thereby enables production-quality rendering at much faster speeds. [19] Now, one group reports in ACS Nano that they have developed an artificial synapse capable of simulating a fundamental function of our nervous system\u2014 the release of inhibitory and stimulatory signals from the same \"pre-synaptic\" terminal. [18] Researchers from France and the University of Arkansas have created an artificial synapse capable of autonomous learning, a component of artificial intelligence. [17] Intelligent machines of the future will help restore memory, mind your children, fetch your coffee and even care for aging parents. [16] Unlike experimental neuroscientists who deal with real-life neurons, computational neuroscientists use model simulations to investigate how the brain functions. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip Category: Artificial Intelligence [303] viXra:1706.0570 [pdf] submitted on 2017-06-30 12:07:02 ### Convolutional Neural Network Authors: George Rajna Comments: 31 Pages. Researchers from Disney Research, Pixar Animation Studios, and the University of California, Santa Barbara have developed a new technology based on artificial intelligence (AI) and deep learning that eliminates this noise and thereby enables production-quality rendering at much faster speeds. [19] Now, one group reports in ACS Nano that they have developed an artificial synapse capable of simulating a fundamental function of our nervous system\u2014 the release of inhibitory and stimulatory signals from the same \"pre-synaptic\" terminal. [18] Researchers from France and the University of Arkansas have created an artificial synapse capable of autonomous learning, a component of artificial intelligence. [17] Intelligent machines of the future will help restore memory, mind your children, fetch your coffee and even care for aging parents. [16] Unlike experimental neuroscientists who deal with real-life neurons, computational neuroscientists use model simulations to investigate how the brain functions. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] Category: Artificial Intelligence [302] viXra:1706.0523 [pdf] submitted on 2017-06-28 09:17:30 ### Artificial Synapse for AI Authors: George Rajna Comments: 30 Pages. Now, one group reports in ACS Nano that they have developed an artificial synapse capable of simulating a fundamental function of our nervous system\u2014 the release of inhibitory and stimulatory signals from the same \"pre-synaptic\" terminal. [18] Researchers from France and the University of Arkansas have created an artificial synapse capable of autonomous learning, a component of artificial intelligence. [17] Intelligent machines of the future will help restore memory, mind your children, fetch your coffee and even care for aging parents. [16] Unlike experimental neuroscientists who deal with real-life neurons, computational neuroscientists use model simulations to investigate how the brain functions. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Category: Artificial Intelligence [301] viXra:1706.0469 [pdf] submitted on 2017-06-25 08:35:27 ### Quantum Machine Learning Computer Hybrids Authors: George Rajna Comments: 28 Pages. Creative Destruction Lab, a technology program affiliated with the University of Toronto's Rotman School of Management in Toronto, Canada hopes to nurture numerous quantum learning machine start-ups in only a few years. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [300] viXra:1706.0468 [pdf] submitted on 2017-06-25 10:31:28 ### Weak AI, Strong AI and Superintelligence Authors: George Rajna Comments: 29 Pages. Should we fear artificial intelligence and all it will bring us? Not so long as we remember to make sure to build artificial emotional intelligence into the technology, according to the website The School of Life. [16] Creative Destruction Lab, a technology program affiliated with the University of Toronto\u2019s Rotman School of Management in Toronto, Canada hopes to nurture numerous quantum learning machine start-ups in only a few years. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \u201cquantum artificial intelligence\u201d. Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries - how a sliced up flatworm can regenerate into new organisms - has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron\u2019s spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [299] viXra:1706.0462 [pdf] submitted on 2017-06-25 02:34:26 ### Brain-Inspired Supercomputing Authors: George Rajna Comments: 48 Pages. IBM and the Air Force Research Laboratory are working to develop an artificial intelligence-based supercomputer with a neural network design that is inspired by the human brain. [28] Researchers have built a new type of \"neuron transistor\"\u2014a transistor that behaves like a neuron in a living brain. [27] Research team led by Professor Hoi-Jun Yoo of the Department of Electrical Engineering has developed a semiconductor chip, CNNP (CNN Processor), that runs AI algorithms with ultra-low power, and K-Eye, a face recognition system using CNNP. [26] Artificial intelligence can improve health care by analyzing data from apps, smartphones and wearable technology. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Category: Artificial Intelligence [298] viXra:1706.0433 [pdf] submitted on 2017-06-23 06:57:24 ### AI and Robots can Help Patients Authors: George Rajna Comments: 45 Pages. McMaster and Ryerson universities today announced the Smart Robots for Health Communication project, a joint research initiative designed to introduce social robotics and artificial intelligence into clinical health care. [26] Artificial intelligence can improve health care by analyzing data from apps, smartphones and wearable technology. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] Category: Artificial Intelligence [297] viXra:1706.0402 [pdf] submitted on 2017-06-20 10:02:53 ### Neuron Transistor Authors: George Rajna Comments: 45 Pages. Researchers have built a new type of \"neuron transistor\"\u2014a transistor that behaves like a neuron in a living brain. [27] Research team led by Professor Hoi-Jun Yoo of the Department of Electrical Engineering has developed a semiconductor chip, CNNP (CNN Processor), that runs AI algorithms with ultra-low power, and K-Eye, a face recognition system using CNNP. [26] Artificial intelligence can improve health care by analyzing data from apps, smartphones and wearable technology. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] Category: Artificial Intelligence [296] viXra:1706.0389 [pdf] submitted on 2017-06-19 04:15:18 ### Artificial Intelligence Health Revolution Authors: George Rajna Comments: 43 Pages. Artificial intelligence can improve health care by analyzing data from apps, smartphones and wearable technology. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] Category: Artificial Intelligence [295] viXra:1706.0387 [pdf] submitted on 2017-06-19 04:54:30 ### K-Eye Face Recognition System Authors: George Rajna Comments: 45 Pages. A research team led by Professor Hoi-Jun Yoo of the Department of Electrical Engineering has developed a semiconductor chip, CNNP (CNN Processor), that runs AI algorithms with ultra-low power, and K-Eye, a face recognition system using CNNP. [26] Artificial intelligence can improve health care by analyzing data from apps, smartphones and wearable technology. [25] Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] Category: Artificial Intelligence [294] viXra:1706.0293 [pdf] submitted on 2017-06-16 06:05:08 ### Computers Reason Like Humans Authors: George Rajna Comments: 40 Pages. Now, researchers at Google's DeepMind have developed a simple algorithm to handle such reasoning\u2014and it has already beaten humans at a complex image comprehension test. [24] A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] Category: Artificial Intelligence [293] viXra:1706.0235 [pdf] submitted on 2017-06-13 02:02:47 ### Deep Learning with Light Authors: George Rajna Comments: 37 Pages. Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Category: Artificial Intelligence [292] viXra:1706.0207 [pdf] submitted on 2017-06-13 11:45:22 ### Neural Networks and Quantum Entanglement Authors: George Rajna Comments: 39 Pages. Specifying a number for each connection and mathematically forgetting the hidden neurons can produce a compact representation of many interesting quantum states, including states with topological characteristics and some with surprising amounts of entanglement. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] Category: Artificial Intelligence [291] viXra:1706.0198 [pdf] submitted on 2017-06-14 08:06:29 ### Robot Write and Play its own Music Authors: George Rajna Comments: 38 Pages. A marimba-playing robot with four arms and eight sticks is writing and playing its own compositions in a lab at the Georgia Institute of Technology. The pieces are generated using artificial intelligence and deep learning. [23] Now, a team of researchers at MIT and elsewhere has developed a new approach to such computations, using light instead of electricity, which they say could vastly improve the speed and efficiency of certain deep learning computations. [22] Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology\u2019s impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron\u2019s spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [290] viXra:1706.0144 [pdf] submitted on 2017-06-11 07:47:04 ### Classical and Quantum Machine Learning Authors: George Rajna Comments: 35 Pages. Physicists have found that the structure of certain types of quantum learning algorithms is very similar to their classical counterparts\u2014a finding that will help scientists further develop the quantum versions. [21] We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Category: Artificial Intelligence [289] viXra:1705.0404 [pdf] submitted on 2017-05-28 12:05:57 ### Using Student Learning Based on Fluency for the Learning Rate in a Deep Convolutional Neural Network Authors: Abien Fred Agarap Comments: 23 Pages. This is a proposal for mathematically determining the learning rate to be used in a deep supervised convolutional neural network (CNN), based on student fluency. The CNN model shall be tasked to imitate how students play the game \u201cPacket Attack\u201d, a form of gamification of information security awareness training, and learn in the same rate as the students did. The student fluency shall be represented by a mathematical function constructed using natural cubic spline interpolation, and its derivative shall serve as the learning rate for the CNN model. If proven right, the results will imply a more human-like rate of learning by machines. Category: Artificial Intelligence [288] viXra:1705.0362 [pdf] submitted on 2017-05-25 03:53:34 ### Artificial Intelligence by Quantum Computing Authors: George Rajna Comments: 34 Pages. We should remain optimistic that quantum computing and AI will continue to improve our lives, but we also should continue to hold companies, organizations, and governments accountable for how our private data is used, as well as the technology's impact on the environment. [20] It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] Category: Artificial Intelligence [287] viXra:1705.0340 [pdf] submitted on 2017-05-22 19:18:05 ### Verifying the Validity of a Conformant Plan is co-NP-Complete Authors: Alban Grastien, Enrico Scala Comments: 3 Pages. The purpose of this document is to show the complexity of verifying the validity of a deterministic conformant plan. We concentrate on a simple version of the conformant planning problem (i.e., one where there is no precondition on the actions and where all conditions are defined as sets of positive or negative facts) in order to show that the complexity does not come from solving a single such formula. Category: Artificial Intelligence [286] viXra:1705.0313 [pdf] submitted on 2017-05-21 09:43:28 ### Rematch of Man vs Machine Authors: George Rajna Comments: 32 Pages. It's man vs machine this week as Google's artificial intelligence programme AlphaGo faces the world's top-ranked Go player in a contest expected to end in another victory for rapid advances in AI. [19] Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron\u2019s spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [285] viXra:1705.0273 [pdf] submitted on 2017-05-18 10:06:56 ### Google Latest Tech Tricks Authors: George Rajna Comments: 31 Pages. Google's computer programs are gaining a better understanding of the world, and now it wants them to handle more of the decision-making for the billions of people who use its services. [18] Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Category: Artificial Intelligence [284] viXra:1705.0223 [pdf] submitted on 2017-05-15 03:07:04 ### A Novel Pandemonium Architecture Based on Visual Topological Invariants and Mental Matching Descriptions Authors: Arturo Tozzi, James F Peters Comments: 13 Pages. A novel daemon-based architecture is introduced to elucidate some brain functions, such as pattern recognition during human perception and mental interpretation of visual scenes. By taking into account the concepts of invariance and persistence in topology, we introduce a Selfridge pandemonium variant of brain activity that takes into account a novel feature, namely, extended feature daemons that, in addition to the usual recognition of short straight as well as curved lines, recognize topological features of visual scene shapes, such as shape interior, density and texture. A series of transformations can be gradually applied to a pattern, in particular to the shape of an object, without affecting its invariant properties, such as its boundedness and connectedness of the parts of a visual scene. We also introduce another Pandemonium implementation: low-level representations of objects can be mapped to higher-level views (our mental interpretations), making it possible to construct a symbolic multidimensional representation of the environment. The representations can be projected continuously to an object that we have seen and continue to see, thanks to the mapping from shapes in our memory to shapes in Euclidean space. A multidimensional vista detectable by the brain (brainscapes) results from the presence of daemons (mind channels) that detect not only ordinary views of the shapes in visual scenes, but also the features of the shapes. Although perceived shapes are 3-dimensional (3+1 dimensional, if we include time), shape features (volume, colour, contour, closeness, texture, and so on) lead to n-dimensional brainscapes, We arrive at 5 as a minimum shape feature space, since every visual shape has at least a contour in space-time. We discuss the advantages of our parallel, hierarchical model in pattern recognition, computer vision and biological nervous system\u2019s evolution. Category: Artificial Intelligence [283] viXra:1705.0217 [pdf] submitted on 2017-05-14 04:25:18 ### Popular Routes Discovery Authors: Tal Ben Yakar Comments: 6 Pages. Finding the optimal driving route has attracted considerable attention in recent years, the problem sounds simple however different companies these days, taxi alternatives companies like Uber and Via trying to find what is the best route to drive find it as a very challenging problem. Ridesharing and maps companies like HERE, navigation companies like waze and public transportation companies like moovit and others. AI robots in addition, need to have the ability to route in the optimal manner. In this work we formulate the problem of finding optimal routes as an optimization problem and come up with a neat, low memory and fast solution to the problem using machine learning algorithms. Category: Artificial Intelligence [282] viXra:1705.0172 [pdf] submitted on 2017-05-10 12:43:05 ### Democratize Artificial Intelligence Authors: George Rajna Comments: 29 Pages. Microsoft on Wednesday unveiled new tools intended to democratize artificial intelligence by enabling machine smarts to be built into software from smartphone games to factory floors. [17] The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] Category: Artificial Intelligence [281] viXra:1705.0108 [pdf] submitted on 2017-05-05 09:20:09 ### Incorrect Moves and Testable States Authors: Dimiter Dobrev Comments: 17 Pages. How do we describe the invisible? Let\u2019s take a sequence: input, output, input, output ... Behind this sequence stands a world and the sequence of its internal states. We do not see the internal state of the world, but only a part of it. To describe that part which is invisible, we will use the concept of \u2018incorrect move\u2019 and its generalization \u2018testable state\u2019. Thus, we will reduce the problem of partial observability to the problem of full observability. Category: Artificial Intelligence [280] viXra:1705.0094 [pdf] submitted on 2017-05-04 04:17:51 ### Rotation Invariance Neural Network Authors: Shiyuan.Li Comments: 7 Pages. Rotation invariance and translate invariance have great values in image recognition. In this paper, we bring a new architecture in convolutional neural network (CNN) to achieve rotation invariance and translate invariance in 2-D symbol recognition. We can also get the position and orientation of the 2-D symbol by the network to achieve detection purpose for multiple non-overlap target. Human being have the ability look at an object by one glance and remember it, we also can use this architecture to achieve this one shot learning. Category: Artificial Intelligence [279] viXra:1705.0027 [pdf] submitted on 2017-05-02 21:38:43 ### Obstacle Detection and Pathfinding for Mobile Robots Authors: Murat Arslan Comments: 116 Pages. In this thesis, obstacle detection via image of objects and then pathfinding problems of NAO humanoid robot is considered. NAO's camera is used to capture the images of world map. The captured image is processed and classified into two classes; area with obstacles and area without obstacles. For classification of images, Support Vector Machine (SVM) is used. After classification the map of world is obtained as area with obstacles and area without obstacles. This map is input for path finding algorithm. In the thesis A* path finding algorithm is used to find path from the start point to the goal. The aim of this work is to implement a support vector machine based solution to robot guidance problem, visual path planning and obstacle avoidance. The used algorithms allow to detect obstacles and find an optimal path. The thesis describe basic steps of navigation of mobile robots. Category: Artificial Intelligence [278] viXra:1704.0353 [pdf] submitted on 2017-04-26 06:56:36 ### Artificial Synapse Authors: George Rajna Comments: 29 Pages. Researchers from France and the University of Arkansas have created an artificial synapse capable of autonomous learning, a component of artificial intelligence. [17] Intelligent machines of the future will help restore memory, mind your children, fetch your coffee and even care for aging parents. [16] Unlike experimental neuroscientists who deal with real-life neurons, computational neuroscientists use model simulations to investigate how the brain functions. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] Category: Artificial Intelligence [277] viXra:1704.0337 [pdf] submitted on 2017-04-26 03:14:10 ### Digital Assistant Authors: George Rajna Comments: 29 Pages. Intelligent machines of the future will help restore memory, mind your children, fetch your coffee and even care for aging parents. [16] Unlike experimental neuroscientists who deal with real-life neurons, computational neuroscientists use model simulations to investigate how the brain functions. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [276] viXra:1704.0308 [pdf] submitted on 2017-04-23 11:14:37 ### 3D Printed Dancing Humanoid Robot \u201cBuddy\u201d for Homecare Authors: Akshay Potnuru, Mohsen Jafarzadeh, Yonas Tadesse Comments: 6 Pages. This paper describes a 3D printed humanoid robot that can perform dancing and demonstrate human-like facial expressions to expand humanoid robotics in entertainment and at the same time to have an assistive role for children and elderly people. The humanoid is small and has an expressive face that is in a comfort zone for a child or an older person. It can maneuver in a day care or home care environment using its wheeled base. This paper discusses on the capabilities of the robot to carry and handle small loads like pills, common measurement tools such as pressure and temperature measurement units. The paper also discusses the use of IP camera for color identification and an Arduino based audio system to synchronize music with dance movements of the robot. Category: Artificial Intelligence [275] viXra:1704.0307 [pdf] submitted on 2017-04-23 11:21:17 ### Humanoid Robot Path Planning with Fuzzy Markov Decision Processes\u200f Authors: Mahdi Fakoor, Amirreza Kosari, Mohsen Jafarzadeh Comments: 11 Pages. In contrast to the case of known environments, path planning in unknown environments, mostly for humanoid robots, is yet to be opened for further development. This is mainly attributed to the fact that obtaining thorough sensory information about an unknown environment is not functionally or economically applicable. This study alleviates the latter problem by resorting to a novel approach through which the decision is made according to fuzzy Markov decision processes (FMDP), with regard to the pace. The experimental results show the efficiency of the proposed method. Category: Artificial Intelligence [274] viXra:1704.0298 [pdf] submitted on 2017-04-22 19:23:44 ### Design and Motion Control of Bioinspired Humanoid Robot Head from Servo Motors Toward Artificial Muscles Authors: Yara Almubarak, Yonas Tadesse Comments: 9 Pages. The potential applications of humanoid robots in social environments, motivates researchers to design, and control biomimetic humanoid robots. Generally, people are more interested to interact with robots that have similar attributes and movements to humans. The head is one of most important part of any social robot. Currently, most humanoid heads use electrical motors, pneumatic actuators, and shape memory alloy (SMA) actuators for actuation. Electrical and pneumatic actuators take most of the space and would cause unsmooth motions. SMAs are expensive to use in humanoids. Recently, in many robotic projects, Twisted and Coiled Polymer (TCP) artificial muscles are used as linear actuators which take up little space compared to the motors. In this paper, we will demonstrate the designing process and motion control of a robotic head with TCP muscles. Servo motors and artificial muscles are used for actuating the head motion, which have been controlled by a cost efficient ARM Cortex-M7 based development board. A complete comparison between the two actuators is presented. Category: Artificial Intelligence [273] viXra:1704.0205 [pdf] submitted on 2017-04-17 01:40:24 ### Formula Analyzer: Find the Formula by Parameters Authors: Artur Eduardovich Sibgatullin Comments: 27 Pages. MIT License, https:\/\/figshare.com\/articles\/Formula_analyzer_Find_the_formula_by_parameters\/4880012 Let it be a formula, e.g.: x + y^2 - z = r. It is usually necessary to find a parameter\u2019s value by knowing others\u2019 ones. However, let\u2019s set another problem to find the formula itself, knowing only its parameters. The solution of such a problem we call reverse computing. For that we'll create an algorithm and accomplish it as a program code. Category: Artificial Intelligence [272] viXra:1704.0113 [pdf] submitted on 2017-04-09 11:21:19 ### Automatic Speech Recognition Authors: George Rajna Comments: 27 Pages. The closer we can get a machine translation to be on par with expert human translation, the happier lots of people struggling with translations will be. [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [271] viXra:1704.0090 [pdf] submitted on 2017-04-07 11:26:30 ### Toward Self-Govern and Self-Protected Data: a Proposal Authors: Kasra Madadipouya Comments: 3 Pages. Unpublished research proposal We live in an era of an explosion of data. The rate of generating data has been increased significantly in the last few years especially by popularization of Web 2.0. In addition to that, our surrounding environments are becoming more dynamics and rapidly emerging as computing systems morph from monolithic and closed entities into globally disaggregated collaborating entities which require sensitive data sharing. As an instance content owners lose full control of their data once it is given away to consumers and hence data can be unlimitedly copied, access, modified and redistributed without data owner awareness. Category: Artificial Intelligence [270] viXra:1704.0089 [pdf] submitted on 2017-04-07 11:40:51 ### Machine Learning Chip Authors: George Rajna Comments: 23 Pages. Google has said the TPU beat Nvidia and Intel. Let's explain that. There is so much to explain. TPU stands for Tensor Processing Unit. This is described by a Google engineer as \"an entirely new class of custom machine learning accelerator.\" [14] Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] New research demonstrates that particles at the quantum level can in fact be seen as behaving something like billiard balls rolling along a table, and not merely as the probabilistic smears that the standard interpretation of quantum mechanics suggests. But there's a catch-the tracks the particles follow do not always behave as one would expect from \"realistic\" trajectories, but often in a fashion that has been termed \"surrealistic.\" [8] Quantum entanglement\u2014which occurs when two or more particles are correlated in such a way that they can influence each other even across large distances\u2014is not an all-or-nothing phenomenon, but occurs in various degrees. The more a quantum state is entangled with its partner, the better the states will perform in quantum information applications. Unfortunately, quantifying entanglement is a difficult process involving complex optimization problems that give even physicists headaches. [7] A trio of physicists in Europe has come up with an idea that they believe would allow a person to actually witness entanglement. Valentina Caprara Vivoli, with the University of Geneva, Pavel Sekatski, with the University of Innsbruck and Nicolas Sangouard, with the University of Basel, have together written a paper describing a scenario where a human subject would be able to witness an instance of entanglement\u2014they have uploaded it to the arXiv server for review by others. [6] The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the relativistic quantum theory. Category: Artificial Intelligence [269] viXra:1704.0022 [pdf] submitted on 2017-04-03 08:49:50 ### Visualizing Scientific Big Data Authors: George Rajna Comments: 32 Pages. Humans are visual creatures: our brain processes images 60,000 times faster than text, and 90 percent of information sent to the brain is visual. Visualization is becoming increasingly useful in the era of big data, in which we are generating so much data at such high rates that we cannot keep up with making sense of it all. In particular, visual analytics\u2014a research discipline that combines automated data analysis with interactive visualizations\u2014has emerged as a promising approach to dealing with this information overload. [18] Neural networks are commonly used today to analyze complex data \u2013 for instance to find clues to illnesses in genetic information. Ultimately, though, no one knows how these networks actually work exactly. [17] Hey Siri, how's my hair?\" Your smartphone may soon be able to give you an honest answer, thanks to a new machine learning algorithm designed by U of T Engineering researchers Parham Aarabi and Wenzhi Guo. [16] Researchers at Lancaster University's Data Science Institute have developed a software system that can for the first time rapidly self-assemble into the most efficient form without needing humans to tell it what to do. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] Category: Artificial Intelligence [268] viXra:1704.0021 [pdf] submitted on 2017-04-03 09:15:46 ### Electronic Synapses Artificial Brain Authors: George Rajna Comments: 33 Pages. Researchers from the CNRS, Thales, and the Universities of Bordeaux, Paris-Sud, and Evry have created an artificial synapse capable of learning autonomously. They were also able to model the device, which is essential for developing more complex circuits. [19] Humans are visual creatures: our brain processes images 60,000 times faster than text, and 90 percent of information sent to the brain is visual. Visualization is becoming increasingly useful in the era of big data, in which we are generating so much data at such high rates that we cannot keep up with making sense of it all. In particular, visual analytics\u2014a research discipline that combines automated data analysis with interactive visualizations\u2014has emerged as a promising approach to dealing with this information overload. [18] Neural networks are commonly used today to analyze complex data \u2013 for instance to find clues to illnesses in genetic information. Ultimately, though, no one knows how these networks actually work exactly. [17] Hey Siri, how's my hair?\" Your smartphone may soon be able to give you an honest answer, thanks to a new machine learning algorithm designed by U of T Engineering researchers Parham Aarabi and Wenzhi Guo. [16] Researchers at Lancaster University's Data Science Institute have developed a software system that can for the first time rapidly self-assemble into the most efficient form without needing humans to tell it what to do. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] Category: Artificial Intelligence [267] viXra:1703.0233 [pdf] submitted on 2017-03-24 10:36:41 ### Parallel Computation and Brain Function Authors: George Rajna Comments: 26 Pages. Unlike experimental neuroscientists who deal with real-life neurons, computational neuroscientists use model simulations to investigate how the brain functions. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [266] viXra:1703.0063 [pdf] submitted on 2017-03-07 09:36:39 ### Human Readable Feature Generation for Natural Language Corpora Authors: Tomasz Dryjanski Comments: 4 Pages. This paper proposes an alternative to the Paragraph Vector algorithm, generating fixed-length vectors of human-readable features for natural language corpora. It extends word2vec retaining its other advantages like speed and accuracy, hence its proposed name is doc2feat. Extracted features are presented as lists of words with their proximity to the particular feature, allowing interpretation and manual annotation. By parameter tuning focus can be made on grammatical aspects of the corpus language, making it useful for linguistic applications. The algorithm can run on variable-length pieces of texts, and provides insight into what features are relevant for text classification or sentiment analysis. The corpus does not have to, and in specific cases should not be, preprocessed with stemming or stop-words removal. Category: Artificial Intelligence [265] viXra:1703.0056 [pdf] submitted on 2017-03-06 13:57:49 ### Quantum Machine Learning to Infinite Dimensions Authors: George Rajna Comments: 27 Pages. Physicists have developed a quantum machine learning algorithm that can handle infinite dimensions\u2014that is, it works with continuous variables (which have an infinite number of possible values on a closed interval) instead of the typically used discrete variables (which have only a finite number of values). [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [264] viXra:1703.0013 [pdf] submitted on 2017-03-02 05:44:53 ### Controlling a Robot Using a Wearable Device (MYO) Authors: Mithileysh Sathiyanarayanan, Tobias Mulling, Bushra Nazir Comments: 6 Pages. IJEDR, 2015, Vol 3, Issue 3 There is a huge demand for military robots in almost all the countries which comes under the field of human computer interaction and artificial intelligence. There are many different ways of operating a robot: self controlled, automatic controlled etc. Also, gesture controlled operation mode is on the rise. This acted as our motivation to develop a gesture controlled robot using MYO armband. The word \u2018MYO\u2019 has created a buzz in the technological world by its astonishing features and its utility in various fields. Its introduction as a armband that can wrap around our arm to control robots with our movements and gestures has opened new wide doors of its experimentation with robotics. This independently working gesture recognition system does not rely on any external sensors (motion capturing system) as it has its sensors embedded in itself which recognizes the gesture commands and acts accordingly. This armband can be worn by soldiers to operate robots to fight against enemies. This work in progress paper illustrates an existing robot designed by us, which can be controlled by hand gestures using a wearable device called as MYO. We would like to investigate more on this and implement, such that the robot can be interfaced with a MYO armband for a successful control. Category: Artificial Intelligence [263] viXra:1703.0012 [pdf] submitted on 2017-03-02 05:49:01 ### Leap Motion Device for Gesture Controlling an Unmanned Ground Vehicle (Robot) Authors: Mithileysh Sathiyanarayanan, Tobias Mulling, Bushra Nazir Comments: 10 Pages. IJEDR, 2016, Vol 4, Issue 4 A new scope of human-computer interaction utilizes the algorithms of computer vision and image processing for detecting the gesture, understanding its objective and making it meaningful for the computer to understand and then interact with the humans. The recent introduction of \"Leap Motion\" is a big revolution in the field of gesture control technology. Using gesture control mode in the field of robotics is also on the rise. This acted as our motivation to develop a gesture controlled robot using a Leap Motion Device that can sense human hands above it and to keep a track of them and aid in navigation. This independently working gesture recognition system does not rely on any external sensors (motion capturing system) as it has its sensors embedded in itself which recognizes the gesture commands and acts accordingly. The soldiers need not wear any physical device on their body (unlike Kinect and\/or MYO Armband) to operate robots to fight against enemies. This work in progress paper illustrates an existing robot designed by us, which can be controlled by hand gestures using a non-wearable (touchless) device called as Leap Motion. Category: Artificial Intelligence [262] viXra:1702.0297 [pdf] submitted on 2017-02-23 18:45:36 ### Some General Results On Overfitting In Machine Learning Authors: Antony Van der Mude Comments: 13 Pages. Overfitting has always been a problem in machine learning. Recently a related phenomenon called \u201coversearching\u201d has been analyzed. This paper takes a theoretical approach using a very general methodology covering most learning paradigms in current use. Overfitting is defined in terms of the \u201cexpressive accuracy\u201d of a model for the data, rather than \u201cpredictive accuracy\u201d. The results show that even if the learner can identify a set of best models, overfitting will cause it to bounce from one model to another. Overfitting is ameliorated by having the learner bound the search space, and bounding is equivalent to using an accuracy (or bias) more restrictive than the problem accuracy. Also, Ramsey\u2019s Theorem shows that every data sequence has an situation where either consistent overfitting or underfitting is unavoidable. We show that oversearching is simply overfitting where the resource used to express a model is the search space itself rather than a more common resource such as a program that executes the model. We show that the smallest data sequence guessing a model defines a canonical resource. There is an equivalence in the limit between any two resources to express the same model space, but it may not be effectively computable. Category: Artificial Intelligence [261] viXra:1702.0275 [pdf] submitted on 2017-02-22 09:32:33 ### Quantum Artificial Biomimetics Authors: George Rajna Comments: 26 Pages. Quantum biomimetics consists of reproducing in quantum systems certain properties exclusive to living organisms. Researchers at University of the Basque Country have imitated natural selection, learning and memory in a new study. The mechanisms developed could give quantum computation a boost and facilitate the learning process in machines. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [260] viXra:1702.0232 [pdf] submitted on 2017-02-18 07:11:56 ### New Materials from Small Data Authors: George Rajna Comments: 22 Pages. Finding new functional materials is always tricky. But searching for very specific properties among a relatively small family of known materials is even more difficult. [14] Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] New research demonstrates that particles at the quantum level can in fact be seen as behaving something like billiard balls rolling along a table, and not merely as the probabilistic smears that the standard interpretation of quantum mechanics suggests. But there's a catch-the tracks the particles follow do not always behave as one would expect from \"realistic\" trajectories, but often in a fashion that has been termed \"surrealistic.\" [8] Quantum entanglement\u2014which occurs when two or more particles are correlated in such a way that they can influence each other even across large distances\u2014is not an all-or-nothing phenomenon, but occurs in various degrees. The more a quantum state is entangled with its partner, the better the states will perform in quantum information applications. Unfortunately, quantifying entanglement is a difficult process involving complex optimization problems that give even physicists headaches. [7] A trio of physicists in Europe has come up with an idea that they believe would allow a person to actually witness entanglement. Valentina Caprara Vivoli, with the University of Geneva, Pavel Sekatski, with the University of Innsbruck and Nicolas Sangouard, with the University of Basel, have together written a paper describing a scenario where a human subject would be able to witness an instance of entanglement\u2014they have uploaded it to the arXiv server for review by others. [6] The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the relativistic quantum theory. Category: Artificial Intelligence [259] viXra:1702.0229 [pdf] submitted on 2017-02-18 02:21:36 ### A.i. Music Duet Authors: George Rajna Comments: 27 Pages. An artificial intelligence experiment has emerged of the most enjoyable kind: It is called \"A.I. Duet.\" [16] Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. Category: Artificial Intelligence [258] viXra:1702.0143 [pdf] submitted on 2017-02-12 10:31:13 ### Human Motion and Language Authors: George Rajna Comments: 26 Pages. Researchers have created a large, open source database to support the development of robot activities based on natural language input. [15] A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [257] viXra:1702.0130 [pdf] submitted on 2017-02-10 09:42:04 ### Artificial Neural Network Authors: George Rajna Comments: 25 Pages. A pair of physicists with ETH Zurich has developed a way to use an artificial neural network to characterize the wave function of a quantum many-body system. [14] A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [256] viXra:1702.0094 [pdf] submitted on 2017-02-07 10:15:12 ### Complex Neutrosophic Soft Set Authors: Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache, Mumtaz Ali, Ganeshsree Selvachandran Comments: 6 Pages. In this paper, we propose the complex neutrosophic soft set model, which is a hybrid of complex fuzzy sets,neutrosophic sets and soft sets. The basic set theoretic operations and some concepts related to the structure of this model are introduced, and illustrated. An example related to a decision making problem involving uncertain and subjective information is presented, to demonstrate the utility of this model. Category: Artificial Intelligence [255] viXra:1702.0010 [pdf] submitted on 2017-02-01 08:18:01 ### Visualize Complex Learning Processes Authors: George Rajna Comments: 29 Pages. Neural networks are commonly used today to analyze complex data \u2013 for instance to find clues to illnesses in genetic information. Ultimately, though, no one knows how these networks actually work exactly. [17] Hey Siri, how's my hair?\" Your smartphone may soon be able to give you an honest answer, thanks to a new machine learning algorithm designed by U of T Engineering researchers Parham Aarabi and Wenzhi Guo. [16] Researchers at Lancaster University's Data Science Institute have developed a software system that can for the first time rapidly self-assemble into the most efficient form without needing humans to tell it what to do. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Category: Artificial Intelligence [254] viXra:1702.0008 [pdf] submitted on 2017-02-01 08:56:06 ### First Passage Under Restart Authors: George Rajna Comments: 31 Pages. Discovering the ways in which many seemingly diverse phenomena are related is one of the overarching goals of scientific inquiry, since universality often allows an insight in one area to be extended to many other areas. [18] Neural networks are commonly used today to analyze complex data \u2013 for instance to find clues to illnesses in genetic information. Ultimately, though, no one knows how these networks actually work exactly. [17] Hey Siri, how's my hair?\" Your smartphone may soon be able to give you an honest answer, thanks to a new machine learning algorithm designed by U of T Engineering researchers Parham Aarabi and Wenzhi Guo. [16] Researchers at Lancaster University's Data Science Institute have developed a software system that can for the first time rapidly self-assemble into the most efficient form without needing humans to tell it what to do. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \u201cquantum artificial intelligence\u201d. Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries - how a sliced up flatworm can regenerate into new organisms - has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron\u2019s spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [253] viXra:1701.0574 [pdf] submitted on 2017-01-22 21:33:04 ### The Relationship Between Agents and Link-Level Acknowledgements Using Mugwump Authors: Thomas Lambert Comments: 8 Pages. In recent years, much research has been devoted to the improvement of architecture; unfortunately, few have explored the emulation of the World Wide Web. In fact, few biologists would disagree with the deployment of evolutionary programming. While this discussion is never a confirmed intent, it is derived from known results. Mugwump, our new framework for hash tables [28], is the solution to all of these challenges. Category: Artificial Intelligence [252] viXra:1701.0559 [pdf] submitted on 2017-01-21 11:05:33 ### AI Systems See the World as Humans Authors: George Rajna Comments: 38 Pages. A Northwestern University team developed a new computational model that performs at human levels on a standard intelligence test. This work is an important step toward making artificial intelligence systems that see and understand the world as humans do. [25] Neuroscience and artificial intelligence experts from Rice University and Baylor College of Medicine have taken inspiration from the human brain in creating a new \"deep learning\" method that enables computers to learn about the visual world largely on their own, much as human babies do. [24] Category: Artificial Intelligence [251] viXra:1701.0530 [pdf] submitted on 2017-01-17 20:03:50 ### Intelligence of Crowd. Authors: Michail Zak Comments: 17 Pages. A new class of dynamical systems with a preset type of interference of probabilities is introduced. It is obtained from the extension of the Madelung equation by replacing the quantum potential with a specially selected feedback from the Liouville equation. It has been proved that these systems are different from both Newtonian and quantum systems, but they can be useful for modeling spontaneous collective novelty phenomena when emerging outputs are qualitatively different from the weighted sum of individual inputs. Formation of language and fast decision-making process as potential applications of the probability interference is discussed. Category: Artificial Intelligence [250] viXra:1701.0516 [pdf] submitted on 2017-01-16 14:14:27 ### Optimal Control Via Self-Generated Stochasticity. Authors: Michail Zak Comments: 19 Pages. Stochastic approach to maximization of a functional constrained by governing equation of a controlled system is introduced and discussed. The idea of the proposed algorithm is the following: represent the functional to be maximized as a limit of a probability density governed by the appropriately selected Liouville equation. Then the corresponding ODE become stochastic, and that sample of the solution which has the largest value will have the highest probability to appear in ODE simulation. Application to optimal control is discussed. Two limitations of optimal control theory - local maxima and possible instability of the optimal solutions - are removed. Special attention is paid to robot motion planning. Category: Artificial Intelligence [249] viXra:1612.0403 [pdf] submitted on 2016-12-30 06:29:12 ### Applications of Machine Learning in Estimating the Minimum Distance of Approach of an NEO Authors: Jayant Mehra Comments: 15 Pages, 6 Figures, 5 Tables Although the current detection techniques have been able to calculate the minimum distance to which a Near Earth Object (NEO) can approach Earth for thousands of NEOs, there are millions of yet undiscovered NEOs which could pose a threat to Planet Earth. An NEO is considered highly dangerous if the minimum distance between it and the centre of the Earth is less than 0.03 AU. However, only a handful NEOs have been detected prior to entering this danger zone. The immense task of asteroid hunting by conventional techniques is further complicated by a high number of false positives and false negatives. In this report, machine learning algorithms are written to predict the minimum distance upto which an NEO can approach the planet and classify NEOs as whether they are in the danger zone or no based on their physical characteristics. In section 4 of the study, an Artificial Neural Network based on the backpropagation algorithm and a Logistic Classification based on Unconstrained Minimisation using the fminunc function are employed to classify NEOs with an accuracy of 92% and 90% respectively. In section 5 of the report, the Levenberg - Marquardt Algorithm based on an Artificial Neural Network is employed to calculate the minimum distance with a regression R value of 0.79 (Value of 1 being the maximum). All the algorithmic systems developed have low false positive and false negative rates Category: Artificial Intelligence [248] viXra:1612.0344 [pdf] submitted on 2016-12-26 10:03:47 ### Advance Artificial Super-Intelligence Authors: Miguel A. Sanchez-Rey Comments: 3 Pages. From FL to AL. Category: Artificial Intelligence [247] viXra:1612.0314 [pdf] submitted on 2016-12-21 07:33:22 ### Spintronics-Based Artificial Intelligence Authors: George Rajna Comments: 45 Pages. Researchers at Tohoku University have, for the first time, successfully demonstrated the basic operation of spintronics-based artificial intelligence. [27] The neural structure we use to store and process information in verbal working memory is more complex than previously understood, finds a new study by researchers at New York University. [26] Surviving breast cancer changed the course of Regina Barzilay's research. The experience showed her, in stark relief, that oncologists and their patients lack tools for data-driven decision making. [25] New research, led by the University of Southampton, has demonstrated that a nanoscale device, called a memristor, could be used to power artificial systems that can mimic the human brain. [24] Scientists at Helmholtz-Zentrum Dresden-Rossendorf conducted electricity through DNA-based nanowires by placing gold-plated nanoparticles on them. In this way it could become possible to develop circuits based on genetic material. [23] Researchers at the Nanoscale Transport Physics Laboratory from the School of Physics at the University of the Witwatersrand have found a technique to improve carbon superlattices for quantum electronic device applications. [22] The researchers have found that these previously underestimated interactions can play a significant role in preventing heat dissipation in microelectronic devices. [21] LCLS works like an extraordinary strobe light: Its ultrabright X-rays take snapshots of materials with atomic resolution and capture motions as fast as a few femtoseconds, or millionths of a billionth of a second. For comparison, one femtosecond is to a second what seven minutes is to the age of the universe. [20] A 'nonlinear' effect that seemingly turns materials transparent is seen for the first time in X-rays at SLAC's LCLS. [19] Leiden physicists have manipulated light with large artificial atoms, so-called quantum dots. Before, this has only been accomplished with actual atoms. It is an important step toward light-based quantum technology. [18] In a tiny quantum prison, electrons behave quite differently as compared to their counterparts in free space. They can only occupy discrete energy levels. Category: Artificial Intelligence [246] viXra:1612.0288 [pdf] submitted on 2016-12-18 09:03:13 ### Neuroscience and Artificial Intelligence Authors: George Rajna Comments: 37 Pages. Neuroscience and artificial intelligence experts from Rice University and Baylor College of Medicine have taken inspiration from the human brain in creating a new \"deep learning\" method that enables computers to learn about the visual world largely on their own, much as human babies do. [24] Category: Artificial Intelligence [245] viXra:1612.0242 [pdf] submitted on 2016-12-14 09:45:02 ### Doctor of Philosophy Thesis in Military Informatics (Openphd) :Lethal Autonomy of Weapons is Designed And\/or Recessive Authors: Nyagudi Musandu Nyagudi Comments: 1 Page. By way of Prior Publications, Practice and Contribution My original contribution to knowledge is : Any weapon that exhibits intended and\/or untended lethal autonomy in targeting and interdiction \u2013 does so by way of design and\/or recessive flaw(s) in its systems of control \u2013 any such weapon is capable of war-fighting and other battle-space interaction in a manner that its Human Commander does not anticipate. A lethal autonomous weapons is therefore independently capable of exhibiting positive or negative recessive norms of targeting in its perceptions of Discrimination between Civilian and Military Objects, Proportionality of Methods and Outcomes, Feasible Precaution before interdiction and its underlying Concepts of Humanity. This marks the completion of an Open PhD ( #openphd ) project done in sui generis form. Category: Artificial Intelligence [244] viXra:1612.0214 [pdf] submitted on 2016-12-12 10:53:08 ### Memory Architecture for AI Authors: George Rajna Comments: 44 Pages. The neural structure we use to store and process information in verbal working memory is more complex than previously understood, finds a new study by researchers at New York University. [26] Surviving breast cancer changed the course of Regina Barzilay's research. The experience showed her, in stark relief, that oncologists and their patients lack tools for data-driven decision making. [25] New research, led by the University of Southampton, has demonstrated that a nanoscale device, called a memristor, could be used to power artificial systems that can mimic the human brain. [24] Scientists at Helmholtz-Zentrum Dresden-Rossendorf conducted electricity through DNA-based nanowires by placing gold-plated nanoparticles on them. In this way it could become possible to develop circuits based on genetic material. [23] Researchers at the Nanoscale Transport Physics Laboratory from the School of Physics at the University of the Witwatersrand have found a technique to improve carbon superlattices for quantum electronic device applications. [22] The researchers have found that these previously underestimated interactions can play a significant role in preventing heat dissipation in microelectronic devices. [21] LCLS works like an extraordinary strobe light: Its ultrabright X-rays take snapshots of materials with atomic resolution and capture motions as fast as a few femtoseconds, or millionths of a billionth of a second. For comparison, one femtosecond is to a second what seven minutes is to the age of the universe. [20] A 'nonlinear' effect that seemingly turns materials transparent is seen for the first time in X-rays at SLAC's LCLS. [19] Leiden physicists have manipulated light with large artificial atoms, so-called quantum dots. Before, this has only been accomplished with actual atoms. It is an important step toward light-based quantum technology. [18] In a tiny quantum prison, electrons behave quite differently as compared to their counterparts in free space. They can only occupy discrete energy levels, much like the electrons in an atom-for this reason, such electron prisons are often called \"artificial atoms\". [17] Category: Artificial Intelligence [243] viXra:1612.0130 [pdf] submitted on 2016-12-08 05:48:31 ### Machine Learning of 2-D Materials Authors: George Rajna Comments: 22 Pages. Machine learning, a field focused on training computers to recognize patterns in data and make new predictions, is helping doctors more accurately diagnose diseases and stock analysts forecast the rise and fall of financial markets. And now materials scientists have pioneered another important application for machine learning\u2014helping to accelerate the discovery and development of new materials. [14] Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. Category: Artificial Intelligence [242] viXra:1612.0030 [pdf] submitted on 2016-12-02 12:28:36 ### Machine Learning Breakthroughs Authors: George Rajna Comments: 47 Pages. Machine Learning Breakthroughs As machine learning breakthroughs abound, researchers look to democratize benefits. [27] Machine-learning system spontaneously reproduces aspects of human neurology. [26] Surviving breast cancer changed the course of Regina Barzilay's research. The experience showed her, in stark relief, that oncologists and their patients lack tools for data-driven decision making. [25] New research, led by the University of Southampton, has demonstrated that a nanoscale device, called a memristor, could be used to power artificial systems that can mimic the human brain. [24] Scientists at Helmholtz-Zentrum Dresden-Rossendorf conducted electricity through DNA-based nanowires by placing gold-plated nanoparticles on them. In this way it could become possible to develop circuits based on genetic material. [23] Researchers at the Nanoscale Transport Physics Laboratory from the School of Physics at the University of the Witwatersrand have found a technique to improve carbon superlattices for quantum electronic device applications. [22] The researchers have found that these previously underestimated interactions can play a significant role in preventing heat dissipation in microelectronic devices. [21] LCLS works like an extraordinary strobe light: Its ultrabright X-rays take snapshots of materials with atomic resolution and capture motions as fast as a few femtoseconds, or millionths of a billionth of a second. For comparison, one femtosecond is to a second what seven minutes is to the age of the universe. [20] A 'nonlinear' effect that seemingly turns materials transparent is seen for the first time in X-rays at SLAC's LCLS. [19] Leiden physicists have manipulated light with large artificial atoms, so-called quantum dots. Before, this has only been accomplished with actual atoms. It is an important step toward light-based quantum technology. [18] In a tiny quantum prison, electrons behave quite differently as compared to their counterparts in free space. They can only occupy discrete energy levels, much like the electrons in an atom-for this reason, such electron prisons are often called \"artificial atoms\". [17] Category: Artificial Intelligence [241] viXra:1612.0022 [pdf] submitted on 2016-12-02 07:17:06 ### Machine-Learning and Human Neurology Authors: George Rajna Comments: 44 Pages. Machine-learning system spontaneously reproduces aspects of human neurology. [26] Surviving breast cancer changed the course of Regina Barzilay's research. The experience showed her, in stark relief, that oncologists and their patients lack tools for data-driven decision making. [25] New research, led by the University of Southampton, has demonstrated that a nanoscale device, called a memristor, could be used to power artificial systems that can mimic the human brain. [24] Scientists at Helmholtz-Zentrum Dresden-Rossendorf conducted electricity through DNA-based nanowires by placing gold-plated nanoparticles on them. In this way it could become possible to develop circuits based on genetic material. [23] Researchers at the Nanoscale Transport Physics Laboratory from the School of Physics at the University of the Witwatersrand have found a technique to improve carbon superlattices for quantum electronic device applications. [22] The researchers have found that these previously underestimated interactions can play a significant role in preventing heat dissipation in microelectronic devices. [21] LCLS works like an extraordinary strobe light: Its ultrabright X-rays take snapshots of materials with atomic resolution and capture motions as fast as a few femtoseconds, or millionths of a billionth of a second. For comparison, one femtosecond is to a second what seven minutes is to the age of the universe. [20] A 'nonlinear' effect that seemingly turns materials transparent is seen for the first time in X-rays at SLAC's LCLS. [19] Leiden physicists have manipulated light with large artificial atoms, so-called quantum dots. Before, this has only been accomplished with actual atoms. It is an important step toward light-based quantum technology. [18] In a tiny quantum prison, electrons behave quite differently as compared to their counterparts in free space. They can only occupy discrete energy levels, much like the electrons in an atom-for this reason, such electron prisons are often called \"artificial atoms\". [17] Category: Artificial Intelligence [240] viXra:1612.0009 [pdf] submitted on 2016-12-01 12:44:05 ### Computer Learns by Watching Video Authors: George Rajna Comments: 28 Pages. In recent years, computers have gotten remarkably good at recognizing speech and images: Think of the dictation software on most cellphones, or the algorithms that automatically identify people in photos posted to Facebook. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [239] viXra:1611.0335 [pdf] submitted on 2016-11-24 10:44:26 ### Kannada Spell Checker with Sandhi Splitter Authors: Akshatha A N, Chandana G Upadhyaya, Rajashekara Murthy S Comments: Number of pages is 7 Spelling errors are introduced in text either during typing, or when the user does not know the correct phoneme or grapheme. If a language contains complex words like sandhi where two or more morphemes join based on some rules, spell checking becomes very tedious. In such situations, having a spell checker with sandhi splitter which alerts the user by flagging the errors and providing suggestions is very useful. A novel algorithm of sandhi splitting is proposed in this paper. The sandhi splitter can split about 7000 most common sandhi words in Kannada language used as test samples. The sandhi splitter was integrated with a Kannada spell checker and a mechanism for generating suggestions was added. A comprehensive, platform independent, standalone spell checker with sandhi splitter application software was thus developed and tested extensively for its efficiency and correctness. A comparative analysis of this spell checker with sandhi splitter was made and results concluded that the Kannada spell checker with sandhi splitter has an improved performance. It is twice as fast, 200 times more space efficient, and it is 90% accurate in case of complex nouns and 50% accurate for complex verbs. Such a spell checker with sandhi splitter will be of foremost significance in machine translation systems, voice processing, etc. This is the first sandhi splitter in Kannada and the advantage of the novel algorithm is that, it can be extended to all Indian languages. Category: Artificial Intelligence [238] viXra:1611.0316 [pdf] submitted on 2016-11-23 08:10:08 ### Minds for Machine Intelligence Authors: George Rajna Comments: 42 Pages. Surviving breast cancer changed the course of Regina Barzilay's research. The experience showed her, in stark relief, that oncologists and their patients lack tools for data-driven decision making. [25] New research, led by the University of Southampton, has demonstrated that a nanoscale device, called a memristor, could be used to power artificial systems that can mimic the human brain. [24] Scientists at Helmholtz-Zentrum Dresden-Rossendorf conducted electricity through DNA-based nanowires by placing gold-plated nanoparticles on them. In this way it could become possible to develop circuits based on genetic material. [23] Researchers at the Nanoscale Transport Physics Laboratory from the School of Physics at the University of the Witwatersrand have found a technique to improve carbon superlattices for quantum electronic device applications. [22] The researchers have found that these previously underestimated interactions can play a significant role in preventing heat dissipation in microelectronic devices. [21] LCLS works like an extraordinary strobe light: Its ultrabright X-rays take snapshots of materials with atomic resolution and capture motions as fast as a few femtoseconds, or millionths of a billionth of a second. For comparison, one femtosecond is to a second what seven minutes is to the age of the universe. [20] A 'nonlinear' effect that seemingly turns materials transparent is seen for the first time in X-rays at SLAC's LCLS. [19] Leiden physicists have manipulated light with large artificial atoms, so-called quantum dots. Before, this has only been accomplished with actual atoms. It is an important step toward light-based quantum technology. [18] In a tiny quantum prison, electrons behave quite differently as compared to their counterparts in free space. They can only occupy discrete energy levels, much like the electrons in an atom-for this reason, such electron prisons are often called \"artificial atoms\". [17] When two atoms are placed in a small chamber enclosed by mirrors, they can simultaneously absorb a single photon. [16] Optical quantum technologies are based on the interactions of atoms and photons at the single-particle level, and so require sources of single photons. Category: Artificial Intelligence [237] viXra:1611.0314 [pdf] submitted on 2016-11-23 08:47:27 ### New AI Algorithm Learns Beyond its Training Authors: George Rajna Comments: 27 Pages. Researchers at Lancaster University's Data Science Institute have developed a software system that can for the first time rapidly self-assemble into the most efficient form without needing humans to tell it what to do. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [236] viXra:1611.0260 [pdf] submitted on 2016-11-17 11:18:04 ### Deng Entropy in Hyper Power Set and Super Power Set Authors: Bingyi Kang, Yong Deng Comments: 18 Pages. Deng entropy has been proposed to handle the uncertainty degree of belief function in Dempster-Shafer framework very recently. In this paper, two new belief entropies based on the frame of Deng entropy for hyper-power sets and super-power sets are respectively proposed to measure the uncertainty degree of more uncertain and more flexible information. Directly, the new entropies based on the frame of Deng entropy in hyper-power sets and super-power sets can be used in the application of DSmT. Category: Artificial Intelligence [235] viXra:1611.0211 [pdf] submitted on 2016-11-14 04:10:17 ### A Variable Order Hidden Markov Model with Dependence Jumps Authors: Anastasios Petropoulos, Stelios Xanthopoulos, Sotirios P. Chatzis Comments: 14 Pages. Hidden Markov models (HMMs) are a popular approach for modeling sequential data, typically based on the assumption of a first- or moderate-order Markov chain. However, in many real-world scenarios the modeled data entail temporal dynamics the patterns of which change over time. In this paper, we address this problem by proposing a novel HMM formulation, treating temporal dependencies as latent variables over which inference is performed. Specifically, we introduce a hierarchical graphical model comprising two hidden layers: on the first layer, we postulate a chain of latent observation-emitting states, the temporal dependencies between which may change over time; on the second layer, we postulate a latent first-order Markov chain modeling the evolution of temporal dynamics (dependence jumps) pertaining to the first-layer latent process. As a result of this construction, our method allows for effectively modeling non-homogeneous observed data, where the patterns of the entailed temporal dynamics may change over time. We devise efficient training and inference algorithms for our model, following the expectation-maximization paradigm. We demonstrate the efficacy and usefulness of our approach considering several real-world datasets. As we show, our model allows for increased modeling and predictive performance compared to the alternative methods, while offering a good trade-off between the resulting increases in predictive performance and computational complexity. Category: Artificial Intelligence [234] viXra:1611.0181 [pdf] submitted on 2016-11-12 07:13:04 ### Finding Patterns in Corrupted Data Authors: George Rajna Comments: 29 Pages. A team, including researchers from MIT's Computer Science and Artificial Intelligence Laboratory, has created a new set of algorithms that can efficiently fit probability distributions to high-dimensional data. [16] Researchers at Lancaster University's Data Science Institute have developed a software system that can for the first time rapidly self-assemble into the most efficient form without needing humans to tell it what to do. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [233] viXra:1611.0177 [pdf] submitted on 2016-11-12 04:50:24 ### Machines Learn by Simply Observing Authors: George Rajna Comments: 28 Pages. It is now possible for machines to learn how natural or artificial systems work by simply observing them, without being told what to look for, according to researchers at the University of Sheffield. [16] Researchers at Lancaster University's Data Science Institute have developed a software system that can for the first time rapidly self-assemble into the most efficient form without needing humans to tell it what to do. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [232] viXra:1611.0174 [pdf] submitted on 2016-11-12 05:46:51 ### Social Emotions Test for Artificial Intelligence Authors: George Rajna Comments: 29 Pages. New evidence from brain studies, including cognitive psychology and neurophysiology research, shows that the emotional assessment of every object, subject, action or event plays an important role in human mental processes. And that means that if we want to create human-like artificial intelligence, we must make it emotionally responsive. But how do we know that such intelligence actually experiences real, human-like emotions? [17] It is now possible for machines to learn how natural or artificial systems work by simply observing them, without being told what to look for, according to researchers at the University of Sheffield. [16] Researchers at Lancaster University's Data Science Institute have developed a software system that can for the first time rapidly self-assemble into the most efficient form without needing humans to tell it what to do. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \u201cquantum artificial intelligence\u201d. Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries - how a sliced up flatworm can regenerate into new organisms - has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron\u2019s spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [231] viXra:1611.0173 [pdf] submitted on 2016-11-12 06:46:28 ### AI System Surfs Web to Improve its Performance Authors: George Rajna Comments: 31 Pages. Of the vast wealth of information unlocked by the Internet, most is plain text. The data necessary to answer myriad questions\u2014about, say, the correlations between the industrial use of certain chemicals and incidents of disease, or between patterns of news coverage and voter-poll results\u2014may all be online. But extracting it from plain text and organizing it for quantitative analysis may be prohibitively time consuming. [18] New evidence from brain studies, including cognitive psychology and neurophysiology research, shows that the emotional assessment of every object, subject, action or event plays an important role in human mental processes. And that means that if we want to create human-like artificial intelligence, we must make it emotionally responsive. But how do we know that such intelligence actually experiences real, human-like emotions? [17] It is now possible for machines to learn how natural or artificial systems work by simply observing them, without being told what to look for, according to researchers at the University of Sheffield. [16] Researchers at Lancaster University's Data Science Institute have developed a software system that can for the first time rapidly self-assemble into the most efficient form without needing humans to tell it what to do. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \u201cquantum artificial intelligence\u201d. Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries - how a sliced up flatworm can regenerate into new organisms - has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron\u2019s spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [230] viXra:1611.0169 [pdf] submitted on 2016-11-12 04:07:06 ### Brain-Inspired Device Authors: George Rajna Comments: 39 Pages. New research, led by the University of Southampton, has demonstrated that a nanoscale device, called a memristor, could be used to power artificial systems that can mimic the human brain. [24] Scientists at Helmholtz-Zentrum Dresden-Rossendorf conducted electricity through DNA-based nanowires by placing gold-plated nanoparticles on them. In this way it could become possible to develop circuits based on genetic material. [23] Researchers at the Nanoscale Transport Physics Laboratory from the School of Physics at the University of the Witwatersrand have found a technique to improve carbon superlattices for quantum electronic device applications. [22] The researchers have found that these previously underestimated interactions can play a significant role in preventing heat dissipation in microelectronic devices. [21] LCLS works like an extraordinary strobe light: Its ultrabright X-rays take snapshots of materials with atomic resolution and capture motions as fast as a few femtoseconds, or millionths of a billionth of a second. For comparison, one femtosecond is to a second what seven minutes is to the age of the universe. [20] A \u2018nonlinear\u2019 effect that seemingly turns materials transparent is seen for the first time in X-rays at SLAC\u2019s LCLS. [19] Leiden physicists have manipulated light with large artificial atoms, so-called quantum dots. Before, this has only been accomplished with actual atoms. It is an important step toward light-based quantum technology. [18] In a tiny quantum prison, electrons behave quite differently as compared to their counterparts in free space. They can only occupy discrete energy levels, much like the electrons in an atom - for this reason, such electron prisons are often called \"artificial atoms\". [17] When two atoms are placed in a small chamber enclosed by mirrors, they can simultaneously absorb a single photon. [16] Optical quantum technologies are based on the interactions of atoms and photons at the single-particle level, and so require sources of single photons that are highly indistinguishable \u2013 that is, as identical as possible. Current single-photon sources using semiconductor quantum dots inserted into photonic structures produce photons that are ultrabright but have limited indistinguishability due to charge noise, which results in a fluctuating electric field. [14] A method to produce significant amounts of semiconducting nanoparticles for light-emitting displays, sensors, solar panels and biomedical applications has gained momentum with a demonstration by researchers at the Department of Energy's Oak Ridge National Laboratory. [13] A source of single photons that meets three important criteria for use in quantum-information systems has been unveiled in China by an international team of physicists. Based on a quantum dot, the device is an efficient source of photons that emerge as solo particles that are indistinguishable from each other. The researchers are now trying to use the source to create a quantum computer based on \"boson sampling\". [11] With the help of a semiconductor quantum dot, physicists at the University of Basel have developed a new type of light source that emits single photons. For the first time, the researchers have managed to create a stream of identical photons. [10] Optical photons would be ideal carriers to transfer quantum information over large distances. Researchers envisage a network where information is processed in certain nodes and transferred between them via photons. [9] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer using Quantum Information. In August 2013, the achievement of \"fully deterministic\" quantum teleportation, using a hybrid technique, was reported. On 29 May 2014, scientists announced a reliable way of transferring data by quantum teleportation. Quantum teleportation of data had been done before but with highly unreliable methods. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron\u2019s spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer with the help of Quantum Information. Category: Artificial Intelligence [229] viXra:1611.0095 [pdf] submitted on 2016-11-08 03:33:30 ### Quantitative Prediction of Electoral Vote for United States Presidential Election in 2016 Authors: Gang Xu Comments: 8 Pages. This work was originally completed by October 22, 2016. The manuscript draft was prepared on November 7, 2016. In this paper I am reporting the quantitative prediction of the electoral vote for United States presidential election in 2016. This quantitative prediction was based on the Google Trends (GT) data that is publicly available on the internet. A simple heuristic statistical model is applied to analyzing the GT data. This is intended to be an experiment for exploring the plausible dependency between the GT data and the electoral vote result of US presidential elections. The model's performance has also been tested by comparing the predicted results and the actual electoral votes in 2004, 2008 and 2012. For the year 2016, the Google Trends data projects that Mr. Trump will win the white house in landslide. This paper serves as a document to put this exploratory experiment in real test, since the actual election result can be compared to the prediction after tomorrow (November 8, 2016). Category: Artificial Intelligence [228] viXra:1611.0086 [pdf] submitted on 2016-11-07 06:27:49 ### Neuromorphic Processor Authors: George Rajna Comments: 22 Pages. Toshiba advances deep learning with extremely low power neuromorphic processor. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [227] viXra:1611.0025 [pdf] submitted on 2016-11-02 08:20:12 ### Machine Learning for Cancer Treatment Authors: George Rajna Comments: 28 Pages. Physicians have long used visual judgment of medical images to determine the course of cancer treatment. A new program package from Fraunhofer researchers reveals changes in images and facilitates this task using deep learning. The experts will demonstrate this software in Chicago from November 27 to December 2 at RSNA, the world's largest radiology meeting. [16] Researchers at Lancaster University's Data Science Institute have developed a software system that can for the first time rapidly self-assemble into the most efficient form without needing humans to tell it what to do. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \u201cquantum artificial intelligence\u201d. Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries - how a sliced up flatworm can regenerate into new organisms - has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron\u2019s spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [226] viXra:1611.0022 [pdf] submitted on 2016-11-02 06:49:11 ### Transforming, Self-Learning Software Authors: George Rajna Comments: 27 Pages. Researchers at Lancaster University's Data Science Institute have developed a software system that can for the first time rapidly self-assemble into the most efficient form without needing humans to tell it what to do. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [225] viXra:1610.0360 [pdf] submitted on 2016-10-30 02:33:09 ### Machine-Learning Decision Rationales Authors: George Rajna Comments: 28 Pages. Researchers from MIT's Computer Science and Artificial Intelligence Laboratory (CSAIL) have devised a way to train neural networks so that they provide not only predictions and classifications but rationales for their decisions. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [224] viXra:1610.0359 [pdf] submitted on 2016-10-30 04:02:16 ### Machine Learning Understand Materials Authors: George Rajna Comments: 21 Pages. Machine learning algorithms are designed to improve as they encounter more data, making them a versatile technology for understanding large sets of photos such as those accessible from Google Images. Elizabeth Holm, professor of materials science and engineering at Carnegie Mellon University, is leveraging this technology to better understand the enormous number of research images accumulated in the field of materials science. [13] With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] New research demonstrates that particles at the quantum level can in fact be seen as behaving something like billiard balls rolling along a table, and not merely as the probabilistic smears that the standard interpretation of quantum mechanics suggests. But there's a catch-the tracks the particles follow do not always behave as one would expect from \"realistic\" trajectories, but often in a fashion that has been termed \"surrealistic.\" [8] Quantum entanglement\u2014which occurs when two or more particles are correlated in such a way that they can influence each other even across large distances\u2014is not an all-or-nothing phenomenon, but occurs in various degrees. The more a quantum state is entangled with its partner, the better the states will perform in quantum information applications. Unfortunately, quantifying entanglement is a difficult process involving complex optimization problems that give even physicists headaches. [7] A trio of physicists in Europe has come up with an idea that they believe would allow a person to actually witness entanglement. Valentina Caprara Vivoli, with the University of Geneva, Pavel Sekatski, with the University of Innsbruck and Nicolas Sangouard, with the University of Basel, have together written a paper describing a scenario where a human subject would be able to witness an instance of entanglement\u2014they have uploaded it to the arXiv server for review by others. [6] The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the relativistic quantum theory. Category: Artificial Intelligence [223] viXra:1610.0336 [pdf] submitted on 2016-10-27 21:31:21 ### Fuzzy Evidential Influence Diagram Evaluation Algorithm Authors: Haoyang Zheng, Yong Deng Comments: 38 Pages. Fuzzy influence diagrams (FIDs) are one of the graphical models that combines the qualitative and quantitative analysis to solve decision-making problems. However, FIDs use an incomprehensive evaluation criteria to score nodes in complex systems, so that many different nodes got the same score, which can not reflect their differences. Based on fuzzy set and Dempster-Shafer (D-S) evidence theory, this paper changes the traditional evaluation system and modifies corresponding algorithm, in order that the influence diagram can more effectively reflect the true situation of the system, and get more practical results. Numerical examples and the real application in supply chain financial system are used to show the efficiency of the proposed influence diagram model. Category: Artificial Intelligence [222] viXra:1610.0314 [pdf] submitted on 2016-10-26 05:16:26 ### Artificial Intelligence Replaces Judges and Lawyers Authors: George Rajna Comments: 20 Pages. An artificial intelligence method developed by University College London computer scientists and associates has predicted the judicial decisions of the European Court of Human Rights (ECtHR) with 79% accuracy, according to a paper published Monday, Oct. 24 in PeerJ Computer Science. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] New research demonstrates that particles at the quantum level can in fact be seen as behaving something like billiard balls rolling along a table, and not merely as the probabilistic smears that the standard interpretation of quantum mechanics suggests. But there's a catch-the tracks the particles follow do not always behave as one would expect from \"realistic\" trajectories, but often in a fashion that has been termed \"surrealistic.\" [8] Quantum entanglement\u2014which occurs when two or more particles are correlated in such a way that they can influence each other even across large distances\u2014is not an all-or-nothing phenomenon, but occurs in various degrees. The more a quantum state is entangled with its partner, the better the states will perform in quantum information applications. Unfortunately, quantifying entanglement is a difficult process involving complex optimization problems that give even physicists headaches. [7] A trio of physicists in Europe has come up with an idea that they believe would allow a person to actually witness entanglement. Valentina Caprara Vivoli, with the University of Geneva, Pavel Sekatski, with the University of Innsbruck and Nicolas Sangouard, with the University of Basel, have together written a paper describing a scenario where a human subject would be able to witness an instance of entanglement\u2014they have uploaded it to the arXiv server for review by others. [6] The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the relativistic quantum theory. Category: Artificial Intelligence [221] viXra:1610.0281 [pdf] submitted on 2016-10-24 04:05:52 ### An Information Volume Measure Authors: Yong Deng Comments: 8 Pages. How to measure the volume of uncertainty information is an open issue. Shannon entropy is used to represent the uncertainty degree of a probability distribution. Given a generalized probability distribution which means that the probability is not only assigned to the basis event space but also the power set of event space. At this time, a so called meta probability space is constructed. A new measure, named as Deng entropy, is presented. The results show that, compared with existing method, Deng entropy is not only better from the aspect of mathematic form, but also has the significant physical meaning. Category: Artificial Intelligence [220] viXra:1610.0249 [pdf] submitted on 2016-10-21 11:35:59 ### New Data Algorithms Authors: George Rajna Comments: 28 Pages. Last year, MIT researchers presented a system that automated a crucial step in big-data analysis: the selection of a \"feature set,\" or aspects of the data that are useful for making predictions. The researchers entered the system in several data science contests, where it outperformed most of the human competitors and took only hours instead of months to perform its analyses. [15] Physicists have shown that quantum effects have the potential to significantly improve a variety of interactive learning tasks in machine learning. [14] A Chinese team of physicists have trained a quantum computer to recognise handwritten characters, the first demonstration of \" quantum artificial intelligence \". Physicists have long claimed that quantum computers have the potential to dramatically outperform the most powerful conventional processors. The secret sauce at work here is the strange quantum phenomenon of superposition, where a quantum object can exist in two states at the same time. [13] One of biology's biggest mysteries-how a sliced up flatworm can regenerate into new organisms-has been solved independently by a computer. The discovery marks the first time that a computer has come up with a new scientific theory without direct human help. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [219] viXra:1610.0169 [pdf] submitted on 2016-10-15 16:49:11 ### Band Gap Estimation Using Machine Learning Techniques Authors: Anantha Natarajan S, R Varadhan, Ezhilvel ME Comments: 3 Pages. The purpose of this study is to build machine learning models to predict the band gap of binary compounds, using its known properties like molecular weight, electronegativity, atomic fraction and the group of the constituent elements in the periodic table. Regression techniques like Linear, Ridge regression and Random Forest were used to build the model. This model can be used by students and researchers in experiments involving unknown band gaps or new compounds. Category: Artificial Intelligence [218] viXra:1610.0142 [pdf] submitted on 2016-10-13 14:04:33 ### Google DeepMind Neural Networks Authors: George Rajna Comments: 24 Pages. A team of researchers at Google's DeepMind Technologies has been working on a means to increase the capabilities of computers by combining aspects of data processing and artificial intelligence and have come up with what they are calling a differentiable neural computer (DNC.) In their paper published in the journal Nature, they describe the work they are doing and where they believe it is headed. To make the work more accessible to the public team members, Alexander Graves and Greg Wayne have posted an explanatory page on the DeepMind website. [13] Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [217] viXra:1610.0110 [pdf] submitted on 2016-10-10 12:21:47 ### Neuro-Inspired Analog Computer Authors: George Rajna Comments: 23 Pages. Researchers have developed a neuro-inspired analog computer that has the ability to train itself to become better at whatever tasks it performs. [13] A small, Santa Fe, New Mexico-based company called Knowm claims it will soon begin commercializing a state-of-the-art technique for building computing chips that learn. Other companies, including HP HPQ-3.45% and IBM IBM-2.10% , have already invested in developing these so-called brain-based chips, but Knowm says it has just achieved a major technological breakthrough that it should be able to push into production hopefully within a few years. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [216] viXra:1610.0074 [pdf] submitted on 2016-10-07 00:22:44 ### Belief Reliability Analysis and Its Application Authors: Haoyang Zheng, Likang Yin, Tian Bian, Yong Deng Comments: 24 Pages. In reliability analysis, Fault Tree Analysis based on evidential networks is an important research topic. However, the existing EN approaches still remain two issues: one is the final results are expressed with interval numbers, which has a relatively high uncertainty to make a final decision. The other is the combination rule is not used to fuse uncertain information. These issues will greatly decrease the efficiency of EN to handle uncertain information. To address these open issues, a new methodology, called Belief Reliability Analysis, is presented in this paper. The combination methods to deal with series system, parallel system, series-parallel system as well as parallel-series system are proposed for reliability evaluation. Numerical examples and the real application in servo-actuation system are used to show the efficiency of the proposed Belief Reliability Analysis methodology. Category: Artificial Intelligence [215] viXra:1610.0029 [pdf] submitted on 2016-10-04 04:31:32 ### Associative Broadcast Neural Network Authors: Aleksei Morozov Comments: 3 Pages. Associative broadcast neural network (ABNN) is an artificial neural network inspired by a hypothesis of broadcasting of neuron's output pattern in a biological neural network. Neuron has wire connections and ether connections. Ether connections are electrical. Wire connections provide a recognition functionality. Ether connections provide an association functionality. Category: Artificial Intelligence [214] viXra:1610.0028 [pdf] submitted on 2016-10-03 13:53:10 ### A New Belief Entropy: Possible Generalization of Deng Entropy, Tsallis Entropy and Shannon Entropy Authors: Bingyi Kang, Yong Deng Comments: 15 Pages. Shannon entropy is the mathematical foundation of information theory, Tsallis entropy is the roots of nonextensive statistical mechanics, Deng entropy was proposed to measure the uncertainty degree of belief function very recently. In this paper, A new entropy H was proposed to generalize Deng entropy, Tsallis entropy and Shannon entropy. The new entropy H can be degenerated to Deng entropy, Tsallis entropy, and Shannon entropy under different conditions, and also can maintains the mathematical properity of Deng entropy, Tsallis entropy and Shannon entropy. Category: Artificial Intelligence [213] viXra:1609.0311 [pdf] submitted on 2016-09-21 07:22:05 ### Artificial Intelligence Discover New Materials Authors: George Rajna Comments: 20 Pages. With the help of artificial intelligence, chemists from the University of Basel in Switzerland have computed the characteristics of about two million crystals made up of four chemical elements. The researchers were able to identify 90 previously unknown thermodynamically stable crystals that can be regarded as new materials. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] New research demonstrates that particles at the quantum level can in fact be seen as behaving something like billiard balls rolling along a table, and not merely as the probabilistic smears that the standard interpretation of quantum mechanics suggests. But there's a catch-the tracks the particles follow do not always behave as one would expect from \"realistic\" trajectories, but often in a fashion that has been termed \"surrealistic.\" [8] Quantum entanglement\u2014which occurs when two or more particles are correlated in such a way that they can influence each other even across large distances\u2014is not an all-or-nothing phenomenon, but occurs in various degrees. The more a quantum state is entangled with its partner, the better the states will perform in quantum information applications. Unfortunately, quantifying entanglement is a difficult process involving complex optimization problems that give even physicists headaches. [7] A trio of physicists in Europe has come up with an idea that they believe would allow a person to actually witness entanglement. Valentina Caprara Vivoli, with the University of Geneva, Pavel Sekatski, with the University of Innsbruck and Nicolas Sangouard, with the University of Basel, have together written a paper describing a scenario where a human subject would be able to witness an instance of entanglement\u2014they have uploaded it to the arXiv server for review by others. [6] The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the relativistic quantum theory. Category: Artificial Intelligence [212] viXra:1609.0238 [pdf] submitted on 2016-09-15 19:25:33 ### Revision on Fuzzy Artificial Potential Field for Humanoid Robot Path Planning in Unknown Environment Authors: Mahdi Fakoor, Amirreza Kosari, Mohsen Jafarzadeh Comments: 10 Pages. Path planning in a completely known environment has been experienced various ways. However, in real world, most humanoid robots work in unknown environments. Robots' path planning by artificial potential field and fuzzy artificial potential field methods are very popular in the field of robotics navigation. However, by default humanoid robots lack range sensors; thus, traditional artificial potential field approaches needs to adopt themselves to these limitations. This paper investigates two different approaches for path planning of a humanoid robot in an unknown environment using fuzzy artificial potential (FAP) method. In the first approach, the direction of the moving robot is derived from fuzzified artificial potential field whereas in the second one, the direction of the robot is extracted from some linguistic rules that are inspired from artificial potential field. These two introduced trajectory design approaches are validated though some software and hardware in the loop simulations and the experimental results demonstrate the superiority of the proposed approaches in humanoid robot real-time trajectory planning problems. Category: Artificial Intelligence [211] viXra:1609.0134 [pdf] submitted on 2016-09-10 11:19:00 ### War Algorithm Accountability Authors: Dustin A. Lewis, Gabriella Blum, Naz K. Modirzadeh Comments: 244 Pages. Compendium on Accountability issues as pertains to Lethal Autonomous Weapons Category: Artificial Intelligence [210] viXra:1609.0133 [pdf] submitted on 2016-09-10 12:00:18 ### Five Hundred Deep Learning Papers, Graphviz and Python Authors: Daniele Ettore Ciriello Comments: 13 Pages. I invested days creating a graph with PyGraphviz to repre- sent the evolutionary process of deep learning\u2019s state of the art for the last twenty-five years. Through this paper I want to show you how and what I obtained. Category: Artificial Intelligence [209] viXra:1609.0126 [pdf] submitted on 2016-09-10 04:54:56 ### Deep Neural Networks Authors: George Rajna Comments: 23 Pages. Nobody understands why deep neural networks are so good at solving complex problems. Now physicists say the secret is buried in the laws of physics. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [208] viXra:1608.0291 [pdf] submitted on 2016-08-24 01:48:59 ### A Comparison of the Generalized minC Combination and the Hybrid DSm Combination Rules Authors: Milan Daniel Comments: 18 Pages. A generalization of the minC combination to DSm hyper-power sets is presented. Both the special formulas for static fusion or dynamic fusion without non-existential constraints and the quite general formulas for dynamic fusion with non-existential constraints are included. Examples of the minC combination on several different hybrid DSm models are presented. A comparison of the generalized minC combination with the hybrid DSm rule is discussed and explained on examples. Category: Artificial Intelligence [207] viXra:1608.0224 [pdf] submitted on 2016-08-20 12:45:56 ### Search for Dynamical Origin of Social Networks Authors: Michail Zak Comments: 33 Pages. The challenge of this work is to re-define the concept of intelligent agent as a building block of social networks by presenting it as a physical particle with additional non-Newtonian properties. The proposed model of an intelligent agent described by a system of ODE coupled with their Liouville equation has been introduced and discussed. Following the Madelung equation that belongs to this class, non-Newtonian properties such as superposition, entanglement, and probability interference typical for quantum systems have been described. Special attention was paid to the capability to violate the second law of thermodynamics, which makes these systems neither Newtonian, nor quantum. It has been shown that the proposed model can be linked to mathematical models of livings as well as to models of AI. The model is presented in two modifications. The first one is illustrated by the discovery of a stochastic attractor approached by the social network; as an application, it was demonstrated that any statistics can be represented by an attractor of the solution to the corresponding system of ODE coupled with its Liouville equation. It was emphasized that evolution to the attractor reveals possible micro-mechanisms driving random events to the final distribution of the corresponding statistical law. Special attention is concentrated upon the power law and its dynamical interpretation: it is demonstrated that the underlying micro- dynamics supports a \u201cviolent reputation\u201d of the power-law statistics. The second modification of the model of social network associated with a decision-making process and applied to solution of NP-complete problems known as being unsolvable neither by classical nor by quantum algorithms. The approach is illustrated by solving a search in unsorted database in polynomial time by resonance between external force representing the address of a required item and the response representing the location of this item. Category: Artificial Intelligence [206] viXra:1608.0187 [pdf] submitted on 2016-08-18 14:02:36 ### Neuromorphic Architecture Authors: George Rajna Comments: 28 Pages. In the future, level-tuned neurons may help enable neuromorphic computing systems to perform tasks that traditional computers cannot, such as learning from their environment, pattern recognition, and knowledge extraction from big data sources. [19] IBM scientists have created randomly spiking neurons using phase-change materials to store and process data. This demonstration marks a significant step forward in the development of energy-efficient, ultra-dense integrated neuromorphic technologies for applications in cognitive computing. [18] An ion trap with four segmented blade electrodes used to trap a linear chain of atomic ions for quantum information processing. Each ion is addressed optically for individual control and readout using the high optical access of the trap. [17] To date, researchers have realised qubits in the form of individual electrons (aktuell.ruhr-uni-bochum.de\/pm2012\/pm00090.html.en). However, this led to interferences and rendered the information carriers difficult to programme and read. The group has solved this problem by utilising electron holes as qubits, rather than electrons. [16] Physicists from MIPT and the Russian Quantum Center have developed an easier method to create a universal quantum computer using multilevel quantum systems (qudits), each one of which is able to work with multiple \"conventional\" quantum elements \u2013 qubits. [15] Precise atom implants in silicon provide a first step toward practical quantum computers. [14] A method to produce significant amounts of semiconducting nanoparticles for light-emitting displays, sensors, solar panels and biomedical applications has gained momentum with a demonstration by researchers at the Department of Energy's Oak Ridge National Laboratory. [13] A source of single photons that meets three important criteria for use in quantum-information systems has been unveiled in China by an international team of physicists. Based on a quantum dot, the device is an efficient source of photons that emerge as solo particles that are indistinguishable from each other. The researchers are now trying to use the source to create a quantum computer based on \"boson sampling\". [11] With the help of a semiconductor quantum dot, physicists at the University of Basel have developed a new type of light source that emits single photons. Category: Artificial Intelligence [205] viXra:1608.0093 [pdf] submitted on 2016-08-08 22:08:41 ### Edge Based Grid Super-Imposition for Crowd Emotion Recognition Authors: Amol S Patwardhan Comments: 6 Pages. Numerous automatic continuous emotion detection system studies have examined mostly use of videos and images containing individual person expressing emotions. This study examines the detection of spontaneous emotions in a group and crowd settings. Edge detection was used with a grid of lines superimposition to extract the features. The feature movement in terms of movement from the reference point was used to track across sequences of images from the color channel. Additionally the video data capturing was done on spontaneous emotions invoked by watching sports events from group of participants. The method was view and occlusion independent and the results were not affected by presence of multiple people chaotically expressing various emotions. The edge thresholds of 0.2 and grid thresholds of 20 showed the best accuracy results. The overall accuracy of the group emotion classifier was 70.9%. Category: Artificial Intelligence [204] viXra:1608.0092 [pdf] submitted on 2016-08-08 22:15:06 ### Human Activity Recognition Using Temporal Frame Decision Rule Extraction Authors: Amol Patwardhan Comments: 4 Pages. Activities of humans and their recognition has many practical and real world applications such as safety, security, surveillance, humanoid assistive robotics and intelligent simulation systems. Numerous human action and emotion recognition systems included analysis of position and geometric features and gesture based co-ordinates to detect actions. There exits additional data and information in the movement and motion based features and temporal and time-sequential series of image and video frames which can be leveraged to detect and extract a certain actions, postures, gestures and expressions. This paper uses dynamic, temporal, time-scale dependent data to compare with decision rules and templates for activity recognition. The human shape boundaries and silhouette is extracted using geometric co-ordinate and centroid model across multiple frames. The extracted shape boundary is transformed to binary state using eigen space mapping and parameter dependent canonical transformation in 3D space dimension. The image blob data frames are down sampled using activity templates to a single candidate reference frame. This candidate frame was compared with the decision rule driven model to associate with an activity class label. The decision rule driven and activity templates method produced 64% recognition accuracy indicating that the method was feasible for recognizing human activities. Category: Artificial Intelligence [203] viXra:1608.0076 [pdf] submitted on 2016-08-08 02:57:31 ### Watson Doctor Authors: George Rajna Comments: 19 Pages. Watson correctly diagnoses woman after doctors were stumped. [12] The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] New research demonstrates that particles at the quantum level can in fact be seen as behaving something like billiard balls rolling along a table, and not merely as the probabilistic smears that the standard interpretation of quantum mechanics suggests. But there's a catch-the tracks the particles follow do not always behave as one would expect from \"realistic\" trajectories, but often in a fashion that has been termed \"surrealistic.\" [8] Quantum entanglement\u2014which occurs when two or more particles are correlated in such a way that they can influence each other even across large distances\u2014is not an all-or-nothing phenomenon, but occurs in various degrees. The more a quantum state is entangled with its partner, the better the states will perform in quantum information applications. Unfortunately, quantifying entanglement is a difficult process involving complex optimization problems that give even physicists headaches. [7] A trio of physicists in Europe has come up with an idea that they believe would allow a person to actually witness entanglement. Valentina Caprara Vivoli, with the University of Geneva, Pavel Sekatski, with the University of Innsbruck and Nicolas Sangouard, with the University of Basel, have together written a paper describing a scenario where a human subject would be able to witness an instance of entanglement\u2014they have uploaded it to the arXiv server for review by others. [6] The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the relativistic quantum theory. Category: Artificial Intelligence [202] viXra:1608.0041 [pdf] submitted on 2016-08-04 08:00:47 ### Combining Infinity Number Of Neural Networks Into One Authors: Bo Tian Comments: 17 Pages. One of the important aspects of a neural network is its generalization property, which is measured by its ability to make correct prediction on unseen samples. One option to improve generalization is to combine results from multiple networks, which is unfortunately a time-consuming process. In this paper, a new approach is presented to combine infinity number of neural networks in analytic way to produce a small, fast and reliable neural network. Category: Artificial Intelligence [201] viXra:1607.0484 [pdf] submitted on 2016-07-25 21:28:15 ### Active Appearance Model Construction: Implementation notes Authors: Nikzad Babaii Rizvandi, Wilfried Philips, Aleksandra Pizurica Comments: 7 Pages. Active Appearance Model (AAM) is a powerful object modeling technique and one of the best available ones in computer vision and computer graphics. This approach is however quite complex and various parts of its implementation were addressed separately by different researchers in several recent works. In this paper, we present systematically a full implementation of the AAM model with pseudo codes for the crucial steps in the construction of this model. Category: Artificial Intelligence [200] viXra:1607.0483 [pdf] submitted on 2016-07-25 21:29:15 ### Active Appearance Model (Aam) from Theory to Implementation Authors: Nikzad Babaii Rizvandi, Aleksandra Piz\u02c7urica, Wilfried Philips Comments: 4 Pages. Active Appearance Model (AAM) is a kind of deformable shape descriptors which is widely used in computer vision and computer graphics. This approach utilizes statistical model obtained from some images in training set and gray-value information of the texture to fit on the boundaries of a new image. In this paper, we describe a brief implementation, apply the method on hand object and finally discuss its performance in compare to Active Shape Model(ASM). Our experiments shows this method is more sensitive to the initialization and slower than ASM. Category: Artificial Intelligence [199] viXra:1607.0459 [pdf] submitted on 2016-07-24 21:30:52 ### Pattern Recognition and Learning in Bistable Cam Networks Authors: Vladimir Chinarov, Martin Dudziak, Yuri Kyrpach Comments: 12 Pages. The present study concerns the problem of learning, pattern recognition and computational abilities of a homogeneous network composed from coupled bistable units. New possibilities for pattern recognition may be realized due to the developed technique that permits a reconstruction of a dynamical system using the distributions of its attractors. In both cases the updating procedure for the coupling matrix uses the minimization of least-mean-square errors between the applied and desired patterns. Category: Artificial Intelligence [198] viXra:1607.0073 [pdf] submitted on 2016-07-07 02:49:14 ### Indian Buffet Process Deep Generative Models Authors: Sotirios P. Chatzis Comments: 16 Pages. Deep generative models (DGMs) have brought about a major breakthrough, as well as renewed interest, in generative latent variable models. However, an issue current DGM formulations do not address concerns the data-driven inference of the number of latent features needed to represent the observed data. Traditional linear formulations allow for addressing this issue by resorting to tools from the field of nonparametric statistics: Indeed, nonparametric linear latent variable models, obtained by appropriate imposition of Indian Buffet Process (IBP) priors, have been extensively studied by the machine learning community; inference for such models can been performed either via exact sampling or via approximate variational techniques. Based on this inspiration, in this paper we examine whether similar ideas from the field of Bayesian nonparametrics can be utilized in the context of modern DGMs in order to address the latent variable dimensionality inference problem. To this end, we propose a novel DGM formulation, based on the imposition of an IBP prior. We devise an efficient Black-Box Variational inference algorithm for our model, and exhibit its efficacy in a number of semi-supervised classification experiments. In all cases, we use popular benchmark datasets, and compare to state-of-the-art DGMs. Category: Artificial Intelligence [197] viXra:1607.0014 [pdf] submitted on 2016-07-01 14:54:48 ### Interval-Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets Authors: Florentin Smarandache Comments: 4 Pages. We have proposed since 1995 the existence of degrees of membership of an element with respect to a neutrosophic set to also be partially or totally above 1 (overmembership), and partially or totally below 0 (undermembership) in order to better describe our world problems [published in 2007]. Category: Artificial Intelligence [196] viXra:1606.0343 [pdf] submitted on 2016-06-30 08:36:33 ### Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset. Similarly for Neutrosophic Over-\/Under-\/Off- Logic, Probability, and Statistics Authors: Florentin Smarandache Comments: 168 Pages. Neutrosophic Over-\/Under-\/Off-Set and -Logic were defined for the first time by Smarandache in 1995 and published in 2007. They are totally different from other sets\/logics\/probabilities. He extended the neutrosophic set respectively to Neutrosophic Overset {when some neutrosophic component is > 1}, Neutrosophic Underset {when some neutrosophic component is < 0}, and to Neutrosophic Offset {when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component > 1 and other neutrosophic component < 0}. This is no surprise with respect to the classical fuzzy set\/logic, intuitionistic fuzzy set\/logic, or classical\/imprecise probability, where the values are not allowed outside the interval [0, 1], since our real-world has numerous examples and applications of over-\/under-\/off-neutrosophic components. Example of Neutrosophic Offset. In a given company a full-time employer works 40 hours per week. Let\u2019s consider the last week period. Helen worked part-time, only 30 hours, and the other 10 hours she was absent without payment; hence, her membership degree was 30\/40 = 0.75 < 1. John worked full-time, 40 hours, so he had the membership degree 40\/40 = 1, with respect to this company. But George worked overtime 5 hours, so his membership degree was (40+5)\/40 = 45\/40 = 1.125 > 1. Thus, we need to make distinction between employees who work overtime, and those who work full-time or part-time. That\u2019s why we need to associate a degree of membership strictly greater than 1 to the overtime workers. Now, another employee, Jane, was absent without pay for the whole week, so her degree of membership was 0\/40 = 0. Yet, Richard, who was also hired as a full-time, not only didn\u2019t come to work last week at all (0 worked hours), but he produced, by accidentally starting a devastating fire, much damage to the company, which was estimated at a value half of his salary (i.e. as he would have gotten for working 20 hours that week). Therefore, his membership degree has to be less that Jane\u2019s (since Jane produced no damage). Whence, Richard\u2019s degree of membership, with respect to this company, was - 20\/40 = - 0.50 < 0. Consequently, we need to make distinction between employees who produce damage, and those who produce profit, or produce neither damage no profit to the company. Therefore, the membership degrees > 1 and < 0 are real in our world, so we have to take them into consideration. Then, similarly, the Neutrosophic Logic\/Measure\/Probability\/Statistics etc. were extended to respectively Neutrosophic Over-\/Under-\/Off-Logic, -Measure, -Probability, -Statistics etc. [Smarandache, 2007]. Category: Artificial Intelligence [195] viXra:1606.0341 [pdf] submitted on 2016-06-30 08:42:46 ### Operators on Single-Valued Neutrosophic Oversets, Neutrosophic Undersets, and Neutrosophic Offsets Authors: Florentin Smarandache Comments: 5 Pages. We have defined Neutrosophic Over-\/Under-\/Off-Set and Logic for the first time in 1995 and published in 2007. During 1995-2016 we presented them to various national and international conferences and seminars. These new notions are totally different from other sets\/logics\/probabilities. We extended the neutrosophic set respectively to Neutrosophic Overset {when some neutrosophic component is > 1}, to Neutrosophic Underset {when some neutrosophic component is < 0}, and to Neutrosophic Offset {when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component > 1 and other neutrosophic component < 0}. This is no surprise since our real-world has numerous examples and applications of over-\/under-\/off-neutrosophic components. Category: Artificial Intelligence [194] viXra:1606.0272 [pdf] submitted on 2016-06-25 19:29:46 ### Self-Controlled Dynamics Authors: Michail Zak Comments: 26 Pages. A new class of dynamical system described by ODE coupled with their Liouville equation has been introduced and discussed. These systems called self-controlled, or self-supervised since the role of actuators is played by the probability produced by the Liouville equation. Following the Madelung equation that belongs to this class, non-Newtonian properties such as randomness, entanglement, and probability interference typical for quantum systems have been described. Special attention was paid to the capability to violate the second law of thermodynamics, which makes these systems neither Newtonian, nor quantum. It has been shown that self-controlled dynamical systems can be linked to mathematical models of livings as well as to models of AI. The central point of this paper is the application of the self-controlled systems to NP-complete problems known as being unsolvable neither by classical nor by quantum algorithms. The approach is illustrated by solving a search in unsorted database in polynomial time by resonance between external force representing the address of a required item and the response representing location of this item. Category: Artificial Intelligence [193] viXra:1606.0181 [pdf] submitted on 2016-06-17 22:41:17 ### Universal Natural Memory Embedding -3 (AI) Authors: Ramesh Chandra Bagadi Comments: 18 Pages. In this research investigation, the author has presented a theory of \u2018Universal Relative Metric That Generates A Field Super-Set To The Fields Generated By Various Distinct Relative Metrics\u2019. Category: Artificial Intelligence [192] viXra:1606.0155 [pdf] submitted on 2016-06-15 07:30:51 ### Universal Natural Memory Embedding - Part Two Authors: Ramesh Chandra Bagadi Comments: 14 Pages. In this research investigation, the author has presented a theory of \u2018The Universal Irreducible Any Field Generating Metric\u2019. Category: Artificial Intelligence [191] viXra:1606.0146 [pdf] submitted on 2016-06-15 00:17:24 ### Universal Natural Memory Embedding - I Authors: Ramesh Chandra Bagadi Comments: 14 Pages. In this research investigation, the author has presented a theory of \u2018Universal Natural Memory Embedding\u2019. Category: Artificial Intelligence [190] viXra:1605.0288 [pdf] submitted on 2016-05-29 02:24:48 ### Syllabic Networks: Measuring the Redundancy of Associative Syntactic Patterns Authors: Bradly Alicea Comments: 6 Pages. 3 figures, 1 table The self-organization and diversity inherent in natural and artificial language can be revealed using a technique called syllabic network decomposition. The topology of such networks are determined by a series of linguistic strings which are broken apart at critical points and then linked together in a non-linear fashion. Small proof-of-concept examples are given using words from the English language. A criterion for connectedness and two statistical parameters for measuring connectedness are applied to these examples. To conclude, we will discuss some applications of this technique, ranging from improving models of speech recognition to bioinformatic analysis and recreational games. Category: Artificial Intelligence [189] viXra:1605.0190 [pdf] submitted on 2016-05-18 06:12:00 ### The Algorithm of the Thinking Machine Authors: Dimiter Dobrev Comments: 13 Pages. Represented at 12 of May 2016 at Faculty of Mathematics and Informatics, University of Sofia. Throughout my life I\u2019ve tried to answer the question \"What is AI?\" and write the program that is AI. I\u2019ve already known the answer to the question \"What is AI?\" for 16 years now but the AI algorithm has eluded me. I\u2019ve come up with individual fragments but something has always been missing to put the puzzle together. Finally, I gathered all the missing pieces and I can introduce you to this algorithm. That is, in this article you will find a sufficiently detailed description of the algorithm of the AI. That sounds so audacious that probably you won\u2019t believe me. Frankly, even I do not fully believe myself. I\u2019ll believe it only when someone writes a program executing this algorithm and when I see that this program actually works. Even if you don\u2019t manage to believe in the importance of this article, I hope that you will like it. Category: Artificial Intelligence [188] viXra:1605.0178 [pdf] submitted on 2016-05-16 10:25:46 ### Artificial Intelligence Replaces Physicists Authors: George Rajna Comments: 19 Pages. The artificial intelligence system's ability to set itself up quickly every morning and compensate for any overnight fluctuations would make this fragile technology much more useful for field measurements, said co-lead researcher Dr Michael Hush from UNSW ADFA. [11] Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] New research demonstrates that particles at the quantum level can in fact be seen as behaving something like billiard balls rolling along a table, and not merely as the probabilistic smears that the standard interpretation of quantum mechanics suggests. But there's a catch-the tracks the particles follow do not always behave as one would expect from \"realistic\" trajectories, but often in a fashion that has been termed \"surrealistic.\" [8] Quantum entanglement\u2014which occurs when two or more particles are correlated in such a way that they can influence each other even across large distances\u2014is not an all-or-nothing phenomenon, but occurs in various degrees. The more a quantum state is entangled with its partner, the better the states will perform in quantum information applications. Unfortunately, quantifying entanglement is a difficult process involving complex optimization problems that give even physicists headaches. [7] A trio of physicists in Europe has come up with an idea that they believe would allow a person to actually witness entanglement. Valentina Caprara Vivoli, with the University of Geneva, Pavel Sekatski, with the University of Innsbruck and Nicolas Sangouard, with the University of Basel, have together written a paper describing a scenario where a human subject would be able to witness an instance of entanglement\u2014they have uploaded it to the arXiv server for review by others. [6] The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the relativistic quantum theory. Category: Artificial Intelligence [187] viXra:1605.0125 [pdf] submitted on 2016-05-12 08:13:31 ### Failure Mode and Effects Analysis Based on D Numbers and Topsis Authors: Tian Bian, Haoyang Zheng, Likang Yin, Yong Deng, Sankaran Mahadevand Comments: 39 Pages. Failure mode and effects analysis (FMEA) is a widely used technique for assessing the risk of potential failure modes in designs, products, process, system or services. One of the main problems of FMEA is to deal with a variety of assessments given by FMEA team members and sequence the failure modes according to the degree of risk factors. The traditional FMEA using risk priority number (RPN) which is the product of occurrence (O), severity (S) and detection (D) of a failure to determine the risk priority ranking order of failure modes. However, it will become impractical when multiple experts give different risk assessments to one failure mode, which may be imprecise or incomplete or the weights of risk factors is inconsistent. In this paper, a new risk priority model based on D numbers, and technique for order of preference by similarity to ideal solution (TOPSIS) is proposed to evaluate the risk in FMEA. In the proposed model, the assessments given by FMEA team members are represented by D numbers, a method can effectively handle uncertain information. TOPSIS method, a novel multi-criteria decision making (MCDM) method is presented to rank the preference of failure modes respect to risk factors. Finally, an application of the failure modes of rotor blades of an aircraft turbine is provided to illustrate the efficiency of the proposed method. Category: Artificial Intelligence [186] viXra:1603.0378 [pdf] submitted on 2016-03-27 16:51:16 ### A Review of Theoretical and Practical Challenges of Trusted Autonomy in Big Data Authors: Hussein A. Abbass, George Leu, Kathryn Merrick Comments: 32 Pages. Despite the advances made in artificial intelligence, software agents, and robotics, there is little we see today that we can truly call a fully autonomous system. We conjecture that the main inhibitor for advancing autonomy is lack of trust. Trusted autonomy is the scientific and engineering field to establish the foundations and ground work for developing trusted autonomous systems (robotics and software agents) that can be used in our daily life, and can be integrated with humans seamlessly, naturally and efficiently. In this paper, we review this literature to reveal opportunities for researchers and practitioners to work on topics that can create a leap forward in advancing the field of trusted autonomy. We focus the paper on the `trust' component as the uniting technology between humans and machines. Our inquiry into this topic revolves around three sub-topics: (1) reviewing and positioning the trust modelling literature for the purpose of trusted autonomy; (2) reviewing a critical subset of sensor technologies that allow a machine to sense human states; and (3) distilling some critical questions for advancing the field of trusted autonomy. The inquiry is augmented with conceptual models that we propose along the way by recompiling and reshaping the literature into forms that enables trusted autonomous systems to become a reality. The paper offers a vision for a Trusted Cyborg Swarm, an extension of our previous Cognitive Cyber Symbiosis concept, whereby humans and machines meld together in a harmonious, seamless, and coordinated manner. Category: Artificial Intelligence [185] viXra:1603.0335 [pdf] submitted on 2016-03-23 10:04:45 ### Conditional Deng Entropy, Joint Deng Entropy and Generalized Mutual Information Authors: Haoyang Zheng, Yong Deng Comments: 16 Pages. Shannon entropy, conditional entropy, joint entropy and mutual information, can estimate the chaotic level of information. However, these methods could only handle certain situations. Based on Deng entropy, this paper introduces multiple new entropy to estimate entropy under multiple interactive uncertain information: conditional Deng entropy is used to calculate entropy under conditional basic belief assignment; joint Deng entropy could calculate entropy by applying joint basic belief assignment distribution; generalized mutual information is applied to estimate the uncertainty of information under knowing another information. Numerical examples are used for illustrating the function of new entropy in the end. Category: Artificial Intelligence [184] viXra:1602.0345 [pdf] submitted on 2016-02-27 04:26:26 ### Biomolecular Parallel Computer Authors: George Rajna Comments: 16 Pages. A study published this week in Proceedings of the National Academy of Sciences reports a new parallel-computing approach based on a combination of nanotechnology and biology that can solve combinatorial problems. [9] The substance that provides energy to all the cells in our bodies, Adenosine triphosphate (ATP), may also be able to power the next generation of supercomputers. The discovery opens doors to the creation of biological supercomputers that are about the size of a book. [8] The one thing everyone knows about quantum mechanics is its legendary weirdness, in which the basic tenets of the world it describes seem alien to the world we live in. Superposition, where things can be in two states simultaneously, a switch both on and off, a cat both dead and alive. Or entanglement, what Einstein called \"spooky action-at-distance\" in which objects are invisibly linked, even when separated by huge distances. [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [183] viXra:1602.0338 [pdf] submitted on 2016-02-27 02:11:05 ### Biological Supercomputers Authors: George Rajna Comments: 15 Pages. The substance that provides energy to all the cells in our bodies, Adenosine triphosphate (ATP), may also be able to power the next generation of supercomputers. The discovery opens doors to the creation of biological supercomputers that are about the size of a book. [8] The one thing everyone knows about quantum mechanics is its legendary weirdness, in which the basic tenets of the world it describes seem alien to the world we live in. Superposition, where things can be in two states simultaneously, a switch both on and off, a cat both dead and alive. Or entanglement, what Einstein called \"spooky action-at-distance\" in which objects are invisibly linked, even when separated by huge distances. [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [182] viXra:1602.0299 [pdf] submitted on 2016-02-24 08:03:56 ### IBM Watson Emotion Analysis Authors: George Rajna Comments: 17 Pages. IBM today announced new and expanded cognitive APIs for developers that enhance Watson's emotional and visual senses, further extending the capabilities of the industry's largest and most diverse set of cognitive technologies and tools. [8] The pursuit of an understanding of the base machinery of the mind led early researchers to anatomical exhaustion. With neuroscience now in the throes of molecular mayhem and a waning biochemical bliss, physics is spicing things up with a host of eclectic quantum, spin, and isotopic novelties. While increases in electron spin content have been linked to anesthetic effects, nuclear spins have recently been implicated in a more rarefied and subtle phenomenon\u2014 neural quantum processing. [7] The hypothesis that there may be something quantum-like about the human mental function was put forward with \" Spooky Activation at Distance \" formula which attempted to model the effect that when a word's associative network is activated during study in memory experiment; it behaves like a quantum-entangled system. The human body is a constant flux of thousands of chemical\/biological interactions and processes connecting molecules, cells, organs, and fluids, throughout the brain, body, and nervous system. Up until recently it was thought that all these interactions operated in a linear sequence, passing on information much like a runner passing the baton to the next runner. However, the latest findings in quantum biology and biophysics have discovered that there is in fact a tremendous degree of coherence within all living systems. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to understand the Quantum Biology. Category: Artificial Intelligence [181] viXra:1602.0289 [pdf] submitted on 2016-02-23 01:09:01 ### Computer Quantum Experiments Authors: George Rajna Comments: 17 Pages. Quantum physicist Mario Krenn and his colleagues in the group of Anton Zeilinger from the Faculty of Physics at the University of Vienna and the Austrian Academy of Sciences have developed an algorithm which designs new useful quantum experiments. As the computer does not rely on human intuition, it finds novel unfamiliar solutions. [10] Researchers at the University of Chicago's Institute for Molecular Engineering and the University of Konstanz have demonstrated the ability to generate a quantum logic operation, or rotation of the qubit, that-surprisingly\u2014is intrinsically resilient to noise as well as to variations in the strength or duration of the control. Their achievement is based on a geometric concept known as the Berry phase and is implemented through entirely optical means within a single electronic spin in diamond. [9] New research demonstrates that particles at the quantum level can in fact be seen as behaving something like billiard balls rolling along a table, and not merely as the probabilistic smears that the standard interpretation of quantum mechanics suggests. But there's a catch-the tracks the particles follow do not always behave as one would expect from \"realistic\" trajectories, but often in a fashion that has been termed \"surrealistic.\" [8] Quantum entanglement\u2014which occurs when two or more particles are correlated in such a way that they can influence each other even across large distances\u2014is not an all-or-nothing phenomenon, but occurs in various degrees. The more a quantum state is entangled with its partner, the better the states will perform in quantum information applications. Unfortunately, quantifying entanglement is a difficult process involving complex optimization problems that give even physicists headaches. [7] A trio of physicists in Europe has come up with an idea that they believe would allow a person to actually witness entanglement. Valentina Caprara Vivoli, with the University of Geneva, Pavel Sekatski, with the University of Innsbruck and Nicolas Sangouard, with the University of Basel, have together written a paper describing a scenario where a human subject would be able to witness an instance of entanglement\u2014they have uploaded it to the arXiv server for review by others. [6] The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the relativistic quantum theory. Category: Artificial Intelligence [180] viXra:1602.0233 [pdf] submitted on 2016-02-18 19:28:27 ### Quantum Decision-Maker Authors: Michail Zak Comments: 24 Pages. A QRN simulating human decision making process is introduced. It consists of quantum recurrent nets generating stochastic processes which represent the motor dynamics, and of classical neural nets describing evolution of probabilities of these processes which represent the mental dynamics. The autonomy of the decision making process is achieved by a feedback from mental to motor dynamics which changes the stochastic matrix based upon the probability distributions. This feedback replaces an unavailable external information by an internal knowledgebase stored in the mental model in the form of probability distributions. As a result, the coupled motor-mental dynamics is described by a nonlinear version of Markov chains which can decrease entropy without an external source of information. Applications to common sense based decisions as well as to evolutionary games are discussed. Category: Artificial Intelligence [179] viXra:1602.0232 [pdf] submitted on 2016-02-18 21:11:43 ### Quantum Model of Emerging Grammars Authors: Michail Zak Comments: 7 Pages. A special class of quantum recurrent nets (QRNs) simulating Markov chains with absorbing states is introduced. The absorbing states are exploited for pattern recognition: each class of patterns is attracted to a unique absorbing state. Due to quantum interference of patterns, each combination of patterns acquires its own meaning: it is attracted to a certain combination of absorbing states which is dierent from those of individual attractions. This fundamentally new eect can be interpreted as formation of a grammar, i.e., a set of rules assigning certain meaning to dierent combinations of patterns. It appears that there exists a class of unitary operators in which each member gives rise to a dierent arti\u00aecial language with associated grammar. Category: Artificial Intelligence [178] viXra:1602.0230 [pdf] submitted on 2016-02-18 22:38:51 ### Quantum-Inspired Teleportation. Authors: Michail.Zak Comments: 11 Pages. Based upon quantum-inspired entanglement in quantum-classical hybrids, a simple algorithm for instantaneous transmissions of non-intentional messages (chosen at random) to remote distances is proposed. A special class of situations when such transmissions are useful is outlined. Application of such a quantum-inspired teleportation, i.e. instantaneous transmission of conditional information on remote distances for security of communications is discussed. Similarities and differences between quantum systems and quantum-classical hybrids are emphasized. Category: Artificial Intelligence [177] viXra:1602.0222 [pdf] submitted on 2016-02-18 04:36:09 ### Neuromorphic Computing Authors: George Rajna Comments: 16 Pages. In the field of neuromorphic engineering, researchers study computing techniques that could someday mimic human cognition. Electrical engineers at the Georgia Institute of Technology recently published a \"roadmap\" that details innovative analog-based techniques that could make it possible to build a practical neuromorphic computer. [9] How does the brain-a lump of 'pinkish gray meat'-produce the richness of conscious experience, or any subjective experience at all? Scientists and philosophers have historically likened the brain to contemporary information technology, from the ancient Greeks comparing memory to a 'seal ring in wax,' to the 19th century brain as a 'telegraph switching circuit,' to Freud's subconscious desires 'boiling over like a steam engine,' to a hologram, and finally, the computer. [8] Discovery of quantum vibrations in 'microtubules' inside brain neurons supports controversial theory of consciousness. The human body is a constant flux of thousands of chemical\/biological interactions and processes connecting molecules, cells, organs, and fluids, throughout the brain, body, and nervous system. Up until recently it was thought that all these interactions operated in a linear sequence, passing on information much like a runner passing the baton to the next runner. However, the latest findings in quantum biology and biophysics have discovered that there is in fact a tremendous degree of coherence within all living systems. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to understand the Quantum Biology. Category: Artificial Intelligence [176] viXra:1602.0077 [pdf] submitted on 2016-02-06 10:58:18 ### Critical Review, Vol. 11, 2015 Authors: Many Authors Comments: 141 Pages. Papers on neutrosophic set and logic Category: Artificial Intelligence [175] viXra:1602.0075 [pdf] submitted on 2016-02-06 11:02:58 ### Neutrosophic Sets and Systems, Vol. 10, 2015 Authors: Many Authors Comments: 107 Pages. Collection of papers on neutrosophics. Category: Artificial Intelligence [174] viXra:1601.0024 [pdf] submitted on 2016-01-04 06:27:06 ### Do People Leave in Matrix? Information, Entropy, Time and Cellular-Automata Authors: Janis Belov Comments: 6 Pages. The paper proves that we leave in Matrix. We show that Matrix was built by the creator. By this we solve the question how everything is built. We prove that the creator is infinite Turing machine or infinite Cellular-automaton. We show that Universe is Cellular-automaton or Turing machine too. And everything in the Universe is built as Cellular-automaton. We show that our Universe was created by \u201cvegetative birth\u201d from another Universe. In other words, there is an infinite Life Tree that is actually the creator itself. In other words, we show that everything is only Information, i.e. the creator. We show that there is no Time, moreover Time is the creator. So, the time is reversible. The arrow of time depends on Information entropy: if entropy increases \u2013 people leave in one world, if the entropy does not increase, in other words, no lost of information occurs \u2013 people step by step become closer to another world \u2013 the world where the creator is sensed directly. Someone can call the creator of Matrix \u2013 God. We would like to recall that Immanuel Kant tried to prove the existence of God but He did not succeed. The main reason for this is that he searched for a proof outside His own mind or His own conscience. Also, he did not take into consideration any evolutionary processes. The given paper tries to fill some gaps and show another way in understanding the reality. It is based on the existing science. In the paper we use mathematical methods in creating scientific concepts. The results are presented in the form Definition \u2013 Theorem to present them in a straightforward way. The immediate consequences of the obtained results are so that in studying the reality it is enough to develop Number Theory, Poetry, Art and other Sciences that are based on the Nature and Inner World of a person. We also give a simple solution for Yang-Mills Mass gap problem: there is no time and mass has no sense since it was invented artificially as an approximation of reality. Category: Artificial Intelligence [173] viXra:1512.0007 [pdf] submitted on 2015-12-02 01:27:54 ### Ontology, Evolving Under the Influence of the Facts Authors: Aleksey A. Demidov Comments: 34 Pages. We propose an algebraic approach to building ontologies which capable of evolution under the influence of new facts and which have some internal mechanisms of validation. For this purpose we build a formal model of the interactions of objects based on cellular automata, and find out the limitations on transactions with objects imposed by this model. Then, in the context of the formal model, we define basic entities of the model of knowledge representation: concepts, samples, properties, and relationships. In this case the formal limitations are induced into the model of knowledge representation in a natural way. Category: Artificial Intelligence [172] viXra:1511.0145 [pdf] submitted on 2015-11-17 06:02:52 ### Which is the Best Belief Entropy? Authors: Liguo Fei, Yong Deng, Sankaran Mahadevan Comments: 4 Pages. In this paper, many numerical examples are designed to compare the existing different belief functions with the new entropy, named Deng entropy. The results illustrate that, among the existing belief entropy functions,Deng entropy is the best alternative due to its reasonable properties. Category: Artificial Intelligence [171] viXra:1511.0144 [pdf] submitted on 2015-11-17 06:09:55 ### Measure Divergence Degree of Basic Probability Assignment Based on Deng Relative Entropy Authors: Liguo Fei, Yong Deng Comments: 15 Pages. Dempster Shafer evidence theory (D-S theory) is more and more extensively applied to information fusion for the advantage dealing with uncertain information. However, the results opposite to common sense are often obtained when combining the different evidence using the Dempster\u2019s combination rules. How to measure the divergence between different evidence is still an open issue. In this paper, a new relative entropy named as Deng relative entropy is proposed in order to measure the divergence between different basic probability assignments (BPAs). The Deng relative entropy is the generalization of Kullback-Leibler Divergence because when the BPA is degenerated as probability, Deng relative entropy is equal to Kullback-Leibler Divergence. Numerical examples are used to illustrate the effectiveness of the proposed Deng relative entropy. Category: Artificial Intelligence [170] viXra:1511.0095 [pdf] submitted on 2015-11-11 13:10:20 ### Robots and Computers 'Consciousness' Authors: George Rajna Comments: 22 Pages. Imagine a world where \"thinking\" robots were able to care for the elderly and people with disabilities. This concept may seem futuristic, but exciting new research into consciousness could pave the way for the creation of intuitive artificial intelligence. [13] A small, Santa Fe, New Mexico-based company called Knowm claims it will soon begin commercializing a state-of-the-art technique for building computing chips that learn. Other companies, including HP HPQ -3.45% and IBM IBM -2.10% , have already invested in developing these so-called brain-based chips, but Knowm says it has just achieved a major technological breakthrough that it should be able to push into production hopefully within a few years. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron\u2019s spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [169] viXra:1511.0020 [pdf] submitted on 2015-11-02 18:50:19 ### Can a Mobile Game Teach Computer Users to Thwart Phishing Attacks? Authors: Nalin Asanka Gamagedara Arachchilage, Steve Love, Carsten Maple Comments: 11 Pages. Usable Security Phishing is an online fraudulent technique, which aims to steal sensitive information such as usernames, passwords and online banking details from its victims. To prevent this, anti-phishing education needs to be considered. This research focuses on examining the effectiveness of mobile game based learning compared to traditional online learning to thwart phishing threats. Therefore, a mobile game prototype was developed based on the design introduced by Arachchilage and Cole [3]. The game design aimed to enhance avoidance behaviour through motivation to thwart phishing threats. A website developed by Anti-Phishing Work Group (APWG) for the public Anti-phishing education initiative was used as a traditional web based learning source. A think-aloud experiment along with a pre- and post-test was conducted through a user study. The study findings revealed that the participants who played the mobile game were better able to identify fraudulent web sites compared to the participants who read the website without any training. Category: Artificial Intelligence [168] viXra:1510.0486 [pdf] submitted on 2015-10-28 20:26:16 ### Embedded System for Waste Management using Fuzzy Logic Authors: Sai Venkatesh Balasubramanian Comments: 6 Pages. The non-degradable wastes such as plastic are a big threat for the environment. Hence an embedded system for Automation in splitting up, disposal and recycling of wastes is the best solution. The entire process is done with Artificial intelligence. The A.I. is provided with the help of Fuzzy logic. In this paper, all the common wastes such as Ferrous & its compounds, Paper, Plastic, Polythene, E-wastes, Bio-degradable wastes, etc. are considered. The input wave is given for detecting the type of wastes. By using the IF \u2026THEN\u2026 ELSE condition it is processed. The image processing has been implemented to check the material. This paper provides the entire design of the embedded system, the entire logic for the automation with the required IF\u2026THEN\u2026.ELSE... codes. The automation used in this paper eliminates a significant amount of entire manual work. Category: Artificial Intelligence [167] viXra:1510.0022 [pdf] submitted on 2015-10-03 02:54:50 ### Nonextensive Deng Entropy Authors: Yong Deng Comments: 9 Pages. In this paper, a generalized Tsallis entropy, named as Nonextensive Deng entropy, is presented. When the basic probability assignment is degenerated as probability, Nonextensive Deng entropy is identical to Tsallis entropy. Category: Artificial Intelligence [166] viXra:1509.0163 [pdf] submitted on 2015-09-18 04:00:41 ### Lie Detection System with Voice Using Bidirectional Associative Memory Algorithm Authors: Bustami; Fadlisyah; Nurdania Delemunte Comments: 07 Pages. Figures :04 Tables : 03, IJCAT.org, Volume 2, Issue 8, August 2015 Lie detection through voice can be detected using the algorithm bidirectional associative memory. This system is a branch of sound processing that can be used to identify the type of sound lies use some verbs like go, roads and move. This study uses an algorithm bidirectional associative memory for the process and the introduction of lie detection training through the sound use of bidirectional associative memory. The system was tested by simulating the training data and test data to generate a percentage of voice recognition and classification of these lies. Experiments performed with several changes in parameter values to obtain the best percentage of recognition and classification. The highest level of recognition contained in the verb \"go\" with up to 90%. Results of this research is a sound that indicated not indicated lies and deceit in the form of values are classified according to the type of sound that is known from the results of calculations of energy use bidirectional associative memory. Category: Artificial Intelligence [165] viXra:1509.0119 [pdf] submitted on 2015-09-13 20:01:24 ### The Maximum Deng Entropy Authors: Bingyi Kang, Yong Deng Comments: 17 Pages Dempster Shafer evidence theory has widely used in many applications due to its advantages to handle uncertainty. Deng entropy, has been proposed to measure the uncertainty degree of basic probability assignment in evidence theory. It is the generalization of Shannon entropy since that the BPA is degenerated as probability, Deng entropy is identical to Shannon entropy. However, the maximal value of Deng entropy has not been disscussed until now. In this paper, the condition of the maximum of Deng entropy has been disscussed and proofed, which is usefull for the application of Deng entropy. Category: Artificial Intelligence [164] viXra:1509.0088 [pdf] submitted on 2015-09-07 18:59:16 ### A Cognitive Architecture for Human-Like and Personable ai Authors: Arvind Chitra Rajasekaran Comments: 4 Pages. In this article we will introduce a cognitive architecture for creating a more human like and personable artificial intelligence. Recent works such as those by Marvin Minsky, Google DeepMind and cognitive models like AMBR, DUAL that aim to propose\/discover an approach to commonsense AI have been promising, since they show that human intelligence can be emulated with a divide and conquer approach on a machine. These frameworks work with an universal model of the human mind and do not account for the variability between human beings. It is these differences between human beings that make communication possible and gives them a sense of identity. Thus, this work, despite being grounded in these methods, will differ in hypothesizing machines that are diverse in their behavior compared to each other and have the ability to express a dynamic personality like a human being. To achieve such individuality in machines, we characterize the various aspects that can be dynamically programmed onto a machine by its human owners. In order to ensure this on a scale parallel to how humans develop their individuality, we first assume a child-like intelligence in a machine that is more malleable and which then develops into a more concrete, mature version. By having a set of tunable inner parameters called aspects which respond to external stimuli from their human owners, machines can achieve personability. The result of this work would be that we will not only be able to bond with the intelligent machines and relate to them in a friendly way, we will also be able to perceive them as having a personality, and that they have their limitations. Just as each human being is unique, we will have machines that are unique and individualistic. We will see how they can achieve intuition, and a drive to find meaning in life, all of which are considered aspects unique to the human mind. Category: Artificial Intelligence [163] viXra:1509.0069 [pdf] submitted on 2015-09-05 10:41:32 ### Startup Breakthrough in Brain-like Computing Authors: George Rajna Comments: 21 Pages. A small, Santa Fe, New Mexico-based company called Knowm claims it will soon begin commercializing a state-of-the-art technique for building computing chips that learn. Other companies, including HP HPQ -3.45% and IBM IBM -2.10% , have already invested in developing these so-called brain-based chips, but Knowm says it has just achieved a major technological breakthrough that it should be able to push into production hopefully within a few years. [12] A team of researchers working at the University of California (and one from Stony Brook University) has for the first time created a neural-network chip that was built using just memristors. In their paper published in the journal Nature, the team describes how they built their chip and what capabilities it has. [11] A team of researchers used a promising new material to build more functional memristors, bringing us closer to brain-like computing. Both academic and industrial laboratories are working to develop computers that operate more like the human brain. Instead of operating like a conventional, digital system, these new devices could potentially function more like a network of neurons. [10] Cambridge Quantum Computing Limited (CQCL) has built a new Fastest Operating System aimed at running the futuristic superfast quantum computers. [9] IBM scientists today unveiled two critical advances towards the realization of a practical quantum computer. For the first time, they showed the ability to detect and measure both kinds of quantum errors simultaneously, as well as demonstrated a new, square quantum bit circuit design that is the only physical architecture that could successfully scale to larger dimensions. [8] Physicists at the Universities of Bonn and Cambridge have succeeded in linking two completely different quantum systems to one another. In doing so, they have taken an important step forward on the way to a quantum computer. To accomplish their feat the researchers used a method that seems to function as well in the quantum world as it does for us people: teamwork. The results have now been published in the \"Physical Review Letters\". [7] While physicists are continually looking for ways to unify the theory of relativity, which describes large-scale phenomena, with quantum theory, which describes small-scale phenomena, computer scientists are searching for technologies to build the quantum computer. The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron\u2019s spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron\/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the Relativistic Quantum Theory and making possible to build the Quantum Computer. Category: Artificial Intelligence [162] viXra:1508.0139 [pdf] submitted on 2015-08-17 14:52:18 ### Themes of Futurism and AI in the Novel BlindSight Authors: Andrew Nassif Comments: 5 Pages. Andrew Nassif reviews the novel BlindSight and how it shows themes of modern Trans-humanism, Futurism, and Artificial Intelligence, as well as its symbolic references in terms of philosophy and human empathy. He reviews it through a philosopher, a physicist, a researcher, and a literary analyst point of view. Category: Artificial Intelligence ## Replacements of recent Submissions [45] viXra:1711.0265 [pdf] replaced on 2017-11-27 03:16:15 ### Revisit Fuzzy Neural Network: Bridging the Gap Between Fuzzy Logic and Deep Learning Authors: Lixin Fan Comments: 76 Pages. This article aims to establish a concrete and fundamental connection between two important elds in artificial intelligence i.e. deep learning and fuzzy logic. On the one hand, we hope this article will pave the way for fuzzy logic researchers to develop convincing applications and tackle challenging problems which are of interest to machine learning community too. On the other hand, deep learning could benefit from the comparative research by re-examining many trail-and-error heuristics in the lens of fuzzy logic, and consequently, distilling the essential ingredients with rigorous foundations. Based on the new findings reported in [41] and this article, we believe the time is ripe to revisit fuzzy neural network as a crucial bridge between two schools of AI research i.e. symbolic versus connectionist [101] and eventually open the black-box of artificial neural networks. Category: Artificial Intelligence [44] viXra:1711.0265 [pdf] replaced on 2017-11-17 16:28:38 ### Revisit Fuzzy Neural Network: Bridging the Gap Between Fuzzy Logic and Deep Learning Authors: Lixin Fan Comments: 76 Pages. This article aims to establish a concrete and fundamental connection between two important fields in artificial intelligence i.e. deep learning and fuzzy logic. On the one hand, we hope this article will pave the way for fuzzy logic researchers to develop convincing applications and tackle challenging problems which are of interest to machine learning community too. On the other hand, deep learning could benefit from the comparative research by re-examining many trail-and-error heuristics in the lens of fuzzy logic, and consequently, distilling the essential ingredients with rigorous foundations. Based on the new findings reported in [38] and this article, we believe the time is ripe to revisit fuzzy neural network as a crucial bridge between two schools of AI research i.e. symbolic versus connectionist [93] and eventually open the black-box of artificial neural networks. Category: Artificial Intelligence [43] viXra:1710.0324 [pdf] replaced on 2017-11-09 05:34:27 ### New Sufficient Conditions of Signal Recovery with Tight Frames Via$l_1$-Analysis Authors: Jianwen Huang, Jianjun Wang, Feng Zhang, Wendong Wang Comments: 18 Pages. The paper discusses the recovery of signals in the case that signals are nearly sparse with respect to a tight frame$D$by means of the$l_1$-analysis approach. We establish several new sufficient conditions regarding the$D$-restricted isometry property to ensure stable reconstruction of signals that are approximately sparse with respect to$D$. It is shown that if the measurement matrix$\\Phi$fulfils the condition$\\delta_{ts}<t\/(4-t)$for$0<t<4\/3$, then signals which are approximately sparse with respect to$D$can be stably recovered by the$l_1$-analysis method. In the case of$D=I$, the bound is sharp, see Cai and Zhang's work \\cite{Cai and Zhang 2014}. When$t=1$, the present bound improves the condition$\\delta_s<0.307$from Lin et al.'s reuslt to$\\delta_s<1\/3$. In addition, numerical simulations are conducted to indicate that the$l_1\\$-analysis method can stably reconstruct the sparse signal in terms of tight frames.\nCategory: Artificial Intelligence\n\n[42] viXra:1709.0108 [pdf] replaced on 2017-09-10 08:24:10\n\n### A New Semantic Theory of Nature Language\n\nAuthors: Kun Xing\n\nFormal Semantics and Distributional Semantics are two important semantic frameworks in Natural Language Processing (NLP). Cognitive Semantics belongs to the movement of Cognitive Linguistics, which is based on contemporary cognitive science. Each framework could deal with some meaning phenomena, but none of them fulfills all requirements proposed by applications. A unified semantic theory characterizing all important language phenomena has both theoretical and practical significance; however, although many attempts have been made in recent years, no existing theory has achieved this goal yet. This article introduces a new semantic theory that has the potential to characterize most of the important meaning phenomena of natural language and to fulfill most of the necessary requirements for philosophical analysis and for NLP applications. The theory is based on a unified representation of information, and constructs a kind of mathematical model called cognitive model to interpret natural language expressions in a compositional manner. It accepts the empirical assumption of Cognitive Semantics, and overcomes most shortcomings of Formal Semantics and of Distributional Semantics. The theory, however, is not a simple combination of existing theories, but an extensive generalization of classic logic and Formal Semantics. It inherits nearly all advantages of Formal Semantics, and also provides descriptive contents for objects and events as fine-gram as possible, descriptive contents which represent the results of human cognition.\nCategory: Artificial Intelligence\n\n[41] viXra:1611.0211 [pdf] replaced on 2016-12-01 04:59:33\n\n### A Variable Order Hidden Markov Model with Dependence Jumps\n\nAuthors: Anastasios Petropoulos, Stelios Xanthopoulos, Sotirios P. Chatzis\n\nHidden Markov models (HMMs) are a popular approach for modeling sequential data, typically based on the assumption of a first- or moderate-order Markov chain. However, in many real-world scenarios the modeled data entail temporal dynamics the patterns of which change over time. In this paper, we address this problem by proposing a novel HMM formulation, treating temporal dependencies as latent variables over which inference is performed. Specifically, we introduce a hierarchical graphical model comprising two hidden layers: on the first layer, we postulate a chain of latent observation-emitting states, the temporal dependencies between which may change over time; on the second layer, we postulate a latent first-order Markov chain modeling the evolution of temporal dynamics (dependence jumps) pertaining to the first-layer latent process. As a result of this construction, our method allows for effectively modeling non-homogeneous observed data, where the patterns of the entailed temporal dynamics may change over time. We devise efficient training and inference algorithms for our model, following the expectation-maximization paradigm. We demonstrate the efficacy and usefulness of our approach considering several real-world datasets. As we show, our model allows for increased modeling and predictive performance compared to the alternative methods, while offering a good trade-off between the resulting increases in predictive performance and computational complexity.\nCategory: Artificial Intelligence\n\n[40] viXra:1611.0211 [pdf] replaced on 2016-11-14 08:01:26\n\n### A Variable Order Hidden Markov Model with Dependence Jumps\n\nAuthors: Anastasios Petropoulos, Stelios Xanthopoulos, Sotirios P. Chatzis\n\nHidden Markov models (HMMs) are a popular approach for modeling sequential data, typically based on the assumption of a first- or moderate-order Markov chain. However, in many real-world scenarios the modeled data entail temporal dynamics the patterns of which change over time. In this paper, we address this problem by proposing a novel HMM formulation, treating temporal dependencies as latent variables over which inference is performed. Specifically, we introduce a hierarchical graphical model comprising two hidden layers: on the first layer, we postulate a chain of latent observation-emitting states, the temporal dependencies between which may change over time; on the second layer, we postulate a latent first-order Markov chain modeling the evolution of temporal dynamics (dependence jumps) pertaining to the first-layer latent process. As a result of this construction, our method allows for effectively modeling non-homogeneous observed data, where the patterns of the entailed temporal dynamics may change over time. We devise efficient training and inference algorithms for our model, following the expectation-maximization paradigm. We demonstrate the efficacy and usefulness of our approach considering several real-world datasets. As we show, our model allows for increased modeling and predictive performance compared to the alternative methods, while offering a good trade-off between the resulting increases in predictive performance and computational complexity.\nCategory: Artificial Intelligence\n\n[39] viXra:1611.0211 [pdf] replaced on 2016-11-14 04:26:58\n\n### A Variable Order Hidden Markov Model with Dependence Jumps\n\nAuthors: Anastasios Petropoulos, Stelios Xanthopoulos, Sotirios P. Chatzis\n\nHidden Markov models (HMMs) are a popular approach for modeling sequential data, typically based on the assumption of a first- or moderate-order Markov chain. However, in many real-world scenarios the modeled data entail temporal dynamics the patterns of which change over time. In this paper, we address this problem by proposing a novel HMM formulation, treating temporal dependencies as latent variables over which inference is performed. Specifically, we introduce a hierarchical graphical model comprising two hidden layers: on the first layer, we postulate a chain of latent observation-emitting states, the temporal dependencies between which may change over time; on the second layer, we postulate a latent first-order Markov chain modeling the evolution of temporal dynamics (dependence jumps) pertaining to the first-layer latent process. As a result of this construction, our method allows for effectively modeling non-homogeneous observed data, where the patterns of the entailed temporal dynamics may change over time. We devise efficient training and inference algorithms for our model, following the expectation-maximization paradigm. We demonstrate the efficacy and usefulness of our approach considering several real-world datasets. As we show, our model allows for increased modeling and predictive performance compared to the alternative methods, while offering a good trade-off between the resulting increases in predictive performance and computational complexity.\nCategory: Artificial Intelligence\n\n[38] viXra:1610.0029 [pdf] replaced on 2016-10-16 08:51:52\n\nAuthors: Aleksei Morozov\n\nAssociative broadcast neural network (aka Ether Neural Network) is an artificial neural network inspired by a hypothesis of broadcasting of neuron's output pattern in a biological neural network. Neuron has wire connections and ether connections. Ether connections are electrical. Wire connections provide a recognition functionality. Ether connections provide an association functionality.\nCategory: Artificial Intelligence\n\n[37] viXra:1607.0073 [pdf] replaced on 2016-07-08 06:55:14\n\n### Indian Buffet Process Deep Generative Models\n\nAuthors: Sotirios P. Chatzis\n\nDeep generative models (DGMs) have brought about a major breakthrough, as well as renewed interest, in generative latent variable models. However, an issue current DGM formulations do not address concerns the data-driven inference of the number of latent features needed to represent the observed data. Traditional linear formulations allow for addressing this issue by resorting to tools from the field of nonparametric statistics: Indeed, nonparametric linear latent variable models, obtained by appropriate imposition of Indian Buffet Process (IBP) priors, have been extensively studied by the machine learning community; inference for such models can been performed either via exact sampling or via approximate variational techniques. Based on this inspiration, in this paper we examine whether similar ideas from the field of Bayesian nonparametrics can be utilized in the context of modern DGMs in order to address the latent variable dimensionality inference problem. To this end, we propose a novel DGM formulation, based on the imposition of an IBP prior. We devise an efficient Black-Box Variational inference algorithm for our model, and exhibit its efficacy in a number of semi-supervised classification experiments. In all cases, we use popular benchmark datasets, and compare to state-of-the-art DGMs.\nCategory: Artificial Intelligence\n\n[36] viXra:1607.0073 [pdf] replaced on 2016-07-08 03:07:07\n\n### Indian Buffet Process Deep Generative Models\n\nAuthors: Sotirios P. Chatzis\n\nDeep generative models (DGMs) have brought about a major breakthrough, as well as renewed interest, in generative latent variable models. However, an issue current DGM formulations do not address concerns the data-driven inference of the number of latent features needed to represent the observed data. Traditional linear formulations allow for addressing this issue by resorting to tools from the field of nonparametric statistics: Indeed, nonparametric linear latent variable models, obtained by appropriate imposition of Indian Buffet Process (IBP) priors, have been extensively studied by the machine learning community; inference for such models can been performed either via exact sampling or via approximate variational techniques. Based on this inspiration, in this paper we examine whether similar ideas from the field of Bayesian nonparametrics can be utilized in the context of modern DGMs in order to address the latent variable dimensionality inference problem. To this end, we propose a novel DGM formulation, based on the imposition of an IBP prior. We devise an efficient Black-Box Variational inference algorithm for our model, and exhibit its efficacy in a number of semi-supervised classification experiments. In all cases, we use popular benchmark datasets, and compare to state-of-the-art DGMs.\nCategory: Artificial Intelligence\n\n[35] viXra:1607.0073 [pdf] replaced on 2016-07-07 10:41:19\n\n### Indian Buffet Process Deep Generative Models\n\nAuthors: Sotirios P. Chatzis\n\nDeep generative models (DGMs) have brought about a major breakthrough, as well as renewed interest, in generative latent variable models. However, an issue current DGM formulations do not address concerns the data-driven inference of the number of latent features needed to represent the observed data. Traditional linear formulations allow for addressing this issue by resorting to tools from the field of nonparametric statistics: Indeed, nonparametric linear latent variable models, obtained by appropriate imposition of Indian Buffet Process (IBP) priors, have been extensively studied by the machine learning community; inference for such models can been performed either via exact sampling or via approximate variational techniques. Based on this inspiration, in this paper we examine whether similar ideas from the field of Bayesian nonparametrics can be utilized in the context of modern DGMs in order to address the latent variable dimensionality inference problem. To this end, we propose a novel DGM formulation, based on the imposition of an IBP prior. We devise an efficient Black-Box Variational inference algorithm for our model, and exhibit its efficacy in a number of semi-supervised classification experiments. In all cases, we use popular benchmark datasets, and compare to state-of-the-art DGMs.\nCategory: Artificial Intelligence\n\n[34] viXra:1607.0073 [pdf] replaced on 2016-07-07 08:09:10\n\n### Indian Buffet Process Deep Generative Models\n\nAuthors: Sotirios P. Chatzis\n\nDeep generative models (DGMs) have brought about a major breakthrough, as well as renewed interest, in generative latent variable models. However, an issue current DGM formulations do not address concerns the data-driven inference of the number of latent features needed to represent the observed data. Traditional linear formulations allow for addressing this issue by resorting to tools from the field of nonparametric statistics: Indeed, nonparametric linear latent variable models, obtained by appropriate imposition of Indian Buffet Process (IBP) priors, have been extensively studied by the machine learning community; inference for such models can been performed either via exact sampling or via approximate variational techniques. Based on this inspiration, in this paper we examine whether similar ideas from the field of Bayesian nonparametrics can be utilized in the context of modern DGMs in order to address the latent variable dimensionality inference problem. To this end, we propose a novel DGM formulation, based on the imposition of an IBP prior. We devise an efficient Black-Box Variational inference algorithm for our model, and exhibit its efficacy in a number of semi-supervised classification experiments. In all cases, we use popular benchmark datasets, and compare to state-of-the-art DGMs.\nCategory: Artificial Intelligence\n\n[33] viXra:1607.0073 [pdf] replaced on 2016-07-07 07:08:00\n\n### Indian Buffet Process Deep Generative Models\n\nAuthors: Sotirios P. Chatzis\n\nDeep generative models (DGMs) have brought about a major breakthrough, as well as renewed interest, in generative latent variable models. However, an issue current DGM formulations do not address concerns the data-driven inference of the number of latent features needed to represent the observed data. Traditional linear formulations allow for addressing this issue by resorting to tools from the field of nonparametric statistics: Indeed, nonparametric linear latent variable models, obtained by appropriate imposition of Indian Buffet Process (IBP) priors, have been extensively studied by the machine learning community; inference for such models can been performed either via exact sampling or via approximate variational techniques. Based on this inspiration, in this paper we examine whether similar ideas from the field of Bayesian nonparametrics can be utilized in the context of modern DGMs in order to address the latent variable dimensionality inference problem. To this end, we propose a novel DGM formulation, based on the imposition of an IBP prior. We devise an efficient Black-Box Variational inference algorithm for our model, and exhibit its efficacy in a number of semi-supervised classification experiments. In all cases, we use popular benchmark datasets, and compare to state-of-the-art DGMs.\nCategory: Artificial Intelligence\n\n[32] viXra:1606.0272 [pdf] replaced on 2016-11-24 16:47:04\n\n### Self-Controlled Dynamics\n\nAuthors: Michail Zak\n\nA new class of dynamical system described by ODE coupled with their Liouville equation has been introduced and discussed. These systems called self-controlled, or self-supervised since the role of actuators is played by the probability produced by the Liouville equation. Following the Madelung equation that belongs to this class, non- Newtonian properties such as randomness, entanglement, and probability interference typical for quantum systems have been described. Special attention was paid to the capability to violate the second law of thermodynamics, which makes these systems neither Newtonian, nor quantum. It has been shown that self-controlled dynamical systems can be linked to mathematical models of livings as well as to models of AI. The central point of this paper is the application of the self-controlled systems to NP-complete problems known as being unsolvable neither by classical nor by quantum algorithms. The approach is illustrated by solving a search in unsorted database in polynomial time by resonance between external force representing the address of a required item and the response representing location of this item.\nCategory: Artificial Intelligence\n\n[31] viXra:1606.0272 [pdf] replaced on 2016-11-24 15:46:24\n\n### Self-Controlled Dynamics\n\nAuthors: Michail Zak\n\nA new class of dynamical system described by ODE coupled with their Liouville equation has been introduced and discussed. These systems called self-controlled, or self-supervised since the role of actuators is played by the probability produced by the Liouville equation. Following the Madelung equation that belongs to this class, non- Newtonian properties such as randomness, entanglement, and probability interference typical for quantum systems have been described. Special attention was paid to the capability to violate the second law of thermodynamics, which makes these systems neither Newtonian, nor quantum. It has been shown that self-controlled dynamical systems can be linked to mathematical models of livings as well as to models of AI. The central point of this paper is the application of the self-controlled systems to NP-complete problems known as being unsolvable neither by classical nor by quantum algorithms. The approach is illustrated by solving a search in unsorted database in polynomial time by resonance between external force representing the address of a required item and the response representing location of this item.\nCategory: Artificial Intelligence\n\n[30] viXra:1605.0190 [pdf] replaced on 2016-05-20 08:36:35\n\n### The Algorithm of the Thinking Machine\n\nAuthors: Dimiter Dobrev\nComments: 15 Pages. Represented at 12 of May, 2016 at Faculty of Mathematics and Informatics, University of Sofia.\n\nIn this article we consider the questions 'What is AI?' and 'How to write a program that satisfies the definition of AI?' It deals with the basic concepts and modules that must be at the heart of this program. The most interesting concept that is discussed here is the concept of abstract signals. Each of these signals is related to the result of a particular experiment. The abstract signal is a function that at any time point returns the probability the corresponding experiment to return true.\nCategory: Artificial Intelligence\n\n[29] viXra:1509.0088 [pdf] replaced on 2015-09-08 18:58:24\n\n### Cognitive Architecture for Personable and Human-Like ai :A Perspective\n\nAuthors: Arvind Chitra Rajasekaran","date":"2018-01-17 23:32:07","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\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2959315776824951, \"perplexity\": 2295.5174873896913}, \"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-2018-05\/segments\/1516084887024.1\/warc\/CC-MAIN-20180117232418-20180118012418-00754.warc.gz\"}"} | null | null |
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Watch the beautiful sun setting with views of the river Vilaine, whilst enjoying al fresco dining with a glass or two of wine to complete your holiday.
You can see many different boats sailing along the river or why not take a leisurely stroll down to the river for a closer look and enjoy the wonderful countryside and wildlife.
Riverview Gites is a successful family run business offering splendid holiday accommodation in a beautiful part of South Brittany, France. We have been established for almost nine years and are situated in Le Grippé, a small, peaceful hamlet situated in the tranquil setting of the Morbihan countryside in Western France. Riverview Gites is privately owned and available for 8 months of the year for holidays or for househunters looking for their own little piece of France.
We can offer beautiful accommodation comprising of two stone cottages with private shared pool and childrens play area. Belle Vue Cottage and Gite du Papillon offer 5 bed and 2 bed accommodation sleeping 12 adults and children. Cider Barn Cottages which are situated about 100 yds away offers 2 two bedroom wood clad cottages. Scrumpy and Woodpecker cottages offer accommodation for 4 persons in each one. Combined we can offer accommodation for 20 adults and children.
Our gites are available for group bookings and events such as birthdays, walking groups, cycling groups and golf enthusiasts. Our Old Cider Barn can be booked for food and drink for such events.
Our main aims are, firstly and most importantly customer satisfaction, along with a friendly and informal relationship with our guests. We pride ourselves on the quality of our accommodation, with each gite decorated internally and externally to a very high standard, each gite providing our guests with all the necessary high quality furnishings and fittings. A superb welcome pack awaits you on arrival to comfort you after your long journey.
Two of our holiday gites have sweeping views of the wonderful river Vilaine valley, viewed from the garden areas whilst enjoying the swimming pool or just relaxing in the sunshine.Whilst the new Cider Barn cotaages have rural views over sweeping fields with a glimpse of the River Vilaine. Each gite is ideally situated for walking, cycling, fishing, and exploring the beautiful Morbihan countryside in South Brittany, or for pure relaxation around the swimming pool and al fresco dining.
Our self catering holiday accommodation is situated in the commune of St Dolay, just 15/20mins from the lively town of Redon with its many bars and restaurants, highly recommended O'Shannons for a pub atmosphere, and La Roche Bernard market town with its beautiful harbour with plenty of places to eat and drink. Highly recommended is Sara B's for wonderful fish and chips, La Douannerie for outside dining and real 'pub grub' and Yackams, a livelier atmosphere and great food. We are ideally placed for visiting the wonderful coastline of Southern Brittany and the many historical towns and tourist attractions which South Brittany has to offer.
We are ideally located for golfing breaks, nearby are the beautiful courses of La Baule and Caden and of course the wonderful Chateau de Bretesche. Each of these offering competetive courses within beautiful surroundings with bars and restaurants for you to enjoy.
This part of Southern Brittany has wonderful countryside and coastlines with so much to offer for an enjoyable and memorable gite holiday.
We hope you enjoy browsing our site and we hope to welcome you sometime in the future.
Le Grippé, St Dolay 56130, Brittany, France. | {
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Neufahrn in Niederbayern este o comună din landul Bavaria, Germania.
Comune din Bavaria | {
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There were quite a few good Blu-ray releases this week but I had to narrow it down to just a few must-haves for your hi-def collection. Yeah I know you want to replace your entire DVD collection with Blu-rays but look how long it took to build up your DVD collection. Get the good ones first.
Many found this film offensive but The Passion of the Christ is a must-have for your hi-def collection. Careful with the kids, this is a film that contains plenty of graphic violence. It's rated R by the MPAA and coincidentally the highest grossing R-rated film the United States (followed by The Matrix Reloaded).
Directed by Mel Gibson, who also co-produced the movie, The Passion of the Christ was nominated for three academy awards including Best Original Score, Best Cinematography, and Best Makeup.
The Blu-ray release comes in two-discs at 1080p resolution/2.35:1 aspect ratio with audio in Aramaic/Latin/Hebrew (this is a non-English film) coded in DTS-HD Master Audio 5.1.
One of Quentin Tarantino's early masterpieces Pulp Fiction is a tale of irony and violence structured in a non-linear narrative. The film won an Academy Award for Best Original Screenplay and was nominated for another six including Best Picture, Best Director and Best Actor.
Pulp Fiction is loaded with stars including Uma Thurman, John Travolta, Christopher Walken, Samuel L. Jackson, and Rosanna Arquette just to name a few.
Video resolution is 1080p with a 2.39:1 aspect ratio. You may notice this import is PAL formatted however the Blu-ray disc is region free and has no problem playing on BD decks or PS3 consoles.
Finally, no Blu-ray Disc collection would be complete without the Academy Award winning Gandhi. The film won eight Academy Awards in 1982 including Best Picture, Best Actor and Best Director.
The biographical film tells the tail of Mohandas Karamchand Gandhi, the spiritual and political leader of India during the first half of the last century who was assassinated in 1948. This is a film that will show off your home theater, with Academy Award winning Cinematography and nominations for Best Sound and Best Film Score.
Video is coded in MPEG-4 AVC with a resolution of 1080p and 2.40:1 aspect ratio. Audio is formatted in Dolby TrueHD 5.1. The Gandhi Blu-ray disc comes with BD Live.
is Pulp Fiction shipping right away? I heard somewhere that it takes longer to ship. | {
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\section{Introduction}
Deep learning-based methods are providing state-of-the-art approaches for various image learning and natural language processing tasks, such as image classification \cite{krizhevsky2012imagenet, he2016deep}, object detection \cite{ren2015faster}, semantic image segmentation \cite{ronneberger2015u}, image synthesis \cite{goodfellow2014generative}, language translation / understanding \cite{hochreiter1997long, young2018recent} and speech synthesis \cite{van2016wavenet}. However, an artifact of many of these models is that regularity priors are hidden in their fundamental neural building blocks, which makes it impossible to apply them directly to irregular data domains. For instance, image convolutional neural networks (CNNs) are based on parametrized 2D convolutional filters with local support, while recurrent neural networks share model parameters across different time stamps. Both architectures share parameters in a way that exploits the symmetries of the underlying data domains.
In order to port deep learners to novel domains, the according parameter sharing schemes reflecting the symmetries in the target data have to be developed \cite{pmlr-v70-ravanbakhsh17a}.
An example are neural architectures for graph data, i.e., data indexed by the vertices of a graph. Graph CNNs define graph convolutional layers by utilizing results from algebraic graph theory for the graph Laplacian \cite{shuman2012emerging, gcn_bruna} and message passing neural networks \cite{scarselli2009graph, mpnn} generalize recurrent neural architectures from chain graphs to general graphs. With these building blocks in place, neural architectures for supervised \cite{gcn_defferrard, mpnn, neurosat}, semi-supervised \cite{gcn_kipf} and generative learning \cite{simonovsky2018graphvae, wang2018graphgan} on graphs have been deployed. These research endeavors fall under the umbrella term of geometric deep learning (GDL) \cite{gdl}.
In this work, we want to open the door for deep learning on set functions, i.e., data indexed by the powerset of a finite set. There are (at least) three ways to do so. First, set functions can be viewed as data indexed by a hypercube graph, which makes graph neural nets applicable.
Second, results from the Fourier analysis of set functions based on the Walsh-Hadamard-transform (WHT) \cite{stobbe, de2008brief, o2014analysis} can be utilized to formulate a convolution for set functions in a similar way to \cite{shuman2012emerging}. Third, \cite{setfctasp} introduces several novel notions of convolution for set functions (powerset convolution) as linear, equivariant functions for different notions of shift on set functions. This derivation parallels the standard 2D-convolution (equivariant to translations) and graph convolutions (equivariant to the Laplacian or adjacency shift) \cite{gsp_overview}. A general theory for deriving new forms of convolutions, associated Fourier transforms and other signal processing tools is outlined in \cite{Pueschel:08a}.
\paragraph{Contributions}
Motivated by the work on generalized convolutions and by the potential utility of deep learning on novel domains, we propose a method-driven approach for deep learning on irregular data domains and, in particular, set functions:
\begin{itemize}
\item We formulate novel powerset CNN architectures by integrating recent convolutions \cite{setfctasp} and proposing novel pooling layers for set functions.
\item As a protoypical application, we consider the set function classification task. Since there is little prior work in this area, we evaluate our powerset CNNs on three synthetic classification tasks (submodularity and spectral properties) and two classification tasks on data derived from real-world hypergraphs \citep{benson2018simplicial}. For the latter, we design classifiers to identify the origin of the extracted subhypergraph. To deal with hypergraph data, we introduce several set-function-based hypergraph representations.
\item We validate our architectures experimentally, and show that they generally outperform the natural fully-connected and graph-convolutional baselines on a range of scenarios and hyperparameter values.
\end{itemize}
\section{Convolutions on Set Functions}
We introduce background and definitions for set functions and associated convolutions. For context and analogy, we first briefly review prior convolutions for 2D and graph data.
From the signal processing perspective, 2D convolutions are linear, shift-invariant (or equivariant) functions on images $s: \mathbb{Z}^2 \to \mathbb{R}; (i, j) \mapsto s_{i,j}$, where the shifts are the translations $T_{(k, l)} s = (s_{i-k, j-l})_{i, j \in \mathbb{Z}^2}$. The 2D convolution thus becomes
\begin{equation}
(h * s)_{i, j} = \sum_{k, l \in \mathbb{Z}^2} h_{k, l} s_{i - k, j - l}.
\end{equation}
Equivariance means that all convolutions commute with all shifts: $h * (T_{(k,l)}s) = T_{(k,l)}(h*s)$.
Convolutions on vertex-indexed graph signals $s: V \to \mathbb{R}; v \mapsto s_v$ are linear and equivariant with respect to the Laplacian shifts $T_{k} s = L^k s$, where $L$ is the graph Laplacian \cite{shuman2012emerging}.
\paragraph{Set functions}
With this intuition in place, we now consider set functions. We fix a finite set $N = \{x_1, \dots, x_n\}$. An associated set function is a signal on the powerset of $N$:
\begin{equation}
s: 2^N \to \mathbb{R}; A \mapsto s_A.
\end{equation}
\paragraph{Powerset convolution} A convolution for set functions is obtained by specifying the shifts to which it is equivariant. The work in \cite{setfctasp} specifies $T_Q s = (s_{A \setminus Q})_{A \subseteq N}$ as one possible choice of shifts for $Q \subseteq N$. Note that in this case the shift operators are parametrized by the monoid $(2^N, \cup)$, since for all $s$
$$
T_Q(T_R s) = (s_{A\setminus R\setminus Q})_{A\subseteq N} =
(s_{A\setminus(R \cup Q)})_{A\subseteq N} = T_{Q\cup R}s,
$$
which implies $T_QT_R = T_{Q\cup R}$. The corresponding linear, shift-equivariant \emph{powerset convolution} is given by \cite{setfctasp} as
\begin{equation}\label{eq:setconv}
(h * s)_A = \sum_{Q \subseteq N} h_Q s_{A \setminus Q}.
\end{equation}
Note that the filter $h$ is itself a set function. Table~\ref{tab:convolutions} contains an overview of generalized convolutions and the associated shift operations to which they are equivariant to.
\paragraph{Fourier transform} Each filter $h$ defines a linear operator $\Phi_h := (h * \cdot)$ obtained by fixing $h$ in \eqref{eq:setconv}. It is diagonalized by the powerset Fourier transform
\begin{equation}
F = \begin{pmatrix}
1 & \phantom{-}0 \\
1 & -1 \\
\end{pmatrix}^{\otimes n} =
\begin{pmatrix}
1 & \phantom{-}0 \\
1 & -1 \\
\end{pmatrix} \otimes \cdots \otimes
\begin{pmatrix}
1 & \phantom{-}0 \\
1 & -1 \\
\end{pmatrix},
\end{equation}
where $\otimes$ denotes the Kronecker product. Note that $F^{-1} = F$ in this case and that the spectrum is also indexed by subsets $B \subseteq N$. In particular, we have
\begin{equation}\label{eq:fr}
F \Phi_h F^{-1} = \mbox{diag}((\tilde h_B)_{B \subseteq N}),
\end{equation}
in which $\tilde h$ denotes the frequency response of the filter $h$ \cite{setfctasp}. We denote the linear mapping from $h$ to its frequency response $\tilde h$ by $\bar{F}$, i.e., $\tilde h = \bar{F} h$.
\paragraph{Other shifts and convolutions}
There are several other possible definitions of set shifts, each coming with its respective convolutions and Fourier transforms \cite{setfctasp}. Two additional examples are $T^{\diamond}_Q s = (s_{A \cup Q})_{A \subseteq N}$ and the symmetric difference $T^{\bullet}_Q s = (s_{(A \setminus Q) \cup (Q \setminus A)})_{A \subseteq N}$ \cite{stobbe}. The associated convolutions are, respectively,
\begin{equation}\label{eq:setconv2}
(h * s)_A = \sum_{Q \subseteq N} h_Q s_{A \cup Q} \text{\hspace{0.3cm} and \hspace{0.3cm}}
(h * s)_A = \sum_{Q \subseteq N} h_Q s_{(A \setminus Q)\cup (Q\setminus A)}.
\end{equation}
\paragraph{Localized filters} Filters $h$ with $h_Q = 0$ for $|Q| > k$ are $k$-localized in the sense that the evaluation of $(h * s)_A$ only depends on evaluations of $s$ on sets differing by at most $k$ elements from $A$. In particular, $1$-localized filters $(h * s)_A = h_{\emptyset} s_A + \sum_{x \in N} h_{\{x\}} s_{A \setminus \{x\}}$ are the counterpart of \emph{one-hop} filters that are typically used in graph CNNs \cite{gcn_kipf}. In contrast to the omnidirectional one-hop graph filters, these one-hop filters have one direction per element in $N$.
\subsection{Applications of Set Functions}
Set functions are of practical importance across a range of research fields. Several optimization tasks, such as cost effective sensor placement \cite{sm:selection}, optimal ad placement \cite{sm:online_max_constraint} and tasks such as semantic image segmentation \cite{sm:sis}, can be reduced to subset selection tasks, in which a set function determines the value of every subset and has to be maximized to find the best one. In combinatorial auctions, set functions can be used to describe bidding behavior. Each bidder is represented as a valuation function that maps each subset of goods to its subjective value to the customer \cite{ca}. Cooperative games are set functions \cite{cgt}. A coalition is a subset of players and a coalition game assigns a value to every subset of players. In the simplest case the value one is assigned to winning and the value zero to losing coalitions. Further, graphs and hypergraphs also admit set function representations:
\begin{definition} \emph{(Hypergraph)}
A hypergraph is a triple $H = (V, E, w)$, where $V = \{v_1, \dots, v_n\}$ is a set of vertices, $E \subseteq (\mathcal{P}(V) \setminus \emptyset)$ is a set of hyperedges and $w: E \to \mathbb{R}$ is a weight function.
\end{definition}
The weight function of a hypergraph is a set function on $V$ by setting $s_A = w_A$ if $A \in E$ and $s_A = 0$ otherwise. Additionally, hypergraphs induce two set functions, namely the hypergraph cut and association score function:
\begin{equation}
\mbox{cut}_A = \sum_{\substack{B \in E, B \cap A \neq \emptyset,\\ B \cap (V \setminus A) \neq \emptyset}} w_B \text{\hspace{0.3cm} and \hspace{0.3cm}} \mbox{assoc}_A = \sum_{B \in E, B \subseteq A} w_B.
\end{equation}
\begin{table}
\footnotesize
\centering
\begin{tabular}{@{}llllll@{}}\toprule
& signal & shifted signal & convolution & reference & CNN \\ \midrule
image & $(s_{i,j})_{i,j}$ & $(s_{i - k,j - l})_{i,j \in \mathbb{Z}}$ & $\sum_{k, l} h_{k,l} s_{i - k, j - l}$ & standard & standard \\
graph Laplacian & $(s_v)_{v \in V}$ & $(L^k s)_{v \in V}$ & $(\sum_{k} h_k L^k s)_v$ & \cite{shuman2012emerging} & \cite{gcn_bruna}\\
graph adjacency & $(s_v)_{v \in V}$ & $(A^k s)_{v \in V}$ & $(\sum_{k} h_k A^k s)_v$ & \cite{gsp} & \cite{such2017robust} \\
group & $(s_g)_{g \in G}$ & $(s_{q^{-1}g})_{g \in G}$ & $\sum_{q} h_q s_{q^{-1}g}$ & \cite{stankovic2005fourier} & \cite{group_inv_cnn}\\
group spherical & $(s_{R})_{R \in \mbox{SO}(3)}$ & $(s_{Q^{-1}R})_{R \in \mbox{SO}(3)}$ & $\int h_Q s_{Q^{-1}R} d\mu(Q)$ & \cite{deepsphere} & \cite{deepsphere}\\
powerset & $(s_A)_{A \subseteq N}$ & $(s_{A\setminus Q})_{A \subseteq N}$ & $\sum_{Q} h_Q s_{A\setminus Q}$ & \cite{setfctasp} & this paper\\ \bottomrule
\\
\end{tabular}
\caption{Generalized convolutions and their shifts.} \label{tab:convolutions}
\end{table}
\subsection{Convolutional Pattern Matching}
The powerset convolution in \eqref{eq:setconv} raises the question of which patterns are ``detected'' by a filter $(h_Q)_{Q \subseteq N}$. In other words, to which signal does the filter $h$ respond strongest when evaluated at a given subset $A$? We call this signal $p^A$ (the pattern matched at position $A$). Formally,
\begin{equation}
p^{A} = \argmax_{s: \|s\| = 1} (h * s)_A.
\end{equation}
For $p^{N}$, the answer is $p^{N} = (1/\|h\|)(h_{N \setminus B})_{B \subseteq N}$. This is because the dot product $\langle h, s^*\rangle$, with ${s^*_A = s_{N \setminus A}}$, is maximal if $h$ and $s^*$ are aligned. Slightly rewriting \eqref{eq:setconv} yields the answer for the general case $A \subseteq N$:
\begin{equation}\label{eq:conv_pattern}
(h * s)_A = \sum_{Q \subseteq N} h_Q s_{A \setminus Q} = \sum_{Q_1 \subseteq A} \underbrace{\left(\sum_{Q_2 \subseteq N\setminus A} h_{Q_1 \cup Q_2}\right)}_{=: h'_{Q_1}} s_{A \setminus Q_1}.
\end{equation}
Namely, \eqref{eq:conv_pattern} shows that the powerset convolution evaluated at position $A$ can be seen as the convolution of a new filter $h'$ with $s$ restricted to the powerset $2^A$ evaluated at position $A$, the case for which we know the answer: $p^A_B = (1/\|h'\|) h'_{A \setminus B}$ if $B \subseteq A$ and $p^A_B = 0$ otherwise.
\begin{example}\emph{(One-hop patterns)} For a one-hop filter $h$, i.e., $(h * s)_A = h_{\emptyset} s_A + \sum_{x \in N} h_{\{x\}} s_{A \setminus \{x\}}$ the pattern matched at position $A$ takes the form
\begin{equation}
p^A_{B} = \begin{cases}
\frac{1}{\|h'\|} (h_{\emptyset} + \sum_{x \in N \setminus A} h_{\{x\}}) & \text{if } B = A, \\
\frac{1}{\|h'\|} h_{\{x\}} & \text{if } B = A \setminus \{x\} \text{ with } x \in A, \\
0 & \text{else.}
\end{cases}
\end{equation}
Here, $h'$ corresponds to the filter restricted to the powerset $2^A$ as in \eqref{eq:conv_pattern}.
\end{example}
Notice that this behavior is different from 1D and 2D convolutions: there the underlying shifts (translations) are invertible and thus the detected patterns are again shifted versions of each other. For example, the 1D convolutional filter $(h_q)_{q \in \mathbb{Z}}$ matches $p^0 = (h_{-q})_{q \in \mathbb{Z}}$ at position $0$ and $p^t = T_{-t} p^0 = (h_{-q + t})_{q \in \mathbb{Z}}$ at position $t$, and, the group convolutional filter $(h_q)_{q \in G}$ matches $p^e = (h_{q^{-1}})_{q \in G}$ at the unit element $e$ and $p^g = T_{g^{-1}} p^e = (h_{gq^{-1}})_{q \in G}$ at position $g$. Since powerset shifts are not invertible, the detected patterns by a filter are not just (set-)shifted versions of each other as shown above.
A similar behavior can be expected with graph convolutions since the Laplacian shift is never invertible and the adjacency shift is not always invertible.
\section{Powerset Convolutional Neural Networks}
\paragraph{Convolutional layers} We define a convolutional layer by extending the convolution to multiple channels, summing up the feature maps obtained by channel-wise convolution as in \cite{gdl}:
\begin{definition}\label{def:sslayer} \emph{(Powerset convolutional layer)} A powerset convolutional layer is defined as follows:
\begin{enumerate}
\item The input is given by $n_c$ set functions $\mathbf{s} = (s^{(1)}, \dots, s^{(n_c)}) \in \mathbb{R}^{2^N \times n_c}$ ;
\item The output is given by $n_f$ set functions $\mathbf{t} = L_{\Gamma}(\mathbf s) = (t^{(1)}, \dots, t^{(n_f)}) \in \mathbb{R}^{2^N \times n_f}$;
\item The layer applies a bank of set function filters $\Gamma = (h^{(i,j)})_{i, j}$, with $i \in \{1, \dots, n_c\}$ and $j \in \{1, \dots, n_f\}$, and a point-wise non-linearity $\sigma$ resulting in
\begin{equation}\label{eq:sslayer}
t^{(j)}_A = \sigma (\sum_{i = 1}^{n_c} (h^{(i, j)} * s^{(i)})_A).
\end{equation}
\end{enumerate}
\end{definition}
\paragraph{Pooling layers} As in conventional CNNs, we define powerset pooling layers to gain additional robustness with respect to input perturbations, and to control the number of features extracted by the convolutional part of the powerset CNN.
From a signal processing perspective, the crucial aspect of the pooling operation is that the pooled signal lives on a valid signal domain, i.e., a powerset.
One way to achieve this is by combining elements of the ground set.
\begin{definition}\emph{(Powerset pooling)}
Let $N'(X)$ be the ground set of size $n - |X| + 1$ obtained by combining all the elements in $X \subseteq N$ into a single element. E.g., for $X = \{x_1, x_2\}$ we get $N'(X) = \{\{x_1, x_2\}, x_3, \dots, x_n \}$. Therefore every subset $X \subseteq N$ defines a pooling operation
\begin{equation}
P^{X}: \mathbb{R}^{2^N} \to \mathbb{R}^{2^{N'(X)}}: (s_A)_{A \subseteq N} \mapsto (s_B)_{B: B \cap X = X\text{ or }B \cap X = \emptyset}.
\end{equation}
\end{definition}
In our experiments we always use $P := P^{\{x_1, x_2\}}$. It is also possible to pool a set function by combining elements of the powerset as in \cite{scheibler2015fast} or by the simple rule $s_B = \max(s_B, s_{B \cup \{x\}})$ for $B \subseteq N \setminus \{x\}$.
Then, a pooling layer is obtained by applying our pooling strategy to every channel.
\begin{definition} \emph{(Powerset pooling layer)} A powerset pooling layer takes $n_c$ set functions as input ${\mathbf{s} = (s^{(1)}, \dots, s^{(n_c)}) \in \mathbb{R}^{2^N \times n_c}}$ and outputs $n_c$ pooled set functions $\mathbf{t} = L_{P}(\mathbf s) = (t^{(1)}, \dots, t^{(n_c)}) \in \mathbb{R}^{2^{N'} \times n_c}$, with $|N'| = |N| - 1$, by applying the pooling operation to every channel
\begin{equation}\label{eq:splayer}
t^{(i)} = P(s^{(i)}).
\end{equation}
\end{definition}
\paragraph{Powerset CNN} A powerset CNN is a composition of several powerset convolutional and pooling layers. Depending on the task, the outputs of the convolutional component can be fed into a multi-layer perceptron, e.g., for classification.
\begin{figure}
\centering
\includegraphics[scale=0.22]{figures/powersetcnn.eps}
\caption{Forward pass of a simple powerset CNN with two convolutional and two pooling layers. Set functions are depicted as signals on the powerset lattice.}
\label{fig:powersetCNN}
\end{figure}
Fig.~\ref{fig:powersetCNN} illustrates a forward pass of a powerset CNN with two convolutional layers, each of which is followed by a pooling layer. The first convolutional layer is parameterized by three one-hop filters and the second one is parameterized by fifteen (three times five) one-hop filters. The filter coefficients were initialized with random weights for this illustration.
\paragraph{Implementation}\footnote{Sample implementations are provided at \url{https://github.com/chrislybaer/Powerset-CNN}.} We implemented the powerset convolutional and pooling layers in Tensorflow~\cite{tensorflow}. Our implementation supports various definitions of powerset shifts, and utilizes the respective Fourier transforms to compute the convolutions in the frequency domain.
\section{Experimental Evaluation}
Our powerset CNN is built on the premise that the successful components of conventional CNNs are domain independent and only rely on the underlying concepts of shift and shift-equivariant convolutions. In particular, if we use only one-hop filters, our powerset CNN satisfies locality and compositionality. Thus, similar to image CNNs, it should be able to learn localized hierarchical features. To understand whether this is useful when applied to set function classification problems, we evaluate our powerset CNN architectures on three synthetic tasks and on two tasks based on real-world hypergraph data.
\paragraph{Problem formulation} Intuitively, our set function classification task will require the models to learn to classify a collection of set functions sampled from some natural distributions. One such example would be to classify (hyper-)graphs coming from some underlying data distributions. Formally, the set function classification problem is characterized by a training set $\{(s^{(i)}, t^{(i)})\}_{i = 1}^m \subseteq (\mathbb{R}^{2^N} \times \mathcal{C})$ composed of pairs (set function, label), as well as a test set. The learning task is to utilize the training set to learn a mapping from the space of set functions $\mathbb{R}^{2^N}$ to the label space $\mathcal{C} = \{1, \dots, k\}$.
\subsection{Synthetic Datasets}
Unless stated otherwise, we consider the ground set $N = \{x_1, \dots, x_{n}\}$ with $n = 10$, and sample $10,000$ set functions per class. We use $80\%$ of the samples for training, and the remaining $20\%$ for testing. We only use one random split per dataset. Given this, we generated the following three synthetic datasets, meant to illustrate specific applications of our framework.
\paragraph{Spectral patterns} In order to obtain non-trivial classes of set functions, we define a sampling procedure based on the Fourier expansion associated with the shift $T_Q s = (s_{A \setminus Q})_{A \subseteq N}$. In particular, we sample Fourier sparse set functions, $s = F^{-1} \hat{s}$ with $\hat{s}$ sparse. We implement this by associating each target ``class'' with a collection of frequencies, and sample normally distributed Fourier coefficients for these frequencies. In our example, we defined four classes, where the Fourier support of the first and second class is obtained by randomly selecting roughly half of the frequencies. For the third class we use the entire spectrum, while for the fourth we use the frequencies that are either in both of class one's and class two's Fourier support, or in neither of them.
\paragraph{\boldmath$k$-junta classification} A $k$-junta \cite{o2014analysis} is a boolean function defined on $n$ variables $x_1, \dots, x_n$ that only depends on $k$ of the variables: $x_{i_1}, \dots, x_{i_k}$. In the same spirit, we call a set function a $k$-junta if its evaluations only depend on the presence or absence of $k$ of the $n$ elements of the ground set:
\begin{definition}\emph{($k$-junta)} A set function $s$ on the ground set $N$ is called a $k$-junta if there exists a subset $N' \subseteq N$, with $|N'| = k$, such that $s(A) = s(A \cap N')$, for all $A \subseteq N$.
\end{definition}
We generate a $k$-junta classification dataset by sampling random $k$-juntas for $k \in \{3, \dots, 7\}$. We do so by utilizing the fact that shifting a set function by $\{x\}$ eliminates its dependency on $x$, i.e., for $A$ with $x \in A$ we have $(T_{\{x\}} s)_A = s_{A \setminus \{x\}} = (T_{\{x\}} s)_{A \setminus \{x\}}$ because $(A \setminus \{x\}) \setminus \{x\} = A \setminus \{x\}$. Therefore, sampling a random $k$-junta amounts to first sampling a random value for every subset $A \subseteq N$ and performing $n - k$ set shifts by randomly selected singleton sets.
\paragraph{Submodularity classification} A set function $s$ is submodular if it satisfies the diminishing returns property
\begin{equation}
\forall A, B \subseteq N\text{ with }A \subseteq B\text{ and }\forall x \in N \setminus B: s_{A \cup \{x\}} - s_A \geq s_{B \cup \{x\}} - s_B.
\end{equation}
In words, adding an element to a small subset increases the value of the set function at least as much as adding it to a larger subset. We construct a dataset comprised of submodular and "almost submodular" set functions. As examples of submodular functions we utilize coverage functions \cite{krause2014submodular} (a subclass of submodular functions that allows for easy random generation). As examples of what we informally call "almost submodular" set functions here, we sample coverage functions and perturb them slightly to destroy the coverage property.
\subsection{Real Datasets}
Finally, we construct two classification tasks based on real hypergraph data. Reference~\cite{benson2018simplicial} provides 19 real-world hypergraph datasets. Each dataset is a hypergraph evolving over time. An example is the DBLP coauthorship hypergraph in which vertices are authors and hyperedges are publications. In the following, we consider classification problems on subhypergraphs induced by vertex subsets of size ten. Each hypergraph is represented by its weight set function $s_A = 1$ if $A \in E$ and $s_A = 0$ otherwise
\begin{definition} \emph{(Induced Subhypergraph \cite{berge1973graphs})}
Let $H = (V, E)$ be a hypergraph. The subset of vertices $V' \subseteq V$ induces a subhypergraph $H' = (V', E')$ with $E' = \{A \cap V': \mbox{ for } A \in E \text{ and } A \cap V' \neq \emptyset\}$
\end{definition}
\paragraph{Domain classification} As we have multiple hypergraphs, an interesting question is whether it is possible to identify from which hypergraph a given subhypergraph of size ten was sampled, i.e., whether it is possible to distinguish the hypergraphs by considering only local interactions. Therefore, among the publicly available hypergraphs in~\cite{benson2018simplicial} we only consider those containing at least 500 hyperedges of cardinality ten (namely, \emph{DAWN}: 1159, \emph{threads-stack-overflow}: 3070, \emph{coauth-DBLP}: 6599, \emph{coauth-MAG-History}: 1057, \emph{coauth-MAG-Geology}: 7704, \emph{congress-bills}: 2952). The \emph{coauth-} hypergraphs are coauthorship hypergraphs, in \emph{DAWN} the vertices are drugs and the hyperedges patients, in \emph{threads-stack-overflow} the vertices are users and the hyperedges questions on threads on \url{stackoverflow.com} and in \emph{congress-bills} the vertices are congresspersons and the hyperedges cosponsored bills. From those hypergraphs we sample all the subhypergraphs induced by the hyperedges of size ten and assign the respective hypergraph of origin as class label. In addition to this dataset (\emph{DOM6}), we create an easier version (\emph{DOM4}) in which we only keep one of the coauthorship hypergraphs, namely \emph{coauth-DBLP}.
\paragraph{Simplicial closure} Reference~\cite{benson2018simplicial} distinguishes between open and closed hyperedges (the latter are called simplices). A hyperedge is called open if its vertices in the 2-section (the graph obtained by making the vertices of every hyperedge a clique) of the hypergraph form a clique and it is not contained in any hyperedge in the hypergraph. On the other hand, a hyperedge is closed if it is contained in one or is one of the hyperedges of the hypergraph. We consider the following classification problem: For a given subhypergraph of ten vertices, determine whether its vertices form a closed hyperedge in the original hypergraph or not.
In order to obtain examples for closed hyperedges, we sample the subhypergraphs induced by the vertices of hyperedges of size ten and for open hyperedges we sample subhypergraphs induced by vertices of hyperedges of size nine extended by an additional vertex. In this way we construct two learning tasks. First, \emph{CON10} in which we extend the nine-hyperedge by choosing the additional vertex such that the resulting hyperedge is open (2952 closed and 4000 open examples). Second, \emph{COAUTH10} in which we randomly extend the size nine hyperedges (as many as there are closed ones) and use \emph{coauth-DBLP} for training and \emph{coauth-MAG-History} \& \emph{coauth-MAG-Geology} for testing.
\subsection{Experimental Setup}
\paragraph{Baselines} As baselines we consider a multi-layer perceptron (MLP) \cite{rosenblatt1961principles} with two hidden layers of size 4096 and an appropriately chosen last layer and graph CNNs (GCNs) on the undirected $n$-dimensional hypercube. Every vertex of the hypercube corresponds to a subset and vertices are connected by an edge if their subsets only differ by one element. We evaluate graph convolutional layers based on the Laplacian shift \cite{gcn_kipf} and based on the adjacency shift \cite{gsp}. In both cases one layer does at most one hop.
\paragraph{Our models} For our powerset CNNs (PCNs) we consider convolutional layers based on one-hop filters of two different convolutions: $(h * s)_A = h_{\emptyset} s_A + \sum_{x \in N} h_{\{x\}} s_{A \setminus \{x\}}$ and ${(h \diamond s)_A = h_{\emptyset} s_A + \sum_{x \in N} h_{\{x\}} s_{A \cup \{x\}}}$. For all types of convolutional layers we consider the following models: three convolutional layers followed by an MLP with one hidden layer of size 512 as illustrated before, a pooling layer after each convolutional layer followed by the MLP, and a pooling layer after each convolutional layer followed by an accumulation step (average of the features over all subsets) as in \cite{mpnn} followed by the MLP. For all models we use 32 output channels per convolutional layer and ReLU \cite{nair2010rectified} non-linearities.
\paragraph{Training} We train all models for 100 epochs (passes through the training data) using the Adam optimizer \cite{kingma2014adam} with initial learning rate 0.001 and an exponential learning rate decay factor of 0.95. The learning rate decays after every epoch. We use batches of size 128 and the cross entropy loss. All our experiments were run on a server with an Intel(R) Xeon(R) CPU @ 2.00GHz with four NVIDIA Tesla T4 GPUs. Mean and standard deviation are obtained by running each experiment 20 times.
\subsection{Results}
Our results are summarized in Table \ref{tab:results}. We report the test classification accuracy in percentages (for models that converged).
\begin{table}
\centering
{\scriptsize
\resizebox{\textwidth}{!}{
\begin{tabular}{@{}lrrr|rrrrrr@{}}\toprule
& {Patterns} & $k$-{Junta} & {Submod.} & {COAUTH10} & {CON10} & {DOM4} & {DOM6}\\\toprule
Baselines \\
\hspace{.3cm}MLP & $46.8 \pm 3.9$ & $43.2 \pm 2.5$ & - & $80.7 \pm 0.2$ & $66.1 \pm 1.8$ & $93.6 \pm 0.2$ & $71.1 \pm 0.3$\\
\hspace{.3cm}$L$-GCN & $52.5 \pm 0.9$ & $69.3 \pm 2.8$ & - & \boldmath$84.7 \pm 0.9$ & \boldmath$67.2 \pm 1.8$ & \boldmath$96.0 \pm 0.2$ & \boldmath$73.7 \pm 0.4$\\
\hspace{.3cm}$L$-GCN pool & $45.0 \pm 1.0$ & $60.9 \pm 1.5$ & - & $83.2 \pm 0.7$ & $65.7 \pm 1.0$ & $93.2 \pm 1.1$ & $71.7 \pm 0.5$\\
\hspace{.3cm}$L$-GCN pool avg. & $42.1 \pm 0.3$ & $64.3 \pm 2.2$ & $82.2 \pm 0.4$ & $56.8 \pm 1.1$ & $64.1 \pm 1.7$ & $88.4 \pm 0.3$ & $62.8 \pm 0.4$\\
\hspace{.3cm}$A$-GCN & \boldmath$65.5 \pm 0.9$ & $95.8 \pm 2.7$ & - & $80.5 \pm 0.7$ & $64.9 \pm 1.8$ & $93.9 \pm 0.3$ & $69.1 \pm 0.5$\\
\hspace{.3cm}$A$-GCN pool & $56.9 \pm 2.2$ & $91.9 \pm 2.1$ & \boldmath$89.8 \pm 1.8$ & $84.1 \pm 0.6$ & $66.0 \pm 1.6$ & $93.8 \pm 0.3$ & $70.7 \pm 0.4$\\
\hspace{.3cm}$A$-GCN pool avg. & $54.8 \pm 0.9$ & \boldmath$95.8 \pm 1.1$ & $84.8 \pm 1.9$ & $64.8 \pm 1.1$ & $65.4 \pm 0.7$ & $92.7 \pm 0.6$ & $67.9 \pm 0.3$\\
Proposed models & & & & & & \\
\hspace{.3cm}$*$-PCN & \boldmath$88.5 \pm 4.3$ & $97.2 \pm 2.3$ & \boldmath$88.6 \pm 0.4$ & $80.6 \pm 0.7$ & $62.8 \pm 2.9$ & $94.1 \pm 0.3$ & $70.5 \pm 0.3$\\
\hspace{.3cm}$*$-PCN pool & $80.9 \pm 0.9$ & $96.0 \pm 1.6$ & $85.1 \pm 1.8$ & $82.6 \pm 0.4$ & $62.9 \pm 2.0$ & $94.0 \pm 0.3$ & $70.2 \pm 0.5$\\
\hspace{.3cm}$*$-PCN pool avg. & $75.9 \pm 1.9$ & $96.5 \pm 0.6$ & $87.0 \pm 1.6$ & $80.6 \pm 0.5$ & $63.4 \pm 3.5$ & $94.4 \pm 0.3$ & $73.0 \pm 0.3$\\
\hspace{.3cm}$\diamond$-PCN & - & \boldmath$97.5 \pm 1.4$ & - & $83.6 \pm 0.4$ & \boldmath$68.7 \pm 1.3$ & $93.7 \pm 0.2$ & $69.9 \pm 0.3$\\
\hspace{.3cm}$\diamond$-PCN pool & - & $96.4 \pm 1.7$ & - & \boldmath$84.8 \pm 0.3$ & $68.2 \pm 0.8$ & $93.6 \pm 0.3$ & $70.3 \pm 0.4$\\
\hspace{.3cm}$\diamond$-PCN pool avg. & $54.8 \pm 1.9$ & $96.6 \pm 0.7$ & $80.9 \pm 2.9$ & $83.3 \pm 0.5$ & $67.0 \pm 2.0$ & \boldmath$94.8 \pm 0.3$ & \boldmath$73.5 \pm 0.5$\\
\bottomrule\\
\end{tabular}
}
}
\caption{Results of the experimental evaluation in terms of test classification accuracy (percentage). The first three columns contain the results from the synthetic experiments and the last four columns the results from the hypergraph experiments. The best-performing model from the corresponding category is in bold.} \label{tab:results}
\end{table}
\paragraph{Discussion} Table \ref{tab:results} shows that in the \emph{synthetic tasks} the powerset convolutional models ($*$-PCNs) tend to outperform the baselines with the exception of $A$-GCNs, which are based on the adjacency graph shift on the undirected hypercube. In fact, the set of $A$-convolutional filters parametrized by our $A$-GCNs is the subset of the powerset convolutional filters associated with the symmetric difference shift \eqref{eq:setconv2} obtained by constraining all filter coefficients for one-element sets to be equal: $h_{\{x_i\}} = c$ with $c \in \mathbb{R}$, for all $i \in \{1, \dots, n\}$. Therefore, it is no surprise that the $A$-GCNs perform well. In contrast, the restrictions placed on the filters of $L$-GCN are stronger, since \cite{gcn_kipf} replaces the one-hop Laplacian convolution $(\theta_0 I + \theta_1 (L - I)) x$ (in Chebyshev basis) with $\theta (2I - L)x$ by setting $\theta = \theta_0 = -\theta_1$.
An analogous trend is not as clearly visible in the tasks derived from \emph{real hypergraph data}. In these tasks, the graph CNNs seem to be either more robust to noisy data, or, to benefit from their permutation equivariance properties. The robustness as well as the permutation equivariance can be attributed to the graph one-hop filters being omnidirectional. On the other hand, the powerset one-hop filters are $n$-directional. Thus, they are sensitive to hypergraph isomorphy, i.e., hypergraphs with same connectivity structure but different vertex ordering are being processed differently.
\paragraph{Pooling} Interestingly, while reducing the hidden state by a factor of two after every convolutional layer, pooling in most cases only slightly decreases the accuracy of the PCNs in the synthetic tasks and has no impact in the other tasks. Also the influence of pooling on the $A$-GCN is more similar to the behavior of PCNs than the one for the $L$-GCN.
\paragraph{Equivariance} Finally, we compare models having a shift-invariant convolutional part (suffix "pool avg.") with models having a shift-equivariant convolutional part (suffix "pool") models. The difference between these models is that the invariant ones have an accumulation step before the MLP resulting in (a) the inputs to the MLP being invariant w.r.t. the shift corresponding to the specific convolutions used and (b) the MLP having much fewer parameters in its hidden layer ($32 \cdot 512$ instead of $2^{10} \cdot 32 \cdot 512$). For the PCNs the effect of the accumulation step appears to be task dependent. For instance, in \emph{$k$-Junta}, \emph{Submod.}, \emph{DOM4} and \emph{DOM6} it is largely beneficial, and in the others it slightly disadvantageous. Similarly, for the GCNs the accumulation step is beneficial in \emph{$k$-Junta} and disadvantageous in \emph{COAUTH10}. A possible cause is that the resulting models are not expressive enough due to the lack of parameters.
\paragraph{Complexity analysis} Consider a powerset convolutional layer \eqref{eq:sslayer} with $n_c$ input channels and $n_f$ output channels. Using $k$-hop filters, the layer is parametrized by $n_p = n_f + n_c n_f \sum_{i = 0}^k {n \choose i}$ parameters ($n_f$ bias terms plus $n_c n_f \sum_{i = 0}^k {n \choose i}$ filtering coefficients). Convolution is done efficiently in the Fourier domain, i.e., $h * s = F^{-1} (\mbox{diag}(\bar{F} h) F s)$, which requires $\frac{3}{2}n 2^n + 2^n$ operations and $2^n$ floats of memory \cite{setfctasp}. Thus, forward as well as backward pass require $\Theta(n_c n_f n 2^n)$ operations and $\Theta(n_c 2^n + n_f 2^n + n_p)$ floats of memory\footnote{The derivation of these results is provided in the supplementary material.}. The hypercube graph convolutional layers are a special case of powerset convolutional layers. Hence, they are in the same complexity class. A $k$-hop graph convolutional layer requires $n_f + n_c n_f (k+1)$ parameters.
\section{Related Work}
Our work is at the intersection of geometric deep learning, generalized signal processing and set function learning. Since each of these areas is broad, due to space limitations, we will only review the work that is most closely related to ours.
\paragraph{Deep learning} Geometric deep learners \cite{gdl} can be broadly categorized into convolution-based approaches \cite{gcn_bruna, gcn_defferrard, gcn_kipf, such2017robust, deepsphere, group_inv_cnn} and message-passing-based approaches \cite{mpnn, neurosat, scarselli2009graph}. The latter assign a hidden state to each element of the index domain (e.g., to each vertex in a graph) and make use of a message passing protocol to learn representations in a finite amount of communication steps. Reference~\cite{mpnn} points out that graph CNNs are a subclass of message passing / graph neural networks (MPNNs). References~\cite{gcn_bruna, gcn_defferrard, gcn_kipf} utilize the spectral analysis of the graph Laplacian \cite{shuman2012emerging} to define graph convolutions, while \cite{such2017robust} makes use of the adjacency shift based convolution \cite{gsp}. Similarly, \cite{group_inv_cnn, deepsphere} utilize group convolutions \cite{stankovic2005fourier} with desirable equivariances.
In a similar vein, in this work we utilize the recently proposed powerset convolutions \cite{setfctasp} as the foundation of a generalized CNN. With respect to the latter reference, which provides the theoretical foundation for powerset convolutions, our contributions are an analysis of the resulting filters from a pattern matching perspective, to define its exact instantiations and applications in the context of neural networks, as well as to show that these operations are practically relevant for various tasks.
\paragraph{Signal processing} Set function signal processing \cite{setfctasp} is an instantiation of algebraic signal processing (ASP) \cite{Pueschel:08a} on the powerset domain. ASP provides a theoretical framework for deriving a complete set of basic signal processing concepts, including convolution, for novel index domains, using as starting point a chosen shift to which convolutions should be equivariant. To date the approach was used for index domains including graphs \citep{gsp, gsp_overview, gsp_fa}, powersets (set functions) \cite{setfctasp}, meet/join lattices \cite{lattice_asp, Wend1911:Sampling}, and a collection of more regular domains, e.g., \cite{Pueschel:07,Sandryhaila:12, Seifert.Hueper:2018b}.
Additionally, there are spectral approaches such as \cite{shuman2012emerging} for graphs and \cite{de2008brief, o2014analysis} for set functions (or, equivalently, pseudo-boolean functions), that utilize analogues of the Fourier transform to port spectral analysis and other signal processing methods to novel domains.
\paragraph{Set function learning} In contrast to the set function classification problems considered in this work, most of existing set function learning is concerned with completing a single partially observed set function \cite{mossel2003learning, stobbe, choi2011almost, sutton2012computing, badanidiyuru2012sketching, balcan11, balcan11sm, sm:dsf, zaheer2017deep}. In this context, traditional methods \cite{mossel2003learning, choi2011almost, sutton2012computing, badanidiyuru2012sketching, balcan11, balcan11sm} mainly differ in the way how the class of considered set functions is restricted in order to be manageable. E.g., \cite{stobbe} does this by considering Walsh-Hadamard-sparse (= Fourier sparse) set functions. Recent approaches \cite{sm:dsf, zaheer2017deep, tschiatschek2018differentiable, djolonga2017differentiable, wang2019satnet, murphy2018janossy} leverage deep learning. Reference~\cite{sm:dsf} proposes a neural architecture for learning submodular functions and \cite{zaheer2017deep, murphy2018janossy} propose architectures for learning multi-set functions (i.e., permutation-invariant sequence functions). References \cite{tschiatschek2018differentiable, djolonga2017differentiable} introduce differentiable layers that allow for backpropagation through the minimizer or maximizer of a submodular optimization problem respectively and, thus, for learning submodular set functions. Similarly, \cite{wang2019satnet} proposes a differentiable layer for learning boolean functions.
\section{Conclusion}
We introduced a convolutional neural network architecture for powerset data. We did so by utilizing novel powerset convolutions and introducing powerset pooling layers. The powerset convolutions used stem from algebraic signal processing theory \cite{Pueschel:08a}, a theoretical framework for porting signal processing to novel domains. Therefore, we hope that our method-driven approach can be used to specialize deep learning to other domains as well. We conclude with challenges and future directions.
\paragraph{Lack of data} We argue that certain success components of deep learning are domain independent and our experimental results empirically support this claim to a certain degree. However, one cannot neglect the fact that data abundance is one of these success components and, for the supervised learning problems on set functions considered in this paper, one that is currently lacking.
\paragraph{Computational complexity} As evident from our complexity analysis and \cite{lu2016practical}, the proposed methodology is feasible only up to about $n = 30$ using modern multicore systems. This is caused by the fact that set functions are exponentially large objects. If one would like to scale our approach to larger ground sets, e.g., to support semisupervised learning on graphs or hypergraphs where there is enough data available, one should either devise methods to preserve the sparsity of the respective set function representations while filtering, pooling and applying non-linear functions, or, leverage techniques for NN dimension reduction like \cite{hackel2018inference}.
\section*{Acknowledgements}
We thank Max Horn for insightful discussions and his extensive feedback, and Razvan Pascanu for feedback on an earlier draft.
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 805223).
\bibliographystyle{plainnat}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,348 |
{"url":"https:\/\/mathhelpboards.com\/threads\/gravitational-acceleration.3152\/","text":"# gravitational acceleration\n\n#### dwsmith\n\n##### Well-known member\nA satellite 300km above the earth's radius would have the same gravitational acceleration magnitude?\n$$\\mathbf{F} = \\frac{GM_{e}m_s}{r^2}\\approx \\frac{GM_e}{r^2}$$\nCorrect?\n\n#### Ackbach\n\n##### Indicium Physicus\nStaff member\nI would write that\n\n$$\\mathbf{F}_{es}=-G\\frac{M_{e}M_{s}}{|\\mathbf{r}_{es}|^{2}}\\, \\hat{\\mathbf{r}}_{es}.$$\n\nHere $\\mathbf{F}_{es}$ is the force exerted on the satellite due to the earth, and $\\mathbf{r}_{es}$ is the radius vector from the earth to the satellite. Therefore, by Newton's Second Law, we have that\n$$\\mathbf{F}_{es}=M_{s}\\mathbf{a}_{s},$$\nand hence\n$$\\mathbf{a}_{s}=-G\\frac{M_{e}}{|\\mathbf{r}_{es}|^{2}}\\, \\hat{\\mathbf{r}}_{es}.$$\n\n#### dwsmith\n\n##### Well-known member\nI would write that\n\n$$\\mathbf{F}_{es}=-G\\frac{M_{e}M_{s}}{|\\mathbf{r}_{es}|^{2}}\\, \\hat{\\mathbf{r}}_{es}.$$\n\nHere $\\mathbf{F}_{es}$ is the force exerted on the satellite due to the earth, and $\\mathbf{r}_{es}$ is the radius vector from the earth to the satellite. Therefore, by Newton's Second Law, we have that\n$$\\mathbf{F}_{es}=M_{s}\\mathbf{a}_{s},$$\nand hence\n$$\\mathbf{a}_{s}=-G\\frac{M_{e}}{|\\mathbf{r}_{es}|^{2}}\\, \\hat{\\mathbf{r}}_{es}.$$\nSo $\\mathbf{a} = -9.8m\/sec^2$?\n\n#### Ackbach\n\n##### Indicium Physicus\nStaff member\nSo $\\mathbf{a} = -9.8m\/sec^2$?\nNo. $|\\mathbf{a}_{g}|=9.8\\,\\text{m\/s}^{2}$ only near the earth's surface. That is, only where the radius vector's length is approximately the radius of the earth. A satellite is going to be much farther away from the earth's center than that. Expect this acceleration to be much lower, in accordance with the inverse square law.\n\n#### MarkFL\n\nStaff member\nI believe we would have:\n\n$\\displaystyle \\mathbf{a}\\approx-9.8\\left(\\frac{r}{r+300} \\right)^2\\frac{\\text{m}}{\\text{s}^2}$\n\nwhere $r$ is the radius of the Earth in km.\n\n#### dwsmith\n\n##### Well-known member\nHow does one account for the sun by finding the magnitude of the gravitational disturbance caused by the Sun?\n\n#### Ackbach\n\n##### Indicium Physicus\nStaff member\nHow does one account for the sun by finding the magnitude of the gravitational disturbance caused by the Sun?\nIt would vary greatly depending on the relative alignment of the earth and the sun compared to the satellite. The total gravitational force on the satellite is just the vector sum of the gravitational forces due to the earth and the sun. I think you'll find that the sun's influence is much less than the earth's despite the sun's immensely greater mass.\n\n#### MarkFL\n\nStaff member\nThe same way you would for the Earth (and find the sum of the forces), except you would have to account for the varying distance of the satellite from the center of the sun since the satellite is orbiting the Earth, and also you would have to account for the varying distance of the Earth from the Sun.\n\nWhile the size of the region around the Earth in which the Earth's gravity is dominant over that of the sun is very small relative to the solar system, it is large compared to the region in which we place artificial satellites.\n\n#### dwsmith\n\n##### Well-known member\nSo the equation would be:\n$$\\mathbf{F}_s = -\\frac{GM_sm_s}{r_s^2}\\hat{\\mathbf{r_s}}$$\nwhere the earth is located at $\\theta = \\pi$ from the sun and the satellite is at $\\beta = \\frac{3\\pi}{2}$ from the earth.\nThis force would be positive since it is moving away from earth?\n$$\\mathbf{a} = \\frac{\\mathbf{F}_{\\text{sunsat}}}{m_{sat}} = -\\frac{GM_{\\text{sun}}}{r^2_{\\text{sunsat}}}\\hat{ \\mathbf{r}}_{\\text{sunsat}}$$\nThen $r_s = \\sqrt{d_{es}^2 + 300^2}$\nCorrect?\nHow do I compare these two(earth and sun on sat) accelerations since I have nothing to compare?\n\nLast edited:\n\n#### topsquark\n\n##### Well-known member\nMHB Math Helper\nSo the equation would be:\n$$\\mathbf{F}_s = -\\frac{GM_sm_s}{r_s^2}\\hat{\\mathbf{r_s}}$$\nwhere the earth is located at $\\theta = \\pi$ from the sun and the satellite is at $\\beta = \\frac{3\\pi}{2}$ from the earth.\nThis force would be positive since it is moving away from earth?\n$$\\mathbf{a} = \\frac{\\mathbf{F}_{\\text{sunsat}}}{m_{sat}} = -\\frac{GM_{\\text{sun}}}{r^2_{\\text{sunsat}}}\\hat{ \\mathbf{r}}_{\\text{sunsat}}$$\nThen $r_s = \\sqrt{d_{es}^2 + 300^2}$\nCorrect?\nHow do I compare these two(earth and sun on sat) accelerations since I have nothing to compare?\nAre you studying this as a class assignment? If so then we need to know of any specifications that your instructor would want. (ie. does your instructor expect you to answer in terms of the angle the Earths's gravitational force on the satellite makes with that of the Sun?)\n\nIf not then I would recommend that you pick the satellite at its closest point to the Sun (The satellite is between the Earth and Sun) and at its furthest point (The Earth is between the satellite and the Sun.) I don't know the exact figures, but I'll bet that you can average these two numbers. (They should be about the same amount anyway.)\n\n-Dan\n\n#### dwsmith\n\n##### Well-known member\nAre you studying this as a class assignment? If so then we need to know of any specifications that your instructor would want. (ie. does your instructor expect you to answer in terms of the angle the Earths's gravitational force on the satellite makes with that of the Sun?)\n\nIf not then I would recommend that you pick the satellite at its closest point to the Sun (The satellite is between the Earth and Sun) and at its furthest point (The Earth is between the satellite and the Sun.) I don't know the exact figures, but I'll bet that you can average these two numbers. (They should be about the same amount anyway.)\n\n-Dan\nThe picture shows it here:\nCode:\n\\begin{center}\n\\begin{tikzpicture}\n\\draw (-4cm,0) -- (4cm,0);\n\\draw (-4cm,0) circle (1.5cm);\n\\draw (-4cm,-1.5cm) -- (4cm,-.2cm);\n\\filldraw[blue] (-4cm,0) circle (.5cm);\n\\filldraw[orange] (4cm,0) circle (1cm);\n\\draw[thick] (-4cm,0) node {Earth};\n\\draw[thick] (4cm,0) node {Sun};\n\\filldraw[gray] (-4cm,-1.5cm) circle (.1cm) node[below = 1pt] {Satellite};\n\\end{tikzpicture}\n\\end{center}\n\n#### topsquark\n\n##### Well-known member\nMHB Math Helper\nThe picture shows it here:\nCode:\n\\begin{center}\n\\begin{tikzpicture}\n\\draw (-4cm,0) -- (4cm,0);\n\\draw (-4cm,0) circle (1.5cm);\n\\draw (-4cm,-1.5cm) -- (4cm,-.2cm);\n\\filldraw[blue] (-4cm,0) circle (.5cm);\n\\filldraw[orange] (4cm,0) circle (1cm);\n\\draw[thick] (-4cm,0) node {Earth};\n\\draw[thick] (4cm,0) node {Sun};\n\\filldraw[gray] (-4cm,-1.5cm) circle (.1cm) node[below = 1pt] {Satellite};\n\\end{tikzpicture}\n\\end{center}\n(Ahem) Is there any other way you can post that? I don't even know how to run the script.\n\n-Dan\n\n#### dwsmith\n\n##### Well-known member\n(Ahem) Is there any other way you can post that? I don't even know how to run the script.\n\n-Dan\n\n#### topsquark\n\n##### Well-known member\nMHB Math Helper\nOkay so you know how to calculate the gravitational force on the satellite from the Earth.\n\n$$F = \\frac{GM_Em}{r^2}$$\n\nThe force from the Sun will be the same idea:\n\n$$F = \\frac{GM_Sm}{r^2}$$\n\nTo get the net result, both forces act in the direction of the objects, toward the Earth or toward the Sun.\n\nOne simplification if you want to go down this road: The distance the satellite is from the Earth is practically negligible so feel free to use the distance from the Earth to the Sun.\n\n-Dan\n\n#### dwsmith\n\n##### Well-known member\nOkay so you know how to calculate the gravitational force on the satellite from the Earth.\n\n$$F = \\frac{GM_Em}{r^2}$$\n\nThe force from the Sun will be the same idea:\n\n$$F = \\frac{GM_Sm}{r^2}$$\n\nTo get the net result, both forces act in the direction of the objects, toward the Earth or toward the Sun.\n\nOne simplification if you want to go down this road: The distance the satellite is from the Earth is practically negligible so feel free to use the distance from the Earth to the Sun.\n\n-Dan\nI have that but how does one compare the two accelerations since I don't have actual numbers?\n\n#### topsquark\n\n##### Well-known member\nMHB Math Helper\nI have that but how does one compare the two accelerations since I don't have actual numbers?\nI can't figure out a way to do it without putting numbers in. Sorry!\n\n-Dan\n\n#### dwsmith\n\n##### Well-known member\nI can't figure out a way to do it without putting numbers in. Sorry!\n\n-Dan\nI can never figure this out since G has so many means in my book. Is G a known constant in this equation?\n\n#### MarkFL\n\nStaff member\nYes, $G$ is Newton's universal gravitational constant, which is given by Wikipedia as:\n\n$\\displaystyle G\\approx6.674\\,\\times\\,10^{-11}\\,\\frac{\\text{N}\\cdot\\text{m}^2}{\\text{kg}^2}$\n\n#### Deveno\n\n##### Well-known member\nMHB Math Scholar\nOkay so you know how to calculate the gravitational force on the satellite from the Earth.\n\n$$F = \\frac{GM_Em}{r^2}$$\n\nThe force from the Sun will be the same idea:\n\n$$F = \\frac{GM_Sm}{r^2}$$\n\nTo get the net result, both forces act in the direction of the objects, toward the Earth or toward the Sun.\n\nOne simplification if you want to go down this road: The distance the satellite is from the Earth is practically negligible so feel free to use the distance from the Earth to the Sun.\n\n-Dan\nthose should be different $r$'s, yes?\n\n#### topsquark\n\n##### Well-known member\nMHB Math Helper\nthose should be different $r$'s, yes?\nYes. The first r will be the distance from the center of the Earth to the satellite and the second r is the distance from the Sun. (No point in worrying about the radius of the Sun, it's very small compared to the distance between the satellite to the Sun.)\n\n-Dan\n\n#### Deveno\n\n##### Well-known member\nMHB Math Scholar\nwell. correct me if i'm wrong, but doesn't this depend on where earth is in its elliptical orbit (aphelion vs. perihelion)?\n\n#### topsquark\n\n##### Well-known member\nMHB Math Helper\nwell. correct me if i'm wrong, but doesn't this depend on where earth is in its elliptical orbit (aphelion vs. perihelion)?\nYes, but given that the distance between the Sun and the satellite is so much larger than the change between aphelion and perihelion it is something that can be ignored if you wish.\n\n-Dan","date":"2020-09-18 07:38:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 2, \"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\": 0.747545063495636, \"perplexity\": 575.3494277242662}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600400187354.1\/warc\/CC-MAIN-20200918061627-20200918091627-00418.warc.gz\"}"} | null | null |
package com.wsn.chapter18javaio;
//package net.mindview.util;
import java.io.*;
import java.util.ArrayList;
import java.util.List;
public class OSExecute {
public static List<String> command(String command) {
List<String> stringList = new ArrayList<>();
boolean err = false;
try {
Process process = new ProcessBuilder(command.split(" ")).start();
BufferedReader results = new BufferedReader(new InputStreamReader(process.getInputStream()));
String s;
while ((s = results.readLine()) != null) {
// System.out.println(s);
stringList.add(s);
}
BufferedReader errors = new BufferedReader(new InputStreamReader(process.getErrorStream()));
// Report errors and return nonzero value
// to calling process if there are problems:
while ((s = errors.readLine()) != null) {
System.err.println(s);
stringList.add(s);
if (!s.contains("¾¯¸æ")) {
err = true;
}
}
} catch (Exception e) {
// Compensate for Windows 2000, which throws an
// exception for the default command line:
if (!command.startsWith("CMD /C"))
command("CMD /C " + command);
else
throw new RuntimeException(e);
}
if (err)
throw new OSExecuteException("Errors executing " + command);
return stringList;
}
} ///:~
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,856 |
package ProjectBackendTestProject1.src.main.java.org.kie.test.project.backend;
public class Bean {
private final int value;
public Bean(int value) {
this.value = value;
}
public int getValue() {
return value*7;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,888 |
\section{Introduction}
Spectra can be used to reveal the chemical composition of the stars, as well as important stellar atmospheric parameters, such as effective temperature (T$_{\mathrm{eff}}$) and [Fe/H]. These parameters are crucial for the characterization of the stars and therefore fundamental to understand their formation and evolution. Furthermore, they influence the properties of the planets forming and orbiting around them \citep{ever13}. However, the spectroscopic analysis to derive these parameters has some difficulties to overcome. One of the main problems is the correct determination of the spectral continuum, which is more problematic in cool and faint stars, such as M dwarfs. Their study is quite difficult and complicated, compared to FGK stars, since in M dwarfs, molecules are the dominant sources of opacity. These molecules create thousands of lines that are poorly known and moreover many of them blend with each other. Therefore, the position of the continuum is hardly identified in their spectra.
Methods which rely on the correct determination of the continuum, work better only for the metal poor and earliest types of M dwarfs \citep{woolf05}. Methods using spectral synthesis have not achieved as precise results as in FGK cases, because of the poor knowledge of many molecular line strengths.
Recently, spectral synthesis in the near infrared has presented advances, as shown by several studies. \citep{one12,lind16,raj18, pass19}.
Regarding these limitations, most attempts for determining effective temperature and metallicity, are done with photometric calibrations \citep{bon05,johnapps09,neves12} or spectroscopic indices \citep{rojas10,rojas12,mann13a}. Metallicity uncertainties range from 0.20 dex using photometric calibrations, to 0.10 dex by using spectroscopic scales in the infrared \citep{rojas12}. For T$_{\mathrm{eff}}$, precisions of 100 K are reported, but significant uncertainties and systematics are still present, ranging from 150 to 300 K. \citep{casag08,rojas12}.
One of the most popular methods to derive atmospheric stellar parameters for FGK stars is by measuring the equivalent widths (EW) of many metal lines of the spectrum.
\citet{neves14} using the MCAL code, measured pseudo EWs in the optical part of the spectrum for 110 M dwarfs observed in the HARPS GTO M dwarf program, by setting a pseudo continuum for each line. They proceeded to the derivation of T$_{\mathrm{eff}}$ and [Fe/H] of these stars applying a calibration based on reference photometric T$_{\mathrm{eff}}$ and [Fe/H] scales that exist for 65 of them from \citet{casag08} and \citet{neves12} respectively. In the first case, the reference T$_{\mathrm{eff}}$ is the average value of the V - J, V - H, and V - K photometric scales as seen in \citet{casag08}, while for [Fe/H] the calculation of its reference values was done using stellar parallaxes, V and Ks magnitudes as described in \citet{neves12}.
Machine learning is an increasingly popular concept in several fields of science. It can be accurate in predicting outcomes without the need of the user explicitly creating a specific model to the problem at hand. The algorithms in machine learning receive input data and by applying statistical analysis, they predict an output value within a reasonable range.
The interest for machine learning algorithms and automatic processes in astronomy is emerging from the increasing volume of survey data \citep{how17}. It can be applied to a wide range of studies, with the input attributes being for example the photometric properties of the sources \citep{dasa19,akras19,rau19,ucci19}.
In our work, we follow the pseudo-EW approach.
We present our tool ODUSSEAS (Observing Dwarfs Using Stellar Spectroscopic Energy-Absorption Shapes), which makes use of the machine learning "scikit learn" package of Python. It offers a quick automatic derivation of T$_{\mathrm{eff}}$ and [Fe/H] for M dwarf stars, by being provided with their 1D spectra and their resolutions.
The main advantage of this tool, compared to other ones that derive stellar parameters such as the MCAL code by \citet{neves14} (which is limited to HARPS range and needs manual adjustment of results for different resolutions), is that it can operate simultaneously in an automatic fashion for spectra of different resolutions and different wavelength ranges in the optical. It is based on a supervised machine learning algorithm, meaning that it is provided with both input and expected output for creating a model.
This input to the machine learning function are the values of the pseudo EWs for 65 HARPS spectra and the expected output are the values of their reference T$_{\mathrm{eff}}$ and [Fe/H] from \citet{casag08} and \citet{neves12} respectively.
After training with a part of these HARPS data, the algorithm produces a model and tests it on the rest of the HARPS data.
It predicts their values and compares them with the reference ones given as expected output. Thus, it examines the accuracy and the precision of the model by using several regression metrics described later. Finally, it applies the model to unknown spectra and estimates their stellar parameters.
In Sect.~\ref{EWsection} we describe how the tool computes the pseudo EWs.
In Sect.~\ref{mlsection} we describe our tool and the flow of its process. We explain the characteristics of the machine learning function and its efficiency regarding different regression types, resolutions and wavelength areas.
In Sect.~\ref{otherspec} we apply our tool to spectra obtained by several spectrographs of various resolutions and we examine the results.
Finally, Sect.~\ref{sum} summarizes the work presented in this paper.
\section{Pseudo-EW measurements}
\label{EWsection}
Since the identification of the continuum is very difficult in the spectra of M dwarfs, we follow the way of setting a pseudo continuum in each absorption line. The method is based on measurements of the pseudo EWs of absorption lines and blended lines in the range between 530 and 690 nm. We have excluded the parts where the activity-sensitive Na doublet and H$\alpha$ lines and strong telluric lines reside. The linelist consists of 4104 features. It is given in the form of left and right boundaries, between which these absorption features are supposed to be created. This method, based on pseudo EWs and the specific linelist, was used by \citet{neves14}.
We have created our own Python version of the method to compute the pseudo EWs.
Our code reads the linelist and the 1D fits files of the stellar spectra. We have set an option for radial-velocity correction of the input spectra by our code, in the case they are shifted. Then, for each line, it identifies the position of the minimum flux of the feature, which is the central absorption wavelength. Starting from it, the code identifies the maximum in each side of this absorption feature, after having cut this spectral area at the range defined by the respective boundaries provided in the linelist. Eventually, it fits the pseudo continuum along the edges of the absorption feature with a straight line and it obtains the pseudo EW by calculating the area between the pseudo continuum and the flux.
Mathematically, the pseudo EW is defined as following, where F$_{pp}$ is the value of the flux between the peaks of the feature (i.e. the pseudo continuum) and F$_{\lambda}$ is the flux of the line at each integration step.
\begin{equation}
{\mathrm{pseudoEW}} = \Sigma \frac{(F_{pp}-F_{\lambda})}{F_{pp}} \Delta{\lambda}
\end{equation}
We present such example in Fig.~\ref{EWs} where we use the star Gl176 and an absorption line at the region around 6530 $\AA$.
\begin{figure}
\centering
\includegraphics[width= \hsize]{pseudoEW.pdf} \\
\caption{Area fitting for the calculation of pseudo EW for a line with central $\lambda$ = 6531.4 $\AA$ of the star Gl176. The position of the pseudo continuum is adjusted accordingly. This pseudo EW is equal to 87 m$\AA$. }
\label{EWs}
\end{figure}
The evaluation of our pseudo-EW measurements, by comparing them with the ones obtained from MCAL code, is presented at Appendix~\ref{A}.
\section{Machine learning on M dwarfs}
\label{mlsection}
We base our tool for the derivation of T$_{\mathrm{eff}}$ and [Fe/H] on the machine learning concept.
The user needs to run two codes. The "HARPS$\_$dataset.py" creates the databases which contain pseudo-EW measurements in different resolutions and the reference stellar parameters. The "ODUSSEAS.py" measures the pseudo EWs of new stellar spectra and derives their unknown T$_{\mathrm{eff}}$ and [Fe/H] via machine learning.
Below, we explain the details of their structure, describing the input parameters and how to use the codes.
\subsection{The HARPS dataset}
Each time the code "HARPS$\_$dataset.py" runs, the outcome is a file which is used later as input to the machine learning algorithm when running "ODUSSEAS.py" for training the machine and testing the generated model. It contains the names of 65 stars of the HARPS M dwarf sample, the central wavelengths of the 4104 absorption features from 530 to 690 nm, their pseudo-EW values according to the resolution we convolve the spectra and their reference values of T$_{\mathrm{eff}}$ and [Fe/H] from \citet{casag08} and \citet{neves12} respectively. All of these 65 spectra have SNR above 100, as reported by \citet{neves14}.
They are presented in Table~\ref{refparam}. The range of the reference stellar parameters is presented in Fig~\ref{range}. Their photometric derivations have uncertainties of 100 K for T$_{\mathrm{eff}}$ and 0.17 dex for [Fe/H], as reported by \citet{casag08} and \citet{neves12} respectively.
\begin{figure}
\centering
\includegraphics[width= \hsize]{65plot.pdf} \\
\caption{The distribution of reference T$_{\mathrm{eff}}$ and [Fe/H] of the 65 stars used to train and test the machine learning models. The cross represents the uncertainties of their photometric derivations, which are 100 K and 0.17 dex respectively.}
\label{range}
\end{figure}
The convolution function we use is the "instrBroadGaussFast" of "pyAstronomy" (\url{https://github.com/sczesla/PyAstronomy}), which applies Gaussian instrumental broadening. The width of the kernel is determined by the resolution.
A description of it can be found at \url{https://www.hs.uni-hamburg.de/DE/Ins/Per/Czesla/PyA/PyA/pyaslDoc/aslDoc/broad.html}.
Since the HARPS spectra have a specific finite resolution, our code calculates the actual resolution to which they need to be convolved by the function, in order to get spectra to the final resolution we really want.
This calculation is done considering the following relation:
\begin{equation}
\sigma _{conv} = \sqrt{\sigma _{final}^{2} - \sigma _{orig}^{2}}
\end{equation}
where $\sigma_{conv}$ corresponds to the resolution to which we need to convolve a spectrum with original resolution of $\sigma_{orig}$, in order to get a final resolution of $\sigma_{final}$.
The settings input by the user are two. a) Choose whether or not to convolve the reference HARPS spectra to the spectral resolution of our new data. We already provide precomputed pseudo EWs for a range of spectral resolutions in widely used spectrographs.
In that case there is no need to convolve again the spectra and recalculate the pseudo EWs.
b) The resolution of the data we want to analyse.
The "HARPS$\_$dataset.py" is presented schematically in Fig.~\ref{HARPSdia}.
\begin{figure*}
\centering
$\begin{array}{c}
\includegraphics[width=11cm]{HARPS_flow.png} \\
\end{array}$
\caption{ The workflow of HARPS$\_$dataset.py}
\label{HARPSdia}
\end{figure*}
\subsection{ODUSSEAS tool}
"ODUSSEAS.py" makes use of two algorithms that we developed: the "New$\_$data.py", for measuring the pseudo EWs of new spectra to analyze, and the "MachineLearning.py" for the derivation of their T$_{\mathrm{eff}}$ and [Fe/H]. The innovative aspect of this tool is the simultaneous predictions for spectra of different resolutions and wavelength ranges.
The user has the option to activate the automatic radial velocity correction for the spectra if they are shifted.
In addition, the user can set the regression type to be used by the machine learning process. The "ridge" is recommended, but also "ridgeCV" and "linear" work at similar level of efficiency as well. We present the efficiency of all the regression types used in Sect.~\ref{mleff}.
The workflow of "New$\_$data.py" is similar to the "HARPS$\_$dataset.py".
It reads the files and resolutions of new spectra and, if needed, it calculates and corrects their radial velocity shift.
In addition, if the original step of a spectrum is not 0.010, i.e. equal to that of the HARPS dataset, it is changed with linear interpolation to this value. Thus, the pseudo EWs are measured in a consistent way.
The files containing the pseudo-EW measurements of each spectrum are then used during the operation of "MachineLearning.py", which returns the values of T$_{\mathrm{eff}}$ and [Fe/H] along with the regression metrics of the models that predicted them.
The diagram of "ODUSSEAS.py" is presented in Fig.~\ref{Mdwarfsdia} showing concisely its inputs, operations and output.
\begin{figure*}
\centering
$\begin{array}{c}
\includegraphics[width= 11cm ]{ODU_flow.png} \\
\end{array}$
\caption{ The workflow of ODUSSEAS.py}
\label{Mdwarfsdia}
\end{figure*}
\subsection{Machine learning function}
\label{mlf}
Here we present in more detail the machine learning function.
The machine learning algorithm operates in a loop for each star separately, as each star may have different wavelength range and different resolution. For each star in the filelist, it loads automatically two files: the HARPS dataset of respective resolution, for training and testing the model, and the pseudo EWs of the star for which we want to derive T$_{\mathrm{eff}}$ and [Fe/H], in order to apply the model and return the stellar parameters. Based on the wavelength range that each spectrum has, a mask is applied on the HARPS dataset for considering the absorption lines in common.
The 65 HARPS stars split into training group consisting of the 70\% of the sample (45 stars) and into testing group consisting of the remaining 30\% of the population (20 stars). With these numbers selected, the machine learning model can be both trained accurately and tested on a sufficient number of stars.
We provide the algorithm with different regression types that can be used: the "linear", the "ridge", the "ridgeCV", the ''multi-task Lasso" and the "multi-task Elastic Net".
All these kinds of models provide an output value by fitting a linear regression to the input values.
The relation between the predicted value \textit{y} (the stellar parameter), the input variables \textit{x} (the pseudo EWs) and the coefficients \textit{w} is expressed as
\begin{equation}
y(w,x)=w_{o}+w_{1}x_{1}+...+w_{p}x_{p}
\end{equation}
The mathematical details of each regression type are described in the official online documentation at \url{https://scikit-learn.org/stable/modules/linear_model.html}.
The performance of machine learning is indicated by the following three kinds of regression metrics that are returned.
The mean absolute error is computed when the model is applied on the test dataset. It corresponds to the expected value of the absolute error loss in the predictions.
In addition, the "explained variance score" is calculated. The best possible value of this score is 1.0. Variance is the expectation of the squared deviation of a random variable from its mean. It measures how far a set of numbers are spread out from their average value.
Furthermore, the "r2 score" computes the coefficient of determination, defined as R$^2$. The coefficient of determination is the proportion of the variance in the dependent variable that is predictable from the independent variables. This score provides a measure of how well future samples are likely to be predicted by the model. Best possible score is 1.0 too. A constant model that always predicts the expected value, disregarding the input features, would get a score of 0.0.
In our case of multi-output, the resulting "explained variance" and "r2" scores are by default the averages with uniform weight of the respective scores for T$_{\mathrm{eff}}$ and [Fe/H].
The mathematical types of those regression metrics are described in their official online address at \url{https://scikit-learn.org/stable/modules/model_evaluation.html#regression-metrics}.
For each star, the tool makes 100 determinations by splitting randomly the train and test groups each time. After these determinations, it returns the average values of T$_{\mathrm{eff}}$ and [Fe/H], the average values of the mean absolute errors of the models, the average scores of machine learning and the dispersion of T$_{\mathrm{eff}}$ and [Fe/H] (measured as the standard deviation).
This iterative process minimizes the possible dependence of the resulting parameters on how the stars from the HARPS dataset are split for training and testing in one single measurement.
Since the reference stars are only 65, which stars end up in the training set could change the results in a measurement. This is the reason we do these multiple runs with shuffling and splitting the reference stars in different train and test groups, and finally we calculate the average values and the dispersion.
The final results are automatically saved in the file called "Parameter$\_$Results.dat".
Moreover, it saves a group of plots with the reference and the predicted parameters of model testing, as well as their differences, as a visualization of the model accuracy. An example is presented at Fig.~\ref{ml}.
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=\hsize]{Teff_test_comparison.pdf} \\
\includegraphics[width=\hsize]{Diff_Teff_test_comparison.pdf} \\
\\
\includegraphics[width=\hsize]{FeH_test_comparison.pdf} \\
\includegraphics[width=\hsize]{Diff_FeH_test_comparison.pdf} \\
\end{array}$
\caption{Demonstration of predictions applying ridge regression. Upper panel: the T$_{\mathrm{eff}}$ values expected (Ref.) and predicted (M.L.) on the test dataset, along with their differences. Lower panel: the [Fe/H] values expected (Ref.) and predicted (M.L.) on the test dataset, along with their differences.}
\label{ml}
\end{figure}
\subsection{Machine learning efficiency}
\label{mleff}
Firstly, we test the regression models mentioned above to find the best one. We use the original spectra of the HARPS dataset to their real resolution of 115000. For 100 runs with each regression type, we measure the scores and the absolute mean errors of the stellar parameters on the test set. We report the average values around which each model tends to result in Table~\ref{reg}.
The "linear", "ridge" and "ridgeCV" work very well in general, having "r2" and "explained variance" scores with average values around 0.93 and 0.94 respectively. The range of these scores, in the 100 runs, is usually from 0.87 to 0.99.
The average uncertainties of those regression types are $\sim$27 K for T$_{\mathrm{eff}}$ and $\sim$0.04 for [Fe/H]. The "ridge" model has slightly greater scores than the "linear" one.
"RidgeCV", which has a built-in cross validation function that applies "leave-one-out" or "k-fold" strategies, does not seem to work better than the classic "ridge" one, at least in this sample of M dwarf measurements. Furthermore, "multi-task Elastic Net" and "multi-task Lasso" give considerably lower scores and higher mean absolute errors. Thus, we suggest "ridge" regression, as it operates best on the spectral values of the M dwarfs.
Secondly, we evaluate the "explained variance" and "r2" scores and the mean absolute errors of the algorithm for different resolutions of the spectra. We do it for the HARPS dataset at its actual resolution of 115000 and we repeat this test for convolved datasets at resolutions of other broadly used spectrographs: 110000 (UVES), 94600 (CARMENES) , 75000 (SOPHIE) and 48000 (FEROS).
This is done to examine the level of machine learning precision towards lower resolutions.
After 100 measurements of each case, we present the average values at Table~\ref{reso}.
To further test the reliability of the method, we examine the efficiency of the machine learning in different wavelength ranges of the spectrum. We divide the linelist, which is from 530 to 690 nm, in four spectral regions and we calculate the respective scores and mean absolute errors. We do this test to check if machine learning works better using the full range or a specific part of the wavelengths.
For this test, we use the case of the convolved data at the resolution of 110000.
The machine learning operates at its best while using the full range of the initial linelist. In addition, regarding the divided areas, we notice that the bluer the part the higher the scores and the lower the mean absolute errors respectively. In general, the results show that we can get highly precise predictions for stars observed at any part of the 530-to-690 nm spectrum. These results are presented in Table~\ref{linelist}.
\begin{table*}
\centering
\caption{The average values of the scores and the mean absolute errors (M.A.E.) for T$_{\mathrm{eff}}$ and [Fe/H] of the test dataset, after 100 runs of each regression type.}
\begin{tabular}{ccccc}
\hline\hline\\
Regression & r2 score & E.V. score & M.A.E. T$_{\mathrm{eff}}$ & M.A.E. [Fe/H] \\
& & & [K] & [dex] \\
\hline\\
Ridge & 0.93 & 0.94 & 27 & 0.037 \\
RidgeCV & 0.93 & 0.94 & 27 & 0.038 \\
Linear & 0.93 & 0.93 & 27 & 0.039 \\
Multi-task Elastic Net & 0.91 & 0.92 & 35 & 0.045 \\
Multi-task Lasso & 0.88 & 0.89 & 41 & 0.056 \\
\hline\\
\end{tabular}
\label{reg}
\end{table*}
\begin{table*}
\centering
\caption{The average values of the scores and the mean absolute errors (M.A.E.) for T$_{\mathrm{eff}}$ and [Fe/H] of the test dataset, after 100 runs of each resolution (using the "ridge" regression).}
\begin{tabular}{ccccc}
\hline\hline\\
Resolution & r2 score & E.V. score & M.A.E. T$_{\mathrm{eff}}$ & M.A.E. [Fe/H] \\
& & & [K] & [dex] \\
\hline\\
real 115000 & 0.93 & 0.94 & 27 & 0.037 \\
conv. 110000 & 0.93 & 0.94 & 28 & 0.038 \\
conv. 94600 & 0.93 & 0.93 & 28 & 0.039 \\
conv. 75000 & 0.93 & 0.93 & 29 & 0.041 \\
conv. 48000 & 0.92 & 0.93 & 30 & 0.043 \\
\hline\\
\end{tabular}
\label{reso}
\end{table*}
\begin{table*}
\centering
\caption{The average values of the scores and the mean absolute errors (M.A.E.) for T$_{\mathrm{eff}}$ and [Fe/H] of the convolved-to-110000 dataset, after 100 runs of each wavelength part of the linelist.}
\begin{tabular}{cccccc}
\hline\hline\\
Wavelength range & Number of lines & r2 score & E.V. score & M.A.E. T$_{\mathrm{eff}}$ & M.A.E. [Fe/H] \\
(nm) & & & & [K] & [dex] \\
\hline\\
530 - 690 & 4104 & 0.93 & 0.94 & 28 & 0.038 \\
530 - 580 & 1300 & 0.92 & 0.93 & 31 & 0.039 \\
580 - 630 & 1300 & 0.91 & 0.91 & 48 & 0.044 \\
630 - 690 & 1504 & 0.89 & 0.90 & 56 & 0.048 \\
\hline\\
\end{tabular}
\label{linelist}
\end{table*}
\section{Derivation of stellar parameters}
\label{otherspec}
We apply our tool to spectra obtained by five widely used instruments of different resolutions: HARPS of 115000, UVES of 110000, CARMENES of 94600, SOPHIE of 75000 and FEROS of 48000.
The spectra were taken from the respective public data archives.
To test the efficiency of our tool on other-than-HARPS instruments, we use spectra from stars in common with the HARPS dataset, so we can compare their results with the reference parameters of the respective HARPS spectra.
To validate further the accuracy of our tool, we proceed to determinations and comparisons on more stars. Finally, we discuss about possible future improvements of our determinations.
\subsection{Resolution and spectral shape}
\label{s1}
We examine the spectral change of M dwarfs according to convolution in different resolutions.
The shapes of M dwarf spectra are different when obtained in lower resolutions.
In general, the lower the resolution, the shallower the absorption lines.
This is illustrated in Fig~\ref{convall} where three lines of Gl176 are shown in detail, for the original HARPS spectrum and the convolved ones to several resolutions.
We also measure these lines and we report their pseudo-EW values in Table~\ref{EWvalues}, to show their differences.
The relative differences can vary, as not only the depth changes but also the location of the pseudo continuum is different in each case. They all confirm that the lower resolution always has lower pseudo-EW values.
This is why we need to convolve the HARPS spectra to the respective resolutions of the new spectra. Consequently, machine learning compares the pseudo EWs of the same resolution and predicts accurately the stellar parameters.
In Fig~\ref{shapeferos} we show the spectral shapes of Gl674 for three different cases: the original HARPS spectrum with resolution 115000, the convolved HARPS spectrum to the resolution of FEROS (48000) and the original FEROS spectrum that is the lowest resolution we examine. We notice that the convolved HARPS spectrum follows the shape of the FEROS one in a consistent way.
In Fig~\ref{hsEW}, we show the comparison of the pseudo EWs of SOPHIE spectrum for Gl908 and the spectrum of the same star by HARPS before and after its convolution. The SOPHIE spectrum, which is of lower resolution, has consistently lower pseudo-EW values than the HARPS one, as expected. After the convolution of HARPS spectrum to the respective resolution, the overall trend of their values become highly compatible.
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=\hsize]{All5resolutions.pdf} \\
\end{array}$
\caption{ The shape of the HARPS original spectrum for Gl176 and convolved in different resolutions. The lower the resolution the swallower the absorption lines.}
\label{convall}
\end{figure}
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=\hsize]{2HARPS1FEROS.pdf} \\
\end{array}$
\caption{ The shape of spectra for Gl674 in original HARPS resolution (blue), FEROS resolution (green) and HARPS convolved to FEROS resolution (orange).}
\label{shapeferos}
\end{figure}
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=\hsize]{SH_original.pdf} \\
\includegraphics[width=\hsize]{SH_convolved.pdf} \\
\end{array}$
\caption{ Upper panel: pseudo-EW values of the original Gl908 spectra for HARPS and SOPHIE. Lower panel: pseudo-EW values of Gl908 after the convolution of HARPS to the resolution of SOPHIE. The units of pseudo EWs are m$\AA$. After the convolution, there is agreement between the identity line (solid green) and the slope (dashed red), with the intersection being close to 0.}
\label{hsEW}
\end{figure}
\begin{table*}
\centering
\caption{The pseudo EWs of three absorption lines for HARPS spectrum Gl176 in different resolutions. The lower the resolution, the smaller the pseudo EW.}
\begin{tabular}{cccc}
\hline\hline\\
Resolution & p.EW of $\lambda$ 6536.67 & p.EW of $\lambda$ 6537.08 & p.EW of $\lambda$ 6537.64 \\
& [m$\AA$] & [m$\AA$] & [m$\AA$] \\
\hline\\
original 115000 & 14.19 & 29.08 & 27.31 \\
convolved 110000 & 14.04 & 28.89 & 26.98 \\
convolved 94600 & 13.39 & 28.27 & 26.35 \\
convolved 75000 & 11.99 & 27.45 & 25.08 \\
convolved 48000 & 7.50 & 22.50 & 20.04 \\
\hline\\
\end{tabular}
\label{EWvalues}
\end{table*}
\subsection{Measurements on different spectrographs}
\label{s2}
We examine the performance of our tool in new spectra.
We show the accuracy of the stellar parameters predicted and the precision for each resolution, by presenting the mean absolute errors of the models and the dispersion of the results, as calculated after the 100 determinations for each spectrum.
For the case of HARPS, we use a HARPS spectrum of Gl643 with SNR = 83, which is not part of the HARPS dataset used in the machine learning. As reference values for this star, we consider its parameters reported by \citet{neves14}.
For the cases of the other instruments, we use a UVES spectrum of Gl846 with SNR = 149, a CARMENES spectrum of Gl514 with SNR = 191, a SOPHIE spectrum of Gl908 with SNR = 90 and a FEROS spectrum of Gl674 with SNR = 61. As reference values to those spectra, we consider the values of the respective HARPS ones in the dataset.
The results of T$_{\mathrm{eff}}$ and [Fe/H] are presented in Table~\ref{newstarspar}. We notice that the parameters of the new spectra are very close to the respective reference values. The differences in T$_{\mathrm{eff}}$ vary up to $\sim$50 K and the differences in [Fe/H] vary up to 0.03 dex. The mean absolute errors of models and the dispersions of values are slightly growing towards lower resolutions.
\subsection{Measurements on different SNR's}
\label{snr}
Here we examine the possible variation of the results regarding different signal-to-noise ratios (SNR) for a given spectrum.
We take the spectrum Gl514 of CARMENES, which has the highest SNR of the ones we examine (equal to 191, as reported in the CARMENES data archive) and we inject amounts of noise which correspond to lower SNR values that we set. Since the final noise is obtained by the quadratic sum of the initial noise and the injected noise, the final SNR values are calculated using the relation below.
\begin{equation}
(\frac{1}{SNR})^{2}_{final}=(\frac{1}{SNR})^{2}_{initial} + (\frac{1}{SNR})^{2}_{injected}
\end{equation}
We create new spectra with final SNR values ranging from 100 to 9.
For each spectrum, we measure the stellar parameters and their dispersion.
Fig.~\ref{snrcomp} illustrates the measurements of the CARMENES spectrum while degrading its SNR.
Overall, the results are similar to the ones of the original spectrum and the differences are kept roughly constant with respect to the reference values.
For SNR values down to 20, we notice that the dispersions are between 17 and 27 K for T$_{\mathrm{eff}}$ and between 0.03 and 0.04 dex for [Fe/H], i.e. at similar levels as those of the original spectrum.
For SNR values below 20, the dispersions start to increase up to $\sim$50 K and up to $\sim$0.07 dex respectively.
Moreover, it seems that there is a slight decrease of the order of 20 K in T$_{\mathrm{eff}}$ and a slight increase of the order of 0.02 dex in [Fe/H] for the spectra with SNR below 20. However, these results are within the uncertainties of the tool.
Therefore, we conclude that our tool works consistently for spectra with SNR above 20. Bellow this SNR, the errors increase significantly.
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=9cm]{multiSNR_Teff.pdf} \\
\includegraphics[width=9cm]{multiSNR_FeH.pdf} \\
\end{array}$
\caption{The average differences and the dispersion of T$_{\mathrm{eff}}$ (upper panel) and [Fe/H] (lower panel) for CARMENES Gl514 spectrum with different values of SNR.}
\label{snrcomp}
\end{figure}
\subsection{Comparison of results between our tool and \citet{neves14}}
\label{oduvn}
Now, we make an overall comparison of our results on a group of HARPS spectra with the ones presented by \citet{neves14}.
For this purpose, we measure 30 HARPS spectra from the initial GTO sample, for which we do not know their parameters from photometry and are not part of the machine learning dataset we use. Based on the information from \citet{neves14}, we have excluded very active stars and stars with SNR lower than 25, below which that method does not apply.
Both methods have been tested and do not work properly for very active or young stars, since the pseudo EWs of such spectra are affected and their parameters can not be determined accurately with the pseudo-EW approach we follow.
Then, we compare the results we get by our tool with the results presented by \citet{neves14}.
The errors of the stellar parameters derived using our tool, are 27 K for T$_{\mathrm{eff}}$ and 0.04 dex for [Fe/H], as the mean absolute errors are measured when the machine learning model is applied on the test dataset. The errors of the calibration by \citet{neves14}, which are quantified from the root mean squared error (RMSE) in that work, are equal to 91 K and 0.08 dex respectively. It is reminded that both methods are tied to the same initial systematic uncertainties of the reference parameters used, which are 100K for T$_{\mathrm{eff}}$ and 0.17 dex for [Fe/H].
The results and their differences are presented in Table~\ref{parall} and Fig.~\ref{paracomp}.
The mean and median difference of T$_{\mathrm{eff}}$ is 11 and 22 K respectively, with a standard deviation of 101 K. Regarding [Fe/H], the mean and median difference is -0.04 dex, with a standard deviation of 0.06 dex.
Work by \citet{neves14} follows a traditional approach, using a least-squares weighted fit to determine parameters. The regression of our tool reduces those errors of T$_{\mathrm{eff}}$ and [Fe/H] from 91 to 27 K and from 0.08 to 0.04 dex respectively. So, our machine learning approach increases significantly the precision of parameter determinations.
In terms of speed, the determination for a star by machine learning, even after the multiple runs with shuffling and splitting again the train/test samples, is a matter of few seconds.
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=\hsize]{TDN.pdf} \\
\includegraphics[width=\hsize]{FDN.pdf}\\
\end{array}$
\caption{T$_{\mathrm{eff}}$ comparison (upper panel) and [Fe/H] comparison (lower panel) between this work and \citet{neves14}.}
\label{paracomp}
\end{figure}
\begin{table*}
\centering
\caption{Stellar parameters of 30 HARPS spectra as calculated by our tool (AA), by \citet{neves14} (Ne14) and their difference. The SNR of those stars are between 28 and 97, as reported by \citet{neves14}.}
\begin{tabular}{ccccccc}
\hline\hline\\
Star & T$_{\mathrm{eff}}$ (AA) & T$_{\mathrm{eff}}$ (Ne14) & T$_{\mathrm{eff}}$ Diff. & [Fe/H] (AA) & [Fe/H] (Ne14) & [Fe/H] Diff. \\
& [$\pm$27 K] & [$\pm$91 K] & [K] & [$\pm$0.04 dex] & [$\pm$0.08 dex] & [dex] \\
\hline\\
CD-44-836A & 3104 & 3032 & 72 & -0.07 & -0.07 & 0.00 \\
G108-21 & 3214 & 3186 & 28 & -0.02 & -0.02 & 0.00 \\
GJ1057 & 2926 & 2916 & 10 & -0.11 & -0.10 & -0.01 \\
GJ1061 & 2772 & 2882 & -110 & -0.25 & -0.09 & -0.16 \\
GJ1065 & 3106 & 3082 & 24 & -0.32 & -0.23 & -0.09 \\
GJ1123 & 2971 & 2779 & 192 & -0.02 & 0.15 & -0.17 \\
GJ1129 & 3037 & 3017 & 20 & -0.02 & 0.05 & -0.07 \\
GJ1236 & 3225 & 3280 & -55 & -0.44 & -0.47 & 0.03 \\
GJ1256 & 2964 & 2853 & 111 & -0.02 & 0.06 & -0.08 \\
GJ1265 & 3020 & 2941 & 79 & -0.28 & -0.20 & -0.08 \\
Gl12 & 3245 & 3239 & 6 & -0.31 & -0.29 & -0.02 \\
Gl145 & 3297 & 3270 & 27 & -0.27 & -0.28 & 0.01 \\
Gl203 & 3174 & 3138 & 36 & -0.31 & -0.22 & -0.09 \\
Gl299 & 3078 & 3373 & -295 & -0.53 & -0.53 & 0.00 \\
Gl402 & 3052 & 2943 & 109 & 0.00 & 0.03 & -0.03 \\
Gl480.1 & 3214 & 3211 & 3 & -0.48 & -0.48 & 0.00 \\
Gl486 & 3096 & 2941 & 155 & -0.02 & 0.03 & -0.05 \\
Gl643 & 3113 & 3102 & 11 & -0.29 & -0.26 & -0.03 \\
Gl754 & 2988 & 3005 & -17 & -0.23 & -0.14 & -0.09 \\
L707-74 & 3250 & 3353 & -103 & -0.39 & -0.38 & -0.01 \\
LHS1134 & 3007 & 2950 & 57 & -0.20 & -0.13 & -0.07 \\
LHS1481 & 3342 & 3510 & -168 & -0.66 & -0.76 & 0.10 \\
LHS1723 & 3031 & 3167 & -136 & -0.29 & -0.24 & -0.05 \\
LHS1731 & 3229 & 3273 & -44 & -0.22 & -0.19 & 0.03 \\
LHS1935 & 3222 & 3181 & 41 & -0.20 & -0.22 & 0.02 \\
LHS337 & 3003 & 3007 & -4 & -0.33 & -0.27 & -0.06 \\
LHS3583 & 3205 & 3236 & -31 & -0.13 & -0.22 & 0.09 \\
LHS3746 & 3111 & 3013 & 98 & -0.17 & -0.13 & -0.04 \\
LHS543 & 3042 & 2872 & 170 & 0.17 & 0.23 & -0.06 \\
LP816-60 & 3030 & 2960 & 70 & -0.11 & -0.07 & -0.04 \\
\hline\\
\end{tabular}
\label{parall}
\end{table*}
\subsection{Estimating total uncertainties }
\label{gauss}
Intrinsic uncertainties exist in the T$_{\mathrm{eff}}$ and [Fe/H] reference values of the HARPS dataset, since their initial photometric derivations have average uncertainties of 100 K and 0.17 dex respectively.
Since these parameters are used as the training values for the machine learning process, we decide to inject these uncertainties by perturbing their values accordingly, in order to see how the final results of the predictions will vary.
Therefore, we create gaussian distributions on the parameters for each HARPS training dataset, increasing the dispersion of distribution on the reference parameters each time with step of 10 K and 0.02 dex, until the uncertainties of 100 K and 0.17 dex. This adds different training values to the machine learning algorithm each time.
For each step, we create 100 gaussian-distributed training datasets. After these runs of machine learning, we calculate the average values of predicted parameters and their dispersion.
In Fig.~\ref{gd}, we present the variations for spectra from the highest resolution (HARPS), the lowest resolution (FEROS) and an intermediate resolution (CARMENES).
The datapoints represent the average difference between the resulting parameters and the reference values, after being calculated with the 100 different datasets. The errorbars are the dispersion of it.
We notice that the average differences from the reference values are almost the same among them, regardless the amount of uncertainty injected to the gaussian distribution.
The average results of T$_{\mathrm{eff}}$ and [Fe/H] for the spectra from all the instruments are presented in Table~\ref{gdstarpar}. We report their maximum errors after considering the maximum gaussian distribution with 100 K and 0.17 dex. Overall, the average values of the parameters remain roughly the same as the ones calculated with no gaussian distribution at all.
The mean absolute errors (M.A.E.) of the machine learning models have grown to values between 65 and 80 K for T$_{\mathrm{eff}}$ and between 0.10 to 0.13 dex for [Fe/H], depending on the resolution of the HARPS dataset.
The dispersion of the derived parameters grows as the resolution of the spectra becomes lower.
Specifically, it is smaller than the injected uncertainties for the HARPS spectrum ($\sim$60 K and $\sim$0.10 dex), while for the spectra from other instruments, it is slightly higher than the uncertainties injected ($\sim$110 to $\sim$130 K and $\sim$0.18 to $\sim$0.22 dex respectively).
In all the cases though, the resulting average values of stellar parameters are very close to their expected values. Differences of T$_{\mathrm{eff}}$ are up to $\sim$40 K and differences of [Fe/H] are up to 0.03 dex, regarding to the expected values.
\begin{table*}
\centering
\caption{The machine learning (M.L.) results of T$_{\mathrm{eff}}$ and [Fe/H], their dispersion (Disp.), the mean absolute errors (M.A.E.) of the models and the reference values (Ref.) for comparison.}
\begin{tabular}{ccccccccccc}
\hline\hline\\
Star & Spec. & Res. & Ref. & M.L. & M.A.E. & Disp. & Ref. & M.L. & M.A.E. & Disp. \\
& & & T$_{\mathrm{eff}}$ & T$_{\mathrm{eff}}$ & T$_{\mathrm{eff}}$ & T$_{\mathrm{eff}}$ & [Fe/H] & [Fe/H] & [Fe/H] & [Fe/H]\\
& & & [K] & [K] & [K] & [K] & [dex] & [dex] & [dex] & [dex] \\
\hline\\
Gl643 & HARPS & 115000 & 3102 & 3113 & 27 & 10 & -0.26 & -0.28 & 0.04 & 0.01 \\
Gl846 & UVES & 110000 & 3682 & 3691 & 28 & 13 & -0.08 & -0.05 & 0.04 & 0.02 \\
Gl514 & CARMENES & 94600 & 3574 & 3547 & 28 & 17 & -0.13 & -0.13 & 0.04 & 0.03 \\
Gl908 & SOPHIE & 75000 & 3587 & 3580 & 28 & 18 & -0.38 & -0.35 & 0.04 & 0.03 \\
Gl674 & FEROS & 48000 & 3284 & 3338 & 30 & 24 & -0.18 & -0.16 & 0.04 & 0.03 \\
\hline\\
\end{tabular}
\label{newstarspar}
\end{table*}
\begin{table*}
\centering
\caption{The machine learning (M.L.) results of T$_{\mathrm{eff}}$ and [Fe/H] after injecting uncertainties with gaussian distributions of 100 K and 0.17 dex in the parameters of the training HARPS datasets, their dispersion (Disp.), the mean absolute errors (M.A.E.) of the models and the reference values (Ref.) for comparison.}
\begin{tabular}{ccccccccccc}
\hline\hline\\
Star & Spec. & Res. & Ref. & M.L. & M.A.E. & Disp. & Ref. & M.L. & M.A.E. & Disp. \\
& & & T$_{\mathrm{eff}}$ & T$_{\mathrm{eff}}$ & T$_{\mathrm{eff}}$ & T$_{\mathrm{eff}}$ & [Fe/H] & [Fe/H] & [Fe/H] & [Fe/H]\\
& & & [K] & [K] & [K] & [K] & [dex] & [dex] & [dex] & [dex] \\
\hline\\
Gl643 & HARPS & 115000 & 3102 & 3126 & 65 & 60 & -0.26 & -0.29 & 0.10 & 0.10 \\
Gl846 & UVES & 110000 & 3682 & 3678 & 68 & 109 & -0.08 & -0.06 & 0.10 & 0.18 \\
Gl514 & CARMENES & 94600 & 3574 & 3545 & 77 & 113 & -0.13 & -0.13 & 0.12 & 0.19 \\
Gl908 & SOPHIE & 75000 & 3587 & 3585 & 78 & 120 & -0.38 & -0.36 & 0.13 & 0.21 \\
Gl674 & FEROS & 48000 & 3284 & 3324 & 80 & 138 & -0.18 & -0.15 & 0.13 & 0.22 \\
\hline\\
\end{tabular}
\label{gdstarpar}
\end{table*}
\begin{figure*}
\centering
$\begin{array}{cc}
\includegraphics[width=9cm]{G_Teff_Harps.pdf} &
\includegraphics[width=9cm]{G_Feh_Harps.pdf} \\
\includegraphics[width=9cm]{G_Teff_Carmenes.pdf} &
\includegraphics[width=9cm]{G_Feh_Carmenes.pdf} \\
\includegraphics[width=9cm]{G_Teff_Feros.pdf} &
\includegraphics[width=9cm]{G_Feh_Feros.pdf} \\
\end{array}$
\caption{The average differences and the dispersions of the results for several amounts of gaussian distribution injected to the reference parameters of the training HARPS datasets. The result of each step is the average outcome from 100 different distributed datasets.}
\label{gd}
\end{figure*}
\subsection{Validation of [Fe/H] determinations by measuring binary systems}
\label{binaries}
Here, we measure [Fe/H] in binary systems, containing M dwarfs which are not part of the reference sample used for machine learning. Thus, we validate our method of [Fe/H] prediction in an independent way. We determine [Fe/H] both in FGK+M and in M+M systems for an even more intrinsic test of [Fe/H] agreement.
The [Fe/H] determinations of eight FGK+M binary systems, from spectra obtained by UVES and FEROS spectrographs, are presented in Table~\ref{FGKM}.
Regarding the FGK stars, their [Fe/H] and respective uncertainties were derived using the methodology described in \citet{sousa} and \citet{santos13}.
The method measures the equivalent widths of FeI and FeII lines and assumes ionization and excitation equilibrium.
It makes use of the radiative transfer code MOOG \citep{sneden} and a grid of Kurucz model atmospheres \citep{kurucz}.
The [Fe/H] values of the respective M dwarf secondaries, derived by ODUSSEAS, are presented along with the total uncertainties of our tool at the resolutions of UVES (0.10 dex) and FEROS (0.13 dex).
All binaries have differences within the uncertainties of the methods.
Furthermore, we proceed to [Fe/H] determinations of stars in five M+M binary systems, measuring their available spectra from the CARMENES public archive.
In Table~\ref{MM}, we present these results along with their own dispersions, since both are estimated by our tool based on the same reference values with the same initial uncertainties.
We notice agreement between the respective members of all the M+M binaries, within the dispersions of their [Fe/H] determinations.
This is a validation that our tool predicts [Fe/H] in a consistent and accurate way.
\begin{table*}
\centering
\caption{[Fe/H] difference between members of FGK+M binary systems.}
\begin{tabular}{ccccccc}
\hline\hline\\
Primary & [Fe/H] & $\sigma$[Fe/H] & Secondary & [Fe/H] & $\sigma$[Fe/H] & [Fe/H] Difference \\
& [dex] & [dex] & & [dex] & [dex] & [dex] \\
\hline\\
Gl100A & -0.29 & 0.05 & Gl100C & -0.29 & 0.13 & 0.00 \\
Gl118.1A & 0.02 & 0.05 & Gl118.1B & 0.05 & 0.10 & 0.03 \\
Gl173.1A & -0.37 & 0.03 & Gl173.1B & -0.29 & 0.10 & 0.08 \\
Gl157A & -0.08 & 0.04 & Gl157B & 0.02 & 0.13 & 0.10 \\
NLTT19073 & 0.08 & 0.03 & NLTT19072 & -0.07 & 0.13 & -0.15 \\
NLTT29534 & 0.00 & 0.03 & NLTT29540 & -0.03 & 0.10 & -0.03 \\
NLTT34137 & -0.12 & 0.05 & NLTT34150 & -0.13 & 0.10 & -0.01\\
NLTT34353 & -0.10 & 0.03 & NLTT34357 & -0.11 & 0.10 & -0.01\\
\hline\\
\end{tabular}
\label{FGKM}
\end{table*}
\begin{table*}
\centering
\caption{[Fe/H] difference between members of M+M binary systems.}
\begin{tabular}{ccccccc}
\hline\hline\\
Primary & [Fe/H] & Disp. & Secondary & [Fe/H] & Disp. & [Fe/H] Difference \\
& [dex] & [dex] & & [dex] & [dex] & [dex] \\
\hline\\
Gl553 & -0.07 & 0.06 & Gl553.1 & -0.10 & 0.07 & 0.03 \\
Gl875 & -0.15 & 0.05 & Gl875.1 & -0.17 & 0.06 & 0.02 \\
Gl617A & 0.04 & 0.04 & Gl617B & -0.08 & 0.08 & 0.12 \\
Gl745A & -0.48 & 0.04 & Gl745B & -0.53 & 0.06 & 0.05 \\
Gl752A & 0.01 & 0.03 & Gl752B & -0.07 & 0.08 & 0.08 \\
\hline\\
\end{tabular}
\label{MM}
\end{table*}
\subsection{Discussion on the reference parameter scales}
\label{referencescales}
Since supervised machine learning determines the parameters based on reference values given to it, their systematics will apply to the results of new stars too.
In this work, we have used the reference T$_{\mathrm{eff}}$ and [Fe/H] photometric scales of \citet{casag08} and \citet{neves12} respectively, as they are derived in a homogeneous way for a sufficiently big number of spectra available to us.
It is important to make a comparison between the reference values we use and values of same stars derived by other recent works, which may be subject to different systematics. Such is \citet{mann15}, with which we share 26 common stars of the 65 ones we use as our reference dataset.
In Table~\ref{scales}, we compare our reference parameters with determinations by \citet{mann15} and report the differences.
These differences are illustrated in Figure~\ref{scalecomp}.
Regarding T$_{\mathrm{eff}}$, we notice that our reference values have a systematic underestimation of 178 K on average with a standard deviation of 73 K. This systematic difference roots back to the different methods of derivation followed. Work by \citet{casag08} is based on the multiple optical-infrared technique (MOITE) for M dwarfs, which is an extension of the infrared flux method (IRFM) as described in \citet{casag06}. On the other hand, determinations by \citet{mann15} are done by comparing the optical spectra with the CFIST suite of the BT-SETTL version of the PHOENIX atmosphere models \citep{allard13}. The detailed description of this method can be found in \citet{mann13b}.
Regarding [Fe/H], we notice no significant systematic difference between the methods of calibration by \citet{neves12} and \citet{mann15}. The average difference is 0.06 dex with a standard deviation of 0.11 dex for the sample of stars in common.
As a potential future improvement of our determinations, we consider the possibility of replacing our reference dataset.
Since new techniques of parameter determination become more accurate and precise and as more spectra will become available to us, their homogeneously derived parameters can be correlated with their pseudo EWs.
Thus, we take into account the creation of an improved reference dataset for our machine learning tool.
\begin{table*}
\centering
\caption{Stellar parameters of 26 stars in common with \citet{mann15} and their difference.}
\begin{tabular}{ccccccc}
\hline\hline\\
Star & T$_{\mathrm{eff}}$ (Ref.) & T$_{\mathrm{eff}}$ (Mann15) & T$_{\mathrm{eff}}$ Diff. & [Fe/H] (Ref.) & [Fe/H] (Mann15) & [Fe/H] Diff. \\
& [$\pm$100 K] & [$\pm$60 K] & [K] & [$\pm$0.17 dex] & [$\pm$0.08 dex] & [dex] \\
\hline\\
Gl54.1 & 2091 & 3056 & -65 & -0.40 & -0.26 & -0.14 \\
Gl87 & 3565 & 3638 & -73 & -0.30 & -0.36 & 0.06 \\
Gl105B & 3054 & 3284 & -230 & -0.14 & -0.12 & -0.02 \\
Gl176 & 3369 & 3680 & -311 & 0.02 & 0.14 & -0.12 \\
Gl205 & 3497 & 3801 & -304 & 0.17 & 0.49 & -0.32 \\
G213 & 3026 & 3250 & -224 & -0.19 & -0.22 & -0.03 \\
Gl250B & 3369 & 3481 & -112 & -0.09 & 0.14 & -0.25 \\
Gl273 & 3107 & 3317 & -210 & -0.05 & -0.11 & 0.06 \\
Gl382 & 3429 & 3623 & -194 & 0.04 & 0.13 & -0.09 \\
Gl393 & 3396 & 3548 & -154 & -0.13 & -0.18 & 0.05 \\
Gl436 & 3277 & 3479 & -202 & 0.01 & 0.01 & 0.00 \\
Gl447 & 2952 & 3192 & -240 & -0.23 & -0.02 & -0.21 \\
Gl514 & 3574 & 3727 & -153 & -0.13 & -0.09 & -0.04 \\
Gl526 & 3545 & 3649 & -104 & -0.18 & -0.31 & 0.13 \\
Gl555 & 2987 & 3211 & -224 & 0.13 & 0.17 & -0.04 \\
Gl581 & 3203 & 3395 & -192 & -0.18 & -0.15 & -0.03 \\
Gl686 & 3542 & 3657 & -115 & -0.29 & -0.25 & -0.04 \\
Gl699 & 3094 & 3228 & -134 & -0.59 & -0.40 & -0.19 \\
Gl701 & 3535 & 3614 & -79 & -0.20 & -0.22 & 0.02 \\
Gl752A & 3336 & 3558 & -222 & 0.04 & 0.10 & -0.06 \\
Gl846 & 3682 & 3848 & -166 & -0.08 & 0.02 & -0.10 \\
Gl849 & 3200 & 3530 & -330 & 0.24 & 0.37 & -0.13 \\
Gl876 & 3059 & 3247 & -188 & 0.14 & 0.17 & -0.03 \\
Gl880 & 3488 & 3720 & -232 & 0.05 & 0.21 & -0.16 \\
Gl887 & 3560 & 3688 & -128 & -0.20 & -0.06 & -0.14 \\
Gl908 & 3587 & 3646 & -59 & -0.38 & -0.45 & -0.07 \\
\hline\\
\end{tabular}
\label{scales}
\end{table*}
\begin{figure}
\centering
$\begin{array}{c}
\includegraphics[width=\hsize]{TRM.pdf} \\
\includegraphics[width=\hsize]{FRM.pdf}\\
\end{array}$
\caption{T$_{\mathrm{eff}}$ comparison (upper panel) and [Fe/H] comparison (lower panel) between the reference values we use and \citet{mann15} for 26 common stars.}
\label{scalecomp}
\end{figure}
\section{Summary}
\label{sum}
We present our machine learning tool ODUSSEAS for the derivation of T$_{\mathrm{eff}}$ and [Fe/H] in M dwarf stars, whose spectra can have different resolutions and wavelength ranges inside the area from 530 to 690 nm. We explain in detail the way it is built and works. We present the results of the tests we perform and we examine its accuracy and precision from very high resolution of 115000 to resolution of 48000. Our tool seems to be reliable, as it operates with high machine learning scores around 0.94 and achieves excellent predictions of significantly high precision with mean absolute errors of $\sim$30 K for T$_{\mathrm{eff}}$ and $\sim$0.04 dex for [Fe/H]. Taking into consideration the intrinsic uncertainties of the reference parameters and perturbing them accordingly, our models have maximum uncertainties of $\sim$80 K for T$_{\mathrm{eff}}$ and $\sim$0.13 dex for [Fe/H], which are within the typical uncertainties for M dwarfs.
Our parameters for spectra from different spectrographs, occurring from the average of 100 determinations, have consistent values with differences within $\sim$50 K and $\sim$0.03 dex from the expected ones. Spectra should have SNR above 20 for optimal predictions.
Our tool is valid for M dwarfs in the intervals 2800 to 4000 K for T$_{\mathrm{eff}}$ and -0.83 to 0.26 dex for [Fe/H], except from very active or young stars.
It can be tested by downloading the files in the webpage \url{https://github.com/AlexandrosAntoniadis/ODUSSEAS}, after reading the README instructions for clarifying the technical details.
\begin{acknowledgements}
This work was supported by FCT/MCTES through national funds and by FEDER - Fundo Europeu de Desenvolvimento Regional through COMPETE2020 - Programa Operacional Competitividade e Internacionalização by these grants: UID/FIS/04434/2019, UIDB/04434/2020 and UIDP/04434/2020; PTDC/FIS-AST/32113/2017 and POCI-01-0145-FEDER-032113; PTDC/FIS-AST/28953/2017 and POCI-
01-0145-FEDER-028953.
A.A.K., S.G.S., E.D.M. and G.D.C.T. acknowledge the support from FCT in the form of the
exploratory projects with references IF/00028/2014/CP1215/CT0002, IF/00849/2015/CP1273/CT0003 and IF/00956/2015/CP1273/CT0002.
S.G.S. and E.D.M. further acknowledge the support from FCT through the Investigador FCT
contracts IF/00028/2014/CP1215/CT0002, IF/00849/2015/CP1273/CT0003 and POCH/FSE (EC).
G.D.C.T. further acknowledges the support from an FCT/Portugal PhD grant with reference
PD/BD/113478/2015.
\end{acknowledgements}
\bibliographystyle{aa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,448 |
Q: Keeping headers as `project` in Xcode framework I'm making an Xcode framework, and I have one public header which accesses a bunch of project headers. But when I import the framework in another project, it throws errors that it can't find the header files that are project. I have referenced objects from the project headers in the public header. How do I keep those headers project but still use the objects from them in the public header?
A: In the public header file use @class to include other interfaces and use #import in the implementation file (.m).
Using @class informs the compiler that the indicated class exists but the compiler will not require it's implementation.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,763 |
{"url":"https:\/\/dsp.stackexchange.com\/questions\/75549\/going-from-an-mfcc-coefficient-to-hz-range","text":"# Going from an MFCC coefficient to Hz range?\n\nI have not worked with MFCCs before, but I am faced with the observation that the 5th MFCC of a signal is actually of interest for me in my speech research. In order to understand why that one is important for me, I need to make sure that I understand what it represents.\n\nIf I know the number of Mel frequency spaced filters applied to the signal, and the sample rate, can I assume that the 5th coefficient is related to the 5th mel spaced filtered regions?\n\nIf not, how do I learn what in the frequency domain is represented by the 5th MFCC?\n\nIn my understanding you need a little more information. You need to know the minimum and maximum frequency and number of coefficients in the MFCC representation.\n\nIf the coefficients are linearly spaced in the mel scale, $$m(f)$$\n\nThe first coefficient is $$m(f_1)$$ the last coefficient is $$m(f_N)$$, and the $$i-$$th coefficient is\n\n$$m(f_i) = m(f_1) + \\frac{i-1}{N-1} (m(f_N) - m(f_1))$$\n\nThen you replace apply the definition of interest for $$m(f)$$, e.g.\n\n$$m(f) = \\frac{1000}{log(2)}log\\left(1 + \\frac{f}{1000}\\right)$$\n\nAnd solve for $$f_i$$.","date":"2021-09-23 09:22:17","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\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 8, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8170269727706909, \"perplexity\": 392.5974524194704}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057417.92\/warc\/CC-MAIN-20210923074537-20210923104537-00411.warc.gz\"}"} | null | null |
\subsection{Feature computation}
The microphone signal is divided into $\nFrames=\ceil{\nSamples/ \hop}$
overlapping frames of length $\frameLength$, where $\hop$ denotes the hop
length. The samples in each $\frameind$\textsuperscript{th} frame are used to construct a vector
of length $\nch \cdot \frameLength$:
\begin{align}
\micvec{\frameind}=
\big[
\timemic{0}{\frameind \hop},\timemic{0}{\frameind \hop + 1}, &\ \ldots \ ,\timemic{0}{\frameind \hop + \frameLength -1},\notag\\
\timemic{1}{\frameind \hop},\timemic{1}{\frameind \hop + 1},&\ \ldots \ ,\timemic{1}{\frameind \hop + \frameLength -1},\\
&\ \ldots\notag\\
\timemic{\nch-1}{\frameind \hop},\timemic{\nch-1}{\frameind \hop + 1},&\ \ldots\ ,\timemic{\nch-1}{\frameind \hop + \frameLength -1}
\Transpose{\big]},\notag
\end{align}
resulting in the time-ordered sequence of $\nFrames$ vectors:
\begin{equation}
\micseq = \Transpose{ \big[ \micvec{0}, \ \micvec{1}, \ \ldots\ ,\ \micvec{\nFrames-1} \big] }.
\end{equation}
The feature computation results in the sequence:
\begin{equation}
\featseq = \Transpose{ \big[ \featvec{0}, \ \featvec{1}, \ \ldots\ ,\ \featvec{\nFrames-1} \big] },
\label{eq:featuresequence}
\end{equation}
where each vector of length $\nFeats$ is defined as:
\begin{equation}
\featvec{\frameind} = \big[
\featcoef{\frameind}{0}, \featcoef{\frameind}{1}, \ \ldots \ , \featcoef{\frameind}{\nFeats - 1}
\Transpose{\big]},
\end{equation}
where $\featcoef{\frameind}{\featind}$ denotes the $\featind^\textrm{th}$ feature
coefficient extracted from the $\frameind^\textrm{th}$ frame. The feature extraction is
typically designed such that $\nFeats < \nch \cdot \frameLength$. Our method
assumes that the information relevant for speech intelligibility prediction,
such as binaural cues and \gls{snr}, is shared among different parts of the
input sequence while information that is only present locally can be discarded.
Therefore, the feature extractor learns to extract sequences $\featseq$ that
maximise the mutual information between the input and output sequences:
\begin{equation}
\mutualInfo{\micseq}{\featseq} = \Sum{\micseq, \featseq}{}{
\jointProba{\micseq}{\featseq}
\logTen{
\frac{\conditionalProba{\micseq}{\featseq}}{\proba{\micseq}}
}
}.
\label{eq:mi}
\end{equation}
To do so, \gls{vq} and \gls{cpc} methods are used to compute the sequence $\featseq$ as a
latent representation of the input sequence $\micseq$.
The computation of these \gls{vqcpc} features consists of three main components:
a non-linear encoder, a \gls{vq} codebook, and an autoregressive aggregator.
First, the non-linear encoder $\encoder{\cdot}$ maps $\micseq$ to an
intermediate latent representation $\latentseq$:
\begin{equation}
\encoder{\micseq} = \latentseq = \Transpose{ \big[ \latentvec{0}, \
\latentvec{1}, \ \ldots\ ,\ \latentvec{\nFrames-1} \big] },
\end{equation}
where $\latentvec{\latentind}$ denotes the $\latentind^{\textrm{th}}$
vector, each of length $\embeddinglength$. \gls{vq} is applied to map each
vector in $\latentseq$ to an embedding vector from a finite codebook $\codebook$
yielding the sequence:
\begin{equation}
\vqseq = \Transpose{ \big[ \vqvec{0}, \ \vqvec{1}, \ \ldots\ ,\ \vqvec{\nFrames-1} \big] },
\end{equation}
where each $\latentind^{\textrm{th}}$ vector $\vqvec{\latentind}$ is computed as:
\begin{equation}\label{eq:codebook}
\vqvec{\latentind} = \quantizer{\latentvec{\latentind}} = \argmin{
|| \latentvec{\latentind} - \codebookvec{\tmpind} ||_{2}
}{
\codebookvec{\tmpind} \in \codebook
}
\end{equation}
Where $\codebookvec{\tmpind}$ denotes the $\tmpind^\textrm{th}$ in the $\nCodebook$
embedding vectors of the codebook. Finally, an autoregressive aggregator
$\aggregator{\cdot}$ is applied to compute each vector from the sequence in
\eref{eq:featuresequence} as:
\begin{equation}
\featvec{\frameind} = \aggregator{\vqvec{\latentind \leq \frameind}}.
\end{equation}
\subsection{VQ-CPC training}
Training of $\encoder{\cdot}$, $\quantizer{\cdot}$ and $\aggregator{\cdot}$ is
conducted end-to-end to maximize the mutual information defined in \eref{eq:mi}.
The proposed approach follows the method in~\cite{oord2019} with
additional loss terms to support the added \gls{vq} codebook~\cite{oord2018}.
To encourage shared information to be encoded, each vector $\featvec{\frameind}$
is used to predict $\vqvec{\frameind+\stepind}$ for up to $\stepind$ steps in
the future.
However, rather than modelling the distribution $\conditionalProba{\micvec{\frameind + \stepind}}{\featvec{\frameind}}$,
the proposed method models the density ratio defined as:
\begin{equation}\label{eq:ratio}
\densityRatio{\stepind}{\micvec{\frameind + \stepind}}{\featvec{\frameind}}
\propto
\frac{
\conditionalProba{\micvec{\frameind + \stepind}}{\featvec{\frameind}}
}{
\proba{\micvec{\frameind + \stepind}}
}.
\end{equation}
The density ratio $ \densityRatio{\stepind}{\micvec{\frameind +
\stepind}}{\featvec{\frameind}}$ may be unnormalized and, in this paper, is
computed as:
\begin{equation}\label{eq:score}
\densityRatio{\stepind}{\micvec{\frameind + \stepind}}{\featvec{\frameind}} =
\Exp{\Transpose{\vqvec{\frameind + \stepind}} \projection{\stepind} \featvec{\frameind} },
\end{equation}
where $\projection{\stepind}$ denotes a learned linear projection.
Using this definition, the encoder and aggregator are trained by minimising the
InfoNCE loss $\mathcal{L}$, based on noise-contrastive estimation and importance
sampling:
\begin{equation}\label{eq:loss}
\mathcal{L} = \beta \cdot \mathcal{L}_\textrm{vq} + \frac{1}{k} \sum_{i=1}^k \mathcal{L}_i,
\end{equation}
where $\mathcal{L}_\textrm{vq}$ denotes the weighted \gls{vq} commitment loss
defined as:
\begin{equation}\label{eq:vq}
\mathcal{L}_\textrm{vq} = \frac{1}{\nFrames}\sum\limits_{\ell = 0}^{\nFrames - 1}
||\latentvec{\latentind} - \textrm{sg}[\codebookvec{\tmpind}]||_2^2
\end{equation}
where $e_i$ is the corresponding embedding vector of $\tilde{z}_\ell$ and
$\textrm{sg}[\cdot]$ is the stop-gradient operator \cite{oord2018} and:
\begin{equation}\label{eq:nce}
\mathcal{L}_k = - \mathop\mathbb{E}\limits_X \left[ \log \frac{\sigma_k(x^+,
c_t)}{\sum_{x_j \in X} \sigma_k(x_j, c_t)} \right],
\end{equation}
where $x_+ \sim p(x_{t+k} \vert
c_t)$ and $x_- \sim p(x_{t+k})$ \cite{oord2019}. The codebook
embedding vectors are updated using \gls{ema} as described in \cite{oord2018}.
\subsection{Intelligibility score predictor}
The computation of the \gls{vqcpc} features does not rely on any assumptions
about the downstream task for which these features are used. In this paper, the
sequence $\featseq$ is input to a predicting function for the purpose of
\gls{si} prediction. Two different predicting functions are considered.
The first considered predicting function uses each vector $\featvec{\frameind}$
as input to a single shared linear layer in order to compute a per-frame score. The
score assigned to the complete sequence is the mean of the scores computed from
each vector. This simple predicting function is used to demonstrate how
easily accessible information about \gls{si} is when using the \gls{vqcpc}
features. This predicting function is referred to as \bsq{Small} in the
remainder of the paper.
The second considered predicting function first builds a global representation
using \gls{seqpool} methods originally used for the classification
of images~\cite{hassani2021}. A global representation is computed by applying
\gls{seqpool} to each vector in the sequence $\featseq$. In this case
\gls{seqpool} inputs each vector $\featvec{\frameind}$ to a linear layer that
outputs a scalar before applying softmax to the computed scalars, forming
weightings for each frame. The weighted sum of each vector is then computed,
forming the global representation. This global representation is then input to
a small \gls{mlp} to compute the estimated speech intelligibility score assigned
to the sequence $\featseq$. This predicting function is referred to as
\bsq{Pool} in the remainder of the paper.
Both Small and Pool are trained to minimise the \gls{mse} between the estimated
and true speech intelligibility scores (see \sref{sec:experiments}).
\subsection{Generated datasets}
For training of both the \gls{vqcpc} model and the predicting functions,
training and development datasets of binaural signals are generated. All
signals have a sampling frequency $\fs=$\, 16\,kHz and are generated as per
\eref{eq:signalmodel}. The clean anechoic speech is extracted from either the
360 hour training set or the 5 hour development set from the LibriSpeech
corpus~\cite{panayotov2015}. Reverberant speech is generated by convolving each
utterance of clean speech with an \gls{rir} randomly selected from the Aachen
Impulse Response Database~\cite{jeub2009}. For each reverberant utterance, two
different noise segments of the same length are selected from the noise signals
in the MUSAN database~\cite{snyder2015}. These two signals are used to generate
the two-channel noise signal of a spherically isotropic noise field using the
method from~\cite{habets2008}. Finally, this generated noise signal is added to
the reverberant signal after being scaled to a chosen \gls{snr}, randomly
selected between -10\,dB and 30\,dB, measured in the first channel according
to~\cite{ITU_T_P56}. In the training and development sets, this process is
repeated three times for each clean speech utterance.
Additionally, two testing datasets are generated, hereafter denoted \bsq{\testA}
and \bsq{\testB}. The signals in \testA{} are generated using the same method
as for the training and development sets but using clean speech from the test
split of the LibriSpeech corpus. The signals in \testB{} are generated by
convolving the speech signals used as target utterances in the first Clarity
Challenge~\cite{barker2021} with \glspl{rir} randomly selected from the binaural
\glspl{rir} available in~\cite{kayser2009} recorded in either a cafeteria or a
courtyard. In this case, two-channel noise signals recorded at the same
location are used and added to the reverberant signals with an \gls{snr}
randomly selected and measured.
A total of 1090.8, 16.2, 5.4 and 10.4 hours of data are generated in the training,
development, \testA{} and \testB{} dataset, respectively. For all signals, the
ground truth is defined as the \gls{dbstoi} computed using the clean
reverberant signal and the noisy reverberant signal as
input~\cite{andersen2018}.
\subsection{Parameters of proposed method}
For training, we use $\micseq$ of length $T=40960$ as input to the encoder
$\encoder{\cdot}$. The encoder has a frame length and hop size of 25~ms and
10~ms respectively. It is implemented as a series of five convolutional blocks,
each consisting of a one-dimensional convolutional layer with 256 filters, a
dropout layer~\cite{srivastava2014}, batch normalisation~\cite{ioffe2015} and
the \gls{relu} activation function. The strides for each block are $[5,4,2,2,2
]$ and the kernel sizes are $[10,8,4,4,4]$. \gls{vq} is applied using a
codebook of 512 vectors of dimensionality 128, with the commitment loss defined
as in~\eqref{eq:vq}.
The aggregator $\aggregator{\cdot}$ is implemented as a two-layer \gls{gru}
\cite{chung2014} with 128 hidden channels. Hence, in our experiments, $\nFeats =
\embeddinglength$.
The InfoNCE loss is computed using $10$ negative samples and $\stepind=$~12
steps.
Augmentation is applied as random channel and polarity swapping, additive noise
and random audio gain.
All resulting sequences $\featseq$ (with
$\nFeats=$\,128) in the training set are used to train the considered
predicting functions. \gls{vqcpc} features are extracted from \testA{} and
\testB{} using the complete \gls{vqcpc} model trained on the training set.
The Small predicting function consists of a single layer mapping each feature
vector of length $\nFeats$ to a single element followed by the sigmoid
activation function. The Pool predictor is implemented as a single shared
linear layer to compute the weighting and a \gls{mlp} with one hidden layer
of
size 2$\nFeats$. The hidden layer uses \gls{relu} as its non-linearity and the
output layer consists of a single element followed by the sigmoid activation
function.
Training and testing of the \gls{vqcpc} model and predicting
functions were implemented in PyTorch~\cite{paszke2019}.
\subsection{Benchmark and figures of merit}
The performance of the proposed \gls{vqcpc} is measured in terms of \gls{lcc},
\gls{srcc} and \gls{mse} between the ground truth and the output of the
predicting functions. The experiments aim to quantify the ability of \gls{vqcpc}
features to represent information useful for speech intelligibility prediction.
To this end, their performance is compared with the use of mel-spectrogram
features (with deltas and double-deltas) that are either extracted from the
first channel, concatenated from both channels or extracted from the
single-channel signal computed using a blind binaural preprocessing
stage~\cite{hauth2020}. These benchmark features are denoted \bsq{Single},
\bsq{Concat} and \bsq{BSIM20}, respectively. In all cases, the mel-spectrogram
features are computed using the same frame length and hop size as the \gls{vqcpc}
features, a \acrshort{fft} of size 512, and 40 mel coefficients. This results in
120 feature coefficients per frame for Single and BSIM20, and 240 feature
coefficients per frame for Concat. For all considered features, the predicting
functions Small and Pool are trained using the same dataset as for \gls{vqcpc}.
The performance of all considered features is additionally compared with the
average non-intrusive single-channel \gls{stoi}~\cite{andersen2017} over the two
channels.
\section{Introduction}
\label{sec:intro}
\input{content/introduction}
\section{Proposed Method}
\label{sec:proposed}
\input{content/proposed}
\section{Experimental Setup}
\label{sec:experiments}
\input{content/experiments}
\section{Results}
\label{sec:results}
\input{content/results}
\section{Conclusion}
\label{sec:conclusion}
\input{content/conclusion}
\pagebreak
\balance
\bibliographystyle{utilities/IEEEbib}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,068 |
Le Dunkerque Grand Littoral Volley-Ball est un club de volley-ball français, évoluant en 2012-2013 au quatrième niveau national (Nationale 2).
Historique
1965 : création de la section volley-ball de l'US Dunkerque
1972 : L'US Dunkerque accède au plus haut niveau national, la Nationale 1
1999 : Champion de France de Nationale 1, accession en Pro B
2000-2001 : obtention de la première place au terme de la saison régulière ; accession en Pro A.
2002-2003 : obtention de la au terme de la saison régulière ; rétrogradation en Pro B.
2003-2004 : échec en finale de Pro B contre Saint-Quentin (2 défaites à 0) ; accession en Pro A.
2004-2005 : obtention de la au terme de la saison régulière ; rétrogradation en Pro B.
2011-2012 : relégation en Nationale 2.
Historique des logos
Palmarès
Championnat de France Pro B (1)
Vainqueur : 2001
Effectifs
Saison 2010-2011 (Ligue B)
Entraîneur : Gaël Bollengier
Saison 2009-2010 (Ligue B)
Entraîneur : Gaël Bollengier
Saison 2008-2009 (Pro B)
Entraîneur : Gaël Bollengier
Saison 2007-2008 (Pro B)
Entraîneur : Gaël Bollengier
Saison 2006-2007 (Pro B)
Entraîneur : Gaël Bollengier
Saison 2005-2006 (Pro B)
Entraîneur : Gabriel Denys
Saison 2004-2005
Entraîneur : Gabriel Denys
Saison 2003-2004 (Pro B)
Entraîneur : Gabriel Denys ()
Saison 2002-2003
Entraîneur : Jean-René Akono ()
Saison 2001-2002
Entraîneur : Jean-René Akono ()
Articles connexes
Liens externes
Site officiel du club
Pro B (volley-ball)
Club de volley-ball en France
Club sportif à Dunkerque
Club sportif fondé en 1965
Dunkerque | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 251 |
'use strict';
const create = (allowedServices) => (event) => {
const serviceLabel = event.entity.service_label;
const planName = event.entity.service_plan_name;
const service = allowedServices[serviceLabel];
return !service || !service.plans.includes(planName);
};
module.exports = create;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,803 |
Q: Remote desktop connection not reflecting last display change until next keypress I've got some really weird behavior remoting from one Windows 7 machine to another. In most apps that I'm using across the remote desktop connection, if I'm typing a word the last letter won't show up until I move the mouse around or hit another key (basically force a repaint). I suppose this could be related to some software running on either the remote or local workstation that is interfering with keypresses. Has anybody seen this before?
A: A refresh problem shouldn't be connected to the presence of a keyboard-hooking software.
Check in Windows Update for an optional update to Remote Desktop on both computers.
Here is a long-shot reference, regarding Auto-Tuning TCP/IP problems:
Remote Desktop slow problem solved
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,146 |
\section{Introduction}
It is well-known that scale invariance is broken by
renormalization and dimensional parameters in quantum field
theories. The concept of scale invariance (and more generally the
conformal symmetry) may still play an important role in high
energy physics. For an asymptotically free theory, such as QCD,
the scale invariance is recovered in the high energy limit. In the
concerned practical physics processes of high energies, breaking
of scale invariance can be systematically incorporated in the
anomalous dimensions of operators using the renormalization group
method \cite{Conformal}. It is indicated that the scale invariance
in the infrared region may be quite different and less known
\cite{BZ}. But the idea of scale invariance is so simple and
attractive that there is no a $priori$ to repel it from our world.
In \cite{Georgi1}, Georgi proposed that a scale invariant stuff
contains no particle, but the so-called unparticle. The unparticle
possesses some properties which are different from that of ordinary
particles. The first aspect is that it has a non-trivial scale
dimension $d_{\cal U}$. The dimension of unparticle is in general
fractional rather than an integral number (the dimension for a
fermion is half-integral). The fractional dimension must come from
some complicated dynamics whose details are unknown at present.
Another aspect is that the free unparticle has no definite mass.
That means that the Lorentz-invariant four-momentum square $P^2$ is
not fixed for a real unparticle. Georgi observed that unparticle
with scale dimension looks like a non-integral number $d_{{\cal U}}$ of
invisible massless particles \cite{Georgi1}. To be consistent with
the present experimental observations, the coupling of unparticle to
the ordinary Standard Model (SM) matter must be sufficiently weak.
However, it may be relevant to the TeV physics and might be explored
at the LHC and ILC. The interactions between the unparticle and the
SM particles are described in the framework of low energy effective
theory and lead to various interesting phenomena. There have been
some phenomenological explorations on possible observable effects
caused by unparticles \cite{Georgi1,Georgi2,CKY,LZ,CG,Liao,DY,ACG}.
The mixing of $K^0-\overline{K^0}$, $D^0-\overline{D^0}$ and
$B_{(s)}^0-\overline{B^0_{(s)}}$ is of fundamental importance to
test the SM and explore new physics beyond the SM. In the scenarios
of new physics, there may exist a flavor-changing neutral current
(FCNC) to result in such a mixing which can only be realized via
loops in the framework of the SM. Thus this observable could be
sensitive to new physics effects. In fact, many authors used to
explore evidence of new physics in $B^0-\overline{B^0}$ (or $B^0_s-\overline{
B^0_s}$) mixing because data about the mixing have been available
for a long while. In the proposed scenario \cite{Georgi1}, the
unparticle can couple to different flavors of quarks and induce FCNC
even at tree level as long as the unparticle is neutral. Thus it
will cause new contributions to the particle-antiparticle mixing,
$B^0-\overline{B^0}$, $D^0-\overline{D^0}$ mixing. Generally, based on physics
conjecture, the energy scale concerning unparticle is high that it
should cause smaller influence on the $K^0-\overline{K^0}$ mixing,
especially the SM contribution to the mixing obviously dominates.
The unparticle effects on $B^0_{(s)}-\overline{B^0_{(s)}}$ mixing
had been studied in \cite{LZ,CG} roughly. Since the
$B^0_{(s)}-\overline{B^0_{(s)}}$ mixing parameter $x_{B_{d(s)}}$ is
large and generally the contributions from the SM dominate, and the
new physics effect if it exists, is less important, thus the
observable is not so sensitive to the new physics. Whereas for the D
system, the SM contribution is confirmed to be sufficiently small,
and the $D^0-\overline{D^0}$ mixing parameter (the SM prediction is
$x_{D}<10^{-3}$ \cite{FGLP}) must not be measured by the present
experiments, if there is no new physics. By contraries, if sizable
mixing is measured, new physics should exist and make main
contributions. It is interesting that recently the $D^0-\overline{D^0}$ has
indeed been measured by the Babar and Belle collaborations
\cite{Babar,Belle}, which may be a signature of existence of new
physics. He and Valencia \cite{He} suggested that the mixing is due
to the FCNC in the up-type-quark sector for non-universal $Z'$ model
and obtained constraints on the model parameters by fitting the
data. Instead, we propose that the unparticle scenario is the new
physics which is responsible for the observable $D^0-\overline{D^0}$ mixing.
In this study, we will investigate the effects of the unparticle
physics on the neutral meson mixing including
$B^0_{(s)}-\overline{B^0_{(s)}}$, $D^0-\overline{D^0}$ and
$K^0-\overline{K^0}$ mixing and constrain the coupling parameter of
the concerned interactions between the unparticles and the SM
quarks.
\section{$M^0-\overline{M^0}$ mixing in unparticle physics}
We start with a brief review about the unparticle scenario. It is
assumed that the scale invariant unparticle fields emerge below an
energy scale $\Lambda_{\cal U}$ which is at the order of TeV
\cite{Georgi1}. The interactions of the unparticle with the SM
particle are described by a low energy effective theory. For our
purpose, the coupling of unparticle to quarks is given by following
the standard strategy to construct effective interactions as
\begin{eqnarray}
\frac{c_S^{q'q}}{\Lambda_{\cal U}^{d_{\cal U}}}\bar q'\gamma_{\mu}(1-\gamma_5)
q\partial^\mu O_{\cal U} +\frac{c_V^{q'q}}{\Lambda_{\cal U}^{d_{\cal U}-1}}\bar
q'\gamma_{\mu}(1-\gamma_5)qO_{\cal U}^\mu+h.c. .
\end{eqnarray}
where $O_{\cal U}$ and $O_{\cal U}^{\mu}$ denote the scalar and vector
unparticle fields, respectively. The $c_S^{q'q}$ and $c_V^{q'q}$ are
dimensionless coefficients and they depend on different flavors in
general. If the $q$ and $q'$ belong to the same up- or down-type
quark sectors, the above effective interactions may induce FCNC
transitions and provide new physics contribution to the neutral
meson mixing. In order to simplify the phenomenological analysis, we
use the same coefficient for all flavors, $c_S^{q'q}\to c_S$ and
$c_V^{q'q}\to c_V$. Relaxing this restriction does not change our
conclusions.
In this study, we are only interested in the effects of the
unparticle field which serves as an intermediate agent in the FCNC
transition, thus it only appears as a propagator with momentum $P$
and scale dimension $d_{\cal U}$. The propagator for the scalar
unparticle field is given by \cite{Georgi2,CKY}
\begin{eqnarray}
\int d^4 x e^{iP\cdot x}\langle 0 |TO_{\cal U}(x)O_{\cal U}(0)|0\rangle &=&
i\frac{A_{d_{\cal U}}}{2{\rm sin}(d_{\cal U}\pi)}\frac{1}{(P^2+i\epsilon)^{2-d_{\cal U}}}
e^{-i(d_{\cal U}-2)\pi},
\end{eqnarray}
where
\begin{eqnarray}
A_{d_{\cal U}}=\frac{16\pi^{5/2}}{(2\pi)^{2d_{\cal U}}}\frac{\Gamma(d_{\cal U}+1/2)}
{\Gamma(d_{\cal U}-1)\Gamma(2d_{\cal U})}.
\end{eqnarray}
The function ${\rm sin}(d_{\cal U}\pi)$ in the denominator implies that
the scale dimension $d_{\cal U}$ cannot be integral for $d_{\cal U}>1$ in
order to avoid singularity. The phase factor $e^{-i(d_{\cal U}-2)\pi}$
provides a CP conserving phase which produces peculiar interference
effects in high energy scattering processes \cite{Georgi2},
Drell-Yan process \cite{CKY} and CP violation in B decays \cite{CG}.
The propagator for the vector unparticle is similarly given by
\begin{eqnarray}
\int d^4 x e^{iP\cdot x}\langle 0 |TO^{\mu}_{\cal U}(x)O^{\nu}_{\cal U}(0)|0\rangle &=&
i\frac{A_{d_{\cal U}}}{2{\rm sin}(d_{\cal U}\pi)}\frac{-g^{\mu\nu}+P^{\mu}P^{\nu}/P^2}
{(P^2+i\epsilon)^{2-d_{\cal U}}}e^{-i(d_{\cal U}-2)\pi},
\end{eqnarray}
where the transverse condition $\partial_\mu O_{\cal U}^\mu=0$ is used.
The neutral meson is denoted by $M^0(q\bar q')$ and its antiparticle
$\overline{M^0} (q' \bar q)$. The mixing occurs via a transition $q\bar q'\to
q' \bar q$ at the quark level. In the SM, these FCNC processes can
only be realized at loop orders. The lowest contribution which
results in the $M^0-\overline{M^0}$ mixing is the box diagrams. With
the unparticle scenario, the FCNC transitions can occur at tree
level and they are depicted in Fig. \ref{fig1}. The double dashed
lines represent the exchanged unparticle fields. There are two
diagrams corresponding to t- and s-channel unparticle-exchanges
which contribute to the $M^0-\overline{M^0}$ mixing.
\begin{figure}[!htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=14cm]{UPmix.eps}
\end{tabular}
\end{center}
\label{figure} \caption{ The $M^0-\overline{M^0}$ mixing in
unparticle physics. The double dashed lines represent the unparticle
fields.} \label{fig1}
\end{figure}
The $M^0-\overline{M^0}$ mixing is usually described by two parameters: the
mass difference $\Delta m_M$ and width difference $\Delta
\Gamma_M$. The unparticle physics modifies $\Delta m_M$ and thus
changes the SM predictions. For the heavy mesons $B_d,~B_s,~D$,
the mass difference $\Delta m_M$ is related to the mixing matrix
element $M_{12}^M$ by
\begin{eqnarray}
\Delta m_M\approx 2|M_{12}^M|=\frac{1}{m_M}\left|\langle
\overline{M^0}|{\cal H}_{\rm eff}(|\Delta F|=2)|M^0\rangle\right|,
\end{eqnarray}
where $|\Delta F|=2$ represents $|\Delta B|=2$ for the $B^0-\overline{B^0}$
mixing and $|\Delta C|=2$ for the $D^0-\overline{D^0}$ mixing. For the D meson
system, the above relation is valid under the assumption of CP
conservation. The effective operators which contribute to $\Delta
F=2$ are
\begin{eqnarray}
Q_1&=&\bar q'\gamma_\mu(1-\gamma_5)q\bar q'\gamma^\mu(1-\gamma_5)q, \nonumber\\
Q_2&=&\bar q'(1-\gamma_5)q\bar q'(1-\gamma_5)q.
\end{eqnarray}
We only keep the operators at the tree level and more operators
would emerge if QCD corrections are taken into account.
It is noted that the transferred momentum square for t- and
s-channels are approximately equal, i.e. $P^2\approx m_M^2$ for
heavy meson system.
Now we are able to give the expressions for the mass difference
$\Delta m_M$. The unparticle physics contribution $\Delta m_M^{{\cal U}}$
is given as
\begin{eqnarray}
\Delta m_M^{\cal U}=\frac{5}{3}\frac{f_M^2\hat B_M}{m_M}\frac{A_{d_{\cal U}}}{2|{\rm
sin}d_{\cal U}\pi|}\left(\frac{m_M}{\Lambda_{\cal U}} \right)^{2d_{\cal U}}|c_S|^2,
\end{eqnarray}
for the scalar unparticle and
\begin{eqnarray}
\Delta m_M^{\cal U}=\frac{f_M^2\hat B_M}{m_M}\frac{A_{d_{\cal U}}}{2|{\rm
sin}d_{\cal U}\pi|}\left(\frac{m_M}{\Lambda_{\cal U}} \right)^{2d_{\cal U}-2}|c_V|^2.
\end{eqnarray}
for the vector unparticle. Note that in the above expression only
the absolute value of the function sin$d_{\cal U}\pi$ exists. Our results
are the same as in \cite{CG} and slightly different from \cite{LZ}
by a constant factor. In the above derivations, we have used the
relations listed below \cite{BS}
\begin{eqnarray}
\langle \overline{M^0}|\bar q'\gamma_\mu(1-\gamma_5)q\bar q'\gamma^\mu(1-\gamma_5)q
|M^0\rangle=\frac{8}{3}f_M^2m_M^2\hat B_M, \nonumber \\
\langle \overline{M^0}|\bar q'(1-\gamma_5)q\bar q'(1-\gamma_5)q
|M^0\rangle=-\frac{5}{3}f_M^2m_M^2\hat B_M.
\end{eqnarray}
where $f_M$ denotes the decay constant and $\hat B_M$ is a
numerical factor which is related to the non-perturbative QCD and
takes different values in various models, but as known, is of
order of unity.
Some comments are in order:
(1) The mass difference is proportional to a meson mass dependent
factor $m_M^{2d_{\cal U}}$ or $m_M^{2d_{\cal U}-2}$ which comes from the
unparticle propagator $\frac{1}{(P^2)^{2-d_{\cal U}}}$. This is a
peculiar effect caused unparticle physics. The propagator for a
heavy particle exchange from other new physics does not depend on
the low energy scale $m_M$ in general.
(2) The above analysis is applicable to $B^0-\overline{B^0}$,
$B_s^0-\overline{B_s^0}$ and $D^0-\overline{D^0}$ mixing. For the K-system,
there are large uncertainties due to long-distance effects and the
approximations which exist in the theoretical calculations. Thus we
will not use the data on $K^0-\overline{K^0}$ mixing to constrain
the unparticle physics parameters.
(3) In this work, following the method commonly adopted in
literature to study new physics effects, we assume that the new
physics beyond the SM which contributes to the mixing is the
unparticle sector. One can write
\begin{eqnarray} \Delta m_{M}^{NP}=\Delta
m_{M}^{exp}-\Delta m_{M}^{SM},
\end{eqnarray}
where $\Delta m_{M}^{NP}$ corresponds to the contribution of new
physics, i.e. the unparticle in this study. The SM prediction on
$\Delta m_{B}$ has already been precise to two-loop order, and the
data are much more accurate than before thanks to the progress in
experimental measurements at Babar and Belle. Therefore by the
deviation between the SM prediction and measured value, we can set a
constraint on the parameters for the unparticle scenario.
Considering an extreme case, let us loosen the above restriction,
namely, we postulate that the mixing $B^0-\overline{B^0}$ is fully due to the
unparticle contribution and see what constraints we would obtain on
the parameters. Later we will show that such constraints are looser
than that from that obtained from $D^0-\overline{D^0}$ mixing. Therefore, one
may not need to take the constraint on the unparticle parameters
from the data of $B^0-\overline{B^0}$ mixing at all.
(4) Because $\frac{\Delta m_{B_s}}{\Delta m_{B_d}}=34\gg 1$,
$B_s^0-\overline{B_s^0}$ mixing provides a looser constraint
compared to the $B^0-\overline{B^0}$ case.
The unknown parameters about the unparticles are: $\Lambda_{\cal U}$,
$d_{\cal U}$ and $c_S(c_V)$. In the numerical results, we fix the value
of $\Lambda_{\cal U}$ by $\Lambda_{\cal U}=1$ TeV. Other input parameters are:
$f_B\sqrt{\hat B}_B=0.2$ GeV \cite{Buras}, $f_D\sqrt{\hat B}_D=0.2$
GeV \cite{BS}, $\Delta m_{B_d}=0.507 ~ps^{-1}$ \cite{PDG}. The
recent experiment carried out by the Belle collaborations sets
$x_D=\frac{\Delta m_D}{\Gamma_D}=(0.80\pm 0.29(stat.)\pm
0.17(syst.))\%$ for the $D^0-\overline{D^0}$ \cite{Belle}. We use
$x_D<10^{-2}$ as the upper bound.
At first, we consider the case with $d_{\cal U}=3/2$ and constrain $c_S$
and $c_V$ from $B^0-\overline{B^0}$ and $D^0-\overline{D^0}$ mixing. Table \ref{t1}
lists the upper bounds for the coupling parameters $c_S$ and $c_V$.
The bounds obtained from $D^0-\overline{D^0}$ are more stringent than that
from $B^0-\overline{B^0}$ especially for the vector coupling $c_V$. This
confirms our expectation in the Introduction. The bounds obtained
from $D^0-\overline{D^0}$ mixing are: $|c_S|<2.1\times 10^{-2}$ and
$|c_V|<5.0\times 10^{-4}$.
\begin{table}[!h]
\caption{The upper bounds of $|c_S|$ and $|c_V|$ with
$\Lambda_{\cal U}=1$ TeV and $d_{\cal U}=3/2$. }\label{t1}
\begin{ruledtabular}
\begin{tabular}{ccc}
& From B-system & From D-system \\ \hline
$|c_S|$ & $3.4\times 10^{-2}$ & $2.1\times 10^{-2}$ \\ \hline
$|c_V|$ & $2.3\times 10^{-3}$ & $5.0\times 10^{-4}$
\end{tabular}
\end{ruledtabular}
\end{table}
Then we consider the case with fixed $c_S$, $c_V$ and study the
dependence of the $D^0-\overline{D^0}$ mixing parameter $x_D$ on the scale
dimension $d_{\cal U}$. Figs. \ref{fig2} and \ref{fig3} plot the
depedence within the parameter range $1<d_{\cal U}<2$. We find that $x_D$
is very sensitive to $d_{\cal U}$ and decreases rapidly to zero as
$d_{\cal U}$ increases.
\begin{figure}[!htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=10cm]{mixingP.eps}
\end{tabular}
\end{center}
\label{figure} \caption{ The $D^0-\overline{D^0}$ mixing parameter $x_D$
versus unparticle scale dimension ($1<d_{\cal U}<2$). The solid line is
given for $|c_S|=1\times 10^{-2}$ and the dashed line for
$|c_S|=2\times 10^{-2}$.} \label{fig2}
\end{figure}
\begin{figure}[!htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=10cm]{mixingV.eps}
\end{tabular}
\end{center}
\label{figure} \caption{ The $D^0-\overline{D^0}$ mixing parameter $x_D$
versus unparticle scale dimension ($1<d_{\cal U}<2$). The solid line is
given for $|c_V|=2\times 10^{-5}$ and the dashed line for
$c_V=5\times 10^{-5}$.} \label{fig3}
\end{figure}
Moreover, we also investigate the case with extending the scale
dimension to the region $2<d_{\cal U}<3$ and depict the dependence of
$x_D$ on $d_{\cal U} $ in Figs. \ref{fig4} and \ref{fig5}. There is no
principal difference compared to the $1<d_{\cal U}<2$ case except a
considerable change for the coupling parameters $c_S$ and $c_V$
which are required to fit the data.
\begin{figure}[!htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=10cm]{mixingP2.eps}
\end{tabular}
\end{center}
\label{figure} \caption{ The $D^0-\overline{D^0}$ mixing parameter $x_D$
versus unparticle scaling dimension ($2<d_{\cal U}<3$). The solid line is
given for $|c_S|=10$ and the dashed line for $|c_S|=20$.}
\label{fig4}
\end{figure}
\begin{figure}[!htb]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=10cm]{mixingV2.eps}
\end{tabular}
\end{center}
\label{figure} \caption{ The $D^0-\overline{D^0}$ mixing parameter $x_D$
versus unparticle scaling dimension $2<d_{\cal U}<3$. The solid line is
given for $|c_V|=2\times 10^{-2}$ and the dashed line for
$|c_V|=5\times 10^{-2}$.} \label{fig5}
\end{figure}
\section{Conclusions}
We have investigated the new physics effects from scale invariant
unparticle sectors on the mixing of $B^0-\overline{B^0}$ and $D^0-\overline{D^0}$. The
exchange of unparticle induces the FCNC transitions at tree level
and provides new contribution to the mass difference of the meson
mass eigenstates. In principle, FCNC transitions may be caused by
other new physics effects which contain heavy massive particles and
break the scale invariance. We observe a peculiar effect caused by
the exchange of unparticle: the mixing parameter depends
non-trivially on the neutral meson mass. This dependence might not
occur for the heavy particle exchange from other new physics. We use
the data on $B^0-\overline{B^0}$ and $D^0-\overline{D^0}$ mixing to constrain the
parameters in unparticle scenario. We find that the $D^0-\overline{D^0}$
mixing provides the most stringent constraint on the coupling of the
scalar and vector unparticles to the SM quarks. The upper bounds we
obtained from $D^0-\overline{D^0}$ mixing are: $|c_S|<2.1\times 10^{-2}$ and
$|c_V|<5.0\times 10^{-4}$ if we set the energy scale $\Lambda_{\cal U}=1$
TeV and scale dimension $d_{\cal U}=3/2$. The dependence of scale
dimension $d_{\cal U}$ shows that the mixing parameter is sensitive to
the scale dimension and decreases rapidly by almost two orders of
magnitude. The obtained parameters may have important effects on CP
violation in B and D decays.
\section*{Acknowledgments}
Z. Wei would like to thank Guo-Huai Zhu and Chuan-Hung Chen for
valuable discussions. This work was supported in part by NNSFC
under contract No. 10475042.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,310 |
Panulirus ornatus е вид десетоного от семейство Palinuridae. Видът не е застрашен от изчезване.
Разпространение и местообитание
Разпространен е в Австралия (Куинсланд, Нов Южен Уелс и Северна територия), Джибути, Египет, Еритрея, Йемен, Кения, Мозамбик, Нова Каледония, Папуа Нова Гвинея, Саудитска Арабия, Соломонови острови, Сомалия, Танзания, Фиджи, Южна Африка (Квазулу-Натал) и Япония (Кюшу, Хоншу и Шикоку).
Обитава крайбрежията и пясъчните и скалисти дъна на морета и коралови рифове в райони с тропически климат. Среща се на дълбочина от 9 до 1000 m, при температура на водата от 4,4 до 26,5 °C и соленост 34,5 – 35,1 ‰.
Описание
Популацията на вида е стабилна.
Източници
Литература
Външни препратки
Panulirus | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,332 |
{"url":"http:\/\/stevecrawford.saao.ac.za\/2015\/08\/21\/how-common-are-the-milky-way-and-andromeda\/","text":"# How common are the Milky Way and Andromeda?\n\nA recent paper by Licquia, Newmann, and Brinchmann\u00a0(2015) reported that the Milky Way, if viewed externally, would have an absolute magnitude of\u00a0Mr5logh=21.00 and a rest frame color of (g-r) = 0.682. \u00a0 \u00a0This would place it very near the red sequence or in the green valley, where galaxies are thought to be transitioning to being quiescent systems. \u00a0Red spirals generally only make up about 6% of the spiral galaxy population as found by Masters et al. (2010), and following their definition, the Milky Way would qualify as a red spiral.\n\nNow, the Milky Way makes up part of the local group, whose other major member is the Andromeda Galaxy. \u00a0Andromeda is also a very red spiral galaxy at a distance\u00a0\u00a0of\u00a00.78 Mpc away from the Milky Way and approaching the Milky Way at ~300 km\/s. \u00a0 Galaxies like the Milky Way and Andromeda are likely to be incredible rare. \u00a0Out of a sample of 130000 galaxies,\u00a0Mutch, Croton, and Poole (2011) only found 997 (0.77%) that were similar to the Milky Way or Andromeda.\n\nOf those galaxies, how many are close together to each other? \u00a0This should be easily answerable using the SDSS CasJobs query. \u00a0 Nonetheless, I\u00a0tried to repeat the query from\u00a0Mutch, Croton, and Poole (2011) of galaxies with spiral structure, stellar masses similar to the Milky Way and Andromeda, and exponential profiles along with adding in a color selection, requiring the galaxies to have a redshift less than 0.1, \u00a0and leaving out the face-on requirement. In the end, I found 5375 galaxies fitting the requirement out of an initial sample of 160000 galaxies with z < 0.1. \u00a0 Compared to the early work, I used a slightly higher redshift cutoff and dropped the face on requirement, which explains the greater number of sources.\u00a0 Of those 5375, 36 pairs exist where the two galaxies are within 1.5 Mpc of each other and have \u00a0a total line of sight velocity difference less than 150 km\/s.\n\nSo while the local group is unique, it would appear roughly about 1 out of every 2200 giant galaxies would be in a local group analog (give or take a factor of 2). \u00a0 \u00a0I have to admit my initial feeling would be that they would be more rare than that. \u00a0Then again, this was a quick calculation and their might easily be several factors that I am missing.\n\nOf course, it might be interesting to follow-up a few of these systems in detail to see what might be in store for the local group. \u00a0\u00a0\u00a0Mutch, Croton, and Poole\u00a0actually suggest that the Milky Way-Andromeda merger will more likely be a dry merger when it happens and maybe some of these systems can give us some ideas about what is in store for the Milky Way.\n\nFor the record, here is the query I used. \u00a0If anyone has better ideas, please feel free to suggest it \u00a0and I\u2019ll be happy to post an update.\n\n-- Find red spiral galaxies similar to the Milky Way and Andromeda\n\nSELECT count(g.objID) -- g.objID, g.ra, g.dec, t04_spiral_a08_spiral_debiased, sp.z, sm.mstellar_median, fiberMag_g - fiberMag_r, log10(1\/expAB_r)\nFROM Galaxy as g\n join dr10.zoo2MainSpecz as z on g.specobjid=z.specobjid\n join SpecObj as sp on g.specobjid=sp.specobjid\n join stellarMassPCAWiscBC03 as sm on g.specobjid=sm.specobjid\n --join Photoz as pz on g.objID = pz.objID\nWHERE\n r < 24 -- r IS NOT deredenned\n and sp.z < 0.1\n and t04_spiral_a08_spiral_debiased > 0.8 --spiral structure is seen\n and sm.mstellar_median > 10.66 and sm.mstellar_median < 11.2 -- only galaxies with stellar masses similar to MW\/M31\n and fiberMag_g - fiberMag_r > 0.63 -- find all red objects (this should be restframe color, but k-corrections should be small)\n and fracDeV_r < 0.5 -- added to remove bulge dominated galaxies\n\nAnd here\u2019s an image of one of the systems that are very close together, however the on-sky separation of some of these systems can be visually quite large (0.78 Mpc at z=0.05 corresponds to roughly 0.2 degrees).\n\nCredit: SDSS\n\nSome Caveats: \u00a0The\u00a0Mutch, Croton, and Poole\u00a0estimate included some assumptions about viewing angle, so there probably are more systems where one object is being viewed edge on. Nonetheless, it goes into much more thorough detail then I do here and also has some very interesting thoughts about the long term evolution of the systems. \u00a0In addition, I\u2019ve done nothing to control for spectroscopic incompleteness. \u00a0 Also my SDSS-fu is very rusty so I could have screwed up the above query and I couldn\u2019t figure out\/remember an easy way to get to rest frame properties from the SDSS database. \u00a0 Also a more thorough job could have been done to decide bound\/un-bound pairs.\n\nReferences and Acknowledge: \u00a0References are many linked from above, but I also made use of NED, astropy, but primarily the data from SDSS. \u00a0 H\/t to @jbprime for the original tweet.","date":"2018-04-26 11:08:23","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\": 0.5917243361473083, \"perplexity\": 1798.4150546030553}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-17\/segments\/1524125948126.97\/warc\/CC-MAIN-20180426105552-20180426125552-00175.warc.gz\"}"} | null | null |
Alexis Hill may refer to:
Alexis Hill (romance writer), romance pseudonym used by Mary Francis Shura (1923–1991)
Alexis Hill, romance pseudonym used by Ruth Glick and Louise Titchener | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,517 |
Do košíka (predobjednávka)
'Call me Ishmael'. So begins Moby-Dick, Herman Melville's epic account of the last voyage of the ill-fated whaling ship Pequod, and its captain's obsessive pursuit of the legendary white whale that maimed him years before. Melville's classic novel has given American literature some of its most iconic characters. Inspired by the real-life ordeal of the crew of the whaling ship Essex--who, in 1819, were set adrift in the heart of the sea for eighty-nine days, after the whale they were hunting stove in their ship's hull--and steeped in the lore and legendry of whaling, Melville's novel is widely regarded as one of the greatest American novels. More than a rousing tale of adventure on the high seas, Moby-Dick is acknowledged today as a fundamental exploration of the ideas and interests that shaped the American experience in the nineteenth century. The text of Moby-Dick in this volume is from the authoritative Northwestern Newberry edition of The Writings of Herman Melville. Moby-Dick is one of Barnes & Noble's leatherbound classics. Each volume features authoritative texts by the world's greatest authors in an exquisitely designed bonded-leather binding, with distinctive gilt edging and a silk-ribbon bookmark.
Confidence Man and Billy Budd
Moby-Dick
Moby Dick Hc
Herman Melville, Christophe Chaboute, John Arcudi
Heidi: The Sisterhood
The Strange Case of Dr Jekyll and Mr Hyde and other stories
Portrait of the Artist as a Young Man
The Besieged City | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,427 |
La contea di Hitchcock (in inglese Hitchcock County) è una contea dello Stato del Nebraska, negli Stati Uniti. La popolazione al censimento del 2000 era di 3.111 abitanti. Il capoluogo di contea è Trenton.
Altri progetti
Collegamenti esterni
Hitchcock | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 647 |
Helmut Poppendick ( – ) was a German medical doctor who served in the SS during World War II. He was an internist and worked in the Medical Doctorate, as Chief of the Personal Staff of the Reich Physician SS and Police. After the war he was a defendant in the Doctors' Trial.
He studied medicine from 1919 to 1926 in Göttingen, Munich, and Berlin. Poppendick received his medical license on 1 February 1928. Then, he worked for four years as a clinical assistant at the First Medical Clinic of Charité in Berlin. From June 1933 to October 1934 he was the assistant medical director at Virchow Hospital in Berlin.
In 1935, he completed training as an expert for "race hygiene" at the Kaiser Wilhelm Institute for Anthropology, Human Genetics and Eugenics. After this, he became the adjutant of the ministerial director Arthur Gütt at the Reich Ministry of the Interior. He was also the chief of staff at the SS Office for Population Politics and Genetic Health Care, which in 1937 became the SS Main Race and Settlement Office. Poppendick was departmental head and staff leader of the Genealogical Office.
At the beginning of World War II, he was drafted as an adjutant to a medical department of the army and took part in the attack on Belgium, France and the Netherlands. In November 1941, Poppendick was accepted into the Waffen-SS. In 1943, Ernst-Robert Grawitz of the Reich Physician SS appointed him to lead his personal staff. Poppendick joined the NSDAP in 1932 (party member No. 998607) and the SS (No. 36345). He reached the rank of Oberführer in the SS.
Poppendick was implicated in a series of medical experiments done on concentration camp prisoners, including the medical experiments done in Ravensbrück. At the American Military Tribunal No. I on 20 August 1947, he was acquitted from being criminally implicated in medical experiments. However, the court ruled in several instances, there was substantial evidence that Poppendick was of illegal human experimentation taking place. Poppendick was sentenced to 10 years imprisonment for being a member of the SS, which had been deemed a criminal organization by the International Military Tribunal. On 31 January 1951, Poppendick's sentence was commuted to time served, and he was released from prison the next day. Later on, Poppendick managed to get his medical services paid by insurance, in Oldenburg.
See also
Doctor's Trial
Nazi human experimentation
References
1902 births
1994 deaths
Nazi Party officials
Physicians in the Nazi Party
Nazi human subject research
SS-Oberführer
People convicted by the United States Nuremberg Military Tribunals
Waffen-SS personnel | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,517 |
\section{Introduction}
With the evolution of the technology for user interaction, the focus of the interface has gradually shifted from being computer-centered to being human-centered. In the last few decades, we have seen a great interest in the use of eye-tracking for Human-Computer Interaction (HCI) \cite{quek1995eyes, zhai1999manual, jacob1990you}. The use of eyes as an input method to track the users' gaze direction provides a natural interface without having touch-based inputs. However, the robustness of the eye-tracking methods depend on various factors such as the sensors used (e.g. head-mounted tracking device or a non-contact tracking device). Eye-gaze for user interaction imposes some restrictions as it can be very volatile and has an `always-on' characteristic. Therefore, it lacks a natural trigger for object selection.
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{samples/bmw-natural-inteaction.jpg}
\caption{Driver makes a pointing gesture to interact with the car. Image courtesy of \cite{bmwnatint}}
\label{fig:teaser}
\end{figure}
Another important and natural source of input for user interaction are mid-air gestures. Since early interactive systems, such as Bolt's seminal work ("Put that there" \cite{bolt1980put}), mid-air pointing gestures have also been of significant interest for user interaction \cite{walter2014cuenesics, schweigert2019eyepointing}. This is because they enable users to point to and reference objects that are too far away to touch in a natural manner, especially with the help of speech commands \cite{sauras2017voge}.
In this paper, we integrate the deictic information from gaze and head pose, along with a specific gesture, i.e., finger pointing gesture, in order to combine the efficiency and naturalness of these modes of interaction, using a late fusion approach. There are two main reasons for this multimodal integration: 1) to compensate the drawbacks in one modality using the other and 2) to enhance the overall performance of the user interaction by taking advantage of the temporal relations between them.
Although the use of multiple inputs for interaction can have numerous applications such as for assistance of people with disabilities, we focus our work on the use of head pose, eye-tracking and finger pointing for applications in the automobile industry. The motivation is driven by the direct application in the BMW Natural Interaction, presented at the Mobile World Congress 2019, that enables genuine multimodal operation for the first time \cite{MWC}.
While driving, the driver's cognitive attention is affected by numerous tasks such as navigating the infotainment system, operating the in-vehicle control units etc. It has been observed that the gestural interfaces are fairly easy to use and they reduce the cognitive load on the driver \cite{riener2012gestural, rumelin2013free}. For this reason, intuitive interactions via free hand pointing gestures are becoming increasingly prevalent for in-vehicle infotainment systems, which is particularly useful for novice users \cite{ahmad2017does}. Furthermore, multimodal interfaces also have a tendency to reduce the driver load while driving \cite{manawadu2017multimodal}. We, therefore, propose an in-vehicle user interaction, that supplements the touch-based inputs, in which the driver is able to operate control modules in the vehicle in a touchless and natural manner that is also comparatively less distracting. In order to identify the desired object or Area-of-Interest (AOI), the user may use a finger pointing gesture, as this type of gesture provides a deictic reference to the various real-world objects, as shown in Figure \ref{fig:teaser}. The action to be performed on the selected object may be provided by speech commands, such as, "what is \textit{that}?" or "close \textit{that} window".
In the context of this paper, we focus on the recognition of the object that the user selects with a combination of gaze, head pose and finger pointing. It has been observed that drivers make relatively large errors in pointing \cite{brand2016pointing, roider2018implementation}, and the integration of gaze improves the accuracy of pointing while driving \cite{roider2018see}. This is because, while driving, the eye-gaze is mainly focused on the road, but momentarily moves and fixates on the target object. In the past, it was observed that gaze anchoring to a target existed for the entire duration of the pointing movement of the finger \cite{neggers2001gaze}. More recently, Ahmed \textit{et al.} show that in most cases, the driver first looks towards the desired object before making the pointing gesture and that there exists a misalignment between the gaze and finger movements \cite{ahmad2017does}. We exploit this relation between the eye-gaze and the finger pointing motion to accurately choose the object located in the entire field-of-view of the driver. Furthermore, we include head pose as well as it is directly linked with eye-gaze \cite{jha2016analyzing}.
\section{Related work}
Multimodal user interaction has a wide variety of applications for in-vehicle functions.
Mitrevska \textit{et al.} demonstrate the use cases of an adaptive multimodal control of in-vehicle functions with the help of an individual modality (speech, gaze or gesture) or a combination of two or more modalities \cite{mitrevska2015siam}.
Apart from the applications, research has shown that the use of multiple input modalities have a potential to outperform systems with a single input modality \cite{liu2018efficient, esteban2005review, turk2014multimodal}. Due to this, the fusion of multiple modalities has been used by many researchers for user interaction. A multimodal technique for selection of objects on the screen, namely the MAGIC pointing, was presented by Zhai \textit{et al.} about two decades ago \cite{zhai1999manual}. This technique allows the user to select objects on a screen by fixating on the target and pressing a regular manual input device to trigger the selection. The Midas-touch problem faced in this technique can be overcome by using mid-air gestures to trigger the selection as presented in \cite{schweigert2019eyepointing, chatterjee2015gaze+, nesselrath2016combining}. Gaze is often used for selection of objects on a screen while using an additional trigger such as using a speech command \cite{maglio2000gaze}. However, selection performed exclusively by gaze is difficult, especially when objects are placed very close to each other \cite{hild2019suggesting}.
EyePointing is an extension to MAGIC pointing that uses finger pointing as a trigger for object selection \cite{schweigert2019eyepointing}. Chatterjee \textit{et al.} demonstrated a better outcome with the integration of gaze and gesture as inputs as compared to systems with only gaze or gesture \cite{chatterjee2015gaze+}. Similarly, Nesselrath \textit{et al.} use a combination of three modalities, speech, gaze and gestures, to initially select objects of the vehicle, e.g., side mirrors or windows, and then use gestures or speech to control these objects \cite{nesselrath2016combining}.
These approaches primary use the gaze information and enhance the naturalness of the user interaction with secondary modality, e.g. gestures or speech. In contrast to this, Sauras-Perez \textit{et al.} propose the use of speech with the finger pointing gesture for selection of Points-of-Interest (POI) while driving a vehicle \cite{sauras2017voge}.
Unlike these approaches, we do not use finger pointing gesture as a trigger for selection, but rather use the finger ray-cast in addition to gaze ray-cast and head pose to fuse them together.
The fusion of the modalities is used for better performance while using speech as the trigger. Each modality is treated equally at the input, and the model learns the weights on its own during the training process.
The problem of object selection inside a car has also been presented by Roider \textit{et al.} who integrate eye gaze with finger pointing gestures in a passive manner using a simple rule-based fusion approach. They have shown that the selection on an in-vehicle display screen achieves increased pointing accuracy over the single modality, i.e., finger pointing \cite{roider2018see}. This experiment is limited to only four objects on a screen adjacent to each other. In our work, we enhance the object selection using a late fusion approach to select a wide range of objects inside the vehicle that lie in the hemisphere in front of the driver. We use an additional modality, i.e. head pose, because the head pose and gaze direction are mostly directly related and are usually considered together for recognizing the visual behaviour \cite{ji2002real, mukherjee2015deep}. Another reason for including head pose is that eyes can be easily occluded when using eye-tracking sensors that are fixed for practical reasons rather than head-mounted. Head pose may prove to be beneficial in such cases.
Head pose and gaze direction are used by Jha and Busso, in their work, to estimate driver's gaze direction \cite{jha2016analyzing}. They use simple linear regression models to predict the direction of gaze. Compared to this, the novelty in our method is that we allow the neural network to learn the relations between the three modalities, and then use linear regression to predict the output.
A driver query system, similar to our experimental setup, is presented by Kang \textit{et al.} in which the smart car is able to figure out where the driver is looking at using visual cues, head pose and speech \cite{kang2015you}. We show that the accuracy of the driver's query can be further improved tremendously by adding finger pointing. However, we do not consider speech as an input to the fusion as it is a very different problem altogether, but rather use speech as a trigger for fusion. The dialogue duration has an impact on the interaction between the car and user \cite{strayer2014measuring}, which we do not explore.
Deep neural networks have been applied to feature fusion for various tasks \cite{ngiam2011multimodal}, but they usually deal with abstract data or abstract features. In \cite{olabiyi2017driver}, Olabiyi uses Deep Recurrent Neural Networks to perform a fusion of sensory inputs to predict driver actions. We use a similar concept and apply Convolutional Neural Networks (CNN) on the sequential (temporal) input data.
To sum up, we extend and combine concepts from various previous research for a more robust performance that can be practically applied in a real car. While many studies have been performed in simulators, only a few are presented in a real-world scenario. For practical reasons, our experiments are performed in a real car to provide the users a more realistic impression, and, therefore, to achieve more significant results than a simulation.
\begin{figure}[t]
\centering
\includegraphics[trim=0 120 0 0,clip, width=\linewidth]{samples/AOI_car.jpg}
\caption{The 12 AOIs in the cockpit }
\label{fig:AOI_car}
\end{figure}
\section{Data collection}
We consider a simple yet a very productive cockpit use case, that is object selection inside the vehicle. We chose to set up our sensors for data collection in a stationary but functional BMW vehicle rather than a simulator to have apt uses in real world. This is why we do not use head-mounted sensors as they may hinder the driver's cognitive abilities while driving.
We do not consider the action to be performed in the context of this work. The selection of objects outside the car is excluded in this work. It is a future extension of this work, and we hope to integrate outside use cases in the future as well, e.g., pointing to buildings or landmarks, and inquiring about them.
\subsection{Apparatus}
There are two types of camera systems used to capture the 3D information of the driver: the Gesture Camera System and the Visual Camera System.
\subsubsection{Gesture Camera System}
The gesture camera, mounted next to the Roof Function Centre of the car, captures hand and finger movements in the 3D space using a Time-of-Flight (ToF) camera. It has a wide Field-of-View so that it covers almost the entire operating zone of the driver. The gesture camera system detects a finger pointing gesture and calculates the vector from the tip of the finger to the base of the finger. The 3D coordinates of the fingertip are used as the finger position.
\subsubsection{Visual Camera System}
The visual camera is a high-definition camera with a built-in technology that evaluates the images of the driver and calculates the required 3D vector data for the head pose and eye-tracking. It uses an actively illuminated infrared (IR) sensor to capture eye and head movements even in low lighting. The camera is placed behind the steering wheel in such a way that eye-occlusion is minimal, and it does not interfere with the driver's attention on the road. The algorithm integrated into the camera system calculates the head rotation as three Euler angles (roll, pitch and yaw), and the 3D coordinates of the estimated centre position of the head. In addition to this, it provides eye position which is the 3D coordinates of the cyclops eye (i.e., centre point between the two human eyes), and the 3D vector coordinates of the eye direction merged together from both the left and the right eye.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{samples/scatter_AOI.png}
\caption{3D scatter plot of the measured AOI points. }
\label{fig:scatter_AOI}
\end{figure}
\subsubsection{Speech command}
Apart from the visual camera and the gesture camera systems, we use a speech command along with the pointing gesture which we implemented with the Wizard-of-Oz (WoZ) method. In order to record the timestamp of the pointing gesture, a secondary person (acting as the wizard) pushes a button manually that stores the timestamp at the instant when the primary user points to an object and says, "what is \textit{that}?" The aim was to note the timestamp when the word "\textit{that}" is said. However, there may be human error involved in measuring this timestamp.
\subsubsection{Selection of AOIs}
Based on the use-cases for the driver interaction, we chose twelve distinct, but closely situated, control modules in the cockpit of the car as Areas-of-Interest (AOIs). They are illustrated with red circles in Figure \ref{fig:AOI_car}. A 3D scatter plot of all the measured points for each of the 12 AOIs is illustrated in Figure \ref{fig:scatter_AOI}. The `x' represent measured points, whereas the `o' shows the mean of the measured point of the AOI.
\subsection{Participants}
In this experiment, we collected data from 22 participants aging between 20 and 40 years old. 5 of these participants were female, and the remaining 17 were male. The drivers were asked to point to the various AOIs in a stationary vehicle and give the command, "what is \textit{that}?" There was no further instruction provided so that the pointing gesture can be as natural as possible. The participants were free to choose either hand and use any finger for pointing. 15 \% samples of pointing samples were performed with left hand while the remaining were with the right hand. About 30 \% of participants wore glasses, 10 \% wore contact lenses, and the remaining had no glasses or lenses.
\subsection{Dataset Statistics}
From each of the 22 drivers, we collected 10 pointing gesture events for each of the 12 AOIs. The entire dataset consisted of 120 samples for every participant where each data sample consists of exactly one gesture pointing event. In total, we had 2640 samples that we collected for the cockpit use case. Due to errors in recording, 60 samples were discarded. Therefore, 2580 samples were used.
The difference between the estimated and actual direction, calculated for each of the modalities, is termed as the estimation error. An error in estimation is considered when the vector direction is outside the visible surface area of the AOIs. The mean and standard deviation of the estimation error for the three modalities in the horizontal direction (azimuth) and the vertical direction (elevation) are shown in Table \ref{Table:AOIs}. It can be seen that the eye direction has relatively large errors for the first three AOIs, while the head direction has relatively large errors for AOIs 10 and 11. The large errors in the elevation angle of the head direction suggest that the head movement in the vertical direction is considerably small even when looking downwards (as almost all of the AOIs lie below the car windscreen).
This is a relatively small dataset, especially when using deep neural networks. However, the outcomes that we achieved from such a small dataset (in section \ref{results}), show a great proficiency in our approach, which may even be enhanced using a larger dataset. Consequently, a much larger dataset that will be used in the future is being collected which incorporates more use cases as well.
\begin{table}[b]
\centering
\footnotesize
\begin{tabularx}{\linewidth}{ c| c c | c c |c c }
\multirow{3}{*}{AOI}
& \multicolumn{2}{c|}{Eye Direction} & \multicolumn{2}{c|}{Head Direction} & \multicolumn{2}{c}{Finger Direction} \\
\hline
& \textbf{Azimuth} & \textbf{Elevation} & \textbf{Azimuth} & \textbf{Elevation} & \textbf{Azimuth} & \textbf{Elevation} \\
& \textbf{M (SD)} & \textbf{M (SD)} & \textbf{M (SD)} & \textbf{M (SD)}& \textbf{M (SD)} & \textbf{M (SD)} \\
\hline
1 & 26° (18°) & 13° (11°) & 5° (7°) & 37° (10°) & 17° (35°) & 15° (17°) \\
2 & 23° (17°) & 11° (11°) & 4° (6°) & 36° (10°) & 11° (22°) & 7° (14°) \\
3 & 25° (16°) & 19° (13°) & 3° (8°) & 46° (11°) & 15° (30°) & 17° (17°) \\
4 & 2° (5°) & 1° (4°) & 2° (5°) & 23° (8°) & 1° (7°) & 4° (8°) \\
5 & 1° (4°) & 1° (5°) & 1° (4°) & 22° (8°) & 1° (5°) & 5° (8°) \\
6 & 5° (5°) & 3° (2°) & 7 (8°) & 21° (8°) & 9° (16°) & 8° (11°) \\
7 & 1° (3°) & 1° (2°) & 3° (7°) & 10° (7°) & 1° (6°) & 1° (5°) \\
8 & 3° (6°) & 3° (6°) & 5° (5°) & 32° (11°) & 3° (10°) & 4° (8°) \\
9 & 2° (4°) & 3° (6°) & 21° (12°) & 33° (14°) & 9° (19°) & 6° (11°) \\
10 & 12° (13°) & 4° (7°) & 34° (14°) & 36° (14°) & 29° (29°) & 26° (23°) \\
11 & 25° (14°) & 9° (11°) & 45° (14°) & 31° (13°) & 27° (28°) & 25° (24°) \\
12 & 17° (18°) & 4° (5°) & 11° (14°) & 15° (7°) & 10° (26°) & 6° (11°)
\end{tabularx}
\caption{Mean (M) and Standard Deviation (SD) of the estimation error (in degrees) of eye, head and finger direction. }
\label{Table:AOIs}
\end{table}
\begin{figure}[t]
\begin{subfigure}
\centering
\includegraphics[trim=0 0 0 40,clip, width=\linewidth]{samples/ray_casting.png}
\caption{The modalities used: gaze and finger ray-cast (left) and head pose (right)}
\label{fig:rays}
\end{subfigure}
\begin{subfigure}
\centering
\includegraphics[trim=30 0 0 0,clip, width=\linewidth]{samples/origin.png}
\caption{Change of origin from centre of front axle of the car (left) to the centre of the driver's seat (right)}
\label{fig:origin}
\end{subfigure}
\end{figure}
\section{Methodology}
Our proposed architecture is given in Figure \ref{fig:methodology}. The camera systems are treated as black boxes that provide input processed from the images. A late fusion is applied on the preprocessed data to estimate the driver's referenced object by matching to the known AOIs.
\begin{figure*}[t]
\includegraphics[width=0.97\linewidth]{samples/Methodology.png}
\caption{A multimodal late fusion architecture}
\label{fig:methodology}
\end{figure*}
\subsection{Preprocessing}
The visual camera and the gesture camera systems provide the head, eye and gesture pose. Pose consists of both position and direction/rotation of the individual modalities. The finger and eye-gaze directions use ray casting while, for the head rotation, we use yaw, pitch and roll as inputs, as shown in Figure \ref{fig:rays}. The direction vectors of eye and finger are normalized to have a unit norm. Due to occlusion of the eyes or the finger, there are some frames with missing data. Occlusion of the eyes mainly occurs when the driver looks downward, and therefore, the eyelids occlude the pupils, or when the pointing arm comes in front of the face. To fill the missing data, we use linear interpolation from the two nearest neighbouring frames. Afterwards, camera calibrations are applied to give real-world coordinates with respect to the origin, i.e., the centre of the front axle of the car, which is the ISO standard \cite{sayers1996standard}. We apply a translation in order to translate the origin point to the centre of the driver's seat as illustrated in Figure \ref{fig:origin}. We observed from experiments that using such a translation makes the learning process in neural networks slightly better, i.e., the translation lead to an increased accuracy.
We evaluated different time intervals to figure out the right interval to use as input. The results are omitted in this paper. It was found out that choosing a small time period of 0.2 seconds at the instant when the WoZ button is pressed provided sufficiently good results. At about 45 fps, this time period amounts to 8 frames from each sample, 4 frames before the noted timestamp by the wizard, and 4 frames after it.
\subsection{Fusion Algorithm}
We use machine learning methods, particularly Deep Neural Networks (DNN), to fuse the three modalities. The motivation behind the choice is that there may be a number of different cases that may be difficult to address with a rule based approach. DNNs, with supervised learning, have a tendency to tackle the different cases on their own, provided that the dataset has a large variance. Additionally, DNNs are easily expandable to add more use cases which we will consider in the future work as well. We present a base model and show the comparison of results to other similar models.
\subsubsection{Ground Truth Definition}
The corner points of the AOIs in the car were measured, w.r.t to the origin (i.e. the centre of the driver's seat), as illustrated in Figure \ref{fig:scatter_AOI}.
We then define the ground truth as the 3D vector (with unit norm) calculated from the origin to the mean point of the measured points. The mean of the measurements reveal to be very close to the actual centre of the AOI (see Figure \ref{fig:scatter_AOI}).
\subsubsection{Base model}
The input to the model, $x$, is a batch of samples of size $b$, such that $x \in \mathbb{R}^{b \times f \times a \times d}$, where $f$ is the number of frames used, $a$ is the number of feature attributes used and $d$ is the number of dimensions in each attribute. We chose 6 feature attributes as inputs: the position and direction of eye, head and finger. Each of these has 3 dimensions, representing a point or a vector in the 3D vector space. The inputs from 8 frames are concatenated together. The base model is a deep Convolutional Neural Network (CNN), consisting of two 2D convolutional layers and one fully connected layer. The output is linearly regressed to give a $(3 \times 1)$ vector for the fused direction.
We use the cosine similarity between the predicted vector and the ground truth vector as the loss function, i.e., the cosine of the angle between the two vectors. The loss function is, thus, given as:
\begin{equation}
\mathcal{L} = \frac{1}{N} \sum^N_{i=1} \text{cos} (\theta_i)
= \frac{1}{N} \sum^N_{i=1} \frac{\textbf{\^y}_i \ . \ \textbf{y}_i}{\|\textbf{\^y}_i \| \ \| \textbf{y} _i\|}
\label{eqn}
\end{equation}
where $ \textbf{ \^y }_i$ is the $i$-th predicted fusion vector, $\textbf{y}_i$ is the $i$-th ground truth vector, $\theta_i$ is the angle between the two 3D vectors, and $N$ is the total number of samples. Therefore, we have $\mathcal{L} \in [-1, 1]$.
\subsubsection{Other models}
The base CNN model is compared with other models to show the performance of the fusion using various approaches. These include a Fully-Connected Neural Network (FC-NN), Recurrent Neural Network (RNN), Support Vector Regression (SVR), and Random Forests (RF) regression. The RNN has 2 LSTM (Long Short-Term Memory) layers and 1 fully connected layer. The FC-NN consists of 3 fully connected layers. Both, FC-NN and RNN, use the cosine similarity as the loss function. Conventional machine learning approaches, namely SVR and RF, are evaluated to figure out if the model complexity with DNNs is an overkill. For the SVR, the polynomial kernel is used with degree 2.
\subsection{Matching Predictions with AOIs}
\label{matching}
In order to identify the desired object by the user, we measure the cosine similarity between the predicted vector and the 12 AOIs separately. The one with the highest cosine similarity is chosen. In other words, the one with the lowest angular deviation from the predicted value is chosen.
\begin{figure*}[t]
\includegraphics[width=\linewidth]{samples/driver_pointing.png}
\caption{Driver points to AOI 5 (multimedia) in the car. The face is blurred due to the privacy policy.}
\label{driver_pointing}
\end{figure*}
\section{Experiments and Results}
\label{results}
In the context of this paper, we report only one use-case, i.e., the cockpit use case. Figure \ref{driver_pointing} shows an example of the driver pointing to one of the AOIs.
Other experiments are in progress for further use cases, and the results will be presented in future publications.
Due to the small size of the dataset, we use cross validation to report the results. A leave-one-out cross-validation (22-fold cross-validation) is used for training and evaluation of the dataset. The dataset is split into three, training, validation and testing. With the leave-one-out cross validation, the test set covers the entire dataset.
The train/test split of the data is user-based, i.e., no sample from the participants in the training set appears in either the validation or the test set and vice versa. This means that each fold of the data contains 120 samples ($\approx$ 5\%) from each driver, respectively.
We train the model for 100 epochs using a batch size of 8 and the Adam optimizer with a learning rate of 0.001. The best performing model on the validation set is selected and tested on the test set in order to avoid overfitting. The test results on the 22 folds are averaged to give the final value. By averaging using the cross-validation, we avoid the bias in the results which may occur due to certain user specific referencing.
\subsection{Metrics}
For the training and test loss, we use the cosine similarity function as shown in Equation \eqref{eqn}. For performance measures, we use two metrics: accuracy and Mean Angular Deviation (MAD).
\subsubsection{Accuracy} We use accuracy to evaluate the classification performance of the models for the 12 AOIs. Accuracy is the percentage of the correctly identified finger pointing samples:
\begin{align}
\text{Accuracy} = \frac{TP}{{N}} \times 100 \%
\end{align}
where $N$ is the total number of predictions and $TP$ is the total number of true predictions (or correct identification) by the model.
\subsubsection{Mean Angular Deviation (MAD)} We define MAD as the mean of the angles between the predicted vectors and the corresponding ground truth vectors in the 3D vector space. This evaluates the performance of the regression output. Consequently, the lower the MAD, the better. Mathematically, we have:
\begin{align}
\text{MAD} = \frac{1}{N} \sum^N_{i=1} \theta_i = \frac{1}{N} \sum^N_{i=1} \text{arccos}\left( \frac{\textbf{\^y}_i \ . \ \textbf{y}_i}{\|\textbf{\^y}_i \| \ \| \textbf{y} _i\|} \right)
\end{align}
\subsection{Ablation Study}
In order to see the effect of each individual modality, we first do an ablation study. The ablation study also includes the removal of the difficult cases of AOIs, and analyzing the results. For all the experiments in the ablation study, the base model (CNN) is used with a 22-fold cross-validation.
\subsubsection{Effect of removing modalities}
The training consists of either one modality (e.g. position and direction of eye only), a combination of two modalities or a fusion of all three modalities. The cross-validation scores are shown in Table \ref{Table:monomodal}. It can be seen that the finger pointing accuracy (64.5\%) is significantly higher than the other two modalities. One possible reason for the low accuracy for the gaze (38\%) might be given by the missing gaze data. In the data, we found 662 samples out of 2640 to have the gaze data missing at the instant when the wizard pushes the trigger. Half of the missing gaze data occurs for the AOI 1, 2 and 3 for which the driver needs to look downwards to the right, while a quarter of these occur for AOI 10 and 11. The effect can also be seen in the confusion matrix shown in Figure \ref{fig:conf_mat} where AOIs 1, 2 and 3 as well as AOIs 10 and 11 have many misclassifications.
From the results, we see that the main contributor is the finger pointing gesture. By adding an additional modality on top of finger, there is a slight increase of about 4\% in performance. Moreover, using all the three modalities, increases the accuracy of further by about 4 - 5\%. In a way, the head pose compensates for the missing eye gaze information and vice versa to improve the monomodal resultant of finger pointing.
\subsubsection{Effect of removing difficult classes}
As it was observed that there was a significant amount of gaze data that was missing for the AOIs 1, 2, and 3, we removed these samples from our dataset to see the effect of gaze. There were 1940 samples that remained in total. The results are shown in Table \ref{Table:monomodal2}.
It is observed that using these 9 AOIs (AOI 4 - 12) only, the eye-gaze accuracy significantly improves from 38\% to 57\%. The MAD also improves with a decrease of 1.6°. This demonstrates that the data collected for the AOIs 1, 2 and 3 has some errors. The accuracy of fusion (of all three modalities) also increases by about 4\%. The confusion matrix for the fusion is shown in Figure \ref{fig:conf_mat_9}. The most misclassifications are seen between AOI 10 and 11 which are very close together on the bottom-left side of the driver. A possible reason may be occlusion of the eyes from the arm when the driver uses the right hand for pointing.
The model accuracy and MAD after further removing AOIs 10 and 11 can be seen in Table \ref{Table:monomodal3}. The confusion matrix, for the model that uses all modalities, is shown in Figure \ref{fig:conf_mat_7}. The accuracy by using only gaze information as input, is increased by 8\% to 65\%, while the accuracy by using all modalities increases by 6\% to 83.9\%.
\subsection{Model Performance on Individual Drivers}
We observed that different driver's point in a very different way. The scatter plot of performance of the CNN based model on the different drivers in the test set is illustrated in Figure \ref{fig:driver_test}. We plotted the accuracy on the `y' axis and MAD on the `x' axis. Therefore, the best point on the plot would be top-left and the worst point would be bottom-right.
We found out that accuracy and MAD of testing the base model on some participants is more than 80\% and less then 3°, respectively. On the other hand, test results on a few other participants demonstrate significantly poor pointing performance, i.e., less than 50\% accuracy and more than 8° MAD. This can be seen at the bottom left corner of the Figure \ref{fig:driver_test}.
Upon removing the two users from the dataset, and performing a 20-fold cross-validation, the model achieved a test accuracy of 75.1\% and a MAD of 5.4°. There is only a slight increase because the dataset becomes even smaller with 20 users.
\begin{figure}[t]
\includegraphics[width=0.89\linewidth]{samples/Confusion_Matrix12.png}
\caption{Confusion matrix for fusion of all three modalities for the 12 classes}
\label{fig:conf_mat}
\end{figure}
\begin{table}[H]
\begin{tabular}{|c | c c | }
\hline
\textbf{Modality} & \textbf{Accuracy} $\uparrow$ & \textbf{MAD} $\downarrow$ \\
\hline\hline
Head & 30.4 \%& 11.7° \\
\hline
Gaze & 38.1 \%& 9.8° \\
\hline
Finger & 64.5 \%& 9.7° \\
\hline\hline
Gaze + Head & 50.5 \%& 7.7° \\
\hline
Finger + Head & 68.5 \%& 6.5° \\
\hline
Finger + Gaze & 69.5 \%& 7.1° \\
\hline\hline
Finger + Gaze + Head & \textbf{73.7 \%} & \textbf{5.2°} \\
\hline
\end{tabular}
\caption{Modality based results using 12 classes (AOIs 1 - 12)}
\label{Table:monomodal}
\end{table}
\begin{figure}[H]
\includegraphics[width=0.883\linewidth]{samples/Confusion_Matrix7.png}
\caption{Confusion matrix for fusion of all three modalities for the 7 classes}
\label{fig:conf_mat_7}
\end{figure}
\begin{figure}[t]
\includegraphics[width=0.89\linewidth]{samples/Confusion_Matrix9.png}
\caption{Confusion matrix for fusion of all three modalities for the 9 classes}
\label{fig:conf_mat_9}
\end{figure}
\begin{table}[th]
\begin{tabular}{|c | c c | }
\hline
\textbf{Modality} & \textbf{Accuracy} $\uparrow$ & \textbf{MAD} $\downarrow$ \\
\hline\hline
Head & 38.5 \% & 11.2° \\
\hline
Gaze & 57.1 \%& 8.2° \\
\hline
Finger & 70.6 \%& 9.7° \\
\hline\hline
Gaze + Head & 64.5 \% & 6.8° \\
\hline
Finger + Head & 72.3 \% & 5.9° \\
\hline
Finger + Gaze & 71.7 \% & 6.6° \\
\hline\hline
Finger + Gaze + Head & \textbf{77.6 \%} & \textbf{4.9°} \\
\hline
\end{tabular}
\caption{Modality based results using 9 classes (AOIs 4 - 12)}
\label{Table:monomodal2}
\end{table}
\begin{table}[th]
\begin{tabular}{|c | c c | }
\hline
\textbf{Modality} & \textbf{Accuracy} $\uparrow$ & \textbf{MAD} $\downarrow$ \\
\hline\hline
Gaze & 65.0 \% & 7.0° \\
\hline\hline
Finger + Gaze + Head & \textbf{83.9 \%} & \textbf{4.1°} \\
\hline
\end{tabular}
\caption{Modality based results using 7 classes (AOIs 4-9, 12)}
\label{Table:monomodal3}
\end{table}
\begin{figure}[H]
\includegraphics[width=0.95\linewidth]{samples/driver_test.jpg}
\caption{Model performance on the individual drivers}
\label{fig:driver_test}
\end{figure}
Upon carefully analyzing the videos, we could not find any substantial reason for the poor performance on some users. One of the possible reasons could be very sunny conditions that caused the drivers to clench their eyes slightly. This resulted in poor quality of gaze data, which had some errors in measurements. Another observation was the alternative use of right or left hand for pointing towards the same AOI. This resulted in different pointing angles because of the position of the arm. However, we can not concretely conclude on any of these reasons. A more extensive study needs to be conducted to explain the reasons behind this user behaviour. The differences in recognition of the pointing direction of the different drivers, shows the need for a personalized fusion approach or the implementation of an online adaptive learning approach.
\subsection{Comparison of Different Models}
In this section we compare the different machine learning models. The results are shown in Table \ref{Table:models}. As can be seen, there is no significant difference between the deep neural network models. However, when compared with SVR, there is a big difference of 19\% in accuracy and a difference of 3° in MAD. When analyzing RF regression against CNN, there is only a small difference of 4\% in accuracy and 1° in MAD. From these results, we can see that even with a small dataset, the deep neural networks are able to learn appropriately, and produce sufficiently good results when compared to conventional machine learning.
\begin{table}[t]
\begin{tabular}{|c || c c || c |}
\hline
\multirow{2}{*}{\textbf{Model}} & \multicolumn{2}{c||}{\textbf{ Linear regression}} & {\textbf{Classification}}\\
\cline{2-4}
& \textbf{Accuracy} $\uparrow$ & \textbf{MAD} $\downarrow$ & \textbf{Accuracy} $\uparrow$ \\
\hline\hline
CNN (base model) & \textbf{73.7 \%} & \textbf{5.1°} & 73.9 \% \\
\hline
RNN & 73.3 \% & 5.9° & 73.0 \%\\
\hline
FC-NN & 72.1 \% & 5.5° & 74.9 \% \\
\hline
SVR / SVM* & 55.0 \% & 8.4° & 71.4 \% * \\
\hline
RF & 69.4 \% & 6.1° & \textbf{76.8 \%} \\
\hline
\end{tabular}
\caption{A comparison of Machine learning models using linear regression (with the cosine similarity loss for DNNs) and classification (using softmax loss for DNNs)}
\label{Table:models}
\end{table}
\subsection{Processing Speed}
We used an Intel Xeon 16 core processor with a Quadro P5000 GPU for evaluating the models. The processing speeds for all the models are presented in Table \ref{Table:speed}. It is to be noted here that SVR and RF are run on the CPU, while the rest use the GPU. SVR appears to be the fastest, processing about 17,800 frames per second (fps) and taking 0.5 milliseconds (ms) to process a sample of 8 frames. RNN, on the other hand, is the slowest taking 15 ms for one sample. Nonetheless, as the neural networks are not very deep, the processing times are practically applicable and fast, especially when using CNN.
\begin{table}[t]
\begin{tabular}{|c | c c c|}
\hline
\textbf{Model} & \textbf{Time per sample} $\downarrow$ & \textbf{Speed} $\uparrow$ & \textbf{Processor}\\
\hline\hline
CNN & 1.1 ms & 7,520 fps & GPU\\
\hline
RNN & 15.4 ms & 520 fps & GPU\\
\hline
FC-NN & 2.2 ms & 2,800 fps & GPU\\
\hline
SVR & 0.5 ms & 17,800 fps & CPU\\
\hline
RF & 4.2 ms & 1,950 fps & CPU\\
\hline
\end{tabular}
\caption{Processing time of Machine learning models}
\label{Table:speed}
\end{table}
\subsection{Alternate Approach Using Softmax Loss}
As we are dealing with a classification problem, we also use the softmax loss instead of cosine similarity. Softmax loss function has properties well suited for classification. Similarly, instead of using SVR and RF with the matching algorithm presented in section \ref{matching}, we use Support Vector Machine (SVM) and RF classifier. The results are shown in the last column of Table \ref{Table:models}. There is only a slight difference in the accuracy of all the different models. Angular deviation is no longer applicable as there is no output vector, but rather probabilities for each class. SVM and RF classifier perform better than SVR and RF regression, respectively, because of intrinsic properties suited for classification. In this case, however, we observe RF classifier to outperform CNN in terms of the classification accuracy. This can be associated with the relatively small dataset when using deep neural networks. We also did not tune the hyper-parameters of the neural networks, which may have also have an impact.
While using the softmax function as loss, there is a very slight increase in accuracy, which shows our approach using regression and the matching algorithm is almost equally good. The reason for using regression is scalability and flexibility for future work when we work with Points-of-Interest (POI) on the outside of the car. In such a case, the number of classes are not known beforehand, and such a classification approach would be difficult to realize.
\section{Conclusion}
In this paper, we presented a unique approach for user interaction inside the car where the driver can select various control modules of the car without a touch based input. To validate our approach, we conducted an experiment in a real but stationary car, and developed a novel approach using different deep neural networks to fuse three modalities for a more robust recognition of the driver's focus of visual attention. Unlike previous research work, we used all the three different modalities, namely finger, head pose and gaze, simultaneously. We have demonstrated that the use of multiple sources of input increases the performance. However, using three modalities instead of two (finger and eye gaze) only results in a negligible enhancement in the recognition accuracy of the driver's selected area-of-interest.
Furthermore, a comparison of deep learning methods and two conventional machine learning methods (namely SVR and RF) to perform the multimodal fusion is presented. We showed that, even with a considerably small dataset, the deep neural networks are able to learn the weights and produce better predictions for the user's referencing direction than the other two methods.
The work presented in this paper is preliminary and is motivated by the BMW Natural Interaction \cite{bmwnatint}. Based on the results achieved in the experiment, we can conclude that there is much potential with this approach for future user experience applications in the automotive industry. Our future work will focus on extending this to various other use cases, especially in the driving case, with a much larger dataset.
\begin{acks}
This work was done in collaboration with the BMW Group. We are grateful to the BMW colleagues for the assistance in the experimental setup and data collection.
\end{acks}
\bibliographystyle{ACM-Reference-Format}
\balance
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,039 |
Q: Android MapActivity cannot retrieve location data I have followed the HelloMapView tutorial for android, and added some code to capture my current location information and display in a textview.
I have tested these code in another program and it worked, it is able to display my longitude and latitude data when the launch that app. However when i copied the code into this MapActivity, it seems that i cannot get the location data.
Here's part of the code:
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.main);
mapView = (MapView) findViewById(R.id.mapview);
mapView.setBuiltInZoomControls(true);
mapOverlays = mapView.getOverlays();
drawable = this.getResources().getDrawable(R.drawable.androidmarker);
itemizedOverlay = new HelloItemizedOverlay(drawable, this);
Button button1 = (Button)this.findViewById(R.id.button1);
tv1 = (TextView) this.findViewById(R.id.textView1);
button1.setOnClickListener(mScan);
latitude = 1.3831625;
longitude = 103.7727321;
LocationManager lm = (LocationManager)getSystemService(Context.LOCATION_SERVICE);
Location location = lm.getLastKnownLocation(LocationManager.GPS_PROVIDER);
if (location == null)
location = lm.getLastKnownLocation(LocationManager.NETWORK_PROVIDER);
if (location != null) {
if (System.currentTimeMillis() - location.getTime() < 3000)
{
longitude = location.getLongitude();
latitude = location.getLatitude();
tv1.setText(Double.toString(longitude));
}
}
lm.requestLocationUpdates(LocationManager.NETWORK_PROVIDER, 2000, 10, locationListener);
}
private final LocationListener locationListener = new LocationListener() {
public void onLocationChanged(Location location) {
longitude = location.getLongitude();
latitude = location.getLatitude();
tv1.setText(Double.toString(latitude));
}
@Override
public void onProviderDisabled(String provider) {
// TODO Auto-generated method stub
}
@Override
public void onProviderEnabled(String provider) {
// TODO Auto-generated method stub
}
@Override
public void onStatusChanged(String provider, int status, Bundle extras) {
// TODO Auto-generated method stub
}
};
A: Did you added proper permissions in Manifest? Like this
<uses-permission android:name="android.permission.ACCESS_FINE_LOCATION" />
Also please check your gps is enabled or working on ur deive.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,577 |
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"redpajama_set_name": "RedPajamaC4"
} | 5,130 |
\section{Performance Modelling Using the Apex-MAP Benchmark}
\label{sec:1}
A simple synthetic benchmark with tunable hardware independent parameters that mimics
the behaviour of typical scientific applications is very useful for the evaluation of
new hardware platforms for a certain job mix.
Mapping application performance data measured on a production system
to specific parameter combinations of the synthetic benchmark
allows to model the performance of a wide spectrum of applications with a
simple approach.
To get insight in the performance patterns of the applications running on
HLRB~II, samples of the most important hardware counters (currently 25
counters) are taken from all processors in 10 minute intervals and are stored
in a huge database at LRZ. Though the measurements do not only include production runs of optimised user
codes, but also badly optimised programs and test runs etc., the results give
a deep insight into LRZ's job mix and the typical performance of the system.
Details about the measurement process, the sampling method, the database
scheme and the data analysis can be found in the LRZ technical report 2006-06~\cite{LRZ}.
\subsection{The Apex-MAP Benchmark}
To synthetically model the performance behaviour of LRZ's application mix we
extended the Apex-MAP benchmark (\emph{A}pplication \emph{pe}rformance
\emph{ch}aracterisation project -- \emph{M}emory \emph{A}ccess \emph{P}robe) originally developed by E. Strohmeier \&
H. Shang from the Future Technology Group at the Lawrence Berkeley National
Lab (LBNL), California~\cite{ApexMap1, ApexMap2}.
The initial idea of the Apex project is the assumption that the performance behaviour of any scientific application can be characterised by a
small set of application-specific and architecture independent performance
factors. Combining these performance factors, synthetic benchmarks that avoid
any hardware specific model can be designed to simulate typical application
performance. Assuming that the combination of memory accesses and
computational intensity is the dominant performance factor,
the Apex-MAP benchmark simulates typical memory access patterns of scientific applications.
Concerning the regularity of the memory access, the original Apex-MAP
benchmark focused on random access patterns inside an allocated memory block.
Our implementation also considers strided access patterns, which are common in many scientific applications.
The benchmark written in the style of Apex-MAP has the following 6 parameters:
\begin{description}[Type 1]
\item[M]{The total size of the allocated memory block \texttt{data} in which
data accesses are simulated, }
\item[L] {the vector length of data access,
(sub-blocks of length $L<M$ starting at \texttt{ind[i]} are accessed in succession), describes the \emph{Spatial Locality}},
\item[$\alpha$] the shape parameter of power distribution function ($0\le\alpha\le1$)
determines the random starting addresses \texttt{ind[i]}, describes the \emph{Temporal
Locality},
\item[S]{the stride width,}
\item[C]{a parameter used to increase the \emph{Computational Intensity} by calling the
subroutine \texttt{compute(C)},}
\item[I]{ the length of the index buffer \texttt{ind[]}.}
\end{description}
In the case of strided access only the parameters M, S and C are
relevant.
The kernel routine for strided access sums up every S-th element of
the allocated memory block \texttt{data[M]}.
\begin{verbatim}
for (int k = 0; k < M/S; k+=1) {
W0 += c0*data[k*S];
W0 += compute(C);
}
\end{verbatim}
To increase the computational intensity,
i.e. the ratio of the number of floating point operations and memory accesses,
we added calls to the subroutine \texttt{compute(C)}:
\begin{verbatim}
double compute(int C){
double s0,s1,s2,s3,s4,s5,s6,s7;
s0=s1=s2=s3=s4=s5=s6=s7=0.;
for(int i=1;i<=C;i++){
dummy(&s0,&s1,&s2,&s3,&s4,&s5,&s6,&s7);
s0+=(x[0]*y[0])+(x[0]*y[1])+(x[0]*y[2])+(x[0]*y[3])+
(x[0]*y[4])+(x[0]*y[5])+(x[0]*y[6])+(x[0]*y[7]);
s1+=(x[1]*y[0])+(x[1]*y[1])+(x[1]*y[2])+(x[1]*y[3])+
(x[1]*y[4])+(x[1]*y[5])+(x[1]*y[6])+(x[1]*y[7]);
...
s7+=(x[7]*y[0])+(x[7]*y[1])+(x[7]*y[2])+(x[7]*y[3])+
(x[7]*y[4])+(x[7]*y[5])+(x[7]*y[6])+(x[7]*y[7]);
}
return s0+s1+s2+s3+s4+s5+s6+s7;
}
\end{verbatim}
Performance is usually a mixture of hardware and compiler properties.
Braces and calls to a \texttt{dummy} routine have been inserted into the \texttt{compute}
routine to assure that the 128 floating point operations in the loop body are
really executed and not cancelled by optimisations of the compiler.
On Itanium the generated assembler code contains 64 consecutive \texttt{fma}
(fused multiply-add) instructions
which make optimal use of the floating point registers.
One 128 Byte cacheline is sufficient to hold the two data arrays \texttt{x[8]}
and \texttt{y[8]}. The \texttt{compute}
routine is thus able to run with nearly peak performance on Itanium.
In the case of random access patterns M, L, $\alpha$,
C and I are the relevant parameters. The kernel routine for random
memory access is:
\begin{verbatim}
for (i = 0; i < I; i++) {
for (k = 0; k < L; k++) {
W0 += c0*data[ind[i]+k];
W0 += compute(C);
}
}
\end{verbatim}
In this mode I subblocks of length L are accessed. The vector length L is the
number of contiguous memory locations accessed in succession
starting at \texttt{ind[i]}. L characterises the spatial locality of the
data access. The starting addresses of the subblocks are kept in
the index buffer \texttt{ind[]}. This access pattern is illustrated in
Fig.~\ref{fig:apex} (a).
\begin{figure}[b]
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.3]{apex.eps}&\includegraphics[scale=0.55]{alpha-dist-crop.eps}\\
(a) & (b)
\end{tabular}
\end{center}
\caption{Random access pattern of Apex-MAP: the left figure illustrates the
indexed random access using the index buffer \texttt{ind[]}. The starting addresses
kept in this array are random numbers drawn from a power distribution
function with a probability distribution shown
on the right for various values of $\alpha$.}
\label{fig:apex}
\end{figure}
The starting addresses are random numbers drawn from a power distribution
function and are defined as follows:
\texttt{ind[j] = (L*pow(drand48(), 1/$\alpha$) * (M/L -1)) $\in$ [0;M-L[}
The parameter $\alpha \in$ [0;1] of this distribution function defines the
temporal reuse of data. Figure~\ref{fig:apex} (b) shows the probability
distribution of the power function \texttt{pow(drand48(), 1/$\alpha$)}.
For $\alpha=1$ the random numbers are just deviates with a uniform probability
distribution, while the smaller $\alpha$ is, the more the distribution
function is peaked near 0 and the higher the temporal reuse of data is. For
$\alpha=0$ always the same starting address is used.
\bigskip
\subsection{Comparison of Apex-MAP with Real Application Performance}
To use Apex-MAP for comparing the average memory bandwidth and the floating point performance
of real applications
the Itanium performance counters \newline
\textit{FP\_OPS\_RETIRED}, \textit{CPU\_OP\_CYCLES\_ALL} and \textit{L3\_MISSES} %
have
been measured and aggregated
for various combinations of the Apex-MAP input parameters.
The L3 cacheline size of the Itanium is 128 Bytes and can hold
16 64-bit (double-precision) values.
The consumed bandwidth between memory
and L3 cache is given by \textit{L3\_MISSES} $\times$ 128 Bytes.
Figure~\ref{fig:counters} shows the number of floating point operations per
cycle \linebreak (\textit{FP\_OPS\_RETIRED / CPU\_OP\_CYCLES\_ALL}) versus the
memory bandwidth, expressed by L3
misses in Bytes/cycle (\textit{L3\_MISSES/CPU\_OP\_CYCLES\_ALL} $\times$ 128
Bytes). Figure~\ref{fig:counters} (a) on the left shows this data for
real applications running on HLRB~II, while Fig.~\ref{fig:counters} (b) on the right
shows data from simulations using the Apex-MAP benchmark with various input parameters.
\begin{figure}[t!]
\begin{tabular}{cc}
\includegraphics[scale=.3,angle=-90]{all-1-2-new.eps}&\includegraphics[scale=.3,angle=-90]{all2-new.eps} \\
(a) & (b)
\end{tabular}
\caption{Comparison of floating point operations per cycle vs. L3 Misses
(in Bytes) per cycle for real applications running on HLRB~II (a)
and for simulations using the Apex-MAP benchmark with various
simulation parameters (b).
}
\label{fig:counters}
\end{figure}
For the left picture the hardware
counters were sampled every 10 minutes on all processors of HLRB~II for approximately 3 days
with a sampling time of 10 seconds. More than 3.2 Mio. samples are taken into account.
The average floating point operations per cycle for this 3-day interval is 0.48, which is equal to
770 MFlops per core and 12\% of the Itanium's peak performance.
The mean for the L3 misses is 0.2 Bytes per cycle.
The parameter space is divided into 32 rectangles of size 0.5 L3 Misses/cycle
$\times$ 0.5 Flops/cycle. The percentage of data points falling into each
rectangle is given.
For the right picture around 23000 different combinations of the Apex-MAP
benchmark are used. This picture includes various runs, using both random and strided memory access
as well as serial and parallel runs using OpenMP to cover the same areas as
the measured data on the left. The range of the simulation parameters for Fig.~\ref{fig:counters} (b) is $L<M=1$ GB,
$0\le\alpha\le 1$, $2\le S\le400$, $0\le C\le1000$, I=50.
The two pictures demonstrate that it is possible to model the performance of real applications
by using suitable combinations of input parameters for the Apex-MAP benchmark.
Comparing the two pictures shows that every region in the left
picture with significant percentage (i.e. above 0.5\%)
of data points can be covered by a specific
combination of Apex-MAP input parameters. In total, the Apex-MAP runs are able
to cover 98.9\% of the measured real-application performance data.
The measurements in the left picture are partly based on MPI-parallelised
programs. Performance counters are only implicitly able to measure the
impact of additional communication overheads, e.g. waiting time for external
data on remote processors. Although Apex-MAP focuses on single processor
performance it is able to mimic the behaviour of parallel applications as long
as the network characteristics stay roughly the same.
\subsection{Modelling LRZ's Application Mix}
It has been shown that Apex-MAP is able to cover the parameter space
that is attained by real applications. It is assumed that Fig.~\ref{fig:counters} (a)
gives a general overview of the application mix running on HLRB~II. A good indication
for this is given by the fact that the mean MFlops-rate for this 3-day
interval is 770 MFlops per core or 12\% of peak performance, which is a good approximation
of the overall mean application performance of HLRB~II (see also~\cite{HLRBPerfMon}).
Therefore the weights associated with each rectangle (percentages in
Fig.~\ref{fig:counters} (a)) are used to model the general application mix.
Besides the weights for each rectangle, the most suitable combinations of input parameters for Apex-MAP needs to be found.
Figure~\ref{fig:stridedVsrandom} shows the achievable combinations of Flops versus L3 Misses for
different versions of the strided access (a) and random access (b) memory patterns using a serial version of the code.
Figure~\ref{fig:stridedVsrandom} (a) visualises the common understanding of the influence of a stride
memory access to performance. Every line corresponds to an increase in computational intensity C. As long as the
computational intensity is low (e.g., C stays small), only 1 Flop per clock cycle is possible (which is
equal to 25\% of peak performance). As the
computational intensity grows larger, the codes are able to run at maximum speed with nearly 4 Flops in every
cycle. As said before, the L3 cacheline size of the Itanium is 128 Bytes;
16 doubles fit in one cacheline. Therefore with an increase in stride along each line from 1 (contiguous access)
to 16 (access only one item per cache line), the performance drops and stays at a minimum for figures
above 16 which always need a new cache line.
\begin{figure}[h!]
\begin{tabular}{cc}
\includegraphics[scale=.3,angle=-90]{stridedMemAccessPattern.ps}&\includegraphics[scale=.3,angle=-90]{randomMemAccessPattern.ps} \\
(a) & (b)
\end{tabular}
\caption{Comparison of floating point operations per cycle vs. L3 Misses
(in Bytes) per cycle for increasing computational intensity C and variations of S for the strided access memory pattern (a)
and variations of L for random access memory patterns (b).
}
\label{fig:stridedVsrandom}
\end{figure}
Figure~\ref{fig:inputValues} shows the data points that have been chosen to model the application
mix of LRZ. The corresponding Apex-MAP input parameters multiplied with the derived weights
are being used to compute an overall performance of the application
mix in MFlops. Running the adapted Apex-MAP benchmark on HLRB~II yields a performance estimate of
898 MFlops per core. This is quite close to the actual application performance on HLRB~II:
14\% deviation from the measured 3-day interval.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=.35,angle=-90]{Apexbm-parameters.ps} \\
\end{center}
\caption{Chosen data points to model the application mix. These data points represent input parameters
of the Apex-MAP benchmark.
}
\label{fig:inputValues}
\end{figure}
\section{Validation Using the EuroBen Mathematical Kernels}
The validation of our Apex-MAP version will be done by using two mathematical kernels
typical for many scientific applications.
Within the EC FP7 funded project "Partnership for Advanced Computing in Europe" (PRACE,~\cite{PRACE}), several
mathematical kernels from the EuroBen benchmark suite~\cite{euroben}
have been chosen as templates for commonly used scientific applications. To validate Apex-MAP two very
distinct codes have been chosen:
\begin{itemize}
\item mod2am, a dense matrix-matrix multiplication,
\item mod2as, a sparse CSR (compressed sparse row) matrix-vector multiplication.
\end{itemize}
Within PRACE these codes have been ported to several new languages and architectures; results will be published
on the PRACE website in deliverable D6.6 and D8.3.2~\cite{PRACDel}.
The PRACE surveys analysed the current standards for parallel programming and their
evolution, PGAS languages, the languages introduced as a consequence of the DARPA HPCS
project and the languages, paradigms and environments
for hardware accelerators.
Performance data has been gathered for various architectures.
The performance of these benchmarks is well known; many different performance runs have been measured,
suitable reference input data sets exists and LRZ was responsible for
the MKL, CUDA and RapidMind ports.
The first benchmark mod2am has a high computational intensity and is well suited for the use of highly multi-threaded
devices. The second benchmark has a low computational intensity and is a template for codes which will benefit
from a higher memory bandwidth. Using these benchmarks will ensure that Apex-MAP is able to model the two extremes
in terms of computational intensity versus memory access and will allow to
validate Apex-MAP for the use on hardware accelerators in the future.
\subsection{mod2am: Dense Matrix-Matrix Multiplication}
Several PRACE implementations of the matrix-matrix multiplication are based on the BLAS Level~3 routine dgemm.
For the x86 implementation the \texttt{cblas\_dgemm} routine from Intel's MKL (Math Kernel Library) has
been used; the CUDA implementation is based on cuBLAS.
The RapidMind implementation uses a code-example from the
RapidMind developer portal \cite{RMcode} for a general matrix-matrix multiplication code
which was slightly adapted. This code is optimised for the use on GPUs.
Figure~\ref{fig:mod2am} shows performance measurements from the PRACE project. It compares
the performance of the CUDA and RapidMind implementations on an Nvidia C1060 GPU,
which is used in Nvidia's Tesla boxes, with the performance of an MKL version on 8 Intel Nehalem EP cores.
The reference input data sets are those from PRACE which operate on quadratic matrices.
They are described in Deliverable D6.6 available
from~\cite{PRACDel} and have been chosen to firstly, represent frequently used problem sizes and secondly, show the dependency
between problem size and performance, especially on hardware accelerators.
The double-precision peak performance of one C1060 (78 GFlops) is comparable to 8 Nehalem cores (80 GFlops).
The diagram shows that the RapidMind implementation is a factor of 4 slower than the highly optimised
cuBLAS library. However, the RapidMind implementation follows roughly the same trend as the CUDA version.
\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{mod2am.eps}
\end{center}
\caption{Comparison of the performance of the dense matrix-matrix multiplications (mod2am, double-precision) for various
matrix sizes using RapidMind's CUDA backend, Nvidia's cuBLAS and Intel's Math Kernel Library.
The peak performance of one C1060 GPU is comparable to 8 Nehalem EP cores (78 vs. 80~GFlops).
}
\label{fig:mod2am}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.7]{mod2as.eps}
\end{center}
\caption{Comparison of the performance of the sparse matrix-vector multiplications (mod2as, double-precision) for various
numbers of rows using RapidMind's CUDA backend, Nvidia CUDA and Intel's MKL.
The peak performance of one C1060 GPU is comparable to 8 Nehalem EP cores (78 vs. 80~GFlops).
}
\label{fig:mod2as}
\end{figure}
\subsection{mod2as: Sparse Matrix-Vector Multiplication}
In the case of the mod2as benchmark the input matrix is stored in the
3-array variation of the CSR (compressed sparse row) format. Using this format
only the nonzero elements of the input matrix are stored in one array, and the
other two arrays contain information to compute the row and the column of the
nonzero elements. The entries of the input matrix are computed using a random number generator
but could be reproduced for several runs by using the same seed.
Figure~\ref{fig:mod2as} compares the performance of the RapidMind implementation
with the CUDA and MKL version. The MKL version makes use of a library call
to\newline \texttt{mkl\_dcsrmv}. The CUDA
implementation is based on the paper "Efficient Sparse Matrix-Vector
Multiplication on CUDA"~\cite{CUDAmod2as}. A description of the RapidMind implementation
can be found in Deliverable D8.3.2 at~\cite{PRACDel}.
Again, the reference input data sets from PRACE have been used; all data sets contain quadratic matrices of
different sizes and fill ratios. The diagram shows that the RapidMind version
is a factor of 3 slower than the optimised CUDA version. The trend of both is very similar.
\subsection{Validation of Apex-MAP}
\begin{figure}[h!]
\begin{tabular}{cc}
\includegraphics[scale=.26,angle=-90]{mod2amWeights.ps}&\includegraphics[scale=.26,angle=-90]{mod2asWeights.ps} \\
(a) & (b)
\end{tabular}
\caption{Floating point operations per cycle versus L3~Misses (in Bytes) as measured by the hardware
counters on HLRB~II for mod2am (a) and mod2as (b) (Step~2). The numbers indicate the resulting weights associated with
each rectangle (Step~3).
}
\label{fig:weightMod2}
\end{figure}
Validating Apex-MAP by using the two mathematical kernels needs several steps:
\begin{enumerate}
\item Measure the performance of mod2am/as on the original hardware (HLRB~II).
\item Measure the hardware counters for mod2am/as on HLRB~II.
\item Generate weights for each rectangle and each kernel.
\item Measure the performance of mod2am/as on the target hardware (Nehalem EP).
\item Run Apex-MAP with the weights for mod2am/as on Nehalem and HLRB~II.
\item Compare the predicted results (Step~5) with the actual results (Steps~1~and~4).
\end{enumerate}
Step~1 yields a mean performance on HLRB~II for all reference input data sets of 5.4~GFlops per core for mod2am (84\% of peak)
and 0.5~GFlops for mod2as (8\% peak).
Figure~\ref{fig:weightMod2} shows the hardware counter measurements done in Step~2 and the derived
weights for the Apex-MAP runs (Step~3): It can be clearly seen, that the dense matrix-matrix multiplication
(a) is compute bound while the sparse matrix-vector multiplication (b) is memory bound.
Step~4 shows an actual performance on the target architecture Nehalem EP of 8.0~GFlops (80\%) for mod2am and 0.9~GFlops (9\%) for mod2as.
The results of Step~5 can be seen in Fig.~\ref{fig:predMod2}; the mean performance predicted by Apex-MAP deviates only slightly from the
measured data on both architectures. The measurements on Nehalem are slightly worse, since the \texttt{compute} routine, which
has been optimised for Itanium is not able to reach peak performance on
Nehalem.
\begin{figure}[h!]
\begin{tabular}{cc}
\includegraphics[scale=.26,angle=-90]{hlrb2-mod2am.ps}&\includegraphics[scale=.26,angle=-90]{hlrb2-mod2as.ps} \\
(a) & (b) \\
\includegraphics[scale=.26,angle=-90]{ice-mod2am.ps}&\includegraphics[scale=.26,angle=-90]{ice-mod2as.ps} \\
(c) & (d)
\end{tabular}
\caption{All figures show the actual performance measured for each reference input data set (red curve) together with
the mean performance measured by the mathematical kernel (green line) and predicted by Apex-MAP (blue line).
The first line shows results on HLRB~II (a,b), the second line results on Nehalem EP (c,d). The left diagrams
are based on mod2am (a,c) and the diagrams on the right on mod2as (b,d).
}
\label{fig:predMod2}
\end{figure}
\section{Conclusion and Outlook}
It has been shown that an adaptation of the Apex-MAP benchmark can be used to
model the application mix in order to use it for benchmarking the suitability
of new architectures for a computing centre. The adapted benchmark has been
validated by using it to predict the performance of two mathematical kernels on
two architectures.
Future work will go mainly into two directions. Firstly we want to investigate in
more detail the quality of the predictions to refine the benchmark and ensure that
it adapts easily to new environments. Secondly we want to use Apex-MAP to investigate
if hardware accelerators are advantageous for our application mix.
Hardware accelerators
like GPUs and the CELL processor with an enormous peak performance
have recently gained much interest in the community.
A programming model that was evaluated at LRZ is the multi-core development
platform RapidMind, which is a tool that allows generating code
for GPUs, the CELL processor and multi-core CPUs with the same source file.
Using a RapidMind version of the Apex-MAP benchmark could offer an easy way to simulate typical application
performance patterns on a broad range of architectures.
RapidMind ports of the two mathematical kernels are already available and could be used to
validate a RapidMind Apex-MAP version.
\section*{Acknowledgment}
This work was financially supported by the KONWIHR-II project ``OMI4papps''
and by the PRACE project funded in part by the EU's 7th Framework Programme
(FP7/2007-2013) under grant agreement no.~RI-211528.
\bibliographystyle{spmpsci}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,160 |
Cordón del Azufre är en ås i Argentina, på gränsen till Chile. Den ligger i den norra delen av landet, km nordväst om huvudstaden Buenos Aires.
Trakten runt Cordón del Azufre är ofruktbar med lite eller ingen växtlighet. Trakten runt Cordón del Azufre är nära nog obefolkad, med mindre än två invånare per kvadratkilometer. I trakten råder ett kallt ökenklimat. Årsmedeltemperaturen i trakten är °C. Den varmaste månaden är januari, då medeltemperaturen är °C, och den kallaste är juli, med °C. Genomsnittlig årsnederbörd är millimeter. Den regnigaste månaden är mars, med i genomsnitt mm nederbörd, och den torraste är augusti, med mm nederbörd.
Källor
Kullar i Argentina
Berg i Argentina 4000 meter över havet eller högre
Kontrollbehov inkommande wikilänkar | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,649 |
{"url":"http:\/\/www.princetonmagazine.com\/palmer-stadium-memories-of-a-magical-time\/","text":"Best place tobuy Valium on line you can find\nBest place toget CBD gummies online you can find\n\n### Palmer Stadium \u2013 Memories Of A Magical Time\n\nBy Anne Levin\n\nPrinceton University hired celebrated architect Henry Janeway Hardenbergh to design its football stadium just over a century ago. During the process, a decision was made to build it out of reinforced concrete instead of costlier masonry. The result was horse shoe shaped Palmer Stadium, which seated 45,725 and had an end zone with an unobstructed view of Carnegie Lake. A mix of Collegiate Gothic ornamentation with a classical Greek plan, it cost $300,000 to build and was completed almost a full month ahead of schedule\u2014well in time for the Tigers to defeat Dartmouth 16-12 in the first official game at the venue on October 24, 1914. But that reinforced concrete turned out to be a bad idea. By the time Palmer Stadium entered its eighth decade, its walls were tumbling down. Orange and black netting installed in some sections was the only thing keeping Tiger fans from getting clobbered by hunks of the crumbling material. Nests of bees, which would swarm the press box if anyone so much as opened a can of sugary soda, lurked in its recesses. So Palmer Stadium was torn down in 1996, after the Tigers played Dartmouth\u2013and lost 24-0\u2013on the field for the last time. The home of Princeton football and track for 82 years, and one of the two oldest college stadiums in the United States, was replaced by a 27,773-seat, state-of-the-art,$45 million structure designed by \u201cstarchitect\u201d Rafael Vinoly. It was built on nearly the same footprint as its predecessor. Unlike at Palmer, where an all-weather-surface track was built in 1978, the newer Princeton Stadium has a separate track outside its confines.\n\nThese days, Tiger fans support their team in safer, more comfortable fashion. But although nearly two decades have passed since they bid the rickety Palmer goodbye, many still look back on it with nostalgia. It was at Palmer, after all, that Heisman trophy winner Dick Kazmaier \u201952 became a Princeton football legend, leading the Tigers to an 18-1 record between 1949 and 1951. Dean Cain, who went on to star in the Superman series, set a record for interceptions at 12 in one season. In all, there were 14 undefeated seasons at Palmer Stadium between 1920 and 1964.\n\n\u201cIt was viewed as holy ground by the football world,\u201d says Gary Walters, who graduated in 1967 and served as Princeton\u2019s athletic director from 1994 until retiring last June. \u201cFor me,\u201d he continues, \u201cI was fortunate enough to see the last undefeated team in 1964, led by Cosmo Iacavazzi. I was friends with these guys. I was vested in their performance.\u201d\n\nPulitzer-Prize-winning author John McPhee, who grew up in Princeton and was the son of the football team\u2019s doctor, admires the current stadium but looks back fondly on the old. \u201cThere are those of us who remember it,\u201d says McPhee, who graduated from Princeton in 1953 and is its Ferris Professor of Journalism. \u201cI roomed with the football players in college. And when I was growing up, I got a big kick out of being the team\u2019s mascot. Those guys of the late 1930s and into the war years were wonderful in the way they related to a kid.\u201d\n\nPalmer Stadium was financed by Edgar Palmer, class of 1903. He named the arena in memory of his father, Stephen S. Palmer. Construction by the George A. Fuller Company took just four months, with workmen divided into two sections. One was assigned to the east side of the stadium and the other worked on the west. According to local lore, there was a friendly competition between the two crews to see who could finish first.\n\nThere are stories about legendary games, played in legendary weather. On November 23, 1935, there was the famous \u201ctwelfth man\u201d snowstorm game in near-blizzard conditions. A crowd of 56,000 people packed the bleachers despite the storm, to watch the Tigers defeat Dartmouth 26-6. What made the game most memorable, though, was something that happened in the fourth quarter. A man who was later identified as a local cook ran out onto the field and took a spot on the Dartmouth line. He was escorted from the field by stadium security after one play.\n\nA hurricane didn\u2019t stop Princeton from defeating Dartmouth again, 13-7, in 1950, despite 80 mile-per-hour winds and gusts reaching 108. Tarpaulins that had covered the field for most of the morning broke from their moorings around noon, and an inch of rain poured onto the field \u201cfrom one 20-yard line to the other and to within a few yards of each sideline,\u201d according to the Princeton Tigers website.\n\n\u201cAtop the stadium, the tar paper roofing was ripped off the press box and water dripped through in increasing quantities. The gusting winds caused the press box and the radio and public address booths to sway noticeably. Nearly 5,000 Tiger faithful braved the elements and watched Princeton complete its perfect season with a 13-7 win against Dartmouth. All three touchdowns were scored by the team driving with the wind.\u201d\n\nThe University\u2019s head football coach Bob Surace, a 1990 graduate who was on the Tigers team from 1986 to 1989, considers himself fortunate to have played at Palmer Stadium. \u201cYou\u2019re walking into a stadium that so many unbelievably great players have played in before you,\u201d he said. \u201cIt\u2019s so special that you\u2019re in the same spot that Dick Kazmaier or Cosmo Iacavazzi or Pink Baker, who played on the Team of Destiny in 1922, came before you. He (Baker) came to almost every practice in my freshman year. And the friendships I still have today with the guys I played with\u2013it\u2019s just such a special place.\u201d\n\nSurace remembers the game against Yale the year before the stadium was torn down. \u201cThe crowd was nearly 40,000 strong,\u201d he said. \u201cIt was full, but some areas of the stadium had been condemned. Back in the 1950s and 60s it was full every week, so it was so special to have it almost full again. We had such a sense of pride.\u201d\n\nA golden anniversary celebration of the 1964 team is planned, Surace says. \u201cUnfortunately Kazmaier passed away last year, but we\u2019re going to honor that undefeated team. The stadium was state of the art in its time. The picures you see in old books of the teams are of a packed stadium. You recognize Princeton was at the level of Alabama or Ohio State. By filling that stadium, so many things on this campus got built. Princeton is partly Princeton because of the gate receipts.\u201d\n\nWalters was on his way to a press conference announcing his appointment to the post of athletic director in 1994 when he was told that if he was asked about the state of Palmer Stadium, he should \u201ctiptoe around\u201d the question. Little did he know that he would spend the first five years of his tenure helping to decide the fate of the venue, and then plan for a new one.\n\n\u201cIt was in incredible disrepair and structurally unsound, I was told, and there was a strong possibility we would have to raze the stadium,\u201d he says. \u201cThat\u2019s what I walked into. It had a tremendous emotional impact on me. It was one of the two or three oldest stadiums in the U.S. at that time. So that issue dominated the first five years of my being director of athletics.\u201d\n\nAn alumni advisory committee including Kazmaier and Iacavazzi was put together. Once it was decided that Palmer would have to go, and Vinoly, who had never designed a football stadium, was hired, Walters set about making sure the new would have a strong tie to the old.\n\n\u201cOne of my charges was to integrate more with the life of the University, and make it more synergistic,\u201d he says. \u201cSome people wanted to move the stadium to the other side of the lake, but I said that wasn\u2019t a good idea. You don\u2019t want to isolate the football field. It was significant that we would be in the footprint of Palmer. We had to have a muscular, aesthetic identity that rivaled the old. For those of us on the alumni committee and the committee to build a new stadium, we were successful in sustaining the memory of Palmer by keeping it where it was. The old Palmer and the reverence we all had for it had great impact on the design of the new. We were very sensitive to the feelings of nostalgia.\u201d\n\nThe late Jeb Stuart, who was editor and publisher of the weekly Princeton newspaper Town Topics, covered high school and University sports. According to his 2008 obituary in the paper, \u201cHe became a fan of Princeton football at age six when he was taken to the press box at the top of Palmer Stadium by his father, who announced the game in progress over the stadium\u2019s public address system. For many years after college Jeb worked beside his father in the announcer\u2019s box as a spotter, following each play through binoculars to feed his father the player\u2019s names and numbers for play-by-play descriptions.\u201d\n\nPrinceton athletics, especially football, likewise played a large role in the childhood of McPhee, \u201cI grew up on stories about Palmer Stadium. It was built of concrete, spread by a method that was novel at the time,\u201d he says. \u201cIt had to do with air pressure, I think. The thing ended up full of holes. And from early on, they had to do dental work on it. My brother worked for a contractor in the summer. He sat in the stadium with a railroad spike in one hand and a hammer in the other. They\u2019d clear out an area and fill it with fresh cement. Either he or his friend hit the sledge of the spike and it went all the way through and down to the ground. They were halfway up. It was honeycombed with holes.\u201d\n\nThe football team was in the stadium only for games during McPhee\u2019s childhood, not for practice. \u201cI went to school at what is now 185 Nassau Street. After soccer practice,\u201d he says, \u201cI\u2019d go down the street to football practice at University Field, where the E-Quad is now, and hang around. There was a wall around the practice field because spies were feared to be coming from Yale or Harvard. There were only about two apertures, and everybody who went in was checked. As the mascot, I went on the field with them on Saturdays.\u201d\n\nWalters recalls that when President John F. Kennedy was assassinated in 1963, the football game was postponed for a week. \u201cThen they played Dartmouth, and lost,\u201d he recalls. \u201cThen in the fall of \u201964, we had a really strong team. The stadium in those days was packed. Everybody went in suits and ties. It was a big social time. The school wasn\u2019t coed yet, and the undergrads brought their dates. That team set the stage for \u201964-\u201965.\u201d\n\nPalmer Stadium has been gone since 1996. But in the centennial year of its construction, it evokes memories for so many. \u201cIn the four major sports at that time, we were dominant,\u201d says Walters of his own college years during Palmer\u2019s heyday. \u201cIt was a magical time.\u201d","date":"2022-11-27 14:23:23","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\": 0.17980876564979553, \"perplexity\": 3810.581316009982}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710409.16\/warc\/CC-MAIN-20221127141808-20221127171808-00529.warc.gz\"}"} | null | null |
\section{Introduction}\label{sec:introduction}}
\IEEEPARstart{T}{he} revolution in deep learning over the last decade has been mainly driven by the confluence of two equally important factors -- the generation of large amounts of data, and the availability of GPUs to train large neural networks. Those contributed to accelerated research in the algorithmic domain by significantly decreasing experiment turn-around time. In recent years, as the computational demands for deep learning have increased, especially in consumer facing domains such as image and speech recognition, there is a trend towards more hardware specialization to improve performance and energy-efficiency. For instance, recent hardware approaches \cite{sijstermans:2018,dally:2018,courbariaux:2014,gupta:2015,han:2015,cchen:2018,jouppi:2018} feature techniques such as reduced precision, aggressive compression schemes and customized systolic data paths aimed at accelerating today's DNNs.
While these efforts are extremely important both from research and commercialization perspectives, the currently available hardware landscape is also somewhat restrictive. This is because GPUs (and their variants) may not lend themselves well to use cases where mini-batching/data parallelism is non-trivial because multiply-accumulate operations are not dominant or when low latency is critical. Therefore, as machine intelligence algorithms continue to evolve, it is unfortunate that promising approaches may be sidelined simply because they do not map well to a GPU, just as backpropagation trailed more conventional machine learning approaches for decades due to the lack of GPUs. One of the prime examples of an algorithm which is not well matched to SIMD architecture is Monte Carlo Tree Search used in the Google Deepmind's AlphaGo system\cite{SilverHuangEtAl16nature}.
This indicates the need for new computer architectures for machine intelligence. However, there is an obvious challenge -- namely, how does one build hardware for algorithms and use cases that may not yet exist? The approach we have taken is to build a hardware system with tremendous flexibility, which will be explored in depth in the rest of the paper. This flexibility extends to the types of algorithms being executed, the portions that need to be off-loaded and accelerated, the model of communication, and the types of parallelism deployed.
\begin{figure*}[!t]
\centering
\includegraphics[width=0.8\textwidth]{figures/card_cube.JPG}
\caption{3D Mesh Topology of a single INC card, with some node numbers shown. Each node contains its own dedicated Zynq System-on-Chip with an ARM Cortex-A9 and FPGA logic. Node 100 (blue shaded cube) is the gateway node to the external Ethernet. Nodes 000 (green shaded cube) and 200 have PCIe Interfaces to communicate with a host computer.}
\label{fig:INC_card}
\end{figure*}
The system, named IBM Neural Computer (INC), is fundamentally a large, highly scalable parallel processing system with compute nodes interconnected in a 3D mesh topology. The total number of compute nodes within the existing single-cage system is 432. Each compute node contains a Xilinx Zynq System-on-Chip (an ARM A9 CPU + FPGA logic on the same die) along with 1GB of dedicated RAM \cite{rajagopalan:2011,crockett:2014,zynq7000}. The availability of FPGA resources on every node allows application-specific processor offload, a feature that is not available on any parallel machine of this scale that we are aware of.
The communication network that realizes the 3D mesh topology is implemented using single-span and multi-span SERDES (Serializer-Deserializer) links connected to the FPGA fabric of the Zynq. It is therefore possible to build tailored hardware network controllers based on the communication mode(s) most suited to the application. The ability to optimize the system performance across application code, middle-ware, system software, and hardware is a key feature of INC.
One may envision that this 3D topology of distributed memory and compute, with the ability to have nodes exchange signals/messages with one another is somewhat reminiscent of works targeting the human brain (most prominently, the SpiNNaker project\cite{furber:2014,painkras:2013}). However, an important distinction is that our goal is more general than computational neuroscience, and extends to machine intelligence in general.
The rest of the paper is organized as follows. Section \ref{sec:xcape9000} begins with an overview of the INC system. Section \ref{sec:comm} discusses inter-node communication mechanisms in INC. Section \ref{sec:diag} contains details on diagnostic and debug capabilities of the INC system, which are crucial in a development environment. Section \ref{sec:conclusion} will conclude the paper.
\begin{figure*}[!t]
\centering
\includegraphics[width=0.8\textwidth]{figures/INC_systems2.JPG}
\caption{Hierarchical organization of the INC system (a) Conceptual design of INC 9000 with 48 cards and 1296 nodes (not yet built), (b) INC 3000 with 16 cards and 432 nodes (operational), (c) Single card with 27 nodes, (d) One node.}
\label{fig:INC_system}
\end{figure*}
\section{INC Overview}\label{sec:xcape9000}
The INC system is designed primarily to be a development platform for emerging machine intelligence algorithms. It is a parallel processing system with a large number of compute nodes organized in a high bandwidth 3D mesh network. The platform is designed to be highly flexible. Within each node is a Xilinx Zynq system-on-chip, which integrates a dual-core Cortex A9 ARM processor and an FPGA on the same die, allowing the system to be reconfigured on a per node basis. Each node also includes 1GB of dedicated DRAM that can be used as program and data space, and is accessible both from the processor and the FPGA. In an eventual at-scale high-performance learning task, we envision that most of the performance critical steps will be offloaded and optimized on the FPGA, with the ARM only providing auxiliary support -- including initialization, diagnostics, output transfer, etc.
While INC is a distributed system in that it is composed of distinct processor+memory nodes interconnected by communication links, it has a unique combination of features not available elsewhere. It is not a multi-FPGA `sea of gates' system \cite{krupnova:2004,assad:2012} whose structure would need to be defined by the logic resident on the FPGA. It has a very well defined structure of compute nodes with a well defined communications network. Therefore it does not carry the performance compromise associated with the need to support a fully-generic interconnect.
It is also different from other distributed systems such as BlueGene \cite{haring:2012}. In addition to the available FPGA offload capability at every node, the communication interfaces are not pre-defined to support a limited set of known use cases. Instead, access to the physical communication links is through the FPGA, and multiple distinct `logical' channels of communication can be established, all utilizing the same underlying SERDES links. In this way, the network interfaces can be designed (and even progressively optimized) to best suit the applications executing on INC.
\subsection{INC card}
The basic building block of the system is an INC card. Each card contains 27 nodes arranged in a 3$\times$3$\times$3 cube.
Figure~\ref{fig:INC_card} shows the 3x3x3 topology of an individual card, along with (XYZ) co-ordinates overlaid to indicate the organization of the 3D mesh. The 27 nodes are placed on the card in a way to minimize the connection lengths between logically adjacent nodes. All 27 nodes on a single card are identical except for some important differences. Node (100) includes an Ethernet port, and can act as a gateway connecting an internal Ethernet network implemented on the FPGAs to a conventional external network. Node (000) is a controller node, and includes a 4 lane PCIe 2.0 connection that can be connected to a host PC. It also has a serial connection that can serve as a console during boot time, or be forwarded to the other nodes on the card. Node (200) is also capable of supporting a PCIe interface, should an application need additional bandwidth.
\subsection{Backplane, Cages and Racks}
In an INC system, individual cards plug into a backplane. Each backplane can support up to 16 cards, and the backplane wiring arranges the 432 nodes of the 16 cards into a 12$\times$12$\times$3 mesh. The backplane and cards are enclosed in an INC card cage (INC 3000 system (Fig.~\ref{fig:INC_system}b)). Connectors on the back side of the backplane allow up to four cages to be connected vertically to build a system of up to 12$\times$12$\times$12 or 1728 nodes (INC 9000 system (Fig.~\ref{fig:INC_system}a)).
The card design supports building an INC system with anywhere from one to 512 cards (13,824 nodes). However, to grow beyond the size of the INC 9000 system, a new backplane design is required.
\subsection{Physical Links}
Each node on a card is connected to its nearest orthogonal neighbors by a single span link composed of two unidirectional serial connections. Each node has six single span links. The nodes on the faces of the cube (i.e. all nodes other than the central (111) node) have single span links that leave the card, and may have nearest neighbors on other cards in the system. In addition to single span links, 6 bi-directional multi-span links allow for more efficient communication in a larger system. Multi-span links connect nodes that are three nodes apart in any one orthogonal direction, and will always begin and terminate on different cards. With a total of 432 links leaving or entering the card, and 1 Giga-byte (GB) per second per link, this amounts to a potential maximum bandwidth of 432 GB per second leaving and entering one card. The bisection bandwidths for the INC 9000 and INC 3000 systems are 864 GB per second and 288 GB per second respectively.
The communications links are pairs of high speed, serial, unidirectional SERDES connections. Each connection only has two wires: the differential data lines. There are no additional lines for handshake signals. The links are controlled by a credit scheme to ensure that overrun errors do not occur and no data is lost. A receiving link sends (via its paired transmit link) a count of how many bytes of data it is willing to receive. A transmitting link will decrement its count as it sends data and never send more data than it holds credits for from the receiver. The receiving side will add to the credit balance as it frees up buffer space. This credit system is implemented entirely in the hardware fabric and does not involve the ARM processor or software.
\subsection{Packet Routing}
The communication network currently supports directed and broadcast packet routing schemes. Features such as multi-cast or network defect avoidance are being considered at the time of writing, and can be included based on application or hardware needs.
In a directed routing mode, a packet originating from the processor complex or the FPGA portion of a compute node is routed to a single destination. Both single-span and multi-span links may be used for the routing, and the packet will be delivered with a minimum number of hops. However, a deterministic routing path is not guaranteed, as each node involved may make a routing decision based on which links happen to be idle at that instant. This implies that in-order delivery of packets is not guaranteed\footnote{This does not preclude applications that need in-order delivery, as reordering can be achieved in either FPGA hardware or in software, or a different packet routing scheme can be devised as necessary.}. The packet routing mechanism is implemented entirely on the FPGA fabric, and the ARM processors may only be involved at the source and destination nodes, if at all.
A broadcast packet radiates out from the source node in all directions and is delivered to every node in the system. Broadcast packets only use the single-span links in the system for simplicity of routing. Depending on which link received a broadcast packet, the receiving node may choose to a) forward to all other links, b) forward to a subset of links, or c) stop forwarding. By choosing the rules for these three scenarios carefully, it is possible to ensure that all nodes in the system receive exactly one copy of the broadcast packet.
\section{Connectivity and Communication}\label{sec:comm}
Multiple virtual channels can be designed to sit atop the underlying packet router logic described in the previous section to give the processor and FPGA logic different virtual or logical interfaces to the communication network. In this section we will review three approaches currently implemented on INC -- Internal Ethernet, Postmaster Direct Memory Access (DMA) and Bridge FIFO.
\subsection{Internal Ethernet}
\begin{figure*}[!t]
\centering
\includegraphics[width=.85\textwidth]{figures/Ethernet_Interface}
\caption{Operation of the virtual internal Ethernet interface: Network packets generated at a source node are initially in its DRAM. At the request of the Ethernet Device driver, a DMA transfer is initiated and the packet is transferred to the router logic facing the SERDES links. The packet will traverse through zero or more intermediate nodes without processor interaction before reaching the destination, where the Ethernet device implemented on the FPGA fabric will raise a hardware interrupt, notifying the driver and thereby the kernel of the new packet to be processed.}
\label{fig:eth_int}
\end{figure*}
One of the virtual interfaces is designed to appear similar to an Ethernet interface\footnote{Note that this is an interface for node to node communication, and is different from the `real' physical Ethernet interface at node (100) which is intended for communicating with the external world.}. While the underlying hardware is quite different from an Ethernet network, this design point is chosen to take advantage of the large amount of standard application software readily available for IP networks such as \texttt{ssh}. A Linux OS and associated device driver running on an ARM processor can then use these applications to communicate with other nodes on the internal network operating as if it were communicating with a real Ethernet device. Similarly, applications that depend on standard parallel software libraries (e.g. Message Passing Interface (MPI) \cite{geist:1996} and its variants) can be easily supported. Using stable, well-established networking applications was also extremely useful during initial debugging of the network hardware and the system software.
The operating mechanism for transmitting packets is conceptualized in Fig.~\ref{fig:eth_int}. During the Transmit Operation, the application passes information to the kernel networking stack, which adds various headers and sends it on to a virtual internal Ethernet interface (ethX).
This interface is owned by the device driver, which manages a set of buffer descriptors that contain information about the size and memory locations of various packets in the DRAM.
The device driver then informs the hardware about the availability of a packet to be transmitted, by setting a status bit.
The actual transfer from the DRAM into the FPGA fabric is a DMA operation, using an AXI-HP bus on the Zynq chip.
Packet receive is conceptually a reverse operation, with the distinction that the device driver has two mechanisms to know of the arrival of a packet on the interface -- one is a hardware interrupt, and the other is a polling mechanism that is far more efficient under high traffic conditions.
Note that while this description assumes applications running in software as the producers and consumers of the packets, the internal Ethernet can also be accessed from other hardware blocks on the FPGA fabric itself, if at all necessary.
The availability of this virtual internal Ethernet also makes it straightforward for any node in the system to communicate with the external world using TCP/IP. This is done by using the physical Ethernet port on node (100) and configuring this node as an Ethernet gateway implementing Network Address Translation (NAT) and port forwarding. One immediate and obvious use of this feature is the implementation of an NFS (network file system) service to save application data from each of the nodes (whose file systems are implemented on the DRAM and are therefore volatile) to a non-volatile external storage medium.
\subsection{Postmaster DMA}
\begin{figure*}[!t]
\centering
\includegraphics[width=.85\textwidth]{figures/Postmaster_int2.JPG}
\caption{Operation of Postmaster DMA, which is a lightweight, high bandwidth interface: An application on the source node (either on the CPU or on the FPGA) writes data to a transmit queue on the FPGA logic. This data is then received on a destination node, where it can be either consumed in the destination FPGA and/or written into a memory mapped region as shown. System software is only involved in the initialization and tear-down process, and is not shown.}
\label{fig:post_int}
\end{figure*}
The postmaster DMA logic provides a method to move small amounts of data between nodes in the system. The function is intended to be used directly by machine intelligence application code on the software or by application hardware modules on the FPGA. It provides a communications channel with much lower overhead than going through the TCP/IP stack.
Postmaster DMA is a tunneled queue model (Fig.~\ref{fig:post_int}), where the processor (or hardware module) sees a queue that can be written at a known, fixed, address. Data written to that queue is transferred to a remote node where it is picked up by a DMA engine and moved to a pre-allocated buffer in system memory. Multiple initiators may send data to the same target. At the target, the received data is stored in a linear stream in the order in which it is received. The Postmaster hardware guarantees that a packet of data from a single initiator is always stored in contiguous locations. To reiterate, it is not necessary that the postmaster DMA be used only for processor-to-processor communication; other FPGA hardware implemented on the source/destination nodes could use it too if necessary (although in some cases this may make the `DMA' superfluous).
Packets from multiple initiators will be interleaved within the single data queue. This model is particularly well-suited to Machine Intelligence applications in which regions or learners are distributed across multiple nodes, and each node generates multiple small outputs during each time step which become the inputs in the next time step. The function of Postmaster is to allow the node to send those outputs to their intended targets as they are generated rather than collect them and send them out as a larger transmission at the end of the time step. In addition to eliminating the burden of aggregating the data, this approach also allows much more overlap of computation and communication.
\subsection{Bridge FIFO}
The bridge FIFO is intended to facilitate direct hardware to hardware communication
between two hardware modules located in separate FPGAs by exposing a regular FIFO interface. The bridge FIFO takes care of assembling the data into network packets and communicating with the packet router logic. Figure \ref{fig:bridge_fifo_figure} presents an implementation of the bridge FIFO.
\begin{figure*}[!t]
\centering
\includegraphics[width=\textwidth]{figures/INC_Bridge_FIFO_figure.pdf}
\caption{Implementation of a Bridge FIFO in the FPGA Logic}
\label{fig:bridge_fifo_figure}
\end{figure*}
The interface is composed of two modules implemented in pairs: the Bridge FIFO transmit and the Bridge FIFO receive. The first one corresponds to the write port of the FIFO while the second corresponds to the read port. They are always implemented in pairs and must be located on the source node for the transmit unit and on the destination node for the receive unit. Together, they form a communication channel.
On the source Node, the Bridge FIFO transmit converts its input (words of data) into network packets. These Bridge FIFO packets are multiplexed with other protocol packets within the Packet Mux unit which transmits all the networks packets to the Packet Router unit. The Packet Mux unit enables coexistence of multiple communication protocols.
On the destination Node, the Packet Router transmits the received network packets to the Packet Demux unit, which separates the various protocol packets and directs them to their corresponding receiver. The Bridge FIFO receive unit receives the packets and converts them back into words of data.
If multiple independent communication channels are required, then multiple pairs of Bridge FIFO transmit and receive can be instantiated. The Bridge FIFO transmits will be multiplexed within the Bridge FIFO mux and the bridge FIFO receives will be demultiplexed within the Bridge FIFO demux.
The Bridge FIFO Mux (respectively Demux) supports up to 32 Bridge FIFO transmit (respectively receive). If more channels are required, then another Bridge FIFO Mux (respectively Demux) must be instantiated.
The Bridge FIFO supports different configurable bit-widths ranging from 7 to 64. If a wider FIFO is needed, then multiple bridge FIFOs must be used in parallel to achieve the required width.
Table \ref{tab:bridge_fifo_table} presents the measured latency when using the bridge FIFO. Number of hops being zero correspond to the case where emitter and receiver are on the same node, where the latency corresponds the the delay incurred by the Bridge FIFO logic alone. The cases with 1, 3 and 6 hops are the best, average and worst case respectively on a single card system (27 nodes arranged in a cubic 3D mesh with 3 nodes on each edge).
\begin{table}[!t]
\renewcommand{\arraystretch}{1.3}
\caption{Communication latency of the bridge FIFO between two nodes. }
\label{tab:bridge_fifo_table}
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
Number of hops (in Node) & 0 & 1 & 3 & 6\\
\hline
Latency (in $\upmu$s) & 0.25 & 1.1 & 2.5 & 4.7\\
\hline
\end{tabular}
\end{table}
\section{Diagnostic Capabilities}\label{sec:diag}
INC features a wide array of diagnostic capabilities built into the hardware platform.
This is especially important in a development platform, as the reconfigurable hardware, the system software and the application software are all concurrently evolving. While some of these have been mentioned in passing in earlier sections, we present a more detailed discussion here.
\subsection{JTAG}
Each INC card has a single JTAG chain that is daisy chained through all 27 Zynq FPGAs. The JTAG chain can be used to access both the individual processors and the FPGAs, as these appear as different devices on chain. Therefore, it can be used for a broad variety of tasks including configuring the FPGAs, loading code, debugging FPGA logic with Xilinx Chipscope, and debugging the ARM code through the ARM debug access port (DAP). This mode of debug is especially useful in ironing out issues during initial system bring up, when other modes may not yet be readily available.
\subsection{Ring Bus and NetTunnel}
The Ring Bus is a sideband communications channel that links all 27 nodes on the escape card.
The bus is implemented as a ring composed of 27 unidirectional point-to-point links.
The topology allows data transfer between any two nodes by forwarding request and write data or read response through the intervening nodes.
The topology also supports broadcast write operations by forwarding a given write command to all nodes on the ring.
The routing of ring traffic is controlled by the hardware with no processor intervention.
The NetTunnel logic is functionally similar to the Ring Bus, but uses the network fabric as the transport as opposed to a dedicated sideband channel. This allows the NetTunnel logic to span the entire system, whereas the Ring Bus is confined to a single card. Note that NetTunnel does not automatically make the Ring Bus superfluous, as the network and router logic can change depending on the demands of the applications. In this scenario, having a dedicated and reliable sideband for communication is particularly useful.
As both of these mechanisms have access to the entire 4GB address space on each node they can reach, they can be used in a wide variety of scenarios. For instance, debugging reconfigurable logic often involves reading a set of hardware registers to determine the current status, active interrupts, errors and so forth. This is especially useful when communication between nodes is involved, as the issue could be at the source, the destination or indeed along links on intermediate nodes. Similarly, checkpoints, statistics or relevant program data may be written into, and subsequently retrieved from, hardware registers. This can be very useful in debugging application or device driver code if \emph{stdout} is not available or is too late, such as in a hang scenario.
\subsection{PCIe Sandbox}
PCIe Sandbox is an interactive utility that runs on a host x86 machine and provides access to the INC system through the PCIe interface on node (000). Using a set of simple commands, a user can read and write to addresses on all nodes in the INC system. PCIe Sandbox also supports a `read all' command that uses the Ring Bus to retrieve data from the same address location on all nodes of the card. Underneath, PCIe Sandbox `translates' these commands into read or write requests on the Ring Bus and NetTunnel mechanisms described above. This abstraction layer proves very useful for rapid debugging.
By reading and writing to registers on a node, special tasks can be accomplished including attaching the UART serial console to a particular node, reading bitstream build IDs for all the nodes, temperature of the card, EEPROM information (which may contain useful information such as USB-UART serial number, MAC ID of the gateway Ethernet interface on (100) etc), and the system configuration (i.e. how many cards are on the system).
Finally, PCIe Sandbox is capable of loading chunks of data into the DRAM of the compute nodes. This is the preferred method for system boot up, as one can broadcast the kernel images and associated device-trees to all the nodes in the system and then write to a boot command register that initiates boot. The same method is used to configure the FPGAs in the system with new bitstreams or to program FLASH chips in the individual nodes.
Programming the FPGAs and FLASH using the PCIe host connection and internal network is much faster than programming over JTAG.
For example, programming 27 FPGAs on a single card over JTAG takes approximately 15 minutes. On the other hand, programming 27 FPGAs on a single card over PCIe takes a couple of seconds, including the data transfer. Similarly, programming 432 FPGAs on 16 cards is nearly identical to programming one card, thanks to the network broadcast capability. It is important to note that JTAG can only work on a single card. The programming speed advantage is even more pronounced for programming the FLASH chips. On one occasion, it took more than 5 hours to program 27 FLASH chips on a single card over JTAG. In contrast, it takes about 2 minutes to program 1, 16, or 432 (or anything in between) FLASH chips over the PCIe interface.
\subsection{ARM Processor}
The ARM processor provides another dimension of per-node debug and diagnostic capability. Getting Linux running on the ARM immediately makes available a rich and varied set of diagnostic utilities. Network utilities such as \texttt{iperf} \cite{tirumala:2005}, \texttt{tcpdump} \cite{tcpdump:2018} and even \texttt{ping} \cite{muuss:2010} were used in debugging the internal Ethernet interface and the associated device driver. Counters on the ARM Performance Monitoring Unit can be readily accessed by \texttt{perf} \cite{weaver:2013} or other utilities to understand bottlenecks in application code and identify candidates for FPGA offload. Standard debuggers such as \texttt{gdb} \cite{gdb:2018} or \texttt{valgrind} \cite{valgrind:2018} could also be cross-compiled and ported, though this is untested at this time.
\section{Conclusions}\label{sec:conclusion}
We presented an overview of the INC machine intelligence platform, and publications on applications will follow shortly. Here we focused on highlighting the main features of the hardware including flexibility, configurability, high bandwidth, highly interconnected node-to-node communication modes, and diagnostic capabilities. The uniqueness of our system is that each of the 432 nodes allows application-specific offload and this feature is not available on any other parallel machine of this scale.
The practical challenges of using the INC systems lie at the interface of hardware and software. Programming directly at the FPGA level is often a daunting task for machine learning practitioners who are used to high-level languages and frameworks. On the other hand, hardware designers are accustomed to working with well-defined specifications and functionality, as circuit and micro-architecture trade-offs are often different as the task being offloaded change. This is a rather significant disconnect, exacerbated by the fast-paced, rapidly evolving field of artificial intelligence.
\ifCLASSOPTIONcompsoc
\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
The authors would like to thank Spike Narayan and Jeff Welser for management support. C. Cox thanks Mike Mastro and Kevin Holland for their support in manufacturing the cards and the system.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,237 |
\section{Introduction}
\label{s_intro}
We start by an informal description of our purpose.
\subsection{Random interlacements in two dimensions}
\label{s_RI}
Random interlacements were introduced by Sznitman in~\cite{Szn10},
motivated by the problem of disconnection of the discrete
torus ${\mathbb Z}_n^d:={\mathbb Z}^d/n{\mathbb Z}^d$ by the trace of simple random walk,
in dimension~$3$ or higher. Detailed accounts can be found in the survey~\cite{CT12} and
the recent book~\cite{DRS14}.
Loosely speaking, the model of random interlacements
in~${\mathbb Z}^d$, $d\geq 3$, is a stationary
Poissonian soup of (transient) doubly infinite simple random
walk trajectories on the integer lattice. There is an additional
parameter~$u>0$ entering the intensity measure of the Poisson
process, the larger $u$~is the more trajectories are thrown in.
The sites of~${\mathbb Z}^d$ that are not touched by the trajectories constitute
the \emph{vacant set}~${\mathcal V}^u$.
The random interlacements are constructed simultaneously for all $u>0$
in such a way that ${\mathcal V}^{u_1}\subset {\mathcal V}^{u_2}$ if $u_1>u_2$.
In fact, the law of the vacant set at level~$u$ can be
uniquely characterized by the following identity:
\begin{equation}
\label{eq_vacant>3}
{\mathbb P}[A\subset {\mathcal V}^u] = \exp\big(-u \mathop{\mathrm{cap}}(A)\big)
\quad \text{for all finite $A\subset{\mathbb Z}^d$},
\end{equation}
where $\mathop{\mathrm{cap}}(A)$ is the \emph{capacity} of~$A$.
Informally, the capacity measures how ``big'' the set
is from the point of view of the walk, see Section~6.5
of~\cite{LL10} for formal definitions,
as well as~\eqref{df_eq_measure}--\eqref{df_cap_trans} below.
At first glance, the title of this section seems to be meaningless,
just because even a single trajectory of two-dimensional
simple random walk a.s.\ visits all sites of~${\mathbb Z}^2$, so the vacant
set would be always empty. Nevertheless, there is also a natural notion of capacity in two dimensions (cf.\ Section~6.6
of~\cite{LL10}),
so one may wonder if there is a way to construct
a decreasing family $({\mathcal V}^\alpha, \alpha>0)$ of random subsets
of~${\mathbb Z}^2$ in such a way that
a formula analogous to~\eqref{eq_vacant>3}
holds for every finite~$A$. This is, however, clearly not possible
since the two-di\-men\-sional capacity of one-point sets equals~$0$.
On the other hand, it turns out to be possible to construct
such a family so that
\begin{equation}
\label{eq_vacant2}
{\mathbb P}[A\subset {\mathcal V}^\alpha] = \exp\big(-\pi\alpha \mathop{\mathrm{cap}}(A)\big)
\end{equation}
holds \emph{for all sets containing the origin}
(the factor~$\pi$ in the exponent is just for
convenience, as explained below).
We present this construction in Section~\ref{s_results}.
To build the interlacements,
we use trajectories of simple random walks
conditioned on never hitting the origin.
Of course, the law of the vacant set is no longer translationally
invariant, but we show that it has the property of
\emph{conditional} translation invariance,
cf.\ Theorem~\ref{t_properties_RI} below.
In addition, we will see that (similarly to the $d\geq 3$
case) the random object we construct has strong connections
to random walks on two-dimensional torus. All this makes
us believe that ``two-dimensional random interlacements'' is
the right term for the object we introduce in this paper.
\subsection{Cover time and late points of simple random walk on
a discrete torus}
\label{s_late}
Consider the simple random walk on the two-dimensional
discrete torus ${\mathbb Z}^2_n$ with the starting point chosen
uniformly at random. Let~${\mathcal T}_n$ be the first moment
when this random walk visits all sites of~${\mathbb Z}^2_n$;
we refer to~${\mathcal T}_n$ as the \emph{cover time}
of the torus. It was shown in~\cite{DPRZ04}
that $\frac{{\mathcal T}_n}{n^2\ln^2 n}\to \frac{4}{\pi}$
in probability; later, this result was refined
in~\cite{D12}, and then even finer results on the first correction to this limit were
obtained in~\cite{BK14} for the similar problem
of covering the (continuous) torus with a Brownian sausage.
The structure of the set of \emph{late points}
(i.e., the set of points that are still unvisited up to
a given time) of the random walk
on the torus is rather well understood in dimensions $d\geq 3$,
see~\cite{B13,MS13},
and also~ \cite{GdH14} for the continuous case.
On the other hand, much remains to be
discovered in two dimensions.
After the heuristic arguments of~\cite{BH91} revealing an intriguing random set,
it was shown
in~\cite{DPRZ06} that this set has interesting fractal-like properties when the
elapsed time is a fraction of the expected cover time. This particular behaviour is induced by
long distance correlations between hitting times due to recurrence.
In this paper, we prove that the law of the uncovered set
around the origin at time $\frac{4\alpha}{\pi}n^2\ln^2 n$
\emph{conditioned} on the event that the origin is uncovered,
is close to the law of two-dimensional random
interlacements at level~$\alpha$ (Theorem~\ref{t_conditional}).
We hope that this result
will lead to other advances in understanding the structure
of the uncovered set.
We now explain why conditioning is necessary to observe a
meaningful point process.
In two dimensions, if we know that simple random walk has
visited a given site by a large time, then it is likely,
by recurrence, that it has has
visited all the nearby sites as well.
This means that a fixed-size window around the origin
will be typically either full (on the event that the origin
was visited)
or empty (if the origin was not yet visited).
Therefore, we need to
condition on a rare event to obtain a nontrivial limit.
As a side note, observe that the two-dimensional random interlacements
relate to the simple random walk on the torus at a time
proportional to the cover time. In higher dimensions, one starts to observe the
``interlacement regime'' already at times below the cover time
by a factor of~$\ln n$.
\medskip
\textbf{Organisation of the paper}: In Section~\ref{s_results}
we construct the model of random interlacements and present some of its properties.
In Section~\ref{s_rw_interl} we formulate a result
relating this model and the vacant
set of the simple random walk on the discrete torus.
We prove some results on the spot
-- when short arguments are available -- postponing the proof
of the other ones to Section~\ref{s_proofs}.
Section~\ref{s_aux} contains
a number of auxiliary facts needed for the proof of the main results.
\section{Definitions and results}
\label{sec:def-res}
We start by defining the two-dimensional
random interlacement process, which involves
some potential-theoretic considerations.
\subsection{Random interlacements: definitions, properties}
\label{s_results}
Let~$\|\cdot\|$ be the Euclidean norm. Define the (discrete)
ball
\[
B(x,r) = \{y\in {\mathbb Z}^2: \|y-x\|\leq r\}
\]
(note that $x$ and $r$ need not be integer), and
abbreviate $B(r):=B(0,r)$.
We write~$x\sim y$ if $x$ and $y$ are neighbours on~${\mathbb Z}^2$.
The (internal) boundary of $A\subset{\mathbb Z}^2$ is defined by
\[
\partial A = \{x\in A: \text{there exists }y\in {\mathbb Z}^2\setminus A
\text{ such that }x\sim y\}.
\]
Let~$(S_n, n\geq 0)$ be two-dimensional simple
random walk. Write~$P_x$ for the law of the walk started from~$x$
and~$E_x$ for the corresponding expectation.
Let
\begin{align}
\tau_0(A) &= \inf\{k\geq 0: S_k\in A\} \label{entrance_t},\\
\tau_1(A) &= \inf\{k\geq 1: S_k\in A\} \label{hitting_t}
\end{align}
be the entrance and the hitting time of the set~$A$ by
simple random walk~$S$ (we use the convention $\inf \emptyset = +{\infty}$).
For a singleton $A=\{x\}$, we will write $\tau_i(A)=\tau_i(x)$, $i=0,1$,
for short.
Define the potential kernel~$a$ by
\begin{equation}
\label{def_a(x)}
a(x) = \sum_{k=0}^\infty\big(P_0[S_k\!=\!0]-P_x[S_k\!=\!0]\big).
\end{equation}
It can be shown that the above series indeed converges
and we have~$a(0)=0$, $a(x)>0$ for $x\neq 0$, and
\begin{equation}
\label{formula_for_a}
a(x) = \frac{2}{\pi}\ln \|x\| + \gamma' + O(\|x\|^{-2})
\end{equation}
as $x\to\infty$, cf.\ Theorem~4.4.4 of~\cite{LL10}
(the value of $\gamma'$ is
known\footnote{$\gamma'=\pi^{-1}(2\gamma+\ln 8)$,
where $\gamma=0.5772156\dots$ is the Euler-Mascheroni constant},
but we will not need it in
this paper).
Also, the function~$a$ is harmonic outside the origin, i.e.,
\begin{equation}
\label{a_harm}
\frac{1}{4}\sum_{y: y\sim x}a(y) = a(x) \quad \text{ for all }
x\neq 0.
\end{equation}
Observe that~\eqref{a_harm} immediately implies
that $a(S_{k\wedge \tau_0(0)})$ is a martingale, we will
repeatedly use this fact in the sequel.
With some abuse of notation, we also consider
the function
\[
a(r)=\frac{2}{\pi}\ln r + \gamma'
\]
of a \emph{real}
argument~$r\geq 1$. The advantage of using this notation is e.g.\ that,
due to~\eqref{formula_for_a}, we may write, as $r\to\infty$,
\begin{equation}
\label{real_a}
\sum_{y\in\partial B(x,r)} \nu(y)a(y) = a(r)
+ O\Big(\frac{\|x\|\vee 1}{r}\Big)
\end{equation}
for \emph{any} probability measure~$\nu$ on $\partial B(x,r)$.
The \emph{harmonic measure} of a finite $A\subset{\mathbb Z}^2$
is
the entrance law ``starting at infinity'',
\begin{equation}
\label{def_hm}
\mathop{\mathrm{hm}}\nolimits_A(x) = \lim_{\|y\|\to\infty}P_y[S_{\tau_1(A)}=x].
\end{equation}
The existence of the above limit
follows from Proposition~6.6.1 of~\cite{LL10}; also, this
proposition together with~(6.44) implies that
\begin{equation}
\label{hm_escape}
\mathop{\mathrm{hm}}\nolimits_A(x) = \frac{2}{\pi}\lim_{R\to \infty}
P_x\big[\tau_1(A)>\tau_1\big(\partial B(R)\big)\big]\ln R .
\end{equation}
Intuitively, \eqref{hm_escape} means that
the harmonic measure at $x\in\partial A$
is proportional to the probability of escaping from~$x$
to a large sphere. Observe also that,
by recurrence of the walk, $\mathop{\mathrm{hm}}\nolimits_A$ is a probability measure
on~$\partial A$.
Now, for a finite set~$A$ containing the origin,
we define its capacity by
\begin{equation}
\label{df_cap2}
\mathop{\mathrm{cap}}(A) = \sum_{x\in A}a(x)\mathop{\mathrm{hm}}\nolimits_A(x);
\end{equation}
in particular, $\mathop{\mathrm{cap}}\big(\{0\}\big)=0$ since $a(0)=0$.
For a set not containing the origin, its capacity is defined
as the capacity of a translate of this set that does contain
the origin. Indeed, it can be shown that the capacity does not depend
on the choice of the translation.
A number of alternative definitions are available,
cf.\ Section~6.6 of~\cite{LL10}.
Intuitively, the capacity of $A$ represents the difference in size between the set $A$ and a single point, as seen from infinity:
From (6.40) and the formula above Proposition 6.6.2 in~\cite{LL10}, we can write
\[
P_x\big[\tau_1\big(\partial B(r)\big) <\tau_1(A)\big]
= \frac{ \ln \|x\| + \frac{\pi \gamma'}{2} - \frac{\pi}{2} \mathop{\mathrm{cap}}(A)
+ \varepsilon_A(x) + \varepsilon_{A,x}'(r)}{\ln r},
\]
where $\varepsilon_A(x)$ vanishes as $\|x\| \to {\infty}$ and $\varepsilon_{A,x}'(r)$ vanishes as $r \to {\infty}$ keeping fixed the other variables.
Observe that, by symmetry,
the harmonic measure of any two-point set is uniform,
so $\mathop{\mathrm{cap}}\big(\{x,y\}\big)=\frac{1}{2}a(y-x)$ for any $x,y\in{\mathbb Z}^2$.
Also, \eqref{real_a} implies that
\begin{equation}
\label{capa_ball}
\mathop{\mathrm{cap}}\big(B(r)\big) = a(r) + O(r^{-1}).
\end{equation}
Let us define another random walk $({\widehat S}_n, n\geq 0)$
on~${\mathbb Z}^2$ (in fact, on~${\mathbb Z}^2\setminus \{0\}$) in the following way:
the transition probability from~$x$ to~$y$
equals $\frac{a(y)}{4a(x)}$ for all $x\sim y$
(this definition does not make sense for $x=0$, but this is
not a problem since the walk~${\widehat S}$ can never enter the origin anyway). The
walk~${\widehat S}$ can be thought of as the Doob $h$-transform
of the simple random walk, under condition of not hitting the origin
(see Lemma~\ref{l_relation_S_hatS} and its proof).
Note that~\eqref{a_harm} implies that the random walk~${\widehat S}$
is indeed well defined, and, clearly, it is
an irreducible Markov chain on~${\mathbb Z}^2\setminus \{0\}$. We
denote by $\widehat{P}_x, \widehat{E}_x$ the probability and expectation
for the random walk~${\widehat S}$ started from~$x \neq 0$.
Let $\widehat{\tau}_0, \widehat{\tau}_1$ be defined as in
\eqref{entrance_t}--\eqref{hitting_t}, but with~${\widehat S}$ in the place of~$S$.
Then, it is straightforward to observe that
\begin{itemize}
\item the walk~${\widehat S}$ is reversible, with the reversible
measure~$\mu_x:=a^2(x)$;
\item in fact, it can be represented as a random walk
on the two-dimensional lattice with conductances (or weights)
$\big(a(x)a(y), x,y\in {\mathbb Z}^2, x\sim y\big)$;
\item $\big(a(x), x\in {\mathbb Z}^2\setminus \{0\}\big)$ is an
excessive measure for~${\widehat S}$ (i.e., for all $y \neq 0$, $\sum_x a(x) \widehat{P}_x({\widehat S}_1=y) \leq a(y)$), with equality failing at the
four neighbours
of the origin. Therefore, by e.g.\
Theorem~1.9 of Chapter~3 of~\cite{R84}, the random walk~${\widehat S}$
is transient;
\item an alternative argument for proving transience is
the following: let~$\mathcal{N}$ be the set of the four
neighbours of the origin. Then, a direct calculation
shows that $1/a({\widehat S}_{k\wedge \widehat{\tau}_0(\mathcal{N})})$ is a
martingale. The transience then follows from Theorem~2.2.2 of~\cite{FMM}.
\end{itemize}
Our next definitions are appropriate for the transient case.
For a finite~$A\subset {\mathbb Z}^2$,
we define the \emph{equilibrium measure}
\begin{equation}
\label{df_eq_measure}
\widehat e_A(x) = \1{x\in A} \widehat{P}_x[\widehat{\tau}_1(A)=\infty]\mu_x,
\end{equation}
and the capacity (with respect to~${\widehat S}$)
\begin{equation}
\label{df_cap_trans}
\mathop{\widehat{\mathrm{cap}}}(A) = \sum_{x\in A}\widehat e_A(x).
\end{equation}
Observe that, since $\mu_0=0$, it holds that
$\mathop{\widehat{\mathrm{cap}}}(A)=\mathop{\widehat{\mathrm{cap}}}(A\cup\{0\})$ for any set~$A\subset {\mathbb Z}^2$.
Now, we use the general construction of random interlacements
on a transient weighted graph introduced in~\cite{T09}.
In the following few lines we briefly summarize this
construction.
Let~$W$ be the space of all doubly infinite nearest-neighbour
transient trajectories in~${\mathbb Z}^2$,
\begin{align*}
W =& \big\{\varrho=(\varrho_k)_{ k\in {\mathbb Z}}:
\varrho_k\sim \varrho_{k+1} \text{ for all }k;\\
&~~~~~~~~~~\text{ the set }
\{m: \varrho_m=y\} \text{ is finite for all }y\in{\mathbb Z}^2 \big\}.
\end{align*}
We say that~$\varrho$ and~$\varrho'$ are equivalent if they
coincide after a time shift, i.e., $\varrho\sim\varrho'$
when there exists~$k$ such that $\varrho_{m+k}=\varrho_m$ for all~$m$.
Then, let $W^*=W/\sim$ be the space of trajectories
modulo time shift, and define~$\chi^*$ to be the canonical
projection from~$W$ to~$W^*$. For a finite $A\subset {\mathbb Z}^2$,
let~$W_A$ be the set of trajectories in~$W$ that intersect~$A$,
and we write~$W^*_A$ for the image of~$W_A$ under~$\chi^*$.
One then constructs the random interlacements as Poisson
point process on $W^*\times {\mathbb R}^+$ with the intensity measure
$\nu\otimes du$, where~$\nu$ is described in the following
way. It is the unique sigma-finite measure on the cylindrical sigma-field of~$W^*$
such that for every finite~$A$
\[
\mathbf{1}_{W^*_A} \cdot \nu = \chi^* \circ Q_A,
\]
where the finite measure~$Q_A$ on~$W_A$ is determined by the
following equality:
\[
Q_A\big[(\varrho_k)_{k\geq 1}\!\in \!F, \varrho_0\!=\!x, (\varrho_{-k})_{k\geq 1}\!\in \!G\big]
= \widehat e_A(x) \cdot \widehat{P}_x[F] \cdot \widehat{P}_x[G\mid \widehat{\tau}_1(A)\!=\!\infty].
\]
The existence and uniqueness of~$\nu$ was shown in
Theorem~2.1 of~\cite{T09}.
\begin{df} \label{def:ri}
For a configuration $\sum_{\lambda}\delta_{(w^*_\lambda,u_\lambda)}$
of the above Poisson process, the {process of random interlacements
at level~$\alpha$} (which will be referred to as RI($\alpha$))
is defined as the set of trajectories with label less than or equal
to~$\pi\alpha$, i.e.,
\[
\sum_{\lambda: u_\lambda\leq \pi\alpha}
\delta_{w^*_\lambda} \;.
\]
\end{df}
Observe that this definition is somewhat unconventional (we used~$\pi\alpha$
instead of just~$\alpha$, as one would normally do), but we will
see below that it is quite reasonable in two dimensions,
since the formulas become generally cleaner.
It is important to have in mind the following ``constructive''
description of random interlacements at level~$\alpha$ ``observed''
on a finite set $A\subset {\mathbb Z}^2$. Namely,
\begin{itemize}
\item take a Poisson($\pi\alpha\mathop{\widehat{\mathrm{cap}}}(A)$) number of particles;
\item place these particles on the boundary of~$A$
independently, with distribution
$\overline{e}_A = \big((\mathop{\widehat{\mathrm{cap}}} A)^{-1}\widehat e_A(x), x\in A\big)$;
\item let the particles perform independent ${\widehat S}$-random walks
(since~${\widehat S}$ is transient, each walk only leaves a finite trace
on~$A$).
\end{itemize}
It is also worth mentioning that the FKG inequality holds
for random interlacements, cf.\ Theorem~3.1 of~\cite{T09}.
The \emph{vacant set} at level $\alpha$,
\[
{\mathcal V}^\alpha = {\mathbb Z}^2 \setminus \bigcup_{\lambda: u_\lambda \leq \pi\alpha} \omega^*_\lambda
({\mathbb Z}),
\]
is the set of lattice points not covered by the random interlacement. It contains the origin by definition. In Figure~\ref{f_simulation} we present
a simulation of the vacant set for different values of the
parameter.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{simulation}
\caption{A realization of the vacant set (dark blue) of RI($\alpha$)
for different values of~$\alpha$.
For $\alpha=1.5$ the only vacant site is the origin.
Also, note that we see the same neighbourhoods
of the origin for $\alpha=1$ and~$\alpha=1.25$;
this is not surprising since just a few new walks enter the picture when
increasing the rate by a small amount.}
\label{f_simulation}
\end{center}
\end{figure}
As a last step, we need to show that we have indeed
constructed the object for which~\eqref{eq_vacant2}
is verified. For this, we need to prove the following fact:
\begin{prop}
\label{p_equalcapa}
For any finite set $A\subset {\mathbb Z}^2$ such that $0\in A$ it holds that
$\mathop{\mathrm{cap}}(A)=\mathop{\widehat{\mathrm{cap}}}(A)$.
\end{prop}
\begin{proof}
Indeed, consider an arbitrary~$x\in\partial A$, $x\neq 0$,
and (large)~$r$ such that $A\subset B(r-2)$.
Write using~\eqref{formula_for_a}
\begin{align*}
\widehat{P}_x\big[\widehat{\tau}_1(A)>\widehat{\tau}_1\big(\partial B(r)\big)\big] & =
\sum_\varrho \frac{a(\varrho_{\text{end}})}{a(x)}
\Big(\frac{1}{4}\Big)^{|\varrho|}\\
&= \big(1+o(1)\big)\frac{\frac{2}{\pi}\ln r}{a(x)}
\sum_\varrho \Big(\frac{1}{4}\Big)^{|\varrho|}\\
&= \big(1+o(1)\big)\frac{\frac{2}{\pi}\ln r}{a(x)}\,
P_x\big[\tau_1(A)>\tau_1\big(\partial B(r)\big)\big],
\end{align*}
where the sums are taken over all trajectories~$\varrho$
that start at~$x$, end at~$\partial B(r)$, and avoid
$A\cup \partial B(r)$ in between;
$\varrho_{\text{end}}\in \partial B(r)$ stands for the
ending point of the trajectory, and~$|\varrho|$ is the
trajectory's length.
Now, we send~$r$ to infinity and use~\eqref{hm_escape}
to obtain that, if $0\in A$,
\begin{equation}
\label{escape_identity}
a(x) \widehat{P}_x[\widehat{\tau}_1(A)=\infty] = \mathop{\mathrm{hm}}\nolimits_A(x).
\end{equation}
Multiplying by~$a(x)$ and summing over
$x\in A$ (recall that $\mu_x=a^2(x)$) we
obtain the expressions in~\eqref{df_cap2} and~\eqref{df_cap_trans}
and thus conclude the proof.
\end{proof}
Together with formula~(1.1) of~\cite{T09}, Proposition~\ref{p_equalcapa} shows the fundamental relation~(\ref{eq_vacant2})
announced in introduction:
for all finite subsets~$A$ of ${\mathbb Z}^2$ containing the origin,
\[
{\mathbb P}[ A \subset {\mathcal V}^\alpha] = \exp\big(-\pi\alpha \mathop{\mathrm{cap}}(A)\big).
\]
As mentioned before, the law of two-dimensional random interlacements
is not translationally invariant,
although it is of course invariant with respect to reflections/rotations
of~${\mathbb Z}^2$ that preserve the origin.
Let us describe some other basic properties of two-dimensional
random interlacements:
\begin{theo}
\label{t_properties_RI}
\begin{itemize}
\item[(i)] For any $\alpha>0$, $x\in{\mathbb Z}^2$,
$A\subset {\mathbb Z}^2$, it holds that
\begin{equation}
\label{properties_RI_i}
{\mathbb P}[A\subset{\mathcal V}^\alpha \mid x\in {\mathcal V}^\alpha]=
{\mathbb P}[-A+x\subset{\mathcal V}^\alpha \mid x\in {\mathcal V}^\alpha].
\end{equation}
More generally, for all $\alpha>0$, $x\in{\mathbb Z}^2 \setminus \{0\}$,
$A\subset {\mathbb Z}^2$, and any
lattice isometry~$M$ exchanging $0$ and $x$, we have
\begin{equation}
\label{properties_RI_i'}
{\mathbb P}[A\subset{\mathcal V}^\alpha \mid x\in {\mathcal V}^\alpha]=
{\mathbb P}[MA \subset{\mathcal V}^\alpha \mid x\in {\mathcal V}^\alpha].
\end{equation}
\item[(ii)] With $\gamma'$ from (\ref{formula_for_a}) we have
\begin{equation}
\label{properties_RI_ii}
{\mathbb P}[x\in {\mathcal V}^\alpha]=\exp\Big(-\pi\alpha \frac{a(x)}{2}
\Big)
=e^{-\gamma'\pi\alpha/2}\|x\|^{-\alpha}\big(1+O(\|x\|^{-2})\big).
\end{equation}
\item[(iii)] For~$A$ such that $0\in A\subset B(r)$
and $x\in{\mathbb Z}^2$ such that $\|x\|\geq 2r$ we have
\begin{equation}
\label{properties_RI_iii}
{\mathbb P}[A\subset{\mathcal V}^\alpha \mid x\in {\mathcal V}^\alpha]=
\exp\Bigg(-\frac{\pi\alpha}{4}\mathop{\mathrm{cap}}(A)
\frac{1+O\big(\frac{r\ln r \ln\|x\|}{\|x\|}\big)}
{1-\frac{\mathop{\mathrm{cap}}(A)}{2a(x)}
+O\big(\frac{r\ln r}{\|x\|}\big)}
\Bigg).
\end{equation}
\item[(iv)] For $x,y\neq 0$, $x\neq y$, we have
${\mathbb P}\big[\{x,y\}\subset {\mathcal V}^\alpha\big]
= \exp\big(-\pi \alpha\Psi\big)$,
where
\[
\Psi = \frac{a(x)a(y)a(x-y)}
{a(x)a(y)+a(x)a(x-y)+a(y)a(x-y)-\frac{1}{2}
\big(a^2(x)+a^2(y)+a^2(x-y)\big)}.
\]
Moreover,
as $s:= \|x\| \to \infty$, $\ln \|y\| \sim \ln s$ and
$\ln \|x-y\|\sim \beta \ln s$ with some $\beta\in [0,1]$, we have
\begin{equation*}
{\mathbb P}\big[\{x,y\}\subset {\mathcal V}^\alpha\big]
= s^{-\frac{4\alpha}{4-\beta}+o(1)},
\end{equation*}
and
polynomially decaying correlations
(cf.~(\ref{def:cor}) in Remark~\ref{rem24} for the definition),
\begin{equation}
\label{eq:cor}
\mathop{\mathrm{Cor}}\big
{\{x\in {\mathcal V}^\alpha\}},
{\{y\in {\mathcal V}^\alpha\}}\big)
= s^{-\frac{\alpha \beta}{4-\beta}+o(1)}.
\end{equation}
\item[(v)] Assume that $\ln \|x\|\sim \ln s$, $\ln r\sim\beta \ln s$ with $\beta<1$.
Then, as $s \to {\infty}$,
\begin{equation}
\label{properties_RI_v}
{\mathbb P}\big[B(x,r)\subset {\mathcal V}^\alpha\big]
= s^{-\frac{2\alpha}{2-\beta}+o(1)}.
\end{equation}
\end{itemize}
\end{theo}
These results invite a few comments.
\begin{rem} \label{rem24}
\begin{enumerate}
\item The statement in (i) describes an invariance property
given that a point is vacant.
We refer to it as the conditional stationarity
\item We can interpret (iii) as follows:
the conditional law of RI($\alpha$)
given that a distant site~$x$ is vacant, is similar -- near the origin --
to the unconditional law of RI($\alpha/4$). Combined with~(i), the similarity holds near~$x$ as well.
Moreover, one can also estimate the ``local rate'' away
from the origin,
see Figure~\ref{f_change_rate}. More specifically,
observe from Lemma~\ref{l_cap_distantball}~(ii) that
$\mathop{\mathrm{cap}}(A_2)\ll \ln s$ with $s={\rm dist}(0,A_2)$ large implies
$\mathop{\mathrm{cap}}\big(\{0\}\cup A_2\big)=\frac{a(s)}{2}(1+o(1))$.
If $x$ is at a much larger distance from the origin than~$A_2$,
say $\ln \|x\| \sim \ln( s^2)$,
then~\eqref{properties_RI_iii} reveals a
``local rate'' equal to $\frac{2}{7}\alpha$,
that is,
${\mathbb P}[A_2\subset{\mathcal V}^\alpha\mid x \in {\mathcal V}^\alpha]=\exp\big(-\frac{2}{7}
\pi\alpha\mathop{\mathrm{cap}}\big(\{0\}\cup A_2\big)(1+o(1))\big)$;
indeed, the expression in the denominator in~\eqref{properties_RI_iii}
equals approximately $1-\frac{\mathop{\mathrm{cap}}(\{0\}\cup A_2)}{2a(x)}
\approx 1 - \frac{a(s)/2}{2a(s^2)}\approx \frac{7}{8}$.
\item
Recall that the correlation between two events~$A$ and~$B$,
\[
\label{def:cor}
\mathop{\mathrm{Cor}}(A,B) = \frac{ {\rm Cov}({\mathbf 1}_A, {\mathbf 1}_ B)}
{[ \mathop{\mathrm{Var}}({\mathbf 1}_A) \mathop{\mathrm{Var}}( {\mathbf 1}_ B)]^{1/2}}\in [-1,1],
\]
can be viewed as the cosine of the angle
between the vectors ${\mathbf 1}_{A}-{\mathbb P}(A),
{\mathbf 1}_B-{\mathbb P}(B)$ in the corresponding Hilbert space $L^2$.
Then,
equation~\eqref{eq:cor} relates the geometry of the lattice
with the geometry of the random point process:
For points~$x$ and~$y$ at large distance~$s$ making a small
angle $s^{1-\beta}$ with the origin, the random variables
$\1{x\in {\mathcal V}^\alpha}$ and
$\1{y\in {\mathcal V}^\alpha}$ make, after centering,
an angle of order $ s^{-\frac{\alpha \beta}{4-\beta}}$
in the space of square integrable random variables.
\item By symmetry, the conclusion of (iv) remains
the same in the situation when
$\ln \|x\|, \ln \|x-y\|\sim \ln s$ and $\ln \|y\|\sim \beta\ln s$.
\end{enumerate}
\end{rem}
\begin{figure}
\begin{center}
\includegraphics{change_rate}
\caption{How the ``local rate'' looks like if we condition
on the event that a ``distant'' site is vacant.}
\label{f_change_rate}
\end{center}
\end{figure}
\begin{proof}[Proof of (i) and (ii)]
To prove~(i), observe that
\[
\mathop{\mathrm{cap}}\big(\{0,x\}\cup A\big) = \mathop{\mathrm{cap}}\big(\{0,x\}\cup (-A+x)\big)
\]
by symmetry. For the second statement in (i),
note that, for $A'=\{0,x\}\cup A$,
it holds that
$
\mathop{\mathrm{cap}}\big(A'\big) =\mathop{\mathrm{cap}}\big(MA'\big) = \mathop{\mathrm{cap}}\big(\{0,x\}\cup MA \big).
$
Item~(ii) follows from the above mentioned fact that
$\mathop{\mathrm{cap}}\big(\{0,x\}\big)=\frac{1}{2}a(x)$
together with~\eqref{formula_for_a}.
\end{proof}
We postpone the proof of other parts of Theorem~\ref{t_properties_RI},
since it requires some estimates for capacities
of various kinds of sets.
We now turn to estimates on the cardinality of the vacant set.
\begin{theo}
\label{t_sizevacant}
\begin{itemize}
\item[(i)] We have
\[
{\mathbb E} \big(\vert {\mathcal V}^\alpha\cap B(r)\vert \big) \sim
\begin{cases}
\frac{2\pi}{2-\alpha}
e^{-\gamma'\pi\alpha/2} \times r^{2-\alpha}, & \text{ for }\alpha < 2,\\
2\pi e^{-\gamma'\pi\alpha/2} \times \ln r, & \text{ for }\alpha = 2,\\
\text{const} , & \text{ for }\alpha > 2.
\end{cases}
\]
\item[(ii)] For $\alpha>1$ it holds that ${\mathcal V}^\alpha$ is
finite a.s. Moreover, ${\mathbb P}\big[{\mathcal V}^\alpha=\{0\}\big]>0$
and ${\mathbb P}\big[{\mathcal V}^\alpha=\{0\}\big]\to 1$ as $\alpha\to\infty$.
\item[(iii)] For $\alpha \in (0,1)$, we have $|{\mathcal V}^\alpha|=\infty$
a.s. Moreover,
\begin{equation}
\label{eq_emptyball}
{\mathbb P}\big[{\mathcal V}^\alpha\cap \big(B(r)\setminus B(r/2)\big)=\emptyset\big]
\leq r^{-2(1-\sqrt{\alpha})^2+o(1)}.
\end{equation}
\end{itemize}
\end{theo}
It is worth noting that the ``phase transition'' at $\alpha=1$
in~(ii) corresponds to the cover time of the torus,
as shown in Theorem~\ref{t_conditional} below.
\begin{proof}[Proof of (i) and (ii) (incomplete, in the latter case)]
Part~(i) immediately follows from Theorem~\ref{t_properties_RI}~(ii).
The proof of the part~(ii) is easy in the case $\alpha>2$.
Indeed, observe first that ${\mathbb E}|{\mathcal V}^\alpha|<\infty$
implies that ${\mathcal V}^\alpha$ itself is a.s.\ finite.
Also, Theorem~\ref{t_properties_RI}~(ii) actually implies
that ${\mathbb E}|{\mathcal V}^\alpha \setminus \{0\}|\to 0$ as $\alpha\to\infty$,
so ${\mathbb P}\big[{\mathcal V}^\alpha=\{0\}\big]\to 1$ by the Chebyshev inequality.
Now, let us prove that, in general, ${\mathbb P}\big[|{\mathcal V}^\alpha|<\infty\big]=1$
implies that ${\mathbb P}\big[{\mathcal V}^\alpha=\{0\}\big]>0$.
Indeed, if~${\mathcal V}^\alpha$ is a.s.\ finite, then one can
find a sufficiently large~$R$ such that
${\mathbb P}\big[|{\mathcal V}^\alpha\cap ({\mathbb Z}^2\setminus B(R))|=0\big]>0$.
Since ${\mathbb P}[x\notin {\mathcal V}^\alpha]>0$ for any $x\neq 0$,
the claim that ${\mathbb P}\big[{\mathcal V}^\alpha=\{0\}\big]>0$
follows from the FKG inequality applied to events
$\{x\notin {\mathcal V}^\alpha\}$, $x\in B(R)$ together
with $\big\{|{\mathcal V}^\alpha\cap ({\mathbb Z}^2\setminus B(R))|=0\big\}$.
\end{proof}
As before, we postpone the proof of part~(iii) and
the rest of part~(ii) of
Theorem~\ref{t_sizevacant}.
Let us remark that we believe that the right-hand side
of~\eqref{eq_emptyball} gives the correct order of decay
of the above probability; we, however, do not have a
rigorous argument at the moment.
Also, note that the question
whether ${\mathcal V}^1$ is a.s.\ finite or not, is open.
Let us now give
a heuristic explanation about the unusual
behaviour of the model for $\alpha \in (1,2)$:
in this non-trivial interval, the vacant set is
a.s.\ finite but its expected size is infinite.
The reason is the following:
the number of ${\widehat S}$-walks that hit~$B(r)$ has Poisson law
with rate of order~$\ln r$ (recall~\eqref{capa_ball}).
Thus, decreasing this number by a
constant factor (with respect to the expectation) has only
a polynomial cost. On the other hand, by doing so, we increase
the probability that a site~$x\in B(r)$ is vacant for all $x\in B(r)$
at once, which increases the expected size of ${\mathcal V}^\alpha\cap B(r)$
by a polynomial factor. It turns out that this effect causes
the actual number of uncovered sites in~$B(r)$ to be typically
of much smaller order then the expected number of uncovered sites
there.
\subsection{Simple random walk on a discrete torus and
its relationship with random interlacements}
\label{s_rw_interl}
Now, we state our results for random walk on the torus.
Let~$(X_k, k\geq 0)$ be simple random walk on~${\mathbb Z}^2_n$
with~$X_0$ chosen uniformly at random.
Define the entrance time
to the site $x\in {\mathbb Z}^2_n$ by
\begin{equation}
\label{eq_defTx}
T_n(x) = \inf\{t\geq 0: X_t=x\},
\end{equation}
and the \emph{cover time} of the torus by
\begin{equation}
\label{eq_defT}
{\mathcal T}_n = \max_{x\in {\mathbb Z}^2_n} T_n(x).
\end{equation}
Let us also define the \emph{uncovered set} at time~$t$,
\begin{equation}
\label{df_U_t}
U_t^{(n)} = \{x\in{\mathbb Z}^2_n : T_n(x) > t\}.
\end{equation}
Denote by $\Upsilon_n:{\mathbb Z}^2 \to {\mathbb Z}^2_n$,
$\Upsilon_n(x,y)=(x \mod n, y\mod n)$,
the natural projection modulo~$n$.
Then, if~$S_0$ were chosen uniformly at random
on any fixed $n\times n$ square,
we can write $X_k=\Upsilon_n (S_k)$.
Similarly,
$B(y,r)\subset {\mathbb Z}^2_n$ is defined by $B(y,r)=\Upsilon_n B(z,r)$,
where $z\in{\mathbb Z}^2$ is such that $\Upsilon_n z = y$.
Let also
\[
t_\alpha:=\frac{4\alpha}{\pi}n^2\ln^2 n
\]
(recall that, as mentioned in Section~\ref{s_late},
$\alpha=1$ corresponds to the leading-order term of the
expected cover time of the torus).
In the following
theorem, we prove that, given that~$0$ is uncovered, the law
of the uncovered set around~$0$ at time~$t_\alpha$
is close to that of RI($\alpha$):
\begin{theo}
\label{t_conditional}
Let $\alpha>0$ and~$A$ be a finite subset of~${\mathbb Z}^2$
such that $0\in A$. Then, we have
\begin{equation}
\label{eq_conditional}
\lim_{n\to\infty}{\mathbb P}[\Upsilon_n A \subset
U_{t_\alpha}^{(n)} \mid 0\in U_{t_\alpha}^{(n)}]
= \exp\big(-\pi\alpha\mathop{\mathrm{cap}}(A)\big).
\end{equation}
\end{theo}
Let us also mention that, for higher dimensions, results
similar to the above theorem have appeared in the literature,
see Theorem~0.1 of~\cite{Szn09-2} and Theorem~1.1 of~\cite{W08}.
Also, stronger ``coupling'' results are available, see
e.g.~\cite{B13,CT15,Szn09-1,TW11}.
The proof of this theorem will be presented in Section~\ref{s_proofs}.
\begin{figure}
\begin{center}
\includegraphics{excursions}
\caption{Excursions (depicted as the solid
pieces of the trajectory) of the SRW on the torus~${\mathbb Z}^2_n$}
\label{f_excursions}
\end{center}
\end{figure}
To give a brief heuristic explanation for~\eqref{eq_conditional},
consider the \emph{excursions} of the random walk~$X$
between $\partial B\big(\frac{n}{3\ln n}\big)$ and $\partial B(n/3)$
up to time~$t_\alpha$.
We hope that Figure~\ref{f_excursions} is self-explanatory;
formal definitions are presented in Section~\ref{s_aux_torus}.
It is possible
to prove that the number of these excursions
is concentrated around
$N=\frac{2\alpha\ln^2 n}{\ln\ln n}$. Also,
one can show that each excursion hits a finite set~$A$ (such that
$0\in A$)
approximately independently of the others (also when
conditioning on $0\in U_{t_\alpha}^{(n)}$), with
probability roughly equal to $p=\frac{\pi\ln\ln n}{2\ln^2 n}\mathop{\mathrm{cap}}(A)$.
This heuristically gives~\eqref{eq_conditional} since
$(1-p)^{N}\approx \exp\big(-\pi\alpha\mathop{\mathrm{cap}}(A)\big)$.
\section{Some auxiliary facts and estimates}
\label{s_aux}
In this section we collect lemmas of all sorts
that will be needed in the sequel.
\subsection{Simple random walk in~${\mathbb Z}^2$}
\label{s_aux_SRW}
First, we recall a couple of basic facts for the exit
probabilities of simple random walk.
\begin{lem}
\label{l_exit_balls}
For all $x, y \in {\mathbb Z}^2$ and $R>0$ with $x \in B(y,R), \|y\|\leq R-2$, we have
\begin{equation}
\label{nothit_0}
P_x\big[\tau_1(0)>\tau_1\big(\partial B(y,R)\big)\big]
= \frac{a(x)}{a(R) + O\big(\frac{\|y\|\vee 1}{R}\big)}\;,
\end{equation}
and for all
$y \in B(r), x \in B(y,R)\setminus B(r)$ with $ r+\|y\| \leq R-2$, we have
\begin{equation}
\label{nothit_r}
P_x\big[\tau_1(\partial B(r))>\tau_1\big(\partial B(y,R)\big)\big] =
\frac{a(x)-a(r)+O(r^{-1})}{a(R)-a(r)
+O\big(\frac{\|y\|\vee 1}{R}+r^{-1}\big)}\;,
\end{equation}
as $r,R\to \infty$.
\end{lem}
\begin{proof}
Both \eqref{nothit_0} and \eqref{nothit_r}
are easily deduced from the following argument:
recall that the sequence $a(S_{k\wedge \tau_0(0)}), k \geq 0, $ is a martingale,
and apply the Optional Stopping Theorem
together with~\eqref{formula_for_a} and~\eqref{real_a}. For the second statement,
with
\[
\tau= \tau_1(\partial B(r)) \wedge \tau_1\big(\partial B(y,R)\big)
\quad \text{ and } \quad q=P_x\big[\tau_1(\partial B(r))>\tau_1
\big(\partial B(y,R)\big)\big],
\]
we write
\begin{align*}
a(x)&= E_x\big[ a(S_{ \tau}); \tau = \tau_1(\partial B(y,R))\big]+
E_x\big[ a(S_{ \tau}); \tau = \tau_1(\partial B(r))\big] \\
&= q \big(a(R)+ O\big(\textstyle\frac{\|y\|\vee 1}{R}\big)\big)
+ (1-q) \big(a(r)+O(r^{-1})\big),
\end{align*}
yielding \eqref{nothit_r}. The first statement has a similar proof.
\end{proof}
We have an estimate for more general sets.
\begin{lem}
\label{l_hit_A}
Let~$A$ be a finite subset of~${\mathbb Z}^2$ such that
$A\subset B(r)$ for some $r > 0$. We have for
$r+1\leq\|x\|\leq R-2$,
$\|x\|+\|y\|\leq R-1$
\begin{equation}
\label{nothit_A}
P_x\big[\tau_1(A)>\tau_1\big(\partial B(y,R)\big)\big]
= \frac{a(x)-\mathop{\mathrm{cap}}(A)+O\big(\frac{r\ln r \ln \|x\|}{\|x\|}\big)}
{a(R)-\mathop{\mathrm{cap}}(A)
+ O\big(\frac{\|y\|\vee 1}{R}+\frac{r\ln r \ln \|x\|}{\|x\|}\big)}.
\end{equation}
\end{lem}
\begin{proof}
First, observe that, for $u \in A$,
\begin{equation}
\label{first_observe}
P_x\big[ S_{\tau_1(A)}=u \big]
= \mathop{\mathrm{hm}}\nolimits_A(u)\big(1+O\big(\textstyle\frac{r\ln \|x\|}{\|x\|}\big)\big).
\end{equation}
Indeed, from~(6.42) and the last display on page~178 of~\cite{LL10}
we obtain
\[
\frac{P_x\big[ S_{\tau_1(A)}=u \big]}{P_{x'}\big[ S_{\tau_1(A)}=u \big]}
= 1+O\big(\textstyle\frac{r\ln \|x\|}{\|x\|}\big)
\]
for all $x'\in \partial B(\|x\|)$. Since~$\mathop{\mathrm{hm}}\nolimits_A(u)$ is a linear combination
of $P_{x'}\big[ S_{\tau_1(A)}=u \big]$, $x'\in \partial B(\|x\|)$,
\eqref{first_observe} follows.
Then, we can write
\begin{align*}
\lefteqn{P_x\big[ S_{\tau_1(A)}= u , \tau_1(A) <
\tau_1\big(\partial B(y,R)\big)\big]}\\
&= P_x\big[ S_{\tau_1(A)} = u \big] - P_x\big[ S_{\tau_1(A)} = u ,
\tau_1(A)>\tau_1\big(\partial B(y,R)\big)\big]\\
& = \mathop{\mathrm{hm}}\nolimits_A(u)\big(1+O\big(\textstyle\frac{r\ln \|x\|}{\|x\|}\big)\big) - P_x\big[\tau_1(A) > \tau_1\big(\partial B(y,R)\big)\big] \\
&\qquad
\times \sum_{z\in \partial B(y,R)}P_z\big[ S_{\tau_1(A)}
= u \big]P_x\big[ S_{\tau_1(\partial B(y,R))} = z \mid \tau_1(A)
> \tau_1\big(\partial B(y,R)\big)\big] \\
&= \mathop{\mathrm{hm}}\nolimits_A(u)\big(1+O\big(\textstyle\frac{r\ln \|x\|}{\|x\|}\big)\big) - P_x\big[\tau_1(A) > \tau_1\big(\partial B(y,R)\big)\big]
\mathop{\mathrm{hm}}\nolimits_A(u)\big(1+O\big(\textstyle\frac{r\ln R}{R}\big)\big) \\
&= \mathop{\mathrm{hm}}\nolimits_A(u)P_x\big[\tau_1(A) < \tau_1\big(\partial B(y,R)\big)\big] \big(1+O\big(\textstyle\frac{r\ln \|x\|}{\|x\|}\big)\big),
\end{align*}
using twice the above observation and also that
$B(\|x\|)\subset B(y,R)$, yielding
\begin{equation}
\label{enter_prob}
P_x\big[ S_{\tau_1(A)}=u \mid \tau_1(A)<\tau_1\big(\partial B(y,R)\big)\big]
= \mathop{\mathrm{hm}}\nolimits_A(u)\big(1+O\big(\textstyle\frac{r\ln \|x\|}{\|x\|}\big)\big).
\end{equation}
Abbreviating $q:=P_x\big[\tau_1(A)>\tau_1\big(\partial B(y,R)\big)\big]$,
we have by the Optional Stopping Theorem and~\eqref{enter_prob}
\begin{align*}
a(x) &= q\big(a(R) + O\big(\textstyle\frac{\|y\|\vee 1}{R}\big)\big)
+ (1-q)\sum_{u\in A\setminus\{0\}}
a(u)\mathop{\mathrm{hm}}\nolimits_A(u) \big(1+O\big(\textstyle\frac{r\ln \|x\|}{\|x\|}\big)\big)\\
&= q\big(a(R) + O\big(\textstyle\frac{\|y\|\vee 1}{R}\big)\big)
+ (1-q)\big(1+O\big(\textstyle\frac{r\ln \|x\|}
{\|x\|}\big)\big)\mathop{\mathrm{cap}}(A),
\end{align*}
and~\eqref{nothit_A} follows (observe also that
$\mathop{\mathrm{cap}}(A)\leq \mathop{\mathrm{cap}}(B(r))=\frac{2}{\pi}\ln r + O(1)$, see~\eqref{capa_ball}).
\end{proof}
\subsection{Simple random walk conditioned on not
hitting the origin}
\label{s_aux_hat_s}
Next, we relate the probabilities of certain events
for the walks~$S$ and~${\widehat S}$. For $M \subset {\mathbb Z}^2$, let~$\Gamma^{(x)}_M$ be the set of all
nearest-neighbour finite
trajectories that start at~$x\in M\setminus\{0\}$
and end when entering~$\partial M$ for the first time;
denote also~$\Gamma^{(x)}_{y,R}=\Gamma^{(x)}_{B(y,R)}$.
For~$A\subset \Gamma^{(x)}_M$ write
$S\in A$ if there exists~$k$ such that
$(S_0,\ldots,S_k)\in A$ (and the same for
the conditional walk~${\widehat S}$). In the next
result we show that $P_x\big[\;\cdot\mid\tau_1(0)>
\tau_1\big(\partial B(R)\big)\big]$
and $\widehat{P}_x[\,\cdot\,]$ are
almost indistinguishable on~$\Gamma^{(x)}_{0,R}$ (that is, the conditional
law of~$S$ almost coincides with the unconditional law
of~${\widehat S}$). A similar result holds for excursions on a
``distant'' (from the origin) set.
\begin{lem}
\label{l_relation_S_hatS}
\begin{itemize}
\item[(i)] Assume $A\subset \Gamma^{(x)}_{0,R}$.
We have
\begin{equation}
\label{eq_relation_S_hatS}
P_x\big[S\in A\mid \tau_1(0)>\tau_1\big(\partial B(R)\big)\big]
=\widehat{P}_x\big[{\widehat S} \in A\big] \big(1+O((R \ln R)^{-1})\big).
\end{equation}
\item[(ii)] Assume that $A\subset \Gamma^{(x)}_M$
and suppose that $0\notin M$, and denote $s=\mathop{\mathrm{dist}}(0,M)$,
$r={\mathop{\mathrm{diam}}}(M)$. Then, for $x \in M$,
\begin{equation}
\label{eq_relation_S_hatS2}
P_x[S\in A]
=\widehat{P}_x[{\widehat S} \in A]\Big(1+O\Big(\frac{r}{s\ln s}\Big)\Big).
\end{equation}
\end{itemize}
\end{lem}
\begin{proof}
Let us prove part~(i).
Assume without loss of generality that no trajectory from~$A$
passes through the origin.
Then, it holds that
\[
\widehat{P}_x[{\widehat S} \in A] = \sum_{\varrho\in A}
\frac{a(\varrho_{\text{end}})}{a(x)}\Big(\frac{1}{4}\Big)^{|\varrho|},
\]
with $|\varrho|$ the length of $\varrho$.
On the other hand, by~\eqref{nothit_0}
\[
P_x[S\in A\mid \tau_1(0)>\tau_1\big(\partial B(R)\big)]
= \frac{a(R) + O(R^{-1})}{a(x)}
\sum_{\varrho\in A}
\Big(\frac{1}{4}\Big)^{|\varrho|}.
\]
Since
$\varrho_{\text{end}}\in\partial B(R)$, we have
$a(\varrho_{\text{end}})=a(R) + O(R^{-1})$,
and so~\eqref{eq_relation_S_hatS} follows.
The proof of part~(ii) is analogous (observe that
$a(y_1)/a(y_2)=1+O\big(\frac{r}{s\ln s}\big)$ for any $y_1, y_2\in M$).
\end{proof}
As observed in Section~\ref{s_RI}, the random walk~${\widehat S}$
is transient.
Next,
we estimate the
probability that the ${\widehat S}$-walk avoids a ball centered
at the origin:
\begin{lem}
\label{l_escape_hatS}
Assume $r \geq 1$ and $\|x\| \geq r+1$. We have
\[
\widehat{P}_x\big[\widehat{\tau}_1\big(B(r)\big)=\infty\big] =
1-\frac{a(r)+O(r^{-1})}{a(x)}.
\]
\end{lem}
\begin{proof}
By Lemma~\ref{l_relation_S_hatS}~(i) we have
\[
\widehat{P}_x\big[\widehat{\tau}_1\big(B(r)\big)=\infty\big] =
\lim_{R\to \infty}
P_x\big[\tau_1(\partial B(r))>\tau_1\big(\partial B(R)\big)
\mid \tau_1(0)>\tau_1\big(\partial B(R)\big)\big].
\]
The claim then follows from~\eqref{nothit_0}--\eqref{nothit_r}.
\end{proof}
\begin{rem}
Alternatively, one can deduce the proof
of Lemma~\ref{l_escape_hatS} from the fact
that $1/a({\widehat S}_{k\wedge \widehat{\tau}_0(\mathcal{N})})$ is a martingale,
together with the Optional Stopping Theorem.
\end{rem}
We will need also an expression for the probability
of avoiding \emph{any} finite set containing the origin:
\begin{lem}
\label{l_avoid_A}
Assume that $0\in A \subset B(r)$, and $\|x\|\geq r+1$.
Then
\begin{equation}
\label{eq_avoid_A}
\widehat{P}_x[\widehat{\tau}_1(A)=\infty] = 1-\frac{\mathop{\mathrm{cap}}(A)}{a(x)}
+O\Big(\frac{r\ln r \ln\|x\|}{\|x\|}\Big).
\end{equation}
\end{lem}
\begin{proof}
Indeed, using Lemmas~\ref{l_hit_A} and~\ref{l_relation_S_hatS}~(i)
together with~\eqref{nothit_0},
we write
\begin{align*}
\widehat{P}_x[\widehat{\tau}_1(A)=\infty] &= \lim_{R\to\infty}
P_x\big[\tau_1(A)>\tau_1\big(\partial B(R)\big)
\mid \tau_1(0)>\tau_1\big(\partial B(R)\big)\big] \\
&=\lim_{R\to\infty} \frac{a(R)+O(R^{-1})}{a(x)}
\times \frac{a(x)-\mathop{\mathrm{cap}}(A)+O\big(\frac{r\ln r \ln \|x\|}{\|x\|}\big)}
{a(R)-\mathop{\mathrm{cap}}(A) + O\big(R^{-1}+\frac{r\ln r \ln \|x\|}{\|x\|}\big)},
\end{align*}
thus obtaining~\eqref{eq_avoid_A}.
\end{proof}
It is also possible to obtain
exact expressions for one-site escape probabilities,
and probabilities of (not) hitting a given site:
\begin{align}
\widehat{P}_x[\widehat{\tau}_1(y)<\infty] &= \frac{a(x)+a(y)-a(x-y)}{2a(x)},
\label{escape_from_site1}\\
\intertext{for $x\neq y$, $x,y\neq 0$ and }
\widehat{P}_x[\widehat{\tau}_1(x)<\infty] &= 1-\frac{1}{2a(x)}
\label{escape_from_site2}
\end{align}
for $x\neq 0$. We temporarily postpone the proof of
\eqref{escape_from_site1}--\eqref{escape_from_site2}.
Observe that, in particular,
we recover from~\eqref{escape_from_site2} the transience of~${\widehat S}$.
Also, observe that~\eqref{escape_from_site1} implies
the following surprising fact:
for any~$x\neq 0$,
\[
\lim_{y\to\infty} \widehat{P}_x[\widehat{\tau}_1(y)<\infty] = \frac{1}{2}.
\]
The above relation leads to the following heuristic explanation for
Theorem~\ref{t_properties_RI}~(iii)
(in the case when~$A$ is fixed and $\|x\|\to{\infty}$). Since the probability
of hitting a distant site is about~$1/2$, by conditioning
that this distant site is vacant, we essentially throw away
three quarters of the trajectories that pass through
a neighbourhood of the origin: indeed, the double-infinite
trajectory has to avoid this distant site two times,
before and after reaching that neighbourhood.
Let us state several other
general estimates, for the probability
of (not) hitting a given set (which is, typically,
far away from the origin),
or, more specifically, a disk:
\begin{lem}
\label{l_escape_from_ball}
Assume that $x\notin B(y,r)$ and $\|y\|>2r\geq 1$.
Abbreviate also $\Psi_1=\|y\|^{-1}r$,
$\Psi_2=\frac{r\ln r\ln\|y\|}{\|y\|}$,
$\Psi_3=r\ln r\big(\frac{\ln\|x-y\|}{\|x-y\|}
+\frac{\ln\|y\|}{\|y\|}\big)$.
\begin{itemize}
\item[(i)] We have
\begin{equation}
\label{eq_escape_from_ball}
\widehat{P}_x\big[\widehat{\tau}_1(B(y,r))<\infty\big]
= \frac{\big(a(y)+O(\Psi_1)\big)\big(a(y)+a(x)-a(x-y)
+O(r^{-1})\big)}{a(x)\big(2a(y)-a(r)+O(r^{-1}+\Psi_1)\big)}.
\end{equation}
\item[(ii)] Consider now any nonempty set $A\subset B(y,r)$.
Then, it holds that
\begin{equation}
\label{eq_escape_from_anyset}
\widehat{P}_x\big[\widehat{\tau}_1(A)<\infty\big]
= \frac{\big(a(y)+O(\Psi_1)\big)\big(a(y)+a(x)-a(x-y)
+O(r^{-1}+\Psi_3
)\big)}
{a(x)\big(2a(y)-\mathop{\mathrm{cap}}(A)+ O(\Psi_2)
\big)}.
\end{equation}
\end{itemize}
\end{lem}
Observe that~\eqref{eq_escape_from_ball} is \emph{not}
a particular case of~\eqref{eq_escape_from_anyset};
this is because~\eqref{nothit_r}
typically provides a more precise estimate than~\eqref{nothit_A}.
\begin{proof}
Fix a (large) $R>0$, such that $R>\max\{\|x\|,\|y\|+r\}+1$. Denote
\begin{align*}
h_1 &= P_x\big[\tau_1(0)<\tau_1\big(\partial B(R)\big)\big],\\
h_2 &= P_x\big[\tau_1\big(B(y,r)\big)<\tau_1\big(\partial B(R)\big)\big],\\
p_1 &= P_x\big[\tau_1(0)<\tau_1\big(\partial B(R)\big)\wedge
\tau_1(B(y,r))\big],\\
p_2 &= P_x\big[\tau_1\big(B(y,r)\big)<\tau_1\big(\partial B(R)\big)\wedge
\tau_1(0)\big],\\
q_{12} &= P_0\big[\tau_1\big(B(y,r)\big)
<\tau_1\big(\partial B(R)\big)\big],\\
q_{21} &= P_\nu\big[\tau_1(0)<\tau_1\big(\partial B(R)\big)\big],
\end{align*}
where~$\nu$ is the entrance measure to~$B(y,r)$ starting from $x$
conditioned on the event
$\big\{\tau_1\big(B(y,r)\big)
<\tau_1\big(\partial B(R)\big)\wedge \tau_1(0)\big\}$,
see Figure~\ref{f_p12}.
\begin{figure}
\begin{center}
\includegraphics[width=0.64\textwidth]{p12}
\caption{On the proof of Lemma~\ref{l_escape_from_ball}}
\label{f_p12}
\end{center}
\end{figure}
Using Lemma~\ref{l_exit_balls}, we obtain
\begin{align}
h_1 &= 1-\frac{a(x)}{a(R)+O(R^{-1})},
\label{expr_h1}\\
h_2 &= 1-\frac{a(x-y)-a(r)+O(r^{-1})}{a(R)-a(r)+O(R^{-1}\|y\|+r^{-1})},
\label{expr_h2}
\end{align}
and
\begin{align}
q_{12} &= 1-\frac{a(y)-a(r)+O(r^{-1})}{a(R)-a(r)+O(R^{-1}\|y\|+r^{-1})},
\label{expr_q12} \\
q_{21} &=1-\frac{a(y)+O(\|y\|^{-1}r)}{a(R)+O(R^{-1}\|y\|)}.
\label{expr_q21}
\end{align}
Then, as a general fact, it holds that
\begin{align*}
h_1 &= p_1 + p_2q_{21},\\
h_2 &= p_2 + p_1q_{12}.
\end{align*}
Solving these equations with respect to $p_1,p_2$, we
obtain
\begin{align}
p_1 &= \frac{h_1-h_2q_{21}}{1-q_{12}q_{21}},
\label{expr_p1}\\
p_2 &= \frac{h_2-h_1q_{12}}{1-q_{12}q_{21}},
\label{expr_p2}
\end{align}
and so, using \eqref{expr_h1}--\eqref{expr_q21}, we write
\begin{align*}
\lefteqn{P_x\big[\tau_1(B(y,r))<\tau_1\big(\partial B(R)\big)
\mid \tau_1(0)>\tau_1\big(\partial B(R)\big)\big]}\\
& = \frac{p_2(1-q_{21})}{1-h_1}
\\
& = \frac{(h_2 - h_1q_{12})(1-q_{21})}{(1-h_1)(1-q_{12}q_{21})}\\
&= \Bigg(\frac{a(x)}{a(R)+O(R^{-1})}
+\frac{a(y)-a(r)+O(r^{-1})}{a(R)-a(r)+O(R^{-1}\|y\|+r^{-1})}\\
&\qquad
- \frac{a(x-y)-a(r)+O(r^{-1})}{a(R)-a(r)+O(R^{-1}\|y\|+r^{-1})}
+O\Big(\frac{\ln\|x-y\|\ln\|y\|}{\ln^2 R}\Big)\Bigg)\\
&\quad \times \frac{a(y)+O(\|y\|^{-1}r)}{a(R)+O(R^{-1}\|y\|)}\times
\Big(\frac{a(x)}{a(R)+O(R^{-1})}\Big)^{-1}
\\
&\quad \times \Bigg(\frac{a(y)-a(r)+O(r^{-1})}
{a(R)-a(r)+O(R^{-1}\|y\|+r^{-1})}
+ \frac{a(y)+O(\|y\|^{-1}r)}{a(R)+O(R^{-1}\|y\|)}
+O\Big(\frac{\ln^2\|y\|}{\ln^2 R}\Big)\Bigg)^{-1}.
\end{align*}
Sending~$R$ to infinity,
we obtain the proof of~\eqref{eq_escape_from_ball}.
To prove~\eqref{eq_escape_from_anyset}, we use the same procedure.
Define $h'_{1,2}$, $p'_{1,2}$, $q'_{12}$, $q'_{21}$ in the same
way but with~$A$ in place of~$B(y,r)$.
It holds that $h'_1=h_1$, $q'_{21}$ is
expressed in the same way as~$q_{21}$
(although $q'_{21}$ and $q_{21}$ are not necessarily equal,
the difference
is only in the error terms $O(\cdot)$)
and, by Lemma~\ref{l_hit_A},
\begin{align*}
h'_2 &= 1-\frac{a(x-y)-\mathop{\mathrm{cap}}(A)
+O\big(\frac{r\ln r\ln\|x-y\|}{\|x-y\|}\big)}
{a(R)-\mathop{\mathrm{cap}}(A)+O\big(R^{-1}\|x-y\|
+\frac{r\ln r\ln\|x-y\|}{\|x-y\|}\big)},\\
q'_{12} &= 1-\frac{a(y)-\mathop{\mathrm{cap}}(A)
+O\big(\frac{r\ln r\ln\|y\|}{\|y\|}\big)}
{a(R)-\mathop{\mathrm{cap}}(A)+O\big(R^{-1}\|y\|
+\frac{r\ln r\ln\|y\|}{\|y\|}\big)}.
\end{align*}
After the analogous calculations, we obtain~\eqref{eq_escape_from_anyset}.
\end{proof}
\begin{proof}[Proof of relations \eqref{escape_from_site1}--\eqref{escape_from_site2}]
Formula~\eqref{escape_from_site2} rephrases~\eqref{escape_identity} with $A=\{0,x\}$.
Identity~\eqref{escape_from_site1} follows from the same proof as in
Lemma~\ref{l_escape_from_ball} (i), using~\eqref{nothit_0}
instead of~\eqref{nothit_r}.
\end{proof}
\subsection{Harmonic measure and capacities}
\label{s_aux_capacities}
Next, we need a formula for calculating the capacity
of three-point sets:
\begin{lem}
\label{l_cap_triangle}
Let $x_1,x_2,x_3\in{\mathbb Z}^2$, and abbreviate $v_1=x_2-x_1$, $v_2=x_3-x_2$,
$v_3=x_1-x_3$. Then, the capacity of the set $A=\{x_1,x_2,x_3\}$
is given by the formula
\begin{equation}
\label{eq_cap_triangle}
\frac{a(v_1)a(v_2)a(v_3)}
{a(v_1)a(v_2)+a(v_1)a(v_3)+a(v_2)a(v_3)-\frac{1}{2}
\big(a^2(v_1)+a^2(v_2)+a^2(v_3)\big)}.
\end{equation}
\end{lem}
\begin{proof}
By Proposition~6.6.3 and Lemma~6.6.4 of~\cite{LL10},
the inverse capacity of~$A$ is equal to the sum of entries
of the matrix
\[
a_A^{-1} =
\begin{pmatrix}
0 & a(v_1) & a(v_3)\\
a(v_1) & 0 & a(v_2)\\
a(v_3) & a(v_2) & 0
\end{pmatrix}^{-1}
=\frac{1}{2}
\begin{pmatrix}
-\frac{a(v_2)}{a(v_1)a(v_3)} & \frac{1}{a(v_1)} &
\frac{1}{a(v_3)}\\
\frac{1}{a(v_1)} & - \frac{a(v_3)}{a(v_1)a(v_2)} &
\frac{1}{a(v_2)}\\
\frac{1}{a(v_3)} & \frac{1}{a(v_2)}
& -\frac{a(v_1)}{a(v_2)a(v_3)}
\end{pmatrix},
\]
and this implies~\eqref{eq_cap_triangle}.
\end{proof}
Before proceeding, let us notice the following immediate consequence
of Lemma~\ref{l_avoid_A}: for any finite $A\subset{\mathbb Z}^2$
such that $0\in A$, we have
\begin{equation}
\label{expr_Cap}
\mathop{\mathrm{cap}}(A) = \lim_{\|x\|\to\infty} a(x)\widehat{P}_x[\widehat{\tau}_1(A)<\infty].
\end{equation}
Next, we need estimates for the ${\widehat S}$-capacity
of a ``distant'' set, and, in particular
of a ball which does not contain the origin.
Recall the notations $\Psi_{1,2,3}$ from Lemma~\ref{l_escape_from_ball}.
\begin{lem}
\label{l_cap_distantball}
Assume that $\|y\|>2r\geq 1$.
\begin{itemize}
\item[(i)]
We have
\begin{equation}
\label{eq_cap_distball}
\mathop{\mathrm{cap}}\big(\{0\}\cup B(y,r)\big)
= \frac{\big(a(y)+O(\Psi_1)\big)\big(a(y)+O(r^{-1})\big)}
{2a(y)-a(r)+O(r^{-1}+\Psi_1)}.
\end{equation}
\item[(ii)] Suppose that $A\subset B(y,r)$. Then
\begin{equation}
\label{eq_cap_distantset}
\mathop{\mathrm{cap}}\big(\{0\}\cup A\big) =\frac{\big(a(y)+O(\Psi_1)\big)
\big(a(y)+O(r^{-1}+\Psi_2 )\big)}
{2a(y)-\mathop{\mathrm{cap}}(A)+ O(\Psi_2)}.
\end{equation}
\end{itemize}
\end{lem}
\begin{proof}
This immediately follows from~\eqref{expr_Cap}
and Lemma~\ref{l_escape_from_ball} (observe that
$a(x)-a(x-y)\to 0$ as $x\to \infty$ and~$\Psi_3$ becomes~$\Psi_2$).
\end{proof}
We also need to compare the harmonic measure on a set
(distant from the origin) to
the entrance measure of the ${\widehat S}$-walk started far away from that set.
\begin{lem}
\label{l_entrance_hat_s}
Assume that $A$ is a finite subset of ${\mathbb Z}^2$,
$0\notin A$, $x\neq 0$, and also that
$2\,{\mathop{\mathrm{diam}}}(A)<\mathop{\mathrm{dist}}(x,A) < \frac{1}{4} \mathop{\mathrm{dist}}(0,A)$.
Abbreviate $u={\mathop{\mathrm{diam}}}(A)$, $s=\mathop{\mathrm{dist}}(x,A)$.
Assume also that $A'\subset {\mathbb Z}^2$ (finite, infinite, or even possibly
empty) is such that $\mathop{\mathrm{dist}}(A,A')\geq s+1$ (for definiteness,
we adopt the convention $\mathop{\mathrm{dist}}(A,\emptyset)=\infty$ for any~$A$).
Then, for $y \in A$, it holds that
\begin{equation}
\label{eq_entrance_hat_s}
\widehat{P}_x\big[{\widehat S}_{\widehat{\tau}_1(A)}=y\mid \widehat{\tau}_1(A)<\infty,
\widehat{\tau}_1(A)<\widehat{\tau}_1(A')\big] = \mathop{\mathrm{hm}}\nolimits_A(y)\Big(1+
O\Big(
\frac{u\ln s}{s}\Big)\Big).
\end{equation}
\end{lem}
\begin{proof}
Let $z_0\in A$ be such that $\|z_0-x\|=s$, and observe that
$A'\cap B(z_0,s) = \emptyset$.
Define the discrete circle $L=\partial B(z_0,s)$;
observe that $x\in L$ and
$\mathop{\mathrm{dist}}(z',A)\geq s/2$ for all $z'\in L$.
Let
\[
\sigma = \sup\big\{0\leq k\leq \widehat{\tau}_1(A) : {\widehat S}_k\in L\big\}
\]
be the \emph{last} time before $\widehat{\tau}_1(A)$ when the trajectory
passes through~$L$. Note also that for all~$z\in L$
(recall~\eqref{enter_prob})
\begin{equation}
\label{harmonic_to_A}
P_z\big[S_{\tau_1(A)}=y \mid \tau_1(A)<\tau_1(L)\big]
= \mathop{\mathrm{hm}}\nolimits_A(y)\Big(1+O\Big(\frac{u\ln s}{s}\Big)\Big).
\end{equation}
Using the Markov property of~${\widehat S}$, we write
\begin{align}
\lefteqn{ \widehat{P}_x\big[\widehat{\tau}_1(A)<\infty,
\widehat{\tau}_1(A)<\widehat{\tau}_1(A'),{\widehat S}_{\widehat{\tau}_1(A)}=y\big] } \nonumber\\
&= \sum_{k\geq 0, z\in L} \widehat{P}_x\big[\widehat{\tau}_1(A)<\infty,
\widehat{\tau}_1(A)<\widehat{\tau}_1(A'),\sigma=k,
{\widehat S}_\sigma=z, {\widehat S}_{\widehat{\tau}_1(A)}=y\big]
\nonumber\\
&=\sum_{k\geq 0, z\in L} \widehat{P}_x\big[{\widehat S}_k=z, {\widehat S}_\ell\notin A\cup A'
\text{ for all }\ell\leq k\big]
\nonumber\\
& \qquad\quad\quad \times
\widehat{P}_z\big[ \widehat{\tau}_1(A)<\infty, \widehat{\tau}_1(A)<\widehat{\tau}_1(A'),
{\widehat S}_{\widehat{\tau}_1(A)}=y,{\widehat S}_\ell\notin L \text{ for all }
\ell\leq \widehat{\tau}_1(A)\big].
\label{conta_sigma}
\end{align}
Abbreviate $r=\mathop{\mathrm{dist}}(0,A)$.
Now, observe that the last term in~\eqref{conta_sigma} only involves
trajectories that lie in $B(z_0,s)$,
and we have $\mathop{\mathrm{dist}}\big(0,B(z_0,s)\big)\geq r/2$.
So, we can use Lemma~\ref{l_relation_S_hatS}~(ii) together
with~\eqref{harmonic_to_A} to write
\begin{align*}
\lefteqn{\widehat{P}_z\big[ \widehat{\tau}_1(A)<\infty, \widehat{\tau}_1(A)<\widehat{\tau}_1(A'),
{\widehat S}_{\widehat{\tau}_1(A)}=y,{\widehat S}_\ell\notin L \text{ for all }
\ell\leq \widehat{\tau}_1(A)\big] } \\
&= P_z\big[S_{\tau_1(A)}=y,S_\ell\notin L \text{ for all }
\ell\leq \tau_1(A)\big]\Big(1+O\Big(\frac{u}{r\ln r}\Big)\Big)\\
&= P_z\big[S_{\tau_1(A)}=y \mid \tau_1(A)<\tau_1(L)\big]
P_z[\tau_1(A)<\tau_1(L)]\Big(1+O\Big(\frac{u}{r\ln r}\Big)\Big)\\
&= \mathop{\mathrm{hm}}\nolimits_A(y)\widehat{P}_z\big[\widehat{\tau}_1(A)<\widehat{\tau}_1(L)\big]\Big(1+
O\Big(\frac{u}{r\ln r}+\frac{u\ln s}{s}\Big)\Big).
\end{align*}
Inserting this back to~\eqref{conta_sigma}, we obtain
\begin{align*}
\lefteqn{ \widehat{P}_x\big[\widehat{\tau}_1(A)<\infty,\widehat{\tau}_1(A)<\widehat{\tau}_1(A'),
{\widehat S}_{\widehat{\tau}_1(A)}=y\big] } \\
&=\mathop{\mathrm{hm}}\nolimits_A(y) \Big(1+
O\Big(\frac{u}{r\ln r}+\frac{u\ln u}{s}\Big)\Big)\\
& \qquad \times \sum_{k\geq 0, z\in L} \widehat{P}_x\big[{\widehat S}_k=z,
{\widehat S}_\ell\notin A\cup A'
\text{ for all }\ell\leq k\big]\widehat{P}_z\big[\widehat{\tau}_1(A)<\widehat{\tau}_1(L)\big]\\
&= \mathop{\mathrm{hm}}\nolimits_A(y) \widehat{P}_z\big[\widehat{\tau}_1(A)<\infty,\widehat{\tau}_1(A)<\widehat{\tau}_1(A')\big]
\Big(1+O\Big(\frac{u}{r\ln r}+\frac{u\ln s}{s}\Big)\Big),
\end{align*}
and this concludes the proof of Lemma~\ref{l_entrance_hat_s}
(observe that the first term in~$O(\cdot)$ is of smaller order
than the second one).
\end{proof}
\subsection{Random walk on the torus and its excursions}
\label{s_aux_torus}
First, we define the entrance time to a set $A\subset{\mathbb Z}^2_n$ by
\[
T_n(A) = \min_{x\in A} T_n(x).
\]
Now, consider two sets $A\subset A'\subset {\mathbb Z}^2_n$, and
suppose that we are only interested in the trace left by the random
walk on the set~$A$. Then,
(apart from the initial piece of the trajectory until
hitting~$\partial A'$
for the first time)
it is enough to know
the excursions of the random walk between the boundaries of~$A$
and~$A'$. By definition, an excursion~$\varrho$ is a simple
random walk path that starts at~$\partial A$ and ends
on its first visit to~$\partial A'$, i.e., $\varrho=(\varrho_0, \varrho_1,\ldots,
\varrho_m)$, where $\varrho_0\in\partial A$, $\varrho_m\in\partial A'$,
$\varrho_k\notin\partial A'$ and $\varrho_k\sim\varrho_{k+1}$ for $k<m$. With some abuse
of notation, we denote by $\varrho_{\text{st}}:=\varrho_0$ and
$\varrho_{\text{end}}:=\varrho_m$ the starting and the ending points
of the excursion.
To define these excursions, consider
the following sequence of stopping times:
\begin{align*}
D_0 &= T_n(\partial A'),\\
J_1 &= \inf\{t> D_0 : X_t \in \partial A\},\\
D_1 &= \inf\{t> J_1 : X_t \in \partial A'\},
\end{align*}
and
\begin{align*}
J_k &= \inf\{t> D_{k-1} : X_t \in \partial A\},\\
D_k &= \inf\{t> J_k : X_t \in \partial A'\},
\end{align*}
for $k\geq 2$. Then,
denote by $Z^{(i)}=(X_{J_i}, \ldots, X_{D_i})$ the $i$th excursion
of~$X$ between~$\partial A$ and~$\partial A'$, for $i\geq 1$.
Also, let $Z^{(0)}=(X_0, \ldots, X_{D_0})$ be the ``initial''
excursion (it is possible, in fact, that it does not intersect
the set~$A$ at all). Recall that $t_\alpha:=\frac{4\alpha}{\pi}n^2\ln^2 n$
and define
\begin{align}
N_\alpha &= \max\{k : J_k\leq t_\alpha\},\label{df_Na}
\\
N'_\alpha &= \max\{k : D_k\leq t_\alpha\}\label{df_N'a}
\end{align}
to be the number of incomplete (respectively, complete)
excursions up to time~$t_\alpha$.
Next, we need also to define the excursions of random
interlacements
in an analogous way. Assume that the trajectories
of the ${\widehat S}$-walks that intersect~$A$ are enumerated according
to their $u$-labels (recall the construction in Section~\ref{s_results}).
For each trajectory from that list (say, the $j$th one,
denoted~${\widehat S}^{(j)}$ and time-shifted in such a way
that ${\widehat S}^{(j)}_k\notin A$ for all $k\leq -1$ and ${\widehat S}^{(j)}_0 \in A$)
define the stopping times
\begin{align*}
{\hat J}_1 &= 0,\\
{\hat D}_1 &= \inf\{t> {\hat J}_1 : {\widehat S}^{(j)}_t \in \partial A'\},
\end{align*}
and
\begin{align*}
{\hat J}_k &= \inf\{t> {\hat D}_{k-1} : {\widehat S}^{(j)}_t \in \partial A\},\\
{\hat D}_k &= \inf\{t> {\hat J}_k : {\widehat S}^{(j)}_t \in \partial A'\},
\end{align*}
for $k\geq 2$. Let~$\ell_j=\inf\{k:{\hat J}_k=\infty\}-1$ be the
number of excursions corresponding to the $j$th trajectory.
The excursions of RI($\alpha$) between~$\partial A$
and~$\partial A'$ are then defined by
\[
{\hat Z}^{(i)}=({\widehat S}^{(j)}_{{\hat J}_m}, \ldots, {\widehat S}^{(j)}_{{\hat D}_m}),
\]
where
$ i = m+\sum_{k=1}^{j-1} \ell_k$,
and $m=1,2,\ldots \ell_j$. We let $R_\alpha$ to be the number of
trajectories intersecting~$A$ and with labels less than~$\alpha \pi$,
and denote ${\hat N}_\alpha = \sum_{k=1}^{R_\alpha} \ell_k$
to be the total number of excursions of RI($\alpha$) between~$\partial A$
and~$\partial A'$.
Observe also that the above construction makes sense with $\alpha=\infty$
as well; we then obtain an infinite sequence of excursions
of RI (=RI($\infty$)) between~$\partial A$
and~$\partial A'$.
Finally, let us
recall a result of~\cite{DPRZ06}
on the number of excursions
for the simple random walk on~${\mathbb Z}^2_n$.
\begin{lem}
\label{l_excursions_torus}
Consider the random variables $J_k,D_k$ defined in this section with
$A=B\big(\frac{n}{3\ln n}\big)$, $A'=B\big(\frac{n}{3}\big)$.
Then, there exist positive constants $\delta_0, c_0 $ such that, for
any $\delta$ with $ \frac{c_0}{\ln n} \leq \delta \leq \delta_0$,
we have
\begin{equation}
\label{eq_excursions_torus_g}
{\mathbb P}\Big[J_k\in\Big((1-\delta)\frac{2 n^2\ln\ln n}{\pi}k,
(1+\delta)\frac{2 n^2\ln\ln n}{\pi}k\Big)\Big]
\geq 1 - \exp\big(-c\delta^2 k\big),
\end{equation}
and the same result holds with~$D_k$ on the place
of~$J_k$.
\end{lem}
\begin{proof}
This is, in fact, a particular case of Lemma~3.2 of~\cite{DPRZ06}.
\end{proof}
We observe that~\eqref{eq_excursions_torus_g} means
that the ``typical'' number of excursions
by time~$s$ is $\frac{\pi s}{2n^2\ln\ln n}$.
In particular, a useful consequence
of Lemma~\ref{l_excursions_torus} is that for all uniformly
positive~$\alpha$ and all large enough~$n$
\begin{equation}
\label{eq_excursions_torus}
{\mathbb P}\Big[(1-\delta)\frac{2\alpha\ln^2 n}{\ln\ln n}\leq N_\alpha
\leq (1+\delta)\frac{2\alpha\ln^2 n}{\ln\ln n}\Big]
\geq 1 - \exp\Big(-c\delta^2\frac{\ln^2 n}{\ln\ln n}\Big),
\end{equation}
and the same result holds with $N'_\alpha$ on the place
of~$N_\alpha$,
where $N_\alpha,N'_\alpha$ are defined
as in~\eqref{df_Na}--\eqref{df_N'a} with
$A=B\big(\frac{n}{3\ln n}\big)$, $A'=B\big(\frac{n}{3}\big)$.
\section{Proofs of the main results}
\label{s_proofs}
We first prove the results related to random interlacements,
and then deal with the connections between random interlacements
and random walk on the torus in Section~\ref{s_proof_torus}.
\subsection{Proofs for random interlacements}
\label{s_proofs_interl}
First of all, we apply some results of Section~\ref{s_aux}
to finish the proof of Theorem~\ref{t_properties_RI}.
\begin{proof}[Proof of Theorem~\ref{t_properties_RI}, parts (iii)--(v).]
Recall the fundamental formula~\eqref{eq_vacant2}
for the random interlacement and the relation~\eqref{formula_for_a}.
Then, the statement~(iv)
follows from Lemma~\ref{l_cap_triangle}
and from (\ref{properties_RI_ii}),
while~(v) is a consequence of Lemma~\ref{l_cap_distantball}~(i).
Finally, observe that, by symmetry, Theorem~\ref{t_properties_RI}~(ii),
and Lemma~\ref{l_cap_distantball}~(ii) we have
\begin{align*}
{\mathbb P}[A\subset{\mathcal V}^\alpha \mid x\in {\mathcal V}^{\alpha}]
&= \exp\Big(-\pi\alpha\big(\mathop{\mathrm{cap}}(A\cup\{x\})-\mathop{\mathrm{cap}}(\{0,x\})\big)\Big)\\
&=\exp\Bigg(-\pi\alpha
\Big(\frac{\big(a(x)+O\big(\frac{r\ln r\ln\|x\|}{\|x\|}\big)\big)^2}
{2a(x)-\mathop{\mathrm{cap}}(A)+O\big(\frac{r\ln r\ln\|x\|}{\|x\|}\big)}
-\frac{a(x)}{2}\Big)\Bigg)\\
&=\exp\Bigg(-\frac{\pi\alpha}{4}\mathop{\mathrm{cap}}(A)
\frac{1+O\big(\frac{r\ln r \ln\|x\|}{\|x\|}\big)}
{1-\frac{\mathop{\mathrm{cap}}(A)}{2a(x)}
+O\big(\frac{r\ln r}{\|x\|}\big)}
\Bigg),
\end{align*}
thus proving the part~(iii).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{t_sizevacant} (iii)]
We start by observing that the first part of~(iii) follows
from the bound~\eqref{eq_emptyball} and Borel-Cantelli.
So, let us concentrate on proving~\eqref{eq_emptyball}.
Recall the following elementary
fact: let~$N$ be a Poisson random variable with parameter~$\lambda$,
and $Y_1,Y_2, Y_3,\ldots$ be independent (also of~$N$) random
variables with exponential distribution ${\mathcal E}(p)$ with parameter~$p$.
Let~$\Theta=\sum_{j=1}^N Y_j$ be the corresponding
compound Poisson random variable. Its Cram\'er transform $b \mapsto \lambda(\sqrt{b}-1)^2$ is easily computed, and Chernov's bound gives,
for all $b>1$,
\begin{equation}
\label{CompPoisson_LD}
{\mathbb P}[\Theta \geq b\lambda p^{-1}] \leq \exp\big(-\lambda(\sqrt{b}-1)^2\big).
\end{equation}
Now, assume that $\alpha<1$.
Fix $\beta\in (0,1)$,
which will be later taken close to~$1$, and fix some set of non-intersecting
disks~$B'_1=B(x_1,r^\beta),\ldots,B'_{k_r}=B(x_{k_r},r^\beta)\subset
B(r)\setminus B(r/2)$, with cardinality $k_r = \frac{1}{4}r^{2(1-\beta)}$.
Denote also $B_j:= B\big(x_j, \frac{r^\beta}{\ln^3 r^\beta}\big)$,
$j=1,\ldots,k_r$.
Before going to the heart of the matter we briefly
sketch the strategy of proof (also,
one may find it helpful to look at Figure~\ref{f_disks}).
We start to show that at least a half of these balls $B_j$ will receive at most
$b\frac{2\alpha\ln^2 r}{3\ln\ln r^\beta}$
excursions from~$\partial B_{j}$ to~$\partial B'_{j}$ up to time $t_\alpha$, where
$b>1$ is a parameter (the above number of excursions is larger than
the typical number of excursions by factor~$b$).
Moreover, using the method of soft local times~\cite{CGPV,SLT}, we couple
such excursions from $RI(\alpha)$ with
a slightly larger number of independent excursions from the ${\widehat S}$-walk:
with overwhelming probability, the trace on
$\cup_j B_j$ of the latter excursion process contains the trace of the former,
so the vacant set~${\mathcal V}^\alpha$ restricted to balls~$B_j$ is smaller than the set
of unvisited points by the independent process. Now, by independence,
it will be possible to estimate the probability for leaving
that many balls partially uncovered, and this will conclude the proof.
\begin{figure}
\begin{center}
\includegraphics{disks}
\caption{On the proof of Theorem~\ref{t_sizevacant} (iii). With
high probability, at least a positive proportion of the inner circles
is not completely covered.}
\label{f_disks}
\end{center}
\end{figure}
Let us observe that the number of ${\widehat S}$-walks in $RI(\alpha)$
intersecting a given disk~$B_{j}$
has Poisson law with parameter
$\lambda=(1+o(1))\frac{2\alpha}{2-\beta}\ln r$.
Indeed, the law is Poisson by construction,
the parameter $\pi \alpha \mathop{\mathrm{cap}} (B_{j} \cup \{0\})$ is found
in~\eqref{eq_vacant2} and
then estimated using
Lemma~\ref{l_cap_distantball} (i).
Next, by Lemma~\ref{l_escape_from_ball} (i), the probability that
the walk ${\widehat S}$ started from any $y\in\partial B'_{j}$
does not hit~$B_{j}$ is $(1+o(1))\frac{3\ln\ln r^\beta}{(2-\beta)\ln r}$.
This depends on the starting point,
however, after the first visit to~$\partial B_{j}$,
each ${\widehat S}$-walk generates a number of excursions
between~$\partial B_{j}$ and~$\partial B'_{j}$ which is dominated by a
geometric law $G(p')$ (supported on $\{1,2,3,\ldots\}$)
with success parameter
$p'=(1+o(1))\frac{3\ln\ln r^\beta}{(2-\beta)\ln r}$.
Recall also that the integer part of ${\mathcal E}(u)$
is geometric $G(1-e^{-u})$.
So, with $p=-\ln (1-p')$, the total number ${\hat N}_\alpha^{(j)}$ of excursions
between~$\partial B_{j}$ and~$\partial B'_{j}$ in RI($\alpha$)
can be dominated by a compound
Poisson law with ${\mathcal E}(p)$ terms in the sum
with expectation
\[
\lambda p^{-1}
= (1+o(1))\frac{2\alpha\ln^2 r}{3\ln\ln r^\beta}.
\]
Then, using~\eqref{CompPoisson_LD}, we obtain for $b>1$
\begin{align}
{\mathbb P}\Big[{\hat N}_\alpha^{(j)}\geq b\frac{2\alpha\ln^2 r}
{3\ln\ln r^\beta}\Big] &\leq
\exp\Big(-(1+o(1))\big(\sqrt{b}-1\big)^2 \frac{2\alpha}{2-\beta}
\ln r\Big) \nonumber\\
&= r^{-(1+o(1))(\sqrt{b}-1)^2 \frac{2\alpha}{2-\beta}}.
\label{LD_numb_exc}
\end{align}
Now, let~$W_b$ be the set
\[
W_b = \Big\{j\leq k_r: {\hat N}_\alpha^{(j)}
< b\frac{2\alpha\ln^2 r}{3\ln\ln r^\beta}\Big\}.
\]
Combining~\eqref{LD_numb_exc} with Markov inequality, we obtain
\begin{equation*}
\frac{k_r}{2} {\mathbb P}\big[|\{1,\ldots,k_r\}\setminus W_b| > k_r/2\big]
\leq {\mathbb E} \big\vert \{1,\ldots,k_r\}\setminus W_b \big\vert
\leq
k_r r^{-(1+o(1))(\sqrt{b}-1)^2
\frac{2\alpha}{2-\beta}},
\end{equation*}
so
\begin{equation}
\label{manydisks}
{\mathbb P}\big[|W_b|\geq k_r/2\big] \geq 1-2r^{-(1+o(1))(\sqrt{b}-1)^2
\frac{2\alpha}{2-\beta}}.
\end{equation}
Assume that~$1<b<\alpha^{-1}$ and~$\beta\in (0,1)$ is close enough to~$1$,
so that $\frac{b\alpha}{\beta^2}<1$.
As in Section~\ref{s_aux_torus}, we denote by
${\hat Z}^{(1),j},\ldots,{\hat Z}^{({\hat N}_\alpha^{(j)}),j}$
the excursions of RI($\alpha$)
between~$\partial B_j$ and~$\partial B'_j$. Also, let
$ {\tilde Z}^{(1),j},{\tilde Z}^{(2),j},{\tilde Z}^{(3),j},\ldots$
be a sequence of i.i.d.\ ${\widehat S}$-excursions
between~$\partial B_j$ and~$\partial B'_j$, started with the law
$\mathop{\mathrm{hm}}\nolimits_{B_j}$; these sequences themselves are also assumed to
be independent.
Abbreviate $m= b\frac{2\alpha\ln^2 r}{3\ln\ln r^\beta}$. Next,
for $j=1,\ldots,k_r$ we consider the events
\[
D_j = \Big\{\big\{{\hat Z}^{(1),j},\ldots,
{\hat Z}^{({\hat N}_\alpha^{(j)}),j}\big\}
\subset \big\{{\tilde Z}^{(1),j}, \ldots,
{\tilde Z}^{((1+\delta)m),j}\big\}\Big\}.
\]
\begin{lem}
\label{l_SLT_coupling}
It is possible to construct the excursions
$\big({\hat Z}^{(k),j}, k=1,\ldots, {\hat N}_\alpha^{(j)}\big)$,
$\big({\tilde Z}^{(k),j}, k=1,2,3,\ldots\big)$, $j=1,\ldots,k_r$,
on a same probability space in such a way that
for a fixed $C'>0$
\begin{equation}
\label{D_j_complementary}
{\mathbb P}\big[D_j^\complement\big]
\leq \exp\Big(- C' \frac{\ln^2 r}{(\ln\ln r)^2}\Big),
\end{equation}
for all $j=1,\ldots,k_r$.
\end{lem}
\begin{proof}
This coupling can be built using the method
of \emph{soft local times} of~\cite{SLT}; in this specific
situation, the exposition of~\cite{CGPV} is better suited.
Roughly speaking, this method consists of using a marked
Poisson point process on $\big(\bigcup_j\partial B_j\big)\times {\mathbb R}_+$,
where the ``marks'' are corresponding excursions; see Section~2 of~\cite{CGPV}
for details. Then, the ${\tilde Z}$-excursions are simply the marks of the
points of the Poisson processes on $\partial B_j\times {\mathbb R}_+$ ordered
according to the second coordinate, so they are independent by
construction. The ${\hat Z}$-excursions are also the marks of the
points of these Poisson processes, but generally taken in a different
order using a special procedure; Figure~1 of~\cite{CGPV} is, hopefully,
self-explanating and may provide some quick insight.
The inequality~\eqref{D_j_complementary} then follows from
Lemma~2.1 of~\cite{CGPV} (observe that, by
Lemma~\ref{l_entrance_hat_s}, the parameter~$v$ in Lemma~2.1 of~\cite{CGPV}
can be anything exceeding $O(1/\ln^2 r)$, so we
choose e.g.\ $v=(\ln\ln r)^{-1}$; also, that lemma is clearly
valid for ${\widehat S}$-excursions as well).
\end{proof}
We continue the proof of part~(iii) of Theorem~\ref{t_sizevacant}. Define
\[
D = \bigcap_{j\leq k_r} D_j;
\]
using~\eqref{D_j_complementary}, we obtain by the union bound the subpolynomial estimate
\begin{equation}
\label{est_D}
{\mathbb P}\big[D^\complement\big]
\leq \frac{1}{4}r^{2(1-\beta)}
\exp\Big(- C' \frac{\ln^2 r}{(\ln\ln r)^2}\Big).
\end{equation}
Let~$\delta>0$ be such that $(1+\delta)\frac{b\alpha}{\beta^2}<1$.
Define the events
\[
{\tilde{\mathcal G}}_j = \big\{B'_j\text{ is completely covered by }
{\tilde Z}^{(1),j}\cup\cdots\cup{\tilde Z}^{((1+\delta)m),j}\big\}.
\]
Then, for all~$j\leq k_r$ it holds that
\begin{equation}
\label{1/5}
{\mathbb P}[{\tilde{\mathcal G}}_j]\leq \frac{1}{5}
\end{equation}
for all large enough~$r$. Indeed, if the $\tilde Z$'s were
independent SRW-excursions,
the above inequality (with any fixed constant in the right-hand
side) could be obtained as in
the proof of Lemma~3.2 of~\cite{CGPV}
(take ${\tilde H}=\mathop{\mathrm{hm}}\nolimits_{B_j}$ and $n=3r^\beta + 1$ there). On the other hand,
Lemma~\ref{l_relation_S_hatS}~(ii) implies that the first~$(1+\delta)m$
${\widehat S}$-excursions can be coupled with SRW-excursions with
high probability, so~\eqref{1/5} holds
for ${\widehat S}$-excursions as well.
Next, define the set
\[
{\widetilde W} = \big\{j\leq k_r : {\tilde{\mathcal G}}_j^\complement
\text{ occurs}\big\}.
\]
Since the events $({\tilde{\mathcal G}}_j, j\leq k_r)$ are independent,
by~\eqref{1/5} we have (recall that $k_r = \frac{1}{4}r^{2(1-\beta)}$)
\begin{equation}
\label{est_widetilde_W}
{\mathbb P}\Big[|{\widetilde W}| \geq \frac{3}{5}k_r\Big]
\geq 1 - \exp\big(-C r^{2(1-\beta)}\big)
\end{equation}
for all~$r$ large enough.
Observe that, by construction, on the event~$D$
we have ${\mathcal V}^\alpha\cap B'_j\neq \emptyset$
for all $j\in {\widetilde W}\cap W_b$.
So, using~\eqref{manydisks}, \eqref{est_widetilde_W},
and~\eqref{est_D}, we obtain
\begin{align*}
\lefteqn{ {\mathbb P}\Big[{\mathcal V}^\alpha\cap \big(B(r)\setminus B(r/2)\big)
=\emptyset\Big]}\\
&\leq
{\mathbb P}\Big[|W_b|< \frac{k_r}{2}\Big] +
{\mathbb P}\big[D^\complement\big]
+ {\mathbb P}\Big[|{\widetilde W}|< \frac{3k_r}{5}\Big]
\\
&\leq
2r^{-(1+o(1))(\sqrt{b}-1)^2
\frac{2\alpha}{2-\beta}}
+\exp\big(-C r^{2(1-\beta)}\big)
+ \frac{1}{4}r^{2(1-\beta)}
\exp\Big(- C' \frac{\ln^2 r}{(\ln\ln r)^2}\Big).
\end{align*}
Since $b\in (1,\alpha^{-1})$ can be arbitrarily close
to~$\alpha^{-1}$ and $\beta\in (0,1)$
can be arbitrarily close to~$1$,
this concludes the proof of (\ref{eq_emptyball}).
\end{proof}
\begin{proof}[Proof of Theorem~\ref{t_sizevacant} (ii)]
To complete the proofs in Section~\ref{s_results}, it remains to prove that $|{\mathcal V}^\alpha|<\infty$
a.s.\ for $\alpha>1$.
First,we establish the following elementary fact.
For $x\in\partial B(2r)$ and $y\in B(r)\setminus B(r/2)$, it holds
\begin{equation}
\label{exc_r_2r}
\widehat{P}_x\big[\widehat{\tau}_1(B(y))<\widehat{\tau}_1\big(B(r\ln r)\big)\big]
= \frac{\ln\ln r}{\ln r}\big(1+o(1)\big).
\end{equation}
Indeed, define the events
\begin{align*}
G_0 &= \{\tau_1(B(0))<\tau_1\big(B(r\ln r)\big)\},\\
G_1 &= \{\tau_1(B(y))<\tau_1\big(B(r\ln r)\big)\};
\end{align*}
then, Lemma~\ref{l_exit_balls} implies that
\[
P_x[G_0] = \frac{\ln\ln r}{\ln r}\big(1+o(1)\big) = P_x[G_1] \big(1+o(1)\big) .
\]
Observe that
\begin{align*}
P_x[G_0 \cap G_1] &= P_x[G_0 \cup G_1] P_x[G_0 \cap G_1\mid G_0 \cup G_1]\\
&\leq \big(P_x[G_0]+P_x[G_1]\big)\big(P_0[G_1]+P_y[G_0]\big)\\
&\leq 4\Big(\frac{\ln\ln r}{\ln r}\Big)^2 \big(1+o(1)\big).
\end{align*}
So,
\begin{align*}
P_x[G_1\mid G_0^\complement]
&= \frac{P_x[G_1]-P_x[G_0\cap G_1]}{1-P_x[G_0]}\\
&= \frac{\ln\ln r}{\ln r}\big(1+o(1)\big),
\end{align*}
and we use Lemma~\ref{l_relation_S_hatS} to conclude
the proof of~\eqref{exc_r_2r}.
Now, the goal is to prove that, for $\alpha>1$
\begin{equation}
\label{covered_ring}
{\mathbb P}\big[\text{there exists }y\in B(r)\setminus B(r/2)
\text{ such that }y\in{\mathcal V}^\alpha\big]
\leq r^{-\frac{\alpha}{2}(1-\alpha^{-1})^2(1+o(1))}.
\end{equation}
This would clearly imply that the set~${\mathcal V}^\alpha$ is a.s.\
finite, since
\begin{equation}
\label{BorCan}
\{{\mathcal V}^\alpha\text{ is infinite}\}
= \big\{{\mathcal V}^\alpha\cap\big(B(2^n)\setminus B(2^{n-1})\big)\neq \emptyset
\text{ for infinitely many }n\big\},
\end{equation}
and the Borel-Cantelli lemma together with~\eqref{covered_ring}
imply that the probability of the latter event equals~$0$.
Let~$N_{\alpha,r}$ be the number of ${\widehat S}$-excursions of RI($\alpha$)
between~$\partial B(r)$ and $\partial B(r\ln r)$.
Analogously to~\eqref{LD_numb_exc} (using Lemma~\ref{l_escape_hatS}
in place of Lemma~\ref{l_escape_from_ball}~(i)),
it is straightforward
to show that, for $b<1$,
\begin{equation}
\label{LD_exc_b<1}
{\mathbb P}\Big[N_{\alpha,r} \leq b\frac{2\alpha\ln^2 r}{\ln\ln r}\Big]
\leq r^{-2\alpha(1-\sqrt{b})^2 (1+o(1))}.
\end{equation}
Now, \eqref{exc_r_2r} implies that for $y\in B(r)\setminus B(r/2)$
\begin{align}
{\mathbb P}\Big[y\text{ is uncovered by first }
b\frac{2\alpha\ln^2 r}{\ln\ln r}\text{ excursions}\Big]
& \leq \Big(1-\frac{\ln\ln r}{\ln r}
(1+o(1))\Big)^{b\frac{2\alpha\ln^2 r}{\ln\ln r}}\nonumber\\
& = r^{-2b\alpha(1+o(1))},
\label{hit_y_in_the_ring}
\end{align}
so, using the union bound,
\begin{equation}
\label{cover_ring_b}
{\mathbb P}\Big[\exists y\in B(r)\setminus B(r/2):
y\in{\mathcal V}^\alpha , N_{\alpha,r} >
b\frac{2\alpha\ln^2 r}{\ln\ln r}
\Big]
\leq r^{-2(b\alpha-1)(1+o(1))}.
\end{equation}
Using~\eqref{LD_exc_b<1} and~\eqref{cover_ring_b} with
$b=\frac{1}{4}\big(1+\frac{1}{\alpha}\big)^2$ we
conclude the proof of~\eqref{covered_ring} and
of Theorem~\ref{t_sizevacant}~(ii).
\end{proof}
\subsection{Proof of Theorem~\ref{t_conditional}}
\label{s_proof_torus}
Let us first give a more detailed
heuristic argument for~\eqref{eq_conditional}.
As usual, we consider the \emph{excursions} of the random walk~$X$
between $\partial B\big(\frac{n}{3\ln n}\big)$ and $\partial B(n/3)$
up to time~$t_\alpha$.
Recall that~$N_\alpha$ denotes the number of these excursions.
Lemma~\ref{l_excursions_torus} shows
that this (random) number is concentrated around
$\frac{2\alpha\ln^2 n}{\ln\ln n}$, with deviation probabilities
of subpolynomial order.
This is for unconditional probabilities, but, since the probability
of the event $\{0\in U_{t_\alpha}^{(n)}\}$ is only polynomially
small (actually, it is $n^{-2\alpha+o(1)}$), the same holds
for the deviation probabilities conditioned on this event.
So, let us just assume for now that the number of the excursions
is \emph{exactly} $\frac{2\alpha\ln^2 n}{\ln\ln n}$ a.s.,
and see where will it lead us.
Assume without restriction of generality that $0\in A$.
Then, Lemmas~\ref{l_exit_balls} and~\ref{l_hit_A} imply that
\begin{itemize}
\item the probability that an excursion hits the origin
is roughly $\frac{\ln\ln n}{\ln (n/3)}$;
\item provided that $\mathop{\mathrm{cap}}(A)\ll \ln n$,
the probability that an excursion hits the set~$A$
is roughly $\frac{\ln\ln n}{\ln (n/3)}
\big(1+\frac{\pi\mathop{\mathrm{cap}}(A)}{2\ln (n/3)}\big)$.
\end{itemize}
So, the \emph{conditional} probability~$p_*$
that an excursion does not hit~$A$ given that it does not
hit the origin is
\[
p_*\approx \frac{1-\frac{\ln\ln n}{\ln (n/3)}
\big(1+\frac{\pi\mathop{\mathrm{cap}}(A)}{2\ln (n/3)}\big)}{1-\frac{\ln\ln n}{\ln (n/3)}}
\approx 1 - \frac{\pi\ln\ln n}{2\ln^2 n}\mathop{\mathrm{cap}}(A),
\]
and then we obtain
\begin{align*}
{\mathbb P}[\Upsilon_n A \subset
U_{t_\alpha}^{(n)} \mid 0\in U_{t_\alpha}^{(n)}]
&\approx p_*^{N_\alpha}\\
&\approx \Big(1 -
\frac{\pi\ln\ln n}{2\ln^2 n}\mathop{\mathrm{cap}}(A)\Big)^{\frac{2\alpha\ln^2 n}{\ln\ln n}}
\\
&\approx \exp\big(-\pi\alpha\mathop{\mathrm{cap}}(A)\big),
\end{align*}
which agrees with the statement of Theorem~\ref{t_conditional}.
However, turning the above heuristics
to a rigorous proof is not an easy task. The reason for this is that,
although~$N_\alpha$ is indeed concentrated around
$\frac{2\alpha\ln^2 n}{\ln\ln n}$, it is not \emph{concentrated enough}:
the probability that~$0$ is not
hit during~$k$ excursions, where~$k$ varies over the ``typical''
values of~$N_\alpha$, changes too much. Therefore,
in the proof of Theorem~\ref{t_conditional} we take a different
route by considering the suitable $h$-transform of the walk,
as explained below.
Define for $x\in{\mathbb Z}^2_n$ and \emph{any}~$t$
\[
h(t,x)=P_x[T_n(0)>t]
\]
(so that $h(t,x)=1$ for $t<0$).
To simplify the notations, let us also assume that~$t_\alpha$
is integer.
We will represent the conditioned random walk
as a time-dependent Markov chain, using
the Doob's $h$-transform.
Indeed, it is well known and easily checked
that the simple random
walk on~${\mathbb Z}_n^2$ conditioned on the event $\{0\in U^{(n)}_{t_\alpha}\}$
is a time-dependent
Markov chain ${\widetilde X}$ with transition
probabilities given by
\begin{equation}
\label{df_h_transf}
{\mathbb P}[{\widetilde X}_{s+1}=y \mid {\widetilde X}_s=x]
= \frac{h(t_\alpha-s-1,y)}{h(t_\alpha-s,x)}\times \frac{1}{4},
\end{equation}
if $x$ and~$y$ are neighbours,
and equal to 0 otherwise. For simpler notations, we do not indicate
the dependence on~$t_\alpha$ in the notation $\tilde X$.
In order to proceed, we need the following fact
(its proof can be skipped on a first reading).
\begin{lem}
\label{l_reg_h}
For all $\lambda \in (0,1/5)$, there exist $c_1>0, n_1 \geq 2$,
$\sigma_1>0$ (depending on~$\lambda$) such that
for all~$n \geq n_1$, $1\leq \beta\leq \sigma_1\ln n$,
$\|x\|,\|y\|\geq \lambda n$,
$|r|\leq \beta n^2$ and all~$s\geq 0$,
\begin{equation}
\label{regularity_h}
\Big|\frac{h(s,x)}{h(s+r,y)}-1\Big| \leq \frac{c_1\beta}{\ln n} \;.
\end{equation}
\end{lem}
\begin{proof}
Denote
\[
h(t, \mu) := P_{\mu}[T_n(0)>t],
\]
where $P_{\mu}[\cdot]$ is the probability for
the simple random walk on ${\mathbb Z}_n^2$ starting from the initial distribution~$\mu$.
Using the local CLT for the two-dimensional SRW (e.g., Theorem~2.3.11 of~\cite{LL10})
it is straightforward to obtain that for a large enough~$\kappa>2$ and all $t>n^2$ and
$x\in{\mathbb Z}^2_n$
\begin{equation}
\label{LCLT}
P_0[X_t = x] \leq \frac{\kappa}{n^2}.
\end{equation}
Let us define the set~${\mathcal M}$ of probability
measures on~${\mathbb Z}^2_n$ in the following way:
\[
{\mathcal M} = \big\{\nu : \nu\big(B(j)\big)\leq 7\kappa j^2n^{-2}
\text{ for all }j\leq \lambda n\big\};
\]
observe that any probability
measure concentrated on a one-point set~$\{x\}$ with $\|x\|\geq \lambda n$
belongs to~${\mathcal M}$.
Assume from now on that~$n$ is odd, so that the simple random walk on~${\mathbb Z}^2_n$
is aperiodic (the case of even~$n$ can be treated essentially
in the same way, with some obvious modifications). Recall that the uniform measure~$\mu_0$ on~${\mathbb Z}^2_n$, i.e., $\mu_0(x)=n^{-2}$ for all~$x\in{\mathbb Z}^2_n$,
is the invariant law of simple random walk $X$ on~${\mathbb Z}^2_n$.
It is straightforward to observe that $\mu_0\in{\mathcal M}$;
moreover, it holds in fact that
$\mu_0\big(B(j)\big)\leq 7 j^2n^{-2}
\text{ for all }j\leq \lambda n$ (i.e., with~$\kappa=1$).
Thus, for any probability measure~$\nu$ such that
$\nu(x)\leq \kappa\mu_0(x)$ for all~$x\in B(\lambda n)$
it holds that $\nu\in{\mathcal M}$. So, \eqref{LCLT} implies that
\begin{equation}
\label{belongs_to_M}
\nu P^{(t)}\in {\mathcal M} \quad \text{ for any }\nu \text{ and all }t\geq n^2.
\end{equation}
Let us abbreviate $\nu P^{(s)}(\cdot)=P_\nu [X_s\in \cdot\,]$.
Recall that the mixing time of~$X$ is of order~$n^2$ (e.g., Theorem~5.5
in~\cite{LPW}).
Using the bound on the separation distance provided by
Lemma~19.3 of~\cite{LPW},
it is clear that for any $\varepsilon\in (0,1)$ one can find large enough~$c'$
(in fact, $c'=O(\ln \varepsilon^{-1})$)
such that for any probability measure~$\nu$
it holds that $\nu P^{(s)}\geq (1-\varepsilon)\mu_0$ for all $s\geq (c'-1) n^2$.
Using~\eqref{belongs_to_M}, we obtain for all $s\geq c' n^2$,
\begin{eqnarray}
\nu P^{(s)}
&=& (1-\varepsilon)\mu_0 P^{(n^2)} +\big( \nu P^{(s\!-\!n^2)}-(1-\varepsilon)\mu_0\big) P^{(n^2)}
\nonumber \\ \label{spectralgap}
&=&
(1-\varepsilon)\mu_0 + \varepsilon\nu', \quad \text{ with } \nu'\in {\mathcal M}.
\end{eqnarray}
We are now going to obtain that there exists some~$c_2>0$
such that for all~$b \in \{1,2,3,\ldots\}$ and all $\nu \in {\mathcal M}$,
\begin{equation}
\label{entre_0_nicht}
h(bn^2,\nu) = P_\nu [T_n(0) > bn^2] \geq 1 - \frac{bc_2}{\ln n}.
\end{equation}
To prove~\eqref{entre_0_nicht}, let us first show that there exists~$c_3=c_3(\lambda)>0$ such that
\begin{equation}
\label{escape_to_lambda_n}
P_\nu\big[T_n(0) < T_n\big(\partial B(\lambda n)\big)\big]
\leq \frac{c_3}{\ln n}
\end{equation}
for all $\nu \in {\mathcal M}$. Abbreviate
$W_j=B\big(\frac{\lambda n}{2^{j-1}}\big)\setminus
B\big(\frac{\lambda n}{2^j}\big)$ and write,
using~\eqref{nothit_0}
\begin{align*}
\sum_{x\in W_j} \nu(x)
P_x\big[T_n(0) < T_n\big(\partial B(\lambda n)\big)\big]
& \leq \sum_{x\in W_j}\nu(x)\Big(1-\frac{a(x)}{a(\lambda n)+O(n^{-1})}\Big)\\
& \leq 7\kappa n^{-2}\times \frac{\lambda^2n^2}{2^{2(j-1)}}\times
\Big(1-\frac{\ln \lambda n - j\ln 2}{\ln \lambda n+O(n^{-1})}\Big)\\
&\leq \frac{1}{\ln\lambda n} \times \frac{7\kappa j\lambda^2\ln 2}{2^{2(j-1)}}.
\end{align*}
Let $j_0$ be such that $\frac{\lambda n}{2^{j_0}}\leq \frac{n}{\sqrt{\ln n}}$.
Write
\begin{align*}
\lefteqn{P_\nu\big[T_n(0) < T_n\big(\partial B(\lambda n)\big)\big]}\\
&\leq \nu\Big(B\Big(\frac{n}{\sqrt{\ln n}}\Big)\Big)
+ \sum_{j=1}^{j_0} \sum_{x\in W_j} \nu(x)
P_x\big[T_n(0) < T_n\big(\partial B(\lambda n)\big)\big]\\
&\leq \frac{7\kappa }{\ln n} + \frac{1}{\ln\lambda n}
\sum_{j=1}^{j_0} \frac{7\kappa j\lambda^2\ln 2}{2^{2(j-1)}}
\end{align*}
for $\nu \in {\mathcal M}$,
which proves~\eqref{escape_to_lambda_n}. To obtain~\eqref{entre_0_nicht},
let us recall that $\nu P^{(n^2)}\in{\mathcal M}$
for any~$\nu$ by~\eqref{belongs_to_M}.
Observe that the number of excursions
by time~$n^2$
between $\partial B(\lambda n)$ and $\partial B(n/3)$
is stochastically bounded by a Geometric random variable
with expectation of constant order. Since (again by~\eqref{nothit_0})
for any $x\in \partial B(\lambda n)$
\[
P_x\big[T_n(0) < T_n\big(\partial B(n/3)\big)\big]
\leq \frac{c_4}{\ln n},
\]
for some $c_4=c_4(\lambda)$,
and, considering a random sum of a geometric number of
independent Bernoulli with parameter $c_4/\ln n$,
using also~\eqref{escape_to_lambda_n} it is not difficult to obtain that
for any $\nu\in {\mathcal M}$
\begin{equation}
\label{for_b_0}
P_\nu [T_n(0) \leq n^2] \leq \frac{c_5}{\ln n}.
\end{equation}
The inequality~\eqref{entre_0_nicht} then follows from~\eqref{for_b_0}
and the union bound,
\begin{equation}
\label{n_entre_pas_0}
P_\nu [T_n(0) \leq k n^2] \leq \sum_{j=0}^{k-1}
P_{\nu P^{(jn^2})} [T_n(0) \leq n^2] \leq \frac{kc_5}{\ln n}.
\end{equation}
Now, let $c' \in \{1,2,3,\ldots\} $
be such that $\varepsilon < 1/3$ in~\eqref{spectralgap}.
Assume also that~$n$ is sufficiently large so that
$\big(1-\frac{c'c_2}{\ln n}\big)^{-1}\leq 2$.
Then, \eqref{entre_0_nicht} implies that for all $s\leq c'n^2$ and
$\nu\in{\mathcal M}$
\[
\frac{h(s,\mu)}{h(s,\nu)}-1 \leq \frac{1}{1-\frac{c'c_2}{\ln n}}-1
=\Big(1-\frac{c'c_2}{\ln n}\Big)^{-1} \times\frac{c'c_2}{\ln n},
\]
and therefore
\begin{align}
\frac{h(s,\mu)}{h(s,\nu)}-1 &\leq \frac{3c'c_2}{\ln n}
\qquad \text{ for any } \nu\in{\mathcal M} \text{ and arbitrary }\mu,
\label{whatweneed}\\
1-\frac{h(s,\mu)}{h(s,\nu)} &\leq \frac{3c'c_2}{\ln n}
\qquad \text{ for any } \mu\in{\mathcal M} \text{ and arbitrary }\nu.
\label{whatweneed2}
\end{align}
We now extend~\eqref{whatweneed}--\eqref{whatweneed2}
from times $s \leq s_0=c'n^2$ to all times using
induction. Let $s_k=(k+1)c'n^2$,
and consider the recursion hypothesis
\[
(H_k) : \qquad \eqref{whatweneed}\ {\rm and } \ \eqref{whatweneed2} \; {\rm hold\ for\ } \ s \leq s_k,
\]
that we just have proved for $k=0$. Assume now $(H_k)$ for some $k$.
Define the event $G_{r,s}=\{X_j\neq 0 \text{ for all }r+1\leq j\leq s\}$,
and write
\begin{align}
h(s+t,\mu) &= P_\mu[G_{t,s+t}]
P_\mu[T_n(0)>t\mid G_{t,s+t}]\nonumber\\
&= h(s,\mu P^{(t)}) P_\mu[T_n(0)>t\mid G_{t,s+t}].
\label{Markov_future}
\end{align}
Abbreviate $t= c'n^2$ for the rest of the proof of the Lemma.
Let us estimate the second term in the right-hand side
of~\eqref{Markov_future}. Let~$\Gamma_{[0,t]}$ be the set
of all nearest-neighbour trajectories on~${\mathbb Z}^2_n$ of length~$t$.
For $\varrho\in \Gamma_{[0,t]}$ we have
$P_\mu[\varrho]=\mu(\varrho_0)\big(\frac{1}{4}\big)^{|\varrho|}$ and
\[
P_\mu[\varrho\mid G_{t,s+t}] = \mu(\varrho_0)\Big(\frac{1}{4}\Big)^{|\varrho|}
\times \frac{h(s,\varrho_{\text{end}})}{h(s, \mu P^{(t)})}
\leq \mu(\varrho_0)\Big(\frac{1}{4}\Big)^{|\varrho|}
\Big(1+\frac{3c'c_2}{\ln n}\Big)
\]
using the relation~\eqref{whatweneed} for $s \leq s_k$.
Summing over $\varrho$ such that $T_n(0)\leq t$ and using~\eqref{entre_0_nicht}, we obtain, for $\mu \in {\mathcal M}$,
\begin{equation}
\label{time_rev_implies}
P_\mu[T_n(0)>t\mid G_{t,s+t}]
\geq 1 - \frac{c'c_2}{\ln n}\Big(1+\frac{3c'c_2}{\ln n}\Big) .
\end{equation}
Now, we use~\eqref{spectralgap} and~\eqref{Markov_future}
to obtain that, with $\mu', \nu'$ defined in~\eqref{spectralgap},
\begin{align*}
\frac{h(s+t,\mu)}{h(s+t,\nu)} &=
\frac{h(s,\mu P^{(t)})P_\mu[T_n(0)>t\mid G_{t,s+t}]}
{h(s,\nu P^{(t)})P_\nu[T_n(0)>t\mid G_{t,s+t}]}\\
&=\frac{\big(1-\varepsilon+\varepsilon\frac{h(s,\mu')}{h(s,\mu_0)}\big)
P_\mu[T_n(0)>t\mid G_{t,s+t}]}
{\big(1-\varepsilon+\varepsilon\frac{h(s,\nu')}{h(s,\mu_0)}\big)
P_\nu[T_n(0)>t\mid G_{t,s+t}]},
\end{align*}
for $s \leq s_k$.
We now use $(H_k)$ for the two ratios of $h$'s in the above expression, we also use~\eqref{time_rev_implies}
for the conditional probability in the denominator -- simply bounding it by~$1$ in the numerator --
to obtain
\begin{align*}
\frac{h(s+t,\mu)}{h(s+t,\nu)}
\leq \frac{\big(1-\varepsilon+\varepsilon\big(1+\frac{3c'c_2}{\ln n}\big)\big)}
{\big(1-\varepsilon+\varepsilon\big(1-\frac{3c'c_2}{\ln n}\big)\big)
\big(1 - \frac{c'c_2}{\ln n}\big(1+\frac{3c'c_2}{\ln n}\big)\big)},
\end{align*}
that is,
\[
\frac{h(s+t,\mu)}{h(s+t,\nu)}-1
\leq (6\varepsilon+1)\frac{c'c_2}{\ln n} + o\big((\ln n)^{-1}\big)
\]
for $\nu\in {\mathcal M}$.
Since $\varepsilon<1/3$, for large enough~$n$ we obtain
that~\eqref{whatweneed}
also holds for all $s\leq s_{k+1}$.
In the same way, we prove the validity of~\eqref{whatweneed2}
for $s\leq s_{k+1}$.
This proves the recursion, which
in turn implies~\eqref{regularity_h} for the case~$r=0$.
To treat the general case, observe that
\begin{equation}
\label{h_decomposition}
h(s+r,y)=h(r,y)h(s,\nu), \quad \text{where }
\nu(\cdot)=P_y[X_r=\cdot\mid T_n(0)>r].
\end{equation}
Note that, by~\eqref{n_entre_pas_0}, we can choose~$\sigma_1$
in such a way that $P_y[T_n(0)>r]\geq \frac{1}{2}$.
Now, without loss of generality, we can assume that $r\geq {\tilde c}n^2$,
where~${\tilde c}$ is such that $2P_y[X_t=\cdot\,]\in{\mathcal M}$ for all
$t\geq {\tilde c}n^2$
(clearly, such~${\tilde c}$ exists; e.g., consider~\eqref{spectralgap}
with $\varepsilon=1/2$). Then, the general case in~\eqref{regularity_h}
follows from~\eqref{n_entre_pas_0} and~\eqref{h_decomposition}.
\end{proof}
Now we are able to prove Theorem~\ref{t_conditional}.
\begin{proof}[Proof of Theorem~\ref{t_conditional}]
Abbreviate $\delta_{n,\alpha} = C\alpha\sqrt{\frac{\ln\ln n}{\ln n}}$
and
\[
I_{\delta_{n,\alpha}} = \Big[(1-\delta_{n,\alpha})
\frac{2\alpha\ln^2 n}{\ln\ln n},
(1+\delta_{n,\alpha})\frac{2\alpha\ln^2 n}{\ln\ln n}\Big].
\]
Let $N_\alpha$ be the number of excursions
between $\partial B\big(\frac{n}{3\ln n}\big)$ and $\partial B(n/3)$
up to time~$t_\alpha$.
It holds that
${\mathbb P}\big[0\in U^{(n)}_{t_\alpha}\big] =n^{-2\alpha+o(1)}$,
see e.g.\ (1.6)--(1.7) in \cite{CGPV}. Then,
observe that~\eqref{eq_excursions_torus} implies that
\begin{align*}
{\mathbb P}\big[N_\alpha\notin I_{\delta_{n,\alpha}} \; \big|\;
0\in U^{(n)}_{t_\alpha}\big]
\leq \frac{{\mathbb P}[N_\alpha\notin I_{\delta_{n,\alpha}}]}
{{\mathbb P}[0\in U^{(n)}_{t_\alpha}]} \leq n^{2\alpha+o(1)}
\times n^{-C'\alpha^2},
\end{align*}
where~$C'$ is a constant that can be made arbitrarily
large by making the constant~$C$ in the definition of~$\delta_{n,\alpha}$
large enough.
So, if~$C$ is large enough, for some $c''>0$ it holds that
\begin{equation}
\label{cond_numb_exc}
{\mathbb P}\big[N_\alpha\in I_{\delta_{n,\alpha}} \;
\big|\; 0\in U^{(n)}_{t_\alpha}\big]
\geq 1 - n^{-c''\alpha}.
\end{equation}
We assume that the set~$A$ is fixed, so that $\mathop{\mathrm{cap}}(A)=O(1)$ and diam$(A)=O(1)$.
In addition, assume without loss of generality that $0\in A$.
Recall that
with~\eqref{cond_numb_exc} we control the number of excursions
between $\partial B\big(\frac{n}{3\ln n}\big)$ and $\partial B(n/3)$
up to time~$t_\alpha$. Now, we estimate the (conditional) probability
that an excursion hits the set~$A$. For this, observe that
Lemmas~\ref{l_exit_balls},
\ref{l_hit_A} and~\ref{l_relation_S_hatS} imply that,
for any $x\in \partial B\big(\frac{n}{3\ln n}\big)$
\begin{align}
\lefteqn{\widehat{P}_x\big[\widehat{\tau}_1(A)>\widehat{\tau}_1(\partial B(n/3))\big]}
\nonumber\\
&= \frac{P_x\big[\tau_1(A)>\tau_1(\partial B(n/3)),
\tau_1(0)> \tau_1(\partial B(n/3))\big]}{P_x\big[\tau_1(0)>
\tau_1(\partial B(n/3))\big]} \big(1+O((n\ln n)^{-1}))\big)
\nonumber\\ \nonumber
& = \frac{a(x)-\mathop{\mathrm{cap}}(A)+O(\frac{\ln^2 n}{n})}
{a(n/3)-\mathop{\mathrm{cap}}(A) +O(\frac{\ln^2 n}{n})}
\times \frac{a(n/3)+O(n^{-1})}{a(x)}
\big(1+O((n\ln n)^{-1}))\big)
\\ \nonumber
& = \frac{1-\frac{\mathop{\mathrm{cap}}(A)}{a(x)}}
{1-\frac{\mathop{\mathrm{cap}}(A)}{a(n/3)}}
\big(1+O(n^{-1}\ln n)\big)\\ \label{eq:decadix}
& = 1 - \frac{\pi}{2}\mathop{\mathrm{cap}}(A)\frac{\ln\ln n}{\ln^2 n}
\big(1+o(1)\big).
\end{align}
Note that the above is for ${\widehat S}$-excursions; we still need
to transfer this result to the conditioned random walk
on the torus.
Recall the notation $\Gamma^{(x)}_{0,R}$ from the
beginning of Section~\ref{s_aux_hat_s}.
Then,
for a fixed $x \in \partial B\big(\frac{n}{3\ln n}\big)$
let us define the set of paths
\[
\Lambda_j = \big\{\varrho \in \Gamma^{(x)}_{0,n/3} :
(j-1)n^2<|\varrho|\leq jn^2\big\}.
\]
It is straightforward to obtain that (since, regardless of the
starting position, after $O(n^2)$ steps the walk goes out
of~$B(n/3)$ with uniformly positive probability)
\begin{equation}
\label{sortir_boule}
\max\big(P_x[\Lambda_j],\widehat{P}_x[\Lambda_j]\big) \leq e^{-cj}
\end{equation}
for some $c>0$.
To extract from~\eqref{eq:decadix} the corresponding formula for the
${\widetilde X}$-excursion, we first observe that, for
$x\in \partial B\big(\frac{n}{3\ln n}\big)$ and $s\geq n^2\sqrt{\ln n}$
\begin{align}
h(s,x) &= P_x[T_n(0)>T_n(\partial B(n/3))]\times
P_x[T_n(0)> s \mid T_n(0)>T_n(\partial B(n/3))] \nonumber\\
&\qquad \qquad \qquad \qquad+ P_x[T_n(\partial B(n/3))\geq T_n(0)>s]
\nonumber\\
&= \frac{a(x)}{a(n/3)+O(n^{-1})}P_x[T_n(0)> s
\mid T_n(0)>T_n(\partial B(n/3))]
+ \psi_{x,s,n}\nonumber\\
& = \frac{a(x)}{a(n/3)+O(n^{-1})}
\sum_{\substack{ y\in\partial B(n/3), \\k\geq 1}} h(s-k,y)
\ell_{y,k}
+ \psi_{x,s,n},
\label{transfer_h}
\end{align}
where $\ell_{y,k}=P_x\big[X_{T_n(\partial B(n/3))}=y,
T_n(\partial B(n/3))=k \mid T_n(0)>T_n(\partial B(n/3))\big]$
and $\psi_{x,s,n}= P_x[T_n(\partial B(n/3))\geq T_n(0)>s]$.
Clearly, by~\eqref{sortir_boule},
$\psi_{x,s,n} \leq \max_x P_x[T_n(\partial B(n/3))>s]
\leq e^{-Cs/n^2}$ for some $C>0$.
Also, we need the following fact: there exist $c_6, c'_6>0$
such that
\begin{equation}
\label{h_lower}
h(s,x) \geq \frac{c'_6}{\ln n} \exp\Big(-\frac{c_6 s}{n^2\ln n}\Big)
\end{equation}
for all~$s$ and all~$x\in{\mathbb Z}^2_n\setminus\{0\}$.
To prove~\eqref{h_lower}, it is enough to observe that
\begin{itemize}
\item a particle
starting from~$x$ will reach $\partial B\big(\frac{n}{3\ln n}\big)$
without hitting~$0$ with probability at least $O\big(\frac{1}{\ln n}\big)$;
\item the number of (possibly incomplete) excursions
between $\partial B\big(\frac{n}{3\ln n}\big)$ and $\partial B(n/3)$ until
time~$s$ does not exceed $\lceil\frac{3s}{n^2 \ln\ln n}\rceil$
with at least constant probability
by Lemma~\ref{l_excursions_torus}; and
\item regardless of the past, each excursion hits~$0$
with probability $\frac{\ln\ln n}{\ln n}(1+o(1))$, by (\ref{nothit_0}).
\end{itemize}
Observe that Lemma~\ref{l_reg_h} and~\eqref{h_lower} imply that
for any $y,y'\in\partial B(n/3)$ and any $t,r\geq 0$
(in the following, $\nu(\cdot) = P_{y'}[X_t=\cdot\mid T_n(0)>t]$)
\begin{equation}
\label{upper_h_over_h}
\frac{h(t,y)}{h(t+r,y')} = \frac{h(t,y)}{h(t,y')h(r,\nu)}
\leq c'' \ln n \times \exp\Big(\frac{c_6 r}{n^2\ln n}\Big).
\end{equation}
Now, going back to~\eqref{transfer_h}
{and setting
$a_j^{(n)} = \frac{c_1j}{\ln n}$ with~$c_1$ from Lemma~\ref{l_reg_h}}, observe that
for \emph{any} $y_0\in\partial B(n/3)$
(recall that $t_\alpha = \frac{4\alpha}{\pi}n^2\ln^2 n$)
\begin{align}
\lefteqn{\sum_{\substack{y\in\partial B(n/3),\\1\leq k\leq t_\alpha}}
h(s-k,y)
\ell_{y,k}}\nonumber\\
&= \sum_{1\leq j \leq \sigma_1\ln n}
\sum_{\substack{y\in\partial B(n/3),\\ (j-1)n^2<k\leq jn^2}} h(s-k,y)
\ell_{y,k}
+\sum_{\substack{y\in\partial B(n/3),\\
\sigma_1n^2\ln n< k \leq n^2\ln^{4/3} n }}
h(s-k,y) \ell_{y,k}
\nonumber\\
& \quad{}+\sum_{\substack{y\in\partial B(n/3),\\
n^2\ln^{4/3} n < k \leq t_\alpha}}
h(s-k,y) \ell_{y,k}
\nonumber\\
&= \sum_{1\leq j \leq \sigma_1\ln n}
\sum_{\substack{y\in\partial B(n/3),\\ (j-1)n^2<k\leq jn^2}}
\ell_{y,k}h(s,y_0)
\Big(1+ O\big(a_j^{(n)}\big)\Big)
\nonumber\\
& \quad{}+\sum_{\substack{y\in\partial B(n/3),\\
\sigma_1n^2\ln n < k \leq n^2\ln^{4/3} n }}
c'' \ell_{y,k}h(s,y_0) \exp\big(c_6 \ln^{1/3} n\big)\ln n
\nonumber\\
& \quad{}+\sum_{\substack{y\in\partial B(n/3),\\
n^2\ln^{4/3} n < k \leq t_\alpha}}
c'' \ell_{y,k}h(s,y_0) \exp\Big(\frac{4\alpha c_6 r \ln n}{\pi}\Big)\ln n
\nonumber\\
&= h(s,y_0) \Bigg(1+O\Big( \sum_{j\geq 1}e^{-cj} a_j^{(n)}\Big)
+ O\big(\exp\big(-(C\sigma_1\ln n-c_6 \ln^{1/3} n)\big)\ln n\big)
\nonumber\\
& \qquad \qquad \qquad {}+ O\Big(\exp\Big(-C\ln^{4/3}n +
\frac{4\alpha c_6 r \ln n}{\pi}\Big)\ln n\Big)\Bigg)
\nonumber\\
&=h(s,y_0) \Big(1+O\Big(\frac{1}{\ln n}\Big)\Big),
\label{regularize_sum_h}
\end{align}
due to Lemma~\ref{l_reg_h}, \eqref{sortir_boule} and~\eqref{upper_h_over_h}.
We plug~\eqref{regularize_sum_h}
into~\eqref{transfer_h}, divide by~$h(s,y_0)$, and use~\eqref{h_lower}
to obtain, for
$x\in \partial B\big(\frac{n}{3\ln n}\big)$ and $s\geq n^2\sqrt{\ln n}$
\[
\frac{h(s,x)}{h(s,y_0)} = \frac{a(x)}{a(n/3)+O(n^{-1})}
\Big(1+O\Big(\frac{1}{\ln n}\Big)\Big)
+\psi'_{s,n},
\]
where $|\psi'_{s,n}|\leq (c'_6)^{-1}\exp\big(-\frac{s}{n^2}
\big(c-\frac{c_6}{\ln n}\big)\big)$. Equivalently,
\begin{equation}
\label{frac_h}
\frac{h(s,y_0)}{h(s,x)} = \frac{a(n/3)}{a(x)}
\Big(1+O\Big(\frac{1}{\ln n}\Big)\Big).
\end{equation}
For $A\subset {\mathbb Z}^2$,
let us define also the hitting times of the
corresponding set on the torus
for the ${\widetilde X}$-walk \emph{after}
a given time~$s$:
\[
\widetilde{T}_n^{(s)}(A) = \min\{k\geq s: {\widetilde X}_k\in \Upsilon_nA\};
\]
we abbreviate $\widetilde{T}_n(A)=\widetilde{T}_n^{(0)}(A)$.
Write
\begin{align*}
\lefteqn{{\mathbb P}\big[\widetilde{T}^{(s)}_n(A)
<\widetilde{T}^{(s)}_n(\partial B(n/3))
\mid {\widetilde X}_s=x\big]}\\
&= \sum_{\varrho} \frac{h(t_\alpha-s-|\varrho|, \varrho_{\text{end}})}{h(t_\alpha-s,x)}
\Big(\frac{1}{4}\Big)^{|\varrho|}\\
&= \Bigg(\sum_{\varrho: \frac{|\varrho|}{n^2} \leq \sqrt{\ln n}}
+ \sum_{\varrho: \sqrt{\ln n}< \frac{|\varrho|}{n^2} \leq \ln^{4/3} n }
+ \sum_{\varrho: \frac{|\varrho|}{n^2} > \ln^{4/3} n}\Bigg)
\frac{h(t_\alpha-s-|\varrho|, \varrho_{\text{end}})}{h(t_\alpha-s,x)}
\Big(\frac{1}{4}\Big)^{|\varrho|},
\end{align*}
where the sums are over paths~$\varrho$ that begin in~$x$,
end on the first visit to~$\partial B(n/3)$, and touch~$A$
without touching~$0$.
Using Lemma~\ref{l_reg_h}, \eqref{sortir_boule} and~\eqref{frac_h}
for the first
sum, and dealing with the second and third sums as in the
derivation of~\eqref{regularize_sum_h},
we obtain
(recall that the term $\widehat{P}_x\big[\widehat{\tau}_1(A)<\widehat{\tau}_1(\partial B(n/3))\big]$
is of order $\frac{\ln\ln n}{\ln^2 n}$)
\begin{align*}
\lefteqn{{\mathbb P}\big[\widetilde{T}^{(s)}_n(A)
<\widetilde{T}^{(s)}_n(\partial B(n/3))
\mid {\widetilde X}_s=x\big]}\\
&= \sum_{\varrho} \frac{a(n/3)}{a(x)}
\Big(\frac{1}{4}\Big)^{|\varrho|}\Big(1+O\Big(\frac{1}{\sqrt{\ln n}}\Big)\Big)
+O\big(\exp(-C\sqrt{\ln n})\big) + O\big(\exp(-C\ln^{-4/3} n)\big)\\
&=\widehat{P}_x\big[\widehat{\tau}_1(A)<\widehat{\tau}_1(\partial B(n/3))\big]
\Big(1+O\Big(\frac{1}{\sqrt{\ln n}}\Big)\Big);
\end{align*}
again, the sum is over all paths~$\varrho$ that begin in~$x$,
end on the first visit to~$\partial B(n/3)$, and touch~$A$
without touching~$0$ (observe that the last equality comes
from the definition of~${\widehat S}$).
So, using (\ref{eq:decadix}), we obtain
for all $s\leq t_\alpha - n^2\sqrt{\ln n}$
and all $x \in \partial B\big(\frac{n}{3 \ln n}\big)$,
\begin{equation}
\label{exc_tilde_hitsA}
{\mathbb P}\big[\widetilde{T}^{(s)}_n(A)>\widetilde{T}^{(s)}_n(\partial B(n/3))
\mid {\widetilde X}_s=x\big]
= 1-\frac{\pi}{2}\mathop{\mathrm{cap}}(A)\frac{\ln\ln n}{\ln^2 n}(1+o(1)).
\end{equation}
Before we are able to conclude the proof of Theorem~\ref{t_conditional}, we
need another step to take care of times close to~$t_\alpha$.
Consider any $x\in\partial B\big(\frac{n}{3\ln n}\big)$
and any~$s\geq 0$ such that $t_\alpha-s\leq n^2\sqrt{\ln n}$.
Then, \eqref{df_h_transf} and~\eqref{h_lower}
together with the fact that $h(\cdot, \cdot)$
is nonincreasing with respect to the first (temporal) argument
imply that
\begin{align}
{\mathbb P}_x\big[({\widetilde X}_0,\ldots,{\widetilde X}_{|\varrho|})=\varrho\big]
&= \frac{h(t_\alpha-s-|\varrho|, \varrho_\text{end})}{h(t_\alpha-s,x)}
{\mathbb P}_x\big[(X_0,\ldots,X_{|\varrho|})=\varrho\big]
\nonumber\\
&\leq c_8 {\mathbb P}_x\big[(X_0,\ldots,X_{|\varrho|})=\varrho\big]
\label{measure_comparison}
\end{align}
for any path~$\varrho$ with $\varrho_0=x$.
Then, similarly to Section~\ref{s_aux_torus}, define
$\tilde{J}_k$ $\tilde{D}_k$ to be the starting and
ending times of $k$th excursion of~${\widetilde X}$
between~$\partial B\big(\frac{n}{3\ln n}\big)$
and~$\partial B(n/3)$, $k\geq 1$. Let
\[
\zeta = \min\{k: \tilde{J}_k\geq t_\alpha-n^2\sqrt{\ln n}\}
\]
be the index of the first ${\widetilde X}$-excursion that
starts after time $t_\alpha-n^2\sqrt{\ln n}$.
Let $\xi'_1,\xi'_2,\xi'_3,\ldots$ be a sequence of i.i.d.\
Bernoulli random variables independent of everything,
with
\[
{\mathbb P}[\xi'_k=1]=1-{\mathbb P}[\xi'_k=0]
=1-\frac{\pi}{2}\mathop{\mathrm{cap}}(A)\frac{\ln\ln n}{\ln^2 n}.
\]
For $k\geq 1$ define
two sequences of random variables
\[
\xi_k = \begin{cases}
\1{{\widetilde X}_j\notin A\text{ for all } j\in[\tilde{J}_k,\tilde{D}_k]},
& \text{ for } k<\zeta,\\
\xi'_k, & \text{ for } k\geq \zeta,
\end{cases}
\]
and
\[
\eta_k = \1{{\widetilde X}_j\notin A\text{ for all } j\in[\tilde{J}_{\zeta+k-1},
\tilde{D}_{\zeta+k-1}]}.
\]
Now, observe that~\eqref{exc_tilde_hitsA} and the
strong Markov property imply that
\begin{equation}
\label{hit_normalexcursion}
{\mathbb P}[\xi_k=1\mid \xi_1,\ldots,\xi_{k-1}] =
1-\frac{\pi}{2}\mathop{\mathrm{cap}}(A)\frac{\ln\ln n}{\ln^2 n}(1+o(1)).
\end{equation}
Also,
the relation~\eqref{measure_comparison}
together with Lemma~\ref{l_hit_A} imply that
\begin{equation}
\label{hit_lastexcursion}
{\mathbb P}[\eta_k=0\mid \eta_1,\ldots,\eta_{k-1}]\leq c_8 \frac{\ln\ln n}{\ln n}.
\end{equation}
Denote
\[
\zeta' = \max\{k: \tilde{J}_k\leq t_\alpha\}.
\]
Then, \eqref{measure_comparison} and Lemma~\ref{l_excursions_torus}
imply that (note that $\pi/2<3$)
\begin{equation}
\label{number_lastexcursions}
{\mathbb P}\Big[\zeta'-\zeta\geq \frac{3\sqrt{\ln n}}{\ln\ln n}\Big]
\leq \exp\Big(-c_9 \frac{\sqrt{\ln n}}{\ln\ln n}\Big).
\end{equation}
Recall the notation~$\delta_{n,\alpha}$ from the
beginning of the proof of this theorem.
We can write (recall~\eqref{df_N'a})
\begin{align*}
\lefteqn{ {\mathbb P}\Big[\xi_k=1 \text{ for all }k\leq
(1-\delta_{n,\alpha})\frac{2\alpha\ln^2 n}{\ln\ln n}
-\frac{3\sqrt{\ln n}}{\ln\ln n}\Big]}\\
&\geq
{\mathbb P}\Big[\widetilde{T}_n(A)>t_\alpha, N'_\alpha \geq (1-\delta_{n,\alpha})
\frac{2\alpha\ln^2 n}{\ln\ln n},
\zeta'-\zeta \leq\frac{3\sqrt{\ln n}}{\ln\ln n}\Big]
\end{align*}
so
\begin{align}
{\mathbb P}\big[\widetilde{T}_n(A)>t_\alpha\big] &\leq
{\mathbb P}\Big[\xi_k=1 \text{ for all }k\leq
(1-\delta_{n,\alpha})\frac{2\alpha'\ln^2 n}{\ln\ln n}
-\frac{3\sqrt{\ln n}}{\ln\ln n}\Big]
\nonumber\\
&\qquad + {\mathbb P}\Big[N'_\alpha < (1-\delta_{n,\alpha})
\frac{2\alpha\ln^2 n}{\ln\ln n}\Big]
+{\mathbb P}\Big[\zeta'-\zeta>\frac{3\sqrt{\ln n}}{\ln\ln n}\Big].
\label{main_upper}
\end{align}
Also,
\begin{align}
\lefteqn{{\mathbb P}\big[\widetilde{T}_n(A)>t_\alpha\big]}
\nonumber\\
&\geq
{\mathbb P}\Big[\xi_k=1 \text{ for all }k\leq
(1+\delta_{n,\alpha})\frac{2\alpha\ln^2 n}{\ln\ln n},
N_\alpha\leq (1+\delta_{n,\alpha})
\frac{2\alpha\ln^2 n}{\ln\ln n},
\nonumber\\
&\qquad\quad
\eta_k=1 \text{ for all }k\leq \frac{3\sqrt{\ln n}}{\ln\ln n},
\zeta'-\zeta \leq \frac{3\sqrt{\ln n}}{\ln\ln n},
{\widetilde X}_0\notin B\Big(\frac{n}{3\ln n}\Big)\Big]
\nonumber\\
&\geq {\mathbb P}\Big[\xi_k=1 \text{ for all }k\leq
(1+\delta_{n,\alpha})\frac{2\alpha\ln^2 n}{\ln\ln n}\Big]
- {\mathbb P}\Big[N_\alpha > (1+\delta_{n,\alpha})
\frac{2\alpha\ln^2 n}{\ln\ln n}\Big]
\nonumber\\
&\quad -
{\mathbb P}\Big[\Big(\eta_k=1 \text{ for all }k\leq
\frac{3\sqrt{\ln n}}{\ln\ln n}\Big)^\complement\Big]
-{\mathbb P}\Big[\zeta'-\zeta > \frac{3\sqrt{\ln n}}{\ln\ln n}\Big]
-O\Big(\frac{1}{\ln^2 n}\Big).
\label{main_lower}
\end{align}
Using~\eqref{hit_normalexcursion},
we obtain that the first terms in the right-hand
sides of~\eqref{main_upper}--\eqref{main_lower} are both
equal to
\[
\Big(1-\frac{\pi}{2}\mathop{\mathrm{cap}}(A)\frac{\ln\ln n}{\ln^2 n}(1+o(1))
\Big)^{\frac{2\alpha \ln^2 n}{\ln \ln n}}
= (1+o(1))\exp\big(-\pi\alpha\mathop{\mathrm{cap}}(A)\big).
\]
The other terms in the right-hand
sides of~\eqref{main_upper}--\eqref{main_lower} are~$o(1)$
due to~\eqref{cond_numb_exc}, \eqref{hit_lastexcursion}, and
\eqref{number_lastexcursions}.
Since, by~\eqref{df_h_transf},
\[
{\mathbb P}[\Upsilon_n A \subset U_{t_\alpha}^{(n)}
\mid 0\in U_{t_\alpha}^{(n)}]
={\mathbb P}\big[\widetilde{T}_n(A)>t_\alpha\big],
\]
the proof of Theorem~\ref{t_conditional} is concluded.
\end{proof}
\section*{Acknowledgements}
The authors are grateful to Caio Alves and Darcy Camargo for
pointing out to us that $\frac{1}{a}$ is a martingale for the
conditioned walk, and to David Belius and Augusto Teixeira
for useful discussions. Darcy Camargo also did the simulations
of the model presented on Figure~\ref{f_simulation}.
The authors thank the referees for their careful
reading of the paper and many valuable comments and suggestions.
Also, the authors thank
the financial support from Franco-Brazilian Scientific Cooperation
program.
S.P.\ and M.V. were partially supported by
CNPq (grants 300886/2008--0 and 301455/2009--0).
The last two authors thank FAPESP (2009/52379--8,
2014/06815--9, 2014/06998--6)
for financial support. F.C.\ is partially supported by
MATH Amsud program
15MATH01-LSBS.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,891 |
I just registered for the MAVT convention to keep up my certification for my degree. For the first time since going to college, I am not a practicing veterinary technician. I am not doing anything with animals—except being a simple cat owner. I don't even have any exotic pets this year—the first for me in eleven years. It's as if my past life with animals never even existed.
Two years ago, I wrote about this in a yet-to-be-published post called "Change". It was a hard-hitter to first realize that I was paying off a loan on a degree I was barely using at that time. It also bruised my pride to not be using my college education—at all.
Some good conversations with Father God that winter changed my perspective. I had perfect peace in the current life I was living. I knew he had called me to this exact position to be a better fit with church ministry, and I was grateful. And I am grateful.
Over and over, God and people have lovingly reminded me, "Rach, who you are is not what you do" (because I often forget). My current position allowed me to take a 10-day mission trip (when I only had 20 hours of vacation time at the time), my boss has consistently honored my request to have Tuesdays off to work at the church and has adjusted my schedule to allow for various church events and outreach opportunities over the past 2+ years.
Who I am is not dependent on whether I am currently using my degree—or if I ever use it again. Who I am has less to do with what I do and far more to do with who I am—who God is making me into. That is what matters to him, and that is what should matter to me too. The mundane, seemingly insignificant aspects of work matter to the degree they affect me—negatively or positively.
As my favorite author and missionary, Helen Roseveare, once wrote (on being whittled arrows in the hands of Jesus), "God can either use me or hide me. The choice is his." How I respond is up to me.
Jesus, you must increase and I must decrease (John 3:30). Help me more fully realize that I am dead and my life is hidden in yours (Col. 3:3, Gal. 2:20) and that you are the only and ultimate Treasure. I want my only boast to be you (Gal. 6:14), and that will never happen if degrees, education, promotions, and positions have a leading place in my heart and life.
He is better. So much better. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,638 |
{"url":"https:\/\/lesslikely.com\/statistics\/s-values\/","text":"# P-Values Are Tough and S-Values Can Help\n\nThe P-value doesn\u2019t have many fans. There are those who don\u2019t understand it, often treating it as a measure it\u2019s not, whether that\u2019s a posterior probability, the probability of getting results due to chance alone, or some other bizarre\/incorrect interpretation.13\n\nThen there are those who dislike it because they think the concept is too difficult to understand or because they see it as a noisy statistic we\u2019re not interested in.\n\nHowever, the groups of people mentioned above aren\u2019t mutually exclusive. Many who dislike and criticize the P-value also do not understand its properties and behavior.\n\n## What is a P-value Anyway?\n\n### Definitions\n\nThe P-value is the probability of getting a result (specifically, a test statistic) at least as extreme as what was observed if every model assumption, in addition to the targeted test hypothesis (usually a null hypothesis), used to compute it were correct.46\n\nKey assumptions are that randomization was employed (sampling, assignment, etc.), there are no uncontrolled sources of bias (programming errors, equipment defects, sparse-data bias) in the results, and the test hypothesis (often the null hypothesis) is correct. Some of these assumptions can be seen in the figure below from,7 which will be discussed more later below.\n\nWe assume all those assumptions to be correct (hence, we \u201ccondition\u201d on them, even though they are often not correct)7 when calculating the P-value, so that any deviation of the data from what was expected under those assumptions would be purely random error. But in reality such deviations could also be the result of assumptions being false, including but not limited to the test hypothesis. For example, in particle physics, neutrinos were found to be faster than light due to the resulting small test statistic\/P-value, but this result was later found to be a result of a loose fiber optic cable that introduced a delay in the timing system.\n\nSo the P-value cannot be the probability of one of these assumptions, such as \u201cthe probability of getting results due to chance alone.\u201d A statement like this is backwards because it\u2019s quantifying one of the assumptions behind the computations of a P-value.\n\nWe assumed this condition to be true (all deviations operating by random error) with several other things, when calculating the P-value, but this does not mean it is actually correct and the calculation of the P-value cannot be the probability of one of those assumptions. It is also worth clarifying that P-values are not probabilities of data, which many like to say to differentiate from probabilities of hypotheses. Rather, P-values are probabilities of \u201cdata features\u201d, such as test statistics (i.e.\u00a0a z-score or $$\\chi^{2}$$ statistic) or can be interpreted as the percentile at which the test statistic falls within the expected distribution for the test statistic assuming all the model assumptions are true.8\n\n### Properties\n\nThe P-value is a random variable and it\u2019s considered to be valid if it\u2019s well calibrated and meets the validity criterion of being uniform under the null hypothesis of no effect, where every value between 0 and 1 is equally likely (see the histogram below). Many frequentist statisticians do not consider P-values to be useful if they fail to meet this validity criterion, hence they do not recognize variants such as the posterior predictive P-values (which concentrate around values such as 0.5, rather than being uniform) to be valid. This validity criterion can also become a problem in certain scenarios such as adaptive clinical trials with repeated testing, where P-values may no longer become calibrated and require special methods to recalibrate them.\n\n## The Different Frameworks Accompanying P-values\n\nThe vast majority of researchers interpret the P-value in a dichotomous way such as being statistically significant or not depending on whether or not observed p (the realization of the random variable) falls below a fixed cutoff level (alpha, which is the maximum tolerable type-I error rate).9\n\nThis decision-making framework (Neyman-Pearson) may be useful in certain scenarios,10 where some sort of randomization is possible and where there is large control over the experimental conditions, with one of the most notable historical examples being Egon Pearson (son of Karl Pearson and coauthor of Jerzy Neyman) using it to improve quality control in industrial settings.\n\nOthers choose to interpret the P-value in a Fisherian way,11,12 as a continuous measure of evidence against the very test hypothesis and entire model (all assumptions) used to compute it (let\u2019s go with this for now, even though there are some problems with this interpretation, more on that below).\n\nThis interpretation as a continuous measure of evidence against the test hypothesis and the entire model used to compute it can be seen in the figure below from.7 In one framework (left panel), we may assume certain assumptions to be true (\u201cconditioning\u201d on them, i.e, use of random assignment), and in the other (right panel), we question all assumptions, hence the \u201cunconditional\u201d interpretation.\n\nThe interpretation of the P-value as a continuous measure of evidence against the test model that produced it shouldn\u2019t be confused with other statistics that serve as support measures. Likelihood ratios and Bayes factors are measures of evidence for a model compared to another model.1315\n\n## Compatibilism To The Rescue\n\nThe P-value is not a measure of evidence for a model (such as the null\/alternative model), it is a continuous measure of the compatibility of the observed data with the model used to compute it.3\n\nIf it\u2019s high, it means the observed data are very compatible with the model used to compute it. If it\u2019s very low, then it indicates that the data are not very compatible with the model used to calculate it, and this low value may be due to random variation and\/or it may be due to a violation of assumptions (such as the null model not being true, not using randomization, a programming error or equipment defect such as that seen with neutrinos, etc.).\n\nLow compatibility of the data with the model can be implied as evidence against the test hypothesis, if we accept the rest of the model used to compute the P-value. Thus, lower P-values from a Fisherian perspective are seen as stronger evidence against the test hypothesis given the rest of the model.\n\n## Many Criticisms Don\u2019t Hold Up\n\nIf we treat the P-value as nothing more or less than a continuous measure of compatibility of the observed data with the model used to compute it (observed p), we won\u2019t run into some of the common misinterpretations such as \u201cthe P-value is the probability of a hypothesis\u201d, or the \u201cprobability of chance alone\u201d, or \u201cthe probability of being incorrect\u201d.3\n\nThus, many of the \u201cproblems\u201d commonly associated with the P-value are not due to the actual statistic itself, but rather researchers\u2019 misinterpretations of what it is and what it means for a study.\n\nThe answer to these misconceptions is compatibilism, with less compatibility (smaller P-values) indicating a poor fit between the data and the test model and hence more evidence against the test hypothesis.\n\nA P-value of 0.04 means that assuming that all the assumptions of the model used to compute the P-value are correct, we won\u2019t get data (a test statistic) at least as extreme as what was observed by random variation more than 4% of the time.\n\nTo many, such low compatibility between the data and the model may lead them to reject the test hypothesis (the null hypothesis).\n\n### Conceptual Mismatch With Direction\n\nIf you recall from above, I wrote that the P-value is seen by many as being a continuous measure of evidence against the test hypothesis and model. Technically speaking, it would be incorrect to define it this way because as the P-value goes up (with the highest value being 1 or 100%), there is less evidence against the test hypothesis since the data are more compatible with the test model. 1 = perfect compatibility of the data with the test model.\n\nAs the P-value gets lower (with the lowest value being 0), there is less compatibility between the data and the model, hence more evidence against the test hypothesis used to compute p.\n\nThus, saying that P-values are measures of evidence against the hypothesis used to compute them is a backward definition. This definition would be correct if higher P-values inferred more evidence against the test hypothesis and vice versa.\n\n### Scaling\n\nAnother problem with P-values and their interpretation is scaling. Since the statistic is meant to be a continuous measure of compatibility (and evidence against the test model + hypothesis), we would hope that differences between P-values are equal (on an additive scale), as this makes it easier to interpret.\n\nFor example, the difference between 0 and 10 dollars is the same as the difference between 90 and 100 dollars. This makes it easy to think about and compare across various intervals.\n\nUnfortunately, this doesn\u2019t apply to the P-value because it is on the inverse-exponential scale. The difference between 0.01 and 0.10 is not the same as the difference between 0.90 and 0.99.\n\nFor example, with a normal distribution (above), a z-score of 0 results in a P-value of 1 (perfect compatibility). If we now move to a z-score of 1, the P-value is 0.31. Thus, we saw a dramatic decrease from a P-value of 1 to 0.31 with one z-score. A 0.69 difference in the P-value.\n\nNow let\u2019s go from a z-score of 1 to a z-score of 2. We saw a difference of 0.69 with the change in one z-score before, so the new P-value must be 0.31 - 0.69 = -0.38 right? No.\u00a0The P-value for a z-score of 2 is 0.045. The P-value for a z-score of 3 is 0.003. Even though we\u2019ve only been moving by one z-score at a time, the changes in P-values don\u2019t remain constant; they become smaller and smaller.\n\nThus, the difference between the P-values of 0.01 and 0.10 in terms of z-scores is substantially larger than the difference between 0.90 and 0.99.\n\nAgain, this makes it difficult to interpret as a statistic across the board, especially as a continuous measure. This can further be seen in the figure from Rafi & Greenland (2020).\n\n## S-values as Cognitive Support Aids\n\nThe issues described above such as the backward definition and the problem of scaling can make it difficult to conceptualize the P-value as being an evidence measure against the test hypothesis and test model. However, these issues can be addressed by taking the negative log of the P-value $$\u2013\\log_{2}(p)$$ , which yields something known as the Shannon information value or surprisal (s) value,6,16 named after Claude Shannon, the father of information theory.17\n\nUnlike the P-value, this value is not a probability but rather a continuous measure of information in bits against the test hypothesis and is taken from the observed test statistic computed by the test model.\n\nIt also provides a highly intuitive way to think about P-values. Imagine that the variable k is always the nearest integer to the calculated value of s. Now, take for example a P-value of 0.05, the S-value for this would be s = $$\u2013\\log_{2}(0.05)$$ which equals 4.3 bits of information embedded in the test statistic, which can be used as evidence against the test hypothesis.\n\nHow much evidence is this? k can help us think about this. The nearest integer to 4.3 is 4. Thus, the data which yield a P-value of 0.05 which results in an s value of 4.3 bits of information is no more surprising than getting all heads in 4 fair coin tosses.\n\nLet\u2019s try another example. Let\u2019s say our study gives us a P-value of 0.005, which would indicate to many very low compatibility between the test model and the observed data; this would yield an s value of $$\u2013\\log_{2}(0.005) = 7.6$$ bits of information. k which is the closest integer to s would be 8. Thus, the data which yield a P-value of 0.005 are no more surprising than getting all heads on 8 fair coin tosses.\n\nUnlike the P-value, the S-value is more intuitive as a measure that provides evidence against the test hypothesis since its value (information against the test hypothesis) increases with less compatibility, whereas it is the opposite for the P-value.\n\n## Examples\n\nLet\u2019s try using some data to see this in action. I\u2019ll simulate some random data in R from a uniform distribution with the following code,\n\nGroupA <- runif(10, 0, 20)\n\nGroupB <- runif(10, 0, 20)\n\n(RandomData <- data.frame(GroupA, GroupB))\n## GroupA GroupB\n## 1 8.853960 18.971568\n## 2 17.854739 12.961634\n## 3 3.564422 14.556913\n## 4 4.957992 5.925018\n## 5 19.805255 4.247317\n## 6 2.596731 9.993899\n## 7 16.455488 16.568982\n## 8 2.205597 6.130026\n## 9 10.548758 12.732344\n## 10 18.777923 17.928943\n\nWe can plot the data and also run an independent samples t-test.\n\nLooks interesting. We can obviously see some differences from the graph. Here\u2019s what our test output gives us,\n\nWelch Two Sample t-test\n\ndata: GroupA and GroupB\n\nt = 1.358, df = 14.856, p-value = 0.1947\n\nalternative hypothesis: true difference in means is not equal to 0\n\n95 percent confidence interval:\n\n-2.137637 9.627015\n\nsample estimates:\n\nmean of GroupA mean of GroupB\n\n10.258502 6.513812\n\nOkay, we cannot reject the test hypothesis (the null hypothesis) at the 5% level and the confidence interval is ridiculously wide. How can I interpret this P-value of 0.1947 more intuitively?\n\nLet\u2019s convert it into an S-value (here\u2019s a calculator I constructed that converts P-values into S-values).\n\n$\u2013\\log_2(0.1947) = 2.36$\n\nS-value= 2.36\n\nThat is 2.36 bits of information against the null hypothesis.\n\nHow would we interpret it within the context of a given confidence interval? The S-value tells us that values within the computed 95% CI: (-2.13, 9.62) have at most 4.3 bits of information against them.\n\nRemember, k is the nearest integer to the calculated value of s and in this case, would be 2.\n\nSo these results (the test statistic) are as surprising as getting all heads in 2 fair coin tosses. Not that surprising.\n\nThe S-value is not meant to replace the P-value, and it isn\u2019t superior to the P-value. It is merely a logarithmic transformation of it that rescales it on an additive scale and tells us how much information is embedded in the test statistic and can be used as evidence against the test hypothesis.\n\nIt is a useful cognitive device that can help us better interpret the information that we get from a calculated P-value.\n\nI\u2019ve constructed a calculator that converts observed p-values into s-values and provides an intuitive way to think about them.\n\nFor a more detailed discussion of S-values, see these articles, in addition to the references below them:\n\n\nCole, S. R., Edwards, J. K., and Greenland, S. (2020), \u201cSurprise!,\u201d American Journal of Epidemiology. https:\/\/doi.org\/10\/gg63md.\n\nRothman, K. J. (2020), \u201cTaken by Surprise,\u201d American Journal of Epidemiology. https:\/\/doi.org\/10\/gg63mf.\n\nRafi, Z., and Greenland, S. (2020), \u201cSemantic and Cognitive Tools to Aid Statistical Science: Replace Confidence and Significance by Compatibility and Surprise,\u201d arXiv:1909.08579 [stat.ME].\n\nGood, I. J. (1956), \u201cThe surprise index for the multivariate normal distribution,\u201d The Annals of Mathematical Statistics, 27, 1130\u20131135. https:\/\/doi.org\/10.1214\/aoms\/1177728079.\n\nBayarri, M. J., and Berger, J. O. (1999), \u201cQuantifying Surprise in the Data and Model Verification,\u201d *Bayesian Statistics*, 6, 53\u201382.\n\nShannon, C. E. (1948), \u201cA mathematical theory of communication,\u201d The Bell System Technical Journal, 27, 379\u2013423. https:\/\/doi.org\/10.1002\/j.1538-7305.1948.tb01338.x.\n\nGreenland, S. (2019), \u201cValid P-values behave exactly as they should: Some misleading criticisms of P-values and their resolution with S-values,\u201d The American Statistician, 73, 106\u2013114. https:\/\/doi.org\/10.1080\/00031305.2018.1529625.\n\n\nAcknowledgment: The analogies and concepts in this blog can be attributed to Sander Greenland and his works (many of which are referenced below) and I thank him for his extensive commentary and corrections on several versions of this article.\n\n### References\n\n1. Gigerenzer G. Statistical Rituals: The Replication Delusion and How We Got There. Advances in Methods and Practices in Psychological Science. 2018;1(2):198-218. doi:10.1177\/2515245918771329\n\n2. Goodman S. A dirty dozen: Twelve p-value misconceptions. Semin Hematol. 2008;45(3):135-140. doi:10.1053\/j.seminhematol.2008.04.003\n\n3. Greenland S, Senn SJ, Rothman KJ, et al. Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. Eur J Epidemiol. 2016;31(4):337-350. doi:10.1007\/s10654-016-0149-3\n\n4. Rafi Z, Greenland S. Semantic and cognitive tools to aid statistical science: Replace confidence and significance by compatibility and surprise. arXiv:190908579 [stat]. July 2020. http:\/\/arxiv.org\/abs\/1909.08579.\n\n5. Greenland S, Senn SJ, Rothman KJ, et al. Statistical tests, P values, confidence intervals, and power: A guide to misinterpretations. European Journal of Epidemiology. 2016;31(4):337-350. doi:10.1007\/s10654-016-0149-3\n\n6. Greenland S. Valid P-values behave exactly as they should: Some misleading criticisms of P-values and their resolution with S-values. The American Statistician. 2019;73(sup1):106-114. doi:10.1080\/00031305.2018.1529625\n\n7. Greenland S, Rafi Z. To Aid Scientific Inference, Emphasize Unconditional Descriptions of Statistics. arXiv:190908583 [stat]. July 2020. http:\/\/arxiv.org\/abs\/1909.08583.\n\n8. Perezgonzalez JD. P-values as percentiles. Commentary on: \u201cNull hypothesis significance tests. A mix\u2013up of two different theories: The basis for widespread confusion and numerous misinterpretations\u201d. Frontiers in Psychology. 2015;6. doi:10.3389\/fpsyg.2015.00341\n\n9. Neyman J, Pearson ES. IX. On the problem of the most efficient tests of statistical hypotheses. Philos Trans R Soc Lond A. 1933;231(694-706):289-337. doi:10.1098\/rsta.1933.0009\n\n10. Lakens D, Adolfi FG, Albers CJ, et al. Justify your alpha. Nature Human Behaviour. 2018;2(3):168-171. doi:10.1038\/s41562-018-0311-x\n\n11. Fisher RA. The Design of Experiments. Oxford, England: Oliver & Boyd; 1935.\n\n12. Fisher R. Statistical Methods and Scientific Induction. J R Stat Soc Series B Stat Methodol. 1955;17(1):69-78.\n\n13. Jeffreys H. Some Tests of Significance, Treated by the Theory of Probability. Math Proc Cambridge Philos Soc. 1935;31(2):203-222. doi:10.1017\/S030500410001330X\n\n14. Jeffreys H. The Theory of Probability. OUP Oxford; 1998.\n\n15. Royall R. Statistical Evidence: A Likelihood Paradigm. CRC Press; 1997.\n\n16. Amrhein V, Trafimow D, Greenland S. Inferential statistics as descriptive statistics: There is no replication crisis if we don\u2019t expect replication. The American Statistician. 2019;73(sup1):262-270. doi:10.1080\/00031305.2018.1543137\n\n17. Shannon CE. A mathematical theory of communication. The Bell System Technical Journal. 1948;27(3):379-423. doi:10.1002\/j.1538-7305.1948.tb01338.x\n\n\u2022 Cite this blog post","date":"2020-09-24 15:56:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8003844022750854, \"perplexity\": 1058.9733865185976}, \"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-2020-40\/segments\/1600400219221.53\/warc\/CC-MAIN-20200924132241-20200924162241-00726.warc.gz\"}"} | null | null |
Home › Reviews › In Theaters › Simon McQuoid's "Mortal Kombat" is not a flawless victory.
Simon McQuoid's "Mortal Kombat" is not a flawless victory.
By Douglas Davidson on April 22, 2021 • ( 1 )
Debuting in 1992, Ed Boon and John Tobias's arcade game Mortal Kombat shook the foundation of popular kulture almost immediately. It wasn't just the karacter design (digitized versions of real people known as "sprites") or the in-game mythos, but the ability to do real damage against your opponents, including brutal fatalities. If not for the row among parents as to the long-term effects of playing violent video games (including kongressional hearings) making the game a pariah and thus making it more enticing to teens, it's possible that Mortal Kombat would never have become the lofty series it is now. The game has gone on to add 10 sequels, several spin-off games, a television series, komics, two live-action movies, two animated features, and a web series. Now, 29 years after the debut of the original arcade game and 26 years since the first live action film hit theaters, the fighters of Earthrealm gather together once more to prevent Outworld from invading in first-time feature director Simon McQuoid's Mortal Kombat. It's obvious from the staging of each fight that McQuoid understands the visual language of the now-beloved franchise, but that doesn't mean it's a flawless victory. Nailing the violence is only half the battle of a Mortal Kombat story; you need to understand the karacters who drive it, something which this film struggles to do in its race to cultivate bloody karnage.
L-R: Joe Taslim as Sub-Zero/Bi-Han and Hiroyuki Sanada as Scorpion/Hanzo Hasashi in New Line Cinema's action adventure MORTAL KOMBAT, a Warner Bros. Pictures release.
**Prepare for Mortal Kombat.**
Long ago, a group of deities known as The Elder Gods split existence into a series of realms which kould only be konverged through victory in kombat. In order to ensure that one realm couldn't forcibly invade another, a tournament (kalled Mortal Kombat) was kreated wherein one realm had to best another 10 consecutive times in order to klaim and merge with it. With nine wins and a 10th tournament on the horizon, Outworld emissary Shang Tsung (Chin Han) sends his minions to secretly murder the champions of Earthrealm, thereby securing victory before the tournament begins. In the midst of all this stands Cole Young (Lewis Tan), an unknown mixed-martial arts fighter who may be the key to Earthrealm's salvation.
L-R: Tadanobu Asano as Lord Raiden and Chin Han as Shang Tsung in New Line Cinema's action adventure MORTAL KOMBAT, a Warner Bros. Pictures release.
**And now, for a taste of things to come.**
Unlike most EoM reviews, this one may kontain light spoilers in order to address certain aspects. For those familiar with the Mortal Kombat series, nothing that komes will surprise, though it may shift expectations. If you're largely unaware of the Mortal Kombat Universe, watch the film then kome back.
Lewis Tan as Cole Young in New Line Cinema's action adventure MORTAL KOMBAT, a Warner Bros. Pictures release.
**Let Mortal Kombat Begin!**
Next to the sprite karacter design, what made Mortal Kombat standout against its arcade kompetitors is the absolute karnage kombatants kan deal unto each other. In the beginning, this meant blood visibly flying off a recently hurt karacter and a violent end to the best of three loser. In recent iterations, the gore factor was amplified thanks to the spontaneous use of attacks kalled X-rays which would present kombatants literally breaking the bones of their opposition. While the violence was present in the '95 Mortal Kombat and its live-action sequel, McQuoid's adaptation is the first one to nail the tone, look, and feel of the violence which is a staple of the series. In the opening sequence, (made available to view online before the movie premiered), set in 17th century Japan, Hiroyuki Sanada's Hanzo Hasashi (more widely known as Scorpion) dispatches several aggressors using a make-shift version of his trademark kunai, a throwing knife attached to a rope or chain. With extreme precision and prejudice, Hanzo slings the kunai through body parts, even katching one opponent in the head, using the rope to yank the individual to the group, head first, before pulling the kunai back to him. While not gratuitous, the violence is bloody and final. There is weight in the kombat, setting up that what komes for the remainder of the runtime employs a similar finality with each kontest. Fans of the series, new and old, will undoubtedly be tickled to see the karacters employing their special attacks with quite a bit of ingenuity, as well as with cinematography designed to invoke the same visceral feeling from the game. Frankly, this is the truest victory for McQuoid's Mortal Kombat. For the first time since the marvelous web series Mortal Kombat: Legacy, fans get the brutal, bone-krunching brawls they krave.
Jessica McNamee as Sonya Blade in New Line Cinema's action adventure MORTAL KOMBAT, a Warner Bros. Pictures release.
Kredit is also due to stunt koordinator Kyle Gardiner (Godzilla vs. Kong) and fight koreographer Chan Griffin (Shazam!) for designing fight sequences that feel authentic to the karacters, paying respect to their respective talents without overclocking or underpowering any of them. Same goes to the actors as each one is entirely believable, something which the opposite kan tarnish any martial arts-based film. It should surprise no-one familiar with the work history of Joe Taslim (The Raid, The Night Comes for Us, Warrior) that he's a natural for the icy villain Sub-Zero or the same for Sanada as the firey Scorpion. Their individual presence on-screen is magnetic, their fight scenes are the best of the entire film, filled with passion and pain. Rounding out the central kast is McNamee, Brooks, Tan, and Josh Lawson (House of Lies) as Black Dragon leader Kano, who make the most of their screentime. Though the film positions Tan's Cole as the audience surrogate and puts him at the center of the narrative, it's McNamee, Brooks, and Lawson who steal the film from under him. It's not that Cole isn't interesting or that Tan isn't kapable, it's that the narrative has too many karacters to establish quickly to do much more than a kursory introduction before karrying on. (We'll get more into that in a moment.) The real surprise with the kasting is Lawson who is the best thing of the entire film. Yep, even better than the pitch-perfect execution of the karacter's signature moves is Lawson's unexpected darkly komic line-delivery. For those who have seen the '95 Mortal Kombat, think Trevor Goddard's performance minus the misogyny. Whatever you think of the performances, though, each of the kast members appears equal to the task in representing the mythical Kombat karacters in terms of look and ferocity. In terms of the action, in most kases, the direction enables the audience to see the respective battles easily, keeping enough distance to make the action easy to track and editing kuts in so that the fisticuffs become more engaging. In the opening sequence, for instance, there's a rhythm to the kuts so that there are wide-klose up-wide-klose up in-sync with the blows. The kut timing adds a layer to the action so that we almost feel the impact of each blow. Later, however, the editing has moments which decrease the tension to a surprising measure as trying to balance more than one immediate konfrontation reduces the immediacy and flow of battle as the kamera tries to show off each engagement equally.
L-R: Mehcad Brooks as Major Jackson "Jax" Briggs and Joe Taslim as Sub-Zero/Bi-Han in New Line Cinema's action adventure "Mortal Kombat," a Warner Bros. Pictures release.
Here's where we really get into the issue with McQuoid's Mortal Kombat and it all comes from script from first-time feature writer Greg Russo and David Callaham (Wonder Woman 1984), based on a story by Oren Uziel (22 Jump Street) and Russo. In the entire franchise, with a few exceptions, Liu Kang (Ludi Lin) is the hero of the tale and he's been shifted in favor of a franchise newbie, Cole. Instead, Liu becomes more a mentor figure and, without getting into details, is adapted in such a way that may seem like an insult to longtime Kombat fans, especially when Liu offers his backstory, better explaining how he joined Earthrealm protector Lord Raiden's (Tadanobu Asano) Order of Light and his relation to fellow fighter Kung Lao (Max Huang). Then there's Special Forces-trained Sonya Blade (Jessica McNamee) and Jax (Mehcad Brooks) who are introduced early in the present-day portion of the film, but who are mostly sidelined in favor of pushing forward Cole. This is partially due to establishing early the deadliness of Sub-Zero, finding a quick way to get Jax to his signature cybernetic arms, and the script's approach to champion selection: a dragon birthmark. The fact that Blade, one of the few female karacters, is often treated as less-than or unworthy by everyone other than Jax and Cole is bound to frustrate and the optics of the sole main Black karacter being given a more dour path to heroism than his storymates is not good. Is it more plausible in a grounded way given his injuries? Yes. Does it give Brooks a chance to shine a bit as an actor? Also yes. But when other karacters mostly laugh off injuries, it feels like a disservice to Jax, a karacter who's an extreme badass. Lastly, there's Raiden, played with the appropriate gravitas by Asano, who the film kan't decide if he's an all-powerful, observant Elder God or if he's ineffectual at best. It just doesn't know what to do with him within the script and, therefore, feels entirely inconsistent in use and presentation. This gets to the heart of the issues with Mortal Kombat: the script seems to be designed, not for flow, function, or sense, but to set up fights and, potentially, a sequel. Rather than use its less than two-hour runtime to allow its karacters to feel real in the hyperreal circumstance, to allow the emotional elements within the story to generate any kind of foothold, the script scampers off to another fight. Say what you will about the '95 Mortal Kombat, but writer Kevin Droney (Highlander TV series) understood that getting to know the karacters set up stakes so that the fights, ridiculous as they are, kontained weight. In this iteration, the fights are the centerpiece but are, mostly, meaningless, especially with the way that the film sets up the events of the story as the run-up to the 10th tournament between realms.
L-R: Ludi Lin as Liu Kang and max Huang as Kung Lao in New Line Cinema's action adventure MORTAL KOMBAT, a Warner Bros. Pictures release.
If all you want out of your Mortal Kombat experience is strong martial arts koupled with kopious gore and viscera, then you're going to dig the film. On this, it delivers and does so well. It's obvious from the finished product that McQuoid understands the visual language of the source and delivers that with aplomb; it's just that the script (a) struggles to make us kare about the karacters, (b) heavily relies on exposition from scene-to-scene, (c) lacks the necessary weight to make anything feel urgent, and (d) refuses to acknowledge the built-in history, resulting in a film that often insults the source material. That said, it's infinitely better than 1997's Mortal Kombat: Annihilation and will (almost to a certainty) make you want to boot up your konsole and start your own in-home tournament right after. In this way, the essence of Mortal Kombat is honored, even if it's not totally respected.
In select theaters April 23rd, 2021.
Available for streaming for 31 days on HBO Max beginning April 23rd, 2021.
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Categories: In Theaters, Reviews, streaming
Tags: action, adaptation, Angus Sampson, Benjamin Wallfisch, Chan Griffin, Chin Han, Damon Herriman, Daniel Nelson, Dave Callaham, Ed Boon, fantasy, Greg Russo, Hiroyuki Sanada, Ian Streetz, Jessica McNamee, Joe Taslim, John Tobias, Josh Lawson, Kris McQuade, Kyle Gardiner, Laura Brent, Lewis Tan, Ludi Lin, martial arts, Matilda Kimber, Max Huang, Mehcad Brooks, Mel Jarnson, Mortal Kombat, Nathan Jones, NetherRealm Studios, New Line Cinema, Oren Uziel, Simon McQuoid, Sisi Stringer, supernatural, Tadanobu Asano, video game, Warner Bros. Pictures
The prophecy complete and the tournament on the horizon, it must mean 2021's "Mortal Kombat" is out on home video. – Elements of Madness | {
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\section{Introduction}
\begin{figure}
\includegraphics[width=0.5 \textwidth]{magnetogram_a}
\includegraphics[width=0.5 \textwidth, bb= 40 20 670 500, clip = true]
{fieldlines_mhs_b}
\caption{Panel a: SUNRISE/IMax magnetogram of a quiet Sun area. The black
rectangular marks the region of interest.
Panel b: Sample field lines for a MHS-model.}
\label{sunrise_mag_full}
\end{figure}
While the corona, at least above active regions,
has a low plasma $\beta$ and is usually modelled by the assumption
of a vanishing Lorentz-force
\citep[see][for an overview of solar force-free fields]{2012LRSP....9....5W},
this is not true in the lower solar atmosphere
\citep[see][for a recent review on magnetic fields in the solar
atmosphere]{2014A&ARv..22...78W}. In the photosphere and lower chromosphere
low and high $\beta$ regions exist side by side and non-magnetic forces
have to be taken into account, to lowest order with a magneto-static model,
where the Lorentz-force is compensated by the gradient of the plasma
pressure and the gravity force.
The most accurate measurements of the
solar magnetic field are available in the photosphere. In active regions
the full magnetic vector can be measured accurately, e.g. with
SDO/HMI, whereas in quiet Sun regions only the line-of-sight or vertical
field is available with sufficient accuracy for a reliable extrapolation,
because in weak field regions there is too much uncertainty in the
transverse field components
\citep[Noise in the Stokes vector translate into an
uncertainty in the inferred values for the magnetic
field, see][]{2011A&A...527A..29B,2012A&A...547A..89B}.
These photospheric measurements are
extrapolated into the solar atmosphere under certain model assumptions, here
a magneto-static approach. The vertical resolution of the model scales
with the horizontal resolution of the photospheric measurements, e.g. about
1400 km for SOHO/MDI-magnetograms and 350 km for SDO/HMI. As the
non-force-free layer containing the photosphere and lower chromosphere is
rather thin (typically less than 2000 km), one can hardly resolve
magnetic structures here for models using SOHO/MDI- or SDO/HMI-magnetograms
as boundary condition. The high resolution
magnetograms from {\sc SUNRISE}/IMaX with a pixel size of only 40 km
allow now to model this layer vertically with about 50 points.
A special class of magneto-static solutions, which allow separable
solutions has been proposed by \cite{low91}. An advantage of this
approach is that the resulting equations are linear
\citep[for nonlinear cases, see][]{neukirch97} and
can be solved effectively by a Fourier transformation
or a Green's function implementation
\citep[see][]{petrie:etal00}.
Separable and linear solutions have been found also
in spherical \citep{bogdan:etal86,neukirch95,al_sphere10}
as well as in cylindrical coordinates \citep[][]{neukirch09,al_cylinder10}.
Especially the solutions found in spherical coordinates have been used
for modelling the global magnetic field of the Sun
\citep[e.g.][]{bagenal:etal91,gibson:etal95,gibson:etal96,zhao:etal00,ruan:etal08}
and other stars \citep[e.g.][]{lanza08,lanza09}.
Usually these models require only the line-of-sight or vertical
photospheric magnetic field as boundary condition and the solutions
contain free parameters and/or free functions.
Nonlinear magneto-static solutions are more demanding numerically
and observationally, because they require photospheric
vector magnetograms as input
\citep[see][for a cartesian and spherical implementation,
respectively]{wiegelmann:etal06,wiegelmann:etal07}.
Within this work we apply the linear magneto-static solutions
proposed by \cite{low91} to a high-resolution magnetogram observed with
{\sc SUNRISE}/IMaX. We outline the paper as follows. In section
\ref{sec:basics} we briefly discuss the basic equations and
model assumptions. Section \ref{sec:data} describes the employed photospheric
magnetograms, which we use as boundary condition for our magneto-static
model in section \ref{sec:results}. In section
\ref{sec:outlook} we finally discuss the prospects and
limitations of this approach and give an outlook for a generalization
of the method towards a non-linear numerical approach.
\section{Basic equations}
\label{sec:basics}
We use the magneto-hydro-static equations
\begin{eqnarray}
{\bf j}\times{\bf B} & = & \nabla P +\rho \nabla \Psi,
\label{forcebal}\\
\nabla \times {\bf B } & = & \mu_0 {\bf j} , \label{ampere} \\
\nabla\cdot{\bf B} & = & 0, \label{solenoidal}
\end{eqnarray}
where ${\bf B}$ is the magnetic field,
${\bf j}$ the electric current density,
$P$ the plasma pressure,
$\rho$ the mass density, $\Psi$ the gravitational potential
and $\mu_0$ the permeability of free space.
To find separable solutions for this set of equations, we
apply the following ansatz
for the electric current density \citep[see][for details]{low91}.
\begin{equation}
\nabla \times {\bf B } = \alpha_0 {\bf B } + f(z) \nabla B_z \times {\bf e_z},
\label{lin_mhs}
\end{equation}
where $\alpha_0$ is the force-free parameter and $f(z)$ is a free function,
which controls the non-magnetic forces.
The first part $\alpha_0 {\bf B }$ corresponds to a field-line-parallel
linear force-free current and the second term
$f(z) \nabla B_z \times {\bf e_z}$ defines a current perpendicular to
the gravitational force
(in the $z$-direction) or, in other words, parallel to the Sun's surface
$(x,y)$.
It is then possible to reduce the MHS equations to a single partial
differential equation
\citep[see e.g.][for a particularly simple formulation]{neukirch:etal99}
that can often be solved by separation of variables.
For convenience we use here \citep[as proposed in][]{low91}
\begin{equation}
f(z)= a \exp(-\kappa z),
\label{lin_mhs_2}
\end{equation}
with a free parameter $a$, which controls the non-magnetic forces in the
photosphere. Obviously, for the choice $a=0$, this approach reduces
to linear force-free fields. Above a certain height in the solar
atmosphere one expects that
the solution becomes approximately force-free, owing to the low
plasma $\beta$ in the solar corona. Consequently $f(z)$ has to decrease
with height and here we choose as a scale height
the distance of the upper chromosphere above the solar surface,
leading to $1/\kappa = 2 Mm$.
With $\kappa$ fixed, our MHS-solution contains two free parameters, $\alpha$
and $a$.
Let us remark that $\kappa$ in equation (\ref{lin_mhs_2})
controls the non-magnetic forces and should not be confused with
the scale height of the plasma pressure.
As boundary conditions we use the measured vertical magnetic field
$B_z(x,y,0)$ in the photosphere. We solve the equations by means of
a Fast-Fourier-Transform method similar to
the linear force-free model developed by
\cite{alissandrakis81}. Different from the linear force-free approach
is that the resulting Schr\"odinger equation for $B_z$ in the Fourier
space contains a Bessel function instead of an exponential function
One finds the following solution for pressure and density
\citep[see][for the derivation]{low91}
\begin{eqnarray}
P & = & \; P_0(z) \; -\frac{1}{2 \mu_0} f(z) B_z^2, \label{pressure} \\
\rho & = & - \frac{1}{g} \frac{d P_0}{dz}
+\frac{1}{\mu_0 g} \left[\frac{d f}{dz}\frac{B_z^2}{2}
+f \, ({\bf B } \cdot \nabla) B_z \right].
\end{eqnarray}
The first term in equation (\ref{pressure}) contains a 1D-solution
(in z-direction), which is independent of the magnetic field and
has to obey $\nabla P = -\rho \nabla \Psi$. The second term
is the disturbance of this 1D-pressure profile by the magnetic field. Here
pressure and density compensate the non-vanishing Lorentz-force.
This disturbance is negative (if $a > 0$), and obtains its highest absolute values
in regions of the highest vertical magnetic field strength $B_z$.
Because the total plasma pressure (sum of both terms) has to
be positive, we get the following in-equality for $P_0(z)$
\begin{equation}
P_0(z) > a \cdot \exp(-k z) \cdot \frac{{\rm Max}(B_z)^2}{2 \mu_0}(z),
\label{P0}
\end{equation}
where $\frac{{\rm Max}(B_z)^2}{2 \mu_0}(z)$ is the maximum at a given height
$z$.
As we will see later, this condition has severe consequences for an
application to data with strong locally enhanced magnetic fields in
the photosphere. To satisfy condition (\ref{P0}) in these regions, the
background pressure $P_0$ has to be so high that the plasma $\beta$ in
weak-field regions (and also on average) becomes unrealistically high,
see Fig. \ref{beta}.
Within this limitation, the choice of $P_0(z)$ has some freedom.
Our choice is given in section \ref{sec4.1}.
\section{Data}
\label{sec:data}
We apply our newly developed code to photospheric magnetic field
measurements taken with the balloon-borne {\sc SUNRISE}
solar observatory in June 2009.
For an overview of the {\sc SUNRISE} mission
and scientific highlights of the first SUNRISE flight
see
\cite{solanki:etal10}, \cite{2011SoPh..268....1B},
\cite{2011SoPh..268..103B}, \cite{2011SoPh..268...35G}.
For a description of the IMaX instrument, we refer to
\cite{martinezpillet:etal11}. The photospheric magnetic field was
computed by inverting the IMaX measurements using the VFISV code as
described in \cite{borrero:etal11}.
The linear force-free extrapolation code, and the particular
case of an $\alpha=0$ potential field has been applied to data
from {\sc SUNRISE}/IMaX before for a single magnetogram by
\cite{wiegelmann:etal10} and to analyse a time series
by \cite{wiegelmann:etal13}.
\cite{chitta14} carried out non-linear force-free
extrapolations from IMaX magnetograms and added vertical
flows at low heights to simulate non-force-free effects
in the photosphere and chromosphere.
Here we apply our newly developed linear MHS-code to a snapshot
of the quiet Sun, observed with {\sc SUNRISE}/IMaX as well. We apply
our code first to the full field-of-view of IMaX, as shown in
Fig. \ref{sunrise_mag_full}
and in a subsequent step
we investigate a subfield (marked with a black rectangular in
Fig. \ref{sunrise_mag_full}) in more detail.
The data set used here was observed in a period of $1.616$ hours starting
at 00:00 UT on 2009 June 9th (image 220 from this series),
see \cite{martinezpillet:etal11}.
\section{Results}
\label{sec:results}
\begin{figure}
\includegraphics[width=0.5 \textwidth,height=4cm]{dP_a}
\includegraphics[width=0.5 \textwidth,height=4cm]{dP_mini_b}
\includegraphics[width=0.5 \textwidth,height=7cm]{beta_cont_c}
\caption{The plasma pressure disturbance
$-\frac{1}{2 \mu_0} f(z) B_z^2$ at a height $z=1 Mm$ for the full IMaX
and the small FOV in panel a) and b), respectively.
Panel c) shows an equi-contour surface for $\beta=1$ in the
small FOV. }
\label{pic_pressure}
\end{figure}
\subsection{Application to the full IMaX-FOV.}
\label{sec4.1}
In our first computation we apply our model to the full
phase-diversity reconstructed IMaX magnetogram of
a quiet Sun region of $37 \times 37$ Mm, which has been resolved
by $936 \times 936$ pixels (pixel size on Sun 40 km),
see Fig. \ref{sunrise_mag_full},
As our main interest lies in the mixed
plasma $\beta$ regions of the photosphere and chromosphere, we extrapolate
up to a height of $z=4$ Mm or 100 pixels. A few sample field lines for a magneto-static
solution with $\alpha=3.0$ and $a=0.5$ is shown in
in Fig. \ref{sunrise_mag_full} b).
In Fig. \ref{pic_pressure} a) we show the pressure
disturbance in the chromosphere at the height $z= 1 {\rm Mm}$
as calculated with the second term
$-\frac{1}{2 \mu_0} f(z) B_z^2$ on the right-hand-side
of Eq. (\ref{pressure}). This term obviously becomes
largest above regions with the highest photospheric field strength,
as seen in the large negative peaks. The total pressure has to be positive
of course and consequently a lower bound for the 1D-background pressure $P_0$
is given by Eq. (\ref{P0}). $P_0$ describes a 1D-equilibrium between
the gravity force and the vertical pressure gradient. One has to solve:
\begin{equation}
\frac{d P_0(z)}{d z} = -g \rho(z)
\end{equation}
for a constant gravity $g$.
Assuming an equation of state of the form
$
P_0=\rho R T
$
we get
\begin{equation}
\frac{d P_0(z)}{d z} = -\frac{g P_0(z)}{R T},
\label{def_P0}
\end{equation}
which leads to the well-known atmospheric exponential decay
$ \propto \exp(-\frac{z}{2H})$, with $H \approx 180 {\rm km }$.
for a constant
Temperature $T=T_0$, which is, however, not realistic for describing
structures reaching from the photosphere through the chromosphere into the
corona. Equation (\ref{def_P0}) can be (numerically) integrated for
any choice of a temperature profile $T(z)$, e.g., from 1D-models of
the solar atmosphere.
Another alternative
is (because we computed already the 3D magnetic field
from Eq. \ref{lin_mhs} and \ref{lin_mhs_2}) to prescribe
the average plasma $\beta(z)$ as a function of z, e.g. from the literature
\citep{gary_01}, leading to
\begin{equation}
P_0(z)=\frac{\beta(z) \, B^2_{\rm ave} }{2 \mu_0}
\end{equation}
where $B_{\rm ave}(z)$ is the horizontally averaged
$B_{z}(x,y,z)$. The allowed ranges for $\beta(z)$
are bounded from below, however, by Eq. \ref{P0}. A choice which
ensures a total positive pressure is obtained by using Eq. \ref{P0}
directly
\begin{equation}
P_0(z) = P_{\epsilon}(z)+ a \cdot \exp(-k z)
\cdot \frac{{\rm Max}(B_z)^2}{2 \mu_0}(z),
\label{max_p0}
\end{equation}
where $P_{\epsilon}(z)$ is the (prescribed) minimum value of
the total pressure at a given height. For $P_{\epsilon}(z)=0$
the total plasma pressure becomes zero at the maximum of $B_z$
and remains positive elsewhere. Taking
this into account we can calculate the full
average plasma $\beta$
(including the pressure disturbance) from Eq.
(\ref{pressure}), as shown in Fig.
\ref{beta} right-most-curve labeled {\it MHS, IMaX FOV} in
Fig. \ref{beta}. The limitations
from Eq. (\ref{P0}) and a magnetogram with some high peak values in
an otherwise weak field region cause values of plasma $\beta$ which
are too high and outside the range given by \cite{gary_01} (dotted curves).
We have to conclude, that the linear MHS-model cannot be applied to the
whole FOV of the SUNRISE magnetogram realistically. The reason is that through
Eq. (\ref{max_p0}) the 1-D background pressure and thereby
the maximum pressure in weak field regions
is coupled with the highest values in the photospheric magnetogram,
which is not very realistic.
\begin{figure}
\includegraphics[width=0.5 \textwidth]{beta1c}
\caption{Plasma $\beta$ in the solar atmosphere. The dotted lines are
taken from \cite{gary_01}. The thin solid line shows the (horizontally averaged)
plasma $\beta$ profile computed with our MHS-model for the full IMaX-FOV
and the thick solid line represents the same for the selected small area.}
\label{beta}
\end{figure}
\subsection{Application to a small part of the FOV}
\begin{figure}
\includegraphics[width=0.5 \textwidth, bb= 80 30 670 500, clip = true]{fieldlines_pot_a}
\includegraphics[width=0.5 \textwidth, bb= 80 30 670 500, clip = true]{fieldlines_ff_b}
\includegraphics[width=0.5 \textwidth, bb= 80 30 670 500, clip = true]{fieldlines_mhs_mini_c}
\caption{Small field of view (rectangular box in fig. \ref{sunrise_mag_full}a).)
a: Potential field, b: Linear force-free field,
c: magneto-static field.}
\label{small_Bf}
\end{figure}
Due to the difficulties of applying the linear MHS-model to a full
magnetogram, we restrict our analysis in the following to the smaller sub-region
marked by the black rectangle in
Fig. \ref{sunrise_mag_full} a). Figure \ref{small_Bf} shows a few sample
field lines for a) a potential field model, b) a linear
force-free model with $\alpha=3$ and c) a magneto-static solution
$\alpha=3, \; a=0.5$.
In the linear force-free case the field lines become sheared compared
with the potential field and for some lines the connectivity changes.
The influence of a non-vanishing Lorentz force (but using the
same value of $\alpha$ as in the linear force-free case) has additional
effects, which seem, however, to be smaller. The maximum heights of
the loops are somewhat reduced and some additional field lines
change their connectivity, e.g. in the MHS-model no lines are
connected with the positive (red) flux region close to the front boundary.
Compared with the potential fields, the number of field lines connecting to
this region was already reduced in the linear-force-free model.
The pressure-disturbance in this smaller FOV is shown in the
center panel of Fig. \ref{pic_pressure}. The absence of strong peaks
in the photospheric field in this region leads to a much smoother
distribution
of the pressure disturbance. We use Eq. \ref{max_p0} to compute
the background pressure and in Fig. \ref{beta} the solid line marked
{\it MHS, local FOV} shows the averaged plasma $\beta$ as a function
of the height. At least in the
photosphere and chromosphere the plasma $\beta$ is within the limitation
given by the dashed lines from the literature \citep{gary_01}.
The true 3D plasma $\beta$ distribution is, however, not a function
of $z$ only, but varies significantly in the horizontal direction.
Fig. \ref{pic_pressure} c) shows the equi-contours
for $\beta=1.0$. As one can see the $\beta=1.0$ surface is by no means
plane-parallel, but strongly corrugated.
This behaviour impacts
methods for extrapolating force-free fields. Traditionally and for
numerical simplicity, one extrapolates from a plane parallel surface
(or the Sun's spherical surface) by assuming that the field above this
lower boundary of the computational domain is force-free. In reality,
however, the force-free domain is bounded below by a corrugated
surface as well. This is also true for planned measurements of the
chromospheric magnetic field vector with Solar-C, so that magnetic field
extrapolation techniques bounded by non-plane-parallel surfaces should
be developed. In the non-force-free region between the photosphere and the
corrugated chromosphere, plasma pressure and gravity must be taken into
account.
\section{Discussion and Outlook}
\label{sec:outlook}
The linear-MHS approach used in this paper has 2 free parameters,
the linear-force-free
parameter $\alpha$ and the force-parameter $a$. Additionally one has to
prescribe, besides the vertical magnetic field component at the lower
boundary, also the height in the solar atmosphere, where the magnetic
field becomes approximately force-free, here $1/\kappa= 2 Mm$.
Applying these solutions to large-scale areas has its limitations.
These are, first of all,
the well-known limitations on $\alpha$, which these solutions
share with linear force-free configurations. Additionally the
pressure-gradient
(which compensates the Lorentz-force) is coupled to the vertical magnetic field.
As a consequence, the pressure disturbance, which is negative, becomes very
large above strong fields in the photosphere.
In order to maintain a positive total pressure,
the background plasma pressure must be so strong that the average
plasma $\beta$ becomes too high.
This limitation of the method has to do with the fact that
the two free parameters
$\alpha$ and $a$ have to be the same in the entire computational
domain. The limitations are similar as for linear force-free
fields, where one has only one free parameter $\alpha$, which
has to be globally constant. While linear force-free
fields cannot be used to model force-free configurations containing
strong current concentrations in part of the domain (leading to localized high
values of alpha), a similar restriction occurs here for the
linear magneto-static approach. Strong magnetic elements
in an otherwise weak field magnetogram cannot be modelled by this class
of MHS solutions.
These limitations do not occur, however, for application to
regions with a smaller field of view, because the assumption
that $\alpha$ and $a$ are constant is naturally more reasonable as smaller
the investigated domain is.
How should one proceed to derive global magneto-static
solutions? One possibility would be to compute the solutions discussed here
only locally (with different values of $\alpha$ and $a$ in different regions)
and to merge these configurations together. This will of course lead to
solutions which are
not entirely self-consistent
and to inconsistencies at the boundaries between the
different regions. Another idea would be to use a numerical scheme, e.g.
an optimization approach as suggested by
\cite{wiegelmann:etal06}, \cite{wiegelmann:etal07} to relax
these merged solutions
towards a self-consistent (nonlinear) MHS-equilibrium.
The methods developed by \cite{wiegelmann:etal06} and
\cite{wiegelmann:etal07}
are both non-linear magneto static codes in cartesian and spherical
geometry, respectively. For the small scale features measured with Sunrise,
one naturally applies the cartesian version.
These codes require photospheric vector magnetograms
as input, which are not available for the investigated quiet Sun region,
because of the poor signal to noise ratio (for horizontal fields) in
weak field regions. Nonlinear approaches
(both force-free and magneto static) are well suited for dealing with
local strong enhancements (e.g. current concentrations and strong flux
elements). It is a weakness of any linear approach, that they cannot deal
with strong localized enhancements of any quantity.
To be able to carry out nonlinear magnetostatic
(or nonlinear force-free) extrapolations,
measurements of the horizontal photospheric
magnetic field, would be helpful.
During the re-flight of SUNRISE in 2013, high resolution vector
magnetograms
of active region(s) have been measured with IMaX and we plan to use
these measurements
for a self-consistent nonlinear magneto-static modelling in our future work.
\begin{acknowledgements}
The German contribution to {\it SUNRISE} is funded by the
Bundesministerium
f\"{u}r Wirtschaft und Technologie through Deutsches Zentrum f\"{u}r Luft-
und Raumfahrt e.V. (DLR), Grant No. 50~OU~0401, and by the Innovationsfonds of
the President of the Max Planck Society (MPG).
The Spanish contribution has
been funded by the Spanish MICINN under projects ESP2006-13030-C06 and
AYA2009-14105-C06 (including European FEDER funds). The HAO contribution was
partly funded through NASA grant number NNX08AH38G.
TN acknowledges support by the U.K.'s Science and Technology
Facilities Council and would like to thank the MPS for its
hospitality during a visit in December 2014.
D.H.N. acknowledges financial support from GA \v{C}R under
grant number 13-24782S. The Astronomical Institute Ond\v{r}ejov is
supported by the project RVO:67985815.
\end{acknowledgements}
\bibliographystyle{aa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,884 |
{"url":"http:\/\/justinbois.github.io\/bootcamp\/2019\/lessons\/l07_intro_to_functions.html","text":"# Lesson 7: Introduction to functions\u00b6\n\nThis document was prepared at Caltech with financial support from the Donna and Benjamin M. Rosen Bioengineering Center.\n\nThis lesson was generated from an Jupyter notebook. You can download the notebook here.\n\nA function is a key element in writing programs. You can think of a function in a computing language much the same way you think of a mathematical function. The function takes in arguments, performs some operation based on the identities of the arguments, and then returns a result. For example, the mathematical function\n\n\\begin{align} f(x, y) = \\frac{x}{y} \\end{align}\n\ntakes arguments $x$ and $y$ and then returns the ratio between the two, $x\/y$. In this lesson, we will learn how to construct functions in Python.\n\n## Basic function syntax\u00b6\n\nFor our first example, we will translate the above function into Python. A function is defined using the def keyword. This is best seen by example.\n\nIn\u00a0[1]:\ndef ratio(x, y):\n\"\"\"The ratio of x to y.\"\"\"\nreturn x \/ y\n\n\nFollowing the def keyword is a function signature which indicates the function's name and its arguments. Just like in mathematics, the arguments are separated by commas and enclosed in parentheses. The indentation following the def line specifies what is part of the function. As soon as the indentation goes to the left again, aligned with def, the contents of the functions are complete.\n\nImmediately following the function definition is the doc string (short for documentation string), a brief description of the function. The first string after the function definition is always defined as the doc string. Usually, it is in triple quotes, as doc strings often span multiple lines.\n\nDoc strings are more than just comments for your code, the doc string is what is returned by the native python function help() when someone is looking to learn more about your function. For example:\n\nIn\u00a0[2]:\nhelp(ratio)\n\nHelp on function ratio in module __main__:\n\nratio(x, y)\nThe ratio of x to y.\n\n\n\nThey are also printed out when you use the ? in a Jupyter notebook or JupyterLab console.\n\nIn\u00a0[3]:\nratio?\n\nSignature: ratio(x, y)\nDocstring: The ratio of x to y.\nType: function\n\n\nYou are free to type whatever you like in doc strings, or even omit them, but you should always have a doc string with some information about what your function is doing. True, this example of a function is kind of silly, since it is easier to type x \/ y than ratio(x, y), but it is still good form to have a doc string. This is worth saying explicitly.\n\nAll functions should have doc strings.\n\nIn the next line of the function, we see a return keyword. Whatever is after the return statement is, you guessed it, returned by the function. Any code after the return is not executed because the function has already returned!\n\n### Calling a function\u00b6\n\nNow that we have defined our function, we can call it.\n\nIn\u00a0[4]:\nratio(5, 4)\n\nOut[4]:\n1.25\nIn\u00a0[5]:\nratio(4, 2)\n\nOut[5]:\n2.0\nIn\u00a0[6]:\nratio(90.0, 8.4)\n\nOut[6]:\n10.714285714285714\n\nIn each case, the function returns a float with the ratio of its arguments.\n\n### Functions need not have arguments\u00b6\n\nA function does not need arguments. As a silly example, let's consider a function that just returns 42 every time. Of course, it does not matter what its arguments are, so we can define a function without arguments.\n\nIn\u00a0[7]:\ndef answer_to_the_ultimate_question_of_life_the_universe_and_everything():\n\"\"\"Simpler program than Deep Thought's, I bet.\"\"\"\nreturn 42\n\n\nWe still needed the open and closed parentheses at the end of the function name. Similarly, even though it has no arguments, we still have to call it with parentheses.\n\nIn\u00a0[8]:\nanswer_to_the_ultimate_question_of_life_the_universe_and_everything()\n\nOut[8]:\n42\n\n### Functions need not return anything\u00b6\n\nJust like they do not necessarily need arguments, functions also do not need to return anything. If a function does not have a return statement (or it is never encountered in the execution of the function), the function runs to completion and returns None by default. None is a special Python keyword which basically means \"nothing.\" For example, a function could simply print something to the screen.\n\nIn\u00a0[9]:\ndef think_too_much():\nprint(\"\"\"Yond Cassius has a lean and hungry look,\nHe thinks too much; such men are dangerous.\"\"\")\n\n\nWe call this function as all others, but we can show that the result it returns is None.\n\nIn\u00a0[10]:\nreturn_val = think_too_much()\n\n# Print a blank line\nprint()\n\n# Print the return value\nprint(return_val)\n\nYond Cassius has a lean and hungry look,\nHe thinks too much; such men are dangerous.\n\nNone\n\n\n## Built-in functions in Python\u00b6\n\nThe Python programming language has several built-in functions. We have already encountered print(), id(), ord(), len(), range(), enumerate(), zip(), and reversed(), in addition to type conversions such as list(). The complete set of built-in functions can be found here. A word of warning about these functions and naming your own.\n\nNever define a function or variable with the same name as a built-in function.\n\nAdditionally, Python has keywords (such as def, for, in, if, True, None, etc.), many of which we have already encountered. A complete list of them is here. The interpreter will throw an error if you try to define a function or variable with the same name as a keyword.\n\n## An example function: reverse complement\u00b6\n\nLet's write a function that does not do something so trivial as computing ratios or giving us the Answer to the Ultimate Question of Life, the Universe, and Everything. We'll write a function to compute the reverse complement of a sequence of DNA. Within the function, we'll use some of our newly acquired iteration skills.\n\nIn\u00a0[11]:\ndef complement_base(base):\n\"\"\"Returns the Watson-Crick complement of a base.\"\"\"\nif base in 'Aa':\nreturn 'T'\nelif base in 'Tt':\nreturn 'A'\nelif base in 'Gg':\nreturn 'C'\nelse:\nreturn 'G'\n\ndef reverse_complement(seq):\n\"\"\"Compute reverse complement of a sequence.\"\"\"\n# Initialize reverse complement\nrev_seq = ''\n\n# Loop through and populate list with reverse complement\nfor base in reversed(seq):\nrev_seq += complement_base(base)\n\nreturn rev_seq\n\n\nNote that we do not have error checking here, which we should definitely do, but we'll cover that in a future lesson. For now, let's test it to see if it works.\n\nIn\u00a0[12]:\nreverse_complement('GCAGTTGCA')\n\nOut[12]:\n'TGCAACTGC'\n\nIt looks good, but we might want to write yet another function to display the template strand (from 5$'$ to 3$'$) above its reverse complement (from 3$'$ to 5$'$). This make it easier to verify.\n\nIn\u00a0[13]:\ndef display_complements(seq):\n\"\"\"Print sequence above its reverse complement.\"\"\"\n# Compute the reverse complement\nrev_comp = reverse_complement(seq)\n\n# Print template\nprint(seq)\n\n# Print \"base pairs\"\nfor base in seq:\nprint('|', end='')\n\n# Print final newline character after base pairs\nprint()\n\n# Print reverse complement\nfor base in reversed(rev_comp):\nprint(base, end='')\n\n# Print final newline character\nprint()\n\n\nLet's call this function and display the input sequence and the reverse complement returned by the function.\n\nIn\u00a0[14]:\nseq = 'GCAGTTGCA'\ndisplay_complements(seq)\n\nGCAGTTGCA\n|||||||||\nCGTCAACGT\n\n\nOk, now it's clear that the result looks good! This example demonstrates an important programming principle regarding functions. We used three functions to compute and display the reverse complement.\n\n1. complement_base() gives the Watson-Crick complement of a given base.\n2. reverse_complement() computes the reverse complement.\n3. display_complements() displays the sequence and the reverse complement.\n\nWe could very well have written a single function to compute the reverse complement with the if statements included within the for loop. Instead, we split this larger operation up into smaller functions. This is an example of modular programming, in which the desired functionality is split up into small, independent, interchangeable modules. This is a very, very important concept.\n\nWrite small functions that do single, simple tasks.\n\n## Pause and think about testing\u00b6\n\nLet's pause for a moment and think about what the complement_base() and reverse_complement() functions do. They do a well-defined operation on string inputs. If we're doing some bioinformatics, we might use these functions over and over again. We should therefore thoroughly test the functions. For example, we should test that reverse_complement('GCAGTTGCA') returns 'TGCAACTGC'. For now, we will proceed without writing tests, but we will soon cover test-driven development, in which your functions are built around tests. For now, I will tell you this: If your functions are not thoroughly tested, you are entering a world of pain. A world of pain. Test your functions.\n\n## Keyword arguments\u00b6\n\nNow let's say that instead of the reverse DNA complement, we want the reverse RNA complement. We could re-write the complement_base() function to do this. Better yet, let's modify it.\n\nIn\u00a0[15]:\ndef complement_base(base, material='DNA'):\n\"\"\"Returns the Watson-Crick complement of a base.\"\"\"\nif base in 'Aa':\nif material == 'DNA':\nreturn 'T'\nelif material == 'RNA':\nreturn 'U'\nelif base in 'TtUu':\nreturn 'A'\nelif base in 'Gg':\nreturn 'C'\nelse:\nreturn 'G'\n\ndef reverse_complement(seq, material='DNA'):\n\"\"\"Compute reverse complement of a sequence.\"\"\"\n# Initialize reverse complement\nrev_seq = ''\n\n# Loop through and populate list with reverse complement\nfor base in reversed(seq):\nrev_seq += complement_base(base, material=material)\n\nreturn rev_seq\n\n\nWe have added a named keyword argument, also known as a named kwarg. The syntax for a named kwarg is\n\nkwarg_name=default_value\n\n\n\nin the def clause of the function definition. In this case, we say that the default material is DNA, but we could call the function with another material (RNA). Conveniently, when you call the function and omit the kwargs, they take on the default value within the function. So, if we wanted to use the default material of DNA, we don't have to do anything different in the function call.\n\nIn\u00a0[16]:\nreverse_complement('GCAGTTGCA')\n\nOut[16]:\n'TGCAACTGC'\n\nBut, if we want RNA, we can use the kwarg. We use the same syntax to call it that we did when defining it.\n\nIn\u00a0[17]:\nreverse_complement('GCAGTTGCA', material='RNA')\n\nOut[17]:\n'UGCAACUGC'\n\n## Calling a function with a splat\u00b6\n\nPython offers another convenient way to call functions. Say a function takes three arguments, a, b, and c, taken to be the sides of a triangle, and determines whether or not the triangle is a right triangle. I.e., it checks to see if $a^2 + b^2 = c^2$.\n\nIn\u00a0[18]:\ndef is_almost_right(a, b, c):\n\"\"\"\nChecks to see if a triangle with side lengths\na, b, and c is right.\n\"\"\"\n\n# Use sorted(), which gives a sorted list\na, b, c = sorted([a, b, c])\n\n# Check to see if it is almost a right triangle\nif abs(a**2 + b**2 - c**2) < 1e-12:\nreturn True\nelse:\nreturn False\n\n\nRemember our warning from before: never use equality checks with floats. We therefore just check to see if the Pythagorean theorem almost holds. The function works as expected.\n\nIn\u00a0[19]:\nis_almost_right(13, 5, 12)\n\nOut[19]:\nTrue\nIn\u00a0[20]:\nis_almost_right(1, 1, 1.4)\n\nOut[20]:\nFalse\n\nNow, let's say we had a tuple with the triangle side lengths in it.\n\nIn\u00a0[21]:\nside_lengths = (13, 5, 12)\n\n\nWe can pass these all in separately by splitting the tuple but putting a * in front of it. A * before a tuple used in this way is referred an unpacking operator, and is referred to by some programmers as a \"splat.\"\n\nIn\u00a0[22]:\nis_almost_right(*side_lengths)\n\nOut[22]:\nTrue\n\nThis can be very convenient, and we will definitely use this feature later in the bootcamp when we do some string formatting.\n\n## Computing environment\u00b6\n\nIn\u00a0[23]:\n%load_ext watermark\n%watermark -v -p jupyterlab\n\nCPython 3.7.3\nIPython 7.1.1\n\njupyterlab 0.35.5","date":"2022-08-13 03:44:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3499026894569397, \"perplexity\": 1505.5623022181232}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882571869.23\/warc\/CC-MAIN-20220813021048-20220813051048-00335.warc.gz\"}"} | null | null |
There's nothing wrong with drinking alcohol from time to time and we're not here to tell you what to drink — 'Late Night, Do It Right' is all about helping you to enjoy your night out and showing you that by making small changes to your night out routine, you can have an active, safe and hangover-free social life.
Remember, food helps slow the absorption of alcohol, stopping it going to your head too quickly, and helping you to avoid that dreaded hangover!
Carbs or protein such as pasta, potatoes and chicken are good to eat before or while you're out drinking. Not only will they help keep you full, but they will also make that unhealthy and fatty takeaway seem less tempting.
If you drink too much too early, you're much more likely to miss out on the proper night. Remember, turning down a drink is much less embarrassing than throwing one up.
Keep an eye on your home pouring, especially when it comes to spirits. Single shot measures are 25ml, which doesn't look like a lot in a glass so don't be fooled into over-pouring.
Avoid top-ups to make it easier for you to keep track of how much you're drinking – it's not rude to say no!
Ever had that sinking feeling in the morning, when you see the pile of receipts for drinks you don't remember buying?
Why not leave your card at home and only take as much cash as you want to spend. Remember though, to make sure you keep some money in a separate pocket for getting yourself home safely too.
A 'spacer' is a non-alcoholic drink that you take in between alcoholic ones - you space them out. That way you slow down your drinking.
You will be surprised how good and refreshing a spacer can be in between alcoholic drinks. On occasions a low-alcohol beer might do.
While on a night out, look out for each other in case someone's getting ahead of themselves. If they are, grab them some water or a soft drink from the bar and encourage them to pace themselves.
You don't want to have to put them in a cab, clean their sick from your shoes or miss out on the night by having to take them home.
If one of you does overdo it, make sure you know the difference between a bit too much and alcohol poisoning, and what to do if it's really serious.
Don't be the bystander who notices inappropriate behaviour and does nothing about it – if it is not acceptable when sober, it is not acceptable when under the influence of alcohol.
The union runs the Raise Your Voice project, offering training in bystander intervention to help you deal with inappropriate behaviour whenever or wherever you encounter it.
If you've done the training and feel confident to intervene in a situation feel free to do so, but when you're on a night out you can also do your bit by raising concerns with venue staff or the police. You can also report any issues you have encountered via the UniSafe App on iLancaster.
Get into the habit of never leaving your drink unattended and don't accept a drink from someone you don't know.
Keep an eye on your drink at all times – don't go off and dance then come back and drink the rest.
Avoid drinking too much alcohol as this will put you in the best position to be alert to anything suspicious and able to look out for your friends.
All good things must come to an end, and when you've had a great night out we want to make sure you make it home safe and sound.
As everybody knows, you could be risking your licence, your freedom and even your life if you get behind the wheel drunk. But you'd be surprised how little you have to drink to be over the drink-drive limit. And the best advice is to just not mix the two – don't drive, even after just one drink, and you'll be in the clear.
And remember, if you've been out at our Sugarhouse nightclub, there's a free bus to bring students back to campus at the end of the night.
If possible it's best to leave the pub or club in pairs or as a group. If someone's disappeared don't assume they've pulled, find out for sure.
Keep a mate with you and try not to spend too much time hanging about at the end of the night.
Drink a glass of water when you get home to re-hydrate – and you'll have a better chance of avoiding a hangover.
Got an early start in the morning? If you've had a few the night before you could still be over the limit well into the morning. If you've got to drive tomorrow, don't go overboard tonight.
Are you interested in supporting others to be safe on a night out or do you just want to find out a bit more about the campaign? | {
"redpajama_set_name": "RedPajamaC4"
} | 7,150 |
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About the Author
Copyright Page
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for Silvey, for whom Eden was no garden
ACKNOWLEDGMENTS
This book has a complicated provenance.
I began to write about race the day after the holiday party in 2005. I am grateful to the universe for bringing me to that awful intersection and pulling me through it.
I began writing poetry after reading Lucille Clifton's Good Woman in 2007. I am grateful to Ms. Clifton for lighting the way.
In 2012 I began an MFA at California College of the Arts where I spent a good deal of time writing about race. I wrote a poem on race in seven voices called "A Day at the Races" for Joseph Lease's class, and am grateful to Joseph for urging me to publish it.
In 2013 I published that poem in a journal cofounded by one of my former students, Tanaya Winder, called As Us, which features the work of indigenous and underrepresented women. I am grateful to Tanaya for giving me my start as a published poet, and for seeing in my nationless mixed-race experience some solidarity with the experience of displaced indigenous people.
In 2013 I adapted that poem into a script for San Francisco's Poets Theater run by Small Press Traffic. I am grateful to Small Press Traffic for the opportunity, to director Brandon Jackson for his vision and tenderness as he staged my words, to Kelsei Wharton for documenting the experience, and to each actor for their careful consciousness: Yvorn "Doc" Aswad, Alonzo Cook, Ashley Hill, Ruth Marks, Estelle Piper, Luke Taylor, and Saroya Whatley.
I continued writing about race at CCA with the urging and support of many classmates and faculty including Stephen Jamal Leeper, Donna de la Perrière, Judith Serin, Dodie Bellamy, and Faith Adiele.
This book emerged from my master's thesis at CCA, which I wrote under the guidance and direction of Juvenal Acosta and Faith Adiele. I am forever grateful to the two of you for all but forcing me to go there.
I am grateful to my brother Stephen Xavier Lythcott for writing about our slave ancestor, Silvey, and to Silvey herself and everyone in between her and me for being and enduring and ultimately giving me life. To my extended family. To my mother and father. To Avery, Sawyer, and my beloved, Dan.
And as ever, I am grateful to the people who turned this manuscript into a book: my editor and my agent, the fierce and fearless Barbara Jones and Kimberly Witherspoon, and their entire teams at Henry Holt and InkWell Management.
IT BEGINS LIKE THIS
I.
"Where are you from?"
"Here."
"No, I mean, where are you from from?"
As a child growing up in the seventies and early eighties in New York, Wisconsin, and Northern Virginia, there was something about my skin color and hair texture that snagged the attention of white children and adults. Their need to make sense of me—to make something of sense out of nonsensical me—was pressing. My existence was a ripple in an otherwise smooth sheet. They needed to iron it down.
[The truth is, I'm not really from here.]
[The truth is, that's not what they were asking.]
II.
The truth is, they were asking, "Why are you so different from what I know? So unclassifiable?"
There's love at first sight. There's American at first sight. And from dozens of "where are you from" interactions with Americans over the years, I've learned that American at first sight is about looks—primarily skin color and hair texture—not nationality.
I am the wooly-haired, medium-brown-skinned offspring typical when Blacks and whites have sex, which was considered illegal activity in seventeen of the fifty "united" states in 1966.
Nineteen sixty-six was the year before the U.S. Supreme Court decided in Loving v. Virginia that the laws preventing interracial marriage were unconstitutional, and 1966 was the year in which my Black father and white mother, an African American doctor and a British teacher who met in West Africa, chose to go ahead and get married anyway. They married in Accra, Ghana. I was born to them in Lagos, Nigeria, in 1967.
I come from people who broke the rules. Chose to live lives outside the box. Chose hope over hate as the arc of history was forced to bend a bit more toward justice. I am the goo in the melting pot.
Rhetorically championed.
Theoretically accepted.
Actually suspect.
In places hated.
Despised.
III.
In the lead-up to the 2008 presidential election a persona stepped to the forefront of public consciousness, that of the "Real American."
More than an individual you want to have a beer with, more than the everyman "Joe the Plumber," the "Real American" is code for an entire era when men like Andy Griffith ran Mayberry or John Wayne swaggered through a western town. When white men cloaked in clothes of real or perceived authority could take what they believed was rightfully theirs with an air of ownership to the opportunity, to the land, to the people, and of belonging at the center of the situation, whatever it was. A time when the word "he" meant all genders. When "normal" and "regular" meant "white."
This fictional character—the Real American—became a talisman, a lifeline to a more halcyon past for some white men and women bewildered by capitalism's demand for low-paid laborers and by the rising tide of legal and regulatory equality that dared to lift others' boats. They looked around at us the others knocking at the door of the hiring manager, the landlord, the admissions dean, the local restaurant. Looked frantically around and began to see fewer—less—of themselves.
Nursed by a milk of white supremacy fed to them as what was natural, right, and good for them, these whites believed the rest of us were interlopers, thieves at the door, threatening to take what was not ours. They grew incensed at the growing number of us others who refused to accept our place at the bottom of America's ladder underneath even the most lowly of whites.
These "Real Americans" found a voice in their candidates, grew in number, became a mob who raised slogans, signs, fists, and arms. Who long to make America great—
normal
regular
white
again.
IV.
These newly emboldened "Real Americans" issue angry orders to the rest of us: "If you don't like it, go back to where you came from."
There is no back to where I came from.
You stole my homeland from me.
Me from my homeland, I mean.
I don't even know where it is.
Literally.
V.
I came from Silvey.
I am the untallied, unpaid, unrepented damages of one of America's founding crimes. I come from people who endured the psycho-cultural genocide of slavery, reconstruction, and Jim Crow. Who began to find a place here really only quite recently amid strides toward effecting a more perfect union, of liberty and justice for all.
I am Silvey's great-great-great-great-granddaughter. She was a slave who worked on a plantation in the late 1700s in Charleston, South Carolina, the harbor town through which close to one in two African slaves entered America over the centuries. Silvey bore three children by her master, Joshua Eden, by which I mean he raped her; there is no consent in slavery. Silvey's daughter Silvia was born in 1785, and Joshua freed Silvey, Silvia, and their other children some years later. Silvia gave birth to a son named Joshua in 1810. Joshua had a son named Joshua Jr., born in 1845. His daughter, Evelyn, was born in 1896. Evelyn bore my father, George, in 1918. And I was born to him in 1967.
The original Americans are the natives whose land was invaded then stolen by the Europeans. Those descended from the Europeans, the ones who came on ships to the New World, like to think they are the original Americans. But I'm from a third set—from those brought here on different ships over different waters, those whose sweat and muscle were the engine of the American economy for over two hundred years, whose blood and tears watered America's ground. I come from them.
I come from people who survived what America did to them.
Ain't I a Real American?
VI.
When the amorphous mob harrumphs about the needs and rights of "Real Americans," they don't picture me. People like me. But is anyone more a product of America than those of us formed by America in an angry war with herself?
This is rhetorical. Theoretical. Of course we are not more than. We're less than, not even equal to. The remainder of an imperfect equation. The child who wasn't supposed to exist. The undesired other. The bastard child of illegitimate rules who dares even to be.
The contradiction of being "less than" in a nation whose forming documents speak of liberty and justice for all plagued me for much of my young adult life.
I'm so American it hurts.
AN AMERICAN CHILDHOOD
I.
As a child growing up in the 1970s, I adored my country, as I imagine most American children do.
II.
When I was three we moved from our Manhattan apartment to a rickety eighteenth-century house my parents rented in an old hamlet known as Snedens Landing in the town of Palisades, New York. Snedens is tucked into the western bank of the Hudson River across from Manhattan and farther north. Its homes nestle along the main road, Washington Spring Road, which meanders through town, then makes a steep, winding descent to the Hudson River below. Alongside the main road and through the backyards of some of the houses runs a stream—the spring said to have provided respite to General George Washington and his troops in 1780, while the nearby woods gave cover to the traitor Benedict Arnold. When the Revolutionary War ended at the Battle of Yorktown in 1781, the British warship HMS Perseverance sailed across the Hudson over to Snedens Landing and fired a seventeen-gun salute in recognition of the brand-new country, the United States of America. Our tiny rented house dated back to these times and, clinging to the hillside, rather looked like an old man, groaning, with an aching back, weathered by weather and time.
By the time I was five my parents had scrounged together enough money to buy a house, this one further up the street on Washington Spring Road, of 1950s modern design, with large living-room windows that looked out at the creek below and onto the thick woods beyond. As a small child I played in that creek with my friend Conrad, a little white boy whose name I mispronounced as "Comrade." Conrad and I would launch leaf and stick boats into the creek, which couldn't have been more than two feet across and three or four inches deep, and then we'd watch delighted and wide-eyed as the current took our little creations down and away and ultimately beyond our sight. I recall feeling a strong pang of worry and hope over whether my little boats would be all right. If they'd get caught up in an eddy or a beaver's dam, get sidetracked and end up on the muddy brown creek bank, or maybe make it to an enormous freedom in the famous river below. That backyard stream was a wonderful laboratory for me and Conrad. One afternoon I lowered my underpants and squatted over the gently flowing water, then stood back with tremendous satisfaction at proving Conrad wrong: poop sinks.
I marched in a Fourth of July parade in Palisades when I was six, my large Afro distinctive among my group of straight-haired peers and adults. I was sporting my Brownie jumper and sash, with its little "Girl Scouts U.S.A." patch stitched in white letters at the top. I drew up my spine and straightened my neck at the honor of wearing that uniform to march with my troop past our town's tiny post office, while our neighbors held ice-cream cones and applauded from the side of the road.
Palisades was where I memorized my first phone numbers, my first address and zip code. As a student at Palisades Elementary School, I jumped to my feet for the Pledge of Allegiance and sang all of the patriotic songs with gusto. My favorites were "This Land Is Your Land," because it contained the name of my state, New York, and "The Star-Spangled Banner," for its unabashedly triumphant violence. On field day at the end of second grade, I ran the fifty-yard dash, and even though I ran diagonally across the field instead of straight ahead I came in third out of twenty and got a lovely little bronze medal the size of a quarter.
When I was seven, my father's growing prominence in the field of public health and academic medicine spurred his departure from the faculty of Columbia University. He moved us to Madison, Wisconsin; he would be Associate Vice Chancellor for Health Services at the University there. It was 1975. Third-grade math was taught by a stern woman named Mrs. Bernard, my first African American teacher, and I took to memorizing my multiplication tables like it was a game I had to win, and did. The following summer I recall the hard work of weaving red, white, and blue streamers through the back and front wheels of my ten-speed bicycle in honor of our nation's Bicentennial. Riding that bicycle in the parade wending through my tree-lined neighborhood, Arbor Hills, in Madison, I felt important, giddy, alive.
A year later we moved to Reston, Virginia, a planned community located just outside of Washington, D.C., boasting a sort of utopian commitment to racial and socioeconomic diversity. President Jimmy Carter had appointed Daddy to be his Assistant Surgeon General with responsibility for running the Health Services Administration in the Department of Health, Education, and Welfare. It was 1977.
On a school field trip to our nation's capital with my fifth-grade classmates, I felt a swell of admiration for America and a surge of pride to be American as I stared up at the gleaming white Washington Monument, heard my voice echo as I walked around Lincoln in his chair, traced my fingers over the bronze plaques. We walked back to our bus in a gaggle and for a few moments were caught in the jumble of people in their gray trench coats trying to hurry down sidewalks to and from their jobs. I stepped to the side so they could pass. Important people worked in this city. I knew my Daddy was one of them.
Back at home in Reston, I had Black friends, Indian friends, and Jewish friends, as well as white friends. There was even another Black family on my street for the very first time in my life, with a daughter named Amanda. Amanda was a few years younger than me, but we could both sense that it was very important to our parents that we become friends. And we did become friends, genuinely, telling each other our secrets, playing board games, and sequestering ourselves behind locked doors to review the girlie magazines our fathers thought they kept well hidden. I felt a mix of wonder and awe as we pawed through page spreads of creamy white skin.
Over the years I did extremely well in school, was a student government representative, sold Girl Scout cookies, and tied a thick yellow ribbon to the strong tree that stood at our curb in honor of the American hostages in Iran.
III.
I adored Daddy. He was fifty when I was born and my childhood coincided with the heyday of his career, which began against all odds amidst the racial hatred of the segregated Jim Crow South. I was his last child of five—the product of his second marriage to my mother—and I knew from the way his eyes twinkled whenever he looked at me that he loved me no matter what. He gave me a variety of nicknames—Old Sport, Knuckle Head—which sounds crude to my grown ears but then, spoken in the butter of his baritone, it felt like melted love. He never had to call for me twice. I came running every single time.
When I was little and skinned my knee, he'd pull me up onto his tall lap, kiss me, and ask with all seriousness how I was going to become Miss America with that scar. I didn't know then that no Black woman had yet been crowned Miss America and that no Black woman would be crowned Miss America until 1983. Instead I heard in Daddy's words that I was beautiful, perhaps the most beautiful girl he'd ever seen.
We all called him "Daddy," even my mother. He was formidable, commanding, gruff, loving, and funny. I hung on to his every word, whether it was "Baby, bring me my cigarettes," or a well-placed retort to the news recited by the anchorman on TV.
Daddy was the protagonist, the lead.
Daddy was the sun.
IV.
Beauty pageants weren't my thing, though. I wanted to be something more like President.
By the end of my junior year in high school (by which time we were back in Wisconsin), I'd been elected vice president of my class for the third year in a row, and in the fall of my senior year, the student council elected me president of that governing body. I was selected for "Badger Girls State"—a statewide program for kids interested in policy and politics held the summer after high school graduation, and was elected senator there. I went on to be one of four presidents of my class at Stanford University, and one of four elected class leaders of my graduating class at Harvard Law School.
I was on track to live the American Dream—through hard work, big dreams, and a bit of luck, to become whoever I wanted.
Mine was in many ways a very American childhood. And, with the buttress of money and influence that came from my father's professional success, it was also a childhood of material comfort that set me up for a privileged life.
BECOMING THE OTHER
I.
Daddy never liked the Fourth of July.
I couldn't understand it, because I adored the parades, songs, and flags, the neighborhood barbecues, the explosion of firecrackers, and the smart looks on everyone's faces that revealed the innate understanding that our country was better—and by extension we the people were better—than the rest of the world.
My mother was the one to inform me of Daddy's opinion about the Fourth, and she did so in a whispered-sideways-glance kind of way with no explanation as to why he felt it. I understood from the way she said it that it had something to do with Daddy's past, his experiences, his Blackness. Her silent "why" bespoke pain too painful to discuss, so I never asked. Didn't think it related to the America I was inhabiting anyway. Didn't think I was Black in the ways he was. Thought America was beyond all that.
I was wrong.
Looking back over the years of even my earliest childhood, the clues were everywhere.
II.
Back in Snedens Landing, I'd begun to sense that something might be wrong with people with dark skin. I lacked the language to describe it and the intellect to analyze it, but I felt the chill of it in my bones, the red-hot heat of it surging up the back of my neck when I was out and about with Daddy.
Daddy was six foot two and lean, with a neat, tightly coiled Afro he kept supple with Afro Sheen, and skin that was dark and crinkly like the top layer of a brownie. On those occasional weekend days when he wasn't traveling or busy at the desk in his den, he'd take me with him on an errand in town, and every now and then to an event in Manhattan. Holding his hand walking down the local street or a bustling city sidewalk, I noticed that some strangers stared at him with eyes that steamed like a cauldron, as if they could brand him like an animal with their searing focus if he dared to look them in the eye. I'd look up at my tall daddy for reassurance, pleading with my small brown eyes to know what was going on, but he gripped my hand tighter, kept his eyes focused straight ahead, pursed his lips tight, and kept walking.
When I walked down the same streets with my white mother, nobody steamed at her that way. The glances she got as a white woman holding the tiny hand of a small brown child were far more subtle. It took a lot longer for me to discern and label those looks as pity and disdain. By choosing to marry my father, she'd crossed a line. By choosing to have me.
III.
I was learning that something might be wrong with me.
I began kindergarten as a four-year-old who would turn five in November. And that year, or maybe it was first grade, I began coming home with my classmates' questions: "What are you?" and "She's your mother?"
When I mentioned these comments to my parents, they responded with all the vociferousness and passion of a political advocacy campaign that I was "Black." It was a deliberate strategy—almost a strategic plan, I could tell—because they'd land heavy glances on each other as they said these things to me. Later I'd learn it's what parents of mixed kids did in the 1970s; there was then no widespread use of the concept "multiracial" or "biracial," and the ill-fated term "mulatto" going back to slave days, denoting the half-and-half mixture of slaves and whites, was considered in poor taste. "Call mixed kids Black," the thinking went, "because the world will see and treat them as Black. They'd better claim it and be proud of it. They'd better know how to defend it. How to hold their heads high. Be Black and proud."
I was not privy to the sociopolitical agenda. I was just bewildered.
Why was I the race of one parent and not the other, effectively denying my mother's contribution to my loosely curling Afro and brown-paper-bag-colored skin that made me look so different from almost everyone else I encountered, including both of my parents as well as my half siblings from Daddy's first marriage to a woman who was half Black Cape Verdean and half White Cape Verdean (from the Portuguese who colonized the island), who'd given my half siblings more easily discernible Black features?
And given the choice between white and Black, why were my parents adamant about labeling me the race that so many people seemed to find problematic? In forcing this "Black" label on me and even bringing it on herself, why was my white mother choosing to lower herself into this pit instead of using her whiteness to lift me and Daddy up and out of it?
IV.
Mom was the Blackest white lady I knew. Maybe even the Blackest person I knew.
Daddy was so immersed in his work and international travel that he was for many months over many years more of a ghostly figure, an absent presence in our daily life. His other children, my siblings, were a generation older than I was, and when they popped in and out of our home for the holidays and an occasional weekend visit, I loved being around them, their enormous Afros, their raucous laughter over stories laced with innuendo that made my mother raise her eyebrows. But I would not come to know them deeply until I was an adult myself. Mom was the one who shouldered the enormous task of bringing Blackness to light.
Born in England, Mom had lived in West Africa for seven years in her twenties, which is where she'd met Daddy, and she'd become a naturalized American citizen in 1968 with white skin tanned a copper brown. She wore dresses made of cloth she'd bought in Nigeria, which made white friends and neighbors here in New York raise their eyebrows, as if she was maybe affiliated with Black activist groups, maybe even the Black Panthers.
She'd read the research that said that Black children need Black dolls to help develop a healthy psychological self. So she filled my childhood bedroom with those dolls, and also read me texts by Black authors, like Nikki Giovanni's poetry collection, Spin a Soft Black Song. She knew the prominent Black thinkers and artists of the time, and could pronounce names like Ntozake Shange accurately and with a little flair, maybe even gusto, like she enjoyed how it felt to make those sounds with her lips and tongue, sounds that prompted bewilderment in other white adults. My mother even went so far as to deny her ethnic ancestry by saying to strangers—such as the white Mormon missionaries who knocked on our door in Snedens Landing in 1975 when I was seven—"Sorry, you have nothing for us. We are a Black family." The missionaries agreed with her and turned away.
I could tell she was putting on an act. Her actions a costume worn by a character in a play designed to transport us to a different reality. But I wasn't transported. I was annoyed. I could see she was trying too hard to convince herself, the world, and me.
One day when I was about six she came home with two picture books she'd bought at a local store. In these days before the "playdate" I had a friend over without her knowing it, and, not wanting to be rude to my friend, my mother acted as if she'd planned all along that one of the books was for my friend and the other for me. My friend and I had immediately reached out for the same book, the one whose cover featured an animal of some kind. As we tugged over who would get that book, my mother gently said the second book was for me. It was The Snowy Day by Ezra Jack Keats, and the drawing on its cover was of a Black child. Why? Why did I have to read those books if my friend, who was white, didn't? Couldn't I have those white dolls my friends had? Didn't we all know they were the more beautiful dolls, those with the bluest of eyes? Wasn't I good enough for those white dolls? Those white books? Wasn't I good enough for the white world?
I would physically bristle when my mother referred to "us" as a "Black family." She seemed to be trying to join a club it seemed rational to want to leave. But so long as I was in it, I wanted to keep out anyone who lacked the chops to belong. I'd fume silently every time she said it, and when she was out of view I rolled my little dark brown eyes.
V.
Of all her efforts to raise a Black daughter well, the most confounding and persistent challenge for my mother was my hair. No matter how Black she tried to be, when it came to my hair she was a stranger in a strange land.
With no one else around to look to for guidance, my hair was a mystery to me as well.
I did not learn for decades that hair is a psychological barometer for Black girls and women—it's alternately our pride or the bane of our existence as we try to locate a sense of self within Western norms of beauty, respond to definitions of "neat and tidy" and "professional," and bear up under humid weather.
I had an Afro of loose brown curls until I was about eight when I decided to grow my hair long enough so that it could be pulled back, reined in, tamped down. In fourth grade in Madison, Wisconsin, I'd sit before the large mirror in my bedroom day after day, and Mom would stand behind me holding a brush like a wizard brandishing a new wand, her eyes round in frightened wonder anticipating both the potential and the disasters that were possible. Stroke by stroke she'd pull my pile of frizz into a ponytail and secure it tight with an elastic holder, the kind with little plastic bobbles that allowed you to wrap the hair tight with the elastic and then use the bobbles to secure one end of the tie to the other. She didn't know the importance of gels and creams and other Black hair care products, so the hair of the ponytail was a frizzy ball of puff.
I grew older. We moved to Reston, Virginia. My hair was an immediate demarcation of difference between me and my friends, me and the magazine image of beautiful, me and normal. From television I knew that Black girls straightened their hair chemically, but my father forbade my mother from doing that to my hair because of the extreme damage he saw done to girls' hair in his youth five decades earlier. I pleaded that today's products were different. But he would hear none of it, and Daddy's word was all but law. Mom had to choose her battles and this was not going be one of them; even if Daddy's perspective was decades out of date, he held the trump card on any conversation related to Blackness. So I used a curling iron to help smooth my hair to make it look more like I felt I should look. The way I wanted to look. More like the hair I saw on Black girls on TV and more like my white friends. Mom would sit with me as I tried to turn my frizzy hair into some kind of smooth, swoopy, flippy design. But it just wouldn't lie flat or flip backward like the hair of Farrah Fawcett, the actress and pinup girl with super hair all my white friends tried to imitate with varying degrees of success. The best I could do was smooth my hair into a longer, sleeker ponytail. I began to use the curling iron every day.
Over time, these efforts to smooth and constrain my hair took their toll. When I was eleven and in sixth grade, the hair at the back of my head began to break off from the damage done by the taut rubber hair ties that held the ponytail. Toward the end of that school year, Mom took me to her hairdresser, a Black woman named Angie, to see what could be done. As I sat in the big vinyl barber's chair and looked in the mirror at all the implements and instruments of beauty in the store, Angie and Mom spoke in conspiratorial tones off to the side and behind me. About me. When it was time to begin the cut, Angie swiveled the chair so my back was to the mirror, then proceeded to cut. All I could see was my mother looking on, pleased. When Angie was done she spun me around to look at my new self in the mirror and I stifled a horrified cry upon seeing the very short Afro that blanketed my head. I was embarrassed to be embarrassed to be wearing the traditional hair of my people, yet I felt as if I'd been assaulted. The next day we had a car wash fund-raiser at my elementary school, and although my tiny breasts were starting to push against my shirt fabric a customer mistook me for a boy.
VI.
In fifth grade at Lake Anne Elementary School in Reston, Virginia, one of my white friends got pulled into the gifted and talented group. She was smart but no smarter than I was, I knew. And now she was getting to do cool projects and puzzles but not me? I went home and mentioned it to my mom, who came to meet with my teacher, Mr. Pulansky, a few days later. Pulansky was not persuaded. So my Mom escalated to the principal, this time insisting that I be tested. They brought in someone from the district to give me an IQ test. Mailed the results to our house. Mom thought I wasn't watching when she opened the envelope, read the results, and squirreled the letter away in a drawer.
I was put in the gifted group soon after and shortly after that Mr. Pulansky announced to our entire class, "Apparently, all it takes to be gifted is for your parent to meet with the principal." But in the privacy of an afternoon home alone, I'd peeked at the letter from the district. The raw score was 99th percentile. As my teacher stood smug at the front of the classroom, it was the first time in my young life I uttered a very silent fuck you.
VII.
Junior high was a game changer. A big school with students from many surrounding towns and an instant infusion of Black kids. LaVerne was a head taller and more developed than the rest in our group, and was dark skinned and strong. She had an opinion on all kinds of things from dating to politics, and a quick tongue when challenged. I admired her. More than that, I yearned for more of her—she, this physical clue to how I might look and behave one day. We both went out for track, and I was fast. LaVerne took notice and gave me a huge hug after my 2:28 time in the 880-yard run. I didn't know then that Daddy had been a great runner in college. I'm not even sure he knew that I had taken up running myself.
At Daddy's urging, Mom signed us up for Jack and Jill, the social organization for middle-class Black families founded in the 1930s when most American Boy and Girl Scout troops refused to let Blacks in. Each J&J family took turns hosting the monthly meetings. When it came time for our turn and Mom and I were setting out plates and napkins and arranging hors d'oeuvres on the buffet table, I was apprehensive over whether our guests would feel we were living the right kind of Black life.
The doorbell began to ring. As I took coats and led people downstairs, I saw parents nod to one another at the paintings in the foyer done by African artists and point to the large wall hanging on the circular stairway as they made their way down into the living room. I overheard Mom answering questions about the sculptures standing in the living-room corners, the figurines on table tops and in bookshelves. I saw people run their finger slowly over the carved ivory tusk that sat on the bookshelf along the far wall. The meeting came and went and, as secretary, I recorded the minutes. When the night was over I'd learned that the artifacts my parents amassed over their seven years in West Africa were an exquisite mask that communicated, particularly to Black adults, that our family was in fact quite Black. Our African art helped us seem more Black American. I had a nagging feeling we were less than we seemed.
Toward the end of seventh grade, LaVerne convinced me and another girl to do a dance in the talent show portion of the upcoming Jack and Jill annual gala. LaVerne choreographed it and she and I practiced over back-to-back weekends, with LaVerne always pointing out where the third girl would go. When the big evening came, both Mom and Daddy were there. Daddy looked out of place mingling with parents he did not know. His being there was Mom's doing, I knew, because his face was reluctant but resigned to the fact of it, like when you roll up your sleeves for a shot at the doctor's office.
When it came time for the talent show portion of the evening, about fifty parents and other family members gathered, and my parents seated themselves in the metal chairs three rows back from the empty space at the front of the room that was our makeshift stage. I stood on the sidelines nervously awaiting our group's turn. We started out fine, but halfway through the song I forgot the steps and looked out of the side of my eye at LaVerne to pantomime what she was doing. I couldn't remember and couldn't recover. I stumbled off to the side and stood there holding my arms against my chest, rocking side to side on my heels, staring at LaVerne and the other girl so as to avoid eye contact with anyone else. It was just a dance, and soon it was over. But it felt like an omen.
On November 4, 1980, I was in eighth grade and about to turn thirteen. It was Election Day, a Tuesday, and that evening I was babysitting someone's kid. As the election results came in, I watched with deepening worry the news of Ronald Reagan's apparent landslide victory over President Carter. I knew this meant my father would lose his job. I knew this meant we'd move. Somewhere. But I didn't know that I'd be saying good-bye to Black friends for a long, long time. And I didn't know how much that would matter.
I would live in eight homes before going off to college, the only child in a very traditionally gendered household of high academic expectations. Moving frequently made me good at conveying myself to others so as to be accepted. But I never really learned to stop trying hard to be whoever they wanted me to be and just be.
In the spring of that final year of living in Northern Virginia, the Jack and Jill adults threw an evening dance party at a fancy hotel for us thirteen- and fourteen-year-olds. I wore the new magenta jewel-toned velour top and Calvin Klein jeans I'd gotten for Christmas, put on the small bit of makeup my mother would allow, and smoothed and curled my hair in a way that looked pretty good. I felt beautiful as I walked into the party, but tentative. I was worried about whether I could dance well enough to blend in.
When the DJ started playing, a few girls and boys began dancing right away, and then more, and then most. But I hung back on the edge of the parquet dance floor, moving my hips and shoulders ever so gently to the thudding disco and R&B music, wanting to be dancing, wanting to feel what my body might do in response to these beats and rhythms, wanting to relax and have fun like all these other kids seemed to be doing, but not wanting to draw attention to myself.
After about twenty minutes, a boy who'd been at the center of the dancers out on the floor broke away from the others and came toward me, kind of dancing as he walked and smiling broadly. He reached for my hand. I felt suddenly shy, which was a feeling I was not accustomed to. I could feel the strong thud of my heart in my chest. I pulled back and shook my head no. He beckoned with his chin, still holding out his hand. Chosen. I was chosen. Was this really happening?
This boy was beautiful, and I knew from our J&J meetings over these past few years that he was kind. But we'd never really spoken. I stepped forward and walked out onto the dance floor behind him. The boy—I don't recall his name, wish I could recall his name—had moves that were three and four times beyond what everyone else had going on, but I just tried to copy what everyone else was doing. He stayed with me. Danced with me. Around me. Kept returning to me with his smiling eyes. Finally I returned his gaze and held it. The song was "Good Times" and when I stopped thinking so hard, my body just did what it wanted to do, which was dance. The DJ put on "Rapper's Delight" next. When that long song was over, I shouted to my new friend, "How'd you learn to dance so well?" "Awww," he replied with his beautiful huge smile. Then he spun away from me to do a signature move. "My Mama taught me."
VIII.
Following Reagan's victory came the routine dismissal of all the previous president's appointees, including my father. By January 1981, Daddy had decided to return us to Wisconsin where he'd take a high-level administrative position at the medical school at the University of Wisconsin–Madison and resume a part-time pediatric practice there.
My parents might have chosen to buy a home in the vibrant, multicultural capital city of Madison, a real city with people from all over the world, as is the hallmark of any university town. Instead they bought the larger plot of land and nicer house their affluence made possible in a development west of Madison in the town of Verona. The development was called Cherrywood—a bit of wishful thinking. It was actually surrounded on all sides by cornfields and had a middle-of-nowhere feel to it.
We moved to Cherrywood in the summer of 1981 and soon met our across-the-street neighbors, the Sullivans, a family of kind white folk originally from rural Vermont with a grown child living back in New England and a younger child named Lisa who was then ten. Unlike both of my parents, neither Mr. nor Mrs. Sullivan was college educated. Mr. Sullivan had worked his way into senior management in the food industry. He'd recently been made president of a company headquartered in Madison and they'd moved to Cherrywood just months before we had. Despite the fact that I was thirteen and headed off to high school, and Lisa was only ten and still in elementary school, we bonded over being outsiders, lonely, and new. Our parents also quickly became fast friends and over the years would become true confidants, dear to each other. They cherished the improbability of being neighbors in the first place, let alone folks who genuinely liked each other.
One weekend evening when we'd lived there about a year, as our fathers fussed over the barbecue in Lisa's backyard and our mothers made side dishes in the kitchen, Lisa and I were upstairs in her small bedroom, sprawled out on her carpeted floor, reading a teen magazine. When we finished leafing through the glossy pages, I smelled dinner penetrating the stale air of the bedroom and suddenly felt very hungry. I rolled over, got up, and headed for the door when Lisa began to speak.
"I'm not supposed to tell you this—"
I paused and turned around. "What?"
"My sister was visiting when you guys first moved here. She saw your dad across the street on his riding lawn mower and said, 'Oooh look, they have a Black gardener.'"
For a split second I stared at Lisa lying there on the floor, my eyes wide, my breath clenched in my throat. Then I shifted my gaze up toward the far wall, keeping my body as still as possible, as if a bad smell had arisen from her and if I just didn't move at all, didn't disturb the air between us, the smell wouldn't come any closer to me. Through her bedroom window I could see a portion of our lawn, large, green, and plush, across the bucolic street.
Daddy loves riding that mower, I thought to myself. Steps on the running board with his left foot and swings that bent right leg over and around, like getting on the back of a horse. He's so proud to have all of this land, these lawns, so excited to think ahead to next spring's gardens. My parents would take a stroll around the property every night, cocktails in hand, admiring the gardenia—Daddy's favorite flower—that grew near the back door, pointing to where the tulips would come up after the cold winter, checking on the three pines they'd planted to give the lawn a bit of character. Our house was bigger than the Sullivans'. My father was the fucking former Assistant Surgeon General of the God Damn United States. I took a deep breath and cocked my head back toward Lisa, who still lay on the floor.
"Swear you won't tell?" she asked. Then she scrambled to stand up. "My parents made me promise not to tell you."
I nodded slightly as if in a trance, feeling the heat rise up my neck. I looked to the walls of Lisa's small bedroom, to the photos of a Vermont childhood tacked up above her desk.
"C'mon," I said, turning toward the bedroom door and opening it. I walked out of her room and strode down the hallway to the stairs that led to the kitchen. I could hear Lisa calling out, her voice sputtering, could hear her footsteps as she scrambled to catch up with me. I smiled at this satisfying feeling, this small bit of control I had in this moment, this choice I had, this chance to say something, and what, or to say nothing at all, as I made my way toward our four parents.
"You have to promise," I heard her say from the stairs, as I reached the kitchen.
"Promise what?" Lisa's mother asked.
"Nothing." I forced a smile, picturing this small-town Vermont family having their secret conversation about mine. The mothers glanced at each other.
I grabbed a bowl of potato salad in one hand and the green salad in the other, and edged my way through the partially open screen door leading to the backyard. I plunked the food on the picnic table on the small patio, then found a place at the end of one bench and sat. I took in the sight of the salads, the hot dogs and burgers, fruit, beers and sodas, the bottles of ketchup and mustard, the freshly picked corn. I listened to the sound of adults in easy conversation, relaxing into their strengthening friendship. My stomach grumbled. I took a cheeseburger and began to eat it, and worked hard to keep my feelings to myself.
I was embarrassed for Daddy, who took such pride in mowing the gorgeous lawn at the large home he'd worked so hard to provide for us. Embarrassed for the Sullivans that they'd had to have an uncomfortable conversation they thought they were keeping safely from us. Embarrassed for Lisa for being so naive—or just young—as to relay this story to me, as if revealing what her ignorant family member had said was a greater gift than keeping it to herself. And maybe it was. I took a bite of my burger, and then another, and another, and then a gulp of iced tea. I looked over at my smiling, innocent parents and was embarrassed, in a way, for all of us.
It was only natural that Lisa's big sister was confused seeing Daddy astride his tractor mower. Most Black people didn't own homes like this. Most Black people were more likely to be someone's gardener than a home owner in a community like this one, weren't they? I watched Mr. Sullivan pop open a can of Leinenkugel's and hand it to my father and then grab one for himself. They shared a hearty laugh over something they wanted to keep between themselves. Was I going to get Lisa in trouble for "telling" what we all knew was true—that Blacks tend to play roles of subservience to whites? No. It was the fact of the Sullivans whispering about it among themselves that made me feel most uncomfortable. I calmed that feeling by eating my burger. I felt safe knowing that none of the grown-ups knew what I knew.
IX.
When we moved to Cherrywood in the summer of 1981, I was an incoming high school freshman. The summer that we moved, my mother decided it was time for her to try to give me some braids, which she'd gleaned was how Black girls my age were supposed to wear their hair. She fashioned it French-style, in two big rows that marched down either side of my head. She still didn't know about the kinds of products that would tame and moisturize my hair—nor did I—so my braids looked like untended crops, a halo of unkempt hair springing up like kudzu around them.
I enrolled at Middleton High School, located one town over from Verona, which did not have its own high school. Middleton High had about twelve hundred other students, many of whom lived on farms in places far more remote than Cherrywood, most of whom had been in school together since kindergarten, and practically all of whom were white. Three of the student body were Black—me, and a pair of siblings who were a sophomore boy and senior girl, the children of the head coach of the University of Wisconsin men's basketball team. I never met the sister, but over the years I did exchange a few brief words with the brother.
I searched for familiar things. Three weeks into the school year I saw a poster announcing freshman class officer elections. I decided to run for class secretary—a role on the team, yet innocuous. A way to ease myself into things. It came time to give our speeches. The entire class of three hundred was in the gym, seated in the bleachers. I stood at the podium in red corduroy pants and a pink short-sleeved shirt with hair like a tumbleweed and began speaking into the mike. My classmates weren't quieting down, though. They kept chattering amongst themselves. I tapped the mike and got their attention. They quieted. "My name is Julie Lythcott," I began.
I came in second out of three vying for the post and one good that came of running was that I got to know people. A classmate named Diana befriended me. She was beautiful, kind, wore big glasses, and was easygoing, with a cursory interest in academics but very plugged in socially. Like a friendly clerk on Ellis Island who might actually take an interest in an immigrant's assimilation, she pulled me over the imaginary line demarcating who was outside and who was inside this new community by calling me at night to be sure we were dressing identically at school the next day, and by inviting me to the boy/girl parties she threw in her basement. She was my closest friend for four years and genuinely liked me for me. I wouldn't have made it through Middleton High School if it hadn't been for Diana.
In our ninth-grade U.S. history class, I felt only the briefest emotional stirring as I read the small paragraph our textbook devoted to slavery. No need to dwell on it. No need to examine it. It was behind us. This was 1981, after all.
Diana loved black-and-white movies depicting the Antebellum South, and often suggested that we watch them together. My intestines twisted at the thought of having to watch, but I couldn't articulate why.
X.
In the spring of my freshman year, I fell hard for a boy named Nick. His skin was practically paper white and covered with freckles, his legs were strong from playing soccer, and he was smart as hell and knew it. Our lockers sat right next to each other, and at the end of school one day, as we packed our backpacks before catching our bus home, we laughed about something that turned out to be so hilarious that we ended up falling to the floor with giggles. It would be common today for a friend to whip out a phone and photograph the moment, but in those days no one carried a camera. Our friend Jenny ran to her nearby locker to get the camera she kept there and came running back to me and Nick, still giggling on the floor. I can't remember what we were laughing about, but I remember feeling in that instant on that shiny cement floor that I was falling in love. Jenny had the film developed at the local drugstore and ordered prints for all three of us.
I pinned the photo to the corkboard on my bedroom wall. One day Daddy noticed it, looked over at me with a loving smile, then shook his head.
"White boys will be your friend," he said with his booming bass voice, "but they'll never date you."
I trembled. Neither then nor ever did I challenge Daddy's authority. His decision to move to this remote area. I never knew how to ask, Why'd you move me to this all-white town?
XI.
The summer after freshman year, I was fourteen, and Diana's aunt took us to a public swimming pool in a nearby town. As we walked along the cement in search of lounge chairs, a stranger going past us in the other direction stopped short, turned around, and said in that round, wide-mouthed, Midwestern twang, "Oooooh mye gaaaaaaaaash. Uuuuure soh taaaaaaaan." I smiled and looked over at the kids splashing in the pool, not wanting to draw further attention to myself. I did not correct her.
XII.
I turned fifteen in the fall of my sophomore year. My hair was longer now, and once again I was straightening it with a curling iron and wearing it down for school pictures and other formal occasions. But I still knew nothing about hair products for Black girls, and however straight and neat my hair was when I left our house in Cherrywood, by midday the humidity in the air made it puff and frizz. On those days, Nick and the other boys I hung out with—the boys I fantasized about as I went to sleep each night—laughed in my face, pointed at my weird hair, and called me Bozo the Clown.
XIII.
I usually went along when Mom went grocery shopping at Cub Foods. The two of us would stand there at the checkout, across the black conveyor belt from the cashier, across the black conveyor belt that contained the groceries for our family, and occasionally, unpredictably, the cashier, whoever it was, would make eye contact with my mother and place his hand on that black conveyor belt wherever there was a discernible gap between one item and the next, trying to demarcate which groceries were hers and which were mine.
"This'll be it for you?"
"No, it's all of this. We're together."
My mom speaking as the eyes of the clerk tell me I cannot possibly belong to my own mother.
XIV.
There was only one Jewish kid at my school: Rachel Klein. Her Dad, Dr. Jacob Klein, was also a physician, and he shared a pediatric practice with Daddy at the university. In a twist of Midwestern irony, here were the Black and the Jew, persecuted throughout history yet rising above in Madison, Wisconsin.
Rachel was one year older than I was and also a member of our school's choir, which was consistently regarded as the very best choir in the state. In the fall of my sophomore year as the holidays approached, Rachel and her parents got themselves in a heap of trouble when they complained to our choir director about the Christian themes at the heart of every single choir song. Our director would have none of it, so the Kleins complained to the school administration, and ultimately to the school board. The result was that our director was forced to include one non-Christian song in each show from then on. Every time we got out the sheet music to rehearse "Ya Ba Bom," a Jewish folk song, our director would glare at Rachel.
I didn't really understand the Kleins' concern. The Christian music was exquisitely beautiful. All great choirs sang it. We were a great choir. Why did they have to try to ruin things for the rest of us?
I didn't yet know about allies. That Rachel needed one. That I'd need allies of my own one day. That maybe I even needed some allies right there and then.
XV.
I spent a lot of time at Diana's house and she at mine. One day during sophomore year when I'd gone over to her house to hang out, I found her in the basement rec room watching a movie on her VCR. It was Gone with the Wind. She looked up at me and said hello, then she turned her gaze back to the television screen and sighed like a Southern belle.
"Wouldn't it have been great to live back then?"
"No?"
"Why not?"
"Because I would have been a slave."
"Oh, but I mean if you weren't Black."
"But I am Black."
"I don't think of you as Black. I think of you as normal."
XVI.
I knew what Diana meant. I felt very un-Black myself, even as my parents continued to insist I was Black, even as I tried to figure out what that meant and to be that person in this white town.
The only Black people I saw on any regular basis were members of my own family—my father and my four siblings who were now in their mid- to late thirties and had lives and families of their own. Like anyone in America, I was bombarded with negative media portrayal and stereotypes about Blacks. And those negative images helped me construct a sense of self.
From The Jeffersons, All in the Family, and Good Times, I knew that Black people seemed to be someone's edgy, hip, funny friend who spoke in some kind of special jargon, who seemed either athletic, or to know how to dance well, or to be lazy, and who greeted other Blacks with a special handshake. From the news, I knew we were associated with poverty and crime. With my parents' constant refrain about me being Black, I thought it was on me to be a great dancer, do my best not to appear to be lazy or badly behaved, and figure out that handshake.
While I was trying to construct this Black self in a completely white world, one day I overheard my mother talking to my visiting sister who was in her late thirties. "I wish Julie had more Black friends," my mother said, sighing. I felt judged. Blamed even. Where was I to find Black friends in an all-white town? Did my own mother think I was prejudiced against Black people?
Was I?
She couldn't have been blaming me, I now know. She was likely blaming herself, her inability to stand up to Daddy who made all the important decisions in our family life such as where we would live. Mom knew I needed Black friends and was confiding in my sister about it. But I was already feeling recalcitrant toward Blackness and my place in it. I heard in my mother's words criticism that maybe I was avoiding Blacks on purpose. I heard in my mother's words my own criticism of my self.
XVII.
When spring of my sophomore year came around, my classmates started making noises about junior prom, a ritual open to all grades at our school despite its name. A few weeks before the dance I got wind that a senior named Rob was thinking of asking me. I knew Rob from choir. He was kind, and smart-boy witty. In a school where choral music was not just appreciated but championed, we all revered the very mature bass voice and heart he brought to his solo in "Swing Low, Sweet Chariot."
But Daddy's admonition clanged like a warning bell in my ears: "White boys will be your friend but they'll never date you." Is Rob so far down the pecking order of white boys that he can't find a single white girl to date? Is something wrong with him? Is he settling for me?
I panicked. I'd been raised to think well of myself. I didn't want to go to the prom as someone's consolation prize.
I decided to prevent that from happening by inviting someone to the prom myself—not a boy I had a crush on; someone who couldn't say no on the basis of race. There was that one Black boy in our school—the son of the head coach of the university's basketball team, Frederick, now a junior—and I felt, given our race, that it was only natural that Frederick and I should go to prom together.
What I knew of Frederick fit the stereotypes I had of Black people. He was the son of a basketball coach and played basketball himself. He had the slangy language. He slouched his lanky body through the halls in a way that was different—which is ordinarily taboo in high school, but with Frederick was appealing, intriguing, as if he didn't give a shit about looking so different from every other boy in the school. From a distance, I admired his easy Blackness. Yet it set him apart, put him out of reach, even for me. In the face of prom looming, I felt an urgent need to try to cross that line.
It was hard to work up the nerve to call him. I had spoken to him maybe once or twice in passing. But the potential awkwardness of going to a fancy dance with someone I hardly knew seemed outweighed by the logic of going to prom with a fellow Black person. A Black date felt like the safest possible choice. If only I could make it happen.
The day after I learned of Rob's plan to ask me to prom, I was ready to do an end run around him by asking Frederick instead. That afternoon after school, I grabbed the phone book off the kitchen counter and brought it upstairs to my bedroom where I could have some privacy. I grabbed my red Trimline phone from my desk and plunked down on my bed with the phone book and the phone. I opened the white pages and thumbed through the Cs until I found Frederick's last name. The address listed was in my town; it had to be the right number.
I could hear my heart beating against my skull, and I began to sweat a little while I rehearsed my opening lines twenty or thirty times. I imagined Frederick on the other end of the line. Would he even know who I was when I said my name? Should I try to speak with a bit of edgy jargon to sound Black enough to him? Did I even know any edgy jargon?
No.
Finally, I pulled the red phone onto my lap, lifted the receiver from the cradle, listened for the dial tone, and dialed the number. It rang three times, then someone answered. "H'lo?" The voice sounded distant. Uninterested. Like the fact of the phone ringing in his house was an annoyance. It must be Frederick, I thought. I began talking. At first he had no idea who I was so we stammered through an awkward back-and-forth where I explained that I was Julie, a sophomore. I knew my "white" voice wasn't going to communicate my race so I said I was the sophomore at school. The Black girl. Now that I knew he knew who I was, I took a deep breath and cut to the chase.
"Hey, so prom's coming up and I figure it's only logical that we go together."
"Yeah? Yeah."
"So—?"
"Yeah, yeah, no that makes sense."
"Cool. We'll go to prom then."
"Yeah."
"Okay great. Thanks. Bye."
The deed was done. But instead of the relief I'd expected would flood me, my blood continued pounding in my head. I was mortified. I'd just asked a guy to the prom whom I didn't even know, all because we both had brown skin. Isn't that racism?
Am I prejudiced? Can Black people even be prejudiced? What the hell kind of person am I?
I was also a little scared. What would it be like to go to the prom with a guy who was for all intents and purposes a stranger to me? And on top of that, a Black guy. What would it be like?
By the next day I was feeling ashamed. I walked through the halls at school preoccupied with thoughts about what I'd done to Frederick. He probably didn't want to go to the prom with some girl he barely knew. He deserved better. So did Rob, the guy I feared was settling for me and was trying to avoid.
I called Frederick back that night and told him a small lie. I told him I'd just heard Rob had been planning to ask me and now I felt badly that I'd jumped the gun.
"So, I guess you and I shouldn't go," I stammered.
"Okay, cool."
I went to prom with Rob dressed in a white Victorian-style dress that suited my mother's sensibilities, with a neckline that rose to the bottom of my chin. Decades later I can see that Rob was brave, even transgressive of social norms in asking me to prom. Maybe even that I was a girl he actually just wanted to take to prom.
Frederick and I never spoke again. His family moved away at the end of that school year. Later I'd learn that his father, who'd been the first Black head coach of any major sport in the Big Ten, had died of cancer just months before my stupid stunt.
XVIII.
At the very end of sophomore year, I started dating Mark, a Mormon boy in my grade whom I'd been crushing on for a while. He was smart, athletic, and cute in a Tom Cruise in Risky Business kind of way, and his conservative ideals made for fiery debates between us. Turns out he'd wanted to ask me to the prom and was kicking himself that Rob got to me first. He never knew about the awkward dance I'd done with Frederick.
"But Mormons don't like Black people," my mother reminded me, harking back to the years the Mormon missionaries visited our door in Snedens Landing when I was seven. She said this repeatedly, even as my relationship with Mark unfolded into months and then years.
She wasn't exactly wrong.
The Mormons have an appalling history on race. For most of their history, they believed—in accordance with their primary text, the Book of Mormon—that God cursed a certain tribe of people with "a skin of blackness." Because of this "truth" from the church's founding, church policy historically forbade men of color from being priests. The policy was finally repealed in 1978 when their church leader—known as the Prophet—received a direct revelation from God that Black men could "hold the priesthood." The Mormons use this 1978 "revelation" as "proof" that church practices do not yield to sociopolitical pressure and change only in response to divine revelation. Which I translate as: Yeah, the country may have bought into equal rights for Blacks in the 1960s, but we Mormons take our cues from God, and God didn't think Black men were equal to white men until 1978.
I started dating Mark five years after God changed his mind about Black people.
That I could envelop myself in a boy whose church believed this about my people was the seed of self-loathing lurking inside me that I could feel but could not yet identify. I spent more time at Mark's house than he did at mine because his parents were kind to me whereas mine were cordial at best to him. When I got invited to church activities, my mother visibly bristled. "You'll convert to that church over my dead body."
XIX.
If I had told my parents about the prom debacle with Frederick, I feared they would have been:
1) sad that I thought Rob's invitation to prom was a sign of him settling;
2) dismayed that I didn't know the one and only Black boy at my school;
3) busy examining my actions and words to discern my motivations (they would have discovered a self-loathing writhing at the heart of it);
4) shown the hazy outline of my wariness toward Black people;
5) unable to ignore the evidence that their interracial child experiment was failing.
They would have wondered what they'd done, whether they could have done anything, whether at that point anything could be done by anyone at all.
But I never told them.
And if, even without this prom fiasco information, they were concerned that their mixed-race child was struggling psychologically, they would have been right.
XX.
In the grand scheme of human existence my father and mother were an improbable couple. Their interracial relationship began in 1962 in West Africa, on the red-brown clay soil of Ghana's capital city, Accra.
My father, George I. Lythcott, was born in 1918 in New York, New York. His father, George Sr., was a Black physician with a medical degree from Boston University who was descended from the Lythcotts of British Guiana. His mother, Evelyn (Wilson) Lythcott, was a descendant of South Carolinian slaves whose father, Joshua, was Postmaster General of Florence, appointed by successive presidents and confirmed by successive senates throughout the late 1800s and into the early 1900s. Evelyn died from tuberculosis in 1920 in Florence. Daddy was not yet two.
Grandfather asked Evelyn's sister-in-law Lillian to look after Daddy until he could pull his life together and figure out a more long-term solution for Daddy's care. Walking up Lillian's pathway to visit his little boy one day, Grandfather could hear Lillian through the screen door shouting at Daddy. "Get your little Black hands off that chair." Evelyn's people were light; Lillian was considered "high yeller." Grandfather urgently sent for his sister Agatha in British Guiana and asked her to move to New York to care for Daddy there.
In 1925 Grandfather got married again to a woman named Corinne—half Black, half Cherokee freedman—and Daddy went to live with Grandfather and Corinne in Tulsa, Oklahoma. Daddy was now eight. Grandfather served the medical needs of Tulsa's Black community, some of whom could pay him with little more than livestock and produce. Corinne doted on Daddy as if he were her own son. She was the only mother he really ever knew.
Daddy came of age in the deeply segregated South. The thriving Black business community in Tulsa's Greenwood neighborhood, known as "the Black Wall Street," had been torched to the ground in 1921 by whites. Hundreds of Blacks were killed in the uprising, most of them lynched, and upward of ten thousand Black residents were left homeless when the damage was tallied. The massacre was labeled a "riot" and a few hundred Blacks were arrested. The lie was perpetuated for decades. In 1996, the Oklahoma state legislature would finally acknowledge that the prosperous Greenwood community had been set upon by white supremacists.
When Daddy was about sixteen, a little girl and her mother were walking along the road just beyond the house, and Daddy's dog, a large Doberman pinscher, got out of the yard, charged over to the young girl, and sank his teeth into her upper thigh. Daddy was terrified—it was his job to make sure the dog was chained up at all times—and he raced first to pull the dog off of the girl and then to get the girl to his father. Grandfather dressed the young girl's wound and then put the girl and her mother in his car to drive them home. The mother described where she lived in a vague manner, and Grandfather at first couldn't make sense of it. He soon realized that the family lived in a makeshift home at the edge of the town dump. He knew it was medically unsafe to send a child with such a bad wound back to that kind of home, but he didn't want to insult the woman, so he simply offered that the girl, Polly, might heal more quickly if he could clean and dress the wound each day, and perhaps she should live back at home with him, his wife Corinne, and Daddy, until the wound healed. The mother agreed. When the wound healed, the mother and Grandfather had a serious conversation about the girl's future. Grandfather officially adopted the little girl, who became my Aunt Polly.
After graduating from high school in Tulsa in 1935, Daddy went way up North to attend Bates College in Lewiston, Maine, where, as he would retell it to me with a chuckle decades later, there was only one Black man in each class, and one Black woman too, so the men would have someone to date.
Daddy ran track for Bates and was dubbed "the Oklahoma Flyer" by the student newspaper for consistently whizzing by his competitors with his lean, six-foot-two frame. In the spring of his freshman year, 1936, he qualified for the U.S. Olympic Trials. Another qualifier, the miler Jesse Owens, would go on to make the U.S. team and to stun Hitler with both his athletic prowess and his Blackness at the summer games in Berlin. Daddy helped pace Owens as he warmed up for his trial—a practice where four of the fastest quarter milers ran just ahead of Owens for one lap each so as to push Owens to run faster. But when it came time to compete in his own right, Daddy pulled his hamstring and had to abandon his Olympic dreams.
Daddy kept running for Bates in his remaining years at college as he pursued premedical studies; he was also quite involved in campus shenanigans. One night he pulled a cart full of hay up the hill above the football stadium, then set it on fire and gave it a push so it would roll back down the hill directly into the wooden stadium, which was quickly set ablaze. (No one complained; they'd been trying to garner support for a new stadium and now they had to build one.)
Another night, he coaxed a horse through the front door of the main administration building and led it up the steps leading to the offices on the second floor. He walked the horse into the president's office, scattered some hay on the floor, and shut the door. Then he crept down the stairs and out of the building, and went back to his dorm room. As any Oklahoma cowboy knows, horses will go up a set of stairs but will not come down. So the next morning when a secretary discovered the horse munching hay in the president's office as it gazed out through the big plate glass window overlooking the college green, the task of getting the horse down would be far more complicated than was the task of bringing him up. They had to hire a glazier to remove the large glass window and then bring in a crane to lift the horse down. Again, no one in the administration ever knew Daddy had done it.
Being one of a handful of Blacks at Bates grated on my father, but so did the stark class differences between himself and many of his peers; he was the son of a physician, yes, but a Black physician serving the Black community wasn't getting rich. Daddy resented the rich kids at Bates who had bicycles enabling them to get around campus and into town rather easily. One winter night Daddy stole a bunch of rich kids' bikes from around campus and threw them into the mountainous banks of snow that blanketed Lewiston each year. When the snow thawed the following spring, bikes kept emerging, rusty and bent, like muddy puddles, on the new, wet, green lawns.
Daddy was an anarchist, some might say. A prankster. A subversive. A rule-breaker. Some would say a thug. I see him as a Black man who in the construct of New England in the 1930s had little agency. He was a man of great intellect, tall and strong, still subject to being called "Boy" by any white man at any moment. He was capable and accomplished, yet subject to being second-guessed or blamed without cause. The pranks were perhaps his most vivid way of retaliation, of pulling the wool over their eyes. I imagine he felt a kind of raucous joy in accomplishing these subversive acts. You think I'm bad? You have no idea how bad I am. Yet each prank was just a brief burst of freedom from a cage. To our family's knowledge no one ever knew my father was behind any of these pranks. Or maybe they knew but didn't want to jeopardize the athletic eligibility of their track star and the recognition Bates College received whenever he ran.
Grandfather expected Daddy to follow in his footsteps and become a physician, but by junior year Daddy had taken a few courses in architecture, political science, and law. The possibility of law school began to tug on his attention. On April 29, 1938, Daddy's twentieth birthday, he sat down at the desk in his dorm room to tackle the difficult task of composing a letter to his father seeking formal permission to study law instead of medicine. He was still working on that letter into the night when his buddies came by to take him out to celebrate his birthday. Daddy and his friends went out and had a good time, and he got back to the dormitory quite late that night. The next morning he awoke to a telegram telling him to come home because his father was dying. Daddy took the first available train and made the long trek from Maine to Oklahoma, but when he arrived on May 1, his father was already gone. Never having gained his father's permission to pursue law, Daddy pursued medicine.
Like his father, Daddy graduated from the medical school at Boston University (1943) and his expertise in pediatrics kept him first in the Boston area, where he and his new wife, Ruth, settled for a time and where their four children were born in 1945, 1946, 1948, and 1950. Ruth began to suffer from mental illness, which grew steadily worse. In 1953 Daddy was called up for military service at Mitchel Air Force Base on Long Island, and after that he set up a private pediatric practice in New York where he treated the children of prominent Blacks, including Jackie Robinson, Nat King Cole, and Harry Belafonte. Because their mother was by this time quite ill, the children went to live in Tulsa with Daddy's stepmother, Corinne.
In 1956 Daddy and Ruth moved to Oklahoma, where Daddy took a position at the medical school at the University of Oklahoma in Oklahoma City. The Pittsburgh Courier reported on November 3, 1956, "For the first time in the history of the institution a Negro has been appointed to the faculty of the University of Oklahoma." It went on, "a specialist in the diseases of infants and children, [Lythcott] was recently appointed clinical assistant in pediatrics in the university's School of Medicine, and became the first and only Negro holding such a position in a Southern university." He received an NIH grant to establish the nation's first well-baby clinic on an Indian reservation nearby. While serving on the faculty Daddy also maintained a private pediatrics practice for Black patients, and, like his father before him, was often paid with things like produce and desserts.
Daddy's children, my half siblings, Ruth, George, Michael, and Stephen, were raised in the Jim Crow South. The civil rights movement began to emerge around them, and in the early years the movement was populated heavily by young people. My sister Ruth, the eldest of Daddy's children, was twelve when she first participated in events organized by the NAACP Youth Council. She was fourteen in 1959 when she decided to participate in sit-ins at a lunch counter at a downtown restaurant. She took our brothers, then ages twelve, eleven, and nine, with her. Daddy knew this was happening and allowed it. One day, the television was on in his office and he saw police dragging Ruthie and Stevie out of a building.
In the early 1960s Oklahoma's School of Medicine was searching for a new dean. The committee mentioned the fact of Daddy to the finalist, who said it would be no problem. But when the finalist became dean, he didn't speak to Daddy for a year. Finally, he summoned Daddy to his office and issued a warning. "I can't have someone on my faculty I can't invite over to my house for dinner." Daddy was being told to quit.
When, in 1962, the U.S. government asked Daddy to join a team headed to Ghana in West Africa to help the Ghanaians establish an organization akin to the National Institutes of Health here in the U.S., he jumped at the opportunity. He'd be Deputy Director of a large team and would focus his own research on a measles vaccine, which would be an important step in his career.
But what of his children? Daddy and Ruth were now separated and she was not well enough to care for the children. Corinne was elderly and growing frail, and couldn't consider making such a huge relocation herself. Men rarely played the role of single parent in those times. A colleague advised Daddy to farm out his kids—now sixteen, fifteen, fourteen, and twelve—to four different families. But breaking the family into bits was an untenable choice for Daddy. Deep down he was convinced that he could handle the challenge. Hell, he was going off to Africa... raising four kids there was doable, had to be doable. He needed to get not just himself but his four children out of the increasingly volatile American South.
Officially a U.S. diplomat, Daddy shipped himself, the kids, their station wagon, and much of their belongings over six thousand miles to West Africa, and moved them into their new home in Ghana's capital city, Accra, in a section of town called Korle-Bu.
For the first time in Daddy's life, skin color wasn't the primary mechanism for evaluating the worth of a human, not the determining factor for whether he'd be allowed or rejected as he tried to make his way in the world. Dark though he was, he was to some Ghanaians "Obruni"—a term in the Twi language for someone not from Africa, a term used, even, for whites. But being Obruni didn't constrain Daddy's options. He felt a psychological freedom unavailable to him in America—and finally began to emerge into himself as a man. This was the state of things when he went to a party one night in Korle-Bu and met Jeannie, the woman who would become my mother.
A few months before Daddy graduated from Bates College in 1939, my mother, Jean Snookes, was born to a white working-class family in Yorkshire, England. Her grandfathers had worked in the coal mines, and her father, a schoolteacher, served in World War II, fighting in the Battle of Britain, leaving behind three very young children—my mother, her older brother, and her younger brother—and a wife barely able to make ends meet with the rations that came from the British government. Mom recalls the air-raid sirens blaring night after night as the Germans tried to locate and bomb Yorkshire's coal mines and steel production facilities, and the nightly ritual of battening the hatches to remove any trace of light from the night sky to conceal themselves from the German pilots. The Britons defeated Hitler so conclusively in the Battle of Britain that Hitler turned his focus elsewhere. Winston Churchill declared, "Never in human history have so many owed so much to so few." When the war finally ended and her father came home, Mom was six years old.
Her family continued to struggle financially. After the war her parents had two more children and when her mother procured a treat for them—such as an apple—they would split it five ways. My mother was a gifted learner who worked tirelessly and was always one of the strongest students in her school, especially in science and math courses, and she rose to the rank of "Head Girl" based on her academic performance. She was the first in her family to go to university and attended the University of Manchester, where she studied honors science. There, she met and fell in love with a young man named Ian Forrester, an Honors Math student who was on the gymnastics team and rode a motorcycle.
In late 1958 Mom was about to turn twenty and was four months pregnant with Ian's child. They were due to be married in a few weeks when Ian fell ill suddenly with severe abdominal pain. The doctor at the University Health Center said liver trouble was going around the university. The pain would be severe but it would soon abate. Take some pills, come back Monday. The doctor had failed to properly diagnose the source of Ian's pain: an adhesion of the appendix.
Mom stayed by Ian's bedside for four days trying to nurse him through this awful pain, but she grew extremely worried—he seemed so very sick. She called upon Ian's landlord, who agreed something was terribly wrong and drove the two of them to Ian's parents' home in the town of Hanley. Although Ian had told them about her and about the baby, this was to be Mom's first time meeting Ian's family.
Ian's parents went to their family doctor and he came back to the house with them. Looking at Ian, the doctor seemed to Mom to be dubious about the university physician's diagnosis, but it being the university physician, the family doctor didn't want to second-guess the situation. He said he would come back the following morning but stressed to Ian's parents that they should come get him if Ian's condition changed at all. (Ian's parents had no phone.)
Ian's condition did change—the terrible pain went away later that day—which his parents took as a good sign. Ian asked his parents if Mom could come up to his bedroom to see him, which caused some consternation since she and Ian were unmarried. But these circumstances threw regular rules into relief and his parents agreed. Mom entered Ian's bedroom and saw his belly blown up like a huge balloon. His skin was an awful gray color. He looked like a shadow of himself, she told me.
The family doctor came as promised early the next morning, took one look at Ian, and raced out the door to call for an ambulance. Doctors operated immediately. Hours later Ian's father took Mom to see him in recovery. She wanted to crawl up into the bed with him, but the sides were pulled up high on his hospital bed to prevent his attempts at escape. She could only stand on her tiptoes to lean over the barrier and kiss him. As she did so, Ian spoke.
"Marry me, kid," he pleaded, using his pet name for her.
"Just as soon as you get out of here, Love, we will do it."
Mom realized only later that Ian had known he was dying, had known he needed to spare her the stigma of being an unwed mother, had needed to marry her right then and there.
She went back to the house with Ian's father. Later that night there was a knock at the door. It was a local policeman relaying a message from the hospital that they must come quickly because Ian was dying. But Mom did not know about the knock at the door or the message, would not know this until the night's events were relayed to her the following day. Ian's parents decided that in her "condition" Mom would not be able to handle what would come next.
Ian's father had gone to the hospital alone that night while Ian's mother stayed home with her three younger children and Mom. He arrived just as a priest emerged from Ian's room. He'd been administering last rites and now looked around and asked, "Who is Jean?" The doctor told Ian's father he could have saved Ian if he'd had him twenty-four hours sooner. When Mom learned this she was shattered. She would blame herself for these elapsed hours for the rest of her life.
In May 1959 Mom gave birth to their baby, whom she named Ian after his father. Her parents bucked the tide of decency and reputation extant at the time by allowing Mom and Little Ian to come live with them and by being emotionally supportive. Big Ian's parents were supportive too—agreeing that the baby would have their last name even though Mom and Ian had not been married. Mom graduated from university the following year with honors, and with her dean's support applied for and received a full scholarship to do a PhD in Saskatoon, Saskatchewan, Canada. The scholarship would pay for everything once she got herself and Little Ian to Canada; the stumbling block was how to find the money to travel there. She went for a loan at her bank but had only her trustworthiness as collateral, which was not enough.
She applied for a few teaching jobs locally but the fact of Little Ian (and his status as a "bastard," which was the prevailing ideology at the time) was a stigma not to be overcome. Finally, after her father pulled some strings in the district in which he'd worked for over ten years, she got a teaching position in Sheffield, found day care for Little Ian, and moved there. The school's headmaster said, "No one is to know about the baby," except, the headmaster said, for his deputy Mr. Braxton, who the headmaster felt needed to know. Mom worked at the Sheffield school for two years. Braxton began to blackmail Mom, forcing her into an unwanted sexual relationship with him in order to keep the news of Little Ian quiet. Little Ian came down with double pneumonia—pneumonia in both lungs—and almost died.
Mom was beginning to give up. She blamed herself for Little Ian's poor health and for not getting Big Ian to the hospital sooner. Braxton demanded to know her whereabouts at all times, who she was with, and what she was up to. She had few prospects for a better life in England and began applying for various teaching fellowships in other countries again, desperate for a better outcome this time.
In 1962, when he was not yet three, Mom gave Little Ian up for adoption to Big Ian's parents, via a formal proceeding. She and Little Ian would not reunite until 1977, when he was eighteen and she was legally allowed to contact him. I was ten. I recall my mother taking me on a walk in our Madison neighborhood to tell me that I had another sibling, in England, and I recall feeling angry to have been kept in the dark on something so important. She quickly set me straight. "How does a mother tell one child she has given up another?" she said. "I needed you to be old enough not to worry whether I might give you up, too." I stared off in the distance blinking away tears.
Shortly after giving Little Ian up for adoption, Mom left England, having accepted a teaching fellowship at Ghana's Pre-Nursing Training School—the first fellowship offered that would pay her airfare to the country on top of providing her a house to live in. She boarded an airplane for the first time in her life for a flight that took her from London thirty-one hundred and fifty-nine miles due south down the Greenwich meridian. She landed in Accra, where she was met by a new colleague, who drove her to her new home at Number Four Nimtree Circle in Korle-Bu. The colleague brought her small suitcase from the trunk, then pulled away as she stood looking up at her new home, a second-floor walk-up over the garage with a circular staircase to one side. There'd be no need for that garage; she didn't know how to drive and was, as she called it, "poor as a church mouse." She was twenty-three.
One evening soon after Mom arrived in the country, the head of the Pre-Nursing Training School had her over for dinner to introduce her to a few more people in the community. At about ten p.m. she was walking the short distance home along the dusty red-brown road when a car pulled alongside her. It was Fran, a nurse on the American medical team in Korle-Bu. Fran had spotted Mom three or four times before on her walk to or from work and had given her a ride, and the two of them had started to become friendly. Fran leaned out of her car window.
"It's too late for you to be walking alone. Let me give you a ride."
Mom got in the car. As Mom's house came into view, Fran slowed down and nodded back toward the trunk of the car. "Actually, we're having a party and I left to go get more beer. I'm headed back there now. You should come."
Mom looked over at her house. A party with the Americans probably required a fancier outfit than she'd worn to the dinner. She had one elegant dress to her name and it was in a closet in her bedroom at the top of those stairs.
She shook her head, said she'd need to change first. But Fran was in a hurry to get back to the party.
"You look fine. C'mon."
Mom gave in. Fran turned the car around and a few minutes later they arrived at the party now in full swing.
Mom was curious about people and gregarious, so she quickly found herself in conversation with other partygoers. A short while later Daddy arrived, making a stir in the crowd because of his high status in the diplomat community and because he had on his arm the sister of the American writer James Baldwin. Baldwin was on an official visit to Accra, and the embassy had tasked Daddy with taking his sister out for the evening before her midnight Pan Am flight back to the States.
Mom and Daddy soon found themselves on the same side of the living room and struck up a conversation. Forty-five minutes later Daddy looked at his watch and realized he had to get Baldwin's sister to the airport. It was eleven fifteen p.m. He asked Mom if she would wait.
"I can't promise. I'm here with Fran. I have to leave when Fran does."
"Well, if you're not here when I get back, can I come see you tomorrow then?"
She didn't hesitate. "Yes."
Daddy left. A short while later Fran was ready to go. When Daddy returned to the party sometime later, Mom was gone.
The next day—a Saturday, Mom rose early. She didn't know when Daddy was coming by and she wanted to be ready. She bathed, put on the special dress she'd wanted to be in the night before, made up her face, dabbed herself with perfume, and fluffed her hair just so. She walked over to the window that faced the road and peeked through the blinds. She could see much of the neighborhood from the second story and saw no sign of Daddy. But it was early—only nine a.m.
Accra is four degrees north of the equator so even the smallest movements can cause a person to sweat, particularly a person unaccustomed to the heat, as Mom was in her first months in Ghana. So she felt it was best to sit as still as possible on the couch and wait for Daddy. She finished a book she'd been reading. Then she began working a crossword in one of the many puzzle and logic books she'd brought with her from England. She got up to stretch her legs and stirred the heavy heat with her movements. The click of her heels on the floor disturbed the silence. She sat down and did more puzzles. Got up to make a sandwich. More puzzles. Hours went by and still she waited, trying to keep cool. Finally, at six p.m., as the sun made its quick drop to the horizon, she gave herself a stern talking-to. What makes you think a man like him—a doctor—would be interested in you?
She woke the next day feeling frustrated, not just over her foolishness in thinking this man might actually want to see her again, but because in spending an entire day waiting for him she'd neglected her housework. She put on a pair of old shorts and a T-shirt and began by making her bed, cleaning the bathroom, and putting dishes away. Then she settled in for a morning of furniture polishing. At about noon she heard the scrunch of tires on her gravel driveway and peeked out the window. It was a huge Mercury station wagon with wings at the back. "It was him, at my house," she'd tell me years later. "I was mortified to be in grubby shorts polishing furniture but here he was. That's the thought that took over. That's what mattered."
Daddy pulled his car to a stop in front of the tree at the end of Mom's driveway. Took the circular stairway two steps at a time and knocked on her door. She welcomed him in and they sat on her couch and talked for hours. Then, just as before, he looked at his watch and knew he had to leave. This time he needed to be home with his four kids. They parted. But they were together from that hot Sunday afternoon in Accra until his death on a Saturday thirty-three years later off the coast of Massachusetts on the island of Martha's Vineyard. My unlikely parents.
Mom soon met Daddy's children—Ruth, George, Michael, and Stephen, who, at sixteen, fifteen, fourteen, and twelve years old, were much closer in age to her than she was to their father. They attended a local boarding school and when Daddy traveled back to the States for six weeks, Mom brought the children the things they needed at school and handled his affairs as a secretary might or perhaps even a wife. Over the ensuing months their attitudes toward her ranged from bemused, to indifferent, to warm, to skeptical, to defiant, and even, at times, to hostile. A year later, Mom was writing regular letters to Daddy's stepmother, Corinne, back in Oklahoma to update her on the family's goings-on. Like Daddy's children, Corinne experienced her own range of emotions about this white woman Daddy was dating who was now writing to her. But on one of Daddy's many trips back to the States, Corinne gave him her ring for when it came time to propose.
He did propose, on a sandy beach along the road to the fishing village of Tema, which President Kwame Nkrumah correctly envisioned would be an international seaport one day. They married in January of 1966 in Accra in a civil ceremony, with only three witnesses, a white man and two white women. The clerk initially mistook the white man to be my mother's intended husband. Then the huge ceiling fan doing its best to keep the air a few degrees above stifling ruffled the pages of the clerk's text and he began reading the procedures for divorce. But they sorted it all out with a chuckle and managed to get married in a ceremony that would have been illegal in seventeen states back in the U.S. at the time.
They moved to Lagos, Nigeria, later that year, when my father joined the historic effort to try to eradicate smallpox and was put in charge of the operation in twenty West African countries. One by one first Ruth and then George had left for college in the States, and then Michael left as well. I was born at the tail end of this adventure, in November 1967, at Lagos Teaching Hospital. Stephen was the only sibling still in Africa when I was born, and he brought me a dark purple fuzzy stuffed animal with long hair. Ruth sent me a Black baby doll. I'd keep both of these toys for years. The West African smallpox effort succeeded a year and a half ahead of schedule and the disease was declared eradicated from the globe in 1980.
When I was four weeks old we headed back to Accra to spend Christmas with my parents' friend, the American Ambassador to Ghana, Franklin Williams, and his wife. Daddy's driver, a Nigerian man named Eric, sped us across the coastal road from Lagos through Dahomey (now Benin), through Togo, and into Ghana, a trek of over three hundred miles. While we were there, a political skirmish broke out in Nigeria related to the Biafran civil war; there were new contentions that children were starving in the Biafra region, and the U.S. government asked Daddy to return to Nigeria immediately to assess the situation. We would have to fly. Daddy hastily drew up the papers Eric would need by way of explanation for himself as he drove alone through the borders of Ghana, Togo, Dahomey, back to Nigeria. Eric dropped us off at the Accra airport. Mom held me in her arms as she and Daddy went through immigration.
"Where's her passport?" spurted the immigration officer, nodding in my direction.
"She's only four weeks old," Daddy stated, with the authority of a pediatrician and the airs of a diplomat.
"She's a person, isn't she?"
As would be the case often in Daddy's life, his rhetoric and authoritative manner would win the day. Though I lacked any identifying evidence of my existence or citizenship, I was permitted to exit Ghana and, a short plane flight later, permitted to enter Nigeria.
America recognized citizenship through either parent, although Britain only recognized it through the father, so British citizenship was never an option for me. Nigeria's rule was that I could claim citizenship up until I was eighteen if I wanted to do so. I never did and really can't say why. Nor did my parents. Indifference? An unwillingness to confront the red tape in a foreign country? A sense that I would never "need" it? A few months after the tense moments at the airport in Accra, I would have my own passport certifying me as American—the only citizenship I've ever had. My parents never dreamed that the American-ness of my citizenship—and that of countless others born to an American outside the U.S.—would ever be the subject of political debate. To them, my being of mixed race might be a contentious issue in my life, but when it came to citizenship, they foresaw no question. I was an American.
XXI.
My American passport took me many places over the years—to visit my British relatives, to a family vacation in Jamaica, and to France when I was fifteen.
The France trip was the summer after my sophomore year of high school, a few months after I'd tried to avoid going to the prom with Rob by inviting Frederick. It was a three-week exchange trip organized by the daughter-in-law of my Auntie Polly (my father's adopted sister). Aunt Polly's daughter-in-law, who taught French at an elite New England prep school, had invited me to join her class since I, too, was studying French in high school. Though the kids were all white, and all affluent, I quickly found a connection with them because they were worldly, familiar with places beyond their hometowns, unlike most of the kids with whom I was attending high school. Some of the kids listened to music on their Sony Walkmans. The Police's new album, Synchronicity, was a huge hit among these kids and allowed them to tune out during our long plane flight to Paris.
We traveled on the red-eye and emerged bleary-eyed into Charles de Gaulle Airport. We then boarded the Paris Métro, which would take us to our youth hostel. The train was packed when we got on and we had to stand in the center, hanging on to the shiny metal poles for balance as the train bumped along the tracks beneath the city. I gripped my purse, my hand on the zipper. The crowd got thicker with every stop and at one point I was jostled by what looked like a tribe of small, ragged children. Moments later, I realized my purse was unzipped and my wallet was gone. My Aunt Polly's daughter-in-law took me to the police station while the rest of our group settled in at the youth hostel. It's not like I'd lost much. I was fifteen. All I had in there were a handful of traveler's checks and a photo of my boyfriend, Mark. But I felt ashamed. I'd been assaulted by, of all people, small white children.
A few days later I stayed behind after our language lesson at a local university to ask the professor a question and then found myself walking back to our youth hostel alone. I came upon a small park where a little white girl of no more than ten was kicking the gravel out of her shoes. As I neared, she stopped what she was doing, looked up at me, and spoke.
"Pourquoi es-tu noire?" (Why are you Black?)
Decades later I would read the work of Frantz Fanon, who had also had a humiliating encounter with a little white French girl. But on that day as a fifteen-year-old walking through Paris I was alone with just my rudimentary French and my fragile sense of self.
"Pourquoi es-tu noire?" she demanded.
"Parce que j'ai de la chance." (Because I am lucky.)
I didn't believe it. But I wanted to. I hoped my words would send this little stranger home with some big questions. Maybe they'd even fuck her up a little bit. I didn't mind. As far as I was concerned, she was every white person who had ever questioned my right to exist, to be a regular person just going through my day without drawing the scrutiny or fascination of others. I didn't want to make excuses or give this little girl a lesson in anthropology. I wanted to fucking shine. I wanted to shine so fucking much that that little white French girl would ache to be me. Ache like me.
XXII.
While I was in France a new family moved into the vacant house next door. Soon after I got home I met their eldest daughter, Stacey, a girl my age who hit me smooth, sweet, and strong, like a shot of single malt scotch.
Stacey was from Alabama. She was white, but she knew Black people. As she told me stories from her childhood in the South, I secretly interrogated her words for every clue, studying what she knew to learn more about my own kind.
She had a stash of Prince's cassette tapes her parents forbade her to listen to. One tape was actually contraband—illegal for distribution in America—but Stacey was the kind of girl who could get her hands on such things. Sitting in my car with Stacey we listened to "Little Red Corvette," "Soft and Wet," "Bambi," "I Wanna Be Your Lover," and my favorite, "Controversy."
Stacey and I pierced our ears in her upstairs bathroom with some blocks of ice and a safety pin. She was my first transgressive friend, transgressing her parents' rules, transgressing rules for girls, transgressing whiteness. She crossed all kinds of borders into the liminal space, which is where she found me—floundering about like I was learning to swim and looking for something stable to hold on to.
XXIII.
By the time junior year began Frederick's family had moved away, making me officially the only Black kid in the twelve-hundred-member student body at Middleton High School. I knew to banter with the boys who mocked my hair and I shrugged off the observations about how un-Black my voice sounded. I worked my way to the top of the school's various ranks:
1. Doing very well in school, I was one of the kids our chemistry teacher teased for being "college-bound."
2. I had a boyfriend.
3. I was on the Pompon team.
4. I was vice president of my class. (My boyfriend Mark was president for the second year in a row, and I couldn't bring myself to run against him.)
This was my normal. If it was a balancing act, you'd barely have seen a wobble. I was coping without knowing I was coping. Like wearing a shirt inside out all day long without ever realizing it.
XXIV.
In the spring of my junior year, 1984, I was sixteen and the Reverend Jesse Jackson was making a serious run for the Democratic nomination for President of the United States. My parents were staunchly liberal and Democrats, and one night at dinner my mom started talking about Jesse Jackson and the headway he was making vis-à-vis the other Democratic candidates. Daddy shook his head. "It'll never happen." He wasn't interested in the oughts and shoulds or even the coulds; he was a realist. Listening to Mom, though, I began to feel a stirring in myself, a lightness from just contemplating the possibility of a big weight being lifted off my shoulders. Our shoulders. Doing the dishes that night, I found myself thinking about whether a Black man being taken seriously as a candidate for President might somehow lift this cloak of presumed inadequacy off of us all.
A few weeks later, when my English teacher asked us to write a persuasive essay, I chose to write about the Jackson campaign. None of my liberal friends thought he should win. None of them thought he could win. I didn't know enough about the various policy points to make a strong case for which candidate was most qualified to lead our nation. Instead, I wrote my paper about the symbolic significance of Jackson's campaign—that it was profoundly powerful for Black people to see a Black person run; that it was making us feel more equal. More human. More American.
When it was my turn to present my paper, I got up out of my desk, walked to the front of the class, and turned around to face my classmates, anticipating the power of the words I had written and my ability to deliver them well with my reading. Oratory, I knew from having given more than my share of speeches as a student leader, was a strength of mine. When I was done presenting, though, there was just silence, and in the silence I heard that neither my classmates nor my English teacher were terribly persuaded by what I'd had to say. In fact, my teacher felt I was so off the mark that he required me to rewrite the essay. When I stood before my classmates for a second time, it was evident from their faux-confused question-asking that maybe I was the one who was confused for thinking that the fact of Jesse Jackson's candidacy meant anything. To anyone. Who mattered.
XXV.
That same year the College Board awarded me a National Merit Commendation for being an "Outstanding Negro Student." In theory it was an honor. But it felt like an insult twice over: my own little Pyrrhic victory.
It was 1984 for Christ's sake. "Negro" was derogatory in our nation's racial lexicon—one step above "Colored," which was one step above the dreaded N-word. We were now Black. Even Afro American. How could the College Board not know this?
And I hardly deserved an award tied to race. A few of my friends were among the "real" National Merit scholars at our school, so this was stark evidence that I hadn't done as well as them on the PSAT; I was just at the top of the heap of Black folks, which was a heap we all knew I didn't belong to.
When a hired photographer came to shoot sports and clubs photos for our school yearbook, all us National Merit people were given a time to pose for photos as well. The other students goofed around in mock self-deprecation over their achievement and I stood off to the side of the group acting as if I hadn't really been invited. The photographer yelled at the rest of them to stand still and pose. Then, with my classmates finally assembled in two lines, the photographer pointed at me.
"You. Are you in this too?"
I swallowed hard and strode over to my classmates, and offered in a jaunty tone: "Here comes the Negro." Then I took my place on my tiptoes behind the second row.
I desperately did not want this attention, this so-called recognition for being a great "Negro" in the eyes of the organization that was every student's gateway to college. I also felt like an impostor for getting an award that should go to a real Black kid, some kid somewhere else who most certainly deserved the recognition more than I did.
XXVI.
We are a college-educated family; when it came time for me to think about applying, it was a question not of whether to go to college but where.
During spring break of my junior year, Mom and I took a road trip out of Wisconsin to the East Coast to visit a dozen or so schools in the span of ten days. After eight hours of driving Interstate 80 east through Illinois, Indiana, and into Ohio, we were ready to pull over for the night, only to learn there were no vacancies—none for one hundred miles due to a big convention, we were told. We pulled into a McDonald's parking lot outside of Cleveland, reclined our seats, locked the car doors, and slept in the car. The next morning, we went back into McDonald's to brush our teeth and wash our faces. There was little I could do in those circumstances with my hair.
By the following night we'd made it all the way to Hanover, New Hampshire, where we stayed in a motel. The next morning, we showered and I put on a pretty drop-waist dress that was lilac with little white flecks, and we headed off for our tour at Dartmouth College. Our tour began in the library, which was full of students at long tables and computer terminals, and I edged my way toward the front of the group so as to talk with the student guide while Mom hung back with the other parents. As we were headed out of the library there was some kind of commotion behind me, and I glanced back to see a Black boy who had leaned too far back in his chair at a computer terminal and was flailing to right himself.
Later, my mom told me that when I'd walked past that boy, she'd seen him do a double take and continue to follow me with his eyes to the point where he'd had to lean way, way back on two legs of his chair. Between the lines of Mom's story I could read her dreams for me: There are Black people out there. Not only that. Maybe Black friends. Maybe even a Black boy who would find me beautiful.
As she spoke I tried to act cool. But I could feel my heartbeat pounding in my veins. We drove from Hanover to Boston, to New York, and to Pennsylvania, visiting school after school after school, and I replayed my mother's story again and again in my head and interlaced it with the image I still had of that boy in my mind. Maybe there'd be boys like him at these colleges. Maybe I'd feel more normal once I got out of Middleton High. In the privacy of my own mind, I blushed.
XXVII.
At the start of my senior year, I was serving my class as vice president for the third year in a row and was also elected president of the student council. The Cosby Show debuted on NBC in September. With the show's father, Cliff Huxtable, being a doctor like Daddy, and the middle daughter, Denise, looking kind of like me, there was finally a fictional family on the TV screen that resembled mine. I was glued to it every Thursday evening, reading it for guidance about how to be someone like me.
I turned seventeen that November, a few weeks after the presidential election that reelected Ronald Reagan. My best friend, Diana, made me a huge birthday locker sign filled with words and images cut from the pages of Tiger Beat, Seventeen, and other teen magazines. She'd woken up extra early to get to school in time to tape it to my locker before my arrival. We did this kind of thing for each other. Her birthday was earlier in November and I'd festooned her locker just two weeks before.
Something about turning seventeen made me want to look like the woman I was becoming. Getting ready for school that morning, I'd pulled the curling iron through my hair over and over again, and smoothed it into a nice, sleek, low ponytail that would hang from the nape of my neck. I'd spent a few extra moments on my makeup, carefully drawing the charcoal eyeliner across my lids, swishing the black mascara along my lashes, contouring my cheeks with chestnut blush, and painting my lips raisin. I'd selected a beautiful black wool dress to wear, a professional cut with long sleeves, a round neck, and the shoulder pads that were the fashion at the time. I'd pulled nylons over my strong calves and thighs, and, to finish the look, I wore a pair of black patent leather pumps that made me three inches taller. Decades later I would read a short story by Julie Orringer that described a middle-aged woman as "no longer ripening but not yet deteriorating." Back at my house in Cherrywood on that November morning, I was ripening. Beautifully. And I knew it.
I drove the snowy route to school, pulled my car into a spot in the far parking lot, got out, and walked on tiptoe toward the main building, deftly avoiding the permanent fixtures of ice and lumps of hard snow that clumped on the asphalt in wintertime in Wisconsin. As I walked up to the main entrance of the school, I saw my reflection in the glass doors, my dark figure silhouetted against the bright white snow behind me.
I entered the school and headed left toward my locker, which was located in the bank reserved for seniors in the central hallway near the administration's offices, conveniently close to everything. Even above the din of student voices and slamming lockers, I could hear my heels clicking with precision on the shiny cement floor.
I could already see the birthday locker sign fifty lockers in front of me, with its five sheets of white paper taped one to the next to the next in a sort of vertical column with shimmering silver ribbons taped to the top and sides spiraling out into the hall. I felt a surge of anticipation of the attention I would get that day. A friend shouted "Happy Birthday" as I made my way down the hall, and I nodded, smiled, and shouted, "Thanks!"
When I got to my locker I stood and admired Diana's creativity, reading from top to bottom all the bits of language and imagery she'd gone to such trouble to cut out and glue on there for me. I opened the locker, put my backpack inside, and pulled out the books I needed for my first two classes. Then I turned and smiled at someone else saying "Happy Birthday," clanged the locker door shut, and twisted the combination lock a few times. I strode down the main corridor toward my first class feeling like I owned the place.
Some unknown minutes later, someone took a thick black marker and wrote "Niger" in three places on my birthday locker sign. Even spelled incorrectly, I knew what they'd meant. I spotted it in late morning during the passing time between classes and immediately my mouth went dry.
I stood with my back against my locker, affecting casual, as the other students opened and shut their locker doors. After an excruciatingly long three minutes of metal scraping on metal and the roar of chatter and movement, and a few more "Happy Birthday's," the hall began to empty as kids went off to their next class. The bell rang, signaling the start of class.
I walked quickly toward the school office, which sat at the crossroads of this hallway and the other main hall. The hallway ramped up just before the intersection, and at the top of it I paused, my chest heaving, my mouth still dry. I had to get my shit together. The glass-walled office was around the corner to the right. I took a deep breath, then drew my spine up straight, smoothed my ponytail, straightened my dress, plastered my most pleasing smile on my face, and strode with casual confidence over to the glass door of the main office and flung it wide open.
As President of the Student Council I knew all the secretaries. Marie sat at the main desk and didn't bat an eyelash over why I was not in class. I asked her if I could borrow a black Magic Marker. She fished around in her drawer. "Will this do?" I took the marker from her with a smile, thanked her, pushed the door open, and walked out of the office. At the intersection I scanned all directions, then turned left and walked quickly back to my locker. When I reached it, I looked down the long hallway once more to make sure no one was coming, and then, as the silence pressed down around me, I took the cap off the marker and began to draw neat black lines over each iteration of the word. I now had three black boxes where the words had been. I turned around and pressed my back into my locker, still clutching the marker, my knees sinking a little bit toward the floor.
At day's end I took the sign home. In the privacy of my bedroom I pulled my senior year scrapbook from the bookshelf above my desk and opened it to the first blank page. There, I pasted my birthday locker sign accordion style, so that it could be completely unfolded to resemble what it had looked like hanging on my locker. Before closing the scrapbook, I took a pair of scissors and, like a surgeon excising tumors, removed the three iterations of the shameful word, then threw them in the trash. I closed the scrapbook and returned it to the shelf containing the recorded history of my childhood.
XXVIII.
Over the Christmas holiday I typed my college applications on a brand-new Apple 2E computer my parents were among the first to buy. In March 1985 the first Internet domain name, "symbolics.com," was registered. In April, I accepted an offer of admission to Stanford University.
A classmate, Harris, had applied to Stanford but had not gotten in. Harris and I were in pre-calculus together (the highest math class at our school), and it was held during the seventh and final period of the day. One day in April, right after the bell rang signaling the end of class, Harris's father walked in, sat down at an empty desk next to mine, and began talking to me in a playful tone.
"Sooo, you got into Stanford?"
I looked up at my friend Harris and silently asked, Why is your dad here? Then I replied. "Yes."
"So, what were your SAT scores?"
I responded.
"Do you think it's fair that you got into Stanford over Harris when his scores were higher than that?"
Harris was not the president of the student council. Our grades were roughly the same. But I had stolen his spot at Stanford with my Blackness.
XXIX.
A Black male sophomore enrolled at Middleton High School that spring. I never met him. Late one day, word spread that the boy had been beaten up in a second-floor bathroom and the N-word was found scrawled on the tiled wall. A shudder ran through the school community—to have both a new Black male and violence against a Black male in our community felt fictional yet inevitable. It was as if the violence were playing out according to a trope, caused by the Black male's arrival or by something he had done consistent with stereotype, instead of racial hatred lurking beneath the surface eager to pounce. I had seen it pounce. It had pounced on me. But there had been no witnesses and I hadn't told.
I was in economics class when I heard the news, and a male friend of mine said he wanted to escort me to and from class and to and from my car. I don't think I've ever been offered a greater act of service in my life.
The school board convened a meeting to "think about these race issues" and invited Daddy to "help." My boyfriend Mark and I were also invited, in our capacity as student body leaders. Together, the three of us walked toward the district building. I could sense from the stiff strength of Daddy's stride and the impatient look on his face that he was frustrated. All of a sudden our Blackness was front and center. What's more, they needed us to be Black. To help them do something. I could tell that Mark was frustrated too—his lips were pursed in a thin line and his normally warm eyes were cold and distant. Later I'd ask him about it and he'd tell me he was frustrated that the conversation had to happen at all; that in contemplating imposing a hate speech code, the school was curtailing civil liberties.
I walked between the two men who mattered most to me and mulled frustrations of my own. Normally Mark and I would hold hands as we walked but I sensed that to do so in these circumstances was to take sides. And I was too old to hold Daddy's hand.
I walked, between the two men who mattered most to me, alone.
XXX.
Unlike in Reston where our Jack and Jill chapter was made up of families who lived within a five-mile radius, in South Central Wisconsin we had to draw a much bigger circle to find enough middle-class Black families to field a monthly meeting.
The absurdity of gathering up all the Black people in the middle of nowhere Wisconsin and trying to call it meaningful grated on my nerves. It was me the Black kid from Verona meeting up with some other Black kid who lived three towns over meeting up with the handful of kids who lived in Madison. Maybe there were seven of us. Somehow, our parents seemed to think, spending a few hours together every few months would—what? Help us find community? Help us be ourselves? One month the Greater Madison group (as we were known) met with the Milwaukee group, which had about twenty-five kids, and we all went to the mall together. What teenager wants to hang out with teenagers who know each other but don't know you? I trudged along from store to store, wishing the charade would end.
In spring of my senior year, the Milwaukee chapter hosted a cotillion. I'd never heard of such a thing so my mother explained: a cotillion was like a debutante ball (I didn't know what that was either) and this one would be a big fancy gathering of Black people with some kind of "coming-out" ceremony for the girls and formal dancing. She spoke of it breathlessly and wide-eyed, as if she was my fairy godmother and this was her big chance to make me part of Black society in one fell swoop.
I loathed the idea of going but could not say no. Could not tell my Mom I had no interest in the artificiality of a formal ritual with a community of people I did not know. Could not remind her that I had a serious boyfriend with whom I'd much rather spend a fancy evening. She was trying so very hard to raise a Black child right. I didn't want to appear not to care about her efforts to salvage me.
Mom disassembled the bright red dress I'd worn in a madrigal group junior year and used her old Singer sewing machine to piece the fabric back together to resemble a ball gown. She bought some stiff rustling fabric that would go underneath the dress and protrude it outward from the waist like a cone. She commandeered my older brother Stephen to "escort" me. I was seventeen. Stephen was thirty-four.
In the late afternoon on the day of the event, resigned to the fact that I have to go to this thing, I do my makeup carefully, then press and press and re-press my hair to try to smooth it into some flippy swoopy design like I know the other Black girls will be sporting. I put on the black character shoes I wear for choir performances, pull on the stiff fabric that goes underneath the dress, and then finally I carefully step into the dress itself and my mother zips it. In the mirror I see that I'm playing a part in a play and am not sure I know my lines.
My brother Stephen arrives at the door in a tuxedo and hands me a corsage. My parents take a few pictures and soon we're off to Milwaukee, an hour and a half due east of Verona. We walk into the hotel lobby and I look around, desperate to find some face familiar from the Jack and Jill mall outing. But all I see is people who smile and laugh with one another and who do not smile and laugh with me. My hair begins its inevitable process of mushrooming into a pouf.
Stephen and I enter the grand ballroom and I make a beeline to an empty round table where I sit and catch my breath. Then I take in the scene swirling around me. Handsome boys in tuxedos dance with girls in elegant department-store dresses. A trio of girls with properly relaxed hair laugh at a food station. A young couple stands with their arms around each other at the foot of the elegant stairway that leads up to God knows where. I can't picture myself doing any of these things, me with my homemade dress, frizzy hair, and my grown man of a brother for a date. A cute boy walks confidently in my direction with a grin spreading across his face and I am about to smile and make eye contact when I see that his destination is a girl a few feet behind me. No one will ask me to dance that night and I already know it.
I go to a nearby food station and come back to our table with a plate loaded with savory hors d'oeuvres. I'm not used to feeling ugly, but that night I feel not only ugly but downright homely. Normally I'd be in clothes I liked and I'd have my hair in a ponytail and I'd look cute, cute enough for someone to date, cute enough to be on the Pompon team. I keep trying to smooth my hair down, pulling at the underneath bits near my ears, smoothing the entire back with the palm of my hand, absentmindedly trying to pull it back into a low bun only to remember and re-remember that I didn't even bring the always-have-on-hand hair tie that might now rescue me. It's like my hair is getting drunk and making a scene and I can't do a damn thing about it.
As the evening becomes a rousing party of rhythmic music punctuated by a voice on a microphone regularly announcing the next stage of this highly scripted debutante coming-out event, I fall rapidly to the bottom of the social ladder. Stephen asks me to dance because he also knows nobody is going to ask me. As we dance he doesn't make eye contact with me and instead gazes over the top of our outstretched clasped hands, which I presume is intended to preserve my dignity. But knowing that he too senses my dignity is falling apart at the seams feels like proof of it, and becomes the ultimate insult.
We return to our table and I keep looking at the black Swatch watch strapped to my wrist. I refuse Stephen's second and third requests to dance. I will the minutes to go by. Finally Stephen gives up and we sit there in silence. When it is nine thirty p.m. I feel we've been there long enough to prove I'd tried to participate, and I tell Stephen I want to go home.
We are largely silent on the long drive home and I cringe at the report he might give my parents. Of the failure at being a Black teenager I had been.
XXXI.
I'd told no one about my locker sign, and I'd go on to tell no one for decades. Not my parents, not the school administration, not my boyfriend Mark, not my best friend, Diana. For more than twenty years, though, the truth of that day hunkered down inside of me and metastasized.
I was the Nigger of my town.
DESPERATE TO BELONG
I.
College, I hoped, would be my chance to make Black friends, even learn how to "be Black." Whatever that would mean. If I was praying for anything in the summer of 1985, then I was praying for that to be true.
I hadn't had my hair cut by anyone but Mom since the disastrous cut in sixth grade that left me looking like a boy. But in the waning days of summer in Wisconsin, I walked into a white salon and asked the stylist to give me a short haircut like Lisa Bonet wore on The Cosby Show—no more than an inch long everywhere except the front, where I wanted it angled upward to form an overhang that would hover above my forehead, defying gravity.
I was preparing myself to join my new college community. I also wanted to go out for the crew team at Stanford—a sport I'd heard you could do without having done it in high school. The early morning workouts would demand easy hair. Something I could wet with the water from the sink, run a bit of gel through, and not have to worry about any longer.
As I watched the stylist snip my curls, watched my childhood hair fall to the floor, I saw the possibility of a new life awaiting where I wouldn't have to worry any longer.
II.
In September of 1985 my parents fly out to Stanford with me and help me move into my dorm, Branner Hall, home to 10 percent of the freshman class and predominantly white. In the late afternoon, we and my 160 dorm mates and their families gather in the huge lounge at the center of the first floor to hear a welcome talk from the faculty member who will be living with us in the dorm and supervising the dorm staff—History Professor Kennell Jackson. As I sit on the dark turquoise rug with my back against a white square column, my knees hugged to my chest, my eyes glued to Professor Jackson, I can't shake the question: What is he?
Kennell is in his midforties, tall, with a bit of a paunch protruding against his crisp white shirt and with enormous feet clad in leather dress shoes at the base of his khaki pants. He has an oblong head, a flat nose, twinkling eyes, and facial hair that covers the entire lower half of his face, sparse, but manicured like a putting green. He is balding, and where he has hair it is short and coiled like his facial hair, and blends well into his skin, which is reddish, like faded terra-cotta.
Kennell had walked to the front of the packed room in an aw-shucks, almost sheepish manner, deliberately casual though, I thought, as if the casualness was a carefully cultivated affect. I'd glanced over at my parents who stood against the far wall, and seen them exchange raised eyebrows and smiles with one another. Maybe they were having the same impression.
When Kennell begins to speak he keeps his elbows at his side and pats the air around him with the long fingers of his large hands, like a pianist plonking all ten fingers on the piano at once, and I hear in his Virginia drawl both words and sounds I'd never heard before. He tells us stories in fragments, and goes off on tangents, and just when I am wondering whether all of my professors in college will be this weird, he ties all of his thoughts together, masterfully. He closes with a look of mischief flashing across his face, eyes crinkling, lips pursed in a wide smile, explaining what we could expect in the coming year as if we students are embarking upon an unprecedented adventure together.
When Kennell finishes speaking we stand in clumps near him waiting to introduce ourselves. The lounge slowly empties, and I meet up with my parents in the dorm's courtyard to begin to say good-bye. I stand on my tippy toes to hug Daddy's tall frame, then lift my head up to receive his kiss as he lowers his head to touch his lips to mine. "You'll be okay, baby," he pronounces, putting his strong hands on my shoulders. I nod quickly and eke out a high-pitched "Yeah." In seventeen years I'd only seen Daddy cry once—when he was using a screwdriver and it slipped off the screw and plunged deep into the palm of his left hand. So as my sixty-seven-year-old father tears up under Palo Alto's cloudless blue sky surrounded by my new classmates and their parents outside of my new home, both he and I know to look away.
I turn to Mom, who, at five feet and half an inch, seems to be about half my father's height and is smiling wide. Any anguish she might have been feeling over sending her baby to college two thousand miles from home seems to have given way to a kind of glee—the same glee she'd shown when the kid almost fell over in his chair in the library at Dartmouth. It's her race strategy look. Her "We Gon' Be Okay" look. She throws her arms open and pulls me into her tight embrace. As she squeezes her final hug into me she whispers, giddily, "Isn't it great you have a Black Resident Fellow?"
It astounds me now to say it but I'd had no idea; until Mom told me Kennell was Black, I was the lightest-colored, most different-looking Black person I'd ever known. Daddy and Mom climb into their rental car and I wave and wave and wave until they drive out of sight. Walking back to my new so-called home, I feel uplifted. Maybe if someone who looks like Professor Jackson belongs within Blackness, well maybe there's a bit of room for me in there too.
III.
The Black community on the Stanford campus was approximately 6 percent of the undergraduate student body at that time, or about 450 students. During new student orientation, I see a flier for a welcome event sponsored by the Black Community Services Center and I decide to go.
I am nervous—conscious that I really only understand how to be myself among white people. Whites are not just what I am used to, they are really all I know. The event is being held at Stanford's African American theme dorm, Ujamaa, all the way across campus from my dorm. Walking up the front path to Ujamaa, I worry about whether I'll be accepted. Know how to behave. Know how—who—to be. How to be a real Black kid.
Through the multipaned window a bunch of kids are gathering. They look happy and comfortable like they might already know one another, even though that isn't possible. I pull open the heavy front door and head in the direction of the noise—the dorm lounge. There are a few empty chairs and spots available on couches but I don't feel like I should take one. I stand off toward a corner, my back against the wall.
There is music and laughter. An air of relaxation. Of exhale. I'd been to a lot of meetings over these first five days at Stanford and this one is by far the most warm. Yet I am overcome by a sensation of being out of place. I feel eyes glance on me, then just as quickly look away. None of these real Black kids has a white parent, I say to myself. None of them was the only Black kid in their high school.
Do I just feel out of place, or am I actually out of place? If I don't belong here, where do I belong?
A series of older students stand at the front of the lounge and proceed to tell us this and that about upcoming events and how to sign up for various clubs and organizations. After the presentations I will myself to go up to the sign-up table to put my name on the list for a few clubs, and while there, finally I try to engage one of the upperclassmen in a conversation. Trying so hard to connect, to be polite, to be liked, I can almost see his body recoil slightly when I speak, as if my effort to belong is so naked as to be pitiable.
I was the Blackest thing at Middleton High School, but having sat among real Black people for the past sixty minutes I'd learned I didn't have a clue about what mattered to Black people, or maybe even what it meant to be a Black person. The music they'd played at the outset was not familiar to me. We didn't have cable television out in Cherrywood, let alone a Black radio station; the only Black music I had listened to was Stacey's mixtape of Prince songs. The pop cultural references that made them laugh or nod in agreement were as meaningless to me as foreign words. They punctuated their conversations with shorthand and slang words, the ease of it like a secret code among them. And I'd learned from the few clues that had come my way in life so far that music, pop culture, and language were the things that were stereotypically Black. At this meeting I was like a kid climbing the ladder to a glorious fort I'd discovered hidden in the trees, and when I got to the top I could see kids playing through the window but when I put my hand on the doorknob, it was locked. Could someone come and let me in, my heart bleated, to no one.
Many years later I would learn that Blackness was less about skin color or hair or language and more, far more, about a lived, conscious committedness to issues that impact Black people, and I would accept my light-colored skin, the sound of my voice, the biracial kink of my hair. I would enter rooms of Black peers and have my smiles returned, and make conversation without feeling shame about my choice of words or manner of speech. I would learn that I had been wrong in perceiving that all Black people thought and acted the same way. And, having set those misperceptions aside, I would be able to locate myself within Blackness and make meaningful connections with Black people. Friends. And I would learn that all the years I thought I was being myself around white people had instead been more like a performance of the self. A performance designed to meet their approval, assuage their concerns, calm their fears, succeed at overcoming their negative judgment. I would one day fully embrace my Black self like a long-lost mother, hold myself in my own arms, singing "Sometimes I Feel Like a Motherless Child," whose lyrics were not meant for me but were nevertheless resoundingly true to my experience.
But on that day in 1985 as an almost eighteen-year-old, my mixed-race ancestry and white-community upbringing are, together, the enormous elephant in the room—perhaps only in my own head, perhaps in the minds of others—and I cannot get past it. Cannot see others around it. And therefore cannot be seen. I am in an unyielding one-way dialogue of apology for my ancestry. For my own existence. I am showing up as what I'd later learn Black people call an "Oreo"—Black on the outside but white on the inside; a label on one far end of the Black identity continuum that starts with Malcolm X and ends with Uncle Tom.
I walk back to my mostly white dorm feeling frantic. I'm not Black enough, and I'm certainly not white, not ever white.
I continue my hunt for clues about how to be. I continue to watch The Cosby Show, then its spin-off, A Different World, with my dorm mates on the large TV that always seems to be on in the Branner Hall lounge. Then, on January 28, 1986, we are watching the launch of the Challenger space shuttle and it blows up in our faces. And for some moments in time, race doesn't matter. We are all just grieving Americans.
IV.
Far from finding my community at college, I found myself on an island. Alone.
Who am I? Who decides that? Where do I belong? Do I belong anywhere? Do I exist at all if no group of humans will claim me?
As I search for familiarity and try to make new friends, these questions are my constant companions. These days of searching become weeks; the weeks become months, and the months become years. I make friends mostly with white people and with the small number of Black kids who also live in my dorm, row on the crew team, participate in theater, do student government, or take the pre-law classes. I would go back to Ujamaa just once in my time at Stanford for a study session in someone's dorm room. It was sophomore or junior year. Walking through the main hallway, I recognized a number of the brown faces and could smile and nod, but I didn't know anyone's name. And they did not know mine. I could not count them as friends. Although I now had more Black friends than in the aggregate years of my childhood—and other friends who were non-white—by staying away from Ujamaa I'd exiled myself from the heart of the Black community on campus.
V.
In November 1985 I turn eighteen. A world away from the familiarity of home, I become, legally, an adult. And as imperfect as the home of Verona, Wisconsin, had been for me, I feel a loss when I can no longer retreat there, to that place where, above all else, I knew my parents loved me. I am becoming a Black woman, treated in the world as such, and lacking the armor of self-love with which to withstand that treatment.
That winter, I bike with a handful of friends to the Stanford Shopping Center, a large shopping mall on the edge of campus which beckons with its expensive stores, gorgeous fountains, outdoor sculpture, and flowers in a perpetual bloom. The open-air pavilions ooze prestige, power, and exclusivity. The regular customers, the mostly white women I'd later come to know as "ladies who lunch," are a permanent part of the façade.
One friend had grown up in Palo Alto and while we are shopping she bumps into a woman she knows from childhood. We stand in our gaggle around the lady and our friend introduces us to her. When she gets to me, the lady takes note.
"Oh you go to Stanford too?" The sound of her voice is intrigue, wonder, even amazement.
"Yes."
"Oh! What team are you on?"
[The truth was I was a walk-on with the crew team.]
[The truth was, that's not what she was trying to get at.]
After we part ways with the lady, I try to laugh with my friends about it. "Can you believe she asked that?" I say casually, seeking their attention over the implied insult that I hadn't gotten into Stanford on my academic merits, and wanting to dismiss it as anomalous at the same time. Some of my friends get it, but not all.
As we make our way through the Stanford Shopping Center, my friends almost certainly forget about the incident, but it stays with me, lurking around every corner, staring back at me from every cosmetics counter mirror and every dressing-room wall.
In the aisles of the various department stores and boutiques, the store clerks' penetrating eyes seem to bore a hole in my back. I try to catch their eye and deliver a reassuring smile that says I'm Black, but I'm not here to steal from you, while smoothing my hair with my hand. But they either won't meet my eyes or no smile is returned. I begin to feel self-conscious about where I put my hands, whether I put them in my pockets or keep them free. I feel not only out of place but unwanted even, perhaps dirty, guilty of something, like my mere presence somehow lessens the value of their clothing, cosmetics, and shoes.
I'd signed up for a credit card from one of the many vendors hawking them at our campus student center. Desperate to refute a clerk's presumption that I am out of place in her store and perhaps up to no good, in the months and years to come I plunk that credit card down time after time after time at these expensive stores, spending more than I can afford to try to buy the certainty that I belonged. As if my purchases can refute their stereotypical assumptions: I'm Black but have money! I'm Black but can afford to shop here! I belong here. See? I belong!
For twenty-five years, as even my husband would come to know, even as I moved into the ranks of the upper middle class and held a position of some prominence in this very town, my steadfast rule was that I would never go to the Stanford Shopping Center unless I was dressed up, and specifically unless my hair was tame by white standards. For decades I worked hard to prevent those questions from coming. From ever penetrating me again.
VI.
Meanwhile, I am keeping a secret.
When in January I'd opened the envelope containing my first quarter grades, I saw a B, a C, and a D on my transcript. My 2.0 GPA was the proof I'd been expecting all along that people like me—Black, female, from the Midwest—in fact did not belong at a place like Stanford. To add insult to injury, the D was in Communications 1—the stereotypical "easy" class at any college.
If my grades get even a hairsbreadth worse, I could flunk out of Stanford.
VII.
On a routine call with my parents in February Daddy asks, "Baby, how'd it go fall quarter?" I begin to cry, and spill the truth, and relief washes over me. It's the volume of reading, I tell them—so many texts to get through really quickly—and papers longer than I'd ever had to write at Middleton High. Daddy and Mom hold on to me over the phone. Tell me they love me. Tell me I can do it. Tell me I have what it takes to succeed. Urge me to go get help.
I get help—ironically, from the very academic advising office I would manage twenty-five years later as dean—from an advisor who asks me about my habits, then tells me that I have what it takes intellectually but my study skills and time management need a lot of help. And she strongly urges me not to choose classes based on what "everyone" is taking and instead to take classes that sound interesting to me. When spring quarter starts in late March, I flip through the course catalog and find a class that sounds right up my alley.
It is a political science class that will survey the history of the doctrines of civil rights and civil liberties in America. The texts look engaging and rigorous. The format is going to be Supreme Court case studies, which excites me because I think I might want to go to law school. And the professor—a young white guy named Jim Steyer—is already peppering his lectures with war stories about his time at the NAACP Legal Defense Fund in New York. And he is cute. Which doesn't hurt. By midway through the quarter every one of the two hundred students in that class know we'd stumbled upon something special. And I, to my great relief, am managing the very intense workload, keeping up with assignments, and making analytical connections at a pretty deep level. I can feel my body healing from the wounds of intellectual inadequacy inflicted by the first quarter.
VIII.
A few weeks earlier, The Color Purple had vied for the Academy Award for Best Picture. Alice Walker's Pulitzer Prize–winning novel was now in the hands of blockbuster director Steven Spielberg, and the cast included Danny Glover, as well as new faces on the film screen—Oprah Winfrey and Whoopi Goldberg. They'd received eleven nominations in all for this violent, haunting, exquisitely sad, redemptive depiction of the lives of Black women. But the white woman's African journey Out of Africa, which began with Meryl Streep saying, "I had a farm in Ahf-ree-kah," won the day. The Color Purple would tie for the dishonor of having the most Oscar nominations without a single win. Two years later, a film about a white woman who got chauffeured everywhere by a Black man played by Morgan Freeman would get the Oscar: Driving Miss Daisy.
IX.
Later that spring, Professor Steyer asks a very tough question in our civil rights class, which was not unusual. What is unusual is that I know exactly what he is getting at and I ache to respond. But to date I'd never raised my hand in a class at Stanford and still don't dare to do so. Besides this is obviously a really complicated question—no one else is raising their hand. My fear of being wrong, of being Black and wrong, silences me even though I know I have a good idea here. Scanning the huge room for potential volunteers, Steyer glances at me. Something in my face must be showing him my brain is working overtime. He nods once at me and raises his eyebrows, signaling that I should speak up. "Well," I begin, clearing my throat and playing with my hair. And then I keep on talking.
Steyer, never one to downplay a dramatic moment, folds his arms across his chest, leans back on one heel, and starts nodding his head vigorously as I talk. So I keep going. My classmates—watching the clear evidence in Steyer's behavior that I am saying good stuff—begin scribbling down what I am saying. I am teaching my classmates. I am speaking from a place grounded in knowledge and bolstered by confidence. With a voice pushing through the brambles out into the clearing.
This is the starting line of my efforts to be better than whites expect a Black person can be. A race I'll run—and try to win—for the next twenty years.
X.
Have I mentioned my roommates? They are a white girl from a very wealthy family in Queens, New York, and a well-to-do Thai girl from Bangkok who goes by the nickname Pinkie. The roommate from Queens tells me her older brother likes to joke that one of us is "Pinkie" and the other is "Blackie."
XI.
"Screw Your Roommate" is an old Stanford tradition. Your dorm throws a party and your roommate sets you up with a blind date from another dorm. When it comes time for Screw Your Roommate in the spring of my freshman year, my roommates pore over the book of pictures of the entire freshman class to find a boy suitable for me.
They point out a picture of a boy in a neighboring dorm. He doesn't look particularly attractive. And from the way they'd skipped from Black face to Black face as they perused the pages of the book, it is clear the only reason they'd chosen him for me was race. It's a throwback to Daddy's old joke about Bates College admitting one Black woman to complement the one Black man in each class, "so they'll have someone to date." Back in the 1930s. And the deliberate race-matching feels like me and Frederick all over again, except this time the matchup is being done to me instead of the other way around.
Inside my head I am screaming what I cannot say, which is, "Wait, aren't I good enough for a white boy?"
I go to the party with this boy who is nice enough but who seems about as interested in me as I am in him. It is impossible to know whether we might have had an interest in each other had the sole reason for our being thrown together in the first place not been race. Had the circumstances not seemed so presumptive, racist, offensive, I might have been able to enjoy myself. And him.
XII.
That first summer after college I return to Wisconsin and live with my parents for the last time.
Mom is a student in her own right now, pursuing at UW Madison the PhD she'd interrupted when Daddy's work took us from Madison to Reston, Virginia, back in 1977. I know I have to get a job, and look through the classified section of the Wisconsin State Journal to find one. I never dreamed of asking Daddy if I could have a job in his office—the kind of gig many of my college classmates were getting that summer.
I come across corn de-tasseling—a sort of quintessential Wisconsin job that entails walking up and down the rows of corn and removing the pollen-producing tassel from the top of each plant and dropping it to the ground, a manner of cross-pollination. It pays minimum wage, $3.35 per hour.
I show up for work on the outskirts of some farm and ride with ten or so other people—all male, all white—in the back of a truck that threatens to jostle us back and forth and into each other as we make our way out to the crop. But I am strong from rowing crew and can keep my body rigid as the ride tosses us around.
I march up and down the rows de-tasseling the stalks with little fatigue. On our water breaks and lunch break my coworkers eye me with uncertainty. I keep to myself. I hadn't realized that I should protect myself from the sun and instead had worn a tank top and shorts because of the heat. At the end of the first day I am considerably darker head to toe and the skin on either side of my tank top straps is blistered a purple brown.
At the end of the second day my coworkers and I come in from the field and stand where the crops end and the dirt road leading back to the main road begins. We form a circle around the foreman, who is telling us what we'll tackle tomorrow. The truck to take us out sits idling. I wipe the sweat from my forehead with the back of my hand, look at the wet dark slime that accumulated there, and have an epiphany. I need to work. But I don't need to do this work. I am a high school graduate, which means I have more options. I quit and get paid right there on the spot: a gross wage of $53.60 netting me $35.90.
I improve my working conditions tremendously by becoming a bus girl at Perkins, a twenty-four-hour diner franchise one step up from a Denny's. I wear a brown skirt, a white blouse with a little brown tie, and a white ruffled apron. My job is to clear, clean, and set tables and mop the bathroom floors. This job also pays $3.35 an hour. But I had moved from the fields to a building where I can go to the bathroom whenever I need to. I shower when I get home, but when Mark and I make out I still smell a bit like fried food.
Early one evening before the dinner rush begins I am walking through the aisles with my gray dirty-dishes bin and pass a Black male customer seated at a table with two other folk. He calls me over. "Aren't you George Lythcott's daughter?" I say yes and smile and make chitchat and he fills in details for his companions. He asks me where I go to college and I tell him. Then I nod in the direction of my manager who is coming toward me, and I sidle over to the nearest table that needs clearing. I put the gray bin on the Formica tabletop and begin piling dishes in.
"She doesn't have to do this work, you know," I hear the customer tell my white manager as I'm wiping the table down. "She'll be a sophomore at Stanford." I feel a strange discomfort as he says this, like I've been outed. I couldn't have explained it then, but I was trying to prove something with this job. I wanted to punch a clock, work hard, and get a paycheck. I already felt an unease from being able to flit through life using my parents' name or that of the university I attended. I didn't want to be given unearned things. A summer spent wearing a uniform and earning minimum wage was a chance to pay some dues that many people who look like me had no choice but to pay.
Yet through my unease I can also hear in his voice that this stranger is happy for me. Proud of me. Proud that I am lifting myself and by extension all of us up and away from this kind of work. He sees me not as a white-talking biracial girl but as a Black kid making it. Maybe the commendation from the College Board wasn't that inaccurate after all.
XIII.
Right before I am to head back to college for the start of sophomore year, Mark leaves on his two-year Mormon mission—a veritable exile from family and friends. I get invited to a party hosted by one of the Black kids in Madison I'd met through Jack and Jill. There, I meet a boy. He is a little darker than me, and a little taller, with trim facial hair and an interest in politics. In conversation about current events with four or five other people, we prove to each other that we care about the same things. We then joke at a level the others don't get. At the end of the evening he asks me out.
We make a date for the following Thursday. This was a real boy-asking-me-out-because-he-likes-me situation, not a racially based prom ask or a consolation prize or a blind date.
We go out. Dinner? A movie? All I remember is being in his car in the hour before my curfew. I felt the need to do things without being asked. To take the initiative. Be aggressive. Be into it. We began kissing. Touching. I felt him through his jeans, unzipped them, and leaned over.
I didn't know where this would go, if anywhere, and it did in fact go nowhere. I barely thought of him again. But I could finally stop berating myself for never having gone out with a Black guy. I used him. Maybe he was using me, too.
XIV.
In the fall of sophomore year, I make the cast of a campus musical, and at rehearsal one day I have trouble mastering a dance step. This embarrasses me. For solace I turn to another Black girl in the cast and gasp in mock self-deprecation, "I should be better at this—I'm Black!" The girl stares at me in cold silence, then abruptly turns away and strides off into the wings, glancing back at me once with an unmistakable look of disgust. I stand there, stammering and gesticulating with my hands, trying to explain what I had really meant.
What did I really mean?
XV.
I fill my course schedule sophomore year with classes in American history, literature, and government on my way to an American Studies major, which not only nourishes me intellectually in a way I'd never experienced before but also provides a foundation for the law degree I plan to pursue after college.
Professor Steyer picks me to be one of only two sophomores in an upper-level seminar on civil rights. We are assigned a case to brief and debate, and I choose Drummond v. Fulton County Department of Family and Children's Services, a case of white foster parents, the Drummonds, seeking to adopt the mixed-race boy they'd been raising since he was one month old.
The case pulls hard on my heart—the white parents, on the one hand, claiming that they can love and raise this Black boy to a healthy and whole adulthood, the Black social workers, on the other hand, claiming that the boy needs to be raised in the Black community by Black parents. The social workers are effectively saying that someone like me could be harmed by being raised in a white community. I could not bear the possible truth of the truth of this so I take the side of the Drummonds.
When I finish my oral argument and get my written brief back, I have two important pieces of feedback from Steyer:
1) Your oral advocacy is fantastic.
2) Your writing needs a lot of work.
XVI.
American studies.
The rules of America as freshly written in 1787 classified my ancestors as chattel—property of a white man—and as three-fifths of a person for political representation purposes. Having it both ways, to suit the white men who ran the South. Later, after the Civil War and the unmet promise of Reconstruction, the laws and policies of America treated us like stray dogs—expecting us to be content with the scraps that America was generous enough to throw our way, tolerating us but not entitling us to come inside the house—or, at least, asking us to remain content with entering through the back door.
This was hardly news to me. The imperfect nature of our so-called union—the unmet promise of "liberty and justice for all"—were playing in the background of my American childhood. The miniseries Roots, which came out when I was ten. The strident conversations between my parents and their liberal friends over cocktails or a barbecue. Daddy's tales of his Jim Crow childhood and young adulthood. My older siblings' accounts of civil rights activism. My mother forcing an indifferent school to test me for giftedness. The debates over school busing, voting rights, and affirmative action that made appearances on the nightly news. Also on the news? The Klan. One night as a child of maybe eight or nine, as I sat with my parents for their evening ritual of watching Walter Cronkite, I saw a cross burning, saw hordes of people with white sheets for robes, white masks with holes cut for eyes, saw a child, someone younger than I was, dressed like that.
But becoming a student of these issues—studying the work of historians, reading Supreme Court decisions, comprehending that the Reagan administration and the Rehnquist Court were rolling back the government's commitment to redressing past discrimination in voting, employment, housing, and college admission—left me heavy with ache. I believed in America right up until then.
Now I felt severed from a love I would never again feel.
Motherless.
Homeless.
America.
I'd thought she wanted me.
XVII.
"Hey-hey, ho-ho. Western Culture's got to go!"
I chant that with the Reverend Jesse Jackson when he comes to campus to support our protest of the university's freshman humanities core requirement: Western Culture, which features only the work of dead white men. Hundreds of us, students of every hue and gender, march with Jesse under his Rainbow Coalition flag. We are the inheritors of the civil rights struggle of his youth. It is our turn to weave some new threads into the American tapestry and narrative.
The faculty senate votes to replace Western Culture with Cultures, Ideas, and Values, which comes with a more diverse set of texts. Reagan's secretary of education, William Bennett, takes to the Wall Street Journal to excoriate Stanford's president, Don Kennedy, for caving to multiculturalists. A few students found the Stanford Review to provide a counter-perspective steeped in conservatism and libertarianism. Looking over my shoulder before doing so, I subscribe.
XVIII.
I am under the influence of many things in college, including:
Malcolm X: teaching separation over integration as the way to our salvation, as adamant about the need for separation of the races as any racist white man.
Amiri Baraka: mocking me in "Poem for HalfWhite College Students"... when you find yourself gesturing like Steve McQueen, check it out, ask / in your black heart who it is you are, and is that image black or white...
The Stanford Review: spelling out my enemy's blueprints and battle plans. But why the annual subscription when I could pick it up from a newspaper kiosk? I tell myself it's because I believe in free speech and their right to exist. But years later, when I can finally interrogate this self, I realize I subscribed because I was scared to death of these unhooded whites printing their disdain for our existence. I thought well if I'm on their subscriber list maybe they'll leave me alone.
The Mormon Church: preaching a different kind of salvation. I visit every Sunday morning by walking out of my dorm and turning left, a twenty-minute walk to a building where I learn how to pray to a God who discriminated against Black men until 1978 and I learn the Missionaries' lessons (although I stop reading the Book of Mormon on page 66 where it says a tribe of people were cursed with a skin of blackness) and when they ask if I am ready to be baptized I say yes and in spring of 1987 I am baptized in a full-water immersion ceremony and at Christmas later that year I will announce to my parents and half siblings that I have joined the Mormon Church and their jaws will drop and their mouths will fill with silence and for once in my life I will have this family's complete attention.
I am an island.
I am on an island.
My family abandoned me on this island.
And I will not be judged.
XIX.
The summer after sophomore year I hold an internship in Washington, D.C., with a public interest law firm located on E Street near Union Station. One of their side projects is DC Statehood, a movement to end the systemic disenfranchisement of DC residents—most of whom are Black—who are taxed by the federal government but have no voting representation in the House of Representatives or the Senate. I attend meetings with impassioned activists, draft language to help our cause, and, of course, I do the obligatory intern things like get coffee and run errands.
One day one of our attorneys leaves something behind at the office and, like any good intern, I volunteer to run back and get it. But I am in a pencil skirt and heels so it is more like a march than a run. Making my way up E Street, I hear the voice of a stranger behind me. "I hope you don't mind my saying this, but you have gorgeous calves." The voice is warm, soft. I know my calves are strong from rowing crew. My quads and biceps, too. I turn around and see a Black man, smiling. I blush. Smile. Thank him. Turn around and keep walking. And keep smiling. This wasn't a creepy whistle from a construction site. I don't feel assaulted by his words. I feel seen.
XX.
The room I sublet that sweltering summer is in a house that has seen better days. Roaches, no air-conditioning, and seventeen roommates in all—mostly students from Stanford and Duke—in a house meant for a single family. The four bedrooms, basement, and sun porch are crammed with our bodies and belongings.
It is 1987, and one particular night a group of us are up late sitting in the hallway on the second floor, talking about President Reagan's decision to slash funds for low-income housing and mental health services, which created what we know thirty years later to be a permanent class of homeless people. Even late in the evening the air is hot and thick with humidity, and fans blast at either end of the crowded hall. Our rhetoric soon turns from criticizing the President to boasting about what we would do if we were President one day. At which point a fellow Stanford classmate turns to me.
"Well at least you don't have to worry about that," he says with a smile.
"Hmmm?"
"You can't be President. You were born in Nigeria."
"But I've always been an American citizen. I've never been anything else."
"Sorry," he says, cutting me off. Was it smugness I saw on his white face or just the confidence that comes from certainty? He looks away.
I scramble to my feet, step over the legs of the students sprawled on the carpeted floor, and walk briskly down the hall to my room. I know exactly where to find the pocket-sized copy of the U.S. Constitution I keep among my political science and history textbooks. I grab it off the shelf and begin flipping through it, searching for the section I know holds the answer. By the time I get back to my housemates splayed on the carpet, I am quoting aloud from Article II:
No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President...
"That's me," I say, tapping the page. "Natural Born. My dad's American and I've never had any citizenship other than American." To me the question was asked and answered. And to my friend as well.
"Sorry, Julie," he says, shaking his head. "Nope. You're not Natural Born."
It was impossible to know that evening who was right. The Supreme Court had not weighed in on whether the requirement "Natural Born" applied to someone like me—born to an American citizen outside the U.S. That question would rear its head on a national stage in 2008 when the Birther Movement arose around the question of Barack Obama's citizenship (despite his birth in Hawaii and his mother's American citizenship), and arose again in 2015 with presidential candidate Senator Ted Cruz having been born in Canada to an American mother and a Cuban father.
But all of that was decades off, unknowable to me or to my fellow interns that sticky evening. This was a conversation among young adults about our wishes and dreams, and I read in my Stanford classmate's declaration not just a statement of fact he believed to be true, but judgment. As if I, already of lower status by virtue of my Blackness, was lowered yet again having been born in the wrong place.
XXI.
One Sunday morning junior year, just weeks after I tell my family I'd joined the Mormon Church, I go to church for the last time.
It's a special service held every now and again where, without warning, the Bishop calls upon a handful of congregants to give a personal testimonial. I am instantly terrified. I know my conversion was half-assed—I'd read less than a quarter of the Book of Mormon because of its line on Blackness but I'd said yes to baptism because the lessons were over and it seemed the inevitable next step and then there's the matter of this boy Mark who introduced me to all of this whom I still love. Now I find myself seated on a pew among real Mormons and I can't shake the nagging feeling that I am going to be called. I don't think I can B.S. my way through it. I start to feel queasy.
The first person begins to speak and I search my mind for scripture that I can thread with my life experience to make some kind of coherent story. The person finishes and the Bishop glances around the room to decide whom to call next. I avoid eye contact by bending to examine the heel of my left shoe. He calls the second person. Phew, not me. Then the third. I look at my watch. Twelve minutes to go. Please God let him not call me. I spend those final twelve minutes half searching for what I might say and half praying this third person will take all the remaining time.
The service ends, and relief courses through my body. I practically bound down the aisle toward the main door where the Bishop stands saying good-bye to worshippers. He stops me gently with his hand. Looks at me. Smiles.
"Julie, you were next, you know."
I did know. I knew it so deeply in my soul that it frightened me. I was so frightened that I never went back.
XXII.
A few months later, I start dating a white, Jewish boy named Dan. His wry humor and gentle manner had made him a fast friend. The mutual attraction is clear by spring. And his handwritten letters mailed every single day over the following summer cement the bond. Dan is now my life partner of close to thirty years. But back at the start we both had family who might try to stand in the way.
In June 1988 Dan and I drive across the country from Stanford to the New York area where my parents had relocated, and where Dan's mother as well as his father and stepmother live. It's a meet-the-parents road trip times three. After the long drive we stop at my folks' place first. I take the steps tentatively toward a house I've never visited before, my new boyfriend in hand.
I know my parents might be disappointed that Dan is white. My mom's fervent, frequently expressed hope that I would have more Black friends despite living in a white town is always in the back of my mind. But they are an interracial couple themselves, and I'm not going to let them balk at the racial difference between me and Dan. My far greater concern is Dan's Jewishness. My childhood was peppered with Daddy's jokes about Jews and other ethnic groups.
Twenty minutes after we arrive at my folks' place, Daddy has sized up Dan. He walks me out of the living room and into the hallway, turns to me, puts his strong finger under my chin, and lifts my face up to meet his aging, watery eyes. Daddy is now seventy years old.
"It's clear he adores you. That's all that matters to me."
XXIII.
Daddy was right. Dan adored me, and three decades later he does still. He loved me when in my deepest self-loathing over being both too Black for whites and not Black enough for Blacks I couldn't even locate a self with which to love myself.
XXIV.
Three decades later I also know that the whiteness of my future spouse was the inevitable outcome of the inertia of those young adult years in which I went after the approval of white strangers from store clerks to faculty to bosses. A white boy, a white husband was the route to the destination I desperately craved without even knowing it: belonging in America.
XXV.
Dan himself was not inevitable. He was a gift.
XXVI.
Dan and I lived in the same dorm. One day that first spring together, before we'd ever introduced each other to our parents, I emerge from the girls' shower on my hall and bump into him. I am wearing a yellow robe and flip-flops, and am gripping my shower caddy. My hair is dripping wet; the corkscrew curls I had not yet blown straight with a hair dryer and pressed smooth with a curling iron are dangling about my head. And while I usually wear a full face of makeup, my skin is bare. Not exactly the way I want to be seen by my new boyfriend.
"You have curly hair." A strange look like bemusement spreads across his face. The memory of white boys teasing me over my hair in high school comes surging back.
"Yeah?" I step back and clutch my robe to my body. I just want to scurry back to my room, shut the door, and emerge when I look presentable. Pretend this whole thing never happened.
"I love it."
What? You wait what?
Damn.
XXVII.
Senior Year. Thanksgiving 1988. Dan is in Florida. I am at my parents' home in New York, waiting for their phone to ring.
Dan's family is gathered for the holiday. Some months ago Dan sent a picture of us to Nana, his grandmother, and his family has heard about it, and they know that what Nana saw in the picture was not a nice girl standing with her grandson but a schvartze, and now everyone is wondering what Nana will say or do when she sees Dan in person.
As a dozen of Dan's family members watch Dan approach the indomitable family matriarch, I sit by the phone in my parents' home, awaiting word, awaiting word that it went okay, that he is okay, that I am okay, hoping he'll say not just that it went okay but that someone in his family finally stood up to this woman instead of making him stand there alone.
XXVIII.
Three weeks later, Dan and I head home for the winter holidays. His father and stepmother throw their annual Christmas party in their Manhattan apartment. My parents are invited, and Dan and I ride to the party in their car. We arrive at the building amid a throng of partygoers who announce breezily to the doorman "Bruce and Judy's." The doorman nods, then gestures to the bank of elevators to the right. However, when Daddy walks through the door, the doorman takes a large step toward him and holds his hand up like a stop sign. I watch Daddy's lips curl in anger as he meets the eyes of this working-class white man. I dart my eyes from Daddy to Dan, who quickly steps up and says, "No. I'm Bruce's son. We're going up." My twenty-year-old white Jewish boyfriend opens a door closed to my seventy-year-old Black father.
XXIX.
Later that evening, I meet Dan's stepsister Emily for the first time.
Emily, a fourteen-year-old white girl not very comfortable hanging out with the Manhattan socialites filling her mother's foyer and living room, is hiding out in her bedroom. That's where, after a half hour of knowing each other, Emily confides a story to me.
"I was raised by a Black nanny named Cathy," she begins. "And one day when I was in kindergarten, I got sent home early..."
For her entire young life Emily had attended elite independent schools on the Upper West Side in Manhattan. One day when she was in kindergarten her teacher had asked the children what they wanted to be when they grew up. "A fire truck," Emily answered. The teacher explained to Emily that she couldn't be a fire truck, she had to be some kind of person. "Okay, then I want to be Black." The teacher said that wasn't possible. "Then I want to be a boy!" The teacher pointed out that this too was impossible. Five-year-old Emily was inconsolable over not being able to achieve any of the things she wanted to become, and the teacher called home. Emily waited on the little cot in the nurse's office. Cathy appeared and held her tight.
In an American literature class at Stanford that fall, I'd read Toni Morrison's The Bluest Eye. I'd related to the little Black girl protagonist who was intrigued by white dolls with blue eyes, but I'd had trouble with the scene about a little white girl seeking refuge in the arms and body of her Black housekeeper. A white girl finding succor in a Black mother figure? I had never heard of such a thing. Could not even imagine it. Before reading Toni Morrison, I had never even read about it in fiction.
Now I'm sitting in the bedroom of a real white girl—a white girl who'd grown up amid considerable privilege in New York—who wanted to be Black? What can this white girl see about Black people that I can't see? How can she want to be me more than I do?
I had never felt the embrace of a Black mother.
I knew of no comfort like Cathy's arms.
XXX.
That June, I graduate from college. The last paper I write is for John Manley's class, The American Dream. Manley is a Marxist. He isn't the first professor to assign me The Autobiography of Malcolm X, but he is the first to force me to confront my own role in my alienation from both America and Black America. I title my paper "Buying into the American Dream—Is It Worth the Price?" It begins:
I sit here at my desk, a soon-to-be-graduating senior at Stanford University, trying to reduce the pile of clutter that covers my desktop in hopes that this will somehow help unclutter my mind. Most of the correspondence can be chucked into a throwaway pile, or can be neatly filed away. But not this particular letter. When I first received it, it angered and frustrated me, and now its presence fills me with remorse as I'm forced to examine the truth about my upbringing. The letter? An invitation to the Black Baccalaureate Ceremony for graduating seniors. The problem? A lifetime of standing on the outskirts of the Black community has taught me that I'm not Black enough to attend.
Rereading this college paper today at close to fifty years of age, holding in my hands the words of my twenty-one-year-old self articulating a concern I'd spend the next twenty years trying to unwind, leaves me breathless. At first it feels like my younger self has made a trip through time to remind me what was clear even then amid what felt like muddy terrain inside me.
I did not attend that Black Baccalaureate Ceremony in 1989. While I had been somewhat politically active as a student, and had met Black student leaders through those efforts, I hadn't attended parties thrown by Black students and had all but exiled myself from Ujamaa. I could not fathom bringing my light brown, white-sounding self to the graduation gathering, not to mention my white mother, whose existence vis-à-vis me and the Black community was becoming an embarrassment to me.
When my parents arrive for graduation, Mom sees the Black Baccalaureate listed in the schedule of weekend activities and asks in a hopeful tone if we'll be going, and I answer her with an angry look. I blame her for my exile, blame her for creating a child who has no sense of belonging to the only people who might possibly claim her as theirs.
XXXI.
That college paper reads less like a time capsule I'm delighted to have unearthed and more like a tomb that was best left buried:
There was a time... when I avoided associating with other blacks, when I tried to emphasize my HALF WHITE/half black status and when I was ashamed to be seen with my daddy.
These feelings about Daddy I don't recall ever feeling despite all I do recall are memorialized in the fading 12-point ink letters of a dot-matrix printer and he is gone now has been gone for twenty-one years and I am ashamed God I am so ashamed not of him but of myself to have felt shame toward this beautiful man for even one fleeting second.
These words. Like quicksand. A trap. Like a truth that swallows itself.
XXXII.
Manley's class was a mirror that showed me things about myself I hadn't seen before. I'd known race and racism and America's preference for whites and whiteness erected a wall between me and whites demarcating white as normal and me as other. But the wall between me and Blacks was there too, though harder to put my hands on or see. Manley's class forced me to see that the higher socioeconomic class that comes with professional success—the access to the good schools, the access to homes in white towns that can come with such status—
if one so chooses—
is a form of passing out of otherness out of darkness into lightness into whiteness.
I did not choose it. No one asked. But there's no question these choices lifted me. And if asked, I'd have said yes lift me with these opportunities. Just maybe not this far.
As loathsome as it was to learn that the engine of the American Dream itself—capitalism—was the invisible hand guiding me away from a people, a community, a tradition, at least now I understood the source of much of my dislocation and unbelonging. That being upper middle class had given me more in common with upper-middle-class whites than with middle-class or working-class or poor Blacks. I graduated from college knowing I was not some freak of nature but an easily predicted data point in our macroeconomic system.
XXXIII.
Dan was a few years behind me in school, so I worked on campus the year following my graduation while waiting for him to finish his degree. I worked at the public service center, in a yearlong position that was always filled by a recent grad. The center's mission was to get more undergraduates to do community service and to commit themselves more broadly to a life of public and community service.
At the start of the new school year I am invited to speak at RA Training—the training for the two hundred or so residential assistants who will serve on staffs in undergraduate dorms in the coming school year. I'd served for two years as a resident assistant on Kennell's staff—regarded by many as the most plum of RA assignments—and it was an honor to be selected as the former RA who would stand at a podium and offer the next crop of campus leaders whatever advice I could muster.
Following my talk, a residence dean named Greg Ricks stands up to announce the next item on the agenda: a workshop on institutional racism. Since RAs work with and support students of all backgrounds, the workshop is a chance to dive deep into some of the research on structural inequality, how to bridge differences, and how to create a space where everyone could be heard. Dean Ricks invites me to stay if I want. I want.
The people setting up the workshop tape six huge pieces of paper to the walls of the large room, demarcating the gathering point for each different racial group. Dean Ricks tells us to go sit by the sign with which we most identify, and talk amongst ourselves about how it feels to be there in that group. We should take notes, he says, because afterward we will gather again as a whole and report out.
I scan the room for the signs and see "White," "Asian," "Chicano/Latino," "Native American." Finally I find the sign that says "Black" and begin to walk toward it. Some folks I know and like are already gathering there. I can choose that. I've always chosen that label, even when the arbiters of Blackness didn't choose me. But as I walk, I see the sixth and final sign: "Mixed/Other." Dean Ricks's instructions pound in my head. "Sit by the sign with which you most identify." I am being given a way to acknowledge my white mother.
What the hell? What the hell.
The mere existence of that sign feels transgressive. Political. Fraught. But it beckons me. I take a deep breath and look from side to side to see who might be noticing the choice I am about to make. I walk toward it.
There are two, maybe three people over there already and they don't look a thing like me. Except for their hair that doesn't match their nose that doesn't match their eyes that doesn't match their lips or the color of their skin. I sit down. A few more students join. There are maybe seven of us in all out of a group of two hundred. The instructions are to sit with our group and write down specific words describing what it feels like to be in a same-race group. But we have a hard time following this instruction because we can hardly get over the fact of our own coexistence.
After about fifteen minutes, Dean Ricks calls us back together to report out on what the prior exercise had felt like. The Black students share words like "safe," "family," "belonging," and "home." The White students say things like "normal," "plain," "why," and "oppressor." When it's time for my group to share our words, we glance around at each other and make strange faces and shrug our shoulders, knowing our words are very different from everything else we've just heard. Our words are "wow," "cool," and "never had this chance before."
After all the groups report out, Ricks tells us to go back to those same-race groups and talk some more. I return to the "Mixed/Other" sign and sit on the ground underneath it and begin talking with the others. We pause as one new person, then another, and another come toward us—people who'd previously sat in the "Black," "Chicano/Latino," and "Asian" groups. About two minutes in, a person comes over from the White group and says "I think I actually belong here." It happens again. And again. By the end of this session, our group has more than doubled in size. Close to twenty people look around at each other and take in the sameness of our myriad differences. People tear up a bit. I may have been one of them. No one is laughing, pointing, or scoffing at us. Or maybe they are but we don't know it. Or maybe we just don't need to look around to find out. We talk about what it is like to be mixed. We nod our heads at each other's stories. We mutts that didn't seem to fit anywhere else have each other because of our deviations from some so-called normal. For an hour that afternoon, we feel powerful. Empowered. We are outing ourselves as mixed-race people.
XXXIV.
The movie Glory came out in December of 1989 to high critical and popular acclaim. The story tells of an all-Black regiment of Civil War soldiers who fought for the Union, and starred Morgan Freeman, Denzel Washington, and Matthew Broderick. Dan and I go one night, and as I sit there in the Cineplex getting drawn deeper and deeper into the drama, something hibernating way down deep in me begins to wake that night. I cannot make the feeling go away.
The story in Glory was not an epiphany. It was just a well-told, well-acted depiction of what it was like to have been a slave, to be freed, to fight for a United States that did not see you as equal, and paid you less than a white even though you wore the same uniform. To fight for a nation whose countrymen called you Nigger.
I was not seeing new things on that screen. I was seeing with eyes that could see more clearly, maybe because now that I'd located a place for myself within Blackness—biracial—I could also locate a place for myself in the larger Black narrative.
Light-skinned mixed-race Black with a white-sounding voice? Yes, and
These are your people.
These are my people. These who suffered so that I could live a life and I have lived a far better life than most. A crystal stair.
What you gonna do about it?
In the theater that night I feel empathy for my ancestors, gratitude for the progress made by prior generations of Black Americans, sheepishness, even some shame for my unearned privilege, and an impulse to do something. To continue the progress of the Black community.
XXXV.
Senior year I'd been too busy with my classes, my responsibilities to the residents on my hall, and my work as a senior class president to apply for law school. I make up for lost time by studying hard for the LSAT and filling out law school applications. I write my personal statement about biracial identity, Black consciousness, and the experience I'd had watching Glory (despite advice from my brother Stephen—a lawyer—who says the essay is too politically risky). In April, I get the incredible news that I am admitted to Harvard Law School. I hunger for the legitimacy a degree from a prestigious East Coast institution will confer upon me.
One of my colleagues at the Haas Center for Public Service had been a Freedom Rider in the summer of 1961. Another would go on to found AmeriCorps. They had taken and were taking paths toward what was good and right rather than toward prestige and money. I don't know whether my colleagues will be impressed or disappointed by my law school news. As word about my intended destination spreads among them, I clarify that I am going because it's a good degree to have and that I'll be doing public service work after I graduate.
I feel genuinely relieved when one of my colleagues, Izzy, claps me on the back and beams over my law school news. Izzy is an older white woman in her fifties who doesn't suffer fools gladly. I am intimidated by her. She knows a lot about a lot of things and her opinion matters to me.
A few weeks after I commit to Harvard, Izzy's husband, Jack, comes to visit the office. He's a retired judge and I know from the way our colleagues speak of him with reverence in their voice that he's had a storied career. I presume he'll be impressed by my law school news. Maybe see us as cut from the same cloth. Maybe, if I make a decent connection with him, he'll help me find a clerkship with a judge when the time comes. I walk across the hall to Izzy's office to meet her husband and feel nervous excitement rising inside me.
Izzy's husband is not what I expected. With his big beard, barrel chest, and stocky legs, he looks more like Santa Claus than like a judge. Izzy introduces us and saves the news about my admittance to Harvard Law School for last, savoring the telling of it like a crescendo. She's proud of me, I can tell.
The judge reacts by leaning his stout body toward me to look me straight in the eye. Then he sinks back on his heels, spreads his arms wide, and guffaws, "Oh, so you're a twofer!" He grabs me by the hand and pulls me out of his wife's office and races ahead of me down the stairs, shouting back at me over his shoulder. "Let's see if you can pass this test." I am good at tests. I have no idea what he's talking about, but as I hit the ground floor I am cautiously excited.
He grabs my hand again as we exit the building and he leads me out to the parking lot. Izzy is bringing up the rear, shouting, "Jack!" She sounds exasperated and apologetic, like a person whose dog likes to sniff people between the legs. The judge walks me over to an old rusty pickup truck. Points at the license plate: XL IXRS. Asks me if I know what it means. It takes me less than a second to decode it—49ers—but my brain can't make sense of what is going on here. Twofer? It was the first time I'd heard the term. Test?
Izzy slaps Jack on the shoulder and scolds him, but he just guffaws again, bending over with his hands on his knees at the hilarity of it all. I can tell I am the butt of his joke but I still do not know why. It seems like a routine Izzy is used to.
For a second or two I stand there in the parking lot watching Jack laugh. Then I think maybe if I laugh too we'll all be in on the joke together. So I laugh, the moment passes, and I feel relief. But I am still confused.
When Jack finally quiets down I turn to Izzy. "Twofer?" She looks at me with a lopsided smile. "Oh, you know. Because you're Black and female. He's just being stupid." She holds her hands up as if to say What can you do? Then she turns away from the pickup truck and walks back toward our office. She does not wait for me to join her. I follow.
I feel like an idiot and so damn naïve for not seeing the naked truth of it in the first place. So is this the law school version of the comment Harris's father made in my high school math class, implying I was stealing an admissions slot with my Blackness—and femaleness? Or is it like the lady at the Stanford Shopping Center who'd presumed that athleticism had gotten me into Stanford? My spidey sense is telling me this guy just might be a racist good old boy asshole. But I have a good deal of respect for judges and lawyers. My mind does not want to go there.
XXXVI.
By the end of the 1989–90 school year, a group of mixed-race students at Stanford had created a new student group, called Spectrum, owing perhaps to the consciousness raised at RA Training that prior fall. I heard from a friend at Harvard that students there had done the same thing, calling their group "Prism." The terms "multiracial" and "biracial" are starting to appear in the media. Nationally, policy makers begin to debate whether the U.S. census could allow people to check multiple race boxes to reflect a heritage of more than one race.
When I hear about these developments in racial classification, I know I've been waiting my whole life for a term like "biracial" or "multiracial" to define the otherwise out-of-bounds nature of my existence. I begin using the terms interchangeably to describe myself, and it feels nourishing, invigorating, like I've received a transplant to replace a diseased organ and have a new lease on life. Finally, checking a box doesn't mean ignoring my mother. Finally, I have a term to explain why I look and sound different from so many other Blacks without that term being derogatory, like "Oreo."
But at the start these new labels draw criticism from Black professors, policy makers, and intellectuals. I hear a Black pundit discuss this on a television news show one day. Blacks are the result of intermixture either in recent times or historically, he reminds his audience, and if mixed-race people identify as multiracial on government forms, the official count of Black people could diminish enough to pull governmental resources away from the Black community where they are much needed. Between the lines of his argument—conveyed with his facial expression, not his words—is the critique that mixed people are just using these labels to distance ourselves from the Black community, maybe to try to be "better" than Black.
Am I?
I don't care about the critique. I'm not thinking about government programs or the extent to which my personal decision might impact a bigger number of people or a more macro set of outcomes. I am desperate for belonging, and I find it with "biracial" and "multiracial."
It is a truth and it is a relief and it is A Chosen Exile, as Allyson Hobbs would title her book on the historical practice of passing. And as with those who passed out of Blackness into whiteness over the centuries prior, I would find that standing on the edge of Blackness where it bleeds lighter and lighter and ultimately starts to look like white came with both privilege and pain.
SELF-LOATHING
I.
What I now know to have been true of myself in my childhood and young adulthood:
1. I hated being Black.
2. I was afraid of Black people.
3. I tried to be what white people valued.
II.
In June 1990, Dan knelt before me on a beach in Half Moon Bay, California, gave me a beautiful ring he'd designed, and asked me to marry him. Later that summer I joined his father's family on a weeklong trip to Harbour Island, one of their favorite vacation spots in the Bahamas. The eight of us stay in a high-end resort right on the beach and I am starting to feel less like a nervous girlfriend who has to work hard to please and more like one of the family.
Tentatively, I begin to let my hair down. I engage Dan's father in conversation about social and political issues—nervously, not because we are on different sides of the political spectrum (we aren't) but because he is a corporate lawyer—in fact, a Harvard Law grad. He makes a game of banter and reasoning. I want to be a player.
Dan and many of his relatives are avid scuba divers and want to get in plenty of dive time on Harbour Island. I'm not a strong swimmer and I've had a fear of fish since early childhood, so I don't want to be anywhere near that, not even to snorkel. Instead I wave good-bye to the divers and stay behind with Dan's stepmother, Judy, and stepsister, Emily, and read novels on the sandy beach in the full sun. When the divers return hours later, I get up to greet them, admiring my darkening skin as I walk toward them, tossing my curly twists of hair now rope-like from being air-dried in the salty Caribbean breeze.
The eight of us gather each night for dinner at a restaurant down the beach. Dan's father has a fabulous sense of humor and he reels off joke after hilarious joke as the tiki torches flicker in the darkening night and we sip drinks thick with rum. Some nights we play poker after dinner, a game my Daddy and brothers had taught me young, and I delight in showing this new family that I know a thing or two about the game. One night, we linger late into the evening, singing whichever songs any of us can think of, full-throated and with harmony.
I miss Dan when he dives. Twenty-nine years later we still don't enjoy being apart, but in those earliest days of togetherness I ached without him. Toward the end of our vacation, I ask if I can go out on the boat with them even if I don't dive. Dan's father says it is no problem.
He reserves time with the local dive master, and the next day, off we go, Dan, his father, his brother, his stepbrother, his stepbrother's friend, me, the dive master, the guy driving the boat, and a few strangers who also booked time on the boat. I am the only person of color. And boy am I. After five days in the strong sun I am a rich chocolate brown.
We board the small boat and spend the thirty-minute ride across the water chitchatting. As we bump along the waves Dan's father tells some fabulous jokes and we all laugh. When the dive master finds the right spot, the boatman slows and then anchors the boat, and the divers shift into gear. It is all de rigueur for these seasoned divers, but between the complicated equipment and the overarching safety issues I don't want to be in the way. So I scoot toward the back corner of the boat to witness their rituals. One by one each diver sits on the edge of the boat and does a backward roll off the side. I yell a good-bye to each. When it is Dan's turn to roll off, he comes over to me and gives me a kiss. Then it's just me and the boatman—a weathered-looking white man. I make small talk with him for a bit and then take a novel out of my bag, pull a hat over my eyes, stretch out my legs, and begin reading.
Forty-five minutes later all of the divers are back, regaling me and each other with stories about what they'd experienced. One had huffed and puffed too much and had had to come up quickly. One had seen a shark. One had gone farther down than he'd ever gone before. I listen intently and ask questions. The trip back is quieter than the trip out, the divers spent. Dan puts his arm around me and I lay my head against his shoulder. Before I know it we are back at the dock.
We gather our things and one by one step onto the dock. The dive master and boatman are huddled in conversation apparently over some paperwork. I am the last to get off, and as I stand on the ledge of the unsteady boat and step tentatively onto the dock, the boatman grabs me hard by the shoulder, his hand like a pincer.
"Wait, you. What do you think you're up to? You didn't pay."
They seem to think I am a freeloader. An urchin who'd just clung to this opportunity for a boat ride. Hadn't they seen me with my family? Hadn't they listened? Hadn't they seen Dan's tenderness toward me?
"What? I, I—I'm with them," I stammer, nodding toward Dan and his family who are trudging away from me toward the boathouse where they'll return their equipment. The boatman still has me by the shoulder. As he shakes me, the diamond on my engagement ring sparkles in the intense Caribbean sun. "I'm his fiancée," I stammer, pointing to my hand and jerking my chin toward my family.
I finally shout for Dan. He is now about twenty feet away and he turns around and looks confused to find that I am not right there behind him. Dan's father hears the commotion and turns to look, too. "He won't... he doesn't," I stammer, waving my hands in small circles, still in the clutch of the boatman's gnarled hand. "He says I didn't pay!"
"She's with us," Dan's father shouts and makes a broad sweep of his hands. The boatman releases me. Dan's father turns and keeps walking.
I am shaking. Dan runs toward me. He looks to the boatman and at me and then back at the boatman, asking with his eyes what is going on, silently demanding an explanation or apology. But both the boatman and the dive master turn away and busy themselves with something on the boat. I grab Dan's hand and step off the boat and I do not look back at the men.
We walk toward the resort hand in hand. I try to explain what had happened, but there is nothing in Dan's life that makes any of this make sense to him.
That night I slept fitfully. They'd treated me like a stowaway. Like a freed Black with no papers. Being some white man's property is what actually spared me from whatever might have happened. Escape and a cage all at once.
III.
After dating almost exclusively white boys all my life, at age twenty-four I marry Dan in a small ceremony in a mansion on the eastern bank of the Hudson River near West Point, about two hours north of Snedens Landing where I'd lived as a small child. In preparation for the wedding, I go to a Black hair salon for the second time in my life—this time on my own terms. Instead of leaving with the short Afro Angie and my mother had conspired to give me in the sixth grade, I emerge with extensions woven into my hair—dark brown tresses made of real human hair that flow long, thick, and supple, which I can flick and let fall or tuck behind my ear. This time I look like any Black girl who knows how to make her hair look the way it is supposed to. Who knows how to make her hair look beautiful.
White hair. White dress. White life.
IV.
In December of my second year of law school, Dan and I want to celebrate his birthday by going out to dinner in Boston's North End. It is 1992. We wedge our car into a parking spot on a narrow street a few blocks away from the restaurant and walk down the sidewalk hand in hand feeling the brisk chill as dusk falls on the night. I'm a little nervous walking these streets. While my law school is located in Cambridge, a cosmopolitan city with people from all over the world, Boston is known as a balkanized city with pockets of deeply embedded racism. We are an interracial couple in the wrong part of town lured by the promise of great Italian food.
On the opposite side of the narrow street, maybe thirty feet or so in front of us, a man walks toward us. As we near him I see him do a double take, which I take as a kind gesture so I return his gaze, begin to smile, and prepare to nod. But he stares back at me with no smile as if he is studying us. I look straight ahead and keep walking and then I look back over at the man, hoping to see we are of no interest to him. But the man is now almost even with us and he is staring straight at me. He passes under a streetlight. I see pale skin, squinted eyes, and an upper lip starting to tremble like the mouth of a growling dog.
I could be wrong. This could be all kinds of things. He could be deranged. Could be mad at something else. Could want to harm us for reasons having nothing to do with race. I just know I need to get out of there.
I squeeze Dan's hand tight and mutter walk faster baby we need to walk faster as I quicken my pace and Dan has no choice but to follow. After twenty brisk paces we round the corner and I turn around and peek in the direction from which we'd just come. The man is walking off into the night.
"What?" Dan asks. "Why did you speed up like that? What were you saying?" He hadn't noticed. Never had to notice. Had not learned to notice.
V.
In 1993 I am a summer associate at the Palo Alto office of the law firm Cooley Godward in what is, at the time, the firm's largest-ever class of summer recruits. Thirty-one law students, each with uncreased briefcases and brand-new suits, jockey to prove our mettle to the partners and to discover the pecking order among ourselves. I am determined to be one of the best.
The only Black partner in the three-hundred-lawyer firm is Tom Jackson. A six-foot-six guy with dark skin, a long stride, and a personality that is alternately captivating and terrifying. You love his hearty chortle unless he is actually laughing at you, which is always a distinct possibility. No one is immune.
He'll come out of his huge corner office and stand near his secretary's carrel and begin to tell a story in a voice loud enough for anyone within forty feet to hear. Lawyers of various ages, and in particular we young ones, come out from behind our desks and stand in our doorways listening, hanging on every word. A fellow summer associate has the great misfortune of having a name one letter off from Tom's—Tom Jackston—and one day the summer associate's biweekly paycheck accidentally gets delivered to Tom the partner. He stands at his secretary's carrel opening his mail and comes upon the paycheck and shouts, "What is this shit?" Then he strides down the hall to the other Tom's office and throws the paycheck at him, laughing that it isn't enough to cover more than a pair of new shoes.
Whenever I hear Tom's heavy footsteps coming down the hall toward my office, my heart starts beating wildly. One day he shows up at my door.
"You busy?" It is more of a bark than a question.
"Um, no?"
"Great. You're coming with me."
He turns around and walks away. I scurry out from behind my desk, grab my briefcase, and race after Tom, who is striding toward the elevators. I catch up to him and we stand waiting in silence. I fidget with the buttons on my double-breasted navy blazer. I am dying to know where we are going—and for how long—but don't dare ask.
We get to his Lexus sport coupe. He opens the passenger door for me. I sit down and he shuts the door and walks around to the other side. I tuck my briefcase in the footwell and begin smoothing my long navy-and-white-striped skirt. "So where are we going?" I finally ask as he eases the car out of the parking lot and onto the main road. "You'll see." I try to affect casual, as if I don't care that I have no idea what is going on. I try, even, to be delighted by it.
When I cross my right leg over my left and settle back into the bucket seat, I notice my right shoe. It is black. But I was sure I was wearing my navy pumps! I uncross my legs and stare at both feet now side by side in the footwell. One is navy and one is black. As I'd soon learn, Tom and I are headed to court for a hearing. Tom is mentoring me and all I have to do is pay attention. But all I can think about is the ribbing I'll get if Tom notices my shoes.
Toward the end of the summer, a huge complaint comes in and it becomes my job to analyze and dismiss every case cited by the other side. I stay at the office until midnight or one a.m. every day for a week. When the memo is done, I leave it on Tom's chair and creep home in the dark of night. The next day I'm in line for lunch at the little café in our office complex. Tom comes up behind me and my heart starts to pound. "That was one helluva memo, kid." That's all he says. It was all he needed to say. I am making it even in this Black partner's eyes. I know he's been counting on me to get this right.
VI.
In spring of my third year of law school I am in Professor Charles Ogletree's coveted criminal justice clinic where, per Massachusetts law, as a third-year law student I can represent defendants in low-level hearings. My clients are battered women and juveniles. "Tree," as we call him, is one of a handful of Blacks on the faculty and I and other students of color gravitate to his warm, intentional mentoring.
I am also working on my thesis—a requirement to graduate—under the direction of Professor Martha Minow, a white woman and an expert in family law who years later will become the school's dean. It is 1994. I choose to write on the particular injustice mixed-race kids face when it comes to moving from foster care to adoptive homes. I argue that transracial adoption is inherently better than the foster care system, which as a rule urges parents not to bond with their foster child. That a permanent, loving, adoptive home of any race is better than the foster care system.
This is the same general topic I'd argued in Steyer's civil rights seminar back in college, but now I have a new weapon in my arsenal—the new labels "multiracial" and "biracial" that are gaining traction among federal and state policy makers. Armed with the new race classification schema I argue that the government has no business deciding which aspect of a multiracial/biracial child's heritage gets preference in adoption placement. As I'd done with the Drummond case back in college, I dismiss the importance of "cultural heritage transmission" for Black kids as pseudoscience. I argue that by keeping mixed-race kids in foster care longer than white kids until a suitable Black adoptive family can be found, the government is denying those mixed kids the equal protection of the law on the basis of their race.
My feeling—based on my own lived experience—was that white parents could raise Black babies just fine. After all, I was raised in a white community largely by my white mother and I was fine. Mom was the one who told the Mormon missionaries at our door that they had nothing to offer us because "we are a Black family." I'd made it up and out of childhood to college and now to graduate school and I knew I hadn't had as much as one drop of this "cultural transmission." And I was fine.
I wrote that thesis furiously, arguing specifically that it wasn't the government's right to call mixed-race kids "Black" and subject them to these "cultural transmission" rituals, which sounded almost like voodoo to me. The counterargument—that mixed kids raised by white parents were not fine—was a mirror too closely held up to my own face. I had to look away.
VII.
That May, my brother Stephen died at age forty-three after a short but intense bout with pneumonia. He'd spent his entire career as a public interest lawyer in Chicago, and when he died was the Vice President of the ACLU of Illinois.
He was the brother I knew best of all my big siblings because he was the last to leave Nigeria for college, and had gone to law school in Wisconsin when Daddy, Mom, and I lived there. He'd been the one to take me to the Jack and Jill cotillion at the end of high school. He was the brother who'd cautioned me against writing my law school application personal statement about my dawning racial consciousness because it might be interpreted the wrong way. He was a confidant of things I wasn't even intending to share. And he loved me anyway.
At Christmas five months before his death, Stephen had given all of us Lythcotts a strange and weighty gift—a printout of the genealogical research he'd done on our family. This was before the Internet could tell you these things with the press of a few buttons. Stephen had walked and talked his way through courthouses and graveyards toward this understanding of our slave ancestor Silvey and her descendants, and had begun to fashion a narrative out of it. A story.
Through Stephen's work I now knew I was a seventh-generation American, descended from folks long buried in unheralded plots in and around Charleston, South Carolina. Thanks to the painstaking work of my beautiful dead brother, I would one day come to know I was Silvey's child. But at the time he gave me this gift I was not interested in the relic of ancestry.
VIII.
One month after Stephen died I graduate from law school. I'd had to slap blinders on my eyes, on my heart, to get from his funeral to my thesis and through final exams. Next up is the bar exam, which I'll take in July.
I'd accepted a job offer with Cooley Godward, which meant I'd be starting my law career in California and would therefore have to take the California Bar Exam, which vies with New York's for the title of being the most difficult to pass. Many people retake it. Many times. To have any chance at passing this three-day exam, you had to study like hell for eight weeks straight.
I was adamant about passing on the first try. The lawyers at Cooley Godward were counting on me to pass. I wasn't going to be that Black person who didn't pass the first time.
Bar study begins a week before graduation. I set up a grid of how many hours I have to study each day and which topics I need to cover. I need no distractions for these eight weeks, including the regular hassle of caring for my impossible hair. So for the third time in my twenty-six years I take myself off to a Black hair salon and for the first time I ask for a real Black hairstyle: braids. Seven hours and $200 later I have them, a gleaming set of thin long ropes that spring out from my scalp like a dome. I shake my head from side to side, enjoying the swish they make against my shoulders and the way they look the same whether I swing them to the left or the right or catch a glimpse of them from the back. Gone are the frizzy bits, the untamable sections. My new braids are as beautiful as they are expensive, and yes, as I'd hoped, they are very simple to care for.
What I hadn't counted on was how differently I'd be treated with these braids. I take the same bus and subway lines to and from my apartment on Massachusetts Avenue in Cambridge. Go to the same grocery store and restaurants. But some white strangers now glance warily at me on the sidewalk, pull their bodies and arms away from me and into themselves as I walk past. At the Star Market a white mother looks over at me and then puts a protective arm around her child while keeping her eyes on me.
It is wild. Like taking a high-power microscope to racism and seeing it writhe and wriggle under the glaring light. And it is depressing.
I know from experience and academic study of these issues that one's life—my life—as a light-skinned biracial Black person is one of relative racial privilege. My skin that in winter wanes from brown paper bag to high yellow and my so-called white way of talking assuages whites who might otherwise have been fearful of me. Seven hours spent getting braids in a Black salon somehow overrides this, and catapults me onto a higher level on the Blackness spectrum. At least in the eyes of some whites in Cambridge, Massachusetts.
For years I'd been trying to be more Black to fit with Black people while simultaneously trying to pass as white enough to avoid the judgment of whites. But these braids are making me seem more Black to whites—which I'd never known was my goal; only in the experience of it do I begin to understand how important this is to me.
I know for the first time that I'd craved to be Black in the eyes of whites. I'd never belonged with or to whites, so being seen as Black by whites was a way to definitively belong somewhere to something and someone, even while my actual relationship to the Black community was still tenuous. In this sense my braids feel like an upgrade. A promotion. An invitation past the bouncer into the club. My braids actively do away with the ambiguity I'd struggled with for most of my conscious life. They look white people in the eye and say, "Yes I'm Black" in ways my own vocal cords and life experience have never articulated. Until sporting the braids, I'd been drawn in pencil. Smudge-able. Erasable. With the braids I am redrawn in ink.
I cherish them for four months through studying for the Bar Exam and a few months beyond, but next up is a law firm in Silicon Valley where Black hair is considered unkempt even in a place boasting "casual Friday" dress. I remove the braids myself and go back to blowing my hair dry and smoothing it straight with an iron.
When I take the braids out, my identity's temporarily strong outline begins to fade. I move west, start work at Cooley, and right before Thanksgiving I learn that I passed the California Bar Exam on the first try.
IX.
I'd wanted to become a lawyer so I could help the underdog. In college, classmates had pinned posters of celebrity crushes to their dorm room walls, and I'd taped up a picture of Thurgood Marshall, who'd argued for the desegregation of American public schools before the Supreme Court and won, and who later served as our nation's first Black Supreme Court Justice. I believed in law as the tool that could help Blacks and people of color more broadly, and all of those who are culturally and systematically disregarded in America. In law school, I'd joined the Civil Rights–Civil Liberties Law Review, and I'd found a public interest faculty mentor in Charles Ogletree.
But when early in my third year my classmates and I were trying to land permanent jobs to start the year following, I couldn't shake the psychological pressure I felt from the law school community suggesting that corporate jobs were most prestigious and therefore most desirable and therefore necessary for someone who looked like me, someone who might want to demonstrate to the world that I was impressive and worthy. I was a middle-of-the-pack student in law school, which made me feel ashamed, and I worried that if I took a public interest job people would think I hadn't been able to land an offer from a prominent law firm. (In reality, the best public interest jobs are just as selective as the firms, if not more so.)
I ignored the personal values that had drawn me to law school and had guided my curricular choices and took a corporate job with Cooley instead. I told myself that because I'd taken out loans to finance my tuition, room, and board, I couldn't afford to go the public interest route, but that was a naked lie. Harvard had a loan repayment program explicitly for those who went into public interest law.
I'd gone to law school to help other people but I took a corporate job to help myself.
X.
It is 1994. I am an intellectual property litigator in Silicon Valley specializing in trademarks at the birth of the commercialized Internet. Netscape goes public in the summer of 1995 and heralds the start of the dot-com era. I am very well paid, well regarded, and being groomed for greater opportunities. There is just one problem; the work is sucking the life out of me one billable hour at a time.
I thought I'd done everything right.
XI.
Late summer 1995. I am twenty-seven. Dan and I are in our third year as a married couple and we rent a tiny house in Silicon Valley. For the past few months a tightening feeling had formed in my stomach every Sunday afternoon as I thought about going into work at the firm the next day. Dan, who was working at an exhibit design firm, was also feeling unhappy in his career.
One weekend evening, he and I sit on the concrete slab back porch of our house and talk. I thought I'd known what I wanted to do with my life, thought I'd charted the right path, but I am far afield from the way I had hoped to feel about my life. I say all of this through scads of tears, blowing my snotty nose, sitting amid mounting piles of tissue.
Then I give myself a talking-to.
You of all people have no right to be miserable you with an upper-middle-class childhood you with loving parents and a great education and all of that does not equal miserable so stop your pathetic crying and get your shit together.
I want to help humans. This I know. I need to work for an organization that helps humans. I think of the law-based public interest organizations I admired—ACLU, Legal Aid—that I fear will now see me as a sellout and reject any effort I might make to join them. I think of how, while I had loved the oral argument of the mock litigation in college and law school, I am growing to despise the actual red tape of litigation, the hoops one has to jump through, the posturing one has to do just to get a case started, the slim chance of any of it ever being heard by a judge. A wistfulness had begun to stir in me as I drove past the Stanford campus while traveling to and from work each day. I can picture the faces of the people who'd believed in me when I was struggling in college. I want to be part of the effort to help young people find their footing and craft a meaningful life. I think maybe I should do admissions work or student affairs.
I begin applying for those kinds of jobs.
XII.
In October of 1995, almost one year to the day after I joined Cooley Godward, Daddy dies. He'd had prostate cancer for years, had felt it coming, and had chosen to die from it. He was seventy-seven and had already exceeded the life expectation for a Black man in America. He knew something would get him, and prostate cancer was it. He died in the house my parents had chosen to be their final home together: a few acres in the woods on Martha's Vineyard off the coast of Massachusetts. There'd been enough time for us all to get there to say good-bye.
After Daddy died I take three weeks off work to help my Mom with the logistics of insurance and banking and bills. I don't ask the firm for permission to stay so long, really. I just stay with my Mom until I feel she can start to walk through her days alone. When I return to work in November, I muddle through the interminably long days and stumble through cloudy thinking. I can't make a to-do list. Can't follow it when I do manage to make one. The holidays come and go. My billable hours dip significantly.
Three months after Daddy died, I confide in my sister-in-law, Stephen's widow, about my inability to get anything done. She encourages me to go to grief counseling. I haven't had so much as an hour of therapy in my life and tend to dismiss it as a crutch for the weak, but I go, if only to make my sister-in-law feel she is helping me.
After just one session with a group of fellow grievers, counseling becomes a biweekly lifeline. There, I begin to talk about my brother's unexpected death as well as Daddy's. For seven months, until the funk dissipates and I begin to feel like my old self again, those twice-monthly group sessions are my religion.
Sometime during those months of therapy, the partner with whom I worked most closely takes me out to lunch, a rare treat. We walk to the restaurant, have a nice meal, and then as we are walking back to the firm she stops on the sidewalk. Turns to me.
"You need to get your hours up."
"I know. It's been hard for me since my dad died."
"I understand. I know what it's like. I lost both my parents when I was young. You work through it, you move on."
"I know. I'm actually in counseling."
"Good. But you need to get your hours up."
XIII.
I go for those student affairs and admissions jobs at Stanford three times over two years and three times I am rejected either for lacking the requisite experience or for being viewed as overqualified given my law degree and experience as a lawyer. With each rejection I feel that an escape hatch has been shut in my face.
XIV.
In the spring of 1996, Dan and I attend his cousin's wedding in the Florida Keys. The rehearsal dinner is at a restaurant with lovely views of the ocean and Dan and I walk down the boardwalk toward it hand in hand. We pass people at outdoor restaurants drinking and laughing in their tropical shirts and white pants. A white man seated at a corner table catches sight of us and begins to stare. I squeeze Dan's hand tight, which is now my cue to him that something is going on. Dan looks in the direction of my gaze and sees it too. We keep walking ahead with purposeful confidence, though feigned. I look back once and see the man continuing to stare at us as we walk.
Dan had learned about the white racist sneer. Life with me was beginning to teach him.
XV.
I take on more pro bono work at the law firm as a way to make my corporate law practice resemble in some small way the reason I'd gotten a law degree in the first place. One case is a huge death penalty appeal with a dozen other lawyers. Another is small—a Black male child is incarcerated at the California Youth Authority for having punched a white child and I am to be his lawyer. It is the middle of 1997.
My mentor for this pro bono litigation work is an acerbic white female litigator at my firm. She is a very busy senior associate and makes it clear that she has little time for me. Frankly I am a little afraid of her so I am happy enough getting to do this on my own. I drive out to Stockton where my client is being held and I interview him. I stay late for many nights to prepare to conduct my first-ever deposition—a deposition of the school administrator who'd broken up the fight between my client and the other child. The week of the deposition I pass my mentor in the hallway. She barks at me. "Ready for that depo?" Yes, I assure her.
The day came. My mentor and I, opposing counsel, the court reporter, and the person being deposed gather in the conference room for the depo. I begin with the opening procedural ritual. I speak nervously. Tentatively. My mentor begins to fidget in her chair. Then she speaks.
"No. No. Are you kidding me? Wait, just—no."
Opposing counsel looks at me with sympathy. Sweat sprouts on my forehead and upper lip. I had spoken all of a dozen words so far. Am I that bad that she needs to be interrupting me like this? The walls of the conference room feel like they are pressing in on me. Blood pounds in my veins.
My mentor speaks to the room. "We're going to postpone." Humiliated, I shake my head and stare at my pile of papers, then gather my things and get the hell out of there. I walk back to my office berating myself for not having practiced more, for not observing more people do a depo, for being so incompetent at the task of trying to stand up for this Black child. But the senior associate had given me all of ninety seconds. I might have gotten into the swing of it if I'd just been given a bit more of a chance.
I left the firm for good a few months later to take a position in the trademarks division at Intel's legal department.
Someone else took over the case. I don't know what happened to this child. But I will never forget his name.
XVI.
A year into my work as a trademark lawyer at Intel, a friend who works as the Dean of Students at Stanford Law School calls to tell me she is going on maternity leave and do I want to try to cover her job for three months? I tell my manager at Intel that I might not want to be a lawyer anymore. That I might want to work in student affairs. She says she values me and agrees to hold my job. "For your sake I hope you love it. For my sake I hope you don't."
Mere days into the job I know I love the work, which boils down to giving a damn about these law students and supporting them as they encounter obstacles in their academic or personal life. I feel like I have my self back. The only gnawing fear is how I will stomach returning to corporate law when the maternity leave ends.
In January of 1999, my friend tells the law school dean that she is not going to return from maternity leave. The dean calls me to say he wants to hire me permanently. And after two and a half years of trying to conceive a child, where sex has felt less like pleasure and more like a physical therapy regime, I am now finally pregnant myself.
XVII.
I am thriving. What I enjoy most about this new work is supporting students of color and students from other underrepresented communities in keeping their nose to the grindstone and their eyes on the prize when life tries to get in the way. I am making less money than I'd made as a lawyer, and the trajectory for future income is nowhere near as high as that of a lawyer, but the knot in my stomach as I think about going to work the next day is gone. And that is worth its weight in gold.
I feel engaged and productive as a problem solver in the lives of humans. We have a little baby boom in the student population with five or six students who themselves are pregnant or whose partners are, and it seems only logical to take an unused storage closet and spruce it up and outfit it to serve as a room for nursing mothers. I also have a fantastic boss and mentor in Paul Brest, an exalted white male law professor who engages me in conversation as if my mind works well and my thoughts genuinely interest him. As if we are equals in the work of helping other humans thrive.
Now in a more relaxed and welcoming work environment, now enjoying myself in my career, now surrounded by many more people of color in the law school student body, and on the faculty and staff, I stop pressing my hair flat and pulling it into a ponytail or bun and begin to let my hair down. Wear it natural. I experiment with the creams and conditioners made for hair like mine that are being developed in response to a burgeoning population of mixed race women, and that I can find via the Internet.
In May of 1999 when I am thick with nine months of pregnancy, waddling up and down stairs, just hoping my water won't break until the school term is over, one of our students takes his life. I'd interacted with him a number of times. He was tall and thin, sweet and funny, and had eyes that shone with curiosity and intellect. Whenever he showed up at my door I knew he'd be warm and respectful, and present a problem he'd already gone to great lengths to resolve before coming to me.
The day after his death his shattered parents come to Paul's office late in the afternoon, and Paul asks me and a few other university officials to be there with him. Together we sit with the parents as disbelief gives way to frustration and when, finally, their anguish releases. We know we'd perhaps known their son in ways they did not, and know just as equally that none of that is relevant now.
The conversation ebbs and flows from the sadness to the practicalities of the young man's belongings, records, and so forth, back to the emotional, and even the existential. Seeing the meeting going long into the night, Paul nudges, then urges, then with his eyes almost begs me to go home. I sit on his couch with my legs neatly tucked beneath me, wearing the black maternity dress I'd put on that morning after hearing the news. I just smile gently at Paul and nod a silent thanks but I'm okay.
Later in trying to be more aware of why exactly I loved the work I was now doing, I'd reflect back on this night sitting with parents experiencing the first shock waves of the unimaginable, and I'd see it clearly. The law partner who'd taken me to lunch implored me to work longer hours despite my grief; here, in total contrast, amid the greatest crisis a school administration can experience, was a boss telling me to go take care of myself. I learned that night that bearing witness to the suffering of another human is the most sacred work we can do.
XVIII.
In June of 1999 my baby boy is born.
We name him Sawyer.
When he gets to be about two, people begin asking him his name instead of directing that question to Dan or me. He is eager to reply.
"August."
"No, not when were you born but what's your name."
"I was born in June. My name is Sawyer. But I wish my parents had called me August."
Sawyer learned to talk in full sentences before he could walk. So these conversations would set strangers back a bit, not only for the logic of it all but for the quality of this tiny person's language.
"August" had been on the short list of names we'd considered, and we'd told him so. I'd loved the name three times over: for its homage to the Black playwright August Wilson, for being the month in which Dan and I were married; for being an adjective I'd hoped would indeed describe my son when he was a grown man one day. But we didn't want people to reduce this magnificent name to Gus or Augie, which we'd also told him. Unwittingly we'd given our son a name shared by fiction's most mischievous white boy. It would be years before I'd be a person who'd lament the lost opportunity to name him after a Black man.
XIX.
Sawyer was beautiful from the first moment I saw him, pale with the slick straight jet black hair of a newborn. In a few months his skin tone became a medium brown and his hair began to grow in medium brown as well with the loop and curl of Blackness. And he was born to a mother who would wear her hair natural from then on.
XX.
After trying for more than two years to conceive Sawyer, my body seemed to have figured out how to be pregnant; in October of 2000, just weeks after starting a new position with Stanford's incoming president, John Hennessy, I got pregnant again without even trying. I wouldn't discover I was pregnant until severe morning sickness began to consume me in December. My daughter, Avery, was born in 2001, six weeks before 9/11.
When we brought Avery home from the hospital, Sawyer—just shy of two years and two months old—looked over at me and said, "For ages, and ages, and ages, and ages I've wanted a baby sister." He bonded with her from the start and she received and returned that love like a reflex. Sixteen years later, they are bonded still.
Having grown up lonely as the only child of my parents, the only young child in an extended family of much older half siblings, the only child on this strange interracial island, witnessing an unyielding bond between my children gives me such hope that if all else fails they will have each other.
XXI.
When I learn I am pregnant with a girl, I feel almost giddy at the thought of teaching her what no one taught me about Black hair:
1. Never take a brush to it.
2. Don't shampoo too often.
3. Condition condition condition.
4. Wet it, and rake through it with fingers or a wide-toothed comb.
5. Use leave-in conditioner to bring out smooth ringlets.
6. Once set, don't you dare touch it until it dries.
I will not let you look like a tumbleweed.
I will not let your hair pouf in humid air.
I will not give you a hairstyle high school boys will tease.
I will buy every single product you need.
It will be my job—my joy—to help you achieve the self-acceptance that eluded me until midlife.
I vow to my unborn daughter: Baby, your mother will get it right.
XXII.
I do not resemble my mother and so? I am desperate for my daughter to resemble me. Like her brother, my girl is a beautiful newborn with pale skin and shiny wet jet-black hair. In the first months of her life, her hair grows in brown and curly too. Unlike her brother though, when her skin begins to darken it seems to pause midway. It is just a pause, I am sure.
Her darkness is coming.
I am sure.
I am wrong.
XXIII.
In occasional quiet moments when my baby girl is asleep in my arms, I gaze down at her and my mind retreats to its dark corner where the most frightening questions hide. She's so light. Am I sure she is mine?
Yes, I have proof, my mind reminds me. When Dan first brought her to me minutes after the Cesarean section, she was swaddled and still a bit wet, and as I gazed upon this lovely little face I noticed a tiny cut of no more than half a centimeter above her left eyebrow, a cut that was still there when we brought her home from the hospital five days later. Thanks to my ob-gyn's careless nick of an otherwise flawless, perfect newborn, I can be certain this girl in my arms is mine.
And there is the matter of her white Daddy. Whom I chose.
Yes, I chose him.
XXIV.
In 2002, when Avery is one and Sawyer is three, I launch a new office on campus and hold a newly created position: dean of freshmen. My job will be to try to build a sense of belonging in our undergraduates, to benefit the students in the immediate term and the university over the long haul. Research shows that students thrive when they feel a sense of belonging, and they feel a sense of belonging if one faculty member or staff person takes an interest in them. My new office will try to increase the likelihood of that happening for all students from the get-go.
I hold this role until 2012. It is joyful work almost every single day. What I love most is showing first-generation students, poor students, students of color, queer students, and anyone who grew up feeling like "the other" that I believe in them, and by extension the university believes in them, even when under the crushing weight of stereotype they don't believe in themselves.
XXV.
By the time Avery is two her skin holds a tint of tan so subtle you might mistake it for sun exposure, while Sawyer, now four, looks much more like me. Avery's soft, curly locks have given way to the wavy, tangle-free stuff of a new Barbie doll, neither too thin nor too thick, almost synthetically sleek. She gets frustrated because it won't hold a braid without a hair tie, and barrettes won't stay in place; mine, on the other hand, ties itself in knots that only certain products, not available at white markets, can untangle.
I sit behind her at the dresser, as my mother did with me, staring at us both in the mirror, as my mother did with me. Slowly I brush through her long brown hair, thinking I'm glad for her that she has what we call good hair. I'm glad she'll have an easy time. But I end every brushstroke with a twirl of the lock around my index finger, trying to tease out a ringlet or two, trying to force a resemblance to me. Trying to bring out the Black.
White parents slather their kids with sunblock, but it isn't cancer I worry about. I hold Avery up to the strong California sun. Darken her. Make her mine.
How will she identify?
How will others see her?
Will she feel the swift kick in the gut when someone says the N-word?
Will she stand up for me?
I brace for some white person to ask if I am her nanny.
XXVI.
When Sawyer and Avery are still quite young I begin my new and improved version of my parents' political advocacy campaign ("You are Black!"):
You are part Black, Eastern European Jew, and Yorkshire coal miner.
Your ancestors were some of the most reviled people in history.
Be proud of that and of them.
You have the right to be here.
You come from people who survived.
I try to say it not as declaration or dare—as my parents' line had sometimes felt to me—but as support, as a kind of reassurance that might become a part of the structure of their developing minds, as ballast when the destabilization comes at them from wherever it might come.
XXVII.
I hold a position among the senior ranks of the Stanford administration: assistant vice provost for undergraduate education and dean of freshmen. Two years into this brand-new role I begin hearing from parents, students, alumni, and colleagues that my new role is making a difference. I am well paid. And my natural, springy coils are part and parcel of my persona. But there is a lingering shadow. A cloak laid across my shoulders I can't seem to shake.
In the outside world my title raises smiling eyebrows and praise. Within academe—not just at Stanford but nationally—the hierarchy of graduate degrees says the PhD is supreme, says that those who went to graduate school to study law, medicine, or business are more doers than thinkers. Beneath the surface of that widely held opinion lies the bias that those without PhDs are not as intelligent as those who have them. It's a bias we can ignore, work hard to contradict, or let eat away at us.
The vice provost to whom I report at Stanford is a PhD-clad faculty member from engineering. When he is considering me as a possibility for the newly vacated role as head of the undergraduate advising office, he runs through the qualifications like a cross-examination.
"You have a JD, right?"
"Right."
"From Harvard."
"Right."
"Okay, good."
This was 2004. I didn't get the promotion then. I would get it in 2009.
In the meantime, the three others who reported to this vice provost—all women, all white, all at least ten years older than I was—had PhDs. At our staff meetings I at times struggle to make myself heard, feel they are overlooking my perspective, feel desperate, even, to make my ideas count as I put forth idea after idea about how to improve the freshman experience from my vantage point as the university's first freshman dean. Occasionally I feel almost paranoid about being excluded from important conversations. Even cloaked in a degree from Harvard Law School, it seems I might not be good enough.
Is it the wrong degree or the wrong skin color? Is my Blackness so Black that it trumps the bona fides of Harvard? Am I less smart? Am I less smart because of my brain or my Blackness? Do I have to be more smart to be considered just as good? Is all of this happening in their eyes or only in mine?
Years later I'd learn I'd not been paranoid.
XXVIII.
In 2004 we created a new freshman orientation program at Stanford called "Three Books," our take on the common book experience most colleges and universities provide for new students. Our program would feature not one but three texts which would be assigned to the students to read over the summer, and the program would culminate in the three authors coming to campus all at once to participate in a moderated conversation about their books and their lives as writers.
A group of us—one faculty member and the rest staff—develop the concept and when the first incarnation is over and done with in September of 2004, we begin brainstorming texts for 2005. The faculty member urges us to consider Possession by A. S. Byatt, so I buy it and begin reading it. But rather quickly, as if meeting a closed door suddenly on my path, I am struck that this text will be a barrier to the many incoming students who had not been exposed to complex works of metafiction in high school. I page through Byatt recalling myself as a seventeen-year-old, who'd been plucked from the relative mediocrity of my Midwestern public high school and had suddenly to confront texts of immense complexity in college. With Byatt in my hands I was seventeen again, trying to immerse myself in the book I'd been assigned as an incoming freshman—The Name of the Rose, by Umberto Eco—a novel I'd struggled to comprehend day after late summer day in Wisconsin back in 1985. I recalled the shame I felt in the Branner lounge as Kennell discussed it with us, at barely being able to read Eco for comprehension let alone mastery. I'd felt marginalized just when the next important chapter of my life was opening in front of me.
As dean I know that many of our incoming students—those who were tops in their high school but whose high school was not relatively rigorous—would feel admitted but not let in to the club of thinkers at Stanford if we slammed Byatt on them as part of New Student Orientation. I had been that kid and I am doing this work in no small part to care about those kids and help pave the way for their ultimate success. Let them become exposed to those texts in the weeks and months to come under the guidance of professors and other instructors, I tell my colleagues. What they need to feel most is a sense of belonging at the start; let's not make them feel excluded. As I sit around our conference room table making this argument on behalf of the next generation of kids, I am aware that I am outing myself as someone who had once struggled with the assigned freshman text. Peeling back this layer of armor, I feel naked. Maybe I am.
XXIX.
I fill Avery's bedroom with stuffed animals as well as Black dolls, like the dolls my mother gave me—alternatives to the definitive white blonde blue-eyed ideal of beauty all around. Psychologists say Black dolls are important for enhancing a Black child's self-esteem.
Or will she have white self-esteem?
I give my daughter these dolls, hold her to the sun, tease a curl around my finger, and talk to her about Silvey.
It is lonely on my island and I want to bring my Avery to me there.
XXX.
On August 29, 2005, Katrina makes landfall and the levees do not hold as the Army knew they would not and the water sweeps life out from under the living.
In New Orleans's Ninth Ward, Black people on rooftops wave signs hastily scrawled on pieces of cardboard: "Help us." The people plead with their bodies and their signs, sure as the helicopters fly over that their government is coming for them. Will help them.
Instead the government flies by.
Over thirty thousand residents stream into the Louisiana Superdome, a building whose roof would leak, whose air-conditioning and refrigeration would fail, where, without enough food, water, restrooms, or restroom supplies, these residents would live for five days. As the Superdome grows more dank with a stench that is a mixture of rotting food, urine, and feces, the government relocates people to the Astrodome, over 350 miles away from their homes, in Houston. The Astrodome and the organizational wherewithal of Houston's local government save the day and save lives. Some evacuees will stay for weeks, some for months.
The former First Lady of the United States, Barbara Bush, takes a tour of the Astrodome on September 5, 2005, when it is brand-new in its role as savior. She chortles, "So many of the people in the arena here, you know, were underprivileged anyway, so this is working very well for them."
Most of us Black folks are Democrats. We believe as Democrats that our government is an organization that will be there for us even when our fellow citizens who see us as other seek to shut us out kick us out shut us down but in late August 2005 we those who live in the Gulf Coast we who have loved ones there we who have no connection to the area but watch on television learn that our government has had no plan for us.
Them Niggers should be grateful, she might as well have said. Here, have a hot dog. We gave you have a damn hot dog. Dog. Be grateful.
Pledge your allegiance.
Stand for it.
Stand.
XXXI.
I had no idea how much he meant to me until he was gone. Professor Kennell Jackson. My freshman year resident fellow who was so light-skinned he didn't even strike me as Black. He died of a degenerative pulmonary disease in November of 2005.
I'd never known quite what to make of Kennell that first year, or what he felt toward me for that matter. He'd glance at us freshmen, shake his head, articulate an observation we couldn't quite understand. He was quirky. Eccentric. Opinionated. Obviously brilliant. There were over 160 of us in his dorm, and that first quarter as I struggled academically, I just hoped to fly below his discerning radar.
By the end of sophomore year, I'd decided I wanted to be a resident assistant. Someone who'd help the new freshmen along. I wanted to serve on the Branner staff, always among the most popular staffing spots. I was selected along with ten others, and being on staff meant weekly meetings with Kennell.
The staff met with him every Monday night, sometimes late into the night if there was a particularly gnarly student situation to try to solve. Once or twice a quarter he'd enlist our help in making homemade chocolate chip cookies for the entire dorm, the scent wafting up to the second and third floors, into the grand lounge in the middle of the first floor, and over to the other side. He knew this was how to get the freshmen to meet one another, to pull them out of their new comfort zones of room and hall. He deployed a dozen stainless steel cookie sheets for the enormous undertaking.
He also wanted the dorm to be an intellectually rich environment and expected we, his staff, would convey that with our behavior and language. So he got annoyed at us when we goofed around, even when it was behind the scenes in his apartment, even when it was during cookie making—except for the times when he wanted to let his hair down and participate in the fun himself, or even start it. I never knew how to predict which Kennell we would get in any given moment. He was chummy with a few of the others on the staff though, joking around, inviting them on an errand to get delicious oranges at his favorite produce market, Sigona's.
I always seemed to be out of step around Kennell. His gruff, judging demeanor delivered his opinion on every big and little thing. Whenever I opened my mouth around him I was afraid of being critiqued, so I opened my mouth as little as possible. Still, secretly I wanted him to like me, to show me with his liking that I was Black enough to be worthy of mattering.
Because I was majoring in American Studies I took a steady diet of American History classes. To know Kennell better, I branched out and took his Introduction to African History class, where I did very well on the papers and made comments in class he appreciated. One day in the lunch line at Branner, Kennell was a person or two ahead of me. I tried to dodge his gaze as I didn't feel like being particularly erudite at that moment, but he was tall and spotted me standing there. I heard him speak to the person between him and me. "I don't think Julie knows how smart she is."
Offended—I took the comment as criticism—I blurted out, "What are you talking about? Yes I do." He smiled, shook his head, and looked away, and I looked away too, not really sure what had happened, trying to replay his words in my head to discern a clearer meaning, but I couldn't.
Later, the Stanford Review would excoriate him for teaching a class on Black Hair, a course that not only acknowledged and explored an issue that mattered well beyond my personal struggles but that also, as it turned out, produced noteworthy research, by students, on the role of black barbers in America's Reconstruction and on other historical aspects of black hair and hair care.
Seventeen or eighteen years after that awkward conversation in the lunch line, I was dean of freshmen. The best resident fellow to ever hold the job at Stanford lay dying at the Stanford Hospital. And if I wasn't in his close circle of friends before, I certainly wasn't in that circle now. Then I heard from a colleague who had been to visit Kennell that day. "Kennell asked where you were."
Really?
I had to do this right.
He'd never asked me to go to Sigona's with him, and I had never in fact been to Sigona's in my life, but now I went of my own volition to Sigona's and selected the two most plump oranges I could find. Then I drove over to the hospital a few streets away. I found the door to his room and looked around but saw no one else I recognized, no one who could provide ballast in case Kennell was in one of his moods. I took a deep breath and rapped my knuckles on the door.
He barked his voice at the intrusion. "Who is it?!" I shuddered and whispered loudly, "It's Julie." I pushed the door slightly open. "Good, good, come in," he said. "I was wondering when you were going to come." A colleague from the English Department was sitting quietly in a chair against the side wall. I began to try to explain why I hadn't come before but didn't know how to finish the sentence. How do you say to someone I didn't think you'd want to see me as you lay dying?
I went over to Kennell's bedside and kissed him on the cheek, a far more intimate gesture than we'd ever exchanged before. I held up an orange and said I'd just been to Sigona's and did he want some? "Yeah, yeah." He smiled, and then he began to cough. I peeled the orange as a nurse came and tended him.
I sat at his bedside and fed him bits of orange sections until he'd had enough, and then I pulled the bedside chair over to the foot of his bed. It was easier to make eye contact from that angle. His long lanky legs filled the entire bed and I asked him if he wanted me to rub his feet. He did. I'd been here before, it seemed, with Stephen and Daddy a decade earlier. And I'd learned from watching those two men die that dying people don't want to be treated like they're dying. And on top of that, this was Kennell. I knew that if my presence was to be useful, I had to bring my A game, talk about what was going on in the world, act nonplussed by the tubes and chirping machines.
The next day when I visited he was asleep. I watched the tubes inflate his withering lungs from my spot at the foot of his bed. I sat with him for a while.
The day following he was awake and alert, and with some urgency he told me he wanted me to organize his memorial service. He had specific ideas for the music, the program, the people I was to invite to speak or perform. I took it all down like he was my boss telling me to plan a project, a task that made me feel for the first time adequate in his eyes, like he'd been putting me through a trial all these years and I'd come out the other side. Like maybe the trial had been mostly in my mind. He'd never been tolerant of emotion. When he said he wanted me to speak on behalf of the RAs, I closed my eyes and scrunched up my mouth so as not to cry.
He died about a week later. We held the service the following January to coincide with the Martin Luther King Day holiday. As I sat in the second pew behind family, pressed shoulder to shoulder with my fellow Branner RAs, listening to others' stories about how he'd impacted their lives, my tears steadily fell. Kennell had been trying to tell me, not just in an awkward conversation in the dining hall years ago but in the years before that and since, Stop stumbling over yourself. Get out there. Go and get it. I see you, girl. You got this. You belong.
XXXII.
After Kennell died, Dan and I filled in as Resident Fellows for the remainder of the 2005–6 school year while the university searched for a permanent replacement for him. We joined the effort to pack up his belongings. The colleague in charge of this effort invited me to take anything that was meaningful to me. I went searching for the cookie pans.
They were in the cabinet under the kitchen sink, a stack of twelve of which I took four. Also under the sink, in the far back corner, there was a large object wrapped in newspaper. I crouched down, reached in and grabbed it, and stood up again holding it in my hands.
The mysterious package was wrapped in eight or ten sheets of newspaper, water-stained in one small spot but otherwise as crisp as the day it was published. What could it be, I wondered, this treasure wrapped with some care yet forgotten under the kitchen sink? To look further felt like prying. But the only person who would care about that was gone.
I pulled back the paper and let it fall to the floor and stood holding an unremarkable object: a medium-sized metal colander with two metal handles, painted reddish-orange flecked with tiny dots of white. I began to imagine that Kennell must have retired it from use when he updated his kitchen décor. Or maybe it was a gift he'd never liked, never used. He had always been particular that way. I decided to keep it and set it on the counter on top of the cookie pans. Then I bent and gathered the newspaper.
The masthead caught my eye. Then the headline. Then the date. A sound forced its way up and out of me, like the gulp of a drowning person or the gasp from sudden injury, such that Dan in the next room overheard and came around the corner to see if I was all right. He found me sitting on the floor, clutching the September 23, 1985, issue of the Stanford Daily—the issue that welcomed my freshman class to campus. I leaned my head back against the cabinet and felt God or Kennell or some existential purpose for all of this comforting me like a blanket.
XXXIII.
A few weeks later something happened that made me realize I might be ashamed to have such a light-skinned daughter. I was thirty-eight.
I was headed to a late afternoon holiday event hosted by a group of mostly Black colleagues, colleagues I had begun to befriend, and with whom I was hoping to make up for the lost years earlier in my life when there were no Black people around. I wanted desperately to go to this party. Not just to go, but to walk in and maybe see a few faces turn and smile and wave me over. They'd invite me to sit. We'd talk. We'd laugh. I'd be accepted as one of them. This was as real a possibility to me as any dream.
But I am on kid-duty that afternoon, which means I will have to bring Sawyer and Avery, now aged six and four, with me to the event. I want to attend so badly, yet I know, in bringing my girl with her light olive skin and shiny, smooth, wavy brown hair, that she will be the tangible evidence that I had not only chosen to marry a white man but that I hadn't been Black enough to pass along Blacker genes to both of my offspring.
By the time I drive up to the event in my minivan and get one kid then the other out of their car seats, a feeling of dread is starting to crawl up my leg bones and into my stomach. Holding a child's hand in each of mine, I fumble with the key fob that automatically shuts the minivan's wide door, and then I begin walking with my little ones through the parking lot toward a meeting room in Tresidder Memorial Union.
We enter the big building and walk the linoleum floor, a familiar walk—I attend meetings here many days a week—but I feel out of place as if I am making this trek for the first time.
We get to the room and through its glass doors I can see my colleagues, smiling. Chitchatting. Someone is playing the piano. Trays of hot food sit on round tables covered in gold cloth. I pushed the glass door open and in a soft voice urge my boy and then my girl to walk through. I bring up the rear behind them, the tiny hairs on the back of my neck stiff as I nudge their small bodies in the direction of the most familiar face in the room.
XXXIV.
Viewing that holiday party scene from a distance of more than ten years, I shudder at having taken my children so close to the deepest cavern of my psyche. Were they blissfully unaware of the anxiety roiling within me? Or could they sense it, the unease flooding me as we walked through the glass doors of that holiday party? Fear comes not only with behavior—fight or flight—but with a scent, strong and acrid. Could my children smell it pouring off me, coating not only their little heads and bodies but the air all around them?
Professionally, I appeared to have taken all the right steps. I had degrees from elite schools. I'd landed prestigious work. I'd done all of this schooling, all of this work, in part so as never to be called Nigger again. But I walked tentatively through my life, unstable, feeling a hollowness inside, as if the very construct of my self was liable to fracture into pieces and fall apart. At any moment I felt I might step on a crack, break my own back.
In the Oak Lounge at Tresidder Memorial Student Union I worried intently about how my Black colleagues might treat my younger child with her pale visage; even a third-rate psychologist would have said the only real harm to my child was lurking within me. And in actuality my colleagues greeted me kindly, cooed over my kids, and no one looked sideways at my girl.
The day after the holiday party I sat down with my laptop, where, propelled by a deep fear of the person I may have become, fueled by an aversion toward the mother it looked like I might be, for the first time in my life I banged out a piece of prose that wasn't for school or work.
"She looks so unlike me, so unlike what I expected..."
Through writing I tried to stare straight into my heart, to examine it, to get closer, and even to hold my heart in my hands. When I did so, what I found was flesh partially covered with a scab still trying to form over a long-festering wound. I took a deep breath, then I poked the scab and picked at it, then pressed hard and watched what happened. The pus oozed out thick as toothpaste. And when it was done oozing, it had formed a word: Nigger.
I wrote that shit down.
I knew the infection of self-loathing was bad and deep, likely to spread to my precious girl child if I didn't find a way to get it out of me. I gave myself permission to tell myself that the birthday locker incident had in fact happened. I dared to tell the truth of it inside my head, dared to put it on the page, dared to write it down. Dared to stare at the word some anonymous white American had called me. And to take a deep breath and see that I still lived.
And why the challenge with Avery? I felt her lightness lessened my Blackness among Blacks; I could never pass as white and now, because of her, I couldn't pass as Black either. This tiny child kicked me deep into a racial crevice, with no ledge to hold on to. I want to drag a Black cloak over my white-looking daughter. To build Black consciousness in a child the world would see as white, by un-hiding her Blackness, by trying to hide her whiteness. So that she'll love the skin her Black ancestors are in, so she will not sit silently, passing, when someone says "Nigger," so when she gets asked the "What are you?" question, she'll claim herself as belonging to me. I even want my girl to be called a Nigger. I want when she hears that for her to know Blackness includes her, too. I want her not to be embarrassed by the Blackness of me as I was once embarrassed by the Blackness of Daddy.
I need a girl child labeled like me so I can feel less alone.
I was extremely fucked up, maybe so fucked up that I might harm my child psychologically. I never wanted Avery to fear that she was anything other than exactly what I'd wanted in a daughter. It was on me to be the mother she deserved. It was on me to work out this race shit.
By the end of the exercise—an essay of some length I revised and revised and revised for months, I'd finally reached an essential truth.
I wasn't ashamed about Avery.
I was ashamed to be me.
EMERGING
I.
My first class of freshmen is graduating in June of 2006, and in April I receive an invitation from Jan Barker-Alexander, the Associate Dean of Students and Director of the Black Community Services Center, to attend the annual "Black Graduation" ceremony.
I hadn't attended my own "Black Grad" seventeen years earlier, then called Black Baccalaureate, for fear of being not Black enough and therefore not welcome. Afraid of showing up with my white mother in tow. How would that go over? But now it seems I am wanted, not only to support the students but to sit up on that large stage in a place of honor with the other Black senior administrators and Black faculty. I feel giddy, like I've been invited to prom by the cutest kid in my class. I feel the nerves that come from doing something important for the first time.
I tell myself to get over it. I am a dean. The graduation isn't about me or my impure ancestry or my fragile history with Blackness. It is about being there to celebrate other people's kids, many of whom I know quite well. I go, having no clue what the experience is about to do to me.
Although it would move to a bigger venue the year following, this year, Black Grad takes place in Memorial Church, a domed sandstone and stained glass structure that sits at the center of campus and feels like its heart. The event begins with drumming, and we, the faculty and staff, wait until all of the guests are seated, then we process down the long aisle two by two. With a couple hundred Black graduates—bachelors, masters, and doctoral candidates—plus families of various sizes, and faculty and staff in attendance, the place is packed to standing room only.
The graduates-to-be sit down in front during the speeches and drumming performance, and then the event everyone waits for—the Kente cloth ceremony—begins. The students rise from their seats and line up in alphabetical order against the far left wall. The parents or other family members do the same on the right. Then one by one by one, as Dean Jan Barker-Alexander reads out the names, the students and parents ascend the left and right steps, respectively, walk across the wide dais toward one another, and meet in the middle. There, the student kneels or bends or stands straight up, depending on their height, to receive the woven strip of Kente cloth. Carefully, the parent drapes the long, thin cloth around their child's neck, like a scarf. Faculty and senior administrators seated on the dais oversee the Kente cloth hooding ceremony like elders. The students and their family members cross directly in front of us.
We don't wear Kente—the traditional embroidered cloth of the Ashanti people of Ghana—because we're Ghanaian. After all, we rarely know whether we are or aren't. We were taken from whatever our home nation was, sold into slavery, brought through the Middle Passage to North America, the Caribbean, and South America. The history and ancestry and culture from whence we came were systematically erased from us, beaten out of our minds if not our bodies, evident only—perhaps, if we can afford to discover it—in our DNA.
Most of us likely are not Ghanaians. But that is not the point of our wearing Kente. Ghana's port city Cape Coast served as a main export site for slaves, a funnel if you will, delivering slaves from the continent into holding pens, onto boats, and over the Atlantic. For many of our African ancestors, regardless of which country they hailed from, Ghana was the last African ground they touched. Wearing Kente is like being in the arms of a long-lost grandmother who has found us and has called us home.
There I sit on the huge stage, facing the crowd of fifteen hundred, wearing the robe representing my highest degree with its black velvet hat and tassel, watching the sons and daughters of the African Diaspora process in front of me, to be greeted with the most wide smiles of love by their parents, to be gifted with the cloth of long-ago ancestry. That alone might have put me over the edge emotionally as I sat watching my first Black Grad, and I did chase tears from my eyes as Jan read not only the names of each graduate, but their degree, which became a call-and-response cadence with the audience's loud applause.
"Bachelor of Arts in History"
"PhD in Chemistry... Stanford's first!"
"Double Major in Economics and African American History"
"Phi. Beta. Kappa." Jan always punctuated that one slowly for maximum effect.
All of this achievement, this excellence, among Blackness is so beautiful to see. I wipe my eyes, repeatedly.
But what really gets me are the family members. A Nigerian couple in exquisite silk robes takes the stage and I'm catapulted back in time. Having been born in Nigeria, outside my immediate family Nigerians were the first people I ever knew. I tear up at the audience's wild applause for these people.
Then comes a single black woman, thin and frail, walking with the effort of age to greet who I presume is her grandchild. I keep pushing tears off my cheeks as the audience continues its thunderous applause. Then I look over to the staircase at the family waiting to go next. It's a white woman and a Black man. I gasp and hold my breath. Oh Jesus, an interracial couple. What will the crowd think of them?
This couple, looking uncannily like my own parents, cross the stage with huge smiles and greet their caramel-skinned boy child who is also grinning. They walk as this crowd continues its thunderous applause. And I'm sitting in my chair on the stage with tears streaming down my face, my nose clogged with snot. I can't even bring myself to clap as I'd been doing for everyone before. A faculty member three seats down passes me his cloth handkerchief, and I clutch it gratefully, desperate to continue to watch before wiping these tears. I manage a few claps as they walk together off the stage. The moment is over and it's the next graduate's turn.
What began to sprout in me that night was a sense that a biracial person could belong within Blackness. The applause for an interracial couple and their kid, willingly, lovingly offered by more than a thousand Blacks, was unambiguous recognition of the existence of interracial families and light brown kids. Not just recognition but approval.
They see me. I'm good enough as is. I don't have to fear I'm not Black enough. I belong.
It felt like the most religious of baptisms.
II.
The following fall at New Student Orientation I go to the Black Community welcome and sit up a bit straighter in my chair than I'd done in years past. Student leaders offer words of welcome to the new students and their families, then Jan gives her remarks to the crowd.
"Welcome to the Black community at Stanford. We are African, we are African American. We are Multi-racial, Biracial, Caribbean. We are from the East West North and South. We are gay, we are straight. We are Muslim, Christian, Jewish, atheist, agnostic. We are first generation educated and we are third generation. We eat sushi, and we eat collard greens too. I don't know what 'acting Black' is, but the one thing that Black means here at Stanford is excellence."
Although she'd been opening like this for years, this time I actually hear her words, and I inhale them deep into me. After Jan speaks, she asks all of us Black staff and faculty in the room to stand up front in a line and introduce ourselves. Now in my fifth year as dean of freshmen, I know a slew of the upperclassmen in the room. When my turn to speak comes, they will show me some love with their applause, I know. As I wait for the mic to be passed to me, I don't think about the "whiteness" of my voice or my shaggy biracial coils, which are now so long they touch my shoulders.
One of our new freshmen that year is from South Central L.A. A student who had graduated high school at the top of his class, who will play football for us, who will champ at the bit to prove to coach Jim Harbaugh just how good he is on the field, and who will go on to play cornerback for the Seahawks and become a beloved figure in Seattle. His name is Richard Sherman.
III.
Back at my office, I'm still in a bit of a tense dance with the colleagues with whom I've tried to press my points about what's right for our freshmen. In November 2006 the vice provost brings in an executive coach to work individually with each of us and to improve our team dynamics. The coach is a woman named Maryellen Myers, a white Buddhist aikido master. Frankly, I look forward to helping her discern what is wrong with each of my colleagues.
IV.
One day the four of us who report to the vice provost—all of us women, the other three white—are meeting in a conference room to strategize over how to present a sticky issue to him. Janine, who is seated next to me at the table, interrupts our conversation and abruptly turns her body ninety degrees to face me. Janine is a thin white woman in her fifties of Eastern European origin with a perpetual expression of mild annoyance and a reputation for being impatient. Junior people walk on eggshells around her. But she seems to like me. More than that, she seems to believe in the changes I am trying to bring about as dean of freshmen. She is tough and fair, and I admire that about her. I am used to being able to spar with such people and win or at least draw.
Now staring at me with a broad smile, Janine widens her eyes in delight, presses her palms down onto her thighs, and blurts out that my hair is "so interesting," even "amazing." Then she reaches out to touch it, at first patting it with both of her hands, then bouncing and lifting it like a beach ball.
I shake my hair to rid it of her hands. I push my seat back, stand up from the table, and back away from her. "This is a thing," I say in a loud voice, both hands up in protest, looking around the room, appealing with my eyes for help from the one woman I hoped knew better. "This is a thing white women do to Black women. Treat us like zoo animals. I'm not a zoo animal. You're not supposed to pet me." My words splatter the room tat-a-tat-tat-tat like ammunition from a machine gun. My tone is emotional yet I am trying to smile so as to make it clear that I am not the bad guy here. My colleagues stare at me, their mouths open.
That day, I become the Angry Black Woman.
V.
In February 2007 Barack Obama announces he is running for President of the United States. I'd watched him speak at the Democratic National Convention three years prior and had physically lifted myself out of my chair that night, jubilant, like when I'm watching my team crush it in a big game.
I am getting restless in my career at Stanford, wondering how much longer the joy of working with the students will outweigh the annoyance of trying to effect change in an environment where change comes slowly. The day after Obama announces, I write a long letter to the folks at his campaign headquarters in Chicago and attach my résumé. Something in that letter piques their interest and that March they fly me to Chicago for a conversation. There I meet Betsy Myers and Analisa Lafontant, two white women overseeing the enormous task of bringing this unknown candidate to the consciousness of a party obsessed with Hillary Clinton. I tell them I want to play a grassroots role in California. They tell me they aren't opening any California offices. I plead on behalf of the largest state in the nation for us to matter to this campaign, and then, shutting up and listening to them, I finally understand: there will be no California campaign until they make it past Iowa, New Hampshire, South Carolina, and Nevada. They offer me a job with Chicago headquarters but I can't imagine uprooting my family or commuting two thousand miles to support such a long shot.
I resign myself to being a volunteer if and when the campaign ever makes it to California. This means staying where I am at least for a while, participating in the coaching, listening to what Maryellen has to say.
VI.
I didn't have a Black mother to teach me how to be in the world. But I found a literary Black mother in the poet Lucille Clifton.
By summer 2007 our Three Books freshman orientation program was in year four, and after the difficult conversations over Possession by Byatt, we'd long since given up trying to select books by committee. For the past few years it had been my job to select a faculty member who would choose the three authors and their books and moderate the event. For the 2007 program, the faculty moderator selected Good Woman by Lucille Clifton to be one of the three texts.
I'd hated poetry for its confounding barriers. Had barely ingested what little of it they fed me back in the English classes at Middleton High School. Couldn't make my way through the obscurity of poetic language, be it Whitman or Dickinson. Could barely make sense of Shakespeare last time I'd tried. Poetry was a locked gate I wasn't interested in trying to open.
But as dean I had to read all three books. I would be meeting the authors for lunch in advance of the program, and I'd have the honor of making opening remarks onstage. I began reading Good Woman out of obligation. An hour later I looked up at the clock. I'd been hooked.
and if the man come to stop me
in my own house
naked in my own window
saying I have offended him
I have offended his
Gods
let him watch my black body
push against my own glass
If she is possible. If these thoughts are possible, this language. Then maybe I am possible?
"Pourquoi es-tu noire?"
Because I am.
VII.
After about nine months of working with the vice provost and his direct reports, Maryellen has conducted a 360 review on each of us, and she is ready to tell me how I am regarded by my colleagues. By now I trust her enough to be able to listen to the feedback:
Too emotional. Too aggressive.
Might as well give me a list of stereotypes of women and Black people and Black women and tell me not to do any of those things.
She lets me continue.
Yes I have a tendency to blurt things out when I get really moved by something or frustrated but my emotion is warranted.
"Is it getting you what you want?"
When I practiced law, my passion and anger could be channeled into useful argument. But in academia? It seemed to just push people away. And then I'm the one who has to apologize.
"I want to know why I'm this way," I plead. "That could take twenty years of therapy," Maryellen says, chuckling. "How about we focus on when you're this way, so you can start to notice the emotion coming and then decide what, if anything, you want to do about it."
What, if anything, I want to do about it. Maryellen isn't siding with stereotype. She is telling me that my power lies in being in charge of my voice.
With Maryellen's help, I begin taking notice of my behavior. When I feel a strong emotion coming, instead of acting on it, I try to pay attention to what I am feeling and where I feel it in my body, and what triggered the feeling, and I write it all down. When these feelings arise in meetings with my colleagues, I have a little code for how to respond: "DDE," which stands for Don't dwell, excel. For weeks my meeting notes are littered with this tiny notation.
Over a few months of this close attention to my self—of mindfulness, some would call it—I begin to be able to sense emotion coming. I can then pause, ask myself what is going on, and tell myself I am okay, while the conversation around me keeps going.
I begin to see that the trigger is a feeling of being overlooked, doubted, or dismissed. I begin to see that my fear that I will be judged as not good enough makes me desperate to prove, constantly, that they are wrong. I begin to see that I can't control anyone else's opinion or behavior. I begin to see that the only thing I can be in control of, if I work hard at it, is myself. With Maryellen's guidance I begin to see that I can love and accept myself regardless of what others may or may not be thinking of me. I can choose whether to speak or not, to be silent or not, to go off on someone or not, rather than let those impulses simply happen to me. As her coaching begins to impact me, I feel renewed. With the help of a white Buddhist aikido master I begin to emerge into a healthy Black self.
VIII.
A day comes when I summon the guts to tell Maryellen one of my most painful secrets: that as a child I hated being Black and was afraid of Black people. This gut-spilling fear-sharing loosens up knots of shame in my psyche. Loosens the muscles not just in my mind but in my soul. Speaking this awful truth out loud through tears kneads the pain out of me. The relief feels astonishingly good.
I wake up the next day no longer feeling the vise grip that asked me to prove I was good enough despite being Black. I look in the mirror and allow myself to see not what whites might see or what they might want to see or what they might want not to see; not conforming to what they admire. To see my actual self.
To see the color of my face and body—paper bag brown in fall and spring, high yellow in winter, milk chocolate in summer—and accept that some in America see me as the "other," and being fine with that.
To see my skin and hair and hear my "white" speech, and decide that it is not up to some committee on Blackness to anoint me as Black.
It took me forty years to stop twisting and turning this way and that in response to how I feared and hoped people of both races would see me.
I drive to work that day having shed the loathing of my Black self and, by extension, of all Black people from my eyes, which had prevented me from really seeing other Black people. I look into the eyes of one, then another, and then another Black person, and I feel my heart swell with feelings like compassion, admiration, love, even desire. As if discovering their existence, their magnificence for the first time. It might as well have been the first time.
Like climbing out of a deep depression, I hadn't known I was this afflicted until I wasn't.
IX.
A history read about in textbooks, literature, poetry, and newspapers, seen in movies and on television, heard in stories, heard in song, becomes mine. I begin.
To feel one with my ancestor, the slave. To know of slavery's systemic dismemberment of Black agency, debasement of Black men, rape of Black women, destruction of the Black family. Know of the wringing of energy and life out of my forebears, and of how they were then thrown out like trash to litter the ground.
To know of the efforts of resistance, rebellion, and escape. To know that those with light skin who passed into the white world left behind community, family, solidarity, and self in joining the white world. To understand that most could not pass and endured wearing the skin God gave them. To know that the promise of God and heaven was at times the only balm.
To know emancipation meant freed from ownership by another human then consigned to a life where skin color equals less than, equals bad, equals thug, equals criminal, equals presumed guilty, equals justifiably frightens whites. Equals death.
To know we've fought and died for America since its inception, on this soil and on foreign soil, have liberated others in the name of America's ideals only to return home and still be called Nigger.
To know of the brief sunlight of Reconstruction. Like Greenwood in Tulsa. To know of the Black leaders including my own ancestors who began to shape a new way forward for us. To know of the rise of the Ku Klux Klan, of Jim Crow, of the uniquely American practice of hanging adults and children from trees, of the economic, social, and psychological re-enslavement of Black people. Of the new enslavement that is mass incarceration.
To know why a rational, educated, hardworking Black man living in the twentieth century hated that false marker of independence, that tribute to a time when our people were chained like dogs and cattle—the Fourth of July.
To see the face of Emmett Till, the child murdered because he may have winked his eye at a white woman, found bloated like a dead frog in a Mississippi stream in 1955. To see Emmett's fourteen-year-old face and see my own. To see my son.
To hear South Carolina's Susan Smith claim in 1994 that a Black man carjacked her vehicle with her two small boys in their car seats in the backseat and see this set off a nationwide manhunt for Black men, and to learn that Susan Smith herself had strapped her boys into their car seats, put her car in neutral, and let it roll into a lake, where her two small boys slowly drowned. To see white people not comprehend the psychological toll this takes on all Black people. This Brutal Imagination, as Cornelius Eady calls it.
To see the face of James Byrd Jr., chained to the back of a pickup truck in Jasper, Texas, in 1998 and dragged along an asphalt road for three miles, still conscious until his head was severed from his body.
To find a home in Black America.
Though later, when Trayvon Martin is murdered, and he looks to me just like my son, to know an even deeper we. A searing pain. A surer Blackness.
X.
I feel rage toward whites. I feel love toward my own people. I try to channel these emotions into something that might help someone.
And so what about this white husband and these quadroon children?
XI.
White Americans, you are infatuated with the Statue of Liberty whose tablet contains words of welcome for all, who did in fact welcome you and your ancestors, and you are simultaneously infatuated with carving lines and borders between who does and does not belong here, with yourselves on one side of the line and the other half of America on the other. You think your whiteness makes you better than the rest of us. You make us your scapegoat. Your excuse for your violent rage.
["It's not all of us, stop saying it's all of us," you say, my white brethren.]
[You want to be treated as an individual instead of a stereotype.]
And I will get out of bed anyway and go out into the streets of America to do my work, to find true love, to raise children who know how to work hard and be kind to others. To speak.
XII.
In late summer of 2007 the Obama campaign finally stakes a claim in California with the grand opening of an office in Oakland—the first office located outside a swing state. When Obama himself comes to the Bill Graham Civic Auditorium in San Francisco that September, the Chicago team invites me to be one of the small handful of people introducing the candidate to his Northern California base.
It is heady, exhilarating, to be tasked with bringing a crowd of thousands to a rolling cadenced frenzy. I get a smiling nod from the candidate as he strolls out. At the conclusion of the event, one of the white women from headquarters takes me aside in the back hallway and thanks me, and then mentions a new constituency group they are forming—Black Women for Obama—and asks if I might want to lead it.
Yes, I desperately want to work for the campaign. But this role? No can do. I lack a connection to the broader Black community and to the network of important Black institutions like the church, sororities, and HBCUs. On top of that, Obama himself is contending with whether folks feel he is Black enough to garner support of Black people. I will not succeed at advancing this crucial cause.
I explain this to the woman.
"Is this about not being Black enough?"
Yes, I tell her.
And I am not ashamed.
For the first time in my life the truth that I am not Black enough for a particular role is just a fact, not a taunt. For the first time in my life, from my position inside Blackness, I consciously reflect upon what the Black people would want and need—deserve—instead of identifying with the white perception of something. Black Women for Obama deserves a better leader than I can be; I can say this plainly to this white woman without feeling inadequate or apologetic, even though I can see from the look in her eyes that she does not understand me.
XIII.
In my fortieth year, I stop letting whites pet me. When they try, without being the Angry Black Woman, I say simply, "Please don't. You shouldn't do that." I can save the Angry Black Woman for far more serious moments. Like when whites fear our unarmed children and kill our unarmed children, and when the system of white justice calls the shooter's fear of our unarmed children's brown skin reasonable and justified.
DECLARING
I.
To survive as a Black person in America, I have to assert that when micro-aggressions penetrate my skin like a parasite, I will not let them burrow deeper into me where they can eat me from the inside out.
What is a micro-aggression?
1. Getting to paw through a Black female colleague's hair
2. Commenting to others about how fascinating you find it
3. Calling the Black woman "angry" or "oversensitive" for minding
4. Not remembering this happened—or
5. Telling us to get over it.
When you feel us like a piece of fabric, it summons a genetic reminder of standing there naked at auction, of being sized up and sold off according to the size of our birthing hips and ripeness of our breasts.
II.
When I am forty-one, I see Barack Obama hoisted onto the shoulders upon shoulders upon shoulders of millions of Black Americans seeking to construct a human column of Blackness so high it can reach the light and maybe diffuse a bit of that light onto us all.
In 2009 President Obama is addressing a joint session of Congress about health care, when white Congressman Joe Wilson, a Republican from South Carolina, interrupts with two shouts of "You lie!"—an act of incivility, a lack of decorum that undermines the very office of the President of the United States. Joe Wilson might as well have called our President "boy."
When in 2012 Clint Eastwood speaks at the Republican National Convention and uses the prop of an empty chair beside him, which he speaks to as if it is Obama, the chair symbolizes the chair underneath the Black man about to be hanged from a Southern tree.
III.
In 2009, the PTA at my kids' elementary school throws a murder mystery party as a fund-raiser. My husband and I volunteer to work at the event, mostly to do our part to help the school, but also to get to know the broader parent community a little better.
Ours is a middle- to upper-middle-class neighborhood in Palo Alto, the heart of Silicon Valley. Stanford University is one of the largest employers in the area, along with big tech companies such as Google, Facebook, Oracle, and Apple. Our neighbors are mostly American whites and first- and second-generation Asian and Indian immigrants, a few European immigrants or expats, and a handful of Blacks, Latinos, and Native Americans. The neighborhood's politics lean heavily liberal. I represented our area at the 2008 Democratic National Convention as an elected delegate for Obama.
The party begins. The theme of the murder mystery is athletes marooned by a plane crash on a South Pacific island. Dan and I carry trays of food and drink from the kitchen to the backyard pool area where tiki torches help create an island feel. Lights float on the surface of the pool. Someone puts on some music. The guests begin to arrive.
One woman comes around the side of the house in Blackface. At first I don't really see what I'm seeing because I'm still bustling about trying to set things up according to the host's plans, and because I would never in a million years expect to see someone show up in Blackface. But there she is, jaunty in her athlete's uniform, skin burnished head to toe in a dark brown shoe polish, playing her assigned role of a Jamaican. Someone puts on some music. A white man assigned to play the part of "Tyrel," a track athlete, wears a huge Afro wig and begins to dance at the center of a circle of the partygoers. Another white man shouts, "Show us your jive dancing." The Blackface woman dances too.
I freeze watching these upper-middle-class white people, neighbors, and even a few friends drinking, laughing, goading the white man doing the mocking dance. The night is loud with music, hot with summer, and wild with this unleashed whiteness. Dan is watching from the other side of the crowd and looks over at me, his eyes wide with disbelief. I scan the crowd and see a Native American professor and his wife looking at each other. There are fifty whites and only a handful of us and they are drinking and laughing and egging each other on, and if I stand up and say something I am not certain enough of them even know my name so as to respect my voice if I turn off the music and shout WHAT THE FUCK ARE YOU PEOPLE DOING. I am more certain that if I turn off the music and say something, they will laugh and continue on with their frat party dancing.
I reel backward, fade away from the taunting crowd, with stomach bile jumping up into my esophagus.
Am I safe here I am not safe here this is not happening here this is happening and I need to leave and I need to leave now this is not happening.
I grab Dan's hand and we walk briskly away from the crowded backyard, through the house, and out the front door. I push the bile back down with deep breaths, fight back tears, ask myself if I dare to ask myself this question aloud.
Is this how white people act when we're not around?
I talk to the Native American professor. I write a letter to the school principal. Someone tells the Blackface woman that her getup offended me. She writes me saying she meant no offense. I Google "blackface" and send her the link. My doorbell rings and I open the door and suck in my breath as she stands there and apologizes in person. As far as I know no one ever said anything to Mister Jive Dancing. I make a point of never again going to parties thrown by white people I do not know well.
IV.
In 2012 Stanford's Human Resources Department is making a video to help orient new employees, and I am invited to participate in it. They want to portray my perspective as a dean and woman of color about the Stanford community.
I sit up straight in my chair under the hot lights and answer the producer's questions. Fifteen minutes later it is done.
"You're so articulate," the producer says, shaking my hand.
"Thanks."
"No, I mean it. You're just, I don't know, somehow incredibly articulate."
"You can't—Are we really going there? You shouldn't say that."
I am forty-four and have been a dean for ten years.
I am old enough not to take this shit anymore. I am old enough to remove the microphone clipped to my lapel, shake her hand while shaking my head, leave this small television studio, and walk confidently back to my office. I am old enough not to get emotional about it.
Holding my shit together is a victory as America works me over.
V.
While Obama is president, cell phone cameras pull back another curtain on what happens when we're not around: horrific violence toward Black people, even children. Cell phones make the whole world a witness. They carry the sounds of Black people crying for help. And dying.
Trayvon Martin, aged seventeen, buys Skittles and an Arizona iced tea at a 7-Eleven and is walking back to a family friend's home in an upper-middle-class white gated community in Sanford, Florida, when a neighborhood watch volunteer deems this unarmed, Skittle-toting hoodie-wearing teenager suspicious and deems that suspicion reasonable grounds to follow Trayvon, stop him, fight with him, shoot him, and kill him.
Darius Simmons, aged thirteen, is retrieving his family's trashcan from the curb one afternoon after school in Milwaukee and an elderly white neighbor shoots him dead in broad daylight. Darius's mother witnesses the entire incident.
Jordan Davis, aged seventeen, is riding in the backseat of a car that stops at a gas station in Jacksonville, Florida, rap music blaring from the car radio. A software developer who was in Jacksonville for a wedding considers the rap music in the car too loud and says so. Then he fires shots at the car, kills Jordan, and returns to his hotel and orders a pizza.
Jonathan Ferrell, aged twenty-four, is injured in a bad car crash in Charlotte, North Carolina. He drags himself out of the vehicle and walks down the street seeking help. When he knocks on the door of the first house he comes to, the white woman who answers decides Jonathan is a menacing Black man. She calls 911. When the cops come he approaches them, thinking they are there to offer him aid. A cop shoots him dead, firing ten times.
Tamir Rice, twelve, is playing with a toy gun in the park near his home in Cleveland, Ohio. He is shot by police and denied CPR and he dies lying on the ground at their feet, his toy gun nearby.
Oscar Grant and Michael Brown and Eric Garner and Freddie Gray, and Philando Castile and Alton Sterling, and Terence Crutcher and Keith Scott
and
VI.
I have been ashamed of America. America should be ashamed. America leveraged a slaveholding disregard for Black and brown skin to power its first industries. America built itself on the back of Blackness as a way to elevate the status of those with lighter skin. America owes Black people a debt of contrition and recompense. A process of truth telling and reconciliation.
VII.
We the people cannot continue to abide the stories of police and civilians on witness stands telling us that in just seeing our Black bodies they were terrified.
You have to be terrified for a justifiable reason.
God gave us this Black and brown skin. The skin God gave us is not a reason for you to be justifiably terrified.
VIII.
We are terrified.
Of you.
IX.
We continue to try to forgive.
To live.
X.
We do our work.
Ohio State professor Michelle Alexander writes of the mass incarceration of Blacks as The New Jim Crow.
Stanford professor Jennifer Eberhardt receives a MacArthur Fellowship, known as the "genius" award, for her research on the learned fear of dark skin. The seeing of brown skin that makes white folks—makes all folks—more likely to pull a trigger. The learned, presumably unlearnable, psychological cancer metastasizing in us all.
Journalist Ta-Nehisi Coates writes Between the World and Me, a letter to his Black son about whiteness—an invention designed to ensure a hierarchy of color—and about how to survive despite it.
Poet Claudia Rankine puts Trayvon's hoodie on the cover of Citizen.
Dan and I have "The Talk" with Sawyer.
XI.
On September 19, 2013, I'm in the lineup to read at the Cat Club, a bar in San Francisco. Jonathan Ferrell has just been murdered by police after surviving a car crash.
here is going on a war there is war on going on here
called him nigger.
called him liar.
hung Clint Eastwood chairs from southern trees.
blue red black
white ninety-nine
one stars and
bars stars and
stripes
invidisible
we may mass the troops amissing Sunday mornings
ammunition stackpiled on a stock of bibles
but just listen to the smalling voice of fearful white people—
i'm telling you:
stand up they are not going to stand for this take this sitting down
XII.
In August 2014 Darren Wilson, a police officer, shoots and kills unarmed eighteen-year-old convenience-store shopper Michael Brown in Ferguson, Missouri.
In October 2014, eighteen- to twenty-two-year-olds at Keene State College in New Hampshire erupt in a massive drunken riot and turn the campus into "a kind of war zone," which includes throwing billiard balls and full bottles of alcohol at the police, pulling street signs out of the ground, setting fires, overturning a car, and reportedly threatening to kill police officers.
No one calls these students thugs. Eighty-seven percent of the students at Keene State College are white. They are "kids" having fun at Halloween. The college president says his students have failed to "pumpkin responsibly."
XIII.
Even dying and in death we deserve no human mercy.
Eric Garner told police "I can't breathe" when they had him in a choke hold for selling cigarettes illegally.
Tamir Rice lay gasping for breath, his toy gun on the ground nearby, and the policemen standing over him did not offer CPR to this twelve-year-old boy they knew by then was only a child with a toy gun.
Trayvon Martin and Michael Brown were left dead on the sidewalk for hours, their bodies unclaimed, the local police do not even lift these boys' bodies off the sidewalk, do not properly care for the corpse.
The mothers frantically call, text, plead Have you seen my son please help me find my son.
XIV.
Trayvon shot no one. Neither did Tamir, nor did Michael. But the white supremacist Dylann Roof, who went to a Bible study at a Black church so that he could shoot people and did shoot people in a church, shot his gun off in a church and killed nine people and then fled, and when he was apprehended, Dylann Roof, the white supremacist, the police got him a Burger King cheeseburger because he was hungry.
The Black family members of the nine Black people slain in cold blood by Dylann Roof said in front of television cameras that they forgive Dylann Roof and they are commended for being able to forgive the white shooter Dylann Roof who opened fire at a Bible study meeting at Mother Emanuel Church and killed nine.
We know forgiveness is all there is in an America where we are not equal. Where we watch as our children are killed because of the color of their skin, and Dylann Roof, a self-professed white supremacist who systematically mowed down nine inside a church, flees the scene and later is apprehended and given a cheeseburger from Burger King, because the poor boy is hungry.
XV.
We watch.
We get up the next morning.
We give birth to baby boys whom Hollywood finds adorable and who show up in commercials and television shows and are coveted by white audiences for their cuteness and ten or fifteen years later we've raised those boys to be men who transition before white eyes into thugs.
Some of us live in middle- and upper-middle-class white communities thinking them safer, thinking them to be the place of arrival, of transcendence. We see Trayvon gunned down in a gated white community because he looked suspicious because his skin color and hoodie made him look suspicious and we gulp down our fear we who think we have passed into a better status with our money and privilege and degrees we gasp knowing we are wrong, know there is no place for us no place that is ours in America.
We have "The Talk" with our sons. Teach our sons how to kowtow to police. How not to draw attention to themselves. How to raise their hands in the air. How not to defend themselves even when they are sure they have done nothing wrong. How not to reach into their pockets for anything, not even to turn off their music. Please, baby, remember: do not reach into your pocket to turn off your music.
We teach them this while trying to also teach them to love themselves and not to be ashamed of their beautiful black bodies. Of their selves.
XVI.
It was inevitable that I would marry a white man. When Dan and I got married in 1992 I made an irrevocable choice that suited me well then.
Decades later, as a middle-aged Black woman and mother, I would examine not only what I'd gained by having a white man on my arm but what I'd irretrievably lost.
I see in my daughter's light skin the possibility of "passing," which I'd studied as an academic concept, a historical relic, in college and law school. What will this racial ambiguity do to her? Who will she be? Where will she locate a self to love? Where will she find belonging?
I see in my son—who looks out at me with soulful dark brown eyes like Trayvon Martin looks at all of us out from under his hoodie—a boy who cannot know what he might have learned if I'd given him a Black father.
XVII.
Identity is in part a response to the version of yourself that gets mirrored back to you. In The Souls of Black Folk, W. E. B. DuBois asks, "How does it feel to be a problem?" As Blacks we have a double consciousness, always looking at ourselves through the eyes of others, always aware that we may be the Nigger in the eyes of the stranger, the coworker, the neighbor, the acquaintance. And in our own eyes as well.
XVIII.
My daughter, Avery, is a teenager now and is more than I dared to dream she might be: smart, beautiful, fiercely sure of herself and also respectful, a ballet dancer, lithe and strong. When she lets down the bun her dance teachers require, her mixed-race hair curls and curls and curls down her back, her skin a light, light brown like butter just starting to cook in the pan.
I want her to be able to answer the "what are you" questions with the kind of educated pride that eluded me as a kid. I want her to be able to bring abject defiance into a situation when necessary. But over the course of her fifteen years, I'm fairly sure no white person has ever felt the need to ask that question of her. Instead, they are surprised to learn that the brown-skinned big brother she adores is in fact her brother, or that I am her mother. Which means white folks see her as one of them.
Crushing me like a tin can under some white foot.
XIX.
Sometimes as I try to raise these children up to love themselves and love others even I still loathe myself in my coffee-brown skin and frizzy hair and flatter nose and at the grocery store which is where I go weekly to get what lies beyond the cocoon of my home and I am muttering something to myself as I walk through the pasta aisle when I spot a middle-aged white man and I make eye contact with him or try but he averts his eyes from me and I realize that while a white man talking to himself in this town is a tech genius in this white man's eyes I am likely homeless or crazy in my ripped Harvard Law School T-shirt I must have gotten from Goodwill and when I get to the checkout lane I try to perform the part of a white person so they don't ask me for ID just like they didn't ask the white person in front of me for ID and I think no one has loathed themselves like I loathe myself and I am ashamed to admit this even to myself or into the air I exhale or to the other brown-skinned people but when I dare to tell it to the brown people for the first time in my life at forty-five after the murder of Trayvon and the acquittal of Zimmerman they look at me and their eyes well with tears and their soul reaches out to touch mine with an invisible hand and for the first time I realize I am not alone when I loathe myself as a Black person. Have never been alone.
XX.
We stand up for ourselves when we can.
XXI.
In 2014 when my girl is thirteen, we are visiting close family friends—a white couple my mother's age whom I refer to as Aunt and Uncle—in their stately home on the East Coast. Aunt Peg and I are drinking a glass of red wine in her gorgeous kitchen. Avery comes to get a glass of water from the tap. Avery leaves the room. Aunt Peg turns to me. "Isn't it great she doesn't look Black?" Aunt Peg takes a sip of her wine.
I feel short of breath. She may as well have said Too bad you and your son are cursed with a skin of blackness. I feel like I am underwater and might not make it to the surface in time.
I love this woman, have giggled with this woman, confided in this woman, and held her confidence once when she poured her deepest concern into me. I am also a guest in her home.
I have no idea what she intends by this comment. Maybe she's just doing her best to express compassion for what Sawyer faces as a Black male, and relief that Avery might, by those standards, have an easier time. In awkward race situations like this I've gained a lot of mileage by assuming the best intentions, by helping the white person not feel uncomfortable, but Am I here to help others not feel uncomfortable? I thought to myself. If I say nothing, then nothing gets said.
"Well no, actually, I mean, she is part Black. It's an important part of her ancestry." Aunt Peg goes on to say how her ancestors were from Scotland and Norway but none of that matters anymore and it is actually better that way. I say, "Yes it would be great if our ancestry 'didn't matter' in that it wasn't a negative in the eyes of others, but it is a negative in the eyes of others, it impacts our experience. And ancestry is who we are, it's who we're from. It's how we got here."
And I continued thinking silently to myself, It's what ties us to something bigger than ourselves, it is our anchor and our stars, and these Black anchors, these Black stars, they matter to me. My slave ancestor Silvey died for me. If she hadn't been raped by that white man I wouldn't be here. And neither would my daughter. We need to honor her. We need to never forget Silvey.
In the Black community we fight an internal war over our skin tone. We scrutinize. We chide each other for being too dark or for not being dark enough. If we're on the light end we pass the paper bag test or we don't. If we don't, we might just as well pass into the white community as our ancestors did during slavery and Jim Crow, exiling ourselves from Black folks, and therefore belonging nowhere and to no one, not even to our own selves. We are, at times, simultaneously ashamed to be Black because of what American society decided Black is and determined to be proud of who we are even in the face of America's hatred of us. We need as many people on our side as possible. It doesn't help when the light-skinned among us hide or even pass to the other side.
We are not turning from Black.
As I think these thoughts Aunt Peg is talking about color blindness.
I've loved this woman and felt loved by this woman for forty years, but that day when she held her wineglass in one hand and casually poked at the biggest button I've got with the other, I had to walk away. If ever there was a time I wanted to punch someone in the face, this was it.
XXII.
On June 10, 2015, the Black Lives Matter movement is front and center in our nation's consciousness as the police killings of Michael Brown in Ferguson, Missouri, Eric Garner on Staten Island in New York, and Freddie Gray in Baltimore, Maryland, are shaking a nation's sense of itself.
My first book, warning against the harms of helicopter parenting, had come out the day before, and a conservative radio talk show host named Laura Ingraham wants to have me on her show. Ingraham is one of the nation's most widely listened-to radio talk show hosts, a mother of three, and is known for "moving books." My publicist is wild over the opportunity. Ingraham is also a rabid Neocon who calls Black people "thugs." I decline.
My publicist pushes back. "This is a massive hit for the book at a critical time in the campaign. The Ingraham people say she only wants to talk to you about parenting."
I'm raising a son who is a human being of worth and value, not a thug, I tell her. Ingraham and those who speak like this over American airwaves put my son's life at risk with their vitriol. I decline.
My editor calls. My book is unlikely to be a New York Times bestseller if I don't do the interview, she tells me. Tells me the team of folks at the publishing house sat around the table and talked about this and they all agree I should do it.
"Are any of the people around your table Black?" I demand. "Are any of them brown? If not, then how about queer? If not that, then how about a religious Jew? You're asking me to do the equivalent of a Jew talking to a Holocaust denier. Find me someone who can relate to that and who wants to try to convince me."
My editor calls again, relays a message from the publisher at the very top of the food chain, who thinks it best that I do the interview.
I decline.
XXIII.
White people.
We win some small victories but America behaves as America does, and we experience small slights and enormous tragedies committed by you.
My nephew is a forty-one-year-old Black man and he was at your house the other day because he and your husband are old friends and he was in town for a meeting so he stayed with us but came over to your place to hang out for a long, long while and he left his shoes behind. (How does a man leave a house without his shoes is the kind of question often left in the wake of my nephew—my nephew who from the airport as he waits for his flight home to New York texted me Can U get my shoes from my friend's house and mail them to me?)
So I drive over to your house, which is in my neighborhood, and it is evening and it is dark and I park my car at the curb and make my way along the stepping-stones of your manicured walk and I ring the doorbell and to the left of the large door is a picture window with drapes only partly drawn against the dark night and from a warm living room your little blonde girl peers out at me and then turns around and tells you something. Then you answer the door and say quite sternly, "How can I help you," and I just want to pick up some fucking shoes left by my nephew at the home of his close friend and his wife but instead I perform.
"Hi, I'm Michael Lythcott's aunt Julie, I'm here—"
"What?"
"Yes, sorry to bother you, but I'm here to pick up my nephew Michael's shoes—
"Your—?"
"Yes, my—Michael, he apparently left his shoes?" I gesture to the pile of shoes visible in the foyer behind you. "He texted you, told you I'd be coming by to pick up his shoes. Or he called you?"
You hear the name of your close friend, my nephew, now for the third time. Your foreboding facial façade gradually falls away into a relaxed smile. "Oh yes of course," you say, stepping back, sweeping your hand across the vestibule of your doorway as if to invite me in, relief visibly slaking off your once-rigid body, and you point at a pile of shoes, where my nephew's lie indistinguishable in the heap of the shoes belonging to your family. And you make some statements about how you love my nephew and I plaster a false smile on my face, which you know is false, and my nephew's shoes are a size twelve and when you hand them to me they leave behind their absence, an absence you will stare at after I leave and even when you take your toe to the corner of your husband's shoe and kick it so it fills the space left by my nephew's you will remember my nephew's shoes.
XXIV.
My son, I look at the faces of Trayvon, Freddie, little Tamir who was all of twelve, and I see you, my son, my precious son, my beautiful Black boy, so smart and bookish and inquisitive and philosophical. I see you grow taller, grow muscles, grow a man's face, and I weep for the future self who will leave this home and discover that in pockets of this great country you are loathed, feared, and worse. My son, you did not ask to be born—I chose you. I asked you to be mine. I gave you a skin of brown.
And you are exquisite beyond measure.
BLACK LIVES MATTER
I.
Trayvon was my Pearl Harbor. The line demarcating before and after. The moment I knew Blackness is the core chord in my life. Because
despite imperfect
whatever my strange history
inadequacies with
as a person, Blackness
a Black person,
a
mother, my
inadequa
Because I am raising a Black son.
He was murdered on February 26, 2012, not in Ferguson but in Sanford, Florida, a neighborhood a lot like mine. I read of it a few days later in a small newspaper, weeks before March 17, which was when the New York Times would pick up the story. The Zimmerman verdict of "not guilty on all counts" came on July 13, 2013 and plunged like a cannonball into the murky self-loathing in my psyche and displaced every bit of that self-loathing, and the water that rushed back in its wake was a torrent of bitter tears and anger, and the calm stillness that followed was pure love. For my people. And for Trayvon.
When I see his face, all I see is my son.
II.
When your very existence defies the rules of the system into which you were born, you don't grow up respecting the rules. You want to fashion new rules where you can be one of the players instead of sitting forever out of bounds. That's the way I see it.
I am on the side of humans mattering. I take an interest in the experience of the other.
Perhaps I would have cared about these things even if I had been born a straight white male. But those were not the genes God asked me to inhabit.
I cannot look at his face. Years dead, years of justice not done, I cannot look at his brown face in that gray hoodie his dark eyes mournful like he'd come back in time as if he knew what was to come.
III.
Yes, sometimes, I regret the choice of a white husband.
To have given my son a father who cannot teach him how to be a Black man in America.
IV.
The more things change, the more they stay the same.
Richard Sherman, a cornerback for the Seattle Seahawks football team, says "thug" is the new N-word.
In the 2014 NFC Championship game between the Seattle Seahawks and the San Francisco 49ers, the final seconds were given over to Richard Sherman, formerly of Stanford, who leapt his body into the air to tip the ball from its intended receiver, Michael Crabtree, thus winning the game for the Seahawks and ending the season for the 49ers. When Sherman went over to shake Crabtree's hands, Crabtree grabbed Sherman's face mask and shoved him away. Then, as the game went to zeroes, Sherman made a "choke" sign at the 49ers quarterback, Colin Kaepernick. Then Sherman made a boastful statement to the ESPN reporter. Then folks got up in arms on social media over how poorly Richard Sherman had behaved. Pundits quickly labeled Richard a thug. Regular people and the media talked about Richard's behavior for weeks.
I take to Facebook to defend Richard, my former student, a man whose character I feel I know. I praise his ascent from the most challenging of beginnings in South Central L.A., I defend the chip he has on his shoulder from having to constantly overcome stereotype, I support his frustration at being undersold in the draft by Jim Harbaugh. A white male professor beloved by many of my former students comments, "Julie, I can't believe you're playing the race card."
It's not a card, I say. Not a game. It's our fucking life.
The following fall the white students at Keene State College who threw full bottles at cops and threatened to kill them later were just kids having a bit of fun.
V.
Richard Sherman is right.
Twitter trolls use coded language so they can spew white supremacist hatred and fly beneath Twitter's language regulations radar. My Facebook feed fills with white men and women casually referring to Black and brown men as thugs. Black and brown men looting stores in Ferguson are thugs, say these white people in my newsfeed, my so-called friends. White kids looting stores and overturning cars in New Hampshire are hooligans and pranksters and "kids who will be kids." Grown-ass white American athletes trashing a bathroom at the Rio Olympics are "kids." But Black and brown equals bad, lower, and you will call them thugs and your newscasters will call them thugs and we will cower in the corner of your imagined reality.
Richard Sherman is Trayvon Martin in a Seahawks jersey. He's straight outta Compton with a 4.2 high school GPA and a football scholarship to Stanford and a successful Stanford student and a professional football player and maybe this is not recognizable to you? That he can be all these things? Black and all these things? And if you don't recognize the Black person at your door as your neighbor or an athlete or an entertainer or as your professor or your president, if you don't recognize this person, you call the cops and when they come to your door, the cops will shoot the unrecognized Black person ten times or was it twelve and it wasn't the car accident that killed Jonathan Ferrell, this Florida A&M University football player, but the gun shot ten times and the 911 phone caller white lady presuming Black equals thug. Jonathan Ferrell's family buried their promising son and that white lady will never go to jail for her criminal behavior. Like Zimmerman's you call her fear of the Black man reasonable. Justified.
I kiss my Black teenaged son good night, night after night, lie awake in bed trying to figure out how to prepare him for life in "post-racial America" night after night after night.
VI.
I drive back into the night from your house with my nephew's shoes suddenly feeling like an impostor in my own neighborhood. Thinking what if instead of me my son had come to claim his cousin's shoes.
VII.
Even the Black man who was our forty-fourth President could not prevent a twelve-year-old Black boy from being gunned down as he played with a toy gun in broad daylight at a local park, could not prevent that Black boy from being denied CPR in the minutes he lay dying on the ground wishing for his mother, could not indict the police officer who felt brown skin meant credible threat.
VIII.
As Ta-Nehisi Coates makes scorchingly plain in Between the World and Me, America herself decided that he and I and people like us are lesser, with our brown skin, curly hair, thicker lips, and flat noses. Our status as the lesser is essential to America's narrative about herself; without us in the role of antagonist, there is no protagonist role for the white citizen. And this was no accident. America created the concept of race to justify enslaved labor, to steal its name, to bend it to its will, to strip it of its dignity.
It them us.
Some whites cling to racist leaders and racism itself so as to assuage themselves that however shitty their life is, at least they are not Black.
IX.
Some think we Blacks have actually gotten more than our due, like we have it easier than them because of antidiscrimination laws in employment, housing, and education. They call our efforts at linguistic inclusion and kindness "political correctness." And they're not about to accept a linguistic phrase—Black Lives Matter—that suggests that we need even more rights. That's how they see it. When in fact:
1. For the group whose historical and unexamined privilege is slowly eroding, the increasing equality of others can feel like oppression.
2. Black kids get shot by police and white kids get warnings. To pretend otherwise is to willingly not see.
X.
In my family we're one degree of separation from white rednecks and Black Panthers. How am I supposed to have that race conversation "we all know we need to have" when I can't even talk to my relatives?
In the quaint days before social media, we could keep our differences largely to ourselves and avoid awkward disagreements with our blood and married kin. When we did find ourselves captive before crazy Uncle So-and-so at the occasional family event, we could either walk away and find someone to commiserate with or choose to stay and defend a different perspective. At least we had to look each other in the eye, and if not in the eye, then at least we had to deliberately look past each other as we walked away, our body language passively communicating a bit about our perspective.
It's all different now, in the age of social media, where on any given day an uncle posts that I should really give Trump a chance, and a sister-in-law harrumphs "all lives matter" as she reposts something written by "Southern and Blessed." I find myself thinking, How the fuck am I related to this person? They're probably thinking the same about me. I want to shake them, shake the lack of disregard for the Black reality out of them, force them to look my Black son in the eyes and say that to his face.
Instead, I respond that we say "Black lives matter" not to mean "only Black lives matter" but to mean "Black lives matter, too" in a time when a Black teenager carrying a bag of Skittles home from the local convenience store is regarded as suspicious and gunned down as he struggles to defend himself, then left dead on the sidewalk, unattended to even in death.
You don't have to worry about that with your son, I remind my sister-in-law, reminding her also of her nephew, my son, whose very right to walk down the street unafraid is in jeopardy as others fear him and those who look like him and take up their guns accordingly.
XI.
You think if given the choice any of us would have asked to be born Black in America? You think we want to be the object of your evident fear as you pass us on streets and crowd away from us on elevators? In the wake of the Zimmerman verdict Questlove wrote so hauntingly about this. He described himself as a six-foot-two, three-hundred-pound Black man, and pleaded, "I mean, what can I do? I have to be somewhere on Earth, correct?"
Correct.
XII.
Sometimes I do wonder where is God in all of this?
I almost vomited when I heard an American doctor thank God for saving him from Ebola. It was the fall of 2014—those terrible few months when the scourge of Ebola had once again reared its head in a few African countries, and we Americans were fearful that, despite our best efforts at isolation, an African plague could invade our borders. A Liberian man named Thomas Eric Duncan had already succumbed to it here in the U.S. while visiting family, after showing up with symptoms summarily disregarded in the Dallas hospital where he sought help. By the time anyone realized he was more than a Black guy with a fever, the disease had consumed him as it does any victim, eating him from the inside out. Liquefying him. The hospital has since apologized to Duncan's family for systemically denying him adequate health care.
But the same tragic fate was not met by this white American doctor—Dr. Kent Brantly—I heard on NPR one day. Brantly had become infected with Ebola while treating patients in Liberia and had been airlifted back home to become the first Ebola patient ever successfully treated in the United States. Emerging as the survivor, the victor, from his intense treatment at Emory Hospital in Atlanta, they held a news conference for him, where he stood behind a podium with his enormous team of doctors and nurses behind him and declared that "God saved my life." And what went unstated but implied was that God didn't give a shit about the 1,350 Africans who had already died of the disease in its recent epidemic to date.
To be an American is to see God's hand in the U.S. health care system, and in the experimental serum known as ZMapp, which Brantly was the first human ever to try. To be an American is to believe God plays favorites, and that of all his children, he favors Americans most of the time.
To be truly devout, though, is to be a family member of one of the nine Blacks murdered during Bible study at Mother Emanuel African Methodist Episcopal Church by self-professed white supremacist Dylann Roof, and to forgive Mr. Roof for killing their loved ones in a house of God where presumably God was watching.
XIII.
Maybe God did give us the choice. Maybe he gathered a group of souls and asked for volunteers. "Now who wants to go down there and inhabit a Black or brown body? Who wants to take that on? Who wants to live a life in America where you may be treated like the scum of the earth? Who will walk among white people and be their opportunity to learn compassion?" And the bravest souls looked around at each other and raised their hands.
XIV.
I'm a middle-aged woman, comfortably upper middle class, with two graduate degrees, a house, two cars, and a retirement account. But my diplomas and dollars are paper shields against the army with their guns and self-righteous vitriol about their status as the inheritors of America. I think it is not unreasonable to feel truly terrified.
Though I am Silvey's child.
One of the original Americans.
I belong here.
And I will not be terrified.
XV.
There is so much I wish I'd asked Daddy before he died. About the birds he and my mother loved to watch and feed I recall only black-capped chickadee, tufted titmouse, robin, cardinal, goldfinch, and sparrow. Of the flowers they cultivated in our front and backyard and visited every evening upon returning home from work, cocktails in hand, I recall only Queen Anne's lace, chicory Cichorium, morning glory, and gardenia. And the things I never even contemplated before he died, such as What does it mean to belong in America? And What does it mean to be Black? And How do we live under the drainage pipe of white supremacy with its drip drip drip of poison into our hair, that oozes down into our eyes, into our nostrils, into our ears, into our mouths, our pores, our bones? How do we coexist with these white people fearing and hating us without fearing and hating ourselves? How do we laugh? How do we stop seeing their fear and their hatred as a mirror that shows us who we are? How do we look into a real mirror and love what we see?
XVI.
In the summer of 2016 I read a white Baltimore police officer's confessional. But it's not what I am expecting.
He's married to a Black woman, and she is pregnant with their first child. During the months of her pregnancy, Freddie Gray is transported in the back of a Baltimore police van, handcuffed, but not tethered with a seat belt, and is unable to stop his body from flying around as the van bounces through the streets of Baltimore. Gray's spinal cord snapped during this ride. Gray died. No one was held responsible.
This Black woman wife of a white male Baltimore officer is pregnant, and neither she nor I nor God can predict the color of her unborn child's skin. Or its gender. And when a son is born to them, a son of the color of brown, which equals Blackness, this white male police officer is quoted in the New York Times as realizing he is now raising a Black son in Baltimore.
Then he confesses that until he had a Black son he saw the young Black kids hanging around on the street corners on his beat as juvenile-delinquents-in-the-making, and now he sees them just as kids. Just kids enjoying the last few days of summer. What one might call "normal."
And in the poignancy of this white man's realization that Blacks deserve to be seen just as normal humans I am reduced to angry, helpless tears. Is this what it's going to take? All the racist white folks need to get some Blacks in their family? Is this why gay marriage took hold so quickly—prejudiced straights had a family member or a close friend who was gay? If whites produce brown progeny, can we once and for all breed the racism, the white supremacy, out of them?
Dear God, can we?
XVII.
I sleep with a white man. But he's different from the officer I read about in the New York Times; my white man never once thought Black kids were thugs.
XVIII.
"Biracial," a term I once courted, turned out to be a fleeting lover. Racial intermixture may be a fine way to root out racism—over decades, perhaps centuries—but what of the biracial child who lives that existence? I've been that tragic mulatto.
I didn't tell my parents about the N-word written on my locker on my seventeenth birthday because I didn't want their pity. Didn't want them to feel badly that they'd given birth to me or plunked me down in an all-white town. Didn't want them to look differently at me. Didn't want to be a victim. But like many victims, I felt I'd brought it upon myself. Instead of telling anyone about it, I let it fester inside of me, let it chase me through college and law school and into the workplace beyond.
I spent most of my life trying not to be your Nigger.
XIX.
In the summer of 2016 a close white friend and I are sitting at a picnic table with a handful of others. She asks me to talk about what I'm writing about in Real American. She listens to a few of my stories then begins to cry and says how hard this is for her to hear. She puts her head down on the picnic table and begins sobbing. I shift from telling about Black pain to putting my hands softly on her shoulder to comfort her in her white pain thinking this is a thing and people write about it and I love her and know her intentions are so solid and she connects with humans so well, which just showed me even more that this is a thing. A real thing.
But time is short and I'd prefer to offer my compassion to the antagonists in the grand narrative of America who manage to get out there every day and hold their heads up. Daring to be a person. Daring to make a go of life. Daring to be an American.
Dear white people,
When you're sad about racism please have the decency not to cry for your selves.
XX.
It comes time to address things with my mother. She is seventy-seven: still strong but more tired now, still very self-reliant and so frustrated when needy. And still beautiful.
In my kitchen one day, I speak to her pointedly with the voice of a woman no longer afraid to confront her past, her accuser, her accused. "How could you choose to live in Verona? How dare you chide me for not having Black friends when you raised me in an all-white town?"
She looks at me and begins to cry. She doesn't try to say my experience wasn't what it was. She tries to reach out to me but I back away and throw my hands in the air.
"He said white boys will be your friends but will never date you," I thunder. "Then why the hell did we live there?" She starts to explain what Daddy was thinking. "It was his truth from the life he'd lived."
"I'm not interested in making this right for Daddy! This was me. My adolescence. If Daddy believed it was okay to plunk me down in an all-white town where 'no boy would date me,' he was wrong. What did that even mean? That I didn't deserve to date? That I didn't deserve to be loved?" I yell this at my mother who, with Daddy gone for more than twenty years, is the only parent I can make listen.
She says she knew it would be a problem for me to grow up without Black people around and wishes she could have stood up to Daddy on that choice. Tears stream down her face and through these tears she says she understands and that she is sorry.
"You can't understand."
She has never walked will never walk in these fucked-up mixed-race American shoes my mother says she does but she cannot understand.
But I believe that she wants to. All I can do is walk toward her and hug her and tell her I know she did the best she could to raise a Black daughter. Because she did.
Being a mother myself, I finally know this. I'll be given hell to pay myself one day from Sawyer or Avery or both.
XXI.
"The arc of the moral universe is long but it bends toward justice," Dr. Martin Luther King Jr. said.
America freed us from 250 years of slavery, set us free upon this land, the only land we knew and yet the land was foreign and hostile, and we were told to go, get on with it, live free. We were promised forty acres and a mule, resources with which to start a life, resources that never came. So we began with little more than our strong bodies to carry us out into a foreign land without family structure, without income, into twelve years of Reconstruction and with it we were allowed to bond ourselves to our own families again, and have names that were our own. We had hope in those years that we could find a bit of land and shelter our families and give our children an education and find love and live free of violence.
But we went about this work with our brown skin, and the band of whites nursed on a milk of white supremacy was hardly ever at bay. Fueled by a seething bitterness that we were daring to try to live as their equals, they donned their hoods and their sheriff's badges and spat on us, hanged us from trees, began the psychological enslavement of terror and then enacted Jim Crow with its rules about where we could and could not go. We were thought so repugnant that white children were not to swim with us.
A childhood friend from my young years in Snedens Landing revealed to me only recently that this once had happened to me. I was seven. It was 1975. We were all swimming in the aboveground pool at her house in Palisades. There was a knock at the door and Jim Wilson's dad was standing on the porch. He'd got wind that a Black child was swimming with his son. He came to claim his white child from the water I was polluting with my presence.
Forty years later, on the day I am fending off the pressure to promote my first book on the radio show of a neoconservative white woman, a white male police officer is called to a pool party in McKinney, Texas, where he grabs a bikini-clad Black teenage girl by the hair and throws her to the ground and pins his knee into her back in broad summer daylight. As she screams, he, called to the scene because Black kids are swimming in a white neighborhood, takes this Black girl in a skimpy bikini down like she is the calf in the rodeo and we can almost hear the cowboys shouting, "Wahoo we got another one." "Blue lives" with black guns clamp Black girls to the green grass, her body still wet with chlorine from her neighbor's pool.
We want to swim. Eat ice cream. Enjoy a barbecue. Go to school. Get a job. Find love. Laugh. Die in peace.
XXII.
You hiding there behind your draperies across the street, it was you acting like Zimmerman who called the cops about a "disturbance" in your neighborhood, you who said there were multiple juveniles who do not live in the area or "have permission to be there," which you know because you guard the white experience and you know who belongs at the pool and who does not.
It was you who saw a Black man getting into a nice car and decided he was stealing it and called the police who trailed him, pulled him over, and pounced five at a time on his twenty-five-year-old Black body, this former student of mine, this man now getting a PhD in Engineering at Northwestern driving his own damn car.
It is you who call your dogs who bring their dogs to bring us down. To keep America white. To buff us out of your existence.
You want to stand your ground.
Which means arm the whites.
XXIII.
In the fall of 2016 a white woman named Kate Riffle Roper posts on Facebook that as the mother of two Black children and three white children, she's seen up close and personal how differently the world treats Black children:
Now my boys look like teenagers. Black teenagers. They are 13. Let me ask you these questions. Do store personnel follow your children when they are picking out their Gatorade flavors? They didn't follow my white kids. Do coffee shop employees interrogate your children about the credit card they are using to pay while you are in the bathroom? They didn't interrogate my white kids. When your kids trick-or-treat, dressed as a Ninja and a Clown, do they get asked who they are with and where they live, door after door? My white kids didn't get asked. Do your kids get pulled out of the TSA line time and again for additional screening? My white kids didn't. Do your kids get treated one way when they are standing alone but get treated a completely different way when you walk up? I mean a completely different way. My white kids didn't. Do shoe sales people ask if your kids' feet are clean before sizing them for shoes? No one asked me that with my white kids. Do complete strangers ask to touch your child's hair? Or ask about their penis size? Or ask if they are "from druggies"? No one did this with my white kids.
My God this was me, and this is now my child, I think, stifling a sob as I read the whole post of this white woman who chose to raise Black sons. Through the tears my heart swells with hope for Roper's two boys fortunate to grow up in such a woke white family.
Around the same time I'm reading this Glenn Beck has a heartfelt change of mind about Black people and says so and with his words tries to turn the larger tide of conservative whites. My heart surges with hope as well.
Then my hope ebbs, and the rage returns. These are not new appeals; we've been saying this for decades, for a century, for the entirety of our time on America's soil. They believe us only when white people vouch for us. Racism will never go away.
XXIV.
In mid-November 2016, I peek out from behind my curtains, each piece of news like a creaky stair bearing the weight of an intruder. It's coming.
"We won. Black lives DON'T matter."
"There's a new sheriff in town and we don't want your kind here."
"Make America white again."
A girl goes to school and sees the word "Nigger" scrawled on her locker.
XXV.
We need white allies.
I hate that we need white allies.
We need white allies.
XXVI.
In December 2016 the prosecution of a white North Charleston, South Carolina, police officer accused of shooting an unarmed Black man in the back ends in a mistrial. The officer shot Walter Scott in the back as Scott was running away and planted his Taser next to Scott's body, claiming Scott had stolen it from him. All this was caught on video. The evidence right there, naked to see if eyes are willing to see. But one juror could not see. One juror refused to see. Will not see that the officer's abject fear of a Black man was unjustifiable. Will unsee that the officer shot down Walter Scott as if he were a wild dog.
Dear America,
What would you have me tell my son?
Don't drive, son?
Don't go to Walgreens, son?
Don't be...
What.
Don't be?
ONWARD
I.
No, I was never white.
A conservative white male friend asks me on Facebook why I call myself Black when I am also white. "I know you love your white mother," he says, like it's evidence supporting his point. A white female writer friend wants me to talk about my whiteness in my memoir as well as my Blackness because I am biracial.
My whiteness?
The one-drop rule invented to preserve white supremacy and to perpetuate the population of slaves as more and more slaves were raped by masters and gave birth to lighter children says whiteness is pure and Blackness is a stain and therefore if you have even one drop of "Black blood" in your lineage, you are Black. Period. It is one of America's oldest rules and it holds to this day.
Yes, I adore my mother, and my white British aunts, uncles, and cousins with their cool accents, tiny cars, wry humor, and fondness for a pint or a cigarette. My relatives on my mother's side embraced my very dark, very tall, very American father back when it was truly an act of transgression to do so. And they show nothing but love to me, as well. But me, white? Never.
Because that store clerk doesn't see me as white. That doorman in the lobby of a high-rise apartment doesn't give me the white right of way. That neighbor in my own neighborhood thinks I am at her door to sell her something or do her harm instead of reclaim my nephew's shoes. I am not white because whites do not see will never see me as one of them.
And because I do not want to be.
II.
It is Christmas 1974. I'd just turned seven. My older siblings—now twenty-nine, twenty-eight, twenty-six, and twenty-four—are visiting us in Snedens Landing for the holidays. It is dinnertime and they are seated at our rectangular dining table wearing full-moon Afros and brightly colored wide-legged pants. They reconnect by jockeying to be the funniest, the most informed, the most profound. Then they tell inside jokes and start giggling about the nicknames they have for each other. I am the small child listening to the language of family, paying attention.
One of my brothers looks over at me. "What's your nickname, Julie?"
I look up. "Bridge over troubled water."
They say we choose our time to be born. They say we choose our family. I am here to live this strange mixed-race Black experience in late twentieth- and early twenty-first-century America.
In the grand scheme of Blackness past and present, owing to light skin and a higher socioeconomic class, mine is a life of privilege that threatens to alienate me from the only people who would ever claim me as theirs. I work to bridge that distance and to never forget whom I'm from. I cannot change what I look like, but I have a choice about how to be.
This I know: Black or white, rich or poor, in the light of day and the dark of night we all cry out in anguish and in ecstasy. We all sweat, bleed, and dream. We all want our children to get home safely. And we all just want to know we matter.
To America.
To someone.
III.
I am drawn like a magnet to the little fuzzy-headed brown-skinned kids with white parents whom I see in restaurants, in airports, in malls. My body wants to go near, my soul wants to be a friend to a child who might be feeling lost on an island.
Instead I smile deeply at the children and at the parent. If the parent doesn't get why I'm smiling, I hope he or she will think, Why is that woman smiling at me? What is she saying with that smile?
I'm saying, Please read up on this. Please have Black friends. Please don't make the mistakes my parents made. Please build some belonging to community. Please give this kid a chance to develop a healthy self.
If the hair is done right I compliment the parent.
IV.
Yes my white friend cry your tears. I know your pain is real as you feel the weight of this history this present lodge in your stomach like a stone.
Go there. Feel it. Hold it. Seek to understand it. Come to me with an open heart and I will show you my own.
V.
In 2015 I take Mom back to Ghana, where she met Daddy, and we hire a driver who drives us to the very beach where Daddy proposed to her in 1965.
Four years after meeting at that party thrown by the Americans in Accra, Daddy took Mom for a drive in his Triumph down what was then called the road to Tema. The top was down. The sun was beginning its plummet into the Atlantic. Daddy pulled over at a little spot where a small rivulet emptied into the sea. He helped Mom out of the car and onto the sand, and they walked toward the ocean. They neared the water. The breeze that always came off the water at that time of day ruffled her hair. Daddy stopped, turned to Mom, spoke some words of love to her, and gave her his mother's ring.
The road to Tema is now called Labadi Beach Road, a two-lane highway littered with trash like a frontage road in any city. Our hired driver, Isaac, a local gentleman whose skin is dark like cobalt, and whose bald head gleams in the bright sun, becomes invested in helping two Americans find the very spot of this long-ago wedding proposal. After driving back and forth along the highway, Mom focuses on a small steel bridge and decides this is indeed the rivulet near where Daddy proposed. Isaac pulls off the road just past the bridge, and we get out of the car. The three of us walk for ten minutes along the hot sand toward the rivulet, toward the ocean.
"Yes, this is the spot. I'm sure of it," Mom tells me.
I can't speak.
I take out my phone and click the camera, framing the picture. The immense Atlantic Ocean roars its constant waves onto the shore behind her. Tears stream down her face. She smiles lopsidedly and points to her ring. I hand my phone to Isaac and jog a few paces toward Mom and I put my arm around her. Isaac takes a picture of this white woman and her light brown daughter, these American women visiting Ghana. And as I steady myself on the sand I have the unmistakable sensation that Daddy had once been here. I wonder if Daddy or Mom had contemplated my existence as they stood, giddy, in the bright sunlight on this vast expanse of unstable ground.
VI.
One day I'm talking with Avery, now fifteen, about the other names we might have given her.
"You shoulda named me Marin."
"That was on Dad's list. I didn't really like it."
"Well, I like it."
"Well, looking back I kinda think maybe I shoulda named you Silvey."
She pauses. And then, "I'll name my own daughter Silvey, Mom."
I know it's my job as her mom not to cry.
VII.
We have "The Talk" a few times with Sawyer as new incidents of violence against Black men and boys come to light. Sawyer's living grandparents, all white, express concern, and I am grateful.
But like any teenager, Sawyer feels immortal. I have a feeling that only his sense of immortality scaffolds him and prevents the weight of these Black deaths from crushing him.
It would be rational to keep him inside. To keep him from the clutch of the strangers who would endanger him. I choose not to keep him inside. I gave him life and I intend to let him live it.
VIII.
This white man I sleep with I love.
This white man who loved my Black hair before I did I love.
This white man to whom I gave Sawyer and Avery I love.
This white man without whom I would not have Sawyer and Avery I love.
I love how he worked part-time his entire career so as to be the primary parent of our children. How he looks with such pride upon Sawyer and Avery. How he develops his consciousness about the Black experience by reading, listening, watching, informing himself so he can be the best possible white father to our Black son. How he loves his daughter just as magnificently as my Daddy loved me.
How he loved me when I did not yet love myself.
And gazes at me with limitless love still.
IX.
In the summer of 1969 we moved from Lagos, Nigeria, to Manhattan. Daddy became a dean at Columbia University's medical school. Mom had just become a naturalized U.S. citizen. I was about eighteen months old.
We moved to a university building overlooking the Hudson River, into an apartment on the twenty-first floor. Lagos was a world away from this Manhattan high-rise in every sense. We'd had a single-family home in Lagos. We'd had next-door neighbors, a family with three little girls, named Dunni, Funmi, and Ronke. We'd had gardens in front, and out back there was a pool and a lawn where I roamed as I took my first steps in this world. Moonflower grew up and along the back fence.
Although I was too young to remember, Mom has told me this story so often over the years, it feels the memory is also mine:
We'd lived in Manhattan for about a week when one morning I wandered into the kitchen where Mom was doing dishes. I tugged on her skirt. "Friends, Mommy, friends," I said, looking up at her.
Mom had never lived in an apartment building before, or in America for that matter, and the when and how of friend-making in this new country, in this tall rectangle of boxed lives, was a bit of a bewilderment to her. But she took one look at me and could see that I was lonely. That of all the things I might be missing from our life in Lagos, what I was missing most was someone to play with.
She dressed me in a lovely little two-piece orange outfit of a tank top with a ruffled hem and pants that tapered at the ankle, and small sandals. She put on a dress made of cloth she'd bought at the bustling Jankara Market in Lagos. She packed a bag of things we'd need for our adventure—snacks, books, and toys. And down the elevator we went. There must be some children who live in this building, Mom thought, and the lobby is where we will find them.
The lobby was spacious and Mom plopped us down on a window seat facing the large bank of elevators forty feet away. I imagine we made quite a picture. She, a gorgeous woman of thirty years, with skin of copper brown stretched taut across her strong, slender frame, dressed in the Dutch wax print cloth worn throughout West Africa. Me, a dark brown toddler with big brown eyes and a fuzzy Afro of medium brown, with my legs in orange pants sticking straight out on the seat. Little shoes pointing to the sky.
We watched and waited. Each time an elevator door opened we looked up, hopeful. A half hour went by. And then, finally, it happened. An elevator opened and there stood a tall, young, white woman with a little girl about my age. Without saying anything to Mom, without even pausing to look at her, I scooched myself off the window seat, hit the ground, and ran toward this child with my arms outstretched, calling, "Friend! Friend!" I never once looked back.
Mom says she realized only in that moment that something terrible could happen. That this white American woman could put a protective arm around her child and gently steer her away from me. That I could be rejected, feel rejected, in the act of trying to make a friend. As I ran across the huge lobby toward this white stranger, with my arms outstretched, calling "Friend," it felt like an eternity to her, my mother recalls.
The little girl responded.
She opened her arms and ran toward me. We met mid-lobby and hugged each other, both of us now saying "Friend" over and over again. My mother, tears running down her cheeks, got up and walked toward the other woman.
Emily and her daughter Gabrielle became our fast friends. Emily introduced Mom to two other women, both of whom had children my age, and we formed a playgroup.
I want it to be this simple.
X.
Well before I knew how to love myself, and well before I had children, my Daddy was gone.
On some evenings when my house has grown quiet and dark while I've worked late into the night and when finally I'm tiptoeing up the stairs to my bedroom, I smell my dead father. I smell him as he was in the childhood of my memory, when the scent of his aerosol-sprayed Afro Sheen and the aroma of his many Marlboro cigarettes formed a musky cloak of oil and wood and safety.
As a child I would hear Daddy's deep bass voice call multiple times a day, "Baby, bring me my cigarettes." He'd glow as I neared him, and next he'd say, "Now give me some sugar." And I'd lean in to plant my tiny kiss on his large, weathered lips, hoping his breath would smell savory like fried eggs and bacon, or sweet from his scotch, and not like the stench of sleep on an unbrushed weekend morning.
When the cancer had inhabited much of his tall, lanky frame and he was finally ready to pass, my mother insisted that it was time to call us children home and permit the kind of emotional letdown he'd forbidden us to have in the five years since his diagnosis.
"But there'll be all that crying," he protested.
"Oh Daddy, they need that, don't you think? Don't you think it's time?"
"I'm not talking about them," he said.
It was prostate cancer, entirely avoidable as a death sentence for him, a doctor who understood the source of his discomfort, knew it was a symptom, an indication, but chose to ignore it, chose not to be a sick person, chose not to incur the pity of others, chose not to buy more time for himself or for us. Five years earlier, when I was just twenty-two, he'd told us of the diagnosis in a letter, and just as bluntly told us not to speak of it to anyone else including him.
My brothers and sister and I got called home on a Friday in early October 1995, late in the morning California time, and we drove or flew in from all over the U.S., my journey from San Francisco to Boston onto the puddle jumper plane to the small Island of Martha's Vineyard being one of the longest. By early Saturday morning we had all arrived on the Island and had made our way to their house in the middle of the woods. There were a dozen of us—we his children, our spouses and partners, our children, and my mother—and we gathered in the cool stillness of the room that had been Daddy's office but was now his dying place, kneeling shoulder to shoulder, lining the stiff hem of his hospital bed with a soft ribbon of family.
The floor we knelt on was covered in a thick, plush, wall-to-wall rug of dark turquoise, like the color of a Caribbean coast where the seafloor falls away to depths below. My parents' artifacts and art procured during years of living in Ghana and Nigeria filled this, his final bedroom. The chiwaras, a pair of antelopes carved into wood from Mali, hung behind his headboard. The carving of Sopona, the god who gave people smallpox, whom my father had courted, negotiated with, and brought to work along with himself and the other Western doctors, rather than in opposition to them, stood on a bookshelf. The big lady from Mali, rising three feet tall, carved of a dark wood and covered in black shoe polish as an additive, naked but for a string of grass-seed beads around her neck, stood by the doorway.
When I entered the room that Saturday morning, I saw Daddy in bed against the far wall looking more like ninety-three than his actual seventy-seven, shriveling before my eyes, almost entirely gray, this tall strong man who had once moved mountains, measles, smallpox, and bad people away; half the size I remembered. I gave him one last kiss, feeling the dryness of his lips, sensing the acrid smell of teeth, mouth, and tongue no longer brushed. "I'm here, Daddy. It's Julie," I bleated, like a small lamb. He couldn't talk or even smile, but his strong right arm lifted his hand a few inches off the bed, and for a long last moment I grasped those strong fingers that had once held my wobbly seat as I learned how to ride a bicycle.
Then I took my place kneeling on that plush rug with the others, attending his final minutes, my hands over his right arm outstretched atop the soft white sheet, the dark skin, the strong fingers. I'd seen him string a bird for the rotisserie with those fingers, seen him accidentally stab his other palm with a screwdriver that slipped off the screw. But now what I saw was the papery skin of a body already too big for the soul that was detaching itself, soon to fade away.
As his breathing turned raspy and jagged, I heard the more audible gasps and fervent prayers of the family, also kneeling, also touching his hands, caressing his feet beneath the sheet, rubbing the strong thighs that once propelled him forward in sprints with Jesse Owens. And then something—a force powerful like a magnet, insistent like a bite on a fishing line—tugged my head upward. And I saw in the air in front of me the dissipating smoke ring from one last Marlboro, or the small, discarded heather-gray feather of an invisible tufted titmouse or black-capped chickadee. And when I looked down again at Daddy he was gone from his body, disappeared into the air, into the room, into the past. His skin and bones lay like a suit of clothing on display at a rummage sale. I didn't need anyone to tell me he was dead.
The hospice nurse who had been keeping quiet in a corner stood and walked toward us, nodding her gentle, knowing gaze, telling us without language that he'd passed on. Then the weeping began, the mournful, awful wailing of survivors who could finally release what we'd been admonished to keep inside us for five years. The cries became a swell, a final song, a symphony of love for our father, grandfather, father-in-law, husband, and a song of anguish for ourselves.
But I wasn't crying. I was perplexed by what I'd seen, caught in numb wonder at that puff of smoke, that feather, that gray wisp of something I'd watched float into the air as he died.
I've only ever dabbled in religious tradition, and never really by choice. As a small child, I was made Presbyterian by my parents. They wanted me to be baptized as some form of Christian, and the Presbyterian church up at the top of Washington Spring Road in Snedens Landing would do fine. And I made myself Mormon as a young adult, really just to prove I could. And so as I sat with my dying father, I'd had no particular religious belief to set a context for me. But when my head was tugged upward and I saw the tuft of something, I knew really, without regard to any religious tradition, that I'd seen my father's soul, or his spirit, or the essence that had animated him these seventy-seven years, leave his body. Seeing this was like discovering that a magician really can perform magic instead of tricking the audience with a sleight of hand. I felt a kind of pure, openhearted, reassuring joy from the knowledge that he was no longer suffering, and I felt wonder at the thought that one day I, too, might pass like this. As others knelt in anguish, I experienced awe over a death that was, in the context of death, still death, yet exquisitely beautiful.
Then the smell came, as the body, no longer clenching muscles, no longer beholden to custom, propriety, or manners, made the final movement of its bowels and released the final flow of urine through the urethra. I quickly regained my senses and was embarrassed for him, my Daddy, this man of tremendous dignity and stature, this former Assistant Surgeon General of the United States, now reduced to skin and bones and shit and piss. We were encouraged to leave the room so the nurse could clean the body and prepare it for its travel to a morgue and then on to a crematorium. We filed out one by one, and as I walked, eyes downcast, I passed the three-foot-tall big lady from Mali who was standing near the door. I touched her shoulder as you touch the knob of a banister to steady yourself, or as you rub the knee of a famous bended copper statue in a museum or stately entryway as a way of permanently recording that you were there.
XI.
The big lady from Mali lives with me now. She stands on a carpeted landing at the halfway point between the first and second stories of my house, where the long run of the staircase turns left before it quickly bends left again and leads to the upstairs bedrooms. On some nights as I creep past her, I smell the musk of Afro Sheen and Marlboros as the oil in her wood exhales a deep sigh into the California night. Or maybe it's Daddy sitting right there beside her, watching me make my way in this life decades past the last time he spoke my name. Maybe it's Daddy, watching with what I hope is pride, the grandchildren with their coltish brown and light brown limbs bolt up the stairs, grandchildren who never had a chance to sit in the lap of this magnificent man. Maybe it's Daddy, watching with what I hope is still unconditional love, the daughter he thought could be Miss America.
ALSO BY JULIE LYTHCOTT-HAIMS
How to Raise an Adult
ABOUT THE AUTHOR
JULIE LYTHCOTT-HAIMS, New York Times bestselling author of How to Raise an Adult, served as dean of freshmen and undergraduate advising at Stanford University, where she received the Dinkelspiel Award for her contributions to the undergraduate experience. She holds a BA from Stanford, a JD from Harvard Law School, and an MFA in writing from California College of the Arts. She is a member of the San Francisco Writers' Grotto and resides in the Bay Area with her partner, their two teenagers, and her mother. You can sign up for email updates here.
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CONTENTS
Title Page
Copyright Notice
Dedication
Acknowledgments
It Begins Like This
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
An American Childhood
Chapter I
Chapter II
Chapter III
Chapter IV
Becoming the Other
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
Chapter VII
Chapter VIII
Chapter IX
Chapter X
Chapter XI
Chapter XII
Chapter XIII
Chapter XIV
Chapter XV
Chapter XVI
Chapter XVII
Chapter XVIII
Chapter XIX
Chapter XX
Chapter XXI
Chapter XXII
Chapter XXIII
Chapter XXIV
Chapter XXV
Chapter XXVI
Chapter XXVII
Chapter XXVIII
Chapter XXIX
Chapter XXX
Chapter XXXI
Desperate to Belong
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
Chapter VII
Chapter VIII
Chapter IX
Chapter X
Chapter XI
Chapter XII
Chapter XIII
Chapter XIV
Chapter XV
Chapter XVI
Chapter XVII
Chapter XVIII
Chapter XIX
Chapter XX
Chapter XXI
Chapter XXII
Chapter XXIII
Chapter XXIV
Chapter XXV
Chapter XXVI
Chapter XXVII
Chapter XXVIII
Chapter XXIX
Chapter XXX
Chapter XXXI
Chapter XXXII
Chapter XXXIII
Chapter XXXIV
Chapter XXXV
Chapter XXXVI
Self-Loathing
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
Chapter VII
Chapter VIII
Chapter IX
Chapter X
Chapter XI
Chapter XII
Chapter XIII
Chapter XIV
Chapter XV
Chapter XVI
Chapter XVII
Chapter XVIII
Chapter XIX
Chapter XX
Chapter XXI
Chapter XXII
Chapter XXIII
Chapter XXIV
Chapter XXV
Chapter XXVI
Chapter XXVII
Chapter XXVIII
Chapter XXIX
Chapter XXX
Chapter XXXI
Chapter XXXII
Chapter XXXIII
Chapter XXXIV
Emerging
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
Chapter VII
Chapter VIII
Chapter IX
Chapter X
Chapter XI
Chapter XII
Chapter XIII
Declaring
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
Chapter VII
Chapter VIII
Chapter IX
Chapter X
Chapter XI
Chapter XII
Chapter XIII
Chapter XIV
Chapter XV
Chapter XVI
Chapter XVII
Chapter XVIII
Chapter XIX
Chapter XX
Chapter XXI
Chapter XXII
Chapter XXIII
Chapter XXIV
Black Lives Matter
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
Chapter VII
Chapter VIII
Chapter IX
Chapter X
Chapter XI
Chapter XII
Chapter XIII
Chapter XIV
Chapter XV
Chapter XVI
Chapter XVII
Chapter XVIII
Chapter XIX
Chapter XX
Chapter XXI
Chapter XXII
Chapter XXIII
Chapter XXIV
Chapter XXV
Chapter XXVI
Onward
Chapter I
Chapter II
Chapter III
Chapter IV
Chapter V
Chapter VI
Chapter VII
Chapter VIII
Chapter IX
Chapter X
Chapter XI
Also by Julie Lythcott-Haims
About the Author
Copyright
REAL AMERICAN. Copyright © 2017 by Julie Lythcott-Haims. All rights reserved. For information, address Henry Holt and Co., 175 Fifth Avenue, New York, N.Y. 10010.
www.henryholt.com
Cover design by Karen Horton
Cover photograph courtesy of the author
The Library of Congress has cataloged the print edition as follows:
Names: Lythcott-Haims, Julie, author.
Title: Real American: a memoir / Julie Lythcott-Haims.
Description: First edition.|New York, New York: Henry Holt and Company, 2017.
Identifiers: LCCN 2017009272|ISBN 9781250137746 (hardcover)|ISBN 9781250137753 (electronic book)
Subjects: LCSH: Lythcott-Haims, Julie.|Racially mixed people—United States—Biography.|Racially mixed people—Race identity—United States.|Racially mixed people—United States—Social conditions.|Race—Social aspects—United States.|United States—Race relations.|New York (N. Y.)—Biography.|Reston (Va.)—Biography.|Palo Alto (Ca.)—Biography.|Stanford (Ca.)—Biography.
Classification: LCC E184.A1 L967 2017|DDC 305.800973—dc23
LC record available at <https://lccn.loc.gov/2017009272>
e-ISBN 9781250137753
First Edition: October 2017
Our e-books may be purchased in bulk for promotional, educational, or business use. Please contact the Macmillan Corporate and Premium Sales Department at (800) 221-7945, extension 5442, or by e-mail at MacmillanSpecialMarkets@macmillan.com.
## Contents
1. Title Page
2. Copyright Notice
3. Dedication
4. Acknowledgments
5. 1. It Begins Like This
1. Chapter I
2. Chapter II
3. Chapter III
4. Chapter IV
5. Chapter V
6. Chapter VI
6. 2. An American Childhood
1. Chapter I
2. Chapter II
3. Chapter III
4. Chapter IV
7. 3. Becoming the Other
1. Chapter I
2. Chapter II
3. Chapter III
4. Chapter IV
5. Chapter V
6. Chapter VI
7. Chapter VII
8. Chapter VIII
9. Chapter IX
10. Chapter X
11. Chapter XI
12. Chapter XII
13. Chapter XIII
14. Chapter XIV
15. Chapter XV
16. Chapter XVI
17. Chapter XVII
18. Chapter XVIII
19. Chapter XIX
20. Chapter XX
21. Chapter XXI
22. Chapter XXII
23. Chapter XXIII
24. Chapter XXIV
25. Chapter XXV
26. Chapter XXVI
27. Chapter XXVII
28. Chapter XXVIII
29. Chapter XXIX
30. Chapter XXX
31. Chapter XXXI
8. 4. Desperate to Belong
1. Chapter I
2. Chapter II
3. Chapter III
4. Chapter IV
5. Chapter V
6. Chapter VI
7. Chapter VII
8. Chapter VIII
9. Chapter IX
10. Chapter X
11. Chapter XI
12. Chapter XII
13. Chapter XIII
14. Chapter XIV
15. Chapter XV
16. Chapter XVI
17. Chapter XVII
18. Chapter XVIII
19. Chapter XIX
20. Chapter XX
21. Chapter XXI
22. Chapter XXII
23. Chapter XXIII
24. Chapter XXIV
25. Chapter XXV
26. Chapter XXVI
27. Chapter XXVII
28. Chapter XXVIII
29. Chapter XXIX
30. Chapter XXX
31. Chapter XXXI
32. Chapter XXXII
33. Chapter XXXIII
34. Chapter XXXIV
35. Chapter XXXV
36. Chapter XXXVI
9. 5. Self-Loathing
1. Chapter I
2. Chapter II
3. Chapter III
4. Chapter IV
5. Chapter V
6. Chapter VI
7. Chapter VII
8. Chapter VIII
9. Chapter IX
10. Chapter X
11. Chapter XI
12. Chapter XII
13. Chapter XIII
14. Chapter XIV
15. Chapter XV
16. Chapter XVI
17. Chapter XVII
18. Chapter XVIII
19. Chapter XIX
20. Chapter XX
21. Chapter XXI
22. Chapter XXII
23. Chapter XXIII
24. Chapter XXIV
25. Chapter XXV
26. Chapter XXVI
27. Chapter XXVII
28. Chapter XXVIII
29. Chapter XXIX
30. Chapter XXX
31. Chapter XXXI
32. Chapter XXXII
33. Chapter XXXIII
34. Chapter XXXIV
10. 6. Emerging
1. Chapter I
2. Chapter II
3. Chapter III
4. Chapter IV
5. Chapter V
6. Chapter VI
7. Chapter VII
8. Chapter VIII
9. Chapter IX
10. Chapter X
11. Chapter XI
12. Chapter XII
13. Chapter XIII
11. 7. Declaring
1. Chapter I
2. Chapter II
3. Chapter III
4. Chapter IV
5. Chapter V
6. Chapter VI
7. Chapter VII
8. Chapter VIII
9. Chapter IX
10. Chapter X
11. Chapter XI
12. Chapter XII
13. Chapter XIII
14. Chapter XIV
15. Chapter XV
16. Chapter XVI
17. Chapter XVII
18. Chapter XVIII
19. Chapter XIX
20. Chapter XX
21. Chapter XXI
22. Chapter XXII
23. Chapter XXIII
24. Chapter XXIV
12. 8. Black Lives Matter
1. Chapter I
2. Chapter II
3. Chapter III
4. Chapter IV
5. Chapter V
6. Chapter VI
7. Chapter VII
8. Chapter VIII
9. Chapter IX
10. Chapter X
11. Chapter XI
12. Chapter XII
13. Chapter XIII
14. Chapter XIV
15. Chapter XV
16. Chapter XVI
17. Chapter XVII
18. Chapter XVIII
19. Chapter XIX
20. Chapter XX
21. Chapter XXI
22. Chapter XXII
23. Chapter XXIII
24. Chapter XXIV
25. Chapter XXV
26. Chapter XXVI
13. 9. Onward
1. Chapter I
2. Chapter II
3. Chapter III
4. Chapter IV
5. Chapter V
6. Chapter VI
7. Chapter VII
8. Chapter VIII
9. Chapter IX
10. Chapter X
11. Chapter XI
14. Also by Julie Lythcott-Haims
15. About the Author
16. Newsletter Sign-up
17. Copyright
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2. Table of Contents
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260. ii
261. ix
| {
"redpajama_set_name": "RedPajamaBook"
} | 9,574 |
Before I joined the NCA staff, I served as director of a Children's Advocacy Center, as well as the Director of Behavioral Health for the umbrella agency which was a primary care clinic. As part of my duties to manage the treatment of children suffering from traumatic stress symptoms, I frequently conducted chart audits, and one of the measures looked at the number of therapy sessions clients were receiving—an important measure to consider when looking at how effective a course of treatment is at achieving its goal of healing the client.
Our clinicians were trained in a variety of effective evidence-based treatments (EBTs): Trauma-Focused Cognitive Behavioral Therapy (TF-CBT), Child and Family Traumatic Stress Intervention (CFTSI), Theraplay, and Cognitive Behavioral Therapy (CBT). While they often provided treatment adhering to these tried-and-true models, sometimes they also offered clients within the same course of treatment an eclectic mix of models. As a clinician, even I was sometimes guilty. Why did we drift from the model for a session or two? Often it's as simple as a crisis the child or family are experiencing or because I, as the clinician, was not adequately prepared for the session.
However, my chart audits continually reminded me that these deviations have a real cost. When clinicians at my CAC used an evidence-based model with fidelity, the average number of sessions delivered were significantly less, ranging from 5-14 sessions, than when the clients were receiving an eclectic mix. On the extreme end, one client received 56 sessions. The consequences for long, meandering courses of treatment that don't stick to the evidence-based model are many. As an illustration of these consequences, I offer a tale of two clients.
The first child came to a clinician at the CAC after a forensic interview, and started a course of TF-CBT. But the sessions began to stray from the treatment model, due, seemingly, to issues related to the parents' recent divorce and the child's behavioral issues. With this particular child, elements of a variety of models were used which significantly increased the number of sessions, and did not result in a treatment model being completed. With the chaos in the family, and mother's persistence about focusing on the behavioral issues, the clinician struggled with keeping the therapy on track. After 28 therapy sessions over a 9-month period, the child dropped out of therapy.
The second child, a 12-year-old, came in for a course of TF-CBT within three weeks of her forensic interview that substantiated her abuse. Owing to the client's relationship to the abuser, and the family dynamics that ensued, this case was no less complicated than the first. However, in this case, the TF-CBT model was followed with fidelity, to include parallel sessions with the parents. 90-minute appointment slots were not always available so we had to be creative in terms of involving the parents in the process. The child and parents attended 11 sessions, and the child successfully discharged. Three months later, I checked in with the parents and the child continued to do well. The family reached out at different times due to other stressors, but the child did not return to regularly scheduled therapy. (That's a good thing!) The child has since graduated high school, is in college and doing well the last time I heard.
As clinicians, it can be difficult to determine the best treatment model when children and families walk through our doors. Parents often are focusing on behavioral manifestations which may guide the direction of a session on that particular day. My suggestion to clinicians that I supervised was to choose an evidence-based model, and stick with it. I always took care to reassure parents that as treatment progressed, behaviors would likely improve.
Evidence shows treatment model fidelity leads to shorter treatment courses, as a recent study of TF-CBT in treating child traumatic stress demonstrated. This matters for a number of reasons. A shorter treatment course means that there's a shorter window for financial, scheduling, and transportation problems that contribute to household stress and cause families and clients to drop out of treatment before achieving their goals. It also means that there's less capacity—simply fewer appointment slots available—for therapists to take on new cases and help more children heal. (Why help one child when you can help three?) Long treatment courses can also contribute to a dependency on therapy with no resolution. For children, successfully discharging from therapy means they can go on leading a developmentally appropriate childhood. Finally, and most importantly, when a child's treatment doesn't follow the model, it can't achieve the goals of the model—helping the child heal from his or her abuse.
Michelle Miller serves as Coordinator for Mental Health Projects at National Children's Alliance. In her 24 years in the field, she has spent 15 years serving as a clinician. She has also served as director of the Butte Child Evaluation Center in Butte, Montana, where she provided both mental health services and forensic interviews for clients. She holds a Ph.D. in psychology and a master's in social work. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,484 |
Q: jquery doesnt recognize div click I have the following code. My issue is that when I click on the div maya, jquery doesn't recognize it and
it will not give me the alert for it.
CSS
<style>
#carousel-single-image {
width: 320px;
height: 400px;
margin: 0 auto;
background: white;
}
#carousel-single-image .touchcarousel-container {
height: 100%;
background: url("../touchcarousel/demo-images/wood-pattern.jpg") repeat;
}
#carousel-single-image .touchcarousel-item {
margin-right: 0;
width: 400px;
height: 400px;
}
#carousel-single-image img {
width: 300px;
height: 360px;
margin: 32px 0 0 8px;
float:left;
position: relative;
display: block;
padding: 3px;
background: transparent;
border: 0;
-webkit-box-shadow: 0 1px 6px rgba(0,0,0,0.6);
-moz-box-shadow: 0 1px 6px rgba(0,0,0,0.6);
box-shadow: 0 1px 6px rgba(0,0,0,0.6);
}
#carousel-single-image .tc-paging-container {
margin-top: -380px;
}
</style>
HTML
<div id="productGallery" style="display:none">
<div id="main-body">
<div id="carousel-single-image" class="touchcarousel minimal-light">
<ul class="touchcarousel-container">
<li class="touchcarousel-item">
<a href="#"> <img data-original="/image1.jpg" />
<span class="maya">CheckClick</span></a>
</li>
<li class="touchcarousel-item">
<a href="#"> <img data-original="/image2.jpg" />
<span class="maya">CheckClick</span></a>
</li>
</ul>
</div>
</div>
</div>
Script
$('.maya').click(function(){
alert("hi");
});
A: Enclose in DOM Ready Event , and your script in the script tags
<script type="text/javascript">
$(function() {
$('.maya').click(function(){
alert("hi");
});
});
</script>
Also make sure the jQuery file is included into your project..
Seems to be working HERE
A: Here is a fiddle which works... I only took out the display none style on your overall container so I can see it http://jsfiddle.net/wcLn4/
this one with your css http://jsfiddle.net/wcLn4/1/ and without display none
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,023 |
# -*- encoding: utf-8 -*-
#
# Author:: Fletcher Nichol (<fnichol@nichol.ca>)
#
# Copyright (C) 2013, Fletcher Nichol
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
require "fileutils"
require "pathname"
require "json"
require "cgi"
require "kitchen/provisioner/chef/policyfile"
require "kitchen/provisioner/chef/berkshelf"
require "kitchen/provisioner/chef/common_sandbox"
require "kitchen/provisioner/chef/librarian"
require "kitchen/util"
require "mixlib/install"
require "mixlib/install/script_generator"
begin
require "chef-config/config"
require "chef-config/workstation_config_loader"
rescue LoadError # rubocop:disable Lint/HandleExceptions
# This space left intentionally blank.
end
module Kitchen
module Provisioner
# Common implementation details for Chef-related provisioners.
#
# @author Fletcher Nichol <fnichol@nichol.ca>
class ChefBase < Base
default_config :require_chef_omnibus, true
default_config :chef_omnibus_url, "https://omnitruck.chef.io/install.sh"
default_config :chef_omnibus_install_options, nil
default_config :run_list, []
default_config :attributes, {}
default_config :config_path, nil
default_config :log_file, nil
default_config :log_level do |provisioner|
provisioner[:debug] ? "debug" : "auto"
end
default_config :profile_ruby, false
# The older policyfile_zero used `policyfile` so support it for compat.
default_config :policyfile, nil
# Will try to autodetect by searching for `Policyfile.rb` if not set.
# If set, will error if the file doesn't exist.
default_config :policyfile_path, nil
# If set to true (which is the default from `chef generate`), try to update
# backend cookbook downloader on every kitchen run.
default_config :always_update_cookbooks, false
default_config :cookbook_files_glob, %w(
README.* VERSION metadata.{json,rb} attributes.rb recipe.rb
attributes/**/* definitions/**/* files/**/* libraries/**/*
providers/**/* recipes/**/* resources/**/* templates/**/*
).join(",")
# to ease upgrades, allow the user to turn deprecation warnings into errors
default_config :deprecations_as_errors, false
# Override the default from Base so reboot handling works by default for Chef.
default_config :retry_on_exit_code, [35, 213]
default_config :multiple_converge, 1
default_config :enforce_idempotency, false
default_config :data_path do |provisioner|
provisioner.calculate_path("data")
end
expand_path_for :data_path
default_config :data_bags_path do |provisioner|
provisioner.calculate_path("data_bags")
end
expand_path_for :data_bags_path
default_config :environments_path do |provisioner|
provisioner.calculate_path("environments")
end
expand_path_for :environments_path
default_config :nodes_path do |provisioner|
provisioner.calculate_path("nodes")
end
expand_path_for :nodes_path
default_config :roles_path do |provisioner|
provisioner.calculate_path("roles")
end
expand_path_for :roles_path
default_config :clients_path do |provisioner|
provisioner.calculate_path("clients")
end
expand_path_for :clients_path
default_config :encrypted_data_bag_secret_key_path do |provisioner|
provisioner.calculate_path("encrypted_data_bag_secret_key", type: :file)
end
expand_path_for :encrypted_data_bag_secret_key_path
#
# New configuration options per RFC 091
# https://github.com/chef/chef-rfc/blob/master/rfc091-deprecate-kitchen-settings.md
#
# Setting product_name to nil. It is currently the pivot point
# between the two install paths (Mixlib::Install::ScriptGenerator and Mixlib::Install)
default_config :product_name
default_config :product_version, :latest
default_config :channel, :stable
default_config :install_strategy, "once"
default_config :platform
default_config :platform_version
default_config :architecture
default_config :download_url
default_config :checksum
deprecate_config_for :require_chef_omnibus do |provisioner|
case
when provisioner[:require_chef_omnibus] == false
Util.outdent!(<<-MSG)
The 'require_chef_omnibus' attribute with value of 'false' will
change to use the new 'install_strategy' attribute with a value of 'skip'.
Note: 'product_name' must be set in order to use 'install_strategy'.
Although this seems counterintuitive, it is necessary until
'product_name' replaces 'require_chef_omnibus' as the default.
# New Usage #
provisioner:
product_name: <chef or chefdk>
install_strategy: skip
MSG
when provisioner[:require_chef_omnibus].to_s.match(/\d/)
Util.outdent!(<<-MSG)
The 'require_chef_omnibus' attribute with version values will change
to use the new 'product_version' attribute.
Note: 'product_name' must be set in order to use 'product_version'
until 'product_name' replaces 'require_chef_omnibus' as the default.
# New Usage #
provisioner:
product_name: <chef or chefdk>
product_version: #{provisioner[:require_chef_omnibus]}
MSG
when provisioner[:require_chef_omnibus] == "latest"
Util.outdent!(<<-MSG)
The 'require_chef_omnibus' attribute with value of 'latest' will change
to use the new 'install_strategy' attribute with a value of 'always'.
Note: 'product_name' must be set in order to use 'install_strategy'
until 'product_name' replaces 'require_chef_omnibus' as the default.
# New Usage #
provisioner:
product_name: <chef or chefdk>
install_strategy: always
MSG
end
end
deprecate_config_for :chef_omnibus_url, Util.outdent!(<<-MSG)
Changing the 'chef_omnibus_url' attribute breaks existing functionality. It will
be removed in a future version.
MSG
deprecate_config_for :chef_omnibus_install_options, Util.outdent!(<<-MSG)
The 'chef_omnibus_install_options' attribute will be replaced by using
'product_name' and 'channel' attributes.
Note: 'product_name' must be set in order to use 'channel'
until 'product_name' replaces 'require_chef_omnibus' as the default.
# Deprecated Example #
provisioner:
chef_omnibus_install_options: -P chefdk -c current
# New Usage #
provisioner:
product_name: chefdk
channel: current
MSG
deprecate_config_for :install_msi_url, Util.outdent!(<<-MSG)
The 'install_msi_url' will be relaced by the 'download_url' attribute.
'download_url' will be applied to Bourne and Powershell download scripts.
Note: 'product_name' must be set in order to use 'download_url'
until 'product_name' replaces 'require_chef_omnibus' as the default.
# New Usage #
provisioner:
product_name: <chef or chefdk>
download_url: http://direct-download-url
MSG
deprecate_config_for :chef_metadata_url, Util.outdent!(<<-MSG)
The 'chef_metadata_url' will be removed. The Windows metadata URL will be
fully managed by using attribute settings.
MSG
# Reads the local Chef::Config object (if present). We do this because
# we want to start bring Chef config and ChefDK tool config closer
# together. For example, we want to configure proxy settings in 1
# location instead of 3 configuration files.
#
# @param config [Hash] initial provided configuration
def initialize(config = {})
super(config)
if defined?(ChefConfig::WorkstationConfigLoader)
ChefConfig::WorkstationConfigLoader.new(config[:config_path]).load
end
# This exports any proxy config present in the Chef config to
# appropriate environment variables, which Test Kitchen respects
ChefConfig::Config.export_proxies if defined?(ChefConfig::Config.export_proxies)
end
def doctor(state)
deprecated_config.each do |attr, msg|
info("**** #{attr} deprecated\n#{msg}")
end
end
# (see Base#create_sandbox)
def create_sandbox
super
sanity_check_sandbox_options!
Chef::CommonSandbox.new(config, sandbox_path, instance).populate
end
# (see Base#init_command)
def init_command
dirs = %w{
cookbooks data data_bags environments roles clients
encrypted_data_bag_secret
}.sort.map { |dir| remote_path_join(config[:root_path], dir) }
vars = if powershell_shell?
init_command_vars_for_powershell(dirs)
else
init_command_vars_for_bourne(dirs)
end
prefix_command(shell_code_from_file(vars, "chef_base_init_command"))
end
# (see Base#install_command)
def install_command
return unless config[:require_chef_omnibus] || config[:product_name]
return if config[:product_name] && config[:install_strategy] == "skip"
prefix_command(sudo(install_script_contents))
end
private
def last_exit_code
"; exit $LastExitCode" if powershell_shell?
end
# @return [Hash] an option hash for the install commands
# @api private
def install_options
add_omnibus_directory_option if instance.driver.cache_directory
project = /\s*-P (\w+)\s*/.match(config[:chef_omnibus_install_options])
{
omnibus_url: config[:chef_omnibus_url],
project: project.nil? ? nil : project[1],
install_flags: config[:chef_omnibus_install_options],
sudo_command: sudo_command,
}.tap do |opts|
opts[:root] = config[:chef_omnibus_root] if config.key? :chef_omnibus_root
[:install_msi_url, :http_proxy, :https_proxy].each do |key|
opts[key] = config[key] if config.key? key
end
end
end
# Verify if the "omnibus_dir_option" has already been passed, if so we
# don't use the @driver.cache_directory
#
# @api private
def add_omnibus_directory_option
cache_dir_option = "#{omnibus_dir_option} #{instance.driver.cache_directory}"
if config[:chef_omnibus_install_options].nil?
config[:chef_omnibus_install_options] = cache_dir_option
elsif config[:chef_omnibus_install_options].match(/\s*#{omnibus_dir_option}\s*/).nil?
config[:chef_omnibus_install_options] << " " << cache_dir_option
end
end
# @return [String] an absolute path to a Policyfile, relative to the
# kitchen root
# @api private
def policyfile
policyfile_basename = config[:policyfile_path] || config[:policyfile] || "Policyfile.rb"
File.join(config[:kitchen_root], policyfile_basename)
end
# @return [String] an absolute path to a Berksfile, relative to the
# kitchen root
# @api private
def berksfile
File.join(config[:kitchen_root], "Berksfile")
end
# @return [String] an absolute path to a Cheffile, relative to the
# kitchen root
# @api private
def cheffile
File.join(config[:kitchen_root], "Cheffile")
end
# Generates a Hash with default values for a solo.rb or client.rb Chef
# configuration file.
#
# @return [Hash] a configuration hash
# @api private
def default_config_rb # rubocop:disable Metrics/MethodLength
root = config[:root_path].gsub("$env:TEMP", "\#{ENV['TEMP']\}")
{
node_name: instance.name,
checksum_path: remote_path_join(root, "checksums"),
file_cache_path: remote_path_join(root, "cache"),
file_backup_path: remote_path_join(root, "backup"),
cookbook_path: [
remote_path_join(root, "cookbooks"),
remote_path_join(root, "site-cookbooks"),
],
data_bag_path: remote_path_join(root, "data_bags"),
environment_path: remote_path_join(root, "environments"),
node_path: remote_path_join(root, "nodes"),
role_path: remote_path_join(root, "roles"),
client_path: remote_path_join(root, "clients"),
user_path: remote_path_join(root, "users"),
validation_key: remote_path_join(root, "validation.pem"),
client_key: remote_path_join(root, "client.pem"),
chef_server_url: "http://127.0.0.1:8889",
encrypted_data_bag_secret: remote_path_join(
root, "encrypted_data_bag_secret"
),
treat_deprecation_warnings_as_errors: config[:deprecations_as_errors],
}
end
# Generates a rendered client.rb/solo.rb/knife.rb formatted file as a
# String.
#
# @param data [Hash] a key/value pair hash of configuration
# @return [String] a rendered Chef config file as a String
# @api private
def format_config_file(data)
data.each.map do |attr, value|
[attr, format_value(value)].join(" ")
end.join("\n")
end
# Converts a Ruby object to a String interpretation suitable for writing
# out to a client.rb/solo.rb/knife.rb file.
#
# @param obj [Object] an object
# @return [String] a string representation
# @api private
def format_value(obj)
if obj.is_a?(String) && obj =~ /^:/
obj
elsif obj.is_a?(String)
%{"#{obj.gsub(/\\/, '\\\\\\\\')}"}
elsif obj.is_a?(Array)
%{[#{obj.map { |i| format_value(i) }.join(', ')}]}
else
obj.inspect
end
end
# Generates the init command variables for Bourne shell-based platforms.
#
# @param dirs [Array<String>] directories
# @return [String] shell variable lines
# @api private
def init_command_vars_for_bourne(dirs)
[
shell_var("sudo_rm", sudo("rm")),
shell_var("dirs", dirs.join(" ")),
shell_var("root_path", config[:root_path]),
].join("\n")
end
# Generates the init command variables for PowerShell-based platforms.
#
# @param dirs [Array<String>] directories
# @return [String] shell variable lines
# @api private
def init_command_vars_for_powershell(dirs)
[
%{$dirs = @(#{dirs.map { |d| %{"#{d}"} }.join(', ')})},
shell_var("root_path", config[:root_path]),
].join("\n")
end
# Load cookbook dependency resolver code, if required.
#
# (see Base#load_needed_dependencies!)
def load_needed_dependencies!
super
if File.exist?(policyfile)
debug("Policyfile found at #{policyfile}, using Policyfile to resolve dependencies")
Chef::Policyfile.load!(logger: logger)
elsif File.exist?(berksfile)
debug("Berksfile found at #{berksfile}, loading Berkshelf")
Chef::Berkshelf.load!(logger: logger)
elsif File.exist?(cheffile)
debug("Cheffile found at #{cheffile}, loading Librarian-Chef")
Chef::Librarian.load!(logger: logger)
end
end
# @return [String] contents of the install script
# @api private
def install_script_contents
# by default require_chef_omnibus is set to true. Check config[:product_name] first
# so that we can use it if configured.
if config[:product_name]
script_for_product
elsif config[:require_chef_omnibus]
script_for_omnibus_version
end
end
# @return [String] contents of product based install script
# @api private
def script_for_product
installer = Mixlib::Install.new({
product_name: config[:product_name],
product_version: config[:product_version],
channel: config[:channel].to_sym,
install_command_options: {
install_strategy: config[:install_strategy],
},
}.tap do |opts|
opts[:shell_type] = :ps1 if powershell_shell?
[:platform, :platform_version, :architecture].each do |key|
opts[key] = config[key] if config[key]
end
if config[:download_url]
opts[:install_command_options][:download_url_override] = config[:download_url]
opts[:install_command_options][:checksum] = config[:checksum] if config[:checksum]
end
if instance.driver.cache_directory
download_dir_option = windows_os? ? :download_directory : :cmdline_dl_dir
opts[:install_command_options][download_dir_option] = instance.driver.cache_directory
end
proxies = {}.tap do |prox|
[:http_proxy, :https_proxy, :ftp_proxy, :no_proxy].each do |key|
prox[key] = config[key] if config[key]
end
# install.ps1 only supports http_proxy
prox.delete_if { |p| [:https_proxy, :ftp_proxy, :no_proxy].include?(p) } if powershell_shell?
end
opts[:install_command_options].merge!(proxies)
end)
config[:chef_omnibus_root] = installer.root
if powershell_shell?
installer.install_command
else
install_from_file(installer.install_command)
end
end
# @return [String] Correct option per platform to specify the the
# cache directory
# @api private
def omnibus_dir_option
windows_os? ? "-download_directory" : "-d"
end
def install_from_file(command)
install_file = "/tmp/chef-installer.sh"
script = ["cat > #{install_file} <<\"EOL\""]
script << command
script << "EOL"
script << "chmod +x #{install_file}"
script << sudo(install_file)
script.join("\n")
end
# @return [String] contents of version based install script
# @api private
def script_for_omnibus_version
installer = Mixlib::Install::ScriptGenerator.new(
config[:require_chef_omnibus], powershell_shell?, install_options)
config[:chef_omnibus_root] = installer.root
installer.install_command
end
# Hook used in subclasses to indicate support for policyfiles.
#
# @abstract
# @return [Boolean]
# @api private
def supports_policyfile?
false
end
# @return [void]
# @raise [UserError]
# @api private
def sanity_check_sandbox_options!
if (config[:policyfile_path] || config[:policyfile]) && !File.exist?(policyfile)
raise UserError, "policyfile_path set in config "\
"(#{config[:policyfile_path]} could not be found. " \
"Expected to find it at full path #{policyfile} " \
end
if File.exist?(policyfile) && !supports_policyfile?
raise UserError, "policyfile detected, but provisioner " \
"#{self.class.name} doesn't support policyfiles. " \
"Either use a different provisioner, or delete/rename " \
"#{policyfile}"
end
end
# Writes a configuration file to the sandbox directory.
# @api private
def prepare_config_rb
data = default_config_rb.merge(config[config_filename.tr(".", "_").to_sym])
data = data.merge(named_run_list: config[:named_run_list]) if config[:named_run_list]
info("Preparing #{config_filename}")
debug("Creating #{config_filename} from #{data.inspect}")
File.open(File.join(sandbox_path, config_filename), "wb") do |file|
file.write(format_config_file(data))
end
prepare_config_idempotency_check(data) if config[:enforce_idempotency]
end
# Writes a configuration file to the sandbox directory
# to check for idempotency of the run.
# @api private
def prepare_config_idempotency_check(data)
handler_filename = "chef-client-fail-if-update-handler.rb"
source = File.join(
File.dirname(__FILE__), %w{.. .. .. support }, handler_filename
)
FileUtils.cp(source, File.join(sandbox_path, handler_filename))
File.open(File.join(sandbox_path, "client_no_updated_resources.rb"), "wb") do |file|
file.write(format_config_file(data))
file.write("\n\n")
file.write("handler_file = File.join(File.dirname(__FILE__), '#{handler_filename}')\n")
file.write "Chef::Config.from_file(handler_file)\n"
end
end
# Returns an Array of command line arguments for the chef client.
#
# @return [Array<String>] an array of command line arguments
# @api private
def chef_args(_config_filename)
raise "You must override in sub classes!"
end
# Returns a filename for the configuration file
# defaults to client.rb
#
# @return [String] a filename
# @api private
def config_filename
"client.rb"
end
# Gives the command used to run chef
# @api private
def chef_cmd(base_cmd)
if windows_os?
separator = [
"; if ($LastExitCode -ne 0) { ",
"throw \"Command failed with exit code $LastExitCode.\" } ;",
].join
else
separator = " && "
end
chef_cmds(base_cmd).join(separator)
end
# Gives an array of command
# @api private
def chef_cmds(base_cmd)
cmd = prefix_command(wrap_shell_code(
[base_cmd, *chef_args(config_filename), last_exit_code].join(" ")
.tap { |str| str.insert(0, reload_ps1_path) if windows_os? }
))
cmds = [cmd].cycle(config[:multiple_converge].to_i).to_a
if config[:enforce_idempotency]
idempotent_cmd = prefix_command(wrap_shell_code(
[base_cmd, *chef_args("client_no_updated_resources.rb"), last_exit_code].join(" ")
.tap { |str| str.insert(0, reload_ps1_path) if windows_os? }
))
cmds[-1] = idempotent_cmd
end
cmds
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,840 |
package org.apache.eagle.common.metric;
import java.io.Serializable;
import java.util.HashMap;
import java.util.Map;
/**
* not thread safe
*/
public class AlertContext implements Serializable{
private Map<String, String> properties = new HashMap<String, String>();
public AlertContext(){
}
public AlertContext(AlertContext context){
this.properties = new HashMap<String, String>(context.properties);
}
public String removeProperty(String name)
{
return properties.remove(name);
}
public AlertContext addProperty(String name, String value){
properties.put(name, value);
return this;
}
public AlertContext addAll(Map<String,String> propHash){
this.properties.putAll(propHash);
return this;
}
public String getProperty(String name){
return properties.get(name);
}
public String toString(){
return properties.toString();
}
public Map<String, String> getProperties(){
return properties;
}
public void setProperties(Map<String, String> properties){
this.properties = properties;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 88 |
\section{Problem definitions}
\label{sec:problems}
We list the definitions of the problems considered in this paper.
\smallskip
\defparproblem{\sc Feedback Vertex Set}{An undirected graph $G$ and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is acyclic?}
\defparproblem{\sc Weighted Feedback Vertex Set}{An undirected graph $G$, a positive integer $k$, a weight function $w:V(G)\rightarrow \mathbb{N}$, and a positive integer $W$.}{$k$}{Is there a set $S\subseteq V(G)$ of size at most $k$ and weight at most $W$ such that $G-S$ is acyclic?}
\defparproblem{\sc Subset Feedback Vertex Set}{An undirected graph $G$, a vertex subset $T\subseteq V(G)$, and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ has no cycle that contains a vertex from $T$?}
\smallskip
Let $\Gamma$ be a finite group with identity element $1_\Gamma$.
A $\Gamma$-labeled graph is a graph $G=(V,E)$ with a labeling $\lambda: E \rightarrow \Gamma$ such that $\lambda(u,v)\lambda(v,u)=1_\Gamma$ for every edge $uv\in E$.
For a cycle $C=(v_1,\dots,v_r,v_1)$, define $\lambda(C) = \lambda(v_1,v_2)\cdot \dots \cdot \lambda(v_r,v_1)$.
\smallskip
\defparproblem{\sc Group Feedback Vertex Set}{A group $\Gamma$, a $\Gamma$-labelled graph $(G,\lambda)$, and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq V(G)$ of size at most $k$ such that every cycle $C$ in $G-S$ has $\lambda(C)=1_\Gamma$?}
\smallskip
\defparproblem{\textsc{Node Unique Label Cover}}{An undirected graph $G=(V,E)$, a finite alphabet $\Sigma$, an integer $k$, and for each edge $e\in E$ and each of its endpoints $v$ a permutation $\psi_{e,v}$ of $\Sigma$ such that if $e=xy$ then $\psi_{e,x} = \psi_{e,v}^{-1}$}{$|\Sigma|+k$}{Is there a vertex subset $S\subset V$ of size at most $k$ and a function $\Psi: V\setminus S \rightarrow \Sigma$ such that for every edge $uv\in E(G-S)$ we have $(\Psi(u),\Psi(v))\in \psi_{uv,u}$?}
For fixed integers $r,\ell \geq 0$, a graph $G$ is called an {\em $(r,\ell)$-{graph}} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\ell$ cliques.
\smallskip
\defparproblem{{{\sc Vertex $(r,\ell)$-Partization}}}{A graph $G$ and a positive integer $k$}{$k$}{Is there a vertex subset $S\subseteq V(G)$ of size at most $k$ such that
$G-S$ is an $(r,\ell)$-{graph}?}
\smallskip
Several special cases of this problem are well known and have been widely studied. For example, $(2,0)$- and $(1,1)$-graphs correspond to bipartite graphs and split graphs respectively.
We note that {\sc Vertex $(r,\ell)$-Partization}\ can be solved in $O(1.1996^{(r+\ell)\cdot n})$ by taking $r$ copies of the input graph, $\ell$ copies of its complement, making all the copies of a same vertex into a clique and computing a maximum independent set of this graph using the algorithm from \cite{XiaoN13}.
This is faster than $O(2^n)$ when $r+\ell\le 3$.
We improve on this algorithm for $r,\ell\le 2$ and $r+\ell\ge 3$.
For the definition of graph classes, including interval graphs, proper interval graphs, block graphs, cluster graphs, we refer to \cite{BrandstadtLS99}.
\defparproblem{\sc Proper Interval Vertex Deletion}{An undirected graph $G$ and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is a proper interval graph?}
\defparproblem{\sc Interval Vertex Deletion}{An undirected graph $G$ and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is an interval graph?}
\defparproblem{\sc Block Graph Vertex Deletion}{An undirected graph $G$ and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is a block graph?}
\defparproblem{\sc Cluster Vertex Deletion}{An undirected graph $G$ and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is a cluster graph?}
\defparproblem{\sc Thread Graph Vertex Deletion} {An undirected graph $G$ and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is of linear rank-width one?}
\defparproblem{\sc Multicut on Trees}{A tree $T$ and a set $\mathcal{R} = \{ \{s_1,t_1\}, \ldots, \{s_r,t_r\}\}$ of pairs of vertices of $T$ called terminals, and a positive integer $k$.}{$k$}{Does there exist a subset
$S\subseteq E(T)$ of size at most $k$ whose removal disconnects each $s_i$ from $t_i$, $i\in [r]$?}
\defparproblem{\sc $d$-Hitting Set}{A family $\mathscr S$ of subsets of size at most $d$ of a universe ${\cal U}$ and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq \cal U$ of size at most $k$ such that $F$ is a hitting set for
$\mathscr S$?}
\defparproblem{\sc Weighted $d$-Hitting Set}{A family $\mathscr S$ of subsets of size at most $d$ of a universe ${\cal U}$, a weight function $w:\mathcal{U}\rightarrow \mathbb{N}$, and positive integers $k$ and $W$.}{$k$}{Does there exist a subset $S\subseteq \cal U$ of size at most $k$ and weight at most $W$ such that $F$ is a hitting set for $\mathscr S$?}
\defparproblem{\sc Min-Ones $d$-Sat}{A propositional formula $F$ in conjunctive normal form where each clause has at most $d$ literals and an integer $k$.}{$k$}{Does $F$ have a satisfying assignment with Hamming weight at most $k$?}
\defparproblem{\sc Weighted $d$-Sat}{A propositional formula $F$ in conjunctive normal form where each clause has at most $d$ literals, a weight function $w:var(F)\rightarrow \mathbb{Z}$, and integers $k$ and $W$.}{$k$}{Is there a set $S\subseteq var(F)$ of size at most $k$ and weight at most $W$ such that $F$ is satisfied by the assignment that sets the variables in $S$ to $1$ and all other variables to $0$?}
\defparproblem{\sc Tournament Feedback Vertex Set}{A tournament $G$ and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is a transitive
tournament?}
\defparproblem{\sc Split Vertex Deletion}{An undirected graph $G$ and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is a split graph?}
\defparproblem{\sc Cograph Vertex Deletion}{An undirected graph $G$ and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is a cograph?}
\defparproblem{\sc Directed Feedback Vertex Set}{A directed graph $G$ and a positive integer $k$.}{$k$}{Does there exist a subset $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is directed acyclic graph?}
\shortversion{
\section{Omitted Proofs}
\label{sec:omittedProofs}
In this section of the appendix, we provide the proofs that are omitted from the main text.
\lemmaTechnical
\proofLemTechnical
\thmComb*
\proofThmComb
\lemSlowBalancedUniversal*
\proofLemSlowBalancedUniversal
\section{Extension to Permissive FPT Subroutines}
\label{subsec:permissive}
\secPermissive
}
\section{Introduction}
\input{introduction.tex}
\section{Combining Random Sampling with FPT Algorithms}\label{sec:randalgo}
\input{randomized-algo.tex}
\section{Efficient Construction of Set-Inclusion-Families}\label{sec:derandomization}
\input{derandomization}
\section{Conclusion and Discusison}\label{sec:concl}
\input{conclusion}
\longversion{
\medskip\noindent\textbf{Acknowledgements.}
Many thanks to Russell Impagliazzo and Meirav Zehavi for insightful discussions.
The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreements n. 267959 and no. 306992.
NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council (ARC) through the ICT Centre of Excellence Program.
Serge Gaspers is the recipient of an ARC Future Fellowship (project number FT140100048) and acknowledges support under the ARC's Discovery Projects funding scheme (project number DP150101134). Lokshtanov is supported by the Beating Hardness by Pre-processing grant under the recruitment programme of the of Bergen Research Foundation.
}
\shortversion{\newpage}
\bibliographystyle{siam}
\subsection{Picking Random Subsets of the Solution.}
This subsection is devoted to the proof of Theorem \ref{thm:main1}.
The theorem will follow from the following lemma, which gives a new randomized algorithm for {\sc $\Phi$-Ex\-ten\-sion}.
\begin{lemma}
\label{lemma:subext}
If there is a constant $c > 1$ and an algorithm for {\sc $\Phi$-Ex\-ten\-sion}\ with running time $c^k N^{{\mathcal{O}}(1)}$, then there is a randomized algorithm for {\sc $\Phi$-Ex\-ten\-sion}\ with running time $(2-\frac{1}{c})^{n-|X|}N^{{\mathcal{O}}(1)}$.
\end{lemma}
\begin{proof}
Let $\cal B$ be an algorithm for {\sc $\Phi$-Ex\-ten\-sion}{} with running time $c^kN^{{\mathcal{O}}(1)}$.
We now give another algorithm, $\cal A$, for the same problem. $\cal A$ is a randomized algorithm and consists of the following two steps for an input instance $(I,X,k')$ with $k'\le k$.
\begin{enumerate}
\shortversion{\setlength{\itemsep}{-2pt}}
\item Choose an integer $t \leq k'$ depending on $c$, $n$, $k'$ and $|X|$, and then select a random subset $Y$ of $U_I\setminus X$ of size $t$. The choice of $t$ will be discussed towards the end of the proof.
\item
Run Algorithm $\cal B$ on the instance $(I, X \cup Y, k'-t)$ and return the
answer.
\end{enumerate}
This completes the description of \longversion{Algorithm }$\cal A$. Its running time is clearly upper bounded by $c^{k'-t} N^{{\mathcal{O}}(1)}$.
If $\cal A$ returns \textsc{yes}\xspace\ for $(I,X,k')$, this is because $\cal B$ returned \textsc{yes}\xspace\ for $(I,X \cup Y, k' - t)$. In this case there exists a set $S \subseteq U_I \setminus (X \cup Y)$ of size at most $k' - t\le k-t$
such that $S\cup X \cup Y \in {\cal F}_I$. Thus, $Y \cup S$ witnesses that $(I,X,k)$
is indeed a \yes-instance\xspace.
Next we lower bound the probability that $\cal A$ returns \textsc{yes}\xspace\ in case there exists a set $S \subseteq U_I \setminus X$ of size exactly $k'$ such that $X \cup S \in {\cal F}_I$.
The algorithm $\cal A$ picks a set $Y$ of size $t$ at random from $U_I \setminus X$. There are ${n-|X| \choose t}$ possible choices for $Y$. If $\cal A$ picks one of the ${k' \choose t}$ subsets of $S$ as $Y$ then $\cal A$ returns \textsc{yes}\xspace. Thus, \longversion{given that there exists a set $S \subseteq U_I \setminus X$ of size $k'$ such that $X \cup S \in {\cal F}_I$, }we have that
\mymath{
\Pr\left[{\cal A} \mbox{ returns \textsc{yes}\xspace} \right] \geq \Pr[Y \subseteq S] = {{k' \choose t}}/{{n - |X| \choose t}}.
}
\noindent
Let $p(k') = {{k' \choose t}}/{{n - |X| \choose t}}$. For each $k'\in\{0,\dots,k\}$, our main algorithm runs ${\cal A}$ independently $1/p(k')$ times with parameter $k'$. The algorithm returns \textsc{yes}\xspace\ \longversion{if any of the runs of}\shortversion{as soon as} ${\cal A}$ return \textsc{yes}\xspace{}.
If $(I,X,k)$ is a \textsc{yes}\xspace-instance,
then the main algorithm returns \textsc{yes}\xspace\ with probability at least $\min_{k' \leq k} \{1-(1-p(k'))^{1/p(k')}\}\geq 1-\frac{1}{e} >\frac{1}{2}$.
Next we upper bound the running time of the main algorithm, which is
\mymath{
\sum_{k' \leq k} \frac{1}{p(k')} \cdot c^{k'-t} N^{{\mathcal{O}}(1)} \leq \max_{k' \leq k} \frac{{n - |X| \choose t}}{{k' \choose t}} \cdot c^{k'-t} N^{{\mathcal{O}}(1)} \leq \max_{k \leq n-|X|} \frac{{n - |X| \choose t}}{{k \choose t}} \cdot c^{k-t} N^{{\mathcal{O}}(1)}.
}
We are now ready to discuss the choice of $t$ in the algorithm ${\cal A}$. The algorithm ${\cal A}$ chooses the value for $t$ that gives the minimum value of $ \frac{{n - |X| \choose t}}{{k' \choose t}} \cdot c^{k'-t}$. Thus\longversion{, for fixed $n$ and $|X|$} the running time of the algorithm is\longversion{ upper bounded by}
\begin{align}\label{eqn:runtimeext}
\max_{0 \leq k \leq n-|X|}\left\{ \min_{0 \leq t \leq k} \left\{ \frac{{n-|X| \choose t}}{{k \choose t}} c^{k-t} N^{{\mathcal{O}}(1)} \right\}\right\}.
\end{align}
We upper bound \longversion{the expression in \eqref{eqn:runtimeext}}\shortversion{this expression} by
$\left(2-\frac{1}{c}\right)^{n-|X|}N^{{\mathcal{O}}(1)}$ in Lemma~\ref{lem:technical}\shortversion{ (see Appendix \ref{sec:omittedProofs})}.
The running time of the algorithm is thus upper bounded by $\left(2-\frac{1}{c}\right)^{n-|X|}N^{{\mathcal{O}}(1)}$\longversion{, completing the proof}.
\end{proof}
\begin{remark}
The proof of Lemma~\ref{lemma:subext} goes through just as well when $\cal B$ is a randomized algorithm. If $\cal B$ is deterministic or has one-sided error (possibly saying {\sc NO}{} whereas it should say \textsc{yes}\xspace{}), then the algorithm of Lemma~\ref{lemma:subext} also has one sided error. If $\cal B$ has two sided error, then the algorithm of Lemma~\ref{lemma:subext} has two sided error as well.
\end{remark}
\longversion{
Now we give the technical lemma that was used to upper bound the running time of the algorithm described in Lemma~\ref{lemma:subext}. }
\newcommand{\lemmaTechnical}{
\begin{restatable}{lemma}{lemTechnical}
\label{lem:technical}
Let $c>1$ be a fixed constant, and \longversion{let }$n$ and $k \leq n$ be non-negative integers. Then,
\begin{eqnarray*}
\max_{0 \leq k \leq n}\left\{ \min_{0 \leq t \leq k} \left\{ \frac{{n \choose t}}{{k \choose t}} c^{k-t} \right\}\right\} \leq \left(2-\frac{1}{c}\right)^{n} n^{{\mathcal{O}}(1)}
\end{eqnarray*}
\end{restatable}
}
\longversion{\lemmaTechnical}
\newcommand{\proofLemTechnical}{%
\begin{proof}
Setting $\mu = \frac{k}{n}$ and $\alpha = \frac{t}{n}$, we have that
\[
\max_{0 \leq k \leq n}\left\{ \min_{0 \leq t \leq k} \left\{ \frac{{n \choose t}}{{k \choose t}} c^{k-t} \right\}\right\} =
\max_{0\leq \mu \leq 1}\left\{ \min_{0\leq \alpha \leq \mu} \left\{ \frac{{n \choose \lceil \alpha n \rceil}}{ {\lceil \mu n \rceil \choose \lceil \alpha n \rceil}} c^{(\mu-\alpha)n} \right\}\right\} \cdot {\mathcal{O}}(1)
\]
The right hand side of the equation above is upper bounded by picking a concrete value of $\alpha$ for every value of $\mu$, rather than minimizing over all $\alpha$. We set $\alpha=\max\left(0, \frac{1-c\mu}{1-c}\right)$. One can show that this choice of $\alpha$ minimizes the internal expression. In particular, this basically guarantees that $\frac{k-t}{n-t} = \frac{1}{c}$, and this is the natural threshold for when to stop sampling (see the discussion in the introduction).
First, consider the case where $\alpha=0$. The expression is upper bounded by $c^{n/c}$ since $1-c\mu\ge 0$.
To show that $\left(2-\frac{1}{c}\right)^n \ge c^{n/c}$, it suffices to show that $2-\frac{1}{c}-c^{1/c}\ge 0$.
But this is so because $2-\frac{1}{c}-c^{1/c}=0$ when $c=1$ and it is increasing with $c$ when $c>1$.
From now on, we assume $\alpha=\frac{1-c\mu}{1-c}>0$.
Thus,
\begin{align}\label{eqn:choseAlpha}
\max_{0\leq \mu \leq 1}\left\{ \min_{0\leq \alpha \leq \mu} \left\{ \frac{ {n \choose \lceil \alpha n \rceil}}{{\lceil \mu n \rceil \choose \lceil \alpha n \rceil}}c^{(\mu-\alpha)n} \right\}\right\}
\leq \max_{\begin{subxarray} 0\leq \mu \leq 1 \\ \alpha=\frac{1-c\mu}{1-c} \end{subxarray}} \left\{ \frac{ {n \choose \lceil \alpha n \rceil}}{{\lceil \mu n \rceil \choose \lceil \alpha n \rceil}}c^{(\mu-\alpha)n} \right\}.
\end{align}
We will also use the following well known bounds on binomial coefficients to simplify our expressions,
\begin{align}\label{eqn:binBound}
\frac{1}{n^{{\mathcal{O}}(1)}}\left[\left(\frac{k}{n}\right)^{-\frac{k}{n}}\left(1 - \frac{k}{n}\right)^{\frac{k}{n} - 1}\right]^n \leq {n \choose k} \leq \left[\left(\frac{k}{n}\right)^{-\frac{k}{n}}\left(1 - \frac{k}{n}\right)^{\frac{k}{n} - 1}\right]^n.
\end{align}
Using the upper bound in Equation~\ref{eqn:choseAlpha} we obtain the following.
\begin{eqnarray}
\frac{ {n \choose \lceil \alpha n \rceil}}{{\lceil \mu n \rceil \choose \lceil \alpha n \rceil}}c^{(\mu-\alpha)n}
&\leq & \left(\frac{\alpha^{-\alpha} (1-\alpha)^{\alpha-1}}{(\frac{\alpha}{\mu})^{-\alpha} (1-\frac{\alpha}{\mu})^{\alpha-\mu} } c^{(\mu-\alpha)}\right)^n n^{{\mathcal{O}}(1)}\nonumber \\
& = & \left(\frac{\alpha^{-\alpha} (1-\alpha)^{\alpha-1}} { \alpha^{-\alpha} \mu^\alpha (\mu-\alpha)^{\alpha -\mu}\mu^{(\mu-\alpha)}} c^{(\mu-\alpha)}\right)^n n^{{\mathcal{O}}(1)} \nonumber \\
& = & \left( (1-\alpha)^{\alpha-1}(\mu-\alpha)^{\mu -\alpha} \mu^{-\mu} c^{(\mu-\alpha)}\right)^n n^{{\mathcal{O}}(1)}
\label{eqn:runtimeA}
\end{eqnarray}
Substituting the value of $\alpha$ in Equation~\ref{eqn:runtimeA}, we get the following as the base of the exponent.
\begin{eqnarray}
& & \Big(\frac{c(1-\mu)}{c-1}\Big)^{ \frac{c(1-\mu)}{1-c}} \Big(\frac{1-\mu}{c-1}\Big)^{ \frac{1-\mu}{c-1}}
\mu^{-\mu} c^{ \frac{1-\mu}{c-1}} \nonumber \\
&=& \Big(\frac{c}{c-1}\Big)^{ \frac{c(1-\mu)}{1-c}+\frac{1-\mu}{c-1}} (1-\mu)^{ \frac{c(1-\mu)}{1-c}+\frac{1-\mu}{c-1}} ~\mu^{-\mu} \nonumber\\
&= & \Big(\frac{c}{c-1}\Big)^{\mu-1} (1-\mu)^{\mu-1} \mu^{-\mu}
\label{eqn:runtimeB}
\end{eqnarray}
The last assertion in Equation~\ref{eqn:runtimeB} follows from the following simplification.
\begin{eqnarray*}
\frac{c(1-\mu)}{1-c}+\frac{1-\mu}{c-1} = \frac{c(1-\mu)-(1-\mu)}{1-c} = \frac{(c-1)(1-\mu)}{1-c}=\mu-1.
\label{eqn:runtimeC}
\end{eqnarray*}
To summarize the above discussion, we have upper bounded the expression in the statement of the lemma by
\begin{eqnarray*}
\max_{0 \leq k \leq n}\left\{ \min_{0 \leq t \leq k} \left\{ \frac{{n \choose t}}{{k \choose t}} c^{k-t} \right\}\right\} \leq \left
[\max_{0 \leq \mu \leq 1} f(\mu)\right]^n \cdot n^{O(1)},
\end{eqnarray*}
where $f(\mu)=\Big(\frac{c}{c-1}\Big)^{\mu-1} (1-\mu)^{\mu-1} \mu^{-\mu}$.
We now turn to upper bounding the maximum of $f(\mu)$. Clearly, $f$ is continuous and differentiable on the interval $[0,1]$ and thus $f$ achieves its maximum at $\mu \in \{0,1\}$ or at a point where the derivative vanishes. Setting $\mu$ to $0$ and to $1$, we get that $f(0)=\frac{c-1}{c}$ and $f(1)=1$, respectively. Next we differentiate $f$ with respect to $\mu$. The product rule for differentiation yields
\begin{eqnarray*}
f'(\mu)&=& f(\mu) \cdot \left( \ln\left( \frac{c}{c-1}\right) + (\ln (1-\mu) +1) -1 -\ln \mu \right).
\end{eqnarray*}
Since $f(\mu) \neq 0$, we have that $f'(\mu) = 0$ if and only if
$$ \ln\left( \frac{c}{c-1}\right) + \ln (1-\mu) -\ln \mu =0, $$
and the unique solution to this equation is $\mu = \frac{c}{2c-1}$.
Substituting this value of $\mu=\frac{c}{2c-1}$ in $f$ we get $f(\mu) = 2-\frac{1}{c}$.
Thus the maximum value $f$ can attain on $\mu \in [0,1]$ is the maximum of $1$, $1-\frac{1}{c}$ and $2-\frac{1}{c}$. Since, $c>1$ this implies that $f(\mu) \leq 2-\frac{1}{c}$, completing the proof.
\end{proof}}
\longversion{\proofLemTechnical}
\noindent
By running the algorithm from Lemma~\ref{lemma:subext} with $X=\emptyset$ and for each value of $k\in \{0,\dots,n\}$, we obtain an algorithm for {\sc $\Phi$-Sub\-set}, and this proves Theorem \ref{thm:main1}.
\subsection{Derandomization}
In this subsection we prove Theorem \ref{thm:main2} by derandomizing the algorithm of Theorem~\ref{thm:main1}, at the cost of a subexponential factor in the running time.
The key tool in our derandomization is a new pseudo-random object, which we call \sepfamwPlural{}, as well as an almost optimal (up to subexponential factors) construction of such objects.
\begin{definition}
Let $U$ be a universe of size $n$ and let $0\leq q\leq p \leq n$. A family $\mathcal{C} \subseteq {U \choose q}$ is an \emph{\sepfam{n}{p}{q}}, if for every set $S \in {U \choose p}$, there exists a set $Y \in \cal C$ such that $Y \subseteq S$.
\end{definition}
\noindent
Let $\kappa(n,p,q)= {{n \choose q}}/{{p \choose q}}$. In Section~\ref{sec:derandomization} (Theorem~\ref{thm:sepfamconstr}) we give a deterministic construction of an \sepfam{n}{p}{q}, $\cal C$, of size at most $\kappa(n,p,q) \cdot 2^{o(n)}$. The running time of the algorithm constructing $\cal C$ is also upper bounded by $\kappa(n,p,q) \cdot 2^{o(n)}$.
The proof of Theorem~\ref{thm:main2} is now almost identical to the proof of Theorem~\ref{thm:main1}. However, in Lemma~\ref{lemma:subext} we replace the sampling step where the algorithm ${\cal A}$ picks a set $Y \subseteq U_I \setminus X$ of size $t$ at random, with a construction of an \sepfam{n-|X|}{k}{t} ${\cal C}$ using Theorem~\ref{thm:sepfamconstr}. Instead of $\kappa(n-|X|,k,t) \cdot n^{O(1)}$ independent repetitions of the algorithm ${\cal A}$, the new algorithm loops over all $Y \in {\cal C}$. The correctness follows from the definition of \sepfamwPlural{}, while the running time analysis is identical to the analysis of Lemma~\ref{lemma:subext}.
\longversion{
\subsection{Extension to Permissive FPT Subroutines}
\label{subsec:permissive}
}
\newcommand{\secPermissive}{
For some of our applications, our results rely on algorithms for permissive variants of the {\sc $\Phi$-Ex\-ten\-sion}\ problem.
Permissive problems were introduced in the context of local search algorithms \cite{MarxS11} and it has been shown that permissive variants can be fixed-parameter tractable even if the strict version is W[1]-hard and the optimization problem is NP-hard \cite{GaspersKOSS12}.
\defoptproblem{\textsc{Permissive} \phiext}{An instance $I$, a set $X \subseteq U_I$, and an integer $k$.}
{If there is a subset $S \subseteq (U_I \setminus X)$ such that $S \cup X \in {\cal F}_I$ and $|S| \leq k$, then answer \textsc{yes}\xspace;\newline
else if $|{\cal F}_I|>0$,
then answer \textsc{yes}\xspace\ or {\sc NO};\newline
else answer {\sc NO}.}
\noindent
We observe that any algorithm solving {\sc $\Phi$-Ex\-ten\-sion}\ also solves \textsc{Permissive} \phiext.
However, using an algorithm for \textsc{Permissive} \phiext\ will only allow us to solve a decision variant of the {\sc $\Phi$-Sub\-set}\ problem, unless it also returns a certificate in case it answers \textsc{yes}\xspace.
\defproblem{\textsc{De\-ci\-sion} \phisub}{An instance $I$}{Is $|{\cal F}_I|>0$?}
\noindent
The proof of Lemma \ref{lemma:subext} can easily be adapted to the \textsc{Permissive} \phiext\ problem.
\begin{lemma}
\label{lemma:subpext}
If there exists a constant $c > 1$ and an algorithm for \textsc{Permissive} \phiext\ with running time $c^k N^{{\mathcal{O}}(1)}$, then there exists a randomized algorithm for \textsc{Permissive} \phiext\ with running time $(2-\frac{1}{c})^{n-|X|}N^{{\mathcal{O}}(1)}$.
\end{lemma}
\noindent
Now, any algorithm for \textsc{Permissive} \phiext\ also solves \textsc{De\-ci\-sion} \phisub. If the algorithm for \textsc{Permissive} \phiext\ also returns a certificate whenever it answers \textsc{yes}\xspace, this also leads to an algorithm for {\sc $\Phi$-Sub\-set}.
Again, these algorithms can be derandomized at the cost of a factor $2^{o(n)}$ in the running time.
\begin{theorem}
\label{thm:permissive}
If there is an algorithm for \textsc{Permissive} \phiext{} with running time $c^kN^{{\mathcal{O}}(1)}$ then there is an algorithm for \textsc{De\-ci\-sion} \phisub\ with running time $(2-\frac{1}{c})^{n+o(n)} N^{{\mathcal{O}}(1)}$.
Moreover, if the algorithm for \textsc{Permissive} \phiext{} computes a certificate whenever it answers \textsc{yes}\xspace, then there is an algorithm for {\sc $\Phi$-Sub\-set}\ with running time $(2-\frac{1}{c})^{n+o(n)} N^{{\mathcal{O}}(1)}$.
\end{theorem}
}
\longversion{\secPermissive}
\subsection{Enumeration and Combinatorial Upper Bounds}
\longversion{In this subsection, we prove Theorems \ref{thm:main3} and \ref{thm:main4} on combinatorial upper bounds and enumeration algorithms.}
\shortversion{We now outline the proofs of Theorems \ref{thm:main3} and \ref{thm:main4} on combinatorial upper bounds and enumeration algorithms (for full proofs see Appendix \ref{sec:omittedProofs}).}
\shortversion{%
For Theorem \ref{thm:main3}, consider the following random process:
\begin{enumerate}
\setlength{\itemsep}{-2pt}
\item Choose an integer $t$ based on $c,$ $n$, and $k$, then randomly sample a subset $X$ of size $t$ from $U_I$.
\item Uniformly at random pick a set $S$ from ${\mathcal F}_{I,X}^{k-t}$, and output $W = X \cup S$. In the special case where ${\mathcal F}_{I,X}^{k-t}$ is empty return the empty set.
\end{enumerate}
An analysis similar to the one in Lemma \ref{lemma:subext} shows that each set in the family $\mathcal{F}_I$ is selected with probability at least $(2 - \frac{1}{c})^{-n} \cdot n^{-{\mathcal{O}}(1)}$. This implies that there are at most $\left(2-\frac{1}{c}\right)^n n^{{\mathcal{O}}(1)}$ such sets.
}
\longversion{\thmComb*}
\newcommand{\proofThmComb}{
\begin{proof}
Let $I$ be an instance and $k \leq n$. We prove that the number of sets in ${\cal F}_I$ of size exactly $k$ is upper bounded by $\left(2-\frac{1}{c}\right)^n n^{{\mathcal{O}}(1)}$. Since $k$ is chosen arbitrarily the bound on $|{\cal F}_I|$ will follow. We describe below a random process that picks a set $W$ of size $k$ from ${\cal F}_I$ as follows.
\begin{enumerate}
\item Choose an integer $t$ based on $c$, $n$, and $k$, then randomly sample a subset $X$ of size $t$ from $U_I$.
\item Uniformly at random pick a set $S$ from ${\mathcal F}_{I,X}^{k-t}$, and output $W = X \cup S$. In the corner case where ${\mathcal F}_{I,X}^{k-t}$ is empty return the empty set.
\end{enumerate}
This completes the description of the process.
For each set $Z \in {\cal F}_I$ of size exactly $k$, let $E_Z$ denote the event that the set $W$ output by the random process above is equal to $Z$. Now we lower bound the probability of the event $E_Z$. We have the following lower bound.
\begin{eqnarray}
\Pr[E_Z]& = & \Pr[X \subseteq Z \wedge S = Z \setminus X] \nonumber \\
& = & \Pr[X \subseteq Z] \times \Pr[S =Z \setminus X~|~X \subseteq Z] \label{eqn:chosingfromasetC} \\
&=&
\frac{{k \choose t}}{{n \choose t}} \times \frac{1}{|{\mathcal F}_{I,X}^{k-t}|}
\nonumber
\end{eqnarray}
Since $\Phi$ is $c$-uniform we have that $|{\mathcal F}_{I,X}^{k-t}| \leq c^{k-t} n^{{\mathcal{O}}(1)}$, hence
$$\Pr[E_Z] \geq \frac{{k \choose t}}{{n \choose t}} c^{-(k-t)} n^{-{\mathcal{O}}(1)}.$$
We are now ready to discuss the choice of $t$ in the random process. The integer $t$ is chosen such that the above expression for $\Pr[E_Z]$ is maximized (or, in other words, it's reciprocal is minimized). By Lemma~\ref{lem:technical} we have that for every $k \leq n$ there exists a $t \leq k$ such that
$$\frac{{k \choose t}}{{n \choose t}} c^{-(k-t)} \geq (2 - \frac{1}{c})^{-n} \cdot n^{-{\mathcal{O}}(1)}.$$
Hence $\Pr[E_Z] \geq (2 - \frac{1}{c})^{-n} \cdot n^{-{\mathcal{O}}(1)}$ for every $Z \in {\cal F}_I$ of size $k$. Since the events $E_Z$ are disjoint for all the different sets $Z \in {\cal F}_I$ we have that
$$\sum_{\begin{subxarray} Z \in {\cal F}_I \\ |Z| = k \end{subxarray}} \Pr[E_{Z}] \leq 1.$$
This, together with the lower bound on $\Pr[E_Z]$ implies that the number of sets in ${\cal F}_I$ of size exactly $k$ is upper bounded by $\left(2-\frac{1}{c}\right)^n n^{{\mathcal{O}}(1)}$, completing the proof.
\end{proof}%
}
\longversion{\proofThmComb}
\longversion{\noindent}%
If the implicit set system $\Phi$ is efficiently $c$-uniform then the proof of Theorem~\ref{thm:main3} can be made constructive by replacing the sampling step by a construction of an \sepfam{n}{k}{t} ${\cal C}$ using Theorem~\ref{thm:sepfamconstr}. For each $X \in {\cal C}$ the algorithm uses the fact that $\Phi$ is efficiently $c$-uniform to loop over all sets $S \in {\mathcal F}_{I,X}^{k-t}$ and output $X \cup S$ for each such $S$. Looping over ${\cal C}$ instead of sampling $X$ incurs a $2^{o(n)}$ overhead in the running time of the algorithm. In order to avoid enumerating duplicates, we also store each set that we output in a trie and for each set that we output, we check first in linear time whether we have already output that set. \shortversion{This proves Theorem \ref{thm:main4}.}
\longversion{\thmEnum*}
\subsection{Weighted problems}
Let us now discuss how our approach can be used for weighted problems.
The {\sc $\Phi$-Sub\-set}\ problem can naturally encode weighted problems: the instance $I$ has some weights associated to the elements of $U_I$ and a target weight $W$, and $\mathcal{F}_I$ contains only sets with weight at most (or at least) $W$.
Recall, though, that the {\sc $\Phi$-Ex\-ten\-sion}\ problem is to decide whether, for an instance $I$, a set $X\subseteq U_I$, and an integer $k$, there exists a subset $S \subseteq U_I \setminus X$ such that $S \cup X \in \mathcal{F}_I$ and $|S| \le k$. That means that the FPT subroutine needs to be able to decide whether there exists such a set $S$ with size at most $k$ and weight at most $W'=W-w(X)$. Importantly, we require the running time to be $c^k N^{{\mathcal{O}}(1)}$, whereas the running time is often upper bounded in terms of $W'$ for weighted problems.
In \cite{AgrawalLKS15}, such a variant of \textsc{Weighted Feedback Vertex Set} was considered, and an algorithm was presented finding a feedback vertex set with size at most $k$ and weight at most $W$, if one exists, in time $3.1681^k n^{{\mathcal{O}}(1)}$.
Shachnai and Zehavi \cite{ShachnaiZ15} gave a $2.168^k N^{{\mathcal{O}}(1)}$ time algorithm for permissive variants of \textsc{Weighted 3-Hitting Set}: if the instance has a hitting set of weight at most $W$ and size at most $k$, the algorithm returns such a hitting set (\textsc{yes}\xspace), else if it has a hitting set of weight at most $W$ it returns such a hitting set (\textsc{yes}\xspace) or {\sc NO}, and otherwise it returns {\sc NO}.
Moreover, by Fomin et al.'s iterative compression approach \cite{FominGKLS10}, the algorithm of Shachnai and Meirav can be extended to $d\ge 4$,
giving algorithms with running times $O((d-0.832)^k N^{{\mathcal{O}}(1)})$.
Using Lemma \ref{thm:permissive} this gives algorithms for \textsc{Weighted} $d$-\textsc{Hitting Set} with running time $(2-\frac{1}{d-0.832})^n m^{{\mathcal{O}}(1)}$.
\todo[inline]{Add: Weighted $d$-Sat}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,012 |
Q: Gray circle next to large notification icon? Im displaying notifications in my app - but for some reason on android version Lolipop its displaying a gray circle next to the large icon like this:
Does anyone have an idea why is this happening?
Here is my code in which I create my notifications:
builder = new NotificationCompat.Builder(context)
// Set Icon
.setSmallIcon(R.drawable.ic_launcher)
.setLargeIcon(icon)
// Set Ticker Message
.setTicker(message)
// Set Title
.setContentTitle(message)
// Set Text
.setContentText(context.getString(R.string.app_name))
// Add an Action Button below Notification
// .addAction(R.drawable.share,
// context.getString(R.string.share), pendingShare)
// Set PendingIntent into Notification
.setContentIntent(contentIntent)
// Dismiss Notification
.setAutoCancel(true)
.setSound(
Uri.parse("android.resource://"
+ context.getPackageName()
+ "/"
+ prefs.getInt(Constants.NOTIF_SOUND,
R.raw.al_affassi_full)));
A: Your notification icon must follow the notification design from here : iconography notifications
Notification icons must be entirely white. Also, the system may scale down and/or darken the icons.
Edit
try with this image (the image is white and is between the ///)
///
///
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,828 |
Čáslavský sněm proběhl v čáslavském kostele sv. Petra a Pavla ve dnech 3.–7. června 1421. Přestože se původně mělo jednat o generální sněm zemí Koruny české, zúčastnili se ho nakonec jen zástupci orebitů, pražanů, táborů, a někteří čeští katoličtí šlechticové. Na sněmu byli přítomni i Zikmundovi zástupci Půta z Častolovic a Aleš Holický ze Šternberka, a dále i přední duchovní Konrád z Vechty, Jan Želivský a Jan z Příbrami. Moravskou šlechtu na sněmu zastupovali Vilém z Pernštejna, Jan z Lomnice, Petr Strážnický z Kravař, Jindřich z Lipé, Jošt z Rosic či Jan Bítovský z Lichtenburka. Jednání se týkalo prosazování čtyř artikul a uznání Zikmunda Lucemburského českým králem.
Průběh a závěry
Jednohlasně bylo přijato prosazování a hájení Čtyř artikul pražských a to i katolíky s podmínkou, že závazek bude platit pouze dokud a pokud se podaří univerzitním mistrům obhájit pražské artikuly před mezinárodním církevním fórem. I přes snahu Zikmundových důvěrníků, kteří přednesli králův list, kde projevil vůli dále jednat o artikulech a smířlivě urovnat spor, byl označen za "zjevného tupitele pravd svatých" a za nehodného trůnu. Bylo to v době po odražení i druhé jím vedené křížové výpravy. Část především moravské šlechty odmítla se na místě vyvázat z poslušnosti krále (někteří z nich byli přítomni jeho korunovaci v létě 1420) a vyžádala si šestitýdenní lhůtu toto učinit (tento slib nakonec nesplnili a na zemském sněmu v Brně ve dnech 11. až 14. června 1421 se od smlouvy s českými kacíři distancovali).
Byla jmenována prozatímní dvacetičlenná zemská vláda, jež měla spravovat věci veřejné do 28. září 1421 pokud do té doby nebude přijat nový král. Ve sboru zasedali:
2 zástupci Starého Města pražského (Jan Kněževský, Lidéř z Radkovic)
2 zástupci Nového Města pražského (Jan Charvát, Pavlík z domu rychtářova)
4 představitelé dalších královských měst (Vacek ze Žatce, Matěj Pražák z Hradce Králové, Petr Hostic z Kouřimi, Franc z Rožmitálu za Kutnou Horu)
2 mluvčí Tábora (Jan Žižka a Zbyněk z Buchova),
5 pánů (Čeněk z Vartenberka, Hynek Krušina z Lichtenburka, Oldřich Vavák z Hradce a katolíci Oldřich z Rožmberka, Jindřich Berka z Dubé)
5 rytířů (Jan Sádlo ze Smilkova, Oněž z Měkovic, Mikuláš Barchovec z Dašic, Jindřich z Boharyně, Milota z Chřenovic na Bohdanči)
Dále byli řešením sporů duchovní povahy pověřeni Jan Želivský a Jan z Příbrami. Na 4. červenec 1421 byla svolána do Prahy synoda, která měla vyřešit fungování církevní správy v Čechách.
Praktické výsledky sněmu byly mizivé. Moravská šlechta od husitů odpadla, prozatímní vláda v podstatě nikdy nefungovala a synoda se vlivem převratu Jana Želivského v Praze nikdy nesešla.
Odkazy
Literatura
Související články
Husitské války
Čtyři artikuly pražské
Externí odkazy
Novodobá freska s čáslavským výjevem na venkovní fasádě domu.J. Mahena 339/17, 286 01 Čáslav-Nové Město, Česko, zdroj: mapy.cz
Zápis welikého sněmu Čáslawského, proti králi Sigmundowi etc.
Husitství
1421
Události v Čáslavi | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,053 |
A Very Jewish Civil War
Editor 4 years ago 1 min read
Recently the only source for Jewish Confederate soldiers, besides considering surnames on rosters, has been using attorney Simon Wolf's 1895 The American Jew as Patriot, Soldier and Citizen, available here.
The National Archives Project will allow Jewish soldiers of the civil war to be identified and will be available online in 2017, including photographs and written documents. Currently, records have only been available from "The American Jew as Patriot, Soldier and Citizen", by Simon Wolfs in 1985, which comprised of a total of 10,000 names; 7,000 of them being Unions and 3,000 Confederates. The only other source in which this information has been available is through last names contained on rosters.
a National Archives project to identify all Jewish soldiers of the Civil War is underway.
Up to now the only source for Jewish Confederate soldiers, besides considering surnames on rosters, has been attorney Simon Wolf's 1895 The American Jew as Patriot, Soldier and Citizen
Altogether about 10,000 names: 7,000 Union and the rest Confederate
"Soon there may be more, as a National Archives project to identify all Jewish soldiers of the Civil War is underway."
https://13thmississippi.com/2015/11/11/a-very-jewish-civil-war/
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Next Otisco mystery: Who stole plaque honoring Civil War veteran, beloved doctor?
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} | 9,591 |
\section{Introduction}
The stochastic multi-armed bandit (MAB) problem is a classical abstraction of the sequential decision-making problem \citep{Thompson1933,Robbins1952,Lai1985}. Best arm identification (BAI) is an instance of the MAB problem, where we consider pure exploration to identify the best treatment arm, a treatment arm that yields the highest expected reward. In this study, we study \emph{contextual BAI with a fixed budget}, with the goal of identifying the best treatment arm minimizing the probability of misidentification after a fixed number of rounds of an adaptive experiments called a \textit{budget}
\citep{Bubeck2009,Bubeck2011,Audibert2010}. To gain efficiency in this task, we can choose a treatment arm based on random variable that characterizes the features of treatment arms and can be observed before drawing one of them. This random variable is referred to as contextual information or covariate. Our setting is a generalization of BAI with a fixed budget \citep{Carpentier2016}. The main focus of this paper is how to employ contextual information for the purpose of efficiently identifying the best-arm maximizing the unconditional mean reward, rather than how to learn optimal context-specific bandit strategies, as studied in the literature of contextual bandit.
In this setting, we develop an asymptotically optimal strategy for BAI with a fixed budget and contextual information under a small-gap regime, where the gaps of expected rewards of the best and suboptimal treatment arms converge to zero. First, we derive lower bounds for the probability of misidentification by extending the distribution-dependent lower bound by \citet{Kaufman2016complexity} to semiparametric setting under the small-gap regime. Then, we propose our BAI strategy, the Contextual RS-AIPW strategy, which consists of a random sampling (RS) rule using an estimated target allocation ratio and
a recommendation rule using an augmented inverse probability weighting (AIPW) estimator. We prove asymptotic optimality of the proposed strategy by showing that an upper bound of its probability of misidentification matches the efficiency lower bound when the budget goes to infinity under the small-gap regime.
Little has been studied on how to make use of covariate information for BAI with a fixed budget.
One of the main research interests in bandit problems is to clarify a tight lower bound on the probability of misidentification, since it enables us to claim that a strategy whose upper bound for the probability of misidentification matches the lower bound is optimal.
\citet{glynn2004large} proposes a strategy based on optimally selected target allocation ratio.
They, however, assume that the optimal target allocation ratio is known in advance, and do not consider the issue of estimating it. Based on the change-of-measure arguments popularized by \citet{Lai1985}, \citet{Kaufman2016complexity} derives a distribution-dependent lower bound on the misidentification probability, which is agnostic to the optimal target allocation ratio.
Despite the seminal result, a strategy with matching upper and lower bounds given an unknown target allocation ratio has not been proposed
\citep{kaufmann2020hdr}. We address this issue by proposing the small-gap regime, where we can ignore an estimation error of the optimal target allocation ratio relative to the probability of misidentification. This small-gap regime is first proposed by \citet{Kato2022small} for two-armed bandits with Gaussian and Bernoulli distributions. This paper refines and generalizes the result for multi-armed bandits with more general distributions and contextual information.
We note that our study is also a pioneering work for BAI with a fixed budget, as even without contextual information, the existence of an asymptotically optimal BAI strategy is unclear. This study addresses this open question by showing an asymptotically optimal strategy under a small-gap regime because BAI with a fixed-budget and without contextual information is a special case of our setting. Furthermore, we show an analytical solution for the target allocation ratio, which also has been also unknown for a long time.
In BAI, there is another setting called BAI with fixed confidence, where the goal is to stop an adaptive experiment as soon as possible when the best treatment can be recommended with a certain confidence. In both settings of BAI with a fixed budget and confidence, there are few studies using contextual information. However, similar problems are frequently considered in the studies of causal inference, which mainly discuss the efficient estimation of causal parameters the gap between expected outcomes of two treatment arms marginalized over the covariate distribution \citep{Laan2008TheCA,Hahn2011}, rather than BAI. The gap is also called the average treatment effect (ATE) in this literature \citep{imbens_rubin_2015}. According to their results, even if the covariates are marginalized, the variance of the estimator can be reduced with the help of covariate information. In BAI with fixed confidence, several recent studies have proposed the use of contextual information to identify a treatment arm with the highest expected reward marginalized over the contextual information \citep{Kato2021Role,Russac2021}. These studies are also based on a similar motivation to causal inference literature. In this study, we also follow the motivation and consider BAI with a fixed budget to identify the best treatment arm marginalized over the contextual information. To the best of our knowledge, our study is the first to consider BAI with contextual information in the fixed-budget setting.
From a technical perspective, we develop a small-gap regime, a large-deviation bound, and semiparametric analysis in a derivation of the lower bound. First, we employ the small-gap regime, which implies a situation in which it is difficult to identify the best treatment arm. As explained above, the asymptotic optimality of BAI strategies under a fixed-budget setting is a long-standing open issue. For example, when the gaps are fixed, \citet{Carpentier2016} shows that upper bounds of BAI strategies cannot match lower bounds conjectured by \citet{Kaufman2016complexity}. We consider that a contributing factor is an estimation error of the optimal target allocation ratio, which affects the probability of misidentification. Under the small-gap regime, we can ignore the estimation error relative to the probability of misidentification because identification of the best treatment arm becomes difficult when the gaps are sufficiently small. Thus, this regime makes the asymptotic optimality argument in fixed-budget BAI tractable by allowing the evaluation of optimal allocation probabilities to be ignored. \citep{Carpentier2016,Ariu2021}.
Second, to evaluate the probability of misidentification, we develop a new large deviation bound for martingales, as existing bounds, such as the ones in Cram\'{e}r theorem \citep{Cramer1938,Cramer1994SurUN} and G\"{a}rtner-Ellis theorem \citep{Gartner1977}, cannot be used for stochastic process as the samples in BAI. The derivation is inspired by the results of \citet{Fan2013,fan2014generalization,Kato2022small}.
Third, we derive the lower bounds by introducing semiparametric analysis. In semiparametric analysis, we can separate parameters into parameters of interest and nuisance parameters, thus enabling us to make a more flexible statistical inference by ignoring unnecessary information. For example, because our target parameter is a treatment arm with the highest marginalized expected reward, the estimation of the distribution of the contextual information is not in our interest.
\paragraph{Organization.} This paper is organized as follows. In Section~\ref{sec:problem_setting}, we formulate our problem. In Section~\ref{sec:lower_bounds}, we derive the general semiparametric lower bounds for contextual BAI and the target allocation ratio.
In Section~\ref{sec:track_aipw}, we propose the contextual RS-AIPW strategy. Then, in Section~\ref{sec:asymp_opt}, we show that the proposed strategy is optimal in a sense that the upper bound for the probability of misidentification matches the lower bound.
We introduce related work in \ref{sec:related} and discuss several topics in Sections~\ref{sec:discuss}. Finally, we present the proof of the semiparametric lower bound in Section~\ref{sec:proof}.
\section{Problem Setting}
\label{sec:problem_setting}
We consider the following setting of BAI with a fixed budget and contextual information. Given a fixed number of rounds $T$, also called a budget, for each round $t = 1,2,\dots, T$, an agent observes a context (covariate) $X_t\in\mathcal{X}$ and chooses a treatment arm $A_t \in [K] = \{1,2,\dots, K\}$, where $\mathcal{X}\subset \mathbb{R}^d$ denotes the context space. Then, the agent immediately receives a reward (or outcome) $Y_t$ linked to the chosen treatment arm $A_t$. This setting is called the bandit feedback or Rubin causal model \citep{Neyman1923,Rubin1974}; that is, a reward in round $t$ is $Y_t= \sum_{a\in[K]}\mathbbm{1}[A_t = a]Y^a_{t}$, where $Y^a_{t}\in\mathbb{R}$ is a potential independent (random) reward, and $Y^1_t,Y^2_t,\dots, Y^K_a$ are conditionally independent given $X_t$. We assume that $X_t$ and $Y^a_{t}$ are independent and identically distributed (i.i.d.) over $t \in [T] = \{1,2,\dots, T\}$. Our goal is to find a treatment arm with the highest expected reward marginalized over the contextual distribution of $X_t$ with a minimal probability of misidentification after observing the reward in the round $T$.
We define our goal formally. Let $P$ be a joint distribution of $(Y^1_t, Y^2_t, \dots, Y^K_t, X_t)$. Because $(Y^1_t, Y^2_t, \dots, Y^K_t, X_t)$ is i.i.d. over $t\in\{1,2,\dots, T\}$, we omit the subscripts and simply denote it as $(Y^1, Y^2, \dots, Y^K, X)$ to make it clear that it is time-independent in some case. The set $(Y^1, Y^2, \dots, Y^K, X)$ is called full-data in semiparametric analysis \citep{Tsiatis2007semiparametric} and potential outcome in causal inference \citep{imbens_rubin_2015}, while the distribution is called a bandit model in the MAB problems. For $P$, let $\mathbb{P}_{P}$, $\mathbb{E}_{P}$, and $\mathrm{Var}_{P}$ be the probability, expectation, and variance in terms of $P$ respectively and $\mu^a(P) = \mathbb{E}_{P}[Y^a] = \mathbb{E}_P[\mu^a(P)(X) ]$ be the expected reward marginalized over the context $X$, where $\mu^a(P)(x) = \mathbb{E}_{P}[Y^a|X=x]$ is the conditional expected reward given $x\in\mathcal{X}$. Let $\mathcal{P}$ be a set of all joint distributions $P$ such that the the best treatment arm $a^*(P)$ uniquely exists; that is, there exists $a^*(P)\in[K]$ such that $\mu^{a^*(P)} > \max_{b \in [K]\backslash a^*(P)} \mu^b$. An algorithms in BAI is called a \emph{strategy}, which recommends a treatment arm $\widehat{a}_T \in [K]$ after sequentially sampling treatment arms in $t = 1,2,\dots,T$. With the sigma-algebras $\mathcal{F}_{t} = \sigma(X_1, A_1, Y_1, \ldots, X_{t}, A_t, Y_t)$, we define a BAI strategy as a pair $ ((A_t)_{t\in[T]}, \widehat{a}_T)$, where
\begin{itemize}
\setlength{\parskip}{0cm}
\setlength{\itemsep}{0cm}
\item the sampling rule chooses a treatment arm $A_t \in [K]$ in each round $t$ based on the past observations $\mathcal{F}_{t-1}$ and observed context $X_t$.
\item the recommendation rule returns an estimator $\widehat{a}_T$ of the best treatment arm $\widehat{a}^*(P)$ based observations up to round $T$. Here, $\widehat{a}_T$ is $\mathcal{F}_T$-measurable.
\end{itemize}
Let $P_0$ be the ``true'' bandit model of the data generating process.
Then, our goal is to find a BAI strategy that minimizes the probability of misidentification $\mathbb{P}_{P_0}( \widehat{a}_T \neq a^*(P_0))$.
\paragraph{Notation.}
For all $a\in[K]$ and $x\in\mathcal{X}$, let $\nu^a(P)(x) = \mathbb{E}_{P}[(Y^a)^2|x]$ and $\mathrm{Var}_P(Y^a|x) = \left(\sigma^a(P)(x)\right)^2$.
For the true bandit model $P_0\in\mathcal{P}$, we denote $\mu^a(P_0) = \mu^a_0$, $\mu^a(P_0)(x) = \mu^a_0(x)$, $\nu^a_0(x) = \nu^a(P_0)(x)$, and $\sigma^a_0(x) = \sigma^a(P_0)(x)$. Let $a^*(P_0) = a^*_0$, $\mu^{a^*_0}_0 = \mu^*_0$, and $\nu^*_0$. For the two Bernoulli distributions with mean parameters $\mu, \mu' \in [0, 1]$, we denote the KL divergence by $d(\mu, \mu') =\mu\log (\mu/\mu') + (1-\mu)\log((1-\mu)/(1-\mu'))$ with the convention that $d(0, 0) = d(1,1) = 0$.
\section{Lower Bounds}
\label{sec:lower_bounds}
In this section, we derive lower bounds for the probability of misidentification $\mathbb{P}_{P_0}( \widehat{a}_T \neq a^*_0)$ under a small gap; that is, $\mu^*_0- \mu^{a}_0 \to 0$ for all $a\in [K]$. Our lower bounds are semiparametric extensions of distribution-dependent lower bounds shown by \citet{Kaufman2016complexity}; hence, we call the lower bounds the semiparametric lower bounds.
First, the following conditions for a class of the bandit model $\mathcal{P}$ are assumed throughout this study.
\begin{assumption} \label{asm:bounded_mean_variance}
For all $P, Q \in \mathcal{P}$ and $a\in[K]$, let $P^a$ and $Q^a$ be the joint distributions of $(Y^a, X)$ of an treatment arm $a$ under $P$ and $Q$, respectively. The distributions $P^a$ and $Q^a$ are mutually absolutely continuous and have density functions with respect to some Lebesgue measure $m$. The potential outcome $Y^a$ has the first and second moments conditioned on $x\in\mathcal{X}$.
There exist known constants $C_{\mu}, C_\nu, C_{\sigma^2} > 0$ such that, for all $P \in \mathcal{P}$, $a \in [K]$, and $x\in\mathcal{X}$, $| \mu^a(P)(x)| \le C_\mu$, $|\nu^a(P)(x)| < C_\nu$ and $\max\{ 1/\left(\sigma^a(P)(x)\right)^2, \left(\sigma^a(P)(x)\right)^2\} \leq C_{\sigma^2}$ for all $x\in\mathcal{X}$.
\end{assumption}
\begin{comment}
assume the followings:
\begin{assumption}
For all $P\in\mathcal{P}$ and expected rewards of each treatment arm, the variance is a positive continuous function with regard to the expected reward and denoted by $\mathrm{Var}_P(Y^a|x) = \left(\sigma^a(P)(x; \mu^a(P))\right)^2 > 0$.
\end{assumption}
\begin{assumption}
There exists a constant $\left(\sigma^a(x)\right)^2 > 0$ such that
for all $P\in\mathcal{P}$, $\left(\sigma^a(P)(x; \mu^a(P))\right)^2 = \left(\sigma^a(x)\right)^2$ as $\mu^{a^*(P)}(P) - \mu^a(P) \to 0$.
\end{assumption}
\end{comment}
For a class of bandit models, we
consider the location-shift class and equal-variance bandit class defined as follows.
\begin{definition}[Location-shift bandit class]
\label{def:ls_bc}
A class of bandit models $\mathcal{P}^{\mathrm{L}}$ is a location-shift bandit class if $\mathcal{P}^{\mathrm{L}} = \left\{P\in\mathcal{P} : \left(\sigma^a(P)(x)\right)^2 = \left(\sigma^a(x)\right)^2\right\}$, where $\sigma^a(x) > 0$ is a constant.
\end{definition}
For simplicity, $\sigma^{a^*_0}$ is denoted by $\sigma^*$.
\begin{definition}[Equal-variance bandit class]
A class of bandit models $\mathcal{P}^{\mathrm{E}}$ is an equal-variance bandit class if $\mathcal{P}^{\mathrm{E}} = \left\{P\in\mathcal{P} : \left(\sigma^{a^*(P)}(P)(x)\right)^2 = \left(\sigma^a(P)(x)\right)^2\ \mathrm{if}\ \mu^{a^*(P)}(P)(x) = \mu^a(P)(x)\right\}$, where $\left(\sigma^{a}(P)(x)\right)^2$ is a continuous function with respect to $\mu^a(P)(x)$.
\end{definition}
For example, Gaussian bandits, which are bandit models whose rewards follow Gaussian distributions, with fixed variances belong to the location-shift bandit class, whereas Bernoulli bandits, which are bandit models whose rewards follow Bernoulli distributions, belong to the equal-variance bandit class.
For the two classes, we derive the semiparametric lower bounds.
To derive the lower bound, we first restrict our BAI strategy to the consistent strategy, which is also considered in \citet{Kaufman2016complexity}.
\begin{definition}[Consistent strategy]\label{def:consistent}
For each $P\in\mathcal{P}$, if $a^*(P)$ is unique, then $\mathbb{P}_P(\widehat{a}_T = a^*(P)) \to 1$ as $T\to \infty$.
\end{definition}
In large deviation efficiency of hypothesis testing, a similar consistency is assumed \citep{Vaart1998}.
Then, we present the semiparametric lower bound for bandit models belonging to the location-shift bandit class.
Let $\mathcal{W}$ be a set of measurable functions such that $w: [K]\times\mathcal{X} \to (0, 1)$ and $\sum_{a\in[K]} w(a|x) = 1$ for all $x\in\mathcal{X}$. We refer to $w\in\mathcal{W}$ an allocation ratio, which can be used to obtain the following semiparametric lower bound.
\begin{theorem}[Semiparametric contextual lower bound for the location-shift bandit class]
\label{thm:semipara_bandit_lower_bound}
Suppose that there exists a constant $C > 0$ such that $\mu^*_0(x) - \mu^a_0(x) = C\left(\mu^*_0 - \mu^a_0 \right)$ for all $x\in\mathcal{X}$.
For any $P_0 \in \mathcal{P}^{\mathrm{L}}$,
under Assumption~\ref{asm:bounded_mean_variance}, any consistent strategies (Definition~\ref{def:consistent}), as $\mu^*_0 - \mu^a_0 \to 0$ for all $a\in [K]$,
\begin{align*}
\limsup_{T \to \infty} -\frac{1}{T}\log\mathbb{P}_{ P_0 }(\widehat{a}_T \neq a^*_0)\leq \sup_{w \in \mathcal{W}}\min_{a\neq a^*_0} \frac{\left(\mu^*_0 - \mu^a_0\right)^2}{2\Omega^{a}_0(w)} + o\left(\left(\mu^*_0 - \mu^a_0\right)^2\right),
\end{align*}
where
\[\Omega^{a}_0(w) = \mathbb{E}_{P_0}\left[\frac{\left(\sigma^*(X_t)\right)^2}{w(a^*_0| X_t)} + \frac{\left(\sigma^a(X_t)\right)^2}{w(a| X_t)} \right].\]
\end{theorem}
The denominator of the first term of the RHS in the lower bound corresponds to a semiparametric analogue of the Cram\'er-Rao lower bound of the asymptotic variance, called the semiparametric efficiency bound \citep{bickel98,Vaart1998}, of the gap (ATE) between two treatment arms $a,b\in[K]$ $a\neq b$ under an allocation ratio $w \in \mathcal{W}$ in the supremum \citep{hahn1998role}\footnote{More precisely, the semiparametric efficiency bound of the asymptotic variance of the ATE between two treatment arms $a,b\in[K]$ $a\neq b$ is given as $\mathbb{E}_{P_0}\left[\frac{\left(\sigma^a_0(X)\right)^2}{w(a| X)} + \frac{\left(\sigma^b_0(X)\right)^2}{w(b| X)} + \big\{(\mu^*_0(X) - \mu^a_0(X)) - (\mu^*_0 - \mu^a_0)\big\}^2\right]$ \citep{hahn1998role}, where $(\Delta_0(X) - \Delta_0)^2$ appears unlike ours. In our derivation of the lower bound, we restrict the alternative hypothesis to a case where $|\mu^*_0(X) - \mu^a_0(X)| \to 0$ as $\mu^*_0 - \mu^a_0 \to 0$. Under this restriction, the term $\big\{(\mu^*_0(X) - \mu^a_0(X)) - (\mu^*_0 - \mu^a_0)\big\}^2$ does not appear in the lower bound explicitly.}. This result implies that the optimal BAI strategy chooses treatment arms so as to reduce the asymptotic variance of estimators for the gaps (ATEs) between the best and suboptimal treatment arms. Here, When the asymptotic variance of the gap estimators is small, the gaps can be estimated more accurately. Theorem \ref{thm:lower_bound2K} will make this implication clearer.
In Theorem~\ref{thm:lower_bound2K}, we show the analytical solution of $\max_{w \in \mathcal{W}}\min_{a\neq a^*_0} \frac{\left(\mu^*_0 - \mu^a_0\right)^2}{2\Omega^{a}_0(w)}$. This solution exists when $C_{\sigma^2}$ is sufficiently small. Therefore, the supremum of the RHS in the lower bound can be replaced with the maximum as
\begin{align}
\label{eq:sup_max}
\sup_{w \in \mathcal{W}}\min_{a\neq a^*_0} \frac{\left(\mu^*_0 - \mu^a_0\right)^2}{2\Omega^{a}_0(w)} = \max_{w \in \mathcal{W}}\min_{a\neq a^*_0} \frac{\left(\mu^*_0 - \mu^a_0\right)^2}{2\Omega^{a}_0(w)}.
\end{align}
Then, the analytical solution of this maximization problem and refined lower bound are shown in the following theorem.
\begin{theorem}
\label{thm:lower_bound2K}
For any $P_0 \in \mathcal{P}^{\mathrm{L}}$, suppose that the same conditions of Theorem~\ref{thm:semipara_bandit_lower_bound} hold, there exists a constant $\Delta_0$ such that $\mu^*_0 - \mu^a_0 \leq \Delta_0$ for all $a\in[K]$, and there exists a constant $C > 0$ such that $\mu^*_0(x) - \mu^a_0(x) = C\left(\mu^*_0 - \mu^a_0 \right)$ for all $x\in\mathcal{X}$. Then, the maximizer in the RHS of \eqref{eq:sup_max} is given as
\begin{align*}
&\widetilde{w}(a^*_0|x) = \frac{\sigma^*(x)}{\sigma^*(x) + \sqrt{\sum_{b\in[K]\backslash\{a^*_0\}}\left(\sigma^b(x)\right)^2}},\\
&\widetilde{w}(a|x) = \Big(1 - \widetilde{w}(a^*_0|x) \Big)\frac{\left(\sigma^a(x)\right)^2}{\sum_{b\in[K]\backslash\{a^*_0\}}\left(\sigma^b(x)\right)^2}\qquad \forall a\in[K]\backslash\{a^*_0\},
\end{align*}
and as $\Delta_0 \to 0$, the semiparametric lower bound is given as
\begin{align*}
&\limsup_{T \to \infty} - \frac{1}{T}\log\mathbb{P}_{ P_0 }(\widehat{a}_T \neq a^*_0)\leq \frac{\Delta^2_0}{2\mathbb{E}_{P_0}\left[\left(\sigma^*(X_t) + \sqrt{\sum_{a\in[K]\backslash\{a^*_0\}}\left(\sigma^a(X_t)\right)^2}\right)^2\right]} + o(\Delta^2_0).
\end{align*}
\end{theorem}
The proof is shown in Appendix~\ref{appdx:lower}.
This result is also consistent with the existing results that many BAI strategies, such as the LUCB strategy \citep{Kalyanakrishnan2012}, choose both an estimated best and second-best treatment arms to discriminate an estimated best treatment arm from the other suboptimal treatment arms with higher probability. Note that in regret minimization, a strategy is usually designed to sample an estimated best treatment arm more and not to sample other suboptimal treatment arms. Furthermore, because all gaps $\mu^*_0 - \mu^a_0$ are assumed to be upper bounded by $\Delta_a$, we can consider a situation where the expected rewards of all suboptimal treatment arms are in $[\mu^* - \Delta_0, \mu^*_0)$. In this situation, a strategy whose probability of misidentification matches the lower bound behaves to accurately estimate the gap between the best and a hypothetical second-best treatment arms, which is constructed as a treatment arm with the conditional variance $\sum_{b\in[K]\backslash\{a^*_0\}}\frac{\left(\sigma^b(X)\right)^2}{w(b| X)}$ for a allocation ratio $w$. Based on this implication obtained from Theorem~\ref{thm:lower_bound2K}, we construct our strategy in Section~\ref{sec:track_aipw}.
When $K=2$, the lower bound is given as
\begin{align*}
&\limsup_{T \to \infty} - \frac{1}{T}\log\mathbb{P}_{ P_0 }(\widehat{a}_T \neq a^*_0)\leq \frac{\Delta^2_0}{\mathbb{E}_{P_0}\left[\Big(\sigma^1(X_t) + \sigma^2(X_t)\Big)^2\right]} + o(\Delta^2_0),
\end{align*}
where the maximizer in the RHS of \eqref{eq:sup_max} is given as
\begin{align*}
\widetilde{w}(1|x) = \frac{\sigma^1(x)}{\sigma^1(x) + \sigma^2(x)},\ \ \ \widetilde{w}(2|x) = \frac{\sigma^2(x)}{\sigma^1(x) + \sigma^2(x)}.
\end{align*}
This result is also compatible with existing studies on efficient ATE estimation via adaptive experiments, as the maximizer $(\widetilde{w}(1|x), \widetilde{w}(2|x))$ is used throughout these studies \citep{Laan2008TheCA,Hahn2011}.
Here, note that an allocation ratio $w$ in the supremum corresponds to an expectation of sampling rule $\frac{1}{T}\sum^T_{t=1}\mathbbm{1}[A_t = a]$ conditioned on $x$ under a alternative hypothesis $Q\in\mathcal{P}$ $Q\neq P_0$, which is used to derive the lower bound. Therefore, the maximizers $\widetilde{w} \in \mathcal{W}$ does not directly imply an allocation ratio used under the true bandit model $P_0\in\mathcal{P}$. However, the maximizer $\widetilde{w}$ still work as conjectures of a target allocation ratio
used in our proposed strategy, although there is no logic at this stage to show that a strategy using the allocation ratio has an upper bound for the probability of misidentification matching to the lower bound.
In Sections~\ref{sec:track_aipw} and \ref{sec:asymp_opt}, we show that by allocation samples following the target allocation ratio, the upper bound for the probability of misidentification in our proposed strategy matches the semiparametric lower bound. Thus, we can confirm that these maximizers correspond to the optimal target allocation ratio.
Finally, we show the semiparametric lower bound for bandit models belonging to the equal-variance bandit class.
\begin{theorem}[Semiparametric contextual lower bound for the equal-variance bandit class]
\label{thm:lower_equal_variance}
For any $P_0 \in \mathcal{P}^{\mathrm{E}}$, suppose that the conditions of Theorem~\ref{thm:semipara_bandit_lower_bound} hold, there exists a constant $\Delta_0$ such that $\mu^*_0 - \mu^a_0 \leq \Delta_0$ for all $a\in[K]$. Then, for any consistent strategy (Definition~\ref{def:consistent}), as $\Delta_0 \to 0$,
\begin{align*}
&\limsup_{T \to \infty} - \frac{1}{T}\log\mathbb{P}_{ P_0 }(\widehat{a}_T \neq a^*_0)\leq \frac{\Delta^2_0}{\mathbb{E}_{P_0}\left[K\left(\sigma^*_0(X)\right)^2\right]} + o(\Delta^2_0),
\end{align*}
where the maximizer in the RHS of \eqref{eq:sup_max} is given as
\begin{align*}
\widetilde{w}(a|x) = 1/K.
\end{align*}
\end{theorem}
This lower bound implies that the uniform-EBM strategy is optimal, where we choose each treatment arm with the same probability (the uniform sampling rule) and recommend a treatment arm with the highest sample average of observed rewards (the empirical best arm (EBA) recommendation rule).
\begin{remark}[Distribution-dependent lower bounds for BAI with a fixed budget without contextual information] We review the distribution-dependent lower bound for BAI with a fixed budget when there is no contextual information \citep{Kaufman2016complexity}. When the potential outcome of each treatment arm $a\in[K]$ follows the Gaussian distributions, the distribution-dependent lower bound is given as
\begin{align*}
-\frac{1}{T}\log\mathbb{P}_{ P_0 }(\widehat{a}_T \neq a^*_0)\leq \frac{\Delta^2_0}{2 \big(\sqrt{\mathrm{Var}_{P_{0}}(Y^1)} + \sqrt{\mathrm{Var}_{P_{0}}(Y^0)}\big)^2}.
\end{align*}
This lower bound can be derived without localization of an alternative hypothesis and small gap ($\Delta_0 \to 0$). Although the optimal target allocation from this lower bound cannot be derived, by using $w^{*}(1) = \frac{\sqrt{\mathrm{Var}_{P_{0}}(Y^1)}}{\sqrt{\mathrm{Var}_{P_{0}}(Y^1)} + \sqrt{\mathrm{Var}_{P_{0}}(Y^0)}}$ and $w^{*}(0) = 1- w^{*}(1)$ as a target allocation ratio, we can find an optimal algorithm whose upper bound matches the lower bound when $\Delta_0 \to 0$.
\end{remark}
\section{Proposed Strategy: the Contextual RS-AIPW Strategy}
\label{sec:track_aipw}
This section presents our strategy, which consists of sampling and recommendation rules. For each $t = 1,2,\dots, T$, our sampling rule randomly chooses a treatment arm with a probability identical to an estimated target allocation ratio. In final round $T$, our recommendation rule recommends a treatment arm with the highest-estimated expected reward. Based on these rules, we refer to this as the contextual RS-AIPW strategy.
\subsection{Target Allocation Ratio}
First, we define a target allocation, which is used to define a sampling rule. We estimate it during an adaptive experiment and employ the estimator as a probability of choosing a treatment arm. We call a target allocation optimal if the upper and lower bounds for the probability of misidentification under our strategy using the allocation ratio match.
We conjecture the optimal target allocation ratio using the results of Section~\ref{sec:lower_bounds}.
In particular, the results of Theorem~\ref{thm:lower_bound2K} yields the following conjectures for the target allocation ratio $w^*\in\mathcal{W}$: for each $x\in\mathcal{X}$,
\begin{align*}
&w^*(a^*_0|x) = \frac{\sigma^*_0(x)}{\sigma^*_0(x) + \sqrt{\sum_{b\in[K]\backslash\{a^*_0\}}\left(\sigma^b_0(x)\right)^2}},\\
&w^*(a|x) = \Big(1 - w^*(a^*_0|x) \Big)\frac{\left(\sigma^a_0(x)\right)^2}{\sum_{b\in[K]\backslash\{a^*_0\}}\left(\sigma^b_0(x)\right)^2}\qquad \forall a\in[K]\backslash\{a^*_0\}.
\end{align*}
Under this conjectured optimal target allocation ratio, we can show that the upper and lower bounds for the probability of misidentification match in Section~\ref{sec:asymp_opt}; hence, this target allocation ratio is optimal.
This target allocation ratio is unknown when the variances are unknown; therefore, it must be estimated via observations during the bandit process.
\subsection{Sampling Rule with Random Sampling (RS) and Estimation}
We provide a sampling rule referred as to a \textit{random sampling} (RS) rule. For $a\in[K]$ and $t\in[T]$, let $\widehat{w}_{t}(a|x)$ be an estimated target allocation ratio at round $t$. In each round $t$, we obtain $\gamma_t$ from the uniform distribution on $[0,1]$ and choose a treatment arm $A_t = 1$ if $\gamma_t \leq \widehat{w}_{t}(1|X_t)$ and $A_t = a$ for $a \geq 2$ if $\gamma_t \in (\sum^{a-1}_{b=1}\widehat{w}_{t}(b|X_t), \sum^a_{b=1}\widehat{w}_{t}(b|X_t)]$.
As an initialization, we choose a treatment arm $A_t$ at round $t \leq K$ and set $\widehat{w}_t(a| x) = 1/K$ for $a\in[K]$ and $x\in\mathcal{X}$.
In a round $t > K$, for all $a\in[K]$, we estimate the target allocation ratio $w^{*}$ using past observations $\mathcal{F}_{t-1}$, such that for all $a\in[K]$ and $x\in\mathcal{X}$, $\widehat{w}_{t}(a|x) > 0$ and $\sum_{a\in[K]} \widehat{w}_{t}(a|x) = 1$.
Then, in round $t$, we choose a treatment arm $a$ with a probability $\widehat{w}_{t}(a|X_t)$.
To construct an estimator $\widehat{w}_{t}(a| x)$ for all $x\in\mathcal{X}$ in each round $t$, we denote a bounded estimator of the conditional expected reward $\mu^a_0(x)$ by $\widehat{\mu}^a_{t}(x)$, that of the conditional expected squared reward $\nu^a_0(x)$ by $\widehat{\nu}^a_{t}(x)$, and that of the conditional variance $\left(\sigma^a_0(x)\right)^2$ by $(\widehat{\sigma}^a_t(x))^2$. All estimators are constructed only from samples up to round $t$. More formally, they are constructed as follows.
For $t=1,2,\dots, K$, we set $\widehat{\mu}^a_{t} = \widehat{\nu}^a_{t} = (\widehat{\sigma}^a_t(x))^2 = 0$.
For $t > K$, we estimate $\mu^a(x)$ and $\nu^a(x)$ using only past samples $\mathcal{F}_{t-1}$ and converge to the true parameter almost surely.
\begin{assumption}
\label{asm:almost_sure_convergence}
For all $a\in[K]$ and $x\in\mathcal{X}$, $\widehat{\mu}^a_{t}(x)$ and $\widehat{\nu}^a_{t}(x)$ are $\mathcal{F}_{t-1}$-measurable, $|\widehat{\mu}^a_{t}(x)| \leq C_\mu$ and $|\widehat{\nu}^a_{t}(x)| \leq C_\nu$, and
\begin{align*}
\widehat{\mu}^a_{t}(x) \xrightarrow{\mathrm{a.s.}} \mu^a(x)\qquad \mathrm{and}\qquad \widehat{\nu}^a_{t}(x) \xrightarrow{\mathrm{a.s.}} \nu^a(x)\qquad \mathrm{as}\ t\to \infty.
\end{align*}
\end{assumption}
For example, we can use nonparametric estimators, such as the nearest neighbor regression estimator and kernel regression estimator, which are prove to converge to the true function almost surely under a bounded sampling probability $\widehat{w}_t$ by \citet{yang2002} and \citet{qian2016kernel}.
As long as these conditions are satisfied, any estimators can be used. Note that we do not assume specific convergence rates for these estimators because we can show the asymptotic optimality without them owing to the unbiasedness of the AIPW estimator \citep{Kato2021adr}. Let $(\widehat{\sigma}^{\dagger a}_t(x))^2 = \widehat{\nu}^a_{t}(x) - \left(\widehat{\mu}^a_{t}(x)\right)^2$ for all $a\in[K]$ and $x\in\mathcal{X}$.
Then, we estimate the variance $\left(\sigma^a_0(x)\right)^2$ for all $a\in[K]$ and $x\in\mathcal{X}$ in a round $t$ as $\left(\widehat{\sigma}^a_{t}(x)\right)^2 =\max\{ \min\{((\widehat{\sigma}^{\dagger a}_t(x))^2,C_{\sigma^2}\},1/C_{\sigma^2}\}$ and define
$\widehat{w}_{t}$ by replacing the variances in $w^*$ with corresponding estimators; that is, for $\widehat{a}_t \in \argmax_{a\in[K]} \widehat{\mu}^a_{t}(x)$,
\begin{align*}
&\widehat{w}(\widehat{a}_t|X_t) = \frac{\widehat{\sigma}^{\widehat{a}_t}_t(X_t)}{\widehat{\sigma}^{\widehat{a}_t}_t(X_t) + \sqrt{\sum_{b\in[K]\backslash\{\widehat{a}_t\}}(\widehat{\sigma}^b_t(X_t))^2}},\\
&\widehat{w}(a|X_t) = \Big(1 - \widehat{w}(\widehat{a}_t|x) \Big)\frac{(\widehat{\sigma}^a_t(X_t))^2}{\sum_{b\in[K]\backslash\{\widehat{a}_t\}}(\widehat{\sigma}^b_t(X_t))^2}\qquad \forall a\in[K]\backslash\{\widehat{a}_t\}.
\end{align*}
If there are multiple elements in $\argmax_{a\in[K]} \widehat{\mu}^a_{t}(x)$, we choose one of them as $\widehat{a}_t$ in some way.
We employ this strategy to apply the large deviation expansion for martingales to the estimator of the expected reward, which is the core of our theoretical analysis in Section~\ref{sec:asymp_opt}.
\subsection{Recommendation Rule with the AIPW Estimator}
The following section presents our recommendation rule.
In the recommendation phase of round $T$, for each $a\in[K]$, we estimate $\mu^a$ for each $a\in[K]$ and recommend the maximum.
To estimate $\mu^a$, the AIPW estimator is defined as
\begin{align}
\label{eq:aipw}
\widehat{\mu}^{\mathrm{AIPW}, a}_{T} =\frac{1}{T} \sum^T_{t=1}\varphi^a\Big(Y_t, A_t, X_t; \widehat{\mu}^a_t, \widehat{w}_t\Big),\qquad \varphi^a(Y_t, A_t, X_t) = \frac{\mathbbm{1}[A_t = a]\big(Y^a_{t}- \widehat{\mu}^a_{t}(X_t)\big)}{\widehat{w}_t(a| X_t)} + \widehat{\mu}^a_{t}(X_t).
\end{align}
In the final round $t=T$, we recommend $\widehat{a}_T \in [K]$ as
\begin{align}
\label{eq:recommend}
\widehat{a}_T = \argmax_{a\in[K]} \widehat{\mu}^{\mathrm{AIPW}, a}_{T}.
\end{align}
The AIPW estimator has the following properties: (i) its components $\{\varphi^a(Y_t, A_t, X_t; \widehat{\mu}^a_t, \widehat{w}_t)\}^T_{t=1}$ are a martingale difference sequence, thereby allowing us to use the large deviation bounds for martingales; (ii) it has the minimal asymptotic variance among the possible estimators.
For instance, we can use other estimators with a martingale property, such as the inverse probability weighting (IPW) estimator \citep{Horvitz1952}, but their asymptotic variance will be larger than that of the AIPW estimator. For the $t$-th element of the sum in the AIPW estimator, we use the nuisance parameters estimated from past observations up to the round $t-1$ to make the sequence in the sum a martingale difference sequence. This technique is often used in semiparametric inference for adaptive experiments \citep{Laan2008TheCA,hadad2019,Kato2020adaptive,Kato2021adr} and also has a similar motivation to double machine learning \citep{ChernozhukovVictor2018Dmlf}. Note that in double machine learning for a doubly robust (DR) estimator, we usually impose specific convergence rates for the estimators of the nuisance parameter, which are not required in our case owing to the unbiasedness of the AIPW estimator (Assumption~\ref{asm:almost_sure_convergence}). Also see \citet{Kato2021adr}.
\begin{remark}
We present the pseudo-code in Algorithm~\ref{alg}. Note that $C_{\mu}$ and $C_{\sigma^2}$ are introduced for technical purposes to bound the estimators. Therefore, any large positive value can be used.
\end{remark}
\begin{remark}[Remark on the sampling rule]
Unlike the sampling rule of \citet{Garivier2016}, our proposed sampling rule does not choose the next treatment arm so that the empirical allocation ratio tracks the optimal target allocation ratio. This is due to the use of martingale properties under the AIPW estimator in the theoretical analysis of the upper bound.
\end{remark}
\begin{remark}[Sampling for stabilization] In the pseudo-code, only the first $K$ rounds are used for initialization. To stabilize the performance, we can increase the number of samplings in initialization, similarly to the forced-sampling approach employed by \citet{Garivier2016}. In Section~\ref{sec:asymp_opt}, to show the asymptotic optimality, we use almost sure convergence of $\widehat{w}_{t}$ to $w^{*}$. As long as $\widehat{w}_{t} \xrightarrow{\mathrm{a.s}} w^{*}$, we can adjust $\widehat{w}_{t}$ appropriately. For instance, we can use $\widetilde{w}_{t} = (1-r_t)\widehat{w}_t(a|X_t) + r_t 1/2$ as the sampling probability instead of $\widehat{w}_t$, where $r_t \to 0$ as $t\to \infty$.
\end{remark}
\begin{remark}[The role of $C_{\sigma^2}$]
Assumption~\ref{asm:bounded_mean_variance} implies that the sampling probability is bounded by a small constant, $1/(2C_{\sigma^2}) \leq w^{*}(a|x) \leq C_{\sigma^2}/2$. Thus, it ensures that the variance of the AIPW estimator is finite. Although the role of this constant appears to be similar to the forced sampling \citep{Garivier2016}, it is substantially different. We can set $C_{\sigma^2}$ sufficiently large so that it is almost negligible in implementation.
\end{remark}
\begin{algorithm}[tb]
\caption{Contextual RS-AIPW strategy}
\label{alg}
\begin{algorithmic}
\STATE {\bfseries Parameter:} Positive constants $C_{\mu}$ and $C_{\sigma^2}$.
\STATE {\bfseries Initialization:}
\FOR{$t=1$ to $K$}
\STATE Draw $A_t=t$. For each $a\in[K]$, set $\widehat{w}_{t}(a|x) = 1/K$.
\ENDFOR
\FOR{$t=K+1$ to $T$}
\STATE Observe $X_t$.
\STATE Construct $\widehat{w}_{t}(1|X_t)$ by using the estimators of the variances.
\STATE Draw $\gamma_t$ from the uniform distribution on $[0,1]$.
\STATE $A_t = 1$ if $\gamma_t \leq \widehat{w}_{t}(1|X_t)$ and $A_t = a$ for $a \geq 2$ if $\gamma_t \in \left(\sum^{a-1}_{b=1}\widehat{w}_{t}(b|X_t), \sum^a_{b=1}\widehat{w}_{t}(b|X_t)\right]$.
\ENDFOR
\STATE Construct $\widehat{\mu}^{\mathrm{AIPW}, a}_{T}$ for each $a\in[K]$ following \eqref{eq:aipw}.
\STATE Recommend $\widehat{a}_T$ following \eqref{eq:recommend}.
\end{algorithmic}
\end{algorithm}
\section{Asymptotic Optimality of the Contextual RS-AIPW Strategy}
\label{sec:asymp_opt}
In this section, we derive the following upper bound of the misspecification probability of the RS-AIPW strategy, which implies that the strategy is asymptotically optimal\footnote{The results are based on \citet{Kato2022small}, presented at the 2022 Asian Meeting of the Econometric Society in East and South-East Asia. This study corrects and refines the previous results.}.
\subsection{Asymptotic Optimality}
We derive the upper bounds for bandit models, where the rewards are sub-exponential rando presents our recommendation rule. m variables.
\begin{assumption}
\label{asm:sub_exp}
For all $P\in\mathcal{P}$ and $a\in[K]$, $X_{t}$ is sub-exponential random variable and $Y^a_{t}$ is conditionally sub-exponential random variable given $X_t = x$; that is, for all $P\in\mathcal{P}$, $a\in[K]$, $t, u, u' > 0$, and $x\in\mathcal{X}$, there are constants $U, U' > 0$ such that $\mathbb{P}_P(|X_t| > u) \leq 2\exp( - u/U)$ and $\mathbb{P}_P(|Y_t| > u| X_t = x) \leq 2\exp( - u'/U')$
\end{assumption}
\begin{theorem}[Upper bound of the RS-AIPW strategy]\label{thm:optimal}
Suppose that Assumptions~\ref{asm:bounded_mean_variance} and \ref{asm:sub_exp} hold. Then, for any $P_{0}\in\mathcal{P}^{\mathrm{L}}$, as $\mu^*_0 - \mu^a_0 \to 0$,
\begin{align*}
&\liminf_{T \to \infty} - \frac{1}{T}\log \mathbb{P}_{P_0}\left(\widehat{a}_T \neq a^*_0\right)\\
&\geq \min_{a\neq a^*_0}\frac{\left(\mu^*_0 - \mu^a_0\right)^2}{2 \mathbb{E}_{P_0}\left[\left(\sigma^*_0(X) + \sqrt{\sum_{b\in[K]\backslash\{a^*_0\}}\left(\sigma^b_0(X)\right)^2}\right)^2 + \left(\mu^*_0(X) - \mu^a_0(X) - (\mu^*_0 - \mu^a_0)\right)^2\right]} - o\left(\left(\mu^*_0 - \mu^a_0\right)^2\right).
\end{align*}
\end{theorem}
This theorem allows us to evaluate the exponentially small probability of misidentification up to the constant term when $\Delta_0\to 0$. Moreover, this result also implies that the estimation error of the target allocation ratio $w^{*}$ is negligible when $\Delta_0\to 0$. This is because the upper bound matches the performance of strategies for Gaussian bandit models developed by \citet{glynn2004large} given the optimal target allocation ratio. This also means that the estimation error of the target allocation ratio is insensitive to the probability of misidentification in situations where identifying the best treatment arm is difficult due to the small gap.
For $\mathcal{P} = \mathcal{P}^\mathrm{L}$, this upper bound matches the lower bounds in Theorems~\ref{thm:lower_bound2K} and \ref{thm:lower_equal_variance} under a small-gap regime.
\begin{corollary}
If we also suppose that there exist constants $\Delta_0, C >0$ such that $\left|\mu^*_0 - \mu^a_0\right| \geq \Delta_0$ and $\left|\mu^*_0(x) - \mu^a_0(x)\right| \leq C\Delta_0$ for all $a\in[K]$ and $x\in\mathcal{X}$, then, as $\Delta_0 \to 0$, \begin{align*}
&\liminf_{T \to \infty} - \frac{1}{T}\log \mathbb{P}_{P_0}\left(\widehat{a}_T \neq a^*_0\right)\geq \min_{a\neq a^*_0}\frac{\Delta^2_0}{2 \mathbb{E}_{P_0}\left[\left(\sigma^*_0(X) + \sqrt{\sum_{b\in[K]\backslash\{a^*_0\}}\left(\sigma^b_0(X)\right)^2}\right)^2\right]} - o\left(\Delta^2_0\right).
\end{align*}
\end{corollary}
Although the upper bound also matches the lower bound in Theorem~ when $\mathcal{P} = \mathcal{P}^\mathrm{E}$, the Uniform-EBM strategy is also obviously optimal.
\subsection{Proof of the Upper Bound}
Owing to the dependency among samples in BAI, it is also difficult to apply the standard large deviation bound \citep{Dembo2009large} to a sample average of some random variable.
For example, G\"{a}rtner-Ellis theorem \citep{Gartner1977,Ellis1984} provides a large deviation bound for dependent samples, but it requires the existence of the cumulant, a logarithmic moment generating function, which is not easily guaranteed for the samples in BAI.
For these problems, we derive a novel Cram\'er-type large deviation bounds for martingales by extending the results of \citet{Grama2000} and \citet{Fan2013,fan2014generalization}. Note that their original large deviation bound is only applicable to martingales whose conditional second moment is bounded deterministically; that is, for some martingale difference sequence $\{W_s\}^n_{s=1}$ of some random variable $W_s$, for any $n > 0$, there exists a real number $0< \epsilon < 1/2$ such that $\mathbb{E}\left[\sum^n_{s=1}\mathbb{E}[W^2_s| \mathcal{F}_{s-1}] - 1\right] \leq \epsilon^2$; then, \citet{Fan2013,fan2014generalization} derive the upper bound for $\mathbb{P}\left(\sum^n_{s=1}W_s > z\right)$, where $\epsilon$ belongs to a range upper bounded by $\epsilon^{-1}$. Thus, their large deviation bound holds when $\mathbb{E}\left[\sum^n_{s=1}\mathbb{E}[W^2_s| \mathcal{F}_{s-1}] - 1\right]$ can be bounded by any $\epsilon$ for any $n > 0$. However, in BAI, we cannot usually bound the conditional second moment for any $\epsilon$ because of the randomness of strategies. This randomness prevents us from applying the original results of \citet{Grama2000} and \citet{Fan2013,fan2014generalization}.
Instead, we consider bounding the conditional second moment for large $T$. We first demonstrate a large deviation bound for martingales by using the mean convergence of an unconditional second moment. Using the novel large deviation bound and AIPW estimator, under our proposed strategy, the upper and lower bounds for probability of misidentification match as the gaps converge to zero.
\subsubsection*{Step~1: Cram\'er's large deviation expansions for the AIPW estimator}
Here, we introduce key elements of our analysis. For each $t\in[T]$, we define the difference variable
\begin{align*}
\xi^a_t &= \frac{\varphi^{a^*_0}\Big(Y_t, A_t, X_t; \widehat{\mu}^{a^*_0}_t, \widehat{w}_t\Big) -\varphi^a\Big(Y_t, A_t, X_t; \widehat{\mu}^a_t, \widehat{w}_t\Big) - (\mu^*_0 - \mu^a_0)}{\sqrt{T \widetilde{V}^a}},
\end{align*}
where $\widetilde{V}^a = \mathbb{E}_{P_0}[(\sigma^*_0(X) + \sqrt{\sum_{a\in[K]\backslash\{a^*_0\}}(\sigma^a_0(X))^2})^2 + (\mu^*_0(X) - \mu^a_0(X) - (\mu^*_0 - \mu^a_0))^2]$.
We also define its sum $Z^a_t = \sum^t_{s=1}\xi^a_{s}$, and a sum of conditional moments $W_t = \sum^t_{s=1}\mathbb{E}_{P_0}[(\xi^a)^2_{s}| \mathcal{F}_{s-1}]$ with initialization $W_0 = 0$. Using the difference variable $\xi^a_t$, we can express the gap estimator as $\sqrt{T}(\widehat{\mu}^{\mathrm{AIPW}, a^*_0}_{T} - \widehat{\mu}^{ \mathrm{AIPW}, a}_T - (\mu^*_0 - \mu^a_0)) / \sqrt{\widetilde{V}^a} = \sum^T_{t=1}\xi^a_t = Z^a_T$. Here, $\left\{\left(\xi^a_t, \mathcal{F}_t\right)\right\}^T_{t=1}$ is a martingale difference sequence (Appendix~\ref{appdx:martingale}),
using the fact that $\widehat{\mu}^a_{t}$ and $\widehat{w}_t(a|X_t)$ are $\mathcal{F}_{t-1}$-measurable random variables.
Let us also define $V_T = \mathbb{E}_{P_0} [ | \sum_{t=1}^T \mathbb{E}_{P_0}[(\xi^a_t)^2 | \mathcal{F}_{t-1}] -1 |]$
and denote the cumulative distribution function of the standard normal distribution by $\Phi(x) = ({\sqrt{2\pi}})^{-1} \int_{-\infty}^x \exp(- {t^2} / {2})\mathrm{d}t$.
We obtain the following theorem on the tail probability of $Z^a_T$:
\begin{theorem}
\label{thm:fan_refine}
Suppose that Assumptions~\ref{asm:bounded_mean_variance} and \ref{asm:sub_exp}, and the following condition hold:\\
Condition~A: $\sup_{1\leq t \leq T}\mathbb{E}_{P_0}[\exp(C_0 \sqrt{T}|\xi^a_t|) \;|\mathcal{F}_{t-1}]\leq C_1$ for some positive constants $C_0,C_1$.\\
Then, for any $\varepsilon > 0$, there exist $T_0, c_1, c_2>0$ such that, for all $T\geq T_0$ and $1\leq u \leq \sqrt{T}\min\{ C_0/4, \sqrt{{3 C_0^2} / ({8 C_1})}\}$,
\begin{align*}
\frac{\mathbb{P}_{P_0}\left(Z^a_T \leq - u\right)}{\Phi(-u)} & \le c_1 u \exp\left(c_2\left( \frac{u^3}{\sqrt{T}} + \frac{u^4}{T} + u^2 (V_T + \varepsilon) + T_0\right) \right),
\end{align*}
where the constants $c_1,c_2$ depend on $C_0$ and $C_1$ but do not depend on $\{(\xi^a_t, \mathcal{F}_t)\}^T_{t=1}$, $u$, and the bandit model $P$.
\end{theorem}
As described by \citet{fan2014generalization}, if $T\mathbb{E}[(\xi^a_t)^2|\mathcal{F}_{t-1}]$ are all bounded from below by a positive constant, Condition~A implies the conditional Bernstein condition: for a positive constant $C$, $|\mathbb{E}[(\xi^a_t)^k|\mathcal{F}_{t-1}]| \leq \frac{1}{2} k!(C/\sqrt{T})^{k-2}\mathbb{E}[(\xi^a_t)^2|\mathcal{F}_{t-1}]$ for all $k\geq 2$ and all $t\in[T]$.
From Theorem~\ref{thm:fan_refine}, if $(\mu^*_0- \mu^a_0) / {\sqrt{\widetilde{V}^a}} \le \min\{ C_0 / 4, \sqrt{{3 C_0^2} / ({8 C_1})}\}$ as $\mu^*_0- \mu^{a}_0 \to 0$.
For $u = \sqrt{T} (\mu^*_0- \mu^a_0) / {\sqrt{\widetilde{V}^a}}$ and
\begin{align*}
&\mathbb{P}_{P_0}\left(Z^a_T \leq - \sqrt{T} (\mu^*_0- \mu^a_0) / {\sqrt{\widetilde{V}^a}}\right) \\
&= \mathbb{P}_{P_0}\left( \sum_{t =1}^T ({\varphi^{a^*_0}\Big(Y_t, A_t, X_t; \widehat{\mu}^{a^*_0}_t, \widehat{w}_t\Big) - \varphi^a\Big(Y_t, A_t, X_t; \widehat{\mu}^a_t, \widehat{w}_t\Big) - \mu^*_0- \mu^a_0})/({\sqrt{T} \widetilde{V}} )\leq - \sqrt{T} (\mu^*_0- \mu^a_0) / {\sqrt{\widetilde{V}^a}} \right)\\
&= \mathbb{P}_{P_0}\left(\widehat{\mu}^{\mathrm{AIPW}, a^*_0}_{T} \le \widehat{\mu}^{\mathrm{AIPW}, a}_{T}\right).
\end{align*}
Then, the probability that we fail to make the correct treatment arm comparison is bounded as
\begin{align*}
&\frac{\mathbb{P}_{P_0}\left(\widehat{\mu}^{\mathrm{AIPW}, a^*_0}_{T} \leq \widehat{\mu}^{\mathrm{AIPW}, a}_{T} \right)}{\Phi\left(-\sqrt{T} (\mu^*_0- \mu^a_0)/\sqrt{\widetilde{V}^a}\right)}\\
&\leq c_1 \sqrt{T} \frac{\mu^*_0- \mu^a_0}{\sqrt{\widetilde{V}^a}} \exp\left(c_2 \left(T\left\{ \left(\frac{\mu^*_0- \mu^a_0}{\sqrt{\widetilde{V}^a}}\right)^3 +\left(\frac{\mu^*_0- \mu^a_0}{\sqrt{\widetilde{V}^a}}\right)^4 +\left(\frac{\mu^*_0- \mu^a_0}{\sqrt{\widetilde{V}^a}}\right)^2(V_T + \varepsilon)\right\}+ T_0\right) \right).
\end{align*}
Here, we provide the proof sketch of Theorem~\ref{thm:fan_refine}. The formal proof is shown in Appendix~\ref{appdx:proof_large_deviation}.
\begin{proof}[Proof sketch of Theorem~\ref{thm:fan_refine}.] Let us define $r_t(\lambda ) = \exp(\lambda \xi^a_t)/\mathbb{E}[\exp(\lambda \xi^a_t)]$. Then, we apply the change-of-measure in \cite{Fan2013, fan2014generalization} to transform the bound. In \citet{Fan2013,fan2014generalization}, the proof is complete up to this procedure. However, in our case, the second moment is also a random variable. Because of the randomness, there remains a term $\mathbb{E}[\exp(\overline{\lambda}(u)\sum^T_{t=1} \xi^a_t)]/(\prod^T_{t=1}\mathbb{E}[\exp(\overline{\lambda}(u) \xi^a_t)])$, where $\overline{\lambda}(u)$ is some positive function of $u$. Therefore, we next consider the bound of the conditional second moment of $\xi^a_t$ to apply $L^r$-convergence theorem (Proposition~\ref{prp:lr_conv_theorem}). With some computation, the proof is complete.
\end{proof}
\subsubsection*{Step~2: Gaussian approximation under a small gap}
Finally, we consider an approximation of the large deviation bound. Here, $\Phi(-u)$ is bounded as $\frac{1}{\sqrt{2 \pi} (1 + u)} \exp(- \frac{u^2}{2})\le \Phi(-u) \le \frac{1}{\sqrt{\pi} (1 + u)} \exp( - \frac{u^2}{2}),\; u\ge 0$ (see \citet[Section 2.2.,][]{Fan2013}).
By combining this bound with Theorem~\ref{thm:fan_refine} and Proposition~\ref{prp:rate_clt} in Appendix~\ref{appdx:prelim}, which shows the rate of convergence in the Central limit theorem (CLT) for $0 \leq u \leq 1$, we have the following corollary.
\begin{corollary}
\label{thm:fan_refine2}
Suppose that Assumptions~\ref{asm:bounded_mean_variance} and \ref{asm:sub_exp}, Condition~A in Theorem~\ref{thm:fan_refine}, and the following conditions hold:\\
Condition~B: $\left(\mu^*_0 - \mu^a_0 \right) / {\sqrt{\widetilde{V}^a}} \le \min\{ C_0/4, \sqrt{3 C_0^2/(8 C_1})\}$;\\
Condition~C: $\lim_{T \to \infty}V_T = 0$.\\
Then, there exist a constant $c>0$ such that
\begin{align*}
\liminf_{T \to \infty} - \frac{1}{T}\log\mathbb{P}_{P_0}\left(\widehat{\mu}^{\mathrm{AIPW}, a^*_0}_{T} \leq \widehat{\mu}^{\mathrm{AIPW}, a}_{T} \right) \geq \frac{(\mu^*_0- \mu^a_0)^2}{2\widetilde{V}^a} - c
\left(\left(\frac{\mu^*_0- \mu^a_0}{\sqrt{\widetilde{V}^a}}\right)^3 + \left(\frac{\mu^*_0- \mu^a_0}{\sqrt{\widetilde{V}^a}}\right)^4 \right).
\end{align*}
\end{corollary}
This approximation can be considered a Gaussian approximation because the probability
is represented by $\exp(-{\left(\mu^*_0- \mu^a_0\right)^2} T / ({2 \widetilde{V}^a}))$.
Condition~B is satisfied as $\mu^*_0- \mu^a_0 \to 0$.
To use Corollary~\ref{thm:fan_refine2}, we need to show that Conditions~A and C hold.
First, the following lemma states that Condition~A holds with the constants $C_0$ and $C_1$, which are universal to the problems in $\mathcal{P}$.
\begin{lemma}
\label{lem:condition1}
Suppose that Assumptions~\ref{asm:bounded_mean_variance} and ~\ref{asm:sub_exp} and hold. For each $C_0 \ge 0$, there exists a positive constant $C_1$ that depends on $ C_0, C_\mu, C_{\sigma^2}$, such that
$\sup_{t \in [T]} \mathbb{E}_{P_0}[\exp(C_0 \sqrt{T} |\xi^a_t|) \;| \mathcal{F}_{t-1}] \le C_1$.
\end{lemma}
With regards to Condition~C, we introduce the following lemma for the convergence of $V_T$, which corresponds to the mean convergence of the variance of the AIPW estimator scaled with $\sqrt{T}$.
\begin{lemma}
\label{lem:condition2}
Suppose that Assumptions~\ref{asm:bounded_mean_variance} and ~\ref{asm:sub_exp} hold. For any $P \in \mathcal{P}$, $\lim_{T \to \infty}V_T = 0$; that is, for any $\delta > 0$, there exists $T_0$ such that for all $T>T_0$, $\mathbb{E}_{P_0} [| \sum_{t=1}^T \mathbb{E}_{P_0}[(\xi^a_t)^2 | \mathcal{F}_{t-1}] - 1 |] \le \delta.$
\end{lemma}
The proofs of Lemma~\ref{lem:condition1} and Lemma~\ref{lem:condition2} are shown in Appendix~\ref{appdx:lem:condition1} and \ref{appdx:lem:condition2}, respectively.
Finally, the proof of Theorem~\ref{thm:optimal} is completed as follows:
\begin{align*}
&\liminf_{T \to \infty} - \frac{1}{T}\log \mathbb{P}_{P_0}(\widehat{a}_T \neq a^*_0) \geq \liminf_{T \to \infty} - \frac{1}{T}\log \sum_{a\neq a^*_0}\mathbb{P}_{P_0}(\widehat{\mu}^{\mathrm{AIPW}, a}_{T} \geq \widehat{\mu}^{\mathrm{AIPW}, a^*_0}_{T})
\\
&\ge \liminf_{T \to \infty} - \frac{1}{T}\log (K-1) \max_{a\neq a^*_0} \mathbb{P}_{P_0}(\widehat{\mu}^{\mathrm{AIPW}, a}_{T} \geq \widehat{\mu}^{\mathrm{AIPW}, a^*_0}_{T})\\
&\geq \min_{a\neq a^*_0} \frac{(\mu^*_0- \mu^a_0)^2}{2 \widetilde{V}^a} - c
\left(\left(\frac{\mu^*_0- \mu^a_0}{\sqrt{\widetilde{V}^a}}\right)^3 + \left(\frac{\mu^*_0- \mu^a_0}{\sqrt{\widetilde{V}^a}}\right)^4 \right).
\end{align*}
\begin{remark}[CLT]
Note that the CLT cannot provide an exponentially small evaluation of the probability of misidentification. It gives an approximation around $1/\sqrt{T}$ of the expected reward, but we are interested in an evaluation with constant deviation from the expected reward. However, when the gap converges to zero with $1/\sqrt{T}$, our large deviation bound gives the CLT for martingale. In this sense, our result is a generalization of the martingale CLT.
\end{remark}
\section{Related work}
\label{sec:related}
\subsection{Additional Literature on BAI}
The stochastic MAB problem is a classical abstraction of the sequential decision-making problem \citep{Thompson1933,Robbins1952,Lai1985}, and BAI is a paradigm of the MAB problem \citep{EvanDar2006,Audibert2010,Bubeck2011}. Though the problem of BAI itself goes back decades, its variants go as far back as the 1950s \citet{bechhofer1968sequential}.
\citet{kaufmann14,Kaufman2016complexity} conjectures distribution-dependent lower bounds for BAI. In the BAI literature, there is another setting, known as BAI with fixed confidence \citep{Jenninson1982,Mannor2004,Kalyanakrishnan2012,wang2021fast}. For the fixed confidence setting, \citet{Garivier2016} solves the problem in the sense that they develop a strategy whose upper bound of the sample complexity, an expected stopping time, matches the distribution-dependent lower bound. The result is further developed by \citet{Degenne2019b} to solve the two-player game by the no-regret saddle point algorithm. Furthermore, \citet{Qin2017}, \citet{Shang2020}, and \citet{Jourdan2022} extend the Top Two Thompson Sampling (TTTS), proposed by \citet{Russo2016} and shows the asymptotic optimality of their strategies in the fixed confidence setting. \citet{wang2021fast} develops Frank-Wolfe-based Sampling (FWS) to characterize the complexity of fixed-confidence BAI with various types of structures among the arms. See \cite{wang2021fast} for techniques in the fixed-confidence setting and a further comprehensive survey.
Recently, \citet{Russo2016}, \citet{Qin2017}, and \citet{Shang2020} propose the Bayesian BAI strategies, which are optimal in the sense of the posterior convergence rate. For some of their methods, although upper bounds of the sample complexity are shown to match the lower bounds of \citet{Kaufman2016complexity} in fixed-confidence BAI, upper bounds for the probability of misidentification do not match that for fixed-budget BAI. Although the rate of the posterior convergence is also optimal in the fixed-budget setting, it does not imply the asymptotic optimality for the probability of misidentification. The difficulty is also suggested by the results of \citet{Kasy2021}, which attempts to show the asymptotic optimality of a variant of the TTTS called exploration sampling. The authors state that the upper bound for the probability of misidentification exploration sampling matches the distribution-dependent lower bound under a large gap by using the arguments of \citet{glynn2004large}. However, \citet{Ariu2021} shows a counterexample based on the results of \citet{Carpentier2016}.
In evaluation, we can use the simple regret. \citet{Bubeck2009} provides a non-asymptotic minimax lower and upper bound of simple regret for bandit models with a bounded support. Following their results, the Uniform-EBM strategy is optimal for bandit models with a bounded support. This result is compatible with Theorem~\ref{thm:lower_equal_variance}, which implies that the uniform sampling is asymptotically optimal for the equal-variance bandit class. Because \citet{Bubeck2009} does not use other parameters, such as variances,
their result does not contradict with Theorem~\ref{thm:semipara_bandit_lower_bound}, which implies that the target allocation ratio using the variances is optimal. Recently, \citet{adusumilli2022minimax} considers another minimax evaluation of BAI with two-armed bandits.
\citet{Komiyama2021} discusses the optimality of Bayesian simple regret minimization, which is closely related to BAI in a Bayesian setting. They showed that parameters with a small gap make a significant contribution to Bayesian simple regret.
\subsection{Literature on Causal Inference}
The framework of bandit problems is closely related to the potential outcome framework of \citep{Neyman1923,Rubin1974}. In causal inference, the gap is often referred to as the average treatment effect, and the estimation is studied in this framework. To estimate the average treatment effect efficiently, \citet{Laan2008TheCA}, \citet{Hahn2011}, \citet{Meehan2018}, \citet{Kato2020adaptive}, and \citet{gupta2021efficient} propose adaptive strategies. The AIPW estimator, which is also referred to as a DR estimator, plays an important role in treatment effect estimation \citep{Robins1994,hahn1998role,bang2005drestimation,dudik2011doubly,Laan2016onlinetml,Luedtke2016}. The AIPW estimator also plays an important role in double/debiased machine learning literature because it mitigates the convergence rate conditions of the nuisance parameters \citep{ChernozhukovVictor2018Dmlf,Ichimura2022}.
In adaptive experiments for efficient ATE estimation, the AIPW estimator has also been used by \citet{Laan2008TheCA} and \citet{Hahn2011}. \citet{Karlan2014} applied the method of \citet{Hahn2011} to test how donors respond to new information regarding the effectiveness of a charity. These studies have been extended by \citet{Meehan2018} and \citet{kato2020efficienttest}.
However, the notion of optimality is based on the analogue of the efficient estimation of the ATE under i.i.d. observations and not complete in adaptive experiments.
When constructing AIPW estimator with samples obtained from adaptive experiments, including BAI strategies, a typical construction is to use sample splitting and martingales \citep{Laan2008TheCA,hadad2019,Kato2020adaptive,Kato2021adr}. \citet{Howard2020TimeuniformNN}, \citet{Kato2020adaptive}, and provide non-asymptotic confidence intervals of the AIPW or DR estimator, which do not bound a tail probability in large deviation as ours. The AIPW estimator is also used in the recent bandit literature, mainly in regret minimization \citep{dimakopoulou2021online,Kim2021}. \citet{hadad2019}, \citet{Bibaut2021}, and \citet{Zhan2021} consider the off-policy evaluation using observations obtained from regret minimization algorithms.
\subsection{Difference between Limit Experiments Frameworks}
\label{app_subsec:diff_limit_dec}
The small-gap regime is inspired by limit experiments framework \citep{LeCam1986,Vaart1998,Hirano2009}.
For a parameter $\theta_0\in\mathbb{R}$ and $n$ i.i.d. observations for a sample size $n$, the limit experiments framework considers local alternatives $\theta = \theta_0 + h/\sqrt{n}$ for a constant $h\in\mathbb{R}$ \citep{Vaart1991,Vaart1998}. Then, we can approximate the statistical experiment by a Gaussian distribution and discuss the asymptotic optimality of statistical procedures under the approximation. \citet{Hirano2009} relates the asymptotic optimality of statistical decision rules \citep{Manski2000,Manski2002,Manski2004,DEHEJIA2005} to the limit experiment framework. This framework is further applied to policy learning, such as \citet{AtheySusan2017EPL}.
Independently, \citet{Armstrong2022} proposes an application of the local asymptotic framework to a setting similar to BAI by replacing the CLT used in the original framework, such as \citet{Vaart1998}, with that for martingales. In their analysis, the gaps converge to zero with $1/\sqrt{T}$, and a class of BAI strategies is restricted for the second moment of the score to converges to a constant, whereas our gaps converge to zero independently of $T$, and a class of BAI strategies is restricted to be consistent.
Here, note that
taking the parameter $\theta = \theta_0 + h/\sqrt{T}$ does not produce the distribution-dependent analysis; that is, the instance is not fixed as $T$ increases. Therefore, a naive application of the distribution-dependent analysis like Proposition~\ref{lem:data_proc_inequality} does not provide a lower bounds for BAI in this setting.
To match the lower bound of \citet{Kaufman2016complexity}, we need to consider the large deviation bound, rather than CLT. In other words, the limit experiment framework first applies a Gaussian approximation and then evaluates the efficiency under that approximation, where efficiency arguments are complete within the Gaussian distribution. In contrast, we derive the lower bounds of an event under the true distribution in our limit decision-making and approximate it by considering the limit of the gap. Therefore, in limit decision-making, we first consider the optimality for the true distribution and find the optimal strategy in the sense that the upper bound matches the lower bound when the gaps converge to zero.
\subsection{Other Related Work}
Our small-gap regime is also inspired by lil'UCB \citep{Jamieson2014}.
\citet{Balsubramani2016} and \citet{Howard2020TimeuniformNN} propose sequential testing using the law of iterated logarithms and discuss the optimality of sequential testing based on the arguments of \citet{Jamieson2014}.
Ordinal optimization has been studied in the operation research community \citep{peng2016myopic, Dohyun2021}, and a modern formulation was established in the 2000s \citep{chen2000,glynn2004large}. Most of these studies consider the estimation of the optimal sampling rule separately from the probability of misidentification.
In addition to \citet{Fan2013,fan2014generalization}, several studies have employed martingales to obtain tight large deviation bounds \citep{Cappe2013,Juneja2019,Howard2020TimeuniformNN,Kaufmann2021}. Some of these studies have applied change-of-measure techniques.
\citet{Tekin2015}, \citet{GuanJiang2018}, and \citet{Deshmukh2018} also consider BAI with contextual information, but their analysis and setting are different from those employed in this study.
\section{Discussion}
\label{sec:discuss}
\subsection{Asymptotic Optimally in BAI with a Fixed Budget}
\citet{Kaufman2016complexity} derives distribution-dependent lower bounds for BAI with a fixed confidence and budget, based on similar change-of-measure arguments to those found in \citet{Lai1985}. In BAI with fixed confidence, \citet{Garivier2016} develops a strategy whose upper bound and lower bounds for the probability of misidentification match. In contrast, in the fixed-budget setting, the existence of a strategy whose upper bound matches the lower bound of \citet{Kaufman2016complexity} was unclear. We consider that this is because the estimation error of an optimal target allocation ratio is negligible in BAI with a fixed budget, unlike BAI with fixed confidence, where we can draw each treatment arm until the strategy satisfies a condition. Furthermore, there are lower bounds different from \citet{Kaufman2016complexity}, such as \citet{Audibert2010}, \citet{Bubeck2011}, and \citet{Carpentier2016}.
\citet{Audibert2010} proposes the UCB-E and Successive Rejects (SR) strategies. Using the complexity terms
$H_1(P) = \sum_{a \in [K] \backslash\{a^*(P)\}} 1/(\Delta^a(P))^2$ and $H_2(P) = \max_{a \in [K] \backslash\{a^*(P)\}} a/(\Delta^a(P))^2$, where $\Delta^a(P) = \mu^{a^*(P)} - \mu^a$, they prove an upper bound for the probabilities of misidentification of the forms $\exp\left( - T/(18 H_1(P_0))\right)$ and $\exp \big( - T/( \log (K) H_2(P_0))\big)$, for UCB-E with the upper bound on $H_1(P_0)$ and SR, respectively.
\citet{Carpentier2016} discusses the optimality of the method proposed by \citet{Audibert2010} by an effect of constant factors in the exponents of certain bandit models. They proved the lower bound on the probability of misidentification of the form: $ \sup_{P\in\mathcal{P}^B}\left\{ \mathbb{P}_{P_0}\left(\widehat{a}_T \neq a^*_0\right) \exp \big(400T/( \log(K) H_1(P))\big)\right\}$, where for all $P\in\mathcal{P}^B$, there exists a constant $B>0$ such that $H_1(P) < B$. Our result does not contradict with the result that found by \citet{Carpentier2016}, as we consider a small-gap regime, rather than the large-gap regime employed by \citet{Carpentier2016}. In the other words, their results are complementary to ours because we consider situations with a small gap.
\subsection{Two-stage Sampling Rule}
Our Contextual RS-AIPW strategy is also applicable to a setting where we can update the sampling rule in batch, ratner than a sequential manner, as well as other BAI strategies in different settings. For example, even in a two-stage setting, where we are allowed to update the sampling rule only once, we can show the asymptotic optimality if the budgets separated into two-stages go to infinity simultaneously. Such a setting has frequently been adopted in the field of economics, such as \citet{Hahn2011} and \citet{Kasy2021}.
\section{Proof of Semiparametric Lower Bound (Theorem~\ref{thm:semipara_bandit_lower_bound})}
\label{sec:proof}
In this section, we provide proof of Theorem~\ref{thm:semipara_bandit_lower_bound}. Our argument is based on a change-of-measure argument, which has been applied to BAI without contextual information \citep{Kaufman2016complexity}. In this derivation, we relate the likelihood ratio to the lower bound. Inspired by \citet{Murphy1997}, we expand the semiparametric likelihood ratio, where the gap parameter $\mu^*_0 - \mu^a_0$ is regarded as a parameter of interest and the other parameters as nuisance parameters. By using a semiparametric efficient score function, we apply a series expansion to the likelihood ratio of the distribution-dependent lower bound around the gap parameter under a bandit model of an alternative hypothesis. Then, when the gap parameter goes to $0$, the lower bound is characterized by the variance of the semiparametric influence function. Our proof is also inspired by \citet{Vaart1998} and \citet{hahn1998role}. Throughout the proof, for simplicity, $\mathcal{P}^{\mathrm{L}}$ is denoted by $\mathcal{P}$.
\subsection{Transportation Lemma}
Our lower bound derivation is based on change-of-measure arguments, which have been extensively used in the bandit literature \citep{Lai1985}.
\cite{Kaufman2016complexity} derives the following result based on change-of-measure argument, which is the principal tool in our lower bound.
Let us define a density of $(Y^1, Y^2, \dots, Y^K, X)$ under a bandit model $P\in\mathcal{P}$ as
\begin{align*}
p(y^1, y^2, \dots, y^K, x) = \prod_{a\in[K]} f^a_{P}(y^a|x)\zeta_{P}(x)
\end{align*}
For the true bandit model $P_0\in\mathcal{P}$, let $f^a_0(y^a|x)\zeta_{P}(x)$ and $\zeta_{P}(x) = \zeta_0(x)$. Let $f^{a^*_0}_P$ be denoted by $f^*_P$.
\begin{proposition}[Lemma~1 in \cite{Kaufman2016complexity}]\label{lem:data_proc_inequality} Suppose that Assumption~\ref{asm:bounded_mean_variance} holds.
Then, for any two bandit model $P,Q\in\mathcal{P}$ with $K$ treatment arms such that for all $a \in [K]$, $f^a_{P}(y^a|x)\zeta_{P}(x)$ and $f^{a}_{Q}(y^a|x)\zeta_{Q}(x)$ are mutually absolutely continuous,
\begin{align*}
&\mathbb{E}_{Q}\left[\sum^T_{t=1} \mathbbm{1}[ A_t = a] \log \left(\frac{f^a_{Q}(Y^a_{t}| X_t)\zeta_{Q}(X_t)}{f^a_{P}(Y^a_{t}| X_t)\zeta_{P}(X_t)}\right) \right]\ge \sup_{\mathcal{E} \in \mathcal{F}_T} d(\mathbb{P}_{Q}(\mathcal{E}),\mathbb{P}_{P}(\mathcal{E})).
\end{align*}
\end{proposition}
Recall that $d(p, q)$ indicates the KL divergence between two Bernoulli distributions with parameters $p, q\in (0, 1)$.
This ``transportation'' lemma provides the distribution-dependent characterization of events under a given bandit model $P$ and corresponding perturbed bandit model $P'$.
Between the true bandit model $P_0 \in \mathcal{P}$ and a bandit model $Q \in \mathcal{P}$, we define the log-likelihood as
\begin{align*}
L_T = \sum^T_{t=1} \sum_{a\in[K]}\mathbbm{1}[ A_t = a] \log \left(\frac{f^a_{Q}(Y^a_{t}| X_t)\zeta_{Q}(X_t)}{f^a_{0}(Y^a_{t}| X_t)\zeta_{0}(X_t)}\right).
\end{align*}
For this log-likelihood ratio, from Lemma~\ref{lem:data_proc_inequality}, between the true model $P_0$, we have
\begin{align*}
\mathbb{E}_{Q}[L_T] \ge \sup_{\mathcal{E} \in \mathcal{F}_T} d(\mathbb{P}_{Q}(\mathcal{E}),\mathbb{P}_{P_{0}}(\mathcal{E})).
\end{align*}
We consider an approximation of $\mathbb{E}_{Q}[L_T]$ under an appropriate alternative hypothesis $Q\in\mathcal{P}$ when the gaps between the expected rewards of the best treatment arm and suboptimal treatment arms are small.
\subsection{Observed-Data Bandit Models}
Next, we define a semiparametric model for observed data $(Y_t, A_t, X_t)$, as we can only observe the triple $(Y_t, A_t, X_t)$ and cannot observe the full-data $(Y^1_t, Y^2_t,\dots, Y^K_t, X_t)$.
For each $x\in\mathcal{X}$, let us define the average allocation ratio under a bandit model $P\in\mathcal{P}$ and a BAI strategy as
\begin{align*}
\frac{1}{T}\sum^T_{t=1}\mathbb{E}_{P}\left[ \mathbbm{1}[A_t = a]| X_t = x\right] = \kappa_{T, P}(a|x)
\end{align*}
This quantity represents the average sample allocation to each treatment arm $a$ under a strategy. Then, we first show the following lemma. We show the proof in Appendix~\ref{appdx:proof:lem_extnd_infinite}.
\begin{lemma}
\label{lem:kauf_lemma_extnd_infinite}
Suppose that Assumption~\ref{asm:bounded_mean_variance} holds. For $P_0, Q, P\in\mathcal{P}$,
\begin{align*}
\frac{1}{T}\mathbb{E}_{P}[L_T] = \sum_{a\in[K]}\mathbb{E}_{P}\left[\mathbb{E}_{P}\left[\log \frac{f^a_{Q}(Y^a_t| X_t)\zeta_{Q}(X)}{f^a_{0}(Y^a_t| X)\zeta_{0}(X)}|X_t\right]\kappa_{T, P}(a| X_t)\right].
\end{align*}
\end{lemma}
Based on Lemma~\ref{lem:kauf_lemma_extnd_infinite}, for some $\kappa \in\mathcal{W}$, we consider the following samples $\{(\overline{Y}_t, \overline{A}_t, X_t)\}^T_{t=1}$, instead of $\{(Y_t, A_t, X_t)\}^T_{t=1}$, generated as
\begin{align*}
\{(\overline{Y}_t, \overline{A}_t, X_t)\}^T_{t=1} \stackrel{\mathrm{i.i.d}}{\sim} \prod_{a\in[K]}\left\{f^a_{P}(y^a| x)\kappa(a| x) \right\}^{\mathbbm{1}[A=a]}\zeta_{P}(x),
\end{align*}
where $\kappa(a| x)(a|x)$ corresponds to the conditional expectation of $\mathbbm{1}[\overline{A}_t = a]$ given $X_t$. The expectation of $L_T$ for $\{(\overline{Y}_t, \overline{A}_t, X_t)\}^T_{t=1}$ on $P$ is identical to that for $\{(Y_t, A_t, X_t)\}^T_{t=1}$ from the result of Lemma~\ref{lem:kauf_lemma_extnd_infinite} when $\kappa = \kappa_{T, P}$. Therefore, to derive the lower bound for $\{(Y_t, A_t, X_t)\}^T_{t=1}$, we consider that for $\{(\overline{Y}_t, \overline{A}_t, X_t)\}^T_{t=1}$.
Note that this data generating process is induced by a full-data bandit model $P\in \mathcal{P}$; therefore, we call it an observed-data bandit model.
Formally, for a bandit model $P\in\mathcal{P}$ and some $\kappa\in\mathcal{W}$, by using a density function of $P$, let $\overline{R}^{\kappa}_P$ be a distribution of an observed-data bandit model $\{(\overline{Y}_t, \overline{A}_t, X_t)\}^T_{t=1}$ with the density given as
\begin{align*}
&\overline{r}^{\kappa}_P(y, d, x) = \overline{r}_P(y, d| x)\overline{r}_P(x) = \prod_{a\in[K]} \left\{f^a_P(y| x)\kappa(a| x)\right\}^{\mathbbm{1}[d=a]}\zeta_P(x).
\end{align*}
We call it an observed-data distribution.
To avoid the complexity of the notation, we will denote $\{(\overline{Y}_t, \overline{A}_t, X_t)\}^T_{t=1}$ as $\{(Y_t, A_t, X_t)\}^T_{t=1}$ in the following arguments. Let $\mathcal{R} = \big\{\overline{R}_P: P \in \mathcal{P}\big\}$ be a set of all observed-data bandit models $\overline{R}_P$. For $P_0\in\mathcal{P}$, let $\overline{R}^{\kappa}_{P_0} = \overline{R}^{\kappa}_0$, and $\overline{r}^{\kappa}_{P_0} = \overline{r}^{\kappa}_{0}$.
\subsection{Parametric Submodels for the Observed-data Distribution and Tangent Set}
\label{sec:para_sub_obs}
The purpose of this section is to introduce parametric submodels for observed-data distribution under the true full-data bandit model $P_0\in\mathcal{P}$, which is indexed by a real-valued parameter and a set of distributions contained in the larger set $\mathcal{R}_{0}$, and define the derivative of a parametric submodel as a preparation for the series expansion of the log-likelihood; that is, we consider approximation of the log-likelihood $L_T = \sum^T_{t=1} \sum_{a\in[K]}\mathbbm{1}[ A_t = a] \log \left(\frac{f^a_{Q}(Y^a_{t}| X_t)\zeta_{Q}(X_t)}{f^a_{0}(Y^a_{t}| X_t)\zeta_{0}(X_t)}\right)$ using $\mu^*_0 - \mu^a_0$, where $Q\in\mathcal{P}$ is an alternative bandit model.
This section consists of the following three parts.
In the first part, we define parametric submodels as \eqref{eq:parametric_submodel} with condition~\eqref{eq:const_ate}. Then, in the following part, we confirm the differentiability \eqref{eq:trans} and define score functions. Finally, we define a set of score functions, called a tangent set in the final paragraph.
By using the parametric submodels and tangent set, in Section~\ref{sec:semiparametric_lratio}, we demonstrate the series expansion of the log-likelihood (Lemma~\ref{lem;taylor_exp_semipara}). In this section and Section~\ref{sec:semiparametric_lratio}, we abstractly provide definitions and conditions for the parametric submodels and do not specify them. However, in Sections~\ref{sec:oberved-data} and \ref{sec:specification-score}, we show a concrete form of the parametric submodel by finding score functions satisfying the conditions imposed in this section.
\paragraph{Definition of parametric submodels for the observed-data distribution}
First, we define parametric submodels for the observed-data distribution $\overline{R}_0$ with the density function $\overline{r}_0(y, d, x)$ by introducing a parameter $\bm{\varepsilon} = (\varepsilon^a)_{a\in[K]\backslash\{a^*_0\}}$ $\varepsilon^a \in \Theta$ with some compact space $\Theta$. We construct our parametric submodels so that the parameter can be interpreted as the gap parameter of a parametric submodel.
For $P\in\mathcal{P}$, we define a set of parametric submodels $\left\{\overline{R}_{\bm{\varepsilon}}: \bm{\varepsilon}\in\Theta^{K-1}\right\} \subset \mathcal{R}$ as follows: for a set of some functions $(g^a)_{a\in[K]\backslash\{a^*_0\}}$ such that $g^a:\mathbb{R}\times [K]\times \mathcal{X} \to \mathbb{R}$, a parametric submodel $\overline{R}_{\bm{\varepsilon}}$ has a density such that for each $a\in[K]\backslash\{a^*_0\}$, $g^a(\phi^d_\tau(y, x), d, x) = 0$ for $d \in [K]\backslash\{a^*_0, a\}$, and
\begin{align}
\label{eq:parametric_submodel}
&\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, a, x) =
\left( 1 + \varepsilon^a g^a(\phi^a_\tau(y, x), a, x) \right)\overline{r}^{\kappa}_0(y, a, x),\nonumber\\
&\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, a^*_0, x) =
\left( 1 + \varepsilon^a g^a(\phi^*_\tau(y, x), a^*_0, x) \right)\overline{r}^{\kappa}_{0}(y, a^*_0, x),
\end{align}
where for a constant $\tau > 0$ and each $d\in[K]$, $\phi^d_\tau:\mathbb{R}\times\mathcal{X}\to (-\tau, \tau)$ is a truncation function such that for $\varepsilon^d < c(\tau)$,
\begin{align*}
&\phi^d_\tau(y, x) = y\mathbbm{1}[|y| < \tau] - \mathbb{E}_{P_0}[Y^d_t\mathbbm{1}[|Y^d_t| < \tau]|X_t = x] + \mu^d_0(x),\qquad |\varepsilon^d g^d\big(\phi^d_\tau(y), d, x\big) | < 1,
\end{align*}
and $c(\tau)$ is some decreasing scalar function with regard to $\tau$ such that for the inverse $c^{-1}(e) = \tau$, $\tau \to \infty$ as $e \to 0$. Let $\phi^{a^*_0}$ be denoted by $\phi^{*}$.
This is a standard construction of parametric submodels with unbounded random variables \citep{Hansen2022}. For $a\in[K]\backslash\{a^*_0\}$,
this parametric submodel must satisfy $\mathbb{E}_{P_0}[g^a(\phi^{A_t}_\tau(Y_t, X_t), A_t, X_t)] = 0$, $\mathbb{E}_{P_0}[(g^a(\phi^{A_t}_\tau(Y_t, X_t), A_t, X_t))^2] < \infty$,
and
\begin{align}
\label{eq:const_ate}
\int \int y \overline{r}^{\kappa}_{\bm{\varepsilon}}(y, a^*_0, x) \mathrm{d}y\mathrm{d}x - \int \int y \overline{r}^{\kappa}_{\bm{\varepsilon}}(y, a, x) \mathrm{d}y\mathrm{d}x = \mu^*_0 - \mu^a_0 + \varepsilon^a \quad.
\end{align}
In Section~\ref{sec:oberved-data}, we specify functions $(g^a)_{a\in[K]\backslash\{a^*_0\}}$ and confirm that the specified $g^a$ satisfies \eqref{eq:const_ate}. Note that the parametric submodels are usually not unique.
For each $a\in[K]\backslash\{a^*_0\}$ and $d\in[K]\backslash\{a^*_0, a\}$, the parametric submodel $\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, d, x)$ is equivalent to $\overline{r}^{\kappa}_{0}(y, d, x)$ when $\varepsilon^a = 0$ for any $(\varepsilon^e)_{e\in[K]\backslash\{a^*_0, a\}}$. Similarly, $\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, a^*_0, x)$ is equivalent to $\overline{r}^{\kappa}_{0}(y, a^*_0, x)$ if there exists at least one $a\in[K]\backslash\{a^*_0\}$ such that $\varepsilon^a = 0$.
For each $a\in[K]\backslash\{a^*_0\}$, let $f^*_{\bm{\varepsilon}}(y| x)$, $f^a_{\bm{\varepsilon}}(y| x) = f^a_{\varepsilon^a}(y| x)$ and $\zeta_{\bm{\varepsilon}}(x)$ be the conditional densities of $Y^{a^*_0}_t$ and $Y^a_t$ given $x$ and the density of $x$, which satisfies \eqref{eq:parametric_submodel} and \eqref{eq:const_ate} as
\begin{align*}
&\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, a, x) = f^a_{\varepsilon^a}(y| x)\kappa(a| x)\zeta_{\bm{\varepsilon}}(x),\\
&\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, a^*_0, x) = f^*_{\bm{\varepsilon}}(y| x)\kappa(a^*_0| x)\zeta_{\bm{\varepsilon}}(x),\\
&\int \int y \left\{f^*_{\bm{\varepsilon}}(y| x) - f^a_{\varepsilon^a}(y| x)\right\}\zeta_{\bm{\varepsilon}}(x) \mathrm{d}y\mathrm{d}x = \mu^*_0 - \mu^a_0 + \varepsilon^a.
\end{align*}
\paragraph{Differentiablity and score functions of the parametric submodels for the observed-data distribution.}
Next, we confirm the differentiablity of $\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, d, x)$.
Because $\sqrt{\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, d, x)}$ is continuously differentiable for every $y, x$ given $d\in[K]$, and $\int \left( \frac{\dot{\overline{r}}_{\bm{\varepsilon}}(y, d, x)}{\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, d, x)}\right)^2\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, d, x) \mathrm{d}m$ are well defined and continuous in $\bm{\varepsilon}$, where $m$ is some reference measure on $(y, d, x)$, from Lemma~7.6 of \citet{Vaart1998}, we see that the parametric submodel has the score function $g^a$ in the $L_2$ sense; that is, the density $\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, d, x)$ is differentiable in quadratic mean (DQM): for $a\in[K]\backslash\{a^*_0\}$, $d\in\{a^*_0, a\}$, and any $(\varepsilon^b)_{b\in[K]\backslash\{a^*_0, a\}}$,
\begin{align}
\label{eq:trans}
&\int\left[ \overline{r}^{\kappa\ 1/2}_{\bm{\varepsilon}}(y, d, x) - \overline{r}^{\kappa\ 1/2}_0(y, d, x) - \frac{1}{2}\varepsilon^a g^a(y, d, x)\overline{r}^{\kappa\ 1/2}_0(y, d, x) \right]^2\mathrm{d}m = o\left(\varepsilon^a\right).
\end{align}
This relationship is derived from
\begin{align*}
&\frac{\partial}{\partial \varepsilon^a}\Big|_{\varepsilon^a = 0}\log \overline{r}^{\kappa}_{\bm{\varepsilon}}(y, a, x) = \frac{g^a(\phi^a_\tau(y, x), a, x)}{ 1 + \varepsilon^a g^a(\phi^a_\tau(y, x), a, x)}
\Big|_{\varepsilon^a = 0}
=
g^a(\phi^a_\tau(y, x), a, x),\\
&\frac{\partial}{\partial \varepsilon^a}\Big|_{\varepsilon^a = 0}\log \overline{r}^{\kappa}_{\bm{\varepsilon}}(y, a^*_0, x) = \frac{g^a(\phi^*_\tau(y, x), a^*_0, x)}{ 1 + \varepsilon^a g^a(\phi^*_\tau(y, x), a^*_0, x)}
\Big|_{\varepsilon^a = a}
=
g^a(\phi^*_\tau(y, x), a^*_0, x),\\
&\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, e, x) = \overline{r}^{\kappa}_0(y, e, x).
\end{align*}
for any $(\varepsilon^b)_{b\in[K]\backslash\{a^*_0, a\}}$.
In the following section, we specify a measurable function $g^a$ satisfying the conditions \eqref{eq:parametric_submodel} and \eqref{eq:const_ate}, which corresponds to a score function of $\overline{r}^{\kappa}_{0}(y, a, x)$ and $\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, a^*_0, x) $ for each $a\in[K]\backslash\{a^*_0\}$. To clarify the relationship between $g^a$ and a score function, for each $a\in[K]\backslash\{a^*_0\}$, and any $(\varepsilon^b)_{b\in[K]\backslash\{a^*_0, a\}}$, we denote the score function as
\begin{align*}
S^a(y, d, x) &= \frac{\partial}{\partial \varepsilon^a}\Big|_{\varepsilon^a=0} \log \overline{r}^{\kappa}_{\bm{\varepsilon}}(y, d, x) =
\mathbbm{1}[d = a^*_0]S^{a, a^*_0}_{f}(y|x) + \mathbbm{1}[d = a]S^{a, a}_{f}(y|x) + S^{a}_{\zeta}(x),\qquad \forall d\in\{a^*_0, a\}\\
S^a(y, d, x) &= 0,\qquad \forall d\in[K]\backslash\{a^*_0, a\},
\end{align*}
where
\begin{align*}
&S^{a, a^*_0}_{f}(y|x) = \frac{\partial}{\partial \varepsilon^a}\Big|_{\varepsilon^a = 0} \log f^*_{\bm{\varepsilon}}(y| x),\qquad S^{a, a}_{f}(y|x) = \frac{\partial}{\partial \varepsilon^a}\Big|_{\varepsilon^a = 0} \log f^a_{\varepsilon^a}(y| x),\qquad S^a_{\zeta}(x) = \frac{\partial}{\partial \varepsilon^a}\Big|_{\varepsilon^a = 0} \log \zeta_{\bm{\epsilon}}(x).
\end{align*}
Note that $S^a(y, d, x) = g^a(\phi^d_\tau(y, x), d, x)$, and $\frac{\partial}{\partial \varepsilon^a} \log\kappa(a| x) = 0$.
\paragraph{Definition of the tangent set.} Recall that parametric submodels and corresponding score functions are not unique. Here, we consider a set of score functions.
For a set of the parametric submodels $\left\{\overline{R}^{\kappa}_{\bm{\varepsilon}}: \bm{\varepsilon}\in\Theta^{K-1}\right\}$, we obtain a corresponding set of score functions $g^a$ in the Hilbert space $L_2(\overline{R}_Q)$, which we call a tangent space of $\mathcal{R}$ at $\overline{R}^{\kappa}_0$ and denote it by $\dot{\mathcal{R}}^a$. Because $\mathbb{E}_{\overline{R}^{\kappa}_0}[(g^a(\phi^{A_t}_\tau(Y_t, X_t), A_t, X_t))^2]$ is automatically finite, the tangent set can be identified with a subset of the Hilbert space $L_2(\overline{R}^{\kappa}_0)$, up to equivalence classes. For our parametric submodels, the tangent set at $\overline{R}^{\kappa}_0$ in $L_2(\overline{R}^{\kappa}_0)$ is given as
\begin{align*}
\dot{\mathcal{R}}^a = \left\{
\mathbbm{1}[d = a^*_0]S^{a, a^*_0}_{f}(y|x) + \mathbbm{1}[d = a]S^{a, a}_{f}(y|x) + S^a_{\zeta}(x)
\Bigg|\ \mbox{\ref{eq:const}}\ \right\},
\end{align*}
where for each $a\in[K]$, \eqref{eq:const} restricts the tangent set at $\overline{R}^{\kappa}_0$ as
\begin{align}
\label{eq:const}
1 =& \int \int y S^{a, a^*_0}_{f}(y| x) f^*_{0}(y| x)\zeta_{0}(x) \mathrm{d}y\mathrm{d}x - \int \int y S^{a, a}_{f}(y| x) f^a_{0}(y| x)\zeta_{0}(x) \mathrm{d}y\mathrm{d}x + \int \left(\mu^*_0(x) - \mu^a_0(x)\right) S^a_{\zeta}(x) \zeta_{0}(x) \mathrm{d}x.
\end{align}
This constraint is derived from the derivative of \eqref{eq:const_ate}, that is, $\frac{\partial}{\partial \varepsilon^a}\Big|_{\varepsilon^a = 0}\left\{\mu^*_0 - \mu^a_0 + \varepsilon^a\right\} = 1$ and
\begin{align*}
&\frac{\partial}{\partial \varepsilon^a}\left\{\int \int y f^*_{\bm{\varepsilon}}(y| x)\zeta_{\bm{\varepsilon}}(x) \mathrm{d}y\mathrm{d}x - \int \int y f^a_{\varepsilon^a}(y| x)\zeta_{\bm{\varepsilon}}(x) \mathrm{d}y\mathrm{d}x\right\}\\
&= \int \int y \left(\frac{\partial}{\partial \varepsilon^a} \log f^*_{\bm{\varepsilon}}(y| x)\right) f^*_{\bm{\varepsilon}}(y| x)\zeta_{\bm{\varepsilon}}(x) \mathrm{d}y\mathrm{d}x + \int \mu^{a^*_0}(x) \left(\frac{\partial}{\partial \varepsilon^a} \log \zeta_{\bm{\varepsilon}}(x)\right) \zeta_{\bm{\varepsilon}}(x) \mathrm{d}x\\
&\ \ \ - \int \int y \left(\frac{\partial}{\partial \varepsilon^a} \log f^a_{\varepsilon^a}(y| x)\right) f^a_{\varepsilon^a}(y| x)\zeta_{\bm{\varepsilon}}(x) \mathrm{d}y\mathrm{d}x - \int \mu^a(x) \left(\frac{\partial}{\partial \varepsilon^a} \log \zeta_{\bm{\varepsilon}}(x)\right) \zeta_{\bm{\varepsilon}}(x) \mathrm{d}x.
\end{align*}
Recall that for each $a\in[K]\backslash\{a^*_0\}$, the parametric submodel is equivalent to $\overline{r}^{\kappa}_{0}(y, a, x)$ when $\varepsilon^a = 0$ for any $(\varepsilon^e)_{e\in[K]\backslash\{a^*_0, a\}}$. Similarly, $\overline{r}^{\kappa}_{\bm{\varepsilon}}(y, a^*_0, x)$ is equivalent to $\overline{r}^{\kappa}_{0}(y, a^*_0, x)$ if there exists at least one $a\in[K]\backslash\{a^*_0\}$ such that $\varepsilon^a = 0$.
This constraint is required because we regard the parameter $\bm{\varepsilon}$ as the mean parameter. A linear space of the tangent set is called a \emph{tangent space}.
\subsection{Alternative Bandit Model}
Then, we define a class of alternative hypotheses. To derive a tight lower bound by applying the change-of-measure arguments, we use an appropriately defined alternative hypothesis. Our alternative hypothesis is defined using the parametric submodel of $P_0$ as follows:
\begin{definition}
Let $\mathrm{Alt}(P_0) \subset \mathcal{P}$ be alternative bandit models such that for all $Q \in \mathrm{Alt}(P_0)$, $a^*(Q) \neq a^*_0$, and $\overline{R}^{\kappa_{T, Q}}_{\bm{\varepsilon}} = \overline{R}^{\kappa_{T, Q}}_Q$, where $\bm{\varepsilon} = (\varepsilon^a)_{a\in[K]\backslash\{a^*_0\}}$, $\varepsilon^a = \left(\mu^{a^*_0}(Q) - \mu^a(Q)\right) - \left(\mu^*_0 - \mu^a_0\right)$.
\end{definition}
This also implies that for all $Q \in \mathrm{Alt}(P_0)$, for all $a\in[K]\backslash\{a^*_0\}$, $\mu^*_0 - \mu^a_0 > 0$ and there exists $a\in[K]\backslash\{a^*_0\}$ such that $\mu^{a^*_0}(Q) - \mu^a (Q) < 0$. Let $\mu^{a^*_0}(Q)$ be denoted by $\mu^*(Q)$.
\subsection{Semiparametric Likelihood Ratio}
\label{sec:semiparametric_lratio}
We consider series expansion of the log-likelihood $L_T$ defined between $P_0\in\mathcal{P}$ and $Q\in\mathrm{Alt}(P_0)$, where $\mathbb{E}_{Q}\left[L_T\right]$ works as a lower bound for the probability of misidentification as shown in Section~\ref{sec:final_step}. We consider an approximation of $L_T$ under a small-gap regime (small $\mu^*_0 - \mu^a_0$), which is upper-bounded by the variance of the score function. Our argument is
inspired by that in \citet{Murphy1997}.
For $a\in [K]\backslash\{a^*_0\}$, let us define
\begin{align*}
L^{a}_T &= \sum^T_{t=1} \left\{\mathbbm{1}[ A_t = a^*_0] \log \left(\frac{f^*_{\bm{\varepsilon}}(Y^{a^*_0}_{t}| X_t)}{f^*_{0}(Y^{a^*_0}_{t}| X_t)}\right) + \mathbbm{1}[ A_t = a] \log \left(\frac{f^a_{\bm{\varepsilon}}(Y^a_{t}| X_t)}{f^a_{0}(Y^a_{t}| X_t)}\right) + \log \left(\frac{\zeta_{\bm{\varepsilon}}(X_t)}{\zeta_{0}(X_t)}\right)\right\}.
\end{align*}
Then, we prove the following lemma:
\begin{lemma}
\label{lem;taylor_exp_semipara}
Suppose that Assumption~\ref{asm:bounded_mean_variance} holds. For $P_0\in\mathcal{P}$, $Q\in\mathrm{Alt}(P_0)$, and each $a\in[K]\backslash\{a^*_0\}$,
\begin{align*}
\frac{1}{T}\mathbb{E}_{Q}\left[L^{a}_T\right] = \frac{\left(\varepsilon^a\right)^2}{2}\mathbb{E}_{P_0}\left[ \left(S^a(Y_t, A_t, X_t)\right)^2\right] + o\left(\left(\varepsilon^a\right)^2\right).
\end{align*}
\end{lemma}
To prove this lemma, for $a\in[K]\backslash\{a^*_)\}$ and $d\in[K]$, we define
\begin{align*}
\ell^{a}_{\bm{\varepsilon}}(y, d, x) &= \mathbbm{1}[d = a^*_0]\log f^*_{\bm{\varepsilon}}(y|x) + \mathbbm{1}[d = a]\log f^a_{\varepsilon^a}(y|x) + \log \zeta_{\bm{\varepsilon}}(x).
\end{align*}
Note that for $e \in [K]\backslash \{a^*_0, a\}$, if $\varepsilon^a = 0$ and $\varepsilon^e = \mu^*_0 - \mu^e_0$, then
\begin{align*}
\ell^{a}_{\bm{\varepsilon}}(y, d, x) &= \mathbbm{1}[d = a^*_0]\log f^*_{0}(y|x) + \mathbbm{1}[d = a]\log f^a_{0}(y|x) + \log \zeta_{0}(x).
\end{align*}
\begin{proof}[Proof of Lemma~\ref{lem;taylor_exp_semipara}]
By using the parametric submodel defined in the previous section, from the series expansion,
\begin{align*}
L^{a}_T &= \sum^T_{t=1} \left\{\mathbbm{1}[ A_t = a^*_0] \log \left(\frac{f^*_{\bm{\varepsilon}}(Y^{a^*_0}_{t}| X_t)}{f^*_{0}(Y^{a^*_0}_{t}| X_t)}\right) + \mathbbm{1}[ A_t = a] \log \left(\frac{f^a_{\bm{\varepsilon}}(Y^a_{t}| X_t)}{f^a_{0}(Y^a_{t}| X_t)}\right) + \log \left(\frac{\zeta_{\bm{\varepsilon}}(X_t)}{\zeta_{0}(X_t)}\right)\right\}\\
&= \sum^T_{t=1}\left\{\frac{\partial}{\partial \varepsilon^a} \Big|_{\varepsilon^a=0} \ell^a_{\bm{\varepsilon}}(Y_t, A_t, X_t)\varepsilon^a + \frac{\partial^2}{\partial (\varepsilon^a)^2}\Big|_{\varepsilon^a=0} \ell^a_{\bm{\varepsilon}}(Y_t, A_t, X_t)\frac{\left(\varepsilon^a\right)^2}{2} + o\left(\left(\varepsilon^a\right)^2\right)\right\}.
\end{align*}
Here, we do not apply the series expansion for $(\varepsilon^b)_{e\in[K]\backslash\{a^*_0, a\}}$, where $\varepsilon^b = \left(\mu^*(Q) - \mu^b(Q)\right) - \left(\mu^*_0 - \mu^b_0\right)$.
Note that
\begin{align*}
&\frac{\partial}{\partial \varepsilon^a } \Big|_{\varepsilon^a = 0} \ell^a_{\bm{\varepsilon}}(y, d, x) = S^a(y, d, x) = g^a(\phi^d_\tau(y, x), d, x)\\
&\frac{\partial}{\partial (\varepsilon^a)^2}\Big|_{\varepsilon^a = 0} \ell^a_{\bm{\varepsilon}}(y, d, x) = - \left(S^a(y, d, x)\right)^2 = - \left(g^a(\phi^d_\tau(y, x), d, x)\right)^2.
\end{align*}
Let $\overline{R}^{\kappa_{T, Q}}_{\bm{\varepsilon}} = \overline{R}_{\bm{\varepsilon}}$, $\overline{r}^{\kappa_{T, Q}}_{\bm{\varepsilon}}(y, d, x) = \overline{r}_{\bm{\varepsilon}}(y, d, x)$, and $\overline{r}^{\kappa_{T, Q}}_{0}(y, d, x) = \overline{r}_{0}(y, d, x)$.
Because the density $\overline{r}_{\bm{\varepsilon}}(y, d, x)$ is DQM \eqref{eq:trans},
\begin{align*}
&\mathbb{E}_{Q}\left[ S^a(Y_t, A_t, X_t)\right] = \mathbb{E}_{\overline{R}_{\bm{\varepsilon}}}\left[ S^a(Y_t, A_t, X_t)\right]\\
&=\mathbb{E}_{\overline{R}_{\bm{\varepsilon}}}\left[ S^a(Y_t, A_t, X_t)\right] - \sum_{d\in\{a^*_0, a\}}\int S^a(y, d, x) \left(1 + \frac{1}{2}\varepsilon^a g^a(\phi^{d}_\tau(y, x), d, x)\right)^2\overline{r}_{0}(y, d, x) \mathrm{d}y\mathrm{d}x\\
&\ \ \ + \sum_{d\in\{a^*_0, a\}}\int S^a(y, d, x) \left(1 + \frac{1}{2}\varepsilon^a g^a(\phi^{d}_\tau(y, x), d, x)\right)^2\overline{r}_{0}(y, d, x) \mathrm{d}y\mathrm{d}x\\
&= \sum_{d\in\{a^*_0, a\}}\int S^a(y, d, x) \left\{\overline{r}_{\bm{\varepsilon}}(y, d, x) - \left(1 + \frac{1}{2}\varepsilon^a g^a(\phi^d_\tau(y, x), d, x)\right)^2\overline{r}_{0}(y, d, x)\right\} \mathrm{d}y\mathrm{d}x\\
&\ \ \ + \sum_{d\in\{a^*_0, a\}}\int S^a(y, d, x) \left(1 + \frac{1}{2}\varepsilon^a g^a(\phi^{d}_\tau(y, x), d, x)\right)^2\overline{r}_{0}(y, d, x) \mathrm{d}y\mathrm{d}x\\
&= o(\varepsilon^a) + \mathbb{E}_{P_0}\left[ S^a(Y_t, A_t, X_t)\right] + \varepsilon^a \mathbb{E}_{P_0}\left[\left( S^a(Y_t, A_t, X_t)\right)^2\right],
\end{align*}
where we used
\begin{align*}
&\sum_{d\in[K]}\int S^a(y, d, x)\overline{r}_{0}(y, d, x) \mathrm{d}y\mathrm{d}x\\
&= \sum_{d\in\{a^*_0, a\}}\int\left\{\mathbbm{1}[d = a^*_0]S^{a, a^*_0}_{f}(y|x) + \mathbbm{1}[d = a]S^{a, a}_{f}(y|x) + S^{a}_{\zeta}(x)\right\}\overline{r}_{0}(y, d, x) \mathrm{d}y\mathrm{d}x.
\end{align*}
Similarly,
\begin{align*}
-\mathbb{E}_{Q}\left[\left( S^a(Y_t, A_t, X_t)\right)^2\right] = o(\varepsilon^a) - \mathbb{E}_{P_0}\left[\left( S^a(Y_t, A_t, X_t)\right)^2\right] + \varepsilon^a \mathbb{E}_{P_0}\left[\left( S^a(Y_t, A_t, X_t)\right)^3\right].
\end{align*}
By using these expansions, we approximate $\mathbb{E}_{Q}\left[L_T\right]$. Here, by definition, $\mathbb{E}_{P_0}\left[ S^a(Y_t, A_t, X_t)\right] = 0$. Then, we approximate the likelihood ratio as follows:
\begin{align*}
\mathbb{E}_{Q}[L^{a}_T]&= \frac{\left(\varepsilon^a\right)^2}{2}T\mathbb{E}_{P_0}\left[ \left(S^a(Y_t, A_t, X_t)\right)^2\right] + o\left(T\left( \varepsilon^a\right)^2\right).
\end{align*}
\end{proof}
\subsection{Observed-Data Semiparametric Efficient Influence Function}
\label{sec:oberved-data}
Our remaining task is to specify the score function $S^a$. Because there can be several score functions for our parametric submodel due to directions of the derivative, we find a parametric submodel that has a score function with the largest variance, called a least-favorable parametric submodel \citep{Vaart1998}.
In this section, instead of the original observed-data bandit model $\overline{R}^{\kappa_{T, Q}}_{\bm{\varepsilon}}$, we consider an alternative observed-data bandit model $\overline{R}^{\kappa_{T, Q}\,\dagger}_{0}$, which is a distribution of $\{(\phi^{A_t}_\tau(Y_t, X_t), A_t, X_t)\}^T_{t=1}$. Let $\overline{R}^{\kappa_{T, Q}\,\dagger}_{\bm{\varepsilon}}$ be parametric submodel defined as well as Section~\ref{sec:para_sub_obs}, $\mathcal{R}^{\kappa_{T,Q}\,\dagger}_{\bm{\varepsilon}}$ be a set of all $\overline{R}^{\kappa_{T, Q}\,\dagger}_{\bm{\varepsilon}}$, and $\overline{r}^{\kappa_{T,Q}\, \dagger}_{\bm{\varepsilon}}(y, d, x) = f^{d, \dagger}_{\varepsilon^d}(y| x)\kappa_{T, Q}(d| x)\zeta_{\bm{\varepsilon}}(x)$. For each $a\in[K]\backslash\{a^*_0\}$, let $S^{a\,\dagger}(y, d, x)$ and $\dot{\mathcal{R}}^{a\,\dagger}$ be a corresponding score function and tangent space, respectively.
As a preparation, we define a parameter $\mu^*(Q) - \mu^a(Q)$ as a function $\psi^a: \mathcal{R}^{\kappa_{T,Q}\,\dagger}_{\bm{\varepsilon}}\to \mathbb{R}$ such that $\psi^a(\overline{R}^{\kappa_{T,Q}\,\dagger}_{\bm{\varepsilon}})=\mu^*_0 - \mu^a_0 + \varepsilon^a$.
The information bound for $\psi^a\left(\overline{R}^{\kappa_{T,Q}\,\dagger}_{\bm{\varepsilon}}\right)$ of interest is called semiparametric efficiency bound.
Let $ \overline{\mathrm{lin}}\dot{\mathcal{R}}^{a\,\dagger}$ be the closure of the tangent space.
Then, $\psi^a\left(\overline{R}^{\kappa_{T,Q}\,\dagger}_{\bm{\varepsilon}}\right) = \mu^*_0 - \mu^a_0 + \varepsilon^a$ is pathwise differentiable relative to the tangent space $\dot{\mathcal{R}}^{a\,\dagger}$ if and only if there exists a function $\widetilde{\psi}^a\in\overline{\mathrm{lin}}\dot{\mathcal{R}}^{a\,\dagger}$ such that
\begin{align*}
&\frac{\partial}{\partial \varepsilon^a}\Big|_{\varepsilon^a=0} \psi^a\left(\overline{R}^{\kappa_{T,Q}\,\dagger}_{\bm{\varepsilon}}\right) \left(= \frac{\partial}{\partial \varepsilon^a}\Big|_{\varepsilon^a=0}\Big\{\mu^*_0 - \mu^a_0 + \varepsilon^a\Big\} = 1\right) = \mathbb{E}_{\overline{R}^{\kappa_{T,Q}\,\dagger}_{\bm{\varepsilon}}}\left[ \widetilde{\psi}^a(Y_t, A_t, X_t) S^{a\,\dagger}(Y_t, A_t, X_t) \right].
\end{align*}
This function $\widetilde{\psi}^a$ is called the \emph{semiparametric influence function}.
Then, we prove the following lemma on the lower bound for $\mathbb{E}_{P_0}\left[ \left(S^a(Y_t, A_t, X_t)\right)^2 \right]$, which is called the semiparametric efficiency bound:
\begin{lemma}
\label{lem:upperbound_semipara}
Any score function $S^{a\,\dagger}\in\dot{\mathcal{R}}^{a\,\dagger}$ satisfies
\begin{align*}
\mathbb{E}_{P_0}\left[ \left(S^{a\,\dagger}(Y_t, A_t, X_t)\right)^2 \right]\geq \frac{1}{\mathbb{E}_{P_0}\left[\left(\widetilde{\psi}^a(Y_t, A_t, X_t)\right)^2\right]}.
\end{align*}
\end{lemma}
\begin{proof}
From the Cauchy-Schwarz inequality, we have
\begin{align*}
&1 = \mathbb{E}_{P_0}\left[ \widetilde{\psi}^a(Y_t, A_t, X_t) S^{a\,\dagger}(Y_t, A_t, X_t) \right]\leq \sqrt{\mathbb{E}_{P_0}\left[\left (\widetilde{\psi}^a(Y_t, A_t, X_t)\right)^2 \right]}\sqrt{\mathbb{E}_{P_0}\left[ \left(S^{a\,\dagger}(Y_t, A_t, X_t)\right)^2 \right]}.
\end{align*}
Therefore,
\begin{align*}
&\sup_{ S^{a\,\dagger} \in\dot{\mathcal{R}}^{a\,\dagger}} \frac{1}{\mathbb{E}_{P_0}\left[ \left(S^{a\,\dagger}(Y_t, A_t, X_t)\right)^2 \right]} \leq \mathbb{E}_{P_0}\left[\left(\widetilde{\psi}^a(Y_t, A_t, X_t)\right)^2\right].
\end{align*}
\end{proof}
For $a\in [K]\backslash\{a^*_0\}$ and $d\in[K]\backslash\{a^*_0, a\}$, let us define a \emph{semiparametric efficient score function} $S^{a}_{\mathrm{eff}}(y, d, x) \in \overline{\mathrm{lin}} \dot{\mathcal{R}}^{a\,\dagger}$ as
\begin{align*}
&S^{a}_{\mathrm{eff}}(y, d, x) = \frac{\widetilde{\psi}^a(y, d, x)}{\mathbb{E}_{P_0}\left[\left(\widetilde{\psi}^a(Y_t, A_t, X_t)\right)^2\right]}.
\end{align*}
Next, we consider finding $\widetilde{\psi}^a \in \overline{\mathrm{lin}} \dot{\mathcal{R}}^{a\,\dagger}$. We can use the result of \citet{hahn1998role}. Let us guess that for each $a\in[K]\backslash\{a^*_0\}$ and $d\in\{a^*_0, a\}$, $\widetilde{\psi}^a(y, d, x)$ is given as follows:
\begin{align}
\label{eq:guess}
\widetilde{\psi}^{a}(y, d, x) = \frac{\mathbbm{1}[d = a](\phi^*_\tau(y, x) - \mu^*_0(x))}{\kappa_{T, Q}(a^*_0| X)} - \frac{\mathbbm{1}[d = a](\phi^a_\tau(y, x) - \mu^a_0(x))}{\kappa_{T, Q}(a| X)} + \mu^*_0(x) - \mu^a_0(x) - \left(\mu^*_0 - \mu^a_0\right).
\end{align}
Then, as shown by \citet{hahn1998role}, the condition $1 = \mathbb{E}_{\overline{R}^{\kappa_{T,Q}\,\dagger}_{\bm{\varepsilon}}}\left[ \widetilde{\psi}^a(Y_t, A_t, X_t) S^a(Y_t, A_t, X_t) \right]$ holds under \eqref{eq:guess} when for each $a\in[K]\backslash\{a^*_0\}$ and $d\in\{a^*_0, a\}$, the semiparametric efficient score functions are given as
\begin{align*}
&S^{a}_{\mathrm{eff}}(y, d, x) = \mathbbm{1}[d = a^*_0]S^{a, a^*_0}_{f, \mathrm{eff}}(y|x) + \mathbbm{1}[d = a]S^{a, a}_{f, \mathrm{eff}}(y|x) + S^{a}_{\zeta, \mathrm{eff}}(x),\\
&S^{a, a^*_0}_{f, \mathrm{eff}}(y|x) = \frac{(\phi^*_\tau(y, x) - \mu^*_0(x))}{\kappa_{T, Q}(a^*_0| X)}/\widetilde{V}^a_0(\kappa_{T, Q}; \tau),\\
&S^{a, a}_{f, \mathrm{eff}}(y|x) = \frac{(\phi^a_\tau(y, x) - \mu^a_0(x))}{\kappa_{T, Q}(a| X)}/\widetilde{V}^a_{0}(\kappa_{T, Q}; \tau),\\
&S^{a}_{\zeta, \mathrm{eff}}(x) = \left(\mu^*_0(x) - \mu^a(x) - \big(\mu^*_0 - \mu^a_0\big)\right)/\widetilde{V}^a_{0}(\kappa_{T, Q}; \tau),
\end{align*}
where
\begin{align*}
&\widetilde{V}^a_{0}(\kappa_{T, Q}; \tau) = \mathbb{E}_{P_0}\left[\frac{\left(\sigma^*_0(X_t; \tau)\right)^2}{\kappa_{T, Q}(a^*_0| X_t)} + \frac{\left(\sigma^a_0(X_t; \tau)\right)^2}{\kappa_{T, Q}(a| X_t)} + \left(\big(\mu^*_0(X_t) - \mu^a_0(X_t)\big) - \big(\mu^*_0 - \mu^a_0\big)\right)^2\right],\\
&\left(\sigma^*_0(X_t; \tau)\right)^2 := \mathbb{E}_{P_0}\left[\left(\phi^*_\tau(Y_t, X_t; \tau) - \mu^*_0(X_t)\right)^2| X_t\right],\\
&\left(\sigma^a_0(X_t; \tau)\right)^2 := \mathbb{E}_{P_0}\left[\left(\phi^a_\tau(Y_t, X_t; \tau) - \mu^a_0(X_t)\right)^2| X_t\right].
\end{align*}
Here, note that for each $d\in[K]$,
\begin{align*}
&\mathbb{E}_{P_0}\left[\left(\phi^d_\tau(Y_t, X_t) - \mu^d_0(X_t)\right)^2\right]\\
&= \mathbb{E}_{P_0}\left[\left(Y^d_t\mathbbm{1}[|Y^d_t| < \tau] - \mathbb{E}_{P_0}[Y^d_t\mathbbm{1}[|Y^d_t| < \tau]|X_t]|X_t] \right)^2\right]\\
&= \mathbb{E}_{P_0}\left[\mathbb{E}_{P_0}\left[\left(Y^d_t\right)^2\mathbbm{1}[|Y^d_t| < \tau]|X_t\right]- \left(\mathbb{E}_{P_0}[Y^d_t\mathbbm{1}[|Y^d_t| < \tau]|X_t]\right)^2\right].
\end{align*}
We also note that $\mathbb{E}_{\overline{R}^{\kappa_{T,Q}\,\dagger}_{\bm{\varepsilon}}}\left[S^a_{\mathrm{eff}}(Y_t, A_t, X_t)\right] = 0$ and \[\mathbb{E}_{\overline{R}^{\kappa_{T,Q}\,\dagger}_{\bm{\varepsilon}}}\left[\Big(S^a_{\mathrm{eff}}(Y_t, A_t, X_t)\Big)^2\right] = \widetilde{V}^a_{0}(\kappa_{T, Q}; \tau) = \left(\mathbb{E}_{\overline{R}^{\kappa_{T,Q}\,\dagger}_{\bm{\varepsilon}}}\left[\Big(\widetilde{\psi}^a(Y_t, A_t, X_t)\Big)^2\right]\right)^{-1}.\]
Summarizing the above arguments, we obtain the following lemma.
\begin{lemma}
\label{lem:semipara_efficient}
For $a\in[K]\backslash\{a^*_0\}$ and $d\in[K]\backslash\{a^*_0, a\}$, the semiparametric efficient influence function is
\begin{align*}
\widetilde{\psi}^a(y, d, x)
&= \frac{\mathbbm{1}[d = a^*_0](\phi^*_\tau(y, x) - \mu^{*}_0(x))}{\kappa_{T, Q}(a^*_0| X)} - \frac{\mathbbm{1}[d = a](\phi^a_\tau(y, x) - \mu^a_0(x))}{\kappa_{T, Q}(a| X)} + \mu^*_0(x) - \mu^a_0(x) - \left(\mu^*_0 - \mu^a_0\right).
\end{align*}
\end{lemma}
We also define the limit of the semiparametric efficient influence function when $\tau \to \infty$ and the variance as
\begin{align*}
\widetilde{\psi}^a_{\infty}(y, d, x) &= \frac{\mathbbm{1}[d = a^*_0](Y^{a^*_0}_t - \mu^{*}_0(x))}{\kappa_{T, Q}(a^*_0| X)} - \frac{\mathbbm{1}[d = a](Y^a_t - \mu^a_0(x))}{\kappa_{T, Q}(a| X)} + \mu^*_0(x) - \mu^a_0(x) - \left(\mu^*_0 - \mu^a_0\right),\\
\widetilde{V}^a_{0}(\kappa_{T, Q}) &= \mathbb{E}_{P_0}\left[\left(\widetilde{\psi}^a_{\infty}(Y_t, A_t, X_t)\right)^2\right] = \mathbb{E}_{P_0}\left[\frac{\left(\sigma^*_0(X_t)\right)^2}{\kappa_{T, Q}(a^*_0| X_t)} + \frac{\left(\sigma^a_0(X_t)\right)^2}{\kappa_{T, Q}(a| X_t)} + \left(\big(\mu^*_0(X_t) - \mu^a_0(X_t)\big) - \big(\mu^*_0 - \mu^a_0\big)\right)^2\right] \\
&= \Omega^{a}_0(\kappa_{T, Q}) + C( \mu^*_0 - \mu^a_0),
\end{align*}
where $C > 0$ is a constant.
\subsection{Specification of the Observed-Data Score Function}
\label{sec:specification-score}
According to Lemma~\ref{lem:upperbound_semipara}, we can conjecture that if we use the semiparametric efficient score function for our score function, we can obtain a tight upper bound for $\mathbb{E}_{P_0}[L^{a}_T]$, which is related to a lower bound for the probability of misidentification. Note that the variance of the semiparametric efficient score function is equivalent to the lower bound in Lemma~\ref{lem:upperbound_semipara}. However, we cannot use the semiparametric efficient score function because it is derived for $\overline{R}^{\kappa_{T, Q}\,\dagger}_{\bm{\varepsilon}}$, rather than $\overline{R}^{\kappa_{T, Q}}_{\bm{\varepsilon}}$.
Furthermore, if we use the semiparametric efficient score function for our score function, the constant \eqref{eq:const_ate} is not satisfied. Therefore, based on our obtained result, we specify our score function, which differs from the semiparametric efficient score function, but they match when $\tau\to \infty$.
We specifiy our score function $S^a= \mathbbm{1}[d = a^*_0]S^{a, a^*_0}_{f}(y|x) + \mathbbm{1}[d = a]S^{a, a}_{f}(y|x) + S^a_{\zeta}(x)$ as follows:
\begin{align*}
&S^{a, a^*_0}_{f}(y|x) = \frac{(\phi^*_\tau(y, x) - \mu^*_0(x))}{\kappa_{T, Q}(a^*_0| X)}/V^a_0(\kappa_{T, Q}; \tau),\\
&S^{a, a}_{f}(y|x) = \frac{(\phi^a_\tau(y, x) - \mu^a_0(x))}{\kappa_{T, Q}(a| X)}/V^a_{0}(\kappa_{T, Q}; \tau),\\
&S^{a}_{\zeta}(x) = (\mu^*_0(x) - \mu^a_0)/V^a_{0}(\kappa_{T, Q}; \tau),
\end{align*}
where
\begin{align}
&V^a_0(\kappa_{T, Q}; \tau) = \widetilde{V}^a_{0}(\kappa_{T, Q}; \tau) + \sum_{d\in\{a^*_0, a\}}\mathbb{E}_{P_0}\left[\frac{ \mu^*_0(X_t)\mathbb{E}_{P_0}[Y^d_t\mathbbm{1}[|Y^d_t| < \tau]|X_t] - \left(\mathbb{E}_{P_0}[Y^d_t\mathbbm{1}[|Y^d_t| < \tau]|X_t]\right)^2}{\kappa_{T, Q}(d| X_t)}\right]\nonumber\\
\label{eq:variance_form}
&=\mathbb{E}_{P_0}\left[\frac{Y^{a^*_0}_t\left(\phi^*_\tau(Y_t, X_t) - \mu^*_0(X_t)\right)}{\kappa_{T, Q}(a^*_0| X_t)} + \frac{Y^a_t\left(\phi^a_\tau(Y_t, X_t) - \mu^a_0(X_t)\right)}{\kappa_{T, Q}(a| X_t)} + \left(\big(\mu^*_0(X_t) - \mu^a_0(X_t)\big) - \big(\mu^*_0 - \mu^a_0\big)\right)^2\right].
\end{align}
Here, note that for $d\in[K]$,
\begin{align*}
\mathbb{E}_{P_0}\left[Y^d_t\left(\phi^d_\tau(Y_t, X_t) - \mu^d_0(X_t)\right)\right]&= \mathbb{E}_{P_0}\left[\left(\left(Y^d_t\right)^2\mathbbm{1}[|Y^d_t| < \tau] - Y^d_t\mathbb{E}_{P_0}[Y^d_t\mathbbm{1}[|Y^d_t| < \tau]|X_t]|X_t] \right)\right]\nonumber\\
&= \mathbb{E}_{P_0}\left[\mathbb{E}_{P_0}\left[\left(Y^d_t\right)^2\mathbbm{1}[|Y^d_t| < \tau]|X_t\right]- \mu^d_0(X_t)\mathbb{E}_{P_0}[Y^d_t\mathbbm{1}[|Y^d_t| < \tau]|X_t]\right].
\end{align*}
We note that $V^a_0(\kappa_{T, Q}; \tau) \to \widetilde{V}^a_{0}(\kappa_{T, Q})$ as $\varepsilon^a \to 0$ and $\tau \to \infty$,.
Then, we confirm that $g^a = S^a$ belongs to $\dot{\mathcal{R}}^a$. Under this score functions, the constraint \eqref{eq:const} is satisfied because
\begin{align*}
&\int \int y S^{a,a^*_0}_{f}(y| x) f^*_0(y| x)\zeta_0(x) \mathrm{d}y\mathrm{d}x - \int \int y S^{a,a}_{f^a}(y| x) f^a_0(y| x)\zeta_0(x) \mathrm{d}y\mathrm{d}x + \int (\mu^*_0(x) - \mu^a_0(x)) S^a_{\zeta}(x) \zeta_{0}(x) \mathrm{d}x\\
&= \frac{\int \int \frac{y\left(\phi^*_\tau(y, x; \tau) - \mu^*_0(x)\right)}{\kappa_{T, Q}(a^*_0| X)} f^*_0(y| x)\zeta_{0}(x) \mathrm{d}y\mathrm{d}x - \int \int \frac{y\left(\phi^a_\tau(y, x; \tau) - \mu^a_0(x)\right)}{\kappa_{T, Q}(a| X)} f^a_0(y| x)\zeta_{0}(x) \mathrm{d}y\mathrm{d}x}{V^a_0(\kappa_{T, Q})}\\
&\ \ \ + \frac{\int \left(\mu^*_0(x) - \mu^a_0(x)\right) \left(\mu^*_0(x) - \mu^a_0(x) - \left(\mu^*_0 - \mu^a_0\right)\right) \zeta_{0}(x) \mathrm{d}x}{V^a_0(\kappa_{T, Q})}\\
& = 1,
\end{align*}
where we used the definition of the variance \eqref{eq:variance_form}.
From the definition of the parametric submodel, we can also confirm that condition \eqref{eq:const_ate} holds for our specified $g^a$:
\begin{align*}
&\int \int y \left(1 + \varepsilon^a g^a(\phi^*_\tau(y, x), a^*_0, x)\right)\overline{r}_0(y, a^*_0, x) \mathrm{d}y\mathrm{d}x - \int \int y \left(1 + \varepsilon^a g^a(\phi^a_\tau(y, x), a, x)\right)\overline{r}_{0}(y, a, x) \mathrm{d}y\mathrm{d}x\\
&=\int \int y \left(1 + \varepsilon^a g^a(\phi^*_\tau(y, x), a^*_0, x)\right)\overline{r}_0(y, a^*_0, x) \mathrm{d}y\mathrm{d}x - \int \int y \left(1 + \varepsilon^a g^a(\phi^a_\tau(y, x), a, x)\right)\overline{r}_{0}(y, a, x) \mathrm{d}y\mathrm{d}x\\
&= \mu^*_0 - \mu^a_0 + \varepsilon^a.
\end{align*}
In summary, from Lemmas~\ref{lem;taylor_exp_semipara}, under our specified score function, we obtain the following lemma:
\begin{lemma}
\label{lem:expansion}
Suppose that Assumption~\ref{asm:bounded_mean_variance} holds. For $P_0\in\mathcal{P}$ and $Q\in\mathrm{Alt}(P_0)$,
\begin{align*}
\mathbb{E}_{\overline{R}_{0}}[L^{a}_T] = \frac{\left(\varepsilon^a\right)^2}{2V^a_{0}(\kappa_{T, Q}; \tau)} + o\left(\left(\varepsilon^a\right)^2\right).
\end{align*}
\end{lemma}
\subsection{Proof of Theorem~\ref{thm:semipara_bandit_lower_bound}}
\label{sec:final_step}
Combining above arguments, we prove Theorem~\ref{thm:semipara_bandit_lower_bound}. First, we derive a lower bound for the probability of misidentification as follows, which is refined later:
\begin{lemma}
\label{lem:semipara_bandit_lower_bound}
Under Assumption~\ref{asm:bounded_mean_variance}, for any $P_0 \in \mathcal{P}$ and $Q \in \mathrm{Alt}(P_0)$, any consistent strategy, and each $\epsilon \in (0, 1)$,
\begin{align*}
&\frac{1}{T}\left(\epsilon \log \frac{\epsilon}{1 -\mathbb{P}_{ P_0 }(\widehat{a}_T \neq a^*_0)} + (1 - \varepsilon) \log \frac{1 - \varepsilon}{\mathbb{P}_{ P_0 }(\widehat{a}_T \neq a^*_0)} \right)\leq \mathbb{E}_{Q}[L_T].
\end{align*}
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{lem:semipara_bandit_lower_bound}] For each $Q \in \mathrm{Alt}(P_0)$, $\mathbb{E}_{Q}[L_T] \ge \sup_{\mathcal{E} \in \mathcal{F}_T} d(\mathbb{P}_{Q}(\mathcal{E}),\mathbb{P}_{P_0}(\mathcal{E}))$ holds from Proposition~\ref{lem:data_proc_inequality}.
Let $\mathcal{E} = \{\widehat{a}_T = a^*_0\}$.
Because we assume that the strategy is consistent for both models and from the definition of $\mathrm{Alt}(P_0)$, for each $\epsilon \in (0, 1)$,
there exists $t_0 (\epsilon)$ such that for all $T \geq t_0 (\epsilon)$, $\mathbb{P}_{Q}(\mathcal{E}) \le \epsilon \le\mathbb{P}_{ P_0 }(\mathcal{E})$.
Then, for all $T \ge t_0(\epsilon)$, $\mathbb{E}_{Q}[L_T]
\ge d (\epsilon, 1 -\mathbb{P}_{P_0}(\widehat{a}_T \neq a^*_0)) = \epsilon \log \frac{\epsilon}{1 -\mathbb{P}_{ P_0 }(\widehat{a}_T \neq a^*_0)} + (1 - \epsilon) \log \frac{1 - \epsilon}{P_0(\widehat{a}_T \neq a^*_0)}$.
The following inequality holds from Lemma~\ref{lem:expansion}:
\begin{align*}
& \frac{1}{T}\left(\epsilon \log \frac{\epsilon}{1 -\mathbb{P}_{ P_0 }(\widehat{a}_T \neq a^*_0)} + (1 - \epsilon) \log \frac{1 - \epsilon}{\mathbb{P}_{P_0}(\widehat{a}_T \neq a^*_0)} \right)\leq \mathbb{E}_{Q}[L_T].
\end{align*}
the proof is complete.
\end{proof}
Then, by refining the lower bound in Lemma~\ref{lem:semipara_bandit_lower_bound}, we prove Theorem~\ref{thm:semipara_bandit_lower_bound}.
\begin{proof}[Proof of Theorem~\ref{thm:semipara_bandit_lower_bound}]
For the inequality in Lemma~\ref{lem:semipara_bandit_lower_bound}, taking the limsup and letting $\epsilon \to 0$,
\begin{align*}
&\limsup_{T\to\infty}-\frac{1}{T}\mathbb{P}_{ P_0 }(\widehat{a}_T \neq a^*_0)\leq \inf_{Q \in \mathrm{Alt}(P_0)}\limsup_{T\to\infty}\mathbb{E}_{Q}[L_T]\\
&\leq \inf_{Q \in \mathrm{Alt}(P_0)}\limsup_{T\to\infty}\sum_{a\in[K]}\mathbb{E}_{Q}\left[\mathbb{E}_{Q}\left[\log \frac{f^a_{Q}(Y^a_t| X_t)\zeta_{Q}(X)}{f^a_{0}(Y^a_t| X)\zeta_{0}(X)}|X_t\right]\kappa_{T, Q}(a| X_t)\right]\\
&\leq \sup_{w \in \mathcal{W}}\inf_{Q \in \mathrm{Alt}(P_0)}\sum_{a\in[K]}\mathbb{E}_{Q}\left[\mathbb{E}_{Q}\left[\log \frac{f^a_{Q}(Y^a_t| X_t)\zeta_{Q}(X)}{f^a_{0}(Y^a_t| X)\zeta_{0}(X)}|X_t\right]w(a| X_t)\right]\\
&= \sup_{w \in \mathcal{W}}\min_{a\in[K]\backslash\{a^*_0\}}\inf_{\substack{Q \in \mathcal{P}\\ \mu^*(Q) - \mu^a(Q) < 0} }\sum_{a\in[K]}\mathbb{E}_{Q}\left[\mathbb{E}_{Q}\left[\log \frac{f^a_{Q}(Y^a_t| X_t)\zeta_{Q}(X)}{f^a_{0}(Y^a_t| X)\zeta_{0}(X)}|X_t\right]w(a| X_t)\right].
\end{align*}
By using $\varepsilon^a = \left(\mu^*(Q) - \mu^a(Q)\right) - \left(\mu^*_0 - \mu^a_0\right) < - \left(\mu^*_0 - \mu^a_0\right)$ for the parametric submodel,
\begin{align}
&\sup_{w \in \mathcal{W}}\min_{a\in[K]\backslash\{a^*_0\}}\inf_{\substack{Q \in \mathcal{P}\\ \mu^*(Q) - \mu^a(Q) < 0} }\sum_{a\in[K]}\mathbb{E}_{Q}\left[\mathbb{E}_{Q}\left[\log \frac{f^a_{Q}(Y^a_t| X_t)\zeta_{Q}(X)}{f^a_{0}(Y^a_t| X)\zeta_{0}(X)}|X_t\right]w(a| X_t)\right]\nonumber\\
&\leq \sup_{w \in \mathcal{W}}\min_{a\in[K]\backslash\{a^*_0\}}\inf_{\substack{\varepsilon^a < - \left(\mu^*_0 - \mu^e_0\right)\nonumber\\
\forall b\in[K]\backslash\{a^*_0, a\}\ \varepsilon^b = 0} }\sum_{a\in[K]}\mathbb{E}_{\overline{R}_{\bm{\varepsilon}}}\left[\mathbb{E}_{\overline{R}_{\bm{\varepsilon}}}\left[\log \frac{f^a_{\bm{\varepsilon}}(Y^a_t| X_t)\zeta_{\bm{\varepsilon}}(X)}{f^a_{0}(Y^a_t| X)\zeta_{0}(X)}|X_t\right]w(a| X_t)\right]\nonumber\\
&\leq \sup_{w \in \mathcal{W}}\min_{a\in[K]\backslash\{a^*_0\}}\inf_{\varepsilon^a < - \left(\mu^{*}_0 - \mu^e_0\right)}\frac{\left(\varepsilon^a\right)^2}{2}\mathbb{E}_{P_0}\left[ \left(S^a(Y_t, A_t, X_t)\right)^2\right] + o\left(\left(\varepsilon^a\right)^2\right)\nonumber\\
&\leq \sup_{w \in \mathcal{W}}\min_{a\in[K]\backslash\{a^*_0\}}\frac{\left(\mu^*_0 - \mu^a_0\right)^2}{2V^a_{0}(w; \tau)} + o\left(\left(\mu^*_0 - \mu^a_0\right)^2\right)\nonumber.
\end{align}
Here, for $\inf_{\varepsilon^a < - \left(\mu^*_0 - \mu^a_0\right)}\frac{\left(\varepsilon^a\right)^2}{2}\mathbb{E}_{P_0}\left[ \left(S^a(Y_t, A_t, X_t)\right)^2\right] $, we set $\varepsilon^a = - \left(\mu^*_0 - \mu^a_0\right)$, which indicates a situation where $\mu^*(Q) - \mu^a(Q)$ is sufficiently close to $0$. Then, after $\mu^*_0 - \mu^a_0 \to 0$, by letting $\tau\to\infty$, we obtain $V^a_{0}(w; \tau) \to \Omega^{a}_0(w)$, which is the semiparametric efficiency bound in Lemmas~\ref{lem:upperbound_semipara} and \ref{lem:semipara_efficient}. The proof is complete.
\end{proof}
\section{Conclusion}
In this study, we considered BAI with a fixed budget and contextual information under a small-gap regime. Subsequently, we derived semiparametric lower bounds for the probability of misidentification by applying semiparametric analysis under the small-gap regime. Then, we proposed the Contextual RS-AIPW strategy. With the help of semiparametric analysis and a new large deviation expansion we developed, we showed that the performance of our proposed Contextual RS-AIPW strategy matches the lower bound under a small gap. We also addressed a long-standing open issue in BAI with a fixed budget; even without contextual information, the existence of an asymptotically optimal BAI strategy was unclear.
Because BAI with a fixed budget and without contextual information is a special case in our setting, we addressed this question. Furthermore, we demonstrated an analytical solution for the target allocation ratio, which has also been unknown for a long time. Thus, our study serves as a breakthrough in the field of BAI with a fixed budget. Our future direction is to develop BAI strategies for various settings, such as linear \citep{Hoffman2014,Liang2019,KatzSamuels2020}, combinatorial \citep{Chen2014}, and policy learning \citep{Kitagawa2018,AtheySusan2017EPL,Zhou2020}.
\bibliographystyle{asa}
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<h2 title="Interface SquidInput.KeyHandler" class="title">Interface SquidInput.KeyHandler</h2>
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<pre>public static interface <a href="../../../../src-html/squidpony/squidgrid/gui/gdx/SquidInput.html#line.35">SquidInput.KeyHandler</a></pre>
<div class="block">A single-method interface used to process "typed" characters, special characters produced by unusual keys, and
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<td class="colLast"><code><span class="memberNameLink"><a href="../../../../squidpony/squidgrid/gui/gdx/SquidInput.KeyHandler.html#handle-char-boolean-boolean-boolean-">handle</a></span>(char key,
boolean alt,
boolean ctrl,
boolean shift)</code>
<div class="block">The only method you need to implement yourself in KeyHandler, this should react to keys such as
'a' (produced by pressing the A key while not holding Shift), 'E' (produced by pressing the E key while
holding Shift), and '←' (left arrow in unicode, also available as a constant in SquidInput, produced by
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<pre>void <a href="../../../../src-html/squidpony/squidgrid/gui/gdx/SquidInput.KeyHandler.html#line.52">handle</a>(char key,
boolean alt,
boolean ctrl,
boolean shift)</pre>
<div class="block">The only method you need to implement yourself in KeyHandler, this should react to keys such as
'a' (produced by pressing the A key while not holding Shift), 'E' (produced by pressing the E key while
holding Shift), and '←' (left arrow in unicode, also available as a constant in SquidInput, produced by
pressing the left arrow key even though that key does not have a default unicode representation). Capital
letters will be capitalized when they are passed to this, but they may or may not have the shift argument as
true depending on how this method was called. Symbols that may be produced by holding Shift and pressing a
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<br>
This can react to the input in whatever way you find appropriate for your game.</div>
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<dd><code>alt</code> - true if the Alt modifier was being held while this key was entered, false otherwise.</dd>
<dd><code>ctrl</code> - true if the Ctrl modifier was being held while this key was entered, false otherwise.</dd>
<dd><code>shift</code> - true if the Shift modifier was being held while this key was entered, false otherwise.</dd>
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Zimbabwe's opposition MDC rejects SADC demands
The 15-nation Southern African Development Community (SADC) said in a resolution Zimbabwe's squabbling political parties should form a unity government immediately to end a stalemate over the allocation of ministries.
But opposition leader Morgan Tsvangirai said he was "shocked and saddened" by the outcome of a summit, which brought together leaders and ministers of SADC countries for more than 12 hours of talks on Zimbabwe's political impasse and the violence in eastern Congo.
"The MDC is shocked and saddened that SADC summit has failed to tackle these key issues … a great opportnity has been missed by SADC to bring an end to the Zimbabwean crisis," Tsvangirai said at a post-summit news conference.
SADC said Tsvangirai did not agree with SADC's call for his Movement for Democratic Change to co-manage Zimbabwe's Home Affairs Ministry with President Robert Mugabe's ruling ZANU-PF.
The resolution calling for joint control of the ministry — which controls Zimbabwe's police and is the main sticking point in the talks — was backed by all 15 members of SADC, said Arthur Mutambara, leader of a breakaway MDC faction.
The SADC said a unity government must be formed.
"We need to form an inclusive government, today or tomorrow," SADC Executive Secretary Tomaz Salamao told reporters late on Sunday night after the summit in South Africa.
"… SADC was asked to rule and SADC took a decision and that's the position of SADC. Now it's up to the parties to implement," he said.
REGIONAL INSTABILITY
Mugabe, in power since 1980, appeared optimistic that an agreement could be reached but Tsvangirai warned of regional instability if the ruling party refused to loosen what he called its illegitimate grip on power.
The old foes have been deadlocked over allocation of important cabinet positions since the September 15 deal, which Zimbabweans hoped would produce a united leadership to revive the ruined economy in the country where inflation is the world's highest and food and fuel shortages widespread.
Control of the Home Affairs Ministry has been one of the main sticking points in implementing the power-sharing deal.
Tsvangirai said co-managing the ministry with the ruling party was unworkable, citing the party's contempt for the MDC.
He said SADC lacked the "courage and decency to look Robert Mugabe in the eyes" and tell him his position was wrong.
Highlighting growing regional impatience, South African President Kgalema Motlanthe said earlier on Sunday the deal offered the only hope for Zimbabwe to ease the economic crisis.
Past SADC meetings have failed to produce a breakthrough.
Although some leaders have taken a tough line on Mugabe, political analysts say SADC does not have the resolve to impose tough measures, such as sanctions, to force an agreement.
The heads of state of Botswana and Zambia, the most outspoken regional critics of Mugabe, did not attend the summit.
Tsvangirai, who would become prime minister under the power-sharing deal, has accused Mugabe's ZANU-PF of trying to seize the lion's share of important ministries and relegating the MDC to the role of junior partner.
Zimbabwe's economic crisis has forced millions of its citizens to flee the country, many of them moving to neighbouring South Africa, Africa's biggest economy.
Zimbabwean state media reported that Mugabe's government would not change its stance on key cabinet positions and the opposition should accept joint control of the interior ministry. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,798 |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-find-the-critical-points-for-f-x-y-x-2-4x-y-2-and-the-local-max-and-m","text":"# How do you find the critical points for f(x, y) = x^2 + 4x + y^2 and the local max and min?\n\nApr 29, 2017\n\nThe point $\\left(- 2 , 0\\right)$ is a minimum for the function.\n\n#### Explanation:\n\nEvaluate the partial derivatives of the first order:\n\n$\\frac{\\partial f}{\\mathrm{dx}} = 2 x + 4$\n\n$\\frac{\\partial f}{\\mathrm{dy}} = 2 y$\n\nso the critical points are the solutions of the equations:\n\n$\\left\\{\\begin{matrix}2 x + 4 = 0 \\\\ 2 y = 0\\end{matrix}\\right.$\n\n$\\left\\{\\begin{matrix}x = - 2 \\\\ y = 0\\end{matrix}\\right.$\n\nWe have therefore a single critical point: $\\left(- 2 , 0\\right)$. To determine the character of the point we have to look at the Hessian matrix:\n\n$\\frac{{\\partial}^{2} f}{{\\mathrm{dx}}^{2}} = 2$\n\n$\\frac{{\\partial}^{2} f}{\\mathrm{dx} \\mathrm{dy}} = 0$\n\n$\\frac{{\\partial}^{2} f}{{\\mathrm{dy}}^{2}} = 2$\n\n$\\left\\mid H \\right\\mid = 2 \\times 2 - 0 = 4$\n\nso we have that:\n\n$\\left\\mid H \\right\\mid > 0$\n\n${\\left[\\frac{{\\partial}^{2} f}{{\\mathrm{dx}}^{2}}\\right]}_{\\left(x , y\\right) = \\left(- 2 , 0\\right)} > 0$\n\nthen the point $\\left(- 2 , 0\\right)$ is a minimum.","date":"2021-06-17 02:00:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 13, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6105497479438782, \"perplexity\": 248.33880433064124}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487626465.55\/warc\/CC-MAIN-20210617011001-20210617041001-00402.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.shaalaa.com\/question-bank-solutions\/how-many-three-digit-numbers-are-there-no-digit-repeated-factorial-n-n-permutations-combinations_53477","text":"# How Many Three-digit Numbers Are There, with No Digit Repeated? - Mathematics\n\nHow many three-digit numbers are there, with no digit repeated?\n\n#### Solution\n\nTotal number of 3-digit numbers = Number of arrangements of 10 numbers, taken 3 at a time =\u00a010P3 =$\\frac{10!}{7!} = 10 \\times 9 \\times 8 = 720$\n\nTotal number of 3-digit numbers, having 0 at its hundred's place =\u00a09P2 =$\\frac{9!}{7!} = 9 \\times 8 = 72$\n\nTotal number of 3-digit numbers with distinct digits =\u00a010P3$-$\u00a09P2\u00a0= 720$-$ 72 = 648\n\nConcept: Factorial N (N!) Permutations and Combinations\nIs there an error in this question or solution?\n\n#### APPEARS IN\n\nRD Sharma Class 11 Mathematics Textbook\nChapter 16 Permutations\nExercise 16.3 | Q 25 | Page 29","date":"2021-04-19 09:19:11","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.23959866166114807, \"perplexity\": 1877.452814287796}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-17\/segments\/1618038879305.68\/warc\/CC-MAIN-20210419080654-20210419110654-00340.warc.gz\"}"} | null | null |
Business Process Automation, or BPA, is the hottest structure to hit businesses across industries now.
In fact, according to the research firm, Gleanster, almost 80 per cent of all top-performing companies are driving marketing strategy and cutting down on repetitive and time-consuming tasks by using business automation system. Whether it's in the form of automated emailers, automated chat responses, automated exchange of information between two APIs, automated communication between your business and any Kiwi bank or IRD, it all works out for the best and is sure to speed up conversions for your business.
Consider this – A potential customer or client contacted you via email or chat (if you use any), but you are not available at the moment. Maybe you're in a different time zone altogether. In this case, you may be late to respond to the potential buyer. But with an automated system, we can set it up to immediately send an introductory email from you to your potential customer. This makes it appear as if you leapt into action almost immediately upon hearing your customer's call/message. Leaving them impressed with your speedy service. Whereas, the old-fashioned way of handling everything on your own means that this potential customer would've lost interest, forgotten about you, and moved on. You've lost a lead right there.
According to a study by Experian Marketing Services, welcome emails have an open rate of nearly 58 per cent. That percentage jumps to 88 per cent if the welcome email is sent within a few seconds. Whereas, promotional emails have an open rate of less than 15 per cent.
A different response can be set up for different scenario depending on what the customer clicks, or the keywords used in the chat system. Each case can be tailor-made to answer a customer's enquiry as if it were you. And all you had to do is follow the conversations via your phone if you want to. You only have to personally respond when you deem it necessary.
Nearly all big corporations use automated system – think UBER, pizza orders, online ticket booking – but, business automation isn't reserved just for big large companies. Small companies with even just about 2-3 employees too, can incorporate automation.
In one of our cases, we found that, in connecting a client with Inland Revenue, we've eradicated the need to import spreadsheets between Excel and Google Sheets eight times a day by simply automating this process.
But of course, some tasks simply should not be automated. These include creative thinking, content creation, design process, and most problem-solving material involving complex algorithm, coding, HTML development, and even testing. A human-touch is still needed for these aspects of a business in order for it to function properly, and possibly with a certain appeal.
With our experience in this field, Business Process Automation can be done in a very short time, but, it's okay to start small and expand it in time.
Most software comes with detailed API documentation, and some, even with examples of how to integrate these APIs. In layman's terms, "make them talk to each other".
While in some cases, it is possible to complete API integration within a very short time, in other's, one may need a higher and more expertise developer in finding a solution – in these cases, the developer may have to rewrite the code before integration.
This is where companies like Tandem NZ steps in, get one of our professional software developers to come up with a solution and fix the issue. Your platform will be stable and with increased efficiency.
Let's break it down to simple terms.
Imagine you stepped into a restaurant, ready to order from the menu, but the kitchen is the "system" where your order is made. So how do you place an order without having to step into the kitchen yourself? Simple. You call upon a waiter who will take your order, head into the kitchen, and come back out with your exact order. That waiter is the link between you and the system. In other words, that waiter is your API.
That is what we specialize in, integrating different APIs – between your business and the other businesses, between your organization and the banking system all over, between your company and IRD. Of course, this also includes integrating other software such as MYOB, XERO, and more into your platform, if that is your business requirement.
Do you want your website to give its users an option to sign up immediately via Facebook or Google? Hence skipping the long process of filing up forms? We can set it up by integrating both side's APIs and simplifying the registration process.
That sounds simple. Can I do it on my own?
While the answer to that is "yes, you can", you ought to keep in mind that most APIs nowadays need some security to keep it safe from bugs, crashing and malware. You basically need an API key, which can be bought, but some need to be developed from scratch, which we have the solutions for.
Not all kinds of integration are called API. Depending on their function, they may be named differently, but let's leave those technical details aside. Their end result is the same – Business Process Automation.
Integrated ERP – This type of integration typically involves running big organizations. Its automation involves handling Project Management Systems, Human Resource Management Systems, and Employee Management Systems to name them broadly.
Integrated HPI – This integration helps the user check the registration number of a vehicle and gives them a complete history of its use, past accidents involved in, mileage used, and more.
Integrated API – This integration enables automatic communication between the company owner and IRD. It can link employees & employers, make payments and send out payslips automatically, and a lot more. It also communicates with all banking systems so the employer can directly make payments.
Integrated API – Almost similar to Your Payroll above, but the communication is more with IRD platform such as registering the business, filing and paying taxes, filing tax returns directly, assigning tax codes to employees which are then hooked on to IRD's platform.
Integrated HPI – Vehicle history and ownership check for the Netherlands.
Integrated POS – machines for making payment via card.
Besides the above, most of our online work have integrations such as – Bank Payment Express, Social Media Plugins, Live Chat, Address Finder, Graph Plugin, as well as automated, scheduled and triggered email systems.
As a signing off, your business should not end with the customer making a purchase. That is actually just the start. Using BPA, after the first series of welcome emails to make them feel special, staying in touch is also important – whether it is to provide offers or business alerts, showing off new products/services, or just to give them information about your product or service, it all helps as a gentle nudge, a quick reminder. It is as effortless as blinking once the emailer is set up. Any activity or a purchase or just an enquiry will trigger the relevant email.
This allows you to keep in touch without remembering to do so, and while making you look like you're an expert who's on top of your game.
The probability of selling to an existing customer is 60 to 70 percent—compared with 5 to 20 per cent for a new prospect, according to the authors of Marketing Metrics.
Tandem NZ can help you to get your business process automated and increase revenue growth by cutting down manual work and processes. Contact us now. For more details on business process automation and web integration services, please visit our site right now. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,912 |
Creamfields 2014
17 years in & the Creamfields festival has grown in stature and quality to become the UK's premier biggest outdoor dance music festival, attracting 195k clubbers and dance music fans across it's three-day event. Last years festival won best Dance Festival at the UK Festival Awards & this year's the fastest selling Creamfields to date & is on course for it's 5th consecutive sell out. The festival has headline live acts confirmed including Avicii, Hardwell, Above & Beyond, Calvin Harris, Laidback Luke & Andy C to name a few.
New State Music & Cream are teaming up to release the Creamfields 2014 album and reflects the calibre of acts at the event across 2 x CD's.
The tracklist includes the headline performing artists & the biggest smashes across electronic dance music with a mix of big room anthems, uplifting trance, deep house and drum & bass mirroring the festival line up.
Album highlights include 5 UK no 1's from Martin Garrix – 'Animals', Ed Sheeran – 'Sing' (Trippy Turtle Mix) (1st time available on a compilation), Avicii – 'Hey Brother', Oliver Heldens ft. Becky Hill – 'Gecko', Duke Dumont – 'I Got U'. And 3 x Top 10's from Avicii's co-produced Coldplay – 'A Sky Full of Stars', Klingande – 'Jubel', Clean Bandit – 'Extraordinary' & Calvin Harris – 'Thinking About You'. Plus future hits from Lilly Wood & Robin Schulz – 'Prayer In C' & Hot Natured – 'Benediction'.
Creamfields 2014 is the essential dance festival album!
A New State release out now on 2CD & Digital
http://wwwcream.co.uk
http://www.twitter.com/officialcream
http://www.facebook.com/officialcream
http://www.newstatemusic.com
http://www.facebook.com/newstatemusic
http://www.twitter.com/newstateemusic | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,818 |
package org.rmatil.sync.core.messaging.chunk;
import org.rmatil.sync.network.core.model.Data;
import org.rmatil.sync.persistence.api.IFileMetaInfo;
import org.rmatil.sync.persistence.core.tree.ITreeStorageAdapter;
import org.rmatil.sync.persistence.core.tree.TreePathElement;
import org.rmatil.sync.persistence.exceptions.InputOutputException;
import org.rmatil.sync.version.api.AccessType;
import org.rmatil.sync.version.api.IObjectStore;
import org.rmatil.sync.version.core.model.PathObject;
import org.rmatil.sync.version.core.model.Sharer;
import java.util.Set;
/**
* Provides access to chunks of a particular file
*/
public class ChunkProvider {
/**
* The storage adapter to access files
*/
protected ITreeStorageAdapter storageAdapter;
/**
* The object store to fetch information about sharers, etc
*/
protected IObjectStore objectStore;
/**
* The path element from which to fetch chunks
*/
protected TreePathElement pathElement;
/**
* @param storageAdapter The storage adapter to access files
* @param objectStore The object store to fetch information of the files, like sharers
* @param pathElement The path element from which to get chunks
*/
public ChunkProvider(ITreeStorageAdapter storageAdapter, IObjectStore objectStore, TreePathElement pathElement) {
this.storageAdapter = storageAdapter;
this.objectStore = objectStore;
this.pathElement = pathElement;
}
/**
* Returns the chunk at the position of the file specified by
* the combination of chunk counter and chunk size.
*
* @param chunkCounter The chunk counter
* @param chunkSize The chunk size
*
* @return The requested chunk
*
* @throws InputOutputException If accessing the storage adapter or the object store failed
* @throws IllegalArgumentException If a chunk counter greater than the total nr of chunks is requested
*/
public Chunk getChunk(long chunkCounter, int chunkSize)
throws InputOutputException, IllegalArgumentException {
IFileMetaInfo fileMetaInfo = this.storageAdapter.getMetaInformation(this.pathElement);
int totalNrOfChunks = 1;
Data data = null;
String checksum = "";
if (fileMetaInfo.isFile()) {
totalNrOfChunks = (int) Math.ceil(fileMetaInfo.getTotalFileSize() / (double) chunkSize);
if (chunkCounter > totalNrOfChunks) {
// maybe the file has changed in the mean time...
throw new IllegalArgumentException("ChunkCounter must be smaller than the total number of chnks");
}
long fileChunkStartOffset = chunkCounter * chunkSize;
// storage adapter trims requests for a too large chunk
byte[] content = this.storageAdapter.read(pathElement, fileChunkStartOffset, chunkSize);
data = new Data(content, false);
checksum = this.storageAdapter.getChecksum(pathElement);
}
PathObject pathObject = this.objectStore.getObjectManager().getObjectForPath(pathElement.getPath());
String owner = pathObject.getOwner();
Set<Sharer> sharers = pathObject.getSharers();
AccessType accessType = pathObject.getAccessType();
return new Chunk(
checksum,
owner,
sharers,
fileMetaInfo.isFile(),
accessType,
chunkCounter,
totalNrOfChunks,
fileMetaInfo.getTotalFileSize(),
data
);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,949 |
To Clare, Arthur and Alfred Wood: a world of
wondrous stories would mean nothing without you.
## ACKNOWLEDGEMENTS
I AM IMMEASURABLY GRATEFUL to my wife Clare anyway, but she not only listened to me as I wittered and wibbled on about this book; she also took the children out so I could write, and fearlessly and methodically undertook the arduous first proofread of this book. I am very, very grateful to you, Clare. Also the greatest of thanks to my family and friends, my colleagues at Bishopsgate Institute and The History Press for their patience, support and trust while I took so damn long writing this book.
Thank you to my friends and those who helped with the book for their advice, guidance, research, encouragement, stories and ears to sound-off into: David V. Barrett, Jason Godwin, Simon Round, Neil Denny, Catherine Halliwell, Sarah Sparkes, Ross McFarlane, Paul Cowdell, Neil Transpontine, Vicky Hill, Caroline Oates, John Rimmer, James Clarke, Matt Brown, Lottie Leedham, Jeremy Harte, Joe McNally, Elizabeth Pinel, Alex Margolis, Mark Pilkington, Johnny Radar, Elizabeth James, Martin Goodson, Tom Oldham, Reena Makwanna, Richard Sanderson, Danielle Sutcliffe, Steven Barrett and those of you I have doubtless forgotten. Thank you, I could not have done it without you.
There is a bibliography at the back of this book, but writing and researching London's urban legends would have been far harder and a lot less fun without the writings of Rodney Dale, Jan Harold Brunvand, Michael Goss, James Hayward, Steve Roud, Antony Clayton and Barbara and David Mikkelson, as well as the editors, writers and contributors of the Fortean Times, Magonia magazine and the Folklore Society Newsletter. Thank you, thank you, thank you!
I owe a debt of gratitude to the London Metropolitan Archive, Royal Society of Architects Archive, the British Library and British Newspaper Library, Bishopsgate Institute Library and Archive, Guildhall Library and Transport for London's Corporate Archives, as well as Clare Norman at Lidl public relations, Tom Artrocker, Jo Tanner at Us Ltd and the Museum of London archaeology department for their patience.
Thank you to the London Fortean Society, the London Cryptozoology Club and the South East London Folklore Society who have let me indulge my obsession for these topics in public.
## CONTENTS | Title
---|---
|
Dedication
|
Acknowledgements
|
Introduction
one | London Phrase and Fable
two | The Hidden Insult
three | The Queen's Head and the Krays' Arms
four | The Genitals of London
five | Legends of Rock
six | New Legends as Old
seven | Legendary Landmarks
eight | The Suicidal Sculptor
nine | The Devils of Cornhill
ten | The Misadventures of Brandy Nan
eleven | Plague Pits
twelve | Subterranean Secrets
thirteen | The Corpse on the Tube
fourteen | The Stranger's Warning
fifteen | Nazis Over London
sixteen | Criminal Lore
seventeen | London Blades
eighteen | The Accidental Theft
nineteen | Concrete Jungle
twenty | The Fantastic Urban Fox
twenty-one | Where the Wild Things Are
twenty-two | Folklore and Fakelore
|
Bibliography
|
Copyright
## INTRODUCTION
* * *
Ghost and other horror stories, political and social commentaries, dirty jokes (hundreds of them!), black humours tales, episodes of revenge, and topical pieces which rely on the audience's shared reaction to AIDS, nuclear warfare, foreigners, etc.
Michael Goss answers his question 'What are urban legends about?' in the article 'Legends for Our Time' in the July 1987 issue of The Unknown.
* * *
THIS IS A book about urban legends and London. The brief for the book is a brilliant one: collect, share and attempt to interpret the funny, scary, filthy and bizarre contemporary legends weaving their way through everyday London life. I hope to have written a book that is about London and urban legends, as well as how they relate to each other. This is a book of London tales that finds strange stories and lost and bizarre truths amongst the folklore. This is a book that looks at urban legends using the capital city of the United Kingdom as a frame whilst not neglecting their ability to travel anywhere there are people, and their talent for adapting very quickly to their environment. It is true that urban legends are universal rather than local, but one way urban legends thrive is by their immediacy: they attach themselves to people and places. It is also true that the temperament of tale-tellers, their audience and the landscape they share shapes their legendary life. Some of the stories in this book are as synonymous with London as mash and liquor with your pie, as people who talk all the way through gigs and having to queue for the swings in the playground. If you are not familiar with the city, this book is a strange introduction, but it can still show you around. You would be just as likely to find yourself standing over a possible plague pit or under a forgotten church gargoyle as in the middle of Trafalgar Square or outside Buckingham Palace. Ideas about London are far more widely spread than ones on urban legends, so I shall spend a few more pages introducing those. But fear not, the city runs through this entire book.
Urban myths are thought of as untrue stories pretending to be true, which they partly are, but I have recently heard many a fallacy or falsehood being denounced as being merely an urban myth. This is not true: there is always some level of narrative within an urban myth or legend. I may have stretched the meaning myself here to include moral panics, delusions and hoaxes, but each of these carries a story within them or are delivered by a fear or belief with a narrative. Urban myths are the stories told by ordinary people to entertain and to communicate a truth, opinion or prejudice through a story. Just as fairy tales explain the dangers of going into the woods at night, tell stories of kings or princesses going to a market in disguise, or why a local rock looks a certain shape, an urban myth will explain the dangers of using the London underground at night, tells a story of a celebrity or princess going to a local pub or bar, or describes why a building or statue is a certain shape. Other urban legends are a mad idea that rocket through the public consciousness, a story just plausible enough to spread: rioters releasing animals from London Zoo; a green patch of land in an overcrowded city lying empty because it hides a deadly secret. Others are even more vague, like the idea that big cats (pumas and panthers) have escaped their rich owners or zoos, and prowl the edges of the city; that urban fox hunts are something that may be useful and real.
There is a mystery to these myths. I have not set out to solve these mysteries, but to offer (hopefully) informed suggestions as to how and why they came to be.
## Origin of Myths
In keeping with something as nebulous as urban myths, the origins of the phrase, along with the term urban legend, are not straightforward. Many believe that the American professor and writer Jan Harold Brunvald coined the phrase 'urban legend', and his books certainly helped popularise the phrase, but the credit could also go to American folklorist, Richard Dorson, who apparently used it in a 1968 essay 'Legends and Tall Tales' (in Our Living Traditions, edited by Tristram P. Coffin). Dorson is the earliest citation in the Oxford English Dictionary, but this does not make him the man who minted the phrase. Dorson used it himself in a 1962 article. Researching the origin in Foaftale News, Charles Clay Doyle and Lara Renee Knight found a New York Times article from 6 December 1925 regarding Europe's population growth: 'Around the subject of population there has been a growth of popular legend hard to remove. Great Britain illustrates the urban legend.' This described a myth of urban life: that it is unhealthy and squalid compared to rural living and is not used to describe contemporary legends and myths. The phrase is old, has multiple uses and its roots are hard to uncover.
Reading Michael Goss's article in the June 1987 small digest magazine The Unknown, the main aspect of urban myths that first captured my imagination was the idea that stories could migrate and adapt themselves as they travel. It was the first time I imagined stories with a life beyond their author. One could read a book written by a deceased writer, but it would still be in their book. But what if the writer was gone, the books, films and songs were forgotten, but somehow their story lived on and found an ever-changing existence travelling around the world?
I began reading magazines like The Unexplained and The Unknown as a precocious pre-adolescent looking for aliens stepping out of UFOs, ghosts drawing themselves out of ancient wallpaper and monsters lurking in the misty night but found something far more humane and fascinating that could contain all of these gaudy wonders. My fascination with the paranormal was enlarged and refined. Years later I was lucky enough to catch Jan Harold Brunvald speak at the Fortean Times gathering, the UnConvention, which reignited my interest.
When American psychologist F.C. Bartlett experimented with how stories change through retelling, his conclusions rang true for urban myths. Details difficult to repeat were smoothed out, and in Bartlett's test story, 'canoes' became 'boats' and 'bush-cats' became 'cats'. Any unusual parts of the story begin to be rationalised and morals formed either through this process or as a reason for telling the story. The experiment was short and contained to a small peergroup, while urban legends are feral and free, but it makes sense that stories survive, not just because they are entertaining but because they carry a central lesson or meaning.
As I have already said, the term urban myth and legend is now used to describe contemporary folk stories. One problem raised by this is that where there are gatherings of people, there are legends, myths and folklore; and London has been an urban environment for around two thousand years. If we were to find a story told by the Romans about a part of London life, would that be an urban myth? Classical myths and legends are often of gods, heroes and supernatural creatures, but urban myths also trouble themselves with royalty and celebrities and wander into the supernatural and paranormal to include ghosts, monsters and stories of miracles. They are not always stories of the common folk.
Other names have been suggested for these tales: Rodney Dale arguably wrote the first acknowledged urban myth book, without using the term, in 1972; The Tumour in the Whale suggested the phrase 'whale tumours', inspired by stories of rationing era whale meat being eaten as a substitute for beef, with its unusual status confirmed with wobbly growths. The phrase did not catch on, but Dale did bring together the phrase 'friend-of-a-friend' and abbreviate it to 'foaf' in order to describe the ever-apocryphal source of an urban myth. David J. Jacobson, in his 1948 book The Affairs of Dame Rumour, and Sir Basil Thomson, in his 1922 book Queer People, both encountered and understood the source of a story as always being just beyond arms' length – they are quoted later in this book describing the process in more depth. Another good term for these stories is the Swedish vandresagn, meaning 'wandering legends' that travel by people sharing stories. Sharing stories is as old as humanity and is still a powerful way for us to express our feelings and innermost thoughts, from epics to emails and campfires to Kindles.
Scott Wood, 2013
## 1
## [LONDON PHRASE
AND FABLE](contents.htm#ch3)
* * *
But the truth is this is not how London is apprehended. It is divided into chapters, the chapters into scenes, the scenes into sentences; it opens to you like a series of rooms, door, passage, door.
Anna Quindlen, Imagined London
* * *
WHY DO WE use that phrase? Why is that statue so strange? Why is that big stone there? Where is that actual place?
There is often a story to answer questions like this about a local landmark. It is as if a large physical object or popular idea deserves to have a neat narrative fixed to it. Once you start examining the origins of a myth or phrase, you can be led down strange alleys and cul-de-sacs chasing old stories and ideas. Things can become confused and leave your thoughts in some disarray or, to use the particularly apt phrase, 'at sixes and sevens'.
The phrase 'at sixes and sevens' is said to have a London origin and refers to a feud between the Merchant Taylors and Merchant Skinners livery companies. Both were founded in the City of London around the same time, so they argued about who should come sixth and who should come seventh in the Order of Precedence, a list of London livery companies organised by age collected from 1515.
'To fall off the wagon' means to succumb to the temptation of alcohol; to be on the wagon is to not drink, so to fall off is to start drinking again. One possible origin of this saying, much-loved on The Robert Elms Show on BBC London Radio, is from when prisoners were taken from Newgate Prison to Tyburn to be hanged. Halfway to their execution there would be a stop where the condemned could have a last drink. One origin of this tradition could be that a 'cup of charity' was bequeathed by Queen Matilda (wife of Henry I). The prisoner would get off the wagon at a tavern at St-Giles-in-the-Fields and have a pint of ale in the cup, and then get back on the wagon to go to Tyburn. He would never drink again. Another version, collected by Snopes, has the last stop at Marble Arch, right by the site of the gallows. A retelling from the Nursery Rhymes: Lyrics, Origins and History website puts a line of dialogue into the story. If the prisoner was offered a second drink, the guard would say, 'No, he's on the wagon.' If they had friends in the crowd they would, perhaps, be pulled off the wagon and rescued. This is falling off the wagon.
The word 'tawdry', which means something that is cheap, low quality and maybe makes you a bit sad to give or receive as a gift, also has its origins in London. St Ethelreda's Church on Ely Place, London's oldest Catholic place of worship, is near London's diamond and jewellery centre, Hatton Garden. I was told by a trustworthy source, a Blue Badge guide no less, that the church gave us the word from the poorer quality trinkets from the area which was said to be a bit 'St Audrey', another version of the name Ethelreda.
All of these phrases make sense on their own, hermetically sealed in their own story. Outside of these are far different possible origins. 'At sixes and sevens' bumbles all the way back to the Old Testament with 'six, yea, seven', meaning an indefinite number and so is unknown and confusing. In the Book of Job is the line that God 'shall deliver thee in six troubles, yea in seven', and it is likely that the Bible has more influence over popular culture than London's livery companies and their lore. Another biblical origin is the story of an error in the King James Bible, in which the sixth commandment is 'Thou Shalt Not Kill' and the seventh 'Thou Shalt Not Commit Adultery'. In the Septuagint version, not committing adultery comes in sixth and the seventh is 'Thou Shalt Not Steal'. Which puts bible scholars at sixes and sevens. Another non-biblical origin is from the French game played with dice called Hazard, where six and seven are the most hazardous numbers to shoot for and anyone attempting it is thought to be careless or confused.
Does tawdry come from St Ethelreda? Perhaps, but not the one in London. St Ethelreda's sits within Ely Place, the site of the palace for the Bishop of Cambridge, and the church started life as the palace's private chapel. An earlier version is that the word comes from St Audrey lace sold on at the fair of St Audrey on the Isle of Ely in Cambridgeshire.
The phrases 'on the wagon' and 'fall off the wagon' evolved probably not along the streets between Newgate and Tyburn but in America. The earliest known version appeared in the 1901 book Mrs Wiggs of the Cabbage Patch, the phrase referring to the water wagons used in America to dampen down dusty roads. The American Temperance movement formed the phrase to describe someone who is not drinking. They felt so strongly about the sinfulness of alcohol that they would rather drink water from the water wagon than let alcohol pass their lips, although now I write it down this explanation sounds just as implausible as the Tyburn theory. The excellent online resource, Snopes, suggests that 'on the wagon' is a derivation of 'following the bandwagon'. The bandwagon is a phrase coined, as far as we know, by the American showman P.T. Barnum to describe what his shows travelled around in, and 'to jump on the bandwagon' means to follow or join the fair. The word only goes back as far as the nineteenth century, with 'on the wagon' still coming to us from the members of the Temperance movement. Any connection to the journey to Tyburn was almost certainly retro-fitted in the twentieth or twenty-first century as speculation made story, or by joining the dots of distance and unrelated ideas to make a pleasing narrative pattern. We can't help but do it; it's impossible for us just to shrug and say 'I don't know' when wondering about such a thing, so we write stories to explain the mysterious origins of phrase and fables.
## 2
## THE HIDDEN INSULT
* * *
I like the idea of infiltrating an area that is
not really exposed to me or my work.
Alexander McQueen
* * *
## Royally Rude
When London-based fashion designer Alexander McQueen was found dead on 11 February 2010, the response was one of shock and grief. His suicide was all over the media and the loss was felt by even the scruffiest of Londoners.
Remembering him in the 2010 obituaries in the 12 December edition of the Observer, Harriet Quick, fashion features director at Vogue, described his collections as 'wildly imaginative' whilst McQueen was a 'shy, sensitive man'. Quick suggested that a McQueen fashion show had such power it could actually affect nature, remembering that 'his shows were frequently accompanied by freak weather', and she describes driving through a hurricane to see a show in New York in 2000. In past times and other lands the sky threatens at war, disaster or the death of a monarch or beloved leader. Alexander McQueen only needed to showcase some expensive clothes that most people could not wear for storms to descend.
Another reason that McQueen was so loved was his background as a working-class east London boy who got to the top of fashion because of his skill, talent and vision. During the remembrance for him in the media a story emerged about a prank McQueen had pulled whilst an apprentice at Savile Row tailors Gieves & Hawkes. The young designer found himself making a jacket (or suit) for the Prince of Wales and could not resist the temptation to put a message within the lining of the jacket. The Radio 4 obituary show Last Word broadcast on 14 February 2010 described the story of McQueen writing 'McQueen was here' in the royal jacket as apocryphal, and his BBC online obituary recounts the story but puts a 'reportedly' ahead of it. Another version, alluded to on the London Design Museum website, is that McQueen actually wrote 'I am a c***' inside Prince Charles' suit. This is much celebrated on the internet, and in the 'Dressed to Thrill' column of the New Yorker on 16 May 2011, Judith Thurman moved the scene of the crime to McQueen's first apprenticeship at Anderson & Sheppherd and reported that the tailors recalled every jacket made for Prince Charles because of the alledged message.
* * *
These stories celebrate McQueen as an enfant terrible of the fashion world, slyly calling the establishment that fed him a c***. Like the world of artists and rock stars, the world of fashion designers is never knowingly under-mythologised and such a story feeds the myth. The swearword jacket story echoes another urban legend I heard about a car belonging to the Queen. Back in the early 1990s, I was working in a warehouse and heard a story in the staff canteen (a comfier and earlier version of the work water cooler as a place to share stories). One car – maybe more than one Rolls-Royce – constructed for the Queen had pornographic magazines secretly hidden in the body work. As well as being another rumour of royalty unwittingly carrying around obscenities, this urban legend also nods to a legend about the cars themselves. Rolls-Royce cars often glided through the popular imagination in the 1970s and '80s and, as with anything, stories followed in their wake. The 'Rolls of legend,' wrote Rodney Dale in The Tumour in the Whale, 'has a sealed bonnet, which must never be opened except at the factory.' Perhaps knowing this legend inspired some factory workers to leave something inside the monarch's car.
There is a more fully formed, muscular American version of the car urban legend picked up by Jan Harold Brunvand in his book The Choking Doberman and Other 'New' Urban Legends. It literally delivers the message of class war in the narrative. A 'wealthy professional man' has ordered a new Cadillac, which is perfect except for a persistent rattling sound when the car is driven, particularly over railway tracks or a bumpy street. After a number of check-ups, the man has the car deconstructed piece by piece and a bottle or tin can, hidden in the body work, is found to be the culprit. Within the recepticle there is an insulting note that reads 'You rich SOB – so you finally found the rattle!'
## Royalty to Celebrity
The Prince of Wales is not the only affluent jacket wearer to be secretly insulted by a sly tailor. Popbitch, the celebrity gossip newsletter and website, told the story in its 2 July 2009 email of footballer Joe Cole's 'beautiful bespoke suit for his wedding'. Cole had recently left West Ham to join Chelsea, and 'someone involved in stitching up the suit' was a West Ham supporter. Knowing the suit was for Cole, the supporter chalked a full West Ham insignia on the lining and 'a few choice words', including 'Judas'.
Football folklore has its own traditions of hidden insults. Scottish memorabilia and tartan scarves can be found under the dugouts and turf of the new Wembley stadium, left at the heart of English football by Scottish construction workers. Outside of London, a similar story is told of the construction of Southampton's new football ground, with some of the builders supporting local rivals Portsmouth. Three football shirts are buried beneath the turf, inscribed bricks are buried in the foundations and seeds were planted in the centre circle which would have, at some point, sprouted to spell out 'Pompey', Portsmouth's nickname. The seed story reminds me of another legend I have heard, set during and after the Second World War, which takes us from football fans to Nazis. A German prisoner of war distinguished himself as a gardener at the English manor house where he had been put to work. The war ended and the time came for the POW to return home, and it was apparently with much sadness that he parted company with the people of the manor house. This sadness was jarred the following spring when a swastika of daffodils sprouted up on the lawn. As we shall see regularly throughout this book, urban legends, like flowers, are often seeded from older growths.
The hidden insult has travelled from royalty to celebrity. This is not surprising considering pop stars, sport stars and television personalities are our newest form of rich aristocracy and are, at present, even more in the public eye than royalty. In the 1990s the rising stars of the Britpop movement – Blur and Oasis – developed a bitter rivalry, sparked off, partly (these things are messy), by Blur releasing the single 'Country House' on the same day (14 August 1995) as Oasis's single 'Roll With It'. Egged on by the media, the race to No. 1 became a class war between the northern working-class lads of Oasis and the southern, middle-class art students of Blur, which peaked with Oasis's songwriter Noel Gallagher wishing death by AIDS on Blur's lead singer Damon Albarn. Once again we refer to the Popbitch newsletter, this time from 21 August 2009. It contained the story of Oasis buying a vintage EMI TG mixing desk from a studio in Australia. A 'famous record producer' heard the band were buying it and carved the name of their rivals Blur on the inside of it. Popbitch reported that the producer said: 'He's always wondered if the Gallaghers ever found his handiwork.'
Back in London, and hidden insults are not only aimed at football teams but whole sporting events. On 28 February 2012, the 150-day mark before the start of the London Olympics, 37ft high, 82ft wide Olympic rings were floated up the Thames. These iconic symbols were to be paraded and displayed all around London to herald the Olympics, at a reported cost of £3.2 million. This generated comments from within London of the cost when the city, along with the rest of Great Britain, was suffering from unemployment, crime and rioting, pay cuts, pay freezes and funding cuts to the arts and libraries. The fanfare of the Olympics and its expense seemed garish and crude in comparison. 'There were better things to have spent this money on,' said the Green Party's 2012 mayoral candidate Jenny Jones. This attitude remained until Danny Boyle's opening ceremony, and a lot of talented Britons achieving medals finally won over even the curmudgeonliest of taxpayers. So, perhaps it was not surprising that the 22 March 2012 edition of Popbitch had the following story: 'We're told that there's something special about one of the rings. Someone involved in their construction had a bit of a downer on the whole Olympics in London thing. So he took a shit inside one of the rings. And then had it welded shut.'
## Finding the Hidden Insult
To be completely fair to Popbitch, they are not some golly-gosh, typo-ridden vacuous scandal rag. The newsletter often writes intelligently about the nature of celebrity and of the media, and they have a passion for music that steps outside the present pop charts. When repeating these urban legends, they are frequently in the company of the BBC, the Guardian and the Design Museum websites. Popbitch have shown themselves to be urban legend savvy when repeating the story of the rich premiership footballer paying off a couple's mortgage so he could take their booking for the wedding venue of his choice. 'It's one of the oldest shaggy dog stories going', they wrote in their 12 April 2012 newsletter before going on to reference an old legend of a female actor and television personality famous for doing on someone's chest, or onto a glass-topped coffee table, what some disgruntled person did into one of the giant Olympic rings. The next week they even produced a desperate email saying the children of the celebrity have 'worked their arses off' to keep her off the internet for fear that she might stumble across the mortifying rumour.
Seeking the hidden insult may provoke similar responses: who does one ask about a rumoured swearword or stool hidden in a public place or on the back of a public figure? I have wondered about contacting Clarence House, the seat for the Prince of Wales, but perhaps they too are working very hard to keep this rumour away from his royal ears. I would not wish to be instrumental in His Royal Highness glaring at his old jackets as his butler dresses him.
The McQueen story's popularity after the designer's death prompted his old company Anderson & Sheppard to issue a statement about the rumour on their blog. It denied the possibility of McQueen writing an insult into a suit or jacket:
Alexander McQueen joined here in 1984 or 1985. He didn't have an introduction I don't think, he just came in to apply in person. The firm's policy at the time was to take young people who had not been to college as they were easier to train. Sixteen was a typical age for apprentices joining the firm.
He worked under a tailor called Cornelius O'Callaghan – one of the best coatmakers that the firm had. Cornelius was known as Con, and was the strictest tailor at the firm. He checked his apprentices' work thoroughly. Despite rumours that things were scribbled inside the lining of a coat for Prince Charles, Con wouldn't have permitted that. And in fact the coat was recalled and checked after the story came out – nothing was found.
The story of the insulting objects hidden inside the Queen's car is true. The car was a Jaguar rather than a Rolls-Royce, and pornographic magazines and a swastika were found behind a seat panel when the car was being bomb-proofed. These, perhaps, were the most offensive things the prankster could think of, as are all of the hidden insults discussed. Even a football shirt hidden in the foundations of a rival's grounds can be viewed as a deadly insult. Stories appeared in the press in June 2001, and one in the Buckingham Post dated 14 June also referenced the McQueen story as 'McQueen Woz Ere' rather than 'was here'.
An unnamed Jaguar employee is quoted as saying; 'It is one of those old traditions where people used to write things behind the seat panel of cars and they were never discovered unless there was an accident. But on this occasion it was not very funny.' The discovery of the magazines and swastika resulted in the dismissal of an unnamed worker. Another unnamed source at the Coventry Evening Telegraph of 13 June 2001 described the insult as a regular prank by apprentices: 'I have never understood if it's for good luck or what, but the person knows the owner of the car will never see it. This one came to light, but normally they never do.'
The story of McQueen's Savile Row prank is set in his early days, perhaps during his apprenticeship at Anderson & Sheppard. Apprentices have rituals and rites and, by their nature, apprentices are young and irreverent. Is there a tradition, particularly when producing something like a car or jacket for rich customers, of hiding something offensive within it?
Unlike the Queen's car, the McQueen story does not have any evidence for it so far. The stories have generally been passed on informally. The Daily Telegraph, writing enthusiastically about Prince Charles' fashion sense on 13 June 2012, claimed that the original was written in the lining of an overcoat by McQueen at Anderson & Sheppard, while a 2011 feature on the Vogue website, dated 11 May 2011, puts the scene of the fashion crime at Gieves & Hawkes, stating that McQueen was embroidering a suit, not making a coat. I have not been able to find a definitive interview with Alexander McQueen in which he states that there is any truth to the rumour. It does not seem like anyone else is referring back to an original article either, as the versions vary so much. There may be one out there somewhere, but the popularity of the myth of this hidden insult is because it perfectly encapsulates who Alexander McQueen was and how he did things.
It is always the underdog that leaves the insult, never a privileged bully hiding a 'kick me' sign on the back of an employee or minion. Even the Oasis v. Blur story relates to a time when middle-class Blur were the chart underdogs to the 'champagne supernova' of Oasis's success. Oasis' album at the time '(What's the Story) Morning Glory' spent ten weeks at No. 1 and sold 16 million copies. Apparently, it was not a member of Blur who hid the message in the desk but a studio producer.
Where is the reality in this? Urban myths often create more questions the deeper you look into them, but each question leads to a truth about our own selves and fears even if they lead away from any actual event. Is the hidden insult regularly concealed on the property of the prosperous by an insubordinate? By its very nature it is hard to tell. There is something known as 'ostentation in folklore'. This describes people hearing a folk story or urban legend and, by copying it, making the story actually real. Had stories of the Queen's car and Prince Charles' jacket inspired a cheeky studio worker and a fed-up artist constructing a giant Olympic monument? And who wants to open up these objects and check?
## 3
## THE QUEEN'S HEAD AND THE KRAYS' ARMS
* * *
Up the stairs to the balcony where King Edward VII, so the foreman told me, liked to have his chair to watch the dancers on the floor below.
Geoffrey Fletcher, The London Nobody Knows
* * *
LONDON IS A city riddled with royalty, with statues of monarchs popping up in all manner of unexpected places (See 'The Suicidal Sculptor' here and 'The Misadventures of Brandy Nan' here for more on those) and the hulking presence of Buckingham Palace at the edge of Green Park is a reminder of our present royal incumbents. From newspapers we know what the younger royals (mostly Prince Harry) like to get up to in the evening, but what about the Queen herself? How does she occupy herself when off duty?
The general idea, I think, is of Her Majesty sitting on a gilded seat watching EastEnders with Prince Philip muttering next to her in his dressing gown. Another suggestion, picked up by Rodney Dale in The Tumour in the Whale, came from a friend-of-a-friend who knew an under-footman at Buckingham Palace who said there is a secret side door and that late at night the Queen emerges and secretly goes window shopping around Piccadilly, Bond Street and Oxford Street.
Stories of royals among us are as old as England itself, and I'm sure everyone around my age remembers the Ladybird book with the image of King Alfred burning the cakes (or loaves) he was asked to mind by a peasant woman. The peasant woman scolded the incompetent kitchen help, without realising it was the king. Alfred was in disguise after fleeing to the Somerset Levels to hide from the Danes.
A more recent rumour was of the myth-magnet Diana, Princess of Wales sneaking out of Kensington Palace in a baseball cap and shades to visit the local newsagent, or simply walk along the high street unharassed. Other word-of-mouth stories had her going out clubbing in a dark wig. In his book A Royal Duty Paul Burrell described buying a long, dark wig and large glasses so that Diana could have a night out in Ronnie Scott's Jazz Club in Soho. She even chatted up the man standing next to her in the queue who, she said to Burrell, didn't have a clue who she was.
Paul Burrell's role as the only man Diana trusted makes these stories difficult to verify and there is a whiff of myth about them. The meaning of them, like the Queen going window shopping, is handily given to us by Diana, marvelling at the freedom of disguise, when she said, 'I can be me in a public place!'
Alfred was in isolation and pondering his fate when the cakes burnt, like Robert the Bruce when he was inspired by a tenacious spider. His story, unlike Diana and Queen Elizabeth's, shows how different he was to ordinary citizens; he was pondering the fate of his nation as the baked goods burned, while the twentieth-century tales suggest that royalty can yearn for something that resembles normal life. Another story of royal otherness, as well as enforcing the ancient advice of always being polite to strangers, is of King James and the tinker. It's a ballad that tells of the king slipping his retinue whilst hunting to go 'in hope of some pass time'. Like a lot of unsupervised men (and incognito Princesses of Wales) with time on their hands, James went to an alehouse and fell into conversation with a tinker over a beer or two. After a while the tinker let slip that he'd heard the king was in the forest, so James got him to jump up on his horse so they could find him. They found James's entourage, and the tinker asked which one was the king. James said it was the only one with a hat on, which was he, and the poor tinker fell to his knees to beg for forgiveness. The king knighted his new drinking buddy, who kept his sack of tinker-tools hanging up in his new, grand hall. The location of the story varies; some claim it is Enfield in north London, where there is a King and Tinker pub that commemorates the story if not any actual event. Norwood in Surrey also claims the story, and there are other, similar stories told about different monarchs in Tamworth and Mansfield.
A recurring theme with these legends is that celebrity increasingly replaces royalty as the subject of the story. When the eccentric and much-loved New Cross pub, the Montague Arms, closed in early 2012, the local blog 'Transpontine' asked for readers' reminiscences of nights and events there. The pub was famous for the blind keyboard player who played cover versions to bemused locals, and coach parties on their way in or out of London. Pete, the keyboard player, would invite members of the audience (including this author) up on stage to sing. One response to Transpontine's request began at this point and was told to one contributor by the pub's former barman, Stan: 'This funny fellah wearing white gloves took to the key board and played the most amazing tunes – 'twas like magic running through his fingertips...'
Who was it? None other than Mr Michael Jackson!
## Criminal Tourism
Michael Jackson and Ronnie and Reggie Kray may not have too many things in common, but all three were said to have visited the Montague Arms. The visit, like the description of a similar visit by the gangsters to Peter Cook's club The Establishment, treats their appearance as almost a celebrity endorsement rather than a demand for money.
London's most famous criminals, from Dick Turpin to the Kray twins, have taken on a legendary status different to the rest of the stories in this book. They are folk heroes who are celebrated for their rough individuality and rule-breaking, and are even thought to be protectors of the common man. One warm Friday night in July 2012, I passed the bus stop opposite Shoreditch Town Hall and a woman, appalled by the hipsters and trendies swaggering past her along this East End street, shouted: 'If only the Krays were still here. They'd sort this out!' Murdering, bully gangsters are now the protectors of the common people who would keep the fey and pretentious out of east London, a vigilante fashion and lifestyle police. I think the attraction to organised criminals like the Krays, the Richardsons and Dick Turpin is that they live successfully by their own rules and not according to the limitations of bureaucracy, government or corporate values. It is imagined that they then become an informal sheriff of their area, not tolerating any crime other than their own.
Whilst not as popular as nearby Jack the Ripper tours, Kray tours have been written so visitors can see the sights of their crime spree. The Blind Beggar pub in Bethnal Green is where Ronnie Kray murdered rival gangster George Cornell. Tourists regularly arrive, looking for the bullet holes from the murder. They should be directed to the Magdela pub on South Hill Park near Hampstead Heath station. Here, a drunk Ruth Ellis shot her boyfriend David Blakely on 10 April 1955 as he left the pub, famously making herself the last woman to be executed in Britain. The four or five bullet holes have been visible since and are regularly referred to in many pub guides. Pubs are, of course, commercial enterprises and keen to use any means to bring people to the site. This could be a ghost, an historical artefact or a story with the evidence for all to see, like a bullet hole, a modern version of the indelible bloodstain that testifies to an ancient murder.
The holes from Ellis' gun are said to be visible in the white wall of the pub and a plaque was hung by them, explaining what the pock-marks on the building are. The plaque has since been stolen or removed; it got the year of the murder wrong, saying that Blakely was shot in 1954. It has been suggested that these marks were enhanced by a previous landlady and may not be linked to the murder at all.
Crime tourism is not new in London. Dick Turpin is one character who seemed to drink and take shelter in pretty much every pub across London except the ones Claude Duval, the Dandy Highwayman, drank in. The London pub most closely associated with Dick Turpin is the Spaniards Inn, which once boasted knives and forks used by him, as well as a small window where the highwayman could be aided and abetted by pub staff who would pass him food, money and drink while he was still in his saddle. Old and New London describes the Coach and Horses pub in Hockley in the Hole, now Ray Street, where a valise marked 'R. Turpin' was found in the cellars along with blank keys used for lock-breaking. Also at the Coach and Horses, still on Ray Street and a backstreet, was said to be a passage from the pub cellar that lead out to the banks of the Fleet river which was used by highwaymen or, as the book calls them, 'minions of the moon'. Turpin also left another unholy relic, a pistol engraved with 'Dick's Friend' in the rafters of the Anchor Inn in Shepperton and another within the walls of Ye Old King's Head in Chigwell.
## Tunnel Visions
There are countless pubs that claim a link to Dick Turpin, but the Dick they are referring to is the romantic, fictional figure and not the actual Richard Turpin, a thuggish burglar and thief. Like the Ellis shooting, the Old Red Lion pub on Whitechapel High Street has a plaque stating 'This is the Old Red Lion where Dick Turpin shot Tom King' after the murder committed here on 1 May 1737. Inside the pub was another plaque with the following inscription: 'It was in the yard of this house that Dick Turpin shot Tom King. Turpin had been traced by the horse to this inn, together with Matthew and Robert King, birds of like feather, by the Bow Street Runners.'
Antony Clayton in Folklore of London says of this plague: 'Apart from the fact that it was Matthew King who was shot, that Matthew's brother's name was John and not Robert and that the Bow Street Runners were founded in 1750, after Turpin's death, this sign was accurate.'
Other London pubs claiming a link to Turpin include the Spotted Dog on Upton Lane, and the Black Lion on Plaistow High Street, with tunnels extending 'over half a mile to emerge very close to Upton Park football ground', which Turpin would scuttle down after stabling Black Bess. Chigwell's Old King's Head has a tunnel that Turpin used to escape from the cellars, presumably after stashing his guns in the wall (Turpin did foolishly risk incriminating himself by signing his equipment). Turpin hid in the Globe Tavern on Bow Street for three days, and the temptation of this legend couldn't resist having him being pursued again by the non-existent Bow Street Runners.
Tunnels for Turpin's escape and stables for Black Bess's rest multiply as often as someone thinks of Turpin or visits an old pub for a drink. There were many more criminals and gangsters in London than the Krays, Richardsons and Dick Turpin, but the further the past gets, and the more romanticised these criminals become, it will be the most famous names that will live on in legend. By 2113 there will pubs called 'The Reggie Kray' or 'The Jack the Ripper' in the East End that will show places these folk heroes killed, or hid, or escaped down a secret tunnel.
Highwaymen are not the only historical celebrities to use secret tunnels, however. Royalty and aristocracy had the means to construct tunnels to cover their clandestine indulgences.
Legends of tunnels and the famous are insistent things that cannot help but insinuate themselves into a discovery. The Argyll Arms on Argyll Street is named after the Duke of Argyll. 'Rumour has it', the pub's website says, 'that a secret tunnel once connected the pub to the duke's mansion.' When staff at Wimbledon Park Golf Club discovered a tunnel in February 2012 the newspaper headline was 'Mysterious tunnels could link golf course with Henry VIII's Wimbledon home', though I think Henry was more a hunting, archery and wrestling man. His daughter, Elizabeth I, shimmied out of the Tower of London during her incarceration in 1554 to take wine in the nearby, but now long-gone, Tiger Tavern. She is also said to have stopped at the Tiger for a drink before heading to Tilbury to speak with the troops before they met the Spanish Armada, and to have used a secret passage that runs from the Old Queen's Head pub in Islington to Canonbury Tower to meet in secret with the Earl of Essex; not that Essex ever lived in Canonbury Tower.
Pubs are public places, a neutral ground with alcohol and comfy chairs and so are ideal places to meet old friends and new people. If urban legend is to be believed, then the great and good of London history were just as keen on a liaison in the pub as highwaymen and gangsters. The Nell of Old Drury has a secret passage running under the road which Charles II used to visit Nell Gwynne. The pub wasn't named after her then, being known at the time as The Lamb. The Red Lion at No. 23 Crown Passage has a tunnel, according to legend, running to No. 79 Pall Mall which Nell used to meet Charles in the pub. When Antony Clayton, an expert on underground London, inquired with the landlady in 2007, he was told of two doors in the cellar facing south in the direction of Pall Mall. (Are they 'his' and 'hers'?) When the Pindar of Wakefield on Gray's Inn Road – now Water Rats – was rebuilt in 1878, an underground tunnel was found heading in the direction of Bagnigge Wells, a pleasure garden where Nell and Charles met up. I've heard speculation that nearly every pub in London with the name Nelson in it was either a place Nelson and Emma Hamilton met, or was started by a wounded sailor pensioned out of the Napoleonic Wars with enough money to start a pub. Attentions in the pub are not always welcome: there is a story of Shakespeare being a regular at the George Inn on Borough High Street and catching the attention of a barmaid. One day Will was in the pub with the keys to the Globe on his person, when the barmaid grabbed the keys and placed them in her cleavage along with the key to her own room, asking the bard which set he desired.
When not meeting for a date in an inn or tavern, the famous did enjoy a drink. Charles Dickens had a reputation for being a furious drinker and countless pubs claim him as a regular, as they do Dr Johnson. Several pubs claim that Christopher Wren ordered them built in order to water the workers building St Paul's Cathedral: amongst those claiming this association are The Salutation on Newgate Street, now gone, and the Old Bell on Fleet Street. Ye Old Watling on Watling Street also claims to have been built for St Paul's workers as well as having an upstairs room in which Wren worked during the project.
Another result of a royal visit is an ordinary place being given a special licence. In the rural areas this could be a passing king with a thirst changing a blacksmiths into a pub so he could get a drink. In London the most famous version is the Castle on Cowcross Street becoming a pawnbroker after George IV found himself at a cock fight at nearby Hockley-in-the-Hole without any cash. The Castle was the nearest pub, so he went in to borrow money from the landlord, using a watch as a deposit. The landlord did not recognise the royal but agreed nonetheless and George won the next bet, redeemed his watch and granted a Royal Warrant to the pub to also trade as a pawnbroker. Three brass balls still hang in the pub as a memorial and a large painting commemorates the event inside. There is another London story of a monarch granting a drinking establishment a special licence after a favour. When Edward III had run out of money, he borrowed some from several City Vintners. Instead of repaying them, he granted them the right to sell wine without a licence. This is why the Boot and Flogger wine bar, tucked down Redcross Way in Borough, can sell wine without a licence: it is owned by the Freemen of the Vintners Company.
It goes without saying that all of these stories should be taken with a fair amount of salt, the most artery hardening one being the story I stumbled upon, saying that the Blind Beggar pub in Bethnal Green, of Kray infamy, was named after the Edinburgh bodysnatcher William Hare who, after getting William Burke executed, found himself in Limehouse where he was thrown into the lime pits. Blinded, he migrated to Bethnal Green to become the famous beggar. The generally agreed story of the Blind Beggar is that he was Henry de Montfort, son of Simon de Montfort, who had been defeated by the son of Henry III, Prince Edward, at the battle of Evesham. Wounded and blind from the battle, Henry lived in disguise as the Blind Beggar of Bethnal Green to escape the attention of Edward, who was now King Edward I. According to this legend the Blind Beggar is another aristocrat incognito amongst ordinary Londoners.
These stories have plenty of meaning; they remind us of the biblical teaching of entertaining strangers as they may be angels or royalty in disguise, and that the lives of the rich, famous and infamous are like ours. They still drink and have sex, and yet are different; they need to build secret tunnels to go and do it. They, like saints and ghosts, bring a mysterious aura to a location, be it a cosy old pub or an unremarkable boozer with a claim to fame. In a city that has always enjoyed the money of tourists and travellers, such a claim or artefact can draw people to a location and feed urban legends for centuries afterwards.
## 4
## THE GENITALS OF LONDON
* * *
To Pee or Not to Pee: An Overview of Electricity Related Deaths,
and Examination of the Question of Whether Peeing on the
Third Rail Can Kill
PowerPoint presentation, medical examiner's
office in Cook County, Illinois
* * *
THERE ARE FEW things less socially acceptable than a stray penis. The penis is chiefly for having sex and urinating, two things that are unacceptable in public. For this reason, no doubt, it is the penis that protrudes into a number of London urban legends, demonstrating the ongoing fascination and awkwardness people feel about it.
So pity the man in the following legend collected by Rodney Dale and written up for his book The Tumour in the Whale. A man rushes into the saloon bar of a City of London pub, puts his hat and briefcase on a table, orders a whisky and tells the barman that he is 'bursting for a pee'. The landlord tells him to go through a doorway and turn left, which the desperate man does, undoing himself on the way. Thinking he is arriving at the toilet the man pulls out his 'apparatus', as it is referred to in the story, but finds himself standing on a platform in the public bar with his private parts on display. The barman sees him, is enraged, and throws the man out onto the street. Our hero returns to the saloon bar to retrieve his hat and briefcase, just as the barman is telling the landlord about what happened. After a shout of 'that's him!', the frustrated man, still not having had his pee, is thrown out onto the street again. Years later the man walks into a pub in Ipswich and sees the former City of London pub landlord behind the bar. 'Don't I know you?' the landlord asks.
A more cautionary tale is told in Paul Screeton's book Mars Bars and Mushy Peas, of the only child of a north London Cypriot family, who is left alone for the first time. Half an hour after his strict parents have left for their holiday in Limassol, the boy is smoking, drinking whisky and masturbating to hard-core porn while naked. If only he had waited longer; his parents soon came home, having forgotten their passports.
In 1978, a couple were caught out having sex in a small two-seater sports car somewhere in Regents Park. The near-naked man suffered a slipped disc, trapping the woman under '200 pounds of pain-racked, immobile man,' said a Dr Brian Richards. In her desperation to be free, the woman began honking the car horn with her foot. A crowd gathered, including women volunteer workers serving tea, while the fire brigade cut away the frame of the car. After the woman is finally helped out of the car and given a coat, she is distraught and asks, 'How am I going to explain to my husband what happened to his car?'
The Tumour in the Whale was published the same year (1978)and carried a similar story. The car was stuck for at least an hour at the end of someone's driveway before the homeowner went to investigate. The woman is more blasé after the rescue workers apologise for having to cut the top of her husband's car away. 'That's all right,' she says. 'It's not my husband.'
In the world of urban legends, getting a penis out in the wrong place can be lethal: I am sure most have heard the story of the man who accidently urinated on the electric rail of the London Underground and was killed by the electric current jumping back up at him. The man is usually drunk, and it is late at night so he thinks he can get away with his public urinating. Sometimes he thinks he is polluting a river, other times he is just so drunk he does not care. There is even a news story from the Daily Mail and Evening Standard newspapers on 22 July 2008 of an unnamed 41-year-old Polish tourist who died whilst urinating on the live rail at Vauxhall rail station.
Many doubt that you could electrocute yourself by weeing on an electric rail. In 2003, the American television programme Mythbusters tested the myth by constructing an anatomically correct man full of yellow liquid. The urine flow was compared to one of the male presenter's actual flow filmed on a high-speed camera. With the mannequin's flow all present and correct, it was released over a live rail. It became apparent that urine does not come out in a continuous stream, but quickly breaks into droplets on the way down, making it, the programme makers said, difficult if not impossible for the current to conduct back up to the penis and hands. Online discussion forums and comments are a useful place to pick up people talking about such ephemeral things as this. Online it was presumed that the initial shock would cause the man to jump, removing the flow from the rail. The voltage of an electric fence or rail line would not kill instantly, unlike in one tragic example in Monsanto of a man urinating on a downed power cable lying unseen in a ditch.
Whilst investigating the possibility, Chicago's The Straight Dope website found that in two cases in America of death by supposed 'electric wee' the victims had both made physical contact with the rail whilst urinating. So while the coroner's report could correctly state that the two people – an adult man and a 14-year-old boy – had died while urinating on an electric rail, it was the physical contact with the live rail during or afterwards that had killed them. It is possible that this is what happened at Vauxhall. The man died around 5.20 p.m. on 12 July, when it was light and there were plenty of witnesses. This is not the late-night lethal release of legend. It was reported in the Metro on 22 July 2008 that the man had gone onto the rail line to relieve himself, so it is possible that he physically touched the live rail while down there, as his body was found slumped over the track. I have not been able to check a coroner's report on the death and, in all honesty, do not wish to read it.
There are many things in this book that the reader should not try, most of them because they would frighten or harm other people, and despite the evidence gathered here that urinating on the electric rail of a train or tube line would not hurt or kill you, please do not try it yourself. No one will be impressed and it is a bit offensive. Wait until you get home, find a public loo or go and urinate in a pub toilet, so long as you have made sure you are undoing yourself in the toilet.
Either at noon, or in the afternoon when the sun shines through the club-shaped balustrades running the length of Westminster Bridge, the top section stretches to cause a row of sunny, phallus shapes to appear. At first I thought it was a digitally enhanced comment on the residents of the Houses of Parliament at the north end of the bridge. Sadly, I have not found the time to linger long on Westminster Bridge seeking illuminated penises, but one brave London member (sorry) of the Snopes message board did go to Westminster Bridge and, at 1.03 p.m., photographed a raft of unfortunate shapes. This was after others had dismissed the image as 'completely unreal IMO [in my opinion]. The contrast between the light penii and the shadow looks wrong.' They were also wrong about the plural of penis.
The story that came with the genuine image was a joke about the Mayor of London, Boris Johnson, closing Westminster Bridge in the afternoons to avoid offending people with the luminous images. As pointed out on the Snopes board, whoever wrote the joke, taken seriously by readers outside the UK, did not know Boris Johnson and his inimitable wit. The website Liveleak attributed the appearance of the penises to the 2007 refurbishment of Westminster Bridge, when the balustrades were installed without thought to how their outlines may look in the long afternoon light.
So far the shape made by the balustrades of Westminster Bridge has only been attributed to perverts though, I am sure someone at some time will put a hidden-insult-style story to this trick of the light. The location near the Houses of Parliament is just too good not to. Let's wait and see...
## 5
## LEGENDS OF ROCK
* * *
I'm in the kitchen with the tombstone blues.
Bob Dylan, 'Tombstone Blues'
* * *
## 'It's taken you so long to
find out you were wrong'
Hopefully everyone now knows that the late actor and quiz-show host of Blockbusters, Bob Holness, did not play the saxophone on Jerry Rafferty's hit song 'Baker Street'. The myth was invented by the author and radio presenter Stuart Maconie for the 'Believe It or Not' column of music paper the NME (the New Musical Express). Another version of the legends origin is that LBC DJ Tommy Boyd claims to have run a 'true or false' question on a quiz about the 'neat and tidy' Bob being able to turn out a raunchy sax break. Things become more confusing when we hear that the actual saxophonist on 'Baker Street', Raphael Ravenscroft, claims to have told a foreign journalist that Bob had played on the song when asked if it was he who had performed it for the twentieth or thirtieth time. Bob himself was said to have encouraged the myth; on one occasion on Blockbusters a question came up about the song 'Baker Street' and Bob winked into the camera and complimented the sax solo. He would also claim to be the guitarist on the Derek and the Dominos song 'Layla', and that he was the person responsible for making Elvis laugh on the notorious live version of 'Are You Lonesome Tonight?' The myth of Bob Holness on 'Baker Street' has begat myths of its own.
## 'Why go to learn the words of fools?'
Another song myth that has not popped its head too high up into mainstream culture yet is the location of Itchycoo Park from The Small Faces' song of the same name. The first time I had a location pointed out to me, I was getting a lift from a work colleague called TBJ, who told me that Itchycoo Park was Altab Ali Park in Whitechapel, previously St Mary's Park and the former site of St Mary's Church. TBJ was a bit of a character, and at work he would tell stories of the characters at his local pub or gym, such as Jimmy Two Times, who was actually a gangster from the film Goodfellas.
I read Altab Ali Park's link to the song again in the 14 September 2012 e-newsletter from the indie music magazine Artrocker, where Tom Artrocker recounts:
I spent a pleasant couple of hours in Whitechapel yesterday. The sun shone as the traffic roared towards the coast, I was there with my team to photograph and video Toy. We took photos in the middle of the traffic's roar, down a dark alley and, traditionalists that we are, against a brick wall. Then we headed a few yards to Altab Ali Park. At which point I pointed out that prior to its re-naming, in honour of a young Bangladeshi murdered by several youths, this was the site of Itchycoo Park, as glorified by The Small Faces.
Small Faces member Ronnie Lane claimed that Little Ilford Park is Itchycoo Park. Tony Calder stated that the park story was invented by himself and the band to get around a BBC ban on the song and its possible drug references. Itchycoo Park was the name of a piece of wasteground in the East End that the band played on as children.
Valentines Park, West Ham and Wanstead Flats have all also been named as possible Itchycoo Park locations, although there is also the possibility that the song was inspired by a pamphlet about Oxford and has nothing to do with east London. Itchycoo Park is a pop music Atlantis or Camelot: it has many locations, some in London.
The name may have migrated from another nearby location: in his 1980 book Rothschild Buildings: Life in an East End Tenement Block 1887–1920, Jerry White describes 'Itchy Park' being Christ Church Gardens. This is the small ground beside Christ Church on Commercial Street, not too far from Altab Ali Park.
According to White's book, the park got its name from the children scratching themselves, like good East End urchins, against the railings of the park while using it as a playground. Tom Artrocker's origin for Itchy Park is even less kind, being from the fleas on the homeless people who used the park in the past and up to the present. The lyrics to 'Itchycoo Park' don't completely match either location; the dreaming spires could be the imposing steeple of Nicholas Hawksmoor's Christ Church, but there is no duck pond, and so no ducks to feed a bun to. We should not get too hung up on treating song lyrics as literal descriptions, however. If song writers are seeking to mirror reality in a lyric it would still be the first thing jettisoned for a pleasing rhyme or suitable mood.
## Liverpool Sunset
Some stories become facts not by being true but by being short, sharp, easily communicable pieces of information. Ray Davies was a chronicler and satiriser of Sixties Britain and The Kinks' song 'Waterloo Sunset' was a description of swinging London, with its characters Terry and Julie meeting at Waterloo station every Friday night based upon the handsome icons of the era Julie Christie and Terence Stamp. Everyone knows that Ray Davies watched from his window as they crossed the river for untold adventures. You know it, I know it, Terence Stamp knew it when interviewed about his retrospective at the nearby British Film Institute in May 2013.
Facts are tricky things and not always based on any actual real occurrence, particularly when the reality of the fact comes from the often volatile mind of a writer or musician. 'Waterloo Sunset', the song stained with the tears of countless Londoners, a group not often given over to sentimentality, started life as a hymn to Mersey-beat called 'Liverpool Sunset'. The Liverpool Echo cheerfully quoted Ray Davies in its 14 May 2010 issue as saying 'Liverpool is my favourite city, and the song was originally called Liverpool Sunset,' going on to proclaim 'London was home, I'd grown up there, but I like to think I could be an adopted Scouser. My heart is definitely there.' It should be noted that Davies was about to play the Liverpool Philharmonic Hall when he gave the interview.
So Terry and Julie may have had very different accents? Probably not as, according to a 'Behind the Song' column in the Independent dated 9 March 1998, once the change was made from Liverpool to Waterloo, Davies could incorporate the scene of countless people flowing out of Waterloo underground like another dirty old river. The couple at the station were not Terence Stamp and Julie Christie, but Davies' young nephew Terry. Davies' brother-in-law had just emigrated to Australia and he imagined Arthur's young son grown-up, back in London and meeting his girlfriend. The Independent article speculated that Julie symbolised England, so may have been based on Julie Christie alone.
Is any of this true? Following the workings of a creative imagination is like trying to jog across a continuous landslide of ideas, images, thoughts and feelings, even long after the creative piece is complete. We will probably never know. If you do meet Terence Stamp however, it may be best not to mention that the Terry in 'Waterloo Sunset' is not him, but Davies' nephew, all grown up and living in a fictional future London. When asked about it by the Evening Standard in an interview about his retrospective, published 2 May 2013, Stamp growled:
My brother Chris [ex-manger of The Who]... told me in the Seventies that when Ray Davies wrote 'Waterloo Sunset' he was thinking of me and Julie Christie. But apparently Ray denies it now. Well, if he says it's not true I don't care. I've believed it all these years...
## 'What's behind the green door?'
A London urban legend, nipped as it was budding, attempted to link an esoteric London location to the rock-and-roll track 'Green Door'. The song is a plea from a desperate man trying to get through a green door and into a midnight party full of laughter and hot piano playing. The protagonist in the song never makes it in; when he tells the unknown revellers that 'Joe' had sent him, they merely laugh at him.
First performed by Jim Lowe and reaching No.1 in the charts in America, 'Green Door' got to No. 2 and No. 8 in the UK. A version by Frankie Vaughn reached No.2, another by Glen Mason reached No. 24 and in 1981 Shakin' Stevens got 'Green Door' to No. 1 for four weeks.
In the Friday, 8 September 2006 'Culture' section of the Guardian Brian Boyd attempted to put the lyrics into a surprising context. The green door of the song was in London, on Bramerton Street off the Kings Road in Chelsea. It was the door to The Gateway, a private lesbian bar or club. The bar was a location for the film The Killing of Sister George, the story of a lesbian love triangle. The story of 'Green Door' is of a man trying to get into a gay women-only bar. When he says 'Joe sent me', he is referring to Joe Meek, the gay British pioneering popstar, which only goes down as a joke with the club's regulars. In his article, Boyd was attempting to put 'Green Door' with other gay pop songs featured in the compilation album From The Closet To The Charts, though the full title of the album, compiled by John Savage, is Queer Noises 61–78: From The Closet To The Charts. 'Green Door' was first a hit in 1956. It is not as popular as 'Waterloo Sunset' or 'Itchycoo Park' and the explanation of their origins, but Boyd's theory did make it far enough to make it into Stephanie Theobald's top five lesbian songs list in the Guardian on 6 March 2007, but this mention comes with a correction and clarification on the website.
The lyrics of 'Green Door' were written by Marvin Moore, with music by Bob Davie and was composed in a four-room apartment they shared in Greenwich Village in New York. As a graduate of the Texas Christian University School of Journalism, it seems Moore would be unfamiliar with the goings-on of 1950s lesbian London, a decade when the majority of Londoners would be unfamiliar with the concept of a 'lesbian London'.
There are more convincing explanations of the meaning of 'Green Door': that the song's original singer, Jim Lowe, was singing about a bar with a green door called The Shack when he went to the University of Missouri, or that it is based on the 1940 novel Behind the Green Door, although whatever is happening behind the door, set in as ski-resort, does not resemble the fun in the song.
Perhaps the most satisfying explanation is that 'Green Door' is a response or shout-back to the song 'Hernando's Hideaway' from the musical The Pajama Game, which describes a secretive nightclub for a 'glass of wine and a fast embrace', and where the password to get in is 'Joe sent us'. The song was a hit the year before 'Green Door'.
## Metal Box
Further off the mainstream radar is the discombobulating electronic music of Richard James, the Aphex Twin. His music can jump from serene to harsh to nausea-inducing. It is a fitting tribute to this that in the early 2000s the large stainless steel box in the centre of the Elephant and Castle roundabout was said to be his home. The box is, in fact, the Michael Faraday Memorial, dedicated to the scientist who was born nearby in Newington Butts. The monument itself contains an electrical substation for the Bakerloo and Northern tube lines and is not a house. The Aphex Twin lived nearby in the slightly more conventional venue of a converted bank.
## Bob Dylan's Crouch End Road Trip
Researching things is great, not only because you find out things you want to know, but that you always happen upon strange and probably apocryphal facts you never knew you needed to look into. A story dated 15 August 1993 in the Independent newspaper tells me that Crouch End once had more curry houses than all of Austria. This does sound possible, although twenty years on and TripAdvisor is listing thirty-one Indian restaurants in Vienna alone.
I stumbled on this while reading up on the connection between megalithic American folk-rocker Bob Dylan and his visits to Crouch End. Apparently he viewed a house there back in 1993 and became a regular at the Shamrat of India curry house. 'I recognised him from the telly,' said the owner at the time, 'but I'm more of a Beatles fan myself.' Bob wasn't getting a lot of love in Crouch End back then – the owner of the local guitar shop said that 'he used to be good, but he's rubbish now.'
According to urban legend a further indignity for Dylan may have happened around the same time. The real untrue story of Bob Dylan in Crouch End begins with his friendship with Dave Stewart of the Eurythmics, who once owned a recording studio called the Crypt, on 145 Crouch Hill. Stewart invited Dylan round, saying that the next time he was in Crouch End he should visit the studio. Dylan was seemingly so keen that he gave the studio's address at the airport so he could go straight there. Unfortunately the taxi driver dropped Dylan off, not at Dave Stewart's grand recording studio in an old church, but on nearby Crouch End Hill, where No.145 was a house. Dylan knocked on the door, asked for Dave and to compound his series of unfortunate events a Dave did live at the house, but was out at the time. Dave's wife said that he would be back soon, so would the mumbling American gentleman like to come in and wait for him? And would he like a cup of tea? Dave the plumber later arrived home and asked his wife if there were any messages for him. She said, 'No, but Bob Dylan's in the living room having a cup of tea.'
Writer Emma Hartley investigated the story for her 'Emma Hartley's Glamour Cave' folk music blog in a post dated, of course, 1 April 2013. She rang the Crypt studio and was told by an Anthony Lerner that he had 'heard it from the man who was Dave Stewart's chief sound engineer at that time'. Emma went out to Crouch End to knock on the door of the house on Crouch End Hill. It was while walking up the hill that she discovered that there is no No. 145. Perhaps the taxi driver was even more cloth-eared than we thought and took Dylan to No. 45 Crouch End Hill, which was, at least in the 1891 census, a residential property. Or perhaps the whole story is made up.
Consoling herself with a drink at Banner's Restaurant, No. 21 Park Road, Crouch End, she spotted a mural on the side of the building showing Bob Dylan asking, 'Don't you know who I am?' Inside, Emma was shown a brass plaque declaring that 'Bob Dylan sat at this table, August 1993'. Apparently, after his ill-fated trip to a possibly non-existent house on Crouch End Hill, Dylan went to console himself with a drink. At the time, however, Banner's alcohol licence did not allow people to have a drink without food, so Bob Dylan was turned down. He asked them 'Do you know who I am?' just so the restaurant staff were sure of who they were denying booze. The response is not recorded, but it seems like Bob Dylan just can't get a break in Crouch End.
## 6
## NEW LEGENDS AS OLD
* * *
They were not history, but legends...
Steve Roud, London Lore
* * *
## The Deptford Jolly Roger
Tucked down a street that's off another street that comes off Creek Road in Deptford is St Nicholas Church. The churchyard is dense and old, and Elizabethan playwright Christopher Marlow is buried somewhere in its grounds. On the gateposts leading into the churchyard is another of the church's famous features: two large decaying, yet still grinning, stone skulls crossed with bones underneath. This striking feature has acquired a legend to suit its visual impact, because these skulls and crossbones are the inspiration for the pirate flag, the Jolly Roger.
Deptford's maritime history is mostly obliterated, save a couple of warehouses and watergates by the river, but it was once the 'King's Yard', having been founded by Henry VIII, remaining a naval and shipping hub until after the Napoleonic Wars. Captain James Cook's ship the HMS Resolution set off from and was refitted at the dockyard, and Sir Francis Drake was knighted by Elizabeth I aboard the Golden Hind at Deptford. By the 1840s ships had become larger, and the shallow, narrow bed of the river made getting to Deptford difficult, closing the area to major shipping.
With over 300 years of naval history, Deptford must have had its fair share of pirates, or would-be pirates, passing St Nicholas Church on their way to their ship, the tavern or even to their execution. Many of them who looked up were inspired by these blood-curdling sculptures enough to incorporate their likeness into their flag and spread the terror of it across the seas.
London bloggers, ever on the lookout for an eye-opening and quirky fact, love the story. An undated 'Summer Strolls' walk around Deptford published by Time Out mentions the legend; even the website of St Nicholas and St Luke's churches repeats the story, although keeps its factual possibility at fingertips' length. The site describes the skull and crossbones flag as a means for British privateer sailors ('freelance' sailors who were paid on commission, working for the British Navy, fighting against the French, Spanish and Dutch ships for the control of the world's trade routes) to hide their nationality by flying the Jolly Roger rather than the English or Union Flag. The Information Britain website even names Henry Morgan, a former British admiral turned privateer who must have been familiar with Deptford, as the St Nicholas parishioner who first got the idea.
The difference between a privateer and a pirate is who benefits from the loot you steal. Henry Morgan was privateering for England and would have flown the English flag as he raided and looted innocent (or enemy) ships.
Historically, the Jolly Roger was not a ubiquitous symbol of piracy and was not adopted as a universal symbol of the pirate's outlaw status. Pirate flags were more in keeping with naval rules of engagement than the attitude of criminals. When one ship attacked another, a red flag was flown to indicate that they were in conflict. If the attacking ship was victorious, it would take the ship and its cargo and take the surviving crew prisoner, or 'give quarter'. To fly a black flag meant to give no quarter: the attacking ship would take no prisoners, so to avoid a slaughter the defending ship had best surrender without a fight. Pirates favoured the black flag, as often this is what would happen; even when outnumbered, enemy ships would surrender to avoid a massacre.
Over time, pirates began to decorate their black flags with personal symbolism in a seventeenth- and eighteenth-century example of 'pimping' something up: Thomas Tew's black flag showed an arm holding a dagger; Edward Teach, the infamous Blackbeard, flew a black flag featuring a skeleton stabbing a heart with an arrow; Bartholomew Roberts, Black Bart, had himself on his flag, holding one side of an hourglass with Death holding the other side. (This was followed by an even more flamboyant flag of himself holding a dagger and a flaming sword, with each foot on a skull). Calico Jack, John Rackam, flew a skull crossed with swords underneath while Henry Every flew a skull in profile with crossed bones beneath; not quite the symbol on the Deptford church. Edward English, born in Ireland, did fly the skull and crossbones, but there are few records of his early life and it is not known whether he visited Deptford.
The skull and crossbones itself is an old symbol that had already graced Spanish graveyards by the time St Nicholas was built. The earliest mention of pirates raising the skull and crossbones comes from a logbook entry dated 6 December 1687. It reads: 'And we put down our white flag, and raised a red flag with a Skull head on it and two crossed bones (all in white and in the middle of the flag), and then we marched on.' There may be a possibility that a pirate, when designing his own flag, thought of St Nicholas in Deptford and copied the design. That pirate did not do it so well, though. As the church's website points out, the skulls wear a laurel-wreath on their heads, probably to signify the victory over death over transient flesh. This wreath has not made it on to any pirate flags. It is still a story loved in Deptford, though. A local pub, the Bird's Nest, has even nicknamed itself the 'pirate pub' due to the Jolly Roger legend.
## The London Stone
History can be hidden in plain sight in the dustiest and busiest locations. Opposite Cannon Street station, set behind a metal grid in front of a branch of WHSmith sits the London Stone. The stone was the centre of a story in 2012 that named it as essential to the survival of London itself. It had been in its approximate location for a millennia or two, but the redevelopment of Cannon Street meant that property company Minerva wanted to move the stone, so that they could demolish the 1960s office block it occupies. The plan was to place it in the corner of its gleaming new Walbrook building on the corner of Cannon Street and Walbook. 'The new dedicated setting will enhance the significance of the asset,' Minerva wrote in 2011, 'and better reveal its significance for current and future generations.' Minerva did not, however, count on newspapers reporting the story with fears that moving the stone would be disaster to London. The Evening Standard, along with other papers, discovered an ancient saying that read: 'So long as the Stone of Brutus is safe, so long will London flourish.'
The London Stone is oolitic limestone and no one knows why it is there or what its original purpose was. It may have been part of a monument in front of the palace of a provincial Roman governor, which lay where Cannon Street is now. It may be a Saxon milestone of some sort: its original location is also the centre of Saxon London when it was re-established by King Alfred in AD 886. The stone has had its name since the twelfth century; an address recorded in a document dated between 1098 and 1108 is 'Eadwaker aet lundene stane'. In his 1598 Survey of London, Stow writes of 'a great stone called London Stone, fixed in the ground very deep, fastened with iron bars.' It is thought that it was damaged in the Great Fire of London in 1666, which possibly reduced it to something near its current size; the stone was placed by the door of the rebuilt St Swithin's Church.
Things start to get strange for the London Stone, as far as we know, from Jack Cade's 1450 rebellion. Cade, invading the City of London, struck the London Stone and declared himself the mayor of London. As John Clark, former Senior Curator (Medieval) at the Museum of London points out, much of our knowledge of this event comes from Shakespeare's dramatic interpretation in Henry VI Part 2. Clark points out that there is no historical precedent for the Lord Mayor having to strike the London Stone, and contemporary chronicles were 'at a loss as to its significance'. In the 1720 update to Stow's Survey of London, John Strype got druids involved for the first time, suggesting that the stone was 'an Object, or Monument, of Heathen Worship'. London poet and mystic William Blake ran with the idea, suggesting that it was a Druid altar stone. Thomas Pennant, in his Some Account of London, suggested that the London Stone could have 'formed part of a druidical circle'. The idea that the stone somehow protected London was first considered in Pennant's Account:
At all times it has been preserved with great care, placed deep in the ground, and strongly fastened with bars of iron. It seems preserved, like the palladium of the city.
This refers to the statue of Pallas Athene that protected the city of Troy. This all fitted very well into the legend first concocted by twelfth-century writer Geoffrey of Monmouth in his History of the Kings of Britain that Brutus of Troy brought the Brutus Stone to Britain after his city's destruction, and ancient kings would swear oaths over it. After all, didn't Jack Cade strike the stone and declare himself ruler of London? Perhaps the London Stone is the stone of Brutus? There is already a Brutus Stone in Totnes where the refugee Trojan landed, but that probably doesn't matter. The first to suggest that the London Stone is the stone of Brutus seems to be Welsh supremacist and language advocate Revd Richard Williams Morgan under his nom de plume Môr Meirion. In 1892, in an article in 'Notes and Queries', a pre-internet user-generated content publication where questions were asked and answered by public correspondence, Morgan is the first to 'discover' the ancient saying 'so long as the Stone of Brutus is safe, so long will London flourish'. In his 1857 book The British Kymry, Or Britons of Cambria, Morgan suggested that the London Stone is the palladium of Troy, possibly dragged under the wooden horse as the city burned. The Newburgh Telegraph of 18 December 1909 described the stone as a 'relic of Homer's days'.
In the twentieth and twenty-first centuries the London Stone was incorporated into a number of ley-lines (lines that can be drawn through a number of sacred sites to decipher arcane truths about the sacred landscape). By 2002 the London Stone was linked to Elizabethan occultist John Dee, in a claim that he believed it had magical powers. It was even given another legendary secret identity as the stone from which King Arthur drew Excalibur. These stories came out suspiciously near to London's bid for the 2012 Olympics.
There is little danger in moving the London Stone: in fact, it was moved from the centre of Cannon Street in 1720 when it became a traffic hazard. It was incorporated into the post-Great Fire St Swithin's Church. As a part of the church furniture, it moved around a little until it ended up in the corner of the church. Come 1884, when the District Line was being constructed, the London Stone was kept where it was, but the earth beneath it was removed. St Swithin's was gutted by Second World War bombing and the stone was moved to its current location in 1961, when the offices at 111 Cannon Street were built. At the time of writing, part of the new Cannon Street gleams under the spring sun while older buildings are being closed down in anticipation of demolition. The London Stone still sits in its tatty 1960s site, accompanied by a huge sign advertising cheap office space.
## The Ravens in the Tower
One of London's most famous pieces of folklore is of the ravens of the Tower of London. Everyone knows, or knows of, the saying 'If the Tower of London ravens are lost or fly away, the Crown will fall and Britain with it.' This is why the wings of the ravens are clipped so they do not fly away. Why would the ravens even be in a book full of upstart urban legends? Could the London Stone legend be a corruption of this famous piece of tradition?
One story of how the ravens gained their power over our nation comes via the first astronomer royal, John Flamstead. As is the way with these things, there are two versions of the story. The first has Flamstead and King Charles II gazing through telescopes in Greenwich when their view is obscured by ravens. They were frustrating Flamstead by either flying in front of the telescopes or defecating on the lens. Charles II said that the ravens must go, but Flamstead told the monarch that it is unlucky to kill a raven and that 'if the ravens left the Tower, the White Tower [the oldest part of the Tower of London] would collapse and a great disaster would befall the Kingdom'. In other versions of the tale it is an obscure soothsayer who warns the king after Flamstead complains about the ravens.
However, when Dr Parnell, official Tower of London historian, went through records of the Tower of London's menagerie he found records of hawks, lions, leopards, monkeys and a polar bear, but no mention of a raven. Dr Parnell along with American writer Boria Sax, who was also interested in the Tower's raven-lore, went through as much literature as possible and found the earliest mention of ravens at the Tower in a supplement called The Pictorial World on the Tower of London from 1883.
Where the legend itself comes from is another story, the earliest written version found only dates as far back as 1955, although the legend was recorded earlier. Natsume Soseki was a Japanese writer sent to study in London in 1900. He visited the Tower and wrote an account that Boria Sax describes as 'phantasmagoric'. Soseki entered the Tower like it was a gothic nightmare, and 'met' the ghosts of Guy Fawkes, Walter Raleigh and Lady Jane Grey. The ghost of Lady Jane tells a child, who can only see three ravens, that there are always five. Soseki writes the following on his encounter with a raven up-close and his thoughts on the Tower's executions: 'Hunching its wings, its black beak protruding, it stares at people. I feel as if the rancour of a hundred years of blood have congealed and taken the form of a bird so as to guard this unhappy place for ever.'
Returning to his lodgings, Soseki is told by his landlord: 'They're sacred ravens. They've been keeping them there since ancient times, and, even if they become one short, they immediately make up the numbers again. There are always five ravens there.'
Soseki's account was not translated into English until much later, but it is possible he picked up a folk-belief gathering around the idea of the ravens. Sax compares the impressionistic way Soseki writes to James Joyce's novel Ulysses, representing the world of London as Joyce represents Dublin, but threading it with fiction. No one at present can ascertain where one ends and the other takes over.
Could the raven myth have been created earlier? It may not be much of a surprise that Dr Parnell did not find ravens in the Tower's menagerie because ravens have long been indigenous to London. To record them at the Tower would be the same as listing the pigeons at London Zoo. Another suggestion for the ravens' presence was that they were a joke gift to the Tower by the 3rd Earl of Dunraven. Dunraven was interested in Celtic raven myths (he had ravens incorporated into his family crest) and must have known the Welsh legends of the Mabinogion. One tells of the hero-giant Bran being fatally wounded in battle and having his head removed and taken to Gwynfryn, a white hill in London with his head placed looking toward France so he could always keep Britain safe. Many think the white hill is the hill the White Tower is built on. Then myth-makers will nod as they tell you the word Bran, in Welsh, means 'crow', which is almost like a raven.
The ritual of the raven numbers is embedded now, whatever its origin. Interviewed for the Fortean Times issue 206, Yeoman Ravenmaster Derek Coyle repeated the Charles II legend and that there must always be six ravens at the Tower by Charles' decree. It may be possible that wandering through early twentieth-century London, Soseki misheard the number as five instead of six. While Coyle does not mention the legend directly, he does say that he keeps twelve birds at other locations to be sent for if numbers at the Tower drop too low. The cynical suspect that the story of the ravens was an extra fable created for tourists that Londoners took to their hearts too. The truth is that there were no ravens in the Tower of London by the end of the Second World War, as some ravens had died in bombing raids and others had pined away or died of shock. By the time the Tower was reopened in 1946, a new set had been found. Their wings are clipped to stop them leaving, not because the Tower may fall if they do, but because it is very difficult to stop ravens from flying away.
## The Lions in the Tower
The ravens in the Tower urban legends may have been inspired by the London Stone story, but also by an earlier animal fable of the Tower. London Zoo started life as the Royal Menagerie at the Tower of London, with the lions being the most popular. John Ashton, writing in his 1883 Social Life in the Reign of Queen Anne, reports that seeing the lions at the Tower was the first thing all those new to London did, and that when three of the four lions died in 1903, it was thought to be a 'dire portent'. The lives of the lions of the Tower were inextricably linked to the lives of the sovereign. In 1603, when one of the lions died just before the death of Queen Elizabeth I, it was seen as portentous. Joseph Addison went to the Tower just after the unsuccessful Jacobite rising of 1715, with a friend sympathetic to the Jacobite cause. The plan was to install James II's son on the throne. The friend asked if any of the lions had fallen ill after the would-be king was defeated at Perth and fled, and was told the lions were in the best of health. Addison wrote: 'I found he was extremely startled, for he had learned from his cradle that the Lions in the Tower were the best judges of the title of our British Kings, and always sympathise with our sovereigns.'
In earlier, less certain centuries, the fate of the nation and its people depended far more greatly on the monarch than now. It may be possible to think that the link between the lions' lives to that of the king transferred itself, once the lions left in 1835, to concerns about the nation connected to the ravens.
Once the lions had moved out of the Tower, they still managed to be an attraction there. Once a year, tickets would go out to people inviting them to the Tower to witness the washing of the lions. The invites were sent out in error in 1860, long after the lions had left the Tower, and the day of the 'annual ceremony' became 1 April. The meeting place was the fictional 'White Gate' to the Tower, and The Chambers Book of Days reported that on the day, cabs 'rattled about Tower Hill all that Sunday morning, vainly endeavouring to discover the White Gate'.
## 7
## LEGENDARY LANDMARKS
* * *
London is a wide place and a long, but rumour
has a wider scope and a longer tongue.
J. Fisher Murray, Physiology of London Life
* * *
WILLIAM KENT, IN his 1951 book Walks in London, recounts a story related to one of London's top tourist spots, St Paul's Cathedral. A boy from Snowdon was at a job interview at local textile manufacturers Hitchcock, Williams & Co., who were founded in St Paul's churchyard. The interviewer asked him if he had ever climbed to the top of the mountain. The boy said he had not, and was told there was 'no vacancy for one who was so unenterprising'. The next day, the boy returned and told his interviewer that he had just climbed up to the ball of St Paul's Cathedral. He asked the man, who worked for years in the shadow of St Paul's dome, whether he had ever done so, and the interviewer had to admit that he had not. The boy's point was taken; he was employed and was 'proved a most profitable servant'.
A nice story about how Londoners, like most people, often don't visit the wonders on their doorstep. Londoners I've known almost take pride in some of the London landmarks they have not visited, although these are often seen as lowbrow tourist places such as Madame Tussauds, the Trocadero and Covent Garden Market. They would be less likely to admit to never going to the Victoria & Albert museum or the Globe. Like the St Paul's story, there is probably a busy life involved too – Londoners live and work in London, and sometimes something in your immediate locale just doesn't seem like a priority. Stories that tell of success from unconventional ingenuity in job interviews always touches anyone who has had to undergo the rigours of interviewing for a position. This is a successful story. So successful, in fact, that it's had an American remake.
Kent goes on to repeat another story which appeared in The Times on 19 August 1950. It told of a Londoner visiting New York for the first time, who was early for meeting a friend. He nervously took the express elevator to the top of the skyscraper his appointment was in and was rewarded with an amazing vista of the city from the roof. Full of admiration, the Londoner told his friend about the view but the friend, a busy 'New Yorker born and bred', smiled superciliously and snapped that he didn't have the time 'for such rubber-necking'.
The Londoner didn't back down and told his friend that he should be ashamed of not taking advantage of the fine things on his doorstep. The New Yorker, with a broader smile, asked his Londoner pal how the view was from the top of St Paul's. It was the Londoner's turn to smile, the story says, as he had passed the cathedral every day on the way to work and had never gone beyond the Whispering Gallery.
Kent doesn't spot this as an urban legend; the term and concept was not around when Kent was writing Walks in London. He does point out that his first story of the Snowdonian interviewee had been published, by him, some time before the New York version appeared. Perhaps other versions existed before the Snowdonian story that tells of the busy lives of Londoners and the things they do not get to do.
## Neil? Kneel!
On 1 September 1983, Los Angeles Times columnist Jack Smith recounted a story of American tourists visiting the Houses of Parliament on a holiday in London. The story goes that they encountered Sir Quentin Hogg, Lord Hailsham, the Keeper of the Woolsack, 'resplendent in the gold and scarlet robes of his office topped by a ceremonial wig'. The pomp of the Houses of Parliament is intimidating to Londoners, so the effect on a corridor of American tourists by this visitation must have been great. Then Lord Hailsham sees, beyond the tourists, his friend the Hon. Neil Matten MP. He shouts his friend's name: 'Neil! Neil!'
The crowd of tourists fall into embarrassed silence and then fall to their knees.
While this story very clearly illustrates American confusion and awkwardness when faced with British parliamentary pomp, it does not illustrate life in the Palace of Westminster. There is no Keeper of the Woolsack in UK Parliament; Sir Quintin Hogg was Lord Chancellor, who sits on the woolsack in the House of Lords and is custodian of the Great Seal, a symbol of the sovereign's approval of state documents. The Lord Chancellor is responsible for the Great Seal unless a Keeper of the Great Seal is appointed. Parliamentary process can baffle anyone beyond its sphere, so there is no shame in confusing the details, but it does put doubt on the story.
There has also never been a Neil Matten in the House of Commons. Neil Marten was the Conservative MP for Banbury between 1959 and 1983. According to Andrew Roth's The MPs' Chart, Marten was a 'pro-commonwealth, anti-EEC... witty, sharp, tense, neat, balding, wartime agent'. Hogg and Marten were Conservatives together and certainly knew each other. The story, with its confused titles and 'scrawled on the back of a beer mat' – misspelling both men's names – has the air of a story told verbally, hastily written down and then repeated without checking any details.
In his Los Angeles Times column, Jack Smith was using this story as a way of tracing an American urban legend. A tale titled 'The Elevator Incident' by Jan Harold Brunvand in The Choking Doberman and Other 'New' Urban Legends, describes a small group of women getting into an elevator in New York. A man gets into the elevator with them and makes the command to 'sit'. The women sit, causing the man to apologise, as he was talking to the dog. At this point the man is revealed to be a celebrity who treats the women to dinner.
## The Waters of Senate House
The imposing University College of London building Senate House keeps going below ground level. Standing at its base, in several places, is a drop showing lower levels of the building created to bring natural light into the basement levels. This gully has water gushing out of it in numerous places, giving it the name of 'the moat'. Researching Senate House folklore for her project 'The Ghosts of Senate House', the artist Sarah Sparks recorded stories of a spring, lake or pond beneath the building. John Stone, the Building Services Technical Officer, took her into a lower basement to show the source of the water.
Water is flowing through a fissure in the wall and collecting on the floor. The duck boards dotted along the tunnel serve as stepping stones and were placed there when the building was first constructed showing that the water was always present. A channel two inches by two inches has been carved into the stone floor to allow the water to flow into a sump pumping the water up to The Moat above.
She then goes on to speculate that:
Geologists, employed to investigate the water, suggest that a spring up to a mile away has been diverted by building work however, this does not account for the fact that the water has been present to a greater or lesser extent since the buildings construction. I speculate that this water may originate from one of the lost rivers of London, possibly the Fleet. John agrees that there may be some truth in this citing that recent excavations of North Block Green unearthed an old conduit.
The Fleet is a mile or two away from Senate House, slurping under Farringdon Road and Farringdon Street through the Fleet Sewer. Speaking to Sarah about the water some months later, she told me an investigation had found that the water was coming from a leaking water main and not some lost spring or river. However, by September 2013 the leak has yet to be found.
The water has been there since before Senate House. Charles Holden, the architect, reported that one of the few problems he had with the construction of the building was a large pocket of water in the building's foundations. It was decided it should be left, as pumping it out could destabilise surrounding buildings as the ground moved to fill the gap left by it. The joists are said to pass through the water and into the clay beneath.
Londoners love the idea of our lost rivers, so it may not be surprising that another one is being used as a way of drawing people into a building. In his book London's Lost Rivers: A Walker's Guide, Tom Bolton described the River Tyburn's appearance in the basement of Gray's Antique Market on Davies Mews off South Molton Lane. The market owners moved into the building in 1977 and found the basement flooded. Claiming the water was the Tyburn, they channelled it into a twee model river with a small bridge and goldfish. The owners of the building take their attraction seriously, putting signs up instructing visitors to not touch the waters of this working river. Tom is not so sure, pointing out that while the Tyburn does flow under South Molton Lane, the river flows through a sewer so would not be fit to be channelled through a building. The water is possibly from groundwater springs that may have fed the Tyburn before it was buried and enclosed.
## 8
## THE SUICIDAL SCULPTOR
* * *
In London, starving workers dine
With old Duke Humphrey; as for wine,
'Twas made by Christ, in 'Auld Lang Syne'
But now he's turned teetotaler.
Woe in London Brimstone Ballads
* * *
## Unknown Stone
If we can be certain of one thing in London, it must be our statues. To be set in stone suggests confidence and permanence, and London's representations of its great and good must be a solid link back to the best of our shared past. 'Dining with Duke Humphrey' is a sweet but sad expression from sixteenth- and seventeenth-century London which means, in short, to be too poor to be able to afford dinner. The homeless and hungry lost scholars would congregate by a memorial of the hospitable Duke Humphrey of Gloucester (1391–1447) in the grounds of St Paul's Cathedral.
This event becomes sadder still when one realises that the cenotaph at the centre of this crowd is not for 'Good Duke Humphrey' but Sir John Beauchamp. London's oldest outdoor indigenous statue (not counting anything ancient, lifted and shipped in from Egypt) is of King Alfred the Great, a bearded and caped figure who is believed to have once stood in the Palace of Westminster and who is now slumming it in Trinity Square in Southwark.
However, the 1910 book Return of Outdoor Memorials in London by the London County Council could not find any reference to who the statue might be, and lists it as Alfred with a question mark by it. The book notes: 'The Secretary of Trinity House states that the Corporation have endeavoured to ascertain the facts in connection with the origin of the statue, but without success.' Understandably, due to the blank drawn about its origins, the statue's status as London's oldest is uncertain too. King Alfred is thought to date back to the fourteenth century. The statute of Queen Elizabeth I that stands on the façade of St Dunstan-in-the-West can also claim to be the oldest, as it was erected during the Queen's reign, either in or around 1586. Nearby are the statues of London's mythical founder, King Lud and his son Tenvantius, who may have first been erected on the gates at Ludgate in 1586.
Another mystery memorial is the 'Eagle Pillar' that stands in Orme Square, just off Bayswater. No one can remember what the double pillar with an eagle on the top is there to represent. The theories are that it was erected by a grateful Mr Orme, who made a fortune selling gravel to Russia; that it is a French eagle in honour of Louis Napoleon's stay on the square, and/or that it commemorates the French Embassy, which once stood at No. 2 Orme Square. Or the eagle could in fact be a phoenix for a fire insurance company; the Geograph website notes that a 'nearby house has birds looking like phoenixes in its frontage'. The final guess in Return of Outdoor Memorials of London is a bit fed up with all the rest; it merely suggests that 'the column is not a memorial at all, but simply an ornament picked up in a builder's yard'.
## Suicidal Sculptors
If we are uncertain about London's oldest stone statue, we're fairly certain that our oldest bronze statue, cast in 1633, is of King Charles I on horseback, which now stands at one end of the Mall by Trafalgar Square. The statue itself was cast just before the start of the Civil War in 1642, and on the outbreak of the war it was taken from its original spot on King Street, Covent Garden, and hidden in the crypt of the church of St Paul. During the interregnum, it was sold to a brazier named John Rivett, who was given orders to break it up. The canny Rivett broke the statue up by making and selling nutcrackers, thimbles and spoons made from the bronze of the dead king's statue. When the Restoration arrived, Rivett was able to provide the new Royalists with the fully intact statue that he had in storage.
This could be the urban legend about the statue of King Charles, but there is another attached which has proven to have far greater longevity and pedigree. In a letter dated 6 December 1725, Cesar de Saussure, from Lausanne, encountered the statue and recorded the story of the sculptor who had been 'almost beside himself with joy and pride' at his creation. However, on taking a closer look at the equestrian statue he realised the sculptor had forgotten to include the girths of the saddle (the strap or belt that goes around the horse that keeps saddle and rider on). The sculptor was so distraught to see his error set in bronze under the king's image that he hanged himself. 'This man was without doubt an Englishman' spat de Saussure, 'this trait depicts his energetic character.'
A community constable told Jeremy Harte of the Folklore Society that the reason the fourth plinth on Trafalgar Square is empty is because a huge equestrian sculpture was planned to be placed on it, and the sculptor was confident it would be his masterwork. The day was set for the unveiling, the sculpture waited under a huge sheet, dignitaries gathered and a band played for the ceremony. The sheet was removed and the crowd began to laugh because the sculptor had left the stirrups off his masterwork. The sculptor was so humiliated he ran down Northumberland Avenue and threw himself into the Thames.
This story has legs, six of them. It has also travelled over to the statue of the Duke of Wellington outside the Royal Exchange in the City. In a letter in the June 2002 issue of FLS News, John Spencer half-remembers having the statue's lack of stirrups pointed out to him by his grandfather and being told that the sculptor only realised his mistake when the king arrived to unveil the statue. Overcome with shame and embarrassment, the sculptor skulked off and shot himself. A year later, John was looking at the statue of George III in Windsor Great Park and overheard a middle-aged man explaining to a boy that the sculptor realised too late that the statue was stirrup-less and so committed suicide.
In reality, the sculptor of the Charles I statue was not an energetic Englishman, but a fellow Frenchman to de Saussure named Hubert Le Sueur. As well as the equestrian bronze, Le Sueur cast busts for England's royalty and aristocracy. Once the English Civil War began, his commissions naturally dried up and he moved back to France to work. He vanished into obscurity afterwards, long after the Charles I statue had been unveiled. The sculptor of the George III statue at Windsor Park portrayed him riding like a Roman, and the Romans did not use stirrups.
The fourth plinth in Trafalgar Square is not empty because of a shameful event involving stirrups. The original plan was for an equestrian statue of William IV to be placed there, but the plan was abandoned due to lack of funds. Another rumour about the plinth is that it is now reserved until after the death of Elizabeth II, so a statue of her can be placed there.
I have heard the legend told about the Maiwand Lion that stands in Forbury gardens in Reading, down the road from Windsor. The sculptor, George Blackall Simonds, is said to have killed himself on realising (after it had been completed) that the lion, one of the world's largest cast-iron statues, was incorrectly represented. Its stance is said to look more like a domestic cat walking than that of a lion.
Farther afield is the story of another enthusiastic English sculptor who threw himself into the Danube when he heard that the lions he had designed for the Chain Bridge in Budapest had been cast without tongues. These Hungarian lions are stone, not metal, and were certainly carved with tongues; it's just that they can only be seen from above.
## Backward Buildings
The eighteenth-century Fort George, on the coast between Nairn and Inverness in Scotland, was apparently designed to be invisible from the sea, but when the architect rode out to view this on completion (why not before?) he could still see one small piece of the fort and so reached for a handy pistol nearby to shoot himself.
The most famous error of this type is the Kelvingrove Art Gallery and Museum in Glasgow, which is said to have been built the wrong way round with a modest entrance for the public at one end and two imposing turrets for the back entrance. It is said that when the architect discovered the error, he leapt to his death from the building. In truth, of course, the building had not been built backwards and the architect probably does not haunt his cursed building. Frank Crocker, however, is said to haunt the hotel he built on Aberdeen Place in NW8. It was built not the wrong way round but in the wrong place. Crocker believed that the terminus for the Great Central Railway would arrive at St John's Wood, and so he went about building the Crown pub and hotel on Aberdeen Place between 1898 and 1899, in anticipation of the masses. It was a fine building with a marble bar and fireplace and guest rooms with imitation Jacobean plaster. Adding another layer to the myth is the Shady Old Lady blog, which says the sly architect of the building managed to get his dig in by including a bust of the Emperor Caracalla in the pub, a Roman emperor known for his 'architectural excesses and his complete insanity'. Caracalla is remembered for his massacres and the exuberant public baths he is said to have commissioned in Rome. However, the London terminus for the line ended at Marylebone, not St John's Wood. And so, ruined financially with nothing to show except a grand hotel with no customers, Crocker jumped out of a high window and the pub's name changed from the Crown to Crockers Folly. The Doctor Johnson pub in Barkingside, east London, has the same story to explain its size: it was built to service the users of a new road in and out of London which never arrived.
As Antony Clayton points out in The Folklore of London, the Doctor Johnson pub is so large because it is an 'improved' public house to serve the growing housing estates on the edge of London. The Crown Hotel, aka 'Crockers Folly', was completed about the same time as Marylebone station, and so was not positioned on Aberdeen Place by mistake. While Frank Crocker died relatively early at the age of 41, his death was of natural causes.
The sculptor or architect's mistake, followed by suicide, is a story that must always be hanging in the air, waiting to attach itself to a large building or statue or when something is out of place or missing, like Charles I's saddle girth. The narrative is then inevitable: the grand project, some hubristic pride, the realisation of the error and then the shameful ending.
## 9
## THE DEVILS OF CORNHILL
* * *
Behind the corpse in the reservoir, behind the ghost on the links,
Behind the lady who dances and the man who madly drinks,
Under the look of fatigue, the attack of migraine and the sigh
There is always another story, there is more than meets the eye.
W.H. Auden, The Secret Is Out
* * *
THE BEST WAY to find the devils of Cornhill is to walk north from London Bridge, up the west side of Gracechurch Street, through the city of London toward Liverpool Street station. Just before you get to the corner shared with Cornhill is one side of St Peter upon Cornhill Church. Its white stone is caked with a layer of grime and three tousle-haired cherubim, with wings round their necks like ruffs, gazing aloof and detachedly across the street. You then turn from Gracechurch Street into Cornhill, glance up, and leering down at you is a red terracotta demon with a dog-like body and a yowling, distended maw. It also has breasts, and a demonic face, its arms and chest are human. On the apex of the building crouches another larger demon, smirking to himself as he watches the passers-by on Cornhill. He looks as if he is preparing to launch himself onto an unsuspecting city worker or passing vicar below. It's a busy street, but I suspect few people feel comfortable loitering at this particular spot.
The story is of a nineteenth-century vicar at St Peter upon Cornhill who noticed that the planned new building on 54-55 Cornhill impinged on the church's land by 1ft. The vicar, or verger, disputed this and successfully stopped the building's construction. The builder, or architect, had to re-draw their plans and, as his revenge, he raised three devils (there's a smaller one beneath the top demon) onto the building overlooking the church and street to glare down on churchgoers and passers-by. In her own retelling of the story, the Shady Old Lady blog says that 'the devil closest to the street apparently bears more than a passing resemblance to the unlucky rector' as an extra twist. The oldest versions of the legend I have found date back to 1950, about fifty-three years after the building's construction. One, from 22 February of that year, is in Peter Jackson's compilation of London Evening News cartoons of London history and ephemera, titled London is Stranger than Fiction. It says: 'Crouching high up on an office building in Cornhill, stone devils glare down at the church of St Peter, below. They were put there by an architect who had just lost a dispute with the church authorities and erected them as a small token of his displeasure.'
William Kent's 1951 book, Walks in London, says: 'If we look across the road at this point we shall see high up on No.54 a devil in stone. A legend says a builder had a feud with the Church and told them to go to the devil. A curse was laid upon him, but defiantly he erected this figure.'
In 1988, estate agents Baker Harris Saunders published details of 54-55 Cornhill when letting it, which included this urban legend as a piece of local colour. 'Legend has it that following a disagreement between the owner of the land and the adjacent church [...] the owner sought retribution by adorning the building with a crouching devil and a chimera.' The truth of the legend is fudged and the leaflet gets the church wrong, claiming the dispute was with nearby St Michael's Cornhill and not neighbouring St Peter's.
The idea of a curse only appears in Kent's book. And was it the landowner, builder or architect who had the devils erected? Is the story true at all? The position of St Peter's is a strange one, with the entrance to the church squashed between offices and a sandwich shop for city workers. To our twenty-first century eyes, having places of commerce built into sacred places seems odd but it was common in the City in earlier times. An image of nearby St Ethelburga's Church on Bishopsgate, held at Bishopsgate Institute, shows shops built into the front of the church. At present, St Stephen's Walbrook and St Mary Woolnoth each have a Starbucks built into their flanks.
The Builder is a trade magazine for the building industry and lists every legal dispute to a building project, but does not mention St Peters or 54-55 Cornhill in its index over that period. The 1889–90 vestry minute book of St Peter Cornhill does list interactions between the church authorities and the architects of the current, devil-infested building, Walker & Runtz. They had recently acquired 54-55 Cornhill on behalf of a client, Mr Hugh H. Gardener, and they had found the original building to be 'somewhat shallow'. On October 1891 they wrote to St Peter's requesting a lease for 578ft of land where the old vestry building and lavatories stood, into which they could extend 54-55 Cornhill in exchange for £290 per annum. Walker & Runtz suggested that this money could be used to build a new vestry on another part of the site. After considerable discussion, the request was passed to the rector and churchwarden, who were 'of the opinion that no sufficient reason has been shown to justify them in recommending the scheme for the favourable consideration of the vestry'.
On 19 April 1892, Walker & Runtz served St Peter's with a party wall notice. Such a notice is given in a dispute over a boundary wall when it encroaches too far onto someone's property. The vestry were concerned enough to consider the cost of moving their wall back. So a dispute took place before the current 54-55 Cornhill was constructed; so far so mythical.
Walker & Runtz applied for the chance to buy 111.5ft of land to the west of the vestry, and this time the proposal was entertained. The money raised from the sale funded a new secure strong room for the church in which to keep their communion plate (before then a warden was taking it home to keep it safe) along with other improvements. After a meeting regarding the sale, the Ecclesiastical Commission made the decision to use the funds for the 'aid of the living'.
Pages 110–119 of the minute book have a report on the whole process of the sale, signed off by the rector on 25 October 1895. Once the Ecclesiastical Commission had approved the sale everything went smoothly: the Corporation of London approved it, and after the old building on 54-55 Cornhill had been demolished, the area of land was re-measured and sold. The report states that St Peter's has benefitted from the sale of ground that was previously used to keep lumber with its new strong room, improved lavatories and cloisters, as well as a fund to aid the living.
And that, as far as St Peter's minute book is concerned, is that, until 9 April 1901 when they received £150 for an increase in height of 54-55 Cornhill. There is no further mention of 54-55 Cornhill, and no mention at all of its devilish decorations.
## Enter the Ceramicist
The man who sculpted the demons was William James Neatby (1860–1910). Neatby was an architect who turned to ceramics, creating some amazing tiles and building façades. He is most famed for the tiles entitled The Chase, which depicts various hunting scenes in the meat hall in Harrods, including speared ducks and captured boars. He could also do symbolic images, such as the Spirit of Literature on the front of the Everard Building in Bristol, a former print works. The spirit, in the form of a woman, has Guttenberg and his successor, William Morris, on either side of her. A grotesque dragon hangs from one drainpipe.
Neatby could also do the bizarre. The Turkey Cafe in Leicester is just that: his signature blending of Art Nouveau and Arts & Crafts-style ceramics with a regal turkey perched at the top, its tail feathers radiating from its rear like sunbeams.
Neatby is a fascinating and mostly forgotten figure, but he is often praised in architectural journals. In The Studio, J. Burnard enthuses about his 'vivid imagination a handicraftsmen who has thoroughly mastered the ways and means of his materials'.
Louise Irvine, writing in the Architectural Review in 1977, declared that 'many terracotta buildings in London and elsewhere reveal his influence and could even be him but, as yet, the necessary documentary evidence has not yet come to light'.
Neatby himself comes across as an enigmatic character; passionate but somehow severe. He could not see the point of impressionism and his style is a robust yet sensual combination of pre-Raphaelite, Art Nouveau, Art & Crafts and more. He used his first wife Emily, described as having 'delicate features and slender figure', as the model for his 'almost burlesque' tile decorations for the Winter Gardens in Blackpool. But Irvine twice tells, in different journals, the story that he was so jealous of other possible suitors for his wife that he kept her locked up at home with the blinds down.
Rather than a jobbing ceramicist for building exteriors, Neatby is often called an artist who is unable to express himself through his commissions, though another article suggests that The Chase was created so quickly for Harrods that he could not have consulted the client too closely. The devils on 54-55 Cornhill follow his trademark images that stop anyone looking at them in their tracks. They also resemble the grotesques he created for the exterior of another London building, the Fox & Anchor pub on Charterhouse. There are foxes on this stunning building, described to me by a Blue Badge guide as London's only Art Nouveau pub, but they have demonic faces and huge yowling maws. They are more similar to the hyena-like creature on Cornhill than a fox.
If not revenge, then why are there three devils on 54-55 Cornhill? One could answer 'why not'? Neatby was an eclectic and evocative artist who, like other designers, decided to put gargoyle-like creatures on one of the façades he was designing. This was his first large piece in London and it is possible that he would want to make a lasting impression. It is a credit to his talent that the impression he makes is still so shocking.
## Demonic Cornhill
If that explanation does not suit some, then I can offer some speculation. Cornhill's fame as the highest point in the City may have inspired Neatby, or the people who commissioned him, to represent the bible story of the Devil tempting Jesus by taking him up high and offering him the world. Perhaps the top demon on 54-55 Cornhill is the Devil waiting on the highest peak to tempt others.
Cornhill has its own demonic history too. There is a satanic legend attached to the previously mentioned St Michael's Cornhill, retold in the Reader's Digest book Folklore, Myths and Legends of Britain, of a stormy night in the sixteenth century and a group of bell-ringers who were horrified by an 'ugly shapen sight' that floated through one window and over to another. The bell-ringers fainted and awoke to find claw marks in the stonework which became known as the Devil's claw-marks. This devilish calling card was annihilated in the Great Fire of London, but it may be that Neatby was referring to this legend with his sculptors. There are also rumours that the self-styled devil worshipping decadents of the Hellfire Club met in the nearby George & Vulture eating house. This side of Cornhill has enough satanic geography for any myth-maker or rogue Blue Badge guide to weave a spooky story.
The problem in trying to prove that something did not happen is that evidence for a non-event is a tenuous and circumstantial thing to try and find. It's like investigating a crime that may not have happened by looking for a gun that is not smoking. It is almost certainly true that the demons of Cornhill are a striking piece of decoration and nothing more; they were not put there as any sort of revenge. The nearest thing I have found to a non-smoking gun on this is an illustration of 54-55 Cornhill which appeared in 29 June 1894 issue of The Architect. It is an architect's drawing of the building, giving it the name 'Tudor Chambers', published after its completion but not from life. The devils on the top and corner of the building are not the robust and grotesque creatures we have now, but winged, dragon-like beasts which are small and unimposing compared to Neatby's devils. This would suggest that Runtz, the architect and so the person with the biggest axe to grind against St Peter's Church, did not plan to have monsters gazing down on to the building. It was the talented ceramicist he employed for the façade that created the demons, for show rather than revenge.
## Satirical Stone Faces
I have a friend who lives on Telegraph Hill, who is a keen local historian of south-east London. One night in the pub he told me that the stone faces carved over all the houses on the hill were representations of the German Royal Family. This did make some sense; the houses on the hill were almost all built between 1877 and 1899 when London had a large German community, many of whom were labourers. Perhaps when faced with hundreds of identical houses to erect, with space for an individual carving, a German stonemason or two couldn't resist chiselling a caricature of the monarchs from home. My friend, known as Neil Transpontine after his blog, sipped his beer again and completed the tale: during the anti-German riots at the start of the First World War, many of these stone faces were defaced by the angry English mob. This is why some houses on Telegraph Hill are missing their royal heads.
I left the story there, until a few years later I found a similar story in Peter Jackson's London is Stranger than Fiction. The entry for 22 February described the southern turret of St Giles' Church in Camberwell. St Giles' is the parish church of Camberwell and the architect of the building was Gilbert Scott. The turret bore gargoyles that were said to represent the political figures Lord Randolph Churchill, Gladstone, Lord Salisbury and Lord John Russell, as well as the politician and abolitionist William Wilberforce. I used to live opposite St Giles' but never noticed these carvings, so I went back to take a look. They may well have been public figures, but now time and rainfall have worn them, and their legend, away. I asked the vicar of St Giles', the Revd Nick George, about them but he knew nothing of the carvings or their history. He did enjoy the interesting story though.
Another stone grotesque with a story to tell is the 'hideous head of an old woman' on the right-hand side of the second window of the western outer wall of Mitcham parish church. The story, collected in James Clark's Strange Mitcham, tells of how the mason carving the corbels to support the church's windows had to do so with constant criticism from an old local woman. Eventually, the old woman found herself immortalised in stone by the mason.
Author William Kent's description of the urban myth origin of the Cornhill Devils is the second earliest after Peter Jackson's. Kent wrote at least a dozen books on London and in The Lost Treasures of London, a 1947 walk through the post-Blitz streets, he recounts a similar story to the Cornhill Devils that is linked to St Luke's Church on Old Street: 'It has sometimes been known locally as "lousy St Luke's" from a tradition regarding the weather vane. The story goes that the builder, peeved by "parsimonious treatment", placed a representation of a louse on the top of the tower.'
Kent went with a pair of field glasses and was 'inclined to accept the story'. This does not make the whole story true, however. Returning after the war to the bomb-damaged church, he noted that the louse was now gone, and presumed it had been taken during the war for its metal. The website for the London Symphony Orchestra, who now use the renovated St Luke's as a venue, has a contradictory quote about the weather vane from local resident John Mason: 'On top of the church there's a brass vane, and people in the area thought it was a louse, that's why they call it Lousy St Luke's... When they took it down I had a look at it; it had a beautiful red eye. After all these years the truth has come out – it's a dragon.'
The story of the revenge of the architect, stonemason or builder who hides an insult in the building he is designing or constructing has travelled across central and south London. It can be used to explain something strange or striking about buildings such as an extra-ugly stone carving, gargoyles with familiar faces or a weather vane that looks like an insect. That the story is best known around the Cornhill Devils is a testament to their visual power and their overbearing presence in such a bustling location.
## 10
## THE MISADVENTURES OF BRANDY NAN
* * *
I suspect there are further legends of this kind.
Jan Harold Brunvand, Curses! Broiled Again!
* * *
QUEEN ANNE (1665–1714), who reigned from 1702 to 1714, is best known for being the last Stuart monarch; the first monarch who had to deal with a two-party parliament system; being monarch during the Act of Union; lending her name to a style of table; and her romantic friendship with Sarah Jennings. In her own time she was known for a different perceived vice: she was known to love a drink and stories were spread of her hiding brandy in a teapot to disguise the amount she drank. She even gained the nickname 'Brandy Nan' for her love of the spirit. In 1712 a statue was erected of her, standing in front of the main entrance to St Paul's Cathedral to commemorate its completion in the ninth year of her reign and, presumably, greet arrivals to St Paul's. Grand intentions are often muddied by the irreverent and, according to the tale, the drink-loving queen's statue was gazing across the road at a pub. A rhyme was composed and scrawled on the statue which went:
Brandy Nan, Brandy Nan, they left you in the lurch,
With your face to the gin shop, your back to the church.
Meanwhile, in Salt Lake City, a bronze statue was erected to Mormon pioneer Brigham Young, standing with his back to the Mormon temple and his hand stretched out toward a bank. In an essay on Mormon humour it is suggested this mistake, if it was a mistake and not another sly designer's comment, provokes a sort of self-deprecating humour amongst Mormons who are required to tithe one tenth of their income to the Church. And that perhaps Young and other elders were more interested in the money of the Church of Latter-day Saints, and that LDS actually stands for 'lay down your silver'. There is even a rhyme that goes with the statue:
There stands Brigham, like a bird on a perch.
His hand to the bank and his back to the church.
In his book Curses! Broiled Again!, Jan Harold Brunvand repeated the Brigham Young song and also found a statue of Scottish poet Robert Burns in Dunedin, New Zealand, standing in The Octagon, the city centre plaza, with his back to St Paul's Anglican Cathedral and facing the 'commercial section of the city'. He correctly writes that he suspects there are more of the same legends elsewhere. I suppose it makes sense not to have a statue facing into a church with its back to arrivals, and with the famous figure looking outwards on to the material world, there is bound to be something inappropriate and unsuitable for their gaze to fall upon. I think it is human nature to comment on the juxtaposition between the two, but the rhyme that links Queen Anne and Brigham Young is a curious one, without pointing out that this must be the only thing linking these two historical figures. I would say the Salt Lake City rhyme follows the London one, though I have not found conclusive evidence for this, and I am sure any book on Salt Lake City urban legends may beg to differ. The Queen Anne statue is older, however: Brigham's bronze went up in 1847 and I am inclined to think that this urban legend migrated from England to America in this instance and not the other way around.
The earliest reference I can find to the 'Brandy Nan' rhyme is from an article of royal nicknames from the Northern Echo on 12 May 1896, which tantalisingly says that 'readers will remember the following lines which a wit of questionable taste bestowed on [Queen Anne's] statue'. The rhyme was well known enough to make it into Brewer's Dictionary of Phrase & Fable, published in 1898.
The original statue did receive a lot of abuse aside from insulting poems. In September 1743 the Universal London Morning Advertiser described the release of John Vaile, who had spent time in an asylum for breaking off the statues sceptre. 'Notes & Queries' from 11 April 1857 remembers that an arm was knocked off the statue in 1780, and was rumoured to have been done by a drunken man (though it may just have fallen off.) It notes that 'the statue of Queen Anne in St Paul's Church Yard seems endowed with the undesirable power of provoking the malice of iconoclasts'.
People still couldn't leave the statue alone in the 1960s, as a Daily Telegraph story from 20 October 1967 states that the statue is still 'a persistent target for vandals and over the years has been robbed of limbs, fingers, orb and sceptre.'
The statue of Queen Anne now has a fence running across her plinth and stands in peace; the only real indignity delivered to her now is how often she is mistaken for that other slightly stout female monarch, Queen Victoria.
## Brandy Nan on the Prowl
The iconoclasts had better watch out. Way west of St Paul's, over on Queen Anne's Gate just off St James' Park, is another statue of Queen Anne that has an even stranger story attached to it than that of a boozy poem and a connection to a Mormon leader. In this quiet corner of affluent London, on the anniversary of Queen Anne's death – 1 August – the statue gets down from its pedestal and walks up and down the road.
The earliest version of the statue moving that I have found was in a cartoon by Peter Jackson for the Evening News. A later version, in the Reader's Digest book Folklore, Myths and Legends of Britain is the source for paranormal researcher James Clarke's account in his book Haunted London. James' account has the addition of the statue's promenade taking place on the stroke of midnight of the anniversary of the queen's death which, he told me, came from the guide of a walking tour he had taken. Always a reliable source.
Everyone loves the terrifying thought of statues coming alive. The earliest account I could find that may relate to Queen Anne's statue moving is from Return of Outdoor Memorials in London and reads as follows: 'The children of the locality were accustomed in their play to call upon the statue, by the name of 'Bloody Queen Mary', to descend from its pedestal, and on receiving, naturally, no response, to assail it with missiles.'
This does not sound to me like an account of a walking statue, but of a game played by children. There may be other and earlier stories of the Queen Anne statue walking around that the compiler of Returns may not have known about, but the children did. Or perhaps it was a game children played before pelting the statue, and the story grew from misreadings of this account and similar ones. But where did the story of the statue moving on an auspicious date come from?
Western Europe's landscape is littered with stone circles, lines of stones and other clusters of stone left from our Neolithic past. At present, these are places of fascination; tourist sites that, for some, hold some ancient peace and wisdom within their bulk and patterns. Earlier peoples had a terror of these giant stones lying on the land and formed different stories of magic to explain them. So the Bulmer Stone in Darlington turns nine times at midday, while the stones that make up the circles of Nine Maids of Belstone Tor and Merry Maids of St Buryan are in fact petrified women who danced on the Sabbath. All of these legendary stones move at some point in time – at cock's crow, at midnight, at midday, on Midsummer Eve or Midwinter Eve or some other time. There are at least three hundred accounts of stones or stone circles moving at a liminal time, like the change of one day to another for Queen Anne, and the legend has managed to make its way from the countryside and into Westminster to attach itself to the statue. This may have been unconscious, as folklore, or conscious, by attaching an older myth to a new object. That statues move like megaliths has also made its way to Bloomsbury, where a recent tale emerged about the stone lions sitting outside the north entrance of the British Museum. Be there at midnight and you will see them stand up and stretch. Ideas are more durable than stone: even before statues wear away, their meaning can become lost and confused but ideas can breed and evolve amongst human humour, wit, error and fear, and can travel in the breath and letters of everyone.
## 11
## PLAGUE PITS
* * *
The Black Death has entered London's folk memory as a founding urban myth; every pothole in the road, every bump in a tube journey, every square or roundabout seems to have a plague story attached to it.
Richard Barnett, Sick City: Two Thousand Years
of Life and Death In London
* * *
WHEN I FIRST moved to the capital I lived in south-west London. I was told that nearby Mortlake had gained its name from the time of the Great Plague. The bodies of plague victims had been sunk into the lake which, forever more, was known as 'The Lake of the Dead': Mort-lake. I was not the only new arrival to London to hear about the city's burial sites for plague victims. A friend fresh from the north west of England and new to south-east London was told that she would need inoculating before going up to the giant plague pit that is Blackheath. Even after decades here it feels like a step cannot be taken in London without crushing someone's bones, and a journey cannot be made without passing by, or through, a plague pit.
Clear pieces of land in the overcrowded city are thought to be where plague pits sit and seethe in the landscape. A work colleague and I were discussing this, and he told me that the 'small green on Caledonian Road and Wynford Street is on the site of a plague pit. That's why it was never built on.' He was told this in the early 1970s when he started a job in that area.
These mass graves are too dangerous to dig into or build over; infection may be lurking beneath the soil waiting for fresh air and a fresh chance to infect people. I read on a web forum that if there is an oval bulging out of an otherwise straight alley behind a line of Victorian houses, it is because there is a plague pit there. These could be seen on maps of nineteenth-century Tottenham, Stoke Newington and Islington before twentieth-century developers blundered over them.
Mount Pond on Clapham Common is a plague pit, as is the triangular piece of land where Champion Hill meets Denmark Hill in Camberwell. Horniman Triangle, the field opposite the Horniman Museum, is a plague pit. The roundabout on the corner of Gypsy Hill and Allen Park is a pit. In Norbury in the 1980s there was a protest against plans to put storage containers on a piece of land thought to be a plague pit.
They are not just a suburban danger, however. In his book Underground London: Travels Beneath the City Streets, Stephen Smith mentions that the Harvey Nichols basement menswear department has a low ceiling as the building cannot be dug any deeper into the ground, for fear of disturbing a pit. In the 1970s and '80s the London Folklore Group's newsletter, London Lore, told of an international bank with an office in the City on Gracechurch Street whose employees thought it was built over the graves of plague victims. The building had its own water supply, which some of the workers in the bank would refuse to drink for fear of infection.
The London Underground has to curve around, drop under or plough straight through the assembled subterranean plague victims previously left in peace. The 1972 Reader's Digest book Folklore, Myths and Legends of Britain lists a tale about the Bakerloo line whilst on the way to St John's Wood from Baker Street. This is now part of the Jubilee line. There is a point between the stations where the ears of passengers 'pop' as the tube tunnel drops to dip underneath a plague pit which sits beneath the Marylebone war memorial. The Piccadilly line between Knightsbridge and South Kensington stations has to bend around Brompton Oratory to avoid a plague pit. Bank station, in the heart of the City, is built on a plague pit or at least will stink 'like an open grave' from fumes wafting up from the pit Liverpool Street station is built into. The Victoria line cut through a plague pit under Green Park in the 1960s, and according to Mike Heffernan on the Unexplained Mysteries website: 'A huge tunnel-boring machine ploughed straight into a long-forgotten plague pit at Green Park, traumatising several brawny construction workers on site.'
Stephen Smith reveals why Muswell Hill does not have a tube station: 'They started to dig a tunnel there and hit a plague pit!' One can also find on the internet the answer to the mystery of why there are far fewer tube stations in south London: because of endless pockets of dead plague victims. Tube drivers using the southbound escape tunnel for runaway trains on the Bakerloo line between Lambeth North and the Elephant and Castle must take care to not hit the end too hard, as a plague pit lies just beyond the walls at the end of the line.
## Blackheath
Blackheath is an open windy space above Greenwich. Despite greater London enveloping all sides of the heath, it does still have a desolate beauty. Houses cluster all along the edge of the hill Blackheath sits on but they avoid the top of it because, according to lore, Blackheath gets its name from being a plague pit for Black Death victims, just as Mortlake is the lake of death. In his book And Did Those Feet: Walking Through 2000 Years of British and Irish History, Charlie Connolly, remembers: 'I grew up in Blackheath in south London, to the casual observer just a great big expanse of grass sliced by a couple of roads. Yet it was a plague pit: during the Great Plague of 1665-6, hundreds of bodies were thrown into pits, scattered with lime and buried.'
Connolly demonstrates part of the attraction to plague pit urban legends here: that the storyteller has hidden knowledge they can share. He can lift the veil from an everyday piece of waste ground or greenery and describe the horror and history behind it. There is a dark glamour there.
On 7 April 2002, Blackheath Hill collapsed with a huge hole appearing across the A2 road so severe it took two years to repair. Remembering the event, a writer on the Tube Professionals Rumour Network website wrote that there was a 'big fuss' as Blackheath is 'another plague pit'. The fuss about the big hole in the A2 was really about the big hole running across a major road. What had collapsed was not a burial pit but the cavern that runs underneath Blackheath and Blackheath Hill. Discovered in 1780, these connecting caverns are thought to be chalk pits or hiding places dug by locals during the Danish wars. They run along and under Blackheath Hill from Maidstone Hill. They are known as 'Jack Cade's Cavern' in local lore, as it is thought the rebel leader Jack Cade hid in them to escape oppressive soldiers. Tours were given from 1850 and, after chandeliers were installed, balls were held in the caverns. They were abandoned in 1853 after a panic, when the lights went out. The caverns were next investigated in 1938 as a possible air-raid shelter. They were found unsuitable and were again sealed and forgotten about until the A2 caved in.
Blackheath is not a plague pit, its name being in use since at least the twelfth century, more than 500 years before the Black Death arrived in Britain. 'Blackheath' is thought to have come from the dark colour of its soil or have evolved from its description as a bleak heath. People are very fond of digging on the heath: as well as whoever dug the chalk pits, Blackheath, like much common land, was used by locals to dig gravel. After the Second World War these gravel pits were used to bury rubble from the Blitz which were then grassed over, causing the heath to lose its rugged, gorse-covered appearance and become the grassy flat space we know today.
The part of Blackheath that survived being built on – much of it didn't – was not because of the dead beneath it, but the living defending their piece of land from enclosure and development.
History being the weird, vast and diverse thing it is, of course, I will have to confess to you that there may still be a burial pit on Blackheath. Between 300-2,000 Cornishmen are in a mass grave somewhere on the heath. These men did not die as victims of the plague, but were killed by soldiers. They were camped up on Blackheath in 1497 on a march to London to protest against the taxes levied by King Henry VII to finance his war with Scotland. Henry sent in the troops and the Cornish rebels got no further than the heath. Local lore speculates that their bodies are buried beneath a mound called Whitfield's Mount. This may or may not be true.
The vast majority of burials in London are not related to the Black Death. However, the idea of the mass, unmarked pits still has such a hold over some imaginations that when we are hurtling through underground London it is always plague victims that keep us company down there.
## Green Park
I have found little written down about London's plague pits outside of repeated pieces of urban legends and the odd ominous nod toward a plague pit in the countless 'Haunted London' and 'Ghosts of London' books. Ghost books will use anything, like any good story-spinner, to set the right atmosphere for their tale. So the stories of the London Underground tunnel meeting a plague pit is in little threads across books, the internet and folklore. One of the clearer stories has already been quoted: the tunnelling of the Victoria line that disturbed a plague pit, while others say that the Jubilee line had to be redirected around a pit under the park.
It is a sign of the poisoned ground of Green Park that flowers will not grow there and, like Blackheath, has a sense of bleakness about it made all the stranger and unsettling for its location in the centre of London. Peter Underwood described Green Park's 'stillness, an air of expectancy, and a sensation of sadness' in his book Haunted London and James Clark mentions the park's 'subdued atmosphere' in his London ghost book, also called Haunted London.
There may well be diseased bodies under the turf of the park, but they are not plague victims. Before the Reformation, the site of St James's Palace was a leper hospital; and this part of its history may have informed the plague myths of the park. The Victoria line, being the first deep tunnel line on the London Underground, does not see daylight at any point. This may also have stoked fears regarding what was down in the earth with the commuters. This has always been a fear related to subterranean travel: when an underground train line was first proposed for London, Dr Cuming held an open-air meeting at Smithfield preaching the apocalypse: 'The forth-coming end of the world would be hastened by the construction of underground railways burrowing into the infernal regions and thereby disturbing the devil.'
King Edward I granted the leper hospital the right to finance itself with an annual May Fair, so giving Mayfair, now one of London's richest areas, its name from an annual charity festival for lepers. Green Park is the former grounds of this London leper colony. Henry VIII, during his marriage to Anne Boleyn, claimed the site for the Crown and St James's Palace was built on the site of the hospital. The grounds were transformed into St James's Park, with the neighbouring ground called Upper St James's Park. However, this very ground (now known as Green Park) bears the mark of its history – flowers will not grow there. There are many theories as to why this is: flowers will not grow there because of the sad virgin leper girls buried beneath it; or perhaps it is because when the hospital was taken by the Crown, Henry VIII had the nuns of the hospital thrown onto the snowy ground of what is now the park. Some believe that there are no flowers in Green Park to mark heaven's displeasure with this cruel story from the Reformation. The irony is, that according to Old and New London, the leper hospital at St James's Park London was struck by the plague, which moved quicker than leprosy to take some of the inhabitants. None of this pestilent history has really affected the ground of Green Park however, because despite the lack of formal beds and gardens, narcissus flowers do bloom in Green Park in the spring.
## Down in the Ground
Where the Dead Men Go
Contacting the Transport for London Corporate Archives, I was told that there are no specific references to plague pits in their records. They had just been through their archive to mark the 150-year anniversary of the London Underground and nothing came up. I wanted to make sure – plague pits really are everywhere in London lore – so I went through the files on the planning and construction of the Victoria line and the Fleet line, which became the Jubilee line, under Green Park.
The digging of the Victoria line is described in some detail in a pamphlet which was given away free when it opened. Miners were employed to work a digging shield that churned through the earth, and then reinforced the tunnels with concrete or steel supports. It must have been uncomfortable and claustrophobic and if they had met with bodies down there, it would have been just as unpleasant as the urban legend describes.
There were no references in the archives to this work uncovering dead bodies. The nearest possible reference I found to the plague was a note stating that there would always be the disinfectant Dettol with the workers constructing the Fleet line.
The plague pits of London are not lost and are not waiting to vomit up the dead onto unsuspecting builders and tunnel diggers. They are mostly well-mapped and their reality is every bit as unsettling and surprising as the urban myths. The mass graves were mainly dug for plague victims during the hottest, highest point of the Great Plague in August 1665. Before then, victims were buried in churchyards or in the grounds of pest houses (specially contained homes for isolating plague sufferers). Carnaby Street is near the sites of two seventeenth-century burial grounds for St James's workhouse. These sites, full and closed by 1733, may have been used as plague pits in the swinging 1660s. Carnaby Street has also been reported as the site of a pest house, the gardens of which extend out across Golden Square and Wardour Street. Maitland, in his History of London, reported that 'some thousands of corpses were buried that died of that dreadful and virulent contagion'.
Beneath the green grass of Charterhouse Square lies part of three pits which dealt with many of the dead. The oldest part, used in 1349, covers the area between Great Sutton Street and Clerkenwell Road. It was known as No-Man's-Land when purchased by the Bishops of London for burying plague victims. The site runs from Clerkenwell Road down to Charterhouse and then into the top of Smithfield. The pits contain an estimated 10,000 bodies. Charterhouse Square has its own plague legend: apparently the schoolboys of Charterhouse School would dare each other to crawl across the square at midnight, when the groans and cries of the dead below could be heard.
The Devonshire Square development in the City is built on a pit once dug into a green field at the upper end of Hand Alley, according to Defoe, as was Hollywell Mount in Shoreditch, which is now a car park. Liverpool Street station and the Broadgate Estate are built on a pit open from 1569 until 1720, which was used for plague victims and other burials. There are two plague pits in the grounds of St Paul's Church, Shadwell; a pest-field (where plague victims were buried in large numbers) in the 'additional ground' of St John's Church Wapping, Whitechapel, had three pits, and St Bride's, off Fleet Street, had a pit which was closed halfway through the 1665 Great Plague. This may be why St Bride's sits so high above the ground. Marsham Street, Horseferry Road and Vincent Street cover a pit which was once part of Tothill Fields in Westminster, Golden Square and the multi-storey Soho car park on Poland Street. Famously, Bunhill Fields, just on the outskirts of the City, was another pit. The 'great plague pit in Finsbury' is under a car park and residential gardens for flats on the corner of Seward Street and Mount Mill. Lille Street Mansions, Normand Park and Fulham Swimming Pool sit on the site of the Lillie Road Pest Field.
There may not be a plague pit stopping the construction of a tube line to Muswell Hill, but a graveyard did prevent the building of houses there. The Queen's Wood was previously known as 'Churchyard Bottom Wood', being the site of an old church burial ground. In 1893 the Ecclesiastical Commissioners planned to sell off some of the wood for development, but local outrage prevented the building – which was sealed with the Highgate Woods Preservation Act of 1897 – going through parliament, allowing the wood's preservation through purchase. There must be many stories like this to explain why a tube line travels a certain way. In reality the older tubes followed the road layout so they could avoid building foundations, basements and crypts, not plague pits. Legal considerations can also affect a tube tunnel's direction. During the construction of the Fleet line, Transport for London set aside millions of pounds for dealing with lease holders who may not have been happy with a train running under their land.
## The Pits
What of the suburban plague pits of Mortlake, Forest Hill, Camberwell, Muswell Hill and all the other sites? It is fair to say that when telling an urban legend, people never consider resources and logistics. With up to 4,000 people dying a day during the worst month of the Great Plague, Londoners did not have the time or energy to cart the wagons of the dead out into the countryside. Until the Great Plague, most plague victims were either buried in their local churchyards or in the grounds of the pest houses where plague sufferers had been confined. These pits are on the outskirts and edges of London (as it was then) and it would have been a huge waste of time and effort to drag the plague wagons out across the countryside to deposit the thousands of dead in Muswell Hill, Camberwell, Forest Hill, Blackheath and beyond. They had no need; the boundaries of the city at that point were on the edge of the City of London and Westminster, Wapping and Shadwell, which is where the dead were buried. In her book Necropolis: London and Its Dead, Catherine Arnold suggests that these legends sprung up after plague victims escaped London and wandered into the countryside of Camberwell and Forest Hill. The rural Surrey folk were familiar with what to do with infections, from their experience of murrain amongst their cattle, and the bodies of plague victims were dragged into holes by long poles and buried. The place of their burial, Arnold speculates, becomes a site of local lore.
People have dug up the dead. With London's long and populous history it would be very strange if digging up the city did not disturb some of the dead. There are still clusters of bones across central London, and often when they are found, the first thought is always that a plague pit has been discovered. This was the belief when bones were dug up in the Main Quad of University College London in 2010. After examination, the 7,394 bone fragments, 6,773 of which were human, were discovered to have a different story. Many of the bones had been cut by saws and scalpels, and many had numbers written on them. The burial was not a fourteenth-century plague pit, but parts of bodies buried there 100 years ago. The date of their burial was traced through a large Bovril jar that was buried with them, and it became clear that they were anatomy specimens that had been disposed of in a pit.
In April 2011, tunnel digging for the new across-London Crossrail line dug up hundreds, if not thousands, of bodies next to Liverpool Street station. These were not plague victims but inmates of the original St Bethlehem Hospital, the asylum known as Bedlam, who were buried in the churchyard.
## 12
## SUBTERRANEAN SECRETS
* * *
Folk-lore means that the soul is sane, but that the universe is wild and full of marvels. Realism means that the world is dull and full of routine, but that the soul is sick and screaming.
G.K. Chesterton
* * *
BETWEEN THE TUNNELS used by amorous and criminal historical figures, and the legends set within the London Underground, are other tales of underground London. The city is so honeycombed with basements, nuclear bunkers and tunnels that it must often ring hollow beneath the feet of its inhabitants. The unseen world is one full of allure, and a natural habitat for secrets.
## The Secret of the Elephant and Castle
Waiting for a bus at the Elephant and Castle on a cold and rainy night must be one of the unspoken rites of passage of contemporary London, like seeing a cast member of EastEnders in the West End or realising that something being 'pop-up' doesn't automatically make it exciting. The asymmetrical jumble of the Elephant and Castle shopping centre, all dirty glass and faded pink plastic, is a strange, sometimes fascinating place. Was it really thought that putting it there would improve the area and the lives of those that inhabit it? 'No', according to writer Nigel Pennick, who believes that the shopping centre was constructed in the 1960s as a cover. In the 1940s an extension of the Jubilee line down to Camberwell Green was planned, before losing out in 1961 to the Victoria line running to Brixton. A mile or so of tunnel was dug and then abandoned, but not without an anarchist group noticing and describing it in their pamphlet London – The Other Underground as a 'government tunnel' linked via other secret passages to the City and Victoria. Pennick decided that during the 1950s, with the chill of the Cold War and fear of nuclear war in the air, these tunnels were converted into nuclear shelters, suggesting that many 1960s redevelopments were a cover for the construction of secret government bunkers. How does one cover up a secret underground government bunker near to Westminster but beneath a disreputable part of south London? You build a giant shopping centre as a cover and hope anarchists and authors do not notice. And that, some say, is why the Elephant and Castle shopping centre is there.
There are people that think that the government still has the power to have secret tunnels and sites across London, and there are endless rumours of a secret tube line for ferrying the Royal Family out of Buckingham Palace in the event of an attack or disaster. These are not new rumours; in 1914 a discussion on the drainage tunnels that run under Greenwich Park gave them the more heroic role of being a possible escape tunnel for Henry VIII, their kinks and bends there to perturb arrows that may be chasing the king as he squeezed down them. Another royal escape route is through the trees in St James's Park. They were arranged during the Second World War, somehow, to ensure that a light aircraft could land in the park to whisk the Royal Family out of Buckingham Palace should Germany successfully invade. This is what I was told by a friend in the Hermit's Cave in Camberwell anyway, but the friend who told me is a bit of a trickster – we were there to discuss his role in the Brentford Griffin hoax.
The MI6 building, Vauxhall Cross, on the south bank of the Thames, has a tunnel running from its basement to Vauxhall station, but in greater dangers it is rumoured that the building has a more drastic self-defence mechanism. Folklorist Martin Goodson found himself on the No. 36 bus going from Peckham to Camberwell and overheard three Camberwell art students talking:
You know that in case of emergency I've heard that it (MI5 building) [actually MI6], can sink down and go under the river.'
General hilarity broke out from the other students.
'No, it's true, it can. I've also heard it can turn black so that it cannot be attacked at night.
The level of hilarity increased, but the student persevered in her conviction that this building is now equipped with some spectacular special effect qualities to protect itself in case of attack.
## 13
## THE CORPSE ON THE TUBE
* * *
I slept with faith and found a corpse in my arms on awakening;
I drank and danced all night with doubt and found her a virgin in the morning.
Aleister Crowley, The Book of Lies
* * *
## Dead Line
The best and most popular place to share an urban legend is some point during an idle chat. During a quiet moment, a work colleague mentioned a story her friend had told her about the London Underground. Someone, a friend of a friend or someone a bit further removed, had been travelling on the Underground late at night – she didn't know which line – in an empty carriage when three people got on and sat opposite her. Two men sat either side of a pale, limp woman. The half-remembered story, as it came across the desks at work, had the traveller being warned away from the two men and their pale companion and for good reason: the woman was dead.
Other versions of the same story can be found on the internet. Web forums are like chatting in the pub or during a tea break at work, but one can converse with like-minded people across the internet. The Unexplained Mysteries forum created a thread in 2007 called 'The Girl on the London Underground', which began with a friend of a friend (an art student) travelling back to her campus from central London late one night. She was alone, except for one other person in the carriage, a man who looked to be in his thirties. Then, three new people board: two men and a woman. The art student decided that the trio looked like drug addicts and avoided making eye-contact with them. Then, the thirty-something man started acting strangely. He walked over to the student and behaved as if he knew her, asking, 'Hi, how are you? I've not spoken to you in a long time,' before leaning into her and whispering, 'Get off at the next stop.'
The student was wary of this, but did not wish to be left alone on the train with what she thought were three drug addicts, so she followed the man off the train and onto the platform. Once they were off the train, the man revealed to the student that the girl in the trio was dead; he had seen the two men drag her onto the train with a pair of scissors embedded in the back of her skull.
A similar story was collected via an anonymous email in 2003 by the Urban Legend Reference Pages, better known as 'Snopes'. This tale came from a work colleague of the sender, whose boyfriend knew or had heard about a girl who got on the tube and, not wanting to sit on her own, sat opposite the three other people in the carriage. Again, this was a woman flanked by two men. The girl started to read, but whenever she looked up the woman was staring directly at her. The girl ignored the stares and at the next station a man boarded the train, looked about the carriage and sat down next to her. This new man then whispered to the girl, 'If you know what's good for you, you'll get off at the next station with me.' Despite feeling threatened by this, the girl presumed there would be other people at the next station and got off. At this point the man revealed himself to be a doctor, who could tell that the staring woman was dead and the two men either side of her were propping her up.
## A Mystery on the Underground
Another version of the previous tale takes place on an intercity train. Two women travellers are stared at intensely by a girl who they find out later had been murdered by her two female companions. Could this be the same story, warped by whispered repetition, as the story Rodney Dale records in his book It's True... It Happened to a Friend (1984)? This version has a girl getting onto a quiet subway carriage and doing her crossword. A man gets on, whom she ignores, and then at the next stop two more men board and sit either side of the first man. As the train stops again, the two men get off, leaving the first man in his seat, although not for long. As the train jolts out of the station the man falls out of his seat with a knife in his back. This time the corpse on the tube becomes one right in front of our unfortunate commuter. The story takes place on the subway rather than the tube, so where could it have happened? I asked Rodney Dale if he could remember his source or the location for this version but sadly, it is a very small section in the book and he could not.
One possible origin for the corpse on the tube legend is a fictional story serial called 'A Mystery on the Underground' by John Oxenham, which appeared in To-Day magazine in 1897. Presented as fake newspaper clippings rather than a conventional narrative, the story begins with men turning up dead while travelling on the District line. Early on in the story, finding the body is like finding the corpse in one version of the urban legend in this book: a lone woman on the tube discovers a man is dead when his body falls off the seat as the train wobbles.
A story about this stroy claims the tube companies were concerned that people would mistake each episode for actual news clippings. There was a mocked-up cartoon, supposedly from Punch, showing shocked people on a crowded stagecoach when they hear a man is still taking the tube. The tube contacted the editor of To-Day, the famed writer Jerome K. Jerome, to complain. Jerome considered pulling the story but, having one more episode to go, which took place on a ship rather than the London Underground, Jerome let it run. There is no mention of any 1897 tube panics in The Times index for that year, and in his autobiography, Jerome does not discuss To-Day any further than his regrets when he had to sell it.
After the initial similarity to the corpse on the tube legend early on in the story, there is no further resemblance to it in 'A Mystery on the Underground'. The murders always take place on a Tuesday and the victims are shot dead in the carriage by an anonymous killer and not led on to the train by shifty characters. If a cause of death is mentioned in the urban legend, it is a stabbing.
## Death on the Tracks
The story of the corpse on the tube is clearly an urban legend, but was it ever more than just a horror story about travelling with strangers in confined spaces? People do die on London's public transport; the TUBEprune, the Tube Professionals' Rumour Network, is a website full of gossip and stories, purportedly from London Underground staff. One section describes two instances when bodies have been found on the tube, though they do both have the air of a story rather than the retelling of an event. The first was when a train arrived at East Finchley station at the end of the morning peak time. The crew inspected the train and found a man slumped in a seat, who they tried to wake. They discovered that the man was dead, and had been for so long that rigor mortis had set in and he was rigid in his seat. The body had to be removed by being laid sideways on a stretcher to prevent it rolling off.
While rigor mortis begins three to four hours after death – so is possible after the morning peak – maximum stiffness does not set in until around twelve hours. It is possible the body was left overnight on the tube, but hopefully not.
Another find was on the eastbound Piccadilly Line at Northfields. A passenger raised the alarm when a man on the packed train seemed 'a bit poorly'. The guard did not wish to delay the train so he persuaded a couple of passengers to help him drag the corpse off the train and left it sitting upright on a bench. The police were called and complained about the disrespectful treatment of a body. The guard then responded with, 'What else could I do, I couldn't delay the train, could I?'
Whether this is a true story or not, or a joke about the far edges of job-worthiness told by Tf L staff, or even a blending of the two, I shall leave up to you to decide. A problem that occurs when one spends a lot of time researching, writing and thinking about urban legends is that you end up doubting every story you hear unless the teller can show you photographs, official documents or the scars. And even then you still doubt.
One person who was almost certainly found dead on the tube was German naval lieutenant-commander and suspected Nazi spy, Franz Rintelen von Kleist. The former Isle of Man internee was found dead on a train at South Kensington tube station in May 1949.
## 14
## THE STRANGER'S WARNING
* * *
That was the way with Man; it had always been that way.
He had carried terror with him. And the thing he was afraid of had always been himself.
Clifford D. Simak Way Station
* * *
ADEADLY STRANGER LURKING in the back of a woman's car has not been the only warning to circle around the internet, purportedly coming from the Metropolitan Police. A fake message warning people not to travel on the London Underground emerged on 24 July 2005. The email claimed that controlled explosions had taken place around 15 July, at Piccadilly Circus and Leicester Square stations. They had not. Much like the fake warning of the stranger hiding on the back seat (See Criminal Lore) persons unknown were inventing police warnings.
The fear of the enemy amongst us, either terrorists or infiltrators, has haunted people for a long time and a particular sort of urban legend has accompanied these fears. In the twenty-first century, urban myths or rumours of the 'Helpful Terrorist' or 'Strangers Warning', have been spread quickly by email and internet forums. Our century has seen terrorist attacks across the world and these have left fear and folklore in their wake. Over the winter of 2001, an email warning of the possibility of an attack started popping into people's inboxes. Below is a typical version taken from the website Snopes.com. It arrived via the girlfriend of a friend of a relative of a friend – even farther removed than the standard 'friend of a friend':
Morning all,
Had a bizarre message from my brother in the early hours of this morning...
His friend's girlfriend was shopping in Harrods on the weekend. There was an Arab man in front of her who was buying a number of things with cash – he was a few pounds short so the girl offered him £3 to cover it.
He thanked her profusely and left. When she left the store the man followed her out and thanked her again and warned her not to travel on the tube today! [1 October 2001]. She was a little thrown by this so she went to the police. The police were very sceptical but in order to eliminate her suspicions gave her the photo-ID book of all known dissidents in the UK. He was on the second page listed as a known terrorist.
This is apparently true and the police are apparently taking this extremely seriously. The most likely time would be rush hour this evening so please avoid it if you can – who knows it may be nothing but is it worth the risk?
PS- I don't have a lot of people's e-mail addresses so obviously please forward this on to anyone and everyone.
I am certain that if the Metropolitan Police suspected that the London Underground was going to be bombed on a specific date, they would not leave it up to members of the public to spread the word via email.
The 'foaf' in this version of the urban legend is generally a woman, which may be because women are considered more kind; it has consistently been a woman throughout the legend's evolution.
The woman gives the mystery man £3 and warns of danger in London, with this legend it is typically the larger fee in London and a lesser fee of 68p at a cash and carry when Birmingham is the potential target for attacks. As well as the London Underground and Birmingham, warnings have been given regarding Coventry, Tamworth, Milton Keynes and Chester. This type of urban folklore has spread to America, with the warning given to a woman about drinking a popular brand of soft drink after 7 September. In an alternative version, a male waiter, after giving a customer change for the phone, is warned to avoid a second popular soft drink after 1 June.
Many tales of the 'Helpful Terrorist' have been borne from the sad events of 11 September 2001, including warnings from boyfriends to their girlfriends not to go near the World Trade Centre or the Pentagon on that day. The story of 9/11 was retrospective and so false but there was a warning for later that year: 'Don't go to any malls on Halloween'. It seems that many urban legends evolve from huge events and are subject to change as the story is carried from person to person.
## The Enemy Within
One of the most disturbing urban legends covers the possibility of the enemy among us, that the person sitting next to us could be a terrorist, or the far more dangerous implication that a group of people living within our community are complicit in the actions of a terrorist cell. It seems unlikely, but some urban rumours have entire communities that are aware of terrorist assaults prior to the actual event.
These rumours can gather such momentum that city authorities have to release statements ensuring the public that they are untrue. Such is the nature of an urban legend, that the repeated telling can give a story credibility although it has no apparent basis in fact, nor any evidence to support it.
In New York, the police released an announcement in January 2003. 'There is no terror plot or threat connected to the rumor that is circulating in New York and in other cities abroad,' said Deputy Police Commissioner Paul Browne. This was apparently in response to a rumour that a cab driver had warned one of his female customers of an impending terror plot.
Snopes, an invaluable aid in gathering rumours, also related an earlier version of the stranger's warning legend. In 2000, a version of this story was circulating in Manchester in which a woman helps a young man out by lending him some money while in the queue of a fast-food restaurant. She is rewarded by being warned to keep out of a local shopping centre in March on that year in a soft Irish accent. The change in stranger is important: the is an urban legend that attaches itself to many different ethnicities.
The rumour of the helpful terrorist goes further back than this century and from terrorism to all out war. During the First and Second World War there were many strange and frightening rumours flying around London and the rest of the UK, and some that seem familiar today. Rumours of the enemy among us came in the shape of supposed fifth columnists walking our streets. At the start of the Blitz, there were fears that German agents were signalling to bombers circling London. One German-Swiss man in Kensington was arrested for smoking a large cigar: 'He was puffing hard to make a big light and pointing it to the sky,' said a witness who feared the smoker was in fact communicating with enemy bombers.
During the Battle of Britain, it was rumoured that Hermann Göring himself had flown in a bombing raid over London. By 1942, this had evolved into Göring being seen sitting in an air-raid shelter in Plymouth after parachuting in to watch the city being blitzed. Just as Adolf Hitler had his eye on London offices and flats and had visited the UK in his youth (See Nazis over London), Osama Bin Laden was often sighted in America after he became public enemy number one. He was usually seen in coffee shops or eating in fast-food restaurants.
The First World War was fertile ground for rumour and folklore, with stories of angels protecting British troops as they retreated from their first engagement at Mons. There was the hope of Russian allies travelling through Britain by train to reinforce the frontline. These warriors were identified by the snow on their boots.
Once the Great War began, bizarre rumours began to spread across the city. Concrete, patented in 1849, became a source of anxiety. Tennis courts were suspected of being secret German machine gun placements, and a factory in Willesden with views of the Crystal Palace that was constructed from concrete was raided by police in 1914, because it was discovered that it had another office located in Leipzig. The owner of Ewell Castle successfully sued the Evening News and the People after the newspapers published a story saying that their concrete-bottomed lake was being capable of mounting five heavy guns, which had the firepower to take out the main railway line into London. Evidence given for the castle owner being a spy included his expensive car. On 3 October 1914 the rumour led to the Daily Mail asking: 'Is it too much to ask that our kid gloved government will ascertain how many German owned factories have been built in this country which incidentally command Woolwich, Dover, Rosyth? A timely inspection might reveal many concrete structures.' Police led a futile search of the abandoned King William tube station when the Railway Magazine suggested there was a cell of enemy agents occupying the location and using it as a base to shoot and bomb their way across the city.
One night in October 1916, a Zeppelin was said to have descended onto Hackney Marshes and a tall man with an eye patch was lowered down in a basket. He got out, asked a couple of bystanders the way to Silvertown and they told him to follow the River Lea until he got to Bow. The man and the Zeppelin then disappeared into the night. The couple then informed the police.
Sir Basil Thomson, head of CID at Scotland Yard during the start of the First World War wrote an informative and entertaining book about his investigations from that time, Queer People. In the book, he tells a tale of a caddish German officer 'being seen in the Haymarket by an English friend; that he returned the salute involuntarily but then changed colour and jumped into a passing taxi, leaving his friend gaping on the pavement.' Other legends have a young girl meeting her fiancé who is an enemy officer in disguise. He forgets himself briefly greeting her with affection, before remembering himself and turning away. Infiltrators have revealed themselves in various ways, such as swearing to themselves in their native tongue.
Around the same time, a familiar story was spread. The following is from Queer People:
The next delusion was that of the grateful German and the Tubes. The commonest form of the story was that an English nurse had brought a German officer back from the door of death, and that in a burst of gratitude he said at parting, 'I must not tell you more, but beware of the Tubes in April [1915].'
Basil Thomson tracked this rumour:
We took the trouble to trace this story from mouth to mouth until we reached the second mistress in a London Board School. She declared that she had had it from the charwoman who cleaned the school, but that lady stoutly denied that she had ever told so ridiculous a story.
The rumour appeared during the Second World War too, but this time the nurse was treating a captured German pilot and was rewarded for her kindness by being advised to carry her gas mask on 15 September. Did a rumour from the cleaner of an unnamed London Board School migrate almost entirely intact to the twenty-first century? Perhaps the reason the character in the tale is always a woman is because the original figure was a female nurse. It is clear that the person given the warning is kind and worthy of the advice. In fact, the latest version of the story does contain a role-reversal. After the shooting of Osama Bin Laden, Paris feared it would suffer reprisal attacks. In the Parisian version of May 2011, one man returns another man's wallet, and is rewarded by being advised not to use the Metro the following day. A legend being told at the same time sees one woman helping another by sparing some change for a parking meter near Carnegie Mellon University – she is told not to attend any rallies for the Tea Party.
Like the mugger hidden on the back seat of a woman's car, or dead eyes staring at you on public transport late at night, this urban myth relates to fear, prejudices or a blind spot in general knowledge where wild speculation can takes its place. The Strangers Warning relates to a very specific situation in which there is a threat of attack by persons unknown on civilians in an urban setting. It is bound within the fear of attack that the Helpful Terrorist story is able to pass from person to person as a truth, especially during times of great anxiety when emotions are running high and people want to believe in the good of a stranger.
The following is an interview with Alexander Walters, who served a term of one year in prison for a hoax bomb alert at Heathrow airport on 15 September 2001. On 26 November 2002, the Guardian reported:
He had been out walking his dog on September 15, when an urge suddenly seized him to phone Heathrow airport on his mobile.
'There is a bomb at the airport,' he told the operator. 'You have exactly one hour.' The call was traced to his phone - which, like him, was in south Wales.
What on earth was he thinking?
'I didn't think at all. I just went for a walk. It was just something that happened so fast that I didn't even know what I was doing until it was too late.
'It wasn't attention-seeking, it was just, I think, a way of letting anger out. I had one or two problems at the time, and obviously I did something really stupid.'
What was he angry about?
'I wouldn't know, it was just a spur of the moment thing. You just totally switch off and do something you shouldn't have done. And then before you know it has caused this huge thing.'
Is this what the author of a hoax email in thinking or feeling as he or she writes it?
## This Era's Enemy
The Great Fire of London, now believed to have been an accident that started at a bakery in Pudding Lane, was long regarded as a piece of Catholic terrorism. Robert Hubert, a French watchmaker living in Romford, confessed to starting the fire, being an 'agent of the Pope' and taking a bribe from the king of France. England was also at war with Holland at the time of the fire and it was feared that 'the French and the Dutch have fire'd the City'. Despite concerns about his mental state, Hubert was hanged at Tyburn on 28 September 1666 for starting the fire, before it emerged that he had in fact arrived in London two days after the fire had begun. During the execution, an effigy of the Pope was burned with a head full of cats that screamed for the pleasure of the crowd as the flames reached them.
The Monument to the Great Fire of London at Fish Street had inscriptions blaming Hubert and 'Popish frenzy, which wrought such horrors, is not yet quenched', which were not permanently removed until 1830.
On Monday, 2 September 1666, the second day of the fire, a maid from Covent Garden called Anne English was arrested after she was reportedly claimed that a group of French men had delivered a warning to her master. The men told him to move his goods as 'within six weeks that house and all the street would be burned to the ground'. She was interrogated at Whitehall, but denied the story stating that she had heard 'that the French and Dutch had kindled the fire in the City.'
The helpful terrorist has an ancestor in a warning passed on before the Gunpowder Plot of 1605. Just before it's failed execution, there was a warning delivered to the Catholic Lord Monteagle who was having dinner in Hoxton when he received a letter pleading with him to 'shift of your attendance at this Parliament' as Parliament was due 'a terrible blow' on 5 November. This is not thought to be a genuine warning from one Catholic to another however. The origin of this warning is suspected to be from Monteagle himself who knew some of those involved in plotting an attack, but did not want to be seen to betray them. Another suggestion is that it was a secret service letter that was designed to be used as evidence against those plotting atrocities. Basil Thomson was a former intelligence officer who was sometimes involved in disseminating misinformation. Perhaps, this urban legend is a covert way of drawing information out of the public. Propaganda has been used as a tool for a long time, but once the story has been released, it is free to be adapted and moulded and is not easy to control. As well as planted rumour, there were hoaxers, liars, old prejudices, people who communicate their ideas through allegory and story, and the people who believe the stories and pass them on.
There is one more tale of an act of kindness gaining insight set in London but the knowledge is of a very different nature. Broadcaster, journalist and wit Nancy Spain (1917–1964) is reported to have seen a ghost on Piccadilly. Spain saw the ghost after she had just left Fortnum & Mason and was looking for a cab. One pulled up in front of her and a woman with red hair got out, fumbling in her purse. Spain was in a hurry and paid the fare for the elderly woman who then went into the store without saying a word. Once Nancy Spain was in the cab the driver said, 'You were caught there, Miss. That old gal could buy both of us. That was Lady C.' Speaking of the incident the next day, Spain was wordlessly given a newspaper by her mother that carried the headline, 'Lady C. Dies in Fire'. The knowledge Spain gained through her generosity was of the world of ghosts and not of terrorism.
Spain apparently saw ghosts in the strangest places; she once encountered the spectre of her friend Bin who had died at the age of twenty-four at a restaurant. Of the event she said, 'Once I am sure I saw her come into a restaurant. She sat down and ordered, of all things, a Scotch Egg. But when I leapt up to say hello she seemed to vanish, leaving a hard, clear line for a second, as a piece of paper does when it burns in the fire.'
The stranger's warning legend is the classic story of entertaining angels unaware, or the fairtytale of a hero or heroine who gained the favour of god through an act of kindness. The story shows how kindness can be rewarded and has the hopeful note that an attack may be avoided.
## 15
## NAZIS OVER LONDON
* * *
German airmen are careful not to bomb breweries and maltings in
Britain because Hitler knows that if Britons go on drinking at the
present rate, we shall lose the war.
Unnamed clergyman from Chester and Warrington
Methodist Synod quoted from The Tumour in the Whale.
* * *
THE GORDON RIOTS of June 1780, London's most violent protest, were inspired, or at least encouraged, by Lord George Gordon's speeches against laws proposing to allow the nation's Catholic citizens the right to buy land, practise medicine, teach and join the House of Commons or Lords. More fuel for the mob's fury came as a result of fearful rumours of 20,000 Jesuits hiding beneath tunnels under the Thames, waiting to take London on orders of the Pope, like the Germans lurking in King William Street Station (See here) .
By 1914, in the prelude to the First World War, it was London's German community who had become the enemies within as London's most violent riots since the Gordon Riots destroyed German shops and homes. The 1914 issue of The Railway Magazine prompted a police investigation of the abandoned King William Street tube station after suggesting it was being used as a base and weapons store for German infiltrators.
## The Nazis' Favourite Landmarks
The Blitz brought new stories. As people died and whole neighbourhoods were devastated, stories swirled around London landmarks to explain how and when they survived. Nelsons Column still stood because Adolf Hitler had taken an interest in it. On his successful occupation of London, he had planned to carry the symbol of British naval might to Germany as a way of underlining his victory.
The 1930s tower of Senate House, University of London's imposing base in Bloomsbury, survived because, according to Graham Greene, it was used as a marker for bombers approaching Kings Cross and St Pancras stations. Senate House was also earmarked to be the base Hitler planned to use as the German central office for ruling Britain after their invasion.
The 1937 Art Deco block of flats Du Cane Court in Balham is quite pleased of its reputation as Hitler's possible home or HQ in London. The Führer even placed spies within the building. Like Senate House, German air crews would use the Du Cane Court as a handy landmark: 'It was turn left at Du Cane Court and then head home for Germany.' Du Cane Court is proud enough of the legend to put the story up on its website but also sheepishly ponders whether the block's architecture may really have attracted the genocidal leader; but 'true or not, the flats were quite an innovation at the time'. Antony Clayton, in The Folklore of London, uncovered stories that the architect of Du Cane Court was a Nazi sympathiser who planned to have the building make a swastika in the middle of South London when viewed from the air. This takes us back to the stealth swastika of the kindly prisoner-of-war German soldier and his gardening surprise (See 'The Hidden Insult' here).
Some South London landmarks that were removed included the golden Goddess of Gaiety statue at the top of Wimbledon Theatre, which was taken down in 1940 and not replaced until 1992, as it was thought to be an excellent guide to German bombers. On the edge of London, and on the top of a hill, St Helier Hospital in Carshalton was painted black during the Second World War so it would not be used as a landmark for incoming German planes.
All of these landmarks fared better than the north tower of the ruined Crystal Palace. Having survived the fire of 1936, which destroyed the rest of the glass building, the tower was destabilised and blown up with dynamite in 1941 because many, including William Kent in his Lost Treasures of London book, thought it was being used as a navigation point for German bombers. Other reasons included to prevent it falling in a bombing raid; presumably a controlled explosion was safer, and the tower's steel was needed for the war effort. This was the line used in a British Pathé news film of the demolition, called Crystal Palace Tower – The End. People were sceptical about the gathering of scrap metal and park railings during the Second World War, thinking that the metal was not and could not have been used for weapons and vehicles. The collecting of metal was thought to be a morale-boosting exercise and the metal was used as ships' ballast, dumped in the Thames Estuary or taken out to sea to be dumped by Canning Town dockers in such great amounts that incoming ships had to be guided in by pilots because the quantities of metal were affecting their compasses.
Another building thought to be spared by the bombers was Winchester Cathedral, as the Nazi propaganda broadcaster Lord Haw-Haw was said to have gone to the school by the cathedral and had asked Field Marshall Goring to spare it in the raids. Another rumour told of Hitler planning to be crowned as king at Winchester Cathedral once Germany was victorious. A retort to the Winchester rumour said, 'Any Coronation dream would obviously have Westminster Abbey as its centre.'
The removal and camouflage of prominent landmarks was perhaps a sensible precaution before and during the Blitz. On the eve of the Second World War, London was preparing for sustained aerial bombardment and for mass burials, stocking up on cardboard coffins, for example; London County Council envisaged mass burials in lime pits. The predictions for an aerial bombardment on London were based on 700 tons of high explosive being released with the casualty rates of 175,000 per week. The destruction of towers, the removal of bright objects from theatres and painting landmark buildings black therefore seems feasible. The estimates were far greater than the actual, still terrible, death toll of the war: the total bombs dropped on Britain were an estimated 64,393 tons, killing 51,509 people.
There is another factor. The architecture of Du Cane Court and Senate House must have linked them to the Nazis and Hitler in the minds of frightened and angry Londoners. Ironically, Senate House was designed to symbolise the future world, having survived the First World War, and it was actually intended to be an international beacon of learning: 'It must not be a replica from the Middle Ages.' Perhaps these rumours of Hitler's interest evolved out of Londoners' suspicions at the modernism of the architecture of Du Cane Court and Senate House and their resemblance more to the Reichstag building than to the British Museum or Natural History Museum. With enemy planes flying overhead and spies rumoured to be everywhere, the Second World War must have felt like no other time to London civilians. Perhaps all of these legends come from Londoners feeling enemy eyes directly on the landmarks of their lives.
## The One and the Many
An earlier British folk tale took place in Dorset and was recorded in 1930. A West Lulworth man remembered a story told to him by a 104-year-old resident of the town, who apparently witnessed Napoleon arriving at Lulworth Cove in August 1804. He arrived at the cove with a companion seeking a place to land for an invasion, but was heard to mutter 'Impossible!' before 'folding his maps and returning to his boat'. Emperor Bonaparte personally taking the time out to inspect land sites on enemy territory seems as likely as Hermann Göring, commander of the Luftwaffe and Adolf Hitler's deputy, braving anti-aircraft fire to take a look over London.
Both stories have a possible origin in fiction. Thomas Hardy claimed to have invented Napoleon's trip to Dorset for a short story in 1882 and was amazed to hear the story repeated back to him by friends. Stories of Göring's midnight flights over London may have been inspired by a fake news report that Göring, who was an ace fighter pilot during the First World War, had piloted a plane over London on 15 September 1941 escorted by two bombers.
However, I think there may be more happening within these legends than just a misinformed regurgitation of fiction. Mother Teresa is quoted as saying, 'If I look at the mass I will never act. If I look at the one, I will.' She was thinking of human responses to suffering: people will give more for an individual rather than a group of people. We are interested in times of war. These stories reflect how people think and feel during a war with an invasive force.
With an event the scale of the Second World War, it would be impossible to imagine the thousands of troops involved in the planned attacks on us. To think of their equipment, how it is maintained and who supplies this aggressive organisation, it is far easier to look to the head of the enemy, the very top, and imagine them taking a very close personal interest in our homes, the beaches they could land on, the buildings they could live in. Somehow, this – Adolf Hitler picking his offices and deciding where he would be crowned – is far easier to imagine and respond to than one nation moving against another.
## 16
## CRIMINAL LORE
* * *
Adults tell fairy tales, to adults, although the maudlinized
and castrated samples in print belie the fact.
Richard Dorson, Folklore and Face
* * *
## Beware all Lady Drivers
It is just before Christmas 2003 and Antony Clayton was checking his email. He found the following warning sent to him on 17 December with the subject line 'Danger when Filling Up at Petrol Stations':
Beware all lady drivers. This is a West End Central Police Crime Prevention information message providing details of local crime and disorder issues. If you have information about any crimes mentioned please contact the Crime Desk at West End Central Police Station on... . We need your help to make Westminster a safer environment.
A woman stopped at a pay at a petrol pump station to get fuel. Once she filled her petrol tank and after paying at the pump and started to leave, the voice of the attendant inside came over the speaker. He told her that something had happened with her card. The lady was confused because the transaction showed complete and approved. She relayed that to him and was getting ready to leave but the attendant, once again, urged her to come in to pay or there would be trouble. She proceeded to go inside and started arguing with attendant about his threat. He told her to calm down and listen carefully. He said that while she was filling her car, a guy slipped into the back seat of her car on the other side and the attendant had already called the police. She became frightened and looked out in time to see her car door open and the guy slip out.
One would hope that warnings from Westminster police would not contain so many typographical errors and banal attempts at drama. The message concludes with a warning:
The report is that the new gang initiation thing is to bring back a woman and her car. One way they are doing this is crawling into the women's cars while they are filling with petrol or at stores at night-time.
Be extra careful going to and from your car at night. If at all possible do not go alone.
1. ALWAYS lock your car doors, even if you are gone for just a second.
2. Check underneath your car when approaching it and check in the back before getting in.
3. Always be aware of your surroundings and of other individuals in your general vicinity, particularly at night.
Antony Clayton got to the end of the warning and knew just what to do. An author of a number of books on London, including The Folklore of London, he submitted the dread message to the Folklore Society News (FLS News) and it appeared in Issue 43, June 2004. The legend itself spread, and arrived to a different reader pretending to be an email sent by Harrow Council Civic Centre which concluded: 'This is a real warning! The alert originated from a London company who had a female employee involved in the above instance.'
In some ways the gang member hiding in a woman's car has a similar plot to the corpse on the tube urban legend. A woman is travelling alone at night; she is in danger, but is unaware of it until a man distracts her by frightening or annoying her until she learns the truth.
There may be, along with the corpse on the tube, some subconscious concern or disapproval of women travelling alone in the urban night, with all its strange but very real dangers. Women are still thought of as more vulnerable than men, so to be plausible these stories may choose a lone woman as the target of the anonymous nocturnal predator.
This is a legend we share with America. Snopes has collected versions via email in 1999 and 2000, and in 2001 a version very much like the London one appeared, minus the Westminster police contact details but with the warning: 'THIS IS TOO SERIOUS ... DO NOT DELETE. PLEASE PASS IT ON!!'
By 2004 the Dublin version named a gang, the 'Westies', who were carrying out these surreptitious attacks, and the message warned that the abducted woman would be taken at knifepoint and gang raped.
The gang initiation aspect is not part of the story yet, but it is easy to see how it became linked. The idea of gang initiation feeds many urban legends, the most popular being the warning to never flash your car headlights back if a strange car flashes them at you. The car has a would-be gang member inside who needs to murder the first driver to flash back at them in order to join. Snopes has a version of the story from 1999, where hidden gang members hamstring a woman and 'remove a body part' to gain entry to the gang. The victim is always a woman. In the 1998 version the attacker is a serial killer, still coasting on the 1990s wave of cultural interest in the mind of serial murderers which drew people to films such as The Silence of the Lambs. Despite Hannibal Lector still haunting the cinema and television, the fear of criminal gangs was soon to sweep the serial killer out of the popular imagination and out of this urban legend.
Like the male rescuer in the corpse on the tube legend, there is a suggested element of danger from the rescuer, until he speaks to the woman (in the earliest version he speaks to the woman's husband when she gets home but happily urban legends have moved on since 1967) and tells her of the dangerous stranger in the back of her car. In The Vanishing Hitchhiker (1983), Brunvald writes: 'In more imaginative sets of these legends the person who spots the dangerous man in the back is a gas station attendant who pretends that a ten dollar bill offered by the woman driver is counterfeit. With this ruse he gets her safely away from her car before calling the police.'
As we have seen in the early twenty-first century London version of this urban legend, with its faulty credit card, the story is the same but the props change over time.
## Child Abductors through History
Another widely travelled abduction legend made it into the FLS News No.37 June 2002 issue. Correspondent Susan Hathaway heard from a work colleague that his wife's 'friend's daughter's college friend' was the mother of a child who had gone missing in a John Lewis store near London. Security was alerted and all doors were sealed to ensure the child did not wander out of the building. A few minutes later the child was found emerging from the toilets with a different coloured dress, a new short haircut and a group of strangers herding her. The Mumsnet internet forum has a thread for sharing and disarming scare stories including this child-danger story. Locations for it included an east London Tesco, a Co-op on the Isle of Wight and other shops in Bristol, Tokyo or Gloucester.
An earlier pre-email version had a ten-year-old boy, not the more usual pre-teenage girl, being accosted in a shopping centre toilet by 'an ethnic gang of youths' and castrated. Another story has a teenage girl going to the toilet in the restaurant of a large shop only to not return after half an hour. As with the girl with the re-dyed hair, the daughter is found just in time as she is being dragged unconscious out of the loo by two 'husky women' – she was being dragged off to be a white slave in the Middle-East.
The version with the castrated boy may have arisen from an earlier generations prejudice against Jewish communities.
The popularity and worldwide dispersal of urban legends involving crime and criminals is easy to imagine. Newspapers in North Wales, Leinster, Shropshire and Plymouth have published denials that child-snatchers are operating in shop toilets in their area. Each story contains a warning about a criminal practice, or the consequence of one moment of a lowered guard.
## 17
## LONDON BLADES
* * *
A Whitechapel Beau: one who dresses with a
needle and thread and undresses with a knife.
Attribute
* * *
## Hidden Blades
Back in September 2010 I took part in an artist's workshop on myth making. I took along a clipping about the dangers of hidden razor blades to illustrate a London version of a popular myth. Two of the group of eight said, 'Oh, you mean like the hidden razor blades in the water slide at Crystal Palace swimming pool?' They were more than happy to join in this violent idea of child-slicing with a broader tradition and share their stories. I came away with an addition to an urban myth and the thought that this sort of thing is probably quite normal for an artist's workshop on myth making, taking place in an abandoned shop at the top of the Elephant and Castle shopping centre.
In his book Urban Legends Uncovered Mark Barber told the same rumour about a nearby waterslide in Walton-on-Thames, Surrey. A Surrey chap himself, Barber told of a popular slide called 'The Black Hole' that children slid down in complete darkness. In 1985 a rumour persisted that 'gangs of youths' were stopping halfway down the slide and planting razorblades stuck down with chewing gum (for an extra unhygienic twist). A 13-year-old girl received serious injuries on her back and legs from using the slide. The popularity of 'The Black Hole' declined, and after a while the park, Barber reports, closed down.
A friend who grew up in south-west London remembered the same rumours being attached to a waterslide in Richmond. Again, it was chewing gum that held the blades in place. A strange message on an email list dedicated to lidos claimed that the 'Wild Waters' flume in Richmond Park was closed due to hidden blades injuring sliders and that a ghost known as 'The Phantom Slider of Richmond' haunted it, describing it as 'the most famous flume haunting in the UK'. I am not sure how serious the message on the lido list is. It does pick up the razor-hysteria which has spread far enough across the world that in America, where this legend is repeated, it is told that waterslides in England are banned because they bristled with hidden blades. American readers: this is not true.
In 2008, a 16-year-old worker in a McDonald's in New Plymouth, New Zealand, was cut by a broken pen whilst cleaning a children's play tunnel. The hidden-blade myth was well known enough that the blame was first put on a razor hidden there by persons unknown and with malicious intent to wound a child.
There's a similarity here to the ubiquitous urban legend of the razor blade hidden in the Halloween apple given to the trick or treating child. This legend transcends location, but is more popular in America. However, I've heard this repeated throughout my life, especially growing up during the 1970s and '80s. In his column 'Halloween Sadists', reprinted in Curses! Broiled Again!, Jan Harold Brunvand looked up the evidence for children being injured as a result of razor blades, syringes and poison hidden in their Halloween booty, and found none at all.
## Cut by Tart Cards
It's not only children who are in danger of getting hurt when out enjoying themselves. There are urban myths of bloody razor blades hidden in the coin return of vending machines and syringes hidden in cinema seats and variations of both. When two women were attacked at a bus stop in Haringey – one on 18 November 2011 and the other on 23 November – by a man with a needle, their first fear was that they had contracted HIV from the assault. At the time of writing neither the attacker has been caught nor has the women's diagnosis, as far as I can find, been made public.
This leads to the story I told the artists in the Elephant and Castle shopping centre. I have a clipping from the now-defunct London Lite newspaper of a police warning issued in north London, of razor blades hidden behind prostitute cards in King's Cross and Euston. The also defunct News of the World printed similar stories of cards with blades in Westminster. 'Tart cards', as they are known, are a familiar part of the central London landscape, and range from a quick description of the prostitute or specific services available written on a blank card in felt-tip to, thanks to cheap digital printing, full colour erotic images and a phone number. I've heard a rumour that one can track changes in the cards in different parts of London: regular sex in the West End and around the train stations, bondage and domination in the legal quarter of Lincolns Inn and things getting kinkier the further into the City of London one goes, the story being that the more affluent and high-powered one is, the more perverted one becomes.
The warning about the 'sex card booby trap' came from PC Dylan Belt of Camden Police. Gangsters were protecting their 'corner of the lucrative sex trade' by hiding traps behind their cards to prevent cleaners and rival gangs from removing them, and members of the public who may be interested in taking a card or two were warned against it. PC Belt is quoted as saying: 'We send the cleaners in and they find cards that have been booby-trapped. It could be with razor blades and they also use an irritant which burns the skin.'
An unnamed spokesman for British Telecom said, 'We will do everything we can to protect people using phone boxes,' which is what you would expect a spokesman to say whatever the danger.
There's a temptation here to read a broader narrative within this phenomena. The forbidden fruit of the apple comes at us all the way from Genesis to Snow White to anonymous villains punishing children for their greed at Halloween and accepting gifts from strangers. I think it is worth noting that in almost every instance of scarring waterslides, HIV-laced syringes and maiming tart cards the victim is not doing something entirely virtuous. They are not all procuring prostitutes, but they are all engaged in something fun, even the innocent use of a vending machine or waterslide. Leisure, it seems, and particularly when it involves sliding down a wet surface or an enthusiastic bite into an ill-gotten apple, has its dangers. In the stiffly moral world of urban myths no one is ever harmed after committing a selfless act.
## The Chelsea Smilers and Friends
Another urban myth that flourished in the 1980s in south London was that of the 'Chelsea Smilers'. The Smilers were a group of Chelsea football fans travelling London in a van with a smiley face painted on the side. They would stop schoolchildren and ask them questions about Chelsea football club. If the children got the questions wrong – perhaps they didn't support Chelsea or, worse, didn't like football – the gang would slice the corners of their mouth. They would then hit the child hard enough to make them scream, which would widen their wounds into a 'smile'. The thug's weapon of choice was a razor blade, knife or the edge of a credit card or phone card. Salt or vinegar was put onto the wounds to make the pain worse. A story of the Smilers' brutality always ended with a warning: the Chelsea Smilers were at another school yesterday, but they were coming to the child's school today.
In his book London Lore, Steve Roud talks about his daughter hearing the story in her school in Croydon in January and February 1989 and, from talking to her and her cousins at other schools, finding the myth at those other schools. By doing this, Roud was able to document the scare story as it spread: 'Many younger ones were in tears, some in hysterics, many refused to come home till their parents came to get them,' Steve writes. 'The children talked of nothing else.'
Roud was able to trace the rumour to where it began in Bexley, around 31 January 1989. It spread across south London, reaching Wandsworth, Merton and Sutton by the first week in March. Soon afterward, the story hit Kent and Surrey. He reported that the panic died down quickly, but the story became a standard playground scare story.
The phantom razor hiders and Chelsea Smilers are not the first imaginary gang to disfigure Londoners. 'The Mohawks', (also know as the Mohocks or Mowhawks) were a gang who wandered 1712 London. They were rumoured to have tattooed innocent faces, put fish hooks in people's cheeks and drag them along with a fishing line, and crush the noses, slit the ears and gouge out the eyes of their victims with 'new invented weapons'. The Mohawks were said to be like a horror-story version of the Bullingdon Club; rich young men who would meet in their clubs, drink to excess then head out, often into St James's Park, to cause havoc.
Another cruel trick of theirs was to put their sword between a man's legs and move it to make the poor chap dance, or to surround a man with their swords and one would stab him in his backside. The man would spin round to face his attacker and then another would stab him from behind. The idea was to keep the unfortunate fellow spinning around like a top.
With all of this cruelty on the street it may be surprising to learn there was was only one Mohawk trial. Sir Mark Cole and Viscount Hinchingbrooke are named in most books as the chief Mohawks. The total number of arrests was seven and the names read like a list of society party attendees: Edward Richard Montague, Lord Hinchingbroke; Sir Mark Cole, baronet; Thomas Fanshawe; Thomas Sydenham, gentleman; Captain John Reading; Captain Robert Beard; Robert Squibb of Lincoln's Inn, gentleman; and Hugh Jones, servant to Sir Mark Cole.
As recorded in Chambers Book of Days this aristocratic crew were put on trial for being 'mohocks'. Their crimes are explained below:
... they had attacked the watch in Devereux Street, slit two persons' noses, cut a woman in the arm with a penknife so as to disable her for life, rolled a woman in a tub down Snow Hill, misused other women in a barbarous manner by setting them on their heads, and overset several coaches and chairs with short clubs, loaded with lead at both ends, expressly made for the purpose.
The defendants claimed that they themselves were vigilante 'scourers' and were out looking for Mohawks. After raiding and wrecking an illegal gambling den, the team heard that the Mohawks were in Devereux Street. On arrival they helped three wounded men, but the nightwatchman John Bouch, an early type of policeman, mistook the rich crime-fighters for Mohawks, attacked them and arrested them.
The jury found them guilty and fined them each three shillings and four pence, which even for the early eighteenth century seems quite cheap for a sadistic night out. It is not clear whether their victims were ever found or if they were invented by the nightwatch, and it doesn't prove much other than a group of privileged men were convicted for a night's misconduct. It does not prove that a conspiracy of Mohawks ever existed. With a lot of rumour and little evidence, the doubts about these stories grew. Jonathan Swift thought the Mohawks were the result of mass hysteria, and Daniel Defoe thought they had the 'air of Grub Street' about them: Grub Street being the home of London's cheaper and more sensationalist publishing and writers at the time – an earlier Fleet Street, if you will.
## Dashing Blades
After the Mohawk moral panic of 1712 came the appearance of Spring-heeled Jack in 1838. Jack was a dark, iron-clawed, fire-breathing figure who would terrify people, often women, walking at night in London before making his getaway by leaping or bouncing over a wall with the aid of his spring-heeled boots. Jack is now thought of as a ghost or demon, some elemental presence spreading fear across London. He has featured in popular culture several times, from penny dreadfuls to comics to the fiction of Philip Pullman, as a supernatural or super-gadget bearing superhero.
The earliest description of Jack appeared in a letter from a Peckham resident to the Lord Mayor of London, published in The Times dated 9 January 1838, describing a dangerous bet laid by an affluent group of men:
The wager has, however, been accepted, and the unmanly villain has succeeded in depriving seven ladies of their senses, two of whom are not likely to recover, but to become burdens to their families. At one house the man rang the bell, and on the servant coming to open the door, this worse than brute stood in no less dreadful figure than a spectre clad most perfectly. The consequence was that the poor girl immediately swooned, and has never from that moment been in her senses. The affair has now been going on for some time, and, strange to say, the papers are still silent on the subject.
To do this, the 'unmanly villain' appeared in villages around London (including Peckham) disguised as 'a ghost, a bear and a devil', and had already left one woman so afraid she could not bear the sight of men. The Peckham resident thought that news or warning of this campaign had not yet appeared in the papers because those involved, being of higher ranks, had sought to keep the stories out of the press. In 1907, Jack was identified as the Marquess of Waterford, an aristocrat with a reputation for cruelty and practical jokes who would hide in dark places in costume, waiting to frighten people.
The identification with the rich may be twofold: firstly there is the idea that those in the higher echelons of society may have contempt for ordinary people and that they gain sport from tormenting and terrorising them. There is also the lack of capture or publicity about the great danger of Spring-heeled Jack. No Mohawk or Spring-heeled villain has ever been captured and shown to the public. This may be because they do not really exist and so are impossible to capture, but those convinced of their reality had other ideas: the Mohawks and Jack are rich and privileged and so escape arrest and publicity through their power.
The reality may be stranger and more sophisticated. Guising was popular in the seventeenth and eighteenth centuries, and dressing up as a ghost and walking the night, looking to frighten people was an almost common adult pastime. As well as Jack there was the Peckham ghost, Plumstead ghost and others. Mike Dash, in his authoritative Spring-heeled Jack: To Victorian Bugaboo from Suburban Ghost, investigated news reports on one of the most famous Spring-heeled Jack cases. As reported in The Times on 22 February 1838, Jane Alsop of Bearbinder Lane, Old Ford, answered a late-night ring at the door. She answered and the man at the door said, 'For God's sake, bring me a light, for we have caught Spring-heeled Jack in the lane.' Jane gave the candle to the man, who she thought was a policeman, but instead of running off with it he threw off his heavy cloak, put the candle to his chest and 'vomited forth a quantity of blue and white flames from his mouth'. Jane saw that the man was wearing a large helmet and that his clothes fitted him very tightly, like a white oilskin. Spring-heeled Jack, as the man was thought to be, darted toward her, catching her by the dress and back of her neck and placed her head under his arm. He began to tear at her dress with his claws and Jane screamed loudly for help. One of her sisters arrived and rescued her.
This account is the heart of the Spring-heeled Jack myth, and the description of the helmet and tight-fitting suit lead researchers in the 1970s to suggest that Jack was an alien running amok in early Victorian London. Mike Dash looked into supplementary accounts of the attack that covered the two investigations the newly formed Metropolitan Police opened to look into it. After a number of interviews, officers Young and Lea concluded that, 'In her fright the young lady had much mistaken the appearance of her assailant.' Two men, a bricklayer named Payne and a carpenter called Millbank, were seen walking away from Jane Alsop's house just after the attack. Millbank was wearing a white hat and a white fustian (heavy woven cloth shooting jacket), which the police thought was Jack's white oilskin. During the investigation, one James Smith, a wheelwright, described an encounter with Millbank and Payne later that evening on the Coborn Road. Millbank, the one in the shooting jacket, pulled the wheel Smith was carrying on his shoulder, and asked him, 'What have you got today to Spring Jack?' Smith replied that he desired Jack to give his wheel back. Smith told the police: 'I have no doubt but that the man Millbank was the person who so frightened the Misses Alsop.'
The myth of Spring-heeled Jack is of a lone monster, either man or a supernatural entity, scaring and assaulting the people of London. One big part of the myth is that it may have been an insane aristocrat. In a talk at the London Ghosts conference of October 2012, Mike Dash suggested that while the main suspect, the Marquess of Waterford, was known to have dressed in a devil costume at a party, this does not mean countless others were doing the same. It seems sensible to suggest that there was not one individual Spring-heeled Jack; this ghost, bear or devil was either a viral idea taken on by many men or something they did – guising in the city – that gained the label of Jack. Some may have been playing practical jokes, others have a more aggressive air to them, and many may have a blend of both.
## Saucy Jack
If Spring-heeled Jack, the Chelsea Smilers and the Mohawks are moral panics, is it possible that another series of actual violent acts have a fictional boogieman attached to them? Is the Jack the Ripper mystery not a mystery at all but a moral panic grown into urban legend and conspiracy theory? I think parallels between the rumours of Mohawks and Spring-heeled Jack and the theories about Jack the Ripper are worth drawing.
That the murders themselves took place is not in doubt; that there was one killer, the enigmatic Jack in his cape and top hat carrying a surgeon's leather bag, is an unproven idea that has developed into a cultural icon. Historian Jan Bondeson wonders in an article in 'History Today' whether the moral panic over the prostitute murders in 1888 created a myth of a single killer. He reports that ripperologists disagree on the number of victims that Jack took, and that two may have been murdered by partners or ex-partners. The violent death of Polly Nichols, Jack the Ripper's first victim, caused a moral outrage, like his Spring-heeled forebear and the Mohawks, and a number of other deaths – Emma Smith, Martha Turner and Rose Mylett – were, at first, also attributed to Jack the Ripper. These deaths have not made it into the 'canonical five' murders for which most ripperologists think Jack the Ripper was responsible; Mary Ann Nichols, Annie Chapman, Elizabeth Stride, Catherine Eddowes and Mary Jane Kelly. Annie Millwood, Ada Wilson and Annie Farmer were all suggested Jack the Ripper victims or survivors, but have since been discounted from the ripper-orthodoxy. Another victim, the aptly named 'Fairy Fey', was allegedly found on 26 December 1887, 'after a stake had been thrust through her abdomen', but there are no records of a murder in Whitechapel over the Christmas period of 1887.
The authorities were unsure whether Rose Myatt had been murdered at all or whether she had choked to death whilst drunk. Writing about the death, Robert Anderson, the officer in charge of the investigation, thought if there had not been a Ripper scare, no one would have thought she had been murdered.
With the mythology of the Ripper has grown the idea that the killer has never been brought to light because of a conspiracy amongst the powerful. Leonard Matters, described by Alan Moore as 'the first ripperologist', in his appendix to From Hell: the Dance of the Gull Catchers, named a Dr Stanley as the Ripper in his book The Mystery of Jack the Ripper, published in 1929. Stanley – not his real name – murdered and mutilated London prostitutes in revenge for his son's death from syphilis before fleeing to Argentina. Dr Stanley was no ordinary doctor, having a large aristocratic practice which no doubt protected him.
Prince Albert Victor, the grandson of Queen Victoria, was named as a possible Ripper suspect in the 1960s, after he was driven mad and angry as a result of catching syphilis from a prostitute. This rumour has evolved into the idea, popularised in Alan Moore's graphic novel and the film it inspired, that the Ripper was Sir William Gull, surgeon to Queen Victoria and a Freemason, another secretive group seen by some to be above the law. The conspiracy now is that Albert Victor had an affair with a woman which the Ripper victims found out about and were murdered by an insane Gull to cover up the truth.
Other suspects include the Duke of Clarence, Sir John Williams, who was obstetrician to Queen Victoria's daughter Princess Beatrice, and sensitive, creative types such as Lewis Carroll and painter Walter Sickert. Each suspect appears in a new book and with the continued growth in popularity of Ripper lore and the deepening of the myth, new and even more unlikely suspects are investigated all the time. After a long look comparing Jack the Ripper crime scene photographs and the paintings of Vincent van Gogh, writer Dale Larner has concluded that van Gogh was, indeed, Jack the Ripper. John Morris takes the idea of Sir John Williams being the Ripper, driven to insanity after not being able to have children, and transfers the crimes to his wife, Lizzie Williams, in his book Jack the Ripper: The Hand of a Woman. Bram Stoker, author of Dracula has never been in the frame for being Jack the Ripper, but The History Press book The Dracula Secrets: Jack the Ripper and the Darkest Sources of Bram Stoker, suggests that Jack the Ripper was, sort-of in a round-about-way, Dracula. That through 'a secret code' found in 'previously unpublished letters', Stoker wove details of Ripper suspect Francis Tumblety into his novel. This 'ripper code' was inspired by Stoker's relationship with Sir Thomas Hall Caine, to whom he dedicated Dracula. Caine also had a relationship with Tumblety, and Tumblety was fingered as a Ripper suspect in the book Jack the Ripper: First American Serial Killer. Tumblety was arrested in 1888 for 'gross indecency', and was possibly gay. Did this drive him to murder and mutilate women? I must confess that I have not read any of the above books; these theories are taken from promotional websites, press releases and news reports, so I have no idea whether each author is sincere, cynically milking the myth for money or undertaking a conceptual exercise in how evidence can be bent into the strangest proofs.
As well as Jan Bondeson, in 1986 Peter Turnbull published his book The Killer Who Never Was, putting forward the no-Ripper hypothesis. Ripperologist and tour guide John G. Bennett published Jack the Ripper: The Making of the Myth in 2011 which, while not denying the single-killer hypothesis, did much to disembowel the countless scabbed-over theories about the original murders. Retired murder-squad detective Trevor Marriott brought his experience into investigating the Jack the Ripper killings and concluded that there was no Jack. If the evocative name Jack the Ripper had not been attached to the Whitechapel killings, the theory would have been forgotten a long time ago. He decided that at least two of the women 'were killed by the same hand' and the others, if they were related at all, were copycat killings. 'The urban myth was created by an overzealous newspaper reporter sending a mysterious letter signed Jack the Ripper. The police certainly never believed in a killer known as Jack the Ripper.'
These theories are a little way from the Mohawks' dangerous rakes, but nearer the fantastical attacks of Spring-heeled Jack. It is the idea of a rich and debauched individual committing murder and mutilation and escaping justice because of their privilege, that all of these blade-wielding figures, fact or fiction, share. The actual, certain evidence for the Whitechapel killings is the bodies of the victims. The theories and the name Jack the Ripper came in the hysteria afterwards, a hysteria that still bends thought. There may have been one murderer, or each killing could have had its own sad story, but the idea of a Victorian killer named Jack the Ripper has such gravity to it that people cannot resist its pull.
## 18
## THE ACCIDENTAL THEFT
* * *
Our old cat died last night
Me wife says to bury it out of sight
But we didn't have a garden;
We was livin' in a flat
So what was I to do with the body of a cat,
Then a big brown paper bag I spied
I put our old dead kittycat inside.
And now I'm off down the street with the body in the bag,
The body in the bag, ta ra ra.
'The Body in the Bag' by Charles O'Hegarty
* * *
JAN HAROLD BRUNVAND often mutters in his books that one should 'never trust a dead cat story'. So consider yourself warned while I tell you the tale of the single 'lady scholar' working at the British Museum who fended off loneliness by sharing her lodgings with a cat. This is a story Brunvand collected for his 1983 book The Vanishing Hitchhiker, which may be why a 'lady scholar' isn't just a scholar. She smuggled the cat into her room, bribed the maid to keep quiet and lived with the cat over winter. In time, though, the cat died and with no garden in which to bury the cat, our scholar neatly and secretly parcelled the cat up to put into the building's incinerator. She was interrupted by the establishment's proprietor and thought, 'this will never do', and so headed off to the British Library. (This would have been when the library was within the British Museum.) She saw a good place along the way to get rid of the cat – a culvert – but this time a policeman came round the corner just when she was about to do the deed.
Some days you just can't get rid of a dead cat, so she took the parcel to the museum and, at lunchtime, was stopped by the guard letting her know that she had forgotten her package. 'This is getting funny,' she thought to herself. She had failed to leave her dead pet on the bus and on the tube, so in desperation rang her friend who told her to come to her, as there was a local pet cemetery in which she could inter the cat.
When she arrived, perhaps filled with guilt at the way she had tried to abandon the cat's remains, she opened the parcel to have one last look at the cat and found... a leg of mutton.
The first part of this legend makes it into Mark Barber's book Urban Legends Uncovered, albeit in a dishevelled state, with a 'young lady' who worked at the British Museum living in a one-bedroom flat with her beloved cat. Not wanting to bin it and with nowhere to bury the cat she set out to inter it in the nearest pet cemetery, which was 10 miles away. She put the cat in a box and the box into a large carrier bag. On the way to this distant animal graveyard she popped into a clothes shop that she did not visit very often, due to it being so far out of central London. While in the shop and looking at a couple of dresses, she put her bag down for a second and when she reached for it again, the bag was gone.
Then there was a disturbance outside the shop: a woman had fainted on the street. Our bereaved cat owner saw that the unconscious woman had her missing bag clutched to her chest, with the head of her dead cat poking out of the top of it. The passed-out woman was a known shoplifter who had been operating in the area for months.
This second version of the dead cat story is as classic an urban myth as babies in microwaves and hairy handed hitchhikers. Its purpose is clear as a revenge fantasy for those who have been robbed, and versions of it appear all over the western world. Whilst dead cats are very popular, often it is a bag of collected dog excrement that is snatched in a park, or a urine sample in a whisky bottle stolen by a thirsty thief. It is a stray old alley cat of a story that crops up, occasionally mangy and reeking, to the party. It is easy to understand the ubiquity of this story, as everyone wishes ill on the person who has snatched their bag or picked their pocket. Some years ago my wife had her bag snatched on Whitechapel High Street while on the way home from a gig. She had been to her dance class before that and the bag contained only her worn dance kit. A few days later, a friend imagined the thief getting back to his crime den with nothing but a used women's dance outfit and his boss making him wear the worthless costume and dance on a table for him as a punishment.
Managing to swap a dead cat for a tasty piece of dead sheep is a different outcome, and has the cat-carrier inadvertently becoming the thief themselves. The constant attempts to dispose of the package end with something far more valuable than a departed pet for one and, presumably, a frustrated roast dinner elsewhere in London for the other.
The mistaken theft crops up a lot in British folklore. A chestnut of a tale that is as common as the dead cat tale is the story of the valuable thing left on the mantelpiece (not the most inspiring title for an urban myth, but please bear with me).
One version of this tale starts with Peter, who is on a business trip in London, discovering that his gold watch is missing while he is travelling back to his hotel on the last tube train.
On the platform is a young man grinning at him and Peter decides that this man must be the thief. He leaps up from his seat and grabs the young man by the lapels of his suit, only for the tube doors to close in front of him, tearing the man's suit lapels off.
Back at his hotel room Peter phones the police to report the theft and then phones his wife to let her know his gold watch is gone. His wife says, 'I'm glad you rang. Did you know you'd left your watch behind on the dresser this morning?'
Other versions have the 'robbed' man wrestling his wallet from the thief, only to find it at home, or the more genteel version with an elderly lady going into town by train with £5. She dozes off during the journey and when she wakes, there is another sleeping woman in the carriage. She then goes into her bag to check her shopping list, finds her £5 is missing and, on impulse, checks the bag of the sleeping woman. There, at the top of the bag, is the £5 note. She removes it quietly and decides not to confront the woman or report her, so leaves her sleeping in the train compartment. With her shopping done, her husband meets her at the station and asks, 'However did you get all that stuff? You left your £5 note on the mantelpiece.'
Both dead cat stories, the accidental theft and the bag-snatcher, are intertwined. In another version the dead cat is taken while the woman has lunch with a friend. The thief faints in the toilets whilst checking her ill-gotten gains and the cat owner finds the theft and the cat package she was trying to lose. As the thief is stretchered away the woman passes the repackaged cat to the paramedics with the words, 'I think this is hers.'
The trope of wandering London looking for a place to leave a dead cat because you do not have a garden is older than this urban legend. It was recalled by Eric Winter in a musical song he recorded in the journal Sing on 5 July 1960. The song 'The Body in the Bag' by Charles O'Hegarty is a cockney music-hall song about a frustrated man who is trying to leave a cat somewhere. The lyrics mirror the troubles of our British Museum lady scholar:
I went off down the street to have a whisky neat
And carefully laid my dead cat underneath my seat.
Then I got down on my hands and knees;
Went halfway through the town,
When the barman stops me,
'Here's your parcel Mr. Brown,'
So I had to thank the silly fool
And give him half a crown
For bringing me the body in the bag.
Having failed all day to get rid of the cat, Mr Brown hears a noise in the bag:
All at once from in the bag
There came a plaintive meow
Say Puss, 'I'm dead no longer,
You needn't bother now.
You've often heard it said
That a cat has got nine lives,
Well, I'm a married Tabby,
One of Tommy's wives
And our families they usually come
In threes, and fours, and fives.'
But there were seven little bodies in the bag!
Another accidental theft is the urban legend about two travellers and a packet of biscuits. Printed versions appeared around 1972/74 and the legends appeared in Folklore from the summer of 1975. A traveller buys a cup of tea and a packet of biscuits in a Joe Lyons corner house, opposite Liverpool Street station (or in a buffet car or station café), and sits down to enjoy them. Also sitting at the table is an African (or Pakistani or West Indian) man, who helps himself to one of the biscuits. Shaken by the effrontery of this, our traveller takes another. The uninvited biscuit-eater takes another and this continues until they are down to the last biscuit. Here the African (West Indian or Pakistani) man breaks the biscuit in two and hands the traveller half. Our traveller loses his temper at this point and hurls abuse at the man. It is only then that he (or sometimes she) realises that his own packet of biscuits is lying unopened on his suitcase, and that he had been helping himself to the other fellow's biscuits.
This story made it as far as Paul Smith's The Book of Nastier Legends published in 1986, where the setting was a café in Southampton, biscuits became the fingers of a Kit-Kat bar and the patient and sharing individual changed from an ethnic minority and possible recent immigrant to an 'outrageously dressed' punk. The message of the story is the same as with the tea house: don't judge people by how they look and, as with all of these stories, double check before confronting someone and do not be so suspicious. Also, take better care of your cat, be it dead or alive.
## 19
## CONCRETE JUNGLE
* * *
If you took the city of Tokyo and turned it upside down and
shook it you would be amazed at the animals that would fall out.
It would pour more than cats and dogs, I tell you.
Yann Martel, Life of PI
* * *
## Rat Land
In London you are never more than a certain distance away from a rat. This is an almost universal indicator of urban filth, London's hidden dangers and the fear and loathing a lot of us have for rats. Each time the idea of rat proximity is repeated the distance varies. A quick Google suggests 6ft, 7ft, 10ft or a metric 5m or 18m. Why do we even think it's possible to have an average distance from a rat? Do London rats outnumber London human beings? This idea seems to come from the 1909 book The Rat Problem by W.R. Boelter who undertook his research by asking country folk whether they thought it was reasonable to say that there was one rat per acre of land. Boelter made an estimated guess at 40 million rats, as there were 40 million acres of cultivated land in Britain at the time. There were also around 40 million people in Britain at the time. Since then, it only seems right to think that the rat population has increased more quickly than the human population, rats must breed like rats after all, and so now they must outnumber us, particularly in our grimy cities.
Luckily, Dr Dave Cowan, leader of the wildlife programme at the Food and Environment Research Agency, has tried to work out the actual person:rat ratio of Britain – both town and country – in a more scientific way. Counting cities, sewers and farms (farms are the most popular rat territories), Dr Cowan calculated that there are 10.2 million rats in Britain. The UK has 60 million human inhabitants, so people outnumber rats by six to one. As for approximate distance from a rat in an urban area you would be, at most, never more than 164ft (50m) away from a rat. Although you may, of course, be much nearer.
## Parakeet Superstars
Unlike the pigs living in London's sewers or the big cats roaming its suburbs, the parakeets of London are not an urban legend in themselves. They have been reported across London, from Twickenham to Boreham Wood to Hither Green. My own visits to open spaces in London, from Kensal Green cemetery to Manor House Gardens in Lee, have been cut through by a flash of green and a sharp parakeet squawk. Ring-necked parakeets cover south and west London, while the Monk Parakeet has colonies in north London. One encounter with the ring-necked variety in February 2011 in Richmond Park was like a Mardi Gras version of Alfred Hitchcock's The Birds; the old oak trees were thick with their bright feathers and delirious parakeet chatter.
How they arrived in London is another story, or rather stories. I first encountered the legend of west London's parakeets in a copy of Time Out from June 2005, which claimed they are all descended from a pair that escaped from Jimi Hendrix's flat in Notting Hill. The two birds were like the guitarist himself: exotic and flamboyant in a cold grey London. On escaping, they went to found a nation of parakeets in London and provide a high-pitched, alien soundtrack to the coo of London's pigeons and chattering of sparrows. In a south London special of the Evening Standard's ES Magazine in 2012, the parakeets were released by Hendrix, rather than escaping, and Surrey Life magazine in December 2011 imagined Hendrix playing 'Little Wing' as they sailed from the window and into Notting Hill.
In its 'Myth Busters' column, the Fortean Times, in January 2010, describes a version that is just that little bit more dramatic, as the parakeets are accidentally released from Hendrix's flat following his death. From medieval to Victorian art, the human soul can be depicted as a dove departing through the window at the moment of death. Perhaps Hendrix's soul couldn't be anything as tame as a single, cooing dove.
Another story tells of the parakeets being the descendants of film stars rather than the pets of a pop star. The same Time Out article repeated the story that the parakeets are descended from some birds that escaped Shepperton Studios during the filming of the 1951 film The African Queen. A friend offered me another version in 2012, by suggesting the parakeets were related to a different African queen, having flown from the set of the 1963 film Antony and Cleopatra.
Yet another account describes a mass escape during the great storm of 1987, when an aviary was damaged in Northdown Park in Kent. If that is not dramatic enough, how about the parakeets being freed from quarantine at Heathrow airport by a storm? Or that a plane fuselage crashed landed on an aviary near Heathrow airport, or that the birds flew to freedom when the tanker that carried them ran aground or capsized? In many urban myths the parakeets of London do not arrive gradually; the story has to be 'disaster!' followed by an 'instant parakeet hoard!'
Another celebrity version of the parakeet origin myth is that they were escapees from the aviaries of King Manuel II, Manuel the Unfortunate (the Portuguese king who ascended to the throne after the assassination of his father and brother and had to flee on 6 October 1910 during Portugal's republican revolution). Manuel landed in Fulwell Park, Twickenham, for his exile where, it has been said, he attempted to recreate Portuguese life, including building an insecure cage for some parakeets. I have not been able to discover whether this story pre-dates the Hendrix one or if they are related. In any case, the parakeets are in London and any slightly exotic figure could be linked to their origins.
The story of the parakeets escaping Shepperton Studios is undone a little when it is pointed out that The African Queen was not filmed there but at Isleworth Studios. Isleworth may be quite near to Shepperton (as the parakeet flies), but no parakeets were imported for the making of the film. Britain has had a long history with these birds. In its factsheet on feral parakeets, the Department for Environment, Food and Rural Affairs states: 'There is a long history of occurrence in GB, with a first record of breeding in Norfolk in 1855. However the present naturalised population dates only from 1969.'
A 1999 census of parakeets, which includes London, describes the birds as 'successfully breeding in the wild in the south east of England since 1969', suggesting there may be a Hendrix link (Hendrix died in 1970). It depends on how long it took the parakeets to get productive before anyone noticed. The census describes the origin of the birds as numerous escapes leading to 'many feral populations'. Although it is a great story to imagine that one famous parakeet owner is the daddy to the birds all around us, there are, of course, hundreds of anonymous parakeet owners who may have lost or released their pets. The book Parrots, by Cyril H. Rogers, says of the Ring-necked parakeet that they are 'probably the most common of all the "Polly Parrots"', so it is perhaps no surprise that enough have escaped to form a breeding population across south London. And no matter how virile the parakeets of Jimi Hendrix may have been, they surely haven't populated all of London with birds.
That all of these stories involve an accident having such a sudden effect on our environment, like a clumsy rock legend, an aviary smashed in an historic storm or a tanker running aground, says a lot more about the human need for narrative than it does about Anglo-Indian parakeets. The numbers and visibility of this relatively new arrival must, it seems, indicate that they come from one event or source: a disaster, an exotic recreation of Africa on a film set, or a rock star opening up his window and sending out bright-coloured birds across the city. The banal explanation of cage and aviary escapees, with long lifespans and fruitful breeding cycles, adapting to our environment does not satisfy.
An article from ES Magazine in 2012 on cosmopolitan south London described the parakeets as 'cheeky birds' and as a 'cheery sight in parks from Greenwich to Brixton to Richmond', while the book Fauna Britannica has the Spickett family of Twickenham describing them as 'invading Benfleet more than thirty years ago' and records their glee at a pair being mobbed out of a spruce tree by the local magpies. The Times, perhaps with its tongue in its cheek, fears that the parakeets are almost a harbinger of environmental and immigration doom. The parakeets are 'the latest, and loudest, evidence of global warming' as well as '[...] further disquieting proofs of shifts in the natural world: ornithologists fear that these parakeets – robust, adaptable and aggressive – will impinge on the habitat of indigenous species such as starlings, kestrels and little owls.'
Fear not for our little owls just yet though. The RSPB's policy on parakeets, last updated in 2009, does not offer any evidence that parakeets are a threat to other species, nor does a 2011 article published in Ibis: The International Journal of Avian Science. Others differ, and in 2011, DEFRA instigated a cull of Monk Parakeets to stop the £1.7 billion-a-year damage they allegedly cause to the British economy.
These out-of-place, bright birds couldn't have as mundane an origin as London's iconic pigeons, which are also feral animals with similar origins. The rock pigeon's native environment is the western coast of Britain. Our London version is descended from domesticated rock doves that have escaped their coops and turned feral on London's streets, buildings and parks. They have been in London for a long time; so long, infact, that they are very much a part of London's landscape. Fourteenth-century Londoners threw so many stones at pigeons that they would break the windows of St Paul's, and Pepys pitied them during the Great Fire of London. They have been with us so long they no longer need a story to explain their presence, unlike the parakeet.
Parakeets have appeared in the UK under much stranger circumstances than being celebrity escapees: in 1895, The Field magazine reported a parakeet sighting in a farmyard in Gledfield, Scotland, two years after another parakeet had visited. No one local had claimed to have lost a parakeet.
## Crack Squirrels and Squirrats
If the poor rat and feral parakeet are abused for the purpose of demonstrating the corrupting effects city living has on nature, then please pity the poor grey squirrels. Dubbed and damned as 'tree rats' for their bird-table raiding ways and being a large immigrant from America that has driven the indigenous red squirrel to the far corners of the kingdom, in summer 2007 the Sun newspaper accused them of a much closer relationship with London's rats. On 31 July it published a photograph of a squirrel with a bare, rat-like tail, taken by central London artist Sia Sumaria. The next day Tom Crew photographed a rat-tailed squirrel on his tree-lined road in Dulwich. Had squirrels and rats interbred to produce 'squirrats' – a fearless urban hybrid?
'The one I saw wasn't afraid of anything and seemed quite tame,' said Mr Crew. 'Most squirrels dart up a tree when you approach them, but this breed is very confident and stood its ground. I've seen a whole family of them in my road. The hairless tail makes them look so strange.'
Unnamed experts suggested that squirrels and rats cannot interbreed and instead of a cross-rodent love-in, the squirrats are simply squirrels with a diseased, bare tail.
Possibly not quite as bad as rutting with rats is the suggestion on the front page of the South London Press on 7 October 2005 that the grey squirrels of Brixton were crackheads. It quoted a local resident who 'did not wish to be named' who had seen an ill-looking squirrel with bloodshot eyes digging in his garden. An hour later, a neighbour informed him that local crack dealers and users had been using his front garden to hide their rocks of crack.
The rest is left to the reader's imagination. Such a weak story was still quirky enough to make the Daily Mirror, the Guardian and the squirrel-damning Sun. By 18 October, Fox News in the US were repeating the story of London's crack squirrels with a mention of a footnote in the South London Press' story that crack squirrels were already a problem in New York and Washington DC. Squirrels on crack even made it into the BBC adult puppet show about urban animals, Mongrels.
So, squirrels of London, please tell us straight, we don't need to know about the rat sex, but do you have a crack problem?
Researching the story for a Fortean Times article, Ben Austwick found a location for the original, unnamed source for the story. On 3 October 2005, a user on the Urban 75 South London web forum began a post that began by matching the report in the South London Press. The user posted about dealers and users hiding their stash and squirrels, which were not ill-looking or bloodshot-eyed, that had been digging in their garden, and then they joked about the squirrel mistaking a rock of crack for a nut or acorn: 'But what if they did? And do I face the prospect of dreaded crack squirrels? Turf wars (flower bed wars) between dealers and squirrels?'
A joke on an internet forum was picked up by a journalist on a local London newspaper who fitted it into a news story. This is why there is no positive sighting of a squirrel chewing on crack; the original poster did not describe it and the journalist chose not to invent that part. Perhaps that would have been too dishonest.
## The Spider in the Supermarket
A popular urban legend from the 1980s, which is currently dormant, is the tale of the squeaking pot plant. It was brought home and, in one version, made squeaking noises when watered, which the woman plant owner thought was the sound of air escaping from the dried pot as the water went in. Then the earth began to shift around the base of the plant, and so she called the police. In turn, the police called the local zoo who removed a large female tarantula and her nest of fifty youngsters. In the London version the spider is discovered at Kew Gardens. Kew's plant inspector, Jim Kessing, said, 'One of our gardeners said it happened to a friend of his son's. He asked me if it was possible. I told him it was – but a bit unlikely.' Tom Kelly, the manager of the Marks & Spencer Oxford Circus branch, lamented in 1985 that 'it's getting beyond a joke. Now we've got an official complaint from the Irish Ministry of Agriculture because someone in Dublin claims one of our people offered a woman £100 to keep it quiet!' Was that in cash or M&S vouchers? This story was so popular that it made the cover of Paul Smith's The Book of Nastier Legends in 1986, passing from person to person in a crowded pub. On the record, Marks & Spencer denied the possibility of illegal immigrant spider families invading London hidden in pot plants, as the African yuccas were all replanted in the Netherlands before arriving in our supermarkets.
Baby spiders feature in another 1970s and '80s urban legend that addresses the danger of travel and foreign lands, with the tale of the girl being bitten while on holiday on the coast of North Africa, or being bitten while on the plane heading home to London. In a Glaswegian version the bites fester until the girl goes to wash her wound and, positioned in front of a mirror to gain maximum horror, her face erupts with baby spiders.
Whilst that story is thankfully not true, a similar legend of tarantulas lurking in supermarket bananas is something that really does happen. On 4 June 2013, Mark Drinkwater was shopping at a Lidl in Sydenham. He reached into a banana box and out came a large spider attached to a bunch of bananas. He told the News Shopper on 13 June:
It was the size of the palm of my hand. It was hairy. It was scary enough. I shook the banana and the tarantula fell back into the box. It probably wasn't very happy having been thrown back in the box. At the time I didn't panic, I was relatively calm, but later I could feel my heart beating through my chest. I decided not to buy the bananas.
He informed Lidl staff that there was a possible tarantula in the box and there followed a fine piece of improvisation: staff located the largest Tupperware bowl they could find and trapped the beast. Mr Drinkwater was put off buying bananas for a few weeks.
I emailed Lidl press and public relations office, and PR Manager Clare Norman confirmed the story and continued it. The 'unidentified spider' (Clare's words) was contained in the shop's disposal freezer while the RSPCA was contacted. They suggested Lidl contact the British Arachnological Society, who advised the staff to keep the spider in the freezer 'for a length of time' until it could be 'subsequently disposed of'.
I had to double check what that meant and it turns out that the best way to deal with a tarantula, which is how the spider was referred to in my second email from Clare Norman, is to freeze it. The freezer was their animal by-products freezer, which presumably, is where all the remains from the butchers' counter go so they do not decay too unpleasantly until they can be disposed of properly. The animal by-products service provider took the contents of the freezer, tarantula and all, and incinerated it. Just in case I did feel sad for the tarantula, who was a long way from home, Clare did reassure me that during incineration 'the energy from it is then used for electricity and other renewable energies.' I now think of the spider whenever I switch on the kettle to make a cup of tea.
## Spontaneous Snakes
A little-known fact about London is that it often sprouts spontaneous snakes. Rodney Dale, in The Tumour in the Whale, tells the tale of a man finding a sleeping snake while strolling through Regents Park. Presuming it had slithered out of nearby London Zoo, the snake keeper was called while other staff members watched over the serpent. Once captured and checked over, the snake was found not to be an escapee. More snakes were found: a London & North Western van driver found a boa constrictor in his van and the son of an MP found a huge snake in a room in his father's London house.
More recently, in 2002, the New Grapes Church band were returning from playing at a wedding in Westminster to their church on John Wilson Street in Woolwich, when they found a 6ft python in their van. The previous people that had hired the van had managed to leave their snake behind.
There was no such simple explanation for the snake recorded by Charles Fort in his 1931 book Lo! A snake appeared on Gower Street, Bloomsbury, in 1920 in the garden of a Dr Michie. And what was the explanation for the rogue reptile? It was a naja haje, an Egyptian cobra, which, therefore, must have been kept by a foreign student staying on Gower Street: 'The oriental snake had escaped from an oriental student.' Fort saw through this though, stating: 'I don't see that oriental students having oriental snakes is any more likely than American students should have American snakes: but there is an association here that will impress some persons.'
## The Penguin Entertained
A penguin popped out of a duffle bag when a boy sat down to his tea after a visit to the zoo. His mother telephoned the zoo, but after a count the zoo keepers found, like the snakes at London Zoo, that they had a full complement. There's a joke about a penguin; the earliest version I've found is in The Tumour in the Whale. A man walking up St John's Wood Road was approached by a penguin. The man found a policeman, asked him what to do with the bird, and was told, 'Take him to the zoo if I were you, Sir.' The next day the policeman saw man and penguin walking again. 'I thought I said you should take the penguin to the zoo,' the policeman said. 'I did,' replied the man, 'and this afternoon we're going to the pictures.'
There is something about a penguin abroad that appeals to people. It may just be down to them being cute yet awkward birds that walk about on two legs. How long had the penguin wandered north London before approaching the young man for a date at the zoo? People do steal penguins from zoos: in 2012 two drunk Welsh tourists stole Dirk the Penguin from Seaworld on Queensland's Gold Coast, and a penguin was taken from Dublin Zoo in summer 2010 and found wandering the streets a few hours later. The question to ask here is which came first, the urban myth or the penguin theft?
## 20
## THE FANTASTIC URBAN FOX
* * *
Although Fox hunting has developed its own special lore and language, the Fox has entered rather little into British Folklore.
Stefan Buczacki, Fauna Britannica
* * *
## Urban Foxes Abound!
There is a reversal to the trend of out-of-place animals invading London story, and that's urban animals invading the greener and more pleasant British countryside. So rather than an urban legend about London, these stories are about how London is seen in other parts of the United Kingdom. In October 2004 the conservative MP for Lichfield, Michael Fabricant, tabled a parliamentary question for the then environment secretary, Margaret Beckett, about the dumping of urban foxes. He had learned of this while visiting Snowdonia. 'With the growing problem of increasing numbers of foxes in our towns and cities, it seems that do-gooders are now transporting live urban foxes from the West Midlands and other conurbations and releasing them into rural Wales where it is thought they will do no harm,' said Fabricant in a 10 October 2004 press release (which is still on his website at the time of writing). 'Instead, they are savaging sheep, poultry, and pets in hill farming country.'
His views on foxes were backed by Nick Smyth of Llwyngwril near Dolgellau, in a letter to the Dysynni and Cambrian News:
A van in a motorway car park was found by acquaintances of ours to be full of urban foxes. When questioned, the driver stated that the animals were being taken to a remote part of the country, where no one lived and no one, in London presumably, had ever heard of, where he said they would do no harm...
Smyth goes on to say that worse still, 'the driver could not pronounce the name of our village.' Smyth reported that farmers had shot more foxes than usual in the past three months, having dispatched 118. 'Somehow the ignorance of these town-dwellers and misguided do-gooders has got to be dispelled,' Mr Smyth muttered.
The supposed ignorant townies and do-gooders were the real issue with these migrant foxes. Rumours of urban fox dumping began over ten years earlier with the Farmers Union of Wales (FUW) issuing a press release entitled 'RSPCA accused of Mass Fox Releases'. It claimed that a farmer helping a lorry out of a ditch discovered a strange cargo. It carried forty-seven urban foxes, which were being transported by the RSPCA from Birmingham to their new pastoral Welsh home. The foolishness of this was outlined by Kim Brake of the FUW as these 'townie' foxes have 'little hope of surviving; and [it is] unfair to farmers who have to pick up the bill in slaughtered lambs.'
Hunt supporters in Cumbria claimed their local fox population had suddenly doubled, and that the new foxes looked and acted differently to the indigenous Lakeland foxes. Ted Bland, a hunt supporter with Lunesdale Foxhounds in Lancashire, claimed to have seen four foxes released from a van, whilst other reports claimed that vans carrying seventy foxes and lorries with up to 200 were heading out of the cities and into farmland ready to unload what was presumably drugged urban foxes.
The stories continued. A broken-down lorry carried ninety-seven urban foxes into Wales, with the men being paid £5 per fox to get them out of the cities. In West Somerset a blue van with no number plates was sighted doing a night-time fox deposit. Possible London foxes were described in Tendring, Essex, as 'not even scared of headlights'. Henry Gibbon said, 'The other day one fox looked down the barrel of my gun as if to say good morning.' These were hardened urban foxes.
From the cities themselves came doubt and enquiry about these fox-ferrying stories. Wild fox welfare charity the Fox Project ridiculed them, and the League Against Cruel Sports investigated, contacting almost every animal shelter in the UK and making extensive enquiries with local authorities: all denied dumping foxes. The BBC's rural news and magazine programme Countryfile put its money where its mouth was by offering a £1,000 reward for information that led to the identification of any animal welfare groups involved in fox smuggling. Not a penny of licence fee money has been paid out.
Why were fox hunters and their followers getting angry about not-so-fresh foxes being delivered to their land to be hunted? The claim was that it swelled the local fox population to levels dangerous to other wildlife, with more hungry, city foxes spreading mange amongst the rural ones, and that the urban foxes, suddenly finding themselves in an unpolluted and open environment, were lost, confused and prone to a swift and painful demise. And not only by hunting dogs.
As John Bryant pointed out in his 'Hunters & Dumpers' article in Issue 73 of the Fortean Times, the 1994 fox-dumping rumours emerged at the same time as Labour MP William MacNamara's private Wild Animals (Protection) Bill was brought before the House of Commons. The bill sought to make law a six-month prison sentence for anyone inflicting unnecessary suffering on a wild animal, and ban the use of dogs to 'kill, injure, pursue or attack'. It was an attempt to outlaw hunting. This attempt failed, but in 2004, with the Smyth and Fabricant fox-myths fuming in the background, the Labour government passed the Hunting Act, making hunting live animals with dogs illegal.
## Urban Fox Hunts
As wild foxes were supposedly saved from being hunted in the countryside, their lives in London became more dangerous. The first mention of an urban fox hunt I have found is from gonzo free-sheet Vice in December 2003. Kid A and Kid B of Lambeth were asked why they hunted foxes, aside from the financial gain. Apparently Lambeth council charged £200 to shoot a fox but the street youth 'only charge a tenner'. 'Fuck it, I'm street. They shit by the swings anyway,' said Kid A in a touching mixture of childlike indignation and lack of empathy. Their preferred method was to drug the foxes' food, wait until the poison affected them and then beat them to death with a bat, or shoot them with a pellet gun.
It's not all anger though; Kid A says, 'I wanna get a fox for a pet anyway. I want to track it back to its lair, get hold of a little cub fox, innit. Take it for walks. Train it to fight.'
Is it just a story? There is blurry photo of two boys on bikes, one with a fox over his shoulder, complete with black bars across their eyes. This would need to have been faked, but such things are not difficult with a stuffed fox prop.
BBC London radio DJ and London enthusiast Robert Elms heard the urban fox-hunt story and would occasionally disappear into a reverie of Mod fox hunting conjecture, cruising London in sharp suits on scooters looking for pesky, pestilent foxes to punish. I think radio has a big influence on the dissemination of urban myths; it's a human voice telling you a story or wondering about a curiosity that is heard by thousands of people. The content of a regular with unscripted and informal dialogue from the presenter and guests is an environment where myths can evolve, and the amount of content makes them difficult to catalogue and reference.
In August 2010, a video was released on YouTube by a group called the Urban Foxhunters, which shows them hunting down a drugged fox and killing it with a cricket bat. The group, claiming to be from around Victoria Park, hated foxes and saw killing them as a public service, describing it as 'a bit unpleasant but it has to be done to keep our streets safe. I have kids and I don't want them being bitten by a diseased vermin scum, what's wrong with that?'
One member, Lone Horseman, wrote on the blog: 'For the record – when we kill these foxes they are dosed up with Xanax, which if you haven't tried it is a trippy anti-anxiety drug. Trust me these fuckers are dying with a smile on their face.'
If this sounds like an absurd Chris Morris-style way of baiting the media, that is because it is. It echoes closely with Vice's 2003 fox hunts in Lambeth, though this could be a coincidence from both parties thinking through the logistics of catching a fox in London. Shortly after, the story appeared right across the British press. Most were appalled by the bludgeoning of a wild animal, 'diseased vermin scum' or not. The Metropolitan Police's wildlife crime unit began to make enquiries, and both the Fox Project and John Bryant, who offers a 'humane deterrence' service for wild animals, each put forward a £1,000 reward for the identity of the group. Meanwhile in the Evening Standard, 'London Diary' columnist Sebastian Shakespeare made another fox-linked political point and enthused:
Those urban fox killers are a perfect (or imperfect) example of Cameron's Big Society in action.
Dave wants to empower communities to do things for themselves. People power, he calls it, redistributing power from the government to the man and woman on the street. 'These are the things you do because it's your passion,' says the PM. Well, you can't accuse the fox killers of lacking passion. There is no denying they are performing a public service. It is about time we learned to be big enough not to have small feelings about foxes. They are pests. And as we now know they have changed their habits and started attacking children.
The 'Big Society' was the plan to cover billions of pounds of cuts to local authorities for essential services by getting passionate volunteers to do them for free. So, while an expensive municipal government offers trained council workers removing foxes with snares and rifles, the Big Society produces a vigilante group with cricket bats and prescription drugs battering a poisoned wild animal to death.
It must be extremely exciting to watch your hoax take on its own life, particularly when it was designed to highlight the media. The urban fox hunting video was produced by film makers Chris Atkins and Johnny Howorth as a response to calls by newspapers and politicians to begin culling urban foxes. These calls were in response to an attack by a fox on nine-month-old twins, Isabella and Lola Koupparis, in their bedroom, near Victoria Park in Hackney. Angry about the attack, one blogger demanded 'Bring Back Fox Hunting Now', and the Labour government's ban on fox hunting was never far from people's thoughts when discussing the urban fox problem. Atkins and Howorth had already produced the film Starsuckers, which was another shot at British media in which they sold fake stories to newspapers about celebrities. They managed to get unverified and ridiculous stories published, including Guy Ritchie giving himself a black eye whilst drunkenly juggling with cutlery, and a friend of Amy Winehouse punching the late singer in the hair after Winehouse had accidently set it on fire. Both are very easy to check, even via a photograph, but still made it into the newspapers. A blog and Facebook group was set up for the urban fox hunters and a video was released, showing a fox being clubbed to death in Victoria Park. Once leaflets started to be distributed around Hackney seeking the 'hunters', and the death threats arrived online, Atkins and Howorth quickly owned up to the Guardian and released a making-of film of the original fox-hunting film, featuring the pair, some friends and a dog called Monty wrapped in fox fur. A stuffed fox was used at the end of the film.
Contemporary British fox legends seem inextricably linked to hunting and how people see foxes. Some people love foxes, urban and otherwise. Many years ago I shared a verdant garden in New Cross with the tenant of another flat, who had foxes queuing up for their nightly sausages. When she ran out of sausages she would feed them bread and jam. Others think of the fox as a ginger wolf, cunning enough to get into your house, vicious enough to attack your children and fearless enough to not be afraid when captured. The attacks on the Koupparis sisters and the story in February 2013 of a fox biting off one-month-old Denny Dolan's thumb in Bromley are shocking and terrifying, but extremely rare, considering the number of foxes in London. Whatever the individual attitudes of journalists toward foxes, a newsmaker will write and print a shocking story such as urban fox attacks, and keep the story in people's minds when foxes do less frightening things such as chewing on a pair of Ugg boots in Putney (January 2013), riding on the Circle line without a ticket (August 2012) and chewing the designer shoes of dancers in the Spiegeltent on the South Bank (September 2012). This will keep tales of fearless and ferocious foxes in the news and provide a space for people who are calling for them to be culled.
## 21
## WHERE THE WILD THINGS ARE
* * *
We're going on a bear hunt.
Michael Rosen, a former Poet Laureate of Hackney
* * *
## Big Cat Country
'About 100 soldiers armed with axes and sticks joined more than 100 policemen and dogs today in a big-game hunt for a roaming leopard.' So began a story in the London Evening News on 17 July 1963 after a lorry driver and motorist saw a leopard in the Shooters Hill area of south-east London. 'I thought it was a dead dog,' said lorry driver David Black. 'When I got up to it, it jumped up and ran off into the wood.' When investigating this sighting the police were surprised by the beast leaping over the bonnet of their patrol car. Trackers found a clawed tree on the south-eastern side of Shooters Hill, near Welling Way, and paw prints in the mud of a dried stream. A local estate, schoolchildren and people in Woolwich Memorial Hospital, all in the area of Oxleas Wood, were warned not to go into the woods.
On 23 July, Jim Green was awoken by loud snarling noises, starting near Kidbrooke Park Road and moving along the course of the River Quaggy. A security sergeant from a nearby RAF station also heard the snarls and investigated with a police officer, seeing 'a big dark animal between 18 and 24 inches high silhouetted against a white cricket screen' at dawn.
Police said they did not know of anyone local who kept wild animals, that there was no circus in the area and no one had reported an escaped animal.
Some urban legends are straight narratives: the 'Corpse on the Tube' and the 'Accidental Theft' are stories that can be told, retold and remodelled according to their teller. Others are composed of ideas that float in the ether, waiting for an event to bring them all together again. In the case of alien big cats (ABCs) stalking Britain, and making it deep into urban London, the story first requires a witness to see an inexplicably large animal in order for the elements to come together. These folk story threads include a cat escaping from one of Britain's incontinent circuses, or a big cat that is more used to slinking about the mansion of a multi-millionaire until it escapes or is released by the bored owner. Under all this is the romantic idea that big carnivorous cats, more suited to the wild expanses of Africa and Asia, are living happily and anonymously in the suburbs and Home Counties.
Between June and August 1994 the Beast of Chiswick was seen twice a week. It was a grey or fawn colour, had a canine body and a kangaroo-like head and lived by tearing open refuse bags and disembowelling pigeons and squirrels. In June 1996 a brown big cat was spotted on a railway embankment in Northolt, prompting Doug Richardson of London Zoo to suggest to the Fortean Times that it was a mountain lion. A similar cat had been seen in Northolt in 1994. Summer 1998 saw the panther-like 'Beast of Ongar', and in 1999 an unidentified big cat was chased up a tree by a dog in Bedfords Park, Havering. In 2002 there were several sightings of a panther around the Plumstead, Bexleyheath and Shooters Hill area, including one sighting on Upton Road in Plumstead, prompting Karen Gardiner to say to the local paper News Shopper in October, 'I feel sorry for it not living in its natural habitat. I'd hate for it to get hurt.' By the time her husband Steve Gardiner was contacted by the Evening Standard on 24 January 2003, he said, 'What I remember about its size was that, as it walked away, its nose disappeared from the edge of one door while its back legs and tail were still visible in the other. Now, that's a big cat.'
The Gardiners' CCTV camera did pick up the 'panther', but after viewing the footage, Danny Bamping of the British Big Cat Society said that the film showed a 'blob', although it was a 'very large blob indeed'.
By 2005 the law of diminished returns seemed to demand that London's ABC encounters get more dramatic. Tony Holder of Sydenham was jumped by a labrador-sized cat in his garden at 2 a.m. Mr Holder had heard his pet cat making strange noises and went out to find it being held down by the beast. The ex-soldier said of the encounter: 'I could see these huge teeth and the whites of its eyes just inches from my face. It was snarling and growling and I really believed it was trying to do some serious damage. I tried to get it off but I couldn't move it, it was heavier than me.'
Perhaps because of his visible wounds (Mr Holder received a scratch on his face and arm and a wound on his finger), this was the first London big cat encounter that prompted a police response for some years. School gates were locked, people were warned not to venture in the local woods, police armed with tasers cordoned off streets and wardens warned dog walkers of the pet-bothering beast as they entered Sydenham Wells Park.
In December 2009, Roger Fleming was chased through nearby Dulwich Woods with his Staffordshire bull terrier puppy under his arm. There is no report of him contacting the police, but he did get in touch with the News Shopper to tell of his race against a big cat. The News Shopper contacted Neil Arnold of Kent Big Cat Research, who said that Mr Fleming 'should have stood his ground, maintained eye contact and backed off slowly – but it's easy to say that. People don't need to panic because big cats won't harm them.'
The response was far less drastic when a panther walked into the living room of Brian Shear, of Nunhead Lane in Zone Two Nunhead, and sat on his sofa – no one panicked at all. Diabetic Mr Shear woke up from a sleep in October 2006 having left his front door open to let in some air after feeling ill, to find that the cat had wandered into his house:
It had green eyes and was between four to five feet long, nose to tail. This was no pussycat. It didn't miaow, it growled. I'd been sitting in my armchair when it walked in. I didn't try to get too close to it because I was concerned it might bite me. I just sat there and talked to it like you would a normal pussy cat. I said, 'Hello puss, where've you been then?' and it just growled. It seemed quite content and I didn't feel threatened. I don't think it would have harmed me. It seemed familiar with humans.
When a recent unnamed New Cross resident was 'freaked out' by seeing a panther early one morning, but the police simply asked her outright if she had been drinking, despite her sighting being in the morning and in clear daylight. She said that she had seen the cat perched over the cover of the bins of her block of flats in Southerngate Way, with its tail hanging down. Neil Arnold commented, perhaps sarcastically, 'New Cross, not far from the station – I just don't know why it would be there.'
There are plenty of witness statements regarding the big cat population of London but very little evidence of their actual existence. Dreams, hoaxes and honest misidentifications are difficult to come by, unless the hoaxer owns up to their jape or the person named in the big cat report decides that they did not actually see, were not attacked or chased by an exotic wild beast.
## One of our Big Cats is Missing
During the 1960s and '70s in London, there were certainly some who kept big cats as 'pets'. The adventures of Christian the lion in Chelsea, bought by John Rendall and Anthony Bourke from Harrods in 1969, were documented on film and in the book A Lion Called Christian, published in 1971. Christian would play football in the side streets of Chelsea and was frequently taken to parties by the duo before they decided that their lion should go to Africa to live wild.
'Christian wasn't the only wild cat in this world,' Rendall told the Guardian in an interview published 28 May 2011. 'His neighbour was a serval cat. There was a chap in Battersea with a puma. John Aspinall had his tigers in Eaton Square and there were cheetahs and cougars roaming around Regent Street.'
The big cats of swinging London were not all fun and games, however; in January 1975 the RSPCA was called when a man left his puma in the back garden of his estranged wife and family in Acton, with a note saying he had nowhere else to keep it. The family were terrified and it took two hours to remove the animal. Another reckless puma episode took place when a man walked into the Farm House pub in South Harrow with his puma on a lead, a story that was reported in the Daily Mirror on 1 November 1974. Uncomfortable with feasting alongside a puma, punters asked the man to leave but the cat then ran riot in the pub, breaking glasses, destroying the bar and tables and, in proper feline fashion, the upholstery. It took fifteen minutes to get the enraged cat out of the pub and into the man's car, where the puma began to attack the upholstery. The police were called, who towed the car, cat and all, away and later charged the man with being 'drunk and incapable'.
Responding to the Acton incident, MP Peter Templemore feared that 'sooner or later someone will get killed' by a loose and barely domesticated big cat. The Dangerous Wild Animals Act became law in 1976, which made keeping a large carnivore, primate, large or venomous reptile or spider illegal without a licence. This registered the animal and allowed the local authority to monitor how the creature was kept. Many believe that with the coming of the Dangerous Wild Animals Act, big cat owners chose to release their pets rather than pay to register them. The legends of the urban big cats comes from the exotic pets of the 1960s and 1970s, and the new law that it is thought encouraged owners to fly-tip their problematic pumas and panthers.
Exotic animals do appear in London; a monitor lizard was believed to have been living in Geraldine Mary Harmsworth Park, next to the Imperial War Museum, for ten weeks before being rescued by the RSPCA in November 2005. In March 2003, a 'four-and-a-half-feet long and very frisky' iguana was found clinging to a tree on Wandsworth common. At first it was thought that Iggy, as RSPCA workers nicknamed the lizard, had escaped and posters were put up around the area. When no one came forward to claim him, it was thought he had been abandoned. During December of 2011, a cold and malnourished lemur was found living on Tooting common.
The lifespan of a big cat is between twelve and fifteen years in the wild and around twenty in captivity, and probably somewhere between these ages if they are living rough on squirrels and discarded takeaways. If the cats were released after 1976, as some believe, they would have died out by the mid-1990s. The urban myth of ABCs alludes to London's big cats either escaping from current homes or circuses, or being the descendants of original pumas and panthers released in the '70s. Having been made homeless, the myth suggests that the cats met in parks, around places like Plumstead or somewhere quiet in Sydenham, to mate. This would have brought about a generation of indigenous and mysterious big cats,which seems improbable. Despite the description of a big-cat infested 1960s London, it seems doubtful to me that there were enough animals to form breeding pairs. Some cryptozoologists date the cat's ancestry further back in time. In his book Kent Urban Legends, Neil Arnold suggests Victorian menageries provided earlier ABC stock and that London's cat community may even have its origins in Roman Britain.
One cat has been captured in the last few years, like our south London lemur and reptiles (See here). On 4 May 2001, Carol Montague was cleaning the house of Alan and Charlotte Newman on Holcroft Avenue in Cricklewood. She looked out of the window to see a large cat, four times bigger than a domestic cat, sitting on the garden fence. Charlotte Newman then came home, saw the cat and locked her Staffordshire bull terrier in the house. At first the police laughed at the report but when two officers arrived later, they confirmed that the animal was not a domestic cat and contacted the RSPCA. The police finally took things more seriously and ten police cars arrived, including one armed response unit. Ray Charter, head zookeeper at London Zoo, identified the cat as a European lynx, an endangered animal, of which there were around 7,000 in 2001. After four hours a senior vet from the London Zoological Society arrived with a tranquiliser gun, just in time for the lynx to make a run for it. She was chased across playing fields and tennis courts and two hours later was cornered in the stairwell of a block of flats on Farm Avenue. She was tranquilised and taken to London Zoo to recuperate.
At the zoo she was nursed back to health, having been found very underweight and with a fracture in her left hind foot, before being sent to Zoo d'Amnéville in France to take part in the European lynx breeding programme. Happily, European lynx numbers are now at 8,000 in Europe, not counting Russia.
No licences had been obtained for lynx ownership for the area and no zoo, circus or anyone else reported a lynx missing.
Catching a misidentification of an urban big cat is a lot harder than catching an actual one. Echoing the August 2012 story of the Essex lion, on 11 March 1994 there were eight reports of a lioness prowling the area around Winchmore Hill. The first report came from a Mrs Lia Bastock, who spotted an animal with 'short golden hair and big padded paws' in Firs Lane strolling along a canal towpath. A 2.5ft-tall cat was reported slinking through local back gardens. The regular carnival arrived, namely a police helicopter along with thirty policemen combing the area and using megaphones to warn people to keep children and pets indoors. London Zoo, as ever, provided a marksman to tranquilise the beast. The cat was captured on camera sunning itself on a garden shed, and Doug Richardson of London Zoo identified the creature as a domestic cat before it was then identified as a ginger tom called Bilbo, owned by Carmel Jarvis. Carmel's cat may not have been the only one after the limelight: the Enfield Advertiser on 16 March 1994 revealed the cat to be Zoe Reid's pet, Twiggy.
The main story to be told about London's phantom cats is in popular mythology and sensational newspaper stories. ABCs are a nationwide phenomenon, not just a London one. Whatever the truth of the mystery of big cat sightings is, the myth is what we carry within ourselves. It is the myth we get close to whenever something strange is seen, and the answer will only come with hard evidence from the outside world.
## We're Going on a Bear Hunt
On 27 December 1981, four boys from Lower Clapton took their dogs out for a walk across Hackney Marshes. Past Millfields Road, near the football pitches, the boys encountered 'a giant great growling hairy thing' – they met a bear in Hackney.
'We were near the football pitches at about five o'clock in the evening when we saw it,' said Darren Willoughby, aged 12. 'It was very close to us, standing on its hind legs and about seven feet tall.'
Once the press began to interview the boys, the stories began to expand. Before meeting the Hackney Bear, the boys had noticed unusual footprints in the snow, which one boy identified as bear tracks.
Following the tracks, the boys met a middle-aged couple walking their own dog and asked them if they'd seen a bear. 'Yes,' the couple replied, 'it's up there.' The couple told the boys to get away, they were near a bear after all, and to further add to the dream-like quality of events, started to throw snowballs at the boys to drive them away. This did not stop them, of course, and Tommy Murray (variously reported as 12 or 13) heard the bear growling, shone his torch on it and saw its profile standing upright in the dark. Tommy's dog, Lassie, did not want to go near it, and neither did they. The boys ran.
The police were impressed with the sincerity of the boys' fear and so launched a hunt across Hackney Marshes and along the banks of the River Lea to find the bear. Inspector Pat Curtis said, 'We do not believe this to be a hoax – we are taking no chances.'
The public were warned to keep off the marshes and not to join in with the police operation. Between fifty and 100 police officers, a police helicopter shining a searchlight over the dark ground, between fourteen and twenty dog handlers, and police marksmen armed with shotguns and handguns spent two cold days searching 8 miles of marshes and waste ground. Four RSPCA workers were concerned enough for the welfare of the bear to bring tranquiliser guns so that it could be subdued rather than assassinated, but it was clear the police were not prepared to take the risk. The Hackney Bear was 'a dangerous animal that can run faster than most men, swim and can climb trees,' Chief Inspector Platten told the press at the time. 'It will be shot dead if spotted.'
On 28 December the police searchers found footprints in the receding snow. Two sets on either side of the Lea and one on an island in the river. RSPCA inspector Derek Knight said, 'If it is a hoax it's an elaborate one. The footprints certainly look like a bear's.' Elsewhere he said, 'Perhaps not a fully grown one, maybe two or four hundredweight. But undoubtedly such a bear would be capable of killing a person.'
London Zoo director Colin Rawlings had a different view, suggesting that the bear could turn nasty 'if bothered by a dog' and that if it was a captive bear that had escaped, it would head towards people and their homes to scavenge for food rather than stay out in the wilds of Hackney. It could have hideouts, making it hard to spot.
When a shed on a local allotment had been forced open and Tommy Murray showed police the claw marks he had found on a tree, Murray told a journalist that he was 'very surprised it has not materialised'.
## The Bear Truth?
The potential claw marks and shed-burglary were the last possible signs of the bear on the marshes and were not enough to continue the search. With disappointment looming, things began to look different. The footprints in the melting snow did not look much like bear prints. Despite a group of children telling police they had seen the bear, by the evening of 29 December police called off the search and declared Hackney Marshes safe for the public. This was not, however, the end of the story.
The next day, the Sun newspaper received a call from the 'Hackney bear', or at least a man named Ron who claimed to be the hoaxer behind the bear scare. Ron was inspired by an earlier bear mystery when two skinned and decapitated carcasses were found in the River Lea near Clapton on 5 December 1981, twenty-two days before the boys' sighting. A jogger had spotted two 'bodies' in the canal and at first it was thought that they were human and the victims of recent 'East End underworld' violence.
Once the bodies were identified as brown bears, a circus that had been near the river two weeks earlier was contacted but was ruled out of any enquiry. These were not the first animal bodies to appear in the Lea; others had included a puma carcass, and it was assumed that an unethical taxidermist was fly-tipping his corpses.
Their heads full of thoughts of Hackney bears, Ron and three friends dreamt up a jape whilst in the pub. They had a bear suit from a fancy dress party, so they drove out to the marshes to leave paw prints and pretend to be a bear on the loose. What Ron had not counted on was frightening the young boys enough into going to the police. 'It was only those kids who were scared,' said a nervous Ron, 'we didn't realise they would take off like that.'
The police had already looked into a fancy dress party at Flamingo Disco on Hackney Marshes on Boxing Day, the day before the first sighting. They were very interested in Ron's confession, scowling 'this has been a very expensive operation'. No doubt they were also embarrassed about the time spent shivering on the marshes whilst searching for a dangerous urban ursine entity. Being dressed up as a bear in public is not a crime as far as I know, and I am still not sure whether Ron wasted police time or whether they managed to do that themselves. There was still doubt whether Ron was telling the truth about the whole caper. A local fancy dress hire shop pointed out that bear costumes do not come with bear feet, so an outfit on its own could not leave the tracks Ron had claimed to. Perhaps he and his friends had only got as far as an idea in the pub and a telephone. Michael Goss, in his thoroughly researched article in The Unknown, in December 1987 to January 1988, suggests that it was the boys themselves, young and perhaps fantasy prone, who dreamt up their bear encounter. No one else came forward to say they saw the bear and the only other evidence were suspicious scratches and paw prints.
The Ron revelation, however real, was the last thing heard about the Hackney bear for a long time. Decades later, on 17 May 2012, the Hackney Citizen reported a sighting of the Beast of Hackney Marshes. On the May Day bank holiday weekend, student Helen Murray was strolling through woodland near Old River Lea, a channel of the River Lea, when something large and shaggy stopped her in her tracks. She grabbed her mobile phone to dial 999 and managed to get two photographs of the creature as it moved through the undergrowth away from her. One image was of its hind quarters disappearing behind a tree, the other shows what looks like a head hunched down low from square shoulders. The two photographs show the hairy lump in the undergrowth but neither communicates movement too well.
'The "Beast of Hackney Marshes" Mystery – Pictures' trumpeted the Hackney Gazette, telling the story of Helen Murray's encounter, bringing newer readers up to date with the Hackney bear story and appealing for explanations. In keeping with the tradition of Ron owning up to being the Hackney bear, and two different domestic cats claiming to be the lion in Winchmore Hill, a dog owner did come forward claiming his pet was the Beast of Hackney Marshes. The dog was a Newfoundland named Willow, a huge, dark, hairy creature owned by Nicole and Paul Winter-Hart, who said they regularly walk her along Hackney Marshes. Paul was previously famous for being in the Brit Pop band Kula Shaker.
'I knew it was her immediately,' said Nicole. 'It's funny because our friends call her "The Beast" and now she's "The Beast of Hackney Marshes"!'
Helen Murray was not convinced by this explanation. While she was quick to say she thought Willow was cute, 'I'm pretty sure it wasn't a dog as it was far too big. And its build wasn't dog-like.'
The explanations for the Hackney bear are many and have become part of its myth.
## 22
## FOLKLORE AND FAKELORE
* * *
'The trouble with fiction', said John Rivers, 'is that
it makes too much sense. Reality never makes sense.'
Aldous Huxley, The Genius And The Goddess
* * *
TOWARD THE END of 2012, videos of wolves in London appeared on YouTube. I came across one, via the Centre for Fortean Zoology, called 'Wolves in Hackney????' It had been uploaded on 17 November 2011 and was a convincing piece of footage of a wolf sighted wandering up Urswick Street in Hackney. It is apparently filmed by a couple hanging out of an upstairs window filming fireworks when one of them looks down and sees the wolf on the street. He shouts 'Oi!' at it while his friend or girlfriend shushes him, but the wolf is off and runs sleekly along the street and into the night.
Other videos appeared, including a man who had seen the remains of a dog smeared across Clapham Cfommon, and one of a couple of women singing and dancing in a living room when something big crashes over their patio. On a fourth video, a group of friends recording a birthday message to a distant friend are disturbed by a wolf in the street. They looked real; the animals intruded on films that looked natural and the creatures in the films were clearly wolves.
It was a thought that was more exciting than it was unlikely, and I shared the first wolf film I saw across Facebook and Twitter. A wise friend quickly let me know that all of the footage showing wolves marauding across London was part of a series of YouTube films to promote a brand of vodka.
This seemed like rather a convoluted way of selling alcohol. The spoof footage was seeded across the internet, people commented and then there was a reveal letting the viewer know that the films and the story attached were created to promote Eristoff Vodka. Travelling Russian show-people Davok (an anagram even I can work out) presented a 'Circus Freakout' in Victoria Park from 1–3 December 2011 with an act called 'Wolves of Vale'. A Davok van crawled through East London with banging beasts within it. Then the shows suffered a set back: 'Wolves of Vale' was cancelled due to unforeseen circumstances. Next, videos appeared on YouTube of Londoners encountering wolves or the carnage left behind by wolves. The story being told is clear: Russian wolves were loose in London. The idea presumably must have been to make the wolves a talking point and then reveal the story in a way that would encourage interested parties to toast the cleverly done spoof with a cup of the vodka they were selling. The main wolf theatre, circus vans with howls, posters and listings would only really have interested anyone in the Hackney area. The Urswick Road video, as of 18 June 2013, has had 28,741 views, about 4,106 views a month, which is better than some 'actual' cryptozoological videos such as 'Wild Black Panther Cat Caught On Video In UK' (which averaged 962 views a month). How effective the wolves were for Eristoff sales is hard to gauge, but the vodka did drop 11 places in the May 2013 edition of 'The Power 100' list of spirits.
I emailed Us Ltd, the creative communications agency behind the campaign, to ask them what they were thinking, and whether they had been inspired by cryptozoolology in an attempt to create a buzz.
Jo Tanner of Us Ltd first responded by saying: 'Great. Love to help! Do we get paid? Royalties?'
I explained how writing local history books works to Jo and suggested the creative team would enjoy talking about their creative process. He responded: 'The idea was put together by a team of people and obviously developed as we went along.'
The thinking behind the wolves campaign was aimed at young people who are often beyond the reach of conventional advertising. '"Traditional-advertising-averse" young target audience,' as Jo put it. The wolves were there to suggest drinking Eristoff vodka to the 'young' people who are interested in 'Twilight-type stuff'. The brand values of Eristoff vodka were described as 'dark and mysterious', and their logo is a wolf. I had already suggested that out-of-place animals and vodka are not an intuitive connection, to which Jo pointed out, 'Well they are when you realise that there's a wolf in the brand's logo based on its roots being Georgian.'
## The Thames Angel
While on her way to meet a friend along the South Bank in May 2006, Jemima Waterhouse, a 16-year-old student from Sheen, saw an angel. 'I felt a sense of calm spreading over me. It was comforting and familiar – a kind of peace that lasted for a while after. It is really hard to put into words, but I guess you could describe it as peace of mind.'
She took a photograph which appeared in the South London Press on 15 September 2006, who described the angel as having some sort of celebrity and gathering a fanbase:
Eerily so far this year four people claim to have seen the angel near the London Eye and an internet cult is growing... These sightings have prompted much online chat about the so-called Angel of the Thames. Already angel walks are being offered along the waterside and Angel T-shirts are available. One angel obsessive – who meets up with other people who have spotted the ghostly figure to share their experiences – thinks it must date back to the fire.
That's the Great Fire of London, which was apparently one of the angel's earliest recorded sightings. Three websites devoted to the Thames Angel appeared in the wake of the article. The Angel of the Thames: Have You Seen the Angel website repeated Jemima's story and wanted to collect more. The Friends of the Thames Angel blog styled itself as the official Thames Angel fan club, complete with parties and a pug dog dressed up as an angel. Thames Angel: A History of the Angel of Promise is all swirly fonts, walking guides in period costume and historical events with a Thames Angel link to them. All three sites pass on the same information in the same way: that the River Thames has a resident angel that appears in times of great strife, or to please or beguile tourists.
Photographs showing a wispy, white-winged figure amongst or behind a group of tourists are placed alongside augmented etchings of a more traditional angel to illustrate a timeline that begins in 1667. There are a lot of images, and in the photographs the angel looks almost the same in all of them.
London's history, from the Great Plague to the Blitz, is alluded to, and the angel is said to have made an appearance during the rebuilding of London after the Great Fire. The most convincing piece of 'evidence' was footage of singer and television presenter David Grant filming on the Thames being distracted by the angel. 'Did you see that?' he asks. 'Did you see it 'cos I thought it looked like ... this is ridiculous, but I think it looked like an angel.' In another piece of footage a reporter from Slovakian Television hassled David Grant to find out what he knew but, the website suggests, he has been 'got to' and did not wish to talk. There is now a conspiracy to prevent people talking about the Thames Angel.
People began to look into these websites and films, and the blog Transpontine, which takes an interest in south London matters, summarised the gaps in the angel story. The Slovakian Television logo was incorrect and the image used as a station ident was a picture of St Basil's in Moscow – the most direct piece of evidence was faked. All the websites were created in 2006 and ceased later that year, and there is no previous mention of the angel before that time. Samuel Pepys did not write about the Thames Angel. This was an etching on the Angel of Promise site of the construction of the Embankment with the angel hovering over it with a suspicious white outline around it. Transpontine found a sharper version of the image on Wikipedia, without the angel – another fake. As early as November 2006, contributors on a James Randi webpage – Randi being the nemesis of those making paranormal claims – looked into the source code of the websites and found in the biggest site, Angel of Promise, the phrase 'Global angels'. The Global Angels Foundation is a charity that aids impoverished and exploited children across the world, founded by Molly Bedingfield in 2004. Molly is the mother of pop stars Natasha and Daniel Bedingfield, and the charity finances its many good works in a number of ways, including sponsoring celebrities in strange challenges. The most chilling I have seen was one in which Bear Grylls was to row 22 miles down the Thames in a bath tub. David Grant, who supposedly saw the angel on film, was a sponsor of Global Angels and hosted their launch at Coutts Banks. All of the contemporary photographs of the angel look like a fuzzy white version of the Global Angels logo.
I have attempted to contact them to discuss the Thames Angel website and their possible connection to it, but so far they have been too busy to respond to my emails and telephone messages.
## The Brentford Griffin
Further west along the Thames is Brentford, famed for its football club, Fuller's Brewery and the 'Brentford Triangle' novels by Robert Rankin. Brentford's other claim to fame is another mythical winged creature, the griffin. Griffins are mythical beasts of a higher order: they have the body of a lion but the head and wings of an eagle-like raptor. On the scale of London's mystery animals, the origin of its parakeets is a diversion and the panthers, pumas and bears are an absurd but romantic idea. The Brentford Griffin, however, is such a strange and ridiculous idea it is a surprise that people even consider it.
Like the Thames Angel, the Griffin has a history. In a letter to the Fortean Times, Issue 110, in May 1998, Martin Collins claimed to have heard stories of the griffin while at school in the 1950s. A family of griffins survived on Brentford Eyot (or Ait), after the first griffin was brought to Brentford by Nell Gwynn who had housed it in The Butts, Brentford. She had been given the griffin as a gift from Charles II. Somehow the griffin fell into the River Brent and was washed away to Brentford Eyot where it lived incognito after being presumed dead. Sir John Banks, a botanist who travelled with Captain Cook, brought another griffin to Brentford, where it was kept in a pagoda in Kew Gardens. After eventually escaping and mating with the first griffin living in the eyot, the griffin dynasty of Brentford was established until at least the 1980s. In the magazine Magonia, dated 19 May 1985, an article on Robert Rankin's 'Great Mysteries of Brentford No. 23: the Gryphon' stated that:
Reports of gryphons [sic] crop up with startling regularity throughout the pages of history. Dr Johnson records one he saw at Brentford's bull Fair: '... it was somewhat smaller than I had expected, but the proprietor assured me it was 'yet young' — it had the body of a lion cub and the neck, head and forelegs of a eagle... curiously formed wings issued from its shoulders.
Johnson was in no doubt that the beast lived 'and was not the product of the gypsies' craft'. No further mention of the gryphon is made in his writings and one wonders what became of it. Possibly it was the same live specimen that my [RR] father saw at Olympia before the war. He was informed it was several hundred years old and was shown old showman's posters as proof.'
The griffin made its first contemporary appearance in March 1983. The Ealing Guardian 14 March 1983 had the front-page headline 'GRIFFIN AT LARGE – Mystery Flying Beast Sighted in Brentford' after one Kevin Chippendale saw the creature.
Mr Chippendale, of Brook Road South, claims to have seen the animal twice, both times in the gasworks:
The first time was last summer when I saw something flying low across the ground in the gasworks. At first I thought it might be a plane, but it was too low and made no noise. I was intrigued to know what it was and as I walked past the Griffin pub realised it looked like the animal on the sign. I saw it again a couple of weeks later in exactly the same place.
More sightings followed. John Baroldi of the Watermans Arts Centre was quoted in the Ealing Gazette dated 15 March as saying that, 'A woman came from the park along the street. She was in an awful state. She had seen a huge bird and was obviously rather shaken by it.'
Robert Rankin, who was poet-in-residence at the Watermans Art Centre at the time, filled in some detail:
It has been a local myth for years. There were sightings of the ones prior to the last year. Previous ones go back to at least before the Second World War. A year ago a jogger called John Olssen reported seeing the bird as he was running by the arts centre. And a woman saw it from the top of a bus.
Miss Angela Keyhoe of Hanwell was on the top deck of the bus. She told the Ealing Gazette she was on a No. 65 bus near the art centre when she saw the griffin perched on top of a gasometer.
That griffins haunt Brentford may not be such a surprise: Fuller's, the local brewery, has a Griffin as a logo and they brew at the Griffin Brewery; there is a Griffin pub and Brentford Football Club's ground is Griffin Park. With the beer associations to this story it seemed fitting to meet someone who knew of the griffin in a pub but I chose one a safe distance from the Thames and Brentford, the Hermit's Cave in Camberwell.
Over pints of London-brewed ale, I learned that Robert Rankin and other locals had planned a festival in the Watermans Art Centre for 13 July 1985. This unfortunately coincided with Live Aid taking place at Wembley Stadium. To rustle up some publicity for the event, Rankin and friends, including a journalist or two, cooked up the story of a griffin being seen around the arts centre. All of the main witnesses were in on the joke; Robert Rankin wrote a historical back story for the griffin and let the idea of the creature loose. But then a strange thing happened. Brentfordians loved the idea of the griffin and took it to their hearts. The letter to the May 1998 edition of the Fortean Times (See page 83) did not, as far as my contact knew, have links with the original jape. When investigating the griffin myth for a lecture in 2003, John Rimmer, of Magonia magazine, went into the local pubs and the drinkers were keen to talk to him about 'their' griffin.
'Is the Brentford Griffin folklore or fakelore?' I asked my contact.
'It started out as fakelore,' he said, 'but it has become a part of Brentford folklore.'
## BIBLIOGRAPHY
THIS BOOK IS made up of countless references from websites, magazine articles and newspaper stories. Where possible I have quoted the source in the text. Urban legend research is an ongoing joy and I am keen to discuss any ideas about any of the urban legends discussed in this book and others. Please contact me on skitster@hotmail.com or at the blog http://living-lore.blogspot.co.uk.
Arnold, Catharine, Necropolis: London and Its Dead (London: Pocket Books, 2007)
Arnold, Neil, Kent Urban Legends: The Phantom Hitch-Hiker and Other Stories (Stroud: The History Press, 2013)
Barber, Mark, Urban Legends: An Investigation into the Truth Behind the Legends (Chichester: Summersdale, 2007)
Bard, Robert, Graveyard London: Lost and Forgotten Burial Grounds (London: Historical Publications, 2008)
Barnett, Richard, Sick City: Two Thousand Years of Life and Death in London (London: Strange Attractor Press, 2008)
Bell, Karl, The Legend of Spring-heeled Jack: Victorian Urban Folklore and Popular Cultures (Woodbridge: Boydell Press, 2012)
Bloom, Clive, Violent London: 2000 Years of Riots, Rebels and Revolts (Pan Macmillan: London, 2004)
Bolton, Tom, London's Lost Rivers: A Walker's Guide (London: Strange Attractor Press, 2011)
Brewer, E. Cobham, Dictionary of Phrase and Fable (London: Cassell, 1909)
Brooks, J. A., Ghosts of London (Norwich: Jarrold, 1995)
Brunvand, Jan Harold, The Choking Doberman and Other 'New' Urban Legends (London: W.W. Norton & Co., 1986)
––––––, The Mexican Pet: More 'New' Urban Legends and Some Old Favorites (London: Penguin, 1986)
––––––, Curses! Broiled Again! The Hottest Urban Legends Going (London: W.W. Norton & Co., 1989)
––––––, The Vanishing Hitchhiker: Urban Legends and their Meanings (London: Pan Books, 1983)
Bucazacki, Stefan, Fauna Britannica (London: Hamlyn, 2002)
Clark, James, Haunted London (Stroud: Tempus Publishing, 2007)
––––––, Mysterious Mitcham (Mitcham: Shadowtime Publishing, 2002)
Clayton, Antony, The Folklore of London (London: Historical Publications, 2008)
Dale, Rodney, The Tumour in the Whale (London: Gerald Duckworth & Co., 1978)
––––––, The Wordsworth Book of Urban Legends (Hertfordshire: Wordsworth War, 2005)
––––––, It's True... It Happened to a Friend (London: Gerald Duckworth & Co., 1984)
Fort, Charles, Lo! (London: John Brown, 1997)
Glinert, Ed, The London Compendium (London: Penguin, 2004)
Halliday, Stephen, Amazing and Extraordinary London Underground Facts (Cincinnati: David & Charles, 2009)
Hart, Edward J., 101 London Oddities (Sussex: J. R. Stallwood Publications, 1994)
Hayward, James, Myths and Legends of the First World War (Stroud: Sutton Publishing, 2002)
––––––, Myths and Legends of the Second World War (Stroud: Sutton Publishing, 2003)
Ibrahim, Mecca, One Stop Short of Barking: Uncovering the London Underground (London: New Holland, 2004)
Jackson, Peter, London Explorer (London: Associated Newspapers, 1953)
––––––, London is Stranger than Fiction (London: Associated Newspaper, 1951)
Jacobson, David J., The Affairs of Dame Rumour (New York: Rinehart, 1948)
Jenkins, Alan C., Wildlife in the City (Exeter: Webb & Bower, 1982)
Jones, Christopher, Subterranean Southwark (London: Past Tense, 2003)
Kempe, David, Living Underground (London: Herbert Press, 1988)
Kent, William, The Lost Treasures of London (London: Phoenix House, 1947)
––––––, Walks in London (London: Staples Press, 1951)
Long, Roger, Historic Inns along the River Thames (Stroud: Sutton Publishing, 2006)
Pyeatt, Samuel Menefee, 'Megalithic Movement: A Study of Thresholds in Time' in Davidson, Hilda Ellis (ed.), Boundaries & Thresholds: Papers from a Colloquium of The Katharine Briggs Club (Stroud: The Thimble Press, 1993)
Roberts, Chris, Football Voodoo: Magic, Superstition and Religion in the Beautiful Game (London: F&M Publications, 2010)
Rogers, Cyril H., Parrots (London: W&G Foyle, 1958)
Roud, Steve, London Lore (London: Random House, 2008)
Screeton, Paul, Mars Bars & Mushy Peas: Urban Legends and the Cult of Celebrity (Loughborough: Heart of Albion, 2008)
Smith, Paul, The Book of Nasty Legends (London: Routledge & Kegan, 1983)
––––––, The Book of Nastier Legends (London: Routledge & Kegan, 1986)
Smith, Stephen, Underground London: Travels Beneath the City Streets (London: Abacus, 2007)
Swinnerton, Jo (ed.), The London Companion (London: Robson, 2004)
Thornbury, Walter, Old and New London Volume II (London: Cassell Petter & Galpin, 1878)
Walford, Edward, London Recollected: Its History, Lore and Legend (London: Alderman, 1985)
White, Jerry, Rothschild Buildings: Life in an East End Tenement Block 1887–1920 (London: Pimlico, 2003)
Willey, Russ, Brewer's Dictionary of London Phrase & Fable (London: Chambers, 2009)
## A Selection of Magazines and Websites
Fortean Times – www.forteantimes.com
FLS News (Folklore Society) – www.folklore-society.com
Magonia – www.magonia.haaan.com
The Unknown
Urban Legend Resource (Snopes) – www.snopes.com
Alexander McQueen obituary: www.news.bbc.co.uk
Design Museum: www.designmuseum.org
'Dressed To Thrill': www.newyorker.com
'Porno magazines found in Queen's Car': www.thefreelibrary.com
'Secrets and lies: Shroud Origins of Giant Swastika': www.msgboard.snopes.com
'We Are Not Amused: Jag Man's Swastika Prank Backfires': www.thefreelibrary.com
Argyll Arms: www.nicholsonspubs.co.uk
King and Tinker pub: www.kingandtinker.co.uk/
Michael Jackson at the Montague Arms: www.transpont.blogspot.co.uk
The Old Watling: www.nicholsonspubs.co.uk/
Travel UK: 'Where Rivals Feared to Tread': www.independent.co.uk
Wimbledon tunnel: www.thisislocallondon.co.uk
'Suicidal Architects' p.7 FLS News, No.37, June 2002
'Suicidal Sculptor' p.13 FLS News, No.57, February 2009
Eagle Pillar: www.geograph.org.uk
The Chain Bridge Lions: www.bridgesofbudapest.com
The Seriousness of Mormon Humour: www.thejazzy.tripod.com
The Devils of Cornhill: www.shadyoldlady.com
## COPYRIGHT
First published in 2013
The History Press
The Mill, Brimscombe Port
Stroud, Gloucestershire, GL5 2QG
www.thehistorypress.co.uk
This ebook edition first published in 2013
All rights reserved
© Scott Wood, 2013
The right of Scott Wood to be identified as the Author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.
This ebook is copyright material and must not be copied, reproduced, transferred, distributed, leased, licensed or publicly performed or used in any way except as specifically permitted in writing by the publishers, as allowed under the terms and conditions under which it was purchased or as strictly permitted by applicable copyright law. Any unauthorised distribution or use of this text may be a direct infringement of the author's and publisher's rights, and those responsible may be liable in law accordingly.
EPUB ISBN 978 0 7524 9380 0
Original typesetting by The History Press
Ebook compilation by RefineCatch Limited, Bungay, Suffolk
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Anaconda is a variety of stud poker. Other names for this game include "pass the trash", "screw your neighbor", "fuck your neighbor", and "3,2,1 left".
Play
Each player is dealt seven cards. They then each select three cards to be passed to the player on their left. These cards are simply set on the table near their left-most opponent. No players get to see their new three cards until everyone has made a pass. Afterward, the players repeat the process, only with two cards, then again with one card. Players then discard two cards to make their best five-card poker hand.
In this version of the game, up to seven people can play, passing out a total of 49 cards and having three left over.
Betting
A round of betting occurs before the first pass of three cards, then again after every card pass is made. Once players have set their hands, one card at a time is exposed, with a round of betting following each card.
Variations
Anaconda can be changed in many ways, such as:
Altering the number of starting cards (six cards is common).
Altering the number of cards passed.
Altering to whom the cards are passed.
Incorporating joker cards.
Including only one betting round & showdown after all passing rounds.
High-low split.
Designating certain cards as wild.
Removing all betting rounds.
External links
Anaconda at Casinocity.com
Stud poker | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,334 |
is Todd and I live in northern Wisconsin, just blocks away from the shores of Lake Superior. It is a beautiful area with lots of clean air, forests, birds – and winter. Yes, winters here are long, snowy and very cold. It's a perfect place to have a bubbling pot of soup on the stove for months on end.
I always loved food and cooking, but it wasn't until I was laid off from my job in 2008 that I started cooking at home with any regularity – and did I have a lot to learn! But as I started bringing armfuls of cookbooks home from the local library and spending each evening cutting up vegetables, sautéing and learning how herbs and spices could bring a dish together that I realized how much I loved to cook. But above all, I came to love – and got good at cooking soup. And it became something more than I imagined. It became a love affair in the kitchen and a reconnection with my childhood.
I really love to make soup and make it almost every week – sometimes twice a week. I love the process of soup making, from choosing a recipe or concept to carrying out the dish: chopping, dicing, sautéing veggies, reducing, thickening, seasoning; all the things that go into making soup I enjoy. Soup of some sort can be found in the cuisine of nearly every culture and country across the globe. Soup making allows me to "travel" via different ingredients and tastes to places I may not ever get to visit directly. Soup keeps me warm during long and cold Wisconsin winters. Whenever I have a pot of soup simmering on the stove, the weather outside never seems quite as ferocious.
Making a big pot of soup on a Saturday or Sunday means my wife and I have plenty of leftovers for lunch in the coming days. You've no doubt heard the old adage, "It tastes even better the next day." Well, most times people are talking about soup when they say this. If you've ever reheated pasta, fish, pizza or steak, you know this isn't true for everything. But with soup, it is!
Lastly, but most importantly, soup is about memories. Some of my earliest and best food memories revolve around soup. When I think back about all the meals I ate as a youngster, a great many of them featured or included soup.
I remember as a kid going on vacation with my grandparents. Back in those days, Howard Johnson's hotels and restaurants were all over the United States, and we often stopped there for a meal. My absolute favorite thing to order from the Howard Johnson's menu was clam chowder. It was thick with clams and utterly delicious. I could never get enough! Little did I know at the time, but there was a reason this chowder was so good. The great French chef Jacques Pepin was behind the scenes in Howard Johnson's kitchens, creating some of the chains most popular dishes and adapting them to large-scale output. One of these dishes was their signature clam chowder.
I grew up in southwestern Ohio, and a mainstay of the dining scene was a family-style restaurant named Bill Knapp's. My memories of eating at Knapp's include countless cups of their awesome chicken noodle soup – I think I ordered it every time I went there! It was a basic but flavorful rendition, and the broth/stock had an almost velvety texture. The noodles were flat and approximately ½-inch wide and a couple inches long. I typically only ever ordered a cup, because I had to save room for biscuits and honey and the fried scallops or fried clams that were my main course. With any luck, I would finish with a piece of chocolate cake. Knapp's always had one on display under glass at the check-out counter, too, in case anyone forgot!
My dad's father didn't cook often, but what he did cook was outstanding. And for me, that meant pancakes in the morning and bean soup for lunch or dinner. I doubt he followed a specific recipe; instead he relied on a hearty smoked ham bone to flavor the navy beans that bubbled slowly on the stove until some of the beans had split from their skins and started to dissolve into the soup, thickening it even more.
My grandfather often baked a loaf of beer bread to serve with the bean soup, or we might have rye bread with butter or a plate of Ritz crackers and perhaps some cheese to eat with the soup. I liked my bean soup with an extra shake of black pepper, while my grandfather would add a dollop or two of ketchup into his. At the time, I thought putting ketchup in bean soup was a weird thing to do, but now I know better!
Chili was another soup I remember fondly from my youth. My mom would make it as would my dad's mother. They were basic, Midwestern chilis, relying on ground beef, chili powder, kidney beans, canned tomatoes, onions and, in my grandmother's case, some green pepper. I don't know why, but it always seemed like a special occasion whenever we were having chili.
When I had Cincinnati-style chili for the first time at the legendary Skyline Chili in the mid-70s, it was a revelation. My first time was after a Cincinnati Reds baseball game, during the era of the Big Red Machine for those baseball fans out there. I had never tasted anything like it before. The Greek-influenced spice blend tasted exotic and very strange compared to the seasonings found in a typical chili seasoning packet, but I took an instant liking to Cincy-style chili. It was the beginning of a life-long love affair with this Queen City specialty and chili in general.
And later, I discovered other soups across the globe that blew my mind and invigorated my taste buds to the point where I thought, there is a lot more to soup than a clunky glob of condensed material in a can. And it was time to spread the good word.
eSoupRecipes is the culmination of my wife and many others telling me, "You should really do something with your soups." So, I offer this blog up to all other soup lovers and those of you who may be new to cooking soups and want to put the best bowl of chili, chicken noodle soup or something entirely new in front of your friends and family. When they take that first spoonful and say, "Wow, this is really good," that's all the thanks you need. I know you can do this because I have done it.
I cordially invite you to join me on this journey across the delicious and rewarding world of soups. As they say, "Soup's on!" Won't you grab a spoon? | {
"redpajama_set_name": "RedPajamaC4"
} | 8,596 |
\section{Introduction}
Consider the complex modified Korteweg-de Vries equation
\begin{equation}\label{1.1}
u_{t}+6|u|^{2}u_{x}+u_{xxx}=0,
\end{equation}
where $u$ is a complex-valued function of $(x,t)\in\mbox{\of R}^2$.
In this paper, we study the orbital stability of the family of periodic
traveling-wave solutions
\begin{equation}\label{1.2}
u=\varphi(x,t)=e^{i\omega (x+(3a+\omega^2)t)}r(x+(a+3\omega^2) t).
\end{equation}
where $r(y)$ is a real-valued $T$-periodic function and $a,
\omega\in\mbox{\of R}$ are parameters. The problem of the stability of
solitary waves for nonlinear dispersive equations goes back to the
works of Benjamin \cite{Be1} and Bona \cite{Bo} (see also
\cite{AlBoHe, W1, W2}). A general approach for investigating the
stability of solitary waves for nonlinear equations having a group
of symmetries was proposed in \cite{GSS}. The existence and
stability of solitary wave solutions for equation (\ref{1.1}) has
been studied in \cite{ZIK}. In contrast to solitary waves for
which stability is well understood, the stability of periodic
traveling waves has received little attention. Recently in \cite{ABS},
the authors developed a complete theory on the stability of cnoidal waves for
the KdV equation. Other new explicit formulae for the periodic traveling
waves based on the Jacobi elliptic functions, together with their
stability, have been obtained in \cite{An2, AnNa, HIK} for the
nonlinear Schr\"odinger equation, modified KdV equation, and
generalized BBM equation. In \cite{Ha}, the stability of periodic traveling
wave solutions of BBM equation which wave profile stays close to the constant
state $u=(c-1)^{1/p}$ is considered.
Our purpose here is to study existence and stability of periodic
traveling wave solutions of equation (\ref{1.1}).
We base our analysis on the invariants $Q(u), P(u),$ and $F(u)$
(see Section 4). Our approach is to verify that $\varphi$ is a minimizer
of a properly chosen functional $M(u)$ which is conservative with respect
to time over the solutions of (\ref{1.1}). We consider the $L^2$-space of
$T$-periodic functions in $x\in\mbox{\of R}$, with a norm $||.||$ and a scalar
product $\langle .,.\rangle$. To establish that the orbit
$${\cal O}=\{e^{i\omega\eta}\varphi(\cdot-\xi,t):\;
\omega\in 2\pi\mbox{\of Z}/T,\; \xi,\eta\in[0,T]\}$$
is stable, we take $u(x,t)=e^{i\omega \eta}\varphi(x-\xi,t)+h(x,t)$,
$h=h_1+ih_2$ and express the leading term of $M(u)-M(\varphi)$ as
$\langle L_1h_1,h_1\rangle+\langle L_2h_2,h_2\rangle$ where
$L_i$ are second-order selfadjoint differential operators in $L^2[0,T]$
with potentials depending on $r$ and satisfying $L_1 r'=L_2r=0$. The proof of
orbital stability requires that zero is the second eigenvalue of $L_1$ and
the first one of $L_2$. Therefore, we are able to establish stability when
$r(y)$ does not oscillate around zero. Sometimes, the waves (\ref{1.2})
with this property are called "dnoidal waves" because $r$ is expressed by
means of the elliptic function $dn(y;k)$.
The paper is organized as follows. In Section 2, we discuss in brief the
correctness of the Cauchy problem for (\ref{1.1}) in periodic Sobolev
spaces $H^s$, $s\in\mbox{\of R}$ (equipped with a norm $||.||_s$). The problem is
locally well-posed for $s>\frac32$ and ill-posed for $s<\frac12$. In
Section 3 we outline the existence and the properties of the periodic
traveling-wave solutions (\ref{1.2}) to (\ref{1.1}), with emphasis on
the case when $r$ does not oscillate around zero. In Section 4 we prove
our main orbital stability result (Theorem \ref{t31}). In the Appendix, we
establish some technical results we need during the proof of our main
theorem.
\section{Cauchy problem}
In this section we discuss the well-posedness of the initial-value problem
for the complex modified Korteweg-de Vries equation in the
periodic case. We take an initial value $u_0(x)$ in a periodic Sobolev
space $H^s=H^s[0,T]$. Local well-posedness means that there exists a
unique solution $u(.,t)$ of (\ref{1.1}) taking values in $H^s$ for a time
interval $[0, t_0)$, it defines a continuous curve in $H^s$ and
depends continuously on the initial data.
The local well-posedness for (\ref{1.1}) in the non-periodic case is
studied in \cite{ZIK} by applying Kato's theory of abstract quasilinear
equations \cite{Ka1, Ka2}. In the periodic case the conditions are
verified in the same way as in \cite{ZIK}, therefore we present
here without proving the following result.
\begin{thm}\label{ts1}
Let $s>\frac{3}{2}$ and $T>0$. For each $u_0(x)\in H^s[0,T]$
there exists
$t_0$ depending only on $||u_{0}||_s$ such that $(\ref{1.1})$
has a unique solution $u(x,t)$, with $u(x,0)=u_0(x)$ and
$$u\in C([0,t_0); H^s)\cap C^{1}([0, t_0); H^{s-3}).$$
Moreover, the mapping $u_0(x)\rightarrow u(x,t)$ is
continuous in the $H^s[0,T]$-norm.
\end{thm}
Sometimes it is more appropriate to consider other version of
well-posedness, for example by strengthening our definition,
requiring that the mapping data-solution is uniformly continuous, that is:
for any $\varepsilon$, there exists $\delta>0$, such that if
$||u_{01}-u_{02}||_s<\delta$, then $||u_{1}-u_{2}||_s<\varepsilon$, with
$\delta=\delta(\varepsilon, M)$, where $||u_{01}||_s\leq
M$ and $||u_{02}||_s\leq M.$ The ill-posedness of some
classical nonlinear dispersive equations (KdV, mKdV, NLS) in
both periodic and non-periodic cases are studied in \cite{BKPS,
BPS, BGT, KPV1}. The approach in these papers is based on the existence
and good properties of the traveling wave solutions associated to the
respective equations.
Below we discuss the problem of the uniform-continuity of
data-solution mapping for (\ref{1.1}) in periodic Sobolev spaces
with small $s$.
\begin{thm}\label{ts11}
The initial value problem for the complex modified Korteweg-de Vries
equation $(\ref{1.1})$ is locally ill-posed for initial data in the
periodic spaces $H^s$ with $s<\frac12$.
\end{thm}
{\bf Proof.} It is easy to see that
\begin{equation}\label{4.2}
u_{N,A}(x,t)=A\exp(i(Nx+(N^{3}-6A^2N)t))
\end{equation}
where $A$ is a real constant and $N$ is a positive integer, solves the
equation (\ref{1.1}) with initial data $u_{0}(x)=A\exp(iNx)$.
For $A=\alpha N^{-s}$ where $\alpha$ is a real parameter, we have
$$\begin{array}{ll}
||u_{0}||_{s} \leq C\alpha^{2} \\
\\
||u_{N,A}(\cdot ,t)||_{s}\leq C\alpha^{2}
\end{array}
$$
with $C>0$. Let $A_{1}=\alpha_{1}N^{-s}$ and $A_{2}=\alpha_{2} N^{-s}$.
For the Sobolev norm of the difference of two initial data, we have
$$\begin{array}{ll}
||u_{A_{1}, N}(0)-u_{A_{2}, N}(0)||_{s}^{2}& =
\sum_{\xi\in\mbox{\of Z}}(1+\xi^{2})^{s}|\widehat{u_{A_{1},N}}(\xi)-
\widehat{u_{A_{2},N}}(\xi)|^{2} \\
\\
& \leq C|\alpha_{1}-\alpha_{2}|^{2} \rightarrow 0 \;
\; \rm{as} \; \; \alpha_{1} \rightarrow \alpha_{2}
\end{array}
$$
On the other hand, we have
$$\begin{array}{cc}
||u_{A_{1}, N}(\cdot ,t)-u_{A_{2}, N}(\cdot ,t)||_{s}^{2}\\
\\ =\sum_{\xi\in\mbox{\of Z}}(1+\xi^{2})^{s}|\widehat{u_{A_{1},N}}(\xi)
-\widehat{u_{A_{2},N}}(\xi)|^{2} \\
\\ =(1+N^{2})^{s}|\alpha_{1}N^{-s}\exp(i(N^{3}
-6\alpha_{1}^{2}N^{-2s+1}))-\alpha_{2}N^{-s}
\exp(i(N^{3}-6\alpha_{2}^{2}N^{-2s+1}))|^{2} \\
\\
\geq C|\alpha_{1}-\alpha_{2}
\exp(i6(\alpha_{1}^{2}-\alpha_{2}^{2})N^{1-2s})|^{2}
\end{array}
$$
Let $s<{\frac{1}{2}}$, and $\alpha_{1}$, $\alpha_{2}$ be chosen so that
$$(\alpha_{1}^{2}-\alpha_{2}^{2})N^{1-2s}=C N^{2\nu},$$
where $\nu >0$ and $2\nu+2s-1<0$. Then for
$t={\frac{\pi}{2}}C^{-1}N^{-2\nu}$, we have
$$||u_{A_{1}, N}(x,t)-u_{A_{2}, N}(x,t)||_{s}^{2}
\geq C(\alpha_{1}^{2}+\alpha_{2}^{2})$$
Note that $t$ can be made arbitrary small by choosing $N$ sufficiently large.
This completes the proof of the theorem. $\Box$
\section{Periodic traveling-wave solutions}
We are looking for traveling-wave solutions for equation (\ref{1.1})
in the form
\begin{equation}\label{2.1}
\varphi (x,t)=e^{i\omega (x+\alpha t)}r(x+\beta t)
\end{equation}
where $\alpha,\beta,\omega\in\mbox{\of R}$ and
$r(y)$ is a smooth real periodic function with a given period $T$.
Substituting (\ref{2.1}) into (\ref{1.1})
and separating real and imaginary parts,
we obtain the following equations
\begin{equation}\label{2.3}
\begin{array}{l}
\beta-3\omega^2=\frac13(\alpha-\omega^2)=a\in\mbox{\of R},\\[2mm]
r''+2r^3+ar=0.\end{array}
\end{equation}
Therefore
\begin{equation}\label{1.3}
u=\varphi(x,t)=e^{i\omega (x+(3a+\omega^2)t)}r(x+(a+3\omega^2) t).
\end{equation}
Integrating once again the second equation in (\ref{2.3}), we obtain
\begin{equation}\label{newton}
r'^2=c-ar^2-r^4,
\end{equation}
hence the periodic solutions are given by the periodic trajectories
$H(r,r')=c$ of the Hamiltonian vector field $dH=0$ where
$$H(x,y)=y^2+x^4+ax^2.$$
Clearly, two cases appear:
\vspace{1ex}
\noindent
1) {\it Global center} $(a\geq 0)$. Then for any $c>0$ the orbit defined by
$H(r,r')=c$ is periodic and oscillates around the center at the origin.
\vspace{1ex}
\noindent
2) {\it Duffing oscillator} $(a<0)$. Then there are two possibilities
\vspace{1ex}
\noindent
2.1) ({\it outer case}): for any $c>0$ the orbit defined by $H(r,r')=c$
is periodic and oscillates around the eight-shaped loop $H(r,r')=0$ through
the saddle at the origin.
\vspace{1ex}
\noindent
2.2) ({\it left and right cases}): for any $c\in(-\frac14a^2,0)$ there are two
periodic orbits defined by $H(r,r')=c$ (the left and right ones). These are
located inside the eight-shaped loop and oscillate around the centers at
$(\mp\sqrt{-a/2},0)$, respectively.
\vspace{1ex}
In cases 1) and 2.1) above, $r(x)$ oscillates around zero and for this reason
we are unable to study stability properties of the wave (\ref{1.3}).
In the rest of the paper, we will consider the left and right cases of
Duffing oscillator.
\vspace{2ex}
\noindent
{\bf Remark.} One could also consider equation (\ref{1.1}) with a minus sign,
$$ u_{t}-6|u|^{2}u_{x}+u_{xxx}=0.$$
It has a traveling-wave solution of the form (\ref{1.3}) where $r$
is a real-valued periodic function of period $T$ satisfying equation
$ r''-2r^3+ar=0.$ Taking $H(x,y)=y^2-x^4+ax^2,\quad a>0$
({\it the Truncated pendulum Hamiltonian}), we see that for
$c\in (0, \frac14a^2)$ the periodic solutions are given by the periodic
trajectories $H(r,r')=c$ of the Hamiltonian vector field $dH=0$ which
oscillate around the center at the origin and are bounded by the separatrix
contour $H(r,r')=\frac14a^2$ connecting the saddles $(\mp\sqrt{a/2},0)$.
Therefore, we are unable to handle this case, too.
\vspace{2ex}
In the left and the right cases, let us denote by $r_0>r_1>0$ the positive
roots of $r^4+ar^2-c=0$. Then, up to a translation, we obtain the respective
explicit formulas
\begin{equation}\label{1.7}
r(z)=\mp r_0 dn(\alpha z; k),\quad k^2=\frac{r_0^2-r_1^2}{r_0^2}
=\frac{a+2r_0^2}{r_0^2},
\quad \alpha=r_0, \quad T=\frac{2K(k)}{\alpha}.
\end{equation}
Here and below $K(k)$ and $E(k)$ are, as usual, the complete elliptic
integrals of the first and second kind in a Legendre form.
By (\ref{1.7}), one also obtains $a=(k^2-2)\alpha^2$ and, finally,
\begin{equation}\label{1.8}
T=\frac{2\sqrt{2-k^2}K(k)}{\sqrt{-a}}, \quad
k\in(0,1), \quad T\in I=\left(\frac{2\pi}{\sqrt{-2a}},\infty\right).
\end{equation}
\vspace{2ex} \noindent {\bf Lemma 3.1.} {\it For any
$a<0$ and $T\in I$, there is a constant $c=c(a)$
such that the periodic traveling-wave solution $(\ref{1.3})$ determined
by $H(r,r')=c(a)$ has a period $T$. The function $c(a)$ is
differentiable.}
\vspace{2ex}
\noindent
{\bf Proof.} The statement follows from the implicit function theorem.
It is easily seen that the period $T$ is a strictly increasing function of $k$:
$$\frac{d}{dk}(\sqrt{2-k^2}K(k))=\frac{(2-k^2)K'-kK}{\sqrt{2-k^2}}=
\frac{K'+E'}{\sqrt{2-k^2}}>0.$$
Given $a$ and $c$ in their range, consider the functions $r_0(a,c)$,
$k(a,c)$ and $T(a,c)$ given by the formulas we derived above.
We obtain
$$\frac{\partial T}{\partial c}=\frac{dT}{dk} \frac{dk}{dc}=
\frac{1}{2k}\frac{dT}{dk}\frac{d(k^2)}{dc}.$$
Further, we have in the left and right cases
$$\frac{d(k^2)}{dc}=\frac{d(k^2)}{d(r_0^2)} \frac{dr_0^2}{dc}
=-\frac{a}{r_0^4(a+2r_0^2)}.$$
We see that $\partial T(a,c)/\partial c\neq 0$, therefore
the implicit function theorem yields the result. $\Box$
\section{Stability}
In this section we prove our main stability result which concerns the
left (right) Duffing oscillator cases. Take $a<0$, $T>2\pi/\sqrt{-2a}$
and determine $c=c(a)$ so that the two orbits given by $H(r,r')=c$
have period $T$. Next, chose $\omega\neq 0$ in (\ref{1.3}) to satisfy
$\omega T/2\pi\in\mbox{\of Z}$. Then $\varphi(x,t)$ is a solution of (\ref{1.1})
having a period $T$ with respect to $x$.
\vspace{2ex}
{\bf 1. Basic statements and reductions}
Take a solution $u(x,t)$ of (\ref{1.1}) of period $T$ in $x$ and
introduce the pseudometric
\begin{equation}\label{3.1}
d(u, \varphi)=\inf_{(\eta, \xi)\in \mbox{\of R}^2}
||u(x,t)-e^{i\omega \eta}\varphi(x-\xi, t)||_{1}.
\end{equation}
The equation (\ref{1.1}) possesses the following conservation laws
$$ Q(u)=i\int_{0}^{T}{\overline{u}_{x}u}dx,\quad
P(u)=\int_{0}^{T}{|u|^{2}}dx,\quad
F(u)=\int_{0}^{T}{(|u_{x}|^{2}-|u|^{4})}dx.$$
Let
$$M(u)=F(u)+(\omega^{2}-a)P(u)-2\omega Q(u).$$
For a fixed $q>0$, we denote
\begin{equation}\label{3.2}
\begin{array}{ll}
d_{q}^{2}(u,\varphi)&=\inf_{(\eta, \xi)\in \mbox{\of R}^2}
\left(||u_{x}(x,t)-e^{i\omega \eta}\varphi_{x}(x-\xi, t)||^2 \right.\\
&\left. +q||u(x,t)-e^{i\omega \eta}\varphi(x-\xi, t)||^2\right).
\end{array}
\end{equation}
Clearly, the infimum in $(\ref{3.1})$ and $(\ref{3.2})$ is attained at some
point $(\eta, \xi)$ in the square $[0,T]\times [0,T]$. Moreover, for
$q\in [q_1,q_2]\subset(0,\infty)$, (\ref{3.2}) is a pseudometric equivalent
to (\ref{3.1}).
Now, we can formulate our main result in the paper.
\begin{thm}\label{t31} Let $\varphi$ be given by $(\ref{1.2})$, with $r\neq 0$.
For each $\varepsilon>0$ there exists $\delta>0$ such that if
$u(x,t)$ is a solution of $(\ref{1.1})$ and
$d(u, \varphi )_{|t=0}<\delta$, then $d(u, \varphi)<\varepsilon$
$\forall t\in [0,\infty)$.
\end{thm}
The crucial step in the proof will be to verify the following statement.
\begin{prop}\label{p31}
There exist positive constants $m, q, \delta_{0}$ such that if $u$
is a solution of $(\ref{1.1})$ such that $P(u)=P(\varphi)$ and
$d_{q}(u,\varphi)<\delta_{0}$, then
\begin{equation}\label{3.3}
M(u)-M(\varphi)\geq md_{q}^{2}(u,\varphi).
\end{equation}
\end{prop}
The proof consists of several steps. The first one concerns the metric
$d_q$ introduced above.
\begin{lem}\label{l31}
The metric $d_{q}(u,\varphi)$ is a continuous function of $t\in
[0, \infty)$.
\end{lem}
\noindent {\bf Proof.} The proof of the lemma is similar to the proof of
Lemmas 1, 2 in \cite{Bo} $\Box$.
\vspace{1ex}
\noindent
We fix $t\in [0,\infty)$ and assume that the minimum in (\ref{3.1})
is attained at the point $(\eta, \xi)=(\eta(t), \xi(t))$. In order
to estimate $\Delta M =M(u)-M(\varphi)$, we set
$$u(x,t)=e^{i\omega \eta}\varphi(x-\xi, t)+h(x,t)$$
and integrating by parts in the terms containing $h_{x}$ and
$\overline{h}_{x}$, we obtain
$$ \begin{array}{ll}
\Delta M &=M(u)-M(\varphi) \\[1mm]
&=2Re\int_0^T e^{i\omega \eta}\left[ -\varphi_{xx}+
(\omega^2-a-2|\varphi |^2)\varphi +2i\omega \varphi_x\right]
\overline{h}dx\\[1mm]
&+\int_0^T\left[|h_{x}|^2+(\omega^2-a-4|\varphi|^2)|h|^2
-2i\omega h\overline{h}_x-2Re\left(e^{-2i\omega \eta}
\overline{\varphi}^2h^2\right)\right]dx\\[1mm]
&-\int_0^T |h|^2\left(4Re\left(e^{i\omega \eta}\varphi
\overline{h}\right)+|h|^2\right)dx\\[1mm]
&=I_{1}+I_{2}+I_{3}.
\end{array}
$$
Note that the boundary terms annihilate by periodicity.
Using that $r(x)$ satisfies equation (\ref{2.3}), we obtain $I_1=0$. Let
$$h=(h_{1}+ih_{2})e^{i\omega(x-\xi+(\omega^{2}+3a)t+\eta)}, $$
where $h_{1}$ and $h_{2}$ are real periodic functions with period $T$.
Then we have
$$ \begin{array}{ll}
|h|^{2}=h_{1}^{2}+h_{2}^{2} \\[1mm]
|h_{x}|^{2}=(h_{1x}-\omega h_{2})^{2}+(h_{2x}+\omega h_{1})^{2}\\[1mm]
\int_{0}^{T}{\overline{h}_{x}h}dx=i\int_{0}^{T}{[h_{2}h_{1x}-h_{1}h_{2x}
-\omega (h_{1}^{2}+h_{2}^{2})]}dx\\[1mm]
Re(h^{2}\overline{\varphi}^{2}e^{-2i\omega \eta})
=r^{2}(h_{1}^{2}-h_{2}^{2}) \end{array}
$$
Thus for $I_2$ we obtain the expression
$$ \begin{array}{ll}
I_{2} &=\int_{0}^{T}{[h_{1x}^{2}-(a+6r^{2})h_{1}^{2}]}dx+
\int_{0}^{T}{[h_{2x}^{2}-(a+2r^{2})h_{2}^{2}]}dx \\[1mm]
&=M_{1}+M_{2}
\end{array}
$$
Introduce in $L^2[0,T]$ the self-adjoint operators $L_1$ and $L_2$
generated by the differential expressions
$$\begin{array}{ll}
L_{1}=-\partial_{x}^{2}-(a+6r^{2}), \\[2mm]
L_{2}=-\partial_{x}^{2}-(a+2r^{2}),
\end{array} $$
with periodic boundary conditions in $[0,T]$.
\vspace{2ex}
{\bf 2. Spectral analysis of the operators $L_{1}$ and $L_{2}$}
Consider in $L^2[0,T]$ the following periodic eigenvalue problems
\begin{equation}\label{3.4}
\left\{
\begin{array}{ll}
L_1\psi=\lambda \psi\quad \mbox{\rm in}\;\,[0,T], \\
\psi(0)=\psi(T), \; \; \psi'(0)=\psi'(T),\\
\end{array} \right.
\end{equation}
\begin{equation}\label{3.5}
\left\{
\begin{array}{ll}
L_2\chi=\lambda \chi \quad \mbox{\rm in}\;\,[0,T], \\
\chi(0)=\chi(T), \; \; \chi'(0)=\chi'(T).\\
\end{array} \right.
\end{equation}
The problems (\ref{3.4}) and (\ref{3.5}) have each a countable infinite
set of eigenvalues $\{ \lambda_{n} \}$ with $\lambda_n \rightarrow \infty$.
We shall denote by $\psi_n$, respectively by $\chi_n$, the eigenfunction
associated to the eigenvalue $\lambda_n$. For the periodic eigenvalue
problems (\ref{3.4}) and (\ref{3.5}) there are associated semi-periodic
eigenvalue problems in $[0,T]$, namely (e.g. for (\ref{3.5}))
\begin{equation}\label{3.6}
\left\{
\begin{array}{ll}
L_2\vartheta=\mu\vartheta \quad \mbox{\rm in}\;\,[0,T],\\
\vartheta(0)=-\vartheta(T), \; \; \vartheta'(0)=-\vartheta'(T).\\
\end{array} \right.
\end{equation}
As in the periodic case, there is a countable infinity set of eigenvalues
$\{\mu_n\}$. Denote by $\vartheta_n$ the eigenfunction associated to the
eigenvalue $\mu_n$. From the Oscillation Theorem \cite{MaWi} we know that
$\lambda_0< \mu_0 \leq \mu_1<\lambda_1 \leq \lambda_2 <\mu_2 \leq \mu_3,
\ldots$, $\lambda_0$ is simple and
$$ \begin{array}{ll}
\rm{(a)} \;\;\chi_0\;\;\mbox{\rm has no zeros on}\;\;[0,T),\\[1mm]
\rm{(b)}\;\;\chi_{2n+1}\;\;\rm{and}\;\;\chi_{2n+2}\;\;
\mbox{\rm have exactly}\;\;2n+2\;\;\mbox{\rm zeros on}\;\;[0,T),\\[1mm]
\rm{(c)}\;\;\vartheta_{2n}\;\;\mbox{\rm and}\;\;\vartheta_{2n+1}\;\;
\mbox{\rm have exactly}\;\;2n+1\;\;\mbox{\rm zeros on}\;\;[0,T).
\end{array}
$$
The intervals $(\lambda_0, \mu_0), (\mu_1, \lambda_1),\ldots$ are called
intervals of stability and the intervals
$(-\infty, \lambda_0), (\mu_0, \mu_1), (\lambda_1, \lambda_2),\ldots$
are called intervals of instability.
\vspace{2ex}
We use now (\ref{1.7}) and (\ref{1.8}) to rewrite operators
$L_1$, $L_2$ in more
appropriate form. From the expression for $r(x)$ from (\ref{1.7})
and the relations
between elliptic functions $sn(x)$, $cn(x)$ and $dn(x)$, we obtain
$$L_1=\alpha^{2}[ -\partial_{y}^{2}+6k^{2} sn^{2}(y)-4-k^2] $$
where $y=\alpha x$.
It is well-known that the first five eigenvalues of
$\Lambda_1=-\partial_{y}^{2}+6k^{2}sn^{2}(y, k)$,
with periodic boundary conditions on $[0, 4K(k)]$, where
$K(k)$ is the complete elliptic integral of the first kind, are
simple. These eigenvalues and corresponding eigenfunctions are:
$$\begin{array}{ll}
\nu_{0}=2+2k^2-2\sqrt{1-k^2+k^4},
& \phi_{0}(y)=1-(1+k^2-\sqrt{1-k^{2}
+k^{4}})sn^{2}(y, k),\\[1mm]
\nu_{1}=1+k^{2}, & \phi_{1}(y)=cn(y, k)dn(y, k)
=sn'(y, k),\\[1mm]
\nu_{2}=1+4k^{2}, & \phi_{2}(y)=sn(y, k)dn(y, k)
=-cn'(y, k),\\[1mm]
\nu_{3}=4+k^{2}, & \phi_{3}(y)=sn(y, k)cn(y, k)
=-k^{-2}dn'(y, k),\\[1mm]
\nu_{4}=2+2k^{2}+2\sqrt{1-k^{2}+k^{4}},
& \phi_{4}(y)=1-(1+k^{2}+\sqrt{1-k^{2}
+k^{4}})sn^{2}(y, k).
\end{array}
$$
It follows that the first three eigenvalues of the operator
$L_1$, equipped with periodic boundary condition on $[0,2K(k)]$
(that is, in the case of left and right family),
are simple and $\lambda_0=\alpha^2(\nu_0-\nu_3)<0, \;
\lambda_1=\alpha^2(\nu_3-\nu_3)=0, \;
\lambda_{2}=\alpha^2(\nu_4-\nu_3)>0$.
The corresponding eigenfunctions are
$\psi_0=\phi_0(\alpha x), \psi_1=r'(x), \psi_2=\phi_4(\alpha x)$.
\vspace{2ex}
Similarly, for the operator $L_2$ we have
$$L_2=\alpha^2[-\partial_y^2+2k^2sn^2(y, k)-k^2]$$
in the case of left and right family. The spectrum of
$\Lambda_2=-\partial_y^2+2k^{2}sn^{2}(y, k)$ is formed
by bands $[k^{2}, 1]\cup [1+k^{2}, +\infty)$. The
first three eigenvalues and the corresponding eigenfunctions with
periodic boundary conditions on $[0, 4K(k)]$ are simple and
$$\begin{array}{ll}
\epsilon_0=k^2, & \theta_0(y)=dn(y, k),\\[1mm]
\epsilon_1=1, & \theta_1(y)=cn(y, k),\\[1mm]
\epsilon_2=1+k^2, & \theta_2(y)=sn(y, k).
\end{array}
$$
>From (\ref{2.3}) it follows that zero is an eigenvalue of $L_2$
and it is the first eigenvalue in the case of left and right family,
with corresponding eigenfunction $r(x)$.
\vspace{2ex}
{\bf 3. The estimate for $M_{2}$}
Below, we will denote by $\langle f,g\rangle=\int_0^T f(x)g(x)dx$
and by $||f||$ the scalar product and the norm in $L^2[0,T]$.
In the formulas that follow, we take $r=r(\bar{x})$ with an argument
$\bar{x}=x-\xi+(a+3\omega)t$. From the previous section, we know that
when considered in $[0,T]$, the operator $L_2$ has an eigenfunction
$r$ corresponding to zero eigenvalue and the rest of the spectrum
is contained in $(\alpha^2,\infty)$.
The derivative of $d_{q}^{2}(u, \varphi)$ with respect to $\eta$
at the point where the minimum is attained is equal to zero.
Together with (\ref{2.3}), this yields
\begin{equation}\label{3.7}
\begin{array}{ll}
0&=-i\omega\int_{0}^{T}{[ e^{i\omega \eta}\varphi_{x}
\overline{h}_{x}- e^{-i\omega \eta}\overline{\varphi}_{x}h_{x}
+q( e^{i\omega \eta}\varphi \overline{h}
- e^{-i\omega \eta}\overline{\varphi}h)]}dx\\[2mm]
&=2\omega Im\int_0^T(-\varphi_{xx}+q\varphi)e^{i\omega\eta}\overline{h}dx\\[2mm]
&=2\omega Im\int_0^T((q+\omega^2+a+2r^2)r-2i\omega r')(h_1-ih_2)dx\\[2mm]
&=-2\omega\int_0^T [(q+\omega^2+a+2r^2)rh_{2}+2\omega r' h_1]dx.
\end{array}
\end{equation}
We set $h_2=\beta r(\bar{x})+\theta$, $\;\int_0^T\theta rdx=0$.
Substituting in (\ref{3.7}), we obtain
$$0=\beta ||r||^2\left(q+\omega^2+a+\frac{2||r^2||^2}{||r||^2}\right)
+2\int_0^T (\theta r^3+\omega r' h_1) dx$$
Using that $\frac{2||r^2||^2}{||r||^2} \geq -a$ (see estimate A of the Appendix),
we obtain the estimate
$$\begin{array}{ll}
|\beta|\,||r||& \displaystyle \leq 2 \frac{\left|
\int_0^T(\theta r^3+\omega r' h_1)dx\right|}
{(q+\omega^2)||r||}\\[5mm]
&\displaystyle \leq \frac{2||r^3||\cdot ||\theta||+2|\omega|\,
||r'||\cdot ||h_1||}{(q+\omega^2) ||r||}\\[3mm]
& \leq m_0(||\theta||+||h_{1}||),
\end{array}
$$
where $m_0=2m_1(a,\omega)/(q+\omega^2)$ and
$$m_1(a,\omega)=\max\limits_{c\in[-\frac14a^2,0]}\left(
\frac{||r^3||}{||r||}, \frac{|\omega|\,||r'||}{||r||},
\frac{3||r^2r'||}{||r'||},\frac{|\omega|\,||ar+2r^3||}{||r'||}\right)$$
(the third and fourth item are included for later use). It is obvious
that the first three fractions are bounded. For the last one, see
estimate D in the Appendix. We will use below that for $a$ and $\omega$
fixed, $m_0\rightarrow 0$ when $q\rightarrow \infty$. Further,
$$||h_{2}||\leq |\beta|\,||r||+||\theta||\leq m_0
(||\theta||+||h_{1}||)+||\theta||=(m_0+1)||\theta||+m_0||h_1||.$$
Hence, we obtain
\begin{equation}\label{3.8}
||\theta||^{2}\geq \frac{||h_2||^2}{2(m_0+1)^2}
-\left( {\frac{m_0}{m_0+1}}\right)^2||h_1||^2.
\end{equation}
Since $L_{2}r=0$ and $\langle \theta, r\rangle=0$, then from the spectral
properties of the operator $L_2$, it follows
$$
M_2=\langle L_2 h_2, h_2\rangle =\langle L_2\theta, \theta\rangle\geq
\alpha^2\langle \theta, \theta\rangle \geq -\frac{a}{2}||\theta||^2.
$$
From here and (\ref{3.8}), one obtains
\begin{equation}\label{3.9}
M_2 \geq \frac{|a|}{4(m_0+1)^2}||h_2||^2
-\frac{|a|m_0^2}{2(m_0+1)^2} ||h_1||^2.
\end{equation}
\vspace{2ex}
{\bf 4. The estimate for $M_{1}$}
First of all, let us note that the operator $L_1$ equipped with
periodic boundary conditions in $[0,T]$ has the following spectral data:
\begin{equation}\label{spl1}
\begin{array}{ll}
\lambda_0=a-2\sqrt{a^2+3c},\quad & \psi_0=6r^2+3a-\lambda_0,\\
\lambda_1=0 & \psi_1=r',\\
\lambda_2=a+2\sqrt{a^2+3c},\quad & \psi_2=6r^2+3a-\lambda_2,
\end{array}\end{equation}
and the rest of the spectrum is contained in $(\lambda_2,\infty)$.
We set
\begin{equation}\label{3.10}
h_1=\gamma_1\psi_0(\bar{x})+\gamma_2r'(\bar{x})
+\theta_1, \; \; r(\bar{x})=\nu \psi_0(\bar{x})+\psi,
\end{equation}
where
\begin{equation}\label{3.10.5}
\langle \theta_{1}, \psi_{0}\rangle= \langle \theta_{1}, r'\rangle=
\langle \psi, \psi_{0}\rangle= \langle \psi_0,r'\rangle=
\langle \psi, r'\rangle=0
\end{equation}
and $\gamma_1$, $\gamma_2$ and $\nu$ are some constants.
By (\ref{3.10.5}), we have
$$M_{1}(h_{1})=\langle L_1h_1, h_1\rangle =\gamma_1^2\lambda_0
\langle \psi_0,\psi_0\rangle+\langle L_1\theta_1, \theta_1\rangle.$$
Therefore, from spectral properties of the operator $L_1$ it follows
\begin{equation}\label{3.11}
M_1(h_1)\geq \gamma_1^2\lambda_0||\psi_0||^2+\lambda_2||\theta_1||^2.
\end{equation}
The fundamental difficulty in the estimate of $M_1$ is the appearance
of the negative term $\gamma_1^2 \lambda_0 ||\psi_0||^2$. Below, we
are going to estimate it. From the condition
$$P(u)=\int_0^T{|h+e^{i\omega \eta}\varphi(x-\xi,t)|^2}dx=P(\varphi)$$
we obtain
$$||h||^2=2Re\int_0^T e^{i\omega \eta}\varphi(x-\xi,t)\overline{h} dx
=-2\int_0^T r h_1 dx.$$
Then using (\ref{3.10}), we have
$$-\frac12||h||^2=\nu \gamma_1||\psi_0||^2
+\int_0^T \psi\theta_1 dx$$
and therefore
\begin{equation}\label{3.12}
\gamma_1^2||\psi_0||^2=\frac{1}{\nu^2 ||\psi_0||^2}\left(\frac12||h||^2
+\int_0^T \psi \theta_1 dx\right)^2.
\end{equation}
From (\ref{3.12}), we obtain
\begin{equation}\label{3.13}
\gamma_1^2||\psi_0||^2\leq \frac{1}{\nu^{2}
||\psi_0||^2}\left({\frac{1+d}{4}}||h||^{4}
+{\frac{d+1}{d}}||\psi||^2||\theta_{1}||^2\right),
\end{equation}
where $d$ is a positive constant which will be fixed later.
Below, we will denote by $C_m$, $D_m$ positive constants,
depending only on $d$ but not on the system parameters $a, c, \omega$.
Using (\ref{3.12}) and (\ref{3.11}), we derive the inequality
\begin{equation}\label{3.14}
\begin{array}{ll}
M_1&\geq \left(\lambda_2+\lambda_0(1+\frac{1}{d})
{\frac{||\psi||^2}{\nu^2||\psi_0||^2}}\right)||\theta_{1}||^{2}
+{\frac{\lambda_0(1+d)}{4\nu^2||\psi_0||^2}}||h||^4 \\[2mm]
&\geq C_1\lambda_2||\theta_1||^2-D_1|a|^\frac12||h||^4
\end{array}
\end{equation}
(see the estimates in point C of the Appendix).
We denote $\vartheta=h_1-\gamma_2r'(\bar{x})=\gamma_1\psi_0(\bar{x})+\theta_1$.
Then from (\ref{3.10.5}), (\ref{3.13}) and the inequalities
$\lambda_2\leq\frac13|\lambda_0|\leq |a|$, we have
$$\begin{array}{rl}
||\vartheta||^2=\gamma_1^2||\psi_0||^2+||\theta_1||^2&
\leq \left(1+\frac{(d+1)||\psi||^2} {d\nu^2||\psi_0||^2}\right)||\theta_1||^2
+\frac{1+d}{4\nu^2||\psi_0||^2}||h||^4\\[2mm]
&\leq C_2||\theta_1||^2+ D_2|a|^{-\frac12}||h||^4.\end{array}$$
Then
$$||\theta_1||^2\geq \frac{||\vartheta||^2}{C_2}
-\frac{D_2||h||^4}{C_2|a|^\frac12}$$
and hence, by (\ref{3.14}) and $\lambda_2\leq |a|$,
\begin{equation}\label{3.15}
\begin{array}{rl}
M_1&\displaystyle\geq \frac{C_1\lambda_2}{C_2}||\vartheta||^2
-\frac{C_1D_2+C_2D_1}{C_2}|a|^\frac12||h||^4\\[3mm]
&= C_3\lambda_2||\vartheta||^2-D_3|a|^\frac12||h||^{4}.
\end{array}
\end{equation}
After differentiating (\ref{3.2}) with respect to $\xi$, we obtain
$$\begin{array}{ll}
0&=2Re\int_0^T{e^{i\omega \eta}(\varphi_{xx}\overline{h}_x
+q\varphi_x\overline{h})}dx
=2Re\int_0^T{e^{i\omega \eta}\left[\varphi_{t}
+(6|\varphi|^{2}+q)\varphi_x\right]\overline{h}}dx\\[2mm]
&=2Re\int_0^T{(h_1-ih_2)[i\omega (3a+\omega^2+6r^2+q)r
+(a+3\omega^2+6r^2+q)r']}dx\\[2mm]
&=2\int_0^T{[(a+3\omega^2+6r^2+q)r'h_1+\omega(3a+\omega^2 +6r^2+q)rh_2]}dx.
\end{array}
$$
From (\ref{3.7}), we have
$$\int_0^T{qrh_2}dx=-\int_0^T[2\omega r'h_1+(a+\omega^2+2r^2)rh_2]dx$$
and replacing in the above equality, we obtain
$$\int_0^T{[(a+\omega^2+6r^2+q)r'h_1+\omega (2a+4r^2)rh_2]}dx=0.$$
Substituting $h_1=\gamma_2r'(\bar{x})+\vartheta$ in the above
equality and using the orthogonality condition
$\langle r',\vartheta\rangle=\langle r', \gamma_1\psi_0
+\theta_1\rangle=0$, we obtain
$$\gamma_2||r'||^2\left(a+\omega^2+q+\frac{6||rr'||^2}
{||r'||^2}\right)+2\int_0^T{[\omega (a+2r^2)rh_2+3r^2r'\vartheta]}dx=0.$$
Using that ${\frac{6||rr'||^2}{||r'||^2}}\geq -a$ (see Appendix),
we further have
$$\begin{array}{ll}
|\gamma_2|\,||r'||& \displaystyle\leq {\frac{2\left| \int_0^T
{[\omega (a+2r^2)rh_2+3r^2r'\vartheta]}dx\right| }
{(\omega^2+q) ||r'||}}\\[5mm]
&\displaystyle \leq 2{\frac{|\omega|\,||ar+2r^3||\cdot||h_2||
+3||r^2r'||\cdot ||\vartheta||}{(\omega^2+q)||r'||}}\\[4mm]
&\leq m_0(||\vartheta||+||h_2||).
\end{array}
$$
Hence
$$||h_1||\leq|\gamma_2|\,||r'||
+||\vartheta||\leq (m_0+1)||\vartheta||+m_0||h_2||,$$
which yields
$$||\vartheta||^{2}\geq \frac{||h_1||^2}{2(m_0+1)^2}
-\left(\frac{m_0}{m_0+1}\right)^2||h_2||^2.$$
Replacing in (\ref{3.15}), we finally obtain
\begin{equation}\label{3.16}
M_{1}\geq \frac{C_3\lambda_2}{2(m_0+1)^2}||h_1||^2
-\frac{C_3\lambda_2m_0^2}{(m_0+1)^2}||h_2||^2-D_3|a|^\frac12|h||^4.
\end{equation}
\vspace{2ex}
{\bf 5. The estimate for $\Delta M$}
From (\ref{3.9}) and (\ref{3.16}), we have
$$ M_1+M_2 \geq \frac{C_3\lambda_2-|a|m_0^2}{2(m_0+1)^2} ||h_1||^2
+\frac{|a|-4C_3\lambda_2m_0^2}{4(m_0+1)^2}||h_2||^2-D_3|a|^\frac12||h||^4.$$
We now fix $q$ so that $|a|m_0^2\leq\frac12C_3\lambda_2$ and assuming that
$C_3\leq\frac12$ (which is no loss of generality), one has also
$4C_3\lambda_2m_0^2\leq\frac12|a|$. Therefore we obtain
$$M_1+M_2\geq C_4\lambda_2(||h_1||^2+||h_{2}||^2)-D_3|a|^\frac12||h||^4=
C_4\lambda_2||h||^2-D_3|a|^\frac12||h||^4$$
where $C_4$ and $D_3$ are absolute constants independent on the parameters
of the system.
On the other hand, estimating directly $I_{2}$ from below (for this purpose
we use its initial formula), we have
$$\begin{array}{ll}
I_2&\geq ||h_{x}||^{2}+\int_{0}^{T}{(\omega^{2}-a-4r^{2})|h|^{2}}dx
-2|\omega|\int_{0}^{T}{|h|\cdot |h_{x}|}dx
-2\int_{0}^{T}{r^{2}|h|^{2}}dx\\[2mm]
&\geq ||h_{x}||^{2}+(\omega^{2}-a+4a)||h||^{2}-2\omega^{2}||h||^{2}
-{\frac{1}{2}}||h_{x}||^{2}+2a||h||^{2}\\[2mm]
&={\frac{1}{2}}||h_{x}||^{2}-(\omega^{2}-5a)||h||^{2}.
\end{array}
$$
Similarly, $|I_{3}|\leq \max(4|a|^\frac12|h|+|h|^2)||h||^2$.
Let $0<m<\frac12$. We have
$$\begin{array}{ll}
\Delta M & =2mI_2+(1-2m)(M_1+M_2)+I_3\\[1mm]
&\geq m||h_{x}||^2-2m(\omega^2-5a)||h||^2
+(1-2m)(C_4\lambda_2||h||^2
-D_3|a|^\frac12||h||^4)\\[1mm]
&-\max (4|a|^\frac12|h|+|h|^2)||h||^2\\[1mm]
&=m||h_{x}||^2+\left[-2m(\omega^2-5a)+
(1-2m)C_4\lambda_2\right]||h||^2\\[1mm]
&-[\max(4|a|^\frac12|h|+|h|^2)+(1-2m)D_3|a|^\frac12||h||^2]||h||^2.
\end{array}
$$
We choose $m$, so that
$$2qm=(1-2m)C_4\lambda_2-2m(\omega^2-5a),\;\;\mbox{\rm i.e.}\;\;
2m=\frac{C_4\lambda_2}{q+\omega^2+5|a|+C_4\lambda_2}<1.$$
From the inequality
$$|h|^2\leq {\frac{1}{T}}\int_0^T{|h|^2}dx +2\left(\int_0^T{|h|^2}dx
\int_0^T{|h_{x}|^2}dx \right)^\frac12$$
we obtain
$$|h|^2\leq {\frac{1}{T}}\int_0^T{|h|^2}dx+\sqrt{q}\int_0^T{|h|^2}dx
+\frac{1}{\sqrt{q}}\int_0^T{|h_{x}|^2}dx.$$
Hence for sufficiently large $q$, we obtain
$$\max |h(x,t)|^2\leq {\frac{2}{\sqrt{q}}}d_{q}^{2}(u, \varphi )$$
and moreover $||h||^2\leq q^{-1}d_{q}^{2}(u, \varphi )$. Consequently we
can choose $\delta_{0}>0$, such that for $d_{q}(u, \varphi )<\delta_{0}$,
we will have
$[\max(4|a|^\frac12|h|+|h|^2)+(1-2m)D_3|a|^\frac12]||h||^2\leq qm$.
Finally, we obtain that if $d_q(u, \varphi )<\delta_0$,
then $\Delta M\geq md_q^2(u, \varphi)$. Proposition \ref{p31} is
completely proved. $\Box$
\vspace{2ex}
{\bf 6. Proof of Theorem \ref{t31}}
We split the proof of our main result
into two steps. We begin with the special case
$P(u)=P(\varphi)$. Assume that $m,q, \delta_{0}$
have been selected according to Proposition \ref{p31}. Since
$\Delta M$ does not depend on $t, t\in [0, \infty)$, there
exists a constant $l$ such that $\Delta M \leq ld^{2}(u,
\varphi)|_{t=0}$. Below, we shall assume without loss of
generality that $l\geq 1, q\geq 1$.
Let
$$\varepsilon >0, \; \; \delta = \min \left( \left(
{\frac{m}{lq}}\right){\frac{\delta_{0}}{2}}, \left(
{\frac{m}{l}}\right)^{1/2}\varepsilon \right) $$
and $d(u, \varphi)|_{t=0}<\delta$. Then
$$d_{q}(u, \varphi)\leq q^{1/2}d(u,
\varphi)|_{t=0}<{\frac{\delta_{0}}{2}} $$
and Lemma \ref{l31} yields that there exists a
$t_{0}>0$ such that $d_{q}(u, \varphi)<\delta_{0}$ if $t\in [0,
t_{0})$. Then, by virtue of Proposition \ref{p31} we have
$$\Delta M \geq md^{2}_{q}(u, \varphi), \; \; t\in [0,
t_{0}).$$
Let $t_{max}$ be the largest value such that
$$\Delta M \geq md^{2}_{q}(u, \varphi), \; \; t\in [0,
t_{max}).$$
We assume that $t_{max}<\infty.$ Then, for $t\in [0,
t_{max}]$ we have
$$d^{2}_{q}(u, \varphi)\leq {\frac{\Delta M}{m}}\leq
{\frac{l}{m}}d^{2}(u,
\varphi)|_{t=0}<{\frac{l}{m}}\delta^{2}\leq
{\frac{\delta^{2}_{0}}{4}}.$$
Applying once again Lemma \ref{l31}, we obtain that there
exists $t_{1}>t_{max}$ such that
$$d_{q}(u, \varphi)<\delta_{0}, \; \; t\in [0,
t_{1}).$$
By virtue of the proposition, this contradicts the
assumption $t_{max}<\infty$. Consequently,
$t_{max}=\infty$,
$$\Delta M \geq md^{2}_{q}(u, \varphi)\geq md^{2}(u,
\varphi), \; \; t\in [0, \infty).$$
Therefore,
$$d^{2}(u, \varphi)\leq {\frac{\Delta M}{m}}\leq
{\frac{l}{m}}\delta^{2}<\varepsilon^{2}, \; \; t\in
[0, \infty), $$
which proves the theorem in the special case.
Now we proceed to remove the restriction
$P(u)=||u||^{2}=||\varphi||^{2}=P(\varphi)$. We have (see (\ref{EK}))
$||\varphi||=(2r_0E(k))^{1/2}$, where $r_0$ is given by (\ref{1.7}).
Below, we are going to apply a perturbation argument, freezing for
a while the period $T$ and the parameters $a,c$ in (\ref{newton}).
We claim there are respective parameter values $a^*, c^*$, and corresponding
$\varphi^*$, $r^*$, $r_0^*$, $k^*$, see (\ref{1.3}), (\ref{newton}) and (\ref{1.7}),
such that $\varphi^*$ has a period $T$ in $x$ and moreover,
$2r_0^*E(k^*)=||u||^2$. By (\ref{1.7}), we obtain the equations
\begin{equation}\label{ift}
\begin{array}{l}
\displaystyle \frac{2K(k^*)}{r_0^*}-T=0,\\
\displaystyle 2r_0^*E(k^*)-||u||^2=0.
\end{array}
\end{equation}
If (\ref{ift}) has a solution $k^*=k^*(T, ||u||)$, $r_0^*=r_0^*(T, ||u||)$,
then the parameter values we need are given by
$$a^*=({k^*}^2-2){r_0^*}^2,\quad c^*=({k^*}^2-1){r_0^*}^4.$$
Moreover, one has $||\varphi^*||=||u||$ and we could use the restricted
result we established above.
As $k=k^*(T, ||\varphi||)$, $r_0=r_0^*(T, ||\varphi||)$, it remains to
apply the implicit function theorem to (\ref{ift}). Since the corresponding
functional determinant reads
$$\left|\begin{array}{cc}\frac{2K'(k^*)}{r_0^*} & -\frac{2K(k^*)}{{r_0^*}^2} \\
2r_0^*E'(k^*) & 2E(k^*)\end{array}\right|=\frac{4}{r_0^*}(KE)'
=\frac{4}{r_0^*}({\textstyle\frac12}\pi+KK')>0$$
(by Legendre's identity), the existence of $a^*$ and $c^*$ with the
needed properties is established.
By (\ref{ift}) and our assumption, we have
\begin{equation}\label{4.20}
\frac{K(k^*)}{r_0^*}=\frac{K(k)}{r_0}=\frac{T}{2}.
\end{equation}
Next, choosing $\eta=2(a^*-a)t$, $\xi=(a-a^*)t$, by (\ref{1.3}) and
(\ref{3.1}) one easily obtains the inequality
$$d^2(\varphi^*, \varphi)\leq
(1+\omega^2)||r^*-r||^2+||{r^*}'-r'||^2.$$
Denote for a while
$\Phi(\rho)=\rho\, dn(z\rho;k(\rho))
=\rho\, dn(y;k)$ where $k=k(\rho)$ is dertermined from
$K(k)=\frac12\rho T$. Then using (\ref{1.7}), we have
$r^*-r=\Phi(r_0^*)-\Phi(r_0)=(r_0^*-r_0)\Phi'(\rho)$ with some appropriate
$\rho$. Moreover,
$$\Phi'(\rho)=dn(y;k)+\rho\left[z\frac{\partial dn}{\partial y} (y;k)
+\frac{T}{2K'(k)}\frac{\partial dn}{\partial k}(y;k)\right]$$
satisfies $|\Phi'(\rho)|\leq C_0$ with a constant $C_0$ independent
on the values with $*$ accent. Hence,
$|r^*-r|\leq C_0|r_0^*-r_0|$. Similarly,
$|{r^*}'-r'|\leq C_1|r_0^*-r_0|$. All this, together with (\ref{4.20}) yields
\begin{equation}\label{4.21}
d(\varphi^*,\varphi)\leq C |r_0^*-r_0|=\frac{2C}{T}|K(k^*)-K(k)|=
\frac{2C}{T}|K'(\kappa)||k^*-k|.
\end{equation}
From the inequalities
$$\left|\, ||\varphi^*||-||\varphi||\,\right|=\left|\,
||u||-||\varphi||\,\right|\leq d(u,\varphi)|_{t=0}<\delta$$
it follows that
$$-\left( 2r_0E(k) \right)^{-1/2}\delta <
(||\varphi||)^{-1}||\varphi^*||-1<\left( 2r_0E(k)
\right)^{-1/2}\delta $$
and, therefore,
$1-\delta_1<{\frac{r_0^*E(k^*)|}{r_0E(k)}}<1+\delta_1$,
i.e. $|r_0^*E(k^*)-r_0E(k)|<r_0E(k)\delta_1$,
where we have denoted $\delta_1=(1+(2r_0E(k))^{-1/2}\delta)^2-1$.
On the other hand, we have (using (\ref{4.20}) again)
\begin{equation}\label{4.22}
|r_0^*E(k^*)-r_0E(k)|=\frac{2}{T}|K(k^*)E(k^*)-K(k)E(k)|=
\frac{2}{T}|(KE)'(\kappa)||k^*-k|\geq C_2|k^*-k|,
\end{equation}
with appropriate $C_2>0$ independent on the values bearing $*$ accent.
In particular, one has $|k^*-k|\leq C_3r_0E(k)\delta_1$.
Thus combining (\ref{4.21}) and (\ref{4.22}), we get
$$d(u, \varphi^*)|_{t=0}\leq d(u,
\varphi)|_{t=0}+d(\varphi,
\varphi^*)|_{t=0}<\delta+Cr_0E(k)\delta_1=\delta_0.$$
Let $\varepsilon >0$. We select $\delta$ (and together, $\delta_0$ and
$\delta_1$) sufficiently small and apply the part of the theorem which has
already been proved to conclude:
$$d(u, \varphi^*)|_{t=0}<\delta_0\Rightarrow
d(u, \varphi^*)<{\frac{\varepsilon}{2}}, \;
\; t\in [0, \infty).$$
Then, choosing an appropriate $\delta>0$, we obtain that
$$d(u, \varphi)\leq d(u,
\varphi^*)+d(\varphi,
\varphi^*)<
{\frac{\varepsilon}{2}}+Cr_0E(k)\delta_1<\varepsilon$$
for all $t\in [0, \infty)$. The theorem is
completely proved. $\Box$
\vspace{2ex}
\section{Appendix}
For $n\in\mbox{\of Z}$ and $c\in(-\frac14a^2,0)$, consider the
line integrals $I_n(c)$ and their derivatives $I'_n(c)$ given by
\begin{equation}\label{in}
I_n(c)=\oint_{H=c}x^nydx,\qquad I_n'(c)=\oint_{H=c}\frac{x^ndx}{2y}
\end{equation}
where one can assume for definiteness that the integration is along the
right oval contained in the level set $\{H=c\}$.
These integrals would be useful because
\begin{equation}\label{red}
\int_0^Tr^n(t)dt=2\int_0^{\frac12T}r^n(t)dt
=2\int_{r_1}^{r_0}\frac{x^ndx}{\sqrt{c-ax^2-x^4}}
=\oint_{H=c}\frac{x^ndx}{y}=2I_n'(c).
\end{equation}
(we applied a change of the variable $r(t)=x$ in the integral
and used equation (\ref{newton})).
The properties of $I_n$ are well known, see e.g. \cite{HIK}
for a recent treatment. Below, we list some facts we are going to use.
\vspace{2ex}
\noindent
{\bf Lemma.} (i) {\it The following identity holds:
$$(n+6)I_{n+3}+(n+3)aI_{n+1}-ncI_{n-1}=0,\quad n\in\mbox{\of Z}$$
which implies}
\begin{equation}\label{recur}
\textstyle
I_3'=-\frac12a I'_1,\quad
I_4'=\frac13cI_0'-\frac23aI_2',\quad
I_6'=-\frac{4}{15}acI_0'+(\frac{8}{15}a^2+\frac35c)I_2'.
\end{equation}
\vspace{1ex}
\noindent
(ii) {\it The integrals $I_0$ and $I_2$ satisfy the system}
$$\begin{array}{l}
4cI'_0-2aI'_2=3I_0,\\
-2acI'_0+(12c+4a^2)I'_2=15I_2.\end{array}$$
\vspace{1ex}
\noindent
(iii) {\it The ratio $R(c)=I_2'(c)/I_0'(c)$ satisfies the Riccati equation
and related system
\begin{equation}\label{ricc}
(8c^2+2a^2c)R'(c)=ac+4cR(c)-aR^2(c),\qquad
\begin{array}{l}\dot{c}=8c^2+2a^2c,\\
\dot{R}=ac+4cR-aR^2,\end{array}\end{equation}
which imply estimates}
\begin{equation}\label{est}
\frac{2c}{a}\leq R(c)\leq\frac{c}{2a}-\frac{3a}{8}.
\end{equation}
\vspace{2ex}
\noindent
The equations in (i)--(iii) are derived in a standard way, see \cite{HIK}
for more details. The estimates (\ref{est}) follow from the fact that,
in the $(c,R)$-plane,
the graph of $R(c)$ coincides with the concave separatrix trajectory of
the system (\ref{ricc}) contained in the triangle with vertices $(0,0)$,
$(-\frac14a^2,-\frac12a)$ and $(0,-\frac38a)$ and connecting the first
two of them.
After this preparation, we turn to prove the estimates we used in the
preceding sections.
\vspace{2ex}
\noindent
{\bf A. The estimate for } $A=\frac{2||r^2||^2}{||r||^2}$.
By (\ref{red}), (\ref{recur}) and the first inequality in (\ref{est}),
we have
$$A=\frac{2\int_0^Tr^4dt}{\int_0^Tr^2dt}=\frac{2I_4'}{I_2'}=
\frac{2cI_0'-4aI_2'}{3I_2'}=\frac{2c}{3}\frac{1}{R}-\frac{4a}{3}\geq-a.$$
\vspace{2ex}
\noindent
{\bf B. The estimate for} $B=\frac{6||rr'||^2}{||r'||^2}$.
By (\ref{newton}), we have as above
$$B=\frac{6\int_0^Tr^2(c-ar^2-r^4)dt}{\int_0^T(c-ar^2-r^4)dt}
=\frac{6(cI_2'-aI_4'-I_6')}{cI_0'-aI_2'-I_4'}$$
$$=\frac65\frac{(2a^2+6c)I'_2-acI'_0}{2cI'_0-aI'_2}
=\frac65\frac{(2a^2+6c)R-ac}{2c-aR}\geq-\frac{12}{5}a.$$
To obtain the last inequality, we used that $4c+a^2\geq 0$
and the second estimate in (\ref{est}).
\vspace{2ex}
\noindent
{\bf C. The estimate for}
$C=\lambda_2+\lambda_0(1+\frac{1}{d})\frac{||\psi||^2}{\nu^2||\psi_0||^2}$.
By (\ref{3.10}) and (\ref{3.10.5}) we have
$$||\psi||^2=||r||^2-\nu^2||\psi_0||^2,\;\;\mbox{\rm where}\;\;
\nu=\frac{\langle r,\psi_0\rangle}{||\psi_0||^2}.$$
Therefore
$$C=\lambda_2+\lambda_0\left(1+\frac{1}{d}\right)
\left(\frac{||r||^2||\psi_0||^2}{\langle r,\psi_0\rangle^2}-1\right).$$
Next,
$$\langle r,\psi_0\rangle=\int_0^T[6r^3+(3a-\lambda_0)r]dt
=12I_3'+(6a-2\lambda_0)I'_1=-2\lambda_0I'_1,$$
$$\begin{array}{rl}
||r||^2||\psi_0||^2 &=\int_0^Tr^2dt\int_0^T(6r^2+3a-\lambda_0)^2dt\\[4mm]
&=4I_2'[36I'_4+12(3a-\lambda_0)I'_2+(3a-\lambda_0)^2I_0']\\[2mm]
&=4I_2'[(12c+(3a-\lambda_0)^2)I'_0+(12a-12\lambda_0)I'_2].
\end{array} $$
By (\ref{1.7}), we have
\begin{equation}\label{EK}
I_2'(c)=\frac12\int_0^Tr^2dt=r_0\int_0^{K(k)}dn^2(t)dt=r_0E(k).
\end{equation}
Making use of the identity $E(k)=\frac12\pi F(\frac12,-\frac12,1,k^2)$
where $F$ is the Gauss hypergeometric function, we obtain an appropriate
expansion to estimate $E$ from above
$$
E(k)=\frac{\pi}{2}\left(1-\frac{k^2}{4}-\frac{3k^4}{64}-\frac{5k^6}{512}
-\ldots,\right), \quad
E^2(k)\leq\frac{\pi^2}{4}\left(1-\frac{k^2}{2}-\frac{k^4}{32}\right)$$
with all removed terms negative. As $I_1'=\frac12\pi$, by (\ref{1.7})
this implies
$$I_2'^2\leq - I_1'^2\frac{a^2+20ar_0^2+4r_0^4}{32r_0^2}.$$
Together with $I_0'I_2'\geq I_1'^2$, this yields
$$\begin{array}{rl}\displaystyle
\displaystyle \frac{||r||^2||\psi_0||^2}{\langle r,\psi_0\rangle^2}-1
&\displaystyle \leq\frac{1}{\lambda_0^2}\left[12c+(3a-\lambda_0)^2
+\frac{3}{8r_0^2}(\lambda_0-a)(a^2+20ar_0^2+4r_0^4)\right]-1\\[3mm]
&=\displaystyle\frac{\lambda_2}{\lambda_0}\left(\frac{\lambda_2-a}{8r_0^2}-1\right)
\leq\frac{\lambda_2}{\lambda_0}\left(\frac{\sqrt3}{8}-1\right)\end{array}$$
where the equality is obtained by direct calculations.
Therefore,
$$C\geq \lambda_2\left(-\frac{1}{d}+\frac{d+1}{d}\frac{\sqrt{3}}{8}\right)
=C_1\lambda_2$$
with $C_1>0$ an absolute constant when $d\geq 4$ is fixed.
As a by-product of our calculations, we easily obtain also the estimate
$$\frac{\lambda_0}{\nu^2||\psi_0||^2}=
\frac{\lambda_0||\psi_0||^2}{\langle r,\psi_0\rangle^2}
\geq \frac{\lambda_2(\frac{\sqrt3}{8}-1)+\lambda_0}{||r||^2}
\geq -D_1|a|^\frac12.$$
\vspace{2ex}
\noindent
{\bf D. The estimate for } $D=\frac{||ar+2r^3||}{||r'||}$.
Making use of statements (i) and (ii) of the Lemma, we have
$$D^2=\frac{\int_0^T(a^2r^2+4ar^4+4r^6)dt}{\int_0^T(c-ar^2-r^4)dt}=
\frac{a^2I_2'+4aI_4'+4I_6'}{cI_0'-aI_2'-I_4'}$$
$$=\frac{4acI_0'+(7a^2+36c)I_2'}{5(2cI_0'-aI_2')}
=\frac{aI_0+6I_2}{I_0}\leq -5a.$$
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,603 |
That companies are besieged with requests to disclose information is not news; nor is the awkward truth that most disclosures (think MD&A in your annual report) are not, ahem, brimming with specifics. This week, columnists Stephen Davis and Jon Lukomnik consider new ways to make your disclosure more manageable. One proposal: that companies admit some audiences might be more important than others. More inside. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,199 |
Le RER nord est un projet de ligne de train urbain à Toulouse sur la ligne ferroviaire Bordeaux-Saint-Jean à Sète-Ville. Elle sera exploitée par la SNCF dans le cadre des TER Occitanie.
La ligne verrait le jour grâce à l'augmentation du nombre de voies à la suite de la construction de la LGV de Bordeaux à Toulouse. Le projet prévoit également le déplacement de la gare de Route-de-Launaguet de quelques centaines de mètres afin de permettre des correspondances avec la station de métro La Vache de la ligne B et de la ligne TAE, qui devrait ouvrir en même temps. Il est envisagé en premier temps un terminus à La Vache - Route-de Launaguet au lieu de la gare Matabiau. En 2017, il pouvait être considéré comme le seul projet de RER encore d'actualité dans l'agglomération toulousaine.
Depuis, les positions semblent évoluer, tant du côté de la Région Occitanie que de celui de Tisséo. Par ailleurs, l'association Rallumons l'Étoile, qui milite pour un RER sur les 5 branches de l'étoile ferroviaire toulousaine, affiche 20 communes adhérentes représentant habitants.
Il est aussi envisagé d'ouvrir une station à Lespinasse, réclamée de longue date par la ville et ses habitants, malgré le refus de la SNCF.
Parcours
Tracé
La ligne sera longue de 21 km et ira de Toulouse (gare Matabiau) jusqu'à Castelnau-d'Estrétefonds.
Stations et correspondances
Offre de services
Fréquence et amplitude
L'aménagement de la ligne permettra d'augmenter la fréquence des trains, jusqu'au quart d'heure en heures de pointes, fréquence comparable aux Transilien les plus fréquents.
Tarification
Il n'est pas encore connu si la ligne sera accessible avec un simple titre de transport urbain Tisséo comme dans le cas de la ligne C, ou si elle utilisera la tarification TER normale (comme pour les lignes D et F). À l'heure actuelle, la ligne fait tout de même partie du réseau de transports urbains et est accessible avec un abonnement Pastel+ jusqu'à la gare de Saint-Jory.
Notes et références
Voir aussi
Articles connexes
TER Occitanie
Transports en commun de Toulouse
Réseau ferroviaire de Toulouse
Ligne C du réseau de transports en commun de Toulouse
Tisséo
LGV de Bordeaux à Toulouse
Liens externes
Aménagement ferroviaire au nord de Toulouse
TER Occitanie | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,794 |
Q: How to make sure that a memory card is compatible to my mac? I want to upgrade my RAM from 4g to 8g.
my mac specs are:
I need 2 4g memory cards, and this is a specific card I saw in a store that should be working, but no one gives a guarantee it will work on mac...they say it works on laptops, but they can't guarantee its working for mac...
This is the card model: G.Skill 1x4GB DDR3 1600Mhz SODIMM
A: I would use something like Crucial's online configurator rather than the vague specs in a shop window.
That way you have a guarantee it will work.
Kingston & Corsair have the same kind of thing, but Crucial gives you the option of downloading an app to do all the work for you.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,048 |
'Tis the season for holiday spending and capitalizing on festive shopping trends! In several holiday themed 2015 auctions, our media partners secured hundreds of joyful products and services from local merchants in their area, offering a risk-free promotional package for each merchant. The holiday-themed auctions utilized NeoFill's nationwide online auction platform, NeoFillBids.com.
With the new year quickly approaching, consumers will be looking to save on amazing, unique gifts for their loved ones! One partner last holiday season offered a $5,000 voucher for a local well-known jeweler. The voucher sold for $2,800, and the entire final bid earnings were contributed to the station's revenue. Another partner secured a Family Four Pack Season Ski Pass, perfect for the season. The item offered four people to ski unlimited times all season, valued at $1,936, and sold for $1,076! A Preferred Living Community provided a One Year Lease for a Two Bedroom Luxury Suite from a local rental community. The retail value of the item was $18,000 and the auction sold for over $8,900. Another media station offered a Fitness and Cycling Weight Machine just in time to meet the New Year Resolutions, valued at $3,199. The final bid price went for $2,921, offering a huge return on investment for the station.
promotional package for each merchant.
• Launch a holiday-themed auction using NeoFill's online auction platform, NeoFillBids.com.
• Offer high-ticketed items to gift-seeking holiday shoppers.
• Provide custom holiday categories to highlight each local merchant in a distinguishable way.
Winter adventures, jewelry, vehicles, family fun sports entertainment, vacation packages, and recreational outdoor attractions are among some of the best winter auction categories. NeoFill's online auction site has assisted hundreds of radio stations generate a new steam of revenue, no matter the season.
NeoFillBids is an Online Auction Platform owned and operated by NeoFill, LLC, an Ohio based digital marketing company, specializing in digital revenue generating programs for companies across America. NeoFillBids is a manageable tool to utilize to attract local businesses, engage your current audience, and earn automated digital revenue. Learn more about how NeoFill can assist any media group, any-where, any season succeed and generate thousands in revenue! | {
"redpajama_set_name": "RedPajamaC4"
} | 9,808 |
Nuveen Investments, an operating division of TIAA Global Asset Management, has announced that it intends to propose a plan of reorganization to the Board of the Nuveen Global Equity Income Fund (JGV).
Nuveen recently announced a plan to accelerate the growth of its global product platform at NWQ Investment Management Company. Assuming necessary Fund Board and shareholder approvals are received, this plan would consolidate a number of existing global equity funds currently managed by Tradewinds Global Investors into funds managed by NWQ Investment Management Company, including that of JGV.
Under the proposed reorganization plan, JGV would be reorganized into the open-end Nuveen NWQ Global Equity Income Fund, reflecting differing approaches between JGV's current mandate and NWQ's global equity income mandate. JGV shareholders would receive Class A shares of the NWQ fund in the reorganization. Nuveen expects to make this proposal at the Board's upcoming meeting in late May. If approved by the Fund's Board, the proposed reorganization is subject to shareholder approval at the fund's annual meeting to be held this fall.
Nuveen provides investment solutions designed to help secure the long-term goals of individual investors and the financial advisors who serve them. Through the expertise and capabilities of TIAA Global Asset Management's high-caliber investment managers, Nuveen is committed to providing world-class consultative services and advice that align with client needs. Funds distributed by Nuveen Securities, LLC, a subsidiary of Nuveen Investments, Inc. Nuveen Investments is an operating division of TIAA Global Asset Management. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,049 |
Jules-Arsène Garnier (* 12. Januar 1847 in Paris; † 25. Dezember 1889 ebenda) war ein französischer Maler.
Garnier war ein Schüler des Historienmalers Jean-Léon Gérôme. Gleich seinem Lehrer fand Garnier seine Sujets in figurenreichen Kultur- und Sittenbildern der Antike; mit einer Vorliebe für die dramatische Greuelszene. Bereits mit 22 Jahren konnte Jules-Arsène Garnier anlässlich der Jahresausstellung des Pariser Salons von 1869 mit zwei Werken debütieren: "Eine Badende" und das lüsterne Nachtbild "Mlle de Sombreuil, ein Glas Blut trinkend".
Mit diesen beiden Gemälden hatte er auch seinen künstlerischen Durchbruch und das "schaudernde" Publikum verlangte weiteres von ihm. Es folgten nun in rascher Folge neben einigen anderen die Sittenstudie "Das Herrenrecht" (1872) und "Le roi s'amuse" (1874). Zu letzterem wurde Garnier durch die Lektüre von Victor Hugo inspiriert. 1876 malte er Strafe der Ehebrecher, ein mittelalterliches Sittenbild von großer koloristischer Wirkung.
Für die große Ausstellung des Pariser Salons von 1877 schuf Garnier "Die Favoritin, welcher der Kopf der eben enthaupteten Konkurrentin gebracht wird". Nach eigenen Angaben war die Lektüre von "Orientales" (Victor Hugo) der Auslöser für die Entstehung dieses Werkes.
1878 stellte Jules-Arsène Garnier sein Werk "Der Befreier" der Öffentlichkeit vor. Nach einer Notiz des "Journal officiel" vom 17. Juni 1877 war damit der Politiker Adolphe Thiers gemeint, welcher im Mai 1871 den Aufstand der Pariser Kommune niederschlagen hatte lassen.
In seinem Spätwerk kommen dann trotz Dramatik auch fast schon humorvolle Darstellungen zur Geltung. 1879 schuf Garnier zum Beispiel "Die Versuchung": ein frommer Einsiedler wird durch zwei nackte Frauengestalten in arge Gewissensnöte gebracht.
Sein Versuch, mit "Verteilung der Fahnen" (14. Juli 1880) auch moderne Stoffe in großem Maßstab zu behandeln, misslang allerdings.
Werke (Auswahl)
Eine Badende (1869)
Mlle de Sombreuil, ein Glas Blut trinkend (1869)
Das Herrenrecht (1878)
Le Droit du Seigneur (1872)
Die Vasallenabgabe (1873)
Die Hinrichtung einer Frau im 16. Jahrhundert (1875)
Die Strafe der Ehebrecher [Le supplice des adultères ] (1876)
Die Favoritin, welcher der Kopf der eben enthaupteten Konkurrentin gebracht wird (1877)
Der Befreier (1878)
Versuchung (1879)
Die Verteilung der Fahnen (1880)
Quellen
Allgemeines Künstlerlexikon – Band XLIX. 2006.
Weblinks
Jules-Arsène Garnier: "Der Befreier" (Le Libérateur du Territoire)
Maler (Frankreich)
Franzose
Geboren 1847
Gestorben 1889
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,613 |
\section{Introduction}
One of the major goals of cosmology is to probe the evolution of structure with
lookback time.
Galaxy clusters play a key role in this ambition because:
\begin{itemize}
\item Their formation by gravitational collapse is understood well enough
to allow comparisons of observations with theoretical
predictions of density evolution.
\item They can be observed out to cosmologically significant distances ($z \simeq 1.0$)
and so provide an important probe of the conditions in dense environments at early
times.
\end{itemize}
Optically selected distant cluster catalogues (e.g. Gunn et al.~1986; Couch et al.~1990)
are plagued by selection effects such as cluster mis-identification due to line
of sight projection effects (Frenk et al.~1990).
X-ray cluster selection greatly reduces such effects and produces cleaner,
statistically better defined samples (Henry~1992).
Two recent surveys measured the
X-ray luminosity function (XLF) and found that there are
fewer X-ray bright clusters at $z \simeq 0.1 - 0.3$ than locally
(Edge et al.~1990; Henry et al.~1992).
This result, if true, is of major consequence to our understanding of cluster
evolution (e.g. Kaiser~1991).
Unfortunately the conclusions are tentative because the surveys become seriously
incomplete at the redshifts where the evolution is seen.
The increased spatial resolution and sensitivity of the ROSAT PSPC detector
compared to previous X-ray imaging satellites and its low background rate mean
that we can measure the cluster XLF at fainter fluxes.
We can therefore test both the claims of cluster evolution and measure
the shape of the local XLF.
Several groups are working towards this goal (e.g. RIXOS, RDCS \& WARPS),
each using different cluster selection methods.
As yet, there is no agreed method for detecting clusters in X-ray
imaging data and so it is important for independent surveys to investigate
different selection algorithms.
A comparison of these surveys is given in Table~\ref{tbl:surveys}.
\begin{table}
\caption{A comparison of distant X-ray cluster surveys.}
\label{tbl:surveys}
\[
\begin{array}{p{0.3\linewidth}c@{\hspace{0.3in}}c} \hline
\noalign{\smallskip} Name & f_{lim}^\dagger/10^{-14} $ erg cm$^{-2} $ s$^{-1} & \Omega/$deg$^{2} \\
\noalign{\smallskip} \hline
\noalign{\smallskip}
This Project & 4 & 14 \\
EMSS & \sim 10 & 40\\
RDCS & 1 & 26\ddagger \\
RIXOS & 3 & 15\ddagger \\
WARPS & 7 & 13\ddagger \\
\noalign{\smallskip} \hline
\end{array}
\]
\begin{list}{}{}
\item[$^{\rm \dagger}$] Flux limits are for the 0.5 - 2.0 keV band.
\item[$^{\rm \ddagger}$] Areal coverage as given in
Rosati 1995, Castander et al.~1995 and Jones et al.~1995.
\end{list}
\end{table}
\section{Source Selection}
\label{sec:source-selec}
To combat the severe contamination expected from stars and AGN at our survey
depths (Stocke et al.~1991), our primary selection criterion is source
extension.
As clusters have significantly harder spectra than the general population of X-ray
sources (Ebeling~1993)
we use source hardness as a secondary selection criterion.
The analysis is restricted to the 0.5-2.0 keV band, to reduce the contamination
from soft sources, and to the central region of the PSPC detector, where
the PSF does not change significantly with off-axis angle.
We are concentrating on the 100 deepest pointings which satisfy:
\begin{itemize}
\item T $\geq$ 10ks
\item $\left| b \right| > 20^0$
\item $\delta < 20^0$
\end{itemize}
Analysis begins with screening out periods of bad aspect error or
high particle background.
We then compute a global estimate for the background and search
for sources using the Cash statistic (Cash~1979).
Sources are tested for extent by comparing the photon distribution
to the PSF, taking care to model both the positional and spectral
dependence of the PSF.
Monte Carlo simulations show that we can reliably use source extent
as a discriminant within the central region ($r \leq 18'$)
and that we expect to be 90\% complete out to $z \simeq 0.5$.
All sources which have an extended profile are selected for optical
imaging and spectroscopy, during which hardness ratios are used to
help identify the X-ray source.
\section{Optical Follow Up}
\label{sec:optical}
We have analysed 50 ROSAT fields and our initial
observing run, using EFOSC on the ESO 3.6m, produced
R band images and spectroscopy for the extended sources in these fields.
These sources can be characterised as:
\begin{itemize}
\item Pairs of stars with small projected separation.
\item Nearby clusters with $z < 0.3$.
\item Imaging suggests a cluster, but
spectroscopic confirmation was not possible. We estimate,
from the brightest galaxy magnitudes, $z = 0.3 - 0.5$.
\item Sources with no obvious identification even after imaging
to $R \simeq 22$.
\end{itemize}
We show, in fig~\ref{fig:cluster}, one of our distant cluster
candidates. Spectroscopy of the central galaxies suggest $z = 0.55$.
Fitting the X-ray spectrum by a Raymond-Smith plasma code, with
T=6 keV and half solar metallicity,
gives L$_X \sim 3 \times 10^{44}$ erg s$^{-1}$ (0.5-2.0keV).
\begin{figure}
\psfig{figure=poster.ps,width=8.6truecm,angle=0}
\caption[]{X-ray contours (lightly smoothed) overlaid on R band image. At $z = 0.55$
1 arcmin equals 440$h_{50}^{-1}$ kpc ($q_0 = 0.5$).
\label{fig:cluster}
}
\end{figure}
\section{Future Work}
\label{sec:future}
The initial goal of this work is to measure the cluster XLF.
We have time allocated to complete the identification and spectroscopy
of our remaining fields.
Time has been applied for to secure identification and redshifs
of our unconfirmed cluster candidates.
As we suspect our unidentified sources may well be distant clusters,
we have applied for K band imaging.
This will allow us to identify clusters and we can use the K-$z$
diagram (Collins \& Mann 1995) to estimate cluster redshifts
with sufficient accuracy for construction of the cluster XLF.
Our collaboration is also using a wavelet-based detection algorithm
to search for clusters in the deepest Northern
ROSAT pointings.
Optical
follow up of these extended sources has begun on the ARC 3.5m telescope
at Apache point.
We will therefore be able to use our two well defined
catalogues to directly compare selection techniques and
assess our completeness.
Future work will investigate the relationship between the hot intracluster
gas and the optical properties of clusters and to study the Butcher-Oemler
effect (Butcher \& Oemler~1984).
Our sample, combined with other distant cluster catalogues
currently being compiled (e.g. RIXOS, RDCS and WARPS), will make an
excellent target list for AXAF and the upcoming 8m class of ground based
telescopes.
\begin{acknowledgements}
This research has made use of data obtained from the Leicester Database
and Archive Service at the Department of Physics and Astronomy, Leicester
University, UK.
\end{acknowledgements}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,614 |
HomeEducationSustainability EducationCOP25Who gets to speak, to tell the story?: Reflections on COP25 Madrid, Spain
By: Professor Rose M. Brewer, U of MN Observer, photo credit UNFCC
As a week two observer of COP25, Madrid, Spain, December 9-13, 2019, it was very clear to me that the most compelling voices were not those in the formal deliberations of parties. The speaking truths voices were those of youth, activists of the Global South, North, Black, Brown, Indigenous peoples. While the corporate and political classes dominated the discussions inside IFEMA, the outside voices, challenging, questioning, most clearly articulated the urgency of the moment. Without mincing words, they asserted that an economic order built on growth and extractivism is at the heart of the climate emergency we face and must be transformed.
As a U of MN observer and scholar activist, the issue of environmental justice guided my observations. Social justice and climate justice are inextricably linked. Those communities bearing the brunt of the current climate devastation within the U.S. and beyond are paying the heaviest price for climate disaster. In the US, these are the frontline, poor, Black, Brown, and Indigenous communities bearing the brunt of the devastation. In the global South, one of the most compelling aspects of the crisis is the climate disasters confronting the small island nations. Their very existence and way of life is at the highest risk. I say this as I spoke with activists and scholars from the African continent who are also confronting climate disaster, a continent on fire.
Complicating the Gender Action Plan
One of the positive turns at COP25 was incorporating the gender action plan in the Paris Guidelines.
The GAP, created under the Lima work programme on gender, seeks to advance women's full, equal and meaningful participation and promote gender-responsive climate policy and the mainstreaming of a gender perspective in the implementation of the Convention and the work of Parties, the secretariat, United Nations entities and all stakeholders at all levels.
Yet the critical insights of Black, women of color, and indigenous feminists articulate an intersectional frame which was not captured in the GAP. To complicate this plan means moving beyond the idea of gender as uniform sameness. In fact, gender(s) is the deeply intersectional realities of race, class indigeneity, sexualities, that articulate gender complexities. This complexity was captured in the press conference held by WEDO, WECAN and the NAACP on Dec 10. The press conference centered a feminist Green New Deal which explicitly articulates an intersectional approach in the plan.
There is much work to be done. Official COP25 delivered little but for those of us committed to climate and social justice, I left with even greater commitment that:
The local and global must be at the heart of our work. This means movement building with those committed to social change, seriously involved in struggles for social and climate justice here and there. We must have an internationalist lens to our work. This also means a fierce green struggle for us living in the US, demanding that not in our name will policies of devastation be foisted on the world. We must be in conversations on how individuals/groups organize, deploy political education, research, and build leadership. As scholars, activists, organizers, radicals, teachers, we must break out of canonic silos and be in deep interdisciplinarity around the scholarship of climate justice – infuse it, change it, build new knowledge(s) from below.
This means serious intellectual work on how the knowledge and political insights of those most impacted by climate devastation in the U.S. can be developed, cultivated and shared. This involves building real political energy and motion. It is the task of deep movement building. But hard questions need to be addressed: How, in fact, is such movement building unfolding in the US? Surely the knowledge of Indigenous peoples must be respected and centered as core. Delving deeply into the meaning and practice of participatory democracy will be key.
Understanding the nature of capitalist extractivism, its form, function, and alternatives is essential. We must bring to the table all who are committed to new visions of society, new relations to nature and all living things. We must organize for the majority of the people of the earth and the planet. What about the solidarity economy? New socialism? Urban gardens/food sovereignty, deep participatory democracy, etc.? What do we know? What do we need to know to build another world? These are the questions we must take on. Our very lives and the earth depend on it.
The Science of Abundance | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,613 |
\section{Introduction}
\label{sec:intro}
The understanding of hadron structure has greatly evolved over the last decades.
The collected knowledge is parametrized by a large number of functions.
Generalized parton distributions (GPDs) are one set of such functions.
They parametrize, e.g., the transverse coordinate distribution of partons
in a fast moving hadron and contain information on how these distributions
depend on the parton or hadron spin direction. Pinning down all these multivariable functions experimentally is unrealistic at present.
Therefore, lattice QCD has to substitute some of the missing experimental data.
With this article we contribute to the effort of various lattice groups to provide some of these needed results
\cite{Hagler:2007xi,Bratt:2010jn,Alexandrou:2011nr,Syritsyn:2011vk,Sternbeck:2012rw,Alexandrou:2013joa,Bali:2013dpa,Alexandrou:2013wka,Ji:2015qla,Bali:2016wqg,Chen:2016utp,Chen:2016fxx,Zhang:2017bzy}.
From the experimental point of view, GPDs play a similarly important role for the description of
exclusive hadronic reactions as parton distribution functions (PDFs) do for inclusive reactions. The most extensively studied channel
is deeply virtual Compton scattering (DVCS), i.e., Compton scattering with a
highly virtual incoming photon and a correspondingly large, spacelike
momentum transfer $Q^2=-q^2$.
One advantage of DVCS is that the GPD matrix element interferes with
the well-known Bethe-Heitler cross section for which the final state photon is emitted from
the scattered lepton. Thus the measured cross sections provide not only information on the
absolute value of the DVCS correlators but also on their signs.
In all generality, including spin effects, the experimental analysis
becomes somewhat involved, as is, e.g., illustrated by the publications~\cite{Airapetian:2001yk,Airapetian:2012pg} of the
\textsc{Hermes} experiment.
For a recent careful theoretical analysis and references to
experimental work see Ref.~\cite{Kumericki:2016ehc}.
The theoretical understanding of GPDs and their moments, the
generalized form factors (GFFs),
has already a long history and is presented in the seminal work
of Refs.~\cite{Dittes:1988xz,Mueller:1998fv,Ji:1996ek,Radyushkin:1996nd,Collins:1996fb}.
More recent reviews can be found in Refs.~\cite{Diehl:2003ny,Belitsky:2005qn}.
The interest in some of the nucleon GPDs (there exist in total eight) is increased by
the fact that they provide information on the elusive orbital angular momentum
of partons in the nucleon.
However, the physical interpretation in this case is not straightforward,
because there exist inequivalent definitions of orbital angular momentum~\cite{Jaffe:1989jz,Ji:1996ek}.
For recent discussions of this topic see, e.g., Refs.~\cite{Leader:2013jra,Ji:2015sio,Engelhardt:2017miy}
and the articles cited therein.
In this article we will not review the many fascinating aspects of GPDs but
concentrate on our lattice calculation of the nucleon GFFs
using well-established techniques for the calculation of Mellin moments of GPDs;
see, e.g., Ref.~\cite{Hagler:2009ni}.
We remark that recently new methods have been proposed
to obtain information on parton distribution functions
(PDFs), distribution amplitudes (DAs), transverse momentum dependent
PDFs (TMDPDFs) and GPDs that is complementary
to the computation of Mellin moments with respect to Bjorken-$x$ from
expectation values of local currents within external states, see, e.g.,
Refs.~\cite{Ji:2013dva,Lin:2014zya,Alexandrou:2015rja,Alexandrou:2016jqi,Bali:2018spj,Braun:2007wv}.
In these approaches Euclidean correlation functions are computed and
then matched within collinear factorization to light cone distribution
functions, employing continuum perturbative QCD. For the example of
DAs~\cite{Bali:2018spj}, some of us are involved in calculations with these new
techniques, using the ``momentum smearing''
technique~\cite{Bali:2016lva} to enable large hadron momenta to be
realized, and found results that are consistent with, but less
accurate than those obtained from the lowest nontrivial
moment. This may change as smaller lattice
spacings and larger computers become available. Here we will only
determine the first $x$-moment, i.e., the second Mellin moment, to
constrain the nucleon GPDs.
This paper is organized as follows.
In Sec.\,\ref{sec:introGPD} we shortly review definitions
and the operator product expansion for Mellin moments of GPDs.
The lattice QCD techniques used to extract GFFs are introduced in
Sec.\,\ref{sec:extration_of_GFFs}
followed by a discussion of the numerical methods in Sec.\,\ref{sec:numerical methods}.
In Secs.\,\ref{sec:results_GFF} and~\ref{sec:chiral_J} we present our results.
Some preliminary findings have been reported in Refs.~\cite{Sternbeck:2012rw,Bali:2013dpa,Bali:2016wqg}.
Finally, we investigate the transverse spin density of the nucleon in
Sec.\,\ref{sec:nucleon_tomography}.
\section{Basic Properties of GPDs}
\label{sec:introGPD}
The starting point is the off-forward nucleon matrix element
\begin{equation}
\label{eq:bilocal-mat_1}
\mathcal{M}_q^\Gamma(x) =
\int^\infty_{-\infty}\!\frac{\mathrm{d}\lambda}{4\pi}\, e^{i\lambda x}
\left\langle N(p^\prime, \sigma^\prime)| O_q^\Gamma(\lambda) |N(p,\sigma) \right\rangle
\end{equation}
of a bilocal operator with quark flavor $q$
\begin{equation}
\label{eq:bilocal-mat_2}
O_q^\Gamma(\lambda) = \bar{q}\left(-\lambda n/2\right) \, \Gamma \ \mathcal{U}_{-\lambda n/2}^{+\lambda n/2} \, q \left(+\lambda n/2\right) \,.
\end{equation}
The Wilson line $\mathcal{U}$ in Eq.~(\ref{eq:bilocal-mat_2}) connects ${-\lambda n/2}$ and $+\lambda n/2$ on the light cone ($n^2=0$).
Depending on the Dirac structure, indicated by the symbol $\Gamma$ in Eqs.~(\ref{eq:bilocal-mat_1}) and (\ref{eq:bilocal-mat_2}),
one can parametrize the matrix element $\mathcal{M}$ in terms of GPDs.
For leading twist these read (see, e.g., Refs.~\cite{Ji:1998pc,Hagler:2009ni}),
\begin{subequations}
\label{eq:gpds}
\begin{align}
\mathcal{M}_q^{\gamma^\mu} = \
&\overline{U}(p^\prime, \sigma^\prime)
\left[
\begin{pmatrix}
\gamma^\mu\\
\frac{ i\sigma^{\mu\nu}\Delta_\nu}{2m_N}
\end{pmatrix}
\!\cdot\!
\begin{pmatrix}
H^q\\
E^q
\end{pmatrix}
\;\;
\right]
\! U(p,\sigma) \, ,\\
\mathcal{M}_q^{\gamma^\mu\gamma_5} = \
&\overline{U}(p^\prime, \sigma^\prime)
\left[
\ \ \
\begin{pmatrix}
\gamma^\mu \gamma_5\\
\frac{ \Delta^\mu \gamma_5 }{2m_N}
\end{pmatrix}
\!\cdot\!
\begin{pmatrix}
\widetilde{H}^q\\
\widetilde{E}^q
\end{pmatrix} \, \;
\right]
\! U(p,\sigma)\, , \\
\mathcal{M}_q^{i \sigma^{\mu\nu}} \! = \
&\overline{U}(p^\prime, \sigma^\prime)
\left[
\begin{pmatrix}
i \sigma^{\mu\nu} \\
\frac{ \gamma^{[\mu} \Delta^{\nu]} }{2m_N} \\
\frac{ \overline{p}^{[\mu} \Delta^{\nu]} }{m_N^2} \\
\frac{ \gamma^{[\mu} \overline{p}^{\nu]} }{m_N} \\
\end{pmatrix}
\!\cdot\!
\begin{pmatrix}
H_T^q\\
E_T^q\\
\widetilde{H}_T^q\\
\widetilde{E}_T^q\\
\end{pmatrix}
\right]
\! U(p,\sigma)\, ,
\end{align}
\end{subequations}
with $\sigma^{\mu\nu} = i\, [ \gamma^\mu , \gamma^\nu ]/2$
and the nucleon spinors
$\overline{U}(p^\prime, \sigma^\prime )$ and $U(p, \sigma)$.
The GPDs, e.g., $H^q$ and $E^q$, and the corresponding tensor structures
$\gamma^\mu$ and $i\sigma^{\mu\nu}\Delta_\nu/(2m_N)$
are written as vectors,
where we apply a standard scalar product to simplify the notation
and introduce the kinematic variables
\begin{align}
\Delta \coloneqq p^\prime - p, \quad \quad
\overline{p} \coloneqq (p^\prime + p)/2 \,.
\end{align}
For the antisymmetrization of indices we use the notation $[\ldots]$,
e.g., $B^{[\mu} C^{\nu]}\coloneqq B^{\mu}C^{\nu} - C^{\nu}
B^{\mu}\eqqcolon\mathsf{A}_{\mu\nu} B^{\mu}C^{\nu}$.
The GPDs are functions of the three variables $(x,\xi,\mathsf{t})$,
such that $H^q = H^q(x,\xi,\mathsf{t})$ etc. We define
\begin{align}
\mathsf{t} \coloneqq \Delta^2 \leq 0, \quad \quad
\xi \coloneqq -\frac{n\cdot \Delta}{2},
\end{align}
where $\mathsf{t}$ is the total momentum transfer squared
which is related to the virtuality $Q^2 = -\mathsf{t}$.
The longitudinal momentum fraction $x$ varies between $-1$ and $1$ and
the skewness $\xi$ between $0$ and $1$. Negative values of
$x$ correspond to plus or minus (depending on the GPD)
times the corresponding antiquark GPD at $-x$.
In this work we restrict ourselves to the isovector case
and therefore we only consider the above eight quark GPDs.
An analogous set of gluonic GPDs exists, which we will not address here.
For a more detailed discussion we refer the reader to Refs.~\cite{Ji:1998pc,Vanderhaeghen:1999xj,Goeke:2001tz,Diehl:2003ny,Belitsky:2005qn,Hagler:2009ni}.
In physical terms (for $|x| > \xi$) GPDs parametrize the
probability amplitude for a hadron to stay intact if
a parton is removed at the light cone point $-\lambda/2$ and replaced by a
parton with different momentum at light cone time $\lambda/2$.
In practice, it is of crucial importance to find
effective parameterizations of GPDs with a minimum number of parameters
which are then fitted to experimental data see, e.g., Ref.~\cite{Kumericki:2016ehc}.
Lattice input in principle allows one to pin down the values of these
parameters; however, at present the accuracy of such studies is for many GPDs
not yet sufficient to make a decisive impact.
As time is analytically continued to imaginary time to enable the
numerical evaluation on the lattice, the light cone loses its meaning.
The operator product expansion (OPE) relates, however, Mellin moments of GPDs to
local matrix elements that are amenable to lattice calculation.
For $H^q$ and $E^q$, for instance, these $x$-moments read (see, e.g., Refs.~\cite{Ji:1998pc,Hagler:2009ni})
\begin{subequations}
\begin{align}
\int_{-1}^{+1}\!\!\mathrm{d}x \, x^{n-1} \, H^q(x, \xi, \mathsf{t})&= \nonumber \\
\sum\limits_{i=0, \,\mathrm{even}}^{n-1}
(-2\xi)^i A^q_{ni}(\mathsf{t}) &+
(-2\xi)^{n} \, C^q_{n0}(\mathsf{t})|_{n=\mathrm{even}}, \\
\int_{-1}^{+1}\!\!\mathrm{d}x \, x^{n-1} \, E^q(x, \xi, \mathsf{t})&= \nonumber \\
\sum
\limits_{i=0 ,\,\mathrm{even}}^{n-1}
(-2\xi)^i B^q_{ni}(\mathsf{t})
&- (-2\xi)^{n} \, C^q_{n0}(\mathsf{t})|_{n=\mathrm{even}}\,,
\end{align}
\end{subequations}
where the real functions
$A^q(\mathsf{t})$,
$B^q(\mathsf{t})$ and
$C^q(\mathsf{t})$ in the $\xi$-expansion on the
rhs are the GFFs.
The case $n=1$ corresponds to the electromagnetic form factors
$F_1^{q}(\mathsf{t}) = A^q_{10}(\mathsf{t})$
and
$F_2^{q}(\mathsf{t}) = B^q_{10}(\mathsf{t})$. For
$n=2$ and $\mathsf{t}=0$ we obtain
the average quark momentum fraction $A^q_{20} = \langle x\rangle_{q^+}$,
where, for this example, we indicated $q^{\pm}=q\pm\bar{q}$. Below
we will drop this distinction since in the case of the vector
and tensor GPDs the even moments automatically give the $q^+$
combination and the odd moments $q^-$, while for axial GPDs it is
the opposite.
In principle one can determine Mellin moments of GPDs for any $n$ on the lattice,
in practice one is restricted to the lowest few $n$.
The reason for this restriction is twofold.
On the one hand the signal to noise ratio becomes worse for an increasing
number of covariant derivatives.
On the other hand as $n$ increases, mixing with lower-dimensional operators
will take place, resulting in divergences that are powers
of the inverse lattice spacing $a^{-1}$.
In this study we focus on the case $n=2$, where such mixing
does not occur.
Similarly to elastic form factors, the respective GFFs are extracted from lattice
calculations of two- and three-point correlation functions where the currents
are the local twist-2 operators,
\begin{subequations}
\label{eq:operators}
\begin{align}
\mathcal{O}^{\mu \nu}_{V,q}(z) &= \mathsf{S}_{\mu\nu}\:\bar{q}(z) \, \gamma^{\mu} i\overleftrightarrow{D}^{\nu}q(z)\,, \\
\mathcal{O}^{\mu \nu}_{A,q}(z) &= \mathsf{S}_{\mu\nu}\:\bar{q}(z) \, \gamma^{\mu}\gamma_5 i \overleftrightarrow{D}^{\nu}q(z)\,,\\
\mathcal{O}^{\mu\nu\rho}_{T,q}(z) &= \mathsf{A}_{\mu\nu}\mathsf{S}_{\nu\rho}\:\bar{q}(z) i\sigma^{\mu\nu} i \overleftrightarrow{D}^{\rho}q(z)\,.
\end{align}
\end{subequations}
Here $\mathsf{S}_{\mu\nu}$ and $\mathsf{A}_{\mu\nu}$ denote symmetrization
(also subtracting traces and dividing by $n!$ for $n$ indices) and
antisymmetrization operators, respectively, and
\begin{align}
\overleftrightarrow{D_\mu} \coloneqq \frac{1}{2}(\overrightarrow{D_\mu}-\overleftarrow{D_\mu})
\end{align}
is the symmetric covariant derivative.
In the continuum we can decompose the matrix elements
\begin{subequations}
\begin{align}
\label{eq:mat_master_v}
\big\langle N(p^\prime,\sigma^\prime)|\mathcal{O}^{\mu\nu}_{V,q}|N(p,\sigma)\big\rangle \! = \! \overline{U}(p^\prime,\sigma^\prime)\mathbb{D}_{V,q}^{\mu \nu} U(p,\sigma)\, , \\
\label{eq:mat_master_a}
\big\langle N(p^\prime,\sigma^\prime)|\mathcal{O}^{\mu\nu}_{A,q}|N(p,\sigma)\big\rangle \! = \! \overline{U}(p^\prime,\sigma^\prime)\mathbb{D}_{A,q}^{\mu \nu} U(p,\sigma)\, , \\
\label{eq:mat_master_t}
\big\langle N(p^\prime,\sigma^\prime)| \mathcal{O}_{T,q}^{\mu\nu\rho}|N(p,\sigma)\big\rangle \! = \! \overline{U}(p^\prime,\sigma^\prime)\mathbb{D}_{T,q}^{\mu \nu \rho} U(p,\sigma) \, ,
\end{align}
\end{subequations}
with the nucleon four-momentum $(p^\mu) = (E_N(\vec{p}\,),\vec{p} \,)$.
In Sec.\,\ref{sec:extration_of_GFFs} we will show how we extract the matrix elements
from the temporal dependence of the three-point correlation functions.
The desired GFFs are contained in the Dirac structures,
\begin{subequations}
\label{eq:Decompositions}
\begin{align}
\label{eq:Decomposition_V}
\mathbb{D}^{\mu\nu}_{V,q}&=
\mathsf{S}_{\mu\nu}\,
\begin{pmatrix}
\gamma^{\mu}\overline{p}^{\nu} \\
i\sigma^{\mu\rho} \Delta_\rho \overline{p}^{\nu} /( 2 m_N) \\
\Delta^{\mu}\Delta^{\nu} / m_N\\
\end{pmatrix}
\cdot
\begin{pmatrix}
A^q_{20}\\
B^q_{20}\\
C^q_{20}
\end{pmatrix} \,, \\
\label{eq:Decomposition_A}
\mathbb{D}^{\mu\nu}_{A,q} &=
\mathsf{S}_{\mu\nu}
\begin{pmatrix}
\gamma^{\mu}\gamma^{5}\overline{p}^{\nu}\\
\gamma_5 \Delta^\mu \overline{p}^{\nu} / (2 m_N)
\end{pmatrix}
\cdot
\begin{pmatrix}
\widetilde{A}^q_{20} \\
\widetilde{B}^q_{20}
\end{pmatrix} \,, \\
\label{eq:Decomposition_T}
\mathbb{D}^{\mu\nu\rho}_{T,q} &=
\mathsf{A}_{\mu\nu}
\mathsf{S}_{\nu\rho}\!
\begin{pmatrix}
i \sigma^{\mu\nu} \overline{p}^{\rho}\\
\gamma^{[\mu}\Delta^{\nu]}\overline{p}^{\rho} /(2 m_N)\\
\overline p^{[\mu} \Delta^{\nu]}\overline{p}^{\rho} / m_N^2 \\
\ \gamma^{[\mu}\overline{p}^{\nu]}\Delta^{\rho} / m_N
\end{pmatrix}
\cdot
\begin{pmatrix}
A^q_{T20}\\
B^q_{T20}\\
\widetilde{A}^q_{T20}\\
\widetilde{B}_{T21}
\end{pmatrix} \,.
\end{align}
\end{subequations}
Some aspects of GFFs have been more intensively discussed in the
literature than others, in particular,
\begin{itemize}
\item As has already been mentioned above,
in the forward limit ($\mathsf{t}=0$), $A^q_{20}$ equals the average
quark momentum fraction.
Similar limits exist for
$\widetilde{A}^q_{20}$ and $A^q_{T20}$ and the polarized and
transversity PDFs, respectively.
\item Furthermore, in this limit $A^q_{20}$ and $B^q_{20}$ add up to twice
the total angular momentum of the quark $q$ plus that of the antiquark
$\bar{q}$ in the nucleon
(the Ji sum rule~\cite{Ji:1996ek}) such that
\begin{equation}
J^{q} = \frac{1}{2}\left[A_{20}^{q}(0) + B^{q}_{20}(0)\right]
\end{equation}
represents the quark contribution to the nucleon spin.
Combining $J^q$ with the quark spin contribution $\tfrac12\Delta\Sigma_q$,
one can also obtain the quark orbital angular momentum
$L_q = J_q-\tfrac12\Delta\Sigma_q$. We remark that this decomposition
is not unique~\cite{Jaffe:1989jz}.
\item The five GFFs $A_{20}$, $B_{20}$, $A_{T20}$, $B_{T20}$ and
$\widetilde{A}_{T20}$ parametrize, after Fourier transformation to
impact parameter space, the first $x$-moment of the transverse spin
density of a quark in a fast-moving nucleon~\cite{Burkardt:2000za}.
\end{itemize}
\section{Extracting generalized form factors}
\label{sec:extration_of_GFFs}
On the lattice, the GFFs are extracted from combinations of hadronic
two- and three-point correlation functions in Euclidean space-time.
The two-point function reads
\begin{align}
\label{eq:2pt_cor}
C^{2\mathrm{pt}}_{\alpha \beta}(t^\prime, \vec{p}^{\,\prime} ) =
\sum_{\vec{x}^{\,\prime}} \! e^{-i\vec{p}^{\,\prime}\!\cdot\vec{x}^{\,\prime}}\left\langle \mathcal{N}_\alpha(t^\prime, \vec{x}^{\,\prime}) \, \overline{\mathcal{N}}_{\!\beta}(0, \vec{0}\,)\right\rangle,
\end{align}
where the nucleon destruction and creation interpolators $\mathcal{N}$ and $\overline{\mathcal{N}}$ are appropriate
combinations of $u$ and $d$ (anti)quark fields
\begin{subequations}
\label{eq:interplators}
\begin{align}
\mathcal{N}_\alpha(t, \vec{x}\,) &= \varepsilon^{abc} u^a_{\alpha}(t, \vec{x}\,)\left[ u^b(t, \vec{x}\,)^{\intercal} \mathsf{C}\,\gamma_5 d^c(t, \vec{x}\,)\right], \\
\overline{\mathcal{N}}_\beta(t, \vec{x}\,) &= \varepsilon^{abc} \left[ \bar{u}^b(t, \vec{x}\,)\, \mathsf{C}\gamma_5 \bar{d}^c(t, \vec{x}\,)^{\intercal}\right] \bar{u}^a_{\beta}(t, \vec{x}\,)\,.
\end{align}
\end{subequations}
$\mathsf{C}$ is the charge conjugation matrix.
The lattice three-point function is expressed as
\begin{align}
\label{eq:3pt_cor}
C^{3\mathrm{pt}}_{\alpha \beta}(\tau,t^{\prime}, \vec{p}^{\, \prime},\vec{p} \,) =
\sum_{\vec{x}^{\, \prime} \vec{z}}
e^{-i\vec{p}^{\, \prime} \! \cdot \vec{x}^{\, \prime}}
e^{+i\vec{z} \cdot ( \vec{p}^{\, \prime} - \vec{p}\,)} \nonumber \\
\times \left\langle \!
\mathcal{N}_\alpha(t^{\prime}, \vec{x}^{\,\prime})
\mathcal{O}(\tau, \vec{z} \,)
\overline{\mathcal{N}}_{\!\beta}(0, \vec{0}\,)\right\rangle \,.
\end{align}
In this work we only consider isovector currents $\mathcal{O}$; therefore,
all quark lines are connected.
To improve the overlap of our interpolators
in Eqs.~\eqref{eq:interplators} with the physical ground state
we employ the combination of APE and Wuppertal (Gauss) smearing techniques
described in Refs.~\cite{Bali:2012av,Bali:2014nma,Bali:2014gha}.
This procedure reduces the impact of excited states substantially.
For the computation of \Eq{eq:3pt_cor},
we use the sequential propagator method~\cite{MAIANI1987420}
which implies fixing the sink time $t^\prime$.
We use the projector
\begin{align}
\label{eq:projection}
\mathbb{P}^\rho=\frac{1}{2}\left(1+\gamma_4\right)\left(-i \gamma^\rho \gamma_5\right)^{1 + \delta_{\rho, 4}}
\end{align}
and contract it with the open spin indices of Eq.~\eqref{eq:3pt_cor}
to realize different spin projections and positive parity.
For $\rho=1,2,3$ we obtain the difference of
the spin polarization with respect to the quantization axis $\rho$, while
$\rho=4$ corresponds to the unpolarized case.
The positive parity projection is only correct for zero momentum;
however, excited state contributions (including states of different
parity for nonvanishing momentum) are exponentially suppressed at
large Euclidean times $\tau$. (The outgoing nucleon is projected onto
zero momentum.)
The definition of the operator $\mathcal{O}$ in \Eq{eq:3pt_cor}
depends on the desired GFF.
For the vector, axial and tensor GFFs at leading \mbox{twist-2}
the operators are given in \Eq{eq:operators}.
On the lattice we construct our operators as linear combinations of
\begin{subequations}
\label{eq:lat_ops}
\begin{align}
\label{eq:lat_vec}
\mathcal{O}^{\mu \nu}_{V,q}(z) &=
\bar{q}(z)\gamma^{\mu} \overleftrightarrow{\nabla}^{\nu}q(z) \, , \\
\mathcal{O}^{\mu \nu}_{A,q}(z) &=
\bar{q}(z)\gamma^{\mu}\gamma_5 \overleftrightarrow{\nabla}^{\nu}q(z) \, , \\
\mathcal{O}^{\mu\nu\rho}_{T,q}(z) &= \bar{q}(z)i\sigma^{\mu\nu}\overleftrightarrow{\nabla}^{\rho}q(z) \, .
\end{align}
\end{subequations}
\begin{table}[tb]
\caption{The renormalization factors used to translate our bare
lattice data to the $\mathsf{\overline{MS}}$ scheme at $\mu=2\,\text{GeV}$, obtained
by reanalyzing the data of Ref.~\cite{Gockeler:2010yr}
\label{tab:renfac}}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{c@{\qquad}c@{\quad}c@{\quad}c@{\quad}}
& $\beta= 5.20$ & $\beta= 5.29$ & $\beta= 5.40$ \\
\hline
$Z_{\mathsf{\overline{MS}}}^{v_{2,a}}$ \, & $1.090 \, (19)$ \, & $1.113 \,(15)$ \, & $1.140 \,(16)$ \\*[0.5ex]
$Z_{\mathsf{\overline{MS}}}^{v_{2,b}}$ \, & $1.096 \, (17)$ \, & $1.117 \,(21)$ \, & $1.143 \,(13)$ \\*[2ex]
$Z_{\mathsf{\overline{MS}}}^{r_{2,a}}$ \, & $1.083 \,(16)$ \, & $1.106 \,(13)$ \, & $1.134 \,(14)$ \\*[0.5ex]
$Z_{\mathsf{\overline{MS}}}^{r_{2,b}}$ \, & $1.118 \,(16)$ \, & $1.138 \,(22)$ \, & $1.163 \,(13)$ \\*[2ex]
$Z_{\mathsf{\overline{MS}}}^{h_{1,a}}$ \, & $1.115 \,(19)$ \, & $1.141 \,(19)$ \, & $1.171 \,(16)$ \\*[0.5ex]
$Z_{\mathsf{\overline{MS}}}^{h_{1,b}}$ \, & $1.129 \,(20)$ \, & $1.154 \,(20)$ \, & $1.184 \,(16)$
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
\begin{table}[tb]
\caption{Relative error of the GFFs for the flavor
combination $u-d$,
induced by the uncertainty of the renormalization constants. This error
turns out to be almost independent of the virtuality.
\label{tab:pe}}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{ccccccccc}
$A_{20}^{u-d}$ & $B_{20}^{u-d}$ & $\widetilde{A}_{20}^{u-d}$ & $\widetilde{B}_{20}^{u-d}$ & $A_{T20}^{u-d}$ & $B_{T20}^{u-d}$ & $\widetilde{A}_{T20}^{u-d}$ & $\widetilde{B}_{T21}^{u-d}$ &$\overline{B}_{T20}^{u-d}$ \\
\hline
0.019 & 0.019 & 0.015 & 0.034 & 0.020 & 0.020 & 0.020 & 0.027 & 0.020
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
In the case of the vector operator we work with multiplets that
transform according to two distinct irreducible representations
of the hypercubic group H(4) labeled as $v_{2,a}$ and $v_{2,b}$.
These are combinations of the operators in \Eq{eq:lat_vec} given by
\begin{align}
\label{eq:ov2a}
\mathcal{O}_{\mu \nu}^{v_{2,a}} &= \mathsf{S}_{\mu\nu}\mathcal{O}_{\mu \nu}^V \quad \mathrm{with} \quad 1 \le \mu < \nu \le 4
\end{align}
and
\begin{subequations}
\label{eq:ov2b}
\begin{align}
\mathcal{O}_{1}^{v_{2,b}} &= \frac{1}{2}(\mathcal{O}_{11}^V + \mathcal{O}_{22}^V - \mathcal{O}_{33}^V - \mathcal{O}_{44}^V) \, ,\\
\mathcal{O}_{2}^{v_{2,b}} &= \frac{1}{\sqrt{2}}(\mathcal{O}_{33}^V-\mathcal{O}_{44}^V)\, ,\\
\mathcal{O}_{3}^{v_{2,b}} &= \frac{1}{\sqrt{2}}(\mathcal{O}_{11}^V-\mathcal{O}_{22}^V) \,,
\end{align}
\end{subequations}
respectively. The renormalized operators read
\begin{align}
\label{eq:z}
\mathcal{O}_{\mathsf{\overline{MS}}}^{v_{2,a\vert b}}(\mu) = Z(\beta, \mu)^{v_{2,a\vert b}}_{\mathsf{\overline{MS}}} \, \mathcal{O}^{v_{2,a\vert b}}(\beta)\,,
\end{align}
where we use $\mu=2\,\text{GeV}$ as the renormalization scale.
Note that the renormalization factors depend on the multiplet, i.e.,
they slightly differ for $v_{2,a}$ and $v_{2,b}$.
Similarly, the axial operators are renormalized with
factors $Z_\mathsf{\overline{MS}}^{r_{2,a}}$ and $Z_\mathsf{\overline{MS}}^{r_{2,b}}$,
substituting $v_{2,a} \mapsto r_{2,a}$, $v_{2,b} \mapsto r_{2,b}$
in Eqs.~\eqref{eq:ov2a}, \eqref{eq:ov2b} and \eqref{eq:z}.
The tensor operators are renormalized with
$Z_\mathsf{\overline{MS}}^{h_{1,a}}$ and $Z_\mathsf{\overline{MS}}^{h_{1,b}}$.
The operator multiplets used in this case are listed in
Appendix~\ref{app:tensor_olc}.
A detailed description of the renormalization procedure, that consists
of first nonperturbatively matching from the lattice to the $\mathsf{RI'\textnormal{-}MOM}$
scheme~\cite{Martinelli:1994ty,Chetyrkin:1999pq} and then
translating perturbatively to the $\mathsf{\overline{MS}}$ scheme,
may be found in Ref.~\cite{Gockeler:2010yr}.
To make the article self-contained we summarize the basic steps
in Appendix~\ref{app:renormproc},
where we also address the error propagation from the renormalization
constants to the GFFs.
The relevant renormalization factors are summarized in \Tab{tab:renfac}.
They result from a reanalysis of the data presented
in Ref.~\cite{Gockeler:2010yr}
and correspond to the physical input $r_0=0.5\,\text{fm}$~\cite{Bali:2012qs}
and $r_0\Lambda^\mathsf{\overline{MS}}=0.789$~\cite{Fritzsch:2012wq}.
\Tab{tab:pe} lists the relative errors on the renormalized
GFFs, associated with the uncertainties in the renormalization constants;
these amount to about~$2\%$.
\begin{table*}
\caption{Parameters of the $N_f=2$ lattice ensembles used in this study.
Latin numerals in the first column serve as ensemble identifiers.
After the number of configurations $N_{\mathrm{conf}}$ we list in
parentheses the number of independent (randomly chosen) source positions
that we average over within each gauge configuration.
Wherever this is indicated by parentheses after the sink-source
separation $t'/a$, a smaller
number of sources was used for this value. For more information about
our setup we refer to Ref.~\cite{Bali:2014nma}.
\label{tab:latsetup}}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{l@{\quad}c@{\quad}c@{\qquad}c@{\qquad}c@{\qquad}c@{\qquad}r@{\qquad}c@{\quad}c@{\quad} c@{\quad} }
Ensemble &
$\beta$ &
$a$\,[fm] &
$\kappa$ &
$V$ &
$m_{\pi}$\,[GeV] &
$m_N$ \,[GeV] &
$Lm_{\pi}$ &
$N_{\mathrm{conf}}$ &
$t'/a$\\ \hline
\crule[mycol1]{0.3cm}{0.3cm} I&5.20&0.081&0.13596&$32^3\times 64$ &0.2795(18)& 1.091(08) &3.69&$1986(4)$&13\\*[1.5ex]
\crule[mycol2]{0.3cm}{0.3cm} II&5.29&0.071&0.13620&$24^3\times 48$ &0.4264(20)& 1.289(15) &3.71&$1999(2)$&15\\
\crule[mycol3]{0.3cm}{0.3cm} III\hfill&&&0.13620&$32^3\times 64$ &0.4222(13)& 1.247(06) &4.90&$1998(2)$&15,17 \\
\crule[mycol4]{0.3cm}{0.3cm} IV&&&0.13632 &$32^3\times 64$ &0.2946(14)& 1.071(11) &3.42&$2023(2)$&7(1),9(1),11(1),13,15,17\\
\crule[mycol5]{0.3cm}{0.3cm} V&&&&$40^3\times 64$ &0.2888(11)& 1.079(09) &4.19&$2025(2)$&15\\
\crule[mycol6]{0.3cm}{0.3cm} VI&&&&$64^3\times 64$ &0.2895(07)& 1.072(05) &6.71&$1232(2)$&15 \\
\crule[mycol7]{0.3cm}{0.3cm} VII&&&0.13640&$48^3\times 64$ &0.1597(15)& 0.968(19) &2.78&$3442(2)$&15 \\
\crule[mycol8]{0.3cm}{0.3cm} VIII&&&&$64^3\times 64$&0.1497(13) &0.944(17) &3.47&$1593(3)$&9(1),12(2),15\\*[1.5ex]
\crule[mycol9]{0.3cm}{0.3cm} IX&5.40 & 0.060 &0.13640&$32^3\times 64$&0.490(02)&1.302(11) &4.81 & $1123(2)$ & 17 \\
\crule[mycol10]{0.3cm}{0.3cm} X&&&0.13647&$32^3\times 64$&0.4262(20)&1.262(09) &4.18&$1999(2)$&17\\
\crule[mycol11]{0.3cm}{0.3cm} XI&&&0.13660&$48^3\times 64$ & 0.2595(09) & 1.010(09) & 3.82 & $2177(2)$ & 17
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table*}
In the following we demonstrate the extraction procedure
for the vector GFFs.
The axial and tensor GFFs are treated analogously.
We start by expanding \Eq{eq:3pt_cor} in terms of energy eigenstates
\begin{widetext}
\begin{align}
\label{eq:c3pt_a}
C^{3\mathrm{pt}}_{\alpha \beta}(\tau,t^{\prime}, \vec{p}^{\, \prime},\vec{p}^{}\,)
=
\mathcal{A}_{\alpha \beta} \cdot
\mathrm{e}^{-E_N(\vec{p}^{\, \prime}) \,(t^{\prime} - \tau)} \,
\mathrm{e}^{-E_N(\vec{p}^{}\,)\,\tau} \, + \, \mathrm{excited\,\,states}\,,
\end{align}
where the ground state amplitude reads
\begin{align}
\label{eq:c3pt_ab}
\mathcal{A}_{\alpha \beta} = \frac{1}{4 \, E_N(\vec{p}^{\, \prime}) E_N(\vec{p}^{}\,)}
\sum \limits_{\sigma^\prime \sigma}
\big\langle 0 | \mathcal{N}_\alpha | N(p^\prime,\sigma^\prime) \big\rangle
\,\big\langle N(p^\prime,\sigma^\prime) |
\mathcal{O}_{\mathsf{\overline{MS}}}^{v_{2,a\vert b}}\,
|
N(p,\sigma) \big\rangle
\,
\big\langle N(p,\sigma)|\overline{\mathcal{N}}_{\!\beta}
| 0 \big\rangle \,.
\end{align}
The exponentials contain the energy of the nucleon as a function of the considered spatial momentum,
the Euclidean operator insertion time $\tau$, and the sink time $t^{\prime}$.
Up to lattice artifacts, the matrix elements of an operator $\mathcal{O}^{\mu\nu}_{V,q}(z)$
can be decomposed according to the Euclidean versions of
Eqs.~\eqref{eq:mat_master_v} and \eqref{eq:Decomposition_V}.
In doing so, it is necessary to distinguish between the two
multiplets $v_{2,a}$
and $v_{2,b}$ [cf.\ Eqs.~\eqref{eq:ov2a} and \eqref{eq:ov2b}].
The decomposition can be written as
\begin{align}
\left\langle N(p^\prime,\sigma^\prime) |
\mathcal{O}_{\mathsf{\overline{MS}}}^{v_{2,a\vert b}}\,
|
N(p,\sigma) \right\rangle = \overline{U}(p^\prime,\sigma^\prime) \, \mathbb{D}_{\mathsf{\overline{MS}}}^{v_{2,a\vert b}} \, U(p,\sigma)\,.
\end{align}
Applying the projection operator $\mathbb{P}^\rho$ [cf. \Eq{eq:projection}]
to $C^{3\mathrm{pt}}$ yields
\begin{align}
\label{eq:c3pt_ampl}
& c^\rho_V(\tau,t^{\prime}, \vec{p}^{\, \prime},\vec{p}^{} \,) \coloneqq \sum_{\alpha,\beta} \mathbb{P}^\rho_{\beta \alpha} C^{3\mathrm{pt}}_{\alpha \beta}(\tau,t',\vec{p}^{\,\prime},\vec{p}^{}\,) =
\sqrt{ Z(\vec{p}^{\,\prime}) \, Z(\vec{p}^{}\,)} \: \mathcal{F}_V \:\mathrm{e}^{-E_N(\vec{p}^{\, \prime}) \,(t^{\prime} - \tau)} \,
\mathrm{e}^{-E_N(\vec{p}\,)\,\tau} \, + \, \mathrm{excited\,\,states}
\end{align}
\end{widetext}
with
\begin{align}
\label{eq:ffunc}
& \mathcal{F}_V = \frac{
\operatorname{tr} \left\{
\mathbb{P}^\rho \,
[ -i \slashed{p}^{\prime} + m_N] \:
\mathbb{D}_{\mathsf{\overline{MS}}}^{v_{2,a\vert b}} \,
[ -i \slashed{p} + m_N] \,
\right\}
}{4 \, E_N(\vec{p}^{\,\prime}) E_N(\vec{p}^{}\,)}
\end{align}
and
$
\slashed{p} \coloneqq i E_N(\vec{p}\,) \gamma_4 + \vec{p} \cdot \vec{\gamma} \,.
$
The $Z$ factors in \Eq{eq:c3pt_ampl} depend on the overlap of our nucleon interpolation operators with the nucleon ground state. They vary with momentum and smearing and can be extracted from the two-point correlation function $C^{2\mathrm{pt}}$.
The right-hand side of \Eq{eq:ffunc} contains the desired GFFs. The prefactors can be computed by inserting the respective Euclidean $\gamma$-matrices.
Here we restrict ourselves to the final momentum $\vec{p}^{\,\prime}=\vec{0}$.
Taking all available combinations of operators [cf.\ Eqs.~\eqref{eq:ov2a} and \eqref{eq:ov2b}],
projections $\mathbb{P}^\rho$ and momenta $\vec{p}$ for a fixed virtuality
\begin{equation}
\label{eq:virtuality}
Q^2=-\mathsf{t} = \left(\vec{p}^{ \, \prime} - \vec{p}^{}\,\right)^2-
\left( \!\!\sqrt{m_N^2 + \vec{p}^{ \, \prime 2} } - \sqrt{m_N^2 + \vec{p}^{\, 2} } \right)^2 \,,
\end{equation}
we obtain a linear system of equations
\begin{align}
\label{eq:eqs_}
\vec{\mathcal{F}}_V = M_V \cdot \vec{g}_V
\end{align}
with the GFF vector $\vec{g}_V=\left(A_{20}(\mathsf{t}),B_{20}(\mathsf{t}),C_{20}(\mathsf{t})\right)^{\intercal}$. The coefficient matrix $M_V$ consists of the
prefactors calculated from \Eq{eq:ffunc} and
$\vec{\mathcal{F}}_V$ is extracted
from a fit of Eqs.~\eqref{eq:c3pt_a} and \eqref{eq:c3pt_ampl} to lattice data for $C^{2\mathrm{pt}}$ and $C^{3\mathrm{pt}}$.
The number of columns of $M_V$ is equal to the number of unknown GFFs (in this case 3),
but the number of rows depends on the available combinations.
In almost all the cases this yields an overdetermined system of equations, meaning
that the number of elements in $\vec{\mathcal{F}}_V$,
denoted with $\dim{\vec{\mathcal{F}}_V}$,
is larger than the number of GFFs.
Note that the individual rows of $M_V$ are either
real or imaginary.\footnote{If a row vanishes, then it does not
restrict the GFF and we remove it from the system of equations.}
For a given ensemble this system of equations has to be solved separately for each
virtuality to yield the GFFs as functions of $\mathsf{t}$.
In the general case we write \Eq{eq:eqs_} as
\begin{align}
\label{eq:eqs}
\vec{\mathcal{F}}_{\Gamma}^{\,q} = M_{\Gamma} \cdot \vec{g}_{\Gamma}^{\,q} \,,
\end{align}
where $\Gamma$ can take the values $V$, $A$, $T$
and $\vec{g}_{\Gamma}^{\,q}$ is the vector of the respective GFFs [cf.~Eqs.~\ref{eq:Decompositions}].
Due to equivalent combinations of momenta and polarizations
most rows in the matrix $M_\Gamma$ are equal or differ by a sign only.
We average the corresponding correlation functions, which improves the
signal-to-noise ratio considerably and reduces the number of equations.
\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{overview_ee.pdf}
\caption{
Overview of the nucleon energies for our ensembles.
We compare the energies $E_N(\vec{p}\,)$ and the errors extracted from a two-exponential fit
shown as black error bars
with the energies $E_N^c$ expected from the continuum dispersion
relation, which are depicted as colored boxes.
}
\label{fig:oee}
\end{figure}
\section{Numerical methods}
\label{sec:numerical methods}
\subsection{Gauge ensembles}
\label{sec:gaugens}
Our analysis is based on the large set of gauge configurations produced by the
QCDSF and the RQCD (Regensburg QCD) Collaborations
using the standard Wilson gauge action with two mass-degenerate nonperturbatively
improved clover fermions; see \Tab{tab:latsetup}.
We have three different lattice spacings 0.081\,fm, 0.071\,fm and 0.060\,fm.
Despite the $\mathcal{O}(a)$ improved action,
we expect discretization effects linear in the
lattice spacing for our matrix elements
since the currents are not improved.
The pion masses range from about 490\,MeV down to 150\,MeV.
In terms of $Lm_\pi$ we cover values from about 3.4 up to 6.7.
\subsection{Fitting two-point correlation functions}
\label{sec:fp2}
We parametrize our two-point correlation functions with a two-exponential fit
ansatz
\begin{subequations}
\label{eq:2pt_cor_fit_2e}
\begin{align}
\label{eq:2pt_cor_fit_2e_a}
C^{2\mathrm{pt}}(t , \vec{p}\,) &=
A(\vec{p}\,) \, \mathrm{e}^{-E_N(\vec{p}\,) \, t} +
X(\vec{p}\,) \, \mathrm{e}^{-Y(\vec{p}\,) \, t}
\intertext{with}
\label{eq:2pt_cor_fit_2e_b}
A(\vec{p}\,) &= Z(\vec{p}\,) \, \frac{E_N(\vec{p}\,) + m_N}{E_N(\vec{p}\,)} ,
\end{align}
\end{subequations}
in order to create bootstrap ensembles for the fit parameters
$A(\vec{p}\,)$, $E_N(\vec{p}\,)$, $X(\vec{p}\,)$ and $Y(\vec{p}\,)$.
To improve the signal, we average over all
momentum combinations which lead to the same $\vec{p}^{\,2}$.
Subsequently, we use \Eq{eq:2pt_cor_fit_2e_b} to fix the overlap factors
$Z(\vec{p}^{\,\prime})$ and $Z(\vec{p}^{}\,)$
which are needed to factor out $\vec{\mathcal{F}}_\Gamma$
from the three-point correlation functions (cf.~\Eq{eq:c3pt_ampl}).
The fit parameters $X(\vec{p}\,)$ and $Y(\vec{p}\,)$ are introduced
in order to parametrize the contributions from excited states.
The parameter $E_N(\vec{p}\,)$ represents the nucleon energy (we do not assume a functional form for the energy).
However, our analysis assumes continuum symmetries.
Therefore we restrict our lattice calculations to momenta whose fitted values for $E_N(\vec{p}\,)$ are
consistent with the continuum dispersion relation (cf.~Fig.~\ref{fig:oee})
\begin{align}
\label{eq:econt}
E^c_N(\vec{p}\,) = \sqrt{m_N^2 + \vec{p}^{\, 2}} \, .
\end{align}
The statistical errors are estimated by virtue of 500 bootstrap ensembles.
We carefully study the fit-range dependence of the fit parameters.
Therefore we consider the start time slices $t_s/a \in \{2,3\}$ and vary the final time slice $t_f/a$.
We find that the impact of $t_s/a$ on the values for the GFFs is rather mild
and therefore we fix $t_s/a = 2$ in the following.
In Fig.~\ref{fig:2ptchoice} we demonstrate how we choose the final time slice $t_f/a$.
We also try single exponential fits and find that they give similar results if
one adjusts the fit ranges appropriately.
However, the resulting errors on $A(\vec{p}\,)$ are larger.
Hence we use the two-exponential fit ansatz for our final analysis.
\begin{figure}[tb]
\centering
\includegraphics[width=0.48\textwidth]{E04_15-2-2.pdf}
\caption{The top panel shows the correlated $\chi^2_{\mathrm{dof}}$ as a
function of the final time slice $t_f/a$ for ensemble IV with $E_N(\vec{p}\,)=1.33\,\mathrm{GeV}$;
the bottom panel shows the uncorrelated normalized statistical error of the fit parameters $E_N$ and $Z$.
For the case shown we select the $t_f/a = 14$ result.
}
\label{fig:2ptchoice}
\end{figure}
\subsection{Three-point correlation functions}
\label{sec:fp3}
For the lattice calculations of three-point functions we use the sequential
source method where we set the outgoing nucleon momentum
$\vec{p}^{\,\prime} = \vec{0}$ for all our ensembles.
We parametrize the data using Eqs.~\eqref{eq:c3pt_a}
and \eqref{eq:c3pt_ampl} with $E_N(\vec{p}^{\, \prime})$ = $m_N$.
The initial energy $E_N(\vec{p}^{}\,)$ is determined from the
continuum dispersion relation (\ref{eq:econt}).
The momentum restriction, which we discussed in the previous section,
translates to a range $0 \le Q^2 < 0.6\, \mathrm{GeV}^2$
for the three-point functions.
With $Z(\vec{p}^{\,\prime})$ and $Z(\vec{p}^{}\,)$ having been determined
from the two-point correlation functions,
the only free parameter left is $\mathcal{F}^{q}_\Gamma$.
To achieve ground state dominance,
one has to make sure that
$aN_T \gg t^{\prime} \gg \tau \gg 0$ [cf.\ \Eq{eq:c3pt_a}].
We consider $\tau\in [\tau_s,\tau_e]$ where
$\tau_s$ is well above zero and $\tau_e$ well below $t^\prime$.
The sink times vary with the ensemble
(see the last column of \Tab{tab:latsetup}).
In Sec.\,\ref{sec:error_estimation} we examine possible excited state contaminations.
\subsection{Determination of the GFFs}
\label{sec:chi_gff}
As explained above, for every
current $\Gamma=V$, $A$ or $T$, quark flavor $q$ and virtuality
$-\mathsf{t}$, we need to solve the linear system \Eq{eq:eqs}, i.e.,
$\vec{\mathcal{F}} = M \cdot \vec{g}$, to extract the
relevant form factors $\vec{g}$ from the vector of inequivalent
matrix elements $\vec{\mathcal{F}}$ that correspond to nonvanishing
rows of $M$. Here
we drop all indices like the quark flavor $q$ and $\Gamma$ for convenience.
In what follows $m$ denotes the number of independent
form factors while $n\geq m$ is the length of $\vec{\mathcal{F}}$.
Consequently, $M$ is a $n\times m$ matrix of maximal rank,
i.e., $\rank(M)=m$.
The determination of the form factors is carried out in two ways.
The first method consists of two steps: First we extract the
ground state nucleon matrix elements $\mathcal{F}_j$ from the
lattice three-point function data $c^\tau_j$, restricted to the
range of insertion times $\tau\in[\tau_s,\tau_e]$,
through the numerical minimization of the $\chi^2$-function
\begin{equation}
\label{eq:chi}
\chi^2\big(\vec{\mathcal{F}}\,\big) =
\sum\limits_{j = 1}^{n}
\sum\limits_{\tau, \tau^\prime =\tau_s }^{\tau_e}
\delta c_{j}^{\tau}
\left[
\mathrm{cov}^{-1}_j
\right]_{\tau\tau^\prime}
\delta c_{j}^{\tau^\prime} \,,
\end{equation}
where $\delta c_j^\tau$ is the difference
\begin{equation}
\label{eq:delta}
\delta c_{j}^{\tau} = c_{j}^{\tau} - \mathcal{F}_j\sqrt{ Z(\vec{p}^{\,\prime}) Z(\vec{p}^{}\,)}
\;\mathrm{e}^{-m_N(t^{\prime} - \tau)} \, \mathrm{e}^{-E_N(\vec{p}\,)\tau}
\end{equation}
between the lattice data and the three-point function
parametrization \Eq{eq:c3pt_ampl}.
The inverse covariance matrix $\mathrm{cov}_j^{-1}$ depends on the
insertion times $\tau$ and $\tau^\prime$. One can easily generalize
the fit to the situation of multiple source-sink distances $t'$ if this
is required or include excited state contributions.
The index $j\in\{1, \ldots, n\}$ runs over all possible polarizations $\rho$
and initial momenta $\vec{p}$ (keeping the virtuality $Q^2$ fixed),
which give nonvanishing contributions.
Once the fit parameters $\mathcal{F}_j$ are determined,
one can minimize
\begin{equation}
\label{eq:bad}
\epsilon^2 =
\left( M \vec{g} - \vec{\mathcal{F}} \right)^2
\end{equation}
to determine the form factors $\vec{g}$.
The total number of parameters for this method is $m+n$ and,
in particular for large virtualities, this number can be quite large (up to 50).
This is not the only problem but
it can happen that the resulting $\epsilon$ value is quite large and
it is not clear how one should deal with such a situation.
Ideally, $\epsilon$ should be zero but this is only possible if
$\vec{\mathcal{F}}$ is in the image of $M$ [cf.~Eq.~\eqref{eq:bad}].
Motivated by this observation, we carry out our fits employing a single step
method, which combines the two subsequent steps into a single
minimization problem, restricting the number of fit parameters to the
relevant degrees of freedom.
We start from the singular value decomposition,
\begin{equation}
\label{eq:SVD}
M = U \cdot \Sigma \cdot V^{\intercal}
\end{equation}
with orthogonal matrices
$U\in\mathbb{R}^{n \times n}$, $V\in \mathbb{R}^{m \times m}$
and the matrix $\Sigma\in \mathbb{R}^{n \times m}$, which
has nonvanishing entries only on the diagonal. The pseudoinverse $\Sigma^+$ is
a $m\times n$ matrix that can easily be
obtained, computing the inverses of the diagonal elements of $\Sigma$.
Each vector $\vec{\mathcal{F}}$
within the image of $M$ can be uniquely expressed
as a linear combination
\begin{equation}
\label{eq:imag}
\vec{\mathcal{F}}(\vec{\alpha}\,) = \sum\limits_{i = 1}^{m} \alpha_i \, \vec{u}^{\,i}
\end{equation}
of the first $m$ column vectors of $U$. Note that $m = \rank({M})$.
Substituting $\vec{\mathcal{F}} \mapsto \vec{\mathcal{F}}(\vec{\alpha}\,)$
in Eq.~\eqref{eq:delta} [and thereby Eq.~\eqref{eq:chi}],
we obtain a modified $\chi^2$-function that depends on the
parameters $\alpha_i$, where $i\in\{1, \ldots, m\}$.
Finally, we convert the extracted vector $\vec{\alpha}$ to the
desired GFF vector,
\begin{align}
\label{eq:SVDI}
\vec{g} =
\left[V\Sigma^+ U^{\intercal}\right]\sum\limits_{i = 1}^{m} \alpha_i \, \vec{u}^{\,i} =
\left[V\Sigma^+\right]\vec{\alpha}\,,
\end{align}
where in the last step $\Sigma^+$ is truncated to a $m\times m$ square matrix.
In \Fig{fig:imag_space} we show for one example on the nearly physical
quark mass ensemble VIII that this method works very well. In this
case eight different lattice channels, listed in
\Tab{tab:operatorcontributions_to_fit}, are well described in terms
of three fit parameters.
\begin{table}[t]
\caption{Individual operator contributions to the fits shown
in \Fig{fig:imag_space}.
The numbers in the legend of \Fig{fig:imag_space} correspond to the
channels below.
We parametrize the spatial lattice momentum $\vec{q}= \hat{k}2\pi/L$
in terms of $\hat{e}_1$,$\hat{e}_2$, and $\hat{e}_3$ which are unit vectors in the three spatial directions. \label{tab:operatorcontributions_to_fit}
}
\begin{center} \begin{ruledtabular}
\begin{tabular}{l@{\quad}c@{\quad}c@{\qquad}c@{\qquad} c@{\qquad} c@{\qquad} }
Channel & $\mathbb{P}^\rho$ & $\mathcal{O}$ &$\hat{k}$ & Channel & \#contrib. \\
\hline
0 & $\mathbb{P}^4$ & $\mathcal{O}_{1\,4}^{v_{2,a}}$ & $\pm2\hat{e}_1$ & imaginary & 2 \\
& & $\mathcal{O}_{2\,4}^{v_{2,a}}$ & $\pm2\hat{e}_2$ & & 2 \\
& & $\mathcal{O}_{3\,4}^{v_{2,a}}$ & $\pm2\hat{e}_3$ & & 2 \\
\hline
1 & & $\mathcal{O}_{1}^{v_{2,b}}$ & $\pm2\hat{e}_1$ & real & 2 \\
& & & $\pm2\hat{e}_2$ & & 2 \\
\hline
2 & & $\mathcal{O}_{2}^{v_{2,b}}$ & $\pm2\hat{e}_1$ & real& 2 \\
& & & $\pm2\hat{e}_2$ & & 2 \\
\hline
3 & & $\mathcal{O}_{3}^{v_{2,b}}$ & $\pm2\hat{e}_1$ & real& 2 \\
& & & $\pm2\hat{e}_2$ & & 2 \\
\hline
4 & $\mathbb{P}^1$ & $\mathcal{O}_{2\,3}^{v_{2,a}}$ & $\pm2\hat{e}_2$, $\pm2\hat{e}_3$ & imaginary & 4 \\
& $\mathbb{P}^2$ & $\mathcal{O}_{1\,3}^{v_{2,a}}$ & $\pm2\hat{e}_1$, $\pm2\hat{e}_3$ & & 4 \\
& $\mathbb{P}^3$ & $\mathcal{O}_{1\,2}^{v_{2,a}}$ & $\pm2\hat{e}_1$, $\pm2\hat{e}_2$ & & 4 \\
\hline
5 & $\mathbb{P}^1$ & $\mathcal{O}_{3\,4}^{v_{2,a}}$ & $\pm2\hat{e}_2$& real & 2 \\
& & $\mathcal{O}_{2\,4}^{v_{2,a}}$ & $\pm2\hat{e}_3$& & 2 \\
& $\mathbb{P}^2$ & $\mathcal{O}_{3\,4}^{v_{2,a}}$ & $\pm2\hat{e}_1$& & 2 \\
& & $\mathcal{O}_{1\,4}^{v_{2,a}}$ & $\pm2\hat{e}_3$& & 2 \\
& $\mathbb{P}^3$ & $\mathcal{O}_{2\,4}^{v_{2,a}}$ & $\pm2\hat{e}_1$& & 2 \\
& & $\mathcal{O}_{1\,4}^{v_{2,a}}$ & $\pm2\hat{e}_2$& & 2 \\
\hline
6 & $\mathbb{P}^4$ & $\mathcal{O}_{1}^{v_{2,b}}$ & $\pm2\hat{e}_3$& real & 2 \\
\hline
7 & $\mathbb{P}^4$ & $\mathcal{O}_{2}^{v_{2,b}}$ & $\pm2\hat{e}_3$& real & 2 \\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
A comparison of the two fit methods shows that the results
are consistent within errors for all GFFs and for all ensembles.
The single step method, however, results in somewhat
smaller statistical errors and a
smoother $Q^2$ dependence, especially for the induced GFFs.
In \Fig{fig:comp_im_da} we directly compare the two methods.
For the final results we only use the single step method.
In \Fig{fig:chi2} we show all $\chi^2_{\mathrm{dof}}$ values of all fits used
in this paper to extract all considered GFFs:
The correlated single step
fits provide a very satisfactory description of the data.
\begin{figure}[t]
\centering
\includegraphics[width=0.47\textwidth]{fit_3pt_image_space.pdf}
\caption{
Fit results using the single step minimization method.
We show ensemble VIII at the virtuality $Q^2 = 0.277\,\mathrm{GeV}^2$
in the vector channel. This corresponds to a
spatial momentum transfer of $2\cdot 2\pi/L$, where we
have averaged over all equivalent lattice directions.
Three fit parameters
$\vec{\alpha} = (\alpha_1, \, \alpha_2, \, \alpha_3)^{\intercal}$
fully describe eight three-point functions.
Colored points lie in the fit range $[\tau_s,\tau_e]$ [cf.\ \Eq{eq:chi}].
On the left we show data for the $u$ quark and on the right
for the $d$ quark~(omitting disconnected contributions). The numbers in the legend
refer to the channels listed in
\Tab{tab:operatorcontributions_to_fit}.
\label{fig:imag_space}}
\end{figure}
\begin{figure}[b]
\centering
\includegraphics[width=0.47\textwidth]{comp_imag_axial_E06_15.pdf}
\caption{
Comparison of single step and two step fit methods for the axial GFFs for ensemble VI. The right panels show $\widetilde{A}_{20}$ and $\widetilde{B}_{20}$ separately for the $u$ and $d$ quark (without disconnected contributions), the left panels for the isovector case.
\label{fig:comp_im_da}}
\end{figure}
\begin{figure}[t]
\includegraphics[width=0.47\textwidth]{chi2.pdf}
\caption{$\chi^2$ distribution of all GFF fits performed for this analysis.}
\label{fig:chi2}
\end{figure}
\subsection{Excited states}
\label{sec:error_estimation}
For some of our ensembles we have three-point function data for
different source-sink separations.
This allows us to analyze the influence of excited states on the GFFs.
Our analysis is based on ensemble IV with five source-sink separations in the
range $t^\prime/a \in [7, 17]$ and on ensemble VIII
with three source-sink separations in the range $t^\prime/a \in [9, 15]$.
In physical units $t^\prime = 15a$ corresponds to about $1\,\mathrm{fm}$.
Ensemble VIII has data for eight values of $Q^2$ and this
ensemble corresponds to an almost physical pion mass.
We show results only for this ensemble, but our findings are consistent for both ensembles.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{overview_ex_NucleonVectorGFF.pdf}
\caption{The vector GFFs vs $Q^2$ for different sink
times $t^{\prime}$ for ensemble VIII. \label{fig:vGFFx}
}
\end{figure}
For the tensor and axial GFFs we find that within statistical
errors the $Q^2$ dependence is not affected by a
variation of $t^\prime$.
Only in the vector case, especially for $A_{20}^{u-d}$,
excited state contaminations are visible (see \Fig{fig:vGFFx}).
We have tried to parametrize these excited-state contributions to the
three-point function with various multiexponential fit ans{\"a}tze.
This, however, introduces additional fit parameters,
in particular the mass and the energy of the first excited state.
The first excitation in the three-point function can be a multihadron state
and hence its energy will in general not be well approximated
by the single particle continuum dispersion relation.
To parametrize excited state contributions clearly several source-sink
separations are required. However, within present statistical
errors little movement is visible for $t'\gtrsim 0.9\,$fm, even
in the $A_{20}$ channel where we achieve the highest accuracy;
see \Fig{fig:vGFFx} for an example.
We therefore have restricted our GFF fits to ranges of $\tau$
where the data
are well described by a single exponential (cf.~Fig.~\ref{fig:imag_space}).
In all the cases $t'$ is larger than 1\,fm.
\begin{figure*}
\centering
\includegraphics[width=0.94\textwidth]{overview_NucleonVecAxvecGFF.pdf}
\caption{The vector and axial GFFs vs $Q^2$. Left: $A_{20}^{u-d}$,
$B_{20}^{u-d}$ and $C_{20}^{u-d}$; right: $\widetilde{A}_{20}^{u-d}$ and $\widetilde{B}_{20}^{u-d}$. All results are for the isovector case and in the $\mathsf{\overline{MS}}$ scheme ($\mu=2\,\text{GeV}$).
\label{fig:vaGFF}}
\bigskip
\includegraphics[width=0.94\textwidth]{overview_NucleonTensorGFF.pdf}
\caption{
The tensor GFFs
$A^{u-d}_{T20}$,
$B^{u-d}_{T20}$,
$\widetilde{A}^{u-d}_{T20}$,
$\widetilde{B}^{u-d}_{T21}$
and the linear combination
$\overline{B}^{u-d}_{T20}$ in
the $\mathsf{\overline{MS}}$ scheme ($\mu=2\,\text{GeV}$).
\label{fig:tGFF}}
\end{figure*}
\section{Nucleon GFFs}
\label{sec:results_GFF}
Below we show results for the nucleon GFFs on a subset of the
ensembles listed
in \Tab{tab:latsetup}. We restrict ourselves to $m_{\pi}<300\,$MeV and
$m_\pi L>3.4$ and analyze the quark mass, volume and lattice spacing
dependence.
All results refer to the $\mathsf{\overline{MS}}$ scheme at $\mu=2\,\text{GeV}$.
\subsection{Vector and axial GFFs}
Results for the vector GFFs, $A_{20}^{u-d}$, $B_{20}^{u-d}$ and
$C_{20}^{u-d}$, are shown in \Fig{fig:vaGFF} (left) as a function of
$Q^2=-\mathsf{t}$. We see that the discretization effects are
negligible within errors (comparing ensembles I and XI, which give
about the same pion mass and a similar value for $Lm_\pi$). Also the
volume dependence (cf.\ V and VI) is small, although there is a slight
trend towards larger values for $B_{20}^{u-d}$ if $Lm_\pi$ increases
from about 4.2 to 6.7. For $A_{20}^{u-d}$ and $C_{20}^{u-d}$ we do not
see any volume dependence within present errors. Similar statements
hold for the quark mass dependence: For $A_{20}^{u-d}$ and
$C_{20}^{u-d}$ it is negligible within errors, but for $B_{20}^{u-d}$
we see a trend towards lower values if the pion mass decreases down to
150 MeV (cf.\ VIII and VI). However, the latter could also be a volume
artifact, since there is also a clear correlation between $Lm_\pi$ and
$B_{20}^{u-d}$ (cf.\ ensembles VIII, V and VI where $Lm_\pi\simeq3.5$,
4.2 and 6.7, respectively). $A_{20}^{u-d}$ and $B_{20}^{u-d}$ have a
roughly linear $Q^2$ dependence for small $Q^2$, and $C_{20}^{u-d}$ is zero
within errors. This agrees with the leading $\mathsf{t}$-dependence
expected from covariant baryon chiral perturbation theory (BChPT, see
below). We remark that also the individual (quark line
connected) $u$ and $d$ quark contributions to $C_{20}^{u-d}$ are zero
within error. So the smallness of this generalized form factor is
not due to an approximate cancellation.
For large $Q^2$ we expect that $A_{20}^{u-d}$ exhibits a
dipolelike $Q^2$-dependence, which we saw in our former study
(cf.\ Fig.~2 of Ref.~\cite{Sternbeck:2012rw}).
Results for the axial GFFs are shown in the right panel of
\Fig{fig:vaGFF}. We see that a change of volume, quark mass or lattice
spacing has almost no effect on the data. Within errors these effects
cannot be resolved. Both form factors grow approximately linearly for
$Q^2\to 0$. For $\widetilde{B}_{20}^{u-d}$ the statistical errors
become larger for $Q^2 \rightarrow 0$ whereas the errors for
$\widetilde{A}_{20}^{u-d}$ are nearly independent of $Q^2$.
\subsection{Tensor GFFs}
Continuing with the tensor GFFs,
we show results for $A^{u-d}_{T20}$, $B^{u-d}_{T20}$,
$\widetilde{A}^{u-d}_{T20}$ and
$\widetilde{B}^{u-d}_{T21}$ in \Fig{fig:tGFF}.
The dominant form factors are $A^{u-d}_{T20}$ and $B^{u-d}_{T20}$.
For the available virtualities $A^{u-d}_{T20}$ rises linearly for $Q^2\to 0$,
while $B^{u-d}_{T20}$ remains more or less constant, well above zero.
Overall, the statistical errors for $A^{u-d}_{T20}$ are smaller than for $B^{u-d}_{T20}$.
Volume, quark mass or lattice spacing effects cannot be resolved within errors.
The other two GFFs, $\widetilde{A}^{u-d}_{T20}$ and $\widetilde{B}^{u-d}_{T21}$,
are smaller in comparison and,
besides a few outliers, are best described by a constant.
However, a final conclusion cannot be drawn as
the statistical errors for both GFFs are rather large.
We also study the linear combination
\begin{align}
\label{eq:Bbar}
\overline{B}^{q}_{T20} = B^{q}_{T20} + 2\widetilde{A}^{q}_{T20}\,,
\end{align}
which corresponds to the combination of GPDs $E_T + 2\tilde{H}_{T}$ that is
related to
the Boer-Mulders function $h_1^\perp$~\cite{Boer:1997nt}.
We find that the statistical error of $\overline{B}^{q}_{T20} $ is
significantly smaller compared to the individual errors of $B^{q}_{T20}$
and $\widetilde{A}^{q}_{T20}$ (see \Fig{fig:tensor_anti}).
We will take advantage of this observation when looking at the transverse
spin of the nucleon in Sec.\,\ref{sec:nucleon_tomography}.
The results for $\overline{B}^{u-d}_{T20}$ are shown with the
tensor GFFs in \Fig{fig:tGFF}
for the same ensembles.
The anticorrelations we find for $\overline{B}^{u-d}_{T20}$ are
present for all ensembles.
\begin{figure}[b]
\centering
\includegraphics[width=0.45\textwidth]{tensor_anti_cor.pdf}
\caption{
Strong anticorrelations between $B^q_{T20}$ and $\widetilde{A}^q_{T20}$
for the example of ensemble VI.
\label{fig:tensor_anti}
}
\end{figure}
\section{Extraction of \texorpdfstring{$\boldsymbol{J^{u-d}}$}{J(u-d)}}
\label{sec:chiral_J}
The GFFs $A_{20}^{u-d}(\mathsf{t})$ and $B^{u-d}_{20}(\mathsf{t})$ are of particular interest since for $\mathsf{t}\to0$ they
are related to the total angular momentum~\cite{Ji:1996ek}
\begin{align}
\label{eq-ji}
J^{u-d} = \frac{1}{2} \left[A_{20}^{u-d}(0) + B_{20}^{u-d} (0) \right].
\end{align}
In order to estimate $J^{u-d}$ at the physical pion mass we analyze our data for $A_{20}^{u-d}(\mathsf{t})$ and
$B^{u-d}_{20}(\mathsf{t})$, employing the BChPT formulas of Ref.~\cite{Wein:2014wma}, which, however, we truncate at order $m_{\pi}^3$,
\begin{align}
\label{eq:chia}
A^{u-d}_{20}(\mathsf{t}, m_\pi) = \left[ 1 - \frac{(1 + 3\,g_A^2) \, m_\pi^2 \, \log( \frac{m_\pi^2}{\mu^2}) }{16\, f_\pi^2\, \pi^2} \right] \, L \, \nonumber \\
+\, m_\pi^2 \, M_2^A \, + m_\pi^3\,M_3^A\,+ \mathsf{t}(\, T_0^A + m_\pi^2 \, T_1^A)
\end{align}
and
\begin{align}
\label{eq:chib}
&B^{u-d}_{20}(\mathsf{t}, m_\pi) =
\frac{ g_A^2 \, m_\pi^2\, \log(\frac{m_\pi^2}{ \mu^2}) }{16\, f_\pi^2 \,\pi^2}L
+ \, \mathsf{t}(\, T_0^B + m_\pi^2 \, T_1^B) \, \nonumber \\
&+\left[ 1 - \frac{(1 + 2 \, g_A^2) \,m_\pi^2 \log(\frac{m_\pi^2}{\mu^2})}{16 \, f_\pi^2 \, \pi^2}\right] L^B \, +m_\pi^2 \, M_2^B \,.
\end{align}
The fit parameters $T_1^A$ and $T_1^B$ are added since our data
extend up to virtualities
$-\mathsf{t} \approx (770\,\mathrm{MeV})^2 \gg m_{\pi}^2$,
however, these terms would naturally appear at the next order of BChPT.
We determine the parameters $(L, M_2^A, M_3^A, T_0^A, T_1^A)$
and $(L, L^B, M_2^B, T_0^B, T_1^B)$ by carrying out
combined fits to our data sets for
$A^{u-d}_{20}(\mathsf{t},\, m_\pi)$ and
$B^{u-d}_{20}(\mathsf{t},\, m_\pi)$.
The remaining parameters in Eqs.~\eqref{eq:chia} and \eqref{eq:chib}
are constrained to \hbox{$g_A = 1.256$}, \hbox{$f_\pi = 92.4\,\mathrm{MeV}$} and \hbox{$\mu = 1.0 \,\mathrm{GeV}$}.
Since it is not clear up to what values of $-\mathsf{t}$ and $m_\pi$ BChPT
is applicable,
we perform fits to all ensembles (set A) as well as fits using only
ensembles
with $m_\pi \le 300 \,\mathrm{MeV}$ (set B).
In \Fig{fig:j} we show the resulting fits for $\mathsf{t}=0$,
where only in the case of $A_{20}^{u-d}$ we can directly compare
to data points.
\begin{figure}[tb]
\centering
\includegraphics[width=0.45\textwidth]{J.pdf}
\caption{
From top to bottom $A_{20}^{u-d}(0)$, $B_{20}^{u-d}(0)$ and $J^{u-d}$ as a function of the pion mass squared.
The vertical solid line marks the physical pion mass; the vertical dashed line
indicates our smallest pion mass. The A-band is from a fit of all our ensembles and the B-band
from a fit where ensembles with $m_\pi > 300 \,\mathrm{MeV}$ are removed. For $A_{20}^{u-d}(0)$ we
have lattice data which are shown in the top panel for comparison.
\label{fig:j}
}
\end{figure}
For set A the fit parameters have smaller statistical errors.
For set B we see that
$A_{20}^{u-d}$ increases with $m_\pi \rightarrow m_{\pi}^{\mathrm{phy}}$. For
both sets we obtain values for $\chi_{\mathrm{dof}}^2$ of about 0.75, hence
we cannot use the $\chi_{\mathrm{dof}}^2$ value to discriminate between the
fit ranges. Instead, one may interpret the difference between fits A and B
as a systematic uncertainty of the parameters. In Fig.~\ref{fig:jpp} we show
our fit for set A as a function of $Q^2$ at two fixed values
of the pion masses ($m_\pi=422\,\text{MeV}$
and $150\,\text{MeV}$, ensembles III and VIII).
Obviously, our ansatz for the $Q^2$ and $m_\pi^2$ dependence describes the lattice data well.
Again, we study
the effect of the uncertainties of the renormalization constants using
the strategy described in Appendix~\ref{sec:zerr}. The final results are
collected in \Tab{tab:res_j}, where we also quote the
total angular momentum $J^{u-d}$. We refrain
from extrapolating to $Q^2=0$ and $m_{\pi}=m_{\pi}^{\mathrm{phy}}$ in the
other cases. Instead, in \Tab{tab:res_others} we give the results
for the form factors where no extrapolation in $Q^2$ is required,
i.e.\ $A_{20}^{u-d}(0)$, $\widetilde{A}_{20}^{u-d}(0)$ and $A_{T20}^{u-d}(0)$,
for our
nearly physical point ensemble VIII. The moment $A_{20}^{u-d}(0)=\langle
x\rangle_{u-d}$ agrees well with the results of the global fits to ensemble
sets A and B and also the helicity and transversity
moments $\widetilde{A}_{20}^{u-d}(0)=\langle x\rangle_{\Delta u -\Delta d}$
and $A_{T20}^{u-d}(0)=\langle x\rangle_{\delta u -\delta d}$ at the physical
point ensemble are in agreement with the global data, see the top right
panel of Fig.~\ref{fig:vaGFF} and the top left panel of Fig.~\ref{fig:tGFF},
respectively.
\begin{figure}[tb]
\centering
\includegraphics[width=0.45\textwidth]{J++.pdf}
\caption{Chiral fit A versus $Q^2$ for two distinct pion masses:
$m_\pi=422\,\text{MeV}$ (green) and $150\,\text{MeV}$ (grey).
The corresponding data points (ensemble III and VIII) are shown as well.
\label{fig:jpp}
}
\end{figure}
Within the errors, our values
agree with the isovector results of Ref.~\cite{Alexandrou:2017oeh}.
\begin{table}[tb]
\caption{
Results for $A_{20}^{u-d}(0,m_\pi)$, $B_{20}^{u-d}(0,m_\pi)$ and $J^{u-d}(m_\pi)$, extrapolated to the physical pion mass $m_\pi^{\mathrm{phy}}$ using the ensemble sets A and B (see the text). The first error is statistical, the second error is due to the uncertainty of the renormalization constants.
\label{tab:res_j}}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{c@{\qquad}cc}
Ensemble selection & A & B \\
\hline
$A_{20}^{u-d}(0, m_{\pi}^{\mathrm{phy}})$ & $0.195 \,(06) \, (03)$ & $0.210\,(08)\,(04)$\\
$B_{20}^{u-d}(0, m_{\pi}^{\mathrm{phy}})$ & $0.271 \,(13) \, (03)$ & $0.287\,(28)\,(04)$\\
$J^{u-d}(m_{\pi}^{\mathrm{phy}})$ & $0.233 \,(07) \, (03)$ & $0.248\,(14)\,(04)$ \\
\end{tabular}
\end{ruledtabular}
\end{center}
\caption{Results for $A_{20}^{u-d}$, $\widetilde{A}_{20}^{u-d}$ and $A_{T20}^{u-d}$
at the nearly physical pion mass $m_\pi=150\,\text{MeV}$ (ensemble VIII). The first error is statistical,
the second error is due to the uncertainty of the renormalization constants.}
\label{tab:res_others}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{c@{\qquad}c}
Ensemble VIII & Value \\
\hline
$A_{20}^{u-d}(0,m_\pi)$ & $0.213 \,(11) \, (04)$ \\
$\widetilde{A}_{20}^{u-d}(0,m_\pi)$ & $0.240 \,(07) \, (03)$ \\
$A_{T20}^{u-d}(0,m_\pi)$ & $0.266 \,(08) \, (04)$
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
\section{Nucleon tomography}
\label{sec:nucleon_tomography}
We use our lattice results for the vector GFFs $A_{20}(\mathsf{t})$, $B_{20}(\mathsf{t})$ and the linear combination $\overline{B}_{T20}(\mathsf{t})$
[cf.~Eq.~(\ref{eq:Bbar})] to investigate the transverse spin density of the nucleon. To this end, we transform these GFFs to the impact parameter space $G(\mathsf{t}) \rightarrow G(\boldsymbol{b}_{\perp}^2)$ with
\begin{align}
\label{eq:gffb}
G\left(\boldsymbol{b}_{\perp}^2\right)=\int\frac{\mathrm{d}^2\boldsymbol{\Delta_{\perp}}}{(2\pi)^2}\;
e^{-i \boldsymbol{b_{\perp}}\cdot \boldsymbol{\Delta_\perp}}\: G\left(\mathsf{t}=-\boldsymbol{\Delta}^2_{\perp}\right)\,,
\end{align}
where we use the $p$-pole ansatz~\cite{Diehl:2005jf,Gockeler2007}
\begin{align}
\label{eq:dipol}
G(\mathsf{t}) = \frac{G_0}{\left(1 - \mathsf{t}/m_p^2 \right)^p}
\end{align}
for the interpolation of our lattice results.
The impact parameter $\boldsymbol{{b_\perp}}$ is defined in the transverse $x$-$y$ plane.
It measures the transverse distance from the ``center of momentum''
\begin{align}
\boldsymbol{R}_\perp = \sum \limits_i \boldsymbol{r}_{i\,\perp} x_i\,,
\quad \sum \limits_i x_i = 1\,,
\end{align}
where $x_i$ is the momentum fraction of the $i$th parton
\cite{Diehl:2005jf,PhysRevD.15.1141}.
We define
\begin{align}
\boldsymbol{b}_\perp \coloneqq ( b_x, \, b_y) \, , \quad \quad
b_{\perp} \coloneqq \sqrt{ \boldsymbol{b}_\perp^2} \, .
\end{align}
To compute the transverse spin density, we also have to evaluate the derivative of
$G(b_\perp^2 )$ with respect to $b_\perp^2$,
\begin{align}
G^{\prime}(b_\perp^2 ) \coloneqq \frac{\partial}{\partial \, b_\perp^2 } \, G(b_\perp^2).
\end{align}
The Fourier transform (\ref{eq:gffb}) of the $p$-pole ansatz (\ref{eq:dipol})
can be expressed in terms of the modified Bessel functions $K_\nu$~\cite{Diehl:2005jf},
\begin{align}
\label{eq:gffb0}
G( b_{\perp}^2 ) &= \frac{
G_0 \, m_p^2 \; (b_{\perp}m_p)^{p-1} \, K_{p-1}( b_{\perp} m_p)
}{
2^p \, \pi\, \Gamma(p)
}\,.
\end{align}
The transverse spin density $\rho^q(x,\boldsymbol{{b_\perp}},\boldsymbol{s_\perp},\boldsymbol{S_\perp})$
describes the probability to find a quark with longitudinal momentum fraction $x$,
flavor $q$ and transverse spin $\boldsymbol{s}_\perp$ at a distance $\boldsymbol{b}_{\perp}$
from the center of momentum of the nucleon with transverse spin $\boldsymbol{S}_\perp$.
The explicit definition in terms of GPDs is given in Eq.\,(8) of Ref.~\cite{Diehl:2005jf}.
Here we consider the two transverse spin combinations,
\begin{subequations}
\label{eq:ss}
\begin{align}
\label{eq:s10_S00}
\boldsymbol{s}_\perp = (1 \,, 0) \quad &\text{and} \quad \boldsymbol{S}_\perp = (0 \,, 0)\,,\\
\boldsymbol{s}_\perp = (0 \,, 0) \quad &\text{and} \quad \boldsymbol{S}_\perp = (1 \,, 0)\,,
\end{align}
\end{subequations}
where the first line describes a transversely polarized quark in an unpolarized nucleon and
the second an unpolarized quark in a transversely polarized nucleon.
In terms of GFFs the first moment of
$\rho^q(x, \boldsymbol{{b_\perp}}, \boldsymbol{s_\perp},\boldsymbol{S_\perp})$
for these spin combinations reads
\begin{align}
\langle\rho\rangle^q(\boldsymbol{b_\perp},\boldsymbol{s_\perp},\boldsymbol{S_\perp}) &=
\int_{-1}^{1}\!\mathrm{d}x\,x \, \rho^q(x, \boldsymbol{{b_\perp}}, \boldsymbol{s_\perp},\boldsymbol{S_\perp})
\nonumber \\
=\frac{1}{2}A_{20}^{q}(b_\perp^2)
&-\frac{ \epsilon^{ij} \, b_\perp^j }{2m_N} \left( s_\perp^i \overline{B}_{T20}^{q\,\prime}(b_\perp^2) + S_\perp^i B_{20}^{q\,\prime}(b_\perp^2) \right) \,.
\label{eq:density_simpl}
\end{align}
For arbitrary spins $\boldsymbol{S}_\perp$ and $\boldsymbol{s}_\perp$
\Eq{eq:density_simpl} will contain additional terms and we refer the reader to Refs.~\cite{Diehl:2005jf,Gockeler2007}.
\begin{figure}[tb]
\centering
\includegraphics[width=0.5\textwidth]{dipol_scan.pdf}
\caption{
The pole mass $m_p$ and $\chi^2_{\mathrm{dof}}$
as a function of the fixed parameter $p$ for ensemble VI.
The colored lines correspond to fits to
$A_{20}^q$,
$B_{20}^q$ and
$\overline{B}_{T20}^q$ from top to bottom
and flavor $q$ from left to right.
\label{fig:dipol_scan}
}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=0.45\textwidth]{tomo_ana_quark_compute_d.pdf}
\caption{
The $p$-dependence of the transverse spin density for a
transversely polarized $d$-quark in an unpolarized nucleon.
The yellow cross indicates the maximum of the density.
The black contour lines are drawn equidistantly with a
difference of 0.05.\label{fig:dens_scan}
}
\end{figure}
We fit the GFFs for ensemble VI to the $p$-pole ansatz \Eq{eq:dipol}.
Due to the limited number of data points at our disposal, where we restricted
ourselves to the kinematic range $-\mathsf{t} \le 0.6 \, \mathrm{GeV}^2$,
we find it impossible to simultaneously determine all three fit parameters, $p$, $m_p$ and $G_0$. In particular the exponent $p$ is strongly correlated with the pole mass $m_p$. This is demonstrated in \Fig{fig:dipol_scan}: An increase of $p$ results in a larger value of $m_p$, whereas $\chi^2_{\mathrm{dof}}$ does not significantly change. Therefore, we cannot constrain $p$.
This arbitrariness means it is difficult to obtain reliable, parametrization
independent results for the moment $\langle\rho\rangle^q(\boldsymbol{b_\perp},\boldsymbol{s_\perp},\boldsymbol{S_\perp})$ as a function of $\boldsymbol{b_\perp}$. This distribution has been studied in the past (see, e.g.,~\cite{Gockeler2007}),
but we find that its shape strongly depends on the value of $p$.
In \Fig{fig:dens_scan} we show
$\langle\rho\rangle^q(\boldsymbol{b_\perp},\boldsymbol{s_\perp},\boldsymbol{S_\perp})$
for $\boldsymbol{s}_\perp = (1,0)$ and $\boldsymbol{S}_\perp = (0, 0)$ for four distinct values of $p$ ranging from
1.45 up to 3.0.
We see that with increasing $p$ the density becomes less localized in
the impact parameter plane and the maximum of the density is shifted away from the center.
This also holds for other spin combinations.
\begin{table}[t]
\caption{The half $\boldsymbol{b_\perp}$-integrated moments
for $p=2$, also shown in Fig.~\ref{fig:rho_pm}. The errors are
statistical. The systematic error of the $p$-dependence is about
0.02.
\label{tab:res_rho_pm}}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{ccc}
& $\boldsymbol{s}_\perp = (1 \,, 0)$ & $\boldsymbol{s}_\perp = (0 \,, 0) $ \\
& $\boldsymbol{S}_\perp = (0 \,, 0)$ & $\boldsymbol{S}_\perp = (1 \,, 0) $ \\
\hline
$\langle\rho\rangle^{u}_{-}$ & 0.312 (26) & 0.403 (12) \\
$\langle\rho\rangle^{u}_{+}$ & 0.688 (26) & 0.597 (12) \\
\hline
$\langle\rho\rangle^{d}_{-}$ & 0.262 (27) & 0.666 (17) \\
$\langle\rho\rangle^{d}_{+}$ & 0.738 (27) & 0.334 (17)
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table}
We discovered that some integrated quantities have a much
milder $p$-dependence, namely the half $\boldsymbol{b_\perp}$-integrated moments
\begin{subequations}
\label{eq:rho_pm}
\begin{align}
\langle\rho\rangle^q_{+}(\boldsymbol{s_\perp},\boldsymbol{S_\perp}) &=
\frac{1}{Z_{\rho}}
\int \limits_{-\infty}^{+\infty}\!
\mathrm{d}b_x \! \!
\int \limits_{0}^{+\infty}\!
\mathrm{d}b_y \,
\langle\rho\rangle^q(\boldsymbol{b_\perp},\boldsymbol{s_\perp},\boldsymbol{S_\perp}) \, ,
\\
\langle\rho\rangle^q_{-}(\boldsymbol{s_\perp},\boldsymbol{S_\perp}) &=
\frac{1}{Z_{\rho}}
\int \limits_{-\infty}^{+\infty}\!
\mathrm{d}b_x \!\!
\int \limits_{-\infty}^{0}\!
\mathrm{d}b_y \,
\langle\rho\rangle^q(\boldsymbol{b_\perp},\boldsymbol{s_\perp},\boldsymbol{S_\perp}) \,,
\end{align}
\end{subequations}
with the normalization factor
\begin{align}
{Z_{\rho}} &=
\int \limits_{-\infty}^{+\infty}\!
\mathrm{d}b_x \,
\int \limits_{-\infty}^{+\infty}\!
\mathrm{d}b_y \, \langle\rho\rangle^q(\boldsymbol{b_\perp},\boldsymbol{s_\perp},\boldsymbol{S_\perp}) \,.
\end{align}
The integrated moment $\langle\rho\rangle^q_{+}(\boldsymbol{s_\perp},\boldsymbol{S_\perp})$
is the probability, weighted with the longitudinal momentum fraction $x$, to find a quark with flavor $q$ in the upper part ($b_y\ge 0$) of the impact parameter space
and $\langle\rho\rangle^q_{-}(\boldsymbol{s_\perp},\boldsymbol{S_\perp})$ is the
$x$-weighted probability to find a quark with flavor $q$ in the lower part ($b_y\le 0$).
These integrated moments are a measure for the asymmetry of the
transverse spin density.
They depend much less on the value of $p$ than
$\langle\rho\rangle^q(\boldsymbol{b_\perp},\boldsymbol{s_\perp},\boldsymbol{S_\perp})$
does.
This is demonstrated in \Fig{fig:densp_scan},
where $\langle\rho\rangle^d_{+}$
and $\langle\rho\rangle^d_{-}$ are shown as functions of $p$ for
the transverse spin combination in
\Eq{eq:s10_S00}.
Doubling $p$, both integrated moments change
by only 5\% and 15\%, respectively.
We find this mild $p$-dependence for all considered transverse spin and flavor
combinations and consider these integrated moments as the better
candidates for reliable lattice estimates.
Our results for $\langle\rho\rangle^q_{\pm}$ for up and down quark
for our two transverse spin combinations [\Eq{eq:ss}]
are shown in \Fig{fig:rho_pm}. The errors shown
are statistical only.
The figure corresponds to the power $p=2$, and one may add
systematic errors of about 0.02 due to the $p$-dependence;
see \Fig{fig:densp_scan}. The numerical values are listed in
\Tab{tab:res_rho_pm}.
We see the probability of a transversely polarized $u$- or $d$-quark in an
unpolarized nucleon is higher ($\sim 70\%$)
in the $b_y>0$ part of the impact parameter space than in the
$b_y<0$ part ($\sim30\%$).
For a transversely polarized nucleon however the probabilities of an
unpolarized $u$- or $d$-quark differ:
The unpolarized $d$-quark is more likely in the
$b_y<0$ part ($67\%$),
while a $u$-quark is more likely in the $b_y>0$ part (60\%)
of the impact parameter space.
\begin{figure}[tb]
\centering
\includegraphics[width=0.45\textwidth]{rhopm_tomo_ana_quark_compute_d.pdf}
\caption{Dependence of $\langle\rho\rangle^d_{+}\left(\boldsymbol{s_\perp},\boldsymbol{S_\perp}\right)$ and
$\langle\rho\rangle^d_{-}\left(\boldsymbol{s_\perp},\boldsymbol{S_\perp}\right)$
on the power $p$ of the pole ansatz. The combination of transverse spins is $\boldsymbol{s_\perp}=(1,0)$ and $\boldsymbol{S_\perp}=(0,0)$. The errors
are statistical only. The systematics due to the uncertainty of the power
$p$ amount to about 0.02.
\label{fig:densp_scan}
}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=0.45\textwidth]{rho_pm.pdf}
\caption{Probability (weighted with $x$) to find a $u$- or $d$-quark in the upper/ lower part ($b_y\lessgtr 0$) of the impact parameter space; left for a transversely polarized quark in an unpolarized nucleon; right for an unpolarized quark in a transversely polarized nucleon.
\label{fig:rho_pm}
}
\end{figure}
\section{Summary}
\label{sec:summary}
We have calculated all quark GFFs, corresponding to operators
with one derivative, of the nucleon GPDs at leading twist-2.
Our lattice calculation includes the dominating connected contributions and
neglects contributions from disconnected diagrams.
The available gauge ensembles cover a wide range of
quark masses and volumes.
However, the three available
lattice spacings only vary from 0.081\,fm down to 0.060\,fm.
Within errors, all GFFs show a mild dependence on the quark mass,
lattice spacing and volume.
We have compared two different fitting strategies for the GFFs and found that
the direct fit method appears to be more reliable.
With this method the number of fit parameters is reduced to the relevant
degrees of freedom. We recommend to use this method in future studies.
The final results for the GFFs are shown in Figs.~\ref{fig:vaGFF}
and~\ref{fig:tGFF}.
We have also studied the total angular momentum and the
transverse spin density of quarks in the nucleon.
Both quantities can be extracted from fits to our GFF data.
For the total angular momentum we obtain a similar estimate in the isovector
case as ETMC in Ref.~\cite{Alexandrou:2017oeh}.
Contributions from disconnected diagrams are not included in our lattice calculation. From Ref.~\cite{Alexandrou:2017oeh} we know that these
are small. Nevertheless, in the isoscalar case they should definitely be taken
into account.
For the second moment of the transverse spin density we have found that
its distribution in impact parameter space strongly depends
on the $\mathsf{t}$-dependence of the GFF data.
The shape of the distribution depends on the value of $p$ that is
used within a $p$-pole ansatz.
High precision data at small and large values of $-\mathsf{t}$ would
be required to eliminate this ambiguity.
For integrated moments this situation improves.
In \Fig{fig:rho_pm} we provide lattice estimates for the
$x$-weighted probabilities of a transversely polarized (unpolarized)
light quark in the upper or lower part of the impact parameter space,
within an unpolarized (transversely polarized) nucleon.
Contributions from higher moments are not yet available but
constitute an interesting object for future study.
\begin{acknowledgments}
The ensembles were generated by RQCD and QCDSF primarily on the QPACE computer~\cite{Baier:2009yq,Nakamura:2011cd}, which was built as part of the
DFG (SFB/TRR 55) project. The authors gratefully acknowledge the Gauss Centre
for Supercomputing e.V.\ (\href{http://www.gauss-centre.eu}{\url{www.gauss-centre.eu}})
for granting computer time on SuperMUC at the Leibniz Supercomputing Centre
(LRZ, \href{http://www.lrz.de}{\url{www.lrz.de}}) for this project.
The BQCD~\cite{Nakamura:2010qh} and CHROMA~\cite{Edwards:2004sx} software
packages were used, along with the locally deflated domain decomposition solver
implementation of openQCD~\cite{Luscher:2012av,openQCD}.
Part of the analysis was also performed on the iDataCool cluster in Regensburg.
Support was provided by the DFG (SFB/TRR 55).
ASt acknowledges support by the BMBF under Grant
No.\ 05P15SJFAA (FAIR-APPA-SPARC) and by the DFG Research
Training Group GRK1523.
We thank Benjamin Gl\"a\ss{}le for software support.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,489 |
Hong Kong's securities regulator ordered on Wednesday nine brokers to freeze client accounts related to suspected manipulation of shares in investment firm China Ding Yi Feng Holdings between 2018 and early 2019. Trading in Ding Yi Feng's shares had been suspended since last week at the direction of the Securities and Futures Commission (SFC). China Ding Yi Feng officials could not be immediately reached for comment.
How Does Investing In Haitong International Securities Group Limited (HKG:665) Impact The Volatility Of Your Portfolio?
Is Haitong International Securities Group Limited (HKG:665) Overpaying Its CEO?
Should You Think About Buying Haitong International Securities Group Limited (HKG:665) Now?
Should You Buy Haitong International Securities Group Limited (HKG:665) At This PE Ratio?
China Vanke Co., the nation's largest listed developer by market value, may price a floating-rate bond on Thursday amid rising funding costs spurred by the rush of issuance from the sector. Vanke is marketing a five-year floating-rate note at 155 basis points over three-month Libor, according to a person familiar with the matter. This could be the first Chinese developer to sell a dollar-denominated FRN through the public syndicated bond market, according to Bloomberg-compiled data. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,543 |
Mr. Jekyll Absinthe (700ml)
Closure: Cork
Absinthe (or Absinth) is an alcoholic drink made with the pounded leaves and flowering tops of one species of wormwood plant (Artemisia absinthium) together with other herbs such as angelica root, fennel, nettles, parsley, balm, sweet flag root and hyssop. Emerald green in colour and usually very bitter, Abisinthe is traditionally poured over a perforated spoonful of sugar into a glass of water – or vice versa. The drink then turns into an opaque white as the essential oils precipitate out of the alcoholic solution.
The Absinthe Story...
Absinthe was invented at Couvet in the Canton of Neuchatel, Switzerland in 1797 by Dr. Pierre Ordinaire. It became popular in the French army in the early part of the 19th century as an antidote to fever. Henri-Louis Pernod later opened the first Absinthe distillery in Switzerland and then moved to a larger premises in Pontarlier, France in 1805. By the 1850's it had become the favourite drink of the upper class. Originally wine based, a blight in 1870's on the vineyards forced manufacturers to use grain alcohol as a base. This increased its affordability and the drink quickly became popular amongst bohemians. Artists and writers like Van Gogh, Baudelaire, and Verlaine, to name a few, believed Absinthe to stimulate creativity. Most days started with a drink of "La fée verte" (the 'green fairy' as it became commonly known), and ended with "l'heure verte (the green hour) as one or two (or more) were taken as an apéritif.
However, in the 1850's, there began to be concern about the results of chronic use. Absinthe was believed to produce a syndrome, called absinthism, which was characterized by addiction, hyperexcitability, and hallucinations. This concern over the health effects of absinthe was amplified by the prevailing belief in Lamarckian theories of heredity. In other words, it was believed that any traits acquired by absinthists would be passed on to their children. Absinthe's association with the bohemian lifestyle also worked to compound fears about its effects, much as has happened with marijuana in America. Absinthe was subsequently banned in many countries in the beginning of the 1900's.
The remains some question as to the active component or components of absinth. Alcohol is obviously one main component. However, another primary candidate is thujone, a monoterpene which is considered a convulsant. Thujone's mechanism of action is not known, although structural similarities between thujone and tetrahydrocannabinol (the active component in marijuana) have led some to hypothesize that both substances have the same site of action in the brain. Thujone makes up 40 to 90% (by weight) of the essence of wormwood, from which absinthe is made. Thus, thujone would appear to be a good candidate for a second active component in absinth.
Absinth today has made a fashionable comeback and is now readily available in the U.S., much of Europe and now Australia. Mr.Jekyll Absinthe is relatively light for the genre (55% alc./vol.) It has a full, rich mouthfeel blending aniseed and fresh citrus notes. Despite the alcoholic strength, the drink is surprisingly smooth. To prepare Absinth the traditional way do the following:
Pour one shot into a glass.
Place a slatted Absinth spoon on the glass. Place one or two sugar lumps on the spoon.
Drip six to eight parts iced water through the sugar and into the absinth.
Stir mixture, which will turn cloudy, and drink.
ALWAYS DRINK IN MODERATION.
Click here to learn more about Liqueurs.
Click here for Classic Cocktails.
Click here to learn more about Absinthe.
La Fee NV Absinthe (700ml)
Kubler Véritable Fée Verte Absinthe (500ml)
Green Fairy Absinth (Absinthe) 500ml
Tamborine Mountain Distillery Moulin Rooz Absinthe...
Pernod Absinthe (700ml) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,382 |
Q: What Are $4$ Sided Shapes Called Again? I apologise for the really basic question. This didn't really fit on any other StackExchange website so the Maths one was the closest one where I could ask.
Really Basic Question- What are $4$ sided shapes called again?
Like how triangles are $3$ sided shapes, octagons are $8$ sided shapes, ... What are the $4$ sided ones called then?
A: Shorter version is "quadrangle".
A: The word quadrilateral is made of the words quad (meaning "four") and lateral (meaning "of sides").
A: Wait, never mind. It's quadrilateral. :)
A: I have a draft article where I 'should' use 'quadrilateral' dozens of times. I refuse, stating up front that I will use 'quad' instead.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,143 |
What to do for the summer holidays?
8 May 2015 by Elinore Court.
Approaching the end of a year at uni or a course always feels like the end of an era and can feel a little daunting. You get so used to living in your chosen city surrounded by friends that 3 months back home can seem a little boring in comparison. Make sure you avoid this by planning in advance and either sorting out a summer job or a fun itinerary to make the most of this summer.
1. Liberty Living has plenty of options for Summer Accommodation across the UK to help you make the most of summer with central locations so that you'll have plenty of options to explore.
For example, if you're in any of the student accommodation options across London then be sure to grab tickets for top festivals like Notting Hill Carnival, British Summer Time so you can catch Taylor Swift and Blur in Hyde Park this June, Wireless festival to see Drake, Avicii, Clean Bandit and so many more – plenty of options for whichever music you love.
If that's not your scene then grab a picnic and plenty of blankets to check out the outdoor cinema at Somerset House or enjoy the sun in Kew Gardens.
2. Summer is also the time to give yourself a break from all the hard work of final exams, making friends and throwing great parties. You don't need me to tell you that your student accommodation is a great place to organise group events quickly – all you need is to knock on everyone's door and make the most of the garden, courtyard or nearby park. With everyone contributing one ingredient you could whip up a great picnic or barbecue with a cheap, portable barbecue box and of course make up a jug (or two) of fruity Pimms to host the perfect summer afternoon.
3. If you got a bit carried away with your student loan throughout the year (don't worry, we've all been there) then speak to Liberty Living about the different options because there is a wide range of accommodation to suit all budgets and other requirements so you can choose what works best for you.
4. Also remember that there are short or long stays available if you're still figuring out your summer holiday plans. In most cities the minimum stay is one week but once you've made your plans then discuss flexibility with the on-site staff so they can help you out.
5. You will still have access to all the on-site facilities. Think no queues in the laundry, a great common room to chill in with no parents nagging at you for watching too much TV and outside areas to chill out in.
6. No packing up your room. If you're staying on in your room or flat for summer then save yourself the tedious effort of packing and sorting everything out. If you're as lazy as me then this is a very tempting option and a great way to save a stressful car journey home.
So there you have it, 5 things to remember to help you get the most out of your summer holidays!
Posted on 8th May 2015 by Elinore Court. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,321 |
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\section{Introduction}
Alzheimer's disease (AD) is the most common cause of dementia \cite{AlzheimersAssociation2019}. AD leads to the atrophy of the brain more quickly than healthy aging and is a progressive neurodegenerative disease, meaning that more and more parts of the brain are damaged over time. The atrophy primarily appears in brain regions such as hippocampus, and it afterwards progresses to the cerebral neocortex. At the same time, the ventricles of the brain as well as cisterns, which are outside of the brain, enlarge \cite{Savva2009}.
Healthy aging also results in changing of the brain, following specific patterns \cite{Alam2014}. Therefore, a possible biomarker used in AD is the estimation of the brain (biological) age of a subject which can then be compared with the subject's real (chronological) age \cite{Franke2019}. People at increased risk can be identified by the deviation between these two ages and early computer-aided detection of possible patients with neurodegenerative disorders can be accomplished. For this reason, a large body of research has focused on estimating brain age, especially using Magnetic Resonance (MR) images, which have long been used successfully in the measurement of brain changes related to age \cite{Good2001}.
Recently, deep learning models have proved to be successful on the task of brain age estimation, providing relatively high accuracy. They are designed to find correlation and patterns in the input data and, in the supervised learning setting, associate that with a label, which in our case is the age of the subject. The models are trained on a dataset of MR images of healthy brains, estimating the expected chronological age of the subject. During training, the difference of the chronological age and the predicted age needs to be as small as possible. During test time, an estimated brain age larger than the subject's chronological age indicates accelerated aging, thus pointing to a possible AD patient.
\begin{figure}[t]
\centering
\includegraphics[width = \textwidth]{figure1_paper15.png}
\caption{Overview of the proposed idea. A noise model is trained with the purpose of adding as much noise as possible to the input. The output is a noise mask, in which noise sampled from the standard normal distribution is added. The result is then added to the input image and is used as an input to a pretrained prediction model with frozen parameters. The aim is to create a noisy image that will maximize the noise level while also not harming the performance of the prediction model.} \label{fig1}
\end{figure}
Convolutional Neural Networks (CNNs) are used with the purpose of an accurate brain age estimation while using the minimum amount of domain information since they can process raw image data with little to no preprocessing required. Many studies provide very accurate results, with mean absolute error (MAE) as low as around 2.2 years \cite{Peng2021,Pardakhti2020,Liu2020}. However, most of these approaches purely focus on minimization of the prediction error while considering the network as a black box. Recent studies have started to try to identify which regions are most informative for age prediction. For example, in \cite{Bintsi2020} the authors provided an age prediction for every patch of the brain instead of whole brain. Although the predictions and results presented in \cite{Bintsi2020} were promising, and the localisation was meaningful, the use of large patches meant that the localisation was not very precise. Similar approaches has been explored, such as \cite{Ballester2021} in which slices of the brain were used instead of patches. In \cite{Popescu}, the authors provided an age estimation per voxel but the accuracy of the voxel-wise estimations dropped significantly. In \cite{Levakov2020}, an ensemble of networks and gradient-based techniques \cite{Smilkov2017} were used in order to provide population-based explanation maps. Finally, 3D explanation maps were provided in \cite{Bass2021}, but the authors used image translation techniques and generative models, such as VAE-GAN networks.
In computer vision, a common approach to investigate black-box models such as CNNs is to use saliency maps, which show which regions the network focuses on, in order to make the prediction. An overview of the existing techniques for explainability of deep learning models in medical image analysis may be found in \cite{Singh}. Gradient-based techniques, such as guided backpropagation \cite{Springenberg2015}, and Grad-CAM \cite{Selvaraju2020}, make their conclusions based on the gradients of the networks with respect to a given input. Grad-CAM is one of the most extensively used approach and usually results in meaningful but very coarse saliency maps. Gradient-based techniques are focused on classification and to our knowledge they do not work as expected for regression task because they detect the features that contributed mostly to the prediction of a specific class. Occlusion-based techniques \cite{Zeiler2014} have also been widely explored in the literature and they can be used both for classification and regression tasks. The idea behind occlusion techniques is very simple: The performance of the network is explored after hiding different patches of the images, with the purpose of finding the ones that affect the performance the most. It is a promising and straightforward approach but bigger patches provide coarse results. On the other hand, the smaller the patches, the greater the computational and time constraints are, which can be a burden in their application.
A recent approach which leverages the advantages of occlusion techniques while also keeping computational and time costs relatively low is U-noise \cite{Koker} which uses the addition of noise in the images pixel-wise, while keeping the performance of the network unchanged with the purpose of understanding where the deep learning models focus in order to do their predictions. The authors created interpretability maps for the task of pancreas segmentation using as input 2D images using noise image occlusion. In this paper, inspired by the work described above \cite{Koker} and the idea that when a voxel is not important for the task, then the addition of noise on this specific voxel will not affect the performance of our network, we make the following contributions:
1) We adapt the architecture of U-noise (Figure \ref{fig1}), which was originally used for pancreas segmentation, for the task of brain age regression and visualise the parts of the brain that played the most important role for the prediction by training a noise model that dynamically adds noise to the image voxel-wise, providing localised and fine-grained importance maps.
2) We extend the U-noise architecture to 3D instead of 2D to accommodate training with three-dimensional volumetric MR images;
3) We propose the use of an autoencoder-based pretraining step on a reconstruction task to facilitate faster convergence of the noise model;
4) We provide a population-based importance map which is generated by aggregating subject-specific importance maps and highlights the regions of the brain that are important for healthy brain aging.
\section{Materials \& Methods}
\subsection{Dataset \& Preprocessing}
We use the UK Biobank (UKBB) dataset \cite{Alfaro-Almagro2018} for estimating brain age and extracting the importance maps. UKBB contains a broad collection of brain MR images, such as T1-weighted, and T2-weighted. The dataset we use in this work includes T1-weighted brain MR images from around 15,000 subjects. The images are provided by UKBB skull-striped and non-linearly registered to the MNI152 space. After removing the subjects lacking the information of age, we end up with 13,750 subjects with ages ranging from 44 to 73 years old, 52,3\% of whom are females and 47.7\% are males. The brain MR images have a size of 182x218x182 but we resize the 3D images to 140x176x140 to remove a large part of the background and at the same time address the memory limitations that arise from the use of 3D data, and normalise them to zero mean and unit variance.
\subsection{Brain Age Estimation}
We firstly train a CNN, the prediction model, $f_{\theta}$, with parameters $\theta$, for the task of brain age regression. We use the 3D brain MR images as input to 3D ResNet-18, similar to the one used in \cite{Bintsi2020}, which uses 3D convolutional layers instead of 2D \cite{He2016}. The network is trained with a Mean Squared Error (MSE) loss and its output is a scalar representing the predicted age of the subject in years.
\subsection{Localisation}
An overview of the proposed idea is shown in Figure \ref{fig1}.
\subsubsection{U-Net Pretraining}
The prediction model and the noise model have identical architectures (2D U-Net \cite{Ronneberger2015U-net:Segmentation}) in \cite{Koker}, as they are both used for image-to-image tasks, which are segmentation and noise mask generation, respectively. Therefore, in that case, the noise model is initialized with the weights of the pretrained prediction model. However, in our case, the main prediction task is not an image-to-image task but rather a regression task and thus, the prediction model's (a 3D ResNet as described above) weights cannot be used to initialize the noise model. Instead, we propose to initially use the noise model $f_{\psi}$ with parameters $\psi$, which has the architecture of a 3D U-Net as a reconstruction model, for the task of brain image reconstruction. By pretraining the noise model with a reconstruction task before using it for the importance map extraction task, we facilitate and accelerate the training of our U-noise model, since the network has already learned features relating to the structure of the brain.
The reconstruction model consists of an encoder and a decoder part. It uses as input the 3D MR volumes and its task is to reconstruct the volumes as well as possible. It is trained with a voxel-wise MSE loss. The number of blocks used, meaning the number of downsample and upsample layers, is 5, while the number of output channels after the first layer is 16.
\subsubsection{Brain Age Importance Map Extraction}
For the extraction of the importance maps, we extend the U-noise architecture \cite{Koker} to 3D. The idea behind U-noise is that if one voxel is important for the prediction task, in our case brain age regression, the addition of noise on it will harm the performance of the prediction model. On the other hand, if a voxel is not relevant for the task, the addition of noise will not affect the performance.
More specifically, after the pretraining phase with the reconstruction loss, the 3D U-Net, i.e. noise model, is used for the task of extraction of a 3D noise mask that provides a noise level for every voxel of the input 3D image. A sigmoid function is applied so the values of the mask are between 0 and 1. The values are scaled to [$v_{min}$, $v_{max}$], where $v_{min}$, $v_{max}$ are hyperparameters. The rescaled mask is then multiplied by $\epsilon \sim \mathcal{N}(0,1) $, which is sampled from the standard normal distribution and the output is added to the input image element-wise in order to extract the noisy image. Given an image $X$, its noisy version can be given by Equation \eqref{eq:1}:
\begin{equation}
X_{noisy} = X + f_{\psi}(X)(v_{max}-v_{min})\epsilon + v_{min},
\label{eq:1}
\end{equation}
We then use the noisy image as input for the already trained prediction model with frozen weights, and see how it affects its performance. The purpose is to maximise the noise level in our mask, while simultaneously keeping the performance of our prediction model as high as possible. In order to achieve that, we use a loss function with two terms, the noise term, which is given by $-log(f_{\psi}(X))$ and motivates the addition of noise for every voxel, and the prediction term, which is an MSE loss whose purpose is to keep the prediction model unchanged. The two loss terms are combined with a weighted sum, which is regulated by the ratio hyperparameter $r$. It is important to note that at this stage the parameters, $\theta$, of the prediction model, $f_{\theta}$, are frozen and it is not being trained. Instead, the loss function $\mathcal{L}$, is driving the training of the noise model $f_{\psi}$. Given an input image $X$ and label $y$, the loss function $\mathcal{L}$ takes the form shown in Equation \eqref{eq:2}
\begin{equation}
\mathcal{L} = \underbrace{(f_{\theta}(X) - y)^2}_\text{prediction term} - r \underbrace{log(f_{\psi}(X))}_\text{noise term}
\label{eq:2}
\end{equation}
The values $v_{min}$, $v_{max}$, as well as $r$ are hyperparameters and are decided based on the performance of our noise model on the validation set.
\section{Results}
\begin{figure}[t]
\centering
\includegraphics[width = \textwidth]{figure2_paper15.png}
\caption{Six different slices of the population-based importance map on top of the normalised average of the brain MR Images of the test set. The most important parts of the brain for the model's prediction are highlighted in red. The most significant regions for the task of brain aging are mesial temporal structures including the hippocampus, brainstem, periventricular and central areas. The results are in agreement with the relevant literature and previous studies.} \label{fig2}
\end{figure}
From the 13,750 3D brain images, 75\% are used for the training set, 10\% for the validation set and 15\% for the test set. All the networks are trained with backpropagation \cite{Rumelhart1986LearningErrors} and adaptive moment estimation (Adam) optimizer \cite{Kingma2015Adam:Optimization} with initial learning rate lr=0.0001, reduced by a factor of 10 every 10 epochs. The experiments are implemented on an NVIDIA Titan RTX using the Pytorch deep-learning library \cite{Paszke2019PyTorch:Library}.
\subsection{Age Estimation}
We train the prediction model for 40 epochs using a batch size of 8 using backpropagation. We use MSE between the chronological and biological age of the subject as a loss function. The model achieves a mean absolute error (MAE) of about 2.4 years on the test set.
\subsection{Population-based importance maps}
The noise model is trained for 50 epochs and with a batch size of 2, on four GPUs in parallel. Different values were tested for hyperparameters $v_{min}$, $v_{max}$ and $r$. The chosen values, for which the network performed the best in the validation set, are $v_{min}=1$, $v_{max}=5$ and $r=0.1$. We average the importance maps for all the subjects of the test set, ending up with a population-based importance map. We use a threshold in order to keep only the top 10\% of the voxels with the lowest tolerated noise levels, meaning the most important ones for brain aging. We then apply a gaussian smoothing filter with a kernel value of 1. Different slices of the 3D population-based importance map are shown in Figure \ref{fig2}. As it can be seen, the areas that are the most relevant for brain aging according to the model's predictions are mesial temporal structures including the hippocampus, brainstem, periventricular and central areas.
\section{Discussion}
Understanding the logic behind a model's decision is very important in assessing trust and therefore in the adoption of the model by the users \cite{Gilpin2019}. For instance, the users should be assured that correct predictions are an outcome of proper problem representation and not of the mistaken use of artifacts. For this reason, some sort of interpretation may be essential in some cases. In the medical domain \cite{Holzinger2017}, the ability of a model to explain the reasoning behind its predictions is an even more important property of a model, as crucial decisions about patient management may need to be made based on its predictions.
In this work, we explored which parts of the brain are important for aging. In order to do so, we made the assumption that unimportant voxels/parts of the brain are not useful for brain age estimation and are not utilized by the prediction model. We trained a prediction model, which accurately estimated brain aging, and a noise model, whose purpose is to increase the noise in the input images voxel-wise, while also keeping the performance of the prediction model unaffected. As can be seen from Figure \ref{fig2}, our importance maps are in agreement with the existing neuropathology literature \cite{Savva2009}. More specifically, it is shown that the hippocampus and parts of the ventricles are where the prediction model focuses to make its decisions.
On the other hand, the differences in the cerebral cortex appear to not be getting captured by the network. In our understanding, there are two reasons behind this. Firstly, the age range of the subjects (44-73 years old) is not large enough for the network to make conclusions. At the same time, the changes in the cerebral cortex are more noticeable after the age of 65 years. In our case, we probably do not have enough subjects in that age range in order to facilitate the network into capturing these differences.
The images that have been used in this study are non-linearly registered, since UKBB provides them ready for use and in the literature more works use the provided preprocessed dataset and therefore, comparison is easier to be done. However, it has been noticed that using non-linearly registered images may lead to the network's missing of subtle changes away from the ventricles, such as cortical changes \cite{Dinsdale2021}.
In the future, a similar experiment will be conducted with linearly registered images instead of non-linearly registered ones because we believe that, although the performance of the prediction model might be slightly lower, the importance maps will not only be focused on the ventricles and the hippocampus, but also on more subtle changes in the cerebral cortex. Additionally, UKBB provides a variety of other non-imaging features, including biomedical and lifestyle measures, and we intend to test our method on related regression and classification tasks, such as sex classification. In the case of classification tasks we will be also comparing against gradient-based interpretability approaches, such as Grad-CAM \cite{Selvaraju2020} and guided backpropagation \cite{Springenberg2015}, since the setting allows for their use.
\section{Conclusion}
In this work, we extend the use of U-noise \cite{Koker} for 3D inputs and brain age regression. We use 3D brain MR images to train a prediction model for brain age and we investigate the parts of the brain that play the most important role for this prediction. In order to do so, we implement a noise model, which aims to add as much noise as possible in the input image, without affecting the performance of the prediction model. We then localise the most important regions for the task, by finding the voxels that are the least tolerant to the addition of noise, which for the task of brain age estimation are mesial temporal structures including the hippocampus and periventricular areas. Moving forward, we plan to test our interpretability method on classification tasks, such as sex classification as well, and compare with gradient-based methods, which are valid for such tasks.
\subsubsection{Acknowledgements} KMB would like to acknowledge funding from the EPSRC Centre for Doctoral Training in Medical Imaging (EP/L015226/1).
\bibliographystyle{splncs04}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,059 |
County Line is een plaats (town) in de Amerikaanse staat Alabama, en valt bestuurlijk gezien onder Blount County en Jefferson County.
Demografie
Bij de volkstelling in 2000 werd het aantal inwoners vastgesteld op 257.
In 2006 is het aantal inwoners door het United States Census Bureau geschat op 263, een stijging van 6 (2,3%).
Geografie
Volgens het United States Census Bureau beslaat de plaats een oppervlakte van
2,5 km², geheel bestaande uit land.
Plaatsen in de nabije omgeving
De onderstaande figuur toont de plaatsen in een straal van 16 km rond County Line.
Externe link
Plaats in Alabama | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,973 |
What to do about Syria
Posted byIan July 27, 2006 February 20, 2016
From the International Herald Tribune Thursday, July 27, 2006
By Thomas L. Friedman (who wrote the book I have, "The World Is Flat".
One wonders what planet US. Secretary of State Condoleezza Rice landed from, thinking she can build an international force to take charge in south Lebanon without going to Damascus and trying to bring the Syrians on board.
Two Syrian officials made no bones about it when I asked their reaction to deploying such a force, without Syrian backing: Do you remember what happened in 1983, each asked, when the Reagan administration tried to impose an Israeli-designed treaty on Lebanon against Syria's will?
I was there, I remember quite well: Hezbollah, no doubt backed by Syria or Iran, debuted its skills for the world by blowing up the US. Embassy in Beirut and the U.S. Marine and French peacekeeping battalions. This is not a knitting circle here.
Can we Americans get the Syrians on board? Can we split Damascus from Tehran? My conversations here • suggest it would be very hard, but worth a shot. It is the most important strategic play we could make, because Syria is the bridge between Iran and Hezbollah. But it would take a high-level, rational dialogue. Rice says we can deal with Syria through normal diplomatic channels. Really?
We've withdrawn our ambassador from Damascus, and the US. diplomats left here are allowed to meet only the Foreign Ministry's director of protocol, whose main job is to ask how you like your Turkish coffee. Syria's ambassador in Washington is similarly isolated.
Is this Syrian regime brutal and ruthless? You bet it is. If the Bush team wants to go to war with Syria, I get that. But the US. boycott of Syria is not intimidating Damascus. (Its economy is still growing, thanks to high oil prices.) So we're left with the worst of all worlds – a hostile Syria that is not afraid of us.
We need to get real on Lebanon. Hezbollah made a reckless mistake in provoking Israel. Shame on Hezbollah for bringing this disaster upon Lebanon by embedding its "heroic" forces amid civilians. I understand Israel's vital need to degrade Hezbollah's rocket network. But Hezbollah's militia, which represents 40 percent of Lebanon, the Shiites, can't be wiped out at a price that Israel, or America's Arab allies, can sustain – if at all.
You can't go into an office in the Arab world today without finding an Arab TV station featuring the daily carnage in Lebanon. It's now the Muzak of the Arab world, and it is toxic for us and our Arab friends.
Despite Hezbollah's bravado, Israel has hurt it and its supporters badly, in a way they will never forget. Point made. It is now time to wind down this war and pull together a deal – a cease-fire, a prisoner exchange, a resumption of the peace effort and an international force to help the Lebanese Army secure the border with Israel – before things spin out of control. Whoever goes for a knockout blow will knock themselves out instead.
Will Syria play? Syrians will tell you that their alliance with Tehran is "a marriage of convenience." Syria is a largely secular country, with a Sunni majority. Its leadership is not comfortable with Iranian Shiite ayatollahs. The Iranians know that, which is why "they keep sending high officials here every few weeks to check on the relationship," a diplomat said.
So uncomfortable are many Syrian Sunnis with the Iran relationship that President Bashar al-Assad has had to allow a surge of Sunni religiosity; last April, a bigger public display was made of
Muhammad's birthday than the Syrian Baath Party's anniversary, which had never happened before.
Syrian officials stress that they formed their alliance with Iran because they felt they had no other option. One top Syrian official said the door with the United States was "not closed from Damascus. [But] when you have only one friend, you stay with him all the time. When you have 10 friends, you stay with each one of them."
What do the Syrians want? They say: respect for their security interests in Lebanon and a resumption of negotiations over the Golan. Syria is also providing support for the Sunni Baathists in Iraq. Much as the Bush team wants to, it can't fight everyone at once and get where it needs to go. There will not be a peace force in south Lebanon unless it's backed by Syria. No one will send troops.
I repeat: I don't know if Syria can be brought around, and we certainly can't do it at Lebanon's expense. But you have to try, with real sticks and real carrots. Syria is not going to calm things in Lebanon, or Iraq, just so the Bush team can then focus on regime change in Damascus. As one diplomat here put it to me, "Turkeys don't vote for Thanksgiving."
Posted byIan July 27, 2006 February 20, 2016 Posted inUncategorized
Who and how are the future terrorists made
Scuba Geek » The worst rebreather diver ever? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,763 |
\section{Introduction}
Given a vector bundle $E \to M$, it is often interesting to look at geometric structures (functions, vector fields, differential forms, etc.) on the total space $E$ that satisfy appropriate compatibility conditions with the vector bundle structure. Such compatibility is often referred to as \emph{linearity} in the literature. Accordingly, one speaks about \emph{linear functions}, \emph{linear vector fields}, \emph{linear differential forms}, etc., on the total space of a vector bundle. In the present paper, we will rather use the terminology ``\emph{fiber-wise linearity}'', to avoid confusion with other types of linearities. Now, the vector bundle structure is completely determined by (the smooth structure on $E$) and the action $h : \mathbb R \times E \to E$ of the monoid $(\mathbb R, \cdot)$ of multiplicative reals by fiber-wise scalar multiplication, $h(t, e) = te$, for all $t \in \mathbb R$, and all $e \in E$ (see, e.g.~\cite{GR2009}). It follows that the fiber-wise linearity of a geometric structure can be usually expressed purely in terms of $h$. For instance, a function $f$ on $E$ is fiber-wise linear if $h_t^\ast f = tf$ for all $t$. Similarly a vector field $X$ (resp.~a differential form $\omega$) on $E$ is fiber-wise linear if $h_t^\ast X = X$ for all $t \neq 0$ (resp.~$h_t^\ast \omega = t\omega$ for all $t$). A fiber-wise linear function is equivalent to a section of the dual vector bundle $E^\ast$, a fiber-wise linear vector fields is a section of the \emph{gauge algebroid} of $E$ (see, e.g, \cite{mackenzie}) and a fiber-wise linear differential $1$-form is equivalent to a section of the first jet bundle $J^1 E \to M$. The latter examples already show that fiber-wise linear structures on $E$ can encode interesting geometric structures on (vector bundles over) $M$. There are even more interesting examples. A fiber-wise linear symplectic structure $\omega$ on $E$ is equivalent to a vector bundle isomorphism $E \cong T^\ast M$. More precisely, there exists a unique fiber-wise linear $1$-form $\theta$ on $E$ such that $\omega = d \theta$, and there exists a unique vector bundle isomorphism $E \cong T^\ast M$ that identifies $\theta$ with the tautological $1$-form on $T^\ast M$, hence $\omega$ with the canonical symplectic structure on $T^\ast M$ (see, e.g., \cite{GR2009}, see also \cite{R2002}). There are more examples: a fiber-wise linear metric is equivalent to an isomorphism $E \cong T^\ast M$ together with a torsion free connection in $TM$ \cite{V20XX}. As a final remarkable example we recall that a fiber-wise linear Poisson structure on $E$ is the same as a Lie algebroid structure on $E^\ast$ \cite{mackenzie}.
In this paper we propose the following definition of \emph{fiber-wise linear scalar differential operator}. An $\mathbb R$-linear differential operator $\Delta : C^\infty (E) \to C^\infty (E)$ of order $q$ is \emph{fiber-wise linear} if $h_t^\ast \Delta = t^{1-q} \Delta$ for all $t \neq 0$. This definition might seem weird at a first glance. However, it is supported by several different facts. For instance, according to our definition, a function and a vector field are fiber-wise linear if and only if they are fiber-wise linear when regarded as a $0$-th order and a first order scalar differential operator, respectively. Moreover the principal symbol of a fiber-wise linear differential operator is a fiber-wise linear symmetric multivector. Another supporting remark is that the Laplacian (acting on functions) of a fiber-wise linear metric is a fiber-wise linear differential operator. Finally, a scalar differential operator $\Delta$ can be \emph{linearized} around a submanifold producing a fiber-wise linear differential operator representing a first order approximation to $\Delta$ in the transverse direction with respect to the submanifold. All these facts suggest that our definition might indeed be the ``correct one''. Our main result is a description of fiber-wise linear differential operators in terms of somehow simpler data. More precisely we prove the following theorem (see Theorem \ref{theor:iso_DO_D_sym} for a more precise statement).
\begin{theorem}\label{theor:0}
Let $E \to M$ be a vector bundle. Then there is a degree inverting $C^\infty (M)$-linear bijection between fiber-wise linear scalar differential operators $\Delta : C^\infty (E) \to C^\infty (E)$ and polynomial derivations of the line bundle $E^\ast \times_M \wedge^{\mathrm{top}} E \to E^\ast$.
\end{theorem}
This theorem is a little surprising because it describes objects of higher order in derivatives (fiber-wise linear differential operators) in terms of objects of order $1$ in derivatives (derivations of an appropriate vector bundle). We hope that this result might be the starting point of a more thorough investigation of \emph{multiplicative differential operators} on Lie groupoids and, at the infinitesimal level, \emph{infinitesimally multiplicative differential operators} on Lie algebroids. Multiplicative (resp.~infinitesimally multiplicative) structures are geometric structures on a Lie groupoid (resp.~Lie algebroid) which are additionally compatible with the groupoid (resp.~algebroid) structure. In the last thirty years, starting from the pioneering works of Weinstein on symplectic groupoids \cite{W1987}, multiplicative structures captured the interest of a large community of people working in Poisson geometry and related fields, and today we have a precise description of several different multiplicative structures and their infinitesimal counterparts: infinitesimally multiplicative structures (see \cite{KS2016} for a survey). However, all the examples investigated so far are of order $1$ in derivatives and it would be interesting to investigate the compatibility of a Lie groupoid/algebroid with structures of higher order in derivatives, e.g.~higher order differential operators. This is a natural issue that might conjecturally lead to new important developments. As infinitesimally multiplicative structures are, in particular, fiber-wise linear structure, this paper might be also considered as a first step in this direction.
The paper is organized as follows. In Section \ref{sec:vector_fields} we recall what does it mean for a vector field on the total space of a vector bundle $E \to M$ to be \emph{fiber-wise linear}, i.e.~compatible with the vector bundle structure. In Section \ref{sec:multivectors} we discuss \emph{fiber-wise linear symmetric multivectors} and we describe them in terms of simpler data. This material is well known to experts (although it is scattered in the literature and it is hard to find a universal reference) and the first two sections are mainly intended to fix our notation. In Section \ref{sec:derivations} we recall what a derivation of a vector bundle $E$ is and introduce what we call $E$-multivectors, a ``derivation analogue'' of plain multivectors. We also discuss fiber-wise linear $E$-multivectors. These objects are not exactly of our primary interest but they play a very useful role in the proofs of our main theorems (Theorems \ref{theor:iso_DO_D_sym} and \ref{theor:linear}). To the best of our knowledge the material in the third section is mostly new. Section \ref{sec:DO} is an extremely compact introduction to linear differential operators on vector bundles, and, in particular, scalar differential operators. Section \ref{sec:FWL_DO} contains our main constructions and results: we define and study fiber-wise linear (scalar) differential operators on the total space of a vector bundle $E$. Somehow surprisingly, fiber-wise linear differential operators on $E$ form a transitive Lie-Rinehart algebra over fiber-wise polynomial functions on $E^\ast$, with abelian isotropies (Theorem \ref{theor:stabilizer}). The reason is ultimately explained by our main result, Theorem \ref{theor:0} above (see also Theorem \ref{theor:iso_DO_D_sym} below). As already announced, $E$-multivectors play a prominent role in the proof. In Section \ref{sec:linear} we discuss the \emph{linearization} of a scalar differential operator $\Delta$ around a submanifold $M$ in a larger manifold. The linearization of $\Delta$ is a fiber-wise linear differential operator on the total space of the normal bundle to $M$, and can be seen as a first order approximation to $\Delta$ in the direction transverse to $M$. The existence of a \emph{linearization construction} strongly supports our definition of fiber-wise linear differential operators.
\section{Core and Linear Vector Fields on a Vector Bundle}\label{sec:vector_fields}
As we mentioned in the introduction, the main aim of the paper is to explain what does it mean for a differential operator on the total space $E$ of a vector bundle $E \to M$ to be compatible with the vector bundle structure. We will reach our definition (Definition \ref{def:FWL_DO}) by stages. We first need to recall what does it mean for a function, a vector field and, more generally, a multivector on $E$, to be compatible with the vector bundle structure. We do this in the present and the next section. We adopt the general philosophy of \cite{GR2009, GR2011} where it is shown that a vector bundle structure is encoded in the fiber-wise scalar multiplication, and compatibility with the vector bundle structure is expressed in terms of such multiplication.
So, let $\pi : E \to M$ be a vector bundle. The fiber-wise scalar multiplication by a real number
\[
h : \mathbb{R} \times E \to E
\]
is an action of the multiplicative monoid of reals $(\mathbb R, \cdot)$. The algebra $C^\infty_{\mathrm{poly}} (E)$ of fiber-wise polynomial functions on the total space $E$ is non-negatively graded:
\[
C^\infty_{\mathrm{poly}} (E) = \bigoplus_{k = 0}^\infty C^\infty (E)_k,
\]
and its $k$-th homogeneous piece $C^\infty (E)_k$ consists of homogeneous polynomial functions of degree $k$, i.e.~functions $f \in C^\infty (E)$ such that
\[
h_t^\ast (f) = t^k f
\]
for all $t \in \mathbb R$. Functions in $C^\infty (E)_0$ are just (pull-backs via the projection $\pi : E \to M$ of) functions on $M$. We call them \emph{core functions} and also denote them by $C_{\mathrm{core}}^\infty (E)$. They form a subalgebra in $C^\infty_{\mathrm{poly}} (E)$. Functions in $C^\infty (E)_1$ are fiber-wise linear (FWL for short in what follows) functions, and identify naturally with sections of the dual vector bundle $E^\ast$. We denote them by $C_{\mathrm{lin}}^\infty (E)$. They form a $C^\infty_{\mathrm{core}}(E)$ submodule in $C^\infty_{\mathrm{poly}} (E)$. We denote by $\ell_\varphi$ the linear function corresponding to the section $\varphi \in \Gamma (E^\ast)$. The terminology ``core function'' (and similarly ``core vector field'', etc., see below) is motivated by the theory of double vector bundles, where ``core sections'' are sections with an appropriate degree with respect to certain actions of $(\mathbb R, \cdot)$ (see, e.g.~\cite{mackenzie}).
A vector field $X \in \mathfrak X (E)$ on $E$ is \emph{fiber-wise polynomial}, or simply \emph{polynomial}, if it maps (fiber-wise) polynomial functions to polynomial functions. Polynomial vector fields $\mathfrak X_{\mathrm{poly}} (E)$ form a (graded) \emph{Lie-Rinehart algebra} over polynomial functions $C^\infty_{\mathrm{poly}} (E)$:
\[
\mathfrak X_{\mathrm{poly}} (E) = \bigoplus_{k = -1}^\infty \mathfrak X (E)_k.
\]
We recall for later purposes that a Lie-Rinehart algebra over a commutative algebra $A$ is a vector space $L$, which is both an $A$-module and a Lie algebra acting on $A$ by derivations with the following two compatibilities:
\begin{itemize}
\item the Lie algebra action map $\rho : L \to \operatorname{Der} A$ is $A$-linear (it is often called the \emph{anchor}), and
\item the Lie bracket $[-,-] : L \times L \to L$ is a bi-derivation, i.e.~it satisfies the following Leibniz rule:
\[
[\lambda, a \mu] = \rho (\lambda) \mu + a [\lambda, \mu],\quad \lambda, \mu \in L, \quad a \in A.
\]
\end{itemize}
Lie-Rinehart algebras are purely algebraic counterparts of Lie algebroids. For more on Lie-Rinehart algebras, see, e.g.~\cite{H2004} and references therein.
Coming back to polynomial vector fields, the $k$-th homogeneous piece $\mathfrak X (E)_k$ of $\mathfrak X_{\mathrm{poly}} (E)$ consists of homogeneous ``polynomial'' vector fields of degree $k$, i.e.~vector fields $f \in \mathfrak X (E)$ such that
\[
h_t^\ast (X) = t^k X
\]
for all $t \neq 0$. Vector fields in $\mathfrak X (E)_{-1}$ are vertical lifts of sections of $E$. We call them \emph{core vector fields} and also denote them by $\mathfrak X_{\mathrm{core}} (E)$. They form an abelian Lie-Rinehart subalgebra (beware over the subalgebra $C^\infty_{\mathrm{core}} (E) = C^\infty (M)$) in $\mathfrak X_{\mathrm{poly}} (E)$. We denote by $e^\uparrow$ the vertical lift of a section $e \in \Gamma(E)$.
Vector fields in $\mathfrak X (E)_0$ are, by definition, fiber-wise linear (FWL) vector fields. They can be equivalently characterized as vector fields preserving linear functions and they satisfy the following property
\[
[X, Y] \in \mathfrak X_{\mathrm{core}} (E), \quad \text{for all $Y \in \mathfrak X_{\mathrm{core}} (E)$}.
\]
We denote FWL vector fields by $\mathfrak X_{\mathrm{lin}} (E)$. They form a Lie-Rinehart subalgebra (over $C^\infty_{\mathrm{core}}(E)$) in $\mathfrak X_{\mathrm{poly}} (E)$.
If $(x^i, u^\alpha)$ are vector bundle coordinates, then a function $f \in C^\infty (E)$ is a core function if and only if $f = f(x)$ and it is a linear functions if and only if, locally, $f = f_\alpha (x) u^\alpha$. Similarly, a vector field $X \in \mathfrak X (E)$ is a core vector field if and only if, locally,
\[
X = X^\alpha (x) \frac{\partial}{\partial u^\alpha}
\]
and it is a linear vector field if and only if, locally,
\[
X = X^i (x) \frac{\partial}{\partial x^i} + X^\alpha_\beta (x) u^\beta \frac{\partial}{\partial u^\alpha}.
\]
\begin{remark}\label{rem:linear_tensor}
There is also a useful notion of \emph{FWL tensor} on $E$. Let $\mathcal T \in \Gamma (T^{\otimes r} E \otimes T^\ast{}^{\otimes s}E)$ be a tensor field of type $(r,s)$. Then $\mathcal T$ is a \emph{FWL tensor} if
\[
h_t^\ast \mathcal T = t^{1-r} \mathcal T
\]
for all $t \neq 0$. Notice that FWL tensors are called \emph{linear tensor fields} in \cite{BD2019}, where they are characterized in terms of the fiber-wise addition in $E$ (rather than via the fiber-wise multiplication $h$ as we do). As an instance, consider a metric $g \in \Gamma (S^2 T^\ast E)$. It is linear if $h_t^\ast g = t g$ for all $t$, and it is easy to see that this is in turn equivalent to $g$ being locally of the form
\[
g = g_{\alpha i} (x) du^\alpha \odot dx^i + g_{\alpha|ij} (x) u^\alpha dx^i \odot dx^j.
\]
Notice that the non-degeneracy condition then implies that the $x$-dependent matrix $\left( g_{\alpha i}\right)$ is invertible. In particular, the dimension of $M$ and the rank of $E$ must agree. Even more, denoting by $T^\pi E = \ker d\pi$ the $\pi$-vertical bundle, the composition
\begin{equation}\label{eq:comp}
E \overset{\cong}{\to} {T^\pi E|_M} \hookrightarrow T E|_M \overset{\flat}{\to} T^\ast E|_M \to T^\ast M
\end{equation}
is a vector bundle isomorphism. Here $\flat : TE \to T^\ast E$ is the musical isomorphism, the second and the fourth arrow are those induced by the canonical direct sum decomposition, $TE|_M = TM \oplus T^\pi E|_M$, and the first arrow is the canonical isomorphism. In other words, a non-degenerate symmetric covariant $2$-tensor $g$ can only exist on the total space of (a vector bundle isomorphic to) the cotangent bundle. Finally, in standard coordinates $(x^i, p_i)$ on $T^\ast M$, $g$ looks like
\begin{equation}\label{eq:g_linear}
g = dp_i \odot dx^i - \Gamma_{ij}^k(x)p_k dx^i \odot dx^j,
\end{equation}
for some appropriate local functions $\Gamma_{ij}^k (x)$. In particular, $g$ is necessarily of split signature. For more on FWL metrics see \cite{V20XX}.
\end{remark}
\section{More on FWL Multivector Fields}\label{sec:multivectors}
The material in this section is well-known to experts, and it is partly folklore, partly scattered in the literature. For this reason it is hard to give precise references (the reader may consult, e.g., \cite[Appendix A]{LV2019} and references therein, although that reference does not cover the same exact material as the following one). In any case, most of the proofs are straightforward and we omit them.
We will need to consider FWL symmetric multivectors. According to Remark \ref{rem:linear_tensor}, a $k$-multivector $P$ on the total space $E$ of a vector bundle $E \to M$ is \emph{FWL} if
\[
h_t^\ast (P) = t^{1-k} P
\]
for all $t \neq 0$. We denote by $\mathfrak X^\bullet_{\mathrm{sym}, \mathrm{lin}}(E)$
FWL symmetric
multivectors.
There is a useful characterization of a FWL $k$-multivectors. Namely, a $k$-multivector $P$ on $E$ is FWL if and only if
\begin{enumerate}
\item $P (f_1, \ldots, f_{k}) \in C^\infty_{\mathrm{lin}}(E)$,
\item $P (f_1, \ldots, f_{k-1}, h_1) \in C^\infty_{\mathrm{core}}(E)$,
\item $P(f_1, \ldots, f_{k-2}, h_1, h_2) = 0$,
\end{enumerate}
for all $f_i \in C^\infty_{\mathrm{lin}}(E)$, and all $h_j \in C^\infty_{\mathrm{core}}(E)$. In particular, a FWL symmetric $k$-multivector $P$ determines a pair of maps $(D_P, l_P)$:
\[
D_P : \underset{\text{$k$ times}}{\underbrace{\Gamma (E^\ast) \times \cdots \times \Gamma (E^\ast)}} \to \Gamma (E^\ast)
\]
and
\[
l_P : \underset{\text{$k-1$ times}}{\underbrace{\Gamma (E^\ast) \times \cdots \times \Gamma (E^\ast)}} \times C^\infty (M) \to C^\infty (M)
\]
via
\[
\begin{aligned}
\ell_{D_P (\varphi_1, \ldots, \varphi_k)} & = P (\ell_{\varphi_1}, \ldots, \ell_{\varphi_k})\\
l_P (\varphi_1, \ldots, \varphi_{k-1}, f) & = P (\ell_{\varphi_1}, \ldots, \ell_{\varphi_k}, f)
\end{aligned}
\]
for all $\varphi_i \in \Gamma (E^\ast)$, and all $f \in C^\infty (M)$. The maps $D_P, l_P$ satisfy the following properties:
\begin{enumerate}
\item $D_P$ is $\mathbb R$-multilinear and symmetric,
\item $l_P$ is $C^\infty (M)$-multilinear and symmetric in the first $(k-1)$-arguments,
\item $D_P (\varphi_1, \ldots, \varphi_{k-1}, f \varphi_k) = f D_P (\varphi_1, \ldots,\varphi_k) + l_P (\varphi_1, \ldots, \varphi_{k-1}, f) \varphi_k$, for $\varphi_i \in \Gamma (E^\ast)$, and $f \in C^\infty (M)$,
\item $l_P$ is a derivation in its last argument.
\end{enumerate}
In particular, $l_P$ can be seen as a vector bundle map $l_P : S^{k-1}E^\ast \to TM$, and we will often write
\[
l_P (\varphi_1, \ldots, \varphi_{k-1}) (f)
\]
instead of $l_P (\varphi_1, \ldots, \varphi_{k-1}, f)$. The assignment $P \mapsto (D_P, l_P)$ establishes a $C^\infty (M)$-linear bijection between FWL symmetric $k$-multivectors on $E$ and \emph{$k$-multiderivations} of $E^\ast$, i.e.~pairs $(D,l)$ consisting of a map $D : \Gamma (E^\ast) \times \cdots \times \Gamma (E^\ast) \to \Gamma (E^\ast)$ and a vector bundle map $l : S^{k-1} E^\ast \to TM$ (equivalently a section of $S^{k-1}E \otimes TM$) satisfying
\[
D (\varphi_1, \ldots, \varphi_{k-1}, f \varphi_k) = f D (\varphi_1, \ldots,\varphi_k) + l (\varphi_1, \ldots, \varphi_{k-1}) (f) \varphi_k.
\]
The map $l$ is sometimes called the \emph{symbol} of $D$ and it is completely determined by $D$. For this reason, we will often refer to $D$ itself as a $k$-multiderivation (see, e.g., \cite{GG2003, CM2008} for a skew-symmetric version of multiderivations).
Recall that there is a natural Poisson bracket $\{-,-\}$ on symmetric multivectors given by the following (Gerstenhaber-type) formula
\begin{equation}\label{eq:Poisson_multiv}
\begin{aligned}
& \{P_1, P_2\} (f_1, \ldots, f_{k_1 + k_2 +1}) \\
& = \sum_{\sigma \in S_{k_2+1, k_1}}P_1 \left( P_2 \left(f_{\sigma(1)}, \ldots, f_{\sigma(k_2+1)}\right), f_{\sigma(k_2+2)}, \ldots, f_{\sigma(k_1 + k_2 +1)}\right) \\
& \quad - \sum_{\sigma \in S_{k_1+1, k_2}}P_2 \left( P_1 \left(f_{\sigma(1)}, \ldots, f_{\sigma(k_1+1)}\right), f_{\sigma(k_1+2)}, \ldots, f_{\sigma(k_1 + k_2 +1)}\right)
\end{aligned}
\end{equation}
for all $(k_1+1)$-multivectors $P_1$, $(k_2+1)$-multivectors $P_2$, and all functions $f_i$, where $S_{k, h}$ denotes $(k, h)$-unshuffles. The Poisson bracket (\ref{eq:Poisson_multiv}) preserves FWL symmetric multivectors and the Poisson bracket $\{P_1, P_2\}$ of the FWL Poisson multivectors $P_1, P_2 \in \mathfrak X^{\bullet}_{\mathrm{sym}, \mathrm{lin}}(E)$ identifies with the obvious \emph{Gerstenhaber-like} bracket $\{ D_1, D_2\}$ of the associated multiderivations $D_1,D_2$.
FWL symmetric multivectors on $E$ do also identify with polynomial vector fields on $E^\ast$. To see this, it is useful to talk about \emph{core multivectors} first. A $k$-multivector $P$ on $E$ is \emph{core} if
\[
h_t^\ast (P) = t^{-k} P
\]
for all $t \neq 0$. Core $k$-multivectors can be characterized as those multivectors $P$ such that
\begin{enumerate}
\item $P (f_1, \ldots, f_{k}) \in C^\infty_{\mathrm{core}}(E) = C^\infty (M)$,
\item $P (f_1, \ldots, f_{k}, h) = 0$,
\end{enumerate}
for all $f_i \in C^\infty_{\mathrm{lin}} (E)$ and $h \in C^\infty_{\mathrm{core}} (E)$, and they form a subalgebra $\mathfrak X^\bullet_{\mathrm{sym}, \mathrm{core}}(E)$ in the associative, commutative algebra $\mathfrak X^\bullet_{\mathrm{sym}}(E)$ (with the symmetric product). More precisely, $\mathfrak X^\bullet_{\mathrm{sym}, \mathrm{core}}(E)$ is the subalgebra spanned by core functions and core vector fields. In particular, $\mathfrak X^\bullet_{\mathrm{sym}, \mathrm{core}}(E)$ identifies with sections $\Gamma (S^\bullet E)$ of the symmetric algebra of $E$ via
\[
(e_1)^\uparrow \odot \cdots \odot (e_k)^\uparrow \mapsto e_1 \odot \cdots \odot e_k.
\]
$e_i \in \Gamma (E)$. In its turn, $\Gamma (S^\bullet E)$ identifies with polynomial functions on $E^\ast$ in the via the (degree preserving) algebra isomorphism
\[
\Gamma (S^\bullet E) \to C^\infty_{\mathrm{poly}} (E^\ast), \quad e_1 \odot \cdots \odot e_q \mapsto \ell_{e_1} \cdots \ell_{e_q},
\]
and, in what follows, we will often understand the latter identifications. Notice that the resulting isomorphism
\[
\mathfrak X^\bullet_{\mathrm{sym}, \mathrm{core}}(E) \to C^\infty_{\mathrm{poly}}(E^\ast), \quad P \mapsto F_P
\]
is given by
\[
F_P (\varphi_x) = \frac{1}{k!}P (\ell_\varphi, \ldots, \ell_\varphi)(x)
\]
$P \in \mathfrak X^k_{\mathrm{sym}, \mathrm{core}}(E)$, $\varphi \in \Gamma (E^\ast)$ and $x \in M$.
We can now go back to FWL symmetric multivectors. The symmetric product of a core symmetric multivector and a FWL one is a FWL multivector, and this turns $\mathfrak X^\bullet_{\mathrm{sym},\mathrm{lin}}(E)$ into a $\Gamma (S^\bullet E)$-module. Now, let $P \in \mathfrak X^\bullet_{\mathrm{sym}, \mathrm{lin}}(E)$. It is easy to see that the Poisson bracket
$H_P = \{P, - \}$ preserves core multivectors. Hence, it is a derivation of the commutative algebra $\mathfrak X^\bullet_{\mathrm{sym}, \mathrm{core}}(E) \cong C^\infty_{\mathrm{poly}} (E^\ast)$. In its turn, $H_P$ extends uniquely to a polynomial vector field, also denoted $H_P$, on $E^\ast$. The assignment $P \mapsto H_P$ establishes a degree inverting isomorphism of Lie algebras, between the Lie algebra of linear symmetric multivectors on $E$ (with the Poisson bracket) and polynomial vector fields on $E^\ast$ (with the commutator). When we equip $\mathfrak X^\bullet_{\mathrm{sym}, \mathrm{lin}}(E)$ with the symmetric product by a core multivector, the latter isomorphism becomes an isomorphism of Lie-Rinehart algebras.
Finally, we remark that linear symmetric multivectors fit in the following short exact sequence
\begin{equation}\label{eq:ses_X_sym_lin}
0 \longrightarrow \Gamma (S^\bullet E \otimes E^\ast) \longrightarrow \mathfrak X^\bullet_{\mathrm{sym}, \mathrm{lin}} (E) \overset{l}{\longrightarrow} \Gamma (S^{\bullet-1} E \otimes TM)\longrightarrow 0
\end{equation}
where the second arrow identifies the section $e_1 \odot \cdots \odot e_k \otimes \varphi$ of $S^k E \otimes E^\ast$ with the FWL $k$-multivector field
$
\ell_{\varphi} e_1^\uparrow \odot \cdots \odot e_k^\uparrow.
$
\section{More on Derivations of a Vector Bundle}\label{sec:derivations}
In this section, for a vector bundle $V \to M$, we introduce a notion of (symmetric) \emph{$V$-multivector} (Definition \ref{def:V-multiv}). To the best of our knowledge this notion is new. It will play a significant role in the description of fiber-wise linear differential operators provided in Section \ref{sec:FWL_DO}. Symmetric $V$-multivectors are in many respect similar to plain symmetric multivectors, so the proofs of most of the statements in this section parallel the proofs of the analogous statements for multivectors and we omit them.
We begin with a vector bundle $E \to M$ and remark that the space $\mathfrak X^1_{\mathrm{lin}}(E) = \mathfrak X_{\mathrm{lin}}(E)$ of linear vector fields is of particular interest. The assignment $X \mapsto D_X$ establishes an isomorphism of Lie-Rinehart algebras (over $C^\infty (M)$) between linear vector fields on $E$ and \emph{derivations} of $E^\ast$, i.e.~$1$-multiderivations. We stress that
\[
X (\ell_\varphi) = \ell_{D_X \varphi}, \quad \varphi \in \Gamma (E^\ast).
\]
In the following, we denote by $\mathfrak D (V)$ the Lie-Rinehart algebra of derivations of a vector bundle $V$. It is the Lie-Rinehart algebra of sections of a Lie algebroid $DV \to M$ whose Lie bracket is the commutator of derivations and whose anchor is the symbol map $D \mapsto l_D$.
Notice also that the assignment $X \mapsto H_X$ (see the last paragraph of the previous section) does also establish an isomorphism of Lie-Rinehart algebras (over $C^\infty (M)$) between linear vector fields on $E$ and linear vector fields on $E^\ast$.
Accordingly we have a canonical Lie algebroid isomorphism $DE \mapsto DE^\ast$, $D \mapsto D^\ast$ which is explicitly given by
\[
\langle D^\ast \varphi, e \rangle = l_D \left(\langle \varphi, e \rangle \right) - \langle \varphi, D e \rangle,
\]
for every $\varphi \in \Gamma (E^\ast)$, $e \in\Gamma (E)$, where $\langle -,-\rangle : E^\ast \otimes E \to \mathbb R_M := M \times \mathbb R$ is the duality pairing. In the following we will simply denote by $D$ the derivation of $E^\ast$ induced by a derivation of $E$ (and vice-versa). It is easy to see that
\[
[X, e^\uparrow] = (D_X e)^\uparrow, \quad e \in \Gamma (E).
\]
More generally, a derivation $D$ of a vector bundle $V$ induces a derivation, also denoted $D$, in each component of the whole (symmetric, resp.~alternating) tensor algebra of $V \oplus V^\ast$. The latter derivation is defined imposing the obvious Leibniz rule with respect to the tensor product and the contraction by an element in the dual.
We are now ready to define the algebra of \emph{$V$-multivectors}, which is a ``derivation analogue'' of the Poisson algebra
of symmetric multivectors. So, let $V \to M$ be a vector
bundle, and consider the graded space $\tilde{\mathfrak D}{}^{\bullet} := \mathfrak
X_{\mathrm{sym}}^{\bullet-1} (M) \otimes \mathfrak D (V)$, where the tensor
product is over functions on $M$. Consider the graded subspace $\mathfrak
D^{\bullet}_{\mathrm{sym}} (V) \subset \tilde{\mathfrak D}{}^\bullet$ consisting of
elements projecting on symmetric multivectors $\mathfrak X_{\mathrm{sym}}
^{\bullet} (M) \hookrightarrow \mathfrak X_{\mathrm{sym}}^{\bullet -1} (M) \otimes
\mathfrak X (M)$ via
\[
\mathrm{id} \otimes l : \tilde{\mathfrak D}{}^\bullet \to \mathfrak X_{\mathrm{sym}}^\bullet (M) \otimes \mathfrak X (M).
\]
We denote by
\[
L : \mathfrak D_{\mathrm{sym}}^{\bullet}(V) \to \mathfrak X_{\mathrm{sym}}^\bullet (M)
\]
the projection. Notice that $\mathfrak D_{\mathrm{sym}}^{\bullet}(V)$ fits in an exact sequence:
\begin{equation}\label{eq:ses_V-derivations}
0 \longrightarrow \mathfrak X^{\bullet - 1}_{\mathrm{sym}} (M) \otimes \Gamma (\operatorname{End} V) \longrightarrow \mathfrak D_{\mathrm{sym}}^{\bullet}(V )\overset{L}{\longrightarrow} \mathfrak X^{\bullet }_{\mathrm{sym}} (M) \longrightarrow 0.
\end{equation}
\begin{definition}\label{def:V-multiv}
Elements in $\mathfrak D_{\mathrm{sym}}^{k}(V)$ are \emph{symmetric $k$-$V$-multivectors}.
\end{definition}
A symmetric $k$-$V$-multivector $D \in \mathfrak D_{\mathrm{sym}}^{k}(V)$ will be often interpreted as an operator
\[
D : C^\infty (M) \times \cdots C^\infty (M) \times \Gamma (V) \to \Gamma (V), \quad (f_1, \ldots, f_{k-1}, v) \mapsto D(f_1, \ldots, f_{k-1}|v).
\]
\begin{lemma}\label{lem:Poisson_D}
The space $\mathfrak D^{\bullet}_{\mathrm{sym}}(V)$ of symmetric $V$-multivectors is a Poisson algebra when equipped with
\begin{enumerate}
\item the associative product given by
\[
\begin{aligned}
& D_1 \cdot D_2 (f_1, \ldots, f_{k_1+ k_2 + 1}|v) \\
& = \sum_{\sigma \in S_{k_1 + 1, k_2}}L_{D_1}(f_{\sigma(1)}, \ldots, f_{\sigma(k_1 + 1)}) D_2(f_{\sigma(k_1+2)}, \ldots, f_{\sigma(k_1 + k_2 +1)}|v) \\
& \quad +
\sum_{\sigma \in S_{k_2 + 1, k_1 }}L_{D_2}(f_{\sigma(1)}, \ldots, f_{\sigma(k_2 + 1)}) D_1(f_{\sigma(k_2+2)}, \ldots, f_{\sigma(k_1 + k_2 +1)}|v)
\end{aligned}
\]
for all $f_i \in C^\infty (M)$, $v \in \Gamma (V)$; and
\item the Lie bracket $\{-,-\}$ given by
\begin{equation}\label{eq:Poisson_V-multivectors}
\{D_1, D_2\} = D_1 \bullet D_2 - D_2 \bullet D_1
\end{equation}
where
\[
D_1 \bullet D_2 : C^\infty (M) \times \cdots \times C^\infty (M) \times \Gamma (V) \to \Gamma(V)
\]
is the operator given by
\[
\begin{aligned}
& D_1 \bullet D_2 \left( f_1, \ldots, f_{k_1 + k_2} |v \right) \\
& =
\sum_{\sigma \in S_{k_1,k_2}} D_1 \left(f_{\sigma(1)}, \ldots, f_{\sigma(k_1)} | D_2 (f_{\sigma (k_1 + 1)}, \ldots, f_{\sigma (k_1 + k_2)}| v) \right) \\
& \quad + \sum_{\sigma \in S_{k_1-1,k_2 + 1}}D_1 \left(f_{\sigma(1)}, \ldots, f_{\sigma(k_1 - 1)}, L_{D_2} (f_{\sigma (k_1)}, \ldots, f_{\sigma (k_1 + k_2)})|v \right)
\end{aligned}
\]
for all $f_i \in C^\infty (M)$, and $v \in \Gamma (V)$.\end{enumerate}
Here $D_1 \in \mathfrak D_{\mathrm{sym}}^{k_1 + 1}(V)$ and $D_2 \in \mathfrak D_{\mathrm{sym}}^{k_2 + 1}(V)$, $D_1 \cdot D_2 \in \mathfrak D_{\mathrm{sym}}^{k_1 + k_2 + 2}(V)$, and $\{D_1, D_2\} \in \mathfrak D_{\mathrm{sym}}^{k_1 + k_2 + 1}(V)$.
The map
\[
L : \mathfrak D_{\mathrm{sym}}^{\bullet}(V) \to \mathfrak X_{\mathrm{sym}}^\bullet (M)
\]
is a surjective Poisson algebra map.
\end{lemma}
\begin{proof}
A long and tedious computation that we omit.
\end{proof}
\begin{remark}
When $V$ is a line bundle, the map $\mathfrak X^{\bullet - 1}_{\mathrm{sym}} (M) \otimes \Gamma (\operatorname{End} V) \to \mathfrak D_{\mathrm{sym}}^{\bullet}(V )$ embeds $\mathfrak X^{\bullet - 1}_{\mathrm{sym}} (M) = \mathfrak X^{\bullet - 1}_{\mathrm{sym}} (M) \otimes C^\infty (M) = \mathfrak X^{\bullet - 1}_{\mathrm{sym}} (M) \otimes \Gamma (\operatorname{End} V) $ into $\mathfrak D_{\mathrm{sym}}^{\bullet}(V)$ as an abelian subalgebra and an ideal.
\end{remark}
There is also a notion of FWL $V$-multivector. In order to discuss it, it is is useful to discuss derivations of pull-back vector bundles first. So, let $V$ be a vector bundle, and consider its pull-back $V_{\mathcal P} := \pi^\ast V$ along a surjective submersion $\pi: \mathcal P \to M$. Clearly, a derivation $D$ of $V_{\mathcal P}$ is completely determined by its symbol and its action on pull-back sections. The restriction $D_M := D|_{\Gamma (V)}$ of $D$ to pull-back sections is a \emph{derivation along $\pi$}, i.e.~it is an $\mathbb R$-linear map $D_M : \Gamma (V) \to \Gamma (V_{\mathcal P})$ and there exists a, necessarily unique, vector field along $\pi$, denoted $l_{D_M} \in \Gamma (\pi^\ast TM)$, fitting in the Leibniz rule
\[
D_M (fv) = \pi^\ast (f) D_M (v) + l_{D_M}(f) v, \quad f \in C^\infty (M), \quad v \in \Gamma (V).
\]
The correspondence $D \mapsto (l_D, D_M)$ establishes a $C^\infty(M)$-linear bijection between derivations $D$ of $V_{\mathcal P}$ and pairs $(X, D_M)$ consisting of a vector field $X \in \mathfrak X (\mathcal P)$ and a derivation along $\pi$ satisfying the following additional compatibility: $d \pi \circ X = l_{D_M}$. When $\mathcal P = E \to M$ is a vector bundle, it makes sense to talk about \emph{polynomial sections of $V_E$}. Namely, $\Gamma (V_E) = C^\infty (E) \otimes \Gamma (V)$, where the tensor product is over $C^\infty (M)$, and multiplicative reals act on $\Gamma (V_E)$ via their action on the first factor. As for functions, we denote by $h^\ast$ this action. A section $v$ of $\Gamma (V_E)$ is polynomial of degree $k$ if $h_t^\ast (v) = t^k v$ for all $t \in \mathbb R$, in other words $v \in C^\infty_{\mathrm{poly}}(E) \otimes \Gamma (V)$. Denote by $\Gamma (V_E)_k$ the space of polynomial sections of degree $k$, and by
\[
\Gamma_{\mathrm{poly}} (V_E) := \bigoplus_{k= 0}^\infty \Gamma (V_E)_k
\]
the space of all polynomial sections. \emph{FWL sections} are degree $1$ sections and they identify with sections of $E^\ast \otimes V$. \emph{Core sections} are degree $0$ sections and they identify simply with sections of $V$. Similarly to vector fields, a derivation $D$ of $V_{E}$ is \emph{polynomial} of degree $k$ if it maps polynomial sections of degree $h$ to polynomial sections of degree $k + h$, in other words
\[
h_t^\ast D = t^k D
\]
for all $t \neq 0$. Polynomial derivations of $V_{E}$ will be denoted $\mathfrak D_{\mathrm{poly}}(V_{E})$ and they correspond to pairs $(X, D_M)$ where $X \in \mathfrak X_{\mathrm{poly}}(E)$ and $D_M$ takes values in polynomial sections.
We can also consider polynomial symmetric $V_E$-multivectors. We will only need core and FWL ones. A symmetric $V_E$-$k$-multivector $D$ is \emph{FWL} (resp.~\emph{core}) if it is polynomial of degree $1-k$ (resp.~$-k$), i.e.
\[
h_t^\ast D = t^{1-k} D\quad \text{(resp.~$h_t^\ast D = t^{-k} D$)}
\]
for all $t \neq 0$. We denote by $\mathfrak D^\bullet_{\mathrm{sym}, \mathrm{lin}} (V_E)$ (resp.~$\mathfrak D^\bullet_{\mathrm{sym}, \mathrm{core}} (V_E))$ the space of FWL (resp.~core) symmetric $V_E$-multivectors. The Lie bracket $\{-,-\}$ on $V$-multivectors preserves FWL ones. Additionally, the projection $L : \mathfrak D^\bullet_{\mathrm{sym}} (V_E) \to \mathfrak X^\bullet_{\mathrm{sym}}(E)$ maps FWL $V$-multivectors to FWL multivectors and we get a short exact sequence of Lie algebras:
\begin{equation}\label{eq:ses_D_sym_lin}
0 \longrightarrow \mathfrak X^{\bullet - 1}_{\mathrm{sym}, \mathrm{lin}} (E) \longrightarrow \mathfrak D^\bullet_{\mathrm{sym}, \mathrm{lin}} (V_E) \overset{L}{\longrightarrow} \mathfrak X^\bullet_{\mathrm{sym}, \mathrm{lin}} (E) \longrightarrow 0.
\end{equation}
\begin{proposition}
A $k$-$V_E$-multivector $D$ is core if and only if
\begin{enumerate}
\item $D (f_1, \ldots, f_{k-1}| v) \in \Gamma_{\mathrm{core}}(V_E)$,
\item $D (f_1, \ldots, f_{k-1}| w) = 0$,
\item $D (f_1, \ldots, f_{k-2}, h | v) = 0$,
\end{enumerate}
for all $f_i \in C^\infty_{\mathrm{lin}} (E)$, $h \in C^\infty_{\mathrm{core}}(E)$, $v \in \Gamma_{\mathrm{lin}}(V_E) := \Gamma (V_E)_1$, and all $w \in \Gamma_{\mathrm{core}}(V_E):= \Gamma (V_E)_0 = \Gamma (V)$.
\end{proposition}
\begin{proof}
Straightforward.
\end{proof}
It easily follows from the above proposition that a core $k$-$V_E$-multivector is completely determined by its symbol. More precisely, the symbol map $D \mapsto L_D$ establishes a one-to-one correspondence between core $k$-$V_E$-multivectors and core multivectors. We conclude that $\mathfrak D^\bullet_{\mathrm{sym}, \mathrm{core}}(V_E) \cong \mathfrak X^\bullet_{\mathrm{sym}, \mathrm{core}}(E) \cong C^\infty_{\mathrm{poly}} (E^\ast)$.
\begin{proposition}
A symmetric $k$-$V_E$-multivector $D$ is FWL if and only if
\begin{enumerate}
\item $D (f_1, \ldots, f_{k-1}| v) \in \Gamma_{\mathrm{lin}}(V_E)$,
\item $D (f_1, \ldots, f_{k-1}| w) \in \Gamma_{\mathrm{core}}(V_E)$,
\item $D (f_1, \ldots, f_{k-2}, h_1 | v) \in \Gamma_{\mathrm{core}}(V_E)$,
\item $D(f_1, \ldots, f_{k-2}, h_{1} | w) = 0$,
\item $D(f_1, \ldots, f_{k-3}, h_{1}, h_2 | v) = 0$,
\end{enumerate}
for all $f_i \in C^\infty_{\mathrm{lin}} (E)$, $h_j \in C^\infty_{\mathrm{core}}(E)$, $v \in \Gamma_{\mathrm{lin}}(V_E) := \Gamma (V_E)_1$, and all $w \in \Gamma_{\mathrm{core}}(V_E):= \Gamma (V_E)_0 = \Gamma (V)$.
\end{proposition}
\begin{proof}
Straightforward.
\end{proof}
In particular, a FWL symmetric $k$-$V$-multivector $D$ determines a map:
\[
\Phi_D : \underset{\text{$k-1$ times}}{\underbrace{\Gamma (E^\ast) \times \cdots \times \Gamma (E^\ast)}} \to \mathfrak D (V)
\]
via
\[
\Phi_D (\varphi_1, \ldots, \varphi_{k-1})(w) = D (\ell_{\varphi_1}, \ldots, \ell_{\varphi_k} | w)
\]
for all $\varphi_i \in \Gamma (E^\ast)$, and all $w \in \Gamma (V)$. The map $\Phi_D$ is $C^\infty (M)$-multilinear and symmetric. Hence, it can be seen as a vector bundle map $\Phi_D : S^{k-1}E^\ast \to DV$, or, equivalently, as a section of $S^{k-1} E \otimes DV$.
\begin{proposition}\label{prop:V_E-multi_pairs}
The assignment $D \mapsto (L_D, \Phi_D)$ establishes a $C^\infty (M)$-linear bijections between FWL symmetric $V_E$-multivectors $D \in \mathfrak D^\bullet_{\mathrm{sym}, \mathrm{lin}} (V_E)$ and pairs $(P, \Phi)$ consisting of a FWL symmetric multivector $P \in \mathfrak X^\bullet_{\mathrm{sym}, \mathrm{lin}} (E)$ and a vector bundle map $\Phi : S^{k-1}E^\ast \to DV$ such that $l_P = l \circ \Phi$.
\end{proposition}
\begin{proof}
Easy and left to the reader.
\end{proof}
According to Proposition \ref{prop:V_E-multi_pairs} we will sometimes call the pair $(L_D, \Phi_D)$ itself a FWL symmetric $V_E$-multivector.
Now, we can combine the exact sequences (\ref{eq:ses_X_sym_lin}) and (\ref{eq:ses_D_sym_lin}) in one exact commutative diagram:
\[
\begin{array}{c}
\xymatrix{ & 0 & 0 & 0 & \\
0 \ar[r] & \Gamma (S^{\bullet -1} E \otimes \operatorname{End} V) \ar[u] \ar[r] & \Gamma (S^{\bullet-1} E \otimes DV) \ar[r]^-{\operatorname{id} \otimes l} \ar[u] & \Gamma (S^{\bullet-1} E \otimes TM) \ar[r] \ar[u] & 0 \\
0 \ar[r] & \Gamma (S^{\bullet -1} E \otimes \operatorname{End} V) \ar@{=}[u] \ar[r] & \mathfrak D^\bullet_{\mathrm{sym}, \mathrm{lin}}(V_E) \ar[u]^\Phi \ar[r]^-L & \mathfrak X_{\mathrm{sym}, \mathrm{lin}}^{\bullet}(E) \ar[u]^l \ar[r] & 0 \\
& 0 \ar[u] \ar[r] & \Gamma (S^{\bullet}E \otimes E^\ast) \ar[u]^-I \ar@{=}[r]
& \Gamma (S^{\bullet}E \otimes E^\ast) \ar[u] \ar[r] & 0 \\
& & 0 \ar[u] & 0 \ar[u] &
}
\end{array}.
\]
We only need to explain the map $I$. To do that, we first remark that $\pi$-vertical vector fields act naturally on sections of $V_E$, via
\[
X (f \otimes v) = X(f) \otimes v
\]
for all $X \in \Gamma (T^\pi E)$, $f \in C^\infty (E)$, and $v \in \Gamma(V)$. Now, take $e_1, \ldots, e_k \in \Gamma (E)$ and $\varphi \in \Gamma (E^\ast)$. Then
\[
I (e_1 \odot \cdots \odot e_k \otimes \varphi)(f_1, \ldots, f_{k-1} | v) = \frac{1}{k!} \cdot \ell_\varphi \sum_{\sigma \in S_{k}} e_{\sigma(1)}^\uparrow (f_1) \cdots e_{\sigma(k-1)}^\uparrow (f_{k-1}) e_{\sigma(k)}^\uparrow (v).
\]
Equivalently, we can interpret $e_1 \odot \cdots \odot e_k$ as a core $V_E$-multivector, via the isomorphism $D^\bullet_{\mathrm{sym}, \mathrm{core}}(V_E) \cong \Gamma (S^\bullet E)$, and then multiply by the FWL function $\ell_\varphi$ to get a FWL $V_E$-multivector.
We conclude this section showing that FWL symmetric $V_E$-multivectors do also identify with polynomial derivations of $V_{E^\ast}$. This is an easy consequence (among other things) of Proposition \ref{prop:V_E-multi_pairs}. Indeed, take $D \in \mathfrak D^\bullet_{\mathrm{sym}, \mathrm{lin}}(V_E)$ and let $(L_D, \Phi_D)$ be the corresponding pair. Denote by $\pi : E^\ast \to M$ the projection. We claim that $\Phi_D$ can be seen as a derivation along $\pi$. Indeed $\Phi_D$ is a section of $S^{\bullet -1} E \otimes DL$ and, by acting on a section $v \in \Gamma (V)$ with the $DL$-factor, we get a section $\Phi_D(v)$ of $S^{\bullet -1} E \otimes V$, i.e.~a polynomial section of $\Gamma (V_{E^\ast})$. In the following, we use this construction to interpret $\Phi_D$ as a derivation along $\pi$. If we do so, the pair $(H_{L_D}, \Phi_D)$ consists of a vector field on $E^\ast$, and a derivation $\Phi_D$ along $\pi$, with the additional property that $d \pi \circ H_{L_D} = l \circ \Phi_D$, hence it corresponds to a (polynomial) derivation $D^\ast$ of the pull-back vector bundle $V_{E^\ast}$. Finally a tedious, but straightforward computation shows that the bijection $D \mapsto D^\ast$ between linear $V_E$-multivectors and polynomial derivations of $V_{E^\ast}$ obtained in this way do also preserve the Lie algebra structures. When we equip $\mathfrak D^\bullet_{\mathrm{sym}, \mathrm{lin}}(E)$ with the product by a core $V_E$-multivector, the latter bijection becomes an isomorphism of Lie-Rinehart algebras. We have thus proved the main result in this section:
\begin{theorem}
Let $E \to M$ and $V \to M$ be vector bundles. The assignment $D \mapsto D^\ast$ establishes a degree inverting isomorphism of Lie-Rinehart algebras over $\mathfrak D^\bullet_{\mathrm{sym}, \mathrm{core}}(E) \cong C^\infty_{\mathrm{poly}} (E^\ast)$ between linear $V_E$-multivectors and polynomial derivations of $V_{E^\ast}$.
\end{theorem}
\section{Differential Operators and Their Symbols}\label{sec:DO}
We finally come to the object of our primary interest: differential operators. This sections is a super-short review of the subject.
Let $V,W \to M$ be vector bundles. A (linear) \emph{differential operator (DO in the following) of order $q$ from $V$ to $W$} is an $\mathbb R$-linear map
\[
\Delta : \Gamma (V) \to \Gamma (W)
\]
such that
\[
[\cdots[[\Delta, f_0], f_1], \cdots, f_q] = 0
\]
for all $f_i \in C^\infty (M)$. In particular, DO of order zero are just vector bundle maps $V \to W$. We denote by $DO_q (V, W)$ the space of order $q$ DOs from $V$ to $W$. Clearly, a DO of order $q$ is also a DO of order $q +1$, and we get the filtration
\[
DO_0 (V, W) = \Gamma (\operatorname{Hom}(V,W)) \subset DO_1 (V, W) \subset \cdots \subset DO_q(V, W) \subset
\]
The union of all $DO_q (V,W)$ will be denoted simply by $DO (V,W)$. A \emph{scalar DO} on $M$ is a DO acting on functions over $M$, i.e.~a DO from the trivial line bundle $\mathbb R_M := M \times \mathbb R$ to itself. We use the symbol $DO_{q}(\mathbb R_E)$ (instead of $DO_{q}(\mathbb R_M, \mathbb R_M)$) for scalar DOs.
The composition of an order $q$ and an order $r$ DO is an order $q + r$ DO. In particular, for all $q$, $DO_q (V,W)$ is a $C^\infty (M)$-module in two different ways: via composition on the left and composition on the right with a function on $M$ (seen as an order $0$ DO). We will consider the first module structure unless otherwise stated. The space $DO(\mathbb R_M)$ is a filtered non-commutative algebra with the composition. It is actually the universal enveloping algebra of the tangent Lie algebroid $TM \to M$. Being an associative algebra, $DO (\mathbb R_M)$ is also a Lie algebra with the commutator. Notice that the commutator of an order $q$ and an order $r$ scalar DO is an order $q + r -1$ scalar DO.
Given an order $q$ DO $\Delta : \Gamma (V) \to \Gamma (W)$ from $V$ to $W$, and functions $f_1, \ldots, f_q$, the nested commutator
\[
[\cdots[\Delta, f_1], \cdots, f_q]
\]
is an order $0$ DO. Additionally, it is a derivation in each of the arguments $f_i$ and it is symmetric in those argument. In this way, we get a map
\[
\sigma : DO_q (V,W) \mapsto \mathfrak \Gamma \left(S^q TM \otimes \operatorname{Hom}(V,W)\right), \quad \Delta \mapsto \sigma (\Delta)
\]
with
\[
\sigma (\Delta) (f_1, \ldots, f_q) = [\cdots[\Delta, f_1], \cdots, f_q].
\]
The map $\sigma$ is called the \emph{symbol} and it fits in a short exact sequence of $C^\infty (M)$-modules
\begin{equation}\label{eq:ses_symbol}
0 \longrightarrow DO_{q-1}(V, W) \longrightarrow DO_{q}(V, W) \overset{\sigma}{\longrightarrow} \Gamma \left(S^q TM \otimes \operatorname{Hom}(V,W)\right) \longrightarrow 0
\end{equation}
where the second arrow is the inclusion.
\begin{example}
Vector fields are first order scalar DOs. Derivations of the vector bundle $V$ are first order DOs $D$ from $V$ to itself such that $\sigma (D)$ belongs to $\mathfrak X (M) \subset \Gamma (TM \otimes \operatorname{End} V)$. Additionally we have $\sigma (D) = l_D$.
\end{example}
For scalar DOs the short exact sequence (\ref{eq:ses_symbol}) becomes
\[
0 \longrightarrow DO_{q-1}(\mathbb R_M) \longrightarrow DO_{q}(\mathbb R_M) \overset{\sigma}{\longrightarrow} \mathfrak X^q_{\mathrm{sym}}(M) \longrightarrow 0.
\]
The symbol of scalar DOs intertwines the commutator with the Poisson bracket (of symmetric multivectors) in the sense that
\[
\sigma \big([\Delta', \Delta]\big) = \big\{ \sigma(\Delta), \sigma (\Delta')\big\}
\]
whenever $\Delta \in DO_q(\mathbb R_M)$ and $\Delta' \in DO_{q'}(\mathbb R_M)$, in which case we take $[\Delta, \Delta'] \in DO_{q+q'-1}(\mathbb R_M)$.
We conclude this short review section commenting briefly on the coordinate description of (scalar) DOs. To do this we first fix our conventions on the multi-index notation for multiple partial derivatives. Let $(x^i)$, $i = 1, \ldots, n$ be variables. A length $k$ multi-index $I$ is a word $I = i_1 \ldots i_k$, with $i_j =1, \ldots, n$, where words are considered modulo permutations of their letters. The length $k$ of a multi-index $I = i_1 \cdots i_k$ is also denoted $|I|$. Words can be composed by concatenation and we also consider the empty multi-index $\varnothing$. If we do so, then multi-indexes are elements in the free abelian monoid spanned by $1, \ldots, n$. The lenght is then a monoid homomorphism. A lenght $k$ multi-index $I = i_1 \ldots i_k$, determines an order $k$ DO
\[
\frac{\partial^{|I|}}{\partial x^I} := \frac{\partial^k}{\partial x^{i_1} \cdots \partial x^{i_k}}.
\]
Now, we go back to manifolds $M$ (and vector bundles over them). Actually, $DO_q (\mathbb R_M)$ (likewise $DO_q (V,W)$) is the $C^\infty (M)$-module of sections of a vector bundle over $M$. If $(x^i)$ are coordinates on $M$, then $DO_q (\mathbb R_M)$ is spanned locally (in the corresponding coordinate neighborhood) by
\[
\frac{\partial^{|I|}}{\partial x^I} , \quad |I| = 0, 1, \ldots, q.
\]
More precisely, locally, every DO $\Delta \in DO_q (\mathbb R_M)$ can be uniquely written in the form
\begin{equation}\label{eq:Delta_loc}
\Delta =\sum_{|I| \leq q} \Delta^I (x) \frac{\partial^{|I|}}{\partial x^I}
\end{equation}
where the $ \Delta^I (x)$ are local functions on $M$. The $\Delta^I (x)$ can be recovered via formulas
\begin{equation}\label{eq:Delta^I}
\Delta^{i_1\cdots i_k} (x) = \frac{1}{(i_1 \cdots i_k)!} [\cdots[\Delta, x^{i_1}], \cdots, x^{i_k}] (1), \quad k = 0, 1, \ldots, q,
\end{equation}
where, for a multi-index $I$, we denoted by $I!$ the product $I[1]! \cdots I[n]!$ where $I[i]$ is the number of times the letter $i$ occurs in $I$.
Finally, if $\Delta$ is an order $q$ scalar DO locally given by (\ref{eq:Delta_loc}), then its symbol $\sigma (\Delta)$ is locally given by
\[
\sigma (\Delta) = \frac{1}{q!} \Delta^{i_1 \cdots i_q} \frac{\partial}{\partial x^{i_1}} \odot \cdots \odot \frac{\partial}{\partial x^{i_q}}.
\]
\section{Core and Fiber-wise Linear Differential Operators}\label{sec:FWL_DO}
This is the main section of the paper. We propose a notion of \emph{FWL} (scalar) \emph{DO} on the total space of a vector bundle. Our definition is partly motivated by the fact that the symbol of a FWL DO is a FWL multivector. It is also motivated by the \emph{linearization construction} discussed in the next section. Yet another motivating little fact is that the Laplacian of a FWL metric is a FWL DO (Example \ref{ex:g_linear}).
Let $E \to M$ be a vector bundle. We have learnt from Sections \ref{sec:vector_fields}, \ref{sec:multivectors} and \ref{sec:derivations} that, given a type $\mathfrak T$ of geometric structures on manifolds (functions, vector fields, tensors, etc.) appropriate notions of \emph{core} and \emph{FWL} structures of the type $\mathfrak T$ on $E$ exist, and these notions can be identified by means of the following recipe: 1) notice that the space $\mathfrak T(E)$ of structures of type $\mathfrak T$ on $E$ is naturally graded (via the action of multiplicative reals on $E$ by fiber-wise scalar multiplication), 2) identify the smallest degree $k$ for which the degree $k$ homogeneous component $\mathfrak T (E)_k$ of $\mathfrak T (E)$ is non-trivial, and 3) put $\mathfrak T_{\mathrm{core}}(E) = \mathfrak T (E)_k$ and $\mathfrak T_{\mathrm{lin}}(E) = \mathfrak T (E)_{k+1}$. A quick check shows that this recipe cooks up the required definitions in all the cases considered so far. Notice that we could make this recipe much more rigorous adopting for the rather vague ``geometric structure of type $\mathfrak T$'' the very precise notion of \emph{natural vector bundle $\mathfrak T$}, but we will not need this level of abstraction.
We adopt the strategy described above to define core and FWL DOs on $E$. Consider the non-commutative algebra $DO(\mathbb R_E)$ of scalar DOs $\Delta : C^\infty (E) \to C^\infty (E)$. We begin noticing that, for each $q$, the space $DO_q (\mathbb R_E)$ of DOs $\Delta : C^\infty (E) \to C^\infty (E)$ of order $q$ is naturally graded:
\[
DO_q (\mathbb R_E) = \bigoplus_{k = -q}^\infty DO_q (\mathbb R_E)_k
\]
where $DO_q (\mathbb R_E)_k$ consists of degree $k$ DOs (of order $q$), i.e.~DOs $\Delta$ such that
\[
h_t^\ast (\Delta) = t^{k} \Delta
\]
for all $t \neq 0$. The smallest degree $k$ for which $DO_q (\mathbb R_E)_k$ is non-trivial is $k = -q$. So, following our recipe, we put
\[
DO_{q, \mathrm{core}} (E) := DO_q (\mathbb R_E)_{-q},
\]
and call them \emph{core DOs}. We also put
\begin{equation}\label{eq:loc_DO_core}
DO_{\mathrm{core}}(E) := \bigoplus_{q} DO_{q, \mathrm{core}} (E).
\end{equation}
Let $(x^i, u^\alpha)$ be vector bundle coordinates on $E$, and let $(x^i, u_\alpha)$ be dual coordinates on $E^\ast$. A DO $F \in DO_q (\mathbb R_E)$ is a core DO if and only if, locally,
\begin{equation}\label{eq:core_DO}
F = \sum_{|A| = q}F^A (x) \frac{\partial^{|A|}}{\partial u^A},
\end{equation}
where $A = \alpha_1 \cdots \alpha_q$ is a lenght $q$ multi-index.
It follows from (\ref{eq:core_DO}) that $DO_{\mathrm{core}}(E) \subset DO (\mathbb R_E)$ is the subalgebra spanned by core functions $C^\infty_{\mathrm{core}}(M)$ and core vector fields $\mathfrak X_{\mathrm{core}}(E)$. Equivalently, it is the universal enveloping algebra of the abelian Lie algebroid $E \Rightarrow M$. Because of the latter description, there is an algebra isomorphism
\[
\Gamma (S^\bullet E) \to DO_{\mathrm{core}}(E),
\]
mapping a monomial
\[
e_1 \odot \cdots \odot e_q, \quad e_i \in \Gamma (E),
\]
to the DO
\[
e_1^\uparrow \circ \cdots \circ e_q^\uparrow.
\]
In its turn, as already mentioned, $\Gamma (S^\bullet E)$ identifies with polynomial functions on $E^\ast$.
In the following we will often identify $DO_{\mathrm{core}}(E)$ with both $\Gamma (S^\bullet E)$ and $C^\infty_{\mathrm{poly}} (E^\ast)$ via the latter isomorphisms. If $F \in DO_{q, \mathrm{core}}(E)$ is locally given by (\ref{eq:loc_DO_core}), then it identifies with
\[
\frac{1}{q!}F^{\alpha_1 \cdots \alpha_q} (x) u_{\alpha_1} \odot \cdots \odot u_{\alpha_q} \in \Gamma (S^q E),
\]
and
\[
\sum_{|A| = q}F^A (x) u_A = \frac{1}{q!}F^{\alpha_1 \cdots \alpha_q} (x) u_{\alpha_1} \cdots u_{\alpha_q}\in C^\infty (E^\ast)_q,
\]
where, for $A = \alpha_1 \cdots \alpha_q$, we denoted by $u_A$ the monomial $u_{\alpha_1} \cdots u_{\alpha_q}$.
We will always consider $DO (\mathbb R_E)$ as a $DO_{\mathrm{core}}(E)$-module with the scalar multiplication given by the \emph{left} composition.
We now pass to \emph{FWL DOs}. Following our recipe again, for each $q$ we put
\[
DO_{q, \mathrm{lin}}(E) := DO_q(\mathbb R_E)_{-q+1}.
\]
\begin{definition}\label{def:FWL_DO}
DOs in $DO_{q, \mathrm{lin}}(E)$ are called \emph{fiber-wise linear differential operators} (FWL DOs) of order $q$.
\end{definition}
For instance, $DO_{0, \mathrm{lin}}(E) = C^\infty_{\mathrm{lin}}(E)$, and $DO_{1, \mathrm{lin}}(E) = \mathfrak X_{\mathrm{lin}}(E) \oplus C^\infty (M)$. It is also clear that $DO_{q, \mathrm{lin}}(E) \supset DO_{q-1, \mathrm{core}}$ for all $q$. More precisely
\[
DO_{q-1, \mathrm{core}} = DO_{q, \mathrm{lin}}(E) \cap DO_{q-1}(\mathbb R_E).
\]
We put
\[
DO_{\mathrm{lin}}(E) := \bigoplus_{q} DO_q(\mathbb R_E)_{-q+1}.
\]
Clearly $DO_{\mathrm{core}}(E) \subseteq DO_{\mathrm{lin}}(E) \subseteq \subseteq DO (\mathbb R_E)$.
A DO $\Delta \in DO_{q}(\mathbb R_E)$ is FWL if and only if, in vector bundle coordinates, it looks like
\begin{equation}\label{eq:loc_DO_stab}
\Delta = \sum_{|A| = q -1} \Delta^{i|A}(x) \frac{\partial^{|A| + 1}}{\partial x^i \partial u^A} + \sum_{|B| = q} \Delta^B_\alpha (x) u^\alpha \frac{\partial^{|B|}}{\partial u^B}
+ \sum_{|C| = q-1} \Delta^C (x) \frac{\partial^{|C|}}{\partial u^C}.
\end{equation}
It is easy to see from this formula that $DO_{\mathrm{lin}}(E) \subset DO (\mathbb R_E)$ is the $DO_{\mathrm{core}}(E)$-submodule spanned by $1$, $C^\infty_{\mathrm{lin}} (E)$ and $\mathfrak X_{\mathrm{lin}}(E)$.
\begin{example}\label{ex:g_linear}
Let $g$ be a metric on $E$, and assume it is FWL. Then, the associated Laplacian operator $\Delta_g : C^\infty (E) \to C^\infty (E)$ is a FWL DO operator (of order $2$). One can see this working in vector bundle coordinates. But there is also a (basically) coordinate free proof that we now illustrate. First of all, from $g$ being FWL, it immediately follows that the inverse tensor $g^{-1}$ is FWL as well. Now, the covariant derivative $\nabla \theta $ of a $1$-form $\theta$ along the Levi-Civita connection $\nabla$ is the covariant $2$-tensor given by the formula:
\begin{equation}\label{eq:cov_der}
\nabla \theta = \frac{1}{2} \left(d \theta + \mathcal L_{\sharp (\theta)} g \right),
\end{equation}
where $\sharp : T^\ast E \to TE$ is the musical isomorphism. Equivalently, the covariant derivative $\nabla_X Y$ of a vector field $Y$ along another vector field $X$ is the vector field $\nabla_X Y$ that acts on functions $f \in C^\infty (M)$ as follows:
\begin{equation}\label{eq:cov_der_2}
\nabla_X Y (f) = X(Y(f)) - \frac{1}{2}(\mathcal L_{\operatorname{grad} f} g)(X,Y)
\end{equation}
where $\operatorname{grad} f = \sharp (df)$ is the \emph{gradient} of $f$. Using (\ref{eq:cov_der}) (or (\ref{eq:cov_der_2})) and the naturality of both the de Rham differential and the Lie derivative, it is easy to see that the covariant derivative of arbitrary tensor fields commutes with the pull back along $h_t$ for all $t \neq 0$. As the Laplacian $\Delta_g f$ of a function $f$ is obtained by contracting the covariant derivative of $df$ with $g^{-1}$, then $\Delta_g$ decreases by one the degree of a homogeneous (fiber-wise polynomial) function. So it is a second order $DO$ of degree $1 - 2 = -1$, i.e.~a FWL DO of order 2, as claimed. It might be also interesting to remark that the Levi-Civita connection of a FWL metric is a FWL connection according to a definition introduced in \cite{PSV2020}.
\end{example}
\begin{lemma}\label{lem:stabilizer}
The space $DO_{\mathrm{lin}}(E)$ of linear DOs is the stabilizer Lie subalgebra of $DO_{\mathrm{core}}(E)$, i.e.~a DO $\Delta \in DO (\mathbb R_E)$ is in $DO_{\mathrm{lin}}(E)$ if and only if $[\Delta, F] \in DO_{\mathrm{core}}(E)$ for all $F \in DO_{\mathrm{core}}(E)$.
\end{lemma}
\begin{proof}
The ``only if part'' of the statement immediately follows from an obvious order/degree argument. For the ``if part'', consider a DO $\Delta$ of order $r$. Locally,
\[
\Delta = \sum_{|I| + |A| \leq r} \Delta^{I|A} (x, u)\frac{\partial^{|I| + |A|}}{\partial x^I \partial u^A}.
\]
Assume that $\Delta$ is in the stabilizer of $DO_{\mathrm{core}}(E)$. We want to show that $\Delta$ is the sum of operators of the form (\ref{eq:loc_DO_stab}) (with possibly varying $q \leq r$). As $x^i$ is a core function for all $i$, the commutator $[\Delta, x^i]$ is a core DO. But
\[
[\Delta, x^i] = \sum_{|J| + |A| \leq r - 1} \left(J[i] +1 \right)\Delta^{Ji|A} (x, u)\frac{\partial^{|J| + |A| - 1}}{\partial x^J \partial u^A},
\]
so it can only be a core DO for all $i$ if 1) $\Delta^{I|A}(x,u) = 0$ for $|I| > 1$, and 2) $\Delta^{i|A}(x,u) = \Delta^{i|A}(x)$. In other words, $\Delta$ is necessarily of the form
\begin{equation}\label{eq:partial_form}
\Delta = \sum_{|A| \leq r -1} \Delta^{i|A}(x) \frac{\partial^{|A| + 1}}{\partial x^i \partial u^A} + \sum_{|B| \leq k} \Delta^B(x,u) \frac{\partial^{|B|}}{\partial u^B}.
\end{equation}
and, to conclude, it is enough to prove that $\Delta^B(x,u)$ is of the form $\Delta^B(x,u) = \Delta^B_\alpha (x) u^\alpha + \Delta^B (x)$. To do this, recall that $\frac{\partial}{\partial u^\alpha}$ is a core vector field for all $\alpha$, hence $\left[\Delta, \frac{\partial}{\partial u^\alpha}\right]$ is a core DO. But, from (\ref{eq:partial_form}),
\[
\left[\Delta, \frac{\partial}{\partial u^\alpha}\right] = - \frac{\partial \Delta^B (x,u)}{\partial u^\alpha} \frac{\partial^{|B|}}{\partial u^B},
\]
which is a core DO for all $\alpha$ if and only if
\[
\frac{\partial^2 \Delta^B (x,u)}{\partial u^\alpha \partial u^\beta} = 0
\]
for all $\alpha, \beta$, i.e.~$\Delta^B(x,u)$ is a (non-necessarily homogeneous) first order polynomial in the variables $u$, as desired.
\end{proof}
It follows from Lemma \ref{lem:stabilizer} that $DO_{\mathrm{lin}}(E)$ is a Lie subalgebra in $DO (\mathbb R_E)$. As already mentioned, it is also a $DO_{\mathrm{core}}$-submodule. Actually, it is a Lie-Rinehart algebra over $DO_{\mathrm{core}}(E)$, the anchor being the adjoint operator $ad : \Delta \mapsto ad(\Delta):= [\Delta, -]$. To see this, first notice that $ad(\Delta)$ is indeed a well-defined derivation of $DO_{\mathrm{core}}(E)$ for all $\Delta \in DO_{\mathrm{lin}}(E)$. Now, take $\Delta, \Delta' \in DO_{\mathrm{lin}}(E)$ and $F, F' \in DO_{\mathrm{core}} (E)$, and compute
\[
\begin{aligned}
{}[\Delta, F \circ \Delta'] & = ad(\Delta)(F) \circ \Delta' + F \circ [\Delta, \Delta'], \\
ad(F \circ \Delta)(F') & = [F \circ \Delta, F'] = F \circ [\Delta, F'] = F \circ ad(\Delta)(F').
\end{aligned}
\]
Finally, from $DO_{\mathrm{core}}(E) \cong C^\infty_{\mathrm{poly}}(E^\ast)$, we see that, for every $\Delta \in DO_{\mathrm{lin}}(E)$, the derivation $ad(\Delta)$ determines a polynomial vector field (of the same degree) on $E^\ast$, also denoted $ad(\Delta)$.
\begin{theorem}\label{theor:stabilizer}
The sequence of Lie-Rinehart algebras
\begin{equation}\label{eq:ex_seq}
0 \longrightarrow DO_{\mathrm{core}}(E) \longrightarrow DO_{\mathrm{lin}}(E) \overset{ad}{\longrightarrow} \mathfrak X_{\mathrm{poly}} (E^\ast) \longrightarrow 0
\end{equation}
is exact.
\end{theorem}
\begin{proof}
First of all, as already remarked, $DO_{\mathrm{core}}(E)$ is in $DO_{\mathrm{lin}}(E)$. Even more, as it is an abelian subalgebra in $DO (\mathbb R_E)$, then it is actually in the kernel of $ad : DO_{\mathrm{lin}}(E) \to \mathfrak X_{\mathrm{poly}}(E^\ast)$. To see that core DOs exhaust the kernel of $ad$ (i.e.~$DO_{\mathrm{core}}(E)$ is its own centralizer), assume that $[\Delta, F] = 0$ for all $F \in DO_{\mathrm{core}}(E)$. Then, exactly the same computation as in the proof of Lemma \ref{lem:stabilizer} shows that $\Delta$ is locally of the form (\ref{eq:loc_DO_stab}) with $\Delta^{i|A} (x) = \Delta^B_{\alpha}(x) = 0$, i.e.~$\Delta \in DO_{\mathrm{core}}(E)$. For the exactness of the sequence (\ref{eq:ex_seq}) it remains to show that the map $ad : DO_{\mathrm{lin}}(E) \to \mathfrak X_{\mathrm{poly}}(E^\ast)$ is surjective. To do that, we work in local coordinates again. So, let $(x^i, u^\alpha)$ be vector bundle coordinates on $E$, and let $(x^i, u_\alpha)$ be dual coordinates on $E^\ast$. It is not hard to see that, if $\Delta$ is locally given by (\ref{eq:loc_DO_stab}),
then the vector field $ad(\Delta)$ is locally given by
\begin{equation}\label{eq:ad_Delta}
ad(\Delta) = \sum_{|A| = q -1} \Delta^{i|A}(x) u_A\frac{\partial}{\partial x^i} - \sum_{|B| = q} \Delta^B_\alpha (x)u_B \frac{\partial}{\partial u_\alpha},
\end{equation}
where, for a multi-index $A = \alpha_1 \cdots \alpha_{s}$, we denoted by $u_A$ the monomial $u_{\alpha_1} \cdots u_{\alpha_s}$ ($s =q, q-1$). As (\ref{eq:ad_Delta}) is the local expression of a generic homogeneous polynomial vector field of degree $q-1$, we are done.
\end{proof}
Our next aim is proving that the Lie-Rinehart algebra $DO_{\mathrm{lin}}(E)$ is canonically isomorphic to the Lie-Rinehart algebra of polynomial derivations of an appropriate line bundle on $E^\ast$. We begin with a simple
\begin{proposition}\label{prop:symbol}
The symbol $\sigma (\Delta)$ of a FWL DO $\Delta \in DO_{q, \mathrm{lin}}(E)$ is a FWL symmetric $q$-multivector field. Every FWL symmetric $q$-multivector field is the symbol of an order $q$ FWL DO.
\end{proposition}
\begin{proof}
The statement immediately follows from (\ref{eq:loc_DO_stab}) and the easy fact that a symmetric $q$-multivector $P$ is FWL if and only if, in vector bundle coordinates, it is of the form
\[
P = P^{i|\alpha_1 \cdots \alpha_{q-1}}(x) \frac{\partial}{\partial x^i} \odot \frac{\partial}{\partial u^{\alpha_1}} \odot \cdots \odot \frac{\partial}{\partial u^{\alpha_{q-1}}} + P^{\beta_1 \cdots \beta_q}_\alpha (x) u^\alpha \frac{\partial}{\partial u^{\beta_1}} \odot \cdots \odot \frac{\partial}{\partial u^{\beta_{q}}}.
\]
\end{proof}
Now let $\Delta \in DO_{q, \mathrm{lin}}(E)$. Notice that the adjoint operator $ad (\Delta)$, seen as a polynomial vector field on $E^\ast$, corresponds exactly to the symbol $\sigma (\Delta)$ via the isomorphism $\mathfrak X^q_{\mathrm{sym}, \mathrm{lin}}(E) \cong \mathfrak X (E^\ast)_{q-1}$. It is also clear that, in view of its coordinate form (\ref{eq:loc_DO_stab}), $\Delta$ is completely determined by $\sigma (\Delta)$ or, equivalently, $ad(\Delta)$, together with the map
\[
\Psi_\Delta : \underset{\text{$q-1$ times}}{\underbrace{\Gamma (E^\ast) \times \cdots \times \Gamma (E^\ast)}} \to C^\infty (M), \quad (\varphi_1, \ldots, \varphi_{q-1}) \mapsto [ \cdots [\Delta, \ell_{\varphi_1}], \cdots, \ell_{\varphi_{q-1}}](1).
\]
The map $\Psi_\Delta$ is clearly well-defined. Additionally, it enjoys the following properties
\begin{enumerate}
\item $\Psi_\Delta$ is symmetric,
\item $\Psi_\Delta$ is a first order DO in each entry.
\end{enumerate}
More precisely, we have the following
\begin{lemma}
The map $\Psi_\Delta$ satisfies
\[
\Psi_\Delta (\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}) = f \Psi_\Delta (\varphi_1, \ldots, \varphi_{q-2}, \varphi_{q-1}) + l_{\sigma (\Delta)}(\varphi_1, \ldots, \varphi_{q-1})(f),
\]
for all $\varphi_i \in \Gamma (E^\ast)$ and $f \in C^\infty (M)$.
\end{lemma}
\begin{proof}
Let $\varphi_i$ and $f$ be as in the statement, and compute
\[
\begin{aligned}
& \Psi_\Delta (\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}) \\
& = [[\cdots [ \Delta, \ell_{\varphi_1}], \cdots, \ell_{\varphi_{q-2}}], f \ell_{\varphi_{q-1}}](1) \\
& = f[[\cdots [ \Delta, \ell_{\varphi_1}], \cdots, \ell_{\varphi_{q-2}}], \ell_{\varphi_{q-1}}](1) + [\cdots [ \Delta, \ell_{\varphi_1}], \cdots, \ell_{\varphi_{q-2}}] (\ell_{\varphi_{q-1}}) \\
& = f\Psi_\Delta (\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}) + [[\cdots [ \Delta, \ell_{\varphi_1}], \cdots, \ell_{\varphi_{q-2}}], f] (\ell_{\varphi_{q-1}})
\end{aligned}
\]
It remains to compute the last summand. So
\[
\begin{aligned}
& [[\cdots [ \Delta, \ell_{\varphi_1}], \cdots, \ell_{\varphi_{q-2}}], f] (\ell_{\varphi_{q-1}}) \\
& = [[[\cdots [ \Delta, \ell_{\varphi_1}], \cdots, \ell_{\varphi_{q-2}}], f],\ell_{\varphi_{q-1}}] + \ell_{\varphi_{q-1}} [[\cdots [ \Delta, \ell_{\varphi_1}], \cdots, \ell_{\varphi_{q-2}}], f] (1) \\
& = l_{\sigma (\Delta)}(\varphi_1, \ldots, \varphi_{q-1})(f),
\end{aligned}
\]
where we used that, from (\ref{eq:loc_DO_stab}) again, the last summand in the second line is necessarily zero. This concludes the proof.
\end{proof}
The data $(\sigma(\Delta), \Psi_\Delta)$ (determine $\Delta$ completely and) can be repackaged in a very useful way. Namely, consider the line bundle
\[
L = \wedge^{\mathrm{top}} E.
\]
Then the pair $(\sigma(\Delta), \Psi_\Delta)$ determines a FWL $q$-$L_E$-multivector in the following way. Recall that a FWL $q$-$L_E$-multivector can be equivalently presented as a pair $(P, \Phi)$ consisting of a FWL symmetric multivector $P \in \mathfrak X^\bullet_{\mathrm{sym}, \mathrm{lin}} (E)$ and a vector bundle map $\Phi : S^\bullet E^\ast \to DL$, such that $l_P = l \circ \Phi$. We claim that we can construct such a pair from the pair $(\sigma(\Delta), \Psi_\Delta)$. Namely, we put
\[
P = \sigma(\Delta)
\]
and define $\Phi = \Phi_\Delta$ by putting
\begin{equation}\label{eq:Phi_Delta}
\Phi_\Delta (\varphi_1, \ldots, \varphi_{q-1} ) (U) := \sigma (\Delta) (\varphi_1, \ldots, \varphi_{q-1}, -) U + \Psi_\Delta (\varphi_1, \ldots, \varphi_{q-1}) U,
\end{equation}
for all $\varphi_i \in \Gamma (E^\ast)$ and $U \in \Gamma (L)$. Equation (\ref{eq:Phi_Delta}) needs some explanations. In the first summand of the rhs, we interpret $\sigma (\Delta)$ as a $q$-multiderivation of $E^\ast$, so, when contracting it with the $q-1$ sections $\varphi_1, \ldots, \varphi_{q-1}$, we get a plain derivation $\sigma (\Delta) (\varphi_1, \ldots, \varphi_{q-1}, -) \in \mathfrak D (E^\ast) \cong \mathfrak D (E)$. As already remarked, derivations of $E$ act on the exterior algebra of $E$. In our case we have
\[
D (e_1 \wedge \cdots \wedge e_{\mathrm{top}}) = \sum_{i} e_1 \wedge \cdots \wedge De_i \wedge \cdots \wedge e_{\mathrm{top}}.
\]
for all $D \in \mathfrak D (E)$, and all $e_i \in \Gamma (E)$.
The next theorem is the main result of the paper.
\begin{theorem}\label{theor:iso_DO_D_sym}
The assignment $\Delta \mapsto (\sigma(\Delta), \Phi_\Delta)$ establishes a degree inverting isomorphism of Lie-Rinehart algebras $A: DO_{\mathrm{lin}}(E) \to \mathfrak D_{\mathrm{sym}, \mathrm{lin}}(L_E)$.
\end{theorem}
\begin{proof}
First of all, we have to show that $\Phi_\Delta$ is well-defined, i.e.~it is symmetric and $C^\infty (M)$-linear in all its arguments. The symmetry is obvious. For the linearity, let $f \in C^\infty (M)$, and compute
\[
\begin{aligned}
& \Phi_\Delta (\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}) (U) \\
& = {\sigma (\Delta) (\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}, -)} U + \Psi_\Delta (\varphi_1, \ldots, \varphi_{q-2}, f\varphi_{q-1}) U.
\end{aligned}
\]
Let us compute the two summands separately. First of all, for every $\varphi \in \Gamma (E^\ast)$,
\[
\sigma (\Delta) (\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}, \varphi) = l_{\sigma(\Delta)}(\varphi_1, \ldots, \varphi_{q-2}, \varphi)(f)\varphi_{q-1} + f \sigma(\Delta)(\varphi_1, \ldots, \varphi_{q-1}, \varphi)
\]
showing that
\[
\sigma (\Delta) (\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}, -) = \varphi_{q-1} \otimes e + f \sigma (\Delta)(\varphi_1, \ldots, \varphi_{q-1}, -),
\]
where $e \in \Gamma (E)$ is the section implicitly defined by
\[
\langle \varphi, e \rangle = l_{\sigma(\Delta)}(\varphi_1, \ldots, \varphi_{q-2}, \varphi)(f)
\]
for all $\varphi \in \Gamma (E^\ast)$. We remark for future use that, in particular,
\begin{equation}\label{eq:varphi_q-1,e}
\langle \varphi_{q-1}, e \rangle = l_{\sigma(\Delta)}(\varphi_1, \ldots, \varphi_{q-1})(f).
\end{equation}
Now
\begin{equation}\label{eq:3}
{\sigma (\Delta) (\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}, -)} U
= (\varphi_{q-1} \otimes e)U + f \sigma (\Delta)(\varphi_1, \ldots, \varphi_{q-1}, -) U.
\end{equation}
The endomorphism $\varphi_{q-1} \otimes e : E^\ast \to E^\ast$ act on $E$ via its dual, which is minus its transpose, hence
\begin{equation}\label{eq:1}
(\varphi_{q-1} \otimes e) U = - \operatorname{trace}(e \otimes \varphi_{q-1}) U = - \langle \varphi_{q-1}, e \rangle U = -l_{\sigma(\Delta)}(\varphi_1, \ldots, \varphi_{q-1})(f) U.
\end{equation}
where we used (\ref{eq:varphi_q-1,e}).
Substituting (\ref{eq:1}) into (\ref{eq:3}), we find
\[
{\sigma (\Delta) (\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}, -)} U = f{\sigma (\Delta)(\varphi_1, \ldots, \varphi_{q-1}, -)} U -l_{\sigma(\Delta)}(\varphi_1, \ldots, \varphi_{q-1})(f) U.
\]
We also have
\[
\Psi_\Delta (\varphi_1, \ldots, \varphi_{q-2}, f\varphi_{q-1}) U = f \Psi_\Delta (\varphi_1, \ldots, \varphi_{q-1}) U + l_{\sigma(\Delta)}(\varphi_1, \ldots, \varphi_{q-1})(f) U,
\]
and putting everything together we find
\[
\begin{aligned}
\Phi_\Delta (\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}) U & = f {\sigma (\Delta)(\varphi_1, \ldots, \varphi_{q-1}, -)} U + f \Psi_\Delta (\varphi_1, \ldots, \varphi_{q-1}) U \\
& = f \Phi_\Delta (\varphi_1, \ldots, \varphi_{q-1}) U.
\end{aligned}
\]
We conclude that $\Phi_\Delta$ is a vector bundle map $S^{q-1} E^\ast \to DL$ as desired. Additionally, the composition $l \circ \Phi_\Delta$ does clearly agree with $\sigma (\Delta)$ so that $(\sigma (\Delta), \Phi_\Delta)$ is indeed a $q$-$L_E$-multivector. It is also clear that $\Psi_\Delta$ can be reconstructed from $(\sigma (\Delta), \Phi_\Delta)$ showing that the correspondence $\Delta \mapsto (\sigma (\Delta), \Phi_\Delta)$ is injective. Next we prove the $DO_{\mathrm{core}}(E)$-linearity. So, take $F \in DO_{p, \mathrm{core}}(E) = \Gamma (S^p E) = C^\infty (E^\ast)_p$, and $\Delta \in DO_{q,\mathrm{lin}}(E)$, so that $F \circ \Delta \in DO_{p + q,\mathrm{lin}}(E)$. We want to show that $(\sigma(F \circ \Delta), \Phi_{F \circ \Delta}) = F \cdot (\sigma (\Delta), \Phi_\Delta)$. To do this we begin noticing that the product $F \cdot (\sigma (\Delta), \Phi_\Delta)$ is the pair $(D', \Phi')$ where $D' \in \mathfrak X^{p+q}_{\mathrm{sym}, \mathrm{lin}}(E)$ is the symmetric multivector that, when interpreted as a multiderivation $D' : \Gamma (E^\ast) \times \cdots \times \Gamma (E^\ast) \to \Gamma (E^\ast)$, is given by
\[
\begin{aligned}
D' (\varphi_1, \ldots, \varphi_{p+q}) & = F\cdot \sigma(\Delta)(\varphi_1, \ldots, \varphi_{p+q}) \\
& = \sum_{\sigma \in S_{p,q}} \left\langle F, \varphi_{\sigma(1)} \odot \cdots \odot \varphi_{\sigma(p)} \right\rangle \sigma(\Delta) (\varphi_{\sigma (p+1)}, \ldots, \varphi_{\sigma(p+q)}),
\end{aligned}
\]
and, similarly, $\Phi ' : S^{p+q-1} E^\ast \to DL$ is the bundle map given by
\[
\begin{aligned}
\Phi' (\varphi_1, \ldots, \varphi_{p+q-1}) & = F \cdot \Phi_\Delta (\varphi_1, \ldots, \varphi_{p+q-1})\\
& = \sum_{\sigma \in S_{p,q-1}} \left\langle F, \varphi_{\sigma(1)} \odot \cdots \odot \varphi_{\sigma(p)} \right\rangle \Phi_\Delta (\varphi_{\sigma (p+1)}, \ldots, \varphi_{\sigma(p+q-1)})
\end{aligned}
\]
for all $\varphi_i \in \Gamma (E^\ast)$. From the properties of the symbol map, we have $\sigma (F \circ \Delta) = F \cdot \sigma (\Delta)$ and it remains to take care of $\Phi_{F \circ \Delta}$. So choose $\varphi_i \in \Gamma (E^\ast)$, and compute $\Phi_{F \circ \Delta}(\varphi_1, \ldots, \varphi_{p+q-1})$. From symmetry, it is enough to choose $\varphi_i = \varphi$ for all $i$ and some $\varphi$. First of all, we have
\[
\begin{aligned}
\Psi_{F \circ \Delta}(\, \underset{\text{\tiny{$p+q-1$ times}}}{\underbrace{\varphi\, ,\, \ldots\, ,\, \varphi}}\, ) & = [ \cdots [ F \circ \Delta, \underset{\text{\tiny{$p+q-1$ times}}}{\underbrace{\varphi], \cdots, \varphi]}}(1) \\
& = \sum_{l+m = p+q-1} \frac{1}{l! m!} [ \cdots [ F, \underset{\text{\tiny{$l$ times}}}{\underbrace{\varphi], \cdots, \varphi]}} \circ [ \cdots [ \Delta, \underset{\text{\tiny{$m$ times}}}{\underbrace{\varphi], \cdots, \varphi]}} (1)
\end{aligned}.
\]
Only the terms with $l = p, p-1$ (hence $m = q-1, q$, respectively) survive, and we get
\begin{align}
\Psi_{F \circ \Delta}(\, \underset{\text{\tiny{$p+q-1$ times}}}{\underbrace{\varphi\, ,\, \ldots\, ,\, \varphi}}\, ) &
= \frac{1}{p! (q-1)!} [ \cdots [ F, \underset{\text{\tiny{$p$ times}}}{\underbrace{\varphi], \cdots, \varphi]}} \circ [ \cdots [ \Delta, \underset{\text{\tiny{$q-1$ times}}}{\underbrace{\varphi], \cdots, \varphi]}} (1) \nonumber \\
& \quad + \frac{1}{(p-1)! q!} [ \cdots [ F, \underset{\text{\tiny{$p-1$ times}}}{\underbrace{\varphi], \cdots, \varphi]}} \circ [ \cdots [ \Delta, \underset{\text{\tiny{$q$ times}}}{\underbrace{\varphi], \cdots, \varphi]}} (1) \nonumber \\
& = \frac{1}{p! (q-1)!} \Big\langle F, \underset{\text{\tiny{$p$ times}}}{\underbrace{\varphi \odot \cdots \odot \varphi}} \Big\rangle \Psi_\Delta \big(\underset{\text{\tiny{$q-1$ times}}}{\underbrace{\varphi, \ldots, \varphi}}\big) \nonumber \\
& \quad + \frac{1}{(p-1)! q!} \Big\langle F, \underset{\text{\tiny{$p-1$ times}}}{\underbrace{\varphi \odot \cdots \odot \varphi}} \odot \sigma(\Delta) \big(\underset{\text{\tiny{$q$ times}}}{\underbrace{\varphi, \ldots, \varphi}}\big)\Big\rangle. \label{eq:Psi_FDelta}
\end{align}
Now, for all $U \in \Gamma (L)$
\[
\Phi_{F \circ \Delta}(\, \underset{\text{\tiny{$p+q-1$ times}}}{\underbrace{\varphi\, ,\, \ldots\, ,\, \varphi}}\, ) U
= {\sigma(F \circ \Delta) (\, \underset{\text{\tiny{$p+q-1$ times}}}{\underbrace{\varphi\, ,\, \ldots\, ,\, \varphi}}\, , -)}U + \Psi_{F \circ \Delta}(\, \underset{\text{\tiny{$p+q-1$ times}}}{\underbrace{\varphi\, ,\, \ldots\, ,\, \varphi}}\, )U .
\]
We already computed the second summand, while the first summand is
\begin{align}
{\sigma(F \circ \Delta) (\, \underset{\text{\tiny{$p+q-1$ times}}}{\underbrace{\varphi\, ,\, \ldots\, ,\, \varphi}}\, , -)}U & = \frac{1}{p! (q-1)!}{\big\langle F, \underset{\text{\tiny{$p$ times}}}{\underbrace{\varphi \odot \cdots \odot \varphi}} \big\rangle \sigma(\Delta) (\, \underset{\text{\tiny{$q-1$ times}}}{\underbrace{\varphi\, ,\, \ldots\, ,\, \varphi}}\, , -)} U\nonumber \\
& \quad + \frac{1}{(p-1)! q!}{\big\langle F, \underset{\text{\tiny{$p-1$ times}}}{\underbrace{\varphi \odot \cdots \odot \varphi}}\, \odot\, - \big\rangle \sigma(\Delta) (\, \underset{\text{\tiny{$q$ times}}}{\underbrace{\varphi , \ldots, \varphi}})}U \nonumber\\
& = \frac{1}{p! (q-1)!}\big\langle F, \underset{\text{\tiny{$p$ times}}}{\underbrace{\varphi \odot \cdots \odot \varphi}} \big\rangle {\sigma(\Delta) (\, \underset{\text{\tiny{$q-1$ times}}}{\underbrace{\varphi\, ,\, \ldots\, ,\, \varphi}}\, , -)}U \nonumber \\
& \quad - \frac{1}{(p-1)! q!}\big\langle F, \underset{\text{\tiny{$p-1$ times}}}{\underbrace{\varphi \odot \cdots \odot \varphi}} \odot \sigma(\Delta) (\, \underset{\text{\tiny{$q$ times}}}{\underbrace{\varphi , \ldots, \varphi}}) \big\rangle U.\label{eq:L_FDelta}
\end{align}
From (\ref{eq:Psi_FDelta}) and (\ref{eq:L_FDelta}) it easily follows that $\Phi_{F \circ \Delta} = F \cdot \Phi_\Delta$ as claimed.
The surjectivity of the map $\Delta \mapsto (\sigma(\Delta), \Phi_\Delta)$ now follows from (local) dimension counting.
It remains to check that the isomorhism $A: DO_{\mathrm{lin}}(E) \to \mathfrak D_{\mathrm{sym}, \mathrm{lin}}(L_E)$ defined in this way is both anchor and bracket preserving. For the anchor, the anchor of $(\sigma (\Delta), \Phi_\Delta)$ is the derivation of $DO_{\mathrm{core}}(E) = \Gamma (S^\bullet E) = C^\infty_{\mathrm{poly}}(E)$ corresponding to the linear multivector $\sigma (\Delta)$, which is exactly $ad(\Delta)$.
For the bracket, as we already discussed $C^\infty_{\mathrm{poly}}(E)$-linearity and compatibility with the anchor, it is enough to discuss the brackets of generators. As already remarked, $DO_{\mathrm{lin}}(E)$ is generated (over $DO_{\mathrm{core}}(E)$) by $1$, $C^\infty_{\mathrm{lin}}(E)$ and $\mathfrak X_{\mathrm{lin}}(E)$. A direct check shows that
\begin{equation}\label{eq:A_generators}
A(1) = (0, 1), \quad A(\ell_{\varphi}) = (- \varphi^\uparrow, 0), \quad A(X) = (X^\ast, D_X)
\end{equation}
for all $\varphi \in \Gamma (E^\ast)$ and all $X \in \mathfrak X_{\mathrm{lin}}(E)$. Here $X^\ast$ and $D_X$ are, respectively, the linear vector field on $E^\ast$, and the derivation of $L = \wedge^{\mathrm{top}}E$ corresponding to $X$. It is now easy to check that the brackets are preserved on these generators, and this concludes the proof.
\end{proof}
Composing with the isomorphism $\mathfrak D^\bullet_{\mathrm{sym}, \mathrm{lin}}(L_E) \cong \mathfrak D (L_{E^\ast})$ we get a (degree inverting) Lie-Rinehart algebra isomorphism
\[
DO_{\mathrm{lin}} (E) \overset{\cong}{\longrightarrow} \mathfrak D (L_{E^\ast})
\]
that we denote by $A$ again.
\begin{remark}
Let $(x^i, u^\alpha)$ be vector bundle coordinates on $E$, and let $(x^i, u_\alpha)$ be dual coordinates on $E^\ast$. Denote by $\mathrm{Vol}_u = u_1 \wedge \cdots \wedge u_{\mathrm{top}}$ the local coordinate generator of $\Gamma(L)$. It is easy to check using, e.g., (\ref{eq:ad_Delta}), (\ref{eq:A_generators}), and the $C^\infty_{\mathrm{poly}}(E^\ast)$-linearity, that, if the operator $\Delta \in DO_{\mathrm{lin}}(E)$ is locally given by (\ref{eq:loc_DO_stab}), then the corresponding derivation $A(\Delta) \in \mathfrak D (L_{E^\ast})$ maps a local section $\lambda = f(x,u) \mathrm{Vol}_u$ of $\Gamma (L_{E^\ast})$ to
\[
\begin{aligned}
& A(\Delta)(\lambda) \\
& = \left( \sum_{|A| = q -1} \Delta^{i|A}(x) u_A\frac{\partial f}{\partial x^i}(x,u) - \sum_{|B| = q} \Delta^B_\alpha (x)u_B \frac{\partial f}{\partial u_\alpha} (x,u) + \sum_{|C| = q-1 }\Delta^C(x)u_C f(x,u) \right) \mathrm{Vol}_u.
\end{aligned}
\]
\end{remark}
\section{Linearization of Differential Operators}\label{sec:linear}
Let $\mathcal E$ be a manifold, let $M \subseteq \mathcal E$ be a submanifold and let $\Delta \in DO (\mathbb R_{\mathcal E})$ be a scalar DO. Denote by $E \to M$ the normal bundle to $M$, i.e.~$E = T\mathcal E|_M /TM$. In this section we show that, under appropriate \emph{linearizability conditions}, the DO $\Delta$ can be \emph{linearized} around $M$ yielding a FWL differential operator $\Delta_{\mathrm{lin}} \in DO_{\mathrm{lin}}(E)$. The DO $\Delta_{\mathrm{lin}}$ represents the first order approximation of $\Delta$ around $M$ in the direction transverse to $M$. This \emph{linearization construction} is a further motivation supporting our definition of FWL DOs.
So let $M \subseteq \mathcal E$ be a submanifold and let $E \to M$ be its normal bundle. We will often consider \emph{adapted coordinates} on $\mathcal E$ around points of $M$, i.e.~coordinates $(X^i, U^\alpha)$ such that $M : \left\{ U^\alpha = 0 \right.$. In particular, the restrictions $(x^i = X^i|_M)$ are coordinates on $M$. From $U^\alpha|_M = 0$ we see that $u^\alpha := dU^\alpha |_M$ are conormal $1$-forms and $(x^i, u^\alpha)$ are vector bundle coordinates on $E$.
We want to explain what does it mean to \emph{linearize} an order $q$ DO operator $\Delta \in DO_q (\mathbb R_{\mathcal E})$ around $M$. We proceed as follows: 1) first, we recall the linearization of a function, 2) second, we discuss the linearization of a symmetric multivector, and, finally 3) we define the linearization of a generic DO. So, let $F \in C^\infty (\mathcal E)$. We say that $F$ is \emph{linearizable} (around $M$) if $F|_M = 0$. In this case, $dF|_M$ is a conormal $1$-form to $M$, i.e.~a section of the conormal bundle $E^\ast \hookrightarrow T^\ast \mathcal E|_M$. Hence it corresponds to a FWL function on $E$. We put $F_{\mathrm{lin}} = \ell_{dF|_M}$ and call it the \emph{linearization} of $F$. For instance, if $(X^i, U^\alpha)$ are adapted coordinates on $\mathcal E$, then the $(U^\alpha)$ are linearizable and the linear fiber coordinates $(u^\alpha)$ on $E$ are their linearizations. If $F$ is any linearizable function on $\mathcal E$, then locally, around a point of $M$, $F(X,U) = F_\alpha(X) U^\alpha + \mathcal O(U^2)$, for some functions $F_\alpha (X)$ of the $(X^i)$ (given by $F_\alpha (X) = \frac{\partial F}{\partial U^\alpha} (X,0)$), and, in this case, $F_{\mathrm{lin}} = F_\alpha(x)u^\alpha$. Notice that every linear function $\varphi \in C^\infty_{\mathrm{lin}}(E)$ is the linearization of a (non-unique) linearizable function $F \in C^\infty (\mathcal E)$: $\varphi = F_{\mathrm{lin}}$.
We now pass to symmetric multivectors. So, let $P$ be a symmetric $q$-multivector on $\mathcal E$. We say that $P$ is \emph{linearizable} if it belongs to the ideal $\mathcal I_M$ in $\mathfrak X^\bullet_{\mathrm{sym}}(\mathcal E)$ spanned by vector fields that are tangent to $M$. In other words $M$ is a \emph{coisotropic submanifold} of $\mathcal E$ with respect to $P$. This notion of linearizable multivector agrees with that of \emph{FWLizable tensor} described in \cite[Definition 5.4]{PSV2020}.
\begin{proposition}\label{prop:lin_multiv}
Let $P \in \mathfrak X^q_{\mathrm{sym}}(\mathcal E)$ be a linearizable symmetric multivector on $\mathcal E$. Then, there exists a unique FWL symmetric $q$-multivector $P_{\mathrm{lin}}$ on $E$ such that
\begin{equation}\label{eq:lin_multiv}
P_{\mathrm{lin}} ((F_1)_{\mathrm{lin}}, \ldots, (F_q)_{\mathrm{lin}}) = P(F_1, \ldots, F_q)_{\mathrm{lin}}
\end{equation}
for all linearizable functions $F_i \in C^\infty (\mathcal E)$. The \emph{linearization} $P \mapsto P_{\mathrm{lin}}$ preserves the Poisson bracket of symmetric multivectors.
\end{proposition}
\begin{proof}
We begin remarking that, as $P \in \mathcal I_M$, the function $P(F_1, \ldots, F_q)$ is clearly linearizable for any choice of linearizable functions $F_i$. Now we want to show that the rhs of (\ref{eq:lin_multiv}) does only depend on $(F_i)_{\mathrm{lin}}$. We do this in coordinates. So, let $(X^i, U^\alpha)$ be adapted coordinates on $\mathcal E$, and let $(x^i, u^\alpha)$ be the associated vector bundle coordinates on $E$. Locally
\begin{equation}\label{eq:P}
P = \sum_{l+m = q}P^{i_1\cdots i_l,\alpha_1 \cdots \alpha_m}(X,U) \frac{\partial}{\partial X^{i_1}} \odot \cdots \odot \frac{\partial}{\partial X^{i_l}} \odot \frac{\partial}{\partial U^{\alpha_1}} \odot \cdots \odot \frac{\partial}{\partial U^{\alpha_m}}.
\end{equation}
Hence
\[
P(F_1, \ldots, F_q) = \sum_{l+m = q} \sum_{\sigma \in S_{l,m}}P^{i_1\cdots i_l,\alpha_1 \cdots \alpha_m}(X,U) \frac{\partial F_{\sigma(1)}}{\partial X^{i_1}} \cdots \frac{\partial F_{\sigma(l)}}{\partial X^{i_l}} \frac{\partial F_{\sigma(l+1)}}{\partial U^{\alpha_1}} \cdots \frac{\partial F_{\sigma(l+m)}}{\partial U^{\alpha_m}}.
\]
It follows from $P \in \mathcal I_M$ that $P^{\alpha_1, \ldots, \alpha_q}(X,0) = 0$. Now compute
\[
P(F_1, \ldots, F_q)_{\mathrm{lin}} = \left(\frac{\partial}{\partial U^\alpha}|_{(x,0)}P(F_1, \ldots, F_q)\right)u^\alpha.
\]
Using that $F_i|_M = 0$, and that $P^{\alpha_1, \ldots, \alpha_q}(X,0) = 0$, we find
\[
\begin{aligned}
\frac{\partial}{\partial U^\alpha}|_{(x,0)}P(F_1, \ldots, F_q) & = \frac{\partial P^{\alpha_1 \cdots \alpha_q}}{\partial U^\alpha}(x,0) (F_1)_{\alpha_1} (x) \cdots (F_q)_{\alpha_q} (x) \\
& \quad + \sum_{\sigma \in S_{1, k-1}} P^{i,\alpha_1 \cdots \alpha_{q-1}}(x,0) \frac{\partial (F_{\sigma(1)})_\alpha}{\partial x^{i}} (x,0) (F_{\sigma(2)})_{\alpha_1} \cdots (F_{\sigma(q)})_{\alpha_{q-1}}
\end{aligned}
\]
which does only depend on the $(F_i)_{\mathrm{lin}}$. As every FWL function is a linearization, this also shows that $P_{\mathrm{lin}}$ is well-defined on linear functions. Finally, $P_{\mathrm{lin}}$ can be uniquely extended to all functions on $E$, as a symmetric $q$-multivector, also denoted by $P_{\mathrm{lin}}$, and locally given by
\[
P_{\mathrm{lin}} = P^{\alpha_1 \cdots \alpha_q}_{\mathrm{lin}}(x, u) \frac{\partial}{\partial u^{\alpha_1}} \odot \cdots \odot \frac{\partial}{\partial u^{\alpha_q}}
+ P^{i,\alpha_1 \cdots \alpha_{q-1}}(x,0) \frac{\partial}{\partial x^{i}} \odot \frac{\partial}{\partial u^{\alpha_1}} \odot \cdots \odot \frac{\partial}{\partial u^{\alpha_{q-1}}}.
\]
In particular $P_{\mathrm{lin}}$ is a linear multivector.
For the last part of the statement, first notice that the ideal $\mathcal I_M$ is preserved by the Poisson bracket, so, if $P,Q \in \mathcal I_M$, then it makes sense to linearize $\{ P, Q\}$. The rest follows easily from Equation (\ref{eq:lin_multiv}).
\end{proof}
\begin{remark}
Proposition \ref{prop:lin_multiv} is a ``symmetric multivector analogue'' of the following well-known fact. Let $(\mathcal P, \pi)$ be a Poisson manifold, let $(T^\ast \mathcal P)_\pi$ be its cotangent algebroid and let $M \subseteq \mathcal P$ be a coisotropic submanifold. Then the conormal bundle $N^\ast M \subseteq T^\ast \mathcal P$ is a subalgebroid $(T^\ast \mathcal P)_\pi$, hence the normal bundle $NM$ is equipped with a linear Poisson structure $\pi_{\mathrm{lin}}$ (see, e.g., \cite[Section 3]{W1988}).
\end{remark}
By definition, the multivector $P_{\mathrm{lin}}$ in Proposition \ref{prop:lin_multiv} is the \emph{linearization} of $P$.
We finally come to a generic differential operator $\Delta \in DO_q (\mathbb R_{\mathcal E})$.
\begin{definition}
An order $q$ DO $\Delta \in DO (\mathbb R_{\mathcal E})$ is \emph{order $q$ linearizable} (around $M$) if its symbol $\sigma (\Delta) \in \mathfrak X^q_{\mathrm{sym}}(\mathcal E)$ is linearizable.
\end{definition}
\begin{remark} \quad
\begin{itemize}
\item An order $q-1$ DO $\Delta \in DO (\mathbb R_{\mathcal E})$ is always order $q$ linearizable, but it might not be order $q-1$ linearizable.
\item A function $F \in C^\infty (\mathcal E) = DO_0 (\mathbb R_{\mathcal E})$ is order $0$ linearizable if and only if $F|_M = 0$, i.e.~$F$ is a linearizable function.
\item A vector field $X \in \mathfrak X (\mathcal E) \subset DO_1 (\mathbb R_{\mathcal E})$ is order $1$ linearizable if and only if it is tangent to $M$, i.e.~it is a linearizable vector field.
\end{itemize}
\end{remark}
Now, let $\Delta \in DO_q (\mathbb R_{\mathcal E})$ be an order $q$ linearizable DO. In order to \emph{linearize} it, we use Theorem \ref{theor:iso_DO_D_sym}. In other words, from $\Delta$ we cook up a $q$-$L_E$-multivector $(P_{\mathrm{lin}}, \Phi)$ of $L_{E^\ast} = E^\ast \times_M L$, with $L = \wedge^\mathrm{top} E$ as in Section \ref{sec:FWL_DO}. The construction is inspired by the proof of Theorem \ref{theor:iso_DO_D_sym} itself. We let $P_{\mathrm{lin}} \in \mathfrak X^q_{\mathrm{sym}, \mathrm{lin}}(E)$ be the linearization of the symbol $P = \sigma (\Delta)$ of $\Delta$. It remains to define the vector bundle map
\[
\Phi : S^{q-1} E^\ast \to DL.
\]
For $\varphi_1, \ldots, \varphi_{q-1} \in \Gamma (E^\ast)$, we define $\Phi (\varphi_1, \ldots, \varphi_{q-1})$ as follows. Pick linearizable functions $F_i \in C^\infty (\mathcal E)$ such that $(F_i)_{\mathrm{lin}} = \ell_{\varphi_i}$ and consider the function on $M$
\[
f_{F_1, \ldots, F_{q-1}} := [[\cdots[\Delta, F_1], \cdots], F_{q-1}](1)|_M.
\]
We put
\[
\Phi (\varphi_1, \ldots, \varphi_{k-1}) (U)= {P_{\mathrm{lin}}(\varphi_1, \ldots, \varphi_{q-1}, -)} U + f_{F_1, \ldots, F_{k-1}} U,
\]
for all $U \in \Gamma (L)$. In the rhs, we interpret $P_{\mathrm{lin}}$ as a multiderivation of $E^\ast$. We have to show that $\Phi$ is well defined, i.e.~$f_{F_1, \ldots, F_{q-1}}$ does only depend on the $\varphi_i$, and $\Phi$ is indeed a vector bundle map $\Phi : S^{q-1} E^\ast \to DL$ as desired, i.e.~it is (symmetric and) $C^\infty (M)$-linear in its arguments. To see that $f_{F_1, \ldots, F_{q-1}}$ does not depend on the choice of the $F_i$, assume that $(F_{q-1})_{\mathrm{lin}} = 0$. Hence we have $F_{q-1}|_M = 0$, and $dF_{q-1}|_M = 0$. Additionally,
\[
\begin{aligned}
f_{F_1, \ldots, F_{q-1}} & = [[\cdots[\Delta, F_1], \cdots], F_{q-1}](1)|_M
\\
& = \left(F_{q-1}[[\cdots[\Delta, F_1], \cdots], F_{q-2}](1) - [[\cdots[\Delta, F_1], \cdots], F_{q-2}](F_{q-1})\right)|_M \\
& = - [[\cdots[\Delta, F_1], \cdots], F_{q-2}](F_{q-1})|_M
\end{aligned}
\]
Now, $[[\cdots[\Delta, F_1], \cdots], F_{q-2}]$ is a second order DO. As the first jet of $F_{q-1}$ vanishes on $M$, the expression $ [[\cdots[\Delta, F_1], \cdots], F_{q-2}](F_{q-1})|_M$ does only depend on the symbol of $[[\cdots[\Delta, F_1], \cdots], F_{q-2}]$ (and the Hessian of $F_{q-1}$). But
\[
\sigma \left( [[\cdots[\Delta, F_1], \cdots], F_{q-2}] \right) = \sigma (\Delta) (F_1, \ldots, F_{q-2}, -,-),
\]
so, if $P = \sigma (\Delta)$ is locally given by (\ref{eq:P}), using that $P^{\alpha_1 \cdots \alpha_q}(X,0) = 0$, we find that, locally,
\[
f_{F_1, \ldots, F_{q-1}} \propto P^{i,\alpha_1 \cdots \alpha_{q-2}\alpha_{q-1}}(x,0) (F_1)_{\alpha_1}(x) \cdots (F_{q-2})_{\alpha_{q-2}}(x) \frac{\partial }{\partial x^{i} } (F_{q-1})_{\alpha_{q-1}}(x) = 0.
\]
We conclude that $\Phi (\varphi_1, \ldots, \varphi_{q-1})$ is well-defined. It remains to see that $\Phi$ is $C^\infty (M)$-linear in one, hence in all, of its arguments. So, take $f \in C^\infty (M)$. The simple formula:
\begin{equation}\label{eq:FG_lin}
(FG)_{\mathrm{lin}} = F|_M G_{\mathrm{lin}}, \quad \text{for all $F \in C^\infty (\mathcal E)$ and all linearizable $G \in C^\infty (\mathcal E)$},
\end{equation}
shows that, when we replace $\varphi_{q-1}$ with $f \varphi_{q-1}$ in $\Phi(\varphi_1, \ldots, \varphi_{q-1})$, we can replace $F_{q-1}$ with $F F_{q-1}$, where $F\in C^\infty (\mathcal F)$ is any function such that $F|_M = f$. So we have
\[
\Phi (\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}) U = {P_{\mathrm{lin}}(\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}, -)}U + f_{F_1, \ldots, F_{q-2}, F F_{q-1}}U.
\]
For the first summand, recall from the proof of Theorem \ref{theor:iso_DO_D_sym} (Equations (\ref{eq:3}), (\ref{eq:1})) that
\[
{P_{\mathrm{lin}}(\varphi_1, \ldots, \varphi_{q-2}, f \varphi_{q-1}, -)}U = f {P_{\mathrm{lin}}(\varphi_1, \ldots, \varphi_{q-2}, \varphi_{q-1}, -)}U - l_{P_{\mathrm{lin}}} (\varphi_1, \ldots, \varphi_{q-1})(f) U
\]
For the second summand, compute
\[
\begin{aligned}
f_{F_1, \ldots, F_{q-2}, F F_{q-1}} & = [[\cdots[\Delta, F_1], \cdots, F_{q-2}], F F_{q-1}](1)|_M \\
& = F [[[\cdots[\Delta, F_1], \cdots], F_{q-2}], F_{q-1}](1)|_M + [[[\cdots[\Delta, F_1], \cdots], F_{q-2}], f](F_{q-1})|_M \\
& = f f_{F_1, \ldots, F_{q-1}} + [[[[\cdots[\Delta, F_1], \cdots], F_{q-2}], F],F_{q-1}](1)|_M \\
& = F f_{F_1, \ldots, F_{q-1}} + \sigma (\Delta) (F_1, \ldots, F_{q-1}, F)|_M.
\end{aligned}
\]
Now, to conclude the proof of the $C^\infty (M)$-multilinearity, it is enough to show that
\begin{equation}\label{eq:sigma|_M_l}
\sigma (\Delta) (F_1, \ldots, F_{q-1}, F)|_M = l_{P_{\mathrm{lin}}} (\varphi_1, \ldots, \varphi_{q-1})(f).
\end{equation}
But this follows easily from the fact that $P_{\mathrm{lin}}$ is the linearization of $\sigma (\Delta)$ and formula (\ref{eq:FG_lin}) again.
Finally, by construction, $l \circ \Phi = l_{P_{\mathrm{lin}}}$, so the pair $(P_{\mathrm{lin}}, \Phi)$ corresponds to a $q$-$L_E$-multivector. We call the linear DO $A^{-1}(P_{\mathrm{lin}}, \Phi)$ the \emph{linearization} of $\Delta$ and denote it $ \Delta_{\mathrm{lin}}$.
The above discussion leads to the following
\begin{theorem}\label{theor:linear}
The assignment $\Delta \mapsto \Delta_{\mathrm{lin}}$ is a well-defined (\emph{linearization}) map from order $q$ linearizable DOs on $\mathcal E$ and order $q$ FWL DOs on $E$. The linearization preserves the commutator of DOs in the following sense: let $\Delta \in DO_{q+1} (\mathcal E)$ and $\Delta' \in DO_{q'+1}(\mathcal E)$ be an order $q + 1$ linearizable and an order $q' +1$ linearizable DO, respectively. Then 1) $[\Delta, \Delta']$ is order $q+q'+1$ linearizable and 2) its linearization is $[\Delta_{\mathrm{lin}}, \Delta{}'_{\mathrm{lin}}]$.
\end{theorem}
\begin{proof}
We only need to prove the last part of the statement. To do that, recall that the commutator $[\Delta, \Delta']$ is a DO of order $q + q' + 1$ and its symbol is the Poisson bracket $\{ \sigma(\Delta), \sigma (\Delta') \}$. Let $\Delta_{\mathrm{lin}} = A^{-1}(P_{\mathrm{lin}}, \Phi)$, and $\Delta'{}_{\mathrm{lin}} = A^{-1}(P'{}_{\mathrm{lin}}, \Phi')$ be their linearizations. As the ideal $\mathcal I_M$ in $\mathfrak X^\bullet (\mathcal E)$ is preserved by the Poisson bracket, then $[\Delta, \Delta']$ is order $q + q' + 1$ linearizable as desired. Let $[\Delta, \Delta']_{\mathrm{lin}} = A^{-1} (P''{}_{\mathrm{lin}}, \Phi'')$ be its linearization. As the linearization of multivectors preserves the Poisson bracket, then $P''{}_{\mathrm{lin}} = \left\{P_{\mathrm{lin}}, P'{}_{\mathrm{lin}}\right\}$. Finally, we have to take care of $\Phi''$. It is a straightforward computation that we sketch to point out the possible subtleties related to the properties of the various objects involved. Recall from the discussion preceding the statement of the theorem that
\[
\Phi (\varphi_1, \ldots, \varphi_{q}) U = {P_{\mathrm{lin}} (\varphi_1, \ldots, \varphi_{q})}U + f_{F_1, \ldots, F_{q}}U
\]
where
\[
f_{F_1, \ldots, F_{q}} = [\cdots [\Delta, F_1], \cdots, F_q](1)|_M,
\]
and likewise for $\Phi', \Phi''$. Here $F_i$ is any linearizable function on $\mathcal E$ such that $(F_i)_{\mathrm{lin}} = \varphi_i$. In the following, to stress that, actually, the function $ f_{F_1, \ldots, F_{q}}$ does only depend on the $\varphi_i$, we write $f_{\varphi_1, \ldots, \varphi_q}$ (instead of $f_{F_1, \ldots, F_{q}}$). Additionally, we call $f$ (resp.~$f', f''$) the \emph{$f$-component} of the FWL DO $\Delta_{\mathrm{lin}}$ (resp.~$\Delta'{}_{\mathrm{lin}}, [\Delta_{\mathrm{lin}}, \Delta'{}_{\mathrm{lin}}]$). We have to show that the $f$-component $f^{[\Delta, \Delta']}$ of $[\Delta, \Delta']_{\mathrm{lin}}$ agrees with $f''$. To do this, we compute $f^{[\Delta, \Delta']}$ explicitly. By symmetry and $\mathbb R$-multilinearity, it is enough to evaluate it on $q + q'$ equal sections $\varphi_1 = \cdots = \varphi_{q+q'} = \varphi \in \Gamma (E^\ast)$. So let $F \in C^\infty (\mathcal E)$ be a linearizable function such that $\varphi = F_{\mathrm{lin}}$. We have
\[
\begin{aligned}
f^{[\Delta, \Delta']}_{\underset{\text{$q+q'$ times}}{\underbrace{\varphi, \ldots, \varphi}}} & = [\cdots [[ \Delta, \Delta'], \underset{\text{$q + q'$ times}}{\underbrace{F], \cdots, F]}}(1)|_M \\
& = \sum_{i + j = q + q'} \frac{1}{i!j!}\bigg[[\cdots [ \Delta, \underset{\text{$i$ times}}{\underbrace{F], \cdots, F]}}, [\cdots [ \Delta', \underset{\text{$j$ times}}{\underbrace{F], \cdots, F]}} \bigg](1)|_M.
\end{aligned}
\]
The only terms that survive are those with $i = q-1, q, q+1 $ (hence $j = q' +1, q', q'-1$ respectively). We call them $T_{q-1}, T_q, T_{q+1}$ respectively. The first one is given by
\[
\begin{aligned}
T_{q-1} & = \frac{1}{(q-1)!(q'+1)!}\bigg[[\cdots [ \Delta, \underset{\text{$q-1$ times}}{\underbrace{F], \cdots, F]}}, [\cdots [ \Delta', \underset{\text{$q' + 1$ times}}{\underbrace{F], \cdots, F]}} \bigg](1)|_M \\
& = \frac{1}{(q-1)!(q'+1)!} f_{\underset{\text{$q - 1$ times}}{\underbrace{\varphi, \ldots, \varphi}}, P'{}_{\mathrm{lin}}(\underset{\text{$q' + 1$ times}}{\underbrace{\varphi, \ldots, \varphi}})}
\end{aligned}.
\]
Similarly the third one is
\[
T_{q+1} = -\frac{1}{(q+1)!(q'-1)!} f'_{\underset{\text{$q' - 1$ times}}{\underbrace{\varphi, \ldots, \varphi}}, P_{\mathrm{lin}}(\underset{\text{$q + 1$ times}}{\underbrace{\varphi, \ldots, \varphi}})}.
\]
To compute $T_q$, we use a simple trick: for every two scalar DOs $\square, \square'$ we have
\[
[\square, \square'](1) = [\square, \square'(1)](1) - [\square', \square(1)](1).
\]
After using this formula we get
\[
\begin{aligned}
T_{q} & = \frac{1}{q!q'!}\bigg[[\cdots [ \Delta, \underset{\text{$q$ times}}{\underbrace{F], \cdots, F]}}, [\cdots [ \Delta', \underset{\text{$q'$ times}}{\underbrace{F], \cdots, F]}} \bigg](1)|_M \\
& = \frac{1}{q!q'!}\sigma(\Delta)\bigg(\underset{\text{$q$ times}}{\underbrace{F, \cdots, F}}, [\cdots [ \Delta', \underset{\text{$q'$ times}}{\underbrace{F], \cdots, F]}} (1)\bigg)|_M \\
& \quad - \frac{1}{q!q'!}\sigma(\Delta')\bigg(\underset{\text{$q'$ times}}{\underbrace{F, \cdots, F}}, [\cdots [ \Delta, \underset{\text{$q$ times}}{\underbrace{F], \cdots, F]}} (1)\bigg)|_M .
\end{aligned}
\]
Now we use that both $\Delta$ and $F$ are linearizable to replace $[\cdots [ \Delta', F], \cdots, F](1)$ with its restriction to $M$ in the first summand (likewise for $\Delta$ in the second summand). We get
\[
\begin{aligned}
T_{q}
& = \frac{1}{q!q'!}\sigma(\Delta)\bigg(\underset{\text{$q$ times}}{\underbrace{F, \cdots, F}}, f'_{\underset{\text{$q'$ times}}{\underbrace{\varphi, \ldots, \varphi}}}\bigg)|_M - \frac{1}{q!q'!}\sigma(\Delta')\bigg(\underset{\text{$q'$ times}}{\underbrace{F, \cdots, F}}, f_{\underset{\text{$q$ times}}{\underbrace{\varphi, \ldots, \varphi}}}\bigg)|_M \\
& = \frac{1}{q!q'!}l_{P_{\mathrm{lin}}}\bigg(\underset{\text{$q$ times}}{\underbrace{\varphi, \cdots, \varphi}}\bigg)\big(f'_{\underset{\text{$q'$ times}}{\underbrace{\varphi, \ldots, \varphi}}}\big) - \frac{1}{q!q'!}l_{P'{}_{\mathrm{lin}}}\bigg(\underset{\text{$q'$ times}}{\underbrace{\varphi, \cdots, \varphi}}\bigg)\big(f_{\underset{\text{$q$ times}}{\underbrace{\varphi, \ldots, \varphi}}}\big) \end{aligned}
\]
where we also used (\ref{eq:sigma|_M_l}). At this point, it can be checked directly, explointing the explicit formula (\ref{eq:Poisson_V-multivectors}) for the Poisson bracket of $L_E$-multivectors, that $T_{q-1} + T_q + T_{q+1}$ agrees with the $f$-component $f''$ of $[\Delta_{\mathrm{lin}}, \Delta'{}_{\mathrm{lin}}]$. We conclude that $[\Delta_{\mathrm{lin}}, \Delta'{}_{\mathrm{lin}}] = [\Delta, \Delta']_{\mathrm{lin}}$ as desired.
\end{proof}
\begin{remark}
Let $\mathcal E$ be the total space of a vector bundle $E \to M$ and interpret $M$ as a submanifold in $\mathcal E = E$ via the zero section. In this situation, the normal bundle $NM$ to $M$ identifies canonically with $E$ itself. Clearly, an order $q$ FWL DO $\Delta \in DO_{q, \mathrm{lin}}(E)$ is automatically order $q$ linearizable and it easily follows from the proof of Theorem \ref{theor:linear} that the vector bundle isomorphism $E \cong NM$ identifies $\Delta$ with its own linearization $\Delta_{\mathrm{lin}}$.
\end{remark}
\textbf{Acknowledgements.} LV is member of the GNSAGA of INdAM.
%
| {
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} | 3,485 |
Kerner may refer to:
Kerner (grape), a variety of white grape
Kerner Commission, established in 1967 by President Lyndon B. Johnson to investigate the causes of race riots in the United States
Kerner Optical, a motion picture visual effects company
Durand–Kerner method, root-finding algorithm for solving polynomial equations in numerical analysis
People
Literature
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Politics
Johann Georg Kerner (1770–1812), political journalist, critical chronicler of the French revolution, brother of Justinus Kerner
Otto Kerner, Jr. (1908–1976), Illinois Governor and judge on the United States Court of Appeals for the Seventh Circuit
Otto Kerner, Sr. (1884–1952), an Illinois Attorney General and judge on the United States Court of Appeals for the Seventh Circuit
Entertainment and media
Debby Kerner, American singer
Nena (Gabriele Susanne Kerner; born 1960), German singer who became famous with the New German Wave songs "Nur geträumt" and "99 Luftballons"
Jordan Kerner, film producer
Johannes B. Kerner, (born 1964), German television broadcaster
Sports
Jonathan Kerner (born 1974), American basketball player
Marlon Kerner (born 1973), American football player
Science and mathematics
Boris Kerner, (born 1947), German physicist
Anton Kerner von Marilaun (1831–1898), Austrian botanist
Other fields
Ian Kerner, American sex counselor
See also
Kerning, adjusting the spaces between typeset letters
Occupational surnames
Surnames of Bavarian origin | {
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In basketball, a free throw is an unopposed attempt to score points from behind the free throw line. The VTB United League's free throw percentage leader is the player with the highest free throw percentage in a given season.
To qualify as a leader for the free throw percentage, a player must play in at least 60 percent of the total number of possible games.
Nando de Colo is the only player in league history to lead the league in free throw percentage multiple times and also to lead the said statistics in consecutive seasons.
Free throw Percentage leaders
Notes
References
Basketball in Lithuania
Basketball in Russia
Free Throw | {
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"Department of Information Technology" means the principal department of state government created under Executive Order 2001-3, MCL 18.41.
All the authority, powers, duties, functions, and responsibilities of the Department of Management and Budget under the Uniform Electronic Transactions Act, 2000 PA 305, MCL 450.831 to 450.849, are transferred by Type II Transfer from the Department of Management and Budget to the Department of Information Technology.
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{"url":"https:\/\/www.auburn.edu\/cosam\/departments\/math\/research\/seminars\/stochastic-seminar.htm","text":"COSAM \u00bb Departments \u00bb Mathematics & Statistics \u00bb Research \u00bb Seminars \u00bb Stochastic Analysis\n\n# Stochastic Analysis\n\nDMS Stochastic Analysis Seminar\nMar 29, 2023 01:10 PM\n352 Parker Hall\n\nSpeaker:\u00a0Yu Gu,\u00a0University of Maryland\n\nTitle:\u00a0KPZ on a large torus\n\nAbstract:\u00a0I will present the recent work with\u00a0Tomasz Komorowski\u00a0and Alex\u00a0Dunlap in which we derived optimal variance bounds on the solution to the KPZ\u00a0equation on a large torus, in certain regimes where the size of the torus\u00a0increases with time. We mostly use the tools from stochastic calculus and I\u00a0will also try to give a heuristic explanation of the 2\/3 and 1\/3 exponents in\u00a0the 1+1 KPZ universality class.\n\nDMS Stochastic Analysis Seminar\nMar 22, 2023 01:10 PM\n352 Parker Hall\n\nSpeaker:\u00a0Cheuk-Yin Lee,\u00a0National Tsing Hua University, Taiwan\n\nTitle:\u00a0Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds\n\nAbstract: In this talk, I will present my joint work with\u00a0David Candil\u00a0and Le Chen\u00a0about nonlinear parabolic SPDEs on a bounded Lipschitz domain driven by a Gaussian noise that is white in time and colored in space, with Dirichlet or Neumann boundary condition. We establish explicit bounds for the moments and correlation function of the solutions under a rough initial condition that is given by a locally finite signed measure. Our focus is on studying how the >moment bounds and related properties of the solutions depend on the rough initial data and the smoothness and geometric property of the domain. For $C^{1,{\\em a}}$-domains with Dirichlet boundary condition, we obtain moment bounds under a weak integrability condition for the initial data which need not be a finite measure. Our results also imply intermittency properties of the solutions.\nDMS Stochastic Analysis Seminar\nMar 15, 2023 01:10 PM\n352 Parker Hall\n\nSpeaker: Dr. Michael Salins, Boston University\n\nTitle: The stochastic heat equation with superlinear forcing\n\nAbstract: I outline some recent results about the stochastic heat equation\u00a0defined on an unbounded spatial domain. In general, the solutions to these\u00a0equations are unbounded in space, which can make the analysis of their behaviors\u00a0difficult. I present global existence and uniqueness results when the equation\u00a0is exposed to superlinear forcing terms in the cases when the forcing term is\u00a0dissipative (pushing away from infinity) and when the superlinear forcing term\u00a0is accretive (pushing toward infinity). I also present a result that proves that\u00a0the laws of the solution have a density in the superlinear dissipative case.\n\nJoint work with Samy Tindel.\n\nDMS Stochastic Analysis Seminar\nMar 01, 2023 01:10 PM\n352 Parker Hall\n\nSpeaker: Dr. Erkan Nane (Auburn)\n\nTitle: Continuity with respect to fractional order for a family of time fractional stochastic heat equations\n\nAbstract: In this talk we present continuity with respect to fractional order of the solution to a certain class of space-time fractional stochastic equations. Our results extend the main results in both [1] and [2].\n\n[1] M. Foondun. Remarks on a fractional-time stochastic equation,\u00a0Proc. Amer. Math. Soc. 149 (2021), 2235-2247.\n[2] D.D. Trong, E. Nane, N.D. Minh, and N.H. Tuan. Continuity of solutions of a class of fractional equations,\u00a0Potential Anal. 49 (2018), no. 3, 423-478.\n\nDMS Stochastic Analysis Seminar\nJan 25, 2023 01:10 PM\n352 Parker Hall\n\nSpeaker: Dr. Le Chen (Auburn)\n\nTitle: Superlinear stochastic heat equation\n\nAbstract: In this talk, we will discuss the superlinear stochastic heat\u00a0equation. It is known that when the forcing term and the diffusion coefficient\u00a0are Lipschitz continuous, there exists a unique random field solution for all\u00a0time, which is called global solution. We explore the existence of a global\u00a0solution when the Lipschitz condition are replaced by certain superlinear growth\u00a0conditions. This gives another instance of the delicate balance between the\u00a0smoothing effect of the heat kernel and the roughening effect of the\u00a0multiplicative noise.\n\nThis talk will be based on a recent work with Jingyu Huang\u00a0and an ongoing project with Mohammud Foondun, Jingyu Huang, and Mickey Salins.\n\nDMS Stochastic Analysis Seminar\nNov 08, 2022 02:30 PM\n356 Parker Hall\n\nSpeaker: Jingyu Huang, University of Birmingham, UK\n\nTitle: Fourier transform method in stochastic differential equation (SPDE)\n\nAbstract:\u00a0We consider the Fourier transform method in stochastic heat equation on $$\\mathbb{R}^d$$\n\n$$\\frac{\\partial \\theta}{\\partial t} = \\frac{1}{2} \\Delta \\theta(t,x) + \\theta(t,x) \\dot{W}(t,x).$$\n\nWe study the existence and uniqueness of the solution under Fourier mode.\n\nThen we apply the similar approach to the turbulent transport of a passive scalar quantity in a stratified, 2-D random velocity field. It is described by the stochastic partial differential equation\n$$\\partial_t \\theta(t,x,y) = \\nu \\Delta \\theta(t,x,y) + \\dot{V}(t,x) \\partial_y \\theta(t,x,y), \\quad t\\ge 0\\:\\: \\text{and}\\:\\: x,y\\in \\mathbb{R},$$\nwhere $$\\dot{V}$$ is some Gaussian noise. We show via a priori bounds that, typically, the solution decays with time. The detailed analysis is based on a probabilistic representation of the solution, which is likely to have other applications as well. This is based on joint work with Davar Khoshnevisan from University of Utah.\n\nDMS Stochastic Analysis Seminar\nNov 01, 2022 02:30 PM\n326 Parker Hall\n\nSpeaker: Prof. Lingjiong Zhu, Florida State University\n\nTitle: Langevin algorithms are core Markov Chain Monte Carlo methods for solving\n\nAbstract: Langevin algorithms are core Markov Chain Monte Carlo methods for solving machine learning problems. These methods arise in several contexts in machine learning and data science including Bayesian (learning) inference problems with high-dimensional models and stochastic non-convex optimization problems including the challenging problems arising in deep learning. In this talk, we illustrate the applications of Langevin algorithms through three examples: (1) Langevin algorithms for non-convex optimization; (2) Decentralized Langevin algorithms; (3) Constrained sampling via penalized Langevin algorithms.\n\nDMS Stochastic Analysis Seminar\nOct 11, 2022 02:30 PM\n326 Parker Hall\n\nSpeaker: Dr. Panqiu XIA\u00a0(Auburn)Title: The moment asymptotics of super-Brownian motionsAbstract: The super-Brownian motion (sBm), or Dawson-Watanabe superprocess, is a typical example of the measure-valued Markov processes. In the spatial dimensional one, the sBm, viewed as a measure on R, is absolutely continuous with respect to the Lebesgue measure. Moreover, the density of this measure is the unique solution to the stochastic heat equation with the diffusion coefficient taking the form of square root. This equation is one of the most important example of the stochastic partial differential equation with non-Lipschitz coefficients. In this talk, I will first give a brief introduction to sBm's and then show some recent results about moment formula and large time and high order moment asymptotics of sBm's.\nDMS Stochastic Analysis Seminar\nSep 27, 2022 02:30 PM\n326 Parker Hall\n\nSpeaker: Prof. Lingjiong Zhu, Florida State UniversityTitle: The Heavy-Tail Phenomenon in stochastic gradient descent (SGD)Abstract: In recent years, various notions of capacity and complexity have been proposed for characterizing the generalization properties of stochastic gradient descent (SGD) in deep learning. Some of the popular notions that correlate well with the performance on unseen data are (i) the flatness of the local minimum found by SGD, which is related to the eigenvalues of the Hessian, (ii) the ratio of the stepsize to the batch-size, which essentially controls the magnitude of the stochastic gradient noise, and (iii) the tail-index, which measures the heaviness of the tails of the network weights at convergence. In this paper, we argue that these three seemingly unrelated perspectives for generalization are deeply linked to each other. We claim that depending on the structure of the Hessian of the loss at the minimum, and the choices of the algorithm parameters, the distribution of the SGD iterates will converge to a heavy-tailed stationary distribution. We rigorously prove this claim in the setting of quadratic optimization: we show that even in a simple linear regression problem with independent and identically distributed data whose distribution has finite moments of all order, the iterates can be heavy-tailed with infinite variance. We further characterize the behavior of the tails with respect to algorithm parameters, the dimension, and the curvature. We then translate our results into insights about the behavior of SGD in deep learning. We support our theory with experiments conducted on synthetic data, fully connected, and convolutional neural networks.\n\nThis is based on the joint work with Mert Gurbuzbalaban and Umut Simsekli.\n\nDMS Stochastic Analysis Seminar\nAug 23, 2022 02:30 PM\n326 Parker Hall\n\nSpeaker: Le Chen\n\nTitle: Matching moment lower bounds for stochastic wave equation\n\nAbstract: The one-dimensional stochastic wave equation with multiplicative space-time white noise has been studied as early as in the Walsh notes in 1980's. The upper bounds for the moment Lyapunov exponents were known in the literature, while to obtain the matching lower bounds has been an open problem for a while. In this talk, we will present a recent joint work (arXiv:2206.10069) with Yuhui Guo and Jian Song from Shandong University, China, where we obtained these matching lower bounds.\n\nDMS Stochastic Analysis Seminar\nApr 26, 2022 12:00 PM\n352 Parker Hall\n\nSpeaker: Antony Pearson\n\nTitle: Adaptive and hybrid classification with domain-dependent digraphs\n\nAbstract: Class cover catch digraph (CCCD) classifiers are a family of nonparametric prototype selection learners. Previous work has demonstrated that CCCD classifiers perform well in the context of class imbalance, whereas state-of-the-art classifiers require resampling or ensemble schemes to achieve similar performance. It is also known that one of the two well-known types of CCCD classifier, the random walk (RW-), performs better than the pure (P-) CCCD classifier in the context of class overlap, i.e., when two classes have substantial similarity. Unfortunately, RW-classifiers suffer from large training time and are less accurate when there is no class overlap. In this work we describe an adaptive decision framework for pure versus random walk classifiers, which may offer superior classification accuracy and sub-cubic computational complexity. We propose a hybrid classifier borrowing the strengths of both types of CCCD classifier that partitions the sample space into a region of high class overlap where a RW-CCCD is trained, and a region in which class supports are separated, where a P-CCCD is trained. The hybrid strategy offers superior classification accuracy compared P-CCCD or RW-CCCD classifiers trained individually, and improved computational complexity over RW-CCCD classifiers.\n\nLast Updated: 09\/21\/2022","date":"2023-03-24 15:10:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 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\": 0.5722854733467102, \"perplexity\": 858.731493451565}, \"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-2023-14\/segments\/1679296945287.43\/warc\/CC-MAIN-20230324144746-20230324174746-00057.warc.gz\"}"} | null | null |
I am giving the following talks and attending the events on composting over the next three months.
9 May Attending NWLDC Composting Road Show at Coalville Market 10-2pm. Part of Intenational Compost Awareness Week activities. Information on Composting, current council offers on bins etc.
10th May. Talk starting at 4pm Composting with Worms at Buckingham Nurseries & Garden Centre.
This SimpleSite has had over 500,000 visits. If you want to promote your hobby, club or other interest SimpleSite offers an effective and cost effective way of reaching people. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,235 |
Nov 26 What Is Markelle Fultz's Future With The 76ers?
Sean Anderson, Ricky Widmer, and Dave Oster take a look at Markelle Fultz's future with the Philadelphia 76ers.
Apr 2 NBA Playoffs: How Does Joel Embiid Injury Hurt The 76ers? | {
"redpajama_set_name": "RedPajamaC4"
} | 8,500 |
Q: Angular Http Get return different in Chrome/Safari vs Firefox I has one Http Get in angular 4, but body response is different in Chrome/Safari vs Firefox
My request is:
this.http.get('https://cors-anywhere.herokuapp.com/https://drive.google.com/uc?export=download&id=0B250MRS8iWM0UFRfc3BBaWRfUlU').subscribe(data => {
// Read the result field from the body response.
console.log(data);
});
My response:
*
*In Chrome/Safari:
*In Firefox:
FIREFOX HEADER:
Anybody can help me? or Any ideas for this problems?
[UPDATED body response:]
+Safari:
1
00:00:24,213 --> 00:00:29,376
Dịch bởi: Nhung Nhung.
2
00:01:23,835 --> 00:01:24,738
Thế nào rồi?
3
00:01:25,130 --> 00:01:27,378
Bà mẹ đang gào khóc
còn ông chú thì đang cáu ầm lên.
4
00:01:28,080 --> 00:01:30,495
- Bà ấy không có chồng à?
- Ly dị, một nách 4 con.
5
00:01:31,297 --> 00:01:33,143
Tôi đoán chắc Cha xứ đang muốn giúp thôi.
6
00:01:33,452 --> 00:01:34,540
Giúp?
+Firefox:
��1
00:00:24,213 --> 00:00:29,376
D�ch b�i: Nhung Nhung.
2
00:01:23,835 --> 00:01:24,738
Th� n�o r�i?
3
00:01:25,130 --> 00:01:27,378
B� m� ang g�o kh�c
c�n �ng ch� th� ang c�u �m l�n.
4
00:01:28,080 --> 00:01:30,495
*
*B� �y kh�ng c� ch�ng �?
*Ly d�, m�t n�ch 4 con.
5
00:01:31,297 --> 00:01:33,143
T�i o�n ch�c Cha x� ang mu�n gi�p th�i.
6
00:01:33,452 --> 00:01:34,540
Gi�p?
7
00:01:37,685 --> 00:01:40,015
Ch�o �ng Burke, h� ang � �ng sau
n�i chuy�n v�i gi[…]
A: From your comments you're using the old Http, you need to do the following:
let url: string = 'https://cors-anywhere.herokuapp.com/https://drive.google.com/uc?export=download&id=0B250MRS8iWM0UFRfc3BBaWRfUlU';
this.http.get(url)
.map((res: Response) => res.json())
.subscribe(data => {
// Read the result field from the body response.
console.log(data);
});
You are missing the map method, which maps the response back to json.
UPDATE to new HttpClientModule:
Add the following to your app.module.ts:
import {HttpClientModule} from '@angular/common/http';
@NgModule({
imports: [
BrowserModule,
HttpClientModule,
],
})
export class MyAppModule {}
Inside your *.component.ts:
import { HttpClient } from '@angular/common/http';
import { Observable } from 'rxjs/Observable';
let url: string = 'https://cors-anywhere.herokuapp.com/https://drive.google.com/uc?export=download&id=0B250MRS8iWM0UFRfc3BBaWRfUlU';
this.http.get<any>(url)
.subscribe((data: any) => {
// Read the result field from the body response.
console.log(data);
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,851 |
\section{Introduction}
Clusters of galaxies are the most massive and recently assembled
structures in the Universe. In the context of the hierarchical growth
of structure in a cold dark matter dominated Universe, clusters are
the repository of copious amounts of the dark matter. Gravitational
lensing, predicted by Einstein's theory of General Relativity, is the
deflection of light rays from distant sources by foreground mass
structures. In its most dramatic manifestation, strong lensing
requires a rare alignment with foreground dense structures and
produces highly distorted, magnified and multiple images of a single
background source (Schneider, Ehlers \& Falco 1992). More commonly,
the observed shapes of background sources viewed via a foreground
cluster lens are systematically elongated, in the so-called weak
lensing regime. Strong and weak lensing offer the most reliable probes
of the distribution of dark matter on various cosmic scales (Blandford
\& Narayan 1992; Mellier 2002; Schneider, Ehlers \& Falco 1992).
Strong lensing studies of the core regions of several clusters
indicate that the dark matter distribution can be represented by a
combination of smoothly distributed, extended cluster mass components and
smaller-scale clumps or subhalos associated with luminous galaxies
(Kneib et al. 1996; Natarajan \& Kneib 1997; Natarajan, Kneib, Smail
\& Ellis 1998). The smooth components have been detected using weak
lensing techniques out to the turn-around radius (typically of the
order of several Mpc) in clusters (Kneib et al. 2003; Gavazzi et al. 2003;
Broadhurst et al. 2005; Bradac et al. 2006; Clowe et al. 2006;
Wittman et al. 2006; Limousin et al. 2007c; Bardeau et al. 2007). To date,
however, attention has largely focused on the lensing derived density
profile of the smooth cluster component, and its agreement with profiles
computed from high resolution numerical simulations of structure
formation in the Universe (Navarro, Frenk \& White 1997; Navarro et
al. 2004; Sand et al. 2004). In fact, the granularity of the dark
matter distribution associated with individual galactic subhalos holds
important clues to the growth and assembly of clusters. Several
earlier studies have explored this issue for the particular case of
Cl\,0024+16 (Tyson, Kochanski \& dell'Antonio 1998; Broadhurst, Huang,
Frye \& Ellis 2000; Jee et al. 2007; Smail et al. 1996).
The detailed mass distribution of clusters and in particular, the
fraction of the total cluster mass associated with individual galaxies
has important consequences for the frequency and nature of galaxy
interactions in clusters (Merritt 1983; Moore et al.\ 1996; Ghigna et
al.\ 1998; Okamato \& Habe 1999). Infalling subhalos suffer a range of
violent fates as the strong gravitational potential of the cluster
tidally strips dark matter and removes baryons via ram-pressure
stripping from them (Cortese et al. 2007). Simulations suggest that
subhalos may not be arranged equally around galaxies of different
morphologies given their varying histories in the cluster environment
(Ghigna et al. 1998; Tormen, Diaferio \& Syer 1998; Springel, White,
Tormen \& Kauffmann 2001). Moreover, subhalos may become tidally
truncated by an amount that will differ substantially over the large
dynamic range in cluster density. Observations of tidal stripping
offer important clues to key questions regarding the growth and
evolution of clusters. How much dark matter is associated with the
subhalos in clusters as a function of radius? To what extent do the
luminous cluster galaxy populations trace the detailed mass
distribution? And, how significant is tidal stripping for the various
morphological galaxy types in the cluster? These are questions we
attempt to answer in this work using observational data and by comparing
with numerical simulations.
To explore cluster galaxy masses, we exploit the technique of
galaxy-galaxy lensing, which was originally proposed as a method to
constrain the masses and spatial extents of field galaxies (Brainerd,
Blandford \& Smail 1996), which has been since extended and developed
over the years to apply inside clusters (Natarajan \& Kneib 1996;
Geiger \& Schneider 1998; Natarajan et al. 1998; 2002a; Natarajan, De
Lucia \& Springel 2007; Limousin et al. 2007a). Previous attempts to
measure the granularity of the dark matter distribution as a function
of cluster-centric radius from observations have had limited
success. Analyzing CFHT (Canada-France-Hawaii Telescope) weak lensing
data from the supercluster MS0302+17, Gavazzi et al.\ (2004) claimed
detection of a radial trend in the extents of dark matter subhalos in
this supercluster region extending out to a few Mpc from the
center. Gavazzi et al. reported that the mass distribution derived
from weak lensing was robustly traced by the luminosity of early-type
galaxies, although their analysis did not include late-type galaxies
or a large-scale smooth component. However, utilizing ground-based
CFHT weak lensing data for a sample of massive clusters at $z = 0.2$,
Limousin et al.\ (2005) did not detect any variation of the dark
matter subhalo masses with cluster-centric radius out to a significant
fraction of the virial radius. The resolution of ground-based data
appears to be inadequate to detect this effect.
In this paper, we present the determination of the mass function of
substructure in Cl\,0024+16 (at $z = 0.39$) in 3 radial bins using
panoramic Hubble Space Telescope {\it HST} imaging data. A high
resolution mass model tightly constrained by current observations is
constructed including individual cluster galaxies and their associated
dark matter subhalos. We show that over a limited mass range we can
successfully construct the mass function of subhalos inside this
cluster as a function of cluster-centric radius. The three bins span
from the center to 5 Mpc (well beyond the the virial radius of 1.7
Mpc) providing us insights into the tidal stripping process. We also
compare properties of the subhalos that host early-type galaxies with
those that host late-type galaxies in Cl\,0024+16. In addition, we
compare the results retrieved from the lensing analysis with results
from the largest cosmological simulation carried out so far - the {\it
Millennium Simulation}. N-body simulations in combination with
the semi-analytic models that we employ in this work are an invaluable
tool for investigating the non-linear growth of structure in detail
and to provide insights into the cluster assembly process.
The outline of this paper is as follows: in \S2, we discuss the
theoretical framework of tidal stripping and galaxy-galaxy lensing in
clusters; in \S3 the observations and modeling are described. The
analysis for Cl\,0024+16 is presented in \S4 including a discussion of
the uncertainties; results and the comparison with clusters in the
Millennium Simulation are described in \S5. We conclude with a
discussion of the implications of our results for the LCDM model and
the future prospects of this work.
\footnote{Throughout this work wherever required we have used the
following values for the cosmological parameters: H$_0$ = 72 km/s/Mpc;
$\Omega_{\rm m}$ = 0.3; $\Omega_{\Lambda}$ = 0.7. At the redshift of
Cl\,0024+16, 1'' = 5.184 kpc.}
\section{Theoretical framework}
\subsection{Tidal stripping and dynamical modification in clusters}
Theoretical studies of cluster formation using simulations and
analytic models predict that there are two key dynamical processes
(Ghigna et al. 1998; Springel, White, Tormen \& Kauffmann 2001; De
Lucia et al. 2004; Moore, Katz, Lake, Dressler \& Oemler 1996; Balogh,
Navarro \& Morris 2000; Merritt 1985) that are relevant to the mass
loss of infalling dark matter subhalos in assembling clusters. The
first process is tidal stripping induced by the interaction of
infalling galaxies and groups with the global tidal field generated by
the smooth dark matter distribution. The second process is
modification to the mass distribution due to high and low velocity
encounters between infalling subhalos (Moore, Katz, Lake, Dressler \&
Oemler 1996).
For the purposes of studying the dynamics of galaxies in clusters we have
partitioned the cluster into 3 distinct regions: the inner
core region where the global tidal field is the strongest, the
transition region where the two above mentioned dynamically transformative
processes occur and finally, the periphery where the dominant
stripping is due to interactions between the infalling galaxies and
groups rather than the global tidal field (Treu et al. 2003). Detailed
study of the properties of cluster galaxies in Cl\,0024+16 by Treu et al. 2003,
find that demarcation into these 3 regions is naturally provided by the
dynamical processes that operate efficiently at various radii from the
cluster center.
In the central region, the gravitational potential of the cluster is
the strongest and tidal stripping is expected to be the dominant
dynamically transformative process. Recent tidal effects are not
expected in the transition region whereas most galaxies inhabiting the
periphery are likely to have never traversed the cluster center. The
galaxies in the outer regions are expected to be modified
predominantly due to local interactions with other nearby galaxies and
groups despite being gravitationally bound to the cluster.
An analytic estimate of the effect of tidal truncation as a function
of cluster-centric radius can be calculated by modeling Cl\,0024+16 as
an isothermal mass distribution and considering the motions of cluster
subhalos in this potential (Merritt 1985). In this framework the tidal
radius of a subhalo hosting a cluster galaxy is given by:
\begin{eqnarray}
R_{\rm tidal} \propto (\frac{\sigma_{\rm gal}}{\sigma_{\rm cluster}})\,r,
\end{eqnarray}
where $R_{\rm tidal}$ is the tidal radius of the subhalo, $\sigma_{\rm
gal}$ is the central velocity dispersion of the galaxy, $\sigma_{\rm
cluster}$ is the velocity dispersion of the cluster and $r$ is the
distance from the cluster center. The current paradigm for structure
formation in the Universe predicts that the masses of infalling
subhalos are a strong function of cluster-centric radius $r$,
indicative of the variation of the strength of tidal stripping from
the periphery (where it is modest) to the inner regions, where it is
severe (Springel, White, Tormen \& Kauffmann 2001; De Lucia et
al. 2004; Moore, Katz, Lake, Dressler \& Oemler 1996; Balogh, Navarro
\& Morris 2000). Mapping the mass function of subhalos directly from
observations offers a powerful way to test these theoretical
predictions.
\subsection{Galaxy-galaxy lensing in clusters}
In this subsection we briefly outline the analysis framework. Details
can be found in several earlier papers (Natarajan \& Kneib 1996;
Natarajan et al. 1998; Natarajan et al. 2002a; 2002b; Natarajan, De
Lucia \& Springel 2007). For the purpose of constraining the
properties of the subhalo population, Cl\,0024+16 is modeled
parametrically as a super-position of smooth large-scale mass
components, which we will refer to with subscript `s' hereafter, and
smaller scale potentials that are associated with bright cluster
members, referred to as perturbers denoted by the subscript
`$p_i$'. Using the same data-set to construct a mass distribution for
Cl\,0024+16 Kneib et al. (2003) found that the best-fit model required
two large-scale components. In our current modeling, we adopt that
parametrization as the prior. In earlier work, our analysis was
limited by data to the inner regions of clusters ($<$ 1 Mpc), and only
to early-type galaxies as perturbers as a consequence (Natarajan et
al. 1998; Natarajan, Kneib \& Smail 2002; Natarajan, De Lucia \&
Springel 2007). With the current data-set we also probe the late-type
cluster member population and statistically constrain parameters that
characterize their dark matter subhalos. There are however, an
insufficent number of late-types in the core region, their numbers
steadily increase with cluster-centric radius. Therefore, in the core
region, we focus on the subhalos of early-types. In effect, the contribution
of late-types in the core region gets inevitably taken into account as
part of the smooth mass distributions. We note here that while we illustrate
our formalism with simple equations to provide insight into our framework,
ultimately the analysis is performed numerically and all the non-linearities
arising in the lensing inversion are taken into account. The gravitational
potential of Cl\,0024+16 is modeled as follows:
\begin{equation}
\phi_{\rm tot} = \Sigma_{n}\,\phi_{\rm s} + \Sigma_i \,\phi_{\rm p_i},
\end{equation}
where the two $\phi_{\rm s}$ ($n=1$ and $n=2$) components represent
the potentials that characterize the smooth component and $\phi_{\rm
p_i}$ are the potentials of the galaxy subhalos treated as
perturbers. The corresponding deflection angle $\alpha_I$ and the
amplification matrix $A^{-1}$ can also be decomposed into independent
contributions from the smooth clumps and perturbers,
\begin{eqnarray}
\alpha_I\,=\,\Sigma_n\,{{\mathbf \nabla}\phi_{\rm s}}\,+\,\Sigma_i \,
{{\mathbf \nabla}\phi_{\rm p_i}},\,\,\\ \nonumber
\,A^{-1}\,=\,I\,-\,\Sigma_n\,{{\mathbf
\nabla\nabla} {\phi_{\rm s}}}\,-\,\Sigma_i \,{{\mathbf \nabla\nabla}
{\phi_{\rm p_i}}}.
\end{eqnarray}
In fact, the amplification matrix can be decomposed as a linear
sum:
\begin{eqnarray}
A^{-1}\,=\,(1\,-\,\Sigma_n\,\kappa_{\rm s}\,-\,\Sigma_i \kappa_{\rm
p})\,I - \Sigma_n\,\gamma_{\rm s}J_{2\theta_{\rm s}} - \Sigma_i \,\gamma_{\rm
p_i}J_{2\theta_{\rm p_i}},
\end{eqnarray}
where $\kappa$ is the magnification
and $\gamma$ the shear. The shear $\gamma$ is written as a complex
number and is used to define the reduced shear $\overline{g}$,
which is the quantity that is measured directly from observations
of the shapes of background galaxies. The reduced shear
can also be further decomposed into contributions from the smooth pieces
and the perturbers:
\begin{eqnarray}
\overline{g_{tot}} = {\overline{\gamma} \over 1-\,\kappa} =
{{\Sigma_n\,\overline\gamma_{\rm s}} + \Sigma_i \,{\overline\gamma_{p_i}} \over
1-\Sigma_n\,\kappa_{\rm s} -\Sigma_i \,\kappa_{p_i}},
\end{eqnarray}
Here $\bar{\gamma}$ is the mean shear of background galaxies in an annulus
around a particular early-type cluster galaxy treated as a local perturber.
In the frame of an individual perturber $j$ (neglecting effect
of perturber $i$ if $i \neq j$), the above simplifies to:
\begin{eqnarray}
{\overline g_{tot}}|_j} = { {\Sigma_n\,{\overline \gamma_{\rm s}}
+{\overline \gamma_{p_j}} \over {1-\Sigma_n\,\kappa_{\rm s} -\kappa_{p_j }}}.
\end{eqnarray}
Restricting our analysis to the weak regime (as mentioned above the analysis
is ultimately performed numerically and includes the effect of strong lensing),
and thereby retaining only the first order terms from the lensing equation for the shape
parameters (e.g. Kneib et al.\ 1996) we have:
\begin{equation}
{\overline g_I}=
{\overline g_S}+{\overline g_{tot}},
\end{equation}
where ${\overline g_I}$ is the distortion
of the image, ${\overline g_S}$ the intrinsic shape of the source,
${\overline g_{\rm tot}}$
is the distortion induced by the lensing potentials (the smooth
component as well as the perturbers). Note that the equations are
outlined here to provide a feel for the technique. The lensing
inversion for the observational data is done numerically taking the
full non-linearities that rise in the strong lensing regime into
account. \footnote{The measured image shape and orientation are used
to construct a complex number whose magnitude is given in terms of the
semi-major axis (a) and semi-minor axis (b) of the image and the
orientation is the phase of the complex number.}
In the local frame of reference of the subhalos, the mean value of the
quantity ${\overline g_I}$ and its dispersion are computed in circular
annuli (at radius $r$ from the perturber centre), assuming a known
value for the smooth cluster component over the area of integration.
In the frame of the perturber, the averaging procedure allows efficient
subtraction of the large-scale component, enabling the extraction of the
shear component induced in the background galaxies only by the local
perturber. The background galaxies are assumed to have intrinsic
ellipticities drawn from a known distribution (see the Appendix for further
details). Schematically the effect of the cluster on the intrinsic
ellipticity distribution of background sources is to cause a coherent
displacement and the presence of perturbers merely adds small-scale
noise to the observed ellipticity distribution. Since we are subtracting
a long-range signal to statistically extract a smaller scale anisotropy
riding on it, we are inherently limited to physical scales on which the
contrast is maximal, i.e. galaxy subhalo scales.
The contribution of the smooth cluster component has 2 effects: it
boosts the shear induced by the perturber which becomes non-negligible
in the cluster center, and it simultaneously dilutes the regular
galaxy-galaxy lensing signal due to the ${\sigma^2_{\overline g_{\rm
s}} / 2}$ term in the dispersion. However, one can in principle
optimize the noise by `subtracting' the measured cluster signal
${\overline g_{\rm s}}$ using a tightly constrained parametric model
for the cluster.
The feasibility of this differenced averaging prescription for
extracting the distortions induced by the possible presence of dark
matter subhalos around cluster galaxies with {\it HST} quality data
has been amply demonstrated in our earlier papers (Natarajan et al.\
1998; 2002a; 2004; 2007). We have also shown with direct comparison to
simulations that the we can reliably recover substructure mass
functions with this technique in the inner 1 Mpc or so of galaxy
clusters. Note here that it is the presence of the underlying
large-scale smooth mass components (with a high value of $\kappa_s$)
that enables the extraction of the weaker signal riding on it.
\section{Observations and modeling}
\subsection{The HST WFPC-2 data-set}
Our dataset comprises a mosaic of 39 sparsely-sampled images of the
rich cluster Cl\,0024+16 ($z = 0.39$) taken by the Wide Field
Planetary Camera-2 on the Hubble Space Telescope ({\it HST}). By applying
lensing techniques to this panoramic imaging dataset, we aim to
characterize the fine scale distribution of dark matter. This unique
dataset extends to the turn-around radius $\simeq$5 Mpc, well beyond
the inner 0.5 - 1 Mpc that has been studied previously. This enables
us to map the detailed dark matter distribution and to calibrate the
tidal stripping effect as a function of distance from the cluster
center. In earlier analysis, we combined strong and weak lensing
constraints to provide an accurate representation of the smooth dark
matter component out to 5 Mpc radius (Kneib et al. 2003). Strong
lensing provides stringent constraints on the mass profile in the
inner region while the detected weak shear constrains the profile out
at large radii (Mellier 2002; Kneib et al. 2003). Non-contiguous,
sparse sampling of the {\it HST} pointings was chosen to maximize radial
coverage. Further details of the data and analyses can be found in
earlier published papers (Kneib et al. 2003; Treu et al. 2003).
We re-iterate here that the WFPC-2 data-set used for this analysis
has been presented already and is described in detailed in earlier works
by our group, including the determination of shapes for the background
galaxies (in Kneib et al. 2003); selection and confirmation of cluster
membership and morphological classifications of cluster galaxies
(in Treu et al. 2003).
Here, we use galaxy-galaxy lensing to detect cluster galaxy subhalos
associated with early-type galaxies and late-type galaxies against the
background of smoothly distributed dark matter in three radial bins.
Using the extensive set of ground-based spectra (Czoske et al. 2001;
Czoske, Moore, Kneib \& Soucail 2002; Moran et al. 2005) and {\it HST}
morphologies (Treu et al. 2003), we first identified early-type and
late-type members to well beyond the virial radius, ($r_{\rm vir} =
1.7\,{\rm Mpc}$), out to $\sim\,5\,{\rm Mpc}$. Details of the data
reduction, cluster membership determination and morphological
classification can be found in Treu et al.(2003).
\subsection{Modeling the cluster Cl\,0024+16}
Cl\,0024+16 is an extremely massive cluster and has a surface mass
density in the inner regions which is significantly higher than the
critical value, therefore produces a number of multiple images of
background sources. By definition, the critical surface mass density
for strong lensing is given by:
\begin{eqnarray}
\Sigma_{\rm crit} = {\frac{c^2}{4 \pi G}} \frac{D_s}{D_d D_{ds}}
\end{eqnarray}
where $D_s$ is the angular diameter distance between the observer and the
source, $D_d$ the angular diameter distance between the observer and
the deflecting lens and $D_{ds}$ the angular diameter distance between
the deflector and the source.
Note that the integrated lensing signal detected is due to all the
mass distributed along the line of sight in a cylinder projected onto
the lens plane. In this and all other cluster lensing work, the
assumption is made that individual clusters dominate the lensing
signal as the probability of encountering two massive rich clusters
along the same line-of-sight is extremely small due to the fact that
these are very rare objects in hierarchical structure formation
models. Cl\,0024+16 is known to have a significant amount of
substructure in velocity space. Czoske et al. (2002) and more recently
Moran et al. (2005) have performed comprehensive redshift surveys of
this cluster and its environs and have enabled the construction of a
3-dimensional picture for this cluster using the $\sim 500$ galaxy
redshifts within about 3 - 5 Mpc from the cluster center. Their
combined data reveal a foreground component of galaxies separated from
the main cluster in velocity space. Both groups argue that this is
likely a remnant of a high-speed collision between the main cluster
and an infalling sub-cluster. The detailed redshift distribution of
cluster members in Cl\,0024+16 is taken carefully into account in our
lensing analysis, starting with a prior that includes 2 large-scale
components to model the smooth mass distribution.
With our current sensitivity limits, galaxy-galaxy lensing within the
cluster provides a determination of the total enclosed mass within an
aperture. We lack sufficient sensitivity to constrain the detailed
mass profile for individual cluster galaxies. With higher resolution
data in the future we hope to be able to obtain constraints on the
slopes of mass profiles within subhalos. In this paper, the subhalos
are modeled as pseudo-isothermal elliptical components (PIEMD models,
derived by Kassiola \& Kovner 1993) centered on galaxies that lie
within a projected radius of out to 5 Mpc from the cluster center and
two NFW profiles are used to model the smooth, large-scale
contribution. We find that the final results obtained for the
characteristics of the subhalos (or perturbers) is largely independent
of the form of the mass distribution used to model the smooth,
large-scale components. A comparison of the best-fit profiles for the
smooth component from lensing with those obtained in high resolution
cosmological N-body simulations has been presented in the work of
Kneib et al. (2003). Combining strong and weak constraints, they were
able to probe the mass profile of the cluster on scales of 0.1-5 Mpc,
thus providing a valuable test of the universal form proposed by
Navarro, Frenk, \& White (NFW) on large scales. We use the best-fit
mass model of Kneib et al. (2003) for the smooth component as a prior
in our analysis, although we allow the parameters like the centroids
of the 2 large-scale components and their velocity dispersion to vary
when obtaining constraints on the subhalos. The 2 NFW components used
as priors are characterized by the following properties: Clump 1: with
$M_{200} = 6.5 \times 10^{14}\,{M_\odot}$; $c = 22_{-5}^{+9} $; $r_{200} =
1.9\,{\rm Mpc} $; $r_s = 88\,{\rm kpc}$ and Clump 2: with $M_{200} =
2.8 \times 10^{14}\,{M_\odot}$; $c = 4_{-1}^{+2}$; $r_{200} = 1.5\,{\rm
Mpc}$; $r_s = 364\,{\rm kpc} $.
To quantify the lensing distortion induced, the individual
galaxy-scale halos are modeled using the PIEMD profile with,
\begin{eqnarray}
\Sigma(R)\,=\,{\Sigma_0 r_0 \over {1 - r_0/r_t}}
({1 \over \sqrt{r_0^2+R^2}}\,-\,{1 \over \sqrt{r_t^2+R^2}}),
\end{eqnarray}
with a model core-radius $r_0$ and a truncation radius $r_t\,\gg\,
r_0$. Correlating the above mass profile with a typical de
Vaucleours light profile (the observed profile for bright early type
galaxies) provides a simple relation between the truncation radius and
the effective radius $R_{\rm e}$, $r_t\sim (4/3) R_{\rm e}$. The
coordinate $R$ is a function of $x$, $y$ and the ellipticity,
\begin{eqnarray}
R^2\,=\,({x^2 \over (1+\epsilon)^2}\,+\,{y^2 \over (1-\epsilon)^2})\,;
\ \ \epsilon= {a-b \over a+b},
\end{eqnarray}
The mass enclosed within an aperture radius $R$ for the $\epsilon = 0$
model is given by:
\begin{equation}
M(R)={2\pi\Sigma_0
r_0 \over {1-{{r_0} \over {r_t}}}}
[\,\sqrt{r_0^2+R^2}\,-\,\sqrt{r_t^2+R^2}\,+\,(r_t-r_0)\,].
\end{equation}
The total mass $M$, is finite with $M\,
\propto \,{\Sigma_0} {r_0} {r_t}$.
The shear is:
\begin{eqnarray}
\gamma(R)\,&=&\,\nonumber \kappa_0[\,-{1 \over \sqrt{R^2 + r_0^2}}\, +\,{2 \over
R^2}(\sqrt{R^2 + r_0^2}-r_0)\,\\ \nonumber &+&\,{1 \over {\sqrt{R^2 +
r_t^2}}}\,-\, {2 \over R^2}(\sqrt{R^2 + r_t^2} - r_t)\,].\\
\end{eqnarray}
In order to relate the light distribution in cluster galaxies to key
parameters of the mass model of subhalos, we adopt a set of physically
motivated scaling laws derived from observations (Brainerd et al.\
1996; Limousin et al. 2005; Halkola et al. 2007):
\begin{eqnarray}
{\sigma_0}\,=\,{\sigma_{0*}}({L \over L^*})^{1 \over 4};\,\,
{r_0}\,=\,{r_{0*}}{({L \over L^*}) ^{1 \over 2}};\,\,
{r_t}\,=\,{r_{t*}}{({L \over L^*})^{\alpha}}.
\end{eqnarray}
The total mass $M$ enclosed within an aperture $r_{t*}$ and
the total mass-to-light ratio $M/L$
then scale with the luminosity as follows for the early-type galaxies:
\begin{eqnarray}
M_{\rm ap}\,\propto\,{\sigma_{0*}^2}{r_{t*}}\,({L \over L^*})^{{1 \over
2}+\alpha},\,\,{M/L}\,\propto\,
{\sigma_{0*}^2}\,{r_{t*}}\left( {L \over L^*} \right )^{\alpha-1/2},
\end{eqnarray}
where $\alpha$ tunes the size of the galaxy halo. In this work
$\alpha$ is taken to be $1/2$. These scaling laws are empirically
motivated by the Faber-Jackson relation for early-type galaxies
(Brainerd, Blandford \& Smail 1996). For late-type cluster members, we
use the analogous Tully-Fisher relation to obtain scalings of
$\sigma_{0*}$ and ${r_{t*}}$ with luminosity. The empirical
Tully-Fisher relation has significantly higher scatter than the
Faber-Jackson relation. In this analysis we do not take the scatter
into account while employing these scaling relations. We assume these
scaling relations and recognize that this could ultimately be a
limitation but the evidence at hand supports the fact that mass traces
light efficiently both on cluster scales (Kneib et al. 2003) and on
galaxy scales (McKay et al. 2001; Wilson et al. 2001; Mandelbaum et
al. 2006). Further explorations of these scaling relations have
recently been presented in Halkola \& Seitz (2007) and Limousin,
Sommer-Larsen, Natarajan \& Milvang-Jensen (2007). The redshift
distribution and intrinsic ellipticity distribution assumed for this
analysis are outlined in the Appendix.
\subsection{The maximum-likelihood method}
Parameters that characterize both the global components and the
subhalos are optimized, using the observed strong lensing features -
positions, magnitudes, geometry of multiple images and measured
spectroscopic redshifts, along with the smoothed shear field as
constraints. With the parameterization presented above, we optimize
and extract values for the central velocity dispersion and the
aperture scale $(\sigma_{0*}, r_{t*})$ for a subhalo hosting a
fiducial $L^*$ cluster galaxy.
Maximum-likelihood analysis is used to obtain significance bounds on
these fiducial parameters that characterize a typical $L^*$ subhalo in the
cluster. The likelihood function of the estimated probability
distribution of the source ellipticities is maximized for a set of
model parameters, given a functional form of the intrinsic ellipticity
distribution measured for faint galaxies. For each `faint' galaxy
$j$, with measured shape $\tau_{\rm obs}$, the intrinsic shape
$\tau_{S_j}$ is estimated in the weak regime by subtracting the
lensing distortion induced by the smooth cluster models and the galaxy
subhalos,
\begin{eqnarray}
\tau_{S_j} \,=\,\tau_{\rm obs_j}\,-{\Sigma_i^{N_c}}\,
{\gamma_{p_i}}\,-\, \Sigma_n\,\gamma_{c},
\end{eqnarray}
where $\Sigma_{i}^{N_{c}}\,{\gamma_{p_i}}$ is the sum of the shear
contribution at a given position $j$ from $N_{c}$ perturbers. This
entire inversion procedure is performed numerically using the code
developed that builds on the ray-tracing routine {\sc lenstool}
written by Kneib (1993). This machinery accurately takes into account
the non-linearities arising in the strong lensing regimeas well. Using a
well-determined `strong lensing' model for the inner-regions along
with the shear field and assuming a known functional form for
$p(\tau_{S})$ the probability distribution for the intrinsic shape
distribution of galaxies in the field, the likelihood for a guessed
model is given by,
\begin{eqnarray}
{\cal L}({{\sigma_{0*}}},{r_{t*}}) =
\Pi_j^{N_{gal}} p(\tau_{S_j}),
\end{eqnarray}
where the marginalization is done over $(\sigma_{0*},r_{t*})$.
We compute ${\cal L}$ assigning the median redshift corresponding to the
observed source magnitude for each arclet. The best fitting model
parameters are then obtained by maximizing the log-likelihood
function $l$ with respect to the parameters ${\sigma_{0*}}$ and ${r_{t*}}$.
Note that the parameters that characterize the smooth component are
also simultaneously optimized. In this work, we perform this
likelihood analysis in each of the 3 radial bins to obtain the set of
$(\sigma_{0*},r_{t*})$ that characterize subhalos in each radial bin.
In summary, the basic steps of our analysis therefore involve lens
inversion, modeling and optimization, which are done using the {\sc
lenstool} software utilities (Kneib 1993; Jullo et al. 2007). These
utilities are used to perform the ray tracing from the image plane to
the source plane with a specified intervening lens. This is achieved
by solving the lens equation iteratively, taking into account the
observed strong lensing features, positions, geometry and magnitudes
of the multiple images. In this case, we also include a constraint on
the location of the critical line (between 2 mirror multiple images)
to tighten the optimization. We fix the core radius of an $L^*$
subhalo to be $0.1\,{\rm kpc}$, as by construction our analysis cannot
constrain this quantity. The measured shear field and the measured
velocity dispersions of early-type galaxies are used as priors in the
likelihood estimation. In addition to the likelihood contours, the
reduced $\chi^2$ for the best-fit model is also found to be robust.
\section{Analysis for Cl\,0024+16}
To detect cluster subhalos, we first select a population of background
galaxies within a magnitude range $23\,<\,I\,<\,26$ (measured in the
F814W filter) and determined their individual shapes to a high degree
of accuracy taking into account the known anisotropy of the point
spread function of the WFPC-2 Camera (Bridle et al. 2002; Kuijken
1999). Details of this procedure and the systematics are described in
detail in Kneib et al. (2003). Shape distortions in this population
were then used to compute the masses of the foreground cluster and its
member subhalos.
To quantify environmental effects on infalling dark matter halos and
noting the three physical regimes discussed earlier, we divided the
cluster into three regions: the central region extending out to $\sim$
600 kpc from the center [core]; the transition region extending out to
$\sim 1.7\,r_{\rm vir} \sim 2.9\,{\rm Mpc}$, [transition] and the
periphery out to $\sim 2.8\,r_{\rm vir} \sim 4.8\,{\rm Mpc}$
[periphery]. These bins partition the cluster into regions of high,
medium and low galaxy number density and dark matter density (Treu et
al. 2003) respectively. The typical surface density of cluster members
in the core region is 120 galaxies per Mpc$^{-2}$; in the transition
region it drops to about 60 galaxies per Mpc$^{-2}$ and in the
periphery it is roughly 50 galaxies per Mpc$^{-2}$ (Treu et al. 2003).
\begin{figure*}
\centerline{\psfig{file=figure1.ps,width=0.8\textwidth}}
\caption{The spatial distribution of morphologically classified
early-type galaxies (red circles) and late-type galaxies (blue
triangles) with measured redshifts in Cl\,0024+16 derived from the
sparsely sampled mosaic using the {\it WFPC-2} Camera aboard the {\it
HST}. The three circles define the radial binning used in our
analysis. The inner-most circle encompasses the {\bf core} region of
the cluster out to 0.6 Mpc, the middle circle the {\bf transition}
region extending out to 2.9 Mpc and the outer circle marks the {\bf
periphery} of the cluster out to 4.8 Mpc. Galaxies plotted here
include spectroscopically confirmed cluster members and galaxies with
secure photometric redshifts in the {\it HST} footprint.}
\end{figure*}
A well defined morphology-density relation is detected in Cl\,0024+16
(Treu et al. 2003; Dressler 1980; Fasano et al. 2000). The fraction
of early-type galaxies declines steeply away from the center, starting
at 70-80\% out to 1 Mpc and decreasing down to 50\% at the
outskirts. In contrast, the late-type galaxy population fraction is
negligible in the center but increases in the transition region and
constitutes 50\% out at the periphery. In fact, Moran et
al. (2007) find that the spirals are kinematically disturbed even
well beyond the virial radius in this cluster. In the core, cluster
membership was defined strictly, and only spectroscopically confirmed
members were used in the galaxy-galaxy lensing analysis. In the
transition region and the periphery, the classification of cluster
members was performed using both spectroscopically and photometrically
determined redshifts. We selected cluster galaxies within
$17\,<\,I\,<\,22$ to ensure comparable degrees of completeness for
both morphological types across all three bins. Our selection
procedure yields 51 early-types in the core; 93 in the transition
region [70 spectroscopically confirmed] and 44 [15 spectroscopically
confirmed] in the periphery. Including early-types from the ground
based survey work (Moran et al. 2007) we have an additional 257
members in the transition region and 294 members in the
periphery. There are a total of 331 late-types (this inventory
includes the {\it HST} mosaic and ground based data) confined to
transition and periphery region. For the early-types in the {\it HST WFPC-2}
mosaic, all 51 in the core are spectroscopically confirmed to be
cluster members, in the outer 2 bins, about 63$\pm$7\% of the
early-types are spectroscopically confirmed, and across all morphologies
$\sim$ 65\% have secure measured redshifts. In addition, we have
redshifts for early-type candidates that lie outside our tiled {\it HST}
mosaic as well photometric redshifts estimates. The radial
distribution of the selected cluster galaxies is shown in
Figure~1. The similarity of the luminosity function of the selected
early-types in the three bins shown in Figure~2 ensures that we have
truly comparable samples with no luminosity bias.
\begin{figure*}
\centerline{\psfig{file=figure2.ps,width=0.7\textwidth}}
\caption{The luminosity function of early-type galaxies in the 3
regions: the number of galaxies per unit area versus magnitude is
shown. It is clear from this plot that there is no systematic
luminosity selection bias with cluster-centric radius for the
early-type cluster members. However, luminosity segregation is
evident in the core region. The luminosity function plotted above
includes spectroscopically confirmed cluster members and those with
secure photometric redshifts in the {\it HST} footprint.}
\end{figure*}
For each radial bin and type, we applied the likelihood analysis
described above to extract the best-fit parameters and significance
bounds for the dark matter halo associated with a fiducial $L^*$
subhalo in the cluster. Gravitational lensing effects are sensitive to
the total mass $M$ enclosed by a subhalo within an aperture $r_{t*}$. To account
for the differing mass-to-light ratios of the early and late-type
galaxies, we utilized the well-known empirical relations between the
velocity dispersion and luminosity for early-types (Faber-Jackson
relation); and equivalently that between the circular velocity and
luminosity (Tully-Fisher relation) for late-type galaxies. We used the
relations determined for Cl\,0024+16 by Moran et al. (2005; 2007) to relate
mass and light in our modeling procedure. An L$^*$ early-type galaxy
and late-type galaxy are assumed to have the same luminosity. The
limitations and systematics of the galaxy-galaxy lensing analysis in
clusters has been described in detail in our all our earlier papers,
below we briefly mention some of the key uncertainties of this method.
The following basic tests were performed for Cl\,0024+16, (i) choosing
random locations (instead of bright, cluster member locations) for the
perturbers; (ii) scrambling the shapes of background galaxies; and
(iii) choosing to associate the perturbers with the faintest (as
opposed to the brightest) galaxies. None of the above yields a
convergent likelihood map, in fact all that is seen in the resultant
2-dimensional likelihood surfaces is noise for all the above test
cases.
While the robustness of our method has been extensively tested and
reported in detail in earlier papers, there are a couple of caveats
and uncertainties inherent to the technique that ought to be
mentioned. In galaxy-galaxy lensing we are only sensitive to a
restricted mass range in terms of secure detection of
substructure. This is due to the fact that we are quantifying a
differential signal above the average tangential shear induced by the
smooth cluster component. Therefore, we are inherently limited by the
average number of distorted background galaxies that lie within the
aperture scale radii of cluster galaxies. This trade-off between
requiring a sufficient number of lensed background galaxies in the
vicinity of the subhalos and the optimum locations for the subhalos
leads us to choose the brightest cluster galaxies in each radial bin.
It is possible that the bulk of the mass in subhalos is in lower mass
clumps, which in this analysis is essentially accounted for as being
part of the smooth components. Also we cannot sensibly quantify the
contribution of close pairs/neighbors individually as it is essentially
a statistical technique.
Our results are robust and we statistically determine the mass of a
dark matter subhalo that hosts an $L^*$ galaxy. Even if we suppose
that the bulk of the dark matter is associated with very low
surface brightness galaxies in clusters, the spatial distribution of
these galaxies is required to be fine-tuned such that these effects do
not show up in the shear field in the any of the 3 regions. In
summary, the principal sources of uncertainty in the above analysis
are (i) shot noise -- we are inherently limited by the finite number
of sources sampled within a few tidal radii of each cluster galaxy;
(ii) the spread in the intrinsic ellipticity distribution of the
source population; (iii) observational errors arising from
uncertainties in the measurement of ellipticities from the images for
the faintest objects and (iv) contamination by foreground galaxies
mistaken as background.
The shot noise is clearly the most significant source of error,
accounting for up to $\sim 50$ per cent; followed by the width of the
intrinsic ellipticity distribution which contributes $\sim 20$ per
cent, and the other sources together contribute $\sim 30$ per cent
(Natarajan, De Lucia \& Springel 2007). This inventory of errors
suggests that the optimal future strategy for such analyses is to go
significantly deeper and wider in terms of the field of view.
\section{Results from galaxy-galaxy lensing}
The fiducial mass of a dark matter subhalo hosting an $L^*$ early-type
galaxy in the central region contained within an aperture of size
$r_{t*} = 45\pm 5\,{\rm kpc}$ is $M =
6.3_{-2.0}^{+2.7}\,\times\,10^{11}\,M_{\odot}$; in the transition
region it increases to $M =
1.3_{-0.6}^{+0.8}\,\times\,10^{12}\,M_{\odot}$, and in the periphery
it increases further to $M =
3.7_{-1.1}^{+1.4}\,\times\,10^{12}\,M_{\odot}$. All error bars
represent 3-$\sigma$ values. These values derived from the likelihood
analysis are shown in Figure~3. The increasing masses of the subhalos
with cluster radius demonstrates that the subhalos that host $L^*$ galaxies
in the inner regions (core and transition) are subject to more severe tidal
truncation than those in the periphery. The mass of a typical subhalo
that hosts an $L^*$ early-type galaxy increases with cluster-centric
radius in concordance with theoretical expectations. The dark matter
subhalo associated with a typical late-type galaxy in the transition
and peripheral region is detected, with an aperture mass of $M =
1.06_{-0.41}^{+0.52}\,\times\,10^{12}\,M_{\odot}$ enclosed within a
radius of $r_{t*} = 25\,\pm 5 {\rm kpc}$ (shown as the solid triangle
in Figure~3). The total mass-to-light ratio for these fiducial subhalos
can also be estimated. The constant total mass-to-light ratio curves
over-plotted on Figure~3, suggest that a typical subhalo hosting an
L$^*$ early-type has a $M/L_{V} \sim 7, 10, 14$ respectively in the
three radial bins and a subhalo hosting an equivalent luminosity
late-type galaxy has a $M/L_V \sim 10$. These values suggest that
galaxies in clusters do possess individual dark matter subhalos that
extend to well beyond the stellar component.
Utilizing the scaling with luminosity provided by the Faber-Jackson
and Tully-Fisher relations, we derived the mass function of subhalos
within each bin (Figure~4). Clearly, the core region where the central
density of the cluster is maximal is expected to be an extreme and
violent environment for infalling galaxies. We interpret our results
to be a consequence of the fact that galaxies in the inner bin are
more tidally truncated as they likely formed earlier and have
therefore had time for many more crossings through the dense cluster
center.
\begin{figure*}
\centerline{\psfig{file=figure3.ps,width=0.8\textwidth}}
\vspace{1cm}
\caption{The fiducial value of the central velocity dispersion
($\sigma_{0*}$) and aperture radius ($r_{t*})$ for an $L^*$
galaxy. These two parameters are chosen in the optimization for the
PIEMD fit to the subhalos. The mass of a subhalo in the context of
this model is proportional to $\sigma_{0*}^2\,r_{t*}$. Over-plotted
are curves of constant mass-to-light ratio in the V-band with values
6, 8, 10, 12 and 14 (increasing from the bottom up). The solid circles
are for subhalos associated with early-type galaxies and the
solid triangle symbol is for the subhalo associated with late-type
galaxies. The plotted error bars are 3-$\sigma$ derived from the
likelihood contours.}
\end{figure*}
These results are in very good agreement with theoretical predictions
wherein galaxies in the inner region are expected to be violently
tidally stripped of their dark matter content, while those in the
periphery are unlikely to have had even a single passage through the
cluster center and therefore be untouched by tidal interactions. A
simple analytic model (Merritt 1985) is used to predict the mass
enclosed within the tidal radius\footnote{The aperture radius $r_t$
that we infer from the lensing analysis is a proxy for the tidal
radius of a dark matter subhalo.} as a function of cluster-centric
distance, and is found to be consistent with our results (solid line
in Figure~6). Our results are also consistent with the findings of Gao
et al.(2004), who found that subhalos closer to the cluster center
retain a smaller fraction of their dark matter. Furthermore, we are
able to quantify the dark matter subhalo masses associated with
late-type galaxies in Cl\,0024+16.
While the mean mass of a dark matter subhalo associated with an
early-type cluster galaxy increases with cluster-centric distance out
to 5 Mpc and they trace the overall spatial distribution of the smooth
mass components robustly. The subhalos associated with late-type
galaxies do not contribute significantly to the total subhalo mass
function at any radius. In fact, it appears that the host subhalos of
late-types do not trace the total dark matter distribution in
clusters. We infer that the mass within 5 Mpc in Cl\,0024+16 is
distributed as follows: $\sim$70\% of the total mass of the cluster is
smoothly distributed, the subhalos associated with early-type galaxies
contribute $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}\,20\%$, and subhalos hosting late-type galaxies
account for the remaining $<\,10\%$.
\subsection{Comparison with N-body simulations}
In this section, we compare the lensing results discussed above with
results from the Millennium Simulation (Springel et al. 2005). The
simulation follows $N = 2160^3$ particles in a box of size
$500\,h^{-1}\,{\rm Mpc}$ on a side, with a particle mass of
$8.6\times10^{8}\,h^{-1}{\rm M}_{\odot}$ (yielding several hundred
particles per subhalo), and with a spatial resolution of $5\,
h^{-1}\,{\rm kpc}$. For each snapshot of the simulation (in total
$64$), substructures within dark matter halos have been identified
using the algorithm {\small SUBFIND} (Springel et al. 2001). We refer
to the original paper for more details on the algorithm. Further
details of the determination of subhalo masses and the biases therein
are discussed in Natarajan, De Lucia \& Springel (2007).
For our comparison with Cl\,0024+16, we have selected all cluster
halos with $M_{200} \ge 8\times10^{14}\,{\rm M}_{\odot}$ from the
simulation box at $z\sim0.4$. A total of $12$ such cluster scale halos
are found. We then use the publicly\footnote{A description of the
publicly available catalogues, and a link to the database can be found
at the following webpage: http://mpa-garching.mpg.de/millennium/}
available results from the semi-analytic model described in De Lucia
\& Blaizot (2007) to select all galaxies in boxes of $10\,h^{-1}\,{\rm
Mpc}$ on a side and centred on the selected halos. We note that the
following nomenclature is used for galaxies in the adopted
semi-analytic model: each FOF group hosts a `central galaxy' (Type 0)
that is located at the position of the most bound particle of the main
halo. All other galaxies attached to dark matter subhalos are labeled
as Type 1 and located at the positions of the most bound particle of
the parent dark matter substructure. Tidal truncation and stripping
can disrupt the substructure down to the resolution limit of the
simulation. A galaxy that is no longer identified with a dark matter
subhalo is labeled as Type 2, and it is assumed not to be affected by
processes that reduce the mass of its parent subhalo. The positions of
Type 2 galaxies are tracked using the position of the most bound
particle of the subhalo before it was disrupted.
We then select all galaxies brighter than $M_K = -18.3$ (this
corresponds to all galaxies brighter than $1/20*L^*$, as $M_{K*} =
-21.37$) and classify as early-types those with $\Delta M = M_B -
M_{bulge} < 0.4$, where $M_B$ is the B-band rest-frame magnitude and
$M_{bulge}$ is the B-band rest frame magnitude of the bulge (Simien \&
de Vaucouleurs 1986). For each simulated cluster, we consider the same
three radial bins used for our lensing analysis and stack the results
for the projections along the $x$, $y$, and $z$ axes. This is done to
mimic as best the projected distances that we employ in our lensing
analysis in the 3 radial bins. In addition, we only consider galaxies
within 1 Mpc (in the redshift direction) from the cluster centre along
the line-of-sight. This choice is motivated by the width of the
measured velocity dispersion histogram in Cl\,0024+16 and therefore
reduces contamination from unassociated structures. The inventory is
as follows: Core region -- the models predict a total of $\sim$ 74
early-types that make the selection cut of which 42 are Type 2
galaxies and 32 are Type 0 \& 1's; Transition region -- the models
predict a total of 83 early-type galaxies that make the selection cut
of which 41 are Type 2 galaxies and 42 are Type 0 \& 1's; Outer region
-- the models predict a total of 22 early-types that make the
selection cut of which 10 are Type 2 galaxies and 12 are Type 0 \&
1's. In contrast, the selection from the observational data of
Cl\,0024+16 yields the following numbers for spectroscopically
confirmed early-types with equivalent selection criteria: Core region
-- 51 early-types; Transition region -- 97 early-types; Outer region
-- 47 early-types.
In Figure~4, we plot the luminosity function of early type galaxies in
the three radial bins considered in this analysis from observations
(thick, solid histograms) and the model (thin solid and dashed
histograms). The thick solid histograms show the luminosity function
of the spectroscopically confirmed early-types in each bin. The thin,
solid histograms show the total model luminosity function for
equivalently selected early-types (this includes Type 1's, Type 0's
and Type 2 galaxies). The dashed histograms show the luminosity
function of Type 2 galaxies only. We note that in all regions the
contribution by number of Type 2 galaxies is comparable to that of
Type 0 and Type 1's. In the core region 58\% of all model early-types
are Type 2's, in the transition region 50\% of all model early-types
are Type 2's and in the outer region 46\% of all model early-types are
Type 2 galaxies.
\begin{figure*}
\centerline{\psfig{file=figure4.ps,width=0.8\textwidth}}
\vspace{1cm}
\caption{The luminosity function of the early-type galaxies in the
three radial bins considered in this analysis. The value of $M_{K*}$
in the K-band is -21.37. Note that the y-axis for the model galaxies
is an averaged number $<N>$ as the total number of selected
early-types is divided by 36 to take into account 12 clusters each
with 3 independent projections. For the observed galaxies the y-axis
denotes the raw number. In all panels, the thick solid histograms
show the luminosity function of spectroscopically confirmed
early-type members from the Cl0024+16 data set. The thin solid
histograms are total luminosity function of early-types in the
simulations including Type 0, Type 1 and Type 2 galaxies. We
separately show the luminosity function of Type 2 early-type
galaxies as the dashed histograms. The fraction of Type 2 galaxies
in the core is 58\%, in the transition region it is 50\% and in the
outer regions it is 46\%.}
\end{figure*}
It is clear from Figure~4 that the luminosity functions of the
early-type galaxies in simulations agree rather well with the observed
ones in all 3 bins. However, we note that the inability of the lensing
analysis to accomodate/distinguish Type 2 galaxies which constitute
roughly half the number of early-types in the model will limit our
analysis. This discrepancy in the total number of early-types that are
hosted in individual dark matter halos in the model versus the lensing
analysis will re-appear when we compare the masses of a typical
subhalo that hosts an $L^*$ early-type galaxy.
In Figure~5 we compare the mass function of dark matter subhalos
obtained from the galaxy-galaxy lensing analysis with that obtained
from averaging the 12 massive clusters (each with 3 independent
projections) in the Millennium Simulation. In the left hand panel, we
plot a direct comparison of the mass functions without taking into
account the discrepancy in number between the observations and the
simulations. In the right hand panel, we scale the observations to
take into account the fraction of Type 2's versus Type 0 \& 1's in
each radial bin. This is done by normalizing the observations to match
the fraction of Type 0s and Type 1s.In order to make a sensible
comparison, we focus on the right hand panel of Figure~5. In the core
region, the mass function from simulations agrees quite nicely with
that determined using the galaxy-galaxy lensing analysis. It is
notable that general shapes of the mass function are in very good
agreement. The agreement between the shape of the mass functions in
transition and outer regions is also good.\footnote{It is worth
mentioning here that in Natarajan, De Lucia \& Springel (2007)
Cl\,0024+16 was the one outlier from the general good agreement. The
lensing determined mass function for Cl\,0024+16 within 1 Mpc was not
in good agreement with the subhalo mass function derived from
simulations. We attribute the current agreement of the mass functions
in the core region ($r < 0.6 Mpc$) to the following 2 key factors (i)
careful selection based on early-type cluster members to mimic the
observations and (ii) more careful classification into Type 1 and Type
2 galaxies and taking their relative numbers into account when
computing the mass function.}
In the inner region, we found in earlier work (Natarajan, De Lucia \&
Springel 2007) that the masses from the simulation tend to be
under-estimated by a factor 2 or so. This offset was found in our
earlier analysis of the core regions in 5 clusters (r < 1 Mpc)
reported in Natarajan, De Lucia \& Springel (2007). The origin of this
offset in the core region has to do primarily with the systematics due
the method employed to determine subhalo masses. The SUBFIND algorithm
used to find dark matter substructures tends to underestimate their
masses (for a more extensive discussion and diagnostic plots see
Figure~3 in Natarajan, De Lucia \& Springel 2007) by a factor of 2 in
the inner regions. Natarajan, De Lucia \& Springel (2007) have
also shown that the cluster-to-cluster variation for simulated halos
is quite large. Therefore, we do not correct for this bias in the current
analysis of Cl\,0024+16.
We note here that the subhalo mass function available from the
simulations and the lensing technique probe a comparable mass range
suggesting that our early-type galaxy selections have been
equivalent. In the periphery, subhalo masses derived from lensing are
of the order of few times $10^{13}\,{M_\odot}$, which is typical of group
masses, suggestive of the presence of infalling groups. Tracking
morphological types and their transformations in this region Treu et
al. (2003) also suggest the prevalence of infalling groups, in
consonance with our lensing results.So we emphasize here, that in the
outskirts of the Cl\,0024+16, it appears that inferred subhalo masses
correspond to group scale masses suggesting that these halos likely
contain other fainter galaxies in addition to the bright, early-type
that we tag in this analysis.
\begin{figure*}
\begin{minipage}{3.5in}
\centerline{\psfig{file=figure5a.ps,width=3.5in}}
\end{minipage} \hspace*{0in}\begin{minipage}{3.5in}
\psfig{file=figure5b.ps,width=3.5in}
\end{minipage}
\caption{Comparison of the mass function determined from galaxy-galaxy
lensing as a function of cluster-centric distance with that determined
from simulated clusters in the Millenium simulation. The solid
histograms are results from the lensing analysis and the dashed ones
are from the Millenium Run. The raw mass function without any
normalization or scaling is shown in the right hand panel, whereas in
the left hand panel the lensing derived mass function is normalized to
compare with the model mass functions.}
\end{figure*}
In Figure~6, we plot the mass of a typical subhalo that hosts an
early-type L$^*$ galaxy as a function of cluster-centric radius
derived from galaxy-galaxy lensing and the simulations. The solid line
in Figure~6 is the trend derived from a simple analytic model of tidal
stripping of galaxies by an isothermal cluster proposed by Merritt
(1985). The dashed-line is the also the analytic model offset
appropriately to compare with the simulation results. The model curve
is nearly identical in slope to the best-fit line going through the
simulation points. The radial trends are in very good agreement
although there is an offset of a factor of $\sim 2.5$ in the mean
value of the subhalo masses. It is likely that systematics in the
lensing also contribute to this discrepancy. The efficiency of tidal
stripping depends on the central density of the cluster. Writing this
out explicitly, we have:$$ M_{\rm lens}/M_{\rm sim} \sim 2.5$$. The
offset in subhalo masses by a factor of $\sim 2.5$ suggests that the
tidal stripping in the averaged simulated clusters is more efficient
than in Cl\,0024+16 as inferred from the lensing data. We note here
that the values of $M_{200}$ for the 12 simulated clusters range from
$\sim\,8\,\times\,10^{14}\,{M_\odot}$ to $\sim\,2\,\times\,
10^{15}\,{M_\odot}$ and the best-fit parameters for Cl\,0024+16 from
observations which consist of a super-position of 2 NFW profiles with
$M_{200} \sim 4 \times 10^{14} {M_\odot}$ and $M_{200} \sim 1.8 \times
10^{14} {M_\odot}$, could partially account for the discrepancy. The
simulated ensemble does not reproduce the observed bi-modal mass
distribution in Cl\,0024+16 which has important dynamical
consequences. No dynamical analog to Cl\,0024+16 was found in the
Millenium Run at $z \sim 0.4$. Therefore it is not surprising that
there is a discrepancy in the inferred mass for a dark matter subhalo
hosting an $L^*$ galaxy. Note that if we correct the subhalo mass in
the inner most bin by a factor of 2 as found in our earlier work, the
agreement gets significantly better in the core region consistent with
our earlier results (Natarajan, De Lucia \& Springel
2007). Regardless, there appears to be an offset despite overall
agreement in the ensemble mass functions (as shown in the right hand
panel of Figure~5). Tidal stripping of dark matter appears to be
more efficient in the simulations compared to estimates from the
lensing data.
We note here that in the Millenium simulation only the dark matter is
followed dynamically but not the baryons. It has been recently argued
that the adiabatic contraction of baryons in the inner regions of
galaxies and clusters is likely to modify density profiles
appreciably. Such modifications will impact the efficiency of tidal
stripping in clusters. This claim has been made in numerical
simulations that include gas cooling and prescriptions for star
formation by Gnedin et al. (2004). Zappacosta et al. (2006) on the
other hand claim using the case of the cluster Abell 2589 that
adiabatic contraction is unimportant for the overall mass distribution
of clusters. A recent study by Limousin et al.(2007b) that examines
the tidal stripping of subhalos in numerical simulations and includes
baryons, find a radial trend in the mass function that is in good
agreement with the lensing derived trend in Cl\,0024+16.
Meanwhile in lensing the systematic arises from the fact that we do
not have measured redshifts for all background sources. While the mass
calibration is most sensitive to the median redshift adopted for the
background galaxies, biases are introduced if the median redshift is
over-estimated or under-estimated.
\begin{figure*}
\centerline{\psfig{file=figure6.ps,width=0.8\textwidth}}
\caption{The variation of the mass of a dark matter subhalo that hosts
an early-type L$^*$ galaxy as a function of cluster centric
radius. The results from the likelihood analysis are used to derive
the subhalo mass for the galaxy-galaxy lensing results and the
counterparts are derived from the Millenium simulation with an
embedded semi-analytic galaxy formation model. This enables selection
of dark matter halos that host a single L$^*$ galaxy akin to our
assumption in the lensing analysis. The solid circles are the data
points from the galaxy-galaxy lensing analysis and the solid squares
are from the Millenium simulation. The upper solid square in the core
region marks the value of the subhalo mass with correction by a factor
of 2 as found in Natarajan, De Lucia \& Natarajan (2007). The solid
triangle is the galaxy-galaxy lensing data point for the subhalo
associated with a late-type $L^*$ galaxy. The radial trend derived
from lensing is in very good agreement with simulations although there
is an offset in the masses which is discussed further in the text.}
\end{figure*}
We note that the galaxy-galaxy lensing technique is sensitive to the
detection of subhalo masses above a threshold value that is determined
by the quality of the observational lensing data. The selection made
in the cluster luminosity function translates into a mass limit. The
contribution of fainter early-types (galaxies fainter than our
selection limit) translates into lower mass subhalos due to the
assumed luminosity scalings. As a consequence, subhalos with lower
masses get included in the mass inventory as constituting the `smooth'
component. The lensing derived mass functions are therefore complete
at the high mass end but are typically incomplete at the low mass
end. The cut-off at the low mass end is hence determined primarily by
the depth of the observational data and the ability to measure shapes
accurately for the faintest background sources. Since galaxy-galaxy
lensing analysis in clusters is inherently statistical, its robustness
is also limited by the ability to accurately pin down the smooth mass
component, subtract it from the observed shear field and then stack
the residuals to characterize the mass of a detectable dark matter
subhalo. In the inner region while the constraints on the smooth
component are tighter due to the presence of strong lensing features,
the smooth component also tends to dominate the overall mass
distribution, so subtracting it is challenging. In the outer regions
while the smooth component is sub-dominant, there are fewer
constraints and the overall value of the shear is significantly lower
as well. These trade-offs cause a varying mass resolution for the
lensing technique as a function of cluster-centric radius. However,
since we assume scaling relations with luminosity, and use the cluster
galaxy luminosity function to determine the mass function, the
detectable limit of subhalo masses is set predominantly by the
magnitude cut adopted for the selected early-type galaxies. Note that
in our comparison with simulations we have restricted ourselves only
to early-types with measured spectroscopic redshifts. Since
spectroscopic follow-up tends to be easier for brighter galaxies, once
again our lensing derived mass function is more complete at the high
mass end and is less so at the low mass end. However, from the
comparison of the mass functions we note that both methods lensing and
the simulations are probing comparable subhalo mass ranges.
\section{Discussion and Conclusions}
Earlier work on galaxy-galaxy lensing in the field has identified a
signal associated with massive halos around typical field galaxies,
extending to beyond 100\,kpc (e.g.\ Brainerd, Blandford \& Smail 1996;
Ebbels et al.\ 2000; Hudson et al.\ 1998; Wilson et al.\ 2001;
Hoekstra et al.\ 2004). In particular, Hoekstra et al.\ (2004) report
the detection of finite truncation radii via weak lensing by galaxies
based on 45.5 deg$^2$ of imaging data from the Red-Sequence Cluster
Survey. Using a truncated isothermal sphere to model the mass in
galaxy halos, they find a best-fit central velocity dispersion for an
$L^*$ galaxy of $\sigma = 136 \pm 5$ kms$^{-1}$ (68\% confidence
limits) and a truncation radius of $185 \pm 30$ kpc. Galaxy-galaxy
lensing results from the analysis of the Sloan Digital Sky Survey data
(Sheldon et al. 2004; Guzik \& Seljak 2002; Mandelbaum et al. 2006)
have contributed to a deeper understanding of the relation between
mass and light. Similar analysis of galaxies in the cores of rich
clusters suggests that the average mass-to-light ratio and spatial
extents of the dark matter halos associated with
morphologically-classified early-type galaxies in these regions may
differ from those of comparable luminosity field galaxies (Natarajan
et al.\ 1998, 2002a). We find that at a given luminosity, galaxies in
clusters have more compact halo sizes and lower masses (by a factor of
2--5) compared to their field counter-parts. The mass-to-light ratios
inferred for cluster galaxies in the V-band are also lower than that
of comparable luminosity field galaxies. This is a strong indication
of the tidal stripping effect of the dense environment on the
properties of dark matter halos. In recent work, using only strong
lensing constraints in the inner regions of the Abell cluster A\,1689
derived from images taken by the Advanced Camera for Surveys (ACS)
aboard {\it HST}, Halkola \& Seitz (2007) also find independently that
the subhalos of cluster galaxies are severely truncated compared to
equivalent luminosity galaxies in the field.
The subhalo mass function represents an important prediction of
hierarchical CDM structure formation models and has been subject of
intense scrutiny since the `satellite crisis' was identified (Moore
et al.\ 1999; Klypin et al.\ 1999). This crisis refers to the fact
that within a radius of $400\,h^{-1}\,{\rm kpc}$, from the Milky Way,
cosmological simulations of structure formation predict $\sim$ 50
dark matter satellites with circular velocities in excess of
$50\,{\rm kms^{-1}}$ and mass greater than
$3\,\times\,10^8\,M_{\odot}$. This number is significantly higher
than the dozen or so satellites actually detected around our
Galaxy. Several explanations have been proposed to resolve this
discrepancy. The missing satellites could for example be identified
with the detected High Velocity Clouds (Kerr \& Sullivan 1969;
Willman et al. 2002; Maller \& Bullock 2004). {\it Warm} or {\it
self-interacting} dark matter could also selectively suppress power
on the small scales, therefore reducing the predicted number of
satellites. The leading hypothesis however remains that the solution
to this problem lies in processes such as heating by a photo-ionizing
background that preferentially suppresses star formation in small
halos at early times (Bullock et al.\ 2000; Benson et al.\ 2002;
Kravtsov et al. 2004). On the scale of galaxy clusters, many more
dark matter structures are expected to be visible, thus making the
comparison with expectation from numerical simulations less affected
by uncertainties in the poorly understood physics of the galaxy
formation. In earlier work, we showed that there is very good
agreement between the lensing observations and the Millenium Run
simulations in the inner 1 Mpc or so for a sample of clusters
(Natarajan, De Lucia \& Springel 2007). Now we are able to extend our
analysis out to 5 Mpc for Cl\,0024+16 due to the unique data-set that
is available. The diagnostic available for comparison here is the
shape of the mass function in each of the three bins. We find that
the overall shape of the subhalo mass functions derived from the 2
independent methods is in very good agreement out to beyond the
virial radius. We find that the mass of a typical subhalo that hosts
an $L^*$ early-type galaxy increases with cluster-centric radius in
concordance with theoretical expectations. However, the estimates of
the mass of a subhalo that hosts an $L^*$ galaxy derived from
simulations is significantly lower than those derived from lensing
observations. The origin of this discrepancy lies in the fact that
tidal stripping appears to be more efficient in the simulations.
Due to the large area probed by this Cl\,0024+16 dataset, we are also
able to constrain the properties of dark matter subhalos associated
with late-type galaxies that preferentially lie in the outer regions
of the cluster (Treu et al. 2003). We report the first detection of
the presence of a dark matter subhalo associated with late-type
galaxies in Cl\,0024+16. While early-type galaxies appear to trace the
overall mass distribution robustly, the subhalos associated with
late-type galaxies do not contribute significantly to the total mass
budget at any radius. In the cluster Cl\,0024+16 within 5 Mpc we find
the following contributions to the total mass: $\sim$70\% of the total
mass of the cluster is smoothly distributed, the subhalos associated
with early-type galaxies contribute $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}\,20\%$, and subhalos
hosting late-type galaxies account for the remaining $<\,10\%$.
The mass resolution of our technique varies slightly with
cluster-centric distance owing to the nature of observational
constraints that dominate the likelihood optimization. While the
strong lensing constraints in the core are the most stringent and
drive the fit in the inner regions, the anisotropy in the shear field
is statistically harder to recover. As we progressively step out in
radius away from the cluster center, the shear of the large scale
smooth component drops, and that of the individual subhalos dominates
but the overall summed shear signal is significantly lower than in the
inner regions. The current analysis is primarily limited by the
quality of the available data. Datasets from the {\it ACS} will allow
mass modeling of lensing clusters at even higher resolutions providing
increasing accuracy enabling better mapping of the lower mass end of
the mass functions of substructure. However, as described above a vast
complement of ground-based observations are also needed for this kind
of comprehensive analysis which is extremely time consuming. Ground
based data provided many important constraints, for instance, the
large number of measured central velocity dispersions for cluster
galaxies (Moran et al. 2007) were used as priors in modeling the
perturbing subhalos that made the optimization more efficient.
Below we summarize the key results on comparison with simulations,
where we mimic-ed the selection process adopted for the observational
data of Cl\,0024+16. Dividing the simulated clusters drawn from the
Millenium Run into 3 equivalent radial bins as the observational data,
we were able to estimate (i) the mass function in each bin and (ii)
the subhalo masses that host $L^*$ early-type galaxies. The shapes of
the lensing derived mass functions are in reasonable agreement
with those derived from simulations when we normalize the lensing
results to the those of the total number of model Type 1's and Type
2's..
Our results provide strong support for the tidal stripping
hypothesis. We also find evidence for the variation in the efficiency
of tidal stripping with cluster-centric radius and morphological type.
We conclude that dark matter in clusters is assembled by the
incorporation of infalling subhalos that are progressively stripped
during their journey through the cluster. The finding of
kinematically disturbed features in the cluster galaxy population by
Moran et al. (2007) corroborates our conclusion. We have significantly
improved on previous ground-based studies as space-based data affords
greater accuracy in shape measurements. Future space-based surveys
coupled with ground-based spectroscopic follow-up will provide an
unprecedented opportunity to follow the cluster assembly process.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,273 |
<?php
namespace Tester\Runner;
use Tester\Environment;
/**
* Single test job.
*/
class Job
{
const
CODE_NONE = -1,
CODE_OK = 0,
CODE_SKIP = 177,
CODE_FAIL = 178,
CODE_ERROR = 255;
/** waiting time between process activity check in microseconds */
const RUN_USLEEP = 10000;
const
RUN_ASYNC = 1,
RUN_COLLECT_ERRORS = 2;
/** @var string test file */
private $file;
/** @var string[] test arguments */
private $args;
/** @var string test output */
private $output;
/** @var string|NULL test error output */
private $errorOutput;
/** @var string[] output headers */
private $headers;
/** @var PhpInterpreter */
private $interpreter;
/** @var resource */
private $proc;
/** @var resource */
private $stdout;
/** @var resource */
private $stderr;
/** @var int */
private $exitCode = self::CODE_NONE;
/**
* @param string test file name
* @return void
*/
public function __construct($testFile, PhpInterpreter $interpreter, array $args = NULL)
{
$this->file = (string) $testFile;
$this->interpreter = $interpreter;
$this->args = (array) $args;
}
/**
* Runs single test.
* @param int RUN_ASYNC | RUN_COLLECT_ERRORS
* @return void
*/
public function run($flags = NULL)
{
putenv(Environment::RUNNER . '=1');
putenv(Environment::COLORS . '=' . (int) Environment::$useColors);
$this->proc = proc_open(
$this->interpreter->getCommandLine()
. ' -n -d register_argc_argv=on ' . \Tester\Helpers::escapeArg($this->file) . ' ' . implode(' ', $this->args),
array(
array('pipe', 'r'),
array('pipe', 'w'),
array('pipe', 'w'),
),
$pipes,
dirname($this->file),
NULL,
array('bypass_shell' => TRUE)
);
list($stdin, $this->stdout, $stderr) = $pipes;
fclose($stdin);
if ($flags & self::RUN_COLLECT_ERRORS) {
$this->stderr = $stderr;
} else {
fclose($stderr);
}
if ($flags & self::RUN_ASYNC) {
stream_set_blocking($this->stdout, 0); // on Windows does not work with proc_open()
if ($this->stderr) {
stream_set_blocking($this->stderr, 0);
}
} else {
while ($this->isRunning()) {
usleep(self::RUN_USLEEP); // stream_select() doesn't work with proc_open()
}
}
}
/**
* Checks if the test is still running.
* @return bool
*/
public function isRunning()
{
if (!is_resource($this->stdout)) {
return FALSE;
}
$this->output .= stream_get_contents($this->stdout);
if ($this->stderr) {
$this->errorOutput .= stream_get_contents($this->stderr);
}
$status = proc_get_status($this->proc);
if ($status['running']) {
return TRUE;
}
fclose($this->stdout);
if ($this->stderr) {
fclose($this->stderr);
}
$code = proc_close($this->proc);
$this->exitCode = $code === self::CODE_NONE ? $status['exitcode'] : $code;
if ($this->interpreter->isCgi() && count($tmp = explode("\r\n\r\n", $this->output, 2)) >= 2) {
list($headers, $this->output) = $tmp;
foreach (explode("\r\n", $headers) as $header) {
$a = strpos($header, ':');
if ($a !== FALSE) {
$this->headers[trim(substr($header, 0, $a))] = (string) trim(substr($header, $a + 1));
}
}
}
return FALSE;
}
/**
* Returns test file path.
* @return string
*/
public function getFile()
{
return $this->file;
}
/**
* Returns script arguments.
* @return string[]
*/
public function getArguments()
{
return $this->args;
}
/**
* Returns exit code.
* @return int
*/
public function getExitCode()
{
return $this->exitCode;
}
/**
* Returns test output.
* @return string
*/
public function getOutput()
{
return $this->output;
}
/**
* Returns test error output.
* @return string|NULL
*/
public function getErrorOutput()
{
return $this->errorOutput;
}
/**
* Returns output headers.
* @return string[]
*/
public function getHeaders()
{
return $this->headers;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,798 |
WORKING LIFE: Lighthouse Futures Trust works with a variety of employers, like John Lewis, as part of its internship programme.
A Leeds charity is helping young people with autism and Learning Difficulties into the world of work.
Lighthouse Futures Trust in Cookridge is running a Supported Internship programme for young people with autistic spectrum condition and/or learning difficulties who are struggling to get on the first rung of the ladder into employment.
ASSISTANCE: The Lighthouse Futures Trust in Cookridge has helped find work placements for young people through its Talent City scheme.
Statistics show that young people on the autistic spectrum have just a 16 per cent chance of employment nationally.
But the group's Talent City programme aims to circumvent those obstacles and to showcase young people's talents and abilities to employers.
The first stage sees students from special schools in Leeds doing one day at week over the academic year at one of the Trust's social enterprises.
HELP: Lighthouse's Caron Munro, left, and Katie Parlett with Jason Clarke, of Lowell.
They can opt for experience in retail at Keepers Coffee and Kitchen on Otley Old Road in Cookridge.
Or the more green-fingered can try their hand at 'Branching Out', the Trust's gardening and grounds maintenance arm. Students have worked at Harewood House, Lotherton Hall, Temple Newsam and with the city council's parks and gardens department.
After completing part one, students can apply to phase two, a more intensive supported internship. They get the chance to work with companies such as Yorkshire Water, KPMG, Johnson & Johnson pharmaceuticals, NHS Primary Care Trust and John Lewis four days a week.
Ms Munro highlighted the case of one young man with selective mutism. His condition meant he would have really struggled in a traditional job interview. But with the trust's help he thrived after going to work for a geological survey company in Leeds logging bore holes.
GARDENING: Branching Out gives people experience in grounds maintenance.
The trust has had some impressive results with its internship programme but is keen to do even more.
They have smashed the average of getting young people into employment and are currently running at 80 per cent outcomes.
If you can help or would like to know more about the Trust's work then email: caron@lighthousefuturestrust.org.uk. | {
"redpajama_set_name": "RedPajamaC4"
} | 577 |
Chisholm NGL (трубопровід для ЗВГ) — трубопровідна система, котра подає зріджені вуглеводневі гази до установки фракціонування в канзаському Конвеї.
Трубопровід призначений для транспортування до Канзасу суміші ЗВГ, отриманої при розробці в Оклахомі нафтогазоносного басейну Анадарко. Він має довжину 202 милі та розрахований на перекачування 42 тисяч барелів на добу.
Chisholm NGL починається на терміналі Кінгфішер, котрий отримує від газопереробних заводів вуглеводневу фракцію та здійснює виділення з неї конденсату в обсягах до 5 тисяч барелів на добу, після чого суміш ЗАГ спрямовується для подальшого фракціонування до Конвею. Термінал Кінгфішер обладнаний ємностями для зберігання 450 тисяч барелів вуглеводнів.
Власниками трубопроводу на паритетних засадах виступають компанії ONEOK та Phillips 66, при цьому остання виконує функцію оператора.
Примітки
Трубопроводи для зріджених вуглеводневих газів | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,140 |
HD Voice - Xtended Reach—new codec management and transcoding capabilities for the company's SBCs—facilitates high-definition service provision.
BURLINGTON, MA, March 18, 2010 — Acme Packet® (NASDAQ: APKT), the leader in session border control solutions, today announced HD Voice - Xtended Reach (HDV-XR), a set of capabilities for its industry-leading Net-Net® session border controllers (SBCs) family that bridge high definition (HD) voice services and applications across IP network borders and ease the transition from standard definition (SD) to HD voice. Support for new HD coder/decoders (codecs) on Acme Packet's Net-Net 9200 platform delivers the transcoding and transrating flexibility needed by fixed and mobile service providers, as well as enterprises and contact centers, to leverage the new generation of codecs found in audio endpoints such as HD-capable mobile handsets, HD telepresence, and IP phones used in audio conferencing and contact center solutions. Additionally, new codec management functions for Acme Packet's entire Net-Net SBC family control codec selection and session routing based on codec to optimize subscriber quality of experience.
HD voice features endpoints equipped with wideband coders/decoders (codecs) deliver CD-quality audio, enabling a much richer communications experience for both wireless and wireline services. Communications-oriented business functions such as audio and video conferencing and customer service can be significantly enhanced by the life-like clarity of HD voice. The high audio quality levels also improve perception in challenging environments such as international phone conversations and noisy venues. Mobile service providers also view HD voice as a driver for the adoption of fixed mobile substitution (FMS), as the single-pair wiring still found in many homes is insufficient for supporting HD phones, many of which require Category 4 wiring or better. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,541 |
BCP vs DR – what's the difference?
BCP vs DR – what's the difference?
It appears as if there is an increase in disasters striking companies around the globe. From something as small as a hacker stealing important information, to as large as a disaster that leaves your premises in ruins, disaster can strike at any time. Many companies are starting to develop plans to prepare for any disasters, two of the most common being Disaster Recovery (DR) and a Business Continuity Plan (BCP).
BCP entered the mainstream just before the year 2000, with the Y2K scare. It's a plan that covers the way a business plans for and maintains critical business functions, directly before, during and after a disaster.
The majority of plans are comprised of activities that ensure maintenance, stability, and recoverability of service. The plan is typically set up on a day-to-day basis, and covers the whole organization. In other words, it's a plan on how to remain operational during and after a disaster.
The main reason companies implement a plan like this is because they wish to remain able to provide their service or product to customers. If something happens and you are not able to deliver to your customers, there is a risk that they will simply go to another company. This will obviously cause you to lose not only customers, but valuable income, some of which may be needed to further recovery.
Disaster Recovery is really more focused on what happens after a disaster. Many times, it's actually a part of the overall continuity plan. While BCP focuses on the whole business, DR plans tend to focus more on the technical side of the business. This includes components such as data backup and recovery, and computer systems.
It's best to think of a BCP as an umbrella policy, with DR as part of it. If companies don't have a DR component of their overall continuity plan, there is a good chance the whole strategy will be either less effective, or useless. On the other hand, DR can actually stand alone, and many companies can do just fine without a full continuity plan.
What should DR and BCP contain?
While these plans are slightly different, they do share the same common goals – to offer support and assistance during a disaster. Therefore, regardless of what type of plan you decide to adopt, there are common elements both need to incorporate in order to be successful.
An operational plan for potential disasters that could happen in your geographical area.
Employee training and cross-training. Your employees should know their role in the plan and be trained in other responsibilities should someone else be unable to perform their role.
A communication plan that includes ways of communicating if networks are down.
Off-site locations for staff and managers to meet and work.
A focus on safety. Foster partnerships and communication with local and emergency response services. Ideally, all employees should at least know basic first aid. Employees who are members of local Emergency Response Teams make great team leaders.
Daily backups of your systems and data. Be sure to also train staff in the testing and recovery of systems.
Training and testing of all employees to practice recovery activities in realistic role-playing scenarios.
Regular audits and updates of your plans to ensure they are still relevant and able to protect your systems and company.
With a plan that is carefully prepared, tested, and updated on a regular basis, you should be able to better weather any disaster. If you are looking for information on how to develop or improve your plans get in touch with us today. | {
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08:30 am Arrive at the Talad Rom-Hop or Tain Market Mea Klong Rail way. It is a long time well-known amazing Mae klong fresh market which sets along sides of the railroad. There are trains passing through about right times a day. When a train is coming, those vendors will close their awnings to make rooms for the train passing it is amazing market and very exciting while a train is running through.
10:00 am Arrived at the Boat pier for Cruise along the canals in a narrow, the Damnoen Saduak floating market, where you will see local vendors in their Thai style canoes, laden with colorful fruits, and vegetables, gently plying their way through the canals looking to sell their goods. You can buy food and fruits.
14.30 pm. Visit to Long Neck Karen village and experience their traditional living.
15.30 pm Time for Elephant Ride this is experience the uniqueness of a ride on the back of an elephant. The ride normally takes half an hour.
16:30 pm Return to Bangkok depending on the traffic.
Pick up: 07:00 am. from your hotel in Bangkok. | {
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