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DoD has started a new Public-Private Talent Exchange program under which its civilian employees can take temporary assignments with certain private sector entities and employees of those entities in turn can be detailed to DoD.
"The PPTE is open to DoD civilian employees at the general schedule grade 12 level and above (or equivalent) and at the federal wage system journeyman level and above. DoD employees and private-sector employees must have knowledge, skills, and abilities to be considered a subject matter expert in their occupational field, perform and meet or exceed all performance standards established at the fully successful level or above," a fact sheet says.
DoD employees further must complete either a confidential or public financial disclosure report, whichever applies; agree to a service obligation on return of twice the length of the assignment; and undergo ethics training and other applicable training. Private sector employees coming into the department would have to do the same; further, they will not be allowed to supervise DoD employees.
The program will be broader than previous exchange programs, it adds, such as one that excludes for-profit private sector organizations and another that is exclusively for IT and cyber workers. | {
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{"url":"http:\/\/www.ck12.org\/book\/Basic-Probability-and-Statistics-A-Full-Course\/section\/8.3\/","text":"<img src=\"https:\/\/d5nxst8fruw4z.cloudfront.net\/atrk.gif?account=iA1Pi1a8Dy00ym\" style=\"display:none\" height=\"1\" width=\"1\" alt=\"\" \/>\n\n# 8.3: Two-Sided Stem-and-Leaf Plots\n\nDifficulty Level: At Grade Created by: CK-12\n\nAs you have learned in an earlier chapter, stem-and-leaf plots are an excellent tool for organizing data. Remember that stem-and-leaf plots are a visual representation of grouped discrete data, but they can also be referred to as a modal representation. This is because by looking at a stem-and-leaf plot, we can determine the mode by quick visual inspection. In the last chapter, you learned about single-sided stem-and-leaf plots. In this lesson, you will learn about two-sided stem-and-leaf plots, which are also often called back-to-back stem-and-leaf plots.\n\nExample 5\n\nThe girls and boys in one of BDF High School's AP English classes are having a contest. They want to see which group can read the most number of books. Mrs. Stubbard, their English teacher, says that the class will tally the number of books each group has read, and the highest mode will be the winner. The following data was collected for the first semester of AP English:\n\nGirls111212171823232324333435444547505151Boys151822222326343535354040424749505051\\begin{align*}& \\text{Girls} \\qquad 11 \\quad 12 \\quad 12 \\quad 17 \\quad 18 \\quad 23 \\quad 23 \\quad 23 \\quad 24 \\quad 33 \\quad 34 \\quad 35 \\quad 44 \\quad 45 \\quad 47 \\quad 50 \\quad 51 \\quad 51\\\\ & \\text{Boys} \\qquad 15 \\quad 18 \\quad 22 \\quad 22 \\quad 23 \\quad 26 \\quad 34 \\quad 35 \\quad 35 \\quad 35 \\quad 40 \\quad 40 \\quad 42 \\quad 47 \\quad 49 \\quad 50 \\quad 50 \\quad 51\\end{align*}\n\na. Draw a two-sided stem-and-leaf plot for the data.\n\nb. Determine the mode for each group.\n\nc. Help Mrs. Stubbard decide which group won the contest.\n\nSolution:\n\na.\n\nb. The mode for the girls is 23 books. It is the number in the girls column that appears most often. The mode for the boys is 35 books. It is the number in the boys column that appears most often.\n\nc. Mrs. Stubbard should decide that the boys group has won the contest.\n\nExample 6\n\nMrs. Cameron teaches AP Statistics at GHI High School. She recently wrote down the class marks for her current grade 12 class and compared it to the previous grade 12 class. The data can be found below. Construct a two-sided stem-and-leaf plot for the data and compare the distributions.\n\n2010 class2009 class70707071727474747475767677787980818282828384858586879398100767676767778787879808082828383838585889195\\begin{align*}\\text{2010 class} \\qquad & 70 \\quad 70 \\quad 70 \\quad 71 \\quad 72 \\quad 74 \\quad 74 \\quad 74 \\quad 74 \\quad 75 \\quad 76 \\quad 76 \\quad 77 \\quad 78 \\quad 79 \\quad 80 \\quad 81\\\\ & 82 \\quad 82 \\quad 82 \\quad 83 \\quad 84 \\quad 85 \\quad 85 \\quad 86 \\quad 87 \\quad 93 \\quad 98 \\quad 100\\\\ \\text{2009 class} \\qquad & 76 \\quad 76 \\quad 76 \\quad 76 \\quad 77 \\quad 78 \\quad 78 \\quad 78 \\quad 79 \\quad 80 \\quad 80 \\quad 82 \\quad 82 \\quad 83 \\quad 83 \\quad 83 \\quad 85 \\\\ & 85 \\quad 88 \\quad 91 \\quad 95\\end{align*}\n\nSolution:\n\nThere is a wide variation in the marks for both years in Mrs. Cameron\u2019s AP Statistics Class. In 2009, her class had marks anywhere from 76 to 95. In 2010, the class marks ranged from 70 to 100. The mode for the 2009 class was 76, but for the 2010 class, it was 74. It would seem that the 2009 class had, indeed, done slightly better than Mrs. Cameron\u2019s current class.\n\nExample 7\n\nThe following data was collected in a survey done by Connor and Scott for their statistics project. The data represents the ages of people who entered into a new hardware store within its first half hour of opening on its opening weekend. The M's in the data represent males, and the F's represent females.\n\n12M\u00a018F15F\u00a015M\u00a010M21F25M21M26F\u00a029F\u00a029F\u00a031M33M35M35M35M41F\u00a042F\u00a042M45M46F\u00a048F\u00a051M51M55F\u00a056M58M59M60M60F\u00a0\u00a061F65M65M66M70M70M71M71M\u00a072M72F\\begin{align*}&12M \\quad \\ 18F \\quad 15F \\quad \\ 15M \\quad \\ 10M \\quad 21F \\quad 25M \\quad 21M\\\\ & 26F \\quad \\ 29F \\quad \\ 29F \\quad \\ 31M \\quad 33M \\quad 35M \\quad 35M \\quad 35M\\\\ & 41F \\quad \\ 42F \\quad \\ 42M \\quad 45M \\quad 46F \\quad \\ 48F \\quad \\ 51M \\quad 51M\\\\ & 55F \\quad \\ 56M \\quad 58M \\quad 59M \\quad 60M \\quad 60F \\quad \\ \\ 61F \\quad 65M\\\\ & 65M \\quad 66M \\quad 70M \\quad 70M \\quad 71M \\quad 71M \\quad \\ 72M \\quad 72F\\end{align*}\n\nConstruct a back-to-back stem-and-leaf plot showing the ages of male customers and the ages of female customers. Compare the distributions.\n\nSolution:\n\nFor the male customers, the ages ranged from 10 to 72. The ages for the male customers were spread out throughout this range, with the mode being age 35. In other words, for the males found to be at the store in the first half hour of opening day, there was no real age category where a concentration of males could be found.\n\nFor the female customers, the ages ranged from 15 to 72, but they were concentrated between 21 and 48. The mode for the ages of the female customers was 29 years of age.\n\n### Notes\/Highlights Having trouble? Report an issue.\n\nColor Highlighted Text Notes\n\nShow Hide Details\nDescription\nTags:\nSubjects:","date":"2016-10-23 02:09:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 3, \"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\": 0.9999982118606567, \"perplexity\": 656.3330532654934}, \"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-2016-44\/segments\/1476988719136.58\/warc\/CC-MAIN-20161020183839-00241-ip-10-171-6-4.ec2.internal.warc.gz\"}"} | null | null |
Neurotrophin-3 is a protein that in humans is encoded by the NTF3 gene.
The protein encoded by this gene, NT-3, is a neurotrophic factor in the NGF (Nerve Growth Factor) family of neurotrophins. It is a protein growth factor which has activity on certain neurons of the peripheral and central nervous system; it helps to support the survival and differentiation of existing neurons, and encourages the growth and differentiation of new neurons and synapses. NT-3 was the third neurotrophic factor to be characterized, after nerve growth factor (NGF) and BDNF (Brain Derived Neurotrophic Factor).
Function
Although the vast majority of neurons in the mammalian brain are formed prenatally, parts of the adult brain retain the ability to grow new neurons from neural stem cells; a process known as neurogenesis. Neurotrophins are chemicals that help to stimulate and control neurogenesis.
NT-3 is unique in the number of neurons it can potentially stimulate, given its ability to activate two of the receptor tyrosine kinase neurotrophin receptors (TrkC and TrkB).
Mice born without the ability to make NT-3 have loss of proprioceptive and subsets of mechanoreceptive sensory neurons.
Mechanism of action
NT-3 binds three receptors on the surface of cells which are capable of responding to this growth factor:
TrkC (pronounced "Track C"), is apparently the "physiologic" receptor, in that it binds with greatest affinity to NT-3.
However, NT-3 is capable of binding and signaling through a TrkC-related receptors called TrkB.
Finally, NT-3 also binds a second-receptor type besides Trk receptors, called the LNGFR (for "low affinity nerve growth factor receptor).
High affinity receptors
TrkC is a receptor tyrosine kinase (meaning it mediates its actions by causing the addition of phosphate molecules on certain tyrosines in the cell, activating cellular signaling).
As mentioned above, there are other related Trk receptors, TrkA and TrkB. Also as mentioned, there are other neurotrophic factors structurally related to NT-3:
NGF (for "Nerve Growth Factor")
BDNF (for "Brain Derived Neurotrophic Factor")
NT-4 (for "Neurotrophin-4")
While TrkB mediates the effects of BDNF, NT-4, and NT-3, TrkA binds and is activated by NGF, and TrkC binds and is activated only by NT-3.
Low affinity receptors
The other NT-3 receptor, the LNGFR, plays a somewhat less clear role. Some researchers have shown the LNGFR binds and serves as a "sink" for neurotrophins.
The crystal structure of NT-3 shows that NT-3 forms a central homodimer around which two glycosylated p75 LNGFR molecules bind symmetrically. The symmetrical binding takes place along the NT-3 interfaces, resulting in a 2:2 ligand-receptor cluster in the center.
Cells which express both the LNGFR and the Trk receptors might therefore have a greater activity – since they have a higher "microconcentration" of the neurotrophin.
It has also been shown, however, that the LNGFR may signal a cell to die via apoptosis – so therefore cells expressing the LNGFR in the absence of Trk receptors may die rather than live in the presence of a neurotrophin.
See also
Tropomyosin receptor kinase B § Agonists
References
Further reading
Neurotrophic factors
Peptide hormones
Growth factors
Developmental neuroscience
Proteins
TrkB agonists | {
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{"url":"http:\/\/quant.stackexchange.com\/tags\/trading-systems\/hot?filter=year","text":"# Tag Info\n\n4\n\nI am not sure Dark Pools (DP) have been created to avoid \"market manipulation\". They have been created by firms because they found an advantage to create them (see Market Microstructure in Practice, L and Laruelle Eds.). The main reasons have been: spare market fees, for DP created by brokers (like UBS MTF); spare market impact, for block pools (like ...\n\n1\n\nA public order book gives traders information not only on the current price of a security, but also the volume and structure of the entire supply and demand schedule. Such information can be used for arbitrage and market manipulation strategies in various ways: Spoofing: Inserting a large limit order as an apparent buy or sell signal which is canceled any ...\n\n1\n\nYour source is not particularly clear about why what they're doing is a Z-score. To give some background, what they're doing is calculating $$\\frac{R-\\mu_{R}}{\\sigma_{R}}$$ where R is the number of runs and the mean and standard deviation are of the number of runs. It's really more of a test statistic than a Z-score per se. The denominator in their formula ...\n\n1\n\nFirstly, I suggest you to use more recognized source to study and compute quantitative finance model or indicators; in such case, for instance, you could take as example the following paper as reference. Precisely there, the authors describe some common errors that one can do in computing the Sortino ratio; although surely you did not do any of them, ...\n\n1\n\nI don't know if there is a standard way of solving the problem, but I solve it thus: Strategy A bought for $C_a$ dollars and sold for $S_a$ dollars for a result of $R_a = S_a - C_a$ over $T_a$ days. Strategy B bought for $C_b$ dollars and sold for $S_b$ dollars for a result of $R_b = S_b - C_b$ over $T_b$ days. Where $C_a$ and $C_b$ is the total sum of ...\n\n1\n\nIn trading you need to make a lot of simple computation of a very large flow of data. FPGA are perfect that for. It is typically FPGA that will host marketfeed handler (see NOVASPARKS website, or ACCELLIZE) ; analytics computations ; risk computation (see ULLINK solution for instance). For more, this generic article is not that bad: Introducing ...\n\n1\n\nFPGA's are used to run the latency sensitive HFT strategies. They can also be used solely for parsing whatever protocol is in use (FIX, ITCH, etc..) and routing the decoded objects to a CPU for number crunching. They can of course be used for anything else but these two uses are what is most common now.\n\n1\n\nI think the problem is not that you optimize a wrong criterion, but the trading strategy itself. Compare this to testing a hypothesis: if you reject at p-value of 1% then the proportion of true discoveries among all discoveries is, say, 70% (high \"expectancy\"). If you reject at 10% then the true discovery proportion is 40% (lower \"expectancy\"), but you make ...\n\n1\n\nUse your total wealth allocated to the trades as denominator. Total wealth allocated would include all collateral. In this way you (or your broker) make sure that the denominator is always positive. Presumably this would also reflect what you really want to track. The only problem that remains is what amount of your wealth needs to be allocated. But this is ...\n\n1\n\nI think the best choice for technical analysis with node is node-talib, a wrapper around TA-Lib. We're using it for some projects and it works ok so far. Here's a list of the indicators you get out of the box: AD Chaikin A\/D Line ADOSC Chaikin A\/D Oscillator ADX Average Directional Movement Index ADXR ...\n\n1\n\nI am using NodeJS for a similar project. There's not a ton of packages on NPM for finance and stocks, so I wrote my own, that might help you get started: Fetching historical stock data, including intraday: https:\/\/www.npmjs.org\/package\/node-activetick Charting, analysing, forecasting the data: https:\/\/www.npmjs.org\/package\/timeseries-analysis You can ...\n\n1\n\nCloud9Trader uses Node.js on the back end and JavaScript across its technology stack, including for writing the trading algorithms themselves. https:\/\/www.cloud9trader.com\n\nOnly top voted, non community-wiki answers of a minimum length are eligible","date":"2015-09-05 03:49:46","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\": 0.5884567499160767, \"perplexity\": 1373.6415849025288}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-35\/segments\/1440645378542.93\/warc\/CC-MAIN-20150827031618-00195-ip-10-171-96-226.ec2.internal.warc.gz\"}"} | null | null |
Home Statistics Filmstars Luke Baines Height, Weight, Age, Body Statistics
Luke Baines Height, Weight, Age, Body Statistics
Luke Baines Quick Info
Height 5 ft 11 in
Date of Birth June 8, 1990
Luke Baines is a well distinguished English-Australian actor who marked his place in the entertainment industry with the role of Tom in The Girl in the Photographs. He has also starred in several TV series such as Home and Away (2004) and The Cooks (2004). Luke has a huge fan base with more than 100k followers on Instagram.
Luke Joseph Baines
Luke Baines in an Instagram selfie as seen in August 2016 (Luke Baines / Instagram)
Hyde, Greater Manchester, United Kingdom
Luke divides his time between Los Angeles, Sydney, and London.
Luke studied performing arts at The McDonald College in North Strathfield.
Mother – Susan Baines
Siblings – Jessica McNamee (Older Sister)
Luke is represented by –
Authentic Talent & Literary Management
Agency for the Performing Arts
5 ft 11 in or 180.5 cm
Girlfriend / Spouse
Luke has dated –
Miranda Rae Mayo – Co-stars Miranda Rae Mayo and Luke Baines dated together for some time. Luke even shared a picture on his Instagram sharing a kiss.
Luke Baines and Miranda Rae Mayo as seen in July 2015 (Luke Baines / Instagram)
Light Brown (Natural)
Pixie Ears
Luke Baines in a selfie as seen in November 2017 (Luke Baines / Instagram)
Brand Endorsements
Luke has done endorsement work for brands such as –
Roark Collective
Best Known For
Starring as Tom in The Girl in the Photographs (2016)
Being cast in several TV shows including as Home and Away (2004)
Appearing on the cover of the Leader Magazine Australia on December 8, 2010
Luke made his theatrical film debut as a waiter in the comedy-biography movie Saving Mr. Banks in 2013.
First TV Show
Luke made his first TV show appearance on the news talk-show Today in September 2004.
Luke Baines Favorite Things
Movie – Hocus Pocus (1993)
Quote – "I am human, therefore nothing human is alien to me." – Terence
Source – PopularTV.com, Twitter
Luke Baines as seen in September 2018 (Luke Baines / Instagram)
Luke Baines Facts
He was born in Hyde, Greater Manchester, but raised in Cronulla, Sydney, Australia.
Luke was 5 years old when he started attending acting lessons.
He loves to eat chocolate.
The meaning of a perfect day for Luke is being on set by day and with friends by night.
At the age of 13, he was rescued by a supermodel who kept him from drowning in the surf.
When he was 3 years old, he got a pair of shoes from "Nike".
In 2008, he auditioned for a role in West End's musical production Spring Awakening.
He was 11 years old when he saw director Wes Craven's movie Scream (1996) after which he began watching every Wes Craven film that was available.
Follow him on Instagram and Twitter.
Featured Image by Luke Baines / Instagram
Luke Baines
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Joseph Sim Height, Weight, Age, Body Statistics | {
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El Pontificio Consejo para la Pastoral de los Emigrantes e Itinerantes (en latín: Pontificium Consilium de Spirituali Migrantium atque Itinerantium Cura; en italiano: Pontificio consiglio della pastorale per i migranti e gli itineranti) fue un organismo de la Curia Romana. Fue suprimido el 1 de enero de 2017 y sus competencias fueron asumidas por el Dicasterio para el Servicio del Desarrollo Humano Integral.
Historia
Con el Motu Proprio Apostolicae Caritatis, del 19 de marzo de 1970, el Papa Pablo VI estableció la Pontificia Commissio de Spirituali Migratorum atque Itinerantiun Cura, con la tarea destinada al estudo y la aplicación de la pastoral para las personas en movimiento: emigrantes, exilados, refugiados, prófugos, pescadores y navegantes marítimos, los que trabajan con los transportes en carretera y en los parques de diversiones, nómadas, circenses, peregrinos y turistas. En fin, para todos aquellos grupos de personas que, de manera diferente, están envueltos en el fenómeno de la movilidad humana, como los estudiantes extranjeros, los operadores y los técnicos, los cuales deben trasladarse de un país a otro para los grandes trabajos o investigaciones científicas a nivel internacional.
Hasta entonces, la competencia para los distintos sectores de la movilidad humana estaba asignada a diversas oficinas junto a las Congregaciones Romanas. En la segunda mitad del siglo XIX, era la Congregación de Propaganda Fide (hoy, Congregación para la Evangelización de los Pueblos), en adelante "el movimiento". Más tarde, especialmente por influencia del Obispo Beato Juan Bautista Scalabrini, fue creada la Oficina de Sanación Espiritual de los Inmigrantes, junto a la Congregación Consistorial. Después de la Segunda Guerra Mundial, en 1952, fue instituido por el Papa Pío XII el "Consejo Superior para las migraciones" junto a la misma Congregación, ahora denominada Congregación para los Obispos.
En el mismo año, y siempre junto al mismo dicasterio se creó la Obra del Apostolatus Maris a favor de los navegantes marítimos. En 1958 el mismo Pío XII confió a tal Congregación la tarea de brindar a asistencia espiritual a los fieles con encargos específicos o actividades a bordo de los aviones, así como de los pasajeros que viajan con tales medios de transporte; a estas instituciones se les dio el nombre de Obra del Apostolatus Coeli et Aëris.
En 1965 fue Pablo VI quien fundó, siempre junto a la Congregación Consistorial, el Secretariado Internacional para la dirección de la Obra del Apostolatus Nomadum, en el intento de llevar consuelo espiritual a una población sin vivienda fija y también para aquellos hombres que viven en condiciones similares.
En 1967 a la Sagrada Congregación para el Clero se le concedió una oficina que debía garantizar la asistencia religiosa para todas aquellas personas que se encontrasen en el ámbito del fenómeno turístico. Pero con el Motu Próprio Apostolicae Caritatis las competencias para los diversos Sectores de la movilidad humana convergieron en la Pontificia Commissio de Spirituali Migratorum atque Itinerantium cura, que fue subordinada a la Congregación para los Obispos. Tal situación terminó con la Constituición Apostólica Pastor Bonus, del 28 de junio de 1988, que también cambió su nombre.
Fue suprimido el 1 de enero de 2017 y sus competencias fueron asumidas por el Dicasterio para el Servicio del Desarrollo Humano Integral.
Atribuciones
Las categorías de personas que, por motivo de sus condiciones de vida, no pueden disfrutar del ministerio ordinario de los párrocos o están privadas de cualquier asistencia, son por lo tanto los migrantes, los exilados, los prófugos y los refugiados, los pescadores y los marineros, los que trabajan en los transportes aéreos, los nómadas, las personas del circo y de los parques de diversiones, los que realizan viajes por motivos de misericordia, de estudio o de diversión, los que trabajan con los transportes terrestres y otras categorías semejantes.
El Pontificio Consejo, "un instrumento en las manos del Papa", dirige la solicitud pastoral de la Iglesia a las necesidades particulares de las personas que se han visto obligadas a abandonar su patria o no la tienen; igualmente, pretende seguir con la debida atención las cuestiones relacionadas en esta materia.
Esto pruemove, por lo tanto, el cuidado pastoral de las personas involucradas en la movilidad humana: proporcionando que las Iglesias locales ofrezcan una eficaz y apropiada asistencia espiritual, también si es necesario mediante adecuadas estructuras pastorales; ejercitando la alta dirección de la Obra del Apostolado del Mar; siguiendo detenidamente los problemas relacionados con la movilidad humana; esforzándose para que el pueblo cristiano adquiera conciencia de las necesidades de las personas implicadas en la movilidad humana, especialmente cuando se celebre el Día Mundial de los Migrantes y los Refugiados; haciendo que el pueblo cristiano manifieste eficazmente su solidaridad en las confrontaciones de las personas en movimiento por las formas en que el mundo se relaciona, a fin de que los viajes hechos por motivos de piedad, de estudio o de diversión contribuyan a la formación moral y religiosa de los fieles.
Además, el Consejo tiene por objeto sustentar, de manera regular y directa, a la Comisión Católica Internacional para las Migraciones, apoyando los objetivos y las iniciativas, participando en los encuentros de Dirección, promoviendo una activa cooperación con este Comité y entre esa y los otros organismos que se interesan por los migrantes y refugiados.
Presidentes
Fuentes
Referencias
Antiguos Pontificios Consejos
Organizaciones religiosas fundadas en 1970
Migración
Organizaciones desaparecidas en 2017 | {
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REVIEW: The Rocky Horror Show at The Alexandra
Posted by Dan Richards at Tuesday, May 14th, 2019 12:37am
Sometimes, when a show's been running for a while, it can become stale - something that's wheeled out to keep the fan's happy but with a complacency that reeks of 'I'm in it for the pay check'. That couldn't be a description any further from the reality of the current UK tour of The Rocky Horror Show.
Since its first appearance at the Royal Court Theatre in 1973, Richard O'Brien's creation has become the world's favourite rock n' roll musical. It's been performed worldwide in over 30 countries, on every continent, and has been translated into more than 20 languages. A humorous tribute to the science fiction and horror B movies of the late 1940s through to the early 1970s, the musical tells the story of a newly engaged couple getting caught in a storm and coming to the home of a mad transvestite scientist unveiling his new creation, a Frankenstein-style monster in the form of an artificially made, fully grown, physically perfect muscle man named Rocky Horror, complete "with blond hair and a tan".
From start to finish, the show is an in your face, confident, well rehearsed phenomenon. With a clever set and a sparkling cast, I defy anybody who watches the production to not instantly be a convert - whether they turn up in stockings or take the more conservative approach!
Joanne Clifton (Strictly Come Dancing, Flashdance) takes to the stage as Janet, the (initially!) timid girlfriend of Brad. Having seen Clifton in Flashdance, I knew she was more than 'a pro from Strictly', but whilst that was a fairly dance-heavy part, Janet is a role that allowed Clifton to flourish. With a handful of shows to her name, you'd be forgiven for thinking Joanne had been performing in musicals for decades. With a confidence and reassurance that shone through, she was welcomed into the Rocky Horror fraternity with open arms.
A1 vocalist, Ben Adams (Flashdance) takes the role of Brad and, in a world where pop-stars have tried (and in many cases, failed!) to cross over to the works of musical theatre, Adams proves it can be done - and done very well. With a strong voice, excellent comedy timing and perfect characterisation, it's evident he's having the tie of his life on stage; something which translates perfectly to the audience.
Top marks, however, have to go to both Duncan James as Frank N Furter and everyone's favourite Brummie, Alison Hammond, as the narrator. In different ways, each became the star of the show. James in the traditional sense with a stunning, flamboyant and, at times, emotional performance as Frank. The role is somewhat iconic and his performance had the hardcore, cult fans of the show on their feet in support; surely the best accolade he could hope for.
Hammond, on the other hand, has possibly the hardest and the best job of the evening. She is without doubt, in her element and put across one of the funniest, most spontaneous performances I've seen in a long time. She had every member of the audience in tears, with proper belly laughs all around the theatre.
Contending with huge amounts of, often spontaneous, audience interaction. It's safe to say the fans at the front of the stalls know the script line for line and are waiting for their cue to shout responses to lines. Alison, with her infectious laugh and larger than life personality, was a brilliant choice for the role and the response from the audience when she first appeared spoke volumes. She loves performing and the crowd loved her.
This isn't a show for the faint hearted and it's always a risk for each cast when they take on the behemoth that is Rocky Horror. But, if you're up for a laugh and ready to party to the Time Warp, you need to grab some tickets before they go. The show itself is a hoot and this cast are a rip-roaring success.
The Rocky Horror Show runs at Birmingham's Alexandra Theatre until Saturday 25th May. For more information, or to get your tickets, head to atgtickets.com/birmingham. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,355 |
#region Copyright
#endregion
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using Kodestruct.Common.Section.Interfaces;
using Kodestruct.Common.Section.SectionTypes;
namespace Kodestruct.Common.Section.Predefined
{
/// <summary>
/// Predefined rectangular section is used for rectangular shapes having known properties
/// from catalog such as precast concrete shapes.
/// </summary>
public class PredefinedSectionRectangle : SectionPredefinedBase, ISectionRectangular, ISliceableShapeProvider
{
public PredefinedSectionRectangle(ISection section)
: base(section)
{
}
public PredefinedSectionRectangle(double Width, double Height, ISection section)
: base(section)
{
this.B = Width;
this.H = Height;
}
public double H { get; set; }
public double B { get; set; }
public ISection GetWeakAxisClone()
{
throw new NotImplementedException();
}
//public override ISection Clone()
//{
// throw new NotImplementedException();
//}
public ISliceableSection GetSliceableShape()
{
SectionRectangular r = new SectionRectangular(this.B, this.H);
return r;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,326 |
using BookLibrary.Data.Models;
using BookLibrary.Web.Infrastructure.Mapping;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Web;
namespace BookLibrary.Web.ViewModels.Home
{
public class BookViewModel : IMapFrom<Book>
{
public int Id { get; set; }
public string Name { get; set; }
public bool IsAvailable { get; set; }
public string RentedBy { get; set; }
public string LastRenter { get; set; }
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,143 |
>>>> add ports to a build.
>>>> ERROR: There are no ports configured in the datastore.
>>>> Can anyone suggest what might be wrong here or what I should check?
>>> Can you provide a dump of your ports and build_ports tables in SQL?
>>> Running sh -x tc ... may also help show what's going on.
>> that to the list?
> Post it to a website if possible, then post the link to the list. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,487 |
Serilog sink to send log events to a web service as Json
###Sample receiver with ServiceStack
Service model:
```c#
using ServiceStack.ServiceHost;
using System;
using System.Collections.Generic;
namespace API.Support.ServiceModels
{
[Route("/support/log", Verbs = "POST", Summary = "Create new log event")]
public class LogEventRequest : IReturnVoid
{
public LogEvent[] Events { get; set; }
}
public class LogEvent
{
public DateTimeOffset Timestamp { get; set; }
public string MessageTemplate { get; set; }
public string Level { get; set; }
public Exception Exception { get; set; }
public string RenderedMessage { get; set; }
public IDictionary<string, object> Properties { get; set; }
}
}
```
Service:
```c#
using API.Support.ServiceModels;
using ServiceStack.ServiceInterface;
namespace API.Support.ServiceInterfaces
{
public class LoggingService : Service
{
public void Post(LogEventRequest request)
{
// Do something with it
}
}
}
```
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,245 |
Girls soccer Group semifinals preview
Lauren Knego
@laurenknego
The NJSIAA Tournament Group semifinals will be held tonight with three Morris County teams in the mix. Kinnelon, Hanover Park and Roxbury will all be competing for berths in the Group finals on Saturday.
Who: Kinnelon vs. Glen Ridge, 7:30 p.m. at Indian Hills High School.
How they got here: Kinnelon (18-5), the No. 6 seed in North 1 Group I, defeated No. 11 Wallkill Valley 1-0, No. 14 Mountain Lakes 3-2, No. 7 New Milford 4-1 and top-seeded Park Ridge 2-1. Glen Ridge (17-6-1), the fifth seed in North 2 Group I, beat No. 12 Dayton 4-1, No. 4 Wood-Ridge 5-1, top-seeded Lyndhurst 1-0 and No. 3 Verona 1-0.
Kinnelon roster: Fr. D Lauren Abderhaldern, Fr. M Haley Angelica, So. M Katie Bonanno, Fr. M/F Tori Civil, Jr. M Kara Connolly, Jr. D Emily Coppa, So. D Sydney Coutts, Jr. D Taylor Coutts, So. M Francesca DeVincentis, Fr. M Kristen Dunn, So. M Olivia Eckstein, So. M/F Tori Falzarano, So. D Angelica Fiuza, Fr. M Daniella Gadaleta, Jr. D Vanessa Gorman (4g), Sr. M Casey Gray (20g, 22a), So. D Paige Hanley, Fr. G Kristina Kalafsky, Jr. G Emily Kapr, Jr. M/F Julia Kleczkowski (9g, 3a), So. G Madison Ledwith, Fr. D Elaine Lewis, So. D Dominique Molee, Fr. M/F Lindsay Naugle, Sr. M Sarah Rosenberg (10g, 10a), So. D Samantha Schaper, Sr. M/F Emily Thiem (10g, 10a), So. M Jessica Tighe, So. F Sam VanZile (9g, 4a), Fr. M Abigail Velisheck, Fr. M Paige Verroca.
Glen Ridge players to watch: Tomison Kennedy (28g, 8a), Sam Sikkerbol (7g, 3a), Shannon Marhan (5g, 7a), Colleen Grady (13a).
What to expect: The Colts have been on a mission to prove themselves one of the better teams in Morris County and the state, and they've been doing a great job of it. Kinnelon had a rough start and was given the No. 15 seed in the Morris County Tournament, which the team felt was an insult. That insult lit a fire under the Colts and they went on to advance all the way to the semifinals before falling to Roxbury, and then finished the season winning its last three games. Kinnelon then went on to win its first sectional title since 2006, outscoring its four opponents 10-4. The upset-minded Colts have a number of offensive weapons, including leading scorer Casey Gray, Sarah Rosenberg and Emily Thiem. Glen Ridge, which shared the Group I title with Shore last season, and won it outright in 2012, also has a number of offensive threats, including leading scorer Tomison Kennedy. The Ridgers have also hit a few bumps in the road this season, and should match up with the Colts rather well. But if this season has been any indication, Kinnelon will come out swinging and will fight to keep its upset streak going, all the way to the Group final.
Who: Hanover Park vs. Ramapo, 5 p.m. at Ridge High School.
How they got here: Hanover Park (12-9-1), the sixth seed in North 2 Group II, defeated No. 11 Parsippany 2-1, No. 14 Caldwell 4-1, No. 2 Morris Tech 4-0 and top-seeded Madison 2-0 to earn its first sectional title in program history. Ramapo (19-3), the top seed in North 1 Group II, beat No. 16 Dover 6-0, No. 9 Pascack Hills 5-0, No. 5 Pequannock 4-0 and third-seeded Ramsey 2-0.
Hanover Park roster: Jr. M/F Taylor Bermingham, Sr. M Erin Cahill, Jr. G Jill Carille, Jr. D/M Ally DeRiggi, Sr. M Courtney Giordano, So. M Jackie Larsen, Sr. F Janine Laudati (5g), Sr. D Sydney Lucas, Jr. D/M Anna Lazur, So. D Gia Maffuci, Jr. D Claire McNally, Jr. M Julia Monteleone, Sr. M Gianna Parlavecchio (13g), Sr. M Maggie Pescatore (6a), Sr. M Rianna Quiogue (4a), Jr. D Laura Romanski, Fr. G Sophia Reynolds, Sr. D Abby Saul, Jr. F Gianna Zarra (11g, 10a).
Ramapo players to watch: Lauren Brzozowski, Sarah Scire, Sommer Cochran.
What to expect: The Hornets have made history this season, advancing to and winning the first sectional final in program history, and now will be playing in their first Group semifinal. At this point, Hanover Park has nothing to lose, which can make it a very dangerous team. The Hornets started the season at 8-1, but then was plagued by injuries. The team has gotten healthy at the right time and in its four state games outscored opponents 12-2 with two shutouts. The defense of Taylor Bermingham, Ali DeRiggi, Gia Maffuci and Rianna Quiogue has been outstanding toward the end of the season and the speed of Anna Lazur in the midfield is a strength. The Hornets' defense will need to hang tough with a seasoned Ramapo side. The Raiders, which won their first sectional title in four years, also have a strong defense, having posted 17 shutouts this season and outscored opponents 17-0 in its run through North 1 Group II. The Raiders will most likely be the toughest team Hanover Park has seen this season, and the Hornets will have to play their best game of the season in order to advance to the Group final.
Who: Roxbury vs. Northern Highlands, 7:30 p.m. at Millburn High School.
How they got here: Roxbury (18-2-1), the top seed in North 2 Group III, defeated No. 16 Iselin Kennedy 7-0, No. 8 Chatham 2-0, No. 4 Somerville 2-1 and No. 3 Nutley 5-0. Northern Highlands (21-1), the No. 1 seed in North 1 Group III, beat No. 16 Teaneck 3-0, No. 8 Paramus 5-0, No. 4 Pascack Valley 2-0 and sixth-seeded Morris Hills 4-0.
Roxbury roster: Sr. Andrea Arias (10a), Jr. Chelsea Artigliere (6g), So. Danielle Dachowski, Fr. Christine Fleming, Jr. Deanna Graziani, So. Jamie Irwin, So. Jackie Katzenberger, Sr. Jamie Katzenberger, So. Shannon Leydon, Fr. Madison Martino (6a), Sr. Mackenzie Mathis, Sr. Caitlin Mertens (8g, 9a), Sr. Paige Monaghan (29g, 9a), Fr. G Sarah Olivero, Sr. Kat Ramage, Sr. Kylie Richardson, Jr. Alyssa Rose (5g, 5a), So. Jess Rudnicky, Fr. Bethany Sansone, Jr. Ryelle Sansone (19a, Jr. Rachel Sclar, Fr. Ashleigh Sarafin (7g, 5a), Jr. Noelle Spinosa, Jr. Stephanie Williams, Jr. Stephanie Wilson.
Northern Highlands players to watch: Sr. Hana Kerner (17g), Jr. Casey Richards (11g, 17a), Jr. Ariel Somple (8g, 10a), So. Eva Hurm (7g), So. Claire Nam (6g, 5a).
What to expect: Roxbury won its second sectional title in three years, last winning the North 2 Group III crown in 2012, where it fell to Northern Highlands in the Group III semifinals. The Highlanders have won the Group III title the last three seasons and this season has outscored opponents 73-3, with its lone loss coming at the hands of Immaculate Heart back on Nov. 3. Northern Highlands also has 20 shutouts this season. Given the amount of shutouts and goals allowed, the Highlanders' defense seems to be a place where opposing offenses go to die. The Gaels will have to play its best soccer of the season if they hope to advance to the Group III final. Roxbury has 15 shutouts this season and has allowed only one goal during their run in the state tournament. The Highlanders will have to focus their attention on senior midfielder/forward Paige Monaghan, who is capable of scoring from almost anywhere on the field. But she's not the only weapon, as Roxbury has a number of players that can sneak through and score, and senior Andrea Arias is a clutch feeder. The Gaels' defense has also been tight this year, allowing only seven goals. Look for this final to be one of the best games played Wednesday night, as it will be a very low-scoring defensive affair.
Staff Writer Lauren Knego: 973-428-6674; lknego@gannett.com | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,187 |
Q: Does the installer use cached downloads? When I use the Nexus 7 installer , if I want to reinstall will the installer use cached downloads or will it redownload the images all over again?
A: The installer will re-use cached downloads.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,596 |
import datetime
from south.db import db
from south.v2 import SchemaMigration
from django.db import models
class Migration(SchemaMigration):
def forwards(self, orm):
# Deleting model 'pagepermission'
db.delete_table('pages_pagepermission')
def backwards(self, orm):
# Adding model 'pagepermission'
db.create_table('pages_pagepermission', (
('user', self.gf('django.db.models.fields.related.ForeignKey')(to=orm['auth.User'])),
('type', self.gf('django.db.models.fields.IntegerField')(default=0)),
('id', self.gf('django.db.models.fields.AutoField')(primary_key=True)),
('page', self.gf('django.db.models.fields.related.ForeignKey')(to=orm['pages.Page'], null=True, blank=True)),
))
db.send_create_signal('pages', ['pagepermission'])
models = {
'auth.group': {
'Meta': {'object_name': 'Group'},
'id': ('django.db.models.fields.AutoField', [], {'primary_key': 'True'}),
'name': ('django.db.models.fields.CharField', [], {'unique': 'True', 'max_length': '80'}),
'permissions': ('django.db.models.fields.related.ManyToManyField', [], {'to': "orm['auth.Permission']", 'symmetrical': 'False', 'blank': 'True'})
},
'auth.permission': {
'Meta': {'ordering': "('content_type__app_label', 'content_type__model', 'codename')", 'unique_together': "(('content_type', 'codename'),)", 'object_name': 'Permission'},
'codename': ('django.db.models.fields.CharField', [], {'max_length': '100'}),
'content_type': ('django.db.models.fields.related.ForeignKey', [], {'to': "orm['contenttypes.ContentType']"}),
'id': ('django.db.models.fields.AutoField', [], {'primary_key': 'True'}),
'name': ('django.db.models.fields.CharField', [], {'max_length': '50'})
},
'auth.user': {
'Meta': {'object_name': 'User'},
'date_joined': ('django.db.models.fields.DateTimeField', [], {'default': 'datetime.datetime(2013, 8, 19, 17, 37, 19, 123021)'}),
'email': ('django.db.models.fields.EmailField', [], {'max_length': '75', 'blank': 'True'}),
'first_name': ('django.db.models.fields.CharField', [], {'max_length': '30', 'blank': 'True'}),
'groups': ('django.db.models.fields.related.ManyToManyField', [], {'to': "orm['auth.Group']", 'symmetrical': 'False', 'blank': 'True'}),
'id': ('django.db.models.fields.AutoField', [], {'primary_key': 'True'}),
'is_active': ('django.db.models.fields.BooleanField', [], {'default': 'True'}),
'is_staff': ('django.db.models.fields.BooleanField', [], {'default': 'False'}),
'is_superuser': ('django.db.models.fields.BooleanField', [], {'default': 'False'}),
'last_login': ('django.db.models.fields.DateTimeField', [], {'default': 'datetime.datetime(2013, 8, 19, 17, 37, 19, 122310)'}),
'last_name': ('django.db.models.fields.CharField', [], {'max_length': '30', 'blank': 'True'}),
'password': ('django.db.models.fields.CharField', [], {'max_length': '128'}),
'user_permissions': ('django.db.models.fields.related.ManyToManyField', [], {'to': "orm['auth.Permission']", 'symmetrical': 'False', 'blank': 'True'}),
'username': ('django.db.models.fields.CharField', [], {'unique': 'True', 'max_length': '255'})
},
'contenttypes.contenttype': {
'Meta': {'ordering': "('name',)", 'unique_together': "(('app_label', 'model'),)", 'object_name': 'ContentType', 'db_table': "'django_content_type'"},
'app_label': ('django.db.models.fields.CharField', [], {'max_length': '100'}),
'id': ('django.db.models.fields.AutoField', [], {'primary_key': 'True'}),
'model': ('django.db.models.fields.CharField', [], {'max_length': '100'}),
'name': ('django.db.models.fields.CharField', [], {'max_length': '100'})
},
'pages.content': {
'Meta': {'object_name': 'Content'},
'body': ('django.db.models.fields.TextField', [], {}),
'creation_date': ('django.db.models.fields.DateTimeField', [], {'default': 'datetime.datetime(2013, 8, 19, 17, 37, 19, 119785)'}),
'id': ('django.db.models.fields.AutoField', [], {'primary_key': 'True'}),
'language': ('django.db.models.fields.CharField', [], {'max_length': '5'}),
'page': ('django.db.models.fields.related.ForeignKey', [], {'to': "orm['pages.Page']"}),
'type': ('django.db.models.fields.CharField', [], {'max_length': '100'})
},
'pages.page': {
'Meta': {'ordering': "['tree_id', 'lft']", 'object_name': 'Page'},
'author': ('django.db.models.fields.related.ForeignKey', [], {'to': "orm['auth.User']"}),
'creation_date': ('django.db.models.fields.DateTimeField', [], {'default': 'datetime.datetime(2013, 8, 19, 17, 37, 19, 120711)'}),
'delegate_to': ('django.db.models.fields.CharField', [], {'max_length': '100', 'null': 'True', 'blank': 'True'}),
'freeze_date': ('django.db.models.fields.DateTimeField', [], {'null': 'True', 'blank': 'True'}),
'id': ('django.db.models.fields.AutoField', [], {'primary_key': 'True'}),
'last_modification_date': ('django.db.models.fields.DateTimeField', [], {}),
'level': ('django.db.models.fields.PositiveIntegerField', [], {'db_index': 'True'}),
'lft': ('django.db.models.fields.PositiveIntegerField', [], {'db_index': 'True'}),
'parent': ('django.db.models.fields.related.ForeignKey', [], {'blank': 'True', 'related_name': "'children'", 'null': 'True', 'to': "orm['pages.Page']"}),
'publication_date': ('django.db.models.fields.DateTimeField', [], {'null': 'True', 'blank': 'True'}),
'publication_end_date': ('django.db.models.fields.DateTimeField', [], {'null': 'True', 'blank': 'True'}),
'redirect_to': ('django.db.models.fields.related.ForeignKey', [], {'blank': 'True', 'related_name': "'redirected_pages'", 'null': 'True', 'to': "orm['pages.Page']"}),
'redirect_to_url': ('django.db.models.fields.CharField', [], {'max_length': '200', 'null': 'True', 'blank': 'True'}),
'rght': ('django.db.models.fields.PositiveIntegerField', [], {'db_index': 'True'}),
'status': ('django.db.models.fields.IntegerField', [], {'default': '0'}),
'template': ('django.db.models.fields.CharField', [], {'max_length': '100', 'null': 'True', 'blank': 'True'}),
'tree_id': ('django.db.models.fields.PositiveIntegerField', [], {'db_index': 'True'})
},
'pages.pagealias': {
'Meta': {'object_name': 'PageAlias'},
'id': ('django.db.models.fields.AutoField', [], {'primary_key': 'True'}),
'page': ('django.db.models.fields.related.ForeignKey', [], {'to': "orm['pages.Page']", 'null': 'True', 'blank': 'True'}),
'url': ('django.db.models.fields.CharField', [], {'unique': 'True', 'max_length': '255'})
}
}
complete_apps = ['pages']
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,418 |
Situated in Hilton Head Island, 102 Baynard Cove features accommodation with a private pool and free WiFi. This property offers access to a terrace. This holiday home includes a living room and a flat-screen TV, an equipped kitchen with a dining area, and 3 bathrooms with a bath. Private parking is available at the holiday home. Cycling can be enjoyed nearby. Bluffton is 28 km from 102 Baynard Cove. The nearest airport is Savannah/Hilton Head International Airport, 81 km from the accommodation.
Please inform 102 Baynard Cove in advance of your expected arrival time. You can use the Special Requests box when booking, or contact the property directly with the contact details provided in your confirmation.
Please note that the check-in and key-collection take place at: 64 Arrow Road, 29928 Hilton Head Island. | {
"redpajama_set_name": "RedPajamaC4"
} | 510 |
I am a Servant Leader, who provides coaching and consulting services to women entrepreneurs by eliminating distractions, both internal and external, in order to fulfill their passion and purpose. By transforming and empowering one life, career, business; I change the lives of many. I recently obtained my coaching certification through the Coach Diversity Institute.
I am Martina Coates-Nance, Founder and CEO of Your Clarity Coach LLC. Your Clarity Coach's mission is to serve women and men like who are seeking transformational CLARITY in their lives, careers and businesses. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,455 |
\section{Introduction}
Systems for translation of controlled language\footnote{For more information
on controlled language, see {\tt http://wwwots.let.ruu.nl/Controlled-languages/}.}
require the source text to be expressed within severe syntactical and lexical
limits. One of the objectives of such
systems is that an author who fully conforms to the imposed restrictions is
rewarded with a reliable and fully automatic translation of his text into one or
more target languages. Therefore a proof of their {\em completeness} is of
great importance. A machine translation system is said to be {\em complete}
if all expressions that are correct according to the source-language grammar
can be translated into the target language.\\
The starting point of this research has been the compositional approach to
machine translation developed in the Rosetta project, \cite{Rosetta-1994}.
An important difference is that Rosetta made use of a rather complex grammar
formalism, M-grammars, for which completeness could not be proven, whereas
the current research focuses on the provability of completeness for relatively
simple grammar formalisms, which may be more appropriate for machine
translation of controlled languages.\\
First sections~\ref{sec-compositional-grammars}
and~\ref{sec-compositional-machine-translation} describe our view and
definitions of respectively compositional grammar and compositional machine
translation.
Section~\ref{sec-completeness} presents the theme of this paper,
viz.\,{\em completeness} of compositional machine translation.
Subsequently section~\ref{sec-completeness-for-CFG} works out completeness
conditions for compositional grammars based on context-free grammars.
These conditions are rather restrictive and may therefore find application
primarily in areas such as controlled languages.
One of the objectives of ongoing research is to relax the conditions.
Section~\ref{sec-discussion} concludes the paper and discusses ongoing and
future research.
\section{Compositional Grammars}
\label{sec-compositional-grammars}%
This section defines {\em compositional grammars}
(subsection \ref{subsec-def-compositional-grammars}),
and the auxiliary notions {\em syntactic derivation tree}
(subsection \ref{subsec-def-syndertree})
and {\em semantic derivation tree} (subsection \ref{subsec-def-semdertree}).
\subsection{A Definition of Compositional Grammars}
\label{subsec-def-compositional-grammars}%
Compositional machine translation assumes that the source language (SL)
and the target language (TL) are defined by means of compositional grammars,
i.e.\,grammars that obey the well-known compositionality principle
(cf.~\cite[p.315ff]{Partee-et-al-1993,Janssen-1986,Gamut-1991}).
Abstracting away from the details of any specific syntactic formalism,
we define a {\em compositional grammar} $G$ as consisting of {\em(i)} a syntactic
component, {\em(ii)} a semantic component, and {\em(iii)} an interpretation from
the syntactic component to the semantic component (cf.\,Montague's Universal
Grammar, \cite{Thomason-1974a}).
Roughly, the syntactic component consists of a set of basic expressions (words),
each having a syntactic category, and a set of syntactic rules that build larger
expressions from basic expressions.
Likewise, the semantic component consists of a set of basic meanings,
each having a semantic category, and a set of semantic rules that build larger
meanings from basic meanings.
The interpretation associates with every basic expression a {\em set} of
basic meanings, and with every syntactic rules a {\em set} of semantic rules.\\
There now follows a more detailed description of these components, which the
eager reader may wish to skip on a first pass.\\
\bullit\
The {\em syntactic component} specifies
a finite set of basic expressions \BE{},
a finite set of syntactic rules \SynR{},
a finite set of syntactic categories \SynKats{},
and a syntactic type-assignment function \SynType{}{\cdot}.
{\em Basic expressions} are, roughly, the smallest meaningful units in a
language (more or less the stems of content words).
{\em Syntactic rules} are operations that recursively build
{\em derived expressions} from basic expressions.
{\em Syntactic categories} describe the syntactic properties of expressions.
Basic expressions~$b$ all have a syntactic category \SynCat{}{b};
syntactic rules restrict their arguments in their categories,
and specify the category of the derived expression they yield.
The {\em syntactic type-assignment function} associates every
syntactic rule~$R$ with a 2-tuple \SynType{}{R} consisting of a so-called
{\em argument list} \SynAL{}{R} of the categories of its arguments
and its resultant category.
The arity \arity{}{R} of a syntactic rule is the number of categories
in the rule's argument list.
We require that all syntactic and semantic rules are total:
They must be applicable for any combination of arguments that matches
their argument lists. Note that this is not a real restriction of
expressiveness: Any partial function can be made into a total function
by an appropriate tuning of the set of categories.
\bullit\
The {\em semantic component} has the same structure as the syntactic component:
It specifies a finite set of basic meanings \BM{},
a finite set of semantic rules \SemR{},
a finite set of semantic categories \SemKats{},
and a semantic type-assignment function \SemType{}{\cdot}.
{\em Basic meanings} are expressions of the semantic domain of some
logical language.
{\em Semantic rules} are operations in the logical language that recursively build
{\em derived meanings} from basic meanings.
For the purpose of compositional translation the choice of this logical language
is not very important. However, the semantic rules must be total.
{\em Semantic categories} describe the semantic properties of semantic expressions.
Basic meanings~$m$ all have a semantic category \SemCat{}{m};
semantic rules restrict their arguments in their semantic categories,
and specify the category of the derived meaning they yield.
The {\em semantic type-assignment function} associates every
semantic rule~$M$ with a 2-tuple \SemType{}{M} consisting of a so-called
{\em argument list} \SemAL{}{M} of the categories of its arguments
and its resultant category.
The arity \arity{}{M} of a semantic rule is the number of categories
in the rule's argument list.
\bullit\
The {\em interpretation}, denoted $\Mean{.}$, associates every basic
expression with a {\em set} of basic meanings, and every syntactic
rule with a {\em set} of semantic rules.
The arities of associated syntactic and semantic rules must match.
Note that our approach differs here from Montague grammar,
in which a basic expression (syntactic rule) is associated with
{\em exactly one} basic meaning (semantic rule).
\subsection{Syntactic Derivation Trees}
\label{subsec-def-syndertree}%
Derivational histories of syntactic expressions are represented
using so-called syntactic derivation trees:
\definition{Syntactic Derivation Tree}
A {\em syntactic derivation tree}~$t$ is either
a tree consisting of a single node~$b$, where~$b$ is the name of
a basic expression, or a tree of the form $R[t_1\kdots t_n]$,
where~$R$ is the name of a syntactic rule, and $t_1\kdots t_n$
is an ordered list of syntactic derivation trees.\\
We define the syntactic category of a syntactic derivation tree~$t$,
denoted {\em SynCat(t)}, to be
the resultant category of its top syntactic rule.
For convenience, we will sometimes annotate syntactic derivation trees
with their syntactic category, e.g. $t:C$.\\
Intuitively one may think of a syntactic derivation tree as
the derivational history of a syntactic expression.
However, not all syntactic derivation trees actually describe
expressions: The definition given above does not require the
syntactic rules to be applicable to their arguments.
This distinction is described by the concept of well-formedness.
\definition{Well-Formedness of Syntactic Derivation Trees}%
\label{def-CFG-wfness}%
A syntactic derivation tree~$t$ is {\em well-formed} if and only if
it consists of a single basic expression or otherwise if all the
syntactic rules in the tree are applicable to their arguments as
specified by tree~$t$, i.e. if and only if for all the syntactic rules
in tree $t$ {\em(i)} the number of arguments (subtrees) matches the
rule's arity, and {\em(ii)} the arguments satisfy any conditions on the
syntactic categories that may be made by the syntactic rule.\\
Since there is generally more than one way to derive an expression,
expressions are in general assigned a {\em set} of corresponding
syntactic derivation trees.
\subsection{Semantic Derivation Trees}
\label{subsec-def-semdertree}%
The meaning of a derived expression is derived in parallel with the syntactic
derivation process. Thus this semantic derivation process may be represented
in a tree with the same geometry as the syntactic derivation tree, but labelled
by basic meanings and semantic rules.
This tree is called a {\em semantic derivation tree}.
\definition{Semantic Derivation Tree}
A {\em semantic derivation tree}~$d$ is either
a tree consisting of a single node~$m$, where $m$ is the name of a basic
meaning, or a tree of the form~$M[d_1\kdots d_n]$, where~$M$ is the
name of a semantic rule, and $d_1\kdots d_n$ is an ordered list of
semantic derivation trees.\\
We define the semantic category of a semantic derivation tree~$d$,
denoted {\em SemCat(d)}, to be
the resultant category of its top semantic rule.
Semantic derivation trees may also be annotated with their semantic
category, e.g. $d:C$.\\
Since every syntactic derivation tree is associated with a {\em set}
of semantic derivation trees, every syntactic derivation tree is
associated with a {\em set} of semantic derivation trees.\\
A semantic derivation tree is well-formed if and only if there is a
corresponding well-formed syntactic derivation tree.
\section{Compositional Machine Translation}
\label{sec-compositional-machine-translation}
In our definition of compositional translation the semantic component is used
as an interlingua: Source- and target-language expressions are
{\em translation-equivalent} if and only if they have at least one well-formed
semantic derivation tree in common.
\definition{Compositional Translation}
For two compositional grammars~$G$ and~$G'$, the {\em compositional translation}
of a source-language utterance~$e$ is a set of target-language utterances,
derived as follows:
\pictura{The Process of Compositional Translation}
{clin95-compositional-translation.ps,height=4cm,width=11cm}
$\bullet$ {\bf Morphosyntactic Analysis} --
Morphosyntactic analysis performs morphological and syntactic analysis
of a SL utterance, yielding the set of all syntactic derivation trees
that correspond to the utterance:\\
\mm{3} $morsynan(e)=\{b\mid b=e,\elem{b}{\BE{}}\}$\\
\mm{3} \mm{5} $\cup\ \{R\boom{t_1\kdots t_n}\mid e=R(e_1\kdots e_n),
\forall i\ (1\leq i\leq n)\ \elem{t_i}{morsynan(e_i)},
\elem{R}{\SynR{}}\}$\\
(`$R(e_1\kdots e_n)$' denotes the result of applying rule $R$
to expressions $e_1\kdots e_n$).\\
$\bullet$ {\bf Semantic Analysis} --
Semantic analysis of a syntactic derivation tree yields the set of all
corresponding semantic derivation trees:\\
\mm{3} $seman(b)=\Mean{b}$\\
\mm{3} $seman(R\boom{t_1\kdots t_n})=\{M\boom{d_1\kdots d_n}\mid
\elem{M}{\Mean{R}}\ \wedge\
\forall i\ (1\leq i\leq n)\ \elem{d_i}{seman(t_i)}\}$\\
$\bullet$ {\bf Semantic Generation} --
Semantic generation from a semantic derivation tree yields the set of all
corresponding syntactic derivation trees:\\
\mm{3} $semgen(m)=\{b\mid\elem{m}{\Mean{b}}\}$\\
\mm{3} $semgen(M\boom{d_1\kdots d_n})=\{R\boom{t_1\kdots t_n}\mid
\elem{M}{\Mean{R}}\ \wedge\
\forall i\ (1\leq i\leq n)\ \elem{t_i}{semgen(d_i)}\}$\\
$\bullet$ {\bf Morphosyntactic Generation} --
Morphosyntactic generation for a {\em well-formed} syntactic derivation tree
produces the corresponding utterance:\\
\mm{3} $morsyngen(b)=b$\\
\mm{3} $morsyngen(R\boom{t_1\kdots t_n})=R(e_1\kdots e_n)$,
where $\forall i\ (1\leq i\leq n)\ \elem{e_i}{morsyngen(t_i)}$
\section{Completeness of Machine Translation}
\label{sec-completeness}
An important question regarding the reliability of compositional translation is
what we call the {\em completeness}\,\footnote{The term `completeness' was taken
from \cite[pp.342-343]{Whitelock-1994}. In the Rosetta framework completeness is
known as `strict isomorphism', and is discussed in~\cite{Landsbergen-1987a}
and~\cite{Rosetta-1994}.} issue:
{\em Can the translation process be guaranteed to produce at least one translation?}
In subsection~\ref{sss-three-levels} we first make this notion of
completeness precise.
Then, in subsection~\ref{sss-guaranteeing-completeness}, we investigate
what conditions must be satisfied to guarantee completeness.
In section~\ref{sec-completeness-for-CFG} conditions are elaborated for
compositional grammars based on context-free grammars.
\subsection{Three Levels of Completeness}\label{sss-three-levels}%
Completeness is about the guaranteed generation of well-formed translations,
given a specific SL and TL grammar, and translation process.
However, this description does not make precise from which stage on the
translation process must be guaranteed to succeed.
Depending on this, one may distinguish (at least) three levels of completeness
(cf.\,fig.\,1):\\
\bullit\ {\bf Utterance Completeness} --
For each well-formed SL utterance, the translation process yields at least one
well-formed TL utterance.\\
\bullit\ {\bf Syntactic Completeness} --
For each syntactic derivation tree of each well-formed SL utterance, the
translation process yields at least one well-formed TL utterance.\\
\bullit\ {\bf Semantic Completeness} --
For each semantic derivation tree of each syntactic derivation tree
of each well-formed SL utterance, the translation process yields at
least one well-formed TL utterance.\\
{\bf Note: } Semantic completeness subsumes syntactic completeness, which
in turn subsumes utterance completeness.\\
Naively, one would like a machine translation system to produce at least one
translation for every SL utterance. This requirement is included in the
definition of utterance completeness above.
However, it is well-known that natural-language utterances are often
ambiguous. For each of its interpretations, such an ambiguous utterance may
have a different translation. Therefore, a machine translation system should
be able to provide at least one translation {\em for each of the
interpretations} of the SL utterance.
Natural-language ambiguity takes on two forms: structural ambiguity and lexical
ambiguity. The notion of syntactic completeness takes
care of the structural ambiguity: It is formulated in terms of structurally
unambiguous syntactic derivation trees.
However, syntactic completeness is still unsatisfactory,
as syntactic derivation trees are often lexically ambiguous. This is
due to the fact that basic expressions may have more than one meaning,
and syntactic rules may have more than one semantic rule associated with them.
What is needed is a formulation of completeness in terms of a structure that
is both structurally and lexically unambiguous. The solution is provided by
the notion of semantic completeness.
Therefore, from now on, the term `completeness' will be taken to refer to
semantic completeness only.
\definition{Completeness}
For a pair of compositional grammars \tuple{G,G'},
compositional translation from~$G$ to~$G'$ is {\em complete} if and only if
for each well-formed semantic derivation tree,
the translation process yields at least one well-formed TL utterance.
\subsection{Guaranteeing Completeness}\label{sss-guaranteeing-completeness}%
The central issue of this paper is the question of how to guarantee completeness.
Or stated in terms of the process of compositional translation described above:
What conditions on the SL and TL grammars are sufficient (and necessary) to
guarantee that, after successful analysis, generation can produce a well-formed
TL expression?
Generation comprises semantic generation and morphosyntactic generation
(cf.\,fig.\,1).\\
{\bf Completeness of Morphosyntactic Generation} --
Morphosyntactic generation evaluates the syntactic derivation trees yielded
by semantic generation by recursive rule application.
As stated in section~\ref{sec-compositional-grammars}, we assume that all
syntactic rules are total for the categories of their arguments.
Rule application therefore succeeds if and only if the arguments are of
the correct categories. To ensure this we must move upstream to
semantic generation.\\
{\bf Completeness of Semantic Generation} --
Semantic generation simply replaces the basic meanings and semantic rules in
the semantic derivation tree with corresponding syntactic elements
of the TL grammar, forming the TL syntactic derivation trees.
An obvious necessary and sufficient condition for completeness of semantic
generation is that there be
{\em at least one} translation-equivalent counterpart in the TL grammar
for each possible semantic element in the SL semantic derivation trees.
A compositional grammar pair satisfying this condition is called a
{\em homomorphic grammar pair} (see also~\cite[p.368]{Rosetta-1994}):
\definition{Grammar Homomorphism}%
\label{def-grammar-homomorphism2}%
A compositional grammar pair \tuple{G,G'} is {\em homomorphic from~$G$ to~$G'$}
if and only if~$G'$ is {\em attuned} to~$G$:
\begin{romanlist}
\item For each SL basic expression~$b$,
for each of the basic meanings~$m$ of~$b$,
there is at least one TL basic expression~$b'$ such that
basic meaning~$m$ is also a basic meaning of~$b'$.
Formally,
$\forall\elem{b}{\BE{}}\ \forall\elem{m}{\Mean{b}}
\ \exists\elem{b'}{\BE{}}\ \ \elem{m}{\Mean{b'}}$.
\item For each SL syntactic rule~$R$,
for each of the semantic rules~$M$ of~$R$,
there is at least one TL syntactic rule~$R'$ such that
semantic rule~$M$ is also a semantic rule of~$R'$.
Formally,
$\forall\elem{R}{\SynR{}}\ \forall\elem{M}{\Mean{R}}
\ \exists\elem{R'}{\SynR{}}\ \elem{M}{\Mean{R'}}$.
\end{romanlist}
However, to demand grammar homomorphism is only a necessary condition
for completeness, and not a sufficient one.
It merely guarantees that for every well-formed SL semantic derivation
tree there is a corresponding TL syntactic derivation tree, and does
not guarantee that this syntactic derivation tree is well-formed.
The next section is about such sufficient conditions for context-free
grammars.
\section{Completeness for CFG-Based Compositional Grammars}
\label{sec-completeness-for-CFG}
This section presents completeness conditions for translation between
compositional grammars based on the context-free grammar (CFG) formalism.
We assume that the reader is familiar with this formalism.
Subsection~\ref{ss-cfg-based-compositional-grammar} explicates how a
compositional grammar can be based on context-free grammars.
Subsections~\ref{ss-m2o-catcor} and~\ref{ss-m2m-catcor} subsequently
develop completeness conditions for such compositional grammars.
\subsection{CFG-Based Compositional Grammar}
\label{ss-cfg-based-compositional-grammar}
A compositional grammar consists of a syntactic component with basic
expressions and syntactic rules, a semantic component with basic
meanings and semantic rules, and an interpretation from the syntactic
component to the semantic component.
Here we model the syntactic component as a CFG. The semantic component
and the interpretation are as defined above.
In the syntactic component we let basic expressions correspond to rewrite rules
that do not have right-hand side (RHS) nonterminals. The rule's RHS corresponds
to the lexical material of the basic expression; the rule's left-hand side
(LHS) nonterminal corresponds to the syntactic category of the basic expression.
We let syntactic rules correspond to rewrite rules that {\em do} have RHS
nonterminals. The type of a syntactic rule is a 2-tuple consisting of a list of
categories of the arguments it expects and the category of the expression it
produces. The list of categories corresponds to an ordered list of the rewrite
rule's RHS
nonterminals; the resultant category corresponds to the rewrite rule's LHS
nonterminal. The operation performed by the syntactic rule is the in-order
concatenation of its RHS terminals and nonterminals, where the nonterminals are
replaced with the lexical material of the expressions which are provided as
arguments to the syntactic rule. An example illustrates this:\\
{\bf Example}\ \ {\em CFG-Based Compositional Grammars}\\
In this example we briefly illustrate CFG-based compositional grammars.
Consider the following table, which shows the syntactic component of a
CFG-based compositional grammar and its interpretation in the semantic
component.
\begin{quote}
\begin{tabular}{llll}
{\em CFG Rewrite Rule}
& {\em Syntactic Rule} & {\em Basic Expression}
& {\em Interpretation}\\
& {\em Name\,:\,Type} & {\em Name\,:\,Category} \\ \hline
$A \raS B\ C$ & $R_1:\tuple{\tuple{B,C},A}$ & & $\{M_1\}$ \\
$A \raS a\ B\ d$ & $R_2:\tuple{\tuple{B},A}$ & & $\{M_{2a},M_{2b}\}$ \\
$A \raS e\ C\ B$ & $R_3:\tuple{\tuple{B,C},A}$ & & $\{M_{3a},M_{3b}\}$ \\
$B \raS b$ & & $b:B$ & $\{m_1\}$ \\
$C \raS c$ & & $c:C$ & $\{m_{2a},m_{2b}\}$
\end{tabular}
\end{quote}
Observe that the order of syntactic categories in the argument list
need not be the same as the order in the rewrite rules (see $R_1,R_3)$.
Syntactic rules $R_1$ and $R_3$ have two arguments. As a consequence
semantic rules $M_1$, $M_{3a}$ and $M_{3b}$ are binary operators.
Syntactic rule $R_2$ and semantic rules $M_{2a}$ and $M_{2b}$ have
one argument.
The notion of well-formedness can be made more precise now:
\definition{CFG-well-formedness}\label{CFG-well-formedness}%
A CFG syntactic derivation tree $t$ is {\em CFG-well-formed} if and only if
it is either the name of a basic expression, or
a tree of the form $R\boom{t_1\kdots t_n}$,
such that {\em(i)} rule $R$'s argument list matches the list of syntactic
categories of the subtrees $t_1\kdots t_n$:
$\SynAL{}{R}=\tuple{\SynCat{}{t_1}\kdots \SynCat{}{t_n}}$,
and {\em(ii)} subtrees $t_1\kdots t_n$ are CFG-well-formed.\\
What about the `translation power' of CFG-based compositional grammars?
The compositional translation method described in
section~\ref{sec-compositional-machine-translation} demands that basic
expressions
of the source language correspond to basic expressions in the target
language, and that the syntactic rules of the source-language
correspond to syntactic rules of the target language with the same arity.
This restricts the translation power considerably. The main degrees of
freedom in the translation relation are the following.
In the syntactic rules, the nonterminals need not occur in the same
order as in the argument list. This allows translation-equivalent
rules to describe word-order differences between languages.
Syntactic rules may also introduce lexical material other than that
of the arguments. This is called {\em syncategorematic introduction}
(cf. syntactic rules $R_2$ and $R_3$ in the example above, where basic
expressions $a$, $d$ and $e$ are left out).
The third degree of freedom relates to the correspondence between categories
of source- and target-language grammars.\\
Subsection~\ref{ss-m2o-catcor} now develops a completeness condition for
CFG-based compositional grammars.
Subsection~\ref{ss-m2m-catcor} then shows that this condition is rather
restrictive and presents a way to relax it.
\subsection{CFG Completeness for Many-to-One Category Correspondence}
\label{ss-m2o-catcor}
In this section we show how a restriction of the correspondence between
syntactic and semantic categories of the target language can lead to
completeness.
First we formally define a restriction of this correspondence.
\definition{N-1 Category Correspondence}
There is an {\em N-1 category correspondence} between a semantic component
and a syntactic component of a compositional grammar if and only if
there is a function $f:\SemKats{}\raS \SynKats{}$ such that:\\
\mm{3}\bullit\ $\forall\elem{m}{\BM{}}\ \ \forall\elem{b}{\BE{}}\ \
\elem{m}{\Mean{b}}\ \implies\ \SynCat{}{b}=f(\SemCat{}{m})$\\
\mm{3}\bullit\ $\forall\elem{M}{\SemR{}}\ \ \forall\elem{R}{\SynR{}}$\\
\mm{5} $\left(\elem{M}{\Mean{R}} \wedge \SemType{}{M}=\tuple{\tuple{c_1\kdots c_n},c}\right)\
\implies\ \SynType{}{R}=\tuple{\tuple{f(c_1)\kdots f(c_n)},f(c)}$\\
The restriction of compositional grammars to such an N-1 category
correspondence together with the grammar homomorphism condition
gives us completeness:
\theorem{CFG Completeness for Many-to-One Category Correspondence}
\label{th-cfg-completeness-1}%
For any CFG-based compositional grammar pair~\tuple{G,G'}, compositional
translation from~$G$ to~$G'$ is {\em complete} if
{\em(i)} the grammar pair is homomorphic from~$G$ to~$G'$, and
{\em(ii)} there is an N-1 category correspondence between the semantic
and the syntactic categories of $G'$.\\
\begin{proof}
As we are concerned with semantic completeness,
we have to prove that for every grammatical SL~utterance, for every one of
its well-formed semantic derivation trees, there exists at least
one grammatical TL utterance.
As we assume it to be trivial that morphosyntactic generation succeeds
for CFG-well-formed syntactic derivation trees, we focus on semantic generation.
We must show that every well-formed semantic derivation tree always
yields at least one {\em CFG-well-formed} TL syntactic derivation tree.
We do this by induction on the depth of the semantic derivation trees.
\inductionbase
A semantic derivation tree of depth~1 is a basic meaning.
Homomorphism from~$G$ to~$G'$ guarantees that there is at least one
TL basic expression that is associated with that basic meaning.
Basic expressions are trivially CFG-well-formed syntactic derivation trees.
\inductionhypothesis
For every well-formed semantic derivation tree derivable in~$G$
which is of depth $m$ or less, compositional translation yields at least one
CFG-well-formed TL syntactic derivation tree in $G'$.
\inductionstep
Assuming the induction hypothesis holds for arbitrary depth~$m$, we must prove
that it also holds for depth~$m+1$.
Every well-formed semantic derivation tree of depth~$m+1$ is of the form
$M[d_1\kdots d_n]:A$, where each subtree~$d_i$ is of the form $M_i[\dots]:A_i$
(see fig.\,2 below).
Because of the given well-formedness of the semantic derivation tree we
know that~$M$ is applicable to its arguments, so that its argument list
\tuple{A_1\kdots A_n} matches the semantic categories of the arguments $A_i$.
Homomorphism guarantees that~$M$ has at least one associated syntactic
rule~$R'$, which has some argument list \tuple{B_1\kdots B_n}.
The induction hypothesis guarantees that every tree~$d_i$ has at least one
CFG-well-formed TL syntactic derivation tree $t_i'=R_i'[\dots]:C_i$ associated
with it. Note that the induction hypothesis says nothing about the categories
$C_i$ of these trees.
\pictura{Induction Step: Generating Syntactic from Semantic Derivation Trees}
{clin-induction-step2.ps,height=4cm,width=12cm}
The remaining question is whether there is at least one TL~syntactic derivation
tree formed in this way which is CFG-well-formed, i.e. for which
(def.\,\ref{def-CFG-wfness}):
{\em(i)} rule~$R'$ is applicable to its arguments, and
{\em(ii)} all subtrees~$t_i'$ are CFG-well-formed.
Condition {\em(ii)} is covered by the induction hypothesis.
Condition {\em(i)} requires that
the argument list of rule~$R'$ matches the syntactic categories of the
subtrees $t'_1\kdots t'_n$:
$\SynAL{}{R'}=\tuple{B_1\kdots B_n}=\tuple{\SynCat{}{t'_1}\kdots \SynCat{}{t'_n}}$.
From the condition in the theorem we know that there is an N-1 category
correspondence~$f$ between the semantic categories and the syntactic categories
of~$G'$.
Because rule~$R'$ is associated with rule~$M$, we know that for all
$1\leq i\leq n$, $B_i=f(A_i)$. Since for all $1\leq i\leq n$, we also know
that tree~$t_i'$ is associated with tree~$d_i$, it holds that $C_i=f(A_i)$.
Since $f$ is a function, it must hold that for all $1\leq i\leq n$,
$B_i=C_i$, so that the argument list of~$R'$ matches the categories
of its arguments.
Therefore, every such rule $R'$ is applicable to its arguments,
so that completeness is guaranteed.
\backupline
\end{proof}
\subsection{Many-to-Many Category Correspondence}
\label{ss-m2m-catcor}
The N-1 category correspondence condition is rather restrictive. It implies
that a semantic category of the source language must be translated into
exactly one syntactic catgory of the target language. We would
like to have a looser category correspondence. For example, consider the
following grammar rules for translating between English and French noun
phrases, where French uses agreement on determiners and nouns:
\begin{center}
\begin{tabular}{ccc}
{\em English Syntax} & {\em Semantics} & {\em French Syntax} \\ \hline
$R_1 : NP\ \ra\ DET\ N$
& $M_1 : \ul{NP}\ \ra\ \ul{DET}\ \ul{N}$
& $R'_{1a} : NP'\ \ra\ DET'_m\ N'_m$ \\
& & $R'_{1b} : NP'\ \ra\ DET'_f\ N'_f$
\end{tabular}
\end{center}
Here we would like to relate semantic category $\ul{DET}$ to syntactic
categories $DET'_m$ and $DET'_f$, and semantic category $\ul{N}$ to
syntactic categories $N'_m$ and $N'_f$.
To be able to do so we could allow every semantic category to be associated
with a number of syntactic categories instead of just one.
This corresponds to an N-N category correspondence.
\definition{N-N Category Correspondence}
There is an {\em N-N category correspondence} between a semantic component
and a syntactic component of a compositional grammar if and only if
there is a function $f:\SemKats{}\raS \SynKats{}$ such that:\\
\mm{3}\bullit\ $\forall\elem{m}{\BM{}}\ \ \forall\elem{b}{\BE{}}\ \
\elem{m}{\Mean{b}}\ \implies\ \elem{\SynCat{}{b}}{f(\SemCat{}{m})}$\\
\mm{3}\bullit\ $\forall\elem{M}{\SemR{}}\ \ \forall\elem{R}{\SynR{}}$\\
\mm{10} $(\elem{M}{\Mean{R}}
\wedge \SemType{}{M}=\tuple{\tuple{c_1\kdots c_n},c})\
\implies\ \SynType{}{R}=\tuple{\tuple{c_1'\kdots c_n'},c'}$,\\
\mm{10} where $\forall i\ (1\leq i\leq n)\ \elem{c_i'}{f(c_i)}$
and \elem{c'}{f(c)}\\
For a semantic category $C$ the set of corresponding syntactic categories
$f(C)$ is called the {\em category correspondence set} of $C$ and is
denoted \~C.\\
For this new situation we must adjust the completeness condition.
Referring to fig.\,2, it now is the case that each syntactic category $C_i$
may be any category in the set $f(A_i)$.
As the induction hypothesis guarantees only one successful translation for
each subtree~$d_i$ -- and it is not known which one -- to guarantee
completeness is to guarantee that there is a syntactic rule $R'$ for
{\em every} argument list in $f(A_1)\times\dots\times f(A_n)$.
This is an unrealistic condition: In the English/French example it
corresponds to the demand that there must be a French syntactic rule for
all four argument lists
$\tuple{DET_m,N_m},\tuple{DET_m,N_f},\tuple{DET_f,N_m},\tuple{DET_f,N_f}$.
But to demand that for example there is a syntactic rule $R'$ that combines
a masculine determiner $DET_m$ and a feminine noun $N_f$, as this would imply,
is nonsensical.
The underlying problem is that the agreement dependencies cannot be expressed
explicitly in the CFG grammar formalism.
The lesson to be learned from this example is that the
dependencies between the categories should be taken into account.
We present a way of encoding information about the dependencies
between categories in CFG-based compositional grammar.
To this end we distinguish two kinds of category correspondence.
\definition{Conjunctive/Disjunctive Correspondence Category}
For a compositional grammar, a semantic category $N$ is a {\em conjunctive
(correspondence) category} if and only if for every well-formed semantic
derivation tree $d$ of category $N$, for {\em every} corresponding category
$N'$ in {\em \~N}, there exists at least one
corresponding well-formed syntactic derivation tree $t'$ of category $N'$.
Any semantic category that is not a conjunctive correspondence category is
called a {\em disjunctive (correspondence) category}.
Semantic categories that have only one syntactic category in their category
correspondence set are trivially conjunctive categories.\\
For example, in the case of the English/French NP rules,
the semantic category $\ul{DET}$ corresponds {\em conjunctively} to categories
{\em DET\,$'_m$} and {\em DET\,$'_f$} (any determiner has both a masculine and
a feminine form), whilst semantic category $\ul{N}$ corresponds
{\em disjunctively} to categories {\em N\,$'_m$} and {\em N\,$'_f$}
(nouns usually have either masculine or feminine gender).
Semantic category $\ul{NP}$ corresponds to only one category, {\em NP}\,$'$,
and is therefore a conjunctive category.\\
How can we use this to establish a condition for completeness?
The key idea is that some of the CFG-well-formed syntactic derivation
trees of some category~$A$ may be guaranteed to translate into at least one
CFG-well-formed TL syntactic derivation tree {\em for all categories}
in~$\tilde{A}$, instead of `for at least one'.
Category~$A$ is then said to {\em correspond conjunctively} to the
categories in~$\tilde{A}$.
As opposed to disjunctive categories, a conjunctive category does not require
every rule~$R'$ to have translation-equivalent variants for all categories
in~$\tilde{A}$.
Thus, the distinction between conjunctive and disjunctive categories
allows for a more realistic condition on the grammars.\\
We adjust the definition of N-N category correspondence, taking into
account the distinction between conjunctive and disjunctive categories.\\
As for the basic meanings and basic expressions:
For every basic meaning $m$, if its category $C$ is a disjunctive
category, there must be at least one associated basic expression $b'$ with
category $C'$ {\em for at least one category} $C'$ in $\tilde{C}$.
If category $C$ of basic meaning $m$ is a conjunctive category, then there
must exist at least one associated basic expression $b'$ with category $C'$
{\em for every category} $C'$ in $\tilde{C}$.\\
As for the semantic and syntactic rules, for every semantic rule $M$
with type \tuple{\tuple{A_1\kdots A_n},A}, we establish conditions on
the syntactic rules with which they are associated.
Again referring to fig.\,2, when generating a syntactic derivation tree
from a semantic derivation tree, for subtrees $d_i$ that have a conjunctive
category $C$ we can guarantee a tree $t_i'$ {\em for every} category in
$\tilde{C}$.
For subtrees $d_i$ that have a disjunctive category $C$ we can guarantee
a tree $t_i$ {\em for only one} category in $\tilde{C}$, and we do not
know which one.
Therefore, we must guarantee that for every tuple\footnote{Consider the
following auxiliary definitions.
For any argument list \tuple{A_1\kdots A_n}, define sets $I_c$ and $I_d$
as consisting of the indices of its conjunctive and disjunctive categories,
respectively.
Define \tuple{A_i\mid\elem{i}{I_c}} and \tuple{A_i\mid\elem{i}{I_d}}
as the corresponding subtuples.}
\elem{D}{{\sf X}_{\elem{i}{I_d}}\ \tilde{A_i}} of the syntactic categories
corresponding to disjunctive categories of $M$,
there exists {\em at least one} syntactic rule $R'$ with type
\tuple{\tuple{B_1\kdots B_n},B} such that:
\begin{bulletlist}
\item The tuple of the syntactic categories corresponding to the disjunctive
categories of the argument list of $M$ is equal to $D$:
\tuple{B_i\mid\elem{i}{I_d}}=$D$.
\item Every syntactic category $B_i$ that corresponds to a conjunctive category
$A_i$ of the argument list of $M$ is in the category correspondence set
of $A_i$:
$\forall \elem{i}{I_c}\ \ \elem{B_i}{\tilde{A_i}}$.
\item In addition, the resultant category $A$ of semantic rule $M$ must be taken
into account. If this is a disjunctive category, then it suffices if the
resultant category $B$ of the syntactic rule $R'$ is in $\tilde{A}$.
If category $A$ is a conjunctive category, then there must be at least
one syntactic rule $R'$ with resultant category $N$ for all categories
$N$ in $\tilde{A}$.
\end{bulletlist}
Using this condition we again obtain completeness:
\theorem{CFG Completeness for Many-to-Many Category Correspondence}
For any CFG-based compositional grammar pair \tuple{G,G'}, compositional
translation from~$G$ to~$G'$ is {\em complete} if
{\em(i)} the grammar pair is homomorphic from~$G$ to~$G'$, and
{\em(ii)} there is an N-N category correspondence between the semantic
and the syntactic categories of $G'$, where every semantic category of $G'$
has been declared conjunctive or disjunctive and the sets of categories
of $G'$ satisfy the condition described above.\\
Because of space limitations we do not include the proof; we trust that the
description of the condition above gives the reader an insight into
how the proof can be given.\\
{\bf Example}\ \ Returning to the English/French example discussed earlier,
we declared $\ul{DET}$ a conjunctive, $\ul{N}$ a disjunctive, and $\ul{NP}$
a conjunctive category. Checking the condition formulated above, this
amounts to the requirement that for every tuple $D$ in
$\{\tuple{N_m'},\tuple{N_f'}\}$, there exists a syntactic rule $R'$ such that
\tuple{B_i\mid\elem{i}{I_d}}=$D$ and
$\forall \elem{i}{I_c}\ \ \elem{B_i}{\tilde{A_i}}$, which is indeed the case.
\section{Conclusion and Future Research}%
\label{sec-discussion}
In this paper we presented the issue of completeness for compositional translation,
and discussed how conditions for compositional translation could be found.
In section~\ref{sec-completeness-for-CFG} we examined the completeness issue for
context-free grammars. We established completeness conditions for grammars with an
N-1 category correspondence. As this condition is rather restrictive, we relaxed
this condition to an N-N category correspondence condition. The first attempt
however led to unrealistic conditions on the grammar rules, so that it was
necessary to introduce the distinction between conjunctive and disjunctive
categories. We adjusted the N-N category correspondence condition accordingly,
and obtained a completeness condition for grammars with an N-N category
correspondence.\\
The central issues in ongoing and future research are
{\em(i)} the completeness issue for some other grammar formalisms,
{\em(ii)} the algebraic formulation of completeness, and
{\em(iii)} polynomial compositional translation.\\
{\bf (i)\ Completeness for Other Grammar Formalisms} --
The definite-clause grammar formalism (DCG, see e.g.\,\cite{Pereira-Shieber-1987})
extends the CFG grammar formalism with attributes added to the nonterminals.
Attributes have a variety of uses, one of the most prominent being the
enforcement of agreement relations.
As for the completeness condition for DCG, we assume the same conditions on the
nonterminals as we did for CFG. In addition, we formulate restrictions on the
use of attributes.
A proof has been established for completeness of grammars that satisfy these
restrictions.\\
Future research will also address the completeness issue for Tree-Adjoining Grammars.
Tree-Adjoining Grammars are interesting because they are somewhat more
expressive than CFG's (they are so-called mildly context-sensitive),
and it enables expressing linguistic phenomena such as long-distance
dependencies.\\
{\bf (ii)\ Algebraic Formulation of Compositional Translation} --
Compositional grammar, compositional translation and the completeness issue
can be formulated algebraically.
Such an algebraic formulation has a number of advantages:
{\em(i)} it abstracts away from the details of specific grammar formalisms,
thus revealing the essentials of compositional translation and completeness,
{\em(ii)} this abstraction provides a basis for the comparison of different
grammar formalisms, and
{\em(iii)} an algebraic formulation gives access to well-investigated
mathematical theory, the results of which may be readily carried over.
I hope to use the algebraic formulation as a basis for the investigation
of the combination of the use of features and completeness.
For other work on algebraic description of natural language, see
\cite{Janssen-1986,Hendriks-1993}. An algebraic view on compositional
translation is presented in \cite[Ch.19]{Rosetta-1994}.\\
{\bf (iii)\ Polynomial Compositional Translation} --
Another line of work is concerned with an extension of the method of compositional
translation for grammar formalisms that use only concatenative operations.
The basic idea here is a generalization of the unit of translation-equivalence
from single elements to combinations of these (polynomials).
This improves `translation power', as it becomes possible to overcome
all kinds of translation problems due to structural divergencies between languages.
For example it becomes possible to relate a structure like $[A\ [B\ C]]$
with a structure like $[A'\ B'\ C']$.
I hope to show that, as polynomially derived algebras are algebras again,
completeness conditions found for compositional translation will carry over
to polynomial compositional translation.
\begin{footnotesize}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,152 |
Archibald Fenner Brockway, Baron Brockway (ur. 1 listopada 1888 w Kalkucie — zm. 28 kwietnia 1988 w Londynie) – brytyjski działacz anytywojenny i polityk związany z Partią Pracy. Wieloletni przewodniczący pacyfistycznej organizacji War Resisters' International.
Przypisy
Urodzeni w 1888
Zmarli w 1988
Politycy Partii Pracy (Wielka Brytania)
Brytyjscy parlamentarzyści 1929–1931
Brytyjscy parlamentarzyści 1950–1951
Brytyjscy parlamentarzyści 1951–1955
Brytyjscy parlamentarzyści 1955–1959
Brytyjscy parlamentarzyści 1959–1964 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,156 |
{"url":"https:\/\/itectec.com\/matlab\/matlab-how-to-format-the-nonlinear-equations-to-find-solutions\/","text":"# MATLAB: How to format the nonlinear equations to find solutions\n\nfunction nonlinear equations\n\nI would like to write a function to solve two nonlinear equations in order to find K and a and use the solved K and a in other function later, that is, K=a*P(K); 1=a*Q(K). P(K)and Q(K) are two complicated expressions with respect to K. How to format this?\n\n\u2022 function ncf = N(x)%ncf = 0.5*(1+erf(x\/sqrt(2)));xr = real(x);xi = imag(x);if abs(xi)>1e-10 error 'imag(x) too large in N(x)'endncf = 0.5*(1+erf(xr\/sqrt(2))) + i*xi.*exp(-0.5*xr.^2)\/sqrt(2*pi);endfunction [K,alpha] = alphaCrit(S,r,sigma,T)x0 = [1 2];sol = fsolve(@(x)fun(x,S,sigma,T),x0)K = sol(1);alpha = sol(2);end function res = fun(x,S,sigma,T)K = x(1);alpha = x(2);d2 = ( log(S) - log(K) + (r-0.5*sigma^2)*T ) \/ (sigma*sqrt(T));res(1) = K - alpha * exp(-r*T)*N(-d2);res(2) = 1 - alpha * exp(-r*T)\/(sqrt(2*pi*sigma^2*T)*K)*exp(-0.5*d2^2);end","date":"2021-05-07 16:46:04","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\": 0.44658342003822327, \"perplexity\": 6881.7294572080145}, \"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-2021-21\/segments\/1620243988796.88\/warc\/CC-MAIN-20210507150814-20210507180814-00310.warc.gz\"}"} | null | null |
Look at one of the best photos of Naomi Watts – it is 1229 picture from all 1245 we have.
Please look for the similar picture if that resolution 752x1049 is less than your mobile device screen resolution. Please be informed that Naomi Watts picture has a resolution of 752x1049. Its size is 122 kilobytes. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,371 |
Q: Nginx config for Slim Framework API routes I have been stuck with this for quite some time now. Basically, i have a routing file in my Slim Framework App which routes my API and then i can access the routes like so: "index.php/api/route". This works fine with apache or php -S. But now when i migrated to an nginx server with php5-fpm, i am facing issues with configuring the site properly. I can access index.php, but anything after is a 404. Checking the logs give me "no such file or folder" or "not a directory". Here's my current config:
server {
listen 80;
listen [::]:80 default_server ipv6only=on;
server_name www.site.com;
root /var/www/site;
index index.php;
error_log /var/log/nginx/site.error.log;
access_log /var/log/nginx/site.access.log;
location ~ \.php$ {
fastcgi_connect_timeout 5s;
fastcgi_read_timeout 10s;
fastcgi_pass unix:/var/run/php5-fpm.sock;
#fastcgi_split_path_info ^(.+\.php)(/.*)$;
fastcgi_index index.php;
include fastcgi_params;
#fastcgi_param SCRIPT_FILENAME $document_root$fastcgi_script_name;
}
}
I tested with the commented out lines, to no success. Any ideas?
A: Nevermind, i was just missing a try_files for the index.php path
location / {
try_files $uri $uri/ /index.php?$query_string;
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,541 |
{"url":"http:\/\/mathhelpforum.com\/calculus\/140167-finding-derivative-complex-fraction-radicals.html","text":"# Math Help - Finding the Derivative - Complex Fraction with Radicals\n\n1. ## Finding the Derivative - Complex Fraction with Radicals\n\nSo I've been doing good with these until this problem.\n\nFind $f'(a)$\n\n$f(x)=\\frac{8}{\\sqrt{x+2}}$\n\n$\\frac{\\frac{8}{\\sqrt{a+h+2}}-\\frac{8}{\\sqrt{a+2}}}{h}$\n\nI began multiplying by a common denominator to get rid of the complex fraction, then multiplied by a conjugate to get rid of the square roots. After that I don't know what to do.\n\nSo I've been doing good with these until this problem.\n\nFind $f'(a)$\n\n$f(x)=\\frac{8}{\\sqrt{x+2}}$\n\n$\\frac{\\frac{8}{\\sqrt{a+h+2}}-\\frac{8}{\\sqrt{a+2}}}{h}$\n\nI began multiplying by a common denominator to get rid of the complex fraction, then multiplied by a conjugate to get rid of the square roots. After that I don't know what to do.\nyou should be up to this point ...\n\n$8 \\lim_{h \\to 0} \\frac{1}{h}\\left(\\frac{-h}{\\sqrt{a+h+2} \\cdot \\sqrt{a+2} \\left[\\sqrt{a+h+2}+\\sqrt{a+2}\\right]}\\right)$\n\nnote that the $h$'s cancel, allowing the limit to be evaluated as ...\n\n$-\\frac{8}{(a+2) \\cdot 2\\sqrt{a+2}} = -\\frac{4}{(a+2)^{\\frac{3}{2}}}$\n\n3. Okay... so the $h$'s cancel out because it's approaching $0$?\n\nIf so, I completely understand now.","date":"2014-09-17 09:36:59","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\": 11, \"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\": 0.8828933835029602, \"perplexity\": 282.7604166103881}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-41\/segments\/1410657123274.33\/warc\/CC-MAIN-20140914011203-00142-ip-10-196-40-205.us-west-1.compute.internal.warc.gz\"}"} | null | null |
Q: Read property in Json in Javascript I have a problem. How can I get refcat in that Json object in javascript. Please find the below code snippet.
{
"request": {
"operation": "Viviendas",
"languague": "es",
"idMunicipio": "5",
"idVia": "196",
"idProvincia": "28",
"pc1": "9115501",
"pc2": "VK6891N",
"numero": "1"
},
"created_http": "Thu Oct 01 09:33:58 GMT+01:00 2015",
"created_server": 20151001092458,
"data": [
{
"portal": "",
"escaleras": [
{
"escalera": "1",
"plantas": [
{
"planta": "00",
"puertas": [
{
"puerta": "01",
"refCat": "9115501VK6891N0001GB",
"superficieVivienda": 60.0,
"superficieTotal": 60.0
}
]
},
{
"planta": "01",
"puertas": [
{
"puerta": "01",
"refCat": "9115501VK6891N0002HZ",
"superficieVivienda": 154.0,
"superficieTotal": 170.0
}
]
},
{
"planta": "02",
"puertas": [
{
"puerta": "01",
"refCat": "9115501VK6891N0003JX",
"superficieVivienda": 161.0,
"superficieTotal": 177.0
}
]
}
]
}
]
}
],
"page": "1",
"pagesTotal": "1",
"status_code": "200",
"status_text": "OK"
}
I tried to fix this with the below method.
alert(json.data[0].escaleras[0].plantas[0].puertas[i].refCat);
But it doesn't work. Can someone please help me with this?
A: The problem is
alert(json.data[0].escaleras[0].plantas[0].puertas[i].refCat);
i is causing the problem...... ^
it should be 0.
alert(json.data[0].escaleras[0].plantas[0].puertas[0].refCat);
var json = {
"request": {
"operation": "Viviendas",
"languague": "es",
"idMunicipio": "5",
"idVia": "196",
"idProvincia": "28",
"pc1": "9115501",
"pc2": "VK6891N",
"numero": "1"
},
"created_http": "Thu Oct 01 09:33:58 GMT+01:00 2015",
"created_server": 20151001092458,
"data": [
{
"portal": "",
"escaleras": [
{
"escalera": "1",
"plantas": [
{
"planta": "00",
"puertas": [
{
"puerta": "01",
"refCat": "9115501VK6891N0001GB",
"superficieVivienda": 60.0,
"superficieTotal": 60.0
}
]
},
{
"planta": "01",
"puertas": [
{
"puerta": "01",
"refCat": "9115501VK6891N0002HZ",
"superficieVivienda": 154.0,
"superficieTotal": 170.0
}
]
},
{
"planta": "02",
"puertas": [
{
"puerta": "01",
"refCat": "9115501VK6891N0003JX",
"superficieVivienda": 161.0,
"superficieTotal": 177.0
}
]
}
]
}
]
}
],
"page": "1",
"pagesTotal": "1",
"status_code": "200",
"status_text": "OK"
};
alert(json.data[0].escaleras[0].plantas[0].puertas[0].refCat);
A: Hi you would have to iterate through the json object to get the data.
Use the following code.
json.data.forEach(function (object) {
object.escaleras.forEach(function (innerObject) {
innerObject.plantas.forEach(function (innerMostObject) {
innerMostObject.puertas.forEach(function (refCatObject) {
console.log(refCatObject.refCat)
})
})
})
})
A: assuming you had a div with the id 'test' , this works. I'd guess it's your indexing logic generating the [i] on 'puertas' that's not correct, or perhaps your 'json' object isn't actually a JSON Object?
example jsfiddle https://jsfiddle.net/p6udz7ao/
copy of example below:
$(document).ready(function () {
var jsonString = '{ "request": { "operation": "Viviendas", "languague": "es" ,"idMunicipio":"5","idVia":"196","idProvincia":"28","pc1":"9115501","pc2":"VK6891N","numero":"1" }, "created_http": "Thu Oct 01 09:33:58 GMT+01:00 2015", "created_server": 20151001092458, "data": [ { "portal": "", "escaleras": [ { "escalera": "1", "plantas": [ { "planta": "00", "puertas": [ { "puerta": "01", "refCat": "9115501VK6891N0001GB", "superficieVivienda": 60.0, "superficieTotal": 60.0 } ] } , { "planta": "01", "puertas": [ { "puerta": "01", "refCat": "9115501VK6891N0002HZ", "superficieVivienda": 154.0, "superficieTotal": 170.0 } ] } , { "planta": "02", "puertas": [ { "puerta": "01", "refCat": "9115501VK6891N0003JX", "superficieVivienda": 161.0, "superficieTotal": 177.0 } ] } ] } ] } ], "page" : "1", "pagesTotal" : "1", "status_code" : "200", "status_text" : "OK" }';
var jsonObject = JSON.parse(jsonString);
var refcattext = jsonObject.data[0].escaleras[0].plantas[0].puertas[0].refCat;
$('#test').text(refcattext);
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,475 |
/**
* Licensed under the Apache License, Version 2.0 (the "License"); you may not
* use this file except in compliance with the License. You may obtain a copy of
* the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS, WITHOUT
* WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the
* License for the specific language governing permissions and limitations under
* the License.
*/
package wicket.contrib.groovy.builder;
/**
* Thrown by UnimplementedComponentBuilder. Might be a bit too specific. A
* WicketComponentBuilderException would probably be just as good.
*
* @author Kevin Galligan
*
*/
public class UnimplementedComponentBuilderException extends WicketComponentBuilderException
{
public UnimplementedComponentBuilderException(String component)
{
super("Component '"+ component +"' unimplemented. You can manually add components by calling 'add(new [ComponentClass])' or 'getCurrent()' and get a direct reference to your Component parent class.");
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,049 |
{"url":"https:\/\/www.lesswrong.com\/posts\/q6d4vtGwt5FE84wk7\/beta-feature-google-docs-like-editing-for-lesswrong-posts","text":"161\n\nTL;DR: LessWrong now has similar features to Google Docs. Warning! Still rough around the edges. To enable the collaborative editor, you must check \"Opt into experimental features\" in your account settings and then press the green \"Share\" button that appears when editing your post.\n\nYou can experiment with commenting and suggesting on this post with this link.\n\nIt's been a loooong time coming[1]\u00a0but at last, we are ready to unveil collaborative editing features for the LessWrong text editor. These features will be familiar to those used to working in Google Docs:\n\n\u2022 Multiple users can edit a document at once\n\u2022 Inline comments (only viewable while in edit mode)\n\u2022 Making and accepting suggested edits\n\u2022 Automatic saving\n\u2022 Version history viewer\n\nSome advantages of using LW-Docs with collaborative editing:\n\n\u2022 LW-Docs supports LaTeX, unlike Google Docs.\n\u2022 If you use entirely LW-Docs, you won't have broken footnotes, unlike with copying from G-Docs.\n\u2022 While writing your post, you'll know what the end result will look like (same font and line width) which helps you optimize paragraphs and layout for looking good when published.\n\u2022 You can continue to get inline feedback and suggestions on your post even after you've copied it over to LessWrong.\n\u2022 These can then be seamlessly integrated into your live post.\n\nHow to enable collaborative editing on your post\n\nStep 1: Opt in to experimental features in your account settings.\n\nStep 2: While editing your post, click the green \"Share\" button in the top right. Enable permissions for some other users. Boom! Your post is now in collaborative mode.\n\nStep 3: Users explicitly shared on your post will receive a notification. Send the url to anyone else you want to grant access.\n\nStep 4: When in collaborative mode, the text of your post is automatically saved. To view how your post will look when published, press the Preview button. To make the current state of your document live, press the Publish button.\n\nThe collaborative editor allows you (and others) to continue editing and commenting on a post even once it's been published. Edits aren't automatically published to the live version. To update the published version to the current state of the document in editing, press \"Publish\".\n\nStep 5: Leave comments and suggestions.\n\nTo enable track changes, use the track changes button in the popup menu (third icon from the right) or set your Mode to Commenting in the header bar.\n\nStep 6: Leave feedback about the feature!\n\nWarning! Rough around the edges\n\nWe're releasing this feature in beta mode because it still requires a few more finishing touches and might be a little confusing to use in places. As above, any feedback is greatly appreciated.\n\nWhy collaborative editing?\n\nWhen developing features, there's always a question of \"does anyone want this?\" In the case of collaborative editing features, there's good evidence of demand from Google Docs. There's a common workflow that goes: (1) write a draft in Google Docs, (2) invite some close friends or collaborators to give feedback, (3) incorporate feedback, (4) copy to LessWrong, (5) publish.\n\nDrawbacks of this workflow are (a) overhead of copying and reformatting the post, (b) enforcing a hard break between the feedback stage and the publication stage, (c) Google Docs does not support LaTeX, (d) valuable comments left in the feedback stage never get published to the wider world.\n\nBy introducing collaborative editing to LessWrong, we address (a), (b), and (c). We haven't yet made it so in-line comments in editing mode can be published in the final version, but we'll look into ways to allow for that. I also expect that we'll add additional features[2]\u00a0to our editor that Google Docs doesn't have, such that users will benefit from the possibility of doing all their writing on LessWrong.\n\nHaving collaborative editing features on LessWrong also lets us start to build programs that rely on easy ways to give people feedback on their drafts. For example, I'd like to run a writing and research workshop for students that involves peer and mentor feedback. With collaborative editing on LessWrong, that will now be much more convenient to do.\n\nIf you're an author, I'd love to hear how collaborative editing features do or don't help you with your workflow, or what you'd really like to see us build. Feel free to comment on this post or message us on Intercom.\n\nThanks and good luck!\n\n1. ^\n\nIt's been ~18 months since the first steps towards this were taken.\n\n2. ^\n\nOne feature I'm particularly excited about is \"link-searching\". In the same way that one can @-mention people on Facebook, I'd like to make it so you can easily link to post and wiki-tags by typing @ or # and then using a few letters to search for the resource you want to link to.\n\n161\n\nNew Comment\n\nI absolutely love this feature set! I really appreciate the markdown to formatting (Obsidian-style), live collaboration and LaTeX rendering. I think this also creates fantastic opportunities for innovations in communicating thoughts through LessWrong. Here's a list of my feedback:\n\nShortcuts: One feature I really enjoy in Google Docs is more deep text editing keyboard shortcuts than the standard set. One of my favourites is the Alt + Shift + Arrow up or down that replaces the current paragraph with the one above or below, effectively moving the paragraph in the text. Obsidian has a keyboard shortcuts editor that is quite nice as well but could have more (see here, page 2). I think shortcuts can make writing as seamless as some code editors (not thinking VIM here, but unintrusive extra editor tooling). Additionally, as I mention elsewhere, some (I just know European) keyboards do not take well to Ctrl + Alt + [] shortcuts like Ctrl + Alt + M, especially on the web.\n\nCollapsible boxes: Having \"fact boxes\" as expandable interactive elements seems like a very good idea as well and relatively cheap to implement. I recommend looking at e.g. Hugo XML syntax for these things (XML is a pain, you can probably figure a better writing UX out).\n\nInteractive documents: I'm more for the Obsidian thought representation but the hierarchy in Roam is quite relevant for communicating thoughts to others. I think a strong representation of this is quite relevant and might be an in-line, minimalist, recursive \"collapsible box\" as described above.\n\nFoot notes: I'd say we already have quite a strong Roam-style organization in the hover pop-ups of LessWrong post links and it would be very nice to have a version of that for foot-notes, given their disparate nature. I.e. hover = shows that specific footnote.\n\nLive preview & 100% keyboard editing: See Obsidian's implementation of this. The markdown to formatting feature is already super awesome is their LaTeX editing that does in-line MathJax with $wrapping and block with$\\$ wrapping. The current LessWrong LaTeX editor requires me to use the cursor, AFAIK. Obsidian is also pretty good at minimalist formatting rulesets for inspiration. Check out their vast plugins library for inspiration as well. I'm sure there's some absolute text editor gold in there.\n\nCollaborative editing extension: This makes or breaks our usage of LessWrong as the editing platform of choice. It would be awesome to have editing group settings, i.e. so I don't have to share every article with Apart Research members but can have a Google Drive-style folder sharing for blog post edits. Otherwise, we have to maintain a collection of links in a weird format somewhere.\n\nI also echo the other comments' feedback points. However overall, absolutely marvelous work! Really looking forward to the developments on this!\n\n(c) Google Docs does not support LaTeX\n\nI'd make this point much more prominent, I almost missed it! This point changes the feature from 'likely much worse than Google Docs + copying later' to 'actually probably worth using' for me\n\nYou're right. Emphasized!\n\nThis looks like a great feature!\n\nSuggestion: For this post, enable at least the inline-comment-mode (and share the link if required) so users can try it. I would love to give feedback on this feature, but don't have an easy way to test it right now.\n\nGood suggestion! Edited into the post, but also here.\n\nThanks! Have tried the feature, loved it, and posted a bunch of feedback :).\n\nFeedback on the editor commenting feature:\n\n\u2022 Love it.\n\u2022 When I try to close the page on which I've submitted comments, the browser (desktop Firefox v97.0.1) sometimes displays the standard warning \"please confirm that you're exiting the page; your changes may not be saved\". After exiting and reloading the page, it appears that everything was indeed saved.\n\u2022 Inline commenting is supposed to be usable via the shortcut Ctrl+Alt+M, but this doesn't work for me. Maybe the shortcut is not implemented correctly; maybe it's a language problem (I use a German keyboard); maybe it's a problem with one of my gazillion Firefox addons...\n\u2022 Apparently you get notifications if someone responds to your comments, but the notification text is wrong. I made a typo suggestion, Ruby replied, and then the notification said: \"Ruby commented on your draft [Beta Feature] Google-Docs-like editing for LessWrong posts\", even though it's neither a draft nor my post.\n\u2022 At the very top of the document, next to \"Share draft with users\", are a bunch of names (like jimrandomh) with \"X\" buttons next to them. In commenting mode, I can ostensibly remove all these users by clicking their \"X\" buttons, but they reappear when I reload the page.\n\u2022 Other related things:\n\u2022 In commenting mode, I can change the text of Ruby's moderation guidelines, though these changes aren't saved.\n\u2022 I see the \"Get Feedback\", \"Move to drafts\", and \"publish changes\" buttons. Pressing any of them displays the errors \"app.operation_not_allowed\" and \"Error submitting form: Post.update\", though the \"get feedback\" button does load Intercom with the default \"Hey MondSemmel,\n\u2022 I can't use the \"track changes\" button (it's grayed out), but I can load the post's Version History, though I somehow only see versions 1.5.1 or newer. (Note: It's probably desirable to make specific past versions inaccessible or invisible, e.g. if you by accident briefly published a version with private information.)\n\u2022 The notifications for getting a reply on your inline comment link to the entire post, not the specific inline comment, so it's currently difficult to find which comment was replied to.\n\nThe shortcuts with Ctrl + Alt generally work sub-optimally for EU keyboards, I believe. Seems worth it to test alternative keyboard layouts for usability - something that most major software companies in the U.S. do not seem to do well ;-)\n\nNice! Can't wait to use it. Will it be possible to let send someone a link to a draft and have them view it without a LessWrong account? Cause one of the reasons I use gdocs is for non-lesswrongers.\n\nI'm afraid that a LessWrong account is required. However, they're very easy to make! You can make one in ~10 seconds.\n\nRight, but some people dislike making accounts, so it would be nice if in the future you could view drafts with a link without an account like you can a gdoc.\n\nI agree. I'll see what we can do.\n\nWriter feedback partially unrelated to these changes: I find myself getting increasingly sold on the roam-style tree-structured writing format. Basically, if you've ever written code, you know why we need indentation sometimes. Sometimes a large number of things pertain to one previous thing, or they're grouped together in some way, and then there is often further nesting inside of that, and more nesting inside that, but it's important that the reader can easily see how it's all structured from an overview.\nAnd that's just how concepts generally are. I'm going to argue that we need tree-structuring in prose just as much as we need it in code or in proofs.\n\nThough I'm not sure. Disclaimer: This all only started when people started getting into Roam, so it's hard to say whether this is going to work out yet.\n\nSo this argument might not be convincing. Regardless, I think this is something we should talk about.\n\nSo, in conventional prose, you'd use naming and the occasional repetition to linearize a tree structure into a flat series of paragraphs, but if you've made anything halfway complex you start to realize it's sort of unnatural to do it that way, it makes everything more verbose, it means you have to cut anything you don't know how to flow in there elegantly, and it makes the overarching structure of the concept less visible.\n\nTree-structuring also seems like a somewhat more reasonable UX for footnotes, for the web. A nested section could be expanded in-line, instead of taking you to this place where all the footnotes are gathered together which is... not a coherent way to group that information (footnotes usually have nothing to do with other footnotes). Expanding a collapsed section is effectively the same as clicking a footnote, but with more reasonable spacial grouping.\n\nWe nest whenever part of the text should be optional.\n\nA traditional writer might say, \"that should be exceedingly rare: Everything in the text should be important, none of it should be optional\". I think that is kinda paternalist hubris, in a way. You don't know what the reader knows or doesn't know, you don't know quite what they need to hear or what they should be allowed to easily skip. If your text has very little structure, if it's been collapsed down to a linear presentation, they're not going to be able to skip any of it, they just have to read it all. Unless you're limiting yourself to saying only the most contrarian or esoteric or entertaining stuff (and those sorts of writers sure do disproportionately flourish, in our scenes), but that's not always what people need.\n\nIt's beneficial if we can allow some form of interactivity and let the reader decide whether a piece of text is for them. Right now people are too passive. When you tell them something they already know they just not along finding satisfaction in agreement. They should be bored. They should follow their boredom. They should be looking for something better to do. They should be asking for clear indications as to whether they can skip this paragraph, but traditional writing formats couldn't fit those in, so they don't know to ask for it.\n\nBut linearized series' of paragraphs really are what people are used to, right now. I suspect that most people would effectively not be able to read tree-structured texts. For instance, you need to have the habit of, on reaching the end of a branch, looking back up the stack to remember its nearest parent context, before proceeding to the next one, or else it just wont make sense. You should develop that habit. It's a good habit to have when you're trying to traverse a complex conceptual structure. Most people don't have it. Even people who have it from programming or reading proofs or reading legal documents, it might not occur to them to apply it to reading prose.\n\nAn example of a document written this way would be the Venture Granters design. My impression is that very few people really read it. Though I'm not sure how whether that was due to the tree structuring or just because I wasn't at the point where I could justify the complexity succinctly, relative to regular retroactive public goods funding, at the time of writing.\n\nAgreed, new ways of ordering thoughts online is an awesome opportunity on LessWrong!\n\nFoot notes: I'd say we already have quite a strong Roam-style organization in the hover pop-ups of LessWrong post links but as you say, it would be very nice to have a version of that for foot-notes, given their disparate nature.\n\nCollapsible boxes: Having \"fact boxes\" as expandable interactive elements seems like a very good idea as well and relatively cheap to implement. I recommend looking at e.g. Hugo XML syntax for these things (XML is a pain, you can probably figure a better writing UX out).\n\nInteractive documents: I'm more for the Obsidian thought representation but the hierarchy in Roam is quite relevant for communicating thoughts to others. I think a strong representation of this is quite relevant and might be an in-line, minimalist, recursive \"collapsible box\" as described above.\n\nAh, yeah link previews are good. I guess the problem with LW's ones that they're difficult to find out about on mobile, the user has to figure out to click and hold, then close the browser popup. I prefer gwern's way, where clicking a link on mobile will only open the preview, and you have to click again to traverse the link. Others have complained about that, though.\n\nI mostly use it from the computer so that missed me but it seems like a very good idea as well!\n\nNice work!\n\nRegarding my own workflow, unfortunately I would still have to go through external software most of the time. This is because most of my posts are latex-heavy, and it's impractical to wait for LessWrong to render a latex-heavy post every time you make and edit and want to see how it came out. Instead, I write the post in LyX[1] while using (inoperative) markdown syntax (e.g. asterisks for italics) and copy it to LessWrong in the end, with appropriate editing (mostly a bunch of \"replace all\").\n\n1. Previously used BaKoMa TeX which is unfortunately no longer supported. \u21a9\ufe0e\n\nThanks for that detail, that's helpful to know!\n\nThis is mostly unrelated and out of scope for your feature, but from time to time I've been wondering about what the minimal workflow for correcting typos would be, for both authors and readers. Some options:\n\n\u2022 Commenting with a list of typos: The current version. Requires making a list while reading, then putting it in a comment; other users might suggest duplicate typos; meta comments like typos detract from other comments; plus they don't get visibly deprecated once addressed by the author; etc.\n\u2022 Typo comments (or suggested edits) \u00e0 la Google Docs or this new LW feature: Typo suggestions include the location of the typo; other users who want to suggest the typo see that it's already been noticed and don't duplicate it; this meta stuff is clearly separated from normal LW comments; etc. Typo suggestions still don't get visibly deprecated once addressed by the author, unless the author can delete comments.\n\u2022 Suggesting changes in a version control system like Git: While forking on Github has tons of overhead, the basic workflow is quite simple: users suggest typo edits to the document, and then the author can confirm or reject them with one button. However, if typo suggestions are associated with users, this might have some weird implications wrt licensing and authorship. (IIRC some entertainment businesses like Wizards of the Coast don't accept user suggestions per company policy, lest the suggesters later claim that their suggestion was implemented without compensation.)\n\nI mainly write on the EA forum, but I'd like to see articles which are in the editing mode all the time - ie anyone can edit. I wonder how big a jump that is from this.\n\nI've written about it here https:\/\/forum.effectivealtruism.org\/posts\/NxWssGagWoQWErRer\/community-posts-a-new-forum-post-type-unofficial-pr-faq?commentId=oJJEn7FLo8uZEnRNF\n\nNit: the link to the account settings in Step 1 has a typo; path ends in acount rather than account.\n\nCongrats on the release! \u00a0The new in-line suggestions are really nice.","date":"2022-07-05 15:54:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.36219120025634766, \"perplexity\": 2227.3136910092735}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104585887.84\/warc\/CC-MAIN-20220705144321-20220705174321-00519.warc.gz\"}"} | null | null |
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#define ENERGEST_CONF_ON 1
#undef UIP_CONF_ROUTER
#define UIP_CONF_ROUTER 0
#undef WITH_SERIAL_LINE_INPUT
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/* Needed for communicating with other nodes outside the LoWPAN,
as UDP checksum is not optional in IPv6. */
#undef UIP_CONF_UDP_CHECKSUMS
#define UIP_CONF_UDP_CHECKSUMS 1
#endif /* WEBSERVER_AJAX_CONF_H_ */
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Just thought I'd let you know I'm in love with your blog. I found it randomly well looking for information on the "30 greatest adventures of all time" list. keep up the good work James.
without any material from the supplements.
I did say that White Box "more closely approximates" OD&D without the supplements, not that it did include bits of them here and there. I'm sure there are other minor examples of supplement-derived material that crept in. It's still much closer to the 3 LBBs than are the S&W Core Rules.
Thanks, James. Feeling a bit better today. | {
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\section{Introduction}
In the last years it has been realized that the (still unknown)
physics at very high energies may not be inaccessible from an
observational point of view. Indeed, trans-Planckian physics may
have left an imprint in the inhomogeneities of the cosmic
microwave background radiation \cite{CMBtrans}, in the evolution of the scale
factor of the universe \cite{scale}, in the propagation of gamma ray bursts \cite{gamma},
etc.
In the absence of a full theory, the theoretical approach to this
problem is phenomenological. One possibility, that we will
consider here, is to assume that the physics at high energies is
such that its main effect is a modification of the dispersion
relation of the quantum fields, thus violating Lorentz symmetry.
Although this is a simplistic approach, it could be useful to
investigate whether the trans-Planckian effects could lead to
observable consequences or not in a given particular situation, by
testing the robustness of the results under changes in the
dispersion relations at very high energies.
The Modified Dispersion Relations (MDR) will obviously affect the
structure of the quantum field theory, in particular its
renormalizability. Having in mind applications to cosmology, in
previous papers \cite{NosUno,NosProc,NosDos}, we have analyzed in
detail the renormalization of free field theories with MDR in flat
Robertson Walker spacetimes. We have shown that the theory can be
renormalized using a generalization of the well known adiabatic
regularization \cite{ad-old,equiv} that is used in theories with
standard dispersion relations. As for the usual case, the
adiabatic expansion of the energy momentum tensor contains
divergent terms that can be written in terms of geometric tensors
in $n$-dimensions, and therefore the theory can be renormalized by
absorbing the infinities into the bare gravitational constants of
the theory. It is remarkable that this can be done whatever the
dispersion relation. This somewhat surprising result could be a
peculiarity of flat Robertson Walker metrics \cite{ted} and/or
valid only for free fields, and therefore it is of interest to
investigate more general situations.
In this paper we extend the adiabatic regularization to the case
of self-interacting fields and anisotropic metrics (Bianchi type
I). We will work within the context of the so called
Einstein-Aether theory \cite{Jacobson}, a covariant theory of gravity in
which the metric is coupled to a dynamical vector field. This
field, that breaks Lorentz invariance dynamically, is also
coupled to the derivatives of the quantum matter fields, leading
to MDR that contain higher powers of the momenta. The specific
model is introduced in Section II.
In Section III we consider a self-interacting scalar field on
Bianchi type I metrics and discuss the renormalization of the
equation for the mean value of the field $\phi_0$. In order to do
this, it will be necessary to compute the mean value of the
fluctuations of the field
$\langle\hat\phi^2\rangle=\langle(\phi-\phi_0)^2\rangle$. We will
calculate explicitly this quantity up to the second adiabatic
order and show that, contrary to what happens for the usual
dispersion relation, the second adiabatic order cannot be entirely
written in terms of the metric and its derivatives, but also
involve the aether field $u_{\mu}$ and its derivatives. This
property of $\langle\hat\phi^2\rangle$ is valid even for free
fields in flat Robertson-Walker spacetimes.
In Section IV we analyze the renormalizability of the
Semiclassical Einstein-Aether Equations (SEAE) for the case of
free scalar fields with MDR in Bianchi type I universes. We
compute $\langle T_{\mu\nu}\rangle$ up to the second adiabatic
order. The zeroth adiabatic order is divergent whatever the
dispersion relation. Being proportional to $g_{\mu\nu}$, the
divergence can be absorbed into a redefinition of the comological
constant. The second adiabatic order is shown to be divergent for
dispersion relations that involve powers of the momenta smaller
than or equal to four. This adiabatic order contains a term
proportional to $G_{\mu\nu}$, that renormalizes Newton's constant.
However, it also contains an additional non-purely geometric term,
proportional to the variation of $(\nabla_{\mu}u^{\mu})^2$. When
this term is divergent, a new counterterm has to be introduced to
renormalize the theory, even if originally not present in the
classical Lagrangian. On the other hand, if it is finite, a
counterterm would be necessary to make the theory consistent
with observations.
In Section V we argue that, for a general metric, the
renormalization of the infinities produced by a quantum free field
satisfying MDR will induce all possible counterterms involving up
to two derivatives of the metric $g_{\mu\nu}$ and the vector
$u_{\mu}$. As shown in Ref. \cite{Jacobsondebil}, the coefficients
of terms like $(\nabla_{\mu}u^{\mu})^2$,
$R_{\mu\nu}u^{\mu}u^{\nu}$, etc, are strongly constrained
observationally by post-Newtonian parameters, and therefore the
counterterms induced by trans-Planckian physics should be fine
tuned to satisfy these constraints.
Throughout the paper we set $c=1$ and adopt the sign convention
denoted (+++) by Misner, Thorne, and Wheeler \cite{MTW}.
\section{The Model}
We work in the frame of a generally covariant theory of gravity
coupled to a dynamical vector field $u^{\mu}$ that breaks local
Lorentz symmetry. The most general action that is quadratic in
derivatives is given by \cite{Jacobson}: \begin{equation} S_{G}=\frac{1}{16\pi
G }\int d^n x \sqrt{-g} (R-2\Lambda+\mathcal{L}_{u}),\label{Sg}\end{equation}
where $g=det(g_{\mu\nu})$, $R$ is the Ricci scalar, $\Lambda$ and
$G$ are the bare cosmological and Newton's constants, and
$\mathcal{L}_{u}$ describe the dynamics of the additional degree
of freedom $u^{\mu}$, \begin{equation}\label{lu}
\mathcal{L}_{u}=-\tilde{\lambda}(g^{\mu\nu}u_{\mu}u_{\nu}+1)-b_1
F_{\mu\nu}F^{\mu\nu}-b_2 (\nabla_{\mu}u^{\mu})^2-b_3
R_{\mu\nu}u^{\mu}u^{\nu}-b_4
u^{\rho}u^{\sigma}\nabla_{\rho}u_{\mu}\nabla_{\sigma}u^{\mu},\end{equation}
where $F_{\mu\nu}=\nabla_{\mu}u_{\nu}-\nabla_{\nu}u_{\mu}$. The
Lagrange multiplier $\tilde{\lambda}$ is introduced to impose the
condition $u_{\mu}u^{\mu}=-1$ and the coefficients $b_i$
($i=1,2,3,4$) are arbitrary. The term
$\nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu}$ coincides with
$(\nabla_{\mu}u^{\mu})^2-R_{\mu\nu}u^{\mu}u^{\nu}$ up to a total
derivative, and hence has been omitted.
We consider a quantum scalar field $\phi$ with a generalized
dispersion relation propagating in a curved space-time with a
classical background metric given by \begin{equation}
ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}\equiv
-(u_{\mu}dx^{\mu})^2+\perp_{\mu\nu}dx^{\mu}dx^{\nu},\end{equation} where
$\mu,\nu= 0,1...n-1$ (with $n$ the space-time dimension) and
$\perp_{\mu\nu}\equiv g_{\mu\nu}+ u_{\mu} u_{\nu}$. The action for
the scalar field can be written as: \begin{equation} S_{\phi}=\int d^n x
\sqrt{-g}
(\mathcal{L}_{\phi}+\mathcal{L}_{cor}+\mathcal{L}_{int}),\end{equation} where
$\mathcal{L}_{\phi}$ is the standard
Lagrangian of a free, massive, minimally coupled scalar field \begin{equation}
\mathcal{L}_{\phi}=-\frac{1}{2}\left[ g^{\mu
\nu}\partial_{\mu}\phi\partial_{\nu}\phi+m^2\phi^2\right],\end{equation}
$\mathcal{L}_{cor}$ is the corrective lagrangian that gives rise
to a generalized dispersion relation \begin{equation}
\mathcal{L}_{cor}=-\sum_{s,p} b_{sp}
(\mathcal{D}^{2s}\phi)(\mathcal{D}^{2p}\phi),\end{equation} where $0< p\leq
s$, $b_{sp}$ are arbitrary coefficients, and
$\mathcal{D}^{2}\phi\equiv\perp_{\mu}^{\lambda}\nabla_{\lambda}\perp_{\gamma}^{\mu}\nabla^{\gamma}\phi$
($\nabla_{\mu}$ is the covariant derivative corresponding to the
metric $g_{\mu\nu}$ and $\perp_{\mu}^{\lambda}\equiv
g^{\lambda\nu}\perp_{\mu\nu}$). The interaction Lagrangian
$\mathcal{L}_{int}$ contains the following terms:
\begin{equation}\label{lintgen} \mathcal{L}_{int}=-\frac{1}{2}[\xi R+\xi_1
F_{\mu\nu}F^{\mu\nu}+\xi_2(\nabla_{\mu}u^{\mu})^2+\xi_3
\nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu}+\xi_4
u^{\rho}u^{\sigma}\nabla_{\rho}u_{\mu}\nabla_{\sigma}u^{\mu}+\xi_5
u^{\mu}u^{\nu}R_{\mu\nu}]\phi^2-\lambda\phi^4,\end{equation} where $\xi$,
$\xi_i$ ($i=1,2,3,4,5$) and $\lambda$ are bare parameters. Note
that, in addition to the self-interaction and the standard
coupling to the Ricci scalar, we have also included couplings
between $\phi^2$ and non-purely geometric terms that involve the
aether field $u_{\mu}$. Note also that, if we assume that the MDR
depart from the usual one at a given scale $M_C$, the coefficients
$b_{sp}$ scale as $b_{sp}\sim M_C^{2(1-s-p)}$.
In the rest of the paper we will consider a four-dimensional
Bianchi type I space-time with line element \begin{equation}
ds^2=-dt^2+\sum_{i=1}^{3}C_{i}(t)dx_i^2=-C(\eta)d\eta^2+\sum_{i=1}^{3}C_{i}(t)dx_i^2,\end{equation}
where $C=(C_1 C_2 C_3)^{1/3}$, $d\eta=dt/C^{1/2}$, and
$u_{\mu}\equiv C^{1/2}(\eta)\delta^{\eta}_{\mu}$. Therefore, in
this frame $F_{\mu\nu}=0$ and $u^{\mu}\nabla_{\mu}u_{\nu}=0$. In
what follows we use primes for denoting derivatives with respect
to the conformal time $\eta$. No sum convention in spatial (latin)
indices is assumed. The generalized dispersion relation takes the
form \begin{equation} \omega^2_k=C(\eta)\left[
m^2+x+2\sum_{s,p}(-1)^{s+p}\,b_{sp}\,x^{(s+p)}\right], \label{dis} \end{equation}
where
$x=\sum_{i=1}^{3}k_i^2/C_i\equiv\sum_{i=1}^{3}x_i\equiv\sum_{i=1}^{3}
x \lambda_i^2$, with $\sum_{i=1}^3\lambda_i^2=1$.
\section{Self-interacting scalar field in Bianchi type I space-times}
In this section we are concerned with the renormalization of the
equation of motion for the expectation value of a self-interacting
scalar field ($\lambda\neq 0$) propagating in a four-dimensional
Bianchi type I space-time. We assume that the state of the system
is such that the expectation value of the field is $\phi_0$. Then,
defining a new quantum field $\hat{\phi}$ as
$\phi=\phi_0+\hat{\phi}$, the equation of motion for $\phi_0$ in
the one-loop approximation is given by \begin{equation}\label{Ecphicero}
\Box\phi_0-\left[ m^2+\xi R+\xi_2(\nabla_{\mu}u^{\mu})^2+\xi_3
\nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu} +\xi_5
R_{\mu\nu}u^{\mu}u^{\nu}+2\sum_{s,p\leq
s}b_{sp}\mathcal{D}^{2(s+p)}+12\lambda\langle\hat{\phi}^2\rangle\right]\phi_0-4\lambda\phi_0^3=0.\end{equation}
The Fourier modes of the scaled field $\chi=C^{1/2}\hat{\phi}$
satisfy \begin{equation} {\chi_k''}+\left[ \omega_k^2+\left(\xi-\frac{1}{6}\right)
CR+Q+\xi_2 C(\nabla_{\mu}u^{\mu})^2+\xi_3 C
\nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu}+\xi_5 C
R_{\mu\nu}u^{\mu}u^{\nu}+12 C\lambda
\phi_0^2\right]\chi_k=0,\label{ecparachi}\end{equation}
with the usual normalization condition \begin{equation} \chi_k
{\chi_k'}^*-\chi_k'\chi_k^*=i\; .\label{nor} \end{equation}
The explicit expressions for the different terms in Eqs.
(\ref{Ecphicero}) and (\ref{ecparachi}) are, in Bianchi type I
metrics,
\begin{subequations}
\begin{align}
&(\nabla_{\mu}u^{\mu})^2=\frac{9 D^2}{4 C},\\
&R_{\mu\nu}u^{\mu}u^{\nu}=- \frac{3}{C}\left[\frac{D'}{2}+2Q\right],\\
&R=\frac{1}{C}\left[ 3 D'+\frac{3}{2}D^2+6Q\right],\\
&\nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu}=\sum_{i=1}^3\frac{d_i^2}{4C}=\frac{3}{4C}(D^2+8Q),\\
&Q=\frac{1}{72}\sum_{i<j}^3 (d_i-d_j)^2,\label{Q}
\end{align}
\end{subequations}
where $d_i=C_i'/C_i$ and $D=\sum_{i=1}^3 d_i/3=C'/C$. Note that
for the metric we are considering\begin{equation} 2
R_{\mu\nu}u^{\mu}u^{\nu}+R=(\nabla_{\mu}u^{\mu})^2-\nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu},\end{equation}
and therefore without loss of generality we can set $\xi_5=0$.
For dispersion relations such that the mean value
$\langle\hat{\phi}^2\rangle$ in Eq. (\ref{Ecphicero}) is
divergent, the infinities must be absorbed into the bare constants
of the theory. To implement the renormalization, we start by
expressing the field modes $\chi_k$ in the well known form \begin{equation}
\chi_k= \frac{1}{\sqrt{ 2 W_k}}\exp\left( -i\int^\eta
W_k(\tilde\eta)d\tilde\eta\right), \label{chi} \end{equation} which allows us to
write \begin{equation} \langle\hat{\phi}^2\rangle =\frac{1}{(2\pi)^3 C}\int
d^3k {|\chi_k|^2}=\frac{1}{(2\pi)^3 C}\int d^3k \frac{1}{2 W_k}.
\end{equation}
Substitution of Eq.
(\ref{chi}) into Eq. (\ref{ecparachi}) yields
\begin{equation} W_k^2 =
\omega_k^2+\left(\xi-\frac{1}{6}\right)
CR+Q+\xi_2C (\nabla_{\mu}u^{\mu})^2+\xi_3
C\nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu}+12\lambda C
\phi^2_0+\frac{5}{16}\frac{[(W_k^2)']^2}{W_k^4}-\frac{1}{4}\frac{(W^2_k)''}{W_k^2}.\label{Weq}
\end{equation} For adiabatic regularization we need the approximate solution
of this non-linear differential equation that is obtained by
assuming that $W^2_k$ is a slowly varying function of $\eta$. In
this adiabatic or WKB approximation the adiabatic order of a term
is given by the number of time derivatives of the metric plus the
power of $\phi_0$ \cite{PazMazzi}. The WKB approximation can be
obtained by solving the Eq.(\ref{Weq}) iteratively \begin{equation} W_k =
^{(0)}W_k + ^{(2)}W_k+...\; ,\end{equation} where the superscript denote the
adiabatic order. To lowest order we have $^{(0)}W_k=\omega_k$. The
second adiabatic order can be computed replacing $W_k$ by $\omega_k$
on the right-hand side of Eq. (\ref{Weq}). Thus,
we straightforwardly obtain
\begin{eqnarray}
^{(2)}W_k^2 & = & C R(\xi-\frac{1 }{6})+Q+\frac{D^2}{16}-\frac{D'}{4}+\xi_2C (\nabla_{\mu}u^{\mu})^2+\xi_3 C\nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu} +12\lambda C \phi^2_0 \nonumber\\
&-&\frac{(f+1)}{4}\sum_{i=1}^3\lambda_i^2\left[ \frac{D d_i}{2}+ d_i^2-d_i' \right]+ \frac{1}{16} \left(\sum_{i=1}^3d_i\lambda_i^2\right)^2 \left[ f^2+6 f-4\dot{f}+5\right],\label{W2}
\end{eqnarray} where we have defined the function
\begin{equation} f\equiv \frac{d\ln\tilde{\omega}_k^2}{d\ln x}-1,\label{f} \end{equation}
with $\tilde{\omega}_k^2\equiv\omega_k^2/C$. We have also used that
\begin{subequations}\label{derivOmega}
\begin{align}
\frac{(\omega_k^2)'}{\omega_k^2}=& D-(f+1)\sum_{i=1}^{3}d_i\lambda_i^2, \\
\frac{(\omega_k^2)''}{\omega_k^2}=&
D'+D^2+(f+1)\sum_{i=1}^{3}\lambda_i^2[d_i^2-2 d_i
D-d_i']+(\dot{f}+f^2+f)\left(\sum_{i=1}^{3}d_i\lambda_i^2\right)^2,
\end{align}\end{subequations} where a dot indicates a derivative with respect to $\ln x$.
We proceed as for the standard dispersion relation, defining the
renormalized expectation value as \begin{equation}
\langle\hat{\phi}^2\rangle_{ren}=\langle\hat{\phi}^2\rangle-\langle\hat{\phi}^2\rangle_{ad2},
\end{equation} with
$\langle\hat{\phi}^2\rangle_{ad2}=\langle\hat{\phi}^2\rangle^{(0)}+\langle\hat{\phi}^2\rangle^{(2)}$,
where again the superscripts indicate the adiabatic order.
We now compute the zeroth adiabatic order of $\langle \hat{\phi}^2
\rangle$ and regularize it by using the fact that the integral of
a total derivative vanishes in dimensional regularization
\cite{Collins}. For this, and in order to avoid the complications
of computing all quantities in $n$-dimensions, we first perform
the angular integrations and then generalize the four-dimensional
integrals to $n$-dimensions by replacing
$d^3k=C^{3/2}d^3y=C^{3/2}y^2dy d\Omega$ ($y_i=k_i/\sqrt{C_i}$) by
$C^{3/2}y^{(n-2)}dy d\Omega$.
Therefore, the zeroth adiabatic order is given by \begin{equation}
\label{rencerophi} \langle\hat{\phi}^2\rangle^{(0)}
=\frac{1}{(2\pi)^3}\int y^{n-2} dy d\Omega \frac{
1}{2\tilde{\omega}_k}=\frac{I_1}{2(2\pi)^2}, \end{equation} where $I_1$ is given
Table \ref{tabla}. Note that the integral $I_1$ is divergent
unless $\omega_k^2$ behaves as $x^{s}$ with $s>3$, for large values of
$x$. This divergence can be absorbed in the bare mass of the
quantum field (see below).
\begin{table}[ht]
\begin{tabular}{|c|c|}
\hline
& \\
$I_0 =\int_0^\infty dx\,x^{\frac{(n-3)}{2}}{\tilde{\omega}_k}$ &
$I_3=\int_{0}^{\infty} dx \frac{x^{\frac{(n-3)}{2}}}{\tilde{\omega}_k^3}$ \\
& \\
\hline
& \\
$I_1=\int_0^\infty dx\,\frac{x^{\frac{(n-3)}{2}}}{\tilde{\omega}_k}$& $I_4=\int_{0}^{\infty} dx \frac{x^{\frac{(n+1)}{2}}}{\tilde{\omega}_k^5} \frac{d^2\tilde{\omega}_k^2}{{dx}^2}$ \\
\hline
& \\
$I_2 = \int_0^\infty
dx\,\frac{x^{\frac{(n+1)}{2}}}{\tilde{\omega}_k^3}\frac{d^2\tilde{\omega}_k^2}{{dx}^2}$
&
$I=\int_0^{+\infty}dx\frac{x^{\frac{(n+3)}{2}}}{\tilde{\omega}_k^3}\frac{d^3\tilde{\omega}_k^2}{{dx}^3}$
\\
& \\
\hline
\end{tabular}
\caption{Explicit expressions for $I_{i}$. To obtain these integrals we have made the change of variables
$x= y^2$ and we have defined $\tilde{\omega}_k=\omega_k/\sqrt{C}$.}\label{tabla}
\end{table}
The second adiabatic order can be written as \begin{equation}
\langle\hat{\phi}^2\rangle^{(2)} =-\frac{\sqrt{C}}{32\pi^3}\int
y^{n-2} dy d\Omega \frac{{}^{(2)}W_k^2 }{\omega_k^3}. \end{equation} The angular
integrations can be performed with the use of the identities
listed in the Appendix A. After some calculations we obtain:
\begin{eqnarray}
\langle\hat{\phi}^2\rangle^{(2)}
&=&-\frac{1}{16\pi^2}\left\{I_3\left[\frac{D^2}{16 C}-\frac{D'}{4
C}+R\left(\xi-\frac{1}{6}\right)+\frac{Q}{C}+ \xi_2
(\nabla_{\mu}u^{\mu})^2+\xi_3
\nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu} +12\lambda
\phi^2_0\right]\right.\nonumber\\ \label{adPhi} &-&\left.\left[\frac{3
D^2}{8 C}+2\frac{Q}{C}-\frac{D'}{4 C}\right]
(J_{1000}+I_3)+\left[\frac{D^2}{16 C}+\frac{ Q}{5 C}\right](J_{2000}+6
J_{1000}-4 J_{0100}+5 I_3)\right\},
\end{eqnarray} where $I_3$ is given in Table \ref{tabla}, and we have defined the integrals
\begin{equation} J _{m n l s}\label{JMNLS}\equiv
\int_{0}^{\infty}dx\frac{x^{\frac{(n-3)}{2}}}{\tilde{\omega}_k^3}
{f}^m\, {\dot{f}}^n\,{\ddot{f}}^l\, {\dddot{f}}^s,\end{equation} with
$m,n,l,s$ integer numbers. As it is shown in the Appendix of
Ref.\cite{NosDos}, this integrals can be expressed in terms of the
ones in Table \ref{tabla} by performing integrations by parts. For
$n\to 4$, we have \cite{NosDos}:
\begin{subequations}\label{jotas}
\begin{align}
J_{1000}=&0, \\
J_{2000}=&\frac{2}{5}I_4\\
J_{0100}=&\frac{3}{5}I_4.
\end{align}\end{subequations}
Then, substituting this results into Eq. (\ref{adPhi}) we arrive
at
\begin{eqnarray} \label{rendosphi}
\langle\hat{\phi}^2\rangle^{(2)}&=&-\frac{I_3}{16\pi^2}\left[
R\left(\xi-\frac{1}{6}\right) +12\lambda \phi^2_0 + \xi_2
(\nabla_{\mu}u^{\mu})^2+\xi_3
\nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu}
\right]\nonumber\\&+&\frac{I_4}{480\pi^2}\left[ (\nabla_{\mu}u^{\mu})^2+2
\nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu} \right].
\end{eqnarray}
This is the main result of this section. The relevant point is
that, in addition of the usual terms proportional to $R$ and
$\phi_0^2$, the second adiabatic order contains terms with two
derivatives of the aether field, which are present even if
$\xi_2=\xi_3=0$. For the standard dispersion relation $I_3$
diverges and $I_4$ vanishes (see Table \ref{tabla}). Therefore,
when $\xi_2=\xi_3=0$ one reobtains the usual result. However, for
any other dispersion relation of the type given in Eq.
(\ref{dis}), $I_3$ and $I_4$ are finite. An interesting point is
that, if we consider a generalized dispersion relation, evaluate
the integral explicitly in four dimensions and {\it then} take the
limit in which the dispersion relation tends to the usual one, a
nonvanishing finite result can be obtained. For example, a
dispersion relation of the form $\omega_k^2=C(x+2 b_{11}x^2)$ yields
\begin{equation} I_4=2 b_{11}\int_{0}^{+\infty}dx (1+2
b_{11}x)^{-\frac{5}{2}}=\frac{4}{3}. \end{equation} Therefore, there is a
finite remnant of the trans-Planckian physics in the second
adiabatic order, even in the limit in which the scale of new
physics is very high $M_C\to\infty$ ($b_{11}\to 0$).
Coming back to the mean value equation (\ref{Ecphicero}), we write
the bare
parameters in terms of the renormalized ones plus the
corresponding to counterterms:
\begin{eqnarray}\label{EcphiceroRe}
&&\Box\phi_0-\left[ m^2_R +\delta m^2+(\xi_R+\delta\xi) R+(\xi_{2R}+\delta\xi_2)(\nabla_{\mu}u^{\mu})^2+(\xi_{3R}+\delta\xi_3) \nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu} \right.\nonumber\\
&&+2\sum_{s,p}b_{sp}\mathcal{D}^{2(s+p)}+\left.12\lambda_R\left(\langle\hat{\phi}^2\rangle_{ren}+\langle\hat{\phi}^2\rangle_{ad2}
\right)\right]\phi_0-4(\lambda_R+\delta\lambda)\phi_0^3=0.
\end{eqnarray}
Introducing Eqs. (\ref{rencerophi}) and (\ref{rendosphi}) into
Eq. (\ref{EcphiceroRe}), we see that the regularized second
adiabatic order $\langle\hat{\phi}^2\rangle_{ad2}$ can be absorbed
into the bare constants by defining counterterms such that
\begin{subequations}\label{contraterminos}
\begin{align}
\delta m^2=&-\frac{6\lambda_R }{(2\pi)^2}I_1, \\
\delta\lambda=&\frac{9\lambda_R^2}{(2\pi)^2} I_3, \\
\delta\xi=&\frac{3\lambda_R}{(2\pi)^2}\left(\xi_R-\frac{1}{6}\right) I_3,\\
\delta\xi_2=&\frac{3\lambda_R\xi_{2R}}{(2\pi)^2} I_3-\frac{\lambda_R}{40\pi^2}I_4, \\
\delta\xi_3=&\frac{3\lambda_R\xi_{3R}}{(2\pi)^2}
I_3-\frac{\lambda_R}{20\pi^2}I_4.
\end{align}\end{subequations}
Note that, even when the parameters $\xi_{2R}$ and $\xi_{3R}$ are set to zero, the corresponding counterterms
arise due to the self-interaction of the scalar field. Note also that by considering the same theory but in a background
flat FRW space-time, it is not possible to distinguish between the
redefinitions of $\xi_2$ and $\xi_3$ proportional
to $I_4$, since in such background we have that \begin{equation}
\nabla_{\mu}u_{\nu}\nabla^{\nu}u^{\mu}=\frac{1}{3}(\nabla_{\mu}u^{\mu})^2.\end{equation}
From the results of this section we conclude that, as long as one
considers the renormalization of the mean value equation in
Bianchi type I spacetimes, and for the class of MDR considered
here, it is enough to subtract the zeroth adiabatic order of
$\langle\hat\phi^2\rangle$, since the second adiabatic order
produce a finite renormalization of the bare constants of the
theory. It would be interesting to check whether this is a general
property, i.e. valid for an arbitrary background, or not. In order
to address this issue, it would be necessary to know the
singularity structure of the two-point function of a quantum field
satisfying MDR for arbitrary values of $g_{\mu\nu}$ and $u_{\mu}$.
This singularity structure could be revealed by a generalized
momentum-space representation of the Green's functions
\cite{bp79,r07}. In any case, the calculation of the second
adiabatic order presented in this section shows that the
interaction terms proportional to $\xi_2$ and $\xi_3$ that appear
in Eq. (\ref{lintgen}) are generated by quantum effects, even if
not present at the classical level. It is likely that the other
interaction terms will also be generated in a more general
background.
\section{On the renormalization of the stress tensor in Bianchi type I space-times}
In this section we focus on the renormalization of the SEAE. We
restrict the analysis to the case of a free scalar field
($\lambda=0$, $\langle\phi\rangle=0$) and, for the sake of
simplicity, we set the parameters $\xi_i=0$ (i=1,2,3,4,5).
The SEAE take the form \begin{equation} G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G
[T_{\mu\nu}^{u_b}+\langle
T_{\mu\nu}^{\tilde{\lambda}_{c}}+T_{\mu\nu}^{\phi}\rangle +
T_{\mu\nu}^{clas}] \end{equation} where $\Lambda$ and $G$ are the bare
cosmological and Newton's constants, $G_{\mu\nu}$ is the Einstein
tensor, $T_{\mu\nu}^{u,(\phi)}=-\frac{2}{\sqrt{-g}}\frac{\delta
S^{u(\phi)}}{\delta g^{\mu\nu}}$, and $T_{\mu\nu}^{u}=
T_{\mu\nu}^{u_b}+T_{\mu\nu}^{\tilde{\lambda}_{c}}$, $
T_{\mu\nu}^{u_b}$ is the stress tensor of the background vector
field while $T_{\mu\nu}^{\tilde{\lambda}_{c}}$ is the additional
contribution due to the modification of the Lagrange multiplier
$\tilde{\lambda}$ arising from the coupling between the scalar
field $\phi$ and $u_{\mu}$. $T_{\mu\nu}^{clas}$ is a stress
tensor coming from classical sources not coupled to the aether
field. As we will compute the mean value of the stress tensor up
to the second adiabatic order, we omit classical terms quadratic
in the curvature (we will comment on this issue in the next
section).
The nontrivial components of the Einstein tensor are, in Bianchi
type I spacetimes:
\begin{subequations}
\begin{align}
G_{\eta\eta}=& 3\left[\frac{D^2}{4}-Q\right], \\
G_{ii}=&- \frac{C_i}{2C}\left[ 3 D'+\frac{3}{2}D^2+6 Q-d_i'-d_i D\right].
\end{align}\label{EinsteinTensorBianchi}
\end{subequations}
The stress tensor corresponding to the background vector field
$u_{\mu}$ can be written as \begin{equation} T_{\mu\nu}^{u_b}=-\frac{b_3}{8\pi
G}G_{\mu\nu}-\frac{b_2}{8\pi G}\tilde{T}_{\mu\nu}^{u}, \end{equation} whose
nonzero components are
\begin{subequations}
\begin{align}
\tilde{T}_{\eta\eta}^{u}=& \frac{9 }{8} D^2, \\
\tilde{T}_{ii}^{u}=& -\frac{3}{2}\frac{C_i}{C}\left[
D'+\frac{D^2}{4}\right].
\end{align}\label{AETHERTensorBianchi}
\end{subequations}
The expectation value of the quantum energy momentum tensor
$T_{\mu\nu}=T_{\mu\nu}^{\phi}+T_{\mu\nu}^{\tilde{\lambda}_c}$ is
given by
\begin{eqnarray}
\nonumber\langle T_{\eta\eta}\rangle &=& \frac{1}{2 C} \int
\frac{d^{3}k}{(2\pi)^{3}} \left\{
|\chi_k'|^2+3 D \left(\xi-\frac{1}{6}\right) (\chi_k'\chi_k^*+\chi_k{\chi_k'}^{*})\right.\\
&+&\left.|\chi_k|^2\left[\omega_k^2-3D^2\left(\xi-\frac{1}{12}\right)+2\xi G_{\eta\eta}\right] \right\},\\
\nonumber \langle T_{ii}\rangle &=& \frac{C_i}{C^2}\int
\frac{d^3 k}{(2\pi)^{3}}
\left\{\left(\frac{1}{2}-2\xi\right) |\chi_k'|^2+\left(\frac{\xi}{2}(2 D+d_i)-\frac{D}{4}\right) (\chi_k'\chi_k^*+\chi_k{\chi_k'}^{*})\right.\\
&-&\left. \xi(\chi_k''\chi_k^*+\chi_k{\chi_k''}^{*})+|\chi_k|^2\left( k_i^2\frac{d\omega_k^2}{
dk_i^2}-\frac{\omega_k^2}{2}+\frac{D^2}{8}\right)+\frac{\xi}{2}
|\chi_k|^2\left[ 2 D'-d_i D+2\frac{C}{C_i} G_{ii}\right]\right\}.
\end{eqnarray}
Using the expression given in Eq. (\ref{chi}) for the modes
$\chi_k$, it can be written as
\begin{eqnarray}
\nonumber\langle T_{\eta\eta}\rangle &=& \frac{1}{2 C} \int
\frac{d^{3}k}{(2\pi)^{3}} \left\{
\frac{[(W_k^2)']^2}{32 W_k^5}-3 D\left(\xi-\frac{1}{6}\right)\frac{(W_k^2)'}{4 W_k^3 }+\frac{W_k}{2}\right.\\
&+&\left.\frac{1}{2 W_k}\left[\omega_k^2-3D^2\left(\xi-\frac{1}{12}\right)+2\xi G_{\eta\eta}\right]\right\},\label{Tetaeta}\\
\nonumber \langle T_{ii}\rangle &=& \frac{C_i}{C^2}\int
\frac{d^3 k}{(2\pi)^{3}}
\left\{\left(\frac{1}{8}-3\xi\right) \frac{[(W_k^2)']^2}{8 W_k^5}+\xi \frac{(W_k^2)''}{4 W_k^3}-\frac{(W_k^2)'}{4 W_k^3}\left(-\frac{D}{4}+\frac{\xi}{2}(2 D+d_i)\right)\right.\\
&+&\left. \frac{W_k}{4}+\frac{1}{2 W_k}\left[ k_i^2\frac{d\omega_k^2}{
dk_i^2}-\frac{\omega_k^2}{2}+\frac{D^2}{8}+\xi D'-\frac{\xi}{2}D
d_i+\xi\frac{C}{C_i} G_{ii}\right]\right\}.\label{PPP}
\end{eqnarray}
Therefore, the zeroth adiabatic order can be expressed in the form
\begin{subequations}
\begin{align}
\langle T_{\eta\eta}\rangle^{(0)} =& \frac{C}{2} \int
\frac{d\Omega dy}{(2\pi)^{3}}
y^{n-2}\tilde{\omega}_k=\frac{C}{2(2\pi)^2}\int_0^{+\infty}dx
x^{\frac{n-3}{2}}\tilde{\omega}_k,\\\label{ordenceroii} \langle
T_{ii}\rangle^{(0)}=& \frac{C_i}{2 } \int \frac{d\Omega
dy}{(2\pi)^{3}} y^{n-2} \lambda_i^2
\frac{y^2}{\tilde{\omega}_k}\frac{d\tilde{\omega}_k^2}{dy^2}=\frac{
C_i}{3 (2\pi)^2}\int_{0}^{+\infty}dx
x^{\frac{n-1}{2}}\frac{d\tilde{\omega}_k}{dx},
\end{align}
\end{subequations}where we have used that $\int d\Omega \lambda_i^2=4\pi/3$.
Then, after an integration by parts in Eq. (\ref{ordenceroii}) we
obtain, as $n\to4$, \begin{equation} \langle
T_{\mu\nu}\rangle^{(0)}=-\frac{I_0}{2(2\pi)^2}g_{\mu\nu},\end{equation} where
$I_0$ is a divergent integral as $n\to 4$ for any of the
dispersion relations given in Eq. (\ref{dis}) (see Table
\ref{tabla}). Hence, this regularized adiabatic order can be
absorbed into a redefinition of the bare cosmological constant
$\Lambda$.
The second adiabatic order of $\langle T_{\mu\nu}\rangle$ can be
written as
\begin{eqnarray}
\langle T_{\eta\eta}\rangle^{(2)} &=& \frac{C}{2} \int
\frac{d\Omega dy}{(2\pi)^{3}}
\frac{y^{(n-2)}}{\tilde{\omega}_k}\left\{
\frac{[(\omega_k^2)']^2}{32 \omega_k^4}-3 D\left(\xi-\frac{1}{6}\right)\frac{(\omega_k^2)'}{4 \omega_k^2}-\frac{3}{2}D^2\left(\xi-\frac{1}{12}\right)+\xi G_{\eta\eta}\right\},\label{TetaetasecOeden}\\
\nonumber \langle T_{ii}\rangle^{(2)} &=& C_i\int \frac{d\Omega
dy}{(2\pi)^{3}}\frac{y^{(n-2)}}{\tilde{\omega}_k}
\left\{\left(\frac{1}{8}-3\xi\right) \frac{[(\omega_k^2)']^2}{8 \omega_k^4}+\xi \frac{(\omega_k^2)''}{4 \omega_k^2}-\frac{(\omega_k^2)'}{4 \omega_k^2}\left(-\frac{D}{4}+\frac{\xi}{2}(2 D+d_i)\right)\right.\\
&+&\left.\frac{^{(2)}W_k^2}{4}\lp1-\lambda_i^2\frac{y^2}{\omega_k^2}\frac{d\omega_k^2}{dy^2}\right)+\frac{D^2}{16}+\frac{\xi}{2} D'-\frac{\xi}{4}D d_i+\xi\frac{C}{2 C_i} G_{ii}\right\},\label{PPPSegorden}
\end{eqnarray} where ${}^{(2)}W_k^2$ is given by the expression in Eq. (\ref{W2}) with $\lambda=\xi_{2}=\xi_{3}=0$. The explicit expressions for $(\omega_k^2)'/{\omega}_k^2$ and $({\omega}_k^2)''/{\omega}_k^2$ are given in Eq. (\ref{derivOmega}).
After performing the angular integrations with the use of the
identities given in the Appendix A and some algebraic
manipulations, we obtain:
\begin{subequations}\label{segordenInt}
\begin{align}
\langle T_{\eta\eta}\rangle^{(2)} &= \frac{1}{(2\pi)^2}[\alpha_1 D^2+\alpha_2 Q],\\
\langle T_{ii}\rangle^{(2)} &= \frac{C_i}{C(2\pi)^2}[\beta_1 D^2+\beta_2 D'+\beta_3 D d_i+\beta_4 Q+\beta_5 d_i^2+\beta_6
d_i'].
\end{align}
\end{subequations}
The coefficients $\alpha_i$ and $\beta_i$ are given in Appendix B,
where it is also shown that using integration by parts they can be expressed in terms of two
of the integrals in Table \ref{tabla}. Thus, we find
\begin{equation}\label{divad2tmunu} \langle
T_{\mu\nu}\rangle^{(2)}=\frac{1}{8\pi^2}\left\{\left[ I_1\left(
\xi-\frac{1}{6}\right) -\frac{I_2}{45}\right] G_{\mu\nu}+\frac{I_2}{30}
\tilde{T}^{u}_{\mu\nu}\right\}.\end{equation}
Note that both $I_1$ and $I_2$ diverge when $\omega_k^2$ behaves as
$x^{s}, s\leq 3$ for large values of $x$. Therefore, in this case
the divergences should be absorbed into the bare constants $G$ and
$b_2$. However, when $s>3$, the second adiabatic order produce
finite renormalizations of both constants.
As in the evaluation of $\langle\hat\phi^2\rangle$ presented in
the previous section, depending on the dispersion relation one
could have a remnant of the trans-Planckian physics in the second
adiabatic order of $\langle T_{\mu\nu}\rangle$. Indeed, while
$I_2$ vanishes for the standard dispersion relation, a non
vanishing (and even divergent) result can be obtained for MDR in
the limit $M_C\to\infty$. For example, for a dispersion relation
of the form $\omega_k^2=C(x+2 b_{22} x^4)$ we find that \begin{equation} I_2=24
b_{22}\int_{0}^{+\infty}dx \frac{x^3}{(1+2 b_{22}
x^3)^{\frac{3}{2}}}=\frac{2^{\frac{8}{3}}}{\sqrt{\pi}b_{22}^{\frac{1}{3}}}\Gamma\lb1/6\right]\Gamma\lb4/3\right],
\end{equation} which diverges as $M_C\to\infty$ ($b_{22}\to 0$).
Eq. (\ref{divad2tmunu}) is the main result of this section. We see
that, for a generalized dispersion relation of the type given in
Eq. (\ref{dis}), not only a redefinition of the Newton's constant
is necessary in order to cancel the divergences of the second
adiabatic order, but also a redefinition of the coefficient $b_2$
which corresponds to the term $(\nabla_{\mu}u^{\mu})^2$ in the
bare Lagrangian of the vector field. The second adiabatic order
contains terms that are non-purely geometric, in the sense that
they cannot be written only in terms of the metric, but also
involve the aether field.
It is noteworthy that for a background flat FRW space-time
$G_{\mu\nu}=3/2 \tilde{T}^{u}_{\mu\nu}$ , thereby, in
Refs.\cite{NosUno,NosDos} it was not possible to realize that a
redefinition of the Newton's constant is not enough for cancelling
the second adiabatic order. In fact, for this particular
space-time $G_{\mu\nu}$ is the unique covariantly conserved tensor
of adiabatic order two that can be derived from an action formed
by combining the vector field $u^{\mu}$, the metric $g_{\mu\nu}$,
and their derivatives.
\section{Discussion}
In this paper we have worked within the context of a generally
covariant theory of gravitation coupled to a dynamical time-like
Lorentz-violating vector field. We considered a quantum scalar
field satisfying MDR, and analyzed the renormalization of the
infinities that arise in the semiclassical theory. In particular,
considering Bianchi type I spacetimes, we have analyzed the
dynamical equation for the expectation value of a self-interacting
scalar field (Section III), and the SEAE for the metric in the
case of a free scalar field (Section IV). With the use of
adiabatic subtraction and dimensional regularization, we have
shown that, in addition to the usual terms required to absorb the
infinities of the second adiabatic orders, it is necessary to
consider more general counterterms that involve the aether field.
This property was not apparent in our previous works
\cite{NosUno,NosDos}, due to the high symmetry of the flat
Robertson Walker metrics.
These results suggest that, in a more general background metric,
any covariant term which can be formed by combining the vector
field $u^{\mu}$, the metric $g_{\mu\nu}$ and up to two of their
derivatives, will appear in the regularized second adiabatic order
of the expectation value of the quantum stress tensor, provided
that the theory contains a scalar field with a generalized
dispersion relation of the type given in Eq. (\ref{dis}). Hence,
in order to absorb the divergences contained in the second
adiabatic order, a bare action as general as the one given in Eq.
(\ref{Sg}) should be considered. Depending on the particular
dispersion relation of the quantum field, the second adiabatic
order may be finite. If this is the case, quantum effects generate
finite renormalizations of the constants appearing in the
classical Lagrangian. As we have also pointed out in Section IV,
this finite renormalizations could be extremely large.
In the weak-field limit, the terms proportional to the constants
$b_i$ in Eq. (\ref{lu}) could have observable consequences. Indeed,
the most general action given in Eq. (\ref{Sg}) has four free
parameters more than general relativity. This theory has been
studied in several contexts, such as of the static weak-field
limit \cite{Jacobsondebil}, the radiation and propagation of the
aether-gravitational waves \cite{Jacobsonwave}, cosmology
\cite{LimCarrolBarrow}, etc., in which stringent constraints on
the parameters have been imposed to make the theory consistent
with observation. For example, in Ref. \cite{Jacobsondebil} it is
shown that for all the PPN parameters to agree with observation,
the four additional parameters of the model must satisfy two
constraint equations with sufficient accuracy (i.e., the
additional four-parameter space of the model has to be practically
reduced to a two-dimensional subspace). In the absence of a known
mechanism to explain why the parameters satisfy precisely such
constraint equations, it seems that quantum effects generate a
fine-tuning problem in the Einstein-Aether theory. This is analogous to
the fine-tuning problem present in the Myers-Pospelov modification
of QED \cite{pipi}.
In this paper we restricted ourselves to the evaluation of the
adiabatic expansion up to the second adiabatic order, in a
particular class of background metrics. By power counting, we
expect the fourth adiabatic order $\langle
T_{\mu\nu}\rangle^{(4)}$ to be finite in these metrics. However,
there could be some subtleties related to the would be
Gauss-Bonnet invariant in four dimensions \cite{NosProc,NosDos}.
In the light of the results obtained in this paper, this issue
should be reexamined. One should compute $\langle
T_{\mu\nu}\rangle^{(4)}$ for a general background metric and
aether field (this could be done by generalizing the
momentum-space representation of the Green's functions). On
dimensional grounds we expect $\langle T_{\mu\nu}\rangle^{(4)}$ to
contain terms proportional to the variation of $R
(\nabla_{\mu}u^{\mu})^2$, $(R_{\mu\nu}u^{\mu}u^{\nu})^2$,
$R_{\mu\nu}u^{\mu}u^{\nu} R$, $R_{\mu\nu\rho\sigma}u^{\mu}u^{\rho}
R^{\nu\sigma}$, etc., in addition to the usual ones: $R^2$,
$R_{\mu\nu}R^{\mu\nu}$ and
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$. Depending on the MDR,
the fourth adiabatic order could be finite or divergent when
expressed in terms of such variations. This fact would define
whether it is necessary or not to subtract the fourth adiabatic
order in a general background, for a given dispersion relation.
Work in this direction is in progress.
\section*{Appendix A: Identities for Bianchi type I space-times}
In this Appendix we briefly summarize some useful formulas
required for the adiabatic regularization of $\langle
\hat{\phi}^2\rangle^{(2)}$ and $\langle T_{\mu\nu}\rangle^{(2)}$
in Bianchi type I space-times.
As we have already mentioned in the text, in order to regularize
the theory we perform the four-dimensional angular integrations
and then generalize the integrals to $n$-dimensions. We rescale
the integration variables $k_i\to y_i=k_i/C_i$ and transform the
volume element $d^{3}y$ from rectangular coordinates to spherical
coordinates $y^2 dy d\Omega$, where $d\Omega$ is the solid angle
element. In terms of $y^2_i=y^2 \lambda_i^2$, the relevant
integrals are of the form \begin{equation} I(i,j,k)= \int d\Omega
\lambda_1^{2i}\lambda_2^{2j}\lambda_3^{2k}, \end{equation} which can be
evaluated by using the fact that they are invariant under
permutations of $\{i,j,k\}$. We provide here a list of the
integrals we have used in this paper (see \cite{Hu} for more
details):
\begin{subequations}\label{ANGULARINT}
\begin{align}
I(0,0,k)=&\frac{4\pi}{2k+1},\\
I(1,1,0)=&\frac{4\pi}{5\times 3},\\
I(1,2,0)=&\frac{4\pi}{7\times 5},\\
I(1,1,1)=&\frac{4\pi}{7\times 5 \times 3}.
\end{align}
\end{subequations}
These results, together with the formula $\sum_{i=1}^3d_i^2=3(8
Q+D^2)$, allow us to derive the following identities:
\begin{subequations}\label{identities}
\begin{align}
\sum_{j=1}^{3}\sum_{k=1}^{3}\int d\Omega d_j d_k \lambda_j^2\lambda_k^2=&4\pi\left( D^2+\frac{16}{5}Q\right),\\
\sum_{j=1}^{3}\sum_{k=1}^{3}\int d\Omega d_j d_k \lambda_i^2\lambda_j^2\lambda_k^2=&\frac{4\pi}{7\times 5}\left[ 5 D^2+4 d_i D +\frac{8}{3}d_i^2+16 Q\right],\\
\sum_{j=1}^{3}\int d\Omega (d_j'+2 d_j D-d_j^2)
\lambda_i^2\lambda_j^2=&\frac{4\pi}{5\times 3}\left[ 2(d_i'+2 d_i
D-d_i^2)+3(D'+D^2-8 Q)\right]\; ,
\end{align}
\end{subequations}
that are useful for the evaluation of $\langle
\hat{\phi}^2\rangle^{(2)}$ and $\langle T_{\mu\nu}\rangle^{(2)}$.
\section*{Appendix B: Regularization of $\langle T_{\mu\nu}\rangle^{(2)}$ in Bianchi type I space-times}
In this Appendix we provide some details for computing the second
adiabatic order of the expectation value of the quantum energy
momentum tensor. The explicit expressions for the coefficients
appearing in Eq. (\ref{segordenInt})are:
\begin{subequations}\label{coeff}
\begin{align}
\alpha_1=&\frac{1}{64}[-4 I_{10}+ I_{20}+4( I_{00}-6\xi I_{00}+6\xi I_{10})],\\
\alpha_2=&\frac{1}{20}[ I_{00}+2 I_{10}+ I_{20}-30\xi I_{00}],\\
\beta_1=&\frac{1}{2240}[52 I_{00}-120 I_{01}-76 I_{10}+20 I_{11}+77 I_{20}-5 I_{30}+280\xi (-I_{00}+2 I_{01}+I_{10}-I_{20})],\\
\beta_2=&\frac{1}{640}[2 I_{00}+39 I_{10}-8 I_{20}-20\xi(22 I_{00}+I_{10})],\\
\beta_3=&\frac{1}{1680}[-8 I_{00}+12 I_{01}-19 I_{10}+12 I_{11}-14 I_{20}-3 I_{30}+210\xi I_{10}],\\
\beta_4=&\frac{1}{140}[-24 I_{01}+3 I_{10}+4 I_{11}+21 I_{20}-I_{30}-14\xi(-8 I_{01}+I_{10}+4 I_{20})+I_{00} (42\xi-19)],\\
\beta_5=&\frac{1}{840}[2 I_{00}+4 I_{01}+3 I_{10}+4 I_{11}-I_{30}],\\
\beta_6=&\frac{1}{120}[-I_{00}-2 I_{10}-I_{20}+30\xi I_{00}],
\end{align}
\end{subequations}
where the integrals $I_{mn}$ are given by \begin{equation}
I_{mn}=\int_{0}^{+\infty} dx
\frac{x^{\frac{n-3}{2}}}{\tilde{\omega}_k} f^{m}\dot{f}^n, \end{equation} with
$m,n=0,1,2,3$.
Let us now sketch the procedure to find relations between these
coefficients in the context of dimensional regularization, which
is completely analogous to the one described in the Appendix of
Ref.\cite{NosDos} for relating the integrals $ J _{m n l s}$ of
Eq. (\ref{JMNLS}). By definition, $I_{00}=I_1$, and
\begin{eqnarray}
I_{10}&=&\int_{0}^{+\infty} dx \frac{x^{\frac{n-3}{2}}}{\tilde{\omega}_k}\left(\frac{x}{\tilde{\omega}_k^2}\frac{d\tilde{\omega}_k^2}{dx}-1\right)=-2\int_{0}^{+\infty} dx x^{\frac{n-1}{2}}\frac{d\tilde{\omega}_k^{-1}}{dx}-I_1\\\nonumber
&=&(n-1)I_1-I_1 \,{ }_{\overrightarrow{(n\to 4)}} 2I_1,
\end{eqnarray}where we have performed an integration by parts and discarded the surface term.
Similarly, one can prove that
\begin{subequations}
\begin{align}
I_{20}=&\frac{2}{3}I_2,\\
I_{30}=&\frac{4}{5}I_2+\frac{8}{15}I,\\
I_{01}=&-2I_1+\frac{1}{3}I_2,\\
I_{11}=&-\frac{2}{15}I_2+\frac{2}{15}I,
\end{align} where the integrals $I_i$ on the right hand side are given in Table \ref{tabla}, and $I$ (which does not appear in the final results) is given by
\begin{equation}
I=\int_{0}^{\infty} dx \frac{x^{\frac{(n+3)}{2}}}{\tilde{\omega}_k^3} \frac{d^3\tilde{\omega}_k^2}{dx^3}.
\end{equation}
\end{subequations}
Replacing these results into Eq. (\ref{coeff}), we obtain:
\begin{subequations}
\begin{align}
\alpha_1=&\frac{3}{8}I_1\left(\xi-\frac{1}{6}\right)+\frac{I_2}{96},\\
\alpha_2=&-\frac{3}{2}I_1\left(\xi-\frac{1}{6}\right)+\frac{I_2}{30},\\
\beta_1=&-\frac{3}{8}I_1\left(\xi-\frac{1}{6}\right)+\frac{I_2}{480},\\
\beta_2=&-\frac{3}{4}I_1\left(\xi-\frac{1}{6}\right)-\frac{I_2}{120},\\
\beta_3=&\beta_6=\frac{1}{4}I_1\left(\xi-\frac{1}{6}\right)-\frac{I_2}{180},\\
\beta_4=&-\frac{3}{2}I_1\left(\xi-\frac{1}{6}\right)+\frac{I_2}{30},\\
\beta_5=& 0.
\end{align}
\end{subequations}
Finally, after substituting these coefficients into Eq.
(\ref{segordenInt}) we arrive at Eq. (\ref{divad2tmunu}).
\begin{acknowledgments}
We would like to thank T. Jacobson and H. Vucetich for useful correspondence and
discussions.
This work has been supported by Universidad de Buenos Aires,
CONICET and ANPCyT.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,310 |
\section{Introduction}\label{intro}
Clustering is the unsupervised task of assigning a categorical value $y_i \in \{1,\ldots,k\}$ to each data point $x_i \in \mathbf{X}$, where no such example categories are given in the training data; i.e., we should map
$\mathbf X= \{x_1,\ldots,x_n\}\mapsto \mathbf Y = \{y_1,\ldots,y_n\}$
with $\bf X$ the input matrix of n data points, each of dimension d; where $y_i = \kappa$ implies that data point $x_i$ is assigned to the $\kappa$-th cluster.
Clustering methods complete this task by measuring similarity (the distance) between training pairs, using a similarity function $s(x_i,x_j) \in \mathbb{R}_+$.
This similarity function should typically reflect subjective criteria fixed by the user. Basically, this means that the user decides what makes a good clustering. As mentioned in \cite{learn}, ``since classes are a high-level abstraction, discovering them automatically is challenging, and perhaps impossible since there are many criteria that could be used to cluster data (e.g., we may equally well cluster objects by colour, size, or shape). Knowledge about some classes is not only a realistic assumption, but also indispensable to narrow down the meaning of clustering". Taking the example of MNIST \cite{MNIST_digits}, one usually groups the same numbers together because these numbers share the highest amount of features (e.g., mutual information based models do that). However one may want to group numbers given their roundness. In this case, we may obtain two clusters, namely straight shaped numbers (i.e., 1, 4,7) and round shaped numbers (i.e., all the others). Both clustering solutions are relevant, since each clustering addresses a different yet possible user subjective criteria (i.e., clustering semantics).
Finding an automated way to derive and incorporate user criteria in a clustering task based on intended semantics can be very hard. Nowadays, the wide availability of shared annotated datasets is a valuable asset and provides examples of possible user criteria. Hence, we argue that, given ``similar'' annotated data, classification logic can be used to derive a user criteria that one can apply to clustering similar non-annotated data. For example, we consider the situation where a human is placed in front of two datasets, each one consisting of letters of a certain alphabet she does not understand. The first dataset is annotated, grouping the same letters together. Only by seeing the first dataset, the person can understand the grouping logic used (grouping same geometrical shapes together) and replicate that logic to the second non annotated dataset and cluster correctly its letters.
In this paper, we are interested in tackling the problem of clustering data when the logic (i.e., user clustering criteria) is encoded into some available labelled datasets.
This raises two main challenges, namely (1) find a solution that works well on the classification task but (2) ensure transferability in its decision mechanism so it is applicable to clustering data from a different domain.
We believe that addressing these challenges calls for the design of a scoring function that should be as general as possible to ensure transferability but is specific enough not to miss the user criteria. More specifically, the scoring function should be a comparing the logic used to produce a certain clustering to the one used to produce clusterings of the already seen training datasets. Using the concept of logic is useful as a logic is general enough to be used on any dataset and specific enough as is it is the main common property shared by all training dataset. Our goal is then to find a suitable metric that retrieves and encapsulate the seen concept for scoring a clustering outcome.
Moreover, modern applications require solutions that are effective when data is of high dimension (i.e., large $d$). While distance-based approaches are broadly used for clustering (e.g., Euclidean distance), we argue that they are not suitable for our problem since they would yield in data specific models in addition to their poor performance in high dimensional spaces due to the curse of dimensionality.
To lower dimensionality, a solution is to perform instance-wise embeddings $x_i \mapsto z_i$, e.g., with an autoencoder. However this mechanism is still domain specific.
To achieve training on more general patterns, we think it is necessary to take the dataset in its entirety. Therefore, instead of learning a metric that compares pairs of data points in a dataset instance (like a similarity measure), a learned metric is applied to sets of data points so comparison is done between sets. The metric can be intuitively understood as a distance between the logic underlying a given clustering and the general logic that was used to produce clusterings in training datasets.
For this, we propose a solution where we use a graph autoencoder \cite{GAE} to embed a set of data points into a vector of chosen dimension. Then, we use the critic part of a Wasserstein GAN (WGAN) \cite{WGAN} to produce a continuous score of the embedded clustering outcome. This critic represents the metric we seek. Thus, our main contributions are:
\vspace{-2mm}
\begin{itemize}
\item We provide a framework for joint metric learning and clustering tasks.
\vspace{-2mm}
\item We show that our proposed solution yields a learned metric that is transferable to datasets of different sizes and dimensions, and across different domains (either vision or tabular) and tasks.
\vspace{-2mm}
\item We obtain results competitive to the state-of-the-art with only a small number of training datasets, relatively simple networks, and no prior knowledge (only an upper bound of the cluster number that can be set to a high value).
\vspace{-6mm}
\item Our method is scalable to large datasets both in terms of number of points or dimensions (e.g the SVHN dataset used in section \ref{sec:experiments}) as it does not have to compute pairwise distances and therefore does not heavily suffer when the number of points or dimensions increase.
\vspace{-2mm}
\item We test the metric on datasets of varying complexity and perform on par with the state-of-the-art while maintaining all the advantages cited above.
\end{itemize}
\section{Related Work}\label{related}
Using auto-encoders before applying classic clustering algorithms resulted in a significant increase of clustering performance, while still being limited by these algorithms capacity.
Deep Embedding Clustering (DEC) \cite{DEC} gets rid of this limitation at the cost of more complex objective functions. It uses an auto-encoder along with a cluster assignment loss as a regularisation. The obtained clusters are refined by minimising the KL-divergence between the distribution of soft labels and an auxiliary target distribution. DEC became a baseline for deep clustering algorithms. Most deep clustering algorithms are based on classical center-based, divergence-based or hierarchical clustering formulations and hence bear limitations like the need for an \textit{a priori} number of clusters.
MPCKMeans \cite{mpckmeans} is more related to metric learning as they use constraints for both metric learning and the clustering objective. However, their learned metrics remain dataset specific and are not transferable.
Constrained Clustering Network (CCN) \cite{transfer_clustering}, learns a metric that is transferable across domains and tasks. Categorical information is reduced to pairwise constraints using a similarity network. Along with the learned similarity function, the authors designed a loss function to regularise the clustering classification. But, using similarity networks only captures local properties instance-wise rather than global geometric properties of dataset clustering. Hence, the learned metric remains non fully transferable, and requires to adapt the loss to the domain to which the metric is transferred to.
In Deep Transfer Clustering (DTC) \cite{learn} and Autonovel \cite{autonovel}, the authors tackle the problem of discovering novel classes in an image collection given labelled examples of other classes. They extended DEC to a transfer learning setting while estimating the number of classes in the unlabelled data. Autonovel uses self-supervised learning to train the representation from scratch on the union of labelled and unlabelled datasets then trains the data representation by optimizing a joint objective function on the labelled and unlabelled subsets of data. We consider these two approaches as our state of the art baselines.
\section{Our Framework}
To restate our objective, we seek an evaluation metric
\begin{equation}\label{eq:map2}
\begin{split}
r : \mathbb R^{\bf n\times d} \times \mathbb {N}^{\bf n}\rightarrow \mathbb{R}\\
(\bf X,\bf y)\mapsto r(\bf X,\bf y)
\end{split}
\end{equation}
where $\bf X \in \mathbb R^{n\times d}$ is a dataset of $n$ points in $d$ dimensions and $\bf y \in \mathbb N^n$ a partition of $\bf X$ (i.e. a clustering of $\bf X$). Metric $r$ should provide a score for \emph{any} labelled dataset of any dimensionality; and in particular this score should be such that $r(\bf{X},\bf y)$ is high when the hamming distance between the ground truth labels $\bf y^*$ and $\bf y$ is small (taking cluster label permutations into account). This would mean that we could perform clustering on any given dataset, simply by solving an optimisation problem even if such a dataset had not been seen before.
Formally stated, our goal is: (1) to produce a metric $r$ that grades the quality of a clustering such that $\bf{y}^*=\argmax_{\bf y} r(\bf X, \bf y)$; (2) Implement an optimisation algorithm that finds $\bf y^*$; (3) use (1) and (2) to perform a clustering on a new unrelated and unlabelled dataset.
We use a collection $\mathcal{D} = \{\mathbf{X}_l,\mathbf{y}_l^*\}_{l=1}^\ell$ of labelled datasets as examples of correctly `clustered' datasets, and learn $r$ such that $\mathbb{E}[r(\mathbf{X},\mathbf{y})]$ is high. In order to make $r$ transferable between datasets, we embed each dataset with its corresponding clustering ($\mathbf X_l,\mathbf y_l)$ into a vector $\mathbf z_l \in \mathbb R^{\bf e}$. More formally, the embedding function is of the form:
\begin{equation}
\begin{split}
g: \, \, & \mathbb R^{\bf n\times d}\times \mathbf Y \rightarrow \mathbb R^\mathbf e \\
& (\bf X,\bf y)\mapsto \bf z
\end{split}
\end{equation}
Therefore, the metric $r$ is actually the composition of two functions $g$ and $c_\theta$ (the scoring function from $\mathbb R^{\bf e}$ to $\mathbb R$). Our training procedure is structured around 3 blocs A, B and C detailed in next sections and depicted in figure \ref{framework} and is summarised in the following main steps:
\vspace{15mm}
\begin{enumerate}[{Bloc A}. step 1]
\item Select a labelled dataset $(\bf{X},\bf{y}^*) \sim\mathcal{D}$
\vspace{-2mm}
\item Given a metric function $r$ (output from bloc B step 2, or initialised randomly), we perform a clustering of dataset $\bf X$: $\mathbf{\hat y} =\argmax_\mathbf{y} r(\mathbf{X},\mathbf{y})$
\end{enumerate}
\vspace{-2mm}
\begin{enumerate}[{Bloc B}. step 1]
\item $\bf y^*$ and $\bf{\hat{y}}$ are represented as graphs where each clique represents a cluster.\vspace{-2mm}
\item Graph convolutional autoencoders perform feature extraction from $\bf \hat{y}$ and $\bf y^*$ and output embeddings $\bf \hat{z}$ and $\bf z^*$
\end{enumerate}
\vspace{-2mm}
\begin{enumerate}[{Bloc C}. step 1]
\item The metric $r$ is modelled by a WGAN critic that outputs evaluations of the clusterings: $r(\bf X,\bf y^*) = c_\theta(\bf z^*)$ and $r(\bf X,\bf \hat{y}) = c_\theta(\bf \hat z)$\vspace{-2mm}
\item Train the model using the error between $r(\bf X,\bf y^*)$ and $r(\bf X,\bf \hat{y})$.
\end{enumerate}
\vspace{-3mm}
\begin{figure}[h!]
\centering
\includegraphics[width=\columnwidth]{Figure.png}
\caption{Our framework's 3 components: the clustering mechanism (A), the GAE (B) and the WGAN (C). (A) takes an unlabelled dataset $\mathbf {X}$ as input and outputs a clustering $\mathbf{\hat{y}}$ that maximises a metric $r$. $\mathbf{\hat{y}}$ is then turned into a graph $\mathcal{G}(\mathbf{X},\mathbf{\hat{y}})$ then into an embedding vector $\mathbf{\hat{z}}$ using (B). Same goes for the correctly labelled dataset, which is embedded as $\mathbf{\hat{z}^*}$. Then, (C), which is the metric itself, evaluates $\mathbf{\hat{z}}$ and $\mathbf{z}^*$ using $c_\theta$ and is trained to produce a new metric $r$ which is then used for (A) in the next iteration.}
\label{framework}
\end{figure}
\begin{comment}
\begin{table}[!ht]
\centering
\caption{Summary of notations}
\begin{tabular}{lp{6.3cm}}
\hline
$\mathbf X$ & A dataset of $n$ points, $x_i \in \mathbb{R}^{\bf d}$\\
$\mathbf y^*$ & True clustering (cluster labels) of $\mathbf X$, $\in \{1,\dots,k\}^n$ \\
$\mathbf y$ & A possible clustering of $\mathbf X$ \\
$\hat{\bf y}$ & Clustering retained after optimisation for a fixed function $r$\\
$\mathcal{M}_{n,m}$ & Set of matrices with $n$ lines and $m$ columns\\
$\mathcal{G}(\mathbf X,\mathbf y)$ & Graph representing a clustered version of $\mathbf X$\\
$A$ & An adjacency matrix \\
$X$ & Feature matrix of $\mathbf X$. $X \in \mathcal{M}_{n,d} $\\
$\bf z^*,\bf\hat{z}$ & Embedding, $\in \mathbb{R}^{\bf e}$, of $(\bf{X},\bf{y}^*)$, and $(\bf{X},\bf{y})$, respectively \\
$r$ & Metric $\mathbb{R}^{\bf n\times d}\times \mathbb N^{\mathbf n} \mapsto \mathbb{R}$, scoring of the clustering \\
CEM & Cross-entropy method\\
$\mathcal{S}$ & Set of all intermediate clustering solutions found through CEM \\[1ex]
\hline
\end{tabular}
\end{table}
\end{comment}
\vspace{-8mm}
\subsection{Clustering mechanism}\label{clustering}
We seek the most suitable optimisation algorithm for clustering given $r$. Considering a neural network that performs the clustering, we need to find its weights $w$ such that the metric is maximised (see equation \eqref{w}). The type of algorithm to use depends on the nature of the metric $r$ to optimise on.
\begin{equation}\label{w}
\text{CEM}_r(\mathbf X)\xrightarrow{\text{finds}} w^* = \argmax_w r(\mathbf{X},\mathbf{y}^w)
\end{equation}
Where $\mathbf y^w$ is a clustering obtained with the weights $w$. The metric is assumed to hold certain properties, discussed in \ref{critic}:
\begin{itemize}
\item \textbf{Unique Maximum:} A unique optimal clustering. $r$ has a unique maximum.
\vspace{-6mm}
\item \textbf{Continuity\footnote{As a reminder, Let $T$ and $U$ be two topological spaces. A function $f:T\mapsto U$ is continuous in the open set definition if for every $t\in T$ and every open set $u$ containing $f(t)$, there exists a neighbourhood $v$ of $t$ such that $f(v)\subset u$.}}: Any two clusterings $\mathbf y$ and $\mathbf y'$ should be similar if $r(\mathbf y)$ and $r(\mathbf y')$ are close in $\mathbb{R}$ space. Hence, $r$ has to satisfy a continuity constraint.
\end{itemize}
There is no guarantee that the best metric for the clustering task is differentiable.
Given the above assumptions, conditions are favourable for evolutionary strategies (ES) to iteratively converge towards the optimal solution. Indeed, if $r$ is continuous and the series $((\mathbf{X},\mathbf{y}_1),\dots,(\mathbf{X},\mathbf{y}_p))$ converges towards $(\mathbf{X},\mathbf{y}^*)$ then $(r(\mathbf{X},\mathbf{y}_1),\dots,r(\mathbf{X},\mathbf{y}_p))$ converges towards $r(\mathbf{X},\mathbf{y}^*)$.
We choose the Cross-Entropy Method (CEM) \cite{CEM}, a popular ES algorithm for its simplicity, to optimise the clustering neural network weights by solving Eq.\eqref{w} (algorithm \ref{Alg:CEM_algo}).
\begin{algorithm}[tb]
\caption{CEM Algorithm}
\label{Alg:CEM_algo}
\begin{algorithmic}
\STATE \textbf{Input:} Dataset $X\in\mathbb R^{\bf{n} \times \bf{d}}$; score function $r$; $\mu \in \mathbb{R}^{\bf{d}}$ and $\sigma \in \mathbb{R}^{\bf{d}}$; elite percentage to retain $p$; $n$ samples of $w_i \sim \mathcal{N}(\mu,\text{diag}(\sigma))$; $T$ number of iterations
\FOR{$\textnormal{iteration}=1$ {\bfseries to} $T$}
\STATE Produce $n$ samples of neural network weights $w_i \sim \mathcal{N(\mu,\text{diag}(\sigma))}$
\STATE Produce clusterings $y_i$ of $X$ using each $w_i$
\STATE Evaluate $r_i = r(X,y_i)$
\STATE Constitute the elite set of $p\%$ best $w_i$
\STATE Fit a Gaussian distribution with diagonal covariance to the elite set and get a new $\mu_t$ and $\sigma_t$
\ENDFOR
\STATE {\bfseries return:} $\mu$, $w^*$
\end{algorithmic}
\end{algorithm}
\subsection{Graph based dataset embedding}
To capture global properties and be transferable across different datasets, we argue that it is necessary to input all the points of a dataset at once. Hence, instead of pairwise similarities between random pairs of points, we propose to get a representation of the relation between a bunch of neighbouring points. Thus, we represent each dataset by a graph structure $\mathcal{G}(\mathbf{X},\mathbf y)$ where each node corresponds to a point in $\mathbf{X}$ and where cliques represent clusters as shown in figure \ref{framework}. This representation takes the form of a feature matrix $X$ and an adjacency matrix $A$.
Using $X$, and $A$, we embed the whole dataset into a vector $\bf z \in \mathbb{R}^{\mathbf e}$. To do so, we use graph autoencoders (GAE). Our implementation is based on \cite{GAE}.
\begin{comment}
Specifically, we have $\{X,A\} \mapsto \bf z$, under the following mechanism:
\begin{equation}
\begin{aligned}
GCN(X,A)= Relu(\Tilde{A}XW_0) = \Bar{X}\\
\end{aligned}
\end{equation}
With $\Tilde{A}$ the symmetrically normalized adjacency matrix and $(W_0,W_1)$ the GCN weight matrices.
\begin{equation}
\begin{aligned}
z=\Tilde{A}\Bar{X}W_1
\end{aligned}
\end{equation}
Finally, the decoder outputs a new adjacency matrix using the sigmoid function $\sigma$:
\begin{equation}
\begin{aligned}
\hat{A}=\sigma(zz^T)
\end{aligned}
\end{equation}
\end{comment}
We obtain $z \in \mathcal{M}_{n,m}$ which is dependent of the shape of the dataset (where $m$ is a user specified hyper-parameter). In order to make it independent from the number of points in $\mathcal{X}$, we turn the matrix $z$ into a square symmetrical one $z \xleftarrow{} z^Tz \in \mathcal{M}_{m,m}$. The final embedding corresponds to a flattened version of the principal triangular bloc of $z^Tz$, which shape is $\mathbf e=(\frac{m+1}{2},1)$. However, the scale of the output still depends on the number of points in the dataset. This could cause an issue when transferring to datasets with a vastly different number of data points. It should therefore require some regularisation; in order to simplify, we decided to use datasets with approximately the same number of points.
\subsection{A critic as a metric}\label{critic}
With embedded vectors of the same shape, we compare the clusterings proposed $\mathbf{\hat{z}}$ and the ground truth ones $\bf z$ using the metric $r$. $r$ is a function mapping an embedding vector $\mathbf z\in \mathbb R^{\mathbf e}$ to $\mathbb{R}$, we therefore parameterise it as:
\begin{equation}\label{large_state_reward}
r_\alpha(\mathbf X,\mathbf y)=r_\alpha(\mathbf z)=\alpha_1\phi_1(\mathbf z)+\alpha_2\phi_2(\mathbf z)+...+\alpha_h\phi_h(\mathbf z)
\end{equation}
Where $\phi_j(\mathbf z)\in \mathbb R$. As per \cite{Russell}, learning a viable metric is possible provided both the following constraints: (1) maximising the difference between the quality of the optimal decision and the quality of the second best; (2) minimising the amplitude of the metric function as using small values encourages the metric function to be simpler, similar to regularisation in supervised learning.
When maximising the metric difference between the two clusterings that have the highest scores, we get a similarity score as in traditional metric learning problems. The problem is formulated by equation \eqref{general_optimization} where $\mathcal{S}$ is a set of solutions (i.e., clustering proposals) found using $r_\alpha$ and $\mathbf{y}^*$ is the true clustering, $\mathbf{y}^{\text{max}}$ is the best solution found in $\mathcal{S}$: $\mathbf{y}^{\text{max}} = \argmax_{\mathbf{y}\in\mathcal S}r_\alpha(\mathbf X, \mathbf{y})$.
\begin{equation}\label{general_optimization}
\begin{aligned}
\min_\alpha r_\alpha(\mathbf X, \mathbf y^*) & -\max_\alpha \min_{\mathbf y'\in \mathcal S\setminus \mathbf y^{\text{max}}} r_\alpha(\mathbf X,\mathbf y^{\text{max}})-r_\alpha(\mathbf X,\mathbf y')\\
& \quad \text{s.t} \quad \mathbf{y}^*=\argmax_{\mathbf{y}\in \mathbf{Y}}r(\mathbf{y})
\end{aligned}
\end{equation}
\begin{algorithm}[h!]
\footnotesize
\caption{Critic2Metric (C2M)}\label{Complete_algo}
\SetAlgoLined
\KwInput{$b$: batch size, $epoch$: number of epochs; $p$: percentage of elite weights to keep; $iteration$: number of CEM iterations; $population$: number of weights to generate; $\mu \in \mathbb{R}^d$: CEM mean; $\sigma \in \mathbb{R}^d$: CEM standard deviation, $\theta$: critic's weights}
\For{$n=1$ {\bfseries to} epoch}{
\For{$k=1$ {\bfseries to} b}{
Sample $(\mathbf X_{k},\mathbf y_k^*) \sim \mathcal D $ a correctly labelled dataset\\
Generate ground truth embeddings $\mathbf z_{(\mathbf X_{k},\mathbf y_k^*)}=GAE(\mathcal{G}(\mathbf X_k,\mathbf y_k^*))$ \\
Initialise clustering neural network weights $\{w_j\}_{j=1}^{population}$ \\
\For{$i=1$ {\bfseries to} iteration}{
\For{$j=1$ {\bfseries to} population}
{Generate clusterings $\mathbf{\hat{y}}_k^{w_j}$ \\
Convert $\mathbf{\hat{y}}_k^{w_j}$ into a graph\\
$\mathbf z_{(\mathbf X_{k},\mathbf {\hat{y}}_k^{w_j})}= GAE(\mathcal{G}(\mathbf X_k,\hat{\mathbf y}_k^{w_j}))$ \\
Evaluate: $r(\mathbf X_k,\hat{\mathbf y}_k^{w_j}) = c_\theta(\mathbf z_{(\mathbf X_{k},\mathbf {\hat{y}}_k^{w_j})})$}
Keep proportion $p$ of best weights $w_p$ \\
$w^* \xleftarrow{} \text{CEM}(w_p, \mu, \sigma)$}
Generate clustering $\mathbf{y}_k^{w^*}$\\
$\mathbf z_{(\mathbf X_{k},\mathbf {\hat{y}}_k^{w^*})} = GAE(\mathcal{G}(\mathbf X_k,\hat{\mathbf y}_k^{w^*}))$ \\
Train critic as in \cite{WGAN} using $\mathbf z_{(\mathbf X_{k},\mathbf {\hat{y}}_k^{w^*})}$ and $\mathbf z_{(\mathbf X_{k},\mathbf y_k^*)}$ \;
}}
\end{algorithm}
\vspace{-5mm}
To solve equation \eqref{general_optimization}, we
use a GAN approach where the clustering mechanism (i.e., CEM) plays the role of the generator while a critic (i.e., metric learning model) plays the role of the discriminator. In a classic GAN, the discriminator only has to discriminate between real and false samples, making it use a cross entropy loss. With this kind of loss, and in our case, the discriminator quickly becomes too strong. Indeed, the score output by the discriminator becomes quickly polarised around 0 and 1.
\vspace{-1mm}
For this reason, we represent $r$ as the critic of a WGAN \cite{WGAN}. This critic scores the realness or fakeness of a given sample while respecting a smoothing constraint. The critic measures the distance between data distribution of the training dataset and the distribution observed in the generated samples. Since WGAN assumes that the optimal clustering provided is unique, the metric solution found by the critic satisfies equation \eqref{general_optimization} constraints. $r$ reaching a unique maximum while being continuous, the assumptions made in section \ref{clustering} are correctly addressed.
To train the WGAN, we use the loss $\mathcal{L}$ in equation \eqref{WGAN_loss} where $\bf \hat{z}$ is the embedding vector of a proposed clustering and $\bf z$ is the embedding vector of the desired clustering. Our framework is detailed in algorithm \ref{Complete_algo}.
\vspace{-1mm}
\begin{equation}\label{WGAN_loss}
\mathcal{L}(\mathbf z^*,\mathbf {\hat{z}})=\max_{\theta}\mathbb{E}_{\mathbf z^*\sim p}[f_\theta(\mathbf z^*)] - \mathbb{E}_{\mathbf {\hat{z}}\sim p(\mathbf {\hat{z})}}[f_\theta(\mathbf {\hat{z}})]
\end{equation}
\section{Experiments}\label{sec:experiments}
\vspace*{-\baselineskip}
\begin{table*}[h!]
\centering
\begin{adjustbox}{width=\columnwidth, center}
\begin{tabular}{||c || c || c || c || c || c || c || c || c || c ||}
\hline
\multicolumn{1}{||c|}{\textbf{Dataset family}} &
\multicolumn{4}{||c|}{Synthetic data} &
\multicolumn{3}{||c|}{MNIST} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}Street view\\ house numbers\end{tabular}} &
\multicolumn{1}{c||}{Omniglot} \\
\hline
\multicolumn{1}{||c|}{\textbf{Dataset}} &
\multicolumn{1}{||c|}{Blob} &
\multicolumn{1}{||c|}{Moon} &
\multicolumn{1}{||c|}{Circles} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}Aniso-\\ tropic\end{tabular} } &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}MNIST-digits\\ \cite{MNIST_digits}\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}letters MNIST\\ \cite{MNIST_letters}\end{tabular} } &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}fashion MNIST\\ \cite{fashion_MNIST}\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}SVHN\\ \cite{SVHN}\end{tabular}} &
\multicolumn{1}{c||}{\begin{tabular}{@{}c@{}}Omniglot\\ \cite{omniglot}\end{tabular} } \\
\hline
\multicolumn{1}{||c|}{\textbf{Snapshot}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{Blobs.PNG}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{Aniso.PNG}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{Circles.PNG}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{Moons.PNG}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{MNIST_example.jpg}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{MNIST_letter.PNG}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{MNIST_fashion.PNG}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{SVHN.png}}} &
\multicolumn{1}{c||}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{Omniglot.PNG}}} \\
\hline
\multicolumn{1}{||c|}{\textbf{\begin{tabular}{@{}c@{}}Feature\\ dimension\end{tabular} }} &
\multicolumn{1}{||c|}{2} &
\multicolumn{1}{||c|}{2} &
\multicolumn{1}{||c|}{2} &
\multicolumn{1}{||c|}{2} &
\multicolumn{1}{||c|}{$28\times 28$} &
\multicolumn{1}{||c|}{$28\times 28$} &
\multicolumn{1}{||c|}{$28\times 28$} &
\multicolumn{1}{||c|}{$32 \times 32$} &
\multicolumn{1}{c||}{$105 \times 105$} \\
\hline
\multicolumn{1}{||c|}{\textbf{\begin{tabular}{@{}c@{}}Maximum number\\ of clusters\end{tabular}}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}9\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}9\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}9\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}9\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{10} &
\multicolumn{1}{||c|}{26} &
\multicolumn{1}{||c|}{10} &
\multicolumn{1}{||c|}{10} &
\multicolumn{1}{c||}{47} \\
\hline
\multicolumn{1}{||c|}{\textbf{Size}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}200\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}200\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}200\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}200\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{60000} &
\multicolumn{1}{||c|}{145600} &
\multicolumn{1}{||c|}{60000} &
\multicolumn{1}{||c|}{73257} &
\multicolumn{1}{c||}{32460} \\
\hline
\end{tabular}
\end{adjustbox}
\caption{Datasets description}
\vspace{-8mm}
\label{tab:dataset}
\end{table*}
For empirical evaluation, we parameterise our framework as follows: The critic (block C in Fig~\ref{framework}) is a 5 layer network of sizes 256, 256, 512, 512, and 1 (output) neurons. All activation functions are LeakyRelu ($\alpha=0.2$) except last layer (no activation). RMSprop optimizer with $0.01$ initial learning rate and a decay rate of $0.95$. The CEM-trained neural network (bloc A in Fig~\ref{framework}) has 1 hidden layer of size 16 with Relu activation, and a final layer of size $k=50$ (the maximum number of clusters). The GAE (bloc B in Fig~\ref{framework}) has 2 hidden layers; sized 32 and 16 for synthetic datasets, and 100 and 50 for real datasets.
We choose datasets based on 3 main criteria: having a similar compatible format; datasets should be large enough to allow diversity in subsampling configurations to guarantee against overfitting; datasets should be similar to the ones used in our identified baseline literature. All used datasets are found in table \ref{tab:dataset}.
For training, we construct $n$ sample datasets and their ground truth clustering, each containing 200 points drawn randomly from a set of 1500 points belonging to the training dataset. Each one of these datasets, along with their clustering is an input to our model. To test the learned metric, we construct 50 new sample datasets from datasets that are different from the training one (e.g., if we train the model on MNIST numbers, we will use datasets from MNIST letters or fashion to test the metric). The test sample datasets contain 200 points each for synthetic datasets and 1000 points each otherwise. The accuracies are then averaged accross the 50 test sample datasets.
To test the ability of the model to learn using only a few samples, we train it using 5 (few shots) and 20 datasets (standard), each containing a random number of clusters. For few shots trainings, we train the critic for 1 epoch and 10 epochs for standard trainings.
To evaluate the clustering, we use Normalised-Mutual Information (NMI) \cite{NMI} and clustering accuracy (ACC) \cite{ACC}. NMI provides a normalised measure that is invariant to label permutations while ACC measures the one-to-one matching of labels. For clustering, we only need that the samples belonging to the same cluster are attributed the same label, independently from the label itself. However, since we want to analyse the behaviour of the metric learned through our framework, we are interested in seeing whether it is permutation invariant or not. Hence, we need the two measures.
\subsection{Results on 2D synthetic datasets}
Analysis on synthetic datasets (see table \ref{tab:dataset}) proves that our model behaves as expected. We do not compare our results to any baseline since existing unsupervised methods are well studied on them.
We train our model using exclusively samples from blobs datasets. We then test the learned metric on the 4 different types of synthetic datasets (blobs, anisotropic, moons and circles). Results are displayed in table \ref{sci-kit_results}. We observe that the model obtains the best score on blobs since it is trained using this dataset. We can also notice that our model achieves high scores for the other types of datasets not included in training.
\begin{table} [h!]
\centering
\begin{tabular}{LCCCC}
\toprule
\multicolumn{1}{l}{Types of datasets} &
\multicolumn{2}{c}{Standard training} &
\multicolumn{2}{c}{Few shots training} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
&
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} \\
\midrule
\text{Blobs} & 98.4\% & 0.980 & 97.3\% & 0.965\\
\text{Anisotropic} & 97.9\% & 0.967 & 97.2\% & 0.945\\
\text{Circles} & 91.7\% & 0.902 & 92.7\% & 0.900\\
\text{Moons} & 92.1\% & 0.929 & 92.8\% & 0.938\\
\bottomrule
\end{tabular}
\caption{Average ACC and NMI on synthetic test datasets.}
\vspace{-5mm}
\label{sci-kit_results}
\end{table}
Our model succeeds in clustering datasets presenting non linear boundaries like circles while blobs datasets used in training are all linearly separable. Hence, the model learns intrinsic properties of training dataset that are not portrayed in the initial dataset structure, and thus that the metric appears to be transferable.
\textbf{Critic's ablation study}. To test if the critic behaves as expected, i.e., grades the clustering proposals proportionally to their quality, we test it on wrongly labelled datasets to see if the score decreases with the number of mislabelled points. We consider 50 datasets from each type of synthetic datasets, create 50 different copies and mislabel a random number of points in each copy. A typical result is displayed in figure \ref{ablation} and shows that the critic effectively outputs an ordering metric as the score increases when the number of mislabelled points decreases, reaching its maximum when there is no mislabelled point. This shows that the metric satisfies the constraints stated in equation \ref{general_optimization}.
\vspace{-1mm}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\columnwidth]{capture.png}
\caption{Metric values (i.e., scores given by the critic) for several clusterings of a dataset. Plots are from an anisotropic dataset (left) and a moons dataset (right). In a 2 cluster case (right), the formula used to compute mislabelled points has been made sensitive to label permutation to verify if permuted labels can fool the critic. The critic assigns a high score either when all the labels match the given ground truth or when all the labels are permuted (which again does not affect the correctness of the clustering)}
\vspace{-2mm}
\label{ablation}
\end{figure}
\vspace{-6mm}
An interesting behaviour is shown in figure \ref{ablation}. Recall that since we are in the context of a clustering problem, we only need for the samples belonging to the same cluster to get the same label, independently from the cluster label itself. Thus, the formula used to compute mislabelled points has been made sensitive to label permutation to verify if permuted labels can fool the critic. For instance, in a 2 clusters case, one can switch the labels of all points in each cluster and still get the maximum score. Switching all labels makes all the points wrongly labelled compared to the given ground truth but nonetheless the clustering itself remains true. This explains the rounded shape in figure \ref{ablation} where the used datasets in the right panel only consisted of 2 clusters. The critic assigns a high score either when all the labels match the given ground truth or when all the labels are permuted (which does not affect the correctness of the clustering).
\vspace{-3mm}
\subsection{Results on MNIST datasets}\label{MNIST_section}
\vspace{-1mm}
MNIST datasets give similar results both in terms of ACC and NMI on all test datasets regardless of the used training dataset (see table \ref{MNIST_result}). Hence, the model effectively capture implicit features that are dataset independent. While standard training shows better results, the few shots training has close performance.
\begin{table}[h!]
\centering
\begin{tabular}{LCCCCCC}
\toprule
\multicolumn{1}{l}{Training Dataset} &
\multicolumn{6}{c}{Testing Dataset} \\
\cmidrule(lr){2-7}
\multicolumn{1}{c}{} &
\multicolumn{2}{c}{Numbers} &
\multicolumn{2}{c}{Letters} &
\multicolumn{2}{c}{Fashion} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
\cmidrule(lr){6-7}
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} \\
\midrule
\text{Numbers (standard)} & 72.3\% & 0.733 & 81.3\% & 0.861 & 65.2\% & 0.792 \\
\text{Numbers (few shots)} & 68.5\% & 0.801 & 79.0\% & 0.821 & 61.8\% & 0.672 \\
\text{Letters (standard)} & 75.9\% & 0.772 & 83.7\% & 0.854 & 67.5\% & 0.800 \\
\text{Letters (few shots)} & 69.8\% & 0.812 & 78.7\% & 0.806 & 60.9\% & 0.641 \\
\text{Fashion (standard)} & 70.6\% & 0.706 & 83.4\% & 0.858 & 72.5\% & 0.762 \\
\text{Fashion (few shots)} & 70.1\% & 0.690 & 82.1\% & 0.834 & 70.7\% & 0.697 \\
\bottomrule
\end{tabular}
\caption{Mean clustering performance on MNIST dataset.}
\label{MNIST_result}
\end{table}
\vspace{-12mm}
\begin{table}[h!]
\centering
\begin{tabular}{LCCCCCC}
\toprule
\multicolumn{1}{l}{Training Dataset} &
\multicolumn{6}{c}{Testing Dataset} \\
\cmidrule(lr){2-7}
\multicolumn{1}{c}{} &
\multicolumn{2}{c}{Numbers} &
\multicolumn{2}{c}{Letters} &
\multicolumn{2}{c}{Fashion} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
\cmidrule(lr){6-7}
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{Best} &
\multicolumn{1}{c}{Top 3} &
\multicolumn{1}{c}{Best} &
\multicolumn{1}{c}{Top 3} &
\multicolumn{1}{c}{Best} &
\multicolumn{1}{c}{Top 3} \\
\midrule
\text{Numbers (standard)} & 78.3\% & 92.5\% & 86.0\% & 97.5\% & 69.2\% & 87.2\%\\
\text{Numbers (few shots)} & 75.8\% & 82.1\% & 83.3\% & 92.0\% & 65.1\% & 83.9\% \\
\text{Letters (standard)} & 77.4\% & 89.2\% & 88.8\% & 96.4\% & 70.2\% & 86.7\%\\
\text{Letters (few shots)} & 73.1\% & 80.6\% & 85.1\% & 91.5\% & 61.0\% & 76.3\% \\
\text{Fashion (standard} & 70.1\% & 83.1\% & 85.0\% & 98.6\% & 76.9\% & 94.7\%\\
\text{Fashion (few shots)} & 67.9\% & 77.4\% & 83.5\% & 95.3\% & 70.2\% & 88.0\%\\
\bottomrule
\end{tabular}
\caption{Critic based performance assessment: Best corresponds to the percentage of times the critic gives the best score to the desired solution. Top 3 is when this solution is among the 3 highest scores.}
\label{MNIST_theoretic}
\vspace{-4mm}
\end{table}
\begin{comment}
\vspace*{-\baselineskip}
\begin{table}[h!]
\begin{adjustwidth}{-3cm}{-3cm}
\begin{subtable}[t]{0.5\linewidth}
\begin{tabular*}{\columnwidth}{LCCCCCC}
\toprule
\multicolumn{1}{l}{Training Dataset} &
\multicolumn{6}{c}{Testing Dataset} \\
\cmidrule(lr){2-7}
\multicolumn{1}{c}{} &
\multicolumn{2}{c}{Numbers} &
\multicolumn{2}{c}{Letters} &
\multicolumn{2}{c}{Fashion} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
\cmidrule(lr){6-7}
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} \\
\midrule
\text{Numbers (standard)} & 72.3\% & 0.733 & 81.3\% & 0.861 & 65.2\% & 0.792 \\
\text{Numbers (few shots)} & 68.5\% & 0.801 & 79.0\% & 0.821 & 61.8\% & 0.672 \\
\text{Letters (standard)} & 75.9\% & 0.772 & 83.7\% & 0.854 & 67.5\% & 0.800 \\
\text{Letters (few shots)} & 69.8\% & 0.812 & 78.7\% & 0.806 & 60.9\% & 0.641 \\
\text{Fashion (standard)} & 70.6\% & 0.706 & 83.4\% & 0.858 & 72.5\% & 0.762 \\
\text{Fashion (few shots)} & 70.1\% & 0.690 & 82.1\% & 0.834 & 70.7\% & 0.697 \\
\bottomrule
\end{tabular*}
\caption{Mean clustering performance on MNIST dataset.}
\label{MNIST_result}
\end{subtable}
\begin{subtable}[t]{0.5\linewidth}
\begin{tabular}{LCCCCCC}
\toprule
\multicolumn{1}{l}{Training Dataset} &
\multicolumn{6}{c}{Testing Dataset} \\
\cmidrule(lr){2-7}
\multicolumn{1}{c}{} &
\multicolumn{2}{c}{Numbers} &
\multicolumn{2}{c}{Letters} &
\multicolumn{2}{c}{Fashion} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
\cmidrule(lr){6-7}
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{Best} &
\multicolumn{1}{c}{Top 3} &
\multicolumn{1}{c}{Best} &
\multicolumn{1}{c}{Top 3} &
\multicolumn{1}{c}{Best} &
\multicolumn{1}{c}{Top 3} \\
\midrule
\text{Numbers (standard)} & 78.3\% & 92.5\% & 86.0\% & 97.5\% & 69.2\% & 87.2\%\\
\text{Numbers (few shots)} & 75.8\% & 82.1\% & 83.3\% & 92.0\% & 65.1\% & 83.9\% \\
\text{Letters (standard)} & 77.4\% & 89.2\% & 88.8\% & 96.4\% & 70.2\% & 86.7\%\\
\text{Letters (few shots)} & 73.1\% & 80.6\% & 85.1\% & 91.5\% & 61.0\% & 76.3\% \\
\text{Fashion (standard} & 70.1\% & 83.1\% & 85.0\% & 98.6\% & 76.9\% & 94.7\%\\
\text{Fashion (few shots)} & 67.9\% & 77.4\% & 83.5\% & 95.3\% & 70.2\% & 88.0\%\\
\bottomrule
\end{tabular}
\caption{Critic based performance assessment: Best corresponds to the percentage of times the critic gives the best score to the desired solution. Top 3 is when this solution is among the 3 highest scores.}
\label{MNIST_theoretic}
\end{subtable}
\caption{Results on MNIST datasets}
\end{adjustwidth}
\vspace{-4mm}
\end{table}
\end{comment}
\vspace{-2mm}
Table \ref{MNIST_theoretic} shows the percentage of times the critic attributes the best score to the desired solution. It shows that ES algorithm choice has a significant impact on the overall performance. Even with a metric that attributes the best score to the desired clustering, the CEM may be stuck in a local optimum and fails to reconstruct back the desired clustering. Hence, a better optimisation can enhance the performance shown in table \ref{MNIST_result} closer to the one presented in table \ref{MNIST_theoretic}.
\subsection{Comparative study}
We compare our approach with baseline methods from the literature
(table \ref{comparative_results}). For some methods, we followed the procedure in \cite{transfer_clustering} and used their backbone neural network as a pairwise similarity metric. Table \ref{Results_SVHN} reports results when training on SVHN and testing on MNIST numbers. We obtain close ACC values to CCN and ATDA \cite{ATDA}. These methods uses Omniglot as an auxiliary dataset to learn a pairwise similarity function, which is not required for our model. Our model only uses a small fraction of SVHN, has shallow networks and does not require any adaptation to its loss function to achieve comparable results. Finally, other cited methods require the number of clusters as an a priori indication. We achieve comparable results without needing this information. When the loss adaptation through Omniglot is discarded (denoted source-only in table \ref{Results_SVHN}), or if the number of clusters is not given, their accuracy falls and our model surpasses them by a margin.
\vspace{-6mm}
\begin{table}[h!]
\begin{subtable}[c]{0.5\textwidth}
\centering
\begin{tabular}{LCC}
\toprule
\text{Method} & \multicolumn{2}{c}{\text{ACC}} \\
\midrule
& \text{Loss Adaptation} & \text{Source Only}\\
\midrule
\text{DANN \cite{DANN}} & 73.9\% & 54.9\%\\
\text{LTR \cite{LTR}} & 78.8\% & 54.9\%\\
\text{ATDA \cite{ATDA}} & 86.2\% & 70.1\%\\
\text{CCN \cite{transfer_clustering}} & 89.1\% & 52\%\\
\text{Ours (standard)} & - & 84.3\% \\
\text{Ours (few shots)} & - & 81.4\% \\
\bottomrule
\end{tabular}
\subcaption{Unsupervised cross-task transfer from SVHN to MNIST digits.}
\vspace{-2mm}
\label{Results_SVHN}
\end{subtable}
\begin{subtable}[c]{0.5\textwidth}
\centering
\begin{tabular}{LCC}
\toprule
\text{Method} & \text{ACC} & \text{NMI} \\
\midrule
\text{k-means} & 18.9\% & 0.464 \\
\text{CSP \cite{CSP}} & 65.4\% & 0.812 \\
\text{MPCK-means \cite{mpckmeans}} & 53.9\% & 0.816 \\
\text{CCN \cite{transfer_clustering}} & 78.18\% & 0.874 \\
\text{DTC \cite{learn}} & 87.0\% & 0.945 \\
\text{Autonovel \cite{autonovel}} & 85.4\% & - \\
\text{Ours (standard)} & 83.4\% & 0.891 \\
\bottomrule
\end{tabular}
\subcaption{Unsupervised cross-task transfer from $\text{Omniglot}_\text{train}$ to $\text{Omniglot}_\text{test}$ ($k=100$ for all).}
\vspace{-2mm}
\label{Omniglot_results}
\end{subtable}
\caption{Comparative clustering performance}
\vspace{-8mm}
\label{comparative_results}
\end{table}
Table \ref{Omniglot_results} reports results when training on $\text{Omniglot}_\text{train}$ and testing on $\text{Omniglot}_\text{test}$. Values are averaged across $20$ alphabets which have $20$ to $47$ letters. We set the maximum number of clusters $k=100$. When the number of clusters is unknown, we get an ACC score relatively close to DTC and Autonovel. Compared to these two approaches, our method bears several significant advantages:
\begin{itemize}
\vspace{-2mm}
\item \textbf{Deep Networks}: DTC and Autonovel used Resnets as a backbone which are very deep networks while we only used shallow networks (2 layers maximum)
\vspace{-6mm}
\item \textbf{Pairwise similarity}: in Autonovel the authors used a pairwise similarity statistic between datasets instances which we aimed to avoid due to its significant computational bottleneck. Moreover, this metric is recalculated after each training epoch, which adds more complexity.
\vspace{-2mm}
\item \textbf{Vision tasks:} While DTC can only handle vision tasks, we present a more general framework which includes vision but also tabular datasets.
\vspace{-2mm}
\item \textbf{Number of classes}: DTC and Autonovel used the labelled dataset as a probe dataset, and estimates the number of classes iteratively, and when the labelled clusters are correctly recovered, they used the ACC metric to keep the best clustering. This approach is effective, but requires access to the labelled dataset at inference time to estimate the number of classes. This is a shortcoming (memory or privacy limitations). Our approach does not require the labelled dataset once the metric is learned. Our metric automatically estimates the number of clusters required to any new unlabelled dataset.
\end{itemize}
\vspace{-2mm}
\section{Conclusion}\label{sec:discussion}
We presented a framework for cross domain/task clustering by learning a transferable metric. This framework consisted of ES methods, and GAE alongside a critic. Our model extracts dataset-independent features from labelled datasets that characterise a given clustering, performs the clustering and grades its quality. We showed successful results using only small datasets and relatively shallow architectures. Moreover, there is more room for improvement. Indeed, since our framework is composed of 3 different blocs (CEM, GAE, critic), overall efficiency can be enhanced by independently improving each bloc (i.e replacing CEM).
In future work, we will study the criteria that determine why some auxiliary datasets are more resourceful than others given a target dataset. In our case, this means to study for instance why using the MNIST letters dataset as training allowed a better performance on Fashion MNIST than when using MNIST numbers. This would allow to deliver a minimum performance guarantee at inference time by creating a transferability measure between datasets.
\textbf{Acknowledgements:} We gratefully acknowledge Orianne Debeaupuis for making the figure. We also acknowledge computing support from
NVIDIA. This work was supported by funds from the French Program "Investissements d'Avenir".
\vspace{-4mm}
\bibliographystyle{splncs04}
\section{Introduction}\label{intro}
Clustering is the unsupervised task of assigning a categorical value $y_i \in \{1,\ldots,k\}$ to each data point $x_i \in \mathbf{X}$, where no such example categories are given in the training data; i.e., we should map
$\mathbf X= \{x_1,\ldots,x_n\}\mapsto \mathbf Y = \{y_1,\ldots,y_n\}$
with $\bf X$ the input matrix of n data points, each of dimension d; where $y_i = \kappa$ implies that data point $x_i$ is assigned to the $\kappa$-th cluster.
Clustering methods complete this task by measuring similarity (the distance) between training pairs, using a similarity function $s(x_i,x_j) \in \mathbb{R}_+$.
This similarity function should typically reflect subjective criteria fixed by the user. Basically, this means that the user decides what makes a good clustering. As mentioned in \cite{learn}, ``since classes are a high-level abstraction, discovering them automatically is challenging, and perhaps impossible since there are many criteria that could be used to cluster data (e.g., we may equally well cluster objects by colour, size, or shape). Knowledge about some classes is not only a realistic assumption, but also indispensable to narrow down the meaning of clustering". Taking the example of MNIST \cite{MNIST_digits}, one usually groups the same numbers together because these numbers share the highest amount of features (e.g., mutual information based models do that). However one may want to group numbers given their roundness. In this case, we may obtain two clusters, namely straight shaped numbers (i.e., 1, 4,7) and round shaped numbers (i.e., all the others). Both clustering solutions are relevant, since each clustering addresses a different yet possible user subjective criteria (i.e., clustering semantics).
Finding an automated way to derive and incorporate user criteria in a clustering task based on intended semantics can be very hard. Nowadays, the wide availability of shared annotated datasets is a valuable asset and provides examples of possible user criteria. Hence, we argue that, given ``similar'' annotated data, classification logic can be used to derive a user criteria that one can apply to clustering similar non-annotated data. For example, we consider the situation where a human is placed in front of two datasets, each one consisting of letters of a certain alphabet she does not understand. The first dataset is annotated, grouping the same letters together. Only by seeing the first dataset, the person can understand the grouping logic used (grouping same geometrical shapes together) and replicate that logic to the second non annotated dataset and cluster correctly its letters.
In this paper, we are interested in tackling the problem of clustering data when the logic (i.e., user clustering criteria) is encoded into some available labelled datasets.
This raises two main challenges, namely (1) find a solution that works well on the classification task but (2) ensure transferability in its decision mechanism so it is applicable to clustering data from a different domain.
We believe that addressing these challenges calls for the design of a scoring function that should be as general as possible to ensure transferability but is specific enough not to miss the user criteria. More specifically, the scoring function should be a comparing the logic used to produce a certain clustering to the one used to produce clusterings of the already seen training datasets. Using the concept of logic is useful as a logic is general enough to be used on any dataset and specific enough as is it is the main common property shared by all training dataset. Our goal is then to find a suitable metric that retrieves and encapsulate the seen concept for scoring a clustering outcome.
Moreover, modern applications require solutions that are effective when data is of high dimension (i.e., large $d$). While distance-based approaches are broadly used for clustering (e.g., Euclidean distance), we argue that they are not suitable for our problem since they would yield in data specific models in addition to their poor performance in high dimensional spaces due to the curse of dimensionality.
To lower dimensionality, a solution is to perform instance-wise embeddings $x_i \mapsto z_i$, e.g., with an autoencoder. However this mechanism is still domain specific.
To achieve training on more general patterns, we think it is necessary to take the dataset in its entirety. Therefore, instead of learning a metric that compares pairs of data points in a dataset instance (like a similarity measure), a learned metric is applied to sets of data points so comparison is done between sets. The metric can be intuitively understood as a distance between the logic underlying a given clustering and the general logic that was used to produce clusterings in training datasets.
For this, we propose a solution where we use a graph autoencoder \cite{GAE} to embed a set of data points into a vector of chosen dimension. Then, we use the critic part of a Wasserstein GAN (WGAN) \cite{WGAN} to produce a continuous score of the embedded clustering outcome. This critic represents the metric we seek. Thus, our main contributions are:
\vspace{-2mm}
\begin{itemize}
\item We provide a framework for joint metric learning and clustering tasks.
\vspace{-2mm}
\item We show that our proposed solution yields a learned metric that is transferable to datasets of different sizes and dimensions, and across different domains (either vision or tabular) and tasks.
\vspace{-2mm}
\item We obtain results competitive to the state-of-the-art with only a small number of training datasets, relatively simple networks, and no prior knowledge (only an upper bound of the cluster number that can be set to a high value).
\vspace{-6mm}
\item Our method is scalable to large datasets both in terms of number of points or dimensions (e.g the SVHN dataset used in section \ref{sec:experiments}) as it does not have to compute pairwise distances and therefore does not heavily suffer when the number of points or dimensions increase.
\vspace{-2mm}
\item We test the metric on datasets of varying complexity and perform on par with the state-of-the-art while maintaining all the advantages cited above.
\end{itemize}
\section{Related Work}\label{related}
Using auto-encoders before applying classic clustering algorithms resulted in a significant increase of clustering performance, while still being limited by these algorithms capacity.
Deep Embedding Clustering (DEC) \cite{DEC} gets rid of this limitation at the cost of more complex objective functions. It uses an auto-encoder along with a cluster assignment loss as a regularisation. The obtained clusters are refined by minimising the KL-divergence between the distribution of soft labels and an auxiliary target distribution. DEC became a baseline for deep clustering algorithms. Most deep clustering algorithms are based on classical center-based, divergence-based or hierarchical clustering formulations and hence bear limitations like the need for an \textit{a priori} number of clusters.
MPCKMeans \cite{mpckmeans} is more related to metric learning as they use constraints for both metric learning and the clustering objective. However, their learned metrics remain dataset specific and are not transferable.
Constrained Clustering Network (CCN) \cite{transfer_clustering}, learns a metric that is transferable across domains and tasks. Categorical information is reduced to pairwise constraints using a similarity network. Along with the learned similarity function, the authors designed a loss function to regularise the clustering classification. But, using similarity networks only captures local properties instance-wise rather than global geometric properties of dataset clustering. Hence, the learned metric remains non fully transferable, and requires to adapt the loss to the domain to which the metric is transferred to.
In Deep Transfer Clustering (DTC) \cite{learn} and Autonovel \cite{autonovel}, the authors tackle the problem of discovering novel classes in an image collection given labelled examples of other classes. They extended DEC to a transfer learning setting while estimating the number of classes in the unlabelled data. Autonovel uses self-supervised learning to train the representation from scratch on the union of labelled and unlabelled datasets then trains the data representation by optimizing a joint objective function on the labelled and unlabelled subsets of data. We consider these two approaches as our state of the art baselines.
\section{Our Framework}
To restate our objective, we seek an evaluation metric
\begin{equation}\label{eq:map2}
\begin{split}
r : \mathbb R^{\bf n\times d} \times \mathbb {N}^{\bf n}\rightarrow \mathbb{R}\\
(\bf X,\bf y)\mapsto r(\bf X,\bf y)
\end{split}
\end{equation}
where $\bf X \in \mathbb R^{n\times d}$ is a dataset of $n$ points in $d$ dimensions and $\bf y \in \mathbb N^n$ a partition of $\bf X$ (i.e. a clustering of $\bf X$). Metric $r$ should provide a score for \emph{any} labelled dataset of any dimensionality; and in particular this score should be such that $r(\bf{X},\bf y)$ is high when the hamming distance between the ground truth labels $\bf y^*$ and $\bf y$ is small (taking cluster label permutations into account). This would mean that we could perform clustering on any given dataset, simply by solving an optimisation problem even if such a dataset had not been seen before.
Formally stated, our goal is: (1) to produce a metric $r$ that grades the quality of a clustering such that $\bf{y}^*=\argmax_{\bf y} r(\bf X, \bf y)$; (2) Implement an optimisation algorithm that finds $\bf y^*$; (3) use (1) and (2) to perform a clustering on a new unrelated and unlabelled dataset.
We use a collection $\mathcal{D} = \{\mathbf{X}_l,\mathbf{y}_l^*\}_{l=1}^\ell$ of labelled datasets as examples of correctly `clustered' datasets, and learn $r$ such that $\mathbb{E}[r(\mathbf{X},\mathbf{y})]$ is high. In order to make $r$ transferable between datasets, we embed each dataset with its corresponding clustering ($\mathbf X_l,\mathbf y_l)$ into a vector $\mathbf z_l \in \mathbb R^{\bf e}$. More formally, the embedding function is of the form:
\begin{equation}
\begin{split}
g: \, \, & \mathbb R^{\bf n\times d}\times \mathbf Y \rightarrow \mathbb R^\mathbf e \\
& (\bf X,\bf y)\mapsto \bf z
\end{split}
\end{equation}
Therefore, the metric $r$ is actually the composition of two functions $g$ and $c_\theta$ (the scoring function from $\mathbb R^{\bf e}$ to $\mathbb R$). Our training procedure is structured around 3 blocs A, B and C detailed in next sections and depicted in figure \ref{framework} and is summarised in the following main steps:
\vspace{15mm}
\begin{enumerate}[{Bloc A}. step 1]
\item Select a labelled dataset $(\bf{X},\bf{y}^*) \sim\mathcal{D}$
\vspace{-2mm}
\item Given a metric function $r$ (output from bloc B step 2, or initialised randomly), we perform a clustering of dataset $\bf X$: $\mathbf{\hat y} =\argmax_\mathbf{y} r(\mathbf{X},\mathbf{y})$
\end{enumerate}
\vspace{-2mm}
\begin{enumerate}[{Bloc B}. step 1]
\item $\bf y^*$ and $\bf{\hat{y}}$ are represented as graphs where each clique represents a cluster.\vspace{-2mm}
\item Graph convolutional autoencoders perform feature extraction from $\bf \hat{y}$ and $\bf y^*$ and output embeddings $\bf \hat{z}$ and $\bf z^*$
\end{enumerate}
\vspace{-2mm}
\begin{enumerate}[{Bloc C}. step 1]
\item The metric $r$ is modelled by a WGAN critic that outputs evaluations of the clusterings: $r(\bf X,\bf y^*) = c_\theta(\bf z^*)$ and $r(\bf X,\bf \hat{y}) = c_\theta(\bf \hat z)$\vspace{-2mm}
\item Train the model using the error between $r(\bf X,\bf y^*)$ and $r(\bf X,\bf \hat{y})$.
\end{enumerate}
\vspace{-3mm}
\begin{figure}[h!]
\centering
\includegraphics[width=\columnwidth]{Figure.png}
\caption{Our framework's 3 components: the clustering mechanism (A), the GAE (B) and the WGAN (C). (A) takes an unlabelled dataset $\mathbf {X}$ as input and outputs a clustering $\mathbf{\hat{y}}$ that maximises a metric $r$. $\mathbf{\hat{y}}$ is then turned into a graph $\mathcal{G}(\mathbf{X},\mathbf{\hat{y}})$ then into an embedding vector $\mathbf{\hat{z}}$ using (B). Same goes for the correctly labelled dataset, which is embedded as $\mathbf{\hat{z}^*}$. Then, (C), which is the metric itself, evaluates $\mathbf{\hat{z}}$ and $\mathbf{z}^*$ using $c_\theta$ and is trained to produce a new metric $r$ which is then used for (A) in the next iteration.}
\label{framework}
\end{figure}
\begin{comment}
\begin{table}[!ht]
\centering
\caption{Summary of notations}
\begin{tabular}{lp{6.3cm}}
\hline
$\mathbf X$ & A dataset of $n$ points, $x_i \in \mathbb{R}^{\bf d}$\\
$\mathbf y^*$ & True clustering (cluster labels) of $\mathbf X$, $\in \{1,\dots,k\}^n$ \\
$\mathbf y$ & A possible clustering of $\mathbf X$ \\
$\hat{\bf y}$ & Clustering retained after optimisation for a fixed function $r$\\
$\mathcal{M}_{n,m}$ & Set of matrices with $n$ lines and $m$ columns\\
$\mathcal{G}(\mathbf X,\mathbf y)$ & Graph representing a clustered version of $\mathbf X$\\
$A$ & An adjacency matrix \\
$X$ & Feature matrix of $\mathbf X$. $X \in \mathcal{M}_{n,d} $\\
$\bf z^*,\bf\hat{z}$ & Embedding, $\in \mathbb{R}^{\bf e}$, of $(\bf{X},\bf{y}^*)$, and $(\bf{X},\bf{y})$, respectively \\
$r$ & Metric $\mathbb{R}^{\bf n\times d}\times \mathbb N^{\mathbf n} \mapsto \mathbb{R}$, scoring of the clustering \\
CEM & Cross-entropy method\\
$\mathcal{S}$ & Set of all intermediate clustering solutions found through CEM \\[1ex]
\hline
\end{tabular}
\end{table}
\end{comment}
\vspace{-8mm}
\subsection{Clustering mechanism}\label{clustering}
We seek the most suitable optimisation algorithm for clustering given $r$. Considering a neural network that performs the clustering, we need to find its weights $w$ such that the metric is maximised (see equation \eqref{w}). The type of algorithm to use depends on the nature of the metric $r$ to optimise on.
\begin{equation}\label{w}
\text{CEM}_r(\mathbf X)\xrightarrow{\text{finds}} w^* = \argmax_w r(\mathbf{X},\mathbf{y}^w)
\end{equation}
Where $\mathbf y^w$ is a clustering obtained with the weights $w$. The metric is assumed to hold certain properties, discussed in \ref{critic}:
\begin{itemize}
\item \textbf{Unique Maximum:} A unique optimal clustering. $r$ has a unique maximum.
\vspace{-6mm}
\item \textbf{Continuity\footnote{As a reminder, Let $T$ and $U$ be two topological spaces. A function $f:T\mapsto U$ is continuous in the open set definition if for every $t\in T$ and every open set $u$ containing $f(t)$, there exists a neighbourhood $v$ of $t$ such that $f(v)\subset u$.}}: Any two clusterings $\mathbf y$ and $\mathbf y'$ should be similar if $r(\mathbf y)$ and $r(\mathbf y')$ are close in $\mathbb{R}$ space. Hence, $r$ has to satisfy a continuity constraint.
\end{itemize}
There is no guarantee that the best metric for the clustering task is differentiable.
Given the above assumptions, conditions are favourable for evolutionary strategies (ES) to iteratively converge towards the optimal solution. Indeed, if $r$ is continuous and the series $((\mathbf{X},\mathbf{y}_1),\dots,(\mathbf{X},\mathbf{y}_p))$ converges towards $(\mathbf{X},\mathbf{y}^*)$ then $(r(\mathbf{X},\mathbf{y}_1),\dots,r(\mathbf{X},\mathbf{y}_p))$ converges towards $r(\mathbf{X},\mathbf{y}^*)$.
We choose the Cross-Entropy Method (CEM) \cite{CEM}, a popular ES algorithm for its simplicity, to optimise the clustering neural network weights by solving Eq.\eqref{w} (algorithm \ref{Alg:CEM_algo}).
\begin{algorithm}[tb]
\caption{CEM Algorithm}
\label{Alg:CEM_algo}
\begin{algorithmic}
\STATE \textbf{Input:} Dataset $X\in\mathbb R^{\bf{n} \times \bf{d}}$; score function $r$; $\mu \in \mathbb{R}^{\bf{d}}$ and $\sigma \in \mathbb{R}^{\bf{d}}$; elite percentage to retain $p$; $n$ samples of $w_i \sim \mathcal{N}(\mu,\text{diag}(\sigma))$; $T$ number of iterations
\FOR{$\textnormal{iteration}=1$ {\bfseries to} $T$}
\STATE Produce $n$ samples of neural network weights $w_i \sim \mathcal{N(\mu,\text{diag}(\sigma))}$
\STATE Produce clusterings $y_i$ of $X$ using each $w_i$
\STATE Evaluate $r_i = r(X,y_i)$
\STATE Constitute the elite set of $p\%$ best $w_i$
\STATE Fit a Gaussian distribution with diagonal covariance to the elite set and get a new $\mu_t$ and $\sigma_t$
\ENDFOR
\STATE {\bfseries return:} $\mu$, $w^*$
\end{algorithmic}
\end{algorithm}
\subsection{Graph based dataset embedding}
To capture global properties and be transferable across different datasets, we argue that it is necessary to input all the points of a dataset at once. Hence, instead of pairwise similarities between random pairs of points, we propose to get a representation of the relation between a bunch of neighbouring points. Thus, we represent each dataset by a graph structure $\mathcal{G}(\mathbf{X},\mathbf y)$ where each node corresponds to a point in $\mathbf{X}$ and where cliques represent clusters as shown in figure \ref{framework}. This representation takes the form of a feature matrix $X$ and an adjacency matrix $A$.
Using $X$, and $A$, we embed the whole dataset into a vector $\bf z \in \mathbb{R}^{\mathbf e}$. To do so, we use graph autoencoders (GAE). Our implementation is based on \cite{GAE}.
\begin{comment}
Specifically, we have $\{X,A\} \mapsto \bf z$, under the following mechanism:
\begin{equation}
\begin{aligned}
GCN(X,A)= Relu(\Tilde{A}XW_0) = \Bar{X}\\
\end{aligned}
\end{equation}
With $\Tilde{A}$ the symmetrically normalized adjacency matrix and $(W_0,W_1)$ the GCN weight matrices.
\begin{equation}
\begin{aligned}
z=\Tilde{A}\Bar{X}W_1
\end{aligned}
\end{equation}
Finally, the decoder outputs a new adjacency matrix using the sigmoid function $\sigma$:
\begin{equation}
\begin{aligned}
\hat{A}=\sigma(zz^T)
\end{aligned}
\end{equation}
\end{comment}
We obtain $z \in \mathcal{M}_{n,m}$ which is dependent of the shape of the dataset (where $m$ is a user specified hyper-parameter). In order to make it independent from the number of points in $\mathcal{X}$, we turn the matrix $z$ into a square symmetrical one $z \xleftarrow{} z^Tz \in \mathcal{M}_{m,m}$. The final embedding corresponds to a flattened version of the principal triangular bloc of $z^Tz$, which shape is $\mathbf e=(\frac{m+1}{2},1)$. However, the scale of the output still depends on the number of points in the dataset. This could cause an issue when transferring to datasets with a vastly different number of data points. It should therefore require some regularisation; in order to simplify, we decided to use datasets with approximately the same number of points.
\subsection{A critic as a metric}\label{critic}
With embedded vectors of the same shape, we compare the clusterings proposed $\mathbf{\hat{z}}$ and the ground truth ones $\bf z$ using the metric $r$. $r$ is a function mapping an embedding vector $\mathbf z\in \mathbb R^{\mathbf e}$ to $\mathbb{R}$, we therefore parameterise it as:
\begin{equation}\label{large_state_reward}
r_\alpha(\mathbf X,\mathbf y)=r_\alpha(\mathbf z)=\alpha_1\phi_1(\mathbf z)+\alpha_2\phi_2(\mathbf z)+...+\alpha_h\phi_h(\mathbf z)
\end{equation}
Where $\phi_j(\mathbf z)\in \mathbb R$. As per \cite{Russell}, learning a viable metric is possible provided both the following constraints: (1) maximising the difference between the quality of the optimal decision and the quality of the second best; (2) minimising the amplitude of the metric function as using small values encourages the metric function to be simpler, similar to regularisation in supervised learning.
When maximising the metric difference between the two clusterings that have the highest scores, we get a similarity score as in traditional metric learning problems. The problem is formulated by equation \eqref{general_optimization} where $\mathcal{S}$ is a set of solutions (i.e., clustering proposals) found using $r_\alpha$ and $\mathbf{y}^*$ is the true clustering, $\mathbf{y}^{\text{max}}$ is the best solution found in $\mathcal{S}$: $\mathbf{y}^{\text{max}} = \argmax_{\mathbf{y}\in\mathcal S}r_\alpha(\mathbf X, \mathbf{y})$.
\begin{equation}\label{general_optimization}
\begin{aligned}
\min_\alpha r_\alpha(\mathbf X, \mathbf y^*) & -\max_\alpha \min_{\mathbf y'\in \mathcal S\setminus \mathbf y^{\text{max}}} r_\alpha(\mathbf X,\mathbf y^{\text{max}})-r_\alpha(\mathbf X,\mathbf y')\\
& \quad \text{s.t} \quad \mathbf{y}^*=\argmax_{\mathbf{y}\in \mathbf{Y}}r(\mathbf{y})
\end{aligned}
\end{equation}
\begin{algorithm}[h!]
\footnotesize
\caption{Critic2Metric (C2M)}\label{Complete_algo}
\SetAlgoLined
\KwInput{$b$: batch size, $epoch$: number of epochs; $p$: percentage of elite weights to keep; $iteration$: number of CEM iterations; $population$: number of weights to generate; $\mu \in \mathbb{R}^d$: CEM mean; $\sigma \in \mathbb{R}^d$: CEM standard deviation, $\theta$: critic's weights}
\For{$n=1$ {\bfseries to} epoch}{
\For{$k=1$ {\bfseries to} b}{
Sample $(\mathbf X_{k},\mathbf y_k^*) \sim \mathcal D $ a correctly labelled dataset\\
Generate ground truth embeddings $\mathbf z_{(\mathbf X_{k},\mathbf y_k^*)}=GAE(\mathcal{G}(\mathbf X_k,\mathbf y_k^*))$ \\
Initialise clustering neural network weights $\{w_j\}_{j=1}^{population}$ \\
\For{$i=1$ {\bfseries to} iteration}{
\For{$j=1$ {\bfseries to} population}
{Generate clusterings $\mathbf{\hat{y}}_k^{w_j}$ \\
Convert $\mathbf{\hat{y}}_k^{w_j}$ into a graph\\
$\mathbf z_{(\mathbf X_{k},\mathbf {\hat{y}}_k^{w_j})}= GAE(\mathcal{G}(\mathbf X_k,\hat{\mathbf y}_k^{w_j}))$ \\
Evaluate: $r(\mathbf X_k,\hat{\mathbf y}_k^{w_j}) = c_\theta(\mathbf z_{(\mathbf X_{k},\mathbf {\hat{y}}_k^{w_j})})$}
Keep proportion $p$ of best weights $w_p$ \\
$w^* \xleftarrow{} \text{CEM}(w_p, \mu, \sigma)$}
Generate clustering $\mathbf{y}_k^{w^*}$\\
$\mathbf z_{(\mathbf X_{k},\mathbf {\hat{y}}_k^{w^*})} = GAE(\mathcal{G}(\mathbf X_k,\hat{\mathbf y}_k^{w^*}))$ \\
Train critic as in \cite{WGAN} using $\mathbf z_{(\mathbf X_{k},\mathbf {\hat{y}}_k^{w^*})}$ and $\mathbf z_{(\mathbf X_{k},\mathbf y_k^*)}$ \;
}}
\end{algorithm}
\vspace{-5mm}
To solve equation \eqref{general_optimization}, we
use a GAN approach where the clustering mechanism (i.e., CEM) plays the role of the generator while a critic (i.e., metric learning model) plays the role of the discriminator. In a classic GAN, the discriminator only has to discriminate between real and false samples, making it use a cross entropy loss. With this kind of loss, and in our case, the discriminator quickly becomes too strong. Indeed, the score output by the discriminator becomes quickly polarised around 0 and 1.
\vspace{-1mm}
For this reason, we represent $r$ as the critic of a WGAN \cite{WGAN}. This critic scores the realness or fakeness of a given sample while respecting a smoothing constraint. The critic measures the distance between data distribution of the training dataset and the distribution observed in the generated samples. Since WGAN assumes that the optimal clustering provided is unique, the metric solution found by the critic satisfies equation \eqref{general_optimization} constraints. $r$ reaching a unique maximum while being continuous, the assumptions made in section \ref{clustering} are correctly addressed.
To train the WGAN, we use the loss $\mathcal{L}$ in equation \eqref{WGAN_loss} where $\bf \hat{z}$ is the embedding vector of a proposed clustering and $\bf z$ is the embedding vector of the desired clustering. Our framework is detailed in algorithm \ref{Complete_algo}.
\vspace{-1mm}
\begin{equation}\label{WGAN_loss}
\mathcal{L}(\mathbf z^*,\mathbf {\hat{z}})=\max_{\theta}\mathbb{E}_{\mathbf z^*\sim p}[f_\theta(\mathbf z^*)] - \mathbb{E}_{\mathbf {\hat{z}}\sim p(\mathbf {\hat{z})}}[f_\theta(\mathbf {\hat{z}})]
\end{equation}
\section{Experiments}\label{sec:experiments}
\vspace*{-\baselineskip}
\begin{table*}[h!]
\centering
\begin{adjustbox}{width=\columnwidth, center}
\begin{tabular}{||c || c || c || c || c || c || c || c || c || c ||}
\hline
\multicolumn{1}{||c|}{\textbf{Dataset family}} &
\multicolumn{4}{||c|}{Synthetic data} &
\multicolumn{3}{||c|}{MNIST} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}Street view\\ house numbers\end{tabular}} &
\multicolumn{1}{c||}{Omniglot} \\
\hline
\multicolumn{1}{||c|}{\textbf{Dataset}} &
\multicolumn{1}{||c|}{Blob} &
\multicolumn{1}{||c|}{Moon} &
\multicolumn{1}{||c|}{Circles} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}Aniso-\\ tropic\end{tabular} } &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}MNIST-digits\\ \cite{MNIST_digits}\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}letters MNIST\\ \cite{MNIST_letters}\end{tabular} } &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}fashion MNIST\\ \cite{fashion_MNIST}\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}SVHN\\ \cite{SVHN}\end{tabular}} &
\multicolumn{1}{c||}{\begin{tabular}{@{}c@{}}Omniglot\\ \cite{omniglot}\end{tabular} } \\
\hline
\multicolumn{1}{||c|}{\textbf{Snapshot}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{Blobs.PNG}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{Aniso.PNG}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{Circles.PNG}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{Moons.PNG}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{MNIST_example.jpg}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{MNIST_letter.PNG}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{MNIST_fashion.PNG}}} &
\multicolumn{1}{||c|}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{SVHN.png}}} &
\multicolumn{1}{c||}{\raisebox{-\totalheight}{\includegraphics[width=20mm, height=20mm]{Omniglot.PNG}}} \\
\hline
\multicolumn{1}{||c|}{\textbf{\begin{tabular}{@{}c@{}}Feature\\ dimension\end{tabular} }} &
\multicolumn{1}{||c|}{2} &
\multicolumn{1}{||c|}{2} &
\multicolumn{1}{||c|}{2} &
\multicolumn{1}{||c|}{2} &
\multicolumn{1}{||c|}{$28\times 28$} &
\multicolumn{1}{||c|}{$28\times 28$} &
\multicolumn{1}{||c|}{$28\times 28$} &
\multicolumn{1}{||c|}{$32 \times 32$} &
\multicolumn{1}{c||}{$105 \times 105$} \\
\hline
\multicolumn{1}{||c|}{\textbf{\begin{tabular}{@{}c@{}}Maximum number\\ of clusters\end{tabular}}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}9\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}9\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}9\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}9\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{10} &
\multicolumn{1}{||c|}{26} &
\multicolumn{1}{||c|}{10} &
\multicolumn{1}{||c|}{10} &
\multicolumn{1}{c||}{47} \\
\hline
\multicolumn{1}{||c|}{\textbf{Size}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}200\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}200\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}200\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{\begin{tabular}{@{}c@{}}200\\ (custom)\end{tabular}} &
\multicolumn{1}{||c|}{60000} &
\multicolumn{1}{||c|}{145600} &
\multicolumn{1}{||c|}{60000} &
\multicolumn{1}{||c|}{73257} &
\multicolumn{1}{c||}{32460} \\
\hline
\end{tabular}
\end{adjustbox}
\caption{Datasets description}
\vspace{-8mm}
\label{tab:dataset}
\end{table*}
For empirical evaluation, we parameterise our framework as follows: The critic (block C in Fig~\ref{framework}) is a 5 layer network of sizes 256, 256, 512, 512, and 1 (output) neurons. All activation functions are LeakyRelu ($\alpha=0.2$) except last layer (no activation). RMSprop optimizer with $0.01$ initial learning rate and a decay rate of $0.95$. The CEM-trained neural network (bloc A in Fig~\ref{framework}) has 1 hidden layer of size 16 with Relu activation, and a final layer of size $k=50$ (the maximum number of clusters). The GAE (bloc B in Fig~\ref{framework}) has 2 hidden layers; sized 32 and 16 for synthetic datasets, and 100 and 50 for real datasets.
We choose datasets based on 3 main criteria: having a similar compatible format; datasets should be large enough to allow diversity in subsampling configurations to guarantee against overfitting; datasets should be similar to the ones used in our identified baseline literature. All used datasets are found in table \ref{tab:dataset}.
For training, we construct $n$ sample datasets and their ground truth clustering, each containing 200 points drawn randomly from a set of 1500 points belonging to the training dataset. Each one of these datasets, along with their clustering is an input to our model. To test the learned metric, we construct 50 new sample datasets from datasets that are different from the training one (e.g., if we train the model on MNIST numbers, we will use datasets from MNIST letters or fashion to test the metric). The test sample datasets contain 200 points each for synthetic datasets and 1000 points each otherwise. The accuracies are then averaged accross the 50 test sample datasets.
To test the ability of the model to learn using only a few samples, we train it using 5 (few shots) and 20 datasets (standard), each containing a random number of clusters. For few shots trainings, we train the critic for 1 epoch and 10 epochs for standard trainings.
To evaluate the clustering, we use Normalised-Mutual Information (NMI) \cite{NMI} and clustering accuracy (ACC) \cite{ACC}. NMI provides a normalised measure that is invariant to label permutations while ACC measures the one-to-one matching of labels. For clustering, we only need that the samples belonging to the same cluster are attributed the same label, independently from the label itself. However, since we want to analyse the behaviour of the metric learned through our framework, we are interested in seeing whether it is permutation invariant or not. Hence, we need the two measures.
\subsection{Results on 2D synthetic datasets}
Analysis on synthetic datasets (see table \ref{tab:dataset}) proves that our model behaves as expected. We do not compare our results to any baseline since existing unsupervised methods are well studied on them.
We train our model using exclusively samples from blobs datasets. We then test the learned metric on the 4 different types of synthetic datasets (blobs, anisotropic, moons and circles). Results are displayed in table \ref{sci-kit_results}. We observe that the model obtains the best score on blobs since it is trained using this dataset. We can also notice that our model achieves high scores for the other types of datasets not included in training.
\begin{table} [h!]
\centering
\begin{tabular}{LCCCC}
\toprule
\multicolumn{1}{l}{Types of datasets} &
\multicolumn{2}{c}{Standard training} &
\multicolumn{2}{c}{Few shots training} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
&
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} \\
\midrule
\text{Blobs} & 98.4\% & 0.980 & 97.3\% & 0.965\\
\text{Anisotropic} & 97.9\% & 0.967 & 97.2\% & 0.945\\
\text{Circles} & 91.7\% & 0.902 & 92.7\% & 0.900\\
\text{Moons} & 92.1\% & 0.929 & 92.8\% & 0.938\\
\bottomrule
\end{tabular}
\caption{Average ACC and NMI on synthetic test datasets.}
\vspace{-5mm}
\label{sci-kit_results}
\end{table}
Our model succeeds in clustering datasets presenting non linear boundaries like circles while blobs datasets used in training are all linearly separable. Hence, the model learns intrinsic properties of training dataset that are not portrayed in the initial dataset structure, and thus that the metric appears to be transferable.
\textbf{Critic's ablation study}. To test if the critic behaves as expected, i.e., grades the clustering proposals proportionally to their quality, we test it on wrongly labelled datasets to see if the score decreases with the number of mislabelled points. We consider 50 datasets from each type of synthetic datasets, create 50 different copies and mislabel a random number of points in each copy. A typical result is displayed in figure \ref{ablation} and shows that the critic effectively outputs an ordering metric as the score increases when the number of mislabelled points decreases, reaching its maximum when there is no mislabelled point. This shows that the metric satisfies the constraints stated in equation \ref{general_optimization}.
\vspace{-1mm}
\begin{figure}[h!]
\centering
\includegraphics[width=0.6\columnwidth]{capture.png}
\caption{Metric values (i.e., scores given by the critic) for several clusterings of a dataset. Plots are from an anisotropic dataset (left) and a moons dataset (right). In a 2 cluster case (right), the formula used to compute mislabelled points has been made sensitive to label permutation to verify if permuted labels can fool the critic. The critic assigns a high score either when all the labels match the given ground truth or when all the labels are permuted (which again does not affect the correctness of the clustering)}
\vspace{-2mm}
\label{ablation}
\end{figure}
\vspace{-6mm}
An interesting behaviour is shown in figure \ref{ablation}. Recall that since we are in the context of a clustering problem, we only need for the samples belonging to the same cluster to get the same label, independently from the cluster label itself. Thus, the formula used to compute mislabelled points has been made sensitive to label permutation to verify if permuted labels can fool the critic. For instance, in a 2 clusters case, one can switch the labels of all points in each cluster and still get the maximum score. Switching all labels makes all the points wrongly labelled compared to the given ground truth but nonetheless the clustering itself remains true. This explains the rounded shape in figure \ref{ablation} where the used datasets in the right panel only consisted of 2 clusters. The critic assigns a high score either when all the labels match the given ground truth or when all the labels are permuted (which does not affect the correctness of the clustering).
\vspace{-3mm}
\subsection{Results on MNIST datasets}\label{MNIST_section}
\vspace{-1mm}
MNIST datasets give similar results both in terms of ACC and NMI on all test datasets regardless of the used training dataset (see table \ref{MNIST_result}). Hence, the model effectively capture implicit features that are dataset independent. While standard training shows better results, the few shots training has close performance.
\begin{table}[h!]
\centering
\begin{tabular}{LCCCCCC}
\toprule
\multicolumn{1}{l}{Training Dataset} &
\multicolumn{6}{c}{Testing Dataset} \\
\cmidrule(lr){2-7}
\multicolumn{1}{c}{} &
\multicolumn{2}{c}{Numbers} &
\multicolumn{2}{c}{Letters} &
\multicolumn{2}{c}{Fashion} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
\cmidrule(lr){6-7}
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} \\
\midrule
\text{Numbers (standard)} & 72.3\% & 0.733 & 81.3\% & 0.861 & 65.2\% & 0.792 \\
\text{Numbers (few shots)} & 68.5\% & 0.801 & 79.0\% & 0.821 & 61.8\% & 0.672 \\
\text{Letters (standard)} & 75.9\% & 0.772 & 83.7\% & 0.854 & 67.5\% & 0.800 \\
\text{Letters (few shots)} & 69.8\% & 0.812 & 78.7\% & 0.806 & 60.9\% & 0.641 \\
\text{Fashion (standard)} & 70.6\% & 0.706 & 83.4\% & 0.858 & 72.5\% & 0.762 \\
\text{Fashion (few shots)} & 70.1\% & 0.690 & 82.1\% & 0.834 & 70.7\% & 0.697 \\
\bottomrule
\end{tabular}
\caption{Mean clustering performance on MNIST dataset.}
\label{MNIST_result}
\end{table}
\vspace{-12mm}
\begin{table}[h!]
\centering
\begin{tabular}{LCCCCCC}
\toprule
\multicolumn{1}{l}{Training Dataset} &
\multicolumn{6}{c}{Testing Dataset} \\
\cmidrule(lr){2-7}
\multicolumn{1}{c}{} &
\multicolumn{2}{c}{Numbers} &
\multicolumn{2}{c}{Letters} &
\multicolumn{2}{c}{Fashion} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
\cmidrule(lr){6-7}
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{Best} &
\multicolumn{1}{c}{Top 3} &
\multicolumn{1}{c}{Best} &
\multicolumn{1}{c}{Top 3} &
\multicolumn{1}{c}{Best} &
\multicolumn{1}{c}{Top 3} \\
\midrule
\text{Numbers (standard)} & 78.3\% & 92.5\% & 86.0\% & 97.5\% & 69.2\% & 87.2\%\\
\text{Numbers (few shots)} & 75.8\% & 82.1\% & 83.3\% & 92.0\% & 65.1\% & 83.9\% \\
\text{Letters (standard)} & 77.4\% & 89.2\% & 88.8\% & 96.4\% & 70.2\% & 86.7\%\\
\text{Letters (few shots)} & 73.1\% & 80.6\% & 85.1\% & 91.5\% & 61.0\% & 76.3\% \\
\text{Fashion (standard} & 70.1\% & 83.1\% & 85.0\% & 98.6\% & 76.9\% & 94.7\%\\
\text{Fashion (few shots)} & 67.9\% & 77.4\% & 83.5\% & 95.3\% & 70.2\% & 88.0\%\\
\bottomrule
\end{tabular}
\caption{Critic based performance assessment: Best corresponds to the percentage of times the critic gives the best score to the desired solution. Top 3 is when this solution is among the 3 highest scores.}
\label{MNIST_theoretic}
\vspace{-4mm}
\end{table}
\begin{comment}
\vspace*{-\baselineskip}
\begin{table}[h!]
\begin{adjustwidth}{-3cm}{-3cm}
\begin{subtable}[t]{0.5\linewidth}
\begin{tabular*}{\columnwidth}{LCCCCCC}
\toprule
\multicolumn{1}{l}{Training Dataset} &
\multicolumn{6}{c}{Testing Dataset} \\
\cmidrule(lr){2-7}
\multicolumn{1}{c}{} &
\multicolumn{2}{c}{Numbers} &
\multicolumn{2}{c}{Letters} &
\multicolumn{2}{c}{Fashion} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
\cmidrule(lr){6-7}
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} &
\multicolumn{1}{c}{ACC} &
\multicolumn{1}{c}{NMI} \\
\midrule
\text{Numbers (standard)} & 72.3\% & 0.733 & 81.3\% & 0.861 & 65.2\% & 0.792 \\
\text{Numbers (few shots)} & 68.5\% & 0.801 & 79.0\% & 0.821 & 61.8\% & 0.672 \\
\text{Letters (standard)} & 75.9\% & 0.772 & 83.7\% & 0.854 & 67.5\% & 0.800 \\
\text{Letters (few shots)} & 69.8\% & 0.812 & 78.7\% & 0.806 & 60.9\% & 0.641 \\
\text{Fashion (standard)} & 70.6\% & 0.706 & 83.4\% & 0.858 & 72.5\% & 0.762 \\
\text{Fashion (few shots)} & 70.1\% & 0.690 & 82.1\% & 0.834 & 70.7\% & 0.697 \\
\bottomrule
\end{tabular*}
\caption{Mean clustering performance on MNIST dataset.}
\label{MNIST_result}
\end{subtable}
\begin{subtable}[t]{0.5\linewidth}
\begin{tabular}{LCCCCCC}
\toprule
\multicolumn{1}{l}{Training Dataset} &
\multicolumn{6}{c}{Testing Dataset} \\
\cmidrule(lr){2-7}
\multicolumn{1}{c}{} &
\multicolumn{2}{c}{Numbers} &
\multicolumn{2}{c}{Letters} &
\multicolumn{2}{c}{Fashion} \\
\cmidrule(lr){2-3}
\cmidrule(lr){4-5}
\cmidrule(lr){6-7}
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{Best} &
\multicolumn{1}{c}{Top 3} &
\multicolumn{1}{c}{Best} &
\multicolumn{1}{c}{Top 3} &
\multicolumn{1}{c}{Best} &
\multicolumn{1}{c}{Top 3} \\
\midrule
\text{Numbers (standard)} & 78.3\% & 92.5\% & 86.0\% & 97.5\% & 69.2\% & 87.2\%\\
\text{Numbers (few shots)} & 75.8\% & 82.1\% & 83.3\% & 92.0\% & 65.1\% & 83.9\% \\
\text{Letters (standard)} & 77.4\% & 89.2\% & 88.8\% & 96.4\% & 70.2\% & 86.7\%\\
\text{Letters (few shots)} & 73.1\% & 80.6\% & 85.1\% & 91.5\% & 61.0\% & 76.3\% \\
\text{Fashion (standard} & 70.1\% & 83.1\% & 85.0\% & 98.6\% & 76.9\% & 94.7\%\\
\text{Fashion (few shots)} & 67.9\% & 77.4\% & 83.5\% & 95.3\% & 70.2\% & 88.0\%\\
\bottomrule
\end{tabular}
\caption{Critic based performance assessment: Best corresponds to the percentage of times the critic gives the best score to the desired solution. Top 3 is when this solution is among the 3 highest scores.}
\label{MNIST_theoretic}
\end{subtable}
\caption{Results on MNIST datasets}
\end{adjustwidth}
\vspace{-4mm}
\end{table}
\end{comment}
\vspace{-2mm}
Table \ref{MNIST_theoretic} shows the percentage of times the critic attributes the best score to the desired solution. It shows that ES algorithm choice has a significant impact on the overall performance. Even with a metric that attributes the best score to the desired clustering, the CEM may be stuck in a local optimum and fails to reconstruct back the desired clustering. Hence, a better optimisation can enhance the performance shown in table \ref{MNIST_result} closer to the one presented in table \ref{MNIST_theoretic}.
\subsection{Comparative study}
We compare our approach with baseline methods from the literature
(table \ref{comparative_results}). For some methods, we followed the procedure in \cite{transfer_clustering} and used their backbone neural network as a pairwise similarity metric. Table \ref{Results_SVHN} reports results when training on SVHN and testing on MNIST numbers. We obtain close ACC values to CCN and ATDA \cite{ATDA}. These methods uses Omniglot as an auxiliary dataset to learn a pairwise similarity function, which is not required for our model. Our model only uses a small fraction of SVHN, has shallow networks and does not require any adaptation to its loss function to achieve comparable results. Finally, other cited methods require the number of clusters as an a priori indication. We achieve comparable results without needing this information. When the loss adaptation through Omniglot is discarded (denoted source-only in table \ref{Results_SVHN}), or if the number of clusters is not given, their accuracy falls and our model surpasses them by a margin.
\vspace{-6mm}
\begin{table}[h!]
\begin{subtable}[c]{0.5\textwidth}
\centering
\begin{tabular}{LCC}
\toprule
\text{Method} & \multicolumn{2}{c}{\text{ACC}} \\
\midrule
& \text{Loss Adaptation} & \text{Source Only}\\
\midrule
\text{DANN \cite{DANN}} & 73.9\% & 54.9\%\\
\text{LTR \cite{LTR}} & 78.8\% & 54.9\%\\
\text{ATDA \cite{ATDA}} & 86.2\% & 70.1\%\\
\text{CCN \cite{transfer_clustering}} & 89.1\% & 52\%\\
\text{Ours (standard)} & - & 84.3\% \\
\text{Ours (few shots)} & - & 81.4\% \\
\bottomrule
\end{tabular}
\subcaption{Unsupervised cross-task transfer from SVHN to MNIST digits.}
\vspace{-2mm}
\label{Results_SVHN}
\end{subtable}
\begin{subtable}[c]{0.5\textwidth}
\centering
\begin{tabular}{LCC}
\toprule
\text{Method} & \text{ACC} & \text{NMI} \\
\midrule
\text{k-means} & 18.9\% & 0.464 \\
\text{CSP \cite{CSP}} & 65.4\% & 0.812 \\
\text{MPCK-means \cite{mpckmeans}} & 53.9\% & 0.816 \\
\text{CCN \cite{transfer_clustering}} & 78.18\% & 0.874 \\
\text{DTC \cite{learn}} & 87.0\% & 0.945 \\
\text{Autonovel \cite{autonovel}} & 85.4\% & - \\
\text{Ours (standard)} & 83.4\% & 0.891 \\
\bottomrule
\end{tabular}
\subcaption{Unsupervised cross-task transfer from $\text{Omniglot}_\text{train}$ to $\text{Omniglot}_\text{test}$ ($k=100$ for all).}
\vspace{-2mm}
\label{Omniglot_results}
\end{subtable}
\caption{Comparative clustering performance}
\vspace{-8mm}
\label{comparative_results}
\end{table}
Table \ref{Omniglot_results} reports results when training on $\text{Omniglot}_\text{train}$ and testing on $\text{Omniglot}_\text{test}$. Values are averaged across $20$ alphabets which have $20$ to $47$ letters. We set the maximum number of clusters $k=100$. When the number of clusters is unknown, we get an ACC score relatively close to DTC and Autonovel. Compared to these two approaches, our method bears several significant advantages:
\begin{itemize}
\vspace{-2mm}
\item \textbf{Deep Networks}: DTC and Autonovel used Resnets as a backbone which are very deep networks while we only used shallow networks (2 layers maximum)
\vspace{-6mm}
\item \textbf{Pairwise similarity}: in Autonovel the authors used a pairwise similarity statistic between datasets instances which we aimed to avoid due to its significant computational bottleneck. Moreover, this metric is recalculated after each training epoch, which adds more complexity.
\vspace{-2mm}
\item \textbf{Vision tasks:} While DTC can only handle vision tasks, we present a more general framework which includes vision but also tabular datasets.
\vspace{-2mm}
\item \textbf{Number of classes}: DTC and Autonovel used the labelled dataset as a probe dataset, and estimates the number of classes iteratively, and when the labelled clusters are correctly recovered, they used the ACC metric to keep the best clustering. This approach is effective, but requires access to the labelled dataset at inference time to estimate the number of classes. This is a shortcoming (memory or privacy limitations). Our approach does not require the labelled dataset once the metric is learned. Our metric automatically estimates the number of clusters required to any new unlabelled dataset.
\end{itemize}
\vspace{-2mm}
\section{Conclusion}\label{sec:discussion}
We presented a framework for cross domain/task clustering by learning a transferable metric. This framework consisted of ES methods, and GAE alongside a critic. Our model extracts dataset-independent features from labelled datasets that characterise a given clustering, performs the clustering and grades its quality. We showed successful results using only small datasets and relatively shallow architectures. Moreover, there is more room for improvement. Indeed, since our framework is composed of 3 different blocs (CEM, GAE, critic), overall efficiency can be enhanced by independently improving each bloc (i.e replacing CEM).
In future work, we will study the criteria that determine why some auxiliary datasets are more resourceful than others given a target dataset. In our case, this means to study for instance why using the MNIST letters dataset as training allowed a better performance on Fashion MNIST than when using MNIST numbers. This would allow to deliver a minimum performance guarantee at inference time by creating a transferability measure between datasets.
\textbf{Acknowledgements:} We gratefully acknowledge Orianne Debeaupuis for making the figure. We also acknowledge computing support from
NVIDIA. This work was supported by funds from the French Program "Investissements d'Avenir".
\vspace{-4mm}
\bibliographystyle{splncs04}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 965 |
<?php
namespace Opifer\ReviewBundle\Form\Type;
use Symfony\Component\Form\AbstractType;
use Symfony\Component\Form\Extension\Core\Type\TextareaType;
use Symfony\Component\Form\Extension\Core\Type\TextType;
use Symfony\Component\Form\FormBuilderInterface;
/**
* Review Type
*/
class ReviewType extends AbstractType
{
/**
* {@inheritdoc}
*/
public function buildForm(FormBuilderInterface $builder, array $options)
{
$builder
->add('author', TextType::class)
->add('title', TextType::class)
->add('content', TextareaType::class)
;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,724 |
I was asked why I wanted to work for the firm and why I wanted to do Law. This was after the initial telephone interview where they asked me about my degree, my previous internships and what I had learnt from there, was I allowed to work in the UK and if I was able to start work within a week if selected.
After the brief talk I was given three scenarios from that days newspaper and was asked to defend clients by making an initial litigation within 90 minuits.
I was also asked how flexible I was with my timings.
I was finally asked whether I was flexible to work in the firms different offices at a short notice.
I was also told about my job role and they verified if I knew all the essential skills.
There was no strict format as there were 8 applicants on that day and we were called into the interviewers office one at a time.
The interviewer was one of the firm's partners.
They also took my permission to have a background check done on me.
The interview along with the case studies lasted just over 2 hours.
WHERE DO I SEE MYSELF IN 4 YEARS TIME?
3 PROBLEM SOLVING QUESTIONS FROM A LITIGATORS POINT OF VIEW. THE SCENARIOS WERE TAKEN FROM THAT DAYS METRO NEWSPAPER.
Be honest and to the point or they'l catch you.
Show that you are willing to learn as the firm intends to teach you on the go rather then you relying on what you learnt at university.
Punctuality is key as one candidate was not interviewed as he was 15 mins late for his scheduled time.
After the breif talk I was given three scenarios from that days newspaper and was asked to defend clients from the scenarios by making an initial litigation within 90 minuits. | {
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\section{Introduction} \label{sec:intro}
Radio-loud AGN drive relativistic jets into their surrounding medium. These jets expand and can form structures extending up to hundreds of kiloparsecs. The emission from AGN jets have been observed over length and energy scales spanning several orders of magnitude, enabling estimates of dynamical parameters such as jet velocities \citep{Kellermann+2004ApJ,Kellermann+2007Ap&SS,Lister+2009AJ,Lister+2016AJ,Lister+2019ApJ,Jorstad+2005AJ,Jorstad+2017ApJ,Angioni+2019A&A}, opening angles \citep{Pushkarev+2009A&A,Pushkarev+2017MNRAS,Algaba+2017ApJ}, and jet power \citep{Godfrey+2013ApJ,Foschini+2019hepr}. The kiloparsec scale morphology of AGN jets is seen to have two distinct categories, despite the wide range of dynamical parameters: the center-brightened or FRI jets, and the edge-brightened or FRII jets \citep{Fanaroff_Riley1974MNRAS}.
The basic morphology of relativistic astrophysical jets has been established by several analytical \citep{Blandford+1974MNRAS,Scheuer1974MNRAS,Begelman+1989ApJ,Falle1991MNRAS,Kaiser+1997MNRAS,Matzner2003MNRAS,Lazzati+2005ApJ} and numerical studies \citep{Duncan+1994ApJ,Marti+1994A&A,Marti+1997ApJ,Komissarov+1998MNRAS}. The accepted picture is that the advancing jet forms an overpressured double bow-shock structure at its head. Material that enters the jet head is pushed out in a direction transverse to that of jet propagation, and forms a hot cocoon around the jet. This cocoon may pressurize the jet and influence its dynamics and state of collimation. The dynamics and state of collimation are the main aspects of the differences between FRI and FRII jets. The origin of this dichotomy has been studied extensively using numerical computations \citep[e.g.,][]{Rossi+2008A&A,Krause+2012MNRAS,Li+2018ApJ}.
\begin{figure}
\centering
\gridline{ \fig{e1e5_hi.png}{0.45\textwidth}{} }
\gridline{ \fig{e1e5_hi_zoomed.png}{0.45\textwidth}{} }
\caption{Snapshots showing an example of our self-similar solutions. \textit{(Top)} Logarithm of the pressure (upper half) and logarithm of the density (lower half) for a jet with $\lambda = 2.5\times 10^{-4}$ and opening angle $\theta_0 = 0.20$, expanding through an ambient medium with density profile $\rho \propto r^{-2}$. It is clearly seen that the jet experiences many recollimation shocks. \textit{(Bottom)} Zoomed-in plot for the two recollimation shocks enclosed in the green rectangle in the top panel.
\label{fig:example}}
\end{figure}
\cite{Bromberg+2011ApJ} \citepalias[hereafter][]{Bromberg+2011ApJ} developed a generalized analytical framework that characterizes non-magnetized jets expanding against a static ambient medium for widely varying dynamical (e.g., non-relativistic vs relativistic jet head velocities) and morphological (e.g., collimated vs freely expanding jets) properties. Their model provides solutions for important dynamical parameters of the jet such as the head velocity, cocoon pressure etc. in terms of power laws. It also provides the criteria that determine the specific nature of the jet dynamics and morphology, which in turn determine the exact form of the power laws. The power law solutions provided by \citetalias{Bromberg+2011ApJ} have been verified numerically by \cite{Mizuta+2013ApJ} and \cite{Harrison+2018MNRAS}. The latter also calibrate the numerical coefficients for the power laws and discuss the applicability of 2D axisymmetric simulations for deriving them.
The aim of this work is to use the model developed by \citetalias{Bromberg+2011ApJ} along with our own numerical calculations to infer dynamical parameters of AGN jets from observational features seen across the known AGN population. We proceed by first testing \citetalias{Bromberg+2011ApJ} 's power law scalings in the context of a relativistic jet for a wide range of jet luminosities and initial opening angles. This is done by numerically calculating self-similar solutions for the system. We then use our findings to obtain quantities and/or morphological properties that can be directly related to observations. We also show that our results can be applied to realistic AGN jets, which aren't necessarily self-similar.
We briefly introduce the jet model and scalings provided in \citetalias{Bromberg+2011ApJ} in Section~\ref{sec:anly}. The numerical method and parameter space adopted in this work are described in Section~\ref{sec:numerical}. The methods of diagnosis we use to analyze our numerical solutions are discussed in Section~\ref{sec:diagnostics}. The results of the numerical calculations and comparison with analytical scalings are presented in Section~\ref{sec:results}. Comparison of our results to observations as well as previous studies and scope for future work are discussed in Section~\ref{sec:discussion}. We provide a brief summary in Section~\ref{sec:conclusion}.
\input{scaling.tex}
\section{Analytical considerations} \label{sec:anly}
A brief qualitative overview of the jet model proposed by \citetalias{Bromberg+2011ApJ} is presented here. The reader is referred to the original work for a detailed explanation, as well as to \cite{Harrison+2018MNRAS} for an excellent quantitative overview. The jet head moves into the ambient medium with speed $\beta_h$. As mentioned before, the jet head expels material to generate a hot pressured cocoon around the jet. The cocoon expands laterally with speed $\beta_c$. It drives a reverse shock into the jet towards its axis that, subject to system parameters, may cause collimation of the jet. The opening angle of the jet is denoted by $\theta_j$, which is generally different from the engine opening angle $\theta_0$ (unless the jet expands freely). The cocoon is expected to have approximately uniform pressure $P_c$ when the jet is collimated, or expands preserving causal contact throughout the jet head. \citetalias{Bromberg+2011ApJ} found that the dynamics of this system can be described in terms of two quantities, the engine opening angle $\theta_0$ and the relativistic jet-to-environment energy density ratio at the jet head ($\Tilde{L}$), given the form of the ambient density profile. They also show that the criteria that determine the dynamical behavior of the jet, for example, whether the jet stays collimated, or whether the jet head maintains an overall causal contact, can be expressed solely by relations between $\Tilde{L}$ and $\theta_0$ (with omission of numerical coefficients of order unity). Once the specific dynamical behavior of the jet is known, the dynamical parameters of the jet (i.e., $\beta_h$, $\beta_c$, $\theta_j$ etc.) also scale as power laws of $\Tilde{L}$ and $\theta_0$. The quantity $\Tilde{L}$ is defined as:
\begin{equation}
\label{eq:Ltilde}
\Tilde{L} \equiv \frac{\rho_j h_j \Gamma_j^2}{\rho_a} \approx \frac{L_j}{\Sigma_j \rho_a(z_h) c^3},
\end{equation}
where $\rho_j$, $h_j$, $\Gamma_j$, $L_j$, $\Sigma_j$ are the jet mass density, specific enthalpy, Lorentz factor, one-sided jet luminosity, and head cross-section area respectively. The ambient mass density at the location of the jet head $z_h$ is given by $\rho_a(z_h)$.
The different regimes of evolution of the jet are dictated by relations between $\Tilde{L}$ and $\theta_0$. For example, the condition (correct up to a factor of order of unity) that the jet remains collimated is given by:
\begin{equation}
\label{eq:coll_con}
\Tilde{L}\lesssim\theta_0^{-4/3},
\end{equation}
We express $\Tilde{L}$ in terms of another dimensionless quantity $\lambda$ \citep{Duffell+2020ApJ} defined as follows:
\begin{equation}
\label{eq:lambda}
\lambda \equiv \frac{L_j}{\rho z_h^2 \theta_0^2 c^3}
\end{equation}
The relation between $\lambda$ and $\Tilde{L}$ is as follows:
\begin{equation}
\label{eq:lambda_ltilde}
\Tilde{L} = \lambda \frac{\theta_0^2}{\pi \theta_j^2}
\end{equation}
For our study we will not control $\theta_j$ but $\theta_0$, the injection angle. Therefore we will describe our results in terms of the parameter $\lambda$ instead of $\Tilde{L}$. A higher value of $\lambda$ thus implies a higher jet power, provided the ambient density profile and injection angle are kept constant. \citetalias{Bromberg+2011ApJ} predict scalings for the jet-cocoon system in a number of different dynamical regimes, the first three of which are listed in Table~\ref{t:lambdascaling}. These regimes can be thought to differ from each other in the velocities of the jet head. We use our self-similar models to test the scalings in these regimes.
\section{Numerical setup} \label{sec:numerical}
We numerically evolve our system in time in accordance with equations of relativistic hydrodynamics in spherical coordinates:
\begin{equation}
\label{eq:SRHD1}
\partial_{\mu} ( \rho u^{\mu} ) = S_D
\end{equation}
\begin{equation}
\label{eq:SRHD2}
\partial_{\mu} ( \rho h u^{\mu} u^{\nu} + P g^{\mu \nu} ) = S^{\nu}
\end{equation}
where $\rho$ is proper density, $P$ is pressure, $u^{\nu}$ is the four-velocity, and $h = 1 + 4P/\rho$ is the specific enthalpy (assuming an adiabatic index, $\Hat{\gamma}=4/3$). We assume our system to be axisymmetric. These equations are scale-free in the sense that there is no length or time scale present. The source terms $S_D$ and $S^{\nu}$ model mass, energy and momentum injection by the engine. All quantities are expressed in normalized or code units, with c = 1. We also evolve a passive scalar field X using:
\begin{equation}
\label{eq:passive_scalar}
\partial_{\mu} ( \rho X u^{\mu} ) = 0
\end{equation}
X advects with the jet and is used to distinguish the material ejected by the engine from the ambient material. We set X=1 for the jet and X=0 for the ambient medium.
The numerical evolution is performed using JET \citep{Duffell+2011ApJS,Duffell+2013ApJ}, a moving-mesh code particularly suitable for modeling relativistic radial outflows over length scales of many orders of magnitude. The moving mesh technique essentially makes the calculation Lagrangian in the radial direction, with highest resolution near the poles. This resolution is necessary to accurately capture high Lorentz factors. The grid size is chosen to have a spatial resolution given by $\Delta\theta = \Delta r/r \sim 0.001$ (The highly resolved flow is apparent in Figure \ref{fig:example}). Only a couple orders of magnitude are resolved in the radial direction at any given time, but the system is evolved over length scales of many orders of magnitude by suitably moving the grid boundaries during calculation.
\subsection{Initial conditions} \label{subsec:IC}
The ambient density profile is modeled by a power-law:
\begin{equation}
\rho(r)=\rho_0\left(\frac{r}{r_0}\right)^{-k},
\end{equation}
where $\rho_0r_0^k$ has been set equal to 1 for all our calculations.
It is known \citep{Komissarov+1997MNRAS, Bromberg+2011ApJ, Marti2019Galax} that a constant-power nonmagnetized relativistic outflow maintains a constant speed of advance for an ambient density profile of $\rho(r)\propto r^{-2}$, while the outflow accelerates or decelerates if the density profile is steeper or shallower, respectively. Accelerating or decelerating solutions are not fully self-similar because they transition between relativistic and non-relativistic regimes. Thus we choose $k=2$ for our self-similar calculations.
Observational studies \citep{Arnaud+1984MNRAS, Blundell+1999AJ, Russell+2015MNRAS} suggest that host environments of AGN have a shallower density profile, corresponding to $k\sim1\mbox{--}1.5$. Hence we also obtain solutions for $k = 1$ and compare them with the self-similar case to investigate how applicable the self-similar solutions are in the context of more realistic interstellar environments.
We perform calculations with k=2 for a number of jet luminosity and injection angle combinations to examine the dependence of our results on $\lambda$ and $\theta_0$. A single combination of jet luminosity and injection angle is chosen to simulate a jet expanding against an ambient medium where k=1. Equation \ref{eq:lambda} implies that $\lambda$ should effectively vary inversely with the jet propagation distance for k=1 provided the jet luminosity and injection angle remain constant. Therefore the effective value of $\lambda$ in this case ($\lambda_{\mathrm{eff}}$) decreases with time as the jet's size increases. This allows us to study the behavior of the jet for a continuous range of $\lambda_{\mathrm{eff}}$.
\subsection{Engine model}
We expect the true engine to operate at scales unresolved in our calculations and be governed by poorly understood physics. Injection is thus simulated at resolved scales using a parameterized model. The source terms in equations \ref{eq:SRHD1} and \ref{eq:SRHD2} completely specify the engine. The engine is turned on at the start of computation and stays on. It is parameterized by a power ($L_0$), injection angle ($\theta_0$), injection radius ($r_0$), Lorentz factor ($\gamma_0$) and baryon loading ($\eta_0$).
The injection radius is set to twice the inner boundary radius of the domain. We have tested the effect of varying this injection radius and find it does not affect our results so long as it is sufficiently larger than the inner boundary radius. We choose a highly relativistic engine by setting the injection Lorentz factor and the injected mass-to-energy ratio to 50 and 0.001 respectively for all of our runs. This is to model a very clean engine where nearly all baryon contamination occurs via interaction with the surrounding medium. The range of $\lambda$ values attained by the jets for a given density profile power law and injection angle are listed in Table~\ref{t:engine}. It can also be seen from the table that the range of $\lambda$ is comparable for both ambient density profiles, i.e., k=2 and k=1.
\input{engine.tex}
The engine is represented using a nozzle function, $g(r, \theta)$, defined as follows:
\begin{equation}
g(r, \theta) \equiv (r/r_0)e^{-(r/r_0)^2}e^{(|cos\,\theta|-1)/\theta_0^2}/N_0
\end{equation}
where $N_0$ is the normalization factor:
\begin{equation}
N_0 = 4\pi r_0^3 \left(1-e^{-2/\theta_0^2}\right)\theta_0^2
\end{equation}
The source terms in equations \ref{eq:SRHD1} and \ref{eq:SRHD2} are expressed in terms of the nozzle function:
\begin{equation}
S^0 = L_0 g(r, \theta)
\end{equation}
\begin{equation}
S^r = S_0\sqrt{1-\gamma_0^{-2}}
\end{equation}
\begin{equation}
S_D = S_0/\eta_0
\end{equation}
\begin{figure*}
\gridline{\fig{e1e7.png}{0.32\textwidth}{(a) $\lambda=2.5\times10^{-6}$}
\fig{e1e6.png}{0.32\textwidth}{(b) $\lambda=2.5\times10^{-5}$}
\fig{e1e5.png}{0.32\textwidth}{(c) $\lambda=2.5\times10^{-4}$}
}
\gridline{
\fig{e1e4.png}{0.32\textwidth}{(d) $\lambda=2.5\times10^{-3}$}
\fig{e1e3.png}{0.32\textwidth}{(e) $\lambda=2.5\times10^{-2}$}
\fig{e1e2.png}{0.32\textwidth}{(f) $\lambda=2.5\times10^{-1}$}
}
\gridline{
\fig{e1e1.png}{0.32\textwidth}{(g) $\lambda=2.5\times10^{0}$}
\fig{e1e0.png}{0.32\textwidth}{(h) $\lambda=2.5\times10^{1}$}
\fig{e1e+1.png}{0.32\textwidth}{(i) $\lambda=2.5\times10^{2}$}
}
\centering
\includegraphics[width=0.8\textwidth]{colorbar_zoo.png}
\caption{Pressure and density maps of self-similar solutions for a single engine opening angle ($\theta_0 = 0.20$ rad) and values of $\lambda$ ranging from $2.5\times10^{-6}$ to $2.5\times10^{2}$. The top and bottom halves of each figure represent pressure and density, respectively. The self-similar solutions range all the way from a non-relativistic forward shock to the ultra-relativistic regime. The colorbar used for plotting the solutions is provided at the bottom. The quantities $\rho_{min}$, $\rho_{max}$, $P_{min}$, and $P_{max}$ denote the minimum and maximum density and the minimum and maximum pressure in the maps, respectively.
\label{fig:zoo}}
\end{figure*}
\section{Diagnostics} \label{sec:diagnostics}
\subsection{Measurement of dynamical parameters}
We measure the jet head advance speed ($v_h$), the cocoon expansion speed ($v_c$), the final jet opening angle ($\theta_j$), and the cocoon pressure ($P_c$) for each of our numerical solutions.
The jet head and the cocoon size increase linearly with time. Thus, $v_h$ and $v_c$ can be measured for a solution at a given time by identifying the jet head or the maximum longitudinal expanse of the shock front, and the maximum lateral expanse of the shock front, respectively. The shock front is identified by noting that the maximum pressure drop along any radial direction should occur at the shock front.
The jet opening angle, $\theta_j$, is calculated by measuring the solid angle $\Omega$ subtended by the jet. We calculate $\theta_j$ following \cite{Duffell-Laskar2018} as:
\begin{equation}
\label{eq:theta_j}
\mathrm{sin}\,\left(\frac{\theta_j}{2}\right) = \frac{ \int (dE/d\Omega) d\Omega }{\sqrt{4\pi \int (dE/d\Omega)^2 d\Omega}}
\end{equation}
We ensure that equation~\ref{eq:theta_j} measures only the angle subtended by the jet by considering the term $dE/d\Omega$ only for those computational zones where the passive scalar, $X = 1$.
\begin{comment}
$\Omega$ is given by \citep{Duffell-Laskar2018}:
\begin{equation}
\Omega = 4\pi E/ E_{iso}
\label{eq:omega}
\end{equation}
where $E$ and $E_{iso}$ are the total energy and the isotropic equivalent energy respectively. They are calculated as follows:
\begin{subequations}
\begin{align}
E &= \int (dE/d\Omega) d\Omega \\
E_{iso} &= 4\pi\frac{\int (dE/d\Omega)^2 d\Omega}{\int (dE/d\Omega) d\Omega}
\end{align}
\label{eq:E_defs}
\end{subequations}
We ensure that equation~\ref{eq:omega} measures only the solid angle subtended by the relativistic outflow by requiring that the term $dE/d\Omega$ in equations~\ref{eq:E_defs} is computated using the energy of only those computational zones which have $\Gamma > 1$. Equations~\ref{eq:omega} and~\ref{eq:E_defs} give us the following formula for $\theta_j$:
\begin{equation}
\mathrm{sin}\,\left(\frac{\theta_j}{2}\right) = \frac{E}{\sqrt{4\pi \int (dE/d\Omega)^2 d\Omega}}
\end{equation}
\end{comment}
The cocoon pressure, $P_c$, is given by the mean pressure of all the cells in our grid corresponding to the cocoon ($0<X<1$), weighted by the cell volume.
\begin{figure*}
\centering
\gridline{\fig{gammabeta_vs_lambda.png}{0.5\textwidth}{(a)}
\fig{vc_vs_lambda.png}{0.5\textwidth}{(b)}
}
\gridline{\fig{tj_vs_lambda.png}{0.5\textwidth}{(c)}
\fig{pc_vs_lambda.png}{0.5\textwidth}{(d)}
}
\caption{Scaling of various dynamic quantities vs $\lambda$. Panels (a)-(d) show the dependence of $\Gamma_h\beta_h$, $\beta_c$, $\theta_j$ and $P_c/\rho_a$ on $\lambda$ respectively. These quantities are the same as in Table~\ref{t:lambdascaling}. The left half of each plot shows the comparison between our result (red dots) and that predicted by \citetalias{Bromberg+2011ApJ} (red solid line) for $\theta_0 = 0.20$. The yellow, blue, and grey shaded regions on plots on the left half indicate the first three dynamical regimes, as in Table \ref{t:lambdascaling}. The right half of each panel show the same scalings for all values of $\theta_0$ used in this work. The purple dashed line and green solid line in the right half of each panel show analytical scalings at $\theta_0=0.10$ and $\theta_0=0.28$ respectively.
\label{fig:scalings}}
\end{figure*}
\subsection{Emissivity maps} \label{subsec:emissivity}
We also calculate synthetic synchrotron emissivity maps from our results to compare features with observations. For each run, we choose a few viewing angles ranging from 0 to $\pi/2$. It is assumed that the emission from the jet is dominated by synchrotron radiation and that the jets are optically thin. We verify the latter assumption by explicitly calculating optical depth maps for our jets using the method followed by \cite{Westhuizen+2019MNRAS}. The value of magnetic field strength is approximated by requiring a sub-equipartition magnetic field, whose energy density is $2\%$ of the internal energy density in our solutions. We note that this produces field strengths of $\sim0.01-0.1$ mG, similar to values predicted by independent observational studies for jets in M87 \citep{Stawarz+2005ApJ} and Cygnus A \citep{Carilli+1996AandARv}. Our optical depth maps show $\tau<0.01$, thus validating the assumption that our jets should be optically thin.
The emissivity from each computational zone is calculated following a prescription similar to that by \cite{Duffell-Kasen2016arXiv}. However, we assume that the magnetic field is only introduced by the engine and is not significant in the ambient medium. Therefore only material originating from the jet can have nonzero emissivity. We implement this by using our passive scalar X. The magnetic field strength is assumed to decrease with distance from the central engine as $B(r)\propto r^{-2}$. This emissivity is multiplied by the following factor to account of relativistic beaming \citep{Urry+1995PASP}:
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{N_vs_lambda.png}
\caption{Scaling of number of knots vs $\lambda$. The black dots denote the number of knots observed in the emissivity maps. The blue dashed line corresponds to the $N = \lambda^{-1/6}$ curve. The dependence on $\theta_0$ is weak and is accounted for by 95\% credible interval shaded in blue.}
\label{fig:Nscaling}
\end{figure}
\begin{equation}
D = \left[\frac{1}{\Gamma(1-\vec{\beta}\cdot\hat{n})}\right]^{2+\alpha}
\end{equation}
where $\Gamma$ and $\vec{\beta}$ are the Lorentz factor and velocity of the cell, $\hat{n}$ is the unit vector along the line of sight, and $\alpha$ is the spectral index for synchrotron emission, taken here to be equal to 0.75. We also considered a constant magnetic field and a distance dependence $B(r)\propto r^{-1}$, and found our results are not significantly affected by this choice.
The emissivities are integrated along the line of sight to produce a 2D map of the jet. A dynamic range (in this case, ratio of the maximum pixel intensity to the minimum pixel intensity) is imposed on the maps for them to resemble observations. We choose a dynamic range of $\sim10^4$, as is typical for VLA extended radio emission images of kpc scale jets.
\section{Results} \label{sec:results}
\subsection{General morphological features}
\begin{figure*}
\centering
\gridline{\fig{emiss_t20e3e6.png}{0.5\textwidth}{(a)}
\fig{emiss_t20e3e4.png}{0.5\textwidth}{(b)}
}
\gridline{
\fig{emiss_t20e3e2.png}{0.5\textwidth}{(c)}
\fig{emiss_t20e3e0.png}{0.5\textwidth}{(d)}
}
\includegraphics[width=0.8\textwidth]{colorbar_emiss.png}
\caption{Panels (a)-(d) show the emissivity contour plots for jets with $\theta_0 = 0.20$ and $\lambda$ = $7.5\times10^{-5}$, $7.5\times10^{-3}$, $7.5\times10^{-1}$ and $7.5\times10^{1}$, respectively. The viewing angle is $90^{\circ}$ in each case. Multiple recollimation shocks are seen in panels (a) and (b), while panels (c) and (d) exhibit one or no recollimation shocks. Each contour level is brighter than the preceding contour level by a factor of $\sqrt{2}$. The brightest contour level is a factor of $10^4$ times brighter than the base contour level for all maps, in accordance with the typical dynamic range of VLA observations. The numbers on the axes denote distances along them in units such that $\mathrm{z_h} = 1$. The quantities $\epsilon_{min}$ and $\epsilon_{max}$ denote the minimum and maximum emissivity per pixel, respectively.
\label{fig:emissivity_maps}}
\end{figure*}
Our fiducial runs, assuming an inverse square power-law atmosphere, show similar qualitative features as predicted by \citetalias{Bromberg+2011ApJ} . Fig.~\ref{fig:example} shows an example plot for a self-similar jet with a low value of $\lambda$. An overpressured cocoon forms around the jet. The jet is seen to be well-collimated, with many recollimation shocks along the axis. The variation of the nature of the jet-cocoon system with increasing $\lambda$ is seen in Fig.~\ref{fig:zoo}. Here we show a subset of our runs with $\theta_0=0.20$ and $k=2$ for a range of engine luminosities. The cocoon seems to be a common feature in all the jets. However, the degree to which jets are collimated decreases with $\lambda$ as the jet head grows wider and the first recollimation shock moves farther from the engine. This trend has been also been reported by previous numerical RHD computations, e.g., \cite{Yates+2018MNRAS}. Almost all the low $\lambda$ jets exhibit multiple recollimation shocks.
\subsection{Scalings of dynamical parameters}
Fig.~\ref{fig:scalings} shows the scalings for jet head velocity, cocoon transverse expansion velocity, jet opening angle, and the ratio of the cocoon pressure to the ambient density at the jet head. The left half of the panels show the scalings with $\lambda$ for $\theta_0=0.2$ radians. They also depict the dynamical regimes from Table~\ref{t:lambdascaling}, which are marked with three different colors (yellow, blue, and gray). It is seen that the transverse expansion velocity of the cocoon ($\beta_c$) scales almost exactly as predicted by \citetalias{Bromberg+2011ApJ} . In contrast, our observed scaling of the jet opening angle ($\theta_j$) appears to deviate from that predicted by \citetalias{Bromberg+2011ApJ} for lower values of $\lambda$. We obtain higher values for $\theta_j$ than is expected by \citetalias{Bromberg+2011ApJ} and $\theta_j$ seems to flatten to a minimum value. It can also be noticed that our measured values of the jet head's four-velocity $\Gamma_h \beta_h$ may indicate a slight deviation from the prediction by \citetalias{Bromberg+2011ApJ} in the intermediate ``Relativistic Head" regime. We further discuss these discrepancies in Section~\ref{subsec:previous_works}.
\subsection{Occurrence of bright spots in the jet}
Bright, non-terminal spots (called knots) are routinely observed in extended emission from AGN jets, at both radio and X-ray wavelengths. Some of the prominent examples are M87 \citep{Marshall2002ApJ}, 3C 273 \citep{Marchenko2017ApJ} and OJ287 \citep{Marscher2011ApJ}. These knots are often attributed to site of particle acceleration in strong shocks in the outflow. These shocks could be formed hydrodynamically by recollimation of the jet by the ambient medium \citep{Komissarov+1998MNRAS}, or from inhomogeneity introduced in the jet by intermittent engine activity \citep{Stawarz2004ApJ}, or by instabilities in the outflow \citep{Micono+1999A&A}. Other alternative theories for knot formation in jets include non-uniform Doppler boosting and sudden large scale expansion in the outflow \citep{Harris2010IJMPD}.
We observe multiple bright spots in some of our synthetic emissivity maps coincident with locations where the recollimation shock(s) converge to the jet axis, resulting in a local pressure maximum. The number of these bright spots or knots can be obtained via visual inspection of the emissivity maps, or by calculating the number of local maxima of pressure along the jet axis. We find both methods to be consistent. The number of knots is found to decrease with increasing $\lambda$, obeying a weak scaling of $N \approx \lambda^{-1/6}$. The effect of varying the injection angle $\theta_0$ is negligible, provided $\lambda$ is held constant. This relationship therefore provides an independent consistency check on the estimated value of $\lambda$ for an AGN jet, using the number of knots observed. A plot of the number of knots in a jet against the jet $\lambda$ value is shown in Fig. \ref{fig:Nscaling}, along with our predicted scaling rule for the same and a 95\% confidence interval.
\subsection{Shapes of AGN jets}
A subset of the synthetic emissivity maps for a range of $\lambda$ is shown in Fig.~\ref{fig:emissivity_maps}. It is seen they exhibit two distinct types of morphology: the low-$\lambda$ systems show a distinct jet structure with one or more knots, while only the bright jet head is visible in high-$\lambda$ systems. This is strongly reminiscent of the Fanaroff-Riley dichotomy. The original suggestion was that center-brightened or FRI galaxies have lower radio luminosities, while the edge-brightened or FRII galaxies have higher radio luminosities. It has been since suggested that the FRI/FRII morphological divide may instead be related to both the jet power and the environmental density (see \cite{Hardcastle+2020NewAR} and references therein). A jet with a given luminosity may remain relativistic and terminate in a bright hotspot in a poor environment, but may decelerate due to entrainment of ISM in a denser environment, resulting in an FRI structure. Our result strongly corroborates with this hypothesis. We find that such a morphological dichotomy should also depend on the injection angle $\theta_0$. This is depicted in Fig.~\ref{fig:morphology_distinction}, where we plot the two different types of morphology shown by our results on a $\theta_0$ vs $\lambda$ diagram. Fig.~\ref{fig:morphology_distinction} also shows the approximate regions most likely to be occupied by the AGN jets in M87 and Cygnus A based on estimates of their $\lambda$ values. This is described in detail in Section~\ref{subsec:observations}.
The demarcation observed in Fig.~\ref{fig:morphology_distinction} can be approximately expressed in terms of $\lambda$ and $\theta_0$, or equivalently in terms of $\Tilde{L}$ and $\theta_j$ as follows:
\begin{equation}
\lambda_{\mathrm{crit}} \theta_0^4 \lesssim 3\times10^{-4} \label{eq:FRC_lambda}
\end{equation}
or,
\begin{equation}
\Tilde{L}_{\mathrm{crit}}^{1/4} \theta_j^3 \lesssim 2\times10^{-3} \label{eq:FRC_ltilde}
\end{equation}
In other words, AGN jets should exhibit an FRI morphology if they satisfy inequalities~\ref{eq:FRC_lambda} or~\ref{eq:FRC_ltilde}, and an FRII morphology otherwise. The region of the parameter space where the FRI/II transition happens corresponds to a collimated jet with a relativistic head. Therefore we assume relations between $\lambda$, $\Tilde{L}$, $\theta_0$ and $\theta_j$ appropriate to that regime while deriving equation~\ref{eq:FRC_ltilde} from equation~\ref{eq:FRC_lambda}.
\subsection{Environments that Break Self-Similarity} \label{sec:k1}
\begin{figure*}
\centering
\includegraphics[width=0.85\textwidth]{shape_distinction.png}
\caption{Overall morphology of all self-similar solutions. Green plus signs represent a completely visible jet morphology, as seen in panels (a) and (b) in Fig.~\ref{fig:emissivity_maps}. Red cross signs indicate morphology dominated by a prominent hotspot with a somewhat visible cocoon but no visible jet. Panel (c) and (d) in Fig.~\ref{fig:emissivity_maps} are examples of such a morphology. The thick black line represents $\lambda_{\mathrm{crit}}(\theta_0)$ (Inequality \ref{eq:FRC_lambda}), the criterion that distinguishes FRI jets from FRII jets. The blue square-hatched and orange cross-hatched regions represent the approximate location that M87 and Cygnus A (respectively) are most likely to occupy on this diagram. }
\label{fig:morphology_distinction}
\end{figure*}
AGN are usually observed in host environments with density profiles shallower than $\rho(r)\propto r^{-2}$ ($k\sim1\mbox{--}1.5$, as explained in Section~\ref{subsec:IC}). It can be seen from equation \ref{eq:lambda} that the value of $\lambda$ should decrease with time for a jet advancing through such an environment. Since the jet head velocity and the jet opening angle depend on $\lambda$, the jet decelerates and changes shape as it advances. This implies AGN jets may not evolve self-similarly.
We investigate whether the scalings and general morphological trends obtained from our self-similar models hold for jets in observed AGN host environments. This is done using a single run, corresponding to the last row in Table \ref{t:engine}, where $k=1$ and $\theta_0=0.2$. The jet starts out with a high $\lambda_{\mathrm{eff}}$ value and is evolved for nine orders of magnitude in time. $\lambda_{\mathrm{eff}}$ is inversely proportional to the jet length $z_h$ in this case according to equation \ref{eq:lambda} since $\rho_a(z_h)\propto z_h^{-1}$. Thus $\lambda_{\mathrm{eff}}$ decreases as the jet grows in size. The duration of evolution is long enough to let $\lambda_{\mathrm{eff}}$ cover a range of values comparable to the range of $\lambda$ for the self-similar models (with $k=2$ and $\theta_0=0.2$).
The jet head position is measured at closely spaced instants of time. The numerical derivative of the head position with respect to time provides us the head velocity as a function of time, or equivalently as a function of $\lambda$. Fig.~\ref{fig:k1_dynamics} shows a comparison between the dependences of the proper head velocity on $\lambda$ for the self-similar jet (k=2) and the decelerating jet (k=1). The self-similar results appear to be applicable to non-self-similar jets (k=1). This suggests that the solution roughly transitions from high-$\lambda$ to low-$\lambda$ solutions consistent with our self-similar results, even though the jet does not evolve self-similarly in this case. This shows that the dynamics of a jet does not significantly depend on how $\lambda$ changes with time. Thus, the instantaneous value of $\lambda$ for a jet is enough to characterize its dynamical properties. Interestingly, the time evolution of $\lambda$, as indicated by the upper x-axis in Fig.~\ref{fig:k1_dynamics} suggests that FRII sources should transform into FRI sources given sufficient time so that $\lambda$ falls below unity. A similar conclusion was reached by \cite{Li+2018ApJ}. We use Inequality~\ref{eq:FRC_lambda} or~\ref{eq:FRC_ltilde} assuming parameters relevant to Cygnus A to deduce what length such a jet has to grow to before such a transition occurs. We find that even though Cygnus A barely satisfies our FRII criterion, it has to grow from $\sim70$ kpc to 700 kpc before we expect it to show FRI morphologies. Hence, we do not expect to observe transition of known FRII sources to FRI.
\section{Discussion} \label{sec:discussion}
\subsection{Comparison with observations} \label{subsec:observations}
We obtain a rough estimate of $\lambda$ for two archetypal nearby FRI and FRII jets, in the radio galaxies M87 and Cygnus A, respectively to see where they lie on Fig. \ref{fig:morphology_distinction}. Equation \ref{eq:lambda} is used to express $\lambda$ as follows:
\begin{equation}
\label{eq:M87-lambda}
\begin{split}
\lambda \approx 2\times10^{-2} \times \left(\frac{L}{10^{43}\,erg\,s^{-1}}\right) \times \left(\frac{\rho}{10^{-28}g\,cm^{-3}}\right)^{-1} \\ \times \left(\frac{z_h}{1.5\,kpc}\right)^{-2} \times \left(\frac{\theta_0}{0.1\,rad}\right)^{-2}
\end{split}
\end{equation}
The value of $\lambda$ for the jet in M87 is calculated to be approximately equal to $\sim2\times10^{-2}$, using observations for the kiloparsec scale jet \citep{Biretta+1995ApJ}, density in the neighborhood of the M87 nucleus \citep{Russell+2015MNRAS}, and jet power estimates \citep{Stawarz+2006MNRAS,Russell+2013MNRAS}. Cygnus A, on the other hand, has a $\lambda$ value of $\sim 9$, calculated from a jet power estimate $\sim 10^{46}\,\mathrm{erg\, s^{-1}}$, jet head density $\sim 10^{-31}\,\mathrm{g\, cm^{-3}}$ \citep[][and references therein]{Godfrey+2013ApJ,Snios+2018ApJ} and a core to hotspot distance of $\sim 70\,\mathrm{kpc}$ \citep{Carilli+1996AandARv}.
The major sources of error in our estimates for $\lambda$ are from uncertainty in jet power estimates and our ignorance about the engine opening angle. We expect each of these to contribute an uncertainty of a factor of few. For example, the jet power values used in this work have been estimated by dividing the amount of mechanical work done by the jet to inflate the radio lobes observed around it by the buoyancy timescale of the lobes \citep{Birzan+2004ApJ}. This estimate neglects radiated power and suffers from time-averaging effects as well as uncertainties in lobe volume or pressure calculation. In case of M87, additional uncertainty of at most a factor of 2 comes from the position of the jet head. We assume it to be co-spatial with the outermost knot (knot C). It's unclear if the jet corresponding to our models extends beyond this region, as seen in high resolution radio images of M87 \citep[e.g., Fig. 1 of][]{Biretta+1995ApJ}. Nonetheless, our results therefore correctly classify the Fanaroff-Riley morphology of the jets in M87 and Cygnus A. This is demonstrated in Fig.~\ref{fig:morphology_distinction}, where we plot the approximate locations of M87 and Cygnus A. The prediction of our classification criterion (Equation~\ref{eq:FRC_lambda}) holds even allowing for the error in jet power and engine opening angle estimates. We calculate whether M87 and Cygnus A satisfy inequality~\ref{eq:FRC_ltilde} as a sanity check, since $\theta_j$ and therefore $\Tilde{L}$ are usually easier to estimate from observations and find our predictions to be consistent.
Our prediction for the number of knots in the jet of M87, from Fig.~\ref{fig:Nscaling} turns out to be $\sim 4$. This is somewhat less than the 6 or 7 resolved knots seen in the kpc scale jet, but may be attributed to the uncertainty in our knowledge of $\lambda$. Moreover, the $\lambda$ value for the jet in Cygnus A is very close to the threshold value, which indicates it may exhibit multiple recollimation shocks or knots characteristic of FRI jets. That is exactly what is seen in the inner kpc scale or parsec scale images of Cygnus A \citep[figure 4 of][]{Carilli+1996AandARv}.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{k1_gammabeta.png}
\caption{Jet head proper velocity ($\Gamma_h\beta_h$) as a function of $\lambda_{\mathrm{eff}}$ for the jet expanding against the k=1 density profile, denoted by the blue solid line. The black dots show $\Gamma_h\beta_h$ for the self-similar (k=2) jets as a function of $\lambda$. The upper x-axis values show the time (in code units) it took the k=1 jet from starting to attain the corresponding value of $\lambda_{\mathrm{eff}}$ on the lower x-axis. }
\label{fig:k1_dynamics}
\end{figure}
\subsection{Comparison with previous works} \label{subsec:previous_works}
Our scaling relations shown in Fig~\ref{fig:scalings} indicate that $\theta_j$ almost stays constant for low values of $\lambda$ as opposed to the $\theta_j \propto \lambda^{1/6}$ scaling predicted by \citetalias{Bromberg+2011ApJ} for $\lambda<\theta_0^2$. Therefore jet collimation appears to be increasingly less effective compared to the analytical expectation as $\lambda$ or $\Tilde{L}$ decreases. The opening angle that the system converges to at low $\lambda$ is seen to be a function of $\theta_0$ and is consistent with the scaling $\theta_0^{1/2}$. We investigated the possibility that this disagreement arose simply due to our inability to resolve the inner zones close to the recollimation shock. \cite{Harrison+2018MNRAS} formulate a criterion for their numerical solutions to ensure the nozzle size is adequate for capturing jet dynamics accurately. They find that the calculations are consistent as long as the first recollimation shock radius and height are much greater than the injection radius and height, respectively. We found all our solutions satisfy this criterion. Additionally, we repeated one of our calculations at a low value of $\lambda$ ($2.5\times10^{-4}$) with an inner boundary and an injection radius both smaller by a factor of 2 than the rest of the runs. The same value was found for $\theta_j$, suggesting the jet is indeed less collimated than expected. One possibility is that in this very low-$\lambda$ regime the solution might revert to entirely non-relativistic scalings (even though the jet core is still relativistic), which would imply a jet and cocoon morphology completely independent of $\lambda$. So far this idea is speculative, but may be worth investigating in a follow-up study.
The jet head's four-velocity $\Gamma_h \beta_h$ has a very shallow dependence on $\lambda$ in the intermediate regime. \citetalias{Bromberg+2011ApJ} predict a very weak scaling in this regime, and our points have sufficient scatter that they are still consistent with \citetalias{Bromberg+2011ApJ} , but they are equally consistent with no dependence on $\lambda$ at all in this regime, only depending on $\theta_0$. It would be interesting to investigate this in more detail in a future study, as very minor changes in the scaling of $\theta_j$ with $\lambda$ would be sufficient to create such a flat scaling.
The recollimation shock ``trains" seen in our solutions have been reported by previous numerical studies of relativistic jets, such as \cite{Perucho+2007MNRAS} and \cite{Saxton2010MNRAS}. The latter show that regularly spaced knots are formed in a supersonic jet moving against constant density ambient medium. However, they predict that this spacing should decrease if the ambient density falls off with distance. We obtain roughly constant knot spacing in our relativistic jet models for a given engine luminosity and opening angle irrespective of the ambient density profile, as is seen in observations \citep{Godfrey2012ApJ}. We also obtain the explicit scaling of the number of knots with the system parameters.
Our solutions exhibit the major dynamical and morphological features seen in long term evolution using numerical RHD computations of both FRI \citep{Perucho+2007MNRAS} and FRII \citep{Perucho+2019MNRAS,Perucho+2022MNRAS} jets. In particular, the zoo of our jet models show a morphological dichotomy that primarily depends on $\lambda$ and weakly on $\theta_0$. The emissivity map of the high $\lambda$ jets is dominated by the jet head, while the low $\lambda$ jets exhibit the complete jet structure, often accompanied by several bright knots on the jet axis. This is consistent with the Fanaroff-Riley dichotomy, which has been suggested to be governed primarily by the jet power in some numerical studies \citep{Li+2018ApJ,Seo+2021ApJ}. We argue that the deciding parameter is $\lambda$, or the ratio of jet power to ambient density. \cite{Li+2018ApJ} also examine the impact of jet speed and the jet to surroundings density contrast and show that for a given jet power, jets with low density in the jet core ($\rho_j$) and/or low Lorentz factor ($\Gamma_j$) should exhibit FRI morphology, while high density and/or high Lorentz factor jets should form FRII morphologies. This is in rough agreement with our results, since $\rho_j$ and $\Gamma_j$ are directly proportional to $\Tilde{L}$ or equivalently $\lambda$, according to equation \ref{eq:Ltilde}.
\subsection{Scope for future work} \label{subsec:future_work}
It would be interesting to study some important aspects of AGN jets in the future which have not been addressed in this work. We don't take into account the role of magnetic fields in collimating the jet.
We consider the well-studied jet in M87 to determine if the magnetic field is dynamically important at kpc scales. \cite{Stawarz+2005ApJ} show using $\gamma$-ray observations that flux from the brightest knot indicates a magnetic field strength of $30\mbox{--}100\,\mathrm{\mu G}$. As mentioned in Section~\ref{subsec:emissivity}, our solutions require an equipartition parameter of ~0.02 to reproduce this value. This implies magnetic fields are not dynamically important for our solutions, which apply to kpc scale jets.
But there may be a deviation from our predictions in Section~\ref{sec:k1} due to magnetic fields for smaller scale jets. Additionally, magnetic fields may play an important role in producing the Fanaroff-Riley dichotomy, as has been suggested by \cite{Tchekhovskoy+2016MNRAS}.
We do not solve for 3D jets. Some numerical studies suggest that there may be some differences in the jet head and the jet spine structure for 2D and 3D models. \cite{Rossi+2008A&A} and \cite{Matsumoto+2021MNRAS} observe that mixing between the jet and the cocoon disrupts the jet spine structure for some of their 3D jet models, especially upstream of the first recollimation shock. Thus these models do not show the recollimation shock train structure that we'd expect from our 2D models. Similar differences pertaining to jet-cocoon mixing and jet head structure are seen by \cite{Harrison+2018MNRAS} who compare 2D and 3D jet calculations in the context of calibrating the analytical model by B11. However, they find that the general morphology of the jet, the cocoon and recollimation shocks remain similar for 2D and 3D jet models. Some other numerical studies also report multiple recollimation shocks in 3D hydrodynamic jet models, e.g., \cite{Zhang+2004ApJ} in the context of a jet propagating through a massive star, \cite{Mukherjee+2018MNRAS} in the context of AGN jets propagating through a turbulent galactic disk and the galactic halo, and \cite{Wang+2008ApJS} for relativistic jets in general. \cite{Rossi+2008A&A} note that formation of multiple recollimation shocks seems more plausible for their relativistic jet models as opposed to non-relativistic ones.
In general, recollimation shocks are seen to be weaker in 3D than in 2D studies, and the conical structure seen at the jet head seems to be purely a 2D artifact. Thus it is unclear if the recollimation shock train seen in our 2D models would remain the same in 3D simulations. It is also unclear how the emissivity map of strong jets with bright heads would look like for 3D solutions, which haven't been calculated in this work.
Lastly, the dynamics and morphology of AGN jets may depart from our constant luminosity jet models if engine variability is taken into account, which is observed for AGN on time scales ranging from minutes to years.
\section{Conclusion} \label{sec:conclusion}
We have obtained self-similar models of relativistic jets expanding against ambient medium for a wide range of system parameters. We have compared these results to the long-term evolution of a jet in a cluster-center like ambient medium (with a density profile $\rho(r) \propto r^{-1}$), where the jet no longer expands self-similarly, and found our self-similar results to remain applicable for suitable definitions of the parameters $\lambda$ and $\theta_0$. Thus, the dynamics of the system should only depend upon the instantaneous values of $\lambda$ and $\theta_0$ and not on the history of evolution as long as the density profile can be modeled as a modest power-law. This eliminates the need for evolving the jet over many orders of magnitude in time, allowing us to infer dynamical parameters of AGN jets by measuring the instantaneous value of $\lambda$ through observations.
The model developed by \citetalias{Bromberg+2011ApJ} has been found to hold for relativistic jets with some minor modifications to the jet opening angle scaling. In particular, the jet opening angle seems to not decrease with $\lambda$ for non-relativistic jets (equivalently, for low values of $\lambda$) but attains a constant value at a low enough $\lambda$. We do not yet have an explanation for this behavior at low $\lambda$, but it suggests an upper limit to the degree of collimation the jet can experience via the surrounding cocoon.
We obtain a criterion for the Fanaroff-Riley classification of AGN jets that is dependent not only on the jet power but also its opening angle and surrounding medium. FRI jets, in addition to being center-brightened, exhibit multiple bright spots called knots along their axis. Our calculations recover this result as a purely hydrodynamical effect, without needing to include magnetic fields. We find that the number of knots in FRI jets scales with $\lambda$ ($N(\lambda)\propto\lambda^{-1/6}$). This provides a consistency check on the idea these knots are recollimation shocks. M87 and Cygnus A seem to be in rough agreement with this check, although agreement in the case of M87 would require the jet power to be on the low end of observationally inferred values.
\acknowledgments
We acknowledge M. Lister, E. Nakar, and K. Blundell for helpful comments. We thank the anonymous referee for useful suggestions. High-resolution calculations were provided in part by the resource Stampede2 through the local allocation awards \#TG-AST190029 and \#TG-AST190033 in the Extreme Science and Engineering Discovery Environment (XSEDE) supported by National Science Foundation grant number ACI-1548562 \citep{xsede}. We also acknowledge access to the resource Bell at the Rosen Center for Advanced Computing (RCAC) of Purdue University \citep{McCartney2014}.
\bibliographystyle{apj}
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Tear severity can vary from minor to drastic. This, in conjunction with previous overuse, can cause micro-tears that accumulate over time. When these minor tears can't repair fast enough to heal, they accumulate and result in a full-blown muscle tear.
Our gentle chiropractic care integrates specific muscle rehabilitation, muscle sequencing, functional movement and lasting change.
If you have chronic pain, weakness or loss of strength, trouble with movement, weight loss issues, achy muscles, or you're a high-performance athlete looking for a very specific analysis of a sport-specific movement, we can put you back in action faster, and make sure you stay pain-free.
Neuromuscular rehabilitation is our best, and fastest, results-based therapy, combining chiropractic and functional rehabilitative exercise. We work on reinforcing positive patterns to strengthen your joints and adjacent muscles. By balancing where your strength comes from your muscles support one another. This treatment heals you and prevents the injury from re-occuring.
When a muscle spasms, it is often a signal that the muscle is weak, but that isn't always the case. Strong muscles can also spasm with in specific muscle spindles.
When these muscle spasms don't release due to poor blood flow and toxins reinforcing the spasm, a tearing or pulling sensation is often followed by pain. The body no longer has its full range of motion, so movement becomes inefficient.
Untreated muscle spasms can lead to a snowball effect. For example, upper back pain is fairly common among Americans. What you may not realize is that muscle pain originating in the upper back can cause headaches. Those headaches, in turn, can lead to debilitating chronic neck pain. Eventually, your neck may be killing you but that pain is masking the true source—your upper back.
Without functional movement chiropractic care to help you discover the true source of your pain, you are likely to treat your neck (where it hurts) rather than your upper back (the source). You may gain temporary relief, but you'll avoid lifelong healing.
Our chiropractic care is uniquely based on functional movement to help you discover the true source of your pain. By looking at your body and mind as a complete system, you'll heal your upper back (the source of your pain), rather than your neck (where you feel the pain). Instead of temporary relief, which is all some doctors provide, I can help you feel better and enjoy lifelong healing.
It may come as a surprise, but headaches are often the result of an underlying injury or accident. As a chiropractor, I can get to the root cause of your headaches, and yes, even get rid of migraines that have plagued you for years.
Craniosacral therapy releases tension in your body and the bones of your cranium and face, giving you immediate relief. Paired with a functional assessment, I can pinpoint the root cause of your pain and permanently cure your chronic headaches.
Yes, chiropractors can treat vertigo and dizziness! We use craniosacral therapy to release tension in the body and bones of your cranium and face. By improving your cerebral spinal fluid flow and the blood flow in your head and neck, we can improve your brain function and effectively treat vertigo and dizziness or correct imbalance issues.
We accept health savings accounts (HSA). We if you would like reimbursement for your insurance we can provide the paperwork to send it to your insurance company to see if it is covered. We currently process Automotive Insurance internally. | {
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} | 7,354 |
Interview with TV Chef Nigella Lawson
F&W talks to Nigella Lawson, host of Food Network's Nigella Feasts.
By Jen Murphy
What was your first big break on food TV? How were you discovered?
"My big break in terms of food programming was completely by chance. I appeared on this Christmas special with a good friend of mine who's a really fabulous food writer, Nigel Slater. And he is very, very, very shy, so he thought he'd feel more comfortable if there was someone there to chat with while he was doing his thing. So I went on to be this cozy person alongside him, and then I got a series of my own after that."
Did someone approach you with the idea for the show, or did you help devise it?
"The idea for the first show came about because I wrote my first book, and a glossy UK magazine wanted to run an excerpt. The book had no pictures, so we did a shoot in my home for the magazine. We really had fun; I had my kids there, and one photo was my feet in shocking-pink kitten heels, and some were of food, and the whole feel of it was warm and messy and fun. A commissioning editor at a network saw this shoot and loved it and wanted to buy an option on the book, and I said, 'I don't want to do TV, forget it.' And he said, 'Please, please, we'll do it the way you want,' and he was very patient and let me do it my way and let me spend so much time on the pilot. When I say 'my way' I mean unscripted, letting me go into the kitchen and babble on and—you know, the whole 'one camera, one house, someone talking.' Back then it was quite new. And it all happened because this very, very nice editor liked the picture of the pink kitten heels and knew it was for him, and he always tells people that."
What are some of the best recipes you've made on-air, and why?
"Sometimes the things that I think make good recipes when I write a book don't translate to TV. TV is very process-driven. It's good in life or in a book to do a recipe in one minute—stick it in the oven and that's the end. But that's not much for the TV viewing public to see. What works for TV recipes is something that transforms in open space, like in a skillet. When I write a book, the words really inhabit the page and get behind the page. But on TV, you want the viewer to have the same experience as you in the kitchen, and you need to unite visually. I do a running commentary and really try to use words in a very precise way and try to be evocative with my language.
I'm lucky because I have a fabulous crew: My cameraman is so good, and he can really get right up close to the food and show how easy and wonderful it is. You see the food so clearly that you can almost smell it, feel it, touch it. For the food to come to life when you're watching TV is the goal. When I lean over, I have a mic on me so it can catch the sounds—every sizzle, crack and pop has to be as clear for the viewers as if they were in my kitchen. When you lose the taste you lose a lot, so the other senses available through the TV medium have to make up for that. It's no good if the viewer feels the camera is too far away or didn't linger long enough on the beautiful mozzarella slices. I did a spaghetti carbonara, and the camera got so close to the cubes of bacon when I was frying them-the smell in heaven must be the smell of frying bacon. The poor viewers: Since they can't smell it, I tantalize them with the sizzle of it coming out of the TV speakers."
Are there any specific recipes that you think worked really well?
"I made a one-pan chicken, sausage and sage bake that is so simple. If I were just to say to you, 'Take the chicken pieces, put them in a freezer bag with the marinade, then put the sausage on,' well, that doesn't sound grabby. But if I do it on TV, the camera gets up close on my hand on the bag, and you can hear the squishing noise when I move the chicken around in the lemon juice, mustard, garlic, Worcestershire sauce and white wine vinegar. So you hear it, and you can see that it's not hard work. And then suddenly you see the sausages go on and it looks exciting, and then the next shot is the dish coming out of the oven looking golden—that's three simple steps. It looks a lot more exciting than just explaining the method. I'm proud of this dish because my sound man had his mother-in-law coming for the weekend, and he cooked it for her without a recipe. That for me is something only TV can do: He watched and saw how simple it was, and he was inspired to do it. Those three simple frames relay a whole lunch."
What distinguishes you from other TV chefs?
"I can only be me, so I think what you get from me is, in a way, negative qualities—and not in disingenuous way. I have no training as a cook. I cook for friends and family, so when I chop something, I chop like a normal person, not quick and not pretty. A great chef can be inspirational but also intimidating: Sometimes they do things so fast you can't see. And sometimes I get stressed because I don't have great dexterity, but in the end, I swallow my pride and say I can't do it in this way. But it tastes good, and it doesn't make a difference in the end if it tastes good. To be honest, your food is there to eat. You can make it look pretty in all beautiful ways, but I love the messiness. I find it warming. And I love big bowls on the table; I don't want to be plating individual dishes."
What is the worst experience you've had on TV? Any disasters?
"I did set fire to something once, and thanks to my lovely, loyal cameraman, the director never noticed. For some reason, I had paper towels and tea towels near a flame, and they caught fire and got knocked onto the stone floor. The cameraman somehow managed to stomp it out with his foot while continuing to shoot, even though the camera weighed a ton. The worst is when I go on talk shows to do demos. It's so frightening—I'm almost weeping and sobbing as I go on. When you go on with someone funny, like David Letterman, I don't know what I'm meant to be doing: It's a cooking demo, you're chatting, and you have to be prepared to let someone make fun of your recipes. That part I got used to—I always say I have an older brother, so I can handle being teased—but I always get butterflies in my tummy. Letterman is the scariest—he's a genius and his show is brilliant, but I find him a bit detached, and I need to have a rapport." | {
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} | 3,146 |
\section{Introduction}
The aim of this paper is to prove the analytic dependence of the solution of a periodic Neumann problem for the Laplace equation, upon joint perturbation of the domain, the periodicity parameters, the Neumann datum, and its integral on the boundary. The domain is obtained as the union of congruent copies of a periodicity cell of edges of length
$q_{11},\ldots,q_{nn}$ with a hole whose shape is the image of a reference domain through a diffeomorphism $\phi$. As Neumann datum we take the projection of a function $g$, defined on the boundary of the reference domain and suitably rescaled, on the space of functions with zero integral on the boundary. As it happens for non-periodic Neumann problems, in order to identify one solution, we impose that the integral of the solution on the boundary is equal to a given real constant $k$. By means of a periodic version of potential theory, we prove that the solution of the problem depends
real analytically
on the `periodicity-domain-Neuman datum-integral' quadruple $((q_{11},\ldots,q_{nn}), \phi,g,k)$.
Many authors have investigated the behavior of the solutions to boundary value problems upon domain perturbations.
We mention, e.g., Henry \cite{He82} and Sokolowski and Zol\'esio \cite{SoZo92} for elliptic domain perturbation problems. Lanza de Cristoforis \cite{La05, La07} has exploited potential theory in order to prove that the solutions of boundary value problems for the Laplace and Poisson equations depend real analytically upon domain perturbation. Moreover, analyticity results for domain perturbation problems for eigenvalues have been obtained for example for the Laplace equation by Lanza de Cristoforis and Lamberti \cite{LaLa04}, for the biharmonic operator by Buoso and Provenzano \cite{BuPr15}, and for the Maxwell's equations by Lamberti and Zaccaron \cite{LaZa21}.
In order to introduce our problem, we fix once for all a natural number
\[
n \in \mathbb{N} \setminus \{0,1\}\,
\]
that represents the dimension of the space.
If $(q_{11},\ldots,q_{nn}) \in \mathopen]0,+\infty[^n$ we define a periodicity cell $Q$ and a matrix $q \in {\mathbb{D}}_{n}^{+}({\mathbb{R}})$ as
\
Q \equiv \prod_{j=1}^n \mathopen]0,q_{jj}[, \quad
q \equiv
\begin{pmatrix}
q_{11} & 0 & \cdots & 0 \\
0 & q_{22} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & q_{nn}
\end{pmatrix},
\
where ${\mathbb{D}}_{n}({\mathbb{R}})$ is the space of $n\times n$ diagonal matrices with real entries and ${\mathbb{D}}_{n}^{+}({\mathbb{R}})$ is the set of elements of
${\mathbb{D}}_{n}({\mathbb{R}})$ with diagonal entries in $]0,+\infty[$. Here we note that we can identify ${\mathbb{D}}_{n}^{+}({\mathbb{R}})$ and $]0,+\infty[^n$.
We denote by $|Q|_n$ the $n$-dimensional measure of the cell $Q$, by $\nu_Q$ the outward unit normal to $\partial Q$, where it exists, and by $q^{-1}$ the inverse matrix of $q$. We find convenient to set
\[
\widetilde{Q}\equiv \mathopen]0,1[^n\, , \qquad \tilde{q} \equiv I_n \,,
\]
where $I_n$ denotes the identity $n\times n$ matrix. Then we introduce the reference domain: we take
\begin{equation}\label{Omega_def}
\begin{split}
&\text{$\alpha \in \mathopen]0,1[$ and a bounded open connected subset $\Omega$ of $\mathbb{R}^{n}$}
\\
&\text{of class $C^{1,\alpha}$ such that $\mathbb{R}^{n}\setminus\overline{\Omega}$ is connected}\, ,
\end{split}
\end{equation}
where the symbol `$\overline{\cdot}$' denotes the closure of a set.
For the definition of sets and functions of the Schauder class $C^{1,\alpha}$ we refer, e.g., to Gilbarg and
Trudinger~\cite{GiTr83}. In order to model our variable domain
we consider a class of diffeomorphisms
${\mathcal{A}}_{ \partial\Omega }^{\widetilde{Q}}$ from $\partial\Omega$ into their images contained in $\widetilde{Q}$ (see \eqref{eq:defA} below).
By the Jordan-Leray separation theorem, if $\phi\in {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}$,
the set
${\mathbb{R}}^{n}\setminus\phi (\partial\Omega)$ has exactly two open
connected components (see, e.g., Deimling \cite[Thm. 5.2, p. 26]{De85}). We denote by ${\mathbb{I}}[\phi]$ the
bounded open connected component of
${\mathbb{R}}^{n}\setminus\phi (\partial\Omega)$.
Since $\phi (\partial\Omega)\subseteq \widetilde{Q}$, a topological argument shows that
$\widetilde{Q}\setminus\overline{\mathbb{I}[\phi]}$
is also connected (cf., e.g., \cite[Theorem A.10]{DaLaMu21}).
We are now in the position to introduce the following two periodic domains (see Figure \ref{fig1}):
\begin{equation*}
\mathbb{S}_{q }[q \mathbb{I}[\phi]] \equiv\bigcup_{z\in {\mathbb{Z}}^{n}}\left(
q z+q {\mathbb{I}}[\phi]
\right), \qquad
\mathbb{S}_{q }[q \mathbb{I}[\phi]]^- \equiv
{\mathbb{R}}^{n}\setminus\overline{\mathbb{S}_{q }[q \mathbb{I}[\phi]]} \,.
\end{equation*}
The set $\mathbb{S}_{q }[q \mathbb{I}[\phi]]^-$ will be the one where we shall set our Neumann problem.
Clearly, a perturbation of $q$ produces a modification of the whole periodicity structure of $\mathbb{S}_{q }[q \mathbb{I}[\phi]]^-$, while a perturbation of $\phi$ induces a change in the shape of the holes $\mathbb{S}_{q }[q \mathbb{I}[\phi]]$.
\begin{figure
\sidecaption
\includegraphics[scale=.18]{qneufig2.pdf}
\caption{The sets $\mathbb{S}_{q}[q\mathbb{I}[\phi]]^-$ (in gray),
$\mathbb{S}_{q}[q\mathbb{I}[\phi]]$ (in white), and $q\phi(\partial\Omega)$ (in black) in case $n=2$.}
\label{fig1}
\end{figure}
If $q \in \mathbb{D}^+_n(\mathbb{R})$, $\phi \in C^{1,\alpha}(\partial\Omega,\mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}$, $g \in C^{0,\alpha}(\partial \Omega)$ and $k \in \mathbb{R}$, we consider the following periodic Neumann problem for the Laplace equation:
\begin{equation}
\label{bvp}
\left\{
\begin{array}{ll}
\Delta u=0 & {\mathrm{in}}\ \mathbb{S}_{q}[q\mathbb{I}[\phi]]^{-}\,,
\\
u(x+qz)=u(x)& \forall x \in \overline{\mathbb{S}_{q}[q\mathbb{I}[\phi]]^{-}}\, , \forall z \in\mathbb{Z}^n\,,\\
\frac{\partial}{\partial \nu_{q\mathbb{I}[\phi]}}u(x)=g\big( \phi^{(-1)}(q^{-1}x)\big)&\\
\qquad \qquad-\frac{1}{\int_{\partial q\mathbb{I}[\phi]}\, d\sigma}\int_{\partial q\mathbb{I}[\phi]} g\big( \phi^{(-1)}(q^{-1}y)\big) \, d\sigma_y& \forall x\in \partial q\mathbb{I}[\phi] \,, \\
\int_{\partial q\mathbb{I}[\phi]}u\, d\sigma=k\, .& \\
\end{array}
\right.
\end{equation}
We note that the function
\[
g\big( \phi^{(-1)}(q^{-1}\cdot)\big)-\frac{1}{\int_{\partial q\mathbb{I}[\phi]}\, d\sigma}\int_{\partial q\mathbb{I}[\phi]} g\big( \phi^{(-1)}(q^{-1}y)\big) \, d\sigma_y
\]
clearly belongs to the space
\[
C^{0,\alpha}(\partial q\mathbb{I}[\phi])_0 \equiv \Big\{\mu \in C^{0,\alpha}(\partial q\mathbb{I}[\phi]) \colon \int_{\partial q\mathbb{I}[\phi]}\mu\, d\sigma=0\Big\}\, .
\]
As a consequence, the solution of problem \eqref{bvp} in the space $C^{1,\alpha}_{q}(\overline{\mathbb{S}_{q}[q\mathbb{I}[\phi]]^{-}})$ of $q$-periodic functions in $\overline{\mathbb{S}_{q}[q\mathbb{I}[\phi]]^-}$ of class $C^{1,\alpha}$ exists and is unique and we denote it by $u[q,\phi,g,k]$ (see \cite[Thm.~12.23]{DaLaMu21}). Our aim is to prove that $u[q,\phi,g,k]$ depends, in a sense that we will clarify, analytically on $(q,\phi,g,k)$ (see Theorem \ref{mainthm}).
Our work originates from Lanza de Cristoforis \cite{La05, La07} on the real analytic dependence
of the solution of the Dirichlet problem for the Laplace and Poisson equations upon domain perturbations. Moreover, this paper can be seen as the Neumann counterpart of \cite{LuMu22}, where the authors have proved analyticity properties for the solution of a periodic Dirichlet problem. An analysis similar to the one of the present paper was also carried out for periodic problems related to physical quantities arising in fluid mechanics and in material science (see \cite{DaLuMuPu21, LuMu20, LuMuPu19}).
\section{Preliminary results}
In order to consider shape perturbations, we introduce a class of diffeomorphisms. Let $\Omega$ be as in \eqref{Omega_def}. Let $\mathcal{A}_{\partial \Omega}$ be the set of functions of class $C^1(\partial\Omega, \mathbb{R}^{n})$ which are injective and whose differential is injective at all points of $\partial\Omega$. The set $\mathcal{A}_{\partial \Omega}$ is well-known to be open
in $C^1(\partial\Omega, \mathbb{R}^{n})$
(see, e.g., Lanza de Cristoforis and Rossi \cite[Lem. 2.5, p. 143]{LaRo04}).
Then we set
\begin{equation}
\label{eq:defA}
{\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}} \equiv \Big\{\phi \in\mathcal{A}_{\partial \Omega} : \phi(\partial\Omega) \subseteq \widetilde{Q}\Big\}.
\end{equation}
In order to analyze our boundary value problem, we are going to exploit periodic layer potentials. To define these operators, it is enough to replace the fundamental solution of the Laplace operator by a $q$-periodic tempered distribution $S_{q,n}$ such that $\Delta S_{q,n}=\sum_{z\in {\mathbb{Z}}^{n}}\delta_{qz}-\frac{1}{|Q|_n}$,
where $\delta_{qz}$ is the Dirac measure with mass in $qz$
(see e.g., \cite[Chapter 12]{DaLaMu21}).
We can take
\begin{equation*}
S_{q,n}(x)=-\sum_{ z\in {\mathbb{Z}}^{n}\setminus\{0\} }
\frac{1}{ |Q|_n4\pi^{2}|q^{-1}z|^{2} }e^{2\pi i (q^{-1}z)\cdot x}
\end{equation*}
in the sense of distributions in $ {\mathbb{R}}^{n}$
(see e.g., Ammari and Kang~\cite[p.~53]{AmKa07}, \cite[\S 12.1]{DaLaMu21}).
Moreover, $S_{q,n}$ is even, real analytic in ${\mathbb{R}}^{n}\setminus q{\mathbb{Z}}^{n}$, and locally integrable in ${\mathbb{R}}^{n}$
(see e.g., \cite[Thm.~12.4]{DaLaMu21}). We now introduce the periodic single layer potential.
Let $\Omega_Q$ be a bounded open subset of ${\mathbb{R}}^{n}$ of class $C^{1,\alpha}$ for some $\alpha\in\mathopen]0,1[$ such that $\overline{\Omega_Q}\subseteq Q$. We define the following two periodic domains:
\begin{equation*}
\mathbb{S}_{q }[\Omega_Q] \equiv\bigcup_{z\in {\mathbb{Z}}^{n}}\left(
q z+\Omega_Q
\right), \qquad
\mathbb{S}_{q }[\Omega_Q]^- \equiv
{\mathbb{R}}^{n}\setminus\overline{\mathbb{S}_{q }[\Omega_Q]} \,
\end{equation*}
and we set
\[
v_q[\partial\Omega_Q, \mu](x)\equiv
\int_{\partial\Omega_Q} S_{q,n}(x-y)\mu(y)\,d\sigma_{y}\qquad\forall x\in {\mathbb{R}}^{n}\,
\]
and
\[
W_q^\ast[\partial\Omega_Q, \mu](x)\equiv
\int_{\partial\Omega_Q}
\nu_{\Omega_Q}(x) \cdot
DS_{q,n}(x-y)\mu(y)\,d\sigma_{y}\qquad\forall x\in \partial\Omega_Q
\]
for all $\mu\in L^{2}(\partial\Omega_Q)$. The symbol $\nu_{\Omega_Q}$ denotes the outward unit normal
field to $\partial\Omega_Q$, $d\sigma$ denotes the area element on $\partial\Omega_Q$ and $DS_{q,n}$ denotes the gradient of $S_{q,n}$.
The function $v_q[\partial\Omega_Q, \mu]$ is called the $q$-periodic single layer potential.
Now let $\mu\in C^{0,\alpha}(\partial\Omega_Q)$. As is well known, $v^+_q[\partial\Omega_Q, \mu]\equiv v_q[\partial\Omega_Q, \mu]_{|\overline{\mathbb{S}_q[\Omega_Q]}}$ belongs to $C_{q}^{1,\alpha}(\overline{\mathbb{S}_{q}[\Omega_Q]})$ and $v^-_q[\partial\Omega_Q, \mu]\equiv v_q[\partial\Omega_Q, \mu]_{|\overline{\mathbb{S}_q[\Omega_Q]^-}}$ belongs to $C_{q}^{1,\alpha}(\overline{\mathbb{S}_{q}[\Omega_Q]^-})$ (see \cite[Thm.~12.8]{DaLaMu21}). Moreover, the following jump formula holds:
\[
\frac{\partial }{\partial \nu_{\Omega_Q}}v_q^\pm[\partial\Omega_Q, \mu] = \mp \frac{1}{2}\mu + W_q^\ast[\partial\Omega_Q, \mu] \qquad \mbox{ on } \partial\Omega_Q.
\]
For a proof of the above formula we refer to \cite[Thm.~12.11]{DaLaMu21}.
Since our approach will be based on integral operators, we need to understand how integrals behave when we perturb the domain of integration. Moreover, we need also to understand the regularity of the normal vector upon domain perturbations. For such reasons, we collect those results in the lemma below (for a proof, see Lanza de Cristoforis and Rossi \cite[p.~166]{LaRo04}).
\begin{lemma}\label{rajacon}
Let $\alpha$, $\Omega$ be as in \eqref{Omega_def}. Then the following statements hold.
\begin{itemize}
\item[(i)] For each $\psi \in C^{1,\alpha}(\partial\Omega, \mathbb{R}^{n})\cap\mathcal{A}_{\partial \Omega}$, there exists a unique
$\tilde \sigma[\psi] \in C^{0,\alpha}(\partial\Omega)$ such that $\tilde \sigma[\psi] > 0$ and
\[
\int_{\psi(\partial\Omega)}\omega(s)\,d\sigma_s= \int_{\partial\Omega}\omega \circ \psi(y)\tilde\sigma[\psi](y)\,d\sigma_y, \qquad \forall \omega \in L^1(\psi(\partial\Omega)).
\]
Moreover, the map $\tilde \sigma[\cdot]$ from $C^{1,\alpha}(\partial\Omega, \mathbb{R}^n)\cap\mathcal{A}_{\partial \Omega} $ to $ C^{0,\alpha}(\partial\Omega)$ is real analytic.
\item[(ii)] The map from $C^{1,\alpha}(\partial\Omega, \mathbb{R}^n)\cap\mathcal{A}_{\partial \Omega} $ to $ C^{0,\alpha}(\partial\Omega, \mathbb{R}^{n})$ which takes $\psi$ to $\nu_{\mathbb{I}[\psi]} \circ \psi$ is real analytic.
\end{itemize}
\end{lemma}
\section{Analyticity of the solution}
Our first goal is to transform problem \eqref{bvp} into an integral equation. In order to analyze the solvability of the obtained integral equation, we need the following lemma.
\begin{lemma}\label{leminteq}
Let $q \in {\mathbb{D}}_{n}^{+}({\mathbb{R}})$. Let $\alpha$, $\Omega$ be as in \eqref{Omega_def}.
Let $\phi\in C^{1,\alpha}(\partial\Omega, \mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}$.
Let $N$ be the map from $C^{0,\alpha}(\partial q\mathbb{I}[\phi])$ to itself, defined by
\[
N[\mu] \equiv \frac{1}{2}\mu+W_{q}^\ast[\partial q\mathbb{I}[\phi], \mu] \qquad \forall \mu \in C^{0,\alpha}(\partial q\mathbb{I}[\phi]).
\]
Then $N$ is a linear homeomorphism from
$C^{0,\alpha}(\partial q\mathbb{I}[\phi])$ to itself. Moreover, $N$ restricts to a linear homeomorphism from
$C^{0,\alpha}(\partial q\mathbb{I}[\phi])_0$ to itself.
\end{lemma}
\begin{proof}
By \cite[Thm.~12.20]{DaLaMu21}, we deduce that $N$ is a linear homeomorphism from
$C^{0,\alpha}(\partial q\mathbb{I}[\phi])$ to itself. By \cite[Prop.~12.15]{DaLaMu21}, we have that $\frac{1}{2}\mu+W_{q}^\ast[\partial q\mathbb{I}[\phi], \mu]$ belongs to $C^{0,\alpha}(\partial q\mathbb{I}[\phi])_0$ if and only if $\mu$ belongs to $C^{0,\alpha}(\partial q\mathbb{I}[\phi])_0$. As a consequence, we also have that $N$ restricts to a linear homeomorphism from
$C^{0,\alpha}(\partial q\mathbb{I}[\phi])_0$ to itself.
\end{proof}
Then, in the following proposition, we show how to convert the Neumann problem into an equivalent integral equation.
\begin{proposition}\label{propAUX}
Let $\alpha$, $\Omega$ be as in \eqref{Omega_def}. Let $q \in {\mathbb{D}}_{n}^{+}({\mathbb{R}})$.
Let $\phi\in C^{1,\alpha}(\partial\Omega, \mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}$.
Let $g \in C^{0,\alpha}(\partial \Omega)$. Let $k \in \mathbb{R}$. Then the boundary value problem
\begin{equation}\label{np}
\left\{
\begin{array}{ll}
\Delta u=0 & {\mathrm{in}}\ \mathbb{S}_{q}[q\mathbb{I}[\phi]]^{-}\,,
\\
u(x+qz)=u(x)& \forall x \in \overline{\mathbb{S}_{q}[q\mathbb{I}[\phi]]^{-}}\, , \forall z \in\mathbb{Z}^n\,,\\
\frac{\partial}{\partial \nu_{q\mathbb{I}[\phi]}}u(x)=g\big( \phi^{(-1)}(q^{-1}x)\big)&\\
\qquad \qquad-\frac{1}{\int_{\partial q\mathbb{I}[\phi]}\, d\sigma}\int_{\partial q\mathbb{I}[\phi]} g\big( \phi^{(-1)}(q^{-1}y)\big) \, d\sigma_y& \forall x\in \partial q\mathbb{I}[\phi] \,, \\
\int_{\partial q\mathbb{I}[\phi]}u\, d\sigma=k\, & \\
\end{array}
\right.
\end{equation}
has a unique solution $u[q,\phi,g,k]$ in $C_{q}^{1,\alpha}(\overline{\mathbb{S}_{q}[q\mathbb{I}[\phi]]^-})$. Moreover,
\begin{equation}\label{solAdd}
\begin{split}
u[q,\phi,&g,k](x)=v_{q}^-[\partial q\mathbb{I}[\phi], \mu](x)\\
&+\frac{1}{\int_{\partial q\mathbb{I}[\phi]}\, d\sigma} \Bigg(k-\int_{\partial q\mathbb{I}[\phi]}v_{q}^-[\partial q\mathbb{I}[\phi], \mu]\, d\sigma\Bigg) \qquad \forall x\in \overline{\mathbb{S}_{q}[q\mathbb{I}[\phi]]^-},
\end{split}
\end{equation}
where $\mu$ is the unique solution in $C^{0,\alpha}(\partial q\mathbb{I}[\phi])_0$ of the integral equation
\begin{equation}\label{intEq}
\begin{split}
\frac{1}{2}\mu(x)&+W_{q}^\ast[\partial q\mathbb{I}[\phi], \mu](x)=g\big( \phi^{(-1)}(q^{-1}x)\big)\\
&-\frac{1}{\int_{\partial q\mathbb{I}[\phi]}\, d\sigma}\int_{\partial q\mathbb{I}[\phi]} g\big( \phi^{(-1)}(q^{-1}y)\big) \, d\sigma_y\qquad \forall x\in \partial q\mathbb{I}[\phi]\, .
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
By \cite[Thm.~12.23]{DaLaMu21} we know that problem \eqref{np} has a unique solution. Moreover, by Lemma \ref{leminteq}, equation \eqref{intEq} has a unique solution $\mu$ which belongs to $C^{0,\alpha}(\partial q\mathbb{I}[\phi])_0$. Then by the properties of the periodic single layer potential, we deduce that the right hand side of \eqref{solAdd} solves problem \eqref{np}.
\end{proof}
In Proposition \ref{propAUX}, we have seen an integral equation on $\partial q\mathbb{I}[\phi]$ equivalent to problem \eqref{bvp}. However, if we want to study the dependence of the solution of the integral equation on the parameters $(q,\phi,g,k)$, it may be convenient to transform the equation on the $(q,\phi)$-dependent set $\partial q\mathbb{I}[\phi]$ into an equation on a fixed domain. We do so in the lemma below.
\begin{lemma}\label{lemSyst}
Let $\alpha$, $\Omega$ be as in \eqref{Omega_def}. Let $q \in {\mathbb{D}}_{n}^{+}({\mathbb{R}})$.
Let $\phi \in C^{1,\alpha}(\partial\Omega,\mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}$. Let $g \in C^{0,\alpha}(\partial \Omega)$. Then the function $\theta\in C^{0,\alpha}(\partial\Omega)$ solves the equation
\begin{equation}\label{intEq1}
\begin{split}
\frac{1}{2}\theta(t) +\int_{q\phi(\partial\Omega)}\,& \nu_{q\mathbb{I}[\phi]}(q\phi(t)) \cdot DS_{q,n}(q\phi(t)-y)\theta\big( \phi^{(-1)}(q^{-1}y)\big)d\sigma_y\\&=g(t)-\frac{1}{\int_{ \partial \Omega }\tilde{\sigma}[q\phi]\, d\sigma}\int_{\partial \Omega } g\tilde{\sigma}[q\phi] \, d\sigma
\qquad \forall t\in \partial\Omega\, ,
\end{split}
\end{equation}
if and only if the function $\mu\in C^{0,\alpha}(\partial q\mathbb{I}[\phi])$, with $\mu$ delivered by
\begin{equation}\label{mudef}
\mu(x)=\theta\big( \phi^{(-1)}(q^{-1}x)\big) \qquad \forall x\in \partial q\mathbb{I}[\phi],
\end{equation}
solves the equation
\
\begin{split}
&\frac{1}{2}\mu(x)+W_{q}^\ast[\partial q\mathbb{I}[\phi], \mu](x)\\
&=g\big( \phi^{(-1)}(q^{-1}x)\big) -\frac{1}{\int_{\partial q\mathbb{I}[\phi]}\, d\sigma}\int_{\partial q\mathbb{I}[\phi]} g\big( \phi^{(-1)}(q^{-1}y)\big) \, d\sigma_y \quad\forall x\in \partial q\mathbb{I}[\phi]\, .
\end{split}
\
Moreover, equation \eqref{intEq1} has a unique solution $\theta$ in $C^{0,\alpha}(\partial\Omega)$ and the function $\mu$ delivered by \eqref{mudef} belongs to $C^{0,\alpha}(\partial q\mathbb{I}[\phi])_0$.
\end{lemma}
\begin{proof}
It is a direct consequence of the theorem of change of variable in integrals, of Lemma \ref{leminteq}, and of the obvious equality
\[
\int_{\partial q\mathbb{I}[\phi]} \Bigg(g\big( \phi^{(-1)}(q^{-1}x)\big) -\frac{1}{\int_{\partial q\mathbb{I}[\phi]}\, d\sigma}\int_{\partial q\mathbb{I}[\phi]} g\big( \phi^{(-1)}(q^{-1}y)\big) \, d\sigma_y\Bigg) d\sigma_x=0\, ,
\]
which implies that
\[
g\big( \phi^{(-1)}(q^{-1}\cdot)\big) -\frac{1}{\int_{\partial q\mathbb{I}[\phi]}\, d\sigma}\int_{\partial q\mathbb{I}[\phi]} g\big( \phi^{(-1)}(q^{-1}y)\big) \, d\sigma_y
\]
is in $C^{0,\alpha}(\partial q\mathbb{I}[\phi])_0$. \end{proof}
%
Our next goal is to study the dependence of the solution of the integral equation \eqref{intEq1} upon $(q,\phi,g)$. We wish to apply the implicit function theorem in Banach spaces. Therefore, having in mind equation \eqref{intEq1}, we introduce the map $\Lambda$ from $ {\mathbb{D}}_{n}^{+}({\mathbb{R}})\times \left(C^{1,\alpha}(\partial\Omega, \mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}\right)\times \big(C^{0,\alpha}(\partial\Omega)\big)^2$ to $C^{0,\alpha}(\partial\Omega)$ by setting
\
\begin{split}
\Lambda&[q,\phi,g,\theta](t) \equiv
\frac{1}{2}\theta(t)\\
& +\int_{q\phi(\partial\Omega)}\,\nu_{q\mathbb{I}[\phi]}(q\phi(t)) \cdot DS_{q,n}(q\phi(t)-y)\theta\big( \phi^{(-1)}(q^{-1}y)\big)d\sigma_y\\
&-g(t)+\frac{1}{\int_{ \partial \Omega }\tilde{\sigma}[q\phi]\, d\sigma}\int_{\partial \Omega } g\tilde{\sigma}[q\phi] \, d\sigma\quad \forall t\in \partial\Omega,
\end{split}
\
for all $(q,\phi,g,\theta)\in{\mathbb{D}}_{n}^{+}({\mathbb{R}})\times\left(C^{1,\alpha}(\partial\Omega, \mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}\right)\times \big(C^{0,\alpha}(\partial\Omega)\big)^2$.
We are now ready to apply the implicit function theorem for real analytic maps in Banach spaces to equation $\Lambda[q,\phi,g,\theta]=0$ and prove that the solution $\theta$ depends analytically on $(q,\phi,g)$.
\begin{proposition}\label{taxi}
Let $\alpha$, $\Omega$ be as in \eqref{Omega_def}.
Then the following statements hold.
\begin{itemize}
\item[(i)] $\Lambda$ is real analytic.
\item[(ii)] For each $(q,\phi,g) \in {\mathbb{D}}_{n}^{+}({\mathbb{R}})\times\left(C^{1,\alpha}(\partial\Omega, \mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}\right) \times C^{0,\alpha}(\partial\Omega)$, there exists a unique
$\theta$ in $C^{0,\alpha}(\partial\Omega)$ such that
\[
\Lambda[q,\phi,g,\theta]=0 \qquad \mbox{ on } \partial\Omega,
\]
and we denote such a function by $\theta[q,\phi,g]$.
\item[(iii)] The map $\theta[\cdot,\cdot,\cdot]$ from ${\mathbb{D}}_{n}^{+}({\mathbb{R}})\times\left(C^{1,\alpha}(\partial\Omega, \mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}\right)\times C^{0,\alpha}(\partial\Omega)$ to $C^{0,\alpha}(\partial\Omega)$ that takes $(q,\phi,g)$ to $\theta[q,\phi,g]$ is real analytic.
\end{itemize}
\end{proposition}
\begin{proof}
By \cite[Thm. 3.2 (ii)]{LuMuPu20}, Lemma \ref{rajacon}, and standard calculus in Banach spaces, we deduce the validity of statement (i). Statement (ii) follows by Lemmas \ref{leminteq} and \ref{lemSyst}. In order to prove (iii), since the analyticity is a local property, it suffices to fix $(q_0,\phi_0,g_0)$ in ${\mathbb{D}}_{n}^{+}({\mathbb{R}})\times\left(C^{1,\alpha}(\partial\Omega, \mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}\right)\times C^{0,\alpha}(\partial\Omega)$ and to show that $\theta[\cdot,\cdot,\cdot]$ is real analytic in a
neighborhood of $(q_0,\phi_0,g_0)$ in the product space ${\mathbb{D}}_{n}^{+}({\mathbb{R}})\times\left(C^{1,\alpha}(\partial\Omega, \mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}\right)\times C^{0,\alpha}(\partial\Omega)$. By standard calculus in normed spaces, the partial
differential $\partial_{\theta}\Lambda[q_0,\phi_0,g_0,\theta[q_0,\phi_0,g_0]]$ of $\Lambda$ at $(q_0,\phi_0,g_0,\theta[q_0,\phi_0,g_0])$ with respect to the variable $\theta$ is delivered by
\begin{align*}
\partial_{\theta}&\Lambda[q_0,\phi_0,g_0,\theta[q_0,\phi_0,g_0]](\psi)(t) &\\
=& \frac{1}{2}\psi(t) +\int_{q_0\phi_0(\partial\Omega)}\,\nu_{q_0\mathbb{I}[\phi_0]}(q_0\phi_0(t)) \cdot DS_{q_0,n}(q_0\phi_0(t)-y)\psi\big( \phi_0^{(-1)}(q_0^{-1}y)\big)d\sigma_y\\
& \hspace{9cm} \forall t\in \partial\Omega,
\end{align*}
for all $\psi \in C^{0,\alpha}(\partial\Omega)$. Lemma \ref{leminteq} together with a change of variable implies that $\partial_{\theta}\Lambda[q_0,\phi_0,g_0,\theta[q_0,\phi_0,g_0]]$
is a linear homeomorphism from $C^{0,\alpha}(\partial\Omega)$ onto $C^{0,\alpha}(\partial\Omega)$.
Finally, by the implicit function theorem for real analytic maps in Banach spaces
(see, e.g., Deimling \cite[Thm. 15.3]{De85})
we deduce that $\theta[\cdot,\cdot,\cdot]$ is real analytic in a neighborhood of $(q_0,\phi_0,g_0)$ in
${\mathbb{D}}_{n}^{+}({\mathbb{R}})\times\left(C^{1,\alpha}(\partial\Omega, \mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}\right)\times C^{0,\alpha}(\partial \Omega)$.
\end{proof}
\begin{remark} \label{repform}
By Lemma \ref{rajacon}, Propositions \ref{propAUX} and \ref{taxi}, we have the following representation formula for the solution $u[q,\phi,g,k]$ of problem \eqref{bvp}:
\begin{align*}
&u[q,\phi,g,k](x) = \int_{\partial\Omega}
S_{q,n}(x-q\phi(s))\theta[q,\phi,g](s)\tilde \sigma[q\phi](s)\,d\sigma_{s}\\
&+\frac{ \Bigg(k-\!\int_{\partial\Omega} \int_{\partial\Omega}
S_{q,n}(q(\phi(t)-\phi(s)))\theta[q,\phi,g](s)\tilde \sigma[q\phi](s)d\sigma_{s}\tilde{\sigma}[q\phi](t)d\sigma_{t}\Bigg)}{\int_{\partial\Omega} \!\tilde \sigma[q\phi]d\sigma}\\
&\forall x\in \overline{\mathbb{S}_{q}[q\mathbb{I}[\phi]]^-},
\end{align*}
for all $(q,\phi,g,k) \in {\mathbb{D}}_{n}^{+}({\mathbb{R}})\times\left(C^{1,\alpha}(\partial\Omega, \mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}\right)\times C^{0,\alpha}(\partial \Omega)\times \mathbb{R}$.
\end{remark}
By exploiting the representation formula of Remark \ref{repform} and the analyticity result for $(q,\phi,g) \mapsto \theta[q,\phi,g]$ of Proposition \ref{taxi}, we are ready to prove our main result on the analyticity of $u[q,\phi,g,k]$ as a map of the variable $(q,\phi,g,k)$.
\begin{theorem}\label{mainthm}
Let $\alpha$, $\Omega$ be as in \eqref{Omega_def}. Let
\[
(q_0, \phi_0,g_0,k_0) \in {\mathbb{D}}_{n}^{+}({\mathbb{R}})\times\left(C^{1,\alpha}(\partial\Omega, \mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}\right)\times C^{0,\alpha}(\partial \Omega)\times \mathbb{R}.
\]
Let $U$ be a bounded open subset of $\mathbb{R}^n$ such that $\overline{U} \subseteq \mathbb{S}_{q_0}[q_0\mathbb{I}[\phi_0]]^-$. Then there exists an open neighborhood $\mathcal{U}$ of $(q_0, \phi_0,g_0,k_0)$ in
$${\mathbb{D}}_{n}^{+}({\mathbb{R}})\times\left(C^{1,\alpha}(\partial\Omega, \mathbb{R}^n) \cap {\mathcal{A}}_{\partial\Omega}^{\widetilde{Q}}\right)\times C^{0,\alpha}(\partial \Omega)\times \mathbb{R}$$ such that the following statements hold.
\begin{itemize}
\item[(i)] $\overline U \subseteq \mathbb{S}_{q}[q\mathbb{I}[\phi]]^-$ for all $(q,\phi,g,k) \in \mathcal{U}$.
\item[(ii)] Let $m \in \mathbb{N}$. Then the map from $\mathcal{U}$ to $C^m(\overline{U})$ which takes $(q,\phi,g,k)$ to the restriction
$u[q,\phi,g,k]_{|\overline{U}}$ of $u[q,\phi,g,k]$ to $\overline{U}$ is real analytic.
\end{itemize}
\end{theorem}
\begin{proof}
We first note that, by taking $\mathcal{U}$ small enough, we can deduce the validity of (i). The validity of (ii) follows by the representation formula of Remark \ref{repform}, by Lemma \ref{rajacon}, by Proposition \ref{taxi}, by the regularity results of \cite{LaMu13} on the analyticity of integral operators with real analytic kernels, and by standard calculus in Banach spaces.
\end{proof}
\begin{acknowledgement}
The authors are members of the `Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni' (GNAMPA) of the `Istituto Nazionale di Alta Matematica' (INdAM). P.L.~and P.M.~acknowledge the support of the Project BIRD191739/19 `Sensitivity analysis of partial differential equations in
the mathematical theory of electromagnetism' of the University of Padova.
P.M.~acknowledges the support of the grant `Challenges in Asymptotic and Shape Analysis - CASA' of the Ca' Foscari University of Venice. P.M.~also acknowledges the support from EU through the H2020-MSCA-RISE-2020 project EffectFact,
Grant agreement ID: 101008140.
\end{acknowledgement}
\input{references}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 96 |
\section{Introduction}\label{sec:introduction}}
\IEEEPARstart{A}{dversarial} examples~\cite{szegedy2013intriguing} are carefully manipulated inputs that appear natural to humans but cause deep models to misbehave.
Recent years have seen multiple methods to generate the underlying manipulative signals (i.e.,~perturbations) for fooling deep models on individual input samples~\cite{szegedy2016rethinking}, \cite{goodfellow2014explaining}, \cite{moosavi2016deepfool}, \cite{dong2018boosting}, \cite{inkawhich2019feature} or
many diverse samples
with a high probability~\cite{moosavi2017universal}, \cite{li2019universal} - termed `universal' perturbations.
The former may also launch `targeted' attacks, where the model predicts the label of attacker's choice for the input adversarial example.
The existence of adversarial examples is being generally perceived as a threat to deep learning~\cite{akhtar2018threat}. Nevertheless, controlled manipulation of model prediction with input perturbations also provides an opportunity of leveraging such signals for analyzing deep models.
In this work, we first introduce a technique to generate manipulative signals to fool deep models into confusing `an entire category of objects' with another target label of our choice, see Fig.~\ref{fig:teaser}(left:top). From the adversarial perspective, the resulting attack
is of high relevance in practical settings. It allows pre-computed perturbations that can change an object's category or a person's identity for a deployed model on-the-fly, where the attacker has also the freedom of choosing the target label, and there is no particular constraint over the input. Concurrently, the convenient control over the manipulative signal in our attack encourages a fresh perspective of seeing adversarial perturbations as a deep model analysis tool.
By appropriately leveraging the perturbation domain and restricting its range, our {\color{black} attack} reveals insightful patterns in the manipulative signal, as shown in Fig.~\ref{fig:teaser} (left:bottom).
Recently, Ilyas et al.~\cite{ilyas2019adversarial} claimed that existing large datasets (e.g.~ImageNet~\cite{deng2009imagenet}) admit to brittle yet highly predictive features that remain imperceptible to humans.
It is argued that deep visual models rely on these non-robust features for high accuracy, which also makes them susceptible to adversarial perturbations.
Reliance of deep models on these apparently incomprehensible features is taken as an indication of misalignment between deep visual representation and human perception~\cite{engstrom2019learning}.
To remove this misalignment, Engstorm et al.~\cite{engstrom2019learning} proposed to learn deep models under a robust optimization framework. However, this entails a significant performance loss for the original model and a drastic increase in the computational complexity of model induction.
\begin{figure*}[t]
\centering
\includegraphics[width = 0.8\textwidth]{Teaser}
\caption{\textbf{Left}: (Top) A single perturbation alters the category of source class samples (Ostrich) to the target label of choice (German Shepherd) while inhibiting the influence of fooling on non-source classes (Jelly fish). The adversarial examples are shown for ResNet-50~\cite{he2016deep}. (Bottom) A variant of the perturbation reveals that the constituting signals exploit correlation between the salient visual features of the incorrect class to fool deep models. The shown perturbations fool VGG-16~\cite{simonyan2014very}. \textbf{Right}: (Top)
Using an image distribution $\mathcal I$, we iteratively generate and refine a perturbation $\boldsymbol{p}$ for a classifier to extract geometric patterns deemed salient for an object category by the classifier. Anchoring the perturbation with different seed images $\boldsymbol{d}_n$ we can allow output variety. (Bottom)
Attacking an adversarially `robust' classifier with our perturbations enables visually appealing image manipulation.}
\label{fig:teaser}
\vspace{-3mm}
\end{figure*}
We find it paradoxal that a representation misaligned with human perception still performs human-meaningful visual tasks with high accuracy. Thus, we leverage our systematically computed perturbations to delve deeper into the composition of these signals with an alternate objective of model `explanation' instead of model `fooling'.
We discover that it is possible to isolate human-understandable visual features of the target label by attacking the classifiers with {\color{black} this modification}, see Fig.~\ref{fig:teaser} (right:top).
Within the context of adversarial perturbations, this observation weakens the argument of misalignment between human perception and deep representation. Rather, it places adversarial perturbations as human-meaningful geometric features of the target label, albeit in a primitive and subtle form.
In both cases of fooling and explaining a deep model, our perturbation estimation {\color{black}algorithms} stochastically maximize the prediction probability of an image distribution's perturbed samples for a given target label. The maximization takes place by iteratively stepping in the Expected gradient direction of the classifier's loss surface \textit{w.r.t.}~the input samples.
The optimization is guided by gradient moments to achieve the ultimate objective more efficiently.
For `fooling' a classifier, we restrict the input image distribution to the natural images of a single object category, termed the source class. Our technique additionally stops the leakage of fooling effects to non-source classes by carefully constraining the perturbation.
For model `explanation', the constraint over the input image distribution is loosened, and the search for perturbation is anchored by a seed image.
We further channel the perturbation signal to focus more on its regions that cause high activity of the neurons in the deeper layers of the classifier.
This procedure is purely based on the intermediate perturbations computed by our algorithm, which keeps our technique model faithful - an appealing property for model explanation methods~\cite{engstrom2019learning}. The computed perturbation eventually provides visualization of the features attributed to the target label by the model.
Besides explaining deep models and highlighting the alignment of deep representation with human perception, the proposed attack also suits the paradigm of performing low-level vision tasks, e.g.~image generation, inpainting and interactive image manipulation, using robust `classifiers'~\cite{santurkar2019computer}, see Fig.~\ref{fig:teaser}(right:bottom).
We perform these tasks to affirm the utility of our technique (and perturbations in general) beyond the adversarial objective by achieving significant visual improvements for these tasks over the original proposal in \cite{santurkar2019computer}.
{\color{black} These are the secondary contributions that this article makes, besides the major contribution of a novel adversarial attack (\S~\ref{sec:AdvP}) that is systematically converted into an attack to explain deep visual models (\S~\ref{sec:ExpPComp}). The overall contributions can be summarized as}:
\vspace{-1mm}
\begin{itemize}
\item {\color{black}We propose a pragmatic adversarial attack} that is able to perform class-specific targeted fooling of deep visual classifiers, {\color{black} and modify it into an attack to interpret deep representation}.
\item For model fooling, our targeted attack spans a whole object category instead of a single image and restricts the fooling effects to the source class.
\item We show effective targeted fooling of VGG-16~\cite{simonyan2014very}, ResNet-50~\cite{he2016deep}, Inception-V3~\cite{szegedy2016rethinking} and MobileNet-V2~\cite{sandler2018mobilenetv2} on ImageNet~\cite{deng2009imagenet}, and ResNet-50 on the large-scale VGG-Face2 dataset~\cite{cao2018vggface2}. We also show that the computed adversarial examples transfer well to the Physical World.
\item {\color{black}With the attack to} explain a model, our perturbations manifest visual features attributed by the model to individual object categories. The computed human-meaningful perturbations for `non-robust' classifiers weaken the argument that deep representation is misaligned with the human perception.
\item We demonstrate visually appealing image generation, inpainting and interactive image manipulation by attacking robust classifiers. Our results affirm the utility of perturbations beyond model fooling.
\end{itemize}
\vspace{-1mm}
This article is a considerable extension of our preliminary work presented in~\cite{ourCVPR}, which proposed an attack for model explanation. Here, we employ a broader framework that additionally allows pragmatic targeted fooling of a given model. It clearly contextualizes
our contribution within the adversarial attacks literature.
The newly introduced `adversarial' perspective of our technique is thoroughly explored on large-scale data models, including experiments for the Physical World.
{\color{black} A parallel treatment of adversarial and explanation optimisation objectives with closely related algorithms} offers clear insights into the potential of perturbations beyond the common adversarial utilities of these signals. This article also extends the discussions in the preliminary work and reviews additional related literature.
\vspace{-3mm}
\section{Related work}
Adversarial perturbations have attracted a considerable interest of computer vision community in the recent years~\cite{akhtar2018threat}.
These signals are commonly seen as a tool to fool deep learning models. Consequently, the existing literature revolves around attacking deep models using input perturbations, or defending them against the resulting attacks. We first discuss the key contributions under this `adversarial' perspective of input perturbations, which divides the literature into the broad categories of adversarial attack and defense schemes. Later, we highlight the contributions that can be categorised as `non-adversarial' approaches, in the sense that the core objective of those methods deviates from the typical fooling/defending of deep models.
\vspace{-3mm}
\subsection{Adversarial perspective}
\subsubsection{Adversarial attacks}
Imperceptible additive input perturbations that can arbitrarily alter the decisions of deep visual models made their first appearance in the seminal work of Szegedy et al.~\cite{szegedy2013intriguing}.
This discovery led to numerous techniques of fooling deep models. Goodfellow et al.~\cite{goodfellow2014explaining} devised the Fast Gradient Sign Method (FGSM) to efficiently compute adversarial perturbations. The FGSM performs a single step gradient ascend over the loss surface of a model \textit{w.r.t.} the input. Later, Kurakin et al.~\cite{kurakin2016adversarial} built on this concept by proposing a mutli-step Iterative FGSM (I-FGSM). The success of gradient based adversarial perturbations resulted in multiple follow-up algorithms, including Momentum I-FGSM (MI-FGSM)~\cite{dong2018boosting}, Variance-Reduced I-FGSM (vr-IGSM)~\cite{wu2018understanding} and Diverse Input I-FGSM (DI\textsuperscript{2}-FGSM) \cite{xie2019improving} etc.
Along the same line, Madry et al.~\cite{madry2017towards} noted that the `projected gradient descent on the negative loss function' strategy originally adopted by Kurakin et al.~\cite{kurakin2016adversarial} results in highly effective attacks.
DeepFool~\cite{moosavi2016deepfool} is another popular attack that computes adversarial perturbations iteratively by linearizing the attacked model's decision boundaries near the input images.
The above-mentioned key contributions triggered a wide interest of the research community in adversarial attacks. More recently, this research direction is seeing techniques for computing adversarial examples that transfer better to black-box models, and computing perturbations with reduced norms for imperceptibility~\cite{shi2019curls}, \cite{rony2019decoupling},
\cite{croce2019sparse},
\cite{dong2019evading}, \cite{yao2019trust}, \cite{dong2019efficient}.
There are also instances of exploiting unusual natural properties of the objects for model fooling~\cite{alcorn2019strike}, \cite{zhao2017generating} and attacking deep learning beyond the image space~\cite{zeng2019adversarial}, including 3D point cloud domain~\cite{xiang2019generating}, human skeletons~\cite{Liu2019AdvAttack} and the real-world~\cite{athalye2017synthesizing}, \cite{rey2019physical}.
For the aforementioned attacks, a perturbation is estimated for a single input sample. Moosavi-Dezfooli et al.~\cite{moosavi2017universal} computed an input perturbation to fool a model into misclassifying `any' image with high probability. Similar `universal' adversarial perturbations also appear in \cite{khrulkov2018art}, \cite{mopuri2017fast}. These attacks are non-targeted, i.e.,~the adversarial input is allowed to be misclassified into any class. The universal perturbations are able to reveal interesting geometric correlations among the decision boundaries of the deep models \cite{moosavi2017universal}, \cite{moosavi2017analysis}. However, these attacks neither restrict the input signals nor the predicted incorrect label for estimating the perturbations~\cite{moosavi2017universal}, \cite{khrulkov2018art}, \cite{mopuri2017fast}.
{\color{black}Hayes and Danezis~\cite{hayes2018learning} proposed Universal Adversarial Networks (UANs), which are generative models that can also compute targeted universal perturbations. Nevertheless, their technique does not consider restricting the scope of the input samples to a specific class, nor it suppresses the adversarial effects of the perturbations on irrelevant classes.
Manipulative signals maybe more revealing by systematically controlling the source and target classes and the effects of perturbations.
This partially motivates our adversarial attack that provides a comprehensive control over the perturbation computation with a transparent algorithm, which is not possible with
any
existing method.}
\vspace{-2mm}
\subsubsection{Adversarial defenses}
The ubiquitous prevalence of adversarial perturbations has naturally led to many defense techniques against the adversarial attacks~\cite{prakash2018deflecting}, \cite{akhtar2018defense}, \cite{raff2019barrage}, \cite{xie2019feature}, \cite{sun2019adversarial}, \cite{qiu2019adversarial}, \cite{jia2019comdefend}, \cite{liu2019detection}. However, adversarial attacks are often later found over-powering the defenses~\cite{carlini2017adversarial}, \cite{athalye2018robustness}. We refer the interested readers to the living document maintained by Carlini et al.~\cite{carlini2019evaluating} for a thorough defense evaluation to avoid such scenarios.
In general, the defense techniques aim at protecting deep models against both image-specific~\cite{prakash2018deflecting} and universal perturbations~\cite{akhtar2018defense}. This is commonly achieved by either detecting the perturbation in an input image, or diluting the adversarial effects of perturbation signals by modifying the model or the input itself. Nevertheless, the existence of a single unbreakable defense for all attacks still eludes the existing literature~\cite{carlini2017adversarial, carlini2016defensive, carlini2017magnet}, \cite{athalye2018robustness}.
Since our main focus is on adversarial attacks, we refer the interested readers to the recent surveys~\cite{akhtar2018threat}, \cite{yuan2019adversarial} for more defense techniques.
\vspace{-2mm}
\subsection{Non-adversarial perspective} Currently, there are also contributions in the literature, albeit very few, that portend the utility of perturbations beyond model fooling. For instance, Tsipras et al.~\cite{tsipras2019robustness} identified the presence of salient visual features of the target class in the perturbation signals that fool `adversarially robust' models, where the models have been robustified by including adversarial samples in the training data. Woods et al.~\cite{woods2019reliable} also made a similar observation for the models robustified with regularized gradients. The existence of salient visual features in perturbations indicate the potential of these signals in model explanation~\cite{madry2017towards, woods2019reliable}. However, their manifestation \textit{uniquely} in the case of robustified models, is interpreted as a misalignment between (non-robust) deep representation and the human perception~\cite{engstrom2019learning, tsipras2019robustness}.
Potentially, the re-alignment is only achievable by making the models robust at a serious cost of performance loss and amplified computational complexity through adversarial training~\cite{engstrom2019learning, tsipras2019robustness}.
It is worth indicating that the idea of using input perturbation \textit{explicitly} for explanation/interpretation also exists in the literature. For instance, Fong et al.~\cite{fong2019understanding} proposed an `extremal' perturbation to identify the regions of an input image which strongly affect model predictions. A similar concept of `meaningful' perturbations was previously introduced by Fong and Vedaldi~\cite{fong2017interpretable}.
{\color{black}Other works that discuss the relation between perturbations and interpretation include~\cite{etmann2019connection},\cite{xu2018structured}, \cite{papernot2016limitations}, \cite{elliott2019adversarial}. However, those works either deal with input-specific interpretations~\cite{papernot2016limitations}, \cite{elliott2019adversarial} that aim at highlighting the important image regions (similar to \cite{fong2019understanding}, \cite{fong2017interpretable}), or they deal with the interpretation of salient regions for input-specific perturbations themselves~\cite{etmann2019connection}, \cite{xu2018structured}.
This is different from our model-centric view of interpretability, for which our attack generates input-agnostic visualizations. Our} perturbation is applied to the whole image, as normally done in the gradient-based adversarial attacks~\cite{goodfellow2014explaining}, \cite{madry2017towards}. {\color{black}Moreover,} we make the perturbation image-agnostic, thereby encoding the `model' information in the computed patterns instead of focusing on individual input samples.
{\color{black}
Besides the perturbation based explanations, Ghorbani et al.~\cite{ghorbani2019towards} developed a framework to automatically identify human-understandable visual concepts deemed important by a model for its prediction. Their technique aggregates related segments across the inputs to identify high-level concepts for model explanation. Zhou et al.~\cite{zhou2018interpretable} decomposed the neural activations of an input into different semantic components that are subsequently used to generate human-meaningful input-specific interpretations. Previously, \cite{yosinski2015understanding} also proposed to visualize network activations to interpret their decisions.
They additionally proposed loss regularization to visualise the activation patterns in the input space preferred by the model. In contrast to \cite{yosinski2015understanding}, we achieve more clear visualisations without any such loss.
In \cite{nguyen2017plug}, Nguyen et al.~introduced a Plug \& Play Generative Network to generate class-conditioned photo-realistic images with the help of class features extracted from a classifier. Such visualizations may be utilized in understanding the concepts learned by the classifier. Model-faithfulness of such images remains an open question though.
Bau et al.~\cite{bau2018gan} specifically focused on studying the internal representation of GANs, proposing a method to visualise it with a segmentation-based method at different levels of abstraction for human-understanding. Though related to model interpretation, these methods do not deal with adversarial attacks on deep learning.
}
\vspace{-2mm}
\section{Problem formulation}
\label{sec:PF}
Let $\boldsymbol{I} \in \mathbb R^m$ denote a sample of a distribution $\mathcal {I}$ over the natural images. Let $\mathcal K (\boldsymbol{I})$ be a deep visual classifier that maps $\boldsymbol{I}$ to its label $\ell_{\text{true}}$. In a typical adversarial setting, the objective is to generate a perturbation $\boldsymbol{p} \in \mathbb R^m$ that satisfies the constraint
\begin{align}
\mathcal K (\boldsymbol{I} + \boldsymbol{p}) \rightarrow \ell_{\text{target}}~~\text{s.t.}~ \ell_{\text{target}} \neq \ell_{\text{true}}, ||\boldsymbol{p}||_{p} \leq \eta,
\label{eq:pert}
\end{align}
where $||.||_{p}$ denotes the $\ell_p$-norm of the vector, restrained by a pre-fixed ${\eta}$. In (\ref{eq:pert}), restricting $\ell_{\text{target}}$ to a pre-defined label results in a targeted adversarial attack.
Based on (\ref{eq:pert}), it is possible to express $\boldsymbol{p}$ as a function over $\boldsymbol{I}$ and $\mathcal{K}(.)$\footnote{Assuming a fixed deterministic algorithm to generate $\boldsymbol{p}$.}. For a given $\mathcal{K}(.)$, computing an image-specific perturbation confines the domain of $\boldsymbol{p}$, say $\text{Dom}(\boldsymbol{p})$ to a single image.
Implying, the information encoded in $\boldsymbol{p}$ is restricted to a single data sample.
{\color{black} This does not make $\boldsymbol{p}$ a good indicator of the general character of the classifier.
This observation also weakens the argument of human perceptual misalignment with deep representation by alluding to image-specific perturbations.}
To truly encode classifier information in the perturbation, this signal needs to be invariant to the input samples, which we can achieve by broadening the domain of $\boldsymbol{p}$.
Incidentally, universal adversarial perturbations~\cite{moosavi2017universal} are computed with a broader domain as per our formulation.
Inline with our observation, those perturbations manifest much more regular geometric patterns as compared to image-specific perturbations. However, they still remain far from salient visual features of objects. This happens because universal perturbations map images to random class labels for the sake of model fooling.
For a given classifier, broadening the perturbation domain with a `targeted' objective is more likely to induce geometric patterns in $\boldsymbol{p}$ that can actually be considered salient features of $\ell_{\text{target}}$, as viewed by the classifier. {\color{black} We also provide further evidence in this regard in \S~8a of the supplementary material of the paper.}
Further to the above argument, we can alternately describe the objective of (\ref{eq:pert}) as maximizing the probability of a perturbed sample being mapped to $\ell_{\text{target}}$ by $\mathcal K(.)$.
For $|\text{Dom}(\boldsymbol{p})|\gg 1$, where $|.|$ is the set cardinally, this maximization must incorporate all the relevant input samples (i.e.,~input image distribution).
Ignoring $\ell_{\text{true}}$ of the input samples for a moment, we define the following broad objective based on (\ref{eq:pert}):
\begin{gather}
\label{eq:comb}
\max \underset{\text{Dom}(\boldsymbol{p})}{P} \big( \mathcal K(\boldsymbol{I} + \boldsymbol{p})\rightarrow \ell_{\text{target}} \big) \geq \gamma,~\text{s.t.} \\ \nonumber
||\boldsymbol{p}||_p \leq \eta, |\text{Dom}(\boldsymbol{p})| \gg \!1,
\end{gather}
where $P(.)$ denotes probability and $\gamma \in [0,1]$ is a pre-defined constant.
To specify the `adversarial' and `explanation' usage of perturbation under (\ref{eq:comb}), we define two separate sets of external constraints. For model fooling, we let
\begin{align*}
\mathcal{C}_0^{f}:&~ \ell_{\text{target}} \neq \ell_{\text{true}},\\
\mathcal{C}_1^{f}:& ~\text{Dom}(\boldsymbol{p}) = \{\boldsymbol{I} | \boldsymbol{I} \sim \mathcal I_{\text{source}} \},\\
\mathcal{C}_2^f:&~ \underset{\text{Dom}^c(\boldsymbol{p})}{P} \big( \mathcal K(\boldsymbol{I} + \boldsymbol{p})\rightarrow \ell_{\text{target}} \big) < \gamma.
\end{align*}
Here, $\mathcal{C}_0^f$ ensures model fooling similar to the external constraint in (\ref{eq:pert}). $\mathcal{C}_1^f$ restricts the input domain to the image distribution $\mathcal{I}_{\text{source}}$ of a pre-defined source class. The constraint $\mathcal{C}_2^f$ mitigates the `leakage' of adversarial effects to the samples of non-source classes, specified by $\text{Dom}^c(\boldsymbol{p})$.
For the adversarial utility of our perturbation, it is more meaningful to have a stronger control over the input domain of this signal because fooling a model on all inputs alike (disregarding their correct labels) is hardly interesting. However, this is not the case for using the perturbation for model explanation. For the later, image-agnostic nature of the task makes the label of input images largely irrelevant. Hence, the external constraint for the `explanation' objective is defined as follows:
\begin{align*}
\mathcal{C}_1^e:&~ \text{Dom}(\boldsymbol{p}) = \{\boldsymbol{I} | \boldsymbol{I} \sim \mathcal{I} \} \cup \boldsymbol{d},
\end{align*}
where $\boldsymbol{d}$ is the `seed' sample that anchors the perturbations computation. Further details on `seed' are provided in \S~\ref{sec:ExpPComp} and \S~\ref{sec:ExpExplain}.
\vspace{-2mm}
\section{Computing the perturbation}
\label{sec:Alg}
{\color{black} The primary contribution of this article is an attack method that is introduced with an adversarial objective (\S~\ref{sec:AdvP}), and then transformed into an attack with model explanation objective (\S~\ref{sec:ExpPComp}).}
For both `adversarial' and `explanation' objectives, our perturbation estimation algorithms and the relevant concepts are closely related. Hence, we first describe the algorithm fully with respect to the adversarial objective. Then, we specify the modifications to the algorithm to achieve the explanation objective.
{\color{black} The secondary contribution of utilizing the model explanation objective of our attack for low-level vision tasks is discussed in \S~\ref{sec:Evaluation} with empirical evidence.}
\vspace{-2mm}
\subsection{Adversarial perturbation computation}
\label{sec:AdvP}
We compute the perturbations for (targeted) fooling of a classifier as shown in Algorithm~\ref{alg:main}. The algorithm optimizes for the objective in (\ref{eq:comb}), while satisfying the constraints $\mathcal{C}_0^f$ to $\mathcal{C}_2^f$. The abstract concept of the algorithm is intuitive. For a given source class, we compute the desired perturbation by taking small steps over the model's cost surface in the directions that increase the log-probability of the target label for the source class. The directions are computed stochastically, and the steps are only taken in the trusted regions that are governed by the first and (raw) second moment estimates of the directions. While computing a direction, we ensure that it also suppresses the prediction of non-source classes as the target label. To bound the perturbation norm, we keep projecting the accumulated signal to the $\ell_p$-ball of the desired norm at each iteration. The text below sequentially explains each step of the algorithm in detail.
Our attack is white-box in nature, hence the algorithm expects the target classifier as one of its inputs. It also requires a set $\mathcal S$ of the source class samples, and a set $\overline{\mathcal S}$ that contains samples of the non-source classes.
These sets are formed by sampling $\text{Dom}(\boldsymbol{p})$ and $\text{Dom}^c(\boldsymbol{p})$ respectively, mentioned in the previous section. Henceforth, we use the newly introduced sets as proxies for input domains for clarity in terms of implementation.
Other input parameters include the desired $\ell_p$-norm `$\eta$' of the perturbation, target label `$\ell_{\text{target}}$', mini-batch size `$b$' for the underlying stochastic optimization, and the desired fooling ratio `$\gamma$' - defined as the percentage of the source class samples predicted as the target class instances.
We momentarily defer the discussion on hyper-parameters `$\beta_1$' and `$\beta_2$' on \textit{line 1} of the algorithm. In a given iteration, we first construct the sets $\mathcal S_s$ and $\mathcal{S}_o$ by randomly sampling the source and non-source classes, respectively. The cardinality of these sets is fixed to `$\frac{b}{2}$' to keep the mini-batch size to `$b$' (\textit{line 3}). Each element of both sets is then perturbed with the current estimate of the perturbation - operation denoted by symbol $\ominus$ on \textit{line 4}. The chosen symbol emphasizes that $\boldsymbol{p}_t$ is subtracted in our algorithm from all the samples to perturb them. The `Clip(.)' function clips the perturbed samples to its valid range, e.g.,~$[0, 255]$ in the case of an 8-bit image representation.
\vspace{1mm}
\noindent{\bf Lemma 4.1:} \textit{For a classifier $\mathcal K(.)$ with cross-entropy cost $\mathcal J(\boldsymbol{\theta}, {\bf s}, \ell)$, the log-probability of an input sample ${\bf s}$ classified as `$\ell$' increases in the direction $-\frac{\nabla_{{\bf s}} \mathcal J(\boldsymbol{\theta}, {\bf s}, \ell)}{||\nabla_{{\bf s}} \mathcal J(\boldsymbol{\theta}, {\bf s}, \ell)||_{\infty}}$, where $\boldsymbol{\theta}$ denotes the model parameters\footnote{The model parameters remain fixed throughout, hence we ignore $\boldsymbol{\theta}$ in Algorithm~\ref{alg:main} and the related text.}.}\\
\noindent{\bf Proof:} We can write $\mathcal J(\boldsymbol{\theta}, {\bf s}, \ell) = - \log\left(\text{P}(\ell| {\bf s})\right)$ for $\mathcal K(.)$. Linearizing the cost and inverting the sign, the log-probability maximizes along $\boldsymbol{\gamma} = -\nabla_{{\bf s}} \mathcal J(\boldsymbol{\theta}, {\bf s}, \ell)$. With $||\boldsymbol{\gamma}||_{\infty} = \max_i|\gamma_i|$, $\ell_{\infty}$-normalization re-scales $\boldsymbol{\gamma}$ in the same direction of increasing $\log\left(\text{P}(\ell | {\bf s})\right)$.
\newcommand{\Exp}[1]{\underset{#1}{\mathbb E}}
\begin{algorithm}[t]
\caption{Attack to fool}
\label{alg:main}
\begin{algorithmic}[1]
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\REQUIRE Classifier $\mathcal K$, source class samples $\mathcal S$, non-source class samples $\overline{\mathcal{S}}$, target label $\ell_{\text{target}}$, perturbation norm $\eta$, mini-batch size $b$, fooling ratio $\gamma$.
\ENSURE Perturbation $\boldsymbol{p} \in \mathbb R^{\text{m}}$.
\STATE Initialize $\boldsymbol{p}_0$, $\boldsymbol{\upsilon}_0$, $\boldsymbol{\omega}_0$ to zero vectors in $\mathbb R^{\text{m}}$ and $t = 0$. Set $\beta_1 = 0.9$, and $\beta_2 = 0.999$.
\WHILE {fooling ratio $< \gamma$}
\STATE $\mathcal S_s \isEq{\text{rand}} \mathcal S$,~$\mathcal S_o \isEq{\text{rand}} \overline{\mathcal S}$ : $|\mathcal S_s| = |\mathcal S_o| = \frac{b}{2}$
\STATE ${\mathcal S_s} \leftarrow \text{Clip} \left ( \mathcal S_s \ominus \boldsymbol{p}_{t} \right)$,~$ {\mathcal S_o} \leftarrow \text{Clip} \left ( \mathcal S_o \ominus \boldsymbol{p}_{t} \right)$
\STATE $t \leftarrow t+1$
\STATE $\delta \leftarrow \frac{\Exp{{\bf s}_i \in \mathcal S_s}[|| \nabla_{{\bf s}_i} \mathcal J({\bf s}_i, \ell_{\text{target}})||_2 ]} {\Exp{{\bf s}_i \in \mathcal S_o}[|| \nabla_{{\bf s}_i} \mathcal J({\bf s}_i, \ell)||_2 ]}$
\STATE $\boldsymbol\xi_t \leftarrow \frac{1}{2}\Big(\Exp{{\bf s}_i \in \mathcal S_s} \big[ \nabla_{{\bf s}_i} \mathcal J({\bf s}_i, \ell_{\text{target}}) \big] +...$ \\
$~\hspace{37mm}\delta \Exp{{\bf s}_i \in \mathcal S_o} \big[ \nabla_{{\bf s}_i} \mathcal J({\bf s}_i, \ell) \big] \Big) $
\STATE $\boldsymbol\upsilon_t \leftarrow \beta_1 \boldsymbol{\upsilon}_{t-1} + (1-\beta_1) \boldsymbol{\xi}_t$
\STATE $\boldsymbol{\omega}_t \leftarrow \beta_2 \boldsymbol{\omega}_{t-1} + (1 - \beta_2) (\boldsymbol{\xi}_t\odot \boldsymbol{\xi}_t)$
\STATE $\boldsymbol{p} \leftarrow \frac{\sqrt{1-\beta_2^t}}{1-\beta_1^t}~\text{diag}\left( \text{diag}(\sqrt{\boldsymbol{\omega}_t})^{-1} \boldsymbol{\upsilon}_t \right)$
\STATE $\boldsymbol{p_t} \leftarrow \boldsymbol{p}_{t-1} +~\frac{\boldsymbol{p}}{|| \boldsymbol{p}||_{\infty}}$
\STATE $\boldsymbol{p_t} \leftarrow \Psi (\boldsymbol{p}_t)$
\ENDWHILE
\STATE return
\end{algorithmic}
\end{algorithm}
\vspace{0.5mm}
Under Lemma 4.1, the algorithm strives to take steps along the cost function's gradient \textit{w.r.t.}~an input ${\bf s}_i$. Since the domain of ${\bf s}_i$ spans multiple samples in our case, we must take steps along the `Expected' direction of those samples. However, we must ensure that the computed direction is not too generic to also cause the log-probability to rise for irrelevant (i.e.~non-source class) samples.
To refrain from the general fooling directions, we nudge the computed direction such that it inhibits the fooling of non-source class samples. \textit{Lines 6 and 7} of the algorithm implement these steps to account for $\mathcal C_2^f$ as follows.
On \textit{line 6}, we estimate the ratio between the Expected norms of the source sample gradients and the non-source sample gradients. Notice that we compute the respective gradients using different prediction labels. In the light of Lemma 4.1, $\nabla_{{\bf s}_i} \mathcal J({\bf s}_i, \ell_{\text{target}}): {\bf s}_i \in \mathcal S_s$ gives us the direction (ignoring the negative sign) to fool the model into predicting label `$\ell_{\text{target}}$' for ${\bf s}_i$, where the sample is from the source class. On the other hand, $\nabla_{{\bf s}_i} \mathcal J({\bf s}_i, \ell): {\bf s}_i \in \mathcal S_o$ provides the direction that improves the model confidence on the correct prediction of ${\bf s}_i$, where the sample is from the non-source class.
The diverse nature of the computed gradients can result in a significant difference between their norms. The scaling factor `$\delta$' on \textit{line 6} is computed to account for that difference in the subsequent steps.
For the $t^{\text{th}}$ iteration, we compute the Expected gradient $\boldsymbol{\xi}_t$ of our mini-batch on \textit{line~7}. At this point, it is worth noting that the effective mini-batch for the underlying stochastic optimization in our algorithm comprises clipped samples in the set $\mathcal S_s \bigcup \mathcal S_o$. The vector $\boldsymbol{\xi}_t$ is computed as the weighted average of the Expected gradients of the source and non-source samples. Under the linearity of the Expectation operator and preservation of the vector direction with scaling, it is straightforward to see that $\boldsymbol{\xi}_t$ encodes the Expected direction to achieve the targeted fooling of the source samples into the label `$\ell_{\text{target}}$', while inhibiting the fooling of non-source samples by increasing their prediction confidence for their correct classes.
Owing to the diversity of the samples in its mini-batch, the algorithm steps in the direction of computed gradient cautiously.
On \textit{line 8} and \textit{line 9}, it respectively estimates the first and the raw second moment (i.e.,~un-centered variance) of the computed gradient using exponential moving averages. The hyper-parameters `$\beta_1$' and `$\beta_2$' decide the decay rates of these averages, whereas $\odot$ denotes the Hadamard product. The use of moving averages as the moment estimates in our algorithm is inspired by the Adam algorithm~\cite{kingma2014adam}, which efficiently performs stochastic optimization. However, instead of using the moving averages of gradients to update the parameters (i.e.~model weights) as in~\cite{kingma2014adam}, we compute those for the Expected gradient and capitalize on the directions for perturbation estimation. Nevertheless, due to the similar physical significance of the hyper-parameters $\beta_1,~\beta_2 \in [0,1)$ in our algorithm and Adam, the performance of both algorithms largely remains insensitive to small changes to the values of these parameters. Following~\cite{kingma2014adam}, we fix $\beta_1 = 0.9, \beta_2 = 0.999$ (\textit{line 1}). We refer to~\cite{kingma2014adam} for further details on the choice of these values for the gradient based stochastic optimization.
In our algorithm, the gradient moment estimates are exploited in stepping along the cost surface. Effectiveness of the moments as stepping guides for stochastic optimization is already well-established~\cite{kingma2014adam}. Briefly ignoring the expression for $\boldsymbol{p}$ on \textit{line~10} of the algorithm, we compute this guide as the ratio between the moment estimates $\frac{\boldsymbol{\upsilon}_t}{\sqrt{\boldsymbol{\omega}_t}}$, where the square-root accounts for $\boldsymbol{\omega}_t$ representing the `second' moment. Note that, we slightly abuse the notation here as both values are vectors. On \textit{line~10}, we use the mathematically correct expression, where diag(.) converts a vector into a diagonal matrix, or a diagonal matrix into a vector, and the inverse is performed element-wise. Another improvement in \textit{line~10} is through the use of the `bias-corrected' ratio of the moment estimates instead. Moving averages are known to get heavily biased at early iterations. This becomes a concern when the algorithm can benefit from well-estimated initial points. In our experiments (\S \ref{sec:AdvExp}), we use our algorithm in that manner. Hence, bias-correction is accounted for in our technique. We provide a detailed derivation to arrive at the expression on \textit{line~10} of Algorithm~\ref{alg:main} in \S A-1 of the supplementary material.
Let us compactly write $\boldsymbol{p} = \frac{\widetilde{\boldsymbol{\upsilon}}_t}{\sqrt{\widetilde{\boldsymbol{\omega}}_t}}$, where
$\sim$
indicates that the vectors are bias-corrected.
It is easy to see that for a large second moment estimate $\widetilde{\boldsymbol{\omega}}$, $\boldsymbol{p}$ shrinks. This is desirable because we eventually take a step along $\boldsymbol{p}$, and a smaller step is preferable along the components that have larger variance.
The perturbation update step on \textit{line 11} of the algorithm further restricts $\boldsymbol{p}$ to unit $\ell_{\infty}$-norm.
To an extent, this relates to computing the gradient's sign in FGSM~\cite{goodfellow2014explaining}. However, most coefficients of $\boldsymbol{p}$ get restricted to smaller values in our case instead of $\pm 1$. As a side remark, we note that simply computing the sign of $\boldsymbol{p}$ for perturbation update eventually nullifies the advantages of the second moment estimate due to the squared terms.
The $\ell_{\infty}$ normalization is able to preserve the required direction in our case, while taking full advantage of the second moment estimate.
{\color{black} Our adversarial attack in Algorithm~\ref{alg:main} not only forces model fooling on the source class samples, it also encourages correct predictions on the non-source class samples. On one hand, this provides a better control over the perturbation scope, on the other, it makes the attack more practical. From the adversarial viewpoint, suspicions of an attack can be minimized by exclusively manipulating a single class, instead of letting the model predict the same label for all the images. The over-generalization of the adversarial effects in the latter case makes the attack less practical.}
\vspace{-2mm}
\subsubsection{Variants of adversarial perturbation}
\label{sec:Unbounded}
Notice that Algorithm~\ref{alg:main} accumulates the signals computed at each iteration.
To restrict the norm of the accumulated perturbation, $\ell_p$-ball projection is performed. The use of different types of balls results in different variants of the algorithm. For the $\ell_{\infty}$-ball projection, we implement $\Psi (\boldsymbol{p}_t) = \text{sign}(\boldsymbol p_t) \odot \text{min}\left(\text{abs}(\boldsymbol p_t), \eta\right)$ on \textit{line 12}. In the case of $\ell_2$-ball projection, we use $\Psi (\boldsymbol{p}_t) = \text{min}\left(1, \frac{\eta}{||\boldsymbol{p}_t||_2}\right) \boldsymbol p_t$. These projections respectively bound the $\ell_{\infty}$ and $\ell_2$ norms of the perturbations. We bound these norms to reduce the perturbation perceptibility, which is in line with the existing literature taking an `adversarial' perspective on perturbations. Differently from the literature, we also employ a variant of our algorithm in which $\Psi (\boldsymbol{p}_t) = \mathbb I(\boldsymbol{p_t})$, where $\mathbb I(.)$ is the identity mapping. In contrast to the typical use of perturbations in adversarial attacks, we employ the perturbations resulting from this `unbounded' variant to explore the classification regions of the target model without restricting the perturbation norm.
{\color{black}Due to our systematic control over the perturbation scope, this variant promises to reveal interesting information about the classification regions of deep models - empirically verified in \S \ref{sec:LUTAU}.}
\vspace{-2mm}
\subsection{Perturbation computation for model explanation}
\label{sec:ExpPComp}
Similar to the adversarial perturbations, we compute perturbations to explain deep representation with a stochastic optimization scheme. {\color{black}The algorithm can be considered a modification of} Algorithm~\ref{alg:main}. Hence, here we only discuss the major differences between the two to avoid repetition. The explanation is kept at a higher level of abstraction for clarity, and only the most relevant concepts are discussed in detail. A complete step-by-step discussion of the algorithm is provided in \S A-2 of the supplementary material.
Recall that the objective for the explanation perturbation is given in (\ref{eq:comb}), followed by the external constraint $\mathcal{C}_1^e$ in~\S \ref{sec:PF}.
To compute the perturbation, we deviate from Algorithm~\ref{alg:main} in three major aspects. (\textbf{i}) Instead of simply estimating the perturbation, we additionally refine it. Details of perturbation refinement are provided in \S \ref{sec:refine}. (\textbf{ii})~The optimization is anchored by a `seed' image. From the implementation viewpoint, the use of a seed image can be conceptualized as replacing the source class in Algorithm~\ref{alg:main} with a single image. However, since there is no constraint over the input class here, the seed as well as the `non-source' class samples (as per Algorithm~\ref{alg:main}) are all forced to be fooled into the same pre-selected target label $\ell_{\text{target}}$.
The overall optimization is anchored with the seed image by weighting the model gradients \textit{w.r.t.}~seed differently as compared to the gradients for the other samples - analogous to \textit{line 7} of Algorithm~\ref{alg:main}. Further details are in the supplementary material. (\textbf{iii}) Before updating the intermediate perturbation in a given iteration, we conduct an additional binary search. We invert the perturbation direction and evaluate if the inverted direction is equally good (or better) than the original direction for our objective in (\ref{eq:comb}). If so, we use the inverted perturbation instead of the original one in the subsequent iteration.
This step is mainly introduced to diversify the patterns encoded in the perturbation. This diversification is now preferred because we are more interested in holistic patterns instead of the individual pixel values under the new objective.
For the convenience of our discussion, we refer to the scheme described above as Algorithm {\color{red}1a} in the text to follow. Algorithm {\color{red}1a} with detailed discussion is provided in the supplementary material.
One aspect worth highlighting here is that
whereas the $\ell_p$-norm of a perturbation is restricted in adversarial settings for \textit{imperceptibility}, this constraint plays a different role for our novel application of perturbation.
In this case, by iterative back-projections on the $\ell_p$-ball, we amplify those geometric patterns in the perturbation that strongly influence $\mathcal{K(.)}$ to predict $\ell_{\text{target}}$ as the label of all the input samples.
With successive back-projections, we let the geometrically salient feature of $\ell_{\text{target}}$ to emerge in our perturbations (Fig.~\ref{fig:refine}) that we subsequently refine iteratively with the method discussed in \S~\ref{sec:refine}.
\vspace{-2mm}
\subsubsection{Perturbation refinement}
\label{sec:refine}
The holistic treatment of perturbation in Algorithm {\color{red}1a} results in an unrestricted spread of energy over the whole perturbation signal.
To achieve finer patterns we let the technique focus more on the relevant regions of input samples with an adaptive filtering mechanism summarized in Algorithm~\ref{alg:filteration}.
A key property of this mechanism is that it upholds the model fidelity of the perturbation by assuming no external priors.
\begin{algorithm}[t]
\caption{Perturbation refinement }
\label{alg:filteration}
\begin{algorithmic}[1]
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\REQUIRE Classifier $\mathcal K$, perturbation $\boldsymbol{p} \in \mathbb R^{\text{m}}$
\ENSURE Refined perturbation $\boldsymbol{p}$
\STATE Initialize $\boldsymbol{f}$ to $\boldsymbol{0} \in \mathbb R^{\text{m}}$ \\Set $\bar{\mathcal K} = $ convolutional base of $\mathcal K$, scale factor $\lambda = 5$
\STATE $\Omega \leftarrow \bar{\mathcal K}(\boldsymbol{p})$ : $\Omega \in \mathbb{R}^{H \times W \times C}$
\STATE $\boldsymbol{a} \leftarrow \frac{1}{C} \sum_{n=1}^{\mathcal{\text{C}}} \Omega^n$
\STATE $ \tau \leftarrow~$ $\Psi(\boldsymbol{a})$
\STATE {\bf if} $\boldsymbol{a}(\text{x,y}) > \tau~~\text{{\bf then}}~~\boldsymbol{a}(\text{x,y}) = \lambda~~\text{{\bf else}}~~ \boldsymbol{a}(\text{x,y})=\text{0}$
\STATE $\boldsymbol{f} \leftarrow~$upsample ($\boldsymbol{a}$) : $\boldsymbol{f} \in \mathbb R^{\text{m}} $
\STATE $\boldsymbol{p} \leftarrow \text{Clip}(\boldsymbol{p} \odot \boldsymbol{f}$)
\STATE return
\end{algorithmic}
\end{algorithm}
To refine the perturbation, the signal is fed to the convolutional base $\bar{\mathcal K}(.)$ of the classifier (\textit{line 2}).
The output $\Omega$ of the base is a set of low resolution 2D signals, which are reduced to an average signal $\boldsymbol{a}$ on \textit{line 3}. This signal captures a rough silhouette of the salient regions in the input perturbation, which makes it a useful spatial filter for our technique. On \textit{line 4}, $\Psi(.)$ computes the Otsu threshold~\cite{otsu1979threshold} for the average signal, that is subsequently used to binarize the image on \textit{line 5}. We empirically set $\lambda = 5$ in this work. The resulting image is up-sampled with bicubic interpolation~\cite{keys1981cubic} on \textit{line~6} to match the dimensions of the input perturbation $\boldsymbol{p}$. The scaled mask is applied to the perturbation, which is subsequently clipped to the valid dynamic range.
The output of Algorithm~\ref{alg:filteration} is a refined perturbation that is again processed by Algorithm~{\color{red}1a} to further highlight any salient patterns that might have been diminished with filtration. The final perturbation is computed by iterating between the two algorithms. In Fig.~\ref{fig:refine}, we show an example perturbation resulting from Algorithm {\color{red}1a} and after refinement by Algorithm~\ref{alg:filteration}.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{refine}
\caption{Visually salient geometric patterns emerge with more iteration of Algorithm~{\color{red}1a} (supplementary material) that are further refined with Algorithm~\ref{alg:filteration}. The refined perturbation is shown after post-refinement 100 iterations of the former. The `Nail' patterns are computed for VGG-16 with $\eta = 10$. We follow~\cite{szegedy2013intriguing} for perturbation visualization.}
\label{fig:refine}
\vspace{-3mm}
\end{figure}
\vspace{-3mm}
\section{Evaluation}
\label{sec:Evaluation}
We thoroughly evaluate our perturbations from both the `adversarial' and `explanation' perspectives. In \S~\ref{sec:AdvExp}, we explore the model fooling efficacy of our perturbations, including a Physical World attack in \S~\ref{sec:Phy}, and also analyze the possibility of model exploration with unbounded adversarial perturbations in \S~\ref{sec:LUTAU}. We demonstrate the model explanation character of our perturbations in \S~\ref{sec:ExpExplain}, with a discussion on leveraging our perturbations for low level computational tasks in \S~\ref{sec:robust}.
\begin{table*}[t!]
\centering
\caption{{Fooling ratios (\%) with $\eta = 15$ for $\ell_{\infty}$ and $4,500$ for $\ell_2$-norm bounded perturbations for ImageNet models.
The label transformations are T$_1$: Airship $\rightarrow$ School Bus, T$_2$: Ostrich $\rightarrow$ Zebra, T$_3$: Lion $\rightarrow$ Orangutang, T$_4$: Bustard $\rightarrow$ Camel, T$_5$: Jelly Fish $\rightarrow$ Killer Wahle, T$_6$: Life Boat $\rightarrow$ White Shark, T$_7$: Scoreboard $\rightarrow$ Freight Car, T$_8$: Pickelhaube $\rightarrow$ Stupa, T$_9$: Space Shuttle $\rightarrow$ Steam Locomotive, T$_{10}$: Rapeseed $\rightarrow$ Butterfly.
{\color{black}Average fooling ratio of the baseline - enhanced \cite{hayes2018learning}, is also provided in parentheses for reference.}
Leakage (last column) is the average fooling of non-source classes into the target label.}}
\vspace{-3mm}
\label{tab:success}
\begin{tabular}{c|l|c|c|c|c|c|c|c|c|c|c||c|c}
\hline
{\bf Bound} & {\bf Model} & {\bf T$_1$} & {\bf T$_2$} & {\bf T$_3$} & {\bf T$_4$} & {\bf T$_5$} & {\bf T$_6$} & {\bf T$_7$} & {\bf T$_8$} & {\bf T$_9$} & {\bf T$_{10}$} & {\bf Avg.} & {\bf Leak.} \\ \hline \hline
\multirow{ 4}{*}{$\ell_{\infty}$-norm} & VGG-16~\cite{simonyan2014very} & 92 & 76 & 80 & 74 & 82 & 78 & 82 & 80 & 74 & 88 & 80.6$\pm$5.8 {\color{black}(62.6$\pm$ 6.4)} & 29.9 \\
& ResNet-50~\cite{he2016deep} & 92 & 78 & 80 & 72 & 76 & 84 & 78 & 76 & 82 & 78 & 79.6$\pm$5.4 {\color{black}(58.2$\pm$ 6.7)} & 31.1 \\
& Inception-V3~\cite{szegedy2016rethinking} & 84 & 60 & 70 & 60 & 68 & 90 & 68 & 62 & 72 & 76 & 71.0$\pm$9.9 {\color{black}(58.6$\pm$ 9.3)} & 24.1 \\
& MobileNet-V2~\cite{sandler2018mobilenetv2} & 92 & 94 & 88 & 78 & 88 & 86 & 74 & 86 & 84 & 94 & 86.4$\pm$6.5 {\color{black}(64.6$\pm$ 5.9)} & 37.1 \\ \hline \hline
\multirow{ 4}{*}{$\ell_{2}$-norm} & VGG-16~\cite{simonyan2014very} & 90 & 84 & 80 & 84 & 94 & 86 & 82 & 92 & 86 & 96 & 87.4$\pm$5.3 {\color{black}(69.8$\pm$ 5.1)} & 30.4\\
& ResNet-50~\cite{he2016deep} & 96 & 94 & 88 & 84 & 90 & 86 & 86 & 94 & 90 & 90 & 89.8$\pm$3.9 {\color{black}(70.4$\pm$ 4.5)} & 38.0\\
& Inception-V3~\cite{szegedy2016rethinking} & 86 & 68 & 62 & 62 & 74 & 72 & 74 & 68 & 66 & 76 & 70.8$\pm$7.2 {\color{black}(56.2$\pm$ 7.7)} & 45.6\\
& MobileNet-V2~\cite{sandler2018mobilenetv2} & 94 & 98 & 92 & 76 & 94 & 92 & 76 & 92 & 92 & 96 & 90.2$\pm$7.7 {\color{black}(73.4$\pm$ 7.3)} & 56.0\\ \hline
\hline
\end{tabular}}
\end{table*}
\vspace{-3mm}
\subsection{Adversarial Experiments}
\label{sec:AdvExp}
\noindent{\bf Setup:} We first demonstrate the success of our targeted attack by fooling VGG-16~\cite{simonyan2014very}, ResNet-50~\cite{he2016deep}, Inception-V3~\cite{szegedy2016rethinking} and MobileNet-V2~\cite{sandler2018mobilenetv2} trained on ImageNet dataset~\cite{deng2009imagenet}.
Our selection of the models is based on their established performance and diversity. We use the training set of ILSVRC2012 for perturbation estimation, whereas the validation set of this data (50 samples per class) is used as our test set. For the non-source classes, we only use the correctly classified samples during training with a lower bound of $60\%$ on the prediction confidence.
This filtration is performed for computational purpose. It still ensures useful gradient directions with fewer non-source samples. We do not filter the source class data.
It is worth emphasizing that the existing works that compute universal perturbations, e.g.,~\cite{moosavi2017universal} only deal with the validation set of ImageNet. Comparatively, our data scope is orders of magnitude larger leading to more confident results.
We compute a perturbation using a \textit{two-step-strategy}. \textbf{First}, we alter Algorithm~\ref{alg:main} to disregard the non-source class data. This is achieved by replacing the non-source class set $\overline{\mathcal S}$ with the source class set $\mathcal S$ and using `$\ell_{\text{target}}$' instead of `$\ell$' for the gradient computation. In the \textbf{second} step, we initialize our algorithm with the perturbation computed in the first step.
This procedure is also adopted for computational gain with a better initialization. In the first step, we let the algorithm run for 100 iterations, while `$\gamma$' is set to $80\%$ in the second step. We additionally ensure at least 100 iterations in the second step. The batch size `$b$' is empirically set to 64 and 128 for the first and second step, respectively. In the text to follow, we discuss the setup details only when those are different from what is already described.
Besides fooling the ImageNet models, we also attack the VGGFace model~\cite{cao2018vggface2} (ResNet-50 architecture) trained on the large-scale VGG-Face2 dataset~\cite{cao2018vggface2}.
For that, 50 random images of an identity are used as the test set, while the remaining images are used for perturbation estimation.
\vspace{1mm}
\noindent{\bf Fooling ImageNet models:}~We randomly choose ten source classes from ImageNet and make another random selection of ten target labels, resulting in ten label transforming (i.e.,~fooling) experiments for a single model. Both $\ell_{\infty}$ and $\ell_2$-norm bounded perturbations are then considered, letting $\eta=15$ and $4,500$ respectively. As will be seen shortly, the perturbations remain imperceptible to quasi-imperceptible for these values. %
We summarize the results of our experiments in Table~\ref{tab:success}. The reported fooling ratios are on test data that is previously unseen by both the targeted model and our algorithm. Successful fooling of the models is apparent from the Table.
The table caption provides the label information for the source~$\rightarrow$~target transformation employing the commonly used nouns. We refer to \S A-3 of the supplementary material for the exact labels and original WordNet IDs of the ImageNet dataset. The last column reports the `Leakage', which is defined as the average fooling ratio of the non-source classes into the target label. Hence, relatively low leakage is desirable, which is observed in the Table.
{\color{black}
We explicitly analyse the success of leakage suppression in \S A-8 of the supplementary material}.
{\color{black} The table also provides baseline results for comparison, which is the average fooling ratio achieved by \cite{hayes2018learning} on the shown transformations. We only report the average value in Table~\ref{tab:success} in parentheses. For the complete results of \cite{hayes2018learning}, we refer to \S A-3 of the supplementary material.}
{\color{black} We emphasize that the baseline results of the Universal Adversarial Network (UAN)~\cite{hayes2018learning} are achieved after considerable enhancement of the original UAN method. Since the adversarial objective of \cite{hayes2018learning} is easier than ours, a direct application of UAN to our problem resulted in only 10-15\% fooling ratios for the transformations considered in Table~\ref{tab:success}. To achieve comparable results, we enhanced UAN by utilizing the $\ell_p$-ball projection concept from our method. Specifically, we replaced the original process of monotonic perturbation norm increment after each iteration (which was also originally used to terminate the model training once a pre-define norm-threshold was achieved) with a projection step. Similar to our algorithm, the projection step allowed UAN to back-project intermediate perturbations to a fixed $\ell_p$-ball. This accumulated the intermediate perturbations more effectively. Since the UAN can not incorporate the leakage suppression that is possible with our method, we used the leakage as the termination criterion for the enhanced-UAN. For any transformation, we stopped the enhanced-UAN training when its leakage reached the corresponding leakage value for our method. Under a fair comparison that uses the same experimental setup for both methods, Table~\ref{tab:success} ascertains that our attack is still considerably stronger than UAN despite a significant enhancement of the latter.}
\begin{figure*}[t]
\centering
\includegraphics[height= 3.5in]{Illus1}
\caption{Representative perturbations and adversarial images for $\ell_{\infty}$-bounded case ($\eta=15$). A row shows perturbations for the same source $\rightarrow$ target fooling of the mentioned models. An adversarial example for each model is also shown for reference (left). Following~\cite{szegedy2013intriguing}, the perturbations are magnified 10x, shifted by 128 and clamped to 0-255 for better visualization.}
\label{fig:illus1}
\vspace{-3mm}
\end{figure*}
In Fig.~\ref{fig:illus1}, we show representative examples of label transformations. The figure includes a sample adversarial example for each network. In our experiments, it was frequently observed that the models show high confidence on the adversarial examples,
as stated in the figure.
We provide further images for both $\ell_{\infty}$ and $\ell_2$-norm perturbations in \S A-4 of the supplementary material. From the images, we can observe that the perturbations are generally not easy to perceive visually.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\textwidth]{Faces}
\caption{Representative face ID switching examples for VGGFace model. Sample clean target ID image is provided for reference. }
\label{fig:face}
\end{figure*}
\begin{table}[t]
\caption{Switching face identities for VGGFace model (\% fooling): The switched identities in the original dataset are, F$_1$: n000234$\rightarrow$ n008779, F$_2$: n000282 $\rightarrow$ n006494, F$_3$: n000314 $\rightarrow$ n007087, F$_4$: n000558 $\rightarrow$ n001800,
F$_5$: n005814 $\rightarrow$ n006402. The $\ell_{\infty}$ and $\ell_2$-norms of the perturbation are upper bounded to 15 and 4,500 respectively.}
\centering
\begin{tabular}{c|c|c|c|c|c|c}\hline
\multicolumn{7}{c}{$\ell_{\infty}$-norm bounded}\\
\hline \hline
F$_1$ & F$_2$ & F$_3$ & F$_4$ & F$_5$ & Avg. & Leak. \\ \hline
88 & 76 & 74 & 86 & 84 & 81.6$\pm$6.2 & 1.9\\ \hline
\multicolumn{7}{c}{$\ell_{2}$-norm bounded} \\
\hline \hline
76 & 80 & 78 & 76 & 84 & 78.8$\pm$3.3 & 1.8 \\ \hline
\end{tabular}
\label{tab:face}
\vspace{-3mm}
\end{table}
\vspace{1mm}
\noindent{\bf Fooling VGGFace model:}
Perhaps the most interesting application of our attack is in switching facial identities i.e. from a specific source identity to a specific target identity.
We demonstrate this by fooling the large-scale VGGFace model~\cite{cao2018vggface2}.
Table~\ref{tab:face} reports the results on five identity switches that are randomly chosen from the VGG-Face2 dataset. Considering the variety of expression, appearance, ambient conditions etc.~for a given subject in VGG-Face2, the results in Table~\ref{tab:face} imply that our perturbations enable an attacker to change their identity on-the-fly with high probability, without worrying about the image capturing conditions. Moreover, leakage of the target label to the non-source classes also remains remarkably low for faces.
Figure~\ref{fig:face} illustrates representative adversarial examples resulting from our algorithm for the face identity switches.
Further images can also be found in \S A-5 of the supplementary material.
The results ascertain successful identity switching on unseen inputs by our technique.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth, height = 1in]{PingPong}
\caption{Adding a single perturbation to a mobile camera video is able to consistently fool VGG-16 with high confidence (conf.) in real-time to misclassify a Coffee mug into a Ping-pong ball. Frames of a random clip are marked with their number.}
\label{fig:PingPong}
\end{figure*}
\subsubsection{Physical World attack}
\label{sec:Phy}
The proposed adversarial attacks have serious implications if they transfer well to the Physical World. To demonstrate this possibility, we select two pragmatic scenarios.
In the first, we observe model predictions on a live webcam stream of printed adversarial images. No enhancement/transformation is applied to the images other than color printing.
Due to geometric and illumination transformations in the Physical World, this setup is considerably more challenging than fooling classifiers on digital scans of the printed adversarial images adopted in~\cite{kurakin2016adversarial}. Despite that, our perturbations are reasonably effective for targeted fooling in the Physical World. Successful fooling is observed even after slight rotations of the images. Further details of our experiments are provided in \S A-6 of the supplementary material and a video of live streaming is also included in the provided material.
In the second scenario, we feed a single perturbation as an additive noise to a live mobile camera video to mimic a real-time attack. In Fig.~\ref{fig:PingPong}, we report the confidence of VGG-16 for classifying a Coffee Mug as a Ping-pong ball for every $50^{\text{th}}$ frame of a random clip in our video. A single perturbation computed by Algorithm~\ref{alg:main} is able to consistently fool the model on all the frames despite varied imaging conditions. This demonstrates the ability of our pre-computed perturbations to launch a real-time attack in practice.
\subsubsection{Explanation character of adversarial perturbations}
\label{sec:LUTAU}
Keeping aside the success of our targeted attack, we find the patterns fooling a deep model on a whole semantic concept interesting in their own right.
Hence, to investigate those, we let the unbounded variant of Algorithm~\ref{alg:main} discussed in \S~\ref{sec:Unbounded} run to achieve 100\% test accuracy and observe the computed perturbation patterns. We notice a repetition of the salient visual features of the target class in the perturbations thus created, see Fig.~\ref{fig:LUTAU-1}.
We also observe that multiple runs of our algorithm resulted in different perturbations, nevertheless those perturbations preserve the characteristic features of the target label.
We refer to \S A-7 of the supplementary material for the corroborating visualizations.
This observation advances the proposition that perturbations with a broader input domain are inherently able to exploit the geometric correlations between the decision boundaries of the classifier~\cite{moosavi2017universal}.
{\color{black}This also entails that the human-understandable patterns emerging in the universal perturbations of \cite{poursaeed2018generative} may also be attributed to the target model instead of the generator network.}
In the supplementary material, we also provide perturbation patterns for the same target but different source classes. Those experiments confirm that the salient visual features of the target class become more pronounced in the perturbations with an increasing visual difference between the source and target classes.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.45\textwidth]{LUTA-U1_1}
\end{center}
\vspace{-3mm}
\caption{Patterns resembling salient visual features of the target class emerge with perturbations allowed to achieve 100\% fooling rate on unseen data without norm restrictions. Representative perturbations for VGG-16 are shown for ImageNet validation set.}
\label{fig:LUTAU-1}
\vspace{-3mm}
\end{figure}
\begin{table*}[t]
\centering
\caption{Average $\ell_2$-norms of the perturbations to achieve $95\%$ fooling on MobileNet-V2~\cite{sandler2018mobilenetv2}.}
\vspace{-3mm}
\label{tab:distance}
\begin{tabular}{l||c|c|c|c|c} \hline
{\bf Target} $\rightarrow$ & Space Shuttle & Steam Locomotive & Airship & School Bus & Life Boat\\
{\bf Source} $\downarrow$ & & & & &\\ \hline\hline
Space Shuttle & - & 4364.4$\pm$81.1 & 4118.3$\pm$74.5 & 4679.4$\pm$179.5 & 5039.1$\pm$230.7 \\
Steam Locomotive & 5406.8$\pm$57.7 & - & 4954.7$\pm$56.5 & 5845.2$\pm$300.4 & 5680.2$\pm$40.0 \\
Airship & 3586.4$\pm$59.4 & 3992.7$\pm$291.1 & - & 3929.5$\pm$50.4 & 3937.8$\pm$33.7 \\
School Bus & 7448.8$\pm$200.9 & 6322.8$\pm$89.5 & 6586.8$\pm$165.1 & - & 5976.5$\pm$112.1 \\
Life Boat & 5290.4$\pm$43.1 & 5173.0$\pm$71.8 & 5121.5$\pm$154.1 & 5690.9$\pm$47.4 & - \\ \hline
\end{tabular}
\end{table*}
We also explore another interesting utility of our unbounded perturbations in terms of exploring the classification regions induced by the deep models.
We employ MobileNet-V2~\cite{sandler2018mobilenetv2}, and let the unbounded perturbations achieve $95\%$ fooling rate on the training samples in each experiment. The bound on fooling rate is for computational reasons only. We choose five ImageNet classes from Table~\ref{tab:success} and convert their labels into each other. We keep the number of training samples the same for each class, i.e.,~965 as allowed by the dataset. In our experiment, the perturbation vector's $\ell_2$-norm is used as the representative distance covered by the source class samples to cross over and stay in the target class region. Experiments are repeated three times and the mean distances are reported in Table~\ref{tab:distance}. Interestingly, the differences between the distances for $A \rightarrow B$ and $B \rightarrow A$ are significant.
On the other hand, we can see particularly lower values for `Airship' and larger values for `School Bus' for all transformations. These observations are explainable under the hypothesis that $w.r.t.$~the remaining classes, the classification region for `Airship' is more like a blob in the high dimensional space that lets the majority of the samples in it move (due to perturbations) more coherently towards other class regions. On the other end, `School Bus' occupies a relatively flat but well-spread region that is farther from `Space Shuttle' as compared to e.g.,~`Life Boat'.
Our algorithm makes the source class samples collectively move towards the target class region of a model with perturbations.
Hence, its iterations also provide a unique opportunity to examine this migration through the classification regions. For Table~\ref{tab:distance} experiment, we monitor the top-1 predictions during the iterations and record the \textit{maximally predicted labels} (excluding the source label) during training. In Fig.~\ref{fig:hops}, we show this information as `max-label hopping' for six representative transformations.
Acute observers will notice that both Table~\ref{tab:distance} and Fig.~\ref{fig:hops} consider `transportation means' as the source and target classes.
This is done intentionally to illustrate the clustering of model classification regions for semantically similar classes.
Notice in Fig.~\ref{fig:hops} that the hopping mostly involves intermediate classes related to transportation/carriage means. Exceptions occur when `School Bus' is the target class. This is inline with our hypothesis that this class has a well-spread region which allows it to attract a variety of intermediate labels as the target when perturbed, including those that live (relatively) far from its main cluster of transportation objects.
The above results demonstrate an innate capacity of our
adversarial perturbations to explore and explain deep models. In the Sections to follow, we verify this potential under the explanation variation of our perturbations, as explained in \S~\ref{sec:ExpPComp}.
\vspace{-2mm}
\subsection{Explanation Experiments}
\label{sec:ExpExplain}
Preliminary results of model {\color{black}interpretation through our attack to explain deep representation} were presented in CVPR2020~\cite{ourCVPR}.
\vspace{1mm}
\noindent{\bf Setup}: Our setup for these experiments is largely similar to the experiments for `Fooling ImageNet models' in \S~\ref{sec:AdvExp}.
However, owing to the modified Algorithm~{\color{red}1a} (supplementary material), we use a single seed image in place of the source class samples.
We randomly choose the seed and another set of 255 samples from the ImageNet validation set to emulate the non-source class samples.
For all the 256 images, we never sample the target class, i.e.~$\ell_{\text{target}}$.
We set $\gamma = 0.8$ and let $\eta = 10$. The value of `$\gamma$' is empirically chosen based on acceptable visual clarity of the salient patterns in the final perturbations. Larger values of this hyper-parameter generally leads to clearer patterns at the cost of higher computation. We keep `$\eta$' comparable to the existing techniques for adversarial perturbation generation~\cite{moosavi2017universal, akhtar2018defense}.
To compute a perturbation, we first let Algorithm~{\color{red}1a} run to achieve its stopping criterion. Then, we apply Algorithm~\ref{alg:filteration} for refinement. Subsequently, Algorithm~{\color{red}1a} is again applied such that a refinement is carried out after every $50^{\text{th}}$ iteration until $300$ iterations.
\begin{figure}[t]
\centering
\includegraphics[width = 0.49\textwidth]{Hops_1}
\caption{Max-label hopping during transformations with proposed unbounded adversarial perturbations. Setup of Table~\ref{tab:distance} is employed for transforming the four labels.}
\label{fig:hops}
\vspace{-3mm}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=0.99\textwidth]{nonRobustImgGen1}
\caption{Well-localized visually salient features of the target class (label given) emerge by accumulating the gradient based perturbations with the explanation objective. The shown perturbations are computed for VGG-16 with ImageNet samples, excluding the target class samples. Perturbations for the same target are generated with different seeds for variety. Further images are also provided in \S A-9 of supplementary material.}
\label{fig:nonRobustIllustration}
\end{figure*}
\vspace{1mm}
\noindent{\bf Salient visual features}:
In Fig.~\ref{fig:nonRobustIllustration}, we show representative examples of the perturbations computed by our technique for VGG-16. The target labels are also given for the shown two explanation perturbations for each class.
Notice the well-localized clear geometric patterns that humans can associate with the target class labels. These patterns emerge without assuming any priors on the perturbation, input image distribution, or the model itself. Firstly, from the figure, it is apparent that our technique can (qualitatively) explain a model in terms of `what human-meaningful semantics are attached to its output neurons?'. This is useful e.g.,~in the settings where an unknown model is available and one must discover the labels of its output layer. Secondly, the perturbations are explaining `what geometric patterns are perceived as the discriminative features of a given class \textit{by the classifier}?'. Interestingly, these patterns align very well with the human perception, and we compute them with the same tool (i.e.,~gradient based perturbation) that is used to promote the argument of misalignment between human perception and deep representation~\cite{engstrom2019learning, tsipras2019robustness}.
\vspace{1mm}
\noindent{\bf Diversity of the salient patterns}: We provide two representative perturbations for each target class in Fig.~\ref{fig:nonRobustIllustration}, where the difference in the perturbations is caused by selecting different seeds. Besides ascertaining the effective role of seed in our algorithm, the diverse patterns that remain visually salient, affirm that the model has learned the general (human-meaningful) semantics for the target label. We emphasize that we have not used any sample of the target class in computing the respective perturbations in Fig.~\ref{fig:nonRobustIllustration}.
The patterns are completely based on the visual model. This also highlights the potential of standard `classifiers' as `generators' of diverse images.
\vspace{1mm}
\noindent{\bf Region specific semantics} {\color{black}{\bf \& Patterns for different models}:
We also successfully explore the possibility of analyzing the model semantics associated with specific image regions in \S A-9 of the supplementary material, where we additionally demonstrate that the manifestation of meaningful patterns in our perturbations is a generic phenomenon across the visual models.}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.9\textwidth]{robustImageGeneration1}
\caption{Image generation by attacking (adversarially) robust ResNet. The generated images are adversarial examples of the shown seeds. The intended class labels are mentioned on the left. We follow the setup of Santurkal et al.~\cite{santurkar2019computer} to generate images with both techniques. See \S A-11 of the supplementary material for further images. {\color{black} The confidence score of the robust classifier on the generated images is also reported.}}
\label{fig:robustImageGeneration}
\vspace{-3mm}
\end{figure*}
To further demonstrate the perceptual alignment of deep representation across different models, we also classify the `perturbations' generated for one model with other models. High confidence of multiple models for the intended target label indicates that the extracted patterns are commonly seen as discriminative visual features of the target class.
Details of our experiments are given in \S A-10 of the supplementary material.
{\color{black} We note that in the above-mentioned tasks, a key requirement is the availability of clear information on the global structure and texture of the image. The shown results extract this information from the used robust models, which are known to encode it more clearly as compared to the non-robust model~\cite{zhang2019interpreting}. Our algorithms contribute towards refining this information by including even more interpretable patterns. To perform similar low-level vision tasks with non-robust models using our method, a more sophisticated external mechanism will be required, which is a future research direction for this work}.
\subsubsection{Leverage in low-level vision tasks}
\label{sec:robust}
Santurakar et al.~\cite{santurkar2019computer} recently showed that (adversarially) robust deep classifiers can be exploited for computer vision tasks beyond classification.
They demonstrated image generation, inpainting and image manipulation etc.~by attacking a robust ResNet with the PGD attack~\cite{madry2017towards}.
The key concept on which Santurakar et al.~capitalized on is the presence of salient visual features in the adversarial perturbations computed with robust classifiers.
Synchronizing this notion with our findings, their study identifies an ideal test-bed for our explanation attack. Successful results for these tasks by our attack cannot only ascertain the implicit model explaining nature of our perturbations, but can also improve the state-of-the-art for the newly found place of the robust classifiers in the broader Machine Learning context.
{\color{black}Note that, the following experiments are not presented to suggest our method as an alternate to the state-of-the-art methods for the specified low-level vision tasks. Instead, the results are shown to demonstrate the explainability character of our method and further the notion of non-adversarial utilities of attack methods.}
To demonstrate improvements in the results, we follow~\cite{santurkar2019computer} closely in terms of the used classifier, perturbation budget and the underlying evaluation procedure. In the experiments to follow, we create the input images for our algorithm by sampling a multivariate Gaussian $\mathcal N(\boldsymbol{\mu}_{I}, \Sigma_{I})$, where $\boldsymbol{\mu}_{I} \in \mathbb R^m$ is the mean value of an image set $\boldsymbol{I}_{i = 1,...,n}\!\!\sim\!\! \mathcal I^{\text{target}}$. Here, $\mathcal I^{\text{target}}$ is the distribution of a target class images, emulated by ImageNet. We compute $\Sigma_I = \mathbb E[(\boldsymbol{I}_i - \boldsymbol{\mu}_{I})^{\intercal} (\boldsymbol{I}_i - \boldsymbol{\mu}_{I})]$.
For computational reasons, the multivariate Gaussian is computed by $4\times$ downsampling of the original images. Random $256$ distribution samples are later upsampled to match the network input and used to create the input images. Out of these images, one random image serves as the seed in an experiment. In the experiments to follow, we do not use the refinement step (i.e.,~Algorithm~\ref{alg:filteration}) where the image processing tasks are performed holistically.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{robustImagePainting2}
\caption{Representative inpainting results. The Masked image is used as the seed. Both methods restore images using the same robust model provided by Santurkar et al.~\cite{santurkar2019computer}, with the same perturbation budget. See \S A-12 of the supplementary material for more images. {\color{black} For each output image, PSNR/SSIM{\color{black}/LPIPS} values are also provided.} {\color{black} For LPIPS, lower values are more desirable.}}
\label{fig:robustInpainting}
\vspace{-2mm}
\end{figure*}
\noindent{\bf Image Generation:}
In Fig.~\ref{fig:robustImageGeneration}, we show representative examples of images generated by our technique and compare those with the results of Santurkal et al.~\cite{santurkar2019computer}. We use the author-provided code for \cite{santurkar2019computer} and strictly follow the guidelines to achieve the best results of their method. In the context of adversarial attacks, the generated images are adversarial examples of the seed images. We show two images per class, generated with the shown seeds for the mentioned target label. Our technique is clearly able to generate more refined and coherent images. Notice the details in the backgrounds as well.
{\color{black} The figure also reports the confidence score of the robust classifier on the generated images. Considering that robust classifiers are able to focus more on the robust object features, higher confidence identifies better image generation.} Theoretically, Santurkar et al.~\cite{santurkar2019computer} used the strongest gradient-based iterative adversarial attack~\cite{madry2017towards} in their method. Hence, our improved performance can be clearly attributed to the model explaining nature of the perturbations computed by our technique. We use the same perturbation budget $\eta = 40$ for both techniques. {\color{black} Overall, for the 1000 images we generated using the two methods, the average confidence score of the robust classifier on our images is $0.793\pm 0.171$, whereas this value for the corresponding images generated by \cite{santurkar2019computer} is $0.712\pm0.212$.}
The variety in the images generated with different seeds, their textural details and clear semantic coherence strengthen the broader concept that robust classifiers are capable of more than simple classification~\cite{santurkar2019computer}. It is a venue worth exploring for the future research.
\vspace{1mm}
\noindent{\bf Inpainting:}
Image inpainting~\cite{bertalmio2000image} restores information in large corrupt regions of images while upholding the perceptual consistency.
For this task, we treat the corrupted image as the seed, where its corrupt region is identified as a binary mask $F \in \{0,1\}^m$. Let $\Im$ be a set containing the seed and samples form our above-mentioned multivariate Gaussian distribution $\mathcal{N}(.)$. Keeping the robust classifier parameters fixed, we minimize the following loss:
\begin{align}
\mathcal L (\boldsymbol{p}) = \mathbb E\big[\mathcal J(\Im_p, \ell_{\text{target}}) + \beta~\big( \boldsymbol{p} \odot (1 - F)\big) \big],
\end{align}
where $\Im_p = \Im \ominus \boldsymbol{p}$, $\mathcal{J}(.)$ is the cross-entropy loss of the classifier and $\beta = 10$ is an empirically chosen scaling factor.
The designed loss function allows the perturbation signal to grow freely for the corrupt region while restricting it in the other regions. This setup is also inspired by Santurkar et al.~\cite{santurkar2019computer}.
In Fig.~\ref{fig:robustInpainting}, we show representative examples of corrupt images restored with our technique and Santurkar et al.~\cite{santurkar2019computer} using the robust ResNet provided by the authors. We use the same perturbation budget $\eta = 21$ for both techniques. The restoration quality of our technique is visibly better. The shown images and mask placements are randomly selected.
{\color{black} For 1000 randomly selected images from the validation set of ImageNet and placing random masks on those images, the average PSNR/SSIM/LIPIS values for our method are $22.31\pm2.09$/$0.87\pm0.07$/$0.188\pm0.079$, whereas these values are $20.13\pm2.53$/$0.79\pm0.11$/$0.191\pm0.093$ for the corresponding images of Santurkar et al.~\cite{santurkar2019computer}.}
\vspace{1mm}
\noindent{\bf Interactive Image Manipulation:}
An interesting recent application of deep neural networks, especially Generative Adversarial Networks~\cite{chen2018sketchygan} is to turn crude sketches into realistic images. Santurkar et al.~\cite{santurkar2019computer} demonstrated the possibility of such interactive image manipulation by adversarially attacking robust classifiers. We advance this concept by demonstrating that our alternate objective of model explanation is more suitable for the problem.
Using the raw sketch as the seed and employing random samples from our multivariate Gaussian, we manipulate the seed similar to image generation. However, this time we also use the refinement process. Representative results of our attack are shown in Fig.~\ref{fig:interactiveImageMan}. Compared to \cite{santurkar2019computer}, images generated with our attack appear to be much more realistic. Such refined manipulation of crude sketches with a classifier affirms the ability of our attack to highlight human-meaningful visual patterns learned by the classifier. {\color{black} For the 100 random images we manipulated with our method, the average confidence score of the robust classifier is 0.79$\pm$0.108, whereas this value is 0.76$\pm$0.110 for the corresponding images manipulated by \cite{santurkar2019computer}}.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{robustInteractiveImage1}
\caption{Representative examples of interactive image manipulation. The seed is a raw image required to be manipulated into an image of the target category. Both techniques use the same robust classifier with perturbation budget 60. See \S A-13 of the supplementary material for more visualizations. {\color{black} The confidence score of the robust classifier on the generated images is also provided.}}
\label{fig:interactiveImageMan}
\vspace{-4mm}
\end{figure}
\section{Conclusion}
{\color{black}We presented a novel attack for targeted fooling of deep visual models on individual object categories while suppressing the adversarial effects of perturbations on irrelevant classes. We modified the attack to also explain deep representation with input perturbations.}
To compute any perturbation, our attack performs a stochastic gradient search on the cost surface of the model to increase the log-probability of a distribution of input images to be classified as a particular target. For fooling, it considers input samples which belong to the same class and suppresses any unintended fooling of any other class. We also show successful fooling results in the Physical World. For model explanation, by iterative back-projection of the gradients and refinement with adaptive attention, our attack finds geometric patterns in the perturbations that are deemed salient by the classifier. We find that these patterns align well with human perception, which weakens the argument of misalignment between human perception and deep representation - in the context of adversarial perturbations. We also demonstrate low-level image manipulation with our technique by attacking robust classifiers. With those experiments, we not only ascertain the model explaining nature of our attack by achieving realistic image generation, inpainting and interactive image manipulation results, we also advance the state-of-the-art of these newly found classifier utilities.
\section{Acknowledgements}
This research was supported by ARC DP190102443 and the GPUs were donated by NVIDIA Corporation.
\balance
\bibliographystyle{IEEEtran}
| {
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U.S. Immigration: National and State Trends and Actions
Data Visualization October 1, 2015
Projects: Immigration and the States Project Tags: State data & Immigration Read time:
An interactive from
Foreign-born residents map
Unauthorized residents map
as a Percentage of State Population Timeline of state and federal action
In the United States, the federal government maintains primary authority over immigrants' admission into and removal from the country. Historically, the states have been largely responsible for the practical aspects of absorbing and integrating immigrants into their communities. But the relationship between the federal government and the states with respect to immigration has become more collaborative, and the states are playing a more active role in creating policies. These developments have resulted in new cooperation and conflicts between the levels of government.
This dynamic coincides with changes in the size and distribution of the nation's foreign-born population over the past three decades. Before 1990, immigrants were largely concentrated in a few states, but today, significant numbers live in all 50 states. This interactive tool illustrates the growth of the foreign-born population in the states from 1980 to 2012 and provides a snapshot of key immigration-related activities at the federal and state levels.
This analysis is not comprehensive, however. It does not include all immigration laws, policies, and other factors that have shaped the relationship between the federal government and the states and does not draw or imply conclusions about causal relationships between population change and federal, state, and local activities. It is intended only to provide historical context for today's discussions on immigration and the states.
Percentage of
Foreign-born
Foreign-born Residents as a Percentage of State Population
Note: Not adjusted for unauthorized immigrants not counted in the census
Source: U.S. Census Bureau, "Historical Census Statistics on the Foreign-Born Population of the United States: 1850 to 2000," Working Paper No. 81.
Unauthorized Immigrants as a Percentage of State Population
Source: Jeffery Passel and Karen A. Woodrow, "Population Division, U.S. Bureau of the Census, Geographic Distribution of Undocumented Immigrants: Estimates of Undocumented Aliens Counted in the 1980 Census by State," International Migration Review 18:3 (1984): 642–71.
The foreign-born population in 1980
Total foreign-born population
14.1 million people, or 6.2 percent of the total U.S. population, were foreign-born, an increase from 4.7 percent in 1970.
61.8 percent lived in five states: California (25.4 percent), New York (17.0 percent), Florida (7.5 percent), Texas (6.1 percent), and Illinois (5.9 percent).
Unauthorized population
An estimated 2.1 million people, or 0.9 percent of the U.S. population, were unauthorized immigrants.
More than 8 in 10 unauthorized immigrants lived in just five states: 49.8 percent lived in California, and 30.9 percent lived in New York, Texas, Illinois, and Florida.
Select state action before 1980
12 states enacted legislation during the 1970s prohibiting employers from knowingly hiring unauthorized workers.
Several states passed laws denying welfare, education, and other public benefits to individuals based on their immigration status.
Select federal action before 1980
The Immigration Act of 1965
Abolished the decades-old national-origins quota system that restricted the number of visas that could be allocated to people of certain national origins. Replaced that quota system with one that allocates permanent resident visas within a preference system for certain close relatives of U.S. citizens and permanent residents (family-based immigration) and for persons with special occupational skills, abilities, or training (employment-based immigration). Immediate relatives of U.S. citizens and some other groups of noncitizens are not subject to numerical limits.
Established annual numerical restrictions of 170,000 permanent resident visas for immigrants from the Eastern Hemisphere and 20,000 per country in that region.
For the first time in U.S. history, the Western Hemisphere was subject to an annual numerical limit of 120,000 permanent resident visas, with no per-country limits.
Amended in 1976 to subject the Western Hemisphere to the limit of 20,000 visas per country and the preference system.
Amended in 1978 to combine the numerical limits for the two hemispheres into a single worldwide cap of 290,000 total permanent resident visas.
Supreme Court decisions
Graham v. Richardson (1971) held that states could not deny state welfare benefits to lawfully residing individuals based on their immigration status.
De Canas v. Bica (1976) found that state laws sanctioning employers for hiring unauthorized workers could be enforced and were not pre-empted by federal law. Congress did not enact legislation regulating the employment of noncitizens until the Immigration Reform and Control Act of 1986.
Note: Adjusted for unauthorized immigrants not counted in the census, ACS, or the March Supplement to the Current Population Survey
Source: Jeffrey Passel and D'Vera Cohn, "Unauthorized Immigrant Totals Rise in 7 States, Fall in 14," Pew Research Center's Hispanic Trends Project (2014), http://www.pewhispanic.org/files/2014/11/2014-11-18_unauthorized-immigration.pdf
19.8 million people, or 7.9 percent of the total U.S. population, were foreign-born, up from 14.1 million (6.2 percent) in 1980.
32.7 percent lived in California, 14.4 percent in New York, and 7.7 percent in Texas.
Approximately 3.5 million people, or 1.4 percent of the total U.S. population, were unauthorized immigrants.
Slightly more than three-quarters of all unauthorized immigrants lived in just five states: 41.4 percent lived in California, and 35.4 percent lived in New York, Texas, Illinois, and Florida.
Select state action, 1980-90
During this period, more than 20 cities and states passed "sanctuary" laws and resolutions. Some declared that Salvadorans and Guatemalans fleeing civil wars in their home countries and denied asylum in the United States could remain in those jurisdictions without fear of arrest for immigration violations. Others prohibited state and local government workers from inquiring about a person's citizenship status or sharing information about an individual's immigration status with federal officials.
Eleven states made English their official language. Three had done so before 1980.
Select federal action, 1980-90
Refugee Act of 1980
Enacted a definition of "refugee" that partially incorporated international standards, created the legal status of "asylum," and established a program to resettle refugees within the United States.
Reduced the worldwide cap on permanent resident visas from 290,000 to 270,000.
Immigration Reform and Control Act of 1986
Created processes by which about 2.7 million unauthorized immigrants who had arrived before 1982 and an additional 1 million unauthorized agricultural workers could obtain legal status.
Created penalties for employers who knowingly hired unauthorized immigrants and a procedure for employers to use for checking the work authorization of new employees.
Established civil rights protections to safeguard lawfully present and U.S.-born workers from discrimination by employers.
Created State Legalization Impact Assistance Grants to reimburse states for costs related to legalization.
Plyler v. Doe (1982) found that states and localities cannot deny K-12 public education to children based on their immigration status.
31.1 million people, or 11.1 percent of the total U.S. population, were foreign-born, up from 19.8 million (7.9 percent) in 1990.
A number of states that traditionally were not home to many immigrants experienced large growth in their foreign-born populations from 1990 to 2000. Two states with exceptional growth were North Carolina, in which the foreign-born population increased from 1.7 percent of the state population in 1990 to 5.3 percent in 2000, and Nevada, where the foreign-born population grew from 8.7 to 15.8 percent of the state population from 1990 to 2000.
8.6 million people, or 3.1 percent of the total U.S. population, were unauthorized immigrants, an increase from 1.4 percent in 1990.
61.9 percent lived in five states (California, New York, Texas, Illinois, and Florida). Only 26.2 percent of this population lived in California in 2000, down from 41.4 percent in 1990.
Unauthorized immigrants made up at least 2 percent of the total population of 22 states and the District of Columbia.
Select state action, 1990-2000
Arizona, California, Florida, New Jersey, New York and Texas sued the federal government for reimbursement of money spent on services—including health care and education—for unauthorized immigrants.
In 1994, California passed a resolution demanding that the federal government compensate the state for its spending on services for unauthorized immigrants and that it enforce immigration laws.
In 1994, California voters passed Proposition 187, which excluded unauthorized immigrants from all state services and benefits, including health care and public education; mandated that service providers report suspected unauthorized immigrants; and required law enforcement officials to check individuals' immigration status at the time of arrest and report unauthorized immigrants to federal authorities.
Several states, including California, Colorado, and Arizona, explicitly barred unauthorized immigrants from eligibility for state driver's licenses. In other states, unauthorized immigrants were effectively ineligible for licenses because they could not provide the required documents, such as Social Security numbers.
Select federal action, 1990-2000
Immigration Act of 1990
Raised the annual cap on permanent resident visas to 675,000 beginning in 1995, with 480,000 allocated for family-based immigration and 140,000 for employment-based immigration.
Created a diversity visa program through which 55,000 visas were made available each year for immigrants from countries with low levels of migration.
Created the H-1B temporary visa for highly skilled workers and the H-2B temporary visa for nonagricultural seasonal workers.
Violent Crime Control and Law Enforcement Act of 1994
Created the State Criminal Alien Assistance Program to reimburse states and localities for costs incurred for incarcerating unauthorized immigrants convicted of crimes.
Illegal Immigration Reform and Immigrant Responsibility Act of 1996
Increased the penalties for noncitizens who commit crimes and created new grounds for excluding and deporting noncitizens.
Provided for additional Border Patrol agents, fencing, and technology along the southern border.
Created an electronic system to allow employers to verify workers' eligibility to work in the U.S.
Prohibited states and local governments from restricting information-sharing between their agencies and federal immigration authorities.
Created the 287(g) program to allow state and local law enforcement agencies to receive immigration law-related training from the federal government and to enforce federal immigration laws.
Personal Responsibility and Work Opportunity Reconciliation Act of 1996
Established a five-year residency requirement for federal means-tested public benefit programs, such as Medicaid, for legal permanent residents and certain other immigrants.
Reaffirmed that unauthorized immigrants are not eligible for any federal public benefits.
Allowed states to provide fully state-funded benefits to immigrants who are ineligible for federal programs.
Antiterrorism and Effective Death Penalty Act of 1996
Authorized state and local police to arrest and detain unauthorized immigrants who have previously been convicted of felonies in the U.S.
In November 1997, a federal judge ruled California's Proposition 187 unconstitutional because it infringed on the federal government's exclusive jurisdiction over matters relating to immigration. After an initial appeal, the case was settled in 1999 and was not heard by a higher court.
Federal courts dismissed lawsuits brought by Arizona, California, Florida, New Jersey, and New York against the federal government seeking reimbursement for state spending on services for unauthorized immigrants. The U.S. Supreme Court declined to hear the cases.
Note: Not adjusted for unauthorized immigrants not counted in the ACS
Source: American Community Survey (ACS) 1-year estimates.
40 million people, or 12.9 percent of the total U.S. population, were foreign-born, up from 31.1 million (11.1 percent) in 2000.
Grew 10 percent or more in all states and the District of Columbia from 2000 to 2010. During the decade, the foreign-born population increased in Alabama from 2.0 percent to 3.5 percent of the state population and in South Carolina from 2.9 percent to 4.7 percent.
Reached 12.2 million people nationwide in 2007 and then declined to 11.4 million, or 3.7 percent of the total U.S. population, by 2010.
In 2010, 55.7 percent of unauthorized immigrants lived in five states: California, New York, Texas, Illinois, and Florida, down from 61.9 percent in 2000; 21.9 percent lived in California in 2010, compared with 26.2 percent in 2000.
In 2003, New Mexico began accepting tax identification numbers to obtain driver's licenses. In 2005, Utah passed legislation allowing unauthorized immigrants to receive one-year driving privilege cards.
In 2005, the governor of Illinois created by executive order the Illinois Office of New Americans, with the mission to "coordinate policies and programs to help newcomers fully assimilate to the state, provide more and better services to the growing number of immigrants living in Illinois, and to study the impact of immigration policy on the state."
In 2008, the governor of Massachusetts signed an executive order launching the New Americans Agenda for Massachusetts. The agenda directs state agencies and community organizations "to develop and deliver a series of policy recommendations that emphasize the positive integration of these communities into the economic and civic life of the Commonwealth."
In 2010, Arizona passed a broad immigration enforcement law, S.B. 1070, which required police to inquire about immigration status during any lawful stop or arrest; made it a misdemeanor to fail to carry proper immigration documents; and made it unlawful to transport, move, conceal, harbor, or shield unauthorized immigrants. Several other states passed similar laws.
Approximately half of the states and the District of Columbia expanded health care coverage to include certain immigrant women and/or children.
More than 10 states passed laws that allow eligible students to pay in-state tuition rates at public postsecondary institutions, regardless of immigration status.
In 2007, the Legal Arizona Workers Act became law, allowing the state to rescind the business licenses of employers who hire unauthorized workers.
Moved many of the immigration functions of the U.S. government from the Immigration and Naturalization Service of the Department of Justice to the newly created Department of Homeland Security.
Created three agencies within the Department of Homeland Security: Customs and Border Protection, Immigration and Customs Enforcement, and U.S. Citizenship and Immigration Services.
Enhanced Border Security and Visa Entry Reform Act of 2002
Implemented new procedures to review visa applications and required that travel and entry documents be machine-readable and tamper-resistant and include biometric identifiers.
REAL ID Act of 2005
Created national standards for driver's licenses. Only licenses that meet the requirements can be accepted as a federal form of identification for certain official purposes, such as boarding an airplane.
Provided that certain noncitizens, excluding unauthorized immigrants, are eligible for REAL ID-compliant licenses. States may choose to issue licenses or other documentation that would allow eligible unauthorized immigrants to drive but could not be used as a federal form of identification.
Secure Fence Act of 2006
Authorized hundreds of additional miles of fencing along the U.S.-Mexico border.
In 2002, a regulatory amendment to the State Children's Health Insurance Program gave states the option to provide prenatal coverage regardless of the immigration status of the mother.
In 2009, the Immigrant Children's Health Improvement Act gave states the option to restore Medicaid for legal immigrant children and pregnant women without a five-year waiting period.
Secure Communities Program, launched 2008
Created to help identify noncitizens in U.S. jails who are deportable under immigration law. Arrestees' fingerprints are submitted to the federal government and checked against immigration databases.
Increased to 40.8 million, or 13.0 percent of the U.S. population, in 2012, slightly higher than in 2010, when it was 40 million (12.9 percent).
The unauthorized population was 11.2 million in 2012, or 3.6 percent of the total U.S. population, essentially unchanged from 11.4 million in 2010.
As of 2012, 56.3 percent of unauthorized immigrants lived in five states (California, Florida, New Jersey, New York, and Texas), with 21.9 percent residing in California. The number of unauthorized immigrants in New Jersey surpassed that of Illinois, making it one of the five states with the highest unauthorized population.
Unauthorized immigrants made up at least 2 percent of the total population in 30 states and the District of Columbia.
From 2010 to 2012, the unauthorized population fell in 19 states, including Alabama, Arizona, California, Illinois, and New York.
By 2012, 30 states explicitly required proof of lawful immigration presence for a person to receive a driver's license, and 18 other states had eligibility requirements that effectively excluded unauthorized immigrants. Only New Mexico and Washington allowed immigrants to obtain licenses regardless of immigration status, and Utah offered a driving privilege card to those who could not prove legal presence.
In 2013, seven states (California, Colorado, Connecticut, Illinois, Maryland, Nevada, and Vermont) and the District of Columbia passed legislation permitting unauthorized immigrants to obtain driver's licenses. (Oregon repealed its law allowing unauthorized immigrants to obtain licenses in 2014.)
In June 2015, Delaware and Hawaii enacted similar laws.
Other states passed laws or clarified policies regarding driver's license eligibility for beneficiaries of the Deferred Action for Childhood Arrival program, a federal initiative that temporarily suspends the deportation of certain people residing unlawfully in the United States who were brought here as children, graduated from U.S. schools, and meet other eligibility criteria.
In January 2014, the Michigan Office for New Americans was created by executive order. The office is intended to help "grow Michigan's economy by attracting global talent to our state and promote the skills, energy, and entrepreneurial spirit of our immigrant communities."
New York created the Office for New Americans in the 2012-13 state budget. The office was codified in state law in August 2014. Its goal is to help immigrants fully participate in civic and economic life.
Deferred Action for Childhood Arrivals (DACA), June 2012
The Department of Homeland Security implemented an initiative to temporarily suspend the deportation of certain people residing unlawfully in the United States who were brought to this country as children, graduated from U.S. schools, and meet other eligibility criteria.
In 2011, the Secure Communities Program was made mandatory for all jurisdictions.
Executive actions, November 2014
The president announced several initiatives to be undertaken by the executive branch, including:
An expansion of the 2012 DACA program and the creation of Deferred Action for Parents of Americans and Lawful Permanent Residents, which allows certain unauthorized immigrants who have U.S. citizen or legal permanent resident children to avoid deportation.
Changes to the federal government's immigration enforcement priorities and the replacement of the Secure Communities Program with the new Priority Enforcement Program, through which information from local law enforcement agencies is shared with federal immigration authorities.
A plan to coordinate with state and local governments to promote integration of all immigrants and, for lawful permanent residents, eventual U.S. citizenship.
A plan to consult with state and local business and labor leaders, public officials, and others to develop recommendations to streamline and improve the legal immigration system.
Whiting v. Chamber of Commerce (2011) found that Arizona's 2007 law penalizing employers who hired unauthorized immigrants was not preempted by federal law and could be enforced.
Arizona v. United States (2012) struck down three provisions of Arizona's S.B. 1070 that conflicted with federal law. The court left a fourth provision open to future legal challenges.
In April 2015, Hawaii's legislature passed a similar law.
Sarah Leiseca Manager, Communications 202.540.6369
State data Immigration
Immigration and the States Project
July 10, 2019 Economic Recovery Has Not Reduced Pension Debt for Many States
July 8, 2019 Native American Leaders and Experts Discuss Ways to Improve Community Health Outcomes
July 2, 2019 Nation's Doctor Says Opioid Epidemic Hits Close to Home | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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package ee.golive.personal.model;
import org.springframework.format.annotation.DateTimeFormat;
import javax.persistence.*;
import java.io.Serializable;
import java.util.Date;
@Entity
@Table(name = "weight")
public class Weight implements Serializable {
@Id
@GeneratedValue(strategy = GenerationType.IDENTITY)
@Basic(optional = false)
@Column(name = "id", unique=true, nullable = false)
private Integer id;
@DateTimeFormat(pattern = "yy-MM-dd")
private Date date_created;
private Double weight;
public Integer getId() {
return id;
}
public void setId(Integer id) {
this.id = id;
}
public Date getDate_created() {
return date_created;
}
public void setDate_created(Date date_created) {
this.date_created = date_created;
}
public Double getWeight() {
return weight;
}
public void setWeight(Double weight) {
this.weight = weight;
}
}
| {
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} | 29 |
These dramatic spikey upper false lashes can be fixed with lash glue (sold separately) along your natural upper lash line to create a high-impact gyaru style big eye effect.
Lash band approximately 35mm wide. Can be trimmed to fit. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,619 |
{"url":"http:\/\/qchu.wordpress.com\/category\/math\/analysis\/operator-algebras\/","text":"Feeds:\nPosts\n\n## Noncommutative probability\n\nThe traditional mathematical axiomatization of probability, due to Kolmogorov, begins with a probability space $P$ and constructs random variables as certain functions $P \\to \\mathbb{R}$. But start doing any probability and it becomes clear that the space $P$ is de-emphasized as much as possible; the real focus of probability theory is on the algebra of random variables. It would be nice to have an approach to probability theory that reflects this.\n\nMoreover, in the traditional approach, random variables necessarily commute. However, in quantum mechanics, the random variables are self-adjoint operators on a Hilbert space $H$, and these do not commute in general. For the purposes of doing quantum probability, it is therefore also natural to look for an approach to probability theory that begins with an algebra, not necessarily commutative, which encompasses both the classical and quantum cases.\n\nHappily, noncommutative probability provides such an approach. Terence Tao\u2019s notes on free probability develop a version of noncommutative probability approach geared towards applications to random matrices, but today I would like to take a more leisurely and somewhat scattered route geared towards getting a general feel for what this formalism is capable of talking about.\n\nBelow all vector spaces are over $\\mathbb{C}$, all algebras are unital, and all algebra homomorphisms preserve units unless otherwise stated. In the context of Banach algebras, the last two assumptions are not standard, but in practice non-unital Banach algebras are studied by adjoining units first, so we do not lose much generality.","date":"2014-03-10 02:26:15","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\": 5, \"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\": 0.6432283520698547, \"perplexity\": 220.18632129694936}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-10\/segments\/1394010554119\/warc\/CC-MAIN-20140305090914-00039-ip-10-183-142-35.ec2.internal.warc.gz\"}"} | null | null |
Ruanda nahm an den Olympischen Spielen 2020 in Tokio teil. Es war die insgesamt zehnte Teilnahme an Olympischen Sommerspielen. Das Comité National Olympique et Sportif du Rwanda nominierte insgesamt fünf Athleten in drei Sportarten.
Teilnehmer nach Sportarten
Leichtathletik
Laufen und Gehen
Radsport
Straße
Schwimmen
Weblinks
Ruanda in der Datenbank von Olympedia.org (englisch)
Ruanda
Sommerspiele 2020 | {
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Saint-Aubin è un comune francese di 712 abitanti situato nel dipartimento dell'Essonne nella regione dell'Île-de-France.
A Saint-Aubin è situato il SOLEIL, un acceleratore di particelle attualmente in fase di costruzione.
Società
Evoluzione demografica
Note
Altri progetti
Collegamenti esterni
Saint-Aubin | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,215 |
University of Dar es Salaam UDSM Prospectus 2021/2022
Download University of Dar es Salaam UDSM Prospectus 2021/2022 PDF for both undergraduate and postgraduate and Get all admission details such as courses, entry requirements, mode of study, fees structure
UDSM Prospectus 2021 is document for UDSM prospective students, University of Dar es Salaam prospectus 2021 contains information about the institution and the UDSM available courses, including advice on how to apply and the benefits of accepting a place. The University of Dar es Salaam (UDSM) have both online and paper versions of their prospectus, and they are divided into an Undergraduate Prospectus and a Postgraduate Prospectus.
The UDSM prospectus usually contains information on the individual courses, the staff (professors), notable alumni, the campus, special facilities (like performance halls for music schools or acting stages for drama schools), how to get in contact with the university, and how to get to the university.
Generally, the University of Dar es Salaam Prospectus comprises all the information a prospective student needs to become a fully admitted student of the University of Dar es Salaam (UDSM).
Here's a University of Dar es Salaam (UDSM) prospectus 2021/2022. You can use the information available in this book to find a suitable course and to apply for admission study at UDSM in 2021/2022 Academic year.
University of Dar es Salaam undergraduate prospectus 2021 pdf
University of Dar es Salaam UDSM postgraduate prospectus 2021 pdf
UDSM Examination Time Table 2022
University of Dar es Salaam UDSM Semester Time Table 2021/2022
University of Dar es Salaam UDSM Result 2021 – ARIS UDSM Results 2021
University of Dar es Salaam Academic Registration Information System – UDSM ARIS
University of Dar es Salaam UDSM Selected applicants 2021/2022
Mkwawa University College of Education MUCE Selected applicants 2021/2022
University of Dar es Salaam UDSM Contacts
University of Dar es Salaam (UDSM) Announcements and Latest Admission News & Notifications
Mkwawa University College of Education almanac 2021/2022 – Academic Calendar 2021/2022 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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Die Floresta ombrófila mista (portugiesisch für regenliebender gemischter Hain) ist eine Ökoregion von subtropisch-immerfeuchten Araukarienwäldern, die sich in Südbrasilien und teilweise in der Provinz Misiones (Argentinien) einst über eine sehr große Fläche erstreckten. Sie sind nicht zu verwechseln mit den Araukarienwäldern der Anden.
Kennzeichnend für die Floresta ombrófila mista ist die Baumart Brasilianische Araukarie (Araucaria angustifolia), die als 45 m hoher Emergent das Kronendach des Lorbeerwaldes überragt. Die Kronenschicht und den größten Anteil der Baumarten machen Laubbaumarten wie Ocotea odorifera (Syn.: Ocotea pretiosa) und Ocotea catharinense, Campomanesia xanthocarpa sowie Mimosa scabrella und Parapiptadenia rigida aus. Evolutionsgeschichtlich bedeutsam sind die Wälder als Relikt früher viel weiter verbreiteter Nadel- und Laubmischwälder, in denen viele für die Flora der Südhalbkugel charakteristische Taxa beheimatet sind.
In diesen Wäldern kommen viele vom Aussterben bedrohte Tier- und Pflanzenarten vor, darunter der Azurblaurabe (Cyanocorax caeruleus) und der Jaguar.
Bemerkenswert ist die ungewöhnlich hohe Artenvielfalt an Froschlurchen (Anura). Bis heute sind hier auf einer Fläche von zirka 300 Quadratkilometern 76 Arten aus 24 Gattungen nachgewiesen, etwa ein Viertel davon ist endemisch.
Diese Waldformation ist Teil des Bioms Mata Atlântica.
Einzelnachweise
Weblinks
Karten der brasilianischen Wälder (portugiesisch)
Pflanzensoziologie | {
"redpajama_set_name": "RedPajamaWikipedia"
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Q: "Error adding attribute" encountered when creating certificate When i create an OpenSSL certificate, i am asked to enter the additional attributes such as country code, state or province name, organization, organization unit name and common name. After entering all these i am asked to enter the common name and email address. i have entered. i have entered it as below
Country Name (2 letter code) [AU]:in
State or Province Name (full name) [Some-State]:tamilnadu
Locality Name (eg, city) []:coimbatore
Organization Name (eg, company) [Internet Widgits Pty Ltd]:abc
Organizational Unit Name (eg, section) []:abcd
Common Name (eg, YOUR name) []:xxxx
After this i am asked to enter the password, i have also entered it. it is shown below:
Please enter the following 'extra' attributes
to be sent with your certificate request
A challenge password []:123ytrewq
After finishing above steps if i press enter to generate the certificate, the following error occurs.
Error adding attribute
7532:error:0D0BA041:asn1 encoding routines:ASN1_STRING_set:malloc failure:./cryp
to/asn1/asn1_lib.c:381:
7532:error:0B08A041:x509 certificate routines:X509_ATTRIBUTE_set1_data:malloc fa
ilure:./crypto/x509/x509_att.c:317:
problems making Certificate Request
Can anyone help me? thanks in advance
A: This is an issue related with OpenSSL 0.9.8h. Using higher versions like v1.1.0f solves the problem.
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Уи́льям Го́вард Стайн (; 25 июня 1911, Нью-Йорк — 2 февраля 1980, там же) — американский биохимик.
Член Национальной академии наук США (1960) и Американской академии искусств и наук.
Биография
Окончил Гарвардский университет (1933), получил степень доктора философии в Медицинском и хирургическом колледже Колумбийского университета (1938). С 1938 года работает в Рокфеллеровском университете (Нью-Йорк), с 1955 года — профессор биохимии. В 1968—1971 годах был редактором журнала Journal of Biological Chemistry.
Основные работы
Основные работы по аналитической химии белков и ферментов. Разработал количественный метод определения аминокислот, основанный на ионообменной хроматографии, впервые установил (совместно с другими) первичную структуру фермента рибонуклеазы, исследовал строение активных центров ферментов.
Нобелевская премия
Нобелевская премия по химии (1972, совместно с С. Муром и К. Анфинсеном).
Примечания
Ссылки
Информация на Нобелевском сайте
Стайн (Stein), Уильям Х. // Лауреаты Нобелевской премии: Энциклопедия: М — Я. — М.: Прогресс, 1992. — С. 435—438.
Выпускники Гарвардского университета
Выпускники Колумбийского университета
Биохимики США
Биологи XX века
Лауреаты Нобелевской премии по химии
Лауреаты Нобелевской премии из США
Члены Национальной академии наук США
Главные редакторы Journal of Biological Chemistry | {
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{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/other-math\/basic-college-mathematics-9th-edition\/chapter-1-whole-numbers-1-4-multiplying-whole-numbers-1-4-exercises-page-40\/98","text":"# Chapter 1 - Whole Numbers - 1.4 Multiplying Whole Numbers - 1.4 Exercises: 98\n\n$(600 \\times 8 \\times 75 \\times 40) = 14,400,000$\n\n#### Work Step by Step\n\nGeneral equation $= a \\times b \\times c \\times d$ 1. Multiply the first two numbers together $(a \\times b)$ $600 \\times 8 = 4800$ New Equation $= (4800 \\times 75 \\times 40)$ 2. Multiply the next two numbers together $((a \\times b) \\times c)$ $4800 \\times 75 = 360,000$ New Equation $= (360,000 \\times 40)$ 3. Multiply the last two numbers together $(((a \\times b) \\times c) \\times d)$ $360,000 \\times 40 = 14,400,000$ Therefore, $(600 \\times 8 \\times 75 \\times 40) = 14,400,000$\n\nAfter you claim an answer you\u2019ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide\u00a0feedback.","date":"2018-04-25 13:19:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5130075216293335, \"perplexity\": 1050.743896974918}, \"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-2018-17\/segments\/1524125947803.66\/warc\/CC-MAIN-20180425115743-20180425135743-00150.warc.gz\"}"} | null | null |
Władysław Bielecki (ur. 1896, zastrzelony w 1942 lub 1943 w Krakowie) – polski rzeźbiarz drzeworytnik, malarz i grafik.
Życiorys
W czasie I wojny światowej walczył w 2 Pułku Ułanów Legionów Polskich, po 1918 wyjechał do Niemiec, gdzie studiował na monachijskiej Akademii Sztuk Pięknych. Po zakończeniu nauki powrócił do Krakowa, gdzie został nauczycielem rzeźby w Szkole Sztuk Pięknych. Był mistrzem drzeworytu i linorytu barwnego, twórcą wielu ekslibrisów, znaczący wpływ na jego twórczość mieli drzeworytnicy japońscy i krakowska szkoła pejzażu.
Tematem jego prac była architektura, pejzaże i zwierzęta, przy tworzeniu których stosował nowatorskie metody ukazujące w sposób eksperymentalny znaczenie naświetlenia.
.
Bibliografia
Władysław Bielecki, artlist.pl
Linki zewnętrzne
Prace Władysława Bieleckiego w bibliotece Polona: lista 2, lista 2
Polscy rzeźbiarze
Polscy graficy
Polscy drzeworytnicy
Współcześni polscy twórcy ekslibrisów
Żołnierze Legionów Polskich 1914–1918
Polscy kolaboranci III Rzeszy
Urodzeni w 1896
Zmarli w XX wieku
Polscy volksdeutsche | {
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Frank Fosco
Mighty Con Special Guest
Penciler/inker who started off with Gary Carlson's independent comic book Megaton. In the pages of that anthology, as his first published comic work, it featured his very own creation, Ethrian. After that Ihe ghosted layouts for, "The Strangers" from Malibu comics. The Strangers gig led to some work as a fill in artist at DC. Frank is most notably known for a 23 issue run on Teenage Mutant Ninja Turtles from Image comics. Frank has done work for Marvel on the "Worlds Greatest Comic Magazine" and back up stories in Erik Larsen's Savage Dragon. | {
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} | 1,315 |
{"url":"http:\/\/crypto.stackexchange.com\/questions\/15822\/zero-knowledge-proof-of-a-product","text":"Zero-knowledge proof of a product\n\nI have non-negative integers $x,y,z$. I'm going to give you commitments $C(x),C(y),C(z)$ to them. Then, I would like to prove in zero knowledge that $xy=z$. I can choose the commitment scheme to make the zero-knowledge proof easy, if that helps. What protocol should I use? Is there a highly efficient protocol for this?\n\nDetails: Assume I know that all three of $x,y,z$ fit in $k$-bit integers, i.e., $x,y,z \\in [0,2^k-1]$, and $k$ is publicly known. Given commitments $c_x,c_y,c_z$, I want a zero-knowledge proof of knowledge that I know $k$-bit non-negative integers $x,y,z$ such that $xy=z$ (over the integers) and $c_x,c_y,c_z$ can be opened to $x,y,z$. It is not sufficient to prove that $xy=z \\pmod q$; I want a proof that the equation holds over the integers. I don't care whether the commitment scheme is information-theoretically hiding or computationally hiding; either is fine.\n\nPreferably, the commitment scheme should also allow me to do range proofs (to prove in zero knowledge that $x,y,z$ are in a particular range).\n\n-\n\nUse the exponential variant of ElGamal, where the plaintext is encoded in the exponent. Elliptic curve ElGamal is fine. In fact, any public key cryptosystem which allows raising ciphertexts to a power such that this operation corresponds homomorphically to multiplication for the plaintext.\n\nYour commitments are $c_x = \\mathsf{E}(x)$; $c_y = \\mathsf{E}(y)$; and $c_z = \\mathsf{E}(z)$. Additionally, you must commit to an encryption of one: $c_1 = \\mathsf{E}(1)$, and prove that it is an encryption of one using a Schnorr proof.\n\nNext, use the Chaum-Pedersen proof of equivalent discrete logarithms to prove the following relation: dlog$_{c_1}c_y =$ dlog$_{c_x}c_z$.\n\nSchnorr proof:\n\nPublic knowledge: $a, b$\n\nPrivate knowledge for $\\mathcal{P}$: $s$\n\nClaim: $a = b^s$\n\n$\\begin{matrix} \\mathcal{P} & & \\mathcal{V} \\\\ u \\xleftarrow{\\$} \\mathbb{Z}_q & & \\\\ v \\leftarrow b^u & & \\\\ & \\xrightarrow{\\quad v \\quad} & \\\\ & & e \\xleftarrow{\\$} \\mathbb{Z}_q \\\\ & \\xleftarrow{\\quad e \\quad} & \\\\ r \\leftarrow se + u \\mod q & & \\\\ & \\xleftarrow{\\quad r \\quad} & \\\\ & & b^r \\stackrel{?}{=} a^ev \\end{matrix}$\n\nThe Chaum-Pedersen proof is very similar:\n\nPublic knowledge: $a_1, a_2, b_1, b_2$\n\nPrivate knowledge for $\\mathcal{P}$: $s$\n\nClaim: $s =$ dlog$_{b_1}a_1 =$ dlog$_{b_2}a_2$\n\n$\\begin{matrix} \\mathcal{P} & & \\mathcal{V} \\\\ u \\xleftarrow{\\$} \\mathbb{Z}_q & & \\\\ v_1 \\leftarrow b_1^u & & \\\\ v_2 \\leftarrow b_2^u & & \\\\ & \\xrightarrow{\\quad v_1,v_2 \\quad} & \\\\ & & e \\xleftarrow{\\$} \\mathbb{Z}_q \\\\ & \\xleftarrow{\\quad e \\quad} & \\\\ r \\leftarrow se + u \\mod q & & \\\\ & \\xleftarrow{\\quad r \\quad} & \\\\ & & b_1^r \\stackrel{?}{=} a_1^ev_1 \\\\ & & b_2^r \\stackrel{?}{=} a_2^ev_2 \\\\ \\end{matrix}$\n\nAt this point, I thought my answer was complete, but the questioner correctly notes: \"This is an excellent start, but it proves that $xy \\equiv z \\pmod q$, rather than that $xy=z$. I think if you choose $q$ to be $>2k$ bits long, and combine it with a range proof of the size of $x,y,z$, though, this might work.\"\n\nIt is nice that you specify that $x, y$ and $z$ are in $[0,2^k]$, because that makes the range proof a little easier. In order to prove this range of $x$ (same for $y$ and $z$), do the following:\n\n\u2022 Commit to $k$ bits of $x$, i.e.: send $\\mathsf{E}(x_i)$ to the verifier, where $x_i$ is the $i$th bit of x.\n\u2022 Prove the commitments are encryptions of either 1 or 0.\n\u2022 The verifier can aggregate the bits homomorphically: $\\mathsf{E}(x) = \\prod_i \\mathsf{E}(x_i)^{2^i}$. Use this as the commitment of $x$.\n\nIt remains to be shown how to prove that an encryption is an encryption either of 1 or else of 0. This can be done using the CDS proof linking technique (link).\n\nFirst, note that in the ElGamal cryptosystem, it is easy to prove that an encryption $\\mathsf{E}(a)$ encrypts a plaintext $a$. The encryption is given by $\\mathsf{E}(a) = (g^y, g^{a+xy})$. Divide $g^a$ out of the second value and then produce a (Chaum-Pedersen) proof that dlog$_g g^y =$ dlog$_{g^x}g^{xy}$.\n\nSecond, note that it is also easy to falsely \"prove\" that an encryption $\\mathsf{E}(a)$ encrypts a different plaintext $b$, provided that you control the challenge $e$ sent by the challenger. The claim to be proved is dlog$_g g^y =$ dlog$_{g^x}g^{-b+a+xy}$. Generate the proof transcript as follows (let $a_1$ correspond to $g^y$, $b_1$ to $g$, $a_2$ to $g^{xy}$ and $b_2$ to $g^x$): $e, r \\xleftarrow{\\$} \\mathbb{Z}_q$;$v_1 \\leftarrow b_1^ra_1^{-e}$;$v_2 \\leftarrow b_2^ra_2^{-e}$. The central idea of the CDS technique is to generate two proof transcripts, one of which is made interactively with the verifier and the other of which was generated beforehand. This is done in such a way that the verifier cannot tell which one of the two proofs was interactively made and which one was generated earlier. In particular, \u2022 The prover generates a proof transcript$T_f$of the false claim:$T_f = (c_f,e_f,t_f)$, and a first message of the true claim:$c_t$. \u2022 The prover sends the two first messages to the verifier:$\\mathcal{P} \\xrightarrow{\\quad c_f, c_t \\quad} \\mathcal{V}$. \u2022 The verifier chooses a random challenge$e \\xleftarrow{\\$} \\mathbb{Z}_q$ and sends it to the prover: $\\mathcal{P} \\xleftarrow{\\quad e \\quad} \\mathcal{V}$.\n\u2022 The prover interpolates on the line defined by the two points $(0,e), (1,e_f)$ and evaluates it in $2$ to obtain $e_t$.\n\u2022 The prover completes the transcript $T_t$ of the proof of the true claim using $e_t$.\n\u2022 The prover sends both transcripts to the verifier: $\\mathcal{P} \\xrightarrow{\\quad T_f, T_t \\quad} \\mathcal{V}$.\n\u2022 The verifier verifies the proof transcripts and verifies that $(0,e), (1,e_f), (2,e_t)$ are on the same straight line.\n\nThe reason this technique works is because the prover can prove only one of the claims. Having committed to the false proof, he has no degrees of freedom left to choose his challenge for the true proof. Thus, the verifier knows that at least one of the transcripts was produced interactively. Obviously, the order of $T_t$ and $T_f$ must be random because otherwise the verifier can easily tell which one is true and which one is false.\n\n-\nThis is an excellent start, but it proves that $xy \\equiv z \\pmod q$, rather than that $xy=z$. I think if you choose $q$ to be $>2k$ bits long, and combine it with a range proof of the size of $x,y,z$, though, this might work. Thank you! \u2013\u00a0D.W. Apr 26 '14 at 18:31\nRight. Range proof. Let met modify my answer to include that. \u2013\u00a0Alan Sz Apr 26 '14 at 18:45\nProof of equality of two logarithms if fine in a group of a hidden order, avoiding $\\pmod{q}$ reduction. \u2013\u00a0Vadym Fedyukovych Jan 1 '15 at 19:49\n\nTo prove that product holds over integers, one would start from commitments with groups of a hidden order. That is, proving party should not know order of the group, which is the case with RSA-like multiplicative group.\n\nConsider Prover responses $\\rho_x = tx + \\alpha_x$, $\\rho_y = ty + \\alpha_y$, $\\rho_z = tz + \\alpha_z$ to Verifier challenge $t$ with initial random coins $\\alpha_x, \\alpha_y, \\alpha_z$ calculated over integers. Consider a polynomial $f(t) = \\rho_x(t) \\rho_y(t) - t \\rho_z(t)$. Degree of this polynomial is quadratic iff relation over secrets $x,y,z$ does not hold. Consider yet another polynomial $F(t) = f(t) - (v_1 t + v_0)$ for $v_1 = x \\alpha_y + y \\alpha_x - \\alpha_z$, $v_0 = \\alpha_x \\alpha_y$.\n\nOne would go for an extended Schnorr-like protocol from probabilistic identity testing $F(t) \\equiv 0$ with Schwartz-Zippel lemma. At the first step of such a protocol, one sends commitments to $v_1$ and $v_0$. For details please take a look at An argument for Hamiltonicity.\n\nFor easy example, consider secrets $x$, $y$, $z$ from $\\mathbb Z_q$ for a large prime $q$. Let $g$, $h$ be generators of a group of order $q$ of Pedersen commitment scheme.\n\nStep 1: Prover chooses $\\alpha_x$, $\\alpha_y$, $\\alpha_z$, $\\beta_1$, $\\beta_2$ at random from $\\mathbb Z_q$, computes and sends commitments to Verifier: $$v_1 = x \\alpha_y + y \\alpha_x - \\alpha_z, \\; v_0 = \\alpha_x \\alpha_y \\\\ D_1 = g^{v_1} h^{\\beta_1}, \\; D_0 = g^{v_1} h^{\\beta_1}$$\n\nStep 2: Verifier chooses a challenge $c$ at random from $\\mathbb Z_q$ and sends it to Prover.\n\nStep 3: Prover computes responses $\\rho_x$, $\\rho_y$, $\\rho_z, \\eta$ and sends them to Verifier: $$\\rho_x = c x + \\alpha_x \\pmod{q} \\\\ \\rho_y = c y + \\alpha_y \\pmod{q} \\\\ \\rho_z = c z + \\alpha_z \\pmod{q} \\\\ \\eta = c \\beta_1 +\\beta_0 \\pmod{q}$$\n\nStep 4: Verifier accepts if $\\rho_x$, $\\rho_y$, $\\rho_z$ open commitments to secrets and $$g^{\\rho_x \\rho_y - c \\rho_z} h^{\\eta} D_1^{-c} D_0^{-1} = 1$$\n\nSoundness of this proof can be derived from observation that, for invalid secrets, verification equation can only hold for at most two values of challenge, to be compared to at most one such value with the well-known proof. It follows, soundness error is $\\frac{2}{q}$.\n\nEfficient range proof from Lagrange 4-squares theorem was introduced at On Diophantine Complexity and Statistical Zero-Knowledge Arguments and was adopted here at ..whether a number is greater than another number without knowing the numbers?. Integer commitment scheme with a group of a hidden order fits range proofs just fine.\n\n-","date":"2016-05-25 18:49:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 1.0000076293945312, \"perplexity\": 4404.7781444819875}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-22\/segments\/1464049275328.0\/warc\/CC-MAIN-20160524002115-00033-ip-10-185-217-139.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/foherab.joeshammas.com\/writing-a-headteacher-reference-angle-88407uu.html","text":"# Writing a headteacher reference angle\n\nClick here for further details. Dinosaur Isle curator, Dr Martin Munt, said: It regularly welcomes schools from across the country on organised trips, and hosts hugely popular fossil walks. The first few days in archives, I felt like everything I was unearthing was a gem, and when I sat down to write, it seemed as if it was all gold.\n\nTiego November 28, at It was all very light and informal and giving back to the community was how Rotary had been sold to her, said Jane. The later biology modules included work on the uniformity and diversity of extinct and living animals and plants; their relationships, anatomy, interactions with each other, and survival strategies.\n\nIt was all about awareness and they had successfully used WhatsApp. Eotyrannus was the first dinosaur page to be replaced, and Iguanodon has now also been updated. But this has now been corrected, and a simple A4 leaflet can now be downloaded print double-sided, and then fold in half to produce a 4 page A5 booklet.\n\nThe early morning had started with heavy rain, but fortunately it had stopped by the time we set out. Due, we hope by the 8th of February. Typically, the cash went to the housewife who could pocket the money without the husband knowing!\n\nWe have unique independent shops bringing in visitors, but car parking charges could kill the centre of the town, and this is troubles me. A scale model was especially commissioned for each of the 26 dinosaurs in the book, all individually designed and constructed by model maker Graham High of Centaur Studios.\n\nYour employment prospects would be instantly better and your transition begin at a much higher economic level. Once the myriad small parts are reassembled and cleaned we may be in a position to provide some answers to what type of sauropod it is, and maybe why some parts were preserved while others were not.\n\nOthers have an anti-authoritarian streak. Blast from the Past - 12th November Saturday the twelfth has been the busiest day of November so far.\n\nThe impact on poverty in developing countries has resulted in Shrewsbury Severn Rotary making a total of 76 loans, helping entrepreneurs, 1, family members and creating jobs. Shrewsbury Severn Rotary Club had received recognition for their contribution to the End Polio Now Campaign where its support had enabled the immunisation of 45, children.\n\nDavid Martill of the University of Portsmouth as a Coloborhynchus, a type not previously found on the Island. They are your lifeline and your best chance of success. Crocus bulbs have been purchased ready for planting around the Abbey this autumn. The bone was originally found in by Steve Hutt and has been carefully conserved.\n\nIt is coated with a brown rocky covering which indicates that it came out of a former river bed that existed here about million years ago. Special Provision Much of the concern which originally led to the Committee's establishment centred around West Indians' fears that their children were being wrongly placed in ESN M schools.\n\nWe need a team effort from everybody - I believe projects get members. Presented by local collectors, societies, Portsmouth University, the Natural History Museum, Island Heritage Service and other luminaries the public were given the opportunity to ask enquiring questions and celebrate some of the rich heritage that goes to make our Island and surrounding area so culturally fascinating.\n\nRev Fox was curate of St. Most cases will end with the same result, regardless of the judge. They are currently on show in our lab whilst we find space in the main museum for them.\n\nUnfortunately, Yorkshire Tea stopped supporting this and now gave a monthly donation. The fermentation eats the sugar and turns it into beer.\n\nOne of our visitors found 8 complete modern Ophiura starfish in ponds on the beach. The first job was to get as near as possible to the building in order to off-load the vehicle. The Committee concludes with a call for comments on this report and further evidence for its main report.\n\nAlso called a stay-back or stoppy-back in Northern England. Through my charity, I want to make a difference to an often overlooked, stigmatised, vulnerable group, a group that may include our friends, husbands, wives, sons, daughters, sisters, mothers or fathers.\n\nAt that time, less than half the houses in the country had electricity.\n\nThis was followed by work in Ireland on behalf of Irish Independent Newspapers where he obtained more than half of the 32 microwave TV licences issued by the Irish government to cover Eire.\n\nRadios were run off batteries that were replaced weekly by a local dealer who took them away and recharged them. He was speaking to members of Shrewsbury Severn Rotary Club, only his third club visit in just nine days after taking office.Writing a dissertation can be one of the hardest tasks a university student has to accomplish \u2013 but it will come to an end.\n\nPhotograph: Randy Faris\/Corbis The sun is shining but many students. View the latest news from Dinosaur Isle, including the latest finds; including Caulkicephalus trimicrodon a new species of pterosaur unique to the Isle of Wight.\n\nThe original is held by Dinosaur Isle. The reference angle is the positive acute angle that can represent an angle of any measure. The reference angle $$\\text{ must be } 90^{\\circ}$$ In radian measure, the reference angle $$\\text{ must be } \\frac{\\pi}{2}$$ Basically, any angle on the x-y plane has a reference angle, which is.\n\nBreaking headlines and latest news from the UK and the World. Exclusives, live updates, pictures, video and comment from The Sun. Rotary Club of Shrewsbury Severn - Welcome!\n\nServing the local, national and international communities. comments. March 10, - am Douglas. I agree it is not corrupt neither it is not fit for purpose. 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Spiros Law, P.C. > Blog > Workers' Compensation > Construction accident on I-57 injures one
Construction accident on I-57 injures one
An accident at a northbound I-57 construction work site hospitalized Brent Harris, 20, on Friday afternoon, August 2, 2013.
Harris, a construction worker, was reported to have been pinned against a barricade by an excavator that turned and struck Harris just as he was walking past. Kevin Delmore, 43, was operating the John Deere excavator at the time of the accident.
Harris was admitted to Carle Foundation Hospital in Urbana after the accident, where he is now listed as in "fair condition". No legal action or fines have been implicated against Delmore or the construction company.
Construction workers have some of the most dangerous jobs in the country, and when they get hurt on the job, they may be eligible for compensation. If you or a loved one has suffered lasting damage from a construction accident, the legal team of Spiros Law, P.C., may be able to help. Call us at (217) 328-2828 to learn more.
Categories: Workers' Compensation
Tags: construction accident, workers' compensation
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Thomasboro
Villa Grove | {
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Fuglsang er en avlsgård under Corselitze. Gården ligger i Toreby Sogn, Musse Herred, Guldborgsund Kommune. Hovedbygningen er opført i 1868-1869 ved J.G. Zinn.
Hovedbygningen, herskabsstalden og havepavillonen blev fredet i 1983, og i 2006 blev fredningen udvidet med havemure, lygter og bro mm.
Fuglsang / Priorskov Godser er på 1731 hektar med Nagelsti Skovgård og Flintingegård. På området findes også Fuglsang Kunstmuseum.
Historie
Godset er en gammel hovedgård, som nævnes i 1368. Den oprindelige befæstede borg lå nogle hundrede meter længere mod nord, hvor voldstedet, Gammelholm stadig kan ses fra landevejen mellem Nykøbing og Nysted. Placeringen i udkanten af sumpene ved Flintinge Ås udmunding i Guldborg Sund, tæt på en af de få vadesteder har givet mulighed for at beherske trafikken mellem Nordøstlolland og egnen omkring Nysted samtidig med, at stedet lå godt for kontrol af det sydlige Guldborg Sund.
Flintinge Å er på trods af sin lidenhed Østlollands største vandløb. Ca. 2 km. mod syd lå en anden befæstet gård ved den langt mindre Frejlev Å. Denne befæstning blev sandsynligvis revet ned i dronning Margrethes 1.s tid med hjælp af landsbyen Frejlevs beboere. Derefter var Fuglsangs nabo i syd den vigtige kongsborg Ålholm.
I det 16. århundrede flyttedes Fuglsang til sin nuværende beliggenhed på en ny, større holm med brede grave om. Dette hus stod til 1840, hvor det blev erstattet af en senklassicistisk hovedbygning. På den tid blev også haven omlagt til det anlæg, som ses i dag. Rolf Viggo de Neergaard overtog godset 1866 efter sine forældre, men på grund af svampeangreb i det kun 26 år gamle hus, måtte han lade det rive ned. Ved hans fætter, arkitekten J. G. Zinn, opførtes så den nuværende hovedbygning i tidens historiske blandingsstil. Huset stod færdigt i 1869
Kulturelt centrum
Rolf Viggo de Neergaard giftede sig i 1885 med Bodil Hartmann. og frem til hans død blev Fuglsang et meget gæstfrit hjem med forbindelser til alle sider af dansk åndsliv. Musikken har altid haft en fremtrædende plads i huset og har det stadig med talrige koncerter året igennem. Både Edvard Grieg og Carl Nielsen blev Neergaards nære venner og kom her ofte. Bodil Neergaards bror, maleren Oluf Hartmann, malede bl.a. de lave kystlandskaber mod Guldborg Sund. Carl Nielsen skrev "Ved en ung kunstners båre" som et minde om Oluf Hartmann, der døde allerede 1910. En anden maler, der havde tilknytning til egnen, var Olaf Rude, som var vokset op på en gård tæt på Fuglsang, og som malede 2 store billeder fra Skejten til Folketingssalen, hvor de hænger på bagvæggen.
Efter Rolf Viggo de Neergaards død i 1915 styrede hans enke Bodil Neergaard den store ejendom alene og organiserede det sociale og kirkelige arbejde, som var begyndt allerede i hendes mands tid. I 1947 overdrog hun det 3000 tdl. store gods til stiftelsen Det Classenske Fideicommis med den hensigt, at hendes hjem og haven skulle være refugium (latin: tilflugtssted), når hun var død. Denne overdragelse var dog, og er stadig, utrolig kontroversiel, idet Rolf Viggos testamente, som sagde at én af hans slægtninge skulle overtage stedet, blev omgået. 1962, indviedes så Refugiet Fuglsang som en selvejende institution med hjemsted i hovedbygningen. Det måtte dog lukke ved årsskiftet 1995 på grund af svigtende tilslutning.
I 1996 stiftedes Fuglsang Musikforening, som afholder 9 koncerter om året i hovedbygningen, og i 1997 flyttede Storstrøms Kammerensemble ind med administration og øvelokale. Andre foreninger som Det Musiske Selskab eller private kan benytte hovedbygningen, der stadig står med den indretning og det inventar, som Rolf og bodil Neergaard udstyrede bygningen med.
I januar 2008 blev Fuglsang Kunstmuseum indviet i nye bygninger tæt på hovedbygningen, og hele området er udset til at være et kulturelt centrum på egnen.
Ejere af Fuglsang
(1368-1390) Anders Syndesen Mule
(1390-1423) Axel Andersen Mule / Sidsel Andersdatter Mule gift Thott
(1423-1446) Mette Jensdatter Thott gift Kabel
(1446-1457) Maribo Kloster
(1457-1477) Oluf Andersen Gjøe
(1477-1520) Sophie Olufsdatter Gjøe gift Venstermand / Dorte Madsdatter Bølle / Birgitte Daa gift Rud
(1520-1554) Knud Rud
(1554-1577) Erik Knudsen Rud
(1577-1630) Corfitz Eriksen Rud
(1630) Helvig Corfitzdatter Rud gift Krabbe
(1630-1645) Gregers Krabbe
(1645) Vibeke Gregersdatter Krabbe gift Daa
(1645-1661) Christen Daa
(1661-1685) Dronning Sophie Amalie
(1685-1726) Kronen
(1726-1727) Christian Carl Gabel
(1727-1759) Abraham Lehn
(1759) Catharina Margaretha Lehn gift Wallmoden
(1759-1789) Christopher Georg von Wallmoden
(1789-1819) Friedrich von Wallmoden
(1819-1835) Peter Johansen de Neergaard
(1835-1849) Johan Ferdinand Petersen de Neergaard
(1849-1866) Enkefru de Neergaard
(1866-1915) Rolf Viggo Johansen de Neergaard
(1915-1947) Ellen Bodil Hartmann gift de Neergaard
(1947-) Det Classenske Fideicommis
Fuglsang trinbræt
Fuglsang fik i 1927 trinbræt på Stubbekøbing-Nykøbing-Nysted Banens strækning Nykøbing-Nysted, som var indviet i 1910. Trinbrættet lå 1½ km nordvest for herregården ved vejkrydset i den østlige ende af Sønderskov. Det havde læskur og læssevej ved sidesporet, der – ret usædvanligt for et trinbræt – havde sporskifte i begge ender. Herfra er der sendt store sukkerroe- og halmtransporter. Persontrafikken på banen blev indstillet i 1961, godstrafikken i 1966.
Galleri
Se også
Fuglsang Kunstmuseum
Det Classenske Fideicommis
Johan Frederik Classen
Hartmann-slægten
Noter
Eksterne kilder/henvisninger
Fuglsang, Priorskov, Corselitze Godser
Fuglsang - fra Dansk Center for Herregårdsforskning
Bodil Neergaard: Minder fra Fuglsang. Gad, 1944
Om Oluf Hartmann og Skejten
Fuglsang musikforening
Realdania
Fuglsangs hjemmeside
pjece om Fuglsang og Skejten
Halvdan Grøndal Hansen: Fuglsang – Bodil Neergårds hjem (eget forlag 2010)
J.P. Trap: Danmark 1955, Kraks Landbrug
Fritz von Bressendorff: Fuglsang - kan svanerne komme tilbage? Skriveforlaget 2017
Herregårde i Guldborgsund Kommune
Herregårde i Maribo Amt
Fredede slotte og herregårde i Danmark
Det Classenske Fideicommis
Fredede bygninger, konstruktioner og anlæg i Guldborgsund Kommune | {
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Home > Collections & Research > Literary Collections
In 2009 Tracy Chevalier published her novel Remarkable Creatures about Lyme Regis's famous fossil-hunter Mary Anning and her friendship with Elizabeth Philpot.
John Fowles, writer, curator, collector. In 1978 when the writer John Fowles became curator of Lyme Regis museum, its fortunes started to change.
Jane Austen (1775 – 1817) was one of Lyme Regis's most famous and best-loved visitors. Her great novel, Persuasion, published in 1818, is in part set in Lyme, making the Dorset town a centre of literary pilgrimage ever since.
Lyme's Literary and Artistic Connections
Lyme Regis has long been a magnet for artists and writers. Many came to this picturesque Dorset town for holidays and were inspired by the character of the town and its environment. Others came here for health reasons. Some came as children and returned in adulthood. | {
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I didn't just run out and start buying Yorkshire Terriers to breed .... I've taken my time and searched not only the US, but other countries, to find just the right Yorkshire Terriers to start my breeding program. It's not been easy to find my dream Yorkies. But I have found a few! And I'll continue looking and may add a couple more. But it will take time, good dreams don't come easy, but I'm excited about my wonderful start and invite you explore my web site and take a look at pictures and pedigrees of my dream Yorkies!
I'm not an inexperienced breeder! I've breed some of the most darling little Chihuahuas and I will always love that breed! I've taken my knowledge of dog care and breeding ethics and have expanded my dream, of breeding little Yorkie puppies.
I will proudly display pictures and pedigrees of each of my Yorkshire Terriers. I will note their characteristics and their temperament. You will also find extended information on my Yorkie puppy nursery page, offering only delightful healthy Yorkie puppies for sale.
Dream Maker Yorkies believes in producing the very best yorkie puppies possible. And because of this we have very strict guidelines that insure the health of our yorkie puppies.
All of our yorkie puppies follow a tried and proven regimen of worming and vaccinations. And we follow this up with a vet check.
We KNOW we produce not only beautiful Yorkshire terrier puppies, we produce healthy ones. And we gladly back this up with a written health guarantee on each yorkie puppy we sell.
We ask a lot of questions before we place one of our special yorkie puppies. You are welcome to do the same. We'd be happy to discuss our routine health care practices with you. So, look through our site, check out our adult yorkies, our adorable yorkie puppies...then e-mail or call with any questions you may have and we'll discuss yorkies!
Website and Images Copyright 2008 © Dream Maker Yorkies. May not be copied or used without permission. All rights reserved.
Website Design by KJ's Designs -Keeping it Simple. Making it Happen. | {
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Sicília
Língua siciliana — língua românica falada na Sicília
Siciliana — dança tradicional barroca
Livraria Siciliano — tradicional rede brasileira de livrarias
Desambiguações de gentílicos | {
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A few months ago this blog reported that, after Italy [see here, and Katposts here], also Greece has now adopted a model of administrative enforcement for online copyright infringement cases.
From Katfriend Katfriend, fellow blogger (iprights.gr), attorney and post-doc researcher Dr Theodoros Chíou (Dr. Theodoros Chíou & Partners, Intellectual Property and Internet Law Office) comes the news that the Greek system has now entered into force.
"Greece has recently become another European country to opt for the introduction of an out-of-court notice and take down (or access blocking /content removal) legal mechanism for illegal online content, based on a procedure carried out in front of and under the responsibility of an new-founded ad hoc administrative authority, namely the "Commission for the notification of online copyright and related rights infringement" (hereinafter: the Commission) οr, transliterated in brief, 'EDPPI'.
The enactment of this mechanism occurred last summer by means of Law 4481/2017 and the introduction of a new analytical Article 66 E in the Greek Copyright Act (Law No. 2121/1993). However, its effective operation has just begun, as the necessary ministerial decisions for the implementation of the provision and the establishment of the Commission were adopted in February 2018.
The use of the new mechanism is voluntary and independent from the judicial protection of rights. The law expressly stipulates that bringing the procedure in front of the Commission does not suspend or prejudices the judicial recourse of the right holders for the same case at the courts.
Similarly, the decision of the Commission does not prevent the parties involved from claiming the protection of their interests in front of courts (in other words, the decision of the Commission does not create a binding res judicata for judicial authorities).
The mechanism may be triggered by the submission in front of the Commission of a formal application by any primary and secondary right holder of copyright or related rights, including CMOs, whose rights are assumed to be infringed.
However, the new mechanism is available in respect of online infringements committed essentially on or via websites (by reference to domain names or IP addresses). Indeed, it is clearly stipulated in the law that the procedure is not applicable to (and, thus, the Commission is not competent for) infringements committed by end-users either through downloading or streaming of protected works, P2P file exchange or cloud storing. On the contrary, the new legal mechanism targets mainly internet intermediaries, which will be the "defendants" within the procedure in front of the Commission.
The Commission is a 3-member administrative authority (but not an independent public authority) whose seat is at the Greek Copyright Organisation (OPI). It is composed of the President of OPI (who presides the Commission), a representative from the Greek Data Protection Authority and a representative from the Greek Telecommunications & Post Commission (who is the secretary of the Commission) with a 3-year mandate. The Commission has decisive power to receive complaints and issue decisions and control their implementation within the frame of the new notice and take down procedure, as prescribed by the legal texts.
· The applicant (or his representative) fills out and submits (either electronically, by email, by post or in person) the model request form provided by OPI accompanied by all mandatory supplementary documentation (as prescribed in the form or requested by the Commission) and any other relevant evidence for establishing his right (and the infringement in case), which includes litigated URLs, identification of content and copyright ownership of the applicant.
· The applicant pays the relevant fee in favour of OPI. The amount can vary from a minimum of EUR 300 to a maximum of EUR 1,000 (+VAT 24%), depending on the number of litigated domain names and type of illegal exploitation (streaming etc.).
· The applicant has made use of relevant notice and take down procedure (if any) which is made available by intermediary service providers involved for the litigated content, without results.
Upon accepting the application, the Commission notifies, by any appropriate means, including by email, within 10 work days from the application date simultaneously all Greek ISPs and, and wherever possible, the web hosting service provider and administrators and/or owners of the websites mentioned in the application. The notification in question mentions at least the rights infringed, the legal provisions that have been infringed and a summary of the assessment of documentation and evidences submitted along with the application.
· submit objections to the Commission within 5 days from the receipt of the notification, along with any relevant evidence proving that no infringement is at stake. After the end of the objection period, the Commission may invite any involved party in the procedure (i.e. the right holder or the recipient(s)) to submit additional elements within 5 days.
After the end of the preliminary stage, the Commission examines the case and issues a decision which shall be issued and communicated equally to the right holder and the recipients of the notification within 40 to 60 work days from the date of submission of the application at the latest.
b. Block the access to illegal content described in the application, by using the most appropriate and effective technical means, which may include IP address or domain name server (DNS) blocking.
The choice depends on the features of the infringement at issue.
It appears that the Commission shall opt for content removal if the website containing the litigated content is hosted on a server located in Greece. The removal shall be permanent, unless a licence from the right holders is obtained. However, the Commission may opt for access blocking, instead of content removal, in case of "national" large scale infringements. Moreover, in case that the litigated website is hosted on server outside Greece, the Commission shall invite ISP's to block the access to the illegal content. In any case, the exact duration of access blocking is decided solely by the Commission, but IP blocking cannot exceed 6 months and DNS blocking (including subdomains) cannot last less than 3 years. Besides, the Commission may ask from ISPs to redirect users who attempt to access the blocked website to a screen with informative message regarding the reasons of the blocking.
It should be noted that the decisional meetings of the Committee are not public and are covered by secrecy while the decisions may be published on the OPI's website at the Committee's sole discretion.
The recipients of the decision shall comply with content removal or access blocking obligation within 3 working day from the date of communication of the Commission's decision. Moreover, the Commission is the competent authority for monitoring the compliance of the recipients with the decisions. In case of non-compliance, the recipients are subject to fines imposed by a separate decision of the Commission which may be between EUR 500 to EUR 1,000 per day of non-compliance, depending on the characteristics of the copyright infringement in question.
This new anti-piracy legal mechanism is undoubtedly a useful tool for right holders who wish to tackle the online infringement of their rights, as it seeks to be not only fast but also practical: it focuses on technical solutions applied by internet intermediary service providers - and especially ISPs – and prescribes quite high fines in case of non-compliance. Of course, its efficiency and success could be assessed only over the months to come and depends on the use made by right holders and the results achieved in practice. | {
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Owner: Antarctic Heritage Trust, New-Zealand. This artefact comes from the collection of c.18,000 artefacts at Terra Nova Hut, the expedition base built by Captain Robert Falcon Scott and party during his 1910-1913 expedition to Antarctica. This iron-alloy tin is heavily corroded; the label and the wrapper are significantly damaged over most of their surfaces as well, stained by corrosion and covered by a thick layer of old marmalade. During initial cleaning it was apparent it would be possible to remove the paper elements from the tin, allowing more comprehensive treatment of the paper, and the opportunity to clean and stabilize the metal (surface corrosion was converted and coated). The deteriorated food content was removed from the tin, and the paper, once washed and repaired, was replaced onto and around the tin. | {
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Paul George's Two-Way Brilliance Has The Thunder Kicking Ass
Photo: Julio Cortez (AP)
Since starting the season 0-4—a stretch highlighted by giving up 131 points to the then-inept Sacramento Kings at home—the Oklahoma City Thunder have molted and become one of the NBA's best teams. Blowout wins over elite teams like the Clippers and Warriors have been peppered in amid a tremendous 17-4 stretch, in which they've established their defense as the best in the NBA by a healthy margin and surged all the way to second in the hyper-competitive Western Conference. The man to thank for all that is Paul George.
The conventional wisdom dictated that George would abandon the dusty confines of Oklahoma after one season and six playoff games with Russell Westbrook in order to compete for a championship alongside, say, LeBron James in his hometown. Instead, George returned to the Thunder as soon as free agency started and settled in on the most expensive team of all time as the running mate to a point guard who is, let's say, aggressive about getting his.
It's working! Really well! George is averaging 24.3 points, 7.8 rebounds, 4.3 assists, and 2.2 steals per game, all of which are career-highs. All of the other advanced metrics point to this campaign as George's best statistical season, which is impressive coming next to Westbrook, a man who likes to have the ball in his hands at all times, always, unconditionally. To his credit, Westbrook has backed down this year, using the rock on only 31.9 percent of Thunder possessions, a rate comparable to his first All-Star season in 2010-11. Westbrook also leads the league in assist percentage, and has cleaned up his shot selection. As he's chilled out, George has stepped up, and their games complement each other perfectly; Westbrook explodes into the lane and creates chaos, George hits shots, cleans shit up, and exploits mismatches.
Last night against the Jazz, Westbrook had one of his increasingly regular bad shooting nights, going 4-for-18 from the floor, 0-for-5 from three, and 4-for-8 from the stripe. No matter: George notched 31 points on just 10 shots, ripping the team that bounced OKC from last year's playoffs for 17 points in the decisive third quarter.
The shooting is good and cool, and George looks very confident this year, but the most important factor in the Thunder's hot streak has been the team's smothering defense. Billy Donovan has talked for years about the sort of team he wants, one built around long athletes who outwork and overwhelm their opponents. His post-Durant squads were weird edifices built in support of Russell Westbrook's irrepressible athleticism, and they were not what anyone would exactly call system-driven.
Now, with George in his second season, Donovan has built the defense he always dreamed of. OKC steals the rock more than any other team, which leads to a good helping of fast break points. In halfcourt sets, Steven Adams's considerable bulk allows George and fellow long guys Terrance Ferguson and Jerami Grant to get aggressive; as the Ringer's Jonathan Tjarks points out, OKC attacks the pick-and-roll better than almost every other team. Grant and his 7-foot-3 wingspan both recently moved into the starting lineup, and he's been great. Consider also that Westbrook is having the best defensive season of his career, and you have a five-man unit that has no holes in the pick-and-roll and is defined by its aggression. They close out with gusto on the three-point line, and OKC's opponents shoot the fourth-fewest threes and the fifth-worst percentage from long-range. This is, as a very long quote from Donovan shows here, the point.
No other team gets more extra possessions per game than OKC. They lead the league in both defensive rating and offensive rebounding percentage, two categories that have tended to be mutually exclusive in recent years. As the league has integrated components from the Rivers-Thibodeau system over the past half-decade, teams have abandoned chasing offensive rebounds, a famous staple of Doc Rivers's Celtics title teams. If you don't crash the glass, you can get back in transition and eliminate some of the easiest points in the game. But if you have plus-rebounders at every starting position and someone as large as Steven Adams fighting on every possession, you're going to get yourself more opportunities by sheer force of physicality.
And they're doing this all without Andre Roberson, one of the NBA's five best on-ball defenders. Roberson went down with an injury last year, and the Thunder only had a league-average defense without him. George can guard every position comfortably, and every pass that comes within his considerable orbit is a steal target. He's probably been the defensive player of the year so far, to say nothing of the fact that he's also been the Thunder's best shooter. This play shows it all; he's so long that he completely obliterates what is normally considered a legitimate passing lane, and once he gets the rock, he can handle and pass and shoot like a superstar.
The successes and failures of the post-Durant Thunder have been almost entirely determined by Russell Westbrook's merciless hold on the franchise. This season, and last night's game in particular, have proven that for the first time since Durant left, the Thunder now have a team that can still play at a high level even when Westbrook's grip fails him, or gets too tight. Nobody is more responsible for that than Paul George.
Staff writer, Deadspin | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,614 |
Locals to play big roles in senior basketball all-star games
Mar 20, 2013 | Uncategorized
WINDSOR — Area high school basketball players — and coaches — will be well represented at Saturday's annual Vermont Basketball Coaches Association North-South senior all-star games at Windsor High School.
Vergennes Union High School seniors Shep Carter, Zach Ouellette and Stanley Salley will play for the North Division I and II boys' team that will meet the South team at 5 p.m. And VUHS coach Peter Quinn will lead the North D-I and D-II boys in that game.
The South team will have a familiar look: Middlebury seniors Connor Collins and Tyler Provencher will play for the South squad, as will Otter Valley senior Ryan Kelley. And MUHS coach Chris Altemose will lead the South team.
One local senior will also play at 3 p.m. in the North-South D-I and II girls' game: The Tigers' Tiffany Danyow will suit up for the South.
The D-III and IV girls' game is set for 11 a.m., and the D-III and IV boys' game is scheduled for 1 p.m.
In addition to the four games, the VBCA also will recognize players of the year, coaches of the year, 1,000-point scorers, Dream Dozen teams of underclassmen, coaches' milestone wins, and a VBCA Hall of Fame inductee.
Among the honorees will be Quinn, as the D-II boys' coach of the year; Quinn and Altemose, for reaching 100 wins; and Mount Abraham junior Ashlie Fay and OV junior Jessica Frazier, as members of the girls' Dream Dozen team.
Tickets will be available at the door at Windsor High School. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,652 |
\section{ INTRODUCTION}
The origin of ultrahigh energy cosmic rays (UHECR) remains one of the most challenging unknowns in astrophysics.
Directional data from the Pierre Auger Observatory (PAO) in the southern hemisphere and the Telescope Array (TA) in the northern hemisphere give hope that we are close to identifying their source.
The Hillas (1984) parameter $E_{\rm max}=ZuBR$ for the maximum attainable cosmic ray (CR) energy expressed in eV, where u, B and R are the characteristic velocity, magnetic field and spatial scale and $Ze$ is the UHECR charge, places
telling constraints on the nature of the source.
As pointed out by many authors (Lovelace, 1976; Waxman, 1995; Blandford,
2000; Lemoine \& Pelletier, 2010), the Hillas parameter is related to the magnetic power $P_{\rm mag}$ of the source:
$ZuBR=Z(u \mu_0 P_{\rm mag})^{1/2}$ where $P_{\rm mag}=uR^2B^2/\mu_0$.
This relation can be inverted to give a lower limit on the magnetic power, $P_{\rm mag}>E_{\rm max}^2 /(Z^2 u \mu _0)$.
The total source power must exceed this because (i) the magnetic energy is only one part of the total energy, (ii) the power channeled through the accelerating structure, probably but not necessarily a shock,
is a fraction $\xi$ of the total source power.
Taking characteristic values suitable for radio galaxies, the total power can be estimated:
$$
P_{\rm total} \sim 2 \times 10^{44} Z_6^{-2} E_{100}^2 (u/0.3c)^{-1}\xi_{0.1}^{-1} \ {\rm erg\ s}^{-1}
\eqno{(1)}
$$
where $Z=6Z_6$, $E_{\rm max}=100 E_{100}\ {\rm EeV}$, $\xi=0.1 \xi_{0.1}$.
Taken together, the Hillas condition and the related power requirement restrict the range of possible sources of UHECR.
The stringent power requirement points to radio galaxies (RG) or gamma-ray bursts (GRB) as possible contenders.
GRB easily reach the required power due to the short time in which energy is released, but there are many issues
that question the viability of GRB as an explanation of UHECR origins.
These issues relate to
limits placed by the expected associated neutrino fluxes, radiation losses during UHECR acceleration and escape, host galaxy metallicity, UHECR composition
and the ability of GRB to provide the entire UHECR flux.
(Hillas 1984, Waxman \& Bahcall 1997, 1998, Aharonian et al 2002,
Ptitsyna \& Troitsky 2010, Dermer 2011, Baerwald et al 2015,
Anchordoqui 2018, Zhang et al 2018, Alves Batista et al 2019, Boncioli 2019, Samuelsson et al 2019; Heinze et al 2020, Rudolph et al 2021).
An additional difficulty with GRB as a producer of UHECR is that relativistic shocks are not good accelerators to ultra-high energies (Bell et al 2018).
Here we focus on RG as a natural source of UHECR because they are large, powerful, and long-lived.
Also, radiative losses are not a strong limitation if acceleration takes place
in the lobes where magnetic and radiation energy densities are relatively small.
RG display features such as shocks, shear flows, and magnetic reconnection that are known to be suitable for CR acceleration.
Cygnus A is a clear candidate for UHECR production since it far exceeds the power requirement (Eichmann et al, 2018),
but its distance of 237Mpc (Persic \& Rephaeli 2020) counts against it because
(i) UHECR from Cygnus A are unlikely to arrive at the Earth anisotropically without scattering since a magnetic field of only $0.03{\rm nG}$ is needed
to make their Larmor radius smaller than the distance to Cygnus A
(ii) even without scattering, the UHECR travel time from Cygnus A is many times the loss time due to the
Greisen–Zatsepin–Kuzmin (GZK; Greisen
1966; Zatsepin \& Kuz'min 1966) effect and photodisintegration
(e.g. Stecker \& Salamon 1999, Hooper, Sarkar \& Taylor 2007).
In fact, powerful FRII galaxies are rare within the local GZK volume (Massaglia 2007, van Velzen et al 2018).
FRI radio galaxies are more populous, and radio galaxies at the FRI/FRII boundary have powers close to $P_{\rm total}$ as given in equation (1).
The most obvious FRI candidate is Centaurus A (Cen A) which is nearby and estimated to have a cavity power of
$10^{44}{\rm erg \ s}^{-1}$ (Hardcastle et al 2009, O'Sullivan et al 2009, Cavagnolo et al 2010, van Velzen et al 2012, Matthews et al 2019).
The case for Cen A is strengthened by the coincidence of Cen A with the brightest UHECR hotspot detected by PAO (Aab et al 2018) and the apparent need for local UHECR sources (Guedes Lang et al 2020).
PAO also detects a weaker hotspot with weaker statistical significance in the direction of the FRI galaxy Fornax A which has similar characteristics to Cen A
(Matthews et al 2018), including evidence of variability (Mackie \& Fabbiano 1998, Maccagni et al 2020).
The arrival directions of UHECRs favour starburst galaxies or AGN as likely sources.
Aab et al (2018) and
Bister et al (2021) find a statistically significant correlation of PAO arrival directions with active galaxies, but find an even stronger correlation with starburst galaxies.
Futher evidence pointing toward starburst galaxies and away from radio galaxies is the lack of a powerful radio galaxy in the direction of the UHECR hotspot
detected by TA in the northern hemisphere.
An analysis of the TA data for associations with starburst galaxies is not statistically significant (Abbasi et al 2018).
However, the TA hotspot lies in the direction of the nearby strong starburst galaxy M82.
Hence starburst galaxies appear to be a strong candidate.
However, their powers reach typically $10^{42}{\rm erg \ s}^{-1}$
and the characteristic wind velocity in a starburst region is $\sim c/100$,
(Heckman, Armus \& Miley 1990; Anchordoqui 2017; Romero,
Muller \& Roth 2018, Matthews et al 2020),
so they fail to reach the power condition (equation 1) for a source of UHECR.
We therefore proceed by ruling out starburst galaxies as a direct source of UHECR despite their statistical correlation with UHECR arrival directions.
The possibility remains that UHECR are produced by GRB associated with starburst galaxies.
Here we explore the possibility that starburst galaxies are instead reflectors of UHECR without themselves being a source of UHECR.
We propose that the environs of starburst galaxies are regions of enhanced scattering that diffusively reflect UHECR back towards the Earth after their initial acceleration by Cen A.
In particular we propose that the TA hotspot is associated with the M81 galaxy group that contains the prominent starburst galaxy M82.
M82 is close by at a distance of 3.7Mpc from the Earth,
which is similar to the distance of Cen A from the Earth.
Starburst galaxies can generate a large magnetic field, $100-300\mu{\rm G}$, in the central kpc
(Domingo-Santamari \& Torres 2005, de Cea del Pozo, Torres \& Marrero 2009, Anchordoqui 2018)
which may be carried 100s kpc into the surrounding medium by the superwind.
In a spherical wind expanding at constant velocity, a frozen-in magnetic field decays as the inverse of the distance.
Hence there is good reason to think that the magnetic field may easily exceed the $\sim 10{\rm nG}$ needed to make the UHECR Larmor radius smaller than the wind radius should it reach 1Mpc.
A magnetic field of $\sim 100{\rm nG}$ at 100 kpc would similarly be sufficient for UHECR to be scattered in a starburst wind.
The TA hotspot extends an angular diameter of $20 -30^\circ$ across the sky (Kawata et al 2015, Anchordoqui 2018),
corresponding to a physical radius of $0.6-1 {\rm Mpc}$ at the distance of M82.
This is consistent with the analysis of di Matteo et al (2019) who
identify the hotspot by drawing around it a circle of radius $15^\circ$ corresponding to $1{\rm Mpc}$.
The size of the hotspot is due to a combination of UHECR scattering and the size of the source, and the
UHECR numbers are not sufficient to clearly define the shape and size of the hotspot.
In our model, we assume that M82 and the M81 group present to UHECR an optically thick sphere of radius 800kpc.
The radius of 800kpc then corresponds to the distance to which the starburst superwind affects the surrounding intergalactic medium to
produce an enhanced magnetic field.
An extension of this radius into the surrounding medium is consistent with recent observations (Wilde et al 2021) given that starburst winds are stronger than typical galactic winds.
Other factors commensurate with an enhanced magnetic field in the M81 group are the presence of a second starburst galaxy NGC3077 in the group
and dynamo action arising from motions as galaxies in the group interact with each other.
Cen A, M82 and the Earth form an approximately isosceles triangle with M82 and Cen A both about 3.7Mpc from the Earth as shown in figure 1.
The corresponding distance between Cen A and M82 is 6.4Mpc.
The UHECR travel time from Cen A to the Earth via M82 is longer by 21Myr,
which means that the UHECR luminosity of the M81 group reflects the luminosity of Cen A 33Myr ago as it would have appeared at the Earth 21Myr ago.
As discussed by Matthews \& Taylor (2021), the time variation of UHECR production by radio galaxies depends on variation in accretion onto the central black hole,
the hydrodynamic response of the jets and lobes, the sensitivity of the acceleration process to the presence of shocks and turbulence, and the residence time of UHECR in the lobes.
Matthews \& Taylor (2021) consider a jet power that varies according to a flicker noise power spectrum with a log-normal distribution of jet powers,
motivated by the chaotic cold accretion model of Gaspari et al. (2017) and numerical simulations of AGN fuelling (Yang \& Reynolds 2016; Beckmann et al 2019);
these studies suggest that variation in jet power by orders of magnitude on a 20Myr timescale is reasonable even before considering the individual history of Cen A.
UHECR production can be especially spasmodic due to the need to exceed the power threshold.
Conditions for UHECR acceleration may exist only during peak periods of activity with CR failing to reach UHECR energies during quiescent periods.
In our picture, Cen A is in a relatively inactive state at the present time with weak UHECR production compared with 20~Myr ago when it was more active. We note that this timescale is comparable to the spectral age of the synchrotron electrons in the giant lobes, estimated to be $\sim25$~Myr by Hardcastle et al (2009).
From repeated application of the inverse square law, the UHECR flux arriving at the Earth via M82
is about $100 \times$ smaller than the flux arriving directly from Cen A if the same number of UHECR are released
in the directions of M82 and the Earth.
This is confirmed by figure 2 below where we show the results of a Monte Carlo calculation.
It is not unreasonable that the UHECR lumnosity of Cen A over longer timescales of 10sMyr should
be greater than its present luminosity.
We have previously suggested (Matthews et al, 2018, 2019) that the giant inflated lobes of Cen A act as UHECR reservoirs
and that the present flux of UHECR direct from Cen A is the leakage from the lobes of UHECR accelerated during past
active episodes.
This would explain how UHECR can now be arriving from Cen A when the present jet power of $\sim 10^{43}{\rm erg \ s}^{-1}$
(Russell et al 2013, Wykes et al 2013, Matthews et al 2018)
falls short of that required by equation (1).
A further indication of past enhanced activity is that
the size and energy content of the large radio lobes is large in comparison with the
present weak radio jets that extend only a small distance into the lobes.
From figure 4 of di Matteo et al 2019, we estimate that the UHECR flux arriving at the Earth via M82 above 53EeV is the equivalent
of an isotropic UHECR source with power $3 \times 10^{39}{\rm erg \ s^{-1}}$ at the position of M82.
Allowing for the factor of 100 discussed above from double application of the inverse square law,
the UHECR power of Cen A as it would have been observed at the Earth 21Myr ago was
$\sim 3 \times 10^{41}{\rm erg \ s^{-1}}$, which is $100-300 \times$ its UHECR power as observed now.
Further indications that the UHECR power of Cen A might have been greater in the past are
(i) the present UHECR (energy greater than 55EeV) power of Cen A, estimated to be $2 \times 10^{39}{\rm erg \ s}^{-1}$ (Joshi et al 2018), is very much smaller than
the Eddington luminosity of $5 \times 10^{45}{\rm erg \ s}^{-1}$
corresponding to its central black hole mass of $5.5 \times 10^7$ solar masses (Capellari et al 2009)
(ii) the large energy content, $10^{59}-10^{60}{\rm erg}$ of the lobes (Wykes et al. 2013; Eilek 2014), suggests a larger energy input in the past
(iii) previous enhanced activity is consistent with a history
of galaxy mergers in the past 1-2 Gyr (Wang et al 2020).
In the next section we explore this model further with the use of Monte Carlo calculation of a burst of UHECR emitted by Cen A,
encountering a sphere of enhanced scattering representing the M81 group, and propagating from there to a detector at the Earth.
\section{ AN ILLUSTRATIVE MONTE CARLO CALCULATION}
We use a Monte Carlo model of small angle UHECR scattering to examine the possibility that the TA hotspot is due
to enhanced scattering by the M81 group of galaxies and principally by the halo of M82.
Given the uncertainty in crucial parameters the model is illustrative rather than definitive.
Many detailed variants of our basic model might be considered.
Our picture is that the lobes of Cen A act as a reservoir of UHECR (Matthews et al 2018, 2019, 2020).
UHECR are injected into the reservoir over a period of $\sim 4 {\rm Myr}$ and varying in time as a Gaussian with $\sigma = 2 {\rm Myr}$.
UHECR then leak out of the reservoir with an exponential decay time of $3 {\rm Myr}$.
The decay time is very uncertain since it depends on UHECR in magnetic field with an unknown structure.
Our decay time of $3{\rm Myr}$ is
chosen to reproduce the observed ratio of the Cen A and M82 hotspots as shown in figure 2.
It is consistent with, if somewhat shorter than, the estimates made by Matthews \& Taylor (2021).
In the calculation, the time $t=0$ corresponds to the peak in UHECR production by Cen A.
The UHECR arriving at the Earth via M82 are delayed $\sim 20{\rm Myr}$ relative to those arriving by direct propagation from Cen A.
We ignore GZK and other losses since they occur on a longer timescale.
We treat the UHECR as monoenergetic with a uniform scattering rate outside the halo of M82.
The scattering rate is chosen to give an approximate match to the rather uncertain angular size of the PAO hotspot coinciding with Cen A.
The monoenergetic assumption is a good representation because the UHECR energy spectrum is steep and dominated by UHECR with energies close to the lower energy cut-off.
Some UHECR arrive directly at the Earth. Other UHECR are scattered by M82 halo to arrive at the Earth from the direction of M82.
We adopt the geometry shown in figure 1 where the distance of both Cen A and M82 from the Earth is 3.7Mpc.
The distance between Cen A and M82 is 6.4Mpc.
The spherical halo of enhanced scattering around M82 has a radius of 800kpc.
\begin{figure}
\includegraphics[angle=0,width=5cm]{Fig1.jpg}
\centering
\caption{
Geometry of Centaurus A, M82 and the Earth.
}
\label{fig:figure1}
\end{figure}
\begin{figure}
\includegraphics[angle=0,width=8.5cm]{Fig2.jpg}
\centering
\caption{
Time evolution of the logarithmic UHECR intensity at the Earth (i) UHECR arriving directly from Centaurus A without passing through
the halo of M82 (blue line) (ii) UHECR arriving at the Earth having passed through the halo of M82 (brown line).
}
\label{fig:figure1}
\end{figure}
A sphere of enhanced scattering represents the halo of M82.
The sphere is placed on the horizontal $z$ axis, thus imposing cylindrical $(r,z)$ symmetry on the calculation.
The enhanced scattering rate in the halo is chosen to be forty times that in the intergalactic medium.
This is sufficient to make the scattering sphere optically thick to UHECR.
The scattering rate determines how far the UHECR penetrate into the sphere before reflection back to the surface.
Its value makes little difference to the results provided optical thickness is maintained.
We add a notional detector at the position of the Earth and analyse how the UHECR
number density and anisotropy evolve in time.
The UHECR density in $(r,z)$ is plotted in figure 3.
The shell of UHECR expanding away from Cen A can be clearly seen.
The number of UHECR reflected by M82 is smaller and is not easily seen in the top row of figure 3
apart from those within the M82 halo.
The bottom row of figure 3 plots only those UHECR that have (i) passed through the halo of M82, and (ii) are not at present within the halo.
The intensity of reflected UHECR is stronger in the direction back towrds Cen A with a broad spread in angle
that encompasses the Earth.
\begin{figure}
\includegraphics[angle=0,width=8.5cm]{Fig3.jpg}
\centering
\caption{
The top row gives the density plotted in $(r,z)$ of all UHECR at times 16.7, 33.3 and 50Myr.
The bottom row is the density of only UHECR that have at some time in their prior history passed
through the halo of M82 but are not at present within the halo.
In each plot the colour scales are scaled to the maximum UHECR density. The absolute scaling can be deduced from Figure 2.
}
\label{fig:figure1}
\end{figure}
Figure 4 plots the angle of the mean streaming velocity of UHECR arriving at the Earth.
The angle is defined relative to the $z$ axis connecting Cen A to M82.
As expected, the angle initially corressponds to arrival from the direction of Cen A and then swings round to
point at M82 when the reflected UHECR arrive.
\begin{figure}
\includegraphics[angle=0,width=8.5cm]{Fig4.jpg}
\centering
\caption{
The angle in degrees of the direction of anisotropy at the Earth. The angle is defined relative to the $z$ axis joining CenA and M82
such that Cen A lies at angle $31^\circ$ and M82 at angle $149 ^\circ$.
}
\label{fig:figure1}
\end{figure}
\section{ ENERGETICS}
The UHECR fluxes from M82 and Cen A are roughly comparable at the present time.
Figure 2 therefore suggests that the flux from Cen A $\sim 20 {\rm Myr}$ ago needs to have been $\sim 200$ times greater than at present
if the flux of UHECR reflected from M82 is to have its present value.
In this section, we examine whether this is reasonable.
From figure 4 of di Matteo et al (2019) we make the very approximate estimate that UHECR with energies above 53EeV arrive at the Earth
from the M82 hotspot with an excess above the background at the rate of $\sim 5 \times 10^{-21} {\rm cm}^{-2} {\rm s}^{-1}$.
This estimate depends heavily on how a line is drawn around the complicated structure and fuzzy extent of the hotspot.
If the arrival rate directly from Cen is 200 times greater than the arrival rate via M82, the UHECR luminosity of Cen A
20 Myr ago would have been approximately $1.4\times 10^{41} {\rm erg \ s}^{-1}$.
This is $2\times 10^{-5}$ of the Eddington luminosity of $7\times 10^{45}{\rm erg \ s}^{-1}$ for a Cen A black hole mass
of $5.5 \times 10^7 {\rm M_\odot}$ (Cappellari et al 2009).
Assuming that 10 percent of the Cen A Eddington luminosity was given to CR with energies above 1GeV, a CR spectral index of -2.3 between 1GeV and 50EeV would provide the required UHECR luminosity of the TA hotspot.
An efficiency of 10 percent and a spectral index of -2.3 is consistent with the efficiency and spectrum of CR production by high velocity shocks
in supernova remnants (Bell et al 2019).
We therefore conclude that energy and power considerations are consistent with a model in which the M82 UHECR hotspot is an echo from Cen A.
\section{ IMPLICATIONS FOR COMPOSITION}
UHECR released into the IGM by Cen A are comprised of a range of energies, and probably also a range of charges $Ze$.
UHECR arriving at the Earth with the same energy may have high different $Z$ and different scattering rates in the intergalactic medium,
and hence different degrees of anisotropy.
An additional complication in our model is that
UHECR arriving via M82 and directly from Cen A have a different history even if they were accelerated at the same time in Cen A.
UHECR with higher $Z$ but the same energy may be contained for longer within the lobes of Cen A in contrast to low $Z$ particles
that escape more easily.
Hence, UHECR arriving via M82 might be expected to have a lower $Z$ than those escaping later and arriving at the Earth directly from Cen A.
It would be consistent with the Matthews \& Taylor (2021) estimates of UHECR escape times from radio galaxies for 50EeV protons to escape in a few Myr but for higher $Z$ nuclei to leak out on the timescale of 20Myr.
If this applies to Cen A, the ratio of the flux from M82 to the direct flux fom Cen A may represent the relative production rates 20Myr ago of
light and intermediate mass UHECR nuclei.
It will be interesting to see whether this conjecture is confirmed by further data and refinements in the analysis of UHECR composition.
\begin{figure}
\includegraphics[angle=0,width=8.5cm]{Fig5.jpg}
\centering
\caption{
Time evolution of the logarithmic UHECR flux at the Earth (i) UHECR arriving directly from Centaurus A without passing through
a starburst halo (blue line) (ii) UHECR arriving at the Earth having passed through the haloes of M82, NGC253 or IC342 (brown line).
}
\label{fig:figure1}
\end{figure}
\begin{figure*}
\includegraphics[angle=0,width=\textwidth]{Fig6.pdf}
\centering
\caption{
All-sky Hammer projection plot in equatorial coordinates (right ascension and declination) of arrival directions of Monte Carlo particles in four time bands of different duration: $15.0<t<15.01{\rm Myr}$ (Top Left),
$30<t<31{\rm Myr}$ (Top Right), $34<t<36{\rm Myr}$ (Bottom Left), $40<t<50{\rm Myr}$ (Bottom Right). The three starburst galaxies included in the calculation (M82, NGC253 \& IC342) are labelled, as is Cen A, and the red lines mark the approximate positions of the TA and PAO excesses in the northern and southern hemispheres, respectively (Di Matteo et al. 2019). The plot is produced using Healpy, a python implementation of the HEALpix scheme (G{\'o}rski et al 2005; Zonca et al 2019). The colour-scale encodes the number of particles per HEALpix pixel, initially calculated with $64\times64$ pixels covering the sky, which has then been smoothed with a Gaussian symmetric beam with full-width at half-maximum of $5^\circ$. The colour-map can be thought of as a linear measure of UHECR intensity at Earth. The $34<t<36{\rm Myr}$ plot provides a reasonable qualitative match with the combined TA and PAO map from Di Matteo et al. (2019; see their figure 4).
}
\label{fig:figure1}
\end{figure*}
\section{ ECHOES FROM OTHER STARBURST GALAXIES}
We now add other strong nearby starburst galaxies to the model to explore their contribution to the UHECR sky.
From Table 4 of Ackermann et al we add NGC253 and IC342 at distances of 2.5 and 3.7Mpc respectively from Earth.
We do not incude the strong starburst galaxy NGC4945 which is part of the Centaurus A group of galaxies and separated from
Cen A by a distance of only 480kpc (Tully et al 2015) which is comparable with the spatial extent of the lobes of Cen A.
UHECR reaching Earth from NGC4945 and Cen A are not separable either in angle on the sky or in their travel times to the Earth.
Any attempt to realistically model CR propagation through NGC4945 would add considerable complexity to the
calculation without making much difference to the results.
The starburst galaxy M83 is also a member of the same group of galaxies as Cen A but at a greater distance from Cen A than NGC4945.
We tried including M83 in our model according to the same formalism adopted for NGC253 and IC342 and found that
it made little difference to the results.
For simplicity and because of the modelling incertainties we also omit M83 from the results presented here.
NG253 and IC342 have radio luminosities at 1.4GHz that are smaller than that of M82 by factors of 0.39 and 0.35 respectively
(Ackermann et al 2012) and we assume that their UHECR reflectivities are smaller by the same ratios.
We model this rather arbitrarily by reducing the areal cross-sections of their haloes.
The radii of their haloes are consequently taken to be $800\times\sqrt {0.39}$ and $800\times \sqrt{0.35}$ kpc respectively.
In other respects we model NGC253 and IC342 in the same way as M82.
Figure 5 is a plot against time of the UHECR flux from Cen A (blue line) and the combined flux at the Earth of UHECR reflected from
the haloes of the three starburst galaxies. Comparison with Figure 2 gives a comparison of the fluxes from NGC253 and IC342
relative to that from M82.
The inclusion of NGC253 and IC342 disrupts the cylindrical symmetry of the calculation and we present the results as
all-sky plots. Figure 6 plots the arrival directions of UHECR in equatorial coodinates in three time bands: (A) $15.0<t<15.01$Myr (Top Left), close to the peak in the direct flux from Cen A
(B) $30<t<31$ Myr (Bottom Left),
(C) $34<t<36$ Myr (Bottom Left), close to the peak in the flux from haloes
(D) $40<t<50$Myr (Bottom Right),
when the direct flux from Cen A has died down.
The choice of different time intervals scales the flux at the different times by the ratios 1:100:200:1000 respectively,
thus allowing the same colour scaling to be used on each plot.
The all-sky plot
uses the Hierarchical Equal Area isoLatitude Pixelization (HEALpix\footnote{http://healpix.sf.net}) scheme, so that Monte Carlo particles are binned according to which HEALpix pixel they fall into, with $64\times 64$ pixels covering the sky.
Comparison with Figure 4 of di Matteo (2019) shows best agreement with the combined TA and PAO data in time band (C) at $t \approx 35{\rm Myr}$.
The actual timing and the ratio between the direct and echoed fluxes depends on the escape time of 3Myr chosen in our calculation.
Nevertheless, figure 6 demonstrates that the Monte Carlo model is able to reproduce the TA and PAO fluxes when we choose our parameters suitably.
The TA and PAO data show UHECR arriving from a wider range of directions on the sky.
These may be reflections from more distant starburst galaxies of UHECR released by Cen A at earlier times.
Reflection may also occur from less prominent star-forming regions in multiple nearby galaxies.
Alternatively, UHECR may originate in other radio galaxies such as Fornax A (Matthews et al 2018) or a combination of multiple sources (Hardcastle et al 2010; Eichmann 2019).
\section{ CONCLUSIONS}
In summary, we have shown that a viable model can be constructed in which the TA hotspot is an echo of earlier UHECR production by Cen A during
a period of greater activity 20Myr ago.
The suggested echo is due to enhanced UHECR scattering by the M81 group of galaxies and
in particular by the halo of the starburst galaxy M82.
Echoes from other starburst galaxies or other regions of enhanced magnetic field may be responsible for UHECR arriving from other parts of the sky.
The model has the potential to reconcile the statistical association of UHECR with starburst galaxies and the inability,
due to their relatively low power,
of starburst galaxies themselves to generate UHECR of sufficient ultra-high energy.
Differences in composition may result from the varied histories of UHECR arriving at the Earth by different paths.
As more data is collected, UHECR may be used to probe the local intergalactic environment and the structure of starburst haloes out to distances of 10-20 Mpc from the Earth.
\section*{Acknowledgments}
We thank Andrew Taylor for helpful discussions.
ARB acknowledges the support of an Emeritus Fellowship from the Leverhulme Trust.
JM acknowledges a Herchel Smith Fellowship at the University of Cambridge. Some of the results in this paper have been derived using the healpy and HEALPix packages (G{\'o}rski et al 2005; Zonca et al 2019). We also gratefully acknowledge the use of Astropy, a community-developed core Python package for astronomy (Astropy Collaboration 2013, 2018), and Matplotlib v3.1.1 (Hunter 2007).
The Monte Carlo calculations were performed using the computing resources provided by STFC Scientific Computing Department's SCARF cluster.
\section*{Data Availability}
The data underlying this article are available from the authors on reasonable request.
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\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,501 |
Тривисмутид никеля — бинарное неорганическое соединение
никеля и висмута
с формулой NiBi,
кристаллы.
Получение
Сплавление стехиометрических количеств чистых веществ:
Физические свойства
Тривисмутид никеля образует кристаллы
ромбической сингонии,
параметры ячейки a = 0,8879 нм, b = 0,40998 нм, c = 1,1483 нм, Z = 4
.
При 4,05 К происходит переход в сверхпроводящее состояние
.
Химические свойства
Разлагается при нагревании:
Примечания
Соединения никеля
никеля | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,045 |
\section{Introduction}\
Recent astrophysical data \cite{exp} strongly indicate that the
universe is accelerating at present. Therefore, it is of paramount
importance to explain why this is happening. Many theories have
been proposed recently that try to address this issue, which has
become the most important problem in cosmology. Although theories
that try to modify Einstein equations \cite{ordishov} constitute a
big part of these attempts, the mainstream explanation for this
problem, however, is known as theories of dark energy. The most
accepted idea indicates that a mysterious dominant component,
dubbed dark energy, which has negative pressure, leads to this
cosmic acceleration, though its nature and cosmological origin
still remain unknown.
The combined analysis of the cosmological observations suggests
that the universe consists of about $70\%$ dark energy, $30\%$
dust matter (cold dark matter plus baryons), and a negligible
percentage of radiation. Although little about dark energy is
known, we can still propose some candidates to describe it. The
most natural candidate to account for this acceleration is the
cosmological constant, by given the problems associated with it we
turn our attention to dynamical dark energy models. The dynamical
nature of dark energy, at least at an effective level, can be
originated from various fields, such as a canonical scalar field
(quintessence) \cite{quint}, a phantom field, that is, a scalar
field with a negative sign of the kinetic term \cite{phant}, or
the combination of quintessence and phantom in a unified model
named quintom \cite{quintom}. The advantage of this combined model
is that although in quintessence the dark energy equation of state
parameter remains always greater than $-1$ and in phantom
cosmology always
smaller than $-1$, in the quintom scenario it can cross $-1$.\\
The general consensus among theorists is that we can not entirely
understand the nature of dark energy until a complete theory of
quantum gravity is established \cite{Witten:2000zk}. The dark
energy problem may be then in essence a problem that belongs to
quantum gravity \cite{Witten:2000zk}. Thus, a complete theory of
quantum gravity should explain the properties of dark energy, such
that the energy density and the equation of state would be
determined definitely and uniquely. However, in spite of the fact
that we still lack a theory of quantum gravity, we can make some
attempts to probe the nature of dark energy by making use of the
holographic principle, which is thought to be present in a final
theory of quantum gravity. In particular, an interesting attempt
within the framework of quantum gravity is the holographic dark
energy (HDE) proposal
\cite{Cohen:1998zx,Hsu:2004ri,Li:2004rb,holoext}, which is
constructed in the light of the holographic principle and
therefore possesses some significant features of an underlying
theory of dark energy. The HDE model has been tested and
constrained by various astronomical observations
\cite{Huang:2004wt,obs3} as well as by the Anthropic Principle
\cite{Huang:2004mx}. Furthermore, the HDE model has been extended
to include the spatial curvature contribution, i.e. the HDE model
in non-flat space \cite{nonflat}.\\
On the other hand, it is usually assumed that both dark energy and
dark energy only couple gravitationally. However, given their
unknown nature and that the underlying symmetry that would set the
interaction to zero is still to be discovered, an entirely
independent behavior between the dark sectors would be very
special indeed. Moreover, since dark energy gravitates, it must be
accreted by massive compact objects such as black holes and, in a
cosmological context, the energy transfer from dark energy to dark
matter may be small but non-vanishing.
Furthermore, the coupling is not only likely but may even be
inevitable \cite{Brax:2006kg}. In addition, it might explain or at
least alleviate the coincidence problem \cite{Phenomenological
interaction, Amendola:2000uh}. It was found that an appropriate
interaction between dark energy and dark matter can influence the
perturbation dynamics and affect the lowest multipoles of the CMB
angular power spectrum \cite{Interaction in perturbation dynamics,
Interaction in the CMB multipoles}. Thus, it could be inferred
from the expansion history of the Universe, as manifested in
several experimental data. Moreover, it was suggested that the
dynamical equilibrium of collapsed structures such as clusters
would be modified due to the coupling between dark energy and dark
matter \cite{Bertolami:2007zm, Kesden}. There is no clear
consensus on the form of the coupling. Most studies on the
interaction between dark sectors rely either on the assumption of
interacting fields from the outset
\cite{Amendola:2000uh,Das:2005yj}, or from phenomenological
requirements \cite{Phenomenological interaction}. The coupling not
only affects the expansion history of the Universe but modifies
the structure formation scenario as well because we no longer have
$\rho_{m} \propto a^{-3}$. The aforesaid interaction has also been
considered from a thermodynamical perspective \cite{Wang:2007ak,
Pavon:2007gt} and has been shown that the second law of
thermodynamics imposes an energy transfer from dark energy to dark
matter.
A pressureless dark matter in interaction with holographic dark
energy is more than just another model to describe an accelerated
expansion of the universe. It provides a unifying view of
different models which are viewed as different realizations of the
interacting HDE model at the perturbative level
\cite{Zimdahl:2007ne}.
In the present paper we extend our recent work \cite{set1} to the
interacting HDE scenario in a non-flat universe. We utilized the
horizon's radius $L$ measured from the sphere of the horizon as
the system's IR cut-off. The organization of our work is as
follows: In section 2 we construct the cosmological scenarios of
non-minimally coupled canonical, phantom and quintom fields, in
the framework of HDE. In section 3 we examine their behavior and
we discuss their cosmological implications. Finally, in section 4
we summarize our results.
\section{ Interacting non-minimally coupled fields of holographic dark energy in non-flat universe}
\subsection{Canonical field} \label{scalar}
We first consider a canonical scalar field with a non-minimal
coupling. This case was partially investigated in \cite{Ito}, and
recently extended in \cite{set1}. The action of the universe is
\begin{equation}
S=\int d^{4}x \sqrt{-g} \left[\frac{1}{2\kappa^{2}}
R-\frac{1}{2}\,\xi_{\phi}\phi^{2} R
-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi
+\chi\cal{L}_{M}\right] \label{actioncan},
\end{equation}
where $\kappa^{2}$ is a gravitational constant. In this action we have
added a canonical scalar field $\phi$, which in non-minimally
coupled to the curvature with coupling parameter $\xi_{\phi}$.
Lastly, the term $\cal{L}_{M}$ represents the matter content of
the universe and the term $\chi$ multiplying it accounts for the
interaction.
The presence of the non-minimal coupling leads to the effective
Newton's constant
\begin{equation}
8\pi
G_{eff}=\kappa^{2}\left(1-\xi_{\phi}\kappa^{2}\phi^{2}\right)^{-1}\label{Geff}\,.
\end{equation}
We proceed now to calculate the equation of state for the HDE
density when there is an interaction between the HDE
$\rho_{\Lambda}$ and Cold Dark Matter(CDM) with $w_{m}=0$. The
continuity equations for the HDE and CDM are
\begin{eqnarray}
\label{2eq1}&& \dot{\rho}_{\rm \Lambda}+3H(1+w_{\rm \Lambda})\rho_{\rm \Lambda} =-Q, \\
\label{2eq2}&& \dot{\rho}_{\rm m}+3H\rho_{\rm m}=Q
\end{eqnarray}
here $H=\dot{a}/a$ is the Hubble parameter. The interaction is
given by the quantity $Q=\Gamma \rho_{\Lambda}$ and describes a
decay of the HDE component into CDM with the decay rate given by
$\Gamma$. Taking the ratio of the two energy densities as
$r_{m}=\rho_{\rm m}/\rho_{\rm \Lambda}$, the above equations lead
to
\begin{equation}
\label{2eq3} \dot{r_{m}}=3Hr_{m}\Big[w_{\rm \Lambda}+
\frac{1+r_{m}}{r_{m}}\frac{\Gamma}{3H}\Big]
\end{equation}
Following Ref.\cite{Kim:2005at}, we define
\begin{eqnarray}\label{eff}
w_\Lambda ^{\rm eff}=w_\Lambda+{{\Gamma}\over {3H}}\;, \qquad w_m
^{\rm eff}=-{1\over r_{m}}{{\Gamma}\over {3H}}\;.
\end{eqnarray}
Thus, the continuity equations can be written in their standard
form as
\begin{equation}
\dot{\rho}_\Lambda + 3H(1+w_\Lambda^{\rm eff})\rho_\Lambda =
0\;,\label{definew1}
\end{equation}
\begin{equation}
\dot{\rho}_m + 3H(1+w_m^{\rm eff})\rho_m = 0\; \label{definew2}.
\end{equation}
We now consider the non-flat Friedmann-Robertson-Walker universe
with line element
\begin{equation}\label{metr}
ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{dr^2}{1-kr^2}+r^2d\Omega^{2}\right).
\end{equation}
where $k$ denotes the curvature of space $k=0,1,-1$ for flat,
closed and open universe, respectively. It must be noticed that a
closed universe with a small positive curvature ($\Omega_k\sim
0.01$) is compatible with observations \cite{ {wmap}, {ws}}. As
usual, we use the Friedmann equation to relate the curvature of
the universe to the energy density. In the interacting case we are
dealing with, the Friedmann equations and the evolution equation
for the scalar field are
\begin{equation}
H^{2}=\frac{\kappa^{2}\left(\rho_{m}+\rho_{\Lambda}+\frac{1}{2}\dot{\phi}^{2}+6\xi_{\phi}
H\phi\dot{\phi}\right)}{3\left(1-\xi_{\phi}\kappa^{2}\phi^{2}\right)}
\label{eqn4}
\end{equation}
\begin{equation}
\dot{\phi}\left[\ddot{\phi}+3H\dot{\phi}+6\xi_{\phi}\left(\dot{H}+2H^{2}\right)\phi\right]=-Q\label{eqn5}
\end{equation}
\begin{equation}
\dot{\rho}_{m}+\dot{\rho}_{\Lambda}+3H\left(\rho_{m}+\rho_{\Lambda}+p_{m}+p_{\Lambda}\right)=0\label{eqn6}.
\end{equation}
In these expressions,
$p_{m}$ and $\rho_{m}$ are the
pressure and density of the matter content of the universe,
respectively. Finally, $p_\Lambda$ and $\rho_\Lambda$ are the corresponding
components of dark energy, which are attributed to the scalar
field.
We define as usual
\begin{equation} \label{2eq9} \Omega_{\rm
m}=\frac{\rho_{m}}{\rho_{cr}}=\frac{ \rho_{\rm
m}}{3M_p^2H^2},\hspace{1cm}\Omega_{\rm
\Lambda}=\frac{\rho_{\Lambda}}{\rho_{cr}}=\frac{ \rho_{\rm
\Lambda}}{3M^2_pH^2},\hspace{1cm}\Omega_{k}=\frac{k}{a^2H^2}
\end{equation}
where $M^2_p=(8\pi G_{eff})^{-1}$.
Now we can rewrite the first
Friedmann equation as
\begin{equation} \label{2eq10} \Omega_{\rm m}+\Omega_{\rm
\Lambda}=1+\Omega_{k}.
\end{equation}
This allows us to express $r_{\rm m}$ and $r_{\rm k}=\rho_{\rm
k}/\rho_{\rm \Lambda}$ in terms of $\Omega_{\rm \Lambda}$ and
$\Omega_{\rm k}$ as
\begin{equation}
\label{2eq11} r_{\rm m}=\frac{1-\Omega_{\rm \Lambda}+\Omega_{\rm
k}}{\Omega_{\rm \Lambda}},~~r_{\rm k}=\frac{\Omega_{\rm
k}}{\Omega_{\rm \Lambda}}.
\end{equation}
In the non-flat universe, our choice for HDE density is
\begin{equation} \label{holoda}
\rho_\Lambda=\frac{3}{\kappa^{2}}(1-\xi_{\phi}\kappa^{2}\phi^{2})L^{-2},
\end{equation}
where $L$ is defined as
\begin{equation}\label{leq}
L=ar(t),
\end{equation}
here, $a$, is the scale factor and $r(t)$ can be obtained from the
following equation
\begin{equation}\label{rdef}
\int_0^{r(t)}\frac{dr}{\sqrt{1-kr^2}}=\int_t^\infty
\frac{dt}{a}=\frac{R_h}{a},
\end{equation}
where $R_h$ is the event horizon. Therefore while $R_h$ is the
radial size of the event horizon measured in the $r$ direction,
$L$ is the radius of the event horizon measured on the sphere of
the horizon. For the closed universe we have (the same calculation
is valid for the open universe by a transformation)
\begin{equation} \label{req}
r(t)=\frac{1}{\sqrt{k}} sin y.
\end{equation}
where $y\equiv \sqrt{k}R_h/a$. As in \cite{set1} we are interested
in extracting power-law solutions of our cosmological model
(\ref{eqn4})-(\ref{eqn6}), in the case of a dark-energy dominated
universe ($\rho_{m},p_{m}\ll 1$). Thus, we are looking for
solutions of the form
\begin{eqnarray}
&&a(t)=a_{0}t^{r}\nonumber\\
&&\phi(t)=\phi_{0}t^{s_{\phi}}.
\end{eqnarray}
The insertion of the second ansatz allows us to express the HDE
density as
\begin{equation} \label{holoda2}
\rho_\Lambda(t)=\frac{3}{\kappa^{2}}(1-\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2t^{2s_{\phi}})L^{-2}.
\end{equation}
Using the definitions
$\Omega_{\Lambda}=\frac{\rho_{\Lambda}}{\rho_{cr}}$ and
$\rho_{cr}=3M_{p}^{2}H^2$, we get
\begin{equation}\label{hl}
HL=\frac{1}{\sqrt{\Omega_{\Lambda}}}.
\end{equation}
Now using Eqs.(\ref{leq}, \ref{rdef}, \ref{req}, \ref{hl}), we
obtain
\begin{equation} \label{ldot}
\dot L= HL+ a \dot{r}(t)=\frac{1}{\sqrt{\Omega_\Lambda}}-cos y.
\end{equation}
By considering the definition of Eq.(\ref{holoda2}) for the HDE
$\rho_{\rm \Lambda}(t)$, and using Eqs.(\ref{hl}, \ref{ldot}) one
can find
\begin{equation}\label{roeq}
\dot{\rho_{\Lambda}}=-2s_{\phi}t^{-1}\varrho_{\Lambda}\left(\frac{\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2t^{2s_{\phi}}}{1-\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2t^{2s_{\phi}}}\right)-2H\left(1-{\sqrt{\Omega_\Lambda}}\cos
y\right)\rho_{\Lambda}.
\end{equation}
If we substitute the above relation into Eq.(\ref{2eq1}) and use
the definition $Q=\Gamma \rho_{\Lambda}$, we arrive at
\begin{equation}\label{stateq}
w_{\rm
\Lambda}(t)=-\frac{8\xi_{\phi}(\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2)}{t^{-2s_{\phi}}-\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2}
-\left(\frac{1}{3}+\frac{2\sqrt{\Omega_{\rm \Lambda}}}{3}\cos
y+\frac{\Gamma}{3H}\right).
\end{equation}
Here as in Ref.\cite{WGA}, we choose the following relation for
the decay rate
\begin{equation}\label{decayeq}
\Gamma=3b^2(1+r_{m})H
\end{equation}
with the coupling constant $b^2$. Using Eq.(\ref{2eq11}), the
above decay rate yields
\begin{equation}\label{decayeq2}
\Gamma=3b^2H\frac{(1+\Omega_{k})}{\Omega_{\Lambda}}.
\end{equation}
Substituting this relation into Eq.(\ref{stateq}), one finds the
HDE equation of state parameter
\begin{equation} \label{3eq4}
w_{\rm
\Lambda}(t)=-\frac{8\xi_{\phi}(\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2)}{t^{-2s_{\phi}}-\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2}
-\frac{1}{3}\left[1+2\sqrt{\Omega_{\rm \Lambda}}\cos
y+\frac{3b^2(1+\Omega_{k})}{\Omega_{\rm \Lambda}}\right].
\end{equation}
According to relation $y\equiv \sqrt{k}R_h/a$, $\cos y=1$ when
$k=0$, so in this case $\Omega_{k}=0$ and therefore, in flat
universe, the HDE equation of state is given by
\begin{equation} \label{3eq401}
w_{\rm
\Lambda}(t)=-\frac{8\xi_{\phi}(\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2)}{t^{-2s_{\phi}}-\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2}
-\frac{1}{3}\left(1+2\sqrt{\Omega_{\rm
\Lambda}}+\frac{3b^2}{\Omega_{\rm \Lambda}}\right) .
\end{equation}
From Eqs.(\ref{eff}, \ref{decayeq2}, \ref{3eq4}), we have the
effective equation of state as
\begin{equation} \label{3eq402}
w_{\rm
\Lambda}^{eff}=-\frac{8\xi_{\phi}(\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2)}{t^{-2s_{\phi}}-\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2}
-\frac{1}{3}\left(1+2\sqrt{\Omega_{\rm \Lambda}}\cos y\right).
\end{equation}or expressed in terms of the redshift $z$ as
\begin{equation} \label{wcanonical}
w_{\rm
\Lambda}^{eff}(z)=-\frac{8\xi_{\phi}(\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2)}{e^{-24\xi_{\phi}ln(1+z)}-\xi_{\phi}\kappa^{2}{{\phi_{0}}}^2}
-\frac{1}{3}\left(1+2\sqrt{\Omega_{\rm \Lambda}}\cos y\right).
\end{equation}
On the
other hand, the effective equation of state for CDM is given
differently by
\begin{equation} \label{wmeff}
\omega^{\rm eff}_{\rm
m}(z)=-\frac{b^{2}(1+\Omega_{k})}{(1+\Omega_{k}-\Omega_{\Lambda})}.
\end{equation}
Now we are in a position to derive two coupled equations whose
solutions determine
the effective equations of state (as in Ref.[24]).
Eq.(\ref{2eq3}) leads to one differential equation for
$\Omega_{\rm \Lambda}$
\begin{equation} \label{diffeqnlambda}
-(1+z)\frac{d \Omega_{\rm \Lambda}}{dz}=-3\Omega_{\rm
\Lambda}(1-\Omega_{\rm \Lambda}+\Omega_{\rm k})\Big(\omega^{\rm
eff}_{\rm \Lambda}-\omega^{\rm eff}_{\rm m}\Big)+\Omega_{\rm
k}\Omega_{\rm \Lambda}(1+3\omega^{\rm eff}_{\rm \Lambda}\Big).
\end{equation}
The remaining differential equation for $\Omega_{\rm k}$ comes
from the derivative of $r_{\rm k}$ in Eq.(\ref{2eq11}) using
Eq.(\ref{2eq10}) as
\begin{equation} \label{omegak}
-(1+z)\frac{d \Omega_{\rm k}}{dz}=-3\Omega_{\rm k}(1-\Omega_{\rm
\Lambda}+\Omega_{\rm k})\Big(\omega^{\rm eff}_{\rm
\Lambda}-\omega^{\rm eff}_{\rm m}\Big)+\Omega_{\rm
k}\Big(1+\Omega_{\rm k}\Big)\Big(1+3\omega^{\rm eff}_{\rm
\Lambda}\Big).
\end{equation}
In order to obtain a solution, we have to solve the above coupled
equations numerically by considering the initial condition at the
present time: $\frac{d \Omega_{\rm
\Lambda}}{dz}|_{z=0}>0,~\Omega^{0}_{\rm \Lambda}=0.73,
\Omega^{0}_{\rm k=1}=0.01$ and $\Omega^{0}_{\rm k=0}=0.0$.
\subsection{Phantom field} \label{phantom}
In this subsection we consider a phantom field with a non-minimal
coupling, that is, a field with an opposite sign in the kinetic
term in the Lagrangian \cite{phant}. Such models are widely used
in order to have $w_\Lambda$ less than $-1$. The action of the universe
in this case is
\begin{equation}
S=\int d^{4}x \sqrt{-g} \left[\frac{1}{2\kappa^{2}}
R-\frac{1}{2}\xi_{\sigma}\sigma^{2} R
+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\sigma
+\chi\cal{L}_{M}\right] \label{actionphan}.
\end{equation}
In this action we have added a phantom field $\sigma$, which in
non-minimally coupled to the curvature with coupling parameter
$\xi_{\sigma}$. Lastly, the term $\cal{L}_{M}$ represents the
matter content of the universe and the term $\chi$ multiplying it
accounts for the interaction.
The presence of the non-minimal
coupling leads to the effective Newton's constant:
\begin{equation}
8\pi
G_{eff}=\kappa^{2}\left(1-\xi_{\sigma}\kappa^{2}\sigma^{2}\right)^{-1}\label{Geff2}\,.
\end{equation}
We shall follow a procedure similar to the one used in the
previous subsection in order to obtain the equation of state.
The cosmological equations and the evolution equation for the
interacting phantom field are given by
\begin{equation}
H^{2}=\frac{\kappa^{2}\left(\rho_{m}+\rho_{\Lambda}-\frac{1}{2}\dot{\sigma}^{2}+6\xi_{\phi}
H\sigma\dot{\sigma}\right)}{3\left(1-\xi_{\phi}\kappa^{2}\sigma^{2}\right)}
\label{eqn4b}
\end{equation}
\begin{equation}
\dot{\sigma}\left[\ddot{\sigma}+3H\dot{\sigma}-6\xi_{\sigma}\left(\dot{H}+2H^{2}\right)\sigma\right]=-Q\label{eqn5b}
\end{equation}
\begin{equation}
\dot{\rho}_{m}+\dot{\rho}_{\Lambda}+3H\left(\rho_{m}+\rho_{\Lambda}+p_{m}+p_{\Lambda}\right)=0\label{eqn6b}
\end{equation}
and the non-flat HDE can be expressed as
\begin{equation} \label{holodaphan}
\rho_\Lambda=\frac{3}{\kappa^{2}}(1-\xi_{\sigma}\kappa^{2}\sigma^{2})L^{-2}
\end{equation}
where we have used the effective nature of the Newton's constant
(\ref{Geff2}).
We examine power-law solutions of equations
(\ref{eqn4b})-(\ref{eqn6b}), in the case of a dark-energy
dominated universe ($\rho_{m},p_{m}\ll 1$). Thus, we impose:
\begin{eqnarray}
&&a(t)=a_{0} t^{r}\nonumber\\
&&\sigma(t)=\sigma_{0}t^{s_{\sigma}} \label{powerlaw2}.
\end{eqnarray}
Using (\ref{holodaphan}) we have that
\begin{equation}\label{holoda3}
\rho_{\Lambda}(t)=\frac{3}{\kappa^{2}}(1-\xi_{\sigma}\kappa^{2}\sigma_{0}^{2}t^{2s_{\sigma}})L^{-2}
\end{equation}
and as in the previous subsection $L$ is defined as in
Eq.(\ref{leq}). By taking the definition of Eq.(\ref{holoda3})
for the HDE density $\rho_{\rm \Lambda}(t)$, and making use of
Eqs.(\ref{hl}, \ref{ldot}) one can obtain
\begin{equation}\label{roeq2}
\dot{\rho_{\Lambda}}=-2s_{\sigma}t^{-1}\varrho_{\Lambda}\left(\frac{\xi_{\sigma}\kappa^{2}{{\sigma_{0}}}^2t^{2s_{\sigma}}}{1-\xi_{\sigma}\kappa^{2}{{\sigma_{0}}}^2t^{2s_{\sigma}}}\right)-2H\left(1-{\sqrt{\Omega_\Lambda}}\cos
y\right)\rho_{\Lambda}.
\end{equation}
Substitution of the above relation into Eq.(\ref{2eq1}) and use of
the definition $Q=\Gamma \rho_{\Lambda}$, yields
\begin{equation}\label{stateq2}
w_{\rm
\Lambda}(t)=\frac{8\xi_{\sigma}(\xi_{\sigma}\kappa^{2}{{\sigma_{0}}}^2)}{t^{-2s_{\sigma}}-\xi_{\sigma}\kappa^{2}{{\sigma_{0}}}^2}
-\left(\frac{1}{3}+\frac{2\sqrt{\Omega_{\rm \Lambda}}}{3}\cos
y+\frac{\Gamma}{3H}\right).
\end{equation}
Inserting Eq.(\ref{decayeq2}) into Eq.(\ref{stateq2}) allows us to
obtain the equation of state parameter
\begin{equation}\label{stateq3}
w_{\rm
\Lambda}(t)=\frac{8\xi_{\sigma}(\xi_{\sigma}\kappa^{2}{{\sigma_{0}}}^2)}{t^{-2s_{\sigma}}-\xi_{\sigma}\kappa^{2}{{\sigma_{0}}}^2}
-\frac{1}{3}\left[1+2\sqrt{\Omega_{\rm \Lambda}}\cos
y+\frac{3b^2(1+\Omega_{k})}{\Omega_{\rm \Lambda}}\right].
\end{equation}
Considering the relation $y\equiv \sqrt{k}R_h/a$, $\cos y=1$ when
$k=0$, i.e. $\Omega_{k}=0$ and therefore, in flat universe, the
HDE equation of state is expressed as
\begin{equation} \label{3eq4011}
w_{\rm
\Lambda}(t)=\frac{8\xi_{\sigma}(\xi_{\sigma}\kappa^{2}{{\sigma_{0}}}^2)}{t^{-2s_{\sigma}}-\xi_{\sigma}\kappa^{2}{{\sigma_{0}}}^2}
-\frac{1}{3}\left(1+2\sqrt{\Omega_{\rm
\Lambda}}+\frac{3b^2}{\Omega_{\rm \Lambda}}\right) .
\end{equation}
From Eqs.(\ref{eff}, \ref{decayeq2}, \ref{stateq3}), we arrive at
the effective equation of state
\begin{equation} \label{wlambda}
w_{\rm
\Lambda}^{eff}=\frac{8\xi_{\sigma}(\xi_{\sigma}\kappa^{2}{{\sigma_{0}}}^2)}{t^{-2s_{\sigma}}-\xi_{\sigma}\kappa^{2}{{\sigma_{0}}}^2}
-\frac{1}{3}\left(1+2\sqrt{\Omega_{\rm \Lambda}}\cos y\right).
\end{equation}Similarly to what was done in the previous subsection, we can express
(\ref{wlambda}) in terms of the redshift $z$ obtaining
\begin{equation} \label{wphantom}
w_{\rm
\Lambda}^{eff}(z)=\frac{8\xi_{\sigma}(\xi_{\sigma}\kappa^{2}{{\sigma_{0}}}^2)}{e^{24\xi_{\sigma}ln(1+z)}-\xi_{\sigma}\kappa^{2}{{\sigma_{0}}}^2}
-\frac{1}{3}\left(1+2\sqrt{\Omega_{\rm \Lambda}}\cos y\right).
\end{equation}
Finally, we insert Eq.(\ref{wphantom}) into
Eqs.(\ref{diffeqnlambda}) and (\ref{omegak}) and solve them
numerically by considering the initial condition at the present
time: $\frac{d \Omega_{\rm \Lambda}}{dz}|_{z=0}>0,~\Omega^{0}_{\rm
\Lambda}=0.73, \Omega^{0}_{\rm k=1}=0.01$ and $\Omega^{0}_{\rm
k=0}=0.0$.
\subsection{Quintom model} \label{quintom}
In this subsection we consider the quintom cosmological scenario
\cite{quintom}, where we consider simultaneously a canonical and a
phantom field, both with non-minimal coupling. As we have stated
in the introduction, this combined cosmological paradigm has been
shown to be capable to describe the crossing of the phantom divide
$w_\Lambda=-1$. The action of this model is given by
\begin{eqnarray}
S=\int d^{4}x \sqrt{-g} \left[\frac{1}{2\kappa^{2}}
R-\frac{1}{2}\xi_{\phi}\phi^{2} R-\frac{1}{2}\xi_{\sigma}\sigma^{2} R-\right.\nonumber\\
\left.
-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\sigma
+\chi\cal{L}_{M}\right], \label{actionquint}
\end{eqnarray}
and the presence of the non-minimal coupling leads to the
effective Newton's constant
\begin{equation}
8\pi
G_{eff}=\kappa^{2}\left[1-\kappa^{2}(\xi_{\phi}\phi^{2}+\xi_{\sigma}\sigma^{2})\right]^{-1}\label{Geff3}\,.
\end{equation}
The cosmological equations and the evolution equation for the
canonical and phantom fields in the interacting case are the
following
\begin{equation}
H^{2}=\frac{\kappa^{2}\left(\rho_{m}+\rho_{\Lambda}+\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\dot{\sigma}^{2}+6\xi_{\phi}H\phi\dot{\phi}+6\xi_{\sigma}H\sigma\dot{\sigma}\right)}{3\left[1-\kappa^{2}\left(\xi_{\phi}\phi^{2}+\xi_{\sigma}\sigma^{2}\right)\right]}\label{eqn4c}
\end{equation}
\begin{equation}
\dot{\phi}\left[\ddot{\phi}+3H\dot{\phi}+6\xi_{\phi}\left(\dot{H}+2H^{2}\right)\phi\right]=-Q\label{eqn5c}
\end{equation}
\begin{equation}
\dot{\sigma}\left[\ddot{\sigma}+3H\dot{\sigma}-6\xi_{\sigma}\left(\dot{H}+2H^{2}\right)\sigma\right]=-Q\label{eqn5c2}
\end{equation}
\begin{equation}
\dot{\rho}_{m}+\dot{\rho}_{\Lambda}+3H\left(\rho_{m}+\rho_{\Lambda}+p_{m}+p_{\Lambda}\right)=0\label{eqn6c}
\end{equation} as usual the non-flat HDE is given by
\begin{equation}
\rho_{\Lambda}=\frac{3}{\kappa^{2}}\left[1-\kappa^{2}\left(\xi_{\phi}\phi^{2}+\xi_{\sigma}\sigma^{2}\right)\right]L^{-2},\label{rhoL3}
\end{equation}
after making use of the effective nature of the Newton's constant
(\ref{Geff3}). We examine power-law solutions of equations
(\ref{eqn4c})-(\ref{eqn6c}), in the case of a dark-energy
dominated universe ($\rho_{m},p_{m}\ll 1$). Thus, we impose:
\begin{eqnarray}
&&a(t)=a_{0} t^{r}\nonumber\\
&&\phi(t)=\phi_{0}t^{s_{\phi}}\nonumber\\
&&\sigma(t)=\sigma_{0}t^{s_{\sigma}} \label{powerlaw3}.
\end{eqnarray}
Substituting into (\ref{rhoL3}) yields
\begin{equation}
\rho_{\Lambda}(t)=\frac{3}{\kappa^{2}}\left[1-\kappa^{2}\left(\xi_{\phi}\phi_{0}^{2}t^{2s_{\phi}}+\xi_{\sigma}\sigma_{0}^{2}t^{2s_{\sigma}}\right)\right]L^{-2}
\label{holoda4}
\end{equation}
and as in the previous subsections $L$ is defined as in
Eq.(\ref{leq}).
Using the definition of Eq.(\ref{holoda4}) for the HDE density
$\rho_{\rm \Lambda}(t)$, and considering Eqs.(\ref{hl},
\ref{ldot}) gives us
\begin{equation}\label{roeq3}
\dot{\rho_{\Lambda}}=-2\left(s_{\phi}+s_{\sigma}\right)t^{-1}\left[\kappa^{2}\left(\xi_{\phi}\phi_{0}^{2}t^{2s_{\phi}}+\xi_{\sigma}\sigma_{0}^{2}t^{2s\sigma}\right)\right]-2H\left(1-\sqrt{\Omega_{\Lambda}}cosy\right)\rho_{\Lambda}.
\end{equation}
The substitution of the above relation into Eq.(\ref{2eq1}) and
the use of the definition $Q=\Gamma \rho_{\Lambda}$, yields
\begin{equation}\label{stateq4}
w_{\rm
\Lambda}(t)=16\left[\frac{\kappa^{2}\left(\xi_{\phi}^{2}\phi_{0}^{2}t^{2s_{\phi}}-\xi_{\sigma}^{2}\sigma_{0}^{2}t^{2s_{\sigma}}\right)}{\kappa^{2}\left(\xi_{\phi}\phi_{0}^{2}t^{2s_{\phi}}+\xi_{\sigma}\sigma_{0}^{2}t^{2s_{\sigma}}\right)-1}\right]
-\left(\frac{1}{3}+\frac{2\sqrt{\Omega_{\rm \Lambda}}}{3}\cos
y+\frac{\Gamma}{3H}\right).
\end{equation}
Inserting Eq.(\ref{decayeq2}) into Eq.(\ref{stateq4}) gives the
equation of state parameter
\begin{equation}\label{stateq5}
w_{\rm
\Lambda}(t)=16\left[\frac{\kappa^{2}\left(\xi_{\phi}^{2}\phi_{0}^{2}t^{2s_{\phi}}-\xi_{\sigma}^{2}\sigma_{0}^{2}t^{2s_{\sigma}}\right)}{\kappa^{2}\left(\xi_{\phi}\phi_{0}^{2}t^{2s_{\phi}}+\xi_{\sigma}\sigma_{0}^{2}t^{2s_{\sigma}}\right)-1}\right]
-\frac{1}{3}\left[1+2\sqrt{\Omega_{\rm \Lambda}}\cos
y+\frac{3b^2(1+\Omega_{k})}{\Omega_{\rm \Lambda}}\right].
\end{equation}
Given the relation $y\equiv \sqrt{k}R_h/a$, $\cos y=1$ when $k=0$,
i.e. $\Omega_{k}=0$ and therefore, in flat universe, the HDE
equation of state is given by
\begin{equation} \label{3eq403}
w_{\rm
\Lambda}(t)=16\left[\frac{\kappa^{2}\left(\xi_{\phi}^{2}\phi_{0}^{2}t^{2s_{\phi}}-\xi_{\sigma}^{2}\sigma_{0}^{2}t^{2s_{\sigma}}\right)}{\kappa^{2}\left(\xi_{\phi}\phi_{0}^{2}t^{2s_{\phi}}+\xi_{\sigma}\sigma_{0}^{2}t^{2s_{\sigma}}\right)-1}\right]
-\frac{1}{3}\left(1+2\sqrt{\Omega_{\rm
\Lambda}}+\frac{3b^2}{\Omega_{\rm \Lambda}}\right) .
\end{equation}
From Eqs.(\ref{eff}, \ref{decayeq2}, \ref{stateq5}), we have the
effective equation of state as
\begin{equation} \label{3eq4022}
w_{\rm
\Lambda}^{eff}=16\left[\frac{\kappa^{2}\left(\xi_{\phi}^{2}\phi_{0}^{2}t^{2s_{\phi}}-\xi_{\sigma}^{2}\sigma_{0}^{2}t^{2s_{\sigma}}\right)}{\kappa^{2}\left(\xi_{\phi}\phi_{0}^{2}t^{2s_{\phi}}+\xi_{\sigma}\sigma_{0}^{2}t^{2s_{\sigma}}\right)-1}\right]
-\frac{1}{3}\left(1+2\sqrt{\Omega_{\rm \Lambda}}\cos y\right).
\end{equation}
that can be expressed in terms of the redshift $z$ as
\begin{equation} \label{wquintom}
w_{\rm
\Lambda}^{eff}(z)=16\left[\frac{\kappa^{2}\left(\xi_{\phi}^{2}\phi_{0}^{2}e^{24\xi_{\phi}ln(1+z)}-\xi_{\sigma}^{2}\sigma_{0}^{2}e^{-24\xi_{\sigma}ln(1+z)}\right)}{\kappa^{2}\left(\xi_{\phi}\phi_{0}^{2}e^{24\xi_{\phi}ln(1+z)}+\xi_{\sigma}\sigma_{0}^{2}e^{-24\xi_{\sigma}ln(1+z)}\right)-1}\right]
-\frac{1}{3}\left(1+2\sqrt{\Omega_{\rm \Lambda}}\cos y\right).
\end{equation}
Finally, as in previous subsections, we insert Eq.(\ref{wquintom})
into Eqs.(\ref{diffeqnlambda}) and (\ref{omegak}) and solve them
numerically by considering the initial condition at the present
time: $\frac{d \Omega_{\rm \Lambda}}{dz}|_{z=0}>0,~\Omega^{0}_{\rm
\Lambda}=0.73, \Omega^{0}_{\rm k=1}=0.01$ and $\Omega^{0}_{\rm
k=0}=0.0$.
\section{Cosmological implications} \label{cosmimpl}
In the previous subsections we have obtained the equation of state
parameter of dark energy, $w_{\rm \Lambda}^{eff}(z)$, in terms of
the coupling parameters $\xi_{\phi}$, $\xi_{\sigma}$ and the
amplitudes $\phi_0$, $\sigma_0$. In the present section we
investigate the cosmological implications for each case.
\subsection{Canonical field} \label{cosmimplcan}
In the case of an interacting canonical field, non-minimally
coupled to gravity, $w_{\rm \Lambda}^{eff}(z)$ is given by the
relation (\ref{wcanonical}). In Fig.1 we depict $w_{\rm
\Lambda}^{eff}(z)$ for two different values of the coupling
$\xi_{\phi}$ and for three different values of the combination
$\kappa^2\phi_0^2$. Note that the physical requirement of an expanding
universe results in an upper limit for $\xi_{\phi}$, namely
$\xi_{\phi}<1/6$ (see \cite{set1}).
As we observe, the value of $w_{\rm \Lambda}^{eff}(z)$ at $z=0$,
that is, its current value $w_{\rm \Lambda0}^{eff}$, decreases as
$\xi_{\phi}$ increases, while its dependence on $\kappa^2\phi_0^2$ is
non-monotonic. However, in this interacting canonical field case
$w_{\rm \Lambda0}^{eff}$ is always greater than $-1$,
independently of the values of $\xi_{\phi}$ and $\kappa^2\phi_0^2$.
This was expected since this case is known to be insufficient to
describe the crossing of the phantom divide $w_{\rm
\Lambda}^{eff}=-1$ from above \cite{Kim:2005at}.
Secondly, we can see that for $\kappa^2\phi_0^2$ of the order of 1 or
below, we obtain a divergence of $w_{\rm \Lambda}^{eff}$. This
behavior is a clear prediction of relation (\ref{wcanonical}),
since it possesses a singularity at
\begin{equation}
z_{s}=-1+\left(\xi_{\phi}\kappa^2\phi_0^2\right)^{-\frac{1}{24\xi_{\phi}}}
\label{singcan}.
\end{equation} Therefore, the combinations of $\xi_{\phi}$
and $\kappa^2\phi_0^2$ that satisfy the equation giving a positive
$z_s$ must be excluded.
Finally, we mention that the effect of
non-flat universe is negligible as the curves for $k=+1,0$ appear
superimposed.
\subsection{Phantom field} \label{cosmimplphan}
In the case of an interacting phantom field, non-minimally coupled
to gravity, $w_{\rm \Lambda}^{eff}(z)$ is given by relation
(\ref{wphantom}). In Fig.2 we depict $w_{\rm \Lambda}^{eff}(z)$
for two different values of the coupling $\xi_{\sigma}$ and for
three different values of the combination $\kappa^2\sigma_0^2$. Note that
in this case the physical requirement of an expanding universe,
results in an upper limit for $\xi_{\sigma}$, namely
$-1/6<\xi_{\sigma}$ (see \cite{set1}).
As we can see, the value of $w_{\rm \Lambda0}^{eff}(z)$ is now a
non-monotonic function of $\xi_{\sigma}$ and $\kappa^2\sigma_0^2$.
Furthermore, we observe that for some particular combinations of
$\xi_{\sigma}$ and $\kappa^2\sigma_0^2$, as a consequence of the
singularity of (\ref{wphantom}), there is a divergence of $w_{\rm
\Lambda}^{eff}(z)$ at
\begin{equation}
z_{s}=-1+\left(\xi_{\sigma}\kappa^2\sigma_0^2\right)^{-\frac{1}{24\xi_{\sigma}}}
\label{singphan}.
\end{equation}
Thus, the combinations of $\xi_{\sigma}$ and $\kappa^2\sigma_0^2$ that
satisfy this transcendental equation giving a positive $z_s$ must
be excluded.
In the case at hand we can see that $w_{\rm \Lambda0}^{eff}(z)$ is
always greater than $-1$, independently of the values of
$\xi_{\sigma}$ and $\kappa^2\sigma_0^2$. This is expected as we cannot
have $w_{\rm \Lambda0}^{eff}(z)<-1$ for a phantom field in
interacting HDE (see \cite{Kim:2006kk}). Furthermore, the effect
of non-flat universe is negligible as the curves for $k=+1,0$
appear superimposed. Therefore, the non-flat universe cannot
induce the phantom phase even if one includes a non-minimal
coupling in the interacting HDE framework.
\subsection{Quintom model} \label{cosmimplquint}
In the case of the combined quintom model, that is, when both the
canonical and phantom fields are considered to be non-minimally
coupled to gravity simultaneously, $w_{\rm \Lambda}^{eff}(z)$ is
given by relation (\ref{wquintom}). In Fig.3 we depict $w_{\rm
\Lambda}^{eff}(z)$ for two different values of the coupling
$\xi_{\phi}$ and for three different combinations of
$\kappa^2\phi_0^2$ and $\kappa^2\sigma_0^2$. Note that in this case the
physical requirement of an expanding universe, results in an upper
limit for $\xi_{\phi}$, namely $\xi_{\phi}<1/6$ (see \cite{set1}).
The value of $w_{\rm \Lambda0}^{eff}(z)$ is a monotonic function
of $\xi_{\phi}$. As in the previous cases, for some particular
combinations of $\xi_{\phi}$, $\kappa^2\phi_0^2$ and $\kappa^2\sigma_0^2$, as
a consequence of (\ref{wquintom}), there is a singularity of
$w_{\rm \Lambda}^{eff}(z)$ at a specific $z_{s}$. The form of the
denominator of (\ref{wquintom}) does not allow for an explicit
expression of $z_{s}$, but a numerical investigation provides the
specific excluded parameter values.
As we can observe in Fig. 3, $w_{\rm \Lambda0}^{eff}(z)$ is
greater than $-1$, independently of the values of $\xi_{\phi}$,
$\kappa^2\phi_0^2$ and $\kappa^2\sigma_0^2$. Once again, the effect of
non-flat universe is negligible as the curves for $k=+1,0$ appear
superimposed. It turns out that not even the quintom scenario with
non-minimal interacting HDE can describe the phantom regime.
\begin{figure}[canonical]
\begin{center}
\includegraphics[width=.8\textwidth]{canonicalclosed}
\end{center}
\caption{{\item $w_{\rm \Lambda}^{eff}$ vs $z$ in the interacting
canonical field case, for $\xi_{\phi}=1/7$, $\xi_{\phi}=1/9$,
$b^{2}=0.01$ and $k=+1,0$, where in each case the combination
$\kappa^2\phi_0^2$ is taken equal to $10$, $1$, $0.1$ respectively.
The curves for $k=+1,0$ appear superimposed showing that the
effect of the non-flat universe is negligible. The divergence of
$w_{\rm \Lambda}^{eff}$ is a direct consequence of the singularity
of (\ref{wcanonical}), and thus the corresponding combinations of
and $\kappa^2\phi_0^2$ must be excluded.}} \label{canonicalfig}
\end{figure}
\begin{figure}[phantom]
\begin{center}
\includegraphics[width=.8\textwidth]{phantomclosed}
\end{center}
\caption{{\item $w_{\rm \Lambda}^{eff}$ vs $z$ in the interacting
phantom field case, for $\xi_{\sigma}=1/4$, $\xi_{\sigma}=1/7$,
$b^{2}=0.01$ and $k=+1,0$, where in each case the combination
$\kappa^2\sigma_0^2$ is taken equal to $3$, $1$, $0.1$ respectively.
The curves for $k=+1,0$ appear superimposed showing that the
effect of the non-flat universe is negligible. The divergence of
$w_{\rm \Lambda}^{eff}$ is a direct consequence of the singularity
of (\ref{wphantom}), and thus the corresponding combinations of
and $\kappa^2\sigma_0^2$ must be excluded.}} \label{phantomfig}
\end{figure}
\begin{figure}[quintom]
\begin{center}
\includegraphics[width=.8\textwidth]{quintomclosed}
\end{center}
\caption{{\item $w_{\rm \Lambda}^{eff}$ vs $z$ in the combined
interacting quintom scenario, for $\xi_{\phi}=1/6^{+}$,
$\xi_{\phi}=1/8$, $b^{2}=0.01$ and $k=+1,0$, where in each case
the combinations $\kappa^2\phi_0^2$ and $\kappa^2\sigma_0^2$ are shown in
the insets. The curves for $k=+1,0$ appear superimposed showing
that the effect of the non-flat universe is negligible. The
divergence of $w_{\rm \Lambda}^{eff}$ is a direct consequence of
the singularity of (\ref{wquintom}), and thus the corresponding
combinations of $\xi_{\phi}$, $\kappa^2\phi_0^2$ and $\kappa^2\sigma_0^2$
must be excluded.}} \label{quintomfig}
\end{figure}
\section{Conclusions}
Currently scalar fields play crucial roles in modern cosmology. In
the inflationary scenario they generate an exponential rate of
evolution of the universe as well as density fluctuations due to
the vacuum energy. It seems that the presence of a non-minimal
coupling (NMC) between scalar field and gravity is also necessary.
There are many theoretical evidences that suggest the
incorporation of an explicit NMC between the scalar field and
gravity in the action \cite{far}. The NMC arises from quantum
corrections and it is required also by the renormalization of the
corresponding field theory. Amazingly, it has been proven that the
phantom divide line crossing of dark energy described by a single
minimally coupled scalar field with general Lagrangian is even
unstable with respect to the cosmological perturbations realized
on the trajectories of the zero measure \cite{vik}. This fact has
motivated a lot of attempts to realize the crossing of the phantom
divide line by a equation of state parameter of the scalar field
used as dark
energy candidate in more complicated frameworks.\\
Studying the interaction between dark energy and ordinary matter
will open the possibility of detecting dark energy. It should be
pointed out that evidence was recently provided by the Abell
Cluster A586 in support of the interaction between dark energy and
dark matter \cite{bert}. However, despite the fact that numerous
works have been carried out, at present there are no stringent
observational bounds on the strength of this interaction
\cite{feng1}. This weakness to set stringent (observational or
theoretical) constraints on the strength of the coupling between
dark energy and dark matter stems from our unawareness of the
nature and origin of the dark components of the Universe. It is
therefore more than obvious that further work is needed in this
direction. Due to this, we have extended our work in \cite{set1}
to the interacting case in this paper. As a result, in the present
paper we have investigated canonical, phantom and quintom models,
with the various fields being non-minimally coupled to gravity, in
the framework of the interacting HDE in non-flat universe. For
this case, the characteristic length is no more the radius of the
event horizon ($R_E$) but the event horizon radius as measured
from the sphere of the horizon ($L$). In each case we have
extracted $w_{\rm \Lambda}^{eff}$, that is, the dark energy
effective equation of state parameter, as a function of the
redshift and used as parameters the couplings and the amplitudes
of the fields. Finally, we have analyzed it in order to obtain its
cosmological implications.
\section{Acknowledgment}The work of M. R. Setare has been supported by Research Institute for Astronomy
and Astrophysics of Maragha, Iran. The work of Alberto
Rozas-Fern\'{a}ndez was supported by DGICYT (Spain) under Research
Project No.~FIS2005-01180.
| {
"redpajama_set_name": "RedPajamaArXiv"
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<property name="secret" value="4e72f0233e6b4594e49ca5eaa59660a4" />
<property name="scope" value="email,user_likes,user_about_me,user_birthday,user_education_history,user_hometown,user_relationship_details,user_location,user_religion_politics,user_relationships,user_work_history,user_website,user_photos,user_events" />
<!-- cas-server-support 3.5.1 / scribe-up 1.1.0 : property name="friendsReturned" value="true" />
<property name="moviesReturned" value="true" />
<property name="musicReturned" value="true" />
<property name="booksReturned" value="true" />
<property name="likesReturned" value="true" />
<property name="albumsReturned" value="true" />
<property name="eventsReturned" value="true" /-->
<!-- cas-server-support 3.5.2 / scribe-up 1.2.0 : -->
<property name="fields" value="id,name,first_name,middle_name,last_name,gender,locale,languages,link,username,third_party_id,timezone,updated_time,verified,bio,birthday,education,email,hometown,interested_in,location,political,favorite_athletes,favorite_teams,quotes,relationship_status,religion,significant_other,website,work,friends,movies,music,books,likes,albums,events" />
</bean>
<bean id="twitter" class="org.scribe.up.provider.impl.TwitterProvider">
<property name="key" value="OPEWaSoTuAe49K4dSoRvNw" />
<property name="secret" value="aKmvleltXAmLKcnlMgzRjTsCnhV3QVMVDh153xJttCo" />
</bean>
<bean id="yahoo" class="org.scribe.up.provider.impl.YahooProvider">
<property name="key" value="dj0yJmk9MzhNZWFkeXQxN09GJmQ9WVdrOVRrbzNVVVJMTkdrbWNHbzlNVGc0TURNek5EVTJNZy0tJnM9Y29uc3VtZXJzZWNyZXQmeD1lNw--" />
<property name="secret" value="2f3146ac90d60b9f79125b1167437b6c64657a0b" />
</bean>
<bean id="github" class="org.scribe.up.provider.impl.GitHubProvider">
<property name="key" value="bb7977b9e7f892c115c3" />
<property name="secret" value="15a6f06329945e1ac5b6f715dcae5314bf88b005" />
</bean>
<bean id="caswrapper" class="org.jasig.cas.support.oauth.provider.impl.CasWrapperProvider20">
<property name="key" value="this_is_the_key" />
<property name="secret" value="this_is_the_secret" />
<property name="serverUrl" value="http://localhost:8080/cas2/oauth2.0" />
</bean>
<bean id="wordpress" class="org.scribe.up.provider.impl.WordPressProvider">
<property name="key" value="251" />
<property name="secret" value="6QCZZBhbokJ2YWfbK2F9vBgvGthtKijwaMRP8x57NWjPW2CXFsJJukt2RWvhlKaQ" />
</bean>
<bean id="oauthConfig" class="org.jasig.cas.support.oauth.OAuthConfiguration">
<property name="loginUrl" value="http://localhost:8080/cas/login" />
<property name="providers">
<list>
<ref bean="facebook" />
<ref bean="twitter" />
<ref bean="yahoo" />
<ref bean="github" />
<ref bean="caswrapper" />
<ref bean="wordpress" />
</list>
</property>
</bean>
<!--
Including this aspectj-autoproxy element will cause spring to automatically
create proxies around any beans defined in this file that match the pointcuts
of any aspects defined in this file.
-->
<aop:aspectj-autoproxy/>
<!--
Declare the TimingAspect that we want to weave into the other beans
defined in this config file.
-->
<bean id="timingAspect" class="org.perf4j.log4j.aop.TimingAspect"/>
<!-- Message source for this context, loaded from localized "messages_xx" files -->
<bean id="messageSource" class="org.springframework.context.support.ResourceBundleMessageSource"
p:basename="messages"/>
<bean
id="servicesManager"
class="org.jasig.cas.services.DefaultServicesManagerImpl">
<constructor-arg index="0" ref="serviceRegistryDao"/>
</bean>
<!--
Job to periodically reload services from service registry.
This job is needed for a clustered CAS environment since service changes
in one CAS node are not known to the other until a reload.
-->
<bean id="serviceRegistryReloaderJobDetail"
class="org.springframework.scheduling.quartz.MethodInvokingJobDetailFactoryBean"
p:targetObject-ref="servicesManager"
p:targetMethod="reload"/>
<bean id="periodicServiceRegistryReloaderTrigger" class="org.springframework.scheduling.quartz.SimpleTriggerBean"
p:jobDetail-ref="serviceRegistryReloaderJobDetail"
p:startDelay="${service.registry.quartz.reloader.startDelay:120000}"
p:repeatInterval="${service.registry.quartz.reloader.repeatInterval:120000}"/>
<bean id="httpClient" class="org.jasig.cas.util.HttpClient"
p:readTimeout="5000"
p:connectionTimeout="5000"/>
<bean id="noRedirectHttpClient" class="org.jasig.cas.util.HttpClient" parent="httpClient"
p:followRedirects="false" />
<bean id="persistentIdGenerator"
class="org.jasig.cas.authentication.principal.ShibbolethCompatiblePersistentIdGenerator"
p:salt="casrocks"/>
<!-- CentralAuthenticationService -->
<bean id="centralAuthenticationService" class="org.jasig.cas.CentralAuthenticationServiceImpl"
p:ticketGrantingTicketExpirationPolicy-ref="grantingTicketExpirationPolicy"
p:serviceTicketExpirationPolicy-ref="serviceTicketExpirationPolicy"
p:authenticationManager-ref="authenticationManager"
p:ticketGrantingTicketUniqueTicketIdGenerator-ref="ticketGrantingTicketUniqueIdGenerator"
p:ticketRegistry-ref="ticketRegistry"
p:servicesManager-ref="servicesManager"
p:persistentIdGenerator-ref="persistentIdGenerator"
p:uniqueTicketIdGeneratorsForService-ref="uniqueIdGeneratorsMap"/>
<bean id="proxy10Handler" class="org.jasig.cas.ticket.proxy.support.Cas10ProxyHandler"/>
<bean id="proxy20Handler" class="org.jasig.cas.ticket.proxy.support.Cas20ProxyHandler"
p:httpClient-ref="httpClient"
p:uniqueTicketIdGenerator-ref="proxy20TicketUniqueIdGenerator"/>
<!-- ADVISORS -->
<bean id="advisorAutoProxyCreator"
class="org.springframework.aop.framework.autoproxy.DefaultAdvisorAutoProxyCreator"/>
<bean id="validationAnnotationBeanPostProcessor" class="org.jasig.cas.util.CustomBeanValidationPostProcessor"
p:afterInitialization="true" />
<!-- The scheduler bean wires up any triggers that define scheduled tasks -->
<bean id="scheduler" class="org.jasig.cas.util.AutowiringSchedulerFactoryBean"/>
<!-- Spring-Json View provides a JsonExceptionResolver exceptions thrown during a controller action -->
<bean id="jsonExceptionResolver" class="org.springframework.web.servlet.view.json.exception.JsonExceptionResolver">
<property name="exceptionView" value="jsonView" />
<property name="errorHandler">
<list>
<bean class="org.springframework.web.servlet.view.json.error.HttpStatusError" p:errorCode="412"/>
<bean class="org.springframework.web.servlet.view.json.error.ModelFlagError"/>
</list>
</property>
<property name="exceptionHandler">
<list>
<bean class="org.springframework.web.servlet.view.json.exception.ExceptionMessageExceptionHandler" />
<bean class="org.springframework.web.servlet.view.json.exception.StackTraceExceptionHandler" />
</list>
</property>
</bean>
</beans> | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,798 |
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Prof. Fathali M. Moghaddam, Psychology
Jessica Kotfila, Ph.D. student, Interdisciplinary Concentration in Cognitive Science, Linguistics
Faculty Steering Committee
Prof. Wayne A. Davis, Philosophy
Prof. Rhonda Dzakpasu, Physics
Prof. Adam Green, Psychology
Prof. Jeff Huang, Biology
Prof. Paul Kainen, Mathematics
Prof. Josef Rauschecker, Neuroscience
Prof. Maximilian Riesenhuber, Neuroscience
Prof. Chandan J. Vaidya, Psychology
Faculty In Cognitive Science
Prof. Rachel Barr, Psychology
Her research focuses on understanding the mechanisms of learning and memory that develop during infancy.
Prof. Heidi Byrnes, German
Her research, scholarship, and teaching focus on adult instructed second language acquisition, particularly at the very advanced level.
Prof. Sandra L. Calvert, Psychology
Her research activities involve the impact of information technologies such as television and computers on children's attention, comprehension, and social behavior.
His research interests are centered in philosophy of mind, philosophy of language, epistemology, and logic, and are focused mainly on the nature of mental states (particularly belief, desire, and thought) and the concept of meaning.
Prof. Rhonda Dzakpasu, Physics and Pharmacology
Her research investigates how excitatory-inhibitory balance modulates the self-organization of in vitro neural circuitry.
Prof. Guinevere Eden, Pediatrics
Her research focus is in neuroscience, particularly developmental dyslexia. Her work focuses on characterizing visual processing in individuals with and without dyslexia using fMRI and in extending this approach to other sensory domains, such as sensorimotor control.
Prof. Rhonda Friedman, Neurology
Her research focuses on deficits in language and cognition in adult neurologic patients with stroke, head injury and dementia.
His research focuses on the neural and molecular genetic processes that underlie human intelligence.
His research aims to understand the mechanisms of glia-neuron interaction, with the goal to devise regenerative and neuroprotective therapies for brain pathologies that occur with aging, injury and disease.
Prof. Darlene V. Howard, Psychology
Her research investigates which cognitive and neural systems decline, and which are spared, in the course of aging.
Prof. Bryce Huebner, Philosophy
His research and teaching cover a host of interdisciplinary questions at the boundaries of philosophy and psychology. Most of his work focuses on issues of cognitive architecture (especially the architecture of the moral mind) and the the ethical dimensions of having the sort of cognitive architecture that we do.
His research interests include topology and geometry of nonlinear approximation (especially in regard to neural networks), theoretical biology (i.e., mathematical notions that may be somehow relevant), and quantum computation and graph theory.
Prof. Jagmeet S. Kanwal, Neuroscience
He studies the functional organization of the brain and the neural coding of sensory information.
Prof. Ruth Kramer, Linguistics
Her research interests are primarily in morphological and syntactic theory, with particular focus on the morphosyntax of noun phrases, and working on Afroasiatic languages.
Prof. Steve Kuhn, Philosophy
His research interests include logic, philosophy of logic, ethics, metaphysics, and the philosophy of language.
Prof. Donna Lardiere, Linguistics
Her research interests include language acquisition, second language acquisition, morphological and syntactic theory, and developmental linguistics.
Prof. Ronald P. Leow, Spanish & Portuguese
His research interests include: teacher education, SLA, psycholinguistics, attention and awareness, and technology in learning.
Prof. David Lightfoot, Linguistics
His research interests include syntactic theory, language acquisition and historical change.
Prof. Alison Mackey, Linguistics
Her major research interests include: second language acquisition, in particular, input and interaction, the roles of attention and working memory in second language development; and second language research methodology.
Prof. Abigail A. Marsh, Psychology
Her research assesses neural substrates and behavioral outcomes related to facets of emotion, with a focus on empathy, altruism, aggression, and nonverbal communication.
Prof. Mark Maloof, Computer Science
His research interests include machine learning, data mining, on-line learning algorithms, concept drift, adaptive software systems, and applications of machine learning and data mining to computer security.
Prof. Janet Mann, Psychology and Biology
Her main interests are in ethological methods, mother-infant relationships and infant development in cetaceans and primates, evolution, and behavioral ecology.
Prof. John Mikhail, Law and Philosophy
His research interests include moral cognition and intuitions of justice and fairness.
His reasearch focuses on the cognitive and cultural foundations of political behavior.
Prof. Paul Portner, Linguistics
His reasearch covers semantics, pragmatics, and the syntax/semantics interface.
His interest is in the functional organization and plasticity of cerebral cortex.
His interest is in computational models, psychophysics and fMRI of high level vision, in particular object recognition, object representation, and attention.
Prof. Cristina Sanz, Spanish & Portuguese
Her focus is the acquisition of second languages and the relationship between bilingualism and cognition.
Prof. Pamela A. Saunders, Neurology
Her research is on communication, aging and Alzheimer's disease, including doctor-patient communication in the older patient population.
Prof. Michael Ullman, Neuroscience
He investigates the neural and psychological bases of language, and relations between language, memory and motor functions.
Prof. Jeffrey Urbach, Physics
His research involves imaging dynamical phenomena in physical systems.
Her research focuses on understanding the cognitive and neural underpinnings of memory and cognitive control.
Prof. Mahendran Velauthapillai, Computer Science
His research focuses on learning theory, bioinformatics, protein linguistics, networks, and online algorithms.
Prof. Elizabeth Zsiga, Linguistics
Her research interests include phonological theory, second language phonology, phonetics (especially articulatory), speech synthesis, and the phonology/phonetics interface. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,181 |
Q: How can I replace within a SELECT the statement from one table to ther other one? I am trying to SELECT two tables an replace from table A an attribute of a value from table B.
Importants is for me that the tables not updated.
table_A
id category_id filename
1 2 apple
2 12 banana
3 453 pineapple
table_B
id category_color category_type
2 red fruit
12 yellow fruit
453 brown fruit
The MYSQL version is
mysql Ver 14.14 Distrib 5.5.62, for debian-linux-gnu (x86_64) using readline 6.3
mysql> SELECT * FROM table_A INNER JOIN table_B ON table_A.category_id = table_B.id SET table_A.category_id = category_name;
ERROR 1064 (42000): You have an error in your SQL syntax; check the manual that corresponds to your MySQL server version for the right syntax to use near 'SET table_A.category_id = category_name' at line 1
The output should like the below example
output
id category_id filename
1 red apple
2 yellow banana
3 brown pineapple
A: SET doesn't belong in a SELECT query. Instead, reference the value from the second table in the field list:
SELECT A.id, B.category_name, A.filename
FROM table_A A
INNER JOIN table_B B ON A.category_id = B.id
Output
id filename category_name
1 apple red
2 banana yellow
3 pineapple brown
Demo on dbfiddle
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,470 |
Суви До () је насеље у општини Липљан, Косово и Метохија, Република Србија.
Географија
Село је у равници, у тзв. Широком пољу. Садашњи главни део села је поред саме косовске железничке пруге, на 2 км северно и северозападно од Липљана, а куће мањег дела села су скоро на двоструко краћем растојању од Липљана. Јужном страном колоније пружа се корито потока Суви До, по коме је село добило име. Село има два дела. На западној страни је била и кула чифличког господара, које је порушена 1912. Делови села немају посебна имена.
Историја
Село се као Соуходол помиње у Грачаничкој повељи краља Милутина из 1321. Крајем 18. века је село као чифлик обновио "Црни бег" из Приштине. Нешто касније су ту на куповину преселио један албански род са Робовачких Појата, па је и он држао чифчије. Али, од обнављања села чифчије су у њему били само Албанции.
Порекло становништва
Колонисти
Из Босанског Петровца
Кртинићи (6 к.)
Дукићи (5 к.)
Балабановићи (4 к.)
Радаковићи (1 к.)
Дрљача (1 к.)
Ромић (2 к.)
Филиповић (2 к.)
Шарац (2 к.)
Радошевић (3 к.)
Бркљач (1 к.)
Даутовић (1 к.)
Вигњевић (1 к.)
Ожеговић (3 к.)
Лабус (2 к.)
Петровић (1 к.)
Мирковић (1 к.)
Ружић (1 к.)
Из Фоче
Мијановић (1 к.).
Из Билеће
Милићевићи (12 к.)
Џелетовићи (4 к.)
Илићи (5 к.)
Вујевић (1 к.)
Грбушић (1 к.)
Мишевић (1 к.)
Јокановић (1 к.)
Бајчетић (1 к.)
Самарџић (3 к.)
Звијер (1 к.)
Дендићи (3 к.)
Милидраговић (1 к.)
Сикимић (1 к.)
Шаренац (1 к.)
Лечић (1 к.)
Из Требиња
Радовић (1 к.)
Милановић (1 к.)
Томашевић (1 к.)
Тарајло (1 к.)
Из Љубиња
Дошлић (1 к.)
Из Карловца (Хрватска)
Папићи (3 к.)
Из Лике
из Зрмање
Сучевић (2 к.)
Ђалић (1 к.)
Заклан (1 к.)
Крњајић (1 к.)
Кричковић (1 к.)
Чанковић (1 к.)
из Удбине
Косановић (1 к.)
из Госпића
Обрадовић (1 к.)
из Лапца
Дошен (2 к.)
Медић (3 к.)
Из Боке
Гуњајевић (1 к.)
Лалошевић (2 к.)
Из Цуца
Кривокапић (1 к.)
Из Његуша
Дуда (1 к.)
Из Никшића
Митровић (1 к.)
Ђалић (1 к.)
Из Даниловграда
Гашовићи (3 к.)
Нешовићи (2 к.)
Јововићи ( 1 к.)
Бешићи (1 к.).
Из Баната
Пиперски (1 к.).
Сви су колонисти досељени од 1920-1923.
Срби муслимани
Хаџовићи (2 к.) Као мухаџири се 1878. иселили из варошице Колашина (Црна Гора) по њеном ослобођењу и настанили се у Бијелом Пољу, где су били чифчије. После неколико година прешли у околину Сјенице. Кренувши 1912. дубље у Турску, затекли се на Косову у балканском рату и зачифличе се у Сувом Долу.
Албанци
Нуховић (1 к.), од фиса Краснића. Досељен крајем 18. века из Скадарске Малесије.
Ашан (5 к.), од фиса Сопа. Преселио се од истоименог рода у Робовачким Појатама око 1820, али су ту били само на зимовању са стоком, јер су лето проводили на Корабу. Породице су из Новог Села у Топојану (код Љуме) довели тек 1915.
Гунцат (1 к.) од фиса Гаша. Досељен из Гунцата (Подрима) око 1870.
Бреговин (2 к.) од фиса Тсача. Мухаџир је из 1878. из Бреговине (Топлица).
Бајрамовић (1 к.) од фиса Битича. Доселио се око 1900. из Мерене (Дреница) за чифчију.
Демографија
Референце
Литература
Општина Липљан
Насељена места на Косову и Метохији
Википројект географија/Насеља у Србији | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,515 |
Q: xterm doesn't run shell script so I'm trying to use this command:
export DISPLAY=:1; /usr/bin/xterm -hold -e /path/to/shscript
Where shscript is:
#!/bin/bash
echo "Restarting ... $(date)" >> /var/log/mw2.txt
if screen -ls | grep -q 'test'; then
screen -X -S test quit
sleep 1000
screen -d -m -S test wine iw4m.exe -dedicated -console +dw_licensefile license.dat +set net_port "28960" +set party_maxplayers 18 +exec server.cfg +map_rotate +set fs_game "mods/tsd"
else
screen -d -m -S test wine iw4m.exe -dedicated -console +dw_licensefile license.dat +set net_port "28960" +set party_maxplayers 18 +exec server.cfg +map_rotate +set fs_game "mods/tsd"
fi
So I want to launch a new xterm window and it to run a shell script.
But whatever shell script I try to use, it doesn't run it. It just goes blank. All other commands work, but when I put a shell script in it, it just goes blank and does nothing. I can't find a solution for this, please help, thank you.
I get this with set -x:
++ date
+ echo 'Restarting ... (date)'
+ grep -q test
+ screen -ls
+ screen -d -m -S test -wine iw4m etc...
A: I think you get exactly what you have asked for...
From screen's man page:
-d -m Start screen in "detached" mode. This creates a new session but
doesn't attach to it. This is useful for system startup
scripts.
On the other hand you use -hold for xterm:
-hold Turn on the hold resource, i.e., xterm will not immediately
destroy its window when the shell command completes. It will
wait until you use the window manager to destroy/kill the win‐
dow, or if you use the menu entries that send a signal, e.g.,
HUP or KILL.
And that's exactly what you see. xterm starts, executes screen which runs, but you don't see the output as screen does not attach to the virtual terminal. Since screen has detached the shell script exits and xterm holds the window for you to destroy it at your leisure.
I bet that if in another xterm you attach to the session with screen -S test you'll see the output.
A: Yep. It was the wine command failing. I changed some stuff, and it works now. I had to put cd /home/MW2 in the script for it to be able to run the iw4m.exe, then screen -S etc.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,478 |
#ifndef XCF_H_INCLUDED
#define XCF_H_INCLUDED
/* GIMP's XCF file properties/structures : */
#include "asvisual.h"
#include "scanline.h"
#ifdef __cplusplus
extern "C" {
#endif
#define XCF_MAX_CHANNELS 4
#define XCF_GRAY_PIX 0
#define XCF_ALPHA_G_PIX 1
#define XCF_RED_PIX 0
#define XCF_GREEN_PIX 1
#define XCF_BLUE_PIX 2
#define XCF_ALPHA_PIX 3
#define XCF_INDEXED_PIX 0
#define XCF_ALPHA_I_PIX 1
#define XCF_COLORMAP_SIZE 768
typedef enum
{
XCF_PROP_END = 0,
XCF_PROP_COLORMAP = 1,
XCF_PROP_ACTIVE_LAYER = 2,
XCF_PROP_ACTIVE_CHANNEL = 3,
XCF_PROP_SELECTION = 4,
XCF_PROP_FLOATING_SELECTION = 5,
XCF_PROP_OPACITY = 6,
XCF_PROP_MODE = 7,
XCF_PROP_VISIBLE = 8,
XCF_PROP_LINKED = 9,
XCF_PROP_PRESERVE_TRANSPARENCY = 10,
XCF_PROP_APPLY_MASK = 11,
XCF_PROP_EDIT_MASK = 12,
XCF_PROP_SHOW_MASK = 13,
XCF_PROP_SHOW_MASKED = 14,
XCF_PROP_OFFSETS = 15,
XCF_PROP_COLOR = 16,
XCF_PROP_COMPRESSION = 17,
XCF_PROP_GUIDES = 18,
XCF_PROP_RESOLUTION = 19,
XCF_PROP_TATTOO = 20,
XCF_PROP_PARASITES = 21,
XCF_PROP_UNIT = 22,
XCF_PROP_PATHS = 23,
XCF_PROP_USER_UNIT = 24,
XCF_PROP_Total = 25
} XcfPropType;
typedef enum
{
XCF_COMPRESS_NONE = 0,
XCF_COMPRESS_RLE = 1,
XCF_COMPRESS_ZLIB = 2,
XCF_COMPRESS_FRACTAL = 3 /* Unused. */
} XcfCompressionType;
typedef enum
{
XCF_RED_CHANNEL,
XCF_GREEN_CHANNEL,
XCF_BLUE_CHANNEL,
XCF_GRAY_CHANNEL,
XCF_INDEXED_CHANNEL,
XCF_ALPHA_CHANNEL,
XCF_AUXILLARY_CHANNEL
} XcfChannelType;
typedef enum
{
XCF_EXPAND_AS_NECESSARY,
XCF_CLIP_TO_IMAGE,
XCF_CLIP_TO_BOTTOM_LAYER,
XCF_FLATTEN_IMAGE
} XcfMergeType;
#define XCF_SIGNATURE "gimp xcf"
#define XCF_SIGNATURE_LEN 8 /* use in strncmp() */
#define XCF_SIGNATURE_FULL "gimp xcf file"
#define XCF_SIGNATURE_FULL_LEN 14 /* use in seek() */
#define XCF_TILE_WIDTH 64
#define XCF_TILE_HEIGHT 64
struct XcfProperty;
struct XcfLayer;
struct XcfChannel;
struct XcfHierarchy;
struct XcfLevel;
struct XcfTile;
typedef struct XcfImage
{
int version;
CARD32 width;
CARD32 height;
CARD32 type ;
CARD8 compression ;
CARD32 num_cols ;
CARD8 *colormap ;
struct XcfProperty *properties ;
struct XcfLayer *layers;
struct XcfChannel *channels;
struct XcfLayer *floating_selection;
struct XcfChannel *selection;
ASScanline scanline_buf[XCF_TILE_HEIGHT];
CARD8 tile_buf[XCF_TILE_WIDTH*XCF_TILE_HEIGHT*6];
}XcfImage;
typedef struct XcfProperty
{
CARD32 id ;
CARD32 len;
CARD8 *data;
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CARD32 buffer[20] ;
struct XcfProperty *next;
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typedef struct XcfLayer
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struct XcfLayer *next;
CARD32 offset ;
/* layer data goes here */
CARD32 width ;
CARD32 height ;
CARD32 type ;
/* we don't give a damn about layer's name - skip it */
struct XcfProperty *properties ;
CARD32 opacity ;
Bool visible ;
Bool preserve_transparency ;
CARD32 mode ;
CARD32 offset_x, offset_y ;
CARD32 hierarchy_offset;
CARD32 mask_offset ;
struct XcfHierarchy *hierarchy ;
struct XcfChannel *mask ;
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typedef struct XcfChannel
{
struct XcfChannel *next;
CARD32 offset ;
/* Channel data goes here */
CARD32 width ;
CARD32 height ;
/* we don't give a damn about layer's name - skip it */
struct XcfProperty *properties ;
CARD32 opacity ;
Bool visible ;
ARGB32 color ;
CARD32 hierarchy_offset;
struct XcfHierarchy *hierarchy ;
}XcfChannel;
typedef struct XcfHierarchy
{
/* layer data goes here */
CARD32 width ;
CARD32 height ;
CARD32 bpp ;
/* we don't give a damn about layer's name - skip it */
struct XcfLevel *levels ;
ASImage *image ;
}XcfHierarchy;
typedef struct XcfLevel
{
struct XcfLevel *next ;
CARD32 offset ;
CARD32 width ;
CARD32 height ;
struct XcfTile *tiles ;
}XcfLevel;
typedef struct XcfTile
{
struct XcfTile *next ;
CARD32 offset ;
CARD32 estimated_size ;
CARD8 *data;
}XcfTile;
union XcfListElem;
typedef struct XcfAnyListElem
{
union XcfListElem *next;
CARD32 offset ;
}XcfAnyListElem;
typedef union XcfListElem{
XcfAnyListElem any;
XcfLayer layer;
XcfChannel channel;
XcfLevel level;
XcfTile tile;
}XcfListElem ;
XcfImage *read_xcf_image( FILE *fp );
void print_xcf_image( XcfImage *xcf_im );
void free_xcf_image( XcfImage *xcf_im );
#ifdef __cplusplus
}
#endif
#endif /* #ifndef XCF_H_INCLUDED */
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{"url":"https:\/\/ask.openstack.org\/en\/question\/69553\/change-default-path-emulator-for-libvirt\/","text":"# Change default (path) emulator for libvirt\n\nHello,\n\nI downloaded the new version of qemu-kvm and I made some changes in the code. I compiled the new version and installed it. The path to the emulators are under \/usr\/local\/bin.\n\nWith libvirt, nova uses the path \/usr\/libexec to find the emulators to be used. I want that nova uses the emulators under \/usr\/local\/bin\/\n\nI am trying three different things without any success:\n\n\u2022 Changing the default emulator path in the libvirt\n\nI actually don't find where I can change the default path. My $PATH has \/usr\/local\/bin as first entry but it seems that libvirt doesn't look at the ENV variable$PATH.\n\n\u2022 Addind a 'emulator' parameter in the XML config file that will be use to spawn the new instance.\n\nI have found that someone already did a small patch but not too much explanation are given. I don't understand where in 'config.py' I can add this option http:\/\/lists.openstack.org\/pipermail\/... I am still looking at the code under nova\/virt\/libvirt. Also it seems that is no emulator is specified in the XML, the default path will be \/usr\/libexec\/qemu-kvm http:\/\/www.redhat.com\/archives\/libvir...\n\n\u2022 Adding the option emulator=\/usr\/local\/bin under [libvirt] in nova.conf\n\nBut this change seems not to be taken into account\n\nEDIT 1 From those 3 directions, I think trying to add the <emulator> parameter in the XML file seems to be the more reasonable but I have some dfficulties to find where to add that in the code under nova\/virt\/libvirt.\n\nCan someone gives me some hints? Thanks a lot :)\n\nedit retag close merge delete\n\nCan you set the $PATH in root user's profile and try? ( 2015-07-08 02:08:32 -0500 )edit My$PATH is already \/usr\/local\/bin:\/usr\/local\/sbin:\/usr\/bin:\/usr\/sbin:\/bin:\/sbin:\/home\/stack\/.local\/bin:\/home\/stack\/bin","date":"2019-10-21 19:21:50","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\": 0.349560022354126, \"perplexity\": 2002.9019652952588}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570987781397.63\/warc\/CC-MAIN-20191021171509-20191021195009-00400.warc.gz\"}"} | null | null |
L'arteriola efferente (in latino arteriola glomerularis efferens) è un'arteriola dell'apparato urinario, situata all'interno del rene, che si origina dalla fusione dei capillari glomerulari (propri del glomerulo renale), per dirigersi verso i capillari peritubulari, irrorando il sistema tubulare del nefrone. Le arteriole efferenti svolgono un importante ruolo nel mantenere la velocità di filtrazione glomerulare, nonostante le fluttuazioni nella pressione sanguigna.
Nei reni dei mammiferi, queste arteriole seguono due percorsi molto diversi, a seconda della posizione in cui si trovano i glomeruli; circa il 15% dei glomeruli si trovano nei pressi del confine tra la corticale e la midollare renale e sono conosciuti come glomeruli iuxtamidollari. Il resto sono glomeruli corticale indifferenziati.
Struttura
Nei glomeruli iuxtamidollari
Le arteriole efferenti dei glomeruli iuxtamidollari formano dei fasci di vasi (arteriole rette spurie) che attraversano la zona midollare esterna per irrorare la zona interna. Le venule di ritorno dal midollo interno (venule rette) si diramano in modo molto regolare a formare una rete mirabile organizzata.
Questa rete è responsabile dell'isolamento osmotico della midollare interna dal resto del rene e permette così l'escrezione di urina ipertonica quando le circostanze lo richiedono. La rete serve anche ad isolare la midollare interna dallo scambio gassoso, dato che il metabolismo in questa zona è anaerobico; i globuli rossi sono ordinariamente deviati dalle arteriole rette con un meccanismo sconosciuto nel plesso capillare che circonda i tubuli della midollare esterna. Il sangue di ritorno dalla zona midollare interna percorre la vena renale.
Nei glomeruli corticali indifferenziati
Le arteriole efferenti dei glomeruli corticali indifferenziati sono più complesse rispetto alle altre arteriole efferenti. Prima di uscire dal glomerulo danno origine a capillari, diventando parte di un ricco complesso di vasi che circondano la porzione corticale dei tubuli renali.
Funzione
L'arteriola efferente, assieme al glomerulo e all'arteriola afferente, forma una rete mirabile arteriosa, ossia una rete di capillari sanguigni interposta fra due arterie, a differenza delle normali reti capillari comprese fra un'arteria e una vena.
Quando i livelli di angiotensina II aumentano a causa di attivazione del sistema renina-angiotensina-aldosterone, la maggior parte delle arterie va incontro a vasocostrizione, al fine di mantenere un'adeguata pressione sanguigna; tuttavia, questo riduce il flusso sanguigno verso i reni. Per compensare, le arteriole efferenti, sempre in risposta ad un aumento dei livelli di angiotensina II, si restringono in misura maggiore rispetto alle altre arterie; quindi non si ha perdita di pressione nei capillari glomerulari e la velocità di filtrazione glomerulare rimane adeguata.
Galleria d'immagini
Bibliografia
Voci correlate
Rene
Glomerulo
Arteriola afferente
Rene | {
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In 1979 the first Winter Camp Awareness outdoor session was held near Lake Tahoe and this year also marked the beginning of the end for the old lodge paper records. Starting in 1979 with the aid of a Radio Shack TRS-80 "supercomputer", Don Wilkinson and the Lodge began the daunting task of converting all of the old Machek N'Gult, Royaneh and Achewon Nimat membership records over into an electronic database. Achewon Nimat was one of the first Lodges in the Country to convert our membership and financial records into an electronic database format. Previously all membership records were contained on a 3 x 8 visual record cards At the 1980 lodge banquet held at the Presidio Officers club in San Francisco, Lodge members feasted on such delicacies as Veal Cordon Bleu and BBQ ground round steak. During the dinner events like blowing up balloons without the use of your hands and keeping a balloon aloft the longest provided fun and amusement for the on-lookers. Winners of these events won a free trip to the 1980 Section Conclave at Camp Parks. Near the end of the year, the Order of the Arrow was saddened by the death of Dr. E Urner Goodman, founder of our Order. Live Oak village hosted a snow trip in Bear Valley at the start of 1981 that was attended by both village and lodge members. The lodge stayed at a three story cabin that could only be reached by hiking up a snow covered road for a mile. Toboggans, inter-tubes and skiing at nearby Mt Reba was the activities for the weekend. A great spaghetti dinner with all the fixins was prepared by village adviser Adrian Stith. In June 1981 the lodge adopted a new yellow bordered flap for service where an arrowman could purchase one flap per ordeal or Mikemosin. NOAC in 1981 was held in Austin, Texas at the University of Austin. Our Lodge had a contingent of about 14 people and we traveled to NOAC in two vans (Bill Parker's Orange Pumpkin and another van). For whatever reason, the Pumpkin didn't have any air conditioning and our trip during the summer took us through Barstow, California (one of the hottest places on Earth). As luck would have it, in addition to no air conditioning the drive train on the Pumpkin broke while we were in Barstow. So the contingent had to wait as Bill looked for a place to get the Pumpkin fixed. Nothing like breaking down in Barstow during August. We made it to Austin on time where we stayed in the 10 story dormitory building at UOA (also not air conditioned). In 1982 with the chapter system failing it was decided to discontinue it altogether in lieu of the village system. Ten years after hosting our first conclave at Fort Cronkite, Achewon Nimat would host its second conclave in 1985 at the San Francisco Presidio with the theme of "Service, Tradition & Honor". Over 125 members attended to help the lodge in its duties and Shepherd Hendrix from Live Oak Village served as the Conclave Chairmen. In September a new unrestricted brotherhood patch was released. The patch used the new updated lodge patch design but would have a red border. In 1986, to help the OA program in Oakland and San Francisco, and to comply with National policy, the lodge re-instituted the chapter system. Live Oak and Golden Acorn Villages merged to form the Oakland Chapter, who called themselves Achewon Tulpe (Strong Turtle). Golden Gate, Mission Trails, and Sierra Villages merged to form the San Francisco Chapter, to be called Royaneh. The three other villages remained the same and were allowed to continue using their Indian names (Tres Ranchos was Amangi Nechochwen, Twin Valley was Seunen Paschengink and Mission Peak remained as Ohlone). The lodge also attained the highest score among all the 69 Western Region Lodges in the Lodge Achievement Program during 1986. In 1990, the lodge celebrated its 25th anniversary and also issued a replica of our original lodge flap with "25" added to it. Requirements were put on the flap, much like the National OA's 75th Anniversary Award. 1990 Also marked the first time in 25 years that the lodge was unable to achieve the National Honor Lodge Award. This began a time of rebuilding for the lodge. The annual lodge fellowship weekend known as the Mikemosin, was renamed to Achiefest in 1992. That same year the lodge also sent a larger contingent to the National Order of the Arrow Conference (NOAC) which was held in Knoxville, Tennessee and issued the first NOAC specific lodge flap. In 1993 the lodge continued to grow and prepare for the Conclave and NOAC. 1994 was a banner year for us as we won our first ever Conclave Award at the Alameda Naval Air Station. The lodge had its first Section Officer in 15 years when Jeremy Davis was elected to the section. We also sent the largest contingent in the section to NOAC. Like good fermented grape juice, 1995 was an even better year. The lodge hosted the W3A Conclave at Camp Parks and won our second consecutive Conclave Award. The attendance was great throughout the year, and continued our streak of National Honor Lodge recognition. As this was our 30th anniversary year, the Lodge issued replicas of the Machek N'Gult and Royaneh flaps to members who met a participation requirement. 1996 was the year in which we ended a 20-year Most Indian Lodge Award drought. To top off yet another great conclave, we again won the Conclave Award, making it three years straight, we had another member become a section officer, Ken Morton, and we sent another large contingent to NOAC at Indiana University. In 1997 Jeremy Davis became Section W3A chief. Achewon Nimat won its fourth Conclave Award in a row and our second consecutive Most Indian Lodge Award, but lost our Indian Handball championship. During this year, 2 more members of our lodge became section officers, Rocky Fernandez and Ed Smith. In the duration of 1998 through 2000 we continued with our streak of winning Conclave and Most Indian Lodge Awards. In 2000, the lodge dance team achieved its long term goal of winning the Indian Dance competition and taking home the coveted (and enormous) Dance Trophy. 2002 proved to be an outstanding year for the lodge. We brought our streak of winning the Conclave Award to nine straight years, when we hosted the W3A Conclave at Camp Royaneh. This same year Dominic Pascucci was elected Western Region Chief, the second national officer in the history of our Lodge (Larry Teshara was National vice chief in 1961 from Royaneh Lodge). In 2003, our streak of winning the Conclave Award was brought to an end when the Lodge placed third at the Conclave at Roaring Camp Railroads in Felton, CA. Matt Griffis who was Lodge Chief in 2001 & 2002 was elected Western Region Chief, marking the first time ever that any lodge in the country has had back-to-back national officers. The lodge commemorated this event by issuing a special lodge flap. In 2004 the lodge took second place at the Conclave, which was held at the Santa Cruz County Fairgrounds in Watsonville. In the summer of 2004 the lodge sent a contingent of 10 members to NOAC, which was held at the Iowa State University. At the conference, the standing Western Region Chief resigned from his position due to personal reasons and the National Chief appointed Matt Griffis to finish the term. For their service to the Order of the Arrow, Dominic Pascucci and Matt Griffis both received the Distinguished Service Award. In 2006 Achewon Nimat Lodge joined the online world when our Web site "AchewonNimat.org" was created and launched by Lodge Adviser Charles Hoffman. 2006 also saw us winning our tenth conclave award when the conclave was located at Cutter Scout Reservation. In 2008 eleven arrowmen from Achie participated in the ArrowCorps 5 project in the Shasta-Trinity Mountains. The arrowmen spent a week working on the Pacific Crest trail and assisting the National Forest service. The following year when Achewon Nimat hosted the 2009 Conclave at Camp Royaneh we took home the coveted Conclave award for an astonishing 11th time. In 2010 during the centennial celebration of the Boy Scouts of America, Achewon Nimat celebrated 45 years as a combined lodge and 66 years since our founding lodges were formed. One of the big events of the year that the lodge was involved with was the Northern California Jamboree that was held at the Alameda County Fairgrounds in Pleasanton. Over 20,000 Scouts, Leaders and the public participated in this three day event to celebrate the 100th anniversary of Scouting. At the Section Conclave held at the Presidio of Monterey, Achewon Nimat Lodge Chief Indy Nelson was voted as the new Section Chief of the W3S section for the 2011-2012 scouting year. Indy would become the 9th member of our lodge to lead the section since its founding in 1965. 2011 was a sad year for our Lodge as we lost two dedicated and longtime great members, Bill Parker and Jim Smith. In honor of their dedication and commitment to the ideals of scouting and the Order of the Arrow, a NOAC Conferenceship Fund was setup in Bill Parkers name. This endowment fund is setup in perpetuity to recognize Bill's passion for NOAC and his years of service to Scouting and the Order of the Arrow. The purpose of the endowment fund is to provide monies toward a youth's participation in the National Order of the Arrow Conference. In October of 2011 after months of discussions, the SFBAC and the Mt. Diablo/Silverado Councils announced that our two great councils would be merging sometime in 2012 pending the outcome of the stakeholders meeting on August 29th, 2012. Along with the merging of the Council so too would Achewon Nimat Lodge and Ut-in Selica Lodge combine their membership into one strong Lodge. Although almost two years of discussion and planning went into the anticipated merger, on August 29th, 91% of the SFBAC voted in favor of the merge while 60% of the voting members of the Mt Diablo Silverado Council voted not to unite. Misleading information distributed by some of the MSDC members upset at the possible merger officially ended the creation of the Golden Gate Area Council. Achewon Nimat would continue to serve as the Order of the Arrow Lodge for the San Francisco Bay Area Council. Also in 2011 the San Francisco Bay Area Council Campership Fund was started with a "seed" gift from Achewon Nimat Lodge. In 2013 at the new Summit Bechtel Jamboree site in West Virginia, a building was dedicated to the Order of the Arrow which contains a giant fireplace that hark's back to the late 1940's when E. Urner Goodman built a stone fireplace at his residence in Bondville, Vermont. What's special about the fireplace is that it contains rocks provided by the different Lodges from across the Country. The rock from our council is inscribed with "Achewon Nimat, San Leandro" and came from an outcropping of rocks located at Wente Scout Reservation near the dam. The fireplace at E. Urner Goodman's Brotherhood Barn next to his home residence in Vermont also contains a special rock from the Oakland Area Council inscribed with the symbol of Camp Dimond-O which closed in 1978. In October of 2013 with the number of active arrowmen in the Lodge decreasing, a number of changes were instituted that affected the entire Lodge. The villages of Amangi Nechochwen and Achewon Tulpe merged to form the Wekemnayon village (meaning New Brotherhood). Wekemnayon was one of the chapters in our lodge in the mid 1970's that encompassed the villages of Live Oak, Golden Acorn, and Charrowood. So it was a good fit to once again use the Wekemnayon name. The other major change was that the number of ordeals was reduced from three ordeals per year to two ordeals per year and the Klondike Derby snow camping adventure was canceled until more arrowmen become active. At the 48th annual banquet, Lodge Chief Hans Mortimer resigned due to school commitments at UC Berkeley. Vice-chief Ryan Shepodd stepped up and was sworn in as the new lodge chief to complete the 2013-2014 lodge year. Early in 2014, the NOAC centennial project for the 100th anniversary of the Order of the Arrow was begun which included creating a booklet to document the history of our lodge. The second portion of the project was the centennial crate to hold wood from our council camps to be burned at NOAC. The ashes would be comingled with the ashes from other lodges and given to attendees at NOAC as a memento. The NOAC centennial crate from Achewon Nimat contained a sampling of wood from each of the council camps where our lodge has held ceremonies since 1944. Redwood to symbolize the majestic tall trees from Camp Royaneh in Sonoma County and Camp Lilienthal in Marin County where Royaneh Lodge was founded. Cedar to symbolize the trees from the hills of Camp Dimond-O near Yosemite and pine to symbolize the local trees from Wente Scout Reservation in Willits, Rancho Los Mochos in the Livermore hills and Camp Dimond in Oakland where Machek N'Gult was founded. 2014 also marked the first time since the early 1990's that unit elections and tap outs would be held at summer camp. The Scout Exec agreed to a plan that would designate one camp staff member as the Order of the Arrow camp representative who would have the responsibility to promote the OA and run elections during summer camp. As 2014 is closed out, a special 50th anniversary banquet honoring the merging of Machek N'Gult Lodge 375 and Royaneh Lodge 282 was planned for the end of the year. On December 13, 2014 exactly 50 years to the day that our two former lodges merged at Goodman's restaurant in Oakland, a similar celebration was held once again but this time at the Council Office with over 100 Arrowmen in attendance. A special issue lodge flap available only at the dinner to commemorate the event was made available to those in attendance. Former lodge advisers, chiefs and arrowmen from both Machek N'Gult and Royaneh were in attendance and the theme of the night was "A Blast to our Past" with memorabilia displays featuring both former lodges. The year 2015 began with the Winter Camp Awareness training session held aboard the USS Hornet in Alameda. Over 260 Scouts and 40 arrowmen from the SFBAC and other councils attended the day-long event. In preparation for the NOAC 2015 event, the Order of the Arrow requested that all former or discontinued lodges provide a rock for the temporary centennial fireplace to be built at NOAC. The rocks would then be transported to the Summit Bechtel Reserve to be incorporated in a permanent fireplace honoring all lodges of the Order of the Arrow. Two rocks were obtained for Machek N'Gult and Royaneh Lodges. The rock for Machek N'Gult was located on the former grounds of Camp Dimond in Oakland near the site of the amphitheater where Machek N'Gult was founded. The rock for Royaneh was located at Camp Royaneh down in the river bed of East Austin Creek where ceremonies were held at the camp. The former lodge names were then inscribed in the two rocks by the Bras & Mattos Monument Company of Hayward. In May the lodge youth leaders attended a weekend retreat at the Youth Retreat Center operated by the Diocese of Oakland in Pleasant Hill. The two day event was used to discuss the operations of the lodge and planning the events of the next year. The site offered a great location to hold the leadership training.
Achewon Nimat Lodge History (continued - page 2) | {
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Q: 12 bytes in 1 blocks are still reachable in loss record I'm using C and wanted to create a doubly linked list that contains char values. I have another file given to use by the teacher to measure the big O notation so I'm sure it's correct. But when I run it, I get a zsh: segmentation fault from the terminal.
I ran it through valgrind and it gave me the error mentioned in the title and mentioned the following function, more specifically the Malloc and strdup command. I'm not sure what's wrong here, it looks fine to me so was hoping to get some help here.
Here's the struct from the header file:
struct node
{
struct node *next;
struct node *prev;
char *value;
};
and here's the function to create a node in the main file:
static struct node *make_node(const char *value)
{
struct node *result = malloc(sizeof(struct node));
result->value = strdup(value);
result -> next = NULL;
result -> prev = NULL;
return result;
}
And for good measure, here's the function that calls make_node
ListPos list_insert(ListPos pos, const char *value)
{
struct node *node = make_node(value);
struct node *before = pos.node->prev;
struct node *after = pos.node;
node->next = after;
after->prev = node;
node->prev = before;
before->next = node;
pos.node = node;
return pos;
}
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You are here: Home » Analysis of Nik Cochran's outside shooting
March 18, 2013 | Andrei Greska | 1 Comment
Analysis of Nik Cochran's outside shooting
In reading Mark's detailed scouting report of Davidson on Monday, I couldn't help but sit in awe of Nik Cochran. Yes, his free-throw shooting is Novakian, going 112-119 this season for a ridiculous 94.1 percent accuracy, but it was his outside shooting that had my eyes bulging.
Cochran has hit a ridiculous 48.5 percent of his 101 shots from beyond the arc, tied for 13th best in the country. Lest you think the sample size was small or simply an aberration, the 101 attempts would be second most on Marquette, only 12 behind Vander Blue and 18 ahead of Jamil Wilson.
That number kept popping into my head because, as most Marquette fans can attest, the Golden Eagles have had problems in the past with hot shooters. Louisville's Mike Marra went 6-10 from 3-point land in the quarterfinals of the Big East Tournament in 2011 en route to an 81-56 dismantling by the Cardinals.
You don't have to look back two years to find another time Marquette was hurt from long distance by a hot shooter. Heck, you don't even have to look back two games. In case it was deleted from your memory during the St. Patrick's Day festivities, Notre Dame's Pat Connaughton lit up Marquette last Thursday, also going 6-10 from 3-point land, snuffing out any and all late runs.
Marquette also allowed an opponent to shoot at least 50 percent from three when taking at least five attempts five times this conference season with Darrun Hilliard, Worrel Clahar, Shabazz Napier, and Lamar Patterson (twice) getting in on the action. That shows that though Marquette does a decent job at defending the three — opponents are only shooting 31.8 percent, the 76th best number in the country — it is susceptible to giving them up in bunches to a hot shooter.
There are a few candidates on the Wildcats who could copy the Connaughton script. Though Chris Czerapowicz and Tyler Kalinosky have taken and made more threes for Davidson than Cochrane has, neither tops 40 percent in accuracy, leaving us with Cochran as the main suspect.
As such, I decided to take a look at every three he has attempted (or at least the ones captured on video at Synergy as some of the games were not available) to see what, if any, patterns would emerge.
The red areas above are where Cochran has been most deadly, which has been just about everywhere. He thrives on the right side of the court though, shooting 66 percent from the baseline and 57 percent from the elbow extended. Those are video game-like numbers.
His weakest spot on the floor is the left elbow where he is hitting at a sub 34 percent rate. The shot from straight away is his second-weakest spot, but at almost 40 percent, I wouldn't call that weak by any stretch of the imagination. Case in point, he can hit from wherever, but forcing him to the left side of the court is the best statistical option.
When looking at the video clips, I also noticed a trend that he very rarely shot off the dribble with almost every three coming off a spot-up jumper that was assisted. Going back to track it the results were so astounding a pie graph was needed for full effect.
With almost 82 percent of Cochran's treys coming off assisted spot up shots, it will be imperative for Marquette's defense to not sag off him much when doubling down low and to not go under screens if at all possible. Granted, I'm sure Buzz was telling his team this throughout the game against Notre Dame, but a few defensive lapses could be the difference between surviving and advancing.
Davidson's quick passing helps to create a lot of the openings for him to shoot (with Jake Cohen in particular finding him on the baseline with some one-touch passing) but they also run a very effective off-ball screen for him.
Screenshots courtesy of Synergy
Be sure to click on the image above to see it enlarged, or keep reading as I break it down frame by frame.
The play begins with all five players on the perimeter in a reverse flying V with the ball at the top of the arc.
Jake Cohen, number 15, will step up to set a pick on the two defenders closest to Cochran.
Cochran will slide left two steps while Cohen sets a screen to allow space for the ball to be delivered.
The defense here was quite bad, as Cohen's defender didn't realize that his teammate was stuck in a screen and didn't help out on Cochran. Against Marquette, should they be playing man-to-man, this would be Chris Otule or Davante Gardner sliding over to cover the shooter. This still leaves you vulnerable if executed correctly, as Cohen will be left matched up against a much smaller defender. Just take a look at that screen he sets, it doesn't get much better than that.
By the time the Vandy defender gets around the screen Cochran has already launched from an area where he shoots almost 54 percent. That's money right there.
The key for Marquette will be to deny passing lanes and pressure the ball handler. As I noted before, Davidson does a great job at working the ball both around and inside and out with precision, turning it over very rarely.
As it pertains to Cochran, having a defender on him will cause him fits, as he simply cannot create his own shot and doesn't blow by many defenders. Whether it's Blue or Junior, someone has to body him up at all times.
One final note is that Cochran has not shown very well against Tournament quality opponents, going 1-8 against Duke, New Mexico and Montana with his only good outing coming against Gonzaga where he hit three of six 3-pointers. That's not to say he will struggle, but rather length can bother him and disrupt his rhythm.
Thursday can't come soon enough.
Tags: Davidson, Marquette, NCAA Tournament, Nik Cochrane
Categories: Analysis, General News, Home, Synergy
← Davidson Scouting Report: All you need to know
History says 3-point shooting will end Marquette early →
Synergy: Where Cohen and Brooks help Davidson the most | Paint Touches - March 19, 2013
[…] night Andrei detailed the type of player Marquette will face in Nik Cochran, an efficient sharpshooter for the Davidson Wildcats. He may be […] | {
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Q: Access query: How to refer to listbox at a form I have Access Datebase comprising mainly to tables and a query. The table tab_Projects contains the fields
*
*ID
*Title
*Status
with several recordsets for different projects.
The field Statusrefers to the second table tab_Status with the fields
*
*ID
*Status
The recordsets for this table are fixed to
|--|-----------------|
|ID|Status |
|--|-----------------|
|0 | "in preparation"|
|1 | "accepted" |
|2 | "declined" |
|3 | "finished" |
|--|-----------------|
For the purpose of a user application with Access-Runtime I have a formular frm_Filter_Projectswith a a listbox listStatus. This listbox recieves the data from the table tab_Status and the user shall use this formular to filter the projects for different statuses. This listbox-elements are multiselectable.
A query shall use this formular and filter the datasets according to the selected elements at frm_Filter_Projects.listStatus
The SQL-code therefor might be something like:
SELECT
tbl_Projekte.ID,
tbl_Projekte.Titel,
tbl_Projekte.Status
FROM tab_Status
INNER JOIN tbl_Projekte ON tab_Status.ID = tbl_Projekte.Status
WHERE tab_Status.ID IN ([Forms]![frm_Filter_Projects]![listStatus].[itemsselected]);
As long as i use
WHERE tab_Status.ID IN (1,3)
everything works in the expected way.
How can I refer to the selected elements at the listbox within a query?
A: WHERE tab_Status.ID IN (...)
cannot be dynamic.
Thus, use code to:
*
*browse the itemsselected collection
*build the static list of the selected items, say (1,3)
*modify the SQL of the query
| {
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{"url":"https:\/\/www.physicsforums.com\/threads\/am-i-allowed-to-use-klv-and-kcl-with-a-capacitor-in-the-circuit.598652\/","text":"# Am I allowed to use KLV and KCL with a capacitor in the circuit?\n\nRelated Electrical Engineering News on Phys.org\nvk6kro\nI.e.\n\nhttp:\/\/img337.imageshack.us\/img337\/2701\/15irk.jpg [Broken]\n\nIs this alright? Is this \"legal\"?\nYou seem to have a short circuit across the capacitor and so across the rest of your circuit apart from the top resistor ( presumably R1)\n\nKirchoff's Laws really only apply to a stable situation, so a charging capacitor would be analysed by taking a \"snapshot\" of the circuit operation.\n\nThe two parallel resistor-diode circuits can be analysed by getting the parallel resistor combination and putting it in series with one diode.\nYou can do this because both diodes will drop 0.7 volts, so you can join their anodes together and no current will flow between points of equal voltage.\n\nFor example, let R1 = 1000 ohms, R2 = 2000 ohms, R3 = 3000 ohms Diode voltage = 0.7 volts.\nParallel R2 and R3 = 1200 ohms call this R4\n\nSo voltage across R1 and R4 = 15 - diode voltage = 14.3 volts.\nCurrent in R1 and R4 = 14.3 Volts \/ (1000 ohms + 1200 ohms) = 0.0065 amps\n\nVoltage across R4 if C wasn't there = IR = 0.0065 * 1200 = 7.8 volts\nR4 plus diode voltage = 8.5 volts\n\nSo this is the maximum voltage that C could charge to, if it didn't have a short circuit on it. :)\n\nLast edited by a moderator:\nGold Member\nYou seem to have a short circuit across the capacitor and so across the rest of your circuit apart from the top resistor ( presumably R1)\nOh yes, there supposed to be a button there to cause it. You're right, the button does cause a short circuit.\n\nThe two parallel resistor-diode circuits can be analysed by getting the parallel resistor combination and putting it in series with one diode.\nYou can do this because both diodes will drop 0.7 volts, so you can join their anodes together and no current will flow between points of equal voltage.\n\nFor example, let R1 = 1000 ohms, R2 = 2000 ohms, R3 = 3000 ohms Diode voltage = 0.7 volts.\nParallel R2 and R3 = 1200 ohms call this R4\n\nSo voltage across R1 and R4 = 15 - diode voltage = 14.3 volts.\nCurrent in R1 and R4 = 14.3 Volts \/ (1000 ohms + 1200 ohms) = 0.0065 amps\n\nVoltage across R4 if C wasn't there = IR = 0.0065 * 1200 = 7.8 volts\nR4 plus diode voltage = 8.5 volts\n\nSo this is the maximum voltage that C could charge to, if it didn't have a short circuit on it. :)\nSo suppose instead of a shortcut there is a switch there or whatever, so there is no short-circuit in fact. Is my method valid in this case?\n\nI appreciate you writing up a way to the solution, but I want to understand whether I have a glimpse of validity in my method before examining yours. :)\n\nvk6kro\nYou have 3 unknowns but two equations, so you need another equation.\n\nBut the equations you have written look OK.\n\nHowever as I tried to show, there is no need to use equations on a trivial problem.\n\nYou can use KCL and KVL on a capacitor. You represent it'd impedance with a phasor or in the laplace domain.\n\nI like Serena\nHomework Helper\nKirchoff's Laws really only apply to a stable situation, so a charging capacitor would be analysed by taking a \"snapshot\" of the circuit operation.\nHuh?\nI though Kirchhoff's laws *always* apply.\n\nI.e.\n\nhttp:\/\/img337.imageshack.us\/img337\/2701\/15irk.jpg [Broken]\n\nIs this alright? Is this \"legal\"?\nIs it legal and valid?\nYes.\n\nLast edited by a moderator:\npsparky\nGold Member\nI agree. Kirchoff's laws and Ohm's law apply to 100% of the cases.\n\nAdding a capacitor, inductor, diode....etc does not change that.\n\nAccepting these laws seems to take time......but it shouldn't.\n\nKVL, KCL and V=IR\n\nJust accept it now.\n\nvk6kro\nI agree. Kirchoff's laws and Ohm's law apply to 100% of the cases.\n\nAdding a capacitor, inductor, diode....etc does not change that.\n\nAccepting these laws seems to take time......but it shouldn't.\n\nKVL, KCL and V=IR\n\nJust accept it now.\nNot really.\n\nIf you have a capacitor that is still charging, you might apply some equations to it for one instant in time, but then it changes as the capacitor charges up some more.\n\nSo, it is better to use logic and derive a Thevenin equivalent circuit which can then charge up the capacitor.\n\nIncidentally, phasors are only relevant for AC circuits. This is DC.\n\nI like Serena\nHomework Helper\nNot really.\nYes. Really.\n\nIf you have a capacitor that is still charging, you might apply some equations to it for one instant in time, but then it changes as the capacitor charges up some more.\nAt any point in time, you have voltages and currents to which KVL and KCL apply.\n\nIncidentally, phasors are only relevant for AC circuits. This is DC.\nThe theory of phasors holds true in DC as well.\nWith DC, you would need to calculate the inverse Laplace (or Fourier) transform to get the proper result.\n\npsparky\nGold Member\nYes. Really.\n\nAt any point in time, you have voltages and currents to which KVL and KCL apply.\n\nThe theory of phasors holds true in DC as well.\nWith DC, you would need to calculate the inverse Laplace (or Fourier) transform to get the proper result.\nYou know....I have to say.\n\n\"I like Serena\"\n\nI like Serena\nHomework Helper\nYou know....I have to say.\n\n\"I like Serena\"\nThat ought to be: I like I like Serena.\n\n--I like ILSe\n\nGold Member\nBasically, while KLV and KCL always stand true, they won't have me to find a solution with respect to time... which is what I have in my case. But I will post it in the homework section later.. I have more relevant concerns when it comes to electronics, so I'll post this exercise later. Thanks a lot, everyone! :)\n\nvk6kro\nBasically, while KLV and KCL always stand true, they won't have me to find a solution with respect to time... which is what I have in my case. But I will post it in the homework section later.. I have more relevant concerns when it comes to electronics, so I'll post this exercise later. Thanks a lot, everyone! :)\nThat's right. It is a question of using the right tool for the job.\nIn this case, you work out a Thevenin equivalent of the circuit apart from the capacitor and then use this to charge the capacitor.\n\nIn the example above, the Thevenin voltage is 8.5 volts and the Thevenin resistance is about 545 ohms.\nSo the time constant with a 500 \u03bcF capacitor is (545 ohms * 0.0005 Farads) or 0.272 seconds.\n\nSo, the capacitor voltage will rise to 0.636 times 8.5 volts (5.4 volts) in 0.272 seconds and it will eventually reach 8.5 volts in about 3 seconds.\n\nI wonder if many of the queries you make could be solved if you became familiar with LTSpice. This is free and very easy to use. Have you tried it?\n\nrbj\nBasically, while KVL and KCL always stand true\nKVL represents conservation of energy in what we call a \"conservative field\" and KCL represents conservation of charge in a system that doesn't allow too much charge to build up at any particular node.\n\nKVL represents conservation of energy in what we call a \"conservative field\" and KCL represents conservation of charge in a system that doesn't allow too much charge to build up at any particular node.\nWhich is why I Like Serena and Psparky are wrong when the say that Kirchoff's Laws are universal. In non-conservative fields such as time-varying electric\/magnetic fields, the voltage between points A and B is not uniquely defined--it depends on the path followed between the points.\n\nThe sum of the voltages around the loop driven by a time-varying magnetic field is non-zero, as shown by Prof. Lewin here:\nOP should ignore this post.\n\nLast edited by a moderator:\nrbj\nWhich is why I Like Serena and Psparky are wrong when the say that Kirchoff's Laws are universal. In non-conservative fields such as time-varying electric\/magnetic fields, the voltage between points A and B is not uniquely defined--it depends on the path followed between the points.\n\nThe sum of the voltages around the loop driven by a time-varying magnetic field is non-zero, as shown by Prof. Lewin here:\nOP should ignore this post.\nyou mean your post? for circuits, KVL and KCL are within $\\epsilon$ of being precisely true.\n\nLast edited by a moderator:\nyou mean your post? for circuits, KVL and KCL are within $\\epsilon$ of being precisely true.\nLumped circuit-theory is already based on approximations of Maxwell's equations (e.g. capacitor current I = C*dV\/dt is derived by ignoring the effect of the time-varying magnetic field in Faraday's Law). So within lumped circuit theory, I'd say KVL and KCL are always true. That's why I said OP could ignore my post. What is this epsilon you mention?\n\nWhich is why I Like Serena and Psparky are wrong when the say that Kirchoff's Laws are universal. In non-conservative fields such as time-varying electric\/magnetic fields, the voltage between points A and B is not uniquely defined--it depends on the path followed between the points.\n\nThe sum of the voltages around the loop driven by a time-varying magnetic field is non-zero, as shown by Prof. Lewin here:\nOP should ignore this post.\nHere we go again. KVL and KCL are universal and always work. Period.\n\nThere are no EM fields in circuit theory and therefore no non-conservative fields either.\n\nThere are no magnetic fields in the inductors of circuit theory. There are no electric fields on a schematic. That's physics, not circuit theory.\n\nLast edited by a moderator:\npsparky\nGold Member\nHere we go again. KVL and KCL are universal and always work. Period.\nIt's truly unbelievable that people try to argue this.\n\nLike I said....for some reason it takes a while to sink in.\n\nAntiphon: Here you go again--I must have missed something! Did you read my follow-up comment, though? I tried to communicate that Kirchoff's Laws apply in lumped circuit theory precisely for the reasons you mentioned.\n\nIf I understand you correctly, you're saying that Kirchoff's laws are \"universal\" because they should only be applied to lumped circuits. So when Prof. Lewin replaces the DC source with a time varying magnetic field to induce a voltage in the loop, he re-applies KVL to a problem that should no longer be solved with KVL? Is this what you're saying?\n\nIt's truly unbelievable that people try to argue this.\n\nLike I said....for some reason it takes a while to sink in.\nIf EM seems that simple to you, then you either know a lot or more likely nothing.\n\nOuabache\nHomework Helper\nFor example, let R1 = 1000 ohms, R2 = 2000 ohms, R3 = 3000 ohms Diode voltage = 0.7 volts.\n:)\nFor the voltage drop across a standard LED, I have seen specs ranging from 1.5 to 5V. Because this depends on the LED used, this parameter should be supplied in this question. I suspect you were thinking of a silicon diode having a typical drop of 0.7 V.\n\nAntiphon: Here you go again--I must have missed something! Did you read my follow-up comment, though? I tried to communicate that Kirchoff's Laws apply in lumped circuit theory precisely for the reasons you mentioned.\n\nIf I understand you correctly, you're saying that Kirchoff's laws are \"universal\" because they should only be applied to lumped circuits. So when Prof. Lewin replaces the DC source with a time varying magnetic field to induce a voltage in the loop, he re-applies KVL to a problem that should no longer be solved with KVL? Is this what you're saying?\n\nYes I saw it but it was at odds with the post above it and I mistook you for two authors.\n\nI humbly recommend editing the post in a case like this but I missed it, sorry.\n\nEdit: professor Lewin violates the rules of the lumped circuit abstraction deliberately to provoke thought among his students.\n\nIn short: any schematic that has any fields on it has left the realm of the lumped circuit theory and stepped back into physics.\n\nIn circuit theory there is voltage and current. There isn't an electric or magnetic vector (E,H) there isn't an electric or magnetic flux (D,B), there are no constitutive relations (permeabilities, permittivities).\n\nLast edited:\nvk6kro\nFor the voltage drop across a standard LED, I have seen specs ranging from 1.5 to 5V. Because this depends on the LED used, this parameter should be supplied in this question. I suspect you were thinking of a silicon diode having a typical drop of 0.7 V.\nAh yes, thanks. I didn't notice the LED symbols.\n\nIt doesn't make any difference to the calculation method, though, unless the LEDs were different colors, in which case they wouldn't have the same voltage.\n\nIf they were different voltages, we wouldn't be able to do this:\n\nhttp:\/\/dl.dropbox.com\/u\/4222062\/diodes.PNG [Broken]\n\nwhich makes the calculations simpler.\n\nLast edited by a moderator:\njim hardy\nGold Member\n2019 Award\nDearly Missed\nRE Lewin's video:\n\nThe professor did not represent with his drawing the circuit he evaluated.\n\nHad he drawn voltage sources in the wires representing \u222be(dot)dl where flux couples them his drawing would be accurate . And Kirchoff would prevail.\n\nAnybody who's worked around magnetics knows your voltmeter leads are one turn. Short them together around an energized transformer core and observe meter.\nHe measured the voltage arriving at his meter not the voltage between A and D.\n\nBut he's an entertaining lecturer.\n\nold jim","date":"2020-10-26 15:47:46","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\": 0.5517393946647644, \"perplexity\": 1162.8955049050421}, \"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-2020-45\/segments\/1603107891428.74\/warc\/CC-MAIN-20201026145305-20201026175305-00624.warc.gz\"}"} | null | null |
Q: Please verify my understanding of extending Facebook Access Tokens I have a an application which will make posts on multiple user profile pages and Fan Pages.
To obtain permission to do this, the app will process a client side auth and obtain the short-lived access token for each user who uses the app.
My app will then immediately exchange that for a 60 day long-lived access token, and store this for future use in a local DB. The app will then be able to update that users profile and pages for up to 60 days, whether the user is logged into FB or not.
The next bit is the important bit:
Each time the user uses my app, my app will test the validity of the current access token, in case the user has changed their password etc, or the 60 days have elapsed.
If the access token is no longer valid, my app will seek to obtain a new one.
If the user is not logged into Facebook at this point, it is my understanding that I will have to prompt a login and force a client side auth, to obtain a new short-lived access token which I will have to exchange for a new 60 day long-lived token.
Is this correct? ie that the user has to login again? ie there is not way to process the access_token update on the server side?
Also, is it the case the the 60 day expiry only applies in respect of user access tokens, and does not apply in respect of page access tokens?
thx
A:
Is this correct? ie that the user has to login again? ie there is not way to process the access_token update on the server side?
No. You need a valid short-lived user access token first, and that you get through the process of client-side login.
Also, is it the case the the 60 day expiry only applies in respect of user access tokens, and does not apply in respect of page access tokens?
Correct, page tokens do not expire by default, if they where obtained using a long-lived user access token.
| {
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\section{Introduction}
\label{sec:intro}
An estimated 60 to 75 percent of the world's population speaks at least two languages~\cite{vince2016amazing}. While somebody is speaking in a foreign language, it has been observed that the person's voice sounds different from when speaking in their mother tongue~\cite{lee2017bilingual}.
With recent trends in globalisation, it has become easier to encounter multilingual scenarios. Therefore the focus on multilingual speaker recognition has become more important~\cite{cieri2007resources, akbacak2007language, ferrer2011promoting, reynolds20172016, matejka2017analysis}.
While the performance of speaker recognition systems has improved significantly due to recent advances in deep learning \cite{he2016deep, snyder2018x, jung2018complete, ravanelli2018speaker, snyder2019speaker, snell2017prototypical, khosla2020supervised, kang2022augmentation} and the availability of large-scale datasets~\cite{nagrani2017voxceleb, chung2018voxceleb2}, the state-of-the-art systems fail easily under the language mismatch condition.
The popular speaker recognition evaluation sets do not consider bilingual scenarios, making it difficult to analyse their effect on speaker recognition performance.
Only a few evaluation datasets consider bilingual scenarios; however, they are collected from controlled environments like phone-call platform~\cite{cieri2007resources} or contain only limited languages~\cite{reynolds20172016}.
The recent VoxCeleb Speaker Recognition Challenge (VoxSRC)~\cite{brown2022voxsrc} contains some bilingual speakers; however, their evaluation datasets remain private.
Hence, to the best of our knowledge, there is no large-scale public evaluation set that takes bilingual speakers into account.
To this end, we introduce a new large-scale bilingual evaluation set derived from VoxCeleb1~\cite{nagrani2017voxceleb}, focusing on bilingual speaking problems. We call this test protocol \texttt{VoxCeleb1-B}. To increase the scale and the diversity compared to the VoxSRC challenge test set~\cite{brown2022voxsrc}, we expand the number of bilingual trials and the number of languages, resulting in a total of 808,574 trials and 15 languages. Moreover, for the first time, we publicly release the manually annotated language labels of VoxCeleb1 to provide an \textit{in the wild} language recognition benchmark.
The ground truth language labels not only help to produce the bilingual speaker recognition evaluation set, but also can be beneficial for other applications such as language identification.
More details of \texttt{VoxCeleb1-B} and the language labels are given in Section \ref{sec:VoxCeleb1-B}.
Using the proposed evaluation protocol, we observe that the existing speaker recognition models do not generalise well to bilingual speakers.
To resolve the language-dependent problem, prior works on multilingual speaker recognition have usually utilised some combination of Probabilistic Linear Discriminant Analysis (PLDA) and scoring functions based on a standard backbone system such as the i-vectors~\cite{ferrer2011promoting, misra2014spoken, matejka2017analysis}. On the other hand, our proposed approach is based on disentangled representation learning. In this field, \cite{kang2022augmentation} uses disentanglement techniques to remove the channel information from speaker embeddings. \cite{parry2022speech} models the speaker-independent speech emotion recognition model by disentangling the speaker information from the emotion representation.
There are various factors constituting a speaker's identity, such as the tone and accent, as well as gender \cite{raj2019probing}, age, nationality~\cite{luu2020leveraging}, emotion~\cite{williams2019disentangling, pappagari2020x} and language~\cite{maiti2020generating}. For the bilingual situation, we hypothesise that language's linguistic characteristics affect models' ability to determine speakers' identity from speech.
In this work, we propose \textit{language-disentangled representation learning}, which removes the language-dependent information from the speaker embeddings.
The two-stream network consists of a language and a speaker classifier. The network is trained to remove the language information from the speaker embeddings.
The language classifier of the proposed network is trained using language pseudo-labels from a network pre-trained on VoxLingua107, and therefore can be trained without additional language labels. This learning strategy can be applied to existing end-to-end models with few modifications.
\input{tab/test_sets}
\input{fig/frameworkfig}
\section{Bilingual speaker recognition test set}
\label{sec:VoxCeleb1-B}
We propose a new large-scale bilingual speaker recognition evaluation protocol extracted from VoxCeleb1 dataset~\cite{nagrani2017voxceleb}, which is one of the widespread benchmark evaluation datasets in the recent speaker recognition field. Most of existing evaluation sets do not focus on the bilingual scenarios. Our speaker recognition evaluation set contains 808,574 trials in total. Half of the trials are intra-speaker cross-lingual (referred to as the \textit{hard positive} pairs) and the remaining trials are inter-speaker monolingual (referred to as the \textit{hard negative} pairs).
\subsection{Obtaining language labels}
To simulate the bilingual scenarios with VoxCeleb1 dataset, it is necessary to have the language labels of the utterances in the test set. In order to economise on human resources, we extract language pseudo-labels and confidence scores using a pre-trained Spoken Language Recognition (SLR) model. In particular, we utilise the model pre-trained on VoxLingua107~\cite{valk2021voxlingua107}, which is a dataset containing 6,628 hours of speech that are divided into 107 languages.
For each language, we randomly sample data across a range of confidence scores. We first manually annotate the language labels of the samples, then determine the confidence value at which incorrect labels start to appear. All of the samples for each language with a confidence below the threshold are manually annotated. 15 languages including English, French, Hindi, German, Spanish, Italian, Afrikaans, Portuguese, Dutch, Korean, Urdu, Swedish, Russian, Chinese, and Arabic are annotated by annotators of various nationalities.
Out of 153,516 utterances in the VoxCeleb1 dataset, languages of 883 utterances are unrecognised and given the \textit{unknown} label. All \textit{unknown} language utterances are excluded from the proposed evaluation list.
\subsection{{\tt VoxCeleb1-B} Evaluation list}
Speaker verification evaluation protocol consists of \textit{positive} and \textit{negative} trials.
Each trial involves an enrollment utterance and a test utterance.
The trial type is decided based on whether the enrollment and the test utterances have the same speaker identity.
To evaluate the robustness of speaker recognition models in the bilingual scenarios, we propose an evaluation list named \texttt{VoxCeleb1-B}, which simulates language-mismatch scenarios with a large amount of cross-lingual trials. Using the speaker and language annotations, we generate \textit{hard positive} pairs by using all the possible intra-speaker cross-lingual combinations of the samples, resulting in 404,287 trials. The inter-speaker monolingual combinations are used to generate \textit{hard negative} pairs. The number of speakers for each language and the number of samples per speaker are limited to 1,000 and 15, respectively, to avoid strong bias to the most frequent languages.
\Cref{tab:testsets} shows the statistics of existing evaluation lists derived from VoxCeleb1, and the proposed \texttt{VoxCeleb1-B}.
The three original VoxCeleb1 test sets and the VoxSRC 2020~\cite{nagrani2020voxsrc} validation set are expected to contain very few cross-lingual positive trials, whereas the VoxSRC 2021~\cite{brown2022voxsrc} contains some cross-lingual trials. \texttt{VoxCeleb1-B} is explicitly designed to contain a large number of cross-lingual trials.
\section{Language-disentangled learning}
\label{sec:language-distentangled learning}
In this section, we describe the proposed language-disentangled representation learning strategy. Our training framework is summarised in \Cref{fig:training_framework}.
There are two main modules in our model: (i) the speaker module, and (ii) the language module. The speaker module consists of a speaker feature extractor, speaker embedding layer and speaker classifier. This module is based on the popular architectures in the speaker recognition field. The language module consists of a language embedding layer and a language classifier. The speaker feature extractor calculates frame-level embeddings ${z_i}$ from the input mel-spectrogram data $x_i \in \mathcal{R}^{{T}\times{F}}$ ($1 \leq i \leq N$), where $T$, $F$ and $N$ are the number of frames, frequency bins, and the size of mini-batch, respectively.
To derive an utterance-level vector ${e_i}$ from the frame-level embeddings ${z_i}$, we adopt Attentive Pooling Layer (APL) which includes self-attentive pooling (SAP)~\cite{cai2018exploring} or attentive statistics pooling (ASP)~\cite{okabe2018attentive} as an embedding layer. In this work, we propose two types of embedding layer structure: (i) single-embedding method, and (ii) multi-embedding method. In both options, two classifiers share the same speaker feature extractor. The embedding process is as follows.
\begin{equation}
{e_i}^{j} = {APL^{j}}(z_i),
\end{equation}
\noindent where $j$ is $j \in \{S, L\}$ and $e^S$, $e^L$ represent the vectors from the speaker and the language embedding layers.
In the multi-embedding method, the classifiers take different embedding vectors, ${e_i}^{S}$ and ${e_i}^{L}$. The reason for proposing the multi-embedding layer is to check the effect of disentanglement with an independent language-specific embedding vector ${e_i}^{L}$. In the single-embedding method, the language module does not have a separate language embedding layer, therefore the input of the language module is ${e_i}^{S}$.
The training process of our framework alternates between two phases within each iteration: (1) language discriminator training, and (2) speaker embedding training. In the first phase, we train the language module to classify the spoken language from the output of the speaker embedding extractor. In the second phase, the speaker embeddings are trained so that it is {\em good} at classifying the speakers, but {\em bad} at classifying the language from the embeddings.
To achieve the latter, we apply a gradient reversal layer (GRL)~\cite{ganin2015unsupervised} to the language module.
\subsection{Language discriminator training}
In this step, we train the language module, while freezing the speaker module. This can be interpreted to train language recognition for only the current state of the speaker representation extracted by the speaker feature extractor of the speaker module. For the input embedding vector ${e_i}^j$, the language classifier is trained using a categorical cross-entropy loss function.
\subsection{Speaker embedding training}
In this step, we train the speaker module, and the language module's parameters are not updated. For speaker recognition, we can select any learning objectives. In this work, we employ learning objective functions such as softmax loss, prototypical loss, and contrastive loss, which have been employed in the previous works~\cite{chung2020defence, kwon2021ins}, while combining them with our proposed learning method. The total loss $L_{total}$ can be formulated as follows.
\begin{equation}
L_{total} = L_{spk} + \lambda L_{slr},
\end{equation}
\noindent where $ L_{spk} $ and $ L_{slr} $ are the losses for training the speaker module and the language module, respectively.
When $L_{total}$ is backpropagated, the GRL makes the gradient value of $L_{slr}$ negative, disturbing the language classifier to be converged. $\lambda$ is a hyperparameter for balancing two losses.
\section{Experiments}
\label{sec:experiments}
\input{tab/exp_results}
\subsection{Input representations and model architecture}
For input representation of the neural network, we use log-mel spectrograms that are extracted with a hamming window, 25ms window size and 10ms stride size.
In this work, we focus on demonstrating the effectiveness of the proposed learning strategy and its compatibility with previous models. Therefore, we employ two existing variants~\cite{chung2020defence, kwon2021ins} of the 34-layer residual network, rename the models as ResNet-S~\cite{chung2020defence} and ResNet-L~\cite{kwon2021ins}. The ResNet-S uses the SAP~\cite{cai2018exploring} and the angular prototypical loss, and the ResNet-L uses the ASP~\cite{okabe2018attentive} and the angular prototypical loss combined with the softmax loss, which is in line with the original papers.
The language recognition module including the gradient reversal layer is similar to~\cite{kang2022augmentation,chung2019delving}, but the difference is that the output size of the last fully connected layer is equal to the number of language classes.
\subsection{Dataset}
We use the development partition of the VoxCeleb2 \cite{chung2018voxceleb2} as the training dataset. In order to train the language module, we use the language pseudo-labels for VoxCeleb2 using an open-source SLR model pre-trained on the VoxLingua107 dataset. For evaluation, we use the three original test sets based on VoxCeleb1~\cite{nagrani2017voxceleb}, the VoxSRC validation sets~\cite{nagrani2020voxsrc, brown2022voxsrc} and VoxCeleb1-B, which is the proposed large-scale bilingual speaking evaluation set.
\subsection{Experimental setting}
Our implementation is based on the PyTorch framework \cite{paszke2019pytorch}. We use the Adam Optimizer \cite{kingma2014adam} to optimise the models. Our experiments are performed on a single NVIDIA A5000 GPU with 24GB memory. The value of an initial learning rate is 0.001 and it is decreased by $5\%$ every 4 epoch after the first 16 epochs. We use the batch size of 400 and 300 for ResNet-S and ResNet-L, respectively. The training takes around 3 days.
\subsection{Curriculum learning}
\label{subsec:curriculum}
Training language-disentangled representations from scratch sometimes results in unstable learning characteristics. To mitigate this effect, we introduce language-disentangled representation learning after applying {\em warm up} for a few epochs, so that the speaker recognition and language recognition model are sufficiently learned from the same speaker embedding. The $\lambda$ value is set to 0 during warming up. In our experiments, curriculum learning is performed for 60 epochs.
\section{Results}
\Cref{tab:results_main} summarises the experimental results. For a more accurate evaluation, we trained all models 3 times with different random seeds, and report the mean and the standard deviation of the three models.
\newpara{The effect of language-disentangled learning.} As shown in \Cref{tab:results_main}, the baselines of each model show weak performance for VoxSRC2021~\cite{brown2022voxsrc} and VoxCeleb1-B, which contain cross-lingual trials. The models trained with language-disentangled representation learning show performance improvement in VoxSRC2021 and VoxCeleb1-B. On the bilingual VoxCeleb1-B, the ResNet-S and ResNet-L models that are most optimised for the bilingual scenarios show improvements of 49\% and 48\% over their respective baselines. We can control the strength of language-disentangling using the $\lambda$ value. In general, it can be seen that the larger the $\lambda$, the better the performance for bilingual scenarios. However, we observe the fact that too high $\lambda$ value makes the model training unstable. As a result, we find that the $\lambda$ value between 0.5 and 1.5 is optimal for most models.
\newpara{Curriculum learning.} The models trained with strong language-disentanglement losses sometimes show weaker performance on the monolingual test sets such as VoxCeleb1. The models trained using curriculum learning~\ref{subsec:curriculum} show better performance on the monolingual test sets, by allowing the model to first learn the effective speaker embeddings. In particular, the ResNet-L model trained using the single-embedding method and curriculum learning shows improvement over the baseline for both monolingual and bilingual test sets.
\newpara{Model capacity and embedding method.} We analyse how model capacity and embedding method affect our proposed learning strategy.
We find that using models with more parameters generally perform better in bilingual situation.
However, when considering performance improvements, both the small and the large model show similar relative performance improvement of around 50\%. It also shows that the small model with multi-embedding method and $\lambda$ of 0.5 outperforms the large models. This means that the outcomes of our proposed learning strategy is not simply due to the model parameters.
For the embedding method, we find that the multi-embedding method is better in the ResNet-S models. We speculate that small models can learn reliably in language-disentangled learning when the two classifiers deal with different embedding vectors.
Summarising these results, it can be inferred that attempts to remove language information directly from speaker embedding vector can be risky when the number of model parameters is small.
\newpara{The use of pseudo-labels.} Training of all experiments are performed with language pseudo-labels of VoxCeleb2.
Nonetheless, our learning strategy works successfully, showing significant performance improvements in bilingual scenarios.
This indicates that it can be cost-effect to use pseudo-labels of specific factor in speaker recognition, which can be extended to other factors of variation that must be removed from speaker embeddings.
\input{tab/exp_slr_results}
\newpara{Language-disentangled speaker representation.} Experiments are performed to verify whether the language-disentangled learning is effective at removing language information from the embeddings.
We extract the speaker embedding vector ${e}^{S}$ from the model trained with our learning strategy, then learn a language recognition model with the input data ${e}^{S}$, and then calculate the accuracy. The structure of language recognition model is same as our language classifier used in disentangled learning. We conduct the training and evaluation with the same dataset as the main experiments. As shown in \Cref{tab:results_extra},
we show that the speaker embeddings trained using language-disentangled learning are {\em less good} at providing useful information to the language classifier trained on top of it. In other words, we confirm that some language information has been removed from the speaker embeddings.
\section{Conclusion}
\label{sec:conclusion}
We have developed strategies to train speaker embeddings that are robust to bilingual speaking scenarios, and proposed an evaluation protocol that takes bilingual speakers into account.
Our large-scale evaluation protocol is designed to analyse speaker recognition performance under bilingual scenarios, and we make this evaluation set publicly available.
We also propose a new learning strategy to resolve the bilingual problem. Our learning strategy disentangles language information from the speaker representation in order to make the embeddings robust to cross-lingual trials.
Our proposed learning strategy can be applied to the existing models with few modifications, and shows significant performance improvements under bilingual scenarios.
\section{Acknowledgements}
\label{sec:acknowledgements}
We would like to thank Jaesung Huh, Icksang Han and Bong-Jin Lee for helpful comments.
| {
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{"url":"https:\/\/math.stackexchange.com\/questions\/3483166\/probability-of-get-the-same-word","text":"# Probability of get the same word\n\nI have this statement:\n\nThere are $$23$$ people where each person will take out of a box with $$365$$ papers written with different words. In addition each person will return the paper to the box. What is the probability that exactly $$2$$ people get the same word?\n\nMy attempt was:\n\nGet the total amount of possible results of choose $$23$$ words of total of $$365$$: $$\\binom{365}{23}_{repitition} = \\frac{387!}{364! \\cdot 23!}$$\n\nGet the case in where exactly $$2$$ people get the same word. Lets suppose a person $$p_i$$ and a person $$p_j$$ get the same word $$w_1$$, Then we have to choose $$21$$ letters out of a total of $$364$$, since it must be repeated exactly $$2$$ times, that is $$\\binom{364}{21} = \\frac{364!}{343! \\cdot 21!}$$\n\nOnce these $$21$$ letters have been chosen and with the two words that are repeated, the cases would remain as: $$w_i -w_i-w_j-w_k-....-w_n$$ and each of these words correspond to 23 people:\n\n$$w_i \\to p_1$$\n\n$$w_i \\to p_2$$\n\n$$w_n \\to p_{23}$$\n\nHowever,person 1 with person 2 will not necessarily be repeated, so we must multiply by $$\\frac{23!}{2!}$$\n\nThen, the prob is $$\\frac{\\binom{364}{21}\\cdot \\frac{23!}{2!}}{\\binom{365}{23}_{repetition}}$$.\n\nHowever, according to the guide, my answer is incorrect and I don't know why. Thanks in advance.\n\n\u2022 The total number of possible selections is $365^{23}$. \u2013\u00a0lulu Dec 20 '19 at 21:55\n\u2022 @lulu True. thus, that is my unique error? \u2013\u00a0Eduardo S. Dec 20 '19 at 21:56\n\u2022 I didn't read beyond that. But: to make that the correct count, we need to consider ordered sets of choices (indexed, say, by an enumeration of the people). Having done that, everything that follows should also reference ordered collections. \u2013\u00a0lulu Dec 20 '19 at 21:59\n\u2022 You also forgot that there are $365$ ways to choose the repeated word. \u2013\u00a0saulspatz Dec 20 '19 at 22:25\n\nCall the people A-W. Each person can have one of 365 words, so the total number of possibilities is $$365^{23}$$. Doing $$\\binom{365}{23}$$ already assumes that the 23 values are distinct.\nTo find the number of ways to choose 23 people, we first decide which pair have the same words. There are $$\\binom{23}{2}$$ ways to do this. After this, we have 22 distinct values for 365 words. The number of ways to do this is $$\\binom{365}{22}\\cdot 22!$$, where the $$\\binom{365}{22}$$ gives the number of ways to choose the 22 distinct values and the $$22!$$ permutes them.","date":"2020-03-31 23:56:07","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\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 27, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8278416991233826, \"perplexity\": 238.74247263309996}, \"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-2020-16\/segments\/1585370504930.16\/warc\/CC-MAIN-20200331212647-20200401002647-00067.warc.gz\"}"} | null | null |
El Malah é um distrito localizado na província de Aïn Témouchent, no noroeste da Argélia. Em 2010, sua população era de habitantes. Foi nomeado após sua capital, El Malah.
Municípios
O distrito está dividido em quatro municípios:
El Malah
Ouled Kihal
Terga
Chaabet El Ham
Distritos da Argélia | {
"redpajama_set_name": "RedPajamaWikipedia"
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🎁 FLASH SALE: Buy 2 or more covers and get FREE SHIPPING!
Black & Hammock Convertible: 54" x 58": Designed for the backseats of All STANDARD cars, trucks & SUVs. Easily convertible between hammock or standard bench coverage. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,788 |
\section{Introduction}
Light Fidelity (LiFi) has emerged as a high speed wireless data access solution using visible light [2]. The light emitting diode (LED) is used as an access point which both illuminates a certain area and provides a wireless connection in that area of illumination. The arrangement of LEDs as a network is called an attocell network. Such an attocell network is centrally monitored and is generally arranged in a deterministic lattice in both one and two dimensions. For downlink access, the signal-to-interference-plus-noise-ratio (SINR) is analyzed at a receiver to measure the system performance. Co-channel interference or simply interference in such networks significantly affects the SINR and hence limits the data rate. Tractable closed form expressions characterizing this interference have been provided in [1] over the attocell geometries. While various multiple access schemes have been used to mitigate interference in conventional wireless networks, the time division multiple access (TDMA) over the LiFi LEDs will be suitable to analyze and implement since the attocell networks are always centrally monitored over a deterministic lattice.
\subsection{Related works}
To the best of the authors' knowledge, due to the absence of tractable closed form expressions for interference and hence the SINR in LiFi attocell networks, the topic of time scheduling to mitigate interference has relatively been untouched. Nonetheless, there have been other approaches to achieve mitigation [3-7], respectively wherein the research focussed on angle diversity receivers and their orientation and some novel resource allocation schemes. All the mentioned works, however significant they actually are, do not consider TDMA over the LEDs to improve the network performance; which actually is an important aspect since the entire attocell network tends to be centrally monitored.
\subsection{Contributions of this work}
\begin{itemize}
\item Since the LiFi LEDs are centrally monitored, the TDMA is implemented over the LEDs to mitigate interference over a two dimensional LiFi attocell network.
\item Using the results in [1] over M-PAM modulated signals, an exact expression \eqref{eqn:good} is derived an for the goodput $G$ of the TDMA system, where $G$ is defined as the product of the rate and the probability of correctness of reception.
\item The existence of an optimum TDMA parameter $K$ is shown, at which $G$ is maximized.
\end{itemize}
\subsection{Arrangement of the paper and notations}
The involved notations are given in Table I.
\begin{table}
\label{t/data1}
\caption{Notations used in the paper}
\label{abc}
\begin{tabular}{ | m{1cm} | m{1.9cm}| m{4cm} |}
\hline
S.No.& NOTATION & MEANING\\
\hline
1 & $z=\sqrt{z_{x}^{2}+z_{y}^{2}}$ & Distance of the PD from the origin. The location is given by Cartesian coordinates $(z_{x},z_{y})$, where $z_x$ and $z_y$ are measured in metres.\\
\hline
2 & $h$ & Height of LED installation.\\
\hline
3 & $m, \beta, \theta_{h}$ & Terms representing the HPSA of the LED.\\
\hline
4 & $a$ & Square lattice edge length or inter LED spacing.\\
\hline
5 & $A_{pd}$ & Area of receiver PD.\\
\hline
6 & $R_{pd}$ & Responsivity of PD.\\
\hline
7 & $(i,j)$ & Indices representing an LED in the network.\\
\hline
8 & $D_{i,j}$ & The distance, on ground, between the PD and an $(i,j)^{th}$ interfering LED.\\
\hline
9 & $\theta_{f}$ & Field-of-View (FOV) of the PD.\\
\hline
10 & $G_{i,j}(z)$ & Channel gain from an $(i,j)^{th}$ LED to a PD at $z$ from origin.\\
\hline
11 & $\gamma(z)$ & The SINR at $z$.\\
\hline
12 & $x_{i,j}(t)$ & Baseband signal from the $(i,j)^{th}$ LED. \\
\hline
13 & $s_{i,j}(t)$ & Intensity modulated signal from $x_{i,j}(t)$.\\
\hline
14 & $\hat{\mathtt{I}}_{u,v}(z)$ & Series approximation for the variance $\sigma_{1}^{2}$ over $(u,v)$ terms.\\
\hline
15 & $\hat{\mathcal{I}}_{u,v}(z)$ & Series approximation for the mean $\mu$ over $(u,v)$ terms.\\
\hline
16 & $(u,v)$ & Number of approximation terms. \\
\hline
17 & $(w,f)$ & Fourier index. \\
\hline
\end{tabular}
\end{table}
The paper is arranged as follows. Section II describes the system model. Section III describes the time scheduling scheme along with the numerical simulations validating the analytical results. The paper concludes with Section IV.
\section{System model}
\subsection{The attocell network}
Consider the two dimensional LiFi attocell network in Fig.\ref{two_dim}. Let the set $\mathbb{S}$ represent an infinite set of LiFi LEDs with $(i,j)\in\mathbb{S}$ indicating the two dimensional index of a particular LiFi LED. All the LEDs are fixed at a height $h$, are separated symmetrically by a distance $a$ and emit at a uniform average optical power $P_{o}$. The photodiode (PD) is assumed to have it's surface always parallel to the ground, i.e., without any orientation towards any LED and is assumed to be located at $(z_{x},z_{y},0)$ from the origin. We neglect any non-linearities of the LED while intensity modulation and assume the field-of-view (FOV) $\theta_{f}$ of the PD to be $\frac{\pi}{2}$ radians. The LEDs have a lambertian emission order $m$ given as $m = -\frac{\ln(2)}{\ln(\cos(\theta_{h}))}$, where $\theta_{h}$ is the half-power-semi-angle (HPSA) of any given LED. Let the PD have a cross section area $A_{pd}$ and the responsivity $R_{pd}$. Let the optical system bandwidth be $W$, and the noise power spectral density at the PD be $N_{o}$. Also, since the entire network is considered to be located in an open area without any opaque obstructions, the works in \cite{nlos1,nlos2,nlos3} will be followed to neglect any multipath and non-line-of-sight components received at the PD. Importantly, this PD tags to the nearest LED, which is considered to be at $(0,0,h)$.
\begin{comment}
\begin{figure}[ht]
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\end{tikzpicture}
}%
\caption{\textit{(One dimension model)} This figure, taken from \cite{atchu2}, shows the infinite one dimension corridor. There are infinite number of LEDs (circular dots) indexed as $i\in\mathbb{S}$ and arranged at an equal interval $a$, all along the corridor, installed at a height $h$. The rectangular dotted regions on ground depict the attocells corresponding to each $i^{th}$ LED above. The user PD (small cuboid) at $z=\sqrt{z_{x}^{2}+z_{y}^{2}}$ $($i.e. $(z_{x},z_{y},0)$ inside one of the attocell$)$, receives data wirelessly from the tagged-LED corresponding to the attocell in which it is located. Here, that attocell is highlighted as dash-dot. All other LEDs are co-channel interferers. Here we assume that the user PD can move anywhere on the ground plane. For a finite network, the interferers are assumed to be located symmetrically on either side of the tagged LED.}
\label{one_dim}
\end{figure}
\end{comment}
\begin{figure}[ht]
\centering
\resizebox{0.5\textwidth}{!}{%
\begin{tikzpicture}
\draw (0,0) -- (0,3) -- (5,8) -- (14,8) -- (14,5);
\draw (14,5) -- (9,0) -- (0,0);
\draw (14,8) -- (9,3) -- (0,3);
\draw (9,0) -- (9,3);
\draw [dotted] (0,0) -- (5,5) -- (14,5);
\draw [dotted] (5,5) -- (5,8);
\draw [line width= 0.03cm,->] (0,0) -- (1.6,0);
\node [below] at (1.6,0) {\large $x$};
\draw [line width= 0.03cm,->] (0,0) -- (1.3,1.3);
\node [below] at (1.3,1.3) {\large $y$};
\draw [fill] (0.5,3.5) circle [radius=0.10];
\draw [fill] (2.5,3.5) circle [radius=0.10];
\draw [fill] (4.5,3.5) circle [radius=0.10];
\node [below] at (4.5,3.5) {\large $(0,-a,h)$};
\draw [fill] (2.5,5.5) circle [radius=0.10];
\node [below] at (2.5,5.5) {\large $(-2a,0,h)$};
\node at (5.5,1.8) {\large Attocell};
\draw [fill] (4.5,5.5) circle [radius=0.10];
\node [below] at (4.5,5.5) {\large $(-a,0,h)$};
\draw [fill] (6.5,5.5) circle [radius=0.10];
\node [below] at (6.5,5.5) {\large $(0,0,h)$};
\node [above] at (6.5,5.5) {\large tagged-LED};
\draw [fill] (8.5,5.5) circle [radius=0.10];
\node [below] at (8.5,5.5) {\large $(a,0,h)$};
\draw [fill] (10.5,5.5) circle [radius=0.10];
\node [below] at (10.5,5.5) {\large $(2a,0,h)$};
\draw [fill] (4.5,7.5) circle [radius=0.10];
\draw [fill] (6.5,7.5) circle [radius=0.10];
\draw [fill] (8.5,7.5) circle [radius=0.10];
\node [below] at (8.5,7.5) {\large $(0,a,h)$};
\draw [fill] (10.5,7.5) circle [radius=0.10];
\draw [fill] (12.5,7.5) circle [radius=0.10];
\draw [dotted] (1,0) -- (6,5);
\draw [dotted] (3,0) -- (8,5);
\draw [dotted] (5,0) -- (10,5);
\draw [dotted] (7,0) -- (12,5);
\draw [dotted] (1.5,1.5) -- (10.5,1.5);
\draw [dotted] (3.5,3.5) -- (12.5,3.5);
\draw [dotted] (6.5,2.5) -- (6.5,5.5);
\node [align=center] at (6.5,2.5) {\small x};
\node [below] at (6.5,2.5) {\large $(0,0,0)$};
\node [left] at (6.5,4.5) {\large $h$};
\draw (7.35,2.99) -- (7.55,2.99) -- (7.65,3.09) -- (7.45,3.09) -- (7.35,2.99);
\draw [fill] (7.35,3.09) -- (7.55,3.09) -- (7.65,3.19) -- (7.45,3.19) -- (7.35,3.09);
\draw (7.35,2.99) -- (7.35,3.09);
\draw (7.55,2.99) -- (7.55,3.09);
\draw (7.65,3.09) -- (7.65,3.19);
\draw (7.45,3.09) -- (7.45,3.19);
\draw [dotted] (6.5,2.5) -- (7.5,3.04);
\node [below] at (8.4,4.04) {\large PD at $(z_{x},z_{y},0)$};
\node [below] at (7,2.77) {\large $z$};
\draw [dashed] (6.5,5.5) -- (7.5,3.04);
\draw [dashed] (6.5,1.5) -- (8.5,3.5) -- (6.5,3.5) -- (4.5,1.5) -- (6.5,1.5);
\node [above] at (5.5,2.5) {\large $a$};
\node [below] at (5.5,1.5) {\large $a$};
\node [above] at (9,8) {\large Infinite two dimension plane};
\end{tikzpicture}
}%
\caption{This figure, taken from \cite{atchu2}, shows the infinite two dimensional model. There are infinite number of LEDs (circular dots) indexed as $(i,j)\in\mathbb{S}$ and arranged symmetrically at regular intervals of $a$ as a uniform square grid, all over the plane, installed at a height $h$. The rectangular dotted regions on ground depict the attocells corresponding to each $(i,j)^{th}$ LED above. The user PD (small cuboid) at $(z_{x},z_{y},0)$ (inside one of the attocell), receives data wirelessly from the tagged-LED corresponding to the attocell in which it is located. Here, that attocell is highlighted as dash-dot. All other LEDs are co-channel interferers. Here we assume that the user PD can move anywhere on the ground plane. For a finite network, the interferers are assumed to be located symmetrically around the tagged LED.}
\label{two_dim}
\end{figure}
\subsection{The modulation scheme}
In LiFi, the intensity modulation and direct detection (IM/DD) schemes are used to form a wireless link \cite{islim}. The data frames $x_{i,j}(t)$ from every $(i,j)^{th}$ LED are intensity modulated to $s_{i,j}(t)$ over the visible light intensities. The single carrier \textit{M}-ary pulse-amplitude-modulation (PAM) scheme is assumed with equiprobable intensity levels; each level with probability $\frac{1}{M}$, and ranging from $A$ to $(2M-1)A$, where $A$ is a constant in watts, as shown in Fig. \ref{pam}.
\begin{figure}[ht]
\centering
\resizebox{0.45\textwidth}{!}{%
\begin{tikzpicture}
\draw [line width= 0.01cm] (0.2,0.5) -- (5,0.5);
\draw [line width= 0.01cm] (7.4,0.5) -- (8.2,0.5);
\draw [fill] (0.2,0.5) circle [radius=0.20];
\node [above] at (0.2,0.8) {\normalsize $A$};
\draw [fill] (2.2,0.5) circle [radius=0.20];
\node [above] at (2.2,0.8) {\normalsize $3A$};
\draw [fill] (4.2,0.5) circle [radius=0.20];
\node [above] at (4.2,0.8) {\normalsize $5A$};
\draw [fill] (5.5,0.5) circle [radius=0.01];
\node [above] at (8.2,0.8) {\normalsize $(2M-1)A$};
\draw [fill] (6,0.5) circle [radius=0.01];
\draw [fill] (6.5,0.5) circle [radius=0.01];
\draw [fill] (8.2,0.5) circle [radius=0.20];
\end{tikzpicture}
}%
\caption{Here, the \textit{M}-PAM constellation of the transmitted intensity at any LED is shown. Each possible point in the constellation is assumed to be equiprobably transmitted with a probability $\frac{1}{M}$. $A$ is a constant in Watts.}
\label{pam}
\end{figure}
It is assumed that $\log_{2}(M)$ bits of information are transmitted over a given time slot from a particular $(i,j)^{th}$ LED. The average optical power $P_{o}$ emitted from this $(i,j)^{th}$ LED is given as
\begin{align*}
P_{o}&=\mathbb{E}(s_{i,j}(t)),\nonumber\\
&=\frac{A}{M}(1+3+...+(2M-1)),\nonumber\\
&=AM,
\label{eqn:pam}
\end{align*}
where $\mathbb{E}(.)$ is the expectation operator over the possible intensity values. So, from \cite[Eqn. 2]{atchu2}, the channel gain $G_{i,j}(z)$ experienced by the optical intensities from the $(i,j)^{th}$ LED in line of sight with the PD is given as
\begin{equation*}
G_{i,j}(z) = \frac{(m+1)A_{pd}h^{m+1}}{2\pi}(D_{i,j}^{2}+h^{2} )^{\frac{-(m+3)}{2}},
\label{eqn:gain2}
\end{equation*}
where $D_{i,j}=\sqrt{(z_{x}+ia)^{2}+(z_{y}+ja)^{2}}$, represents the horizontal distance between the PD and the $i^{th}$ LED at $(ia,ja,h)$. All the LEDs, saving the tagged LED at $(0,0,h)$, are termed as co-channel interferers. Now, the total received current $I(z,t)$ at the PD, during a time slot $t$ is given as
\[ I(z,t) =s_{0}(t) G_{0}(z) R_{pd} + \mathcal{I}_{\infty}(z) + n(t),\]
where $\mathcal{I}_{\infty}(z)=\sum_{(i,j) \in \mathbb{S}\setminus (0,0)}s_{i,j}(t)G_{i,j}(z)R_{pd}$ is the co-channel interference current, and $n(t)\sim\mathcal{N}(0,\sigma^{2})$ has a power spectral density $N_{o}$ such that the variance of the noise process $\sigma^{2}=N_{o}W$. Also, the co-channel interference term $s_{i,j}(t) G_{i,j}(z) R_{pd}$, is a uniformly distributed random variable over the \textit{M}-PAM intensity levels whose mean is given as
\begin{align*}
I_{i,j}(z)&=\mathbb{E}[s_{i,j}(t) G_{i,j}(z) R_{pd}],\nonumber\\
&=AMG_{i,j}(z)R_{pd},
\end{align*}
and the variance is given as $\frac{A^{2}(M^{2}-1)}{3}G_{i,j}^{2}(z) R_{pd}^{2}$. Summing up over a large number of interfering LEDs, the sum $\sum_{i,j \in \mathbb{S}\setminus 0}s_{i,j}(t) G_{i,j}(z) R_{pd}$ is approximated to converge in distribution to a Gaussian random variable with a mean of
\begin{align}
\mu&=\mathbb{E}[\mathcal{I}_{\infty}(z)],\nonumber\\
&=AM\sum_{(i,j) \in \mathbb{S}\setminus (0,0)}G_{i,j}(z)R_{pd},\nonumber\\
&= \sum_{(i,j) \in \mathbb{S}\setminus (0,0)}T_{1}(D_{i,j}^{2} + h^{2} )^{\frac{-(m+3)}{2}},
\label{eqn:int1}
\end{align}
and a variance of
\begin{align}
\sigma_{1}^{2}&= \frac{A^{2}(M^{2}-1)}{3}\sum_{(i,j) \in \mathbb{S}\setminus (0,0)}G_{i,j}^{2}(z) R_{pd}^{2},\nonumber\\
&=\sum_{(i,j) \in \mathbb{S}\setminus (0,0)} T_{2}(D_{i,j}^{2} + h^{2} )^{-(m+3)},
\label{eqn:var}
\end{align}
where $T_{1}=\frac{AMR_{pd}A_{pd}(m+1)h^{m+1}}{2\pi}$ and $T_{2}=\frac{A^{2}(M^{2}-1)R_{pd}^{2}A^{2}_{pd}(m+1)^{2}h^{2(m+1)}}{12\pi^{2}}$. So, the received interference plus noise $n(t)$ at the PD is normally distributed as $\mathcal{N}(\mu,\sigma^{2}+\sigma_{1}^{2})$. Using the interference characterization in \cite{atchu2} by considering till $(u,v)$ Fourier terms, we give a closed form expression for mean $\mu$ in \eqref{eqn:int1} as
\begin{align}
\mu & \approx \hat {\mathcal{I}}_{u,v}(z), \nonumber\\
&=T_{1} \bigg( \frac{h^{2-\beta}\pi}{a^{2}\big(\frac{\beta}{2}-1\big)}-\frac{1}{(z^{2}+h^{2})^{\frac{\beta}{2}}}+ \sum_{(w,f)\in\mathbb{A}} g_{1}(w,f)\bigg),
\label{eqn:intapp1}
\end{align}
where $\mathbb{A} \triangleq \{\mathbb{Z}^{2}\cap([0,u]\times[0,v])\}\setminus(0,0)$ over the set of integers $\mathbb{Z}^{2}, u$ and $v$; $\beta=m+3$ and $\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\mathrm{d} t$ is the standard gamma function. Also,
\begin{align*}
g_{1}(w,f)=\frac{\mathbb{K}_{\frac{\beta}{2}-1}\bigg(\frac{2\pi h\sqrt{f^{2}+w^{2}}}{a}\bigg)\cos\big(\frac{2\pi wz_{x}}{a}\big)\cos\big(\frac{2\pi fz_{y}}{a}\big)}{\bigg(\frac{h}{2\pi\sqrt{f^{2}+w^{2}}}\bigg)^{\frac{\beta}{2}-1}2^{\frac{\beta}{2}-4} a^{\frac{\beta}{2}+1}\frac{\Gamma\big(\frac{\beta}{2}\big)}{\pi}},
\end{align*}
where $\mathbb{K}_{\tau}(y)=\frac{\Gamma(\tau+\frac{1}{2})(2y^{\tau})}{\sqrt{\pi}}\int_{0}^{\infty}\frac{\cos(t)\mathrm{d} t}{(t^{2}+y^{2})^{v+\frac{1}{2}}}$, is the modified Bessel function of second kind.
Similarly, $\sigma_{1}^{2}$ in \eqref{eqn:var} can be approximated to a closed form expression as
\begin{align}
\sigma_{1}^{2} & \approx \hat {\mathtt{I}}_{u,v}(z), \nonumber\\
&=T_{2} \bigg(\frac{h^{2-2\beta}\pi}{a^{2}(\beta-1)}-\frac{1}{(z^{2}+h^{2})^{\beta}}+ \sum_{(w,f)\in\mathbb{A}} g_{2}(w,f)\bigg),
\label{eqn:intapp1a}
\end{align}
where,
\begin{align*}
g_{2}(w,f)=\frac{\mathbb{K}_{\beta-1}\bigg(\frac{2\pi h\sqrt{f^{2}+w^{2}}}{a}\bigg)\cos\big(\frac{2\pi wz_{x}}{a}\big)\cos\big(\frac{2\pi fz_{y}}{a}\big)}{\bigg(\frac{h}{2\pi\sqrt{f^{2}+w^{2}}}\bigg)^{\beta-1}2^{\beta-4} a^{\beta+1}\frac{\Gamma(\beta)}{\pi}}.
\end{align*}
\subsection{The Probability of Error}
For an \textit{M}-PAM IM/DD scheme, the probability of error depends on two factors, namely, the distance $d$ between two adjacent constellation points of \textit{M}-PAM, and the interference plus noise at the PD. We neglect any non-linearities of the PD while reception. The tagged LED at $(0,0,h)$ transmits data to the PD over $M$ equiprobably different intensity levels. If there were no effect of interference or noise at the PD, then every $l^{th}$ intensity level would just suffer through a channel gain $G_{0,0}(z)$, and hence be received as $(2l-1)AG_{0,0}(z)$ at the PD. So, the adjacent distance $d$ between two constellation points would be $2AG_{0,0}(z)$, as shown in Fig. \ref{pam2}. \\ \par
\begin{figure}[ht]
\centering
\resizebox{0.45\textwidth}{!}{%
\begin{tikzpicture}
\draw [line width= 0.01cm] (0.2,0.5) -- (5,0.5);
\draw [line width= 0.01cm] (7.4,0.5) -- (8.2,0.5);
\draw [fill] (0.2,0.5) circle [radius=0.20];
\node [above] at (0.2,0.8) {\normalsize $AG_{0,0}(z)$};
\draw [fill] (2.2,0.5) circle [radius=0.20];
\node [below] at (2.2,0.2) {\normalsize $3AG_{0,0}(z)$};
\draw [fill] (4.2,0.5) circle [radius=0.20];
\node [above] at (4.2,0.8) {\normalsize $5AG_{0,0}(z)$};
\draw [fill] (5.5,0.5) circle [radius=0.01];
\node [below] at (8.2,0.2) {\normalsize $(2M-1)AG_{0,0}(z)$};
\draw [fill] (6,0.5) circle [radius=0.01];
\draw [fill] (6.5,0.5) circle [radius=0.01];
\draw [fill] (8.2,0.5) circle [radius=0.20];
\draw [dotted] (1.2,0) -- (1.2,1.2);
\draw [dotted] (3.2,0) -- (3.2,1.2);
\draw [dotted] (2.2,0.5) -- (2.2,2);
\draw [dotted] (4.2,0.5) -- (4.2,2);
\node at (3.2,1.9) {\normalsize $d$};
\draw[arrows=->](3.5,1.9)--(4.2,1.9);
\draw[arrows=->](2.9,1.9)--(2.2,1.9);
\end{tikzpicture}
}%
\caption{Here, the \textit{M}-PAM constellation of the received intensity at the PD is shown when the effect of noise and interference is not considered. The intensities suffer through a channel gain $G_{0,0}(z)$.}
\label{pam2}
\end{figure}
If the noise and the interference were present at the PD, the received symbols would be in error. For a given constellation point, the per symbol probability of error $P_{s}$ can be calculated using the mean $(\mu)$ and variance$(\sigma_{1}^{2})$ respectively in \eqref{eqn:int1} and \eqref{eqn:var} as
\begin{align*}
P_{s}&=\mathbb{P}\bigg[\mathcal{I}_{\infty}(z)+n(t)>\frac{R_{pd}d}{2}\bigg],\nonumber\\
&=Q\Bigg(\frac{\frac{R_{pd}d}{2}-\mu}{\sqrt{\sigma^{2}+\sigma_{1}^{2}}}\Bigg),\nonumber\\
&\stackrel{(a)}{=}Q\Bigg(\frac{\frac{R_{pd}d}{2}-\hat{\mathcal{I}}_{u,v}(z)}{\sqrt{\sigma^{2}+\hat{\mathtt{I}}_{u,v}(z)}}\Bigg),
\end{align*}
where $(a)$ follows the approximation in \eqref{eqn:intapp1} and \eqref{eqn:intapp1a}. Assuming the leftmost and the rightmost constellation points have neighbours at infinity, the total symbol error $P_{e}$, using the union bound is given as
\begin{align}
P_{e}&\leq \frac{2}{M} Q\Bigg(\frac{\frac{R_{pd}d}{2}-\hat{\mathcal{I}}_{u,v}(z)}{\sqrt{\sigma^{2}+\hat{\mathtt{I}}_{u,v}(z)}}\Bigg)+\frac{2}{M}\sum_{j=2}^{M-1}Q\Bigg(\frac{\frac{R_{pd}d}{2}-\hat{\mathcal{I}}_{u,v}(z)}{\sqrt{\sigma^{2}+\hat{\mathtt{I}}_{u,v}(z)}}\Bigg),\nonumber\\
&=\frac{2(M-1)}{M} Q\Bigg(\frac{\frac{R_{pd}d}{2}-\hat{\mathcal{I}}_{u,v}(z)}{\sqrt{\sigma^{2}+\hat{\mathtt{I}}_{u,v}(z)}}\Bigg).
\label{eqn:probe}
\end{align}
\subsection{The Rate Expression}
The SINR is calculated using the electrical powers received at the PD as
\begin{align}
\gamma(z)&=\frac{T_{1}^{2}(z^{2}+h^{2})^{-m-3}}{\sigma_{1}^{2}+\sigma^{2}}, \nonumber\\
&\stackrel{(b)}{=}\frac{T_{1}^{2}(z^{2}+h^{2})^{-m-3}}{\hat{\mathtt{I}}_{u,v}(z)+\sigma^{2}},
\label{eqn:sinr}
\end{align}
where $(b)$ follows from \eqref{eqn:intapp1a}.
So, the rate $R$ with the units of bits/s/Hz is given as
\begin{align}
R&=\log_{2}(1+\gamma(z)),\nonumber\\
&\stackrel{(c)}{=}\log_{2}\bigg(1+\frac{T_{1}^{2}(z^{2}+h^{2})^{-m-3}}{\hat{\mathtt{I}}_{u,v}(z)+\sigma^{2}}\bigg),
\label{eqn:sinrl1r}
\end{align}
where $(c)$ follows from \eqref{eqn:sinr}. \\ \par
We now proceed to understand how to improve the performance of the system by applying the TDMA over the LEDs.
\section{The TDMA over the LEDs}
A generalized top view of the infinite 2D plane of Fig.\ref{two_dim} is shown in Fig.\ref{two_dim1}. Out of the infinite number of LEDs over the plane, if every $K^{2}$ LEDs ($K=3$ in Fig.\ref{two_dim1}) are square-symmetrically grouped together with $K$ LEDs on each side of the square, and only one out of those $K^{2}$ LEDs now acts as a LiFi source for a time period of $\frac{1}{K^{2}}$, then the following two things happen. Firstly, in that duration of $\frac{1}{K^2}$, every LiFi source in the infinite plane is separated by a distance of $Ka$ along both the edges of the virtual square; and secondly, the other remaining $K^2-1$ LEDs consecutively take up the LiFi ability over the rest of the duration of $\frac{(K^2-1)}{K^2}$. \par
\begin{figure}[ht]
\centering
\resizebox{0.35\textwidth}{!}{%
\begin{tikzpicture}
\node at (2.8,0) {\tiny$a$}
\draw[arrows=->](3.25,0)--(3.05,0);
\draw[arrows=->](2.35,0)--(2.55,0);
\node at (0.8,0) {\tiny$Ka$}
\draw[arrows=->](1.05,0)--(1.55,0);
\draw[arrows=->](0.55,0)--(0.05,0);
\node at (-0.2,1.0) {\tiny$Ka$}
\draw[arrows=->](-0.2,1.25)--(-0.2,1.75);
\draw[arrows=->](-0.2,0.75)--(-0.2,0.25);
\node at (3.3,0.5) {\tiny$a$}
\draw[arrows=->](3.3,0.05)--(3.3,0.25);
\draw[arrows=->](3.3,0.95)--(3.3,0.75);
\draw [dotted,line width= 0.025cm] (-0.4,0.25) -- (3.6,0.25);%
\draw [dotted,line width= 0.01cm] (-0.4,0.75) -- (3.6,0.75);
\draw [dotted,line width= 0.01cm] (-0.4,1.25) -- (3.6,1.25);
\draw [dotted,line width= 0.025cm] (-0.4,1.75) -- (3.6,1.75);%
\draw [dotted,line width= 0.01cm] (-0.4,2.25) -- (3.6,2.25);
\draw [dotted,line width= 0.01cm] (-0.4,2.75) -- (3.6,2.75);
\draw [dotted,line width= 0.025cm] (-0.4,3.25) -- (3.6,3.25)
\draw [dotted,line width= 0.025cm] (0.05,-0.1) -- (0.05,3.65);%
\draw [dotted,line width= 0.01cm] (0.55,-0.1) -- (0.55,3.65);
\draw [dotted,line width= 0.01cm] (1.05,-0.1) -- (1.05,3.65);
\draw [dotted,line width= 0.025cm] (1.55,-0.1) -- (1.55,3.65)
\draw [dotted,line width= 0.01cm] (2.05,-0.1) -- (2.05,3.65);
\draw [dotted,line width= 0.01cm] (2.55,-0.1) -- (2.55,3.65);
\draw [dotted,line width= 0.025cm] (3.05,-0.1) -- (3.05,3.65)
\draw [fill] (-0.2,0) circle [radius=0.05];
\draw [fill] (1.8,0) circle [radius=0.11];
\draw [fill] (0.3,0.5) circle [radius=0.05];
\draw [fill] (0.8,0.5) circle [radius=0.05];
\draw [fill] (1.3,0.5) circle [radius=0.05];
\draw [fill] (1.8,0.5) circle [radius=0.05];
\draw [fill] (2.3,0.5) circle [radius=0.05];
\draw [fill] (2.8,0.5) circle [radius=0.05];
\draw [fill] (0.3,1.0) circle [radius=0.05];
\draw [fill] (0.8,1.0) circle [radius=0.05];
\draw [fill] (1.3,1.0) circle [radius=0.05];
\draw [fill] (1.8,1.0) circle [radius=0.05];
\draw [fill] (2.3,1.0) circle [radius=0.05];
\draw [fill] (2.8,1.0) circle [radius=0.05];
\draw [fill] (0.3,1.5) circle [radius=0.11];
\draw [fill] (0.8,1.5) circle [radius=0.05];
\draw [fill] (1.3,1.5) circle [radius=0.05];
\draw [fill] (1.8,1.5) circle [radius=0.11];
\draw [fill] (2.3,1.5) circle [radius=0.05];
\draw [fill] (2.8,1.5) circle [radius=0.05];
\draw [fill] (3.3,1.5) circle [radius=0.11];
\draw [fill] (-0.2,2.0) circle [radius=0.05];
\draw [fill] (0.3,2.0) circle [radius=0.05];
\draw [fill] (0.8,2.0) circle [radius=0.05];
\draw [fill] (1.3,2.0) circle [radius=0.05];
\draw [fill] (1.8,2.0) circle [radius=0.05];
\draw [fill] (2.3,2.0) circle [radius=0.05];
\draw [fill] (2.8,2.0) circle [radius=0.05];
\draw [fill] (3.3,2.0) circle [radius=0.05];
\draw [fill] (-0.2,2.5) circle [radius=0.05];
\draw [fill] (0.3,2.5) circle [radius=0.05];
\draw [fill] (0.8,2.5) circle [radius=0.05];
\draw [fill] (1.3,2.5) circle [radius=0.05];
\draw [fill] (1.8,2.5) circle [radius=0.05];
\draw [fill] (2.3,2.5) circle [radius=0.05];
\draw [fill] (2.8,2.5) circle [radius=0.05];
\draw [fill] (3.3,2.5) circle [radius=0.05];
\draw [fill] (-0.2,3) circle [radius=0.05];
\draw [fill] (0.3,3) circle [radius=0.11];
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}%
\caption{\textit{Generalized top view of the infinite 2D attocell network for a duration of $\frac{1}{K^2}$}. Here, $K=3$, or $K^2=9$ LEDs are symmetrically grouped together into the virtual square groups. Each group is bordered by boldly dotted lines, and is represented by a collation of $K^2$ smaller square attocells. In the $\frac{1}{K^2}$ duration, only one of the $K^2$ LEDs is LiFi capable and is represented by a larger filled circle. All other non-LiFi LEDs are represented by relatively smaller filled circles. For a given attocell length $a$, the adjacent separation between the LiFi LEDs now becomes $Ka$.}
\label{two_dim1}
\end{figure}
Given this time division multiplexing, the PD will experience the signal power from the tagged LED only for a duration of $\frac{1}{K^2}$. Whereas, for the rest of $\frac{K^2-1}{K^2}$ duration, it doesn't experience any. During this time period $\frac{1}{K^2}$, because the separation between every neighbouring LiFi source becomes $Ka$, the expressions in \eqref{eqn:probe} and \eqref{eqn:sinrl1r} modify respectively as
\begin{align}
P'_{e}=\frac{2(M-1)}{M} Q\Bigg(\frac{\frac{R_{pd}d}{2}-\hat{\mathcal{I}}'_{u,v}(z)}{\sqrt{\sigma^{2}+\hat{\mathtt{I}}'_{u,v}(z)}}\Bigg),
\label{eqn:probe1}
\end{align}
\begin{equation}
R'=\frac{1}{K^2}\log_{2}\bigg(1+\frac{T_{1}^{2}(z^{2}+h^{2})^{-m-3}}{\hat{\mathtt{I}}'_{u,v}(z)+\sigma^{2}}\bigg),
\label{eqn:sinrl1r1}
\end{equation}
where $\hat{\mathcal{I}}'_{u,v}(z)=\hat{\mathcal{I}}_{u,v}(z)|_{a\rightarrow Ka}$ and $\hat{\mathtt{I}}'_{u,v}(z)=\hat{\mathtt{I}}_{u,v}(z)|_{a\rightarrow Ka}$.
Using the modified expressions, \eqref{eqn:probe1} and \eqref{eqn:sinrl1r1}, relative to this TDMA over the LEDs, an exact expression for the goodput $G$ of the system can now be expressed as
\begin{align}
G&=R'\times(1-P'_{e}),\nonumber\\
&=\frac{1}{K^2}\log_{2}\bigg(1+\frac{T_{1}^{2}(z^{2}+h^{2})^{-m-3}}{\hat{\mathtt{I}}'_{u,v}(z)+\sigma^{2}}\bigg)\nonumber\\
&\ \ \ \ \ \ \ \ \ \ \times\Bigg(1-\frac{2(M-1)}{M} Q\Bigg(\frac{\frac{R_{pd}d}{2}-\hat{\mathcal{I}}'_{u,v}(z)}{\sqrt{\sigma^{2}+\hat{\mathtt{I}}'_{u,v}(z)}}\Bigg)\Bigg).
\label{eqn:good}
\end{align}
\subsection{Optimal time scheduling}
It is of significant importance to exploit this time scheduling to mitigate the effect of interference and maximize the system performance. Evidently, as the TDMA parameter $K$ increases, the LiFi LEDs are separated by a distance of $Ka$ and therefore the co-channel interference also considerably reduces. By analyzing the modified terms $\hat{\mathcal{I}}'_{u,v}(z)$ and $\hat{\mathtt{I}}'_{u,v}(z)$, it can be seen that when $K$ gets bigger and for $\beta>2$,
\[\mathbb{K}_{\frac{\beta}{2}-1}\bigg(\frac{2\pi h\sqrt{f^2+w^2}}{Ka}\bigg) \ll \mathbb{K}_{\beta-1}\bigg(\frac{2\pi h\sqrt{f^2+w^2}}{Ka}\bigg). \]
This means the denominator inside the $Q$ function in \eqref{eqn:probe1} reduces more rapidly than the numerator, implying that the probability of error $P'_{e}$ dwindles rapidly over this increase in $K$. Moreover, the rate $R'$ is also bound to increase due to a decrease in interference but only upto a certain value $K$. The reason lies in the fact that $K^2$, which resides as a denominator in the expression for $R'$, dominates the logarithmic behaviour of the rest of the expression. This work essentially focusses on maximizing the goodput $G$ because it represents a joint optimization between the $P'_{e}$ and the rate $R'$ of the time scheduled system. That is to say, the system must be optimally scheduled such that the $P'_{e}$ is relatively minimized upto the extent that the rate $R'$ is maximized. The optimum value of $K$ is given as
\[ K^{*}=\arg\max \limits_{K}G. \]
This value can easily be evaluated for its existence by performing numerical simulations over the characterization for $G$ using \eqref{eqn:good}. However, it is to note that the $K$ at which $R'$ may become maximum may not be equal to $K^{*}$. This discussion can be further understood with the following numerical simulations.
\subsection{Numerical analysis}
All numerical simulations have been performed using the parameters in Table II.
\begin{table}
\caption{Parameters considered for numerical simulations}
\label{abc}
\begin{tabular}{ | m{3cm} | m{1cm}| m{2cm} | m{1.0cm} |}
\hline
Parameter& Symbol & Value & Unit \\
\hline
Temperature of Operation & $T$ & $300$ & K \\
\hline
Noise power spectral density at Photodiode & $N_{o}$ & $4.14\times10^{-21}$ & A$^{2}$Hz$^{-1}$ \\
\hline
Modulation bandwidth of LED & $W$ & $40\times10^{6}$ & Hz \\
\hline
Area of Photodiode & $A_{pd}$ & $10^{-4}$ & m$^{2}$ \\
\hline
Responsivity of PD & $R_{pd}$ & $0.1$ & AW$^{-1}$ \\
\hline
Order of the PAM & $M$ & $8$ & \\
\hline
Optical power constant & $A$ & $1$ & W\\
\hline
Field Of View of PD & $\theta_{f}$ & $90$ & degrees \\
\hline
Half power semi angle of the LEDs & $\theta_{h}$ & $60$ & degrees \\
\hline
\end{tabular}
\end{table}
The LEDs are assumed to be installed at different values of height to LED separation ratio $h/a\in\{3,5,7\}$. From \cite{atchu2}, it is sufficient to consider the interference summation order of $u=v=2$, which itself is a good approximation over the considered values of $K\in\{1, 2, 3, ... , 15\}$. First of all, the behaviour of $P'_{e}$ and $R'$ against $K$ is observed in Fig.\ref{pe1} and \ref{rate1}. The graph for goodput is then drawn in Fig.\ref{good1}.
\subsubsection{Probability of error $P'_{e}$ (Fig.\ref{pe1})} When the value of $K$ increases, the probability of error reduces rapidly. For both numerical and analytical simulations over different values of $h/a$. Importantly, the numerical simulations performed without any of the approximations made in this paper tightly bound with the analytical ones, which actually validates the latter. The drop in $P'_{e}$ generally starts after $K>2$ and for any given $K$, the $P'_{e}$ is always lesser for a lesser value of $h/a$.
\begin{figure}[ht]
\centering
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\caption{The variation of probability of error $P'_{e}$, for both numerical and analytical simulations, is plotted against the TDMA parameter $K$ for different values of the ratio $h/a$ of LED installation. Here, the PD is assumed to be located at $z_x=z_y=0$.}
\label{pe1}
\end{figure}
\subsubsection{Rate $R'$ (Fig.\ref{rate1})} As mentioned earlier, it can be seen that the rate $R'$ attains a peak value at a particular value of $K$. It improves till the peak and decreases thereafter, which clearly implies the effect of a logarithmic increase due to decrease in interference before the peak, and the dominant effect of $K^2$ in the denominator of \eqref{eqn:sinrl1r1} after the peak. Also, the numerical simulations performed without any of the approximations made in this paper tightly bound with the analytical ones, which actually validates the latter. Parallelly, the rate is always higher for a lower value of $h/a$ for any given value of $K$.
\begin{figure}[ht]
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\caption{The variation of rate $R'$, for both numerical and analytical simulations, is plotted against the TDMA parameter $K$ for different values of the ratio $h/a$ of LED installation. Here, the PD is assumed to be located at $z_x=z_y=0$.}
\label{rate1}
\end{figure}
\subsubsection{Goodput $G$ (Fig.\ref{good1})} It was observed in Fig.\ref{pe1} and \ref{rate1} that though the rate $R'$ may have been maximized at a particular value of $K$, the $P'_{e}$ was still large enough to cause a decrease in the system performance. For example, at $h/a=3$, and at $K=5$ where the rate was maximized to $R'=0.43\times10^{7}$bits/s/Hz, the value of $P'_{e}=0.6$ was still large enough, and it was desirable to achieve a lower value. So, it is essential to perform a trade-off between the allowable probability of error and the rate that can be achieved; and this is taken care of by the goodput $G$ of the system. The value of $K^{*}(=\arg\max \limits_{K}G)$ is higher than that obtained to maximize the rate. It occurs between $K=6$ and $8$ for different values of $h/a$ considered. Here too, the numerical simulations performed without any of the approximations made in this paper tightly bound with the analytical ones, which actually validates the latter. Also, $G$ is always higher for a lower value of $h/a$ at any given $K$.
\begin{figure}[ht]
\centering
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\caption{The variation of Goodput $G=R'\times(1-P'_{e})$, for both numerical and analytical simulations, is plotted against the TDMA parameter $K$ for different values of the ratio $h/a$ of LED installation. Here, the PD is assumed to be located at $z_x=z_y=0$.}
\label{good1}
\end{figure}
Above all, the fact is clearly established that co-channel interference can be mitigated and the performance of a LiFi attocell network can be improved with TDMA over the LEDs, and there exists an optimal value of $K$ at which the goodput $G$, an evidently appropriate parameter, can be maximized.
\section{Conclusion}
This work uses the interference approximation in \cite{atchu2} to provide closed form expressions for the goodput $G$ of a time scheduled LiFi system in its downlink. It was shown through analytical simulations that the co-channel interference can be mitigated and the performance of a LiFi attocell network can be improved with TDMA over the LEDs. Moreover, the existence of an optimum scheduling parameter $K$ was shown at which $G$ was maximized.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,960 |
Weekly Greeting
stage&art
By Chris Murphy May 20, 2013 Read More →
George Lucas Grand Marshall 2013 Cruise Parade
Modesto, USA- The members of The North Modesto Kiwanis Club are proud to announce that Modesto's Favorite Son and legendary film director and producer, George Lucas, will be the Grand Marshall of their 2013 Modesto American Graffiti Car Show and Festival parade. The parade will take place June 7, 2013 at 7:00. There will be an honorary presentation at 6:30 pm at 10 th and I St. This event will be webcast live at www.modestoview.com
" We are excited to be able to provide a venue for the citizens of Modesto to honor Mr. Lucas " explains John Sanders, spokesperson for the Kiwanis Club that presents this event. "American Graffiti, his movie about growing up in Modesto, was released in 1973 and is still popular the world over. We citizens of Modesto are proud of what he has done not only for his hometown, but for people everywhere. We know that his fans will enjoy lining the parade route and letting him know how much they appreciate and enjoy his many accomplishments".
His career was set in motion by the success of the film American Graffiti. The cruising of 1962 on 10th and 11th streets and the people of Modesto inspired the film that is now listed in the "Top 100 Films" and allowed Mr. Lucas to go on to create his many films including Star Wars and, most recently, Red Tails. Mr. Lucas recently sold LucasFilm to Disney and will now focus on his educational foundation and small films.
The Modesto American Graffiti Car Show and Festival will be celebrating its 15th year in 2013. This is also the 40th anniversary of the release of American Graffiti, the movie. Over 1,000 cars participated in the 2012 car show and over 800 in the Friday evening parade. The Modesto Historic Cruise Route was officially opened at the beginning ceremonies of the 2012 parade. It provides a walking tour of Modesto, its history of cruising, music, and the part played by George Lucas. Lucasfilm and the Lucas family worked closely on the Historic Cruise Route and George Lucas made a promotional video for the tourist and historic attraction. Mr. Lucas also provided video for the Modesto Chamber of Commerce's 100th anniversary celebration.
"We are honored that George Lucas will be a part of our 2013 event and it shows how important his "growing up" years in Modesto were as a part of his career. Given his hectic schedule and limited private time, this is a rare opportunity for all of us to honor him. This will be a very positive event for our City of Modesto and its citizens", said Sanders.
The Modesto American Graffiti Car Show and Festival received the Modesto Area Music Association MAMA Award for Best Large Event for 2012. The 2013 car show and festival will be held on June 7, 8, 9, 2013. For more information leading up to the show: www.northmodestokiwanis.org
Posted in: event calendar, featured, Graffiti, news
About the Author: Chris Murphy
Chris Murphy is the President and CEO of Sierra Pacific Warehouse Group and Publisher and Founder of ModestoView Inc. Chris worked globally in the cycling industry returning to Modesto in 1996. He is also the founder of the Modesto Historic Graffiti Cruise Route, Legends of the Cruise Walk of Fame, Modesto Rockin' Holiday, the Modesto Music History Organization and co-founder of the Modesto Area Music Association. Chris is married to his artist wife Rebecca since 1985 and has two daughters Madison and Abigail, both graduating from Modesto High and UC Berkeley. He is lead singer and guitarist for his band, Third Party that donates their performances to non-profits.
Modesto View Supporters
MV Calendar
MV Tweets
Tweets by @modestoview
Central Valley TV Tweets
Tweets by @CentralValleyTV
#featured #art #featured #community #Featured #music #featured #news #modestousa all ages american graffiti art bar Blue Monday blues buzz Cars comedy comic con community cosplay drink entertainment event events featured fitness food fun gallo center graffiti Graffiti Summer jazz local mama metal MJC modesto Modesto Area Music Modesto Entertainment Modesto Event Modesto Events Modesto Music Modesto USA music news rock rockabilly sfoutsidelands
© 2019 Modestoview. WordPress Theme by Solostream. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 335 |
{"url":"http:\/\/gmatclub.com\/forum\/gmat-problem-solving-ps-140\/index-500.html?sk=a&sd=a","text":"Find all School-related info fast with the new School-Specific MBA Forum\n\n It is currently 13 Oct 2015, 14:45\n\n# Events & Promotions\n\n###### Events & Promotions in June\nOpen Detailed Calendar\n\n# GMAT Problem Solving (PS)\n\n Question banks Downloads My Bookmarks Reviews Important topics Go to page Previous \u00a0 \u00a01\u00a0\u00a0...\u00a0\u00a09\u00a0\u00a0\u00a010\u00a0\u00a0\u00a011\u00a0\u00a0\u00a012\u00a0\u00a0\u00a013\u00a0\u00a0...\u00a0\u00a0213\u00a0 \u00a0 Next Search for:\nTopics Author Replies \u00a0 Views Last post\n\nAnnouncements\n\n37\n 150 Hardest and easiest questions for PS \u00a0 Tags:\nBunuel\n\n2\n\n4051\n\n21 Aug 2015, 04:20\n\n512\n GMAT PS Question Directory by Topic & Difficulty\nbb\n\n0\n\n194842\n\n22 Feb 2012, 10:27\n\nTopics\n\n5\n For all positive integers x, f(x)=f(x+1). If f(3)=15, then what is f(7\nBunuel\n\n6\n\n955\n\n03 Feb 2015, 20:04\n\n3\n If x<0,y>0, and |x|>|y|, which of the following must be true?\nBunuel\n\n7\n\n1173\n\n13 Jun 2015, 15:14\n\n2\n A business school plans a new wing that will cost $16 million to build Bunuel 8 595 16 Feb 2015, 05:09 4 O is the center of the semicircle. If angle BCD = 30 and BC = Bunuel 6 615 23 Mar 2015, 04:22 2 From a total of 5 boys and 4 girls, how many 4-person committees can Bunuel 5 746 31 Mar 2015, 04:24 6 In the sequence gn defined for all positive integer values of n, g1 = Bunuel 5 399 15 Jun 2015, 01:14 4 The number of people who purchased book A is twice the number of peopl Bunuel 7 495 17 Aug 2015, 09:01 6 In the figure above, point O is the center of the semicircle, and PQ Bunuel 5 204 27 Sep 2015, 10:44 1 If 11 persons meet at a reunion and each person shakes hands exactly o Bunuel 4 398 09 Dec 2014, 06:00 2 A glucose molecule contains 6 carbon, 12 hydrogen and 6 oxygen atoms. Bunuel 5 1035 09 Jan 2015, 06:09 5 How many ways are there to arrange the letters in the word Tennessee? Bunuel 5 783 09 Feb 2015, 05:02 5 In terms of x and y, what is the area of the shaded region? Bunuel 8 804 25 Mar 2015, 02:26 3 In how many ways can 16 different gifts be divided among four children Bunuel 4 740 31 Mar 2015, 02:14 9 Sn = n^2 + 5n + 94 and K = S6 \u2013 S5 + S4 \u2013 S3 + S2 \u2013 S1. What is the Bunuel 5 396 12 Jul 2015, 03:27 3 A computer retailer sells only TFT and LCD style monitors. According Bunuel 7 388 17 Aug 2015, 08:55 3 Train A leaves New York at 7:00 am traveling to Boston at 80mph. Train Bunuel 2 200 27 Sep 2015, 10:23 6 A total of j dollars is given to Abby, Bill, and Carla and divided equ Bunuel 4 720 12 Nov 2014, 03:02 8 At the beginning of year 1, an investor puts p dollars into an investm Bunuel 5 684 18 Nov 2014, 07:08 1 The Malibu Country Club needs to drain its pool for refinishing. The h Bunuel 2 483 09 Feb 2015, 05:05 4 A is the center of the circle, and the length of AB is . The blue shad Bunuel 7 648 26 Mar 2015, 21:59 1 How many three-digit numbers are there such that all three digits are Bunuel 4 574 31 Mar 2015, 11:19 5 12!\/(3^4*5!*2^6)= Bunuel 4 324 15 Jun 2015, 01:28 2 A \u201cpalindromic integer\u201d is an integer that remains the same when its Bunuel 2 135 27 Sep 2015, 10:25 5 The harmonic mean of two numbers x and y, symbolized as h(x, y), is de Bunuel 4 763 12 Nov 2014, 03:10 1 The figure above shows three tangent circles (they touch at exactly on Bunuel 3 546 09 Jan 2015, 06:11 1 A factory produces x widgets per day. The factory's fixed costs are$8\nBunuel\n\n3\n\n574\n\n09 Feb 2015, 05:07\n\n4\n If the area of the outer square is 4x2, then the area of the inner squ\nBunuel\n\n15\n\n1139\n\n03 Apr 2015, 06:20\n\n12\n Line k is in the rectangular coordinate system. If the x-intercept of\nBunuel\n\n9\n\n626\n\n08 Apr 2015, 09:01\n\n7\n The volume of a cone is S \u00d7 H\/3, where S is the area of the base and H\nBunuel\n\n4\n\n305\n\n17 Aug 2015, 08:44\n\n For all numbers a and b, the operation && is defined by a&&b = (a + 2\nBunuel\n\n2\n\n158\n\n29 Sep 2015, 19:49\n\n8\n If ab = x and a\/b = y, and ab does not equal zero, then\nBunuel\n\n7\n\n889\n\n18 Nov 2014, 07:22\n\n5\n Currently bananas cost 50 cents\/pound. Due to a disease affecting the\nBunuel\n\n7\n\n567\n\n09 Jan 2015, 06:03\n\n4\n In five football games thus far this season, Barry has run for 98, 107\nBunuel\n\n8\n\n539\n\n09 Feb 2015, 05:08\n\n4\n What is the value of x?\nBunuel\n\n8\n\n521\n\n24 Mar 2015, 06:38\n\n10\n In the xy-coordinate system, the distance between the point (0,0) and\nBunuel\n\n7\n\n738\n\n31 Mar 2015, 11:17\n\n5\n What is the sum of the cubes of the first ten positive integers?\nBunuel\n\n4\n\n366\n\n15 Jun 2015, 02:17\n\n1\n The smallest 3 digit positive integer obtained by adding two positive\nBunuel\n\n2\n\n195\n\n20 Aug 2015, 06:35\n\n In triangle ABC, DB and DC are angle bisectors and the angle BAC = 60\u00b0\nBunuel\n\n6\n\n571\n\n09 Jan 2015, 05:46\n\n293\n Baker's Dozen\nBunuel\n\n152\n\n59480\n\n26 Sep 2015, 09:14\n\n2\n In the figure below, \u2220ADE = 60\u00b0, \u2220EFC = 40\u00b0, and \u2220DAE = 55\u00b0. If AB ||\nBunuel\n\n5\n\n656\n\n23 Mar 2015, 10:14\n\n4\n The points A(0, 0), B(0, 4a - 5) and C(2a + 1, 2a + 6) form a triangle\nBunuel\n\n3\n\n496\n\n30 Mar 2015, 02:37\n\n Jacob marks his goods up by 75% and then offers a discount on the mark\nBunuel\n\n3\n\n219\n\n20 Aug 2015, 07:01\n\n2\n Of the 14,210 employees of the anvil factory, 2\/7 are journeymen. If\nBunuel\n\n3\n\n160\n\n27 Sep 2015, 10:25\n\n23\n Machines A, B, and C can either load nails into a bin or\nBunuel\n\n16\n\n4496\n\n16 Dec 2014, 21:39\n\n2\n Machine A can make 350 widgets in 1 hour, and machine B can make 250 w\nBunuel\n\n7\n\n846\n\n11 Jul 2015, 00:11\n\n3\n In the xy-coordinate system, line k passes through points (-5m, 0) and\nBunuel\n\n4\n\n449\n\n30 Mar 2015, 02:39\n\n4\n In the figure above, ABCD is a rectangle inscribed in a circle. Angle\nBunuel\n\n7\n\n445\n\n17 Aug 2015, 08:54\n\n3\n Shaun invests a certain sum of money at 18.75% p.a. simple interest.\nBunuel\n\n11\n\n263\n\n20 Aug 2015, 05:01\n\n3\n What is the greatest power that 5 can be raised to so that the resulti\nBunuel\n\n4\n\n178\n\n27 Sep 2015, 09:56\n\n5\n If 3x \u2013 2y \u2013 z = 32 + z and \u221a(3x) - \u221a(2y + 2z) = 4, what is the value\nBunuel\n\n2\n\n984\n\n10 Nov 2014, 00:32\n\n Question banks Downloads My Bookmarks Reviews Important topics Go to page Previous \u00a0 \u00a01\u00a0\u00a0...\u00a0\u00a09\u00a0\u00a0\u00a010\u00a0\u00a0\u00a011\u00a0\u00a0\u00a012\u00a0\u00a0\u00a013\u00a0\u00a0...\u00a0\u00a0213\u00a0 \u00a0 Next Search for:\n Who is online In total there are 2 users online :: 0 registered, 0 hidden and 2 guests (based on users active over the past 15 minutes) Users browsing this forum: No registered users and 2 guests Statistics Total posts 1394196 | Total topics 171272 | Active members 415571 | Our newest member gnahc\n\n Powered by phpBB \u00a9 phpBB Group and phpBB SEO Kindly note that the GMAT\u00ae test is a registered trademark of the Graduate Management Admission Council\u00ae, and this site has neither been reviewed nor endorsed by GMAC\u00ae.","date":"2015-10-13 22:45:00","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\": 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\": 0.2958674430847168, \"perplexity\": 4189.442918590594}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-40\/segments\/1443738017788.94\/warc\/CC-MAIN-20151001222017-00118-ip-10-137-6-227.ec2.internal.warc.gz\"}"} | null | null |
Syrphoctonus rubeoauratilis är en stekelart som beskrevs av Diller 1982. Syrphoctonus rubeoauratilis ingår i släktet Syrphoctonus och familjen brokparasitsteklar. Inga underarter finns listade i Catalogue of Life.
Källor
Brokparasitsteklar
rubeoauratilis | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,198 |
Worldwide Experience
Andre Pretorius
John Gransbury
Scott Johnstone
Bjorn Baker
Planning and Investigation
Design and Documentation
System Evaluation and Operation
Racing & Equine
Urban > Resort World Sentosa
Resort World Sentosa
In its first three years of opening in 2010 over 45 million people visited Resorts World on Sentosa Island in Singapore. It is clearly a special place and no wonder it has won prestigous travel awards ever since. This 49 hectare site is packed with world-class entertainment, aquariums, and 6 unique hotels. It even boasts a casino, waterpark, convention centre, museum and Universal Studios. Resorts World Sentosa is wholly owned by Genting Singapore, a company of the Genting Group.
HydroPlan designed and oversaw construction in 6 stages on this extensive site with final completion in 2014. Working closely with the client and contractor ensured the best outcomes in a challenging and rewarding project.
<!-feature_project->
Feature Project
Royal Adelaide Golf Club
One of Australia's top ten ranked courses, The Royal Adelaide Golf Club is situated on the coastal sand belt between Adelaide's CBD and the beaches of Gulf St Vincent. This private links style course which hosted the Australian Open in the late 1990's, was designed by Dr Alistair McKenzie and follows the natural sand dune terrain of the area.
The existing irrigation system had reached its use-by date and had undergone many upgrades and modifications over a twenty five year life.
The brief for a new irrigation system had an emphasis on water efficiency and was to take maximum advantage of a recently developed stormwater harvesting system, wetland complex and aquifer storage and recovery system within the course.
HydroPlan was commissioned in early 2009 to design an automatic system, to be installed in two stages. The club was in the process of a major redevelopment of the 17th hole and irrigation was installed on the new layout along with modifications to the 16th hole during winter 2009.
The balance of the system was installed during the winter and spring of 2010.
Features of the system include a new 500 kL tank and doubling the above ground storage to approximately 1 ML which is managed with an intricate system of motorised valves. This allows Superintendent Nathan Bennett the flexibility to store stormwater separately to bore water, and change water supplies automatically mid cycle to avoid irrigating greens with the more saline bore water. The existing variable frequency pump station was completely rebuilt with increased capacity.
Toro valve-in-head sprinklers are individually controlled by a network of VP series satellites and SitePro software using a combination of radio and hardwire communications.
Fairway sprinklers are arranged in multiple rows with a combination of full and part circle versions ensuring that irrigation is mostly contained within the fairway alignments, avoiding wasteful overspray onto the rough.
Most irrigation for the greens had been upgraded during recent years so they were integrated into the new pipework and control system.
All new sprinkler positions were precisely located by HydroPlan using state of the art GPS survey equipment, which was also employed to accurately plot as-constructed drawings. All pipework is electro-fusion welded MDPE poly, with some PVC laterals on tees and clubhouse surrounds.
HydroPlan was also commissioned to manage the tender and installation process on the club's behalf. As-constructed drawings were updated on a weekly basis, providing the club with highly accurate records of infrastructure locations.
The Royal Adelaide Golf Club have taken firm action to minimise overall water consumption and dramatically reduce the reliance on ground water reserves by irrigating with harvested storm-water that previously ended up in the nearby gulf.
http://www.royaladelaidegolf.com.au
One of Australia's top ten ranked courses, The Royal Adelaide Golf Club is situated on th... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,279 |
Ukraine's Military Told How the M113 Armored Personnel Carrier Is Better In Combat Than the Soviet BMP
The YPR-765 of the Armed Forces of Ukraine during the battles in Bakhmut, December 2022 / Photo credits ArmyInform
The superiority of the American vehicle over the Soviet-made counterparts was shown during battles in Bakhmut
ArmyInform dedicated the article to the M113 armored personnel carrier, to be accurate to its Dutch variation YPR-765. Such armored vehicles are used by one of the units of the Armed Forces of Ukraine, which holds the front in the Bakhmut area.
In particular, Ukraine's fighters say that the YPR-765 successfully passed the "test drive" in combat conditions. The vehicle helps to effectively dispose of the enemy, successfully performs the task of delivering the infantry or evacuating the wounded, has high cross-country ability and excellent maneuverability.
Read more: How Excalibur Army Modernizes the T-72 For Ukraine: Almost a Hundred Such Vehicles Are Expected to Be Done (Video)
For exapmle, even with sharp turns with the M113 and its modifications, the track does not fly off, unlike Soviet-made infantry fighting vehicles.
"Compared to the Soviet infantry fighting vehicles, these APC's are much more maneuverable, easier to operate, they're equipped with an automatic transmission, and easy to maintain. The armor can easily withstand a hit from a large-caliber machine gun and even a grenade launcher," ArmyInform journalists cite a driver-mechanic with call sign "Lotus".
The M113 has 3 crewmembers and are capable of carrying up to 12 soldiers, armed with a 12.7 mm machine gun. According to open sources data, during the entire war, the Armed Forces of Ukraine received at least 200 units of the M113 armored personnel carriers in various modifications, including American, Australian, Lithuanian and Dutch vehicles.
Read more: Money Matter No More When it Comes to Weapons: Polish General Staff Describes Situation on Arms Market
TAGS Defense ExpressWar
'The Ice Has Broken', Leopard 2 Tanks Go to Ukraine: How Many German Tanks Reinforce the Ukrainian Army
Europe's Defense Industry Depends On Chinese Tungsten, Which Is Transported Through the Territory of russia | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,088 |
define([
'find/app/page/search/filters/indexes/index-item-view',
'backbone'
], function(IndexItemView, Backbone) {
describe('IndexItemView with a valid database', function() {
beforeEach(function() {
this.model = new Backbone.Model({
name: 'Wikipedia',
deleted: false
});
this.view = new IndexItemView({model: this.model});
this.view.render();
});
it('is not disabled', function() {
expect(this.view.$el).not.toHaveClass('disabled-index');
});
it('is disabled once the database has been deleted', function() {
this.model.set('deleted', true);
this.view.updateDeleted();
expect(this.view.$el).toHaveClass('disabled-index');
});
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,828 |
Detectives: 15-year-old arrested after admitting to sexually battering 7-year-old girl
By: WFTS Webteam
WESLEY CHAPEL, Fla. — Pasco County Detectives arrested a 15-year-old boy from Wesley Chapel who they say admitted to sexually battering a 7-year-old girl.
Detectives say, starting at an unknown time in 2017 up until this week, the defendant would repeatedly rape the victim.
The victim told detectives the defendant did this to her, "a lot" and that in the past she asked him to stop.
A babysitter found the suspect with the victim alone in a closet and asked the victim what happened. When the victim told her, she immediately contacted the girl's mother who then contacted the sheriff's office.
When questioned by detectives, the defendant first told them he started exposing himself to the victim and grabbing her butt around a year and a half ago. He denied any other allegations.
When detectives confronted the defendant with what the victim said, they say he slumped his head before admitting that the victim was telling the truth. The defendant told deputies he would touch the victim with his genitals as well as place his genitals in the victim's mouth. However, the defendant denied the rape allegations, detectives say.
After his admission, the defendant was placed under arrest and transported to a Juvenile Assessment Center.
He is facing charges for sexual battery of a victim under the age of 12. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,540 |
package org.http4s
import cats.kernel.laws.discipline._
import org.http4s.laws.discipline.arbitrary._
final class RequestPreludeSuite extends Http4sSuite {
checkAll("Hash[RequestPrelude]", HashTests[RequestPrelude].hash)
checkAll("Order[RequestPrelude]", OrderTests[RequestPrelude].order)
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,704 |
Activist's Daughter Denied Scholarship
A CIF Award* for the 21-year-old daughter of an environmental and racial justice activist who runs a podcast that highlights local issues and amplifies the voices of those experiencing oppression.… Read More
Professor Fired for Speaking Out
$2000 for childcare for the two-year-old son whose dad is a professor and racial justice activist. The dad had a series of disagreements with his department and university about his fundraising for… Read More
Hunger Strikers Retaliated Against
$12750 for therapy, childcare and educational support for eight children, ages two to 12, from four families whose fathers all participated in a hunger strike at a detention center to protest their… Read More
Location Withheld
Campus Organizer Targeted
$1000 for school tuition for the seven-year-old daughter whose mom is a recognized leader at her college. They have both been threatened by those opposed to racial justice events the mom has… Read More
Police Target Mother
$1300 for dance lessons and summer camp for the 13-year-old daughter whose mother fought to reallocate funding from the police to mental health services. As a result, she was harassed online by local… Read More
Family Threatened by KKK
$2100 for a computer and a CIF award for two daughters, ages 16 and 19, whose activist mom received death threats from the Klan for her efforts to desegregate a public housing complex.
Teacher Threatened
$3000 for childcare and school tuition for two brothers, ages four and 16, whose father was targeted after advocating for fair treatment of a student who reacted physically to a classmate's racist… Read More
Mother Fired for Advocating for Anti-Racist Workplace
$3200 for recreational activities for two children, ages eight and 13, whose mom was fired after she advocated for an anti-racist framework at her job. She was subsequently harassed, and her home was… Read More
Moms Blacklisted
$8350 for a computer, therapy, athletic expenses, and cultural programming for six children, ages 10 to 19, from two families, whose mothers were both targeted for their anti-racist organizing in the… Read More
GA, OK
Fathers Targeted for Anti-Racist Activism
$11,800 for tuition, tutoring, and recreational activities for eight kids, ages nine to 16, from four families, whose dads faced repression for their racial justice organizing. Two lost jobs, one was… Read More
CO, GA
Dad Punished for Anti-Prison Art
$3070 for music lessons and recreational activities for the eight and 12-year-old daughters whose incarcerated father advocates for prison abolition in his poetry and writing. His views have resulted… Read More
Prisoners' Rights/ U.S. Political Prisoners
Former Political Prisoner Faces Harassment
$4000 for school tuition for the 13-year-old twins whose father was incarcerated for anti-apartheid efforts and continues to face ongoing targeting for his work to free political prisoners.
Mother Threatened for Helping Prisoners
$4500 for cultural programs for three children, ages seven to 18, whose mom created an alternative program for inmates at a federal penitentiary. She was accused of conspiring with a political… Read More
Sons of Political Prisoners Targeted
$7500 for school tuition and an at home nature project for five kids, ages 11 to 17, from two families whose fathers are both children of political prisoners. Their activism on behalf of political… Read More
GA, VA
Adoptive Dad Thwarted by Felonies
$1000 for sports expenses for the 14-year-old son whose father has been arrested upwards of 100 times for his direct actions related to peace and social justice. He has two felony convictions for… Read More
Father Fights to Stop Witch Hunts
$2000 for sports programs for the 15-year-old daughter whose father, a member of a peace and international solidarity group, had his home raided and was threatened with arrest but refuses to testify… Read More
Father and Son Separated
$3000 for travel costs for the 15-year-old son whose soldier father refused deployment to Iraq, fled to Canada, and became active in the Canadian anti-war movement. After being deported back to the U… Read More
Support for Children of Torture Survivors
$4600 for a computer, recreational activities, a CIF award, and an Attica grant** for two siblings, ages 16 and 18, whose father was arrested at a peaceful demonstration and faced additional charges… Read More | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,197 |
# QUANTUM THEORY
CANNOT HURT
YOU
A Guide to the Universe
Marcus Chown
To Patrick, who, when I'm down and wondering why everyone seems to be against me, consoles me by saying: "That's because you're a complete bastard, Marcus!"
# CONTENTS
Title Page
Dedication
Foreword
**PART ONE: SMALL THINGS**
1 Breathing in Einstein
2 Why God Plays Dice with the Universe
3 The Schizophrenic Atom
4 Uncertainty and the Limits of Knowledge
5 The Telepathic Universe
6 Identicalness and the Roots of Diversity
**PART TWO: BIG THINGS**
7 The Death of Space and Time
8 _E =mc_ 2 and the Weight of Sunshine
9 The Force of Gravity Does Not Exist
10 The Ultimate Rabbit Out of a Hat
Glossary
Further Reading
Index
Acknowledgements
About the Author
By the Same Author
Copyright
# FOREWORD
One of the following is true:
* Every breath you take contains an atom breathed out by Marilyn Monroe.
* There is a liquid that can run uphill.
* You age faster at the top of a building than at the bottom.
* An atom can be in many different places at once, the equivalent of you being in New York and London at the same time.
* The entire human race would fit in the volume of a sugar cube.
* One per cent of the static on a television tuned between stations is the relic of the Big Bang.
* Time travel is not forbidden by the laws of physics.
* A cup of coffee weighs more when it is hot than when it is cold.
* The faster you travel, the slimmer you get.
No, I'm joking. They are all true!
As a science writer I am constantly amazed by how much stranger science is than science fiction, how much more incredible the Universe is than anything we could possibly have invented. Despite this, however, very few of the extraordinary discoveries of the past century seem to have trickled through into the public consciousness.
The two towering achievements of the past 100 years are quantum theory, our picture of atoms and their constituents, and Einstein's general theory of relativity, our picture of space, time, and gravity. Between them the two explain virtually everything about the world and about us. In fact, it can be argued that quantum theory has actually _created_ the modern world, not only explaining why the ground beneath our feet is solid and why the Sun shines but also making possible computers and lasers and nuclear reactors. Relativity may not be as ubiquitous in the everyday world. Nevertheless, it has taught us that there are things called black holes from which nothing, not even light, can escape; that the Universe has not existed forever but was born in a titanic explosion called the Big Bang; and that time machines—remarkably—may be possible.
Although I have read many popular accounts of these topics, the explanations have often left me baffled, even with my science background. I can only guess what it must be like for nonscientists.
Einstein said: "Most of the fundamental ideas of science are essentially simple and may, as a rule, be expressed in a language comprehensible to everyone." All my experience tells me he was right. My idea in writing this book was to try to help ordinary people understand the principal ideas of 21st-century physics. All I had to do was identify the key ideas behind quantum theory and relativity, which turn out to be deceptively simple, and then show how absolutely everything else follows from them logically and unavoidably.
Easier said than done. Quantum theory in particular is a patchwork of fragments, accrued over the past 80 years, that nobody seems to have sewn together into a seamless garment. What's more, crucial pieces of the theory, such as "decoherence"—which explains why atoms but not people can be in two places at once—seem to be beyond the power of physicists to communicate in any intelligible way. After corresponding with many "experts," and beginning to think that decoherence should be renamed "incoherence," it dawned on me that maybe the experts didn't completely understand it themselves. In a way this was liberating. Since a coherent picture seemed not to exist, I realised that I had to piece together my own from insights gleaned from different people. Because of this, many of the explanations given here you will not find anywhere else. I hope they help lift some of the fog that surrounds the key ideas of modern physics and that you can begin to appreciate what a breathtakingly amazing Universe we find ourselves in.
PART ONE
# SMALL THINGS
# BREATHING IN EINSTEIN
HOW WE DISCOVERED THAT EVERYTHING IS MADE OF ATOMS AND THAT ATOMS ARE MOSTLY EMPTY SPACE
_A hydrogen atom in a cell at the end of my nose was once part of an elephant's trunk_.
Jostein Gaarder
_We never had any intention of using the weapon. But they were such a_ _terribly troublesome race. They insisted on seeing us as the "enemy" despite_ _all our efforts at reassurance. When they fired their entire nuclear_ _stockpile at our ship, orbiting high above their blue planet, our patience_ _simply ran out._
_The weapon was simple but effective. It squeezed out all the empty_ _space from matter._
_As the commander of our Sirian expedition examined the shimmering_ _metallic cube, barely 1 centimetre across, he shook his primary_ _head despairingly. Hard to believe that this was all that was left of the_ _"human race"!_
If the idea of the entire human race fitting into the volume of a sugar cube sounds like science fiction, think again. It is a remarkable fact that 99.9999999999999 per cent of the volume of ordinary matter is empty space. If there were some way to squeeze all the empty space out of the atoms in our bodies, humanity would indeed fit into the space occupied by a sugar cube.
The appalling emptiness of atoms is only one of the extraordinary characteristics of the building blocks of matter. Another, of course, is their size. It would take 10 million atoms laid end to end to span the width of a single full stop on this page, which raises the question, how did we ever discover that everything is made of atoms in the first place?
The idea that everything is made of atoms was actually first suggested by the Greek philosopher Democritus in about 440 BC.1 Picking up a rock—or it may have been a branch or a clay pot—he asked himself the question: "If I cut this in half, then in half again, can I go on cutting it in half forever?" His answer was an emphatic _no_. It was inconceivable to him that matter could be subdivided forever. Sooner or later, he reasoned, a tiny grain of matter would be reached that could be cut no smaller. Since the Greek for "uncuttable" was " _a-tomos_ ," Democritus called the hypothetical building blocks of all matter "atoms."
Since atoms were too small to be seen with the senses, finding evidence for them was always going to be difficult. Nevertheless, a way was found by the 18th-century Swiss mathematician Daniel Bernoulli. Bernoulli realised that, although atoms were impossible to observe directly, it might still be possible to observe them indirectly. In particular, he reasoned that if a large enough number of atoms acted together, they might have a big enough effect to be obvious in the everyday world. All he needed was to find a place in nature where this happened. He found one—in a "gas."
Bernoulli imagined a gas like air or steam as a collection of billions upon billions of atoms in perpetual frenzied motion like a swarm of angry bees. This vivid picture immediately suggested an explanation for the "pressure" of a gas, which kept a balloon inflated or pushed against the piston of a steam engine. When confined in any container, the atoms of a gas would drum relentlessly on the walls like hailstones on a tin roof. Their combined effect would be to create a jittery force that, to our coarse senses, would seem like a constant force pushing back the walls.
But Bernoulli's microscopic explanation of pressure provided more than a convenient mental picture of what was going on in a gas. Crucially, it led to a specific prediction. If a gas were squeezed into half its original volume, the gas atoms would need to fly only half as far between collisions with the container walls. They would therefore collide twice as frequently with those walls, doubling the pressure. And if the gas were squeezed into a third of its volume, the atoms would collide three times as frequently, trebling the pressure. And so on.
Exactly this behaviour was observed by the English scientist Robert Boyle in 1660. It confirmed Bernoulli's picture of a gas. And since Bernoulli's picture was of tiny grainlike atoms flying hither and thither through empty space, it bolstered the case for the existence of atoms. Despite this success, however, definitive evidence for the existence of atoms did not come until the beginning of the 20th century. It was buried in an obscure phenomenon called Brownian motion.
Brownian motion is named after Robert Brown, a botanist who sailed to Australia on the Flinders expedition of 1801. During his time down under, he classified 4,000 species of antipodean plants; in the process, he discovered the nucleus of living cells. But he is best remembered for his observation in 1827 of pollen grains suspended in water. To Brown, squinting through a magnifying lens, it seemed as if the grains were undergoing a curious jittery motion, zigzagging their way through the liquid like drunkards lurching home from a pub.
Brown never solved the mystery of the wayward pollen grains. That breakthrough had to wait for Albert Einstein, aged 26 and in the midst of the greatest explosion of creativity in the history of science. In his "miraculous year" of 1905, not only did Einstein overthrow Newton, supplanting Newtonian ideas about motion with his special theory of relativity, but he finally penetrated the 80-year-old mystery of Brownian motion.
The reason for the crazy dance of pollen grains, according to Einstein, was that they were under continual machine-gun bombardment by tiny water molecules. Imagine a giant inflatable rubber ball, taller than a person, being pushed about a field by a large number of people. If each person pushes in their own particular direction, without any regard for the others, at any instant there will be slightly more people on one side than another. This imbalance is enough to cause the ball to move erratically about the field. Similarly, the erratic motion of a pollen grain can be caused by slightly more water molecules bombarding it from one side than from another.
Einstein devised a mathematical theory to describe Brownian motion. It predicted how far and how fast the average pollen grain should travel in response to the relentless battering it was receiving from the water molecules all around. Everything hinged on the size of the water molecules, since the bigger they were the bigger would be the imbalance of forces on the pollen grain and the more exaggerated its consequent Brownian motion.
The French physicist Jean Baptiste Perrin compared his observations of water-suspended "gamboge" particles, a yellow gum resin from a Cambodian tree, with the predictions of Einstein's theory. He was able to deduce the size of water molecules and hence the atoms out of which they were built. He concluded that atoms were only about one 10-billionth of a metre across—so small that it would take 10 million, laid end to end, to span the width of a full stop.
Atoms were so small, in fact, that if the billions upon billions of them in a single breath were spread evenly throughout Earth's atmosphere, every breath-sized volume of the atmosphere would end up containing several of those atoms. Put another way, every breath you take contains at least one atom breathed out by Albert Einstein—or Julius Caesar or Marilyn Monroe or even the last Tyrannosaurus Rex to walk on Earth!
What is more, the atoms of Earth's "biosphere" are constantly recycled. When an organism dies, it decays and its constituent atoms are returned to the soil and the atmosphere to be incorporated into plants that are later eaten by animals and humans. "A carbon atom in my cardiac muscle was once in the tail of a dinosaur," writes Norwegian novelist Jostein Gaarder in _Sophie's_ World.
Brownian motion was the most powerful evidence for the existence of atoms. Nobody who peered down a microscope and saw the crazy dance of pollen grains under relentless bombardment could doubt that the world was ultimately made from tiny, bulletlike particles. But watching jittery pollen grains—the effect of atoms—was not the same as actually _seeing_ atoms. This had to wait until 1980 and the invention of a remarkable device called the scanning tunnelling microscope (STM).
The idea behind the STM, as it became known, was very simple. A blind person can "see" someone's face simply by running a finger over it and building up a picture in their mind. The STM works in a similar way. The difference is that the "finger" is a finger of metal, a tiny stylus reminiscent of an old-fashioned gramophone needle. By dragging the needle across the surface of a material and feeding its up-and-down motion into a computer, it is possible to build up a detailed picture of the undulations of the atomic terrain.2
Of course, there is a bit more to it than that. Although the principle of the invention was simple, there were formidable practical difficulties in its realisation. For instance, a needle had to be found that was fine enough to "feel" atoms. The Nobel Prize committee certainly recognised the difficulties. It awarded Gerd Binnig and Heinrich Rohrer, the IBM researchers behind the STM, the 1986 Nobel Prize for Physics.
Binnig and Rohrer were the first people in history to actually "see" an atom. Their STM images were some of the most remarkable in the history of science, ranking alongside that of Earth rising above the gray desolation of the Moon or the sweeping spiral staircase of DNA. Atoms looked like tiny footballs. They looked like oranges, stacked in boxes, row on row. But most of all they looked like the tiny hard grains of matter that Democritus had seen so clearly in his mind's eye, 2,400 years before. No one else has ever made a prediction that far in advance of experimental confirmation.
But only one side of the atom was revealed by the STM. As Democritus himself had realised, atoms were a lot more than simply tiny grains in ceaseless motion.
## NATURE'S LEGO BRICKS
Atoms are nature's Lego bricks. They come in a variety of different shapes and sizes, and by joining them together in any number of different ways, it is possible to make a rose, a bar of gold, or a human being. Everything is in the combinations.
The American Nobel Prize winner Richard Feynman said: "If in some cataclysm all of scientific knowledge were destroyed and only one sentence passed on to succeeding generations, what statement would convey the most information in the fewest words?" He was in no doubt: "Everything is made of atoms."
The key step in proving that atoms are nature's Lego bricks was identifying the different kinds of atoms. However, the fact that atoms were far too small to be perceived directly by the senses made the task every bit as formidable as proving that atoms were tiny grains of matter in ceaseless motion. The only way to identify different types of atoms was to find substances that were made exclusively out of atoms of a single kind.
In 1789 the French aristocrat Antoine Lavoisier compiled a list of substances that he believed could not, by any means, be broken down into simpler substances. There were 23 "elements" in Lavoisier's list. Though some later turned out not to be elements, many—including gold, silver, iron, and mercury—were indeed elemental. Within 40 years of Lavoisier's death at the guillotine in 1794, the list of elements had grown to include close to 50. Nowadays, we know of 92 naturally occurring elements, from hydrogen, the lightest, to uranium, the heaviest.
But what makes one atom different from another? For instance, how does a hydrogen atom differ from a uranium atom? The answer would come only by probing their internal structures. But atoms are so fantastically small. It seemed impossible that anyone would ever find a way to look inside one. But one man did—a New Zealander named Ernest Rutherford. His ingenious idea was to use atoms to look inside other atoms.
## THE MOTH IN THE CATHEDRAL
The phenomenon that laid bare the structure of atoms was radioactivity, discovered by the French chemist Henri Becquerel in 1896. Between 1901 and 1903, Rutherford and the English chemist Frederick Soddy found strong evidence that a radioactive atom is simply a heavy atom that is seething with excess energy. Inevitably, after a second or a year or a million years, it sheds this surplus energy by expelling some kind of particle at high speed. Physicists say it disintegrates, or "decays," into an atom of a slightly lighter element.
One such decay particle was the alpha particle, which Rutherford and the young German physicist Hans Geiger demonstrated was simply an atom of helium, the second lightest element after hydrogen.
In 1903, Rutherford had measured the speed of alpha particles expelled from atoms of radioactive radium. It was an astonishing 25,000 kilometres per second—100,000 times faster than a present-day passenger jet. Here, Rutherford realised, was a perfect bullet to smash into atoms and reveal what was deep inside.
The idea was simple. Fire alpha particles into an atom. If they encountered anything hard blocking their way, they would be deflected from their path. By firing thousands upon thousands of alpha particles and observing how they were deflected, it would be possible to build up a detailed picture of the interior of an atom.
Rutherford's experiment was carried out in 1909 by Geiger and a young New Zealand physicist called Ernest Marsden. Their "alpha-scattering" experiment used a small sample of radium, which fired off alpha particles like microscopic fireworks. The sample was placed behind a lead screen containing a narrow slit, so a thread-thin stream of alpha particles emerged on the far side. It was the world's smallest machine gun, rattling out microscopic bullets.
In the firing line Geiger and Marsden placed a sheet of gold foil only a few thousand atoms thick. It was insubstantial enough that all the alpha particles from the miniature machine gun would pass through. But it was substantial enough that, during their transit, some would pass close enough to gold atoms to be deflected slightly from their path.
At the time of Geiger and Marsden's experiment, one particle from inside the atom had already been identified. The electron had been discovered by the British physicist "J. J." Thomson in 1895. Ridiculously tiny particles—each about 2,000 times smaller than even a hydrogen atom—had turned out to be the elusive particles of electricity. Ripped free from atoms, they surged along a copper wire amid billions of others, creating an electric current.
The electron was the first subatomic particle. It carried a negative electric charge. Nobody knows exactly what electric charge is, only that it comes in two forms: negative and positive. Ordinary matter, which consists of atoms, has no net electrical charge. In ordinary atoms, then, the negative charge of the electrons is always perfectly balanced by the positive charge of something else. It is a characteristic of electrical charge that unlike charges attract each other whereas like charges repel each other. Consequently, there is a force of attraction between an atom's negatively charged electrons and its positively charged something else. It is this attraction that glues the whole thing together.
Not long after the discovery of the electron, Thomson used these insights to concoct the first-ever scientific picture of the atom. He visualised it as a multitude of tiny electrons embedded "like raisins in a plum pudding" in a diffuse ball of positive charge. It was Thomson's plum pudding model of the atom that Geiger and Marsden expected to confirm with their alpha-scattering experiment.
They were to be disappointed.
The thing that blew the plum pudding model out of the water was a rare but remarkable event. One out of every 8,000 alpha particles fired by the miniature machine gun actually bounced back from the gold foil!
According to Thomson's plum pudding model, an atom consisted of a multitude of pin-prick electrons embedded in a diffuse globe of positive charge. The alpha particles that Geiger and Marsden were firing into this flimsy arrangement, on the other hand, were unstoppable subatomic express trains, each as heavy as around 8,000 electrons. The chance of such a massive particle being wildly deflected from its path was about as great as that of a real express train being derailed by a runaway dolls pram. As Rutherford put it: "It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you!"
Geiger and Marsden's extraordinary result could only mean that an atom was not a flimsy thing at all. Something buried deep inside it could stop a subatomic express train dead in its tracks and turn it around. That something could only be a tiny nugget of positive charge sitting at the dead centre of an atom and repelling the positive charge of an incoming alpha particle. Since the nugget was capable of standing up to a massive alpha particle without being knocked to kingdom come, it too must be massive. In fact, it must contain almost all of the mass of an atom.
Rutherford had discovered the atomic nucleus.
The picture of the interior of the atom that emerged was as unlike Thomson's plum pudding picture as was possible to imagine. It was a miniature solar system in which negatively charged electrons were attracted to the positive charge of the nucleus and orbited it like planets around the Sun. The nucleus had to be at least as massive as an alpha particle—and probably a lot more so—for the nucleus with which it collided not to be kicked out of the atom. It therefore had to contain more than 99.9 per cent of the atom's mass.3
The nucleus was very, very tiny. Only if nature crammed a large positive charge into a very small volume could a nucleus exert a repulsive force so overwhelming that it could make an alpha particle execute a U-turn. What was most striking about Rutherford's vision of an atom was, therefore, its appalling emptiness. The playwright Tom Stoppard put it beautifully in his play _Hapgood:_ "Now make a fist, and if your fist is as big as the nucleus of an atom then the atom is as big as St Paul's, and if it happens to be a hydrogen atom then it has a single electron flitting about like a moth in an empty cathedral, now by the dome, now by the altar."
Despite its appearance of solidity, the familiar world was actually no more substantial than a ghost. Matter, whether in the form of a chair, a human being, or a star, was almost exclusively empty space. What substance an atom possessed resided in its impossibly small nucleus—100,000 times smaller than a complete atom.
Put another way, matter is spread extremely thinly. If it were possible to squeeze out all the surplus empty space, matter would take up hardly any room at all. In fact, this is perfectly possible. Although an easy way to squeeze the human race down to the size of a sugar cube probably does not exist, a way does exist to squeeze the matter of a massive star into the smallest volume possible. The squeezing is done by tremendously strong gravity, and the result is a neutron star. Such an object packs the enormous mass of a body the size of the Sun into a volume no bigger than Mount Everest.4
## THE IMPOSSIBLE ATOM
Rutherford's picture of the atom as a miniature solar system with tiny electrons flitting about a dense atomic nucleus like planets around the Sun was a triumph of experimental science. Unfortunately, it had a slight problem. It was totally incompatible with all known physics!
According to Maxwell's theory of electromagnetism—which described all electrical and magnetic phenomena—whenever a charged particle accelerates, changing its speed or direction of motion, it gives out electromagnetic waves—light. An electron is a charged particle. As it circles a nucleus, it perpetually changes its direction; so it should act like a miniature lighthouse, constantly broadcasting light waves into space. The problem is that this would be a catastrophe for any atom. After all, the energy radiated as light has to come from somewhere, and it can only come from the electron itself. Sapped continually of energy, it should spiral ever closer to the centre of the atom. Calculations showed that it would collide with the nucleus within a mere hundred-millionth of a second. By rights, atoms should not exist.
But atoms do exist. We and the world around us are proof enough of that. Far from expiring in a hundred-millionth of a second, atoms have survived intact since the earliest times of the Universe almost 14 billion years ago. Some crucial ingredient must be missing from Rutherford's picture of the atom. That ingredient is a revolutionary new kind of physics—quantum theory.
1 Some of these ideas were covered in my earlier book, _The Magic Furnace_ (Vintage, London, 2000). Apologies to those who have read it. In my defense, it is necessary to know some basic things about the atom in order to appreciate the chapters that follow on quantum theory, which is essentially a theory of the atomic world.
2 Of course, there is no way a needle can actually feel a surface like a human finger can. However, if the needle is charged with electricity and placed extremely close to a conducting surface, a minuscule but measurable electric current leaps the gap between the tip of the needle and the surface. It is known as a "tunnelling current", and it has a crucial property that can be exploited: the size of the current is extraordinarily sensitive to the width of the gap. If the needle is moved even a shade closer to the surface, the current grows very rapidly; if it is pulled away a fraction, the current plummets. The size of the tunnelling current therefore reveals the distance between the needle tip and the surface. It gives the needle an artificial sense of touch.
3 Eventually, physicists would discover that the nucleus contains two particles: positively charged protons and uncharged, or neutral, neutrons. The number of protons in a nucleus is always exactly balanced by an equal number of electrons in orbit about it. The difference between atoms is in the number of protons in their nuclei (and consequently the number of electrons in orbit). For instance, hydrogen has one proton in its nucleus and uranium a whopping 92.
4 See Chapter 4, "Uncertainty and the Limits of Knowledge."
# WHY GOD PLAYS DICE WITH THE UNIVERSE
HOW WE DISCOVERED THAT THINGS IN THE WORLD OF ATOMS HAPPEN FOR NO REASON AT ALL
_A philosopher once said, "It is necessary for the very existence of science that the same conditions always produce the same results." Well, they don't!_
Richard Feynman
_It's 2025 and high on a desolate mountain top a giant 100-metre telescope_ _tracks around the night sky. It locks onto a proto-galaxy at the edge_ _of the observable Universe and feeble light, which has been travelling_ _through space since long before Earth was born, is concentrated by the_ _telescope mirror onto ultrasensitive electronic detectors. Inside the telescope_ _dome, seated at a control panel not unlike the console of the_ _starship_ Enterprise, _the astronomers watch a fuzzy image of the galaxy_ _swim into view on a computer monitor. Someone turns up a loudspeaker_ _and a deafening crackle fills the control room. It sounds like machine_ _gun fire; it sounds like rain drumming on a tin roof. In fact, it is the_ _sound of tiny particles of light raining down on the telescope's detectors_ _from the very depths of space._
To these astronomers, who spend their careers straining to see the weakest sources of light in the Universe, it is a self-evident fact that light is a stream of tiny bulletlike particles—photons. Not long ago, however, the scientific community had to be dragged kicking and screaming to an acceptance of this idea. In fact, it's fair to say that the discovery that light comes in discrete chunks, or quanta, was the single most shocking discovery in the history of science. It swept away the comfort blanket of pre-20th-century science and exposed physicists to the harsh reality of an _Alice in Wonderland_ universe where things happen because they happen, with utter disregard for the civilised laws of cause and effect.
The first person to realise that light was made of photons was Einstein. Only by imagining it as a stream of tiny particles could he make sense of a phenomenon known as the photoelectric effect. When you walk into a supermarket and the doors open for you automatically, they are being controlled by the photoelectric effect. Certain metals, when exposed to light, eject particles of electricity—electrons. When incorporated into a photocell, such a metal generates a small electric current as long as a light beam is falling on it. A shopper who breaks the beam chokes off the current, signalling the supermarket doors to swish aside.
One of the many peculiar characteristics of the photoelectric effect is that, even if a very weak light is used, the electrons are kicked out of the metal instantaneously—that is, with no delay whatsoever.1 This is inexplicable if light is a wave. The reason is that a wave, being a spread-out thing, will interact with a large number of electrons in the metal. Some will inevitably be kicked out after others. In fact, some of the electrons could easily be emitted 10 minutes or so after light is shone on the metal.
So how is it possible that the electrons are kicked out of the metal instantaneously? There is only one way—if each electron is kicked out of the metal by _a single particle of light_.
Even stronger evidence that light consists of tiny bulletlike particles comes from the Compton effect. When electrons are exposed to X-rays—a high-energy kind of light—they recoil in exactly the way they would if they were billiard balls being struck by other billiard balls.
On the surface, the discovery that light behaves like a stream of tiny particles may not appear very remarkable or surprising. But it is. The reason is that there is also abundant and compelling evidence that light is something as different from a stream of particles as it is possible to imagine—a wave.
## RIPPLES ON A SEA OF SPACE
At the beginning of the 19th century, the English physician Thomas Young, famous for decoding the Rosetta stone independently of the Frenchman Jean François Champollion, took an opaque screen, made two vertical slits in it very close together, and shone light of a single colour onto them. If light were a wave, he reasoned, each slit would serve as a new source of waves, which would spread out on the far side of the screen like concentric ripples on a pond.
A characteristic property exhibited by waves is interference. When two similar waves pass through each other, they reinforce each other where the crest of one wave coincides with the crest of another, and they cancel each other out where the crest of one coincides with the trough of the other. Look at a puddle during a rain shower and you will see the ripples from each raindrop spreading out and "constructively" and "destructively" interfering with each other.
In the path of the light emerging from his two slits Young interposed a second, white, screen. He immediately saw a series of alternating dark and light vertical stripes, much like the lines on a supermarket bar code. This interference pattern was irrefutable evidence that light was a wave. Where the light ripples from the two slits were in step, matching crest for crest, the light was boosted in brightness; where they were out of step, the light was cancelled out.
Using his "double slit" apparatus, Young was able to determine the wavelength of light. He discovered it was a mere thousandth of a millimetre—far smaller than the width of a human hair—explaining why nobody had guessed light was a wave before.
For the next two centuries, Young's picture of light as ripples on a sea of space reigned supreme in explaining all known phenomena involving light. But by the end of the 19th century, trouble was brewing. Although few people noticed at first, the picture of light as a wave and the picture of the atom as a tiny mote of matter were irreconcilable. The difficulty was at the interface, the place where light meets matter.
## TWO FACES OF A SINGLE COIN
The interaction between light and matter is of crucial importance to the everyday world. If the atoms in the filament of a bulb did not spit out light, we could not illuminate our homes. If the atoms in the retina of your eye did not absorb light, you would be unable to read these words. The trouble is that the emission and absorption of light by atoms are impossible to understand if light is a wave.
An atom is a highly localised thing, confined to a tiny region of space, whereas a light wave is a spread-out thing that fills a large amount of space. So, when light is absorbed by an atom, how does such a big thing manage to squeeze into such a tiny thing? And when light is emitted by an atom, how does such a small thing manage to cough out such a big thing?
Common sense says that the only way light can be absorbed or emitted by a small localised thing is if it too is a small, localised thing. "Nothing fits inside a snake like another snake," as the saying goes. Light, however, is known to be a wave. The only way out of the conundrum was for physicists to throw up their hands in despair and grudgingly accept that light is both a wave and a particle. But surely something cannot be simultaneously localised and spreadout? In the everyday world, this is perfectly true. Crucially, however, we are not talking about the everyday world; we are talking about the microscopic world.
The microscopic world of atoms and photons turns out to be nothing like the familiar realm of trees and clouds and people. Since it is a domain millions of times smaller than the realm of familiar objects, why should it be? Light really is both a particle and a wave. Or more correctly, light is "something else" for which there is no word in our everyday language and nothing to compare it with in the everyday world. Like a coin with two faces, all we can see are its particlelike face and its wavelike face. What light _actually_ is is as unknowable as the colour blue is to a blind man.
Light sometimes behaves like a wave and sometimes like a stream of particles. This was an extremely difficult thing for the physicists of the early 20th century to accept. But they had no choice; it was what nature was telling them. "On Mondays, Wednesdays and Fridays, we teach the wave theory and on Tuesdays, Thursdays and Saturdays the particle theory," joked the English physicist William Bragg in 1921.
Bragg's pragmatism was admirable. Unfortunately, it was not enough to save physics from disaster. As Einstein first realised, the dual wave-particle nature of light was a catastrophe. It was not just impossible to visualise, it was completely incompatible with all physics that had gone before.
## WAVING GOODBYE TO CERTAINTY
Take a window. If you look closely you can see a faint reflection of your face. This is because glass is not perfectly transparent. It transmits about 95 per cent of the light striking it while reflecting the remaining 5 per cent. If light is a wave, this is perfectly easy to understand. The wave simply splits into a big wave that goes through the window and a much smaller wave that comes back. Think of the bow wave from a speedboat. If it encounters a half-submerged piece of driftwood, a large part of the wave continues on its way while a small part doubles back on itself.
But while this behaviour is easy to understand if light is a wave, it is extremely difficult to understand if light is a stream of identical bulletlike particles. After all, if all the photons are identical, it stands to reason that each should be affected by the window in an identical way. Think of David Beckham taking a free kick over and over again. If the soccer balls are identical and he kicks each one in exactly the same way, they will all curl through the air and hit the same spot at the back of goal. It's hard to imagine the majority of the balls peppering the same spot while a minority flies off to the corner flag.
How, then, is it possible that a stream of absolutely identical photons can impinge on a window and 95 per cent can go right through while 5 per cent come back? As Einstein realised, there is only one way: if the word "identical" has a very different meaning in the microscopic world than in the everyday world—a diminished, cut-down meaning.
In the microscopic domain, it turns out, identical things do not behave in identical ways in identical circumstances. Instead, they merely have an identical _chance_ of behaving in any particular way. Each individual photon arriving at the window has exactly the same _chance_ of being transmitted as any of its fellows—95 per cent—and exactly the same _chance_ of being reflected—5 per cent. There is absolutely no way to know for certain what will happen to a given photon. Whether it is transmitted or reflected is entirely down to random chance.
In the early 20th century, this unpredictability was something radically new in the world. Imagine a roulette wheel and a ball jouncing around as the wheel spins. We think of the number the ball comes to rest on when the wheel finally halts as inherently unpredictable. But it is not—not really. If it were possible to know the initial trajectory of the ball, the initial speed of the wheel, the way the air currents changed from instant to instant in the casino, and so on, the laws of physics could be used to predict with 100 per cent certainty where the ball will end up. The same is true with the tossing of a coin. If it were possible to know how much force is applied in the flipping, the exact shape of the coin, and so on, the laws of physics could predict with 100 per cent certainty whether the coin will come down heads or tails.
Nothing in the everyday world is fundamentally unpredictable; nothing is truly random. The reason we cannot predict the outcome of a game of roulette or of the toss of a coin is that there is simply too much information for us to take into account. But in principle—and this is the key point—there is nothing to prevent us from predicting both.
Contrast this with the microscopic world of photons. It matters not the slightest how much information we have in our possession. It is impossible to predict whether a given photon will be transmitted or reflected by a window—even in principle. A roulette ball does what it does for a reason—because of the interplay of myriad subtle forces. A photon does what it does for no reason whatsoever! The unpredictability of the microscopic world is fundamental. It is truly something new under the Sun.
And what is true of photons turns out to be true of all the denizens of the microscopic realm. A bomb detonates because its timer tells it to or because a vibration disturbs it or because its chemicals have suddenly become degraded. An unstable, or "radioactive," atom simply detonates. There is absolutely no discernible difference between one that detonates at this moment and an identical atom that waits quietly for 10 million years before blowing itself to pieces. The shocking truth, which stares you in the face every time you look at a window, is that the whole Universe is founded on random chance. So upset was Einstein by this idea that he stuck out his lip and declared: "God does not play dice with the Universe!"
The trouble is He does. As British physicist Stephen Hawking has wryly pointed out: "Not only does God play dice with the Universe, he throws the dice where we cannot see them!"
When Einstein received the Nobel Prize for Physics in 1921 it was not for his more famous theory of relativity but for his explanation of the photoelectric effect. And this was no aberration on the part of the Nobel committee. Einstein himself considered his work on the "quantum" the only thing he ever did in science that was truly revolutionary. And the Nobel committee completely agreed with him.
Quantum theory, born out of the struggle to reconcile light and matter, was fundamentally at odds with all science that had gone before. Physics, pre-1900, was basically a recipe for predicting the future with absolute certainty. If a planet is in a particular place now, in a day's time it will have moved to another place, which can be predicted with 100 per cent confidence by using Newton's laws of motion and the law of gravity. Contrast this with an atom flying through space. Nothing is knowable with certainty. All we can ever predict is its probable path, its probable final position.
Whereas quantum is based on uncertainty, the rest of physics is based on certainty. To say this is a problem for physicists is a bit of an understatement! "Physics has given up on the problem of trying to predict what would happen in a given circumstance," said Richard Feynman. "We can only predict the odds."
All is not lost, however. If the microworld were totally unpredictable, it would be a realm of total chaos. But things are not this bad. Although what atoms and their like get up to is intrinsically unpredictable, it turns out that the unpredictability is at least predictable!
## PREDICTING THE UNPREDICTABILITY
Think of the window again. Each photon has a 95 per cent chance of being transmitted and a 5 per cent chance of being reflected. But what determines these probabilities?
Well, the two different pictures of light—as a particle and as a wave—must produce the same outcome. If half the wave goes through and half is reflected, the only way to reconcile the wave and particle pictures is if each individual particle of light has a 50 per cent probability of being transmitted and a 50 per cent probability of being reflected. Similarly, if 95 per cent of the wave is transmitted and 5 per cent is reflected, the corresponding probabilities for the transmission and reflection of individual photons must be 95 per cent and 5 per cent.
To get agreement between the two pictures of light, the particlelike aspect of light must somehow be "informed" about how to behave by its wavelike aspect. In other words, in the microscopic domain, waves do not simply behave like particles; those particles behave like waves as well! There is perfect symmetry. In fact, in a sense this statement is all you need to know about quantum theory (apart from a few details). Everything else follows unavoidably. All the weirdness, all the amazing richness of the microscopic world, is a direct consequence of this wave-particle "duality" of the basic building blocks of reality.
But how exactly does light's wavelike aspect inform its particle-like aspect about how to behave? This is not an easy question to answer.
Light reveals itself either as a stream of particles or as a wave. We never see both sides of the coin at the same time. So when we observe light as a stream of particles, there is no wave in existence to inform those particles about how to behave. Physicists therefore have a problem in explaining the fact that photons do things—for instance, fly through windows—as if directed by a wave.
They solve the problem in a peculiar way. In the absence of a real wave, they imagine an abstract wave—a mathematical wave. If this sounds ludicrous, this was pretty much the reaction of physicists when the idea was first proposed by the Austrian physicist Erwin Schrödinger in the 1920s. Schrödinger imagined an abstract mathematical wave that spread through space, encountering obstacles and being reflected and transmitted, just like a water wave spreading on a pond. In places where the height of the wave was large, the probability of finding a particle was highest, and in locations where it was small, the probability was lowest. In this way Schrödinger's wave of probability christened the wave function, informed a particle what to do, and not just a photon—any microscopic particle, from an atom to a constituent of an atom like an electron.
There is a subtlety here. Physicists could make Schrödinger's picture accord with reality only if the probability of finding a particle at any point was related to the square of the height of the probability wave at that point. In other words, if the probability wave at some point in space is twice as high as it is at another point in space, the particle is four times as likely to be found there than at the other place.
The fact that it is the square of the probability wave and not the probability wave itself that has real physical meaning to this day causes debate about whether the wave is a real thing lurking beneath the skin of the world or just a convenient mathematical device for calculating things. Most but not all people favour the latter.
The probability wave is crucially important because it makes a connection between the wavelike aspect of matter and familiar waves of all kinds, from water waves to sound waves to earthquake waves. All obey a so-called wave equation. This describes how they ripple through space and allows physicists to predict the wave height at any location at any time. Schrödinger's great triumph was to find the wave equation that described the behaviour of the probability wave of atoms and their like.
By using the Schrödinger equation, it is possible to determine the probability of finding a particle at any location in space at any time. For instance, it can be used to describe photons impinging on the obstacle of a windowpane and to predict the 95 per cent probability of finding one on the far side of the pane. In fact, the Schrödinger equation can be used to predict the probability of any particle, be it a photon or an atom, doing just about anything. It provides the crucial bridge to the microscopic world, allowing physicists to predict everything that happens there—if not with 100 per cent certainty, at least with predictable uncertainty!
Where is all this talk of probability waves leading? Well, the fact that waves behave like particles in the microscopic world leads unavoidably to the realisation that the microscopic world dances to an entirely different tune than that of the everyday world. It is governed by random unpredictability. This in itself was a shocking, confidence-draining blow to physicists and their belief in a predictable, clockwork universe. But this, it turns out, is only the beginning. Nature had many more shocks in store. The fact that waves not only behave as particles but also that those particles behave as waves leads to the realisation that all the things that familiar waves, like water waves and sound waves, can do, so too can the probability waves that inform the behaviour of atoms, photons, and their kin.
So what? Well, waves can do an awful lot of different things. And each of these things turns out to have a semi-miraculous consequence in the microscopic world. The most straightforward thing waves can do is exist as superpositions. Remarkably, this enables an atom to be in two places at once, the equivalent of you being in London and New York at the same time.
1 Another interesting characteristic of the photoelectric effect is that no electrons at all are emitted by the metal if it is illuminated by light with a wavelength—a measure of the distance between successive wave crests—above a certain threshold. This, as Einstein realised, is because photons of light have an energy that goes down with increasing wavelength. And below a certain wavelength the photons have insufficient energy to kick an electron out of the metal.
# THE SCHIZOPHRENIC ATOM
HOW AN ATOM CAN BE IN MANY PLACES AT ONCE AND DO MANY THINGS AT ONCE
_If you imagine the difference between an abacus and the world's fastest supercomputer, you would still not have the barest inkling of how much more powerful a quantum computer could be compared with the computers we have today_.
Julian Brown
_It's 2041. A boy sits at a computer in his bedroom. It's not an ordinary computer. It's a quantum computer. The boy gives the computer a task... and instantly it splits into thousands upon thousands of versions of itself, each of which works on a separate strand of the problem. Finally, after just a few seconds, the strands come back together and a single answer flashes on the computer display. It's an answer that all the normal computers in the world put together would have taken a trillion trillion years to find. Satisfied, the boy shuts the computer down and goes out to play, his night's homework done._
Surely, no computer could possibly do what the boy's computer has just done? Not only could a computer do such a thing, crude versions are already in existence today. The only thing in serious dispute is whether such a quantum computer merely behaves like a vast multiplicity of computers or whether, as some believe, it literally exploits the computing power of multiple copies of itself existing in parallel realities, or universes.
The key property of a quantum computer—the ability to do many calculations at once—follows directly from two things that waves—and therefore microscopic particles such as atoms and photons, which behave like waves—can do. The first of those things can be seen in the case of ocean waves.
On the ocean there are both big waves and small ripples. But as anyone who has watched a heavy sea on a breezy day knows, you can also get big, rolling waves with tiny ripples superimposed on them. This is a general property of all waves. If two different waves can exist, so too can a combination, or superposition, of the waves. The fact that superpositions can exist is pretty innocuous in the everyday world. However, in the world of atoms and their constituents, its implications are nothing short of earth-shattering.
Think again of a photon impinging on a windowpane. The photon is informed about what to do by a probability wave, described by the Schrödinger equation. Since the photon can either be transmitted or reflected, the Schrödinger equation must permit the existence of two waves—one corresponding to the photon going through the window and another corresponding to the photon bouncing back. Nothing surprising here. However, remember that, if two waves are permitted to exist, a superposition of them is also permitted to exist. For waves at sea such a combination is nothing out of the ordinary. But here the combination corresponds to something quite extraordinary—the photon being both transmitted and reflected. In other words, the photon can be on both sides of the windowpane simultaneously!
And this unbelievable property follows unavoidably from just two facts: that photons are described by waves and that superpositions of waves are possible.
This is no theoretical fantasy. In experiments it is actually possible to observe a photon or an atom being in two places at once—the everyday equivalent of you being in San Francisco and Sydney at the same time. (More accurately, it is possible to observe the _consequences_ of a photon or an atom being in two places at once.) And since there is no limit to the number of waves that can be superposed, a photon or an atom can be in three places, 10 places, a million places at once.
But the probability wave associated with a microscopic particle does more than inform it where it could be _located_. It informs it _how_ _to behave_ in all circumstances—telling a photon, for instance, whether or not to be transmitted or reflected by a pane of glass. Consequently, atoms and their like can not only be in many places at once, they can _do many things at once_ , the equivalent of you cleaning the house, walking the dog, and doing the weekly supermarket shopping all at the same time. This is the secret behind the prodigious power of a quantum computer. It exploits the ability of atoms to do many things at once, to do many calculations at once.
## DOING MANY THINGS AT ONCE
The basic elements of a conventional computer are transistors. These have two distinct voltage states, one of which is used to represent the binary digit, or bit, "0", the other to represent a "1." A row of such zeros and ones can represent a large number, which in the computer can be added, subtracted, multiplied, and divided by another large number.1 But in a quantum computer the basic elements—which may be single atoms—can be in a superposition of states. In other words, they can represent a zero and a one simultaneously. To distinguish them from normal bits, physicists call such schizophrenic entities quantum bits, or qubits.
One qubit can be in two states (0 or 1), two qubits in four (00 or 01 or 10 or 11), three qubits in eight, and so on. Consequently, when you calculate with a single qubit, you can do two calculations simultaneously, with two qubits four calculations, with three eight, and so on. If this doesn't impress you, with 10 qubits you could do 1,024 calculations all at once, with 100 qubits 100 billion billion billion! Not surprisingly, physicists positively salivate at the prospect of quantum computers. For some calculations, they could massively outperform conventional computers, making conventional personal computers appear positively retarded.
But for a quantum computer to work, wave superpositions are not sufficient on their own. They need another essential wave ingredient: interference.
The interference of light observed by Thomas Young in the 18th century was the key observation that convinced everyone that light was a wave. When, at the beginning of the 20th century, light was also shown to behave like a stream of particles, Young's double slit experiment assumed a new and unexpected importance—as a means of exposing the central peculiarity of the microscopic world.
## INTERFERENCE IS THE KEY
In the modern incarnation of Young's experiment, a double slit in an opaque screen is illuminated with light, which is undeniably a stream of particles. In practice, this means using a light source so feeble that it spits out photons one at a time. Sensitive detectors at different positions on the second screen count the arrival of photons. After the experiment has been running for a while, the detectors show something remarkable. Some places on the screen get peppered with photons while other places are completely avoided. What is more, the places that are peppered by photons and the places that are avoided alternate, forming vertical stripes—exactly as in Young's original experiment.
But wait a minute! In Young's experiment the dark and light bands are caused by interference. And a fundamental feature of interference is that it involves the mingling of two sets of waves from the same source—the light from one slit with the light from the other slit. But in this case the photons are arriving at the double slit one at a time. Each photon is completely alone, with no other photon to mingle with. How, then, can there be any interference? How can it know where its fellow photons will land?
There would appear to be only one way—if each photon somehow goes through both slits simultaneously. Then it can interfere with itself. In other words, each photon must be in a superposition of two states—one a wave corresponding to a photon going through the left-hand slit and the other a wave corresponding to a photon going through the right-hand slit.
The double slit experiment can be done with photons or atoms or any other microscopic particles. It shows graphically how the behaviour of such particles—where they can and cannot strike the second screen—is orchestrated by their wavelike alter ego. But this is not all the double slit experiment demonstrates. Crucially, it shows that the individual waves that make up a superposition are not passive but can actively interfere with each other. It is this ability of the individual states of a superposition to interfere with each other that is the absolute key to the microscopic world, spawning all manner of weird quantum phenomena.
Take quantum computers. The reason they can carry out many calculations at once is because they can exist in a superposition of states. For instance, a 10-element quantum computer is simultaneously in 1,024 states and can therefore carry out 1,024 calculations at once. But all the parallel strands of a calculation are of absolutely no use unless they get woven together. Interference is the means by which this is accomplished. It is the means by which the 1,024 states of the superposition can interact and influence each other. Because of interference, the single answer coughed out by the quantum computer is able to reflect and synthesise what was going on in all those 1,024 parallel calculations.
Think of a problem divided into 1,024 separate pieces and one person working on each piece. For the problem to be solved, the 1,024 people must communicate with each other and exchange results. This is what interference makes possible in a quantum computer.
An important point worth making here is that, although superpositions are a fundamental feature of the microscopic world, it is a curious property of reality that they are never actually observed. All we ever see are the consequences of their existence—what results when the individual waves of a superposition _interfere_ with each other. In the case of the double slit experiment, for instance, all we ever see is an interference pattern, from which we infer that an electron was in a superposition in which it went through both slits simultaneously. It is impossible to actually _catch_ an electron going through both slits at once. This is what was meant by the earlier statement that it is possible only to observe the _consequences_ of an atom being in two places at once, not it actually being in two places at once.
## MULTIPLE UNIVERSES
The extraordinary ability of quantum computers to do enormous numbers of calculations simultaneously poses a puzzle. Though practical quantum computers are currently at a primitive stage, manipulating only a handful of qubits, it is nevertheless possible to imagine a quantum computer that can do billions, trillions, or quadrillions of calculations simultaneously. In fact, it is quite possible that in 30 or 40 years we will be able to build a quantum computer that can do more calculations simultaneously than there are particles in the Universe. This hypothetical situation poses a sticky question: Where exactly will such a computer be doing its calculations? After all, if such a computer can do more calculations simultaneously than there are particles in the Universe, it stands to reason that the Universe has insufficient computing resources to carry them out.
One extraordinary possibility, which provides a way out of the conundrum, is that a quantum computer does its calculations in parallel realities or universes. The idea goes back to a Princeton graduate student named Hugh Everett III, who, in 1957, wondered why quantum theory is such a brilliant description of the microscopic world of atoms but we never actually see superpositions. Everett's extraordinary answer was that each state of the superposition exists in a totally separate reality. In other words, there exists a multiplicity of realities—a _multiverse_ —where all possible quantum events occur.
Although Everett proposed his "Many Worlds" idea long before the advent of quantum computers, it can shed some helpful light on them. According to the Many Worlds idea, when a quantum computer is given a problem, it splits into multiple versions of itself, each living in a separate reality. This is why the boy's quantum personal computer at the start of this chapter split into so many copies. Each version of the computer works on a strand of the problem, and the strands are brought together by interference. In Everett's picture, therefore, interference has a very special significance. It is the all-important _bridge_ between separate universes, the means by which they interact and influence each other.
Everett had no idea _where_ all the parallel universes were located. And, frankly, nor do the modern-day proponents of the Many Worlds idea. As Douglas Adams wryly observed in _The Hitchhiker's Guide to_ _the Galaxy:_ "There are two things you should remember when dealing with parallel universes. One, they're not really parallel, and two, they're not really universes!"
Despite such puzzles, half a century after Everett proposed the Many Worlds idea, it is undergoing an upsurge in popularity. An increasing number of physicists, most notably David Deutsch of the University of Oxford, are taking it seriously. "The quantum theory of parallel universes is not some troublesome, optional interpretation emerging from arcane theoretical considerations," says Deutsch in his book, _The Fabric of Reality_. "It is _the_ explanation—the only one that is tenable—of a remarkable and counterintuitive reality."
If you go along with Deutsch—and the Many Worlds idea predicts exactly the same outcome for every conceivable experiment as more conventional interpretations of quantum theory—then quantum computers are something radically new under the Sun. They are the very first machines humans have ever built that exploit the resources of multiple realities. Even if you do not believe the Many Worlds idea, it still provides a simple and intuitive way of imagining what is going on in the mysterious quantum world. For instance, in the double slit experiment, it is not necessary to imagine a single photon going through both slits simultaneously and interfering with itself. Instead, a photon going through one slit interferes with another photon going through the other slit. What other photon, you may ask? A photon in a neighbouring universe, of course!
## WHY ARE ONLY SMALL THINGS QUANTUM?
Quantum computers are extremely difficult to build. The reason is that the ability of the individual states of a quantum superposition to interfere with each other is destroyed, or severely degraded, by the environment. This destruction can be vividly seen in the double slit experiment.
If some kind of particle detector is used to spot a particle going through one of the slits, the interference stripes on the screen immediately vanish, to be replaced by more or less uniform illumination. The act of observing which slit the particle goes through is all that is needed to destroy the superposition in which it goes through both slits simultaneously. And a particle going through one slit only is as likely to exhibit interference as you are to hear the sound of one hand clapping.
What has really happened here is that an attempt has been made to locate, or measure, the particle by the outside world. Knowledge of the superposition by the outside world is all that is needed to destroy it. It is almost as if quantum superpositions are a secret. Of course, once the world knows about the secret, the secret no longer exists!
Superpositions are _continually_ being measured by their environment. And it takes only a single photon to bounce off a superposition and take information about it to the rest of the world to destroy the superposition. This process of natural measurement is called decoherence. It is the ultimate reason we do not see weird quantum behaviour in the everyday world.2 Although naively we may think of quantum behaviour as a property of small things like atoms but not of big things like people and trees, this is not necessarily so. Quantum behaviour is actually a property of isolated things. The reason we see it in the microscopic world but not in the everyday world is simply because it is easier to isolate a small thing from its surroundings than a big thing.
The price of quantum schizophrenia is therefore isolation. As long as a microscopic particle like an atom can remain isolated from the outside world, it can do many different things at once. This is not difficult in the microscopic world, where quantum schizophrenia is an everyday phenomenon. However, in the large-scale world in which we live, it is nearly impossible, with countless quadrillions of photons bouncing off every object every second.
Keeping a quantum computer isolated from its surroundings is the main obstacle facing physicists in trying to construct such a machine. So far, the biggest quantum computer they have managed to build has been composed of only 10 atoms, storing 10 qubits. Keeping 10 atoms isolated from their surroundings for any length of time takes all their ingenuity. If a single photon bounces off the computer, 10 schizophrenic atoms instantly become 10 ordinary atoms.
Decoherence illustrates a limitation of quantum computers not often publicised amid the hype surrounding such devices. To extract an answer, someone from the outside world—you—must interact with it, and this necessarily destroys the superposition. The quantum computer reverts to being an ordinary computer in a single state. A 10-qubit machine, instead of spitting out the answers to 1,024 separate calculations, spits out just one.
Quantum computers are therefore restricted to parallel calculations that output only a single answer. Consequently, only a limited number of problems are suited to solution by quantum computer, and much ingenuity is required to find them. They are not, as is often claimed, the greatest thing since sliced bread. Nevertheless, when a problem is found that plays to the strengths of a quantum computer, it can massively outperform a conventional computer, calculating in seconds what otherwise might take longer than the lifetime of the Universe.
On the other hand, decoherence, which is the greatest enemy of those struggling to build quantum computers, is also their greatest friend. It is because of decoherence, after all, that the giant superposition of a quantum computer with all its mutually interfering strands is finally destroyed; it is only by being destroyed—reduced to a single state representing a single answer—that anything useful comes out of such a machine. The world of the quantum is indeed a paradoxical one!
1 Binary was invented by the 17th-century mathematician Gottfried Leibniz. It is a way of representing numbers as a strings of zeros and ones. Usually, we use decimal, or base 10. The right-hand digit represents the ones, the next digit the tens, the next the 10 × 10s, and so on. So, for instance, 9,217 means 7 + 1 × 10 + 2 × (10 × 10) + 9 × (10 × 10 × 10). In binary, or base 2, the right-hand digit represents the ones, the next digit the twos, the next the 2 × 2s, and so on. So for instance, 1101 means 1 + 0 × 2 + 1 × (2 × 2) + 1 × (2 × 2 × 2), which in decimal is 13.
2 I am totally aware that all this talk of quantumness being a "secret" that is destroyed if the rest of the world learns about it is a complete fudge. But it is sufficient for our discussion here. Decoherence, the means by which the quantum world, with its schizophrenic superpositions, becomes the everyday world where trees and people are never in two places at once, is a can of worms with which the experts are still wrestling. For a real explanation, see Chapter 5, "The Telepathic Universe."
# UNCERTAINTY AND THE LIMITS OF KNOWLEDGE
WHY WE CAN NEVER KNOW ALL WE WOULD LIKE TO KNOW ABOUT ATOMS AND WHY THIS FACT MAKES ATOMS POSSIBLE
_Passing farther through the quantum land our travelers met quite a lot of other interesting phenomena, such as quantum mosquitoes, which could scarcely be located at all, owing to their small mass._
George Gamow
_He must be going mad. Only moments before he had parked his shiny_ _red Ferrari in the garage. He had even stood there on the driveway, admiring_ _his pride and joy until the last possible moment, as the automatic_ _door swung shut. But then as he crunched across the gravel to his_ _front door there had been a curious rustling of the air, a faint tremor of_ _the ground. He had wheeled round. And there, squatting back on his_ _driveway, in front of the still-locked garage doors, was his beautiful red_ _Ferrari!_
Such Houdini-like feats of escapology are never of course seen in the everyday world. In the realm of the ultrasmall, however, they are a common occurrence. One instant an atom can be locked up in a microscopic prison; the next it has shed its shackles and slipped away silently into the night.
This miraculous ability to escape escape-proof prisons is entirely due to the wavelike face of microscopic particles, which enables atoms and their constituents to do all the things that waves can do. And one of the many things waves can do is penetrate apparently impenetrable barriers. This is not an obvious or well-known wave property. But it can be demonstrated by a light beam travelling through a block of glass and trying to escape into the air beyond.
The key thing is what happens at the edge of the glass block, the boundary where the glass meets the air. If the light happens to strike the boundary at a shallow angle, it gets reflected back into the glass block and fails to escape into the air beyond. In effect, it is imprisoned in the glass. However, something radically different happens if another block of glass is brought close to the boundary, leaving a small gap of air between the two blocks. Just as before, some of the light is reflected back into the glass. But—and this is the crucial thing—some of the light now leaps the air gap and travels into the second glass block.
The parallel between the Ferrari escaping its garage and the light escaping the block of glass may not be very obvious. However, for all intents and purposes, the air gap should be just as impenetrable a barrier to the light as the garage walls are to the Ferrari.
The reason the light wave can penetrate the barrier and escape from the block of glass is that a wave is not a localised thing but something spread out through space. So when the light waves strike the glass-air boundary and are reflected back into the glass, they are not actually reflected from the exact boundary of the glass. Instead, they penetrate a short distance into the air beyond. Consequently, if they encounter another block of glass before they can turn back, they can continue on their way. Place a second glass block within a hair's breadth of the first and, hey presto, the light jumps the air gap and escapes its prison.
This ability to penetrate an apparently impenetrable barrier is common to all types of waves, from light waves to sound waves to the probability waves associated with atoms. It therefore manifests itself in the microscopic world. Arguably, the most striking example is the phenomenon of alpha decay in which an alpha particle breaks out of the apparently escape-proof prison of an atomic nucleus.
## BREAKING OUT OF A NUCLEUS
An alpha particle is the nucleus of a helium atom. An unstable, or radioactive, nucleus sometimes spits out an alpha particle in a desperate attempt to turn itself into a lighter and more stable nucleus. The process poses a big puzzle, however. By rights, an alpha particle should not be able to get out of a nucleus.
Think of an Olympic high jumper penned in by a 5-metre-high metal fence. Even though he is one of the best high jumpers in the world, there is no way he can jump over a fence that high. No human being alive has sufficient strength in their legs. Well, an alpha particle inside an atomic nucleus finds itself in a similar position. The barrier that pens it in is created by the nuclear forces that operate inside a nucleus, but it is just as impenetrable a barrier to the alpha particle as the solid metal fence is to the high jumper.
Contrary to all expectations, however, alpha particles do escape from atomic nuclei. And their escape is entirely due to their wavelike face. Like light waves trapped in a glass block, they can penetrate an apparently impenetrable barrier and slip away quietly into the outside world.
This process is called quantum tunnelling and alpha particles are said to "tunnel" out of an atomic nucleus. Tunnelling is actually an instance of a more general phenomenon known as uncertainty, which puts a fundamental limit on what we can and cannot know about the microscopic world. The double slit experiment is an excellent demonstration of uncertainty.
## THE HEISENBERG UNCERTAINTY PRINCIPLE
The reason a microscopic particle like an electron can go through both slits in the screen simultaneously is that it can exist as a superposition of two waves—one wave corresponding to the particle going through one slit and the other to the particle going through the other slit. But that is not sufficient to guarantee that its schizophrenic behaviour will be noticed. For that to happen, an interference pattern must appear on the second screen. But this, of course, requires the individual waves in the superposition to interfere. The fact that interference is a crucial ingredient for the electron to exhibit weird quantum behaviour turns out to have profound implications for what nature permits us to know about the electron.
Say in the double slit experiment we try to locate the slit each electron goes through. If we succeed, the interference pattern on the second screen disappears. After all, interference requires that two things mingle. If the electron and its associated probability wave go through only one slit, there is only one thing.
How, in practice, could we locate which slit an electron goes through? Well, to make the double slit experiment a bit easier to visualise, think of an electron as a bullet from a machine gun and the screen as a thick metal sheet with two vertical parallel slits. When bullets are fired at the screen, some enter the slits and go through. Think of the slits as deep channels cut through the thick metal. The bullets ricochet off the internal walls of the channels and by this means reach the second screen. They can obviously hit any point on the second screen. But, for simplicity, imagine they end up at the midpoint of the second screen. Also for simplicity, say that at this point the probability waves associated with the bullets interfere constructively, so it is a place that gets peppered with lots of bullets.
Now, when a bullet ricochets off the inside of a slit, it causes the metal screen to recoil in the opposite direction. It's the same if you are playing tennis and a fast serve ricochets off your racquet. Your racquet recoils in the opposite direction. Crucially, the recoil of the screen can be used to deduce which slit a bullet goes through. After all, if the screen moves to the left, the bullet must have gone through the left-hand slit; if it moves to the right, it must have been the right-hand slit.
However, we know that if we locate which slit a bullet goes through, it destroys the interference pattern on the second screen. This is straightforward to understand from the wave point of view. We are as unlikely to see one thing interfere with itself as we are to hear the sound of one hand clapping. But how do we make sense of things from the equally valid particle point of view?
Remember that the interference pattern on the second screen is like a supermarket bar code. It consists of vertical "stripes" where no bullets hit, alternating with vertical stripes where lots of bullets hit. For simplicity, think of the stripes as black and white. The key question therefore is: From the bullet's point of view, what would it take to destroy the interference pattern?
The answer is a little bit of sideways jitter. If each bullet, instead of flying unerringly towards a black stripe, possesses a little sideways jitter in its trajectory so that it can hit either the black stripe or an adjacent white stripe, this will be sufficient to "smear out" the interference pattern. Stripes that were formerly white will become blacker, and stripes that were formerly black will become whiter. The net result will be a uniform gray. The interference pattern will be smeared out.
Because it must be impossible to tell whether a given bullet will hit a black stripe or an adjacent white stripe (or vice versa), the jittery sideways motion of each bullet must be entirely unpredictable. And all this must come to pass for no other reason than that we are locating which slit each bullet goes through by the recoil of the screen.
In other words, the very act of pinning down the location of a particle like an electron adds unpredictable jitter, making its velocity uncertain. And the opposite is true as well. The act of pinning down the velocity of a particle makes its location uncertain. The first person to recognise and quantify this effect was the German physicist Werner Heisenberg, and it is called the Heisenberg uncertainty principle in his honour.
According to the uncertainty principle, it is impossible to know both the location and the velocity of a microscopic particle with complete certainty. There is a trade-off, however. The more precisely its location is pinned down, the more uncertain is its velocity. And the more precisely its velocity is pinned down, the more uncertain its location.
Imagine if this constraint also applied to what we could know about the everyday world. If we had precise knowledge of the speed of a jet aeroplane, we would not be able to tell whether it was over London or New York. And if we had precise knowledge of the location of the aeroplane, we would be unable to tell whether it was cruising at 1,000 kilometres per hour or 1 kilometre per hour—and about to plummet out of the sky.
The uncertainty principle exists to _protect_ quantum theory. If you could measure the properties of atoms and their like better than the uncertainty principle permits, you would destroy their wave behaviour—specifically, interference. And without interference, quantum theory would be impossible. Measuring the position and velocity of a particle with greater accuracy than the uncertainty principle dictates must therefore be impossible. Because of the Heisenberg uncertainty principle, when we try to look closely at the microscopic world, it starts to get fuzzy, like a newspaper picture that has been overmagnified. Infuriatingly, nature does not permit us to measure precisely all we would like to measure. There is a limit to our knowledge.
This limit is not simply a quirk of the double slit experiment. It is fundamental. As Richard Feynman remarked: "No one has ever found (or even thought of) a way around the uncertainty principle. Nor are they ever likely to."
It is because alpha particles have a wavelike character that they can escape the apparently escape-proof prison of an atomic nucleus.
However, the Heisenberg uncertainty principle makes it possible to understand the phenomenon from the particle point of view.
## GOING WHERE NO HIGH JUMPER HAS GONE BEFORE
Recall that an alpha particle in a nucleus is like an Olympic high jumper corralled by a 5-metre-high fence. Common sense says that it is moving about inside the nucleus with insufficient speed to launch itself over the barrier. But common sense applies only to the everyday world, not to the microscopic world. Ensnared in its nuclear prison, the alpha particle is very localised in space—that is, its position is pinned down with great accuracy. According to the Heisenberg uncertainty principle, then, its velocity must necessarily be very uncertain. It could, in other words, be much greater than we think. And if it is greater, then, contrary to all expectations, the alpha particle can leap out of the nucleus—a feat comparable to the Olympic high jumper jumping the 5-metre fence.
Alpha particles emerge into the world outside their prison as surprisingly as the Ferrari emerged into the world outside its garage. And this "tunnelling" is due to the Heisenberg uncertainty principle. But tunnelling is a two-way process. Not only can subatomic particles like alpha particles tunnel out of a nucleus, they can tunnel into it too. In fact, such tunnelling in reverse helps explain a great mystery: why the Sun shines.
## TUNNELLING IN THE SUN
The Sun generates heat by gluing together protons—the nuclei of hydrogen atoms—to make the nuclei of helium atoms.1 This nuclear fusion produces as a by-product a dam burst of nuclear binding energy, which ultimately emerges from the Sun as sunlight.
But hydrogen fusion has a problem. The force of attraction that glues together protons—the "strong nuclear force"—has an extremely short range. For two protons in the Sun to come under its influence and be snapped together, they must pass extremely close to each other. But two protons, by virtue of their similar electric charge, repel each other ferociously. To overcome this fierce repulsion, the protons must collide at enormous speed. In practice, this requires the core of the Sun, where nuclear fusion goes on, to be at an extremely high temperature.
Physicists calculated the necessary temperature in the 1920s, just as soon as it was suspected that the Sun was running on hydrogen fusion. It turned out to be roughly 10 billion degrees. This, however, posed a problem. The temperature at the heart of the Sun was known to be only about 15 million degrees—roughly a thousand times lower. By rights, the Sun should not be shining at all. Enter the German physicist Fritz Houtermans and the English astronomer Robert Atkinson.
When a proton in the core of the Sun approaches another proton and is pushed back by its fierce repulsion, it is just as if it encounters a high brick wall surrounding the second proton. At the 15 million degrees temperature in the heart of the Sun, the proton would appear to be moving far too slowly to jump the wall. However, the Heisenberg uncertainty principle changes everything.
In 1929, Houtermans and Atkinson carried out the relevant calculations. They discovered that the first proton can tunnel through the apparently impenetrable barrier around the second proton and successfully fuse with it even at the ultralow temperature of 15 million degrees. What is more, this explains perfectly the observed heat output of the Sun.
The night after Houtermans and Atkinson did the calculation, Houtermans reportedly tried to impress his girlfriend with a line that nobody in history had used before. As they stood beneath a perfect moonless sky, he boasted that he was the only person in the world who knew why the stars were shining. It must have worked. Two years later, Charlotte Riefenstahl agreed to marry him. (Actually, she married him twice, but that's another story.)
Sunlight apart, the Heisenberg uncertainty principle explains something much closer to home: the very existence of the atoms in our bodies.
## UNCERTAINTY AND THE EXISTENCE OF ATOMS
By 1911 the Cambridge experiments of New Zealand physicist Ernest Rutherford had revealed the atom as resembling a miniature solar system. Tiny electrons flitted about a compact atomic nucleus much like planets around the Sun. However, according to Maxwell's theory of electromagnetism, an orbiting electron should radiate light energy and, within a mere hundred-millionth of a second, spiral into the nucleus. "Atoms," as Richard Feynman pointed out, "are completely impossible from the classical point of view." But atoms do exist. And the explanation comes from quantum theory.
An electron cannot get too close to a nucleus because, if it did, its location in space would be very precisely known. But according to the Heisenberg uncertainty principle, this would mean that its velocity would be very uncertain. It could become enormously huge.
Imagine an angry bee in a shrinking box. The smaller the box gets, the angrier the bee and the more violently it batters itself against the walls of its prison. This is pretty much the way an electron behaves in an atom. If it were squeezed into the nucleus itself, it would acquire an enormous speed—far too great to stay confined in the nucleus.
The Heisenberg uncertainty principle, which explains why electrons do not spiral into their nuclei, is therefore the ultimate reason why the ground beneath our feet is solid. But the principle does more than simply explain the existence of atoms and the solidity of matter. It explains why atoms are so big—or at least so much bigger than the nuclei at their cores.
## WHY ATOMS ARE SO BIG
Recall that a typical atom is about 100,000 times bigger than the nucleus at its centre. Understanding why there is such a fantastic amount of empty space in atoms requires being a bit more precise about the Heisenberg uncertainty principle. Strictly speaking, it says that it is a particle's position and momentum—rather than just its velocity—that cannot simultaneously be determined with 100 per cent certainty.
The momentum of a particle is the product of its mass and velocity. It's really just a measure of how difficult it is to stop something that is moving. A train, for instance, has a lot of momentum compared to a car, even if the car is going faster. A proton in an atomic nucleus is about 2,000 times more massive than an electron. According to the Heisenberg uncertainty principle, then, if a proton and an electron are confined in the same volume of space, the electron will be moving about 2,000 times faster.
Already, we get an inkling of why the electrons in an atom must have a far bigger volume to fly about in than the protons and neutrons in the nucleus. But atoms are not just 2,000 times bigger than their nuclei; they are more like 100,000 times bigger. Why?
The answer is that an electron in an atom and a proton in a nucleus are not in the grip of the same force. While the nuclear particles are held by the powerful "strong nuclear" force, the electrons are held by the much weaker electric force. Think of the electrons flying about the nucleus attached to gossamer threads of elastic while the protons and the neutrons are constrained by elastic 50 times thicker. Here is the explanation for why the atom is a whopping 100,000 times bigger than the nucleus.
But the electrons in an atom do not orbit at one particular distance from the nucleus. They are permitted to orbit at a range of distances. Explaining this requires resorting to yet another wave picture—this one involving organ pipes!
## OF ATOMS AND ORGAN PIPES
There are always many different ways of looking at things in the quantum world, each a glimpse of a truth that is frustratingly elusive. One way is to think of the probability waves associated with an atom's electrons as being like sound waves confined to an organ pipe. It is not possible to make just any note with the organ pipe. The sound can vibrate in only a limited number of different ways, each with a definite pitch, or frequency.
This turns out to be a general property of waves, not just sound waves. In a confined space they can exist only at particular, definite frequencies.
Now think of an electron in an atom. It behaves like a wave. And it is gripped tightly by the electrical force of the atomic nucleus. This may not be exactly the same as being trapped in a physical container. However, it confines the electron wave as surely as the wall of an organ pipe confines a sound wave. The electron wave can therefore exist at only certain frequencies.
The frequencies of the sound waves in an organ pipe and of the electron waves in an atom depend on the characteristics of the organ pipe—a small organ pipe, for instance, produces higher-pitched notes than a big organ pipe—and on the characteristics of the electrical force of the atomic nucleus. In general, though, there is lowest, or fundamental, frequency and a series of higher-frequency "overtones."
A higher-frequency wave has more peaks and troughs in a given space. It is choppier, more violent. In the case of an atom, such a wave corresponds to a faster-moving, more energetic electron. And a faster-moving, more energetic electron is able to defy the electrical attraction of the nucleus and orbit farther away.
The picture that emerges is of an electron that is permitted to orbit at only certain special distances from the nucleus. This is quite unlike our solar system where a planet such as Earth could, in principle, orbit at any distance whatsoever from the Sun.
This property highlights another important difference between the microscopic world of atoms and the everyday world. In the everyday world, all things are continuous—a planet can orbit the Sun anywhere it likes, people can be any weight they like—whereas things in the microscopic world are discontinuous—an electron can exist in only certain orbits around a nucleus, light and matter can come in only certain indivisible chunks. Physicists call the chunks quanta—which is why the physics of the microscopic world is known as quantum theory.
The innermost orbit of an electron in an atom is determined by the Heisenberg uncertainty principle—by its hornetlike resistance to being confined in a small space. But the Heisenberg uncertainty principle does not simply prevent small things like atoms from shrinking without limit—ultimately explaining the solidity of matter. It also prevents far bigger things from shrinking without limit. The far bigger things in question are stars.
## UNCERTAINTY AND STARS
A star is a giant ball of gas held together by the gravitational pull of its own matter. That pull is constantly trying to shrink the star and, if unopposed, would very quickly collapse it down to the merest speck—a black hole. For the Sun this would take less than half an hour. Since the Sun is very definitely not shrinking down to a speck, there must be another force counteracting gravity. There is. It comes from the hot matter inside. The Sun—along with every other normal star—is in a delicate state of balance, with the inward force of gravity exactly matched by the outward force of its hot interior.
This balance, however, is temporary. The outward force can be maintained only while there is fuel to burn and keep the star hot. Sooner or later, the fuel will run out. For the Sun this will occur in about another 5 billion years. When this happens, gravity will be king. Unopposed, it will crush the star, shrinking it ever smaller.
But all is not lost. In the dense, hot environment inside a star, frequent and violent collisions between high-speed atoms strip them of their electrons, creating a plasma, a gas of atomic nuclei mixed in with a gas of electrons. It is the tiny electrons that unexpectedly come to the rescue of the fast-shrinking star. As the electrons in the star's matter are jammed ever closer together, they buzz about ever more violently because of the Heisenberg uncertainty principle. They batter anything trying to confine them, and this collective battering results in a tremendous outward force. Eventually, it is enough to slow and halt the shrinkage of the star.
A new balance is struck with the inward pull of gravity balanced not by the outward force of the star's hot matter but by the naked force of its electrons. Physicists call it degeneracy pressure. But it's just a fancy term for the resistance of electrons to being squeezed too close together. A star supported against gravity by electron pressure is known as a white dwarf. Little more than the size of Earth and occupying about a millionth of the star's former volume, a white dwarf is an enormously dense object. A sugarcube of its matter weighs as much as a car!
One day the Sun will become a white dwarf. Such stars have no means of replenishing their lost heat. They are nothing more than stellar embers, cooling inexorably and gradually fading from view. But the electron pressure that prevents white dwarfs from shrinking under their own gravity has its limits. The more massive a star, the stronger its self-gravity. If the star is massive enough, its gravity will be powerful enough to overcome even the stiff resistance of the star's electrons.
In fact, the star is sabotaged from both outside and inside. The stronger the gravity of a star, the more it squeezes the gas inside. And the more a gas is squeezed, the hotter it gets, as anyone who has used a bicycle pump knows. Since heat is nothing more than the microscopic jiggling of matter, the electrons inside the star fly about ever faster—so fast, in fact, that the effects of relativity become important.2 The electrons get more massive rather than much faster, which means they are less effective at battering the walls of their prison.
The star suffers a double whammy—crushed by stronger gravity and simultaneously robbed of the ability to fight back. The two effects combine to ensure that the heaviest a white dwarf can be is a mere 40 per cent more massive than the Sun. If a star is heavier than this "Chandrasekhar limit", electron pressure is powerless to halt its headlong collapse and it just goes on shrinking.
But, once again, all is not lost. Eventually, the star shrinks so much that its electrons, despite their tremendous aversion to being confined in a small volume, are actually squeezed into the atomic nuclei. There they react with protons to form neutrons, so that the whole star becomes one giant mass of neutrons.
Recall that all particles of matter—not just electrons—resist being confined because of the Heisenberg uncertainty principle. Neutrons are thousands of times more massive than electrons. They therefore have to be squeezed into a volume thousands of times smaller to begin to put up significant resistance. In fact, they have to be squeezed together until they are virtually touching before they finally halt the shrinkage of the star.
A star supported against gravity by neutron degeneracy pressure is known as a neutron star. In effect, it is a huge atomic nucleus with all the empty space squeezed out of its matter. Since atoms are mostly empty space, with their nuclei 100,000 times smaller than their surrounding cloud of orbiting electrons, neutron stars are 100,000 times smaller than a normal star. This makes them only about 15 kilometres across, not much bigger than Mount Everest. So dense is a neutron star that a sugarcube of its matter weighs as much as the entire human race. (This, of course, is an illustration of just how much empty space there is in all of us. Squeeze it all out and humanity would fit in your hand.)
Such stars are thought to form violently in supernova explosions. While the outer regions of a star are blown into space, the inner core shrinks to form a neutron star. Neutron stars, being tiny and cold, ought to be difficult to spot. However, they are born spinning very fast and produce lighthouse beams of radio waves that flash around the sky. Such pulsating neutron stars, or simply pulsars, semaphore their existence to astronomers.
## UNCERTAINTY AND THE VACUUM
White dwarfs and neutron stars apart, perhaps the most remarkable consequence of the Heisenberg uncertainty principle is the modern picture of empty space. It simply cannot be empty!
The Heisenberg uncertainty principle can be reformulated to say that it is impossible to simultaneously measure the energy of a particle and the interval of time for which it has been in existence. Consequently, if we consider what happens in a region of empty space in a very tiny interval of time, there will be a large uncertainty in the energy content of that region. In other words, energy can appear out of nothing!
Now, mass is a form of energy.3 This means that mass too can appear out of nothing. The proviso is that it can appear only for a mere split second before disappearing again. The laws of nature, which usually prevent things from appearing out of nothing, appear to turn a blind eye to events that happen too quickly. It's rather like a teenager's dad not noticing his son has borrowed the car for the night as long as it gets put back in the garage before daybreak.
In practice, mass is conjured out of empty space in the form of microscopic particles of matter. The quantum vacuum is actually a seething morass of microscopic particles such as electrons popping into existence and then vanishing again.4 And this is no mere theory. It actually has observable consequences. The roiling sea of the quantum vacuum actually buffets the outer electrons in atoms, very slightly changing the energy of the light they give out.5
The fact that the laws of nature permit something to come out of nothing has not escaped cosmologists, people who think about the origin of the Universe. Could it be, they wonder, that the entire Universe is nothing more than a quantum fluctuation of the vacuum? It's an extraordinary thought.
1 See Chapter 8, " _E = mc_ 2 and the Weight of Sunshine."
2 See Chapter 7, "The Death of Space and Time."
3 See Chapter 8, " _E = mc_ 2 and the Weight of Sunlight."
4 Actually, every particle created is created alongside its antiparticle, a particle with opposite properties. So a negatively charged electron is always created with a positively charged positron.
5 This effect is called the Lamb shift.
# THE TELEPATHIC UNIVERSE
HOW ATOMS CAN INFLUENCE EACH OTHER INSTANTLY EVEN WHEN ON OPPOSITE SIDES OF THE UNIVERSE
_Beam me up, Mr. Scott_.
Captain James T. Kirk
_A coin is spinning. The coin is in a strong box sitting in the mud at the_ _bottom of the deepest ocean trench. Don't ask what has set the coin spinning_ _or what is keeping it spinning. This isn't a well-thought-out story!_ _The point is that there is an identical spinning coin in an identical box_ _sitting on a cold moon in a distant galaxy on the far side of the Universe._
_The first coin comes down heads. Instantaneously, without the merest_ _split-second of delay, its cousin 10 billion light-years from Earth_ _comes down tails._
The coin on Earth could equally well have come down tails and its distant cousin heads. This is not important. The significant thing is that the coin on the far side of the Universe _knows_ instantly the state of its distant terrestrial cousin—and does the opposite.
But how can it possibly know? The cosmic speed limit in our Universe is the speed of light.1 Since the coins are separated by 10 billion light-years, information about the state of one coin must take a minimum of 10 billion years to reach the other. Yet they know about each other in a split second.
This kind of "spooky action at a distance" turns out to be one of the most remarkable features of the microscopic world. It so upset Einstein that he declared that quantum theory must be wrong. In fact, Einstein was wrong.
In the past 20 years, physicists have observed the behaviour of coins that are separated by large distances. The coins are quantum coins, and the distances are not of course as large as the width of the Universe.2 Nevertheless, the experiments have successfully demonstrated that atoms and their kin can indeed communicate instantaneously, in total violation of the speed-of-light barrier.
Physicists have christened this weird kind of quantum telepathy nonlocality. The best way to understand it is by considering a peculiar particle property called spin.
## SPOOKY ACTION AT A DISTANCE
Spin is unique to the microscopic world. Particles that possess spin behave as if they are rotating like tiny spinning tops. Only they aren't actually spinning! Once again, we come up against the fundamental ungraspability of the microscopic world. The spin of particles, like their inherent unpredictability, is something with no direct analogue in the everyday world. Microscopic particles can have different amounts of spin. The electron happens to carry the minimum quantity. This permits it to spin in two possible ways. Think of it as spinning either clockwise or anticlockwise (although of course it isn't actually spinning at all!).
If two electrons are created together—the first with clockwise spin, the second with anticlockwise spin—their spins cancel. Physicists say their total spin is zero. Of course, the pair of electrons can also have a total spin of zero if the first electron has an anticlockwise spin and the second a clockwise spin.
Now, there is a law of nature that says the total spin of such a system can never change. (It's actually called the law of conservation of angular momentum.) So once the pair of electrons has been created with a total spin of zero, the pair's spin must remain zero as long as the pair remains in existence.
Nothing out of the ordinary here. However, there is another way to create two electrons with a total spin of zero. Recall that, if two states of a microscopic system are possible, then a superposition of the two is also possible. This means it is possible to create a pair of electrons that are simultaneously clockwise-anticlockwise and anticlockwise-clockwise.
So what? Well, remember that such a superposition can exist only as long as the pair of electrons is isolated from its environment. The moment the outside world interacts with it—and that interaction could be someone checking to see what the electrons are doing—the superposition undergoes decoherence and is destroyed. Unable to exist any longer in their schizophrenic state, the electrons plump for being either clockwise-anticlockwise or anticlockwise-clockwise.
Still nothing out of the ordinary (at least for the microscopic world!). However, imagine that, after the electrons are created in their schizophrenic state, they remain isolated and nobody looks at them. Instead, one electron is taken away in a box to a faraway place. Only then does someone finally open the box and observe the spin of the electron.
If the electron at the faraway place turns out to have a clockwise spin, then instantaneously the other electron must stop being in its schizophrenic state and assume an anticlockwise spin. The total spin, after all, must always remain zero. If, on the other hand, the electron turns out to be spinning anticlockwise, its cousin must instantaneously assume a clockwise spin.
It does not matter if one electron is in a steel box half-buried on the seafloor and the other is in a box on the far side of the Universe. One electron will respond instantaneously to the other's state. This is not merely some esoteric theory. Instantaneous influence has actually been observed in the laboratory.
In 1982, Alain Aspect and his colleagues at the University of Paris South created pairs of photons and sent members of each pair to detectors separated by a distance of 13 metres. The detectors measured the polarisation of the photons, a property related to their spin. Aspect's team showed that measuring the polarisation of photons at one detector affected the polarisation measured at the other detector. The influence that travelled between the detectors did so in less than 10 nanoseconds. Crucially, this was a quarter of the time a light beam would have taken to bridge the 13-metre gap.
At the bare minimum, some kind of influence travelled between the detectors at four times the speed of light. If the technology had made it possible to measure an even smaller time interval, Aspect could have shown the ghostly influence to be even faster. Quantum theory was right. And Einstein—bless him—was wrong.
Nonlocality could never happen in the ordinary, nonquantum world. An air mass might split into two tornadoes, one spinning clockwise and the other anticlockwise. But that's the way they would stay—spinning in opposite directions—until finally they both ran out of steam. The crucial difference in the microscopic, quantum world is that the spins of particles are undetermined until the instant they are observed. And, before the spin of one electron in the pair is observed, it is totally unpredictable. It has a 50 per cent chance of being clockwise and a 50 per cent chance of being anticlockwise (once again we come up against the naked randomness of the microworld). But even though there is no way of knowing the spin of one electron until it is observed, the spin of the other electron must settle down to being opposite instantaneously—no matter how far away the other particle happens to be.
## ENTANGLEMENT
At the heart of nonlocality is the tendency of particles that interact with each other to become entwined, or "entangled", so that the properties of one are forever dependent on the properties of the other. In the case of the pair of electrons, it is their spins that become dependent on each other. In a very real sense, entangled particles cease to have a separate existence. Like a much-in-love couple, they become a weird joined-at-the-hip entity. No matter how far apart they are pulled, they remain forever connected.
The weirdest manifestation of entanglement is, without doubt, nonlocality. In fact, it would seem that if we could harness it we could create an instantaneous communications system. With it we could phone the other side of the world with no time delay. In fact, we could phone the other side of the Universe with no time delay! No longer would we need to be inconvenienced by the pesky speed-of-light barrier.
Frustratingly, however, nonlocality cannot be harnessed to create an instantaneous communications system. Attempts to use the spin of particles to send a message across large distances might use one direction of spin to code for a "0" and the other for a "1." However, to know that you were sending a "0" or a "1," you would have to check the spin of the particle. But checking kills the superposition, which is essential for the instantaneous effect. If you sent a message without first looking, you could be only 50 per cent sure of sending a "1," a level of uncertainty that effectively scrambles any meaningful message.
So although instantaneous influence is a fundamental feature of our Universe, it turns out that nature does exactly what is required to make it unusable for sending real information. This is how it permits the speed-of-light barrier to be broken without actually being broken. What nature gives with one hand it cruelly takes away with the other.
## TELEPORTATION
Arguably, the sexiest potential use of entanglement involves taking an object and sending a complete description of the object to a faraway place so that a suitably clever machine at the other end can construct a perfect copy. This is of course the recipe for the _Star Trek_ transporter, which routinely "beamed" crew members back and forth between planet and ship.
The technology to reconstruct a solid object merely from the information describing it is of course way beyond our current technological capabilities. But, actually, the idea of creating a perfect copy of an object at a remote location founders on something much more basic than this. According to the Heisenberg uncertainty principle, it is impossible to perfectly describe an object—the positions of all its atoms, the electrons in each of those atoms, and so on. Without such knowledge, however, how can an exact copy ever be assembled?
Entanglement, remarkably, offers a way out. The reason is that entangled particles behave like a single indivisible entity. At some level, they _know_ each other's deepest secrets.
Say we have a particle, P, and we want to make a perfect copy, P*. It stands to reason that in order to do this it is necessary to know P's properties. However, according to the Heisenberg uncertainty principle, if we measure one particular property of P precisely—say its location—we inevitably lose all knowledge of some other property—in this case, its velocity. Nevertheless, this crippling limitation can be circumvented by an ingenious use of entanglement.
Take another particle, A, which is similar to both P and P*. The important thing is that A and P* are an entangled pair. Now, entangle A with P and make a measurement of the pair together. This will tell us about some property of P. According to the Heisenberg uncertainty principle, however, the measurement will inevitably involve us losing knowledge of some other property of P.
But all is not lost. Because P* was entangled with A, it retains knowledge about A. And because A was entangled with P, it retains knowledge about P. This means that P*, though it has never been in touch with P, nevertheless knows its secrets. Furthermore, when the measurement was made on A and P together and information about some property of P seemed to be lost, instantaneously it became available to A's partner, P*. This is the miracle of entanglement.
Since we already know about the other properties of P, obtained from A, we now have all we need to make sure P* has _exactly_ the attributes of P.3 Thus we have exploited entanglement to circumvent the restrictions of the Heisenberg uncertainty principle.
The amazing thing is that, although we have exploited entanglement to make a particle P* with the exact properties of P, at no time did we ever possess any information about the missing property of P! It was transmitted out of our sight through the ghostly connections of entanglement.4
Calling this scheme teleportation is a bit of a cheeky exaggeration since it solves only one of the many problems in making a _Star_ _Trek_ transporter. The researchers of course knew this. But they also knew a thing or two about how to grab newspaper headlines!
As it happens, the Achilles' heel of the _Star Trek_ transporter turns out to be neither pinning down the position, and so on, of every atom in a person's body nor assembling a copy of the person from that information. It's actually _transmitting_ the sheer volume of information needed to describe a person across space. Billions of times more information is needed than for the reconstruction of a two-dimensional TV image. The obvious way to send the information is as a series of binary "bits"—dots and dashes. If the information is to be sent in a reasonable time, the pulses must obviously be short. But ultrashort pulses are possible only with ultra-high-energy light. As science fiction writer Arthur C. Clarke has pointed out, beaming up Captain Kirk could easily take more energy than there is in a small galaxy of stars!
Teleportation and nonlocality aside, the most mind-blowing consequence of entanglement is what it means for the Universe as a whole. At one time, all particles in the Universe were in the same state because all particles were together in the Big Bang. Consequently, all particles in the Universe are to some extent entangled with each other.
There is a ghostly web of quantum connections crisscrossing the Universe and coupling you and me to every last bit of matter in the most distant galaxy. We live in a telepathic universe. What this actually means physicists have not yet figured out.
Entanglement may also help explain the outstanding question posed by quantum theory: Where does the everyday world come from?
## WHERE DOES THE EVERYDAY WORLD COME FROM?
According to quantum theory, weird superpositions of states are not only possible but guaranteed. An atom can be in many places at once or do many things at once. It is the interference between these possibilities that leads directly to many of the bizarre phenomena observed in the microscopic world. But why is it that, when large numbers of atoms club together to form everyday objects, those objects never display quantum behaviour? For instance, trees never behave as if they are in two places at once and no animal behaves as if it is a combination of a frog and a giraffe.
The first attempt to explain the conundrum was made in Copenhagen in the 1920s by quantum pioneer Niels Bohr. The Copenhagen Interpretation, in effect, divides the Universe into two domains, ruled by different laws. On the one hand, there is the domain of the very small, which is ruled by quantum theory, and on the other there is the domain of the very big, ruled by normal, or classical, laws. According to the Copenhagen Interpretation, it is when a quantum object like an atom interacts with a classical object that it is forced to stop being in a schizophrenic superposition and start behaving sensibly. The classical object could be a detecting device or even a human being.
But what exactly does a classical object do to stop a quantum object from being quantum? Even more importantly, what constitutes a classical object? After all, an eye is just a big collection of atoms, which individually obey quantum theory. This turns out to be the Achilles' heel of the Copenhagen Interpretation and the reason it has always appeared to many to be a very unsatisfactory explanation of where the everyday world comes from.
The Copenhagen Interpretation divides the universe, arbitrarily, into two domains, only one of which is governed by quantum theory. This in itself is very defeatist. After all, if quantum theory is a fundamental description of reality, surely it should apply everywhere—to the atomic world and the everyday world. The idea that it is a universal theory is, in a nutshell, what physicists believe today.
It turns out we never observe a quantum system directly. We only observe its effect on its environment. This may be a measuring device or a human eye or, in general, the universe. For instance, the light from an object impinges on the retina of the eye and makes an impression there. What the observer _knows_ is inseparable from what the observer _is_. Now, if quantum theory applies everywhere, we have a quantum object observing, or recording, another quantum object. The central question can therefore be restated: Why do weird schizophrenic states fail to impress themselves on, or _entangle_ themselves with, the environment, whereas everyday one-place-at-one-time states do? An example may help.
If a high-speed subatomic particle flies through the air, it knocks electrons from any atoms it passes close to. Imagine it was possible to see a 10-centimetre-long portion of its track. And, say in that 10 centimetres the particle has a 50 per cent chance of interacting with one electron, kicking it out of its parent atom.
The particle, therefore, either knocks out an electron or doesn't knock out an electron. But because the event of knocking out an electron is a quantum event, there is another possibility—the superposition of the two events. The particle both knocks out an electron and doesn't knock out an electron! The question is: Why, when this event entangles itself with the environment, does it not leave an indelible impression? As luck would have it, it is possible to actually _see_ an electron ejection event with an ingenious device known as a cloud chamber.
Clouds form in the air when a drop in temperature causes water droplets to condense out of water vapour. But this process happens rapidly only if there are things like dust particles in the air that act as "seeds" around which water droplets can grow. Now the seed—and this is the key to the cloud chamber's operation—need not be as big as a dust grain. In fact, it need be only a single atom that has lost an electron—an ion.
A cloud chamber is a box filled with water vapour with a window in its side to look through. Crucially, the water vapour is ultrapure, so there are no seeds about which the vapour can condense. The vapour is held in a state in which it is absolutely desperate to form droplets, but it is frustrated because there are no seeds. Enter a high-speed subatomic particle. Where it knocks an electron out of an atom, a water droplet will instantly grow around the ion. The droplet is small but big enough to see through the window of the cloud chamber if properly illuminated.
So what would you see if you looked through the window? The answer is of course just one of the possibilities—either a single water droplet or no water droplet. You would never see a superposition of both—a ghostly droplet, hovering half in existence and half out of existence. The question is, what happens in the cloud chamber to prevent it from recording this superposition?
Take the event in which a water droplet forms. It was triggered by a single ionised atom. The same atom exists in the event in which no droplet formed. It just does not get ionised, so no water droplet forms around it. Say, this atom is painted red in both cases to make it stand out (forget the fact that you can't paint an atom!).
Now, in the event a droplet forms, zoom in on an atom near the red atom. Water is denser than water vapour; the atoms are closer together. Consequently, the atom in question will be closer to the red atom than it is in the event in which no water droplet forms. For this reason, the probability wave representing the atom in the first event only partially overlaps with the probability wave of the same atom in the second event. Say, for example, that their waves only half overlap.
Now take a second atom in the first event. It too will be closer in the first case than in the second. Once again, their probability waves will only half overlap. If we now consider the probability wave representing the two atoms together, it will overlap only one-quarter with the second case, since ½ × ½ = ¼.
See where this is going? Say the water droplet contains a million atoms, which actually corresponds to a very small droplet. How much will the probability wave representing a million atoms in the first event overlap with the probability wave representing a million atoms in the second event? The answer is ½ × ½ × ½ ×... a million times. This is an extraordinarily small number. There will therefore be essentially zero overlap.
But if two waves don't overlap at all, how can they interfere? The answer is, of course, they cannot. Interference, however, is at the root of all quantum phenomena. If interference between the two events is impossible, we see either one event or the other but never the effect of one event mingling with the other, the essence of quantumness.
Probability waves that do not overlap and so cannot interfere are said to have lost coherence, or to have _decohered_. Decoherence is the ultimate reason why the record of a quantum event in the environment, which always consists of a lot of atoms, is never quantum. In the case of the cloud chamber, the "environment" is the million atoms around the ionised/nonionised atom. In general, however, the environment consists of the countless quadrillions of atoms in the Universe. Decoherence is therefore hugely effective at destroying any overlap between the probability waves of events entangled with the environment. And since that's the only way we can experience them—what the observer _knows_ is inseparable from what the observer _is_ —we never directly see quantum behaviour.
1 See Chapter 7, "The Death of Space and Time."
2 In fact, the quantum coins have to be created together, then separated, to show spooky action at a distance, which is another reason the tale of coins on different sides of the Universe shouldn't be taken too seriously. As pointed out, it isn't a well-thought-out story. It exists merely to convey one amazing truth and one amazing truth only—that quantum theory permits objects to influence each other instantaneously, even when on opposite sides of the Universe.
3 The information on the original particle, P, must be transmitted by ordinary means—that is, slower than the speed of light, the cosmos's speed limit. So even if P and P* are far apart, the creation of P*—the perfect copy of P—is not instantaneous, despite the fact that communication between the entangled particles, A and P, is instantaneous.
4 It is worth emphasising that, even with entanglement, the most you can ever do is make a copy of an object at the expense of destroying the original. Making a copy and keeping the original is impossible.
# IDENTICALNESS AND THE ROOTS OF DIVERSITY
HOW THE BEWILDERING VARIETY OF THE EVERYDAY WORLD STEMS FROM THE FACT THAT YOU CANNOT TATTOO AN ELECTRON
_I woke up one morning and all of my stuff had been stolen, and replaced by exact duplicates._
Steven Wright
_They came from far and wide to see it—the river that ran uphill. It_ _flowed past the fishing port, climbed through the close-packed houses,_ _before meandering up the sheep-strewn hillside to the craggy summit_ _overlooking the town. Startled seagulls bobbed on it. Excited children_ _ran beside it. And at picnic tables outside pubs all along the river's lower_ _reaches, daytrippers sat transfixed by this wonder of nature as beer crept_ _steadily up the sides of their beer glasses and quietly emptied itself onto_ _the ground._
Surely, there is no liquid that can defy gravity like this and run uphill? Remarkably, there is. It's yet another consequence of quantum theory.
Atoms and their kin can do many impossible things before breakfast. For instance, they can be in two or more places at once, penetrate impenetrable barriers, and _know_ about each other instantly even when on different sides of the Universe. They are also totally unpredictable, doing things for no reason at all—perhaps the most shocking and unsettling of all their characteristics.
All of these phenomena ultimately come down to the wave-particle character of electrons, photons, and their like. But the peculiar dual nature of microscopic objects is not the only thing that makes them profoundly different from everyday objects. There is something else: their _indistinguishability_. Every electron is identical to every other electron, every photon is identical to every other photon, and so on.1
At first sight this may not seem a very remarkable property. But think of objects in the everyday world. Although two cars of the same model and colour appear the same, in reality they are not. A careful inspection would reveal that they differ slightly in the evenness of their paint, in the air pressure in their tires, and in a thousand other minor ways.
Contrast this with the world of the very small. Microscopic particles cannot be scratched or marked in any way. You cannot tattoo an electron! They are utterly indistinguishable.2 The same is true of photons and all other denizens of the microscopic world. This indistinguishability is truly something new under the Sun. And it has remarkable consequences for both the microscopic world and the everyday world. In fact, it is fair to say that it is the reason the world we live in is possible.
## THINGS YOU CAN'T TELL APART INTERFERE
Recall that all the bizarre behaviour in the microscopic world, such as an atom's ability to be in many places at once, comes down to interference. In the double slit experiment, for example, it is the interference between the wave corresponding to a particle going through the left-hand slit and the wave corresponding to the particle going through the right-hand slit that produces the characteristic pattern of alternating dark and light stripes on the second screen.
Recall also that if you were to set up some means of determining which slit each particle goes through—enabling you to distinguish between the two alternative events—the interference stripes disappear because of decoherence. Interference, it turns out, happens only if the alternative events are _indistinguishable_ —in this case, the particle going through one slit and the particle going through the other slit.
In the case of the double slit experiment, the alternative events are indistinguishable just as long as nobody looks. But identical particles, such as electrons, raise the possibility of entirely new kinds of indistinguishable events.
Think of a teenage boy who plans to go out clubbing with his girlfriend, who happens to have an identical twin sister. Unbeknown to him, his girlfriend decides to stay in and watch TV and sends her twin in her place. Because the two girls appear identical to the boy (although they are not of course identical at the microscopic level), the events of going clubbing with his girlfriend and going clubbing with his girlfriend's sister are indistinguishable.
Events such as this one, which are indistinguishable simply because they involve apparently indistinguishable things, have no serious consequences in the wider world (apart from allowing identical twin girls to run rings around their boyfriends). However, in the microscopic world, they have truly profound consequences. Why? Because events that are indistinguishable—for any reason whatsoever—are able to interfere with each other.
## THE COLLISION OF IDENTICAL THINGS
Take two atomic nuclei that collide. Any such collision—and this particular point will have to be taken on trust—can be looked at from a point of view in which the nuclei fly in from opposite directions, hit, then fly back out in opposite directions. In general, the in and out directions are not the same. Think of a clock face. If the nuclei fly into the collision point from, say, 9:00 and 3:00, they might fly out toward 4:00 and 10:00. Or 1:00 and 7:00. Or any other pair of directions, as long as the directions are opposite each other.
An experimenter could tell which direction the two nuclei ricochet by placing detectors at opposite sides of the imaginary clock face and then moving them around the rim together. Say the detectors are placed at 4:00 and 10:00. In this case, there are two possible ways the nuclei can get to the detectors. They could strike each other with a glancing blow so that the one coming from 9:00 hits the detector at 4:00 and the one coming from 3:00 hits the one at 10:00. Or they could hit head on, so that the one coming from 9:00 bounces back almost the way it came and hits the detector at 10:00 and the one coming from 3:00 bounces back almost the way it came and hits the detector at 4:00.
The directions of 4:00 and 10:00 are in no way special. Wherever the two detectors are positioned, there will be two alternative ways the nuclei can get to them. Call them events A and B.
What happens if the two nuclei are different? Say the one that flies in from 9:00 is a nucleus of carbon and the one that flies in from 3:00 is a nucleus of helium. Well, in this case, it is always possible to distinguish between events A and B. After all, if a carbon nucleus is picked up by the detector at 10:00, it is obvious that event A occurred; if it is picked up by the detector at 3:00, it must have been event B instead.
What happens, however, if the two nuclei are the same? Say each is a nucleus of helium? Well, in this case, it is impossible to distinguish between events A and B. A helium nucleus that is picked up by the detector in the direction of 10:00 could have got there by either route, and the same is true for a helium nucleus picked up in the direction of 4:00. Events A and B are now indistinguishable. And if two events in the microscopic world are indistinguishable, the waves associated with them interfere.
In the collision of two nuclei, interference makes a huge difference. For instance, it is possible that the two waves associated with the two indistinguishable collision events destructively interfere, or cancel each other out, in the direction of 10:00 and 4:00. If so the detectors will pick up no nuclei at all, no matter how many times the experiment is repeated. It is also possible that the two waves constructively interfere, or reinforce each other, in the direction of 10:00 and 4:00. In this case, the detectors will pick up an unusually large number of nuclei.
In general, because of interference, there will be certain outward directions in which the waves corresponding to events A and B cancel each other and certain outward directions in which they reinforce each other. So if the experiment is repeated thousands of times and the ricocheting nuclei are picked up by detectors all around the rim of the imaginary clock face, the detectors will see a tremendous variation in the number of nuclei arriving. Some detectors will pick up many nuclei, while others will pick up none at all.
But this is dramatically different from the case when the nuclei are different. Then there is no interference, and the detectors will pick up nuclei ricocheting in all directions. There will be no places around the clock face where nuclei are not seen.
This striking difference between the outcomes of the experiment when the nuclei are the same and when they are different is not because of the difference in masses of the nuclei of carbon and helium, although this has a small effect. It is truly down to whether collision events A and B are distinguishable or not.
If this kind of thing happened in the real world, think what it would mean. A red bowling ball and a blue bowling ball that are repeatedly collided together would ricochet in all possible directions. But everything would be changed merely by painting the red ball blue so the two balls were indistinguishable. Suddenly, there would be directions in which the balls ricocheted far more often than when they were different colours and directions in which they never, ever ricocheted.
This fact, that events involving identical particles in the microscopic world can interfere with each other, may seem little more than a quantum quirk. But it isn't. It is the reason why there are 92 different kinds of naturally occurring atoms rather than just 1. In short, it is responsible for the variety of the world we live in. Understanding why, however, requires appreciating one more subtlety of the process in which identical particles collide.
## TWO TRIBES OF PARTICLES
Recall the case in which the nuclei are different—a carbon nucleus and a helium nucleus—and consider again the two possible collision events. In one, the nuclei strike each other with a glancing blow, and in the other they hit head on and bounce back almost the way they came. What this means is that, for the nucleus that comes in at 9:00, there is a wave corresponding to it going out at 4:00 and a wave corresponding to it going out at 10:00.
The key thing to understand here is that the probability of an event is not related to the height of the wave associated with that event but to the square of the height of the wave. The probability of the 4:00 event is therefore the square of the wave height in the direction of 4:00 and the probability of the 10:00 event is the square of the wave height in the direction of 10:00. It is here that the crucial subtlety comes in.
Say the wave corresponding to the nucleus that flies out at 10:00 is flipped by the collision, so that its troughs become its peaks and its peaks become its troughs. Would it make any difference to the probability of the event? To answer this, consider a water wave—a series of alternating peaks and troughs. Think of the average level of the water as corresponding to a height equal to zero so that the height of the peaks is a positive number—say plus 1 metre—and the height of the troughs is a negative number—minus 1 metre. Now it makes no difference whether you square the height of a peak or the height of a trough since 1 × 1 = 1 and –1 × –1 also equals 1. Consequently, flipping the probability wave associated with a ricocheting nucleus makes no difference to the event's probability.
But is there any reason to believe that one wave might get flipped? Well, the 10:00 collision and the 4:00 collision are very different events. In one, the trajectory of the nucleus hardly changes whereas in the other it is turned violently back on itself. It is at least plausible that the 10:00 wave might get flipped.
Just because something is plausible does not mean it actually happens. True. In this case, however, it does! Nature has two possibilities available to it: It can flip the wave of one collision event or it can leave it alone. It turns out that it avails itself of both.
But how would we ever know about a probability wave getting flipped? After all, the only thing an experimenter can measure is the number of nuclei picked up by a detector which depends on the probability of a particular collision event. But this is determined by the square of the wave height, which is the same whether the wave is flipped or not. It would seem that what actually happens to the probability wave in the collision is hidden from view.
If the colliding particles are different, this is certainly true. But, crucially, it is not if they are identical. The reason is that the waves corresponding to indistinguishable events interfere with each other. And in interference it matters tremendously whether or not a wave is flipped before it combines with another wave. It could mean the difference between peaks and troughs coinciding or not, between the waves cancelling or boosting each other.
What happens then in the collision of identical particles? Well, this is the peculiar thing. For some particles—for instance, photons—everything is the same as it is for identical helium nuclei. The waves that correspond to the two alternative collision events interfere with each other normally. However, for other particles—for instance, electrons—things are radically different. The waves corresponding to the two alternative collision events interfere, but only after one has been flipped.
Nature's basic building blocks turn out to be divided into two tribes. On the one hand, there are particles whose waves interfere with each other in the normal way. These are known as bosons. They include photons and gravitons, the hypothetical carriers of the gravitational force. And, on the other hand, there are particles whose waves interfere with one wave flipped. These are known as fermions. They include electrons, neutrinos, and muons.
Whether particles are fermions or bosons—that is, whether or not they indulge in waveflipping—turns out to depend on their spin. Recall that particles with more spin than others behave as if they are spinning faster about their axis (although in the bizarre quantum world particles that possess spin are not actually spinning!). Well, it turns out that there is a basic indivisible chunk of spin, just like there is a basic indivisible chunk of everything in the microscopic world. For historic reasons, this "quantum" of spin is 1/2 unit (don't worry what the units are). Bosons have integer spin—0 units, 1 unit, 2 units, and so on—and fermions have "half-integer" spin—1/2 unit, 3/2 units, 5/2 units, and so on.
Why do particles with half-integer spin indulge in waveflipping, whereas particles with integer spin do not? This, of course, is a very good question. But it brings us to the end of what can easily be conveyed without opaque mathematics. Richard Feynman at least came clean about this: "This seems to be one of the few places in physics where there is a rule which can be stated very simply but for which no one has found an easy explanation. It probably means that we do not have a complete understanding of the fundamental principles involved."
Feynman, who worked on the atomic bomb and won the 1965 Nobel Prize for Physics, was arguably the greatest physicist of the postwar era. If you find the ideas of quantum theory a little difficult, you are therefore in very good company. It is fair to say that, 80-odd years after the birth of quantum theory, physicists are still waiting for the fog to lift so that they can clearly see what it is trying to tell us about fundamental reality. As Feynman himself said: "I think I can safely say that nobody understands quantum mechanics."
Brushing the spin mystery under the carpet, we come finally to the point of all this—the implication of waveflipping for fermions such as electrons.
Instead of two helium nuclei, think of two electrons, each of which collides with another particle. After the collision, they ricochet in almost the same direction. Call the electrons A and B and call the directions 1 and 2 (even though they are almost the same direction). Exactly as in the case of two identical nuclei, there are two indistinguishable possibilities. Electron A could ricochet in direction 1 and electron B in direction 2, or electron A could ricochet in direction 2 and electron B in direction 1.
Since electrons are fermions, the wave corresponding to one possibility will be flipped before it interferes with the wave corresponding to the other possibility. Crucially, however, the waves for the two possibilities are identical, or pretty identical. After all, we are talking about two identical particles doing almost identical things. But if you add two identical waves—one of which has been flipped—the peaks of one will exactly match the troughs of the other. They will completely cancel each other out. In other words, the probability of two electrons ricocheting in exactly the same direction is zero. It is completely impossible!
This result is actually far more general than it appears. It turns out that two electrons are not only forbidden from ricocheting in the same direction, they are forbidden from doing the same thing, period. This prohibition, known as the Pauli exclusion principle, after Austrian physicist Wolfgang Pauli, turns out to be the ultimate reason for the existence of white dwarfs. While it is certainly true that an electron cannot be confined in too small a volume of space, this still does not explain why all the electrons in a white dwarf do not simply crowd together in exactly the same small volume. The Pauli exclusion principle provides the answer. Two electrons cannot be in the same quantum state. Electrons are hugely antisocial and avoid each other like the plague.
Think of it this way. Because of the Heisenberg uncertainty principle, there is a minimum-sized "box" in which an electron can be squeezed by the gravity of a white dwarf. However, because of the Pauli exclusion principle, each electron demands a box to itself. These two effects, working in concert, give an apparently flimsy gas of electrons the necessary "stiffness" to resist being squeezed by a white dwarf's immense gravity.
Actually, there is yet another subtlety here. The Pauli exclusion principle prevents two fermions from doing the same thing if they are identical. But electrons have a way of being different from each other because of their spin. One can behave as if it is spinning clockwise and one as if it is spinning anticlockwise.3 Because of this property of electrons, _two_ electrons are permitted to occupy the same volume of space. They may be unsociable, but they are not complete loners! White dwarfs are hardly everyday objects. However, the Pauli exclusion principle has much more mundane implications. In particular, it explains why there are so many different types of atoms and why the world around us is the complex and interesting place it is.
## WHY ATOMS AREN'T ALL THE SAME
Recall that, just as sound waves confined in an organ pipe can vibrate in only restricted ways, so too can the waves associated with an electron confined in an atom. Each distinct vibration corresponds to a possible orbit for an electron at a particular distance from the central nucleus and with a particular energy. (Actually, of course, the orbit is merely the most probable place to find an electron since there is no such thing as a 100 per cent certain path for an electron or any other microscopic particle.)
Physicists and chemists number the orbits. The innermost orbit, also known as the ground state, is numbered 1, and orbits successively more distant from the nucleus are numbered 2, 3, 4, and so on. The existence of these quantum numbers, as they are called, emphasises yet again how everything in the microscopic world—even the orbits of electrons—comes in discrete steps with no possibility of intermediate values.
Whenever an electron "jumps" from one orbit to another orbit closer to the nucleus, the atom loses energy, which is given out in the form of a photon of light. The energy of the photon is exactly equal to the difference in energy of two orbits. The opposite process involves an atom absorbing a photon with an energy equal to the difference in energy of two orbits. In this case, an electron jumps from one orbit to another orbit farther from the nucleus.
This picture of the "emission" and "absorption" of light explains why photons of only special energies—corresponding to special frequencies—are spat out and swallowed by each kind of atom. The special energies are simply the energy differences between the electron orbits. It is because there is a limited number of permitted orbits that there is a restricted number of orbital "transitions."
But things are not quite this simple. The electron waves that are permitted to vibrate inside an atom can be quite complex three-dimensional things. They may correspond to an electron that is not only most likely to be found at a certain distance from the nucleus but more likely to be found in some directions rather than others. For instance, an electron wave might be bigger over the north and south poles of the atom than in other directions. An electron in such an orbit would most likely be found over the north and south poles.
Describing a direction in three-dimensional space requires two numbers. Think of a terrestrial globe where a latitude and longitude are required. Similarly, in addition to the numbers specifying its distance from the nucleus, an electron wave whose height changes with direction requires two more quantum numbers to describe it. This makes a total of three. In recognition of the fact that electron orbits are totally unlike more familiar orbits—for instance, the orbits of planets around the Sun—they are given a special name: orbitals.
The precise shape of electron orbitals turns out to be crucially important in determining how different atoms stick together to make molecules such as water and carbon dioxide. Here, the key electrons are the outermost ones. For instance, an outer electron from one atom might be shared with another atom, creating a chemical bond. Where exactly the outermost electron is clearly plays an important role. If, for example, it has its highest probability of being found above the atom's north and south poles, the atom will most easily bond with atoms to its north or south.
The science that concerns itself with all the myriad ways in which atoms can join together is chemistry. Atoms are the ultimate Lego bricks. By combining them in different ways, it is possible to make a rose or a gold bar or a human being. But exactly how the Lego bricks combine to create the bewildering variety of objects we see in the world around us is determined by quantum theory.
Of course, an obvious requirement for the existence of a large number of combinations of Lego bricks is that there be more than one kind of brick. Nature in fact uses 92 Lego bricks. They range from hydrogen, the lightest naturally occurring atom, to uranium, the heaviest. But why are there so many different atoms? Why are they not all the same? Once again, it all comes down to quantum theory.
## WHY ATOMS ARE NOT ALL THE SAME
Electrons trapped in the electric force field of a nucleus are like footballs trapped in a steep valley. By rights they should run rapidly downhill to the lowest possible place—the innermost orbital. But if this was what the electrons in atoms really did, all atoms would be roughly the same size. More seriously, since the outermost electrons determine how an atom bonds, all atoms would bond in exactly the same way. Nature would have only one kind of Lego brick to play with and the world would be a very dull place indeed.
What rescues the world from being a dull place is the Pauli exclusion principle. If electrons were bosons, it is certainly true that an atom's electrons would all pile on top of each other in the innermost orbital. But electrons are not bosons. They are fermions. And fermions abhor being crowded together.
This is how it works. Different kinds of atoms have different numbers of electrons (always of course balanced by an equal number of protons in their nuclei). For instance, the lightest atom, hydrogen, has one electron and the heaviest naturally occurring atom, uranium, has 92. In this discussion the nucleus is not important. Focus instead on the electrons. Imagine starting with a hydrogen atom and then adding electrons, one at a time.
The first available orbit is the innermost one, nearest the nucleus. As electrons are added, they will first go into this orbit. When it is full and can take no more electrons, they will pile into the next available orbit, farther away from the nucleus. Once that orbit is full, they will fill the next most distant one. And so on.
All the orbitals at a particular distance from the nucleus—that is, with different directional quantum numbers—are said to make up a shell. The maximum number of electrons that can occupy the innermost shell turns out to be two—one electron with clockwise spin and one with anticlockwise spin. A hydrogen atom has one electron in this shell and an atom of helium, the next biggest atom, has two.
The next biggest atom is lithium. It has three electrons. Since there is no more room in the innermost shell, the third electron starts a new shell farther out from the nucleus. The capacity of this shell is eight. For atoms with more than 10 electrons, even this shell is all used up, and another begins to fill up yet farther from the nucleus.
The Pauli exclusion principle, by forbidding more than two electrons from being in the same orbital—that is, from having the same quantum numbers—is the reason that atoms are different from each other. It is also responsible for the rigidity of matter. "It is the fact that electrons cannot get on top of each other that makes tables and everything else solid," said Richard Feynman.
Since the manner in which an atom behaves—its very identity—depends on its outer electrons, atoms with similar numbers of electrons in their outermost shells tend to behave in a similar way. Lithium, with three electrons, has one electron in its outer shell. So too does sodium, with 11 electrons. Lithium and sodium therefore bond with similar kinds of atoms and have similar properties.
So much for fermions, which are subject to the Pauli exclusion principle. What about bosons? Well, since such particles are not governed by the exclusion principle, they are positively gregarious. And this gregariousness leads to a host of remarkable phenomena, from lasers to electrical currents that flow forever to liquids that flow uphill.
## WHY BOSONS LIKE TO BE TOGETHER WITH THEIR MATES
Say two boson particles fly into a small region of space. One hits an obstruction in its path and ricochets; the other hits a second obstruction and ricochets. It doesn't matter what the obstructing bodies are; they may be nuclei or anything else. The important thing here is the direction in which they ricochet, which is the same for both.
Call the particles A and B, and call the directions they ricochet in 1 and 2 (even if they are almost the same direction!). There are two possibilities. One is that particle A ends up in direction 1 and particle B ends up in direction 2. The other is that A ends up in direction 2 and B in direction 1. Because A and B are schizophrenic denizens of the microscopic world, there is a wave corresponding to A going in direction 1 and to B in direction 2. And there is also a wave corresponding to A going in direction 2 and to B in direction 1.
If the two bosons are different particles there can be no interference between them. So the probability that a detector picks up the two ricocheting particles is simply the square of the height of the first wave plus the square of the height of the second wave, since the probability of anything happening in the microscopic world is always the square of the height of the wave associated with it. Well, it turns out—and this will have to be taken on trust—that the two probabilities are roughly the same. So the overall probability simply is twice the probability of each event happening individually.
Say the waves have a height of 1 for both processes. This would mean that if they were squared and added to get the probability for both processes, it would be (1 × 1) + (1 × 1) = 2. Now a probability of 1 corresponds to 100 per cent, so a probability of 2 is clearly ridiculous! But bear with this. It is still possible to make a comparison of probabilities, which is where all this is leading.
Now, say the two bosons are identical particles. In this case, the two possibilities—A going in direction 1 and B in direction 2, and A going in direction 2 and B in direction 1—are indistinguishable. And because they are indistinguishable, the waves associated with them can interfere with each other. Their combined height is 1 + 1. The probability for both processes is therefore (1 + 1) × (1 + 1) = 4.
This is twice as big as when the bosons were not identical. In other words, if two bosons are identical, they are twice as likely to ricochet in the same direction as if they were different. Or to put it another way, a boson is twice as likely to ricochet in a particular direction if another boson ricochets in that direction too.
The more bosons there are the more significant the effect. If _n_ bosons are present, the probability that one more particle will ricochet in the same direction is _n_ \+ 1 times bigger than if no other bosons are present. Talk about herd behaviour! The mere presence of other bosons doing something greatly increases the probability that one more will do the same thing.
This gregariousness turns out to have important practical applications—for instance, in the propagation of light.
## LASERS AND LIQUIDS THAT RUN UPHILL
All the processes so far considered have involved particles colliding and ricocheting in a particular direction. But that is not essential. The arguments used could apply equally well to the creation of particles—for instance, the "creation" of photons by atoms that emit light.
Photons are bosons, so the probability that an atom will emit a photon in a particular direction with a particular energy is increased by a factor of _n_ \+ 1 if there are already n photons flying in that direction with that energy. Each new photon emitted increases the chance of another photon being emitted. Once there are thousands, even millions, flying through space together, the probability of new photons being emitted is enormously enhanced.
The consequences are dramatic. Whereas a normal light source like the Sun produces a chaotic mixture of photons of all different energies, a laser generates an unstoppable tide of photons that surge through space in perfect lockstep. Lasers, however, are far from the only consequence of the gregariousness of bosons. Take liquid helium, which is composed of atoms that are bosons.
Helium-4, the second most common atom in the Universe, is one of nature's most peculiar substances.4 It was the only element to have been discovered on the Sun before it was discovered on Earth, and it has the lowest boiling point of any liquid, –269 degrees Celsius. In fact, it is the only liquid that never freezes to become a solid, at least not at normal atmospheric pressure. All these things, however, pale into insignificance beside the behaviour of helium below about –271 degrees Celsius. Below this "lambda point," it becomes a superfluid.
Usually, a liquid resists any attempt to move one part relative to another. For instance, treacle resists when you stir it with a spoon and water resists when you try to swim through it. Physicists call this resistance viscosity. It is really just liquid friction. But whereas we are used to friction between solids moving relative to each other—for instance, the friction between a car's tyres and the road—we are not familiar with the friction between parts of a liquid moving relative to each other. Treacle, because it resists strongly, is said to have a high viscosity, or simply to be very viscous.
Clearly, viscosity can manifest itself only when one part of a liquid moves differently from the rest. At the microscopic level of atoms, this means that it must be possible to knock some liquid atoms into states that are different from those occupied by other liquid atoms.
In a liquid at normal temperature, the atoms can be in many possible states in each of which they jiggle about at different speeds. But as the temperature falls, they become more and more sluggish and fewer and fewer states are open to them. Despite this effect, however, not all atoms will be in the same state, even at the lowest temperatures.
But things are different for a liquid of bosons such as liquid helium. Remember, if there are already _n_ bosons in a particular state, the probability of another one entering the state is _n_ \+ 1 bigger than if there were no other particles in the state. And for liquid helium, with countless helium atoms, _n_ is a very large number indeed. Consequently, there comes a time, as liquid helium is cooled to sufficiently low temperatures, when all the helium atoms suddenly try to crowd into the same state. It's called the Bose-Einstein condensation.
With all the helium atoms in the same state, it is impossible—or at least extremely difficult—for one part of the liquid to move differently from another part. If some atoms are moving along, all the atoms have to move along together. Consequently, the liquid helium has no viscosity whatsoever. It has become a superfluid.
In superfluid liquid helium there is a kind of rigidity to the motion of the atoms. It is very hard to make the liquid do anything because you either have to get all of its atoms to do the thing together or they simply do not do the thing at all. For instance, if you put water in a bucket and spin the bucket about its axis, the water will end up spinning with the bucket. This is because the bucket drags around the water atoms—strictly speaking, the water _molecules_ —that are in direct contact with the sides, and these in turn drag around the atoms farther from the sides, and so on, until the entire body of water is turning with the bucket. Clearly, for the water to get to the state in which it is spinning along with the bucket, different parts of the liquid must move relative to each other. But as just pointed out, this is very hard for a superfluid. All the atoms move together or they do not move at all. Consequently, if superfluid liquid helium is put in a bucket and the bucket is spun, it has no means open to it to attain the spin of the bucket. Instead, the superfluid helium stays stubbornly still while the bucket spins.
The cooperative motion of atoms in superfluid liquid helium leads to even more bizarre phenomena. For instance, the superfluid can flow through impossibly small holes that no other liquid can flow through. It is also the only liquid that can flow uphill. Interestingly, helium has a rare, lightweight cousin. Helium-3 turns out to be a normal, boring liquid. The reason is that helium-3 particles are fermions. And superfluidity is a property solely of bosons.
Actually, this isn't entirely true. The microscopic world is full of surprising phenomena. And in a special case, fermions can behave like bosons!
## ELECTRIC CURRENTS THAT RUN FOREVER
The special case, when fermions behave like bosons, is that of an electric current in a metal. Because the outermost electrons of metal atoms are very loosely bound, they can break free. If a voltage is then applied between the ends of the metal by a battery, all the countless liberated electrons will surge through the material as an electric current.5
Electrons are, of course, fermions, which means they are antisocial. Imagine a ladder, with the rungs corresponding to ever higher energy states. Electrons would fill up the rungs two at a time from the bottom (bosons would happily crowd on the lowest rungs). The need for a separate rung for each pair of electrons means that the electrons in a metal have far more energy on average than might be naively expected.
But something really weird happens when a metal is cooled to close to absolute zero, the lowest possible temperature. Usually, each electron travels through the metal entirely independently of all other electrons. However, as the temperature falls, the metal atoms vibrate ever more sluggishly. Although they are thousands of times more massive than electrons, the attractive electrical force between an electron and a metal atom is enough to tug the atom toward it as the electron passes by.6 The tugged atom, in turn, tugs on another electron. In this way, one electron attracts another through the intermediary of the metal atom.
This effect radically changes the nature of the current flowing through the metal. Instead of being composed of single electrons, it is composed of paired-up electrons known as Cooper pairs. But the electrons in each Cooper pair spin in an opposite manner and cancel out. Consequently, Cooper pairs are bosons!
A Cooper pair is a peculiar thing. The electrons that make it up may not even be close to each other in the metal. There could easily be thousands of other electrons between one member of a Cooper pair and its partner. This is just a curious detail, however. The key thing is that Cooper pairs are bosons. And at the ultralow temperature of the superconductor all the bosons crowd into the same state. They therefore behave as a single, irresistible entity. Once they are flowing en masse, it is extremely difficult to stop them.
In a normal metal an electrical current is resisted by nonmetal, impurity atoms, which get in the way of electrons, obstructing their progress through the metal. But whereas an impurity atom can easily hinder an electron in a normal metal, it is nearly impossible for it to hinder a Cooper pair in a superconductor. This is because each Cooper pair is in lockstep with billions upon billions of others. An impurity atom can no more thwart this flow than a single soldier can stop the advance of an enemy army. Once started, the current in a superconductor will flow forever.
1 Since photons come with different _wavelengths_ , we are of course talking here about photons _with the same wavelength_ being identical to each other.
2 John Wheeler and Richard Feynman once came up with an interesting suggestion for why electrons are utterly indistinguishable—because there is only one electron in the Universe! It weaves backwards and forwards in time like a thread going back and forth through a tapestry. We see the multitude of places where the thread goes through the fabric of the tapestry and mistakenly attribute each to a separate electron.
3 Physicists call two alternatives spin "up" and spin "down." But that is just a technicality.
4 Helium-4 has four particles in its nucleus—two protons and two neutrons. It has a less common cousin, helium-3, which has the same number of protons but one fewer neutron.
5 Why then doesn't a metal fall apart? The full explanation requires quantum theory. But, simplistically, the stripped, or conduction, electrons form a negatively charged cloud permeating the metal. It is the attraction between this cloud and the positively charged electron-stripped metal ions that glues the metal together.
6 Strictly speaking, the atoms are positive ions, the name given to atoms that have lost electrons.
PART TWO
# BIG THINGS
# THE DEATH OF SPACE AND TIME
HOW WE DISCOVERED THAT LIGHT IS THE ROCK ON WHICH THE UNIVERSE IS FOUNDED AND TIME AND SPACE ARE SHIFTING SANDS
_When a man sits with a pretty girl for an hour, it seems like a minute. But let him sit on a hot stove for a minute—it's longer than an hour. That's relativity!_
Albert Einstein
_It's the most peculiar 100 metres anyone has ever seen. As the sprinters_ _explode out of their starting blocks and get into their stride, it seems to_ _the spectators in the grandstand that the runners get ever slimmer. Now,_ _as they dash past the cheering crowd, they appear as flat as pancakes._ _But that's not the most peculiar thing—not by a long shot. The arms_ _and legs of the athletes are pumping in ultraslow motion, as if they are_ _running not through air but through molasses. Already, the crowd is_ _beginning to slow-hand-clap. Some people are even ripping up their tickets_ _and angrily tossing them into the air. At this pathetic rate of progress,_ _it could take an hour for the sprinters to reach the finishing tape. Disgusted_ _and disappointed, the spectators get up from their seats and, one_ _by one, traipse out of the stadium._
This scene seems totally ridiculous. But, actually, it is wrong in essentially only one detail—the speed of the sprinters. If they could run 10 million times faster, this is exactly what everyone would see. When objects fly past at ultrahigh speed, space shrinks while time slows down.1 It's an inevitable consequence of one thing—the impossibility of ever catching up with a light beam.
Naively, you might think that the only thing that is not catch-upable is something travelling at infinite speed. Infinity, after all, is defined as the biggest number imaginable. Whatever number you think of, infinity is bigger. So if there were something that could travel infinitely fast, it is clear you could never get abreast of it. It would represent the ultimate cosmic speed limit.
Light travels tremendously fast—300,000 kilometres per second in empty space—but this is far short of infinite speed. Nevertheless, you can never catch up with a light beam, no matter how fast you travel. In our universe, for reasons nobody completely understands, the speed of light plays the role of infinite speed. It represents the ultimate cosmic speed limit.
The first person to recognise this peculiar fact was Albert Einstein. Reputedly at the age of only 16, he asked himself: What would a beam of light look like if you could catch up with it?
Einstein could ask such a question and hope to answer it only because of a discovery made by the Scottish physicist James Clerk Maxwell. In 1868, Maxwell summarised all known electrical and magnetic phenomena—from the operation of electric motors to the behaviour of magnets—with a handful of elegant mathematical equations. The unexpected bonus of Maxwell's equations was that they predicted the existence of a hitherto unsuspected wave, a wave of electricity and magnetism.
Maxwell's wave, which propagated through space like a ripple spreading on a pond, had a very striking feature. It travelled at 300,000 kilometres per second—the same as the speed of light in empty space. It was too much of a coincidence. Maxwell guessed—correctly—that the wave of electricity and magnetism was none other than a wave of light. Nobody, apart perhaps from the electrical pioneer Michael Faraday, had the slightest inkling that light was connected with electricity and magnetism. But there it was, written indelibly in Maxwell's equations: light was an electromagnetic wave.
Magnetism is an invisible force field that reaches out into the space surrounding a magnet. The magnetic field of a bar magnet, for instance, attracts nearby metal objects such as paperclips. Nature also boasts an electric field, an invisible force field that extends into the space around a body that is electrically charged. The electric field of a plastic comb rubbed against a nylon sweater, for instance, can pick up small scraps of paper.
Light, according to Maxwell's equations, is a wave rippling through these invisible force fields, much like a wave rippling through water. In the case of a water wave, the thing that changes as the wave passes by is the level of the water, which goes up and down, up and down. In the case of light, it is the strength of the magnetic and electric force fields, which grow and die, grow and die. (Actually, one field grows while the other dies, and vice versa, but that's not important here.)
Why go into such gory detail about what an electromagnetic wave is? The answer is because it is necessary in order to understand Einstein's question: What would a light beam look like if you could catch up with it?
Say you are driving a car on a motorway and you catch up with another car travelling at 100 kilometres per hour. What does the other car look like as you come abreast of it? Obviously, it appears stationary. If you wind down your window, you may even be able to shout to the other driver above the noise of the engine. In exactly the same way, if you could catch up with a light beam, it ought to appear stationary, like a series of ripples frozen on a pond.
However—and this is the key thing noticed by the 16-year-old Einstein—Maxwell's equations have something important to say about a frozen electromagnetic wave, one in which the electric and magnetic fields never grow or fade but remain motionless forever. No such thing exists! A stationary electromagnetic wave is an impossibility.
Einstein, with his precocious question, had put his finger on a paradox, or inconsistency, in the laws of physics. If you were able to catch up with a beam of light, you would see a stationary electromagnetic wave, which is impossible. Since seeing impossible things is, well, impossible, you can never catch up with a light beam! In other words, the thing that is uncatchable—the thing that plays the role of infinite speed in our Universe—is light.
## FOUNDATION STONES OF RELATIVITY
The uncatchability of light can be put another way. Imagine that the cosmic speed limit really is infinity (though, of course, we now know it isn't). And say for instance, a missile is fired from a fighter plane that can fly at infinite speed. Is the speed of the missile relative to someone standing on the ground infinity plus the speed of the plane? If it is, the missile's speed relative to the ground is greater than infinity. But this is impossible since infinity is the biggest number imaginable. The only thing that makes sense is that the speed of the missile is still infinitely fast. In other words, its speed does not depend on the speed of its source—the speed of the fighter plane.
It follows that in the real Universe, where the role of infinite speed is played by the speed of light, the speed of light does not depend on the motion of its source either. It's the same—300,000 kilometres per second—no matter how fast the light source is travelling.
The speed of light's lack of dependence on the motion of its source is one of the two pillars on which Einstein, in his "miraculous year" of 1905, proceeded to build a new and revolutionary picture of space and time—his "special" theory of relativity. The other one—equally important—is the principle of relativity.
In the 17th century the great Italian physicist Galileo noticed that the laws of physics are unaffected by relative motion. In other words, they appear the same, no matter how fast you are moving relative to someone else. Think of standing in a field and throwing a ball to a friend 10 metres away. Now imagine you are on a moving train instead and throwing the ball to your friend, who is standing 10 metres along the aisle. The ball in both cases loops between you on a similar trajectory. In other words, the path the ball follows takes no account of the fact that you are in a field or on a train barrelling along at, say 120 kilometres per hour.
In fact, if the windows of the train are blacked out, and the train has such brilliant suspension that it is vibration free, you will be unable to tell from the motion of the ball—or any other object inside the train, for that matter—whether or not the train is moving. For reasons nobody knows, the laws of physics are the same no matter what speed you are travelling, as long as that speed remains constant.
When Galileo made this observation, the laws he had in mind were the laws of motion that govern such things as the trajectory of cannonballs flying through the air. Einstein's audacious leap was to extend the idea to all laws of physics, including the laws of optics that govern the behaviour of light. According to his principle of relativity, all laws appear the same for observers moving with constant speed relative to each other. In a blacked-out train, in other words, you could not tell even from the way light was reflected back and forth whether or not the train was moving.
By combining the principle of relativity with the fact that the speed of light is the same irrespective of the motion of its source, it is possible to deduce another remarkable property of light. Say you are travelling towards a source of light at high speed. At what speed does the light come towards you? Well, remember there is no experiment you can do to determine whether it is you or the light source that is moving (recall the blacked-out train). So an equally valid point of view is to assume that you are stationary and the light source is moving towards you. But remember, the speed of light does not depend on the speed of its source. It always leaves the source at precisely 300,000 kilometres per second. Since you are stationary, therefore, the light must arrive at precisely 300,000 kilometres per second.
Consequently, not only is the speed of light independent of the motion of its source, it is also independent of the motion of anyone observing the light. In other words, everyone in the Universe, no matter how fast they are moving, always measures exactly the same speed of light—300,000 kilometres per second.
What Einstein set out to answer in his special theory of relativity was how, in practice, everyone can end up measuring precisely the same speed for light. It turns out there is only one way: If space and time are totally different from what everyone thinks they are.
## SHRINKING SPACE, STRETCHY TIME
Why do space and time come into things? Well, the speed of anything—light included—is the distance in space a body travels in a given interval of time. Rulers are commonly used to measure distance and clocks to measure time. Consequently, the question—how can everyone, no matter what their state of motion, measure the same speed of light?—can be put another way. What must happen to everyone's rulers and clocks so that, when they measure the distance light travels in a given time, they always get a speed of exactly 300,000 kilometres per second?
This, in a nutshell, is special relativity—a "recipe" for what must happen to space and time so that everyone in the Universe agrees on the speed of light.
Think of a spaceship firing a laser beam at a piece of space debris that happens to be flying toward it at 0.75 times the speed of light. The laser beam cannot hit the debris at 1.75 times the speed of light because that is impossible; it must hit it at exactly the speed of light. The only way this can happen is if someone observing the events and estimating the distance that the arriving light travels in a given time either underestimates the distance or overestimates the time.
In fact, as Einstein discovered, they do both. To someone watching the spaceship from outside, moving rulers shrink and moving clocks slow down. Space "contracts" and time "dilates," and they contract and dilate in exactly the manner necessary for the speed of light to come out as 300,000 kilometres per second for everyone in the Universe. It's like some huge cosmic conspiracy. The constant thing in our Universe isn't space or the flow of time but the speed of light. And everything else in the Universe has no choice but to adjust itself to maintain light in its preeminent position.
Space and time are both relative. Lengths and time intervals become significantly warped at speeds approaching the speed of light. One person's interval of space is not the same as another person's interval of space. One person's interval of time is not the same as another person's interval of time.
Time, it turns out, runs at different rates for different observers, depending on how fast they are moving relative to each other. And the discrepancy between the ticking of their clocks gets greater the speedier the motion. The faster you go, the slower you age!2 It's a truth that has been hidden from us for most of human history for the simple reason that the slowing down of time is apparent only at speeds approaching that of light, and the speed of light is so enormous that a supersonic jet, by comparison, flies at a snail's pace across the sky. If the speed of light had instead been only 30 kilometres per hour, it would not have taken a genius like Einstein to discover the truth. The effects of special relativity such as time dilation and length contraction would be glaringly obvious to the average 5-year-old.
As with time, so with space. The spatial distance between any two bodies is different for different observers, depending on how fast they are moving relative to each other. And the discrepancy between their rulers gets greater the faster the motion. "The faster you go, the slimmer you are," said Einstein.3 Once again, this would be self-evident if we lived our lives travelling close to the speed of light. But living as we do in nature's slow lane, we cannot see the truth—that space and time are shifting sand, the unvarying speed of light the bedrock on which the Universe is built.
(If you think relativity is hard, take heart from the words of Einstein: "The hardest thing in the world to understand is income tax!" Ignore, however, the words of Israel's first president, Chaim Weizmann, who, after a sea voyage with the great scientist in 1921, said: "Einstein explained his theory to me every day and, on my arrival, I was fully convinced that he understood it!")
Can anything travel faster than light? Well, nothing can catch up with a beam of light. But the possibility exists that there are "subatomic" particles that live their lives permanently travelling faster than light. Physicists call such hypothetical particles tachyons. If tachyons exist, perhaps in the far future we can find a way to change the atoms of our bodies into tachyons and then back again. Then we too could travel faster than light.
One of the problems with tachyons, however, is that from the point of view of certain moving observers, a body travelling faster than light could appear to be travelling back in time! There is a limerick that goes like this:
_A rocket explorer named Wright_ ,
_Once travelled much faster than light_.
_He set out one day, in a relative way_ ,
_And returned on the previous night!_
Anonymous
Time travel scares the living daylights out of physicists because it raises the possibility of paradoxes, events that lead to logical contradictions like you going back in time and killing your grandfather. If you killed your grandfather before he conceived your mother, goes the argument, how could you have been born to go back in time to kill your grandfather? Some physicists, however, think that some as-yet-undiscovered law of physics intervenes to prevent any paradoxical things from happening, and so time travel may be possible.
## THE MEANING OF RELATIVITY
But what does relativity mean in a nuts-and-bolts sense? Well, say it were possible for you to travel to the nearest star and back at 99.5 per cent of the speed of light. Since Alpha Centauri is about 4.3 light-years from Earth, those left on Earth will see you return after about 9 years, assuming a brief stopover to see the sights. From your point of view, however, the distance to Alpha Centauri will be shrunk by 10 times because of relativity. Consequently, the round-trip will take only nine-tenths of a year, or about 11 months. Say you departed on your journey on your twenty-first birthday, waved off from the spaceport by your identical twin brother. When you arrived back home, now almost 22 years old, your twin would be 30!4
How would your stay-at-home twin make sense of this state of affairs? Well, he would assume that you had been living in slow motion throughout your journey. And, sure enough, if it were somehow possible for him to observe you inside your spaceship, he would see you moving as if through treacle, with all the shipboard clocks crawling around 10 times slower than normal. Your twin will correctly attribute this to the time dilation of relativity. But to you all the clocks and everything else on board will appear to be moving at perfectly normal speed. This is the magic of relativity.
Of course, the more rapidly you travelled to Alpha Centauri and back, the greater the discrepancy between your age and your twin's. Travel fast enough and far enough across the Universe and you will return to find that your twin is long dead and buried. Even faster and you will find that Earth itself has dried up and died. In fact, if you travelled within a whisker of the speed of light, time would go so slowly for you that you could watch the entire future history of the Universe flash past you like a movie in fast-forward. "The possibility of visiting the future is quite awesome to anyone who learns about it for the first time," says Russian physicist Igor Novikov.
We do not yet have the ability to travel to the nearest star and back at close to the speed of light (or even 0.01 per cent of the speed of light). Nevertheless, time dilation is detectable—just—in the everyday world. Experiments have been carried out in which super-accurate atomic clocks are synchronised and separated, one being flown around the world on an airplane while the other stays at home. When the clocks are reunited, the experimenters find that the around-the-world clock has registered the passage of marginally less time than its stay-at-home counterpart. The shorter time measured by the moving clock is precisely what is predicted by Einstein.
The slowing of time affects astronauts too. As Novikov points out in his excellent book, _The River of Time_ : "When the crew of the Soviet Salyut space station returned to Earth in 1988 after orbiting for a year at 8 kilometres a second, they stepped into the future by one hundredth of a second."
The time dilation effect is minuscule because airplanes and spacecraft travel at only a tiny fraction of the speed of light. However, it is far greater for cosmic-ray muons, subatomic particles created when cosmic rays—superfast atomic nuclei from space—slam into air molecules at the top of Earth's atmosphere.
The key thing to know about muons is that they have tragically short lives and, on average, disintegrate, or decay after a mere 1.5 millionths of a second. Since they streak down through the atmosphere at more than 99.92 per cent of the speed of light, this means that they should travel barely 0.5 kilometres before self-destructing. This is not far at all when it is realised that cosmic-ray muons are created about 12.5 kilometres up in the air. Essentially none, therefore, should reach the ground.
Contrary to all expectations, however, every square metre of Earth's surface is struck by several hundred cosmic-ray muons every second. Somehow, these tiny particles manage to travel 25 times farther than they have any right to. And it is all because of relativity.
The time experienced by a speeding muon is not the same as the time experienced by someone on Earth's surface. Think of a muon as having an internal alarm clock that tells it when to decay. At 99.92 per cent of the speed of light, the clock slows down by a factor of about 25, at least to an observer on the ground. Consequently, cosmic-ray muons live 25 times longer than they would if stationary—time enough to travel all the way to the ground before they disintegrate. Cosmic-ray muons on the ground owe their very existence to time dilation.
What does the world look like from a muon's point of view? Or come to think of it, from the point of view of the space-faring twin or the atomic clock flown round the world? Well, from the point of view of all of these, time flows perfectly normally. Each, after all, is stationary with respect to itself. Take the muon. It still decays after 1.5 millionths of a second. From its point of view, however, it is standing still and it's Earth's surface that is approaching at 99.92 per cent of the speed of light. It therefore sees the distance it has to travel shrink by a factor of 25, enabling it to reach the ground even in its ultrashort lifetime.
The great cosmic conspiracy between time and space works whatever way you look at it.
## WHY RELATIVITY HAD TO BE
The behaviour of space and time at speeds approaching that of light is indeed bizarre. However, it need not have been a surprise to anyone. Although our everyday experience in nature's slow lane has taught us that one person's interval of time is another person's interval of time and that one person's interval of space is another person's interval of space, our belief in both of these things is in fact based on a very rickety assumption.
Take time. You can spend a lifetime trying futilely to define it. Einstein, however, realised that the only useful definition is a practical one. We measure the passage of time with watches and clocks. Einstein therefore said: "Time is what a clock measures." (Sometimes, it takes a genius to state the obvious!)
If everyone is going to measure the same interval of time between two events, this is equivalent to saying that their clocks run at the same rate. But as everyone knows, this never quite happens. Your alarm clock may run a little slow, your watch a little fast. We overcome these problems by, now and then, synchronising them. For instance, we ask someone the right time and, when they tell us, we correct our watch accordingly. Or we listen for the time signal "pips" on the BBC. But in using the pips, we make a hidden assumption. The assumption is that it takes no time at all for the radio announcement to travel to our radio. Consequently, when we hear the radio announcer say it is 6 a.m., it is 6 a.m.
A signal that takes no time at all travels infinitely fast. The two statements are entirely equivalent. But as we know, there is nothing in our Universe that can travel with infinite speed. On the other hand, the speed of radio waves—a form of light invisible to the naked eye—is so huge compared to all human distances that we notice no delay in their travel to us from the transmitter. Our assumption that the radio waves travel infinitely fast, although false, is not a bad one in the circumstances. But what happens if the distance from the transmitter is very large indeed? Say the transmitter is on Mars.
When Mars is at its closest, the signal takes 5 minutes to fly across space to Earth. If, when we hear the announcer on Mars say it is 6 a. m., we set our clock to 6 a. m., we will be setting it to the wrong time. The way around this is obviously to take into account the 5-minute time delay and, when we hear 6 a. m., set our clock to 6:05.
Everything, of course, hinges on knowing the time it takes for the signal to travel from Earth to Mars. In practice this can be done by bouncing a radio signal from Earth off Mars and picking up the return signal. If it takes 10 minutes for the round-trip, then it must take 5 minutes to travel from the spaceship to Earth.
The lack of an infinitely fast means of sending signals is not, therefore, a problem in itself for synchronising everyone's clocks. It can still be done by bouncing light signals back and forth and taking into account the time delays. The trouble is that this works perfectly only if everyone is stationary with respect to everyone else. In reality, everyone in the Universe is moving with respect to everyone else. And the minute you start bouncing light signals between observers who are moving, the peculiar constancy of the speed of light starts to wreak havoc with common sense.
Say there is a spaceship travelling between Earth and Mars and say it is moving so fast that, by comparison, Earth and Mars appear stationary. Imagine that, as before, you send a radio signal to Mars, which bounces off the planet and which you then pick up back on Earth. The round-trip takes 10 minutes, so, as before, you deduce that the signal arrived at Mars after only 5 minutes. Once again, if you pick up a time signal from Mars, saying it is 6 a. m., you will deduce from the time delay that it is really 6:05.
Now consider the spaceship. Assume that at the instant you send your radio signal to Mars, it sets off at its full speed to Mars. At what time does an observer on the spaceship see the radio signal arrive at Mars?
Well, from the observer's point of view, Mars is approaching, so the radio signal has a shorter distance to travel. But the speed of the signal is the same for you and for the observer on the spaceship. After all, that's the central peculiarity of light—it has exactly the same speed for everyone.
Speed, remember, is simply the distance something travels in a given time. So if the observer on the spaceship sees the radio signal travel a shorter distance and still measures the same speed, the observer must measure a shorter time too. In other words, the observer deduces that the radio signal arrives at Mars earlier than you deduce it does. To the observer, clocks on Mars tick more slowly. If the observer picks up a time signal from Mars, saying it is 6 a. m., the observer will correct it using a shorter time delay and conclude it is, say 6:03, not the 6:05 you conclude.
The upshot is that two observers who are moving relative to each other never assign the same time to a distant event. Their clocks always run at different speeds. What is more, this difference is absolutely fundamental—no amount of ingenuity in synchronising clocks can ever get around it.
## SHADOWS OF SPACE-TIME
The slowing of time and the shrinking of space is the price that must be paid so that everyone in the Universe, no matter what their state of motion, measures the same speed of light. But this is only the beginning.
Say there are two stars and a space-suited figure is floating in the blackness midway between them. Imagine that the two stars explode and the floating figure sees them detonate simultaneously—two blinding flashes of light on either side of him. Now picture a spaceship travelling at enormous speed along the line joining the two stars. The spaceship passes by the space-suited figure just as he sees the two stars explode. What does the pilot of the spaceship see?
Well, since the ship is moving towards one star and away from the other, the light from the star it is approaching will arrive before the light from the star it is receding from. The two explosions will therefore not appear simultaneously. Consequently, even the concept of simultaneity is a casualty of the constancy of the speed of light. Events that one observer sees as simultaneous are not simultaneous to another observer moving with respect to the first.
The key thing here is that the exploding stars are separated by an interval of space. Events that one person sees separated by only space, another person sees separated by space and time—and vice versa. Events one person sees separated only by time, another person sees separated by time and space.
The price of everyone measuring the same speed of light is therefore not only that the time of someone moving past you at high speed slows down while their space shrinks but that some of their space appears to you as time and some of their time appears to you as space. One person's interval of space is another person's interval of space and time. And one person's interval of time is another person's interval of time and space. The fact that space and time are interchangeable in this way tells us something remarkable and unexpected about space and time. Fundamentally, they are same thing—or at least different sides of the same coin.
The person who first saw this—more clearly even than Einstein himself—was Einstein's former mathematics professor Hermann Minkowski, a man famous for calling his student a "lazy dog" who would never amount to anything. (To his eternal credit, he later ate his words.) "From now on," said Minkowski, "space of itself and time of itself will sink into mere shadows and only a kind of union between them will survive."
Minkowski christened this peculiar union of space and time "space-time." Its existence would be blatantly obvious to us if we lived our lives travelling at close to the speed of light. Living as we do in nature's ultraslow lane, however, we never experience the seamless entity. All we glimpse instead are its space and time facets.
As Minkowski put it, space and time are like shadows of spacetime. Think of a stick suspended from the ceiling of a room so that it can spin around its middle and point in any direction like a compass needle. A bright light casts a shadow of the stick on one wall while a second bright light casts a shadow of the object on an adjacent wall. We could, if we wanted, call the size of the stick's shadow on one wall its "length" and the size of its shadow on the other wall its width. What then happens as the stick swings around?
Clearly, the size of the shadow on each wall changes. As the length gets smaller, the width gets bigger, and vice versa. In fact, the length appears to change into the width and the width into the length—just as if they are aspects of the same thing.
Of course, they are aspects of the same thing. The length and width are not fundamental at all. They are simply artifacts of the direction from which we choose to observe the stick. The fundamental thing is the stick itself, which we can see simply by ignoring the shadows on the wall and walking up to it at the centre of the room.
Well, space and time are much like the length and width of the stick. They are not fundamental at all but are artifacts of our viewpoint—specifically, how fast we are travelling. But though the fundamental thing is space-time, this is apparent only from a viewpoint travelling close to the speed of light, which of course is why it is not obvious to any of us in our daily lives.
Of course, the stick-and-shadow analogy, like all analogies, is helpful only up to a point. Whereas the length and width of the stick are entirely equivalent, this is not quite true of the space and the time facets of space-time. Though you can move in any direction you like in space, as everyone knows you can only move in one direction in time.
The fact that space-time is solid reality and space and time the mere shadows raises a general point. Like shipwrecked mariners clinging to rocks in a wild sea, to make sense of the world we search desperately for things that are unchanging. We identify things like distance and time and mass. But later, we discover that the things we identified as unchanging are unchanging only from our limited viewpoint. When we widen our perspective on the world we discover that other things we never suspected are the invariant things. So it is with space and time. When we see the world from a high-speed vantage point, we see neither space nor time but the seamless entity of spacetime.
Actually, we should long ago have guessed that space and time are inextricably entwined. Think of the Moon. What is it like now, at this instant? The answer is that we can never know. All we can ever know is what it was like 1¼ seconds ago, which is the time it takes light from the Moon to fly across the 400,000 kilometres to Earth. Now think of the Sun. We cannot know what it is like either, only what it was like 8½ minutes ago. And for the nearest star system, Alpha Centauri, it is even worse. We have to make do with a picture that by the time we see it is already 4.3 years out of date.
The point is that, although we think of the Universe we see through our telescopes as existing now, this is a mistaken view. We can never know what the Universe is like at this instant. The farther across space we look, the farther back in time we see. If we look far enough across space we can actually see close to the Big Bang itself, 13.7 billion years back in time. Space and time are inextricably bound together. The Universe we see "out there" is not a thing that extends in space but a thing that extends in space-time.
The reason we have been hoodwinked into thinking of space and time as separate things is that light takes so little time to travel human distances that we rarely notice the delay. When you are talking with someone, you see them as they were a billionth of a second earlier. But this interval is unnoticeable because it is 10 million times shorter than any event that can be perceived by the human brain. It is no wonder that we have come to believe that everything we perceive around us exists now. But "now" is a fictitious concept, which becomes obvious as soon as we contemplate the wider universe, where distances are so great that it takes light billions of years to span them.
The space-time of the Universe can be thought of as a vast map. All events—from the creation of the Universe in the Big Bang to your birth at a particular time and place on Earth—are laid out on it, each with its unique space-time location. The map picture is appropriate because time, as the flip side of space, can be thought of as an additional spatial dimension. But the map picture poses a problem. If everything is laid out—preordained almost—there is no room for the concepts of past, present, and future. As Einstein remarked: "For us physicists, the distinction between past, present, and future is only an illusion."
It is a pretty compelling illusion, though. Nevertheless, the fact remains that the concepts of past, present, and future do not figure at all in special relativity, one of our most fundamental descriptions of reality. Nature appears not to need them. Why we do is one of the great unsolved mysteries.
## _E = mc_ 2 AND ALL THAT
The special theory of relativity does more than profoundly change our ideas of space and time. It changes our ideas about a host of other things too. The reason is that all the basic quantities of physics are founded on space and time. If, as relativity tells us, space and time are malleable, blurring one into the other as the speed of light is approached, then so too are the other entities—momentum and energy, electric fields and magnetic fields. Like space and time, which merge into the seamless medium of space-time, they too are inextricably tied together in the interests of keeping the speed of light constant.
Take electricity and magnetism. It turns out that, just as one person's space is another person's time, one person's magnetic field is another person's electric field. Electric and magnetic fields are crucial to both generators that make electrical currents and motors that turn electric currents into motion. "The rotating armatures of every generator and every motor in this age of electricity are steadily proclaiming the truth of the relativity theory to all who have ears to hear," wrote the physicist Leigh Page in the 1940s. Because we live in a slow-motion world, we are hoodwinked into believing that electric and magnetic fields have separate existences. But just like space and time, they are merely different faces of the same coin. In reality there is only a seamless entity: the electromagnetic field.
Two other quantities that turn out to be different faces of the same coin are energy and momentum.5 And in this unlikely connection is hidden perhaps the greatest surprise of relativity—that mass is a form of energy. The discovery is encapsulated in the most famous, and least understood, formula in all of science: _E = mc_ 2.
1 Strictly speaking, each runner will also appear to rotate, so the spectators will see some of the far side of each of them—the side facing away from the grandstand, which would normally be hidden. This peculiar effect is known as relativistic aberration, or relativistic beaming. However, it is beyond the scope of this book.
2 To be precise, a stationary observer sees time slow down for a moving observer by a factor γ, where γ = 1/√(1 – ( _v_ 2/ _c_ 2)) and _v_ and _c_ are the speed of the moving observer and the speed of light, respectively. At speeds close to _c_ , γ becomes enormous and time for a moving observer slows almost to a standstill!
3 To be precise, a stationary observer sees the length of a moving body shrink by a factor γ, where γ = 1/√(1 – ( _v_ 2/ _c_ 2)) and v and c are the speed of the moving observer and the speed of light, respectively. At speeds close to _c_ , γ becomes enormous and a body becomes as flat as a pancake in the direction of its motion!
4 Actually, there is a subtle flaw in this argument. Since motion is relative, it is perfectly justifiable for your Earth-bound twin to assume that it is Earth that receded from your spacecraft at 99.5 per cent of the speed of light. However, this viewpoint leads to the opposite conclusion than before—that time slows for your twin relative to you. Clearly, time cannot run slowly for each of you, with respect to the other. The resolution of this twin paradox, as it is known, is to realise that your spaceship actually has to slow down and reverse its motion at Alpha Centauri. Because of this deceleration, the two points of view—your spaceship moving or Earth moving—are not really equivalent and interchangeable.
5 The momentum of a body is a measure of how much effort is required to stop it. For instance, an oil tanker, even though it may be moving only a few kilometres an hour, is far harder to stop than a Formula 1 racing car going 200 kilometres per hour. We say the oil tanker has more momentum.
# _E = MC_2 AND THE WEIGHT OF SUNSHINE
HOW WE DISCOVERED THAT ORDINARY MATTER CONTAINS A MILLION TIMES THE DESTRUCTIVE POWER OF DYNAMITE
_Photons have mass?!? I didn't even know they were Catholic._
Woody Allen
_It's the biggest set of bathroom scales imaginable. And, oh, yes, it's heat_ _resistant too. It's so big in fact that it can weigh a whole star. And today_ _it's weighing the nearest star of all: our Sun. The digital display has just_ _come to rest and it's registering 2 × 10 27_ _tons. That's 2 followed by 27_ _zeroes—2,000 million million million million tons. But wait a minute,_ _something's wrong. The scales are superaccurate. That's another remarkable_ _thing about them, in addition to their size and heat resistance! But_ _every second, when the display is refreshed, it reads 4 million tons less_ _than it did the previous second. What's going on? Surely the Sun isn't_ _really getting lighter—by the weight of a good-sized supertanker—every_ _single second?_
Ah, but it is! The Sun is losing heat-energy, radiating it into space as sunlight. And energy actually _weighs_ something.1 So the more sunlight the Sun gives out, the lighter it gets. Mind you, the Sun is big and we're only talking about it losing about a _10-million-_ _million_ - _millionth_ of a per cent of its mass per second. That's hardly more than 0.1 per cent of its mass since its birth.
The fact that energy does indeed weigh something can be seen vividly from the behaviour of a comet. The tail of a comet always points away from the Sun just like a windsock points away from the gathering storm.2 What do the two have in common? Both are being pushed by a powerful wind. In the case of the windsock, it's a wind of air; in the case of the comet tail, a wind of light streaming outward from the Sun.
The windsock is being hit by air molecules in their countless trillions. It is this relentless bombardment that is pushing the fabric and causing it to billow outward. The story is pretty much the same out in deep space. The comet tail is being battered by countless tiny particles of light. It is the machine-gun bombardment of these photons that is causing the glowing cometary gases to billow across tens of millions of kilometres of empty space.3
But there is an important difference between the windsock being struck by air molecules and the comet's tail being hit by photons. The air molecules are solid grains of matter. They thud into the material of the windsock like tiny bullets, and this is why the windsock recoils. But photons are not solid matter. They actually have no mass. How then can they be having a similar effect to air molecules, which do?
Well, one thing photons certainly do have is energy. Think of the heat that sunlight deposits on your skin when you sunbathe on a summer's day. The inescapable conclusion is that the energy must actually _weigh_ something.4
This turns out to be a direct consequence of the uncatchability of light. Because the speed of light is terminally out of reach, no material body can ever be accelerated to the speed of light, no matter how hard it is pushed. The speed of light, recall, plays the role of infinite speed in our Universe. Just as it would take an infinite amount of energy to accelerate a body to infinite speed, it would take an infinite amount of energy to push one to light speed. In other words, the reason that getting to the speed of light is impossible is because it would take more energy than is contained in the Universe.
What would happen, however, if you were to push a mass closer and closer to the speed of light? Well, since the ultimate speed is unattainable, the body would have to become harder and harder to push as you get it closer and closer to the ultimate speed.
Being hard to push is the same as having a big mass. In fact, the mass of a body is defined by precisely this property—how hard it is to push it. A loaded refrigerator which is difficult to budge, is said to have a large mass, whereas a toaster, which is easy to budge, is said to have a small mass. It follows therefore that, if a body gets harder to push as it approaches the speed of light, it must get more massive. In fact, if a material body was ever to attain the speed of light itself, it would acquire an infinite mass, which is just another way of saying its acceleration would take an infinite amount of energy. Whatever way you look at it, it's an impossibility.
Now, it is a fundamental law of nature that energy can neither be created or destroyed, only transformed from one guise into another. For instance, electrical energy changes into light energy in a lightbulb; sound energy changes into the energy of motion of a vibrating diaphragm in a microphone. What, then, happens to the energy put into pushing a body that is moving close to the speed of light? Hardly any of the energy can go into increasing the body's speed since a body moving at close to the speed of light is already travelling within a whisker of the ultimate speed limit.
The only thing that increases as the body is pushed harder and harder is its mass. This, then, must be where all the energy goes. But, recall, energy can only be changed from one form into another. The inescapable conclusion, discovered by Einstein, is therefore that mass itself is a _form of energy_. The formula for the energy locked up in a chunk of matter of mass, _m_ , is given by perhaps the most famous equation in all of science: _E = mc_ 2, where _c_ is the scientists' shorthand for the speed of light.
The connection between energy and mass is perhaps the most remarkable of all the consequences of Einstein's special theory of relativity. And like the connection between space and time, it is a two-way thing. Not only is mass a form of energy, but energy has an effective mass. Put crudely, _energy weighs something_.
Sound energy, light energy, electrical energy—any form of energy you can think of—they all weigh something. When you warm up a pot of coffee, you add heat-energy to it. But heat-energy weighs something. Consequently, a cup of coffee weighs slightly more when hot than when cold. The operative word here is slightly. The difference in weight is far too small to measure. In fact, it is far from obvious that energy has a weight, which is of course why it took the genius of Einstein to first notice it. Nevertheless, one form of energy at least—the energy of sunlight—does reveal its mass when it interacts with a comet.
Light can push the tail of a comet because light energy weighs something. Photons have an effective mass by virtue of their energy.
Another familiar form of energy is energy of motion. If you step into the path of a speeding cyclist, you will be left in no doubt that such a thing exists. Energy of motion, like all other forms of energy, weighs something. So you weigh marginally more when you are running than when you are walking.
It is energy of motion that explains why the photons of sunlight can push a comet tail. An explanation is needed because they actually have no _intrinsic_ _mass_. If they did, after all, they would be unable to travel at the speed of light, a speed that is forbidden to all bodies with mass. What light has instead is an effective mass—a mass by virtue of the fact that it has energy of motion.
The existence of energy of motion also explains why a cup of coffee is heavier when hot than when cold. Heat is microscopic motion. The atoms in a liquid or solid jiggle about, while the atoms in a gas fly hither and thither. Because the atoms in a cup of hot coffee are jiggling faster than the atoms in a stone-cold cup, they possess more energy of motion. Consequently, the coffee weighs more.
## RABBITS OUT OF HATS
So much for energy having an equivalent mass, or weighing something. The fact that mass is a form of energy also has profound implications. Since one form of energy can be converted into another, mass-energy can be transformed into other forms of energy and, conversely, other forms of energy can be changed into mass-energy.
Take the latter process. If mass-energy can be made out of other forms of energy, it follows that mass can pop into existence where formerly no mass existed. This is exactly what happens in giant particle accelerators, or atom smashers. At CERN, the European centre for particle physics near Geneva, for instance, subatomic particles—the building blocks of atoms—are whirled around a subterranean racetrack and slammed together at speeds approaching that of light. In the violent smash-up, the tremendous energy of motion of the particles is converted into mass-energy—the mass of new particles that physicists wish to study. At the collision point, these particles appear apparently out of nothing, like rabbits out of a hat.
This phenomenon is an instance of one type of energy changing into mass-energy. But what about mass-energy changing into another type of energy? Does that happen? Yes, all the time.
## A MILLION TIMES THE DESTRUCTIVE POWER OF DYNAMITE
Think of a piece of burning coal. Because the heat it gives out weighs something, the coal gradually loses mass. So if it were possible to collect and weigh all the products of burning—the ash, the gases given off, and so on—they would turn out to weigh less than the original lump of coal.
The amount of mass-energy turned into heat-energy when coal burns is so small as to be essentially unmeasurable. Nevertheless, there is a place in nature where a significant mass is converted into other forms of energy. It was identified by the English physicist Francis Aston in 1919 while he was "weighing" atoms.
Recall that each of the 92 naturally occurring atoms contains a nucleus made from two distinct subatomic particles—the proton and neutron.5 Since the masses of these two nucleons are essentially the same, the nucleus, at least as far as its weight is concerned, can be thought of as being made from a single building block. Think of it as a Lego brick. Hydrogen, the lightest nucleus, is therefore made from one Lego brick; uranium, the heaviest, is made from 238 Lego bricks.
Now, there had been a suspicion since the beginning of the 19th century that perhaps the Universe had started out with only one kind of atom—the simplest, hydrogen. Since that time, all the other atoms have somehow been built up from hydrogen, by the process of sticking together hydrogen Lego bricks. The evidence for the idea, which had been proposed by a London physician named William Prout in 1815, was that an atom like lithium appeared to weigh exactly six times as much as hydrogen, an atom like carbon exactly 12 times as much, and so on.
However, when Aston compared the masses of different kinds of atoms more precisely with an instrument he invented called a mass spectrograph, he discovered something different. Lithium in fact weighed a shade less than six hydrogen atoms; carbon weighed a shade less than 12 hydrogen atoms. The biggest discrepancy was helium, the second lightest atom. Since a helium nucleus was assembled from four Lego bricks, by rights it should weigh four times as much as a hydrogen atom. Instead, it weighed 0.8 per cent less than four hydrogen atoms. It was like putting four 1-kilogram bags of sugar on a set of scales and finding that they weighed almost 1 per cent less than 4 kilograms!
If all atoms had indeed been assembled out of hydrogen atom Lego bricks, as Prout strongly suspected, Aston's discovery revealed something remarkable about atom building. During the process, a significant amount of mass-energy went AWOL.
Mass-energy, like all forms of energy, cannot be destroyed. It can only be changed from one form into another, ultimately the lowest form of energy—heat-energy. Consequently, if 1 kilogram of hydrogen was converted into 1 kilogram of helium, 8 grams of mass-energy would be converted into heat-energy. Amazingly, this is a million times more energy than would be liberated by burning 1 kilogram of coal!
This factor of a million did not go unnoticed by astronomers. For millennia, people had wondered what kept the Sun burning. In the 5th century BC, the Greek philosopher Anaxagoras had speculated that the Sun was "a red-hot ball of iron not much bigger than Greece." Later, in the 19th century, the age of coal, physicists had naturally wondered whether the Sun was a giant lump of coal. It would have to be the mother of all lumps of coal! They found, however, that if it was a lump of coal, it would burn out in about 5,000 years. The trouble is that the evidence from geology and biology is that Earth—and by implication the Sun—is at least a million times older. The inescapable conclusion is that the Sun is drawing on an energy source a million times more concentrated than coal.
The man who put two and two together was English astronomer Arthur Eddington. The Sun, he guessed, was powered by atomic, or nuclear, energy. Deep in its interior it was sticking together the atoms of the lightest substance, hydrogen, to make atoms of the second lightest, helium. In the process, mass-energy was being turned into heat and light energy. To maintain the Sun's prodigious output, 4 million tons of mass—the equivalent of a million elephants—was being destroyed every second. Here, at last, was the ultimate source of sunlight.
This discussion conveniently skirts over the matter of why making a heavy atom out of a light atom converts so much mass-energy into other forms of energy. A digression may help.
Imagine you are walking past a house and a slate falls from the roof and hits you on the head. Energy is released in this process. For instance, there is the whack as the slate hits your head—sound energy. Maybe it even knocks you over. Then there is heat energy. If you could measure the temperature of the slate and your head very accurately, you would find they were slightly warmer than before.
Where did all this energy come from? The answer is from gravity. Gravity is a force of attraction between any two massive bodies. In this case, the gravity between Earth and the slate pulls them closer together.
Now, what would happen if gravity was twice as strong as it is? Clearly, the slate would be pulled towards Earth faster. It would make a bigger noise when it hit, create more heat, and so on. In short, more energy would be released. What if gravity were 10 times stronger? Well, even more energy would be unleashed. Now, what if gravity was 10,000 trillion trillion trillion times stronger? Obviously, a mind-bogglingly huge amount of energy would be released by the crashing slate (and the combination of Earth and slate would be lighter, like the helium atom).
But isn't this just fantasy? Surely, there is no force that is 10 trillion trillion trillion times stronger than gravity? Well, there is, and it is operating in each and every one of us at this very moment! It is called the nuclear force, and it is the glue that holds together the nuclei of atoms.
Imagine what would happen if you took the nuclei of two light atoms and let them fall together under the nuclear force rather like the slate and Earth falling together under gravity. The collision would be tremendously violent and an enormous amount of energy would be liberated—a million times more energy than would be released by burning the same weight of coal.
Atom building is not only the source of the Sun's energy. It is also the source of the energy of the hydrogen bomb. For that's all H-bombs do—slam together hydrogen nuclei (normally, a heavy cousin of hydrogen, but that's another story) to make nuclei of helium. The helium nuclei are lighter than the combined weight of their hydrogen building blocks, and the missing mass reappears as the tremendous heat energy of the nuclear fireball. The destructive power of a 1-megaton hydrogen bomb—about 50 times greater than the one that devastated Hiroshima—comes from the destruction of little more than a kilogram of mass. "If only I had known, I should have become a watchmaker!" said Einstein, reflecting on his role in the development of the nuclear bomb.
## TOTAL CONVERSION OF MASS INTO ENERGY
Even though Einstein downgraded mass, showing that it was merely one among countless other forms of energy, it is special in one way: It is the most concentrated form of energy known. In fact, the equation _E = mc_ 2 encapsulates this fact. The physicists' symbol for the speed of light, _c_ , is a big number—300 million metres per second. Squaring it—multiplying it by itself—creates an even bigger number. Applying the formula to 1 kilogram of matter shows that it contains 9 × 1016 joules of energy—enough to lift the entire population of the world into space!
Of course, to get this kind of energy out of a kilogram of matter, it would be necessary to convert it entirely into another form of energy—that is, to destroy all of its mass. The nuclear processes in the Sun and a hydrogen bomb liberate barely 1 per cent of the energy locked up in matter. However, it turns out that nature can do far better than this.
Black holes are regions of space where gravity is so strong that light itself cannot escape—hence their blackness. They are the remnant left behind when a massive star dies, shrinking catastrophically in size until they literally wink out of existence. As matter swirls down into a black hole, like water down a plug hole, it rubs against itself, heating itself to incandescence. Energy is unleashed as both light and heat. In the special case when a black hole is spinning at its maximum possible rate, the liberated energy is equivalent to 43 per cent of the mass of the matter swirling in. This means that, pound for pound, the in-fall of matter onto a black hole is 43 times more efficient at generating energy than the nuclear processes powering the Sun or an H-bomb.
And this isn't just theory. The Universe contains objects called quasars, the superbright cores of newborn galaxies. Even our own Milky Way galaxy may have had a quasar in its heart in its wayward youth 10 billion years ago. The puzzling thing about quasars is that they often pump out the light energy of 100 normal galaxies—that's 10 million million suns—and from a tiny region smaller than our solar system. All that energy cannot be coming from stars; it would be impossible to squeeze 10 million million suns into such a small volume of space. It can only come from a giant black hole sucking in matter. Astronomers, therefore, firmly believe that quasars contain "supermassive" black holes—up to 3 billion times the mass of the Sun—that are steadily gobbling whole stars. But even black holes can convert barely half of the mass of matter into other forms of energy. Is there a process that can convert all of the mass into energy? The answer is _yes_. Matter actually comes in two types—matter and antimatter. It is not necessary to know anything about antimatter other than the fact that, when matter and antimatter meet, the two destroy, or annihilate each other, with 100 per cent of their mass-energy flashing instantly into other forms of energy.
Now, our Universe, for a reason nobody knows, appears to be made almost entirely of matter. This is a deep puzzle because, when tiny amounts of antimatter are made in the laboratory, their birth is always accompanied by an equal amount of matter. Because there is essentially no antimatter in the Universe, if we want any we have to make it. It's difficult. Not only do you have to put in a lot of energy to make it—as much energy as you are likely to get out!—but it tends to annihilate as soon as it meets ordinary matter, so it's difficult to accumulate a lot of it. So far scientists have managed to collect less than a billionth of a gram.
Nevertheless, if the problem of making antimatter in quantities could be cracked, we would have at our disposal the most powerful energy source imaginable. The problem with all spacecraft is that they have to take their fuel along with them. But that fuel weighs a lot. So fuel is needed to lift the fuel into space. The _Saturn V_ rocket, for instance, weighed 3,000 tons and all that weight—mostly fuel—was needed to take two men to the surface of the Moon and return them safely to Earth. Antimatter offers a way out. A spacecraft would require hardly any antimatter to fuel it because antimatter contains such a tremendous amount of energy pound for pound. If we ever manage to travel to the stars, we will have to squeeze every last drop of energy out of matter. Just as in _Star Trek_ , we will have to build starships powered by antimatter.
1 I am using the word weight here the way it is used in everyday life as synonymous with mass. Strictly speaking, weight is equivalent to the force of gravity.
2 A comet is a giant interplanetary snowball. Billions of such bodies are believed to orbit in the deep freeze beyond the outermost planet. Occasionally, one is nudged by the gravity of a passing star and falls toward the Sun. As it heats up, its surface cracks, and buckles, and boils off into the vacuum to form a long, glowing tail of gas.
3 Actually, the tail of a comet is pushed by a combination of the light from the Sun and the solar wind, the million-mile-an-hour hurricane of subatomic particles—mostly hydrogen nuclei—that streams out from the Sun.
4 Strictly speaking, the thing photons possess is momentum. In other words, it takes an effort to stop them. This effort is provided by the comet's tail, which recoils as a result.
5 Except, of course, the most common isotope of hydrogen, the nucleus of which consists simply of one proton and no neutrons.
# THE FORCE OF GRAVITY DOES NOT EXIST
HOW WE DISCOVERED THE TRUTH ABOUT GRAVITY AND CAME FACE TO FACE WITH BLACK HOLES, WORMHOLES, AND TIME MACHINES
_The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly the thought struck me: If a man falls freely, he does not feel his own weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity._
Albert Einstein
_They are 20-year-old twin sisters. They work in the same skyscraper in_ _Manhattan. One is an assistant in a boutique at street level, the other a_ _waitress at the High Roost restaurant on the 52nd floor. It's 8:30 a.m._ _They come through the revolving doors into the foyer and go their separate_ _ways. One heads across the marble expanse to the ground-floor_ _shopping mall; the other sprints into the mouth of the high-speed elevator_ _just before the doors swish shut._
_The hands of the clock above the elevator spin around. Now it's 5:30_ _p.m. On the ground floor the shop-assistant twin stares up at the big red_ _indicator light as it counts down the floors. With a "ding," the doors_ _burst open and out comes her waitress sister... an 85-year-_ _old bent_ _figure clutching a silver zimmer frame!_
If you think this scenario is pure fantasy, think again. It's an exaggeration, granted, but it's an exaggeration of the truth. You really do age more slowly on the ground floor of a building than on the top floor. It's an effect of Einstein's "general" theory of relativity, the framework he came up with in 1915 to fix the shortcomings of his special theory of relativity.
The problem with the special theory of relativity is that, well, it is _special_. It relates what one person sees when looking at another person moving at constant speed relative to them, revealing that the moving person appears to shrink in the direction of their motion while their time slows down, effects that become ever more marked as they approach the speed of light. But motion at constant speed is of a very special kind. Bodies in general change their speed with time—for instance, a car accelerates away from traffic lights or NASA's space shuttle slows when it reenters Earth's atmosphere.
The question Einstein therefore set out to answer after he published his special theory of relativity in 1905 was: What does one person see when looking at another person accelerating relative to them? The answer, which took him more than a decade to obtain, was contained in the "general" theory of relativity, arguably the greatest contribution to science by a single human mind.
When Einstein embarked on his quest, one problem in particular worried him: what to do about Newton's law of gravity. Although it had stood unchallenged for almost 250 years, it was clear to Einstein that it was fundamentally incompatible with the special theory of relativity. According to Newton, every massive body tugs on every other massive body with an attractive force called gravity. For instance, there is a gravitational pull between Earth and each and every one of us; it keeps our feet glued firmly to the ground. There is a gravitational pull between the Sun and Earth, which keeps Earth trapped in orbit around the Sun. Einstein did not object to this idea. His difficulty was with the speed of gravity.
Newton assumed that the force of gravity acts instantaneously—that is, the Sun's gravity reaches out across space to Earth and Earth feels the tug of that gravity without any delay. Consequently, if the Sun were to vanish at this very moment—an unlikely scenario!—Earth would notice the absence of the Sun's gravity instantly and promptly fly off into interstellar space.
An influence that can cross the gulf between the Sun and Earth in no time at all must travel infinitely fast—instantaneous travel and infinite speed are completely equivalent. However, as Einstein discovered, nothing—and that necessarily includes gravity—can travel faster than light. Since light takes just over eight minutes to travel between the Sun and Earth, it follows that, if the Sun were to vanish suddenly, Earth would continue merrily in its orbit for at least eight and a bit minutes before spinning off to the stars.
Newton's tacit assumption that gravity reaches out across space at infinite speed is not the only serious flaw in his theory of gravity. He also assumed that the force of gravity is generated by mass. Einstein, however, discovered that all forms of energy have an effective mass, or weigh something. Consequently, all forms of energy—not just mass-energy—must be sources of gravity.
The challenge facing Einstein was, therefore, to incorporate the ideas of the special theory of relativity into a new theory of gravity and, at the same time, to generalise the special theory of relativity to describe what the world looked like to an accelerated person. It was as he contemplated these gargantuan challenges that a lightbulb lit up in Einstein's head. He realised, to his surprise and delight, that the two tasks were one and the same.
## THE ODD THING ABOUT GRAVITY
To understand the connection it is necessary to appreciate a peculiar property of gravity. All bodies, irrespective of their mass, fall at the same rate. A peanut, for instance, picks up speed just as quickly as a person. This behaviour was first noticed by the 17th-century Italian scientist Galileo. In fact, Galileo is reputed to have demonstrated the effect by taking a light object and a heavy object and dropping them together from the top of the Leaning Tower of Pisa. Reportedly, they hit the ground at the same time.
On Earth the effect is obscured because objects with a large surface area are preferentially slowed by their passage through the air. Nevertheless, Galileo's experiment can be carried out in a place where there is no air resistance to mess things up—the Moon. In 1972, _Apollo 15_ commander Dave Scott dropped a hammer and a feather together. Sure enough, they hit the lunar soil at exactly the same time.
What is peculiar about this phenomenon is that, usually, the way in which a body moves in response to a force depends on its mass. Imagine a wooden stool and a loaded refrigerator standing on an ice rink, where there is no friction to confuse things. Now imagine that someone pushes the refrigerator and the stool with exactly the same force. The stool, being less massive than the refrigerator will obviously budge more easily and pick up speed more quickly.
What happens, however, if the stool and the refrigerator are acted on by the force of gravity? Say someone tips them both off the roof of a 10-story building? In this case, as Galileo himself would have predicted, the stool will not pick up speed faster than the refrigerator. Despite their wildly different masses, the stool and the refrigerator will accelerate towards the ground at exactly the same rate.
Now, perhaps you appreciate the central peculiarity of gravity. A big mass experiences a bigger force of gravity than a small mass, and that force is in direct proportion to its mass, so the big mass accelerates at exactly the same rate as the small mass. But how does gravity adjust itself to the mass it is acting on? It was Einstein's genius to realise that it does so in an incredibly simple and natural way—a way, furthermore, that has profound implications for our picture of gravity.
## THE EQUIVALENCE OF GRAVITY AND ACCELERATION
Say an astronaut is in a room accelerating upwards at 9.8 metres per second per second, which is the acceleration gravity imparts to falling bodies near Earth's surface. Think of the room as a cabin in a spacecraft whose rocket engines have just started firing. Now, say the astronaut takes a hammer and a feather, holds them out from him at the same height above the floor of the cabin, then lets them go simultaneously. What happens? Well, the hammer and feather meet the floor of course. How this event is interpreted, though, depends entirely on the particular viewpoint.
Assuming the spacecraft is far away from the gravity of any big masses like planets, the hammer and the feather are weightless. So if we look into the spacecraft from outside with some kind of X-ray vision, we see the two objects hanging motionless. However, because the spacecraft is accelerating upward, we see the floor of the cabin racing up to meet the hammer and the feather. When it strikes them, furthermore, it strikes them both simultaneously.
Say the astronaut has amnesia and has entirely forgotten he is in a spacecraft. The portholes, in addition, are blacked out so there is nothing to tell him where he is. How does he interpret what he sees?
Well, the astronaut maintains that the hammer and the feather have fallen under gravity. After all, they have done the one thing a hammer and a feather experiencing gravity would do—they have fallen at the same rate and hit the ground at the same time (ignoring air resistance of course). The astronaut is further convinced that gravity is responsible for what he has seen by the fact that his feet appear to be glued to the floor just as they would be if he was in a room on Earth's surface. In fact, everything the astronaut experiences turns out to be indistinguishable from what he would experience if he was on Earth's surface.
Of course, it could be a coincidence. Einstein, however, was convinced he had stumbled onto a deep truth about nature. Gravity is indeed indistinguishable from acceleration, and the reason for that could not be simpler. Gravity is acceleration! This realisation, which Einstein later called "the happiest thought of my life," convinced him that the search for a theory of gravity and for a theory that described accelerated motion were one and the same thing.
Einstein elevated the indistinguishability of gravity and acceleration to a grand principle of physics, which he christened the principle of equivalence. The principle of equivalence recognises that gravity is not like other forces. In fact, it is not even a real force. We are all like the amnesiac astronaut in the blacked-out spacecraft. We do not realise that our surroundings are accelerating and so have to find some other way to explain away the fact that rivers flow downhill and apples fall from trees. The only way is to invent a fictitious force—gravity.
## THE FORCE OF GRAVITY DOES NOT EXIST!
The idea that gravity is a fictitious force may sound a little far-fetched. However, in other everyday situations, we are perfectly happy to invent forces to make sense of what happens to us. Say you are a passenger in a car that is racing round a sharp corner in the road. You appear to be flung outward and, to explain why, you invent a force—centrifugal force. In reality, however, no such force exists.
All massive bodies, once set in motion, have a tendency to keep travelling at constant speed in a straight line.1 Because of this property, known as inertia, unrestrained objects inside the car, including a passenger like you, continue to travel in the same direction the car was travelling before it rounded the bend. The path followed by the car door however, is a curve. It should be no surprise, then, that you find yourself jammed up against a door. But the car door has merely come to meet you in the same way that the floor of the accelerating spacecraft came up to meet the hammer and feather.2 There is no force.
Centrifugal force is known as an inertial force. We invent it to explain our motion because we choose to ignore the truth—that our surroundings are moving relative to us. But, really, our motion is just a result of our inertia, our natural tendency to keep moving in a straight line. It was Einstein's great insight to realise that gravity too is an inertial force. "Can gravitation and inertia be identical?" asked Einstein. "This question leads directly to my theory of gravity."
According to Einstein, we concoct the force of gravity to explain away the motion of apples falling from trees and planets circling the Sun because we ignore the truth—that our surroundings are accelerating relative to us. In reality, things move merely as a result of their inertia. The force of gravity does not exist!
But wait a minute. If the motion we attribute to the force of gravity is actually just the result of inertia, that must mean that bodies like Earth are really just flying through space at constant speed in straight lines. That's patently ridiculous! Earth is circling the Sun and not flying in a straight line, right? Not necessarily. It all depends on how you define a straight line.
## GRAVITY IS WARPED SPACE
A straight line is the shortest path between two points. This is certainly true on a flat piece of paper. But what about on a curved surface—for instance, the surface of Earth? Think of a plane flying the shortest route between London and New York. What path does it take? To someone looking down from space, it is obvious—a curved path. Think of a hiker trekking between two points in a hilly landscape. What path does the hiker take? To someone looking down on the hiker from a vantage point so high that the undulations of the landscape cannot be seen, the path of the hiker wiggles back and forth in the most tortuous manner.
Contrary to expectations, then, the shortest path between two points is not always a straight line. In fact, it is only a straight line on a very special kind of surface—a flat one. On a curved surface like Earth's, the shortest route between two points is always a curve. In light of this point, mathematicians have generalised the concept of a straight line to include curved surfaces. They define a geodesic to be the shortest path between two points on any surface, not just a flat one.
What has all this got to do with gravity? The connection, it turns out, is light. It is a characteristic property of light that it always takes the shortest route between two points. For instance, it takes the shortest path from these words you are reading to your eyes.
Now think back to the amnesiac astronaut in his accelerating, blacked-out spacecraft. Bored of experimenting with a hammer and feather, he gets out a laser and places it on a shelf on the left-hand wall of his cabin, at a height of say 1.5 metres. He then crosses to the right-hand wall of the cabin and, with a marker pen, draws a red line, also at a height of 1.5 metres. Finally, the astronaut turns on the laser so that its beam stabs horizontally across the cabin. Where does it strike the right-hand wall?
It stands to reason that, since the astronaut has fired the beam horizontally, it will hit the wall exactly on the red line. So does it? The answer is _no!_
While the light is in flight across the cabin, the floor of the spacecraft is all the time being boosted by the rocket motors. Consequently, the floor is moving steadily upward to meet the beam. As the light gets closer and closer to the right-hand wall, the floor gets closer and closer to the light. Or from the point of view of the astronaut, the light gets closer and closer to the floor. Clearly, when the beam hits the right-hand wall, it hits it below the red line. The astronaut sees the light beam curving steadily downward as it crosses the cabin.
Now light, remember, always takes the shortest path between two points. The shortest path on something that is flat is a straight line, whereas the shortest path on something that is curved is a curve. What then are we to make of the fact that the light beam follows a curved trajectory across the spacecraft cabin? There is only one possible inter-pretation: The space inside the cabin is in some sense curved.
Now, you can argue that this is just an illusion caused by the accelerating spacecraft. The crucial point, however, is that the astronaut has no way of knowing that he is in an accelerating spacecraft. He could just as well be experiencing gravity in a room on Earth's surface. Acceleration and gravity are indistinguishable. This is the principle of equivalence. What the experiment with the laser beam is actually demonstrating—and this shows the tremendous power of the principle of equivalence—is that light in the presence of gravity follows a curved trajectory. Or to put it another way, gravity bends the path of light.
Gravity bends light because space, in the presence of gravity, is somehow curved. In fact, this is all gravity turns out to be—curved space.
What exactly do we mean by curved space? It is easy to visualise a curved surface like the surface of Earth. But that is because it has only two directions, or dimensions—north-south and east-west. Space is a bit more complicated than that. In addition to three space dimensions—north-south, east-west, and up-down—there is one time dimension—past-future. As Einstein showed, however, space and time are really just aspects of the same thing, so it is more accurate to think of there being four "space-time" dimensions.
Four-dimensional space-time is impossible for us to visualise since we live in a world of three-dimensional objects. This means that the curvature, or warpage, of four-dimensional space-time is doubly impossible to visualise. But that's what gravity is: the warpage of four-dimensional space-time.
Fortunately, we can get some idea of what this means. Imagine a race of ants that spends its entire existence on the two-dimensional surface of a taut trampoline. The ants can only see what happens on the surface and have no concept whatsoever of the space above and below the trampoline—the third dimension. Now imagine that you or I—mischievous beings from the third dimension—put a cannonball on the trampoline. The ants discover that when they wander near the cannonball their paths are mysteriously bent towards it. Quite reasonably, they explain their motion by saying that the cannonball is exerting a force of attraction on them. Perhaps they even call the force gravity.
However, from the God-like vantage point of the third dimension, it is clear the ants are mistaken. There is no force attracting them to the cannonball. Instead, the cannonball has made a valleylike depression in the trampoline, and this is the reason the paths of the ants are bent towards it.
Einstein's genius was to realise that we are in a remarkably similar position to the ants on the trampoline. The path of Earth as it travels through space is constantly bent towards the Sun, so much so that the planet traces out a near-circular orbit. Quite reasonably, we explain away this motion by saying that the Sun exerts a force of attraction on Earth—the force of gravity. However, we are mistaken. If we could see things from the God-like perspective of the fourth dimension—something that is as impossible for us to do as it is for the ants to see things from the third dimension—we would see there is no such force. Instead, the Sun has created a valleylike depression in the four-dimensional space-time in its vicinity, and the reason Earth follows a near-circular path around it is because this is the shortest possible path through the warped space.
There is no force of gravity. Earth is merely following the straightest possible line through space-time. It is because space-time near the Sun is warped that that line happens to be a near-circular orbit. According to physicists Raymond Chiao and Achilles Speliotopoulos: "In general relativity, no 'gravitational force' exists. What we normally associate with the force of gravity on a particle is not a force at all: The particle is simply travelling along the 'straightest' possible path in curved space-time."
A body travelling along the "straightest" possible path through space-time is in free fall. And, since it is in free fall, it experiences no gravity. Earth is in free fall around the Sun. Consequently, we do not feel the Sun's gravity on Earth. The astronauts on the International Space Station are in free fall around Earth. Consequently, they do not feel Earth's gravity.3
Gravity arises only when a body is prevented from following its natural motion. Our natural motion is free fall towards the centre of Earth. The ground thwarts us, however, so we feel its force on our bodies, which we interpret as gravity. Just as centrifugal force is what we feel when a cornering car prevents us from following the natural motion in a straight line, the force of gravity is what we feel when our surroundings prevent us from following our natural motion along a geodesic.
Probably, it seems unnecessarily complicated to view massive bodies as moving under their own inertia through warped space-time rather than simply moving under the influence of a universal force of gravitational attraction. However, the two pictures are not equivalent. Einstein's is superior. For a start, the thing that is warped is not merely space but the space-time of special relativity. The picture, therefore, automatically incorporates the peculiar interplay between space and time necessary to keep the speed of light a constant. Einstein's picture also predicts new things.
Think of those ants on the trampoline. There are more things you can do with the material of the trampoline than merely depress it with a heavy mass like a cannonball. For instance, you could shake one corner up and down. This would cause ripples in the fabric to spread outwards across the trampoline like ripples on the surface of a pond. In the same way, the vibration of a large mass like a black hole out in space can generate ripples in the "fabric" of space-time. Such gravitational waves have yet to be detected directly, but their existence is a unique prediction of Einstein's theory.
The fact that waves can ripple through space-time suggests that space is not the empty, passive medium imagined by Newton. Instead, it is an active medium with real properties. Matter does not simply pull on other matter across empty space, as Newton imagined. Matter distorts space-time, and it is this distorted space-time that in turn affects other matter. As John Wheeler put it: "Mass tells spacetime how to warp and warped space-time tells mass how to move."
The distortion of space-time caused by a massive body takes time to propagate to another mass, just as the distortion of the trampoline by another cannonball takes time to reach the corners of the trampoline. Because of this, gravity—warped space-time—acts only after a delay, in perfect accord with the cosmic speed limit set by the speed of light.
The fact that space-time has some of the qualities of a real medium like air or water has implications for large bodies like planets and stars. When they rotate on their axes, they actually drag spacetime around with them. NASA has measured the effect, known as frame dragging, with an orbiting space experiment called Gravity Probe B. Frame dragging is tiny in the case of Earth but overwhelming in the case of a rapidly spinning black hole. Such a body sits at the eye of a great tornado of spinning space-time. Anyone falling into the black hole would be whirled around with the tornado, which no power in the Universe could defy.
## THE RECIPE OF GENERAL RELATIVITY
Einstein's novel take on gravity is now clear. Masses—for instance, stars like the Sun—warp the space-time around them. Other masses—for instance, planets like Earth—then fly freely under their own inertia through the warped space-time. The paths they follow are curved because these are the shortest possible paths in warped space. This is it. This is the general theory of relativity.
The devil, however, is in the details. We know how a massive body like a planet moves in warped space. It takes the shortest possible path. But how precisely does a mass like the Sun warp the space-time around it? It took Einstein more than a decade to find out, and the details would fill a textbook as big as a phone directory. However, Einstein's starting point for the general theory of relativity is not difficult to appreciate. It is none other than the principle of equivalence.
Recall again the hammer and the feather in the blacked-out spacecraft. To the astronaut, they appeared to fall to the floor under gravity. To someone watching the experiment from outside the spacecraft, however, it was obvious that the hammer and the feather were hanging in midair and that the floor of the cabin was accelerating upwards to meet them. They were completely weightless.
This observation is of key importance. A body falling freely in gravity feels no gravity. Imagine you are in an elevator and someone cuts the cable. As it falls, you are weightless; you feel no gravity.
"The breakthrough came suddenly one day," Einstein wrote in 1907. "I was sitting on a chair in my patent office in Bern. Suddenly the thought struck me: If a man falls freely, he does not feel his own weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity."
What is the significance of a freely falling body feeling no gravity? Well, if it experiences no gravity—or acceleration, since the two are the same—then its behaviour is described entirely by Einstein's special theory of relativity. Here then is the crucial point of contact—the all-important bridge—between the special theory of relativity and the theory of gravity sought by Einstein.
The observation that a freely falling body does not feel its weight and is therefore described by special relativity suggests a crude way to extend special relativity to a body experiencing gravity. Think of a friend standing on Earth and very obviously experiencing gravity pressing his or her feet to the ground. You can observe your friend from any point of view you like—from hanging upside down from a nearby tree or from an aeroplane flying past. But one point of view provides a big payoff. If you imagine things from a point of view that is in free fall, then you will be weightless, subject to no acceleration. Since you feel no acceleration, you are justified in using the special theory of relativity to describe your friend.
But special relativity relates what the world looks like to people moving at constant speed relative to each other and your friend is accelerating upwards relative to you. That's true. But if you do not mind a lot of laborious calculation, you can imagine your friend travelling at constant speed, a second, say then at a slightly higher constant speed for the next second, and so on. It's not perfect, but you can approximate your friend's acceleration as a series of rapid steps up in speed. For each speed you simply use special relativity to tell you what is happening to the space and time of your friend.
According to special relativity, time slows down for a moving observer. It therefore follows that time slows down for your friend since your friend is moving relative to you. But wait. Your friend is moving relative to you because he or she is experiencing gravity. It follows that gravity must slow down time! This should not be too much of a surprise. After all, if gravity is simply the warpage of space-time, it stands to reason that if we are experiencing gravity, our space and our time must be distorted in some way.
The other thing that follows from thinking about your friend standing on Earth's surface is that if gravity were stronger—say your friend was standing on a more massive planet—his or her speed relative to you in free fall would get faster quicker. According to special relativity, the faster someone moves, the more their time slows down. Consequently, the stronger the gravity someone is experiencing, the more their time slows down. What this means is that if you work on the ground floor of an office building, you age more slowly than your colleagues who work on the top floor. Why? Because, being closer to Earth, you experience a more powerful pull, and time slows down in stronger gravity.
Earth's gravity, however, is very weak. After all, you can hold your arm out in front of you and not even the gravity of the entire Earth can force you to drop it. The weakness of Earth's gravity means that the difference in the flow rate of time between the ground and top floors of even the tallest building is nearly impossible to measure. The opening scene, with the twin sisters aging at vastly different rates in their skyscraper workplace, is therefore a gross exaggeration. Nevertheless, there are places in the Universe with far stronger gravity.
One place is the surface of a white dwarf star, where the gravity is much stronger even than the Sun's. Einstein's theory of gravity predicts that time for these stars should pass slightly slower than for us. Testing such a prediction might seem impossible. However, nature has very conveniently provided us with "clocks" on the surfaces of white dwarfs. The clocks are actually atoms.
Atoms give out light. Light is actually a wave that undulates up and down like a wave on water, and atoms of particular elements such as sodium or hydrogen give out light that is unique to the element, undulating a characteristic number of times a second. These undulations can be thought of as the ticks of a clock. (In fact, the second is defined in terms of the undulations of light given out by a particular type of atom.)
How does this property of atoms help us see the effect of gravity on time? Well, with our telescopes we can pick up the light from atoms on white dwarfs. We can then compare the number of undulations per second of the light from, say, hydrogen on a white dwarf, with the number of undulations per second of hydrogen on Earth. What we find is that there are fewer undulations per second in the light from a white dwarf. Light is more sluggish. Time runs slower!4 We are seeing a direct confirmation of Einstein's general theory of relativity.
And there are stars known as neutron stars with even stronger gravity than that of white dwarfs. As a result of the strong gravity, time on the surface of a neutron star progresses one and a half times more slowly than on Earth.
## THE CONSEQUENCES OF GENERAL RELATIVITY
Time dilation is only one of the novel predictions of Einstein's general theory of relativity. Another, already touched on, is the existence of gravitational waves. We know they exist because astronomers have observed pairs of stars, which include at least one neutron star, losing energy as they spiral in towards each other. This puzzling loss of energy can be explained only if it is being carried away by gravitational waves.
The race is now on to detect gravitational waves directly. As they pass by, they should alternately stretch and squeeze space. Experiments designed to detect them therefore use giant "rulers," many kilometres long. The rulers are made of light, but the idea is simple—to detect the change in length of the rulers as a gravitational wave ripples past.
Another prediction of Einstein's theory, so far passed over without comment, is the bending of light by gravity. The reason for this bending, of course, is that light must negotiate the warped terrain of four-dimensional space-time. Although Newton's law of gravity predicts no such effect, it does when combined with the special relativistic idea that all forms of energy—including light—have an effective mass. As light passes a massive body like the Sun, it therefore feels the tug of gravity and is bent slightly from its course.
Of course, special relativity is incompatible with Newton's law of gravity, so this light-bending prediction has to be taken with a pinch of salt. In fact, the correct theory—general relativity—predicts that the path of light will be bent by twice as much.
This extra factor of two serves to highlight something subtle about the principle of equivalence. Recall the experiment in which the astronaut fired the laser horizontally across his spacecraft and noticed that the beam was bent downwards. Because there was no way he could know he was not experiencing gravity in a room on Earth's surface, it was possible to deduce that gravity bends the path of light. Well, there is a little lie in here. You see, it turns out that it is possible for the astronaut to tell whether he is in a rocket or on Earth's surface.
In the accelerating rocket, the force that pins the astronaut's feet to the floor pulls him vertically downwards—wherever he stands in the cabin. On Earth's surface, however, it matters where you stand because gravity always pulls things towards the centre of Earth. Consequently, gravity pulls in one direction in England but in the opposite direction in New Zealand—to the English, the New Zealanders are upside down, and vice versa. Now, the direction of the pull of gravity does not change too much from one side of a room to another. Nevertheless, with sensitive-enough measuring instruments, our astronaut could always detect the change and tell whether he was in a rocket accelerating out in space or on Earth's surface.
Surely, this invalidates the principle of equivalence and brings the whole edifice of general relativity tumbling down? Well, you might think so. However, to construct a theory of gravity it is sufficient only that the principle of equivalence apply in tiny volumes of space, and in extremely tiny, localised volumes of space you can never detect changes in the direction of gravity.
What has this got to do with Einstein's theory predicting twice the light deflection of Newton's? Well, we have established that the laser beam will be bent downwards as it traverses a room on Earth's surface, and this amount turns out to be roughly what Newtonian gravity predicts. Now imagine that the room is in free fall—say it has been dropped from an aeroplane—and the astronaut carries out the same experiment. In free fall, remember, there is no gravity. So the light beam should travel horizontally across the room and not be bent at all. But not all parts of the room are in a perfect state of free fall. Because Earth's gravity pulls in one direction from one corner of the room and from a different direction from the other corner, gravity is not perfectly cancelled out as the room falls through the air. Because of this, what the astronaut actually sees is the light beam bent downwards by roughly the same amount as in the room on Earth's surface. The two effects add together to give twice the light bending predicted by Newton's theory of gravity plus special relativity.
So if the light from a distant star passes close to the Sun on its way to Earth, its trajectory should be bent about twice as sharply as Newton would have predicted. Such an effect would cause the position of a star to shift slightly relative to other stars. Though impossible to see in the glare of daylight, it is observable during a total eclipse when the Moon blots out the bright solar disc. Such an eclipse was due to occur on May 29, 1919, and the English astronomer Arthur Eddington travelled to the island of Principe off the coast of West Africa to see it. His photographs confirmed that starlight was indeed deflected by the Sun's gravity by exactly the amount predicted by the general theory of relativity.
Eddington's observations made Einstein's reputation as "the man who proved Newton wrong." But it was not the end of general relativity's successful predictions. Newton had demonstrated theoretically that the planets orbited the Sun not in circles but in ellipses—squashed circles. He proved that this was a direct consequence of the fact that the force of gravity drops off in strength with a so-called inverse-square law. In other words, when you are twice as far away from the Sun, the force of gravity is four times as weak; three times as far away, it is nine times as weak; and so on.
Relativity changes everything. For a start, all forms of energy, not just mass-energy, generate gravity. Now gravity itself is a form of energy. Think of a warped trampoline and how much elastic energy that contains. Since gravity is a form of energy, the gravity of the Sun itself creates gravity! It's a tiny effect and most of the Sun's gravity still comes from its mass. Nevertheless, close in to the Sun, where gravity is strong, there is a small extra contribution from gravity itself. Consequently, any body orbiting there feels a gravitational tug greater than expected from the inverse square law.
Now—and this is the point—planets follow elliptical orbits only if they are being tugged by a force obeying an inverse-square law of force. This was Newton's discovery. Relativity predicts that the force does not obey an inverse-square law. In fact, there are other effects that also cause a departure from Newtonian gravity, like the fact that gravity takes time to travel across space. The gravity that a moving planet feels at any moment therefore depends on its position at an earlier time and, because of this, is not directed towards the dead centre of the Sun. The upshot is that planets do not follow elliptical paths that repeat but rather elliptical paths which gradually change their orientation in space, tracing out a rosette-like pattern. This is not noticeable far from the Sun. The biggest effect is close in, where gravity is strongest.
Sure enough, there is something odd about the orbit of the innermost planet, Mercury. For some time before Einstein published his theory of gravity in 1915, astronomers had been puzzled by the fact that Mercury's orbit gradually traces out a rosette pattern in space. Most of this effect is due to the gravitational pull of Venus and Jupiter. The odd thing, however, is that Mercury's orbit would still be tracing out a rosette pattern _even if Venus and Jupiter were not there._ It is a tiny effect. Although Mercury orbits the Sun once every 88 days, a rosette is traced out only once every _3 million years_. Remarkably, this is exactly what Einstein's theory predicted. Using general relativity, he could explain every last detail of Mercury's orbit. With yet another successful prediction under its belt, there could be no doubt that Einstein had discovered the correct theory of gravity.5
## THE PECULIARITIES OF GENERAL RELATIVITY
General relativity is a fantastically elegant theory. Nevertheless, it is tremendously difficult to apply to real situations—for instance, to find the warpage of space-time caused by a given distribution of mass. The reason is that the theory is rather circular. Matter tells space-time how to warp. Then warped space-time tells matter how to move. The matter, which has just moved, tells space-time how to change its warpage. And so on, ad infinitum. There's a kind of chicken-and-egg paradox at the heart of the theory. Physicists call it nonlinearity, and nonlinearity is a tough nut for theorists to crack.
One manifestation of nonlinearity already mentioned is the fact that gravity is a source of gravity. Well, if gravity can make more gravity, that extra gravity can make a little more gravity, and so on. Fortunately, gravity is so weak that this is not normally a runaway process and the gravity generated by a massive body is usually well behaved—usually, but not always.
Some very massive stars end their lives in a spectacular way. Usually, a star is prevented from being crushed by its own gravity by the pressure of the hot gas in its interior pushing outwards. But this outward pressure only exists while the star is generating heat. When it runs out of all possible fuels, it shrinks. Usually, some other form of pressure intervenes to make a white dwarf or a neutron star, superdense stellar embers. However, if the star is very massive and its gravity is very strong, nothing can stop the star from shrinking down to a point. As far as physicists know, such stars literally vanish from existence. However, they leave something behind: their gravity.
What we are talking about here are black holes, perhaps the most bizarre of all the predictions of general relativity. A black hole is a region of space-time where gravity is so strong that not even light can escape it—hence its blackness. And "region of space-time" is the operative phrase, for the mass of the star has gone.
How can you have gravity without mass? Well, gravity arises not just from mass but from all forms of energy. In the case of the black hole, its own gravity creates more gravity and that extra gravity creates more gravity... so the hole regenerates itself like a man holding himself in midair by his boot straps. From the space-time point of view, a black hole is literally a hole. Whereas a star like the Sun creates a mere dimple in the surrounding space-time, a black hole produces a bottomless well into which matter falls but can never escape again.
As Nobel Prize-winning physicist Subrahmanyan Chandrasekhar observed: "The black holes of nature are the most perfect macroscopic objects there are in the universe: The only elements in their construction are our concepts of space and time."6
Because of their ultrastrong gravity, black holes reveal the most dramatic effects of general relativity. Surrounding them is a surface known as an event horizon. This marks the point of no return for objects straying too close to the black hole. If you moved in close to the event horizon, you could see the back of your head since light from behind you would be bent all the way around the hole before reaching your eyes. If you could somehow hover just outside the event horizon, time would flow so slowly for you that you could in theory watch the entire future of the Universe flash past you like a movie in fast-forward!
The fact that time runs far more slowly in the strong gravity of a black hole than elsewhere in the Universe has an intriguing consequence. Imagine you are far away from a black hole and you have a friend lingering close to it. Because of the marked difference in the flow of time for both of you, while you go from Monday to Friday, your friend progresses only from Monday to Tuesday. This means that, if you could find some way to spirit yourself over to your friend's location, you could go from Friday back to Tuesday. You could travel back in time!
It turns out that there is in fact a way to spirit yourself from one location to another. Einstein's theory of relativity permits the existence of "wormholes," tunnel-like shortcuts through space-time. By entering one mouth of such a wormhole and exiting a mouth near your friend, it would indeed be possible to go back in time from Friday to Tuesday.
The trouble with wormholes is that they snap shut in an instant unless held open by matter with repulsive gravity. Nobody knows whether such "exotic matter" exists in the Universe. Nevertheless, the extraordinary fact remains that Einstein's theory of gravity does not rule out the possibility of time travel.
There are a few differences, however, between the kind of "time machine" permitted by general relativity and the type described by science fiction writers like H. G. Wells. For one thing, you have to travel a distance through space to travel a distance through time. You cannot simply sit still in a time machine, pull a lever, and find yourself in 1066. And a second important difference is that you cannot go back to a time before your time machine was built. So if you want to go on a dinosaur safari, building a time machine today will not help. You will have to find one built and abandoned by extraterrestrials (or some very smart dinosaurs) 65 million years ago!
To theorists the possibility of time machines is very unsettling. If time travel is possible, all sorts of impossible situations, or "paradoxes," raise their ugly heads. The most famous is the grandfather paradox in which a man goes back in time and shoots his grandfather before he conceives the man's mother. The problem is, if he shoots his grandfather, how can he ever be born to go back in time and do the dirty deed?!
Embarrassing questions like this have prompted the English physicist Stephen Hawking to propose the Chronology protection conjecture. Basically, it's just a fancy name for an outright ban on time travel. According to Hawking, some as-yet-unknown law of physics must intervene to prevent time travel. He has no cast-iron evidence of such a law but simply asks: "Where are the tourists from the future?"
Einstein himself did not believe that time travel was possible, despite the fact that his theory of gravity predicted it. He was wrong, however, about two other predictions of his theory. He did not believe that black holes were possible, and today we have compelling evidence that they exist. And he did not believe what his theory was trying to tell him about the origin of the Universe—that it began in a Big Bang.
1 This is not at all obvious on Earth, where frictional forces act to slow a moving body. However, it is apparent in the empty vacuum of space.
2 It is worth pointing out that acceleration does not just mean a change in speed. It can also mean a change in direction. So a car travelling around a bend—even at constant speed—is accelerating.
3 Most people assume that astronauts orbiting Earth are weightless because there is no gravity in space. However, at the 500-kilometre-or-so height of the International Space Station, gravity is only about 15 per cent weaker than on Earth's surface. The real reason astronauts are weightless is that they and their spacecraft are in free fall just as surely as someone in an elevator when the cable breaks. The difference is that they never hit the ground. Why? Because Earth is round and, as fast as they fall toward the surface, the surface curves away from them. They, therefore, fall forever in a circle.
4 For technical reasons, this effect is known as the gravitational red shift.
5 Or at least a workable theory for the time being, since even general relativity is not thought to be the last word on gravity.
6 The term "black hole" was coined by John Wheeler in 1965. Before 1965 there were very few scientific papers on such objects. Afterward, the field exploded. The term has even entered everyday language. People often talk about things disappearing down a bureaucratic black hole. The term is a perfect illustration of the importance of getting the right words to describe a phenomenon in science. If they paint a vivid picture in people's minds, researchers are attracted to the subject.
# THE ULTIMATE RABBIT OUT OF A HAT
HOW WE LEARNED THAT THE UNIVERSE HAS NOT EXISTED FOREVER BUT WAS BORN IN A TITANIC EXPLOSION 13.7 BILLION YEARS AGO
_A white rabbit is pulled out of a top hat. Because it is an extremely large rabbit, the trick takes billions of years_.
Jostein Gaarder
_They are high-tech glasses. Merely by twiddling a knob on the frame,_ _you can "tune" them to see all kinds of light normally invisible to the_ _human eye. You take them outside on a cold, starry night and start twiddling._
_The first thing you see is the sky in ultraviolet, light pumped out by_ _stars much hotter than the Sun. Some familiar stars have vanished, and_ _some new ones have swum into view, shrouded in misty nebulosity. The_ _most striking feature of the sky, however, is the same as it was for the_ _naked-eye sky. It's mostly black._
_You twiddle on._
_Now you're seeing X-rays, high-energy light radiated by gas heated_ _to hundreds of thousands of degrees as it swirls down onto exotic objects_ _like black holes. Once again, the most striking feature of the sky is that it_ _is mostly black._
_You twiddle back the other way, zipping back through ultraviolet_ _light and visible light to infrared light, given out by objects much colder_ _than the Sun. Now the sky is peppered by stellar embers—stars so_ _recently born they are still swathed in shimmering placental gas and_ _bloated red giants in their death throes. But despite the fact that the sky_ _is lit by a new population of stars, its most striking thing remains the_ _same. It is mostly black._
_You twiddle on. Now you are seeing microwaves—the kind of light_ _used for radar, mobile phones, and microwave ovens. But something_ _odd is happening. The sky is getting brighter. Not just bits of it—all of it!_
_You take off the glasses, rub your eyes, and put them back on. But_ _nothing has changed. Now the whole sky, from horizon to horizon, is_ _glowing a uniform, pearly white. You twiddle further, but the sky just_ _gets brighter and brighter. The whole of space seems to be glowing. It's_ _like being inside a giant lightbulb._
Are the glasses malfunctioning? No, they are working perfectly. What you are seeing is the cosmic background radiation, the relic of the fireball in which the Universe was born 13.7 billion years ago. Incredibly, it still permeates every pore of space, greatly cooled by the expansion of the Universe so that it now appears as low-energy microwaves rather than visible light. Believe it or not, the cosmic background radiation accounts for an astonishing 99 per cent of the light in today's Universe. It is incontrovertible proof that the Universe began in a titanic explosion—the Big Bang.
The cosmic background radiation was discovered in 1965. But the realisation that there had been a Big Bang actually came earlier. In fact, the first step was taken by Einstein.
## THE ULTIMATE SCIENCE
Einstein's theory of gravity—the general theory of relativity—describes how every chunk of matter pulls on every other chunk of matter. The biggest collection of matter we know of is the Universe. Never one to shy away from the really big problems in science, Einstein in 1916 applied his theory of gravity to the whole of creation. In doing so he created cosmology—the ultimate science—which deals with the origin, evolution, and ultimate fate of the Universe.
Although the ideas behind Einstein's theory of gravity are deceptively simple, the mathematical apparatus is not. Working out exactly how a particular distribution of matter warps space-time is very hard indeed. It was not until 1962, for instance—almost half a century after Einstein published his general theory of relativity—that New Zealand physicist Roy Kerr calculated the distortion of space-time caused by a realistic, spinning, black hole.
Figuring out how the whole Universe warps space-time would have been impossible without making some simplifying assumptions about how its matter is spread throughout space. Einstein assumed that it makes no difference where in the Universe an observer happens to be. In other words, he assumed that the Universe has the same gross properties wherever you are located and, from wherever you are located, it looks roughly the same in every direction.
Astronomical observations since 1916 have actually shown these assumptions to be well founded. The Universe's building blocks—which Einstein and everyone else were unaware of at the time—are galaxies, great islands of stars like our own Milky Way. And modern telescopes do indeed show them to be scattered pretty evenly around the Universe, so the view from one galaxy is much the same as the view from any other.
Einstein's conclusion, after applying his theory to the Universe as a whole, was that its overall space-time must be warped. Warped space-time, however, causes matter to move. This is the central mantra of general relativity. Consequently, the Universe could not possibly be still. This dismayed Einstein. Like Newton before him, he fervently believed the Universe to be static, its constituent bodies—now known to be galaxies—suspended essentially motionless in the void.
A static universe was appealing because it remained the same for all time. There was no need to address sticky questions about where the Universe came from or where it was going. It had no beginning. It had no end. The reason the Universe was the way it was was because that was the way it had always been.
According to Newton, for the Universe to be static, one condition had to be satisfied: matter had to extend infinitely in all directions. In such a neverending cosmos, each body has just as many bodies on one side, pulling it one way with their gravity, as on the opposite side, pulling it the other way. Like a rope being pulled by two equally strong tug-of-war teams, it therefore remains motionless.
However, according to Einstein's theory of gravity, the Universe was finite, not infinite. Its space-time curved back on itself—the four-dimensional equivalent of the two-dimensional surface of a basketball. In such a Universe the gravitational tug-of-war is at no point perfectly balanced. Because every body tries to pull every other body toward it, the Universe shrinks uncontrollably.
To salvage the idea of a static Universe, Einstein had to resort to mutilating his elegant theory. He added a mysterious force of cosmic repulsion, which pushed apart the objects in the Universe. He hypothesised that it had a significant effect only on bodies that were enormously far apart, explaining why it had not been noticed before in Earth's neighbourhood. By precisely counteracting the force of gravity that was perpetually trying to drag bodies together, the cosmic repulsion kept the Universe forever static.
## THE EXPANDING UNIVERSE
Einstein's instincts turned out to be wrong. In 1929, Edwin Hubble—the American astronomer responsible for discovering that the Universe's building blocks were galaxies—announced a dramatic new discovery. The galaxies were flying apart from each other like pieces of cosmic shrapnel. Far from being static, the Universe was growing in size. As soon as Einstein learned of Hubble's discovery of the expanding universe, he renounced his cosmic repulsion, calling it the biggest blunder he ever made in his life.1 Einstein's mysterious repulsive force could never have kept the galaxies hanging motionless in space. As Arthur Eddington pointed out in 1930, a static cosmos is inherently unstable, like a knife balanced on its point. The merest nudge would be enough to set it expanding or contracting.
Others did not make the same mistake as Einstein. In 1922 the Russian physicist Aleksandr Friedmann applied Einstein's theory of gravity to the Universe and correctly concluded that it must either be contracting or expanding. Five years later the same conclusion was reached independently by the Belgian Catholic priest Georges-Henri Lemaître.
As John Wheeler has said: "Einstein's description of gravity as curvature of space-time led directly to that greatest of all predictions: The Universe itself is in motion." It is ironic that Einstein himself missed the message in his own theory.
## THE BIG BANG UNIVERSE
Since the Universe is expanding, one conclusion is inescapable: it must have been smaller in the past. By imagining the expansion running backwards, like a movie in reverse, astronomers deduce that 13.7 billion years ago all of Creation was squeezed into the tiniest of tiny volumes. The lesson of the receding galaxies is that the Universe, though old, has not existed forever. There was a beginning to time. A mere 13.7 billion years ago, all matter, energy, space, and time fountained into existence in a titanic explosion—the Big Bang.
The cosmic expansion turns out to obey a remarkably simple law: Every galaxy is rushing away from the Milky Way with a speed that is in direct proportion to its distance. So a galaxy that is twice as far away as another is receding twice as fast, one 10 times as far away 10 times as fast, and so on. This relation, known as Hubble's law, turns out to be unavoidable in any universe that grows in size while continuing to look the same from every galaxy.
Imagine a cake with raisins in it. If you could shrink in size and sit on any raisin, the view will always be the same. Furthermore, if the cake is put in an oven and expands, or rises, not only will you see all the other raisins recede from you but you will see them recede with speeds in direct proportion to their distance from you. It matters not at all what raisin you sit on. The view will always be the same. (The tacit assumption here is that it is a big cake, so that you are always far from the edge.) Galaxies in an expanding universe are like raisins in a rising cake.
It follows that, just because we see all the galaxies flying away from us, we should not assume that we are at the centre of the Universe and that the Big Bang happened in our cosmic backyard. Were we to be in any galaxy other than the Milky Way, we would see the same thing—all the other galaxies fleeing from us. The Big Bang did not happen here, or over there, or at any one point in the Universe. It happened in all places simultaneously. "In the universe, no centre or circumference exists, but the centre is everywhere," said the 16thcentury philosopher Giordano Bruno.
The Big Bang is a bit of a misnomer. It was totally unlike any explosion with which we are familiar. When a stick of dynamite detonates, for instance, it explodes outwards from a localised point and the debris expands into preexisting space. The Big Bang did not happen at a single point and there was no preexisting void! Everything—space, time, energy, and matter—came into being in the Big Bang and began expanding everywhere at once.
## THE HOT BIG BANG
Whenever you squeeze something into a smaller volume—for in-stance, air into a bicycle pump—it gets hot. The Big Bang was therefore a hot Big Bang. The first person to realise this was the Ukrainian-American physicist George Gamow. In the first few moments after the Big Bang, he reasoned, the Universe was reminiscent of the blisteringly hot fireball of a nuclear explosion.2
But whereas the heat and light of a nuclear fireball dissipate into the atmosphere so that, hours or days after the explosion, they are all gone, this was not true of the heat and light of the Big Bang fireball. Since the Universe, by definition, is all there is, there was simply nowhere for it to go. The "afterglow" of the Big Bang was instead bottled up in the Universe forever. This means it should still be around today, not as visible light—since it would have been greatly cooled by the expansion of the Universe since the Big Bang—but as microwaves, an invisible form of light characteristic of very cold bodies.3
Gamow did not believe it would be possible to distinguish this microwave afterglow from other sources of light in today's Universe. However, he was mistaken. As his research students Ralph Alpher and Robert Herman realised, the relic heat of the Big Bang would have two unique features that would make it stand out. First, because it came from the Big Bang, and the Big Bang happened everywhere simultaneously, the light should be coming equally from every direction in the sky. And, second, its spectrum—the way the brightness of the light changed with the light's energy—would be that of a "black body." It's not necessary to know what a black body is, only that a black body spectrum is a unique "fingerprint."
Although Alpher and Herman predicted the existence of the afterglow of the Big Bang—the cosmic microwave background radiation—in 1948, it was not discovered until 1965 and then totally by accident. Arno Penzias and Robert Wilson, two young astronomers at Bell Labs at Holmdel in New Jersey, were using a horn-shaped microwave antenna formerly used for communicating with _Telstar_ , the first modern communications satellite, when they picked up a mysterious hiss of microwave "static" coming equally from every direction in the sky. Over the following months as they puzzled over the signal, they variously thought that it might be radio static from nearby New York City, atmospheric nuclear tests, or even pigeon droppings coating the interior of their microwave horn. In fact, they had made the most important cosmological discovery since Hubble found that the Universe was expanding. The afterglow of creation was powerful evidence that our Universe had indeed begun in a hot, dense state—a Big Bang—and had been growing in size and cooling ever since.
Penzias and Wilson did not accept the Big Bang origin of their mysterious static for at least two years. Nevertheless, for the discovery of the afterglow of creation, they carried off the 1978 Nobel Prize for Physics.
The cosmic background radiation is the oldest "fossil" in creation. It comes to us directly from the Big Bang, carrying with it precious information about the state of the Universe in its infancy, almost 13.7 billion years ago. The cosmic background is also the coldest thing in nature—only 2.7 degrees above absolute zero, the lowest possible temperature (–270 degrees Celsius).
The cosmic background radiation is actually one of the most striking features of our Universe. When we look up at the night sky, its most obvious feature is that it is mostly black. However, if our eyes were sensitive to microwave light rather than visible light, we would see something very different. Far from being black, the entire sky, from horizon to horizon, would be white, like the inside of a lightbulb. Even billions of years after the event, all of space is still glowing softly with relic heat of the Big Bang fireball.
In fact, every sugarcube-sized region of empty space contains 300 photons of the cosmic background radiation. Ninety-nine per cent of all the photons in the Universe are tied up in it, with a mere 1 per cent in starlight. The cosmic background radiation is truly ubiquitous. If you tune your TV between stations, 1 per cent of the "snow" on the screen is the relic static of the Big Bang.
## DARKNESS AT NIGHT
The fact that the Universe began in a Big Bang explains another great mystery—why the night sky is dark. The German astronomer Johannes Kepler, in 1610, was the first to realise this was a puzzle.
Think of a forest of regularly spaced pine trees going on forever. If you ran into the forest in a straight line, sooner or later you would bump into a tree. Similarly, if the Universe is filled with regularly spaced stars and goes on forever, your gaze will alight on a star no matter which direction you look out from Earth. Some of those stars will be distant and faint. However, there will be more distant stars than nearby ones. In fact—and this is the crucial point—the number of stars will increase in such a way that it exactly compensates for their faintness. In other words, the stars at a certain distance from Earth will contribute just as much light in total as the ones twice as far away, three times away, four times away, and so on. When all the light arriving at Earth is added up, the result will therefore be an infinite amount of light!
This is clearly nonsensical. Stars are not pointlike; they are tiny discs. So nearby stars blot out some of the light of more distant stars just as nearby pine trees block out more distant pine trees. But even taking this effect into account, the conclusion seems inescapable that the entire sky should be "papered" with stars, with no gaps in between. Far from being dark at night, the night sky should be as bright as the surface of a typical star. A typical star is a red dwarf, a star glowing like a dying ember. Consequently, the sky at midnight should be glowing blood red. The puzzle of why it isn't was popularised in the early 19th century by the German astronomer Heinrich Olbers and is known as Olbers' paradox in his honour.
The way out of Olbers' paradox is the realisation that the Universe has not in fact existed forever but was born in a Big Bang. Since the moment of creation, there has been only 13.7 billion years for the light of distant stars to reach us. So the only stars and galaxies we see are those that are near enough that their light has taken less than 13.7 billion years to get to us. Most of the stars and galaxies in the Universe are so far away that their light will take more than 13.7 billion years to reach us. The light of these objects is still on its way to Earth.
Therefore, the main reason the sky at night is dark is that the light from most of the objects in the Universe has yet to reach us. Ever since the dawn of human history, the fact that the Universe had a beginning has been staring us in the face in the darkness of the night sky. We have simply been too stupid to realise it.
Of course, if we could wait another billion years, we would see stars and galaxies so far away that their light has taken 14.7 billion years to get here. The question therefore arises of whether, if we lived many trillions of years in the future when the light from many more stars and galaxies had time to reach us, the sky at night would be red. The answer turns out to be no. The reasoning of Kepler and Olbers is based on an incorrect assumption—that stars live forever. In fact, even the longest-lived stars will use up all their fuel and burn out after about 100 billion years. This is long before enough light has arrived at Earth to make the sky red.
## DARK MATTER
The Big Bang has enormous explanatory power. Nevertheless, it has serious problems. For one it is difficult to understand where galaxies like our Milky Way came from.
The fireball of the Big Bang was a mix of particles of matter and light. The matter would have affected the light. For instance, if the matter had curdled into clumps, this would be reflected in the afterglow of the Big Bang—it would not be uniform all over the sky today but would be brighter in some places than others. The fact that the afterglow is even all around the sky means that matter in the fireball of the Big Bang must have been spread about extremely smoothly. But we know that it could not be spread completely smoothly. After all, today's Universe is clumpy, with galaxies of stars and clusters of galaxies and great voids of empty space in between. At some point, therefore, the matter in the Universe must have gone from being smoothly distributed throughout space to being clumpy. And the start of this process should be visible in the cosmic background radiation.
Sure enough, in 1992, very slight variations in the brightness of the afterglow of the Big Bang were discovered by NASA's Cosmic Background Explorer Satellite, COBE. These cosmic ripples—one of the scientists involved was more picturesque in likening them to "the face of God"—showed that, about 450,000 years after the Big Bang, some parts of the Universe were a few thousandths of a per cent denser than others. Somehow, these barely noticeable clumps of matter—the "seeds" of structure—had to grow to form the great clusters of galaxies we see in today's Universe. But there is a problem.
Clumps of matter grow to become bigger clumps because of gravity. Basically, if a region has slightly more matter than a neighbouring region, its stronger gravity will ensure that it will steal yet more matter from its neighbour. Just as the richer get richer and the poor get poorer, the denser regions of the Universe will get ever denser until, eventually, they become the galaxies we see around us today. The problem the theorists noticed was that 13.7 billion years was not enough time for gravity to make today's galaxies out of the tiny clumps of matter seen by the COBE satellite. The only way they could do it was if there was much more matter in the Universe than was tied up in visible stars.
Actually, there was strong evidence for missing matter close to home. Spiral galaxies like our own Milky Way are like giant whirlpools of stars, only their stars turn out to be whirling about their centres far too fast. By rights, they should fly off into intergalactic space just as you would be flung off a merry-go-round that someone had spun too fast. The extraordinary explanation that the world's astronomers have come up with is that galaxies like our Milky Way actually contain about 10 times as much matter as is visible in stars. They call the invisible matter dark matter. Nobody knows what it is. However, the extra gravity of the dark matter holds the stars in their orbits and stops them from flying off into intergalactic space.
If the Universe as a whole contains 10 times as much dark matter as ordinary matter, the extra gravity is just enough to turn the clumps of matter seen by COBE into today's galaxy clusters in the 13.7 billion years since the Universe was born. The Big Bang picture is saved.4 The price is the addition of a lot of dark matter, whose identity nobody knows—well, almost, nobody. In the words of Douglas Adams in _Mostly Harmless_ : "For a long period of time there was much speculation and controversy about where the so-called 'missing matter' of the Universe had gotten to. All over the Galaxy the science departments of all the major universities were acquiring more and elaborate equipment to probe and search the hearts of distant galaxies, and then the very centre and the very edges of the whole Universe, but when eventually it was tracked down it turned out in fact to be all the stuff which the equipment had been packed in!"
## INFLATION
The fact that the standard Big Bang picture does not provide enough time for matter to clump into galaxies is not the only problem with the scenario. There is another, arguably more serious, one. It concerns the smoothness of the cosmic background radiation.
Things reach the same temperature when heat travels from a hot body to a cold body. For instance, if you put your cold hand on a hot water bottle, heat will flow from the bottle until your hand reaches the same temperature. The cosmic background radiation is basically all at the same temperature. This means that, as the early Universe grew in size, and some bits lagged behind others in temperature, heat always flowed into them from a warmer bit, equalising the temperature.
The problem arises if you imagine the expansion of the Universe running backwards like a movie in reverse. At the time that the cosmic background radiation last had any contact with matter—about 450,000 years after the Big Bang—bits of the Universe that today are on opposite sides of the sky were too far apart for heat to flow from one to the other. The maximum speed it could flow is the speed of light, and the 450,000 years the Universe had been in-existence was simply not long enough. So how is it that the cosmic background radiation is the same temperature everywhere today?
Physicists have come up with an extraordinary answer. Heat could have flowed back and forth throughout the Universe, equalising the temperature, only if the early Universe was much smaller than our backward-running movie would imply. If regions were much closer together than expected, there would have been plenty of time for heat to flow from hot to cold regions and equalise the temperature. But if the Universe was much smaller earlier on, it must have put on a big spurt of growth to get to its present size.
According to the theory of inflation, the Universe "inflated" during its first split-second of existence, undergoing a phenomenally violent expansion. What drove the expansion was a peculiar property of the vacuum of empty space, although that's still hazy to physicists. The point is that there was this enormously fast expansion, which very quickly ran out of steam, and then the more sedate expansion that we see today took over. If the normal Big Bang expansion is likened to the explosion of a stick of dynamite, inflation can be likened to a nuclear explosion. "The standard Big Bang theory says nothing about what banged, why it banged or what happened before it banged," says inflation pioneer Alan Guth. Inflation is at least an at-tempt to address such questions.
With inflation plus dark matter tagged on, the Big Bang scenario can be rescued. In fact, when astronomers talk of the Big Bang these days, they often mean the Big Bang plus inflation plus dark matter. However, inflation and dark matter are not as well-founded ideas as the Big Bang. Beyond any doubt, we know that the Universe began in a hot dense state and has been expanding and cooling ever since—the Big Bang scenario. That inflation happened is still not certain, and nobody has yet discovered the identity of dark matter.
One of the pluses of inflation is that it provides a possible explanation of the origins of structures such as galaxies in today's Universe. For such structures to have formed, there must have been some kind of unevenness in the Universe at a very early stage. That primordial roughness could have been caused by so-called quantum fluctuations. Basically, the laws of microscopic physics cause extremely small regions of space and matter to jiggle about restlessly like water in a boiling saucepan. Such fluctuations in the density of matter were minuscule—smaller even than present-day atoms. However, the phenomenal expansion of space caused by inflation would actually have enhanced them, blowing them up to noticeable size. Bizarrely, the largest structures in today's Universe—great clusters of galaxies—may have been spawned by "seeds" smaller than atoms!
Inflation, however, predicts something about our Universe that does not seem to accord with the facts. Currently, the Universe is expanding. However, the gravity of all the matter in the Universe is acting to brake the expansion. There are two main possibilities. One is that the Universe contains sufficient matter to eventually slow and reverse its expansion, causing the Universe to collapse back down to a Big Crunch, a sort of mirror image of the Big Bang in which the Universe was born. The other is that it contains insufficient matter and goes on expanding forever. Inflation predicts that the Universe should be balanced on the knife edge between these two possibilities. It will continue expanding, but slowing down all the time, and finally running out of steam only in the infinite future. For this to happen, the Universe must have what is known as the critical mass. The problem is that, even when all the matter in the Universe—visible matter and dark matter—is added up, it amounts to only about a third of the critical mass. Inflation, it would seem, is a nonstarter. Well, that's how it seemed—until a sensational discovery was made in 1998.
## DARK ENERGY
Two teams were observing "supernovas"—exploding stars—in distant galaxies. One team was led by American Saul Perlmutter and the other by Australians Nick Suntzeff and Brian Schmidt. Supernovas are exploding stars that often outshine their parent galaxy and so can be seen at great distances out into the Universe. The kind the two teams of astronomers were looking at were known as Type Ia supernovas. They have the property that, when they detonate, they always shine with the same peak luminosity. So if you see one that is fainter than another, you know it is farther away.
What the astronomers saw, however, was that the ones that were farther away were fainter than they ought to be, taking into account their distance from Earth. The only way to explain what they were seeing was that the Universe's expansion had speeded up since the stars exploded, pushing them farther away than expected and making them appear fainter.
It was a bombshell dropped into the world of science. The sole force affecting the galaxies ought to be their mutual gravitational pull. That should be braking the expansion, not speeding it up.
The only thing that could be accelerating things was space itself. Contrary to all expectations, it could not be empty. It must contain some kind of weird stuff unknown to science—"dark energy"—that was exerting a kind of cosmic repulsion on the Universe, countering gravity and driving the galaxies apart.
Physicists are totally at sea when it comes to understanding dark energy. Their best theory—quantum mechanics—predicts an energy associated with empty space that is 1 followed by 123 zeroes bigger than Perlmutter observed! Nobel laureate Steven Weinberg has described this as "the worst failure of an order-of-magnitude estimate in the history of science."
Despite this embarrassment, the dark energy has at least one desirable consequence. Recall that inflation requires the Universe to have the critical mass but that all the matter in the Universe adds up to only about a third of the critical mass. Well all forms of energy, as Einstein discovered, have an effective mass. And that includes the dark energy. In fact, it turns out to account for about two-thirds of the critical mass, so that the Universe has exactly the critical mass—just what is predicted by inflation.
Although nobody knows what the dark energy is, one possibility is that it is associated with the repulsive force of empty space proposed by Einstein. In science, it seems, all things begin and end with Einstein. His biggest mistake may yet turn out to be his biggest success.
It is worth stressing, however, that the Big Bang, for all its successes, is still basically a description of how our Universe has evolved from a superdense, superhot state to its present state, with galaxies, stars, and planets. How it all began is still shrouded in mystery.
## TO THE SINGULARITY AND BEYOND
Imagine the expansion of the Universe running backwards again like a movie in reverse. As the Universe shrinks down to a speck, its matter content becomes ever more compressed and ever hotter. In fact, there is no limit to this process. At the instant the Universe's expansion began—the moment of its birth—it was infinitely dense and infinitely hot. Physicists call the point when something skyrockets to infinity a singularity. According to the standard Big Bang picture, the Universe was therefore born in a singularity.
The other place where Einstein's theory of gravity predicts a singularity is at the heart of a black hole. In this case the matter of a catastrophically shrinking star eventually becomes compressed into zero volume and therefore becomes infinitely dense and infinitely hot. "Black holes," as someone once said, "are where God divided by zero."5
A singularity is a nonsense. When such a monstrous entity pops up in a theory of physics, it is telling us that the theory—in this case, Einstein's theory of gravity—is faulty. We are stretching it beyond the domain where it has anything sensible to say about the world. This is not surprising. General relativity is a theory of the very large. In its earliest stages, however, the Universe was smaller than an atom. And the theory of the atomic realm is quantum theory.
Normally, there is no overlap between these two towering monuments of 20th-century physics. However, they come into conflict at the heart of black holes and at the birth of the Universe. If we are ever going to understand how the Universe came into being, we are going to have to find a better description of reality than Einstein's theory of gravity. We need a quantum theory of gravity.
The task of finding such a theory is formidable because of the fundamental incompatibility between general relativity and quantum theory. General relativity, like every theory of physics before it, is a recipe for predicting the future. If a planet is here now, in a day's time it will have moved over there, by following this path. All these things are predictable with 100 per cent certainty. Quantum theory, however, is a recipe for predicting probabilities. If an atom is flying through space, all we can predict is its probable final position, its probable path. Quantum theory therefore undermines the very foundation stones of general relativity.
Currently, physicists are trying to discover the elusive quantum theory of gravity by a number of routes. Undoubtedly, the one getting the most publicity is superstring theory, which views the fundamental building blocks of matter not as pointlike particles but as ultra-tiny pieces of "string." The string—superconcentrated mass-energy—can vibrate just like a violin string, and each distinct vibration "mode" corresponds to a fundamental particle such as an electron or a photon.
What excites string theorists is that some form of gravity—although not necessarily general relativity—is automatically contained within string theory. One slight complication is that the strings of string theory vibrate in a 10-dimensional world, which means there have to exist an additional six space dimensions too small for us to have noticed. Another problem is that string theory involves such horrendously complicated mathematics that it has so far proved impossible to make a prediction with it that can be tested against reality.
No one knows how close or how far away we are to possessing a quantum theory of gravity. But without it there is no hope of travelling those last tantalising steps back to the beginning of the Universe. However, some of the things that must happen along the route are clear.
Think of the expansion of the Universe in reverse again. At first the Universe will shrink at the same rate in all directions. This is because the Universe is pretty much the same in all directions. But _pretty_ _much the same_ is not the same as _exactly the same_. Undoubtedly, there will be slightly more galaxies in one direction than another. In the early stages of the contraction this imbalance will have no noticeable effect. However, as the Universe shrinks down to zero volume, such matter irregularities will become ever more magnified. So when the body shrinks to zero volume, the final stages of the collapse will be wildly chaotic. Gravity—warped space-time—will vary wildly depending on the direction from which the singularity is approached by an in-falling body.
Very close to the singularity, the warpage of space-time will become so violent and chaotic that space and time will actually shatter, splitting into myriad droplets. Concepts like "before" and "after" now lose all meaning. So too do concepts like "distance" and "direction." An impenetrable fog blocks the view ahead. It shrouds the mysterious domain of quantum gravity, where no theory yet exists to act as our guide.
But deep in that fog lie the answers to science's most pressing questions. Where did the Universe come from? Why did it burst into being in a Big Bang 13.7 billion years ago? What, if anything, existed before the Big Bang?
The fervent hope is that, when at last we manage to mesh together our theory of the very small with our theory of the very large, we will find the answers to these questions. Then we will come face to face with the ultimate question: How could something have come from nothing? "It is enough to hold a stone in your hand," wrote Jostein Gaarder in _Sophie's World_. "The universe would have been equally incomprehensible if it had only consisted of that one stone the size of an orange. The question would be just as impenetrable: Where did this stone come from?"
1 See _My World Line_ by George Gamow (New York, 1970), in which the author writes of Einstein: "He remarked [to me] that the introduction of the cosmological term was the biggest blunder he ever made in his life."
2 The Big Bang was named by the English astronomer Fred Hoyle during a BBC radio programme in 1949. The great irony is that Hoyle, to the day he died, never believed in the Big Bang.
3 And of magnetrons, which power microwave ovens and radar transmitters.
4 Actually, there is thought to be between 6 and 7 times as much dark matter as ordinary matter. This is because the stars account for only about half the ordinary matter. The rest, which may be in the form of dim gas clouds between the galaxies, has not yet been identified.
5 Actually, there is a subtle distinction between the singularities at the heart of a black hole and the Big Bang. The former is a singularity in time and the latter a singularity in space.
# GLOSSARY
ABSOLUTE ZERO Lowest temperature attainable. As a body is cooled, its atoms move more and more sluggishly. At absolute zero, equivalent to –273.15 on the Celsius scale, they cease to move altogether. (Actually, this is not entirely true since the Heisenberg uncertainty principle produces a residual jitter even at absolute zero.)
ACCRETION DISC CD-shaped disc of in-swirling matter that forms around a strong source of gravity such as a black hole. Since gravity weakens with distance from its source, matter in the outer portion of the disc orbits more slowly than in the inner portion. Friction between regions where matter is travelling at different speeds heats the disc to millions of degrees. Quasars are thought to owe their prodigious brightness to ferociously hot accretion discs surrounding "supermassive" black holes.
ALPHA CENTAURI The nearest star system to the Sun. It consists of three stars and is 4.3 light-years distant.
ALPHA DECAY The spitting out of a high-speed alpha particle by a large, unstable nucleus in an attempt to turn itself into a lighter, stable nucleus.
ALPHA PARTICLE A bound state of two protons and two neutrons—essentially a helium nucleus—that rockets out of an unstable nucleus during radioactive alpha decay.
ANTHROPIC PRINCIPLE The idea that the Universe is the way it is because, if it was not, we would not be here to notice it. In other words, the fact of our existence is an important scientific observation.
ANTIMATTER Term for a large accumulation of antiparticles. Anti-protons, antineutrons, and positrons can in fact come together to make anti-atoms. And there is nothing in principle to rule out the possibility of antistars, antiplanets, or antilife. One of the greatest mysteries of physics is why we appear to live in a Universe made solely of matter when the laws of physics seem to predict a pretty much 50/ 50 mix of matter and antimatter.
ANTIPARTICLE Every subatomic particle has an associated antiparticle with opposite properties, such as electrical charge. For instance, the negatively charged electron is twinned with a positively charged antiparticle known as the positron. When a particle and its antiparticle meet, they self-destruct, or "annihilate," in a flash of high-energy light, or gamma rays.
ATOM The building block of all normal matter. An atom consists of a nucleus orbited by a cloud of electrons. The positive charge of the nucleus is exactly balanced by the negative charge of the electrons. An atom is about one 10-millionth of a millimetre across.
ATOMIC ENERGY See Nuclear Energy.
ATOMIC NUCLEUS The tight cluster of protons and neutrons (a single proton in the case of hydrogen) at the centre of an atom. The nucleus contains more than 99.9 per cent of the mass of an atom.
BIG BANG The titanic explosion in which the Universe is thought to have been born 13.7 billion years ago. "Explosion" is actually a misnomer since the Big Bang happened everywhere at once and there was no preexisting void into which the Universe erupted. Space, time, and energy all came into being in the Big Bang.
BIG BANG THEORY The idea that the Universe began in a superdense, superhot state 13.7 billion years ago and has been expanding and cooling ever since.
BIG CRUNCH If there is enough matter in the Universe, its gravity will one day halt and reverse the Universe's expansion so that it shrinks down to a Big Crunch. This is a sort of mirror image of the Big Bang.
BLACK BODY A body that absorbs all the heat that falls on it. The heat is shared among the atoms in such a way that the heat radiation it gives out takes no account of what the body is made of but depends solely on its temperature and has a characteristic and easily recognisable form. The stars are approximate black bodies.
BLACK HOLE The grossly warped space-time left behind when a massive body's gravity causes it to shrink down to a point. Nothing, not even light, can escape—hence a black hole's blackness. The Universe appears to contain at least two distinct types of black hole—stellar-sized black holes that form when very massive stars can no longer generate internal heat to counterbalance the gravity trying to crush them and "supermassive" black holes. Most galaxies appear to have a supermassive black hole in their heart. They range from millions of times the mass of the Sun in our Milky Way to billions of solar masses in the powerful quasars.
BOSE-EINSTEIN CONDENSATION Phenomenon in which all the microscopic particles in a body suddenly crowd into the same state. The particles must be bosons and the temperature must generally be very low. Helium atoms, for instance, crowd into the same state below–271 degrees Celsius, turning liquid helium into a superfluid.
BOSON A microscopic particle with integer spin—that is, 0 units, 1 unit, 2 units, and so on. By virtue of their spin, such particles are hugely gregarious, participating in collective behaviour that leads to lasers, superfluids, and superconductors.
BOYLE'S LAW The observation that the volume of a gas is inversely proportional to its pressure—that is, doubling the pressure halves the volume.
BROWNIAN MOTION The random, jittery motion of a large body under machine-gun bombardment from smaller bodies. The most famous instance is of pollen grains zigzagging through water as they are repeatedly hit by water molecules. The phenomenon, discovered by botanist Robert Brown in 1827 and triumphantly explained by Einstein in 1905, was powerful proof of the existence of atoms.
CAUSALITY The idea that a cause always precedes an effect. Causality is a much-cherished principle in physics. However, quantum events such as the decay of atoms appear to be effects with no prior cause.
CHANDRASEKHAR LIMIT The largest possible mass for a white dwarf. It depends on a star's chemical composition, but for a white dwarf made of helium it is about 44 per cent more massive than the Sun. For a star bigger than this, the electron degeneracy pressure inside is insufficient to prevent gravity from crushing the star farther.
CHARGE-COUPLED DEVICE (CCD) Supersensitive electronic light detector that can register close to 100 per cent of the light that falls on it. Since photographic plates register a mere 1 per cent, CCDs allow a telescope to perform as well as a telescope with 100 times the light-collecting area.
CHEMICAL BOND The "glue" that sticks atoms together to make molecules.
CHRONOLOGY PROTECTION CONJECTURE The stricture that time travel is impossible. No one has yet managed to prove it—in fact, the laws of physics appear to permit time travel—but physicists such as Stephen Hawking remain convinced that some, as-yet-undiscovered law of nature forbids time machines.
CLASSICAL PHYSICS Nonquantum physics. In effect, all physics before 1900 when the German physicist Max Planck first proposed that energy might come in discrete chunks, or quanta. Einstein was the first to realise that this idea was totally incompatible with all physics that had gone before.
CLOSED TIME-LIKE CURVE (CTC) Region of space-time so dramatically warped that time loops back on itself in much the same way that space loops back on itself on an athletics track. A CTC, in common parlance, is a time machine. It is permitted to exist by the current laws of physics.
COMET Small icy body—usually mere kilometres across—that orbits a star. Most comets orbit the Sun beyond the outermost planets in an enormous cloud known as the Oort Cloud. Like asteroids, comets are builders' rubble left over from the formation of the planets.
COMPTON EFFECT The recoil of an electron when exposed to high-energy light just as if the electron is a tiny billiard ball struck by another tiny billiard ball. The effect is a graphic demonstration that light is ultimately made of tiny bulletlike particles, or photons.
CONDUCTOR A material through which an electrical current can flow.
CONSERVATION LAW Law of physics that expresses the fact that a quantity can never change. For instance, the conservation of energy states that energy can never be created or destroyed, only converted from one form to another. For example, the chemical energy of petrol can be converted into the energy of motion of a car.
CONSERVATION OF ENERGY Principle that energy can never be created or destroyed, only converted from one form to another.
COOPER PAIR Two electrons with opposite spin that pair up in some metals at extremely low temperature. Cooper pairs, unlike individual electrons, are bosons. Consequently, they can crowd into the same state, moving together in lockstep through the metal like an irresistible army on the move. The electrical current in such a "superconductor" can run forever.
COPERNICAN PRINCIPLE The idea that there is nothing special about our position in the Universe, in either space or time. This is a generalised version of Copernicus's recognition that Earth is not in a special position at the centre of the solar system but is just another planet circling the Sun.
COSMIC BACKGROUND RADIATION The "afterglow" of the Big Bang fireball. Incredibly, it still permeates all of space 13.7 billion years after the event, a tepid radiation corresponding to a temperature of –270 degrees Celsius.
COSMIC RAYS High-speed atomic nuclei, mostly protons, from space. Low-energy ones come from the Sun; high-energy ones probably come from supernovas. The origin of ultra-high-energy cosmic rays, particles millions of times more energetic than anything we can currently produce on Earth, is one of the great unsolved puzzles of astronomy.
COSMOLOGY The ultimate science. The science whose subject matter is the origin, evolution, and fate of the entire Universe.
COSMOS Another word for Universe.
DARK ENERGY Mysterious "material" with repulsive gravity. Discovered unexpectedly in 1998, it is invisible, fills all of space and appears to be pushing apart the galaxies and speeding up the expansion of the Universe. Nobody has much of a clue what it is.
DARK MATTER Matter in the Universe that gives out no light. Astronomers know it exists because the gravity of the invisible stuff bends the paths of visible stars and galaxies as they fly through space. There is between 6 and 7 times as much dark matter in the Universe as ordinary, light-emitting matter. The identity of the dark matter is the outstanding problem of astronomy.
DECOHERENCE The mechanism that destroys the weird quantum nature of a body—so that, for instance, it appears localised rather than in many different places simultaneously. Decoherence occurs if the outside world gets to "know" about the body. The knowledge may be taken away by a single photon of light or an air molecule that bounces off the body. Since big bodies like tables are continually struck by photons and air molecules and cannot remain isolated from their surroundings for long, they lose their ability to be in many places at once in a fantastically short time—far too short for us to notice.
DEGENERACY PRESSURE The bee-in-a-box-like pressure exerted by electrons squeezed into a small volume of space. A consequence of the Heisenberg uncertainty principle, it arises because a microscopic particle whose location is known very well necessarily has a large uncertainty in its velocity. The degeneracy pressure of electrons prevents white dwarfs from shrinking under their own gravity, whereas the degeneracy pressure of neutrons does the same thing for neutron stars.
DENSITY The mass of an object divided by its volume. Air has a low density, and iron has a high density.
DIMENSION An independent direction in space-time. The familiar world around us has three space dimensions (east–west, north–south, up-down) and one of time (past-future). Superstring theory requires the Universe to have six extra space dimensions. These differ radically from the other dimensions because they are rolled up very small.
DOUBLE SLIT EXPERIMENT Experiment in which microscopic particles are shot at a screen with two closely spaced, parallel slits cut in it. On the far side of the screen, the particles mingle, or "interfere," with each other to produce a characteristic "interference pattern" on a second screen. The bizarre thing is that the pattern forms even if the particles are shot at the slits one at a time, with long gaps between—in other words, when there is no possibility of them mingling with each other. This result, claimed Richard Feynman, highlighted the "central mystery" of quantum theory.
ELECTRIC CHARGE A property of microscopic particles that comes in two types—positive and negative. Electrons, for instance, carry a negative charge and protons a positive charge. Particles with the same charge repel each other, while particles with unlike charge attract. ELECTRIC CURRENT A river of charged particles, usually electrons, that can flow through a conductor.
ELECTRIC FIELD The field of force that surrounds an electric charge.
ELECTROMAGNETIC FORCE One of the four fundamental forces of nature. It is responsible for gluing together all ordinary matter, including the atoms in our bodies and the atoms in the rocks beneath our feet.
ELECTROMAGNETIC WAVE A wave that consists of an electric field that periodically grows and dies, alternating with a magnetic field that periodically dies and grows. An electromagnetic wave is generated by a vibrating electric charge and travels through space at the speed of light.
ELECTRON Negatively charged subatomic particle typically found orbiting the nucleus of an atom. As far as anyone can tell, it is a truly elementary particle, incapable of being subdivided.
ELEMENT A substance that cannot be reduced any further by chemical means. All atoms of a given element possess the same number of protons in their nucleus. For instance, all atoms of hydrogen have one proton, all atoms of chlorine have 17, and so on.
ENERGY A quantity that is almost impossible to define! Energy can never be created or destroyed, only converted from one form to another. Among the many familiar forms are heat energy, energy of motion, electrical energy, and sound energy.
ENTANGLEMENT The intermingling of two or more microscopic particles so that they lose their individuality and in many ways be-have as a single entity.
EVENT HORIZON The one-way "membrane" that surrounds a black hole. Anything that falls through—whether matter or light—can never get out again.
EXOTIC MATTER Hypothetical matter with repulsive gravity.
EXPANDING UNIVERSE The fleeing of the galaxies from each other in the aftermath of the Big Bang.
FERMION A microscopic particle with half-integer spin—that is, ½ unit, 3/2 units, 5/2 units, and so on. By virtue of their spin, such particles shun each other. Their unsociability is the reason that atoms exist and the ground beneath our feet is solid.
FRAME DRAGGING The dragging around of space-time by a massive rotating body. The effect is very small—though potentially measurable—in the vicinity of Earth but enormous near a fast-rotating black hole. Such a black hole sits at the eye of a tornado of whirling space-time.
FUNDAMENTAL FORCE One of the four basic forces that are believed to underlie all phenomena. The four forces are the gravitational force, electromagnetic force, strong force, and weak force. The strong suspicion among physicists is that these forces are actually facets of a single superforce. In fact, experiments have already shown the electromagnetic and weak forces to be different sides of the same coin.
FUNDAMENTAL PARTICLE One of the basic building blocks of all matter. Currently, physicists believe there are six different quarks and six different leptons, making a total of 12 truly fundamental particles. The hope is that the quarks will turn out to be merely different faces of the leptons.
FUSION See Nuclear Fusion.
GALAXY One of the basic building blocks of the Universe. Galaxies are great islands of stars. Our own island, the Milky Way, is spiral in shape and contains about 200,000 million stars.
GAS Collection of atoms that fly about through space like a swarm of tiny bees.
GENERAL THEORY OF RELATIVITY Einstein's theory of gravity that shows gravity to be nothing more than the warpage of spacetime. The theory incorporates several ideas that were not incorporated in Newton's theory of gravity. One was that nothing, not even gravity, can travel faster than light. Another was that all forms of energy have mass and so are sources of gravity. Among other things, the theory predicted black holes, the expanding Universe, and that gravity would bend the path of light.
GEODESIC The shortest path between two points in warped, or curved, space.
GRAVITATIONAL FORCE The weakest of the four fundamental forces of nature. Gravity is approximately described by Newton's universal law of gravity but more accurately described by Einstein's theory of gravity—the general theory of relativity. General relativity breaks down at the singularity at the heart of a black hole and the singularity at the birth of the Universe. Physicists are currently looking for a better description of gravity. The theory, already dubbed quantum gravity, will explain gravity in terms of the exchange of particles called gravitons.
GRAVITATIONAL LIGHT BENDING The bending of the trajectory of light that passes by a massive body. Because the space in the vicinity of such a body is warped like a valley, the light has no choice but to travel along a curved path.
GRAVITATIONAL RED SHIFT The loss of energy as light climbs out of the valley in space-time around a massive celestial body. Since the "colour" of light is related to its energy, with red light having less energy than blue light, astronomers talk of light being shifted to the red end of the spectrum or "red-shifted."
GRAVITATIONAL WAVE A ripple spreading out through spacetime. Gravitational waves are generated by violent motions of mass, such as the merger of black holes. Because they are weak, they have not yet been detected directly.
GRAVITY See Gravitational Force.
HALF-LIFE The time it takes half the nuclei in a radioactive sample to disintegrate. After one half-life, half the atoms will be left; after two half-lives, a quarter; after three, an eighth, and so on. Half-lives can vary from the merest split-second to many billions of years.
HEISENBERG UNCERTAINTY PRINCIPLE A principle of quantum theory that there are pairs of quantities such as a particle's location and speed that cannot simultaneously be known with absolute precision. The uncertainty principle puts a limit on how well the product of such a pair of quantities can be known. In practice, this means that if the speed of a particle is known precisely, it is impossible to have any idea where the particle is. Conversely, if the location is known with certainty, the particle's speed is unknown. By limiting what we can know, the Heisenberg uncertainty principle imposes "fuzziness" on nature. If we look too closely, everything blurs like a newspaper picture dissolving into meaningless dots.
HELIUM Second lightest element in nature and the only one to have been discovered on the Sun before it was discovered on Earth. Helium is the second most common element in the Universe after hydrogen, accounting for about 10 per cent of all atoms.
HORIZON The Universe has a horizon much like the horizon that surrounds a ship at sea. The reason for the Universe's horizon is that light has a finite speed and the Universe has been in existence for only a finite time. This means that we only see objects whose light has had time to reach us since the Big Bang. The observable universe is therefore like a bubble centred on Earth, with the horizon being the surface of the bubble. Every day the Universe gets older (by one day), so every day the horizon expands outwards and new things become visible, just like ships coming over the horizon at sea.
HORIZON PROBLEM The problem that far-flung parts of the Universe that could never have been in contact with each other, even in the Big Bang, have almost identical properties such as density and temperature. Technically, they were always beyond each other's horizon. The theory of inflation provides a way for such regions to have been in contact in the Big Bang and so can potentially solve the horizon problem.
HYDROGEN The lightest element in nature. A hydrogen atom consists of a single proton orbited by a single electron. Close to 90 per cent of all atoms in the Universe are hydrogen atoms.
HYDROGEN BURNING The fusion of hydrogen into helium accompanied by the liberation of large quantities of nuclear binding energy. This is the power source of the Sun and most stars.
HYDROSTATIC EQUILIBRIUM The state in which the gravitational force trying to crush a star is perfectly balanced by the force of its hot gas pushing outwards.
INERTIA The tendency for a massive body, once set in motion, to keep on moving, at constant speed in a straight line in unwarped space and along a geodesic in warped space. Nobody knows the origin of inertia.
INERTIAL FORCE A force we invent to explain a motion that is actually due to nothing more than inertia. A good example is centrifugal force. There is no such force flinging us outwards in a car rounding a sharp corner. We are simply continuing to move in a straight line because of our inertia, and the interior of the car, because it is moving along a curved path, intercepts us.
INFLATION, THEORY OF Idea that in the first split-second of its creation the Universe underwent a fantastically fast expansion. In a sense, inflation preceded the conventional Big Bang explosion. If the Big Bang is likened to the explosion of a grenade, inflation was like the explosion of an H-bomb. Inflation can solve some problems with the Big Bang theory such as the horizon problem.
INFRARED Type of invisible light that is given out by warm bodies.
INTERFERENCE The ability of two waves passing through each other to mingle, reinforcing where their peaks coincide and cancelling where the peaks of one coincide with the troughs of another.
INTERFERENCE PATTERN Pattern of light and dark stripes that appears on a screen illuminated by light from two sources. The pattern is due to the light from the two sources reinforcing at some places on the screen and cancelling at other places.
INTERSTELLAR MEDIUM The tenuous gas and dust floating between the stars. In the vicinity of the Sun this gas comprises about one hydrogen atom in every 3 cubic centimetres, making it a vacuum far better than anything achievable on Earth.
INTERSTELLAR SPACE The space between the stars.
ION An atom or molecule that has been stripped of one or more of its orbiting electrons and so has a net positive electrical charge.
ISOTOPE One possible form of an element. Isotopes are distinguishable by their differing masses. For instance, chlorine comes in two stable isotopes, with a mass of 35 and 37. The mass difference is due to a differing number of neutrons in their nuclei. For instance, chlorine-35 contains 18 neutrons and chlorine-37 contains 20 neutrons. (Both contain the same number of protons—17—since this determines the identity of an element.)
JOULE The standard scientific unit of energy. The energy of motion of a flying cricket ball is about 10 joules; the chemical energy provided by a single slice of bread is about 100,000 joules; and the electrical energy of a lightning discharge is about 10 billion joules.
LAMBDA POINT Temperature below which liquid helium begins to turn into a superfluid.
LASER Light source in which the gregarious nature of photons—bosons—comes to the fore. Specifically, the more photons there are passing through a material the greater the probability that other atoms will emit others with the same properties. The result is an avalanche of photons all travelling in lockstep.
LIGHT, CONSTANCY OF The peculiarity that in our Universe the speed of light in empty space is always the same, irrespective of the speed of the source of light or of anyone observing the light. This is one of two cornerstones of Einstein's special theory of relativity, the other being the principle of relativity.
LIGHT, SPEED OF The cosmic speed limit—300,000 kilometres per second.
LIGHT BENDING See Gravitational Light Bending.
LIGHT-YEAR Convenient unit for expressing distances in the Universe. It is simply the distance that light travels in one year in a vacuum, which turns out to be 9.46 trillion kilometres.
LORENTZ CONTRACTION The contraction of a body moving relative to an "observer." The observer sees the body shrink in the direction of its motion. The effect is noticeable only when the body is moving close to the speed of light with respect to the observer.
LUMINOSITY The total amount of light pumped into space each second by a celestial body such as a star.
MAGNETIC FIELD The field of force that surrounds a magnet.
MANY WORLDS The idea that quantum theory describes everything, not simply the microscopic world of atoms and their constituents. Since quantum theory permits an atom to be in two places at once, this must mean that a table can be in two places at once. According to the Many Worlds idea, however, the mind of the person observing the table splits into two—one that perceives the table to be in one place and another that perceives it to be in another. The two minds exist in separate realities, or universes.
MASS A measure of the amount of matter in a body. Mass is the most concentrated form of energy. A single gram contains the same amount of energy as 100,000 tonnes of dynamite.
MAXWELL'S EQUATIONS OF ELECTROMAGNETISM The handful of elegant equations, written down by James Clerk Maxwell in 1868, that neatly summarise all electrical and magnetic phenomena. The equations reveal that light is an electromagnetic wave.
MILKY WAY Our galaxy.
MOLECULE Collection of atoms glued together by electromagnetic forces. One atom, carbon, can link with itself and other atoms to make a huge number of molecules. For this reason, chemists divide molecules into "organic"—those based on carbon—and "inorganic"—the rest.
MOMENTUM The momentum of a body is a measure of how much effort is required to stop it. For instance, an oil tanker, even though it may be going at only a few kilometres an hour, is far harder to stop than a Formula 1 racing car going 200 kilometres per hour. The oil tanker is said to have more momentum.
MOMENTUM, CONSERVATION OF Principle that momentum can never be created or destroyed.
MULTIVERSE Hypothetical enlargement of the cosmos in which our Universe turns out to be one among an enormous number of separate and distinct universes. Most universes are dead and uninteresting. Only in a tiny subset do the laws of physics promote the emergence of stars, planets, and life.
MUON Short-lived subatomic particle that behaves like a heavy version of the electron.
NEUTRINO Neutral subatomic particle with a very small mass that travels very close to the speed of light. Neutrinos, of which there are three kinds, hardly ever interact with matter. However, when created in huge numbers, they can blow a star apart in a supernova.
NEUTRON One of the two main building blocks of the atomic nucleus at the centre of atoms. Neutrons have essentially the same mass as protons but carry no electrical charge. They are unstable outside of a nucleus and disintegrate in about 10 minutes.
NEUTRON STAR A star that has shrunk under its own gravity to such an extent that most of its material has been compressed into neutrons. Typically, such a star is only 20 to 30 kilometres across. A sugar cube of neutron star stuff would weigh as much as the entire human race.
NEWTON'S UNIVERSAL LAW OF GRAVITY The idea that all bodies pull on each other across space with a force that depends on the product of their individual masses and the inverse square of their distance apart. In other words, if the distance between the bodies is doubled, the force becomes four times weaker; if it is tripled, nine times weaker; and so on. Newton's theory of gravity is perfectly good for everyday applications but turns out to be an approximation. Einstein provided an improvement in the general theory of relativity.
NONLOCALITY The spooky ability of objects subject to quantum theory to continue to "know" about each other's state even when separated by a large distance.
NUCLEAR ENERGY The excess energy released when one atomic nucleus changes into another atomic nucleus.
NUCLEAR FUSION The welding together of two light nuclei to make a heavier nucleus, a process that results in the liberation of nuclear binding energy. The most important fusion process for human beings is the gluing together of hydrogen nuclei to make helium in the core of the Sun since its by-product is sunlight.
NUCLEAR REACTION Any process that converts one type of atomic nucleus into another type of atomic nucleus.
NUCLEON Umbrella term used for protons and neutrons, the two building blocks of the atomic nucleus.
NUCLEUS See Atomic Nucleus.
PARTICLE ACCELERATOR Giant machine, often in the shape of a circular racetrack, in which subatomic particles are accelerated to high speed and smashed into each other. In such collisions the energy of motion of the particles becomes available to create new particles.
PARTICLE PHYSICS The quest to discover the fundamental building blocks and fundamental forces of nature.
PAULI EXCLUSION PRINCIPLE The prohibition on two microscopic particles (fermions) sharing the same quantum state. The Pauli exclusion stops electrons, which are fermions, from piling on top of each other and, consequently, explains the existence of different atoms and of the variety of the world around us.
PHOTOCELL A practical device that exploits the photoelectric effect. The interruption of an electric current when a body breaks the light beam falling on a metal is used to control something—for instance, an automatic door at the entrance to a supermarket.
PHOTOELECTRIC EFFECT The ejection of electrons from the surface of a metal by photons striking the metal.
PHOTON Particle of light.
PHYSICS, LAWS OF The fundamental laws that orchestrate the be-havior of the Universe.
PLANCK ENERGY The superhigh energy at which gravity becomes comparable in strength to the other fundamental forces of nature.
PLANCK LENGTH The fantastically tiny length scale at which gravity becomes comparable in strength to the other fundamental forces of nature. The Planck length is a trillion trillion times smaller than an atom. It corresponds to the Planck energy. Small distances are synonymous with high energies because of the wave nature of matter.
PLASMA An electrically charged gas of ions and electrons.
POSITRON Antiparticle of the electron.
PRECESSION OF THE PERIHELION OF MERCURY The fact that the orbit of Mercury, the planet closest to the Sun, does not follow a straightforward elliptical orbit but rather an elliptical orbit whose nearest point to the Sun gradually moves around the Sun, resulting in the planet tracing out a rosettelike pattern. The explanation is that the gravity of the Sun weakens with distance from the Sun more slowly than in the case of Newtonian gravity, which uniquely predicts elliptical orbits. It weakens more slowly because, in the Einsteinian picture, gravity itself is a source of more gravity.
PRINCIPLE OF EQUIVALENCE The idea that gravity and acceleration are indistinguishable.
PROTON One of the two main building blocks of the nucleus. Protons carry a positive electrical charge, equal and opposite to that of electrons.
PULSAR A rapidly rotating neutron star that sweeps an intense beam of radio waves around the sky much like a lighthouse.
QED See Quantum Electrodynamics.
QUANTUM The smallest chunk into which something can be divided. Photons, for instance, are quanta of the electromagnetic field.
QUANTUM COMPUTER A machine that exploits the fact that quantum systems such as atoms can be in many different states at once to carry out many calculations at once. The best quantum computers can manipulate only a handful of binary digits, or bits, but in principle such computers could massively outperform conventional computers.
QUANTUM ELECTRODYNAMICS Theory of how light interacts with matter. The theory explains almost everything about the everyday world, from why the ground beneath our feet is solid to how a laser works, from the chemistry of metabolism to the operation of computers.
QUANTUM INDISTINGUISHABILITY The inability to distinguish between two quantum events. These may be indistinguishable, for instance, because they involve identical particles or simply because the events are not observed. The crucial thing, however, is that the probability waves associated with indistinguishable events interfere. This leads to all manner of quantum phenomena.
QUANTUM NUMBER A number that specifies a microscopic property that comes in chunks such as the spin or orbital energy of an electron.
QUANTUM PROBABILITY The chance, or probability, of a microscopic event. Although nature prohibits us from knowing things with certainty, it nevertheless permits us to know the probabilities with certainty.
QUANTUM SUPERPOSITION Situation in which a quantum object such as an atom is in more than one state at a time. It might, for instance, be in many places simultaneously. It is the interaction, or "interference," between the individual states in the superposition that is the basis of all quantum weirdness. Decoherence prevents such interaction and therefore destroys quantum behaviour.
QUANTUM THEORY The theory of objects isolated from their surroundings. Because it is very hard to isolate a big object, the theory is essentially a theory of the microscopic world of atoms and their constituents.
QUANTUM TUNNELLING The apparently miraculous ability of microscopic particles to escape their prisons. For instance, an alpha particle can tunnel through the barrier penning it in the nucleus, the equivalent of a high jumper jumping a 4-metre-high wall. Tunnelling is yet another consequence of the wavelike character of microscopic particles.
QUANTUM UNPREDICTABILITY The unpredictability of microscopic particles. Their behaviour is unpredictable even in principle. Contrast this with the unpredictability of a coin toss. It is unpredictable only in practice. In principle, if we knew the shape of the coin, the force exerted on it, the air currents around it, and so on, we could predict the outcome.
QUANTUM VACUUM The quantum picture of empty space. Far from empty, it seethes with ultra-short-lived microscopic particles that are permitted by the Heisenberg uncertainty principle to blink into existence and blink out again.
QUASAR A galaxy that derives most of its energy from matter heated to millions of degrees as it swirls into a central giant black hole. Quasars can generate as much light as a hundred normal galaxies from a volume smaller than the solar system, making them the most powerful objects in the Universe.
QUBIT A quantum bit, or binary digit. Whereas a normal bit can only represent a "0" or a "1," a qubit can exist in a superposition of the two states, representing a "0" and a "1" simultaneously. Because strings of qubits can represent a large number of numbers simultaneously, they can be used to do a large number of calculations simultaneously.
RADIOACTIVE DECAY The disintegration of unstable heavy atomic nuclei into lighter, stabler atomic nuclei. The process is accompanied by the emission of either alpha particles, beta particles, or gamma rays.
RADIOACTIVITY Property of atoms that undergo radioactive decay.
RADIUM Highly unstable, or radioactive, element discovered by Marie Curie in 1898.
RELATIVITY, GENERAL THEORY OF Einstein's generalisation of his special theory of relativity. General relativity relates what one person sees when looking at another person accelerating relative to them. Because acceleration and gravity are indistinguishable—the principle of equivalence—general relativity is also a theory of gravity.
RELATIVITY, PRINCIPLE OF The observation that all the laws of physics are the same for observers moving at constant speed with respect to each other.
RELATIVITY, SPECIAL THEORY OF Einstein's theory that relates what one person sees when looking at another person moving at constant speed relative to them. It reveals, among other things, that the moving person appears to shrink in the direction of their motion while their time slows down, effects that become ever more marked as they approach the speed of light.
SCANNING TUNNELLING MICROSCOPE (STM) A device that drags an ultrafine needle across the surface of a material and converts the up-and-down motion of the needle into an image of the atomic landscape of the surface.
SCHRÖDINGER EQUATION Equation that governs the way in which the probability wave, or wave function, describing, say a subatomic particle, changes with time.
SIMULTANEITY The idea that events that appear to happen at the same time for one person should appear to happen at the same time for everyone in the Universe. Special relativity shows that this idea is mistaken.
SINGULARITY Location where the fabric of space-time ruptures and so cannot be understood by Einstein's theory of gravity, the general theory of relativity. There was a singularity—a point where quantities such as temperature skyrocketed to infinity—at the beginning of the Universe. There is also one in the centre of every black hole.
SOLAR SYSTEM The Sun and its family of planets, moons, comets, and other assorted rubble.
SPACE-TIME In the general theory of relativity, space and time are seen to be essentially the same thing. They are therefore treated as a single entity—space-time. It is the warpage of space-time that turns out to be gravity.
SPECTRAL LINE Atoms and molecules absorb and give out light at characteristic wavelengths. If they swallow more light than they emit, the result is a dark line in the spectrum of a celestial object. Conversely, if they emit more than they swallow, the result is a bright line.
SPECTRUM The separation of light into its constituent "rainbow" colours.
SPIN Quantity with no everyday analog. Loosely speaking, subatomic particles with spin behave as if they are tiny spinning tops (only they are not spinning at all!).
STAR A giant ball of gas that replenishes the heat it loses to space by means of nuclear energy generated in its core.
STRING THEORY See Superstring Theory.
STRONG NUCLEAR FORCE The powerful short-range force that holds protons and neutrons together in an atomic nucleus.
SUBATOMIC PARTICLE A particle smaller than an atom, such as an electron or a neutron.
SUN The nearest star.
SUPERCONDUCTOR A material that, when cooled to ultralow temperatures, conducts an electrical current forever—that is, with no resistance. This ability is connected with a change in the conducting particles from fermions to bosons. Specifically, electrons (fermions) pair up to form Cooper pairs (bosons).
SUPERFLUID A fluid that, below a critical temperature, develops bizarre properties such as the ability to flow uphill and squeeze through impossibly small holes. The best example is liquid helium, which becomes a superfluid below a temperature of 2.17 degrees above absolute zero. Superfluid liquid helium owes its weirdness to quantum theory and the fact that helium atoms are bosons.
SUPERNOVA A cataclysmic explosion of a massive star. A supernova may, for a short time, outshine an entire galaxy of 100 billion ordinary stars. It is thought to leave behind a highly compressed neutron star or even a black hole.
SUPERSTRING THEORY Theory which postulates that the fundamental ingredients of the Universe are tiny strings of matter. The strings vibrate in a space-time of 10 dimensions. The great payoff of this idea is that it may be able to unite, or "unify," quantum theory and the general theory of relativity.
TACHYON Hypothetical particle that lives its life permanently travelling faster than light.
TELEPORTATION The clever use of entanglement to pin down the exact state of a microscopic particle, in apparent violation of what is permitted by the Heisenberg uncertainty principle. This enables the information necessary to reconstruct the state of the particle to be sent to a remote site.
TEMPERATURE The degree of hotness of a body. Related to the energy of motion of the particles that compose it.
THERMODYNAMICS, SECOND LAW OF The decree that entropy, or microscopic disorder of a body, cannot ever decrease. This is equivalent to saying that heat can never flow from a cold to a hot body.
TIME DILATION The slowing down of time for an observer moving close to the speed of light or experiencing strong gravity.
TIME LOOP See Closed Time-Like Curve.
TIME MACHINE See Closed Time-Like Curve.
TIME TRAVEL Travel into the past or future—in the case of the future, at a rate of more than 1 year per year.
TIME TRAVEL PARADOX Nonsensical situation that time travel appears to permit. The most famous is the grandfather paradox in which someone goes back in time and shoots their grandfather before he conceives their mother. How then could they have been born to go back in time and commit the act?
TOTAL ECLIPSE OF THE SUN The coverage of the Sun by the disc of the Moon when the Moon moves between the Sun and Earth.
TWIN PARADOX The paradox that arises when someone travels at close to light speed to, say, Alpha Centauri and back while their twin stays at home. According to special relativity, the space-travelling twin ages less. However, from another point of view, it is Earth that receded from the space-travelling twin at close to the speed of light and therefore the stay-at-home-twin who ages less. The paradox is resolved by realising that the two situations are not equivalent. The space-travelling twin must undergo a deceleration and an acceleration at the turnaround at Alpha Centauri, and accelerations require general relativity not special relativity.
ULTRAVIOLET Type of invisible light that is given out by very hot bodies which is responsible for sunburn.
UNCERTAINTY PRINCIPLE See Heisenberg Uncertainty Principle.
UNIFICATION The idea that at extremely high energy the four fundamental forces of nature are one, united in a single theoretical framework.
UNIVERSE All there is. This is a flexible term once used for what we now call the solar system. Later, it was used for what we call the Milky Way. Now it is used for the sum total of all the galaxies, of which there appear to be about 100 billion within the observable Universe.
UNIVERSE, EXPANSION OF The fleeing of the galaxies from each other in the aftermath of the Big Bang.
UNIVERSE, OBSERVABLE All we can see out to the Universe's horizon.
URANIUM The heaviest naturally occurring element.
VIRTUAL PARTICLE Subatomic particle that has a fleeting existence, popping into being and popping out again according to the constraint imposed by the Heisenberg uncertainty principle.
VISCOSITY The internal friction of a liquid. Treacle has high viscosity and water has low viscosity.
WAVE FUNCTION A mathematical entity that contains all that is knowable about a quantum object such as an atom. The wave function changes in time and space according to the Schrödinger equation.
WAVELENGTH The distance for a wave to go through a complete oscillation cycle.
WAVE-PARTICLE DUALITY The ability of a subatomic particle to behave as a localised billiard ball-like particle or a spread-out wave.
WEAK NUCLEAR FORCE The second force experienced by protons and neutrons in an atomic nucleus, the other being the strong nuclear force. The weak nuclear force can convert a neutron into a proton and so is involved in beta decay.
WHITE DWARF A star that has run out of fuel and that gravity has compressed until it is about the size of Earth. A white dwarf is supported against further shrinkage by electron degeneracy pressure. A sugar cube of white dwarf material weighs about as much as a family car.
WORMHOLE A tunnel through space-time that connects widely spaced regions and so provides a shortcut.
X-RAYS A high-energy form of light.
# FURTHER READING
#### ATOMS AND QUANTUM THEORY
_Quantum: A Guide for the Perplexed_ , by Jim Al-Khalili (Weidenfeld & Nicolson, London, 2003).
_Taming the Atom_ , by Hans Christian von Baeyer (Penguin, London, 1994).
_Minds, Machines, and the Multiverse,_ by Julian Brown (Little Brown, New York, 2000).
_The Magic Furnace_ , by Marcus Chown (Vintage, London, 2000).
_The Fabric of Reality_ , by David Deutsch (Penguin, London, 1997).
_Thirty Years That Shook Physics,_ by George Gamow (Dover, New York, 1985).
_The Great Physicists from Galileo to Einstein_ , by George Gamow (Dover, New York, 1988).
_The New Quantum Universe,_ by Tony Hey and Patrick Walters, 2nd edition (Cambridge University Press, Cambridge, England, 2004).
_The Feynman Lectures on Physics,_ edited by Robert Leighton et al. (Addison-Wesley, New York, 1989).
#### RELATIVITY AND COSMOLOGY
_Afterglow of Creation_ , by Marcus Chown (University Science Books, Sausalito, California, 1994).
_The Universe Next Door,_ by Marcus Chown (Headline, London, 2002).
_Cosmology_ , by Edward Harrison (Cambridge University Press, Cambridge, England, 1991).
_The River of Time_ , by Igor Novikov (Cambridge University Press, Cambridge, England, 1998).
_Einstein's Legacy,_ by Julian Schwinger (Scientific American Library, New York, 1986).
_The Physical Universe_ , by Frank Shu (University Science Books, Sausalito, California, 1982).
# INDEX
#### **A**
* Acceleration
* curvature of space and,
* gravity and,
* Adams, Douglas, ,
* Aging
* general theory of relativity,
* gravity effects,
* special theory of relativity, ,
* Allen, Woody,
* Alpha particle
* decay, ,
* definition,
* escape from nucleus, ,
* scattering studies, ,
* tunnelling,
* Alpher, Ralph,
* Anaxagoras,
* Antimatter, _n_ ,
* Aspect, Alain,
* Aston, Francis, ,
* Atkinson, Robert,
* Atmosphere,
* Atomic theory
* atomic decay, ,
* chemical properties, ,
* duration of atoms, ,
* nature of light, ,
* origins and development,
* quantum theory and,
* size of atoms, ,
* structure and properties of atoms, , ,
* structure and properties of matter, ,
* types of atoms,
* uncertainty principle,
* Atomic weight, ,
* Attraction, atomic,
#### **B**
* Becquerel, Henri,
* Beginning of Universe. _See_ Big Bang
* Bernoulli, Daniel,
* Big Bang
* cosmic background radiation as evidence of, , ,
* * distribution of matter after, ,
* entanglement and,
* expansion of universe after,
* explanatory power,
* general theory of relativity and, ,
* inflation theory,
* singularity at moment of,
* visibility of stars and,
* Big Crunch,
* Binary calculations, _n_
* Binnig, Gerd,
* Biosphere,
* Black body,
* Black holes
* definition and properties, , ,
* Einstein's theory,
* energy conversion in,
* event horizon,
* gravity in,
* in quasars,
* singularity in,
* space-time distortions in, ,
* Bohr, Niels,
* Bose-Einstein condensation,
* Bosons
* behaviour in presence of other bosons,
* definition,
* fermions behaving as,
* Boyle, Robert,
* Bragg, William,
* Breathing, shared atoms in,
* Brown, Julian,
* Brown, Robert,
* Brownian motion, ,
* Bruno, Giordano,
#### **C**
* Carbon,
* Centrifugal force,
* Chance, in quantum theory,
* Chandrasekhar, Subrahmanyan,
* Chandrasekhar limit,
* Charge, atomic, , _n_
* Chemistry, atomic basis of, ,
* Chiao, Raymond,
* Chronology protection conjecture,
* Cloud chamber,
* Comets, , ,
* Compton effect,
* Computers, quantum. _See_ Quantum computers
* Contraction of universe,
* Cooper pairs,
* Copenhagen Interpretation,
* Cosmic Background Explorer Satellite,
* Cosmic background radiation,
* Big Bang theory and, , , ,
* discovery,
* distribution of, , ,
* temperature, ,
* Cosmology,
* Critical mass, ,
#### **D**
* Dark matter, ,
* Decay
* atomic,
* of organic matter,
* Decoherence, ,
* Degeneracy pressure, ,
* Democritus, ,
* Deutsch, David,
* * Double slit experiment
* design,
* indistinguishability and,
* interference pattern, , ,
* particle physics of, , ,
* significance of, , ,
* uncertainty principle and,
* Duration of atoms, ,
#### **E**
* Eclipse, solar,
* Eddington, Arthur, , ,
* Einstein, Albert, , , , , , , , , , , , , , , , , , , , ,
* Electric force, ,
* Electricity
* atomic theory, ,
* electric current,
* electrical charge,
* photoelectric effect,
* Electromagnetism
* light and, ,
* special theory of relativity,
* theory of, , ,
* Electrons,
* atomic structure, , ,
* in Cooper pairs,
* discovery,
* ejection event,
* indistinguishability,
* nucleus and, , ,
* orbitals, , , , ,
* Pauli exclusion principle, ,
* in photoelectric effect,
* properties, ,
* spin,
* in stars,
* uncertainty principle,
* velocity,
* vibration, ,
* wave frequency,
* _See also_ Fermions
* Elements, ,
* _E=mc_, ,
* Emptiness
* of matter, , ,
* quantum vacuum,
* Energy
* atomic weight and,
* in black holes,
* dark energy,
* in electron orbit jump,
* in empty space,
* gravity effects,
* hydrogen bomb,
* mass and, ,
* mass converted into,
* of motion,
* to reach speed of light,
* as source of gravity, ,
* of stars,
* transformation of, , ,
* weight of, ,
* Entanglement,
* Equivalence, Principle of, ,
* Event horizon,
* Everett, Hugh, III,
* Exclusion principle. _See_ Pauli exclusion principle
* Expanding universe
* discovery of,
* future of,
* rate,
* _See also_ Inflation of Universe
#### **F**
* Faraday, Michael,
* * Fermions
* boson behaviour in,
* definition,
* probability waveflipping and,
* Feynman, Richard, , , , , _n_ , ,
* Fourth dimension,
* Frame dragging,
* Free fall, , ,
* Frequencies, wave,
* Friction, , _n_
* Friedman, Aleksandr,
#### **G**
* Gaarder, Jostein, , , ,
* Galaxies
* dark matter of,
* distribution,
* in expanding universe,
* Galileo, ,
* Gamboge particles,
* Gamow, George, , _n_ ,
* Gas, pressure of,
* Geiger, Hans, , ,
* General theory of relativity
* bending of light in,
* cosmological application, ,
* goals, ,
* gravity waves in,
* nonlinearity of,
* principles of,
* quantum theory and,
* real-world implications, ,
* time travel and,
* Geodesics,
* Gravitational red shift, _n_
* Gravitons,
* Gravity
* acceleration and,
* bending of light by,
* of black holes, ,
* creation of gravity by, ,
* of dark matter,
* effects on time,
* experience of,
* frame dragging,
* general theory of relativity,
* as inertial force,
* mass and,
* Newtonian conceptualisation,
* particle carriers of,
* in production of energy,
* quantum theory of,
* repulsive force and,
* sources of,
* speed of,
* as warped space,
* waves, ,
* without mass,
* Ground state,
* Guth, Alan,
#### **H**
* Hawking, Stephen, ,
* Heisenberg, Werner,
* Helium
* alpha particles, ,
* liquid,
* nuclear fusion,
* properties,
* structure, , _n_ ,
* Helium-,
* Herman, Robert,
* Houtermans, Fritz,
* Hoyle, Fred, _n_
* Hubble, Edwin,
* Hubble's law,
* * Hydrogen,
* atomic structure, _n_ ,
* nuclear fusion, ,
* Hydrogen bomb,
#### **I**
* Indistinguishability of microscopic objects
* electron spin and,
* interference and, ,
* Pauli exclusion principle and,
* significance of, ,
* Inertia,
* Inflation of Universe,
* Interference
* decoherence and,
* evidence of,
* indistinguishability and,
* Many Worlds idea and,
* obstacles to,
* particle physics, , ,
* pattern smearing,
* superposition and,
* uncertainty principle,
* Inverse-square law,
* Ions, , _n_
#### **K**
* Kepler, Johannes,
* Kerr, Roy,
#### **L**
* Lamb shift, _n_
* Lasers,
* Lavoisier, Antoine,
* Leibniz, Gottfried, _n_
* Lemaître, Georges-Henri,
* Light
* ability to penetrate matter,
* atomic theory and,
* bending of, by gravity,
* boson behaviour and,
* cosmic background radiation,
* curvature of space and,
* dual wave-particle nature,
* effect of gravity of time and,
* as electromagnetic wave, ,
* interference,
* mass of,
* as particle phenomenon,
* photoelectric effect,
* as wave phenomenon,
* _See also_ Speed of light
* Liquids, behaviour of,
* Lithium, , ,
* Location of particle, uncertainty principle,
#### **M**
* Many Worlds idea,
* Marsden, Ernest, ,
* Mass
* in empty space,
* as form of energy, ,
* gravity and, ,
* of protons,
* speed and,
* transformation into energy,
* _See also_ Matter
* Mass spectrograph,
* Matter
* antimatter and,
* atomic structure, , ,
* creation in empty space, ,
* critical mass, ,
* dark matter, ,
* * distribution in universe,
* light and,
* properties of, determinants of,
* in space-time distortion,
* teleportation,
* _See also_ Mass
* Maxwell, James Clerk,
* Maxwell's wave,
* Mercury,
* Metals
* electric current in,
* superconductors,
* Microwave radiation. _See_ Cosmic background radiation
* Minkowski, Hermann,
* Momentum, ,
* Multiple universes, ,
* Multiverse. _See_ Multiple universes
* Muons,
#### **N**
* Neutrinos,
* Neutron star, , , ,
* Neutrons,
* Newtonian physics, , , , , , , ,
* Night sky, ,
* Nobel Prize, , ,
* Nonlocality, ,
* Novikov, Igor,
* Nuclear force,
* Nuclear fusion, ,
* Nucleus, atomic
* alpha particle escape from, ,
* collision of nuclei,
* electron (s) and, , ,
* nuclear force in,
* structure and properties, , ,
#### **O**
* Observation
* of atomic behaviour and properties, , ,
* effect of gravity of time and,
* electron ejection event,
* instantaneous influence and,
* principle of relativity, , ,
* probability wave flipping,
* of quantum behaviour, ,
* of speed of light,
* of superposition, , ,
* superposition of atoms,
* uncertainty principle of,
* visibility of stars,
* Olbers, Heinrich,
* Olbers' paradox,
* Orbitals, electron, , , ,
* properties of atoms and,
* Orbits, planetary,
#### **P**
* Page, Leigh,
* Parallel realities, ,
* Particle physics
* creation of mass-energy,
* entanglement,
* light, ,
* momentum,
* probability waveflipping,
* quantum theory,
* spin,
* uncertainty principle,
* Pauli, Wolfgang,
* Pauli exclusion principle,
* Penzias, Arno,
* Perlmutter, Saul,
* Perrin, Jean Baptiste,
* * Photoelectric effect, ,
* Photons
* cosmic background radiation,
* creation of, ,
* discovery,
* dual wave-particle nature,
* effective mass, ,
* indistinguishability,
* predictability of behaviour of,
* superposition, , ,
* _See also_ Bosons
* Planetary orbits,
* Pollen grains,
* Predictability of phenomenon,
* Pressure,
* Probability
* boson behaviour,
* in quantum theory,
* Probability wave, , ,
* of bosons,
* decoherence,
* flipping of,
* Protons
* atomic structure, _n_
* mass,
* nuclear fusion,
* tunnelling,
* Prout, William,
* Pulsars,
#### **Q**
* Quanta,
* Quantum bits,
* Quantum computers
* application,
* conceptual basis,
* current status, ,
* design challenges, ,
* operation in multiple universes,
* power of, , ,
* qubits,
* role of interference in, ,
* Quantum fluctuations,
* Quantum numbers, ,
* Quantum theory
* atomic theory and,
* behaviour of large objects in,
* chemical properties and,
* of entanglement,
* existence of multiple universes in,
* Feynman on,
* future prospects,
* general theory of relativity and,
* of gravity,
* of particle spin,
* probability in,
* purpose,
* quanta in,
* significance of,
* of spooky action at distance,
* uncertainty principle in,
* wave-particle duality in,
* Quantum vacuum,
* Quasars,
* Qubits. _See_ Quantum bits
#### **R**
* Radio waves,
* Radioactivity,
* Radium,
* Red dwarf star,
* Reflectivity, , ,
* Relativistic aberration/beaming, _n_
* Relativity
* principle of,
* _See also_ General theory of relativity; Special theory of relativity
* * Riefenstahl, Charlotte,
* Rohrer, Heinrich,
* Rutherford, Ernest, , ,
#### **S**
* Scanning tunnelling microscope,
* Schmidt, Brian,
* Schrödinger, Erwin,
* Schrödinger equation, ,
* Scott, Dave,
* Shell, atomic,
* Simultaneity,
* Singularity, ,
* Size of atoms, , , ,
* Soddy, Frederick,
* Sodium,
* Space
* curvature of,
* emptiness of, , , ,
* general theory of relativity,
* relationship to time,
* special theory of relativity, ,
* Special theory of relativity, ,
* energy and mass relationship in,
* gravity in,
* implications for physics,
* principles of, ,
* real-world implications,
* shortcomings,
* simultaneity,
* Speed
* of alpha particles,
* of gravity,
* instantaneous influence,
* of light. _See_ Speed of light
* mass and,
* of muons,
* of radio waves,
* theory of relativity,
* _See also_ Acceleration; Velocity of particle
* Speed of light
* as maximum limit of speed, , ,
* particle speed in excess of,
* principle of relativity,
* special theory of relativity, ,
* speed of light source and,
* spooky action at a distance and, , ,
* Speliotopoulos, Achilles,
* Spin, particle
* in Cooper pairs,
* electron distinguishability,
* integer/half-integer,
* probability waveflipping and,
* properties of particles,
* spooky action at a distance and,
* Spooky action at distance,
* Stars. _See_ Sun/stars
* Stoppard, Tom,
* Strong nuclear force, ,
* Sun/stars
* darkness of night sky,
* energy of,
* life cycle, ,
* light from,
* neutron stars, , , ,
* physics of,
* red dwarfs,
* supernovas, ,
* warping of space-time by,
* weight of,
* white dwarfs, , ,
* Suntzeff, Nick,
* Superconductors,
* Superfluids,
* Supernova, ,
* * Superposition
* conceptual basis,
* decoherence,
* definition,
* interference in, ,
* observability, , , ,
* in quantum computing, ,
* Superstring theory,
#### **T**
* Tachyons,
* Teleportation,
* Television static,
* Temperature
* Big Bang,
* cosmic background radiation, ,
* electric current in metal, ,
* weight and, ,
* Thomson, J. J., ,
* Time
* in black holes,
* concept of past and future,
* four dimensional space-time,
* general theory of relativity,
* gravity effects,
* relationship to space,
* special theory of relativity, ,
* synchronisation,
* travel in, , _n_
* Tunnelling,
* current, _n_
* in nuclear fusion,
* Twin paradox, _n_
#### **U**
* Uncertainty principle,
* atomic structure and, ,
* conceptual basis,
* empty space and,
* entanglement and,
* Pauli exclusion principle and,
* physics of stars and,
* tunnelling and,
* Uranium, _n_ ,
#### **V**
* Velocity of particle
* exceeding speed of light,
* uncertainty principle,
* Vibration
* electron behaviour and properties, ,
* superstring theory,
* Viscosity,
* Volume of matter, , ,
#### **W**
* Wave equation,
* Wave phenomenon
* ability to penetrate matter,
* electron properties,
* gravity waves, ,
* interference in,
* light as, , ,
* Maxwell's electromagnetic theory,
* in quantum theory,
* in quantum tunnelling,
* superposition, , ,
* vibration,
* Wavelength of light,
* Weight
* of atoms, ,
* of energy, ,
* motion and,
* temperature and, ,
* * Weizmann, Chaim,
* Wheeler, John, _n_ , , _n_ ,
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# ACKNOWLEDGEMENTS
My thanks to the following people who either helped me directly, inspired me, or simply encouraged me during the writing of this book: My dad, Karen, Sara Menguc, Jeffrey Robbins, Neil Belton, Henry Volans, Rachel Marcus, Moses Cardona, Brian Clegg, Professor Tony Hey, Kate Oldfield, Vivien James, Brian May, Dr. Bruce Bassett, Dr. Larry Schulman, Dr. Wojciech Zurek, Sir Martin Rees, Allison Chown, Colin Wellman, Rosie and Tim Chown, Patrick O'Halloran, Julie and Dave Mayes, Stephen Hedges, Sue O'Malley, Sarah Topalian, Dr. David Deutsch, Alexandra Feacham, Nick Mayhew-Smith, Elisabeth Geake, Al Jones, David Hough, Fred Barnum, Pam Young, Roy Perry, Hazel Muir, Stuart and Nikki Clark, Simon Ings, Barry Fox, Spencer Bright, Karen Gunnell, Jo Gunnell, Pat and Brian Chilver, Stella Barlow, Silvano Mazzon, Barbara Pell and David, Julia Bateson, Anne Ursell, Barbara Kiser, Dottie Friedli, Jon Holland, Martin Dollard, Sylvia and Sarah Kefyalew, Matilda and Dennis and Amanda and Andrew Buckley, Diane and Peter and Ciaran and Lucy Tomlin, Eric Gourlay, Paul Brandford. It goes without saying, I hope, that none of these people are responsible for any errors.
# About the Author
Marcus Chown is an award-winning writer and broadcaster. Formerly a radio astronomer at the California Institute of Technology in Pasadena, he is currently cosmology consultant of the weekly science magazine _New Scientist_.
His previous book, _The Never-Ending Days of Being Dead_ , was called 'a limousine among popular science vehicles' by the _Guardian_ , 'a masterpiece' by _Astronomy Now_ , and described as 'like being at a party... with an almost perfect DJ' in the _Independent_. It is also available as an ebook. Marcus Chown has also written a work for children, _Felicity Frobisher and the Three-Headed Aldebaran Dust Devil_.
# _By the Same Author_
Afterglow of Creation
The Magic Furnace
The Universe Next Door
The Never-Ending Days of Being Dead
# Copyright
First published in the United Kingdom in 2007
by Faber and Faber Limited
Bloomsbury House
74–77 Great Russell Street
London WC1B 3DA
This ebook edition first published in 2008
All rights reserved
© Marcus Chown, 2007
The right of Marcus Chown to be identified as author of this work has been asserted in accordance with Section 77 of 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.
ISBN 978–0–571–24601–4
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Q: How to bypass IndexError Here is my situation: My code parses out data from HTML tables that are within emails. The roadblock I'm running into is that some of these tables have blank empty rows right in the middle of the table, as seen in the photo below. This blank space causes my code to fail (IndexError: list index out of range) since it attempts to extract text from the cells.
Is it possible to say to Python: "ok, if you run into this error that comes from these blank rows, just stop there and take the rows you have acquired text from so far and execute the rest of the code on those"...?
That might sound like a dumb solution to this problem but my project involves me taking data from only the most recent date in the table anyway, which is always amongst the first few rows, and always before these blank empty rows.
So if it is possible to say "if you hit this error, just ignore it and proceed" then I would like to learn how to do that. If it's not then I'll have to figure out another way around this. Thanks for any and all help.
The table with the gap:
My code:
from bs4 import BeautifulSoup, NavigableString, Tag
import pandas as pd
import numpy as np
import os
import re
import email
import cx_Oracle
dsnStr = cx_Oracle.makedsn("sole.nefsc.noaa.gov", "1526", "sole")
con = cx_Oracle.connect(user="user", password="password", dsn=dsnStr)
def celltext(cell):
'''
textlist=[]
for br in cell.findAll('br'):
next = br.nextSibling
if not (next and isinstance(next,NavigableString)):
continue
next2 = next.nextSibling
if next2 and isinstance(next2,Tag) and next2.name == 'br':
text = str(next).strip()
if text:
textlist.append(next)
return (textlist)
'''
textlist=[]
y = cell.find('span')
for a in y.childGenerator():
if isinstance(a, NavigableString):
textlist.append(str(a))
return (textlist)
path = 'Z:\\blub_2'
for filename in os.listdir(path):
file_path = os.path.join(path, filename)
if os.path.isfile(file_path):
html=open(file_path,'r').read()
soup = BeautifulSoup(html, 'lxml') # Parse the HTML as a string
table = soup.find_all('table')[1] # Grab the second table
df_Quota = pd.DataFrame()
for row in table.find_all('tr'):
columns = row.find_all('td')
if columns[0].get_text().strip()!='ID': # skip header
Quota = celltext(columns[1])
Weight = celltext(columns[2])
price = celltext(columns[3])
print(Quota)
Nrows= max([len(Quota),len(Weight),len(price)]) #get the max number of rows
IDList = [columns[0].get_text()] * Nrows
DateList = [columns[4].get_text()] * Nrows
if price[0].strip()=='Package':
price = [columns[3].get_text()] * Nrows
if len(Quota)<len(Weight):#if Quota has less itmes extend with NaN
lstnans= [np.nan]*(len(Weight)-len(Quota))
Quota.extend(lstnans)
if len(price) < len(Quota): #if price column has less items than quota column,
val = [columns[3].get_text()] * (len(Quota)-len(price)) #extend with
price.extend(val) #whatever is in
#price column
#if len(DateList) > len(Quota): #if DateList is longer than Quota,
#print("it's longer than")
#value = [columns[4].get_text()] * (len(DateList)-len(Quota))
#DateList = value * Nrows
if len(Quota) < len(DateList): #if Quota is less than DateList (due to gap),
stu = [np.nan]*(len(DateList)-len(Quota)) #extend with NaN
Quota.extend(stu)
if len(Weight) < len(DateList):
dru = [np.nan]*(len(DateList)-len(Weight))
Weight.extend(dru)
FinalDataframe = pd.DataFrame(
{
'ID':IDList,
'AvailableQuota': Quota,
'LiveWeightPounds': Weight,
'price':price,
'DatePosted':DateList
})
df_Quota = df_Quota.append(FinalDataframe, ignore_index=True)
#df_Quota = df_Quota.loc[df_Quota['DatePosted']=='5/20']
df_Q = df_Quota['DatePosted'].iloc[0]
df_Quota = df_Quota[df_Quota['DatePosted'] == df_Q]
print (df_Quota)
for filename in os.listdir(path):
file_path = os.path.join(path, filename)
if os.path.isfile(file_path):
with open(file_path, 'r') as f:
pattern = re.compile(r'Sent:.*?\b(\d{4})\b')
email = f.read()
dates = pattern.findall(email)
if dates:
print("Date:", ''.join(dates))
#cursor = con.cursor()
#exported_data = [tuple(x) for x in df_Quota.values]
#sql_query = ("INSERT INTO ROUGHTABLE(species, date_posted, stock_id, pounds, money, sector_name, ask)" "VALUES (:1, :2, :3, :4, :5, 'NEFS 2', '1')")
#cursor.executemany(sql_query, exported_data)
#con.commit()
#cursor.close()
#con.close()
A: Use try: ... except: ...:
try:
#extract data from table
except IndexError:
#execute rest of program
A: continue is the keyword to use for skipping empty/problem rows. IndexError is thanks to the attempt to access columns[0] on an empty columns list. So just skip to next row when there is an exception.
for row in table.find_all('tr'):
columns = row.find_all('td')
try:
if columns[0].get_text().strip()!='ID':
# Rest as above in original code.
except IndexError:
continue
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Branding yourself is your first step to getting on top of the digital world. Ask yourself: Why does your brand exist? What does your brand do? What is your brand's story? What kind of message are you trying to send to the audience? Your brand is your logo, your attitude, how you present your products, your business cards and your website. Now you might be asking, how do you build a superior brand? Here are some tips to get you started.
To have a great brand, you have to know what branding is and how it works. A brand is anything that separates one thing from another. To create your brand, you choose a unique name and image for your product. The goal with branding is to establish a significant, and if possible, unique presence in the market that attracts and retains your audience.
Why does your brand exist? Your brand is your identity as a business. It is therefore important to identify the primary product, service, resource, special ability, etc. you have to offer others. What makes your product or service unique? Perhaps you're business offers a unique solution or service that differs you from your competitors. This "edge" should be conveyed as part of your brand. If you aren't clear about your business, brand our product, then neither will your audience. You have to know your brand before trying to explain or express it to others.
What are you particularly gifted at delivering or creating? This is your specialty. A good way to start is to think of five key skills that you offer. What are your most important values as a business? Keep in mind that creating and building your unique brand is an ongoing process. Anticipate refining your specialties as your company expands and grows.
Networking is one of the best ways to become known in your industry. Social Media platforms can assist you in managing and staying connected with others in your industry. Add and follow people and businesses with the same interests as you to help build recognition amongst peers. By forming networking relationships with people in your audience, you can grow your business and your brand long-term. After all, word of mouth recommendations are always more valuable to a business.
Consistency is a critical in creating a brand. Think about the most successful companies you know. Chances are, you remember a logo, the product presentation, website, business cards are all "speaking" their brand. By communicating your brand to your customers with a consistent "tone of voice" makes your brand and your company memorable. It will make things seem familiar and reliable to the audience and creates an unspoken trust relationship. Consistency also helps your customers to have a consistent expectation and value of your product or service.
In short, branding is your business identity that goes with whatever medium you use, print, digital, radio or face to face. The more consistent your brand is, the more recognizable and trusted it will be in a customer's eyes. Your brand should express your specialization, goals and values of your company. More than anything, branding can be that little something that makes your business a successful business. | {
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Installing Adobe Photoshop and then cracking it is easy and simple. The first step is to download and install the Adobe Photoshop software on your computer. Then, you need to locate the installation.exe file and run it. Once the installation is complete, you need to locate the patch file and copy it to your computer. The patch file is usually available online, and it is used to unlock the full version of the software. Once the patch file is copied, you need to run it and then follow the instructions on the screen. Once the patching process is complete, you will have a fully functional version of Adobe Photoshop on your computer. To make sure that the software is running properly, you should check the version number to ensure that the crack was successful. And that's it – you have now successfully installed and cracked Adobe Photoshop!
Also forget whatever you think you know about using Lightroom. It doesn't have the kind of depth and precision Photoshop does and couldn't be more inconvenient for working with images. Lightroom 5 and Elements 11 are just too complicated for the average photographer.
Adobe is starting to offer more classes for Photoshop than Lightroom, and developing a solid tutorial base is a good move. The question remains, however, whether developers are going to recognize Photoshop's inherent hierarchy in the same way that Lightroom's has been recognized. And, of course, how many people are still going to be running 32-bit versions of Windows XP?
For my money, Photoshop is still the Best of Breed in the photo editing landscape. Lightroom remains a very solid alternative, but there's nothing quite like Photoshop in terms of depth, simplicity and power. The competition in this war isn't worthy, it's either Photoshop or Lightroom.
The Opacity slider enables you to choose from a range of opacity levels and, as the parameters of the alpha channel change, the effect changes. I don't want to get into a discussion of how alpha channels work, only that they are the way to go with a digital image. Not an easy concept to explain but, when done right, the image looks great, too. Your Photos stack is now accessible from within Photoshop, which is a great way to keep your own private pages organized.
Swiping with the Apple Pencil to access the Batch Export tool was a bit like using a magic wand. No photos move like that. A small, neat change happens on every swipe, and there's no highlighting, no pauses, no doubt. I tend to save my images in spaced, click-free intervals. Assuming my photos are aligned, I don't notice the gaps from one command to the next, and I don't have to think twice. I'm used to it by now.
Now, it wouldn't be realistic to guarantee that you'd be a Photoshop wiz at this point — but that isn't what this guide designed to do. We hope we've provided you with the understanding you'll need to use the powerful tools in Photoshop in a timely, efficient, non-hair-pulling manner, so that you can elevate your visual content game, like, today.
Adobe Photoshop is the flagship product of Adobe and has been used by professional photo editors everywhere from companies to universities all over the world. Adobe Photoshop is an incredibly powerful tool, capable of creating some of the most stunning images you can possibly imagine.
And you cant deny the impact Photoshop has had on the photography field! Marketing campaigns heavily use with images that have been created by Photoshop.
Adobe Photoshop is a bit of an intimidating piece of software, but it wont have you guessing like the old days when you didn't know how to open your power shot camera.
The basic task of the Photoshop user is to help in the production of photos by making them presentable. In this task, the user makes use of the power of the software – which comes by its versatility, ability to perform several functions, and time efficiency.
What does a photo editor do that other software does not?
Whatdoes a photo editor do of what other software does not?
With the use of the Photoshop, the photo editor offers the opportunity of saving several photos at once and subsequently, allows for a quick comparison between the photos.
If you are used to the other photo editing software, it is quite likely that you will be interested in Photoshop because of its wide range of features. It can, for example, create different effects that are interesting, resulting in the creation of photo frames, or convert photos to other formats, including JPG, PNG, and TIFF, and then provide you with the necessary crafting.
Adobe Photoshop is a very powerful tool – and with a bit of practice, your creativity will not have to be held back any longer.
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The Content Aware Fill feature helps produce professional-looking images with seamless replacement and color matching. Instead of replacing an object with a new object that's identical in appearance, Content Aware Fill detects the area of the picture that needs filling and replaces it with a new color that matches the fill that surrounds the selected object. This makes it easier to replace objects with a new object that's identical in appearance.
Pixel Perfect is an advanced enhancement that can be applied to multiple effects applied to any layer. Apply the program to a web layer and it will automatically adjust the appearance of the layer to the surroundings.
The list of Adobe Photoshop features are too big to tell you about: It has improved the quality along edges in objects, improved copy&paste support, an alternate true-to-life rendering mode for high-resolution output and others.
The Adjustments Panel is an extremely powerful feature that's simply a window that shows the available adjustment presets. It's usually located in the bottom right corner of the screen, but it can also be minimized to occupy none of the screen space and show only its bottom border. This panel offers a huge collection of adjustment presets, including levels and curves, adjustments for color, saturation or hue, and more.
For even easier image editing, Adobe added a new feature called Guides. It creates a grid by which you can align and align the path you're creating, whether you're filling in with layers or creating an outline. You can still work with the path by drawing around it with the usual techniques of using a pencil, tracing or erasing. Guides allow you to see where you are in the image and give instantaneous feedback.
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This Article is a "PC Perspective" Tom's Hardware Review article. For more reviews, previews, rumors, speculation, and news about Adobe Photoshop, check out the " PC Perspective Adobe " category on the site.
The latest Photoshop CC (Creative Cloud) update adds a new Pixel Sense tool, improvements to the healing tool, and support for layer groups. It also features new blending modes, the ability to add or remove points, the revamped Magic Wand, and more. More details on these features and changes can be found in the Photoshops release notes and on the Updates section on the Photoshop CC page.
The new Photoshop features, identified as beta, include a new default behavior that makes colors in images and video match the actual color under certain circumstances, and the ability to create and edit video in the editor itself. It also features the ability to edit a single layer or selected Photoshop documents without having to open a separate Photoshop document. Among other things, Photoshop now lets you create a "project" and save it as a Set of Actions, which you can then apply to different photos or video projects. The app also has a real-time feature that automatically detects the correct settings for an image and corrects them—a function that is usually not realized until after the image has been saved.
Adobe Photoshop is a professional and powerful photo editing tool. It empowers the users to enhance their photos. It supports all the latest formats like JPG, GIF, PNG, PSD and Adobe PDF. The new version of Photoshop CS6 is available for all the platforms like Windows, Mac, Linux, iOS and Android.
Switching back to "true" PS would lock you into Windows, and we're not sure that's a good idea now. Still, until there's a better alternative, if you do need to go back, don't be afraid to try Elements. It's a free upgrade for anyone who owns Photoshop, and it's a surprisingly capable and usable program in its own right.
Like Photoshop before it, Photoshop CC 2018 features APIs based on DirectX, OpenGL, and Metal that allow 3D elements and content to be imported into the layer or adjustment panels, including content from 3D applications such as SketchBook Pro, LightWave, and Cinema 4D. Anyone working with 3D files will have access to a broad range of 3D operations that were prior only available in legacy 3D content with new features including:
Powerful Object Selection with multi-object selection. You can work with multiple objects at once, or select individual parts of the image, and activate separate tools to operate on them. You can also do detailed segmentation.
Improved Object Masking. You can now edit the mask itself – not just interact with it and show the mask.
New object collective operations that grouped objects into a bounding box and new options to let you transform objects together. You can also now use Face Fit, and use pivot points to make transformations on objects.
New object views. Now you can view 3D content in a 2D workspace. You can keep working with your new object selection, and add and remove objects in 3D views that appear as real 2D content but allow you to see your 3D selections and all their associated tools and settings.
New Advanced Object Options Page (Preview-only), that lets you switch back and forth between modes of object selection and edit tools. The option page allows you to see three different displays with three different selection views (flat, wireframe, and 3D aspects views).
New Content-Aware Fill options. Content-Aware Fill now works with 3D artwork, even where the boundary of the content is not connected. The algorithm detects the boundaries and fills surfaces with the content of other objects, as if they're full 3D objects.
New workflow features that streamline the most common editing tasks, such as using the object precision, and use of a remapping box to make 3D remappings easier to visualize and work with.
New Filters & Adjustments with new virtual 3D Layers, and access to the full range of adjustments and filters available in Photoshop.
New 3D preview panel in the Layers panel. You can switch between viewing a flattened image of the 3D content and a 3D preview of the content inside Photoshop.
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The free to use Paintshop Pro professional giving you incredible 2D/3D drawing, painting and photo editing tools. No need for expensive or complicated programs. This is a quick and easy to use image editor that completely extends your creative possibilities. You can use Adobe Photoshop CC or any photo editing software, but this is a special RAW image editing software. You can use Adobe Photoshop to edit your images with all standard tools. The
Another free form, if you want to design web, how you can design web so easy. It has a set of graphic tools for web designers. In this process, you do not need any specialized software. And using the camera function, you can make your design (web or bill board) with pictures and illustration.
Adobe Photoshop has developed a lot of features since it was first released in 1987. However, not all of those features have made it out into the normal Photoshop world. It is perhaps not as popular as photoshop. There are a number of features in Photoshop which make it stand-out from a background. But the most important feature in photoshop is not the features built into it, those are obviously much more important. What is the most important feature is the user experience.
Photoshop is a powerful tool and has a wide range of features. It provides unlimited customization for text, image and other objects. Photoshop does not allow all images to be edited on-the-fly. The editing tools are not always obvious but they provide fine control. To ensure the best results, a design step is needed for structural issues such as assets, cleanness & reputation, etc.
The pen tool is undoubtedly one of the most useful tools for graphic designers. It is a pen tool and it works like a pencil. In fact, similar to a pencil, you can add or delete things on the image. It is very important when working with such images. However, it is not so easy to find a solution for the color of the pen tool. Now, if you have a solution for color tool, you will be absolutely thrilled when you finally come across this. This is especially important for artists who rely on Photoshop to add and edit part of their creativity. If they are not found easily, it's time to wait for some of the top new Photoshop feature for 2019.
The fusion tool is one of the top tools used in graphic design. It is a tool that works like a pencil, but has some interesting and useful features. It was first seen in the days of Photoshop 3.5 as an extension of another tool. After that, it appears as "Pencil" tool in Photoshop 5.0. Now, with some minor changes, it has gained the ability to make some cool changes on an image. In addition to that, the tiny length of this tool allows you to tweak with one pixel at a time in the image. And if properly used, this can remove some distortion in the image and bring a higher level of clarity in the image. And finally, it is dark and to some extent destructive to the image. So, for those who are interested in some creative uses for this tool, it's definitely a great one. However, I do not recommend using it on tables or chairs.
When it comes to printing, we probably store the same companies. There is very little or no difference when it comes to printing processes and materials. However, there is a big difference when it comes to the inks that these companies use to print. These inks are generally designed for the specific paper or presentation media you are using for your project. So, what's a good way to find the right paper or presentation material? Actually, it may give you a lot of trouble until you find it. And sometimes, you have a difficult time finding it in a local store.
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In addition, Photoshop has improved the workflow for mobile and web with a new responsive editing experience, new templating capabilities for detailed adjustments, and new mobile app enhancements for working on smaller devices.
Adobe MAX 2019 also introduces new features for Mageworks: Capture One, DNG Converter and Magicshop, and the premiere of new features, experiences and Adobe Sensei powered tools for Adobe Media & Advertising Cloud. To learn more about these advancements visit: http://events.adobe.com/mx/sessions/?immortals=120856.
The newest edition of Photoshop updated copy-paste support from Illustrator to Photoshop, making it easier to move text layers and other typographic properties. Adobe has also improved the quality along edges in objects in its Sky Replacement feature. More enhancements include the addition of multithreaded and GPU compositing options for faster performance, the ability to search cloud documents in recents and improvements in Photoshop's saving preferences.
Photoshop is about to gain serious firepower. In this month's Download, you'll see new effects, layers, and editing tools; gain brand new tools for social media and more edgy adjustments; and get slightly less than a dozen new Quick Presets, so you can get the job done quickly and efficiently.
What do Randy Raney, Dennis Layton, Dustin Pike, and Brett Lott have in common? They're members of the Pixelscope Facebook Group , where they gather to discuss a variety of Adobe Photoshop issues. In this Facebook group , the Pixelscope crew got together for a live show to chat about Photoshop, share their experiences, and help others get the most out of this popular tool. They go through Photoshop issues from 90% of the questions that have come through Pixelscope group and other Photoshop Facebook groups.
Adobe has added new features to Edit flattened previews to make it easy to see how edits will look on an image in its original state. You can also see the newly improved smart guide when you edit the flattened image in the original (non-flattened) state.
The new N-level sort list is available in the Content Aware Fill dialog. It lets you quickly sort out different content assets by letter, number or word in one click. Exact Match is a new feature that lets you identify likely duplicates and suggest an alternative for your selected content. The solution is instantly applied to the content in the workspace while you finish your edits. More improvements include performance enhancements for layers and a couple of refinements to the Smart Objects dialog.
Photoshop is a complex creative tool packed with hundreds of features. The most complex thing about Photoshop is to know which features work for what purpose. In this post, we'll describe some of the most used features of Photoshop CC and which are the most useful for video editing, still images, graphics, web design, etc. The post will give you a heads-up on what Photoshop features will be most useful for your next Photoshop project.
Yes, an eCommerce CMS can be perfectly integrated with your eCommerce website to showcase your products. It will help you to display your products in a professional way. For more eCommerce e-theme ideas take a look at our post on how to Create Stunning Ecommerce Websites for Beginners. This post will help you get started with the basics, such as setting up the store, adding your shopping cart and choosing the right eCommerce CMS:
•Adobe Camera RAW, a leading digital image processing application for professional photographers, now includes support for adding adjustments to images in a browser-based application. A new Adobe Component is used to handle the exchange of data and processing abilities between the browser-based application and Photoshop.
•The Adobe desktop app includes an updated version of the Content-Aware Fill tool, which used to require a licensing change that would prevent new users from using the tool. Now, users can launch an app for Content-Aware Fill and simply select "Auto-Make Fill" to add a content-aware smart request to the selected object in an image.
"We leveraged Adobe's extensive research and development expertise to ensure key innovations like our new Share for Review feature are intuitive and easy to use," said Shantanu Narayen, president and CEO of Adobe. "These exciting new features bring breakthrough capabilities to the Photoshop desktop app, and we're expecting them to be warmly embraced by the creative community."
Image-editing customers around the world can expect to experience Photoshop on the web and on the device of their choice at the maximum speed. The web application is designed to be quick for home users, and give mobile users the same rich editing capabilities on smaller screens as desktop users. Adobe Photoshop is available in both a desktop and Mac version, and at the time of the announcement, the desktop version was available across the globe.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,682 |
Лей () — денежная единица, являющаяся названием национальных валют Румынии и Молдавии — соответственно румынского лея и молдавского лея.
Кроме того, название «лей» носили следующие обращавшиеся параллельно с румынским леем денежные единицы:
военный лей — денежные знаки, выпущенные в период германо-австрийской оккупации Румынии во время Первой мировой войны;
лей Командования Красной армии — денежные знаки, выпускавшиеся советским военным командованием в Румынии в 1944 году;
а также не выпущенный в обращение лей Института внешнего финансирования.
Этимология
На русский язык названия молдавской и румынской валют обычно транслитерируются как лей, хотя слово происходит от и означает «лев». Стоящий на задних лапах геральдический лев был изображён на голландских талерах, которые впервые были отчеканены в 1575 году и в XVII веке получили широкое распространение в Османской империи, в том числе на Балканах, а также во многих других странах. В Голландии эти монеты назывались «левендальдер», «левендаальдер» или «львиный талер» (), на Руси — левками или левковыми талерами, в Румынии — леями (львами). Отсюда и происходят современные названия румынского и молдавского леев.
Банкноты и монеты, номинированные в леях
Примечания
Источники
Ссылки
Национальный Банк Молдовы, Национальная валюта
Национальный Банк Румынии, Монеты и банкноты
Лей | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,100 |
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