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William Playfair William Playfair (22 September 1759 – 11 February 1823), a Scottish engineer and political economist, served as a secret agent on behalf of Great Britain during its war with France.[1] The founder of graphical methods of statistics,[2] Playfair invented several types of diagrams: in 1786 the line, area and bar chart of economic data, and in 1801 the pie chart and circle graph, used to show part-whole relations.[3] As a secret agent, Playfair reported on the French Revolution and organized a clandestine counterfeiting operation in 1793 to collapse the French currency. William Playfair Born(1759-09-22)September 22, 1759 Benvie, Forfarshire, Scotland Died11 February 1823(1823-02-11) (aged 63) London, England Known forinventor of statistical graphs, writer on political economy, and secret agent for Great Britain FamilyJohn Playfair (brother) James Playfair (brother) William Henry Playfair (nephew) Biography William Playfair was born in 1759 in Scotland. He was the fourth son (named after his grandfather) of the Reverend James Playfair of the parish of Liff & Benvie near the city of Dundee in Scotland; his notable brothers were architect James Playfair and mathematician John Playfair. His father died in 1772 when he was 13, leaving the eldest brother John to care for the family and his education. After his apprenticeship with Andrew Meikle, the inventor of the threshing machine, Playfair became draftsman and personal assistant to James Watt at the Boulton and Watt steam engine manufactory in Soho, Birmingham.[4] Playfair had a variety of careers. He was in turn a millwright, engineer, draftsman, accountant, inventor, silversmith, merchant, investment broker, economist, statistician, pamphleteer, translator, publicist, land speculator, convict, banker, ardent royalist, editor, blackmailer and journalist. On leaving Watt's company in 1782, he set up a silversmithing business and shop in London, which failed. In 1787 he moved to Paris, taking part in the storming of the Bastille two years later. After the French revolution, Playfair played a role in the Scioto Land sale to French settlers in the Ohio River Valley.[1] He returned to London in 1793, where he opened a "security bank", which also failed. From 1775 he worked as a writer and pamphleteer and did some engineering work.[4] In the 1790s, Playfair informed the British government on events in France and proposed various clandestine operations to bring down the French government. At the end of the 1790s he was imprisoned for debt in the Fleet Prison, being released in 1802.[1] Work Ian Spence and Howard Wainer in 2001 describe Playfair as "engineer, political economist and scoundrel" while "Eminent Scotsmen" calls him an "ingenious mechanic and miscellaneous writer".[5] It compares his career with the glorious one of his older brother John Playfair, the distinguished Edinburgh mathematics professor, and draws a moral about the importance of "steadiness and consistency of plan" as well as of "genius". Bruce Berkowitz in 2018 provides a detailed portrait of Playfair as an "ambitious, audacious, and woefully imperfect British patriot" who undertook the "most complex covert operation anyone had ever conceived".[1] Bar chart Two decades before Playfair's first achievements, in 1765 Joseph Priestley had created the innovation of the first timeline charts, in which individual bars were used to visualise the life span of a person, and the whole can be used to compare the life spans of multiple persons. According to James R. Beniger and Robyn (1978) "Priestley's timelines proved a commercial success and a popular sensation, and went through dozens of editions".[6] These timelines directly inspired Playfair's invention of the bar chart, which first appeared in his Commercial and Political Atlas, published in 1786. According to Beniger and Robyn (1978) "Playfair was driven to this invention by a lack of data. In his Atlas he had collected a series of 34 plates about the import and export from different countries over the years, which he presented as line graphs or surface charts: line graphs shaded or tinted between abscissa and function. Because Playfair lacked the necessary series data for Scotland, he graphed its trade data for a single year as a series of 34 bars, one for each of 17 trading partners".[6] In this bar chart Scotland's imports and exports from and to 17 countries in 1781 are represented. "This bar chart was the first quantitative graphical form that did not locate data either in space, as had coordinates and tables, or time, as had Priestley's timelines. It constitutes a pure solution to the problem of discrete quantitative comparison".[6] The idea of representing data as a series of bars had earlier (14th century) been published by Jacobus de Sancto Martino and attributed to Nicole Oresme. Oresme used the bars to generate a graph of velocity against continuously varying time. Playfair's use of bars was to generate a chart of discrete measurements.[7] Graphics Playfair, who argued that charts communicated better than tables of data, has been credited with inventing the line, bar, area, and pie charts. His time-series plots are still presented as models of clarity. Playfair first published The Commercial and Political Atlas in London in 1786. It contained 43 time-series plots and one bar chart, a form apparently introduced in this work. It has been described as the first major work to contain statistical graphs. Playfair's Statistical Breviary, published in London in 1801, contains what is generally credited as the first pie chart.[8][9][10] From 1809 until 1811, he published the massive "British Family Antiquity, Illustrative of the Origin and Progress of the Rank, honours and personal merit of the nobility of the United Kingdom. Accompanied with an Elegant Set of Chronological Charts." The work was 9 large volumes in 11 parts; Volume six contained a suite of 12 plates of which 10 are in two states, coloured and uncoloured, and 9 large folding tables, partly hand coloured. This was an important work on genealogy published in a very limited edition. In it, Playfair sought to show the true character and heroism of the British nobility and that the Monarchy, particularly the British Monarchy, is the true defender of liberty. The volumes are separated into the peerage and baronetage of England, Scotland and Ireland. Counterfeiting operation In 1793 Playfair as secret agent devised a clandestine plan that he presented to Henry Dundas, who was Home Secretary soon to become Britain's Secretary of State for War. Playfair proposed to "fabricate one hundred millions of assignats (the French currency) and spread them in France by every means in my power." He saw the counterfeiting plan as the lesser of two evils: "That there are two ways of combatting the French nation the forces of which are measured by men and money. Their assignats are their money and it is better to destroy this paper founded upon an iniquitous extortion and a villainous deception than to shed the blood of men." Playfair forged the assignats at Haughton Castle in Northumberland and distributed them according to an elaborate plan. The plan apparently worked: by 1795 the French assignat had become worthless and the ensuing chaos undermined the French government. Playfair never told anyone about the operation.[1] Playfair cycle The following quotation, known as the "Playfair cycle," has achieved notoriety as it pertains to the "Tytler cycle": :...wealth and power have never been long permanent in any place. ...they travel over the face of the earth, something like a caravan of merchants. On their arrival, every thing is found green and fresh; while they remain all is bustle and abundance, and, when gone, all is left trampled down, barren, and bare.[11] Works • 1785. The Increase of Manufactures, Commerce, and Finance, with the Extension of Civil Liberty, Proposed in Regulations for the Interest of Money. London: G.J. & J. Robinson. • 1786. The Commercial and Political Atlas: Representing, by Means of Stained Copper-Plate Charts, the Progress of the Commerce, Revenues, Expenditure and Debts of England during the Whole of the Eighteenth Century. • 1787. Joseph and Benjamin, a Conversation Translated from a French Manuscript. London: J. Murray. • 1793. Thoughts on the Present State of French Politics, and the Necessity and Policy of Diminishing France, for Her Internal Peace, and to Secure the Tranquillity of Europe. London: J. Stockdale. • 1793. A general view of the actual force and resources of France, in January, M. DCC. XCIII: to which is added, a table, shewing the depreciation of assignats, arising from their increase in quantity. J. Stockdale. • 1796. The History of Jacobinism, Its Crimes, Cruelties and Perfidies: Comprising an Inquiry into the Manner of Disseminating, under the Appearance of Philosophy and Virtue, Principles which are Equally Subversive of Order, Virtue, Religion, Liberty and Happiness. Vol. I.. Philadelphia: W. Cobbett. • 1796. For the Use of the Enemies of England, a Real Statement of the Finances and Resources of Great Britain • 1798. Lineal arithmetic, Applied to Shew the Progress of the Commerce and Revenue of England During the Present Century. A. Paris. • 1799. Stricture on the Asiatic Establishments of Great Britain, With a View to an Enquiry into the True Interests of the East India Company. Bunney & Gold. • 1801. Statistical Breviary; Shewing, on a Principle Entirely New, the Resources of Every State and Kingdom in Europe. London: Wallis. • 1805. An Inquiry into the Permanent Causes of the Decline and Fall of Powerful and Wealthy Nations. London: Greenland & Norris. • 1805. European commerce, shewing new and secure channels of trade with the continent of Europe... • 1805. Statistical Account of the United States of America by D. F. Donnant. London: J. Whiting. William Playfair, Trans. • 1807. European Commerce, Shewing New and Secure Channels of Trade with the Continent of Europe. Vol. I.. Philadelphia: J. Humphreys. • 1808. Inevitable Consequences of a Reform in Parliament • 1809. A Fair and Candid Address to the Nobility and Baronets of the United Kingdom; Accompanied with Illustrations and Proofs of the Advantage of Hereditary Rank and Title in a Free Country • 1811. British Family Antiquity: Index to the 9 Volumes of William Playfair's Family Antiquity of the British Nobility • 1813. Outlines of a Plan for a New and Solid Balance of Power in Europe. J. Stockdale. • 1814. Political Portraits in This New Æra, Vol. II . London: C. Chapple. • 1816. Supplementary Volume to Political Portraits in This New Æra. London: C. Chapple. • 1818. The History of England, from the Revolution in 1688 to the Death of George II. Vol. II. R. Scholey. • 1819. France as it Is, Not Lady Morgan's France, Vol. I. London: C. Chapple. • 1820. France as it Is, Not Lady Morgan's France, Vol. II. London: C. Chapple. References 1. Berkowitz, Bruce (2018). Playfair: The True Story of the British Secret Agent Who Changed How We See the World. ISBN 978-1-942695-04-2. 2. Paul J. FitzPatrick (1960). "Leading British Statisticians of the Nineteenth Century". In: Journal of the American Statistical Association, Vol. 55, No. 289 (Mar. 1960), pp. 38–70. 3. Michael Friendly (2008). "Milestones in the history of thematic cartography, statistical graphics, and data visualization" Archived 26 September 2018 at the Wayback Machine. pp 13–14. Retrieved 7 July 2008. 4. Ian Spence and Howard Wainer (1997). "Who Was Playfair?". In: Chance 10, p. 35–37. 5. Ian Spence and Howard Wainer (2001). "William Playfair". In: Statisticians of the Centuries. C.C. Heyde and E. Seneta (eds.) New York: Springer. pp. 105–110. 6. James R. Beniger and Dorothy L. Robyn (1978). "Quantitative graphics in statistics: A brief history". In: The American Statistician. 32: pp. 1–11. 7. Der, Geoff; Everitt, Brian S. (2014). A Handbook of Statistical Graphics Using SAS ODS. Chapman and Hall - CRC. p. 4. ISBN 978-1-584-88784-3. William Playfair, for example, is often credited with inventing the bar chart (see Chapter 3) in the last part of the 18th century, although a Frenchman, Nicole Oresme, used a bar chart in a 14th century publication, The Latitude of Forms to plot the velocity of a constantly accelerating object against time. But it was Playfair who popularized the idea of graphic depiction of quantitative information. 8. Edward R. Tufte (2001). The Visual Display of Quantitative Information. Cheshire, CT: Graphics Press, p. 44. 9. Ian Spence (2005). "No Humble Pie: The Origins and Usage of a statistical Chart" Archived 20 March 2007 at the Wayback Machine. In: Journal of Educational and Behavioral Statistics. Winter 2005, 30 (4), 353–368. 10. Playfair, William; Wainer, Howard; Spence, Ian (2005). Playfair's Commercial and Political Atlas and Statistical Breviary. Cambridge University Press. ISBN 9780521855549. 11. William Playfair (1807). An Inquiry into the Permanent Causes of the Decline and Fall of Powerful and Wealthy Nations, p. 102. External links Wikimedia Commons has media related to William Playfair. • Playfair, William (1759–1823) at oxforddnb.com • William PLAYFAIR b. 22 September 1759 - d. 11 February 1823 at statprob.com • "Biographical Dictionary of Eminent Scotsmen" • Works by William Playfair at Project Gutenberg • Works by or about William Playfair at Internet Archive Visualization of technical information Fields • Biological data visualization • Chemical imaging • Crime mapping • Data visualization • Educational visualization • Flow visualization • Geovisualization • Information visualization • Mathematical visualization • Medical imaging • Molecular graphics • Product visualization • Scientific visualization • Software visualization • Technical drawing • User interface design • Visual culture • Volume visualization Image types • Chart • Diagram • Engineering drawing • Graph of a function • Ideogram • Map • Photograph • Pictogram • Plot • Sankey diagram • Schematic • Skeletal formula • Statistical graphics • Table • Technical drawings • Technical illustration People Pre-19th century • Edmond Halley • Charles-René de Fourcroy • Joseph Priestley • Gaspard Monge 19th century • Charles Dupin • Adolphe Quetelet • André-Michel Guerry • William Playfair • August Kekulé • Charles Joseph Minard • Luigi Perozzo • Francis Amasa Walker • John Venn • Oliver Byrne • Matthew Sankey • Charles Booth • Georg von Mayr • John Snow • Florence Nightingale • Karl Wilhelm Pohlke • Toussaint Loua • Francis Galton Early 20th century • Edward Walter Maunder • Otto Neurath • W. E. B. Du Bois • Henry Gantt • Arthur Lyon Bowley • Howard G. Funkhouser • John B. Peddle • Ejnar Hertzsprung • Henry Norris Russell • Max O. Lorenz • Fritz Kahn • Harry Beck • Erwin Raisz Mid 20th century • Jacques Bertin • Rudolf Modley • Arthur H. Robinson • John Tukey • Mary Eleanor Spear • Edgar Anderson • Howard T. Fisher Late 20th century • Borden Dent • Nigel Holmes • William S. Cleveland • George G. Robertson • Bruce H. McCormick • Catherine Plaisant • Stuart Card • Pat Hanrahan • Edward Tufte • Ben Shneiderman • Michael Friendly • Howard Wainer • Clifford A. Pickover • Lawrence J. Rosenblum • Thomas A. DeFanti • George Furnas • Sheelagh Carpendale • Cynthia Brewer • Miriah Meyer • Jock D. Mackinlay • Alan MacEachren • David Goodsell • Michael Maltz • Leland Wilkinson • Alfred Inselberg Early 21st century • Ben Fry • Hans Rosling • Christopher R. 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William Raoul Reagle Transue William Raoul Reagle Transue (January 31, 1937 – December 17, 2008) was an American mathematician and topologist. He is the son of mathematician William Reagle Transue and Monique Serpette who moved from her native France to the US in 1936. Bill, as he was known, earned his bachelor's degree from Harvard University in 1958, and his Ph.D. in mathematics from The University of Georgia in 1967 under Billy Joe Ball. He was a professor of mathematics at Auburn University from 1967 until his retirement over 30 years later. References • William Raoul Reagle Transue at the Mathematics Genealogy Project Authority control International • VIAF Academics • MathSciNet • Mathematics Genealogy Project
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William S. Massey William Schumacher Massey (August 23, 1920[1] – June 17, 2017) was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact couples, and wrote several textbooks, including A Basic Course in Algebraic Topology (ISBN 0-387-97430-X). William Schumacher Massey Born(1920-08-20)August 20, 1920 Granville, Illinois, United States DiedJune 17, 2017(2017-06-17) (aged 96) Hamden, Connecticut, U.S. NationalityAmerican Alma materUniversity of Chicago Princeton University Known forMassey product Blakers–Massey theorem Exact couple SpouseEthel H. Massey Children3 Scientific career FieldsTopology InstitutionsBrown University Yale University ThesisClassification of mappings of an (n + 1)-dimensional space into an n-sphere (1948) Doctoral advisorNorman Steenrod Military career AllegianceUnited States Service/branchUnited States Navy Years of service1942–1945 Life William Massey was born in Granville, Illinois, in 1920, the son of Robert and Alma Massey, and grew up in Peoria. He was an undergraduate student at the University of Chicago. After serving as a meteorologist aboard aircraft carriers in the United States Navy for 4 years during World War II, he received a Ph.D. degree from Princeton University in 1949.[2] His dissertation, entitled Classification of mappings of an $(n+1)$-dimensional space into an n-sphere, was written under the direction of Norman Steenrod. He spent two additional years at Princeton as a post-doctoral research assistant.[3] He then taught for ten years on the faculty of Brown University. In 1958 he was elected to the American Academy of Arts and Sciences.[4] From 1960 till his retirement he was a professor at Yale University. He died on June 17, 2017, in Hamden, Connecticut. He had 23 PhD students, including Donald Kahn, Larry Smith, and Robert Greenblatt. Selected works • Algebraic topology: an introduction. NY: Harcourt, Brace & World. 1967; xix+261 pp.{{cite book}}: CS1 maint: postscript (link) 4th corrected printing. 1977. • Homology and cohomology theory. Monographs and Textbooks in Pure and Applied Mathematics. Vol. 46. New York: Marcel Dekker. 1978; xiv+412 pp.{{cite book}}: CS1 maint: postscript (link)[5] • Singular homology theory. Graduate Texts in Mathematics. Springer-Verlag. 1980; xii+265 pp.{{cite book}}: CS1 maint: postscript (link)[6] • A basic course in algebraic topology. Springer. 1991. ISBN 9780387974309. 3rd corrected printing. 1997. • Massey, William S. (1952). "Exact couples in algebraic topology. I, II". Annals of Mathematics. Second Series. 56: 363–396. doi:10.2307/1969805. JSTOR 1969805. MR 0052770. See also • Blakers–Massey theorem • Exact couple • Massey product External links • Address at Yale References 1. Massey, William S. "Indiana, Marriages, 1811–195". familysearch.org. Retrieved 1 November 2013. 2. "William Massey obituary". New Haven Register. June 20, 2017. Retrieved July 8, 2022. 3. William S. Massey at the Mathematics Genealogy Project 4. "In Memoriam: William S. Massey, 1920–2017". math.yale.edu. Department of Mathematics, Yale University. June 30, 2017. Retrieved July 8, 2022. 5. Ewing, John H. (1979). "Review: Homology and cohomology theory by W. S. Massey" (PDF). Bulletin of the American Mathematical Society. New Series. 1 (6): 985–989. doi:10.1090/s0273-0979-1979-14707-4. 6. Vick, James W. (1981). "Review: Singular homology theory by W. S. Massey" (PDF). Bulletin of the American Mathematical Society. New Series. 4 (2): 229–233. doi:10.1090/s0273-0979-1981-14892-8. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
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William Shaw (mathematician) William Shaw (born 14 May 1958) is a British mathematician, and formerly professor of the mathematics and computation of risk at University College London.[1][2] He is a consultant on financial derivatives, an author of a primary book on using Mathematica to model financial derivatives, formerly co-Editor-in-Chief of the journal Applied Mathematical Finance. Shaw studied at King's College, Cambridge, where he studied mathematics; he was Wrangler and earned a B.A. in 1980. In 1981 he won the Mayhew Prize[3] for his performance on the Cambridge Mathematical Tripos. In 1984 he received a D.Phil. (PhD) in mathematical physics from Wolfson College, Oxford. From 1984 to 1987 he was a research fellow at Clare College, Cambridge and C.L.E. Moore Instructor at the Massachusetts Institute of Technology. From 1987 to 1990, he worked for Smith Associates in Guildford, and ECL in Henley-on Thames. From 1991 to 2002 he was a lecturer in mathematics at Balliol College, Oxford. In 2002 he moved to St Catherine's College, Oxford, where he was University Lecturer in financial mathematics. In 2006 he moved to a Professorship at King's College London and in 2011 to a Professorship at UCL. He returned to the financial industry in 2012 and remained a visiting professor at UCL until 2017. Books • Applied Mathematica: Getting Started, Getting it Done by W.T. Shaw and J. Tigg. Addison-Wesley, 1993. • Modelling Financial Derivatives with Mathematica by W.T. Shaw, Cambridge University Press, 1998. • Complex Analysis with Mathematica by W.T. Shaw, Cambridge University Press, 2006. References 1. "Professor William T. Shaw". University College London. 2. "Profile: William T. Shaw". ResearchGate. 3. Mayhew Prize External links • William Shaw's former UCL web-page • Entry in Mathematics Genealogy Project • LinkedIn profile Authority control: Academics • MathSciNet • Mathematics Genealogy Project • zbMATH
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William James Sidis William James Sidis (/ˈsaɪdɪs/; April 1, 1898 – July 17, 1944) was an American child prodigy with exceptional mathematical and linguistic skills. He wrote the book The Animate and the Inanimate, published in 1925 (written around 1920), in which he speculated about the origin of life in the context of thermodynamics. William James Sidis Sidis at his Harvard graduation (1914) Born(1898-04-01)April 1, 1898 Manhattan, New York City, U.S. DiedJuly 17, 1944(1944-07-17) (aged 46) Boston, Massachusetts, U.S. Other names • John W. Shattuck • Frank Folupa • Parker Greene • Jacob Marmor Alma materHarvard University Rice Institute Harvard Law School Notable work • The Animate and the Inanimate (1925) • The Tribes and the States (c. 1935) His father, psychiatrist Boris Sidis raised his son according to certain principles as he wished his son to be gifted. Sidis became famous first for his precocity and later for his eccentricity and withdrawal from public life. Eventually, he avoided mathematics altogether, writing on other subjects under a number of pseudonyms. He entered Harvard at age 11 and, as an adult, was said to have an extremely high IQ, and to be conversant in about 25 languages and dialects. Some of these statements have not been verified, but many of his contemporaries, including Norbert Wiener, Daniel Frost Comstock and William James had agreed that he was extremely intelligent. Biography Parents and upbringing (1898–1908) Sidis was born to Jewish emigrants from Ukraine,[1] on April 1, 1898, in New York City. His father, Boris Sidis, had emigrated in 1887 to escape political and anti-semitic persecution.[2]: 2–4  His mother, Sarah (Mandelbaum) Sidis, and her family had fled the pogroms in the late 1880s.[2]: 7  Sarah attended Boston University and graduated from its School of Medicine in 1897.[3] William was named after his godfather, Boris's friend and colleague, the American philosopher William James. Boris was a psychiatrist and published numerous books and articles, performing pioneering work in abnormal psychology. He was a polyglot, and his son William would also become one at a young age. Sidis's parents believed in nurturing a precocious and fearless love of knowledge, although their methods of parenting were criticized in the media and retrospectively.[4][2]: 281, Epilogue  Sidis could read The New York Times at 18 months.[2]: 23  By age eight, he had reportedly taught himself eight languages (Latin, Greek, French, Russian, German, Hebrew, Turkish, and Armenian) and invented another, which he called "Vendergood". Harvard University and college life (1909–1914) Although the university had previously refused to let his father enroll him at age 9 because he was still a child, Sidis set a record in 1909 by becoming the youngest person to enroll at Harvard University. In early 1910, Sidis's mastery of higher mathematics was such that he lectured the Harvard Mathematical Club on four-dimensional bodies, attracting nationwide attention.[5][6] Notable child prodigy cybernetics pioneer Norbert Wiener who also attended Harvard at the time and knew Sidis, later stated in his book Ex-Prodigy: "The talk would have done credit to a first or second-year graduate student of any age...talk represented the triumph of the unaided efforts of a very brilliant child."[7] MIT physics professor Daniel F. Comstock was full of praises: "Karl Friedrich Gauss is the only example in history, of all prodigies, whom Sidis resembles. I predict that young Sidis will be a great astronomical mathematician. He will evolve new theories and invent new ways of calculating astronomical phenomena. I believe he will be a great mathematician, the leader in that science in the future."[2] Sidis began taking a full-time course load in 1910 and earned his Bachelor of Arts degree, cum laude, on June 18, 1914, at age 16.[8] Shortly after graduation, he told reporters "I want to live the perfect life. The only way to live the perfect life is to live it in seclusion". He granted an interview to a reporter from the Boston Herald. The paper reported Sidis's vows to remain celibate and never to marry, as he said women did not appeal to him. Later he developed a strong affection for Martha Foley, one year older than him. He later enrolled at Harvard Graduate School of Arts and Sciences. Teaching and further education (1915–1919) After a group of Harvard students threatened Sidis physically, his parents secured him a job at the William Marsh Rice Institute for the Advancement of Letters, Science, and Art (now Rice University) in Houston, Texas, as a mathematics teaching assistant. He arrived at Rice in December 1915 at the age of 17. He was a graduate fellow working toward his doctorate. Sidis taught three classes: Euclidean geometry, non-Euclidean geometry, and freshman math (he wrote a textbook for the Euclidean geometry course in Greek).[2]: 112  After less than a year, frustrated with the department, his teaching requirements, and his treatment by students older than himself, Sidis left his post and returned to New England. When a friend later asked him why he had left, he replied, "I never knew why they gave me the job in the first place—I'm not much of a teacher. I didn't leave: I was asked to go." Sidis abandoned his pursuit of a graduate degree in mathematics and enrolled at the Harvard Law School in September 1916, but withdrew in good standing in his final year in March 1919.[9] Politics and arrest (1919–1921) In 1919, shortly after his withdrawal from law school, Sidis was arrested for participating in a socialist May Day parade in Boston that turned violent. He was sentenced to 18 months in prison under the Sedition Act of 1918 by Roxbury Municipal Court Judge Albert F. Hayden. Sidis's arrest featured prominently in newspapers, as his early graduation from Harvard had garnered considerable local celebrity status. During the trial, Sidis stated that he had been a conscientious objector to the World War I draft, was a socialist, and did not believe in a god like the "big boss of the Christians," but rather in something that is in a way apart from a human being.[10][11] He later developed his own libertarian philosophy based on individual rights and "the American social continuity".[12][13] His father arranged with the district attorney to keep Sidis out of prison before his appeal came to trial; his parents, instead, held him in their sanatorium in New Hampshire for a year. They took him to California, where he spent another year.[14] At the sanatorium, his parents set about "reforming" him and threatened him with transfer to an insane asylum.[14] Later life (1921–1944) After returning to the East Coast in 1921, Sidis was determined to live an independent and private life. He only took work running adding machines or other fairly menial tasks. He worked in New York City and became estranged from his parents. It took years before he was cleared legally to return to Massachusetts, and he was concerned for years about his risk of arrest. He obsessively collected streetcar transfers, wrote self-published periodicals, and taught small circles of interested friends his version of American history. In 1933, Sidis passed a Civil Service exam in New York, but scored a low ranking of 254.[15] In a private letter, Sidis wrote that this was "not so encouraging".[15] In 1935, he wrote an unpublished manuscript, The Tribes and the States, which traces Native American contributions to American democracy.[16] In 1944, Sidis won a settlement from The New Yorker for an article published in 1937.[17] He had alleged it contained many false statements.[18] Under the title "Where Are They Now?", James Thurber pseudonymously described Sidis's life as lonely, in a "hall bedroom in Boston's shabby South End".[19] Lower courts had dismissed Sidis as a public figure with no right to challenge personal publicity. He lost an appeal of an invasion of privacy lawsuit at the United States Court of Appeals for the Second Circuit in 1940 over the same article. Judge Charles Edward Clark expressed sympathy for Sidis, who claimed that the publication had exposed him to "public scorn, ridicule, and contempt" and caused him "grievous mental anguish [and] humiliation," but found that the court was not disposed to "afford to all the intimate details of private life an absolute immunity from the prying of the press".[20] Sidis died from a cerebral hemorrhage in 1944 in Boston at age 46.[21] Publications and research From writings on cosmology, to Native American history, to Notes on the Collection of Transfers, and several purported lost texts on anthropology, philology, and transportation systems, Sidis covered a broad range of subjects. Some of his ideas concerned cosmological reversibility[22] and "social continuity".[23] In The Animate and the Inanimate (1925), Sidis predicted the existence of regions of space where the second law of thermodynamics operates in reverse to the temporal direction experienced in our local area. Everything outside of what we call a galaxy would be such a region. Sidis said that the matter in this region would not generate light. Sidis's The Tribes and the States (c. 1935) employs the pseudonym "John W. Shattuck", purporting to give a 100,000-year history of the Settlement of the Americas, from prehistoric times to 1828.[24] In this text, he suggests that "there were red men at one time in Europe as well as in America".[25] Sidis was a "peridromophile", a term he coined for people fascinated with transportation research and streetcar systems. He wrote a treatise on streetcar transfers under the pseudonym of "Frank Folupa" that identified means of increasing public transport usage.[26] For this work, in 1926 he was invited to speak at the inaugural "genius meeting" hosted by Winifred Sackville Stoner's League for Fostering Genius in Tuckahoe, New York.[27] In 1930, Sidis received a patent for a rotary perpetual calendar that took into account leap years.[28] The Animate and the Inanimate Sidis wrote The Animate and the Inanimate to elaborate his thoughts on the origin of life, cosmology, and the potential reversibility of the second law of thermodynamics through Maxwell's Demon, among other things. It was published in 1925;[29] however, it has been suggested that Sidis was working on the theory as early as 1916.[30] One motivation for the theory appears to be to explain psychologist and philosopher William James's "reserve energy" theory which proposed that a "reserve energy" could be used by people subjected to extreme conditions. Sidis's own "forced prodigy" upbringing was a result of testing the theory. The work is one of the few that Sidis did not write under a pseudonym. In The Animate and the Inanimate, Sidis states that the universe is infinite, and contains sections of "negative tendencies" where[31] the laws of physics are reversed, juxtaposed with "positive tendencies", which swap over epochs of time. Sidis states that there was no "origin of life", but that life has always existed and that it has only changed through evolution. Sidis adopted Eduard Pflüger's cyanogen based life theory, and Sidis cites "organic" things such as almonds (his example) that have cyanogen that does not kill. Because cyanogen is normally highly toxic, almonds are a strange anomaly. Sidis describes his theory as a fusion of the mechanistic model of life and the vitalist model of life, as well as entertaining the notion of life coming to earth from asteroids (as advanced by Lord Kelvin and Hermann von Helmholtz). Sidis also states that functionally speaking, stars are "alive" and undergo an eternally repeating light-dark cycle, reversing the second law in the dark portion of the cycle.[32] Sidis's theory was ignored upon release,[17] only to be found in an attic in 1979. Upon this discovery, Buckminster Fuller (who was a classmate of Sidis) commented on The Animate and the Inanimate:[33] Imagine my excitement and joy on being handed this xerox of Sidis's 1925 book, in which he clearly predicts the black hole. In fact, I find his whole book, The Animate and the Inanimate to be a fine cosmological piece. I find him focusing on the same subjects that fascinate me, and coming to about the same conclusions as those I have published in SYNERGETICS, and will be publishing in SYNERGETICS Volume II, which has already gone to the press. As a Harvard man of a generation later, I hope you will become as excited as I am at this discovery that Sidis did go on after college to do the most magnificent thinking and writing." — Buckminster Fuller Vendergood language Sidis created a constructed language called Vendergood in his second book, the Book of Vendergood, which he wrote at the age of 8. The language was mostly based on Latin and Greek, but also drew on German and French and other Romance languages.[2] It distinguished between eight moods: indicative, potential, imperative absolute, subjunctive, imperative, infinitive, optative, and Sidis's own strongeable.[2]: 41  One of its chapters is titled "Imperfect and Future Indicative Active". Other parts explain the origin of Roman numerals. It uses base 12 instead of base 10: • eis – 'one' • duet – 'two' • tre – 'three' • guar – 'four' • quin – 'five' • sex – 'six' • sep – 'seven' • oo (oe?) – 'eight' • non – 'nine' • ecem – 'ten' • elevenos – 'eleven' • dec – 'twelve' • eidec (eis, dec) – 'thirteen' Most of the examples are presented in the form of tests: 1. 'Do I love the young man?' = Amevo (-)ne the neania? 2. 'The bowman obscures.' = The toxoteis obscurit. 3. 'I am learning Vendergood.' = (Euni) disceuo Vendergood. 4. 'What do you learn?' (sing.). = Quen diseois-nar? 5. 'I obscure ten farmers.' = Obscureuo ecem agrieolai.[2]: 42–43  The Tribes and the States The Tribes and the States outlines the history of the Native Americans, focusing on the Northeastern tribes and continuing up to the mid-19th century. It was written around 1935 but never completed, and remained unpublished at the time of Sidis's death. Sidis wrote the history under the pseudonym "John W. Shattuck". Much of the history was taken from wampum belts; Sidis explained, "The weaving of wampum belts is a sort of writing by means of belts of colored beads, in which the various designs of beads denoted different ideas according to a definitely accepted system, which could be read by anyone acquainted with wampum language, irrespective of what the spoken language is. Records and treaties are kept in this manner, and individuals could write letters to one another in this way."[34] Much of the book is centered on the influence of Native Americans on migrating Europeans and the effect of Native Americans on the formation of the United States. It describes the origination of the federations that were to be an important event to the Founding Fathers. Legacy After his death, Helena Sidis said that her brother had an IQ reported in Abraham Sperling's 1946 book Psychology for the Millions as "the very highest that had ever been obtained",[35] but some of his biographers, such as Amy Wallace, exaggerated his IQ.[15] Sperling wrote:[35] Helena Sidis told me that a few years before his death, her brother Bill took an intelligence test with a psychologist. His score was the very highest that had ever been obtained. In terms of I. Q., the psychologist related that the figure would be between 250 and 300. Late in life William Sidis took general intelligence tests for Civil Service positions in New York and Boston. His phenomenal ratings are a matter of record. It has been acknowledged that Helena and William's mother Sarah had a reputation for exaggerated statements about the Sidis family.[15] Helena had falsely stated that the Civil Service exam William took in 1933 was an IQ test and that his ranking of 254 was an IQ score of 254.[15] It is speculated that the number "254" was actually William's placement on the list after he passed the Civil Service exam, as he stated in a letter sent to his family.[36] Helena also said that "Billy knew all the languages in the world, while my father only knew 27. I wonder if there were any Billy didn't know."[15] This statement was not backed by any source outside the Sidis family, and Sarah Sidis also made the improbable statement in her 1950 book The Sidis Story that William could learn a language in just one day.[15] Boris Sidis had once dismissed tests of intelligence as "silly, pedantic, absurd, and grossly misleading".[37] Regardless of the exaggerations, Sidis was judged by contemporaries such as MIT Physics professor Daniel Frost Comstock and American mathematician Norbert Wiener (who wrote about Sidis in his autobiography) to have had genuine ability.[38][2]: 54 [39] Sidis's life and work, particularly his ideas about Native Americans, are extensively discussed in Robert M. Pirsig's book Lila: An Inquiry into Morals (1991).[40] Sidis is also discussed in Ex-Prodigy, an autobiography by mathematician Norbert Wiener (1894–1964), who was a prodigy himself.[41] The Danish author Morten Brask wrote a novel that was a fictional account based on Sidis's life; The Perfect Life of William Sidis was published in Denmark in 2011. Another novel based on his biography was published by the German author Klaus Caesar Zehrer in 2017.[42] In education discussions The debate about Sidis's manner of upbringing occurred within a larger discourse about the best way to educate children. Newspapers criticized Boris Sidis's child-rearing methods. Most educators of the day believed that schools should expose children to common experiences to create good citizens. Most psychologists thought intelligence was hereditary, a position that precluded early childhood education at home.[43] The difficulties Sidis encountered in dealing with the social structure of a collegiate setting may have shaped opinion against allowing such children to rapidly advance through higher education in his day. Research indicates that a challenging curriculum can relieve social and emotional difficulties commonly experienced by gifted children.[44] Embracing these findings, several colleges incorporated procedures for early entrance. The Davidson Institute for Talent Development has developed a guidebook on the topic.[45] Sidis was portrayed derisively in the The New York Times in 1909, as "a wonderfully successful result of a scientific forcing experiment".[4] His mother later maintained that newspaper accounts of her son bore little resemblance to him. Bibliography • The Animate and the Inanimate (1925) • The Tribes and the States (c. 1935) (PDF file) References 1. Heinze, Andrew R. (2006). Jews and the American Soul: Human Nature in the Twentieth Century. Princeton University Press. ISBN 978-0-691-12775-0. 2. Wallace, Amy (1986). The prodigy: a biography of William James Sidis, the world's greatest child prodigy. London: Macmillan. ISBN 978-0333432235. 3. "History of Homeopathy and Its Institutions in America By William Harvey King, M.D., LL.D. Presented by Sylvain Cazalet". Homeoint.org. Retrieved May 25, 2011. 4. "Sidis Could Read at Two Years Old; Youngest Harvard Undergraduate Under Father's Scientific Forcing Process Almost from Birth. Good Typewriter at Four; At 5 Composed Text Book on Anatomy, in Grammar School at 6, Then Studied German, French, Latin, and Russian". The New York Times. October 18, 1909. p. 7. 5. Montour, Kathleen (April 1977). "William James Sidis, the broken twig". American Psychologist. 32 (4): 265–279. doi:10.1037/0003-066X.32.4.265. 6. "Wonderful Boys of History Compared With Sidis. All Except Macaulay Showed Special Ability in Mathematics. Instances of Boys Having 'Universal Genius'". The New York Times. January 16, 1910. p. SM11. Retrieved November 26, 2014. 7. Renselle, Doug. "A Review of Kathleen Montour's William James Sidis, The Broken Twig". Quantonics.com. Retrieved February 13, 2020. 8. "Harvard College, 1952". Retrieved November 26, 2014 – via Sidis.net. 9. "Harvard Transcripts". Retrieved May 25, 2011 – via Sidis.net. 10. "Sidis Gets Year and Half in Jail". Boston Herald. May 14, 1919. Retrieved January 12, 2018. 'Do you believe in a god?' 'No.' Atty. Connolly then asked the court what God he meant, whereupon Judge Hayden replied, God Almighty. Here Sidis said that the kind of a God that he did not believe in was the 'big boss of the Christians,' adding that he believed in something that is in a way apart from a human being. 11. Mahony, Dan. "Frequently Asked Questions About W. J. Sidis". Retrieved January 12, 2018. Was he religious? 'He espoused no religion, but said that... the kind of a God he did not believe in was the "big boss of the Christians", adding that he believed in something that is in a way apart from a human being (Boston Herald, May 14, 1919).' 12. Sidis, William James (June 1938). "Libertarian". Continuity News. Cambridge, Massachusetts (2): 4. 13. Sidis, William James. "The Concept of Rights". American Independence Society. Retrieved November 26, 2014. {{cite journal}}: Cite journal requires |journal= (help) 14. "Railroading in the Past". Retrieved May 25, 2011 – via Sidis.net. 15. "The Logics – Was William James Sidis the Smartest Man on Earth". Thelogics.org. Archived from the original on December 20, 2014. Retrieved November 26, 2014. 16. Johansen, Bruce E. (Fall 1989). "William James Sidis' 'Tribes and States': An Unpublished Exploration of Native American Contributions to Democracy". Northeast Indian Quarterly. 6 (3): 24–29 – via eric.ed.gov. 17. Bates, Stephen (2011). "The Prodigy and the Press: William James Sidis, Anti-Intellectualism, and Standards of Success". J&MC Quarterly. 88 (2): 374–397. doi:10.1177/107769901108800209. ISSN 1077-6990. S2CID 145637498. 18. "Sidis vs New Yorker". Sidis.net. February 29, 2008. Retrieved May 25, 2011. 19. LaMay, Craig L. (2003). Journalism and the Debate Over Privacy. LEA's Communication Series. Mahwah, New Jersey: Lawrence Erlbaum Associates, Inc. p. 63. ISBN 978-0-8058-4626-3. 20. Seitz, Robert N. (2002). "Review of Amy Wallace, The Prodigy (1986)". High IQ News. Archived from the original on June 2, 2008. Retrieved February 5, 2016. 21. Smith, Shirley (July 19, 1944). "Letter to the Editor". Boston Traveler. Retrieved May 25, 2011 – via Sidis.net. 22. Sidis, William James (1925). "The Animate and the Inanimate". Boston: The Gorham Press. {{cite journal}}: Cite journal requires |journal= (help) 23. Sidis, William James. "Continuity News". Archived from the original on May 7, 2016. Retrieved August 12, 2008 – via Sidis.net. 24. "The Tribes and the States, Table of Contents". Sidis.net. Retrieved May 25, 2011. 25. "The Tribes and the States, Native American history". Sidis.net. Retrieved May 25, 2011. 26. "Notes on the Collection of Transfers". June 20, 1926. Retrieved May 25, 2011 – via Sidis.net. 27. Bates, Stephen (June 20, 1926). "Youthful Prodigies at Genius Meeting" (PDF). The New York Times. p. 8. Retrieved January 16, 2023. 28. U.S. Patent 1784117A, Perpetual Calendar, December 9, 1930 29. "The Animate and the Inanimate". Sidis.net. Retrieved August 23, 2019. 30. "Letter to Huxley". Sidis.net. Retrieved August 23, 2019. 31. "The Animate and the Inanimate : William James Sidis". Archived from the original on December 10, 2000. Retrieved August 29, 2019. 32. "ANIM11". October 24, 2000. Archived from the original on October 24, 2000. Retrieved August 29, 2019. 33. "Bucky Ltr". March 3, 2001. Archived from the original on March 3, 2001. Retrieved August 29, 2019. 34. William James Sidis, 'The Tribes And The States: 100,000-Year History of North America' (via sidis.net) 35. Sperling, Abraham Paul (1947) [July 1946]. Psychology for the Millions. New York: Frederick Fell. pp. 332–339. Retrieved November 26, 2014. 36. Gowdy, Larry Neal (October 20, 2013). "Myths, Facts, Lies, and Humor About William James Sidis – Part One". thelogics.org. Retrieved March 4, 2016. A letter written by William Sidis stated that he had taken a civil service exam, that he passed the state clerical exam, and that he was number 254 on the list; "not so encouraging". It may never be known if Sidis actually did take an IQ test, and it may never be known if the 250–300 number arrived from Sidis's placement in the job pool. 37. "Foundations of Normal and Abnormal psychology". Sidis.net. Retrieved May 25, 2011. 38. Manley, Jared L.; (James Thurber) (August 14, 1937). "Where Are They Now? April Fool!". The New Yorker. pp. 22–26. Retrieved February 13, 2020 – via sidis.net. 39. Pirsig, Robert M. (1991). Lila. p. 55. Retrieved February 13, 2020 – via sidis.net. 40. "Lila: An Inquiry into Morals". barnesandnoble.com. Barnes & Noble. Retrieved April 1, 2019. 41. Ex-Prodigy. 1964. ISBN 978-0262230117. Retrieved April 1, 2019. {{cite book}}: |website= ignored (help) 42. Zehrer, Klaus Cäsar (2017). Das Genie (in German). Zürich: Diogenes Verlag. ISBN 978-3-257-06998-3. 43. Kett, Joseph F. (1978). "Curing the Disease of Precocity". The American Journal of Sociology. 84 (suppl): S183–S211. doi:10.1086/649240. ISSN 0002-9602. JSTOR 3083227. S2CID 144509596. 44. Neihart, Maureen; Reis, Sally M.; Robinson, Nancy M.; Moon, Sidney M., eds. (2002). The Social and Emotional Development of Gifted Children: What Do We Know. National Association for Gifted Children (Prufrock Press, Inc.). pp. 286–287. 45. Considering the Options: A Guidebook for Investigating Early College Entrance (PDF). Print.ditd.org. Retrieved November 26, 2014. Sources • Wallace, Amy (1986). The Prodigy: a Biography of William James Sidis, America's Greatest Child Prodigy. New York: E.P. Dutton & Co. ISBN 0-525-24404-2. External links • Sidis Archives at Sidis.net • Article about William James Sidis at "The Straight Dope" Authority control International • FAST • ISNI • VIAF National • Germany • Israel • United States • Netherlands Other • SNAC
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William S. Burnside William Snow Burnside (20 December 1839 – 11 March 1920) was an Irish mathematician whose entire career was spent at Trinity College Dublin (TCD). He is chiefly remembered for the book The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms (1881)[1] and his long tenure as Erasmus Smith's Professor of Mathematics at TCD . He is sometimes confused with his rough contemporary, the English mathematician William Burnside.[2] William S. Burnside Born(1839-12-20)December 20, 1839 DiedMarch 11, 1920(1920-03-11) (aged 80) NationalityIrish Academic background Alma materTrinity College Dublin Academic work DisciplineMathematics InstitutionsTrinity College Dublin William Snow Burnside was born at Corcreevy House, near Fivemiletown, Tyrone, to William Smyth Burnside (1810–1884, Chancellor of Clogher Cathedral) and Anne Henderson (1808–1881).[3] He studied mathematics under George Salmon at TCD (BA 1861, MA 1866, Fellowship 1871), and taught there until his retirement in 1917. He served as Erasmus Smiths's Professor of Mathematics for many decades (1879–1913), and co-authored the influential 1881 book The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms with his TCD colleague Arthur William Panton (1843–1906). It ran to at least 7 editions, and was reissued by Dover Books in 1960. TCD awarded him DSc in 1891. He lived one and a half miles away from campus, on Raglan Road, and was allegedly "the last man to regularly arrive in College on horseback"[4] References 1. co-authored with Arthur William Panton 2. William Snow Burnside Obituary, Irish Times, 13 March 1920 3. Burnside Family Genealogy Library Ireland 4. The Collected Papers of William Burnside: Commentary on Burnside's life by William Burnside Authority control International • ISNI • VIAF National • Israel • United States • Sweden
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William Spence (mathematician) William Spence (born 31 July 1777 in Greenock, Scotland – died 22 May 1815 in Glasgow, Scotland) was a Scottish mathematician who published works on the fields of logarithmic functions, algebraic equations and their relation to integral and differential calculus respectively. William Spence Born31 July 1777 Greenock,Scotland Died20 May 1815 (aged 37) Glasgow, Scotland Parents • Ninian Spence (father) • Sarah Townsend (mother) Notes A D D Craik, The 'Mathematical Essays' of William Spence (1777–1815), Historia Mathematica 40: 4 (2013), 386–422. Image accredited to the Watt Institute, Inverclyde Council Early life, family, and personal life Spence was the second son to Ninian Spence and his wife Sarah Townsend. Ninian Spence ran a coppersmith business, and the Spence family were a prominent family in Greenock at the time.[1][2] From an early age, Spence was characterised as having a docile and reasonable nature, with him being mature for his age.[3] At school he formed a life-long friendship with John Galt, who documented much of his life and his works posthumously.[3][4] Despite having received a formal education until he was a teenager, Spence never attended university, instead he moved to Glasgow where he lodged with a friend of his fathers, learning the skills of a manufacturer.[1] Two years after his father's death in 1795, Spence returned to Greenock in 1797.[1] With the support of Galt and others, he established a small literary society, wherein once a month they read a range of essays on varying subjects, this society met frequently until 1804.[5] After this, Spence visited many places in England, he lived in London for a few months where, in 1809, he published his first work.[1] In 1814, he published his second work, getting married in the same year – Spence intended to live in London, and began his journey back before becoming ill, having travelled as far as Glasgow, he died in his sleep due to illness.[1][5] Spence held an interest in musical composition, and played the flute.[4] Published works Spence published An Essay on the Theory of the Various Orders of Logarithmic Transcendents: With an Inquiry Into Their Applications to the Integral Calculus and the Summation of Series in 1809.[6] Throughout his work, he displayed a familiarity with the work of Lagrange and Arbogast, which is notable since at the time very few were familiar with their works.[1][7] In his preface he derived the binomial theorem and mainly focused on the properties and analytic applications of the series:[1][6][7] $\pm x/1^{n}-x^{2}/2^{n}\pm x^{3}/3^{n}-...$ which he denoted with $L_{n}(1\pm x)$.[6] He went on further to derive nine general properties of this function in a table.[6] Spence also wrote on presenting analytical mathematics without the need of demonstrating the practical applications of such work.[6] Spence continues to write that the functions $L_{n}(1\pm x)$ can be expressed as iterations of the previous n term: $L_{1}(1\pm x)=\int \pm dx/1\pm x$, $L_{2}(1\pm x)=\int (dx/x)L_{1}(1\pm x)$, . . . , $L_{n}(1\pm x)=\int (dx/x)L_{n-1}(1\pm x)$ for all values of x.[6] Spence goes on to calculate the values of: $L_{2}(x)=-\int _{0}^{x}{\frac {\ln(1-t)}{t}}\operatorname {d} \!t$ to nine decimal places, in a table, for all integer values of $1+x$ from 1 to 100, the first ever of its kind.[1][6] These functions became known as the polylogarithm functions, with this particular case often called Spence's function after Spence. Later on he also created a similar table for $\tan ^{-1}x$.[6][7] Spence published his last work, Outlines of a theory of Algebraical Equations, deduced from the principles of Harriott, and extended to the fluxional or differential calculus was published in 1814.[8] In which he took a systematic approach to solving equations up to the fourth degree using symmetrical functions of the roots.[7][8] After Spence's death, John Herschel edited Mathematical Essays by the late William Spence, which was published in 1819, with John Galt writing a biography on Spence.[3][9] Legacy Spence's work was noted to be remarkable at the time, with John Herschel, his acquaintance and one of Britain's leading mathematicians at the time, had referenced it in one of his later publications Consideration of various points of analysis, which prompted Herschel to edit Spence's manuscripts.[1][10] Spence was held in such high regard by Galt, and later Herschel that they published a collection of his individual essays in 1819.[1][11] Posthumously, his work was met with appreciation from his contemporaries, with a review in the ninety-fourth number of the Quarterly Review (reproduced in Galt's The Literary and Miscellanies of John Galt, Volume 1) that described his first work in 1809 as: " [The] first formal essay in our language on any distinct and considerable branch of the integral calculus, which has appeared since… Hellinsʼs papers on the ‘Rectification of the Conic Sections".[1][12][13] References 1. Craik, Alex D.D. (October 2013). "Polylogarithms, functional equations and more: The elusive essays of William Spence (1777–1815)". Historia Mathematica. 40 (4): 386–422. doi:10.1016/j.hm.2013.06.002. 2. "Greenock - Towns - Scottish Directories - National Library of Scotland". digital.nls.uk. Retrieved 24 June 2022. 3. Galt, J. (May 1819). "THE LATE MR. WILLIAM SPENCE". The Monthly Magazine. 47 (325): 373–375. ProQuest 4520067. 4. Spence, William (1819). Mathematical Essays, by the Late William Spence, Esq. Edited by John F. W. Herschel, Esq. With a Biographical Sketch of the Author. Thomas and George Underwood, 32, Fleet Street. OCLC 1021878949. 5. Galt, John (1833). The autobiography of John Galt. Key & Biddle. 6. Spence, William (1809). An Essay on the Theory of the Various Orders of Logarithmic Transcendents: With an Inquiry Into Their Applications to the Integral Calculus and the Summation of Series. John Murray and Archibald Constable and Company. OCLC 10156665. 7. "William Spence – Biography". Maths History. Retrieved 24 June 2022. 8. Spence, William (1814). Outlines of a theory of Algebraical Equations, deduced from the principles of Harriott, and extended to the fluxional or differential calculus. OCLC 1063204490. 9. Spence, William (1819). Mathematical Essays, by the Late William Spence, Esq. Edited by John F. W. Herschel, Esq. With a Biographical Sketch of the Author. Thomas and George Underwood, 32, Fleet Street. OCLC 1021878949. 10. "XXII. Consideration of various points of analysis". Philosophical Transactions of the Royal Society of London. 104: 440–468. 31 December 1814. doi:10.1098/rstl.1814.0023. S2CID 111328500. 11. "John Herschel Correspondence". historydb.adlerplanetarium.org. Retrieved 28 June 2022. 12. Galt, John (1834). The Literary Life and Miscellanies of John Galt. W. Blackwood. 13. "XVIL. Of the rectification of the conic sections". Philosophical Transactions of the Royal Society of London. 92: 448–476. 31 December 1802. doi:10.1098/rstl.1802.0020. S2CID 110222385.
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William Thomson (mathematician) Sir William Thomson FRSE LLD (1856–1947) was a 19th/20th century Scottish mathematician and physicist primarily working as a university administrator in South Africa. Life He was born on New Year's Eve, 31 December 1856, in the village of Kirkton of Mailler in Perthshire. He was educated at Perth Academy then studied mathematics and physics at the University of Edinburgh. He graduated with a BSc and MA in 1878 and began assisting in lectures at the University.[1] In 1882, aged 26, he was elected a Fellow of the Royal Society of Edinburgh. His proposers were George Chrystal, Peter Guthrie Tait, Alexander Crum Brown and Sir William Turner.[2] In 1883 he succeeded Prof George Gordon as Professor of Mathematics at Stellenbosch University in South Africa.[3] In 1895 he succeeded Rev James Cameron as University Registrar at the University of the Cape of Good Hope. In 1918 he transferred to the same role in the newly created University of South Africa and in 1922 moved to the University of the Witwatersrand. He was knighted by King George V in 1922 for services to university education. He retired in 1928 and died in Simonstown near Cape Town on 6 August 1947. Family In 1884 he married Annie Catherine van der Riet. They had two daughters. Publications • Mensuration in 9th edition of Encyclopædia Britannica (1878) • Introduction to Determinants (1881) • Algebra for the Use of Schools and Colleges (1886) • Textbook of Geometrical Deductions (1891) • Elementary Algebra (1901) References 1. "S2A3 Biographical Database of Southern African Science". s2a3.org.za. Retrieved 14 December 2018. 2. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0 902 198 84 X. 3. "Thomson_William summary". www-history.mcs.st-andrews.ac.uk. Retrieved 14 December 2018. Authority control International • VIAF National • United States Academics • MathSciNet • zbMATH
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William Thurston William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. William Thurston Thurston in 1991 Born William Paul Thurston (1946-10-30)October 30, 1946 Washington, D.C., United States DiedAugust 21, 2012(2012-08-21) (aged 65) Rochester, New York, United States NationalityAmerican Alma materNew College of Florida University of California, Berkeley Known forThurston's geometrization conjecture Thurston's theory of surfaces Milnor–Thurston kneading theory AwardsFields Medal (1982) Oswald Veblen Prize in Geometry (1976) Alan T. Waterman Award (1979) National Academy of Sciences (1983) Doob Prize (2005) Leroy P. Steele Prize (2012). Scientific career FieldsMathematics InstitutionsCornell University University of California, Davis Mathematical Sciences Research Institute University of California, Berkeley Princeton University Massachusetts Institute of Technology Institute for Advanced Study ThesisFoliations of three-manifolds which are circle bundles (1972) Doctoral advisorMorris Hirsch Doctoral studentsRichard Canary Benson Farb David Gabai William Goldman Richard Kenyon Steven Kerckhoff Yair Minsky Igor Rivin Oded Schramm Richard Schwartz Danny Calegari Thurston was a professor of mathematics at Princeton University, University of California, Davis, and Cornell University. He was also a director of the Mathematical Sciences Research Institute. Early life and education William Thurston was born in Washington, D.C., to Margaret Thurston (née Martt), a seamstress, and Paul Thurston, an aeronautical engineer.[1] William Thurston suffered from congenital strabismus as a child, causing issues with depth perception.[1] His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones.[1] He received his bachelor's degree from New College in 1967 as part of its inaugural class.[1][2] For his undergraduate thesis, he developed an intuitionist foundation for topology.[3] Following this, he received a doctorate in mathematics from the University of California, Berkeley under Morris Hirsch, with his thesis Foliations of Three-Manifolds which are Circle Bundles in 1972.[1][4] Career After completing his Ph.D., Thurston spent a year at the Institute for Advanced Study,[1][5] then another year at the Massachusetts Institute of Technology as an assistant professor.[1] In 1974, Thurston was appointed a full professor at Princeton University.[1][6] He returned to Berkeley in 1991 to be a professor (1991-1996) and was also director of the Mathematical Sciences Research Institute (MSRI) from 1992 to 1997.[1][7] He was on the faculty at UC Davis from 1996 until 2003, when he moved to Cornell University.[1] Thurston was an early adopter of computing in pure mathematics research.[1] He inspired Jeffrey Weeks to develop the SnapPea computing program.[1] During Thurston's directorship at MSRI, the institute introduced several innovative educational programs that have since become standard for research institutes.[1] His Ph.D. students include Danny Calegari, Richard Canary, David Gabai, William Goldman, Benson Farb, Richard Kenyon, Steven Kerckhoff, Yair Minsky, Igor Rivin, Oded Schramm, Richard Schwartz, William Floyd, and Jeffrey Weeks.[8] Research Foliations His early work, in the early 1970s, was mainly in foliation theory. His more significant results include: • The proof that every Haefliger structure on a manifold can be integrated to a foliation (this implies, in particular, that every manifold with zero Euler characteristic admits a foliation of codimension one). • The construction of a continuous family of smooth, codimension-one foliations on the three-sphere whose Godbillon–Vey invariant (after Claude Godbillon and Jacques Vey) takes every real value. • With John N. Mather, he gave a proof that the cohomology of the group of homeomorphisms of a manifold is the same whether the group is considered with its discrete topology or its compact-open topology. In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that it led to an exodus from the field, where advisors counselled students against going into foliation theory,[9] because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6[10]). The geometrization conjecture Main article: Geometrization conjecture His later work, starting around the mid-1970s, revealed that hyperbolic geometry played a far more important role in the general theory of 3-manifolds than was previously realised. Prior to Thurston, there were only a handful of known examples of hyperbolic 3-manifolds of finite volume, such as the Seifert–Weber space. The independent and distinct approaches of Robert Riley and Troels Jørgensen in the mid-to-late 1970s showed that such examples were less atypical than previously believed; in particular their work showed that the figure-eight knot complement was hyperbolic. This was the first example of a hyperbolic knot. Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the figure-eight knot complement. He showed that the figure-eight knot complement could be decomposed as the union of two regular ideal hyperbolic tetrahedra whose hyperbolic structures matched up correctly and gave the hyperbolic structure on the figure-eight knot complement. By utilizing Haken's normal surface techniques, he classified the incompressible surfaces in the knot complement. Together with his analysis of deformations of hyperbolic structures, he concluded that all but 10 Dehn surgeries on the figure-eight knot resulted in irreducible, non-Haken non-Seifert-fibered 3-manifolds. These were the first such examples; previously it had been believed that except for certain Seifert fiber spaces, all irreducible 3-manifolds were Haken. These examples were actually hyperbolic and motivated his next theorem. Thurston proved that in fact most Dehn fillings on a cusped hyperbolic 3-manifold resulted in hyperbolic 3-manifolds. This is his celebrated hyperbolic Dehn surgery theorem. To complete the picture, Thurston proved a hyperbolization theorem for Haken manifolds. A particularly important corollary is that many knots and links are in fact hyperbolic. Together with his hyperbolic Dehn surgery theorem, this showed that closed hyperbolic 3-manifolds existed in great abundance. The hyperbolization theorem for Haken manifolds has been called Thurston's Monster Theorem, due to the length and difficulty of the proof. Complete proofs were not written up until almost 20 years later. The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds. Thurston was next led to formulate his geometrization conjecture. This gave a conjectural picture of 3-manifolds which indicated that all 3-manifolds admitted a certain kind of geometric decomposition involving eight geometries, now called Thurston model geometries. Hyperbolic geometry is the most prevalent geometry in this picture and also the most complicated. The conjecture was proved by Grigori Perelman in 2002–2003.[11][12] Density conjecture Thurston and Dennis Sullivan generalized Lipman Bers' density conjecture from singly degenerate Kleinian surface groups to all finitely generated Kleinian groups in the late 1970s and early 1980s.[13][14] The conjecture states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups, and was independently proven by Ohshika and Namazi–Souto in 2011 and 2012 respectively.[13][14] Orbifold theorem In his work on hyperbolic Dehn surgery, Thurston realized that orbifold structures naturally arose. Such structures had been studied prior to Thurston, but his work, particularly the next theorem, would bring them to prominence. In 1981, he announced the orbifold theorem, an extension of his geometrization theorem to the setting of 3-orbifolds. Two teams of mathematicians around 2000 finally finished their efforts to write down a complete proof, based mostly on Thurston's lectures given in the early 1980s in Princeton. His original proof relied partly on Richard S. Hamilton's work on the Ricci flow. Awards and honors In 1976, Thurston and James Harris Simons shared the Oswald Veblen Prize in Geometry.[1] Thurston received the Fields Medal in 1982 for "revolutioniz[ing] [the] study of topology in 2 and 3 dimensions, showing interplay between analysis, topology, and geometry" and "contribut[ing] [the] idea that a very large class of closed 3-manifolds carry a hyperbolic structure."[15][16] In 2005, Thurston won the first American Mathematical Society Book Prize, for Three-dimensional Geometry and Topology. The prize "recognizes an outstanding research book that makes a seminal contribution to the research literature".[17] He was awarded the 2012 Leroy P. Steele Prize by the American Mathematical Society for seminal contribution to research. The citation described his work as having "revolutionized 3-manifold theory".[18] Personal life Thurston and his first wife, Rachel Findley, had three children: Dylan, Nathaniel, and Emily.[6] Dylan was a MOSP participant (1988–90)[19] and is a mathematician at Indiana University Bloomington.[20] Thurston had two children with his second wife, Julian Muriel Thurston: Hannah Jade and Liam.[6] Thurston died on August 21, 2012, in Rochester, New York, of a sinus mucosal melanoma that was diagnosed in 2011.[6][21][7] Selected publications • William Thurston, The geometry and topology of three-manifolds, Princeton lecture notes (1978–1981). • William Thurston, Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, New Jersey, 1997. x+311 pp. ISBN 0-691-08304-5 • William Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds. Ann. of Math. (2) 124 (1986), no. 2, 203–246. • William Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357–381. • William Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431 • Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio V. F.; Paterson, Michael S.; Thurston, William P. Word Processing in Groups. Jones and Bartlett Publishers, Boston, Massachusetts, 1992. xii+330 pp. ISBN 0-86720-244-0[22] • Eliashberg, Yakov M.; Thurston, William P. Confoliations. University Lecture Series, 13. American Mathematical Society, Providence, Rhode Island and Providence Plantations, 1998. x+66 pp. ISBN 0-8218-0776-5 • William Thurston, On proof and progress in mathematics. Bull. Amer. Math. Soc. (N.S.) 30 (1994) 161–177 • William P. Thurston, "Mathematical education". Notices of the AMS 37:7 (September 1990) pp 844–850 See also • Automatic group • Cannon–Thurston map • Circle packing theorem • Hyperbolic volume • Hyperbolic Dehn surgery • Thurston boundary • Milnor–Thurston kneading theory • Misiurewicz–Thurston points • Nielsen–Thurston classification • Normal surface • Orbifold notation • Thurston norm • Thurston's double limit theorem • Thurston elliptization conjecture • Thurston's geometrization conjecture • Thurston's height condition • Thurston's orbifold theorem • Earthquake theorem References 1. Gabai, David; Kerckhoff, Steven (2015). "William P. Thurston, 1946–2012" (PDF). Notices of the American Mathematical Society. 62 (11): 1318–1332. doi:10.1090/noti1300. Archived (PDF) from the original on 2022-10-09. 2. Kelley, Susan (Aug 24, 2012). "World-renowned mathematician William Thurston dies at 65". Retrieved 2023-01-11. 3. See p. 3 in Laudenbach, François; Papadopoulos, Athanase (2019). "W. P. Thurston and French mathematics". arXiv:1912.03115 [math.GT]. 4. "William Thurston – the Mathematics Genealogy Project". 5. "Institute for Advanced Study: A Community of Scholars". Ias.edu. Retrieved 2013-09-06. 6. Leslie Kaufman (August 23, 2012). "William P. Thurston, Theoretical Mathematician, Dies at 65". New York Times. p. B15. 7. "William P. Thurston, 1946-2012". American Mathematical Society. August 22, 2012. Retrieved March 25, 2022. 8. "William Thurston – the Mathematics Genealogy Project". 9. "The Mathematical Legacy of William Thurston (1946–2012)". 10. Thurston, William P. (April 1994). "On Proof and Progress in Mathematics". Bulletin of the American Mathematical Society. 30 (2): 161–177. arXiv:math/9404236. Bibcode:1994math......4236T. doi:10.1090/S0273-0979-1994-00502-6. 11. Perelman, Grisha (2003-03-10). "Ricci flow with surgery on three-manifolds". arXiv:math/0303109. 12. Kleiner, Bruce; Lott, John (2008-11-06). "Notes on Perelman's papers". Geometry & Topology. 12 (5): 2587–2855. arXiv:math/0605667. doi:10.2140/gt.2008.12.2587. ISSN 1364-0380. 13. Namazi, Hossein; Souto, Juan (2012). "Non-realizability and ending laminations: Proof of the density conjecture". Acta Mathematica. 209 (2): 323–395. doi:10.1007/s11511-012-0088-0. ISSN 0001-5962. S2CID 10138438. 14. Ohshika, Ken'ichi (2011). "Realising end invariants by limits of minimally parabolic, geometrically finite groups". Geometry and Topology. 15 (2): 827–890. arXiv:math/0504546. doi:10.2140/gt.2011.15.827. ISSN 1364-0380. S2CID 14463721. Archived from the original on May 25, 2014. Retrieved March 24, 2022. 15. "William P. Thurston, 1946–2012". 30 August 2012. Retrieved 18 August 2014. 16. "Fields Medals and Nevanlinna Prize 1982". mathunion.org. International Mathematical Union. 17. "William P. Thurston Receives 2005 AMS Book Prize". Retrieved 2008-06-26. 18. "AMS prize booklet 2012" (PDF). Archived (PDF) from the original on 2022-10-09. 19. "YEAR 1990" (PDF). USAMO Archive. Retrieved 30 January 2023. 20. Thurston, Dylan P., ed. (2020). What's Next? The Mathematical Legacy of William P. Thurston. Princeton University Press. ISBN 978-0-691-16776-3. 21. "Department mourns loss of friend and colleague, Bill Thurston", Cornell University 22. Reviews of Word Processing in Groups: B. N. Apanasov, Zbl 0764.20017; Gilbert Baumslag, Bull. AMS, doi:10.1090/S0273-0979-1994-00481-1; D. E. Cohen, Bull LMS, doi:10.1112/blms/25.6.614; Richard M. Thomas, MR1161694 Further reading • Gabai, David; Kerckhoff, Steve (Coordinating Editors). "William P. Thurston, 1946–2012" (part 2), Notices of the American Mathematical Society, January 2015, Volume 63, Number 1, pp. 31–41. External links • Media related to William Thurston at Wikimedia Commons Wikiquote has quotations related to William Thurston. • William Thurston at the Mathematics Genealogy Project • O'Connor, John J.; Robertson, Edmund F., "William Thurston", MacTutor History of Mathematics Archive, University of St Andrews • Thurston's page at Cornell • Tribute and remembrance page at Cornell • Etienne Ghys : La géométrie et la mode • "Landau Lectures | Prof. Thurston | Part 1 | 1995/6". YouTube. Hebrew University of Jerusalem. April 8, 2014. • "Landau Lectures | Prof. Thurston | Part 2 | 1995/6". YouTube. Hebrew University of Jerusalem. April 8, 2014. • "Landau Lectures | Prof. Thurston | Part 3 | 1995/6". YouTube. Hebrew University of Jerusalem. April 8, 2014. • "The Mystery of 3-Manifolds - William Thurston". YouTube. PoincareDuality. November 27, 2011. 2010 Clay Research Conference • Goldman, William (May 9, 2013). "William Thurston: A Mathematical Perspective". YouTube. UMD Mathematics. William Goldman (U. of Maryland), Collloquium, Department of Mathematics, Howard University, 25 January 2013 Fields Medalists • 1936  Ahlfors • Douglas • 1950  Schwartz • Selberg • 1954  Kodaira • Serre • 1958  Roth • Thom • 1962  Hörmander • Milnor • 1966  Atiyah • Cohen • Grothendieck • Smale • 1970  Baker • Hironaka • Novikov • Thompson • 1974  Bombieri • Mumford • 1978  Deligne • Fefferman • Margulis • Quillen • 1982  Connes • Thurston • Yau • 1986  Donaldson • Faltings • Freedman • 1990  Drinfeld • Jones • Mori • Witten • 1994  Bourgain • Lions • Yoccoz • Zelmanov • 1998  Borcherds • Gowers • Kontsevich • McMullen • 2002  Lafforgue • Voevodsky • 2006  Okounkov • Perelman • Tao • Werner • 2010  Lindenstrauss • Ngô • Smirnov • Villani • 2014  Avila • Bhargava • Hairer • Mirzakhani • 2018  Birkar • Figalli • Scholze • Venkatesh • 2022  Duminil-Copin • Huh • Maynard • Viazovska • Category • Mathematics portal Recipients of the Oswald Veblen Prize in Geometry • 1964 Christos Papakyriakopoulos • 1964 Raoul Bott • 1966 Stephen Smale • 1966 Morton Brown and Barry Mazur • 1971 Robion Kirby • 1971 Dennis Sullivan • 1976 William Thurston • 1976 James Harris Simons • 1981 Mikhail Gromov • 1981 Shing-Tung Yau • 1986 Michael Freedman • 1991 Andrew Casson and Clifford Taubes • 1996 Richard S. Hamilton and Gang Tian • 2001 Jeff Cheeger, Yakov Eliashberg and Michael J. Hopkins • 2004 David Gabai • 2007 Peter Kronheimer and Tomasz Mrowka; Peter Ozsváth and Zoltán Szabó • 2010 Tobias Colding and William Minicozzi; Paul Seidel • 2013 Ian Agol and Daniel Wise • 2016 Fernando Codá Marques and André Neves • 2019 Xiuxiong Chen, Simon Donaldson and Song Sun Authority control International • FAST • ISNI • VIAF National • Norway • France • BnF data • Catalonia • Germany • Israel • United States • Japan • Czech Republic • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • SNAC • IdRef
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William A. Veech William A. Veech was the Edgar O. Lovett Professor of Mathematics at Rice University[1] until his death. His research concerned dynamical systems; he is particularly known for his work on interval exchange transformations, and is the namesake of the Veech surface. He died unexpectedly on August 30, 2016 in Houston, Texas.[2] Education Veech graduated from Dartmouth College in 1960,[1] and earned his Ph.D. in 1963 from Princeton University under the supervision of Salomon Bochner.[1][3] Contributions An interval exchange transformation is a dynamical system defined from a partition of the unit interval into finitely many smaller intervals, and a permutation on those intervals. Veech and Howard Masur independently discovered that, for almost every partition and every irreducible permutation, these systems are uniquely ergodic, and also made contributions to the theory of weak mixing for these systems.[4] The Rauzy–Veech–Zorich induction map, a function from and to the space of interval exchange transformations is named in part after Veech: Rauzy defined the map, Veech constructed an infinite invariant measure for it, and Zorich strengthened Veech's result by making the measure finite.[5] The Veech surface and the related Veech group are named after Veech, as is the Veech dichotomy according to which geodesic flow on the Veech surface is either periodic or ergodic.[6] Veech played a role in the Nobel-prize-winning discovery of buckminsterfullerene in 1985 by a team of Rice University chemists including Richard Smalley. At that time, Veech was chair of the Rice mathematics department, and was asked by Smalley to identify the shape that the chemists had determined for this molecule. Veech answered, "I could explain this to you in a number of ways, but what you've got there, boys, is a soccer ball."[7][8] Veech is the author of A Second Course in Complex Analysis (W. A. Benjamin, 1967; Dover, 2008, ISBN 9780486462943).[9][10][11] Awards and honors In 2012, Veech became one of the inaugural fellows of the American Mathematical Society.[12] References 1. Faculty profile, Rice University, retrieved 2015-03-01. 2. Todd, Hannah. "Former math department chair passes away". Rice Thresher. Retrieved 29 September 2016. 3. William A. Veech at the Mathematics Genealogy Project 4. Hunt, B. R.; Kaloshin, V. Yu. (2010), "Prevalence", in Broer, H.; Takens, F.; Hasselblatt, B. (eds.), Handbook of Dynamical Systems, Volume 3, Elsevier, pp. 43–88, ISBN 9780080932262. See in particular p. 51. 5. Bufetov, Alexander I. (2006), "Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of abelian differentials", Journal of the American Mathematical Society, 19 (3): 579–623, arXiv:math/0506222, doi:10.1090/S0894-0347-06-00528-5, MR 2220100, S2CID 15755696. 6. Smillie, John; Weiss, Barak (2008), "Veech's dichotomy and the lattice property", Ergodic Theory and Dynamical Systems, 28 (6): 1959–1972, doi:10.1017/S0143385708000114, MR 2465608, S2CID 42112090. 7. Edelson, Edward (August 1991), "Buckyball: the magic molecule", Popular Science: 52–57, 87. The quote is on p. 55. 8. Ball, Philip (1996), Designing the Molecular World: Chemistry at the Frontier, Princeton Science Library, Princeton University Press, p. 46, ISBN 9780691029009. 9. Review of A Second Course in Complex Analysis by E. Hille, MR0220903. 10. Wenzel, H., "W. A. Veech, A Second Course in Complex Analysis", Book Reviews, Journal of Applied Mathematics and Mechanics, 48 (7): 502–503, Bibcode:1968ZaMM...48..502W, doi:10.1002/zamm.19680480725. 11. Stenger, Allen (April 24, 2008), "A Second Course in Complex Analysis, William A. Veech", MAA Reviews, Mathematical Association of America. 12. List of Fellows of the American Mathematical Society, retrieved 2015-03-01. Authority control International • ISNI • VIAF National • Israel • United States • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
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William Wallace (mathematician) William Wallace FRSE MInstCE FRAS LLD (23 September 1768 – 28 April 1843) was a Scottish mathematician and astronomer who invented the eidograph (an improved pantograph). Life Wallace was born at Dysart in Fife, the son of Alexander Wallace, a leather manufacturer, and his wife, Janet Simson.[1] He received his school education in Dysart and Kirkcaldy. In 1784 his family moved to Edinburgh, where he himself was set to learn the trade of a bookbinder.[2] In 1790 he appears as "William Wallace, bookbinder" living and trading at Cowgatehead, at the east end of the Grassmarket.[3] His taste for mathematics had already developed itself, and he made such use of his leisure hours that before the completion of his apprenticeship he had made considerable acquirements in geometry, algebra and astronomy. He was further assisted in his studies by John Robison (1739–1805) and John Playfair, to whom his abilities had become known.[2] After various changes of situation, dictated mainly by a desire to gain time for study, he became assistant teacher of mathematics in the Perth Academy in 1794. This post he exchanged in 1803 for a mathematical mastership in the Royal Military College at Great Marlow, in which post he continued after it moved to Sandhurst, with a recommendation by Playfair.[2] In 1804 he was elected a Fellow of the Royal Society of Edinburgh.[2] His proposers were John Playfair, Thomas Charles Hope and William Wright.[4] In 1819 he was chosen to succeed John Playfair in the chair of mathematics at Edinburgh. Playfair's second chair (in Natural Philosophy) was taken by John Leslie. Wallace developed a reputation for being an excellent teacher. Among his students was Mary Somerville. In 1838 he retired from the university due to ill health.[2] In his final years he lived at 6 Lauriston Lane on the south side of Edinburgh.[5] He died in Edinburgh aged 74 and was buried in Greyfriars Kirkyard. The grave lies on the north-facing retaining wall in the centre of the northern section. Mathematical contributions In his earlier years Wallace was an occasional contributor to Leybourne's Mathematical Repository and the Gentleman's Mathematical Companion. Between 1801 and 1810 he contributed articles on "Algebra", "Conic Sections", "Trigonometry", and several others in mathematical and physical science to the fourth edition of the Encyclopædia Britannica, and some of these were retained in subsequent editions from the fifth to the eighth inclusive. He was also the author of the principal mathematical articles in the Edinburgh Encyclopædia, edited by David Brewster. He also contributed many important papers to the Transactions of the Royal Society of Edinburgh.[2] He mainly worked in the field of geometry and in 1799 became the first to publish the concept of the Simson line, which erroneously was attributed to Robert Simson.[6] In 1807 he proved a result about polygons with an equal area, that later became known as the Bolyai–Gerwien theorem.[7] His most important contribution to British mathematics however was, that he was one of the first mathematicians introducing and promoting the advancement of the continental European version of calculus in Britain.[6] Other works Wallace also worked in astronomy and invented the eidograph, a mechanical device for scaling drawings.[6][8] Books • A Geometrical Treatise on the Conic Sections with an Appendix Containing Formulae for their Quadrature. (1838) • Geometrical Theorems and Analytical Formulae with their application to the Solution of Certain Geodetical Problems and an Appendix. (1839) Family Wallace was married to Janet Kerr (1775–1824). His daughter, Margaret Wallace, married the mathematician Thomas Galloway. His sons included the Rev Alexander Wallace (1803–1842) and Archibald C. Wallace (1806–1830). He also appears to have had a son William Wallace (1784–1864) when William was aged only 16. The mother is not clear. References 1. "William Wallace - Biography". Maths History. 2. Chisholm 1911. 3. Williamson's Edinburgh Directory 1790 4. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 978-0-902198-84-5. 5. Edinburgh and Leith Post Office directory 1835-36 6. O'Connor, John J.; Robertson, Edmund F., "William Wallace (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews 7. Ian Stewart: From Here to Infinity. Oxford University Press 1996 (3. edition), ISBN 978-0-19-283202-3, p. 169 (restricted online copy, p. 169, at Google Books) 8. Gerard L'Estrange Turner: Nineteenth-Century Scientific Instruments. University of California Press 1983, ISBN 0-520-05160-2, p. 280 (online copy, p. 280, at Google Books) Sources • This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Wallace, William". Encyclopædia Britannica. Vol. 28 (11th ed.). Cambridge University Press. p. 278. External links • O'Connor, John J.; Robertson, Edmund F., "William Wallace (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews • Short biographical note on William Wallace in the Gazetteer for Scotland Authority control International • FAST • ISNI • VIAF National • Germany • Israel • United States • Netherlands • Poland Academics • zbMATH People • Deutsche Biographie Other • SNAC • IdRef
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William Whiston William Whiston (9 December 1667 – 22 August 1752) was an English theologian, historian, natural philosopher, and mathematician, a leading figure in the popularisation of the ideas of Isaac Newton. He is now probably best known for helping to instigate the Longitude Act in 1714 (and his attempts to win the rewards that it promised) and his important translations of the Antiquities of the Jews and other works by Josephus (which are still in print). He was a prominent exponent of Arianism and wrote A New Theory of the Earth. William Whiston Born(1667-12-09)9 December 1667 Norton-juxta-Twycross, Leicestershire, England Died22 August 1752(1752-08-22) (aged 84) Lyndon, Rutland, England NationalityEnglish Alma materClare College, Cambridge Known forTranslating the works of Josephus, catastrophism, isoclinic maps, work on longitude Scientific career FieldsMathematics, theology InstitutionsClare College, Cambridge Academic advisorsIsaac Newton Robert Herne Notable studentsJames Jurin InfluencesDavid Gregory Isaac Newton Signature Whiston succeeded his mentor Newton as Lucasian Professor of Mathematics at the University of Cambridge. In 1710 he lost the professorship and was expelled from the university as a result of his unorthodox religious views. Whiston rejected the notion of eternal torment in hellfire, which he viewed as absurd, cruel, and an insult to God. What especially pitted him against church authorities was his denial of the doctrine of the Trinity, which he believed had pagan origins. Early life and career Whiston was born to Josiah Whiston (1622–1685) and Katherine Rosse (1639–1701) at Norton-juxta-Twycross, in Leicestershire, where his father was rector. His mother was daughter of the previous rector at Norton-juxta-Twycross, Gabriel Rosse. Josiah Whiston was a presbyterian, but retained his rectorship after the Stuart Restoration in 1660. William Whiston was educated privately, for his health, and so that he could act as amanuensis to his blind father.[1][2] He studied at Queen Elizabeth Grammar School at Tamworth, Staffordshire. After his father's death, he entered Clare College, Cambridge as a sizar in 1686. He applied himself to mathematical study, was awarded the degree of Bachelor of Arts (BA) (1690), and AM (1693), and was elected Fellow in 1691 and probationary senior Fellow in 1693.[1][3] William Lloyd ordained Whiston at Lichfield in 1693. In 1694, claiming ill health, he resigned his tutorship at Clare to Richard Laughton, chaplain to John Moore, the bishop of Norwich, and swapped positions with him. He now divided his time between Norwich, Cambridge and London. In 1698 Moore gave him the living of Lowestoft where he became rector. In 1699 he resigned his Fellowship of Clare College and left to marry.[1] Whiston first met Isaac Newton in 1694 and attended some of his lectures, though he first found them, by his own admission, incomprehensible. Encouraged after reading a paper by David Gregory on Newtonian philosophy, he set out to master Newton's Principia mathematica thereafter. He and Newton became friends.[1] In 1701 Whiston resigned his living to become Isaac Newton's substitute, giving the Lucasian lectures at Cambridge.[2] He succeeded Newton as Lucasian professor in 1702. There followed a period of joint research with Roger Cotes, appointed with Whiston's patronage to the Plumian professorship in 1706. Students at the Cotes–Whiston experimental philosophy course included Stephen Hales, William Stukeley, and Joseph Wasse.[4] Newtonian theologian In 1707 Whiston was Boyle lecturer; this lecture series was at the period a significant opportunity for Newton's followers, including Richard Bentley and Samuel Clarke, to express their views, especially in opposition to the rise of deism.[5] The "Newtonian" line came to include, with Bentley, Clarke and Whiston in particular, a defence of natural law by returning to the definition of Augustine of Hippo of a miracle (a cause of human wonderment), rather than the prevailing concept of a divine intervention against nature, which went back to Anselm. This move was intended to undermine arguments of deists and sceptics.[6] The Boyle lectures dwelt on the connections between biblical prophecies, dramatic physical events such as floods and eclipses, and their explanations in terms of science.[7] On the other hand, Whiston was alive to possible connections of prophecy with current affairs: the War of the Spanish Succession, and later the Jacobite rebellions.[8] Whiston supported a qualified biblical literalism: the literal meaning should be the default, unless there was a good reason to think otherwise.[9] This view again went back to Augustine. Newton's attitude to the cosmogony of Thomas Burnet reflected on the language of the Genesis creation narrative; as did Whiston's alternative cosmogony. Moses as author of Genesis was not necessarily writing as a natural philosopher, nor as a law-giver, but for a particular audience.[10] The new cosmogonies of Burnet, Whiston and John Woodward were all criticised for their disregard of the biblical account, by John Arbuthnot, John Edwards and William Nicolson in particular.[11] The title for Whiston's Boyle lectures was The Accomplishment of Scripture Prophecies. Rejecting typological interpretation of biblical prophecy, he argued that the meaning of a prophecy must be unique. His views were later challenged by Anthony Collins.[12] There was a more immediate attack by Nicholas Clagett in 1710.[13] One reason prophecy was topical was the Camisard movement that saw French exiles ("French prophets") in England. Whiston had started writing on the millenarianism that was integral to the Newtonian theology, and wanted to distance his views from theirs, and in particular from those of John Lacy.[14] Meeting the French prophets in 1713, Whiston developed the view that the charismatic gift of revelation could be demonic possession.[15] Tensions with Newton It is no longer assumed that Whiston's Memoirs are completely trustworthy on the matter of his personal relations with Newton. One view is that the relationship was never very close, Bentley being more involved in Whiston's appointment to the Lucasian chair; and that it deteriorated as soon as Whiston began to write on prophecy, publishing Essay on the Revelation of St John (1706).[14] This work proclaimed the millennium for the year 1716.[16] Whiston's 1707 edition of Newton's Arithmetica Universalis did nothing to improve matters. Newton himself was heavily if covertly involved in the 1722 edition, nominally due to John Machin, making many changes.[17] In 1708–9 Whiston was engaging Thomas Tenison and John Sharp as archbishops in debates on the Trinity. There is evidence from Hopton Haynes that Newton reacted by pulling back from publication on the issue;[18] his antitrinitarian views, from the 1690s, were finally published in 1754 as An Historical Account of Two Notable Corruptions of Scripture. Whiston was never a Fellow of the Royal Society. In conversation with Edmond Halley he blamed his reputation as a "heretick". Also, though, he claimed Newton had disliked having an independent-minded disciple; and was unnaturally cautious and suspicious by nature.[19] Expelled Arian Whiston's route to rejection of the Nicene Creed, the historical orthodox position against Arianism, began early in his tenure of the Lucasian chair as he followed hints from Samuel Clarke. He read also in Louis Ellies Dupin, and the Explication of Gospel Theism (1706) of Richard Brocklesby.[20] His study of the Apostolic Constitutions then convinced him that Arianism was the creed of the early church.[2] The general election of 1710 brought the Tories solid political power for a number of years, up to the Hanoverian succession of 1714. Their distrust of theological innovation had a direct impact on Whiston, as well as others of similar views. His heterodoxy was notorious.[21] In 1710 he was deprived of his professorship and expelled from the university.[2] The matter was not allowed to rest there: Whiston tried to get a hearing before Convocation. He did have defenders even in the high church ranks, such as George Smalridge.[22] For political reasons, this development would have been divisive at the time. Queen Anne made a point of twice "losing" the papers in the case.[23] After her death in 1714 the intended hearing was allowed to drop.[24] The party passions of these years found an echo in Henry Sacheverell's attempt to exclude Whiston from his church of St Andrew's, Holborn, taking place in 1719.[24][25] "Primitive Christianity" Whiston founded a society for promoting primitive Christianity, lecturing in support of his theories in halls and coffee-houses at London, Bath, and Tunbridge Wells.[2] Those he involved included Thomas Chubb,[26] Thomas Emlyn,[27] John Gale,[28] Benjamin Hoadley,[29] Arthur Onslow,[29] and Thomas Rundle.[30] There were meetings at Whiston's house from 1715 to 1717; Hoadley avoided coming, as did Samuel Clarke, though invited.[31] A meeting with Clarke, Hoadley, John Craig and Gilbert Burnet the younger had left these leading latitudinarians unconvinced about Whiston's reliance on the Apostolical Constitutions.[32] Franz Wokenius wrote a 1728 Latin work on Whiston's view of primitive Christianity.[33] His challenge to the teachings of Athanasius meant that Whiston was commonly considered heretical on many points. On the other hand, he was a firm believer in supernatural aspects of Christianity. He defended prophecy and miracle. He supported anointing the sick and touching for the king's evil. His dislike of rationalism in religion also made him one of the numerous opponents of Hoadley's Plain Account of the Nature and End of the Sacrament. He was fervent in his views of ecclesiastical government and discipline, derived from the Apostolical Constitutions.[2] Around 1747, when his clergyman began to read the Athanasian Creed, which Whiston did not believe in, he physically left the church and the Anglican communion, becoming a Baptist.[2] By the 1720s, some dissenters and early Unitarians viewed Whiston as a role model.[1] Lecturer and popular author Whiston began lecturing on natural philosophy in London. He gave regular courses at coffee houses, particularly Button's, and also at the Censorium, a set of riverside meeting rooms in London run by Richard Steele.[34] At Button's, he gave courses of demonstration lectures on astronomical and physical phenomena, and Francis Hauksbee the younger worked with him on experimental demonstrations. His passing remarks on religious topics were sometimes objected to, for example by Henry Newman writing to Steele.[35][36] His lectures were often accompanied by publications. In 1712, he published, with John Senex, a chart of the Solar System showing numerous paths of comets.[37] In 1715, he lectured on the total solar eclipse of 3 May 1715 (which fell in April Old Style in England); Whiston lectured on it at the time, in Covent Garden, and later, as a natural event and as a portent.[38] By 1715 Whiston had also become adept at newspaper advertising.[39] He frequently lectured to the Royal Society. Longitude In 1714, he was instrumental in the passing of the Longitude Act, which established the Board of Longitude. In collaboration with Humphrey Ditton he published A New Method for Discovering the Longitude, both at Sea and Land,[40] which was widely referenced and discussed. For the next forty years he continued to propose a range of methods to solve the longitude reward, which earned him widespread ridicule, particularly from the group of writers known as the Scriblerians.[41][42] In one proposal for using magnetic dip to find longitude he produced one of the first isoclinic maps of southern England in 1719 and 1721. In 1734, he proposed using the eclipses of Jupiter's satellites.[43] Broader natural philosophy Whiston's A New Theory of the Earth from its Original to the Consummation of All Things (1696) was an articulation of creationism and flood geology. It held that the global flood of Noah had been caused by a comet. The work obtained the praise of John Locke, who classed the author among those who, if not adding much to our knowledge, "At least bring some new things to our thoughts."[2] He was an early advocate, along with Edmond Halley, of the periodicity of comets; he also held that comets were responsible for past catastrophes in Earth's history. In 1736, he caused widespread anxiety among London's citizens when he predicted the world would end on 16 October that year because a comet would hit the earth.[44] William Wake as Archbishop of Canterbury officially denied this prediction to calm the public. There was no consensus within the Newtonians as to how far mechanical causes could be held responsible for key events of sacred history: John Keill was at the opposite extreme to Whiston in minimising such causes.[45] As a natural philosopher, Whiston's speculations respected no boundary with his theological views. He saw the creation of man as an intervention in the natural order. He picked up on Arthur Ashley Sykes's advice to Samuel Clarke to omit an eclipse and earthquake mentioned by Phlegon of Tralles from future editions of Clarke's Boyle lectures, these events being possibly synchronous with Christ's crucifixion. Whiston published The Testimony of Phlegon Vindicated in 1732.[46] Views The series of Moyer Lectures often made Whiston's unorthodox views a particular target.[47] Whiston held that Song of Solomon was apocryphal and that the Book of Baruch was not.[2] He modified the biblical Ussher chronology, setting the Creation at 4010 BCE.[48] He challenged Newton's system of The Chronology of Ancient Kingdoms Amended (1728). Westfall absolves Whiston of the charge that he pushed for the posthumous publication of the Chronology just to attack it, commenting that the heirs were in any case looking to publish manuscripts of Newton, who died in 1727.[49] Whiston's advocacy of clerical monogamy is referenced in Oliver Goldsmith's novel The Vicar of Wakefield. His last "famous discovery, or rather revival of Dr Giles Fletcher, the Elder's," which he mentions in his autobiography, was the identification of the Tatars with the lost tribes of Israel.[2] Personal life Whiston married Ruth, daughter of George Antrobus, his headmaster at Tamworth school. He had a happy family life and died in Lyndon Hall, Rutland, at the home of his son-in-law, Samuel Barker, on 22 August 1752.[1] He was survived by his children Sarah, William, George, and John.[50] Works Whiston's later life was spent in continual controversy: theological, mathematical, chronological, and miscellaneous. He vindicated his estimate of the Apostolical Constitutions and the Arian views he had derived from them in his Primitive Christianity Revived (5 vols., 1711–1712). In 1713 he produced a reformed liturgy. His Life of Samuel Clarke appeared in 1730.[2] In 1727 he published a two volume work called Authentik Record belonging to the Old and New Testament. This was a collection of translations and essays on various deuterocanonical books, pseudepigrapha and other essays with a translation if relevant.[2] Whiston translated the complete works of Josephus into English, and published them along with his own notes and dissertations under the title The Genuine Works of Flavius Josephus the Jewish Historian in 1737. This translation was based on the same Greek edition of Josephus' works used by Siwart Haverkamp in his prior translation.[51] The text on which Whiston's translation of Josephus is based is, reputedly, one which had many errors in transcription.[52] In 1745 he published his Primitive New Testament (on the basis of Codex Bezae and Codex Claromontanus). Whiston left memoirs (3 vols., 1749–1750). These do not contain the account of the proceedings taken against him at Cambridge for his antitrinitarianism, which was published separately at the time.[2] Editions • New theory of the Earth. London: Robert Roberts. 1696. • New theory of the Earth (in German). Frankfurt am Main: Christian Gottlieb Ludwig. 1713. See also • Noah's Flood • Catastrophism • Biblical prophecy • Dorsa Whiston, named after him References 1. "Whiston, William". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/29217. (Subscription or UK public library membership required.) 2. One or more of the preceding sentences incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Whiston, William". Encyclopædia Britannica. Vol. 28 (11th ed.). Cambridge University Press. p. 597. 3. "Whiston, William (WHSN686W)". A Cambridge Alumni Database. University of Cambridge. 4. Knox, Kevin C. (6 November 2003). From Newton to Hawking: A History of Cambridge University's Lucasian Professors of Mathematics. Cambridge University Press. p. 145. ISBN 978-0-521-66310-6. Retrieved 28 May 2013. 5. Shank, J. B. (2008). The Newton Wars and the Beginning of the French Enlightenment. University of Chicago Press. pp. 128–29. ISBN 978-0-226-74947-1. Retrieved 22 May 2013. 6. Shaw, Jane (2006). Miracles in Enlightenment England. Yale University Press. pp. 31, 171. ISBN 978-0-300-11272-6. Retrieved 22 May 2013. 7. Andrew Pyle (editor), The Dictionary of Seventeenth Century British Philosophers (2000), Thoemmes Press (two volumes), article Whiston, William, p. 875. 8. Sara Schechner; Sara Schechner Genuth (1999). Comets, popular culture, and the birth of modern cosmology. Princeton University Press. p. 292. ISBN 978-0-691-00925-4. Retrieved 23 May 2013. 9. Kidd, Colin (1999). British Identities before Nationalism. Cambridge University Press. p. 45. ISBN 978-1-139-42572-8. Retrieved 22 May 2013. 10. Poole, William (2010). The World Makers: Scientists of the Restoration and the Search for the Origins of the Earth. Peter Lang. p. 68. ISBN 978-1-906165-08-6. Retrieved 22 May 2013. 11. Stephen Gaukroger; John Schuster; John Sutton (2002). Descartes' Natural Philosophy. Taylor & Francis. pp. 176–77. ISBN 978-0-203-46301-7. Retrieved 23 May 2013. 12. Henk J. M. Nellen, ed. (1994). Hugo Grotius, Theologian: Essays in Honour of G. H. M. Posthumus Meyjes. Brill. p. 195. ISBN 978-90-04-10000-8. Retrieved 22 May 2013. 13. Stephen, Leslie, ed. (1887). "Clagett, Nicholas (1654–1727)" . Dictionary of National Biography. Vol. 10. London: Smith, Elder & Co. 14. Jed Zachary Buchwald; Mordechai Feingold (2012). Newton and the origin of civilization. Princeton University Press. p. 336. ISBN 978-0-691-15478-7. Retrieved 22 May 2013. 15. J.E. Force; S. Hutton (2004). Newton and Newtonianism: New Studies. Springer. p. 179 note 102. ISBN 978-1-4020-1969-2. Retrieved 25 May 2013. 16. Jacob, Margaret C. (1976). The Newtonians and the English Revolution 1689–1720. Harvester Press. pp. 132–33. 17. D. T. Whiteside, ed. (2008). The Mathematical Papers of Isaac Newton. Cambridge University Press. p. 14. ISBN 978-0-521-04584-1. Retrieved 22 May 2013. 18. J.E. Force; S. Hutton (2004). Newton and Newtonianism: New Studies. Springer. p. 109. ISBN 978-1-4020-1969-2. Retrieved 25 May 2013. 19. Richard H. Popkin, ed. (1999). The Pimlico History of Western Philosophy. Pimlico. p. 427. ISBN 0-7126-6534-X. 20. Wiles, Maurice (1996). Archetypal Heresy: Arianism Through the Centuries. Oxford University Press. pp. 94–. ISBN 978-0-19-826927-4. Retrieved 22 May 2013. 21. Gibson, William (2004). The Enlightenment Prelate: Benjamin Hoadly, 1767–1761. James Clarke & Co. pp. 121–23. ISBN 978-0-227-67978-4. Retrieved 22 May 2013. 22. William Gibson; Robert G.. Ingram (2005). Religious Identities in Britain: 1660– 1832. Ashgate Publishing, Ltd. pp. 47–48. ISBN 978-0-7546-3209-2. Retrieved 22 May 2013. 23. William Gibson; William Gibson (2002). The Church of England 1688–1832: Unity and Accord. Taylor & Francis. p. 81. ISBN 978-0-203-13462-7. Retrieved 22 May 2013. 24. Lee, Sidney, ed. (1900). "Whiston, William" . Dictionary of National Biography. Vol. 61. London: Smith, Elder & Co. 25. Steele, John M. (2012). Ancient Astronomical Observations and the Study of the Moon's Motion (1691–1757). Springer. p. 24. ISBN 978-1-4614-2149-8. Retrieved 22 May 2013. 26. Probyn, Clive. "Chubb, Thomas". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/5378. (Subscription or UK public library membership required.) 27. McLachlan, H. J. "Emlyn, Thomas". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/8793. (Subscription or UK public library membership required.) 28. Benedict, Jim. "Gale, John". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/10292. (Subscription or UK public library membership required.) 29. Force, James E. (2002). William Whiston: Honest Newtonian. Cambridge University Press. p. 27. ISBN 978-0-521-52488-9. Retrieved 21 May 2013. 30. Acheson, Alan R. "Rundle, Thomas". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/24279. (Subscription or UK public library membership required.) 31. Sheehan, Jonathan (2005). "The" Enlightenment Bible: Translation, Scholarship, Culture. Princeton University Press. p. 35 note 21. ISBN 978-0-691-11887-1. Retrieved 22 May 2013. 32. Gibson, William (2004). The Enlightenment Prelate: Benjamin Hoadly, 1767-1761. James Clarke & Co. p. 122. ISBN 978-0-227-67978-4. Retrieved 22 May 2013. 33. Wokenius, Franz (1728). Christianismus primaevus quem Guil. Whistonus modo non-probando restituendum dictitat sed Apostolus Paulus breviter quasi in tabula depinxit ... Retrieved 25 May 2013. 34. Margaret C. Jacob; Larry Stewart (2009). Practical Matter: Newton's Science in the Service of Industry and Empire 1687–1851. Harvard University Press. p. 64. ISBN 978-0-674-03903-2. Retrieved 21 May 2013. 35. O'Connor, John J.; Robertson, Edmund F., "London Coffee houses and mathematics", MacTutor History of Mathematics Archive, University of St Andrews 36. Stewart, Larry. "Hauksbee, Francis". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/12619. (Subscription or UK public library membership required.) 37. Thomas Hockey; Katherine Bracher; Marvin Bolt; Virginia Trimble; Richard Jarrell; JoAnn Palmeri; Jordan D. Marché; Thomas Williams; F. Jamil Ragep, eds. (2007). Biographical Encyclopedia of Astronomers. Springer. p. 1213. ISBN 978-0-387-30400-7. Retrieved 25 May 2013. 38. Knox, Kevin C. (2003). From Newton to Hawking: A History of Cambridge University's Lucasian Professors of Mathematics. Cambridge University Press. p. 162. ISBN 978-0-521-66310-6. Retrieved 22 May 2013. 39. Wigelsworth, Jeffrey R. (2010). Selling Science in the Age of Newton: Advertising and the Commoditization of Knowledge. Ashgate Publishing, Ltd. p. 137. ISBN 978-1-4094-2310-2. Retrieved 25 May 2013. 40. Ditton, William Whiston; Ditton, Humphrey (1714). A New Method for Discovering the Longitude, both at Sea and Land. John Phillips. Retrieved 15 April 2015. 41. For example, Jonathan Swift's 1714 "Ode, to Musick. On the Longitude", including numerous references to bepissing and beshitting upon both Whiston and Ditton. 42. S.D. Snobelen, "William Whiston: Natural Philosopher, Prophet, Primitive Christian" (Cambridge Univ. PhD Thesis, 2000) 43. Mr Whiston's Project for finding the Longitude (MSS/79/130.2), Board of Longitude project, University of Cambridge Digital Library 44. "This Month in Physics History". Retrieved 16 October 2018. 45. Poole, William (2010). The World Makers: Scientists of the Restoration and the Search for the Origins of the Earth. Peter Lang. p. 72. ISBN 978-1-906165-08-6. Retrieved 25 May 2013. 46. Force, James E. (1985). William Whiston: Honest Newtonian. Cambridge University Press. p. 181 note 128. ISBN 978-0-521-26590-4. Retrieved 25 May 2013. 47. J.E. Force; S. Hutton (2004). Newton and Newtonianism: New Studies. Springer. p. 102. ISBN 978-1-4020-1969-2. Retrieved 25 May 2013. 48. Davis A. Young; Ralph Stearley (2008). The Bible, Rocks and Time: Geological Evidence for the Age of the Earth. InterVarsity Press. p. 67. ISBN 978-0-8308-2876-0. Retrieved 23 May 2013. 49. Westfall, Richard S. (1983). Never at Rest: A Biography of Isaac Newton. Cambridge University Press. pp. 815 note 112. ISBN 978-0-521-27435-7. Retrieved 25 May 2013. 50. Farrell, Maureen (1981). William Whiston. New York: Arno Press. pp. 46–47. 51. "The genuine works of Flavius Josephus the Jewish historian". University of Chicago. Retrieved 16 June 2023. 52. Josephus (1981). Josephus Complete Works. Translated by William Whiston. Grand Rapids, Michigan: Kregel Publications. p. xi (Foreword). ISBN 0-8254-2951-X. Further reading • Farrell, Maureen (1981). William Whiston. New York: Arno Press. • Force, James E. (2002). William Whiston: Honest Newtonian. Cambridge: Cambridge University Press. • Rouse Ball, W. W. (2009) [1889]. A History of the Study of Mathematics at Cambridge University. Cambridge University Press. pp. 83–85. ISBN 978-1-108-00207-3. External links • Media related to William Whiston at Wikimedia Commons • Biography of William Whiston at the LucasianChair.org, the homepage of the Lucasian Chair of Mathematics at Cambridge University • Bibliography for William Whiston Archived 10 May 2012 at the Wayback Machine at the LucasianChair.org the homepage of the Lucasian Chair of Mathematics at Cambridge University • Whiston's MacTutor Biography • Works by William Whiston at Project Gutenberg • Works by or about William Whiston at Internet Archive • Works by William Whiston at LibriVox (public domain audiobooks) • William Whiston at the Mathematics Genealogy Project • "Account of Newton", Collection of Authentick Records (1728), pp. 1070–1082 • "The Works of Flavius Josephus" translated by William Whiston • "William Whiston and the Deluge" by Immanuel Velikovsky • "Whiston's Flood" • Whiston biography at Chambers' Book of Days • Some of Whiston's views on biblical prophecy Archived 25 April 2013 at the Wayback Machine • "William Whiston, The Universal Deluge, and a Terrible Specracle" by Roomet Jakapi • Collection of Authentick Records by Whiston at the Newton Project Archived 4 October 2006 at the Wayback Machine • William Whiston, 1667–1752 Archived 29 September 2007 at the Wayback Machine • Collection of William Whiston portraits at England's National Portrait Gallery • Primitive New Testament • William Whiston | Portraits From the Past • A New Theory of the Earth (1696) – full digital facsimile at Linda Hall Library Lucasian Professors of Mathematics • Isaac Barrow (1664) • Isaac Newton (1669) • William Whiston (1702) • Nicholas Saunderson (1711) • John Colson (1739) • Edward Waring (1760) • Isaac Milner (1798) • Robert Woodhouse (1820) • Thomas Turton (1822) • George Biddell Airy (1826) • Charles Babbage (1828) • Joshua King (1839) • George Stokes (1849) • Joseph Larmor (1903) • Paul Dirac (1932) • James Lighthill (1969) • Stephen Hawking (1979) • Michael Green (2009) • Michael Cates (2015) Authority control International • FAST • ISNI • VIAF National • Spain • 2 • France • BnF data • Germany • Israel • Belgium • United States • Sweden • Czech Republic • Australia • Greece • Netherlands • Poland • Portugal • Vatican Academics • CiNii • MathSciNet • Mathematics Genealogy Project People • Deutsche Biographie • Trove Other • SNAC • IdRef
Wikipedia
William Whyburn William Marvin Whyburn (12 November 1901 – 5 May 1972) was an American mathematician who worked on ordinary differential equations. His work focussed on a multitude of topics including, boundary value problems, properties of Green’s function and properties of Green’s matrix.[1] William Whyburn Born(1901-11-12)12 November 1901 Lewisville, Texas Died5 May 1972(1972-05-05) (aged 70) Greenville, North Carolina Resting placeChinn's Chapel Cemetery, Copper Canyon, Denton County, Texas, USA NationalityAmerican SpouseMarie Barfield ChildrenWilla Whyburn, Clifton Whyburn Biography Early life William Marvin Whyburn was born in Lewisville, Texas, on 12 November 1901.[1] He was the son of farmers Thomas Whyburn and Eugenia Elizabeth McLeod. He had a brother Gordon Whyburn, also a mathematician who primarily studied topology. He attended Bethel School where he would study until he was 14, when he sat an entrance exam and was accepted into North Texas State College in 1916.[2] Undergraduate education Whyburn studied for four years at North Texas State College, where after he would study mathematics at the University of Texas. He attained his Bachelor of Arts degree in 1922 and went on to achieve a Master of Arts degree in mathematics the following year.[2] In 1923 Whyburn married Marie Barfield. Marie was also a student at the University of Texas. Together they had two children, Willa Whyburn and Clifton Whyburn. Clifton also studied mathematics.[2] Undergraduate teaching While studying at North Texas State College (1918–1920) Whyburn taught at different schools in Denton County. One of the students taught by Whyburn was famous mathematician Samuel S. Wilks.[2] In 1923/24 Whyburn taught full-time at South Park Junior College, Beaumont then he held an assistant professor role at Texas A&M College the years after. Again in 1925/26 Whyburn was an associate professor at the Texas Technological College in Lubbock, Texas. Whyburn was given the Louis Lipsitz fellowship for the academic year 1926/27, which allowed him to study full-time.[2] Postgraduate career Whyburn continued to study at the University of Texas for his Ph.D. under the supervision of his advisor Hyman Joseph Ettlinger.[3] After the publication of his thesis Linear Boundary Value Problems for Ordinary Differential Equations and Their Associated Difference Equations he was awarded his doctorate in June 1927.[4] Additionally, in the three years before this publication Whyburn published three other papers, two of which were on Green’s function. Whyburn published two more papers in 1927 before spending the 1927/28 academic year at Harvard university as a National Research Fellow.[2] Whyburn was assigned as an Assistant Professor of Mathematics at the University of California, Los Angeles in 1928. Ten years later Whyburn was made a full professor in 1938 as well as being the chairman for the Mathematics Department for a seven-year tenure beginning in 1937.[2] Whyburn was the chairman of the supervisors committee for Engineering, Science, Management War Training Programs during the second world war. Throughout the war he wrote a paper and a book about mathematics as its applied in war. Whyburn was given the role of president of the Texas Technological College in 1944. In this role he would help improve the educational profile of the school to other major educational bodies such as the American Association of Universities and American Association of University Women. As a result of his work, the college gained recognition from governmental agencies, reflecting his presidential impact.[2] In 1948 Whyburn resigned from his position at the Texas Technological College as he was appointed Kenan Professor of Mathematics at the University of North Carolina at Chapel Hill, where he would further be appointed as chairman of the Mathematics department. After serving three years as Vice President for research from 1957-1960, Whyburn retired in 1967. He was then appointed as the Frensley Professor of Mathematics at the Southern Methodist University in Dallas. Whyburn retired from this position in 1970 before working a part-time teaching position at East Carolina University, North Carolina.[1][2] As a teacher Whyburn was focussed on the students perception and put them first. He would be methodical in how he approached different students and their areas of postgraduate research whilst supervising. He supervised the Ph.D. of the following students: Leonard P. Burton, Albert Deal, Bertram Drucker, Garett Etgen, Paul Herwitz, Sandra Hilt, A. Keith Hinds, Nathaniel Macon, Edward J. Pellicciaro, Tullio Pignani, Clay Campbell Ross, David Showalter and Frank Stellard.[3] Whyburn died of a heart attack on 5 May 1972 in Greenville, North Carolina.[5][6]  He is buried in Chinn's Chapel Cemetery, Copper Canyon, Texas[7] Selected publications • "An Extension of the Definition of the Green's Function in One Dimension"(1924)[8] • "Second-Order Differential Systems With Integral and k-Point Boundary Conditions" (1928)[9] • "Functional Properties of the Solutions of Differential Systems" (1930)[10] • "Differential Equations with General Boundary Conditions" (1942)[11] • "A Nonlinear Boundary Value Problem For Second Order Differential Systems" (1955)[12] • "Complexes of Differential Systems" (1972)[13] References 1. "William Whyburn - Biography". Maths History. Retrieved 2022-06-16. 2. Reid, W.T. (1973). "William M. Whyburn" (PDF). Bulletin of the American Mathematical Society. 3. "William Whyburn - The Mathematics Genealogy Project". www.genealogy.math.ndsu.nodak.edu. Retrieved 2022-07-02. 4. "Mathematics Alumni". web.ma.utexas.edu. Retrieved 2022-07-02. 5. North Carolina, State Board of Health (June 12, 1972). "Certificate of Death: William Marvin Whyburn". 6. "Prof. William Whyburn Dead; Mathematician at Chapel Hill". The New York Times. 1972-05-07. ISSN 0362-4331. Retrieved 2022-07-02. 7. "William Marvin Whyburn (1901-1972)". www.findagrave.com. Retrieved 2022-06-30. 8. Whyburn, W. M. (1924). "An Extension of the Definition of the Green's Function in One Dimension". Annals of Mathematics. 26 (1/2): 125–130. doi:10.2307/1967748. ISSN 0003-486X. JSTOR 1967748. MR 1502681 – via jstor.org. 9. Whyburn, William M. (1928). "Second-Order Differential Systems With Integral and k-Point Boundary Conditions" (PDF). Transactions of the American Mathematical Society. 30 (4): 630–640. MR 1501451 – via ams.org. 10. Whyburn, William M. (1930). "Functional Properties of the Solutions of Differential Systems" (PDF). Transactions of the American Mathematical Society. 32 (3): 502–508. doi:10.1090/S0002-9947-1930-1501548-9. MR 1501548 – via ams.org. 11. Whyburn, William M. (1942). "Differential Equations with General Boundary Conditions" (PDF). Bulletin of the American Mathematical Society. 48 (10): 692–704. doi:10.1090/S0002-9904-1942-07760-3. MR 0007192. S2CID 51822059 – via ams.org. 12. Whyburn, William M. (1955). "A nonlinear boundary value problem for second order differential systems". Pacific Journal of Mathematics. 5: 147–160. doi:10.2140/pjm.1955.5.147. MR 0069368 – via projecteuclid.org. 13. Whyburn, William M.; Williams, Richard K. (1972-03-01). "Complexes of differential systems". Journal of Differential Equations. 11 (2): 299–306. Bibcode:1972JDE....11..299W. doi:10.1016/0022-0396(72)90046-0. ISSN 0022-0396. MR 0294752. 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Wikipedia
William Wilson Hunter Sir William Wilson Hunter KCSI CIE (15 July 1840 – 6 February 1900)[1] was a Scottish historian, statistician, a compiler and a member of the Indian Civil Service. Sir William Wilson Hunter Born(1840-07-15)15 July 1840 Glasgow, Scotland, UK Died6 February 1900(1900-02-06) (aged 59) Oaken Holt, England, UK NationalityBritish Alma materUniversity of Glasgow Scientific career FieldsHistory, statistics InstitutionsIndian Civil Service University of Calcutta He is most known for The Imperial Gazetteer of India on which he started working in 1869, and which was eventually published in nine volumes in 1881, then fourteen, and later as a twenty-six volume set after his death. Early life and education William Wilson Hunter was born on 15 July 1840 in Glasgow, Scotland, to Andrew Galloway Hunter, a Glasgow manufacturer. He was the second of his father's three sons. In 1854 he started his education at the 'Quaker Seminary' at Queenswood, Hampshire and a year later he joined The Glasgow Academy. He was educated at the University of Glasgow (BA 1860), Paris and Bonn, acquiring a knowledge of Sanskrit, LL.D., before passing first in the final examination for the Indian Civil Service in 1862.[2] Career He reached Bengal Presidency in November 1862 and was appointed assistant magistrate and collector of Birbhum, in the lower provinces of Bengal, where he began collecting local traditions and records, which formed the materials for his publication, entitled The Annals of Rural Bengal,[2] which influenced the historical romances of Bankim Chandra Chattopadhyay.[3] He also compiled A Comparative Dictionary of the Non-Aryan Languages of India, a glossary of dialects based mainly upon the collections of Brian Houghton Hodgson, which according to the Encyclopædia Britannica Eleventh Edition, "testifies to the industry of the writer but contains much immature philological speculation".[2] In 1869 Lord Mayo, the then governor-general, asked Hunter to submit a scheme for a comprehensive statistical survey of India. The work involved the compilation of a number of local gazetteers, in various stages of progress, and their consolidation in a condensed form upon a single and uniform plan.[2] There was unhappiness with the scope and completeness of the earlier surveys conducted by administrators such as Buchanan, and Hunter determined to model his efforts on the Ain-i-Akbari and Description de l'Égypte. Hunter said that "It was my hope to make a memorial of England's work in India, more lasting, because truer and more complete, than these monuments of Mughal Empire and of French ambition."[4] In response to Mayo's question on 30 May 1871 of whether the Indian Muslims are "bound by their religion to rebel against the Queen" Hunter completed his influential work The Indian Musalmans in mid-June 1871 and later published it as a book in mid-August of the same year.[5][6] In it, Hunter concluded that the majority of the Indian Muslim scholars rejected the idea of rebelling against the Government because of their opinion that the condition for religious war, i.e. the absence of protection and liberty between Muslims and infidel rulers, did not exist in British India; and that "there is no jihad in a country where protection is afforded".[7] In 1872 Hunter published his history of Orissa. The third International Sanitary Conference held at Constantinople in 1866 declared Hindu and Muslim pilgrimages to be 'the most powerful of all the causes which conduce to the development and propagation of Cholera epidemics'. Hunter echoing the view described the 'squalid pilgrim army of Jagannath' as[8] with its rags and hair and skin freighted with vermin and impregnated with infection, may any year slay thousands of the most talented and beautiful of our age in Vienna, London, or Washington. He embarked on a series of tours throughout the country,[4] and he supervised the A Statistical Account of Bengal (20 volumes, 1875–1877)[9] and a similar work for Assam (2 volumes, 1879).[10] Hunter wrote that Under this system, the materials for the whole of British India have now been collected, in several Provinces the work of compilation has rapidly advanced, and everywhere it is well in hand. During the same period the first Census of India has been taken, and furnished a vast accession to our knowledge of the people. The materials now amassed form a Statistical Survey of a continent with a population exceeding that of all Europe, Russia excepted."[11] The statistical accounts, covering the 240 administrative districts, comprised 128 volumes and these were condensed into the nine volumes of The Imperial Gazetteer of India, which was published in 1881.[4] The Gazetteer was revised in later series, the second edition comprising 14 volumes published between 1885 and 1887, while the third comprised 26 volumes, including an atlas, and was published in 1908 under the editorship of Herbert Hope Risley, William Stevenson Meyer, Richard Burn and James Sutherland Cotton.[12] Again according to the Encyclopædia Britannica Eleventh Edition, Hunter "adopted a transliteration of vernacular place-names, by which means the correct pronunciation is ordinarily indicated; but hardly sufficient allowance was made for old spellings consecrated by history and long usage."[13] Hunter's own article on India was published in 1880 as A Brief History of the Indian Peoples, and has been widely translated and utilized in Indian schools. A revised form was issued in 1895, under the title of The Indian Empire: its People, History and Products. Hunter later said that Nothing is more costly than ignorance. I believe that, in spite of its many defects, this work will provide a memorable episode in the long battle against ignorance; a breakwater against the tide of prejudice and false opinions flowing down upon us from the past, and the foundation for a truer and wider knowledge of India in time to come. Its aim has been not literary graces, nor scientific discovery, nor antiquarian research; but an earnest endeavour to render India better governed, because better understood.[4] Hunter contributed the articles "Bombay", "Calcutta", "Dacca", "Delhi" and "Mysore" to the 9th edition (1875–89) of the Encyclopædia Britannica.[14] In 1882 Hunter, as a member of the governor-general's council, presided over the Commission on Indian Education; in 1886 he was elected vice-chancellor of the University of Calcutta. In 1887 he retired from the service, was created KCSI, and settled at Oaken Holt, near Oxford.[15] He was on the governing body of Abingdon School from 1895 until his death in 1900.[16] On 13 March 1889 Philip Lyttelton Gell the then Secretary to the Delegates of the Clarendon Press, wrote to Hunter about a project which has been for some time under the consideration of the Delegates, to publish a series giving the salient features of Indian History in the Biographies of successive Generals and Administrators.[17] Gell arranged the publication of the series by June 1889; with Hunter receiving £75 for each volume, and the author £25. Gell's experience of the earlier unsaleable Sacred Books of the East and financial constraints forced the Rulers of India to end at 28 volumes in spite of Hunter's disappointment about the same.[18] Hunter himself contributed the volumes on Dalhousie (1890)[19] and Mayo (1891)[20] to the series. He had previously written an official Life of Lord Mayo, which was published on 19 November 1875 in two volumes with a second edition appearing in 1876.[21] He also wrote a weekly article on Indian affairs for The Times. But the great task to which he applied himself on his settlement in England was a history upon a large scale of the British Dominion in India, two volumes of which only had appeared when he died, carrying the reader barely down to 1700. He was much hindered by the confused state of his materials, a portion of which he arranged and published in 1894 as Bengal Manuscript Records, in three volumes.[15] Hunter dedicated his 1892 work Bombay 1885 to 1890: A Study in Indian Administration to Florence Nightingale.[22] His later works include the novel titled The Old Missionary (1895, described on the title-page as "revised from The Contemporary Review"),[23] and The Thackerays in India (1897). John F. Riddick describes Hunter's The Old Missionary as one of the "three significant works" produced by Anglo-Indian writers on Indian missionaries along with The Hosts of the Lord (1900) by Flora Annie Steel and Idolatry (1909) by Alice Perrin.[24] In the winter of 1898–1899, in consequence of the fatigue incurred in a journey to the Caspian and back, on a visit to the sick-bed of one of his two sons, Hunter was stricken down by a severe attack of influenza, which affected his heart. He died at Oaken Holt on 6 February 1900.[15] S. C. Mittal believes that Hunter "represented the official mind of the bureaucratic Victorian historians in India", of whom James Talboys Wheeler and Alfred Comyn Lyall were other examples.[25] Bibliography Works • A Comparative Dictionary of the Languages of India and High Asia: With a Dissertation. Based on the Hodgson Lists, Official Records, and Mss. Trübner and Company. 1868. • Annals of Rural Bengal. Smith, Elder & Co. 1868. • The Indian Musalmans: Are They Bound in Conscience to Rebel Against the Queen?. Trübner and Company. 1871.[26] • Orissa, Or, The Vicissitudes of an Indian Province Under Native and British Rule. Smith, Elder and Company. 1872. • A Statistical Account of Bengal. London: Trübner & Co. 1875–1879. (20 volumes) • A Statistical Account of Assam. 1879. (2 volumes) • A Brief history of the Indian peoples. Oxford: Clarendon Press. 1880. • The Imperial Gazetteer of India. 1908–1909. (3rd ed. 26 vols; 1st ed. 9 vols, 1881; 2nd ed. 14 vols, 1885–87) • The Indian Empire: Its People, History, and Products London (Second ed.). Trübner & Co. 1886. ISBN 9788120615816. • Bombay, 1885-1890: A Study in Indian Administration. Frowde. 1892. • The Marquess of Dalhousie. 1894. • State Education for the People in America, Europe, India, and Australia: With Papers on the Education of Women, Technical Instruction, and Payment by Results. C. W. Bardeen. 1895. • The Thackerays in India and Some Calcutta Graves. London: Henry Frowde. 1897.[27] • Williams Jackson, A. V., ed. (1906). History of India: From the first European settlements to the founding of the English East India Company . History of India. Vol. 6. London: Grolier Society. • Williams Jackson, A. V., ed. (1907). History of India: The European struggle for Indian supremacy in the seventeenth century . History of India. Vol. 7. London: Grolier Society. Works about Hunter • Francis Henry Bennett Skrine (1901). Life of Sir William Wilson Hunter, K.C.S.I. Longmans, Green. See also • The Imperial Gazetteer of India • Hunterian transliteration • Census of India prior to independence References 1. "Obituary: Sir William Wilson Hunter, K. C. S. I., C. I. E.". The Geographical Journal. 15 (3): 289–290. March 1900. JSTOR 1774698. 2. Chisholm 1911, p. 945. 3. Chatterjee, Rimi B. (2004). "'Every Line for India': The Oxford University Press and the Rise and Fall of the Rulers of India Series". In Gupta, Abhijit; Chakravorty, Swapan (eds.). Print Areas: Book History in India. Permanent Black. pp. 77, 93. ISBN 978-81-7824-082-4. 4. Marriott, John (2003). The other empire: metropolis, India and progress in the colonial imagination. Manchester University Press. p. 209. ISBN 978-0-7190-6018-2. Retrieved 7 December 2011. 5. v. L. B. (1872). "De Mohammedanen in Hindostan. —Our Indian Musalmans: Are they bound in conscience to rebel against the Queen? by W. W. Hunter". Bijdragen tot de Taal-, Land- en Volkenkunde van Nederlandsch-Indië. 18 (2): 121–122. JSTOR 25736656. 6. Ali, M. Mohar (1980). "Hunter's "Indian Musalmans": A Re-Examination of Its Background". The Journal of the Royal Asiatic Society of Great Britain and Ireland. 112 (1): 30–51. doi:10.1017/S0035869X00135889. JSTOR 25211084. S2CID 154830629. 7. Bonney, R. (2004) Jihad: From Qur'an to Bin Laden, Hampshire: Palgrave Macmillan, pp. 193-194 8. Thomas R. Metcalf (27 February 1997). Ideologies of the Raj. Cambridge University Press. p. 175. ISBN 978-0-521-58937-6. 9. "A Statistical Account of Bengal by W. W. Hunter". The North American Review. 127 (264): 339–342. September–October 1878. JSTOR 25100678. 10. "Hunter, Sir William Wilson". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/14237. (Subscription or UK public library membership required.) 11. Nicholas B. Dirks (2003). Castes of Mind: Colonialism and the Making of Modern India. Permanent Black. p. 199. ISBN 978-81-7824-072-5. 12. Henry Scholberg (1970). The District Gazetteers of British India: A Bibliography. Zug, Switzerland: Inter Documentation Company. ISBN 9780800212650. 13. Chisholm 1911, pp. 945–946. 14. Important Contributors to the Britannica, 9th and 10th Editions. 1902encyclopedia.com. Retrieved 20 April 2018. 15. Chisholm 1911, p. 946. 16. "School Notes" (PDF). The Abingdonian. 17. Chatterjee, Rimi B. (2004). "'Every Line for India': The Oxford University Press and the Rise and Fall of the Rulers of India Series". In Gupta, Abhijit; Chakravorty, Swapan (eds.). Print Areas: Book History in India. Permanent Black. p. 81. ISBN 978-81-7824-082-4. 18. Chatterjee, Rimi B. (2004). "'Every Line for India': The Oxford University Press and the Rise and Fall of the Rulers of India Series". In Gupta, Abhijit; Chakravorty, Swapan (eds.). Print Areas: Book History in India. Permanent Black. pp. 81, 87. ISBN 978-81-7824-082-4. 19. H. P. (1891). "The Marquess of Dalhousie by William Wilson Hunter". Revue Historique. 47 (2): 387–393. JSTOR 40938228. 20. H. P. (1892). "The Earl of Mayo by William Wilson Hunter". Revue Historique. 48 (2): 387–400. JSTOR 40939452. 21. Satish Chandra Mittal (1 January 1996). India Distorted: A Study of British Historians on India. M.D. Publications Pvt. Ltd. p. 199. ISBN 978-81-7533-018-4. 22. Florence Nightingale (6 December 2007). Florence Nightingale on Social Change in India: Collected Works of Florence Nightingale. Wilfrid Laurier University Press. p. 841. ISBN 978-0-88920-495-9. 23. "The Old Missionary". The Spectator. 5 October 1895. p. 19. Retrieved 23 December 2014. 24. John F. Riddick (1 January 2006). The History of British India: A Chronology. Greenwood Publishing Group. p. 179. ISBN 978-0-313-32280-8. 25. Mittal, Satish Chandra (1996). India Distorted: A Study of British Historians on India. Vol. 2. M.D. Publications. p. 170. ISBN 9788175330184. 26. "In the book 'The Indian Musalmans' by William Wilson Hunter, the author has mentioned the Indians Muslims who rebelled against the British empire as 'Wahhabis', so did only Salafis/Ahlul Hadith fight against the British? – Quora". quora.com. Retrieved 27 March 2021. 27. "Review: The Thackerays in India and Some Calcutta Graves by Sir William Wilson Hunter". The Athenæum (3613): 111–112. 23 January 1897. Attribution: • This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Hunter, Sir William Wilson". Encyclopædia Britannica. Vol. 13 (11th ed.). Cambridge University Press. pp. 945–946. External links • Works by William Wilson Hunter at Project Gutenberg • Works by or about William Wilson Hunter at Internet Archive Vice chancellors of the University of Calcutta • James William Colvile • William Ritchie • Claudius James Erskine • Henry Sumner Maine • W. S. Seton-Karr • Edward Clive Bayley • Arthur Hobhouse • William Markby • Alexander Arbuthnot • Arthur Wilson • H. J. Reynolds • C. P. Ilbert • William Wilson Hunter • William Comer Petheram • Gooroodass Banerjee • Jones Quain Pigot • Alfred Woodley Croft • E. J. 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Sen • Sushil Kumar Mukherjee • Ramendra Kumar Podder • Santosh Bhattacharyya • Bhaskarananda Ray Chaudhuri • Rathindra Narayan Basu • Asis Kumar Banerjee • Suranjan Das • Sugata Marjit • Sonali Chakravarti Banerjee Authority control International • FAST • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Australia • Greece • Netherlands • Poland • Portugal • Vatican People • Deutsche Biographie • Trove Other • SNAC • IdRef
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William Woolsey Johnson William Woolsey Johnson (1841–1927) was an American mathematician, who was one of the founders of the American Mathematical Society. William Woolsey Johnson Picture by Thomas Eakins Born(1841-06-23)June 23, 1841 Owego (NY) DiedMay 14, 1927(1927-05-14) (aged 85) Baltimore (MD) Resting placeWoodlawn Cemetery (Baltimore) 39.325200°N 76.726093°W / 39.325200; -76.726093 Alma materYale University Scientific career FieldsMathematics InstitutionsUnited States Naval Academy St. John's College Kenyon College Life and work Johnson, son of a farmer of Tioga County, New York, studied at Yale University where he received his BA in 1862. After two years serving in the Nautical Almanac Office in Cambridge, Massachusetts, he began his academic career as assistant professor in the Naval Academy in Newport, Rhode Island, but soon transferred to Annapolis, Maryland, from 1864 to 1869. In 1870 he was appointed professor of mathematics at Kenyon College and since 1872 at St. John's College (Annapolis).[1] In 1881 he returned to the Naval Academy as full professor where he remained until his retirement in 1921. He served as one of the five members of the Council of the American Mathematical Society for the 1892–1893 term[2] and he was one of the impulsors of the birth of the Bulletin of the Society[3] and one of his main first contributors. Johnson is mainly remembered by his books on differential calculus, basing it on related rates.[4] He is also known to be the first on probing the conditions of solvability of the 15 puzzle.[5] Selected publications Articles • Johnson, W. Woolsey (1891). "Octonary numeration". Bull. Amer. Math. Soc. 1: 1–6. doi:10.1090/S0002-9904-1891-00015-2. • Johnson, W. Woolsey (1892). "The mechanical axioms or laws of motion". Bull. Amer. Math. Soc. 1 (6): 129–139. doi:10.1090/S0002-9904-1892-00051-1. • Johnson, W. Woolsey (1893). "On Peters's formula for probable error". Bull. Amer. Math. Soc. 2 (4): 57–61. doi:10.1090/S0002-9904-1893-00107-9. • Johnson, W. Woolsey (1893). "A case of non-euclidian geometry". Bull. Amer. Math. Soc. 2 (7): 158–161. doi:10.1090/S0002-9904-1893-00130-4. • Johnson, W. Woolsey (1894). "Gravitation and absolute units of force". Bull. Amer. Math. Soc. 3 (8): 197–199. doi:10.1090/S0002-9904-1894-00210-9. • Johnson, W. Woolsey (1895). "Kinetic stability of central orbits". Bull. Amer. Math. Soc. 1 (8): 193–196. doi:10.1090/S0002-9904-1895-00272-4. • Johnson, W. Woolsey (1906). "Note on the numerical transcendents Sn and sn =Sn-1". Bull. Amer. Math. Soc. 12 (10): 477–482. doi:10.1090/S0002-9904-1906-01374-X. Books • A treatise on ordinary and partial differential equations. 1881. 3rd edition. 1893. • An elementary treatise on the differential calculus, founded on the method of rates or fluxions. 1889. • The theory of errors and method of least squares. 1893. • Theoretical mechanics. An elementary treatise. 1901. References 1. Dwight 1874, p. 298. 2. Austin, Barry & Berman 2010, p. 145. 3. Fiske 2000, p. 5. 4. Austin, Barry & Berman 2010, p. 144. 5. Hendrixson 2011, p. 6. Bibliography • Austin, Bill; Barry, Don; Berman, David (2010). "The Lengthening Shadow: The Story of Related Rates". In Caren L. Diefenderfer; Roger B. Nelsen (eds.). The Calculus Collection: A Resource for AP and Beyond. Mathematical Association of America. pp. 139–148. ISBN 978-0-88385-761-8. • Dwight, Benjamin Woodbridge (1874). The History of the Descendants of John Dwight of Dedham, Mass. John F. Trow & Son. pp. 298. • Fiske, Thomas S. (2000). "Mathematical progress in America". Bulletin of the AMS. 37 (1): 3–8. doi:10.1090/S0273-0979-99-00799-5. ISSN 0273-0979. PMID 17781208. • Hendrixson, Lisa Rose (2011). Variations of the 15 Puzzle. Ohio Link (Thesis). External links • O'Connor, John J.; Robertson, Edmund F. "William Woolsey Johnson". MacTutor History of Mathematics Archive. University of St Andrews. • William Woolsey Johnson papers. Manuscripts and Archives Repository, Yale University. Authority control International • FAST • ISNI • VIAF National • Germany • Israel • United States • Japan • Czech Republic • Australia • Netherlands Academics • CiNii • MathSciNet • zbMATH People • Trove Other • IdRef
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William Jones (mathematician) William Jones, FRS (1675 – 1 July 1749[1]) was a Welsh mathematician, most noted for his use of the symbol π (the Greek letter Pi) to represent the ratio of the circumference of a circle to its diameter. He was a close friend of Sir Isaac Newton and Sir Edmund Halley. In November 1711, he became a Fellow of the Royal Society, and was later its vice-president.[2] William Jones Portrait of William Jones by William Hogarth, 1740 National Portrait Gallery Born1675 Llanfihangel Tre'r Beirdd, Isle of Anglesey Died3 July 1749 (aged 73–74) London, England Part of a series of articles on the mathematical constant π 3.1415926535897932384626433... Uses • Area of a circle • Circumference • Use in other formulae Properties • Irrationality • Transcendence Value • Less than 22/7 • Approximations • Madhava's correction term • Memorization People • Archimedes • Liu Hui • Zu Chongzhi • Aryabhata • Madhava • Jamshīd al-Kāshī • Ludolph van Ceulen • François Viète • Seki Takakazu • Takebe Kenko • William Jones • John Machin • William Shanks • Srinivasa Ramanujan • John Wrench • Chudnovsky brothers • Yasumasa Kanada History • Chronology • A History of Pi In culture • Indiana Pi Bill • Pi Day Related topics • Squaring the circle • Basel problem • Six nines in π • Other topics related to π Biography William Jones was born the son of Siôn Siôr (John George Jones) and Elizabeth Rowland in the parish of Llanfihangel Tre'r Beirdd, about 4 miles (6.4 km) west of Benllech on the Isle of Anglesey in Wales. He attended a charity school at Llanfechell, also on the Isle of Anglesey, where his mathematical talents were spotted by the local landowner Lord Bulkeley, who arranged for him to work in a merchant's counting-house in London.[3] His main patrons were the Bulkeley family of north Wales, and later the Earl of Macclesfield.[4] Jones initially served at sea, teaching mathematics on board Navy ships between 1695 and 1702, where he became very interested in navigation and published A New Compendium of the Whole Art of Navigation in 1702,[3] dedicated to a benefactor John Harris.[5] In this work he applied mathematics to navigation, studying methods of calculating position at sea. After his voyages were over he became a mathematics teacher in London, both in coffee houses and as a private tutor to the son of the future Earl of Macclesfield and also the future Baron Hardwicke. He also held a number of undemanding posts in government offices with the help of his former pupils. Jones published Synopsis Palmariorum Matheseos in 1706, a work which was intended for beginners and which included theorems on differential calculus and infinite series. This used π for the ratio of circumference to diameter, following earlier abbreviations for the Greek word periphery (περιφέρεια) by William Oughtred and others.[6][7][8][9][10] His 1711 work Analysis per quantitatum series, fluxiones ac differentias introduced the dot notation for differentiation in calculus.[11] He was noticed and befriended by two of Britain's foremost mathematicians – Edmund Halley and Sir Isaac Newton – and was elected a fellow of the Royal Society in 1711. He later became the editor and publisher of many of Newton's manuscripts and built up an extraordinary library that was one of the greatest collections of books on science and mathematics ever known, and only recently fully dispersed.[12] He married twice, firstly the widow of his counting-house employer, whose property he inherited on her death, and secondly, in 1731, Mary, the 22-year-old daughter of cabinet-maker George Nix, with whom he had two surviving children. His son, also named William Jones and born in 1746, was a renowned philologist who established links between Latin, Greek and Sanskrit, leading to the concept of the Indo-European language group.[13] References 1. Roberts, Gareth Ffowc (2020). Cyfri'n Cewri. University Press Wales. p. 57. ISBN 978-1786835949. 2. "Library and Archive catalogue". Royal Society. Retrieved 1 November 2010. 3. "Jones biography". University of St. Andrews. Retrieved 12 December 2010. 4. Cyfri'n Cewri by Gareth Ffowc Roberts; University of Wales Press (2020); p. 14. 5. William Jones (1702). A New Compendium of the Whole Art of Navigation. Retrieved 3 February 2011. 6. Jones, William (1706). Synopsis Palmariorum Matheseos : or, a New Introduction to the Mathematics. pp. 243, 263. 7. Rothman, Patricia (7 July 2009). "William Jones and his Circle: The Man who invented Pi". History Today. Retrieved 6 October 2017. 8. Roberts, Gareth Ffowc (14 March 2015). "Pi Day 2015: meet the man who invented π". The Guardian. ISSN 0261-3077. Retrieved 6 October 2017. 9. Bogart, Steven. "What is pi, and how did it originate?". Scientific American. Archived from the original on 6 October 2017. Retrieved 6 October 2017. 10. Archibald, R. C. (1921). "Historical Notes on the Relation $e^{-(\pi /2)}=i^{i}$". The American Mathematical Monthly. 28 (3): 121. doi:10.2307/2972388. JSTOR 2972388. It was probably suggested to Jones by Oughtred who employed the symbol in a different sense. 11. Garland Hampton Cannon (1990). The life and mind Oriental Jones. Retrieved 3 February 2011. 12. "How a farm boy from Wales gave the world pi". The Conversation. Retrieved 14 March 2017. 13. Roberts, Gareth Ffowc (14 March 2015). "Pi Day 2015: meet the man who invented π". The Guardian. Retrieved 14 March 2015. External links • William Jones and other important Welsh mathematicians • William Jones and his Circle: The Man who invented Pi • Pi Day 2015: meet the man who invented π Sir Isaac Newton Publications • Fluxions (1671) • De Motu (1684) • Principia (1687) • Opticks (1704) • Queries (1704) • Arithmetica (1707) • De Analysi (1711) Other writings • Quaestiones (1661–1665) • "standing on the shoulders of giants" (1675) • Notes on the Jewish Temple (c. 1680) • "General Scholium" (1713; "hypotheses non fingo" ) • Ancient Kingdoms Amended (1728) • Corruptions of Scripture (1754) Contributions • Calculus • fluxion • Impact depth • Inertia • Newton disc • Newton polygon • Newton–Okounkov body • Newton's reflector • Newtonian telescope • Newton scale • Newton's metal • Spectrum • Structural coloration Newtonianism • Bucket argument • Newton's inequalities • Newton's law of cooling • Newton's law of universal gravitation • post-Newtonian expansion • parameterized • gravitational constant • Newton–Cartan theory • Schrödinger–Newton equation • Newton's laws of motion • Kepler's laws • Newtonian dynamics • Newton's method in optimization • Apollonius's problem • truncated Newton method • Gauss–Newton algorithm • Newton's rings • Newton's theorem about ovals • Newton–Pepys problem • Newtonian potential • Newtonian fluid • Classical mechanics • Corpuscular theory of light • Leibniz–Newton calculus controversy • Newton's notation • Rotating spheres • Newton's cannonball • Newton–Cotes formulas • Newton's method • generalized Gauss–Newton method • Newton fractal • Newton's identities • Newton polynomial • Newton's theorem of revolving orbits • Newton–Euler equations • Newton number • kissing number problem • Newton's quotient • Parallelogram of force • Newton–Puiseux theorem • Absolute space and time • Luminiferous aether • Newtonian series • table Personal life • Woolsthorpe Manor (birthplace) • Cranbury Park (home) • Early life • Later life • Apple tree • Religious views • 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William of Soissons William of Soissons; French: Guillaume de Soissons; was a French logician who lived in Paris in the 12th century. He belonged to a school of logicians, called the Parvipontians.[1] William of Soissons fundamental logical problem and solution William of Soissons[2] seems to have been the first one to answer the question, "Why is a contradiction not accepted in logic reasoning?" by the principle of explosion. Exposing a contradiction was already in the ancient days of Plato a way of showing that some reasoning was wrong, but there was no explicit argument as to why contradictions were incorrect. William of Soissons gave a proof in which he showed that from a contradiction any assertion can be inferred as true.[1] In example from: It is raining (P) and it is not raining (¬P) you may infer that there are trees on the moon (or whatever else)(E). In symbolic language: P & ¬P → E. If a contradiction makes anything true then it makes it impossible to say anything meaningful: whatever you say, its contradiction is also true. C. I. Lewis's reconstruction of his proof William's contemporaries compared his proof with a siege engine (12th century).[3] Clarence Irving Lewis[4] formalized this proof as follows:[5] Proof V  : or &  : and →  : inference P  : proposition ¬ P  : denial of P P &¬ P : contradiction. E  : any possible assertion (Explosion). (1) P &¬ P → P (If P and ¬ P are both true then P is true) (2) P → P∨E (If P is true then P or E is true) (3) P &¬ P → P∨E (If P and ¬ P are both true then P or E are true (from (2)) (4) P &¬ P → ¬P (If P and ¬ P are both true then ¬P is true) (5) P &¬ P → (P∨E) &¬P (If P and ¬ P are both true then (P∨E) is true (from (3)) and ¬P is true (from (4))) (6) (P∨E) &¬P → E (If (P∨E) is true and ¬P is true then E is true) (7) P &¬ P → E (From (5) and (6) one after the other follows (7)) Acceptance and criticism in later ages In the 15th century this proof was rejected by a school in Cologne. They didn't accept step (6).[6] In 19th-century classical logic, the Principle of Explosion was widely accepted as self-evident, e.g. by logicians like George Boole and Gottlob Frege, though the formalization of the Soissons proof by Lewis provided additional grounding for the Principle of Explosion. References 1. Graham Priest, 'What's so bad about contradictions?' in Priest, Beall and Armour-Garb, The Law of Non-Contradiction, p. 25, Clarendon Press, Oxford, 2011. 2. His writings are lost, see: The Metalogicon of John Salisbury. A Twelfth-Century Defense of the Verbal and Logical Arts of the Trivium, translated with an Introduction and Notes by Daniel D. McGarry, Gloucester (Mass.), Peter Smith, 1971, Book II, Chapter 10, pp. 98-99. 3. William Kneale and Martha Kneale, The Development of Logic, Clarendon Press Oxford, 1962, p. 201. 4. C. I. Lewis and C. H. Langford, Symbolic Logic, New York, The Century Co, 1932. 5. Christopher J. Martin, William’s Machine, Journal of Philosophy, 83, 1986, pp. 564 – 572. In particular p. 565 6. "Paraconsistent Logic (Stanford Encyclopedia of Philosophy)". Plato.stanford.edu. Retrieved 2017-12-18.
Wikipedia
William Oughtred William Oughtred (5 March 1574 – 30 June 1660),[1] also Owtred, Uhtred, etc., was an English mathematician and Anglican clergyman.[2][3] After John Napier invented logarithms and Edmund Gunter created the logarithmic scales (lines, or rules) upon which slide rules are based, Oughtred was the first to use two such scales sliding by one another to perform direct multiplication and division. He is credited with inventing the slide rule in about 1622.[4] He also introduced the "×" symbol for multiplication and the abbreviations "sin" and "cos" for the sine and cosine functions.[5] William Oughtred William Oughtred engraving by Wenceslaus Hollar Born5 March 1574 Eton, Buckinghamshire, England Died30 June 1660(1660-06-30) (aged 86) Albury, Surrey, England EducationEton College Alma materKing's College, Cambridge Known for • Slide rule • Multiplication "×" sign Scientific career FieldsMathematician InstitutionsKing's College, Cambridge Notable students • John Wallis • Christopher Wren • Richard Delamain • Seth Ward Clerical life Education The son of Benjamin Oughtred of Eton in Buckinghamshire (now part of Berkshire), William was born there on 5 March 1574/75 and was educated at Eton College, where his father, a writing-master, was one of his teachers.[6] Oughtred had a passion for mathematics, and would often stay awake at nights to learn while others were sleeping.[7] He then attended King's College, Cambridge, where he graduated BA in 1596/97 and MA in 1600, holding a fellowship in the college from 1595 to 1603.[8] He composed a Funeral Ode in Latin for Sir William More of Loseley Park in 1600.[9] Rector at Guildford and at Shalford Admitted to holy orders, he left the University of Cambridge about 1603, when as "Master" William Oughtred he held the rectorate of St Mary's Church, Guildford, Surrey.[10] At the presentation of the lay patron George Austen, gent., he was instituted as vicar at Shalford near Wonersh, in the neighbourhood of Guildford in western Surrey, on 2 July 1605.[11] On 20 February 1606, at Shalford, Oughtred married Christsgift Caryll, a member of the Caryll family seated at Great Tangley Hall at Shalford.[12] The Oughtreds had twelve children, William, Henry, Henry (the first Henry died as a baby), Benjamin, Simon, Margaret, Judith, Edward, Elizabeth, Anne, George, and John. Two of the sons, Benjamin and John, shared their father's interest in instruments and became watchmakers.[13] Oughtred's wife was a niece of Simon Caryll of Tangley and his wife Lady Elizabeth Aungier (married 1607), daughter of Sir Francis Aungier. Oughtred was a witness to Simon Caryll's will, made 1618,[14][15] and through two further marriages Elizabeth remained matriarch and dowager of Great Tangley until her death in about 1650.[16][17] Elizabeth's brother Gerald, 2nd Baron Aungier of Longford, was married to Jane, daughter of Sir Edward Onslow of Knowle, Surrey in 1638. Oughtred praised Gerald (whom he taught) as a man of great piety and learning, skilled in Latin, Greek, Hebrew and other oriental languages.[18][19] In January 1610 Sir George More, patron of Compton church adjacent to Loseley Park, granted the advowson (right of presentation of the minister) to Oughtred, when it should next fall vacant, though Oughtred was not thereby empowered to present himself to the living.[20] This was soon after Sir George More became reconciled to the marriage of his daughter Anne to the poet John Donne, which had occurred secretly in 1601. Rector of Albury Oughtred was presented by Sir (Edward) Randall (lord of the manor) to the rectory of Albury, near Guildford in Surrey and instituted on 16 October 1610,[21] vacating Shalford on 18 January 1611.[22] In January 1615/16 Sir George More re-granted the advowson of Compton church (still occupied) in trust to Roger Heath and Simon Caryll, to present Oughtred himself, or any other person whom Oughtred should nominate, when the vacancy should arise.[23] Soon afterwards Oughtred was approached by John Tichborne seeking his own nomination, and entering an agreement to pay him a sum of money upon certain days. Before this could be completed the incumbent died (November 1618), and Oughtred sought for himself to be presented, preaching several times at Compton, having the first fruits sequestered to his use, and, after four months, asking the patron to present him. However, Tichborne offered to complete the agreed payment at once, and was accordingly presented by the trustees in May 1619 (Simon Caryll dying in that year): but before he could be admitted, the Crown interposed a different candidate because the contract between Oughtred and Tichborne was deemed by Sir Henry Yelverton clearly to be Simoniacal.[20] Oughtred therefore remained at Albury,[24] serving as rector there for fifty years.[25][26] William Lilly, that celebrated astrologer, knew Oughtred and claimed in his autobiography to have intervened on his behalf to prevent his ejection from his living by Parliament in 1646: "About this time, the most famous mathematician of all Europe, Mr. William Oughtred, parson of Aldbury in Surry, was in danger of sequestration by the Committee of or for plundered ministers; (Ambo-dexters they were;)[27] several inconsiderable articles were deposed and sworn against him, material enough to have sequestered him, but that, upon his day of hearing, I applied myself to Sir Bolstrode Whitlock, and all my own old friends, who in such numbers appeared in his behalf, that though the chairman and many other Presbyterian members were stiff against him, yet he was cleared by the major number."[28] Of his portrait (aged 73, 1646) engraved by Wenceslas Hollar, prefixed to the Clavis Mathematica, John Evelyn remarked that it "extreamly resembles him", and that it showed "that calm and placid Composure, which seemed to proceed from, and be the result of some happy ἕυρησις and Invention".[29] William Oughtred died at Albury in 1660, a month after the restoration of Charles II. A staunch supporter of the royalty, he is said to have died of joy at the knowledge of the return of the King. He was buried in Old St Peter and St Paul's Church, Albury.[30] Autobiographical information is contained in his address "To the English gentrie" in his Just Apologie of c. 1634.[31] Mathematician Oughtred developed his interest in mathematics early in life, and devoted whatever spare time his academic studies allowed him to it. Among the short tracts added to the 1647/48 editions of the Clavis Mathematica was one describing a natural and easy way of delineating sun-dials upon any surface, however positioned, which the author states he invented in his 23rd year (1597/98), which is to say, during his fellowship at King's College, Cambridge. His early preoccupation was to find a portable instrument or dial by which to find the hour, he tried various contrivances, but never to his satisfaction. "At last, considering that all manner of questions concerning the first motions were performed most properly by the Globe itself, rectified to the present elevation by the help of a moveable Azimuth; he projected the Globe upon the plane of the Horizon, and applied to it at the center, which was therein the Zenith, an Index with projected degrees, for the moveable Azimuth."[2] This projection answered his search, but then he had to invent theorems, problems and methods to calculate sections and intersections of large circles, which he could not find by instruments, not having access to any of sufficient size. In this way he drew out his findings, presenting one example to Bishop Thomas Bilson (who had ordained him), and another, in about 1606, to a certain noble lady, for whom he wrote notes for its use. In London, in spring 1618, Oughtred visited his friend Henry Briggs at Gresham College, and was introduced to Edmund Gunter, Reader in Astronomy, then occupying Dr Brooks's rooms. He showed Gunter his "Horizontall Instrument", who questioned him closely about it and spoke very approvingly. Soon afterwards Gunter sent him a print taken from a brass instrument made by Elias Allen, after Oughtred's written instructions (which Allen preserved).[2] When, in 1632, Richard Delamain the elder claimed that invention for himself,[32][33] William Robinson wrote to Oughtred: "I cannot but wonder at the indiscretion of Rich. Delamain, who being conscious to himself that he is but the pickpurse of another man's wit, would thus inconsiderately provoke and awake a sleeping lion..."[34] Around 1628 he was appointed by the Earl of Arundel to instruct his son William Howard in mathematics.[24] Some of Oughtred's mathematical correspondence survives, and is printed in Bayle's General Dictionary,[2] and (with some editorial omissions restored) in Dr Rigaud's Correspondence of Scientific Men.[35] William Alabaster wrote to him in 1633 to propose the quadrature of the circle by consideration of the fourth chapter of the Book of Ezekiel.[36] In 1634 he corresponded with the French architect François Derand, and (among others) with Sir Charles Cavendish (1635), Johannes Banfi Hunyades (1637), William Gascoigne (1640)[37] and Dr John Twysden, M.D. (1650).[38] Oughtred offered free mathematical tuition to pupils, among them Richard Delamain and Jonas Moore, and his teaching influenced a generation of mathematicians. Seth Ward resided with Oughtred for six months to learn contemporary mathematics, and the physician Charles Scarborough also stayed at Albury: John Wallis and Christopher Wren corresponded with him.[39] Another Albury pupil was Robert Wood, who helped him to see the Clavis through the press.[40] Isaac Newton's high opinion of Oughtred is expressed in his letter of 1694 to Nathaniel Hawes, where he quotes him extensively, calling him "a Man whose judgement (if any man's) may safely be relyed upon... that very good and judicious man, Mr Oughtred".[41] The first edition of John Wallis's foundational text on infinitesimal calculus, Arithmetica Infinitorum (1656), carries a long letter of dedication to William Oughtred.[42] Publications Clavis Mathematicæ (1631) William Oughtred's most important work was first published in 1631, in Latin, under the title Arithemeticæ in Numeris et Speciebus Institutio, quae tum Logisticæ, tum Analyticæ, atque adeus totius Mathematicæ quasi Clavis est (i.e. "The Foundation of Arithmetic in Numbers and Kinds, which is as it were the Key of the Logistic, then of the Analytic, and so of the whole Mathematic(s)"). It was dedicated to William Howard, son of Oughtred's patron Thomas Howard, 14th Earl of Arundel.[43] This is a textbook on elementary algebra. It begins with a discussion of the Hindu-Arabic notation of decimal fractions and later introduces multiplication and division sign abbreviations of decimal fractions. Oughtred also discussed two ways to perform long division and introduced the "~" symbol, in terms of mathematics, expressing the difference between two variables. Clavis Mathematicae became a classic, reprinted in several editions. It was used as a textbook by John Wallis and Isaac Newton among others. A concise work, it argued for a less verbose style in mathematics, and greater dependence on symbols. Drawing on François Viète (though not explicitly), Oughtred also innovated freely with symbols, introducing not only the multiplication sign as now used universally,[44] but also the proportion sign (double colon ::).[45] The first edition, 1631, contained 20 chapters and 88 pages including algebra and various fundamentals of mathematics.[46] The work was recast for the New Key, which appeared first in an English edition of 1647, The Key of the Mathematicks New Forged and Filed, dedicated to Sir Richard Onslow and to his son Arthur Onslow (son and grandson of Sir Edward), and then in a Latin edition of 1648, entitled Clavis Mathematica Denuo Limata, sive potius Fabricata (i.e. "The Mathematical Key Newly Filed, or rather Made"), in which the preface was removed and the book was reduced by one chapter. In the English Foreword, Oughtred explains that the intention had always been to provide the ingenious reader with an Ariadne's thread through the intricate labyrinth of these studies, but that his earlier, highly compressed style had been found difficult by some, and was now further elucidated.[47] These editions contained additional tracts on the resolution of adfected equations proposed in numbers, and other materials necessary for the use of decimal parts and logarithms, as well as his work on delineating sundials.[48] The last lifetime edition (third) was in 1652, and posthumous editions (as Clavis Mathematicæ: i.e. "The Key of Mathematic(s)") appeared in 1667 and 1693 (Latin), and in 1694 (English). The work gained popularity around 15 years after it first appeared, as mathematics took a greater role in higher education. Wallis wrote the introduction to his 1652 edition, and used it to publicise his skill as cryptographer;[49] in another, Oughtred promoted the talents of Wren. The Circles of Proportion and the Horizontal Instrument (1632) This work[50] was used by Oughtred in manuscript before it was edited for publication by his pupil, William Forster.[51] Here Oughtred introduced the abbreviations for trigonometric functions. It contains his description and instructions for the use of his important invention, the slide rule, a mechanical means of finding logarithmic results.[52] Two of Oughtred's students, William Forster and Richard Delamaine the elder, are concerned with the story of this book.[54] As instructor to the Earl of Arundel's son, Oughtred had the use of a room in Arundel House, the Earl's residence in the Strand, in London. He gave free instruction there to Richard Delamaine, whom he found to be too dependent on mathematical instruments to get a proper grasp of the theory behind them. Another student of his, Forster, who came to him as a beginner during the 1620s, was therefore taught without reference to instruments so that he should have a true grounding.[55] However, during the long vacation of 1630 Forster (who taught mathematics from a house in St Clement Danes churchyard, on the Westminster side of Temple Bar, in the same locality as Elias Allen's shop), while staying with Oughtred at Albury, asked him about Gunter's Ruler, and was shown two instruments used by his master, including Oughtred's circular slide rule.[56] Oughtred then said to Forster: "... the true way of Art is not by Instruments, but by Demonstration: and ... it is a preposterous course of vulgar Teachers, to begin with Instruments, and not with the Sciences, and so in-stead of Artists, to make their Schollers only doers of tricks, and as it were Iuglers: to the despite of Art, losse of precious time, and betraying of willing and industrious wits, unto ignorance, and idlenesse. ... the use of Instruments is indeed excellent, if a man be an Artist: but contemptible, being set and opposed to Art. And lastly, ... he meant to commend to me, the skill of Instruments, but first he would have me well instructed in the Sciences."[56] Forster obtained Oughtred's permission to translate, edit and publish the description, explanations and instructions which Oughtred had in manuscript, finishing his work in 1632.[56] Meanwhile Delamaine, who had also been shown the instruments, and had copied a text sent by Oughtred to his instrument-maker Elias Allen, was writing-up his own description and account. Delamaine came off the press first, in two separate tracts,[57] claiming himself to be the inventor, and dedicating the prior treatise to King Charles I. He went so far as to show his page-proofs to Oughtred as they were being prepared, and dismissed his objections,[55] printing some derogatory comments aimed at Forster and Oughtred in his Foreword. Forster, who dedicated The Circles of Proportion to the famous intellectual Sir Kenelm Digby, observed that another person had hastily anticipated Oughtred's publication.[56] It was left to Oughtred himself to publish his Just Apologie explaining the priority of his inventions and writings, and showing the behaviour of Delamaine.[55][58] It is stated in Cajori's book that John Napier was the first person ever to use to the decimal point and comma, but Bartholomaeus Pitiscus was really the first to do so.[59] Trigonometria, with Canones Sinuum (1657) Trigonometria, Hoc est, Modus Computandi Triangulorum Latera & Angulos was a collection compiled from Oughtred's papers by Richard Stokes and Arthur Haughton.[60] It contains about 36 pages of writing. Here the abbreviations for the trigonometric functions are explained in further detail consisting of mathematical tables.[7] It carries a frontispiece portrait of Oughtred similar to that by Wenceslas Hollar, but re-engraved by William Faithorne, and depicted as aged 83, and with a short epigram by "R.S." beneath. Longer verses addressed to Oughtred are prefixed by Christopher Wase. Opuscula Mathematica (1677) A miscellaneous collection of his hitherto unpublished mathematical papers (in Latin) was edited and published by his friend Sir Charles Scarborough in 1677.[61][62] The treatises contained are on these subjects: • Institutiones Mechanicæ. • De variis corporum generibus gravitate et magnitudine comparatis. • Automata • Quæstiones Diophanti Alexandrini Lib. 3 • De Triangulis planis rectangulis • De Divisione Superficierum • Musicæ Elementa • De Propugnaculorum Munitionibus • Sectiones Angulares Inventions Slide rule Oughtred's invention of the slide rule consisted of taking a single "rule", already known to Gunter, and simplifying the method of employing it. Gunter required the use of a pair of dividers to lay off distances on his rule; Oughtred made the step of sliding two rules past each other to achieve the same ends.[63] His original design of some time in the 1620s was for a circular slide rule; but he was not the first into print with this idea, which was published by Delamain in 1630. The conventional design of a sliding middle section for a linear rule was an invention of the 1650s.[64] Double horizontal sun dial At the age of 23, Oughtred invented the double horizontal sundial, now named the Oughtred type after him.[65] A short description The description and use of the double Horizontall Dyall (16 pp.) was added to a 1653 edition (in English translation) of the pioneer book on recreational mathematics, Récréations Mathématiques (1624) by Hendrik van Etten, a pseudonym of Jean Leurechon.[66] The translation itself is no longer attributed to Oughtred, but (probably) to Francis Malthus.[67] Universal equinoctial ring dial Oughtred also invented the Universal equinoctial ring dial.[68] Occult interests According to his contemporaries, Oughtred had interests in alchemy and astrology.[69] The Hermetic science remained a philosophical touchstone among many reputable scientists of his time, and his student Thomas Henshaw copied a Diary and "Practike" given to him by his teacher.[70] He was well-acquainted with the astrologer William Lilly who, as noted above, helped to prevent his ejection from his living in 1646. John Aubrey: Astrology and Geomancy John Aubrey states that (despite their political differences) Sir Richard Onslow, son of Sir Edward, also defended Oughtred against ejection in 1646. He adds that Oughtred was an astrologer, and successful in the use of natal astrology, and used to say that he did not know why it should be effective, but believed that some "genius" or "spirit" assisted. According to Aubrey, Elias Ashmole possessed the original copy in Oughtred's handwriting of his rational division of the twelve houses of the zodiac. Oughtred penned an approving testimonial, dated 16 October 1659, to the foot of the English abstract of The Cabal of the Twelve Houses Astrological by "Morinus" (Jean-Baptiste Morin) which George Wharton inserted in his Almanac for 1659.[71] Aubrey suggests that Oughtred was happy to allow the country people to believe that he was capable of conjuring. Aubrey himself had seen a copy of Christopher Cattan's work on Geomancy[72] annotated by Oughtred.[73] He reported that Oughtred had told Bishop Ward and Elias Ashmole that he had received sudden intuitions or solutions to problems when standing in particular places, or leaning against a particular oak or ash tree, "as if infused by a divine genius", after having pondered those problems unsuccessfully for months or years.[74] Elias Ashmole: Freemasonry Oughtred was well-known to Elias Ashmole, as Ashmole stated in a note to Lilly's autobiographical sketch: "This gentleman I was very well acquainted with, having lived at the house over-against his, at Aldbury in Surrey, three or four years. E.A."[28] The biography of Ashmole in the Biographia Britannica (1747)[75] called forth the supposition that Oughtred was a participant in Ashmole's admission to freemasonry in 1646. Friedrich Nicolai, in both sections of his Essay (on the Templar and Masonic Orders) of 1783, associated Oughtred, Lilly, Wharton and other Astrologers in the formation of the order of Free and Accepted Masons in Warrington and London.[76] The statement was reinforced through repetition by Thomas De Quincey,[77] and elaborated by Jean-Marie Ragon,[78] but was debunked in A.G. Mackey's History of Freemasonry (1906).[79] Ashmole noted that he paid a visit to "Mr. Oughtred, the famous mathematician", on 15 September 1654, about three weeks after the Astrologers' Feast of that year.[80] John Evelyn: Millenarianism Oughtred expressed millenarian views to John Evelyn in 1655: "Came that renowned mathematician, Mr. Oughtred, to see me, I sending my coach to bring him to Wotton, being now very aged. Among other discourse, he told me he thought water to be the philosopher's first matter, and that he was well persuaded of the possibility of their elixir; he believed the sun to be a material fire, the moon a continent, as appears by the late selenographers; he had strong apprehensions of some significant event to happen the following year, from the calculation of difference with the diluvian period; and added that it might possibly be to convert the Jews by our Saviour's visible appearance, or to judge the world; and therefore, his word was, Parate in occursum;[81] he said Original Sin was not met with in the Greek Fathers, yet he believed the thing; this was from some discourse on Dr. Taylor's late book, which I had lent him."[82] Oughtred Society Oughtred's name is remembered in the Oughtred Society, a group formed in the United States in 1991 for collectors of slide rules. It produces the twice-yearly Journal of the Oughtred Society and holds meetings and auctions for its members.[83][84] References 1. Smith, David Eugene (1923). History of Mathematics. Vol. 1. p. 392. ISBN 9780486204291. 2. 'Oughtred (William)', in P. Bayle, translated and revised by J.P. Bernard, T. Birch and J. Lockman, A General Dictionary, Historical and Critical, (James Bettenham, for G. Strachan and J. Clarke, London 1734/1739), Vol. VIII, pp. 77-86 (Google). 3. F. Willmoth, 'Oughtred, William (bap. 1575, d. 1660)', Oxford Dictionary of National Biography (2004); superseding J.B. Mullinger, 'Oughtred, William (1575-1660)', Dictionary of National Biography (1885-1900), vol. 42. 4. Smith, David E. (1958). History of Mathematics. Courier Corporation. p. 205. ISBN 9780486204307. 5. Florian Cajori (1919). A History of Mathematics. Macmillan. p. 157. cajori william-oughtred multiplication. 6. Wallis, P.J. (1968). "William Oughtred's 'Circles of Proportion' and 'Trigonometries'". Transactions of the Cambridge Bibliographical Society. 4 (5): 372–382. JSTOR 41154471. 7. F. Cajori, William Oughtred, a Great Seventeenth-Century Teacher of Mathematics (Open Court Publishing Company, Chicago 1916), pp. 12-14 (Internet Archive). 8. "Oughtred, William (OTRT592W)". A Cambridge Alumni Database. University of Cambridge. 9. 'Funeral ode by William Outhred', Surrey History Centre, ref. 6729/7/129 (Discovery Catalogue). 10. Church of England Clergy database, Liber Cleri detail, CCEd Record ID: 199392, from British Library Harleian MS 595. 11. Church of England Clergy database, Episcopal Register of Thomas Bilson (Winchester), Appointment Record ("Owtred") CCEd Record ID: 59030. 12. ODNB, and see Aubrey's Brief Lives, Ed. Oliver Lawson Dick (Ann Arbor, Michigan 1962), pp. 222–224. 13. J.J. O'Connor and E.F. Robertson, "Biographies: William Oughtred", MacTutor History of Mathematics archive, Last Update, 2017 (University of St Andrews). 14. London Metropolitan Archive ref. DW/PA/5/1619/22. 15. William "Owtred" also witnessed the first codicil (8 November 1620) to the Will of George Austen of Shalford (P.C.C. 1621, Dale quire). 16. W. Bruce Bannerman (ed.), The Visitations of the county of Surrey taken in the years 1530, 1572 and 1623, Harleian Society Vol. XLIII (1899), pp. 88–89 (Internet Archive). 17. Cf. Will of John Machell, Gentleman of Wonersh (P.C.C. 1647, Fine quire); Will of Elizabeth Machell of Wonersh (P.C.C. 1650, Pembroke quire). 18. E.W. Brayley, J. Britton and E.W Brayley jnr., A Topographical History of Surrey (G. Willis, London 1850), vol. II, p. 49 (Google). 19. Gerald is sometimes referred to as "Gerard", but the name derives from his mother's family of Fitzgerald, seated at Hatchlands Park in East Clandon. 20. "Yelverton, Sir Henry. Opinion that simony was involved in the contract between William Oughtred and John Tichborne for presentation to the church of Compton in Surrey", Papers of the More family of Loseley Park, Surrey, 1488-1682 (bulk 1538-1630), Surrey History Centre, ref. Z/407/Lb.668.4 (Discovery Catalogue). View original document at Folger Shakespeare Library. 21. Church of England Clergy database, Episcopal Register of Thomas Bilson (Winchester), Appointment Record, CCEd Record ID: 59103. The advowson was in the lord of the manor, who was Sir Edward. 22. Church of England Clergy database, Episcopal Register of Thomas Bilson (Winchester), Vacancy Evidence Record ("Outhred"), CCEd Record ID: 59115 23. This was just one year after the ordination of John Donne (23 January 1615) and 6 months before his institution as rector of Sevenoaks (12 July 1616). The Clergy database, CCEd Ordination record ID 119823, and Appointment Evidence Record ID 114560. 24. Chisholm, Hugh, ed. (1911). "Oughtred, William" . Encyclopædia Britannica. Vol. 20 (11th ed.). Cambridge University Press. p. 378. 25. J. and J.A. Venn, Alumni Cantabrigienses Part 1 Vol. III (Cambridge University Press 1924), p. 288 (Internet Archive) (appointment 1610). 26. "Parishes: Albury", in H.E. Malden (ed.), A History of the County of Surrey, Volume 3 (V.C.H./HMSO, London 1911), pp. 72-77 (British History Online): "was rector from 1610 to 1660". 27. "Ambo-dexters", in the figurative meaning, i.e. their allegiances swayed according to their own advantage. 28. William Lilly's History of his Life and Times, from the year 1602 to 1681 (Published London 1715), Reprint (Charles Baldwyn, London 1822), pp. 135-37 (Internet Archive). 29. J. Evelyn, Numismata: A Disccourse of Medals, Ancient and Modern... To which is added, A Discourse concerning Physiognomy (Benjamin Tooke, London 1697), p. 341 (Google). 30. "Parishes: Albury", in H.E. Malden (ed.), A History of the County of Surrey, Volume 3 (V.C.H./HMSO, London 1911), pp. 72-77 (British History Online, accessed 6 December 2018). 31. (W. Oughtred), To the English gentrie, and all others studious of the mathematicks which shall bee readers hereof. The just apologie of Wil: Oughtred, against the slaunderous insimulations of Richard Delamain, in a pamphlet called Grammelogia, or the Mathematicall Ring, or Mirifica Logarithmorum Projectio Circularis (A. Mathewes, London ?1634). Full text at Umich/eebo. Extracts in F. Cajori (1915) (Further reading). 32. M. Selinger, Teaching Mathematics (1994), p. 142. 33. "The Galileo Project". Galileo.rice.edu. Retrieved 31 October 2012. 34. 'VII: W. Robinson to Oughtred', in S.J. Rigaud (ed.), Correspondence of Scientific Men of the Seventeenth Century, 2 vols (University Press, Oxford 1841), I pp. 11-14 (Google). 35. Letters II-XXXVI, in S.J. Rigaud (ed.), Correspondence of Scientific Men of the Seventeenth Century, 2 vols (University Press, Oxford 1841), I, pp. 3-92 (Google). 36. Bayle, General Dictionary, VIII, p. 80, note, col. b (Google). 37. "DSpace at Cambridge: Letter from William Gascoigne to William Oughtred". Dspace.cam.ac.uk. 13 June 2007. Retrieved 31 October 2012. {{cite journal}}: Cite journal requires |journal= (help) 38. "Janus: Oughtred, William (? 1574-1660) mathematician". Janus.lib.cam.ac.uk. Retrieved 31 October 2012. 39. H.M. Pycior, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra Through the Commentaries on Newton's Universal Arithmetick (1997), p. 42. 40. T.C. Barnard, Cromwellian Ireland: English Government and Reform in Ireland 1649–1660 (2000), p. 223. 41. 'Appendix. No. XXV. Newton to Hawes', in J. Edleston (ed.), Correspondence of Sir Isaac Newton and Professor Cotes, Including Letters of Other Eminent Men (John W. Parker, London/John Deighton, Cambridge 1850), pp. 279-92, at pp. 291-92 (Google). 42. J. Wallis, Arithmetica Infinitorum, sive Nova Methodus Inquirendi in Curvisineorum Quadraturam (Typis Leon: Lichfield Academiae Typographi, Impensis Tho. Robinson, Oxford 1656), unpaginated front matter (Internet Archive). 43. (Londini, apud Thomam Harperum, 1631): see full page-views at Google. 44. F. Cajori, 'The cross X as a symbol for multiplication', Nature, Vol. XCIV (1914), Abstract, pp. 363-64 (journal's webpage). 45. Helena Mary Pycior, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra Through the Commentaries on Newton's Universal Arithmetick (1997), p. 48. 46. Cajori, Florian (1915). "The Works of William Oughtred". The Monist. 25 (3): 441–466. doi:10.5840/monist191525315. JSTOR 27900548. 47. Bayle, 'Oughtred (William)', General Dictionary, VIII, p. 78, note col. a (Google). 48. G.[W.] Oughtred, Clavis Mathematica Denuo Limata, sive potius Fabricata (Londini, Excudebat Thomas Harper, sumptibus Thome Whitakeri, apud quem venales sunt in Cœmiterio D. Pauli, 1648); Full page views at Google. 49. "Oxford Figures, Chapter 1: 800 years of mathematical traditions". Mathematical Institute – University of Oxford. 17 September 2007. Archived from the original on 26 October 2012. Retrieved 31 October 2012. 50. W. Oughtred, ed. W. Forster, The Circles of Proportion and the Horizontall Instrument. The former shewing the maner how to work proportions both simple and compound: and the ready and easy resolving of quæstions both in arithmetic, geometrie, & astronomie: and is newly increased with an additament for navigation. All which rules may also be wrought with the penne by arithmetic, and the canon of triangles. The latter teaching how to work most quæstions, which may be performed by the globe: and to delineat dialls upon any kind of plaine. Both invented, and the uses of both written in Latine by Mr. W. O. Translated into English and set forth for the publique benefit by William Forster First issue (Printed for Elias Allen maker of these and all other Mathematical Instruments and are to be sold at his shop over against St Clements church without Temple Bar, London 1632); Second issue (Printed by Augustine Mathewes, and are to bee sold by Nic: Bourne at the Royall Exchange, London 1633), full text at Umich/eebo. 51. Stephen, Leslie, ed. (1889). "Forster, William (fl.1632)" . Dictionary of National Biography. Vol. 20. London: Smith, Elder & Co. 52. Ball, W. W. Rouse (1917). "Review of William Oughtred: a great Seventeenth-century Teacher of Mathematics". Science Progress (1916-1919). 11 (44): 694–695. JSTOR 43426914. 53. The captions "Versus Septentrionem", "versus Meridiem", suggest the opposite orientation, but the river Thames (to the south) is clearly visible in the "Septentrio" scene. 54. A.J. Turner, 'William Oughtred, Richard Delamain and the Horizontal Instrument in seventeenth-century England', Annali dell'Istituto e Museo di storia della scienza di Firenze vol. 6 pt. 2 (1981), pp. 99-125. 55. (W. Oughtred), To the English gentrie, and all others studious of the mathematicks which shall bee readers hereof. The Just Apologie of Wil: Oughtred, against the slaunderous insimulations of Richard Delamain, in a pamphlet called Grammelogia, or The Mathematicall Ring, or Mirifica Logarithmorum Projectio Circularis (A. Mathewes, London ?1634). Full text at Umich/eebo. 56. "To the Honourable and Renowned for vertue, learning, and true valour, Sir Kenelme Digbye, Knight (dated 1632)", in W. Oughtred, ed. W. Forster, The Circles of Proportion and the Horizontall Instrvment (1632, second issue 1633), unpaginated front matter; also here (Umich/eebo). 57. R. Delamaine, ''Grammelogia, or, The mathematicall ring extracted from the logarythmes, and projected circular (printed 1631), full text at Umich/eebo: R. Delamaine, The Making, Description, and Use of a small portable Instrument called a Horizontall Quadrant (London: Printed [by Thomas Cotes] for Richard Hawkins and are to be sold at his shop in Chancery lane neere Sarjants Inne, 1632), full text at Umich/eebo. 58. F.J. Swetz, 'Mathematical Treasure: Oughtred's Defense of His Slide Rule', Convergence (Online Periodical of the Mathematical Association of America), August 2018 (MAA). 59. L.C. Karpinsky, 'Review: William Oughtred, a Great Seventeenth-Century Teacher of Mathematics, by F. Cajori', The American Mathematical Monthly, Vol. 24 part 1, pp. 29-30 (jstor, open pdf). Cajori is also chastised for his having mis-spelled the name of Erasmus O. Schreckenfuchs. 60. W. Oughtred, ed. R. Stokes and A. Haughton, Trigonometria, Hoc est, Modus Computandi Triangulorum Latera & Angulos Ex Canone Mathematico traditus et demonstratus... Una cum Tabulis Sinuum, Tangent & Secant, &c. (Londini, Typis R. & L.W. Leybourn, Impensis Thomæ Johnson, apud quem væneunt sub signo Clavis Aureæ in Cœmiterio S. Pauli, 1657); full page views at Internet Archive. 61. 'William Oughtred', in O. Manning and Bray, The History and Antiquities of the County of Surrey (John White/John Nichols and Son, London 1804-14), II, pp. 132-33 (Google). 62. W. Oughtred, ed. C. Scarborough, Gulielmi Oughtredi Ætonensis, quondam Collegii Regalis Cantabrigia Socii, Opuscula Mathematica hactenus inedita (E Theatro Sheldoniano, Oxford 1677); Full pageviews at Google. 63. "Slide Rules". Hpmuseum.org. Retrieved 31 October 2012. 64. "The slide rule – a forgotten tool". Powerhouse Museum Collection. Retrieved 31 October 2012. 65. "Harvard University – Department of History of Science". Dssmhi1.fas.harvard.edu. Archived from the original on 20 February 2012. Retrieved 31 October 2012. 66. W. Oughtred, 'The Description, and use of the double Horizontall Diall', in H. van Etten, Mathematicall recreations. Or, A collection of many problemes, extracted out of the ancient and modern philosophers (William Leake, London 1653), unnumbered pages, full text at Umich/eebo. 67. Heefer, Albrecht. "Récréations Mathématiques (1624) A Study on its Authorship, Sources and Influence" (PDF). logica.ugent.be. 68. "Royal Museums Greenwich". 69. Keith Thomas, Religion and the Decline of Magic (1973), p. 322 and 452n. 70. D.R. Dickson, 'Thomas Henshaw and Sir Robert Paston's pursuit of the Red Elixir: an early collaboration between Fellows of the Royal Society', Notes and Records of the Royal Society of London, Vol. 51, No. 1 (Jan. 1997), pp. 57-76, at pp. 67-72. 71. 'The Cabal of the Twelve Houses Astrological', collected in J. Gadbury (ed.), The Works of that Late Most Excellent Philosopher and Astronomer, Sir George Wharton, bar. collected into one volume (M.H. for John Leigh, London 1683), pp. 189-208, at p. 208 (Google). 72. La Geomance du Seigneur Christofe de Cattan, Gentilhomme Genevoys. Livre non moins plaisant et recreatif. Avec la roüe de Pythagoras (Gilles Gilles, Paris 1558). Full text (page views) at Internet Archive. 73. Oughtred may have possessed the English translation by Francis Sparry, The Geomancie of Maister Christopher Catton, a Booke no lesse pleasant and recreative, then of a wittie invention (London 1591). 74. 'William Oughtred, 1575-1660', in R. Barber (ed.), John Aubrey - Brief Lives: A selection based upon existing contemporary portraits (Folio Society, London 1975), 232-37. 75. W. Oldys (ed.), Biographia Britannica: or, the Lives of the most eminent persons who have flourished in Great Britain and Ireland, 6 vols (W. Innys (etc.), London 1747-1766), I, pp. 223-36, at p. 224, note E, 'Collections relating to the history of Free-Masons', and pp. 228-29 (Google). 76. C.F. Nicolai, Versuch über die Beschuldigungen welche dem Tempelherrenorden gemacht worden, und über dessen Geheimniß: Nebst einem Anhange über das Entstehen der Freymaurergesellschaft (Nicolai, Berlin und Stettin 1782), Theil I, at p. 188; Theil II, pp. 191-196 (Google). 77. T. De Quincey, 'Historico-Critical Inquiry into the Origins of the Rosicrucians and the Free-Masons', in D. Masson (ed.), The Collected Writings of Thomas De Quincey, New and Enlarged Edition, Vol. XIII: Tales and Prose Phantasies (Adam and Charles Black, Edinburgh 1890), at pp. 425-26 (Google). 78. J.M. Ragon, Orthodoxie maçonnique: suivie de la Maçonnerie occulte et de l'initiation hermétique (E. Dentu, Paris 1853), pp. 28-33, pp. 99-108, and passim (Google). 79. A.G. Mackey, The History of Freemasonry, 2 volumes (The Masonic History Company, New York and London 1906), II, pp. 306, 316-18 (Internet Archive). 80. 'The Life of Elias Ashmole, Esq.', in The Lives of Those Eminent Antiquaries Elias Ashmole, Esquire, and Mr William Lilly, Written by Themselves (T. Davies, London 1774), at p. 321 (Google). 81. I.e. "Praeparare in occursum Dei tui, Israel" (Book of Amos, Chapter IV, v. 12): "Prepare to meet thy God, O Israel". 82. 'Entry for 28 August 1655', in W. Bray (ed.), The Diary of John Evelyn, with a Biographical introduction by the editor, and a special introduction by Richard Garnett, LL.D., 2 vols (M. Walter Dunne, New York and London 1901), I, pp. 305-06 (Internet Archive, Retrieved 5 December 2018). 83. "The Oughtred Society". The Oughtred Society. Retrieved 18 March 2015. 84. "Brochure" (PDF). The Oughtred Society. Retrieved 18 March 2015. Further reading • Cajori, Florian (1916). William Oughtred: A Great Seventeenth-Century Teacher of Mathematics. The Open Court Publishing Company. • Florian Cajori (1915), "The Life of William Oughtred", The Open Court, Vol. XXIX no. 8 (Chicago, August 1915), p. 711, pp. 449-59 (pdf) • Jacqueline Anne Stedall, Ariadne's Thread: The Life and Times of Oughtred's Clavis, Annals of Science, Volume 57, Issue 1 January 2000, pp. 27–60. doi:10.1080/000337900296290 External links • Media related to William Oughtred at Wikimedia Commons • O'Connor, John J.; Robertson, Edmund F., "William Oughtred", MacTutor History of Mathematics Archive, University of St Andrews • Galileo Project page • The Oughtred Society inspired by Oughtred and dedicated to the history and preservation of slide rules. • Answers.com article with additional material on Oughtred. • Account of Oughtred by John Aubrey • William Oughtred's "Key of the Mathematics" (John Salusbury's English translation of Oughtred's "Clavis Mathematicae"). Authority control International • FAST • ISNI • VIAF National • Germany • Italy • Israel • Belgium • United States • Czech Republic • Korea • Netherlands • Portugal • Vatican Academics • MathSciNet • zbMATH People • Deutsche Biographie Other • SNAC • IdRef
Wikipedia
Williams's p + 1 algorithm In computational number theory, Williams's p + 1 algorithm is an integer factorization algorithm, one of the family of algebraic-group factorisation algorithms. It was invented by Hugh C. Williams in 1982. It works well if the number N to be factored contains one or more prime factors p such that p + 1 is smooth, i.e. p + 1 contains only small factors. It uses Lucas sequences to perform exponentiation in a quadratic field. It is analogous to Pollard's p − 1 algorithm. Algorithm Choose some integer A greater than 2 which characterizes the Lucas sequence: $V_{0}=2,V_{1}=A,V_{j}=AV_{j-1}-V_{j-2}$ where all operations are performed modulo N. Then any odd prime p divides $\gcd(N,V_{M}-2)$ whenever M is a multiple of $p-(D/p)$, where $D=A^{2}-4$ and $(D/p)$ is the Jacobi symbol. We require that $(D/p)=-1$, that is, D should be a quadratic non-residue modulo p. But as we don't know p beforehand, more than one value of A may be required before finding a solution. If $(D/p)=+1$, this algorithm degenerates into a slow version of Pollard's p − 1 algorithm. So, for different values of M we calculate $\gcd(N,V_{M}-2)$, and when the result is not equal to 1 or to N, we have found a non-trivial factor of N. The values of M used are successive factorials, and $V_{M}$ is the M-th value of the sequence characterized by $V_{M-1}$. To find the M-th element V of the sequence characterized by B, we proceed in a manner similar to left-to-right exponentiation: x := B y := (B ^ 2 − 2) mod N for each bit of M to the right of the most significant bit do if the bit is 1 then x := (x × y − B) mod N y := (y ^ 2 − 2) mod N else y := (x × y − B) mod N x := (x ^ 2 − 2) mod N V := x Example With N=112729 and A=5, successive values of $V_{M}$ are: V1 of seq(5) = V1! of seq(5) = 5 V2 of seq(5) = V2! of seq(5) = 23 V3 of seq(23) = V3! of seq(5) = 12098 V4 of seq(12098) = V4! of seq(5) = 87680 V5 of seq(87680) = V5! of seq(5) = 53242 V6 of seq(53242) = V6! of seq(5) = 27666 V7 of seq(27666) = V7! of seq(5) = 110229. At this point, gcd(110229-2,112729) = 139, so 139 is a non-trivial factor of 112729. Notice that p+1 = 140 = 22 × 5 × 7. The number 7! is the lowest factorial which is multiple of 140, so the proper factor 139 is found in this step. Using another initial value, say A = 9, we get: V1 of seq(9) = V1! of seq(9) = 9 V2 of seq(9) = V2! of seq(9) = 79 V3 of seq(79) = V3! of seq(9) = 41886 V4 of seq(41886) = V4! of seq(9) = 79378 V5 of seq(79378) = V5! of seq(9) = 1934 V6 of seq(1934) = V6! of seq(9) = 10582 V7 of seq(10582) = V7! of seq(9) = 84241 V8 of seq(84241) = V8! of seq(9) = 93973 V9 of seq(93973) = V9! of seq(9) = 91645. At this point gcd(91645-2,112729) = 811, so 811 is a non-trivial factor of 112729. Notice that p−1 = 810 = 2 × 5 × 34. The number 9! is the lowest factorial which is multiple of 810, so the proper factor 811 is found in this step. The factor 139 is not found this time because p−1 = 138 = 2 × 3 × 23 which is not a divisor of 9! As can be seen in these examples we do not know in advance whether the prime that will be found has a smooth p+1 or p−1. Generalization Based on Pollard's p − 1 and Williams's p+1 factoring algorithms, Eric Bach and Jeffrey Shallit developed techniques to factor n efficiently provided that it has a prime factor p such that any kth cyclotomic polynomial Φk(p) is smooth.[1] The first few cyclotomic polynomials are given by the sequence Φ1(p) = p−1, Φ2(p) = p+1, Φ3(p) = p2+p+1, and Φ4(p) = p2+1. References 1. Bach, Eric; Shallit, Jeffrey (1989). "Factoring with Cyclotomic Polynomials" (PDF). Mathematics of Computation. American Mathematical Society. 52 (185): 201–219. doi:10.1090/S0025-5718-1989-0947467-1. JSTOR 2008664. • Williams, H. C. (1982), "A p+1 method of factoring", Mathematics of Computation, 39 (159): 225–234, doi:10.2307/2007633, JSTOR 2007633, MR 0658227 External links • P + 1 factorization method at Prime Wiki Number-theoretic algorithms Primality tests • AKS • APR • Baillie–PSW • Elliptic curve • Pocklington • Fermat • Lucas • Lucas–Lehmer • Lucas–Lehmer–Riesel • Proth's theorem • Pépin's • Quadratic Frobenius • Solovay–Strassen • Miller–Rabin Prime-generating • Sieve of Atkin • Sieve of Eratosthenes • Sieve of Pritchard • Sieve of Sundaram • Wheel factorization Integer factorization • Continued fraction (CFRAC) • Dixon's • Lenstra elliptic curve (ECM) • Euler's • Pollard's rho • p − 1 • p + 1 • Quadratic sieve (QS) • General number field sieve (GNFS) • Special number field sieve (SNFS) • Rational sieve • Fermat's • Shanks's square forms • Trial division • Shor's Multiplication • Ancient Egyptian • Long • Karatsuba • Toom–Cook • Schönhage–Strassen • Fürer's Euclidean division • Binary • Chunking • Fourier • Goldschmidt • Newton-Raphson • Long • Short • SRT Discrete logarithm • Baby-step giant-step • Pollard rho • Pollard kangaroo • Pohlig–Hellman • Index calculus • Function field sieve Greatest common divisor • Binary • Euclidean • Extended Euclidean • Lehmer's Modular square root • Cipolla • Pocklington's • Tonelli–Shanks • Berlekamp • Kunerth Other algorithms • Chakravala • Cornacchia • Exponentiation by squaring • Integer square root • Integer relation (LLL; KZ) • Modular exponentiation • Montgomery reduction • Schoof • Trachtenberg system • Italics indicate that algorithm is for numbers of special forms
Wikipedia
William Ruggles William Ruggles (September 5, 1797 – September 10, 1877) was a professor at George Washington University.[1][2] Biography William Ruggles was born in Rochester, Massachusetts, about fifty miles south of present-day Boston, on Tuesday September 5, 1797. He was the son of Elisha Ruggles and Mary Clap who also parented six other children: Nathaniel, Micah, Henry, Charles, James, and Lucy. William was the second youngest child in his family. Not much is known about his childhood growing up in Massachusetts until he enrolled in Brown University; where he later graduated from, at the age of twenty-three, in the class of 1820. Two years after graduating from Brown University, Ruggles became a tutor at Columbian College. On February 9, 1821 Congress chartered Columbian College, a nonsectarian school but with Baptist sponsorship that would not become the George Washington University until January 23, 1904. In 1824, two years after he became a tutor, Ruggles became a Professor of Mathematics and Natural Philosophy, a position he would retain until 1865. In 1827 William Ruggles became the chair of both mathematics and natural philosophy. In 1865 Ruggles was made professor emeritus. He continued to lecture on political economy and civil polity from 1865 to 1874. He died at Schooley's Mountain, New Jersey, on Monday, September 10, 1877, five days after his eightieth birthday.[1][2] Legacy Ruggles was a man of many admirable characteristics including loyalty, conscientiousness, and morality, which are shown not only through the words of those who describe him but through his actions as well. The best example of William Ruggles' loyalty was shown in his dedication to the Columbian College itself. Until 1985, no person had a longer stay at the George Washington University. Ruggles came to the George Washington University in 1822 and stayed until his death, a record at the time, of fifty-five and a half years, which has only been surpassed once by Elmer Louis Kayser who came to the university as a student in 1914 and stayed until his eventual death in 1985. William Ruggles was involved at the Columbian College (George Washington University) from almost the beginning of the Columbian College. He lived through the drab years of 1826 through 1842 where mounting debt and pressure threatened to shut down the Columbian College and rejoiced harder than anyone when that debt was lifted. He served under the first six presidents of the institution and even served as acting president three times. He saw a large student body dwindle down to a handful of students due to the US civil war. Since the college buildings were being used as a hospital in the war efforts, classes were taught in the homes of professors. Lastly, Professor Ruggles had the enjoyment of being a contemporary to University President Welling. Ruggles was the owner of over thirty historical and archival books which are now located in the Gelman Library Archives at George Washington University. The subjects include theology, philosophy, psychology, economics, mathematics (algebra and geometry), and chemistry, all of which Ruggles either taught or was passionate about. Ruggles was described by President Welling, the second president of the George Washington University: When the Board was tardy in paying salaries or when it embarked on some policy he opposed, his resignation was always forthcoming .... A member of no religious denomination, but dealing almost exclusively with Baptists, he observed on the basis of attitudes, that he was perhaps the better Christian .... [he was] a man of great conscientiousness, high intelligence and blameless character. His excellent portrait in the University collection suggests a very wise man who perhaps was not always loved but who was respected. — President Welling The Board of Trustees, in adopting resolutions in appreciation of his services, declared Ruggles in similar words to Welling, saying, "We hereby testify and record our exulted sense of the virtues which adorned his private character, the unselfish zeal he brought to the performance of all his duties and the inestimable value of the manifold and multiform services which he rendered to the College during the long period of his connection with its history." William Ruggles was noted for his generous contributions to charities and missionary bequests. His accomplishments and contributions were honored and recognized when he received an honorary LL.D. from Brown University in 1852, the same school that he graduated from more than thirty years prior Upon reviewing letters that Ruggles had written, it can be seen that he was full of charity towards his students. While he may be described as someone who was "not always loved but who was always respected," it can be seen that Ruggles did have great affection towards the students he taught. On November 24, 1837, Ruggles wrote a letter to Joel R. Poinsett about the character of a young man named John D. Kuntz, who was expecting to make appointment as a cadet at West Point Military Academy. In the letter Ruggles goes on to praise the young boy and even go as far as to call him, "a young man of sound principles." William Ruggles' folders were filled with many such letters, expressing his kind opinions on the young students that he respected and cared for. Although it is noted that Ruggles was a man of spirit, he was a member of no religion despite the fact that he dealt almost exclusively with the Baptist church. Ruggles shared very many correspondences with Reverend Elon Galusha with letters dating from November 1825 all the way until April 1832, while it is unclear what the two wrote about it clear that these letters dealt with Ruggles faith. In a letter preserved to a different Reverend, Ruggles states that, "Oh, whatever else is taken from me, may I have a share in the great inheritance purchased by Jesus Christ for those who love him and are regenerated by the Holy Spirit!" Although William Ruggles died 1877, a reprint of his obituary appeared in the Faculty Newsletter volume 2, number 1, Spring 1965 as part of a series of called GW Footnotes. It was titled the 55-Year Professor and was the first in a series of anecdotes from the university's past written by the university historian. Ruggles Prizes The Ruggles Prizes are awarded annually for excellence in mathematics to a candidate for a bachelor's degree. The prizes were initiated in 1859, and consists of two gold medals. They are awarded "upon examination to the best two scholars in Mathematics."[3] References 1. Elmer Louis Kayser (1970). Bricks without straw. Appleton-Century-Crofts. At the end of 1873, Professor William Ruggles asked to be relieved of the duties of his chair and they were ... 1877, at Schooley's Mountain, New Jersey 2. Lyle Slovick (2006-12-11). "William Ruggles". The GW and Foggy Bottom Historical Encyclopedia. Retrieved 2008-05-13. Born in Rochester, Massachusetts, September 5, 1797, he died at Schooley's Mountain, New Jersey, September 10, 1877. Ruggles was connected with the College and University for fifty-five years. A graduate of Brown in the class of 1820, he became a tutor in Columbian College two years later and Professor of Mathematics and Natural Philosophy from 1824 to 1865 when he was made Professor Emeritus in those fields. 3. George Washington University Bulletin. George Washington University. 1905. The Ruggles Prizes, for excellence in Mathematics, founded by Professor William Ruggles, LL. D., consist of two gold medals, annually awarded upon examination to the best two scholars in Mathematics. ... Authority control International • FAST • VIAF National • United States Other • SNAC
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William Wales (astronomer) William Wales (1734? – 29 December 1798) was a British mathematician and astronomer who sailed on Captain Cook's second voyage of discovery, then became Master of the Royal Mathematical School at Christ's Hospital and a Fellow of the Royal Society. Early life Wales was born around 1734 to John and Sarah Wales and was baptised in Warmfield (near the West Yorkshire town of Wakefield) that year. As a youth, according to the historian John Cawte Beaglehole, Wales travelled south in the company of a Mr Holroyd, who became a plumber in the service of George III.[1] By the mid-1760s, Wales was contributing to The Ladies' Diary. In 1765 he married Mary Green, sister of the astronomer Charles Green.[1] In 1765, Wales was employed by the Astronomer Royal Nevil Maskelyne as a computer, calculating ephemerides that could be used to establish the longitude of a ship, for Maskelyne's Nautical Almanac.[2] 1769 transit of Venus and wintering at Hudson Bay As part of the plans of the Royal Society to make observations of the June 1769 transit of Venus, which would lead to an accurate determination of the astronomical unit (the distance between the Earth and the Sun), Wales and an assistant, Joseph Dymond, were sent to Prince of Wales Fort on Hudson Bay to observe the transit,[3] with the pair being offered a reward of £200 for a successful conclusion to their expedition.[1] Other Royal Society expeditions associated with the 1769 transit were Cook's first voyage to the Pacific, with observations of the transit being made at Tahiti, and the expedition of Jeremiah Dixon and William Bayly to Norway. Due to winter pack ice making the journey impossible during the winter months, Wales and Dymond were obliged to begin their journey in the summer of 1768, setting sail on 23 June. Ironically, Wales when volunteering to make a journey to observe the transit, had requested that he be sent to a more hospitable location.[4] The party arrived at Prince of Wales Fort in August 1768.[5] Due to the scarcity of building materials at the chosen site, the party had to bring not only astronomical instruments, but the materials required for the construction of living quarters.[5] On their arrival, the pair constructed two "Portable Observatories", which had been designed by the engineer John Smeaton.[6] Construction work occupied the pair for a month and then they settled in for the long winter season. When the day of the transit, 3 June 1769, finally arrived, the pair were lucky to have a reasonably clear day and they were able to observe the transit at around local midday. However, the two astronomers' results for the time of first contact, when Venus first appeared to cross the disc of the Sun, differed by 11 seconds; the discrepancy was to prove a cause of upset for Wales.[4] They were to stay in Canada for another three months before making the return voyage to England, thus becoming the first scientists to spend the winter at Hudson Bay.[7] On his return, Wales was still upset by the difference in the observations and refused to present his findings to the Royal Society until March 1770; however, his report of the expedition, including the astronomical results as well as other climatic and botanical observations, met with approval and he was invited by James Cook to join his next expedition.[4] Captain Cook's second circumnavigation voyage Wales and William Bayly were appointed by the Board of Longitude to accompany James Cook on his second voyage of 1772–75,[3] with Wales accompanying Cook aboard the Resolution. Wales' brother-in-law Charles Green, had been the astronomer appointed by the Royal Society to observe the 1769 transit of Venus but had died during the return leg of Cook's first voyage.[8] The primary objective of Wales and Bayly was to test Larcum Kendall's K1 chronometer, based on the H4 of John Harrison.[8] Wales compiled a log book of the voyage, recording locations and conditions, the use and testing of the instruments entrusted to him, as well as making many observations of the people and places encountered on the voyage.[9] Later life Following his return, Wales was commissioned in 1778 to write the official astronomical account of Cook's first voyage.[10] Wales became Master of the Royal Mathematical School at Christ's Hospital and was elected a Fellow of the Royal Society in 1776.[2][7] Amongst Wales' pupils at Christ's Hospital were Samuel Taylor Coleridge and Charles Lamb.[5] It has been suggested that Wales' accounts of his journeys might have influenced Coleridge when writing his poem The Rime of the Ancient Mariner.[11] The writer Leigh Hunt, another of Wales' pupils, remembered him as "a good man, of plain simple manners, with a heavy large person and a benign countenance".[12] He was appointed as Secretary of the Board of Longitude in 1795, serving in that position until his death in 1798.[10][13] He was nominated by the First Lord of the Admiralty, Earl Spencer, and his appointment confirmed 5 December 1795.[14] Recognition of his work During his voyage of 1791–95, George Vancouver, who had studied astronomy under Wales as a midshipman on HMS Resolution during Cook's second circumnavigation, named Wales Point, a cape at the entrance to Portland Inlet on the coast of British Columbia, in honour of his tutor; the name was later applied to the nearby Wales Island by an official at the British Hydrographic Office.[15] In his journal, Vancouver recorded his gratitude and indebtedness to Wales's tutelage "for that information which has enabled me to traverse and delineate these lonely regions."[16] Wales featured on a New Hebrides (now Vanuatu) postage stamp of 1974 commemorating the 200th anniversary of Cook's discovery of the islands.[8] The asteroid 15045 Walesdymond, discovered in 1998, was named after Wales and Dymond.[17] Works by William Wales • "Journal of a voyage, made by order of the Royal Society, to Churchill River, on the North-west Coast of Hudson's Bay". Philosophical Transactions of the Royal Society of London. 60: 109–136. 1771. • The Method of Finding the Longitude by timekeepers London: 1794. See also • European and American voyages of scientific exploration • Wales, Wendy (2015). Captain Cook’s Computer: the life of William Wales, F.R.S. (1734-1798). Hame House. ISBN 978-09933758-0-4. Notes 1. Wendy Wales. "William Wales' First Voyage". Cook's Log. Captain Cook Society. Retrieved 10 September 2009. 2. Mary Croarken (September 2002). "Providing longitude for all – The eighteenth-century computers of the Nautical Almanac". Journal for Maritime Research. Retrieved 6 August 2009. 3. "William Wales". State Library of New South Wales. Retrieved 6 August 2009. 4. Hudon, Daniel (February 2004). "A (Not So) Brief History of the Transits of Venus". Journal of the Royal Astronomical Society of Canada. 98 (1): 11–13. Retrieved 18 February 2022. 5. Fernie, J. Donald (September–October 1998). "Transits, Travels and Tribulations, IV: Life on the High Arctic". American Scientist. 86 (5): 422. doi:10.1511/1998.37.3396. 6. Steven van Roode. "Historical observations of the transit of Venus". Retrieved 10 August 2009. 7. Glyndwr Williams. "Wales, William". Dictionary of Canadian Biography Online. Retrieved 6 August 2009. 8. "William Wales". Ian Ridpath. Retrieved 6 August 2009. 9. Wales, William. "Log book of HMS 'Resolution'". Cambridge Digital Library. Retrieved 28 May 2013. 10. Orchison, Wayne (2007). Hockey, Thomas A. (ed.). The Biographical Encyclopedia of Astronomers: A-L. p. 1189. ISBN 978-0-387-31022-0. 11. Christopher Ondaatje (15 March 2002). "From Fu Man Chu to a grizzly end". Times Higher Education. Retrieved 11 August 2009. 12. Hunt, Leigh (1828). Lord Byron and some of his comtemporanies with recollections of the author's life and of his visit to Italy. Colburn. p. 352. 13. The Philosophical Transactions of the Royal Society of London, from Their Commencement, in 1665, to the Year 1800: 1763–1769. Royal Society. 1809. p. 683. 14. "Papers of the Board of Longitude : Confirmed minutes of the Board of Longitude, 1780-1801 (5 December 1795)". Cambridge Digital Library. Retrieved 15 January 2017. 15. "Wales Island Cannery". Porcher Island Cannery. Retrieved 10 August 2009. 16. "Captain George Vancouver". Discover Vancouver. Retrieved 10 August 2009. 17. "15045 Walesdymond (1998 XY21)". JPL Small-Body Database Browser. Retrieved 10 August 2009. Sources • Who's Who in Science (Marquis Who's Who Inc, Chicago Ill. 1968) ISBN 0-8379-1001-3 • Francis Lucian Reid "William Wales (ca. 1734–1798): playing the astronomer", Studies in History and Philosophy of Science, 39 (2008) 170–175 External links • "Wales, William" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900. • Journal of a Voyage, Made by Order of the Royal Society, to Churchill River, on the North-West Coast of Hudson's Bay; Of Thirteen Months Residence in That Country; and of the Voyage Back to England; In the Years 1768 and 1769: By William Wales • Extracts of William Wales's Journal kept on his voyage aboard HMS Resolution • Full digitised version of Wales' Logbook from his voyage on HMS Resolution • The Original Astronomical Observations, Made in the Course of a Voyage...in the Resolution and Adventure – Results of Wales' work published in 1777 • Article on Wales compiled for Captain Cook Society • The Transit of William Wales Educational comic book produced by the Hudson's Bay Company for Canadian high school students Captain James Cook Voyages • First voyage (1769 transit of Venus observed from Tahiti) • Second voyage • Third voyage (Death of James Cook) Vessels • HMS Adventure • HMS Discovery • HMS Eagle • HMS Endeavour • HMS Grenville • HMS Pembroke • HMS Resolution Associates • Joseph Banks • William Bayly • William Bligh • Alexander Buchan • James Burney • Charles Clerke • James Colnett • Alexander Dalrymple • Georg Forster • Johann Reinhold Forster • Tobias Furneaux • John Gore • Charles Green • Zachary Hickes • James King • John Ledyard • David Nelson • Omai • Hugh Palliser • Sydney Parkinson • Nathaniel Portlock • Edward Riou • Henry Roberts • David Samwell • Daniel Solander • Herman Spöring • William Taylor • James Trevenen • William Wales • John Watts • John Webber • Thomas Willis Artworks • Zoffany's Death of Cook • Statue in Christchurch • Statue in The Mall, London • Hawaii Sesquicentennial half dollar Books • An Account of the Voyages • A Journal of a Voyage to the South Seas • Characteres generum plantarum • A Voyage Round the World • Observations Made During a Voyage Round the World Related • Birthplace Museum • Cooks' Cottage • James Cook Collection: Australian Museum • Memorial Museum • Puhi Kai Iti / Cook Landing Site Authority control International • FAST • ISNI • VIAF National • Chile • France • BnF data • Germany • Italy • Belgium • United States • Czech Republic • Australia • Netherlands Artists • ULAN People • Trove Other • SNAC • IdRef
Wikipedia
Willmore conjecture In differential geometry, the Willmore conjecture is a lower bound on the Willmore energy of a torus. It is named after the English mathematician Tom Willmore, who conjectured it in 1965.[2] A proof by Fernando Codá Marques and André Neves was announced in 2012 and published in 2014.[1][3] Willmore energy Main article: Willmore energy Let v : M → R3 be a smooth immersion of a compact, orientable surface. Giving M the Riemannian metric induced by v, let H : M → R be the mean curvature (the arithmetic mean of the principal curvatures κ1 and κ2 at each point). In this notation, the Willmore energy W(M) of M is given by $W(M)=\int _{M}H^{2}\,dA.$ It is not hard to prove that the Willmore energy satisfies W(M) ≥ 4π, with equality if and only if M is an embedded round sphere. Statement Calculation of W(M) for a few examples suggests that there should be a better bound than W(M) ≥ 4π for surfaces with genus g(M) > 0. In particular, calculation of W(M) for tori with various symmetries led Willmore to propose in 1965 the following conjecture, which now bears his name For every smooth immersed torus M in R3, W(M) ≥ 2π2. In 1982, Peter Wai-Kwong Li and Shing-Tung Yau proved the conjecture in the non-embedded case, showing that if $f:\Sigma \to S^{3}$ is an immersion of a compact surface, which is not an embedding, then W(M) is at least 8π.[4] In 2012, Fernando Codá Marques and André Neves proved the conjecture in the embedded case, using the Almgren–Pitts min-max theory of minimal surfaces.[3][1] Martin Schmidt claimed a proof in 2002,[5] but it was not accepted for publication in any peer-reviewed mathematical journal (although it did not contain a proof of the Willmore conjecture, he proved some other important conjectures in it). Prior to the proof of Marques and Neves, the Willmore conjecture had already been proved for many special cases, such as tube tori (by Willmore himself), and for tori of revolution (by Langer & Singer).[6] References 1. Marques, Fernando C.; Neves, André (2014). "Min-max theory and the Willmore conjecture". Annals of Mathematics. 179: 683–782. arXiv:1202.6036. doi:10.4007/annals.2014.179.2.6. MR 3152944. 2. Willmore, Thomas J. (1965). "Note on embedded surfaces". Analele Ştiinţifice ale Universităţii "Al. I. Cuza" din Iaşi, Secţiunea I a Matematică. 11B: 493–496. MR 0202066. 3. Frank Morgan (2012) "Math Finds the Best Doughnut", The Huffington Post 4. Li, Peter; Yau, Shing Tung (1982). "A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces". Inventiones Mathematicae. 69 (2): 269–291. doi:10.1007/BF01399507. MR 0674407. 5. Schmidt, Martin U. (2002). "A proof of the Willmore conjecture". arXiv:math/0203224. 6. Langer, Joel; Singer, David (1984). "Curves in the hyperbolic plane and mean curvature of tori in 3-space". The Bulletin of the London Mathematical Society. 16 (5): 531–534. doi:10.1112/blms/16.5.531. MR 0751827.
Wikipedia
Willmore energy In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is defined to be the integral of the square of the mean curvature minus the Gaussian curvature. It is named after the English geometer Thomas Willmore. Definition Expressed symbolically, the Willmore energy of S is: ${\mathcal {W}}=\int _{S}H^{2}\,dA-\int _{S}K\,dA$ where $H$ is the mean curvature, $K$ is the Gaussian curvature, and dA is the area form of S. For a closed surface, by the Gauss–Bonnet theorem, the integral of the Gaussian curvature may be computed in terms of the Euler characteristic $\chi (S)$ of the surface, so $\int _{S}K\,dA=2\pi \chi (S),$ which is a topological invariant and thus independent of the particular embedding in $\mathbb {R} ^{3}$ that was chosen. Thus the Willmore energy can be expressed as ${\mathcal {W}}=\int _{S}H^{2}\,dA-2\pi \chi (S)$ An alternative, but equivalent, formula is ${\mathcal {W}}={1 \over 4}\int _{S}(k_{1}-k_{2})^{2}\,dA$ where $k_{1}$ and $k_{2}$ are the principal curvatures of the surface. Properties The Willmore energy is always greater than or equal to zero. A round sphere has zero Willmore energy. The Willmore energy can be considered a functional on the space of embeddings of a given surface, in the sense of the calculus of variations, and one can vary the embedding of a surface, while leaving it topologically unaltered. Critical points A basic problem in the calculus of variations is to find the critical points and minima of a functional. For a given topological space, this is equivalent to finding the critical points of the function $\int _{S}H^{2}\,dA$ since the Euler characteristic is constant. One can find (local) minima for the Willmore energy by gradient descent, which in this context is called Willmore flow. For embeddings of the sphere in 3-space, the critical points have been classified:[1] they are all conformal transforms of minimal surfaces, the round sphere is the minimum, and all other critical values are integers greater than 4$\pi $. They are called Willmore surfaces. Willmore flow The Willmore flow is the geometric flow corresponding to the Willmore energy; it is an $L^{2}$-gradient flow. $e[{\mathcal {M}}]={\frac {1}{2}}\int _{\mathcal {M}}H^{2}\,\mathrm {d} A$ where H stands for the mean curvature of the manifold ${\mathcal {M}}$. Flow lines satisfy the differential equation: $\partial _{t}x(t)=-\nabla {\mathcal {W}}[x(t)]\,$ where $x$ is a point belonging to the surface. This flow leads to an evolution problem in differential geometry: the surface ${\mathcal {M}}$ is evolving in time to follow variations of steepest descent of the energy. Like surface diffusion it is a fourth-order flow, since the variation of the energy contains fourth derivatives. Applications • Cell membranes tend to position themselves so as to minimize Willmore energy.[2] • Willmore energy is used in constructing a class of optimal sphere eversions, the minimax eversions. See also • Willmore conjecture Notes 1. Bryant, Robert L. (1984), "A duality theorem for Willmore surfaces", Journal of Differential Geometry, 20 (1): 23–53, doi:10.4310/jdg/1214438991, MR 0772125. 2. Müller, Stefan; Röger, Matthias (May 2014). "Confined structures of least bending energy". Journal of Differential Geometry. 97 (1): 109–139. doi:10.4310/jdg/1404912105. References • Willmore, T. J. (1992), "A survey on Willmore immersions", Geometry and Topology of Submanifolds, IV (Leuven, 1991), River Edge, NJ: World Scientific, pp. 11–16, MR 1185712.
Wikipedia
Wilson matrix Wilson matrix is the following $4\times 4$ matrix having integers as elements:[1][2][3][4][5] $W={\begin{bmatrix}5&7&6&5\\7&10&8&7\\6&8&10&9\\5&7&9&10\end{bmatrix}}$ This is the coefficient matrix of the following system of linear equations considered in a paper by J. Morris published in 1946:[6] ${\text{(S1)}}\quad {\begin{aligned}5x+7y+6z+5u&=23\\7x+10y+8z+7u&=32\\6x+8y+10z+9u&=33\\5x+7y+9z+10u&=31\end{aligned}}$ Morris ascribes the source of the set of equations to one T. S. Wilson but no details about Wilson have been provided. The particular system of equations was used by Morris to illustrate the concept of ill-conditioned system of equations. The matrix $W$ has been used as an example and for test purposes in many research papers and books over the years. John Todd has referred to $W$ as “the notorious matrix W of T. S. Wilson”.[1] Properties 1. $W$ is a symmetric matrix. 2. $W$ is positive definite. 3. The determinant of $W$ is $1$. 4. The inverse of $W$ is $W^{-1}={\begin{bmatrix}68&-41&-17&10\\-41&25&10&-6\\-17&10&5&-3\\10&-6&-3&2\end{bmatrix}}$ 5. The characteristic polynomial of $W$ is $\lambda ^{4}-35\lambda ^{3}+146\lambda ^{2}-100\lambda +1$. 6. The eigenvalues of $W$ are $\quad 0.01015004839789187,\quad 0.8431071498550294,\quad 3.858057455944953,\quad 30.28868534580213$. 7. Since $W$ is symmetric, the 2-norm condition number of $W$ is $\kappa _{2}(W)=({\text{max eigen value}})/({\text{min eigen value}})=30.28868534580213/0.01015004839789187=2984.09270167549$. 8. The solution of the system of equations $(S1)$ is $x=y=z=u=1$. 9. The Cholesky factorisation of $W$ is $W=R^{T}R$ where $R={\begin{bmatrix}{\sqrt {5}}&{\frac {7}{\sqrt {5}}}&{\frac {6}{\sqrt {5}}}&{\sqrt {5}}\\0&{\frac {1}{\sqrt {5}}}&-{\frac {2}{\sqrt {5}}}&0\\0&0&{\sqrt {2}}&{\frac {3}{\sqrt {2}}}\\0&0&0&{\frac {1}{\sqrt {2}}}\end{bmatrix}}$. 10. $W$ has the factorisation $W=LDL^{T}$ where $L={\begin{bmatrix}1&0&0&0\\{\frac {7}{5}}&1&0&0\\{\frac {6}{5}}&-2&1&0\\1&0&{\frac {3}{2}}&1\end{bmatrix}},\quad D={\begin{bmatrix}5&0&0&0\\0&{\frac {1}{5}}&0&0\\0&0&2&0\\0&0&0&{\frac {1}{2}}\end{bmatrix}}$. 11. $W$ has the factorisation $W=Z^{T}Z$ with $Z$ as the integer matrix[7] $Z={\begin{bmatrix}2&3&2&2\\1&1&2&1\\0&0&1&2\\0&0&1&1\end{bmatrix}}$. Research problems spawned by Wilson matrix A consideration of the condition number of the Wilson matrix has spawned several interesting research problems relating to condition numbers of matrices in certain special classes of matrices having some or all the special features of the Wilson matrix. In particular, the following special classes of matrices have been studied:[1] 1. $S=$ the set of $4\times 4$ nonsingular, symmetric matrices with integer entries between 1 and 10. 2. $P=$ the set of $4\times 4$ positive definite, symmetric matrices with integer entries between 1 and 10. An exhaustive computation of the condition numbers of the matrices in the above sets has yielded the following results: 1. Among the elements of $S$, the maximum condition number is $7.6119\times 10^{4}$ and this maximum is attained by the matrix ${\begin{bmatrix}2&7&10&10\\7&10&10&9\\10&10&10&1\\10&9&1&10\end{bmatrix}}$. 2. Among the elements of $P$, the maximum condition number is $3.5529\times 10^{4}$ and this maximum is attained by the matrix ${\begin{bmatrix}9&1&1&5\\1&10&1&9\\1&1&10&1\\5&9&1&10\end{bmatrix}}$. References 1. Nick Higham (June 2021). "What Is the Wilson Matrix?". What Is the Wilson Matrix?. Retrieved 24 May 2022. 2. Nicholas J. Higham and Matthew C. Lettington (2022). "Optimizing and Factorizing the Wilson Matrix". The American Mathematical Monthly. 129 (5): 454–465. doi:10.1080/00029890.2022.2038006. S2CID 233322415. Retrieved 24 May 2022. (An eprint of the paper is available here) 3. Cleve Moler. "Reviving Wilson's Matrix". Cleve’s Corner: Cleve Moler on Mathematics and Computing. MathWorks. Retrieved 24 May 2022. 4. Carl Erik Froberg (1969). Introduction to Numerical Analysis (2 ed.). Reading, Mass.: Addison-Wesley. 5. Robert T Gregory and David L Karney (1978). A Collection of Matrices for Testing Computational Algorithms. Huntington, New York: Robert Krieger Publishing Company. p. 57. 6. J Morris (1946). "An escalator process for the solution of linear simultaneous equations". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 37:265 (265): 106–120. doi:10.1080/14786444608561331. Retrieved 19 May 2022. 7. Nicholas J. Higham, Matthew C. Lettington, Karl Michael Schmidt (2021). "nteger matrix factorisations, superalgebras and the quadratic form obstruction". Linear Algebra and Its Applications. 622: 250–267. doi:10.1016/j.laa.2021.03.028. S2CID 232146938.{{cite journal}}: CS1 maint: multiple names: authors list (link)
Wikipedia
Wilson polynomials In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by James A. Wilson (1980) that generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined in terms of the generalized hypergeometric function and the Pochhammer symbols by $p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}{}_{4}F_{3}\left({\begin{matrix}-n&a+b+c+d+n-1&a-t&a+t\\a+b&a+c&a+d\end{matrix}};1\right).$ See also • Askey–Wilson polynomials are a q-analogue of Wilson polynomials. References • Wilson, James A. (1980), "Some hypergeometric orthogonal polynomials", SIAM Journal on Mathematical Analysis, 11 (4): 690–701, doi:10.1137/0511064, ISSN 0036-1410, MR 0579561 • Koornwinder, T.H. (2001) [1994], "Wilson polynomials", Encyclopedia of Mathematics, EMS Press
Wikipedia
Wilson prime In number theory, a Wilson prime is a prime number $p$ such that $p^{2}$ divides $(p-1)!+1$, where "$!$ !} " denotes the factorial function; compare this with Wilson's theorem, which states that every prime $p$ divides $(p-1)!+1$. Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson,[1] although it had been stated centuries earlier by Ibn al-Haytham.[2] Wilson prime Named afterJohn Wilson No. of known terms3 First terms5, 13, 563 OEIS index • A007540 • Wilson primes: primes $p$ such that $(p-1)!\equiv -1\ (\operatorname {mod} {p^{2}})$ The only known Wilson primes are 5, 13, and 563 (sequence A007540 in the OEIS). Costa et al. write that "the case $p=5$ is trivial", and credit the observation that 13 is a Wilson prime to Mathews (1892).[3][4] Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer,[5][3][6] but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem.[3][7][8] If any others exist, they must be greater than 2 × 1013.[3] It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval $[x,y]$ is about $\log \log _{x}y$.[9] Several computer searches have been done in the hope of finding new Wilson primes.[10][11][12] The Ibercivis distributed computing project includes a search for Wilson primes.[13] Another search was coordinated at the Great Internet Mersenne Prime Search forum.[14] Generalizations Wilson primes of order n Wilson's theorem can be expressed in general as $(n-1)!(p-n)!\equiv (-1)^{n}\ {\bmod {p}}$ for every integer $n\geq 1$ and prime $p\geq n$. Generalized Wilson primes of order n are the primes p such that $p^{2}$ divides $(n-1)!(p-n)!-(-1)^{n}$. It was conjectured that for every natural number n, there are infinitely many Wilson primes of order n. The smallest generalized Wilson primes of order $n$ are: 5, 2, 7, 10429, 5, 11, 17, ... (The next term > 1.4 × 107) (sequence A128666 in the OEIS) Near-Wilson primes pB 1282279+20 1306817−30 1308491−55 1433813−32 1638347−45 1640147−88 1647931+14 1666403+99 1750901+34 1851953−50 2031053−18 2278343+21 2313083+15 2695933−73 3640753+69 3677071−32 3764437−99 3958621+75 5062469+39 5063803+40 6331519+91 6706067+45 7392257+40 8315831+3 8871167−85 9278443−75 9615329+27 9756727+23 10746881−7 11465149−62 11512541−26 11892977−7 12632117−27 12893203−53 14296621+2 16711069+95 16738091+58 17879887+63 19344553−93 19365641+75 20951477+25 20972977+58 21561013−90 23818681+23 27783521−51 27812887+21 29085907+9 29327513+13 30959321+24 33187157+60 33968041+12 39198017−7 45920923−63 51802061+4 53188379−54 56151923−1 57526411−66 64197799+13 72818227−27 87467099−2 91926437−32 92191909+94 93445061−30 93559087−3 94510219−69 101710369−70 111310567+22 117385529−43 176779259+56 212911781−92 216331463−36 253512533+25 282361201+24 327357841−62 411237857−84 479163953−50 757362197−28 824846833+60 866006431−81 1227886151−51 1527857939−19 1636804231+64 1686290297+18 1767839071+8 1913042311−65 1987272877+5 2100839597−34 2312420701−78 2476913683+94 3542985241−74 4036677373−5 4271431471+83 4296847931+41 5087988391+51 5127702389+50 7973760941+76 9965682053−18 10242692519−97 11355061259−45 11774118061−1 12896325149+86 13286279999+52 20042556601+27 21950810731+93 23607097193+97 24664241321+46 28737804211−58 35525054743+26 41659815553+55 42647052491+10 44034466379+39 60373446719−48 64643245189−21 66966581777+91 67133912011+9 80248324571+46 80908082573−20 100660783343+87 112825721339+70 231939720421+41 258818504023+4 260584487287−52 265784418461−78 298114694431+82 A prime $p$ satisfying the congruence $(p-1)!\equiv -1+Bp\ (\operatorname {mod} {p^{2}})$ with small $|B|$ can be called a near-Wilson prime. Near-Wilson primes with $B=0$ are bona fide Wilson primes. The table on the right lists all such primes with $|B|\leq 100$ from 106 up to 4×1011.[3] Wilson numbers A Wilson number is a natural number $n$ such that $W(n)\equiv 0\ (\operatorname {mod} {n^{2}})$, where $W(n)=\pm 1+\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}{k},$ and where the $\pm 1$ term is positive if and only if $n$ has a primitive root and negative otherwise.[15] For every natural number $n$, $W(n)$ is divisible by $n$, and the quotients (called generalized Wilson quotients) are listed in OEIS: A157249. The Wilson numbers are 1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158, ... (sequence A157250 in the OEIS) If a Wilson number $n$ is prime, then $n$ is a Wilson prime. There are 13 Wilson numbers up to 5×108.[16] See also • PrimeGrid • Table of congruences • Wall–Sun–Sun prime • Wieferich prime • Wolstenholme prime References 1. Edward Waring, Meditationes Algebraicae (Cambridge, England: 1770), page 218 (in Latin). In the third (1782) edition of Waring's Meditationes Algebraicae, Wilson's theorem appears as problem 5 on page 380. On that page, Waring states: "Hanc maxime elegantem primorum numerorum proprietatem invenit vir clarissimus, rerumque mathematicarum peritissimus Joannes Wilson Armiger." (A man most illustrious and most skilled in mathematics, Squire John Wilson, found this most elegant property of prime numbers.) 2. O'Connor, John J.; Robertson, Edmund F. "Abu Ali al-Hasan ibn al-Haytham". MacTutor History of Mathematics Archive. University of St Andrews. 3. Costa, Edgar; Gerbicz, Robert; Harvey, David (2014). "A search for Wilson primes". Mathematics of Computation. 83 (290): 3071–3091. arXiv:1209.3436. doi:10.1090/S0025-5718-2014-02800-7. MR 3246824. S2CID 6738476. 4. Mathews, George Ballard (1892). "Example 15". Theory of Numbers, Part 1. Deighton & Bell. p. 318. 5. Lehmer, Emma (April 1938). "On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson" (PDF). Annals of Mathematics. 39 (2): 350–360. doi:10.2307/1968791. JSTOR 1968791. Retrieved 8 March 2011. 6. Beeger, N. G. W. H. (1913–1914). "Quelques remarques sur les congruences $r^{p-1}\equiv 1\ (\operatorname {mod} {p^{2}})$ et $(p-1)!\equiv -1\ (\operatorname {mod} {p^{2}})$". The Messenger of Mathematics. 43: 72–84. 7. Wall, D. D. (October 1952). "Unpublished mathematical tables" (PDF). Mathematical Tables and Other Aids to Computation. 6 (40): 238. doi:10.2307/2002270. JSTOR 2002270. 8. Goldberg, Karl (1953). "A table of Wilson quotients and the third Wilson prime". J. London Math. Soc. 28 (2): 252–256. doi:10.1112/jlms/s1-28.2.252. 9. The Prime Glossary: Wilson prime 10. McIntosh, R. (9 March 2004). "WILSON STATUS (Feb. 1999)". E-Mail to Paul Zimmermann. Retrieved 6 June 2011. 11. Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997). "A search for Wieferich and Wilson primes". Math. Comput. 66 (217): 433–449. Bibcode:1997MaCom..66..433C. doi:10.1090/S0025-5718-97-00791-6. See p. 443. 12. Ribenboim, P.; Keller, W. (2006). Die Welt der Primzahlen: Geheimnisse und Rekorde (in German). Berlin Heidelberg New York: Springer. p. 241. ISBN 978-3-540-34283-0. 13. "Ibercivis site". Archived from the original on 2012-06-20. Retrieved 2011-03-10. 14. Distributed search for Wilson primes (at mersenneforum.org) 15. see Gauss's generalization of Wilson's theorem 16. Agoh, Takashi; Dilcher, Karl; Skula, Ladislav (1998). "Wilson quotients for composite moduli" (PDF). Math. Comput. 67 (222): 843–861. Bibcode:1998MaCom..67..843A. doi:10.1090/S0025-5718-98-00951-X. Further reading • Crandall, Richard E.; Pomerance, Carl (2001). Prime Numbers: A Computational Perspective. Springer-Verlag. p. 29. ISBN 978-0-387-94777-8. • Pearson, Erna H. (1963). "On the Congruences (p − 1)! ≡ −1 and 2p−1 ≡ 1 (mod p2)" (PDF). Math. Comput. 17: 194–195. External links • The Prime Glossary: Wilson prime • Weisstein, Eric W. "Wilson prime". MathWorld. • Status of the search for Wilson primes Prime number classes By formula • Fermat (22n + 1) • Mersenne (2p − 1) • Double Mersenne (22p−1 − 1) • Wagstaff (2p + 1)/3 • Proth (k·2n + 1) • Factorial (n! ± 1) • Primorial (pn# ± 1) • Euclid (pn# + 1) • Pythagorean (4n + 1) • Pierpont (2m·3n + 1) • Quartan (x4 + y4) • Solinas (2m ± 2n ± 1) • Cullen (n·2n + 1) • Woodall (n·2n − 1) • Cuban (x3 − y3)/(x − y) • Leyland (xy + yx) • Thabit (3·2n − 1) • Williams ((b−1)·bn − 1) • Mills (⌊A3n⌋) By integer sequence • Fibonacci • Lucas • Pell • Newman–Shanks–Williams • Perrin • Partitions • Bell • Motzkin By property • Wieferich (pair) • Wall–Sun–Sun • Wolstenholme • Wilson • Lucky • Fortunate • Ramanujan • Pillai • Regular • Strong • Stern • Supersingular (elliptic curve) • Supersingular (moonshine theory) • Good • Super • Higgs • Highly cototient • Unique Base-dependent • Palindromic • Emirp • Repunit (10n − 1)/9 • Permutable • Circular • Truncatable • Minimal • Delicate • Primeval • Full reptend • Unique • Happy • Self • Smarandache–Wellin • Strobogrammatic • Dihedral • Tetradic Patterns • Twin (p, p + 2) • Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …) • Triplet (p, p + 2 or p + 4, p + 6) • Quadruplet (p, p + 2, p + 6, p + 8) • k-tuple • Cousin (p, p + 4) • Sexy (p, p + 6) • Chen • Sophie Germain/Safe (p, 2p + 1) • Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...) • Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...) • Balanced (consecutive p − n, p, p + n) By size • Mega (1,000,000+ digits) • Largest known • list Complex numbers • Eisenstein prime • Gaussian prime Composite numbers • Pseudoprime • Catalan • Elliptic • Euler • Euler–Jacobi • Fermat • Frobenius • Lucas • Somer–Lucas • Strong • Carmichael number • Almost prime • Semiprime • Sphenic number • Interprime • Pernicious Related topics • Probable prime • Industrial-grade prime • Illegal prime • Formula for primes • Prime gap First 60 primes • 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 • 23 • 29 • 31 • 37 • 41 • 43 • 47 • 53 • 59 • 61 • 67 • 71 • 73 • 79 • 83 • 89 • 97 • 101 • 103 • 107 • 109 • 113 • 127 • 131 • 137 • 139 • 149 • 151 • 157 • 163 • 167 • 173 • 179 • 181 • 191 • 193 • 197 • 199 • 211 • 223 • 227 • 229 • 233 • 239 • 241 • 251 • 257 • 263 • 269 • 271 • 277 • 281 List of prime numbers
Wikipedia
Wilson quotient The Wilson quotient W(p) is defined as: $W(p)={\frac {(p-1)!+1}{p}}$ If p is a prime number, the quotient is an integer by Wilson's theorem; moreover, if p is composite, the quotient is not an integer. If p divides W(p), it is called a Wilson prime. The integer values of W(p) are (sequence A007619 in the OEIS): W(2) = 1 W(3) = 1 W(5) = 5 W(7) = 103 W(11) = 329891 W(13) = 36846277 W(17) = 1230752346353 W(19) = 336967037143579 ... It is known that[1] $W(p)\equiv B_{2(p-1)}-B_{p-1}{\pmod {p}},$ $p-1+ptW(p)\equiv pB_{t(p-1)}{\pmod {p^{2}}},$ where $B_{k}$ is the k-th Bernoulli number. Note that the first relation comes from the second one by subtraction, after substituting $t=1$ and $t=2$. See also • Fermat quotient References 1. Lehmer, Emma (1938). "On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson". Annals of Mathematics. 39 (2): 350–360. doi:10.2307/1968791. JSTOR 1968791. External links • MathWorld: Wilson Quotient
Wikipedia
Wilson's theorem In algebra and number theory, Wilson's theorem states that a natural number n > 1 is a prime number if and only if the product of all the positive integers less than n is one less than a multiple of n. That is (using the notations of modular arithmetic), the factorial $(n-1)!=1\times 2\times 3\times \cdots \times (n-1)$ satisfies $(n-1)!\ \equiv \;-1{\pmod {n}}$ exactly when n is a prime number. In other words, any number n is a prime number if, and only if, (n − 1)! + 1 is divisible by n.[1] History This theorem was stated by Ibn al-Haytham (c. 1000 AD),[2] and, in the 18th century, by the English mathematician John Wilson.[3] Edward Waring announced the theorem in 1770, although neither he nor his student Wilson could prove it. Lagrange gave the first proof in 1771.[4] There is evidence that Leibniz was also aware of the result a century earlier, but he never published it.[5] Example For each of the values of n from 2 to 30, the following table shows the number (n − 1)! and the remainder when (n − 1)! is divided by n. (In the notation of modular arithmetic, the remainder when m is divided by n is written m mod n.) The background color is blue for prime values of n, gold for composite values. Table of factorial and its remainder modulo n $n$$(n-1)!$ (sequence A000142 in the OEIS) $(n-1)!\ {\bmod {\ }}n$ (sequence A061006 in the OEIS) 211 322 462 5244 61200 77206 850400 9403200 103628800 11362880010 12399168000 1347900160012 1462270208000 15871782912000 1613076743680000 172092278988800016 183556874280960000 19640237370572800018 201216451004088320000 2124329020081766400000 22510909421717094400000 23112400072777760768000022 24258520167388849766400000 256204484017332394393600000 26155112100433309859840000000 274032914611266056355840000000 28108888694504183521607680000000 2930488834461171386050150400000028 3088417619937397019545436160000000 Proofs The proofs (for prime moduli) below use the fact that the residue classes modulo a prime number are a field—see the article prime field for more details.[6] Lagrange's theorem, which states that in any field a polynomial of degree n has at most n roots, is needed for all the proofs. Composite modulus If n is composite it is divisible by some prime number q, where 2 ≤ q ≤ n − 2. Because $q$ divides $n$, let $n=qk$ for some integer $k$. Suppose for the sake of contradiction that $(n-1)!$ were congruent to −1 (mod n) where n is composite. Then (n-1)! would also be congruent to −1 (mod q) as $(n-1)!\equiv -1\ ({\text{mod}}\ n)$ implies that $(n-1)!=nm-1=(qk)m-1=q(km)-1$ for some integer $m$ which shows (n-1)! being congruent to -1 (mod q). But (n − 1)! ≡ 0 (mod q) by the fact that q is a term in (n-1)! making (n-1)! a multiple of q. A contradiction is now reached. In fact, more is true. With the sole exception of 4, where 3! = 6 ≡ 2 (mod 4), if n is composite then (n − 1)! is congruent to 0 (mod n). The proof is divided into two cases: First, if n can be factored as the product of two unequal numbers, n = ab, where 2 ≤ a < b ≤ n − 2, then both a and b will appear in the product 1 × 2 × ... × (n − 1) = (n − 1)! and (n − 1)! will be divisible by n. If n has no such factorization, then it must be the square of some prime q, q > 2. But then 2q < q2 = n, both q and 2q will be factors of (n − 1)!, and again n divides (n − 1)!. Elementary proof The result is trivial when p = 2, so assume p is an odd prime, p ≥ 3. Since the residue classes (mod p) are a field, every non-zero a has a unique multiplicative inverse, a−1. Lagrange's theorem implies that the only values of a for which a ≡ a−1 (mod p) are a ≡ ±1 (mod p) (because the congruence a2 ≡ 1 can have at most two roots (mod p)). Therefore, with the exception of ±1, the factors of (p − 1)! can be arranged in disjoint pairs such that product of each pair is congruent to 1 modulo p. This proves Wilson's theorem. For example, for p = 11, one has $10!=[(1\cdot 10)]\cdot [(2\cdot 6)(3\cdot 4)(5\cdot 9)(7\cdot 8)]\equiv [-1]\cdot [1\cdot 1\cdot 1\cdot 1]\equiv -1{\pmod {11}}.$ Proof using Fermat's little theorem Again, the result is trivial for p = 2, so suppose p is an odd prime, p ≥ 3. Consider the polynomial $g(x)=(x-1)(x-2)\cdots (x-(p-1)).$ g has degree p − 1, leading term xp − 1, and constant term (p − 1)!. Its p − 1 roots are 1, 2, ..., p − 1. Now consider $h(x)=x^{p-1}-1.$ h also has degree p − 1 and leading term xp − 1. Modulo p, Fermat's little theorem says it also has the same p − 1 roots, 1, 2, ..., p − 1. Finally, consider $f(x)=g(x)-h(x).$ f has degree at most p − 2 (since the leading terms cancel), and modulo p also has the p − 1 roots 1, 2, ..., p − 1. But Lagrange's theorem says it cannot have more than p − 2 roots. Therefore, f must be identically zero (mod p), so its constant term is (p − 1)! + 1 ≡ 0 (mod p). This is Wilson's theorem. Proof using the Sylow theorems It is possible to deduce Wilson's theorem from a particular application of the Sylow theorems. Let p be a prime. It is immediate to deduce that the symmetric group $S_{p}$ has exactly $(p-1)!$ elements of order p, namely the p-cycles $C_{p}$. On the other hand, each Sylow p-subgroup in $S_{p}$ is a copy of $C_{p}$. Hence it follows that the number of Sylow p-subgroups is $n_{p}=(p-2)!$. The third Sylow theorem implies $(p-2)!\equiv 1{\pmod {p}}.$ Multiplying both sides by (p − 1) gives $(p-1)!\equiv p-1\equiv -1{\pmod {p}},$ that is, the result. Applications Primality tests In practice, Wilson's theorem is useless as a primality test because computing (n − 1)! modulo n for large n is computationally complex, and much faster primality tests are known (indeed, even trial division is considerably more efficient). Used in the other direction, to determine the primality of the successors of large factorials, it is indeed a very fast and effective method. This is of limited utility, however. Quadratic residues Using Wilson's Theorem, for any odd prime p = 2m + 1, we can rearrange the left hand side of $1\cdot 2\cdots (p-1)\ \equiv \ -1\ {\pmod {p}}$ to obtain the equality $1\cdot (p-1)\cdot 2\cdot (p-2)\cdots m\cdot (p-m)\ \equiv \ 1\cdot (-1)\cdot 2\cdot (-2)\cdots m\cdot (-m)\ \equiv \ -1{\pmod {p}}.$ This becomes $\prod _{j=1}^{m}\ j^{2}\ \equiv (-1)^{m+1}{\pmod {p}}$ or $(m!)^{2}\equiv (-1)^{m+1}{\pmod {p}}.$ We can use this fact to prove part of a famous result: for any prime p such that p ≡ 1 (mod 4), the number (−1) is a square (quadratic residue) mod p. For this, suppose p = 4k + 1 for some integer k. Then we can take m = 2k above, and we conclude that (m!)2 is congruent to (−1) (mod p). Formulas for primes Wilson's theorem has been used to construct formulas for primes, but they are too slow to have practical value. p-adic gamma function Wilson's theorem allows one to define the p-adic gamma function. Gauss' generalization Gauss’ generalization of Wilson’s Theorem states that if $n$ is four, an odd prime power, or twice an odd prime power, then the product of relatively prime integers less than itself add one is divisible by $n$. It goes further to say that otherwise, the same product subtract one is divisible by $n$. To state Gauss' Generalization of Wilson's Theorem, we use the Euler's totient function, denoted $\varphi (n)$, which is defined as the number of positive integers less than or equal to $n$ which are also relatively prime with $n$. Let's call such numbers $a_{i}$, where $\gcd(a_{i},n)=1$. Gauss proved given an odd prime $p$ and some integer $\alpha >0$, then $\prod _{k=1}^{\varphi (n)}\!\!a_{k}\ ={\begin{cases}-1{\pmod {n}}&{\text{if }}n=4,\;p^{\alpha },\;2p^{\alpha }\\\;\;\,1{\pmod {n}}&{\text{otherwise}}\end{cases}}$. First, let's note this is the proof for cases $n>2$, since the results are trivial for $n=\{1,2\}$. For all $a_{i}$, we know there exist some $a_{j}$, where $i\neq j$ and $\gcd(a_{j},n)=1$, such that $a_{i}a_{j}=1$. This allows us to pair each of the elements together with its inverse. We are left now with $a_{i}$ being its own inverse. So in other words $a_{i}$ is a root of $f(x)=x^{2}-1$ in $\mathbb {Z} /n\mathbb {Z} $, and $f(x)=(x-1)(x+1)$, in the polynomial ring $\mathbb {Z} /n\mathbb {Z} [X]$. If $a_{i}$ is a root, it follows that $n-a_{i}$ is also a root. Our objective is to show that the number of roots is divisible by four, unless $n=4,n=p^{\alpha }$, or $n=2p^{\alpha }$. Let's consider $n=2$. Then we notice we have one root since $1\equiv -1{\pmod {2}}$. Consider $n=4$. Then, it is clear there are two roots, specifically, $x\equiv 1{\pmod {4}}$ and $x\equiv -1{\pmod {4}}$. Say $n=p^{\alpha }$. It is again clear there are two solutions. We now consider $n=2^{\beta },\beta >2$. If one of the factors of $f(x)$ is divisible by 2, so is the other. Take the factor $(x+1)$ to be divisible by $2^{1}$. Then, it follows that there are 4 distinct roots of $f(x)$, namely $x\equiv 1{\pmod {n}}$, $x\equiv 1+2^{\beta -1}{\pmod {n}}$, $x\equiv -1{\pmod {n}}$, and $x\equiv -1-2^{\beta -1}{\pmod {n}}$, when $n=2^{\beta },\beta >2$. Finally, let's look at the general case where $n=2^{\beta }p_{1}{^{\gamma _{1}}}p_{2}{^{\gamma _{2}}}\dots p_{k}{^{\gamma _{k}}}$. We find 2 roots of $f(x)$ over each $\mathbb {Z} /2^{\beta }\mathbb {Z} $ and $\mathbb {Z} /p_{r}^{\gamma _{r}}\mathbb {Z} $, except when $\beta >2$. Using the Chinese remainder theorem, we find that when $n$ is not divisible by 2, we have a total of $2^{k}$ solutions of $f(x)$. Assuming $\beta =1$, in $\mathbb {Z} /2\mathbb {Z} $, we have one root, so we still have a total of $2^{k}$ solutions. When $\beta =2$, we have 2 roots in $\mathbb {Z} /4\mathbb {Z} $, so there are a total of $2^{k+1}$ roots of $f(x)$. For all cases where $\beta >2$, there are 4 roots in $\mathbb {Z} /2^{\beta }\mathbb {Z} $ with a total of $2^{k+2}$ solutions. This shows that the number of roots are divisible by 4, unless $n=4,n=p^{\alpha }$, or $n=2p^{\alpha }$. Say $a_{i}$ is a root of $f(x)$ in $\mathbb {Z} /n\mathbb {Z} $. Then $a_{i}(n-a_{i})=-a_{i}^{2}=-1$. So, if the number of roots of $f(x)$ is divisible by 4, then we can say the product of the roots if 1. Otherwise, we can say the product is -1. So we can conclude that $\prod _{k=1}^{\varphi (n)}\!\!a_{k}\ ={\begin{cases}-1{\pmod {n}}&{\text{if }}n=4,\;p^{\alpha },\;2p^{\alpha }\\\;\;\,1{\pmod {n}}&{\text{otherwise}}\end{cases}}$. See also • Wilson prime • Table of congruences Notes 1. The Universal Book of Mathematics. David Darling, p. 350. 2. O'Connor, John J.; Robertson, Edmund F., "Abu Ali al-Hasan ibn al-Haytham", MacTutor History of Mathematics Archive, University of St Andrews 3. Edward Waring, Meditationes Algebraicae (Cambridge, England: 1770), page 218 (in Latin). In the third (1782) edition of Waring's Meditationes Algebraicae, Wilson's theorem appears as problem 5 on page 380. On that page, Waring states: "Hanc maxime elegantem primorum numerorum proprietatem invenit vir clarissimus, rerumque mathematicarum peritissimus Joannes Wilson Armiger." (A man most illustrious and most skilled in mathematics, Squire John Wilson, found this most elegant property of prime numbers.) 4. Joseph Louis Lagrange, "Demonstration d'un théorème nouveau concernant les nombres premiers" (Proof of a new theorem concerning prime numbers), Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres (Berlin), vol. 2, pages 125–137 (1771). 5. Giovanni Vacca (1899) "Sui manoscritti inediti di Leibniz" (On unpublished manuscripts of Leibniz), Bollettino di bibliografia e storia delle scienze matematiche ... (Bulletin of the bibliography and history of mathematics), vol. 2, pages 113–116; see page 114 (in Italian). Vacca quotes from Leibniz's mathematical manuscripts kept at the Royal Public Library in Hanover (Germany), vol. 3 B, bundle 11, page 10: Original : Inoltre egli intravide anche il teorema di Wilson, come risulta dall'enunciato seguente: "Productus continuorum usque ad numerum qui antepraecedit datum divisus per datum relinquit 1 (vel complementum ad unum?) si datus sit primitivus. Si datus sit derivativus relinquet numerum qui cum dato habeat communem mensuram unitate majorem." Egli non giunse pero a dimostrarlo. Translation : In addition, he [Leibniz] also glimpsed Wilson's theorem, as shown in the following statement: "The product of all integers preceding the given integer, when divided by the given integer, leaves 1 (or the complement of 1?) if the given integer be prime. If the given integer be composite, it leaves a number which has a common factor with the given integer [which is] greater than one." However, he didn't succeed in proving it. See also: Giuseppe Peano, ed., Formulaire de mathématiques, vol. 2, no. 3, page 85 (1897). 6. Landau, two proofs of thm. 78 References The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. • Gauss, Carl Friedrich; Clarke, Arthur A. (1986), Disquisitiones Arithemeticae (2nd corrected ed.), New York: Springer, ISBN 0-387-96254-9(translated into English){{citation}}: CS1 maint: postscript (link). • Gauss, Carl Friedrich; Maser, H. (1965), Untersuchungen über hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (2nd ed.), New York: Chelsea, ISBN 0-8284-0191-8(translated into German){{citation}}: CS1 maint: postscript (link). • Landau, Edmund (1966), Elementary Number Theory, New York: Chelsea. • Ore, Øystein (1988). Number Theory and its History. Dover. pp. 259–271. ISBN 0-486-65620-9. External links • "Wilson theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Weisstein, Eric W. "Wilson's Theorem". MathWorld. • Mizar system proof: http://mizar.org/version/current/html/nat_5.html#T22 • [1] 1. Ohana, Andrew. "A Generalization of Wilson's Theorem" (PDF).
Wikipedia
Wim Blok Willem Johannes "Wim" Blok (1947–2003) was a Dutch logician who made major contributions to algebraic logic, universal algebra, and modal logic. His important achievements over the course of his career include "a brilliant demonstration of the fact that various techniques and results that originated in universal algebra can be used to prove significant and deep theorems in modal logic."[1] Blok began his career in 1973 as an algebraist investigating the varieties of interior algebras at the University of Illinois at Chicago. Following the 1976 completion of his Ph.D. on that topic, he continued on to study more general varieties of modal algebras. As an algebraist, Blok "was recognised by the modal logic community as one of the most influential modal logicians" by the end of the 1970s.[2] He published many papers in the Reports on Mathematical Logic, served as a member on their editorial board, and was one of their guest editors.[1] Along with Don Pigozzi, Wim Blok co-authored the monograph Algebraizable Logics, which began the field now known as abstract algebraic logic.[3] He died in a car accident on November 30, 2003.[1] See also • Abstract algebraic logic • Blok–Esakia isomorphism • Leibniz operator References 1. Font, Josep Maria (May 2006). "In Memory of Wim Blok". Reports on Mathematical Logic (Special Issue). CiteSeerX 10.1.1.103.2741. 2. Rautenberg, W.; Wolter, F.; Zakharyaschev, M. (June 2006). "Willem Blok and Modal Logic". Studia Logica. 83 (1–3): 15–30. doi:10.1007/s11225-006-8296-2. S2CID 17091670. Retrieved June 24, 2016 – via ResearchGate. 3. Raftery, James G. (March 2004). "Willem Blok's Work in Algebraic Logic". Studia Logica. 76 (2): 155–160. doi:10.1023/B:STUD.0000032083.45504.62. JSTOR 20016583. S2CID 37825139. Authority control International • ISNI • VIAF National • Norway • Germany • Israel • United States • Czech Republic • Netherlands Academics • CiNii • zbMATH • 2 Other • IdRef
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Wiman's sextic In mathematics, Wiman's sextic is a degree 6 plane curve with four nodes studied by Anders Wiman (1896). It is given by the equation (in homogeneous coordinates) $x^{6}+y^{6}+z^{6}+(x^{2}+y^{2}+z^{2})(x^{4}+y^{4}+z^{4})=12x^{2}y^{2}z^{2}$ Its normalization is a genus 6 curve with automorphism group isomorphic to the symmetric group S5. References • Inoue, Naoki; Kato, Fumiharu (2005), "On the geometry of Wiman's sextic", Journal of Mathematics of Kyoto University, 45 (4): 743–757, doi:10.1215/kjm/1250281655, ISSN 0023-608X, MR 2226628 • Wiman, A. (1896), "Zur Theorie der endlichen Gruppen von birationalen Transformationen in der Ebene", Mathematische Annalen, 48 (1–2): 195–240, doi:10.1007/BF01446342, S2CID 123516972
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Winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of turns. For certain open plane curves, the number of turns may be non-integer. The winding number depends on the orientation of the curve, and it is negative if the curve travels around the point clockwise. Not to be confused with Map winding number. Mathematical analysis → Complex analysis Complex analysis Complex numbers • Real number • Imaginary number • Complex plane • Complex conjugate • Unit complex number Complex functions • Complex-valued function • Analytic function • Holomorphic function • Cauchy–Riemann equations • Formal power series Basic theory • Zeros and poles • Cauchy's integral theorem • Local primitive • Cauchy's integral formula • Winding number • Laurent series • Isolated singularity • Residue theorem • Conformal map • Schwarz lemma • Harmonic function • Laplace's equation Geometric function theory People • Augustin-Louis Cauchy • Leonhard Euler • Carl Friedrich Gauss • Jacques Hadamard • Kiyoshi Oka • Bernhard Riemann • Karl Weierstrass •  Mathematics portal Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics (such as in string theory). Intuitive description Suppose we are given a closed, oriented curve in the xy plane. We can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the winding number of the curve is equal to the total number of counterclockwise turns that the object makes around the origin. When counting the total number of turns, counterclockwise motion counts as positive, while clockwise motion counts as negative. For example, if the object first circles the origin four times counterclockwise, and then circles the origin once clockwise, then the total winding number of the curve is three. Using this scheme, a curve that does not travel around the origin at all has winding number zero, while a curve that travels clockwise around the origin has negative winding number. Therefore, the winding number of a curve may be any integer. The following pictures show curves with winding numbers between −2 and 3: $\cdots $                −2 −1 0                $\cdots $ 1 2 3 Formal definition Let $\gamma :[0,1]\to \mathbb {C} \setminus \{a\}$ :[0,1]\to \mathbb {C} \setminus \{a\}} be a continuous closed path on the plane minus one point. The winding number of $\gamma $ around $a$ is the integer ${\text{wind}}(\gamma ,a)=s(1)-s(0),$ where $(\rho ,s)$ is the path written in polar coordinates, i.e. the lifted path through the covering map $p:\mathbb {R} _{>0}\times \mathbb {R} \to \mathbb {C} \setminus \{a\}:(\rho _{0},s_{0})\mapsto a+\rho _{0}e^{i2\pi s_{0}}.$ The winding number is well defined because of the existence and uniqueness of the lifted path (given the starting point in the covering space) and because all the fibers of $p$ are of the form $\rho _{0}\times (s_{0}+\mathbb {Z} )$ (so the above expression does not depend on the choice of the starting point). It is an integer because the path is closed. Alternative definitions Winding number is often defined in different ways in various parts of mathematics. All of the definitions below are equivalent to the one given above: Alexander numbering A simple combinatorial rule for defining the winding number was proposed by August Ferdinand Möbius in 1865[1] and again independently by James Waddell Alexander II in 1928.[2] Any curve partitions the plane into several connected regions, one of which is unbounded. The winding numbers of the curve around two points in the same region are equal. The winding number around (any point in) the unbounded region is zero. Finally, the winding numbers for any two adjacent regions differ by exactly 1; the region with the larger winding number appears on the left side of the curve (with respect to motion down the curve). Differential geometry In differential geometry, parametric equations are usually assumed to be differentiable (or at least piecewise differentiable). In this case, the polar coordinate θ is related to the rectangular coordinates x and y by the equation: $d\theta ={\frac {1}{r^{2}}}\left(x\,dy-y\,dx\right)\quad {\text{where }}r^{2}=x^{2}+y^{2}.$ Which is found by differentiating the following definition for θ: $\theta (t)=\arctan {\bigg (}{\frac {y(t)}{x(t)}}{\bigg )}$ By the fundamental theorem of calculus, the total change in θ is equal to the integral of dθ. We can therefore express the winding number of a differentiable curve as a line integral: ${\text{wind}}(\gamma ,0)={\frac {1}{2\pi }}\oint _{\gamma }\,\left({\frac {x}{r^{2}}}\,dy-{\frac {y}{r^{2}}}\,dx\right).$ The one-form dθ (defined on the complement of the origin) is closed but not exact, and it generates the first de Rham cohomology group of the punctured plane. In particular, if ω is any closed differentiable one-form defined on the complement of the origin, then the integral of ω along closed loops gives a multiple of the winding number. Complex analysis Winding numbers play a very important role throughout complex analysis (c.f. the statement of the residue theorem). In the context of complex analysis, the winding number of a closed curve $\gamma $ in the complex plane can be expressed in terms of the complex coordinate z = x + iy. Specifically, if we write z = reiθ, then $dz=e^{i\theta }dr+ire^{i\theta }d\theta $ and therefore ${\frac {dz}{z}}={\frac {dr}{r}}+i\,d\theta =d[\ln r]+i\,d\theta .$ As $\gamma $ is a closed curve, the total change in $\ln(r)$ is zero, and thus the integral of $ {\frac {dz}{z}}$ is equal to $i$ multiplied by the total change in $\theta $. Therefore, the winding number of closed path $\gamma $ about the origin is given by the expression[3] ${\frac {1}{2\pi i}}\oint _{\gamma }{\frac {dz}{z}}\,.$ More generally, if $\gamma $ is a closed curve parameterized by $t\in [\alpha ,\beta ]$, the winding number of $\gamma $ about $z_{0}$, also known as the index of $z_{0}$ with respect to $\gamma $, is defined for complex $z_{0}\notin \gamma ([\alpha ,\beta ])$ as[4] $\mathrm {Ind} _{\gamma }(z_{0})={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {d\zeta }{\zeta -z_{0}}}={\frac {1}{2\pi i}}\int _{\alpha }^{\beta }{\frac {\gamma '(t)}{\gamma (t)-z_{0}}}dt.$ This is a special case of the famous Cauchy integral formula. Some of the basic properties of the winding number in the complex plane are given by the following theorem:[5] Theorem. Let $\gamma :[\alpha ,\beta ]\to \mathbb {C} $ :[\alpha ,\beta ]\to \mathbb {C} } be a closed path and let $\Omega $ be the set complement of the image of $\gamma $, that is, $\Omega :=\mathbb {C} \setminus \gamma ([\alpha ,\beta ])$ :=\mathbb {C} \setminus \gamma ([\alpha ,\beta ])} . Then the index of $z$ with respect to $\gamma $, $\mathrm {Ind} _{\gamma }:\Omega \to \mathbb {C} ,\ \ z\mapsto {\frac {1}{2\pi i}}\oint _{\gamma }{\frac {d\zeta }{\zeta -z}},$ is (i) integer-valued, i.e., $\mathrm {Ind} _{\gamma }(z)\in \mathbb {Z} $ for all $z\in \Omega $; (ii) constant over each component (i.e., maximal connected subset) of $\Omega $; and (iii) zero if $z$ is in the unbounded component of $\Omega $. As an immediate corollary, this theorem gives the winding number of a circular path $\gamma $ about a point $z$. As expected, the winding number counts the number of (counterclockwise) loops $\gamma $ makes around $z$: Corollary. If $\gamma $ is the path defined by $\gamma (t)=a+re^{int},\ \ 0\leq t\leq 2\pi ,\ \ n\in \mathbb {Z} $, then $\mathrm {Ind} _{\gamma }(z)={\begin{cases}n,&|z-a|<r;\\0,&|z-a|>r.\end{cases}}$ Topology In topology, the winding number is an alternate term for the degree of a continuous mapping. In physics, winding numbers are frequently called topological quantum numbers. In both cases, the same concept applies. The above example of a curve winding around a point has a simple topological interpretation. The complement of a point in the plane is homotopy equivalent to the circle, such that maps from the circle to itself are really all that need to be considered. It can be shown that each such map can be continuously deformed to (is homotopic to) one of the standard maps $S^{1}\to S^{1}:s\mapsto s^{n}$, where multiplication in the circle is defined by identifying it with the complex unit circle. The set of homotopy classes of maps from a circle to a topological space form a group, which is called the first homotopy group or fundamental group of that space. The fundamental group of the circle is the group of the integers, Z; and the winding number of a complex curve is just its homotopy class. Maps from the 3-sphere to itself are also classified by an integer which is also called the winding number or sometimes Pontryagin index. Turning number One can also consider the winding number of the path with respect to the tangent of the path itself. As a path followed through time, this would be the winding number with respect to the origin of the velocity vector. In this case the example illustrated at the beginning of this article has a winding number of 3, because the small loop is counted. This is only defined for immersed paths (i.e., for differentiable paths with nowhere vanishing derivatives), and is the degree of the tangential Gauss map. This is called the turning number, rotation number,[6] rotation index[7] or index of the curve, and can be computed as the total curvature divided by 2π. Polygons Further information: Density (polytope) § Polygons In polygons, the turning number is referred to as the polygon density. For convex polygons, and more generally simple polygons (not self-intersecting), the density is 1, by the Jordan curve theorem. By contrast, for a regular star polygon {p/q}, the density is q. Space curves Turning number cannot be defined for space curves as degree requires matching dimensions. However, for locally convex, closed space curves, one can define tangent turning sign as $(-1)^{d}$, where $d$ is the turning number of the stereographic projection of its tangent indicatrix. Its two values correspond to the two non-degenerate homotopy classes of locally convex curves.[8] [9] Winding number and Heisenberg ferromagnet equations The winding number is closely related with the (2 + 1)-dimensional continuous Heisenberg ferromagnet equations and its integrable extensions: the Ishimori equation etc. Solutions of the last equations are classified by the winding number or topological charge (topological invariant and/or topological quantum number). Applications Point in polygon Further information: Point in polygon § Winding number algorithm A point's winding number with respect to a polygon can be used to solve the point in polygon (PIP) problem – that is, it can be used to determine if the point is inside the polygon or not. Generally, the ray casting algorithm is a better alternative to the PIP problem as it does not require trigonometric functions, contrary to the winding number algorithm. Nevertheless, the winding number algorithm can be sped up so that it too, does not require calculations involving trigonometric functions.[10] The sped-up version of the algorithm, also known as Sunday's algorithm, is recommendable in cases where non-simple polygons should also be accounted for. See also • Argument principle • Coin rotation paradox • Linking coefficient • Nonzero-rule • Polygon density • Residue theorem • Schläfli symbol • Topological degree theory • Topological quantum number • Twist (mathematics) • Wilson loop • Writhe References 1. Möbius, August (1865). "Über die Bestimmung des Inhaltes eines Polyëders". Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften, Mathematisch-Physische Klasse. 17: 31–68. 2. Alexander, J. W. (April 1928). "Topological Invariants of Knots and Links". Transactions of the American Mathematical Society. 30 (2): 275–306. doi:10.2307/1989123. JSTOR 1989123. 3. Weisstein, Eric W. "Contour Winding Number". MathWorld. Retrieved 7 July 2022. 4. Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill. p. 201. ISBN 0-07-054235-X. 5. Rudin, Walter (1987). Real and Complex Analysis (3rd ed.). McGraw-Hill. p. 203. ISBN 0-07-054234-1. 6. Abelson, Harold (1981). Turtle Graphics: The Computer as a Medium for Exploring Mathematics. MIT Press. p. 24. 7. Do Carmo, Manfredo P. (1976). "5. Global Differential Geometry". Differential Geometry of Curves and Surfaces. Prentice-Hall. p. 393. ISBN 0-13-212589-7. 8. Feldman, E. A. (1968). "Deformations of closed space curves". Journal of Differential Geometry. 2 (1): 67–75. doi:10.4310/jdg/1214501138. S2CID 116999463. 9. Minarčík, Jiří; Beneš, Michal (2022). "Nondegenerate homotopy and geometric flows". Homology, Homotopy and Applications. 24 (2): 255–264. doi:10.4310/HHA.2022.v24.n2.a12. S2CID 252274622. 10. Sunday, Dan (2001). "Inclusion of a Point in a Polygon". Archived from the original on 26 January 2013. External links • Winding number at PlanetMath. Topology Fields • General (point-set) • Algebraic • Combinatorial • Continuum • Differential • Geometric • low-dimensional • Homology • cohomology • Set-theoretic • Digital Key concepts • Open set / Closed set • Interior • Continuity • Space • compact • Connected • Hausdorff • metric • uniform • Homotopy • homotopy group • fundamental group • Simplicial complex • CW complex • Polyhedral complex • Manifold • Bundle (mathematics) • Second-countable space • Cobordism Metrics and properties • Euler characteristic • Betti number • Winding number • Chern number • Orientability Key results • Banach fixed-point theorem • De Rham cohomology • Invariance of domain • Poincaré conjecture • Tychonoff's theorem • Urysohn's lemma • Category •  Mathematics portal • Wikibook • Wikiversity • Topics • general • algebraic • geometric • Publications Authority control: National • Germany
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Windmill graph In the mathematical field of graph theory, the windmill graph Wd(k,n) is an undirected graph constructed for k ≥ 2 and n ≥ 2 by joining n copies of the complete graph Kk at a shared universal vertex. That is, it is a 1-clique-sum of these complete graphs.[1] Windmill graph The Windmill graph Wd(5,4). Verticesn(k – 1) + 1 Edgesnk(k − 1)/2 Radius1 Diameter2 Girth3 if k > 2 Chromatic numberk Chromatic indexn(k – 1) NotationWd(k,n) Table of graphs and parameters Properties It has n(k – 1) + 1 vertices and nk(k − 1)/2 edges,[2] girth 3 (if k > 2), radius 1 and diameter 2. It has vertex connectivity 1 because its central vertex is an articulation point; however, like the complete graphs from which it is formed, it is (k – 1)-edge-connected. It is trivially perfect and a block graph. Special cases By construction, the windmill graph Wd(3,n) is the friendship graph Fn, the windmill graph Wd(2,n) is the star graph Sn and the windmill graph Wd(3,2) is the butterfly graph. Labeling and colouring The windmill graph has chromatic number k and chromatic index n(k – 1). Its chromatic polynomial can be deduced from the chromatic polynomial of the complete graph and is equal to $x\prod _{i=1}^{k-1}(x-i)^{n}.$ The windmill graph Wd(k,n) is proved not graceful if k > 5.[3] In 1979, Bermond has conjectured that Wd(4,n) is graceful for all n ≥ 4.[4] Through an equivalence with perfect difference families, this has been proved for n ≤ 1000. [5] Bermond, Kotzig, and Turgeon proved that Wd(k,n) is not graceful when k = 4 and n = 2 or n = 3, and when k = 5 and n = 2.[6] The windmill Wd(3,n) is graceful if and only if n ≡ 0 (mod 4) or n ≡ 1 (mod 4).[7] Gallery References 1. Gallian, J. A. (3 January 2007). "A dynamic survey of graph labeling" (PDF). Electronic Journal of Combinatorics. DS6: 1–58. MR 1668059. 2. Weisstein, Eric W. "Windmill Graph". MathWorld. 3. Koh, K. M.; Rogers, D. G.; Teo, H. K.; Yap, K. Y. (1980). "Graceful graphs: some further results and problems". Congressus Numerantium. 29: 559–571. MR 0608456. 4. Bermond, J.-C. (1979). "Graceful graphs, radio antennae and French windmills". In Wilson, Robin J. (ed.). Graph theory and combinatorics (Proc. Conf., Open Univ., Milton Keynes, 1978). Research notes in mathematics. Vol. 34. Pitman. pp. 18–37. ISBN 978-0273084358. MR 0587620. OCLC 757210583. 5. Ge, G.; Miao, Y.; Sun, X. (2010). "Perfect difference families, perfect difference matrices, and related combinatorial structures". Journal of Combinatorial Designs. 18 (6): 415–449. doi:10.1002/jcd.20259. MR 2743134. S2CID 120800012. 6. Bermond, J.-C.; Kotzig, A.; Turgeon, J. (1978). "On a combinatorial problem of antennas in radioastronomy". In Hajnal, A.; Sos, Vera T. (eds.). Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. I. Colloquia mathematica Societatis János Bolyai. Vol. 18. North-Holland. pp. 135–149. ISBN 978-0-444-85095-9. MR 0519261. 7. Bermond, J.-C.; Brouwer, A. E.; Germa, A. (1978). "Systèmes de triplets et différences associées". Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976). Colloques internationaux du Centre National de la Recherche Scientifique. Vol. 260. Éditions du Centre national de la recherche scientifique. pp. 35–38. ISBN 978-2-222-02070-7. MR 0539936.
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Wine/water paradox The wine/water paradox is an apparent paradox in probability theory. It is stated by Michael Deakin as follows: A mixture is known to contain a mix of wine and water in proportions such that the amount of wine divided by the amount of water is a ratio $x$ lying in the interval $1/3\leq x\leq 3$ (i.e. 25-75% wine). We seek the probability, $P^{\ast }$ say, that $x\leq 2$. (i.e. less than or equal to 66%.) The core of the paradox is in finding consistent and justifiable simultaneous prior distributions for $x$ and ${\frac {1}{x}}$.[1] Calculation This calculation is the demonstration of the paradoxical conclusion when making use of the principle of indifference. To recapitulate, We do not know $x$, the wine to water ratio. When considering the numbers above, it is only known that it lies in an interval between the minimum of one quarter wine over three quarters water on one end (i.e. 25% wine), to the maximum of three quarters wine over one quarter water on the other (i.e. 75% wine). In term of ratios, $ x_{\mathrm {min} }={\frac {1/4}{3/4}}={\frac {1}{3}}$ resp. $ x_{\mathrm {max} }={\frac {3/4}{1/4}}=3$. Now, making use of the principle of indifference, we may assume that $x$ is uniformly distributed. Then the chance of finding the ratio $x$ below any given fixed threshold $x_{t}$, with $x_{\mathrm {min} }<x_{t}<x_{\mathrm {max} }$, should linearly depend on the value $x_{t}$. So the probability value is the number $\operatorname {Prob} \{x\leq x_{t}\}={\frac {x_{t}-x_{\mathrm {min} }}{x_{\mathrm {max} }-x_{\mathrm {min} }}}={\frac {1}{8}}(3x_{t}-1).$ As a function of the threshold value $x_{t}$, this is the linearly growing function that is $0$ resp. $1$ at the end points $ x_{\mathrm {min} }$ resp. the larger $ x_{\mathrm {max} }$. Consider the threshold $x_{t}=2$, as in the example of the original formulation above. This is two parts wine vs. one part water, i.e. 66% wine. With this we conclude that $\operatorname {Prob} \{x\leq 2\}={\frac {1}{8}}(3\cdot 2-1)={\frac {5}{8}}$. Now consider $y={\frac {1}{x}}$, the inverted ratio of water to wine but the equivalent wine/water mixture threshold. It lies between the inverted bounds. Again using the principle of indifference, we get $\operatorname {Prob} \{y\geq y_{t}\}={\frac {x_{\mathrm {max} }(1-x_{\mathrm {min} }\,y_{t})}{x_{\mathrm {max} }-x_{\mathrm {min} }}}={\frac {3}{8}}(3-y_{t})$. This is the function which is $0$ resp. $1$ at the end points ${\tfrac {1}{x_{\mathrm {min} }}}$ resp. the smaller $ {\tfrac {1}{x_{\mathrm {max} }}}$. Now taking the corresponding threshold $ y_{t}={\frac {1}{x_{t}}}={\frac {1}{2}}$ (also half as much water as wine). We conclude that $\operatorname {Prob} \left\{y\geq {\tfrac {1}{2}}\right\}={\frac {3}{8}}{\frac {3\cdot 2-1}{2}}={\frac {15}{16}}={\frac {3}{2}}{\frac {5}{8}}$. The second probability always exceeds the first by a factor of $ {\frac {x_{\mathrm {max} }}{x_{t}}}\geq 1$. For our example the numbers is $ {\frac {3}{2}}$. Paradoxical conclusion Since $ y={\frac {1}{x}}$, we get ${\frac {5}{8}}=\operatorname {Prob} \{x\leq 2\}=P^{*}=\operatorname {Prob} \left\{y\geq {\frac {1}{2}}\right\}={\frac {15}{16}}>{\frac {5}{8}}$, a contradiction. References 1. Deakin, Michael A. B. (December 2005). "The Wine/Water Paradox: background, provenance and proposed resolutions". Australian Mathematical Society Gazette. 33 (3): 200–205.
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Winifred Sargent Winifred Lydia Caunden Sargent (8 May 1905 – October 1979) was an English mathematician. She studied at Newnham College, Cambridge and carried out research into Lebesgue integration, fractional integration and differentiation and the properties of BK-spaces. Winifred Sargent Born(1905-05-08)8 May 1905 Ambergate, England Died(1979-10-00)October 1979 London, England Alma materNewnham College, Cambridge Scientific career FieldsMathematics, Numerical integration, Functional analysis InfluencesLancelot Stephen Bosanquet Early life Sargent was born into a Quaker family, daughter of Henry Sargent and Edith, his second wife, growing up in Fritchley, Derbyshire. She attended Ackworth School, a private school for Quakers, from 1915 to 1919. She then won a scholarship to attend The Mount School, York, another Quaker school, and later the Herbert Strutt School. In 1923, while there, she won a Derby scholarship, a State Scholarship, and a Mary Ewart scholarship to attend Newnham College, Cambridge and study mathematics in 1924. While at Newnham she won further awards: an Arthur Hugh Clough Scholarship in 1927, a Mary Ewart Travelling Scholarship and a Goldsmiths Company Senior Studentship both in 1928. She graduated with a First class degree and remained at Cambridge conducting research but was unsatisfied by her progress and left to teach mathematics at Bolton High School. Academic career Sargent's first publication was in 1929, On Young's criteria for the convergence of Fourier series and their conjugates, published in the Mathematical Proceedings of the Cambridge Philosophical Society. In 1931 she was appointed an Assistant Lecturer at Westfield College and became a member of the London Mathematical Society in January 1932.[1] in 1936 she moved to Royal Holloway, University of London, at the time both women's colleges. In 1939 she became a doctoral student of Lancelot Bosanquet, but World War II broke out, preventing his formal supervision from continuing. In 1941 Sargent was promoted to lecturer at Royal Holloway, moving to Bedford College in 1948. She served on the Mathematical Association teaching committee from 1950 to 1954.[2] In 1954 she was awarded the degree of Sc.D. (Doctor of Science) by Cambridge and was given the title of Reader. While at the University of London she supervised Alan J. White in 1959.[3][4] Bosanquet started a weekly seminar in mathematics in 1947, which Sargent attended without absence for twenty years until her retirement in 1967. She rarely presented at it, and did not attend mathematical conferences, despite being a compelling speaker. Mathematical results Much of Sargent's mathematical research involved studying types of integral, building on work done on Lebesgue integration and the Riemann integral. She produced results relating to the Perron and Denjoy integrals and Cesàro summation. Her final three papers consider BK-spaces or Banach coordinate spaces, proving a number of interesting results.[5] For example, her 1936 paper[6] proves a version of Rolle's theorem for Denjoy–Perron integrable functions using different techniques from the standard proofs:[7] as in much of Dr. Sargent's work, the arguments are pushed as far as they will go and counter examples given to show that the results are the best possible. Her 1953 paper[8] established several important results on summability kernels and is referenced in two textbooks on functional analysis.[9] Her papers in 1950 and 1957 contributed to fractional integration and differentiation theory.[10] In her obituary, her work is described as being:[11] marked by its exceptional lucidity, its exactness of expression and by the decisiveness of her results. She made important contributions to a field in which the complexity of the structure can only be revealed by subtle arguments. Papers • Sargent, Miss W. L. C. (1929). "On Young's criteria for the convergence of Fourier series and their conjugates". Mathematical Proceedings of the Cambridge Philosophical Society. 25 (1): 26–30. Bibcode:1929PCPS...25...26S. doi:10.1017/S030500410001851X. • Sargent, W. L. C. (1935). "The Borel derivates of a function". Proceedings of the London Mathematical Society. Second Series. 38 (1): 180–196. doi:10.1112/plms/s2-38.1.180. • Sargent, W. L. C. (1936). "On the Cesàro derivates of a function". Proceedings of the London Mathematical Society. Second Series. 40 (1): 235–254. doi:10.1112/plms/s2-40.1.235. • Sargent, W. L. C. (1942a). "A descriptive definition of Cesàro–Perron integrals". Proceedings of the London Mathematical Society. Second Series. 47 (1): 212–247. doi:10.1112/plms/s2-47.1.212. • Sargent, W. L. C. (1942b). "On sufficient conditions for a function integrable in the Cesàro–Perron sense to be monotonic". The Quarterly Journal of Mathematics. Oxford Series. 12 (1): 148–153. doi:10.1093/qmath/os-12.1.148. • Sargent, W. L. C. (1946a). "On the order of magnitude of the Fourier coefficients of a function integrable in the CλL sense". Journal of the London Mathematical Society. First Series. 21 (3): 198–203. doi:10.1112/jlms/s1-21.3.198. • Sargent, W. L. C. (1946b). "A mean value theorem involving Cesàro means". Proceedings of the London Mathematical Society. Second Series. 49 (1): 227–240. doi:10.1112/plms/s2-49.3.227. • Sargent, W. L. C. (1948a). "On the integrability of a product". Journal of the London Mathematical Society. First Series. 23 (1): 28–34. doi:10.1112/jlms/s1-23.1.28. hdl:10338.dmlcz/127918. • Sargent, W. L. C. (1948b). "On the summability (C) of allied series and the existence of $(CP)\int \limits _{0}^{\pi }{\frac {f(x+t)-f(x-t)}{t}}\,dt$". Proceedings of the London Mathematical Society. Second Series. 50 (1): 330–348. doi:10.1112/plms/s2-50.5.330. • Sargent, W. L. C. (1949). "On fractional integrals of a function integrable in the Cesàro-Perron sense". Proceedings of the London Mathematical Society. Second Series. 51 (1): 46–80. doi:10.1112/plms/s2-51.1.46. • Sargent, W. L. C. (1950a). "On linear functionals in spaces of conditionally integrable functions". The Quarterly Journal of Mathematics. Oxford Second Series. 1 (1): 288–298. Bibcode:1950QJMat...1..288S. doi:10.1093/qmath/1.1.288. hdl:10338.dmlcz/140532. • Sargent, W. L. C. (1950b). "On the continuity (C) and integrability (CP) of fractional integrals". Proceedings of the London Mathematical Society. Second Series. 52 (1): 253–270. doi:10.1112/plms/s2-52.4.253. • Sargent, W. L. C. (1950c). "On generalized derivatives and Cesàro–Denjoy integrals". Proceedings of the London Mathematical Society. Second Series. 52 (1): 365–376. doi:10.1112/plms/s2-52.5.365. • Sargent, W. L. C. (1951a). "Some properties of Cλ-continuous functions". Journal of the London Mathematical Society. First Series. 26 (2): 116–121. doi:10.1112/jlms/s1-26.2.116. • Sargent, W. L. C. (1951b). "On the integrability of a product (II)". Journal of the London Mathematical Society. First Series. 26 (4): 278–285. doi:10.1112/jlms/s1-26.4.278. • Sargent, W. L. C. (1951c). "2213. On the differentiation of a function of a function". The Mathematical Gazette. 35 (312): 121–122. doi:10.2307/3609346. JSTOR 3609346. • Sargent, W. L. C. (1952a). "On the summability of infinite integrals". Journal of the London Mathematical Society. First Series. 27 (4): 401–413. doi:10.1112/jlms/s1-27.4.401. • Sargent, W. L. C. (1952b). "Book review: Éléments de Mathématiques. XII by N. Bourbaki". The Mathematical Gazette. 36 (317): 216–217. doi:10.2307/3608266. JSTOR 3608266. • Sargent, W. L. C. (1952b). "Book review: Vorlesungen über Fouriersche Integrale by S. Bochner". The Mathematical Gazette. 36 (317): 217–218. doi:10.2307/3608268. JSTOR 3608268. • Sargent, W. L. C. (1953). "On some theorems of Hahn, Banach and Steinhaus". Journal of the London Mathematical Society. First Series. 28 (4): 438–451. doi:10.1112/jlms/s1-28.4.438. • Sargent, W. L. C. (1954). "Book review: Volume and Integral by W. W. Rogosinski". The Mathematical Gazette. 38 (323): 67. doi:10.2307/3609800. JSTOR 3609800. • Sargent, W. L. C. (1955). "On the transform $y_{x}(s)=\int \limits _{0}^{\infty }x(t)k_{s}(t)dt$". Journal of the London Mathematical Society. First Series. 30 (4): 401–416. doi:10.1112/jlms/s1-30.4.401. • Sargent, W. L. C. (1957a). "Some summability factor theorems for infinite integrals". Journal of the London Mathematical Society. First Series. 32 (4): 387–396. doi:10.1112/jlms/s1-32.4.387. • Sargent, W. L. C. (1957b). "On some cases of distinction between integrals and series". Proceedings of the London Mathematical Society. Third Series. 7 (1): 249–264. doi:10.1112/plms/s3-7.1.249. • Sargent, W. L. C. (1960). "Some sequence spaces related to the lp spaces". Journal of the London Mathematical Society. First Series. 35 (2): 161–171. doi:10.1112/jlms/s1-35.2.161. • Sargent, W. L. C. (1961). "Some analogues and extensions of Marcinkiewicz's interpolation problem". Proceedings of the London Mathematical Society. Third Series. 11 (1): 457–468. doi:10.1112/plms/s3-11.1.457. • Sargent, W. L. C. (1964). "On sectionally bounded BK-spaces". Mathematische Zeitschrift. 83 (1): 57–66. doi:10.1007/BF01111108. S2CID 119753065. • Sargent, W. L. C. (1966). "On compact matrix transformations between sectionally bounded BK-spaces". Journal of the London Mathematical Society. First Series. 41 (1): 79–87. doi:10.1112/jlms/s1-41.1.79. Notes 1. Dixon 1932, p. 81. 2. j. t. c (1950). "Report of the Meeting of the Teaching Committee. 5th January 1950". The Mathematical Gazette. 34 (307): 5–7. doi:10.1017/S0025557200023469. JSTOR 3610867., p. 6. 3. White 1961, p. 319. 4. "Alan J. White". Mathematics Genealogy Project. Department of Mathematics, North Dakota State University. Retrieved 15 October 2015. 5. Sargent 1961, Sargent 1964, Sargent 1966. 6. Sargent 1936, pp. 239–240. 7. Eggleston 1981, pp. 173–174. 8. Sargent 1953. 9. Swartz, Charles (1992). An introduction to functional analysis. CRC Press. pp. 102–104. ISBN 978-0824786434. and Orlicz, Władysław (1992). Linear functional analysis. World Scientific Publishing. p. 125. ISBN 978-9810208530. 10. Sargent 1950a, Sargent 1950b and Sargent 1957a. 11. Eggleston 1981, p. 175. References • Dixon, Prof. A. C. (1932). "Records of proceedings at meetings. Session November, 1931–June, 1932". Journal of the London Mathematical Society. First Series. 7 (2): 81–82. doi:10.1112/jlms/s1-7.2.81. • Eggleston, H.G. (1981). "Winifred L. C. Sargent". Bulletin of the London Mathematical Society. 13 (2): 173–176. doi:10.1112/blms/13.2.173. • O'Connor, JJ; Robertson, EF. "Winifred Lydia Caunden Sargent Biography". MacTutor History of Mathematics archive. University of St Andrews. Retrieved 13 October 2015. • Ogilvie, Marilyn; Harvey, Joy (2000). The Biographical Dictionary of Women in Science: Pioneering Lives From Ancient Times to the Mid-20th Century. Routledge. pp. 1152–1153. ISBN 978-0415920384. • White, A. J. (1961). "On the restricted Cesàro summability of double Fourier series". Transactions of the American Mathematical Society. 99 (2): 308–319. doi:10.2307/1993402. JSTOR 1993402. Authority control: Academics • MathSciNet • Mathematics Genealogy Project • zbMATH
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Winifred Asprey Winifred "Tim" Alice Asprey (April 8, 1917 – October 19, 2007) was an American mathematician and computer scientist. She was one of only around 200 women to earn PhDs in mathematics from American universities during the 1940s, a period of women's underrepresentation in mathematics at this level.[1] She was involved in developing the close contact between Vassar College and IBM that led to the establishment of the first computer science lab at Vassar.[1] Winifred Asprey Born(1917-04-08)April 8, 1917 Sioux City, Iowa DiedOctober 19, 2007(2007-10-19) (aged 90) Poughkeepsie (town), New York Alma materVassar College University of Iowa Scientific career FieldsMathematics, computer science Doctoral advisorEdward Wilson Chittenden InfluencesGrace Hopper Family Asprey was born in Sioux City, Iowa; her parents were Gladys Brown Asprey, Vassar class of 1905, and Peter Asprey Jr.[2] She had two brothers, actinide and fluorine chemist Larned B. Asprey (1919–2005), a signer of the Szilárd petition, and military historian and writer Robert B. Asprey (1923–2009) who dedicated several of his books to his sister Winifred.[3][4] Education and work Asprey attended Vassar College in Poughkeepsie, New York, where she earned her undergraduate degree in 1938. As a student there, Asprey met Grace Hopper, the "First Lady of Computing," who taught mathematics at the time. After graduating, Asprey taught at several private schools in New York City and Chicago before going on to earn her MS and PhD degrees from the University of Iowa in 1942 and 1945, respectively.[2] Her doctoral advisor was the topologist Edward Wilson Chittenden.[5] Asprey returned to Vassar College as a professor. By then, Grace Hopper had moved to Philadelphia to work on UNIVAC (Universal Automatic Computer) project. Asprey became interested in computing and visited Hopper to learn about the foundations of computer architecture.[2] Asprey believed that computers would be an essential part of a liberal arts education. At Vassar, Asprey taught mathematics and computer science for 38 years and was the chair of the mathematics department from 1957 until her retirement in 1982.[6] She created the first Computer Science courses at Vassar, the first being taught in 1963, and secured funds for the college's first computer, making Vassar the second college in the nation to acquire an IBM System/360 computer in 1967.[7] Asprey connected with researchers at IBM and other research centers and lobbied for computer science at Vassar. In 1989, due to her contributions, the computer center she started was renamed the Asprey Advanced Computation Laboratory.[2] References 1. Margaret Anne Marie Murray (2001). Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America. MIT Press. ISBN 0-262-63246-2. 2. "Winifred "Tim" Asprey, computer pioneer and longtime professor at Vassar College, dies at 90". Vassar Office of College Relations. 3. "Dr. Larned "Larry" Brown Asprey". Obituaries. Albuquerque Journal(. March 11, 2005. 4. "New College Receives Gift from Estate of Robert B. Asprey". New College of Florida. June 15, 2009. Retrieved December 10, 2009. 5. Winifred Asprey at the Mathematics Genealogy Project 6. "Scientists in the News". Science. American Association for the Advancement of Science. 125 (3257): 1080–1081. 1957. doi:10.1126/science.125.3257.1077. JSTOR 1752434. PMID 17756202. 7. "Winifred Asprey: Into the Future". Vassar Office of College Relations. Archived from the original on May 2, 2016. Retrieved January 31, 2014. External links • Profile at Vassar College Innovators Gallery • Winifred Asprey Papers at Vassar College Archives and Special Collections Library Authority control International • ISNI • VIAF National • United States • Australia Academics • Mathematics Genealogy Project People • Trove
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Peter Winkler Peter Mann Winkler is a research mathematician, author of more than 125 research papers in mathematics[1] and patent holder in a broad range of applications, ranging from cryptography to marine navigation.[2] His research areas include discrete mathematics, theory of computation and probability theory. He is currently a professor of mathematics and computer science at Dartmouth College.[3] Peter Winkler studied mathematics at Harvard University and later received his PhD in 1975 from Yale University under the supervision of Angus McIntyre.[4] He has also served as an assistant professor at Stanford, full professor and chair at Emory and as a mathematics research director at Bell Labs and Lucent Technologies.[2] He was visiting professor at the Technische Universität Darmstadt.[5] He has published three books on mathematical puzzles: Mathematical Puzzles: A connoisseur's collection (A K Peters, 2004, ISBN 978-1-56881-201-4), Mathematical Mind-Benders (A K Peters, 2007, ISBN 978-1-56881-336-3), and Mathematical Puzzles (A K Peters, 2021, ISBN 978-0-36720-693-2). And he is widely considered to be a pre eminent scholar in this domain. He was the Visiting Distinguished Chair for Public Dissemination of Mathematics at the National Museum of Mathematics (MoMath), gave topical talks at the Gathering 4 Gardner conferences, and wrote novel papers related to some of these puzzles. Winkler's book Bridge at the Enigma Club[6] was a runner up for the 2011 Master Point Press Book Of The Year award.[7] Also in 2011, Winkler received the David P. Robbins Prize of the Mathematical Association of America as coauthor of one of two papers[8] in the American Mathematical Monthly. Paul Erdős anecdote According to a story included in Chapter One of "The Man Who Loved Only Numbers / The Story of Paul Erdös and the Search for Mathematical Truth",[9] Paul Erdős attended the bar mitzvah celebration for Peter Winkler's twins, and Winkler's mother-in-law tried to throw Erdős out. [Quote:] "Erdös came to my twins' bar mitzvah, notebook in hand," said Peter Winkler, a colleague of Graham's at AT&T. "He also brought gifts for my children--he loved kids--and behaved himself very well. But my mother-in-law tried to throw him out. She thought he was some guy who wandered in off the street, in a rumpled suit, carrying a pad under his arm. It is entirely possible that he proved a theorem or two during the ceremony."[9] References 1. Publication list from Winkler's home page at Dartmouth. 2. Information listed on Peter Winkler's homepage at Dartmouth. 3. Dartmouth mathematics faculty listing. 4. Peter Winkler at the Mathematics Genealogy Project. 5. "Humboldt network profile of Peter Winkler". www.humboldt-foundation.de. Retrieved 2019-09-17.{{cite web}}: CS1 maint: url-status (link) 6. The Bridge World Bookstore Bridge at the Enigma Club by Peter Winkler 7. The 2011 Master Point Press Book Of The Year Award 2014 IPBA Handbook, p. 176 8. "Overhang", American Mathematical Monthly, vol. 116, January 2009 (Online) "Maximum Overhang", American Mathematical Monthly, vol. 116, December 2009 (Online) 9. Hoffman, Paul (15 July 1998). The Man Who Loved Only Numbers / The Story of Paul Erdös and the Search for Mathematical Truth. "The Man Who Loved Only Numbers" was published in hardcover by: Hyperion Books and a later edition was published by The New York Times Book Company. ISBN 0-7868-6362-5. Retrieved November 23, 2017.{{cite book}}: CS1 maint: url-status (link) External links • Peter Winkler at the Mathematics Genealogy Project Authority control International • ISNI • VIAF • 2 National • Germany • Israel • United States • Czech Republic Academics • DBLP • MathSciNet • Mathematics Genealogy Project • ORCID • Scopus • zbMATH Other • IdRef
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Winnie Li Wen-Ch'ing (Winnie) Li (Chinese: 李文卿; born December 25, 1948) is a Taiwanese-American mathematician and a Distinguished Professor of Mathematics at Pennsylvania State University.[1] She is a number theorist, with research focusing on the theory of automorphic forms and applications of number theory to coding theory and spectral graph theory. In particular, she has applied her research results in automorphic forms and number theory to construct efficient communication networks called Ramanujan graphs and Ramanujan complexes. Wen-Ch'ing Li 李文卿 Born (1948-12-25) December 25, 1948 Awards • Chern Prize (2010) • Fellow, American Mathematical Society (2012) • Noether Lecture (2015) Academic background Alma materNational Taiwan University, University of California, Berkeley Doctoral advisorAndrew Ogg Academic work DisciplineMathematics InstitutionsHarvard University, University of Illinois at Chicago, Pennsylvania State University Professional career Li did her undergraduate studies at National Taiwan University, graduating in 1970;[1][2] at NTU, she was a classmate of other notable female mathematicians Fan Chung, Sun-Yung Alice Chang and Jang-Mei Wu.[3] She earned a doctorate from the University of California, Berkeley in 1974, under the supervision of Andrew Ogg.[1][2][4] Before joining the PSU faculty in 1979, she was a Benjamin Pierce assistant professor at Harvard University for 3.5 years from 1974 to 1977, and a tenure-track assistant professor at the University of Illinois at Chicago from 1978 to 1979.[1][2] She was also the director of the National Center of Theoretical Sciences in Taiwan from 2009 to 2014.[1][2] Awards and honors In 2010, Li was the winner of the Chern Prize, given every three years to an outstanding Chinese mathematician.[5] In 2012 she became a fellow of the American Mathematical Society.[6] She was chosen to give the 2015 Noether Lecture.[7] References 1. Winnie Li Named Distinguished Professor, Pennsylvania State University, 30 January 2012, retrieved 2013-02-02. 2. Staff biography, National Center of Theoretical Sciences, retrieved 2013-02-02. 3. Fan Chung Graham, Association for Women in Mathematics, retrieved 2013-02-02. 4. Wen-Ching Winnie Li at the Mathematics Genealogy Project 5. Wen Ching Li Awarded the 2010 Chern Prize in Mathematics, Pennsylvania State University, 30 January 2012, retrieved 2013-02-02. 6. List of Fellows of the American Mathematical Society, retrieved 2013-01-27. 7. “Modular Forms for Congruence and Noncongruence Subgroups” External links • Home page • Ramdorai, Sujatha (January 2015). "Interview with Wen-Ching (Winnie) Li" (PDF). Asia Pacific Mathematics Newsletter. pp. 18–21. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • 2 • zbMATH People • Deutsche Biographie Other • IdRef
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Winning Ways for Your Mathematical Plays Winning Ways for Your Mathematical Plays (Academic Press, 1982) by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy is a compendium of information on mathematical games. It was first published in 1982 in two volumes. The first volume introduces combinatorial game theory and its foundation in the surreal numbers; partizan and impartial games; Sprague–Grundy theory and misère games. The second volume applies the theorems of the first volume to many games, including nim, sprouts, dots and boxes, Sylver coinage, philosopher's phutball, fox and geese. A final section on puzzles analyzes the Soma cube, Rubik's Cube, peg solitaire, and Conway's Game of Life. A republication of the work by A K Peters split the content into four volumes. Editions • 1st edition, New York: Academic Press, 2 vols., 1982; vol. 1, hardback: ISBN 0-12-091150-7, paperback: ISBN 0-12-091101-9; vol. 2, hardback: ISBN 0-12-091152-3, paperback: ISBN 0-12-091102-7. • 2nd edition, Wellesley, Massachusetts: A. K. Peters Ltd., 4 vols., 2001–2004; vol. 1: ISBN 1-56881-130-6; vol. 2: ISBN 1-56881-142-X; vol. 3: ISBN 1-56881-143-8; vol. 4: ISBN 1-56881-144-6. Games mentioned in the book This is a partial list of the games mentioned in the book. Note: Misère games not included • Hackenbush • Blue-Red Hackenbush • Blue-Red-Green Hackenbush (Introduced as Hackenbush Hotchpotch in the book) • Childish Hackenbush • Ski-Jumps • Toads-and-Frogs • Cutcake • Maundy Cake • (2nd Unnamed Cutcake variant by Dean Hickerson) • Hotcake • Coolcakes • Baked Alaska • Eatcake • Turn-and-Eatcake • Col • Snort • Nim (Green Hackenbush) • Prim • Dim • Lasker's Nim • Seating Couples • Northcott's Game (Poker-Nim) • The White Knight • Wyt Queens (Wythoff's Game) • Kayles • Double Kayles • Quadruple Kayles • Dawson's Chess • Dawson's Kayles • Treblecross • Grundy's Game • Mrs. Grundy • Domineering • No Highway • De Bono's L-Game • Snakes-and-Ladders (Adders-and-Ladders) • Jelly Bean Game • Dividing Rulers Reviews • Games[1] See also • On Numbers and Games by John H. Conway, one of the three coauthors of Winning Ways References 1. https://archive.org/details/games-32-1982-October/page/n57/mode/2up
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Winograd schema challenge The Winograd schema challenge (WSC) is a test of machine intelligence proposed in 2012 by Hector Levesque, a computer scientist at the University of Toronto. Designed to be an improvement on the Turing test, it is a multiple-choice test that employs questions of a very specific structure: they are instances of what are called Winograd schemas, named after Terry Winograd, professor of computer science at Stanford University.[1] On the surface, Winograd schema questions simply require the resolution of anaphora: the machine must identify the antecedent of an ambiguous pronoun in a statement. This makes it a task of natural language processing, but Levesque argues that for Winograd schemas, the task requires the use of knowledge and commonsense reasoning.[2] Nuance Communications announced in July 2014 that it would sponsor an annual WSC competition, with a prize of $25,000 for the best system that could match human performance.[3] However, the prize is no longer offered. Background The Winograd Schema Challenge was proposed in the spirit of the Turing test. Proposed by Alan Turing in 1950, the Turing test plays a central role in the philosophy of artificial intelligence. Turing proposed that, instead of debating whether a machine can think, the science of AI should be concerned with demonstrating intelligent behavior, which can be tested. But the exact nature of the test Turing proposed has come under scrutiny, especially since an AI chatbot named Eugene Goostman claimed to pass it in 2014. One of the major concerns with the Turing test is that a machine could easily pass the test with brute force and/or trickery, rather than true intelligence.[4] The Winograd schema challenge was proposed in 2012 in part to ameliorate the problems that came to light with the nature of the programs that performed well on the test.[5] Turing's original proposal was what he called the imitation game, which involves free-flowing, unrestricted conversations in English between human judges and computer programs over a text-only channel (such as teletype). In general, the machine passes the test if interrogators are not able to tell the difference between it and a human in a five-minute conversation.[4] Weaknesses of the Turing test The performance of Eugene Goostman exhibited some of the Turing test's problems. Levesque identifies several major issues,[2] summarized as follows:[6] • Deception: The machine is forced to construct a false identity, which is not part of intelligence. • Conversation: A lot of interaction may qualify as "legitimate conversation"—jokes, clever asides, points of order—without requiring intelligent reasoning. • Evaluation: Humans make mistakes and judges often would disagree on the results. Winograd schemas The key factor in the WSC is the special format of its questions, which are derived from Winograd schemas. Questions of this form may be tailored to require knowledge and commonsense reasoning in a variety of domains. They must also be carefully written not to betray their answers by selectional restrictions or statistical information about the words in the sentence. Origin The first cited example of a Winograd schema (and the reason for their name) is due to Terry Winograd:[7] The city councilmen refused the demonstrators a permit because they [feared/advocated] violence. The choices of "feared" and "advocated" turn the schema into its two instances: The city councilmen refused the demonstrators a permit because they feared violence. The city councilmen refused the demonstrators a permit because they advocated violence. The schema challenge question is, "Does the pronoun 'they' refer to the city councilmen or the demonstrators?" Switching between the two instances of the schema changes the answer. The answer is immediate for a human reader but proves difficult to emulate in machines. Levesque[2] argues that knowledge plays a central role in these problems: the answer to this schema has to do with our understanding of the typical relationships between and behavior of councilmen and demonstrators. Since the original proposal of the Winograd schema challenge, Ernest Davis, a professor at New York University, has compiled a list of over 140 Winograd schemas from various sources as examples of the kinds of questions that should appear on the Winograd schema challenge.[8] Formal description A Winograd schema challenge question consists of three parts: 1. A sentence or brief discourse that contains the following: • Two noun phrases of the same semantic class (male, female, inanimate, or group of objects or people), • An ambiguous pronoun that may refer to either of the above noun phrases, and • A special word and alternate word, such that if the special word is replaced with the alternate word, the natural resolution of the pronoun changes. 2. A question asking the identity of the ambiguous pronoun, and 3. Two answer choices corresponding to the noun phrases in question. A machine will be given the problem in a standardized form which includes the answer choices, thus making it a binary decision problem. Advantages The Winograd schema challenge has the following purported advantages: • Knowledge and commonsense reasoning are required to solve them. • Winograd schemas of varying difficulty may be designed, involving anything from simple cause-and-effect relationships to complex narratives of events. • They may be constructed to test reasoning ability in specific domains (e.g., social/psychological or spatial reasoning). • There is no need for human judges.[5] Pitfalls One difficulty with the Winograd schema challenge is the development of the questions. They need to be carefully tailored to ensure that they require commonsense reasoning to solve. For example, Levesque[5] gives the following example of a so-called Winograd schema that is "too easy": The women stopped taking pills because they were [pregnant/carcinogenic]. Which individuals were [pregnant/carcinogenic]? The answer to this question can be determined on the basis of selectional restrictions: in any situation, pills do not get pregnant, women do; women cannot be carcinogenic, but pills can. Thus this answer could be derived without the use of reasoning, or any understanding of the sentences' meaning—all that is necessary is data on the selectional restrictions of pregnant and carcinogenic. Activity In 2016 and 2018, Nuance Communications sponsored a competition, offering a grand prize of $25,000 for the top scorer above 90% (for comparison, humans correctly answer to 92–96% of WSC questions[9]). However, nobody came close to winning the prize in 2016 and the 2018 competition was cancelled for lack of prospects;[10] the prize is no longer offered.[11] The Twelfth International Symposium on the Logical Formalizations of Commonsense Reasoning was held on March 23–25, 2015 at the AAAI Spring Symposium Series at Stanford University, with a special focus on the Winograd schema challenge. The organizing committee included Leora Morgenstern (Leidos), Theodore Patkos (The Foundation for Research & Technology Hellas), and Robert Sloan (University of Illinois at Chicago).[12] The 2016 Winograd Schema Challenge was run on July 11, 2016 at IJCAI-16. There were four contestants. The first round of the contest was to solve PDPs—pronoun disambiguation problems, adapted from literary sources, not constructed as pairs of sentences.[13] The highest score achieved was 58% correct, by Quan Liu et al, of the University of Science and Technology, China.[14] Hence, by the rules of that challenge, no prizes were awarded, and the challenge did not proceed to the second round. The organizing committee in 2016 was Leora Morgenstern, Ernest Davis, and Charles Ortiz.[15] In 2017, a neural association model designed for commonsense knowledge acquisition achieved 70% accuracy on 70 manually selected problems from the original 273 Winograd schema dataset.[16] In June 2018, a score of 63.7% accuracy was achieved on the full dataset using an ensemble of recurrent neural network language models,[17] marking the first use of deep neural networks that learn from independent corpora to acquire common sense knowledge. In 2019 a score of 90.1%, was achieved on the original Winograd schema dataset by fine-tuning of the BERT language model with appropriate WSC-like training data to avoid having to learn commonsense reasoning.[9] The general language model GPT-3 achieved a score of 88.3% without specific fine-tuning in 2020.[18] A more challenging, adversarial "Winogrande" dataset of 44,000 problems was designed in 2019. This dataset consists of fill-in-the-blank style sentences, as opposed to the pronoun format of previous datasets.[9] A version of the Winograd schema challenge is one part of the GLUE (General Language Understanding Evaluation) benchmark collection of challenges in automated natural-language understanding.[19] References 1. Ackerman, Evan (29 July 2014). "Can Winograd Schemas Replace Turing Test for Defining Human-level AI". IEEE Spectrum. Retrieved 29 October 2014. 2. Levesque, H. J. (2014). "On our best behaviour". Artificial Intelligence. 212: 27–35. doi:10.1016/j.artint.2014.03.007. 3. "Nuance announces the Winograd Schemas Challenge to Advance Artificial Intelligence Innovation". Business Wire. 28 July 2014. Retrieved 9 November 2014. 4. Turing, Alan (October 1950). "Computing Machinery and Intelligence" (PDF). Mind. LIX (236): 433–460. doi:10.1093/mind/LIX.236.433. Retrieved 28 October 2014. 5. Levesque, Hector; Davis, Ernest; Morgenstern, Leora (2012). The Winograd Schema Challenge. Proceedings of the Thirteenth International Conference on Principles of Knowledge Representation and Reasoning. 6. Michael, Julian (18 May 2015). The Theory of Correlation Formulas and Their Application to Discourse Coherence (Thesis). UT Digital Repository. p. 6. hdl:2152/29979. 7. Winograd, Terry (January 1972). "Understanding Natural Language" (PDF). Cognitive Psychology. 3 (1): 1–191. doi:10.1016/0010-0285(72)90002-3. Retrieved 4 November 2014. 8. Davis, Ernest. "A Collection of Winograd Schemas". cs.nyu.edu. NYU. Retrieved 30 October 2014. 9. Sakaguchi, Keisuke; Le Bras, Ronan; Bhagavatula, Chandra; Choi, Yejin (2019). "WinoGrande: An Adversarial Winograd Schema Challenge at Scale". arXiv:1907.10641 [cs.CL]. 10. Boguslavsky, I.M.; Frolova, T.I.; Iomdin, L.L.; Lazursky, A.V.; Rygaev, I.P.; Timoshenko, S.P. (2019). "Knowledge-based approach to Winograd Schema Challenge" (PDF). Proceedings of the International Conference of Computational Linguistics and Intellectual Technologies. Moscow. The prize could not be awarded to anybody. Most of the participants showed a result close to the random choice or even worse. The second competition scheduled for 2018 was canceled due to the lack of prospective participants. 11. "Winograd Schema Challenge". CommonsenseReasoning.org. Retrieved 24 January 2020. 12. "AAAI 2015 Spring Symposia". Association for the Advancement of Artificial Intelligence. Retrieved 1 January 2015. 13. Davis, Ernest; Morgenstern, Leora; Ortiz, Charles (Fall 2017). "The First Winograd Schema Challenge at IJCAI-16". AI Magazine. 14. Liu, Quan; Jiang, Hui; Ling, Zhen-Hua; Zhu, Xiaodan; Wei, Si; Hu, Yu (2016). "Commonsense Knowledge Enhanced Embeddings for Solving Pronoun Disambiguation Problems in Winograd Schema Challenge". arXiv:1611.04146 [cs.AI]. 15. Morgenstern, Leora; Davis, Ernest; Ortiz, Charles L. (March 2016). "Planning, Executing, and Evaluating the Winograd Schema Challenge". AI Magazine. 37 (1): 50–54. doi:10.1609/aimag.v37i1.2639. ISSN 0738-4602. 16. Liu, Quan; Jiang, Hui; Evdokimov, Andrew; Ling, Zhen-Hua; Zhu, Xiaodan; Wei, Si; Hu, Yu (2017). "Cause-Effect Knowledge Acquisition and Neural Association Model for Solving A Set of Winograd Schema Problems". Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence: 2344–2350. doi:10.24963/ijcai.2017/326. ISBN 9780999241103. 17. Trinh, Trieu H.; Le, Quoc V. (26 September 2019). "A Simple Method for Commonsense Reasoning". arXiv:1806.02847 [cs.AI]. 18. Brown, Tom B.; Mann, Benjamin; Ryder, Nick; Subbiah, Melanie; Kaplan, Jared; Dhariwal, Prafulla; Neelakantan, Arvind; Shyam, Pranav; Sastry, Girish; Askell, Amanda; Agarwal, Sandhini; Herbert-Voss, Ariel; Krueger, Gretchen; Henighan, Tom; Child, Rewon; Ramesh, Aditya; Ziegler, Daniel M.; Wu, Jeffrey; Winter, Clemens; Hesse, Christopher; Chen, Mark; Sigler, Eric; Litwin, Mateusz; Gray, Scott; Chess, Benjamin; Clark, Jack; Berner, Christopher; McCandlish, Sam; Radford, Alec; et al. (2020). "Language Models are Few-Shot Learners". arXiv:2005.14165 [cs.CL]. 19. "GLUE Benchmark". GlueBenchmark.com. Retrieved 30 July 2019. External links • Website for the contest sponsored by Nuance Communications • https://arxiv.org/abs/2201.02387
Wikipedia
Hash chain A hash chain is the successive application of a cryptographic hash function to a piece of data. In computer security, a hash chain is a method used to produce many one-time keys from a single key or password. For non-repudiation, a hash function can be applied successively to additional pieces of data in order to record the chronology of data's existence. Definition A hash chain is a successive application of a cryptographic hash function $h$ to a string $x$. For example, $h(h(h(h(x))))$ gives a hash chain of length 4, often denoted $h^{4}(x)$ Applications Leslie Lamport[1] suggested the use of hash chains as a password protection scheme in an insecure environment. A server which needs to provide authentication may store a hash chain rather than a plain text password and prevent theft of the password in transmission or theft from the server. For example, a server begins by storing $h^{1000}(\mathrm {password} )$ which is provided by the user. When the user wishes to authenticate, they supply $h^{999}(\mathrm {password} )$ to the server. The server computes $h(h^{999}(\mathrm {password} ))=h^{1000}(\mathrm {password} )$ and verifies this matches the hash chain it has stored. It then stores $h^{999}(\mathrm {password} )$ for the next time the user wishes to authenticate. An eavesdropper seeing $h^{999}(\mathrm {password} )$ communicated to the server will be unable to re-transmit the same hash chain to the server for authentication since the server now expects $h^{998}(\mathrm {password} )$. Due to the one-way property of cryptographically secure hash functions, it is infeasible for the eavesdropper to reverse the hash function and obtain an earlier piece of the hash chain. In this example, the user could authenticate 1000 times before the hash chain were exhausted. Each time the hash value is different, and thus cannot be duplicated by an attacker. Binary hash chains Binary hash chains are commonly used in association with a hash tree. A binary hash chain takes two hash values as inputs, concatenates them and applies a hash function to the result, thereby producing a third hash value. The above diagram shows a hash tree consisting of eight leaf nodes and the hash chain for the third leaf node. In addition to the hash values themselves the order of concatenation (right or left 1,0) or "order bits" are necessary to complete the hash chain. Winternitz chains Winternitz chains (also known as function chains[2]) are used in hash-based cryptography. The chain is parameterized by the Winternitz parameter w (number of bits in a "digit" d) and security parameter n (number of bits in the hash value, typically double the security strength,[3] 256 or 512). The chain consists of $2^{w}$ values that are results of repeated application of a one-way "chain" function F to a secret key sk: $sk,F(sk),F(F(sk)),...,F^{2^{w-1}}(sk)$. The chain function is typically based on a standard cryptographic hash, but needs to be parameterized ("randomized"[4]), so it involves few invocations of the underlying hash.[5] In the Winternitz signature scheme a chain is used to encode one digit of the m-bit message, so the Winternitz signature uses approximately $mn/w$ bits, its calculation takes about $2^{w}m/w$ applications of the function F.[3] Note that some signature standards (like Extended Merkle signature scheme, XMSS) define w as the number of possible values in a digit, so $w=16$ in XMSS corresponds to $w=4$ in standards (like Leighton-Micali Signature, LMS) that define w in the same way as above - as a number of bits in the digit.[6] Hash chain vs. blockchain A hash chain is similar to a blockchain, as they both utilize a cryptographic hash function for creating a link between two nodes. However, a blockchain (as used by Bitcoin and related systems) is generally intended to support distributed agreement around a public ledger (data), and incorporates a set of rules for encapsulation of data and associated data permissions. See also • Challenge–response authentication • Hash list – In contrast to the recursive structure of hash chains, the elements of a hash list are independent of each other. • One-time password • Key stretching • Linked timestamping – Binary hash chains are a key component in linked timestamping. • X.509 References 1. L. Lamport, “Password Authentication with Insecure Communication”, Communications of the ACM 24.11 (November 1981), pp 770-772. 2. Hülsing 2013b, pp. 18–20. 3. Buchmann et al. 2011, p. 2. 4. Hülsing 2013b. 5. RFC 8391 6. NIST SP 800-208, Recommendation for Stateful Hash-Based Signature Schemes, p. 5 Sources • Buchmann, Johannes; Dahmen, Erik; Ereth, Sarah; Hülsing, Andreas; Rückert, Markus (2011). "On the Security of the Winternitz One-Time Signature Scheme" (PDF). Lecture Notes in Computer Science. Vol. 6737. Springer Berlin Heidelberg. pp. 363–378. doi:10.1007/978-3-642-21969-6_23. eISSN 1611-3349. ISBN 978-3-642-21968-9. ISSN 0302-9743. • Hülsing, Andreas (2013b). Practical Forward Secure Signatures using Minimal Security Assumptions (PDF) (PhD). TU Darmstadt. • Hülsing, Andreas (2013a). "W-OTS+ – Shorter Signatures for Hash-Based Signature Schemes" (PDF). Progress in Cryptology – AFRICACRYPT 2013. Lecture Notes in Computer Science. Vol. 7918. Springer Berlin Heidelberg. pp. 173–188. doi:10.1007/978-3-642-38553-7_10. eISSN 1611-3349. ISBN 978-3-642-38552-0. ISSN 0302-9743.
Wikipedia
Wireworld Wireworld, alternatively WireWorld, is a cellular automaton first proposed by Brian Silverman in 1987, as part of his program Phantom Fish Tank. It subsequently became more widely known as a result of an article in the "Computer Recreations" column of Scientific American.[1] Wireworld is particularly suited to simulating transistors, and is Turing-complete. Rules A Wireworld cell can be in one of four different states, usually numbered 0–3 in software, modeled by colors in the examples here: 1. empty (black), 2. electron head (blue), 3. electron tail (red), 4. conductor (yellow). As in all cellular automata, time proceeds in discrete steps called generations (sometimes "gens" or "ticks"). Cells behave as follows: • empty → empty, • electron head → electron tail, • electron tail → conductor, • conductor → electron head if exactly one or two of the neighbouring cells are electron heads, otherwise remains conductor. Wireworld uses what is called the Moore neighborhood, which means that in the rules above, neighbouring means one cell away (range value of one) in any direction, both orthogonal and diagonal. These simple rules can be used to construct logic gates (see below). Applications Entities built within Wireworld universes include Langton's Ant (allowing any Langton's Ant pattern to be built within Wireworld)[2] and the Wireworld computer, a Turing-complete computer implemented as a cellular automaton.[3] See also • von Neumann's cellular automaton References 1. Dewdney, A K (January 1990). "Computer recreations: The cellular automata programs that create Wireworld, Rugworld and other diversions". Scientific American. 262 (1): 146–149. JSTOR 24996654. Retrieved 2 December 2018. 2. Nyles Heise. "Wireworld". Archived from the original on 2011-02-04. 3. Mark Owen. "The Wireworld Computer". External links • Wireworld on Rosetta Code • The Wireworld computer in Java • No Wires (contains an interactive Wireworld widget)
Wikipedia
Wirtinger's inequality for functions In the mathematical field of analysis, the Wirtinger inequality is an important inequality for functions of a single variable, named after Wilhelm Wirtinger. It was used by Adolf Hurwitz in 1901 to give a new proof of the isoperimetric inequality for curves in the plane. A variety of closely related results are today known as Wirtinger's inequality, all of which can be viewed as certain forms of the Poincaré inequality. For other inequalities named after Wirtinger, see Wirtinger's inequality. Theorem There are several inequivalent versions of the Wirtinger inequality: • Let y be a continuous and differentiable function on the interval [0, L] with average value zero and with y(0) = y(L). Then $\int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{4\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x,$ and equality holds if and only if y(x) = c sin 2π(x − α)/L for some numbers c and α.[1] • Let y be a continuous and differentiable function on the interval [0, L] with y(0) = y(L) = 0. Then $\int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x,$ and equality holds if and only if y(x) = c sin πx/L for some number c.[1] • Let y be a continuous and differentiable function on the interval [0, L] with average value zero. Then $\int _{0}^{L}y(x)^{2}\,\mathrm {d} x\leq {\frac {L^{2}}{\pi ^{2}}}\int _{0}^{L}y'(x)^{2}\,\mathrm {d} x.$ and equality holds if and only if y(x) = c cos πx/L for some number c.[2] Despite their differences, these are closely related to one another, as can be seen from the account given below in terms of spectral geometry. They can also all be regarded as special cases of various forms of the Poincaré inequality, with the optimal Poincaré constant identified explicitly. The middle version is also a special case of the Friedrichs inequality, again with the optimal constant identified. Proofs The three versions of the Wirtinger inequality can all be proved by various means. This is illustrated in the following by a different kind of proof for each of the three Wirtinger inequalities given above. In each case, by a linear change of variables in the integrals involved, there is no loss of generality in only proving the theorem for one particular choice of L. Fourier series Consider the first Wirtinger inequality given above. Take L to be 2π. Since Dirichlet's conditions are met, we can write $y(x)={\frac {1}{2}}a_{0}+\sum _{n\geq 1}\left(a_{n}{\frac {\sin nx}{\sqrt {\pi }}}+b_{n}{\frac {\cos nx}{\sqrt {\pi }}}\right),$ and the fact that the average value of y is zero means that a0 = 0. By Parseval's identity, $\int _{0}^{2\pi }y(x)^{2}\,\mathrm {d} x=\sum _{n=1}^{\infty }(a_{n}^{2}+b_{n}^{2})$ and $\int _{0}^{2\pi }y'(x)^{2}\,\mathrm {d} x=\sum _{n=1}^{\infty }n^{2}(a_{n}^{2}+b_{n}^{2})$ and since the summands are all nonnegative, the Wirtinger inequality is proved. Furthermore it is seen that equality holds if and only if an = bn = 0 for all n ≥ 2, which is to say that y(x) = a1 sin x + b1 cos x. This is equivalent to the stated condition by use of the trigonometric addition formulas. Integration by parts Consider the second Wirtinger inequality given above.[1] Take L to be π. Any differentiable function y(x) satisfies the identity $y(x)^{2}+{\big (}y'(x)-y(x)\cot x{\big )}^{2}=y'(x)^{2}-{\frac {d}{dx}}{\big (}y(x)^{2}\cot x{\big )}.$ Integration using the fundamental theorem of calculus and the boundary conditions y(0) = y(π) = 0 then shows $\int _{0}^{\pi }y(x)^{2}\,\mathrm {d} x+\int _{0}^{\pi }{\big (}y'(x)-y(x)\cot x{\big )}^{2}\,\mathrm {d} x=\int _{0}^{\pi }y'(x)^{2}\,\mathrm {d} x.$ This proves the Wirtinger inequality, since the second integral is clearly nonnegative. Furthermore, equality in the Wirtinger inequality is seen to be equivalent to y′(x) = y(x) cot x, the general solution of which (as computed by separation of variables) is y(x) = c sin x for an arbitrary number c. There is a subtlety in the above application of the fundamental theorem of calculus, since it is not the case that y(x)2 cot x extends continuously to x = 0 and x = π for every function y(x). This is resolved as follows. It follows from the Hölder inequality and y(0) = 0 that $|y(x)|=\left|\int _{0}^{x}y'(x)\,\mathrm {d} x\right|\leq \int _{0}^{x}|y'(x)|\,\mathrm {d} x\leq {\sqrt {x}}\left(\int _{0}^{x}y'(x)^{2}\,\mathrm {d} x\right)^{1/2},$ which shows that as long as $\int _{0}^{\pi }y'(x)^{2}\,\mathrm {d} x$ is finite, the limit of 1/x y(x)2 as x converges to zero is zero. Since cot x < 1/x for small positive values of x, it follows from the squeeze theorem that y(x)2 cot x converges to zero as x converges to zero. In exactly the same way, it can be proved that y(x)2 cot x converges to zero as x converges to π. Functional analysis Consider the third Wirtinger inequality given above. Take L to be 1. Given a continuous function f on [0, 1] of average value zero, let Tf) denote the function u on [0, 1] which is of average value zero, and with u′′ + f = 0 and u′(0) = u′(1) = 0. From basic analysis of ordinary differential equations with constant coefficients, the eigenvalues of T are (kπ)−2 for nonzero integers k, the largest of which is then π−2. Because T is a bounded and self-adjoint operator, it follows that $\int _{0}^{1}Tf(x)^{2}\,\mathrm {d} x\leq \pi ^{-2}\int _{0}^{1}f(x)Tf(x)\,\mathrm {d} x={\frac {1}{\pi ^{2}}}\int _{0}^{1}(Tf)'(x)^{2}\,\mathrm {d} x$ for all f of average value zero, where the equality is due to integration by parts. Finally, for any continuously differentiable function y on [0, 1] of average value zero, let gn be a sequence of compactly supported continuously differentiable functions on (0, 1) which converge in L2 to y′. Then define $y_{n}(x)=\int _{0}^{x}g_{n}(z)\,\mathrm {d} z-\int _{0}^{1}\int _{0}^{w}g_{n}(z)\,\mathrm {d} z\,\mathrm {d} w.$ Then each yn has average value zero with yn′(0) = yn′(1) = 0, which in turn implies that −yn′′ has average value zero. So application of the above inequality to f = −yn′′ is legitimate and shows that $\int _{0}^{1}y_{n}(x)^{2}\,\mathrm {d} x\leq {\frac {1}{\pi ^{2}}}\int _{0}^{1}y_{n}'(x)^{2}\,\mathrm {d} x.$ It is possible to replace yn by y, and thereby prove the Wirtinger inequality, as soon as it is verified that yn converges in L2 to y. This is verified in a standard way, by writing $y(x)-y_{n}(x)=\int _{0}^{x}{\big (}y_{n}'(z)-g_{n}(z){\big )}\,\mathrm {d} z-\int _{0}^{1}\int _{0}^{w}(y_{n}'(z)-g_{n}(z){\big )}\,\mathrm {d} z\,\mathrm {d} w$ and applying the Hölder or Jensen inequalities. This proves the Wirtinger inequality. In the case that y(x) is a function for which equality in the Wirtinger inequality holds, then a standard argument in the calculus of variations says that y must be a weak solution of the Euler–Lagrange equation y′′(x) + y(x) = 0 with y′(0) = y′(1) = 0, and the regularity theory of such equations, followed by the usual analysis of ordinary differential equations with constant coefficients, shows that y(x) = c cos πx for some number c. To make this argument fully formal and precise, it is necessary to be more careful about the function spaces in question.[2] Spectral geometry In the language of spectral geometry, the three versions of the Wirtinger inequality above can be rephrased as theorems about the first eigenvalue and corresponding eigenfunctions of the Laplace–Beltrami operator on various one-dimensional Riemannian manifolds:[3] • the first eigenvalue of the Laplace–Beltrami operator on the Riemannian circle of length L is 4π2/L2, and the corresponding eigenfunctions are the linear combinations of the two coordinate functions. • the first Dirichlet eigenvalue of the Laplace–Beltrami operator on the interval [0, L] is π2/L2 and the corresponding eigenfunctions are given by c sin πx/L for arbitrary nonzero numbers c. • the first Neumann eigenvalue of the Laplace–Beltrami operator on the interval [0, L] is π2/L2 and the corresponding eigenfunctions are given by c cos πx/L for arbitrary nonzero numbers c. These can also be extended to statements about higher-dimensional spaces. For example, the Riemannian circle may be viewed as the one-dimensional version of either a sphere, real projective space, or torus (of arbitrary dimension). The Wirtinger inequality, in the first version given here, can then be seen as the n = 1 case of any of the following: • the first eigenvalue of the Laplace–Beltrami operator on the unit-radius n-dimensional sphere is n, and the corresponding eigenfunctions are the linear combinations of the n + 1 coordinate functions.[4] • the first eigenvalue of the Laplace–Beltrami operator on the n-dimensional real projective space (with normalization given by the covering map from the unit-radius sphere) is 2n + 2, and the corresponding eigenfunctions are the restrictions of the homogeneous quadratic polynomials on Rn + 1 to the unit sphere (and then to the real projective space).[5] • the first eigenvalue of the Laplace–Beltrami operator on the n-dimensional torus (given as the n-fold product of the circle of length 2π with itself) is 1, and the corresponding eigenfunctions are arbitrary linear combinations of n-fold products of the eigenfunctions on the circles.[6] The second and third versions of the Wirtinger inequality can be extended to statements about first Dirichlet and Neumann eigenvalues of the Laplace−Beltrami operator on metric balls in Euclidean space: • the first Dirichlet eigenvalue of the Laplace−Beltrami operator on the unit ball in Rn is the square of the smallest positive zero of the Bessel function of the first kind J(n − 2)/2.[7] • the first Neumann eigenvalue of the Laplace−Beltrami operator on the unit ball in Rn is the square of the smallest positive zero of the first derivative of the Bessel function of the first kind Jn/2.[7] Application to the isoperimetric inequality In the first form given above, the Wirtinger inequality can be used to prove the isoperimetric inequality for curves in the plane, as found by Adolf Hurwitz in 1901.[8] Let (x, y) be a differentiable embedding of the circle in the plane. Parametrizing the circle by [0, 2π] so that (x, y) has constant speed, the length L of the curve is given by $\int _{0}^{2\pi }{\sqrt {x'(t)^{2}+y'(t)^{2}}}\,\mathrm {d} t$ and the area A enclosed by the curve is given (due to Stokes theorem) by $-\int _{0}^{2\pi }y(t)x'(t)\,\mathrm {d} t.$ Since the integrand of the integral defining L is assumed constant, there is ${\frac {L^{2}}{2\pi }}-2A=\int _{0}^{2\pi }{\big (}x'(t)^{2}+y'(t)^{2}+2y(t)x'(t){\big )}\,\mathrm {d} t$ which can be rewritten as $\int _{0}^{2\pi }{\big (}x'(t)+y(t){\big )}^{2}\,\mathrm {d} t+\int _{0}^{2\pi }{\big (}y'(t)^{2}-y(t)^{2}{\big )}\,\mathrm {d} t.$ The first integral is clearly nonnegative. Without changing the area or length of the curve, (x, y) can be replaced by (x, y + z) for some number z, so as to make y have average value zero. Then the Wirtinger inequality can be applied to see that the second integral is also nonnegative, and therefore ${\frac {L^{2}}{4\pi }}\geq A,$ which is the isoperimetric inequality. Furthermore, equality in the isoperimetric inequality implies both equality in the Wirtinger inequality and also the equality x′(t) + y(t) = 0, which amounts to y(t) = c1 sin(t – α) and then x(t) = c1 cos(t – α) + c2 for arbitrary numbers c1 and c2. These equations mean that the image of (x, y) is a round circle in the plane. References 1. Hardy, Littlewood & Pólya 1952, Section 7.7. 2. Brezis 2011, pp. 511–513, 576–578. 3. Chavel 1984, Sections I.3 and I.5. 4. Stein & Weiss 1971, Chapter IV.2. 5. Chavel 1984, p. 36. 6. Chavel 1984, Section II.2. 7. Chavel 1984, Theorem II.5.4. 8. Hardy, Littlewood & Pólya 1952, Section 7.7; Hurwitz 1901. • Brezis, Haim (2011). Functional analysis, Sobolev spaces and partial differential equations. Universitext. New York: Springer. doi:10.1007/978-0-387-70914-7. ISBN 978-0-387-70913-0. MR 2759829. Zbl 1220.46002. • Chavel, Isaac (1984). Eigenvalues in Riemannian geometry. Pure and Applied Mathematics. Vol. 115. Orlando, FL: Academic Press. doi:10.1016/s0079-8169(08)x6051-9. ISBN 0-12-170640-0. MR 0768584. Zbl 0551.53001. • Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952). Inequalities (Second edition of 1934 original ed.). Cambridge University Press. MR 0046395. Zbl 0047.05302. • Hurwitz, A. (1901). "Sur le problème des isopérimètres". Comptes Rendus des Séances de l'Académie des Sciences. 132: 401–403. JFM 32.0386.01. • Stein, Elias M.; Weiss, Guido (1971). Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series. Vol. 32. Princeton, NJ: Princeton University Press. MR 0304972. Zbl 0232.42007.
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Wirtinger's representation and projection theorem In mathematics, Wirtinger's representation and projection theorem is a theorem proved by Wilhelm Wirtinger in 1932 in connection with some problems of approximation theory. This theorem gives the representation formula for the holomorphic subspace $\left.\right.H_{2}$ of the simple, unweighted holomorphic Hilbert space $\left.\right.L^{2}$ of functions square-integrable over the surface of the unit disc $\left.\right.\{z:|z|<1\}$ of the complex plane, along with a form of the orthogonal projection from $\left.\right.L^{2}$ to $\left.\right.H_{2}$. Wirtinger's paper [1] contains the following theorem presented also in Joseph L. Walsh's well-known monograph [2] (p. 150) with a different proof. If $\left.\right.\left.F(z)\right.$ is of the class $\left.\right.L^{2}$ on $\left.\right.|z|<1$, i.e. $\iint _{|z|<1}|F(z)|^{2}\,dS<+\infty ,$ where $\left.\right.dS$ is the area element, then the unique function $\left.\right.f(z)$ of the holomorphic subclass $H_{2}\subset L^{2}$, such that $\iint _{|z|<1}|F(z)-f(z)|^{2}\,dS$ is least, is given by $f(z)={\frac {1}{\pi }}\iint _{|\zeta |<1}F(\zeta ){\frac {dS}{(1-{\overline {\zeta }}z)^{2}}},\quad |z|<1.$ The last formula gives a form for the orthogonal projection from $\left.\right.L^{2}$ to $\left.\right.H_{2}$. Besides, replacement of $\left.\right.F(\zeta )$ by $\left.\right.f(\zeta )$ makes it Wirtinger's representation for all $f(z)\in H_{2}$. This is an analog of the well-known Cauchy integral formula with the square of the Cauchy kernel. Later, after the 1950s, a degree of the Cauchy kernel was called reproducing kernel, and the notation $\left.\right.A_{0}^{2}$ became common for the class $\left.\right.H_{2}$. In 1948 Mkhitar Djrbashian[3] extended Wirtinger's representation and projection to the wider, weighted Hilbert spaces $\left.\right.A_{\alpha }^{2}$ of functions $\left.\right.f(z)$ holomorphic in $\left.\right.|z|<1$, which satisfy the condition $\|f\|_{A_{\alpha }^{2}}=\left\{{\frac {1}{\pi }}\iint _{|z|<1}|f(z)|^{2}(1-|z|^{2})^{\alpha -1}\,dS\right\}^{1/2}<+\infty {\text{ for some }}\alpha \in (0,+\infty ),$ and also to some Hilbert spaces of entire functions. The extensions of these results to some weighted $\left.\right.A_{\omega }^{2}$ spaces of functions holomorphic in $\left.\right.|z|<1$ and similar spaces of entire functions, the unions of which respectively coincide with all functions holomorphic in $\left.\right.|z|<1$ and the whole set of entire functions can be seen in.[4] See also • Jerbashian, A. M.; V. S. Zakaryan (2009). "The Contemporary Development in M. M. Djrbashian Factorization Theory and Related Problems of Analysis". Izv. NAN of Armenia, Matematika (English translation: Journal of Contemporary Mathematical Analysis). 44 (6). References 1. Wirtinger, W. (1932). "Uber eine Minimumaufgabe im Gebiet der analytischen Functionen". Monatshefte für Mathematik und Physik. 39: 377–384. doi:10.1007/bf01699078. S2CID 120529823. 2. Walsh, J. L. (1956). "Interpolation and Approximation by Rational Functions in the Complex Domain". Amer. Math. Soc. Coll. Publ. XX. Ann Arbor, Michigan: Edwards Brothers, Inc. 3. Djrbashian, M. M. (1948). "On the Representability Problem of Analytic Functions" (PDF). Soobsch. Inst. Matem. I Mekh. Akad. Nauk Arm. SSR. 2: 3–40. 4. Jerbashian, A. M. (2005). "On the Theory of Weighted Classes of Area Integrable Regular Functions". Complex Variables. 50 (3): 155–183. doi:10.1080/02781070500032846. S2CID 218556016. Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C*-algebra • Spectrum of a C*-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory •  Mathematics portal • Category • Commons
Wikipedia
Wirtinger derivatives In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators[1]), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions on complex domains. These operators permit the construction of a differential calculus for such functions that is entirely analogous to the ordinary differential calculus for functions of real variables.[2] Historical notes Early days (1899–1911): the work of Henri Poincaré Wirtinger derivatives were used in complex analysis at least as early as in the paper (Poincaré 1899), as briefly noted by Cherry & Ye (2001, p. 31) and by Remmert (1991, pp. 66–67).[3] As a matter of fact, in the third paragraph of his 1899 paper,[4] Henri Poincaré first defines the complex variable in $\mathbb {C} ^{n}$ and its complex conjugate as follows ${\begin{cases}x_{k}+iy_{k}=z_{k}\\x_{k}-iy_{k}=u_{k}\end{cases}}\qquad 1\leqslant k\leqslant n.$ Then he writes the equation defining the functions $V$ he calls biharmonique,[5] previously written using partial derivatives with respect to the real variables $x_{k},y_{q}$ with $k,q$ ranging from 1 to $n$, exactly in the following way[6] ${\frac {d^{2}V}{dz_{k}\,du_{q}}}=0$ This implies that he implicitly used definition 2 below: to see this it is sufficient to compare equations 2 and 2' of (Poincaré 1899, p. 112). Apparently, this paper was not noticed by early researchers in the theory of functions of several complex variables: in the papers of Levi-Civita (1905), Levi (1910) (and Levi 1911) and of Amoroso (1912) all fundamental partial differential operators of the theory are expressed directly by using partial derivatives respect to the real and imaginary parts of the complex variables involved. In the long survey paper by Osgood (1966) (first published in 1913),[7] partial derivatives with respect to each complex variable of a holomorphic function of several complex variables seem to be meant as formal derivatives: as a matter of fact when Osgood expresses the pluriharmonic operator[8] and the Levi operator, he follows the established practice of Amoroso, Levi and Levi-Civita. The work of Dimitrie Pompeiu in 1912 and 1913: a new formulation According to Henrici (1993, p. 294), a new step in the definition of the concept was taken by Dimitrie Pompeiu: in the paper (Pompeiu 1912), given a complex valued differentiable function (in the sense of real analysis) of one complex variable $g(z)$ defined in the neighbourhood of a given point $z_{0}\in \mathbb {C} ,$ he defines the areolar derivative as the following limit ${{\frac {\partial g}{\partial {\bar {z}}}}(z_{0})}\mathrel {\overset {\mathrm {def} }{=}} \lim _{r\to 0}{\frac {1}{2\pi ir^{2}}}\oint _{\Gamma (z_{0},r)}g(z)\mathrm {d} z,$ where $\Gamma (z_{0},r)=\partial D(z_{0},r)$ is the boundary of a disk of radius $r$ entirely contained in the domain of definition of $g(z),$ i.e. his bounding circle.[9] This is evidently an alternative definition of Wirtinger derivative respect to the complex conjugate variable:[10] it is a more general one, since, as noted a by Henrici (1993, p. 294), the limit may exist for functions that are not even differentiable at $z=z_{0}.$[11] According to Fichera (1969, p. 28), the first to identify the areolar derivative as a weak derivative in the sense of Sobolev was Ilia Vekua.[12] In his following paper, Pompeiu (1913) uses this newly defined concept in order to introduce his generalization of Cauchy's integral formula, the now called Cauchy–Pompeiu formula. The work of Wilhelm Wirtinger The first systematic introduction of Wirtinger derivatives seems due to Wilhelm Wirtinger in the paper Wirtinger 1927 in order to simplify the calculations of quantities occurring in the theory of functions of several complex variables: as a result of the introduction of these differential operators, the form of all the differential operators commonly used in the theory, like the Levi operator and the Cauchy–Riemann operator, is considerably simplified and consequently easier to handle. The paper is deliberately written from a formal point of view, i.e. without giving a rigorous derivation of the properties deduced. Formal definition Despite their ubiquitous use,[13] it seems that there is no text listing all the properties of Wirtinger derivatives: however, fairly complete references are the short course on multidimensional complex analysis by Andreotti (1976, pp. 3–5),[14] the monograph of Gunning & Rossi (1965, pp. 3–6),[15] and the monograph of Kaup & Kaup (1983, p. 2,4)[16] which are used as general references in this and the following sections. Functions of one complex variable Definition 1. Consider the complex plane $\mathbb {C} \equiv \mathbb {R} ^{2}=\{(x,y)\mid x,y\in \mathbb {R} \}$ (in a sense of expressing a complex number $z=x+iy$ for real numbers $x$ and $y$). The Wirtinger derivatives are defined as the following linear partial differential operators of first order: ${\begin{aligned}{\frac {\partial }{\partial z}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x}}-i{\frac {\partial }{\partial y}}\right)\\{\frac {\partial }{\partial {\bar {z}}}}&={\frac {1}{2}}\left({\frac {\partial }{\partial x}}+i{\frac {\partial }{\partial y}}\right)\end{aligned}}$ Clearly, the natural domain of definition of these partial differential operators is the space of $C^{1}$ functions on a domain $\Omega \subseteq \mathbb {R} ^{2},$ but, since these operators are linear and have constant coefficients, they can be readily extended to every space of generalized functions. Functions of n > 1 complex variables Definition 2. Consider the Euclidean space on the complex field $\mathbb {C} ^{n}=\mathbb {R} ^{2n}=\left\{\left(\mathbf {x} ,\mathbf {y} \right)=\left(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n}\right)\mid \mathbf {x} ,\mathbf {y} \in \mathbb {R} ^{n}\right\}.$ The Wirtinger derivatives are defined as the following linear partial differential operators of first order: ${\begin{cases}{\frac {\partial }{\partial z_{1}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{1}}}-i{\frac {\partial }{\partial y_{1}}}\right)\\\qquad \vdots \\{\frac {\partial }{\partial z_{n}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{n}}}-i{\frac {\partial }{\partial y_{n}}}\right)\\\end{cases}},\qquad {\begin{cases}{\frac {\partial }{\partial {\bar {z}}_{1}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{1}}}+i{\frac {\partial }{\partial y_{1}}}\right)\\\qquad \vdots \\{\frac {\partial }{\partial {\bar {z}}_{n}}}={\frac {1}{2}}\left({\frac {\partial }{\partial x_{n}}}+i{\frac {\partial }{\partial y_{n}}}\right)\\\end{cases}}.$ As for Wirtinger derivatives for functions of one complex variable, the natural domain of definition of these partial differential operators is again the space of $C^{1}$ functions on a domain $\Omega \subset \mathbb {R} ^{2n},$ and again, since these operators are linear and have constant coefficients, they can be readily extended to every space of generalized functions. Relation with complex differentiation Wirtinger derivatives are closely related with complex differentiation (differentiation with respect to a complex variable $z=x+iy$ where $x$ and $y$ are real variables). The first Wirtinger derivative in the definition 1 is really differentiation with respect to $z$. For a complex function $f(z)=u(z)+iv(z)$ which is complex differentiable (equivalent to satisfying the Cauchy-Riemann equations ${\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}},{\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}$), ${\begin{aligned}{\frac {\partial f}{\partial z}}&={\frac {1}{2}}\left({\frac {\partial f}{\partial x}}-i{\frac {\partial f}{\partial y}}\right)\\&={\frac {1}{2}}\left({\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}-i{\frac {\partial u}{\partial y}}+{\frac {\partial v}{\partial y}}\right)\\&={\frac {\partial u}{\partial x}}+i{\frac {\partial v}{\partial x}}={\frac {\partial f}{\partial x}}\end{aligned}}$ where the 3rd equality uses the Cauchy-Riemann equations. Because the complex derivative is independent of the choice of a path in differentiation, the first Wirtinger derivative is the complex derivative. The second Wirtinger derivative is also related with complex differentiation; ${\frac {\partial f}{\partial {\bar {z}}}}=0$ is equivalent to the Cauchy-Riemann equations in a complex form. Basic properties In the present section and in the following ones it is assumed that $z\in \mathbb {C} ^{n}$ is a complex vector and that $z\equiv (x,y)=(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})$ where $x,y$ are real vectors, with n ≥ 1: also it is assumed that the subset $\Omega $ can be thought of as a domain in the real euclidean space $\mathbb {R} ^{2n}$ or in its isomorphic complex counterpart $\mathbb {C} ^{n}.$ All the proofs are easy consequences of definition 1 and definition 2 and of the corresponding properties of the derivatives (ordinary or partial). Linearity Lemma 1. If $f,g\in C^{1}(\Omega )$ and $\alpha ,\beta $ are complex numbers, then for $i=1,\dots ,n$ the following equalities hold ${\begin{aligned}{\frac {\partial }{\partial z_{i}}}\left(\alpha f+\beta g\right)&=\alpha {\frac {\partial f}{\partial z_{i}}}+\beta {\frac {\partial g}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}\left(\alpha f+\beta g\right)&=\alpha {\frac {\partial f}{\partial {\bar {z}}_{i}}}+\beta {\frac {\partial g}{\partial {\bar {z}}_{i}}}\end{aligned}}$ Product rule Lemma 2. If $f,g\in C^{1}(\Omega ),$ then for $i=1,\dots ,n$ the product rule holds ${\begin{aligned}{\frac {\partial }{\partial z_{i}}}(f\cdot g)&={\frac {\partial f}{\partial z_{i}}}\cdot g+f\cdot {\frac {\partial g}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}(f\cdot g)&={\frac {\partial f}{\partial {\bar {z}}_{i}}}\cdot g+f\cdot {\frac {\partial g}{\partial {\bar {z}}_{i}}}\end{aligned}}$ This property implies that Wirtinger derivatives are derivations from the abstract algebra point of view, exactly like ordinary derivatives are. Chain rule This property takes two different forms respectively for functions of one and several complex variables: for the n > 1 case, to express the chain rule in its full generality it is necessary to consider two domains $\Omega '\subseteq \mathbb {C} ^{m}$ and $\Omega ''\subseteq \mathbb {C} ^{p}$ and two maps $g:\Omega '\to \Omega $ and $f:\Omega \to \Omega ''$ having natural smoothness requirements.[17] Functions of one complex variable Lemma 3.1 If $f,g\in C^{1}(\Omega ),$ and $g(\Omega )\subseteq \Omega ,$ then the chain rule holds ${\begin{aligned}{\frac {\partial }{\partial z}}(f\circ g)&=\left({\frac {\partial f}{\partial z}}\circ g\right){\frac {\partial g}{\partial z}}+\left({\frac {\partial f}{\partial {\bar {z}}}}\circ g\right){\frac {\partial {\bar {g}}}{\partial z}}\\{\frac {\partial }{\partial {\bar {z}}}}(f\circ g)&=\left({\frac {\partial f}{\partial z}}\circ g\right){\frac {\partial g}{\partial {\bar {z}}}}+\left({\frac {\partial f}{\partial {\bar {z}}}}\circ g\right){\frac {\partial {\bar {g}}}{\partial {\bar {z}}}}\end{aligned}}$ Functions of n > 1 complex variables Lemma 3.2 If $g\in C^{1}(\Omega ',\Omega )$ and $f\in C^{1}(\Omega ,\Omega ''),$ then for $i=1,\dots ,m$ the following form of the chain rule holds ${\begin{aligned}{\frac {\partial }{\partial z_{i}}}\left(f\circ g\right)&=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial z_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial z_{i}}}\\{\frac {\partial }{\partial {\bar {z}}_{i}}}\left(f\circ g\right)&=\sum _{j=1}^{n}\left({\frac {\partial f}{\partial z_{j}}}\circ g\right){\frac {\partial g_{j}}{\partial {\bar {z}}_{i}}}+\sum _{j=1}^{n}\left({\frac {\partial f}{\partial {\bar {z}}_{j}}}\circ g\right){\frac {\partial {\bar {g}}_{j}}{\partial {\bar {z}}_{i}}}\end{aligned}}$ Conjugation Lemma 4. If $f\in C^{1}(\Omega ),$ then for $i=1,\dots ,n$ the following equalities hold ${\begin{aligned}{\overline {\left({\frac {\partial f}{\partial z_{i}}}\right)}}&={\frac {\partial {\bar {f}}}{\partial {\bar {z}}_{i}}}\\{\overline {\left({\frac {\partial f}{\partial {\bar {z}}_{i}}}\right)}}&={\frac {\partial {\bar {f}}}{\partial z_{i}}}\end{aligned}}$ See also • CR–function • Dolbeault complex • Dolbeault operator • Pluriharmonic function Notes 1. See references Fichera 1986, p. 62 and Kracht & Kreyszig 1988, p. 10. 2. Some of the basic properties of Wirtinger derivatives are the same ones as the properties characterizing the ordinary (or partial) derivatives and used for the construction of the usual differential calculus. 3. Reference to the work Poincaré 1899 of Henri Poincaré is precisely stated by Cherry & Ye (2001), while Reinhold Remmert does not cite any reference to support his assertion. 4. See reference (Poincaré 1899, pp. 111–114) 5. These functions are precisely pluriharmonic functions, and the linear differential operator defining them, i.e. the operator in equation 2 of (Poincaré 1899, p. 112), is exactly the n-dimensional pluriharmonic operator. 6. See (Poincaré 1899, p. 112), equation 2': note that, throughout the paper, the symbol $d$ is used to signify partial differentiation respect to a given variable, instead of the now commonplace symbol ∂. 7. The corrected Dover edition of the paper (Osgood 1913) harv error: no target: CITEREFOsgood1913 (help) contains much important historical information on the early development of the theory of functions of several complex variables, and is therefore a useful source. 8. See Osgood (1966, pp. 23–24): curiously, he calls Cauchy–Riemann equations this set of equations. 9. This is the definition given by Henrici (1993, p. 294) in his approach to Pompeiu's work: as Fichera (1969, p. 27) remarks, the original definition of Pompeiu (1912) does not require the domain of integration to be a circle. See the entry areolar derivative for further information. 10. See the section "Formal definition" of this entry. 11. See problem 2 in Henrici 1993, p. 294 for one example of such a function. 12. See also the excellent book by Vekua (1962, p. 55), Theorem 1.31: If the generalized derivative $\partial _{\bar {z}}w\in $$L_{p}(\Omega )$, p > 1, then the function $w(z)$ has almost everywhere in $G$ a derivative in the sense of Pompeiu, the latter being equal to the Generalized derivative in the sense of Sobolev $\partial _{\bar {z}}w$. 13. With or without the attribution of the concept to Wilhelm Wirtinger: see, for example, the well known monograph Hörmander 1990, p. 1,23. 14. In this course lectures, Aldo Andreotti uses the properties of Wirtinger derivatives in order to prove the closure of the algebra of holomorphic functions under certain operations: this purpose is common to all references cited in this section. 15. This is a classical work on the theory of functions of several complex variables dealing mainly with its sheaf theoretic aspects: however, in the introductory sections, Wirtinger derivatives and a few other analytical tools are introduced and their application to the theory is described. 16. In this work, the authors prove some of the properties of Wirtinger derivatives also for the general case of $C^{1}$ functions: in this single aspect, their approach is different from the one adopted by the other authors cited in this section, and perhaps more complete. 17. See Kaup & Kaup 1983, p. 4 and also Gunning 1990, p. 5: Gunning considers the general case of $C^{1}$ functions but only for p = 1. References Andreotti 1976, p. 5 and Gunning & Rossi 1965, p. 6, as already pointed out, consider only holomorphic maps with p = 1: however, the resulting formulas are formally very similar. References Historical references • Amoroso, Luigi (1912), "Sopra un problema al contorno", Rendiconti del Circolo Matematico di Palermo (in Italian), 33 (1): 75–85, doi:10.1007/BF03015289, JFM 43.0453.03, S2CID 122956910. "On a boundary value problem" (free translation of the title) is the first paper where a set of (fairly complicate) necessary and sufficient conditions for the solvability of the Dirichlet problem for holomorphic functions of several variables is given. • Cherry, W.; Ye, Z. (2001), Nevanlinna's theory of value distribution: the second main theorem and its error terms, Springer Monographs in Mathematics, Berlin: Springer Verlag, pp. XII+202, ISBN 978-3-540-66416-1, MR 1831783, Zbl 0981.30001. • Fichera, Gaetano (1969), "Derivata areolare e funzioni a variazione limitata", Revue Roumaine de Mathématiques Pures et Appliquées (in Italian), XIV (1): 27–37, MR 0265616, Zbl 0201.10002. "Areolar derivative and functions of bounded variation" (free English translation of the title) is an important reference paper in the theory of areolar derivatives. • Levi, Eugenio Elia (1910), "Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse", Annali di Matematica Pura ed Applicata, s. III (in Italian), XVII (1): 61–87, doi:10.1007/BF02419336, JFM 41.0487.01, S2CID 122678686. "Studies on essential singular points of analytic functions of two or more complex variables" (English translation of the title) is an important paper in the theory of functions of several complex variables, where the problem of determining what kind of hypersurface can be the boundary of a domain of holomorphy. • Levi, Eugenio Elia (1911), "Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse", Annali di Matematica Pura ed Applicata, s. III (in Italian), XVIII (1): 69–79, doi:10.1007/BF02420535, JFM 42.0449.02, S2CID 120133326. "On the hypersurfaces of the 4-dimensional space that can be the boundary of the domain of existence of an analytic function of two complex variables" (English translation of the title) is another important paper in the theory of functions of several complex variables, investigating further the theory started in (Levi 1910). • Levi-Civita, Tullio (1905), "Sulle funzioni di due o più variabili complesse", Rendiconti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 5 (in Italian), XIV (2): 492–499, JFM 36.0482.01. "On the functions of two or more complex variables" (free English translation of the title) is the first paper where a sufficient condition for the solvability of the Cauchy problem for holomorphic functions of several complex variables is given. • Osgood, William Fogg (1966) [1913], Topics in the theory of functions of several complex variables (unabridged and corrected ed.), New York: Dover, pp. IV+120, JFM 45.0661.02, MR 0201668, Zbl 0138.30901. • Peschl, Ernst (1932), "Über die Krümmung von Niveaukurven bei der konformen Abbildung einfachzusammenhängender Gebiete auf das Innere eines Kreises. Eine Verallgemeinerung eines Satzes von E. Study.", Mathematische Annalen (in German), 106: 574–594, doi:10.1007/BF01455902, JFM 58.1096.05, MR 1512774, S2CID 127138808, Zbl 0004.30001, available at DigiZeitschriften. • Poincaré, H. (1899), "Sur les propriétés du potentiel et sur les fonctions Abéliennes", Acta Mathematica (in French), 22 (1): 89–178, doi:10.1007/BF02417872, JFM 29.0370.02. • Pompeiu, D. (1912), "Sur une classe de fonctions d'une variable complexe", Rendiconti del Circolo Matematico di Palermo (in French), 33 (1): 108–113, doi:10.1007/BF03015292, JFM 43.0481.01, S2CID 120717465. • Pompeiu, D. (1913), "Sur une classe de fonctions d'une variable complexe et sur certaines équations intégrales", Rendiconti del Circolo Matematico di Palermo (in French), 35 (1): 277–281, doi:10.1007/BF03015607, S2CID 121616964. • Vekua, I. N. (1962), Generalized Analytic Functions, International Series of Monographs in Pure and Applied Mathematics, vol. 25, London–Paris–Frankfurt: Pergamon Press, pp. xxx+668, MR 0150320, Zbl 0100.07603 • Wirtinger, Wilhelm (1927), "Zur formalen Theorie der Funktionen von mehr komplexen Veränderlichen", Mathematische Annalen (in German), 97: 357–375, doi:10.1007/BF01447872, JFM 52.0342.03, S2CID 121149132, available at DigiZeitschriften. In this important paper, Wirtinger introduces several important concepts in the theory of functions of several complex variables, namely Wirtinger's derivatives and the tangential Cauchy-Riemann condition. Scientific references • Andreotti, Aldo (1976), Introduzione all'analisi complessa (Lezioni tenute nel febbraio 1972), Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni (in Italian), vol. 24, Rome: Accademia Nazionale dei Lincei, p. 34, archived from the original on 2012-03-07, retrieved 2010-08-28. Introduction to complex analysis is a short course in the theory of functions of several complex variables, held in February 1972 at the Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni "Beniamino Segre". • Fichera, Gaetano (1986), "Unification of global and local existence theorems for holomorphic functions of several complex variables", Memorie della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, 8, 18 (3): 61–83, MR 0917525, Zbl 0705.32006. • Gunning, Robert C.; Rossi, Hugo (1965), Analytic Functions of Several Complex Variables, Prentice-Hall series in Modern Analysis, Englewood Cliffs, N.J.: Prentice-Hall, pp. xiv+317, ISBN 9780821869536, MR 0180696, Zbl 0141.08601. • Gunning, Robert C. (1990), Introduction to Holomorphic Functions of Several Variables. Volume I: Function Theory, Wadsworth & Brooks/Cole Mathematics Series, Belmont, California: Wadsworth & Brooks/Cole, pp. xx+203, ISBN 0-534-13308-8, MR 1052649, Zbl 0699.32001. • Henrici, Peter (1993) [1986], Applied and Computational Complex Analysis Volume 3, Wiley Classics Library (Reprint ed.), New York–Chichester–Brisbane–Toronto–Singapore: John Wiley & Sons, pp. X+637, ISBN 0-471-58986-1, MR 0822470, Zbl 1107.30300. • Hörmander, Lars (1990) [1966], An Introduction to Complex Analysis in Several Variables, North–Holland Mathematical Library, vol. 7 (3rd (Revised) ed.), Amsterdam–London–New York–Tokyo: North-Holland, ISBN 0-444-88446-7, MR 1045639, Zbl 0685.32001. • Kaup, Ludger; Kaup, Burchard (1983), Holomorphic functions of several variables, de Gruyter Studies in Mathematics, vol. 3, Berlin–New York: Walter de Gruyter, pp. XV+349, ISBN 978-3-11-004150-7, MR 0716497, Zbl 0528.32001. • Kracht, Manfred; Kreyszig, Erwin (1988), Methods of Complex Analysis in Partial Differential Equations and Applications, Canadian Mathematical Society Series of Monographs and Advanced Texts, New York–Chichester–Brisbane–Toronto–Singapore: John Wiley & Sons, pp. xiv+394, ISBN 0-471-83091-7, MR 0941372, Zbl 0644.35005. • Martinelli, Enzo (1984), Introduzione elementare alla teoria delle funzioni di variabili complesse con particolare riguardo alle rappresentazioni integrali, Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e Loro Applicazioni (in Italian), vol. 67, Rome: Accademia Nazionale dei Lincei, pp. 236+II, archived from the original on 2011-09-27, retrieved 2010-08-24. "Elementary introduction to the theory of functions of complex variables with particular regard to integral representations" (English translation of the title) are the notes form a course, published by the Accademia Nazionale dei Lincei, held by Martinelli when he was "Professore Linceo". • Remmert, Reinhold (1991), Theory of Complex Functions, Graduate Texts in Mathematics, vol. 122 (Fourth corrected 1998 printing ed.), New York–Berlin–Heidelberg–Barcelona–Hong Kong–London–Milan–Paris–Singapore–Tokyo: Springer Verlag, pp. xx+453, ISBN 0-387-97195-5, MR 1084167, Zbl 0780.30001 ISBN 978-0-387-97195-7. A textbook on complex analysis including many historical notes on the subject. • Severi, Francesco (1958), Lezioni sulle funzioni analitiche di più variabili complesse – Tenute nel 1956–57 all'Istituto Nazionale di Alta Matematica in Roma (in Italian), Padova: CEDAM – Casa Editrice Dott. Antonio Milani, pp. XIV+255, Zbl 0094.28002. Notes from a course held by Francesco Severi at the Istituto Nazionale di Alta Matematica (which at present bears his name), containing appendices of Enzo Martinelli, Giovanni Battista Rizza and Mario Benedicty. An English translation of the title reads as:-"Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the Istituto Nazionale di Alta Matematica in Rome".
Wikipedia
Wirtinger inequality (2-forms) In mathematics, the Wirtinger inequality, named after Wilhelm Wirtinger, is a fundamental result in complex linear algebra which relates the symplectic and volume forms of a hermitian inner product. It has important consequences in complex geometry, such as showing that the normalized exterior powers of the Kähler form of a Kähler manifold are calibrations. For other inequalities named after Wirtinger, see Wirtinger's inequality. Statement Consider a real vector space with positive-definite inner product g, symplectic form ω, and almost-complex structure J, linked by ω(u, v) = g(J(u), v) for any vectors u and v. Then for any orthonormal vectors v1, ..., v2k there is $(\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(v_{1},\ldots ,v_{2k})\leq k!.$ There is equality if and only if the span of v1, ..., v2k is closed under the operation of J.[1] In the language of the comass of a form, the Wirtinger theorem (although without precision about when equality is achieved) can also be phrased as saying that the comass of the form ω ∧ ⋅⋅⋅ ∧ ω is equal to k!.[1] Proof k = 1 In the special case k = 1, the Wirtinger inequality is a special case of the Cauchy–Schwarz inequality: $\omega (v_{1},v_{2})=g(J(v_{1}),v_{2})\leq \|J(v_{1})\|_{g}\|v_{2}\|_{g}=1.$ According to the equality case of the Cauchy–Schwarz inequality, equality occurs if and only if J(v1) and v2 are collinear, which is equivalent to the span of v1, v2 being closed under J. k > 1 Let v1, ..., v2k be fixed, and let T denote their span. Then there is an orthonormal basis e1, ..., e2k of T with dual basis w1, ..., w2k such that $\iota ^{\ast }\omega =\sum _{j=1}^{k}\omega (e_{2j-1},e_{2j})w_{2j-1}\wedge w_{2j},$ where ι denotes the inclusion map from T into V.[2] This implies $\underbrace {\iota ^{\ast }\omega \wedge \cdots \wedge \iota ^{\ast }\omega } _{k{\text{ times}}}=k!\prod _{i=1}^{k}\omega (e_{2i-1},e_{2i})w_{1}\wedge \cdots \wedge w_{2k},$ which in turn implies $(\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(e_{1},\ldots ,e_{2k})=k!\prod _{i=1}^{k}\omega (e_{2i-1},e_{2i})\leq k!,$ where the inequality follows from the previously-established k = 1 case. If equality holds, then according to the k = 1 equality case, it must be the case that ω(e2i − 1, e2i) = ±1 for each i. This is equivalent to either ω(e2i − 1, e2i) = 1 or ω(e2i, e2i − 1) = 1, which in either case (from the k = 1 case) implies that the span of e2i − 1, e2i is closed under J, and hence that the span of e1, ..., e2k is closed under J. Finally, the dependence of the quantity $(\underbrace {\omega \wedge \cdots \wedge \omega } _{k{\text{ times}}})(v_{1},\ldots ,v_{2k})$ on v1, ..., v2k is only on the quantity v1 ∧ ⋅⋅⋅ ∧ v2k, and from the orthonormality condition on v1, ..., v2k, this wedge product is well-determined up to a sign. This relates the above work with e1, ..., e2k to the desired statement in terms of v1, ..., v2k. Consequences Given a complex manifold with hermitian metric, the Wirtinger theorem immediately implies that for any 2k-dimensional embedded submanifold M, there is $\operatorname {vol} (M)\geq {\frac {1}{k!}}\int _{M}\omega ^{k},$ where ω is the Kähler form of the metric. Furthermore, equality is achieved if and only if M is a complex submanifold.[3] In the special case that the hermitian metric satisfies the Kähler condition, this says that 1/k!ωk is a calibration for the underlying Riemannian metric, and that the corresponding calibrated submanifolds are the complex submanifolds of complex dimension k.[4] This says in particular that every complex submanifold of a Kähler manifold is a minimal submanifold, and is even volume-minimizing among all submanifolds in its homology class. Using the Wirtinger inequality, these facts even extend to the more sophisticated context of currents in Kähler manifolds.[5] See also • Gromov's inequality for complex projective space • Systolic geometry Notes 1. Federer 1969, Section 1.8.2. 2. McDuff & Salamon 2017, Lemma 2.4.5. 3. Griffiths & Harris 1978, Section 0.2. 4. Harvey & Lawson 1982. 5. Federer 1969, Section 5.4.19. References • Federer, Herbert (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. Berlin–Heidelberg–New York: Springer-Verlag. doi:10.1007/978-3-642-62010-2. ISBN 978-3-540-60656-7. MR 0257325. Zbl 0176.00801. • Griffiths, Phillip; Harris, Joseph (1978). Principles of algebraic geometry. Pure and Applied Mathematics. New York: John Wiley & Sons. ISBN 0-471-32792-1. MR 0507725. Zbl 0408.14001. • Harvey, Reese; Lawson, H. Blaine, Jr. (1982). "Calibrated geometries". Acta Mathematica. 148: 47–157. doi:10.1007/BF02392726. MR 0666108. Zbl 0584.53021.{{cite journal}}: CS1 maint: multiple names: authors list (link) • McDuff, Dusa; Salamon, Dietmar (2017). Introduction to symplectic topology. Oxford Graduate Texts in Mathematics (Third edition of 1995 original ed.). Oxford: Oxford University Press. doi:10.1093/oso/9780198794899.001.0001. ISBN 978-0-19-879490-5. MR 3674984. Zbl 1380.53003. • Wirtinger, W. (1936). "Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde in euklidischer und Hermitescher Maßbestimmung". Monatshefte für Mathematik und Physik. 44: 343–365. doi:10.1007/BF01699328. MR 1550581. Zbl 0015.07602.
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Wirtinger presentation In mathematics, especially in group theory, a Wirtinger presentation is a finite presentation where the relations are of the form $wg_{i}w^{-1}=g_{j}$ where $w$ is a word in the generators, $\{g_{1},g_{2},\ldots ,g_{k}\}.$ Wilhelm Wirtinger observed that the complements of knots in 3-space have fundamental groups with presentations of this form. Preliminaries and definition A knot K is an embedding of the one-sphere S1 in three-dimensional space R3. (Alternatively, the ambient space can also be taken to be the three-sphere S3, which does not make a difference for the purposes of the Wirtinger presentation.) The open subspace which is the complement of the knot, $S^{3}\setminus K$ is the knot complement. Its fundamental group $\pi _{1}(S^{3}\setminus K)$ is an invariant of the knot in the sense that equivalent knots have isomorphic knot groups. It is therefore interesting to understand this group in an accessible way. A Wirtinger presentation is derived from a regular projection of an oriented knot. Such a projection can be pictured as a finite number of (oriented) arcs in the plane, separated by the crossings of the projection. The fundamental group is generated by loops winding around each arc. Each crossing gives rise to a certain relation among the generators corresponding to the arcs meeting at the crossing. Wirtinger presentations of high-dimensional knots More generally, co-dimension two knots in spheres are known to have Wirtinger presentations. Michel Kervaire proved that an abstract group is the fundamental group of a knot exterior (in a perhaps high-dimensional sphere) if and only if all the following conditions are satisfied: 1. The abelianization of the group is the integers. 2. The 2nd homology of the group is trivial. 3. The group is finitely presented. 4. The group is the normal closure of a single generator. Conditions (3) and (4) are essentially the Wirtinger presentation condition, restated. Kervaire proved in dimensions 5 and larger that the above conditions are necessary and sufficient. Characterizing knot groups in dimension four is an open problem. Examples For the trefoil knot, a Wirtinger presentation can be shown to be $\pi _{1}(\mathbb {R} ^{3}\backslash {\text{trefoil}})=\langle x,y\mid yxy=xyx\rangle .$ See also • Knot group Further reading • Rolfsen, Dale (1990), Knots and links, Mathematics Lecture Series, vol. 7, Houston, TX: Publish or Perish, ISBN 978-0-914098-16-4, section 3D • Kawauchi, Akio (1996), A survey of knot theory, Birkhäuser, doi:10.1007/978-3-0348-9227-8, ISBN 978-3-0348-9953-6 • Hillman, Jonathan (2012), Algebraic invariants of links, Series on Knots and Everything, vol. 52, World Scientific, doi:10.1142/9789814407397, ISBN 9789814407397 • Livingston, Charles (1993), Knot Theory, The Mathematical Association of America
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Wirtinger sextic In mathematics, the Wirtinger plane sextic curve, studied by Wirtinger, is a degree 6 genus 4 plane curve with double points at the 6 vertices of a complete quadrilateral. References • Coble, Arthur B. (1929), Algebraic geometry and theta functions, American Mathematical Society Colloquium Publications, vol. 10, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1010-1, MR 0733252 • Dolgachev, Igor. (2012), Classical Algebraic Geometry:a modern view, Cambridge.: Cambridge University Press, ISBN 978-1-107-01765-8, MR 2964027
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Witness (mathematics) In mathematical logic, a witness is a specific value t to be substituted for variable x of an existential statement of the form ∃x φ(x) such that φ(t) is true. Examples For example, a theory T of arithmetic is said to be inconsistent if there exists a proof in T of the formula "0 = 1". The formula I(T), which says that T is inconsistent, is thus an existential formula. A witness for the inconsistency of T is a particular proof of "0 = 1" in T. Boolos, Burgess, and Jeffrey (2002:81) define the notion of a witness with the example, in which S is an n-place relation on natural numbers, R is an (n+1)-place recursive relation, and ↔ indicates logical equivalence (if and only if): S(x1, ..., xn) ↔ ∃y R(x1, . . ., xn, y) "A y such that R holds of the xi may be called a 'witness' to the relation S holding of the xi (provided we understand that when the witness is a number rather than a person, a witness only testifies to what is true)." In this particular example, the authors defined s to be (positively) recursively semidecidable, or simply semirecursive. Henkin witnesses In predicate calculus, a Henkin witness for a sentence $\exists x\,\varphi (x)$ in a theory T is a term c such that T proves φ(c) (Hinman 2005:196). The use of such witnesses is a key technique in the proof of Gödel's completeness theorem presented by Leon Henkin in 1949. Relation to game semantics The notion of witness leads to the more general idea of game semantics. In the case of sentence $\exists x\,\varphi (x)$ the winning strategy for the verifier is to pick a witness for $\varphi $. For more complex formulas involving universal quantifiers, the existence of a winning strategy for the verifier depends on the existence of appropriate Skolem functions. For example, if S denotes $\forall x\,\exists y\,\varphi (x,y)$ then an equisatisfiable statement for S is $\exists f\,\forall x\,\varphi (x,f(x))$. The Skolem function f (if it exists) actually codifies a winning strategy for the verifier of S by returning a witness for the existential sub-formula for every choice of x the falsifier might make. See also • Certificate (complexity), an analogous concept in computational complexity theory References • George S. Boolos, John P. Burgess, and Richard C. Jeffrey, 2002, Computability and Logic: Fourth Edition, Cambridge University Press, ISBN 0-521-00758-5. • Leon Henkin, 1949, "The completeness of the first-order functional calculus", Journal of Symbolic Logic v. 14 n. 3, pp. 159–166. • Peter G. Hinman, 2005, Fundamentals of mathematical logic, A.K. Peters, ISBN 1-56881-262-0. • J. Hintikka and G. Sandu, 2009, "Game-Theoretical Semantics" in Keith Allan (ed.) Concise Encyclopedia of Semantics, Elsevier, ISBN 0-08095-968-7, pp. 341–343
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Disjunction and existence properties In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005). Definitions • The disjunction property is satisfied by a theory if, whenever a sentence A ∨ B is a theorem, then either A is a theorem, or B is a theorem. • The existence property or witness property is satisfied by a theory if, whenever a sentence (∃x)A(x) is a theorem, where A(x) has no other free variables, then there is some term t such that the theory proves A(t). Related properties Rathjen (2005) lists five properties that a theory may possess. These include the disjunction property (DP), the existence property (EP), and three additional properties: • The numerical existence property (NEP) states that if the theory proves $(\exists x\in \mathbb {N} )\varphi (x)$, where φ has no other free variables, then the theory proves $\varphi ({\bar {n}})$ for some $n\in \mathbb {N} {\text{.}}$ Here ${\bar {n}}$ is a term in $T$ representing the number n. • Church's rule (CR) states that if the theory proves $(\forall x\in \mathbb {N} )(\exists y\in \mathbb {N} )\varphi (x,y)$ then there is a natural number e such that, letting $f_{e}$ be the computable function with index e, the theory proves $(\forall x)\varphi (x,f_{e}(x))$. • A variant of Church's rule, CR1, states that if the theory proves $(\exists f\colon \mathbb {N} \to \mathbb {N} )\psi (f)$ then there is a natural number e such that the theory proves $f_{e}$ is total and proves $\psi (f_{e})$. These properties can only be directly expressed for theories that have the ability to quantify over natural numbers and, for CR1, quantify over functions from $\mathbb {N} $ to $\mathbb {N} $. In practice, one may say that a theory has one of these properties if a definitional extension of the theory has the property stated above (Rathjen 2005). Results Non-examples and examples Almost by definition, a theory that accepts excluded middle while having independent statements does not have the disjunction property. So all classical theories expressing Robinson arithmetic do not have it. Most classical theories, such as Peano arithmetic and ZFC in turn do not validate the existence property either, e.g. because they validate the least number principle existence claim. But some classical theories, such as ZFC plus the axiom of constructibility, do have a weaker form of the existence property (Rathjen 2005). Heyting arithmetic is well known for having the disjunction property and the (numerical) existence property. While the earliest results were for constructive theories of arithmetic, many results are also known for constructive set theories (Rathjen 2005). John Myhill (1973) showed that IZF with the axiom of replacement eliminated in favor of the axiom of collection has the disjunction property, the numerical existence property, and the existence property. Michael Rathjen (2005) proved that CZF has the disjunction property and the numerical existence property. Freyd and Scedrov (1990) observed that the disjunction property holds in free Heyting algebras and free topoi. In categorical terms, in the free topos, that corresponds to the fact that the terminal object, $\mathbf {1} $, is not the join of two proper subobjects. Together with the existence property it translates to the assertion that $\mathbf {1} $ is an indecomposable projective object—the functor it represents (the global-section functor) preserves epimorphisms and coproducts. Relationship between properties There are several relationship between the five properties discussed above. In the setting of arithmetic, the numerical existence property implies the disjunction property. The proof uses the fact that a disjunction can be rewritten as an existential formula quantifying over natural numbers: $A\vee B\equiv (\exists n)[(n=0\to A)\wedge (n\neq 0\to B)]$. Therefore, if $A\vee B$ is a theorem of $T$, so is $\exists n\colon (n=0\to A)\wedge (n\neq 0\to B)$. Thus, assuming the numerical existence property, there exists some $s$ such that $({\bar {s}}=0\to A)\wedge ({\bar {s}}\neq 0\to B)$ is a theorem. Since ${\bar {s}}$ is a numeral, one may concretely check the value of $s$: if $s=0$ then $A$ is a theorem and if $s\neq 0$ then $B$ is a theorem. Harvey Friedman (1974) proved that in any recursively enumerable extension of intuitionistic arithmetic, the disjunction property implies the numerical existence property. The proof uses self-referential sentences in way similar to the proof of Gödel's incompleteness theorems. The key step is to find a bound on the existential quantifier in a formula (∃x)A(x), producing a bounded existential formula (∃x<n)A(x). The bounded formula may then be written as a finite disjunction A(1)∨A(2)∨...∨A(n). Finally, disjunction elimination may be used to show that one of the disjuncts is provable. History Kurt Gödel (1932) stated without proof that intuitionistic propositional logic (with no additional axioms) has the disjunction property; this result was proven and extended to intuitionistic predicate logic by Gerhard Gentzen (1934, 1935). Stephen Cole Kleene (1945) proved that Heyting arithmetic has the disjunction property and the existence property. Kleene's method introduced the technique of realizability, which is now one of the main methods in the study of constructive theories (Kohlenbach 2008; Troelstra 1973). See also • Constructive set theory • Heyting arithmetic • Law of excluded middle • Realizability • Existential quantifier References • Peter J. Freyd and Andre Scedrov, 1990, Categories, Allegories. North-Holland. • Harvey Friedman, 1975, The disjunction property implies the numerical existence property, State University of New York at Buffalo. • Gerhard Gentzen, 1934, "Untersuchungen über das logische Schließen. I", Mathematische Zeitschrift v. 39 n. 2, pp. 176–210. • Gerhard Gentzen, 1935, "Untersuchungen über das logische Schließen. II", Mathematische Zeitschrift v. 39 n. 3, pp. 405–431. • Kurt Gödel, 1932, "Zum intuitionistischen Aussagenkalkül", Anzeiger der Akademie der Wissenschaftischen in Wien, v. 69, pp. 65–66. • Stephen Cole Kleene, 1945, "On the interpretation of intuitionistic number theory," Journal of Symbolic Logic, v. 10, pp. 109–124. • Ulrich Kohlenbach, 2008, Applied proof theory, Springer. • John Myhill, 1973, "Some properties of Intuitionistic Zermelo-Fraenkel set theory", in A. Mathias and H. Rogers, Cambridge Summer School in Mathematical Logic, Lectures Notes in Mathematics v. 337, pp. 206–231, Springer. • Michael Rathjen, 2005, "The Disjunction and Related Properties for Constructive Zermelo-Fraenkel Set Theory", Journal of Symbolic Logic, v. 70 n. 4, pp. 1233–1254. • Anne S. Troelstra, ed. (1973), Metamathematical investigation of intuitionistic arithmetic and analysis, Springer. External links • Intuitionistic Logic by Joan Moschovakis, Stanford Encyclopedia of Philosophy
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Witt algebra In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, and the Lie algebra of derivations of the ring C[z,z−1]. The Witt algebra is not directly related to the Witt ring of quadratic forms, or to the algebra of Witt vectors. There are some related Lie algebras defined over finite fields, that are also called Witt algebras. The complex Witt algebra was first defined by Élie Cartan (1909), and its analogues over finite fields were studied by Witt in the 1930s. Basis A basis for the Witt algebra is given by the vector fields $L_{n}=-z^{n+1}{\frac {\partial }{\partial z}}$, for n in $\mathbb {Z} $. The Lie bracket of two basis vector fields is given by $[L_{m},L_{n}]=(m-n)L_{m+n}.$ This algebra has a central extension called the Virasoro algebra that is important in two-dimensional conformal field theory and string theory. Note that by restricting n to 1,0,-1, one gets a subalgebra. Taken over the field of complex numbers, this is just the Lie algebra ${\mathfrak {sl}}(2,\mathbb {C} )$ of the Lorentz group $\mathrm {SO} (3,1)$. Over the reals, it is the algebra sl(2,R) = su(1,1). Conversely, su(1,1) suffices to reconstruct the original algebra in a presentation.[1] Over finite fields Over a field k of characteristic p>0, the Witt algebra is defined to be the Lie algebra of derivations of the ring k[z]/zp The Witt algebra is spanned by Lm for −1≤ m ≤ p−2. Images n = -1 Witt vector field n = 0 Witt vector field n = 1 Witt vector field n = -2 Witt vector field n = 2 Witt vector field n = -3 Witt vector field See also • Virasoro algebra • Heisenberg algebra References 1. D Fairlie, J Nuyts, and C Zachos (1988). Phys Lett B202 320-324. doi:10.1016/0370-2693(88)90478-9 • Élie Cartan, Les groupes de transformations continus, infinis, simples. Ann. Sci. Ecole Norm. Sup. 26, 93-161 (1909). • "Witt algebra", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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Witt vector cohomology In mathematics, Witt vector cohomology was an early p-adic cohomology theory for algebraic varieties introduced by Serre (1958). Serre constructed it by defining a sheaf of truncated Witt rings Wn over a variety V and then taking the inverse limit of the sheaf cohomology groups Hi(V, Wn) of these sheaves. Serre observed that though it gives cohomology groups over a field of characteristic 0, it cannot be a Weil cohomology theory because the cohomology groups vanish when i > dim(V). For Abelian varieties Serre (1958b) showed that one could obtain a reasonable first cohomology group by taking the direct sum of the Witt vector cohomology and the Tate module of the Picard variety. References • Serre, J.P. (1958), "Sur la topologie des variétés algébriques en caractéristique p", 1958 Symposium internacional de topología algebraica, Mexico City: Universidad Nacional Autónoma de México and UNESCO, pp. 24–53, MR 0098097 • Serre, Jean-Pierre (1958b), "Quelques propriétés des variétés abéliennes en caractéristique p", Amer. J. Math., 80: 715–739, doi:10.2307/2372780, MR 0098100
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Witten conjecture In algebraic geometry, the Witten conjecture is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by Edward Witten in the paper Witten (1991), and generalized in Witten (1993). Witten's original conjecture was proved by Maxim Kontsevich in the paper Kontsevich (1992). Witten's motivation for the conjecture was that two different models of 2-dimensional quantum gravity should have the same partition function. The partition function for one of these models can be described in terms of intersection numbers on the moduli stack of algebraic curves, and the partition function for the other is the logarithm of the τ-function of the KdV hierarchy. Identifying these partition functions gives Witten's conjecture that a certain generating function formed from intersection numbers should satisfy the differential equations of the KdV hierarchy. Statement Suppose that Mg,n is the moduli stack of compact Riemann surfaces of genus g with n distinct marked points x1,...,xn, and Mg,n is its Deligne–Mumford compactification. There are n line bundles Li on Mg,n, whose fiber at a point of the moduli stack is given by the cotangent space of a Riemann surface at the marked point xi. The intersection index 〈τd1, ..., τdn〉 is the intersection index of Π c1(Li)di on Mg,n where Σdi = dimMg,n = 3g – 3 + n, and 0 if no such g exists, where c1 is the first Chern class of a line bundle. Witten's generating function $F(t_{0},t_{1},\ldots )=\sum \langle \tau _{0}^{k_{0}}\tau _{1}^{k_{1}}\cdots \rangle \prod _{i\geq 0}{\frac {t_{i}^{k_{i}}}{k_{i}!}}={\frac {t_{0}^{3}}{6}}+{\frac {t_{1}}{24}}+{\frac {t_{0}t_{2}}{24}}+{\frac {t_{1}^{2}}{24}}+{\frac {t_{0}^{2}t_{3}}{48}}+\cdots $ encodes all the intersection indices as its coefficients. Witten's conjecture states that the partition function Z = exp F is a τ-function for the KdV hierarchy, in other words it satisfies a certain series of partial differential equations corresponding to the basis $\{L_{-1},L_{0},L_{1},\ldots \}$ of the Virasoro algebra. Proof Kontsevich used a combinatorial description of the moduli spaces in terms of ribbon graphs to show that $\sum _{d_{1}+\cdots +d_{n}=3g-3+n}\langle \tau _{d_{1}},\ldots ,\tau _{d_{n}}\rangle \prod _{1\leq i\leq n}{\frac {(2d_{i}-1)!!}{\lambda _{i}^{2d_{i}+1}}}=\sum _{\Gamma \in G_{g,n}}{\frac {2^{-|X_{0}|}}{|{\text{Aut}}\Gamma |}}\prod _{e\in X_{1}}{\frac {2}{\lambda (e)}}$ Here the sum on the right is over the set Gg,n of ribbon graphs X of compact Riemann surfaces of genus g with n marked points. The set of edges e and points of X are denoted by X 0 and X1. The function λ is thought of as a function from the marked points to the reals, and extended to edges of the ribbon graph by setting λ of an edge equal to the sum of λ at the two marked points corresponding to each side of the edge. By Feynman diagram techniques, this implies that F(t0,...) is an asymptotic expansion of $\log \int \exp(i{\text{tr}}X^{3}/6)d\mu $ as Λ lends to infinity, where Λ and Χ are positive definite N by N hermitian matrices, and ti is given by $t_{i}={\frac {-{\text{tr }}\Lambda ^{-1-2i}}{1\times 3\times 5\times \cdots \times (2i-1)}}$ and the probability measure μ on the positive definite hermitian matrices is given by $d\mu =c_{\Lambda }\exp(-{\text{tr}}X^{2}\Lambda /2)dX$ where cΛ is a normalizing constant. This measure has the property that $\int X_{ij}X_{kl}d\mu =\delta _{il}\delta _{jk}{\frac {2}{\Lambda _{i}+\Lambda _{j}}}$ which implies that its expansion in terms of Feynman diagrams is the expression for F in terms of ribbon graphs. From this he deduced that exp F is a τ-function for the KdV hierarchy, thus proving Witten's conjecture. Generalizations The Witten conjecture is a special case of a more general relation between integrable systems of Hamiltonian PDEs and the geometry of certain families of 2D topological field theories (axiomatized in the form of the so-called cohomological field theories by Kontsevich and Manin), which was explored and studied systematically by B. Dubrovin and Y. Zhang, A. Givental, C. Teleman and others. The Virasoro conjecture is a generalization of the Witten conjecture. References • Cornalba, Maurizio; Arbarello, Enrico; Griffiths, Phillip A. (2011), Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 268, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-69392-5, ISBN 978-3-540-42688-2, MR 2807457 • Kazarian, M. E.; Lando, Sergei K. (2007), "An algebro-geometric proof of Witten's conjecture", Journal of the American Mathematical Society, 20 (4): 1079–1089, arXiv:math/0601760, Bibcode:2007JAMS...20.1079K, doi:10.1090/S0894-0347-07-00566-8, ISSN 0894-0347, MR 2328716 • Kontsevich, Maxim (1992), "Intersection theory on the moduli space of curves and the matrix Airy function", Communications in Mathematical Physics, 147 (1): 1–23, Bibcode:1992CMaPh.147....1K, doi:10.1007/BF02099526, ISSN 0010-3616, MR 1171758 • Lando, Sergei K.; Zvonkin, Alexander K. (2004), Graphs on surfaces and their applications (PDF), Encyclopaedia of Mathematical Sciences, vol. 141, Berlin, New York: Springer-Verlag, ISBN 978-3-540-00203-1, MR 2036721 • Witten, Edward (1991), "Two-dimensional gravity and intersection theory on moduli space", Surveys in differential geometry (Cambridge, MA, 1990), vol. 1, Bethlehem, PA: Lehigh Univ., pp. 243–310, ISBN 978-0-8218-0168-0, MR 1144529 • Witten, Edward (1993), "Algebraic geometry associated with matrix models of two-dimensional gravity", in Goldberg, Lisa R.; Phillips, Anthony V. (eds.), Topological methods in modern mathematics (Stony Brook, NY, 1991), Proceedings of the symposium in honor of John Milnor's sixtieth birthday held at the State University of New York, Stony Brook, New York, June 14–21, 1991., Houston, TX: Publish or Perish, pp. 235–269, ISBN 978-0-914098-26-3, MR 1215968 Topics in algebraic curves Rational curves • Five points determine a conic • Projective line • Rational normal curve • Riemann sphere • Twisted cubic Elliptic curves Analytic theory • Elliptic function • Elliptic integral • Fundamental pair of periods • Modular form Arithmetic theory • Counting points on elliptic curves • Division polynomials • Hasse's theorem on elliptic curves • Mazur's torsion theorem • Modular elliptic curve • Modularity theorem • Mordell–Weil theorem • Nagell–Lutz theorem • Supersingular elliptic curve • Schoof's algorithm • Schoof–Elkies–Atkin algorithm Applications • Elliptic curve cryptography • Elliptic curve primality Higher genus • De Franchis theorem • Faltings's theorem • Hurwitz's automorphisms theorem • Hurwitz surface • Hyperelliptic curve Plane curves • AF+BG theorem • Bézout's theorem • Bitangent • Cayley–Bacharach theorem • Conic section • Cramer's paradox • Cubic plane curve • Fermat curve • Genus–degree formula • Hilbert's sixteenth problem • Nagata's conjecture on curves • Plücker formula • Quartic plane curve • Real plane curve Riemann surfaces • Belyi's theorem • Bring's curve • Bolza surface • Compact Riemann surface • Dessin d'enfant • Differential of the first kind • Klein quartic • Riemann's existence theorem • Riemann–Roch theorem • Teichmüller space • Torelli theorem Constructions • Dual curve • Polar curve • Smooth completion Structure of curves Divisors on curves • Abel–Jacobi map • Brill–Noether theory • Clifford's theorem on special divisors • Gonality of an algebraic curve • Jacobian variety • Riemann–Roch theorem • Weierstrass point • Weil reciprocity law Moduli • ELSV formula • Gromov–Witten invariant • Hodge bundle • Moduli of algebraic curves • Stable curve Morphisms • Hasse–Witt matrix • Riemann–Hurwitz formula • Prym variety • Weber's theorem (Algebraic curves) Singularities • Acnode • Crunode • Cusp • Delta invariant • Tacnode Vector bundles • Birkhoff–Grothendieck theorem • Stable vector bundle • Vector bundles on algebraic curves
Wikipedia
Witten zeta function In mathematics, the Witten zeta function, is a function associated to a root system that encodes the degrees of the irreducible representations of the corresponding Lie group. These zeta functions were introduced by Don Zagier who named them after Edward Witten's study of their special values (among other things).[1][2] Note that in,[2] Witten zeta functions do not appear as explicit objects in their own right. Definition If $G$ is a compact semisimple Lie group, the associated Witten zeta function is (the meromorphic continuation of) the series $\zeta _{G}(s)=\sum _{\rho }{\frac {1}{(\dim \rho )^{s}}},$ where the sum is over equivalence classes of irreducible representations of $G$. In the case where $G$ is connected and simply connected, the correspondence between representations of $G$ and of its Lie algebra, together with the Weyl dimension formula, implies that $\zeta _{G}(s)$ can be written as $\sum _{m_{1},\dots ,m_{r}>0}\prod _{\alpha \in \Phi ^{+}}{\frac {1}{\langle \alpha ^{\lor },m_{1}\lambda _{1}+\cdots +m_{r}\lambda _{r}\rangle ^{s}}},$ where $\Phi ^{+}$ denotes the set of positive roots, $\{\lambda _{i}\}$ is a set of simple roots and $r$ is the rank. Examples • $\zeta _{SU(2)}(s)=\zeta (s)$, the Riemann zeta function. • $\zeta _{SU(3)}(s)=\sum _{x=1}^{\infty }\sum _{y=1}^{\infty }{\frac {1}{(xy(x+y)/2)^{s}}}.$ Abscissa of convergence If $G$ is simple and simply connected, the abscissa of convergence of $\zeta _{G}(s)$ is $r/\kappa $, where $r$ is the rank and $\kappa =|\Phi ^{+}|$. This is a theorem due to Alex Lubotzky and Michael Larsen.[3] A new proof is given by Jokke Häsä and Alexander Stasinski [4] which yields a more general result, namely it gives an explicit value (in terms of simple combinatorics) of the abscissa of convergence of any "Mellin zeta function" of the form $\sum _{x_{1},\dots ,x_{r}=1}^{\infty }{\frac {1}{P(x_{1},\dots ,x_{r})^{s}}},$ where $P(x_{1},\dots ,x_{r})$ is a product of linear polynomials with non-negative real coefficients. Singularities and values of the Witten zeta function associated to SU(3) $\zeta _{SU(3)}$ is absolutely convergent in $\{s\in \mathbb {C} ,\Re (s)>2/3\}$, and it can be extended meromorphicaly in $\mathbb {C} $. Its singularities are in ${\Bigl \{}{\frac {2}{3}}{\Bigr \}}\cup {\Bigl \{}{\frac {1}{2}}-k,k\in \mathbb {N} {\Bigr \}},$ and all of those singularities are simple poles.[5] In particular, the values of $\zeta _{SU(3)}(s)$ are well defined at all integers, and have been computed by Kazuhiro Onodera.[6] At $s=0$, we have $\zeta _{SU(3)}(0)={\frac {1}{3}},$ and $\zeta _{SU(3)}'(0)=\log(2^{4/3}\pi ).$ Let $a\in \mathbb {N} ^{*}$ be a positive integer. We have $\zeta _{SU(3)}(a)={\frac {2^{a+2}}{1+(-1)^{a}2}}\sum _{k=0}^{[a/2]}{2a-2k-1 \choose a-1}\zeta (2k)\zeta (3a-k).$ If a is odd, then $\zeta _{SU(3)}$ has a simple zero at $s=-a,$ and $\zeta _{SU(3)}'(-a)={\frac {2^{-a+1}(a!)^{2}}{(2a+1)!}}\zeta '(-3a-1)+2^{-a+2}\sum _{k=0}^{(a-1)/2}{a \choose 2k}\zeta (-a-2k)\zeta '(-2a+2k).$ If a is even, then $\zeta _{SU(3)}$ has a zero of order $2$ at $s=-a,$ and $\zeta _{SU(3)}''(-a)=2^{-a+2}\sum _{k=0}^{a/2}{a \choose 2k}\zeta '(-a-2k)\zeta '(-2a+2k).$ References 1. Zagier, Don (1994), "Values of Zeta Functions and Their Applications", First European Congress of Mathematics Paris, July 6–10, 1992, Birkhäuser Basel, pp. 497–512, doi:10.1007/978-3-0348-9112-7_23, ISBN 9783034899123 2. Witten, Edward (October 1991). "On quantum gauge theories in two dimensions". Communications in Mathematical Physics. 141 (1): 153–209. doi:10.1007/bf02100009. ISSN 0010-3616. S2CID 121994550. 3. Larsen, Michael; Lubotzky, Alexander (2008). "Representation growth of linear groups". Journal of the European Mathematical Society. 10 (2): 351–390. arXiv:math/0607369. doi:10.4171/JEMS/113. ISSN 1435-9855. S2CID 9322647. 4. Häsä, Jokke; Stasinski, Alexander (2019). "Representation growth of compact linear groups". Transactions of the American Mathematical Society. 372 (2): 925–980. doi:10.1090/tran/7618. 5. Romik, Dan (2017). "On the number of $n$-dimensional representations of $\operatorname{SU}(3)$, the Bernoulli numbers, and the Witten zeta function". Acta Arithmetica. 180 (2): 111–159. doi:10.4064/aa8455-3-2017. ISSN 0065-1036. 6. Onodera, Kazuhiro (2014). "A functional relation for Tornheim's double zeta functions". Acta Arithmetica. 162 (4): 337–354. arXiv:1211.1480. doi:10.4064/aa162-4-2. ISSN 0065-1036. S2CID 119636956.
Wikipedia
Witting polytope In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: 3{3}3{3}3{3}3, and Coxeter diagram . It has 240 vertices, 2160 3{} edges, 2160 3{3}3 faces, and 240 3{3}3{3}3 cells. It is self-dual. Each vertex belongs to 27 edges, 72 faces, and 27 cells, corresponding to the Hessian polyhedron vertex figure. Witting polytope Schläfli symbol3{3}3{3}3{3}3 Coxeter diagram Cells240 3{3}3{3}3 Faces2160 3{3}3 Edges2160 3{} Vertices240 Petrie polygon30-gon van Oss polygon90 3{4}3 Shephard groupL4 = 3[3]3[3]3[3]3, order 155,520 Dual polyhedronSelf-dual PropertiesRegular Symmetry Its symmetry by 3[3]3[3]3[3]3 or , order 155,520.[1] It has 240 copies of , order 648 at each cell.[2] Structure The configuration matrix is:[3] $\left[{\begin{smallmatrix}240&27&72&27\\3&2160&8&8\\8&8&2160&3\\27&72&27&240\end{smallmatrix}}\right]$ The number of vertices, edges, faces, and cells are seen in the diagonal of the matrix. These are computed by the order of the group divided by the order of the subgroup, by removing certain complex reflections, shown with X below. The number of elements of the k-faces are seen in rows below the diagonal. The number of elements in the vertex figure, etc., are given in rows above the digonal. L4 k-facefkf0f1f2f3k-figure Notes L3( ) f0 2402772273{3}3{3}3L4/L3 = 216*6!/27/4! = 240 L2L13{ } f1 32160883{3}3L4/L2L1 = 216*6!/4!/3 = 2160 3{3}3 f2 88216033{ } L33{3}3{3}3 f3 277227240( )L4/L3 = 216*6!/27/4! = 240 Coordinates Its 240 vertices are given coordinates in $\mathbb {C} ^{4}$: (0, ±ωμ, -±ων, ±ωλ) (-±ωμ, 0, ±ων, ±ωλ) (±ωμ, -±ων, 0, ±ωλ) (-±ωλ, -±ωμ, -±ων, 0) (±iωλ√3, 0, 0, 0) (0, ±iωλ√3, 0, 0) (0, 0, ±iωλ√3, 0) (0, 0, 0, ±iωλ√3) where $\omega ={\tfrac {-1+i{\sqrt {3}}}{2}},\lambda ,\nu ,\mu =0,1,2$. The last 6 points form hexagonal holes on one of its 40 diameters. There are 40 hyperplanes contain central 3{3}3{4}2, figures, with 72 vertices. Witting configuration Coxeter named it after Alexander Witting for being a Witting configuration in complex projective 3-space:[4] $\left[{\begin{smallmatrix}40&12&12\\2&240&2\\12&12&40\end{smallmatrix}}\right]$ or $\left[{\begin{smallmatrix}40&9&12\\4&90&4\\12&9&40\end{smallmatrix}}\right]$ The Witting configuration is related to the finite space PG(3,22), consisting of 85 points, 357 lines, and 85 planes.[5] Related real polytope Its 240 vertices are shared with the real 8-dimensional polytope 421, . Its 2160 3-edges are sometimes drawn as 6480 simple edges, slightly less than the 6720 edges of 421. The 240 difference is accounted by 40 central hexagons in 421 whose edges are not included in 3{3}3{3}3{3}3.[6] The honeycomb of Witting polytopes The regular Witting polytope has one further stage as a 4-dimensional honeycomb, . It has the Witting polytope as both its facets, and vertex figure. It is self-dual, and its dual coincides with itself.[7] Hyperplane sections of this honeycomb include 3-dimensional honeycombs . The honeycomb of Witting polytopes has a real representation as the 8-dimensional polytope 521, . Its f-vector element counts are in proportion: 1, 80, 270, 80, 1.[8] The configuration matrix for the honeycomb is: L5 k-facefkf0f1f2f3f4k-figure Notes L4( ) f0 N240216021602403{3}3{3}3{3}3L5/L4 = N L3L13{ } f1 380N2772273{3}3{3}3L5/L3L1 = 80N L2L23{3}3 f2 88270N883{3}3L5/L2L2 = 270N L3L13{3}3{3}3 f3 27722780N33{}L5/L3L1 = 80N L43{3}3{3}3{3}3 f4 24021602160240N( )L5/L4 = N Notes 1. Coxeter Regular Convex Polytopes, 12.5 The Witting polytope 2. Coxeter, Complex Regular Polytopes, p.134 3. Coxeter, Complex Regular polytopes, p.132 4. Alexander Witting, Ueber Jacobi'sche Functionen kter Ordnung Zweier Variabler, Mathemematische Annalen 29 (1887), 157-70, see especially p.169 5. Coxeter, Complex regular polytopes, p.133 6. Coxeter, Complex Regular Polytopes, p.134 7. Coxeter, Complex Regular Polytopes, p.135 8. Coxeter Regular Convex Polytopes, 12.5 The Witting polytope References • Coxeter, H. S. M. and Moser, W. O. J.; Generators and Relations for Discrete Groups (1965), esp pp 67–80. • Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, second edition (1991). pp. 132–5, 143, 146, 152. • Coxeter, H. S. M. and Shephard, G.C.; Portraits of a family of complex polytopes, Leonardo Vol 25, No 3/4, (1992), pp 239–244
Wikipedia
Witt group In mathematics, a Witt group of a field, named after Ernst Witt, is an abelian group whose elements are represented by symmetric bilinear forms over the field. For Witt groups in the theory of algebraic groups, see Witt vector. Definition Fix a field k of characteristic not equal to two. All vector spaces will be assumed to be finite-dimensional. We say that two spaces equipped with symmetric bilinear forms are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a hyperbolic plane, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector.[1] Each class is represented by the core form of a Witt decomposition.[2] The Witt group of k is the abelian group W(k) of equivalence classes of non-degenerate symmetric bilinear forms, with the group operation corresponding to the orthogonal direct sum of forms. It is additively generated by the classes of one-dimensional forms.[3] Although classes may contain spaces of different dimension, the parity of the dimension is constant across a class and so rk : W(k) → Z/2Z is a homomorphism.[4] The elements of finite order in the Witt group have order a power of 2;[5][6] the torsion subgroup is the kernel of the functorial map from W(k) to W(kpy), where kpy is the Pythagorean closure of k;[7] it is generated by the Pfister forms $\langle \!\langle w\rangle \!\rangle =\langle 1,-w\rangle $ with $w$ a non-zero sum of squares.[8] If k is not formally real, then the Witt group is torsion, with exponent a power of 2.[9] The height of the field k is the exponent of the torsion in the Witt group, if this is finite, or ∞ otherwise.[8] Ring structure The Witt group of k can be given a commutative ring structure, by using the tensor product of quadratic forms to define the ring product. This is sometimes called the Witt ring W(k), though the term "Witt ring" is often also used for a completely different ring of Witt vectors. To discuss the structure of this ring we assume that k is of characteristic not equal to 2, so that we may identify symmetric bilinear forms and quadratic forms. The kernel of the rank mod 2 homomorphism is a prime ideal, I, of the Witt ring[4] termed the fundamental ideal.[10] The ring homomorphisms from W(k) to Z correspond to the field orderings of k, by taking signature with respective to the ordering.[10] The Witt ring is a Jacobson ring.[9] It is a Noetherian ring if and only if there are finitely many square classes; that is, if the squares in k form a subgroup of finite index in the multiplicative group of k.[11] If k is not formally real, the fundamental ideal is the only prime ideal of W[12] and consists precisely of the nilpotent elements;[9] W is a local ring and has Krull dimension 0.[13] If k is real, then the nilpotent elements are precisely those of finite additive order, and these in turn are the forms all of whose signatures are zero;[14] W has Krull dimension 1.[13] If k is a real Pythagorean field then the zero-divisors of W are the elements for which some signature is zero; otherwise, the zero-divisors are exactly the fundamental ideal.[5][15] If k is an ordered field with positive cone P then Sylvester's law of inertia holds for quadratic forms over k and the signature defines a ring homomorphism from W(k) to Z, with kernel a prime ideal KP. These prime ideals are in bijection with the orderings Xk of k and constitute the minimal prime ideal spectrum MinSpec W(k) of W(k). The bijection is a homeomorphism between MinSpec W(k) with the Zariski topology and the set of orderings Xk with the Harrison topology.[16] The n-th power of the fundamental ideal is additively generated by the n-fold Pfister forms.[17] Examples • The Witt ring of C, and indeed any algebraically closed field or quadratically closed field, is Z/2Z.[18] • The Witt ring of R is Z.[18] • The Witt ring of a finite field Fq with q odd is Z/4Z if q ≡ 3 mod 4 and isomorphic to the group ring (Z/2Z)[F*/F*2] if q ≡ 1 mod 4.[19] • The Witt ring of a local field with maximal ideal of norm congruent to 1 modulo 4 is isomorphic to the group ring (Z/2Z)[V] where V is the Klein 4-group.[20] • The Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 is (Z/4Z)[C2] where C2 is a cyclic group of order 2.[20] • The Witt ring of Q2 is of order 32 and is given by[21] $\mathbf {Z} _{8}[s,t]/\langle 2s,2t,s^{2},t^{2},st-4\rangle .$ Invariants Certain invariants of a quadratic form can be regarded as functions on Witt classes. We have seen that dimension mod 2 is a function on classes: the discriminant is also well-defined. The Hasse invariant of a quadratic form is again a well-defined function on Witt classes with values in the Brauer group of the field of definition.[22] Rank and discriminant We define a ring over K, Q(K), as a set of pairs (d, e) with d in K*/K* 2 and e in Z/2Z. Addition and multiplication are defined by: $(d_{1},e_{1})+(d_{2},e_{2})=((-1)^{e_{1}e_{2}}d_{1}d_{2},e_{1}+e_{2})$ $(d_{1},e_{1})\cdot (d_{2},e_{2})=(d_{1}^{e_{2}}d_{2}^{e_{1}},e_{1}e_{2}).$ Then there is a surjective ring homomorphism from W(K) to this obtained by mapping a class to discriminant and rank mod 2. The kernel is I2.[23] The elements of Q may be regarded as classifying graded quadratic extensions of K.[24] Brauer–Wall group The triple of discriminant, rank mod 2 and Hasse invariant defines a map from W(K) to the Brauer–Wall group BW(K).[25] Witt ring of a local field Let K be a complete local field with valuation v, uniformiser π and residue field k of characteristic not equal to 2. There is an injection W(k) → W(K) which lifts the diagonal form ⟨a1,...an⟩ to ⟨u1,...un⟩ where ui is a unit of K with image ai in k. This yields $W(K)=W(k)\oplus \langle \pi \rangle \cdot W(k)$ identifying W(k) with its image in W(K).[26] Witt ring of a number field Let K be a number field. For quadratic forms over K, there is a Hasse invariant ±1 for every finite place corresponding to the Hilbert symbols. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the signatures coming from real embeddings.[27] We define the symbol ring over K, Sym(K), as a set of triples (d, e, f ) with d in K*/K* 2, e in Z/2 and f a sequence of elements ±1 indexed by the places of K, subject to the condition that all but finitely many terms of f are +1, that the value on acomplex places is +1 and that the product of all the terms in f in +1. Let [a, b] be the sequence of Hilbert symbols: it satisfies the conditions on f just stated.[28] We define addition and multiplication as follows: $(d_{1},e_{1},f_{1})+(d_{2},e_{2},f_{2})=((-1)^{e_{1}e_{2}}d_{1}d_{2},e_{1}+e_{2},[d_{1},d_{2}][-d_{1}d_{2},(-1)^{e_{1}e_{2}}]f_{1}f_{2})$ $(d_{1},e_{1},f_{1})\cdot (d_{2},e_{2},f_{2})=(d_{1}^{e_{2}}d_{2}^{e_{1}},e_{1}e_{2},[d_{1},d_{2}]^{1+e_{1}e_{2}}f_{1}^{e_{2}}f_{2}^{e_{1}})\ .$ Then there is a surjective ring homomorphism from W(K) to Sym(K) obtained by mapping a class to discriminant, rank mod 2, and the sequence of Hasse invariants. The kernel is I3.[29] The symbol ring is a realisation of the Brauer-Wall group.[30] Witt ring of the rationals The Hasse–Minkowski theorem implies that there is an injection[31] $W(\mathbf {Q} )\rightarrow W(\mathbf {R} )\oplus \prod _{p}W(\mathbf {Q} _{p})\ .$ We make this concrete, and compute the image, by using the "second residue homomorphism" W(Qp) → W(Fp). Composed with the map W(Q) → W(Qp) we obtain a group homomorphism ∂p: W(Q) → W(Fp) (for p = 2 we define ∂2 to be the 2-adic valuation of the discriminant, taken mod 2). We then have a split exact sequence[32] $0\rightarrow \mathbf {Z} \rightarrow W(\mathbf {Q} )\rightarrow \mathbf {Z} /2\oplus \bigoplus _{p\neq 2}W(\mathbf {F} _{p})\rightarrow 0\ $ which can be written as an isomorphism $W(\mathbf {Q} )\cong \mathbf {Z} \oplus \mathbf {Z} /2\oplus \bigoplus _{p\neq 2}W(\mathbf {F} _{p})\ $ where the first component is the signature.[33] Witt ring and Milnor's K-theory Let k be a field of characteristic not equal to 2. The powers of the ideal I of forms of even dimension ("fundamental ideal") in $W(k)$ form a descending filtration and one may consider the associated graded ring, that is the direct sum of quotients $I^{n}/I^{n+1}$. Let $\langle a\rangle $ be the quadratic form $ax^{2}$ considered as an element of the Witt ring. Then $\langle a\rangle -\langle 1\rangle $ is an element of I and correspondingly a product of the form $\langle \langle a_{1},\ldots ,a_{n}\rangle \rangle =(\langle a_{1}\rangle -\langle 1\rangle )\cdots (\langle a_{n}\rangle -\langle 1\rangle )$ is an element of $I^{n}.$ John Milnor in a 1970 paper [34] proved that the mapping from $(k^{*})^{n}$ to $I^{n}/I^{n+1}$ that sends $(a_{1},\ldots ,a_{n})$ to $\langle \langle a_{1},\ldots ,a_{n}\rangle \rangle $ is multilinear and maps Steinberg elements (elements such that for some $i$ and $j$ such that $i\neq j$ one has $a_{i}+a_{j}=1$) to zero. This means that this mapping defines a homomorphism from the Milnor ring of k to the graded Witt ring. Milnor showed also that this homomorphism sends elements divisible by 2 to zero and that it is surjective. In the same paper he made a conjecture that this homomorphism is an isomorphism for all fields k (of characteristic different from 2). This became known as the Milnor conjecture on quadratic forms. The conjecture was proved by Dmitry Orlov, Alexander Vishik and Vladimir Voevodsky[35] in 1996 (published in 2007) for the case ${\textrm {char}}(k)=0$, leading to increased understanding of the structure of quadratic forms over arbitrary fields. Grothendieck-Witt ring The Grothendieck-Witt ring GW is a related construction generated by isometry classes of nonsingular quadratic spaces with addition given by orthogonal sum and multiplication given by tensor product. Since two spaces that differ by a hyperbolic plane are not identified in GW, the inverse for the addition needs to be introduced formally through the construction that was discovered by Grothendieck (see Grothendieck group). There is a natural homomorphism GW → Z given by dimension: a field is quadratically closed if and only if this is an isomorphism.[18] The hyperbolic spaces generate an ideal in GW and the Witt ring W is the quotient.[36] The exterior power gives the Grothendieck-Witt ring the additional structure of a λ-ring.[37] Examples • The Grothendieck-Witt ring of C, and indeed any algebraically closed field or quadratically closed field, is Z.[18] • The Grothendieck-Witt ring of R is isomorphic to the group ring Z[C2], where C2 is a cyclic group of order 2.[18] • The Grothendieck-Witt ring of any finite field of odd characteristic is Z ⊕ Z/2Z with trivial multiplication in the second component.[38] The element (1, 0) corresponds to the quadratic form ⟨a⟩ where a is not a square in the finite field. • The Grothendieck-Witt ring of a local field with maximal ideal of norm congruent to 1 modulo 4 is isomorphic to Z ⊕ (Z/2Z)3.[20] • The Grothendieck-Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 it is Z' ⊕ Z/4Z ⊕ Z/2Z.[20] Grothendieck-Witt ring and motivic stable homotopy groups of spheres Fabien Morel[39][40] showed that the Grothendieck-Witt ring of a perfect field is isomorphic to the motivic stable homotopy group of spheres π0,0(S0,0) (see "A¹ homotopy theory"). Witt equivalence Two fields are said to be Witt equivalent if their Witt rings are isomorphic. For global fields there is a local-to-global principle: two global fields are Witt equivalent if and only if there is a bijection between their places such that the corresponding local fields are Witt equivalent.[41] In particular, two number fields K and L are Witt equivalent if and only if there is a bijection T between the places of K and the places of L and a group isomorphism t between their square-class groups, preserving degree 2 Hilbert symbols. In this case the pair (T, t) is called a reciprocity equivalence or a degree 2 Hilbert symbol equivalence.[42] Some variations and extensions of this condition, such as "tame degree l Hilbert symbol equivalence", have also been studied.[43] Generalizations Main article: L-theory Witt groups can also be defined in the same way for skew-symmetric forms, and for quadratic forms, and more generally ε-quadratic forms, over any *-ring R. The resulting groups (and generalizations thereof) are known as the even-dimensional symmetric L-groups L2k(R) and even-dimensional quadratic L-groups L2k(R). The quadratic L-groups are 4-periodic, with L0(R) being the Witt group of (1)-quadratic forms (symmetric), and L2(R) being the Witt group of (−1)-quadratic forms (skew-symmetric); symmetric L-groups are not 4-periodic for all rings, hence they provide a less exact generalization. L-groups are central objects in surgery theory, forming one of the three terms of the surgery exact sequence. See also • Reduced height of a field Notes 1. Milnor & Husemoller (1973) p. 14 2. Lorenz (2008) p. 30 3. Milnor & Husemoller (1973) p. 65 4. Milnor & Husemoller (1973) p. 66 5. Lorenz (2008) p. 37 6. Milnor & Husemoller (1973) p. 72 7. Lam (2005) p. 260 8. Lam (2005) p. 395 9. Lorenz (2008) p. 35 10. Lorenz (2008) p. 31 11. Lam (2005) p. 32 12. Lorenz (2008) p. 33 13. Lam (2005) p. 280 14. Lorenz (2008) p. 36 15. Lam (2005) p. 282 16. Lam (2005) pp. 277–280 17. Lam (2005) p.316 18. Lam (2005) p. 34 19. Lam (2005) p.37 20. Lam (2005) p.152 21. Lam (2005) p.166 22. Lam (2005) p.119 23. Conner & Perlis (1984) p.12 24. Lam (2005) p.113 25. Lam (2005) p.117 26. Garibaldi, Merkurjev & Serre (2003) p.64 27. Conner & Perlis (1984) p.16 28. Conner & Perlis (1984) p.16-17 29. Conner & Perlis (1984) p.18 30. Lam (2005) p.116 31. Lam (2005) p.174 32. Lam (2005) p.175 33. Lam (2005) p.178 34. Milnor, John Willard (1970), "Algebraic K-theory and quadratic forms", Inventiones Mathematicae, 9 (4): 318–344, doi:10.1007/BF01425486, ISSN 0020-9910, MR 0260844 35. Orlov, Dmitry; Vishik, Alexander; Voevodsky, Vladimir (2007), "An exact sequence for K*M/2 with applications to quadratic forms", Annals of Mathematics, 165 (1): 1–13, arXiv:math/0101023, doi:10.4007/annals.2007.165.1 36. Lam (2005) p. 28 37. Garibaldi, Merkurjev & Serre (2003) p.63 38. Lam (2005) p.36, Theorem 3.5 39. , On the motivic stable π0 of the sphere spectrum, In: Axiomatic, Enriched and Motivic Homotopy Theory, pp. 219–260, J.P.C. Greenlees (ed.), 2004 Kluwer Academic Publishers. 40. Fabien Morel, A1-Algebraic topology over a field. Lecture Notes in Mathematics 2052, Springer Verlag, 2012. 41. Perlis, R.; Szymiczek, K.; Conner, P.E.; Litherland, R. (1994). "Matching Witts with global fields". In Jacob, William B.; et al. (eds.). Recent advances in real algebraic geometry and quadratic forms. Contemp. Math. Vol. 155. Providence, RI: American Mathematical Society. pp. 365–387. ISBN 0-8218-5154-3. Zbl 0807.11024. 42. Szymiczek, Kazimierz (1997). "Hilbert-symbol equivalence of number fields". Tatra Mt. Math. Publ. 11: 7–16. Zbl 0978.11012. 43. Czogała, A. (1999). "Higher degree tame Hilbert-symbol equivalence of number fields". Abh. Math. Sem. Univ. Hamburg. 69: 175–185. doi:10.1007/bf02940871. Zbl 0968.11038. References • Conner, Pierre E.; Perlis, Robert (1984). A Survey of Trace Forms of Algebraic Number Fields. Series in Pure Mathematics. Vol. 2. World Scientific. ISBN 9971-966-05-0. Zbl 0551.10017. • Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003). Cohomological invariants in Galois cohomology. University Lecture Series. Vol. 28. Providence, RI: American Mathematical Society. ISBN 0-8218-3287-5. Zbl 1159.12311. • Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023. • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001 • Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001. • Milnor, John; Husemoller, Dale (1973). Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73. Springer-Verlag. ISBN 3-540-06009-X. Zbl 0292.10016. • Witt, Ernst (1936), "Theorie der quadratischen Formen in beliebigen Korpern", Journal für die reine und angewandte Mathematik, 176 (3): 31–44, Zbl 0015.05701 Further reading • Balmer, Paul (2005). "Witt groups". In Friedlander, Eric M.; Grayson, D. R. (eds.). Handbook of K-theory. Vol. 2. Springer-Verlag. pp. 539–579. ISBN 3-540-23019-X. Zbl 1115.19004. External links • Witt rings in the Springer encyclopedia of mathematics
Wikipedia
William Kingdon Clifford William Kingdon Clifford FRS (4 May 1845 – 3 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics,[1] geometry,[2] and computing.[3] Clifford was the first to suggest that gravitation might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression mind-stuff. William Clifford William Kingdon Clifford (1845–1879) Born4 May 1845 (1845-05-04) Exeter, Devon, England Died3 March 1879 (1879-03-04) (aged 33) Madeira, Portugal NationalityEnglish Alma materKing's College London Trinity College, Cambridge Known forClifford algebra Clifford's circle theorems Clifford's theorem Clifford torus Clifford–Klein form Clifford parallel Bessel–Clifford function Dual quaternion Elements of Dynamic SpouseLucy Clifford (1875–1879) Scientific career FieldsMathematics Philosophy InstitutionsUniversity College London Doctoral studentsArthur Black InfluencesGeorg Friedrich Bernhard Riemann Nikolai Ivanovich Lobachevsky Biography Born at Exeter, William Clifford showed great promise at school. He went on to King's College London (at age 15) and Trinity College, Cambridge, where he was elected fellow in 1868, after being second wrangler in 1867 and second Smith's prizeman.[4][5] Being second was a fate he shared with others who became famous scientists, including William Thomson (Lord Kelvin) and James Clerk Maxwell. In 1870, he was part of an expedition to Italy to observe the solar eclipse of 22 December 1870. During that voyage he survived a shipwreck along the Sicilian coast.[6] In 1871, he was appointed professor of mathematics and mechanics at University College London, and in 1874 became a fellow of the Royal Society.[4] He was also a member of the London Mathematical Society and the Metaphysical Society. Clifford married Lucy Lane on 7 April 1875, with whom he had two children.[7] Clifford enjoyed entertaining children and wrote a collection of fairy stories, The Little People.[8] Death and legacy In 1876, Clifford suffered a breakdown, probably brought on by overwork. He taught and administered by day, and wrote by night. A half-year holiday in Algeria and Spain allowed him to resume his duties for 18 months, after which he collapsed again. He went to the island of Madeira to recover, but died there of tuberculosis after a few months, leaving a widow with two children. Clifford and his wife are buried in London's Highgate Cemetery, near the graves of George Eliot and Herbert Spencer, just north of the grave of Karl Marx. The academic journal Advances in Applied Clifford Algebras publishes on Clifford's legacy in kinematics and abstract algebra. Mathematics "Clifford was above all and before all a geometer." — Henry John Stephen Smith[4] The discovery of non-Euclidean geometry opened new possibilities in geometry in Clifford's era. The field of intrinsic differential geometry was born, with the concept of curvature broadly applied to space itself as well as to curved lines and surfaces. Clifford was very much impressed by Bernhard Riemann’s 1854 essay "On the hypotheses which lie at the bases of geometry".[9] In 1870, he reported to the Cambridge Philosophical Society on the curved space concepts of Riemann, and included speculation on the bending of space by gravity. Clifford's translation[10][11] of Riemann's paper was published in Nature in 1873. His report at Cambridge, "On the Space-Theory of Matter", was published in 1876, anticipating Albert Einstein's general relativity by 40 years. Clifford elaborated elliptic space geometry as a non-Euclidean metric space. Equidistant curves in elliptic space are now said to be Clifford parallels. Clifford's contemporaries considered him acute and original, witty and warm. He often worked late into the night, which may have hastened his death. He published papers on a range of topics including algebraic forms and projective geometry and the textbook Elements of Dynamic. His application of graph theory to invariant theory was followed up by William Spottiswoode and Alfred Kempe.[12] Algebras In 1878, Clifford published a seminal work, building on Grassmann's extensive algebra.[13] He had succeeded in unifying the quaternions, developed by William Rowan Hamilton, with Grassmann's outer product (aka the exterior product). He understood the geometric nature of Grassmann's creation, and that the quaternions fit cleanly into the algebra Grassmann had developed. The versors in quaternions facilitate representation of rotation. Clifford laid the foundation for a geometric product, composed of the sum of the inner product and Grassmann's outer product. The geometric product was eventually formalized by the Hungarian mathematician Marcel Riesz. The inner product equips geometric algebra with a metric, fully incorporating distance and angle relationships for lines, planes, and volumes, while the outer product gives those planes and volumes vector-like properties, including a directional bias. Combining the two brought the operation of division into play. This greatly expanded our qualitative understanding of how objects interact in space. Crucially, it also provided the means for quantitatively calculating the spatial consequences of those interactions. The resulting geometric algebra, as he called it, eventually realized the long sought goal[lower-roman 1] of creating an algebra that mirrors the movements and projections of objects in 3-dimensional space.[14] Moreover, Clifford's algebraic schema extends to higher dimensions. The algebraic operations have the same symbolic form as they do in 2 or 3-dimensions. The importance of general Clifford algebras has grown over time, while their isomorphism classes - as real algebras - have been identified in other mathematical systems beyond simply the quaternions.[15] The realms of real analysis and complex analysis have been expanded through the algebra H of quaternions, thanks to its notion of a three-dimensional sphere embedded in a four-dimensional space. Quaternion versors, which inhabit this 3-sphere, provide a representation of the rotation group SO(3). Clifford noted that Hamilton's biquaternions were a tensor product $H\otimes C$ of known algebras, and proposed instead two other tensor products of H: Clifford argued that the "scalars" taken from the complex numbers C might instead be taken from split-complex numbers D or from the dual numbers N. In terms of tensor products, $H\otimes D$ produces split-biquaternions, while $H\otimes N$ forms dual quaternions. The algebra of dual quaternions is used to express screw displacement, a common mapping in kinematics. Philosophy As a philosopher, Clifford's name is chiefly associated with two phrases of his coining, mind-stuff and the tribal self. The former symbolizes his metaphysical conception, suggested to him by his reading of Baruch Spinoza,[4] which Clifford (1878) defined as follows:[17] That element of which, as we have seen, even the simplest feeling is a complex, I shall call Mind-stuff. A moving molecule of inorganic matter does not possess mind or consciousness ; but it possesses a small piece of mind-stuff. When molecules are so combined together as to form the film on the under side of a jelly-fish, the elements of mind-stuff which go along with them are so combined as to form the faint beginnings of Sentience. When the molecules are so combined as to form the brain and nervous system of a vertebrate, the corresponding elements of mind-stuff are so combined as to form some kind of consciousness; that is to say, changes in the complex which take place at the same time get so linked together that the repetition of one implies the repetition of the other. When matter takes the complex form of a living human brain, the corresponding mind-stuff takes the form of a human consciousness, having intelligence and volition. — "On the Nature of Things-in-Themselves" (1878) Regarding Clifford's concept, Sir Frederick Pollock wrote: Briefly put, the conception is that mind is the one ultimate reality; not mind as we know it in the complex forms of conscious feeling and thought, but the simpler elements out of which thought and feeling are built up. The hypothetical ultimate element of mind, or atom of mind-stuff, precisely corresponds to the hypothetical atom of matter, being the ultimate fact of which the material atom is the phenomenon. Matter and the sensible universe are the relations between particular organisms, that is, mind organized into consciousness, and the rest of the world. This leads to results which would in a loose and popular sense be called materialist. But the theory must, as a metaphysical theory, be reckoned on the idealist side. To speak technically, it is an idealist monism.[4] Tribal self, on the other hand, gives the key to Clifford's ethical view, which explains conscience and the moral law by the development in each individual of a 'self,' which prescribes the conduct conducive to the welfare of the 'tribe.' Much of Clifford's contemporary prominence was due to his attitude toward religion. Animated by an intense love of his conception of truth and devotion to public duty, he waged war on such ecclesiastical systems as seemed to him to favour obscurantism, and to put the claims of sect above those of human society. The alarm was greater, as theology was still unreconciled with Darwinism; and Clifford was regarded as a dangerous champion of the anti-spiritual tendencies then imputed to modern science.[4] There has also been debate on the extent to which Clifford's doctrine of 'concomitance' or 'psychophysical parallelism' influenced John Hughlings Jackson's model of the nervous system and, through him, the work of Janet, Freud, Ribot, and Ey.[18] Ethics In his 1877 essay, The Ethics of Belief, Clifford argues that it is immoral to believe things for which one lacks evidence.[19] He describes a ship-owner who planned to send to sea an old and not well built ship full of passengers. The ship-owner had doubts suggested to him that the ship might not be seaworthy: "These doubts preyed upon his mind, and made him unhappy." He considered having the ship refitted even though it would be expensive. At last, "he succeeded in overcoming these melancholy reflections." He watched the ship depart, "with a light heart…and he got his insurance money when she went down in mid-ocean and told no tales."[19] Clifford argues that the ship-owner was guilty of the deaths of the passengers even though he sincerely believed the ship was sound: "[H]e had no right to believe on such evidence as was before him."[lower-roman 2] Moreover, he contends that even in the case where the ship successfully reaches the destination, the decision remains immoral, because the morality of the choice is defined forever once the choice is made, and actual outcome, defined by blind chance, doesn't matter. The ship-owner would be no less guilty: his wrongdoing would never be discovered, but he still had no right to make that decision given the information available to him at the time. Clifford famously concludes with what has come to be known as Clifford's principle: "it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence."[19] As such, he is arguing in direct opposition to religious thinkers for whom 'blind faith' (i.e. belief in things in spite of the lack of evidence for them) was a virtue. This paper was famously attacked by pragmatist philosopher William James in his "Will to Believe" lecture. Often these two works are read and published together as touchstones for the debate over evidentialism, faith, and overbelief. Premonition of relativity Though Clifford never constructed a full theory of spacetime and relativity, there are some remarkable observations he made in print that foreshadowed these modern concepts: In his book Elements of Dynamic (1878), he introduced "quasi-harmonic motion in a hyperbola". He wrote an expression for a parametrized unit hyperbola, which other authors later used as a model for relativistic velocity. Elsewhere he states:[20] The geometry of rotors and motors…forms the basis of the whole modern theory of the relative rest (Static) and the relative motion (Kinematic and Kinetic) of invariable systems.[lower-roman 3] This passage makes reference to biquaternions, though Clifford made these into split-biquaternions as his independent development. The book continues with a chapter "On the bending of space", the substance of general relativity. Clifford also discussed his views in On the Space-Theory of Matter in 1876. In 1910, William Barrett Frankland quoted the Space-Theory of Matter in his book on parallelism: "The boldness of this speculation is surely unexcelled in the history of thought. Up to the present, however, it presents the appearance of an Icarian flight."[21] Years later, after general relativity had been advanced by Albert Einstein, various authors noted that Clifford had anticipated Einstein. Hermann Weyl (1923), for instance, mentioned Clifford as one of those who, like Bernhard Riemann, anticipated the geometric ideas of relativity.[22] In 1940, Eric Temple Bell published The Development of Mathematics, in which he discusses the prescience of Clifford on relativity:[23] Bolder even than Riemann, Clifford confessed his belief (1870) that matter is only a manifestation of curvature in a space-time manifold. This embryonic divination has been acclaimed as an anticipation of Einstein's (1915–16) relativistic theory of the gravitational field. The actual theory, however, bears but slight resemblance to Clifford's rather detailed creed. As a rule, those mathematical prophets who never descend to particulars make the top scores. Almost anyone can hit the side of a barn at forty yards with a charge of buckshot. John Archibald Wheeler, during the 1960 International Congress for Logic, Methodology, and Philosophy of Science (CLMPS) at Stanford, introduced his geometrodynamics formulation of general relativity by crediting Clifford as the initiator.[24] In The Natural Philosophy of Time (1961), Gerald James Whitrow recalls Clifford's prescience, quoting him in order to describe the Friedmann–Lemaître–Robertson–Walker metric in cosmology.[25] Cornelius Lanczos (1970) summarizes Clifford's premonitions:[26] [He] with great ingenuity foresaw in a qualitative fashion that physical matter might be conceived as a curved ripple on a generally flat plane. Many of his ingenious hunches were later realized in Einstein's gravitational theory. Such speculations were automatically premature and could not lead to anything constructive without an intermediate link which demanded the extension of 3-dimensional geometry to the inclusion of time. The theory of curved spaces had to be preceded by the realization that space and time form a single four-dimensional entity. Likewise, Banesh Hoffmann (1973) writes:[27] Riemann, and more specifically Clifford, conjectured that forces and matter might be local irregularities in the curvature of space, and in this they were strikingly prophetic, though for their pains they were dismissed at the time as visionaries. In 1990, Ruth Farwell and Christopher Knee examined the record on acknowledgement of Clifford's foresight.[28] They conclude that "it was Clifford, not Riemann, who anticipated some of the conceptual ideas of General Relativity." To explain the lack of recognition of Clifford's prescience, they point out that he was an expert in metric geometry, and "metric geometry was too challenging to orthodox epistemology to be pursued."[28] In 1992, Farwell and Knee continued their study of Clifford and Riemann:[29] [They] hold that once tensors had been used in the theory of general relativity, the framework existed in which a geometrical perspective in physics could be developed and allowed the challenging geometrical conceptions of Riemann and Clifford to be rediscovered. Selected writings • 1872. On the aims and instruments of scientific thought, 524–41. • 1876 [1870]. On the Space-Theory of Matter.[30][31] • 1877. "The Ethics of Belief." Contemporary Review 29:289.[19][32] • 1878. Elements of Dynamic: An Introduction to the Study of Motion And Rest In Solid And Fluid Bodies.[33] • Book I: "Translations" • Book II: "Rotations" • Book III: "Strains" • 1878. "Applications of Grassmann's Extensive Algebra." American Journal of Mathematics 1(4):353.[34] • 1879: Seeing and Thinking[35]—includes four popular science lectures:[4] • "The Eye and the Brain" • "The Eye and Seeing" • "The Brain and Thinking" • "Of Boundaries in General" • 1879. Lectures and Essays I & II, with an introduction by Sir Frederick Pollock.[36] • 1881. "Mathematical fragments" (facsimiles).[37] • 1882. Mathematical Papers, edited by Robert Tucker, with an introduction by Henry J. S. Smith.[38] • 1885. The Common Sense of the Exact Sciences, completed by Karl Pearson.[39][4] • 1887. Elements of Dynamic 2.[40] • 1885 copy of "The Common Sense of the Exact Sciences" • Title page of an 1885 copy of "The Common Sense of the Exact Sciences" • Table of contents page for an 1885 copy of "The Common Sense of the Exact Sciences" • First page of an 1885 copy of "The Common Sense of the Exact Sciences" Quotations "I…hold that in the physical world nothing else takes place but this variation [of the curvature of space]." — Mathematical Papers (1882) "There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture—that it came to him from outside, and that he did not consciously create it from within." — "Some of the conditions of mental development" (1882), lecture to the Royal Institution "It is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence." — The Ethics of Belief (1879) [1877] "If a man, holding a belief which he was taught in childhood or persuaded of afterwards, keeps down and pushes away any doubts which arise about it in his mind, purposely avoids the reading of books and the company of men that call in question or discuss it, and regards as impious those questions which cannot easily be asked without disturbing it—the life of that man is one long sin against mankind." — The Ethics of Belief (1879) [1877] "I was not, and was conceived. I loved and did a little work. I am not and grieve not." — Epitaph See also • Bessel–Clifford function • Clifford's principle • Clifford analysis • Clifford gates • Clifford bundle • Clifford module • Clifford number • Motor • Rotor • Simplex • Split-biquaternion • Will to Believe Doctrine References Notes 1. "I believe that, so far as geometry is concerned, we need still another analysis which is distinctly geometrical or linear and which will express situation directly as algebra expresses magnitude directly." Leibniz, Gottfried. 1976 [1679]. "Letter to Christian Huygens (8 September 1679)." In Philosophical Papers and Letters (2nd ed.). Springer. 2. The italics are in the original. 3. This passage is immediately followed by a section on "The bending of space." However, according to the preface (p.vii), this section was written by Karl Pearson Citations 1. Doran, Chris; Lasenby, Anthony (2007). Geometric Algebra for Physicists. Cambridge, England: Cambridge University Press. p. 592. ISBN 9780521715959. 2. Hestenes, David (2011). "Grassmann's legacy". Grassmann's Legacy in From Past to Future: Graßmann's Work in Context, Petsche, Hans-Joachim, Lewis, Albert C., Liesen, Jörg, Russ, Steve (ed). Basel, Germany: Springer. pp. 243–260. doi:10.1007/978-3-0346-0405-5_22. ISBN 978-3-0346-0404-8. 3. Dorst, Leo (2009). Geometric Algebra for Computer Scientists. Amsterdam: Morgan Kaufmann. p. 664. ISBN 9780123749420. 4. Chisholm 1911, p. 506. 5. "Clifford, William Kingdon (CLFT863WK)". A Cambridge Alumni Database. University of Cambridge. 6. Chisholm, M. (2002). Such Silver Currents. Cambridge: The Lutterworth Press. p. 26. ISBN 978-0-7188-3017-5. 7. Stephen, Leslie; Pollock, Frederick (1901). Lectures and Essays by the Late William Kingdon Clifford, F.R.S. Vol. 1. New York: Macmillan and Company. p. 20. 8. Eves, Howard W. (1969). In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes. Vol. 3–4. Prindle, Weber and Schmidt. pp. 91–92. 9. Riemann, Bernhard. 1867 [1854]. "On the hypotheses which lie at the bases of geometry" (Habilitationsschrift), translated by W. K. Clifford. – via School of Mathematics, Trinity College Dublin. 10. Clifford, William K. 1873. "On the hypotheses which lie at the bases of geometry." Nature 8:14–17, 36–37. 11. Clifford, William K. 1882. "Paper #9." P. 55–71 in Mathematical Papers. 12. Biggs, Norman L.; Lloyd, Edward Keith; Wilson, Robin James (1976). Graph Theory: 1736-1936. Oxford University Press. p. 67. ISBN 978-0-19-853916-2. 13. Clifford, William (1878). "Applications of Grassmann's extensive algebra". American Journal of Mathematics. 1 (4): 350–358. doi:10.2307/2369379. JSTOR 2369379. 14. Hestenes, David. "On the Evolution of Geometric Algebra and Geometric Calculus". 15. Dechant, Pierre-Philippe (March 2014). "A Clifford algebraic framework for Coxeter group theoretic computations". Advances in Applied Clifford Algebras. 14 (1): 89–108. arXiv:1207.5005. Bibcode:2012arXiv1207.5005D. doi:10.1007/s00006-013-0422-4. S2CID 54035515. 16. Frontispiece of Lectures and Essays by the Late William Kingdon Clifford, F.R.S., vol 2. 17. Clifford, William K. 1878. "On the Nature of Things-in-Themselves." Mind 3(9):57–67. doi:10.1093/mind/os-3.9.57. JSTOR 2246617. 18. Clifford, C. K., and G. E. Berrios. 2000. "Body and Mind." History of Psychiatry 11(43):311–38. doi:10.1177/0957154x0001104305. PMID 11640231. 19. Clifford, William K. 1877. "The Ethics of Belief." Contemporary Review 29:289. 20. Clifford, William K. 1885. Common Sense of the Exact Sciences. London: Kegan Paul, Trench and Co. p. 214. 21. Frankland, William Barrett. 1910. Theories of Parallelism. Cambridge: Cambridge University Press. pp. 48–49. 22. Weyl, Hermann. 1923. Raum Zeit Materie. Berlin: Springer-Verlag. p. 101 23. Bell, Eric Temple. 1940. The Development of Mathematics. pp. 359–60. 24. Wheeler, John Archibald. 1962 [1960]. "Curved empty space as the building material of the physical world: an assessment." In Logic, Methodology, and Philosophy of Science, edited by E. Nagel. Stanford University Press. 25. Whitrow, Gerald James. 1961. The Natural Philosophy of Time (1st ed.). pp. 246–47.—1980 [1961]. The Natural Philosophy of Time (2nd ed.). pp. 291. 26. Lanczos, Cornelius. 1970. Space Through the Ages: The Evolution of Geometrical Ideas from Pythagoras to Hilbert and Einstein. Academic Press. p. 222. 27. Hoffmann, Banesh. 1973. "Relativity." Dictionary of the History of Ideas 4:80. Charles Scribner's Sons. 28. Farwell, Ruth, and Christopher Knee. 1990. Studies in History and Philosophy of Science 21:91–121. 29. Farwell, Ruth, and Christopher Knee. 1992. "The Geometric Challenge of Riemann and Clifford." Pp. 98–106 in 1830–1930: A Century of Geometry, edited by L. Boi, D. Flament, and J. Salanskis. Lecture Notes in Physics 402. Springer Berlin Heidelberg. ISBN 978-3-540-47058-8. doi:10.1007/3-540-55408-4_56. 30. Clifford, William K. 1876 [1870]. "On the Space-Theory of Matter." Proceedings of the Cambridge Philosophical Society 2:157–58. OCLC 6084206. OL 20550270M. proceedingscamb06socigoog at the Internet Archive 31. Clifford, William K. 2007 [1870]. "On the Space-Theory of Matter." P. 71 in Beyond Geometry: Classic Papers from Riemann to Einstein, edited by P. Pesic. Mineola: Dover Publications. Bibcode:2007bgcp.book...71K. 32. Clifford, William K. 1886 [1877]. "The Ethics of Belief" (full text). Lectures and Essays (2nd ed.), edited by L. Stephen and F. Pollock. Macmillan and Co. – via A. J. Burger (2008). 33. Clifford, William K. 1878. Elements of Dynamic: An Introduction to the Study of Motion And Rest In Solid And Fluid Bodies I, II, & III. London: MacMillan and Co. – via Internet Archive. 34. Clifford, William K. 1878. "Applications of Grassmann's Extensive Algebra." American Journal of Mathematics 1(4):353. doi:10.2307/2369379. 35. Clifford, William K. 1879. Seeing and Thinking. London: Macmillan and Co. 36. Clifford, William K. 1901 [1879]. Lectures and Essays I (3rd ed.), edited by L. Stephen and F. Pollock. New York: The Macmillan Company. 37. Clifford, William K. 1881. "Mathematical Fragments" (facsimile). London: Macmillan Company. Located at University of Bordeaux. Science and Technology Library. FR 14652. 38. Clifford, William K. 1882. Mathematical Papers, edited by R. Tucker, introduction by H. J. S. Smith. London: MacMillan and Co. – via Internet Archive. 39. Clifford, William K. 1885. The Common Sense of the Exact Sciences, completed by K. Pearson. London: Kegan, Paul, Trench, and Co. 40. Clifford, William K. 1996 [1887]. "Elements of Dynamic" 2. In From Kant to Hilbert: A Source Book in the Foundations of Mathematics, edited by W. B. Ewald. Oxford. Oxford University Press. • This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Clifford, William Kingdon". Encyclopædia Britannica. Vol. 6 (11th ed.). Cambridge University Press. p. 506. Further reading • Chisholm, M. (1997). "William Kingdon Clifford (1845-1879) and his wife Lucy (1846-1929)". Advances in Applied Clifford Algebras. 7S: 27–41. (The on-line version lacks the article's photographs.) • Chisholm, M. (2002). Such Silver Currents - The Story of William and Lucy Clifford, 1845-1929. Cambridge, UK: The Lutterworth Press. ISBN 978-0-7188-3017-5. • Farwell, Ruth; Knee, Christopher (1990). "The End of the Absolute: a nineteenth century contribution to General Relativity". Studies in History and Philosophy of Science. 21 (1): 91–121. Bibcode:1990SHPSA..21...91F. doi:10.1016/0039-3681(90)90016-2. • Macfarlane, Alexander (1916). Lectures on Ten British Mathematicians of the Nineteenth Century. New York: John Wiley and Sons. Lectures on Ten British Mathematicians of the Nineteenth Century. (See especially pages 78–91) • Madigan, Timothy J. (2010). W.K. Clifford and "The Ethics of Belief Cambridge Scholars Press, Cambridge, UK 978-1847-18503-7. • Penrose, Roger (2004). The Road to Reality: A Complete Guide to the Laws of the Universe. Alfred A. Knopf. ISBN 9780679454434. (See especially Chapter 11) • Stephen, Leslie; Pollock, Frederick (1879). Lectures and Essays by the Late William Kingdon Clifford, F.R.S. Vol. 1. New York: Macmillan and Company. • Stephen, Leslie; Pollock, Frederick (1879). Lectures and Essays by the Late William Kingdon Clifford, F.R.S. Vol. 2. New York: Macmillan and Company. External links • Works by William Kingdon Clifford at Project Gutenberg • William and Lucy Clifford (with pictures) • O'Connor, John J.; Robertson, Edmund F., "William Kingdon Clifford", MacTutor History of Mathematics Archive, University of St Andrews • Works by or about William Kingdon Clifford at Internet Archive • Works by William Kingdon Clifford at LibriVox (public domain audiobooks) • Clifford, William Kingdon, William James, and A.J. Burger (Ed.), The Ethics of Belief. • Joe Rooney William Kingdon Clifford, Department of Design and Innovation, the Open University, London. 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Without loss of generality Without loss of generality (often abbreviated to WOLOG, WLOG[1] or w.l.o.g.; less commonly stated as without any loss of generality or with no loss of generality) is a frequently used expression in mathematics. The term is used to indicate the assumption that follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the proof in general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic.[2] As a result, once a proof is given for the particular case, it is trivial to adapt it to prove the conclusion in all other cases. In many scenarios, the use of "without loss of generality" is made possible by the presence of symmetry.[3] For example, if some property P(x,y) of real numbers is known to be symmetric in x and y, namely that P(x,y) is equivalent to P(y,x), then in proving that P(x,y) holds for every x and y, one may assume "without loss of generality" that x ≤ y. There is no loss of generality in this assumption, since once the case x ≤ y ⇒ P(x,y) has been proved, the other case follows by interchanging x and y : y ≤ x ⇒ P(y,x), and by symmetry of P, this implies P(x,y), thereby showing that P(x,y) holds for all cases. On the other hand, if neither such a symmetry nor another form of equivalence can be established, then the use of "without loss of generality" is incorrect and can amount to an instance of proof by example – a logical fallacy of proving a claim by proving a non-representative example.[4] Example Consider the following theorem (which is a case of the pigeonhole principle): If three objects are each painted either red or blue, then there must be at least two objects of the same color. A proof: Assume, without loss of generality, that the first object is red. If either of the other two objects is red, then we are finished; if not, then the other two objects must both be blue and we are still finished. The above argument works because the exact same reasoning could be applied if the alternative assumption, namely, that the first object is blue, were made, or, similarly, that the words 'red' and 'blue' can be freely exchanged in the wording of the proof. As a result, the use of "without loss of generality" is valid in this case. See also • Up to • Mathematical jargon References 1. "Without Loss of Generality". Art of Problem Solving. Retrieved 2019-10-21. 2. Chartrand, Gary; Polimeni, Albert D.; Zhang, Ping (2008). Mathematical Proofs / A Transition to Advanced Mathematics (2nd ed.). Pearson/Addison Wesley. pp. 80–81. ISBN 978-0-321-39053-0. 3. Dijkstra, Edsger W. (1997). "WLOG, or the misery of the unordered pair (EWD1223)". In Broy, Manfred; Schieder, Birgit (eds.). Mathematical Methods in Program Development (PDF). NATO ASI Series F: Computer and Systems Sciences. Vol. 158. Springer. pp. 33–34. doi:10.1007/978-3-642-60858-2_9. 4. "An Acyclic Inequality in Three Variables". www.cut-the-knot.org. Retrieved 2019-10-21. External links • WLOG at PlanetMath. • "Without Loss of Generality" by John Harrison - discussion of formalizing "WLOG" arguments in an automated theorem prover.
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Wold's theorem In statistics, Wold's decomposition or the Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the Wiener–Khinchin theorem), named after Herman Wold, says that every covariance-stationary time series $Y_{t}$ can be written as the sum of two time series, one deterministic and one stochastic. This article is about the theorem as used in time series analysis. For an abstract mathematical statement, see Wold decomposition. Formally $Y_{t}=\sum _{j=0}^{\infty }b_{j}\varepsilon _{t-j}+\eta _{t},$ where: • $Y_{t}$ is the time series being considered, • $\varepsilon _{t}$ is an uncorrelated sequence which is the innovation process to the process $Y_{t}$ – that is, a white noise process that is input to the linear filter $\{b_{j}\}$. • $b$ is the possibly infinite vector of moving average weights (coefficients or parameters) • $\eta _{t}$ is a deterministic time series, such as one represented by a sine wave. The moving average coefficients have these properties: 1. Stable, that is square summable $\sum _{j=1}^{\infty }|b_{j}|^{2}$ < $\infty $ 2. Causal (i.e. there are no terms with j < 0) 3. Minimum delay 4. Constant ($b_{j}$ independent of t) 5. It is conventional to define $b_{0}=1$ This theorem can be considered as an existence theorem: any stationary process has this seemingly special representation. Not only is the existence of such a simple linear and exact representation remarkable, but even more so is the special nature of the moving average model. Imagine creating a process that is a moving average but not satisfying these properties 1–4. For example, the coefficients $b_{j}$ could define an acausal and non-minimum delay model. Nevertheless the theorem assures the existence of a causal minimum delay moving average that exactly represents this process. How this all works for the case of causality and the minimum delay property is discussed in Scargle (1981), where an extension of the Wold Decomposition is discussed. The usefulness of the Wold Theorem is that it allows the dynamic evolution of a variable $Y_{t}$ to be approximated by a linear model. If the innovations $\varepsilon _{t}$ are independent, then the linear model is the only possible representation relating the observed value of $Y_{t}$ to its past evolution. However, when $\varepsilon _{t}$ is merely an uncorrelated but not independent sequence, then the linear model exists but it is not the only representation of the dynamic dependence of the series. In this latter case, it is possible that the linear model may not be very useful, and there would be a nonlinear model relating the observed value of $Y_{t}$ to its past evolution. However, in practical time series analysis, it is often the case that only linear predictors are considered, partly on the grounds of simplicity, in which case the Wold decomposition is directly relevant. The Wold representation depends on an infinite number of parameters, although in practice they usually decay rapidly. The autoregressive model is an alternative that may have only a few coefficients if the corresponding moving average has many. These two models can be combined into an autoregressive-moving average (ARMA) model, or an autoregressive-integrated-moving average (ARIMA) model if non-stationarity is involved. See Scargle (1981) and references there; in addition this paper gives an extension of the Wold Theorem that allows more generality for the moving average (not necessarily stable, causal, or minimum delay) accompanied by a sharper characterization of the innovation (identically and independently distributed, not just uncorrelated). This extension allows the possibility of models that are more faithful to physical or astrophysical processes, and in particular can sense ″the arrow of time.″ References • Anderson, T. W. (1971). The Statistical Analysis of Time Series. Wiley. • Nerlove, M.; Grether, David M.; Carvalho, José L. (1995). Analysis of Economic Time Series (Revised ed.). San Diego: Academic Press. pp. 30–36. ISBN 0-12-515751-7. • Scargle, J. D. (1981). Studies in astronomical time series analysis. I – Modeling random processes in the time domain. Astrophysical Journal Supplement Series. Vol. 45. pp. 1–71. • Wold, H. (1954) A Study in the Analysis of Stationary Time Series, Second revised edition, with an Appendix on "Recent Developments in Time Series Analysis" by Peter Whittle. Almqvist and Wiksell Book Co., Uppsala. 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Wold's decomposition In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator. This article is about the general mathematical result. For the application to time series analysis, see Wold's theorem. In time series analysis, the theorem implies that any stationary discrete-time stochastic process can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a moving average process. Details Let H be a Hilbert space, L(H) be the bounded operators on H, and V ∈ L(H) be an isometry. The Wold decomposition states that every isometry V takes the form $V=(\oplus _{\alpha \in A}S)\oplus U$ for some index set A, where S is the unilateral shift on a Hilbert space Hα, and U is a unitary operator (possible vacuous). The family {Hα} consists of isomorphic Hilbert spaces. A proof can be sketched as follows. Successive applications of V give a descending sequences of copies of H isomorphically embedded in itself: $H=H\supset V(H)\supset V^{2}(H)\supset \cdots =H_{0}\supset H_{1}\supset H_{2}\supset \cdots ,$ where V(H) denotes the range of V. The above defined Hi = Vi(H). If one defines $M_{i}=H_{i}\ominus H_{i+1}=V^{i}(H\ominus V(H))\quad {\text{for}}\quad i\geq 0\;,$ then $H=(\oplus _{i\geq 0}M_{i})\oplus (\cap _{i\geq 0}H_{i})=K_{1}\oplus K_{2}.$ It is clear that K1 and K2 are invariant subspaces of V. So V(K2) = K2. In other words, V restricted to K2 is a surjective isometry, i.e., a unitary operator U. Furthermore, each Mi is isomorphic to another, with V being an isomorphism between Mi and Mi+1: V "shifts" Mi to Mi+1. Suppose the dimension of each Mi is some cardinal number α. We see that K1 can be written as a direct sum Hilbert spaces $K_{1}=\oplus H_{\alpha }$ where each Hα is an invariant subspaces of V and V restricted to each Hα is the unilateral shift S. Therefore $V=V\vert _{K_{1}}\oplus V\vert _{K_{2}}=(\oplus _{\alpha \in A}S)\oplus U,$ which is a Wold decomposition of V. Remarks It is immediate from the Wold decomposition that the spectrum of any proper, i.e. non-unitary, isometry is the unit disk in the complex plane. An isometry V is said to be pure if, in the notation of the above proof, ∩i≥0 Hi = {0}. The multiplicity of a pure isometry V is the dimension of the kernel of V*, i.e. the cardinality of the index set A in the Wold decomposition of V. In other words, a pure isometry of multiplicity N takes the form $V=\oplus _{1\leq \alpha \leq N}S.$ In this terminology, the Wold decomposition expresses an isometry as a direct sum of a pure isometry and a unitary operator. A subspace M is called a wandering subspace of V if Vn(M) ⊥ Vm(M) for all n ≠ m. In particular, each Mi defined above is a wandering subspace of V. A sequence of isometries The decomposition above can be generalized slightly to a sequence of isometries, indexed by the integers. The C*-algebra generated by an isometry Consider an isometry V ∈ L(H). Denote by C*(V) the C*-algebra generated by V, i.e. C*(V) is the norm closure of polynomials in V and V*. The Wold decomposition can be applied to characterize C*(V). Let C(T) be the continuous functions on the unit circle T. We recall that the C*-algebra C*(S) generated by the unilateral shift S takes the following form C*(S) = {Tf + K | Tf is a Toeplitz operator with continuous symbol f ∈ C(T) and K is a compact operator}. In this identification, S = Tz where z is the identity function in C(T). The algebra C*(S) is called the Toeplitz algebra. Theorem (Coburn) C*(V) is isomorphic to the Toeplitz algebra and V is the isomorphic image of Tz. The proof hinges on the connections with C(T), in the description of the Toeplitz algebra and that the spectrum of a unitary operator is contained in the circle T. The following properties of the Toeplitz algebra will be needed: 1. $T_{f}+T_{g}=T_{f+g}.\,$ 2. $T_{f}^{*}=T_{\bar {f}}.$ 3. The semicommutator $T_{f}T_{g}-T_{fg}\,$ is compact. The Wold decomposition says that V is the direct sum of copies of Tz and then some unitary U: $V=(\oplus _{\alpha \in A}T_{z})\oplus U.$ So we invoke the continuous functional calculus f → f(U), and define $\Phi :C^{*}(S)\rightarrow C^{*}(V)\quad {\text{by}}\quad \Phi (T_{f}+K)=\oplus _{\alpha \in A}(T_{f}+K)\oplus f(U).$ One can now verify Φ is an isomorphism that maps the unilateral shift to V: By property 1 above, Φ is linear. The map Φ is injective because Tf is not compact for any non-zero f ∈ C(T) and thus Tf + K = 0 implies f = 0. Since the range of Φ is a C*-algebra, Φ is surjective by the minimality of C*(V). Property 2 and the continuous functional calculus ensure that Φ preserves the *-operation. Finally, the semicommutator property shows that Φ is multiplicative. Therefore the theorem holds. References • Coburn, L. (1967). "The C*-algebra of an isometry". Bull. Amer. Math. Soc. 73 (5): 722–726. doi:10.1090/S0002-9904-1967-11845-7. • Constantinescu, T. (1996). Schur Parameters, Factorization and Dilation Problems. Operator Theory, Advances and Applications. Vol. 82. Birkhäuser. ISBN 3-7643-5285-X. • Douglas, R. G. (1972). Banach Algebra Techniques in Operator Theory. Academic Press. ISBN 0-12-221350-5. • Rosenblum, Marvin; Rovnyak, James (1985). Hardy Classes and Operator Theory. Oxford University Press. ISBN 0-19-503591-7. 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Wolf Prize in Mathematics The Wolf Prize in Mathematics is awarded almost annually[lower-alpha 1] by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. According to a reputation survey conducted in 2013 and 2014, the Wolf Prize in Mathematics is the third most prestigious international academic award in mathematics, after the Abel Prize and the Fields Medal.[1][2] Until the establishment of the Abel Prize, it was probably the closest equivalent of a "Nobel Prize in Mathematics", since the Fields Medal is awarded every four years only to mathematicians under the age of 40. The Wolf Prize includes a monetary award of $100,000. Laureates Year Name Nationality Citation 1978 Israel Gelfand  Soviet Union for his work in functional analysis, group representation, and for his seminal contributions to many areas of mathematics and its applications. Carl L. Siegel  Germany for his contributions to the theory of numbers, theory of several complex variables, and celestial mechanics. 1979 Jean Leray  France for pioneering work on the development and application of topological methods to the study of differential equations. André Weil  France for his inspired introduction of algebraic-geometric methods to the theory of numbers. 1980 Henri Cartan  France for pioneering work in algebraic topology, complex variables, homological algebra and inspired leadership of a generation of mathematicians. Andrey Kolmogorov  Soviet Union for deep and original discoveries in Fourier analysis, probability theory, ergodic theory and dynamical systems. 1981 Lars Ahlfors  Finland for seminal discoveries and the creation of powerful new methods in geometric function theory. Oscar Zariski  United States creator of the modern approach to algebraic geometry, by its fusion with commutative algebra. 1982 Hassler Whitney  United States for his fundamental work in algebraic topology, differential geometry and differential topology. Mark Krein  Soviet Union for his fundamental contributions to functional analysis and its applications. 1983/84 Shiing-Shen Chern  China  United States for outstanding contributions to global differential geometry, which have profoundly influenced all mathematics. Paul Erdős  Hungary for his numerous contributions to number theory, combinatorics, probability, set theory and mathematical analysis, and for personally stimulating mathematicians the world over. 1984/85 Kunihiko Kodaira  Japan for his outstanding contributions to the study of complex manifolds and algebraic varieties. Hans Lewy  United States for initiating many, now classic and essential, developments in partial differential equations. 1986 Samuel Eilenberg  Poland  United States for his fundamental work in algebraic topology and homological algebra. Atle Selberg  Norway for his profound and original work on number theory and on discrete groups and automorphic forms. 1987 Kiyoshi Itō  Japan for his fundamental contributions to pure and applied probability theory, especially the creation of the stochastic differential and integral calculus. Peter Lax  Hungary  United States for his outstanding contributions to many areas of analysis and applied mathematics. 1988 Friedrich Hirzebruch  Germany for outstanding work combining topology, algebraic geometry and differential geometry, and algebraic number theory; and for his stimulation of mathematical cooperation and research. Lars Hörmander  Sweden for fundamental work in modern analysis, in particular, the application of pseudo-differential operators and Fourier integral operators to linear partial differential equations. 1989 Alberto Calderón  Argentina for his groundbreaking work on singular integral operators and their application to important problems in partial differential equations. John Milnor  United States for ingenious and highly original discoveries in geometry, which have opened important new vistas in topology from the algebraic, combinatorial, and differentiable viewpoint. 1990 Ennio de Giorgi  Italy for his innovating ideas and fundamental achievements in partial differential equations and calculus of variations. Ilya Piatetski-Shapiro  Israel for his fundamental contributions in the fields of homogeneous complex domains, discrete groups, representation theory and automorphic forms. 1991 No award 1992 Lennart Carleson  Sweden for his fundamental contributions to Fourier analysis, complex analysis, quasi-conformal mappings and dynamical systems. John G. Thompson  United States for his profound contributions to all aspects of finite group theory and connections with other branches of mathematics. 1993 Mikhail Gromov  Russia  France for his revolutionary contributions to global Riemannian and symplectic geometry, algebraic topology, geometric group theory and the theory of partial differential equations; Jacques Tits  Belgium  France for his pioneering and fundamental contributions to the theory of the structure of algebraic and other classes of groups and in particular for the theory of buildings. 1994/95 Jürgen Moser  Switzerland  United States for his fundamental work on stability in Hamiltonian mechanics and his profound and influential contributions to nonlinear differential equations. 1995/96 Robert Langlands  Canada for his path-blazing work and extraordinary insight in the fields of number theory, automorphic forms and group representation. Andrew Wiles  United Kingdom for spectacular contributions to number theory and related fields, major advances on fundamental conjectures, and for settling Fermat's Last Theorem. 1996/97 Joseph B. Keller  United States for his profound and innovative contributions, in particular to electromagnetic, optical, and acoustic wave propagation and to fluid, solid, quantum and statistical mechanics. Yakov G. Sinai  Russia for his fundamental contributions to mathematically rigorous methods in statistical mechanics and the ergodic theory of dynamical systems and their applications in physics. 1998 No award 1999 László Lovász  Hungary  United States for his outstanding contributions to combinatorics, theoretical computer science and combinatorial optimization. Elias M. Stein  United States for his contributions to classical and Euclidean Fourier analysis and for his exceptional impact on a new generation of analysts through his eloquent teaching and writing. 2000 Raoul Bott  Hungary for his deep discoveries in topology and differential geometry and their applications to Lie groups, differential operators and mathematical physics. Jean-Pierre Serre  France for his many fundamental contributions to topology, algebraic geometry, algebra, and number theory and for his inspirational lectures and writing. 2001 Vladimir Arnold  Ukraine for his deep and influential work in a multitude of areas of mathematics, including dynamical systems, differential equations, and singularity theory. Saharon Shelah  Israel for his many fundamental contributions to mathematical logic and set theory, and their applications within other parts of mathematics. 2002/03 Mikio Sato  Japan for his creation of algebraic analysis, including hyperfunction theory and microfunction theory, holonomic quantum field theory, and a unified theory of soliton equations. John Tate  United States for his creation of fundamental concepts in algebraic number theory. 2004 No award 2005 Gregory Margulis  Russia for his monumental contributions to algebra, in particular to the theory of lattices in semi-simple Lie groups, and striking applications of this to ergodic theory, representation theory, number theory, combinatorics, and measure theory. Sergei Novikov  Russia for his fundamental and pioneering contributions to algebraic and differential topology, and to mathematical physics, notably the introduction of algebraic-geometric methods. 2006/07 Stephen Smale  United States for his groundbreaking contributions that have played a fundamental role in shaping differential topology, dynamical systems, mathematical economics, and other subjects in mathematics. Hillel Furstenberg  United States  Israel for his profound contributions to ergodic theory, probability, topological dynamics, analysis on symmetric spaces and homogeneous flows. 2008 Pierre Deligne  Belgium for his work on mixed Hodge theory; the Weil conjectures; the Riemann-Hilbert correspondence; and for his contributions to arithmetic. Phillip A. Griffiths  United States for his work on variations of Hodge structures; the theory of periods of abelian integrals; and for his contributions to complex differential geometry. David B. Mumford  United States for his work on algebraic surfaces; on geometric invariant theory; and for laying the foundations of the modern algebraic theory of moduli of curves and theta functions. 2009 No award 2010 Shing-Tung Yau  United States for his work in geometric analysis that has had a profound and dramatic impact on many areas of geometry and physics. Dennis P. Sullivan  United States for his innovative contributions to algebraic topology and conformal dynamics. 2011 No award 2012 Michael Aschbacher  United States for his work on the theory of finite groups. Luis Caffarelli  Argentina for his work on partial differential equations. 2013 George D. Mostow  United States for his fundamental and pioneering contribution to geometry and Lie group theory. Michael Artin  United States for his fundamental contributions to algebraic geometry, both in commutative and noncommutative. 2014 Peter Sarnak  South Africa  United States for his deep contributions in analysis, number theory, geometry, and combinatorics. 2015 James G. Arthur  Canada for his monumental work on the trace formula and his fundamental contributions to the theory of automorphic representations of reductive groups. 2016 No award 2017 Richard Schoen  United States for his contributions to geometric analysis and the understanding of the interconnectedness of partial differential equations and differential geometry. Charles Fefferman  United States for his contributions in a number of mathematical areas including complex multivariate analysis, partial differential equations and sub-elliptical problems. 2018 Alexander Beilinson  United States for their work that has made significant progress at the interface of geometry and mathematical physics. Vladimir Drinfeld  Russia  United States 2019 Jean-Francois Le Gall  France for his several deep and elegant contributions to the theory of stochastic processes. Gregory Lawler  United States for his comprehensive and pioneering research on erased loops and random walks.[3] 2020 Simon K. Donaldson  United Kingdom for their contributions to differential geometry and topology.[4] Yakov Eliashberg  United States 2021 No award 2022 George Lusztig  Romania  United States for his groundbreaking contributions to representation theory and related areas.[5] 2023 Ingrid Daubechies  Belgium  United States for her work in wavelet theory and applied harmonic analysis.[6] Laureates per country Below is a chart of all laureates per country (updated to 2023 laureates). Some laureates are counted more than once if have multiple citizenship. Country Number of laureates  United States 30  Soviet Union /  Russia 9  France 7  Hungary 5  Israel 3  Japan 3  Belgium 3  Germany 2  United Kingdom 2  Canada 2  Argentina 2  Sweden 2  South Africa 1  Poland 1  Italy 1  China 1  Norway 1  Finland 1  Romania 1  Ukraine 1 Notes 1. The Wolf Foundation website describes the prize as annual; however, some prizes are split across years, while in some years no prize is awarded. See also • List of mathematics awards References 1. IREG Observatory on Academic Ranking and Excellence. IREG List of International Academic Awards (PDF). Brussels: IREG Observatory on Academic Ranking and Excellence. Retrieved 3 March 2018. 2. Zheng, Juntao; Liu, Niancai (2015). "Mapping of important international academic awards". Scientometrics. 104: 763–791. doi:10.1007/s11192-015-1613-7. 3. Wolf Prize 2019 - Mathematics 4. Wolf Prize 2020 - Mathematics 5. Wolf Prize 2022 - Mathematics 6. Wolf Prize 2023 - Mathematics External links • "The Wolf Foundation Prize in Mathematics". • "Huffingtonpost Israel-Wolf-Prizes 2012". Huffington Post. 10 January 2012. • "Jerusalempost Israel-Wolf-Prizes 2013". • Israel-Wolf-Prizes 2015 • Jerusalempost Wolf Prizes 2017 • Jerusalempost Wolf Prizes 2018 • Wolf Prize 2019 Wolf Foundation Prizes • Agriculture • Arts • Chemistry • Mathematics • Medicine • Physics Laureates of the Wolf Prize in Mathematics 1970s • Israel Gelfand / Carl L. Siegel (1978) • Jean Leray / André Weil (1979) 1980s • Henri Cartan / Andrey Kolmogorov (1980) • Lars Ahlfors / Oscar Zariski (1981) • Hassler Whitney / Mark Krein (1982) • Shiing-Shen Chern / Paul Erdős (1983/84) • Kunihiko Kodaira / Hans Lewy (1984/85) • Samuel Eilenberg / Atle Selberg (1986) • Kiyosi Itô / Peter Lax (1987) • Friedrich Hirzebruch / Lars Hörmander (1988) • Alberto Calderón / John Milnor (1989) 1990s • Ennio de Giorgi / Ilya Piatetski-Shapiro (1990) • Lennart Carleson / John G. Thompson (1992) • Mikhail Gromov / Jacques Tits (1993) • Jürgen Moser (1994/95) • Robert Langlands / Andrew Wiles (1995/96) • Joseph Keller / Yakov G. Sinai (1996/97) • László Lovász / Elias M. Stein (1999) 2000s • Raoul Bott / Jean-Pierre Serre (2000) • Vladimir Arnold / Saharon Shelah (2001) • Mikio Sato / John Tate (2002/03) • Grigory Margulis / Sergei Novikov (2005) • Stephen Smale / Hillel Furstenberg (2006/07) • Pierre Deligne / Phillip A. Griffiths / David B. Mumford (2008) 2010s • Dennis Sullivan / Shing-Tung Yau (2010) • Michael Aschbacher / Luis Caffarelli (2012) • George Mostow / Michael Artin (2013) • Peter Sarnak (2014) • James G. Arthur (2015) • Richard Schoen / Charles Fefferman (2017) • Alexander Beilinson / Vladimir Drinfeld (2018) • Jean-François Le Gall / Gregory Lawler (2019) 2020s • Simon K. Donaldson / Yakov Eliashberg (2020) • George Lusztig (2022) • Ingrid Daubechies (2023)  Mathematics portal
Wikipedia
Wolfe conditions In the unconstrained minimization problem, the Wolfe conditions are a set of inequalities for performing inexact line search, especially in quasi-Newton methods, first published by Philip Wolfe in 1969.[1][2] In these methods the idea is to find $\min _{x}f(\mathbf {x} )$ for some smooth $f\colon \mathbb {R} ^{n}\to \mathbb {R} $. Each step often involves approximately solving the subproblem $\min _{\alpha }f(\mathbf {x} _{k}+\alpha \mathbf {p} _{k})$ where $\mathbf {x} _{k}$ is the current best guess, $\mathbf {p} _{k}\in \mathbb {R} ^{n}$ is a search direction, and $\alpha \in \mathbb {R} $ is the step length. The inexact line searches provide an efficient way of computing an acceptable step length $\alpha $ that reduces the objective function 'sufficiently', rather than minimizing the objective function over $\alpha \in \mathbb {R} ^{+}$ exactly. A line search algorithm can use Wolfe conditions as a requirement for any guessed $\alpha $, before finding a new search direction $\mathbf {p} _{k}$. Armijo rule and curvature A step length $\alpha _{k}$ is said to satisfy the Wolfe conditions, restricted to the direction $\mathbf {p} _{k}$, if the following two inequalities hold: ${\begin{aligned}{\textbf {i)}}&\quad f(\mathbf {x} _{k}+\alpha _{k}\mathbf {p} _{k})\leq f(\mathbf {x} _{k})+c_{1}\alpha _{k}\mathbf {p} _{k}^{\mathrm {T} }\nabla f(\mathbf {x} _{k}),\\[6pt]{\textbf {ii)}}&\quad {-\mathbf {p} }_{k}^{\mathrm {T} }\nabla f(\mathbf {x} _{k}+\alpha _{k}\mathbf {p} _{k})\leq -c_{2}\mathbf {p} _{k}^{\mathrm {T} }\nabla f(\mathbf {x} _{k}),\end{aligned}}$ with $0<c_{1}<c_{2}<1$. (In examining condition (ii), recall that to ensure that $\mathbf {p} _{k}$ is a descent direction, we have $\mathbf {p} _{k}^{\mathrm {T} }\nabla f(\mathbf {x} _{k})<0$, as in the case of gradient descent, where $\mathbf {p} _{k}=-\nabla f(\mathbf {x} _{k})$, or Newton–Raphson, where $\mathbf {p} _{k}=-\mathbf {H} ^{-1}\nabla f(\mathbf {x} _{k})$ with $\mathbf {H} $ positive definite.) $c_{1}$ is usually chosen to be quite small while $c_{2}$ is much larger; Nocedal and Wright give example values of $c_{1}=10^{-4}$ and $c_{2}=0.9$ for Newton or quasi-Newton methods and $c_{2}=0.1$ for the nonlinear conjugate gradient method.[3] Inequality i) is known as the Armijo rule[4] and ii) as the curvature condition; i) ensures that the step length $\alpha _{k}$ decreases $f$ 'sufficiently', and ii) ensures that the slope has been reduced sufficiently. Conditions i) and ii) can be interpreted as respectively providing an upper and lower bound on the admissible step length values. Strong Wolfe condition on curvature Denote a univariate function $\varphi $ restricted to the direction $\mathbf {p} _{k}$ as $\varphi (\alpha )=f(\mathbf {x} _{k}+\alpha \mathbf {p} _{k})$. The Wolfe conditions can result in a value for the step length that is not close to a minimizer of $\varphi $. If we modify the curvature condition to the following, ${\textbf {iii)}}\quad {\big |}\mathbf {p} _{k}^{\mathrm {T} }\nabla f(\mathbf {x} _{k}+\alpha _{k}\mathbf {p} _{k}){\big |}\leq c_{2}{\big |}\mathbf {p} _{k}^{\mathrm {T} }\nabla f(\mathbf {x} _{k}){\big |}$ then i) and iii) together form the so-called strong Wolfe conditions, and force $\alpha _{k}$ to lie close to a critical point of $\varphi $. Rationale The principal reason for imposing the Wolfe conditions in an optimization algorithm where $\mathbf {x} _{k+1}=\mathbf {x} _{k}+\alpha \mathbf {p} _{k}$ is to ensure convergence of the gradient to zero. In particular, if the cosine of the angle between $\mathbf {p} _{k}$ and the gradient, $\cos \theta _{k}={\frac {\nabla f(\mathbf {x} _{k})^{\mathrm {T} }\mathbf {p} _{k}}{\|\nabla f(\mathbf {x} _{k})\|\|\mathbf {p} _{k}\|}}$ is bounded away from zero and the i) and ii) conditions hold, then $\nabla f(\mathbf {x} _{k})\rightarrow 0$. An additional motivation, in the case of a quasi-Newton method, is that if $\mathbf {p} _{k}=-B_{k}^{-1}\nabla f(\mathbf {x} _{k})$, where the matrix $B_{k}$ is updated by the BFGS or DFP formula, then if $B_{k}$ is positive definite ii) implies $B_{k+1}$ is also positive definite. Comments Wolfe's conditions are more complicated than Armijo's condition, and a gradient descent algorithm based on Armijo's condition has a better theoretical guarantee than one based on Wolfe conditions (see the sections on "Upper bound for learning rates" and "Theoretical guarantee" in the Backtracking line search article). See also • Backtracking line search References 1. Wolfe, P. (1969). "Convergence Conditions for Ascent Methods". SIAM Review. 11 (2): 226–235. doi:10.1137/1011036. JSTOR 2028111. 2. Wolfe, P. (1971). "Convergence Conditions for Ascent Methods. II: Some Corrections". SIAM Review. 13 (2): 185–188. doi:10.1137/1013035. JSTOR 2028821. 3. Nocedal, Jorge; Wright, Stephen (1999). Numerical Optimization. p. 38. 4. Armijo, Larry (1966). "Minimization of functions having Lipschitz continuous first partial derivatives". Pacific J. Math. 16 (1): 1–3. doi:10.2140/pjm.1966.16.1. Further reading • "Line Search Methods". Numerical Optimization. Springer Series in Operations Research and Financial Engineering. 2006. pp. 30–32. doi:10.1007/978-0-387-40065-5_3. ISBN 978-0-387-30303-1. • "Quasi-Newton Methods". Numerical Optimization. Springer Series in Operations Research and Financial Engineering. 2006. pp. 135–163. doi:10.1007/978-0-387-40065-5_6. ISBN 978-0-387-30303-1. Optimization: Algorithms, methods, and heuristics Unconstrained nonlinear Functions • Golden-section search • Interpolation methods • Line search • Nelder–Mead method • Successive parabolic interpolation Gradients Convergence • Trust region • Wolfe conditions Quasi–Newton • Berndt–Hall–Hall–Hausman • Broyden–Fletcher–Goldfarb–Shanno and L-BFGS • Davidon–Fletcher–Powell • Symmetric rank-one (SR1) Other methods • Conjugate gradient • Gauss–Newton • Gradient • Mirror • Levenberg–Marquardt • Powell's dog leg method • Truncated Newton Hessians • Newton's method Constrained nonlinear General • Barrier methods • Penalty methods Differentiable • Augmented Lagrangian methods • Sequential quadratic programming • Successive linear programming Convex optimization Convex minimization • Cutting-plane method • Reduced gradient (Frank–Wolfe) • Subgradient method Linear and quadratic Interior point • Affine scaling • Ellipsoid algorithm of Khachiyan • Projective algorithm of Karmarkar Basis-exchange • Simplex algorithm of Dantzig • Revised simplex algorithm • Criss-cross algorithm • Principal pivoting algorithm of Lemke Combinatorial Paradigms • Approximation algorithm • Dynamic programming • Greedy algorithm • Integer programming • Branch and bound/cut Graph algorithms Minimum spanning tree • Borůvka • Prim • Kruskal Shortest path • Bellman–Ford • SPFA • Dijkstra • Floyd–Warshall Network flows • Dinic • Edmonds–Karp • Ford–Fulkerson • Push–relabel maximum flow Metaheuristics • Evolutionary algorithm • Hill climbing • Local search • Parallel metaheuristics • Simulated annealing • Spiral optimization algorithm • Tabu search • Software
Wikipedia
Wolfe duality In mathematical optimization, Wolfe duality, named after Philip Wolfe, is type of dual problem in which the objective function and constraints are all differentiable functions. Using this concept a lower bound for a minimization problem can be found because of the weak duality principle.[1] Mathematical formulation For a minimization problem with inequality constraints, ${\begin{aligned}&{\underset {x}{\operatorname {minimize} }}&&f(x)\\&\operatorname {subject\;to} &&g_{i}(x)\leq 0,\quad i=1,\dots ,m\end{aligned}}$ the Lagrangian dual problem is ${\begin{aligned}&{\underset {u}{\operatorname {maximize} }}&&\inf _{x}\left(f(x)+\sum _{j=1}^{m}u_{j}g_{j}(x)\right)\\&\operatorname {subject\;to} &&u_{i}\geq 0,\quad i=1,\dots ,m\end{aligned}}$ where the objective function is the Lagrange dual function. Provided that the functions $f$ and $g_{1},\ldots ,g_{m}$ are convex and continuously differentiable, the infimum occurs where the gradient is equal to zero. The problem ${\begin{aligned}&{\underset {x,u}{\operatorname {maximize} }}&&f(x)+\sum _{j=1}^{m}u_{j}g_{j}(x)\\&\operatorname {subject\;to} &&\nabla f(x)+\sum _{j=1}^{m}u_{j}\nabla g_{j}(x)=0\\&&&u_{i}\geq 0,\quad i=1,\dots ,m\end{aligned}}$ is called the Wolfe dual problem.[2] This problem employs the KKT conditions as a constraint. Also, the equality constraint $\nabla f(x)+\sum _{j=1}^{m}u_{j}\nabla g_{j}(x)$ is nonlinear in general, so the Wolfe dual problem may be a nonconvex optimization problem. In any case, weak duality holds.[3] See also • Lagrangian duality • Fenchel duality References 1. Philip Wolfe (1961). "A duality theorem for non-linear programming". Quarterly of Applied Mathematics. 19 (3): 239–244. doi:10.1090/qam/135625. 2. "Chapter 3. Duality in convex optimization" (PDF). October 30, 2011. Retrieved May 20, 2012. 3. Geoffrion, Arthur M. (1971). "Duality in Nonlinear Programming: A Simplified Applications-Oriented Development". SIAM Review. 13 (1): 1–37. doi:10.1137/1013001. JSTOR 2028848.
Wikipedia
Wolfgang Haken Wolfgang Haken (June 21, 1928 – October 2, 2022) was a German American mathematician who specialized in topology, in particular 3-manifolds. Wolfgang Haken Haken in 2008 Born(1928-06-21)June 21, 1928 Berlin, Germany DiedOctober 2, 2022(2022-10-02) (aged 94) Champaign, Illinois Alma materKiel University Occupation(s)Mathematician, professor Known forSolving the four-color theorem Biography Haken was born on June 21, 1928, in Berlin, Germany. His father was Werner Haken, a physicist who had Max Planck as a doctoral thesis advisor.[1] In 1953, Haken earned a Ph.D. degree in mathematics from Christian-Albrechts-Universität zu Kiel (Kiel University) and married Anna-Irmgard von Bredow, who earned a Ph.D. degree in mathematics from the same university in 1959. In 1962, they left Germany so he could accept a position as visiting professor at the University of Illinois at Urbana-Champaign. He became a full professor in 1965, retiring in 1998. In 1976, together with colleague Kenneth Appel at the University of Illinois at Urbana-Champaign, Haken solved the four-color problem: they proved that any planar graph can be properly colored using at most four colors. Haken has introduced several ideas, including Haken manifolds, Kneser-Haken finiteness, and an expansion of the work of Kneser into a theory of normal surfaces. Much of his work has an algorithmic aspect, and he is a figure in algorithmic topology. One of his key contributions to this field is an algorithm to detect whether a knot is unknotted. In 1978, Haken delivered an invited address at the International Congress of Mathematicians in Helsinki.[2] He was a recipient of the 1979 Fulkerson Prize of the American Mathematical Society for his proof with Appel of the four-color theorem.[3] Haken died in Champaign, Illinois, on October 2, 2022, aged 94.[4] Family Haken's eldest son, Armin, proved that there exist propositional tautologies that require resolution proofs of exponential size.[5] Haken's eldest daughter, Dorothea Blostein, is a professor of computer science, known for her discovery of the master theorem for divide-and-conquer recurrences. Haken’s second son, Lippold, is the inventor of the Continuum Fingerboard. Haken’s youngest son, Rudolf, is a professor of music, who established the world's first Electric Strings university degree program at the University of Illinois at Urbana-Champaign.[6] Wolfgang is the cousin of Hermann Haken, a physicist known for laser theory and synergetics. See also • Unknotting problem References 1. Werner Haken, Beitrag zur Kenntnis der thermoelektrischen Eigenschaften der Metallegierungen. Accessed May 6, 2019 2. International Congress of Mathematicians 1978. International Mathematical Union. Accessed May 29, 2011 3. Delbert Ray Fulkerson Prize, American Mathematical Society. Accessed May 29, 2011 4. "Wolfgang Haken's obituary". news-gazette.com. October 13, 2022. Archived from the original on October 14, 2022. Retrieved October 14, 2022. 5. Avi Wigderson, Mathematics and Computation, March 27 2018, footnote at Theorem 6.11 6. University of Illinois Electric Strings Degree Program Accessed November 15, 2022 • Haken, W. "Theorie der Normalflachen." Acta Math. 105, 245–375, 1961. • Ilya Kapovich (2016). "Wolfgang Haken: A biographical sketch". Illinois Journal of Mathematics. 60 (1): iii–ix. External links • Wolfgang Haken memorial website • Wolfgang Haken at the Mathematics Genealogy Project • Haken's faculty page at the University of Illinois at Urbana-Champaign • Wolfgang Haken biography from World of Mathematics • Lippold Haken's life story • Haken, Armin (1985), "The intractability of resolution", Theoretical Computer Science, 39: 297–308, doi:10.1016/0304-3975(85)90144-6 • Appel, Kenneth; Haken, Wolfgang (1989), Every Planar Map is Four Colorable, AMS, p. xv, ISBN 0-8218-5103-9 Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef
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Wolfgang Krieger Wolfgang Krieger (born 3 June 3, 1940, Garmisch-Partenkirchen) is a German mathematician, specializing in analysis. Krieger studied mathematics and physics at the Ludwig-Maximilians-Universität München from 1959, where he obtained his doctorate in 1968 under Elmar Thoma with the thesis Über Maßklassen.[1] Krieger studied at Harvard University from 1962 to 1965, earning a master's degree in 1964. From 1966 to 1968 he was a research assistant in Munich; and from 1968 assistant professor, from 1970 associate professor and from 1972 full professor at Ohio State University. For the academic year 1973–1974 he was a visiting professor at the University of Göttingen. In 1974, until his retirement in 2006, he was a professor at Heidelberg University. From 1985 to 1987 he was Dean of the Faculty of Mathematics.[2] His research deals with ergodic theory, dynamical systems, and operator algebras. Cuntz-Krieger algebras, introduced in 1980, are named after him and Joachim Cuntz.[3] Krieger was a visiting scholar at IHES and Paris VI University in 1977–1978, at the University of Ottawa in 1982–1983, at the Almaden Research Center of IBM in 1988–1989, and at the Hebrew University of Jerusalem in 2005–2006. In 1997 he was a Fellow of the Japan Society for the Promotion of Science. He was elected a Fellow of the American Mathematical Society in 2012. He was an invited speaker with the talk On Generators in Ergodic Theory at the ICM in Vancouver in 1974. Selected publications • On non-singular transformations of a measure space. I, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 11, No. 2, 1969, pp. 83–97. doi:10.1007/BF00531811 • On entropy and generators of measure-preserving transformations, Transactions of the American Mathematical Society, Vol. 149, 1970, pp. 453–464. doi:10.1090/S0002-9947-1970-0259068-3 • On ergodic flows and the isomorphism of factors, Mathematical Annals, Vol. 223, 1976, pp. 19–70 doi:10.1007/BF01360278 • with Alain Connes: Measure space automorphisms, the normalizers of their full groups, and approximate finiteness, Journal of Functional Analysis, Vol. 24, 1977, pp. 336–352 doi:10.1016/0022-1236(77)90062-3 • with Joachim Cuntz: A class of C*-algebras and topological Markov chains, Inventiones Mathematicae, Vol. 56, 1980, pp. 251–268. doi:10.1007/BF01390048 • On the Subsystems of Topological Markov Chains, Ergodic Theory and Dynamical Systems, Vol. 2, 1982, pp. 195–202 doi:10.1017/S0143385700001516 • with Mike Boyle: Periodic points and automorphisms of the shift, Trans. Amer. Math. Soc., Vol. 302, 1987, pp. 125–149 doi:10.1090/S0002-9947-1987-0887501-5 • with Brian Marcus and Selim Tuncel: On automorphisms of Markov chains, Trans. Amer. Math. Soc., Vol. 333, 1992, pp. 531–565 doi:10.1090/S0002-9947-1992-1099353-3 References 1. Wolfgang Krieger at the Mathematics Genealogy Project 2. Heidelberg Scholarly Lexicon 3. Nasr-Isfahani, Alireza (2017). "Algebraic Cuntz-Krieger algebras". arXiv:1708.01780 [math.RA]. External links • Prof. Dr. Wolfgang Krieger, math.uni-heidelberg.de Authority control International • ISNI • VIAF National • Germany Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie
Wikipedia
Wolfgang Walter Wolfgang Ludwig Walter (2 May 1927 – 26 June 2010) was a German mathematician, who specialized in the theory of differential equations. His textbook[2] on ordinary differential equations became a standard graduate text on the subject at many institutions. Wolfgang Ludwig Walter Born(1927-05-02)2 May 1927 Schwäbisch Gmünd, Weimar Republic Died26 June 2010(2010-06-26) (aged 83) Karlsruhe, Germany NationalityGerman Alma materUniversity of Tübingen Scientific career FieldsMathematics InstitutionsUniversity of Karlsruhe Doctoral advisorErich Kamke, Hellmuth Kneser[1] Biography Wolfgang Walter was born in 1927 in Schwäbisch Gmünd, Baden-Württemberg. His school studies were interrupted in 1943 when he was drafted into the army. He served as a soldier on the Eastern Front, was subsequently wounded and later interned as a prisoner of war by US troops. In 1946 after his release he completed his school education and in the years 1947–1952 studied mathematics and physics at the University of Tübingen, where he stayed on to study for his PhD under Erich Kamke and Hellmuth Kneser, defending his thesis in 1956.[3] In 1986–1992 Walter held the post of president of GAMM, the German society of applied mathematics and mechanics.[4] He died in Karlsruhe in 2010 at the age of 83. Works • Walter, Wolfgang (1998). Ordinary Differential Equations. Springer. ISBN 978-0387984599. • Walter, Wolfgang (2012). Differential and Integral Inequalities. Springer; Softcover reprint of the original 1st ed. 1970 edition. ISBN 978-3642864070. References 1. Wolfgang Walter. Mathematical Genealogy Project 2. Walter, Wolfgang (1998). Ordinary Differential Equations. Springer. ISBN 978-0387984599. 3. Wolfgang Reichel. In Memoriam Wolfgang Walter (1927–2010), Jahresber Dtsch Math-Ver, 2011. (in German) 4. R.M. Redheffer's 66th birthday tribute to Wolfgang Walter, World Scientific Publishing Company, 1994. Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • Belgium • United States • Czech Republic • Netherlands • Poland Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef
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WolframAlpha WolframAlpha (/ˈwʊlf.rəm-/ WUULf-rəm-) is an answer engine developed by Wolfram Research.[3] It answers factual queries by computing answers from externally sourced data.[4][5] WolframAlpha Type of site Answer engine OwnerWolframAlpha LLC Created byWolfram Research URLwww.wolframalpha.com CommercialYes RegistrationOptional LaunchedMay 18, 2009 (2009-05-18)[1] (official launch) May 15, 2009 (2009-05-15)[2] (public launch) Current statusActive Written inWolfram Language WolframAlpha was released on May 18, 2009, and is based on Wolfram's earlier product Wolfram Mathematica, a technical computing platform.[1] WolframAlpha gathers data from academic and commercial websites such as the CIA's The World Factbook, the United States Geological Survey, a Cornell University Library publication called All About Birds, Chambers Biographical Dictionary, Dow Jones, the Catalogue of Life,[3] CrunchBase,[6] Best Buy,[7] and the FAA to answer queries.[8] A Spanish language version was launched in 2022.[9] Technology Overview Users submit queries and computation requests via a text field. WolframAlpha then computes answers and relevant visualizations from a knowledge base of curated, structured data that come from other sites and books. It can respond to particularly phrased natural language fact-based questions. It displays its "Input interpretation" of such a question, using standardized phrases. It can also parse mathematical symbolism and respond with numerical and statistical results. Development WolframAlpha is written in the Wolfram Language, a general multi-paradigm programming language, and implemented in Mathematica. Wolfram language is proprietary and not commonly used by developers.[10] Usage WolframAlpha was used to power some searches in the Microsoft Bing and DuckDuckGo search engines but is no longer used to provide search results.[11][12] For factual question answering, WolframAlpha was used by Apple's Siri and Amazon Alexa for math and science queries but is no longer operational within those services.[13][14] WolframAlpha data types became available in July 2020 within Microsoft Excel, but the Microsoft-Wolfram partnership ended nearly two years later, in 2022, in favor of Microsoft Power Query data types.[15] WolframAlpha functionality in Microsoft Excel will end in June 2023.[16] History Launch preparations for WolframAlpha began on May 15, 2009 at 7 p.m. CDT and were broadcast live on Justin.tv. The plan was to publicly launch the service a few hours later. However, there were issues due to extreme load. The service officially launched on May 18, 2009,[17] receiving mixed reviews.[18][19] In 2009, WolframAlpha advocates pointed to its potential, some stating that how it determines results is more important than current usefulness.[18] WolframAlpha was free at launch, but later Wolfram Research attempted to monetize the service by launching an iOS application with a cost of $50, while the website itself was free.[20] That plan was abandoned after criticism.[21] On February 8, 2012, WolframAlpha Pro was released,[22] offering users additional features for a monthly subscription fee.[22][23] Some high-school and college students use WolframAlpha to cheat on math homework, though Wolfram Research says the service helps students understand math with its problem-solving capabilities.[24] Copyright claims InfoWorld published an article warning readers of the potential implications of giving an automated website proprietary rights to the data it generates.[25] Free software advocate Richard Stallman also opposes recognizing the site as a copyright holder and suspects that Wolfram Research would not be able to make this case under existing copyright law.[26] See also • Commonsense knowledge problem • Strong AI • Watson (computer) References 1. The Wolfram|Alpha Launch Team (May 8, 2009). "So Much for A Quiet Launch". Wolfram|Alpha Blog. Wolfram Alpha. Retrieved February 9, 2013.{{cite web}}: CS1 maint: multiple names: authors list (link) 2. The Wolfram|Alpha Launch Team (May 12, 2009). "Going Live—and Webcasting It". Wolfram|Alpha Blog. Wolfram Alpha. Retrieved February 9, 2013.{{cite web}}: CS1 maint: multiple names: authors list (link) 3. Bobbie Johnson (May 21, 2009). "Where does Wolfram Alpha get its information?". The Guardian. Retrieved March 8, 2013. 4. "About Wolfram|Alpha: Making the World's Knowledge Computable". wolframalpha.com. Retrieved November 25, 2015. 5. Johnson, Bobbie (March 9, 2009). "British search engine 'could rival Google'". The Guardian. UK: Guardian News and Media. Retrieved February 9, 2013. 6. Dillet, Romain (September 7, 2012). "Wolfram Alpha Makes CrunchBase Data Computable Just In Time For Disrupt SF". TechCrunch. Retrieved February 9, 2013. 7. Golson, Jordan (December 16, 2011). "Wolfram Delivers Siri-Enabled Shopping Results From Best Buy". MacRumors. Retrieved February 9, 2013. 8. Barylick, Chris (November 19, 2011). "Wolfram Alpha search engine now tracks flight paths, trajectory information". Engadget. Retrieved February 9, 2013. 9. "Wolfram Alpha Spanish Announcement". Wolfram Alpha. Wolfram Research. July 22, 2022. Retrieved July 22, 2022. 10. "TIOBE Index". TIOBE. Retrieved October 6, 2022. 11. Krazit, Tom (August 21, 2009). "Bing strikes licensing deal with Wolfram Alpha". CNET. Archived from the original on October 23, 2013. Retrieved February 9, 2013. 12. The Wolfram|Alpha Team (April 18, 2011). "Wolfram|Alpha and DuckDuckGo Partner on API Binding and Search Integration". Wolfram|Alpha Blog. Wolfram Alpha. Retrieved February 9, 2013.{{cite web}}: CS1 maint: multiple names: authors list (link) 13. "Alexa gets access to Wolfram Alpha's knowledge engine". TechCrunch. December 20, 2018. Retrieved March 17, 2021. 14. "Alexa Can Now Answer Those Tricky Math Questions". News18. December 26, 2018. 15. "Excel Data Types with Wolfram End of Support FAQ". support.microsoft.com. Retrieved August 15, 2022. 16. "Microsoft is killing Money in Excel along with Wolfram Alpha data types". XDA. May 31, 2022. Retrieved August 15, 2022. 17. "Wolfram 'search engine' goes live". BBC News. May 18, 2009. Retrieved February 9, 2013. 18. Spivack, Nova (March 7, 2009). "Wolfram Alpha is Coming – and It Could be as Important as Google". Nova Spivack – Minding the Planet. Retrieved February 9, 2013. 19. Singel, Ryan (May 18, 2009). "Wolfram|Alpha Fails the Cool Test". Wired. Retrieved February 9, 2013. 20. "Nice Try, Wolfram Alpha. Still Not Paying $50 For Your App". TechCrunch. December 3, 2009. Retrieved August 15, 2022. 21. "Nice Try, Wolfram Alpha. Still Not Paying $50 For Your App". TechCrunch. December 3, 2009. Retrieved August 15, 2022. 22. Wolfram, Stephen (February 8, 2012). "Announcing Wolfram|Alpha Pro". Wolfram|Alpha Blog. Wolfram Alpha. Retrieved February 9, 2013. 23. "Step-by-Step Math". 24. Biddle, Pippa. "AI Is Making It Extremely Easy for Students to Cheat | Backchannel". Wired. ISSN 1059-1028. Retrieved October 6, 2022. 25. McAllister, Neil (July 29, 2009). "How Wolfram Alpha could change software". InfoWorld. Retrieved February 28, 2012. 26. Stallman, Richard (August 4, 2009). "How Wolfram Alpha's Copyright Claims Could Change Software". Access 2 Knowledge (Mailing list). Archived from the original on April 28, 2013. Retrieved February 17, 2012. External links • Official website Wolfram Research Products • Computable Document Format • Mathematica • GridMathematica • MathWorld • WolframAlpha • Wolfram Demonstrations Project • Wolfram Language • Wolfram SystemModeler People • Stephen Wolfram • Conrad Wolfram • Theodore Gray • Eric Weisstein • Ed Pegg Jr. Virtual assistants Active • AliGenie • Alexa • Alice • Bixby • Viv • Braina • Celia • Clova • Cortana • Google Assistant • Maluuba • Mycroft • Siri • Voice Mate • Watson • WolframAlpha • Xiaoice Discontinued • BlackBerry Assistant • Google Now • M • Microsoft Agent • Microsoft Bob • Microsoft Voice Command • Ms. Dewey • Mya • Office Assistant (Clippy) • S Voice • Speaktoit Assistant • Tafiti • Vlingo
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Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-sized) interactive programmes called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hosted by Wolfram Research, whose stated goal is to bring computational exploration to a large population. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Parents' Choice Award in 2008. Technology The Demonstrations run in Mathematica 6 or above and in Wolfram CDF Player which is a free modified version of Wolfram's Mathematica [1] and available for Windows, Linux and macOS[2] and can operate as a web browser plugin. They typically consist of a very direct user interface to a graphic or visualization, which dynamically recomputes in response to user actions such as moving a slider, clicking a button, or dragging a piece of graphics. Each Demonstration also has a brief description of the concept. Demonstrations are now easily embeddable into any website or blog.[3] Each Demonstration page includes a snippet of JavaScript code in the Share section of the sidebar. Topics The website is organized by topic: for example, science,[4] mathematics, computer science, art, biology, and finance. They cover a variety of levels, from elementary school mathematics to much more advanced topics such as quantum mechanics and models of biological organisms. The site is aimed at both educators and students, as well as researchers who wish to present their ideas to the broadest possible audience. Process Wolfram Research's staff organizes and edits the Demonstrations, which may be created by any user of Mathematica, then freely published[5] and freely downloaded. The Demonstrations are open-source, which means that they not only demonstrate the concept itself but also show how to implement it. Alternatives The use of the web to transmit small interactive programs is reminiscent of Sun's Java applets, Adobe's Flash, and the open-source Processing. However, those creating Demonstrations have access to the algorithmic and visualization capabilities of Mathematica making it more suitable for technical demonstrations. The Demonstrations Project also has similarities to user-generated content websites like Wikipedia and Flickr. Its business model is similar to Adobe's Acrobat and Flash strategy of charging for development tools but providing a free reader. References 1. Math Games MAA Online, May 1, 2007. 2. Adventures with the Wolfram Demonstrations Project John Wass, Scientific Computing 3. Kaurov, Vitaliy. "Add a Wolfram Demonstration to Your Site in One Easy Step". 4. Molecular Wolfram Demonstrations ScienceBase 5. Throwing beanbags in Mathematica 6, Scientific Computing, May 17, 2007. External links • Official site Wolfram Research Products • Computable Document Format • Mathematica • GridMathematica • MathWorld • WolframAlpha • Wolfram Demonstrations Project • Wolfram Language • Wolfram SystemModeler People • Stephen Wolfram • Conrad Wolfram • Theodore Gray • Eric Weisstein • Ed Pegg Jr.
Wikipedia
MathWorld MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein.[2][3] It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign.[3] MathWorld Type of businessPrivate Type of site Internet encyclopedia project Available inEnglish OwnerWolfram Research, Inc. Created byEric W. Weisstein[1] URLmathworld.wolfram.com LaunchedNovember 1999 (1999-11) Current statusActive History Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled "CRC Concise Encyclopedia of Mathematics." The free online version became only partially accessible to the public. In 1999 Weisstein went to work for Wolfram Research, Inc. (WRI), and WRI renamed the Math Treasure Trove to MathWorld and hosted it on the company's website without access restrictions. CRC lawsuit In 2000, CRC Press sued Wolfram Research Inc. (WRI), WRI president Stephen Wolfram, and author Eric W. Weisstein, due to what they considered a breach of contract: that the MathWorld content was to remain in print only. The site was taken down by a court injunction.[4] The case was later settled out of court, with WRI paying an unspecified amount and complying with other stipulations. Among these stipulations is the inclusion of a copyright notice at the bottom of the website and broad rights for the CRC Press to produce MathWorld in printed book form. The site then became once again available free to the public. This case made a wave of headlines in online publishing circles. The PlanetMath project was a result of MathWorld's being unavailable.[5] See also • List of online encyclopedias • Mathematica References 1. Eric Weisstein (2007). "Making MathWorld". Mathematica Journal. 10 (3). Archived from the original on 2012-07-09. Retrieved 2010-08-22. 2. "Wolfram MathWorld : the web's most extensive mathematics resource | WorldCat.org". www.worldcat.org. Retrieved 2023-03-28. 3. "Making MathWorld « The Mathematica Journal". Retrieved 2023-03-28. 4. "CRC Press, LLC v. Wolfram Research, Inc., 149 F. Supp. 2d 500 | Casetext Search + Citator". casetext.com. Retrieved 2023-03-28. 5. Corneli, Joseph (2011). "The PlanetMath Encyclopedia" (PDF). ITP 2011 Workshop on Mathematical Wikis (MathWikis 2011) Nijmegen, Netherlands, August 27, 2011. External links • Official website Wolfram Research Products • Computable Document Format • Mathematica • GridMathematica • MathWorld • WolframAlpha • Wolfram Demonstrations Project • Wolfram Language • Wolfram SystemModeler People • Stephen Wolfram • Conrad Wolfram • Theodore Gray • Eric Weisstein • Ed Pegg Jr.
Wikipedia
Wolstenholme number A Wolstenholme number is a number that is the numerator of the generalized harmonic number Hn,2. Not to be confused with Wolstenholme prime. The first such numbers are 1, 5, 49, 205, 5269, 5369, 266681, 1077749, ... (sequence A007406 in the OEIS). These numbers are named after Joseph Wolstenholme, who proved Wolstenholme's theorem on modular relations of the generalized harmonic numbers. References • Weisstein, Eric W. "WolstenholmeNumber". MathWorld. Classes of natural numbers Powers and related numbers • Achilles • Power of 2 • Power of 3 • Power of 10 • Square • Cube • Fourth power • Fifth power • Sixth power • Seventh power • Eighth power • Perfect power • Powerful • Prime power Of the form a × 2b ± 1 • Cullen • Double Mersenne • Fermat • Mersenne • Proth • Thabit • Woodall Other polynomial numbers • Hilbert • Idoneal • Leyland • Loeschian • Lucky numbers of Euler Recursively defined numbers • Fibonacci • Jacobsthal • Leonardo • Lucas • Padovan • Pell • Perrin Possessing a specific set of other numbers • Amenable • Congruent • Knödel • Riesel • Sierpiński Expressible via specific sums • Nonhypotenuse • Polite • Practical • Primary pseudoperfect • Ulam • Wolstenholme Figurate numbers 2-dimensional centered • Centered triangular • Centered square • Centered pentagonal • Centered hexagonal • Centered heptagonal • Centered octagonal • Centered nonagonal • Centered decagonal • Star non-centered • Triangular • Square • Square triangular • Pentagonal • Hexagonal • Heptagonal • Octagonal • Nonagonal • Decagonal • Dodecagonal 3-dimensional centered • Centered tetrahedral • Centered cube • Centered octahedral • Centered dodecahedral • Centered icosahedral non-centered • Tetrahedral • Cubic • Octahedral • Dodecahedral • Icosahedral • Stella octangula pyramidal • Square pyramidal 4-dimensional non-centered • Pentatope • Squared triangular • Tesseractic Combinatorial numbers • Bell • Cake • Catalan • Dedekind • Delannoy • Euler • Eulerian • Fuss–Catalan • Lah • Lazy caterer's sequence • Lobb • Motzkin • Narayana • Ordered Bell • Schröder • Schröder–Hipparchus • Stirling first • Stirling second • Telephone number • Wedderburn–Etherington Primes • Wieferich • Wall–Sun–Sun • Wolstenholme prime • Wilson Pseudoprimes • Carmichael number • Catalan pseudoprime • Elliptic pseudoprime • Euler pseudoprime • Euler–Jacobi pseudoprime • Fermat pseudoprime • Frobenius pseudoprime • Lucas pseudoprime • Lucas–Carmichael number • Somer–Lucas pseudoprime • Strong pseudoprime Arithmetic functions and dynamics Divisor functions • Abundant • Almost perfect • Arithmetic • Betrothed • Colossally abundant • Deficient • Descartes • Hemiperfect • Highly abundant • Highly composite • Hyperperfect • Multiply perfect • Perfect • Practical • Primitive abundant • Quasiperfect • Refactorable • Semiperfect • Sublime • Superabundant • Superior highly composite • Superperfect Prime omega functions • Almost prime • Semiprime Euler's totient function • Highly cototient • Highly totient • Noncototient • Nontotient • Perfect totient • Sparsely totient Aliquot sequences • Amicable • Perfect • Sociable • Untouchable Primorial • Euclid • Fortunate Other prime factor or divisor related numbers • Blum • Cyclic • Erdős–Nicolas • Erdős–Woods • Friendly • Giuga • Harmonic divisor • Jordan–Pólya • Lucas–Carmichael • Pronic • Regular • Rough • Smooth • Sphenic • Størmer • Super-Poulet • Zeisel Numeral system-dependent numbers Arithmetic functions and dynamics • Persistence • Additive • Multiplicative Digit sum • Digit sum • Digital root • Self • Sum-product Digit product • Multiplicative digital root • Sum-product Coding-related • Meertens Other • Dudeney • Factorion • Kaprekar • Kaprekar's constant • Keith • Lychrel • Narcissistic • Perfect digit-to-digit invariant • Perfect digital invariant • Happy P-adic numbers-related • Automorphic • Trimorphic Digit-composition related • Palindromic • Pandigital • Repdigit • Repunit • Self-descriptive • Smarandache–Wellin • Undulating Digit-permutation related • Cyclic • Digit-reassembly • Parasitic • Primeval • Transposable Divisor-related • Equidigital • Extravagant • Frugal • Harshad • Polydivisible • Smith • Vampire Other • Friedman Binary numbers • Evil • Odious • Pernicious Generated via a sieve • Lucky • Prime Sorting related • Pancake number • Sorting number Natural language related • Aronson's sequence • Ban Graphemics related • Strobogrammatic • Mathematics portal
Wikipedia
Wolstenholme prime In number theory, a Wolstenholme prime is a special type of prime number satisfying a stronger version of Wolstenholme's theorem. Wolstenholme's theorem is a congruence relation satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician Joseph Wolstenholme, who first described this theorem in the 19th century. Not to be confused with Wolstenholme number. Wolstenholme prime Named afterJoseph Wolstenholme Publication year1995[1] Author of publicationMcIntosh, R. J. No. of known terms2 Conjectured no. of termsInfinite Subsequence ofIrregular primes First terms16843, 2124679 Largest known term2124679 OEIS index • A088164 • Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4) Interest in these primes first arose due to their connection with Fermat's Last Theorem. Wolstenholme primes are also related to other special classes of numbers, studied in the hope to be able to generalize a proof for the truth of the theorem to all positive integers greater than two. The only two known Wolstenholme primes are 16843 and 2124679 (sequence A088164 in the OEIS). There are no other Wolstenholme primes less than 109.[2] Definition Unsolved problem in mathematics: Are there any Wolstenholme primes other than 16843 and 2124679? (more unsolved problems in mathematics) Wolstenholme prime can be defined in a number of equivalent ways. Definition via binomial coefficients A Wolstenholme prime is a prime number p > 7 that satisfies the congruence ${2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}},$ where the expression in left-hand side denotes a binomial coefficient.[3] In comparison, Wolstenholme's theorem states that for every prime p > 3 the following congruence holds: ${2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}.$ Definition via Bernoulli numbers A Wolstenholme prime is a prime p that divides the numerator of the Bernoulli number Bp−3.[4][5][6] The Wolstenholme primes therefore form a subset of the irregular primes. Definition via irregular pairs Main article: Irregular prime A Wolstenholme prime is a prime p such that (p, p–3) is an irregular pair.[7][8] Definition via harmonic numbers A Wolstenholme prime is a prime p such that[9] $H_{p-1}\equiv 0{\pmod {p^{3}}}\,,$ i.e. the numerator of the harmonic number $H_{p-1}$ expressed in lowest terms is divisible by p3. Search and current status The search for Wolstenholme primes began in the 1960s and continued over the following decades, with the latest results published in 2007. The first Wolstenholme prime 16843 was found in 1964, although it was not explicitly reported at that time.[10] The 1964 discovery was later independently confirmed in the 1970s. This remained the only known example of such a prime for almost 20 years, until the discovery announcement of the second Wolstenholme prime 2124679 in 1993.[11] Up to 1.2×107, no further Wolstenholme primes were found.[12] This was later extended to 2×108 by McIntosh in 1995 [5] and Trevisan & Weber were able to reach 2.5×108.[13] The latest result as of 2007 is that there are only those two Wolstenholme primes up to 109.[14] Expected number of Wolstenholme primes It is conjectured that infinitely many Wolstenholme primes exist. It is conjectured that the number of Wolstenholme primes ≤ x is about ln ln x, where ln denotes the natural logarithm. For each prime p ≥ 5, the Wolstenholme quotient is defined as $W_{p}{=}{\frac {{2p-1 \choose p-1}-1}{p^{3}}}.$ Clearly, p is a Wolstenholme prime if and only if Wp ≡ 0 (mod p). Empirically one may assume that the remainders of Wp modulo p are uniformly distributed in the set {0, 1, ..., p–1}. By this reasoning, the probability that the remainder takes on a particular value (e.g., 0) is about 1/p.[5] See also • Wieferich prime • Wall–Sun–Sun prime • Wilson prime • Table of congruences Notes 1. Wolstenholme primes were first described by McIntosh in McIntosh 1995, p. 385 2. Weisstein, Eric W., "Wolstenholme prime", MathWorld 3. Cook, J. D., Binomial coefficients, retrieved 21 December 2010 4. Clarke & Jones 2004, p. 553. 5. McIntosh 1995, p. 387. 6. Zhao 2008, p. 25. 7. Johnson 1975, p. 114. 8. Buhler et al. 1993, p. 152. 9. Zhao 2007, p. 18. 10. Selfridge and Pollack published the first Wolstenholme prime in Selfridge & Pollack 1964, p. 97 (see McIntosh & Roettger 2007, p. 2092). 11. Ribenboim 2004, p. 23. 12. Zhao 2007, p. 25. 13. Trevisan & Weber 2001, p. 283–284. 14. McIntosh & Roettger 2007, p. 2092. References • Buhler, J.; Crandall, R.; Ernvall, R.; Metsänkylä, T. (1993), "Irregular Primes and Cyclotomic Invariants to Four Million" (PDF), Mathematics of Computation, 61 (203): 151–153, Bibcode:1993MaCom..61..151B, doi:10.2307/2152942, JSTOR 2152942 Archived 12 November 2010 at WebCite • Clarke, F.; Jones, C. (2004), "A Congruence for Factorials" (PDF), Bulletin of the London Mathematical Society, 36 (4): 553–558, doi:10.1112/S0024609304003194, S2CID 120202453 Archived 2 January 2011 at WebCite • Johnson, W. (1975), "Irregular Primes and Cyclotomic Invariants" (PDF), Mathematics of Computation, 29 (129): 113–120, doi:10.2307/2005468, JSTOR 2005468 Archived 20 December 2010 at WebCite • McIntosh, R. J. (1995), "On the converse of Wolstenholme's Theorem" (PDF), Acta Arithmetica, 71 (4): 381–389, doi:10.4064/aa-71-4-381-389 Archived 8 November 2010 at WebCite • McIntosh, R. J.; Roettger, E. L. (2007), "A search for Fibonacci-Wieferich and Wolstenholme primes" (PDF), Mathematics of Computation, 76 (260): 2087–2094, Bibcode:2007MaCom..76.2087M, doi:10.1090/S0025-5718-07-01955-2 Archived 10 December 2010 at WebCite • Ribenboim, P. (2004), "Chapter 2. How to Recognize Whether a Natural Number is a Prime", The Little Book of Bigger Primes, New York: Springer-Verlag New York, Inc., ISBN 978-0-387-20169-6 Archived 24 November 2010 at WebCite • Selfridge, J. L.; Pollack, B. W. (1964), "Fermat's last theorem is true for any exponent up to 25,000", Notices of the American Mathematical Society, 11: 97 • Trevisan, V.; Weber, K. E. (2001), "Testing the Converse of Wolstenholme's Theorem" (PDF), Matemática Contemporânea, 21 (16): 275–286, doi:10.21711/231766362001/rmc2116 Archived 10 December 2010 at WebCite • Zhao, J. (2007), "Bernoulli numbers, Wolstenholme's theorem, and p5 variations of Lucas' theorem" (PDF), Journal of Number Theory, 123: 18–26, doi:10.1016/j.jnt.2006.05.005, S2CID 937685Archived 12 November 2010 at WebCite • Zhao, J. (2008), "Wolstenholme Type Theorem for Multiple Harmonic Sums" (PDF), International Journal of Number Theory, 4 (1): 73–106, doi:10.1142/s1793042108001146 Archived 27 November 2010 at WebCite Further reading • Babbage, C. (1819), "Demonstration of a theorem relating to prime numbers", The Edinburgh Philosophical Journal, 1: 46–49 • Krattenthaler, C.; Rivoal, T. (2009), "On the integrality of the Taylor coefficients of mirror maps, II", Communications in Number Theory and Physics, 3 (3): 555–591, arXiv:0907.2578, Bibcode:2009arXiv0907.2578K, doi:10.4310/CNTP.2009.v3.n3.a5 • Wolstenholme, J. (1862), "On Certain Properties of Prime Numbers", The Quarterly Journal of Pure and Applied Mathematics, 5: 35–39 External links • Caldwell, Chris K. Wolstenholme prime from The Prime Glossary • McIntosh, R. J. Wolstenholme Search Status as of March 2004 e-mail to Paul Zimmermann • Bruck, R. Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients • Conrad, K. The p-adic Growth of Harmonic Sums interesting observation involving the two Wolstenholme primes Prime number classes By formula • Fermat (22n + 1) • Mersenne (2p − 1) • Double Mersenne (22p−1 − 1) • Wagstaff (2p + 1)/3 • Proth (k·2n + 1) • Factorial (n! ± 1) • Primorial (pn# ± 1) • Euclid (pn# + 1) • Pythagorean (4n + 1) • Pierpont (2m·3n + 1) • Quartan (x4 + y4) • Solinas (2m ± 2n ± 1) • Cullen (n·2n + 1) • Woodall (n·2n − 1) • Cuban (x3 − y3)/(x − y) • Leyland (xy + yx) • Thabit (3·2n − 1) • Williams ((b−1)·bn − 1) • Mills (⌊A3n⌋) By integer sequence • Fibonacci • Lucas • Pell • Newman–Shanks–Williams • Perrin • Partitions • Bell • Motzkin By property • Wieferich (pair) • Wall–Sun–Sun • Wolstenholme • Wilson • Lucky • Fortunate • Ramanujan • Pillai • Regular • Strong • Stern • Supersingular (elliptic curve) • Supersingular (moonshine theory) • Good • Super • Higgs • Highly cototient • Unique Base-dependent • Palindromic • Emirp • Repunit (10n − 1)/9 • Permutable • Circular • Truncatable • Minimal • Delicate • Primeval • Full reptend • Unique • Happy • Self • Smarandache–Wellin • Strobogrammatic • Dihedral • Tetradic Patterns • Twin (p, p + 2) • Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …) • Triplet (p, p + 2 or p + 4, p + 6) • Quadruplet (p, p + 2, p + 6, p + 8) • k-tuple • Cousin (p, p + 4) • Sexy (p, p + 6) • Chen • Sophie Germain/Safe (p, 2p + 1) • Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...) • Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...) • Balanced (consecutive p − n, p, p + n) By size • Mega (1,000,000+ digits) • Largest known • list Complex numbers • Eisenstein prime • Gaussian prime Composite numbers • Pseudoprime • Catalan • Elliptic • Euler • Euler–Jacobi • Fermat • Frobenius • Lucas • Somer–Lucas • Strong • Carmichael number • Almost prime • Semiprime • Sphenic number • Interprime • Pernicious Related topics • Probable prime • Industrial-grade prime • Illegal prime • Formula for primes • Prime gap First 60 primes • 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 • 23 • 29 • 31 • 37 • 41 • 43 • 47 • 53 • 59 • 61 • 67 • 71 • 73 • 79 • 83 • 89 • 97 • 101 • 103 • 107 • 109 • 113 • 127 • 131 • 137 • 139 • 149 • 151 • 157 • 163 • 167 • 173 • 179 • 181 • 191 • 193 • 197 • 199 • 211 • 223 • 227 • 229 • 233 • 239 • 241 • 251 • 257 • 263 • 269 • 271 • 277 • 281 List of prime numbers
Wikipedia
Timeline of women in mathematics This is a timeline of women in mathematics. Timeline Early Common Era Before 350: Pandrosion, a Greek Alexandrine mathematician known for an approximate solution to doubling the cube and a simplified exact solution to the construction of the geometric mean.[1] c. 350–370 until 415: The lifetime of Hypatia, a Greek Alexandrine Neoplatonist philosopher in Egypt who was the first well-documented woman in mathematics.[2] 18th Century 1748: Italian mathematician Maria Agnesi published the first book discussing both differential and integral calculus, called Instituzioni analitiche ad uso della gioventù italiana.[3][4] 1759: French mathematician Émilie du Châtelet's translation and commentary on Isaac Newton’s work Principia Mathematica was published posthumously; it is still considered the standard French translation.[5] c. 1787 – 1797: Self-taught Chinese astronomer Wang Zhenyi published at least twelve books and multiple articles on astronomy and mathematics.[6] 19th Century 1827: French mathematician Sophie Germain saw her theorem, known as Germain's Theorem, published in a footnote of a book by the mathematician Adrien-Marie Legendre.[7][8] In this theorem Germain proved that if x, y, and z are integers and if x5 + y5 = z5 then either x, y, or z must be divisible by 5. Germain's theorem was a major step toward proving Fermat's Last Theorem for the case where n equals 5.[7] 1829: The first public examination of an American girl in geometry was held.[9] 1858: Florence Nightengale became the first female member of the Royal Statistical Society.[10] 1873: Sarah Woodhead of Britain became the first woman to take, and to pass, the Cambridge Mathematical Tripos Exam.[11] 1874: Russian mathematician Sofia Kovalevskaya became the first woman in modern Europe to gain a doctorate in mathematics, which she earned from the University of Göttingen in Germany.[12] 1880: Charlotte Angas Scott of Britain obtained special permission to take the Cambridge Mathematical Tripos Exam, as women were not normally allowed to sit for the exam. She came eighth on the Tripos of all students taking them, but due to her sex, the title of "eighth wrangler," a high honour, went officially to a male student.[13] At the ceremony, however, after the seventh wrangler had been announced, all the students in the audience shouted her name. Because she could not attend the award ceremony, Scott celebrated her accomplishment at Girton College where there were cheers and clapping at dinner, and a special evening ceremony where the students sang "See the Conquering Hero Comes", and she received an ode written by a staff member, and was crowned with laurels.[13] 1885: Charlotte Angas Scott became the first British woman to receive a doctorate in mathematics, which she received from the University of London.[14] 1886: Winifred Edgerton Merrill became the first American woman to earn a PhD in mathematics, which she earned from Columbia University.[15] 1888: The Kovalevskaya top, one of a brief list of known examples of integrable rigid body motion, was discovered by Sofia Kovalevskaya.[16][17] 1889: Sofia Kovalevskaya was appointed as the first female professor in Northern Europe, at the University of Stockholm.[12][18] 1890: Philippa Fawcett of Britain[19] became the first woman to obtain the top score in the Cambridge Mathematical Tripos Exam. Her score was 13 per cent higher than the second highest score. When the women's list was announced, Fawcett was described as "above the senior wrangler", but she did not receive the title of senior wrangler, as at that time only men could receive degrees and therefore only men were eligible for the Senior Wrangler title.[20][21] 1891: Charlotte Angas Scott of Britain became the first woman to join the American Mathematical Society, then called the New York Mathematical Society.[22] 1894: Charlotte Angas Scott of Britain became the first woman on the first Council of the American Mathematical Society.[23] 1897: Four women attended the inaugural International Congress of Mathematicians in Zurich in 1897 - Charlotte Angas Scott, Iginia Massarini, Vera von Schiff, and Charlotte Wedell.[24] 20th Century 1911: Swedish mathematician Louise Petrén-Overton became the first woman in Sweden with a doctorate in mathematics.[25] 1913: American mathematician Mildred Sanderson earned her PhD for a thesis that included an important theorem about modular invariants.[26] 1918: German mathematician Emmy Noether published Noether's (first) theorem, which states that any differentiable symmetry of the action of a physical system has a corresponding conservation law.[27] 1927: American mathematician Anna Pell-Wheeler became the first woman to present a lecture at the American Mathematical Society Colloquium.[28][29] 1930: Cecilia Kreiger became the first woman to earn a PhD in mathematics in Canada, at the University of Toronto.[30] 1930s: British mathematician Mary Cartwright proved her theorem, now known as Cartwright's theorem, which gives an estimate for the maximum modulus of an analytic function that takes the same value no more than p times in the unit disc. To prove the theorem she used a new approach, applying a technique introduced by Lars Ahlfors for conformal mappings.[31] 1943: Euphemia Haynes became the first African-American woman to earn a Ph.D. in mathematics, which she earned from Catholic University of America.[32] 1944: Helen Walker became the first female president of the American Statistical Association.[33] 1949: American mathematician Gertrude Mary Cox became the first woman elected into the International Statistical Institute.[34] Also, Maria Laura Lopes obtained her PhD in Mathematics, being the first woman to obtain the title in Brazil. 1951: Mary Cartwright of Britain became the first female president of the Mathematical Association.[35][31] 1956: American mathematician Gladys West began collecting data from satellites at the Naval Surface Warfare Center Dahlgren Division. Her calculations directly impacted the development of accurate GPS systems.[36] 1960s 1960 and 1966: British mathematician Lucy Joan Slater published two books about the hypergeometric functions from the Cambridge University Press.[37][38] 1961: Mary Cartwright of Britain became the first woman to be President of the London Mathematical Society.[39] 1962: American mathematician Mina Rees became the first person to receive the Award for Distinguished Service to Mathematics from the Mathematical Association of America.[40] 1963: Grace Alele-Williams became the first Nigerian woman to earn a Ph.D when she defended her thesis in Mathematics Education at the University of Chicago (U.S.)[41][42] 1964: Mary Cartwright of Britain became the first woman to be given the Sylvester Medal of the Royal Society.[39][43] 1965: Scottish mathematician Elizabeth McHarg became the first female president of the Edinburgh Mathematical Society.[44][45] 1966: American mathematician Mary L. Boas published Mathematical Methods in the Physical Sciences, which was still widely used in college classrooms as of 1999.[46][47][48] 1968: Mary Cartwright of Britain became the first woman to be given the De Morgan Medal, the London Mathematical Society’s premier award.[49][43] 1970s 1970: American mathematician Mina Rees became the first female president of the American Association for the Advancement of Science.[50] 1971: American mathematician Mary Ellen Rudin constructed the first Dowker space.[51][52] 1971: The Association for Women in Mathematics (AWM) was founded. It is a professional society whose mission is to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity for and the equal treatment of women and girls in the mathematical sciences. It is incorporated in the state of Massachusetts.[53] 1971: The American Mathematical Society established its Joint Committee on Women in the Mathematical Sciences (JCW), which later became a joint committee of multiple scholarly societies.[54] 1973: American mathematician Jean Taylor published her dissertation on "Regularity of the Singular Set of Two-Dimensional Area-Minimizing Flat Chains Modulo 3 in R3" which solved a long-standing problem about length and smoothness of soap-film triple function curves.[55] 1974: American mathematician Joan Birman published the book Braids, Links, and Mapping Class Groups. It has become a standard introduction, with many of today's researchers having learned the subject through it.[56] 1975: American mathematician Julia Robinson became the first female mathematician elected to the National Academy of Sciences.[57][58] 1975: Stella Cunliffe became the first female president of the Royal Statistical Society.[10] 1976-1977: Marjorie Rice, an amateur American mathematician, discovered four new types of tessellating pentagons in 1976 and 1977.[59][60][61] 1979: American mathematician Dorothy Lewis Bernstein became the first female president of the Mathematical Association of America.[62] 1979: American mathematician Mary Ellen Rudin became the first woman to present the Mathematical Association of America’s Earle Raymond Hedrick Lectures, intended to showcase skilled expositors and enrich the understanding of instructors of college-level mathematics.[52][28] 1980s 1980: Joséphine Guidy Wandja, from the Ivory Coast, became the first African woman to earn a doctorate in mathematics.[63][64] 1981: Canadian-American mathematician Cathleen Morawetz became the first woman to give the Gibbs Lecture of the American Mathematical Society.[65] 1981: American mathematician Doris Schattschneider became the first female editor of Mathematics Magazine, a refereed bimonthly publication of the Mathematical Association of America.[66][67] 1982: Rebecca Walo Omana became the first female mathematics professor in the Democratic Republic of the Congo.[68][69] 1983: American mathematician Julia Robinson was elected the first female president of the American Mathematical Society for the term of 1983-1984 (but was unable to complete her term as she was suffering from leukemia),[58][70] and became the first female mathematician to be awarded a MacArthur Fellowship.[28] 1986: European Women in Mathematics (EWM) was founded as an organization in 1986 by Bodil Branner, Caroline Series, Gudrun Kalmbach, Marie-Françoise Roy, and Dona Strauss, inspired by the activities of the Association for Women in Mathematics in the USA.[71] It is the "first and best known" of several organizations devoted to women in mathematics in Europe.[72] 1987: Eileen Poiani became the first female president of Pi Mu Epsilon.[73] 1988: American mathematician Doris Schattschneider became the first woman to present the Mathematical Association of America’s J. Sutherland Frame Lectures.[28][74] 1990s 1992: Australian mathematician Cheryl Praeger became the first female President of the Australian Mathematical Society.[75] 1992: American mathematician Gloria Gilmer became the first woman to deliver a major National Association of Mathematicians lecture (it was the Cox-Talbot address).[76] 1995: American mathematician Margaret Wright became the first female president of the Society for Industrial and Applied Mathematics.[28][77] 1995: Israeli-Canadian mathematician Leah Edelstein-Keshet became the first female president of the Society for Mathematical Biology.[78] 1995: Ina Kersten became the president of the German Mathematical Society, which meant she was the first woman to head the society.[79][80] 1996: American mathematician Joan Birman became the first woman to receive the Mathematical Association of America’s Chauvenet Prize.[81][28] 1996: Katherine Heinrich became the first female President of the Canadian Mathematical Society.[82] 1996: Ioana Dumitriu, a New York University sophomore from Romania, became the first woman to be named a Putnam Fellow.[83] Putnam Fellows are the top five (or six, in case of a tie) scorers on The William Lowell Putnam Mathematical Competition.[84][85] 1998: Melanie Wood became the first female American to make the U.S. International Math Olympiad Team. She won silver medals in the 1998 and 1999 International Mathematical Olympiads.[86] 2000s 2002: Susan Howson became the first woman to be given the Adams Prize, given annually by the University of Cambridge to a British mathematician under the age of 40.[87] 2002: Melanie Wood became the first American woman and second woman overall to be named a Putnam Fellow in 2002. Putnam Fellows are the top five (or six, in case of a tie) scorers on William Lowell Putnam Mathematical Competition.[84][85] 2004: American Melanie Wood became the first woman to win the Frank and Brennie Morgan Prize for Outstanding Research in Mathematics by an Undergraduate Student. It is an annual award given to an undergraduate student in the US, Canada, or Mexico who demonstrates superior mathematics research.[88][86] 2004: American Alison Miller became the first female gold medal winner on the U.S. International Mathematical Olympiad Team.[89][90] 2006: Polish-Canadian mathematician Nicole Tomczak-Jaegermann became the first woman to win the CRM-Fields-PIMS prize.[91][92][93] 2006: Stefanie Petermichl, a German mathematical analyst then at the University of Texas at Austin, became the first woman to win the Salem Prize, an annual award given to young mathematicians who have worked in Raphael Salem's field of interest, chiefly topics in analysis related to Fourier series.[94][28] She shared the prize with Artur Avila.[95][28] 2006: When Olga Gil Medrano became president of the Royal Spanish Mathematical Society in 2006, she was the first woman elected to that position.[96] 2010s 2011: Belgian mathematician Ingrid Daubechies became the first female president of the International Mathematical Union.[97] 2012: Latvian mathematician Daina Taimina became the first woman to win the Euler Book Prize, for her 2009 book Crocheting Adventures with Hyperbolic Planes.[98][99] 2012: The Working Committee for Women in Mathematics, Chinese Mathematical Society (WCWM-CMS) was founded; it is a national non-profit academic organization in which female mathematicians who are engaged in research, teaching, and applications of mathematics can share their scientific research through academic exchanges both in China and abroad.[100] It is one of the branches of the Chinese Mathematical Society (CMS).[100] 2013: The African Women in Mathematics Association was founded. This professional organization with over 300 members promotes mathematics to African women and girls and supports female mathematicians.[101][102] 2014: Maryam Mirzakhani became the first woman as well as the first Iranian to be awarded the Fields Medal, which she was awarded for "her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces."[103][104][105] That year the Fields Medal was also awarded to Martin Hairer, Manjul Bhargava, and Artur Avila.[106] It is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, and is often viewed as the greatest honor a mathematician can receive.[107][108] 2016: French mathematician Claire Voisin received the CNRS Gold medal, the highest scientific research award in France.[109] 2016: The London Mathematical Society's Women in Mathematics Committee was awarded the Royal Society's inaugural Athena Prize.[110] 2017: Nouzha El Yacoubi became the first female president of the African Mathematical Union.[111] 2019: American mathematician Karen Uhlenbeck became the first woman to win the Abel Prize, with the award committee citing "the fundamental impact of her work on analysis, geometry and mathematical physics."[112] 2019: Marissa Kawehi Loving became the first Native Hawaiian woman to earn a PhD in mathematics when she graduated from the University of Illinois Urbana-Champaign in 2019. In addition to being Native Hawaiian, she is also black, Japanese, and Puerto Rican.[113] 2020s 2022: Maryna Viazovska was awarded the Fields Medal in July 2022, making her the second woman (after Maryam Mirzakhani) and the first Ukrainian to be awarded it.[114][115] That year the Fields Medal was also awarded to Hugo Duminil-Copin, June Huh, and James Maynard.[115] The Fields Medal is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union, and is often viewed as the greatest honor a mathematician can receive.[107][108] 2023: Ingrid Daubechies was awarded the Wolf Prize in Mathematics in February 2023, becoming the first woman to receive this award.[116] See also • List of women in mathematics • Timeline of women in mathematics in the United States • Timeline of mathematics • Timeline of mathematical innovation in South and West Asia References 1. 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Wikipedia
Female education in STEM Female education in STEM refers to child and adult female representation in the educational fields of science, technology, engineering, and mathematics (STEM). In 2017, 33% of students in STEM fields were women. The organization UNESCO has stated that this gender disparity is due to discrimination, biases, social norms and expectations that influence the quality of education women receive and the subjects they study.[1] UNESCO also believes that having more women in STEM fields is desirable because it would help bring about sustainable development.[1] Current status of girls and women in STEM education Overall trends in STEM education Gender differences in STEM education participation are already visible in early childhood care and education in science- and math-related play, and become more pronounced at higher levels of education. Girls appear to lose interest in STEM subjects with age, particularly between early and late adolescence.[1] This decreased interest affects participation in advanced studies at the secondary level and in higher education.[1] Female students represent 35% of all students enrolled in STEM-related fields of study at this level globally. Differences are also observed by disciplines, with female enrollment lowest in engineering, manufacturing and construction, natural science, mathematics and statistics and ICT fields. Significant regional and country differences in female representation in STEM studies can be observed, though, suggesting the presence of contextual factors affecting girls’ and women's engagement in these fields. Women leave STEM disciplines in disproportionate numbers during their higher education studies, in their transition to the world of work and even in their career cycle.[1][3][4][5][6][7] Learning achievement in STEM education Data on gender differences in learning achievement present a complex picture, depending on what is measured (subject, knowledge acquisition against knowledge application), the level of education/age of students, and geographic location. Overall, women's participation has been increasing, but significant regional variations exist. For example, where data are available in Africa, Latin America and the Caribbean, the gender gap is largely in favor of boys in mathematics achievement in secondary education. In contrast, in the Arab States, girls perform better than boys in both subjects in primary and secondary education. As with the data on participation, national and regional variations in data on learning achievement suggest the presence of contextual factors affecting girls’ and women's engagement in these fields. Girls’ achievement seems to be stronger in science than mathematics and where girls do better than boys, the score differential is up to three times higher than where boys do better.[8] Girls tend to outperform boys in certain sub-topics such as biology and chemistry but do less well in physics and earth science. The gender gap has fallen significantly in science in secondary education among TIMSS trend countries: 14 out of 17 participating countries had no gender gap in science in 2015, compared to only one in 1995. However, the data are less well known outside of these 17 countries. The gender gap in boys' favor is slightly bigger in mathematics but improvements over time in girls’ favor are also observed in certain countries, despite the important regional variations. Gender differences are observed within mathematical sub-topics with girls outperforming boys in topics such as algebra and geometry but doing less well in "number". Girls’ performance is stronger in assessments that measure knowledge acquisition than those measuring knowledge application. Country coverage in terms of data availability is quite limited while data are collected at a different frequency and against different variables in the existing studies. There are large gaps in our knowledge of the situation in low- and middle-income countries in sub-Saharan Africa, Central Asia, and South and West Asia, particularly at the secondary level.[1][4][5][9][10][11][12][13] Factors influencing girls' and women's participation and achievement in STEM education According to UNESCO, there are multiple and overlapping factors which influence girls' and women's participation, achievement and progression in STEM studies and careers, all of which interact in complex ways, including: • Individual level: biological factors that may influence individuals’ abilities, skills, and behaviour such as brain structure and function, hormones, genetics, and cognitive traits like spatial and linguistic skills. It also considers psychological factors, including self-efficacy, interest and motivation. • Family and peer level: parental beliefs and expectations, parental education and socioeconomic status, and other household factors, as well as peer influences. • School level: factors within the learning environment, including teachers’ profile, experience, beliefs and expectations, curricula, learning materials and resources, teaching strategies and student teacher interactions, assessment practices, and the overall school environment. • Societal level: social and cultural norms related to gender equality, and gender stereotypes in the media.[1] Individual level Individual level The question of whether there are differences in cognitive ability between men and women has long been a topic of debate among researchers and scholars. Some studies have found no differences in the neural mechanism of learning based on sex. Loss of interest has been the major reason cited for girls opting out of STEM. However, some have stated that this choice is influenced heavily by the socialisation process and stereotyped ideas about gender roles, including stereotypes about gender and STEM. Gender stereotypes that communicate the idea that STEM studies and careers are male domains can negatively affect girls' interest, engagement, and achievement in STEM and discourage them from pursuing STEM careers. Girls who assimilate such stereotypes have lower levels of self-efficacy and confidence in their ability than boys.[15] Self-efficacy affects both STEM education outcomes and aspirations for STEM careers to a considerable extent. In recent years, more women have been majoring in STEM, although we still continue to witness vast imbalances between men and women studying math, engineering, or science.[16] Family and peer level Parents, including their beliefs and expectations, play an important role in shaping girls' attitudes towards, and interest in, STEM studies. Parents with traditional beliefs about gender roles and who treat girls and boys unequally can reinforce stereotypes about gender and ability in STEM. Parents can also have a strong influence on girls' STEM participation and learning achievement through the family values, environment, experiences, and encouragement that they provide. Some research finds that parents’ expectations, particularly the mother's expectations, have more influence on the higher education and career choices of girls than those of boys.[1] Higher socio-economic status and parental educational qualifications are associated with higher scores in mathematics and science for both girls and boys. Girls' science performance appears to be more strongly associated with mothers' higher educational qualifications, and boys' with their fathers'. Family members with STEM careers can also influence girls’ STEM engagement. The broader socio-cultural context of the family can also play a role. Factors such as ethnicity, language used at home, immigrant status, and family structure may also have an influence on girls' participation and performance in STEM. Peers can also impact on girls’ motivation and feeling of belonging in STEM education. Influence of female peers is a significant predictor of girls' interest and confidence in mathematics and science.[9] School level Qualified teachers with specialisation in STEM can positively influence girls' performance and engagement with STEM education and their interest in pursuing STEM careers. Female STEM teachers often have stronger benefits for girls, possibly by acting as role models and by helping to dispel stereotypes about sex-based STEM ability. Teachers' beliefs, attitudes, behaviours, and interactions with students, as well as curricula and learning materials, can all play a role as well. Opportunities for real-life experiences with STEM, including hands-on practice, apprenticeships, career counselling, and mentoring can expand girls' understanding of STEM studies and professions and maintain interest. Assessment processes and tools that are gender-biased or include gender stereotypes may negatively affect girls' performance in STEM. Girls' learning outcomes in STEM can also be compromised by psychological factors such as mathematics or test anxiety.[1][4][6][9] The confidence of a female teacher in STEM subjects also has a strong impact on how well female students will perform in those subjects in the elementary school classroom. For example, female elementary teachers with anxiety around math will negatively affect the achievement of their female students in math.[17] Correlations have been found between gender bias in female elementary students and their achievement in mathematics. Those who had lower achievement over time have also been found to believe that boys are inherently better at mathematics than girls.[17] Societal level Cultural and social norms influence girls’ perceptions about their abilities, roles in society and career and life aspirations. The degree of gender equality in wider society influences girls' participation and performance in STEM. In countries with greater gender equality, girls tend to have more positive attitudes and confidence about mathematics, and the gender gap in achievement in the subject is smaller. Additionally, in some countries there were more women receiving computer science degrees than men.[18] That was primarily because a computer science degree was seen as indoor work. When the job title was adjusted to sound less masculine and more geared towards relationship building, females appeared to be more likely to enter the STEM field. Gender stereotypes portrayed in the media are internalised by children and adults and affect the way they view themselves and others. Media can perpetuate or challenge gender stereotypes about STEM abilities and careers.[19] Effects of gender disparities The prolonged consequence of consistent gendered stereotypes relating to women's inability to become successful in the field of STEM is the development of a fixed mindset that they are not sufficiently equipped to think critically or contribute valuable ideas in careers in fields that currently employ predominantly male workers. Stepping into a workplace where men outnumber women, knowing that male co-workers expect lower capabilities from a woman, significantly undermines women's skills and performance in their jobs. This in part is due to the heuristic representativeness – when people do not look the part, others are more critical of them. In a heavily male populated environment, men are more critical of women because they do not appear how the abstract representation in STEM fields typically appear. A study demonstrating the effects of construal level priming conditions between men and women, concluded that high construal levels facilitate the use of representativeness heuristic. In contrast, low construal conditions portrayed a decrease in the use of representativeness heuristic.[15] Possible solutions to reduce gender gap • Inclusive STEM approaches such as Problem-Based Learning (PBL) and personalization of learning could generate solutions to lower gender disparities in STEM.[20] • Students' intellectual engagement and success can develop and improve as a result of the instructor's gender. Gender disparities decrease when a course is taught by a female instructor.[21] • Increasing awareness about gender biases in STEM careers can also reduce the gender gap.[22] Sources  This article incorporates text from a free content work. (license statement/permission). Text taken from Cracking the code: girls' and women's education in science, technology, engineering and mathematics (STEM)​, 23, 37, 46, 49, 56, 58, UNESCO, UNESCO. To learn how to add open license text to Wikipedia articles, please see this how-to page. For information on reusing text from Wikipedia, please see the terms of use. References 1. Cracking the code: Girls' and women's education in science, technology, engineering and mathematics (STEM) (PDF). UNESCO. 2017. ISBN 978-92-3-100233-5. 2. Mullis, I. V. S., Martin, M. O., Foy, P. and Hooper, M. (2016). "TIMSS Advanced 2015 International Results in Advanced Mathematics and Physics". TIMSS & PIRLS International Study Center website. Archived from the original on 2017-02-15. Retrieved 2 June 2017.{{cite web}}: CS1 maint: multiple names: authors list (link) 3. "STEM and Gender Advancement (SAGA) | United Nations Educational, Scientific and Cultural Organization". www.unesco.org. Retrieved 2017-10-12. 4. PISA 2015 Results (Volume I): Excellence and Equity in Education. Paris: OECD. 2016. 5. "TIMSS ADVANCED 2015 INTERNATIONAL RESULTS REPORT – TIMSS 2015 INTERNATIONAL RESULTS REPORT". timssandpirls.bc.edu. Retrieved 2017-10-12. 6. UIS. "UIS Statistics". data.uis.unesco.org. Retrieved 2017-10-12. 7. Science and Engineering Indicators 2014. Arlington: National Science Board. 2014. 8. UNESCO (2017). Cracking the code girls' and women's education in science, technology, engineering and mathematics (STEM). Paris: Unesco. ISBN 978-92-3-100233-5. OCLC 1113762987. 9. Mullis, I. V. S., Martin, M. O. and Loveless, T. (2016). International Trends in Mathematics and Science Achievement, Curriculum and Instruction. Boston: 20 Years of TIMSS.{{cite book}}: CS1 maint: multiple names: authors list (link) 10. Gender Inequality in Learning Achievement in Primary Education. What can TERCE Tell us?. Santiago: UNESCO. 2016. 11. PASEC 2014: Education System Performance in Francophone Sub-Saharan Africa. Dakar: PASEC. 2015. 12. Salto, M. (2011). Trends in the Magnitude and Direction of Gender Differences in Learning Outcomes. SACMEQ. 13. Fraillon, J., Ainley, J., Schulz, W., Friedman, T. and Gebhardt, E. (2014). Preparing for Life in a Digital Age. The IEA International Computer and Information Literacy Study (ICILS) Report. Melbourne: ICILS and Springer Open.{{cite book}}: CS1 maint: multiple names: authors list (link) 14. Catherine André/VoxEurop/EDJNet; Marzia Bona/OBC Transeuropa/EDJNet (19 April 2018). "The ICT sector is booming. But are women missing out?". Retrieved 27 August 2018. 15. "Supplemental Material for The Effects of Construal Level on Heuristic Reasoning: The Case of Representativeness and Availability". Decision. 2014-12-22. doi:10.1037/dec0000021.supp. ISSN 2325-9965. 16. Sonnert, Gerhard; Fox, Mary Frank; Adkins, Kristen (December 2007). "Undergraduate Women in Science and Engineering: Effects of Faculty, Fields, and Institutions Over Time". Social Science Quarterly. 88 (5): 1333–1356. doi:10.1111/j.1540-6237.2007.00505.x. ISSN 0038-4941. 17. Beilock, Sian L.; Gunderson, Elizabeth A.; Ramirez, Gerardo; Levine, Susan C.; Smith, Edward E. (February 5, 2010). "Female Teachers' Math Anxiety Affects Girls' Math Achievement". Proceedings of the National Academy of Sciences. 107 (5): 1860–1863. Bibcode:2010PNAS..107.1860B. doi:10.1073/pnas.0910967107. JSTOR 40536499. PMC 2836676. PMID 20133834. 18. El-Hout, Mona; Garr-Schultz, Alexandra; Cheryan, Sapna (January 2021). "Beyond biology: The importance of cultural factors in explaining gender disparities in STEM preferences". European Journal of Personality. 35 (1): 45–50. doi:10.1177/0890207020980934. ISSN 0890-2070. S2CID 231606736. 19. Beasley, Maya (Summer 2012). "Why they leave: the impact of stereotype threat on the attrition of women and minorities from science, math and engineering majors". Social Psychology of Education. 15 (4): 427–448. doi:10.1007/s11218-012-9185-3. S2CID 2470487. 20. Zuo, Huifang; LaForce, Melanie; Ferris, Kaitlyn; Noble, Elizabeth (2019-07-18). "Revisiting Race and Gender Differences in STEM: Can Inclusive STEM High Schools Reduce Gaps?". European Journal of STEM Education. 4 (1). doi:10.20897/ejsteme/5840. ISSN 2468-4368. S2CID 199136661. 21. Solanki, Sabrina M.; Xu, Di (August 2018). "Looking Beyond Academic Performance: The Influence of Instructor Gender on Student Motivation in STEM Fields". American Educational Research Journal. 55 (4): 801–835. doi:10.3102/0002831218759034. ISSN 0002-8312. S2CID 149379494. 22. "Introduction: Women in science: Why so few?", Athena Unbound, Cambridge University Press, pp. 1–4, 2000-10-19, doi:10.1017/cbo9780511541414.001, ISBN 9780521563802, retrieved 2021-11-28
Wikipedia
Woo circles In geometry, the Woo circles, introduced by Peter Y. Woo, are a set of infinitely many Archimedean circles. Construction Form an arbelos with the two inner semicircles tangent at point C. Let m denote any nonnegative real number. Draw two circles, with radii m times the radii of the smaller two arbelos semicircles, centered on the arbelos ground line, also tangent to each other at point C and with radius m times the radius of the corresponding small arbelos arc. Any circle centered on the Schoch line and externally tangent to the circles is a Woo circle.[1] See also • Schoch circles References 1. Thomas Schoch (2007). "Arbelos - The Woo Circles". Retrieved 2008-06-05.
Wikipedia
Woodall's conjecture In the mathematics of directed graphs, Woodall's conjecture is an unproven relationship between dicuts and dijoins. It was posed by Douglas Woodall in 1976.[1] Unsolved problem in mathematics: Does the minimum number of edges in a dicut of a directed graph always equal the maximum number of disjoint dijoins? (more unsolved problems in mathematics) Statement A dicut is a partition of the vertices into two subsets such that all edges that cross the partition do so in the same direction. A dijoin is a subset of edges that, when contracted, produces a strongly connected graph; equivalently, it is a subset of edges that includes at least one edge from each dicut.[2] If the minimum number of edges in a dicut is $k$, then there can be at most $k$ disjoint dijoins in the graph, because each one must include a different edge from the smallest dicut. Woodall's conjecture states that, in this case, it is always possible to find $k$ disjoint dijoins. That is, any directed graph the minimum number of edges in a dicut equals the maximum number of disjoint dijoins that can be found in the graph (a packing of dijoins).[2][1] Partial results It is a folklore result that the theorem is true for directed graphs whose minimum dicut has two edges.[2] Any instance of the problem can be reduced to a directed acyclic graph by taking the condensation of the instance, a graph formed by contracting each strongly connected component to a single vertex. Another class of graphs for which the theorem has been proven true are the directed acyclic graphs in which every source vertex (a vertex without incoming edges) has a path to every sink vertex (a vertex without outgoing edges).[3][4] Related results A fractional weighted version of the conjecture, posed by Jack Edmonds and Rick Giles, was refuted by Alexander Schrijver.[5][6][2] In the other direction, the Lucchesi–Younger theorem states that the minimum size of a dijoin equals the maximum number of disjoint dicuts that can be found in a given graph.[7][8] References 1. Woodall, D. R. (1978), "Menger and König systems", in Alavi, Yousef; Lick, Don R. (eds.), Theory and applications of graphs (Proc. Internat. Conf., Western Mich. Univ., Kalamazoo, Mich., 1976), Lecture Notes in Mathematics, vol. 642, Berlin: Springer, pp. 620–635, doi:10.1007/BFb0070416, MR 0499529 2. Abdi, Ahmad; Cornuéjols, Gérard; Zlatin, Michael (2022), On packing dijoins in digraphs and weighted digraphs, arXiv:2202.00392 3. Schrijver, A. (1982), "Min-max relations for directed graphs", Bonn Workshop on Combinatorial Optimization (Bonn, 1980), Annals of Discrete Mathematics, vol. 16, North-Holland, pp. 261–280, MR 0686312 4. Feofiloff, P.; Younger, D. H. (1987), "Directed cut transversal packing for source-sink connected graphs", Combinatorica, 7 (3): 255–263, doi:10.1007/BF02579302, MR 0918396 5. Edmonds, Jack; Giles, Rick (1977), "A min-max relation for submodular functions on graphs", Studies in integer programming (Proc. Workshop, Bonn, 1975), Annals of Discrete Mathematics, vol. 1, North-Holland, Amsterdam, pp. 185–204, MR 0460169 6. Schrijver, A. (1980), Bachem, Achim; Grötschel, Martin; Korte, Bernhard (eds.), "A counterexample to a conjecture of Edmonds and Giles", Discrete Mathematics, 32 (2): 213–215, doi:10.1016/0012-365X(80)90057-6, MR 0592858 7. Lovász, László (1976), "On two minimax theorems in graph", Journal of Combinatorial Theory, Series B, 21 (2): 96–103, doi:10.1016/0095-8956(76)90049-6, MR 0427138 8. Lucchesi, C. L.; Younger, D. H. (1978), "A minimax theorem for directed graphs", Journal of the London Mathematical Society, Second Series, 17 (3): 369–374, doi:10.1112/jlms/s2-17.3.369, MR 0500618 External links • Feofiloff, Paulo (November 30, 2005), Woodall’s conjecture on Packing Dijoins: a survey (PDF) • "Woodall's conjecture", Open Problem Garden, April 5, 2007
Wikipedia
Woodall number In number theory, a Woodall number (Wn) is any natural number of the form $W_{n}=n\cdot 2^{n}-1$ for some natural number n. The first few Woodall numbers are: 1, 7, 23, 63, 159, 383, 895, … (sequence A003261 in the OEIS). History Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917,[1] inspired by James Cullen's earlier study of the similarly defined Cullen numbers. Woodall primes Unsolved problem in mathematics: Are there infinitely many Woodall primes? (more unsolved problems in mathematics) Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents n for which the corresponding Woodall numbers Wn are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ... (sequence A002234 in the OEIS); the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... (sequence A050918 in the OEIS). In 1976 Christopher Hooley showed that almost all Cullen numbers are composite.[2] In October 1995, Wilfred Keller published a paper discussing several new Cullen primes and the efforts made to factorise other Cullen and Woodall numbers. Included in that paper is a personal communication to Keller from Hiromi Suyama, asserting that Hooley's method can be reformulated to show that it works for any sequence of numbers n · 2n + a + b, where a and b are integers, and in particular, that almost all Woodall numbers are composite.[3] It is an open problem whether there are infinitely many Woodall primes. As of October 2018, the largest known Woodall prime is 17016602 × 217016602 − 1.[4] It has 5,122,515 digits and was found by Diego Bertolotti in March 2018 in the distributed computing project PrimeGrid.[5] Restrictions Starting with W4 = 63 and W5 = 159, every sixth Woodall number is divisible by 3; thus, in order for Wn to be prime, the index n cannot be congruent to 4 or 5 (modulo 6). Also, for a positive integer m, the Woodall number W2m may be prime only if 2m + m is prime. As of January 2019, the only known primes that are both Woodall primes and Mersenne primes are W2 = M3 = 7, and W512 = M521. Divisibility properties Like Cullen numbers, Woodall numbers have many divisibility properties. For example, if p is a prime number, then p divides W(p + 1) / 2 if the Jacobi symbol $\left({\frac {2}{p}}\right)$ is +1 and W(3p − 1) / 2 if the Jacobi symbol $\left({\frac {2}{p}}\right)$ is −1. Generalization A generalized Woodall number base b is defined to be a number of the form n × bn − 1, where n + 2 > b; if a prime can be written in this form, it is then called a generalized Woodall prime. The smallest value of n such that n × bn − 1 is prime for b = 1, 2, 3, ... are[6] 3, 2, 1, 1, 8, 1, 2, 1, 10, 2, 2, 1, 2, 1, 2, 167, 2, 1, 12, 1, 2, 2, 29028, 1, 2, 3, 10, 2, 26850, 1, 8, 1, 42, 2, 6, 2, 24, 1, 2, 3, 2, 1, 2, 1, 2, 2, 140, 1, 2, 2, 22, 2, 8, 1, 2064, 2, 468, 6, 2, 1, 362, 1, 2, 2, 6, 3, 26, 1, 2, 3, 20, 1, 2, 1, 28, 2, 38, 5, 3024, 1, 2, 81, 858, 1, 2, 3, 2, 8, 60, 1, 2, 2, 10, 5, 2, 7, 182, 1, 17782, 3, ... (sequence A240235 in the OEIS) As of November 2021, the largest known generalized Woodall prime with base greater than 2 is 2740879 × 322740879 − 1.[7] b Numbers n such that n × bn − 1 is prime[6] OEIS sequence 3 1, 2, 6, 10, 18, 40, 46, 86, 118, 170, 1172, 1698, 1810, 2268, 4338, 18362, 72662, 88392, 94110, 161538, 168660, 292340, 401208, 560750, 1035092, ... A006553 4 1, 2, 3, 5, 8, 14, 23, 63, 107, 132, 428, 530, 1137, 1973, 2000, 7064, 20747, 79574, 113570, 293912, ..., 1993191, ... A086661 5 8, 14, 42, 384, 564, 4256, 6368, 21132, 27180, 96584, 349656, 545082, ... A059676 6 1, 2, 3, 19, 20, 24, 34, 77, 107, 114, 122, 165, 530, 1999, 4359, 11842, 12059, 13802, 22855, 41679, 58185, 145359, 249987, ... A059675 7 2, 18, 68, 84, 3812, 14838, 51582, ... A242200 8 1, 2, 7, 12, 25, 44, 219, 252, 507, 1155, 2259, 2972, 4584, 12422, 13905, 75606, ... A242201 9 10, 58, 264, 1568, 4198, 24500, ... A242202 10 2, 3, 8, 11, 15, 39, 60, 72, 77, 117, 183, 252, 396, 1745, 2843, 4665, 5364, 524427, ... A059671 11 2, 8, 252, 1184, 1308, 1182072, ... A299374 12 1, 6, 43, 175, 821, 910, 1157, 13748, 27032, 71761, 229918, 549721, 866981, 1405486, ... A299375 13 2, 6, 563528, ... A299376 14 1, 3, 7, 98, 104, 128, 180, 834, 1633, 8000, 28538, 46605, 131941, 147684, 433734, 1167708, ... A299377 15 2, 10, 14, 2312, 16718, 26906, 27512, 41260, 45432, 162454, 217606, 1527090, ... A299378 16 167, 189, 639, ... A299379 17 2, 18, 20, 38, 68, 3122, 3488, 39500, ... A299380 18 1, 2, 6, 8, 10, 28, 30, 39, 45, 112, 348, 380, 458, 585, 17559, 38751, 43346, 46984, 92711, ... A299381 19 12, 410, 33890, 91850, 146478, 189620, 280524, ... A299382 20 1, 18, 44, 60, 80, 123, 429, 1166, 2065, 8774, 35340, 42968, 50312, 210129, 663703, ... A299383 See also • Mersenne prime - Prime numbers of the form 2n − 1. References 1. Cunningham, A. J. C; Woodall, H. J. (1917), "Factorisation of $Q=(2^{q}\mp q)$ and $(q\cdot {2^{q}}\mp 1)$", Messenger of Mathematics, 47: 1–38. 2. Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. Vol. 104. Providence, RI: American Mathematical Society. p. 94. ISBN 0-8218-3387-1. Zbl 1033.11006. 3. Keller, Wilfrid (January 1995). "New Cullen primes". Mathematics of Computation. 64 (212): 1739. doi:10.1090/S0025-5718-1995-1308456-3. ISSN 0025-5718. Keller, Wilfrid (December 2013). "Wilfrid Keller". www.fermatsearch.org. Hamburg. Archived from the original on February 28, 2020. Retrieved October 1, 2020. 4. "The Prime Database: 8508301*2^17016603-1", Chris Caldwell's The Largest Known Primes Database, retrieved March 24, 2018 5. PrimeGrid, Announcement of 17016602*2^17016602 - 1 (PDF), retrieved April 1, 2018 6. List of generalized Woodall primes base 3 to 10000 7. "The Top Twenty: Generalized Woodall". primes.utm.edu. Retrieved 20 November 2021. Further reading • Guy, Richard K. (2004), Unsolved Problems in Number Theory (3rd ed.), New York: Springer Verlag, pp. section B20, ISBN 0-387-20860-7. • Keller, Wilfrid (1995), "New Cullen Primes" (PDF), Mathematics of Computation, 64 (212): 1733–1741, doi:10.2307/2153382, JSTOR 2153382. • Caldwell, Chris, "The Top Twenty: Woodall Primes", The Prime Pages, retrieved December 29, 2007. External links • Chris Caldwell, The Prime Glossary: Woodall number, and The Top Twenty: Woodall, and The Top Twenty: Generalized Woodall, at The Prime Pages. • Weisstein, Eric W. "Woodall number". MathWorld. • Steven Harvey, List of Generalized Woodall primes. • Paul Leyland, Generalized Cullen and Woodall Numbers Prime number classes By formula • Fermat (22n + 1) • Mersenne (2p − 1) • Double Mersenne (22p−1 − 1) • Wagstaff (2p + 1)/3 • Proth (k·2n + 1) • Factorial (n! ± 1) • Primorial (pn# ± 1) • Euclid (pn# + 1) • Pythagorean (4n + 1) • Pierpont (2m·3n + 1) • Quartan (x4 + y4) • Solinas (2m ± 2n ± 1) • Cullen (n·2n + 1) • Woodall (n·2n − 1) • Cuban (x3 − y3)/(x − y) • Leyland (xy + yx) • Thabit (3·2n − 1) • Williams ((b−1)·bn − 1) • Mills (⌊A3n⌋) By integer sequence • Fibonacci • Lucas • Pell • Newman–Shanks–Williams • Perrin • Partitions • Bell • Motzkin By property • Wieferich (pair) • Wall–Sun–Sun • Wolstenholme • Wilson • Lucky • Fortunate • Ramanujan • Pillai • Regular • Strong • Stern • Supersingular (elliptic curve) • Supersingular (moonshine theory) • Good • Super • Higgs • Highly cototient • Unique Base-dependent • Palindromic • Emirp • Repunit (10n − 1)/9 • Permutable • Circular • Truncatable • Minimal • Delicate • Primeval • Full reptend • Unique • Happy • Self • Smarandache–Wellin • Strobogrammatic • Dihedral • Tetradic Patterns • Twin (p, p + 2) • Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …) • Triplet (p, p + 2 or p + 4, p + 6) • Quadruplet (p, p + 2, p + 6, p + 8) • k-tuple • Cousin (p, p + 4) • Sexy (p, p + 6) • Chen • Sophie Germain/Safe (p, 2p + 1) • Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...) • Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...) • Balanced (consecutive p − n, p, p + n) By size • Mega (1,000,000+ digits) • Largest known • list Complex numbers • Eisenstein prime • Gaussian prime Composite numbers • Pseudoprime • Catalan • Elliptic • Euler • Euler–Jacobi • Fermat • Frobenius • Lucas • Somer–Lucas • Strong • Carmichael number • Almost prime • Semiprime • Sphenic number • Interprime • Pernicious Related topics • Probable prime • Industrial-grade prime • Illegal prime • Formula for primes • Prime gap First 60 primes • 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 • 23 • 29 • 31 • 37 • 41 • 43 • 47 • 53 • 59 • 61 • 67 • 71 • 73 • 79 • 83 • 89 • 97 • 101 • 103 • 107 • 109 • 113 • 127 • 131 • 137 • 139 • 149 • 151 • 157 • 163 • 167 • 173 • 179 • 181 • 191 • 193 • 197 • 199 • 211 • 223 • 227 • 229 • 233 • 239 • 241 • 251 • 257 • 263 • 269 • 271 • 277 • 281 List of prime numbers Classes of natural numbers Powers and related numbers • Achilles • Power of 2 • Power of 3 • Power of 10 • Square • Cube • Fourth power • Fifth power • Sixth power • Seventh power • Eighth power • Perfect power • Powerful • Prime power Of the form a × 2b ± 1 • Cullen • Double Mersenne • Fermat • Mersenne • Proth • Thabit • Woodall Other polynomial numbers • Hilbert • Idoneal • Leyland • Loeschian • Lucky numbers of Euler Recursively defined numbers • Fibonacci • Jacobsthal • Leonardo • Lucas • Padovan • Pell • Perrin Possessing a specific set of other numbers • Amenable • Congruent • Knödel • Riesel • Sierpiński Expressible via specific sums • Nonhypotenuse • Polite • Practical • Primary pseudoperfect • Ulam • Wolstenholme Figurate numbers 2-dimensional centered • Centered triangular • Centered square • Centered pentagonal • Centered hexagonal • Centered heptagonal • Centered octagonal • Centered nonagonal • Centered decagonal • Star non-centered • Triangular • Square • Square triangular • Pentagonal • Hexagonal • Heptagonal • Octagonal • Nonagonal • Decagonal • Dodecagonal 3-dimensional centered • Centered tetrahedral • Centered cube • Centered octahedral • Centered dodecahedral • Centered icosahedral non-centered • Tetrahedral • Cubic • Octahedral • Dodecahedral • Icosahedral • Stella octangula pyramidal • Square pyramidal 4-dimensional non-centered • Pentatope • Squared triangular • Tesseractic Combinatorial numbers • Bell • Cake • Catalan • Dedekind • Delannoy • Euler • Eulerian • Fuss–Catalan • Lah • Lazy caterer's sequence • Lobb • Motzkin • Narayana • Ordered Bell • Schröder • Schröder–Hipparchus • Stirling first • Stirling second • Telephone number • Wedderburn–Etherington Primes • Wieferich • Wall–Sun–Sun • Wolstenholme prime • Wilson Pseudoprimes • Carmichael number • Catalan pseudoprime • Elliptic pseudoprime • Euler pseudoprime • Euler–Jacobi pseudoprime • Fermat pseudoprime • Frobenius pseudoprime • Lucas pseudoprime • Lucas–Carmichael number • Somer–Lucas pseudoprime • Strong pseudoprime Arithmetic functions and dynamics Divisor functions • Abundant • Almost perfect • Arithmetic • Betrothed • Colossally abundant • Deficient • Descartes • Hemiperfect • Highly abundant • Highly composite • Hyperperfect • Multiply perfect • Perfect • Practical • Primitive abundant • Quasiperfect • Refactorable • Semiperfect • Sublime • Superabundant • Superior highly composite • Superperfect Prime omega functions • Almost prime • Semiprime Euler's totient function • Highly cototient • Highly totient • Noncototient • Nontotient • Perfect totient • Sparsely totient Aliquot sequences • Amicable • Perfect • 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Sarah Woodhead Sarah Woodhead (1851–1912) was the first woman to take and pass a Tripos examination. In particular, she was the first woman to take, and to pass, the Mathematical Tripos exam, which she did in 1873.[1] Education Woodhead’s family had long belonged to the Society of Friends, so she was able to attend Ackworth School, a Quaker school that accepted daughters of Friends as well as their sons.[2] Woodhead later studied at Girton College, the first women's college to be founded at Oxford or Cambridge. As the physical college had yet to be built, she attended courses set up by Girton founder Emily Davies at Benslow House, Hitchin. In 1873, Woodhead took the same Mathematical Tripos examination as the male students, having already gained a first at Part I, and was classed equivalent to Senior Optime in mathematics. She was the first woman to take, and to pass, the Mathematical Tripos exam.[3] This also made her the very first of the first three women to complete any Tripos at Girton College.[4] The three "honorary" (rather than actual) graduates became known as "Woodhead, Cook and Lumsden, the Girton Pioneers".[5] Later life and death Woodhead married architect Christopher Corbett, after which she ran her own school in Bolton. She then became the second headmistress of Bolton School, known then as Bolton High School for Girls.[6] After her husband moved the family back to Manchester to take over his family firm, she found employment as an inspector of schools. Widowed in her fifties, she moved to Harrogate and died there in July, 1908, aged fifty-seven.[7] See also • Philippa Fawcett, the first woman to obtain the top score on the Mathematical Tripos. References 1. Jensen-Vallin, Jacqueline A.; Beery, Janet L.; Mast, Maura B.; Greenwald, Sarah J., eds. (2018). Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America. Springer. p. "Sarah+woodhead"+tripos+1873&pg=PA8 8. ISBN 978-3-319-88303-8. 2. Woodhead, David L., Sarah Woodhead: Trail-Blazer - Quaker Girl & Pioneer, self (see Girton College Library) 3. Jensen-Vallin, Jacqueline A.; Beery, Janet L.; Mast, Maura B.; Greenwald, Sarah J., eds. (2018). Women in Mathematics: Celebrating the Centennial of the Mathematical Association of America. Springer. p. "Sarah+woodhead"+tripos+1873&pg=PA8 8. ISBN 978-3-319-88303-8. 4. Tuker, Mildred Anna Rosalie; Matthison, William (1907), Cambridge, A. - and C. Black, p. 321 5. Megson, Barbara; Lindsay, Jean Olivia (1961), Girton College, 1869–1959: an informal history, W. Heffer for the Girton Historical and Political Society, p. 19 6. Stephen, Barbara (1932), Girton College 1869–1932, Cambridge University Press, p. 194. 7. Campion, Val (2008), Pioneering Women: The Origins of Girton College in Hitchin, Hitchin Historical Society Publication, p. 49
Wikipedia
Robert Woodhouse Robert Woodhouse FRS (28 April 1773 – 23 December 1827) was a British mathematician and astronomer. Robert Woodhouse FRS Born(1773-04-28)28 April 1773 Norwich, Norfolk, England Died23 December 1827(1827-12-23) (aged 54) Cambridge, England NationalityBritish Alma materCambridge University AwardsFirst Smith's Prize (1795) Scientific career FieldsMathematics and astronomy Biography Early life and education Robert Woodhouse was born on 28 April 1773 in Norwich, Norfolk, the son of Robert Woodhouse, linen draper, and Judith Alderson, the daughter of a Unitarian minister from Lowestoft.[1] Robert junior was baptised at St George's Church, Colegate, Norwich, on 19 May, 1773.[2] A younger son, John Thomas Woodhouse, was born in 1780. The brothers were educated at the Paston School in North Walsham, 24 kilometres (15 mi) north of Norwich.[1] In May 1790 Woodhouse was admitted to Gonville and Caius College, Cambridge,[1] the college where Paston pupils were traditionally sent.[3] In 1795 he graduated as the Senior Wrangler (ranked first among the mathematics undergraduates at the university), and took the First Smith's Prize.[1] He obtained his Master's degree at Cambridge in 1798.[4] Marriage and career at Cambridge Woodhouse was a fellow of the college from 1798 to 1823,[note 1] after which he resigned so as to be able to marry Harriet, the daughter of William Wilkin, a Norwich architect.[1][4] They were married on 20 February 1823; the marriage produced a son, also named Robert.[1][4] Harriet Woodhouse died at Cambridge on 31 March 1826.[6] Woodhouse was elected a Fellow of the Royal Society on 16 December 1802.[7] His earliest work, entitled the Principles of Analytical Calculation, was published at Cambridge in 1803.[8] In this he explained the differential notation and strongly pressed the employment of it; but he severely criticised the methods used by continental writers, and their constant assumption of non-evident principles.[9] In 1809 Woodhouse published a textbook covering planar trigonometry and spherical trigonometry and the next year a historical treatise on the calculus of variations and isoperimetrical problems. He next produced an astronomy; of which the first book (usually bound in two volumes), on practical and descriptive astronomy, was issued in 1812, and the second book, containing an account of the treatment of physical astronomy by Pierre-Simon Laplace and other continental writers, was issued in 1818.[8] Woodhouse became the Lucasian Professor of Mathematics in 1820, but the small income caused him to resign the professorship in 1822 and instead accept the better paid post as the Plumian professor in the university.[1][10] As Plumian Professor he was responsible for installing and adjusting the transit instruments and clocks at the Cambridge Observatory.[11] Woodhouse did not exercise much influence on the majority of his contemporaries, and the movement might have died away for the time being if it had not been for the advocacy of George Peacock, Charles Babbage, and John Herschel, who formed the Analytical Society, with the object of advocating the general use in the university of analytical methods and of the differential notation.[12] Woodhouse was the first director of the newly built observatory at Cambridge, a post he held until his death in 1827.[1] On his death in Cambridge he was buried in Caius College chapel.[7] Notes 1. Woodhouse's younger brother John was also a fellow at the college.[5] References 1. O'Connor, John J.; Robertson, Edmund F. (2005). "Robert Woodhouse". MacTutor. University of St Andrews. Retrieved 24 September 2021. 2. Robert Woodhouse in "the Norfolk, England, Transcripts of Church of England Baptism, Marriage and Burial Registers, 1600-1935, FamilySearch (Robert Woodhouse). Citing Archdeacon Transcripts 1600-1812. (registration required) 3. "The History of Paston College". Paston College. Retrieved 24 September 2021. 4. "Robert Woodhouse (WDHS790R)". A Cambridge Alumni Database. University of Cambridge. Retrieved 24 September 2021. 5. Becher 2004. 6. "Biographies T - Z". Institute of Astronomy, Cambridge. Retrieved 24 September 2021. 7. "Record: Woodhouse; Robert (1773 - 1827)". The Royal Society. Retrieved 24 September 2021. 8. Rouse Ball 1912, p. 440. 9. Becher 1980, pp. 389–400. 10. Guicciardini 1989, p. 126. 11. Woodhouse 1825, pp. 418–428. 12. Rouse Ball 1912, pp. 440–441. Sources Wikimedia Commons has media related to Robert Woodhouse (mathematician). Wikiquote has quotations related to Robert Woodhouse. • Becher, Harvey W. (1980). "Woodhouse, Babbage, Peacock and Modern Algebra". Historia Mathematica. 7 (4): 389–400. doi:10.1016/0315-0860(80)90003-8. • Becher, Harvey W. (2004). "Woodhouse, Robert". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/29926. OCLC 56568095. Retrieved 24 September 2021. (Subscription or UK public library membership required.) (subscription may be required or content may be available in libraries that are in the UK) • Guicciardini, Niccolò (1989). "Robert Woodhouse". The Development of Newtonian Calculus in Britain 1700–1800. New York: Cambridge University Press. ISBN 0-521-36466-3. • Rouse Ball, W.W. (1912). A Short Account Of The History Of Mathematics. London: MacMillan and Co. Ltd. OCLC 844389098. • Woodhouse, Robert (1825). "Some account of the transit instrument made by Mr. Dollond, and lately put up at the Cambridge Observatory". Philosophical Transactions of the Royal Society of London: 418–428. Further reading • Harman, P. M. (1988). "Newton to Maxwell: The 'Principia' and British Physics". Notes and Records of the Royal Society of London. 42 (1: Newton's 'Principia' and Its Legacy): 75–96. Bibcode:1988npl..conf...75H. doi:10.1098/rsnr.1988.0008. JSTOR 531370. S2CID 122622492 – via JSTOR. • Johnson, W. (1995). "Contributors to Improving the Teaching of Calculus in Early 19th-Century England". Notes and Records of the Royal Society of London. 49 (1): 93–103. doi:10.1098/rsnr.1995.0006. JSTOR 531886. S2CID 145534544 – via JSTOR. External links • Facsimile of Woodhouse's certificate of election to the Royal Society Works • 1803: Principles of Analytical Calculation • 1809: A Treatise on Plane and Spherical Trigonometry (5th edition 1827) • 1810: A Treatise on Isoperimetric Problems and the Calculus of Variations • 1818: An Elementary Treatise on Physical Astronomy, volume 1 • 1818: An Elementary Treatise on Astronomy, volume 2 • 1821: A Treatise on Astronomy, Theoretical and Practical Lucasian Professors of Mathematics • Isaac Barrow (1664) • Isaac Newton (1669) • William Whiston (1702) • Nicholas Saunderson (1711) • John Colson (1739) • Edward Waring (1760) • Isaac Milner (1798) • Robert Woodhouse (1820) • Thomas Turton (1822) • George Biddell Airy (1826) • Charles Babbage (1828) • Joshua King (1839) • George Stokes (1849) • Joseph Larmor (1903) • Paul Dirac (1932) • James Lighthill (1969) • Stephen Hawking (1979) • Michael Green (2009) • Michael Cates (2015) Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Netherlands Other • IdRef
Wikipedia
Woodin cardinal In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number $\lambda $ such that for all functions $f:\lambda \to \lambda $ there exists a cardinal $\kappa <\lambda $ with $\{f(\beta )\mid \beta <\kappa \}\subseteq \kappa $ and an elementary embedding $j:V\to M$ from the Von Neumann universe $V$ into a transitive inner model $M$ with critical point $\kappa $ and $V_{j(f)(\kappa )}\subseteq M.$ An equivalent definition is this: $\lambda $ is Woodin if and only if $\lambda $ is strongly inaccessible and for all $A\subseteq V_{\lambda }$ there exists a $\lambda _{A}<\lambda $ which is $<\lambda $-$A$-strong. $\lambda _{A}$ being $<\lambda $-$A$-strong means that for all ordinals $\alpha <\lambda $, there exist a $j:V\to M$ which is an elementary embedding with critical point $\lambda _{A}$, $j(\lambda _{A})>\alpha $, $V_{\alpha }\subseteq M$ and $j(A)\cap V_{\alpha }=A\cap V_{\alpha }$. (See also strong cardinal.) A Woodin cardinal is preceded by a stationary set of measurable cardinals, and thus it is a Mahlo cardinal. However, the first Woodin cardinal is not even weakly compact. Consequences Woodin cardinals are important in descriptive set theory. By a result[1] of Martin and Steel, existence of infinitely many Woodin cardinals implies projective determinacy, which in turn implies that every projective set is Lebesgue measurable, has the Baire property (differs from an open set by a meager set, that is, a set which is a countable union of nowhere dense sets), and the perfect set property (is either countable or contains a perfect subset). The consistency of the existence of Woodin cardinals can be proved using determinacy hypotheses. Working in ZF+AD+DC one can prove that $\Theta _{0}$ is Woodin in the class of hereditarily ordinal-definable sets. $\Theta _{0}$ is the first ordinal onto which the continuum cannot be mapped by an ordinal-definable surjection (see Θ (set theory)). Mitchell and Steel showed that assuming a Woodin cardinal exists, there is an inner model containing a Woodin cardinal in which there is a $\Delta _{4}^{1}$-well-ordering of the reals, ◊ holds, and the generalized continuum hypothesis holds.[2] Shelah proved that if the existence of a Woodin cardinal is consistent then it is consistent that the nonstationary ideal on $\omega _{1}$ is $\aleph _{2}$-saturated. Woodin also proved the equiconsistency of the existence of infinitely many Woodin cardinals and the existence of an $\aleph _{1}$-dense ideal over $\aleph _{1}$. Hyper-Woodin cardinals A cardinal $\kappa $ is called hyper-Woodin if there exists a normal measure $U$ on $\kappa $ such that for every set $S$, the set $\{\lambda <\kappa \mid \lambda $ is $<\kappa $-$S$-strong$\}$ is in $U$. $\lambda $ is $<\kappa $-$S$-strong if and only if for each $\delta <\kappa $ there is a transitive class $N$ and an elementary embedding $j:V\to N$ with $\lambda ={\text{crit}}(j),$ $j(\lambda )\geq \delta $, and $j(S)\cap H_{\delta }=S\cap H_{\delta }$. The name alludes to the classical result that a cardinal is Woodin if and only if for every set $S$, the set $\{\lambda <\kappa \mid \lambda $ is $<\kappa $-$S$-strong$\}$ is a stationary set. The measure $U$ will contain the set of all Shelah cardinals below $\kappa $. Weakly hyper-Woodin cardinals A cardinal $\kappa $ is called weakly hyper-Woodin if for every set $S$ there exists a normal measure $U$ on $\kappa $ such that the set $\{\lambda <\kappa \mid \lambda $ is $<\kappa $-$S$-strong$\}$ is in $U$. $\lambda $ is $<\kappa $-$S$-strong if and only if for each $\delta <\kappa $ there is a transitive class $N$ and an elementary embedding $j:V\to N$ with $\lambda ={\text{crit}}(j)$, $j(\lambda )\geq \delta $, and $j(S)\cap H_{\delta }=S\cap H_{\delta }.$ The name alludes to the classic result that a cardinal is Woodin if for every set $S$, the set $\{\lambda <\kappa \mid \lambda $ is $<\kappa $-$S$-strong$\}$ is stationary. The difference between hyper-Woodin cardinals and weakly hyper-Woodin cardinals is that the choice of $U$ does not depend on the choice of the set $S$ for hyper-Woodin cardinals. Notes and references 1. A Proof of Projective Determinacy 2. W. Mitchell, Inner models for large cardinals (2012, p.32). Accessed 2022-12-08. Further reading • Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3. • For proofs of the two results listed in consequences see Handbook of Set Theory (Eds. Foreman, Kanamori, Magidor) (to appear). Drafts of some chapters are available. • Ernest Schimmerling, Woodin cardinals, Shelah cardinals and the Mitchell-Steel core model, Proceedings of the American Mathematical Society 130/11, pp. 3385–3391, 2002, online • Steel, John R. (October 2007). "What is a Woodin Cardinal?" (PDF). Notices of the American Mathematical Society. 54 (9): 1146–7. Retrieved 2008-01-15.
Wikipedia
Word-representable graph In the mathematical field of graph theory, a word-representable graph is a graph that can be characterized by a word (or sequence) whose entries alternate in a prescribed way. In particular, if the vertex set of the graph is V, one should be able to choose a word w over the alphabet V such that letters a and b alternate in w if and only if the pair ab is an edge in the graph. (Letters a and b alternate in w if, after removing from w all letters but the copies of a and b, one obtains a word abab... or a word baba....) For example, the cycle graph labeled by a, b, c and d in clock-wise direction is word-representable because it can be represented by abdacdbc: the pairs ab, bc, cd and ad alternate, but the pairs ac and bd do not. The word w is G's word-representant, and one says that that w represents G. The smallest (by the number of  vertices) non-word-representable graph is the wheel graph W5, which is the only non-word-representable graph on 6 vertices. The definition of a word-representable graph works both in labelled and unlabelled cases since any labelling of a graph is equivalent to any other labelling. Also, the class of word-representable graphs is hereditary. Word-representable graphs generalise several important classes of graphs such as circle graphs, 3-colorable graphs and comparability graphs. Various generalisations of the theory of word-representable graphs accommodate representation of any graph. History Word-representable graphs were introduced by Sergey Kitaev in 2004 based on joint research with Steven Seif[1] on the Perkins semigroup, which has played an important role in semigroup theory since 1960.[2] The first systematic study of word-representable graphs was undertaken in a 2008 paper by Kitaev and Artem Pyatkin,[3] starting development of the theory. One of key contributors to the area is Magnús M. Halldórsson.[4][5][6] Up to date, 35+ papers have been written on the subject, and the core of the book[2] by Sergey Kitaev and Vadim Lozin is devoted to the theory of word-representable graphs. A quick way to get familiar with the area is to read one of the survey articles.[7][8][9] Motivation to study the graphs According to,[2] word-representable graphs are relevant to various fields, thus motivating to study the graphs. These fields are algebra, graph theory, computer science, combinatorics on words, and scheduling. Word-representable graphs are especially important in graph theory, since they generalise several important classes of graphs, e.g. circle graphs, 3-colorable graphs and comparability graphs. Early results It was shown in [3] that a graph G is word-representable if it is k-representable for some k, that is, G can be represented by a word having k copies of each letter. Moreover, if a graph is k-representable then it is also (k + 1)-representable. Thus, the notion of the representation number of a graph, as the minimum k such that a graph is word-representable, is well-defined. Non-word-representable graphs have the representation number ∞. Graphs with representation number 1 are precisely the set of complete graphs, while graphs with representation number 2 are precisely the class of circle non-complete graphs. In particular, forests (except for single trees on at most 2 vertices), ladder graphs and cycle graphs have representation number 2. No classification for graphs with representation number 3 is known. However, there are examples of such graphs, e.g. Petersen's graph and prisms. Moreover, the 3-subdivision of any graph is 3-representable. In particular, for every graph G there exists a 3-representable graph H that contains G as a minor.[3] A graph G is permutationally representable if it can be represented by a word of the form p1p2...pk, where pi is a permutation. On can also say that G is permutationally k-representable. A graph is permutationally representable iff it is a comparability graph.[1] A graph is word-representable implies that the neighbourhood of each vertex is permutationally representable (i.e. is a comparability graph).[1] Converse to the last statement is not true.[4] However, the fact that the neighbourhood of each vertex is a comparability graph implies that the Maximum Clique problem is polynomially solvable on word-representable graphs.[5][6] Semi-transitive orientations Semi-transitive orientations provide a powerful tool to study word-representable graphs. A directed graph is semi-transitively oriented iff it is acyclic and for any directed path u1→u2→ ...→ut, t ≥ 2, either there is no edge from u1 to ut or all edges ui → uj exist for 1 ≤ i < j ≤ t. A key theorem in the theory of word-representable graphs states that a graph is word-representable iff it admits a semi-transitive orientation.[6] As a corollary to the proof of the key theorem one obtain an upper bound on word-representants: Each non-complete word-representable graph G is 2(n − κ(G))-representable, where κ(G) is the size of a maximal clique in G.[6] As an immediate corollary of the last statement, one has that the recognition problem of word-representability is in NP. In 2014, Vincent Limouzy observed that it is an NP-complete problem to recognise whether a given graph is word-representable.[2][7] Another important corollary to the key theorem is that any 3-colorable graph is word-representable. The last fact implies that many classical graph problems are NP-hard on word-representable graphs.   Overview of selected results Non-word-representable graphs Wheel graphs W2n+1, for n ≥ 2, are not word-representable and W5 is the minimum (by the number of vertices) non-word-representable graph. Taking any non-comparability graph and adding an apex (a vertex connected to any other vertex), we obtain a non-word-representable graph, which then can produce infinitely many non-word-representable graphs.[2] Any graph produced in this way will necessarily have a triangle (a cycle of length 3), and a vertex of degree at least 5. Non-word-representable graphs of maximum degree 4 exist [10] and non-word-representable triangle-free graphs exist.[5] Regular non-word representable graphs also exist.[2] Non-isomorphic non-word-representable connected graphs on at most eight vertices were first enumerated by Heman Z.Q. Chen. His calculations were extended in,[11] where it was shown that the numbers of non-isomorphic non-word-representable connected graphs on 5−11 vertices are given, respectively, by 0, 1, 25, 929, 54957, 4880093, 650856040. This is the sequence A290814 in the Online Encyclopaedia of Integer Sequences (OEIS). Operations on graphs and word-representability Operations preserving word-representability are removing a vertex, replacing a vertex with a module, Cartesian product, rooted product, subdivision of a graph, connecting two graphs by an edge and gluing two graphs in a vertex.[2] The operations not necessarily preserving word-representability are taking the complement, taking the line graph, edge contraction,[2] gluing two graphs in a clique of size 2 or more,[12] tensor product, lexicographic product and strong product.[13] Edge-deletion, edge-addition and edge-lifting with respect to word-representability (equivalently, semi-transitive orientability) are studied in.[13] Graphs with high representation number While each non-complete word-representable graph G is 2(n − κ(G))-representable, where κ(G) is the size of a maximal clique in G,[6] the highest known representation number is floor(n/2) given by crown graphs with an all-adjacent vertex.[6] Interestingly, such graphs are not the only graphs that require long representations.[14] Crown graphs themselves are shown to require long (possibly longest) representations among bipartite graphs.[15] Computational complexity Known computational complexities for problems on word-representable graphs can be summarised as follows:[2][7] PROBLEM COMPLEXITY deciding word-representability NP-complete Dominating Set NP-hard Clique Covering NP-hard Maximum Independent Set NP-hard Maximum Clique in P approximating the graph representation number within a factor n1−ε for any ε > 0 NP-hard Representation of planar graphs Triangle-free planar graphs are word-representable.[6] A K4-free near-triangulation is 3-colourable if and only if it is word-representable;[16] this result generalises studies in.[17][18] Word-representability of face subdivisions of triangular grid graphs is studied in [19] and word-representability of triangulations of grid-covered cylinder graphs is studied in.[20] Representation of split graphs Word-representation of split graphs is studied in.[21][12] In particular,[21] offers a characterisation in terms of forbidden induced subgraphs of word-representable split graphs in which vertices in the independent set are of degree at most 2, or the size of the clique is 4, while a computational characterisation of word-representable split graphs with the clique of size 5 is given in.[12] Also, necessary and sufficient conditions for an orientation of a split graph to be semi-transitive are given in,[21] while in [12] threshold graphs are shown to be word-representable and the split graphs are used to show that gluing two word-representable graphs in any clique of size at least 2 may, or may not result in a word-representable graph, which solved a long-standing open problem. Graphs representable by pattern avoiding words A graph is p-representable if it can be represented by a word avoiding a pattern p. For example, 132-representable graphs are those that can be represented by words w1w2...wn where there are no 1 ≤ a < b < c ≤ n such that wa < wc < wb. In [22] it is shown that any 132-representable graph is necessarily a circle graph, and any tree and any cycle graph, as well as any graph on at most 5 vertices, are 132-representable. It was shown in [23] that not all circle graphs are 132-representable, and that 123-representable graphs are also a proper subclass of the class of circle graphs. Generalisations A number of generalisations [24][25][26] of the notion of a word-representable graph are based on the observation by Jeff Remmel that non-edges are defined by occurrences of the pattern 11 (two consecutive equal letters) in a word representing a graph, while edges are defined by avoidance of this pattern. For example, instead of the pattern 11, one can use the pattern 112, or the pattern, 1212, or any other binary pattern where the assumption that the alphabet is ordered can be made so that a letter in a word corresponding to 1 in the pattern is less than that corresponding to 2 in the pattern. Letting u be an ordered binary pattern we thus get the notion of a u-representable graph. So, word-representable graphs are just the class of 11-representable graphs. Intriguingly, any graph can be u-represented assuming u is of length at least 3.[27] Another way to generalise the notion of a word-representable graph, again suggested by Jeff Remmel, is to introduce the "degree of tolerance" k for occurrences of a pattern p defining edges/non-edges. That is, we can say that if there are up to k occurrence of p formed by letters a and b, then there is an edge between a and b. This gives the notion of a k-p-representable graph, and k-11-representable graphs are studied in.[28] Note that 0-11-representable graphs are precisely word-representable graphs. The key results in [28] are that any graph is 2-11-representable and that word-representable graphs are a proper subclass of 1-11-representable graphs. Whether or not any graph is 1-11-representable is a challenging open problem. For yet another type of relevant generalisation, Hans Zantema suggested the notion of a k-semi-transitive orientation refining the notion of a semi-transitive orientation.[14] The idea here is restricting ourselves to considering only directed paths of length not exceeding k while allowing violations of semi-transitivity on longer paths. Open problems Open problems on word-representable graphs can be found in,[2][7][8][9] and they include: • Characterise (non-)word-representable planar graphs. • Characterise word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs [16]). • Classify graphs with representation number 3. (See [29] for the state-of-the-art in this direction.) • Is the line graph of a non-word-representable graph always non-word-representable? • Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter? • Is it true that out of all bipartite graphs crown graphs require longest word-representants? (See [15] for relevant discussion.) • Characterise word-representable graphs in terms of (induced) forbidden subgraphs. • Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)? Literature The list of publications to study representation of graphs by words contains, but is not limited to 1. Ö. Akgün, I.P. Gent, S. Kitaev, H. Zantema. Solving computational problems in the theory of word-representable graphs. Journal of Integer Sequences 22 (2019), Article 19.2.5. 2. P. Akrobotu, S. Kitaev, and Z. Masárová. On word-representability of polyomino triangulations. Siberian Adv. Math. 25 (2015), 1−10. 3. B. Broere. Word representable graphs, 2018. Master thesis at Radboud University, Nijmegen. 4. B. Broere and H. Zantema. "The k-cube is k-representable," J. Autom., Lang., and Combin. 24 (2019) 1, 3-12. 5. J. N. Chen and S. Kitaev. On the 12-representability of induced subgraphs of a grid graph, Discussiones Mathematicae Graph Theory, to appear 6. T. Z. Q. Chen, S. Kitaev, and A. Saito. Representing split graphs by words, arXiv:1909.09471 7. T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of face subdivisions of triangular grid graphs, Graphs and Combin. 32(5) (2016), 1749−1761. 8. T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of triangulations of grid-covered cylinder graphs, Discr. Appl. Math. 213 (2016), 60−70. 9. G.-S. Cheon, J. Kim, M. Kim, and S. Kitaev. Word-representability of Toeplitz graphs, Discr. Appl. Math., to appear. 10. G.-S. Cheon, J. Kim, M. Kim, and A. Pyatkin. On k-11-representable graphs. J. Combin. 10 (2019) 3, 491−513. 11. I. Choi, J. Kim, and M. Kim. On operations preserving semi-transitive orient ability of graphs, Journal of Combinatorial Optimization 37 (2019) 4, 1351−1366. 12. A. Collins, S. Kitaev, and V. Lozin. New results on word-representable graphs, Discr. Appl. Math. 216 (2017), 136−141. 13. A. Daigavane, M. Singh, B.K. George. 2-uniform words: cycle graphs, and an algorithm to verify specific word-representations of graphs. arXiv:1806.04673 (2018). 14. M. Gaetz and C. Ji. Enumeration and extensions of word-representable graphs. Lecture Notes in Computer Science 11682 (2019) 180−192. In R. Mercas, D. Reidenbach (Eds) Combinatorics on Words. WORDS 2019. 15. M. Gaetz and C. Ji. Enumeration and Extensions of Word-representants, arXiv:1909.00019. 16. M. Gaetz and C. Ji. Enumeration and Extensions of Word-representants, Combinatorics on words, 180-192, Lecture Notes in Comput. Sci., 11682, Springer, Cham, 2019. 17. A. Gao, S. Kitaev, and P. Zhang. On 132-representable graphs. Australasian J. Combin. 69 (2017), 105−118. 18. M. Glen. Colourability and word-representability of near-triangulations, Pure and Applied Mathematics, to appear, 2019. 19. M. Glen. On word-representability of polyomino triangulations & crown graphs, 2019. PhD thesis, University of Strathclyde. 20. M. Glen and S. Kitaev. Word-Representability of Triangulations of Rectangular Polyomino with a Single Domino Tile, J. Combin.Math. Combin. Comput. 100, 131−144, 2017. 21. M. Glen, S. Kitaev, and A. Pyatkin. On the representation number of a crown graph, Discr. Appl. Math. 244, 2018, 89−93. 22. M.M. Halldórsson, S. Kitaev, A. Pyatkin On representable graphs, semi-transitive orientations, and the representation numbers, arXiv:0810.0310 (2008). 23. M.M. Halldórsson, S. Kitaev, A. Pyatkin (2010) Graphs capturing alternations in words. In: Y. Gao, H. Lu, S. Seki, S. Yu (eds), Developments in Language Theory. DLT 2010. Lecture Notes Comp. Sci. 6224, Springer, 436−437. 24. M.M. Halldórsson, S. Kitaev, A. Pyatkin (2011) Alternation graphs. In: P. Kolman, J. Kratochvíl (eds), Graph-Theoretic Concepts in Computer Science. WG 2011. Lecture Notes Comp. Sci. 6986, Springer, 191−202. 25. M. Halldórsson, S. Kitaev and A. Pyatkin. Semi-transitive orientations and word-representable graphs, Discr. Appl. Math. 201 (2016), 164−171. 26. M. Jones, S. Kitaev, A. Pyatkin, and J. Remmel. Representing Graphs via Pattern Avoiding Words, Electron. J. Combin. 22 (2), Res. Pap. P2.53, 1−20 (2015). 27. S. Kitaev. On graphs with representation number 3, J. Autom., Lang. and Combin. 18 (2013), 97−112. 28. S. Kitaev. A comprehensive introduction to the theory of word-representable graphs. In: É. Charlier, J. Leroy, M. Rigo (eds), Developments in Language Theory. DLT 2017. Lecture Notes Comp. Sci. 10396, Springer, 36−67. 29. S. Kitaev. Existence of u-representation of graphs, Journal of Graph Theory 85 (2017) 3, 661−668. 30. S. Kitaev, Y. Long, J. Ma, H. Wu. Word-representability of split graphs, arXiv:1709.09725 (2017). 31. S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015. ISBN 978-3-319-25859-1. 32. S. Kitaev and A. Pyatkin. On representable graphs, J. Autom., Lang. and Combin. 13 (2008), 45−54. 33. S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296. 34. S. Kitaev and A. Pyatkin. On semi-transitive orientability of triangle-free graphs, arXiv:2003.06204v1. 35. S. Kitaev and A. Saito. On semi-transitive orientability of Kneser graphs and their complements, Discrete Math., to appear. 36. S. Kitaev, P. Salimov, C. Severs, and H. Úlfarsson (2011) On the representability of line graphs. In: G. Mauri, A. Leporati (eds), Developments in Language Theory. DLT 2011. Lecture Notes Comp. Sci. 6795, Springer, 478−479. 37. S. Kitaev and S. Seif. Word problem of the Perkins semigroup via directed acyclic graphs, Order 25 (2008), 177−194. 38. E. Leloup. Graphes représentables par mot. Master Thesis, University of Liège, 2019 39. Mandelshtam. On graphs representable by pattern-avoiding words, Discussiones Mathematicae Graph Theory 39 (2019) 375−389. 40. С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53. Software Software to study word-representable graphs can be found here: 1. M. Glen. Software to deal with word-representable graphs, 2017. Available at https://personal.cis.strath.ac.uk/sergey.kitaev/word-representable-graphs.html. 2. H. Zantema. Software REPRNR to compute the representation number of a graph, 2018. Available at https://www.win.tue.nl/~hzantema/reprnr.html. References 1. S. Kitaev and S. Seif. Word problem of the Perkins semigroup via directed acyclic graphs, Order 25 (2008), 177−194. 2. S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015. ISBN 978-3-319-25859-1 3. S. Kitaev and A. Pyatkin. On representable graphs, J. Autom., Lang. and Combin. 13 (2008), 45−54. 4. M.M. Halldórsson, S. Kitaev, A. Pyatkin (2010) Graphs capturing alternations in words. In: Y. Gao, H. Lu, S. Seki, S. Yu (eds), Developments in Language Theory. DLT 2010. Lecture Notes Comp. Sci. 6224, Springer, 436−437. 5. M.M. Halldórsson, S. Kitaev, A. Pyatkin (2011) Alternation graphs. In: P. Kolman, J. Kratochvíl (eds), Graph-Theoretic Concepts in Computer Science. WG 2011. Lecture Notes Comp. Sci. 6986, Springer, 191−202. 6. M. Halldórsson, S. Kitaev and A. Pyatkin. Semi-transitive orientations and word-representable graphs, Discr. Appl. Math. 201 (2016), 164−171. 7. S. Kitaev. A comprehensive introduction to the theory of word-representable graphs. In: É. Charlier, J. Leroy, M. Rigo (eds), Developments in Language Theory. DLT 2017. Lecture Notes Comp. Sci. 10396, Springer, 36−67. 8. S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296. 9. С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53 10. A. Collins, S. Kitaev, and V. Lozin. New results on word-representable graphs, Discr. Appl. Math. 216 (2017), 136–141. 11. Ö. Akgün, I.P. Gent, S. Kitaev, H. Zantema. Solving computational problems in the theory of word-representable graphs. Journal of Integer Sequences 22 (2019), Article 19.2.5. 12. T. Z. Q. Chen, S. Kitaev, and A. Saito. Representing split graphs by words, arXiv:1909.09471 13. I. Choi, J. Kim, and M. Kim. On operations preserving semi-transitive orient ability of graphs, Journal of Combinatorial Optimization 37 (2019) 4, 1351−1366. 14. Ö. Akgün, I.P. Gent, S. Kitaev, H. Zantema. Solving computational problems in the theory of word-representable graphs. Journal of Integer Sequences 22 (2019), Article 19.2.5. 15. M. Glen, S. Kitaev, and A. Pyatkin. On the representation number of a crown graph, Discr. Appl. Math. 244, 2018, 89–93. 16. M. Glen. Colourability and word-representability of near-triangulations, Pure and Applied Mathematics, to appear, 2019. 17. P. Akrobotu, S. Kitaev, and Z. Masárová. On word-representability of polyomino triangulations. Siberian Adv. Math. 25 (2015), 1−10. 18. M. Glen and S. Kitaev. Word-Representability of Triangulations of Rectangular Polyomino with a Single Domino Tile, J. Combin.Math. Combin. Comput. 100, 131−144, 2017. 19. T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of face subdivisions of triangular grid graphs, Graphs and Combin. 32(5) (2016), 1749−1761. 20. T. Z. Q. Chen, S. Kitaev, and B. Y. Sun. Word-representability of triangulations of grid-covered cylinder graphs, Discr. Appl. Math. 213 (2016), 60−70. 21. S. Kitaev, Y. Long, J. Ma, H. Wu. Word-representability of split graphs, arXiv:1709.09725 (2017). 22. A. Gao, S. Kitaev, and P. Zhang. On 132-representable graphs. Australasian J. Combin. 69 (2017), 105−118. 23. Mandelshtam. On graphs representable by pattern-avoiding words, Discussiones Mathematicae Graph Theory 39 (2019) 375−389. 24. M. Jones, S. Kitaev, A. Pyatkin, and J. Remmel. Representing Graphs via Pattern Avoiding Words, Electron. J. Combin. 22 (2), Res. Pap. P2.53, 1−20 (2015). 25. M. Gaetz and C. Ji. Enumeration and extensions of word-representable graphs. Lecture Notes in Computer Science 11682 (2019) 180−192. In R. Mercas, D. Reidenbach (Eds) Combinatorics on Words. WORDS 2019. 26. M. Gaetz and C. Ji. Enumeration and Extensions of Word-representants, arXiv:1909.00019. 27. S. Kitaev. Existence of u-representation of graphs, Journal of Graph Theory 85 (2017) 3, 661−668. 28. G.-S. Cheon, J. Kim, M. Kim, and A. Pyatkin. On k-11-representable graphs. J. Combin. 10 (2019) 3, 491−513. 29. S. Kitaev. On graphs with representation number 3, J. Autom., Lang. and Combin. 18 (2013), 97−112.
Wikipedia
Word (group theory) In group theory, a word is any written product of group elements and their inverses. For example, if x, y and z are elements of a group G, then xy, z−1xzz and y−1zxx−1yz−1 are words in the set {x, y, z}. Two different words may evaluate to the same value in G,[1] or even in every group.[2] Words play an important role in the theory of free groups and presentations, and are central objects of study in combinatorial group theory. Definitions Let G be a group, and let S be a subset of G. A word in S is any expression of the form $s_{1}^{\varepsilon _{1}}s_{2}^{\varepsilon _{2}}\cdots s_{n}^{\varepsilon _{n}}$ where s1,...,sn are elements of S, called generators, and each εi is ±1. The number n is known as the length of the word. Each word in S represents an element of G, namely the product of the expression. By convention, the unique[3] identity element can be represented by the empty word, which is the unique word of length zero. Notation When writing words, it is common to use exponential notation as an abbreviation. For example, the word $xxy^{-1}zyzzzx^{-1}x^{-1}\,$ could be written as $x^{2}y^{-1}zyz^{3}x^{-2}.\,$ This latter expression is not a word itself—it is simply a shorter notation for the original. When dealing with long words, it can be helpful to use an overline to denote inverses of elements of S. Using overline notation, the above word would be written as follows: $x^{2}{\overline {y}}zyz^{3}{\overline {x}}^{2}.\,$ Reduced words Any word in which a generator appears next to its own inverse (xx−1 or x−1x) can be simplified by omitting the redundant pair: $y^{-1}zxx^{-1}y\;\;\longrightarrow \;\;y^{-1}zy.$ This operation is known as reduction, and it does not change the group element represented by the word. Reductions can be thought of as relations (defined below) that follow from the group axioms. A reduced word is a word that contains no redundant pairs. Any word can be simplified to a reduced word by performing a sequence of reductions: $xzy^{-1}xx^{-1}yz^{-1}zz^{-1}yz\;\;\longrightarrow \;\;xyz.$ The result does not depend on the order in which the reductions are performed. A word is cyclically reduced if and only if every cyclic permutation of the word is reduced. Operations on words The product of two words is obtained by concatenation: $\left(xzyz^{-1}\right)\left(zy^{-1}x^{-1}y\right)=xzyz^{-1}zy^{-1}x^{-1}y.$ Even if the two words are reduced, the product may not be. The inverse of a word is obtained by inverting each generator, and reversing the order of the elements: $\left(zy^{-1}x^{-1}y\right)^{-1}=y^{-1}xyz^{-1}.$ The product of a word with its inverse can be reduced to the empty word: $zy^{-1}x^{-1}y\;y^{-1}xyz^{-1}=1.$ You can move a generator from the beginning to the end of a word by conjugation: $x^{-1}\left(xy^{-1}z^{-1}yz\right)x=y^{-1}z^{-1}yzx.$ Generating set of a group Main article: Generating set of a group A subset S of a group G is called a generating set if every element of G can be represented by a word in S. When S is not a generating set for G, the set of elements represented by words in S is a subgroup of G, known as the subgroup of G generated by S and usually denoted $\langle S\rangle $. It is the smallest subgroup of G that contains the elements of S. Normal forms A normal form for a group G with generating set S is a choice of one reduced word in S for each element of G. For example: • The words 1, i, j, ij are a normal form for the Klein four-group with S = {i,  j}  and 1 representing the empty word (the identity element for the group). • The words 1, r, r2, ..., rn-1, s, sr, ..., srn-1 are a normal form for the dihedral group Dihn with S = {s,  r}  and 1 as above. • The set of words of the form xmyn for m,n ∈ Z are a normal form for the direct product of the cyclic groups ⟨x⟩ and ⟨y⟩ with S = {x,  y}. • The set of reduced words in S are the unique normal form for the free group over S. Relations and presentations Main article: Presentation of a group If S is a generating set for a group G, a relation is a pair of words in S that represent the same element of G. These are usually written as equations, e.g. $x^{-1}yx=y^{2}.\,$ A set ${\mathcal {R}}$ of relations defines G if every relation in G follows logically from those in ${\mathcal {R}}$ using the axioms for a group. A presentation for G is a pair $\langle S\mid {\mathcal {R}}\rangle $, where S is a generating set for G and ${\mathcal {R}}$ is a defining set of relations. For example, the Klein four-group can be defined by the presentation $\langle i,j\mid i^{2}=1,\,j^{2}=1,\,ij=ji\rangle .$ Here 1 denotes the empty word, which represents the identity element. Free groups Main article: Free group If S is any set, the free group over S is the group with presentation $\langle S\mid \;\rangle $. That is, the free group over S is the group generated by the elements of S, with no extra relations. Every element of the free group can be written uniquely as a reduced word in S. See also • Word problem (mathematics) • Word problem for groups Notes 1. for example, fdr1 and r1fc in the group of square symmetries 2. for example, xy and xzz−1y 3. Uniqueness of identity element and inverses References • Epstein, David; Cannon, J. W.; Holt, D. F.; Levy, S. V. F.; Paterson, M. S.; Thurston, W. P. (1992). Word Processing in Groups. AK Peters. ISBN 0-86720-244-0.. • Novikov, P. S. (1955). "On the algorithmic unsolvability of the word problem in group theory". Trudy Mat. Inst. Steklov (in Russian). 44: 1–143. • Robinson, Derek John Scott (1996). A course in the theory of groups. Berlin: Springer-Verlag. ISBN 0-387-94461-3. • Rotman, Joseph J. (1995). An introduction to the theory of groups. Berlin: Springer-Verlag. ISBN 0-387-94285-8. • Schupp, Paul E; Lyndon, Roger C. (2001). Combinatorial group theory. Berlin: Springer. ISBN 3-540-41158-5. • Solitar, Donald; Magnus, Wilhelm; Karrass, Abraham (2004). Combinatorial group theory: presentations of groups in terms of generators and relations. New York: Dover. ISBN 0-486-43830-9. • Stillwell, John (1993). Classical topology and combinatorial group theory. Berlin: Springer-Verlag. ISBN 0-387-97970-0.
Wikipedia
Word Processing in Groups Word Processing in Groups is a monograph in mathematics on the theory of automatic groups; these are a type of abstract algebra whose operations are defined by the behavior of finite automata. The book's authors are David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Mike Paterson, and William Thurston. Widely circulated in preprint form, it formed the foundation of the study of automatic groups even before its 1992 publication by Jones and Bartlett Publishers (ISBN 0-86720-244-0).[1][2][3] Topics The book is divided into two parts, one on the basic theory of these structures and another on recent research, connections to geometry and topology, and other related topics.[1] The first part has eight chapters. They cover automata theory and regular languages, and the closure properties of regular languages under logical combinations; the definition of automatic groups and biautomatic groups; examples from topology and "combable" structure in the Cayley graphs of automatic groups; abelian groups and the automaticity of Euclidean groups; the theory of determining whether a group is automatic, and its practical implementation by Epstein, Holt, and Sarah Rees; extensions to asynchronous automata; and nilpotent groups.[1][2][4] The second part has four chapters, on braid groups, isoperimetric inequalities, geometric finiteness, and the fundamental groups of three-dimensional manifolds.[1][4] Audience and reception Although not primarily a textbook, the first part of the book could be used as the basis for a graduate course.[1][4] More generally, reviewer Gilbert Baumslag recommends it "very strongly to everyone who is interested in either group theory or topology, as well as to computer scientists." Baumslag was an expert in a related but older area of study, groups defined by finite presentations, in which research was eventually stymied by the phenomenon that many basic problems are undecidable. Despite tracing the origins of automatic groups to early 20th-century mathematician Max Dehn, he writes that the book studies "a strikingly new class of groups" that "conjures up the fascinating possibility that some of the exploration of these automatic groups can be carried out by means of high-speed computers" and that the book is "very likely to have a great impact".[2] Reviewer Daniel E. Cohen adds that two features of the book are unusual, and welcome: First, that the mathematical results that it presents all have names, not just numbers, and second, that the cost of the book is low.[3] Years later, in 2009, mathematician Mark V. Lawson wrote that despite its "odd title" the book made automata theory, once the domain of computer scientists, respectable among mathematicians, and that it became part of "a quiet revolution in the diplomatic relations between mathematics and computer science".[5] References 1. Apanasov, B. N., "Review of Word Processing in Groups", zbMATH, Zbl 0764.20017 2. Baumslag, Gilbert (1994), "Review of Word Processing in Groups", Bulletin of the American Mathematical Society, New Series, 31 (1): 86–91, doi:10.1090/S0273-0979-1994-00481-1, MR 1568123 3. Cohen, D. E. (November 1993), "Review of Word Processing in Groups", Bulletin of the London Mathematical Society, 25 (6): 614–616, doi:10.1112/blms/25.6.614 4. Thomas, Richard M. (1993), "Review of Word Processing in Groups", Mathematical Reviews, MR 1161694 5. Lawson, Mark V. (December 2009), "Review of A Second Course in Formal Languages and Automata Theory by Jeffrey Shallit", SIAM Review, 51 (4): 797–799, JSTOR 25662348
Wikipedia
Combinatorics on words Combinatorics on words is a fairly new field of mathematics, branching from combinatorics, which focuses on the study of words and formal languages. The subject looks at letters or symbols, and the sequences they form. Combinatorics on words affects various areas of mathematical study, including algebra and computer science. There have been a wide range of contributions to the field. Some of the first work was on square-free words by Axel Thue in the early 1900s. He and colleagues observed patterns within words and tried to explain them. As time went on, combinatorics on words became useful in the study of algorithms and coding. It led to developments in abstract algebra and answering open questions. Definition Combinatorics is an area of discrete mathematics. Discrete mathematics is the study of countable structures. These objects have a definite beginning and end. The study of enumerable objects is the opposite of disciplines such as analysis, where calculus and infinite structures are studied. Combinatorics studies how to count these objects using various representations. Combinatorics on words is a recent development in this field that focuses on the study of words and formal languages. A formal language is any set of symbols and combinations of symbols that people use to communicate information.[1] Some terminology relevant to the study of words should first be explained. First and foremost, a word is basically a sequence of symbols, or letters, in a finite set.[1] One of these sets is known by the general public as the alphabet. For example, the word "encyclopedia" is a sequence of symbols in the English alphabet, a finite set of twenty-six letters. Since a word can be described as a sequence, other basic mathematical descriptions can be applied. The alphabet is a set, so as one would expect, the empty set is a subset. In other words, there exists a unique word of length zero. The length of the word is defined by the number of symbols that make up the sequence, and is denoted by |w|.[1] Again looking at the example "encyclopedia", |w| = 12, since encyclopedia has twelve letters. The idea of factoring of large numbers can be applied to words, where a factor of a word is a block of consecutive symbols.[1] Thus, "cyclop" is a factor of "encyclopedia". In addition to examining sequences in themselves, another area to consider of combinatorics on words is how they can be represented visually. In mathematics various structures are used to encode data. A common structure used in combinatorics is the tree structure. A tree structure is a graph where the vertices are connected by one line, called a path or edge. Trees may not contain cycles, and may or may not be complete. It is possible to encode a word, since a word is constructed by symbols, and encode the data by using a tree.[1] This gives a visual representation of the object. Major contributions The first books on combinatorics on words that summarize the origins of the subject were written by a group of mathematicians that collectively went by the name of M. Lothaire. Their first book was published in 1983, when combinatorics on words became more widespread.[1] Patterns Patterns within words A main contributor to the development of combinatorics on words was Axel Thue (1863–1922); he researched repetition. Thue's main contribution was the proof of the existence of infinite square-free words. Square-free words do not have adjacent repeated factors.[1] To clarify, "dining" is not square-free since "in" is repeated consecutively, while "baggage" is square-free, its two "ag" factors not being adjacent. Thue proves his conjecture on the existence of infinite square-free words by using substitutions. A substitution is a way to take a symbol and replace it with a word. He uses this technique to describe his other contribution, the Thue–Morse sequence, or Thue–Morse word.[1] Thue wrote two papers on square-free words, the second of which was on the Thue–Morse word. Marston Morse is included in the name because he discovered the same result as Thue did, yet they worked independently. Thue also proved the existence of an overlap-free word. An overlap-free word is when, for two symbols x and y, the pattern xyxyx does not exist within the word. He continues in his second paper to prove a relationship between infinite overlap-free words and square-free words. He takes overlap-free words that are created using two different letters, and demonstrates how they can be transformed into square-free words of three letters using substitution.[1] As was previously described, words are studied by examining the sequences made by the symbols. Patterns are found, and they are able to be described mathematically. Patterns can be either avoidable patterns, or unavoidable. A significant contributor to the work of unavoidable patterns, or regularities, was Frank Ramsey in 1930. His important theorem states that for integers k, m≥2, there exists a least positive integer R(k,m) such that despite how a complete graph is colored with two colors, there will always exist a solid color subgraph of each color.[1] Other contributors to the study of unavoidable patterns include van der Waerden. His theorem states that if the positive integers are partitioned into k classes, then there exists a class c such that c contains an arithmetic progression of some unknown length. An arithmetic progression is a sequence of numbers in which the difference between adjacent numbers remains constant.[1] When examining unavoidable patterns sesquipowers are also studied. For some patterns x,y,z, a sesquipower is of the form x, xyx, xyxzxyx, .... This is another pattern such as square-free, or unavoidable patterns. Coudrain and Schützenberger mainly studied these sesquipowers for group theory applications. In addition, Zimin proved that sesquipowers are all unavoidable. Whether the entire pattern shows up, or only some piece of the sesquipower shows up repetitively, it is not possible to avoid it.[1] Patterns within alphabets Necklaces are constructed from words of circular sequences. They are most frequently used in music and astronomy. Flye Sainte-Marie in 1894 proved there are 22n−1−n binary de Bruijn necklaces of length 2n. A de Bruijn necklace contains factors made of words of length n over a certain number of letters. The words appear only once in the necklace.[1] In 1874, Baudot developed the code that would eventually take the place of Morse code by applying the theory of binary de Bruijn necklaces. The problem continued from Sainte-Marie to Martin in 1934, who began looking at algorithms to make words of the de Bruijn structure. It was then worked on by Posthumus in 1943.[1] Language hierarchy Main article: Chomsky hierarchy Possibly the most applied result in combinatorics on words is the Chomsky hierarchy, developed by Noam Chomsky. He studied formal language in the 1950s.[2] His way of looking at language simplified the subject. He disregards the actual meaning of the word, does not consider certain factors such as frequency and context, and applies patterns of short terms to all length terms. The basic idea of Chomsky's work is to divide language into four levels, or the language hierarchy. The four levels are: regular, context-free, context-sensitive, and computably enumerable or unrestricted.[2] Regular is the least complex while computably enumerable is the most complex. While his work grew out of combinatorics on words, it drastically affected other disciplines, especially computer science.[3] Word types Sturmian words Main article: Sturmian word Sturmian words, created by François Sturm, have roots in combinatorics on words. There exist several equivalent definitions of Sturmian words. For example, an infinite word is Sturmian if and only if it has n+1 distinct factors of length n, for every non-negative integer n.[1] Lyndon word Main article: Lyndon word A Lyndon word is a word over a given alphabet that is written in its simplest and most ordered form out of its respective conjugacy class. Lyndon words are important because for any given Lyndon word x, there exists Lyndon words y and z, with y<z, x=yz. Further, there exists a theorem by Chen, Fox, and Lyndon, that states any word has a unique factorization of Lyndon words, where the factorization words are non-increasing. Due to this property, Lyndon words are used to study algebra, specifically group theory. They form the basis for the idea of commutators.[1] Visual representation Cobham contributed work relating Eugène Prouhet's work with finite automata. A mathematical graph is made of edges and nodes. With finite automata, the edges are labeled with a letter in an alphabet. To use the graph, one starts at a node and travels along the edges to reach a final node. The path taken along the graph forms the word. It is a finite graph because there are a countable number of nodes and edges, and only one path connects two distinct nodes.[1] Gauss codes, created by Carl Friedrich Gauss in 1838, are developed from graphs. Specifically, a closed curve on a plane is needed. If the curve only crosses over itself a finite number of times, then one labels the intersections with a letter from the alphabet used. Traveling along the curve, the word is determined by recording each letter as an intersection is passed. Gauss noticed that the distance between when the same symbol shows up in a word is an even integer.[1] Group theory Walther Franz Anton von Dyck began the work of combinatorics on words in group theory by his published work in 1882 and 1883. He began by using words as group elements. Lagrange also contributed in 1771 with his work on permutation groups.[1] One aspect of combinatorics on words studied in group theory is reduced words. A group is constructed with words on some alphabet including generators and inverse elements, excluding factors that appear of the form aā or āa, for some a in the alphabet. Reduced words are formed when the factors aā, āa are used to cancel out elements until a unique word is reached.[1] Nielsen transformations were also developed. For a set of elements of a free group, a Nielsen transformation is achieved by three transformations; replacing an element with its inverse, replacing an element with the product of itself and another element, and eliminating any element equal to 1. By applying these transformations Nielsen reduced sets are formed. A reduced set means no element can be multiplied by other elements to cancel out completely. There are also connections with Nielsen transformations with Sturmian words.[1] Considered problems One problem considered in the study of combinatorics on words in group theory is the following: for two elements x,y of a semigroup, does x=y modulo the defining relations of x and y. Post and Markov studied this problem and determined it undecidable. Undecidable means the theory cannot be proved.[1] The Burnside question was proved using the existence of an infinite cube-free word. This question asks if a group is finite if the group has a definite number of generators and meets the criteria xn=1, for x in the group.[1] Many word problems are undecidable based on the Post correspondence problem. Any two homomorphisms $g,h$ with a common domain and a common codomain form an instance of the Post correspondence problem, which asks whether there exists a word $w$ in the domain such that $g(w)=h(w)$. Post proved that this problem is undecidable; consequently, any word problem that can be reduced to this basic problem is likewise undecidable.[1] Other applications Combinatorics on words have applications on equations. Makanin proved that it is possible to find a solution for a finite system of equations, when the equations are constructed from words.[1] See also • Fibonacci word • Kolakoski sequence • Levi's lemma • Partial word • Shift space • Word metric • Word problem (computability) • Word problem (mathematics) • Word problem for groups • Young–Fibonacci lattice References 1. Berstel, Jean; Dominique Perrin (April 2007). "The origins of combinatorics on words". European Journal of Combinatorics. 28 (3): 996–1022. doi:10.1016/j.ejc.2005.07.019. 2. Jäger, Gerhard; James Rogers (July 2012). "Formal language theory: refining the Chomsky hierarchy". Philosophical Transactions of the Royal Society B. 367 (1598): 1956–1970. doi:10.1098/rstb.2012.0077. PMC 3367686. PMID 22688632. 3. Morales-Bueno, Rafael; Baena-Garcia, Manuel; Carmona-Cejudo, Jose M.; Castillo, Gladys (2010). "Counting Word Avoiding Factors". Electronic Journal of Mathematics and Technology. 4 (3): 251. Further reading • The origins of combinatorics on words, Jean Berstel, Dominique Perrin, European Journal of Combinatorics 28 (2007) 996–1022 • Combinatorics on words – a tutorial, Jean Berstel and Juhani Karhumäki. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 79:178–228, 2003. • Combinatorics on Words: A New Challenging Topic, Juhani Karhumäki. • Choffrut, Christian; Karhumäki, Juhani (1997). "Combinatorics of words". In Rozenberg, Grzegorz; Salomaa, Arto (eds.). Handbook of formal languages. Vol. 1. Springer. CiteSeerX 10.1.1.54.3135. ISBN 978-3-540-60420-4. • Lothaire, M. (1983), Combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 17, Addison-Wesley Publishing Co., Reading, Mass., ISBN 978-0-201-13516-9, MR 0675953, Zbl 0514.20045 • Lothaire, M. (2002), Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 90, Cambridge University Press, ISBN 978-0-521-81220-7, MR 1905123, Zbl 1001.68093 • "Infinite words: automata, semigroups, logic and games", Dominique Perrin, Jean Éric Pin, Academic Press, 2004, ISBN 978-0-12-532111-2. • Lothaire, M. (2005), Applied combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 105, Cambridge University Press, ISBN 978-0-521-84802-2, MR 2165687, Zbl 1133.68067 • "Algorithmic Combinatorics on Partial Words", Francine Blanchet-Sadri, CRC Press, 2008, ISBN 978-1-4200-6092-8. • Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009), Combinatorics on words. Christoffel words and repetitions in words, CRM Monograph Series, vol. 27, Providence, RI: American Mathematical Society, ISBN 978-0-8218-4480-9, Zbl 1161.68043 • "Combinatorics of Compositions and Words", Silvia Heubach, Toufik Mansour, CRC Press, 2009, ISBN 978-1-4200-7267-9. • "Word equations and related topics: 1st international workshop, IWWERT '90, Tübingen, Germany, October 1–3, 1990 : proceedings", Editor: Klaus Ulrich Schulz, Springer, 1992, ISBN 978-3-540-55124-9 • "Jewels of stringology: text algorithms", Maxime Crochemore, Wojciech Rytter, World Scientific, 2003, ISBN 978-981-02-4897-0 • Berthé, Valérie; Rigo, Michel, eds. (2010). Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge: Cambridge University Press. ISBN 978-0-521-51597-9. Zbl 1197.68006. • Berstel, Jean; Reutenauer, Christophe (2011). Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications. Vol. 137. Cambridge: Cambridge University Press. ISBN 978-0-521-19022-0. Zbl 1250.68007. • "Patterns in Permutations and Words", Sergey Kitaev, Springer, 2011, ISBN 9783642173325 • "Distribution Modulo One and Diophantine Approximation", Yann Bugeaud, Cambridge University Press, 2012, ISBN 9780521111690 External links Wikimedia Commons has media related to Combinatorics on words. • Jean Berstel's page • Tero Harju's page • Guy Melançon's page
Wikipedia
Word problem for groups In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element. More precisely, if A is a finite set of generators for G then the word problem is the membership problem for the formal language of all words in A and a formal set of inverses that map to the identity under the natural map from the free monoid with involution on A to the group G. If B is another finite generating set for G, then the word problem over the generating set B is equivalent to the word problem over the generating set A. Thus one can speak unambiguously of the decidability of the word problem for the finitely generated group G. The related but different uniform word problem for a class K of recursively presented groups is the algorithmic problem of deciding, given as input a presentation P for a group G in the class K and two words in the generators of G, whether the words represent the same element of G. Some authors require the class K to be definable by a recursively enumerable set of presentations. History Throughout the history of the subject, computations in groups have been carried out using various normal forms. These usually implicitly solve the word problem for the groups in question. In 1911 Max Dehn proposed that the word problem was an important area of study in its own right,[1] together with the conjugacy problem and the group isomorphism problem. In 1912 he gave an algorithm that solves both the word and conjugacy problem for the fundamental groups of closed orientable two-dimensional manifolds of genus greater than or equal to 2.[2] Subsequent authors have greatly extended Dehn's algorithm and applied it to a wide range of group theoretic decision problems.[3][4][5] It was shown by Pyotr Novikov in 1955 that there exists a finitely presented group G such that the word problem for G is undecidable.[6] It follows immediately that the uniform word problem is also undecidable. A different proof was obtained by William Boone in 1958.[7] The word problem was one of the first examples of an unsolvable problem to be found not in mathematical logic or the theory of algorithms, but in one of the central branches of classical mathematics, algebra. As a result of its unsolvability, several other problems in combinatorial group theory have been shown to be unsolvable as well. It is important to realize that the word problem is in fact solvable for many groups G. For example, polycyclic groups have solvable word problems since the normal form of an arbitrary word in a polycyclic presentation is readily computable; other algorithms for groups may, in suitable circumstances, also solve the word problem, see the Todd–Coxeter algorithm[8] and the Knuth–Bendix completion algorithm.[9] On the other hand, the fact that a particular algorithm does not solve the word problem for a particular group does not show that the group has an unsolvable word problem. For instance Dehn's algorithm does not solve the word problem for the fundamental group of the torus. However this group is the direct product of two infinite cyclic groups and so has a solvable word problem. A more concrete description In more concrete terms, the uniform word problem can be expressed as a rewriting question, for literal strings.[10] For a presentation P of a group G, P will specify a certain number of generators x, y, z, ... for G. We need to introduce one letter for x and another (for convenience) for the group element represented by x−1. Call these letters (twice as many as the generators) the alphabet $\Sigma $ for our problem. Then each element in G is represented in some way by a product abc ... pqr of symbols from $\Sigma $, of some length, multiplied in G. The string of length 0 (null string) stands for the identity element e of G. The crux of the whole problem is to be able to recognise all the ways e can be represented, given some relations. The effect of the relations in G is to make various such strings represent the same element of G. In fact the relations provide a list of strings that can be either introduced where we want, or cancelled out whenever we see them, without changing the 'value', i.e. the group element that is the result of the multiplication. For a simple example, take the presentation {a | a3}. Writing A for the inverse of a, we have possible strings combining any number of the symbols a and A. Whenever we see aaa, or aA or Aa we may strike these out. We should also remember to strike out AAA; this says that since the cube of a is the identity element of G, so is the cube of the inverse of a. Under these conditions the word problem becomes easy. First reduce strings to the empty string, a, aa, A or AA. Then note that we may also multiply by aaa, so we can convert A to aa and convert AA to a. The result is that the word problem, here for the cyclic group of order three, is solvable. This is not, however, the typical case. For the example, we have a canonical form available that reduces any string to one of length at most three, by decreasing the length monotonically. In general, it is not true that one can get a canonical form for the elements, by stepwise cancellation. One may have to use relations to expand a string many-fold, in order eventually to find a cancellation that brings the length right down. The upshot is, in the worst case, that the relation between strings that says they are equal in G is an Undecidable problem. Examples The following groups have a solvable word problem: • Automatic groups, including: • Finite groups • Negatively curved (aka. hyperbolic) groups • Euclidean groups • Coxeter groups • Braid groups • Geometrically finite groups • Finitely generated free groups • Finitely generated free abelian groups • Polycyclic groups • Finitely generated recursively absolutely presented groups,[11] including: • Finitely presented simple groups. • Finitely presented residually finite groups • One relator groups[12] (this is a theorem of Magnus), including: • Fundamental groups of closed orientable two-dimensional manifolds. • Combable groups • Autostackable groups Examples with unsolvable word problems are also known: • Given a recursively enumerable set A of positive integers that has insoluble membership problem, ⟨a,b,c,d | anban = cndcn : n ∈ A⟩ is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble[13] • Every finitely generated group with a recursively enumerable presentation and insoluble word problem is a subgroup of a finitely presented group with insoluble word problem[14] • The number of relators in a finitely presented group with insoluble word problem may be as low as 14 [15] or even 12.[16][17] • An explicit example of a reasonable short presentation with insoluble word problem is given in Collins 1986:[18][19] ${\begin{array}{lllll}\langle &a,b,c,d,e,p,q,r,t,k&|&&\\&p^{10}a=ap,&pacqr=rpcaq,&ra=ar,&\\&p^{10}b=bp,&p^{2}adq^{2}r=rp^{2}daq^{2},&rb=br,&\\&p^{10}c=cp,&p^{3}bcq^{3}r=rp^{3}cbq^{3},&rc=cr,&\\&p^{10}d=dp,&p^{4}bdq^{4}r=rp^{4}dbq^{4},&rd=dr,&\\&p^{10}e=ep,&p^{5}ceq^{5}r=rp^{5}ecaq^{5},&re=er,&\\&aq^{10}=qa,&p^{6}deq^{6}r=rp^{6}edbq^{6},&pt=tp,&\\&bq^{10}=qb,&p^{7}cdcq^{7}r=rp^{7}cdceq^{7},&qt=tq,&\\&cq^{10}=qc,&p^{8}ca^{3}q^{8}r=rp^{8}a^{3}q^{8},&&\\&dq^{10}=qd,&p^{9}da^{3}q^{9}r=rp^{9}a^{3}q^{9},&&\\&eq^{10}=qe,&a^{-3}ta^{3}k=ka^{-3}ta^{3}&&\rangle \end{array}}$ Partial solution of the word problem The word problem for a recursively presented group can be partially solved in the following sense: Given a recursive presentation P = ⟨X|R⟩ for a group G, define: $S=\{\langle u,v\rangle :u{\text{ and }}v{\text{ are words in }}X{\text{ and }}u=v{\text{ in }}G\ \}$ then there is a partial recursive function fP such that: $f_{P}(\langle u,v\rangle )={\begin{cases}0&{\text{if}}\ \langle u,v\rangle \in S\\{\text{undefined/does not halt}}\ &{\text{if}}\ \langle u,v\rangle \notin S\end{cases}}$ More informally, there is an algorithm that halts if u=v, but does not do so otherwise. It follows that to solve the word problem for P it is sufficient to construct a recursive function g such that: $g(\langle u,v\rangle )={\begin{cases}0&{\text{if}}\ \langle u,v\rangle \notin S\\{\text{undefined/does not halt}}\ &{\text{if}}\ \langle u,v\rangle \in S\end{cases}}$ However u=v in G if and only if uv−1=1 in G. It follows that to solve the word problem for P it is sufficient to construct a recursive function h such that: $h(x)={\begin{cases}0&{\text{if}}\ x\neq 1\ {\text{in}}\ G\\{\text{undefined/does not halt}}\ &{\text{if}}\ x=1\ {\text{in}}\ G\end{cases}}$ Example The following will be proved as an example of the use of this technique: Theorem: A finitely presented residually finite group has solvable word problem. Proof: Suppose G = ⟨X|R⟩ is a finitely presented, residually finite group. Let S be the group of all permutations of N, the natural numbers, that fixes all but finitely many numbers then: 1. S is locally finite and contains a copy of every finite group. 2. The word problem in S is solvable by calculating products of permutations. 3. There is a recursive enumeration of all mappings of the finite set X into S. 4. Since G is residually finite, if w is a word in the generators X of G then w ≠ 1 in G if and only of some mapping of X into S induces a homomorphism such that w ≠ 1 in S. Given these facts, algorithm defined by the following pseudocode: For every mapping of X into S If every relator in R is satisfied in S If w ≠ 1 in S return 0 End if End if End for defines a recursive function h such that: $h(x)={\begin{cases}0&{\text{if}}\ x\neq 1\ {\text{in}}\ G\\{\text{undefined/does not halt}}\ &{\text{if}}\ x=1\ {\text{in}}\ G\end{cases}}$ This shows that G has solvable word problem. Unsolvability of the uniform word problem The criterion given above, for the solvability of the word problem in a single group, can be extended by a straightforward argument. This gives the following criterion for the uniform solvability of the word problem for a class of finitely presented groups: To solve the uniform word problem for a class K of groups, it is sufficient to find a recursive function $f(P,w)$ that takes a finite presentation P for a group G and a word $w$ in the generators of G, such that whenever G ∈ K: $f(P,w)={\begin{cases}0&{\text{if}}\ w\neq 1\ {\text{in}}\ G\\{\text{undefined/does not halt}}\ &{\text{if}}\ w=1\ {\text{in}}\ G\end{cases}}$ Boone-Rogers Theorem: There is no uniform partial algorithm that solves the word problem in all finitely presented groups with solvable word problem. In other words, the uniform word problem for the class of all finitely presented groups with solvable word problem is unsolvable. This has some interesting consequences. For instance, the Higman embedding theorem can be used to construct a group containing an isomorphic copy of every finitely presented group with solvable word problem. It seems natural to ask whether this group can have solvable word problem. But it is a consequence of the Boone-Rogers result that: Corollary: There is no universal solvable word problem group. That is, if G is a finitely presented group that contains an isomorphic copy of every finitely presented group with solvable word problem, then G itself must have unsolvable word problem. Remark: Suppose G = ⟨X|R⟩ is a finitely presented group with solvable word problem and H is a finite subset of G. Let H* = ⟨H⟩, be the group generated by H. Then the word problem in H* is solvable: given two words h, k in the generators H of H*, write them as words in X and compare them using the solution to the word problem in G. It is easy to think that this demonstrates a uniform solution of the word problem for the class K (say) of finitely generated groups that can be embedded in G. If this were the case, the non-existence of a universal solvable word problem group would follow easily from Boone-Rogers. However, the solution just exhibited for the word problem for groups in K is not uniform. To see this, consider a group J = ⟨Y|T⟩ ∈ K; in order to use the above argument to solve the word problem in J, it is first necessary to exhibit a mapping e: Y → G that extends to an embedding e*: J → G. If there were a recursive function that mapped (finitely generated) presentations of groups in K to embeddings into G, then a uniform solution of the word problem in K could indeed be constructed. But there is no reason, in general, to suppose that such a recursive function exists. However, it turns out that, using a more sophisticated argument, the word problem in J can be solved without using an embedding e: J → G. Instead an enumeration of homomorphisms is used, and since such an enumeration can be constructed uniformly, it results in a uniform solution to the word problem in K. Proof that there is no universal solvable word problem group Suppose G were a universal solvable word problem group. Given a finite presentation P = ⟨X|R⟩ of a group H, one can recursively enumerate all homomorphisms h: H → G by first enumerating all mappings h†: X → G. Not all of these mappings extend to homomorphisms, but, since h†(R) is finite, it is possible to distinguish between homomorphisms and non-homomorphisms, by using the solution to the word problem in G. "Weeding out" non-homomorphisms gives the required recursive enumeration: h1, h2, ..., hn, ... . If H has solvable word problem, then at least one of these homomorphisms must be an embedding. So given a word w in the generators of H: ${\text{If}}\ w\neq 1\ {\text{in}}\ H,\ h_{n}(w)\neq 1\ {\text{in}}\ G\ {\text{for some}}\ h_{n}$ ${\text{If}}\ w=1\ {\text{in}}\ H,\ h_{n}(w)=1\ {\text{in}}\ G\ {\text{for all}}\ h_{n}$ Consider the algorithm described by the pseudocode: Let n = 0 Let repeatable = TRUE while (repeatable) increase n by 1 if (solution to word problem in G reveals hn(w) ≠ 1 in G) Let repeatable = FALSE output 0. This describes a recursive function: $f(w)={\begin{cases}0&{\text{if}}\ w\neq 1\ {\text{in}}\ H\\{\text{undefined/does not halt}}\ &{\text{if}}\ w=1\ {\text{in}}\ H.\end{cases}}$ The function f clearly depends on the presentation P. Considering it to be a function of the two variables, a recursive function $f(P,w)$ has been constructed that takes a finite presentation P for a group H and a word w in the generators of a group G, such that whenever G has soluble word problem: $f(P,w)={\begin{cases}0&{\text{if}}\ w\neq 1\ {\text{in}}\ H\\{\text{undefined/does not halt}}\ &{\text{if}}\ w=1\ {\text{in}}\ H.\end{cases}}$ But this uniformly solves the word problem for the class of all finitely presented groups with solvable word problem, contradicting Boone-Rogers. This contradiction proves G cannot exist. Algebraic structure and the word problem There are a number of results that relate solvability of the word problem and algebraic structure. The most significant of these is the Boone-Higman theorem: A finitely presented group has solvable word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. It is widely believed that it should be possible to do the construction so that the simple group itself is finitely presented. If so one would expect it to be difficult to prove as the mapping from presentations to simple groups would have to be non-recursive. The following has been proved by Bernhard Neumann and Angus Macintyre: A finitely presented group has solvable word problem if and only if it can be embedded in every algebraically closed group What is remarkable about this is that the algebraically closed groups are so wild that none of them has a recursive presentation. The oldest result relating algebraic structure to solvability of the word problem is Kuznetsov's theorem: A recursively presented simple group S has solvable word problem. To prove this let ⟨X|R⟩ be a recursive presentation for S. Choose a ∈ S such that a ≠ 1 in S. If w is a word on the generators X of S, then let: $S_{w}=\langle X|R\cup \{w\}\rangle .$ There is a recursive function $f_{\langle X|R\cup \{w\}\rangle }$ such that: $f_{\langle X|R\cup \{w\}\rangle }(x)={\begin{cases}0&{\text{if}}\ x=1\ {\text{in}}\ S_{w}\\{\text{undefined/does not halt}}\ &{\text{if}}\ x\neq 1\ {\text{in}}\ S_{w}.\end{cases}}$ Write: $g(w,x)=f_{\langle X|R\cup \{w\}\rangle }(x).$ Then because the construction of f was uniform, this is a recursive function of two variables. It follows that: $h(w)=g(w,a)$ is recursive. By construction: $h(w)={\begin{cases}0&{\text{if}}\ a=1\ {\text{in}}\ S_{w}\\{\text{undefined/does not halt}}\ &{\text{if}}\ a\neq 1\ {\text{in}}\ S_{w}.\end{cases}}$ Since S is a simple group, its only quotient groups are itself and the trivial group. Since a ≠ 1 in S, we see a = 1 in Sw if and only if Sw is trivial if and only if w ≠ 1 in S. Therefore: $h(w)={\begin{cases}0&{\text{if}}\ w\neq 1\ {\text{in}}\ S\\{\text{undefined/does not halt}}\ &{\text{if}}\ w=1\ {\text{in}}\ S.\end{cases}}$ The existence of such a function is sufficient to prove the word problem is solvable for S. This proof does not prove the existence of a uniform algorithm for solving the word problem for this class of groups. The non-uniformity resides in choosing a non-trivial element of the simple group. There is no reason to suppose that there is a recursive function that maps a presentation of a simple groups to a non-trivial element of the group. However, in the case of a finitely presented group we know that not all the generators can be trivial (Any individual generator could be, of course). Using this fact it is possible to modify the proof to show: The word problem is uniformly solvable for the class of finitely presented simple groups. See also • Combinatorics on words • SQ-universal group • Word problem (mathematics) • Reachability problem • Nested stack automata (have been used to solve the word problem for groups) Notes 1. Dehn 1911. 2. Dehn 1912. 3. Greendlinger, Martin (June 1959), "Dehn's algorithm for the word problem", Communications on Pure and Applied Mathematics, 13 (1): 67–83, doi:10.1002/cpa.3160130108. 4. Lyndon, Roger C. (September 1966), "On Dehn's algorithm", Mathematische Annalen, 166 (3): 208–228, doi:10.1007/BF01361168, hdl:2027.42/46211, S2CID 36469569. 5. Schupp, Paul E. (June 1968), "On Dehn's algorithm and the conjugacy problem", Mathematische Annalen, 178 (2): 119–130, doi:10.1007/BF01350654, S2CID 120429853. 6. Novikov, P. S. (1955), "On the algorithmic unsolvability of the word problem in group theory", Proceedings of the Steklov Institute of Mathematics (in Russian), 44: 1–143, Zbl 0068.01301 7. Boone, William W. (1958), "The word problem" (PDF), Proceedings of the National Academy of Sciences, 44 (10): 1061–1065, Bibcode:1958PNAS...44.1061B, doi:10.1073/pnas.44.10.1061, PMC 528693, PMID 16590307, Zbl 0086.24701 8. Todd, J.; Coxeter, H.S.M. (1936). "A practical method for enumerating cosets of a finite abstract group". Proceedings of the Edinburgh Mathematical Society. 5 (1): 26–34. doi:10.1017/S0013091500008221. 9. Knuth, D.; Bendix, P. (2014) [1970]. "Simple word problems in universal algebras". In Leech, J. (ed.). Computational Problems in Abstract Algebra: Proceedings of a Conference Held at Oxford Under the Auspices of the Science Research Council Atlas Computer Laboratory, 29th August to 2nd September 1967. Springer. pp. 263–297. ISBN 9781483159423. 10. Rotman 1994. 11. Simmons, H. (1973). "The word problem for absolute presentations". J. London Math. Soc. s2-6 (2): 275–280. doi:10.1112/jlms/s2-6.2.275. 12. Lyndon, Roger C.; Schupp, Paul E (2001). Combinatorial Group Theory. Springer. pp. 1–60. ISBN 9783540411581. 13. Collins & Zieschang 1990, p. 149. 14. Collins & Zieschang 1993, Cor. 7.2.6. sfn error: no target: CITEREFCollinsZieschang1993 (help) 15. Collins 1969. 16. Borisov 1969. 17. Collins 1972. 18. Collins 1986. 19. We use the corrected version from John Pedersen's A Catalogue of Algebraic Systems References • Boone, W.W.; Cannonito, F.B.; Lyndon, Roger C. (1973). Word problems : decision problems and the Burnside problem in group theory. Studies in logic and the foundations of mathematics. Vol. 71. North-Holland. ISBN 9780720422719. • Boone, W. W.; Higman, G. (1974). "An algebraic characterization of the solvability of the word problem". J. Austral. Math. Soc. 18: 41–53. doi:10.1017/s1446788700019108. • Boone, W. W.; Rogers Jr, H. (1966). "On a problem of J. H. C. Whitehead and a problem of Alonzo Church". Math. Scand. 19: 185–192. doi:10.7146/math.scand.a-10808. • Borisov, V. V. (1969), "Simple examples of groups with unsolvable word problem", Akademiya Nauk SSSR. Matematicheskie Zametki, 6: 521–532, ISSN 0025-567X, MR 0260851 • Collins, Donald J. (1969), "Word and conjugacy problems in groups with only a few defining relations", Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 15 (20–22): 305–324, doi:10.1002/malq.19690152001, MR 0263903 • Collins, Donald J. (1972), "On a group embedding theorem of V. V. Borisov", Bulletin of the London Mathematical Society, 4 (2): 145–147, doi:10.1112/blms/4.2.145, ISSN 0024-6093, MR 0314998 • Collins, Donald J. (1986), "A simple presentation of a group with unsolvable word problem", Illinois Journal of Mathematics, 30 (2): 230–234, doi:10.1215/ijm/1256044631, ISSN 0019-2082, MR 0840121 • Collins, Donald J.; Zieschang, H. (1990), Combinatorial group theory and fundamental groups, Springer-Verlag, p. 166, MR 1099152 • Dehn, Max (1911), "Über unendliche diskontinuierliche Gruppen", Mathematische Annalen, 71 (1): 116–144, doi:10.1007/BF01456932, ISSN 0025-5831, MR 1511645, S2CID 123478582 • Dehn, Max (1912), "Transformation der Kurven auf zweiseitigen Flächen", Mathematische Annalen, 72 (3): 413–421, doi:10.1007/BF01456725, ISSN 0025-5831, MR 1511705, S2CID 122988176 • Kuznetsov, A.V. (1958). "Algorithms as operations in algebraic systems". Izvestia Akad. Nauk SSSR Ser Mat. 13 (3): 81. • Miller, C.F. (1991). "Decision problems for groups — survey and reflections". Algorithms and Classification in Combinatorial Group Theory. Mathematical Sciences Research Institute Publications. Vol. 23. Springer. pp. 1–60. doi:10.1007/978-1-4613-9730-4_1. ISBN 978-1-4613-9730-4. • Nyberg-Brodda, Carl-Fredrik (2021), "The word problem for one-relation monoids: a survey", Semigroup Forum, 103 (2): 297–355, arXiv:2105.02853, doi:10.1007/s00233-021-10216-8 • Rotman, Joseph (1994), An introduction to the theory of groups, Springer-Verlag, ISBN 978-0-387-94285-8 • Stillwell, J. (1982). "The word problem and the isomorphism problem for groups". Bulletin of the AMS. 6: 33–56. doi:10.1090/s0273-0979-1982-14963-1.
Wikipedia
Word problem (mathematics) In computational mathematics, a word problem is the problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word problem for groups, but there are many other instances as well. A deep result of computational theory is that answering this question is in many important cases undecidable.[1] Background and motivation In computer algebra one often wishes to encode mathematical expressions using an expression tree. But there are often multiple equivalent expression trees. The question naturally arises of whether there is an algorithm which, given as input two expressions, decides whether they represent the same element. Such an algorithm is called a solution to the word problem. For example, imagine that $x,y,z$ are symbols representing real numbers - then a relevant solution to the word problem would, given the input $(x\cdot y)/z\mathrel {\overset {?}{=}} (x/z)\cdot y$, produce the output EQUAL, and similarly produce NOT_EQUAL from $(x\cdot y)/z\mathrel {\overset {?}{=}} (x/x)\cdot y$. The most direct solution to a word problem takes the form of a normal form theorem and algorithm which maps every element in an equivalence class of expressions to a single encoding known as the normal form - the word problem is then solved by comparing these normal forms via syntactic equality.[1] For example one might decide that $x\cdot y\cdot z^{-1}$ is the normal form of $(x\cdot y)/z$, $(x/z)\cdot y$, and $(y/z)\cdot x$, and devise a transformation system to rewrite those expressions to that form, in the process proving that all equivalent expressions will be rewritten to the same normal form.[2] But not all solutions to the word problem use a normal form theorem - there are algebraic properties which indirectly imply the existence of an algorithm.[1] While the word problem asks whether two terms containing constants are equal, a proper extension of the word problem known as the unification problem asks whether two terms $t_{1},t_{2}$ containing variables have instances that are equal, or in other words whether the equation $t_{1}=t_{2}$ has any solutions. As a common example, $2+3\mathrel {\overset {?}{=}} 8+(-3)$ is a word problem in the integer group ℤ, while $2+x\mathrel {\overset {?}{=}} 8+(-x)$ is a unification problem in the same group; since the former terms happen to be equal in ℤ, the latter problem has the substitution $\{x\mapsto 3\}$ as a solution. History One of the most deeply studied cases of the word problem is in the theory of semigroups and groups. A timeline of papers relevant to the Novikov-Boone theorem is as follows:[3][4] • 1910 (1910): Axel Thue poses a general problem of term rewriting on tree-like structures. He states "A solution of this problem in the most general case may perhaps be connected with unsurmountable difficulties".[5][6] • 1911 (1911): Max Dehn poses the word problem for finitely presented groups.[7] • 1912 (1912): Dehn presents Dehn's algorithm, and proves it solves the word problem for the fundamental groups of closed orientable two-dimensional manifolds of genus greater than or equal to 2.[8] Subsequent authors have greatly extended it to a wide range of group-theoretic decision problems.[9][10][11] • 1914 (1914): Axel Thue poses the word problem for finitely presented semigroups.[12] • 1930 (1930) – 1938 (1938): The Church-Turing thesis emerges, defining formal notions of computability and undecidability.[13] • 1947 (1947): Emil Post and Andrey Markov Jr. independently construct finitely presented semigroups with unsolvable word problem.[14][15] Post's construction is built on Turing machines while Markov's uses Post's normal systems.[3] • 1950 (1950): Alan Turing shows the word problem for cancellation semigroups is unsolvable,[16] by furthering Post’s construction. The proof is difficult to follow but marks a turning point in the word problem for groups.[3]: 342  • 1955 (1955): Pyotr Novikov gives the first published proof that the word problem for groups is unsolvable, using Turing’s cancellation semigroup result.[17][3]: 354  The proof contains a "Principal Lemma" equivalent to Britton's Lemma.[3]: 355  • 1954 (1954) – 1957 (1957): William Boone independently shows the word problem for groups is unsolvable, using Post's semigroup construction.[18][19] • 1957 (1957) – 1958 (1958): John Britton gives another proof that the word problem for groups is unsolvable, based on Turing's cancellation semigroups result and some of Britton's earlier work.[20] An early version of Britton's Lemma appears.[3]: 355  • 1958 (1958) – 1959 (1959): Boone publishes a simplified version of his construction.[21][22] • 1961 (1961): Graham Higman characterises the subgroups of finitely presented groups with Higman's embedding theorem,[23] connecting recursion theory with group theory in an unexpected way and giving a very different proof of the unsolvability of the word problem.[3] • 1961 (1961) – 1963 (1963): Britton presents a greatly simplified version of Boone's 1959 proof that the word problem for groups is unsolvable.[24] It uses a group-theoretic approach, in particular Britton's Lemma. This proof has been used in a graduate course, although more modern and condensed proofs exist.[25] • 1977 (1977): Gennady Makanin proves that the existential theory of equations over free monoids is solvable.[26] The word problem for semi-Thue systems The accessibility problem for string rewriting systems (semi-Thue systems or semigroups) can be stated as follows: Given a semi-Thue system $T:=(\Sigma ,R)$ and two words (strings) $u,v\in \Sigma ^{*}$, can $u$ be transformed into $v$ by applying rules from $R$? Note that the rewriting here is one-way. The word problem is the accessibility problem for symmetric rewrite relations, i.e. Thue systems.[27] The accessibility and word problems are undecidable, i.e. there is no general algorithm for solving this problem.[28] This even holds if we limit the systems to have finite presentations, i.e. a finite set of symbols and a finite set of relations on those symbols.[27] Even the word problem restricted to ground terms is not decidable for certain finitely presented semigroups.[29][30] The word problem for groups Main article: Word problem for groups Given a presentation $\langle S\mid {\mathcal {R}}\rangle $ for a group G, the word problem is the algorithmic problem of deciding, given as input two words in S, whether they represent the same element of G. The word problem is one of three algorithmic problems for groups proposed by Max Dehn in 1911. It was shown by Pyotr Novikov in 1955 that there exists a finitely presented group G such that the word problem for G is undecidable.[31] The word problem in combinatorial calculus and lambda calculus Main article: Combinatory logic § Undecidability of combinatorial calculus One of the earliest proofs that a word problem is undecidable was for combinatory logic: when are two strings of combinators equivalent? Because combinators encode all possible Turing machines, and the equivalence of two Turing machines is undecidable, it follows that the equivalence of two strings of combinators is undecidable. Alonzo Church observed this in 1936.[32] Likewise, one has essentially the same problem in (untyped) lambda calculus: given two distinct lambda expressions, there is no algorithm which can discern whether they are equivalent or not; equivalence is undecidable. For several typed variants of the lambda calculus, equivalence is decidable by comparison of normal forms. The word problem for abstract rewriting systems The word problem for an abstract rewriting system (ARS) is quite succinct: given objects x and y are they equivalent under ${\stackrel {*}{\leftrightarrow }}$?[29] The word problem for an ARS is undecidable in general. However, there is a computable solution for the word problem in the specific case where every object reduces to a unique normal form in a finite number of steps (i.e. the system is convergent): two objects are equivalent under ${\stackrel {*}{\leftrightarrow }}$ if and only if they reduce to the same normal form.[33] The Knuth-Bendix completion algorithm can be used to transform a set of equations into a convergent term rewriting system. The word problem in universal algebra In universal algebra one studies algebraic structures consisting of a generating set A, a collection of operations on A of finite arity, and a finite set of identities that these operations must satisfy. The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras.[1] The word problem on free Heyting algebras is difficult.[34] The only known results are that the free Heyting algebra on one generator is infinite, and that the free complete Heyting algebra on one generator exists (and has one more element than the free Heyting algebra). The word problem for free lattices Example computation of x∧z ~ x∧z∧(x∨y) x∧z∧(x∨y)≤~x∧z by 5. since x∧z≤~x∧z by 1. since x∧z=x∧z x∧z≤~x∧z∧(x∨y) by 7. since x∧z≤~x∧z and x∧z≤~x∨y by 1. since x∧z=x∧z by 6. since x∧z≤~x by 5. since x≤~x by 1. since x=x The word problem on free lattices and more generally free bounded lattices has a decidable solution. Bounded lattices are algebraic structures with the two binary operations ∨ and ∧ and the two constants (nullary operations) 0 and 1. The set of all well-formed expressions that can be formulated using these operations on elements from a given set of generators X will be called W(X). This set of words contains many expressions that turn out to denote equal values in every lattice. For example, if a is some element of X, then a ∨ 1 = 1 and a ∧ 1 = a. The word problem for free bounded lattices is the problem of determining which of these elements of W(X) denote the same element in the free bounded lattice FX, and hence in every bounded lattice. The word problem may be resolved as follows. A relation ≤~ on W(X) may be defined inductively by setting w ≤~ v if and only if one of the following holds: 1.   w = v (this can be restricted to the case where w and v are elements of X), 2.   w = 0, 3.   v = 1, 4.   w = w1 ∨ w2 and both w1 ≤~ v and w2 ≤~ v hold, 5.   w = w1 ∧ w2 and either w1 ≤~ v or w2 ≤~ v holds, 6.   v = v1 ∨ v2 and either w ≤~ v1 or w ≤~ v2 holds, 7.   v = v1 ∧ v2 and both w ≤~ v1 and w ≤~ v2 hold. This defines a preorder ≤~ on W(X), so an equivalence relation can be defined by w ~ v when w ≤~ v and v ≤~ w. One may then show that the partially ordered quotient set W(X)/~ is the free bounded lattice FX.[35][36] The equivalence classes of W(X)/~ are the sets of all words w and v with w ≤~ v and v ≤~ w. Two well-formed words v and w in W(X) denote the same value in every bounded lattice if and only if w ≤~ v and v ≤~ w; the latter conditions can be effectively decided using the above inductive definition. The table shows an example computation to show that the words x∧z and x∧z∧(x∨y) denote the same value in every bounded lattice. The case of lattices that are not bounded is treated similarly, omitting rules 2 and 3 in the above construction of ≤~. Example: A term rewriting system to decide the word problem in the free group Bläsius and Bürckert [37] demonstrate the Knuth–Bendix algorithm on an axiom set for groups. The algorithm yields a confluent and noetherian term rewrite system that transforms every term into a unique normal form.[38] The rewrite rules are numbered incontiguous since some rules became redundant and were deleted during the algorithm run. The equality of two terms follows from the axioms if and only if both terms are transformed into literally the same normal form term. For example, the terms $((a^{-1}\cdot a)\cdot (b\cdot b^{-1}))^{-1}\mathrel {\overset {R2}{\rightsquigarrow }} (1\cdot (b\cdot b^{-1}))^{-1}\mathrel {\overset {R13}{\rightsquigarrow }} (1\cdot 1)^{-1}\mathrel {\overset {R1}{\rightsquigarrow }} 1^{-1}\mathrel {\overset {R8}{\rightsquigarrow }} 1$, and $b\cdot ((a\cdot b)^{-1}\cdot a)\mathrel {\overset {R17}{\rightsquigarrow }} b\cdot ((b^{-1}\cdot a^{-1})\cdot a)\mathrel {\overset {R3}{\rightsquigarrow }} b\cdot (b^{-1}\cdot (a^{-1}\cdot a))\mathrel {\overset {R2}{\rightsquigarrow }} b\cdot (b^{-1}\cdot 1)\mathrel {\overset {R11}{\rightsquigarrow }} b\cdot b^{-1}\mathrel {\overset {R13}{\rightsquigarrow }} 1$ share the same normal form, viz. $1$; therefore both terms are equal in every group. As another example, the term $1\cdot (a\cdot b)$ and $b\cdot (1\cdot a)$ has the normal form $a\cdot b$ and $b\cdot a$, respectively. Since the normal forms are literally different, the original terms cannot be equal in every group. In fact, they are usually different in non-abelian groups. Group axioms used in Knuth–Bendix completion A1$1\cdot x$$=x$ A2$x^{-1}\cdot x$$=1$ A3    $(x\cdot y)\cdot z$$=x\cdot (y\cdot z)$ Term rewrite system obtained from Knuth–Bendix completion R1$1\cdot x$$\rightsquigarrow x$ R2$x^{-1}\cdot x$$\rightsquigarrow 1$ R3$(x\cdot y)\cdot z$$\rightsquigarrow x\cdot (y\cdot z)$ R4$x^{-1}\cdot (x\cdot y)$$\rightsquigarrow y$ R8$1^{-1}$$\rightsquigarrow 1$ R11$x\cdot 1$$\rightsquigarrow x$ R12$(x^{-1})^{-1}$$\rightsquigarrow x$ R13$x\cdot x^{-1}$$\rightsquigarrow 1$ R14$x\cdot (x^{-1}\cdot y)$$\rightsquigarrow y$ R17    $(x\cdot y)^{-1}$$\rightsquigarrow y^{-1}\cdot x^{-1}$ See also • Conjugacy problem • Group isomorphism problem References 1. Evans, Trevor (1978). "Word problems". Bulletin of the American Mathematical Society. 84 (5): 790. doi:10.1090/S0002-9904-1978-14516-9. 2. Cohen, Joel S. (2002). Computer algebra and symbolic computation: elementary algorithms. Natick, Mass.: A K Peters. pp. 90–92. ISBN 1568811586. 3. Miller, Charles F. (2014). Downey, Rod (ed.). "Turing machines to word problems" (PDF). Turing's Legacy: 330. doi:10.1017/CBO9781107338579.010. hdl:11343/51723. ISBN 9781107338579. Retrieved 6 December 2021. 4. Stillwell, John (1982). "The word problem and the isomorphism problem for groups". Bulletin of the American Mathematical Society. 6 (1): 33–56. doi:10.1090/S0273-0979-1982-14963-1. 5. Müller-Stach, Stefan (12 September 2021). "Max Dehn, Axel Thue, and the Undecidable". p. 13. arXiv:1703.09750 [math.HO]. 6. Steinby, Magnus; Thomas, Wolfgang (2000). "Trees and term rewriting in 1910: on a paper by Axel Thue". Bulletin of the European Association for Theoretical Computer Science. 72: 256–269. CiteSeerX 10.1.1.32.8993. MR 1798015. 7. Dehn, Max (1911). "Über unendliche diskontinuierliche Gruppen". Mathematische Annalen. 71 (1): 116–144. doi:10.1007/BF01456932. ISSN 0025-5831. MR 1511645. S2CID 123478582. 8. Dehn, Max (1912). "Transformation der Kurven auf zweiseitigen Flächen". Mathematische Annalen. 72 (3): 413–421. doi:10.1007/BF01456725. ISSN 0025-5831. MR 1511705. S2CID 122988176. 9. Greendlinger, Martin (June 1959). "Dehn's algorithm for the word problem". Communications on Pure and Applied Mathematics. 13 (1): 67–83. doi:10.1002/cpa.3160130108. 10. Lyndon, Roger C. (September 1966). "On Dehn's algorithm". Mathematische Annalen. 166 (3): 208–228. doi:10.1007/BF01361168. hdl:2027.42/46211. S2CID 36469569. 11. Schupp, Paul E. (June 1968). "On Dehn's algorithm and the conjugacy problem". Mathematische Annalen. 178 (2): 119–130. doi:10.1007/BF01350654. S2CID 120429853. 12. Power, James F. (27 August 2013). "Thue's 1914 paper: a translation". arXiv:1308.5858 [cs.FL]. 13. See History of the Church–Turing thesis. The dates are based on On Formally Undecidable Propositions of Principia Mathematica and Related Systems and Systems of Logic Based on Ordinals. 14. Post, Emil L. (March 1947). "Recursive Unsolvability of a problem of Thue" (PDF). Journal of Symbolic Logic. 12 (1): 1–11. doi:10.2307/2267170. JSTOR 2267170. S2CID 30320278. Retrieved 6 December 2021. 15. Mostowski, Andrzej (September 1951). "A. Markov. Névožmoinost' nékotoryh algoritmov v téorii associativnyh sistém (Impossibility of certain algorithms in the theory of associative systems). Doklady Akadémii Nauk SSSR, vol. 77 (1951), pp. 19–20". Journal of Symbolic Logic. 16 (3): 215. doi:10.2307/2266407. JSTOR 2266407. 16. Turing, A. M. (September 1950). "The Word Problem in Semi-Groups With Cancellation". The Annals of Mathematics. 52 (2): 491–505. doi:10.2307/1969481. JSTOR 1969481. 17. Novikov, P. S. (1955). "On the algorithmic unsolvability of the word problem in group theory". Proceedings of the Steklov Institute of Mathematics (in Russian). 44: 1–143. Zbl 0068.01301. 18. Boone, William W. (1954). "Certain Simple, Unsolvable Problems of Group Theory. I". Indagationes Mathematicae (Proceedings). 57: 231–237. doi:10.1016/S1385-7258(54)50033-8. 19. Boone, William W. (1957). "Certain Simple, Unsolvable Problems of Group Theory. VI". Indagationes Mathematicae (Proceedings). 60: 227–232. doi:10.1016/S1385-7258(57)50030-9. 20. Britton, J. L. (October 1958). "The Word Problem for Groups". Proceedings of the London Mathematical Society. s3-8 (4): 493–506. doi:10.1112/plms/s3-8.4.493. 21. Boone, William W. (1958). "The word problem" (PDF). Proceedings of the National Academy of Sciences. 44 (10): 1061–1065. Bibcode:1958PNAS...44.1061B. doi:10.1073/pnas.44.10.1061. PMC 528693. PMID 16590307. Zbl 0086.24701. 22. Boone, William W. (September 1959). "The Word Problem". The Annals of Mathematics. 70 (2): 207–265. doi:10.2307/1970103. JSTOR 1970103. 23. Higman, G. (8 August 1961). "Subgroups of finitely presented groups". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 262 (1311): 455–475. Bibcode:1961RSPSA.262..455H. doi:10.1098/rspa.1961.0132. S2CID 120100270. 24. Britton, John L. (January 1963). "The Word Problem". The Annals of Mathematics. 77 (1): 16–32. doi:10.2307/1970200. JSTOR 1970200. 25. Simpson, Stephen G. (18 May 2005). "A Slick Proof of the Unsolvability of the Word Problem for Finitely Presented Groups" (PDF). Retrieved 6 December 2021. 26. "Subgroups of finitely presented groups". Mathematics of the USSR-Sbornik. 103 (145): 147–236. 13 February 1977. doi:10.1070/SM1977v032n02ABEH002376. 27. Matiyasevich, Yuri; Sénizergues, Géraud (January 2005). "Decision problems for semi-Thue systems with a few rules". Theoretical Computer Science. 330 (1): 145–169. doi:10.1016/j.tcs.2004.09.016. 28. Davis, Martin (1978). "What is a Computation?" (PDF). Mathematics Today Twelve Informal Essays: 257–259. doi:10.1007/978-1-4613-9435-8_10. ISBN 978-1-4613-9437-2. Retrieved 5 December 2021. 29. Baader, Franz; Nipkow, Tobias (5 August 1999). Term Rewriting and All That. Cambridge University Press. pp. 59–60. ISBN 978-0-521-77920-3. • Matiyasevich, Yu. V. (1967). "Простые примеры неразрешимых ассоциативных исчислений" [Simple examples of undecidable associative calculi]. Doklady Akademii Nauk SSSR (in Russian). 173 (6): 1264–1266. ISSN 0869-5652. • Matiyasevich, Yu. V. (1967). "Simple examples of undecidable associative calculi". Soviet Mathematics. 8 (2): 555–557. ISSN 0197-6788. 30. Novikov, P. S. (1955). "On the algorithmic unsolvability of the word problem in group theory". Trudy Mat. Inst. Steklov (in Russian). 44: 1–143. 31. Statman, Rick (2000). "On the Word Problem for Combinators". Rewriting Techniques and Applications. Lecture Notes in Computer Science. 1833: 203–213. doi:10.1007/10721975_14. ISBN 978-3-540-67778-9. 32. Beke, Tibor (May 2011). "Categorification, term rewriting and the Knuth–Bendix procedure". Journal of Pure and Applied Algebra. 215 (5): 730. doi:10.1016/j.jpaa.2010.06.019. 33. Peter T. Johnstone, Stone Spaces, (1982) Cambridge University Press, Cambridge, ISBN 0-521-23893-5. (See chapter 1, paragraph 4.11) 34. Whitman, Philip M. (January 1941). "Free Lattices". The Annals of Mathematics. 42 (1): 325–329. doi:10.2307/1969001. JSTOR 1969001. 35. Whitman, Philip M. (1942). "Free Lattices II". Annals of Mathematics. 43 (1): 104–115. doi:10.2307/1968883. JSTOR 1968883. 36. K. H. Bläsius and H.-J. Bürckert, ed. (1992). Deduktionsssysteme. Oldenbourg. p. 291.; here: p.126, 134 37. Apply rules in any order to a term, as long as possible; the result doesn't depend on the order; it is the term's normal form. Authority control International • FAST National • Germany • 2 • Israel • United States
Wikipedia
Ambigram An ambigram is a calligraphic or typographic design with multiple interpretations as written words.[2] Alternative meanings are often yielded when the design is transformed or the observer moves, but they can also result from a shift in mental perspective.[3] The term was coined by Douglas Hofstadter in 1983–1984.[3][4] Most often, ambigrams appear as visually symmetrical words. When flipped, they remain unchanged, or they mutate to reveal another meaning. "Half-turn" ambigrams undergo a point reflection (180-degree rotational symmetry) and can be read upside down, while mirror ambigrams have axial symmetry and can be read through a reflective surface like a mirror. Many other types of ambigrams exist.[5] Ambigrams can be constructed in various languages and alphabets, and the notion often extends to numbers and other symbols. It is a recent interdisciplinary concept, combining art, literature, mathematics, cognition, and optical illusions. Drawing symmetrical words constitutes also a recreational activity for amateurs. Numerous ambigram logos are famous, and ambigram tattoos have become increasingly popular. There are methods to design an ambigram, a field in which some artists have become specialists. Etymology The word ambigram was coined in 1983 by Douglas Hofstadter, an American scholar of cognitive science best known as the Pulitzer Prize-winning author of the book Gödel, Escher, Bach.[6][2][4] Hofstadter describes ambigrams as "calligraphic designs that manage to squeeze in two different readings."[7] "The essence is imbuing a single written form with ambiguity".[8] An ambigram is a visual pun of a special kind: a calligraphic design having two or more (clear) interpretations as written words. One can voluntarily jump back and forth between the rival readings usually by shifting one's physical point of view (moving the design in some way) but sometimes by simply altering one's perceptual bias towards a design (clicking an internal mental switch, so to speak). Sometimes the readings will say identical things, sometimes they will say different things.[2] — Douglas Hofstadter Hofstadter attributed the origin of the word ambigram to conversations among a small group of friends during 1983–1984.[4] Prior to Hofstadter's terminology, other names were used to refer to ambigrams. Among them, the expressions "vertical palindromes" by Dmitri Borgmann[9] (1965) and Georges Perec,[10][11] "designatures" (1979),[12] "inversions" (1980) by Scott Kim,[13][14] or simply "upside-down words" by John Langdon and Robert Petrick.[14] Ambigram was added to the Oxford English Dictionary in March 2011,[5][15] and to the Merriam-Webster dictionary in September 2020.[3][16] Scrabble included the word in its database in November 2022.[17][18][19] History Many ambigrams can be described as graphic palindromes. The first Sator square palindrome was found in the ruins of Pompeii, meaning it was created before the Eruption of Mount Vesuvius in 79 AD. A sator square using the mirror writing for the representation of the letters S and N was carved in a stone wall in Oppède (France) between the Roman Empire and the Middle Ages,[21] thus producing a work made up of 25 letters and 8 different characters, 3 naturally symmetrical (A, T, O), 3 others decipherable from left to right (R, P, E), and 2 others from right to left (S, N). This engraving is therefore readable in four directions.[22] Although the term is recent, the existence of mirror ambigrams has been attested since at least the first millennium. They are generally palindromes stylized to be visually symmetrical. In ancient Greek, the phrase "ΝΙΨΟΝ ΑΝΟΜΗΜΑΤΑ ΜΗ ΜΟΝΑΝ ΟΨΙΝ" (wash the sins, not only the face), is a palindrome found in several locations, including the site of the church Hagia Sophia in Turkey.[20][23] It is sometimes turned into a mirror ambigram when written in capital letters with the removal of spaces, and the stylization of the letter Ν (Ν). A boustrophedon is a type of bi-directional text, mostly seen in ancient manuscripts and other inscriptions. Every other line of writing is flipped or reversed, with reversed letters. Rather than going left-to-right as in modern European languages, or right-to-left as in Arabic and Hebrew, alternate lines in boustrophedon must be read in opposite directions. Also, the individual characters are reversed, or mirrored. This two-way writing system reveals that modern ambigrams can have quite ancient origins, with an intuitive component in some minds. Mirror writing in Islamic calligraphy flourished during the early modern period, but its origins may stretch as far back as pre-Islamic mirror-image rock inscriptions in the Hejaz.[24] The earliest known non-natural rotational ambigram dates to 1893 by artist Peter Newell.[25] Although better known for his children's books and illustrations for Mark Twain and Lewis Carroll, he published two books of reversible illustrations, in which the picture turns into a different image entirely when flipped upside down. The last page in his book Topsys & Turvys contains the phrase The end, which, when inverted, reads Puzzle. In Topsys & Turvys Number 2 (1902), Newell ended with a variation on the ambigram in which The end changes into Puzzle 2. In March 1904 the Dutch-American comic artist Gustave Verbeek used ambigrams in three consecutive strips of The UpsideDowns of old man Muffaroo and little lady Lovekins.[26] His comics were ambiguous images, made in such a way that one could read the six-panel comic, flip the book and keep reading. From June to September 1908, the British monthly The Strand Magazine published a series of ambigrams by different people in its "Curiosities" column.[27] Of particular interest is the fact that all four of the people submitting ambigrams believed them to be a rare property of particular words. Mitchell T. Lavin, whose "chump" was published in June, wrote, "I think it is in the only word in the English language which has this peculiarity," while Clarence Williams wrote, about his "Bet" ambigram, "Possibly B is the only letter of the alphabet that will produce such an interesting anomaly."[27][28] Characteristics Natural ambigrams In the Latin alphabet, many letters are symmetrical glyphs. The capital letters B, C, D, E, H, I, K, O, and X have a horizontal symmetry axis. This means that all words that can be written using only these letters are natural lake reflection ambigrams. For example, BOOK, CHOICE, or DECIDE. The lowercase letters l, o, s, x and z are rotationally symmetrical, while pairs such as b/q, d/p, m/w, n/u, and in some typefaces h/y and a/e, are rotations of each other. Thus, the words "sos", "pod", "suns", "yeah", "swims", "dollop", or "passed" form natural rotational ambigrams. More generally, a "natural ambigram" is a word that possesses one or more symmetries when written in its natural state, requiring no typographic styling. The words "bud", "bid", or "mom", form natural mirror ambigrams when reflected over a vertical axis, as does "ليبيا", the name of the country Libya in Arabic. The words "HIM", "TOY, "TOOTH" or "MAXIMUM", in all capitals, form natural mirror ambigrams when their letters are stacked vertically and reflected over a vertical axis. The uppercase word "OHIO" can flip a quarter to produce a 90° rotational ambigram when written in serif style (with large "feet" above and below the "I"). Like all strobogrammatic numbers, 69 is a natural rotational ambigram. Patterns in nature are visible regularities of form found in the natural world.[29] Similarly, patterns in ambigrams are regularities found in graphemes. As a consequence to this "natural" property, some shapes appear more or less appropriate to handle for the designer. Ambigram candidates can become "almost natural", when all the letters except maybe one or two are symmetrically cooperative, for example the word "awesome" possesses 5 compatible letters (the central s that flips around itself, and the couples a/e and w/m). Single words or several words A symmetrical ambigram can be called "homogram" (contraction of "homo-ambigram") when it remains unchanged after reflection, and "heterogram" when it transforms.[30][31] In the most common type of ambigram, the two interpretations arise when the image is rotated 180 degrees with respect to each other (in other words, a second reading is obtained from the first by simply rotating the sheet). Single word ambigrams Douglas Hofstadter coined the word "homogram" to define an ambigram with identical letters.[30][31] In this case, the first half of the word turns into the last half.[14] • Ambigram "Wikipedia", drawn by French artist Jean-Claude Pertuzé, 180° rotational symmetry. • "Candy", 180° symmetrical ambigram. • "Cloud", vertical axis mirror ambigram with a cloud occupying negative space in the letter O. • "Doug", hypocorism for Douglas Hofstadter, the "father" of the ambigram concept. Several words A symmetrical ambigram is called "heterogram"[30][31] (contraction of "hetero-ambigram") when it gives another word. Visually, a heterogram ambigram is symmetrical only when both versions of the pairing are shown together. The aesthetical appearance is more difficult to design, when a changing ambigram aims to be revealed in one way only, alternatively or separately, because symmetry generally enhances elegance. Technically, there are twice more combinations of letters involved in a hetero-ambigram than in a homo-ambigram. For example, the 180° rotational ambigram "yeah" contains only two pairs of letters: y/h and e/a, whereas the heterogram "yeah / good" contains four : y/d, e/o, a/o, and h/g. A single word ambigram cannot be hetero-, but a multiple words ambigram can be homo- type if the letters overlapse, like in "upsidedown" written attached, for example. The ambigram saying "upsidedown" one way and "upsidedown" again the other way, means it is a two words homogram. But the ambigram saying "upside" one way and "down" after rotation, means it is a two words heterogram. There is no limitation to the number of words potentially associable, and full ambigram sentences have even been published.[10][14] • "Ambigram / Wikipedia", hetero- type.[32] • "True flag", self-referential flag, horizontal axis mirror hetero- type. • Two words ambigram "Stay Here". • Two words ambigram "Real / Fake" showing alternatively one version of the pair. Types Ambigrams are exercises in graphic design that play with optical illusions, symmetry and visual perception. Some ambigrams feature a relationship between their form and their content. Ambigrams usually fall into one of several categories. 180° rotational ambigrams "Half-turn" ambigrams or point reflection ambigrams, commonly called "upside-down words", are 180° rotational symmetrical calligraphies.[6] We can read them right side up or upside down, or both. Rotation ambigrams are the most common type of ambigrams for good reason. When a word is turned upside down, the top halves of the letters turn into the bottom halves. And because our eyes pay attention primarily to the top halves of letters when we read, that means that you can essentially chop off the top half of a word, turn it upside down, and glue it to itself to make an ambigram. [...][14] — Scott Kim • Rotating ambigram "Say Yes", half-turn type with 8 occurrences of the same pattern. The phrase itself is a phonetic palindrome. • Point reflection ambigram merci. • "Home / Away", 180° rotational hetero-ambigram. • "Lift", half-turn ambigram logo. Mirror ambigrams A mirror ambigram, or reflection ambigram, is a design that can be read when reflected in a mirror vertically, horizontally, or at 45 degrees,[14] giving either the same word or another word or phrase. Vertical axis reflection ambigrams When the reflecting surface is vertical (like a mirror for example), the calligraphic design is a vertical axis mirror ambigram. The "museum" ambigram is almost natural with mirror symmetry, because the first two letters are easily exchanged with the last two, and the lowercase letter e can be transformed into s by a fairly obvious typographical acrobatics.[34] Vertical axis mirror ambigrams find clever applications in mirror writing (or specular writing), that is formed by writing in the direction that is the reverse of the natural way for a given language, such that the result is the mirror image of normal writing: it appears normal when it is reflected in a mirror. For example, the word "ambulance" could be read frontward and backward in a vertical axis reflective ambigram. Following this idea, the French artist Patrice Hamel created a mirror ambigram saying "entrée" (entrance, in French) one way, and "sortie" (exit) the other way, displayed in the giant glass façade of the Gare du Nord in Paris, so that the travelers coming in read entrance, and those leaving read way out.[35] Horizontal axis reflection ambigrams When the reflecting surface is horizontal (like a mirroring lake for example), the calligraphic design is a horizontal axis mirror ambigram. The book Ambigrams Revealed features several creations of this type, like the word "Failure" mirroring in the water of a pond to give "Success", or "Love" changing into "Lust".[14] Figure-ground ambigrams In a figure / ground ambigram, letters fit together so the negative space around and between one word spells another word.[14] In Gestalt psychology, figure–ground perception is known as identifying a figure from the background. For example, black words on a printed paper are seen as the "figure", and the white sheet as the "background". In ambigrams, the typographic space of the background is used as negative space to form new letters and new words. For example, inside a capital H, one can easily insert a lowercase i. The oil painting You & Me (US) by John Langdon (1996) belongs to this category. The word "me" fills the space between the letters of "you".[36] Ambigram tessellations With Escher-like tessellations associated to word patterns, ambigrams can be oriented in three, four, and up to six directions via rotational symmetries of 120°, 90° and 60° respectively,[37] such as those created by French artist Alain Nicolas.[38] Some words can also transform in the negative space, but the multiplication of constraints often has the effect of reducing either the readability or the complexity of the designed words. Ambigram tessellations are sorts of word puzzles, in which geometry set the rules.[38] • Tessellation build with the natural ambigram "Yeah". • 3-directional ambigram "Serie" (series, in French), tessellation using a 120° rotational symmetry. Created from a hexagon. Media related to Ambigram tessellations at Wikimedia Commons. Chain ambigrams A chain ambigram is a design where a word (or sometimes words) are interlinked, forming a repeating chain.[14] Letters are usually overlapped: a word will start partway through another word. Sometimes chain ambigrams are presented in the form of a circle. For example, the chain "...sunsunsunsun..." can flip upside down, but not the word "sun" alone, written horizontally. A chain ambigram can be constituted of one to several elements. A single element ambigram chain is like a snake eating its own tail. A two-elements ambigram chain is like a snake eating the neighbor's tail with the neighbor eating the first snake's, and so on. Scott Kim's "Infinity" works, and that of John Langdon "Chain reaction", are also self-referential, since the first is infinite in the literal sense of the word, and the second, both reversible at 180° and interfering around the letter O, evokes a chain reaction.[14] Spinonyms A spinonym is a type of ambigram in which a word is written using the same glyph repeated in different orientations.[14] WEB is an example of a word that can easily be made into a spinonym. • MBE (Motor Bike Expo) spinonym logo. The same glyph is repeated in three different orientations. • Spinonym "neun 9" (German for nine), five times the same glyph repeated in different orientations. • "Happy new year" spinonym, the same glyph in different orientations shapes the twelve letters of the sentence. Perceptual shift ambigrams Perceptual shift ambigrams, also called "oscillation" ambigrams, are designs with no symmetry but can be read as two different words depending on how the curves of the letters are interpreted.[14] These ambigrams work on the principle of rabbit-duck-style ambiguous images. For example Douglas Hofstadter expresses the dual nature of light as revealed by physics with his perceptual shift ambigram Wave / Particle. 90° rotational ambigrams "Quarter-turn" ambigrams or 90° rotational ambigrams turn clockwise or counterclockwise to express different meanings.[2] For example, the letter U can turn into a C and reciprocally, or the letters M or W into an E.[14] Totem ambigrams The Alabama A&M University has a totem mirror ambigram logo. Words crossing or totem ambigram "Hot dog", vertical axis reflection symmetry. A totem ambigram is an ambigram whose letters are stacked like a totem, most often offering a vertical axis mirror symmetry. This type helps when several letters fit together, but hardly the whole word. For example, in the Maria monogram, the letters M, A and I are individually symmetrical, and the pairing R/A is almost naturally mirroring. When adequately stacked, the 5 letters produce a nice totem ambigram, whereas the whole name "Maria" would not offer the same cooperativeness. The ambigrammist artist John Langdon designed several totemic assemblages, such as the word "METRO" composed of the symmetrical letter M, then section ETR, and below O; or the sentence "THANK YOU", vertical assembly of T, H, A, then of the symmetric NK couple, then finally Y, O, U.[39] Fractal ambigrams In mathematics, a fractal is a geometrical shape that exhibits invariance under scaling. A piece of the whole, if enlarged, has the same geometrical features as the entire object itself. A fractal ambigram is a sort of space-filling ambigrams where the tiled word branches from itself and then shrinks in a self-similar manner, forming a fractal.[40] In general, only a few letters are constrainted in a fractal ambigram. The other letters don't need to look like any other, and thus can be shaped freely. 3-dimensional ambigrams A 3D ambigram is a design where an object is presented that will appear to read several letters or words when viewed from different angles. Such designs can be generated using constructive solid geometry, a technique used in solid modeling, and then physically constructed with the rapid prototyping method. 3-dimensional ambigram sculptures can also be achieved in plastic arts. They are volume ambigrams. The original 1979 edition of Hofstadter's Gödel, Escher, Bach featured two 3-D ambigrams on the cover.[41] Complex ambigrams Complex ambigrams are ambigrams involving more than one symmetry, or satisfying the criteria for several types. For example, a complex ambigram can be both rotational and mirror with a 4-fold dihedral symmetry. Or a spinonym that reads upside down is also a complex ambigram. • The logo Oxo has a 4-fold dihedral symmetry (mirror and 180° rotational ambigram). • The famous DJ Étienne de Crécy has a complex ambigram logo "EDC", mirroring through a horizontal axis, and figure-ground type with a power plug pictogram inserted in the negative space. • 4-fold dihedral symmetrical ambigram (mirror and rotational) "Dig hole, Die". Symbols Other languages Ambigram 곰 / 문 (Bear / Door, in Korean), 180° rotational symmetry. The word "বাংলা" (Bangla or Bengali, in Bengali), half-turn ambigram. Ambigrams exist in many languages. With the Latin alphabet, they generally mix lowercase and uppercase letters. But words can also be symmetrical in other alphabets, like Arabic, Bengali, Cyrillic, Greek, and even in Chinese characters and Japanese kanji. In Korean, 곰 (bear) and 문 (door), 공 (ball) and 운 (luck), or 물 (water) and 롬 (ROM) form a natural rotational ambigram. Some syllables like 응 (yes), 표 (ticket/signage) or 를 (object particle), and words like "허리피라우" (straighten your back) also make full ambigrams. The han character meaning "hundred" is written 百, that makes a natural 90° rotational ambigram when the glyph makes a quarter turn counterclockwise, one sees "100".[42] Media related to Ambigrams by language at Wikimedia Commons. Numbers Although not totally symmetrical, the Sochi 2014 (Olympic games) official logo offers mirror and rotational symmetries, linking the numbers to the letters like an ambigram. Rio 2016 (Olympic games), half-turn rotational ambigram logo containing letters and digits. An ambigram of numbers, or numeral ambigram, contains numerical digits, like 1, 2, 3...[14] In mathematics, a palindromic number (also known as a numeral palindrome) is a number that remains the same when its digits are reversed through a vertical axis (but not necessarily visually). The palindromic numbers containing only 1, 8, and 0, constitute natural numeric ambigrams (visually symmetrical through a mirror). Also, because the glyph 2 is graphically the mirror image of 5, it means numbers like 205 or 85128 are natural numeral mirror ambigrams. Though not palindromic in the mathematical sense, they read frontward and backward like real ambigrams. A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated 180 degrees. The numeral looks the same right-side up and upside down (e.g., 69, 96, 1001).[43][44][45] Some dates are natural numeral ambigrams.[46] In March 1961, artist Norman Mingo created an upside-down cover for Mad magazine featuring an ambigram of the current year. The title says "No matter how you look at it... it's gonna be a Mad year. 1961, the first upside-down year since 1881."[47] Tuesday, 22 February 2022, was a palindrome and ambigram date called "Twosday" because it contained reversible 2 (two).[48][49][50] Ambigrams of numbers receive most attention in the realm of recreational mathematics.[2][51] Ambigrams with numbers sometimes combine letters and numerical digits. Because the number 5 is approximately shaped like the letter S, the number 6 like a lowercase b, the number 9 like the letter g, it is possible to play on these similarities to design ambigrams. A good example is the Sochi 2014 (Olympic games) logo where the four glyphs contained in 2014 are exact symmetries of the four letters S, o, i and h, individually.[52] Other symbols As alphabet letters are glyphs used in the writing systems to express the languages visually, other symbols are also used in the world to code other fields, like the prosigns in the Morse code or the musical notes in music. Similarly to the ambigrams of letters, the ambigrams with other symbols are generally visually symmetrical, either point reflective or reflective through an axis. The international Morse code distress signal SOS   ▄ ▄ ▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄ ▄ ▄  is a natural ambigram constituted of dots and dashes. It flips upside down or through a mirror. In morse code, the letter P coded   ▄ ▄▄▄ ▄▄▄ ▄  and the letter R coded   ▄ ▄▄▄ ▄  are individually symmetrical, like many other letters and numbers. Also, the letter G coded   ▄▄▄ ▄▄▄ ▄  is the exact reverse of the letter W coded   ▄ ▄▄▄ ▄▄▄ . Thus, the combination   ▄▄▄ ▄▄▄ ▄  /   ▄ ▄▄▄ ▄▄▄  coding the pairing G/W constitutes a natural ambigram. Consequently, meaningful natural ambigrams written in morse code certainly exist, like for example the words "wog"   ▄ ▄▄▄ ▄▄▄   ▄▄▄ ▄▄▄ ▄▄▄    ▄▄▄ ▄▄▄ ▄ , "Dou"   ▄▄▄ ▄ ▄    ▄▄▄ ▄▄▄ ▄▄▄    ▄ ▄ ▄▄▄  or "mom"   ▄▄▄ ▄▄▄    ▄▄▄ ▄▄▄ ▄▄▄    ▄▄▄ ▄▄▄ .[6][53][54] In music, the interlude from Alban Berg's opera Lulu is a palindrome, thus the score made up of musical notes is almost symmetrical through a vertical axis.[55] In biology, researchers study the ambigrammatic property of narnaviruses by using visual representations of the symmetrical sequences.[29][1][56] Fields Calligraphy and typography Calligraphic color-reversal ambigram Soul of Laos, published in the book Ambigrams Revealed.[14] Calligraphic design Danke (thanks, in German) and half-turn ambigram. Instead of simply writing them, ambigram lettering covers the art of drawing letters. In ambigram calligraphy, each letter acts as an illustration, each letter is created with attention to detail and has a unique role within a composition. Lettering ambigrams do not translate into combinations of alphabet letters that can be used like a typeface, since they are created with a specific candidate in mind. The calligrapher, graffiti writer and graphic designer Niels Shoe Meulman created several rotational ambigrams like the number "fifty",[14] the names "Shoe / Patta",[14] and the opposition "Love / Fear".[57] The cover of the 7th volume of the typography book Typism is an ambigram drawn by Nikita Prokhorov.[58] The American type designer Mark Simonson designed poetic and humorous ambigrams, such as the words "Revelation", "Typophile", and the symbiosis "Drink / Drunk".[14] The last one makes a visual pun when printed on a shot glass, sold commercially.[59] Logos The rotational logo New Man created by Raymond Loewy in 1968 is a natural ambigram. The online two-sided marketplace for residential cleaning Handy has a 180° rotational ambigram logo. Sun (Microsystems) logo designed by Vaughan Pratt in 1982, chain ambigram, spinonym, 90° and 180° rotational symmetries. Nissin (Foods) ambigram visual identity (half-turn). Since they are visually striking, and sometimes surprising, ambigram words find large application in corporate logos and wordmarks, setting the visual identity of many organizations, trademarks and brands.[60] In 1968[61] or 1969, Raymond Loewy designed the rotational New Man ambigram logo.[62][63][64] The mirror ambigram DeLorean Motor Company logo, designed by Phil Gibbon, was first used in 1975.[65][66][67] Robert Petrick designed the invertible Angel logo[14] in 1976. The logo Sun (Microsystems) designed by professor Vaughan Pratt[68] in 1982 fulfills the criteria of several types: chain ambigram, spinonym, 90° and 180° rotational symmetries. The Swedish pop group ABBA owns a mirror ambigram logo stylized AᗺBA with a reversed B, designed by Rune Söderqvist[69] in 1976.[70] The Ventura logo of the Visitors & Convention Bureau's board, in California, cost US$25,000 and was created in 2014 by the DuPuis group. It uses a 180° rotational symmetry.[71][72] Other famous ambigram logos include: the insurance company Aviva;[73] the acronym CRD (Capital Regional District) in the Canadian province of British Columbia;[74] the American multinational corporation DXC Technology; the two-sided marketplace for residential cleaning Handy;[75][76] the brand name of French premium high-speed train services InOui;[77] the French company specializing in ticketing and passenger information systems IXXI; the century-old brand Maoam of the confectionery manufacturer Haribo;[78] the American industrial rock band NIͶ; the Japanese food company Nissin; the biotechnology company Noxxon Pharma, founded in 1997; the online travel agency Opodo in 2001;[79] the brand of food products OXO[80] born in 1899; the video game Pod; the American developer and manufacturer of audio products Sonos;[81] the American professional basketball team Phoenix Suns;[82][83] the German manufacturer of adhesive products UHU; the quadruple symmetrical logo UA from the American clothing brand Under Armour ; the Canadian corporation mandated to operate intercity passenger rail service VIA in 1978;[84] the American international broadcaster VOA, born in 1942; and the Malaysian mobile virtual network operator XOX. The student edition of the Tesco Clubcard used 180° rotational symmetry.[85] Visual communication Because they are visual puns,[2] ambigrams generally attract attention, and thus can be used in visual communication to broadcast a marketing or political message. In France, a mirror ambigram "Penelope / benevole" legible through a horizontal axis became a meme on the web after its diffusion on Wikimedia Commons.[86] Penelope Fillon, wife of French politician and former Prime Minister of France François Fillon, is suspected of having received wages for a fictitious job. Ironically, her name through the mirror becomes benevole (voluntary in French), suggesting dedication for a free service. Shared tens of thousands of times on the social networks, this humorous ambigram made the buzz via several French,[87] Belgian[88][89] and Swiss[86] medias. Ambigrams are regularly used by communication agencies such as Publicis to engage the reader or the consumer through two-way messages.[90] Thus, in 2021, male first names transformed into female first names are included in a Swiss advertising campaign aimed at raising awareness about gender equality. An intriguing catchphrase typography upside down invites the reader to rotate the magazine, in which the first names "Michael" or "Peter" are transformed into "Nathalie" or "Alice".[91][92] In 2015 iSmart's logo on one of its travel chargers went viral because the brand's name turned out to be a natural ambigram that read "+Jews!" upside down. The company noted that "...we learned a powerful lesson of what not to do when creating a logo." [93] Cinema posters sometimes seduce observers with ambigram titles, such as that of Tenet by Christopher Nolan, by central symmetry.[22] or Anna by Luc Besson around a vertical axis,[94][95] • Ambigram meme "Penelope / benevole" with a political message. • Half-turn traffic sign using a directional arrow symbol to display alternatively "Station / Toilets". • Visual pun "Avoid the plane" to attract attention towards the environmental impact of aviation. • A practical application of mirror ambigrams in a banner reading "Idaplatz fest" front and back (Zurich, 2008). Comics Ambigrams in comics The Upside Downs of Little Lady Lovekins and Old Man Muffaroo by Gustave Verbeek containing ambigram sentences in 1904. Another frame. The American artist and writer Peter Newell published a rotational ambigram in 1893 saying "Puzzle / The end" in the book containing reversible illustrations Topsys & Turvys.[25] In March 1904 the Dutch-American comic artist Gustave Verbeek used ambigrams in three consecutive strips of The UpsideDowns of old man Muffaroo and little lady Lovekins.[26] His comics were ambiguous images, made in such a way that one could read the six-panel comic, flip the book and keep reading. In The Wonderful Cure of the Waterfall (13 March 1904) an Indian medicine man says 'Big waters would make her very sound', while when flipped the medicine man turns into an Indian woman who says 'punos dery, eay apew poom, serlem big'. Which is explained as, 'poor deary' several foreign words that meant that she would call the 'Serlem Big'. The next comic called At the House of the Writing Pig (20 March 1904), where two ambigram word balloons are featured. The first features an angry pig trying to make the main protagonist leave by showing a sign that says; 'big boy go away, dis am home of mr h hog', up side down it reads 'Boy yew go away. We sip. Home of hog pig.' The protagonist asks the pig if it wants a big bun, upon which it replies 'Why big buns? Am mad u!', which flips into 'In pew we sang big hym'. Finally in The Bad Snake and the Good Wizard (1904 Mar 27) there are two more ambigrams. The first turns 'How do you do' into the name of a wizard called 'Opnohop Moy', the second features a squirrel telling the protagonist 'Yes further on' only to inform it that there are 'No serpents here' on his way back. In a 2012 Swedish remake of the book,[96] the artist Marcus Ivarsson redraws The Bad Snake and the Good Wizard in his own style. He removes the squirrel, but keeps the other ambigram. 'How do you do' is replaced by 'Nejnej' (Swedish for no) and the wizard is now called 'Laulau'. Media related to Ambigrams by Gustave Verbeek at Wikimedia Commons. Oubapo, workshop of potential comic book art, is a comics movement which believes in the use of formal constraints to push the boundaries of the medium. Étienne Lécroart, cartoonist, is a founder and key member of Oubapo association, and has composed cartoons that could be read either horizontally, vertically, or in diagonal, and vice versa, sometimes including appropriate ambigrams.[97] Drawings and paintings Ambigram "¡OHO!" published by Rex Whistler in 1946. Ambigram "¡OHO!" with reversible faces by Rex Whistler created before 1944. A young woman transforms into a grandmother. The British painter, designer and illustrator Rex Whistler, published in 1946 a rotational ambigram "¡OHO!" for the cover of a book gathering reversible drawings.[98] The artist John Langdon, specialist of ambigrams,[60] designed many color paintings featuring ambigrams of all kinds, figure-ground, rotational, mirror or totem. Among other influences, he particularly admires M. C. Escher's drawings.[14] The Canadian artist Kelly Klages painted several acrylics on canvas with ambigram words and sentences referring to famous writers' novels written by William Shakespeare or Agatha Christie, such as Third Girl, The Tempest, After the Funeral, The Hollow, Reformation, Sherlock Holmes, and Elephants Can Remember.[99] Sculptures Mia Florentine Weiss "Love Hate" sculpture in Munich, Germany, in 2020. "Now / Won" installation in front of the Reichstag building, Berlin, Germany, 2017. The German conceptual artist Mia Florentine Weiss built a sculptural ambigram Love Hate,[100] that has traveled Europe as a symbol of peace and change of perspective.[101] Depending on which side the viewer looks at it, the sculpture says "Love" or "Hate". A similar concept was installed in front of the Reichstag building in Berlin with the words "Now / Won". Both sculptures are mirror type ambigrams, symmetrical around a vertical axis.[102] The Swiss sculptor Markus Raetz made several three-dimensional ambigram works, featuring words generally with related meanings, such as YES-NO (2003),[103] ME-WE (2004, 2010),[104] OUI-NON (2000–2002) in French,[105][106] SI–NO (1996)[107] and TODO-NADA (1998) in Spanish[108][109] These are anamorphic works, which change in appearance depending on the angle of view of the observer. The OUI–NON ambigram is installed on the Place du Rhône, in Geneva, Switzerland, at the top of a metal pole. Physically, the letters have the appearance of iron twists. With the perspective, this work demonstrates that reality can be ambiguous.[106] Some ambigram sculptures by the French conjurer Francis Tabary are reversible by a half-turn rotation, and can therefore be exhibited on a support in two different ways.[110][111] Palindromes Ambigrams are sorts of visual palindromes.[112] Some words turn upside down, others are symmetrical through a mirror. Natural ambigram palindromes exist, like the words "wow", "malayalam"[113] (Dravidian language), or the biotechnology company Noxxon that possesses a palindromic name associated to a rotational ambigram logo. But some words are natural ambigrams, though not palindromes in the literary acception, like "bud" for example, because b and d are different letters. As a result, some words and sentences are good candidates for ambigrammists, but not for palindromists, and reciprocally, since the constraint slightly differ. Authors of ambigrams also benefit from a certain flexibility by playing on the typeface and graphical adjustments to influence the reading of their visual palindromes. Oulipo, workshop of potential literature, seeks to create works using constrained writing techniques. Georges Perec, French novelist and member of the Oulipo group, designed a rotational ambigram, that he called "vertical palindrome".[10] Sibylline, the sentence "Andin Basnoda a une épouse qui pue" in French means "Andin Basnoda has a smelly wife". Perec did not care about punctuation spaces, but his creation flips easily with a classical font like Arial. Visual palindromes sometimes perfectly illustrate literary contents. The American author Dan Brown incorporated John Langdon's designs into the plot of his bestseller Angels & Demons, and his fictional character Robert Langdon's surname was a homage to the ambigram artist.[114] The fantasy novel Abarat, written and illustrated by Clive Barker, features an ambigram of the title on its cover.[115] Calligrams Reflective calligram hat in Alevism forming a human face with Arabic letters. Oslo Climbing Club official logo[116] "OK" (acronym for Oslo Klatreklubb) 90° rotational ambigram showing a human silhouette vertically. A calligram is text arranged in such a way that it forms a thematically related image. It can be a poem, a phrase, a portion of scripture, or a single word. The visual arrangement can rely on certain use of the typeface, calligraphy or handwriting. The image created by the words illustrates the text by expressing visually what it says, or something closely associated. In Islamic calligraphy, symmetrical calligrams appear in ancient and modern periods, forming mirror ambigrams in Arabic language.[24] The word "OK" turned 90° counterclockwise evokes a human icon, with the letter O forming the head and the letter K the arms and the legs. The Norwegian Climbing Club Oslo Klatreklubb (acronym "OK") borrowed the concept of this natural calligram for their official logo.[116] Semantics As described by Douglas Hofstadter, ambigrams are visual puns having two or more (clear) interpretations as written words.[2] Multilingual ambigrams can be read one way in a language, and another way in a different language or alphabet.[14] Multi-lingual ambigrams can occur in all of the various types of ambigrams, with multi-lingual perceptual shift ambigrams being particularly striking. Like certain anagrams with providential meanings such as "Listen / Silent" or "The eyes / They see", ambigrams also sometimes take on a timely sense, for example "up" becomes the abbreviation "dn", very naturally by rotation of 180°.[117] But on the other hand, it happens that the luck of the letters makes things bad. This is the case with the weird anagram "Santa / Satan", as it is with a rotational ambigram that has gone viral because of the paradoxical and unintentional message it expresses. Spotted in 2015 on a metal medal marketed without bad intention, the text "hope" displays upside down with a fairly obvious reading "Adolf", first name of the Nazi leader situated at the antipodes of optimism. This coincidence photographed by an Internet user was relayed by several media and constitutes an ambiguous image.[118][119] Mathematics Recreational mathematics is carried out for entertainment rather than as a strictly research and application-based professional activity.[51] An ambigram magic square exists, with the sums of the numbers in each row, each column, and both main diagonals the same right side up and upside down (180° rotational design). Numeral ambigrams also associate with alphabet letters. A "dissection" ambigram of "squaring the circle" was achieved in a puzzle where each piece of the word "circle" fits inside a perfect square.[2] Burkard Polster, professor of mathematics in Melbourne[120] conducted researches on ambigrams and published several books dealing with the topic, including Eye Twisters, Ambigrams & Other Visual Puzzles to Amaze and Entertain.[121] In the abstract Mathemagical Ambigrams, Polster performs several ambigrams closely related to his realm, like the words "algebra", "geometry", "math", "maths", or "mathematics".[2] Message written with the digits "07734" upside down. Calculator spelling is an unintended characteristic of the seven-segment display traditionally used by calculators, in which, when read upside-down, the digits resemble letters of the Latin alphabet. Also, palindromic numbers and strobogrammatic numbers sometimes attract attention of mathematician ambigrammists.[44][43] Ambigram tessellations and 3D ambigrams are two types particularly fun for the mathematician in geometry. Word patterns in tessellations can start from 35 different fundamental polygons, such as the rhombus, the isosceles right triangle, or the parallelogram.[37] Word puzzles are used as a source of entertainment, but can additionally serve an educational purpose. The American puzzle designer Scott Kim published several ambigrams in Scientific American in Martin Gardner's "Mathematical Games" column, among them long sentences like "Martin Gardner's celebration of mind" turning into "Physics, patterns and prestidigitation".[14] Duality and analogy In the word "ambigram", the root ambi- means "both" and is a popular prefix in a world of dualities, such as day/night, left/right, birth/death, good/evil.[14] In Wordplay: The Philosophy, Art, and Science of Ambigrams,[60] John Langdon mentions the yin and yang symbol as one of his major influences to create upside down words. Ambigrams are mentioned in Metamagical Themas, an eclectic collection of articles that Douglas Hofstadter wrote for the popular science magazine Scientific American during the early 1980s.[8] Seeking the balance point of analogies is an aesthetic exercise closely related to the aesthetically pleasing activity of doing ambigrams, where shapes must be concocted that are poised exactly at the midpoint between two interpretations. But seeking the balance point is far more than just aesthetic play; it probes the very core of how people perceive abstractions, and it does so without their even knowing it. It is a crucial aspect of Copycat research.[8] — Douglas Hofstadter Cognition and psychology Legibility is an important aspect in successful ambigrams. It concerns the ease with which a reader decodes symbols. If the message is lost or difficult to perceive, an ambigram does not work.[7] Readability is related to perception, or how our brain interprets the forms we see through our eyes.[122] Symmetry in ambigrams generally improves the visual appearance of the calligraphic words.[14] Hermann Rorschach, inventor of the Rorschach Test notices that asymmetric figures are rejected by many subjects. Symmetry supplies part of the necessary artistic composition.[123] For many amateurs, designing ambigrams represents a recreational activity, where serendipity can play a fertile role, when the author makes an unplanned fortunate discovery.[2][27] Magic Ambigram "Magic / Dream", with a handheld pattern giving a reversed shadow. "incredible!" Magical ambigram.[124] In magic, ambigrams work like visual illusions, revealing an unexpected new message from a particular written word.[125] In the first series of the British show Trick or Treat, the show's host and creator Derren Brown uses cards with rotational ambigrams.[126][127] These cards can read either 'Trick' or 'Treat'. Ambiguous images, of which ambigrams are a part, cause ambiguity in different ways. For example, by rotational symmetry, as in the Illusion of The Cook by Giuseppe Arcimboldo (1570);[128] sometimes by a figure-ground ambivalence as in Rubin vase; by perceptual shift as in the rabbit–duck illusion, or through pareidolias; or again, by the representation of impossible objects, such as Necker cube or Penrose triangle. For all these types of images, certain ambigrams exist, and can be combined with visuals of the same type. John Langdon designed a figure-ground ambigram "optical illusion" with the two words "optical" and "illusion", one forming the figure and the other the background. "Optical" is easier to see initially but "illusion" emerges with longer observation.[129] Tattooing Mirror ambigram tattoos on wrists "Love / Eros". Handmade ambigram in tattoo "New York / Rich Man", right side up and upside down. 180° rotational ambigram tattoo "No religion". Ambigram tattoo Texas / Sexy, 180° rotational symmetry. One of the most dynamic sectors that harbors ambigrams is tattooing. Because they possess two ways of reading, ambigram tattoos inked on the skin benefit from a "mind-blowing" effect. On the arm, sleeve tattoos flip upside-down, on the back or jointly on two wrists they are more striking with a mirror symmetry. A large range of scripts and fonts is available. Experienced ambigram artists can create an optical illusion with a complex visual design.[130] In 2015, an ambigram tattoo went viral following an advertising campaign developed by the Publicis group two years earlier. The Samaritans of Singapore organization, active in suicide prevention, has a 180° reversible "SOS" ambigram logo, acronym of its name and homonym of the famous SOS distress signal. In 2013, this center orders advertisements that could be inserted in magazines to make readers aware of the problem of depression among young people, and the communication agency notices the symmetrical aspect of the logo. As a result, it begins to produce several ambigrammatic visuals, staged in photographic contexts, where sentences such as "I'm fine", "I feel fantastic" or "Life is great" turn into "Save me", "I'm falling apart", and "I hate myself". Readers noticing this logo placed at the upper left corner of the page with an upside-down typographical catchphrase rotate the newspaper and visualize the double calligraphed messages, which call out with the SOS.[90][131] These ads are so influential that Bekah Miles, an American student herself coming out of a severe depression, chooses to use the "I'm fine / Save me" ambigram to get a tattoo on her thigh. Posted on Facebook, the two-sided photography immediately appeals to many young people, impressed or sensitive to this difficulty.[132][133] To educate its students, George Fox University in the United States then relays the optical illusion in its official journal, through a video totaling more than three million views[134] and the information is also reproduced in several local media and international organizations, thus helping to popularize this famous two-way tattoo.[135][136] Less fortunate, another teenage girl, aged 16, committed suicide, with her also this ambigram found on a note in her room, "I'm fine / Save me", reversible calligraphy today printed on badges and bracelets, for educational purposes.[137] Clothing and fashion Adidas marketed a line of sneakers called "Bounce", with an ambigram typography printed inside the shoe. Several clothing brands, such as Helly Hansen (HH), Under Armour (UA), or New Man, raise an ambigram logo as their visual identity.[64] Mirror ambigrams are also sometimes placed on T-shirts, towels and hats, while socks are more adapted to rotational ambigrams. The conceptual artist Mia Florentine Weiss marketed T-shirts and other products with her mirror ambigram Love Hate.[138][101] Likewise, the city of Ventura in California sells sweatshirts, caps, jackets, and other fashion accessories printed with its rotational ambigram logo.[139] • Rotational and reflective ambigram "Ideal", printed on a T-shirt. • "Zen Yes" embroidered on a blue T-shirt with a meditation symmetrical pictogram. • Helly Hansen, Norwegian manufacturer and retailer of clothing and sports equipment, has an ambigram logo. Accessories The CD cover of the thirteenth studio album Funeral by American rapper Lil Wayne features a 180° rotational ambigram reading "Funeral / Lil Wayne".[140] The special edition paper sleeve (CD with DVD) of the solo album Chaos and Creation in the Backyard by Paul McCartney features an ambigram of the singer's name.[141] The Grateful Dead have used ambigrams several times, including on their albums Aoxomoxoa and American Beauty. Although the words spelled by most ambigrams are relatively short in length, one DVD cover for The Princess Bride movie creates a rotational ambigram out of two words "Princess Bride," whether viewed right side up or upside down. [142] The cover of the studio album Create/Destroy/Create by rock band Goodnight, Sunrise is an ambigram composition constituted of two invariant words, "create" and "destroy", designed by Polish artist Daniel Dostal.[143] The reversible shot glass containing a changing message "Drink / Drunk", created by the typographer Mark Simonson was manufactured and sold in the market.[59] The concept of reversible sign that some merchants use through their windows to indicate that the store is sometimes "open", sometimes "closed", was inaugurated at the beginning of the 2000s, by a rotational ambigram "Open / Closed" developed by David Holst.[34] Creating ambigrams Different ambigram artists, sometimes called ambigrammists,[8][14] may create distinctive ambigrams from the same words, differing in both style and form. Handmade designs There are no universal guidelines for creating ambigrams, and different ways of approaching problems coexist. A number of books suggest methods for creation, including WordPlay,[60] Eye Twisters,[121] and Ambigrams Revealed,[14] in English. Ambigram generators Computerized methods to automatically create ambigrams have been developed. Most of them function on the simplified principle of mapping a single letter to another single letter. Because of this weakness, most of them can only map a word to itself or to another word that is the same length and do not combine letters. Thus, the generated ambigrams are in general of poor quality when compared to hand made ambigrams. More sophisticated techniques employ databases of thousands of curves to create complex ambigrams. Some ambigram generators are free, while some others require payment. Artists John Langdon and Scott Kim each believed that they had invented ambigrams in the 1970s.[144] Douglas Hofstadter Douglas Hofstadter coined the term.[2] To explain visually the numerous types of possible ambigrams, Hofstadter created many pieces with different constraints and symmetries.[145] Hofstadter has had several exhibitions of his artwork in various university galleries. According to Scott Kim, Hofstadter once created a series of 50 ambigrams on the name of all the states in the US.[14] In 1987 a book of 200 of his ambigrams, together with a long dialogue with his alter ego Egbert G. Gebstadter on ambigrams and creativity, was published in Italy.[4] John Langdon John Langdon is a self-taught artist, graphic designer and painter, who started designing ambigrams in the late 1960s and early 70s. Lettering specialist, Langdon is a professor of typography and corporate identity at Drexel University in Philadelphia.[146] John Langdon produced a mirror image logo "Starship" in 1972[147] or 1975, that was sold to the rock band Jefferson Starship. Langdon's ambigram book Wordplay was published in 1992. It contains about 60 ambigrams. Each design is accompanied by a brief essay that explores the word's definition, its etymology, its relationship to philosophy and science, and its use in everyday life.[60] Ambigrams became more popular as a result of Dan Brown incorporating John Langdon's designs into the plot of his bestseller, Angels & Demons, and the DVD release of the Angels & Demons movie contains a bonus chapter called "This is an Ambigram". Langdon also produced the ambigram that was used for some versions of the book's cover.[144] Brown used the name Robert Langdon for the hero in his novels as an homage to John Langdon.[114][148] Blacksmith Records, the music management company and record label, possesses a rotational ambigram logo[149] designed by John Langdon.[150] Scott Kim Scott Kim is one of the best-known masters of the art of ambigrams.[63] He is an American puzzle designer and artist who published in 1981 a book called Inversions with ambigrams of many types.[13][148] Other artists Nikita Prokhorov is a graphic artist, typographer and professional ambigrammist. His book Ambigrams Revealed showcases ambigram designs of all types, from all around the world.[14][151] Born in 1946, Alain Nicolas is a specialist of figurative and ambigram tessellations. 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Further reading • Hofstadter, Douglas R., "Metafont, Metamathematics, and Metaphysics: Comments on Donald Knuth's Article 'The Concept of a Meta-Font'" Scientific American (August 1982) (republished, with a postscript, as chapter 13 in the book Metamagical Themas ISBN 978-0-553-34683-1) • Hofstadter, Douglas R., Ambigrammi: Un microcosmo ideale per lo studio della creativita (Ambigrams: An Ideal Microworld for the Study of Creativity), Hopefulmonster Editore Firenze (1987) (in Italian) ISBN 978-88-7757-006-2 • Langdon, John, Wordplay: Ambigrams and Reflections on the Art of Ambigrams, Harcourt Brace (1992, republished 2005) ISBN 978-0-15-198454-1 • Polster, Burkard, Eye Twisters: Ambigrams & Other Visual Puzzles to Amaze and Entertain, Constable (2008) ISBN 978-1-4027-5798-3 External links • Ambigrams at Curlie Optical illusions (list) Illusions • Afterimage • Ambigram • Ambiguous image • Ames room • Autostereogram • Barberpole • Bezold • Café wall • Checker shadow • Chubb • Cornsweet • Delboeuf • Ebbinghaus • Ehrenstein • Flash lag • Fraser spiral • Gravity hill • Grid • Hering • Impossible trident • Jastrow • Lilac chaser • Mach bands • McCollough • Müller-Lyer • Necker cube • Oppel-Kundt • Orbison • Penrose stairs • Penrose triangle • Peripheral drift • Poggendorff • Ponzo • Rubin vase • Sander • Schroeder stairs • Shepard tables • Spinning dancer • Ternus • Vertical–horizontal • White's • Wundt • Zöllner Popular culture • Op art • Trompe-l'œil • Spectropia (1864 book) • Ascending and Descending (1960 drawing) • Waterfall (1961 drawing) • The dress (2015 photograph) Related • Accidental viewpoint • Auditory illusions • Tactile illusions • Temporal illusion
Wikipedia
Names of large numbers Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-English-speaking areas, including continental Europe and Spanish-speaking countries in Latin America. These naming procedures are based on taking the number n occurring in 103n+3 (short scale) or 106n (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix -illion. Names of numbers above a trillion are rarely used in practice; such large numbers have practical usage primarily in the scientific domain, where powers of ten are expressed as 10 with a numeric superscript. However, these somewhat rare names are considered acceptable for approximate statements. For example, the statement "There are approximately 7.1 octillion atoms in an adult human body" is understood to be in short scale of the table below (and is only accurate if referring to short scale rather than long scale). Indian English does not use millions, but has its own system of large numbers including lakhs and crores.[1] English also has many words, such as "zillion", used informally to mean large but unspecified amounts; see indefinite and fictitious numbers. Standard dictionary numbers x Name (SS/LS, LS) SS (103x+3) LS (106x, 106x+3) Authorities AHD4[2] CED[3] COD[4] OED2[5] OEDweb[6] RHD2[7] SOED3[8] W3[9] HM[10] 1Million 106106 ✓✓✓✓✓✓✓✓✓ Milliard 109 ✓✓✓✓✓✓ 2Billion 1091012 ✓✓✓✓✓✓✓✓✓ 3Trillion 10121018 ✓✓✓✓✓✓✓✓✓ 4Quadrillion 10151024 ✓✓✓✓✓✓✓✓✓ 5Quintillion 10181030 ✓✓ ✓✓✓✓✓✓ 6Sextillion 10211036 ✓✓ ✓✓✓✓✓✓ 7Septillion 10241042 ✓✓ ✓✓✓✓✓✓ 8Octillion 10271048 ✓✓ ✓✓✓✓✓✓ 9Nonillion 10301054 ✓✓ ✓✓✓✓✓✓ 10Decillion 10331060 ✓✓ ✓✓✓✓✓✓ 11Undecillion 10361066 ✓✓   ✓ ✓✓ 12Duodecillion 10391072 ✓✓   ✓ ✓✓ 13Tredecillion 10421078 ✓✓   ✓ ✓✓ 14Quattuordecillion 10451084 ✓✓   ✓ ✓✓ 15Quindecillion 10481090 ✓✓   ✓ ✓✓ 16Sexdecillion 10511096 ✓✓   ✓ ✓✓ 17Septendecillion 105410102 ✓✓   ✓ ✓✓ 18Octodecillion 105710108 ✓✓   ✓ ✓✓ 19Novemdecillion 106010114 ✓✓   ✓ ✓✓ 20Vigintillion 106310120 ✓✓ ✓✓✓✓✓✓ 100Centillion 1030310600 ✓✓ ✓✓✓  ✓ Usage: • Short scale: US, English Canada, modern British, Australia, and Eastern Europe • Long scale: French Canada, older British, Western & Central Europe Apart from million, the words in this list ending with -illion are all derived by adding prefixes (bi-, tri-, etc., derived from Latin) to the stem -illion.[11] Centillion[12] appears to be the highest name ending in -"illion" that is included in these dictionaries. Trigintillion, often cited as a word in discussions of names of large numbers, is not included in any of them, nor are any of the names that can easily be created by extending the naming pattern (unvigintillion, duovigintillion, duo­quinqua­gint­illion, etc.). Name Value Authorities AHD4CEDCODOED2OEDnewRHD2SOED3W3UM Googol 10100 ✓✓✓✓✓✓✓✓✓ Googolplex10googol (1010100) ✓✓✓✓✓✓✓✓✓ All of the dictionaries included googol and googolplex, generally crediting it to the Kasner and Newman book and to Kasner's nephew (see below). None include any higher names in the googol family (googolduplex, etc.). The Oxford English Dictionary comments that googol and googolplex are "not in formal mathematical use". Usage of names of large numbers Some names of large numbers, such as million, billion, and trillion, have real referents in human experience, and are encountered in many contexts. At times, the names of large numbers have been forced into common usage as a result of hyperinflation. The highest numerical value banknote ever printed was a note for 1 sextillion pengő (1021 or 1 milliard bilpengő as printed) printed in Hungary in 1946. In 2009, Zimbabwe printed a 100 trillion (1014) Zimbabwean dollar note, which at the time of printing was worth about US$30.[13] Names of larger numbers, however, have a tenuous, artificial existence, rarely found outside definitions, lists, and discussions of how large numbers are named. Even well-established names like sextillion are rarely used, since in the context of science, including astronomy, where such large numbers often occur, they are nearly always written using scientific notation. In this notation, powers of ten are expressed as 10 with a numeric superscript, e.g. "The X-ray emission of the radio galaxy is 1.3×1045 joules." When a number such as 1045 needs to be referred to in words, it is simply read out as "ten to the forty-fifth". This is easier to say and less ambiguous than "quattuordecillion", which means something different in the long scale and the short scale. When a number represents a quantity rather than a count, SI prefixes can be used—thus "femtosecond", not "one quadrillionth of a second"—although often powers of ten are used instead of some of the very high and very low prefixes. In some cases, specialized units are used, such as the astronomer's parsec and light year or the particle physicist's barn. Nevertheless, large numbers have an intellectual fascination and are of mathematical interest, and giving them names is one way people try to conceptualize and understand them. One of the earliest examples of this is The Sand Reckoner, in which Archimedes gave a system for naming large numbers. To do this, he called the numbers up to a myriad myriad (108) "first numbers" and called 108 itself the "unit of the second numbers". Multiples of this unit then became the second numbers, up to this unit taken a myriad myriad times, 108·108=1016. This became the "unit of the third numbers", whose multiples were the third numbers, and so on. Archimedes continued naming numbers in this way up to a myriad myriad times the unit of the 108-th numbers, i.e. $(10^{8})^{(10^{8})}=10^{8\cdot 10^{8}},$ and embedded this construction within another copy of itself to produce names for numbers up to $((10^{8})^{(10^{8})})^{(10^{8})}=10^{8\cdot 10^{16}}.$ Archimedes then estimated the number of grains of sand that would be required to fill the known universe, and found that it was no more than "one thousand myriad of the eighth numbers" (1063). Since then, many others have engaged in the pursuit of conceptualizing and naming numbers that have no existence outside the imagination. One motivation for such a pursuit is that attributed to the inventor of the word googol, who was certain that any finite number "had to have a name". Another possible motivation is competition between students in computer programming courses, where a common exercise is that of writing a program to output numbers in the form of English words. Most names proposed for large numbers belong to systematic schemes which are extensible. Thus, many names for large numbers are simply the result of following a naming system to its logical conclusion—or extending it further. Origins of the "standard dictionary numbers" The words bymillion and trimillion were first recorded in 1475 in a manuscript of Jehan Adam. Subsequently, Nicolas Chuquet wrote a book Triparty en la science des nombres which was not published during Chuquet's lifetime. However, most of it was copied by Estienne de La Roche for a portion of his 1520 book, L'arismetique. Chuquet's book contains a passage in which he shows a large number marked off into groups of six digits, with the comment: Ou qui veult le premier point peult signiffier million Le second point byllion Le tiers point tryllion Le quart quadrillion Le cinqe quyllion Le sixe sixlion Le sept.e septyllion Le huyte ottyllion Le neufe nonyllion et ainsi des ault's se plus oultre on vouloit preceder (Or if you prefer the first mark can signify million, the second mark byllion, the third mark tryllion, the fourth quadrillion, the fifth quyillion, the sixth sixlion, the seventh septyllion, the eighth ottyllion, the ninth nonyllion and so on with others as far as you wish to go). Adam and Chuquet used the long scale of powers of a million; that is, Adam's bymillion (Chuquet's byllion) denoted 1012, and Adam's trimillion (Chuquet's tryllion) denoted 1018. The googol family The names googol and googolplex were invented by Edward Kasner's nephew Milton Sirotta and introduced in Kasner and Newman's 1940 book Mathematics and the Imagination[14] in the following passage: The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely 1 with one hundred zeroes after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "googolplex." A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. This is a description of what would happen if one tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance. The googolplex is, then, a specific finite number, equal to 1 with a googol zeros after it. Value Name Authority 10100GoogolKasner and Newman, dictionaries (see above) 10googol = 1010100GoogolplexKasner and Newman, dictionaries (see above) John Horton Conway and Richard K. Guy[15] have suggested that N-plex be used as a name for 10N. This gives rise to the name googolplexplex for 10googolplex = 101010100. Conway and Guy[15] have proposed that N-minex be used as a name for 10−N, giving rise to the name googolminex for the reciprocal of a googolplex, which is written as 10-(10100). None of these names are in wide use. The names googol and googolplex inspired the name of the Internet company Google and its corporate headquarters, the Googleplex, respectively. Extensions of the standard dictionary numbers This section illustrates several systems for naming large numbers, and shows how they can be extended past vigintillion. Traditional British usage assigned new names for each power of one million (the long scale): 1,000,000 = 1 million; 1,000,0002 = 1 billion; 1,000,0003 = 1 trillion; and so on. It was adapted from French usage, and is similar to the system that was documented or invented by Chuquet. Traditional American usage (which was also adapted from French usage but at a later date), Canadian, and modern British usage assign new names for each power of one thousand (the short scale.) Thus, a billion is 1000 × 10002 = 109; a trillion is 1000 × 10003 = 1012; and so forth. Due to its dominance in the financial world (and by the US dollar), this was adopted for official United Nations documents. Traditional French usage has varied; in 1948, France, which had originally popularized the short scale worldwide, reverted to the long scale. The term milliard is unambiguous and always means 109. It is seldom seen in American usage and rarely in British usage, but frequently in continental European usage. The term is sometimes attributed to French mathematician Jacques Peletier du Mans circa 1550 (for this reason, the long scale is also known as the Chuquet-Peletier system), but the Oxford English Dictionary states that the term derives from post-Classical Latin term milliartum, which became milliare and then milliart and finally our modern term. Concerning names ending in -illiard for numbers 106n+3, milliard is certainly in widespread use in languages other than English, but the degree of actual use of the larger terms is questionable. The terms "Milliarde" in German, "miljard" in Dutch, "milyar" in Turkish, and "миллиард," milliard (transliterated) in Russian, are standard usage when discussing financial topics. For additional details, see billion and long and short scale. The naming procedure for large numbers is based on taking the number n occurring in 103n+3 (short scale) or 106n (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix -illion. In this way, numbers up to 103·999+3 = 103000 (short scale) or 106·999 = 105994 (long scale) may be named. The choice of roots and the concatenation procedure is that of the standard dictionary numbers if n is 9 or smaller. For larger n (between 10 and 999), prefixes can be constructed based on a system described by Conway and Guy.[15] Today, sexdecillion and novemdecillion are standard dictionary numbers and, using the same reasoning as Conway and Guy did for the numbers up to nonillion, could probably be used to form acceptable prefixes. The Conway–Guy system for forming prefixes: Units Tens Hundreds 1 Un N Deci NX Centi 2 Duo MS Viginti N Ducenti 3 Tre (*) NS Triginta NS Trecenti 4 Quattuor NS Quadraginta NS Quadringenti 5 Quinqua NS Quinquaginta NS Quingenti 6 Se (*) N Sexaginta N Sescenti 7 Septe (*) N Septuaginta N Septingenti 8 Octo MX Octoginta MX Octingenti 9 Nove (*) Nonaginta Nongenti (*) ^ When preceding a component marked S or X, "tre" changes to "tres" and "se" to "ses" or "sex"; similarly, when preceding a component marked M or N, "septe" and "nove" change to "septem" and "novem" or "septen" and "noven". Since the system of using Latin prefixes will become ambiguous for numbers with exponents of a size which the Romans rarely counted to, like 106,000,258, Conway and Guy co-devised with Allan Wechsler the following set of consistent conventions that permit, in principle, the extension of this system indefinitely to provide English short-scale names for any integer whatsoever.[15] The name of a number 103n+3, where n is greater than or equal to 1000, is formed by concatenating the names of the numbers of the form 103m+3, where m represents each group of comma-separated digits of n, with each but the last "-illion" trimmed to "-illi-", or, in the case of m = 0, either "-nilli-" or "-nillion".[15] For example, 103,000,012, the 1,000,003rd "-illion" number, equals one "millinillitrillion"; 1033,002,010,111, the 11,000,670,036th "-illion" number, equals one "undecillinilli­septua­ginta­ses­centilli­sestrigint­illion"; and 1029,629,629,633, the 9,876,543,210th "-illion" number, equals one "nonillise­septua­ginta­octingentillitres­quadra­ginta­quingentillideciducent­illion".[15] The following table shows number names generated by the system described by Conway and Guy for the short and long scales.[16] Base -illion (short scale) Base -illion (long scale) Value US, Canada and modern British (short scale) Traditional British (long scale) Traditional European (Peletier) (long scale) SI Symbol SI Prefix 1 1 106 Million Million Million M Mega- 2 1 109 Billion Thousand million Milliard G Giga- 3 2 1012 Trillion Billion Billion T Tera- 4 2 1015 Quadrillion Thousand billion Billiard P Peta- 5 3 1018 Quintillion Trillion Trillion E Exa- 6 3 1021 Sextillion Thousand trillion Trilliard Z Zetta- 7 4 1024 Septillion Quadrillion Quadrillion Y Yotta- 8 4 1027 Octillion Thousand quadrillion Quadrilliard R Ronna- 9 5 1030 Nonillion Quintillion Quintillion Q Quetta- 10 5 1033 Decillion Thousand quintillion Quintilliard 11 6 1036 Undecillion Sextillion Sextillion 12 6 1039 Duodecillion Thousand sextillion Sextilliard 13 7 1042 Tredecillion Septillion Septillion 14 7 1045 Quattuordecillion Thousand septillion Septilliard 15 8 1048 Quindecillion Octillion Octillion 16 8 1051 Sedecillion Thousand octillion Octilliard 17 9 1054 Septendecillion Nonillion Nonillion 18 9 1057 Octodecillion Thousand nonillion Nonilliard 19 10 1060 Novendecillion Decillion Decillion 20 10 1063 Vigintillion Thousand decillion Decilliard 21 11 1066 Unvigintillion Undecillion Undecillion 22 11 1069 Duovigintillion Thousand undecillion Undecilliard 23 12 1072 Tresvigintillion Duodecillion Duodecillion 24 12 1075 Quattuor­vigint­illion Thousand duodecillion Duodecilliard 25 13 1078 Quinvigintillion Tredecillion Tredecillion 26 13 1081 Sesvigintillion Thousand tredecillion Tredecilliard 27 14 1084 Septemvigintillion Quattuordecillion Quattuordecillion 28 14 1087 Octovigintillion Thousand quattuordecillion Quattuordecilliard 29 15 1090 Novemvigintillion Quindecillion Quindecillion 30 15 1093 Trigintillion Thousand quindecillion Quindecilliard 31 16 1096 Untrigintillion Sedecillion Sedecillion 32 16 1099 Duotrigintillion Thousand sedecillion Sedecilliard 33 17 10102 Trestrigintillion Septendecillion Septendecillion 34 17 10105 Quattuor­trigint­illion Thousand septendecillion Septendecilliard 35 18 10108 Quintrigintillion Octodecillion Octodecillion 36 18 10111 Sestrigintillion Thousand octodecillion Octodecilliard 37 19 10114 Septentrigintillion Novendecillion Novendecillion 38 19 10117 Octotrigintillion Thousand novendecillion Novendecilliard 39 20 10120 Noventrigintillion Vigintillion Vigintillion 40 20 10123 Quadragintillion Thousand vigintillion Vigintilliard 50 25 10153 Quinquagintillion Thousand quinvigintillion Quinvigintilliard 60 30 10183 Sexagintillion Thousand trigintillion Trigintilliard 70 35 10213 Septuagintillion Thousand quintrigintillion Quintrigintilliard 80 40 10243 Octogintillion Thousand quadragintillion Quadragintilliard 90 45 10273 Nonagintillion Thousand quin­quadra­gint­illion Quin­quadra­gint­illiard 100 50 10303 Centillion Thousand quinquagintillion Quinquagintilliard 101 51 10306 Uncentillion Unquinquagintillion Unquinquagintillion 110 55 10333 Decicentillion Thousand quin­quinqua­gint­illion Quin­quinqua­gint­illiard 111 56 10336 Undecicentillion Ses­quinqua­gint­illion Ses­quinqua­gint­illion 120 60 10363 Viginticentillion Thousand sexagintillion Sexagintilliard 121 61 10366 Unviginticentillion Unsexagintillion Unsexagintillion 130 65 10393 Trigintacentillion Thousand quinsexagintillion Quinsexagintilliard 140 70 10423 Quadra­gintacent­illion Thousand septuagintillion Septuagintilliard 150 75 10453 Quinqua­gintacent­illion Thousand quin­septua­gint­illion Quin­septua­gint­illiard 160 80 10483 Sexagintacentillion Thousand octogintillion Octogintilliard 170 85 10513 Septuagintacentillion Thousand quinoctogintillion Quinoctogintilliard 180 90 10543 Octogintacentillion Thousand nonagintillion Nonagintilliard 190 95 10573 Nonagintacentillion Thousand quinnonagintillion Quinnonagintilliard 200 100 10603 Ducentillion Thousand centillion Centilliard 300 150 10903 Trecentillion Thousand quinqua­gintacent­illion Quinqua­gintacent­illiard 400 200 101203 Quadringentillion Thousand ducentillion Ducentilliard 500 250 101503 Quingentillion Thousand quinqua­gintaducent­illion Quinqua­gintaducent­illiard 600 300 101803 Sescentillion Thousand trecentillion Trecentilliard 700 350 102103 Septingentillion Thousand quinqua­gintatrecent­illion Quinqua­gintatrecent­illiard 800 400 102403 Octingentillion Thousand quadringentillion Quadringentilliard 900 450 102703 Nongentillion Thousand quinqua­ginta­quadringent­illion Quinqua­ginta­quadringent­illiard 1000 500 103003 Millinillion [17] Thousand quingentillion Quingentilliard Value Name Equivalent US, Canadian and modern British (short scale) Traditional British (long scale) Traditional European (Peletier) (long scale) 10100 Googol Ten duotrigintillion Ten thousand sedecillion Ten sedecilliard 1010100 Googolplex [1] Ten trillitrestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centillitrestrigintatre­centilli­trestrigintatre­centillitrestrigintatre­centilliduotrigintatre­centillion [2] Ten thousand milli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentillion [2] Ten milli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexagintasescentilli­sesexaginta­sescentilliard ^[1] Googolplex's short scale name is derived from it equal to ten of the 3,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​333,​332nd "-illion"s (This is the value of n when 10 × 10(3n + 3) = 1010100) ^[2] Googolplex's long scale name (both traditional British and traditional European) is derived from it being equal to ten thousand of the 1,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666,​666th "-illion"s (This is the value of n when 10,000 × 106n = 1010100). Binary prefixes The International System of Quantities (ISQ) defines a series of prefixes denoting integer powers of 1024 between 10241 and 10248.[18] Power Value ISQ symbol ISQ prefix 1 10241 Ki Kibi- 2 10242 Mi Mebi- 3 10243 Gi Gibi- 4 10244 Ti Tebi- 5 10245 Pi Pebi- 6 10246 Ei Exbi- 7 10247 Zi Zebi- 8 10248 Yi Yobi- Other named large numbers used in mathematics, physics and chemistry • Avogadro number • Graham's number • Skewes's number • Steinhaus–Moser notation • TREE(3) • Rayo's number See also • -yllion – Mathematical notation • Asaṃkhyeya – Buddhist name for a large number • Chinese numerals – Words and characters used to denote numbers in Chinese • History of large numbers • Indefinite and fictitious numbers • Indian numbering system – Indic methods of naming large numbers • Knuth's up-arrow notation – Method of notation of very large integers • Law of large numbers – Averages of repeated trials converge to the expected value • List of numbers – Notable numbers • Long and short scale – Two meanings of "billion" and "trillion"Pages displaying short descriptions of redirect targets • Metric prefix – Order of magnitude indicator • Names of small numbers – Usage and derivation of names • Number names – Word or phrase which describes a numerical quantity • Number prefix – Prefix derived from numerals or other numbersPages displaying short descriptions of redirect targets • Orders of magnitude – Scale of numbers with a fixed ratioPages displaying short descriptions of redirect targets • Orders of magnitude (data) – Computer data measurements and scales. • Orders of magnitude (numbers) – Scale of numbers of interest arranged from small to large • Power of 10 – Ten raised to an integer power References 1. Bellos, Alex (2011). Alex's Adventures in Numberland. A&C Black. p. 114. ISBN 978-1-4088-0959-4. 2. The American Heritage Dictionary of the English Language (4th ed.). Houghton Mifflin. 2000. ISBN 0-395-82517-2. 3. "Collins English Dictionary". HarperCollins. 4. "Cambridge Dictionaries Online". Cambridge University Press. 5. The Oxford English Dictionary (2nd ed.). Clarendon Press. 1991. ISBN 0-19-861186-2. 6. "Oxford English Dictionary". Oxford University Press. 7. The Random House Dictionary of the English Language (2nd ed.). Random House. 1987. 8. Brown, Lesley; Little, William (1993). The New Shorter Oxford English Dictionary. Oxford University Press. ISBN 0198612710. 9. Webster, Noah (1981). Webster's Third New International Dictionary of the English Language, Unabridged. Merriam-Webster. ISBN 0877792011. 10. Rowlett, Russ. "How Many? A Dictionary of Units of Measures". Russ Rowlett and the University of North Carolina at Chapel Hill. Archived from the original on 1 March 2000. Retrieved 25 September 2022. 11. Emerson, Oliver Farrar (1894). The History of the English Language. Macmillan and Co. p. 316. 12. "Entry for centillion in dictionary.com". dictionary.com. Retrieved 25 September 2022. 13. "Zimbabwe rolls out Z$100tr note". BBC News. 16 January 2009. Retrieved 25 September 2022. 14. Kasner, Edward; Newman, James (1940). Mathematics and the Imagination. Simon and Schuster. ISBN 0-486-41703-4. 15. Conway, J. H.; Guy, R. K. (1998). The Book of Numbers. Springer Science & Business Media. pp. 15–16. ISBN 0-387-97993-X. 16. Fish. "Conway's illion converter". Retrieved 1 March 2023. 17. Stewart, Ian (2017). Infinity: A Very Short Introduction. Oxford University Press. p. 20. ISBN 978-0-19-875523-4. 18. "IEC 80000-13:2008". International Organization for Standardization. 15 April 2008. Retrieved 25 September 2022. Large numbers Examples in numerical order • Thousand • Ten thousand • Hundred thousand • Million • Ten million • Hundred million • Billion • Trillion • Quadrillion • Quintillion • Sextillion • Septillion • Octillion • Nonillion • Decillion • Eddington number • Googol • Shannon number • Googolplex • Skewes's number • Moser's number • Graham's number • TREE(3) • SSCG(3) • BH(3) • Rayo's number • Transfinite numbers Expression methods Notations • Scientific notation • Knuth's up-arrow notation • Conway chained arrow notation • Steinhaus–Moser notation Operators • Hyperoperation • Tetration • Pentation • Ackermann function • Grzegorczyk hierarchy • Fast-growing hierarchy Related articles (alphabetical order) • Busy beaver • Extended real number line • Indefinite and fictitious numbers • Infinitesimal • Largest known prime number • List of numbers • Long and short scales • Number systems • Number names • Orders of magnitude • Power of two • Power of three • Power of 10 • Sagan Unit • Names • History
Wikipedia
Workshop on Geometric Methods in Physics The Workshop on Geometric Methods in Physics (WGMP) is a conference on mathematical physics focusing on geometric methods in physics . It is organized each year since 1982 in the village of Białowieża, Poland. It is organized by the Chair of Mathematical Physics of Faculty of Mathematics, University of Białystok. Its founder and main organizer is Anatol Odzijewicz.[1] Workshop on Geometric Methods in Physics StatusActive GenreMathematics conference FrequencyAnnual Location(s)Białowieża CountryPoland Years active1982–present Inaugurated1982 (1982) FounderAnatol Odzijewicz Most recent19 June – 25 June 2022 Next eventJune – July 2023 ActivityActive Organised byUniversity of Białystok Websitewgmp.uwb.edu.pl WGMP takes place in its home venue, in the heart of the Białowieża National Park. A number of social events, including campfire, an excursion to the Białowieża forest and a banquet, are usually organized during the week. Notable participants In the past, Workshops were attended by scientists including: Roy Glauber, Francesco Calogero, Ludvig Faddeev, Martin Kruskal, es:Bogdan Mielnik, Emma Previato, Stanisław Lech Woronowicz, Vladimir E. Zakharov, Dmitry Anosov, de:Gérard Emch, George Mackey, fr:Moshé Flato, Daniel Sternheimer, Tudor Ratiu, Simon Gindikin, Boris Fedosov, pl:Iwo Białynicki-Birula, Jędrzej Śniatycki, Askolʹd Perelomov, Alexander Belavin, Yvette Kosmann-Schwarzbach, pl:Krzysztof Maurin, Mikhail Shubin, Kirill Mackenzie.[2] Special sessions Many times special sessions were scheduled within the programme of the Workshop. In the year 2016 there was a session "Integrability and Geometry" financed by National Science Foundation.[3][4] In the year 2017 there was a session dedicated to the memory and scientific achievements of S. Twareque Ali, long time participant and co-organizer of the Workshop. In the year 2018 there was a session dedicated to scientific achievements of prof. Daniel Sternheimer on the occasion of his 80th birthday. In the previous years, there were sessions dedicated to other prominent mathematicians and physicists such as S.L. Woronowicz, G. Emch, B. Mielnik, F. Berezin.[5] School on Geometry and Physics Since 2012 the Workshop is accompanied by a School on Geometry and Physics, which is targeted at young researchers and graduate students. During the School several courses by leading experts in mathematical physics take place. Proceedings Starting at 1992, after the Workshop a volume of proceedings is published. In the recent years it was published in the series Trends in Mathematics by Birkhäuser.[6] In 2005 a commemorative tome Twenty Years of Bialowieza: A Mathematical Anthology. Aspects of Differential Geometric Methods in Physics was published by World Scientific.[7] References 1. Webpage of Faculty of Mathematics, University of Białystok 2. Voronov, Theodore; Ali, Syed Twareque; Goliński, Tomasz (March 2010). "The Białowieża Meetings on Geometric Methods in Physics: Thirty Years of Success and Inspiration" (PDF). European Mathematical Society Newsletter. No. 75. 3. NSF Award Abstract 4. Integrability and Geometry at WGMP 2016. Post-conference materials. 5. Berceanu, Stefan (August 2013). "Berezin la Białowieża XXX – o perspectivă personală" (PDF). Curierul de Fizica. No. 75. 6. List of proceedings volumes, WGMP webpage 7. Ali, Syed Twareque; Emch, Gerard G.; Odzijewicz, Anatol; Sclichenmaier, Martin; Woronowicz, Stanisław Lech, eds. (2005). Twenty Years of Bialowieza: A Mathematical Anthology. Aspects of Differential Geometric Methods in Physics. World Scientific Monograph Series in Mathematics. Vol. 8. World Scientific. doi:10.1142/5744. ISBN 978-981-256-146-6. Further reading • Voronov, Theodore; Ali, Syed Twareque; Goliński, Tomasz (March 2010). "The Białowieża Meetings on Geometric Methods in Physics: Thirty Years of Success and Inspiration" (PDF). European Mathematical Society Newsletter. No. 75. External links • Conference webpage • Workshop on Geometric Methods in Physics on Facebook
Wikipedia
World Education Games The World Education Games[1] is a global online event for all schools and students around the world and is held semi annually during the month of October. It is the expanded format of what was once known as World Maths Day but it now includes World Literacy Day and World Science Day too. It is organized by the 3PLearning and sponsored by Microsoft,[2] UNICEF,[3] 3P Learning[2] and MACQUARIE.[4] The World Maths Day holds the 'Guinness World Record' for the Largest Online Maths Competition in 2010.[5][6] Its Global Ambassador is 'Scott Flansburg' aka the Human Calculator[7]. World Education Games Logo of World Education Games GenreInternational Event for Students around the Globe FrequencyBiennial Location(s)Worldwide Inaugurated2007 Most recentOctober 2015 Next eventMarch 2018 ParticipantsOpen to any student 4-18 years Attendance5,960,862 students from 240 Countries Patron(s)Microsoft UNICEF 3P Learning Macquarie Group Organised by3P Learning Websiteworldeducationgames.com Its inception with the expanded format was in 2012 when 5,960,862 students from 240 countries and territories around the world competed with each other. In 2013, it was held March 5–7. The World Education Games had taken place October 13 through 15, 2015,[8] where over 6 Million students joined worldwide from over 20,000 schools in 159 countries and raised over $100,000 which will help send 33,000 students to school.[9] History The World Education Games is a major free online educational competition-style event, hosted by the global e-learning provider 3P Learning (creators of subscription-based e-learning platforms designed primarily for schools - such as Mathletics, Spellodrome and IntoScience). The World Education Games had its origins purely as a mathematics-based event, then known as World Maths Day in 2007. The event was powered by 3P Learning's flagship online learning resource, Mathletics. In 2011, the event expanded to include a second subject (World Spelling Day, renamed World Literacy Day in 2013), followed a year later by a third subject (World Science Day) and at which point the event took on the fully encompassing World Education Games[10] name and branding. Since 2012 The World Education Games has been collaborating with UNICEF[11] in the framework of a program called "School in a Box"[12] that supports the development of education in regions that are affected by various disasters and poverty. Rules Participation in the games is open to all students from any country and is free. Registration is required and an access to the internet is a must. Students are matched according to their age and grade levels or abilities if such is requested by their teachers. Students play randomly against other students from all over the world. Students answer as many questions as possible during the allotted time for each game. Correct answers get points, wrong answers no points and three wrong answers end the game prematurely. Each student plays and scored for the points accumulated during the first 20 games only. Results are announced after counting all the points and after the organizers had communicated with the parents/ teachers of possible winners and ensured that they had participated in the Games under their own registered accounts. Multiple registrations and/ or playing under someone's else account is a violation punishable by annulling the results of everyone involved. Winners are the students who score the highest points in their grade level in each competition separately and in total. Schools are also awarded for receiving the highest point-average in each grade level as long as at least 10 students had participated from that school. Top 100 participants also get their achievements listed on the Hall of Fame.[13] World Education Games Ambassadors Students who win their regional lead-up events in their countries are hand-picked to become ambassadors.[14] No.AmbassadorCountry 1Alexander Y  United Kingdom 2Alexandra B  Australia 3Amy M  United Kingdom 4Anna A  Russia 5Creedon C  Canada 6Ellie E  United Kingdom 7Emmanuel M  Mexico 8Fatima Y  Pakistan 9Geoffrey M  Canada 10 Gerania R  United States 11Hui Qing L  United Kingdom 12 Imaan C  United Kingdom 13Kalliopi C[15]  Australia 14 Luke W  United States 15Maathangi A  United Arab Emirates 16Meeral N  Pakistan 17 Melina S  United States 18Michael Murray  United States 19Musaab H  Canada 20 Steve Jobs  United Kingdom 21 Peyton H  Canada 22 Remi L  Australia 23 Mariam K[16]  Australia 24 Samuel O[17]  Nigeria 25 Thomas  United Kingdom 26 Tristan G  Australia 27 Ursula H  Australia 28 Vikayra G  South Africa Prizes[18] Platinum Prizes • A glittering award ceremony to celebrate the winners will be held in November 2015 at the Sydney Opera House. • The student with the highest total World Education Games score (in each of the age categories) will be invited to attend the award ceremony to receive their medal. • Winning students will be flown to Sydney, Australia, along with one parent, to attend the ceremony. The trip includes flights, accommodation and a VIP Sydney tour. Trophies • Trophies will be awarded to the top scoring school in each of the three World Education Games events. • Trophies will be awarded across each of the ten year/grade categories. • Trophies will be specially engraved with the details of the winning schools. Medals • Medals will be awarded to the Top Scoring Students in each of the three World Education Games events. • Medals will be awarded across each of the ten year/grade categories. • Students finishing in first, second and third place in each of the age categories will receive a gold, silver or bronze medal. • Medals will be specially engraved with the details of the winning students. The winners of each group are awarded a 'minted gold medal' and the top ten in each group receive 'gold medals'. There are also various other prizes including trophies and certificates. A full list of winners including top ten in each category is available at the official website of World Education Games. The complete list of various prizes and cups over the years can be found in the official website of World Education Games Winners by countries Literacy Maths Science WEG 2015 Pending Pending Pending Pending 2014 No Games No Games No Games No Games 2013[19]  Malaysia  Turkey  Malaysia  Malaysia 2012 [20]  United Kingdom  Australia  Malaysia  United Kingdom Winners by names 4-7 yrs 8-10 yrs 11-13 yrs 14-18 yrs 2015 Pending Pending Pending Pending 2014 No Games No Games No Games No Games 2013[19] Sandali Rajapakse Salcombe Prep School, UK Vihangi Rajapakse Salcombe Prep School, UK Sachin Kumar Mital Canadian International School, Hong Kong & Shoaib hassan Beaconhouse School System, Mandi, Bahauddin, Pakistan Danial bin Muhammad Syafiq Cempaka Schools, CH, Malaysia 2012[20] Sandali Rajapakse Salcombe Prep School, UK Oliver Papillo Balwyn Primary School, Australia Sharan Maiya The Glasgow Academy, UK Malayandi P Cempaka Schools DH, Malaysia Winners of individual events World Literacy Day World Maths Day World Science Day 2015 11-13 yrs: Sydny Lum Shen Li 11-13 yrs: Sydny Lum Shen Li 11-13 yrs: Sydny Lum Shen Li 2014 No Games No Games No Games 2013[19] 4-7 yrs: Sandali Rajapakse, Salcombe Prep School, UK 8-10 yrs: Alastair Gibson, Hexham Middle School, UK 11-13 yrs: Sydny Lum Shen Li 14-18 yrs: Kianna Wan, Team Canada, Canada 4-7yrs : Sandali Rajapakse, Salcombe Prep School, UK 8-10 yrs: Rohith Niranjan, Global Indian International School, Japan 11-13 yrs: Ata Çağın Kolbaşı, Ata College, Izmir, Turkey 14-18 yrs: Husnain Ali Abid, FFC Grammar School, Pakistan 4-7 yrs: Sandali Rajapakse, Salcombe Prep School, UK 8-10 yrs: Chiacia Putri Effendy, Cahaya Harapan Sejahtera, Indonesia 11-13 yrs: Aryan Saju, The British Al Khubairat, UAE 14-18yrs: Danial Bin Muhammad Syafiq, Cempaka Schools CH, Malaysia 2012[20] 4-7 yrs: Sandali Rajapakse, Salcombe Prep School, UK 8-10 yrs: Dylan.C, Linn Primary School, UK 11-13 yrs: Edryna Syfinaz Z A, Cempaka Schools DH, Malaysia 14-18 yrs: Phoebe M, Sha Tin College, Hong Kong 4-7yrs: Yousuf Mohammad, Orbit International School, Saudi Arabia 8-10 yrs: Darshan.S, Indian Public School, India 11-13 yrs: Moosa FerozeTarrar, Beaconhouse School System, Pakistan 14-18 yrs: Kaya Genc, Southport College, Australia 4-7 yrs: Ashwati. N, Christ the Sower School, UK 8-10 yrs: Derek.L, Monterey Ridge Elementary School, USA 11-13 yrs: Sharan Maiya, The Glasgow Academy, UK 14-18yrs: Malayandi P, Cempaka Schools DH, Malaysia 2011 4-7 yrs: Vihangi Rajapakse, Salcombe Prep School, UK 8-10 yrs: Dylan.C, Linn Primary School, UK 11-13 yrs: George.W, Team United Kingdom, UK 14-18 yrs: Phoebe M, Sha Tin College, Hong Kong 4-7yrs: Eric Z, Team Australia, Australia 8-10 yrs: Mason F, Team New Zealand, New Zealand 11-13 yrs: Kaya Genc, The Southport School, Australia 14-18 yrs: David Andersen, Fraser Coast Anglican College, Australia No Games 2010 No Games 5-8yrs: Rohith Niranjan, Team Japan, Japan 9-13 yrs: Kaya Genc, The Southport School, Australia 14-18 yrs: David Andersen, Fraser Coast Anglican College, Australia No Games 2009 No Games 5-8yrs : N.S, The Sikh International School, Thailand 9-13 yrs: Kaya Genc, The Southport School, Australia 14-18 yrs: David Andersen, Fraser Coast Anglican College, Australia No Games 2008 No Games All Ages: Tatiana Devendranath, Haileybury College, Australia No Games 2007 No Games All Ages: Stefan L, Christian Alliance P.C. Lau Memorial International School, Hong Kong No Games See also • Mathletics • 3P Learning References 1. "World Education Games – home". 3P Learning. Retrieved 2015-09-25. 2. Games, The World Education. "The World Education Games Brings Together 8MM Students From 30,000 Schools in 200 Countries and Challenges Kids in Math, Literacy and Science". www.prnewswire.com. Retrieved 2015-09-24. 3. "3P Learning". 4. "3P Learning Pty Ltd (Organizations on EdSurge)". EdSurge. Retrieved 2015-09-24. 5. "The Worlds Largest Online Maths Competition". Archived from the original on March 7, 2012. 6. "Top Ten Facts About Maths". 7. "Fastest human calculator". Guinness World Records. Retrieved 2015-10-17. 8. "World Education Games to Return in 2015". 9. "The World Education Games Brings Together 8MM Students From 30,000 Schools in 200 Countries and Challenges Kids in Math, Literacy and Science". Reuters. 2015-09-21. Archived from the original on 2015-09-30. Retrieved 2015-09-29. 10. Games, The World Education. "The World Education Games Brings Together 8MM Students From 30,000 Schools in 200 Countries and Challenges Kids in Math, Literacy and Science". www.prnewswire.com. Retrieved 2015-09-25. 11. "Support UNICEF". support.unicef.org. Retrieved 2015-10-27. 12. "World Education Games". secured.unicef.org.au. Retrieved 2015-10-27. 13. "World Education Games 2015". worldeducationgames.com. Retrieved 2015-10-26. 14. "Hall of Fame Home". World Maths Day 2018. Retrieved 2018-06-03. 15. Media, Fairfax Regional. "Mandurah student crowned ambassador for games". Retrieved 2015-09-29. 16. "Teen wants to show positive side of Islam". 17. "African Ambassador makes his debut speech!". 3P Learning. Retrieved 2015-09-29. 18. "WEG Trophies and Medals". 3P Learning. Retrieved 2015-09-24. 19. "World Education Games 2013 Winners" (PDF). 20. "The 2012 World Education Games Results Fact Sheet" (PDF). External links • Official website • The World Education Games Blog • Всемирные Образовательные Игры (in Russian).
Wikipedia
World Maths Day World Maths Day (World Math Day in American English) is an online international mathematics competition, powered by Mathletics (a learning platform from 3P Learning, the same organisation behind Reading Eggs and Mathseeds).[1] Smaller elements of the wider Mathletics program effectively power the World Maths Day event. World Maths Day GenreInternational Event Years active16 Inaugurated2007 Founder3P Learning Most recent2022 Next event8 March 2023 Attendance5,960,862 students from 240 Countries Organised by3P Learning Websitewww.worldmathsday.com The first World Maths Day started in 2007.[2] Despite these origins, the phrases "World Maths Day" and "World Math Day" are trademarks, and not to be confused with other competitions such as the International Mathematical Olympiad or days such as Pi Day. In 2010, World Maths Day created a Guinness World Record for the Largest Online Maths Competition.[3][4] The next World Maths Day will take place on the 8th of March 2023. Overview Open to all school-aged students (4 to 18 years old), World Maths Day involves participants playing 20 × 60-second games, with the platform heavily based on "Live Mathletics" found in Mathletics. The contests involve mental maths problems appropriate for each age group, which test the accuracy and speed of the students as they compete against other students across the globe. The simple but innovative idea of combining the aspects of multi-player online gaming with maths problems has contributed to its popularity around the world. There will be 10 Year group divisions for students to compete in from Kindergarten to Year 9 and above. An online Hall of Fame will track points throughout the competition with prizes to be awarded to the top students and schools.[5] The Champions Challenge is a new addition to the 2021 competition. Top Year/Grade 9 and above World Maths Day student come together to compete in a knockout tournament. As part of the challenge, students will have their event live streamed, bringing mathematics and Esports together.[6] History The inaugural World Maths Day was held on March 13, 2007. 287,000 students from 98 countries answered 38,904,275 questions. The student numbers and the participating countries have steadily increased in the following years. In 2009, 1.9 million students took part in World Maths Day. In 2011, World Maths Day sets a Guinness World Record for the Largest Online Maths Competition,[7] with almost 500 million maths questions answered during the event. In 2012, 3P Learning launched the World Education Games. Over 5.9 Million students from 240 Countries and Territories around the world registered to take part, with World Maths Day being the biggest attraction. In 2013, it was held between 5–7 March and the awards were presented at the Sydney Opera House to the Champions. In 2015, there were participants from 150 countries. US, UK and Australia all had over 1 million registrations. The 2019 World Maths Day event was combined with a social media competition, where students around the world were encouraged to dress up in a maths-themed outfit to celebrate maths. Entries included famous mathematicians, an aerial shot of students forming a pi symbol, and human calculators.[8] In 2022, World Maths Day celebrates 15 years in the making. Awards A number of awards are offered to the students who take part and for those who do well in the event. Additionally the champions and the top ten students in the world are awarded gold medals every year. There are also a number of national lead-up events in different regions around the world which are also based on the Mathletics format. Champions The individual gold medal winners through the years are listed below: 2007 Results All Ages[9] 1Stefan L, Christian Alliance P.C.Lau Memorial International School, Hong Kong 2Kelvin H, Taunton School, United Kingdom 3Ana Catarina V, CLIP, Portugal 4Simone C, Newlands Intermediate, New Zealand 5Maoki G, International Christian Academy of Nagoya, Japan 6Shoaib Akram S, Beaconhouse School, Cambridge Branch, Pakistan 7Joshua S, Dulwich College, Shanghai, China 8Yiannis Z, The English School Nicosia, Cyprus 9Nicolae F, Mark Twain International School, Romania 10Ross R, Team Australia, Australia 2008 Results All Ages 1Tatiana D, Haileybury College, Australia 2Rock T, George Heriot's School, UK 3Kaya G, The McDonald College, Australia 4Chris T, The English College, UAE 5TK, Garden International School, Malaysia 6Joshua S, Dulwich College, Shanghai, China 7Abhishek C, International Pioneer School, Thailand 8Crystal L, Team Australia, Australia 9Pratyush G, Delia School of Canada, Hong Kong 10AS, Team Thailand, Thailand 2009 Results 5-8 Years9-13 Years14-18 Years 1N S, Thai Sikh International School, ThailandKaya G, Team Australia, AustraliaDavid A, Fraser Coast Anglican College, Australia 2Dushyant S, International Pioneers School, ThailandDavid M, Aloha College, Marbella, SpainM G, Izmir OzelI Isikkent lisesi, Turkey 3N K, International Pioneers School, ThailandShoaib A, Team Pakistan, PakistanCarlos D, Amman National School, Jordan 4Alexander B, Bechtel Elementary School, USADante M, Canterbury School, SpainThevaa C, ACS Ipoh School, Malaysia 5O C, St Paul's Convent School, Hong KongLarksana Y, Ontario International Institute, CanadaE K, Tevitol High School, Turkey 6Nico A, Bechtel Elementary School, USACaleb L, Australian International School, UAESaptarshi C, Bangladesh International School Riyadh, Saudi Arabia 7Rico C, Clearwater Bay School, Hong KongK H, Dalat International School,Sultan V, Prague British School, Czech 8Eric S, Clearwater Bay School, Hong KongM S, Indian International School, JapanMiguel B, Escola Secundaria Jorge Peixinho, Portugal 9Eda K, Galliard Primary School, UKWasif S, Gordon A Brown Middle School, CanadaTatiana D, Team Australia, Australia 10M N, Bechtel Elementary School, USAH K, British School of Bucharest, RomaniaJ S, Inti University College, Malaysia 2010 Results 5-8 Years9-13 Years14-18 Years 1Vivek R, Our Lady of Lourdes Park Lodge, Northern IrelandKaya G, The Southport School, AustraliaDavid A, Fraser Coast Anglican College, Australia 2Yizhen Y, Team Singapore, SingaporeCaleb L, Australian International School, UAETatiana D, Team Australia, Australia 3Yan Tung Jovanna Y, St Paul's Convent School, Hong KongBrody H, Team Australia, AustraliaArthur T, Team Hong Kong, Hong Kong 4Sik Chee Harriet C, St Paul's Convent School, Hong KongSharan M, Hamilton College, United KingdomKai Yuan Y, Wesley Methodist School, Malaysia 5Tien-erh H, Team Malaysia, MalaysiaSatvik T, St Georges School Cologne, GermanyFrancis L, Cempaka Schools, Malaysia 6Jimin J, Lake Highland Preparatory SchoolDavid M, Aloha College, Marbella, SpainJake C, Sha Tin College, Hong Kong 7Daniel Newton F, Master Brain Academy, UKSyed Ali R, Beaconhouse School, Middle Branch, PakistanByung hee C, Saint Louis School, USA 8Max W, Helen Wilson Public School, CanadaSai M, India International School, JapanLee Y, Cempaka Schools, Malaysia 9Aidan S, West Leeming Primary School, AustraliaHarish S, India International School, JapanEdwin See Jun H, Cempaka Schools, Malaysia 10Sum yin Tracy M, St Paul's Convent School, Hong KongNaunidh S, Thai Sikh International School, ThailandMohammed Shaan R, Slough Grammar School, London, U.K 2011 Results 4-7 Years8-10 Years11-13 Years14-18 Years 1Eric Z, Team Australia, AustraliaMason F, Team New Zealand, New ZealandKaya G, The Southport School, AustraliaDavid A, Fraser Coast Anglican College, Australia 2Vihangi R, Salcombe Prep School, England, U.KSai M, India International School, JapanDavid M, Aloha College, Marbella, SpainTham C, Team Malaysia, Malaysia 3Evan M, Stanley Bay School, New ZealandEdwin V, St Joseph's School, New ZealandSatvik R, St George's The English International, Cologne, GermanyTiger Z, Team Malaysia, Malaysia 4Andrey M, Laude San Pedro International College, SpainSachin Kumar M, Canadian International School, Hong KongHarish S, Team India, IndiaYeoh K, Team Malaysia, Malaysia 5Aditya C, Team United States, USAMuhammad Abdul Mannan, Thorncliffe PS, CanadaMoosa Feroze T, BeaconHouse School System, PakistanEdwin See Jun H, Cempaka Schools, Malaysia 6Zahid B, Team Pakistan, PakistanWillem E, Remarkables Primary School, New ZealandSharan M, The Glasgow Academy, UK Lim C, Cempaka Schools, Malaysia 7Michael Z, Holy Family Primary School, AustraliaGordon C, German Swiss International School, Hong KongAhsan A, Beaconhouse School Mandi Bahauddin, PakistanAaron T, Sha Tin College, Hong Kong 8Jovanka Vienna S, Bunda Mulia International School, IndonesiaThomas P, Goulburn Street Primary, AustraliaChong Seng K, Independent SchoolSiddharth P, Team United States, USA 9Alldon Garren Tan T, Rosyth School, SingaporeMax W, Team Canada, CanadaAaron H, Team Australia, AustraliaMalayandi P, Cempaka Schools, Malaysia 10Ali S, Team Pakistan, PakistanDaniel Newton F, Master Brain Academy, UKAnna S, British International School of Ljubljana, SloveniaKaan Aykurt, Team Turkey, Turkey 2012 Results 4-7 Years8-10 Years11-13 Years14-18 Years 1Yousuf M, Team Saudi Arabia, Saudi ArabiaDarshan S, The Indian Public School, IndiaMoosa Feroze T, Beaconhouse School System, PakistanKaya G, Team Australia, Australia 2Sandali R, Salcombe Prep School, England, U.KOliver P, Balwyn Primary School, AustraliaHusnain Ali Abid, FFC Grammar H/S School, PakistanOsama Shahid, Beaconhouse School Gujranwala, Pakistan 3Daksh C, Team India, IndiaThomas P, Goulburn Street Primary, AustraliaKarl H, Team Australia, AustraliaZhe W, Team United States, USA 4Joy J, Team India, IndiaRohith N, Team Japan, JapanDavid M, Aloha College, Marbella, SpainMohammed Shaan R, Slough Grammar School, London, UK 5Austin M, AHES School, USARishabh K, Team India, IndiaJoseph T, Team Australia, AustraliaAisya A, Cempaka Schools DH, Malaysia 6Douglas G, Pitt Island School, New ZealandDaniel Newton F, Master Brain Academy, UKSachin Kumar M, Canadian International School, Hong KongFrancis L, Cempaka Schools DH, Malaysia 7Joyel G, Ghyllside Primary School, UKEric Z, Palmerston District Primary School, AustraliaAaron H, Team Australia, AustraliaAngela M, Team Macedonia, Macedonia 8Alison K, Canberra Grammar School, AustraliaLeon H, Team Australia, AustraliaEdwin V, Team New Zealand, New ZealandAaron T, Sha Tin College, Hong Kong 9Archie G, Pitt Island School, New ZealandVihangi R, Salcombe Prep School, England, U.KSharan M, The Glasgow Academy, UKEdwin See Jun H, Cempaka Schools DH, Malaysia 10Leow Z, Sri Tenby School, MalaysiaKangan M, Australian International SchoolFilip Szary, Team England, UKBelinda C, Green Bay High School, New Zealand 2013 Results 4-7 Years[10] 8-10 Years[10]11-13 Years[10]14-18 Years[10] 1Sandali R, Salcombe Prep School, England, U.KRohith N, Global Indian International School, JapanKedar H, Liverpool Public School, AustraliaHusnain Ali A, FFC Grammar School, Pakistan 2Becky L, Undercliffe Public School, AustraliaVihangi R, Salcombe Prep School, England, UKMeet S, SN Kansagra, IndiaAaron H, Team Seaford, Australia 3Beykent Doga T, Beykent Doga College, TurkeyRohidh M, Riyadh, Saudi ArabiaMuhammad Abdul Mannan, Toronto, Canada Edwin See Jun H, Team Malaysia, Malaysia 4Martin E, Izmir Ata Koleji, TurkeyMartin E, Izmir Ata Koleji, TurkeyChoong M, Cempaka School CH, Malaysia Low C, Cempaka School CH, Malaysia 5Tuna Y, Izmir Ata Koleji, TurkeyYousuef M, Orbit International School (Khobar), Saudi ArabiaSachin Kumar M, Canadian International School, Hong KongHussain A, Bloomfield Hall Upper School, Pakistan 6Tapkac D, Izmir Ata Koleji, TurkeyAditya C, New Albany Elementary, Albany, USAFilip S, Team England, England Panayioti K, St Spyridon College, Australia 7Abeeha Saud K, Beaconhouse School, Mandi B, PakistanUgo Dos R, American International School of Bucharest, Romania Hassan Feroze T, Beaconhouse School Mandi B, PakistanHassan Ali A, FFC Grammar School, Pakistan 8Usman A, Millennium School Mirpur, Pakistan & Tuna C, Izmir Ata Koleji I, TurkeyDaniyal N, Australian International School, UAEKarl H, Riverina Anglican College, Australia & Willem E, Remarkables Primary School, New ZealandMohammed Shaan R, Team England, England, U.K 9Evan Manning, Team New Zealand, New ZealandApoorv Agrawal, Anubhuti School, India Danial B, Cempaka School CH, Malaysia 10Bahar F, Izmir Ata Koleji, TurkeyChloe Isabella Tsang, Chinese International School, Hong KongBenjamin H, McCarthy Catholic College, AustraliaYi Shuen L, Cempaka Schools DH, Malaysia 2015 Event The 2015 event was held on October 13-October 15, 2015. There were 10 ages categories: 1 each for grades K-8, and one for grades 9+. The game limit was dropped to 20 games per student. It is possible to play further, but these do not count to ones personal total, only the event total. 169 Million points were scored across Maths, Literacy and Science. Grade KGrade 1Grade 2Grade 3Grade 4Grade 5Grade 6Grade 7Grade 8Grade 9+ 1Jinansh D, Genius Kid, IndiaSarah R, Team PakistanAhmed Feroze T, Team PakistanMuhammad S, Beaconhouse School System, Mandi Bahauddin, PakistanAbeeha S, Beaconhouse School System, Mandi Bahauddin, PakistanAustin M, Steuart Weller, USABonnie L, Undercliffe Public School, AustraliaHashir Feroze T, Team PakistanDara H, Team AustraliaAli Saud K, Beaconhouse School System, Mandi Bahauddin, Pakistan 2Vilaxi S Geniud Kid, IndiaYashdeep S Genius Kid, IndiaBilge Kaan S Takev Karsiyaka, TurkeyEmmanuel A Ladybird Nursery/Primary School, NigeriaKainat F Beaconhouse School System, Mandi Bahauddin, PakistanFilbert Ephraim WMGC New Life Christian AcademySydny L Cempaka DamansaraIman F Team Pakistan, PakistanJayden L Cempaka DamansaraBenjamin H Team Australia Official National Mathletics Challenges leading up to World Maths Day Throughout the year Mathletics host several National Mathletics challenges in the lead up to World Maths Day. These challenges and the winners list are as follows: 2010 Results The American Math Challenge :Winner- Alek K, Haddonfields schools, Null. The Australian Maths Challenge :Winner- Parker C, Home Education, Queensland The Canadian Math Challenge :Winner- Shekar S, North Kipling Junior Middle School, ON. The European Schools Maths Challenge:Winner- Anna S, British International School of Ljubljana, Slovenia. The Middle East Schools Maths Challenge:Winner- Zakria Y, Australian International School, UAE. The NZ Maths Challenge :Winner- Vlad B, St Mary's School, Christchurch. The South African Maths Challenge :Winner- Jaden D, Wilton House, GT. The UK Four Nations Maths Challenge :Winner- Sharan Maiya, Glasgow Academy, Scotland. 2011 Results The American Math Challenge :Winner- Sayan Das, Team USA, Minnesota. The Australian Maths Challenge :Winner- Tatiana Devendranath, Team Australia, VIC. The Canadian Math Challenge :Winner- Tom.L, MPS, Etobicoke. The European Schools Maths Challenge:Winner- The Middle East Schools Maths Challenge:Winner- . The NZ Maths Challenge :Winner- Thomas Graydon, Pitt Island School. The Pakistan Maths Challenge: Winner- Dilsher A, The International School of Choueifat. The South African Maths Challenge :Winner- Jaden D, Team ZAF. The UK Four Nations Maths Challenge :Winner- Sharan Maiya, Glasgow Academy, Scotland, United Kingdom. 2012 Results The American Math Challenge :Winner- Zhe W, Team USA, Massachusetts The Latin American Math Challenge :Winner- Adriana Donis, Colegio Internacional Montessori, Guatemala The Australian Maths Challenge :Winner- Aaron Herrmann,, Seaford 6-12 School, South Australia The Canadian Math Challenge :Winner- Hanting C, Maywood Community School, Canada The European Schools Maths Challenge:Winner- Filip Szary, Team Poland The Middle East Schools Maths Challenge:Winner- Pushp raj P, MES Indian School, Qatar The NZ Maths Challenge :Winner- Willem Ebbinge, Remarkables Primary School, Otago The Pakistan Maths Challenge: Winner- Husnain Ali Abid, FFC Grammar H/S School, Punjab The South African Maths Challenge :Winner- Bradley P, Merrifield College, Eastern Cape The UK Four Nations Maths Challenge :Winner- Ryan Conlan, Team GBR, Scotland 2014 results The Nigerian Maths Challenge Winner Ayomide Adebanjo, Xplanter Private School, Lagos References 1. "Learn About 3P Learning - 15 Years of Edtech Excellence". 3P Learning. Retrieved 2023-01-03. 2. "Mathletics Announce World Maths Day 2018". influencing.com. Retrieved 2022-03-03. 3. "The Largest Online Maths Competition". Archived from the original on March 7, 2012. Retrieved March 22, 2012. 4. "Top ten facts about maths". Express. March 6, 2013. Retrieved January 16, 2014. 5. mathletics_admin (2021-04-22). "[Press Release] Mathletes get ready… World Maths Day is back! - Mathletics Japan" (in Japanese). Retrieved 2022-03-03. 6. Learning, 3P. "Mathletes get ready - World Maths Day is back". www.prnewswire.com (Press release). Retrieved 2022-03-03. 7. "Largest maths competition". Guinness World Records. 3 March 2010. Retrieved 2021-11-24. 8. "World Maths Day 2020". www.woodsprimaryschool.com. Retrieved 2022-03-03. 9. "History-2007". Archived from the original on 2013-04-08. Retrieved 2013-04-07. 10. "World Education Games 2013 Winners" (PDF). External links • Official website
Wikipedia
Ranking A ranking is a relationship between a set of items such that, for any two items, the first is either "ranked higher than", "ranked lower than", or "ranked equal to" the second.[1] In mathematics, this is known as a weak order or total preorder of objects. It is not necessarily a total order of objects because two different objects can have the same ranking. The rankings themselves are totally ordered. For example, materials are totally preordered by hardness, while degrees of hardness are totally ordered. If two items are the same in rank it is considered a tie. By reducing detailed measures to a sequence of ordinal numbers, rankings make it possible to evaluate complex information according to certain criteria.[2] Thus, for example, an Internet search engine may rank the pages it finds according to an estimation of their relevance, making it possible for the user quickly to select the pages they are likely to want to see. Analysis of data obtained by ranking commonly requires non-parametric statistics. Strategies for handling ties It is not always possible to assign rankings uniquely. For example, in a race or competition two (or more) entrants might tie for a place in the ranking.[3] When computing an ordinal measurement, two (or more) of the quantities being ranked might measure equal. In these cases, one of the strategies below for assigning the rankings may be adopted. A common shorthand way to distinguish these ranking strategies is by the ranking numbers that would be produced for four items, with the first item ranked ahead of the second and third (which compare equal) which are both ranked ahead of the fourth.[4] These names are also shown below. Standard competition ranking ("1224" ranking) In competition ranking, items that compare equal receive the same ranking number, and then a gap is left in the ranking numbers. The number of ranking numbers that are left out in this gap is one less than the number of items that compared equal. Equivalently, each item's ranking number is 1 plus the number of items ranked above it. This ranking strategy is frequently adopted for competitions, as it means that if two (or more) competitors tie for a position in the ranking, the position of all those ranked below them is unaffected (i.e., a competitor only comes second if exactly one person scores better than them, third if exactly two people score better than them, fourth if exactly three people score better than them, etc.). Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 2 ("joint second"), C also gets ranking number 2 ("joint second") and D gets ranking number 4 ("fourth"). This method is called "Low" by IBM SPSS[5] and "min" by the R programming language[6] in their methods to handle ties. Modified competition ranking ("1334" ranking) Sometimes, competition ranking is done by leaving the gaps in the ranking numbers before the sets of equal-ranking items (rather than after them as in standard competition ranking). The number of ranking numbers that are left out in this gap remains one less than the number of items that compared equal. Equivalently, each item's ranking number is equal to the number of items ranked equal to it or above it. This ranking ensures that a competitor only comes second if they score higher than all but one of their opponents, third if they score higher than all but two of their opponents, etc. Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 3 ("joint third"), C also gets ranking number 3 ("joint third") and D gets ranking number 4 ("fourth"). In this case, nobody would get ranking number 2 ("second") and that would be left as a gap. This method is called "High" by IBM SPSS[5] and "max" by the R programming language[6] in their methods to handle ties. Dense ranking ("1223" ranking) In dense ranking, items that compare equally receive the same ranking number, and the next items receive the immediately following ranking number. Equivalently, each item's ranking number is 1 plus the number of items ranked above it that are distinct with respect to the ranking order. Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 2 ("joint second"), C also gets ranking number 2 ("joint second") and D gets ranking number 3 ("Third"). This method is called "Sequential" by IBM SPSS[5] and "dense" by the R programming language[7] in their methods to handle ties. Ordinal ranking ("1234" ranking) In ordinal ranking, all items receive distinct ordinal numbers, including items that compare equal. The assignment of distinct ordinal numbers to items that compare equal can be done at random, or arbitrarily, but it is generally preferable to use a system that is arbitrary but consistent, as this gives stable results if the ranking is done multiple times. An example of an arbitrary but consistent system would be to incorporate other attributes into the ranking order (such as alphabetical ordering of the competitor's name) to ensure that no two items exactly match. With this strategy, if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first") and D gets ranking number 4 ("fourth"), and either B gets ranking number 2 ("second") and C gets ranking number 3 ("third") or C gets ranking number 2 ("second") and B gets ranking number 3 ("third"). In computer data processing, ordinal ranking is also referred to as "row numbering". This method corresponds to the "first", "last", and "random" methods in the R programming language[6] to handle ties. Fractional ranking ("1 2.5 2.5 4" ranking) Items that compare equal receive the same ranking number, which is the mean of what they would have under ordinal rankings; equivalently, the ranking number of 1 plus the number of items ranked above it plus half the number of items equal to it. This strategy has the property that the sum of the ranking numbers is the same as under ordinal ranking. For this reason, it is used in computing Borda counts and in statistical tests (see below). Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B and C each get ranking number 2.5 (average of "joint second/third") and D gets ranking number 4 ("fourth"). Here is an example: Suppose you have the data set 1.0, 1.0, 2.0, 3.0, 3.0, 4.0, 5.0, 5.0, 5.0. The ordinal ranks are 1, 2, 3, 4, 5, 6, 7, 8, 9. For v = 1.0, the fractional rank is the average of the ordinal ranks: (1 + 2) / 2 = 1.5. In a similar manner, for v = 5.0, the fractional rank is (7 + 8 + 9) / 3 = 8.0. Thus the fractional ranks are: 1.5, 1.5, 3.0, 4.5, 4.5, 6.0, 8.0, 8.0, 8.0 This method is called "Mean" by IBM SPSS[5] and "average" by the R programming language[6] in their methods to handle ties. Statistics This section is an excerpt from Ranking (statistics).[edit] In statistics, ranking is the data transformation in which numerical or ordinal values are replaced by their rank when the data are sorted. For example, the numerical data 3.4, 5.1, 2.6, 7.3 are observed, the ranks of these data items would be 2, 3, 1 and 4 respectively. For example, the ordinal data hot, cold, warm would be replaced by 3, 1, 2. In these examples, the ranks are assigned to values in ascending order. (In some other cases, descending ranks are used.) Ranks are related to the indexed list of order statistics, which consists of the original dataset rearranged into ascending order. Sports This section is an excerpt from Standings (sports).[edit] In sports, standings, rankings, or league tables group teams of a particular league, conference, or division in a chart based on how well each is doing in a particular season of a sports league or competition. These lists are generally published in newspapers and other media, as well as the official web sites of the sports leagues and competitions. Education League tables are used to compare the academic achievements of different institutions. College and university rankings order institutions in higher education by combinations of factors. In addition to entire institutions, specific programs, departments, and schools are ranked. These rankings usually are conducted by magazines, newspapers, governments and academics. For example, league tables of British universities are published annually by The Guardian, The Independent, The Sunday Times, and The Times. The primary aim of these rankings is to inform potential applicants about British universities based on a range of criteria. Similarly, in countries like India, league tables are being developed and a popular magazine, Education World, published them based on data from TheLearningPoint.net. It is complained that the ranking of England's schools to rigid guidelines that fail to take into account wider social conditions actually makes failing schools even worse. This is because the most involved parents will then avoid such schools, leaving only the children of non-ambitious parents to attend.[8] Business In business, league tables list the leaders in investment banking activity, enabling people to quickly analyze financial data. Companies which collect this kind of data include Dealogic, whose league tables are rankings of investment banks in terms of the dollar volume of deals that investment banks work on; Bloomberg L.P., whose league tables provide an overview of top underwriters and legal advisers to securities deals, as well as fees netted from these transactions; and Thomson Reuters, whose league tables list the top financiers in a particular industry. Applications The rank methodology based on some specific indices is one of the most common systems used by policy makers and international organizations in order to assess the socio-economic context of the countries. Some notable examples include the Human Development Index (United Nations), Doing Business Index (World Bank), Corruption Perceptions Index (Transparency International), and Index of Economic Freedom (the Heritage Foundation). For instance, the Doing Business Indicator of the World Bank measures business regulations and their enforcement in 190 countries. Countries are ranked according to ten indicators that are synthesized to produce the final rank. Each indicator is composed of sub-indicators; for instance, the Registering Property Indicator is composed of four sub-indicators measuring time, procedures, costs, and quality of the land registration system. These kinds of ranks are based on subjective criteria for assigning the score. Sometimes, the adopted parameters may produce discrepancies with the empirical observations, therefore potential biases and paradox may emerge from the application of these criteria.[9] Other examples • In politics, rankings focus on the comparison of economic, social, environmental and governance performance of countries. • In relation to credit standing, the ranking of a security refers to where that particular security would stand in a wind up of the issuing company, i.e., its seniority in the company's capital structure. For instance, capital notes are subordinated securities; they would rank behind senior debt in a wind up. In other words, the holders of senior debt would be paid out before subordinated debt holders received any funds. • Search engines rank web pages by their expected relevance to a user's query using a combination of query-dependent and query-independent methods. Query-independent methods attempt to measure the estimated importance of a page, independent of any consideration of how well it matches the specific query. Query-independent ranking is usually based on link analysis; examples include the HITS algorithm, PageRank and TrustRank. Query-dependent methods attempt to measure the degree to which a page matches a specific query, independent of the importance of the page. Query-dependent ranking is usually based on heuristics that consider the number and locations of matches of the various query words on the page itself, in the URL or in any anchor text referring to the page. • In webometrics, it is possible to rank institutions according to their presence in the web (number of webpages) and the impact of these contents, such as the Webometrics Ranking of World Universities. • In video gaming, players may be given a ranking. To "rank up" is to achieve a higher ranking relative to other players, especially with strategies that do not depend on the player's skill. • The TrueSkill ranking system is a skill based ranking system for Xbox Live developed at Microsoft Research. • A bibliogram ranks common noun phrases in a piece of text. • In language, the status of an item (usually through what is known as "downranking" or "rank-shifting") in relation to the uppermost rank in a clause; for example, in the sentence "I want to eat the cake you made today", "eat" is on the uppermost rank, but "made" is downranked as part of the nominal group "the cake you made today"; this nominal group behaves as though it were a single noun (i.e., I want to eat it), and thus the verb within it ("made") is ranked differently from "eat". • Academic journals are sometimes ranked according to impact factor; the number of later articles that cite articles in a given journal. See also • League table • Ordinal data • Percentile rank • Rating (disambiguation) References 1. "Definition of RANKING". 2. Malara, Zbigniew; Miśko, Rafał; Sulich, Adam. "Wroclaw University of Technology graduates' career paths". {{cite journal}}: Cite journal requires |journal= (help) 3. Sulich, Adam. "The young people's labour market and crisis of integration in European Union". Retrieved 2017-03-04. 4. "The Data School - How to Rank by Group in Alteryx - Part 1 - Standard Competition, Dense, Ordinal Ranking". www.thedataschool.co.uk. Retrieved 2023-07-23. 5. "Rank Cases: Ties". www.ibm.com. Retrieved 2023-07-23. 6. "rank function - RDocumentation". www.rdocumentation.org. Retrieved 2023-07-23. 7. "R: Fast Sample Ranks". search.r-project.org. Retrieved 2023-07-23. 8. Chris Roberts, Heavy Words Lightly Thrown: The Reason Behind Rhyme, Thorndike Press, 2006 (ISBN 0-7862-8517-6) 9. RIEDS, Italian Review of Economics Demography and Statistics (2014). "World Bank Doing Business Project and the statistical methods based on ranks: the paradox of the time indicator". Rieds - Rivista Italiana di Economia, Demografia e Statistica - the Italian Journal of Economic, Demographic and Statistical Studies. 68 (1): 79–86. External links Wikimedia Commons has media related to Rankings. Look up ranking in Wiktionary, the free dictionary. • RANKNUM, a Matlab function to compute the five types of ranks • Matlab Toolbox with functions to compute ranks • TrueSkill Ranking System • Ranking Library written in Ruby • List of Global Development Indexes and Rankings Statistics • Outline • Index Descriptive statistics Continuous data Center • Mean • Arithmetic • Arithmetic-Geometric • Cubic • Generalized/power • Geometric • Harmonic • Heronian • Heinz • Lehmer • Median • Mode Dispersion • Average absolute deviation • Coefficient of variation • Interquartile range • Percentile • Range • Standard deviation • Variance Shape • Central limit theorem • Moments • Kurtosis • L-moments • Skewness Count data • Index of dispersion Summary tables • Contingency table • Frequency distribution • Grouped data Dependence • Partial correlation • Pearson product-moment correlation • Rank correlation • Kendall's τ • Spearman's ρ • Scatter plot Graphics • Bar chart • Biplot • Box plot • Control chart • Correlogram • Fan chart • Forest plot • Histogram • Pie chart • Q–Q plot • Radar chart • Run chart • Scatter plot • Stem-and-leaf display • Violin plot Data collection Study design • Effect size • Missing data • Optimal design • Population • Replication • Sample size determination • Statistic • Statistical power Survey methodology • Sampling • Cluster • Stratified • Opinion poll • Questionnaire • Standard error Controlled experiments • Blocking • Factorial experiment • Interaction • Random assignment • Randomized controlled trial • Randomized experiment • Scientific control Adaptive designs • Adaptive clinical trial • Stochastic approximation • Up-and-down designs Observational studies • Cohort study • Cross-sectional study • Natural experiment • Quasi-experiment Statistical inference Statistical theory • Population • Statistic • Probability distribution • Sampling distribution • Order statistic • Empirical distribution • Density estimation • Statistical model • Model specification • Lp space • Parameter • location • scale • shape • Parametric family • Likelihood (monotone) • Location–scale family • Exponential family • Completeness • Sufficiency • Statistical functional • Bootstrap • U • V • Optimal decision • loss function • Efficiency • Statistical distance • divergence • Asymptotics • Robustness Frequentist inference Point estimation • Estimating equations • Maximum likelihood • Method of moments • M-estimator • Minimum distance • Unbiased estimators • Mean-unbiased minimum-variance • Rao–Blackwellization • Lehmann–Scheffé theorem • Median unbiased • Plug-in Interval estimation • Confidence interval • Pivot • Likelihood interval • Prediction interval • Tolerance interval • Resampling • Bootstrap • Jackknife Testing hypotheses • 1- & 2-tails • Power • Uniformly most powerful test • Permutation test • Randomization test • Multiple comparisons Parametric tests • Likelihood-ratio • Score/Lagrange multiplier • Wald Specific tests • Z-test (normal) • Student's t-test • F-test Goodness of fit • Chi-squared • G-test • Kolmogorov–Smirnov • Anderson–Darling • Lilliefors • Jarque–Bera • Normality (Shapiro–Wilk) • Likelihood-ratio test • Model selection • Cross validation • AIC • BIC Rank statistics • Sign • Sample median • Signed rank (Wilcoxon) • Hodges–Lehmann estimator • Rank sum (Mann–Whitney) • Nonparametric anova • 1-way (Kruskal–Wallis) • 2-way (Friedman) • Ordered alternative (Jonckheere–Terpstra) • Van der Waerden test Bayesian inference • Bayesian probability • prior • posterior • Credible interval • Bayes factor • Bayesian estimator • Maximum posterior estimator • Correlation • Regression analysis Correlation • Pearson product-moment • Partial correlation • Confounding variable • Coefficient of determination Regression analysis • Errors and residuals • Regression validation • Mixed effects models • Simultaneous equations models • Multivariate adaptive regression splines (MARS) Linear regression • Simple linear regression • 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Wikipedia
Worldly cardinal In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank Vκ is a model of Zermelo–Fraenkel set theory.[1] Relationship to inaccessible cardinals By Zermelo's theorem on inaccessible cardinals, every inaccessible cardinal is worldly. By Shepherdson's theorem, inaccessibility is equivalent to the stronger statement that (Vκ, Vκ+1) is a model of second order Zermelo-Fraenkel set theory.[2] Being worldly and being inaccessible are not equivalent; in fact, the smallest worldly cardinal has countable cofinality and therefore is a singular cardinal.[3] The following are in strictly increasing order, where ι is the least inaccessible cardinal: • The least worldly κ. • The least worldly κ and λ (κ<λ, and same below) with Vκ and Vλ satisfying the same theory. • The least worldly κ that is a limit of worldly cardinals (equivalently, a limit of κ worldly cardinals). • The least worldly κ and λ with Vκ ≺Σ2 Vλ (this is higher than even a κ-fold iteration of the above item). • The least worldly κ and λ with Vκ ≺ Vλ. • The least worldly κ of cofinality ω1 (corresponds to the extension of the above item to a chain of length ω1). • The least worldly κ of cofinality ω2 (and so on). • The least κ>ω with Vκ satisfying replacement for the language augmented with the (Vκ,∈) satisfaction relation. • The least κ inaccessible in Lκ(Vκ); equivalently, the least κ>ω with Vκ satisfying replacement for formulas in Vκ in the infinitary logic L∞,ω. • The least κ with a transitive model M⊂Vκ+1 extending Vκ satisfying Morse–Kelley set theory. • (not a worldly cardinal) The least κ with Vκ having the same Σ2 theory as Vι. • The least κ with Vκ and Vι having the same theory. • The least κ with Lκ(Vκ) and Lι(Vι) having the same theory. • (not a worldly cardinal) The least κ with Vκ and Vι having the same Σ2 theory with real parameters. • (not a worldly cardinal) The least κ with Vκ ≺Σ2 Vι. • The least κ with Vκ ≺ Vι. • The least infinite κ with Vκ and Vι satisfying the same L∞,ω statements that are in Vκ. • The least κ with a transitive model M⊂Vκ+1 extending Vκ and satisfying the same sentences with parameters in Vκ as Vι+1 does. • The least inaccessible cardinal ι. References 1. Hamkins (2014). 2. Kanamori (2003), Theorem 1.3, p. 19. 3. Kanamori (2003), Lemma 6.1, p. 57. • Hamkins, Joel David (2014), "A multiverse perspective on the axiom of constructibility", Infinity and truth, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 25, Hackensack, NJ: World Sci. Publ., pp. 25–45, arXiv:1210.6541, Bibcode:2012arXiv1210.6541H, MR 3205072 • Kanamori, Akihiro (2003), The Higher Infinite, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag External links • Worldly cardinal in Cantor's attic
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Worldwide Online Olympiad Training The Worldwide Online Olympiad Training (WOOT) program was established in 2005 by Art of Problem Solving,[1] with sponsorship from Google and quantitative hedge fund giant D. E. Shaw & Co., in order to meet the needs of the world's top high school math students. Sponsorship allowed free enrollment for students of the Mathematical Olympiad Program (MOP). D.E. Shaw continued to sponsor enrollment of those students for the 2006-2007 year of WOOT. Program The focus on the WOOT program is taking already excellent pre-college students deeper into their studies of elementary mathematics, with a focus on proof-writing. • Numerous exams are given over the course of the program and graded by undergraduates at MIT and Harvard. Feedback on proofs is returned to students electronically within a couple weeks of exam submission. These tests are styled after the American Invitational Mathematics Examination (AIME) and the International Mathematics Olympiad (IMO) • Online classes are given throughout the school year on topics such as bridging ideas between different areas of math (using algebraic tactics on number theory problems), combinatorial geometry, inequalities (such as the Cauchy–Schwarz inequality), invariants, and proof writing. • Problem sets are given out on a private LaTeX-enabled WOOT phpbb message board where students post proofs, discuss problem solving tactics, and review each other's solutions. • A private, LaTeX-enabled chatroom allows students to discuss problems at any time. Students During the first year (2005–2006) of the WOOT program, a little over 100 students participated, over 90% of whom were among the fewer than 500 qualifiers for the 2006 United States of America Mathematics Olympiad (USAMO), including most of the competition's 12 "winners." Several participants from the United States and other countries won medals at the 2006 IMO held in Slovenia. Instructors WOOT students (WOOTers) are guided by veterans of national and international mathematics competitions such as IMO medalists, winners of the USAMO, a former Westinghouse competition winner, a Canadian Math Olympiad winner, perfect scorers on the AIME, perfect scorers on the American High School Mathematics Examination (now the American Mathematics Competitions), and a perfect scorer at the national MathCounts competition. Funding The first year of the program was sponsored by Google and D. E. Shaw & Co. Subsequent years have been sponsored by:[2] • Alameda Research • Citadel • Hudson River Trading • IMC Financial Markets • Jane Street Capital • Susquehanna International Group • Two Sigma References 1. "WOOT". Art of Problem Solving. Archived from the original on 2023-03-22. Retrieved 2023-03-22. 2. "WOOT: Worldwide Online Olympiad Training". Art of Problem Solving. Archived from the original on 2022-04-06. Retrieved 2023-03-22.{{cite web}}: CS1 maint: unfit URL (link)
Wikipedia
Wright omega function In mathematics, the Wright omega function or Wright function,[note 1] denoted ω, is defined in terms of the Lambert W function as: $\omega (z)=W_{{\big \lceil }{\frac {\mathrm {Im} (z)-\pi }{2\pi }}{\big \rceil }}(e^{z}).$ Uses One of the main applications of this function is in the resolution of the equation z = ln(z), as the only solution is given by z = e−ω(π i). y = ω(z) is the unique solution, when $z\neq x\pm i\pi $ for x ≤ −1, of the equation y + ln(y) = z. Except on those two rays, the Wright omega function is continuous, even analytic. Properties The Wright omega function satisfies the relation $W_{k}(z)=\omega (\ln(z)+2\pi ik)$. It also satisfies the differential equation ${\frac {d\omega }{dz}}={\frac {\omega }{1+\omega }}$ wherever ω is analytic (as can be seen by performing separation of variables and recovering the equation $\ln(\omega )+\omega =z$), and as a consequence its integral can be expressed as: $\int w^{n}\,dz={\begin{cases}{\frac {\omega ^{n+1}-1}{n+1}}+{\frac {\omega ^{n}}{n}}&{\mbox{if }}n\neq -1,\\\ln(\omega )-{\frac {1}{\omega }}&{\mbox{if }}n=-1.\end{cases}}$ Its Taylor series around the point $a=\omega _{a}+\ln(\omega _{a})$ takes the form : $\omega (z)=\sum _{n=0}^{+\infty }{\frac {q_{n}(\omega _{a})}{(1+\omega _{a})^{2n-1}}}{\frac {(z-a)^{n}}{n!}}$ where $q_{n}(w)=\sum _{k=0}^{n-1}{\bigg \langle }\!\!{\bigg \langle }{\begin{matrix}n+1\\k\end{matrix}}{\bigg \rangle }\!\!{\bigg \rangle }(-1)^{k}w^{k+1}$ in which ${\bigg \langle }\!\!{\bigg \langle }{\begin{matrix}n\\k\end{matrix}}{\bigg \rangle }\!\!{\bigg \rangle }$ is a second-order Eulerian number. Values ${\begin{array}{lll}\omega (0)&=W_{0}(1)&\approx 0.56714\\\omega (1)&=1&\\\omega (-1\pm i\pi )&=-1&\\\omega (-{\frac {1}{3}}+\ln \left({\frac {1}{3}}\right)+i\pi )&=-{\frac {1}{3}}&\\\omega (-{\frac {1}{3}}+\ln \left({\frac {1}{3}}\right)-i\pi )&=W_{-1}\left(-{\frac {1}{3}}e^{-{\frac {1}{3}}}\right)&\approx -2.237147028\\\end{array}}$ Plots • Plots of the Wright omega function on the complex plane • z = Re(ω(x + i y)) • z = Im(ω(x + i y)) • ω(x + i y) Notes 1. Not to be confused with the Fox–Wright function, also known as Wright function. References • "On the Wright ω function", Robert Corless and David Jeffrey
Wikipedia
Writhe In knot theory, there are several competing notions of the quantity writhe, or $\operatorname {Wr} $. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amount of "coiling" of a mathematical knot (or any closed simple curve) in three-dimensional space and assumes real numbers as values. In both cases, writhe is a geometric quantity, meaning that while deforming a curve (or diagram) in such a way that does not change its topology, one may still change its writhe.[1] Writhe of link diagrams In knot theory, the writhe is a property of an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings. A direction is assigned to the link at a point in each component and this direction is followed all the way around each component. For each crossing one comes across while traveling in this direction, if the strand underneath goes from right to left, the crossing is positive; if the lower strand goes from left to right, the crossing is negative. One way of remembering this is to use a variation of the right-hand rule. Positive crossing Negative crossing For a knot diagram, using the right-hand rule with either orientation gives the same result, so the writhe is well-defined on unoriented knot diagrams. The writhe of a knot is unaffected by two of the three Reidemeister moves: moves of Type II and Type III do not affect the writhe. Reidemeister move Type I, however, increases or decreases the writhe by 1. This implies that the writhe of a knot is not an isotopy invariant of the knot itself — only the diagram. By a series of Type I moves one can set the writhe of a diagram for a given knot to be any integer at all. Writhe of a closed curve Writhe is also a property of a knot represented as a curve in three-dimensional space. Strictly speaking, a knot is such a curve, defined mathematically as an embedding of a circle in three-dimensional Euclidean space, $\mathbb {R} ^{3}$. By viewing the curve from different vantage points, one can obtain different projections and draw the corresponding knot diagrams. Its writhe $\operatorname {Wr} $ (in the space curve sense) is equal to the average of the integral writhe values obtained from the projections from all vantage points.[2] Hence, writhe in this situation can take on any real number as a possible value.[1] In a paper from 1961,[3] Gheorghe Călugăreanu proved the following theorem: take a ribbon in $\mathbb {R} ^{3}$, let $\operatorname {Lk} $ be the linking number of its border components, and let $\operatorname {Tw} $ be its total twist. Then the difference $\operatorname {Lk} -\operatorname {Tw} $ depends only on the core curve of the ribbon,[2] and $\operatorname {Wr} =\operatorname {Lk} -\operatorname {Tw} $. In a paper from 1959,[4] Călugăreanu also showed how to calculate the writhe Wr with an integral. Let $C$ be a smooth, simple, closed curve and let $\mathbf {r} _{1}$ and $\mathbf {r} _{2}$ be points on $C$. Then the writhe is equal to the Gauss integral $\operatorname {Wr} ={\frac {1}{4\pi }}\int _{C}\int _{C}d\mathbf {r} _{1}\times d\mathbf {r} _{2}\cdot {\frac {\mathbf {r} _{1}-\mathbf {r} _{2}}{\left|\mathbf {r} _{1}-\mathbf {r} _{2}\right|^{3}}}$. Numerically approximating the Gauss integral for writhe of a curve in space Since writhe for a curve in space is defined as a double integral, we can approximate its value numerically by first representing our curve as a finite chain of $N$ line segments. A procedure that was first derived by Michael Levitt[5] for the description of protein folding and later used for supercoiled DNA by Konstantin Klenin and Jörg Langowski[6] is to compute $\operatorname {Wr} =\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {\Omega _{ij}}{4\pi }}=2\sum _{i=2}^{N}\sum _{j<i}{\frac {\Omega _{ij}}{4\pi }}$, where $\Omega _{ij}/{4\pi }$ is the exact evaluation of the double integral over line segments $i$ and $j$; note that $\Omega _{ij}=\Omega _{ji}$ and $\Omega _{i,i+1}=\Omega _{ii}=0$.[6] To evaluate $\Omega _{ij}/{4\pi }$ for given segments numbered $i$ and $j$, number the endpoints of the two segments 1, 2, 3, and 4. Let $r_{pq}$ be the vector that begins at endpoint $p$ and ends at endpoint $q$. Define the following quantities:[6] $n_{1}={\frac {r_{13}\times r_{14}}{\left|r_{13}\times r_{14}\right|}},\;n_{2}={\frac {r_{14}\times r_{24}}{\left|r_{14}\times r_{24}\right|}},\;n_{3}={\frac {r_{24}\times r_{23}}{\left|r_{24}\times r_{23}\right|}},\;n_{4}={\frac {r_{23}\times r_{13}}{\left|r_{23}\times r_{13}\right|}}$ Then we calculate[6] $\Omega ^{*}=\arcsin \left(n_{1}\cdot n_{2}\right)+\arcsin \left(n_{2}\cdot n_{3}\right)+\arcsin \left(n_{3}\cdot n_{4}\right)+\arcsin \left(n_{4}\cdot n_{1}\right).$ Finally, we compensate for the possible sign difference and divide by $4\pi $ to obtain[6] ${\frac {\Omega }{4\pi }}={\frac {\Omega ^{*}}{4\pi }}{\text{sign}}\left(\left(r_{34}\times r_{12}\right)\cdot r_{13}\right).$ In addition, other methods to calculate writhe can be fully described mathematically and algorithmically, some of them outperform method above (which has quadratic computational complexity, by having linear complexity).[6] Applications in DNA topology DNA will coil when twisted, just like a rubber hose or a rope will, and that is why biomathematicians use the quantity of writhe to describe the amount a piece of DNA is deformed as a result of this torsional stress. In general, this phenomenon of forming coils due to writhe is referred to as DNA supercoiling and is quite commonplace, and in fact in most organisms DNA is negatively supercoiled.[1] Any elastic rod, not just DNA, relieves torsional stress by coiling, an action which simultaneously untwists and bends the rod. F. Brock Fuller shows mathematically[7] how the “elastic energy due to local twisting of the rod may be reduced if the central curve of the rod forms coils that increase its writhing number”. See also • DNA supercoiling • Linking number • Ribbon theory • Twist (mathematics) • Winding number References 1. Bates, Andrew (2005). DNA Topology. Oxford University Press. pp. 36–37. ISBN 978-0-19-850655-3. 2. Cimasoni, David (2001). "Computing the writhe of a knot". Journal of Knot Theory and Its Ramifications. 10 (387): 387–395. arXiv:math/0406148. doi:10.1142/S0218216501000913. MR 1825964. S2CID 15850269. 3. Călugăreanu, Gheorghe (1961). "Sur les classes d'isotopie des nœuds tridimensionnels et leurs invariants". Czechoslovak Mathematical Journal (in French). 11 (4): 588–625. doi:10.21136/CMJ.1961.100486. MR 0149378. 4. Călugăreanu, Gheorghe (1959). "L'intégrale de Gauss et l'analyse des nœuds tridimensionnels" (PDF). Revue de Mathématiques Pure et Appliquées (in French). 4: 5–20. MR 0131846. 5. Levitt, Michael (1986). "Protein Folding by Restrained Energy Minimization and Molecular Dynamics". Journal of Molecular Biology. 170 (3): 723–764. CiteSeerX 10.1.1.26.3656. doi:10.1016/s0022-2836(83)80129-6. PMID 6195346. 6. Klenin, Konstantin; Langowski, Jörg (2000). "Computation of writhe in modeling of supercoiled DNA". Biopolymers. 54 (5): 307–317. doi:10.1002/1097-0282(20001015)54:5<307::aid-bip20>3.0.co;2-y. PMID 10935971. 7. Fuller, F. Brock (1971). "The writhing number of a space curve". Proceedings of the National Academy of Sciences of the United States of America. 68 (4): 815–819. Bibcode:1971PNAS...68..815B. doi:10.1073/pnas.68.4.815. MR 0278197. PMC 389050. PMID 5279522. Further reading • Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1 Knot theory (knots and links) Hyperbolic • Figure-eight (41) • Three-twist (52) • Stevedore (61) • 62 • 63 • Endless (74) • Carrick mat (818) • Perko pair (10161) • (−2,3,7) pretzel (12n242) • Whitehead (52 1 ) • Borromean rings (63 2 ) • L10a140 • Conway knot (11n34) Satellite • Composite knots • Granny • Square • Knot sum Torus • Unknot (01) • Trefoil (31) • Cinquefoil (51) • Septafoil (71) • Unlink (02 1 ) • Hopf (22 1 ) • Solomon's (42 1 ) Invariants • Alternating • Arf invariant • Bridge no. • 2-bridge • Brunnian • Chirality • Invertible • Crosscap no. • Crossing no. • Finite type invariant • Hyperbolic volume • Khovanov homology • Genus • Knot group • Link group • Linking no. • Polynomial • Alexander • Bracket • HOMFLY • Jones • Kauffman • Pretzel • Prime • list • Stick no. • Tricolorability • Unknotting no. and problem Notation and operations • Alexander–Briggs notation • Conway notation • Dowker–Thistlethwaite notation • Flype • Mutation • Reidemeister move • Skein relation • Tabulation Other • Alexander's theorem • Berge • Braid theory • Conway sphere • Complement • Double torus • Fibered • Knot • List of knots and links • Ribbon • Slice • Sum • Tait conjectures • Twist • Wild • Writhe • Surgery theory • Category • Commons
Wikipedia
Wronskian In the mathematics of a square matrix, the Wronskian (or Wrońskian) is a determinant introduced by the Polish mathematician Józef Hoene-Wroński (1812). It is used in the study of differential equations, where it can sometimes show linear independence of a set of solutions. Differential equations Scope Fields • Natural sciences • Engineering • Astronomy • Physics • Chemistry • Biology • Geology Applied mathematics • Continuum mechanics • Chaos theory • Dynamical systems Social sciences • Economics • Population dynamics List of named differential equations Classification Types • Ordinary • Partial • Differential-algebraic • Integro-differential • Fractional • Linear • Non-linear By variable type • Dependent and independent variables • Autonomous • Coupled / Decoupled • Exact • Homogeneous / Nonhomogeneous Features • Order • Operator • Notation Relation to processes • Difference (discrete analogue) • Stochastic • Stochastic partial • Delay Solution Existence and uniqueness • Picard–Lindelöf theorem • Peano existence theorem • Carathéodory's existence theorem • Cauchy–Kowalevski theorem General topics • Initial conditions • Boundary values • Dirichlet • Neumann • Robin • Cauchy problem • Wronskian • Phase portrait • Lyapunov / Asymptotic / Exponential stability • Rate of convergence • Series / Integral solutions • Numerical integration • Dirac delta function Solution methods • Inspection • Method of characteristics • Euler • Exponential response formula • Finite difference (Crank–Nicolson) • Finite element • Infinite element • Finite volume • Galerkin • Petrov–Galerkin • Green's function • Integrating factor • Integral transforms • Perturbation theory • Runge–Kutta • Separation of variables • Undetermined coefficients • Variation of parameters People List • Isaac Newton • Gottfried Leibniz • Jacob Bernoulli • Leonhard Euler • Józef Maria Hoene-Wroński • Joseph Fourier • Augustin-Louis Cauchy • George Green • Carl David Tolmé Runge • Martin Kutta • Rudolf Lipschitz • Ernst Lindelöf • Émile Picard • Phyllis Nicolson • John Crank Definition The Wronskian of two differentiable functions f  and g is $W(f,g)=fg'-gf'$. More generally, for n real- or complex-valued functions f1, …, fn, which are n – 1 times differentiable on an interval I, the Wronskian $W(f_{1},\ldots ,f_{n})$ is a function on $x\in I$ defined by $W(f_{1},\ldots ,f_{n})(x)=\det {\begin{bmatrix}f_{1}(x)&f_{2}(x)&\cdots &f_{n}(x)\\f_{1}'(x)&f_{2}'(x)&\cdots &f_{n}'(x)\\\vdots &\vdots &\ddots &\vdots \\f_{1}^{(n-1)}(x)&f_{2}^{(n-1)}(x)&\cdots &f_{n}^{(n-1)}(x)\end{bmatrix}}.$ This is the determinant of the matrix constructed by placing the functions in the first row, the first derivatives of the functions in the second row, and so on through the $(n-1)^{\text{th}}$ derivative, thus forming a square matrix. When the functions fi are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel's identity, even if the functions fi are not known explicitly. (See below.) The Wronskian and linear independence If the functions fi are linearly dependent, then so are the columns of the Wronskian (since differentiation is a linear operation), and the Wronskian vanishes. Thus, one may show that a set of differentiable functions is linearly independent on an interval by showing that their Wronskian does not vanish identically. It may, however, vanish at isolated points.[1] A common misconception is that W = 0 everywhere implies linear dependence, but Peano (1889) pointed out that the functions x2 and |x| · x have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0.[lower-alpha 1] There are several extra conditions which combine with vanishing of the Wronskian in an interval to imply linear dependence. • Maxime Bôcher observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent.[3] • Bôcher (1901) gave several other conditions for the vanishing of the Wronskian to imply linear dependence; for example, if the Wronskian of n functions is identically zero and the n Wronskians of n – 1 of them do not all vanish at any point then the functions are linearly dependent. • Wolsson (1989a) gave a more general condition that together with the vanishing of the Wronskian implies linear dependence. Over fields of positive characteristic p the Wronskian may vanish even for linearly independent polynomials; for example, the Wronskian of xp and 1 is identically 0. Application to linear differential equations In general, for an $n$th order linear differential equation, if $(n-1)$ solutions are known, the last one can be determined by using the Wronskian. Consider the second order differential equation in Lagrange's notation: $y''=a(x)y'+b(x)y$ where $a(x)$, $b(x)$ are known, and y is the unknown function to be found. Let us call $y_{1},y_{2}$ the two solutions of the equation and form their Wronskian $W(x)=y_{1}y'_{2}-y_{2}y'_{1}$ Then differentiating $W(x)$ and using the fact that $y_{i}$ obey the above differential equation shows that $W'(x)=aW(x)$ Therefore, the Wronskian obeys a simple first order differential equation and can be exactly solved: $W(x)=C~e^{A(x)}$ where $A'(x)=a(x)$ and $C$ is a constant. Now suppose that we know one of the solutions, say $y_{2}$. Then, by the definition of the Wronskian, $y_{1}$ obeys a first order differential equation: $y'_{1}-{\frac {y'_{2}}{y_{2}}}y_{1}=-W(x)/y_{2}$ and can be solved exactly (at least in theory). The method is easily generalized to higher order equations. Generalized Wronskians For n functions of several variables, a generalized Wronskian is a determinant of an n by n matrix with entries Di(fj) (with 0 ≤ i < n), where each Di is some constant coefficient linear partial differential operator of order i. If the functions are linearly dependent then all generalized Wronskians vanish. As in the single variable case the converse is not true in general: if all generalized Wronskians vanish, this does not imply that the functions are linearly dependent. However, the converse is true in many special cases. For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent. Roth used this result about generalized Wronskians in his proof of Roth's theorem. For more general conditions under which the converse is valid see Wolsson (1989b). History The Wronskian was introduced by Józef Hoene-Wroński (1812) and given its current name by Thomas Muir (1882, Chapter XVIII). See also • Variation of parameters • Moore matrix, analogous to the Wronskian with differentiation replaced by the Frobenius endomorphism over a finite field. • Alternant matrix • Vandermonde matrix Notes 1. Peano published his example twice, because the first time he published it, an editor, Paul Mansion, who had written a textbook incorrectly claiming that the vanishing of the Wronskian implies linear dependence, added a footnote to Peano's paper claiming that this result is correct as long as neither function is identically zero. Peano's second paper pointed out that this footnote was nonsense.[2] Citations 1. Bender, Carl M.; Orszag, Steven A. (1999) [1978], Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, New York: Springer, p. 9, ISBN 978-0-387-98931-0 2. Engdahl, Susannah; Parker, Adam (April 2011). "Peano on Wronskians: A Translation". Convergence. Mathematical Association of America. doi:10.4169/loci003642. Retrieved 2020-10-08. 3. Engdahl, Susannah; Parker, Adam (April 2011). "Peano on Wronskians: A Translation". Convergence. Mathematical Association of America. Section "On the Wronskian Determinant". doi:10.4169/loci003642. Retrieved 2020-10-08. The most famous theorem is attributed to Bocher, and states that if the Wronskian of $n$ analytic functions is zero, then the functions are linearly dependent ([B2], [BD]). [The citations 'B2' and 'BD' refer to Bôcher (1900–1901) and Bostan and Dumas (2010), respectively.] References • Bôcher, Maxime (1900–1901). "The Theory of Linear Dependence". Annals of Mathematics. Princeton University. 2 (1/4): 81–96. doi:10.2307/2007186. ISSN 0003-486X. JSTOR 2007186. • Bôcher, Maxime (1901), "Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence" (PDF), Transactions of the American Mathematical Society, Providence, R.I.: American Mathematical Society, 2 (2): 139–149, doi:10.2307/1986214, ISSN 0002-9947, JFM 32.0313.02, JSTOR 1986214 • Bostan, Alin; Dumas, Philippe (2010). "Wronskians and Linear Independence". American Mathematical Monthly. Taylor & Francis. 117 (8): 722–727. arXiv:1301.6598. doi:10.4169/000298910x515785. ISSN 0002-9890. JSTOR 10.4169/000298910x515785. S2CID 9322383. • Hartman, Philip (1964), Ordinary Differential Equations, New York: John Wiley & Sons, ISBN 978-0-89871-510-1, MR 0171038, Zbl 0125.32102 • Hoene-Wroński, Józef (1812), Réfutation de la théorie des fonctions analytiques de Lagrange, Paris • Muir, Thomas (1882), A Treatise on the Theorie of Determinants., Macmillan, JFM 15.0118.05 • Peano, Giuseppe (1889), "Sur le déterminant wronskien.", Mathesis (in French), IX: 75–76, 110–112, JFM 21.0153.01 • Rozov, N. Kh. (2001) [1994], "Wronskian", Encyclopedia of Mathematics, EMS Press • Wolsson, Kenneth (1989a), "A condition equivalent to linear dependence for functions with vanishing Wronskian", Linear Algebra and Its Applications, 116: 1–8, doi:10.1016/0024-3795(89)90393-5, ISSN 0024-3795, MR 0989712, Zbl 0671.15005 • Wolsson, Kenneth (1989b), "Linear dependence of a function set of m variables with vanishing generalized Wronskians", Linear Algebra and Its Applications, 117: 73–80, doi:10.1016/0024-3795(89)90548-X, ISSN 0024-3795, MR 0993032, Zbl 0724.15004 Matrix classes Explicitly constrained entries • Alternant • Anti-diagonal • Anti-Hermitian • Anti-symmetric • Arrowhead • Band • Bidiagonal • Bisymmetric • Block-diagonal • Block • Block tridiagonal • Boolean • Cauchy • Centrosymmetric • Conference • Complex Hadamard • Copositive • Diagonally dominant • Diagonal • Discrete Fourier Transform • Elementary • Equivalent • Frobenius • Generalized permutation • Hadamard • Hankel • Hermitian • Hessenberg • Hollow • Integer • Logical • Matrix unit • Metzler • Moore • Nonnegative • Pentadiagonal • Permutation • Persymmetric • Polynomial • Quaternionic • Signature • Skew-Hermitian • Skew-symmetric • Skyline • Sparse • Sylvester • Symmetric • Toeplitz • Triangular • Tridiagonal • Vandermonde • Walsh • Z Constant • Exchange • Hilbert • Identity • Lehmer • Of ones • Pascal • Pauli • Redheffer • Shift • Zero Conditions on eigenvalues or eigenvectors • Companion • Convergent • Defective • Definite • Diagonalizable • Hurwitz • Positive-definite • Stieltjes Satisfying conditions on products or inverses • Congruent • Idempotent or Projection • Invertible • Involutory • Nilpotent • Normal • Orthogonal • Unimodular • Unipotent • Unitary • Totally unimodular • Weighing With specific applications • Adjugate • Alternating sign • Augmented • Bézout • Carleman • Cartan • Circulant • Cofactor • Commutation • Confusion • Coxeter • Distance • Duplication and elimination • Euclidean distance • Fundamental (linear differential equation) • Generator • Gram • Hessian • Householder • Jacobian • Moment • Payoff • Pick • Random • Rotation • Seifert • Shear • Similarity • Symplectic • Totally positive • Transformation Used in statistics • Centering • Correlation • Covariance • Design • Doubly stochastic • Fisher information • Hat • Precision • Stochastic • Transition Used in graph theory • Adjacency • Biadjacency • Degree • Edmonds • Incidence • Laplacian • Seidel adjacency • Tutte Used in science and engineering • Cabibbo–Kobayashi–Maskawa • Density • Fundamental (computer vision) • Fuzzy associative • Gamma • Gell-Mann • Hamiltonian • Irregular • Overlap • S • State transition • Substitution • Z (chemistry) Related terms • Jordan normal form • Linear independence • Matrix exponential • Matrix representation of conic sections • Perfect matrix • Pseudoinverse • Row echelon form • Wronskian •  Mathematics portal • List of matrices • Category:Matrices
Wikipedia
Wu's method of characteristic set Wenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept of characteristic set introduced in the late 1940s by J.F. Ritt. It is fully independent of the Gröbner basis method, introduced by Bruno Buchberger (1965), even if Gröbner bases may be used to compute characteristic sets.[1][2] Wu's method is powerful for mechanical theorem proving in elementary geometry, and provides a complete decision process for certain classes of problem. It has been used in research in his laboratory (KLMM, Key Laboratory of Mathematics Mechanization in Chinese Academy of Science) and around the world. The main trends of research on Wu's method concern systems of polynomial equations of positive dimension and differential algebra where Ritt's results have been made effective.[3][4] Wu's method has been applied in various scientific fields, like biology, computer vision, robot kinematics and especially automatic proofs in geometry.[5] Informal description Wu's method uses polynomial division to solve problems of the form: $\forall x,y,z,\dots I(x,y,z,\dots )\implies f(x,y,z,\dots )\,$ where f is a polynomial equation and I is a conjunction of polynomial equations. The algorithm is complete for such problems over the complex domain. The core idea of the algorithm is that you can divide one polynomial by another to give a remainder. Repeated division results in either the remainder vanishing (in which case the I implies f statement is true), or an irreducible remainder is left behind (in which case the statement is false). More specifically, for an ideal I in the ring k[x1, ..., xn] over a field k, a (Ritt) characteristic set C of I is composed of a set of polynomials in I, which is in triangular shape: polynomials in C have distinct main variables (see the formal definition below). Given a characteristic set C of I, one can decide if a polynomial f is zero modulo I. That is, the membership test is checkable for I, provided a characteristic set of I. Ritt characteristic set A Ritt characteristic set is a finite set of polynomials in triangular form of an ideal. This triangular set satisfies certain minimal condition with respect to the Ritt ordering, and it preserves many interesting geometrical properties of the ideal. However it may not be its system of generators. Notation Let R be the multivariate polynomial ring k[x1, ..., xn] over a field k. The variables are ordered linearly according to their subscript: x1 < ... < xn. For a non-constant polynomial p in R, the greatest variable effectively presenting in p, called main variable or class, plays a particular role: p can be naturally regarded as a univariate polynomial in its main variable xk with coefficients in k[x1, ..., xk−1]. The degree of p as a univariate polynomial in its main variable is also called its main degree. Triangular set A set T of non-constant polynomials is called a triangular set if all polynomials in T have distinct main variables. This generalizes triangular systems of linear equations in a natural way. Ritt ordering For two non-constant polynomials p and q, we say p is smaller than q with respect to Ritt ordering and written as p <r q, if one of the following assertions holds: (1) the main variable of p is smaller than the main variable of q, that is, mvar(p) < mvar(q), (2) p and q have the same main variable, and the main degree of p is less than the main degree of q, that is, mvar(p) = mvar(q) and mdeg(p) < mdeg(q). In this way, (k[x1, ..., xn],<r) forms a well partial order. However, the Ritt ordering is not a total order: there exist polynomials p and q such that neither p <r q nor p >r q. In this case, we say that p and q are not comparable. The Ritt ordering is comparing the rank of p and q. The rank, denoted by rank(p), of a non-constant polynomial p is defined to be a power of its main variable: mvar(p)mdeg(p) and ranks are compared by comparing first the variables and then, in case of equality of the variables, the degrees. Ritt ordering on triangular sets A crucial generalization on Ritt ordering is to compare triangular sets. Let T = { t1, ..., tu} and S = { s1, ..., sv} be two triangular sets such that polynomials in T and S are sorted increasingly according to their main variables. We say T is smaller than S w.r.t. Ritt ordering if one of the following assertions holds 1. there exists k ≤ min(u, v) such that rank(ti) = rank(si) for 1 ≤ i < k and tk <r sk, 2. u > v and rank(ti) = rank(si) for 1 ≤ i ≤ v. Also, there exists incomparable triangular sets w.r.t Ritt ordering. Ritt characteristic set Let I be a non-zero ideal of k[x1, ..., xn]. A subset T of I is a Ritt characteristic set of I if one of the following conditions holds: 1. T consists of a single nonzero constant of k, 2. T is a triangular set and T is minimal w.r.t Ritt ordering in the set of all triangular sets contained in I. A polynomial ideal may possess (infinitely) many characteristic sets, since Ritt ordering is a partial order. Wu characteristic set The Ritt–Wu process, first devised by Ritt, subsequently modified by Wu, computes not a Ritt characteristic but an extended one, called Wu characteristic set or ascending chain. A non-empty subset T of the ideal ⟨F⟩ generated by F is a Wu characteristic set of F if one of the following condition holds 1. T = {a} with a being a nonzero constant, 2. T is a triangular set and there exists a subset G of ⟨F⟩ such that ⟨F⟩ = ⟨G⟩ and every polynomial in G is pseudo-reduced to zero with respect to T. Wu characteristic set is defined to the set F of polynomials, rather to the ideal ⟨F⟩ generated by F. Also it can be shown that a Ritt characteristic set T of ⟨F⟩ is a Wu characteristic set of F. Wu characteristic sets can be computed by Wu's algorithm CHRST-REM, which only requires pseudo-remainder computations and no factorizations are needed. Wu's characteristic set method has exponential complexity; improvements in computing efficiency by weak chains, regular chains, saturated chain were introduced[6] Decomposing algebraic varieties An application is an algorithm for solving systems of algebraic equations by means of characteristic sets. More precisely, given a finite subset F of polynomials, there is an algorithm to compute characteristic sets T1, ..., Te such that: $V(F)=W(T_{1})\cup \cdots \cup W(T_{e}),$ where W(Ti) is the difference of V(Ti) and V(hi), here hi is the product of initials of the polynomials in Ti. See also • Regular chain • Mathematics-Mechanization Platform References 1. Corrochano, Eduardo Bayro; Sobczyk, Garret, eds. (2001). Geometric algebra with applications in science and engineering. Boston, Mass: Birkhäuser. p. 110. ISBN 9780817641993. 2. P. Aubry, D. Lazard, M. Moreno Maza (1999). On the theories of triangular sets. Journal of Symbolic Computation, 28(1–2):105–124 3. Hubert, E. Factorisation Free Decomposition Algorithms in Differential Algebra. Journal of Symbolic Computation, (May 2000): 641–662. 4. Maple (software) package diffalg. 5. Chou, Shang-Ching; Gao, Xiao Shan; Zhang, Jing Zhong. Machine proofs in geometry. World Scientific, 1994. 6. Chou S C, Gao X S; Ritt–Wu's decomposition algorithm and geometry theorem proving. Proc of CADE, 10 LNCS, #449, Berlin, Springer Verlag, 1990 207–220. • P. Aubry, M. Moreno Maza (1999) Triangular Sets for Solving Polynomial Systems: a Comparative Implementation of Four Methods. J. Symb. Comput. 28(1–2): 125–154 • David A. Cox, John B. Little, Donal O'Shea. Ideals, Varieties, and Algorithms. 2007. • Hua-Shan, Liu (24 August 2005). "WuRittSolva: Implementation of Wu-Ritt Characteristic Set Method". Wolfram Library Archive. Wolfram. Retrieved 17 November 2012. • Heck, André (2003). Introduction to Maple (3. ed.). New York: Springer. pp. 105, 508. ISBN 9780387002309. • Ritt, J. (1966). Differential Algebra. New York, Dover Publications. • Dongming Wang (1998). Elimination Methods. Springer-Verlag, Wien, Springer-Verlag • Dongming Wang (2004). Elimination Practice, Imperial College Press, London ISBN 1-86094-438-8 • Wu, W. T. (1984). Basic principles of mechanical theorem proving in elementary geometries. J. Syst. Sci. Math. Sci., 4, 207–35 • Wu, W. T. (1987). A zero structure theorem for polynomial equations solving. MM Research Preprints, 1, 2–12 • Xiaoshan, Gao; Chunming, Yuan; Guilin, Zhang (2009). "Ritt-Wu's characteristic set method for ordinary difference polynomial systems with arbitrary ordering". Acta Mathematica Scientia. 29 (4): 1063–1080. CiteSeerX 10.1.1.556.9549. doi:10.1016/S0252-9602(09)60086-2. External links • wsolve Maple package • The Characteristic Set Method
Wikipedia
Wu–Yang dictionary In topology and high energy physics, the Wu–Yang dictionary refers to the mathematical identification that allows to translate back and forth between the concepts of gauge theory and those of differential geometry. It was devised by Tai Tsun Wu and C. N. Yang in 1975 when studying the relation between electromagnetism and fiber bundle theory.[1] This dictionary has been credited as bringing mathematics and theoretical physics closer together.[2] A crucial example of the success of the dictionary is that it allowed to understand Paul Dirac's monopole quantization in terms of Hopf fibrations.[3] History In 1975, theoretical physicists Tsun Wu and C. N. Yang working in Stony Brook University, published a paper on the mathematical framework of electromagnetism and the Aharonov–Bohm effect in terms of fiber bundles. A year later, mathematician Isadore Singer came to visit and brought a copy back to the University of Oxford.[2][4][5] Singer showed the paper to Michael Atiyah and other mathematicians, sparking a close collaboration between physicists and mathematicians.[2] Yang also recounts a conversation that he had with one of the mathematicians that founded fiber bundle theory, Shiing-Shen Chern:[2] In 1975, impressed with the fact that gauge fields are connections on fiber bundles, I drove to the house of Shiing-Shen Chern in El Cerrito, near Berkeley. (I had taken courses with him in the early 1940s when he was a young professor and I an undergraduate student at the National Southwest Associated University in Kunming, China. That was before fiber bundles had become important in differential geometry and before Chern had made history with his contributions to the generalized Gauss–Bonnet theorem and the Chern classes.) We had much to talk about: friends, relatives, China. When our conversation turned to fiber bundles, I told him that I had finally learned from Jim Simons the beauty of fiber-bundle theory and the profound Chern-Weil theorem. I said I found it amazing that gauge fields are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world. I added ‘this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere.’ He immediately protested, ‘No, no. These concepts were not dreamed up. They were natural and real.' Description Summarized version The Wu-Yang dictionary relates terms in particle physics with terms in mathematics, specifically fiber bundle theory. Many versions and generalization of the dictionary exist. Here is an example of a dictionary, which puts each physics term next to its mathematical analogue:[6] Physics Mathematics Potential Connection Field tensor (interaction) Curvature Field tensor-potential relation Structural equation Gauge transformation Change of bundle coordinates Gauge group Structure group Original version for electromagnetism Wu and Yang considered the description of an electron traveling around a cylinder in the presence of a magnetic field inside the cylinder (outside the cylinder the field vanishes i.e. $f_{\mu \nu }=0$). According to the Aharonov–Bohm effect, the interference patterns shift by a factor $\exp(-i\Omega /\Omega _{0})$, where $\Omega $ is the magnetic flux and $\Omega _{0}$ is the magnetic flux quantum. For two different fluxes a and b, the results are identical if $\Omega _{a}-\Omega _{b}=N\Omega _{0}$, where $N$ is an integer. We define the operator $S_{ab}$ as the operator that brings the electron wave function from one configuration to the other $\psi _{b}=S_{ba}\psi _{a}$. For an electron that takes a path from point P to point Q, we define the phase factor as $\Phi _{PQ}=\exp \left(-{\frac {i}{\Omega _{0}}}\int _{P}^{Q}A_{\mu }\mathrm {d} x^{\mu }\right)$, where $A_{\mu }$ is the electromagnetic four-potential. For the case of a SU2 gauge field, we can make the substitution $A_{\mu }=ib_{\mu }^{k}X_{k}$, where $X_{k}=-i\sigma _{k}/2$ are the generators of SU2, $\sigma _{k}$ are the Pauli matrices. Under these concepts, Wu and Yang showed the relation between the language of gauge theory and fiber bundles, was codified in following dictionary:[2][7][8] Wu–Yang dictionary (1975) Gauge field terminology Bundle terminology Gauge (or global gauge) Principal coordinate fiber bundle Gauge type Principal fiber bundle Gauge potential $b_{\mu }^{k}$ Connection on principal fiber bundle $S_{ba}$ Transition function Phase factor $\Phi _{QP}$ Parallel displacement Field strength $f_{\mu \nu }^{k}$ Curvature Source $J_{\mu }^{k}$ ? Electromagnetism Connection in a U1(1) bundle Isotopic spin gauge field Connection in a SU2 bundle Dirac's monopole quantization Classification in a U1(1) bundle according to first Chern class Electromagnetism without monopole Connection on a trivial a U1(1) bundle Electromagnetism with monopole Connection on a nontrivial a U1(1) bundle See also • 't Hooft–Polyakov monopole • Wu–Yang monopole References 1. Wu, Tai Tsun; Yang, Chen Ning (1975-12-15). "Concept of nonintegrable phase factors and global formulation of gauge fields". Physical Review D. 12 (12): 3845–3857. doi:10.1103/PhysRevD.12.3845. ISSN 0556-2821. 2. Poo, Mu-ming; Chao, Alexander Wu (2020-01-01). "Conversation with Chen-Ning Yang: reminiscence and reflection". National Science Review. 7 (1): 233–236. doi:10.1093/nsr/nwz113. ISSN 2095-5138. PMC 8288855. PMID 34692035. 3. Woit, Peter (5 April 2008). "Stony Brook Dialogues in Mathematics and Physics". Not even wrong blog. Retrieved 2023-03-14. 4. Wells, Raymond O'Neil; Weyl, Hermann (1988). The Mathematical Heritage of Hermann Weyl. American Mathematical Soc. ISBN 978-0-8218-1482-6. 5. Freed, Daniel S. (2021). "Isadore Singer Transcended Mathematical Boundaries". Quanta Magazine. 6. Zeidler, Eberhard (2008-09-03). Quantum Field Theory II: Quantum Electrodynamics: A Bridge between Mathematicians and Physicists. Springer Science & Business Media. ISBN 978-3-540-85377-0. 7. Boi, Luciano (2004). "Geometrical and topological foundations of theoretical physics: from gauge theories to string program". International Journal of Mathematics and Mathematical Sciences. 2004 (34): 1777–1836. doi:10.1155/S0161171204304400. ISSN 0161-1712. 8. Wells, Raymond O'Neil; Weyl, Hermann (1988). The Mathematical Heritage of Hermann Weyl. American Mathematical Soc. ISBN 978-0-8218-1482-6.
Wikipedia
Cynthia Wyels Cynthia Jean Wyels is an American mathematician whose interests include linear algebra, combinatorics, and mathematics education, and who is known for her research in graph pebbling and radio coloring of graphs. She is a professor of mathematics at California State University Channel Islands (CSUCI) in Camarillo, California,[1] where she also co-directs the Alliance for Minority Participation.[2] Education and Career Wyels did her undergraduate studies at Pomona College, and earned a master's degree from the University of Michigan.[1] She completed her Ph.D. in mathematics from the University of California, Santa Barbara in 1994; her dissertation, Isomorphism Problems In A Matrix Setting, was supervised by Morris Newman.[1][3] She has taught mathematics at Weber State University and the United States Military Academy, and was chair of mathematics at California Lutheran University before moving to CSUCI.[1][4] Awards In 2012, Wyels was a winner of the Deborah and Franklin Haimo Awards for Distinguished College or University Teaching of Mathematics, given by the Mathematical Association of America to recognize teaching excellence that extends beyond a single institution. Her award citation particularly recognized her mentorship of Mexican and first-generation college students through the Research Experiences for Undergraduates program and through personal donations to education in Mexico, and her foundation of a mentorship program at CSUCI.[5] In 2017, the Society for the Advancement of Chicanos/Hispanics and Native Americans in Science gave Wyels their distinguished mentor award.[6] She received the CSUCI UndocuAlly of the Year award in 2017-18.[7] References 1. "Cynthia J. Wyels", Faculty Profiles, California State University Channel Islands, retrieved 2018-04-29 2. Estrada, Roxanne (January 26, 2012), "Educator's passion for math a real plus", Thousand Oaks Acorn 3. Cynthia Wyels at the Mathematics Genealogy Project 4. Bush, Sharon Raiford (November 29, 2015), Mathematics Professor Helps Students Excel In Societal Areas, CBS Los Angeles 5. "MAA Prizes Presented in Boston" (PDF), Notices of the American Mathematical Society, 59 (5): 680–683, May 2012 6. Gazette staff (November 6, 2017), "CSUCI Professor of Mathematics wins national mentoring award", The Fillmore Gazette 7. "Cynthia J. Wyels - Faculty Biographies- CSU Channel Islands". ciapps.csuci.edu. Retrieved 2020-10-22. Authority control: Academics • MathSciNet • Mathematics Genealogy Project
Wikipedia
Navier–Stokes equations The Navier–Stokes equations (/nævˈjeɪ stoʊks/ nav-YAY STOHKS) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes). Claude-Louis Navier George Gabriel Stokes Part of a series on Continuum mechanics $J=-D{\frac {d\varphi }{dx}}$ Fick's laws of diffusion Laws Conservations • Mass • Momentum • Energy Inequalities • Clausius–Duhem (entropy) Solid mechanics • Deformation • Elasticity • linear • Plasticity • Hooke's law • Stress • Finite strain • Infinitesimal strain • Compatibility • Bending • Contact mechanics • frictional • Material failure theory • Fracture mechanics Fluid mechanics Fluids • Statics · Dynamics • Archimedes' principle · Bernoulli's principle • Navier–Stokes equations • Poiseuille equation · Pascal's law • Viscosity • (Newtonian · non-Newtonian) • Buoyancy · Mixing · Pressure Liquids • Adhesion • Capillary action • Chromatography • Cohesion (chemistry) • Surface tension Gases • Atmosphere • Boyle's law • Charles's law • Combined gas law • Fick's law • Gay-Lussac's law • Graham's law Plasma Rheology • Viscoelasticity • Rheometry • Rheometer Smart fluids • Electrorheological • Magnetorheological • Ferrofluids Scientists • Bernoulli • Boyle • Cauchy • Charles • Euler • Fick • Gay-Lussac • Graham • Hooke • Newton • Navier • Noll • Pascal • Stokes • Truesdell The Navier–Stokes equations mathematically express momentum balance and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density.[1] They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing viscous flow. The difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity into account while the Euler equations model only inviscid flow. As a result, the Navier–Stokes are a parabolic equation and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never completely integrable). The Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations, they can be used to model and study magnetohydrodynamics. The Navier–Stokes equations are also of great interest in a purely mathematical sense. Despite their wide range of practical uses, it has not yet been proven whether smooth solutions always exist in three dimensions—i.e., whether they are infinitely differentiable (or even just bounded) at all points in the domain. This is called the Navier–Stokes existence and smoothness problem. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1 million prize for a solution or a counterexample.[2][3] Flow velocity The solution of the equations is a flow velocity. It is a vector field—to every point in a fluid, at any moment in a time interval, it gives a vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time. It is usually studied in three spatial dimensions and one time dimension, although two (spatial) dimensional and steady-state cases are often used as models, and higher-dimensional analogues are studied in both pure and applied mathematics. Once the velocity field is calculated, other quantities of interest such as pressure or temperature may be found using dynamical equations and relations. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid, although for visualization purposes one can compute various trajectories. In particular, the streamlines of a vector field, interpreted as flow velocity, are the paths along which a massless fluid particle would travel. These paths are the integral curves whose derivative at each point is equal to the vector field, and they can represent visually the behavior of the vector field at a point in time. General continuum equations See also: Cauchy momentum equation § Conservation form The Navier–Stokes momentum equation can be derived as a particular form of the Cauchy momentum equation, whose general convective form is ${\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}={\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {g} .$ By setting the Cauchy stress tensor $ {\boldsymbol {\sigma }}$ to be the sum of a viscosity term $ {\boldsymbol {\tau }}$ (the deviatoric stress) and a pressure term $ -p\mathbf {I} $ (volumetric stress), we arrive at Cauchy momentum equation (convective form) $\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \,\mathbf {g} $ where • $ {\frac {\mathrm {D} }{\mathrm {D} t}}$ is the material derivative, defined as $ {\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla $, • $ \rho $ is the (mass) density, • $ \mathbf {u} $ is the flow velocity, • $ \nabla \cdot \,$ is the divergence, • $ p$ is the pressure, • $ t$ is time, • $ {\boldsymbol {\tau }}$ is the deviatoric stress tensor, which has order 2, • $ \mathbf {g} $ represents body accelerations acting on the continuum, for example gravity, inertial accelerations, electrostatic accelerations, and so on. In this form, it is apparent that in the assumption of an inviscid fluid – no deviatoric stress – Cauchy equations reduce to the Euler equations. Assuming conservation of mass we can use the mass continuity equation (or simply continuity equation), ${\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \,\mathbf {u} )=0$ to arrive at the conservation form of the equations of motion. This is often written:[4] Cauchy momentum equation (conservation form) ${\frac {\partial }{\partial t}}(\rho \,\mathbf {u} )+\nabla \cdot (\rho \,\mathbf {u} \otimes \mathbf {u} )=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \,\mathbf {g} $ where $ \otimes $ is the outer product: $\mathbf {u} \otimes \mathbf {v} =\mathbf {u} \mathbf {v} ^{\mathrm {T} }.$ The left side of the equation describes acceleration, and may be composed of time-dependent and convective components (also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity). All non-relativistic balance equations, such as the Navier–Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a constitutive relation. By expressing the deviatoric (shear) stress tensor in terms of viscosity and the fluid velocity gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier–Stokes equations below. Convective acceleration See also: Cauchy momentum equation § Convective acceleration A significant feature of the Cauchy equation and consequently all other continuum equations (including Euler and Navier–Stokes) is the presence of convective acceleration: the effect of acceleration of a flow with respect to space. While individual fluid particles indeed experience time-dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle. Compressible flow Remark: here, the deviatoric stress tensor is denoted $ {\boldsymbol {\sigma }}$ (instead of $ {\boldsymbol {\tau }}$ as it was in the general continuum equations and in the incompressible flow section). The compressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:[5] • the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient $ \nabla \mathbf {u} $. • the stress is linear in this variable: $ {\boldsymbol {\sigma }}\left(\nabla \mathbf {u} \right)=\mathbf {C} :\left(\nabla \mathbf {u} \right)$ :\left(\nabla \mathbf {u} \right)} , where $ \mathbf {C} $ is the fourth-order tensor representing the constant of proportionality, called the viscosity or elasticity tensor, and : is the double-dot product. • the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently $ \mathbf {V} $ is an isotropic tensor; furthermore, since the stress tensor is symmetric, by Helmholtz decomposition it can be expressed in terms of two scalar Lamé parameters, the second viscosity $ \lambda $ and the dynamic viscosity $ \mu $, as it is usual in linear elasticity: Linear stress constitutive equation (expression used for elastic solid) ${\boldsymbol {\sigma }}=\lambda (\nabla \cdot \mathbf {u} )\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}$ where $ \mathbf {I} $ is the identity tensor, $ {\boldsymbol {\varepsilon }}\left(\nabla \mathbf {u} \right)\equiv {\frac {1}{2}}\nabla \mathbf {u} +{\frac {1}{2}}\left(\nabla \mathbf {u} \right)^{T}$ is the rate-of-strain tensor and $ \nabla \cdot \mathbf {u} $ is the divergence (i.e. rate of expansion) of the flow. So this decomposition can be explicitly defined as: ${\boldsymbol {\sigma }}=\lambda (\nabla \cdot \mathbf {u} )\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right).$ Since the trace of the rate-of-strain tensor in three dimensions is: $\operatorname {tr} ({\boldsymbol {\varepsilon }})=\nabla \cdot \mathbf {u} .$ The trace of the stress tensor in three dimensions becomes: $\operatorname {tr} ({\boldsymbol {\sigma }})=(3\lambda +2\mu )\nabla \cdot \mathbf {u} .$ So by alternatively decomposing the stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics:[6] ${\boldsymbol {\sigma }}=\left(\lambda +{\tfrac {2}{3}}\mu \right)\left(\nabla \cdot \mathbf {u} \right)\mathbf {I} +\mu \left(\nabla \mathbf {u} +\left(\nabla \mathbf {u} \right)^{\mathrm {T} }-{\tfrac {2}{3}}\left(\nabla \cdot \mathbf {u} \right)\mathbf {I} \right)$ Introducing the bulk viscosity $ \zeta $, $\zeta \equiv \lambda +{\tfrac {2}{3}}\mu ,$ we arrive to the linear constitutive equation in the form usually employed in thermal hydraulics:[5] Linear stress constitutive equation (expression used for fluids) ${\boldsymbol {\sigma }}=\zeta (\nabla \cdot \mathbf {u} )\mathbf {I} +\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]$ Both second viscosity $ \zeta $ and dynamic viscosity $ \mu $ need not be constant – in general, they depend on two thermodynamics variables if the fluid contains a single chemical species, say for example, pressure and temperature. Any equation that makes explicit one of these transport coefficient in the conservation variables is called an equation of state.[7] The most general of the Navier–Stokes equations become Navier–Stokes momentum equation (convective form) $\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} \right)=-\nabla p+\nabla \cdot \left\{\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]+\zeta (\nabla \cdot \mathbf {u} )\mathbf {I} \right\}+\rho \mathbf {g} .$ Apart from its dependence of pressure and temperature, the second viscosity coefficient also depends on the process, that is to say, the second viscosity coefficient is not just a material property. For instance, in the case of a sound wave with a definitive frequency that alternatively compresses and expands a fluid element, the second viscosity coefficient depends on the frequency of the wave. This dependence is called the dispersion. In some cases, the second viscosity $ \zeta $ can be assumed to be constant in which case, the effect of the volume viscosity $ \zeta $ is that the mechanical pressure is not equivalent to the thermodynamic pressure:[8] as demonstrated below. $\nabla \cdot (\nabla \cdot \mathbf {u} )\mathbf {I} =\nabla (\nabla \cdot \mathbf {u} ),$ ${\bar {p}}\equiv p-\zeta \,\nabla \cdot \mathbf {u} ,$ However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves,[9] where second viscosity coefficient becomes important) by explicitly assuming $ \zeta =0$. The assumption of setting $ \zeta =0$ is called as the Stokes hypothesis.[10] The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory,;[11] for other gases and liquids, Stokes hypothesis is generally incorrect. With the Stokes hypothesis, the Navier–Stokes equations become Navier–Stokes momentum equation (convective form) $\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} \right)=-\nabla p+\nabla \cdot \left\{\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]\right\}+\rho \mathbf {g} .$ If the dynamic viscosity μ is also assumed to be constant, the equations can be simplified further. By computing the divergence of the stress tensor, since the divergence of tensor $ \nabla \mathbf {u} $ is $ \nabla ^{2}\mathbf {u} $ and the divergence of tensor $ \left(\nabla \mathbf {u} \right)^{\mathrm {T} }$ is $ \nabla \left(\nabla \cdot \mathbf {u} \right)$, one finally arrives to the compressible (most general) Navier–Stokes momentum equation:[12] Navier–Stokes momentum equation (convective form) $\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} \right)=-\nabla p+\mu \,\nabla ^{2}\mathbf {u} +{\tfrac {1}{3}}\mu \,\nabla (\nabla \cdot \mathbf {u} )+\rho \mathbf {g} .$ where $ {\frac {\mathrm {D} }{\mathrm {D} t}}$ is the material derivative. The left-hand side changes in the conservation form of the Navier–Stokes momentum equation: Navier–Stokes momentum equation (conservation form) ${\frac {\partial }{\partial t}}(\rho \,\mathbf {u} )+\nabla \cdot (\rho \,\mathbf {u} \otimes \mathbf {u} )=-\nabla p+\mu \,\nabla ^{2}\mathbf {u} +{\tfrac {1}{3}}\mu \,\nabla (\nabla \cdot \mathbf {u} )+\rho \mathbf {g} .$ Bulk viscosity is assumed to be constant, otherwise it should not be taken out of the last derivative. The convective acceleration term can also be written as $\mathbf {u} \cdot \nabla \mathbf {u} =(\nabla \times \mathbf {u} )\times \mathbf {u} +{\tfrac {1}{2}}\nabla \mathbf {u} ^{2},$ where the vector $ (\nabla \times \mathbf {u} )\times \mathbf {u} $ is known as the Lamb vector. For the special case of an incompressible flow, the pressure constrains the flow so that the volume of fluid elements is constant: isochoric flow resulting in a solenoidal velocity field with $ \nabla \cdot \mathbf {u} =0$.[13] Incompressible flow The incompressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:[5] • the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient $ \nabla \mathbf {u} $. • the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently $ {\boldsymbol {\tau }}$ is an isotropic tensor; furthermore, since the deviatoric stress tensor can be expressed in terms of the dynamic viscosity $ \mu $: Stokes' stress constitutive equation (expression used for incompressible elastic solids) ${\boldsymbol {\tau }}=2\mu {\boldsymbol {\varepsilon }}$ where ${\boldsymbol {\varepsilon }}={\tfrac {1}{2}}\left(\mathbf {\nabla u} +\mathbf {\nabla u} ^{\mathrm {T} }\right)$ is the rate-of-strain tensor. So this decomposition can be made explicit as:[5] Stokes's stress constitutive equation (expression used for incompressible viscous fluids) ${\boldsymbol {\tau }}=\mu \left(\nabla \mathbf {u} +\nabla \mathbf {u} ^{\mathrm {T} }\right)$ Dynamic viscosity μ need not be constant – in incompressible flows it can depend on density and on pressure. Any equation that makes explicit one of these transport coefficient in the conservative variables is called an equation of state.[7] The divergence of the deviatoric stress is given by: $\nabla \cdot {\boldsymbol {\tau }}=2\mu \nabla \cdot {\boldsymbol {\varepsilon }}=\mu \nabla \cdot \left(\nabla \mathbf {u} +\nabla \mathbf {u} ^{\mathrm {T} }\right)=\mu \,\nabla ^{2}\mathbf {u} $ because $ \nabla \cdot \mathbf {u} =0$ for an incompressible fluid. Incompressibility rules out density and pressure waves like sound or shock waves, so this simplification is not useful if these phenomena are of interest. The incompressible flow assumption typically holds well with all fluids at low Mach numbers (say up to about Mach 0.3), such as for modelling air winds at normal temperatures.[14] the incompressible Navier–Stokes equations are best visualized by dividing for the density:[15] Incompressible Navier–Stokes equations (convective form) ${\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\mathbf {g} .$ If the density is constant throughout the fluid domain, or, in other words, if all fluid elements have the same density, $ \rho =\rho _{0}$, then we have Incompressible Navier–Stokes equations (convective form) ${\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-\nabla \left({\frac {p}{\rho _{0}}}\right)+\mathbf {g} ,$ where $ \nu ={\frac {\mu }{\rho _{0}}}$ is called the kinematic viscosity. A laminar flow example Velocity profile (laminar flow): $u_{x}=u(y),\quad u_{y}=0,\quad u_{z}=0$ for the x-direction, simplify the Navier–Stokes equation: $0=-{\frac {\mathrm {d} P}{\mathrm {d} x}}+\mu \left({\frac {\mathrm {d} ^{2}u}{\mathrm {d} y^{2}}}\right)$ Integrate twice to find the velocity profile with boundary conditions y = h, u = 0, y = −h, u = 0: $u={\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}y^{2}+Ay+B$ From this equation, substitute in the two boundary conditions to get two equations: ${\begin{aligned}0&={\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}h^{2}+Ah+B\\0&={\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}h^{2}-Ah+B\end{aligned}}$ Add and solve for B: $B=-{\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}h^{2}$ Substitute and solve for A: $A=0$ Finally this gives the velocity profile: $u={\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}\left(y^{2}-h^{2}\right)$ It is well worth observing the meaning of each term (compare to the Cauchy momentum equation): $\overbrace {{\vphantom {\frac {}{}}}\underbrace {\frac {\partial \mathbf {u} }{\partial t}} _{\text{Variation}}+\underbrace {{\vphantom {\frac {}{}}}(\mathbf {u} \cdot \nabla )\mathbf {u} } _{\text{Divergence}}} ^{\text{Inertia (per volume)}}=\overbrace {{\vphantom {\frac {\partial }{\partial }}}\underbrace {{\vphantom {\frac {}{}}}-\nabla w} _{\begin{smallmatrix}{\text{Internal}}\\{\text{source}}\end{smallmatrix}}+\underbrace {{\vphantom {\frac {}{}}}\nu \nabla ^{2}\mathbf {u} } _{\text{Diffusion}}} ^{\text{Divergence of stress}}+\underbrace {{\vphantom {\frac {}{}}}\mathbf {g} } _{\begin{smallmatrix}{\text{External}}\\{\text{source}}\end{smallmatrix}}.$ The higher-order term, namely the shear stress divergence $ \nabla \cdot {\boldsymbol {\tau }}$, has simply reduced to the vector Laplacian term $ \mu \nabla ^{2}\mathbf {u} $.[16] This Laplacian term can be interpreted as the difference between the velocity at a point and the mean velocity in a small surrounding volume. This implies that – for a Newtonian fluid – viscosity operates as a diffusion of momentum, in much the same way as the heat conduction. In fact neglecting the convection term, incompressible Navier–Stokes equations lead to a vector diffusion equation (namely Stokes equations), but in general the convection term is present, so incompressible Navier–Stokes equations belong to the class of convection–diffusion equations. In the usual case of an external field being a conservative field: $\mathbf {g} =-\nabla \varphi $ by defining the hydraulic head: $h\equiv w+\varphi $ one can finally condense the whole source in one term, arriving to the incompressible Navier–Stokes equation with conservative external field: ${\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-\nabla h.$ The incompressible Navier–Stokes equations with conservative external field is the fundamental equation of hydraulics. The domain for these equations is commonly a 3 or less dimensional Euclidean space, for which an orthogonal coordinate reference frame is usually set to explicit the system of scalar partial differential equations to be solved. In 3-dimensional orthogonal coordinate systems are 3: Cartesian, cylindrical, and spherical. Expressing the Navier–Stokes vector equation in Cartesian coordinates is quite straightforward and not much influenced by the number of dimensions of the euclidean space employed, and this is the case also for the first-order terms (like the variation and convection ones) also in non-cartesian orthogonal coordinate systems. But for the higher order terms (the two coming from the divergence of the deviatoric stress that distinguish Navier–Stokes equations from Euler equations) some tensor calculus is required for deducing an expression in non-cartesian orthogonal coordinate systems. The incompressible Navier–Stokes equation is composite, the sum of two orthogonal equations, ${\begin{aligned}{\frac {\partial \mathbf {u} }{\partial t}}&=\Pi ^{S}\left(-(\mathbf {u} \cdot \nabla )\mathbf {u} +\nu \,\nabla ^{2}\mathbf {u} \right)+\mathbf {f} ^{S}\\\rho ^{-1}\,\nabla p&=\Pi ^{I}\left(-(\mathbf {u} \cdot \nabla )\mathbf {u} +\nu \,\nabla ^{2}\mathbf {u} \right)+\mathbf {f} ^{I}\end{aligned}}$ where $ \Pi ^{S}$ and $ \Pi ^{I}$ are solenoidal and irrotational projection operators satisfying $ \Pi ^{S}+\Pi ^{I}-1$ and $ \mathbf {f} ^{S}$ and $ \mathbf {f} ^{I}$ are the non-conservative and conservative parts of the body force. This result follows from the Helmholtz theorem (also known as the fundamental theorem of vector calculus). The first equation is a pressureless governing equation for the velocity, while the second equation for the pressure is a functional of the velocity and is related to the pressure Poisson equation. The explicit functional form of the projection operator in 3D is found from the Helmholtz Theorem: $\Pi ^{S}\,\mathbf {F} (\mathbf {r} )={\frac {1}{4\pi }}\nabla \times \int {\frac {\nabla ^{\prime }\times \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} V',\quad \Pi ^{I}=1-\Pi ^{S}$ with a similar structure in 2D. Thus the governing equation is an integro-differential equation similar to Coulomb and Biot–Savart law, not convenient for numerical computation. An equivalent weak or variational form of the equation, proved to produce the same velocity solution as the Navier–Stokes equation,[17] is given by, $\left(\mathbf {w} ,{\frac {\partial \mathbf {u} }{\partial t}}\right)=-{\bigl (}\mathbf {w} ,\left(\mathbf {u} \cdot \nabla \right)\mathbf {u} {\bigr )}-\nu \left(\nabla \mathbf {w} :\nabla \mathbf {u} \right)+\left(\mathbf {w} ,\mathbf {f} ^{S}\right)$ :\nabla \mathbf {u} \right)+\left(\mathbf {w} ,\mathbf {f} ^{S}\right)} for divergence-free test functions $ \mathbf {w} $ satisfying appropriate boundary conditions. Here, the projections are accomplished by the orthogonality of the solenoidal and irrotational function spaces. The discrete form of this is eminently suited to finite element computation of divergence-free flow, as we shall see in the next section. There one will be able to address the question "How does one specify pressure-driven (Poiseuille) problems with a pressureless governing equation?". The absence of pressure forces from the governing velocity equation demonstrates that the equation is not a dynamic one, but rather a kinematic equation where the divergence-free condition serves the role of a conservation equation. This all would seem to refute the frequent statements that the incompressible pressure enforces the divergence-free condition. Strong form Consider the incompressible Navier–Stokes equations for a Newtonian fluid of constant density $ \rho $ in a domain $\Omega \subset \mathbb {R} ^{d}\quad (d=2,3)$ with boundary $\partial \Omega =\Gamma _{D}\cup \Gamma _{N},$ being $ \Gamma _{D}$ and $ \Gamma _{N}$ portions of the boundary where respectively a Dirichlet and a Neumann boundary condition is applied ($ \Gamma _{D}\cap \Gamma _{N}=\emptyset $):[18] ${\begin{cases}\rho {\dfrac {\partial \mathbf {u} }{\partial t}}+\rho (\mathbf {u} \cdot \nabla )\mathbf {u} -\nabla \cdot {\boldsymbol {\sigma }}(\mathbf {u} ,p)=\mathbf {f} &{\text{ in }}\Omega \times (0,T)\\\nabla \cdot \mathbf {u} =0&{\text{ in }}\Omega \times (0,T)\\\mathbf {u} =\mathbf {g} &{\text{ on }}\Gamma _{D}\times (0,T)\\{\boldsymbol {\sigma }}(\mathbf {u} ,p){\hat {\mathbf {n} }}=\mathbf {h} &{\text{ on }}\Gamma _{N}\times (0,T)\\\mathbf {u} (0)=\mathbf {u} _{0}&{\text{ in }}\Omega \times \{0\}\end{cases}}$ $ \mathbf {u} $ is the fluid velocity, $ p$ the fluid pressure, $ \mathbf {f} $ a given forcing term, ${\hat {\mathbf {n} }}$ the outward directed unit normal vector to $ \Gamma _{N}$, and $ {\boldsymbol {\sigma }}(\mathbf {u} ,p)$ the viscous stress tensor defined as:[18] ${\boldsymbol {\sigma }}(\mathbf {u} ,p)=-p\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}(\mathbf {u} ).$ Let $ \mu $ be the dynamic viscosity of the fluid, $ \mathbf {I} $ the second-order identity tensor and $ {\boldsymbol {\varepsilon }}(\mathbf {u} )$ the strain-rate tensor defined as:[18] ${\boldsymbol {\varepsilon }}(\mathbf {u} )={\frac {1}{2}}\left(\left(\nabla \mathbf {u} \right)+\left(\nabla \mathbf {u} \right)^{\mathrm {T} }\right).$ The functions $ \mathbf {g} $ and $ \mathbf {h} $ are given Dirichlet and Neumann boundary data, while $ \mathbf {u} _{0}$ is the initial condition. The first equation is the momentum balance equation, while the second represents the mass conservation, namely the continuity equation. Assuming constant dynamic viscosity, using the vectorial identity $\nabla \cdot \left(\nabla \mathbf {f} \right)^{\mathrm {T} }=\nabla (\nabla \cdot \mathbf {f} )$ and exploiting mass conservation, the divergence of the total stress tensor in the momentum equation can also be expressed as:[18] ${\begin{aligned}\nabla \cdot {\boldsymbol {\sigma }}(\mathbf {u} ,p)&=\nabla \cdot \left(-p\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}(\mathbf {u} )\right)\\&=-\nabla p+2\mu \nabla \cdot {\boldsymbol {\varepsilon }}(\mathbf {u} )\\&=-\nabla p+2\mu \nabla \cdot \left[{\tfrac {1}{2}}\left(\left(\nabla \mathbf {u} \right)+\left(\nabla \mathbf {u} \right)^{\mathrm {T} }\right)\right]\\&=-\nabla p+\mu \left(\Delta \mathbf {u} +\nabla \cdot \left(\nabla \mathbf {u} \right)^{\mathrm {T} }\right)\\&=-\nabla p+\mu {\bigl (}\Delta \mathbf {u} +\nabla \underbrace {(\nabla \cdot \mathbf {u} )} _{=0}{\bigr )}=-\nabla p+\mu \,\Delta \mathbf {u} .\end{aligned}}$ Moreover, note that the Neumann boundary conditions can be rearranged as:[18] ${\boldsymbol {\sigma }}(\mathbf {u} ,p){\hat {\mathbf {n} }}=\left(-p\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}(\mathbf {u} )\right){\hat {\mathbf {n} }}=-p{\hat {\mathbf {n} }}+\mu {\frac {\partial {\boldsymbol {u}}}{\partial {\hat {\mathbf {n} }}}}.$ Weak form In order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation[18] $\rho {\frac {\partial \mathbf {u} }{\partial t}}-\mu \Delta \mathbf {u} +\rho (\mathbf {u} \cdot \nabla )\mathbf {u} +\nabla p=\mathbf {f} $ multiply it for a test function $ \mathbf {v} $, defined in a suitable space $ V$, and integrate both members with respect to the domain $ \Omega $:[18] $\int \limits _{\Omega }\rho {\frac {\partial \mathbf {u} }{\partial t}}\cdot \mathbf {v} -\int \limits _{\Omega }\mu \Delta \mathbf {u} \cdot \mathbf {v} +\int \limits _{\Omega }\rho (\mathbf {u} \cdot \nabla )\mathbf {u} \cdot \mathbf {v} +\int \limits _{\Omega }\nabla p\cdot \mathbf {v} =\int \limits _{\Omega }\mathbf {f} \cdot \mathbf {v} $ Counter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem:[18] ${\begin{aligned}-\int \limits _{\Omega }\mu \Delta \mathbf {u} \cdot \mathbf {v} &=\int _{\Omega }\mu \nabla \mathbf {u} \cdot \nabla \mathbf {v} -\int \limits _{\partial \Omega }\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}\cdot \mathbf {v} \\\int \limits _{\Omega }\nabla p\cdot \mathbf {v} &=-\int \limits _{\Omega }p\nabla \cdot \mathbf {v} +\int \limits _{\partial \Omega }p\mathbf {v} \cdot {\hat {\mathbf {n} }}\end{aligned}}$ Using these relations, one gets:[18] $\int \limits _{\Omega }\rho {\dfrac {\partial \mathbf {u} }{\partial t}}\cdot \mathbf {v} +\int \limits _{\Omega }\mu \nabla \mathbf {u} \cdot \nabla \mathbf {v} +\int \limits _{\Omega }\rho (\mathbf {u} \cdot \nabla )\mathbf {u} \cdot \mathbf {v} -\int \limits _{\Omega }p\nabla \cdot \mathbf {v} =\int \limits _{\Omega }\mathbf {f} \cdot \mathbf {v} +\int \limits _{\partial \Omega }\left(\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}-p{\hat {\mathbf {n} }}\right)\cdot \mathbf {v} \quad \forall \mathbf {v} \in V.$ In the same fashion, the continuity equation is multiplied for a test function q belonging to a space $ Q$ and integrated in the domain $ \Omega $:[18] $\int \limits _{\Omega }q\nabla \cdot \mathbf {u} =0.\quad \forall q\in Q.$ The space functions are chosen as follows: ${\begin{aligned}V=\left[H_{0}^{1}(\Omega )\right]^{d}&=\left\{\mathbf {v} \in \left[H^{1}(\Omega )\right]^{d}:\quad \mathbf {v} =\mathbf {0} {\text{ on }}\Gamma _{D}\right\},\\Q&=L^{2}(\Omega )\end{aligned}}$ Considering that the test function v vanishes on the Dirichlet boundary and considering the Neumann condition, the integral on the boundary can be rearranged as:[18] $\int \limits _{\partial \Omega }\left(\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}-p{\hat {\mathbf {n} }}\right)\cdot \mathbf {v} =\underbrace {\int \limits _{\Gamma _{D}}\left(\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}-p{\hat {\mathbf {n} }}\right)\cdot \mathbf {v} } _{\mathbf {v} =\mathbf {0} {\text{ on }}\Gamma _{D}\ }+\int \limits _{\Gamma _{N}}\underbrace {{\vphantom {\int \limits _{\Gamma _{N}}}}\left(\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}-p{\hat {\mathbf {n} }}\right)} _{=\mathbf {h} {\text{ on }}\Gamma _{N}}\cdot \mathbf {v} =\int \limits _{\Gamma _{N}}\mathbf {h} \cdot \mathbf {v} .$ Having this in mind, the weak formulation of the Navier–Stokes equations is expressed as:[18] ${\begin{aligned}&{\text{find }}\mathbf {u} \in L^{2}\left(\mathbb {R} ^{+}\;\left[H^{1}(\Omega )\right]^{d}\right)\cap C^{0}\left(\mathbb {R} ^{+}\;\left[L^{2}(\Omega )\right]^{d}\right){\text{ such that: }}\\[5pt]&\quad {\begin{cases}\displaystyle \int \limits _{\Omega }\rho {\dfrac {\partial \mathbf {u} }{\partial t}}\cdot \mathbf {v} +\int \limits _{\Omega }\mu \nabla \mathbf {u} \cdot \nabla \mathbf {v} +\int \limits _{\Omega }\rho (\mathbf {u} \cdot \nabla )\mathbf {u} \cdot \mathbf {v} -\int \limits _{\Omega }p\nabla \cdot \mathbf {v} =\int \limits _{\Omega }\mathbf {f} \cdot \mathbf {v} +\int \limits _{\Gamma _{N}}\mathbf {h} \cdot \mathbf {v} \quad \forall \mathbf {v} \in V,\\\displaystyle \int \limits _{\Omega }q\nabla \cdot \mathbf {u} =0\quad \forall q\in Q.\end{cases}}\end{aligned}}$ Discrete velocity With partitioning of the problem domain and defining basis functions on the partitioned domain, the discrete form of the governing equation is $\left(\mathbf {w} _{i},{\frac {\partial \mathbf {u} _{j}}{\partial t}}\right)=-{\bigl (}\mathbf {w} _{i},\left(\mathbf {u} \cdot \nabla \right)\mathbf {u} _{j}{\bigr )}-\nu \left(\nabla \mathbf {w} _{i}:\nabla \mathbf {u} _{j}\right)+\left(\mathbf {w} _{i},\mathbf {f} ^{S}\right).$ It is desirable to choose basis functions that reflect the essential feature of incompressible flow – the elements must be divergence-free. While the velocity is the variable of interest, the existence of the stream function or vector potential is necessary by the Helmholtz theorem. Further, to determine fluid flow in the absence of a pressure gradient, one can specify the difference of stream function values across a 2D channel, or the line integral of the tangential component of the vector potential around the channel in 3D, the flow being given by Stokes' theorem. Discussion will be restricted to 2D in the following. We further restrict discussion to continuous Hermite finite elements which have at least first-derivative degrees-of-freedom. With this, one can draw a large number of candidate triangular and rectangular elements from the plate-bending literature. These elements have derivatives as components of the gradient. In 2D, the gradient and curl of a scalar are clearly orthogonal, given by the expressions, ${\begin{aligned}\nabla \varphi &=\left({\frac {\partial \varphi }{\partial x}},\,{\frac {\partial \varphi }{\partial y}}\right)^{\mathrm {T} },\\[5pt]\nabla \times \varphi &=\left({\frac {\partial \varphi }{\partial y}},\,-{\frac {\partial \varphi }{\partial x}}\right)^{\mathrm {T} }.\end{aligned}}$ Adopting continuous plate-bending elements, interchanging the derivative degrees-of-freedom and changing the sign of the appropriate one gives many families of stream function elements. Taking the curl of the scalar stream function elements gives divergence-free velocity elements.[19][20] The requirement that the stream function elements be continuous assures that the normal component of the velocity is continuous across element interfaces, all that is necessary for vanishing divergence on these interfaces. Boundary conditions are simple to apply. The stream function is constant on no-flow surfaces, with no-slip velocity conditions on surfaces. Stream function differences across open channels determine the flow. No boundary conditions are necessary on open boundaries, though consistent values may be used with some problems. These are all Dirichlet conditions. The algebraic equations to be solved are simple to set up, but of course are non-linear, requiring iteration of the linearized equations. Similar considerations apply to three-dimensions, but extension from 2D is not immediate because of the vector nature of the potential, and there exists no simple relation between the gradient and the curl as was the case in 2D. Pressure recovery Recovering pressure from the velocity field is easy. The discrete weak equation for the pressure gradient is, $(\mathbf {g} _{i},\nabla p)=-\left(\mathbf {g} _{i},\left(\mathbf {u} \cdot \nabla \right)\mathbf {u} _{j}\right)-\nu \left(\nabla \mathbf {g} _{i}:\nabla \mathbf {u} _{j}\right)+\left(\mathbf {g} _{i},\mathbf {f} ^{I}\right)$ where the test/weight functions are irrotational. Any conforming scalar finite element may be used. However, the pressure gradient field may also be of interest. In this case, one can use scalar Hermite elements for the pressure. For the test/weight functions $ \mathbf {g} _{i}$ one would choose the irrotational vector elements obtained from the gradient of the pressure element. Non-inertial frame of reference The rotating frame of reference introduces some interesting pseudo-forces into the equations through the material derivative term. Consider a stationary inertial frame of reference $ K$ , and a non-inertial frame of reference $ K'$, which is translating with velocity $ \mathbf {U} (t)$ and rotating with angular velocity $ \Omega (t)$ with respect to the stationary frame. The Navier–Stokes equation observed from the non-inertial frame then becomes Navier–Stokes momentum equation in non-inertial frame $\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=-\nabla {\bar {p}}+\mu \,\nabla ^{2}\mathbf {u} +{\tfrac {1}{3}}\mu \,\nabla (\nabla \cdot \mathbf {u} )+\rho \mathbf {g} -\rho \left[2\mathbf {\Omega } \times \mathbf {u} +\mathbf {\Omega } \times (\mathbf {\Omega } \times \mathbf {x} )+{\frac {\mathrm {d} \mathbf {U} }{\mathrm {d} t}}+{\frac {\mathrm {d} \mathbf {\Omega } }{\mathrm {d} t}}\times \mathbf {x} \right].$ Here $ \mathbf {x} $ and $ \mathbf {u} $ are measured in the non-inertial frame. The first term in the parenthesis represents Coriolis acceleration, the second term is due to centrifugal acceleration, the third is due to the linear acceleration of $ K'$ with respect to $ K$ and the fourth term is due to the angular acceleration of $ K'$ with respect to $ K$. Other equations The Navier–Stokes equations are strictly a statement of the balance of momentum. To fully describe fluid flow, more information is needed, how much depending on the assumptions made. This additional information may include boundary data (no-slip, capillary surface, etc.), conservation of mass, balance of energy, and/or an equation of state. Continuity equation for incompressible fluid Regardless of the flow assumptions, a statement of the conservation of mass is generally necessary. This is achieved through the mass continuity equation, given in its most general form as: ${\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0$ or, using the substantive derivative: ${\frac {\mathrm {D} \rho }{\mathrm {D} t}}+\rho (\nabla \cdot \mathbf {u} )=0.$ For incompressible fluid, density along the line of flow remains constant over time, ${\frac {\mathrm {D} \rho }{\mathrm {D} t}}=0.$ Therefore divergence of velocity is always zero: $\nabla \cdot \mathbf {u} =0.$ Stream function for incompressible 2D fluid Taking the curl of the incompressible Navier–Stokes equation results in the elimination of pressure. This is especially easy to see if 2D Cartesian flow is assumed (like in the degenerate 3D case with $ u_{z}=0$ and no dependence of anything on $ z$), where the equations reduce to: ${\begin{aligned}\rho \left({\frac {\partial u_{x}}{\partial t}}+u_{x}{\frac {\partial u_{x}}{\partial x}}+u_{y}{\frac {\partial u_{x}}{\partial y}}\right)&=-{\frac {\partial p}{\partial x}}+\mu \left({\frac {\partial ^{2}u_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}u_{x}}{\partial y^{2}}}\right)+\rho g_{x}\\\rho \left({\frac {\partial u_{y}}{\partial t}}+u_{x}{\frac {\partial u_{y}}{\partial x}}+u_{y}{\frac {\partial u_{y}}{\partial y}}\right)&=-{\frac {\partial p}{\partial y}}+\mu \left({\frac {\partial ^{2}u_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}u_{y}}{\partial y^{2}}}\right)+\rho g_{y}.\end{aligned}}$ Differentiating the first with respect to $ y$, the second with respect to $ x$ and subtracting the resulting equations will eliminate pressure and any conservative force. For incompressible flow, defining the stream function $ \psi $ through $u_{x}={\frac {\partial \psi }{\partial y}};\quad u_{y}=-{\frac {\partial \psi }{\partial x}}$ results in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum and mass conservation condense into one equation: ${\frac {\partial }{\partial t}}\left(\nabla ^{2}\psi \right)+{\frac {\partial \psi }{\partial y}}{\frac {\partial }{\partial x}}\left(\nabla ^{2}\psi \right)-{\frac {\partial \psi }{\partial x}}{\frac {\partial }{\partial y}}\left(\nabla ^{2}\psi \right)=\nu \nabla ^{4}\psi $ where $ \nabla ^{4}$ is the 2D biharmonic operator and $ \nu $ is the kinematic viscosity, $ \nu ={\frac {\mu }{p}}$. We can also express this compactly using the Jacobian determinant: ${\frac {\partial }{\partial t}}\left(\nabla ^{2}\psi \right)+{\frac {\partial \left(\psi ,\nabla ^{2}\psi \right)}{\partial (y,x)}}=\nu \nabla ^{4}\psi .$ This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter. Note that the equation for creeping flow results when the left side is assumed zero. In axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describe the velocity components of an incompressible flow with one scalar function. The incompressible Navier–Stokes equation is a differential algebraic equation, having the inconvenient feature that there is no explicit mechanism for advancing the pressure in time. Consequently, much effort has been expended to eliminate the pressure from all or part of the computational process. The stream function formulation eliminates the pressure but only in two dimensions and at the expense of introducing higher derivatives and elimination of the velocity, which is the primary variable of interest. Properties Nonlinearity The Navier–Stokes equations are nonlinear partial differential equations in the general case and so remain in almost every real situation.[21][22] In some cases, such as one-dimensional flow and Stokes flow (or creeping flow), the equations can be simplified to linear equations. The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the turbulence that the equations model. The nonlinearity is due to convective acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity. An example of convective but laminar (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging nozzle. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood.[23] Turbulence Turbulence is the time-dependent chaotic behaviour seen in many fluid flows. It is generally believed that it is due to the inertia of the fluid as a whole: the culmination of time-dependent and convective acceleration; hence flows where inertial effects are small tend to be laminar (the Reynolds number quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly.[24] The numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation or direct numerical simulation. Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately. To counter this, time-averaged equations such as the Reynolds-averaged Navier–Stokes equations (RANS), supplemented with turbulence models, are used in practical computational fluid dynamics (CFD) applications when modeling turbulent flows. Some models include the Spalart–Allmaras, k–ω, k–ε, and SST models, which add a variety of additional equations to bring closure to the RANS equations. Large eddy simulation (LES) can also be used to solve these equations numerically. This approach is computationally more expensive—in time and in computer memory—than RANS, but produces better results because it explicitly resolves the larger turbulent scales. Applicability Together with supplemental equations (for example, conservation of mass) and well-formulated boundary conditions, the Navier–Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agree with real world observations. The Navier–Stokes equations assume that the fluid being studied is a continuum (it is infinitely divisible and not composed of particles such as atoms or molecules), and is not moving at relativistic velocities. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the continuous fluids modeled by the Navier–Stokes equations. For example, capillarity of internal layers in fluids appears for flow with high gradients.[25] For large Knudsen number of the problem, the Boltzmann equation may be a suitable replacement.[26] Failing that, one may have to resort to molecular dynamics or various hybrid methods.[27] Another limitation is simply the complicated nature of the equations. Time-tested formulations exist for common fluid families, but the application of the Navier–Stokes equations to less common families tends to result in very complicated formulations and often to open research problems. For this reason, these equations are usually written for Newtonian fluids where the viscosity model is linear; truly general models for the flow of other kinds of fluids (such as blood) do not exist.[28] Application to specific problems The Navier–Stokes equations, even when written explicitly for specific fluids, are rather generic in nature and their proper application to specific problems can be very diverse. This is partly because there is an enormous variety of problems that may be modeled, ranging from as simple as the distribution of static pressure to as complicated as multiphase flow driven by surface tension. Generally, application to specific problems begins with some flow assumptions and initial/boundary condition formulation, this may be followed by scale analysis to further simplify the problem. Parallel flow Assume steady, parallel, one-dimensional, non-convective pressure-driven flow between parallel plates, the resulting scaled (dimensionless) boundary value problem is: ${\frac {\mathrm {d} ^{2}u}{\mathrm {d} y^{2}}}=-1;\quad u(0)=u(1)=0.$ The boundary condition is the no slip condition. This problem is easily solved for the flow field: $u(y)={\frac {y-y^{2}}{2}}.$ From this point onward, more quantities of interest can be easily obtained, such as viscous drag force or net flow rate. Radial flow Difficulties may arise when the problem becomes slightly more complicated. A seemingly modest twist on the parallel flow above would be the radial flow between parallel plates; this involves convection and thus non-linearity. The velocity field may be represented by a function f(z) that must satisfy: ${\frac {\mathrm {d} ^{2}f}{\mathrm {d} z^{2}}}+Rf^{2}=-1;\quad f(-1)=f(1)=0.$ This ordinary differential equation is what is obtained when the Navier–Stokes equations are written and the flow assumptions applied (additionally, the pressure gradient is solved for). The nonlinear term makes this a very difficult problem to solve analytically (a lengthy implicit solution may be found which involves elliptic integrals and roots of cubic polynomials). Issues with the actual existence of solutions arise for $ R>1.41$ (approximately; this is not √2), the parameter $ R$ being the Reynolds number with appropriately chosen scales.[29] This is an example of flow assumptions losing their applicability, and an example of the difficulty in "high" Reynolds number flows.[29] Convection A type of natural convection that can be described by the Navier–Stokes equation is the Rayleigh–Bénard convection. It is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. Exact solutions of the Navier–Stokes equations Some exact solutions to the Navier–Stokes equations exist. Examples of degenerate cases—with the non-linear terms in the Navier–Stokes equations equal to zero—are Poiseuille flow, Couette flow and the oscillatory Stokes boundary layer. But also, more interesting examples, solutions to the full non-linear equations, exist, such as Jeffery–Hamel flow, Von Kármán swirling flow, stagnation point flow, Landau–Squire jet, and Taylor–Green vortex.[30][31][32] Note that the existence of these exact solutions does not imply they are stable: turbulence may develop at higher Reynolds numbers. Under additional assumptions, the component parts can be separated.[33] A two-dimensional example For example, in the case of an unbounded planar domain with two-dimensional — incompressible and stationary — flow in polar coordinates (r,φ), the velocity components (ur,uφ) and pressure p are:[34] ${\begin{aligned}u_{r}&={\frac {A}{r}},\\u_{\varphi }&=B\left({\frac {1}{r}}-r^{{\frac {A}{\nu }}+1}\right),\\p&=-{\frac {A^{2}+B^{2}}{2r^{2}}}-{\frac {2B^{2}\nu r^{\frac {A}{\nu }}}{A}}+{\frac {B^{2}r^{\left({\frac {2A}{\nu }}+2\right)}}{{\frac {2A}{\nu }}+2}}\end{aligned}}$ where A and B are arbitrary constants. This solution is valid in the domain r ≥ 1 and for A < −2ν. In Cartesian coordinates, when the viscosity is zero (ν = 0), this is: ${\begin{aligned}\mathbf {v} (x,y)&={\frac {1}{x^{2}+y^{2}}}{\begin{pmatrix}Ax+By\\Ay-Bx\end{pmatrix}},\\p(x,y)&=-{\frac {A^{2}+B^{2}}{2\left(x^{2}+y^{2}\right)}}\end{aligned}}$ A three-dimensional example For example, in the case of an unbounded Euclidean domain with three-dimensional — incompressible, stationary and with zero viscosity (ν = 0) — radial flow in Cartesian coordinates (x,y,z), the velocity vector v and pressure p are: ${\begin{aligned}\mathbf {v} (x,y,z)&={\frac {A}{x^{2}+y^{2}+z^{2}}}{\begin{pmatrix}x\\y\\z\end{pmatrix}},\\p(x,y,z)&=-{\frac {A^{2}}{2\left(x^{2}+y^{2}+z^{2}\right)}}.\end{aligned}}$ There is a singularity at x = y = z = 0. A three-dimensional steady-state vortex solution A steady-state example with no singularities comes from considering the flow along the lines of a Hopf fibration. Let $ r$ be a constant radius of the inner coil. One set of solutions is given by:[35] ${\begin{aligned}\rho (x,y,z)&={\frac {3B}{r^{2}+x^{2}+y^{2}+z^{2}}}\\p(x,y,z)&={\frac {-A^{2}B}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{3}}}\\\mathbf {u} (x,y,z)&={\frac {A}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{2}}}{\begin{pmatrix}2(-ry+xz)\\2(rx+yz)\\r^{2}-x^{2}-y^{2}+z^{2}\end{pmatrix}}\\g&=0\\\mu &=0\end{aligned}}$ for arbitrary constants $ A$ and $ B$. This is a solution in a non-viscous gas (compressible fluid) whose density, velocities and pressure goes to zero far from the origin. (Note this is not a solution to the Clay Millennium problem because that refers to incompressible fluids where $ \rho $ is a constant, and neither does it deal with the uniqueness of the Navier–Stokes equations with respect to any turbulence properties.) It is also worth pointing out that the components of the velocity vector are exactly those from the Pythagorean quadruple parametrization. Other choices of density and pressure are possible with the same velocity field: Other choices of density and pressure Another choice of pressure and density with the same velocity vector above is one where the pressure and density fall to zero at the origin and are highest in the central loop at z = 0, x2 + y2 = r2: ${\begin{aligned}\rho (x,y,z)&={\frac {20B\left(x^{2}+y^{2}\right)}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{3}}}\\p(x,y,z)&={\frac {-A^{2}B}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{4}}}+{\frac {-4A^{2}B\left(x^{2}+y^{2}\right)}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{5}}}.\end{aligned}}$ In fact in general there are simple solutions for any polynomial function f where the density is: $\rho (x,y,z)={\frac {1}{r^{2}+x^{2}+y^{2}+z^{2}}}f\left({\frac {x^{2}+y^{2}}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{2}}}\right).$ Viscous three-dimensional periodic solutions Two examples of periodic fully-three-dimensional viscous solutions are described in.[36] These solutions are defined on a three-dimensional torus $\mathbb {T} ^{3}=[0,L]^{3}$ and are characterized by positive and negative helicity respectively. The solution with positive helicity is given by: ${\begin{aligned}u_{x}&={\frac {4{\sqrt {2}}}{3{\sqrt {3}}}}\,U_{0}\left[\,\sin(kx-\pi /3)\cos(ky+\pi /3)\sin(kz+\pi /2)-\cos(kz-\pi /3)\sin(kx+\pi /3)\sin(ky+\pi /2)\,\right]e^{-3\nu k^{2}t}\\u_{y}&={\frac {4{\sqrt {2}}}{3{\sqrt {3}}}}\,U_{0}\left[\,\sin(ky-\pi /3)\cos(kz+\pi /3)\sin(kx+\pi /2)-\cos(kx-\pi /3)\sin(ky+\pi /3)\sin(kz+\pi /2)\,\right]e^{-3\nu k^{2}t}\\u_{z}&={\frac {4{\sqrt {2}}}{3{\sqrt {3}}}}\,U_{0}\left[\,\sin(kz-\pi /3)\cos(kx+\pi /3)\sin(ky+\pi /2)-\cos(ky-\pi /3)\sin(kz+\pi /3)\sin(kx+\pi /2)\,\right]e^{-3\nu k^{2}t}\end{aligned}}$ where $k=2\pi /L$ is the wave number and the velocity components are normalized so that the average kinetic energy per unit of mass is $U_{0}^{2}/2$ at $t=0$. The pressure field is obtained from the velocity field as $p=p_{0}-\rho _{0}\|{\boldsymbol {u}}\|^{2}/2$ (where $p_{0}$ and $\rho _{0}$ are reference values for the pressure and density fields respectively). Since both the solutions belong to the class of Beltrami flow, the vorticity field is parallel to the velocity and, for the case with positive helicity, is given by $\omega ={\sqrt {3}}\,k\,{\boldsymbol {u}}$. These solutions can be regarded as a generalization in three dimensions of the classic two-dimensional Taylor-Green Taylor–Green vortex. Wyld diagrams Wyld diagrams are bookkeeping graphs that correspond to the Navier–Stokes equations via a perturbation expansion of the fundamental continuum mechanics. Similar to the Feynman diagrams in quantum field theory, these diagrams are an extension of Keldysh's technique for nonequilibrium processes in fluid dynamics. In other words, these diagrams assign graphs to the (often) turbulent phenomena in turbulent fluids by allowing correlated and interacting fluid particles to obey stochastic processes associated to pseudo-random functions in probability distributions.[37] Representations in 3D Note that the formulas in this section make use of the single-line notation for partial derivatives, where, e.g. $ \partial _{x}u$ means the partial derivative of $ u$ with respect to $ x$, and $ \partial _{y}^{2}f_{\theta }$ means the second-order partial derivative of $ f_{\theta }$ with respect to $ y$. A 2022 paper provides a less costly, dynamical and recurrent solution of the Navier-Stokes equation for 3D turbulent fluid flows. On suitably short time scales, the dynamics of turbulence is deterministic.[38] Cartesian coordinates From the general form of the Navier–Stokes, with the velocity vector expanded as $ \mathbf {u} =(u_{x},u_{y},u_{z})$, sometimes respectively named $ u$, $ v$, $ w$, we may write the vector equation explicitly, ${\begin{aligned}x:\ &\rho \left({\partial _{t}u_{x}}+u_{x}\,{\partial _{x}u_{x}}+u_{y}\,{\partial _{y}u_{x}}+u_{z}\,{\partial _{z}u_{x}}\right)\\&\quad =-\partial _{x}p+\mu \left({\partial _{x}^{2}u_{x}}+{\partial _{y}^{2}u_{x}}+{\partial _{z}^{2}u_{x}}\right)+{\frac {1}{3}}\mu \ \partial _{x}\left({\partial _{x}u_{x}}+{\partial _{y}u_{y}}+{\partial _{z}u_{z}}\right)+\rho g_{x}\\\end{aligned}}$ ${\begin{aligned}y:\ &\rho \left({\partial _{t}u_{y}}+u_{x}{\partial _{x}u_{y}}+u_{y}{\partial _{y}u_{y}}+u_{z}{\partial _{z}u_{y}}\right)\\&\quad =-{\partial _{y}p}+\mu \left({\partial _{x}^{2}u_{y}}+{\partial _{y}^{2}u_{y}}+{\partial _{z}^{2}u_{y}}\right)+{\frac {1}{3}}\mu \ \partial _{y}\left({\partial _{x}u_{x}}+{\partial _{y}u_{y}}+{\partial _{z}u_{z}}\right)+\rho g_{y}\\\end{aligned}}$ ${\begin{aligned}z:\ &\rho \left({\partial _{t}u_{z}}+u_{x}{\partial _{x}u_{z}}+u_{y}{\partial _{y}u_{z}}+u_{z}{\partial _{z}u_{z}}\right)\\&\quad =-{\partial _{z}p}+\mu \left({\partial _{x}^{2}u_{z}}+{\partial _{y}^{2}u_{z}}+{\partial _{z}^{2}u_{z}}\right)+{\frac {1}{3}}\mu \ \partial _{z}\left({\partial _{x}u_{x}}+{\partial _{y}u_{y}}+{\partial _{z}u_{z}}\right)+\rho g_{z}.\end{aligned}}$ Note that gravity has been accounted for as a body force, and the values of $ g_{x}$, $ g_{y}$, $ g_{z}$ will depend on the orientation of gravity with respect to the chosen set of coordinates. The continuity equation reads: $\partial _{t}\rho +\partial _{x}(\rho u_{x})+\partial _{y}(\rho u_{y})+\partial _{z}(\rho u_{z})=0.$ When the flow is incompressible, $ \rho $ does not change for any fluid particle, and its material derivative vanishes: $ {\frac {\mathrm {D} \rho }{\mathrm {D} t}}=0$. The continuity equation is reduced to: $\partial _{x}u_{x}+\partial _{y}u_{y}+\partial _{z}u_{z}=0.$ Thus, for the incompressible version of the Navier–Stokes equation the second part of the viscous terms fall away (see Incompressible flow). This system of four equations comprises the most commonly used and studied form. Though comparatively more compact than other representations, this is still a nonlinear system of partial differential equations for which solutions are difficult to obtain. Cylindrical coordinates A change of variables on the Cartesian equations will yield[14] the following momentum equations for $ r$, $ \phi $, and $ z$[39] ${\begin{aligned}r:\ &\rho \left({\partial _{t}u_{r}}+u_{r}{\partial _{r}u_{r}}+{\frac {u_{\varphi }}{r}}{\partial _{\varphi }u_{r}}+u_{z}{\partial _{z}u_{r}}-{\frac {u_{\varphi }^{2}}{r}}\right)\\&\quad =-{\partial _{r}p}\\&\qquad +\mu \left({\frac {1}{r}}\partial _{r}\left(r{\partial _{r}u_{r}}\right)+{\frac {1}{r^{2}}}{\partial _{\varphi }^{2}u_{r}}+{\partial _{z}^{2}u_{r}}-{\frac {u_{r}}{r^{2}}}-{\frac {2}{r^{2}}}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +{\frac {1}{3}}\mu \partial _{r}\left({\frac {1}{r}}{\partial _{r}\left(ru_{r}\right)}+{\frac {1}{r}}{\partial _{\varphi }u_{\varphi }}+{\partial _{z}u_{z}}\right)\\&\qquad +\rho g_{r}\\[8px]\end{aligned}}$ ${\begin{aligned}\varphi :\ &\rho \left({\partial _{t}u_{\varphi }}+u_{r}{\partial _{r}u_{\varphi }}+{\frac {u_{\varphi }}{r}}{\partial _{\varphi }u_{\varphi }}+u_{z}{\partial _{z}u_{\varphi }}+{\frac {u_{r}u_{\varphi }}{r}}\right)\\&\quad =-{\frac {1}{r}}{\partial _{\varphi }p}\\&\qquad +\mu \left({\frac {1}{r}}\ \partial _{r}\left(r{\partial _{r}u_{\varphi }}\right)+{\frac {1}{r^{2}}}{\partial _{\varphi }^{2}u_{\varphi }}+{\partial _{z}^{2}u_{\varphi }}+{\frac {2}{r^{2}}}{\partial _{\varphi }u_{r}}-{\frac {u_{\varphi }}{r^{2}}}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r}}\partial _{\varphi }\left({\frac {1}{r}}{\partial _{r}\left(ru_{r}\right)}+{\frac {1}{r}}{\partial _{\varphi }u_{\varphi }}+{\partial _{z}u_{z}}\right)\\&\qquad +\rho g_{\varphi }\\[8px]\end{aligned}}$ :\ &\rho \left({\partial _{t}u_{\varphi }}+u_{r}{\partial _{r}u_{\varphi }}+{\frac {u_{\varphi }}{r}}{\partial _{\varphi }u_{\varphi }}+u_{z}{\partial _{z}u_{\varphi }}+{\frac {u_{r}u_{\varphi }}{r}}\right)\\&\quad =-{\frac {1}{r}}{\partial _{\varphi }p}\\&\qquad +\mu \left({\frac {1}{r}}\ \partial _{r}\left(r{\partial _{r}u_{\varphi }}\right)+{\frac {1}{r^{2}}}{\partial _{\varphi }^{2}u_{\varphi }}+{\partial _{z}^{2}u_{\varphi }}+{\frac {2}{r^{2}}}{\partial _{\varphi }u_{r}}-{\frac {u_{\varphi }}{r^{2}}}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r}}\partial _{\varphi }\left({\frac {1}{r}}{\partial _{r}\left(ru_{r}\right)}+{\frac {1}{r}}{\partial _{\varphi }u_{\varphi }}+{\partial _{z}u_{z}}\right)\\&\qquad +\rho g_{\varphi }\\[8px]\end{aligned}}} ${\begin{aligned}z:\ &\rho \left({\partial _{t}u_{z}}+u_{r}{\partial _{r}u_{z}}+{\frac {u_{\varphi }}{r}}{\partial _{\varphi }u_{z}}+u_{z}{\partial _{z}u_{z}}\right)\\&\quad =-{\partial _{z}p}\\&\qquad +\mu \left({\frac {1}{r}}\partial _{r}\left(r{\partial _{r}u_{z}}\right)+{\frac {1}{r^{2}}}{\partial _{\varphi }^{2}u_{z}}+{\partial _{z}^{2}u_{z}}\right)\\&\qquad +{\frac {1}{3}}\mu \partial _{z}\left({\frac {1}{r}}{\partial _{r}\left(ru_{r}\right)}+{\frac {1}{r}}{\partial _{\varphi }u_{\varphi }}+{\partial _{z}u_{z}}\right)\\&\qquad +\rho g_{z}.\end{aligned}}$ The gravity components will generally not be constants, however for most applications either the coordinates are chosen so that the gravity components are constant or else it is assumed that gravity is counteracted by a pressure field (for example, flow in horizontal pipe is treated normally without gravity and without a vertical pressure gradient). The continuity equation is: ${\partial _{t}\rho }+{\frac {1}{r}}\partial _{r}\left(\rho ru_{r}\right)+{\frac {1}{r}}{\partial _{\varphi }\left(\rho u_{\varphi }\right)}+{\partial _{z}\left(\rho u_{z}\right)}=0.$ This cylindrical representation of the incompressible Navier–Stokes equations is the second most commonly seen (the first being Cartesian above). Cylindrical coordinates are chosen to take advantage of symmetry, so that a velocity component can disappear. A very common case is axisymmetric flow with the assumption of no tangential velocity ($ u_{\phi }=0$), and the remaining quantities are independent of $ \phi $: ${\begin{aligned}\rho \left({\partial _{t}u_{r}}+u_{r}{\partial _{r}u_{r}}+u_{z}{\partial _{z}u_{r}}\right)&=-{\partial _{r}p}+\mu \left({\frac {1}{r}}\partial _{r}\left(r{\partial _{r}u_{r}}\right)+{\partial _{z}^{2}u_{r}}-{\frac {u_{r}}{r^{2}}}\right)+\rho g_{r}\\\rho \left({\partial _{t}u_{z}}+u_{r}{\partial _{r}u_{z}}+u_{z}{\partial _{z}u_{z}}\right)&=-{\partial _{z}p}+\mu \left({\frac {1}{r}}\partial _{r}\left(r{\partial _{r}u_{z}}\right)+{\partial _{z}^{2}u_{z}}\right)+\rho g_{z}\\{\frac {1}{r}}\partial _{r}\left(ru_{r}\right)+{\partial _{z}u_{z}}&=0.\end{aligned}}$ Spherical coordinates |In spherical coordinates, the $ r$, $ \phi $, and $ \theta $ momentum equations are[14] (note the convention used: $ \theta $ is polar angle, or colatitude,[40] $ 0\leq \theta \leq \pi $): ${\begin{aligned}r:\ &\rho \left({\partial _{t}u_{r}}+u_{r}{\partial _{r}u_{r}}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{r}}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{r}}-{\frac {u_{\varphi }^{2}+u_{\theta }^{2}}{r}}\right)\\&\quad =-{\partial _{r}p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{r}}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{r}}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{r}}\right)-2{\frac {u_{r}+{\partial _{\theta }u_{\theta }}+u_{\theta }\cot \theta }{r^{2}}}-{\frac {2}{r^{2}\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +{\frac {1}{3}}\mu \partial _{r}\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{r}\\[8px]\end{aligned}}$ ${\begin{aligned}\varphi :\ &\rho \left({\partial _{t}u_{\varphi }}+u_{r}{\partial _{r}u_{\varphi }}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{\varphi }}+{\frac {u_{r}u_{\varphi }+u_{\varphi }u_{\theta }\cot \theta }{r}}\right)\\&\quad =-{\frac {1}{r\sin \theta }}{\partial _{\varphi }p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{\varphi }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{\varphi }}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{\varphi }}\right)+{\frac {2\sin \theta {\partial _{\varphi }u_{r}}+2\cos \theta {\partial _{\varphi }u_{\theta }}-u_{\varphi }}{r^{2}\sin ^{2}\theta }}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r\sin \theta }}\partial _{\varphi }\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{\varphi }\\[8px]\end{aligned}}$ :\ &\rho \left({\partial _{t}u_{\varphi }}+u_{r}{\partial _{r}u_{\varphi }}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{\varphi }}+{\frac {u_{r}u_{\varphi }+u_{\varphi }u_{\theta }\cot \theta }{r}}\right)\\&\quad =-{\frac {1}{r\sin \theta }}{\partial _{\varphi }p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{\varphi }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{\varphi }}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{\varphi }}\right)+{\frac {2\sin \theta {\partial _{\varphi }u_{r}}+2\cos \theta {\partial _{\varphi }u_{\theta }}-u_{\varphi }}{r^{2}\sin ^{2}\theta }}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r\sin \theta }}\partial _{\varphi }\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{\varphi }\\[8px]\end{aligned}}} ${\begin{aligned}\theta :\ &\rho \left({\partial _{t}u_{\theta }}+u_{r}{\partial _{r}u_{\theta }}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{\theta }}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{\theta }}+{\frac {u_{r}u_{\theta }-u_{\varphi }^{2}\cot \theta }{r}}\right)\\&\quad =-{\frac {1}{r}}{\partial _{\theta }p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{\theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{\theta }}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{\theta }}\right)+{\frac {2}{r^{2}}}{\partial _{\theta }u_{r}}-{\frac {u_{\theta }+2\cos \theta {\partial _{\varphi }u_{\varphi }}}{r^{2}\sin ^{2}\theta }}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r}}\partial _{\theta }\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{\theta }.\end{aligned}}$ :\ &\rho \left({\partial _{t}u_{\theta }}+u_{r}{\partial _{r}u_{\theta }}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{\theta }}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{\theta }}+{\frac {u_{r}u_{\theta }-u_{\varphi }^{2}\cot \theta }{r}}\right)\\&\quad =-{\frac {1}{r}}{\partial _{\theta }p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{\theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{\theta }}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{\theta }}\right)+{\frac {2}{r^{2}}}{\partial _{\theta }u_{r}}-{\frac {u_{\theta }+2\cos \theta {\partial _{\varphi }u_{\varphi }}}{r^{2}\sin ^{2}\theta }}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r}}\partial _{\theta }\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{\theta }.\end{aligned}}} Mass continuity will read: ${\partial _{t}\rho }+{\frac {1}{r^{2}}}\partial _{r}\left(\rho r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }(\rho u_{\varphi })}+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(\sin \theta \rho u_{\theta }\right)=0.$ These equations could be (slightly) compacted by, for example, factoring $ {\frac {1}{r^{2}}}$ from the viscous terms. However, doing so would undesirably alter the structure of the Laplacian and other quantities. Navier–Stokes equations use in games The Navier–Stokes equations are used extensively in video games in order to model a wide variety of natural phenomena. Simulations of small-scale gaseous fluids, such as fire and smoke, are often based on the seminal paper "Real-Time Fluid Dynamics for Games"[41] by Jos Stam, which elaborates one of the methods proposed in Stam's earlier, more famous paper "Stable Fluids"[42] from 1999. Stam proposes stable fluid simulation using a Navier–Stokes solution method from 1968, coupled with an unconditionally stable semi-Lagrangian advection scheme, as first proposed in 1992. More recent implementations based upon this work run on the game systems graphics processing unit (GPU) as opposed to the central processing unit (CPU) and achieve a much higher degree of performance.[43][44] Many improvements have been proposed to Stam's original work, which suffers inherently from high numerical dissipation in both velocity and mass. An introduction to interactive fluid simulation can be found in the 2007 ACM SIGGRAPH course, Fluid Simulation for Computer Animation.[45] See also • Adhémar Jean Claude Barré de Saint-Venant • Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy of equations • Boltzmann equation • Cauchy momentum equation • Cauchy stress tensor • Chapman–Enskog theory • Churchill–Bernstein equation • Coandă effect • Computational fluid dynamics • Continuum mechanics • Convection–diffusion equation • Derivation of the Navier–Stokes equations • Einstein–Stokes equation • Euler equations • Hagen–Poiseuille flow from the Navier–Stokes equations • Millennium Prize Problems • Non-dimensionalization and scaling of the Navier–Stokes equations • Pressure-correction method • Primitive equations • Rayleigh–Bénard convection • Reynolds transport theorem • Stokes equations • Supersonic flow over a flat plate • Vlasov equation Citations 1. McLean, Doug (2012). "Continuum Fluid Mechanics and the Navier-Stokes Equations". Understanding Aerodynamics: Arguing from the Real Physics. John Wiley & Sons. pp. 13–78. ISBN 9781119967514. The main relationships comprising the NS equations are the basic conservation laws for mass, momentum, and energy. To have a complete equation set we also need an equation of state relating temperature, pressure, and density... 2. "Millennium Prize Problems—Navier–Stokes Equation", claymath.org, Clay Mathematics Institute, March 27, 2017, retrieved 2017-04-02 3. Fefferman, Charles L. "Existence and smoothness of the Navier–Stokes equation" (PDF). claymath.org. Clay Mathematics Institute. Archived from the original (PDF) on 2015-04-15. Retrieved 2017-04-02. 4. Batchelor (1967) pp. 137 & 142. 5. Batchelor (1967) pp. 142–148. 6. Chorin, Alexandre E.; Marsden, Jerrold E. (1993). A Mathematical Introduction to Fluid Mechanics. p. 33. 7. Batchelor (1967) p. 165. 8. Landau & Lifshitz (1987) pp. 44–45, 196 9. White (2006) p. 67. 10. Stokes, G. G. (2007). On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. 11. Vincenti, W. G., Kruger Jr., C. H. (1975). Introduction to physical gas dynamic. Introduction to physical gas dynamics/Huntington. 12. Batchelor (1967) pp. 147 & 154. 13. Batchelor (1967) p. 75. 14. See Acheson (1990). 15. Abdulkadirov, Ruslan; Lyakhov, Pavel (2022-02-22). "Estimates of Mild Solutions of Navier–Stokes Equations in Weak Herz-Type Besov–Morrey Spaces". Mathematics. 10 (5): 680. doi:10.3390/math10050680. ISSN 2227-7390. 16. Batchelor (1967) pp. 21 & 147. 17. Temam, Roger (2001), Navier–Stokes Equations, Theory and Numerical Analysis, AMS Chelsea, pp. 107–112 18. Quarteroni, Alfio (2014-04-25). Numerical models for differential problems (Second ed.). Springer. ISBN 978-88-470-5522-3. 19. Holdeman, J. T. (2010), "A Hermite finite element method for incompressible fluid flow", Int. J. Numer. Methods Fluids, 64 (4): 376–408, Bibcode:2010IJNMF..64..376H, doi:10.1002/fld.2154, S2CID 119882803 20. Holdeman, J. T.; Kim, J. W. (2010), "Computation of incompressible thermal flows using Hermite finite elements", Comput. Meth. Appl. Mech. Eng., 199 (49–52): 3297–3304, Bibcode:2010CMAME.199.3297H, doi:10.1016/j.cma.2010.06.036 21. Potter, M.; Wiggert, D. C. (2008). Fluid Mechanics. Schaum's Outlines. McGraw-Hill. ISBN 978-0-07-148781-8. 22. Aris, R. (1989). Vectors, Tensors, and the basic Equations of Fluid Mechanics. Dover Publications. ISBN 0-486-66110-5. 23. Parker, C. B. (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). ISBN 0-07-051400-3. 24. Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3 25. Gorban, A.N.; Karlin, I. V. (2016), "Beyond Navier–Stokes equations: capillarity of ideal gas", Contemporary Physics (Review article), 58 (1): 70–90, arXiv:1702.00831, Bibcode:2017ConPh..58...70G, doi:10.1080/00107514.2016.1256123, S2CID 55317543 26. Cercignani, C. (2002), "The Boltzmann equation and fluid dynamics", in Friedlander, S.; Serre, D. (eds.), Handbook of mathematical fluid dynamics, vol. 1, Amsterdam: North-Holland, pp. 1–70, ISBN 978-0444503305 27. Nie, X.B.; Chen, S.Y.; Robbins, M.O. (2004), "A continuum and molecular dynamics hybrid method for micro-and nano-fluid flow", Journal of Fluid Mechanics (Research article), 500: 55–64, Bibcode:2004JFM...500...55N, doi:10.1017/S0022112003007225, S2CID 122867563 28. Öttinger, H.C. (2012), Stochastic processes in polymeric fluids, Berlin, Heidelberg: Springer Science & Business Media, doi:10.1007/978-3-642-58290-5, ISBN 9783540583530 29. Shah, Tasneem Mohammad (1972). "Analysis of the multigrid method". NASA Sti/Recon Technical Report N. 91: 23418. Bibcode:1989STIN...9123418S. 30. Wang, C. Y. (1991), "Exact solutions of the steady-state Navier–Stokes equations", Annual Review of Fluid Mechanics, 23: 159–177, Bibcode:1991AnRFM..23..159W, doi:10.1146/annurev.fl.23.010191.001111 31. Landau & Lifshitz (1987) pp. 75–88. 32. Ethier, C. R.; Steinman, D. A. (1994), "Exact fully 3D Navier–Stokes solutions for benchmarking", International Journal for Numerical Methods in Fluids, 19 (5): 369–375, Bibcode:1994IJNMF..19..369E, doi:10.1002/fld.1650190502 33. http://www.claudino.webs.com/Navier%20Stokes%20Equations.pps 34. Ladyzhenskaya, O. A. (1969), The Mathematical Theory of viscous Incompressible Flow (2nd ed.), p. preface, xi 35. Kamchatno, A. M. (1982), Topological solitons in magnetohydrodynamics (PDF), archived (PDF) from the original on 2016-01-28 36. Antuono, M. (2020), "Tri-periodic fully three-dimensional analytic solutions for the Navier–Stokes equations", Journal of Fluid Mechanics, 890, Bibcode:2020JFM...890A..23A, doi:10.1017/jfm.2020.126, S2CID 216463266 37. McComb, W. D. (2008), Renormalization methods: A guide for beginners, Oxford University Press, pp. 121–128, ISBN 978-0-19-923652-7 38. Georgia Institute of Technology (August 29, 2022). "Physicists uncover new dynamical framework for turbulence". Proceedings of the National Academy of Sciences of the United States of America. Phys.org. 119 (34): e2120665119. doi:10.1073/pnas.2120665119. PMC 9407532. PMID 35984901. S2CID 251693676. 39. de' Michieli Vitturi, Mattia, Navier–Stokes equations in cylindrical coordinates, retrieved 2016-12-26 40. Eric W. Weisstein (2005-10-26), Spherical Coordinates, MathWorld, retrieved 2008-01-22 41. Stam, Jos (2003), Real-Time Fluid Dynamics for Games (PDF), S2CID 9353969, archived from the original (PDF) on 2020-08-05 42. Stam, Jos (1999), Stable Fluids (PDF), archived (PDF) from the original on 2019-07-15 43. Harris, Mark J. (2004), "38", GPUGems - Fast Fluid Dynamics Simulation on the GPU 44. Sander, P.; Tatarchuck, N.; Mitchell, J.L. (2007), "9.6", ShaderX5 - Explicit Early-Z Culling for Efficient Fluid Flow Simulation, pp. 553–564 45. Robert Bridson; Matthias Müller-Fischer. "Fluid Simulation for Computer Animation". www.cs.ubc.ca. General references • Acheson, D. J. (1990), Elementary Fluid Dynamics, Oxford Applied Mathematics and Computing Science Series, Oxford University Press, ISBN 978-0-19-859679-0 • Batchelor, G. K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, ISBN 978-0-521-66396-0 • Currie, I. G. (1974), Fundamental Mechanics of Fluids, McGraw-Hill, ISBN 978-0-07-015000-3 • V. Girault and P. A. Raviart. Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag, 1986. • Landau, L. D.; Lifshitz, E. M. (1987), Fluid mechanics, vol. Course of Theoretical Physics Volume 6 (2nd revised ed.), Pergamon Press, ISBN 978-0-08-033932-0, OCLC 15017127 • Polyanin, A. D.; Kutepov, A. M.; Vyazmin, A. V.; Kazenin, D. A. (2002), Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Taylor & Francis, London, ISBN 978-0-415-27237-7 • Rhyming, Inge L. (1991), Dynamique des fluides, Presses polytechniques et universitaires romandes • Smits, Alexander J. (2014), A Physical Introduction to Fluid Mechanics, Wiley, ISBN 0-47-1253499 • Temam, Roger (1984): Navier–Stokes Equations: Theory and Numerical Analysis, ACM Chelsea Publishing, ISBN 978-0-8218-2737-6 • White, Frank M. (2006), Viscous Fluid Flow, McGraw-Hill, ISBN 978-0-07-124493-0 External links • Simplified derivation of the Navier–Stokes equations • Three-dimensional unsteady form of the Navier–Stokes equations Glenn Research Center, NASA Topics in continuum mechanics Divisions • Solid mechanics • Fluid mechanics • Acoustics • Vibrations • Rigid body dynamics Laws and Definitions Laws • Conservation of mass • Conservation of momentum • Navier-Stokes • Bernoulli • Poiseuille • Archimedes • Pascal • Conservation of energy • Entropy inequality Definitions • Stress • Cauchy stress • Stress measures • Deformation • Small strain • Antiplane shear • Large strain • Compatibility Solid and Structural mechanics Solids • Elasticity • linear • Hooke's law • Transverse isotropy • Orthotropy • hyperelasticity • Membrane elasticity • Equation of state • Hugoniot • JWL • hypoelasticity • Cauchy elasticity • Viscoelasticity • Creep • Concrete creep • Plasticity • Rock mass plasticity • Viscoplasticity • Yield criterion • Bresler-Pister • Contact mechanics • Frictionless • Frictional Material failure theory • Drucker stability • Material failure theory • Fatigue • Fracture mechanics • J-integral • Compact tension specimen • Damage mechanics • Johnson-Holmquist Structures • Bending • Bending moment • Bending of plates • Sandwich theory Fluid mechanics Fluids • Fluid statics • Fluid dynamics • Navier–Stokes equations • Bernoulli's principle • Poiseuille equation • Buoyancy • Viscosity • Newtonian • Non-Newtonian • Archimedes' principle • Pascal's law • Pressure • Liquids • Surface tension • Capillary action Gases • Atmosphere • Boyle's law • Charles's law • Gay-Lussac's law • Combined gas law Plasma Acoustics • Acoustic theory • Aeroacoustics Rheology • Viscoelasticity • Smart fluids • Magnetorheological • Electrorheological • Ferrofluids • Rheometry • Rheometer Scientists • Bernoulli • Boyle • Cauchy • Charles • Euler • Gay-Lussac • Hooke • Pascal • Newton • Navier • Stokes Awards • Eringen Medal • William Prager Medal Authority control National • Spain • France • BnF data • Germany • Israel • United States • Czech Republic Other • IdRef
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Willem Abraham Wythoff Willem Abraham Wythoff, born Wijthoff (Dutch pronunciation: [ʋɛithɔf]), (6 October 1865 – 21 May 1939) was a Dutch mathematician. Willem Abraham Wythoff Born Willem Abraham Wijthoff (1865-10-06)6 October 1865 Amsterdam Died21 May 1939(1939-05-21) (aged 73) Amsterdam NationalityDutch Alma materUniversity of Amsterdam Known forWythoff's game, Wythoff construction, Wythoff symbol Scientific career FieldsMathematics Doctoral advisorDiederik Korteweg Biography Wythoff was born in Amsterdam to Anna C. F. Kerkhoven and Abraham Willem Wijthoff,[1] who worked in a sugar refinery.[2] He studied at the University of Amsterdam, and earned his Ph.D. in 1898 under the supervision of Diederik Korteweg.[3] Contributions Wythoff is known in combinatorial game theory and number theory for his study of Wythoff's game, whose solution involves the Fibonacci numbers.[2] The Wythoff array, a two-dimensional array of numbers related to this game and to the Fibonacci sequence, is also named after him.[4][5] In geometry, Wythoff is known for the Wythoff construction of uniform tilings and uniform polyhedra and for the Wythoff symbol used as a notation for these geometric objects. Selected publications • Wythoff, W. A. (1905–1907), "A modification of the game of nim", Nieuw Archief voor Wiskunde, 2: 199–202. • Wythoff, W. A. (1918), "A relation between the polytopes of the C600-family", Proceedings of the Section of Sciences, Koninklijke Akademie van Wetenschappen te Amsterdam, 20: 966–970, Bibcode:1918KNAB...20..966W. References 1. "Gezinsblad van Willem Abraham Wijthoff". www.humanitarisme.nl. Retrieved Oct 12, 2022. 2. Stakhov, Alexey; Stakhov, Alekseĭ Petrovich; Olsen, Scott Anthony (2009), The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science, K & E Series on Knots and Everything, vol. 22, World Scientific, pp. 129–130, ISBN 9789812775825. 3. Willem Abraham Wythoff at the Mathematics Genealogy Project 4. Kimberling, Clark (1995), "The Zeckendorf array equals the Wythoff array" (PDF), Fibonacci Quarterly, 33 (1): 3–8. 5. Morrison, D. R. (1980), "A Stolarsky array of Wythoff pairs", A Collection of Manuscripts Related to the Fibonacci Sequence (PDF), Santa Clara, Calif: The Fibonacci Association, pp. 134–136. External links Media related to Willem Wijthoff at Wikimedia Commons • Kimberling, Clark, Willem Abraham Wythoff (1865–1939) number-theorist Authority control: Academics • MathSciNet • Mathematics Genealogy Project • zbMATH
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