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Robert Remak (mathematician) Robert Erich Remak (14 February 1888 – 13 November 1942) was a German mathematician. He is chiefly remembered for his work in group theory (Remak decomposition). His other interests included algebraic number theory, mathematical economics and geometry of numbers. Robert Remak was the son of the neurologist Ernst Julius Remak and the grandson of the embryologist Robert Remak. He was murdered in the Holocaust. Robert Remak Born14 February 1888 Berlin, German Empire Died13 November 1942 (1942-11-14) (aged 54) Auschwitz-Birkenau, German-occupied Poland Known forRemak decomposition Scientific career FieldsMathematics Biography Robert Remak was born in Berlin. He studied at Humboldt University of Berlin under Ferdinand Georg Frobenius and received his doctorate in 1911. His dissertation, Über die Zerlegung der endlichen Gruppen in indirekte unzerlegbare Faktoren ("On the decomposition of a finite group into indirect indecomposable factors") established that any two decompositions of a finite group into a direct product are related by a central automorphism. A weaker form of this statement, uniqueness, was first proved by Joseph Wedderburn in 1909. Later the theorem was generalized by Wolfgang Krull and Otto Schmidt to some classes of infinite groups and became known as the Krull–Schmidt theorem or the Krull–Remak–Schmidt theorem. Although the dissertation was first submitted in 1911, it was rejected several times and Remak did not obtain his Habilitation until 1929. In the meantime, he wrote several papers on the geometry of numbers. Between 1929 and 1933 Remak lectured as a Privatdozent at Humboldt University. In the 1929 essay Kann die Volkwirtschaftslehre eine exakte Wissenschaft werden? ("Can economics become an exact science?"), Remak analyzed price formation in socialist and capitalist economies. He also anticipated the role played by digital computers in numerical solution of systems of linear equations. Remak's analysis may have influenced John von Neumann, who was a fellow lecturer in Berlin, but most of it has not been translated into English and it remains little known and appreciated in the English-speaking world.[1] In 1932 Remak published a paper giving a lower bound for the regulator of an algebraic number field in terms of the numbers r1 and r2 of real embeddings and pairs of complex embeddings. He went on to investigate relations between the regulator and the discriminant of an algebraic number field, isolating an important class of CM-fields ("fields with unit defect"). His last two papers on the subject appeared in Compositio Mathematica in 1952 and 1954, more than ten years after his death. After the Nazis seized power in 1933 and the Civil Service Law was passed a few months later, Remak, who was of Jewish ancestry, lost his right to teach in September 1933. He was arrested on Kristallnacht, 9 November 1938, and was interned at Sachsenhausen concentration camp for several weeks. After an unsuccessful campaign by his wife to secure a permission for him to emigrate to the United States, he was released and permitted to leave for Amsterdam. In 1942, however, he was arrested by the German occupational authorities in the Netherlands and deported to Auschwitz, where he was murdered.[2] Notes 1. Kurz and Salvadori, pp 40–46. 2. Emmer, Emmer (2004). Mathematics and culture I. Axel Springer AG. p. 59. ISBN 978-3-540-01770-7. Bibliography • Harald Hagemann: Robert Remak. In: Neue Deutsche Biographie. Band 21. Duncker & Humblot, Berlin 2003, ISBN 3-428-11202-4, p. 410ff. • Heinz D.Kurz and Neri Salvadori, von Neumann's 'growth model' and the classical tradition. In Understanding "classical" economics: studies in long-period theory, Routledge studies in the history of economics, 2003. ISBN 978-0-415-15871-8 • Uta C. Merzbach, Robert Remak and the estimation of units and regulators. Amphora, 481–522, Birkhäuser, Basel, 1992 MR1192337 • Reinhard Siegmund-Schultze: Dokumente zur Geschichte der Mathematik. Quellen und Studien zur Emigration einer Wissenschaft. Band 10: Mathematiker auf der Flucht vor Hitler. Vieweg, Wiesbaden 1998, ISBN 3-528-06993-7 External links • O'Connor, John J.; Robertson, Edmund F., "Robert Remak", MacTutor History of Mathematics Archive, University of St Andrews • Robert Remak at the Mathematics Genealogy Project • Willy Tiabou, Christoph Bichlmeier: Verfolgte Mathematiker Authority control International • ISNI • VIAF National • Germany • United States Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef	Wikipedia
Robert Riley (mathematician) Robert F. Riley (December 22, 1935–March 4, 2000[1]) was an American mathematician. He is known for his work in low-dimensional topology using computational tools and hyperbolic geometry, being one of the inspirations for William Thurston's later breakthroughs in 3-dimensional topology.[2] Career Riley earned a bachelor's degree in mathematics from MIT in 1957; shortly thereafter he dropped out of the graduate program and went on to work in industry, eventually moving to Amsterdam in 1966. In 1968 he took a temporary position at the University of Southampton.[3] He defended his Ph.D. at this institution in 1980, under the nominal direction of David Singerman.[4] For the next two years he occupied a postdoctoral position in Boulder where William Thurston was employed at the time, before moving on to Binghamton University as a professor.[3] Mathematical work Riley's research was in geometric topology, especially in knot theory, where he mostly studied representations of knot groups. Early on, following work of Ralph Fox, he was interested in morphisms to finite groups. Later on in Southampton, considering $\mathrm {SL} _{2}(\mathbb {C} )$-representations sending peripheral elements to parabolics led him to discover the hyperbolic structure on the complement of the figure-eight knot and some others.[5][6] This was one of the few examples of hyperbolic 3-manifolds that were available at the time, and as such it was one of the motivations which led to William Thurston's geometrisation conjecture, which includes as a particular case a criterion for a knot complement to support a hyperbolic structure.[7] One notable feature of Riley's work is that it relied much on the assistance of a computer.[8] Selected publications R75a. Riley, Robert (1975a). "Discrete parabolic representations of link groups". Mathematika. 22 (2): 141–150. doi:10.1112/S0025579300005982. MR 0425946. R75b. Riley, Robert (1975b). "A quadratic parabolic group". Math. Proc. Cambridge Philos. Soc. 77 (2): 281–288. Bibcode:1975MPCPS..77..281R. doi:10.1017/S0305004100051094. MR 0412416. R13. Riley, Robert (2013). "A personal account of the discovery of hyperbolic structures on some knot complements". Expositiones Mathematicae. 31 (2): 104–115. arXiv:1301.4601. doi:10.1016/j.exmath.2013.01.003. MR 3057120. S2CID 119319528. Notes 1. "Deaths–Robert Freed Riley" (PDF). Notices of the American Mathematical Society. 47 (6): 679. 2000. 2. Thurston 1982, p. 360. 3. Brin, Jones & Singerman 2013. 4. Robert Riley at the Mathematics Genealogy Project 5. Riley 1975a. 6. Riley 1975b. 7. Thurston 1982. 8. Riley 2013. References • Thurston, William P. (1982). "Three dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. 6 (3): 357–382. doi:10.1090/S0273-0979-1982-15003-0. • Brin, Matthew G.; Jones, Gareth A.; Singerman, David (2013). "Commentary on Robert Riley's article "A personal account of the discovery of hyperbolic structures on some knot complements"". Expositiones Mathematicae. 31 (2): 99–103. arXiv:1301.4599. doi:10.1016/j.exmath.2013.01.002. MR 3057119. S2CID 119568843. Authority control: Academics • MathSciNet • Mathematics Genealogy Project • zbMATH	Wikipedia
Robert Russell (Irish mathematician) Robert Russell (c. 1858–18 May 1938) was an Irish mathematician and academic at Trinity College Dublin (TCD), who served as Erasmus Smith's Professor of Mathematics (1917–1921). Robert Russell was born in Portadown, Armagh, and was educated at Santry School, Portarlington. He attended TCD, became a Scholar in 1877, and won the Brooke Prize, Bishop Law's Prize, McCullagh Prize, and Madden Prize.[1] He was awarded BA in mathematics (1880), became a Fellow a few years later, and got his MA (1888). In 1887, he was elected a member of the London Mathematical Society.[2] He spent his whole career at TCD, at various times serving as Junior Bursar, Junior Dean, Registrar of Chambers,[3] and from the early 1920s on, Senior Bursar.[4] He was Donegall Lecturer in Mathematics (1904–1907), Erasmus Smith's Professor of Mathematics (1917–1921), and became Senior Fellow in 1920. Selected papers • Geometry of Surfaces Derived from Cubics, 26 June 1899 • Ruler Constructions in Connexion with Cubic Curves, 24 April 1893 • On a Theorem in Higher Algebra, The Quarterly Journal of Pure and Applied Mathematics, Volume 21, 23 May 2016 References 1. TCD Bursar Dead, Obituary of Robert Russell, Evening Herald, 5 May 1938, p. 14 2. Proceedings of the London Mathematical Society, Volume 18 7 April 1887, p. 288 3. The Dublin university calendar online TCD 4. REProfessors And Lecturers Of The University: Erasmus Smith's Professor of Mathematics: 1917 Robert Russell, M.A.F3 The Dublin University Calendar	Wikipedia
Robert Sinclair MacKay Robert Sinclair MacKay FRS FInstP FIMA (born 1956) is a British mathematician and professor at the University of Warwick. He researches dynamical systems, the calculus of variations, Hamiltonian dynamics and applications to complex systems in physics, engineering, chemistry, biology and economics.[4][1] Robert MacKay FRS FInstP FIMA Born Robert Sinclair MacKay Carshalton, Surrey[1] Alma mater • University of Cambridge (BA) • Princeton University (PhD) Awards • FRS (2000)[2] • FInstP (2000) • FIMA (2003) Scientific career Institutions • University of Warwick • Queen Mary University of London • Institut des Hautes Études Scientifiques ThesisRenormalisation in area preserving maps (1982) Academic advisors • John M. Greene • Martin David Kruskal[3] Websitehomepages.warwick.ac.uk/~masfu/ Education MacKay was educated at Newcastle High School, leaving in 1974. He completed his Bachelor of Arts degree with first class honours in mathematics at Trinity College, Cambridge in 1977, and completed Part III of the tripos with distinction in 1978. He obtained his PhD in astrophysical sciences in 1982 from the Plasma Physics Laboratory at Princeton University for research supervised by John M. Greene and Martin David Kruskal.[3][5] Career and research Between 1982 and 1995, MacKay held postdoctoral research positions at Queen Mary College, London,[1] the Institut des Hautes Etudes Scientifiques, and the University of Warwick. From 1995 to 2000 he was Professor of Nonlinear Dynamics in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, Director of the Nonlinear Centre, and Fellow of Trinity College. In 2000 he returned to Warwick as Professor of Mathematics and Director of Mathematical Interdisciplinary Research. Awards and honours MacKay was awarded the Stefanos Pnevmatikos International Award in 1992.[6] He was elected a Fellow of the Royal Society (FRS) in 2000.[2] In 2012 he was elected President of the Institute of Mathematics and its Applications.[7] Personal life MacKay was born to Donald MacCrimmon MacKay and Valerie MacKay (née Wood) in 1956.[8] His younger brother David J. C. MacKay FRS was the Regius Professor of Engineering at the University of Cambridge. References 1. "Curriculum Vitae: Robert Sinclair MacKay" (PDF). University of Warwick. Archived from the original (PDF) on 25 March 2016. 2. "Professor Robert MacKay FRS". London: Royal Society. Archived from the original on 17 November 2015. One or more of the preceding sentences incorporates text from the royalsociety.org website where: "All text published under the heading 'Biography' on Fellow profile pages is available under Creative Commons Attribution 4.0 International License." --"Royal Society Terms, conditions and policies". Archived from the original on 25 September 2015. Retrieved 9 March 2016.{{cite web}}: CS1 maint: bot: original URL status unknown (link) 3. Robert Sinclair MacKay at the Mathematics Genealogy Project 4. An interview with Robert MacKay, CIM Bulletin 31, 15–18, 2012, by J Lopes Dias 5. MacKay, Robert Sinclair (1982). Renormalisation in area preserving maps (PhD thesis). Princeton University. OCLC 18798421. ProQuest 303253194. 6. "Recipients". University of Crete. Retrieved 29 May 2018. 7. "IMA Council". Institute of Mathematics and its Applications. Retrieved 20 March 2013. Fellows of the Royal Society elected in 2000 Fellows • Michael Akam • James Binney • Brice Bosnich • Cyrus Chothia • Peter Cresswell • Alan Davison • John Douglas Denton • Warren Ewens • Michael Fasham • Michael Anthony John Ferguson • Chris Frith • Michel Goedert • Don Grierson • Peter Gavin Hall • Alexander Halliday • Andrew Bruce Holmes • Roy Jackson • Bruce Arthur Joyce • Simon Barry Laughlin • Peter Francis Leadlay • Anthony Charles Legon • Robert Glanville Lloyd • Robert Sinclair MacKay • Thomas John Martin • Kiyoshi Nagai • Stuart Parkin • Ole Holger Petersen • M. S. Raghunathan • T. V. Ramakrishnan • Michael Alfred Robb • Janet Rossant • Patricia Simpson • Harry Smith • Peter Somogyi • Martin Sweeting • Brian Douglas Sykes • James Till • Paul Townsend • Alan Andrew Watson • Ian Wilson • John Henry Woodhouse • Adrian Wyatt Foreign • Grigory Barenblatt • Ronald Breslow • Harry B. Gray • Erwin Hahn • Martin Karplus • Mitsuhiro Yanagida Honorary • John Maddox Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States Academics • DBLP • MathSciNet • Mathematics Genealogy Project • Scopus • zbMATH Other • IdRef	Wikipedia
Robert Thomas Seeley Robert Thomas Seeley (born February 26, 1932, in Bryn Mawr, Pennsylvania, United States–died November 30, 2016, in Newton, Massachusetts) was a mathematician who worked on pseudo differential operators and the heat equation approach to the Atiyah–Singer index theorem.[1][2] Seeley did his undergraduate studies at Haverford College, and earned his Ph.D. from the Massachusetts Institute of Technology in 1959, under the supervision of Alberto Pedro Calderón.[3] He taught at Harvey Mudd College and then in 1962 joined the faculty of Brandeis University. In 1972 he moved to the University of Massachusetts Boston; he retired as an emeritus professor.[2] In 2012 he became a fellow of the American Mathematical Society.[4] References 1. "ROBERT THOMAS SEELEY's Obituary". Boston Globe. December 2016. Retrieved 2017-12-18. 2. , U. Mass. Boston Mathematics, retrieved 2016-12-01. 3. Robert Thomas Seeley at the Mathematics Genealogy Project 4. List of Fellows of the American Mathematical Society, retrieved 2013-07-14. Authority control International • ISNI • VIAF National • Germany • Israel • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Robert Tucker (mathematician) Robert Tucker (1832–1905) was an English mathematician, who was secretary of the London Mathematical Society for more than 30 years. Robert Tucker Born(1832-04-26)26 April 1832 Walworth, Surrey, England Died29 January 1905(1905-01-29) (aged 72) Worthing, England Alma materSt. John's College, Cambridge Scientific career FieldsMathematics InstitutionsUniversity College London InfluencesWilliam Kingdon Clifford Life and work Son of a soldier who fought in the Peninsular War, Tucker studied at St. John's College, Cambridge, where he was 35th wrangler in 1855.[1] He mastered mathematics at University College London from 1865 to 1899. He is known by the now known as Tucker circles, a family of circles invariant on parallel displacing.[2] He is also known by his edition of the Mathematical Papers of William Kingdon Clifford in 1882. Tucker acted as secretary of the London Mathematical Society from 1867 to 1902.[3] He was also a collector of mathematician's photographs. His collection, named Tucker collection is preserved by the London Mathematical Society at De Morgan house.[4] References 1. Rice, Wilson & Gardner 1995, p. 420. 2. Weisstein, MathWorld. 3. Rice, Wilson & Gardner 1995, p. 413. 4. O'Connor & Robertson, MacTutor History of Mathematics. Bibliography • Clifford, William Kingdon (2007). William Tucker (ed.). Mathematical Papers. AMS Chelsea Publishing. ISBN 978-0-8218-4252-2. • Rice, Adrian C.; Wilson, Robin J.; Gardner, J. Helen (1995). "From Student Club to National Society: The Founding of the London Mathematical Society in 1865". Historia Mathematica. 22 (4): 402–421. doi:10.1006/hmat.1995.1032. ISSN 0315-0860. External links • O'Connor, John J.; Robertson, Edmund F., "Robert Tucker (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews • "Tucker Collection of Photographs". London Mathematical Society. Retrieved 21 May 2017. • Weisstein, Eric W. "Tucker Circles". MathWorld. Retrieved 21 May 2017. Authority control International • ISNI • VIAF National • Norway • France • BnF data • Israel • United States • Czech Republic • Australia • Netherlands Academics • zbMATH Other • IdRef	Wikipedia
Robert Moody Robert Vaughan Moody, OC FRSC (/ˈmuːdi/; born November 28, 1941) is a Canadian mathematician. He is the co-discover of Kac–Moody algebra,[1] a Lie algebra, usually infinite-dimensional, that can be defined through a generalized root system. Robert Moody Robert Moody (left) Born (1941-11-28) November 28, 1941 NationalityCanadian Alma materUniversity of Toronto University of Saskatchewan AwardsCoxeter–James Prize (1978) Jeffery–Williams Prize (1995) Wigner Medal (1996) CRM-Fields-PIMS prize (1998) Scientific career FieldsMathematics InstitutionsUniversity of Saskatchewan University of Alberta Doctoral advisorMaria Wonenburger "Almost simultaneously in 1967, Victor Kac in the USSR and Robert Moody in Canada developed what was to become Kac–Moody algebra. Kac and Moody noticed that if Wilhelm Killing's conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." - A. J. Coleman[2] Born in Great Britain, he received a Bachelor of Arts in Mathematics in 1962 from the University of Saskatchewan, a Master of Arts in Mathematics in 1964 from the University of Toronto, and a Ph.D. in Mathematics in 1966 from the University of Toronto. In 1966, he joined the Department of Mathematics as an assistant professor in the University of Saskatchewan. In 1970, he was appointed an associate professor and a professor in 1976. In 1989, he joined the University of Alberta as a professor in the Department of Mathematics. In 1999, he was made an Officer of the Order of Canada.[3] In 1980, he was made a fellow of the Royal Society of Canada. In 1996 Moody and Kac were co-winners of the Wigner Medal.[4] Selected works • Moody, R. V. (1967). "Lie algebras associated with generalized Cartan matrices". Bull. Amer. Math. Soc. 73 (2): 217–222. doi:10.1090/s0002-9904-1967-11688-4. MR 0207783. • Moody, R. V. (1975). "Macdonald identities and Euclidean Lie algebras". Proc. Amer. Math. Soc. 48 (1): 43–52. doi:10.1090/s0002-9939-1975-0442048-2. MR 0442048. • with S. Berman: Berman, S.; Moody, R. V. (1979). "Lie algebra multiplicities". Proc. Amer. Math. Soc. 76 (2): 223–228. doi:10.1090/s0002-9939-1979-0537078-x. MR 0537078. • with J. Patera: Moody, R. V.; Patera, J. (1982). "Fast recursion formula for weight multiplicities". Bull. Amer. Math. Soc. (N.S.). 7 (1): 237–242. doi:10.1090/s0273-0979-1982-15021-2. MR 0656202. • with Bremner & Patera: Tables of weight space multiplicities, Marcel Dekker 1983 • with A. Pianzola: Moody, R. V.; Pianzola, A. (1989). "On infinite root systems". Trans. Amer. Math. Soc. 315 (2): 661–696. doi:10.1090/s0002-9947-1989-0964901-8. MR 0964901. • with S. Kass, J. Patera, & R. Slansky: Affine Lie Algebras, weight multiplicities and branching rules, 2 vols., University of California Press 1991 vol. 1 books.google • with Pianzola: Lie algebras with triangular decompositions, Canadian Mathematical Society Series, John Wiley 1995[5] • with Baake & Grimm: Die verborgene Ordnung der Quasikristalle, Spektrum, Feb. 2002; What is Aperiodic Order?, Eng. trans. on arxiv.org Notes 1. Stephen Berman, Karen Parshall Victor Kac and Robert Moody — their paths to Kac–Moody-Algebras, Mathematical Intelligencer, 2002, Nr.1 2. Coleman, A. John, "The Greatest Mathematical Paper of All Time," The Mathematical Intelligencer, vol. 11, no. 3, pp. 29–38. 3. "Robert V. Moody Appointed Officer of the Order of Canada" (PDF). Newsletter of the Pacific Institute for the Mathematical Sciences. Vol. 4, no. 1. Winter 2000. p. 1. 4. Jackson, Allyn (Dec 1995). "Kac and Moody Receive Wigner Medal" (PDF). Notices of the AMS. 42 (12): 1543–1544. 5. Seligman, George B. (1996). "Review: Lie algebras with triangular decompositions, by Robert B. Moody and Arturo Pianzola" (PDF). Bull. Amer. Math. Soc. (N.S.). 33 (3): 347–349. doi:10.1090/s0273-0979-96-00653-2. References • Robert Moody at the Mathematics Genealogy Project • "Robert Vaughan Moody's Home Page". • "Robert Vaughan Moody Curriculum Vitae". Retrieved March 7, 2006. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • Belgium • United States • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • SNAC • IdRef	Wikipedia
Robert W. Brooks Robert Wolfe Brooks (Washington, D.C., September 16, 1952 – Montreal, September 5, 2002) was a mathematician known for his work in spectral geometry, Riemann surfaces, circle packings, and differential geometry. He received his Ph.D. from Harvard University in 1977; his thesis, The smooth cohomology of groups of diffeomorphisms, was written under the supervision of Raoul Bott. He worked at the University of Maryland (1979–1984), then at the University of Southern California, and then, from 1995, at the Technion in Haifa.[1] Work In an influential paper (Brooks 1981), Brooks proved that the bounded cohomology of a topological space is isomorphic to the bounded cohomology of its fundamental group.[2] Honors • Alfred P. Sloan fellowship • Guastella fellowship Selected publications • Brooks, Robert (1981). "Some remarks on bounded cohomology". Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978). Ann. of Math. Stud. Vol. 97. Princeton, N.J.: Princeton Univ. Press. pp. 53–63. MR 0624804. • Brooks, Robert (1981). "A relation between growth and the spectrum of the Laplacian". Mathematische Zeitschrift. 178 (4): 501–508. doi:10.1007/BF01174771. MR 0638814. S2CID 122114581. • Brooks, Robert (1981). "The fundamental group and the spectrum of the Laplacian". Commentarii Mathematici Helvetici. 56 (4): 581–598. doi:10.1007/BF02566228. MR 0656213. S2CID 121175762. • Brooks, Robert (1988). "Constructing isospectral manifolds". American Mathematical Monthly. 95 (9): 823–839. doi:10.1080/00029890.1988.11972094. MR 0967343. Reviewer Maung Min-Oo for MathSciNet wrote: "This is a well written survey article on the construction of isospectral manifolds which are not isometric with emphasis on hyperbolic Riemann surfaces of constant negative curvature."[3] • Brooks, Robert, "Form in Topology", The Magicians of Form, ed. by Robert M. Weiss. Laurelhurst Publications, 2003. References 1. Buser, Peter (2005). "On the mathematical work of Robert Brooks". Geometry, spectral theory, groups, and dynamics. Contemp. Math. Vol. 387. Providence, RI: Amer. Math. Soc. pp. 1–35. ISBN 9780821885642. MR 2179784. 2. Ivanov, Nikolai V. (1987). "Foundations of the theory of bounded cohomology". Journal of Mathematical Sciences. 37 (3): 1090–1115. doi:10.1007/BF01086634. MR 0806562. S2CID 122503635. 3. MR967343 External links • Memorial page (Technion) • Robert W. Brooks at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Latvia • Australia • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Robert Wallace (minister) Robert Wallace (7 January 1697 – 29 July 1771) was a minister of the Church of Scotland and writer on population. Life He was the only son of Margaret Stewart, wife of Rev Matthew Wallace, the parish minister of Kincardine-in-Menteith[1] (west of Stirling), where he was born on 7 January 1697. Educated at Stirling grammar school, he then attended the University of Edinburgh in 1711, and acted for a time (1720) as assistant to James Gregory, the University professor of mathematics. He was one of the founders of the Rankenian Club in 1717.[2] On 31 July 1722 Wallace was licensed as a preacher by the presbytery of Dunblane, Perthshire, and he was presented by the Marquis of Annandale to the parish of Moffat, Dumfriesshire, in August 1723. In 1733 he became minister of New Greyfriars, Edinburgh. He offended the government of 1736 by declining to read from his pulpit the proclamation against the Porteous rioters. On 30 August 1738 he was translated to the New North (St Giles). In 1742, on a change of ministry at Westminster, he regained influence, and was entrusted for five years with the management of church business and the distribution of ecclesiastical patronage. From a suggestion of John Mathison of the High Kirk, St Giles, Wallace, togerther with Alexander Webster of the Tolbooth St Giles, developed the Ministers' Widows' Fund.[2] On 12 May 1743 Wallace was elected Moderator of the General Assembly of the Church of Scotland. The Assembly approved the Widows' Fund. scheme, and at the end of the year he submitted it in London to Robert Craigie, the Lord Advocate, who saw it into legislation.[2] In June 1744 Wallace was appointed a Chaplain in Ordinary to King George II in Scotland and Dean of the Chapel Royal. He received the honorary degree of D.D. from the University of Edinburgh on 13 March 1759, and died on 29 July 1771.[2] His position at New Church, St Giles was filled by Rev William Gloag. Works Wallace published in 1753 a Dissertation on the Numbers of Mankind in Ancient and Modern Times. It contained criticism of the chapter on the Populousness of Ancient Nations in David Hume's Political Discourses. The work was translated into French under the supervision of Montesquieu, and it was republished in an English edition with a memoir in 1809. In 1758 appeared Wallace's Characteristics of the Present State of Great Britain. In Various Prospects of Mankind, Nature, and Providence (1761), he recurred to his population theories, and was believed (by William Hazlitt and Thomas Noon Talfourd) to have influenced Robert Malthus.[2] Family In October 1726 Wallace married Helen Turnbull, daughter of Rev George Turnbull, minister of Tyninghame in East Lothian. She died on 9 February 1776, leaving two sons: • Rev Dr Matthew Wallace DD vicar of Tenterden in Kent • George Wallace (1727-1805) was known as an advocate and writer.[2] • Elizabeth, all of whom died unmarried. Notes Wikisource has original works by or about: Robert Wallace 1. Fasti Ecclesiae Scoticanae; by Hew Scott 2. Lee, Sidney, ed. (1899). "Wallace, Robert (1697-1771)" . Dictionary of National Biography. Vol. 59. London: Smith, Elder & Co. Attribution This article incorporates text from a publication now in the public domain: Lee, Sidney, ed. (1899). "Wallace, Robert (1697-1771)". Dictionary of National Biography. Vol. 59. London: Smith, Elder & Co. Authority control International • FAST • ISNI • VIAF National • Spain • France • BnF data • Germany • Italy • Israel • Belgium • United States • Sweden • Netherlands • Vatican People • Trove Other • SNAC • IdRef	Wikipedia
Robert Wedderburn (statistician) Robert William Maclagan Wedderburn (1947–1975) was a Scottish statistician who worked at the Rothamsted Experimental Station. He was co-developer, with John Nelder, of the generalized linear model methodology,[1] and then expanded this subject to develop the idea of quasi-likelihood.[2] Wedderburn was born in Edinburgh, where he attended Fettes College, then studied for a degree and a diploma in statistics at the University of Cambridge.[3] He died aged 28 of anaphylactic shock from an insect bite while on a canal holiday.[4] "His colleagues remember him as someone of engaging diffidence, who would nonetheless hold his own in argument when he was sure he was right (as he usually was)," wrote John Nelder in Wedderburn's obituary.[3] References 1. Nelder, John A; Wedderburn, Robert W (1972). "Generalized linear models". Journal of the Royal Statistical Society, Series A. Royal Statistical Society. 135 (3): 370–384. doi:10.2307/2344614. JSTOR 2344614. 2. Wedderburn, RWM (1974). "Quasi-likelihood functions, generalized linear models, and the Gauss—Newton method". Biometrika. 61 (3): 439–447. doi:10.1093/biomet/61.3.439. 3. Nelder, J.A. (1975). "Robert William MacLagan Wedderburn, 1947–1975". Journal of the Royal Statistical Society, Series A. Royal Statistical Society. 138 (4): 587. JSTOR 2345239. 4. Senn, Stephen (2003). "A conversation with John Nelder". Statistical Science. 18 (1): 118–131. doi:10.1214/ss/1056397489. Authority control: Academics • zbMATH	Wikipedia
Robert Wood (mathematician) Robert Wood or Woods (1622?–1685) was an English mathematician. Life Born at Pepperharrow, near Godalming in Surrey, in 1621 or 1622, was the son of Robert Wood (d. 1661), rector of Pepperharrow. He was educated at Eton College, and matriculated from New Inn Hall on 3 July 1640. Obtaining one of the Eton postmasterships at Merton College in 1642, he graduated B. A. from that college on 18 March 1646–7, proceeded M.A. on 14 July 1649, and was elected a fellow of Lincoln College by order of the parliamentary commissioners, on 19 September 1650, in the place of Thankfull Owen. After studying physic for six years he was licensed to practise by convocation on 10 April 1656. He associated with the ‘Oxford club’ around John Wilkins of Wadham College. On a visit to Samuel Hartlib in 1658 he described how he had been assigned a task related to the cataloguing of the Bodleian Library, one of the interests of the time of the ‘club’, which was a precursor to the Royal Society. Wood was elected a Fellow of the Royal Society, but much later (6 April 1681). Wood had contacted Hartlib in 1656 with a scheme for currency reform to decimal coinage, and was drawn into the Hartlib circle of correspondents.[1][2] He went to Ireland and became a retainer of Henry Cromwell, who dispatched him to Scotland to ascertain the state of affairs there. On his return to England he became one of the first fellows of the Durham College founded by Oliver Cromwell. He was a prominent supporter of the Commonwealth, and a frequenter of the Rota Club formed by James Harrington. On the Restoration he was deprived of his fellowship at Lincoln College and returned to Ireland, where he professed loyalty, graduated M.D., and became chancellor of the diocese of Meath. He purchased an estate in Ireland, which, he afterwards sold in order to buy one at Sherwill in Essex. On his return to England he became mathematical master at Christ's Hospital, but after some years he resigned the post and paid a third visit to Ireland, where he was made a commissioner of the revenue, and finally accountant-general. This office he retained until his death, at Dublin, on 9 April 1685. He was buried in St. Michael's Church. He married Miss Adams, by whom he had three daughters Catherine, Martha, and Frances. Works He was the author of A New Al-moon-ac for Ever; or a Rectified Account of Time, London, 1680; and of another tract, entitled The Times Mended; or a Rectified Account of Time by a New Luni-Solar Year; the true way to Number our Days, London, 1681. In these treatises, which were dedicated to the Order of the Garter, and sometimes accompanied by a single folio sheet entitled Novus Annus Luni-solaris,'he proposed to rectify the year so that the first day of the month should always be within a day of the change of the moon, while by a system of compensations the length of the year should be kept within a week of the period of rotation round the sun. Wood translated the greater part of William Oughtred's Clavis Mathematica into English; he had been one of Oughtred's pupils.[3] He published two papers in the Philosophical Transactions in 1681. Notes 1. Toby Christopher Barnard, Cromwellian Ireland (2000), p. 224. 2. Margery Purver, The Royal Society: Concept and Creation (1967), p. 125 and note 109. 3. "The Galileo Project". References • This article incorporates text from a publication now in the public domain: "Wood, Robert (1622?-1685)". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900. Authority control International • ISNI • VIAF National • Germany • United States Other • SNAC	Wikipedia
Roberto Conti (mathematician) Roberto Conti (23 April 1923 – 30 August 2006) was an Italian mathematician, who contributed to the theory of ordinary differential equations[2] and the development of the comparison method.[3] Roberto Conti Born(1923-04-29)29 April 1923 Florence, Italy Died30 August 2006(2006-08-30) (aged 83) Florence, Italy NationalityItalian Alma materScuola Normale Superiore di Pisa Known forComparison method Scientific career FieldsMathematics Institutions • University of Florence, • University of Catania Doctoral advisorGiovanni Sansone[1] Biography Roberto Conti was born in Florence on 29 April 1923. He obtained his M.Sc. and Ph.D. in mathematics from the Scuola Normale Superiore di Pisa, under the supervision, respectively, of Leonida Tonelli (replaced, after his premature death, by Emilio Baiada) and Giovanni Sansone. Conti’s M.Sc. and Ph.D. dissertations dealt with translation surfaces (possibly a topic suggested by Tonelli after knowing about some Russian works) and particular aspects of the Cauchy problem. Later he held the position of research assistant to the chair of Sansone at the University of Florence. Their collaboration was fruitful and resulted in numerous articles, as well as the book (Sansone & Conti 1964),(Sansone & Conti 1964) originally published in Italian, which was translated into a number of languages and became one of the standard texts on the subject in the 1960s.[2] In 1956 Conti became full professor at the University of Catania, holding the chair of mathematical analysis until 1958, when he returned to Florence.[4] In 1963-1964 he held a visiting professorship at the Research Institute for Advanced Studies (RIAS) in Baltimore, Maryland. His research focused on several topics, which often overlapped in the time and contributed to motivate each other. A leading theme was constituted by functional analysis and its applications to the theory of ordinary differential equations, dynamical systems and control systems.[2] An impulse to do research on control systems was most probably given by his discussions with the Russian engineer and mathematician Nicolas Minorsky, who first proposed application of control theory to the automatic steering of ships. Minorsky, after the Russian revolution, moved to U.S. and later to southern France: during that period he came frequently to Florence, to give seminars and exchange mathematical ideas with Sansone and Conti. On another hand, Conti’s contributions to the development of the comparison method for the qualitative analysis of differential equations were particularly prominent.[3] He was a corresponding member of the Accademia dei Lincei since 1983 and a full member since 1994,[5] a foreign honorary member of the Romanian Academy since 1997[6] and was also a member of the editorial board of the Journal of Differential Equations since its inception in 1964 until his death in 2006.[2] Selected works Books and book chapters • Sansone, Giovanni; Conti, Roberto (1956), Equazioni differenziali non lineari, Monografie matematiche del Consiglio Nazionale delle Ricerche (in Italian), vol. 3, Roma: Edizioni Cremonese, pp. XIX+647, MR 0088607, Zbl 0075.26803, translated in English as Sansone, Giovanni; Conti, Roberto (1964), Nonlinear Differential Equations (PDF), International Series of Monographs in Pure and Applied Mathematics, vol. 67, translated by Diamond, Ainsley H., Oxford–London–Edinburg–New York–Paris–Frankfurt: Pergamon Press, pp. XIII+533, ISBN 978-0080101941, MR 0177153, Zbl 0128.08403 • Reissig, Rolf; Sansone, Giovanni; Conti, Roberto (1964), Qualitative Theorie nichtlineare Differentialgleichungen [Qualitative Theory of Nonlinear Differential Equations] (in German), Roma: Edizioni Cremonese, pp. XXXII+382, MR 0158121, Zbl 0114.04302. • Reissig, Rolf; Sansone, Giovanni; Conti, Roberto (1969), Nichtlineare Differentialgleichungen höherer Ordnung, Monografie matematiche del Consiglio Nazionale delle Ricerche (in German), vol. 16, Roma: Edizioni Cremonese, pp. XV+738, MR 0241749, Zbl 0172.10801, translated in English as Reissig, Rolf; Sansone, Giovanni; Conti, Roberto (1974), Non-linear differential equations of higher order, Monographs and Textbooks on Pure and Applied Mathematics, Leyden: Noordhoff International Publishing, pp. xiii+669, ISBN 90-01-75270-5, MR 0344556, Zbl 0275.34001. • Conti, Roberto (1977), Problemi di Controllo e di Controllo Ottimale, Collezione di matematica applicata (in Italian), vol. 6 (3 ed.), Torino: Unione Tipografico-Editrice Torinese, p. 239, ISBN 9788802019468 • Conti, Roberto (1977), Linear Differential Equations and Control, Institutiones Mathematicae, vol. 1, Rome / London–New York: Istituto Nazionale di Alta Matematica / Academic Press, p. 174, ISBN 978-0123636010, MR 0513642, Zbl 0356.34007. • Conti, Roberto; Galeotti, Marcello (2000), "Chapter 2: Totally Bounded Cubic Systems in ℝ2", in Macki, J. W.; Zecca, P. (eds.), Dynamical Systems. Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 19-26, 2000, Lecture Notes in Mathematics, vol. 1822, Springer, pp. 103–172, doi:10.1007/978-3-540-45204-1_2, ISBN 3-540-40786-3, MR 2051723, Zbl 1048.37043. Articles • Conti, Roberto (1955), "Sulla stabilità dei sistemi di equazioni differenziali lineari" [On the stability of systems of linear differential equations] (PDF), Rivista di Matematica della Università di Parma, (1) (in Italian), 6: 5–35, MR 0076953, Zbl 0067.31402. • Conti, Roberto (1956), "Limitazioni "in ampiezza" delle soluzioni di un sistema di equazioni differenziali e applicazioni" ["Amplitude" limitations of the solutions to a system of differential equations and their applications], Bollettino dell'Unione Matematica Italiana, Serie III (in Italian), 11 (3): 334–349, MR 0080829, Zbl 0071.30601. • Conti, Roberto (1956a), "Sulla prolungabilità delle soluzioni di un sistema di equazioni differenziali ordinarie" [On the continuability of solutions to a system of ordinary differential equations], Bollettino dell'Unione Matematica Italiana, Serie III (in Italian), 11 (4): 510–514, MR 0083628, Zbl 0072.30403. • Conti, Roberto (1957), "Sulla t∞–similitudine tra matici e l'equivalenza asintotica dei sistemi differenziali lineari" [On the t∞–similitude between matrices and the asymptotic equivalence of linear differential systems] (PDF), Rivista di Matematica della Università di Parma, (1) (in Italian), 8: 43–47, MR 0098870, Zbl 0089.29702. • Conti, Roberto (1965), "Contributions to linear control theory", Journal of Differential Equations, 1 (4): 427–445, Bibcode:1965JDE.....1..427C, doi:10.1016/0022-0396(65)90003-3, MR 0227518, Zbl 0152.08401 • Conti, Roberto (1966), "On the boundedness of solutions of ordinary differential equations" (PDF), Funkcialaj Ekvacioj, 9: 23–26, MR 0080829, Zbl 0071.30601. • Conti, Roberto (1967), "Recent trends in the theory of boundary value problems for ordinary differential equations", Bollettino dell'Unione Matematica Italiana, Serie III, 22 (2): 135–178, MR 0218650, Zbl 0154.09101 • Conti, Roberto (1968), "On ordinary differential equations with interface conditions", J. Differential Equations, 4 (1): 4–11, Bibcode:1968JDE.....4....4C, doi:10.1016/0022-0396(68)90045-4. • Conti, Roberto (1983), "Return sets of a linear control process", J. Optim. Theory Appl., 41 (1): 37–53, doi:10.1007/BF00934435, ISSN 0022-3239, S2CID 122611719. • Conti, Roberto (1991), "Strictly stable linear ordinary differential equations and similarity" (PDF), Rivista di Matematica della Università di Parma, (4), 17: 217–220, MR 1174950, Zbl 0758.34005. • Conti, Roberto (1998), "Centers of planar polynomial systems. A review", Le Matematiche, 53 (2): 207–240, MR 1710759, Zbl 1156.34315. • Conti, Roberto; Galeotti, Marcello (2002), "Totally bounded differential polynomial systems in ℝ2", Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Serie IX, 13 (2): 91–99, MR 1949482, Zbl 1108.34311. Notes 1. Roberto Conti at the Mathematics Genealogy Project 2. See (Villari 2007). 3. See (Corduneanu 2009). 4. (Marino 2008, p. 2). 5. According to the Lincean yearbook (2012, p. 439), he was elected "membro corrispondente" on July 30, 1983, and "membro nazionale" on November 30, 1994. 6. MEMBRII ACADEMIEI ROMÂNE, Membrii AR din străinătate (in Romanian) References • Accademia Nazionale dei Lincei (2012), Annuario dell'Accademia Nazionale dei Lincei 2012 – CDX dalla Sua Fondazione (PDF) (in Italian), Roma: Accademia Nazionale dei Lincei, p. 734. The "Yearbook" of the renowned Italian scientific institution, including an historical sketch of its history, the list of all past and present members as well as a wealth of information about its academic and scientific activities. • Corduneanu, Constantin (2009), "The Contribution of R. Conti to the Comparison Method in Differential Equations", Libertas Mathematica, XXIX: 113–116, MR 2663488, Zbl 1187.34042. • Marino, Mario (2008), "In Memoria del Prof. Giuseppe Santagati" [In memory of Prof. Giuseppe Santagati], Bollettino dell'Accademia Gioenia (in Italian), 41 (369): 1–7. • Bacciotti, Andrea; Pandolfi, Luciano (2007), "Commemorazione di Roberto Conti" (PDF), Bollettino della Unione Matematica Italiana, Sezione A, la Matematica Nella Società e Nella Cultura (in Italian), Dipartimento di Scienze Matematiche "Giuseppe Luigi Lagrange", Politecnico di Torino, 8 (16–A): 585–602, retrieved 21 May 2016. • Villari, Gabriele Jr. (2007), "In Memoriam Roberto Conti (1923–2006)", Journal of Differential Equations, 234 (1): 337–338, doi:10.1016/j.jde.2006.11.006, MR 2298975, Zbl 1168.01346. External links • Roberto Conti at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • France • BnF data • Catalonia • Germany • Italy • Israel • United States • Sweden • Czech Republic • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Roberto Longo (mathematician) Roberto Longo (born 9 May 1953) is an Italian mathematician, specializing in operator algebras and quantum field theory.[1] Roberto Longo Born(1953-05-09)9 May 1953 Rome, Italy NationalityItalian Alma materSapienza University of Rome Awards • Humboldt Prize (2016) • Medaglia dei XL (2021) Scientific career FieldsMathematics Institutions • University of Pennsylvania • University of California, Berkeley • University of Rome Tor Vergata ThesisTomita-Takesaki modular structure for AFD von Neumann algebras (1975) Academic advisorsSergio Doplicher Education and career Roberto Longo graduated in Mathematics[2] at the Sapienza University of Rome in 1975 under the supervision of the mathematical physicist Sergio Doplicher.[3] From 1975 to 1977 Longo was a predoctoral fellow of the Consiglio Nazionale delle Ricerche and later assistant professor at the Sapienza University of Rome, where he became an associate professor in 1980. In 1987 he was nominated full professor of functional analysis at the University of Rome Tor Vergata and since 2010 he is the director of the Center for Mathematics and Theoretical Physics in Rome. Between 1978 and 1979 he was visiting scholar at the University of Pennsylvania and the University of California, Berkeley. He has been a visiting professor in numerous research centers, including the CNRS in Marseille, the Mathematical Sciences Research Institute in Berkeley, California, the Harvard University, MIT, and the University of Göttingen.[4] Longo is an expert in the theory of operator algebras and its applications to quantum field theory. His work influenced the structural analysis of quantum field theory, especially of conformal field theory, and opened up to new developments in model building methods of interest in local quantum physics.[5] Roberto Longo is known in particular for his work with Sergio Doplicher on split inclusions of von Neumann algebras[6] and for having solved, independently with Sorin Popa, the Stone-Weierstrass conjecture for factorial states.[7] He also found the relationship between the statistical dimension[8] and the Jones index.[9] In a work with Yasuyuki Kawahigashi, Longo classified the discrete series of conformal chiral networks of von Neumann algebras.[10] Together with Vincenzo Morinelli and Karl-Henning Rehren, he also showed that particles with infinite spin cannot appear in a local theory.[11] His most recent works concern entropy and information for infinite quantum systems.[12] Honors and awards In 1994 Longo was an invited speaker at the International Congress of Mathematicians in Zurich.[13] He was invited speaker at the International Congress on Mathematical Physics in 1981 in Berlin,[14] in 1988 in Swansea,[15] in 1994 in Paris,[16] and in 2003 in Lisbon.[17] In 2004 he was Andrejewski Lecturer in Göttingen.[18] He was a plenary speaker at the International Congress of Mathematicians in 2009 in Prague[19] and at Strings 2018 in Okinawa.[20] In 2013 he was elected a Fellow of the American Mathematical Society[21] and in 2021 a member of the Academia Europaea.[22] He was awarded in 2016 the Humboldt Research Award[5][23] and in 2021 the XL medal from the Accademia Nazionale delle Scienze detta dei XL for his in-depth and innovative research in operator algebras and in conformal field theory.[24] In 2013 the conference Mathematics and Quantum Physics at the Lincei National Academy was dedicated to him on the occasion of his 60th birthday.[25] In 2008[26] and in 2015[27] he received two Advanced Grants from the European Research Council.[28] In 2018 he was member of the sectional panel Mathematical Physics of the International Congress of Mathematicians in Rio de Janeiro.[29] Selected publications • Hislop, Peter D.; Longo, Roberto (1982). "Modular structure of the local algebras associated with the free massless scalar field theory". Communications in Mathematical Physics. 84 (1): 71–85. Bibcode:1982CMaPh..84...71H. doi:10.1007/BF01208372. S2CID 119856852. • Doplicher, S.; Longo, R. (1984). "Standard and split inclusions of von Neumann algebras". Inventiones Mathematicae. 75 (3): 493–536. Bibcode:1984InMat..75..493D. doi:10.1007/BF01388641. ISSN 0020-9910. S2CID 121819323. • Longo, Roberto (1984). "Solution of the factorial Stone-Weierstrass conjecture. An application of the theory of standard split W-inclusions". Inventiones Mathematicae. 76 (1): 145–155. Bibcode:1984InMat..76..145L. doi:10.1007/BF01388497. S2CID 122775835. • Longo, Roberto (1989). "Index of subfactors and statistics of quantum fields. I". Communications in Mathematical Physics. 126 (2): 217–247. Bibcode:1989CMaPh.126..217L. doi:10.1007/BF02125124. S2CID 122420613. • Longo, Roberto (1994). "A duality for Hopf algebras and for subfactors. I". Communications in Mathematical Physics. 159 (1): 133–150. Bibcode:1994CMaPh.159..133L. doi:10.1007/BF02100488. ISSN 0010-3616. S2CID 123347247. • Guido, Daniele; Longo, Roberto (1996). "The conformal spin and statistics theorem". Communications in Mathematical Physics. 181 (1): 11–35. arXiv:hep-th/9505059. Bibcode:1996CMaPh.181...11G. doi:10.1007/BF02101672. S2CID 17663903. • Longo, Roberto; Roberts, John E. (1997). "A Theory of Dimension". K-Theory. 11 (2): 103–159. arXiv:funct-an/9604008. Bibcode:1996funct.an..4008L. doi:10.1023/A:1007714415067. S2CID 119581477. • Kawahigashi, Yasuyuki; Longo, Roberto; Müger, Michael (2001). "Multi-Interval Subfactors and Modularity of Representations in Conformal Field Theory". Communications in Mathematical Physics. 219 (3): 631–669. arXiv:math/9903104. Bibcode:2001CMaPh.219..631K. doi:10.1007/PL00005565. S2CID 15568392. • Kawahigashi, Yasuyuki; Longo, Roberto (2004). "Classification of local conformal nets: Case c < 1". Annals of Mathematics. 160 (2): 493–522. arXiv:math-ph/0201015. doi:10.4007/annals.2004.160.493. S2CID 7145642. • Kac, Victor G.; Longo, Roberto; Xu, Feng (2004). "Solitons in affine and permutation orbifolds". Communications in Mathematical Physics. 253 (3): 723–764. arXiv:math/0312512. doi:10.1007/s00220-004-1160-1. S2CID 14298134. • Longo, Roberto; Witten, Edward (2011). "An Algebraic Construction of Boundary Quantum Field Theory". Communications in Mathematical Physics. 303 (1): 213–232. arXiv:1004.0616. Bibcode:2011CMaPh.303..213L. doi:10.1007/s00220-010-1133-5. S2CID 115174261. • Longo, Roberto; Morinelli, Vincenzo; Rehren, Karl-Henning (2016). "Where Infinite Spin Particles Are Localizable". Communications in Mathematical Physics. 345 (2): 587–614. arXiv:1505.01759. Bibcode:2016CMaPh.345..587L. doi:10.1007/s00220-015-2475-9. S2CID 119569712. • Carpi, Sebastiano; Kawahigashi, Yasuyuki; Longo, Roberto; Weiner, Mihály (2018). "From vertex operator algebras to conformal nets and back". Memoirs of the American Mathematical Society. 254 (1213). arXiv:1503.01260. doi:10.1090/memo/1213. S2CID 119618626. • Ciolli, Fabio; Longo, Roberto; Ruzzi, Giuseppe (2019). "The information in a wave". Communications in Mathematical Physics. 379 (3): 979–1000. arXiv:1906.01707. Bibcode:2019CMaPh.379..979C. doi:10.1007/s00220-019-03593-3. S2CID 174799030. See also • Axiomatic quantum field theory • Local quantum physics • Operator algebra • Quantum field theory References 1. "Roberto Longo". Department of Mathematics, University of Rome Tor Vergata. Retrieved February 19, 2021. 2. The academic degree is Laurea in Matematica and was the highest academic title at the epoch. 3. The thesis is Longo, Roberto (1975). Tomita-Takesaki modular structure for AFD von Neumann algebras (Thesis). Rome. Roberto Longo at the Mathematics Genealogy Project 4. "Roberto Longo, Curriculum Vitae" (PDF). University of Rome Tor Vergara. Retrieved February 19, 2021. 5. "Humboldt Foundation member page of Roberto Longo". humboldt-foundation.de. Retrieved February 19, 2021. 6. Doplicher, S.; Longo, R. (1984). "Standard and split inclusions of von Neumann algebras". Inventiones Mathematicae. 75 (3): 493–536. Bibcode:1984InMat..75..493D. doi:10.1007/BF01388641. ISSN 0020-9910. S2CID 121819323. 7. Longo, Roberto (1984). "Solution of the factorial Stone-Weierstrass conjecture. An application of the theory of standard split W-inclusions". Inventiones Mathematicae. 76 (1): 145–155. Bibcode:1984InMat..76..145L. doi:10.1007/BF01388497. S2CID 122775835. 8. Doplicher, Sergio; Haag, Rudolf; Roberts, John E. (1971). "Local observables and particle statistics. 1". Communications in Mathematical Physics. 23 (3): 199–230. Bibcode:1971CMaPh..23..199D. doi:10.1007/BF01877742. S2CID 189833852. 9. Longo, Roberto; Roberts, John E. (1997). "A Theory of Dimension". K-Theory. 11 (2): 103–159. arXiv:funct-an/9604008. Bibcode:1996funct.an..4008L. doi:10.1023/A:1007714415067. S2CID 119581477. 10. Kawahigashi, Yasuyuki; Longo, Roberto (2004). "Classification of local conformal nets: Case c < 1". Annals of Mathematics. 160 (2): 493–522. arXiv:math-ph/0201015. doi:10.4007/annals.2004.160.493. S2CID 7145642. 11. Longo, Roberto; Morinelli, Vincenzo; Rehren, Karl-Henning (2016). "Where Infinite Spin Particles Are Localizable". Communications in Mathematical Physics. 345 (2): 587–614. arXiv:1505.01759. Bibcode:2016CMaPh.345..587L. doi:10.1007/s00220-015-2475-9. S2CID 119569712. 12. Ciolli, Fabio; Longo, Roberto; Ruzzi, Giuseppe (2019). "The information in a wave". Communications in Mathematical Physics. 379 (3): 979–1000. arXiv:1906.01707. Bibcode:2019CMaPh.379..979C. doi:10.1007/s00220-019-03593-3. S2CID 174799030. 13. "ICM Plenary and Invited Speakers". mathunion.org. Retrieved February 19, 2021. Longo, Roberto (1995). "Von Neumann algebras and quantum field theory". Proceedings of the International Congress of Mathematicians, 1994, Zürich. Vol. 2. pp. 1281–1291. 14. Longo, Roberto (1981). "Modular Automorphisms of Local Algebras in Quantum Field Theory". 6th International Conference on Mathematical Physics - Congress of Association for Mathematical Physics. Berlin. pp. 372–373.{{cite book}}: CS1 maint: location missing publisher (link) 15. Longo, Roberto (1988). "Inclusions of Von Neumann Algebrass and Quantum Field Theory". IX International Conference on Mathematical Physics (IAMP). Swansea. pp. 472–474.{{cite book}}: CS1 maint: location missing publisher (link) 16. Longo, Roberto (1994). "Inclusions of von Neumann algebras and superselection structures". 11th International Conference on Mathematical Physics (ICMP-11). Paris. pp. 342–351.{{cite book}}: CS1 maint: location missing publisher (link) 17. "ICMP 2003 schedule". XIV ICMP Lisbon. Retrieved February 19, 2021. 18. "Mathematisches Institut - Andrejewski Vorlesung". uni-math.gwdg.de (in German). Retrieved February 23, 2021. 19. "Plenary Speakers". ICMP Prague. Retrieved February 19, 2021. 20. "Modular theory and Bekenstein's bound by Robert Longo, partly based on a joint work with Feng Xu" (PDF). Okinawa, OIST, Strings 2018. Retrieved February 19, 2021. 21. "List of Fellows of the American Mathematical Society". American Mathematical Society. Retrieved February 19, 2021. Jackson, Allyn (2013). "Fellows of the AMS: Inaugural Class" (PDF). Notices of the AMS. 60 (5): 631–637. 22. "Longo's member page of Academia Europæa". Academia Europæa. Retrieved July 20, 2021. 23. "Humboldt-Preisträger". Uni Inform, Göttingen (in German): 5. April 2015. 24. "A Roberto Longo la Medaglia dei XL per la Matematica 2021". uniroma2.it (in Italian). Retrieved July 20, 2021. 25. "Mathematics and Quantum Physics Conference". cmtp.uniroma2.it. Retrieved February 19, 2021. 26. "Scienze, Roma trionfa negli Advanced grants". Il Sole 24 Ore, Roma (in Italian) (40). October 15, 2008. "Advanced Grants: Operator Algebras and Conformal Field Theory". europa.eu. Retrieved February 25, 2021. 27. "Advanced Grants: Quantum Algebraic Structures and Models". europa.eu. Retrieved February 25, 2021. 28. "Un matematico di Tor Vergata vincitore dell'"Advanced Grants"". maddmaths.simai.eu (in Italian). Retrieved February 23, 2021. 29. "Sectional Panels for ICM 2018" (PDF). mathunion.org/icm/past-icms. Retrieved February 19, 2021. External links Wikimedia Commons has media related to Roberto Longo. • Roberto Longo at the Mathematics Genealogy Project. • Roberto Longo at zbMATH. • "Center for Mathematics and Theoretical Physics webpage". Center for Mathematics and Theoretical Physics. Retrieved February 21, 2021. • "Longo's personal webpage". uniroma2.it. Retrieved February 21, 2021. • "Longo's seminar in the conference about the 50 years of algebraic quantum field theory". gwdg.de. July 30, 2009. Retrieved February 19, 2021. • Longo, Roberto (July 6, 2016). "KMS states in Conformal Field Theory". YouTube. Hausdorff Center for Mathematics. Retrieved February 19, 2021. • Longo, Roberto (2017). "Noncommutative Geometry: State of the Art and Future Prospects, Alain Connes 70th Birthday". mat.uniroma2.it. Shanghai. Retrieved February 21, 2021. • Longo, Roberto (June 12, 2017). "Discussion about the Landauer principle (and bound)". Isaac Newton Institute for Mathematical Sciences. Retrieved February 21, 2021. • Longo, Roberto (May 5, 2020). "The Information in a Wave". YouTube. Retrieved February 21, 2021. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Italy • Israel • United States Academics • Google Scholar • MathSciNet • Mathematics Genealogy Project • ORCID • ResearcherID • Scopus • zbMATH Other • IdRef	Wikipedia
Roberto Markarian Roberto Markarian Abrahamian (12 December 1946) is a Uruguayan mathematician of Armenian descent, expert in dynamical systems and chaos theory. Biography He started studying at the University of the Republic in the 1960s. During the civic-military dictatorship he was arrested due to political reasons. Later he went to Brazil, where he graduated from the Federal University of Rio Grande do Sul. Later on, his degree was validated in Uruguay. Markarian served as rector of the University of the Republic (2014-2018).[1] He is brother of the football coach Sergio Markarian. References 1. "University of the Republic: last rectors". Universidad de la República (in Spanish). Retrieved 13 October 2022.	Wikipedia
Roberts's triangle theorem Roberts's triangle theorem, a result in discrete geometry, states that every simple arrangement of $n$ lines has at least $n-2$ triangular faces. Thus, three lines form a triangle, four lines form at least two triangles, five lines form at least three triangles, etc. It is named after Samuel Roberts, a British mathematician who published it in 1889.[1][2] Statement and example The theorem states that every simple arrangement of $n$ lines in the Euclidean plane has at least $n-2$ triangular faces. Here, an arrangement is simple when it has no two parallel lines and no three lines through the same point.[1] One way to form an arrangement of $n$ lines with exactly $n-2$ triangular faces is to choose the lines to be tangent to a semicircle. For lines arranged in this way, the only triangles are the ones formed by three lines with consecutive points of tangency. As the $n$ lines have $n-2$ consecutive triples, they also have $n-2$ triangles.[1] Proof Branko Grünbaum found the proof in Roberts's original paper "unconvincing",[3][4] and credits the first correct proof of Roberts's theorem to Robert W. Shannon, in 1979.[1][5] He presents instead the following more elementary argument, first published in Russian by Alexei Belov.[1][6] It depends implicitly on a simpler version of the same theorem, according to which every simple arrangement of three or more lines has at least one triangular face. This follows easily by induction from the fact that adding a line to an arrangement cannot decrease the number of triangular faces: if the line cuts an existing triangle, one of the resulting two pieces is again a triangle. On the other hand, although the bound of Roberts's theorem increases with each added line, the number of triangles in any particular arrangement may sometimes remain unchanged.[1] If the $n$ given lines are all moved without changing their slopes, their new positions can be described by a system of $n$ real numbers, the offsets of each line from its original position. For each triangular face, there is a linear equation on the offsets of its three lines that, if satisfied, causes the face to retain its original area. If there could be fewer than $n-2$ triangles, then (because there would be more variables than equations constraining them) it would be possible to fix two of the lines in place and find a simultaneous linear motion of all remaining lines, keeping their slopes fixed, that preserves all of the triangle areas. Such a motion must pass through arrangements that are not simple, for instance when one of the moving lines passes over the crossing point of the two fixed lines. At the time when the moving lines first form a non-simple arrangement, three or more lines meet at a point. Just before these lines meet, this subset of lines would have a triangular face (also present in the original arrangement) whose area approaches zero. But this contradicts the invariance of the face areas. The contradiction shows that the assumption that there are fewer than $n-2$ triangles cannot be true.[1][6] Related results Whereas Roberts's theorem concerns the fewest possible triangles made by a given number of lines, the related Kobon triangle problem concerns the largest number possible.[3] The two problems differ already for $n=5$, where Roberts's theorem guarantees that three triangles will exist, but the solution to the Kobon triangle problem has five triangles.[1] Roberts's theorem can be generalized from simple line arrangements to some non-simple arrangements, to arrangements in the projective plane rather than in the Euclidean plane, and to arrangements of hyperplanes in higher-dimensional spaces.[5] Beyond line arrangements, the same bound as Roberts's theorem holds for arrangements of pseudolines.[7] References 1. Grünbaum, Branko (1998), "How many triangles?" (PDF), Geombinatorics, 8 (1): 154–159, MR 1633757 2. Roberts, Samuel (November 1887), "On the figures formed by the intercepts of a system of straight lines in a plane, and on analogous relations in space of three dimensions", Proceedings of the London Mathematical Society, s1-19 (1): 405–422, doi:10.1112/plms/s1-19.1.405 3. Fejes Tóth, L. (1975), "A combinatorial problem concerning oriented lines in the plane", Research Problems, The American Mathematical Monthly, 82 (4): 387–389, doi:10.1080/00029890.1975.11993840, JSTOR 2318414, MR 1537693 4. Grünbaum, Branko (1972), Arrangements and Spreads, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, vol. 10, Providence, Rhode Island: American Mathematical Society, p. 26, ISBN 9780821816592, MR 0307027 5. Shannon, R. W. (1979), "Simplicial cells in arrangements of hyperplanes", Geometriae Dedicata, 8 (2): 179–187, doi:10.1007/BF00181486, MR 0538524, S2CID 119681116 6. Belov, A. Ya. (1992), "A problem in combinatorial geometry", Uspekhi Matematicheskikh Nauk, 47 (3): 151–152, doi:10.1070/RM1992v047n03ABEH000898, MR 1185304, S2CID 250734782 7. Felsner, S.; Kriegel, K. (1999), "Triangles in Euclidean arrangements", Discrete & Computational Geometry, 22 (3): 429–438, doi:10.1007/PL00009471, MR 1706582, S2CID 16696505	Wikipedia
Robin Gandy Robin Oliver Gandy (22 September 1919 – 20 November 1995) was a British mathematician and logician.[4] He was a friend, student, and associate of Alan Turing, having been supervised by Turing during his PhD at the University of Cambridge,[1] where they worked together.[5][6][7] Robin Gandy Born Robin Oliver Gandy (1919-09-22)22 September 1919 Rotherfield Peppard, Oxfordshire, England Died20 November 1995(1995-11-20) (aged 76) Oxford, England NationalityBritish EducationAbbotsholme School Alma materUniversity of Cambridge (PhD) Known forRecursion theory Scientific career FieldsMathematical logic Institutions • University of Manchester • University of Leicester • University of Leeds • Stanford University • University of California, Los Angeles • University of Oxford ThesisOn Axiomatic Systems in Mathematics and Theories in Physics (1953) Doctoral advisorAlan Turing[1][2] Doctoral students • Martin Hyland[3] • Jane Kister[2] • Jeff Paris[2] Education and early life Robin Gandy was born in the village of Rotherfield Peppard, Oxfordshire, England.[4] He was the son of Thomas Hall Gandy (1876–1948), a general practitioner, and Ida Caroline née Hony (1885–1977), a social worker and later an author.[8] He was a great-great-grandson of the architect and artist Joseph Gandy (1771–1843). Educated at Abbotsholme School in Derbyshire, Gandy took two years of the Mathematical Tripos, at King's College, Cambridge, before enlisting for military service in 1940. During World War II he worked on radio intercept equipment at Hanslope Park, where Alan Turing was working on a speech encipherment project, and he became one of Turing's lifelong friends and associates. In 1946, he completed Part III of the Mathematical Tripos, then began studying for a PhD under Turing's supervision. He completed his thesis, On axiomatic systems in mathematics and theories in Physics, in 1952.[1] He was a member of the Cambridge Apostles.[9] Career and research Gandy held positions at the University of Leicester, the University of Leeds, and the University of Manchester. He was a visiting associate professor at Stanford University from 1966 to 1967, and held a similar position at University of California, Los Angeles in 1968. In 1969, he moved to Wolfson College, Oxford, where he became Reader in Mathematical Logic. Gandy is known for his work in recursion theory. His contributions include the Spector–Gandy theorem, the Gandy Stage Comparison theorem, and the Gandy Selection theorem. He also made a significant contribution to the understanding of the Church–Turing thesis, and his generalisation of the Turing machine is called a Gandy machine.[10] Gandy died in Oxford, England on 20 November 1995.[4][11] Legacy The Robin Gandy Buildings, a pair of accommodation blocks at Wolfson College, Oxford, are named after Gandy.[12][13] A one-day centenary Gandy Colloquium was held on 22 February 2020 at the College in Gandy's honour, including contributions by some of his students;[14][15] the speakers were Marianna Antonutti Marfori (Munich), Andrew Hodges (Oxford), Martin Hyland (Cambridge), Jeff Paris (Manchester), Göran Sundholm (Leiden), Christine Tasson (Paris), and Philip Welch (Bristol). References 1. Gandy, Robin Oliver (1953). On axiomatic systems in mathematics and theories in physics. repository.cam.ac.uk (PhD thesis). University of Cambridge. doi:10.17863/CAM.16125. EThOS uk.bl.ethos.590164. 2. Robin Gandy at the Mathematics Genealogy Project 3. Hyland, John Martin Elliott (1975). Recursion Theory on the Countable Functionals. bodleian.ox.ac.uk (DPhil thesis). University of Oxford. EThOS uk.bl.ethos.460247. 4. Yates, Mike (24 November 1995). "Obituary: Robin Gandy". The Independent. Retrieved 1 January 2012. 5. Hodges, Andrew (1983). Alan Turing: The Enigma. Simon & Schuster. ISBN 0-671-49207-1. 6. "Notices". The Bulletin of Symbolic Logic. 2 (1): 121–125. March 1996. doi:10.1017/s1079898600007988. JSTOR 421052. S2CID 246638427. 7. Moschovakis, Yannis & Yates, Mike (September 1996). "In Memoriam: Robin Oliver Gandy, 1919–1995". The Bulletin of Symbolic Logic. 2 (3): 367–370. doi:10.1017/s1079898600007873. JSTOR 420996. S2CID 120785678. 8. "Ida Gandy - Writer". Aldbourne Heritage Centre. Retrieved 7 April 2021. 9. "Wolfson College salutes Robin Gandy on his centenary \| Wolfson College, Oxford". 10. Wilfried Sieg, 2005, Church without dogma: axioms for computability, Carnegie Mellon University 11. Robin Gandy — The Alan Turing Scrapbook, archived at Archive.Today 12. "Accommodation types – Robin Gandy Buildings". UK: Wolfson College, Oxford. Retrieved 14 April 2020. 13. "Robin Gandy Buildings, Wolfson". Flickr. 6 April 2008. Retrieved 1 January 2012. 14. "The Gandy Colloquium". UK: Wolfson College, Oxford. 22 February 2020. Retrieved 14 April 2020. 15. Isaacson, Daniel (2020). "Wolfson College salutes Robin Gandy on his centenary". UK: Wolfson College, Oxford. Retrieved 14 April 2020. Authority control International • ISNI • VIAF National • Norway • Catalonia • Germany • Israel • United States • Czech Republic • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
C. Robin Graham Charles Robin Graham is professor emeritus of mathematics at the University of Washington, known for a number of contributions to the field of conformal geometry and CR geometry; his collaboration with Charles Fefferman on the ambient construction has been particularly widely cited. The GJMS operators are, in part, named for him.[1] He is a 2012 Fellow of the American Mathematical Society.[2] Graham received his Ph.D. from Princeton University in 1981, under the direction of Elias Stein.[3] Major publications • Fefferman, Charles; Graham, C. Robin. Conformal invariants. The mathematical heritage of Élie Cartan (Lyon, 1984). Astérisque 1985, Numéro Hors Série, 95–116. • Graham, C. Robin; Jenne, Ralph; Mason, Lionel J.; Sparling, George A.J. Conformally invariant powers of the Laplacian. I. Existence. J. London Math. Soc. (2) 46 (1992), no. 3, 557–565. • Fefferman, Charles; Graham, C. Robin. The Ambient Metric. Annals of Mathematics Studies 178, Princeton University Press, 2012.[4] References 1. Baum, Helga; Juhl, Andreas (2011). "Section 1.3: GJMS-operators and Branson's Q-curvatures". Conformal Differential Geometry: Q-Curvature and Conformal Holonomy. Oberwolfach Seminars. Vol. 40. Springer. pp. 21ff. ISBN 9783764399092. 2. Choi, Rose (November 1, 2012). "AMS Fellows Named". Department of Mathematics, University of Washington. Retrieved 2020-10-22. 3. C. Robin Graham at the Mathematics Genealogy Project 4. Reviews of The Ambient Metric: Andreas Cap, Zbl 1243.53004; Michael G. Eastwood, MR2858236; Michael G. Eastwood, Bull. AMS, doi:10.1090/S0273-0979-2013-01435-6; Rod Gover, SIAM Rev., doi:10.1137/130973478 Authority control: Academics • MathSciNet • Mathematics Genealogy Project	Wikipedia
Robin Hartshorne Robin Cope Hartshorne (/ˈhɑːrts.hɔːrn/ HARTS-horn; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry. Robin Hartshorne Born (1938-03-15) March 15, 1938 Boston, United States NationalityAmerican Alma materPrinceton University Harvard University Phillips Exeter Academy Known forAlgebraic Geometry Hartshorne ellipse AwardsLeroy P. Steele Prize (1979) Fellow of the American Mathematical Society (2012) Scientific career FieldsMathematics InstitutionsUniversity of California, Berkeley Harvard University ThesisConnectedness of the Hilbert scheme (1963) Doctoral advisorJohn Coleman Moore Oscar Zariski Doctoral studentsMei-Chu Chang Lawrence Ein David Gieseker Mark Gross Arthur Ogus Career Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University[1] (under the name Robert C. Hartshorne[2]). He received a Ph.D. in mathematics from Princeton University in 1963 after completing a doctoral dissertation titled Connectedness of the Hilbert scheme under the supervision of John Coleman Moore and Oscar Zariski.[3][4] He then became a Junior Fellow at Harvard University, where he taught for several years.[5] In 1972, he was appointed to the faculty at the University of California, Berkeley,[5] where he is a Professor Emeritus as of 2020.[6] Hartshorne is the author of the text Algebraic Geometry.[7][8] Awards In 1979, Hartshorne was awarded the Leroy P. Steele Prize for "his expository research article Equivalence relations on algebraic cycles and subvarieties of small codimension, Proceedings of Symposia in Pure Mathematics, volume 29, American Mathematical Society, 1975, pp. 129-164; and his book Algebraic geometry, Springer-Verlag, Berlin and New York, 1977."[9] In 2012, Hartshorne became a fellow of the American Mathematical Society.[10] Personal life Hartshorne attended high school at Phillips Exeter Academy, graduating in 1955. Hartshorne is married to Edie Churchill and has two sons and an adopted daughter.[5] He is a mountain climber and amateur flute and shakuhachi player.[5] Selected publications • Foundations of Projective Geometry, New York: W. A. Benjamin, 1967; • Ample Subvarieties of Algebraic Varieties, New York: Springer-Verlag. 1970; • Algebraic Geometry, New York: Springer-Verlag, 1977;[11] corrected 6th printing, 1993. GTM 52, ISBN 0-387-90244-9 • Families of Curves in P3 and Zeuthen's Problem. Vol. 617. American Mathematical Society, 1997. • Geometry: Euclid and Beyond, New York: Springer-Verlag, 2000;[12] corrected 2nd printing, 2002;[13] corrected 4th printing, 2005. ISBN 0-387-98650-2 • Local Cohomology: A Seminar Given by A. Grothendieck, Harvard University. Fall, 1961. Vol. 41. Springer, 2006. (lecture notes by R. Hartshorne) • Deformation Theory, Springer-Verlag, GTM 257, 2010, ISBN 978-1-4419-1595-5[14] See also • Hartshorne ellipse References 1. Gallian, Joseph A. (October 1989). "Fifty Years of Putnam Trivia". The American Mathematical Monthly. 96 (8): 711–713. doi:10.2307/2324720. JSTOR 2324720. Retrieved December 10, 2021. 2. "List of Previous Putnam Winners" (PDF). Mathematical Association of America. Retrieved December 30, 2020. 3. Hartshorne, Robin (1963). Connectedness of the Hilbert scheme. 4. Robin Hartshorne at the Mathematics Genealogy Project 5. Algebraic Geometry. Retrieved December 30, 2020. {{cite book}}: \|website= ignored (help) 6. "Robin C. Hartshorne". University of California, Berkeley. Retrieved December 30, 2020. 7. Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157, Zbl 0367.14001 8. Shatz, Stephen S. (1979), "Review: Robin Hartshorne, Algebraic geometry", Bull. Amer. Math. Soc., New Series, 1 (3): 553–560, doi:10.1090/S0273-0979-1979-14618-4 9. "Prize: Leroy P. Steele Prize (1970 - 1992)". American Mathematical Society. Retrieved December 30, 2020. 10. List of Fellows of the American Mathematical Society, retrieved 2013-01-19. 11. Shatz, Stephen S. (1979). "Review of Algebraic geometry by Robin Hartshorne". Bulletin of the American Mathematical Society. New Series. 1: 553–560. doi:10.1090/S0273-0979-1979-14618-4. 12. Henderson, David W. (2002). "Review of Geometry: Euclid and beyond by Robin Hartshorne" (PDF). Bulletin of the American Mathematical Society. New Series. 39: 563–571. doi:10.1090/S0273-0979-02-00949-7. 13. Seddighin, Morteza (21 April 2004). "Review of Geometry: Euclid and Beyond by R. Hartshorne". MAA Reviews, Mathematical Association of America, maa.org. 14. Zaldivar, Felipe (9 March 2010). "Review of Deformation Theory by R. Hartshorne". MAA Reviews, Mathematical Association of America, maa.org. External links Wikiquote has quotations related to Robin Hartshorne. • Home page at the University of California at Berkeley • Hartshorne's Paintings Authority control International • ISNI • VIAF National • Norway • France • BnF data • Catalonia • Germany • Israel • United States • Japan • Czech Republic • Netherlands Academics • CiNii • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef	Wikipedia
Robin Thomas (mathematician) Robin Thomas (August 22, 1962 – March 26, 2020) was a mathematician working in graph theory at the Georgia Institute of Technology. Robin Thomas Born(1962-08-22)August 22, 1962[1] Prague, Czechoslovakia DiedMarch 26, 2020(2020-03-26) (aged 57)[2] Alma materCharles University AwardsFulkerson Prize Karel Janeček Foundation Neuron Prize Scientific career FieldsMathematics InstitutionsGeorgia Institute of Technology Doctoral advisorJaroslav Nešetřil Thomas received his doctorate in 1985 from Charles University in Prague, Czechoslovakia (now the Czech Republic), under the supervision of Jaroslav Nešetřil.[3] He joined the faculty at Georgia Tech in 1989, and became a Regents' Professor there,[4][5] briefly serving as the department Chair. On March 26, 2020, he died of Amyotrophic Lateral Sclerosis at the age of 57 after 12 years of struggle with the illness.[2][6] Awards Thomas was awarded the Fulkerson Prize for outstanding papers in discrete mathematics twice,[7] in 1994 as co-author of a paper on the Hadwiger conjecture,[8] and in 2009 for the proof of the strong perfect graph theorem.[9] In 2011 he was awarded the Karel Janeček Foundation Neuron Prize for Lifetime Achievement in Mathematics.[10] In 2012 he became a fellow of the American Mathematical Society.[11] He was named a SIAM Fellow in 2018.[12] The January 2023 issue of the Journal of Combinatorial Theory, Series B was a tribute to his work.[13] References 1. Cremation Society of Georgia: Robin Thomas 2. Fortnow, Lance (March 28, 2020). "Robin Thomas". Computational Complexity. 3. Robin Thomas at the Mathematics Genealogy Project. 4. Author biography from Berg, Deborah E.; Norine, Serguei; Su, Francis Edward; Thomas, Robin; Wollan, Paul (2008). "Voting in agreeable societies". arXiv:0811.3245 [math.CO].. 5. Robin Thomas Earns Distinction, Named Regents' Professor, Georgia Tech College of Computing, June 24, 2010. 6. Robin Thomas tribute, Georgia Tech Mathematics, April 7, 2020. 7. Fulkerson Prize: Official site with award details. 8. Robertson, Neil; Seymour, Paul; Thomas, Robin (1993), "Hadwiger's conjecture for K6-free graphs", Combinatorica, 13 (3): 279–361, doi:10.1007/BF01202354, S2CID 9608738. 9. Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2006), "The strong perfect graph theorem", Annals of Mathematics, 164 (1): 51–229, arXiv:math/0212070, doi:10.4007/annals.2006.164.51, S2CID 119151552. 10. Karel Janeček Foundation 2011 Neuron Prize winners (in Czech) Archived 2012-12-23 at the Wayback Machine 11. List of Fellows of the American Mathematical Society, retrieved 2013-08-27. 12. "SIAM Announces Class of 2018 Fellows", SIAM News, March 29, 2018 13. Krivelevich, Michael; Mohar, Bojan (2023). "Editorial \| Robin Thomas (1962–2020)". Journal of Combinatorial Theory, Series B. doi:10.1016/j.jctb.2022.10.001. External links • Personal homepage of Robin Thomas • Liu, Chun-Hung (June 2022). "Legacy of Robin Thomas" (PDF). Notices of the American Mathematical Society. 69 (6): 966–977. doi:10.1090/noti2497. Authority control International • ISNI • VIAF National • Germany • Israel • United States • Czech Republic • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Robin Wilson (mathematician) Robin James Wilson (born 5 December 1943) is an emeritus professor in the Department of Mathematics at the Open University, having previously been Head of the Pure Mathematics Department and Dean of the Faculty.[1] He was a stipendiary lecturer at Pembroke College, Oxford[2] and, as of 2006, Gresham Professor of Geometry at Gresham College, London, where he has also been a visiting professor.[3] On occasion, he teaches at Colorado College in the United States.[4] He is also a long standing fellow of Keble College, Oxford. Professor Robin Wilson Born Robin James Wilson (1943-12-05) 5 December 1943 London, England Alma materUniversity College School, Hampstead, London University of Oxford (Balliol College) University of Pennsylvania Spouse Joy Crispin (m. 1968) Scientific career FieldsGraph theory InstitutionsOpen University, Pembroke College, Oxford, Gresham College Doctoral advisorNesmith Ankeny Doctoral studentsAmanda Chetwynd Professor Wilson is a son of former British Prime Minister Harold Wilson and his wife, Mary. Early life and education Wilson was born in 1943 to the politician Harold Wilson, who later became Prime Minister, and his wife the poet Mary Wilson (née Baldwin). He has a younger brother, Giles, who in his 50s gave up a career as a teacher to be a train driver.[5] Wilson attended University College School in Hampstead, North London. He achieved a BA First Class Honours in Mathematics from Balliol College, Oxford, an MA from the University of Pennsylvania, a PhD from the University of Pennsylvania (1965–1968) and a BA First Class Honours in Humanities with Music from the Open University. In a Guardian interview in 2008, Wilson spoke of the fact he grew up known to everyone primarily as a son of the Labour Party leader and Prime Minister Harold Wilson: "I hated the attention and I still dislike being introduced as Harold Wilson's son. I feel uncomfortable talking about it to strangers even now."[6] Mathematics career Wilson's academic interests lie in graph theory, particularly in colouring problems, e.g. the four colour problem, and algebraic properties of graphs. He also researches the history of mathematics, particularly British mathematics and mathematics in the 17th century and the period 1860 to 1940, and the history of graph theory and combinatorics. In 1974, he won the Lester R. Ford Award from the Mathematical Association of America for his expository article An introduction to matroid theory.[7][8] Due to his collaboration on a 1977 paper[9] with the Hungarian mathematician Paul Erdős, Wilson has an Erdős number of 1. In July 2008, he published a study of the mathematical work of Lewis Carroll, the creator of Alice's Adventures in Wonderland and Through the Looking-Glass — Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life (Allen Lane, 2008. ISBN 978-0-7139-9757-6). From January 1999 to September 2003, Wilson was editor-in-chief of the European Mathematical Society Newsletter.[10] He is past President of the British Society for the History of Mathematics.[11] Other interests He has strong interests in music, including the operas of Gilbert and Sullivan, and is the co-author (with Frederic Lloyd) of Gilbert and Sullivan: The Official D'Oyly Carte Picture History.[12] In 2007, he was a guest on Private Passions, the biographical music discussion programme on BBC Radio 3.[13] Personal life Wilson is married and has twin daughters.[14] Publications Wilson has written or edited about thirty books, including popular books on sudoku and the Four Color Theorem: • Oxford's Savilian Professors of Geometry: The First 400 Years (editor), Oxford University Press, 2022: ISBN 978-0-19-886903-0 • Number Theory: A Very Short Introduction, Oxford University Press, 2020: ISBN 978-0-19-879809-5 • The Turing Guide (with Jack Copeland, Jonathan Bowen, Mark Sprevak, et al.), Oxford University Press, 2017: ISBN 978-0198747826 (hardcover), ISBN 978-0198747833 (paperback)[15] • Combinatorics: A Very Short Introduction, Oxford University Press, 2016: ISBN 978-0-19-872349-3 • Combinatorics: Ancient & Modern (with John Watkins), Oxford University Press, 2013: ISBN 0-19-965659-2 • The Great Mathematicians (with Raymond Flood), Arcturus Publishing Ltd, 2011: ISBN 1-84837-902-1 • Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life, Allen Lane, 2008: ISBN 978-0-7139-9757-6 • Hidden Word Sudoku, Infinite Ideas Limited 2005: ISBN 1-904902-74-X • How to Solve Sudoku, Infinite Ideas Limited 2005: ISBN 1-904902-62-6 • Sherlock Holmes in Babylon and Other Tales of Mathematical History (co-edited with Marlow Anderson and Victor J. Katz), The Mathematical Association of America, 2004: ISBN 0-88385-546-1 • Mathematics and Music: From Pythagoras to Fractals (co-edited with John Fauvel & Raymond Flood), Oxford University Press, 2003: ISBN 0-19-851187-6 • Four Colours Suffice: How the Map Problem Was Solved, Allen Lane (Penguin), 2002: ISBN 0-7139-9670-6 • Stamping through Mathematics, Springer, 2001: ISBN 0-387-98949-8 • Oxford Figures: 800 Years of the Mathematical Sciences (with John Fauvel & Raymond Flood), Oxford: Clarendon Press, 2000: ISBN 0-19-852309-2 • Graphs and Applications: An Introductory Approach (with Joan Aldous), Springer, 2000: ISBN 1-85233-259-X • Mathematical Conversations: Selections from the Mathematical Intelligencer (with J. Gray), Springer, 2000: ISBN 0-387-98686-3 • An Atlas of Graphs (with Ronald Read), Oxford: Clarendon Press, 1998: ISBN 0-19-853289-X (paperback edition, 2002: ISBN 0-19-852650-4) • Graph Theory, 1736–1936 (with Norman L. Biggs and Keith Lloyd), Oxford: Clarendon Press, 1976: ISBN 0-19-853901-0 References 1. "Prof Robin Wilson". UK: Open University, Department of Mathematics And Statistics. Retrieved 8 December 2013. 2. Pembroke College website 3. "Professor Robin Wilson". Gresham College. Retrieved 8 December 2013. 4. "Block Visitors" (PDF). Countable Bits. The Colorado College Department of Mathematics and Computer Science. 8 (1). May 2015. Retrieved 23 June 2017. 5. "Son of former PM Harold Wilson swaps teaching for a career as train driver". London Evening Standard. 20 November 2006. Retrieved 11 December 2019. 6. Crace, John (6 October 2008). "Interview: Robin Wilson, mathematics professor, on his passions and father". The Guardian. ISSN 0261-3077. Retrieved 16 February 2019. 7. Paul R. Halmos – Lester R. Ford Awards, Mathematical Association of America 8. Wilson, R. J. (1973). "An introduction to matroid theory". Amer. Math. Monthly. 80 (5): 500–525. CiteSeerX 10.1.1.599.5103. doi:10.2307/2319608. JSTOR 2319608. 9. Erdős, P.; Wilson, Robin J. (1977). "On the chromatic index of almost all graphs". Journal of Combinatorial Theory. Series B. 23 (2–3): 255–257. doi:10.1016/0095-8956(77)90039-9. 10. European Mathematical Society Newsletter, No 49, September 2003, ISSN 1027-488X 11. "Professor Robin Wilson". Open University. Retrieved 8 December 2013. 12. Knopf, 1984. ISBN 978-0-394-54113-6 13. BBC Radio 3 14. John Crace (7 October 2008). "Serious showman". The Guardian. Retrieved 8 December 2013. 15. Robinson, Andrew (4 January 2017). "The Turing Guide: Last words on an enigmatic codebreaker?". New Scientist. External links • Robin Wilson's Page at the Open University • Robin Wilson's entry in the Faculty of Mathematics and Computing at the Open University • Lectures by Robin Wilson at Gresham College • Robin Wilson's entry at the Mathematics Genealogy Project • Robin Wilson at IMDb Harold Wilson Premierships • 1964–1970 • 1974–1976 General elections • 1964 • 1966 • 1970 • February 1974 • October 1974 Party elections • 1960 leadership • 1962 deputy leadership • 1963 leadership Shadow Cabinet elections • 1952 • 1953 • 1954 • 1955 • 1956 • 1957 • 1958 • 1959 • 1960 Referendum • 1975 European Communities membership Constituencies • Ormskirk • Huyton Resignation Honours • 1970 • 1976 Family • Mary Wilson (wife) • Robin Wilson (son) • Harold Seddon (Uncle) Media and popular culture • Mrs. Wilson's Diary • Yesterday's Men (TV documentary, 1971) • The Lavender List (TV, 2006) • Longford (TV, 2006) • Made in Dagenham (Film, 2010) • The Audience (Play, 2013) • The Crown (TV, 2019) • Stonehouse (TV, 2023) Related articles • Rhodesia's Unilateral Declaration of Independence • Beira Patrol • D-notice affair • Gnomes of Zürich • Harold Wilson conspiracy theories • Clockwork Orange • Harry letters affair • In Place of Strife • Wilson Doctrine • Bibliography • Category Authority control International • ISNI • VIAF National • Norway • France • BnF data • Catalonia • Germany • Israel • Belgium • United States • Sweden • Latvia • Japan • Czech Republic • Australia • Greece • Korea • Netherlands • Poland Academics • CiNii • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Robinson's joint consistency theorem Robinson's joint consistency theorem is an important theorem of mathematical logic. It is related to Craig interpolation and Beth definability. The classical formulation of Robinson's joint consistency theorem is as follows: Let $T_{1}$ and $T_{2}$ be first-order theories. If $T_{1}$ and $T_{2}$ are consistent and the intersection $T_{1}\cap T_{2}$ is complete (in the common language of $T_{1}$ and $T_{2}$), then the union $T_{1}\cup T_{2}$ is consistent. A theory $T$ is called complete if it decides every formula, meaning that for every sentence $\varphi ,$ the theory contains the sentence or its negation but not both (that is, either $T\vdash \varphi $ or $T\vdash \neg \varphi $). Since the completeness assumption is quite hard to fulfill, there is a variant of the theorem: Let $T_{1}$ and $T_{2}$ be first-order theories. If $T_{1}$ and $T_{2}$ are consistent and if there is no formula $\varphi $ in the common language of $T_{1}$ and $T_{2}$ such that $T_{1}\vdash \varphi $ and $T_{2}\vdash \neg \varphi ,$ then the union $T_{1}\cup T_{2}$ is consistent. See also • Łoś–Vaught test References • Boolos, George S.; Burgess, John P.; Jeffrey, Richard C. (2002). Computability and Logic. Cambridge University Press. p. 264. ISBN 0-521-00758-5. • Robinson, Abraham, 'A result on consistency and its application to the theory of definition', Proc. Royal Academy of Sciences, Amsterdam, series A, vol 59, pp 47-58. Mathematical logic General • Axiom • list • Cardinality • First-order logic • Formal proof • Formal semantics • Foundations of mathematics • Information theory • Lemma • Logical consequence • Model • Theorem • Theory • Type theory Theorems (list) & Paradoxes • Gödel's completeness and incompleteness theorems • Tarski's undefinability • Banach–Tarski paradox • Cantor's theorem, paradox and diagonal argument • Compactness • Halting problem • Lindström's • Löwenheim–Skolem • Russell's paradox Logics Traditional • Classical logic • Logical truth • Tautology • Proposition • Inference • Logical equivalence • Consistency • Equiconsistency • Argument • Soundness • Validity • Syllogism • Square of opposition • Venn diagram Propositional • Boolean algebra • Boolean functions • Logical connectives • Propositional calculus • Propositional formula • Truth tables • Many-valued logic • 3 • Finite • ∞ Predicate • First-order • list • Second-order • Monadic • Higher-order • Free • Quantifiers • Predicate • Monadic predicate calculus Set theory • Set • Hereditary • Class • (Ur-)Element • Ordinal number • Extensionality • Forcing • Relation • Equivalence • Partition • Set operations: • Intersection • Union • Complement • Cartesian product • Power set • Identities Types of Sets • Countable • Uncountable • Empty • Inhabited • Singleton • Finite • Infinite • Transitive • Ultrafilter • Recursive • Fuzzy • Universal • Universe • Constructible • Grothendieck • Von Neumann Maps & Cardinality • Function/Map • Domain • Codomain • Image • In/Sur/Bi-jection • Schröder–Bernstein theorem • Isomorphism • Gödel numbering • Enumeration • Large cardinal • Inaccessible • Aleph number • Operation • Binary Set theories • Zermelo–Fraenkel • Axiom of choice • Continuum hypothesis • General • Kripke–Platek • Morse–Kelley • Naive • New Foundations • Tarski–Grothendieck • Von Neumann–Bernays–Gödel • Ackermann • Constructive Formal systems (list), Language & Syntax • Alphabet • Arity • Automata • Axiom schema • Expression • Ground • Extension • by definition • Conservative • Relation • Formation rule • Grammar • Formula • Atomic • Closed • Ground • Open • Free/bound variable • Language • Metalanguage • Logical connective • ¬ • ∨ • ∧ • → • ↔ • = • Predicate • Functional • Variable • Propositional variable • Proof • Quantifier • ∃ • ! • ∀ • rank • Sentence • Atomic • Spectrum • Signature • String • Substitution • Symbol • Function • Logical/Constant • Non-logical • Variable • Term • Theory • list Example axiomatic systems (list) • of arithmetic: • Peano • second-order • elementary function • primitive recursive • Robinson • Skolem • of the real numbers • Tarski's axiomatization • of Boolean algebras • canonical • minimal axioms • of geometry: • Euclidean: • Elements • Hilbert's • Tarski's • non-Euclidean • Principia Mathematica Proof theory • Formal proof • Natural deduction • Logical consequence • Rule of inference • Sequent calculus • Theorem • Systems • Axiomatic • Deductive • Hilbert • list • Complete theory • Independence (from ZFC) • Proof of impossibility • Ordinal analysis • Reverse mathematics • Self-verifying theories Model theory • Interpretation • Function • of models • Model • Equivalence • Finite • Saturated • Spectrum • Submodel • Non-standard model • of arithmetic • Diagram • Elementary • Categorical theory • Model complete theory • Satisfiability • Semantics of logic • Strength • Theories of truth • Semantic • Tarski's • Kripke's • T-schema • Transfer principle • Truth predicate • Truth value • Type • Ultraproduct • Validity Computability theory • Church encoding • Church–Turing thesis • Computably enumerable • Computable function • Computable set • Decision problem • Decidable • Undecidable • P • NP • P versus NP problem • Kolmogorov complexity • Lambda calculus • Primitive recursive function • Recursion • Recursive set • Turing machine • Type theory Related • Abstract logic • Category theory • Concrete/Abstract Category • Category of sets • History of logic • History of mathematical logic • timeline • Logicism • Mathematical object • Philosophy of mathematics • Supertask Mathematics portal	Wikipedia
Gary Robinson Gary Robinson is an American software engineer and mathematician[2] and inventor notable for his mathematical algorithms to fight spam.[3] In addition, he patented a method to use web browser cookies to track consumers across different web sites, allowing marketers to better match advertisements with consumers.[4][5] The patent was bought by DoubleClick, and then DoubleClick was bought by Google.[6][7] He is credited as being one of the first to use automated collaborative filtering technologies to turn word-of-mouth recommendations into useful data.[2] Gary Robinson Born (1956-02-06) February 6, 1956 Bronxville, NY, US EducationBard College;Courant Institute[1] OccupationComputer programmer EmployerEmergent Music LLC[1] Known forSpamBayes, SpamAssassin, Recommendation engine, Collaborative filtering TitleChief Technology officer[1] WebsiteGaryRobinson.net Algorithms to identify spam In 2003, Robinson's article in Linux Journal detailed a new approach to computer programming perhaps best described as a general purpose classifier which expanded on the usefulness of Bayesian filtering. Robinson's method used math-intensive algorithms combined with Chi-square statistical testing to enable computers to examine an unknown file and make intelligent guesses about what was in it.[8] The technique had wide applicability; for example, Robinson's method enabled computers to examine a file and guess, with much greater accuracy, whether it contained pornography, or whether an incoming email to a corporation was a technical question or a sales-related question.[9] The method became the basis for anti-spam techniques used by Tim Peters and Rob Hooft of the influential SpamBayes project.[10][11] Spamming is the abuse of electronic messaging systems to send unsolicited, undesired bulk messages.[12] SpamBayes assigned probability scores to both spam and ham (useful emails) to guess intelligently whether an incoming email was spam; the scoring system enabled the program to return a value of unsure if both the spam and ham scores were high.[8] Robinson's method was used in other anti-spam projects such as SpamAssassin.[13][14][15] Robinson commented in Linux Journal on how fighting spam was a collaborative effort: The approach described here truly has been a distributed effort in the best open-source tradition. Paul Graham, an author of books on Lisp, suggested an approach to filtering spam in his on-line article, "A Plan for Spam". I took his approach for generating probabilities associated with words, altered it slightly and proposed a Bayesian calculation for dealing with words that hadn't appeared very often ... an approach based on the chi-square distribution for combining the individual word probabilities into a combined probability (actually a pair of probabilities—see below) representing an e-mail. Finally, Tim Peters of the Spambayes Project proposed a way of generating a particularly useful spamminess indicator based on the combined probabilities. All along the way the work was guided by ongoing testing of embodiments written in Python by Tim Peters for Spambayes and in C by Greg Louis of the Bogofilter Project. The testing was done by a number of people involved with those projects. — Gary Robinson, 2003.[11] In 1996, Robinson patented a method to help marketers focus their online advertisements to consumers. He explained: As far as I have been able to tell, it's the very first patent ... to mention using web browser cookies to track consumers across different web sites and build a profile of their interests in order to determine what ads to show them ... There was an aspect in the way browser cookies were implemented that allowed them to be used ... I hired programmers to do the programming to actually test it ... the hypothesis turned out to be correct. — Gary B. Robinson, 2014 Entrepreneurial activity In 2010, Robinson was the chief technology officer at FlyFi, an online music service owned by Maine-based[16] Emergent Discovery which uses his anti-spam programming techniques along with collaborative filtering technologies to help make music recommendations to web users.[17][18] His blog Gary Robinson's Rants has been quoted by others in the computer and online music industries[17] and cited by academic papers.[12][19][20] Robinson helped develop recommendation engine technology which applies high-power mathematical techniques using software algorithms to have a computer guess intelligently about what a consumer might like.[21] For example, if a consumer likes music by artists such as the Beach Boys, Bob Dylan and the Talking Heads, the computer software will match these preferences with a much larger dataset of other consumers who also like those three artists but which cumulatively has much greater musical knowledge than the single consumer. Accordingly, the computer will find music that the user might like but hasn't been exposed to, and therefore hopefully offer intelligent recommendations, in a process which has come to be called knowledge management.[2] But the mathematics behind such comparisons can become quite complex and involved. Robinson studied mathematics at Bard College and graduated in 1979 and studied further at the Courant Institute of New York University.[1] In the 1980s, Robinson worked on an entrepreneurial start-up dating service called 212-Romance which used similar computer algorithms to match singles romantically.[2][22] The New York City-based voice mail dating service created community-based automated recommendations and used collaborative filtering technologies which Robinson developed further in other capacities. References 1. "Gary Robinson". 2010-09-18. Retrieved 2010-09-18. I make the music recommendation technology at FlyFi — Where I grew up Bronxville, NY — Companies I've worked for Athenium, OLI Systems, Lambda Technology — Schools I've attended Bard College; Courant Institute of Mathematical Sciences {{cite web}}: External link in \|quote= (help) 2. Matthew French, May 20, 2002, Boston Business Journal, Romantic beginnings have worldwide effect, Retrieved August 6, 2016, "... Gary Robinson ... a mathematician by training ... first automated collaborative filtering applications ..." 3. "SpamBayes Project Page". SpamBayes. 2010-09-18. Retrieved 2010-09-18. Gary Robinson provided a lot of the serious maths and theory, as well as his essay on "how to do it better" (see the background page for a link). 4. US 5918014 A, Application number US 08/774,180, Publication date Jun 29, 1999, Filing date Dec 26, 1996, Automated collaborative filtering in world wide web advertising, "... This invention combines techniques for: determining the subject's community, and determining which ads to show ... to determine whether a given individual should be in the subject's community is gleaned from the individual's activities ... Means are provided to track a consumer's activities ... e.g. by means of "cookies"..." 5. Patent Buddy, Gary B Robinson Inventor, Patent years: 1999, 2001, "... Automated collaborative filtering in world wide web advertising ..." 6. TechCrunch, Apr 13, 2007 by Michael Arrington, Breaking: Google Spends $3.1 Billion To Acquire DoubleClick, Accessed March 12, 2014, "... About 20 minutes ago Google announced that they have agreed to acquired DoubleClick for $3.1 billion in cash ..." 7. Bill Slawski, Apr 14, 2007, SEO by the Sea, Doubleclick + Google: Looking at Some of the Doubleclick Patent Filings, Accessed March 12, 2014, "... smart ad box showing on a page that displays different advertisements to users over time, based upon a recommendations system. ..." 8. "Background Reading". SpamBayes project. 2010-09-18. Retrieved 2010-09-18. Sharpen your pencils, this is the mathematical background (such as it is).* The paper that started the ball rolling: Paul Graham's A Plan for Spam.* Gary Robinson has an interesting essay suggesting some improvements to Graham's original approach.* Gary Robinson's Linux Journal article discussed using the chi squared distribution. {{cite news}}: External link in \|quote= (help) 9. Ben Kamens, Fog Creek Publishing, Bayesian Filtering: Beyond Binary Classification Archived 2015-09-24 at the Wayback Machine, Retrieved February 7, 2015, "... Of these, Robinson's technique ... borrowed from R.A. Fischer's combination of probabilities into a chi-squared distribution, has been extensively tested and is used by the most successful filters, including SpamBayes. Robinson provides ample theoretical justification for this improvement in practical accuracy over the original filters ..." 10. T.A. Meyer and B Whateley (2010-09-18). "SpamBayes: Effective open-source, Bayesian based, email classification system". Massey University, Auckland, New Zealand. Retrieved 2010-09-18. G. Robinson, "Spam Detection", [online] 2002, ... G. Robinson, "Instructions for Training to Exhaustion", (Gary' Longer Rants), [online] 2004, (see page 8) 11. Gary Robinson (Mar 1, 2003). "A Statistical Approach to the Spam Problem: Using Bayesian statistics to detect an e-mail's spamminess". Linux Journal. Retrieved 2010-09-18. This article discusses one of many possible mathematical foundations for a key aspect of spam filtering—generating an indicator of "spamminess" from a collection of tokens representing the content of an e-mail. 12. David Anderson (September 2006). "Statistical Spam Filtering — EECS595, Fall 2006". Retrieved 2010-09-18. Gary Robinson proposes an improved method for calculating the word value of a token W. His method modifies Graham's by adding a confidence factor to scale the word value by the amount of historical data that is available for the token. Let N be ... 13. The SpamAssassin Project. "train SpamAssassin's Bayesian classifier". SpamAssassin website. Retrieved 2010-09-18. Gary Robinson's f(x) and combining algorithms, as used in SpamAssassin 14. "Credits — the Perl Programming Language — Algorithms". Perl. 2010-09-18. Retrieved 2010-09-18. Algorithms: The Bayesian-style text classifier used by SpamAssassin's BAYES rules is based on an approach outlined by Gary Robinson. Thanks, Gary! 15. "Installation". Ubuntu manuals. 2010-09-18. Archived from the original on 2010-09-29. Retrieved 2010-09-18. Gary Robinson's f(x) and combining algorithms, as used in SpamAssassin 16. "Contact "Emergent Discovery"". Emergent Discovery. 2010-10-14. Retrieved 2010-10-14. Emergent Discovery — 565 Congress Street — Suite 201 —Portland, ME 04101 17. Kevin Dangoor (April 30, 2002). "Gary Robinson's Three Steps to Freedom". BlueSkyOnMars. Retrieved 2010-09-18. Gary Robinson, the head of Emergent Music has an article on his blog about the Three Steps To Freedom. His opinion on this definitely counts, because EM might very well be the future of music. I'm going to chime in with my thoughts here and copy them over to EM's forum as well. 18. "Management Team". FlyFi. 2010-09-18. Retrieved 2010-09-18. Gary Robinson, CTO, is both a musician and leader in the "recommendation engine" field. Gary's background reflects his pioneering work in mathematics, technology and collaborative filtering. 19. Gary Robinson (2006-01-30). "Request for Your Input Regarding Three Steps To Freedom: THE 3 STEPS TO FREEDOM". Gary Robinson's Rants: Rants on spam, business, digital music, patents, and other assorted random stuff. Retrieved 2010-09-18. So, as a "thought experiment," I have imagined the following path to creating an alternative music industry. 20. "FlyFi iTunes Helper 2.0.0.1 for Mac". CNet. 2010-09-18. Retrieved 2010-09-18. The FlyFi iTunes Helper sends the contents of your iTunes data file (a behind the scenes part of your iTunes library) to FlyFi server to be analyzed. By looking at your iTunes music, which is one of the best reflections of your musical tastes, FlyFi can make better new music suggestion. FlyFi can also use this information to better serve other members. 21. "Management Team". Emergent Discovery. 2010-09-18. Retrieved 2010-09-18. Gary Robinson, CTO, is a leader in the "recommendation engine" field. Gary's background reflects his pioneering work in mathematics, technology and collaborative filtering. For instance, as a Research Director at ActiveState, Gary's work on spam detection is now being widely adopted by the anti-spam industry, including such leading filters as SpamAssassin (PC Magazine's Editor's Choice for spam filtering), SpamSieve (MacWorld's Software of the Year) and SpamBayes (PC World's Editor's Choice for spam filtering). 22. "New York Magazine". Sep 12, 1988. Retrieved 2010-09-18. (ad for 212-Romance on left side of page) External links • Gary Robinson's Rants blog • Automated collaborative filtering patent	Wikipedia
Robinson–Schensted correspondence In mathematics, the Robinson–Schensted correspondence is a bijective correspondence between permutations and pairs of standard Young tableaux of the same shape. It has various descriptions, all of which are of algorithmic nature, it has many remarkable properties, and it has applications in combinatorics and other areas such as representation theory. The correspondence has been generalized in numerous ways, notably by Knuth to what is known as the Robinson–Schensted–Knuth correspondence, and a further generalization to pictures by Zelevinsky. The simplest description of the correspondence is using the Schensted algorithm (Schensted 1961), a procedure that constructs one tableau by successively inserting the values of the permutation according to a specific rule, while the other tableau records the evolution of the shape during construction. The correspondence had been described, in a rather different form, much earlier by Robinson (Robinson 1938), in an attempt to prove the Littlewood–Richardson rule. The correspondence is often referred to as the Robinson–Schensted algorithm, although the procedure used by Robinson is radically different from the Schensted algorithm, and almost entirely forgotten. Other methods of defining the correspondence include a nondeterministic algorithm in terms of jeu de taquin. The bijective nature of the correspondence relates it to the enumerative identity $\sum _{\lambda \in {\mathcal {P}}_{n}}(t_{\lambda })^{2}=n!$ where ${\mathcal {P}}_{n}$ denotes the set of partitions of n (or of Young diagrams with n squares), and tλ denotes the number of standard Young tableaux of shape λ. The Schensted algorithm The Schensted algorithm starts from the permutation σ written in two-line notation $\sigma ={\begin{pmatrix}1&2&3&\cdots &n\\\sigma _{1}&\sigma _{2}&\sigma _{3}&\cdots &\sigma _{n}\end{pmatrix}}$ where σi = σ(i), and proceeds by constructing sequentially a sequence of (intermediate) ordered pairs of Young tableaux of the same shape: $(P_{0},Q_{0}),(P_{1},Q_{1}),\ldots ,(P_{n},Q_{n}),$ where P0 = Q0 are empty tableaux. The output tableaux are P = Pn and Q = Qn. Once Pi−1 is constructed, one forms Pi by inserting σi into Pi−1, and then Qi by adding an entry i to Qi−1 in the square added to the shape by the insertion (so that Pi and Qi have equal shapes for all i). Because of the more passive role of the tableaux Qi, the final one Qn, which is part of the output and from which the previous Qi are easily read off, is called the recording tableau; by contrast the tableaux Pi are called insertion tableaux. Insertion The basic procedure used to insert each σi is called Schensted insertion or row-insertion (to distinguish it from a variant procedure called column-insertion). Its simplest form is defined in terms of "incomplete standard tableaux": like standard tableaux they have distinct entries, forming increasing rows and columns, but some values (still to be inserted) may be absent as entries. The procedure takes as arguments such a tableau T and a value x not present as entry of T; it produces as output a new tableau denoted T ← x and a square s by which its shape has grown. The value x appears in the first row of T ← x, either having been added at the end (if no entries larger than x were present), or otherwise replacing the first entry y > x in the first row of T. In the former case s is the square where x is added, and the insertion is completed; in the latter case the replaced entry y is similarly inserted into the second row of T, and so on, until at some step the first case applies (which certainly happens if an empty row of T is reached). More formally, the following pseudocode describes the row-insertion of a new value x into T.[1] 1. Set i = 1 and j to one more than the length of the first row of T. 2. While j > 1 and x < Ti, j−1, decrease j by 1. (Now (i, j) is the first square in row i with either an entry larger than x in T, or no entry at all.) 3. If the square (i, j) is empty in T, terminate after adding x to T in square (i, j) and setting s = (i, j). 4. Swap the values x and Ti, j. (This inserts the old x into row i, and saves the value it replaces for insertion into the next row.) 5. Increase i by 1 and return to step 2. The shape of T grows by exactly one square, namely s. Correctness The fact that T ← x has increasing rows and columns, if the same holds for T, is not obvious from this procedure (entries in the same column are never even compared). It can however be seen as follows. At all times except immediately after step 4, the square (i, j) is either empty in T or holds a value greater than x; step 5 re-establishes this property because (i, j) now is the square immediately below the one that originally contained x in T. Thus the effect of the replacement in step 4 on the value Ti, j is to make it smaller; in particular it cannot become greater than its right or lower neighbours. On the other hand the new value is not less than its left neighbour (if present) either, as is ensured by the comparison that just made step 2 terminate. Finally to see that the new value is larger than its upper neighbour Ti−1, j if present, observe that Ti−1, j holds after step 5, and that decreasing j in step 2 only decreases the corresponding value Ti−1, j. Constructing the tableaux The full Schensted algorithm applied to a permutation σ proceeds as follows. 1. Set both P and Q to the empty tableau 2. For i increasing from 1 to n compute P ← σi and the square s by the insertion procedure; then replace P by P ← σi and add the entry i to the tableau Q in the square s. 3. Terminate, returning the pair (P, Q). The algorithm produces a pair of standard Young tableaux. Invertibility of the construction It can be seen that given any pair (P, Q) of standard Young tableaux of the same shape, there is an inverse procedure that produces a permutation that will give rise to (P, Q) by the Schensted algorithm. It essentially consists of tracing steps of the algorithm backwards, each time using an entry of Q to find the square where the inverse insertion should start, moving the corresponding entry of P to the preceding row, and continuing upwards through the rows until an entry of the first row is replaced, which is the value inserted at the corresponding step of the construction algorithm. These two inverse algorithms define a bijective correspondence between permutations of n on one side, and pairs of standard Young tableaux of equal shape and containing n squares on the other side. Properties One of the most fundamental properties, but not evident from the algorithmic construction, is symmetry: • If the Robinson–Schensted correspondence associates tableaux (P, Q) to a permutation σ, then it associates (Q, P) to the inverse permutation σ−1. This can be proven, for instance, by appealing to Viennot's geometric construction. Further properties, all assuming that the correspondence associates tableaux (P, Q) to the permutation σ = (σ1, ..., σn). • In the pair of tableaux (P′, Q′) associated to the reversed permutation (σn, ..., σ1), the tableau P′ is the transpose of the tableau P, and Q′ is a tableau determined by Q, independently of P (namely the transpose of the tableau obtained from Q by the Schützenberger involution). • The length of the longest increasing subsequence of σ1, ..., σn is equal to the length of the first row of P (and of Q). • The length of the longest decreasing subsequence of σ1, ..., σn is equal to the length of the first column of P (and of Q), as follows from the previous two properties. • The descent set {i : σi > σi+1} of σ equals the descent set {i : i+1 is in a row strictly below the row of i} of Q. • Identify subsequences of π with their sets of indices. It is a theorem of Greene that for any k ≥ 1, the size of the largest set that can be written as the union of at most k increasing subsequences is λ1 + ... + λk. In particular, λ1 equals the largest length of an increasing subsequence of π. • If σ is an involution, then the number of fixed points of σ equals the number of columns of odd length in λ. See also • Viennot's geometric construction, which provides a diagrammatic interpretation of the correspondence. • Plactic monoid: the insertion process can be used to define an associative product of Young tableaux with entries between 1 and n, which is referred to as the Plactic monoid of order n. Notes 1. Adapted from D. E. Knuth (1973), The Art of Computer Programming, vol. 3, pp. 50–51 References • Fulton, William (1997), Young Tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, ISBN 978-0-521-56144-0, MR 1464693. • Knuth, Donald E. (1970), "Permutations, matrices, and generalized Young tableaux", Pacific Journal of Mathematics, 34: 709–727, doi:10.2140/pjm.1970.34.709, MR 0272654 • Robinson, G. de B. (1938), "On the Representations of the Symmetric Group", American Journal of Mathematics, 60 (3): 745–760, doi:10.2307/2371609, JSTOR 2371609, Zbl 0019.25102. • Sagan, B. E. (2001), The Symmetric Group, Graduate Texts in Mathematics, vol. 203, New York: Springer-Verlag, ISBN 0-387-95067-2. • Schensted, C. (1961), "Longest increasing and decreasing subsequences", Canadian Journal of Mathematics, 13: 179–191, doi:10.4153/CJM-1961-015-3, MR 0121305. • Stanley, Richard P. (1999), Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, ISBN 978-0-521-56069-6, MR 1676282. • Zelevinsky, A. V. (1981), "A generalization of the Littlewood–Richardson rule and the Robinson–Schensted–Knuth correspondence", Journal of Algebra, 69 (1): 82–94, doi:10.1016/0021-8693(81)90128-9, MR 0613858. Further reading • Green, James A. (2007). Polynomial representations of GLn. Lecture Notes in Mathematics. Vol. 830. With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, J. A. Green and M. Schocker (2nd corrected and augmented ed.). Berlin: Springer-Verlag. ISBN 3-540-46944-3. Zbl 1108.20044. External links • van Leeuwen, M.A.A. (2001) [1994], "Robinson–Schensted correspondence", Encyclopedia of Mathematics, EMS Press • Williams, L., Interactive animation of the Robinson-Schensted algorithm	Wikipedia
Robinson–Schensted–Knuth correspondence In mathematics, the Robinson–Schensted–Knuth correspondence, also referred to as the RSK correspondence or RSK algorithm, is a combinatorial bijection between matrices A with non-negative integer entries and pairs (P,Q) of semistandard Young tableaux of equal shape, whose size equals the sum of the entries of A. More precisely the weight of P is given by the column sums of A, and the weight of Q by its row sums. It is a generalization of the Robinson–Schensted correspondence, in the sense that taking A to be a permutation matrix, the pair (P,Q) will be the pair of standard tableaux associated to the permutation under the Robinson–Schensted correspondence. The Robinson–Schensted–Knuth correspondence extends many of the remarkable properties of the Robinson–Schensted correspondence, notably its symmetry: transposition of the matrix A results in interchange of the tableaux P,Q. The Robinson–Schensted–Knuth correspondence Introduction The Robinson–Schensted correspondence is a bijective mapping between permutations and pairs of standard Young tableaux, both having the same shape. This bijection can be constructed using an algorithm called Schensted insertion, starting with an empty tableau and successively inserting the values σ1,…,σn of the permutation σ at the numbers 1,2,…n; these form the second line when σ is given in two-line notation: $\sigma ={\begin{pmatrix}1&2&\ldots &n\\\sigma _{1}&\sigma _{2}&\ldots &\sigma _{n}\end{pmatrix}}$. The first standard tableau P is the result of successive insertions; the other standard tableau Q records the successive shapes of the intermediate tableaux during the construction of P. The Schensted insertion easily generalizes to the case where σ has repeated entries; in that case the correspondence will produce a semistandard tableau P rather than a standard tableau, but Q will still be a standard tableau. The definition of the RSK correspondence reestablishes symmetry between the P and Q tableaux by producing a semistandard tableau for Q as well. Two-line arrays The two-line array (or generalized permutation) wA corresponding to a matrix A is defined[1] as $w_{A}={\begin{pmatrix}i_{1}&i_{2}&\ldots &i_{m}\\j_{1}&j_{2}&\ldots &j_{m}\end{pmatrix}}$ in which for any pair (i,j) that indexes an entry Ai,j of A, there are Ai,j columns equal to ${\tbinom {i}{j}}$, and all columns are in lexicographic order, which means that 1. $i_{1}\leq i_{2}\leq i_{3}\cdots \leq i_{m}$, and 2. if $i_{r}=i_{s}\,$ and $r\leq s$ then $j_{r}\leq j_{s}$. Example The two-line array corresponding to $A={\begin{pmatrix}1&0&2\\0&2&0\\1&1&0\end{pmatrix}}$ is $w_{A}={\begin{pmatrix}1&1&1&2&2&3&3\\1&3&3&2&2&1&2\end{pmatrix}}$ Definition of the correspondence By applying the Schensted insertion algorithm to the bottom line of this two-line array, one obtains a pair consisting of a semistandard tableau P and a standard tableau Q0, where the latter can be turned into a semistandard tableau Q by replacing each entry b of Q0 by the b-th entry of the top line of wA. One thus obtains a bijection from matrices A to ordered pairs,[2] (P,Q) of semistandard Young tableaux of the same shape, in which the set of entries of P is that of the second line of wA, and the set of entries of Q is that of the first line of wA. The number of entries j in P is therefore equal to the sum of the entries in column j of A, and the number of entries i in Q is equal to the sum of the entries in row i of A. Example In the above example, the result of applying the Schensted insertion to successively insert 1,3,3,2,2,1,2 into an initially empty tableau results in a tableau P, and an additional standard tableau Q0 recoding the successive shapes, given by $P\quad =\quad {\begin{matrix}1&1&2&2\\2&3\\3\end{matrix}},\qquad Q_{0}\quad =\quad {\begin{matrix}1&2&3&7\\4&5\\6\end{matrix}},$ and after replacing the entries 1,2,3,4,5,6,7 in Q0 successively by 1,1,1,2,2,3,3 one obtains the pair of semistandard tableaux $P\quad =\quad {\begin{matrix}1&1&2&2\\2&3\\3\end{matrix}},\qquad Q\quad =\quad {\begin{matrix}1&1&1&3\\2&2\\3\end{matrix}}.$ Direct definition of the RSK correspondence The above definition uses the Schensted algorithm, which produces a standard recording tableau Q0, and modifies it to take into account the first line of the two-line array and produce a semistandard recording tableau; this makes the relation to the Robinson–Schensted correspondence evident. It is natural however to simplify the construction by modifying the shape recording part of the algorithm to directly take into account the first line of the two-line array; it is in this form that the algorithm for the RSK correspondence is usually described. This simply means that after every Schensted insertion step, the tableau Q is extended by adding, as entry of the new square, the b-th entry ib of the first line of wA, where b is the current size of the tableaux. That this always produces a semistandard tableau follows from the property (first observed by Knuth[2]) that for successive insertions with an identical value in the first line of wA, each successive square added to the shape is in a column strictly to the right of the previous one. Here is a detailed example of this construction of both semistandard tableaux. Corresponding to a matrix $A={\begin{pmatrix}0&0&0&0&0&0&0\\0&0&0&1&0&1&0\\0&0&0&1&0&0&0\\0&0&0&0&0&0&1\\0&0&0&0&1&0&0\\0&0&1&1&0&0&0\\0&0&0&0&0&0&0\\1&0&0&0&0&0&0\\\end{pmatrix}}$ one has the two-line array $w_{A}={\begin{pmatrix}2&2&3&4&5&6&6&8\\4&6&4&7&5&3&4&1\end{pmatrix}}.$ The following table shows the construction of both tableaux for this example Inserted pair${\tbinom {2}{4}}$${\tbinom {2}{6}}$${\tbinom {3}{4}}$${\tbinom {4}{7}}$${\tbinom {5}{5}}$${\tbinom {6}{3}}$${\tbinom {6}{4}}$${\tbinom {8}{1}}$ P ${\begin{matrix}4\end{matrix}}$ ${\begin{matrix}4&6\end{matrix}}$ ${\begin{matrix}4&4\\6\end{matrix}}$ ${\begin{matrix}4&4&7\\6\end{matrix}}$ ${\begin{matrix}4&4&5\\6&7\end{matrix}}$ ${\begin{matrix}3&4&5\\4&7\\6\end{matrix}}$ ${\begin{matrix}3&4&4\\4&5\\6&7\end{matrix}}$ ${\begin{matrix}1&4&4\\3&5\\4&7\\6\end{matrix}}$ Q ${\begin{matrix}2\end{matrix}}$ ${\begin{matrix}2&2\end{matrix}}$ ${\begin{matrix}2&2\\3\end{matrix}}$ ${\begin{matrix}2&2&4\\3\end{matrix}}$ ${\begin{matrix}2&2&4\\3&5\end{matrix}}$ ${\begin{matrix}2&2&4\\3&5\\6\end{matrix}}$ ${\begin{matrix}2&2&4\\3&5\\6&6\end{matrix}}$ ${\begin{matrix}2&2&4\\3&5\\6&6\\8\end{matrix}}$ Combinatorial properties of the RSK correspondence The case of permutation matrices If $A$ is a permutation matrix then RSK outputs standard Young Tableaux (SYT), $P,Q$ of the same shape $\lambda $. Conversely, if $P,Q$ are SYT having the same shape $\lambda $, then the corresponding matrix $A$ is a permutation matrix. As a result of this property by simply comparing the cardinalities of the two sets on the two sides of the bijective mapping we get the following corollary: Corollary 1: For each $n\geq 1$ we have $\sum _{\lambda \vdash n}(t_{\lambda })^{2}=n!$ where $\lambda \vdash n$ means $\lambda $ varies over all partitions of $n$ and $t_{\lambda }$ is the number of standard Young tableaux of shape $\lambda $. Symmetry Let $A$ be a matrix with non-negative entries. Suppose the RSK algorithm maps $A$ to $(P,Q)$ then the RSK algorithm maps $A^{T}$ to $(Q,P)$, where $A^{T}$ is the transpose of $A$.[1] In particular for the case of permutation matrices, one recovers the symmetry of the Robinson–Schensted correspondence:[3] Theorem 2: If the permutation $\sigma $ corresponds to a triple $(\lambda ,P,Q)$, then the inverse permutation, $\sigma ^{-1}$, corresponds to $(\lambda ,Q,P)$. This leads to the following relation between the number of involutions on $S_{n}$ with the number of tableaux that can be formed from $S_{n}$ (An involution is a permutation that is its own inverse):[3] Corollary 2: The number of tableaux that can be formed from $\{1,2,3,\ldots ,n\}$ is equal to the number of involutions on $\{1,2,3,\ldots ,n\}$. Proof: If $\pi $ is an involution corresponding to $(P,Q)$, then $\pi =\pi ^{-}$ corresponds to $(Q,P)$; hence $P=Q$. Conversely, if $\pi $ is any permutation corresponding to $(P,P)$, then $\pi ^{-}$ also corresponds to $(P,P)$; hence $\pi =\pi ^{-}$. So there is a one-one correspondence between involutions $\pi $ and tableaux $P$ The number of involutions on $\{1,2,3,\ldots ,n\}$ is given by the recurrence: $a(n)=a(n-1)+(n-1)a(n-2)\,$ Where $a(1)=1,a(2)=2$. By solving this recurrence we can get the number of involutions on $\{1,2,3,\ldots ,n\}$, $I(n)=n!\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {1}{2^{k}k!(n-2k)!}}$ Symmetric matrices Let $A=A^{T}$ and let the RSK algorithm map the matrix $A$ to the pair $(P,P)$, where $P$ is an SSYT of shape $\alpha $.[1] Let $\alpha =(\alpha _{1},\alpha _{2},\ldots )$ where the $\alpha _{i}\in N$ and $\sum \alpha _{i}<\infty $. Then the map $A\longmapsto P$ establishes a bijection between symmetric matrices with row($A$) $=\alpha $ and SSYT's of type $\alpha $. Applications of the RSK correspondence Cauchy's identity The Robinson–Schensted–Knuth correspondence provides a direct bijective proof of the following celebrated identity for symmetric functions: $\prod _{i,j}(1-x_{i}y_{j})^{-1}=\sum _{\lambda }s_{\lambda }(x)s_{\lambda }(y)$ where $s_{\lambda }$ are Schur functions. Kostka numbers Fix partitions $\mu ,\nu \vdash n$, then $\sum _{\lambda \vdash n}K_{\lambda \mu }K_{\lambda \nu }=N_{\mu \nu }$ where $K_{\lambda \mu }$ and $K_{\lambda \nu }$ denote the Kostka numbers and $N_{\mu \nu }$ is the number of matrices $A$, with non-negative elements, with row($A$) $=\mu $ and column($A$) $=\nu $. References 1. Stanley, Richard P. (1999). Enumerative Combinatorics. Vol. 2. New York: Cambridge University Press. pp. 316–380. ISBN 0-521-55309-1. 2. Knuth, Donald E. (1970). "Permutations, matrices, and generalized Young tableaux". Pacific Journal of Mathematics. 34 (3): 709–727. doi:10.2140/pjm.1970.34.709. MR 0272654. 3. Knuth, Donald E. (1973). The Art of Computer Programming, Vol. 3: Sorting and Searching. London: Addison–Wesley. pp. 54–58. ISBN 0-201-03803-X. • Brualdi, Richard A. (2006). Combinatorial matrix classes. Encyclopedia of Mathematics and Its Applications. Vol. 108. Cambridge: Cambridge University Press. pp. 135–162. ISBN 0-521-86565-4. Zbl 1106.05001. Donald Knuth Publications • The Art of Computer Programming • "The Complexity of Songs" • Computers and Typesetting • Concrete Mathematics • Surreal Numbers • Things a Computer Scientist Rarely Talks About • Selected papers series Software • TeX • Metafont • MIXAL (MIX • MMIX) Fonts • AMS Euler • Computer Modern • Concrete Roman Literate programming • WEB • CWEB Algorithms • Knuth's Algorithm X • Knuth–Bendix completion algorithm • Knuth–Morris–Pratt algorithm • Knuth shuffle • Robinson–Schensted–Knuth correspondence • Trabb Pardo–Knuth algorithm • Generalization of Dijkstra's algorithm • Knuth's Simpath algorithm Other • Dancing Links • Knuth reward check • Knuth Prize • Knuth's up-arrow notation • Man or boy test • Quater-imaginary base • -yllion • Potrzebie system of weights and measures	Wikipedia
Robinson arithmetic In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by R. M. Robinson in 1950.[1] It is usually denoted Q. Q is almost PA without the axiom schema of mathematical induction. Q is weaker than PA but it has the same language, and both theories are incomplete. Q is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable. Axioms The background logic of Q is first-order logic with identity, denoted by infix '='. The individuals, called natural numbers, are members of a set called N with a distinguished member 0, called zero. There are three operations over N: • A unary operation called successor and denoted by prefix S; • Two binary operations, addition and multiplication, denoted by infix + and ·, respectively. The following axioms for Q are Q1–Q7 in Burgess (2005, p. 42) (cf. also the axioms of first-order arithmetic). Variables not bound by an existential quantifier are bound by an implicit universal quantifier. 1. Sx ≠ 0 • 0 is not the successor of any number. 2. (Sx = Sy) → x = y • If the successor of x is identical to the successor of y, then x and y are identical. (1) and (2) yield the minimum of facts about N (it is an infinite set bounded by 0) and S (it is an injective function whose domain is N) needed for non-triviality. The converse of (2) follows from the properties of identity. 3. y=0 ∨ ∃x (Sx = y) • Every number is either 0 or the successor of some number. The axiom schema of mathematical induction present in arithmetics stronger than Q turns this axiom into a theorem. 4. x + 0 = x 5. x + Sy = S(x + y) • (4) and (5) are the recursive definition of addition. 6. x·0 = 0 7. x·Sy = (x·y) + x • (6) and (7) are the recursive definition of multiplication. Variant axiomatizations The axioms in Robinson (1950) are (1)–(13) in Mendelson (2015, pp. 202–203). The first 6 of Robinson's 13 axioms are required only when, unlike here, the background logic does not include identity. The usual strict total order on N, "less than" (denoted by "<"), can be defined in terms of addition via the rule x < y ↔ ∃z (Sz + x = y). Equivalently, we get a definitional conservative extension of Q by taking "<" as primitive and adding this rule as an eighth axiom; this system is termed "Robinson arithmetic R" in Boolos, Burgess & Jeffrey (2002, Sec 16.4). A different extension of Q, which we temporarily call Q+, is obtained if we take "<" as primitive and add (instead of the last definitional axiom) the following three axioms to axioms (1)–(7) of Q: • ¬(x < 0) • x < Sy ↔ (x < y ∨ x = y) • x < y ∨ x = y ∨ y < x Q+ is still a conservative extension of Q, in the sense that any formula provable in Q+ not containing the symbol "<" is already provable in Q. (Adding only the first two of the above three axioms to Q gives a conservative extension of Q that is equivalent to what Burgess (2005, p. 56) calls Q*. See also Burgess (2005, p. 230, fn. 24), but note that the second of the above three axioms cannot be deduced from "the pure definitional extension" of Q obtained by adding only the axiom x < y ↔ ∃z (Sz + x = y).) Among the axioms (1)–(7) of Q, axiom (3) needs an inner existential quantifier. Shoenfield (1967, p. 22) gives an axiomatization that has only (implicit) outer universal quantifiers, by dispensing with axiom (3) of Q but adding the above three axioms with < as primitive. That is, Shoenfield's system is Q+ minus axiom (3), and is strictly weaker than Q+, since axiom (3) is independent of the other axioms (for example, the ordinals less than $\omega ^{\omega }$ forms a model for all axioms except (3) when Sv is interpreted as v + 1). Shoenfield's system also appears in Boolos, Burgess & Jeffrey (2002, Sec 16.2), where it is called the "minimal arithmetic" (also denoted by Q). A closely related axiomatization, that uses "≤" instead of "<", may be found in Machover (1996, pp. 256–257). Metamathematics On the metamathematics of Q see Boolos, Burgess & Jeffrey (2002, chpt. 16), Tarski, Mostowski & Robinson (1953), Smullyan (1991), Mendelson (2015, pp. 202–203) and Burgess (2005, §§1.5a, 2.2). The intended interpretation of Q is the natural numbers and their usual arithmetic in which addition and multiplication have their customary meaning, identity is equality, Sx = x + 1, and 0 is the natural number zero. Any model (structure) that satisfies all axioms of Q except possibly axiom (3) has a unique submodel ("the standard part") isomorphic to the standard natural numbers (N, +, ·, S, 0). (Axiom (3) need not be satisfied; for example the polynomials with non-negative integer coefficients forms a model that satisfies all axioms except (3).) Q, like Peano arithmetic, has nonstandard models of all infinite cardinalities. However, unlike Peano arithmetic, Tennenbaum's theorem does not apply to Q, and it has computable non-standard models. For instance, there is a computable model of Q consisting of integer-coefficient polynomials with positive leading coefficient, plus the zero polynomial, with their usual arithmetic. A notable characteristic of Q is the absence of the axiom scheme of induction. Hence it is often possible to prove in Q every specific instance of a fact about the natural numbers, but not the associated general theorem. For example, 5 + 7 = 7 + 5 is provable in Q, but the general statement x + y = y + x is not. Similarly, one cannot prove that Sx ≠ x. [2] A model of Q that fails many of the standard facts is obtained by adjoining two distinct new elements a and b to the standard model of natural numbers and defining Sa = a, Sb = b, x + a = b and x + b = a for all x, a + n = a and b + n = b if n is a standard natural number, x·0 = 0 for all x, a·n = b and b·n = a if n is a non-zero standard natural number, x·a = a for all x except x = a, x·b = b for all x except x = b, a·a = b, and b·b = a.[3] Q is interpretable in a fragment of Zermelo's axiomatic set theory, consisting of extensionality, existence of the empty set, and the axiom of adjunction. This theory is S' in Tarski, Mostowski & Robinson (1953, p. 34) and ST in Burgess (2005, pp. 90–91, 223). See general set theory for more details. Q is a finitely axiomatized first-order theory that is considerably weaker than Peano arithmetic (PA), and whose axioms contain only one existential quantifier. Yet like PA it is incomplete and incompletable in the sense of Gödel's incompleteness theorems, and essentially undecidable. Robinson (1950) derived the Q axioms (1)–(7) above by noting just what PA axioms are required [4] to prove that every computable function is representable in PA.[5] The only use this proof makes of the PA axiom schema of induction is to prove a statement that is axiom (3) above, and so, all computable functions are representable in Q. [6][7][8] The conclusion of Gödel's second incompleteness theorem also holds for Q: no consistent recursively axiomatized extension of Q can prove its own consistency, even if we additionally restrict Gödel numbers of proofs to a definable cut.[9][10][11] The first incompleteness theorem applies only to axiomatic systems defining sufficient arithmetic to carry out the necessary coding constructions (of which Gödel numbering forms a part). The axioms of Q were chosen specifically to ensure they are strong enough for this purpose. Thus the usual proof of the first incompleteness theorem can be used to show that Q is incomplete and undecidable. This indicates that the incompleteness and undecidability of PA cannot be blamed on the only aspect of PA differentiating it from Q, namely the axiom schema of induction. Gödel's theorems do not hold when any one of the seven axioms above is dropped. These fragments of Q remain undecidable, but they are no longer essentially undecidable: they have consistent decidable extensions, as well as uninteresting models (i.e., models which are not end-extensions of the standard natural numbers). See also • Gentzen's consistency proof • Gödel's incompleteness theorem • List of first-order theories • Peano axioms • Presburger arithmetic • Skolem arithmetic • Second-order arithmetic • Set-theoretic definition of natural numbers • General set theory References 1. Robinson 1950. 2. Burgess 2005, p. 56. 3. Boolos, Burgess & Jeffrey 2002, Sec 16.4. 4. Mendelson 2015, p. 188, Proposition 3.24. 5. A function $f$ is said to be representable in $\operatorname {PA} $ if there is a formula $\phi $ such that for all $x_{1},\cdots ,x_{k},y$ $f({\vec {x}})=y\Longleftrightarrow \operatorname {PA} \vdash \phi ({\vec {x}},y),$ $f({\vec {x}})\neq y\Longleftrightarrow \operatorname {PA} \vdash \lnot \phi ({\vec {x}},y).$ 6. Odifreddi 1989. 7. Mendelson 2015, p. 203, Proposition 3.33. 8. Rautenberg 2010, p. 246. 9. Bezboruah & Shepherdson 1976. 10. Pudlák 1985. 11. Hájek & Pudlák 1993, p. 387. Bibliography • Bezboruah, A.; Shepherdson, John C. (June 1976). "Gödel's Second Incompleteness Theorem for Q". Journal of Symbolic Logic. 41 (2): 503–512. doi:10.2307/2272251. JSTOR 2272251. • Boolos, George; Burgess, John P.; Jeffrey, Richard (2002). Computability and Logic (4th ed.). Cambridge University Press. ISBN 0-521-00758-5. • Burgess, John P. (July 2005). Fixing Frege. Princeton University Press. ISBN 978-0691122311. • Hájek, Petr; Pudlák, Pavel (1993). Metamathematics of first-order arithmetic (2nd ed.). Springer-Verlag. • Jones, James P.; Shepherdson, John C. (1983). "Variants of Robinson's essentially undecidable theoryR". Archiv für mathematische Logik und Grundlagenforschung. 23: 61–64. doi:10.1007/BF02023013. S2CID 2659126. • Lucas, John R. Conceptual Roots of Mathematics. Routledge. • Machover, Moshé (1996). Set Theory, Logic, and Their Limitation. Cambridge University Press. • Mendelson, Elliott (2015). Introduction to Mathematical Logic (6th ed.). Chapman & Hall. ISBN 9781482237726. • Odifreddi, Piergiorgio (1989). Classical recursion theory, Vol. 1 (The Theory of Functions and Sets of Natural Numbers). ISBN 9780444894830. {{cite book}}: \|journal= ignored (help) • Pudlák, Pavel (June 1985). "Cuts, consistency statements and interpretations". Journal of Symbolic Logic. 50 (2): 423–441. doi:10.2307/2274231. JSTOR 2274231. S2CID 30289163. • Rautenberg, Wolfgang (2010). A Concise Introduction to Mathematical Logic (3rd ed.). New York: Springer Science+Business Media. doi:10.1007/978-1-4419-1221-3. ISBN 978-1-4419-1220-6.. • Robinson, R. M. (1950). "An Essentially Undecidable Axiom System". Proceedings of the International Congress of Mathematics: 729–730. • Shoenfield, Joseph R. (1967). Mathematical logic. Addison Wesley. (Reprinted by Association for Symbolic Logic and A K Peters in 2000). • Smullyan, Raymond (1991). Gödel's Incompleteness Theorems. Oxford University Press. • Tarski, Alfred; Mostowski, A.; Robinson, R. M. (1953). Undecidable theories. North Holland. • Vaught, Robert L. (1966). "On a Theorem of Cobham Concerning Undecidable Theories". Studies in Logic and the Foundations of Mathematics. 44: 14–25. doi:10.1016/S0049-237X(09)70566-X. ISBN 9780804700962. Mathematical logic General • Axiom • list • Cardinality • First-order logic • Formal proof • Formal semantics • Foundations of mathematics • Information theory • Lemma • Logical consequence • Model • Theorem • Theory • Type theory Theorems (list) & Paradoxes • Gödel's completeness and incompleteness theorems • Tarski's undefinability • Banach–Tarski paradox • Cantor's theorem, paradox and diagonal argument • Compactness • Halting problem • Lindström's • Löwenheim–Skolem • Russell's paradox Logics Traditional • Classical logic • Logical truth • Tautology • Proposition • Inference • Logical equivalence • Consistency • Equiconsistency • Argument • Soundness • Validity • Syllogism • Square of opposition • Venn diagram Propositional • Boolean algebra • Boolean functions • Logical connectives • Propositional calculus • Propositional formula • Truth tables • Many-valued logic • 3 • Finite • ∞ Predicate • First-order • list • Second-order • Monadic • Higher-order • Free • Quantifiers • Predicate • Monadic predicate calculus Set theory • Set • Hereditary • Class • (Ur-)Element • Ordinal number • Extensionality • Forcing • Relation • Equivalence • Partition • Set operations: • Intersection • Union • Complement • Cartesian product • Power set • Identities Types of Sets • Countable • Uncountable • Empty • Inhabited • Singleton • Finite • Infinite • Transitive • Ultrafilter • Recursive • Fuzzy • Universal • Universe • Constructible • Grothendieck • Von Neumann Maps & Cardinality • Function/Map • Domain • Codomain • Image • In/Sur/Bi-jection • Schröder–Bernstein theorem • Isomorphism • Gödel numbering • Enumeration • Large cardinal • Inaccessible • Aleph number • Operation • Binary Set theories • Zermelo–Fraenkel • Axiom of choice • Continuum hypothesis • General • Kripke–Platek • Morse–Kelley • Naive • New Foundations • Tarski–Grothendieck • Von Neumann–Bernays–Gödel • Ackermann • Constructive Formal systems (list), Language & Syntax • Alphabet • Arity • Automata • Axiom schema • Expression • Ground • Extension • by definition • Conservative • Relation • Formation rule • Grammar • Formula • Atomic • Closed • Ground • Open • Free/bound variable • Language • Metalanguage • Logical connective • ¬ • ∨ • ∧ • → • ↔ • = • Predicate • Functional • Variable • Propositional variable • Proof • Quantifier • ∃ • ! • ∀ • rank • Sentence • Atomic • Spectrum • Signature • String • Substitution • Symbol • Function • Logical/Constant • Non-logical • Variable • Term • Theory • list Example axiomatic systems (list) • of arithmetic: • Peano • second-order • elementary function • primitive recursive • Robinson • Skolem • of the real numbers • Tarski's axiomatization • of Boolean algebras • canonical • minimal axioms • of geometry: • Euclidean: • Elements • Hilbert's • Tarski's • non-Euclidean • Principia Mathematica Proof theory • Formal proof • Natural deduction • Logical consequence • Rule of inference • Sequent calculus • Theorem • Systems • Axiomatic • Deductive • Hilbert • list • Complete theory • Independence (from ZFC) • Proof of impossibility • Ordinal analysis • Reverse mathematics • Self-verifying theories Model theory • Interpretation • Function • of models • Model • Equivalence • Finite • Saturated • Spectrum • Submodel • Non-standard model • of arithmetic • Diagram • Elementary • Categorical theory • Model complete theory • Satisfiability • Semantics of logic • Strength • Theories of truth • Semantic • Tarski's • Kripke's • T-schema • Transfer principle • Truth predicate • Truth value • Type • Ultraproduct • Validity Computability theory • Church encoding • Church–Turing thesis • Computably enumerable • Computable function • Computable set • Decision problem • Decidable • Undecidable • P • NP • P versus NP problem • Kolmogorov complexity • Lambda calculus • Primitive recursive function • Recursion • Recursive set • Turing machine • Type theory Related • Abstract logic • Category theory • Concrete/Abstract Category • Category of sets • History of logic • History of mathematical logic • timeline • Logicism • Mathematical object • Philosophy of mathematics • Supertask Mathematics portal	Wikipedia
Robinson compass mask A Robinson compass mask is a type of compass mask used for edge detection. It has eight major compass orientations,[1] each will extract the edges in respect to its direction. A combined use of compass masks of different directions could detect the edges from different angles. Technical explanation The Robinson compass mask[2] is defined by taking a single mask and rotating it to form eight orientations: ${\text{North:}}{\begin{bmatrix}-1&0&1\\-2&0&2\\-1&0&1\end{bmatrix}}$ ${\text{North West:}}{\begin{bmatrix}0&1&2\\-1&0&1\\-2&-1&0\end{bmatrix}}$ ${\text{West:}}{\begin{bmatrix}1&2&1\\0&0&0\\-1&-2&-1\end{bmatrix}}$ ${\text{South West:}}{\begin{bmatrix}2&1&0\\1&0&-1\\0&-1&-2\end{bmatrix}}$ ${\text{South:}}{\begin{bmatrix}1&0&-1\\2&0&-2\\1&0&-1\end{bmatrix}}$ ${\text{South East:}}{\begin{bmatrix}0&-1&-2\\1&0&-1\\2&1&0\end{bmatrix}}$ ${\text{East:}}{\begin{bmatrix}-1&-2&-1\\0&0&0\\1&2&1\end{bmatrix}}$ ${\text{North East:}}{\begin{bmatrix}-2&-1&0\\-1&0&1\\0&1&2\end{bmatrix}}$ The direction axis[3] is the line of zeros in the matrix. Robinson compass mask is similar to kirsch compass masks, but is simpler to implement. Since the matrix coefficients only contains 0, 1, 2, and are symmetrical, only the results of four masks[4] need to be calculated, the other four results are the negation of the first four results. An edge, or contour is an tiny area with neighboring distinct pixel values. The convolution of each mask with the image would create a high value output where there is a rapid change of pixel value, thus an edge point is found. All the detected edge points would line up as edges. Example An example of Robinson compass masks applied to the original image. Obviously, the edges in the direction of the mask is enhanced. References 1. Babu C.R., Sreenivasa Reddy E., Prabhakara Rao B. (2015) Age Group Classification of Facial Images Using Rank Based Edge Texture Unit (RETU). In: Mandal J., Satapathy S., Kumar Sanyal M., Sarkar P., Mukhopadhyay A. (eds) Information Systems Design and Intelligent Applications. Advances in Intelligent Systems and Computing, vol 340. Springer, New Delhi 2. Dr. Borislav D Dimitrov, Dr. Vishal Goyal, Mr. Nehinbe Joshua, Mr. Vassilis Papataxiarhis "International Journal of Computer Science Issues(Volume 8, Issue1)", IJCSI PUBLICATION, Jan 2011, 3. S Edy Victor Haryanto, M. Y. Mashor, A. S. Abdul Nasir, H. Jaafar, "A fast and accurate detection of Schizont plasmodium falciparum using channel color space segmentation method", Cyber and IT Service Management (CITSM) 2017 5th International Conference on, pp. 1-4, 2017. 4. "Computer imaging: Digital image analysis and processing (Second ed.)" by Scott E Umbaugh, ISBN 978-1-4398-0206-9(2010)	Wikipedia
Ursescu theorem In mathematics, particularly in functional analysis and convex analysis, the Ursescu theorem is a theorem that generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle. Ursescu Theorem The following notation and notions are used, where ${\mathcal {R}}:X\rightrightarrows Y$ is a set-valued function and $S$ is a non-empty subset of a topological vector space $X$: • the affine span of $S$ is denoted by $\operatorname {aff} S$ and the linear span is denoted by $\operatorname {span} S.$ • $S^{i}:=\operatorname {aint} _{X}S$ denotes the algebraic interior of $S$ in $X.$ • ${}^{i}S:=\operatorname {aint} _{\operatorname {aff} (S-S)}S$ denotes the relative algebraic interior of $S$ (i.e. the algebraic interior of $S$ in $\operatorname {aff} (S-S)$). • ${}^{ib}S:={}^{i}S$ if $\operatorname {span} \left(S-s_{0}\right)$ is barreled for some/every $s_{0}\in S$ while ${}^{ib}S:=\varnothing $ otherwise. • If $S$ is convex then it can be shown that for any $x\in X,$ $x\in {}^{ib}S$ if and only if the cone generated by $S-x$ is a barreled linear subspace of $X$ or equivalently, if and only if $\cup _{n\in \mathbb {N} }n(S-x)$ is a barreled linear subspace of $X$ • The domain of ${\mathcal {R}}$ is $\operatorname {Dom} {\mathcal {R}}:=\{x\in X:{\mathcal {R}}(x)\neq \varnothing \}.$ • The image of ${\mathcal {R}}$ is $\operatorname {Im} {\mathcal {R}}:=\cup _{x\in X}{\mathcal {R}}(x).$ For any subset $A\subseteq X,$ ${\mathcal {R}}(A):=\cup _{x\in A}{\mathcal {R}}(x).$ • The graph of ${\mathcal {R}}$ is $\operatorname {gr} {\mathcal {R}}:=\{(x,y)\in X\times Y:y\in {\mathcal {R}}(x)\}.$ • ${\mathcal {R}}$ is closed (respectively, convex) if the graph of ${\mathcal {R}}$ is closed (resp. convex) in $X\times Y.$ • Note that ${\mathcal {R}}$ is convex if and only if for all $x_{0},x_{1}\in X$ and all $r\in [0,1],$ $r{\mathcal {R}}\left(x_{0}\right)+(1-r){\mathcal {R}}\left(x_{1}\right)\subseteq {\mathcal {R}}\left(rx_{0}+(1-r)x_{1}\right).$ • The inverse of ${\mathcal {R}}$ is the set-valued function ${\mathcal {R}}^{-1}:Y\rightrightarrows X$ defined by ${\mathcal {R}}^{-1}(y):=\{x\in X:y\in {\mathcal {R}}(x)\}.$ For any subset $B\subseteq Y,$ ${\mathcal {R}}^{-1}(B):=\cup _{y\in B}{\mathcal {R}}^{-1}(y).$ • If $f:X\to Y$ is a function, then its inverse is the set-valued function $f^{-1}:Y\rightrightarrows X$ obtained from canonically identifying $f$ with the set-valued function $f:X\rightrightarrows Y$ defined by $x\mapsto \{f(x)\}.$ • $\operatorname {int} _{T}S$ is the topological interior of $S$ with respect to $T,$ where $S\subseteq T.$ • $\operatorname {rint} S:=\operatorname {int} _{\operatorname {aff} S}S$ is the interior of $S$ with respect to $\operatorname {aff} S.$ Statement Theorem[1] (Ursescu) — Let $X$ be a complete semi-metrizable locally convex topological vector space and ${\mathcal {R}}:X\rightrightarrows Y$ be a closed convex multifunction with non-empty domain. Assume that $\operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)$ is a barrelled space for some/every $y\in \operatorname {Im} {\mathcal {R}}.$ Assume that $y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})$ and let $x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right)$ (so that $y_{0}\in {\mathcal {R}}\left(x_{0}\right)$). Then for every neighborhood $U$ of $x_{0}$ in $X,$ $y_{0}$ belongs to the relative interior of ${\mathcal {R}}(U)$ in $\operatorname {aff} (\operatorname {Im} {\mathcal {R}})$ (that is, $y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)$). In particular, if ${}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing $ then ${}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).$ Corollaries Closed graph theorem Closed graph theorem — Let $X$ and $Y$ be Fréchet spaces and $T:X\to Y$ be a linear map. Then $T$ is continuous if and only if the graph of $T$ is closed in $X\times Y.$ Proof For the non-trivial direction, assume that the graph of $T$ is closed and let ${\mathcal {R}}:=T^{-1}:Y\rightrightarrows X.$ It is easy to see that $\operatorname {gr} {\mathcal {R}}$ is closed and convex and that its image is $X.$ Given $x\in X,$ $(Tx,x)$ belongs to $Y\times X$ so that for every open neighborhood $V$ of $Tx$ in $Y,$ ${\mathcal {R}}(V)=T^{-1}(V)$ is a neighborhood of $x$ in $X.$ Thus $T$ is continuous at $x.$ Q.E.D. Uniform boundedness principle Uniform boundedness principle — Let $X$ and $Y$ be Fréchet spaces and $T:X\to Y$ be a bijective linear map. Then $T$ is continuous if and only if $T^{-1}:Y\to X$ is continuous. Furthermore, if $T$ is continuous then $T$ is an isomorphism of Fréchet spaces. Proof Apply the closed graph theorem to $T$ and $T^{-1}.$ Q.E.D. Open mapping theorem Open mapping theorem — Let $X$ and $Y$ be Fréchet spaces and $T:X\to Y$ be a continuous surjective linear map. Then T is an open map. Proof Clearly, $T$ is a closed and convex relation whose image is $Y.$ Let $U$ be a non-empty open subset of $X,$ let $y$ be in $T(U),$ and let $x$ in $U$ be such that $y=Tx.$ From the Ursescu theorem it follows that $T(U)$ is a neighborhood of $y.$ Q.E.D. Additional corollaries The following notation and notions are used for these corollaries, where ${\mathcal {R}}:X\rightrightarrows Y$ is a set-valued function, $S$ is a non-empty subset of a topological vector space $X$: • a convex series with elements of $S$ is a series of the form $ \sum _{i=1}^{\infty }r_{i}s_{i}$ where all $s_{i}\in S$ and $ \sum _{i=1}^{\infty }r_{i}=1$ is a series of non-negative numbers. If $ \sum _{i=1}^{\infty }r_{i}s_{i}$ converges then the series is called convergent while if $\left(s_{i}\right)_{i=1}^{\infty }$ is bounded then the series is called bounded and b-convex. • $S$ is ideally convex if any convergent b-convex series of elements of $S$ has its sum in $S.$ • $S$ is lower ideally convex if there exists a Fréchet space $Y$ such that $S$ is equal to the projection onto $X$ of some ideally convex subset B of $X\times Y.$ Every ideally convex set is lower ideally convex. Corollary — Let $X$ be a barreled first countable space and let $C$ be a subset of $X.$ Then: 1. If $C$ is lower ideally convex then $C^{i}=\operatorname {int} C.$ 2. If $C$ is ideally convex then $C^{i}=\operatorname {int} C=\operatorname {int} \left(\operatorname {cl} C\right)=\left(\operatorname {cl} C\right)^{i}.$ Related theorems Simons' theorem Simons' theorem[2] — Let $X$ and $Y$ be first countable with $X$ locally convex. Suppose that ${\mathcal {R}}:X\rightrightarrows Y$ is a multimap with non-empty domain that satisfies condition (Hwx) or else assume that $X$ is a Fréchet space and that ${\mathcal {R}}$ is lower ideally convex. Assume that $\operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)$ is barreled for some/every $y\in \operatorname {Im} {\mathcal {R}}.$ Assume that $y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})$ and let $x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right).$ Then for every neighborhood $U$ of $x_{0}$ in $X,$ $y_{0}$ belongs to the relative interior of ${\mathcal {R}}(U)$ in $\operatorname {aff} (\operatorname {Im} {\mathcal {R}})$ (i.e. $y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)$). In particular, if ${}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing $ then ${}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).$ Robinson–Ursescu theorem The implication (1) $\implies $ (2) in the following theorem is known as the Robinson–Ursescu theorem.[3] Robinson–Ursescu theorem[3] — Let $(X,\\|\,\cdot \,\\|)$ and $(Y,\\|\,\cdot \,\\|)$ be normed spaces and ${\mathcal {R}}:X\rightrightarrows Y$ be a multimap with non-empty domain. Suppose that $Y$ is a barreled space, the graph of ${\mathcal {R}}$ verifies condition condition (Hwx), and that $(x_{0},y_{0})\in \operatorname {gr} {\mathcal {R}}.$ Let $C_{X}$ (resp. $C_{Y}$) denote the closed unit ball in $X$ (resp. $Y$) (so $C_{X}=\{x\in X:\\|x\\|\leq 1\}$). Then the following are equivalent: 1. $y_{0}$ belongs to the algebraic interior of $\operatorname {Im} {\mathcal {R}}.$ 2. $y_{0}\in \operatorname {int} {\mathcal {R}}\left(x_{0}+C_{X}\right).$ 3. There exists $B>0$ such that for all $0\leq r\leq 1,$ $y_{0}+BrC_{Y}\subseteq {\mathcal {R}}\left(x_{0}+rC_{X}\right).$ 4. There exist $A>0$ and $B>0$ such that for all $x\in x_{0}+AC_{X}$ and all $y\in y_{0}+AC_{Y},$ $d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq B\cdot d(y,{\mathcal {R}}(x)).$ 5. There exists $B>0$ such that for all $x\in X$ and all $y\in y_{0}+BC_{Y},$ $d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq {\frac {1+\left\\|x-x_{0}\right\\|}{B-\left\\|y-y_{0}\right\\|}}\cdot d(y,{\mathcal {R}}(x)).$ See also • Closed graph theorem – Theorem relating continuity to graphs • Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs • Open mapping theorem (functional analysis) – Condition for a linear operator to be open • Surjection of Fréchet spaces – Characterization of surjectivity • Uniform boundedness principle – A theorem stating that pointwise boundedness implies uniform boundedness • Webbed space – Space where open mapping and closed graph theorems hold Notes 1. Zălinescu 2002, p. 23. 2. Zălinescu 2002, p. 22-23. 3. Zălinescu 2002, p. 24. References • Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive. • Baggs, Ivan (1974). "Functions with a closed graph". Proceedings of the American Mathematical Society. 43 (2): 439–442. doi:10.1090/S0002-9939-1974-0334132-8. ISSN 0002-9939. Convex analysis and variational analysis Basic concepts • Convex combination • Convex function • Convex set Topics (list) • Choquet theory • Convex geometry • Convex metric space • Convex optimization • Duality • Lagrange multiplier • Legendre transformation • Locally convex topological vector space • Simplex Maps • Convex conjugate • Concave • (Closed • K- • Logarithmically • Proper • Pseudo- • Quasi-) Convex function • Invex function • Legendre transformation • Semi-continuity • Subderivative Main results (list) • Carathéodory's theorem • Ekeland's variational principle • Fenchel–Moreau theorem • Fenchel-Young inequality • Jensen's inequality • Hermite–Hadamard inequality • Krein–Milman theorem • Mazur's lemma • Shapley–Folkman lemma • Robinson-Ursescu • Simons • Ursescu Sets • Convex hull • (Orthogonally, Pseudo-) Convex set • Effective domain • Epigraph • Hypograph • John ellipsoid • Lens • Radial set/Algebraic interior • Zonotope Series • Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Duality • Dual system • Duality gap • Strong duality • Weak duality Applications and related • Convexity in economics 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 Topological vector spaces (TVSs) Basic concepts • Banach space • Completeness • Continuous linear operator • Linear functional • Fréchet space • Linear map • Locally convex space • Metrizability • Operator topologies • Topological vector space • Vector space Main results • Anderson–Kadec • Banach–Alaoglu • Closed graph theorem • F. Riesz's • Hahn–Banach (hyperplane separation • Vector-valued Hahn–Banach) • Open mapping (Banach–Schauder) • Bounded inverse • Uniform boundedness (Banach–Steinhaus) Maps • Bilinear operator • form • Linear map • Almost open • Bounded • Continuous • Closed • Compact • Densely defined • Discontinuous • Topological homomorphism • Functional • Linear • Bilinear • Sesquilinear • Norm • Seminorm • Sublinear function • Transpose Types of sets • Absolutely convex/disk • Absorbing/Radial • Affine • Balanced/Circled • Banach disks • Bounding points • Bounded • Complemented subspace • Convex • Convex cone (subset) • Linear cone (subset) • Extreme point • Pre-compact/Totally bounded • Prevalent/Shy • Radial • Radially convex/Star-shaped • Symmetric Set operations • Affine hull • (Relative) Algebraic interior (core) • Convex hull • Linear span • Minkowski addition • Polar • (Quasi) Relative interior Types of TVSs • Asplund • B-complete/Ptak • Banach • (Countably) Barrelled • BK-space • (Ultra-) Bornological • Brauner • Complete • Convenient • (DF)-space • Distinguished • F-space • FK-AK space • FK-space • Fréchet • tame Fréchet • Grothendieck • Hilbert • Infrabarreled • Interpolation space • K-space • LB-space • LF-space • Locally convex space • Mackey • (Pseudo)Metrizable • Montel • Quasibarrelled • Quasi-complete • Quasinormed • (Polynomially • Semi-) Reflexive • Riesz • Schwartz • Semi-complete • Smith • Stereotype • (B • Strictly • Uniformly) convex • (Quasi-) Ultrabarrelled • Uniformly smooth • Webbed • With the approximation property • Mathematics portal • Category • Commons	Wikipedia
Robion Kirby Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology. Together with Laurent C. Siebenmann he developed the Kirby–Siebenmann invariant for classifying the piecewise linear structures on a topological manifold. He also proved the fundamental result on the Kirby calculus, a method for describing 3-manifolds and smooth 4-manifolds by surgery on framed links. Along with his significant mathematical contributions, he has over 50 doctoral students and his problem list. Robion Kirby Kirby in 2009 Born (1938-02-25) February 25, 1938 Chicago, Illinois, US Alma materUniversity of Chicago Known forKirby–Siebenmann class Kirby calculus AwardsOswald Veblen Prize in Geometry (1971) NAS Award for Scientific Reviewing (1995) Scientific career FieldsMathematics InstitutionsUniversity of California, Berkeley Doctoral advisorEldon Dyer[1] Doctoral students • Selman Akbulut • Stephen Bigelow • Tim Cochran • David Gauld • Robert Gompf • Elisenda Grigsby • Tomasz Mrowka • Yongbin Ruan • Martin Scharlemann He received his Ph.D. from the University of Chicago in 1965. He soon became an assistant professor at UCLA. While there he developed his "torus trick" which enabled him to solve, in dimensions greater than four (with additional joint work with Siebenmann), four of John Milnor's seven most important problems in geometric topology.[2] In 1971, he was awarded the Oswald Veblen Prize in Geometry by the American Mathematical Society. In 1995 he became the first mathematician to receive the NAS Award for Scientific Reviewing from the National Academy of Sciences for his problem list in low-dimensional topology.[3] He was elected to the National Academy of Sciences in 2001. In 2012 he became a fellow of the American Mathematical Society.[4] Kirby is also the President of Mathematical Sciences Publishers, a small non-profit academic publishing house that focuses on mathematics and engineering journals. Books • Kirby, Robion C.; Siebenmann, Laurence C. (1977). Foundational Essays on Topological Manifolds, Smoothings, and Triangulations (PDF). Annals of Mathematics Studies. Vol. 88. Princeton, NJ: Princeton University Press. ISBN 0-691-08191-3. MR 0645390. • Kirby, Robion C. (1989). The topology of 4-manifolds. Lecture Notes in Mathematics. Vol. 1374. Berlin, New York: Springer-Verlag. doi:10.1007/BFb0089031. ISBN 978-3-540-51148-9. MR 1001966.[5] References 1. Robion Kirby at the Mathematics Genealogy Project 2. S. Ferry. Lecture notes in geometric topology (PDF). 3. "NAS Award for Scientific Reviewing". National Academy of Sciences. Archived from the original on 18 March 2011. Retrieved 27 February 2011. 4. List of Fellows of the American Mathematical Society, retrieved 2013-01-27. 5. Taylor, Lawrence R. (1991). "Review: Robion C. Kirby, The topology of 4 manifolds". Bull. Amer. Math. Soc. (N.S.). 24 (2): 466–471. doi:10.1090/s0273-0979-1991-16068-4. External links • Kirby's home page. • Kirby's list of problems in low dimensional topology. (This is a large 380 page gzipped ps file.) • Biographical notes from the Proceedings of the Kirbyfest in honour of his 60th birthday in 1998. • Robion Kirby at the Mathematics Genealogy Project • Video Lectures by Kirby at Edinburgh 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 • WorldCat National • France • BnF data • Germany • Israel • United States • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • Scopus • zbMATH Other • IdRef	Wikipedia
Robust fuzzy programming Robust fuzzy programming (ROFP) is a powerful mathematical optimization approach to deal with optimization problems under uncertainty. This approach is firstly introduced at 2012 by Pishvaee, Razmi & Torabi[1] in the Journal of Fuzzy Sets and Systems. ROFP enables the decision makers to be benefited from the capabilities of both fuzzy mathematical programming and robust optimization approaches. At 2016 Pishvaee and Fazli[2] put a significant step forward by extending the ROFP approach to handle flexibility of constraints and goals. ROFP is able to achieve a robust solution for an optimization problem under uncertainty. Definition of robust solution Robust solution is defined as a solution which has "both feasibility robustness and optimality robustness; Feasibility robustness means that the solution should remain feasible for (almost) all possible values of uncertain parameters and flexibility degrees of constraints and optimality robustness means that the value of objective function for the solution should remain close to optimal value or have minimum (undesirable) deviation from the optimal value for (almost) all possible values of uncertain parameters and flexibility degrees on target value of goals".[2] Classification of ROFP methods As fuzzy mathematical programming is categorized into Possibilistic programming and Flexible programming, ROFP also can be classified into:[2] 1. Robust possibilistic programming (RPP) 2. Robust flexible programming (RFP) 3. Mixed possibilistic-flexible robust programming (MPFRP) The first category is used to deal with imprecise input parameters in optimization problems while the second one is employed to cope with flexible constraints and goals. Also, the last category is capable to handle both uncertain parameters and flexibility in goals and constraints. From another point of view, it can be said that different ROFP models developed in the literature can be classified in three categories according to degree of conservatism against uncertainty. These categories include:[1] 1. Hard worst case ROFP 2. Soft worst case ROFP 3. Realistic ROFP Hard worst case ROFP has the most conservative nature among ROFP methods since it provides maximum safety or immunity against uncertainty. Ignoring the chance of infeasibility, this method immunizes the solution for being infeasible for all possible values of uncertain parameters. Regarding the optimality robustness, this method minimizes the worst possible value of objective function (min-max logic). On the other hand, Soft worst case ROFP method behaves similar to hard worst case method regarding optimality robustness, however does not satisfy the constraints in their extreme worst case. Lastly, realistic method establishes a reasonable trade-off between the robustness, the cost of robustness and other objectives such as improving the average system performance (cost-benefit logic). Applications ROFP is successfully implemented in different practical application areas such as the following ones. • Supply chain management such as the work by Pishvaee et al.[1] which addresses the design of a social responsible supply chain network under epistemic uncertainty. • Healthcare management such as the works by Zahiri et al.[3] and Mousazadeh et al.[4] which consider the planning of an organ transplantation network and a pharmaceutical supply chain, respectively. • Energy planning such as Bairamzadeh et al.[5] which uses a multi-objective possibilistic programming model to deal with the design of a bio-ethanol production-distribution network. Also in another research, Zhou et al.[6] developed a robust possibilistic programming model to deal with the planning problem of municipal electric power system. • Sustainability such as Xu and Huang[7] which employ ROFP to cope with an air quality management problem. References 1. Pishvaee, M. S.; Razmi, J.; Torabi, S. A. (2012-11-01). "Robust possibilistic programming for socially responsible supply chain network design: A new approach". Fuzzy Sets and Systems. Theme : Operational Research. 206: 1–20. doi:10.1016/j.fss.2012.04.010. 2. Pishvaee, Mir Saman; Fazli Khalaf, Mohamadreza (2016-01-01). "Novel robust fuzzy mathematical programming methods". Applied Mathematical Modelling. 40 (1): 407–418. doi:10.1016/j.apm.2015.04.054. 3. Zahiri, Behzad; Tavakkoli-Moghaddam, Reza; Pishvaee, Mir Saman (2014-08-01). "A robust possibilistic programming approach to multi-period location–allocation of organ transplant centers under uncertainty". Computers & Industrial Engineering. 74: 139–148. doi:10.1016/j.cie.2014.05.008. 4. Mousazadeh, M.; Torabi, S. A.; Zahiri, B. (2015-11-02). "A robust possibilistic programming approach for pharmaceutical supply chain network design". Computers & Chemical Engineering. 82: 115–128. doi:10.1016/j.compchemeng.2015.06.008. 5. Bairamzadeh, Samira; Pishvaee, Mir Saman; Saidi-Mehrabad, Mohammad (2015-12-22). "Multiobjective Robust Possibilistic Programming Approach to Sustainable Bioethanol Supply Chain Design under Multiple Uncertainties". Industrial & Engineering Chemistry Research. 55 (1): 237–256. doi:10.1021/acs.iecr.5b02875. 6. Zhou, Y.; Li, Y.P.; Huang, G.H. (2015-12-15). "A robust possibilistic mixed-integer programming method for planning municipal electric power systems". International Journal of Electrical Power & Energy Systems. 73: 757–772. doi:10.1016/j.ijepes.2015.06.009. 7. Xu, Ye; Huang, Guohe (2015-10-15). "Development of an Improved Fuzzy Robust Chance-Constrained Programming Model for Air Quality Management". Environmental Modeling & Assessment. 20 (5): 535–548. doi:10.1007/s10666-014-9441-3.	Wikipedia
Robust geometric computation In mathematics, specifically in computational geometry, geometric nonrobustness is a problem wherein branching decisions in computational geometry algorithms are based on approximate numerical computations, leading to various forms of unreliability including ill-formed output and software failure through crashing or infinite loops. For instance, algorithms for problems like the construction of a convex hull rely on testing whether certain "numerical predicates" have values that are positive, negative, or zero. If an inexact floating-point computation causes a value that is near zero to have a different sign than its exact value, the resulting inconsistencies can propagate through the algorithm causing it to produce output that is far from the correct output, or even to crash. One method for avoiding this problem involves using integers rather than floating point numbers for all coordinates and other quantities represented by the algorithm, and determining the precision required for all calculations to avoid integer overflow conditions. For instance, two-dimensional convex hulls can be computed using predicates that test the sign of quadratic polynomials, and therefore may require twice as many bits of precision within these calculations as the input numbers. When integer arithmetic cannot be used (for instance, when the result of a calculation is an algebraic number rather than an integer or rational number), a second method is to use symbolic algebra to perform all computations with exactly represented algebraic numbers rather than numerical approximations to them. A third method, sometimes called a "floating point filter", is to compute numerical predicates first using an inexact method based on floating-point arithmetic, but to maintain bounds on how accurate the result is, and repeat the calculation using slower symbolic algebra methods or numerically with additional precision when these bounds do not separate the calculated value from zero. References • Mei, Gang; Tipper, John C.; Xu, Nengxiong (2014), "Numerical robustness in geometric computation: an expository summary", Applied Mathematics & Information Sciences, 8 (6): 2717–2727, doi:10.12785/amis/080607, MR 3228669 • Sharma, Vikram; Yap, Chee K. (2017), "Robust geometric computation" (PDF), in Goodman, Jacob E.; O'Rourke, Joseph; Tóth, Csaba D. (eds.), Handbook of Discrete and Computational Geometry, CRC Press Series on Discrete Mathematics and its Applications (3rd ed.), CRC Press, pp. 1189–1223, MR 1730191 • Shewchuk, Jonathan (April 15, 2013), Lecture Notes on Geometric Robustness (PDF)	Wikipedia
Robust principal component analysis Robust Principal Component Analysis (RPCA) is a modification of the widely used statistical procedure of principal component analysis (PCA) which works well with respect to grossly corrupted observations. A number of different approaches exist for Robust PCA, including an idealized version of Robust PCA, which aims to recover a low-rank matrix L0 from highly corrupted measurements M = L0 +S0.[1] This decomposition in low-rank and sparse matrices can be achieved by techniques such as Principal Component Pursuit method (PCP),[1] Stable PCP,[2] Quantized PCP,[3] Block based PCP,[4] and Local PCP.[5] Then, optimization methods are used such as the Augmented Lagrange Multiplier Method (ALM[6]), Alternating Direction Method (ADM[7]), Fast Alternating Minimization (FAM[8]), Iteratively Reweighted Least Squares (IRLS [9][10][11]) or alternating projections (AP[12][13][14]). Algorithms Non-convex method The 2014 guaranteed algorithm for the robust PCA problem (with the input matrix being $M=L+S$) is an alternating minimization type algorithm.[12] The computational complexity is $O\left(mnr^{2}\log {\frac {1}{\epsilon }}\right)$ where the input is the superposition of a low-rank (of rank $r$) and a sparse matrix of dimension $m\times n$ and $\epsilon $ is the desired accuracy of the recovered solution, i.e., $\\|{\widehat {L}}-L\\|_{F}\leq \epsilon $ where $L$ is the true low-rank component and ${\widehat {L}}$ is the estimated or recovered low-rank component. Intuitively, this algorithm performs projections of the residual onto the set of low-rank matrices (via the SVD operation) and sparse matrices (via entry-wise hard thresholding) in an alternating manner - that is, low-rank projection of the difference the input matrix and the sparse matrix obtained at a given iteration followed by sparse projection of the difference of the input matrix and the low-rank matrix obtained in the previous step, and iterating the two steps until convergence. This alternating projections algorithm is later improved by an accelerated version, coined AccAltProj.[13] The acceleration is achieved by applying a tangent space projection before project the residue onto the set of low-rank matrices. This trick improves the computational complexity to $O\left(mnr\log {\frac {1}{\epsilon }}\right)$ with a much smaller constant in front while it maintains the theoretically guaranteed linear convergence. Another fast version of accelerated alternating projections algorithm is IRCUR.[14] It uses the structure of CUR decomposition in alternating projections framework to dramatically reduces the computational complexity of RPCA to $O\left(\max\{m,n\}r^{2}\log(m)\log(n)\log {\frac {1}{\epsilon }}\right)$ Convex relaxation This method consists of relaxing the rank constraint $rank(L)$ in the optimization problem to the nuclear norm $\\|L\\|_{*}$ and the sparsity constraint $\\|S\\|_{0}$ to $\ell _{1}$-norm $\\|S\\|_{1}$. The resulting program can be solved using methods such as the method of Augmented Lagrange Multipliers. Deep-learning augmented method Some recent works propose RPCA algorithms with learnable/training parameters.[15] Such a learnable/trainable algorithm can be unfolded as a deep neural network whose parameters can be learned via machine learning techniques from a given dataset or problem distribution. The learned algorithm will have superior performance on the corresponding problem distribution. Applications RPCA has many real life important applications particularly when the data under study can naturally be modeled as a low-rank plus a sparse contribution. Following examples are inspired by contemporary challenges in computer science, and depending on the applications, either the low-rank component or the sparse component could be the object of interest: Video surveillance Given a sequence of surveillance video frames, it is often required to identify the activities that stand out from the background. If we stack the video frames as columns of a matrix M, then the low-rank component L0 naturally corresponds to the stationary background and the sparse component S0 captures the moving objects in the foreground.[1][16] Face recognition Images of a convex, Lambertian surface under varying illuminations span a low-dimensional subspace.[17] This is one of the reasons for effectiveness of low-dimensional models for imagery data. In particular, it is easy to approximate images of a human's face by a low-dimensional subspace. To be able to correctly retrieve this subspace is crucial in many applications such as face recognition and alignment. It turns out that RPCA can be applied successfully to this problem to exactly recover the face.[1] See also • L1-norm principal component analysis Surveys • Robust PCA [16] • Dynamic RPCA [18] • Decomposition into Low-rank plus Additive Matrices [19] • Low-rank models[20] Books, journals and workshops Books • T. Bouwmans, N. Aybat, and E. Zahzah. Handbook on Robust Low-Rank and Sparse Matrix Decomposition: Applications in Image and Video Processing, CRC Press, Taylor and Francis Group, May 2016. (more information: http://www.crcpress.com/product/isbn/9781498724623) • Z. Lin, H. Zhang, "Low-Rank Models in Visual Analysis: Theories, Algorithms, and Applications", Academic Press, Elsevier, June 2017. (more information: https://www.elsevier.com/books/low-rank-models-in-visual-analysis/lin/978-0-12-812731-5) Journals • N. Vaswani, Y. Chi, T. Bouwmans, Special Issue on “Rethinking PCA for Modern Datasets: Theory, Algorithms, and Applications”, Proceedings of the IEEE, 2018. • T. Bouwmans, N. Vaswani, P. Rodriguez, R. Vidal, Z. Lin, Special Issue on “Robust Subspace Learning and Tracking: Theory, Algorithms, and Applications”, IEEE Journal of Selected Topics in Signal Processing, December 2018. Workshops • RSL-CV 2015: Workshop on Robust Subspace Learning and Computer Vision in conjunction with ICCV 2015 (For more information: http://rsl-cv2015.univ-lr.fr/workshop/) • RSL-CV 2017: Workshop on Robust Subspace Learning and Computer Vision in conjunction with ICCV 2017 (For more information: http://rsl-cv.univ-lr.fr/2017/) • RSL-CV 2021: Workshop on Robust Subspace Learning and Computer Vision in conjunction with ICCV 2021 (For more information: https://rsl-cv.univ-lr.fr/2021/) Sessions • Special Session on "Online Algorithms for Static and Dynamic Robust PCA and Compressive Sensing" in conjunction with SSP 2018. (More information: https://ssp2018.org/) Resources and libraries Websites • Background Subtraction Website • DLAM Website • Documentation from the University of Illinois - Archive Link Libraries The LRS Library (developed by Andrews Sobral) provides a collection of low-rank and sparse decomposition algorithms in MATLAB. The library was designed for moving object detection in videos, but it can be also used for other computer vision / machine learning tasks. Currently the LRSLibrary offers more than 100 algorithms based on matrix and tensor methods. References 1. Emmanuel J. Candes; Xiaodong Li; Yi Ma; John Wright (2009). "Robust Principal Component Analysis?". Journal of the ACM. 58 (3): 1–37. doi:10.1145/1970392.1970395. S2CID 7128002. 2. J. Wright; Y. Peng; Y. Ma; A. Ganesh; S. Rao (2009). "Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Matrices by Convex Optimization". Neural Information Processing Systems, NIPS 2009. 3. S. Becker; E. Candes, M. Grant (2011). "TFOCS: Flexible First-order Methods for Rank Minimization". Low-rank Matrix Optimization Symposium, SIAM Conference on Optimization. 4. G. Tang; A. Nehorai (2011). "Robust principal component analysis based on low-rank and block-sparse matrix decomposition". 2011 45th Annual Conference on Information Sciences and Systems. pp. 1–5. doi:10.1109/CISS.2011.5766144. ISBN 978-1-4244-9846-8. S2CID 17079459.{{cite book}}: CS1 maint: date and year (link) 5. B. Wohlberg; R. Chartrand; J. Theiler (2012). "Local principal component pursuit for nonlinear datasets". 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). pp. 3925–3928. doi:10.1109/ICASSP.2012.6288776. ISBN 978-1-4673-0046-9. S2CID 2747520.{{cite book}}: CS1 maint: date and year (link) 6. Z. Lin; M. Chen; L. Wu; Y. Ma (2013). "The Augmented Lagrange Multiplier Method for Exact Recovery of Corrupted Low-Rank Matrices". Journal of Structural Biology. 181 (2): 116–27. arXiv:1009.5055. doi:10.1016/j.jsb.2012.10.010. PMC 3565063. PMID 23110852. 7. X. Yuan; J. Yang (2009). "Sparse and Low-Rank Matrix Decomposition via Alternating Direction Methods". Optimization Online. 8. P. Rodríguez; B. Wohlberg (2013). "Fast principal component pursuit via alternating minimization". 2013 IEEE International Conference on Image Processing. pp. 69–73. doi:10.1109/ICIP.2013.6738015. ISBN 978-1-4799-2341-0. S2CID 5726914.{{cite book}}: CS1 maint: date and year (link) 9. C. Guyon; T. Bouwmans; E. Zahzah (2012). "Foreground Detection via Robust Low Rank Matrix Decomposition including Spatio-Temporal Constraint". International Workshop on Background Model Challenges, ACCV 2012. 10. C. Guyon; T. Bouwmans; E. Zahzah (2012). "Foreground Detection via Robust Low Rank Matrix Factorization including Spatial Constraint with Iterative Reweighted Regression". International Conference on Pattern Recognition, ICPR 2012. 11. C. Guyon; T. Bouwmans; E. Zahzah (2012). "Moving Object Detection via Robust Low Rank Matrix Decomposition with IRLS scheme". International Symposium on Visual Computing, ISVC 2012. 12. P., Netrapalli; U., Niranjan; S., Sanghavi; A., Anandkumar; P., Jain (2014). "Non-convex robust PCA". Advances in Neural Information Processing Systems. 27: 1107–1115. arXiv:1410.7660. Bibcode:2014arXiv1410.7660N. 13. Cai, H.; Cai, J.-F.; Wei, K. (2019). "Accelerated alternating projections for robust principal component analysis". The Journal of Machine Learning Research. 20 (1): 685–717. arXiv:1711.05519. Bibcode:2017arXiv171105519C. 14. Cai, H.; Hamm, K.; Huang, L.; Li, J.; Wang, T. (2021). "Rapid Robust Principal Component Analysis: CUR Accelerated Inexact Low Rank Estimation". IEEE Signal Processing Letters. 28: 116–120. arXiv:2010.07422. Bibcode:2021ISPL...28..116C. doi:10.1109/LSP.2020.3044130. S2CID 222378834. 15. Cai, H.; Liu, J.; Yin, W. (2021). "Learned Robust PCA: A Scalable Deep Unfolding Approach for High-Dimensional Outlier Detection". Advances in Neural Information Processing Systems. 34: 16977–16989. arXiv:2110.05649. Bibcode:2021arXiv211005649C. 16. T. Bouwmans; E. Zahzah (2014). "Robust PCA via Principal Component Pursuit: A Review for a Comparative Evaluation in Video Surveillance". Computer Vision and Image Understanding. 122: 22–34. doi:10.1016/j.cviu.2013.11.009. 17. R. Basri; D. Jacobs. "Lambertian reflectance and linear subspaces". {{cite journal}}: Cite journal requires \|journal= (help) 18. N. Vaswani; T. Bouwmans; S. Javed; P. Narayanamurthy (2017). "Robust PCA and Robust Subspace Tracking". Preprint. 35 (4): 32–55. arXiv:1711.09492. Bibcode:2018ISPM...35d..32V. doi:10.1109/MSP.2018.2826566. S2CID 3691367. 19. T. Bouwmans; A. Sobral; S. Javed; S. Jung; E. Zahzahg (2015). "Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset". Computer Science Review. 23: 1–71. arXiv:1511.01245. Bibcode:2015arXiv151101245B. doi:10.1016/j.cosrev.2016.11.001. S2CID 10420698. 20. Z. Lin (2016). "A Review on Low-Rank Models in Data Analysis". Big Data and Information Analytics. 1 (2): 139–161. doi:10.3934/bdia.2016001. External links • LRSLibrary	Wikipedia
Robustification Robustification is a form of optimisation whereby a system is made less sensitive to the effects of random variability, or noise, that is present in that system's input variables and parameters. The process is typically associated with engineering systems, but the process can also be applied to a political policy, a business strategy or any other system that is subject to the effects of random variability. Clarification on definition Robustification as it is defined here is sometimes referred to as parameter design or robust parameter design (RPD) and is often associated with Taguchi methods. Within that context, robustification can include the process of finding the inputs that contribute most to the random variability in the output and controlling them, or tolerance design. At times the terms design for quality or Design for Six Sigma (DFFS) might also be used as synonyms Principles Robustification works by taking advantage of two different principles. Non-linearities Consider the graph below of a relationship between an input variable x and the output Y, for which it is desired that a value of 7 is taken, of a system of interest. It can be seen that there are two possible values that x can take, 5 and 30. If the tolerance for x is independent of the nominal value, then it can also be seen that when x is set equal to 30, the expected variation of Y is less than if x were set equal to 5. The reason is that the gradient at x = 30 is less than at x = 5, and the random variability in x is suppressed as it flows to Y. This basic principle underlies all robustification, but in practice there are typically a number of inputs and it is the suitable point with the lowest gradient on a multi-dimensional surface that must be found. Non-constant variability Consider a case where an output Z is a function of two inputs x and y that are multiplied by each other. Z = x y For any target value of Z there is an infinite number of combinations for the nominal values of x and y that will be suitable. However, if the standard deviation of x was proportional to the nominal value and the standard deviation of y was constant, then x would be reduced (to limit the random variability that will flow from the right hand side of the equation to the left hand side) and y would be increased (with no expected increase random variability because the standard deviation is constant) to bring the value of Z to the target value. By doing this, Z would have the desired nominal value and it would be expected that its standard deviation would be at a minimum: robustified. By taking advantage of the two principles covered above, one is able to optimise a system so that the nominal value of a systems output is kept at its desired level while also minimising the likelihood of any deviation from that nominal value. This is despite the presence of random variability within the input variables. Methods There are three distinct methods of robustification, but a combination that provides the best in results, resources, and time can be used. Experimental The experimental approach is probably the most widely known. It involves the identification of those variables that can be adjusted and those variables that are treated as noises. An experiment is then designed to investigate how changes to the nominal value of the adjustable variables can limit the transfer of noise from the noise variables to the output. This approach is attributed to Taguchi and is often associated with Taguchi methods. While many have found the approach to provide impressive results, the techniques have also been criticised for being statistically erroneous and inefficient. Also, the time and effort required can be significant. Another experimental method that was used for robustification is the Operating Window. It was developed in the United States before the wave of quality methods from Japan came to the West, but still remains unknown to many.[1] In this approach, the noise of the inputs is continually increased as the system is modified to reduce sensitivity to that noise. This increases robustness, but also provides a clearer measure of the variability that is flowing through the system. After optimisation, the random variability of the inputs is controlled and reduced, and the system exhibits improved quality. Analytical The analytical approach relies initially on the development of an analytical model of the system of interest. The expected variability of the output is then found by using a method like the propagation of error or functions of random variables.[2] These typically produce an algebraic expression that can be analysed for optimisation and robustification. This approach is only as accurate as the model developed and it can be very difficult if not impossible for complex systems. The analytical approach might also be used in conjunction with some kind of surrogate model that is based on the results of experiments or numerical simulations of the system. Numerical In the numerical approach a model is run a number of times as part of a Monte Carlo simulation or a numerical propagation of errors to predict the variability of the outputs. Numerical optimisation methods such as hill climbing or evolutionary algorithms are then used to find the optimum nominal values for the inputs. This approach typically requires less human time and effort than the other two, but it can be very demanding on computational resources during simulation and optimization. See also • Sensitivity analysis Footnotes 1. See Clausing (2004) reference for more details 2. See the 'Probabilistic Design' link in the external links for more information. References • Clausing (1994) Total Quality Development: A Step-By-Step Guide to World-Class Concurrent Engineering. American Society of Mechanical Engineers. ISBN 0-7918-0035-0 • Clausing, D. (2004) Operating Window: An Engineering Measure for Robustness Technometrics. Vol. 46 [1] pp. 25–31. • Siddall (1982) Optimal Engineering Design. CRC. ISBN 0-8247-1633-7 • Dodson, B., Hammett, P., and Klerx, R. (2014) Probabilistic Design for Optimization and Robustness for Engineers John Wiley & Sons, Inc. ISBN 978-1-118-79619-1 External links • Probabilistic design	Wikipedia
Rokhlin's theorem In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, orientable, closed 4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class $w_{2}(M)$ vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group $H^{2}(M)$, is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952. Examples • The intersection form on M $Q_{M}\colon H^{2}(M,\mathbb {Z} )\times H^{2}(M,\mathbb {Z} )\rightarrow \mathbb {Z} $ is unimodular on $\mathbb {Z} $ by Poincaré duality, and the vanishing of $w_{2}(M)$ implies that the intersection form is even. By a theorem of Cahit Arf, any even unimodular lattice has signature divisible by 8, so Rokhlin's theorem forces one extra factor of 2 to divide the signature. • A K3 surface is compact, 4 dimensional, and $w_{2}(M)$ vanishes, and the signature is −16, so 16 is the best possible number in Rokhlin's theorem. • A complex surface in $\mathbb {CP} ^{3}$ of degree $d$ is spin if and only if $d$ is even. It has signature $(4-d^{2})d/3$, which can be seen from Friedrich Hirzebruch's signature theorem. The case $d=4$ gives back the last example of a K3 surface. • Michael Freedman's E8 manifold is a simply connected compact topological manifold with vanishing $w_{2}(M)$ and intersection form $E_{8}$ of signature 8. Rokhlin's theorem implies that this manifold has no smooth structure. This manifold shows that Rokhlin's theorem fails for the set of merely topological (rather than smooth) manifolds. • If the manifold M is simply connected (or more generally if the first homology group has no 2-torsion), then the vanishing of $w_{2}(M)$ is equivalent to the intersection form being even. This is not true in general: an Enriques surface is a compact smooth 4 manifold and has even intersection form II1,9 of signature −8 (not divisible by 16), but the class $w_{2}(M)$ does not vanish and is represented by a torsion element in the second cohomology group. Proofs Rokhlin's theorem can be deduced from the fact that the third stable homotopy group of spheres $\pi _{3}^{S}$ is cyclic of order 24; this is Rokhlin's original approach. It can also be deduced from the Atiyah–Singer index theorem. See Â genus and Rochlin's theorem. Robion Kirby (1989) gives a geometric proof. The Rokhlin invariant Since Rokhlin's theorem states that the signature of a spin smooth manifold is divisible by 16, the definition of the Rokhlin invariant is deduced as follows: For 3-manifold $N$ and a spin structure $s$ on $N$, the Rokhlin invariant $\mu (N,s)$ in $\mathbb {Z} /16\mathbb {Z} $ is defined to be the signature of any smooth compact spin 4-manifold with spin boundary $(N,s)$. If N is a spin 3-manifold then it bounds a spin 4-manifold M. The signature of M is divisible by 8, and an easy application of Rokhlin's theorem shows that its value mod 16 depends only on N and not on the choice of M. Homology 3-spheres have a unique spin structure so we can define the Rokhlin invariant of a homology 3-sphere to be the element $\operatorname {sign} (M)/8$ of $\mathbb {Z} /2\mathbb {Z} $, where M any spin 4-manifold bounding the homology sphere. For example, the Poincaré homology sphere bounds a spin 4-manifold with intersection form $E_{8}$, so its Rokhlin invariant is 1. This result has some elementary consequences: the Poincaré homology sphere does not admit a smooth embedding in $S^{4}$, nor does it bound a Mazur manifold. More generally, if N is a spin 3-manifold (for example, any $\mathbb {Z} /2\mathbb {Z} $ homology sphere), then the signature of any spin 4-manifold M with boundary N is well defined mod 16, and is called the Rokhlin invariant of N. On a topological 3-manifold N, the generalized Rokhlin invariant refers to the function whose domain is the spin structures on N, and which evaluates to the Rokhlin invariant of the pair $(N,s)$ where s is a spin structure on N. The Rokhlin invariant of M is equal to half the Casson invariant mod 2. The Casson invariant is viewed as the Z-valued lift of the Rokhlin invariant of integral homology 3-sphere. Generalizations The Kervaire–Milnor theorem (Kervaire & Milnor 1960) states that if $\Sigma $ is a characteristic sphere in a smooth compact 4-manifold M, then $\operatorname {signature} (M)=\Sigma \cdot \Sigma {\bmod {1}}6$. A characteristic sphere is an embedded 2-sphere whose homology class represents the Stiefel–Whitney class $w_{2}(M)$. If $w_{2}(M)$ vanishes, we can take $\Sigma $ to be any small sphere, which has self intersection number 0, so Rokhlin's theorem follows. The Freedman–Kirby theorem (Freedman & Kirby 1978) states that if $\Sigma $ is a characteristic surface in a smooth compact 4-manifold M, then $\operatorname {signature} (M)=\Sigma \cdot \Sigma +8\operatorname {Arf} (M,\Sigma ){\bmod {1}}6$. where $\operatorname {Arf} (M,\Sigma )$ is the Arf invariant of a certain quadratic form on $H_{1}(\Sigma ,\mathbb {Z} /2\mathbb {Z} )$. This Arf invariant is obviously 0 if $\Sigma $ is a sphere, so the Kervaire–Milnor theorem is a special case. A generalization of the Freedman-Kirby theorem to topological (rather than smooth) manifolds states that $\operatorname {signature} (M)=\Sigma \cdot \Sigma +8\operatorname {Arf} (M,\Sigma )+8\operatorname {ks} (M){\bmod {1}}6$, where $\operatorname {ks} (M)$ is the Kirby–Siebenmann invariant of M. The Kirby–Siebenmann invariant of M is 0 if M is smooth. Armand Borel and Friedrich Hirzebruch proved the following theorem: If X is a smooth compact spin manifold of dimension divisible by 4 then the Â genus is an integer, and is even if the dimension of X is 4 mod 8. This can be deduced from the Atiyah–Singer index theorem: Michael Atiyah and Isadore Singer showed that the Â genus is the index of the Atiyah–Singer operator, which is always integral, and is even in dimensions 4 mod 8. For a 4-dimensional manifold, the Hirzebruch signature theorem shows that the signature is −8 times the Â genus, so in dimension 4 this implies Rokhlin's theorem. Ochanine (1980) proved that if X is a compact oriented smooth spin manifold of dimension 4 mod 8, then its signature is divisible by 16. References • Freedman, Michael; Kirby, Robion (1978), "A geometric proof of Rochlin's theorem", Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 85–97, Proceedings of Symposia in Pure Mathematics, vol. XXXII, Providence, Rhode Island: American Mathematics Society, ISBN 0-8218-1432-X, MR 0520525 • Kirby, Robion (1989), The topology of 4-manifolds, Lecture Notes in Mathematics, vol. 1374, Springer-Verlag, doi:10.1007/BFb0089031, ISBN 0-387-51148-2, MR 1001966 • Kervaire, Michel A.; Milnor, John W. (1960), "Bernoulli numbers, homotopy groups, and a theorem of Rohlin", Proceedings of the International Congress of Mathematicians, 1958, New York: Cambridge University Press, pp. 454–458, MR 0121801 • Kervaire, Michel A.; Milnor, John W. (1961), "On 2-spheres in 4-manifolds", Procedings of the National Academy of Sciences, vol. 47, pp. 1651–1657, MR 0133134 • Matsumoto, Yoichirou (1986), An elementary proof of Rochlin's signature theorem and its extension by Guillou and Marin (PDF) • Michelsohn, Marie-Louise; Lawson, H. Blaine (1989), Spin geometry, Princeton, New Jersey: Princeton University Press, ISBN 0-691-08542-0, MR 1031992 (especially page 280) • Ochanine, Serge, Signature modulo 16, invariants de Kervaire généralisés et nombres caractéristiques dans la K-théorie réelle, Mém. Soc. Math. France 1980/81, no. 5, MR 1809832 • Rokhlin, Vladimir A., New results in the theory of four-dimensional manifolds, Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221–224. MR0052101 • Scorpan, Alexandru (2005), The wild world of 4-manifolds, American Mathematical Society, ISBN 978-0-8218-3749-8, MR 2136212 • Szűcs, András (2003), "Two Theorems of Rokhlin", Journal of Mathematical Sciences, 113 (6): 888–892, doi:10.1023/A:1021208007146, MR 1809832, S2CID 117175810	Wikipedia
R. Tyrrell Rockafellar Ralph Tyrrell Rockafellar (born February 10, 1935) is an American mathematician and one of the leading scholars in optimization theory and related fields of analysis and combinatorics. He is the author of four major books including the landmark text "Convex Analysis" (1970),[1] which has been cited more than 27,000 times according to Google Scholar and remains the standard reference on the subject, and "Variational Analysis" (1998, with Roger J-B Wets) for which the authors received the Frederick W. Lanchester Prize from the Institute for Operations Research and the Management Sciences (INFORMS). Ralph Tyrrell Rockafellar R. Tyrrell ("Terry") Rockafellar in 1977 Born (1935-02-10) February 10, 1935 Milwaukee, Wisconsin, U.S. Alma materHarvard University Known forConvex analysis Monotone operator Calculus of variation Stochastic programming Oriented matroid AwardsDantzig Prize of SIAM and MPS 1982 von Neumann citation of SIAM 1992 Frederick W. Lanchester Prize of INFORMS 1998 John von Neumann Theory Prize of INFORMS 1999 Doctor Honoris Causa: Groningen, Montpellier, Chile, Alicante Scientific career FieldsMathematical optimization InstitutionsUniversity of Washington 1966- University of Florida (adjunct) 2003- University of Texas, Austin 1963–1965 ThesisConvex Functions and Dual Extremum Problems (1963) Doctoral advisorGarrett Birkhoff Notable studentsPeter Wolenski Francis Clarke InfluencesAlbert W. Tucker Werner Fenchel Roger J-B Wets InfluencedRoger J-B Wets He is professor emeritus at the departments of mathematics and applied mathematics at the University of Washington, Seattle. Early life and education Ralph Tyrrell Rockafellar was born in Milwaukee, Wisconsin.[2] He is named after his father Ralph Rockafellar, with Tyrrell being his mother’s maiden name. Since his mother was fond of the name Terry, the parents adopted it as a nickname for Tyrrell and soon everybody referred to him as Terry.[3] Rockafellar is a distant relative of the American business magnate and philanthropist John D. Rockefeller. They both can trace their ancestors back to two brothers named Rockenfelder that came to America from the Rhineland-Pfaltz region of Germany in 1728. Soon the spelling of the family name evolved, resulting in Rockafellar, Rockefeller, and many other versions of the name.[4] Rockafellar moved to Cambridge, Massachusetts to attend Harvard College in 1953. Majoring in mathematics, he graduated from Harvard in 1957 with summa cum laude. He was also elected for the Phi Beta Kappa honor society. Rockafellar was a Fulbright Scholar at the University of Bonn in 1957–58 and completed a Master of Science degree at Marquette University in 1959. Formally under the guidance of Professor Garrett Birkhoff, Rockafellar completed his Doctor of Philosophy degree in mathematics from Harvard University in 1963 with the dissertation “Convex Functions and Dual Extremum Problems.” However, at the time there was little interest in convexity and optimization at Harvard and Birkhoff was neither involved with the research nor familiar with the subject.[5] The dissertation was inspired by the duality theory of linear programming developed by John von Neumann, which Rockafellar learned about through volumes of recent papers compiled by Albert W. Tucker at Princeton University.[6] Rockafellar’s dissertation together with the contemporary work by Jean-Jacques Moreau in France are regarded as the birth of convex analysis. Career After graduating from Harvard, Rockafellar became Assistant Professor of Mathematics at the University of Texas, Austin, where he also was affiliated with the Department of Computer Science. After two years, he moved to University of Washington in Seattle where he filled joint positions in the Departments of Mathematics and Applied Mathematics from 1966 to 2003 when he retired. He is presently Professor Emeritus at the university. He has held adjunct positions at the University of Florida and Hong Kong Polytechnic University. Rockafellar was a visiting professor at the Mathematics Institute, Copenhagen (1964), Princeton University (1965–66), University of Grenoble (1973–74), University of Colorado, Boulder (1978), International Institute of Applied Systems Analysis, Vienna (1980–81), University of Pisa (1991), University of Paris-Dauphine (1996), University of Pau (1997), Keio University (2009), National University of Singapore (2011), University of Vienna (2011), and Yale University (2012). Rockafellar received the Dantzig Prize from the Society for Industrial and Applied Mathematics (SIAM) and the Mathematical Optimization Society in 1982, delivered the 1992 John von Neumann Lecture, received with Roger J-B Wets the Frederick W. Lanchester Prize from the Institute for Operations Research and the Management Sciences (INFORMS) in 1998 for the book “Variational Analysis.” In 1999, he was awarded the John von Neumann Theory Prize from INFORMS. He was elected to the 2002 class of Fellows of INFORMS.[7] He is the recipient of honorary doctoral degrees from University of Groningen (1984), University of Montpellier (1995), University of Chile (1998), and University of Alicante (2000). The Institute for Scientific Information (ISI) lists Rockafellar as a highly cited researcher.[8] Research Rockafellar’s research is motivated by the goal of organizing mathematical ideas and concepts into robust frameworks that yield new insights and relations.[9] This approach is most salient in his seminal book "Variational Analysis" (1998, with Roger J-B Wets), where numerous threads developed in the areas of convex analysis, nonlinear analysis, calculus of variation, mathematical optimization, equilibrium theory, and control systems were brought together to produce a unified approach to variational problems in finite dimensions. These various fields of study are now referred to as variational analysis. In particular, the text dispenses of differentiability as a necessary property in many areas of analysis and embraces nonsmoothness, set-valuedness, and extended real-valuedness, while still developing far-reaching calculus rules. Contributions to Mathematics The approach of extending the real line with the values infinity and negative infinity and then allowing (convex) functions to take these values can be traced back to Rockafellar’s dissertation and, independently, the work by Jean-Jacques Moreau around the same time. The central role of set-valued mappings (also called multivalued functions) was also recognized in Rockafellar’s dissertation and, in fact, the standard notation ∂f(x) for the set of subgradients of a function f at x originated there. Rockafellar contributed to nonsmooth analysis by extending the rule of Fermat, which characterizes solutions of optimization problems, to composite problems using subgradient calculus and variational geometry and thereby bypassing the implicit function theorem. The approach broadens the notion of Lagrange multipliers to settings beyond smooth equality and inequality systems. In his doctoral dissertation and numerous later publications, Rockafellar developed a general duality theory based on convex conjugate functions that centers on embedding a problem within a family of problems obtained by a perturbation of parameters. This encapsulates linear programming duality and Lagrangian duality, and extends to general convex problems as well as nonconvex ones, especially when combined with an augmentation. Contributions to Applications Rockafellar also worked on applied problems and computational aspects. In the 1970s, he contributed to the development of the proximal point method, which underpins several successful algorithms including the proximal gradient method often used in statistical applications. He placed the analysis of expectation functions in stochastic programming on solid footing by defining and analyzing normal integrands. Rockafellar also contributed to the analysis of control systems and general equilibrium theory in economics. Since the late 1990s, Rockafellar has been actively involved with organizing and expanding the mathematical concepts for risk assessment and decision making in financial engineering and reliability engineering. This includes examining the mathematical properties of risk measures and coining the terms "conditional value-at-risk," in 2000 as well as "superquantile" and "buffered failure probability" in 2010, which either coincide with or are closely related to expected shortfall. Selected publications Books • Rockafellar, R. T. (1997). Convex analysis. Princeton landmarks in mathematics (Reprint of the 1970 Princeton mathematical series 28 ed.). Princeton, NJ: Princeton University Press. pp. xviii+451. ISBN 978-0-691-01586-6. MR 1451876. • Rockafellar, R. T. (1974). Conjugate duality and optimization. Lectures given at the Johns Hopkins University, Baltimore, Md., June, 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16. Society for Industrial and Applied Mathematics, Philadelphia, Pa. vi+74 pp. • Rockafellar, R. T. (1981). The theory of subgradients and its applications to problems of optimization. Convex and nonconvex functions. Heldermann Verlag, Berlin. vii+107 pp. ISBN 3-88538-201-6 • Rockafellar, R. T. (1984). Network Flows and Monotropic Optimization. Wiley. • Rockafellar, R. T.; Wets, Roger J-B (2005) [1998]. Variational analysis (PDF). Grundlehren der mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences). Vol. 317 (third corrected printing ed.). Berlin: Springer-Verlag. pp. xiv+733. doi:10.1007/978-3-642-02431-3. ISBN 978-3-540-62772-2. MR 1491362. Retrieved 12 March 2012. • Dontchev, A. L.; Rockafellar, R. T. (2009). Implicit functions and solution mappings. A view from variational analysis. Springer Monographs in Mathematics. Springer, Dordrecht. xii+375 pp. ISBN 978-0-387-87820-1. Papers • Rockafellar, R. T. (1967). Monotone processes of convex and concave type. Memoirs of the American Mathematical Society, No. 77 American Mathematical Society, Providence, R.I. i+74 pp. • Rockafellar, R. T. (1969). "The Elementary Vectors of a Subspace of $R^{N}$" (1967)" (PDF). In R. C. Bose and T. A. Dowling (ed.). Combinatorial Mathematics and its Applications. The University of North Carolina Monograph Series in Probability and Statistics. Chapel Hill, North Carolina: University of North Carolina Press. pp. 104–127. MR 0278972. • Rockafellar, R. T. (1970). "On the maximal monotonicity of subdifferential mappings". Pacific J. Math. 33: 209–216. doi:10.2140/pjm.1970.33.209. • Rockafellar, R. T. (1973). "The multiplier method of Hestenes and Powell applied to convex programming". J. Optimization Theory Appl. 12 (6): 555–562. doi:10.1007/bf00934777. S2CID 121931445. • Rockafellar, R. T. (1974). "Augmented Lagrange multiplier functions and duality in nonconvex programming". SIAM J. Control. 12 (2): 268–285. doi:10.1137/0312021. • Rockafellar, R. T. (1976). "Augmented Lagrangians and applications of the proximal point algorithm in convex programming". Math. Oper. Res. 1 (2): 97–116. CiteSeerX 10.1.1.298.6206. doi:10.1287/moor.1.2.97. • Rockafellar, R. T. (1993). "Lagrange multipliers and optimality". SIAM Rev. 35 (2): 183–238. doi:10.1137/1035044. (1992 John von Neumann Lecture) • Rockafellar, R. T.; Wets, Roger J-B (1991). "Scenarios and policy aggregation in optimization under uncertainty" (PDF). Math. Oper. Res. 16 (1): 119–147. doi:10.1287/moor.16.1.119. S2CID 32457406. • Rockafellar, R. T.; Uryasev, S. (2000). "Optimization of conditional value-at-risk". Journal of Risk. 2 (3): 493–517. doi:10.21314/JOR.2000.038. S2CID 854622. • Rockafellar, R. T.; Uryasev, S.; Zabarankin, M. (2006). "Generalized deviations in risk analysis". Finance and Stochastics. 10: 51–74. doi:10.1007/s00780-005-0165-8. S2CID 12632322. • Rockafellar, R. T.; Royset, J. O. (2010). "On buffered failure probability in design and optimization of structures". Reliability Engineering and System Safety. 95 (5): 499–510. doi:10.1016/j.ress.2010.01.001. S2CID 1653873. • Rockafellar, R. T.; Uryasev, S. (2013). "The fundamental risk quadrangle in risk management, optimization and statistical estimation". Surveys in Operations Research and Management Science. 18 (1–2): 33–53. doi:10.1016/j.sorms.2013.03.001. See also • Convex analysis (c.f. Werner Fenchel) • Convex function • Characteristic function (convex analysis) • Closed convex function • Convex conjugate • Epigraph (mathematics) • Fenchel conjugate • Legendre–Fenchel transformation • Proper convex function • Subdifferential • Subgradient • Convex set • Carathéodory's theorem • Convex cone • Duality (mathematics) • Monotone operator (Cyclic decomposition of maximal monotone operator) • Oriented matroids (realizable OMs and applications) • Carathéodory's theorem (convex hull) • Lemma of Farkas • Monotropic programming • Tucker, Albert W. • Set-valued analysis • Pompeiu–Hausdorff distance • Mordukhovich, Boris • Stochastic programming • Variational analysis and Control theory • Epigraph (mathematics) • Wets, Roger J-B Notes 1. Rockafeller, Ralph Tyrell (12 January 1997). Convex Analysis: (PMS-28) (Princeton Landmarks in Mathematics and Physics, 18). ISBN 978-0691015866. 2. Kalte, Pamela M.; Nemeh, Katherine H.; Schusterbauer, Noah (2005). Q - S. ISBN 9780787673987. 3. Rockafellar, R.T. "About my name". Personal webpage. Retrieved 7 August 2020. 4. Rockafellar, R.T. "About my name". Personal webpage. Retrieved 7 August 2020. 5. "An Interview with R. Tyrrell Rockafellar" (PDF). SIAG/Opt News and Views. 15 (1). 2004. 6. "An Interview with R. Tyrrell Rockafellar" (PDF). SIAG/Opt News and Views. 15 (1). 2004. 7. Fellows: Alphabetical List, Institute for Operations Research and the Management Sciences, archived from the original on 2019-05-10, retrieved 2019-10-09 8. In the Institute for Scientific Information highly cited researcher list, Rockafellar's author id is "A0071-2003-A". 9. "An Interview with R. Tyrrell Rockafellar" (PDF). SIAG/Opt News and Views. 15 (1). 2004. References • Aardal, Karen (July 1995). "Optima interview Roger J.-B. (sic.) Wets" (PDF). Optima: Mathematical Programming Society Newsletter: 3–5. • "An Interview with R. Tyrrell Rockafellar" (PDF). SIAG/Opt News and Views. 15 (1). 2004. • Wets, Roger J-B (23 November 2005), Wets, Roger J-B (ed.), "Foreword", Special Issue on Variational Analysis, Optimization, and their Applications (Festschrift for the 70th Birthday of R. Tyrrell Rockafellar), Mathematical Programming, Berlin and Heidelberg: Springer Verlag, 104 (2): 203–204, doi:10.1007/s10107-005-0612-5, ISSN 0025-5610, S2CID 39388358 External links • Homepage of R. Tyrrell Rockafellar at the University of Washington. • R. Tyrrell Rockafellar at the Mathematics Genealogy Project • Biography of R. Tyrrell Rockafellar from the Institute of Operations Research and the Management Sciences John von Neumann Lecturers • Lars Ahlfors (1960) • Mark Kac (1961) • Jean Leray (1962) • Stanislaw Ulam (1963) • Solomon Lefschetz (1964) • Freeman Dyson (1965) • Eugene Wigner (1966) • Chia-Chiao Lin (1967) • Peter Lax (1968) • George F. Carrier (1969) • James H. Wilkinson (1970) • Paul Samuelson (1971) • Jule Charney (1974) • James Lighthill (1975) • René Thom (1976) • Kenneth Arrow (1977) • Peter Henrici (1978) • Kurt O. Friedrichs (1979) • Keith Stewartson (1980) • Garrett Birkhoff (1981) • David Slepian (1982) • Joseph B. Keller (1983) • Jürgen Moser (1984) • John W. Tukey (1985) • Jacques-Louis Lions (1986) • Richard M. Karp (1987) • Germund Dahlquist (1988) • Stephen Smale (1989) • Andrew Majda (1990) • R. Tyrrell Rockafellar (1992) • Martin D. Kruskal (1994) • Carl de Boor (1996) • William Kahan (1997) • Olga Ladyzhenskaya (1998) • Charles S. Peskin (1999) • Persi Diaconis (2000) • David Donoho (2001) • Eric Lander (2002) • Heinz-Otto Kreiss (2003) • Alan C. Newell (2004) • Jerrold E. Marsden (2005) • George C. Papanicolaou (2006) • Nancy Kopell (2007) • David Gottlieb (2008) • Franco Brezzi (2009) • Bernd Sturmfels (2010) • Ingrid Daubechies (2011) • John M. Ball (2012) • Stanley Osher (2013) • Leslie Greengard (2014) • Jennifer Tour Chayes (2015) • Donald Knuth (2016) • Bernard J. Matkowsky (2017) • Charles F. Van Loan (2018) • Margaret H. Wright (2019) • Nick Trefethen (2020) • Chi-Wang Shu (2021) • Leah Keshet (2022) John von Neumann Theory Prize 1975–1999 • George Dantzig (1975) • Richard Bellman (1976) • Felix Pollaczek (1977) • John F. Nash / Carlton E. Lemke (1978) • David Blackwell (1979) • David Gale / Harold W. Kuhn / Albert W. Tucker (1980) • Lloyd Shapley (1981) • Abraham Charnes / William W. Cooper / Richard J. Duffin (1982) • Herbert Scarf (1983) • Ralph Gomory (1984) • Jack Edmonds (1985) • Kenneth Arrow (1986) • Samuel Karlin (1987) • Herbert A. Simon (1988) • Harry Markowitz (1989) • Richard Karp (1990) • Richard E. Barlow / Frank Proschan (1991) • Alan J. Hoffman / Philip Wolfe (1992) • Robert Herman (1993) • Lajos Takacs (1994) • Egon Balas (1995) • Peter C. Fishburn (1996) • Peter Whittle (1997) • Fred W. Glover (1998) • R. Tyrrell Rockafellar (1999) 2000–present • Ellis L. Johnson / Manfred W. Padberg (2000) • Ward Whitt (2001) • Donald L. Iglehart / Cyrus Derman (2002) • Arkadi Nemirovski / Michael J. Todd (2003) • J. Michael Harrison (2004) • Robert Aumann (2005) • Martin Grötschel / László Lovász / Alexander Schrijver (2006) • Arthur F. Veinott, Jr. (2007) • Frank Kelly (2008) • Yurii Nesterov / Yinyu Ye (2009) • Søren Asmussen / Peter W. Glynn (2010) • Gérard Cornuéjols (2011) • George Nemhauser / Laurence Wolsey (2012) • Michel Balinski (2013) • Nimrod Megiddo (2014) • Vašek Chvátal / Jean Bernard Lasserre (2015) • Martin I. Reiman / Ruth J. Williams (2016) • Donald Goldfarb / Jorge Nocedal (2017) • Dimitri Bertsekas / John Tsitsiklis (2018) • Dimitris Bertsimas / Jong-Shi Pang (2019) • Adrian Lewis (2020) • Alexander Shapiro (2021) • Vijay Vazirani (2022) Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • United States • Japan • Czech Republic • Netherlands • Poland Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Rocket City Math League Rocket City Math League (RCML) is a student-run mathematics competition in the United States. Run by students at Virgil I. Grissom High School in Huntsville, Alabama, RCML gets its name from Huntsville's nickname as the "Rocket City".[1] RCML was started in 2001 and has been annually sponsored by the Mu Alpha Theta Math Honor Society. The competition consists of three individual rounds and a team round that was added in 2008. It is divided into five divisions named for NASA programs: Explorer (pre-algebra), Mercury (algebra I), Gemini (geometry), Apollo (algebra II), and Discovery (comprehensive). Individual rounds Each of the 3 individual rounds consists of a 10 question test with a 30-minute time limit. Out of the 10 questions, there are four 1-point questions, three 2-point questions, two 3-point questions, and one 4-point question, with the more difficult questions having larger point values. The maximum score on an individual test is 20, and individual tests often contain many interesting space-themed questions. Team round The team round is divided into a senior division and a junior division that take separate tests for the team round. It consists of a 15 question test with a 30-minute time limit, in which team members work together to get as many correct answers as possible. Out of the 15 questions, there are five 1-point questions, four 2-point questions, three 3-point questions, two 4-point questions, and one 5-point question, making the maximum score on the team test a 35. Sources • http://www.rocketcitymath.org Notes 1. Check the "RCML About". page External links • http://www.mualphatheta.org/Contests/RocketCity.aspx • http://www.artofproblemsolving.com/Wiki/index.php/Rocket_City_Math_League	Wikipedia
Rocket science (finance) "Rocket science" in finance is a metaphor for activity carried out by specialised quantitative staff to provide detailed output from mathematical modeling and computational simulations to support investment decisions. Their work depends on use of complex mathematical models implemented in sophisticated IT environments.[1][2] finance rocket scientist Wall street, site of most finance rocket science activity Occupation Occupation type profession Activity sectors banking, financial market, economics Description Competenciesmacroeconomics, microeconomics, statistics, information technologies, pure mathematics Education required math-based For instance, a firm that invests its money in funds of investment is thought to have a result that depends on a mix of scientific questions and hazards. Different decisions in how to divide the financial resources into the funds lead to different sets of probabilities of return. Advising the investor about the consequences of each possible decision in the risk-return context is one of the typical roles of a rocket scientist. Core activities Although the financial rocket science is found mostly in banks and financial enterprises,[3] this area is emerging in firms with other kinds of core activities.[4] The reasons why firms may have to hire these professionals vary and may be related to the core itself, or to auxiliary areas. An example of the first case is the one of an insurance firm that needs to calculate sets of probabilities of expenses, from probabilities of accident of its customers. The other one is advising about what the need every organization, or even every individual, has for properly deciding what to do with the money. Goals The goal of a financial rocket scientist is to survey the high administration of a firm with the most precise scenario possible of the result probabilities in choosing decisions like investing, trading and borrowing.[5] It means that not only how to invest money is a problem to rocket scientists, but also matters like pricing assets, creating new products or managing debts. Skills The skills required of a financial rocket scientist are broadly based. These include knowledge of microeconomics, macroeconomics,[6] pure mathematics,[2] statistics, information technologies and financial market practice.[7] The microeconomics knowledge is necessary because the firm itself is an entity subject to microeconomics laws.[8] Macroeconomics are needed to evaluate the response of groups of entities to a wide range of external factors and influences.[9] Pure mathematics and statistics are required to solve the problems arising from questions submitted to the tech workers. Finally, financial market practice is needed to determine the possible decisions built into the financial models. Also, skill with Information Technology is required to prepare effective data-entry into complex computer systems. Some concepts and tools found in this area are the Pareto optimum, the Value at Risk, and the Monte Carlo simulation.[7] It's not rocket science Although this phrase, very often found in sources of every kind, seems to deny the ontological existence of the theme here in approach, it actually means context-dependently that some matter is not difficult to understand in certain level of depth. The term Rocket Science was originally derived from WWII rocket developments by Warner Von Braun and later by the NASA aerospace engineering program in the 1960s with the objective of reaching the moon. It was later coined in 1995 by the child of comedian Davis Mathews [10] Similar activities Some professions or activities are similar but not identified with rocket science, and they are lied by common issues and relations mean-goal, these areas are basically the financial engineering and risk analysis, according to some sources.[11][12][13] Nonetheless, a second and smaller set of sources found in this research identifies them as a unique matter.[14][15] See also • Computational finance • Mathematical finance • Financial modeling • Securities • Trading • Derivatives • Investment management References 1. "Definition of 'Rocket Scientist'". Investopedia. Retrieved 25 March 2012. 2. "Rocket scientist". The Free Financial Dictionary. Retrieved 26 March 2012. 3. Morrison, John (18 April 2009). "Is SAP Bank Analyser too Complex?". asymptotics. Retrieved 27 March 2012. 4. "MIT Sloan Team Introduces 'Rocket Science' to Fast-Fashion Retailing". MIT Sloan Management. 3 October 2007. Archived from the original on 8 August 2012. Retrieved 27 March 2012. 5. Davis, Evan (14 January 2009). "The Rocket Scientists of Finance". BBC News. Retrieved 26 March 2012. 6. these two areas differ one from the other in that the former studies the behavior of firms and families from an internal point of view, as the latter deals with a national scope and concepts as inflation and employment 7. "The Rocket Scientists Of Wall Street". Investopedia. Retrieved 26 March 2012. 8. Chorafas, Dimitris. N. (2007). Risk Management in Finance Services - Risk Control, Stress Test, Models (1 ed.). Oxford: Elsevier. p. 104. ISBN 9780750683043. 9. Endicott, Jared Row. "The Disputability of Macroeconomic Knowledge". Realizing Resonance. Archived from the original on 1 February 2013. Retrieved 27 March 2012. 10. "It's not Rocket Science". The Phrase Finder. Retrieved 30 March 2012. 11. "Risk Management & Financial Engineering". Rotman. Retrieved 30 March 2012. 12. "Advanced Financial Engineering Mathematics Applied to Algorithmic Trading of Stocks & Commodities". Meyer Analytics. Retrieved 30 March 2012. 13. "Financial Engineering:Science or Myth?". CSI Wall Street. Retrieved 30 March 2012. 14. "From Rocket Scientists to Financial Engineers". Digital Library. 2002. Retrieved 30 March 2012. 15. "Option Pricing Theory". Riskglossary. Retrieved 1 April 2012. General areas of finance • Capital management • Computational finance • Experimental finance • Financial economics • Financial engineering • Financial institutions • Financial management • Financial markets • Financial technology (Fintech) • Investment management • Mathematical finance • Personal finance • Public finance • Quantitative behavioral finance • Quantum finance • Statistical finance	Wikipedia
A. Rod Gover Ashwin Rod Gover[1][2] is a New Zealand mathematician and a Fellow of the Royal Society of New Zealand. He is currently employed as a Professor of Pure Mathematics at the University of Auckland in New Zealand.[2] He is the PhD students' Adviser for the Department of Mathematics and is the head of the Analysis, Geometry and Topology Research Group at the University of Auckland.[3] Rod Gover Born Ashwin Rod Gover NationalityNew Zealander Known forInvariant theory problems, operator classification problem Scientific career FieldsMathematics, differential geometry, theoretical physics ThesisA Geometrical Construction of Conformally Invariant Differential Operators (1989) Doctoral advisorMichael Eastwood Lane P. Hughston Education and career Gover received his secondary education at Tauranga Boys' College, where he was Head Boy and Dux. He earned a Bachelor of Science with Honours and Master of Science in physics at Canterbury University and a Doctor of Philosophy (DPhil) in Mathematics in 1989 at Oxford. He joined the University of Auckland as a lecturer in 1999, before being promoted to Senior Lecturer in 2001, Associate Professor in 2005, and Professor in 2008.[4] Research areas His current main research areas are • Differential geometry and its relationship to representation theory • Applications to analysis on manifolds, PDE theory and Mathematical Physics • Conformal, CR and related structures He has published work on a range of topics including integral transforms and their applications to representation theory and quantum groups. His main area of specialisation is the class of parabolic differential geometries. Tractor calculus is important for treating geometries in this class, and a current theme of his work is the further development of this calculus, its relationship to other geometric constructions and tools, as well as its applications to the construction and understanding of local and global geometric invariants and natural differential equations.[3] A list of his publications can be found here. References 1. "A. Rod Gover". Institute for Advanced Study. Retrieved 30 September 2016. 2. "Professor Rod Gover". University of Auckland. Retrieved 4 February 2023. 3. "Rod Gover – Research – MathsDept". University of Auckland. Retrieved 30 September 2016. 4. "Kudos for Maths expert". East & Bays Courier. 16 July 2008. Retrieved 30 September 2016 – via PressReader. Authority control Academics • MathSciNet • Mathematics Genealogy Project Other • IdRef	Wikipedia
Rod group In mathematics, a rod group is a three-dimensional line group whose point group is one of the axial crystallographic point groups. This constraint means that the point group must be the symmetry of some three-dimensional lattice. Table of the 75 rod groups, organized by crystal system or lattice type, and by their point groups: Triclinic 1 p1 2 p1 Monoclinic/inclined 3 p211 4 pm11 5 pc11 6 p2/m11 7 p2/c11 Monoclinic/orthogonal 8 p112 9 p1121 10 p11m 11 p112/m 12 p1121/m Orthorhombic 13 p222 14 p2221 15 pmm2 16 pcc2 17 pmc21 18 p2mm 19 p2cm 20 pmmm 21 pccm 22 pmcm Tetragonal 23 p4 24 p41 25 p42 26 p43 27 p4 28 p4/m 29 p42/m 30 p422 31 p4122 32 p4222 33 p4322 34 p4mm 35 p42cm, p42mc 36 p4cc 37 p42m, p4m2 38 p42c, p4c2 39 p4/mmm 40 p4/mcc 41 p42/mmc, p42/mcm Trigonal 42 p3 43 p31 44 p32 45 p3 46 p312, p321 47 p3112, p3121 48 p3212, p3221 49 p3m1, p31m 50 p3c1, p31c 51 p3m1, p31m 52 p3c1, p31c Hexagonal 53 p6 54 p61 55 p62 56 p63 57 p64 58 p65 59 p6 60 p6/m 61 p63/m 62 p622 63 p6122 64 p6222 65 p6322 66 p6422 67 p6522 68 p6mm 69 p6cc 70 p63mc, p63cm 71 p6m2, p62m 72 p6c2, p62c 73 p6/mmm 74 p6/mcc 75 p63/mmc, p63/mcm The double entries are for orientation variants of a group relative to the perpendicular-directions lattice. Among these groups, there are 8 enantiomorphic pairs. See also • Point group • Crystallographic point group • Space group • Line group • Frieze group • Layer group References • Hitzer, E.S.M.; Ichikawa, D. (2008), "Representation of crystallographic subperiodic groups by geometric algebra" (PDF), Electronic Proc. Of AGACSE, Leipzig, Germany (3, 17–19 Aug. 2008), archived from the original (PDF) on 2012-03-14 • Kopsky, V.; Litvin, D.B., eds. (2002), International Tables for Crystallography, Volume E: Subperiodic groups, International Tables for Crystallography, vol. E (5th ed.), Berlin, New York: Springer-Verlag, doi:10.1107/97809553602060000105, ISBN 978-1-4020-0715-6 External links • "Subperiodic Groups: Layer, Rod and Frieze Groups" on Bilbao Crystallographic Server • Nomenclature, Symbols and Classification of the Subperiodic Groups, V. Kopsky and D. B. Litvin	Wikipedia
Roderick Melnik Roderick Melnik is a Canadian-Australian[1] mathematician and scientist, internationally known[2] for his research in applied mathematics, numerical analysis, and mathematical modeling for scientific and engineering applications. Roderick Melnik CitizenshipAustralia Alma materKiev State University AwardsNSERC Tier I Canada Research Chair Scientific career FieldsApplied Mathematics, Nanoscale Systems, Mathematical Modelling in Science and Engineering Biography Melnik is a Tier I Canada Research Chair in Mathematical Modeling and Professor at Wilfrid Laurier University in Waterloo, Canada. His other affiliations include the University of Waterloo and University of Guelph. Education and career He earned his Ph.D. at Kiev State University in the late 1980s. According to the Mathematics Genealogy Project, his scientific ancestors include A. Tikhonov and other outstanding mathematicians and scientists. Before moving to Canada as a Tier I Canada Research Chair, Melnik gained a worldwide reputation in mathematical modelling and applied mathematics,[3] while working in Europe, Australia, and the United States. Awards and honors Melnik is a recipient of many fellowships and awards,[4] including the Andersen fellowship at Syddansk Universitet in Denmark, the Isaac Newton Institute visiting fellowship at the University of Cambridge in England, the Ikerbasque Fellowship in Spain, the fellowship of the Institute of Advanced Studies at the University of Bologna in Italy, and others. He is a life member of the Canadian Applied and Industrial Mathematics Society. Melnik is the director of the Laboratory of Mathematical Modeling for New Technologies (M2NeT Lab) in Waterloo, Ontario, Canada. Research In his early works Melnik studied fully coupled hyperbolic-elliptic models applied in dynamic piezoelectricity theory. Such models, originally proposed by W. Voigt in 1910, have found many applications, and Melnik was the first to rigorously prove well-posedness of a large class of such models in the dynamic case.[5] The piezoelectric effect itself, captured by such models, was discovered in 1880 by Pierre and Jacques Curie. Mathematical models describing this effect in time-dependent situations are based on initial-boundary value problems for coupled systems of partial differential equations. The mathematical and computational analysis of such coupled systems has been in the focus of many Melnik's works. In the 1990s he extended his scientific interests to applications of mathematics in semiconductor and other advanced technologies, including smart and bio-inspired materials technologies, where in collaboration with A. Roberts and their students he pioneered computationally efficient low-dimensional reductions of complex time-dependent nonlinear mathematical models. His other important contributions at that time included fundamental problems in control theory and dynamic system evolution, as well as a range of problems in industrial & applied mathematics and numerical analysis. Melnik is an expert in computational and applied mathematics with a number of important results in the coupled field theory as applied in physics, biology, and engineering. He is a leading computational analyst,[3] well known for his contributions to the analysis of coupled multiscale phenomena, processes, and systems. His recent significant contributions are in the analysis of mutual influence between quantum and classical effects in complex systems, in particular in the study of coupled effects in low-dimensional nanostructures,[6] as well as in bio-inspired engineering and biological systems.[7] References 1. Canada in Australia, March 2004, ISSN 1446-7291. 2. Barbara Aggerholm, Championing science, The Record, June 16, 2004. 3. Rindy Metcalf, New professor brings ideas, The Tech Talk, January 23, 2004. 4. Af Bente Dalgaard, At bringe verden til SDU, Ny Viden, No. 1, January 2007. 5. Masayuki Akamatsu & Gen Nakamura, Applicable Analysis, 81, 2002. 6. Unraveling coupled multiscale phenomena in quantum dot nanostructures , Nanotechnology, 20, 125402, 2009 7. Coarse-graining RNA nanostructures for MD simulations , Physical Biology, 7, 036001, 2010. External links • Roderick Melnik at the Mathematics Genealogy Project • Laurier Science Research Centre, M2NeT Laboratory • Wilfrid Laurier University, Faculty of Science • Research at the Guelph-Waterloo Institute of Physics • Roderick Melnik at the TRRA Accelerate Innovation Authority control National • Germany Academics • DBLP • Google Scholar • MathSciNet • Mathematics Genealogy Project • ORCID • Scopus Other • IdRef	Wikipedia
Rodger's method Rodger's method is a statistical procedure for examining research data post hoc following an 'omnibus' analysis (e.g., after an analysis of variance – anova). The various components of this methodology were fully worked out by R. S. Rodger in the 1960s and 70s, and seven of his articles about it were published in the British Journal of Mathematical and Statistical Psychology between 1967 and 1978.[1][2][3][4][5][6][7] Statistical procedures for finding differences between groups, along with interactions between the groups that were included in an experiment or study, can be classified along two dimensions: 1) were the statistical contrasts that will be evaluated decided upon prior to collecting the data (planned) or while trying to figure out what those data are trying to reveal (post hoc), and 2) does the procedure use a decision-based (i.e., per contrast) error rate or does it instead use an experiment-wise error rate. Rodger's method, and some others, are classified according to these dimensions in the table below. Table 1: Some multiple comparison procedures Planned contrastsPost hoc contrasts Decision-based error rate t testsDuncan's method Rodger's method Experiment-wise error rate Bonferroni's inequality Dunnett's method Newman–Keuls method Tukey's range method Scheffé's method Statistical power In early 1990s, one set of researchers made this statement about their decision to use Rodger's method: “We chose Rodger’s method because it is the most powerful post hoc method available for detecting true differences among groups. This was an especially important consideration in the present experiments in which interesting conclusions could rest on null results” (Williams, Frame, & LoLordo, 1992, p. 43).[8] The most definitive evidence for the statistical power advantage that Rodger's method possesses (as compared with eight other multiple comparison procedures) is provided in a 2013 article by Rodger and Roberts.[9] Type 1 error rate Statistical power is an important consideration when choosing what statistical procedure to use, but it isn't the only important one. All statistical procedures permit researchers to make statistical errors and they are not all equal in their ability to control the rate of occurrence of several important types of statistical error. As Table 1 shows, statisticians can't agree on how error rate ought to be defined, but particular attention has been traditionally paid to what are called 'type 1 errors' and whether or not a statistical procedure is susceptible to type 1 error rate inflation. On this matter, the facts about Rodger's method are straightforward and unequivocal. Rodger's method permits an absolutely unlimited amount of post hoc data snooping and this is accompanied by a guarantee that the long run expectation of type 1 errors will never exceed the commonly used rates of either 5 or 1 percent. Whenever a researcher falsely rejects a true null contrast (whether it is a planned or post hoc one) the probability of that being a type 1 error is 100%. It is the average number of such errors over the long run that Rodger's method guarantees cannot exceed Eα = 0.05 or 0.01. This statement is a logical tautology, a necessary truth, that follows from the manner in which Rodger's method was originally conceived and subsequently built. Type 1 error rate inflation is statistically impossible with Rodger's method, but every statistical decision a researcher makes that might be a type 1 error will either actually be one or it won't. Decision-based error rate The two features of Rodger's method that have been mentioned thus far, its increased statistical power and the impossibility of type 1 error rate inflation when using it, are direct by-products of the decision-based error rate that it utilizes. "An error occurs, in the statistical context, if and only if a decision is made that a specified relationship among population parameters either is, or is not, equal to some number (usually, zero), and the opposite is true. Rodger’s very sensible, and cogently argued, position is that statistical error rate should be based exclusively on those things in which errors may occur, and that (necessarily, by definition) can only be the statistical decisions that researchers make" (Roberts, 2011, p. 69).[10] Implied true population means There is a unique aspect of Rodger's method that is statistically valuable and is not dependent on its decision-based error rate. As Bird stated: "Rodger (1965, 1967a, 1967b, 1974) explored the possibility of examining the logical implications of statistical inferences on a set of J − 1 linearly independent contrasts. Rodger’s approach was formulated within the Neyman-Pearson hypothesis-testing framework [...] and required that the test of each contrast Ψi (i = 1, ... , J − 1) should result in a ‘decision’ between the null hypothesis (iH0: Ψi = 0) and a particular value δi specified a priori by the alternative hypothesis (iH1: Ψi = δi). Given the resulting set of decisions, it is possible to determine the implied values of all other contrasts" (Bird, 2011, p. 434).[11] The statistical value that Rodger derived from the ‘implication equation’ that he invented is prominently displayed in the form of 'implied means' that are logically implied, and mathematically entailed, by the J − 1 statistical decisions that the user of his method makes. These implied true population means constitute a very precise statement about the outcome of one's research, and assist other researchers in determining the size of effect that their related research ought to seek. Whither Rodger’s method? Since the inception of Rodger's method, some researchers who use it have had their work published in prestigious scientific journals, and this continues to happen. Nevertheless, it is fair to currently conclude that “Rodger’s work on deduced inference has been largely ignored” (Bird, 2011, p. 434). Bird uses implication equations, similar to Rodger's, to deduce interval inferences concerning any contrasts not included in an analysis from the upper and lower limits of confidence intervals on J − 1 linearly independent planned contrasts; a procedure that Rodger himself opposes.[12] A very different desired outcome for Rodger's method was conveyed in this statement by Roberts: "Will Rodger’s method continue to be used by only a few researchers, become extinct, or supplant most or all of the currently popular post hoc procedures following ANOVA? This article and the SPS computer program constitute an attempted intervention in the competition for dominance and survival that occurs among ideas. My hope is that the power and other virtues of Rodger’s method will become much more widely known and that, as a consequence, it will become commonly used. ... Better ideas and the ‘mousetraps’ they are instantiated in, ought, eventually, to come to the fore" (Roberts, 2011, p. 78). The possible futures for Rodger's method mentioned in the two previous paragraphs are therefore not exhaustive, and the possibilities on a more comprehensive list are no longer mutually exclusive. References 1. Rodger, R. S. (1974). Multiple contrasts, factors, error rate and power. British Journal of Mathematical and Statistical Psychology, 27, 179–198. 2. Rodger, R. S. (1975a). The number of non-zero, post hoc contrasts from ANOVA and error-rate I. British Journal of Mathematical and Statistical Psychology, 28, 71–78. 3. Rodger, R. S. (1975b). Setting rejection rate for contrasts selected post hoc when some nulls are false. British Journal of Mathematical and Statistical Psychology, 28, 214–232. 4. Rodger, R. S. (1978). Two-stage sampling to set sample size for post hoc tests in ANOVA with decision-based error rates. British Journal of Mathematical and Statistical Psychology, 31, 153–178. 5. Rodger, R. S. (1969). Linear hypotheses in 2xa frequency tables. British Journal of Mathematical and Statistical Psychology, 22, 29–48. 6. Rodger, R. S. (1967a). Type I errors and their decision basis. British Journal of Mathematical and Statistical Psychology, 20, 51–62. 7. Rodger, R. S. (1967b). Type II errors and their decision basis. British Journal of Mathematical and Statistical Psychology, 20, 187–204. 8. Williams, D. A., Frame, K. A., & LoLordo, V. M. (1992). Discrete signals for the unconditioned stimulus fail to overshadow contextual or temporal conditioning. Journal of Experimental Psychology: Animal Behavior Processes, 18(1), 41–55. 9. Rodger, R.S. and Roberts, M. (2013). Comparison of power for multiple comparison procedures. Journal of Methods and Measurement in the Social Sciences, 4(1), 20–47. 10. Roberts, M. (2011). Simple, Powerful Statistics: An instantiation of a better ‘mousetrap’. Journal of Methods and Measurement in the Social Sciences, 2(2), 63–79. 11. Bird, K. D. (2011). Deduced inference in the analysis of experimental data. Psychological Methods, 16(4), 432–443. 12. Rodger, R. S. (2012). Paired comparisons, confusion, constraint, contradictions, and confidence intervals. Unpublished manuscript. External links Wikiversity has learning resources about Rodger's Method • Wikiversity: Rodger's Method (the Wikiversity explication that includes mathematical formulas, matrices, and a numerical illustration) • Simple, Powerful Statistics (SPS) (download website for a free, Windows-based computer program that makes using Rodger's method accessible to all researchers)	Wikipedia
Rodion Kuzmin Rodion Osievich Kuzmin (Russian: Родион Осиевич Кузьмин, 9 November 1891, Riabye village in the Haradok district – 24 March 1949, Leningrad) was a Soviet mathematician, known for his works in number theory and analysis.[1] His name is sometimes transliterated as Kusmin. He was an Invited Speaker of the ICM in 1928 in Bologna.[2] Rodion Kuzmin Rodion Kusmin, circa 1926 Born(1891-10-09)9 October 1891 Riabye village in the Haradok district Died24 March 1949(1949-03-24) (aged 57) Leningrad NationalityRussian Alma materSaint Petersburg State University nee Petrograd University Known forGauss–Kuzmin distribution, number theory and mathematical analysis. Scientific career FieldsMathematics InstitutionsPerm State University, Tomsk Polytechnic University, Saint Petersburg State Polytechnical University Doctoral advisorJames Victor Uspensky Selected results • In 1928, Kuzmin solved[3] the following problem due to Gauss (see Gauss–Kuzmin distribution): if x is a random number chosen uniformly in (0, 1), and $x={\frac {1}{k_{1}+{\frac {1}{k_{2}+\cdots }}}}$ is its continued fraction expansion, find a bound for $\Delta _{n}(s)=\mathbb {P} \left\{x_{n}\leq s\right\}-\log _{2}(1+s),$ where $x_{n}={\frac {1}{k_{n+1}+{\frac {1}{k_{n+2}+\cdots }}}}.$ Gauss showed that Δn tends to zero as n goes to infinity, however, he was unable to give an explicit bound. Kuzmin showed that $\|\Delta _{n}(s)\|\leq Ce^{-\alpha {\sqrt {n}}}~,$ where C,α > 0 are numerical constants. In 1929, the bound was improved to C 0.7n by Paul Lévy. • In 1930, Kuzmin proved[4] that numbers of the form ab, where a is algebraic and b is a real quadratic irrational, are transcendental. In particular, this result implies that Gelfond–Schneider constant $2^{\sqrt {2}}=2.6651441426902251886502972498731\ldots $ is transcendental. See Gelfond–Schneider theorem for later developments. • He is also known for the Kusmin-Landau inequality: If $f$ is continuously differentiable with monotonic derivative $f'$ satisfying $\Vert f'(x)\Vert \geq \lambda >0$ (where $\Vert \cdot \Vert $ denotes the Nearest integer function) on a finite interval $I$, then $\sum _{n\in I}e^{2\pi if(n)}\ll \lambda ^{-1}.$ Notes 1. Venkov, B. A.; Natanson, I. P. "R. O. Kuz'min (1891–1949) (obituary)". Uspekhi Matematicheskikh Nauk. 4 (4): 148–155. 2. Kuzmin, R. "Sur un problème de Gauss." In Atti del Congresso Internazionale dei Matematici: Bologna del 3 al 10 de settembre di 1928, vol. 6, pp. 83–90. 1929. 3. Kuzmin, R.O. (1928). "On a problem of Gauss". Dokl. Akad. Nauk SSSR: 375–380. 4. Kuzmin, R. O. (1930). "On a new class of transcendental numbers". Izvestiya Akademii Nauk SSSR (Math.). 7: 585–597. External links • Rodion Kuzmin at the Mathematics Genealogy Project (The chronology there is apparently wrong, since J. V. Uspensky lived in USA from 1929.) Authority control International • ISNI • VIAF National • Germany • Israel • United States • Czech Republic • Netherlands • Poland Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Rodney Hill Rodney Hill FRS[1] (11 June 1921 – 2 February 2011)[2] was an applied mathematician and a former Professor of Mechanics of Solids at Gonville and Caius College, Cambridge. Rodney Hill Born(1921-06-11)11 June 1921 Died2 February 2011(2011-02-02) (aged 89) AwardsFRS[1] Scientific career FieldsPlasticity InstitutionsUniversity of Cambridge Career In 1953 he was appointed Professor of Applied Mathematics at the University of Nottingham. His 1950 The Mathematical Theory of Plasticity work[3] forms the foundation of plasticity theory. Hill is widely regarded as among the foremost contributors to the foundations of solid mechanics over the second half of the 20th century. His early work was central to founding the mathematical theory of plasticity. This deep interest led eventually to general studies of uniqueness and stability in nonlinear continuum mechanics, work which has had a profound influence on the field of solid mechanics—theoretical, computational and experimental alike—over the past decades. Hill was the founding editor of the Journal of the Mechanics and Physics of Solids, still among the principal journals in the field.[4] Recognition Hill's work is recognized worldwide for its concise style of presentation and exemplary standards of scholarship. Publisher Elsevier, in collaboration with IUTAM, established a quadrennial award in the field of solid mechanics, known as the Rodney Hill Prize, first presented at ICTAM in Adelaide in August 2008. The prize consists of a plaque and a cheque for US$25,000. Its first recipient is Michael Ortiz, for his contribution to nonconvex plasticity and deformation microstructures (California Institute of Technology, USA).[5][6] Hill won the Royal Medal in 1993 for his contribution to the theoretical mechanics of soil and the plasticity of solids.[7] He was elected a Fellow of the Royal Society (FRS) in 1961.[1][8] He was awarded an Honorary Degree (Doctor of Science) by the University of Bath in 1978. Death He died on 2 February 2011.[1] References 1. Sewell, Michael J. (2015). "Rodney Hill. 11 June 1921 — 2 February 2011". Biographical Memoirs of Fellows of the Royal Society. 61: 161–181. doi:10.1098/rsbm.2014.0024. ISSN 0080-4606. 2. "Obituaries: Professor Rodney Hill". The Telegraph. 8 March 2011. Retrieved 25 March 2011. 3. Hill R., The Mathematical Theory of Plasticity, Oxford University Press, 1950. 4. JMPS. https://www.sciencedirect.com/journal/journal-of-the-mechanics-and-physics-of-solids. {{cite web}}: Missing or empty \|title= (help) 5. "ICTAM 2008". Retrieved 18 December 2008. 6. "Rodney Hill Prize for Solid Mechanics (pdf) –" (PDF). Retrieved 18 December 2008. 7. "Royal Medal Winners: 2007 – 1990". Retrieved 6 December 2008. 8. "Fellows". Royal Society. Retrieved 19 November 2010. Authority control International • FAST • ISNI • VIAF National • France • BnF data • Germany • Italy • United States • Japan • Czech Republic • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • Scopus • zbMATH Other • IdRef	Wikipedia
Rodrigo Bañuelos Rodrigo Bañuelos is an American mathematician and a professor of mathematics at Purdue University in West Lafayette, Indiana. His research is in probability and its applications to harmonic analysis and spectral theory. Early life, education, and career Bañuelos was born in La Masita in the state of Zacatecas, Mexico.[1] When he was 15, Bañuelos, his mother, grandmother, and six siblings moved to Pasadena, California.[2] In 1978, Bañuelos received a B.A. in mathematics from the University of California, Santa Cruz. In 1980, he received a M.A.T. in mathematics with a California High School Teaching Credential form the University of California, Davis.[3] In 1984, Bañuelos was awarded a Ph.D. in mathematics by the University of California, Los Angeles. He wrote his dissertation "Martingale Transforms, Related Singular Integrals, and AP-Weights" under the supervision of Richard Timothy Durrett.[4] According to MathSciNet, Bañuelos has authored or co-authored 102 articles in mathematical journals and books, which appeared in various journals. Bañuelos has served on several editorial boards, including the Annals of Probability, Transactions of the AMS, Probability and Mathematical Statistics, Revista Matemática Iberoamericana, Latin American Journal of Probability and Mathematical Statistics, Potential Analysis, Annals of Probability and the Latin American Journal of Probability and Mathematical Statistics. He has served on numerous committees of the AMS. [3] Book Rodrigo Bañuelos and Charles N. Moore, Probabilistic Behavior of Harmonic Functions, Birkhäuser, 1999, ISBN 978-3-0348-8728-1. Honors and awards • 1984–1986 Bantrell Research Fellow at California Institute of Technology • 1986–1989 National Science Foundation (NSF) Postdoctoral Fellow at University of Illinois at Urbana-Champaign and Purdue University • 1989–1994 NSF Presidential Young Investigator[5] • 2000 Elected Fellow of the Institute of Mathematical Statistics[6] • 2004 Blackwell-Tapia Prize in Mathematics[7] • 2009 Outstanding Latino Award, Purdue Latino Faculty and Staff Association • 2013 Elected Fellow of the American Mathematical Society[8] • 2017 Elected Fellow of the Association for Women in Mathematics in the inaugural class[9] • 2018 Martin Luther King Jr. Dreamer Award, Purdue University[10] • 2022 AMS Award for Distinguished Public Service[11] • 2023 Class of SIAM Fellows[12] References 1. "Rodrigo Bañuelos". Math Alliance: The National Alliance for Doctoral Studies in the Mathematical Sciences. 7 May 2019. Retrieved 14 January 2021. 2. "Rodrigo Bañuelos, PhD". SACNAS. 22 December 2020. Retrieved 14 January 2021. 3. "- Rodrigo Bañuelos CV". Department of Mathematics, Purdue University. Retrieved 14 January 2021. 4. Rodrigo Bañuelos at the Mathematics Genealogy Project 5. "NSF Award Search: Award#8957316". Presidential Young Investigator Award. Retrieved 14 January 2021. 6. "Honored IMS Fellows". Institute of Mathematical Statistics. 22 July 2020. Retrieved 14 January 2021. 7. "Blackwell-Tapia Conference and Prize Presentation". IPAM. 14 August 2014. Retrieved 14 January 2021. 8. "2013 Class of Fellows of the AMS". American Mathematical Society. Retrieved 14 January 2021. 9. "The AWM Fellows Program: 2018 Class of AWM Fellows". Association for Women in Mathematics. Retrieved 14 January 2021. 10. "Dreamer Award - Diversity Resource Office - Purdue University". www.purdue.edu. Retrieved 2021-12-05. 11. "News from the AMS". American Mathematical Society. Retrieved 2021-12-05. 12. "SIAM Announces Class of 2023 Fellows". SIAM News. Retrieved 2023-04-08. External links • Official website • Rodrigo Bañuelos' Author profile on MathSciNet • Rodrigo Bañuelos publications indexed by Google Scholar Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • United States • Netherlands • Poland Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Rodrigues' rotation formula In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3), the group of all rotation matrices, from an axis–angle representation. In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra so(3) to its Lie group SO(3). Not to be confused with the Euler–Rodrigues parameters and The Euler–Rodrigues formula for 3D rotation. This formula is variously credited to Leonhard Euler, Olinde Rodrigues, or a combination of the two. A detailed historical analysis in 1989 concluded that the formula should be attributed to Euler, and recommended calling it "Euler's finite rotation formula."[1] This proposal has received notable support,[2] but some others have viewed the formula as just one of many variations of the Euler–Rodrigues formula, thereby crediting both.[3] Statement If v is a vector in ℝ3 and k is a unit vector describing an axis of rotation about which v rotates by an angle θ according to the right hand rule, the Rodrigues formula for the rotated vector vrot is $\mathbf {v} _{\mathrm {rot} }=\mathbf {v} \cos \theta +(\mathbf {k} \times \mathbf {v} )\sin \theta +\mathbf {k} ~(\mathbf {k} \cdot \mathbf {v} )(1-\cos \theta )\,.$ The intuition of the above formula is that the first term scales the vector down, while the second skews it (via vector addition) toward the new rotational position. The third term re-adds the height (relative to ${\textbf {k}}$) that was lost by the first term. An alternative statement is to write the axis vector as a cross product a × b of any two nonzero vectors a and b which define the plane of rotation, and the sense of the angle θ is measured away from a and towards b. Letting α denote the angle between these vectors, the two angles θ and α are not necessarily equal, but they are measured in the same sense. Then the unit axis vector can be written $\mathbf {k} ={\frac {\mathbf {a} \times \mathbf {b} }{\|\mathbf {a} \times \mathbf {b} \|}}={\frac {\mathbf {a} \times \mathbf {b} }{\|\mathbf {a} \|\|\mathbf {b} \|\sin \alpha }}\,.$ This form may be more useful when two vectors defining a plane are involved. An example in physics is the Thomas precession which includes the rotation given by Rodrigues' formula, in terms of two non-collinear boost velocities, and the axis of rotation is perpendicular to their plane. Derivation Let k be a unit vector defining a rotation axis, and let v be any vector to rotate about k by angle θ (right hand rule, anticlockwise in the figure). Using the dot and cross products, the vector v can be decomposed into components parallel and perpendicular to the axis k, $\mathbf {v} =\mathbf {v} _{\parallel }+\mathbf {v} _{\perp }\,,$ where the component parallel to k is $\mathbf {v} _{\parallel }=(\mathbf {v} \cdot \mathbf {k} )\mathbf {k} $ called the vector projection of v on k, and the component perpendicular to k is $\mathbf {v} _{\perp }=\mathbf {v} -\mathbf {v} _{\parallel }=\mathbf {v} -(\mathbf {k} \cdot \mathbf {v} )\mathbf {k} =-\mathbf {k} \times (\mathbf {k} \times \mathbf {v} )$ called the vector rejection of v from k. The vector k × v can be viewed as a copy of v⊥ rotated anticlockwise by 90° about k, so their magnitudes are equal but directions are perpendicular. Likewise the vector k × (k × v) a copy of v⊥ rotated anticlockwise through 180° about k, so that k × (k × v) and v⊥ are equal in magnitude but in opposite directions (i.e. they are negatives of each other, hence the minus sign). Expanding the vector triple product establishes the connection between the parallel and perpendicular components, for reference the formula is a × (b × c) = (a · c)b − (a · b)c given any three vectors a, b, c. The component parallel to the axis will not change magnitude nor direction under the rotation, $\mathbf {v} _{\parallel \mathrm {rot} }=\mathbf {v} _{\parallel }\,,$ only the perpendicular component will change direction but retain its magnitude, according to ${\begin{aligned}\left\|\mathbf {v} _{\perp \mathrm {rot} }\right\|&=\left\|\mathbf {v} _{\perp }\right\|\,,\\\mathbf {v} _{\perp \mathrm {rot} }&=\cos(\theta )\mathbf {v} _{\perp }+\sin(\theta )\mathbf {k} \times \mathbf {v} _{\perp }\,,\end{aligned}}$ and since k and v∥ are parallel, their cross product is zero k × v∥ = 0, so that $\mathbf {k} \times \mathbf {v} _{\perp }=\mathbf {k} \times \left(\mathbf {v} -\mathbf {v} _{\parallel }\right)=\mathbf {k} \times \mathbf {v} -\mathbf {k} \times \mathbf {v} _{\parallel }=\mathbf {k} \times \mathbf {v} $ and it follows $\mathbf {v} _{\perp \mathrm {rot} }=\cos(\theta )\mathbf {v} _{\perp }+\sin(\theta )\mathbf {k} \times \mathbf {v} \,.$ This rotation is correct since the vectors v⊥ and k × v have the same length, and k × v is v⊥ rotated anticlockwise through 90° about k. An appropriate scaling of v⊥ and k × v using the trigonometric functions sine and cosine gives the rotated perpendicular component. The form of the rotated component is similar to the radial vector in 2D planar polar coordinates (r, θ) in the Cartesian basis $\mathbf {r} =r\cos(\theta )\mathbf {e} _{x}+r\sin(\theta )\mathbf {e} _{y}\,,$ where ex, ey are unit vectors in their indicated directions. Now the full rotated vector is $\mathbf {v} _{\mathrm {rot} }=\mathbf {v} _{\parallel \mathrm {rot} }+\mathbf {v} _{\perp \mathrm {rot} }\,,$ By substituting the definitions of v∥rot and v⊥rot in the equation results in ${\begin{aligned}\mathbf {v} _{\mathrm {rot} }&=\mathbf {v} _{\parallel }+\cos(\theta )\,\mathbf {v} _{\perp }+\sin(\theta )\,\mathbf {k} \times \mathbf {v} \\&=\mathbf {v} _{\parallel }+\cos(\theta )\left(\mathbf {v} -\mathbf {v} _{\parallel }\right)+\sin(\theta )\,\mathbf {k} \times \mathbf {v} \\&=\cos(\theta )\,\mathbf {v} +(1-\cos \theta )\mathbf {v} _{\parallel }+\sin(\theta )\,\mathbf {k} \times \mathbf {v} \\&=\cos(\theta )\,\mathbf {v} +(1-\cos \theta )(\mathbf {k} \cdot \mathbf {v} )\mathbf {k} +\sin(\theta )\,\mathbf {k} \times \mathbf {v} \end{aligned}}$ Matrix notation Representing v and k × v as column matrices, the cross product can be expressed as a matrix product ${\begin{bmatrix}(\mathbf {k} \times \mathbf {v} )_{x}\\(\mathbf {k} \times \mathbf {v} )_{y}\\(\mathbf {k} \times \mathbf {v} )_{z}\end{bmatrix}}={\begin{bmatrix}k_{y}v_{z}-k_{z}v_{y}\\k_{z}v_{x}-k_{x}v_{z}\\k_{x}v_{y}-k_{y}v_{x}\end{bmatrix}}={\begin{bmatrix}0&-k_{z}&k_{y}\\k_{z}&0&-k_{x}\\-k_{y}&k_{x}&0\end{bmatrix}}{\begin{bmatrix}v_{x}\\v_{y}\\v_{z}\end{bmatrix}}\,.$ By K, denote the "cross-product matrix" for the unit vector k, $\mathbf {K} =\left[{\begin{array}{ccc}0&-k_{z}&k_{y}\\k_{z}&0&-k_{x}\\-k_{y}&k_{x}&0\end{array}}\right]\,.$ That is to say, $\mathbf {K} \mathbf {v} =\mathbf {k} \times \mathbf {v} $ for any vector v. (In fact, K is the unique matrix with this property. It has eigenvalues 0 and ±i). It follows that iterating the cross product is equivalent to multiplying by the cross-product matrix on the left; specifically: $\mathbf {K} (\mathbf {K} \mathbf {v} )=\mathbf {K} ^{2}\mathbf {v} =\mathbf {k} \times (\mathbf {k} \times \mathbf {v} )\,.$ The previous rotation formula in matrix language is therefore ${\begin{aligned}\mathbf {v} _{\mathrm {rot} }&=\mathbf {v} \cos \theta +(\mathbf {k} \times \mathbf {v} )\sin \theta +\mathbf {k} ~(\mathbf {k} \cdot \mathbf {v} )(1-\cos \theta )\\&=\mathbf {v} \cos \theta +(\mathbf {k} \times \mathbf {v} )\sin \theta +(\mathbf {v} -\mathbf {v} _{\perp })(1-\cos \theta )\\&=\mathbf {v} \cos \theta +(\mathbf {k} \times \mathbf {v} )\sin \theta +(\mathbf {v} +\mathbf {k} \times (\mathbf {k} \times \mathbf {v} ))(1-\cos \theta )\\&=\mathbf {v} \cos \theta +(\mathbf {k} \times \mathbf {v} )\sin \theta +(\mathbf {v} +\mathbf {K} (\mathbf {K} \mathbf {v} ))(1-\cos \theta )\\&=\mathbf {v} (\cos \theta +1-\cos \theta )+(\mathbf {k} \times \mathbf {v} )\sin \theta +\mathbf {K} (\mathbf {K} \mathbf {v} )(1-\cos \theta )\end{aligned}}$ So we have: $\mathbf {v} _{\mathrm {rot} }=\mathbf {v} +(\sin \theta )\mathbf {K} \mathbf {v} +(1-\cos \theta )\mathbf {K} ^{2}\mathbf {v} \,,\quad \\|\mathbf {K} \\|_{2}=1\,.$ Note the coefficient of the leading term is now 1, in this notation: see the Lie-Group discussion below. Factorizing the v allows the compact expression $\mathbf {v} _{\mathrm {rot} }=\mathbf {R} \mathbf {v} $ where $\mathbf {R} =\mathbf {I} +(\sin \theta )\mathbf {K} +(1-\cos \theta )\mathbf {K} ^{2}$ is the rotation matrix through an angle θ counterclockwise about the axis k, and I the 3 × 3 identity matrix.[4] This matrix R is an element of the rotation group SO(3) of ℝ3, and K is an element of the Lie algebra ${\mathfrak {so}}(3)$ generating that Lie group (note that K is skew-symmetric, which characterizes ${\mathfrak {so}}(3)$). In terms of the matrix exponential, $\mathbf {R} =\exp(\theta \mathbf {K} )\,.$ To see that the last identity holds, one notes that $\mathbf {R} (\theta )\mathbf {R} (\phi )=\mathbf {R} (\theta +\phi ),\quad \mathbf {R} (0)=\mathbf {I} \,,$ characteristic of a one-parameter subgroup, i.e. exponential, and that the formulas match for infinitesimal θ. For an alternative derivation based on this exponential relationship, see exponential map from ${\mathfrak {so}}(3)$ to SO(3). For the inverse mapping, see log map from SO(3) to ${\mathfrak {so}}(3)$. The Hodge dual of the rotation $\mathbf {R} $ is just $\mathbf {R} ^{}=-\sin(\theta )\mathbf {k} $ which enables the extraction of both the axis of rotation and the sine of the angle of the rotation from the rotation matrix itself, with the usual ambiguity, ${\begin{aligned}\sin(\theta )&=\sigma \left\|\mathbf {R} ^{}\right\|\\[3pt]\mathbf {k} &=-{\frac {\sigma \mathbf {R} ^{}}{\left\|\mathbf {R} ^{}\right\|}}\end{aligned}}$ where $\sigma =\pm 1$. The above simple expression results from the fact that the Hodge duals of $\mathbf {I} $ and $\mathbf {K} ^{2}$ are zero, and $\mathbf {K} ^{*}=-\mathbf {k} $. When applying the Rodrigues' formula, however, the usual ambiguity could be removed with an extended form of the formula.a See also • Axis angle • Rotation (mathematics) • SO(3) and SO(4) • Euler–Rodrigues formula References 1. Cheng, Hui; Gupta, K. C. (March 1989). "An Historical Note on Finite Rotations". Journal of Applied Mechanics. American Society of Mechanical Engineers. 56 (1): 139–145. Retrieved 2022-04-11. 2. Fraiture, Luc (2009). "A History of the Description of the Three-Dimensional Finite Rotation". The Journal of the Astronautical Sciences. Springer. 57: 207–232. Retrieved 2022-04-15. 3. Dai, Jian S. (October 2015). "Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections". Mechanism and Machine Theory. Elsevier. 92: 144–152. Retrieved 2022-04-14. 4. Belongie, Serge. "Rodrigues' Rotation Formula". mathworld.wolfram.com. Retrieved 2021-04-07. • Leonhard Euler, "Problema algebraicum ob affectiones prorsus singulares memorabile", Commentatio 407 Indicis Enestoemiani, Novi Comm. Acad. Sci. Petropolitanae 15 (1770), 75–106. • Olinde Rodrigues, "Des lois géométriques qui régissent les déplacements d'un système solide dans l'espace, et de la variation des coordonnées provenant de ces déplacements considérés indépendants des causes qui peuvent les produire", Journal de Mathématiques Pures et Appliquées 5 (1840), 380–440. online. • Friedberg, Richard (2022). "Rodrigues, Olinde: "Des lois géométriques qui régissent les déplacements d'un systéme solide...", translation and commentary". arXiv:2211.07787. • Don Koks, (2006) Explorations in Mathematical Physics, Springer Science+Business Media,LLC. ISBN 0-387-30943-8. Ch.4, pps 147 et seq. A Roundabout Route to Geometric Algebra • ^a Liang, Kuo Kan (2018). "Efficient conversion from rotating matrix to rotation axis and angle by extending Rodrigues' formula". arXiv:1810.02999 [cs]. External links • Johan E. Mebius, Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations., arXiv General Mathematics 2007. • For another descriptive example see: http://chrishecker.com/Rigid_Body_Dynamics#Physics_Articles, Chris Hecker, physics section, part 4. "The Third Dimension" – on page 3, section ``Axis and Angle, http://chrishecker.com/images/b/bb/Gdmphys4.pdf	Wikipedia
Rogemar Mamon Rogemar Sombong Mamon, CSci, CMath, FIMA, FHEA, FRSA is a Canadian mathematician,[1] quant, and academic. He is a co-editor of the IMA Journal of Management Mathematics published by Oxford University Press since 2009.[2] Mamon is known for his contributions to the developments and applications of regime-switching framework useful in economic, financial and actuarial modeling. Majority of his works promote regime-switching paradigms modulated by either discrete- or continuous-time hidden Markov models (HMM). A recurrent theme of his research is dynamic parameter estimation via HMM filtering recursions. He also made contributions in the areas of derivative pricing, asset allocation, risk measurement, filtering to remove noise from data as well as inverse problems in quantitative finance. He was the lead editor of the handbook Hidden Markov Models in Finance, published by Springer.[3][4] In 2010, he and two co-authors won the Society of Actuaries Award for the Best Paper published in the North American Actuarial Journal.[5] Since 2006, he has taught, conducted research and held administrative roles at the University of Western Ontario, and garnered recognitions for excellence in teaching and research.[6][7] Previously, he held academic positions at Brunel University, London, UK; University of British Columbia; University of Waterloo; and University of Alberta. He spent short-term research visits[8] at several institutions including the Isaac Newton Institute for Mathematical Sciences, University of Cambridge, England; Maxwell Institute for Mathematical Sciences, Scotland; Centre for Mathematical Physics and Stochastics, University of Aarhus, Denmark; Institute for Mathematics and its Applications, University of Minnesota, USA; University of Adelaide, Australia; University of Wollongong in New South Wales, Australia; and Centro de Investigacion en Matematicas, Mexico. Mamon holds professional designations conferred by various British learned societies.[9] He is a Fellow and Chartered Mathematician of the Institute of Mathematics and its Applications; Chartered Scientist of the Science Council; and Fellow of the Higher Education Academy. He is also a Fellow of the Royal Society of Arts and the Royal Statistical Society, and was an elected member of the London Mathematical Society. He began PhD studies in Mathematical Finance at the University of Alberta, and completed his dissertation during a research visit at the University of Adelaide, Australia.[10] He was supervised by Robert J. Elliott making him a mathematical descendant of Godfrey Harold Hardy, Sir Isaac Newton and Galileo Galilei.[1] References 1. Mathematics Genealogy Project. http://genealogy.math.ndsu.nodak.edu/id.php?id=60661&fChrono=1. 2. IMA Journal of Mathematics, accessed March 29, 2014. 3. R.S. Mamon and R.J. Elliott (Eds.), 2007, Hidden Markov Models in Finance, Springer's International Series in Operations Research and Management Science, Vol. 104, XX, 188 p. 11, illus., Hardcover ISBN 978-0-387-71081-5 4. , springer.com; accessed March 29, 2014. 5. Annual Prize Announcement, Society of Actuaries' North American Actuarial Journal (NAAJ), vol 14, #2. 6. Science honours its own, 30 September 2010, Western News, University of Western Ontario, London, Canada. http://communications.uwo.ca/western_news/stories/2010/September/science_honours_its_own-lite.html. 7. Marquis Who's Who in Science and Engineering 2005-2006; ISBN 978-0-8379-5764-7. 8. Website of Department of Statistical and Actuarial Sciences, University of Western Ontario, stats.uwo.ca; accessed March 29, 2014. 9. Marquis Who’s Who in the World 2008. ISBN 0837911397, ISBN 978-0837911397. 10. Editorial: Philippine Canadian Times of Alberta, Vol. 2, No. 4, Fall 2000, p. 4. Authority control International • ISNI • VIAF National • United States Academics • MathSciNet • Mathematics Genealogy Project • zbMATH	Wikipedia
Roger Apéry Roger Apéry (French: [apeʁi]; 14 November 1916, Rouen – 18 December 1994, Caen) was a French mathematician most remembered for Apéry's theorem, which states that ζ(3) is an irrational number. Here, ζ(s) denotes the Riemann zeta function. Roger Apéry Born(1916-11-14)14 November 1916 Rouen, France Died18 December 1994(1994-12-18) (aged 78) Caen, France NationalityFrench EducationLycée Louis-le-Grand Alma materÉcole normale supérieure OccupationMathematician Biography Apéry was born in Rouen in 1916 to a French mother and Greek father. His childhood was spent in Lille until 1926, when the family moved to Paris, where he studied at the Lycée Ledru-Rollin and the Lycée Louis-le-Grand. He was admitted at the École normale supérieure in 1935. His studies were interrupted at the start of World War II; he was mobilized in September 1939, taken prisoner of war in June 1940, repatriated with pleurisy in June 1941, and hospitalized until August 1941. He wrote his doctoral thesis in algebraic geometry under the direction of Paul Dubreil and René Garnier in 1947. In 1947 Apéry was appointed Maître de conférences (lecturer) at the University of Rennes. In 1949 he was appointed Professor at the University of Caen, where he remained until his retirement. In 1979 he published an unexpected proof of the irrationality of ζ(3), which is the sum of the inverses of the cubes of the positive integers. An indication of the difficulty is that the corresponding problem for other odd powers remains unsolved. Nevertheless, many mathematicians have since worked on the so-called Apéry sequences to seek alternative proofs that might apply to other odd powers (Frits Beukers, Alfred van der Poorten, Marc Prévost, Keith Ball, Tanguy Rivoal, Wadim Zudilin, and others). Apéry was active in politics and for a few years in the 1960s was president of the Calvados Radical Party of the Left. He abandoned politics after the reforms instituted by Edgar Faure after the 1968 revolt, when he realised that university life was running against the tradition he had always upheld. Personal life Apéry married in 1947 and had three sons, including mathematician François Apéry. His first marriage ended in divorce in 1971. He then remarried in 1972 and divorced in 1977. In 1994, Apéry died from Parkinson's disease after a long illness in Caen. He was buried next to his parents at the Père Lachaise Cemetery in Paris. His tombstone has a mathematical inscription stating his theorem. $1+{\frac {1}{8}}+{\frac {1}{27}}+{\frac {1}{64}}+\cdots \neq {\frac {p}{q}}$ See also • Apéry's constant • Basel problem External links • Apéry, François (1996). "Roger Apéry, 1916-1994: A Radical Mathematician". The Mathematical Intelligencer. 18 (2): 54–61. doi:10.1007/BF03027295. S2CID 120113351. • van der Poorten, Alfred (1979). "A proof that Euler missed ... Apéry's proof of the irrationality of ζ(3)". The Mathematical Intelligencer. 1 (4): 195–203. doi:10.1007/BF03028234. S2CID 121589323. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Netherlands Academics • MathSciNet • zbMATH Other • IdRef	Wikipedia
Roger C. Alperin Roger Charles Alperin (January 8, 1947 – November 21, 2019) was an American mathematician, best known for his work in group theory, including its connections with geometry and topology. He was a professor at the University of Oklahoma and at San Jose State University. Education and career Alperin was born on January 8, 1947, in Cambridge, Massachusetts.[1] He received a bachelor's degree from the University of Chicago,[2] and his PhD from Rice University in 1973. His thesis was supervised by Stephen M. Gersten, and was titled Whitehead Torsion of Finite Abelian Groups.[3] After temporary positions at Brown University, Haverford College, and Washington University in St. Louis, Alperin took a permanent position at the University of Oklahoma in 1978.[4] He was eventually promoted to full professor at the University of Oklahoma, but resigned his position to move to California in 1987.[4] Upon moving to California, he found a position at San Jose State University, which he held until his retirement in 2015.[4] Alperin died on November 21, 2019, at his home in Carlsbad, California.[4] Research Alperin's work on real trees in the 80s (partly joint with Hyman Bass and Kenneth Moss) helped to stimulate interest in these objects, and helped establish them as a basic tool in geometric group theory.[4] Alperin has also done foundational work on the mathematical theory of origami.[1] References 1. "In Memoriam: Roger Charles Alperin (1947 - 2019)" (PDF). International Journal of Geometry. 9 (1): 42–43. 2020. Retrieved December 22, 2020. 2. American Men & Women of Science (22nd ed.). Thomson Gale. 2005. ISBN 0787674001. 3. Roger C. Alperin at the Mathematics Genealogy Project 4. Farb, Benson; Shalev, Peter (December 2020). "Roger Alperin". Notices of the American Mathematical Society. American Mathematical Society. 67 (11): 1768–1769. doi:10.1090/noti2189. External links • Roger C. Alperin publications indexed by Google Scholar Mathematics of paper folding Flat folding • Big-little-big lemma • Crease pattern • Huzita–Hatori axioms • Kawasaki's theorem • Maekawa's theorem • Map folding • Napkin folding problem • Pureland origami • Yoshizawa–Randlett system Strip folding • Dragon curve • Flexagon • Möbius strip • Regular paperfolding sequence 3d structures • Miura fold • Modular origami • Paper bag problem • Rigid origami • Schwarz lantern • Sonobe • Yoshimura buckling Polyhedra • Alexandrov's uniqueness theorem • Blooming • Flexible polyhedron (Bricard octahedron, Steffen's polyhedron) • Net • Source unfolding • Star unfolding Miscellaneous • Fold-and-cut theorem • Lill's method Publications • Geometric Exercises in Paper Folding • Geometric Folding Algorithms • Geometric Origami • A History of Folding in Mathematics • Origami Polyhedra Design • Origamics People • Roger C. Alperin • Margherita Piazzola Beloch • Robert Connelly • Erik Demaine • Martin Demaine • Rona Gurkewitz • David A. Huffman • Tom Hull • Kôdi Husimi • Humiaki Huzita • Toshikazu Kawasaki • Robert J. Lang • Anna Lubiw • Jun Maekawa • Kōryō Miura • Joseph O'Rourke • Tomohiro Tachi • Eve Torrence Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef	Wikipedia
Roger Horn Roger Alan Horn (born January 19, 1942) is an American mathematician specializing in matrix analysis. He was research professor of mathematics at the University of Utah. He is known for formulating the Bateman–Horn conjecture with Paul T. Bateman on the density of prime number values generated by systems of polynomials.[3] His books Matrix Analysis and Topics in Matrix Analysis, co-written with Charles R. Johnson, are standard texts in advanced linear algebra.[4][5][6] Roger Alan Horn Born (1942-01-19) January 19, 1942[1] NationalityAmerican Alma materCornell University Stanford University Known forMatrix analysis Bateman-Horn conjecture SpouseSusan Horn Scientific career FieldsMathematics InstitutionsUniversity of Santa Clara Johns Hopkins University University of Maryland, Baltimore County University of Utah ThesisInfinitely Divisible Matrices, Kernels, and Functions (1967) Doctoral advisorDonald C. Spencer, Charles Loewner InfluencesGene Golub[2] Career Roger Horn graduated from Cornell University with high honors in mathematics in 1963,[7] after which he completed his PhD at Stanford University in 1967. Horn was the founder and chair of the Department of Mathematical Sciences at Johns Hopkins University from 1972 to 1979.[8] As chair, he held a series of short courses for a monograph series published by the Johns Hopkins Press. He invited Gene Golub and Charles Van Loan to write a monograph, which later became the seminal Matrix Computations text book.[9] He later joined the Department of Mathematics at the University of Utah as research professor. In 2007, the journal Linear Algebra and its Applications published a special issue in honor of Roger Horn.[10] He was Editor of The American Mathematical Monthly during 1997–2001. Personal life In 1987, Horn submitted testimony to the US Senate Subcommittee on Transportation regarding the 1987 Maryland train collision which killed his 16-year-old daughter Ceres who was returning to Princeton University from the family home in Baltimore for her freshman year fall term final exams.[11] Bibliography • Horn, Roger Alan; Johnson, Charles Royal (2018) [1985]. Matrix Analysis (2nd ed.). Cambridge University Press. ISBN 978-0-521-54823-6. • Horn, Roger Alan; Johnson, Charles Royal (1991). Topics in Matrix Analysis. Cambridge University Press. ISBN 0-521-46713-6. • Garcia, Stephan Ramon; Horn, Roger Alan (2023) [2017]. Matrix Mathematics: A Second Course in Linear Algebra (2nd ed.). Cambridge University Press. ISBN 978-1-108-83710-1. References 1. Bhatia, Rajendra; Kittaneh, Fuad; Mathias, Roy; Zhan, Xingzhi (2007). "Special issue dedicated to Roger Horn" (PDF). Linear Algebra and Its Applications. 424 (1): 1–2. doi:10.1016/j.laa.2007.02.014. Retrieved 16 April 2022. 2. Higham, Nick; Golub, Gene. "In His Own Words". SIAM News. Society for Industrial and Applied Mathematics. Retrieved 13 February 2017. 3. Bateman, Paul T.; Horn, Roger A. (1962), "A heuristic asymptotic formula concerning the distribution of prime numbers", Mathematics of Computation, 16 (79): 363–367, doi:10.2307/2004056, JSTOR 2004056, MR 0148632 4. Horn, Roger A.; Johnson, Charles R. (1990-02-23). Topics in Matrix Analysis. ISBN 0521386322. 5. "Topics in Matrix Analysis: Roger A. Horn, Charles R. Johnson: 9780521467131: Amazon.com: Books". Amazon. Retrieved 27 October 2014. 6. Marcus, Marvin (1992). "Review: Topics in Matrix Analysis, by Roger A. Horn and Charles R. Johnson". Bull. Amer. Math. Soc. (N.S.). 27 (1): 191–198. doi:10.1090/s0273-0979-1992-00296-3. 7. "The Class of 1963". The Cornell Daily Sun. 79 (151): 20. 7 June 1963. Retrieved 13 February 2017. 8. "Department History". Department of Applied Mathematics & Statistics. Johns Hopkins University. Retrieved 13 February 2017. 9. Chan, Raymond H.; Greif, Chen; O'Leary, Dianne P. (2007). Milestones in Matrix Computation: Selected Works of Gene H. Golub, with commentaries. Oxford: Oxford Univ. Press. p. 10. ISBN 978-0199206810. 10. "Special Issue in honor of Roger Horn". Linear Algebra and Its Applications. 424 (1): 1–338. 1 July 2007. doi:10.1016/j.laa.2007.02.014. 11. Remarks on Transportation Safety, Based on Testimony to the Senate Subcommittee on Transportation, Committee on Appropriations April 9 and May 13, 1987. 1 January 1989. pp. 415–423. Authority control International • ISNI • VIAF National • Germany • Israel • United States • Czech Republic • Netherlands Academics • CiNii • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Roger J-B Wets Roger Jean-Baptiste Robert Wets (born February 1937) is a "pioneer" in stochastic programming[2] and a leader in variational analysis who publishes as Roger J-B Wets. His research, expositions, graduate students, and his collaboration with R. Tyrrell Rockafellar have had a profound influence on optimization theory, computations, and applications.[2][3][4] Since 2009, Wets has been a distinguished research professor at the mathematics department of the University of California, Davis.[5][6] Roger J-B Wets BornFebruary 1937 (age 86) Belgium[1] Alma materUniversité libre de Bruxelles University of California, Berkeley AwardsFrederick W. Lanchester Prize (1997) Scientific career Fieldsstochastic programming ThesisProgramming under uncertainty (1965) Doctoral advisorGeorge Dantzig David Blackwell Websitewww.math.ucdavis.edu/~rjbw/ Schooling and positions Roger Wets attended high school in Belgium, after which he worked for his family while earning his Licence in applied economics from Université de Bruxelles (Brussels, Belgium) in 1961.[7] He was encouraged by Jacques H. Drèze to study optimization with George Dantzig at the program in operations research at the University of California, Berkeley.[8] Dantzig and mathematician–statistician David Blackwell jointly supervised Wets's dissertation.[6][9] In 1965 Wets befriended R. Tyrrell Rockafellar, whom Wets introduced to stochastic optimization, starting a collaboration of many decades.[10] He worked at Boeing Scientific Research Labs, 1964–1970 and was Ford Professor at the University of Chicago, 1970–1972 before being appointed Professor at the Mathematics Department of the University of Kentucky and then University Research Professor (1977–78).[5] While at the International Institute for Applied Systems Analysis (IIASA) in Austria, during 1980–1984,[5] he led research in decision-making in uncertainty, returning as an acting leader in 1985–1987; during that time, Wets and Rockafellar developed the progressive-hedging algorithm for stochastic programming.[2][4] The University of California, Davis named him Professor (1984–1997), Distinguished Professor, and Distinguished Research Professor of Mathematics (2009–).[5] Awards and contributions Wets was awarded a George B. Dantzig Prize for "original research that has had a major impact on the field of mathematical programming" by the Society for Industrial and Applied Mathematics (SIAM) and the Mathematical Programming Society (MPS, now the Mathematical Optimization Society).[4] In 1994, the Dantzig Prize was awarded to Wets and also to the French pioneer in nonsmooth computational-optimization, Claude Lemaréchal.[4] Wets's contributions included developing set-valued analysis, including metric spaces of sets, which he used to study the convergence of epigraphs; Wets's ideas of epigraphical convergence was used to study the convergence iterative methods of stochastic optimization and has had applications in the approximation theory of statistics.[2][4][6][11] A metric theory of finite-dimensional epigraphical convergence ("cosmic convergence") appears in Variational analysis.[11] Wets and his coauthor R. Tyrrell Rockafellar were awarded the 1997 Frederick W. Lanchester Prize by the Institute for Operations Research and the Management Sciences (INFORMS) for their monograph Variational Analysis, which was published in November 1997 and copyrighted in 1998.[3][11] With Rockafellar, Wets proposed, studied, and implemented the progressive-hedging algorithm for stochastic programming. Besides his theoretical and computational contributions, Wets has worked with applications on lake ecology (IIASA), finance (Frank Russel investment system), and developmental economics (World Bank). He also consulted with the development of professional stochastic-optimization software (IBM).[4] See also • Pompeiu–Hausdorff distance References 1. John Simon Guggenheim Memorial Foundation (1981). Reports of the President and the Treasurer - John Simon Guggenheim Memorial Foundation. John Simon Guggenheim Memorial Foundation. 2. Anonymous (2004, p. 1) 3. Anonymous (1998, p. 1) 4. Dantzig Prize Committee (1994, p. 5) 5. Wets (2011) 6. Wets (2011b) 7. Aardal (1995, p. 3) 8. In an interview with Karen Aardal, Wets stated that he believed that he took the only course in operations research available (at that time) in Western Europe. The instructor was Jacques H. Drèze at the Université Catholique de Louvain. Because of Drèze's suggestion to study with Dantzig at Berkeley, Wets credits Drèze as being responsible for getting him into the field of optimization. (Aardal 1995, p. 3) 9. Programming under uncertainty. (1965) Wets, Roger Jean-Baptiste Robert. Abstract Thesis (PhD in Engineering Science)--Univ. of California, Jan. 1965. Bibliography: l. 81-88. Publisher [Berkeley] Repository OCLC's Experimental Thesis Catalog (United States) 10. Vicente (2004, p. 11) 11. Rockafellar & Wets (2005) Sources • Aardal, Karen (July 1995). "Optima interview Roger J.-B. (sic.) Wets" (PDF). Optima: Mathematical Programming Society Newsletter. Mathematical Programming Society. 46: 3–5. • Anonymous, COSP (1 November 2004). Roger J-B Wets (PDF). Pioneers in Stochastic Programming. Committee on Stochastic Programming (COSP). Archived from the original (PDF) on 3 March 2012. Retrieved 12 March 2012. • Anonymous, INFORMS (1998). "Roger J-B Wets, Past awards: 1977 Frederick W. Lanchester Prize: Winner". Institute for Operations Research and the Management Sciences (INFORMS). Archived from the original on 24 May 2012. Retrieved 12 March 2012. • Rockafellar, R. Tyrrell; Wets, Roger J-B (2005) [1996]. Variational Analysis (PDF). Grundlehren der mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences). Vol. 317 (third corrected printing ed.). Berlin: Springer-Verlag. pp. xiv+733. doi:10.1007/978-3-642-02431-3. ISBN 978-3-540-62772-2. MR 1491362. Retrieved 12 March 2012. • Vicente, L .N. (2004). "An Interview with R. Tyrrell Rockafellar" (PDF). SIAG/Opt News and Views. Society for Industrial and Applied Mathematics (SIAM), Special Interest Group in Optimization. 15 (1): 9–14. Retrieved 12 March 2012. • Dantzig Prize Committee (1994). "Citation of Roger Wets (for the George Dantzig Prize, 1994)" (PDF). Optima: Mathematical Programming Society Newsletter. Mathematical Programming Society. 44: 5. Retrieved 12 March 2012. • Wets, Roger J-B (31 December 2011). "ROGER J-B WETS (Curriculum Vitae)" (PDF). Department of Mathematics, University of California, Davis. Retrieved 12 March 2012. • Wets, Roger J-B (31 December 2011b). "ROGER J-B WETS : Biography-Summary" (PDF). Department of Mathematics, University of California, Davis. Retrieved 12 March 2012. External links • Homepage of Roger J-B Wets at the Mathematics Department of the University of California, Davis. Contains biography, research overviews, lectures and presentations. 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Roger Paman Roger Paman (c. 1700 - 1748) was an 18th-century English mathematician. Roger Paman Bornc. 1700 (1700) England Died1748 (aged 47–48) Unknown Known forOpposition to Berkeley Scientific career FieldsMathematics Life and work Very little is known about the life of Roger Paman. He cited a professor of St John's College, Cambridge as mentor, so it is possible that he studied at Cambridge University.[1] As he explains in the preface of his only published book, he participated in George Anson's voyage around the world (1740–1744), but he returned to England in 1742 with one of two ships which returned before the end of the expedition.[2] Before embarking on the expedition, he left his mathematical papers to philosopher David Hartley.[3] His papers were presented to the Royal Society in 1742 and, as a result, he was named fellow of the institution in 1743.[1] In 1745 his book "The Harmony of the Ancient and Modern Geometry Asserted" was published in London. The book is a reply to the mathematical concepts of George Berkeley in his book The Analyst (1734).[4] The originality of this book lies in the development of all the calculus in finite differences, in order to avoid the paradoxes of infinity explained by Berkeley.[5] Paman presents in his book the concepts of minimaius and maximinus, a far antecedent of the mathematical concepts of infimum and supremum.[6][7] References 1. Smestad Chapter 5.1 2. Appleby, page 200. 3. Jesseph, page 285. 4. Breidert, pages 119–123. 5. Jesseph, page 287. 6. Smestad, chapter 5.3.1 7. Jesseph, pages 287–290. Bibliography • Appleby, John H (2001). "Mapping Russia: Farquharson, Delisle and the Royal Society". Notes and Records of the Royal Society of London. 55 (2): 191–204. doi:10.1098/rsnr.2001.0138. ISSN 0035-9149. • Breidert, Wolfgang (1989). George Berkeley 1685–1753. Birkhauser. doi:10.1007/978-3-0348-7248-5. ISBN 978-3-0348-7249-2. • Jesseph, Douglas M (1993). Berkeley's Philosophy of Mathematics. University of Chicago Press. ISBN 978-0-226-39897-6. External links • O'Connor, John J.; Robertson, Edmund F., "Roger Paman", MacTutor History of Mathematics Archive, University of St Andrews • Smestad, Bjorn (1995). "Foundations for fluxions". University of Oslo. Archived from the original on 4 November 2004. Retrieved 6 January 2015. Doctoral Thesis Authority control International • ISNI • VIAF National • Germany • United States • Netherlands Other • IdRef	Wikipedia
Roger Guesnerie Roger Guesnerie is an economist born in France in 1943. He is currently the Chaired Professor of Economic Theory and Social Organization of the Collège de France, Director of Studies at the École des hautes études en sciences sociales, and the chairman of the board of directors of the Paris School of Economics. Roger Guesnerie Born1943 (age 79–80) France NationalityFrench Academic career InstitutionCollège de France École des hautes études en sciences sociales Paris School of Economics FieldEconomic theory Macroeconomics Public economics School or tradition Mathematical economics Alma materÉcole Polytechnique École nationale des ponts et chaussées Doctoral advisor Jean-Jacques Laffont Doctoral students Thomas Piketty AwardsPresident, Econometric Society (1996) President, French Association of Economic Sciences (2002–2003) President, European Economic Association (1994) Foreign Honorary Member of the American Economic Association Foreign Honorary Member, American Academy of Arts and Sciences CNRS Silver Medal Chevalier de l'Ordre National du Mérite Chevalier de la Légion d'honneur Information at IDEAS / RePEc Career Guesnerie studied at École Polytechnique and the École Nationale des Ponts et Chaussées, and received his doctorate in economics from the University of Toulouse in 1982. He has taught at the London School of Economics, the École Polytechnique, and at Harvard University.[1] Guesnerie has published widely in economics, including in public economics, in the theory of incentives and economic mechanisms, and in the theory of general economic equilibrium. Honors and responsibilities Guesnerie has been elected president of several scholarly societies, notably the French Association of Economic Sciences (2002–2003), the Econometric Society (1996), and the European Economic Association (1994). Guesnerie has been elected as a foreign honorary member of the American Economic Association and as a foreign member of the American Academy of Arts and Sciences. He has served as co-editor of Econometrica (1984–1989) and as foreign editor of the Review of Economic Studies. In France, Guesnerie's research has been recognized with the CNRS Silver Medal; he has been declared to be a Chevalier de l'Ordre National du Mérite and Chevalier de la Légion d'honneur. Publications Books • Roger Guesnerie and Henry Tulkens, 2008, The Design of Climate Policy, MIT Press.[2] • "Assessing Rational Expectations 2: Eductive stability in economics", MIT Press, 2005, 453p.[3] • "Assessing Rational Expectations: Sunspot multiplicity and economic fluctuations", MIT Press, 2001, 319 p.ISBN 978-0-262-26279-8 • "A contribution to the pure theory of taxation", Cambridge University Press, 1995, 301 pages[4] Papers • Guesnerie, Roger; Roberts, Kevin W.S. (January 1984). "Effective policy tools and quantity controls" (PDF). Econometrica. 52 (1): 59–86. doi:10.2307/1911461. JSTOR 1911461. • Guesnerie, Roger (1975). "Pareto optimality in non-convex economies". Econometrica. 43 (1): 1–29. doi:10.2307/1913410. JSTOR 1913410. MR 0443877. with "Errata". Econometrica. Vol. 43, no. 5–6. 1975. p. 1010. doi:10.2307/1911353. JSTOR 1911353. MR 0443878. Donald J. Brown credited Guesnerie's "seminal" paper with the "major methodological innovation in the general equilibrium analysis of firms with pricing rules", "the introduction of the methods of nonsmooth analysis, as a [synthesis] of global analysis (differential topology) and [of] convex analysis."[5] This paper introduced cone of interior displacements of Dubovickii and Miljutin into economics.[6][7] • "General equilibrium when Some firms follow special pricing rules", (with Egbert Dierker and W. Neuefeind), Econometrica, 53, 6, 1985 This paper stimulated a subfield of economics, devoted to pricing rules, as discussed by Jacques Drèze: "Starting with a paper in Econometrica by Dierker, Guesnerie and Neuefeind (1985), a theory of general equilibrium has developed for economies with non-convex production sets, where firms follow well-defined pricing rules. In particular, existence theorems of increasing generality cover (to some extent, because of various differences in assumptions) the case of Ramsey-Boiteux pricing. Those interested primarily in applications might express skepticism, perhaps even horrified skepticism, upon realizing that 90 pages of a serious economics journal—a 1988 issue of The Journal of Mathematical Economics—were devoted to existence proofs of equilibrium in non-convex economies, under alternative formulations of the assumption that marginal cost pricing entails bounded losses at normalized prices. Still, I think that economic research must cover the whole spectrum from concrete applications to that level of abstraction."[8] • Guesnerie, Roger; Roberts, Kevin W.S. (February–March 1987). "Minimum wage legislation as a second best policy". European Economic Review. 31 (1–2): 490–498. doi:10.1016/0014-2921(87)90067-5. • Guesnerie, Roger (1989). "First-best allocation of resources with nonconvexities in production". In Cornet, Bernard; Tulkens, Henry (eds.). Contributions to Operations Research and Economics: The twentieth anniversary of CORE (Papers from the symposium held in Louvain-la-Neuve, January 1987). Cambridge, MA: MIT Press. pp. 99–143. ISBN 978-0-262-03149-3. MR 1104662. References 1. "Roger Guesnerie: (English) Curriculum Vitae". Paris School of Economics. Retrieved 8 December 2019. 2. R. Guesnerie; Henry Tulkens (2008). The Design of Climate Policy. MIT Press. ISBN 978-0-262-07302-8. 3. Roger Guesnerie (2005). Assessing Rational Expectations 2: "eductive" Stability in Economics. MIT Press. ISBN 978-0-262-26290-3. 4. Roger Guesnerie (12 November 1998). A Contribution to the Pure Theory of Taxation. Cambridge University Press. ISBN 978-0-521-62956-0. 5. Page 1967: Brown, Donald J. (1991). "Equilibrium analysis with non-convex technologies". In Hildenbrand, Werner; Sonnenschein, Hugo (eds.). Handbook of mathematical economics, Volume IV. Handbooks in Economics. Vol. 1. Amsterdam: North-Holland Publishing Co. pp. 1963–1995. ISBN 978-0-444-87461-0. MR 1207195. 6. 1965. A.J. Dubovickii and A. Miljutin, Extremum problems in the presence of restrictions. Zh. Vychisl. Mat. Fiz. 5 (1965), pp. 395–453. USSR Comp. Math. and Math. Physics 5 (1965), pp. 1–80. 7. Page 495: Mordukhovich, Boris S. (2006). "8 Applications to economics". Variational analysis and generalized differentiation II: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences). Vol. 331. Springer. pp. 461–505. MR 2191745. 8. Drèze, Jacques H. (1995). "Forty years of public economics: A personal perspective". Journal of Economic Perspectives. Vol. 9, no. 2. pp. 111–130. External links • Homepage of Roger Guesnerie (Paris School of Economics) Presidents of the European Economic Association 1986–2000 • Jacques Drèze (1986) • János Kornai (1987) • Edmond Malinvaud (1988) • Anthony Atkinson (1989) • Agnar Sandmo (1990) • Assar Lindbeck (1991) • Martin Hellwig (1992) • Mervyn King (1993) • Roger Guesnerie (1994) • Louis Philps (1995) • David Newbery (1996) • Reinhard Selten (1997) • Jean-Jacques Laffont (1998) • Partha Dasgupta (1999) • James A. Mirrlees (2000) 2001–present • Jean Tirole (2001) • J. Peter Neary (2002) • Torsten Persson (2003) • Richard Blundell (2004) • Mathias Dewatripont (2005) • Andreu Mas-Colell (2006) • Guido Tabellini (2007) • Ernst Fehr (2008) • Nicholas Stern (2009) • Timothy Besley (2010) • Christopher Pissarides (2011) • Jordi Galí (2012) • Manuel Arellano (2013) • Orazio Attanasio (2014) • Rachel Griffith (2015) • Fabrizio Zilibotti (2016) • Philippe Aghion (2017) • Eliana La Ferrara (2018) • Kjetil Storesletten (2019) • Per Krusell (2020) • Silvana Tenreyro (2021) Presidents of the Econometric Society 1931–1950 • Irving Fisher (1931–1934) • François Divisia (1935) • Harold Hotelling (1936–1937) • Arthur Bowley (1938–1939) • Joseph Schumpeter (1940–1941) • Wesley Mitchell (1942–1943) • John Maynard Keynes (1944–1945) • Jacob Marschak (1946) • Jan Tinbergen (1947) • Charles Roos (1948) • Ragnar Frisch (1949) • Tjalling Koopmans (1950) 1951–1975 • R. G. D. Allen (1951) • Paul Samuelson (1952) • René Roy (1953) • Wassily Leontief (1954) • Richard Stone (1955) • Kenneth Arrow (1956) • Trygve Haavelmo (1957) • James Tobin (1958) • Marcel Boiteux (1959) • Lawrence Klein (1960) • Henri Theil (1961) • Franco Modigliani (1962) • Edmond Malinvaud (1963) • Robert Solow (1964) • Michio Morishima (1965) • Herman Wold (1966) • Hendrik Houthakker (1967) • Frank Hahn (1968) • Leonid Hurwicz (1969) • Jacques Drèze (1970) • Gérard Debreu (1971) • W. M. Gorman (1972) • Roy Radner (1973) • Don Patinkin (1974) • Zvi Griliches (1975) 1976–2000 • Hirofumi Uzawa (1976) • Lionel McKenzie (1977) • János Kornai (1978) • Franklin M. Fisher (1979) • J. Denis Sargan (1980) • Marc Nerlove (1981) • James A. Mirrlees (1982) • Herbert Scarf (1983) • Amartya K. Sen (1984) • Daniel McFadden (1985) • Michael Bruno (1986) • Dale Jorgenson (1987) • Anthony B. Atkinson (1988) • Hugo Sonnenschein (1989) • Jean-Michel Grandmont (1990) • Peter Diamond (1991) • Jean-Jacques Laffont (1992) • Andreu Mas-Colell (1993) • Takashi Negishi (1994) • Christopher Sims (1995) • Roger Guesnerie (1996) • Robert E. Lucas, Jr. (1997) • Jean Tirole (1998) • Robert B. Wilson (1999) • Elhanan Helpman (2000) 2001–present • Avinash Dixit (2001) • Guy Laroque (2002) • Eric Maskin (2003) • Ariel Rubinstein (2004) • Thomas J. Sargent (2005) • Richard Blundell (2006) • Lars Peter Hansen (2007) • Torsten Persson (2008) • Roger B. Myerson (2009) • John H. Moore (2010) • Bengt Holmström (2011) • Jean-Charles Rochet (2012) • James J. Heckman (2013) • Manuel Arellano (2014) • Robert Porter (2015) • Eddie Dekel (2016) • Drew Fudenberg (2017) • Tim Besley (2018) • Stephen Morris (2019) • Orazio Attanasio (2020) • Pinelopi Koujianou Goldberg (2021) • Guido Tabellini (2022) Authority control International • ISNI • VIAF National • Norway • Spain • France • BnF data • Germany • Israel • United States • Latvia • Korea • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Roger Wolcott Richardson Roger Wolcott Richardson (30 May 1930 – 15 June 1993) was a mathematician noted for his work in representation theory and geometry. He was born in Baton Rouge, Louisiana, and educated at Louisiana State University, Harvard University and University of Michigan, Ann Arbor where he obtained a Ph.D. in 1958 under the supervision of Hans Samelson. After a postdoc appointment at Princeton University, he accepted a faculty position at the University of Washington in Seattle. He emigrated to the United Kingdom in 1970, taking up a chair at Durham University. In 1978 he moved to the Australian National University in Canberra, where he stayed as faculty until his death. Richardson's best known result states that if P is a parabolic subgroup of a reductive group, then P has a dense orbit on its nilradical, i.e., one whose closure is the whole space.[1] This orbit is now universally known as the Richardson orbit.[2] Publications • Nijenhuis, Albert; Richardson Jr., Roger W. (1966). "Cohomology and deformations in graded Lie algebras". Bulletin of the American Mathematical Society. 72 (1): 1–29. doi:10.1090/s0002-9904-1966-11401-5. MR 0195995. See also • Prehomogeneous vector space External links • Roger Wolcott Richardson at the Mathematics Genealogy Project • Mathematical Reviews analysis References 1. Richardson, R. W. (1974). "Conjugacy Classes in Parabolic Subgroups of Semisimple Algebraic Groups". Bulletin of the London Mathematical Society. 6: 21–24. doi:10.1112/blms/6.1.21. 2. Gus I. Lehrer, Roger Wolcott Richardson 1930–1993, Historical Records of Australian Science, Volume 11 Number 4 (1997) Authority control International • FAST • ISNI • VIAF National • Germany • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • SNAC	Wikipedia
Rogers–Szegő polynomials In mathematics, the Rogers–Szegő polynomials are a family of polynomials orthogonal on the unit circle introduced by Szegő (1926), who was inspired by the continuous q-Hermite polynomials studied by Leonard James Rogers. They are given by $h_{n}(x;q)=\sum _{k=0}^{n}{\frac {(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}}}x^{k}$ Not to be confused with Rogers polynomials. where (q;q)n is the descending q-Pochhammer symbol. Furthermore, the $h_{n}(x;q)$ satisfy (for $n\geq 1$) the recurrence relation[1] $h_{n+1}(x;q)=(1+x)h_{n}(x;q)+x(q^{n}-1)h_{n-1}(x;q)$ with $h_{0}(x;q)=1$ and $h_{1}(x;q)=1+x$. References 1. Vinroot, C. Ryan (12 July 2012). "An enumeration of flags in finite vector spaces". The Electronic Journal of Combinatorics. 19 (3). doi:10.37236/2481. • Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719 • Szegő, Gábor (1926), "Beitrag zur theorie der thetafunktionen", Sitz Preuss. Akad. Wiss. Phys. Math. Ki., XIX: 242–252, Reprinted in his collected papers	Wikipedia
Rogers polynomials In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by Rogers (1892, 1893, 1894) in the course of his work on the Rogers–Ramanujan identities. They are q-analogs of ultraspherical polynomials, and are the Macdonald polynomials for the special case of the A1 affine root system (Macdonald 2003, p.156). Not to be confused with Rogers–Szegő polynomials. Askey & Ismail (1983) and Gasper & Rahman (2004, 7.4) discuss the properties of Rogers polynomials in detail. Definition The Rogers polynomials can be defined in terms of the q-Pochhammer symbol and the basic hypergeometric series by $C_{n}(x;\beta \|q)={\frac {(\beta ;q)_{n}}{(q;q)_{n}}}e^{in\theta }{}_{2}\phi _{1}(q^{-n},\beta ;\beta ^{-1}q^{1-n};q,q\beta ^{-1}e^{-2i\theta })$ ;\beta ^{-1}q^{1-n};q,q\beta ^{-1}e^{-2i\theta })} where x = cos(θ). References • Askey, Richard; Ismail, Mourad E. H. (1983), "A generalization of ultraspherical polynomials", in Erdős, Paul (ed.), Studies in pure mathematics. To the memory of Paul Turán., Basel, Boston, Berlin: Birkhäuser, pp. 55–78, ISBN 978-3-7643-1288-6, MR 0820210 • Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719 • Macdonald, I. G. (2003), Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, vol. 157, Cambridge University Press, doi:10.1017/CBO9780511542824, ISBN 978-0-521-82472-9, MR 1976581 • Rogers, L. J. (1892), "On the expansion of some infinite products", Proc. London Math. Soc., 24 (1): 337–352, doi:10.1112/plms/s1-24.1.337, JFM 25.0432.01 • Rogers, L. J. (1893), "Second Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc., 25 (1): 318–343, doi:10.1112/plms/s1-25.1.318 • Rogers, L. J. (1894), "Third Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc., 26 (1): 15–32, doi:10.1112/plms/s1-26.1.15	Wikipedia
Rogers–Ramanujan identities In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by Leonard James Rogers (1894), and were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof (Rogers & Ramanujan 1919). Issai Schur (1917) independently rediscovered and proved the identities. Definition The Rogers–Ramanujan identities are $G(q)=\sum _{n=0}^{\infty }{\frac {q^{n^{2}}}{(q;q)_{n}}}={\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}=1+q+q^{2}+q^{3}+2q^{4}+2q^{5}+3q^{6}+\cdots $ (sequence A003114 in the OEIS) and $H(q)=\sum _{n=0}^{\infty }{\frac {q^{n^{2}+n}}{(q;q)_{n}}}={\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}=1+q^{2}+q^{3}+q^{4}+q^{5}+2q^{6}+\cdots $ (sequence A003106 in the OEIS). Here, $(a;q)_{n}$ denotes the q-Pochhammer symbol. Combinatorial interpretation Consider the following: • ${\frac {q^{n^{2}}}{(q;q)_{n}}}$ is the generating function for partitions with exactly $n$ parts such that adjacent parts have difference at least 2. • ${\frac {1}{(q;q^{5})_{\infty }(q^{4};q^{5})_{\infty }}}$ is the generating function for partitions such that each part is congruent to either 1 or 4 modulo 5. • ${\frac {q^{n^{2}+n}}{(q;q)_{n}}}$ is the generating function for partitions with exactly $n$ parts such that adjacent parts have difference at least 2 and such that the smallest part is at least 2. • ${\frac {1}{(q^{2};q^{5})_{\infty }(q^{3};q^{5})_{\infty }}}$ is the generating function for partitions such that each part is congruent to either 2 or 3 modulo 5. The Rogers–Ramanujan identities could be now interpreted in the following way. Let $n$ be a non-negative integer. 1. The number of partitions of $n$ such that the adjacent parts differ by at least 2 is the same as the number of partitions of $n$ such that each part is congruent to either 1 or 4 modulo 5. 2. The number of partitions of $n$ such that the adjacent parts differ by at least 2 and such that the smallest part is at least 2 is the same as the number of partitions of $n$ such that each part is congruent to either 2 or 3 modulo 5. Alternatively, 1. The number of partitions of $n$ such that with $k$ parts the smallest part is at least $k$ is the same as the number of partitions of $n$ such that each part is congruent to either 1 or 4 modulo 5. 2. The number of partitions of $n$ such that with $k$ parts the smallest part is at least $k+1$ is the same as the number of partitions of $n$ such that each part is congruent to either 2 or 3 modulo 5. Modular functions If q = e2πiτ, then q−1/60G(q) and q11/60H(q) are modular functions of τ. Applications The Rogers–Ramanujan identities appeared in Baxter's solution of the hard hexagon model in statistical mechanics. Ramanujan's continued fraction is $1+{\frac {q}{1+{\frac {q^{2}}{1+{\frac {q^{3}}{1+\cdots }}}}}}={\frac {G(q)}{H(q)}}.$ Relations to affine Lie algebras and vertex operator algebras James Lepowsky and Robert Lee Wilson were the first to prove Rogers–Ramanujan identities using completely representation-theoretic techniques. They proved these identities using level 3 modules for the affine Lie algebra ${\widehat {{\mathfrak {sl}}_{2}}}$. In the course of this proof they invented and used what they called $Z$-algebras. Lepowsky and Wilson's approach is universal, in that it is able to treat all affine Lie algebras at all levels. It can be used to find (and prove) new partition identities. First such example is that of Capparelli's identities discovered by Stefano Capparelli using level 3 modules for the affine Lie algebra $A_{2}^{(2)}$. See also • Rogers polynomials • Continuous q-Hermite polynomials References • Rogers, L. J.; Ramanujan, Srinivasa (1919), "Proof of certain identities in combinatory analysis.", Cambr. Phil. Soc. Proc., 19: 211–216, Reprinted as Paper 26 in Ramanujan's collected papers • Rogers, L. J. (1892), "On the expansion of some infinite products", Proc. London Math. Soc., 24 (1): 337–352, doi:10.1112/plms/s1-24.1.337, JFM 25.0432.01 • Rogers, L. J. (1893), "Second Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc., 25 (1): 318–343, doi:10.1112/plms/s1-25.1.318 • Rogers, L. J. (1894), "Third Memoir on the Expansion of certain Infinite Products", Proc. London Math. Soc., 26 (1): 15–32, doi:10.1112/plms/s1-26.1.15 • Schur, Issai (1917), "Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche", Sitzungsberichte der Berliner Akademie: 302–321 • W.N. Bailey, Generalized Hypergeometric Series, (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge. • George Gasper and Mizan Rahman, Basic Hypergeometric Series, 2nd Edition, (2004), Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. ISBN 0-521-83357-4. • Bruce C. Berndt, Heng Huat Chan, Sen-Shan Huang, Soon-Yi Kang, Jaebum Sohn, Seung Hwan Son, The Rogers-Ramanujan Continued Fraction, J. Comput. Appl. Math. 105 (1999), pp. 9–24. • Cilanne Boulet, Igor Pak, A Combinatorial Proof of the Rogers-Ramanujan and Schur Identities, Journal of Combinatorial Theory, Ser. A, vol. 113 (2006), 1019–1030. • Slater, L. J. (1952), "Further identities of the Rogers-Ramanujan type", Proceedings of the London Mathematical Society, Series 2, 54 (2): 147–167, doi:10.1112/plms/s2-54.2.147, ISSN 0024-6115, MR 0049225 • James Lepowsky and Robert L. Wilson, Construction of the affine Lie algebra $A_{1}^{(1)}$, Comm. Math. Phys. 62 (1978) 43-53. • James Lepowsky and Robert L. Wilson, A new family of algebras underlying the Rogers-Ramanujan identities, Proc. Natl. Acad. Sci. USA 78 (1981), 7254-7258. • James Lepowsky and Robert L. Wilson, The structure of standard modules, I: Universal algebras and the Rogers-Ramanujan identities, Invent. Math. 77 (1984), 199-290. • James Lepowsky and Robert L. Wilson, The structure of standard modules, II: The case $A_{1}^{(1)}$, principal gradation, Invent. Math. 79 (1985), 417-442. • Stefano Capparelli, Vertex operator relations for affine algebras and combinatorial identities, Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick. 1988. 107 pp. External links • Weisstein, Eric W. "Rogers-Ramanujan Identities". MathWorld. • Weisstein, Eric W. "Rogers-Ramanujan Continued Fraction". MathWorld.	Wikipedia
Albanese variety In mathematics, the Albanese variety $A(V)$, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve. Precise statement The Albanese variety is the abelian variety $A$ generated by a variety $V$ taking a given point of $V$ to the identity of $A$. In other words, there is a morphism from the variety $V$ to its Albanese variety $\operatorname {Alb} (V)$, such that any morphism from $V$ to an abelian variety (taking the given point to the identity) factors uniquely through $\operatorname {Alb} (V)$. For complex manifolds, André Blanchard (1956) defined the Albanese variety in a similar way, as a morphism from $V$ to a torus $\operatorname {Alb} (V)$ such that any morphism to a torus factors uniquely through this map. (It is an analytic variety in this case; it need not be algebraic.) Properties For compact Kähler manifolds the dimension of the Albanese variety is the Hodge number $h^{1,0}$, the dimension of the space of differentials of the first kind on $V$, which for surfaces is called the irregularity of a surface. In terms of differential forms, any holomorphic 1-form on $V$ is a pullback of translation-invariant 1-form on the Albanese variety, coming from the holomorphic cotangent space of $\operatorname {Alb} (V)$ at its identity element. Just as for the curve case, by choice of a base point on $V$ (from which to 'integrate'), an Albanese morphism $V\to \operatorname {Alb} (V)$ is defined, along which the 1-forms pull back. This morphism is unique up to a translation on the Albanese variety. For varieties over fields of positive characteristic, the dimension of the Albanese variety may be less than the Hodge numbers $h^{1,0}$ and $h^{0,1}$ (which need not be equal). To see the former note that the Albanese variety is dual to the Picard variety, whose tangent space at the identity is given by $H^{1}(X,O_{X}).$ That $\dim \operatorname {Alb} (X)\leq h^{1,0}$ is a result of Jun-ichi Igusa in the bibliography. Roitman's theorem If the ground field k is algebraically closed, the Albanese map $V\to \operatorname {Alb} (V)$ can be shown to factor over a group homomorphism (also called the Albanese map) $CH_{0}(V)\to \operatorname {Alb} (V)(k)$ from the Chow group of 0-dimensional cycles on V to the group of rational points of $\operatorname {Alb} (V)$, which is an abelian group since $\operatorname {Alb} (V)$ is an abelian variety. Roitman's theorem, introduced by A.A. Rojtman (1980), asserts that, for l prime to char(k), the Albanese map induces an isomorphism on the l-torsion subgroups.[1][2] The constraint on the primality of the order of torsion to the characteristic of the base field has been removed by Milne[3] shortly thereafter: the torsion subgroup of $\operatorname {CH} _{0}(X)$ and the torsion subgroup of k-valued points of the Albanese variety of X coincide. Replacing the Chow group by Suslin–Voevodsky algebraic singular homology after the introduction of Motivic cohomology Roitman's theorem has been obtained and reformulated in the motivic framework. For example, a similar result holds for non-singular quasi-projective varieties.[4] Further versions of Roitman's theorem are available for normal schemes.[5] Actually, the most general formulations of Roitman's theorem (i.e. homological, cohomological, and Borel–Moore) involve the motivic Albanese complex $\operatorname {LAlb} (V)$ and have been proven by Luca Barbieri-Viale and Bruno Kahn (see the references III.13). Connection to Picard variety The Albanese variety is dual to the Picard variety (the connected component of zero of the Picard scheme classifying invertible sheaves on V): $\operatorname {Alb} V=(\operatorname {Pic} _{0}V)^{\vee }.$ For algebraic curves, the Abel–Jacobi theorem implies that the Albanese and Picard varieties are isomorphic. See also • Intermediate Jacobian • Albanese scheme • Motivic Albanese Notes & references 1. Rojtman, A. A. (1980). "The torsion of the group of 0-cycles modulo rational equivalence". Annals of Mathematics. Second Series. 111 (3): 553–569. doi:10.2307/1971109. ISSN 0003-486X. JSTOR 1971109. MR 0577137. 2. Bloch, Spencer (1979). "Torsion algebraic cycles and a theorem of Roitman". Compositio Mathematica. 39 (1). MR 0539002. 3. Milne, J. S. (1982). "Zero cycles on algebraic varieties in nonzero characteristic : Rojtman's theorem". Compositio Mathematica. 47 (3): 271–287. 4. Spieß, Michael; Szamuely, Tamás (2003). "On the Albanese map for smooth quasi-projective varieties". Mathematische Annalen. 325: 1–17. arXiv:math/0009017. doi:10.1007/s00208-002-0359-8. S2CID 14014858. 5. Geisser, Thomas (2015). "Rojtman's theorem for normal schemes". Mathematical Research Letters. 22 (4): 1129–1144. arXiv:1402.1831. doi:10.4310/MRL.2015.v22.n4.a8. S2CID 59423465. • Barbieri-Viale, Luca; Kahn, Bruno (2016), On the derived category of 1-motives, Astérisque, vol. 381, SMF, arXiv:1009.1900, ISBN 978-2-85629-818-3, ISSN 0303-1179, MR 3545132 • Blanchard, André (1956), "Sur les variétés analytiques complexes", Annales Scientifiques de l'École Normale Supérieure, Série 3, 73 (2): 157–202, doi:10.24033/asens.1045, ISSN 0012-9593, MR 0087184 • Griffiths, Phillip; Harris, Joe (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. pp. 331, 552. ISBN 978-0-471-05059-9. • Igusa, Jun-ichi (1955). "A fundamental inequality in the theory of Picard varieties". Proceedings of the National Academy of Sciences of the United States of America. 41 (5): 317–20. Bibcode:1955PNAS...41..317I. doi:10.1073/pnas.41.5.317. PMC 528086. PMID 16589672. • Parshin, Aleksei N. (2001) [1994], "Albanese_variety", Encyclopedia of Mathematics, EMS Press	Wikipedia
Rokhlin lemma In mathematics, the Rokhlin lemma, or Kakutani–Rokhlin lemma is an important result in ergodic theory. It states that an aperiodic measure preserving dynamical system can be decomposed to an arbitrary high tower of measurable sets and a remainder of arbitrarily small measure. It was proven by Vladimir Abramovich Rokhlin and independently by Shizuo Kakutani. The lemma is used extensively in ergodic theory, for example in Ornstein theory and has many generalizations. Terminology Rokhlin lemma belongs to the group mathematical statements such as Zorn's lemma in set theory and Schwarz lemma in complex analysis which are traditionally called lemmas despite the fact that their roles in their respective fields are fundamental. Statement of the lemma Lemma: Let $T\colon X\to X$ be an invertible measure-preserving transformation on a standard measure space $\textstyle (X,\Sigma ,\mu )$ with $\textstyle \mu (X)=1$. We assume $\textstyle T$ is (measurably) aperiodic, that is, the set of periodic points for $\textstyle T$ has zero measure. Then for every integer $\textstyle n\in \mathbb {N} $ and for every $\textstyle \varepsilon >0$, there exists a measurable set $\textstyle E$ such that the sets $\textstyle E,TE,\ldots ,T^{n-1}E$ are pairwise disjoint and such that $\textstyle \mu (E\cup TE\cup \cdots \cup T^{n-1}E)>1-\varepsilon $. A useful strengthening of the lemma states that given a finite measurable partition $\textstyle P$, then $\textstyle E$ may be chosen in such a way that $\textstyle T^{i}E$ and $\textstyle P$ are independent for all $\textstyle 0\leq i<n$.[1] A topological version of the lemma Let $\textstyle (X,T)$ be a topological dynamical system consisting of a compact metric space $\textstyle X$ and a homeomorphism $\textstyle T:X\rightarrow X$. The topological dynamical system $\textstyle (X,T)$ is called minimal if it has no proper non-empty closed $\textstyle T$-invariant subsets. It is called (topologically) aperiodic if it has no periodic points ($T^{k}x=x$ for some $x\in X$ and $k\in \mathbb {Z} $ implies $k=0$). A topological dynamical system $\textstyle (Y,S)$ is called a factor of $\textstyle (X,T)$ if there exists a continuous surjective mapping $\textstyle \varphi :X\rightarrow Y$ which is equivariant, i.e., $\textstyle \varphi (Tx)=S\varphi (x)$ for all $\textstyle x\in X$. Elon Lindenstrauss proved the following theorem:[2] Theorem: Let $\textstyle (X,T)$ be a topological dynamical system which has an aperiodic minimal factor. Then for integer $\textstyle n\in \mathbb {N} $ there is a continuous function $\textstyle f\colon X\rightarrow \mathbb {R} $ such that the set $\textstyle E=\{x\in X\mid f(Tx)\neq f(x)+1\}$ satisfies $\textstyle E,TE,\ldots ,T^{n-1}E$ are pairwise disjoint. Gutman proved the following theorem:[3] Theorem: Let $(X,T)$ be a topological dynamical system which has an aperiodic factor with the small boundary property. Then for every $\varepsilon >0$, there exists a continuous function $f\colon X\rightarrow \mathbb {R} $ such that the set $\textstyle E=\{x\in X\mid f(Tx)\neq f(x)+1\}$ satisfies $\operatorname {ocap} (\textstyle E)<\varepsilon $, where $\operatorname {ocap} $ denotes orbit capacity. Further generalizations • There are versions for non-invertible measure preserving transformations.[4][5] • Donald Ornstein and Benjamin Weiss proved a version for free actions by countable discrete amenable groups.[6] • Carl Linderholm proved a version for periodic non-singular transformations.[7] References 1. Shields, Paul (1973). The theory of Bernoulli shifts (PDF). Chicago Lectures in Mathematics. Chicago, Illinois and London: The University of Chicago Press. pp. Chapter 3. 2. Lindenstrauss, Elon (1999-12-01). "Mean dimension, small entropy factors and an embedding theorem". Publications Mathématiques de l'IHÉS. 89 (1): 227–262. doi:10.1007/BF02698858. ISSN 0073-8301. S2CID 2413058. 3. Gutman, Yonatan. "Embedding ℤk-actions in cubical shifts and ℤk-symbolic extensions." Ergodic Theory and Dynamical Systems 31.2 (2011): 383-403. 4. Kornfeld, Isaac (2004). "Some old and new Rokhlin towers". Contemporary Mathematics. 356: 145–169. doi:10.1090/conm/356/06502. ISBN 9780821833131. 5. Avila, Artur; Candela, Pablo (2016). "Towers for commuting endomorphisms, and combinatorial applications". Annales de l'Institut Fourier. 66 (4): 1529–1544. doi:10.5802/aif.3042. 6. Ornstein, Donald S.; Weiss, Benjamin (1987-12-01). "Entropy and isomorphism theorems for actions of amenable groups". Journal d'Analyse Mathématique. 48 (1): 1–141. doi:10.1007/BF02790325. ISSN 0021-7670. S2CID 120653036. 7. Ionescu Tulcea, Alexandra (1965-01-01). "On the Category of Certain Classes of Transformations in Ergodic Theory". Transactions of the American Mathematical Society. 114 (1): 261–279. doi:10.2307/1994001. JSTOR 1994001. Notes • Vladimir Rokhlin. A "general" measure-preserving transformation is not mixing. Doklady Akademii Nauk SSSR (N.S.), 60:349–351, 1948. • Shizuo Kakutani. Induced measure preserving transformations. Proc. Imp. Acad. Tokyo, 19:635–641, 1943. • Benjamin Weiss. On the work of V. A. Rokhlin in ergodic theory. Ergodic Theory and Dynamical Systems, 9(4):619–627, 1989. • Isaac Kornfeld. Some old and new Rokhlin towers. Contemporary Mathematics, 356:145, 2004. See also Rokhlin's lemma should not be confused with Rokhlin's theorem.	Wikipedia
Roland Andrew Sweet Roland Andrew Sweet (born March 14, 1940 in St. Petersburg, Florida – April 15, 2019) was an American mathematician and computer scientist. He is known for his software contributions[1] exploiting computer vectorization on Cray super computers including CRAYFISHPAK,[2] multigrid solvers for elliptic problems, vectorized versions of the fast Fourier transforms, parallelized versions of the cyclic reduction algorithm, preconditioned conjugate gradient methods and numerous others. Research and career He was a son of Fred and Blanche (Aubin) Sweet. After graduating from St. Petersburg High School in 1958, he served for two years in the U.S. Navy. He studied at St. Petersburg Junior College and then obtained a BS in Mathematics from Florida State University in 1963. He obtained Ph. D. from the Computer Science department at Purdue University and he joined Computer Science Department at Cornell University in 1967 as an associate professor. He joined the University of Colorado's Mathematics Department as a tenure-track professor in 1970. At that time he was consulting at the National Center for Atmospheric Research. He received tenure in 1974 and continued to teach as an associate professor until 1980. He left to take a position at the National Bureau of Standards in Gaithersburg, Maryland. Two years later he transferred to a position at the National Bureau of Standards labs in Boulder and rejoined the Mathematics Department at University of Colorado in Denver as a full professor and Director of the Computational Mathematics Group. He retired from the university in 1996. In 1998 he moved to Seattle to work on a digital image compression project for a small start-up firm, LizardTech, and later to McKinney, Texas, where he worked on programming and software projects.[3] References 1. Sweet, Roland A. "A generalized cyclic reduction algorithm." SIAM Journal on Numerical Analysis 11, no. 3 (1974): 506–520. 2. Swarztrauber, P., Roland A. Sweet, and John C. Adams. "FISHPACK: Efficient FORTRAN subprograms for the solution of elliptic partial differential equations." UCAR Publication, July (1999). 3. 2019 Obituary	Wikipedia
Roland Bulirsch Roland Zdeněk Bulirsch (10 November 1932 – 21 September 2022) was a German mathematician specialising in numerical analysis. He studied and taught at the Technical University of Munich, and taught internationally as visiting professor. He was co-author of the reference book Introduction to Numerical Analysis, and president of the edition of the works by Johannes Kepler. He received honorary doctorates from international universities, and several awards. Roland Bulirsch Born(1932-11-10)10 November 1932 Liberec, Czechoslovakia Died21 September 2022(2022-09-21) (aged 89) Gauting, Germany EducationTechnische Hochschule München OccupationMathematician OrganizationTechnical University of Munich Known forBulirsch–Stoer algorithm Life and career Bulirsch was born in Liberec (Reichenberg) on 10 November 1932.[1] He had to leave Czechoslovakia in 1946. In 1947, he became an apprentice as a machinist with Siemens-Schuckert in Nuremberg, completing in 1951. He achieved the Abitur in Nördlingen in 1954,[1] and then studied mathematics and physics at the Technical University of Munich to 1959,[2] earning his Ph.D. there in 1961, supervised by Klaus Samelson, and his habilitation in mathematics in 1965.[1] He taught as associated professor at the University of California, San Diego, from 1967 to 1969, and as professor of applied mathematics at the University of Cologne from 1969. He joined the faculty in Munich in 1973.[2] He was elected a member of the Bavarian Academy of Sciences and Humanities in 1991. In 1998 he became president of the edition of the works by Johannes Kepler. He was emerited in 2001.[1] Bulirsch specialised in numerical analysis.[3] He is the author (with Josef Stoer) of Introduction to Numerical Analysis, a standard reference for the theory of numerical methods, and has also authored numerous other books and articles. The book From Nano to Space: Applied Mathematics Inspired by Roland Bulirsch is a tribute to his work.[3] The Bulirsch–Stoer algorithm is named after him and Stoer. Bulirsch received honorary doctorates from the University of Hamburg, the Technical University of Liberec, the National Technical University of Athens and the University of Viên Toán Hoc in Hanoi. He was awarded a medal from the Union of Czech Mathematicians and Physicists, and a medal from the Charles University in Prague. In 1998, he received the Bavarian Maximilian Order for Science and Art, and the following year the Liebig-Denkmünze, the highest award of the Heimatkreis Reichenberg. He was elected to the Sudetendeutsche Akademie der Wissenschaften und Künste the same year. In 2012, he received the Großer Sudetendeutscher Kulturpreis.[4] Personal life Bulirsch and his wife Waltraut had two daughters. She died in 2020.[5] Bulirsch died in Gauting on 21 September 2022 at age 89.[6] References 1. Bauer, Friedrich L.; Bode, Arndt (November 2007). "Zum 75. Geburtstag von Roland Bulirsch" (PDF). badw.de (in German). Retrieved 23 September 2022. 2. "Prof. Dr. Dr. h.c. Roland Bulirsch - Ordinarius" (in German). Technical University of Munich. 2022. Retrieved 23 September 2022. 3. Breitner, Michael; Denk, Georg; Rentrop, Peter (2008). "Roland Bulirsch - 75th Birthday". From Nano to Space: 1–2. doi:10.1007/978-3-540-74238-8_1. ISBN 978-3-540-74237-1. 4. "Großer Sudetendeutscher Kulturpreis 2012 an Prof. Roland Zdeněk Bulirsch" (in German). Reichenberg. 8 May 2012. Retrieved 27 September 2022. 5. "Waltraut Bulirsch" (death notice) (in German). Münchner Merkur. 17 February 2020. Retrieved 22 September 2022. 6. "Univ.-Prof. em. Dr. rer.nat. Dr. h.c. mult. Roland Zdeněk Bulirsch (Bulíř)". bulirsch.eu (in German). 21 September 2020. Retrieved 23 September 2022. External links • Official website Authority control International • FAST • ISNI • VIAF National • France • BnF data • Catalonia • Germany • Israel • United States • Czech Republic • Netherlands • Poland Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie • Trove Other • IdRef	Wikipedia
Roland Fraïssé Roland Fraïssé (French: [ʁɔlɑ̃ fʁajse]; 12 March 1920 – 30 March 2008[1]) was a French mathematical logician. Roland Fraïssé Born(1920-03-12)12 March 1920 Bressuire, France Died30 March 2008(2008-03-30) (aged 88) Marseille, France NationalityFrench Alma materUniversity of Paris Known forEhrenfeucht–Fraïssé games, Fraïssé limit Scientific career FieldsMathematics InstitutionsUniversity of Provence ThesisSur quelques classifications des systèmes de relations (1953) Doctoral advisorRené de Possel Life Fraïssé received his doctoral degree from the University of Paris in 1953. In his thesis,[2][3] Fraïssé used the back-and-forth method to determine whether two model-theoretic structures were elementarily equivalent. This method of determining elementary equivalence was later formulated as the Ehrenfeucht–Fraïssé game. Fraïssé worked primarily in relation theory. Another of his important works was the Fraïssé construction of a Fraïssé limit of finite structures. He also formulated Fraïssé's conjecture on order embeddings, and introduced the notion of compensor in the theory of posets.[4] Most of his career was spent as Professor at the University of Provence in Marseille, France. Selected publications • Sur quelques classifications des systèmes de relations, thesis, University of Paris, 1953; published in Publications Scientifiques de l'Université d'Alger, series A 1 (1954), 35–182. • Cours de logique mathématique, Paris: Gauthier-Villars Éditeur, 1967; second edition, 3 vols., 1971–1975; tr. into English and ed. by David Louvish as Course of Mathematical Logic, 2 vols., Dordrecht: Reidel, 1973–1974. • Theory of relations, tr. into English by P. Clote, Amsterdam: North-Holland, 1986; rev. ed. 2000. References 1. Rogics08 – Décès de Roland Fraïssé – Message de Maurice Pouzet et Gérard Lopez, retrieved 22 May 2008. 2. Sur une nouvelle classification des systèmes de relations, Roland Fraïssé, Comptes Rendus 230 (1950), 1022–1024. 3. Sur quelques classifications des systèmes de relations, Roland Fraïssé, thesis, Paris, 1953; published in Publications Scientifiques de l'Université d'Alger, series A 1 (1954), 35–182. 4. Petits posets : dénombrement, représentabilité par cercles et compenseurs, Roland Fraïssé and Nik Lygeros, Comptes Rendus de l'Académie des Sciences, Série I 313 (1991), no. 7, 417–420 Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • Belgium • United States • Czech Republic • Netherlands Academics • CiNii • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Roland Richardson Roland George Dwight Richardson (born May 14, 1878, Dartmouth, Nova Scotia; died July 17, 1949, Antigonish, Nova Scotia) was a prominent Canadian-American mathematician chiefly known for his work building the math department at Brown University and as Secretary of the American Mathematical Society.[1] Early life Richardson was the son of George J. Richardson (1828–1898), a teacher, and Rebecca Newcomb Richardson (1837–1923). The family lived in several different towns in Nova Scotia during Richardson's youth. After completing high school, Richardson taught school in the small village of Margaretsville, Nova Scotia. In 1896 Richardson entered Acadia University; after graduating in 1898, he returned to his teaching job in Margaretsville. From 1899 to 1902 he was the principal of the high school in tiny Westport, Nova Scotia. There he met his future wife Louise MacHattie, whom he married in 1908. Career in mathematics In 1902 Richardson entered Yale University, earning an AB in 1903 and a Masters in 1904. He became an instructor at Yale in the Math department and began research under Professor James Pierpont. In 1906 Richardson was awarded a PhD by Yale for his thesis on "Improper Multiple Integrals". In 1907 he was appointed assistant professor of mathematics at Brown University, with the stipulation that he first spend a study year in Gottingen, Germany. By 1915 Richardson had become a full professor and the head of the mathematics department at Brown. In 1926 he was also given the position of Dean of the Graduate School at Brown. Under Roland's leadership Brown's graduate program was recognized when Brown was elected to the elite Association of American Universities in 1933.[2] Richardson was the Secretary of the American Mathematical Society in 1921 and held the job until 1940. During his time, Raymond Clare Archibald wrote in his article on Richardson, "No American mathematician was more widely known among his colleagues and the careers of scores of them were notably promoted by his time-consuming activities in their behalf."[2] He was credited with helping many European mathematicians concerned about conditions in Europe move to America during the 1930s.[3] At the start of World War II Richardson organized accelerated applied mathematics courses at Brown for servicemen as the "Program of Advanced Instruction and Research in Applied Mechanics", recruiting German mathematician William Prager to lead it.[4] This led to the founding of a new "Quarterly of Applied Mathematics" edited at Brown in 1943. After the war the program was converted into a new graduate division of applied mathematics. From 1943 to 1946 he was a member of the applied mathematics panel of the National Defense Research Committee. Family and death Richardson died while on a fishing trip to his native Nova Scotia and was buried in Camp Hill Cemetery in Halifax. Richardson and his wife had one child, George Webdell Richardson (b. July 7, 1920).[5] Recognition Richardson received a number of honorary degrees. Acadia University awarded him a Doctor of Civil Law in 1931, Lehigh University gave him an LLD in 1941, and Brown University an LLD on his retirement in 1948. Richardson was elected a member of the American Academy of Arts and Sciences in 1914 and served as vice president 1945–9. References 1. Mitchell, Martha (1993). "Richardson, Roland G. D.". Encyclopedia Brunoniana. Providence, RI: Brown University Library. Retrieved 29 September 2022. 2. Archibald, Raymond Clare (1950). "R. G. D. Richardson, 1878-1949". Bulletin of the American Mathematical Society. 56 (3): 256–265. doi:10.1090/S0002-9904-1950-09376-8. Retrieved 29 September 2022. 3. Reingold, Nathan (1981). "Refugee mathematicians in the United States of America, 1933–1941: Reception and reaction". Annals of Science. 38 (3): 313–338. doi:10.1080/00033798100200251. 4. Abikoff, William; Sibner, Robert J. (2015). "Bers – From Graduate Student to Quasiconformal Mapper". In Keen, Linda; Kra, Irwin; Rodríguez, Rubí E. (eds.). Lipman Bers, a Life in Mathematics. American Mathematical Society. p. 57. ISBN 9781470420567. 5. O'Connor, J.J.; Robertson, E.F. (November 2019). "Roland George Dwight Richardson". MacTutor History of Mathematics archive. Retrieved 29 September 2022. Authority control International • FAST • VIAF National • Germany • United States Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie	Wikipedia
Roland Weitzenböck Roland Weitzenböck (26 May 1885 – 24 July 1955) was an Austrian mathematician working on differential geometry who introduced the Weitzenböck connection. He was appointed professor of mathematics at the University of Amsterdam in 1923 at the initiative of Brouwer, after Hermann Weyl had turned down Brouwer’s offer. Roland Weitzenböck Left to right: Diederik Korteweg, Roland Weitzenböck, Remmelt Sissingh, 1926 in Amsterdam Born(1885-05-26)26 May 1885 Kremsmünster, Austria-Hungary Died24 July 1955(1955-07-24) (aged 70) Zelhem, Netherlands NationalityAustrian Alma materUniversity of Vienna Known forWeitzenböck connection Scientific career FieldsMathematics InstitutionsUniversity of Amsterdam Doctoral advisorWilhelm Wirtinger Gustav Ritter von Escherich Doctoral studentsGeorge Griss Biography Roland Weitzenböck was born in Kremsmünster, Austria-Hungary. He studied from 1902 to 1904 at the Imperial and Royal Technical Military Academy (now HTL Vienna) and was a captain in the Austrian army. He then studied at the University of Vienna, where he graduated in 1910 with the dissertation Zum System von 3 Strahlenkomplexen im 4-dimensionalen Raum (The system of 3-rays complexes in 4-dimensional space). After further studies at Bonn and Göttingen, he became professor at the University of Graz in 1912. After Army service in World War I, he became Professor of Mathematics at the Karl-Ferdinand University in Prague in 1918. In 1923 Weitzenböck took a position of professor of mathematics at the University of Amsterdam, where he stayed until 1945. He settled in Blaricum, where he became a fully accepted member of the community. He was a man of few words, without observable political views. Appearances are often, however, deceptive, and in this case the solid imperturbable exterior hid a considerable amount of frustration resulting from the disastrous course of the First World War. As so many German and Austrian ex-service men, Weitzenböck became a hard-core revanchist, and an implacable enemy of France. But whereas Brouwer actively campaigned for the rehabilitation of German scientists, Weitzenböck refrained from political activity. However, after the ‘Anschluss’ of Austria in 1938, he started to vent his approval of Hitler’s policies in private conversations. Weitzenböck was elected member of the Royal Netherlands Academy of Arts and Sciences (KNAW) in May 1924, but suspended in May 1945 because of his attitude during the war. Weitzenböck had been a member of the National Socialist Movement in the Netherlands. In 1923 Weitzenböck published a modern monograph on the theory of invariants on manifolds that included tensor calculus. In the Preface of this monograph one can read an offensive acrostic. One finds that the first letter of the first word in the first 21 sentences spell out: NIEDER MIT DEN FRANZOSEN (down with the French). He also published papers on torsion. In fact, in his paper "Differential Invariants in Einstein’s Theory of Tele-parallelism" Weitzenböck[1] had given a supposedly complete bibliography of papers on torsion without mentioning Élie Cartan. Weitzenböck died in Zelhem, Netherlands in 1955. His doctoral students include G. F. C. Griss, Daniel Rutherford and Max Euwe. Publications • Komplex-Symbolik. Eine Einführung in die analytische Geometrie mehrdimensionaler Räume, Leipzig: Göschen, 1908[2] • Invariantentheorie, Groningen: Noordhoff, 1923 • Der vierdimensionale Raum, Die Wissenschaft, Sammlung von Einzeldarstellungen aus den Gebieten der Naturwissenschaft und der Technik... Bd. 80, Braunschweig: F. Vieweg & Sohn, 1929 • Neuere Arbeiten zur algebraischen Invariantentheorie. Differentialinvarianten. Enzyklopädie der mathematischen Wissenschaften, III, Bd.3, Teubner 1921 • Differentialinvarianten in der Einsteinschen Theorie des Fernparallelismus, Sitzungsberichte Preußische Akademie der Wissenschaften, Phys.-Math.Klasse, 1928, S.466 See also • Weitzenböck's identity • Weitzenböck's inequality Notes 1. Weitzenböck, R. (1921), Neuere Arbeiten der algebraischen Invariantentheorie. Differentialinvarianten. in: Encyclopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, vol. III.E.1., Leipzig: Teubner 2. Moore, C. L. E. (1911). "Review: Komplex-Symbolik. Eine Einführung in die analytische Geometrie mehrdimensionaler Räume. Von Roland Weitzenböck". Bull. Amer. Math. Soc. 17 (7): 368–369. doi:10.1090/S0002-9904-1911-02080-8. External links • Roland Weitzenböck at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Norway • Germany • Israel • United States • Czech Republic • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH People • Netherlands • Deutsche Biographie Other • IdRef	Wikipedia
Rolando Chuaqui Rolando Basim Chuaqui Kettlun (December 30, 1935–April 23, 1994[1]) was a Chilean mathematician who worked on the foundations of probabilities and foundations of mathematics. Throughout his lifetime, he published two books and over 50 journal articles in mathematics and logic.[1] He also spearheaded the creation and expansion of mathematics departments across multiple Chilean universities. Rolando Chuaqui Chuaqui in 1973 Born(1935-12-30)December 30, 1935 Santiago, Chile DiedApril 23, 1994(1994-04-23) (aged 58) Santiago, Chile Alma materUniversity of California, Berkeley Scientific career ThesisA Definition of Probability Based on Equal Likelihood (1965) Doctoral advisorDavid Blackwell Biography Chuaqui entered the University of Chile in 1953 to study medicine. He obtained a Ph.D. in Logic and the Methodology of Science, an interdisciplinary program between the Department of Mathematics and Department of Philosophy, from the University of California, Berkeley in 1965. His doctoral advisor was David Blackwell. Chuaqui returned to Chile after graduating, serving as a professor at the University of Chile and then the Pontifical Catholic University of Chile. During his time at the Pontifical Catholic, he advised three doctoral students.[2] Chuaqui held several visiting positions, including at UCLA (1967), Princeton University (1970), University of São Paulo (1971 and 1982), University of California, Berkeley (1973–74), University of Campinas (1976, 1977 and 1978), Stanford University (1984), and San José State University (1986-89). He was a long-term collaborator of Patrick Suppes, with whom he worked on non-standard analysis and measurement in sciences.[3] In 1986, he proposed a mathematical formulation for pragmatic truth.[4] Honors and awards He was awarded a Guggenheim Fellowship in Mathematics in 1983.[5] Since 1999, a series of annual research conferences in Chile, known as the Jornadas Rolando Chuaqui Kettlun, is held in his memory.[6][7] The Pontifical Catholic University of Chile also has a building named after him, which houses its Department of Mathematics.[8] References 1. "Rolando Chuaqui Kettlun". Proyecciones (Antofagasta). 13 (1). 1994. doi:10.22199/S07160917.1994.0001.00001. Retrieved 3 March 2022. 2. "Rolando Chuaqui - The Mathematics Genealogy Project". genealogy.math.ndsu.nodak.edu. Retrieved 3 March 2022. 3. Life and Work of Rolando Chuaqui Kettlun (in Spanish), Jornados Rolando Chuaqui Kettlun, retrieved 2017-03-25. 4. Mikenberg, Irene; da Costa, Newton C. A.; Chuaqui, Rolando (March 1986). "Pragmatic truth and approximation to truth". Journal of Symbolic Logic. 51 (1): 201–221. doi:10.2307/2273956. 5. "Rolando Chuaqui Kettlun". John Simon Guggenheim Foundation. Retrieved 2 March 2022. 6. Jornadas Rolando Chuaqui Kettlun: filosofía y ciencias, retrieved 2017-03-25. 7. Cordero, Alberto. "Philosophy of Science in Latin America". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy (Winter 2016 ed.). 8. "Facultad de Matemáticas:: Edificio Rolando Chuaqui". www.mat.uc.cl. Retrieved 2 March 2022. Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • United States • Netherlands • Poland Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Rolph Ludwig Edward Schwarzenberger Rolph Ludwig Edward Schwarzenberger (7 February 1936 – 29 February 1992) was a British mathematician at the University of Warwick who worked on vector bundles (where he introduced jumping lines), crystallography, and mathematics education. He was President of the Mathematical Association in 1983–1984. Publications • Hirzebruch, Friedrich (1995) [1956], Schwarzenberger, R. L. E. (ed.), Topological methods in algebraic geometry, Classics in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-58663-0, MR 1335917 Schwarzenberger translated this book into English and added a long appendix on later developments. • Schwarzenberger, R. L. E. (1972), Topics in differential topology, Publications of the Ramanujan Institute, vol. 3, Ramanujan Institute, University of Madras, Madras, MR 0405443 • Schwarzenberger, Rolf L. E. (1980), n-dimensional crystallography, Research Notes in Mathematics, vol. 41, Boston, Mass.: Pitman (Advanced Publishing Program), ISBN 978-0-8224-8468-4, MR 0586945 • Schwarzenberger, R. L. E. (1984), "The importance of mistakes: the 1984 presidential address", The Mathematical Gazette, 68 (445): 159–172, doi:10.2307/3616337, ISSN 0025-5572, JSTOR 3616337, MR 0769804 References • Brown, Margaret (1993), "Rolph Schwarzenberger (1936–1992): An Appreciation", The Mathematical Gazette, 77 (478): 95–97, doi:10.1017/S0025557200152386, ISSN 0025-5572, JSTOR 3619271 • "Obituary. Professor Rolph Schwarzenberger", International Journal of Mathematical Education in Science and Technology, 23 (2): 337, 1992, doi:10.1080/0020739920230217, ISSN 0020-739X, MR 1162426 • Rolph Ludwig Edward Schwarzenberger at the Mathematics Genealogy Project Authority control International • VIAF National • Germany Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie	Wikipedia
Euler angles The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.[1] They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general basis in 3-dimensional linear algebra. Classic Euler angles usually take the inclination angle in such a way that zero degrees represent the vertical orientation. Alternative forms were later introduced by Peter Guthrie Tait and George H. Bryan intended for use in aeronautics and engineering in which zero degrees represent the horizontal position. Chained rotations equivalence Any target orientation can be reached, starting from a known reference orientation, using a specific sequence of intrinsic rotations, whose magnitudes are the Euler angles of the target orientation. This example uses the z-x′-z″ sequence. Euler angles can be defined by elemental geometry or by composition of rotations. The geometrical definition demonstrates that three composed elemental rotations (rotations about the axes of a coordinate system) are always sufficient to reach any target frame. The three elemental rotations may be extrinsic (rotations about the axes xyz of the original coordinate system, which is assumed to remain motionless), or intrinsic (rotations about the axes of the rotating coordinate system XYZ, solidary with the moving body, which changes its orientation with respect to the extrinsic frame after each elemental rotation). In the sections below, an axis designation with a prime mark superscript (e.g., z″) denotes the new axis after an elemental rotation. Euler angles are typically denoted as α, β, γ, or ψ, θ, φ. Different authors may use different sets of rotation axes to define Euler angles, or different names for the same angles. Therefore, any discussion employing Euler angles should always be preceded by their definition. Without considering the possibility of using two different conventions for the definition of the rotation axes (intrinsic or extrinsic), there exist twelve possible sequences of rotation axes, divided in two groups: • Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) • Tait–Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z). Tait–Bryan angles are also called Cardan angles; nautical angles; heading, elevation, and bank; or yaw, pitch, and roll. Sometimes, both kinds of sequences are called "Euler angles". In that case, the sequences of the first group are called proper or classic Euler angles. Classic Euler angles Left: A gimbal set, showing a z-x-z rotation sequence. External frame shown in the base. Internal axes in red color. Right: A simple diagram showing similar Euler angles. Geometrical definition The axes of the original frame are denoted as x, y, z and the axes of the rotated frame as X, Y, Z. The geometrical definition (sometimes referred to as static) begins by defining the line of nodes (N) as the intersection of the planes xy and XY (it can also be defined as the common perpendicular to the axes z and Z and then written as the vector product N = z × Z). Using it, the three Euler angles can be defined as follows: • $\alpha $ (or $\varphi $) is the signed angle between the x axis and the N axis (x-convention – it could also be defined between y and N, called y-convention). • $\beta $ (or $\theta $) is the angle between the z axis and the Z axis. • $\gamma $ (or $\psi $) is the signed angle between the N axis and the X axis (x-convention). Euler angles between two reference frames are defined only if both frames have the same handedness. Conventions by intrinsic rotations Intrinsic rotations are elemental rotations that occur about the axes of a coordinate system XYZ attached to a moving body. Therefore, they change their orientation after each elemental rotation. The XYZ system rotates, while xyz is fixed. Starting with XYZ overlapping xyz, a composition of three intrinsic rotations can be used to reach any target orientation for XYZ. Euler angles can be defined by intrinsic rotations. The rotated frame XYZ may be imagined to be initially aligned with xyz, before undergoing the three elemental rotations represented by Euler angles. Its successive orientations may be denoted as follows: • x-y-z or x0-y0-z0 (initial) • x′-y′-z′ or x1-y1-z1 (after first rotation) • x″-y″-z″ or x2-y2-z2 (after second rotation) • X-Y-Z or x3-y3-z3 (final) For the above-listed sequence of rotations, the line of nodes N can be simply defined as the orientation of X after the first elemental rotation. Hence, N can be simply denoted x′. Moreover, since the third elemental rotation occurs about Z, it does not change the orientation of Z. Hence Z coincides with z″. This allows us to simplify the definition of the Euler angles as follows: • α (or φ) represents a rotation around the z axis, • β (or θ) represents a rotation around the x′ axis, • γ (or ψ) represents a rotation around the z″ axis. Conventions by extrinsic rotations Extrinsic rotations are elemental rotations that occur about the axes of the fixed coordinate system xyz. The XYZ system rotates, while xyz is fixed. Starting with XYZ overlapping xyz, a composition of three extrinsic rotations can be used to reach any target orientation for XYZ. The Euler or Tait–Bryan angles (α, β, γ) are the amplitudes of these elemental rotations. For instance, the target orientation can be reached as follows (note the reversed order of Euler angle application): • The XYZ system rotates about the z axis by γ. The X axis is now at angle γ with respect to the x axis. • The XYZ system rotates again, but this time about the x axis by β. The Z axis is now at angle β with respect to the z axis. • The XYZ system rotates a third time, about the z axis again, by angle α. In sum, the three elemental rotations occur about z, x and z. Indeed, this sequence is often denoted z-x-z (or 3-1-3). Sets of rotation axes associated with both proper Euler angles and Tait–Bryan angles are commonly named using this notation (see above for details). If each step of the rotation acts on the rotating coordinate system XYZ, the rotation is intrinsic (Z-X'-Z''). Intrinsic rotation can also be denoted 3-1-3. Signs, ranges and conventions Angles are commonly defined according to the right-hand rule. Namely, they have positive values when they represent a rotation that appears clockwise when looking in the positive direction of the axis, and negative values when the rotation appears counter-clockwise. The opposite convention (left hand rule) is less frequently adopted. About the ranges (using interval notation): • for α and γ, the range is defined modulo 2π radians. For instance, a valid range could be [−π, π]. • for β, the range covers π radians (but can't be said to be modulo π). For example, it could be [0, π] or [−π/2, π/2]. The angles α, β and γ are uniquely determined except for the singular case that the xy and the XY planes are identical, i.e. when the z axis and the Z axis have the same or opposite directions. Indeed, if the z axis and the Z axis are the same, β = 0 and only (α + γ) is uniquely defined (not the individual values), and, similarly, if the z axis and the Z axis are opposite, β = π and only (α − γ) is uniquely defined (not the individual values). These ambiguities are known as gimbal lock in applications. There are six possibilities of choosing the rotation axes for proper Euler angles. In all of them, the first and third rotation axes are the same. The six possible sequences are: 1. z1-x′-z2″ (intrinsic rotations) or z2-x-z1 (extrinsic rotations) 2. x1-y′-x2″ (intrinsic rotations) or x2-y-x1 (extrinsic rotations) 3. y1-z′-y2″ (intrinsic rotations) or y2-z-y1 (extrinsic rotations) 4. z1-y′-z2″ (intrinsic rotations) or z2-y-z1 (extrinsic rotations) 5. x1-z′-x2″ (intrinsic rotations) or x2-z-x1 (extrinsic rotations) 6. y1-x′-y2″ (intrinsic rotations) or y2-x-y1 (extrinsic rotations) Precession, nutation and intrinsic rotation Precession, nutation, and intrinsic rotation (spin) are defined as the movements obtained by changing one of the Euler angles while leaving the other two constant. These motions are not expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where the first angle moves the line of nodes around the external axis z, the second rotates around the line of nodes N and the third one is an intrinsic rotation around Z, an axis fixed in the body that moves. The static definition implies that: • α (precession) represents a rotation around the z axis, • β (nutation) represents a rotation around the N or x′ axis, • γ (intrinsic rotation) represents a rotation around the Z or z″ axis. If β is zero, there is no rotation about N. As a consequence, Z coincides with z, α and γ represent rotations about the same axis (z), and the final orientation can be obtained with a single rotation about z, by an angle equal to α + γ. As an example, consider a top. The top spins around its own axis of symmetry; this corresponds to its intrinsic rotation. It also rotates around its pivotal axis, with its center of mass orbiting the pivotal axis; this rotation is a precession. Finally, the top can wobble up and down; the inclination angle is the nutation angle. The same example can be seen with the movements of the earth. Though all three movements can be represented by a rotation operator with constant coefficients in some frame, they cannot be represented by these operators all at the same time. Given a reference frame, at most one of them will be coefficient-free. Only precession can be expressed in general as a matrix in the basis of the space without dependencies of the other angles. These movements also behave as a gimbal set. If we suppose a set of frames, able to move each with respect to the former according to just one angle, like a gimbal, there will exist an external fixed frame, one final frame and two frames in the middle, which are called "intermediate frames". The two in the middle work as two gimbal rings that allow the last frame to reach any orientation in space. Tait–Bryan angles The second type of formalism is called Tait–Bryan angles, after Peter Guthrie Tait and George H. Bryan. It is the convention normally used for aerospace applications, so that zero degrees elevation represents the horizontal attitude. Tait–Bryan angles represent the orientation of the aircraft with respect to the world frame. When dealing with other vehicles, different axes conventions are possible. Definitions The definitions and notations used for Tait–Bryan angles are similar to those described above for proper Euler angles (geometrical definition, intrinsic rotation definition, extrinsic rotation definition). The only difference is that Tait–Bryan angles represent rotations about three distinct axes (e.g. x-y-z, or x-y′-z″), while proper Euler angles use the same axis for both the first and third elemental rotations (e.g., z-x-z, or z-x′-z″). This implies a different definition for the line of nodes in the geometrical construction. In the proper Euler angles case it was defined as the intersection between two homologous Cartesian planes (parallel when Euler angles are zero; e.g. xy and XY). In the Tait–Bryan angles case, it is defined as the intersection of two non-homologous planes (perpendicular when Euler angles are zero; e.g. xy and YZ). Conventions The three elemental rotations may occur either about the axes of the original coordinate system, which remains motionless (extrinsic rotations), or about the axes of the rotating coordinate system, which changes its orientation after each elemental rotation (intrinsic rotations). There are six possibilities of choosing the rotation axes for Tait–Bryan angles. The six possible sequences are: • x-y′-z″ (intrinsic rotations) or z-y-x (extrinsic rotations) • y-z′-x″ (intrinsic rotations) or x-z-y (extrinsic rotations) • z-x′-y″ (intrinsic rotations) or y-x-z (extrinsic rotations) • x-z′-y″ (intrinsic rotations) or y-z-x (extrinsic rotations) • z-y′-x″ (intrinsic rotations) or x-y-z (extrinsic rotations): the intrinsic rotations are known as: yaw, pitch and roll • y-x′-z″ (intrinsic rotations) or z-x-y (extrinsic rotations) Signs and ranges Tait–Bryan convention is widely used in engineering with different purposes. There are several axes conventions in practice for choosing the mobile and fixed axes, and these conventions determine the signs of the angles. Therefore, signs must be studied in each case carefully. The range for the angles ψ and φ covers 2π radians. For θ the range covers π radians. Alternative names These angles are normally taken as one in the external reference frame (heading, bearing), one in the intrinsic moving frame (bank) and one in a middle frame, representing an elevation or inclination with respect to the horizontal plane, which is equivalent to the line of nodes for this purpose. For an aircraft, they can be obtained with three rotations around its principal axes if done in the proper order. A yaw will obtain the bearing, a pitch will yield the elevation and a roll gives the bank angle. Therefore, in aerospace they are sometimes called yaw, pitch, and roll. Notice that this will not work if the rotations are applied in any other order or if the airplane axes start in any position non-equivalent to the reference frame. Tait–Bryan angles, following z-y′-x″ (intrinsic rotations) convention, are also known as nautical angles, because they can be used to describe the orientation of a ship or aircraft, or Cardan angles, after the Italian mathematician and physicist Gerolamo Cardano, who first described in detail the Cardan suspension and the Cardan joint. Angles of a given frame A common problem is to find the Euler angles of a given frame. The fastest way to get them is to write the three given vectors as columns of a matrix and compare it with the expression of the theoretical matrix (see later table of matrices). Hence the three Euler Angles can be calculated. Nevertheless, the same result can be reached avoiding matrix algebra and using only elemental geometry. Here we present the results for the two most commonly used conventions: ZXZ for proper Euler angles and ZYX for Tait–Bryan. Notice that any other convention can be obtained just changing the name of the axes. Proper Euler angles Assuming a frame with unit vectors (X, Y, Z) given by their coordinates as in the main diagram, it can be seen that: $\cos(\beta )=Z_{3}.$ And, since $\sin ^{2}x=1-\cos ^{2}x,$ for $0<x<\pi $ we have $\sin(\beta )={\sqrt {1-Z_{3}^{2}}}.$ As $Z_{2}$ is the double projection of a unitary vector, $\cos(\alpha )\cdot \sin(\beta )=-Z_{2},$ $\cos(\alpha )=-Z_{2}/{\sqrt {1-Z_{3}^{2}}}.$ There is a similar construction for $Y_{3}$, projecting it first over the plane defined by the axis z and the line of nodes. As the angle between the planes is $\pi /2-\beta $ and $\cos(\pi /2-\beta )=\sin(\beta )$, this leads to: $\sin(\beta )\cdot \cos(\gamma )=Y_{3},$ $\cos(\gamma )=Y_{3}/{\sqrt {1-Z_{3}^{2}}},$ and finally, using the inverse cosine function, $\alpha =\arccos \left(-Z_{2}/{\sqrt {1-Z_{3}^{2}}}\right),$ $\beta =\arccos \left(Z_{3}\right),$ $\gamma =\arccos \left(Y_{3}/{\sqrt {1-Z_{3}^{2}}}\right).$ Tait–Bryan angles Assuming a frame with unit vectors (X, Y, Z) given by their coordinates as in this new diagram (notice that the angle theta is negative), it can be seen that: $\sin(\theta )=-X_{3}$ As before, $\cos ^{2}x=1-\sin ^{2}x,$ for $-\pi /2<x<\pi /2$ we have $\cos(\theta )={\sqrt {1-X_{3}^{2}}}.$ in a way analogous to the former one: $\sin(\psi )=X_{2}/{\sqrt {1-X_{3}^{2}}}.$ $\sin(\phi )=Y_{3}/{\sqrt {1-X_{3}^{2}}}.$ Looking for similar expressions to the former ones: $\psi =\arcsin \left(X_{2}/{\sqrt {1-X_{3}^{2}}}\right),$ $\theta =\arcsin(-X_{3}),$ $\phi =\arcsin \left(Y_{3}/{\sqrt {1-X_{3}^{2}}}\right).$ Last remarks Note that the inverse sine and cosine functions yield two possible values for the argument. In this geometrical description, only one of the solutions is valid. When Euler angles are defined as a sequence of rotations, all the solutions can be valid, but there will be only one inside the angle ranges. This is because the sequence of rotations to reach the target frame is not unique if the ranges are not previously defined.[2] For computational purposes, it may be useful to represent the angles using atan2(y, x). For example, in the case of proper Euler angles: $\alpha =\operatorname {atan2} (Z_{1},-Z_{2}),$ $\gamma =\operatorname {atan2} (X_{3},Y_{3}).$ Conversion to other orientation representations Main article: Rotation formalisms in three dimensions § Conversion formulae between formalisms Euler angles are one way to represent orientations. There are others, and it is possible to change to and from other conventions. Three parameters are always required to describe orientations in a 3-dimensional Euclidean space. They can be given in several ways, Euler angles being one of them; see charts on SO(3) for others. The most used orientation representation are the rotation matrices, the axis-angle and the quaternions, also known as Euler–Rodrigues parameters, which provide another mechanism for representing 3D rotations. This is equivalent to the special unitary group description. Expressing rotations in 3D as unit quaternions instead of matrices has some advantages: • Concatenating rotations is computationally faster and numerically more stable. • Extracting the angle and axis of rotation is simpler. • Interpolation is more straightforward. See for example slerp. • Quaternions do not suffer from gimbal lock as Euler angles do. Regardless, the rotation matrix calculation is the first step for obtaining the other two representations. Rotation matrix Any orientation can be achieved by composing three elemental rotations, starting from a known standard orientation. Equivalently, any rotation matrix R can be decomposed as a product of three elemental rotation matrices. For instance: $R=X(\alpha )Y(\beta )Z(\gamma )$ is a rotation matrix that may be used to represent a composition of extrinsic rotations about axes z, y, x, (in that order), or a composition of intrinsic rotations about axes x-y′-z″ (in that order). However, both the definition of the elemental rotation matrices X, Y, Z, and their multiplication order depend on the choices taken by the user about the definition of both rotation matrices and Euler angles (see, for instance, Ambiguities in the definition of rotation matrices). Unfortunately, different sets of conventions are adopted by users in different contexts. The following table was built according to this set of conventions: 1. Each matrix is meant to operate by pre-multiplying column vectors $ {\begin{bmatrix}x\\y\\z\end{bmatrix}}$ (see Ambiguities in the definition of rotation matrices) 2. Each matrix is meant to represent an active rotation (the composing and composed matrices are supposed to act on the coordinates of vectors defined in the initial fixed reference frame and give as a result the coordinates of a rotated vector defined in the same reference frame). 3. Each matrix is meant to represent, primarily, a composition of intrinsic rotations (around the axes of the rotating reference frame) and, secondarily, the composition of three extrinsic rotations (which corresponds to the constructive evaluation of the R matrix by the multiplication of three truly elemental matrices, in reverse order). 4. Right handed reference frames are adopted, and the right hand rule is used to determine the sign of the angles α, β, γ. For the sake of simplicity, the following table of matrix products uses the following nomenclature: 1. 1, 2, 3 represent the angles α, β and γ, i.e. the angles corresponding to the first, second and third elemental rotations respectively. 2. X, Y, Z are the matrices representing the elemental rotations about the axes x, y, z of the fixed frame (e.g., X1 represents a rotation about x by an angle α). 3. s and c represent sine and cosine (e.g., s1 represents the sine of α). Proper Euler anglesTait–Bryan angles $X_{1}Z_{2}X_{3}={\begin{bmatrix}c_{2}&-c_{3}s_{2}&s_{2}s_{3}\\c_{1}s_{2}&c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}\\s_{1}s_{2}&c_{1}s_{3}+c_{2}c_{3}s_{1}&c_{1}c_{3}-c_{2}s_{1}s_{3}\end{bmatrix}}$ $X_{1}Z_{2}Y_{3}={\begin{bmatrix}c_{2}c_{3}&-s_{2}&c_{2}s_{3}\\s_{1}s_{3}+c_{1}c_{3}s_{2}&c_{1}c_{2}&c_{1}s_{2}s_{3}-c_{3}s_{1}\\c_{3}s_{1}s_{2}-c_{1}s_{3}&c_{2}s_{1}&c_{1}c_{3}+s_{1}s_{2}s_{3}\end{bmatrix}}$ $X_{1}Y_{2}X_{3}={\begin{bmatrix}c_{2}&s_{2}s_{3}&c_{3}s_{2}\\s_{1}s_{2}&c_{1}c_{3}-c_{2}s_{1}s_{3}&-c_{1}s_{3}-c_{2}c_{3}s_{1}\\-c_{1}s_{2}&c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{1}c_{2}c_{3}-s_{1}s_{3}\end{bmatrix}}$ $X_{1}Y_{2}Z_{3}={\begin{bmatrix}c_{2}c_{3}&-c_{2}s_{3}&s_{2}\\c_{1}s_{3}+c_{3}s_{1}s_{2}&c_{1}c_{3}-s_{1}s_{2}s_{3}&-c_{2}s_{1}\\s_{1}s_{3}-c_{1}c_{3}s_{2}&c_{3}s_{1}+c_{1}s_{2}s_{3}&c_{1}c_{2}\end{bmatrix}}$ $Y_{1}X_{2}Y_{3}={\begin{bmatrix}c_{1}c_{3}-c_{2}s_{1}s_{3}&s_{1}s_{2}&c_{1}s_{3}+c_{2}c_{3}s_{1}\\s_{2}s_{3}&c_{2}&-c_{3}s_{2}\\-c_{3}s_{1}-c_{1}c_{2}s_{3}&c_{1}s_{2}&c_{1}c_{2}c_{3}-s_{1}s_{3}\end{bmatrix}}$ $Y_{1}X_{2}Z_{3}={\begin{bmatrix}c_{1}c_{3}+s_{1}s_{2}s_{3}&c_{3}s_{1}s_{2}-c_{1}s_{3}&c_{2}s_{1}\\c_{2}s_{3}&c_{2}c_{3}&-s_{2}\\c_{1}s_{2}s_{3}-c_{3}s_{1}&c_{1}c_{3}s_{2}+s_{1}s_{3}&c_{1}c_{2}\end{bmatrix}}$ $Y_{1}Z_{2}Y_{3}={\begin{bmatrix}c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{1}s_{2}&c_{3}s_{1}+c_{1}c_{2}s_{3}\\c_{3}s_{2}&c_{2}&s_{2}s_{3}\\-c_{1}s_{3}-c_{2}c_{3}s_{1}&s_{1}s_{2}&c_{1}c_{3}-c_{2}s_{1}s_{3}\end{bmatrix}}$ $Y_{1}Z_{2}X_{3}={\begin{bmatrix}c_{1}c_{2}&s_{1}s_{3}-c_{1}c_{3}s_{2}&c_{3}s_{1}+c_{1}s_{2}s_{3}\\s_{2}&c_{2}c_{3}&-c_{2}s_{3}\\-c_{2}s_{1}&c_{1}s_{3}+c_{3}s_{1}s_{2}&c_{1}c_{3}-s_{1}s_{2}s_{3}\end{bmatrix}}$ $Z_{1}Y_{2}Z_{3}={\begin{bmatrix}c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}&c_{1}s_{2}\\c_{1}s_{3}+c_{2}c_{3}s_{1}&c_{1}c_{3}-c_{2}s_{1}s_{3}&s_{1}s_{2}\\-c_{3}s_{2}&s_{2}s_{3}&c_{2}\end{bmatrix}}$ $Z_{1}Y_{2}X_{3}={\begin{bmatrix}c_{1}c_{2}&c_{1}s_{2}s_{3}-c_{3}s_{1}&s_{1}s_{3}+c_{1}c_{3}s_{2}\\c_{2}s_{1}&c_{1}c_{3}+s_{1}s_{2}s_{3}&c_{3}s_{1}s_{2}-c_{1}s_{3}\\-s_{2}&c_{2}s_{3}&c_{2}c_{3}\end{bmatrix}}$ $Z_{1}X_{2}Z_{3}={\begin{bmatrix}c_{1}c_{3}-c_{2}s_{1}s_{3}&-c_{1}s_{3}-c_{2}c_{3}s_{1}&s_{1}s_{2}\\c_{3}s_{1}+c_{1}c_{2}s_{3}&c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{1}s_{2}\\s_{2}s_{3}&c_{3}s_{2}&c_{2}\end{bmatrix}}$ $Z_{1}X_{2}Y_{3}={\begin{bmatrix}c_{1}c_{3}-s_{1}s_{2}s_{3}&-c_{2}s_{1}&c_{1}s_{3}+c_{3}s_{1}s_{2}\\c_{3}s_{1}+c_{1}s_{2}s_{3}&c_{1}c_{2}&s_{1}s_{3}-c_{1}c_{3}s_{2}\\-c_{2}s_{3}&s_{2}&c_{2}c_{3}\end{bmatrix}}$ These tabular results are available in numerous textbooks.[3] For each column the last row constitutes the most commonly used convention. To change the formulas for passive rotations (or find reverse active rotation), transpose the matrices (then each matrix transforms the initial coordinates of a vector remaining fixed to the coordinates of the same vector measured in the rotated reference system; same rotation axis, same angles, but now the coordinate system rotates, rather than the vector). The following table contains formulas for angles α, β and γ from elements of a rotation matrix $R$.[4] Proper Euler angles Tait–Bryan angles $X_{1}Z_{2}X_{3}$ ${\begin{aligned}\alpha &=\arctan \left({\frac {R_{31}}{R_{21}}}\right)\\\beta &=\arccos \left(R_{11}\right)\\\gamma &=\arctan \left({\frac {R_{13}}{-R_{12}}}\right)\end{aligned}}$ $X_{1}Z_{2}Y_{3}$ ${\begin{aligned}\alpha &=\arctan \left({\frac {R_{32}}{R_{22}}}\right)\\\beta &=\arcsin \left(-R_{12}\right)\\\gamma &=\arctan \left({\frac {R_{13}}{R_{11}}}\right)\end{aligned}}$ $X_{1}Y_{2}X_{3}$ ${\begin{aligned}\alpha &=\arctan \left({\frac {R_{21}}{-R_{31}}}\right)\\\beta &=\arccos \left(R_{11}\right)\\\gamma &=\arctan \left({\frac {R_{12}}{R_{13}}}\right)\end{aligned}}$ $X_{1}Y_{2}Z_{3}$ ${\begin{aligned}\alpha &=\arctan \left({\frac {-R_{23}}{R_{33}}}\right)\\\beta &=\arcsin \left(R_{13}\right)\\\gamma &=\arctan \left({\frac {-R_{12}}{R_{11}}}\right)\end{aligned}}$ $Y_{1}X_{2}Y_{3}$ ${\begin{aligned}\alpha &=\arctan \left({\frac {R_{12}}{R_{32}}}\right)\\\beta &=\arccos \left(R_{22}\right)\\\gamma &=\arctan \left({\frac {R_{21}}{-R_{23}}}\right)\end{aligned}}$ $Y_{1}X_{2}Z_{3}$ ${\begin{aligned}\alpha &=\arctan \left({\frac {R_{13}}{R_{33}}}\right)\\\beta &=\arcsin \left(-R_{23}\right)\\\gamma &=\arctan \left({\frac {R_{21}}{R_{22}}}\right)\end{aligned}}$ $Y_{1}Z_{2}Y_{3}$ ${\begin{aligned}\alpha &=\arctan \left({\frac {R_{32}}{-R_{12}}}\right)\\\beta &=\arccos \left(R_{22}\right)\\\gamma &=\arctan \left({\frac {R_{23}}{R_{21}}}\right)\end{aligned}}$ $Y_{1}Z_{2}X_{3}$ ${\begin{aligned}\alpha &=\arctan \left({\frac {-R_{31}}{R_{11}}}\right)\\\beta &=\arcsin \left(R_{21}\right)\\\gamma &=\arctan \left({\frac {-R_{23}}{R_{22}}}\right)\end{aligned}}$ $Z_{1}Y_{2}Z_{3}$ ${\begin{aligned}\alpha &=\arctan \left({\frac {R_{23}}{R_{13}}}\right)\\\beta &=\arctan \left({\frac {\sqrt {1-R_{33}^{2}}}{R_{33}}}\right)\\\gamma &=\arctan \left({\frac {R_{32}}{-R_{31}}}\right)\end{aligned}}$ $Z_{1}Y_{2}X_{3}$ ${\begin{aligned}\alpha &=\arctan \left({\frac {R_{21}}{R_{11}}}\right)\\\beta &=\arcsin \left(-R_{31}\right)\\\gamma &=\arctan \left({\frac {R_{32}}{R_{33}}}\right)\end{aligned}}$ $Z_{1}X_{2}Z_{3}$ ${\begin{aligned}\alpha &=\arctan \left({\frac {R_{13}}{-R_{23}}}\right)\\\beta &=\arccos \left(R_{33}\right)\\\gamma &=\arctan \left({\frac {R_{31}}{R_{32}}}\right)\end{aligned}}$ $Z_{1}X_{2}Y_{3}$ ${\begin{aligned}\alpha &=\arctan \left({\frac {-R_{12}}{R_{22}}}\right)\\\beta &=\arcsin \left(R_{32}\right)\\\gamma &=\arctan \left({\frac {-R_{31}}{R_{33}}}\right)\end{aligned}}$ Properties See also: Charts on SO(3) and Quaternions and spatial rotation The Euler angles form a chart on all of SO(3), the special orthogonal group of rotations in 3D space. The chart is smooth except for a polar coordinate style singularity along β = 0. See charts on SO(3) for a more complete treatment. The space of rotations is called in general "The Hypersphere of rotations", though this is a misnomer: the group Spin(3) is isometric to the hypersphere S3, but the rotation space SO(3) is instead isometric to the real projective space RP3 which is a 2-fold quotient space of the hypersphere. This 2-to-1 ambiguity is the mathematical origin of spin in physics. A similar three angle decomposition applies to SU(2), the special unitary group of rotations in complex 2D space, with the difference that β ranges from 0 to 2π. These are also called Euler angles. The Haar measure for SO(3) in Euler angles is given by the Hopf angle parametrisation of SO(3), ${\textrm {d}}V\propto \sin \beta \cdot {\textrm {d}}\alpha \cdot {\textrm {d}}\beta \cdot {\textrm {d}}\gamma $,[5] where $(\beta ,\alpha )$ parametrise $S^{2}$, the space of rotation axes. For example, to generate uniformly randomized orientations, let α and γ be uniform from 0 to 2π, let z be uniform from −1 to 1, and let β = arccos(z). Geometric algebra Other properties of Euler angles and rotations in general can be found from the geometric algebra, a higher level abstraction, in which the quaternions are an even subalgebra. The principal tool in geometric algebra is the rotor $\mathbf {R} =[\cos(\theta /2)-Iu\sin(\theta /2)]$ where $\theta =$angle of rotation, $\mathbf {u} $ is the rotation axis (unitary vector) and $\mathbf {I} $ is the pseudoscalar (trivector in $\mathbb {R} ^{3}$) Higher dimensions It is possible to define parameters analogous to the Euler angles in dimensions higher than three.[6] In four dimensions and above, the concept of "rotation about an axis" loses meaning and instead becomes "rotation in a plane." The number of Euler angles needed to represent the group SO(n) is n(n − 1)/2, equal to the number of planes containing two distinct coordinate axes in n-dimensional Euclidean space. In SO(4) a rotation matrix is defined by two unit quaternions, and therefore has six degrees of freedom, three from each quaternion. Applications Vehicles and moving frames Their main advantage over other orientation descriptions is that they are directly measurable from a gimbal mounted in a vehicle. As gyroscopes keep their rotation axis constant, angles measured in a gyro frame are equivalent to angles measured in the lab frame. Therefore, gyros are used to know the actual orientation of moving spacecraft, and Euler angles are directly measurable. Intrinsic rotation angle cannot be read from a single gimbal, so there has to be more than one gimbal in a spacecraft. Normally there are at least three for redundancy. There is also a relation to the well-known gimbal lock problem of mechanical engineering.[7] When studying rigid bodies in general, one calls the xyz system space coordinates, and the XYZ system body coordinates. The space coordinates are treated as unmoving, while the body coordinates are considered embedded in the moving body. Calculations involving acceleration, angular acceleration, angular velocity, angular momentum, and kinetic energy are often easiest in body coordinates, because then the moment of inertia tensor does not change in time. If one also diagonalizes the rigid body's moment of inertia tensor (with nine components, six of which are independent), then one has a set of coordinates (called the principal axes) in which the moment of inertia tensor has only three components. The angular velocity of a rigid body takes a simple form using Euler angles in the moving frame. Also the Euler's rigid body equations are simpler because the inertia tensor is constant in that frame. Crystallographic texture In materials science, crystallographic texture (or preferred orientation) can be described using Euler angles. In texture analysis, the Euler angles provide a mathematical depiction of the orientation of individual crystallites within a polycrystalline material, allowing for the quantitative description of the macroscopic material.[9] The most common definition of the angles is due to Bunge and corresponds to the ZXZ convention. It is important to note, however, that the application generally involves axis transformations of tensor quantities, i.e. passive rotations. Thus the matrix that corresponds to the Bunge Euler angles is the transpose of that shown in the table above.[10] Others Euler angles, normally in the Tait–Bryan convention, are also used in robotics for speaking about the degrees of freedom of a wrist. They are also used in electronic stability control in a similar way. Gun fire control systems require corrections to gun-order angles (bearing and elevation) to compensate for deck tilt (pitch and roll). In traditional systems, a stabilizing gyroscope with a vertical spin axis corrects for deck tilt, and stabilizes the optical sights and radar antenna. However, gun barrels point in a direction different from the line of sight to the target, to anticipate target movement and fall of the projectile due to gravity, among other factors. Gun mounts roll and pitch with the deck plane, but also require stabilization. Gun orders include angles computed from the vertical gyro data, and those computations involve Euler angles. Euler angles are also used extensively in the quantum mechanics of angular momentum. In quantum mechanics, explicit descriptions of the representations of SO(3) are very important for calculations, and almost all the work has been done using Euler angles. In the early history of quantum mechanics, when physicists and chemists had a sharply negative reaction towards abstract group theoretic methods (called the Gruppenpest), reliance on Euler angles was also essential for basic theoretical work. Many mobile computing devices contain accelerometers which can determine these devices' Euler angles with respect to the earth's gravitational attraction. These are used in applications such as games, bubble level simulations, and kaleidoscopes. See also • 3D projection • Axis-angle representation • Conversion between quaternions and Euler angles • Davenport chained rotations • Euler's rotation theorem • Gimbal lock • Quaternion • Quaternions and spatial rotation • Rotation formalisms in three dimensions • Spherical coordinate system References 1. Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478) PDF 2. Gregory G. Slabaugh, Computing Euler angles from a rotation matrix 3. E.g. Appendix I (p. 483) of: Roithmayr, Carlos M.; Hodges, Dewey H. (2016), Dynamics: Theory and Application of Kane's Method (1st ed.), Cambridge University Press, ISBN 978-1107005693 4. Henderson, D. M. (1977-06-09). "Euler angles, quaternions, and transformation matrices for space shuttle analysis": 12–24. {{cite journal}}: Cite journal requires \|journal= (help) 5. Yershova, A.; Jain, S.; Lavalle, S. M.; Mitchell, J. C. (2010). "Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration". The International Journal of Robotics Research. 29 (7). Section 8 – Derivation of Hopf parametrisation. doi:10.1177/0278364909352700. PMC 2896220. PMID 20607113. 6. (in Italian) A generalization of Euler Angles to n-dimensional real spaces 7. The relation between the Euler angles and the Cardan suspension is explained in chap. 11.7 of the following textbook: U. Krey, A. Owen, Basic Theoretical Physics – A Concise Overview, New York, London, Berlin, Heidelberg, Springer (2007) . 8. Liss KD, Bartels A, Schreyer A, Clemens H (2003). "High energy X-rays: A tool for advanced bulk investigations in materials science and physics". Textures Microstruct. 35 (3/4): 219–52. doi:10.1080/07303300310001634952. 9. Kocks, U.F.; Tomé, C.N.; Wenk, H.-R. (2000), Texture and Anisotropy: Preferred Orientations in Polycrystals and their effect on Materials Properties, Cambridge, ISBN 978-0-521-79420-6 10. Bunge, H. (1993), Texture Analysis in Materials Science: Mathematical Methods, Cuvillier Verlag, ASIN B0014XV9HU Bibliography • Biedenharn, L. C.; Louck, J. D. (1981), Angular Momentum in Quantum Physics, Reading, MA: Addison–Wesley, ISBN 978-0-201-13507-7 • Goldstein, Herbert (1980), Classical Mechanics (2nd ed.), Reading, MA: Addison–Wesley, ISBN 978-0-201-02918-5 • Gray, Andrew (1918), A Treatise on Gyrostatics and Rotational Motion, London: Macmillan (published 2007), ISBN 978-1-4212-5592-7 • Rose, M. E. (1957), Elementary Theory of Angular Momentum, New York, NY: John Wiley & Sons (published 1995), ISBN 978-0-486-68480-2 • Symon, Keith (1971), Mechanics, Reading, MA: Addison-Wesley, ISBN 978-0-201-07392-8 • Landau, L.D.; Lifshitz, E. M. (1996), Mechanics (3rd ed.), Oxford: Butterworth-Heinemann, ISBN 978-0-7506-2896-9 External links Wikimedia Commons has media related to Euler angles. • "Euler angles", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Weisstein, Eric W. "Euler Angles". MathWorld. • David Eberly. Euler Angle Formulas, Geometric Tools • An interactive tutorial on Euler angles available at https://www.mecademic.com/en/how-is-orientation-in-space-represented-with-euler-angles • EulerAngles – an iOS app for visualizing in 3D the three rotations associated with Euler angles • Orientation Library – "orilib", a collection of routines for rotation / orientation manipulation, including special tools for crystal orientations • Online tool to convert rotation matrices available at rotation converter (numerical conversion) • Online tool to convert symbolic rotation matrices (dead, but still available from the Wayback Machine) symbolic rotation converter • Rotation, Reflection, and Frame Change: Orthogonal tensors in computational engineering mechanics, IOP Publishing • Euler Angles, Quaternions, and Transformation Matrices for Space Shuttle Analysis, NASA Authority control: National • Germany	Wikipedia
Rolle's theorem In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero. The theorem is named after Michel Rolle. Part of a series of articles about Calculus • Fundamental theorem • Limits • Continuity • Rolle's theorem • Mean value theorem • Inverse function theorem Differential Definitions • Derivative (generalizations) • Differential • infinitesimal • of a function • total Concepts • Differentiation notation • Second derivative • Implicit differentiation • Logarithmic differentiation • Related rates • Taylor's theorem Rules and identities • Sum • Product • Chain • Power • Quotient • L'Hôpital's rule • Inverse • General Leibniz • Faà di Bruno's formula • Reynolds Integral • Lists of integrals • Integral transform • Leibniz integral rule Definitions • Antiderivative • Integral (improper) • Riemann integral • Lebesgue integration • Contour integration • Integral of inverse functions Integration by • Parts • Discs • Cylindrical shells • Substitution (trigonometric, tangent half-angle, Euler) • Euler's formula • Partial fractions • Changing order • Reduction formulae • Differentiating under the integral sign • Risch algorithm Series • Geometric (arithmetico-geometric) • Harmonic • Alternating • Power • Binomial • Taylor Convergence tests • Summand limit (term test) • Ratio • Root • Integral • Direct comparison • Limit comparison • Alternating series • Cauchy condensation • Dirichlet • Abel Vector • Gradient • Divergence • Curl • Laplacian • Directional derivative • Identities Theorems • Gradient • Green's • Stokes' • Divergence • generalized Stokes Multivariable Formalisms • Matrix • Tensor • Exterior • Geometric Definitions • Partial derivative • Multiple integral • Line integral • Surface integral • Volume integral • Jacobian • Hessian Advanced • Calculus on Euclidean space • Generalized functions • Limit of distributions Specialized • Fractional • Malliavin • Stochastic • Variations Miscellaneous • Precalculus • History • Glossary • List of topics • Integration Bee • Mathematical analysis • Nonstandard analysis Standard version of the theorem If a real-valued function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f (a) = f (b), then there exists at least one c in the open interval (a, b) such that $f'(c)=0.$ This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem. History Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem.[1] The name "Rolle's theorem" was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846.[2] Examples First example For a radius r > 0, consider the function $f(x)={\sqrt {r^{2}-x^{2}}},\quad x\in [-r,r].$ Its graph is the upper semicircle centered at the origin. This function is continuous on the closed interval [−r, r] and differentiable in the open interval (−r, r), but not differentiable at the endpoints −r and r. Since f (−r) = f (r), Rolle's theorem applies, and indeed, there is a point where the derivative of f is zero. The theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval. Second example If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold. Consider the absolute value function $f(x)=\|x\|,\qquad x\in [-1,1].$ Then f (−1) = f (1), but there is no c between −1 and 1 for which the f ′(c) is zero. This is because that function, although continuous, is not differentiable at x = 0. The derivative of f changes its sign at x = 0, but without attaining the value 0. The theorem cannot be applied to this function because it does not satisfy the condition that the function must be differentiable for every x in the open interval. However, when the differentiability requirement is dropped from Rolle's theorem, f will still have a critical number in the open interval (a, b), but it may not yield a horizontal tangent (as in the case of the absolute value represented in the graph). Generalization The second example illustrates the following generalization of Rolle's theorem: Consider a real-valued, continuous function f on a closed interval [a, b] with f (a) = f (b). If for every x in the open interval (a, b) the right-hand limit $f'(x^{+}):=\lim _{h\to 0^{+}}{\frac {f(x+h)-f(x)}{h}}$ and the left-hand limit $f'(x^{-}):=\lim _{h\to 0^{-}}{\frac {f(x+h)-f(x)}{h}}$ exist in the extended real line [−∞, ∞], then there is some number c in the open interval (a, b) such that one of the two limits $f'(c^{+})\quad {\text{and}}\quad f'(c^{-})$ is ≥ 0 and the other one is ≤ 0 (in the extended real line). If the right- and left-hand limits agree for every x, then they agree in particular for c, hence the derivative of f exists at c and is equal to zero. Remarks • If f is convex or concave, then the right- and left-hand derivatives exist at every inner point, hence the above limits exist and are real numbers. • This generalized version of the theorem is sufficient to prove convexity when the one-sided derivatives are monotonically increasing:[3] $f'(x^{-})\leq f'(x^{+})\leq f'(y^{-}),\qquad x<y.$ Proof of the generalized version Since the proof for the standard version of Rolle's theorem and the generalization are very similar, we prove the generalization. The idea of the proof is to argue that if f (a) = f (b), then f must attain either a maximum or a minimum somewhere between a and b, say at c, and the function must change from increasing to decreasing (or the other way around) at c. In particular, if the derivative exists, it must be zero at c. By assumption, f is continuous on [a, b], and by the extreme value theorem attains both its maximum and its minimum in [a, b]. If these are both attained at the endpoints of [a, b], then f is constant on [a, b] and so the derivative of f is zero at every point in (a, b). Suppose then that the maximum is obtained at an interior point c of (a, b) (the argument for the minimum is very similar, just consider −f ). We shall examine the above right- and left-hand limits separately. For a real h such that c + h is in [a, b], the value f (c + h) is smaller or equal to f (c) because f attains its maximum at c. Therefore, for every h > 0, ${\frac {f(c+h)-f(c)}{h}}\leq 0,$ hence $f'(c^{+}):=\lim _{h\to 0^{+}}{\frac {f(c+h)-f(c)}{h}}\leq 0,$ where the limit exists by assumption, it may be minus infinity. Similarly, for every h < 0, the inequality turns around because the denominator is now negative and we get ${\frac {f(c+h)-f(c)}{h}}\geq 0,$ hence $f'(c^{-}):=\lim _{h\to 0^{-}}{\frac {f(c+h)-f(c)}{h}}\geq 0,$ where the limit might be plus infinity. Finally, when the above right- and left-hand limits agree (in particular when f is differentiable), then the derivative of f at c must be zero. (Alternatively, we can apply Fermat's stationary point theorem directly.) Generalization to higher derivatives We can also generalize Rolle's theorem by requiring that f has more points with equal values and greater regularity. Specifically, suppose that • the function f is n − 1 times continuously differentiable on the closed interval [a, b] and the nth derivative exists on the open interval (a, b), and • there are n intervals given by a1 < b1 ≤ a2 < b2 ≤ ⋯ ≤ an < bn in [a, b] such that f (ak) = f (bk) for every k from 1 to n. Then there is a number c in (a, b) such that the nth derivative of f at c is zero. The requirements concerning the nth derivative of f can be weakened as in the generalization above, giving the corresponding (possibly weaker) assertions for the right- and left-hand limits defined above with f (n − 1) in place of f. Particularly, this version of the theorem asserts that if a function differentiable enough times has n roots (so they have the same value, that is 0), then there is an internal point where f (n − 1) vanishes. Proof The proof uses mathematical induction. The case n = 1 is simply the standard version of Rolle's theorem. For n > 1, take as the induction hypothesis that the generalization is true for n − 1. We want to prove it for n. Assume the function f satisfies the hypotheses of the theorem. By the standard version of Rolle's theorem, for every integer k from 1 to n, there exists a ck in the open interval (ak, bk) such that f ′(ck) = 0. Hence, the first derivative satisfies the assumptions on the n − 1 closed intervals [c1, c2], …, [cn − 1, cn]. By the induction hypothesis, there is a c such that the (n − 1)st derivative of f ′ at c is zero. Generalizations to other fields Rolle's theorem is a property of differentiable functions over the real numbers, which are an ordered field. As such, it does not generalize to other fields, but the following corollary does: if a real polynomial factors (has all of its roots) over the real numbers, then its derivative does as well. One may call this property of a field Rolle's property. More general fields do not always have differentiable functions, but they do always have polynomials, which can be symbolically differentiated. Similarly, more general fields may not have an order, but one has a notion of a root of a polynomial lying in a field. Thus Rolle's theorem shows that the real numbers have Rolle's property. Any algebraically closed field such as the complex numbers has Rolle's property. However, the rational numbers do not – for example, x3 − x = x(x − 1)(x + 1) factors over the rationals, but its derivative, $3x^{2}-1=3\left(x-{\tfrac {1}{\sqrt {3}}}\right)\left(x+{\tfrac {1}{\sqrt {3}}}\right),$ does not. The question of which fields satisfy Rolle's property was raised in (Kaplansky 1972).[4] For finite fields, the answer is that only F2 and F4 have Rolle's property.[5][6] For a complex version, see Voorhoeve index. See also • Mean value theorem • Intermediate value theorem • Linear interpolation • Gauss–Lucas theorem References 1. Besenyei, A. (September 17, 2012). "A brief history of the mean value theorem" (PDF). 2. See Cajori, Florian (1999). A History of Mathematics. p. 224. ISBN 9780821821022. 3. Artin, Emil (1964) [1931], The Gamma Function, translated by Butler, Michael, Holt, Rinehart and Winston, pp. 3–4 4. Kaplansky, Irving (1972), Fields and Rings 5. Craven, Thomas; Csordas, George (1977), "Multiplier sequences for fields", Illinois J. Math., 21 (4): 801–817, doi:10.1215/ijm/1256048929 6. Ballantine, C.; Roberts, J. (January 2002), "A Simple Proof of Rolle's Theorem for Finite Fields", The American Mathematical Monthly, Mathematical Association of America, 109 (1): 72–74, doi:10.2307/2695770, JSTOR 2695770 Further reading • Leithold, Louis (1972). The Calculus, with Analytic Geometry (2nd ed.). New York: Harper & Row. pp. 201–207. ISBN 0-06-043959-9. • Taylor, Angus E. (1955). Advanced Calculus. Boston: Ginn and Company. pp. 30–37. External links • "Rolle theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Rolle's and Mean Value Theorems at cut-the-knot. • Mizar system proof: http://mizar.org/version/current/html/rolle.html#T2 Wikimedia Commons has media related to Rolle's theorem.	Wikipedia
Rom Varshamov Rom Rubenovich Varshamov (Russian Ром Рубенович Варшамов; Born April 9, 1927, in Tbilisi; Died August 24, 1999, in Moscow) was a Soviet Armenian mathematician who worked in Coding theory, especially on error-correcting codes and Number theory. Rom Rubenovich Varshamov Born9 April 1927 (1927-04-09) Tbilisi, Georgian Socialist Soviet Republic Died24 August 1999 (1999-08-25) (aged 72) Moscow Alma materTbilisi State University Known forCoding Theory Scientific career FieldsMathematician Doctoral advisorArnold Walfisz Varshamov studied in Tbilisi with Arnold Walfisz (where he was Georgian students’ champion in the 100 metres), as well as in Tomsk. After that, he was a researcher in Moscow at the Steklov Institute of Mathematics with Ivan Matveyevich Vinogradov, especially on Number theory and Coding theory, and the Ministry of Radio Engineering (working in Cryptography). In 1957 he proved the Gilbert-Varshamov bound for linear codes (independently of Edgar Gilbert who proved the non-linear part). From 1968 he worked in Yerevan and was director of the Computer Centre (now Institute for Informatics and Automation Problems[1]) of the Academy of Sciences of the Armenian SSR. He was author and co-author of more than 25 scientific articles[2] and also a member of the Armenian National Academy of Sciences.[3] Selected bibliography • Varshamov, R. R.: Estimate of the number of signals in error correcting codes (Russian), Dokl. Akad. Nauk SSSR 117, 739–741, 1957 (English Translation in I. F. Blake: Algebraic Coding Theory: History and Development, Dowden, Hutchinson & Ross, 1973, pp 68–71) • Varshamov, R. R.: A class of codes for symmetric channels and a problem from the additive theory of numbers, IEEE Trans. Inf. Theory 19, 92–95, 1973 • Varshamov, R. R.: On a method in the theory of reducibility of polynomials over a finite field, Sov. Math., Dokl. 44, No.1, 194–199, 1992; translation from Dokl. Akad. Nauk SSSR 319, No.4, 787-791, 1991 References 1. Institute for Informatics and Automation Problems of NAS RA. "Web page". 2. Zentralblatt MATH. "List of Rom Varshamov's publications". 3. Armenian National Academy of Sciences. "Page of Rom Varshamov". External links • Varshamov on mathnet.ru • Article in Golos Armenii, April 10, 2007, electronic version No. 142 of 26.12.09 Authority control International • VIAF Academics • DBLP • zbMATH	Wikipedia
Roman abacus The Ancient Romans developed the Roman hand abacus, a portable, but less capable, base-10 version of earlier abacuses like those that were used by the Greeks and Babylonians.[1] Origin The Roman abacus was the first portable calculating device for engineers, merchants, and presumably tax collectors. It greatly reduced the time needed to perform the basic operations of arithmetic using Roman numerals. Karl Menninger said: For more extensive and complicated calculations, such as those involved in Roman land surveys, there was, in addition to the hand abacus, a true reckoning board with unattached counters or pebbles. The Etruscan cameo and the Greek predecessors, such as the Salamis Tablet and the Darius Vase, give us a good idea of what it must have been like, although no actual specimens of the true Roman counting board are known to be extant. But language, the most reliable and conservative guardian of a past culture, has come to our rescue once more. Above all, it has preserved the fact of the unattached counters so faithfully that we can discern this more clearly than if we possessed an actual counting board. What the Greeks called psephoi, the Romans called calculi. The Latin word calx means 'pebble' or 'gravel stone'; calculi are thus little stones (used as counters).[2] Both the Roman abacus and the Chinese suanpan have been used since ancient times. With one bead above and four below the bar, the systematic configuration of the Roman abacus is comparable to the modern Japanese soroban, although the soroban was historically derived from the suanpan. Layout The Late Roman hand abacus shown here as a reconstruction contains seven longer and seven shorter grooves used for whole number counting, the former having up to four beads in each, and the latter having just one. The rightmost two grooves were for fractional counting. The abacus was made of a metal plate where the beads ran in slots. The size was such that it could fit in a modern shirt pocket. \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|X\| CCC\|ƆƆƆ CC\|ƆƆ C\|Ɔ C X I Ө \| \| --- --- --- --- --- --- --- --- S \|O\| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| Ɔ \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \| \| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| \|O\| 2 \|O\| \|O\| \|O\| The lower groove marked I indicates units, X tens, and so on up to millions. The beads in the upper shorter grooves denote fives (five units, five tens, etc.), resembling a bi-quinary coded decimal place value system. Computations are made by means of beads which, we believe, would have been slid up and down the grooves to indicate the value of each column. The upper slots contained a single bead while the lower slots contained four beads, the only exceptions being the two rightmost columns, column 2 marked Ө and column 1 with three symbols down the side of a single slot or beside three separate slots with Ɛ, 3 or S or a symbol like the £ sign but without the horizontal bar beside the top slot, a backwards C beside the middle slot and a 2 symbol beside the bottom slot, depending on the example abacus and the source which could be Friedlein,[3] Menninger[2] or Ifrah.[4] These latter two slots are for mixed-base math, a development unique to the Roman hand abacus[5] described in following sections. The longer slot with five beads below the Ө position allowed for the counting of 1/12 of a whole unit called an uncia (from which the English words inch and ounce are derived), making the abacus useful for Roman measures and Roman currency. The first column was either a single slot with 4 beads or 3 slots with one, one and two beads respectively top to bottom. In either case, three symbols were included beside the single slot version or one symbol per slot for the three slot version. Many measures were aggregated by twelfths. Thus the Roman pound ('libra'), consisted of 12 ounces (unciae) (1 uncia = 28 grams). A measure of volume, congius, consisted of 12 heminae (1 hemina = 0.273 litres). The Roman foot (pes), was 12 inches (unciae) (1 uncia = 2.43 cm). The actus, the standard furrow length when plowing, was 120 pedes. There were however other measures in common use - for example the sextarius was two heminae. The as, the principal copper coin in Roman currency, was also divided into 12 unciae. Again, the abacus was ideally suited for counting currency. Symbols and usage The first column was arranged either as a single slot with three different symbols or as three separate slots with one, one and two beads or counters respectively and a distinct symbol for each slot. It is most likely that the rightmost slot or slots were used to enumerate fractions of an uncia and these were, from top to bottom, 1/2 s, 1/4 s and 1/12 s of an uncia. The upper character in this slot (or the top slot where the rightmost column is three separate slots) is the character most closely resembling that used to denote a semuncia or 1/24. The name semuncia denotes 1/2 of an uncia or 1/24 of the base unit, the As. Likewise, the next character is that used to indicate a sicilicus or 1/48 of an As, which is 1/4 of an uncia. These two characters are to be found in the table of Roman fractions on page 75 of Graham Flegg's[6] book. Finally, the last or lower character is most similar but not identical to the character in Flegg's table to denote 1/144 of an As, the dimidio sextula, which is the same as 1/12 of an uncia. This is however even more strongly supported by Gottfried Friedlein[3] in the table at the end of the book which summarizes the use of a very extensive set of alternative formats for different values including that of fractions. In the entry in this table numbered 14 referring back to (Zu) 48, he lists different symbols for the semuncia (1/24), the sicilicus (1/48), the sextula (1/72), the dimidia sextula (1/144), and the scriptulum (1/288). Of prime importance, he specifically notes the formats of the semuncia, sicilicus and sextula as used on the Roman bronze abacus, "auf dem chernan abacus". The semuncia is the symbol resembling a capital "S", but he also includes the symbol that resembles a numeral three with horizontal line at the top, the whole rotated 180 degrees. It is these two symbols that appear on samples of abacus in different museums. The symbol for the sicilicus is that found on the abacus and resembles a large right single quotation mark spanning the entire line height. The most important symbol is that for the sextula, which resembles very closely a cursive digit 2. Now, as stated by Friedlein, this symbol indicates the value of 1/72 of an As. However, he stated specifically in the penultimate sentence of section 32 on page 23, the two beads in the bottom slot each have a value of 1/72. This would allow this slot to represent only 1/72 (i.e. 1/6 × 1/12 with one bead) or 1/36 (i.e. 2/6 × 1/12 = 1/3 × 1/12 with two beads) of an uncia respectively. This contradicts all existing documents that state this lower slot was used to count thirds of an uncia (i.e. 1/3 and 2/3 × 1/12 of an As. This results in two opposing interpretations of this slot, that of Friedlein and that of many other experts such as Ifrah,[4] and Menninger[2] who propose the one and two thirds usage. There is however a third possibility. If this symbol refers to the total value of the slot (i.e. 1/72 of an as), then each of the two counters can only have a value of half this or 1/144 of an as or 1/12 of an uncia. This then suggests that these two counters did in fact count twelfths of an uncia and not thirds of an uncia. Likewise, for the top and upper middle, the symbols for the semuncia and sicilicus could also indicate the value of the slot itself and since there is only one bead in each, would be the value of the bead also. This would allow the symbols for all three of these slots to represent the slot value without involving any contradictions. A further argument which suggests the lower slot represents twelfths rather than thirds of an uncia is best described by the figure above. The diagram above assumes for ease that one is using fractions of an uncia as a unit value equal to one. If the beads in the lower slot of column I represent thirds, then the beads in the three slots for fractions of 1/12 of an uncia cannot show all values from 1/12 of an uncia to 11/12 of an uncia. In particular, it would not be possible to represent 1/12, 2/12 and 5/12. Furthermore, this arrangement would allow for seemingly unnecessary values of 13/12, 14/12 and 17/12. Even more significant, it is logically impossible for there to be a rational progression of arrangements of the beads in step with unit increasing values of twelfths. Likewise, if each of the beads in the lower slot is assumed to have a value of 1/6 of an uncia, there is again an irregular series of values available to the user, no possible value of 1/12 and an extraneous value of 13/12. It is only by employing a value of 1/12 for each of the beads in the lower slot that all values of twelfths from 1/12 to 11/12 can be represented and in a logical ternary, binary, binary progression for the slots from bottom to top. This can be best appreciated by reference to the figure below. Alternative usages of the beads in the lower slot It can be argued that the beads in this first column could have been used as originally believed and widely stated, i.e. as ½, ¼ and ⅓ and ⅔, completely independently of each other. However this is more difficult to support in the case where this first column is a single slot with the three inscribed symbols. To complete the known possibilities, in one example found by this author, the first and second columns were transposed. It would not be unremarkable if the makers of these instruments produced output with minor differences, since the vast number of variations in modern calculators provide a compelling example. What can be deduced from these Roman abacuses, is the undeniable proof that Romans were using a device that exhibited a decimal, place-value system, and the inferred knowledge of a zero value as represented by a column with no beads in a counted position. Furthermore, the biquinary-like nature of the integer portion allowed for direct transcription from and to the written Roman numerals. No matter what the true usage was, what cannot be denied by the very format of the abacus is that if not yet proven, these instruments provide very strong arguments in favour of far greater facility with practical mathematics known and practised by the Romans in this authors view. The reconstruction of a Roman hand abacus in the Cabinet,[7] supports this. The replica Roman hand abacus at,[8] shown alone here,[9] plus the description of an Roman abacus on page 23 of [10] provides further evidence of such devices. References 1. Keith F. Sugden (1981) A HISTORY OF THE ABACUS. Accounting Historians Journal: Fall 1981, Vol. 8, No. 2, pp. 1-22. 2. Menninger, Karl (1992). Number Words and Number Symbols: A Cultural History of Numbers, German to English translation, M.I.T., 1969, Dover Publications, p 315. 3. Friedlein, Gottfried, Die Zahlzeichen und das elementare rechnen der Griechen und Römer und des Christlichen Abendlandes vom 7. bis 13. Jahrhundert (Erlangen, 1869) 4. Ifrah, Georges, "The Universal History of Numbers" ISBN 1-86046-324-X 5. Stephenson, Steve. "The Roman Hand-Abacus". Retrieved 2007-07-04. 6. Flegg, Graham, "Numbers, Their History and Meaning" ISBN 0-14-022564-1 7. des Médailles, Bibliothèque nationale 8. Abacus-Online-Museum of Jörn Lütjens 9. Replica Roman Hand Abacus 10. Die Zahlzeichen und das elementare Rechnen der Griechen und Römer und des christlichen Further reading • Stephenson, Stephen K. (July 7, 2010), Ancient Computers, IEEE Global History Network, retrieved 2011-07-02 • Stephenson, Stephen K. (2011), Ancient Computers, Part I - Rediscovery, Amazon.com, ASIN B004RH3J7S	Wikipedia
Romanian Master of Mathematics and Sciences The Romanian Master of Mathematics and Sciences (formerly known as the Romanian Masters in Mathematics) is an annual competition for students at the pre-university level, held in Bucharest, Romania. The contestants compete individually, in four different sections: mathematics, physics, chemistry and computer science. The participating teams (national and local teams) can have up to four students for each section (plus two coaches: a leader and a deputy leader). The contest follows the same structure as IMO and IPhO and is usually held at the end of February. Romanian Master of Mathematics and Sciences Founded2008 RegionWorld Current champions United States – mathematics WebsiteRMMS website History The first Romanian Master in Mathematics was held in 2008 and has been initiated by Prof. Severius Moldoveanu and Prof. Radu Gologan.[1] In 2010[2] Physics was also added as a section, therefore the name changed to RMMS. At the beginning, the competition structure had been 4 problems in 5 hours, but also in 2010, it was changed to 6 problems over 2 days, with 4.5 hours of exam each day. The first country that won the competition was the United Kingdom. The 4th edition was held between 23–28 of February 2011 and included also Chemistry and Computer Science. The 5th edition, held in 2012 was only for Physics and Mathematics. The current champion team in Mathematics is the United States of America. Teams reaching the top three in mathematics TeamTitlesRunners-upThird placeTop 3 finishes United States 6 (2011, 2013, 2016, 2018, 2019, 2023) 1 (2015) 4 (2009, 2010, 2012, 2020) 11 Russia 4 (2010, 2015, 2020, 2021) 4 (2008, 2012, 2013, 2018) 1 (2011) 9 China 3 (2009, 2012, 2021) 1 (2010) 2 (2015, 2017) 6 United Kingdom 1 (2008) 3 (2011, 2016, 2017) 1 (2013) 5 South Korea 1 (2017) 1 (2019) 0 2 Romania 1 (2012) 1 (2023) 0 2 Serbia 0 2 (2009, 2019) 1 (2008) 3 Hungary 0 2 (2011, 2018) 0 2 Ukraine 0 1 (2020) 1 (2018) 2 Israel 0 1 (2023) 1 (2019) 2 Poland 0 0 2 (2016, 2021) 2 * = teams finished equal points Organizers The contest is organised at the Tudor Vianu National College of Computer Science in collaboration with the Sector 1 town council. As a host, Tudor Vianu has the right to have its own team entering the contest in each section, thus participating against countries. References 1. Romanian Master of Mathematics 2008 website 2. Romanian Master of Mathematics and Sciences 2010 website	Wikipedia
Romanov's theorem In mathematics, specifically additive number theory, Romanov's theorem is a mathematical theorem proved by Nikolai Pavlovich Romanov. It states that given a fixed base b, the set of numbers that are the sum of a prime and a positive integer power of b has a positive lower asymptotic density. Romanov's theorem TypeTheorem FieldAdditive number theory Conjectured byAlphonse de Polignac Conjectured in1849 First proof byNikolai Pavlovich Romanov First proof in1934 Statement Romanov initially stated that he had proven the statements "In jedem Intervall (0, x) liegen mehr als ax Zahlen, welche als Summe von einer Primzahl und einer k-ten Potenz einer ganzen Zahl darstellbar sind, wo a eine gewisse positive, nur von k abhängige Konstante bedeutet" and "In jedem Intervall (0, x) liegen mehr als bx Zahlen, weiche als Summe von einer Primzahl und einer Potenz von a darstellbar sind. Hier ist a eine gegebene ganze Zahl und b eine positive Konstante, welche nur von a abhängt".[1] These statements translate to "In every interval $(0,x)$ there are more than $\alpha x$ numbers which can be represented as the sum of a prime number and a k-th power of an integer, where $\alpha $ is a certain positive constant that is only dependent on k" and "In every interval $(0,x)$ there are more than $\beta x$ numbers which can be represented as the sum of a prime number and a power of a. Here a is a given integer and $\beta $ is a positive constant that only depends on a" respectively. The second statement is generally accepted as the Romanov's theorem, for example in Nathanson's book.[2] Precisely, let $d(x)={\frac {\left\vert \{n\leq x:n=p+2^{k},p\ {\textrm {prime,}}\ k\in \mathbb {N} \}\right\vert }{x}}$ and let ${\underline {d}}=\liminf _{x\to \infty }d(x)$, ${\overline {d}}=\limsup _{x\to \infty }d(x)$. Then Romanov's theorem asserts that ${\underline {d}}>0$.[3] History Alphonse de Polignac wrote in 1849 that every odd number larger than 3 can be written as the sum of an odd prime and a power of 2. (He soon noticed a counterexample, namely 959.)[4] This corresponds to the case of $a=2$ in the original statement. The counterexample of 959 was, in fact, also mentioned in Euler's letter to Christian Goldbach,[5] but they were working in the opposite direction, trying to find odd numbers that cannot be expressed in the form. In 1934, Romanov proved the theorem. The positive constant $\beta $ mentioned in the case $a=2$ was later known as Romanov's constant.[6] Various estimates on the constant, as well as ${\overline {d}}$, has been made. The history of such refinements are listed below.[3] In particular, since ${\overline {d}}$ is shown to be less than 0.5 this implies that the odd numbers that cannot be expressed this way has positive lower asymptotic density. Refinements of ${\overline {d}}$ and ${\underline {d}}$ Year Lower bound on ${\underline {d}}$ Upper bound on ${\overline {d}}$ Prover Notes 1950 $0.5-5.06\times 10^{-80}$[lower-alpha 1] Paul Erdős ;[7] First proof of infinitely many odd numbers that are not of the form $2^{k}+p$ through an explicit arithmetic progression 2004 0.0868 Chen, Xun [8] 2006 0.0933 0.49094093[lower-alpha 2] Habsieger, Roblot ;[9] Considers only odd numbers; not exact, see note 2006 0.093626 Pintz ;[6] originally proved 0.9367, but an error was found and fixing it would yield 0.093626 2010 0.0936275 Habsieger, Sivak-Fischler [10] 2018 0.107648 Elsholtz, Schlage-Puchta 1. Exact value is $0.5-{\frac {1}{2^{241}\times 3\times 5\times 7\times 13\times 17\times 241}}$. 2. The value cited is 0.4909409303984105956480078184, which is just approximate. Generalisations Analogous results of Romanov's theorem has been proven in number fields by Riegel in 1961.[11] In 2015, the theorem was also proven for polynomials in finite fields.[12] Also in 2015, an arithmetic progression of Gaussian integers that are not expressible as the sum of a Gaussian prime and a power of 1+i is given.[13] References 1. Romanoff, N. P. (1934-12-01). "Über einige Sätze der additiven Zahlentheorie". Mathematische Annalen (in German). 109 (1): 668–678. doi:10.1007/BF01449161. ISSN 1432-1807. S2CID 119938116. 2. Nathanson, Melvyn B. (2013-03-14). Additive Number Theory The Classical Bases. Springer Science & Business Media. ISBN 978-1-4757-3845-2. 3. Elsholtz, Christian; Schlage-Puchta, Jan-Christoph (2018-04-01). "On Romanov's constant". Mathematische Zeitschrift. 288 (3): 713–724. doi:10.1007/s00209-017-1908-x. ISSN 1432-1823. S2CID 125994504. 4. de Polignac, A. (1849). "Recherches nouvelles sur les nombres premiers" [New research on prime numbers]. Comptes rendus (in French). 29: 397–401. 5. L. Euler, Letter to Goldbach. 16-12-1752. 6. Pintz, János (2006-07-01). "A note on Romanov's constant". Acta Mathematica Hungarica. 112 (1): 1–14. doi:10.1007/s10474-006-0060-6. ISSN 1588-2632. 7. Erdős, Paul (1950). "On Integers of the form $2^{k}+p$ and some related problems" (PDF). Summa Brasiliensis Mathematicae. 2: 113–125. S2CID 17379721. Archived from the original (PDF) on 2019-02-28. 8. Chen, Yong-Gao; Sun, Xue-Gong (2004-06-01). "On Romanoff's constant". Journal of Number Theory. 106 (2): 275–284. doi:10.1016/j.jnt.2003.11.009. ISSN 0022-314X. 9. Habsieger, Laurent; Roblot, Xavier-Franc¸ois (2006). "On integers of the form $p+2^{k}$". Acta Arithmetica. 1: 45–50. doi:10.4064/aa122-1-4. 10. Habsieger, Laurent; Sivak-Fischler, Jimena (2010-12-01). "An effective version of the Bombieri–Vinogradov theorem, and applications to Chen's theorem and to sums of primes and powers of two". Archiv der Mathematik. 95 (6): 557–566. doi:10.1007/s00013-010-0202-5. ISSN 1420-8938. S2CID 120510181. 11. Rieger, G. J. (1961-02-01). "Verallgemeinerung zweier Sätze von Romanov aus der additiven Zahlentheorie". Mathematische Annalen (in German). 144 (1): 49–55. doi:10.1007/BF01396540. ISSN 1432-1807. S2CID 121911723. 12. Shparlinski, Igor E.; Weingartner, Andreas J. (2015-10-30). "An explicit polynomial analogue of Romanoff's theorem". arXiv:1510.08991 [math.NT]. 13. Madritsch, Manfred G.; Planitzer, Stefan (2018-01-08). "Romanov's Theorem in Number Fields". arXiv:1512.04869 [math.NT].	Wikipedia
Romanovski polynomials In mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky[1] (Romanovski in French transcription) within the context of probability distribution functions in statistics. They form an orthogonal subset of a more general family of little-known Routh polynomials introduced by Edward John Routh[2] in 1884. The term Romanovski polynomials was put forward by Raposo,[3] with reference to the so-called 'pseudo-Jacobi polynomials in Lesky's classification scheme.[4] It seems more consistent to refer to them as Romanovski–Routh polynomials, by analogy with the terms Romanovski–Bessel and Romanovski–Jacobi used by Lesky for two other sets of orthogonal polynomials. In some contrast to the standard classical orthogonal polynomials, the polynomials under consideration differ, in so far as for arbitrary parameters only a finite number of them are orthogonal, as discussed in more detail below. The differential equation for the Romanovski polynomials The Romanovski polynomials solve the following version of the hypergeometric differential equation ${\begin{aligned}&s(x){R_{n}^{(\alpha ,\beta )}}''(x)+t_{1}^{(\alpha ,\beta )}(x){R_{n}^{(\alpha ,\beta )}}'(x)+\lambda _{n}R_{n}^{(\alpha ,\beta )}(x)=0,\\[4pt]&\qquad x\in (-\infty ,+\infty ),\quad s(x)=\left(1+x^{2}\right),\quad t_{1}^{(\alpha ,\beta )}(x)=2\beta x+\alpha ,\quad \lambda _{n}=-n(2\beta +n-1).\end{aligned}}$ (1) Curiously, they have been omitted from the standard textbooks on special functions in mathematical physics[5][6] and in mathematics[7][8] and have only a relatively scarce presence elsewhere in the mathematical literature.[9][10][11] The weight functions are $w^{(\alpha ,\beta )}(x)=\left(1+x^{2}\right)^{\beta -1}\exp \left(-\alpha \operatorname {arccot} x\right);$ (2) they solve Pearson's differential equation $[s(x)w(x)]'=t(x)w(x),\quad s(x)=1+x^{2},$ (3) that assures the self-adjointness of the differential operator of the hypergeometric ordinary differential equation. For α = 0 and β < 0, the weight function of the Romanovski polynomials takes the shape of the Cauchy distribution, whence the associated polynomials are also denoted as Cauchy polynomials[12] in their applications in random matrix theory.[13] The Rodrigues formula specifies the polynomial R(α,β) n (x) as $R_{n}^{(\alpha ,\beta )}(x)=N_{n}{\frac {1}{w^{(\alpha ,\beta )}(x)}}{\frac {\mathrm {d} ^{n}}{\mathrm {d} x^{n}}}\left(w^{(\alpha ,\beta )}(x)s(x)^{n}\right),\quad 0\leq n,$ (4) where Nn is a normalization constant. This constant is related to the coefficient cn of the term of degree n in the polynomial R(α,β) n (x) by the expression $N_{n}={\frac {(-1)^{n}n!\,c_{n}}{\prod _{k=0}^{n-1}\lambda _{n}^{(k)}}},\quad \lambda _{n}=-n\left({t_{n}^{(\alpha ,\beta )}}'+{\tfrac {1}{2}}(n-1)s''(x)\right),$ (5) which holds for n ≥ 1. Relationship between the polynomials of Romanovski and Jacobi As shown by Askey this finite sequence of real orthogonal polynomials can be expressed in terms of Jacobi polynomials of imaginary argument and thereby is frequently referred to as complexified Jacobi polynomials.[14] Namely, the Romanovski equation (1) can be formally obtained from the Jacobi equation,[15] ${\begin{aligned}&\left(1-x^{2}\right){P_{n}^{(\gamma ,\delta )}}''(x)+t_{1}^{(\gamma ,\delta )}(x){P_{n}^{(\gamma ,\delta )}}'(x)+\lambda _{n}P_{n}^{(\gamma ,\delta )}(x)=0,\\[4pt]&\qquad t_{1}^{(\gamma ,\delta )}(x)=\delta -\gamma -(\gamma +\delta +2)x,\quad \lambda _{n}=n(n+\gamma +\delta +1),\quad x\in \left[-1,1\right],\end{aligned}}$ (6) via the replacements, for real x, $x\to ix,\quad {\frac {\mathrm {d} }{{\mathrm {d} }x}}\to -i{\frac {\mathrm {d} }{{\mathrm {d} }x}},\quad \gamma =\delta ^{\ast }=\beta -1+{\frac {\alpha i}{2}},$ (7) in which case one finds $R_{n}^{(\alpha ,\beta )}(x)=i^{n}P_{n}^{\left(\beta -1+{\frac {i}{2}}\alpha ,\beta -1-{\frac {i}{2}}\alpha \right)}(ix),$ (8) (with suitably chosen normalization constants for the Jacobi polynomials). The complex Jacobi polynomials on the right are defined via (1.1) in Kuijlaars et al. (2003)[16] which assures that (8) are real polynomials in x. Since the cited authors discuss the non-hermitian (complex) orthogonality conditions only for real Jacobi indexes the overlap between their analysis and definition (8) of Romanovski polynomials exists only if α = 0. However examination of this peculiar case requires more scrutiny beyond the limits of this article. Notice the invertibility of (8) according to $P_{n}^{(\alpha ,\beta )}(x)=(-i)^{n}R_{n}^{\left(i(\alpha -\beta ),{\frac {1}{2}}(\alpha +\beta )+1)\right)}(-ix),$ (9) where, now, P(α,β) n (x) is a real Jacobi polynomial and $R_{n}^{\left(i(\alpha -\beta ),{\frac {1}{2}}(\alpha +\beta )+1)\right)}(-ix)$ would be a complex Romanovski polynomial. Properties of Romanovski polynomials Explicit construction For real α, β and n = 0, 1, 2, ..., a function R(α,β) n (x) can be defined by the Rodrigues formula in Equation (4) as $R_{n}^{(\alpha ,\beta )}(x)\equiv {\frac {1}{w^{(\alpha ,\beta )}(x)}}{\frac {{\rm {d}}^{n}}{{\rm {d}}x^{n}}}\left(w^{(\alpha ,\beta )}(x)s(x)^{n}\right),$ (10) where w(α,β) is the same weight function as in (2), and s(x) = 1 + x2 is the coefficient of the second derivative of the hypergeometric differential equation as in (1). Note that we have chosen the normalization constants Nn = 1, which is equivalent to making a choice of the coefficient of highest degree in the polynomial, as given by equation (5). It takes the form $c_{n}={\frac {1}{n!}}\prod _{k=0}^{n-1}{\bigl (}2\beta (n-k)+n(n-1)-k(k-1){\bigr )},\quad n\geq 1.$ (11) Also note that the coefficient cn does not depend on the parameter α, but only on β and, for particular values of β, cn vanishes (i.e., for all the values $\beta ={\frac {k(k-1)-n(n-1)}{2(n-k)}}$ where k = 0, ..., n − 1). This observation poses a problem addressed below. For later reference, we write explicitly the polynomials of degree 0, 1, and 2, ${\begin{aligned}R_{0}^{(\alpha ,\beta )}(x)&=1,\\[6pt]R_{1}^{(\alpha ,\beta )}(x)&={\frac {1}{w^{(\alpha ,\beta )}(x)}}\left(w'^{(\alpha ,\beta )}(x)s(x)+s'(x)w^{(\alpha ,\beta )}(x)\right)\\[6pt]&=t^{(\alpha ,\beta )}(x)=2\beta x+\alpha ,\\[6pt]R_{2}^{(\alpha ,\beta )}(x)&={\frac {1}{w^{(\alpha ,\beta )}(x)}}{\frac {\mathrm {d} }{{\mathrm {d} }x}}\left(s^{2}(x)w'^{(\alpha ,\beta )}(x)+2s(x)s'(x)w^{(\alpha ,\beta )}(x)\right)\\&={\frac {1}{w^{(\alpha ,\beta )}(x)}}{\frac {\mathrm {d} }{{\mathrm {d} }x}}\left(s(x)w^{(\alpha ,\beta )}(x)\left(t^{(\alpha ,\beta )}(x)+s'(x)\right)\right)\\[6pt]&=\left(2x+t^{(\alpha ,\beta )}(x)\right)t^{(\alpha ,\beta )}(x)+\left(2+t'^{(\alpha ,\beta )}(x)\right)s(x)\\[6pt]&=(2\beta +1)(2\beta +2)x^{2}+2(2\beta +1)\alpha x+\left(2\beta +\alpha ^{2}+2\right),\end{aligned}}$ which derive from the Rodrigues formula (10) in conjunction with Pearson's ODE (3). Orthogonality The two polynomials, R(α,β) m (x) and R(α,β) n (x) with m ≠ n, are orthogonal,[3] $\int _{-\infty }^{+\infty }w^{(\alpha ,\beta )}(x)R_{m}^{(\alpha ,\beta )}(x)R_{n}^{(\alpha ,\beta )}(x)=0$ (12) if and only if, $m+n<1-2\beta .$ (13) In other words, for arbitrary parameters, only a finite number of Romanovski polynomials are orthogonal. This property is referred to as finite orthogonality. However, for some special cases in which the parameters depend in a particular way on the polynomial degree infinite orthogonality can be achieved. This is the case of a version of equation (1) that has been independently encountered anew within the context of the exact solubility of the quantum mechanical problem of the trigonometric Rosen–Morse potential and reported in Compean & Kirchbach (2006).[17] There, the polynomial parameters α and β are no longer arbitrary but are expressed in terms of the potential parameters, a and b, and the degree n of the polynomial according to the relations, ${\begin{aligned}\alpha \to \alpha _{n}={\frac {2b}{n+1+a}},\quad \beta \to \beta _{n}=-(a+n+1)+1,\quad n=0,1,2,\ldots ,\infty .\end{aligned}}$ (14) Correspondingly, λn emerges as λn = −n(2a + n − 1), while the weight function takes the shape $\left(1+x^{2}\right)^{-(a+n+1)}\exp \left(-{\frac {2b}{n+a+1}}\operatorname {arccot} x\right).$ Finally, the one-dimensional variable, x, in Compean & Kirchbach (2006)[17] has been taken as $x=\cot \left({\frac {r}{d}}\right),$ where r is the radial distance, while $d$ is an appropriate length parameter. In Compean & Kirchbach[17] it has been shown that the family of Romanovski polynomials corresponding to the infinite sequence of parameter pairs, $\left(\alpha _{1},\beta _{1}\right),\left(\alpha _{2}\beta _{2}\right),\ldots ,\left(\alpha _{n}\beta _{n}\right),\ldots ,\quad n\longrightarrow \infty ,$ (15) is orthogonal. Generating function In Weber (2007)[18] polynomials Q(αn, βn + n) ν (x) , with βn + n = −a, and complementary to R(αn, βn) n (x) have been studied, generated in the following way: $Q_{\nu }^{\left(\alpha _{n},\beta _{n}+n\right)}(x)={\frac {1}{w^{\left(\alpha _{n},\beta _{n}+n-\nu \right)}}}{\frac {{\mathrm {d} }^{\nu }}{{\mathrm {d} }x^{\nu }}}w^{\left(\alpha _{n},\beta _{n}\right)}(x)\left(1+x^{2}\right)^{n}.$ (16) In taking into account the relation, $w^{\left(\alpha _{n},\beta _{n}\right)}(x)\left(1+x^{2}\right)^{\delta }=w^{\left(\alpha _{n},\beta _{n}+\delta \right)}(x),$ (17) Equation (16) becomes equivalent to ${\begin{aligned}Q_{\nu }^{\left(\alpha _{n},\beta _{n}+n\right)}(x)&={\frac {1}{w^{\left(\alpha _{n},\beta _{n}+n-\nu \right)}}}{\frac {{\mathrm {d} }^{\nu }}{{\mathrm {d} }x^{\nu }}}w^{\left(\alpha _{n},\beta _{n}+n-\nu \right)}(x)\left(1+x^{2}\right)^{\nu }\\[4pt]&=R_{\nu }^{\left(\alpha _{n},\beta _{n}+n-\nu \right)}(x),\end{aligned}}$ (18) and thus links the complementary to the principal Romanovski polynomials. The main attraction of the complementary polynomials is that their generating function can be calculated in closed form.[19] Such a generating function, written for the Romanovski polynomials based on Equation (18) with the parameters in (14) and therefore referring to infinite orthogonality, has been introduced as $G^{\left(\alpha _{n},\beta _{n}\right)}(x,y)=\sum _{\nu =0}^{\infty }R_{\nu }^{\left(\alpha _{n},\beta _{n}+n-\nu \right)}(x){\frac {y^{\nu }}{\nu !}}.$ !}}.} (19) The notational differences between Weber[18] and those used here are summarized as follows: • G(αn, βn)(x,y) here versus Q(x,y;α,−a) there, α there in place of αn here, • a = −βn − n, and • Q(α,−a) ν (x) in Equation (15) in Weber[18] corresponding to R(αn, βn + n − ν) ν (x) here. The generating function under discussion obtained in Weber[18] now reads: $G^{(\alpha _{n},\beta _{n})}(x,y)=\left(1+x^{2}\right)^{-\beta _{n}-n+1}\exp \left(\alpha _{n}\operatorname {arccot} x\right)\left(1+\left(x+y\left(1+x^{2}\right)\right)^{2}\right)^{-\left(-\beta _{n}-n+1\right)}\exp \left(-\alpha _{n}\operatorname {arccot} \left(x+y\left(1+x^{2}\right)\right)\right.$ (20) Recurrence relations Recurrence relations between the infinite orthogonal series of Romanovski polynomials with the parameters in the above equations (14) follow from the generating function,[18] $\nu {\bigl (}\nu +1-2(\beta _{n}+n){\bigr )}R_{\nu -1}^{\left(\alpha _{n},\beta _{n}+n-\nu +1\right)}(x)+{\frac {\mathrm {d} }{{\mathrm {d} }x}}R_{\nu }^{\left(\alpha _{n},\beta _{n}+n-\nu \right)}(x)=0,$ (21) and $R_{\nu +1}^{\left(\alpha _{n},\beta _{n}+n-\nu -1\right)}(x)={\bigl (}\alpha _{n}-2x(-\beta _{n}-n+\nu +1){\bigr )}R_{\nu }^{\left(\alpha _{n},\beta _{n}+n-\nu \right)}-\nu \left(1+x^{2}\right){\bigl (}2(-\beta _{n}-n)+\nu +1{\bigr )}R_{\nu -1}^{\left(\alpha _{n},\beta _{n}+n-\nu +1\right)},$ (22) as Equations (10) and (23) of Weber (2007)[18] respectively. See also • Associated Legendre functions • Gaussian quadrature • Gegenbauer polynomials • Legendre rational functions • Turán's inequalities • Legendre wavelet • Jacobi polynomials • Legendre polynomials • Spherical harmonics • Trigonometric Rosen–Morse potential References 1. Romanovski, V. (1929). "Sur quelques classes nouvelles de polynomes orthogonaux". C. R. Acad. Sci. Paris (in French). 188: 1023–1025. 2. Routh, E. J. (1884). "On some properties of certain solutions of a differential equation of second order". Proc. London Math. Soc. 16: 245. doi:10.1112/plms/s1-16.1.245. 3. Raposo, A. P.; Weber, H. J.; Álvarez Castillo, D. E.; Kirchbach, M. (2007). "Romanovski polynomials in selected physics problems". Cent. Eur. J. Phys. 5 (3): 253–284. arXiv:0706.3897. Bibcode:2007CEJPh...5..253R. doi:10.2478/s11534-007-0018-5. S2CID 119120266. 4. Lesky, P. A. (1996). "Endliche und unendliche Systeme von kontinuierlichen klassischen Orthogonalpolynomen". Z. Angew. Math. Mech. (in German). 76 (3): 181. Bibcode:1996ZaMM...76..181L. doi:10.1002/zamm.19960760317. 5. Abramowitz, M.; Stegun, I. (1972). Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (2nd ed.). New York, NY: Dover. ISBN 978-0-486-61272-0. 6. Nikiforov, Arnol'd F.; Uvarov, Vasilii B. (1988). Special Functions of Mathematical Physics: A Unified Introduction with Applications. Basel: Birkhäuser Verlag. ISBN 978-0-8176-3183-3. 7. Szego, G. (1939). Orthogonal Polynomials. Colloquium Publications. Vol. 23 (1st ed.). Providence, RI: American Mathematical Society. ISBN 978-0-8218-1023-1. 8. Ismail, Mourad E. H. (2005). Classical and Quantum Orthogonal Polynomials in One Variable. With two chapters by Walter V. Assche. Cambridge: Cambridge University Press. ISBN 978-0-521-78201-2. 9. Askey, R. (1987). "An integral of Ramanujan and orthogonal polynomials". Journal of the Indian Mathematical Society. 51 (1–2): 27. 10. Askey, R. (1989). "Beta integrals and the associated orthogonal polynomials". In Alladi, Krishnaswami (ed.). Number Theory, Madras 1987: Proceedings of the International Ramanujan Centenary Conference, Held at Anna University, Madras, India, December 21, 1987. Lecture Notes in Math. Vol. 1395. Berlin: Springer-Verlag. pp. 84–121. doi:10.1007/BFb0086401. ISBN 978-3-540-51595-1. 11. Zarzo Altarejos, A. (1995). Differential Equations of the Hypergeometric Type (PhD) (in Spanish). Faculty of Science, University of Granada. 12. Witte, N. S.; Forrester, P. J. (2000). "Gap probabilities in finite and scaled Cauchy random matrix ensembles". Nonlinearity. 13 (6): 13–1986. arXiv:math-ph/0009022. Bibcode:2000Nonli..13.1965W. doi:10.1088/0951-7715/13/6/305. S2CID 7151393. 13. Forrester, P. J. (2010). Log-Gases and Random Matrices. London Mathematical Society Monographs. Princeton: Princeton University Press. ISBN 978-0-691-12829-0. 14. Cotfas, N. (2004). "Systems of orthogonal polynomials defined by hypergeometric type equations with application to quantum mechanics". Cent. Eur. J. Phys. 2 (3): 456–466. arXiv:math-ph/0602037. Bibcode:2004CEJPh...2..456C. doi:10.2478/bf02476425. S2CID 15594058. 15. Weisstein, Eric W. "Jacobi Differential Equation". MathWorld. 16. Kuijlaars, A. B. J.; Martinez-Finkelshtein, A.; Orive, R. (2005). "Orthogonality of Jacobi polynomials with general parameters". Electron. Trans. Numer. Anal. 19: 1–17. arXiv:math/0301037. Bibcode:2003math......1037K. 17. Compean, C. B.; Kirchbach, M. (2006). "The trigonometric Rosen–Morse potential in supersymmetric quantum mechanics and its exact solutions". J. Phys. A: Math. Gen. 39 (3): 547–558. arXiv:quant-ph/0509055. Bibcode:2006JPhA...39..547C. doi:10.1088/0305-4470/39/3/007. S2CID 119742004. 18. Weber, H. J. (2007). "Connection between Romanovski polynomials and other polynomials". Central European Journal of Mathematics. 5 (3): 581. arXiv:0706.3153. doi:10.2478/s11533-007-0014-4. S2CID 18728079. 19. Weber, H. J. (2007). "Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula". Central European Journal of Mathematics. 5 (2): 415–427. arXiv:0706.3003. doi:10.2478/s11533-007-0004-6. S2CID 115166725.	Wikipedia
Romberg's method In numerical analysis, Romberg's method[1] is used to estimate the definite integral $\int _{a}^{b}f(x)\,dx$ by applying Richardson extrapolation[2] repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points. The integrand must have continuous derivatives, though fairly good results may be obtained if only a few derivatives exist. If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate. The method is named after Werner Romberg (1909–2003), who published the method in 1955. Method Using $ h_{n}={\frac {(b-a)}{2^{n}}}$, the method can be inductively defined by ${\begin{aligned}R(0,0)&=h_{1}(f(a)+f(b))\\R(n,0)&={\tfrac {1}{2}}R(n-1,0)+h_{n}\sum _{k=1}^{2^{n-1}}f(a+(2k-1)h_{n})\\R(n,m)&=R(n,m-1)+{\tfrac {1}{4^{m}-1}}(R(n,m-1)-R(n-1,m-1))\\&={\frac {1}{4^{m}-1}}(4^{m}R(n,m-1)-R(n-1,m-1))\end{aligned}}$ where $n\geq m$ and $m\geq 1\,$. In big O notation, the error for R(n, m) is:[3] $O\left(h_{n}^{2m+2}\right).$ The zeroeth extrapolation, R(n, 0), is equivalent to the trapezoidal rule with 2n + 1 points; the first extrapolation, R(n, 1), is equivalent to Simpson's rule with 2n + 1 points. The second extrapolation, R(n, 2), is equivalent to Boole's rule with 2n + 1 points. The further extrapolations differ from Newton-Cotes formulas. In particular further Romberg extrapolations expand on Boole's rule in very slight ways, modifying weights into ratios similar as in Boole's rule. In contrast, further Newton-Cotes methods produce increasingly differing weights, eventually leading to large positive and negative weights. This is indicative of how large degree interpolating polynomial Newton-Cotes methods fail to converge for many integrals, while Romberg integration is more stable. By labelling our $ O(h^{2})$ approximations as $ A_{0}{\big (}{\frac {h}{2^{n}}}{\big )}$ instead of $ R(n,0)$, we can perform Richardson extrapolation with the error formula defined below: $\int _{a}^{b}f(x)\,dx=A_{0}{\bigg (}{\frac {h}{2^{n}}}{\bigg )}+a_{0}{\bigg (}{\frac {h}{2^{n}}}{\bigg )}^{2}+a_{1}{\bigg (}{\frac {h}{2^{n}}}{\bigg )}^{4}+a_{2}{\bigg (}{\frac {h}{2^{n}}}{\bigg )}^{6}+\cdots $ Once we have obtained our $ O(h^{2(m+1)})$ approximations $ A_{m}{\big (}{\frac {h}{2^{n}}}{\big )}$, we can label them as $ R(n,m)$. When function evaluations are expensive, it may be preferable to replace the polynomial interpolation of Richardson with the rational interpolation proposed by Bulirsch & Stoer (1967). A geometric example To estimate the area under a curve the trapezoid rule is applied first to one-piece, then two, then four, and so on. After trapezoid rule estimates are obtained, Richardson extrapolation is applied. • For the first iteration the two piece and one piece estimates are used in the formula (4 × (more accurate) − (less accurate))/3 The same formula is then used to compare the four piece and the two piece estimate, and likewise for the higher estimates • For the second iteration the values of the first iteration are used in the formula (16(more accurate) − less accurate))/15 • The third iteration uses the next power of 4: (64 (more accurate) − less accurate))/63 on the values derived by the second iteration. • The pattern is continued until there is one estimate. Number of piecesTrapezoid estimatesFirst iterationSecond iterationThird iteration (4 MA − LA)/3(16 MA − LA)/15(64 MA − LA)/63 10(4×16 − 0)/3 = 21.333...(16×34.667 − 21.333)/15 = 35.556...(64×42.489 − 35.556)/63 = 42.599... 216(4×30 − 16)/3 = 34.666...(16×42 − 34.667)/15 = 42.489... 430(4×39 − 30)/3 = 42 839 • MA stands for more accurate, LA stands for less accurate Example As an example, the Gaussian function is integrated from 0 to 1, i.e. the error function erf(1) ≈ 0.842700792949715. The triangular array is calculated row by row and calculation is terminated if the two last entries in the last row differ less than 10−8. 0.77174333 0.82526296 0.84310283 0.83836778 0.84273605 0.84271160 0.84161922 0.84270304 0.84270083 0.84270066 0.84243051 0.84270093 0.84270079 0.84270079 0.84270079 The result in the lower right corner of the triangular array is accurate to the digits shown. It is remarkable that this result is derived from the less accurate approximations obtained by the trapezium rule in the first column of the triangular array. Implementation Here is an example of a computer implementation of the Romberg method (in the C programming language): #include <stdio.h> #include <math.h> void print_row(size_t i, double R) { printf("R[%2zu] = ", i); for (size_t j = 0; j <= i; ++j) { printf("%f ", R[j]); } printf("\n"); } /* INPUT: (f) : pointer to the function to be integrated a : lower limit b : upper limit max_steps: maximum steps of the procedure acc : desired accuracy OUTPUT: Rp[max_steps-1]: approximate value of the integral of the function f for x in [a,b] with accuracy 'acc' and steps 'max_steps'. / double romberg(double (f)(double), double a, double b, size_t max_steps, double acc) { double R1[max_steps], R2[max_steps]; // buffers double Rp = &R1[0], Rc = &R2[0]; // Rp is previous row, Rc is current row double h = b-a; //step size Rp[0] = (f(a) + f(b))h0.5; // first trapezoidal step print_row(0, Rp); for (size_t i = 1; i < max_steps; ++i) { h /= 2.; double c = 0; size_t ep = 1 << (i-1); //2^(n-1) for (size_t j = 1; j <= ep; ++j) { c += f(a + (2j-1) * h); } Rc[0] = hc + .5Rp[0]; // R(i,0) for (size_t j = 1; j <= i; ++j) { double n_k = pow(4, j); Rc[j] = (n_kRc[j-1] - Rp[j-1]) / (n_k-1); // compute R(i,j) } // Print ith row of R, R[i,i] is the best estimate so far print_row(i, Rc); if (i > 1 && fabs(Rp[i-1]-Rc[i]) < acc) { return Rc[i]; } // swap Rn and Rc as we only need the last row double rt = Rp; Rp = Rc; Rc = rt; } return Rp[max_steps-1]; // return our best guess } Here is an example of a computer implementation of the Romberg method in the Javascript programming language. /** * INPUTS * func = integrand, function to be integrated * a = lower limit of integration * b = upper limit of integration * nmax = number of partitions, n = 2^nmax * tol_ae = maximum absolute approximate error acceptable (should be >= 0) * tol_rae = maximum absolute relative approximate error acceptable (should be >= 0) * OUTPUTS * integ_value = estimated value of integral / function auto_integrator_trap_romb_hnm(func, a, b, nmax, tol_ae, tol_rae) { if (typeof a !== 'number') { throw new TypeError('<a> must be a number'); } if (typeof b !== 'number') { throw new TypeError('<b> must be a number'); } if (!Number.isInteger(nmax) \|\| nmax<1) { throw new TypeError('<nmax> must be an integer greater than or equal to one.'); } if ((typeof tol_ae !== 'number') \|\| tol_ae < 0) { throw new TypeError('<tole_ae> must be a number greater than or equal to zero'); } if ((typeof tol_rae !== 'number') \|\| tol_rae <= 0) { throw new TypeError('<tole_ae> must be a number greater than or equal to zero'); } var h = b - a; // initialize matrix where the values of integral are stored var Romb = []; // rows for (var i = 0; i < nmax+1; i++) { Romb.push([]); for (var j = 0; j < nmax+1; j++) { Romb[i].push(math.bignumber(0)); } } // calculating the value with 1-segment trapezoidal rule Romb[0][0] = 0.5 h * (func(a)+func(b)); var integ_val = Romb[0][0]; for (var i = 1; i <= nmax; i++) { // updating the value with double the number of segments // by only using the values where they need to be calculated // See https://autarkaw.org/2009/02/28/an-efficient-formula-for-an-automatic-integrator-based-on-trapezoidal-rule/ h = h / 2; var integ = 0; for (var j=1; j<=2*i-1; j+=2) { var integ=integ+func(a+jh) } Romb[i][0] = 0.5Romb[i-1][0] + integh; // Using Romberg method to calculate next extrapolatable value // See https://young.physics.ucsc.edu/115/romberg.pdf for (k = 1; k <= i; k++) { var addterm = Romb[i][k-1] - Romb[i-1][k-1] addterm = addterm/(4*k-1.0) Romb[i][k] = Romb[i][k-1] + addterm //Calculating absolute approximate error var Ea = math.abs(Romb[i][k] - Romb[i][k-1]) //Calculating absolute relative approximate error var epsa = math.abs(Ea/Romb[i][k]) 100.0; //Assigning most recent value to the return variable integ_val = Romb[i][k]; // returning the value if either tolerance is met if (epsa < tol_rae \|\| Ea < tol_ae) { return integ_val; } } } // returning the last calculated value of integral whether tolerance is met or not return integ_val; } References 1. Romberg 1955 2. Richardson 1911 3. Mysovskikh 2002 • Richardson, L. F. (1911), "The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations, with an Application to the Stresses in a Masonry Dam", Philosophical Transactions of the Royal Society A, 210 (459–470): 307–357, doi:10.1098/rsta.1911.0009, JSTOR 90994 • Romberg, W. (1955), "Vereinfachte numerische Integration", Det Kongelige Norske Videnskabers Selskab Forhandlinger, Trondheim, 28 (7): 30–36 • Thacher Jr., Henry C. (July 1964), "Remark on Algorithm 60: Romberg integration", Communications of the ACM, 7 (7): 420–421, doi:10.1145/364520.364542 • Bauer, F.L.; Rutishauser, H.; Stiefel, E. (1963), Metropolis, N. C.; et al. (eds.), "New aspects in numerical quadrature", Experimental Arithmetic, high-speed computing and mathematics, Proceedings of Symposia in Applied Mathematics, AMS (15): 199–218 • Bulirsch, Roland; Stoer, Josef (1967), "Handbook Series Numerical Integration. Numerical quadrature by extrapolation", Numerische Mathematik, 9: 271–278, doi:10.1007/bf02162420 • Mysovskikh, I.P. (2002) [1994], "Romberg method", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer-Verlag, ISBN 1-4020-0609-8 • Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 4.3. Romberg Integration", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8 External links • ROMBINT – code for MATLAB (author: Martin Kacenak) • Free online integration tool using Romberg, Fox–Romberg, Gauss–Legendre and other numerical methods • SciPy implementation of Romberg's method • Romberg.jl — Julia implementation (supporting arbitrary factorizations, not just $2^{n}+1$ points)	Wikipedia
Ron Goldman (mathematician) Ronald Neil Goldman is a Professor of Computer Science at Rice University in Houston, Texas. Professor Goldman received his B.S. in Mathematics from the Massachusetts Institute of Technology in 1968 and his M.A. and Ph.D. in Mathematics from Johns Hopkins University in 1973.[1] Goldman's current research interests lie in the mathematical representation, manipulation, and analysis of shape using computers. His work includes research in computer-aided geometric design, solid modeling, computer graphics, and splines. He is particularly interested in algorithms for polynomial and piecewise polynomial curves and surfaces, and he is currently investigating applications of algebraic and differential geometry to geometric modeling. He has published over a hundred articles in journals, books, and conference proceedings on these and related topics. Before returning to academia, Goldman worked for 10 years in industry solving problems in computer graphics, geometric modeling, and computer aided design. He served as a mathematician at Manufacturing Data Systems Inc., where he helped to implement one of the first industrial solid modeling systems. Later he worked as a senior design engineer at Ford Motor Company, enhancing the capabilities of their corporate graphics and computer-aided design software. From Ford he moved on to Control Data Corporation, where he was a principal consultant for the development group devoted to computer-aided design and manufacture. His responsibilities included database design, algorithms, education, acquisitions, and research. Goldman left Control Data Corporation in 1987 to become an associate professor of computer science at the University of Waterloo in Ontario, Canada. He joined the faculty at Rice University in Houston, Texas as a professor of computer science in July 1990. Selected publications • Goldman, Ron (2002). Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling. Morgan Kaufmann. ISBN 9781558603547. • Goldman, Ron (2009). An Integrated Introduction to Computer Graphics and Geometric Modeling. CRC Press. ISBN 9781439803349. References 1. Ron Goldman at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Netherlands Academics • DBLP • Mathematics Genealogy Project • ORCID Other • IdRef	Wikipedia
Rona Gurkewitz Rona Gurkewitz is an American mathematician and computer scientist, known for her work on modular origami.[1][2] She is a professor emerita of computer science at Western Connecticut State University,[3] and the former head of the department of computer science there.[2] Origami Gurkewitz became interested in origami after meeting origami pioneer Lillian Oppenheimer at a dinner party and becoming a regular visitor to Oppenheimer's origami get-togethers.[2] She has written several books on origami, exhibited works at international origami shows,[2] supplied a piece for the set design of the premiere of the Rajiv Joseph play Animals Out of Paper,[4] and has made modular origami quilts as well as polyhedra.[2] Books With retired mechanical engineer Bennett Arnstein,[2] Gurkewitz is the coauthor of books including: • 3D Geometric Origami: Modular Origami Polyhedra (Dover, 1996)[5] • Multimodular Origami Polyhedra: Archimedeans, Buckyballs and Duality (Dover, 2002)[6] • Beginner's Book of Modular Origami Polyhedra: The Platonic Solids (Dover, 2008) With Arnstein and Lewis Simon, she is a coauthor of the second edition of the book Modular Origami Polyhedra (Dover, 1999), extended from the first edition by Arnstein and Simon.[7] References 1. "Origami: Doing the Math Without the Numbers", Republican-American, 6 January 2009 – via Mathematical Association of America 2. Overton, Penelope (11 January 2009), "Conn. origami master hooked on 'geometry without numbers'", Telegram & Gazette 3. Computer Science Faculty and Staff, Western Connecticut State University, retrieved 2020-08-27 4. Gluckman, Neil (11 August 2008), "Origami, More Than Paper Critters", Art Around Town, New York Sun 5. Reviews of 3D Geometric Origami: Modular Origami Polyhedra: • Plummer, Robert (December 1996), The Mathematics Teacher, 89 (9): 782, JSTOR 27970022{{citation}}: CS1 maint: untitled periodical (link) • Barnette, David (1997), Mathematical Reviews, MR 1375920{{citation}}: CS1 maint: untitled periodical (link) • Cannon, Mary Ellen (May 1997), Mathematics Teaching in the Middle School, 2 (6): 444–445, JSTOR 41181638{{citation}}: CS1 maint: untitled periodical (link) • Blackwell, Joan (March 1999), "Review", School Science and Mathematics, Wiley, 99 (3): 160, doi:10.1111/j.1949-8594.1999.tb17467.x, ProQuest 195202376 6. Reviews of Multimodular Origami Polyhedra: Archimedeans, Buckyballs and Duality: • Murphey, Bonnie (January 2004), Mathematics Teaching in the Middle School, 9 (5): 288, JSTOR 41181919{{citation}}: CS1 maint: untitled periodical (link) • Kessler, Charlotte (January 2004), The Mathematics Teacher, 97 (1): 78, JSTOR 20871510{{citation}}: CS1 maint: untitled periodical (link) 7. Reviews of Modular Origami Polyhedra (2nd ed.): • Böhm, Johannes, "none", zbMATH, Zbl 1059.00005 • Johnston, Christopher (September 2002), Mathematics Teaching in the Middle School, 8 (1): 59, 62, JSTOR 41181231{{citation}}: CS1 maint: untitled periodical (link) External links • Rona Gurkewitz' Modular Origami Polyhedra Systems Page Mathematics of paper folding Flat folding • Big-little-big lemma • Crease pattern • Huzita–Hatori axioms • Kawasaki's theorem • Maekawa's theorem • Map folding • Napkin folding problem • Pureland origami • Yoshizawa–Randlett system Strip folding • Dragon curve • Flexagon • Möbius strip • Regular paperfolding sequence 3d structures • Miura fold • Modular origami • Paper bag problem • Rigid origami • Schwarz lantern • Sonobe • Yoshimura buckling Polyhedra • Alexandrov's uniqueness theorem • Blooming • Flexible polyhedron (Bricard octahedron, Steffen's polyhedron) • Net • Source unfolding • Star unfolding Miscellaneous • Fold-and-cut theorem • Lill's method Publications • Geometric Exercises in Paper Folding • Geometric Folding Algorithms • Geometric Origami • A History of Folding in Mathematics • Origami Polyhedra Design • Origamics People • Roger C. Alperin • Margherita Piazzola Beloch • Robert Connelly • Erik Demaine • Martin Demaine • Rona Gurkewitz • David A. Huffman • Tom Hull • Kôdi Husimi • Humiaki Huzita • Toshikazu Kawasaki • Robert J. Lang • Anna Lubiw • Jun Maekawa • Kōryō Miura • Joseph O'Rourke • Tomohiro Tachi • Eve Torrence	Wikipedia
Ronald C. Read Ronald Cedric Read (19 December 1924 – 7 January 2019) was a British mathematician, latterly a professor emeritus of mathematics at the University of Waterloo, Canada. He published many books[1] and papers, primarily on enumeration of graphs, graph isomorphism, chromatic polynomials, and particularly, the use of computers in graph-theoretical research. A majority of his later work was done in Waterloo. Read received his Ph.D. (1959) in graph theory from the University of London.[2] R. C. Read Born Ronald Cedric Read (1924-12-19)19 December 1924 Croydon, England Died7 January 2019(2019-01-07) (aged 94) Oakville, Ontario, Canada CitizenshipBritish Alma materUniversity of Cambridge and University of London Scientific career FieldsGraph theory InstitutionsUniversity of Waterloo Doctoral studentsJorge Urrutia and William Lawrence Kocay Life and career Ronald Read served in the Royal Navy during World War II, then completed a degree in mathematics at the University of Cambridge before joining the University College of the West Indies (later the University of the West Indies) in Jamaica as the second founding member of the Mathematics Department there. In 1970 he moved his family to Canada to take up a post as Professor of Mathematics at the University of Waterloo, Ontario, Canada. While in Jamaica he became interested in cave exploration, and in 1957 he founded the Jamaica Caving Club. He had a lifelong interest in the making of string figures and is the inventor of the Olympic Flag String Figure on YouTube. He was an accomplished musician and played many instruments including violin, viola, cello, double bass, piano, guitar, lute, and many early music instruments, some of which he also built. He had diplomas in Theory and in Composition from the Royal Conservatory of Music in Toronto, Canada, and composed four works for orchestra and several pieces for smaller groups.[3] Read died in January 2019 at the age of 94.[4] Selected papers • An Introduction to Chromatic Polynomials. Journal of Combinatorial Theory 4 (1968) 52 - 71. • Every One A Winner; or How to avoid isomorphism search when cataloguing combinatorial configurations. Annals of Discrete Mathematics 2, North-Holland Publishing Company (1978) 107-120. • (With P. Rosenstiehl) On the Principal Edge Tripartition of a Graph. Annals of Discrete Mathematics 3, North-Holland Publishing Company, (1978) 195-226. • (With W. T. Tutte), Chromatic Polynomials. Selected Topics in Graph Theory, Vol. 3 (1988) 15-42. • (with G. F. Royle) Chromatic Roots of Families of Graphs. Graph Theory, Combinatorics and Applications. John Wiley (1991) 1009 - 1029 • Prospects for Graph-theoretical Algorithms. Annals of Discrete Mathematics 55 (1993) 201 - 210. Books • Read, Ronald C. (1965). Tangrams : 330 puzzles. New York: Dover Publications, Inc. ISBN 0-486-21483-4. OCLC 1136293. • Read, Ronald C. (1972). A mathematical background for economists and social scientists. Englewood Cliffs, N.J.: Prentice-Hall. ISBN 0-13-560987-9. OCLC 327336. • Read, Ronald C. (1972). Graph theory and computing. Claude Berge. New York: Academic Press. ISBN 0-12-583850-6. OCLC 525261. • Pólya, G.; Read, Ronald C. (1987). Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds. New York, NY: Springer New York. ISBN 978-1-4612-4664-0. OCLC 840279750. • Read, Ronald C.; Wilson, Robin J. (1998). An atlas of graphs. Robin J. Wilson. Oxford: Clarendon Press. ISBN 0-19-853289-X. OCLC 40647141. See also • List of University of Waterloo people References 1. - Books on Amazon 2. - His Ph.D. 3. - Published music 4. Ronald (Ron) Cedric READ Authority control International • ISNI • VIAF National • Israel • United States • Czech Republic Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH	Wikipedia
Ronald Fedkiw Ronald Paul "Ron" Fedkiw (born February 27, 1968) is a full professor in the Stanford University department of computer science and a leading researcher in the field of computer graphics, focusing on topics relating to physically based simulation of natural phenomena and machine learning. His techniques have been employed in many motion pictures. He has earned recognition at the 80th Academy Awards and the 87th Academy Awards as well as from the National Academy of Sciences. Ronald Fedkiw Born (1968-02-27) February 27, 1968[1] Buffalo, New York, US NationalityAmerican Alma materUCLA (Ph.D., 1996) Known forPhysics-based techniques for visual simulations AwardsNational Academy of Sciences, 80th Academy Awards, 87th Academy Awards Scientific career FieldsComputer graphics, Computer vision, Machine learning, Computational physics InstitutionsStanford University (2000-) Doctoral advisorStanley Osher His first Academy Award was awarded for developing techniques that enabled many technically sophisticated adaptations including the visual effects in 21st century movies in the Star Wars, Harry Potter, Terminator, and Pirates of the Caribbean franchises. Fedkiw has designed a platform that has been used to create many of the movie world's most advanced special effects since it was first used on the T-X character in Terminator 3: Rise of the Machines. His second Academy Award was awarded for computer graphics techniques for special effects for large scale destruction. Although he has won an Oscar for his work, he does not design the visual effects that use his technique. Instead, he has developed a system that other award-winning technicians and engineers have used to create visual effects for some of the world's most expensive and highest-grossing movies. Early life and family Fedkiw was born in Buffalo, New York, in 1968.[1] He received his Ph.D. in applied mathematics from UCLA in 1996.[2] His dissertation was chaired by Stanley Osher.[1] He completed postdoctoral studies both at UCLA in Mathematics and at Caltech in Aeronautics before joining the Stanford Computer Science Department.[2] Fedkiw has two daughters: Brittany and Briana.[2] Career Fedkiw began working in the movie industry in 1998, working for a company that produced 3-D water simulations. The algorithms they used were known as Navier-Stokes equations.[3] Fedkiw is now a full professor in the department of computer science at Stanford where he researches computational physics.[3][4] Fedkiw serves on the editorial boards of Journal of Computational Physics and the Journal of Scientific Computing. He has published Level Set Methods and Dynamic Implicit Surfaces (Springer 2002, ISBN 0-387-95482-1) along with Stanley Osher. Since 2000, Fedkiw has been a consultant with Industrial Light & Magic receiving screen credits for work on Terminator 3: Rise of the Machines, Star Wars: Episode III – Revenge of the Sith and Poseidon.[5] In addition, he has worked on all three Pirates of the Caribbean and some Harry Potter movies.[6] Fedkiw's techniques have made possible the renderings of the sea in the Pirates movies and the dragon's flaming breath in Harry Potter and the Goblet of Fire.[7] They have also made possible the rushing floodwaters in Evan Almighty and were first used with T-X in Terminator 3.[8] Fedkiw feels the best result of the use of his techniques was the sinking ship shots in Poseidon.[9] Pirates of the Caribbean: Dead Man's Chest won the Academy Award for Visual Effects at the 79th Academy Awards awarded on February 25, 2007, and Poseidon was also nominated that year in that category.[10] Among the applications that Fedkiw's math engine is responsible for is the tentacles of Davy Jones (pictured left) in the Academy Award-winning Dead Man's Chest.[5] On February 9, 2008, in the Academy Scientific and Technical Award ceremony at the Beverly Wilshire Hotel in Beverly Hills, California, Fedkiw was awarded an 80th Academy Award for Technical Achievement for the development of the Industrial Light & Magic (ILM) fluid simulation system.[11][12] He shared the award with Nick Rasmussen and Frank Losasso Petterson.[3] Fedkiw does physics-based simulation that enable better water effects. Previous representations had varying levels of success. They often did well at surface representation, but were less efficient at smaller particles such as breaking waves. Fedkiw's team's innovative "particle level set method" allows both smooth surfaces and water breakdown renderings including water spray.[9] Fedkiw has worked with Industrial Light & Magic, Pixar Animation Studios, Intel Corporation, Honda and Sony Pictures Imageworks.[5] Fedkiw commented that when he was informed that he would be presented his award by Jessica Alba he was quoted by the Associated Press as follows: "They said I got 60 seconds so I might just spend the last 15 realizing I'm 10 feet away from the most beautiful woman on the planet . . . and no restraining order this time.".[12] On February 7, 2015, he received a second Academy Scientific and Technical Award for the development of the ILM PhysBAM Destruction System.[13] Fedkiw and his colleagues have designed a C++ code library for Physics Based Modelling (PhysBAM http://physbam.stanford.edu) that facilitates the creation of better special effects for movies, including water, smoke, fire, cloth, rigid bodies and deformable bodies. Fedkiw often receives screen credit for consulting with special effects engineers, technicians and movie executives. His research has focused on the design of new computational algorithms that can be used for many purposes, including computational fluid dynamics and soft-body dynamics, computer graphics, computer vision and computational biomechanics. In fact, the system can also be used for a range of applications from motion capture to rendering, but Fedkiw's main emphasis is on physics-based simulation.[9] Fedkiw has described his work as follows: It is an exhaustive task to prescribe the motion of every degree of freedom in a piece of clothing or a crashing wave. . .Since these motions are governed by physical processes, it can be difficult to make these phenomena appear natural. Thus, physically based simulation has become quite popular in the special effects industry. The same class of tools useful for computational fluid dynamics is also useful for sinking a ship on the big screen.[5] Awards • Presidential Early Career Award for Scientists and Engineers • National Academy of Sciences: Award for Initiatives in Research • SIGGRAPH: Significant New Researcher Award (2005) Book • S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer-Verlag, New York (2002). See also • Industrial Light & Magic • Level-set method Notes 1. "A Survey of Chemically Reacting, Compressible Flow". 1997. CiteSeerX 10.1.1.75.8892. 2. "Ron Fedkiw". Stanford.edu. Retrieved October 17, 2008. 3. "The Man Behind Amazing Movie Simulations". Beta site for NSF - National Science Foundation. Retrieved July 21, 2022. 4. Orenstein, David (January 16, 2008). "Professor wins Academy Award for cyber-fluids". Stanford University. Retrieved July 21, 2022. 5. Levy, Dawn (February 19, 2007). "Computer scientist reveals the math and science behind blockbuster movies". Stanford University. Retrieved February 13, 2008. 6. "Computer Scientist Makes Splash With Academy Award For Fluid Simulation". ScienceDaily. Retrieved July 21, 2022. 7. "Ron Fedkiw to be awarded an Oscar for his SFX work". FILMdetail. January 31, 2008. Retrieved February 13, 2008. 8. "Stanford computer scientist makes splash with Academy Award for fluid simulation" (Press release). Stanford University. January 16, 2008. Retrieved February 15, 2008. 9. Dada, Kamil (January 22, 2008). "CS prof wins film award". Stanford Daily. Archived from the original on February 9, 2008. Retrieved February 13, 2008. 10. "Academy of Motion Picture Arts and Sciences:2006 (79th) VISUAL EFFECTS". Academy of Motion Picture Arts and Sciences. Retrieved February 13, 2008. 11. "Awards for Ronald Fedkiw". IMDb.com, Inc. Retrieved February 13, 2008. 12. Pearson, Ryan (February 11, 2008). "Alba dazzles nerds at tech Oscars". USA Today. Retrieved February 13, 2008. 13. Abate, Tom (February 5, 2015). "Stanford professor shares Academy Award for software to digitize destruction". Stanford.edu. Retrieved September 5, 2017. External links • Ron Fedkiw Homepage at Stanford • Ronald Fedkiw at IMDb • Fedkiw at Association for Computing Machinery • Ron Fedkiw at New York Times • Ronald Fedkiw at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Sweden • Latvia • Czech Republic • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Ronald Graham Ronald Lewis Graham (October 31, 1935 – July 6, 2020)[1] was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years".[2] He was president of both the American Mathematical Society and the Mathematical Association of America, and his honors included the Leroy P. Steele Prize for lifetime achievement and election to the National Academy of Sciences. Ronald Graham Graham in 1998 Born Ronald Lewis Graham (1935-10-31)October 31, 1935 Taft, California, U.S. DiedJuly 6, 2020(2020-07-06) (aged 84) San Diego, California, U.S. Alma mater • Univ. of Alaska (BS) • UC Berkeley (PhD) Known for • Coffman–Graham algorithm • Erdős–Graham problem • Graham's number • Graham scan • Graham–Pollak theorem • Optimal job scheduling notation Spouse Fan Chung (m. 1983) Awards • Pólya Prize (1971) • Nat. Acad. Sci. (1985) • Steele Prize (2003) Scientific career Fields • Mathematics • Computer science Institutions • Bell Labs • AT&T Labs • Cal-(IT)2 • UCSD ThesisOn Finite Sums of Rational Numbers (1962) Doctoral advisorDerrick Henry Lehmer After graduate study at the University of California, Berkeley, Graham worked for many years at Bell Labs and later at the University of California, San Diego. He did important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness,[3] and many topics in mathematics are named after him. He published six books and about 400 papers, and had nearly 200 co-authors, including many collaborative works with his wife Fan Chung and with Paul Erdős. Graham has been featured in Ripley's Believe It or Not! for being not only "one of the world's foremost mathematicians", but also an accomplished trampolinist and juggler. He served as president of the International Jugglers' Association.[3][4][5] Biography Graham was born in Taft, California, on October 31, 1935;[6] his father was an oil field worker and later merchant marine. Despite Graham's later interest in gymnastics, he was small and non-athletic.[7] He grew up moving frequently between California and Georgia, skipping several grades of school in these moves, and never staying at any one school longer than a year.[1][7] As a teenager, he moved to Florida with his then-divorced mother, where he went to but did not finish high school. Instead, at the age of 15 he won a Ford Foundation scholarship to the University of Chicago, where he learned gymnastics but took no mathematics courses.[1] After three years, when his scholarship expired, he moved to the University of California, Berkeley, officially as a student of electrical engineering but also studying number theory under Derrick Henry Lehmer,[1] and winning a title as California state trampoline champion.[7] He enlisted in the United States Air Force in 1955, when he reached the age of eligibility,[8] left Berkeley without a degree, and was stationed in Fairbanks, Alaska, where he finally completed a bachelor's degree in physics in 1959 at the University of Alaska Fairbanks.[1] Returning to the University of California, Berkeley for graduate study, he received his Ph.D. in mathematics in 1962. His dissertation, supervised by Lehmer, was On Finite Sums of Rational Numbers.[9] While a graduate student, he supported himself by performing on trampoline in a circus,[8] and married Nancy Young, an undergraduate mathematics student at Berkeley; they had two children.[1] After completing his doctorate, Graham went to work in 1962 at Bell Labs and later as Director of Information Sciences at AT&T Labs, both in New Jersey. In 1963, at a conference in Colorado, he met the prolific Hungarian mathematician Paul Erdős (1913–1996),[1] who became a close friend and frequent research collaborator. Graham was chagrined to be beaten in ping-pong by Erdős, then already middle-aged; he returned to New Jersey determined to improve his game, and eventually became Bell Labs champion and won a state title in the game.[1] Graham later popularized the concept of the Erdős number, a measure of distance from Erdős in the collaboration network of mathematicians;[10][8] his many works with Erdős include two books of open problems[B1][B5] and Erdős's final posthumous paper.[A15] Graham divorced in the 1970s; in 1983 he married his Bell Labs colleague and frequent coauthor Fan Chung.[1] While at Bell Labs, Graham also took a position at Rutgers University as University Professor of Mathematical Sciences in 1986, and served a term as president of the American Mathematical Society from 1993 to 1994. He became Chief Scientist of the labs in 1995.[1] He retired from AT&T in 1999 after 37 years of service there,[11] and moved to the University of California, San Diego (UCSD), as the Irwin and Joan Jacobs Endowed Professor of Computer and Information Science.[1][8] At UCSD, he also became chief scientist at the California Institute for Telecommunications and Information Technology.[8][5] In 2003–04, he was president of the Mathematical Association of America.[1] Graham died of bronchiectasis[12] on July 6, 2020, aged 84, in La Jolla, California.[6][13] Contributions Graham made important contributions in multiple areas of mathematics and theoretical computer science. He published about 400 papers, a quarter of those with Chung,[14] and six books, including Concrete Mathematics with Donald Knuth and Oren Patashnik.[B4] The Erdős Number Project lists him as having nearly 200 coauthors.[15] He was the doctoral advisor of nine students, one each at the City University of New York and Rutgers University while he was at Bell Labs, and seven at UC San Diego.[9] Notable topics in mathematics named after Graham include the Erdős–Graham problem on Egyptian fractions, the Graham–Rothschild theorem in the Ramsey theory of parameter words and Graham's number derived from it, the Graham–Pollak theorem and Graham's pebbling conjecture in graph theory, the Coffman–Graham algorithm for approximate scheduling and graph drawing, and the Graham scan algorithm for convex hulls. He also began the study of primefree sequences, the Boolean Pythagorean triples problem, the biggest little polygon, and square packing in a square. Graham was one of the contributors to the publications of G. W. Peck, a pseudonymous mathematical collaboration named for the initials of its members, with Graham as the "G".[16] Number theory Graham's doctoral dissertation was in number theory, on Egyptian fractions,[7][9] as is the Erdős–Graham problem on whether, for every partition of the integers into finitely many classes, one of these classes has a finite subclass whose reciprocals sum to one. A proof was published by Ernie Croot in 2003.[17] Another of Graham's papers on Egyptian fractions was published in 2015 with Steve Butler and (nearly 20 years posthumously) Erdős; it was the last of Erdős's papers to be published, making Butler his 512th coauthor.[A15][18] In a 1964 paper, Graham began the study of primefree sequences by observing that there exist sequences of numbers, defined by the same recurrence relation as the Fibonacci numbers, in which none of the sequence elements is prime.[A64] The challenge of constructing more such sequences was later taken up by Donald Knuth and others.[19] Graham's 1980 book with Erdős, Old and new results in combinatorial number theory, provides a collection of open problems from a broad range of subareas within number theory.[B1] Ramsey theory The Graham–Rothschild theorem in Ramsey theory was published by Graham and Bruce Rothschild in 1971, and applies Ramsey theory to combinatorial cubes in combinatorics on words.[A71a] Graham gave a large number as an upper bound for an instance of this theorem, now known as Graham's number, which was listed in the Guinness Book of Records as the largest number ever used in a mathematical proof,[20] although it has since then been surpassed by even larger numbers such as TREE(3).[21] Graham offered a monetary prize for solving the Boolean Pythagorean triples problem, another problem in Ramsey theory; the prize was claimed in 2016.[22] Graham also published two books on Ramsey theory.[B2][B3] Graph theory The Graham–Pollak theorem, which Graham published with Henry O. Pollak in two papers in 1971 and 1972,[A71b][A72a] states that if the edges of an $n$-vertex complete graph are partitioned into complete bipartite subgraphs, then at least $n-1$ subgraphs are needed. Graham and Pollak provided a simple proof using linear algebra; despite the combinatorial nature of the statement and multiple publications of alternative proofs since their work, all known proofs require linear algebra.[23] Soon after research in quasi-random graphs began with the work of Andrew Thomason, Graham published in 1989 a result with Chung and R. M. Wilson that has been called the "fundamental theorem of quasi-random graphs", stating that many different definitions of these graphs are equivalent.[A89a][24] Graham's pebbling conjecture, appearing in a 1989 paper by Chung,[25] concerns the pebbling number of Cartesian products of graphs. As of 2019, it remains unsolved.[26] Packing, scheduling, and approximation algorithms Graham's early work on job shop scheduling[A66][A69] introduced the worst-case approximation ratio into the study of approximation algorithms, and laid the foundations for the later development of competitive analysis of online algorithms.[27] This work was later recognized to be important also for the theory of bin packing,[28] an area that Graham later worked in more explicitly.[A74] The Coffman–Graham algorithm, which Graham published with Edward G. Coffman Jr. in 1972,[A72b] provides an optimal algorithm for two-machine scheduling, and a guaranteed approximation algorithm for larger numbers of machines. It has also been applied in layered graph drawing.[29] In a survey article on scheduling algorithms published in 1979, Graham and his coauthors introduced a three-symbol notation for classifying theoretical scheduling problems according to the system of machines they are to run on, the characteristics of the tasks and resources such as requirements for synchronization or non-interruption, and the performance measure to be optimized.[A79] This classification has sometimes been called "Graham notation" or "Graham's notation".[30] Discrete and computational geometry Graham scan is a widely used and practical algorithm for convex hulls of two-dimensional point sets, based on sorting the points and then inserting them into the hull in sorted order.[31] Graham published the algorithm in 1972.[A72c] The biggest little polygon problem asks for the polygon of largest area for a given diameter. Surprisingly, as Graham observed, the answer is not always a regular polygon.[A75a] Graham's 1975 conjecture on the shape of these polygons was finally proven in 2007.[32] In another 1975 publication, Graham and Erdős observed that for packing unit squares into a larger square with non-integer side lengths, one can use tilted squares to leave an uncovered area that is sublinear in the side length of the larger square, unlike the obvious packing with axis-aligned squares.[A75b] Klaus Roth and Bob Vaughan proved that uncovered area at least proportional to the square root of the side length may sometimes be needed; proving a tight bound on the uncovered area remains an open problem.[33] Probability and statistics In nonparametric statistics, a 1977 paper by Persi Diaconis and Graham studied the statistical properties of Spearman's footrule, a measure of rank correlation that compares two permutations by summing, over each item, the distance between the positions of the item in the two permutations.[A77] They compared this measure to other rank correlation methods, resulting in the "Diaconis–Graham inequalities" $I+E\leq D\leq 2I$ where $D$ is Spearman's footrule, $I$ is the number of inversions between the two permutations (a non-normalized version of the Kendall rank correlation coefficient), and $E$ is the minimum number of two-element swaps needed to obtain one permutation from the other.[34] The Chung–Diaconis–Graham random process is a random walk on the integers modulo an odd integer $p$, in which at each step one doubles the previous number and then randomly adds zero, $1$, or $-1$ (modulo $p$). In a 1987 paper, Chung, Diaconis, and Graham studied the mixing time of this process, motivated by the study of pseudorandom number generators.[A87][35] Juggling Graham became a capable juggler beginning at age 15, and was practiced in juggling up to six balls.[4] (Although a published photo shows him juggling twelve balls,[5] it is a manipulated image.[3]) He taught Steve Mills, a repeat winner of the International Jugglers' Association championships, how to juggle, and his work with Mills helped inspire Mills to develop the Mills' Mess juggling pattern. As well, Graham made significant contributions to the theory of juggling, including a sequence of publications on siteswaps. In 1972 he was elected president of the International Jugglers' Association.[4] Awards and honors In 2003, Graham won the American Mathematical Society's annual Leroy P. Steele Prize for Lifetime Achievement. The prize cited his contributions to discrete mathematics, his popularization of mathematics through his talks and writing, his leadership at Bell Labs, and his service as president of the society.[2] He was one of five inaugural winners of the George Pólya Prize of the Society for Industrial and Applied Mathematics, sharing it with fellow Ramsey theorists Klaus Leeb, Bruce Rothschild, Alfred Hales, and Robert I. Jewett.[36] He was also one of two inaugural winners of the Euler Medal of the Institute of Combinatorics and its Applications, the other being Claude Berge.[37] Graham was elected to the National Academy of Sciences in 1985.[38] In 1999 he was inducted as an ACM Fellow "for seminal contributions to the analysis of algorithms, in particular the worst-case analysis of heuristics, the theory of scheduling, and computational geometry".[39] He became a Fellow of the Society for Industrial and Applied Mathematics in 2009; the fellow award cited his "contributions to discrete mathematics and its applications".[40] In 2012 he became a fellow of the American Mathematical Society.[41] Graham was an invited speaker at the 1982 International Congress of Mathematicians (held 1983 in Warsaw),[13] speaking on "Recent developments in Ramsey theory".[A84] He was twice Josiah Willard Gibbs Lecturer, in 2001 and 2015.[13] The Mathematical Association of America awarded him both the Carl Allendoerfer Prize for his paper "Steiner Trees on a Checkerboard" with Chung and Martin Gardner in Mathematics Magazine (1989),[A89b][42] and the Lester R. Ford Award for his paper "A whirlwind tour of computational geometry" with Frances Yao in the American Mathematical Monthly (1990).[A90][43] His book Magical Mathematics with Persi Diaconis[B6] won the Euler Book Prize.[44] The proceedings of the Integers 2005 conference was published as a festschrift for Ron Graham's 70th birthday.[45] Another festschrift, stemming from a conference held in 2015 in honor of Graham's 80th birthday, was published in 2018 as the book Connections in discrete mathematics: a celebration of the work of Ron Graham.[46] Selected publications Books B1. Old and new results in combinatorial number theory. With Paul Erdős. Monographie 28, L'Enseignement Mathématique, 1980.[47] B2. Ramsey Theory. With Bruce Rothschild and Joel Spencer. Wiley, 1980; 2nd ed., 1990.[48] B3. Rudiments of Ramsey Theory. American Mathematical Society, 1981; 2nd ed., with Steve Butler, 2015.[49] B4. Concrete Mathematics: a foundation for computer science. With Donald Knuth and Oren Patashnik. Addison-Wesley, 1989; 2nd ed., 1994.[50] B5. Erdős on Graphs. His legacy of unsolved problems. With Fan Chung. A K Peters, 1998.[51] B6. Magical Mathematics: the mathematical ideas that animate great magic tricks. With Persi Diaconis. Princeton University Press, 2011.[52] Edited volumes V1. Handbook of Combinatorics. Edited with Martin Grötschel and László Lovász. MIT Press, 1995.[53] V2. The mathematics of Paul Erdős. Edited with Jaroslav Nešetřil. 2 volumes. Springer, 1997; 2nd ed., 2013.[54] Articles A64. Graham, Ronald L. (1964). "A Fibonacci-like sequence of composite numbers" (PDF). Mathematics Magazine. 37 (5): 322–324. doi:10.2307/2689243. JSTOR 2689243. MR 1571455. Zbl 0125.02103. A66. Graham, R. L. (1966). "Bounds for certain multiprocessing anomalies" (PDF). Bell System Technical Journal. 45 (9): 1563–1581. doi:10.1002/j.1538-7305.1966.tb01709.x. Zbl 0168.40703. A69. Graham, R. L. (1969). "Bounds on multiprocessing timing anomalies" (PDF). SIAM Journal on Applied Mathematics. 17 (2): 416–429. doi:10.1137/0117039. MR 0249214. Zbl 0188.23101. A71a. Graham, R. L.; Rothschild, B. L. (1971). "Ramsey's theorem for n-parameter sets" (PDF). Transactions of the American Mathematical Society. 159: 257–292. doi:10.1090/S0002-9947-1971-0284352-8. JSTOR 1996010. MR 0284352. Zbl 0233.05003. A71b. Graham, R. L.; Pollak, H. O. (1971). "On the addressing problem for loop switching" (PDF). Bell System Technical Journal. 50 (8): 2495–2519. doi:10.1002/j.1538-7305.1971.tb02618.x. MR 0289210. Zbl 0228.94020. A72a. Graham, R. L.; Pollak, H. O. (1972). "On embedding graphs in squashed cubes". Graph theory and applications (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1972; dedicated to the memory of J. W. T. Youngs) (PDF). Lecture Notes in Mathematics. Vol. 303. pp. 99–110. MR 0332576. Zbl 0251.05123. A72b. Coffman, E. G. Jr.; Graham, R. L. (1972). "Optimal scheduling for two-processor systems" (PDF). Acta Informatica. 1 (3): 200–213. doi:10.1007/bf00288685. MR 0334913. S2CID 40603807. Zbl 0248.68023. A72c. Graham, R. L. (1972). "An efficient algorithm for determining the convex hull of a finite planar set" (PDF). Information Processing Letters. 1 (4): 132–133. doi:10.1016/0020-0190(72)90045-2. Zbl 0236.68013. A74. Johnson, D. S.; Demers, A.; Ullman, J. D.; Garey, M. R.; Graham, R. L. (1974). "Worst-case performance bounds for simple one-dimensional packing algorithms" (PDF). SIAM Journal on Computing. 3 (4): 299–325. doi:10.1137/0203025. MR 0434396. Zbl 0297.68028. A75a. Graham, R. L. (1975). "The largest small hexagon" (PDF). Journal of Combinatorial Theory. Series A. 18 (2): 165–170. doi:10.1016/0097-3165(75)90004-7. MR 0360353. Zbl 0299.52006. A75b. Erdős, P.; Graham, R. L. (1975). "On packing squares with equal squares" (PDF). Journal of Combinatorial Theory. Series A. 19: 119–123. doi:10.1016/0097-3165(75)90099-0. MR 0370368. Zbl 0324.05018. A77. Diaconis, Persi; Graham, R. L. (1977). "Spearman's footrule as a measure of disarray". Journal of the Royal Statistical Society. 39 (2): 262–268. doi:10.1111/j.2517-6161.1977.tb01624.x. JSTOR 2984804. MR 0652736. Zbl 0375.62045. A79. Graham, R. L.; Lawler, E. L.; Lenstra, J. K.; Rinnooy Kan, A. H. G. (1979). "Optimization and approximation in deterministic sequencing and scheduling: a survey" (PDF). Annals of Discrete Mathematics. 5: 287–326. doi:10.1016/S0167-5060(08)70356-X. ISBN 9780080867670. MR 0558574. Zbl 0411.90044. A84. Graham, R. L. (1984). "Recent developments in Ramsey theory" (PDF). Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983). Warsaw: PWN. pp. 1555–1567. MR 0804796. Zbl 0572.05009. A87. Chung, F. R. K.; Diaconis, Persi; Graham, R. L. (1987). "Random walks arising in random number generation" (PDF). Annals of Probability. 15 (3): 1148–1165. doi:10.1214/aop/1176992088. JSTOR 2244046. MR 0893921. Zbl 0622.60016. A89a. Chung, F. R. K.; Graham, R. L.; Wilson, R. M. (1989). "Quasi-random graphs" (PDF). Combinatorica. 9 (4): 345–362. doi:10.1007/BF02125347. MR 1054011. S2CID 17166765. Zbl 0715.05057. A89b. Chung, Fan; Gardner, Martin; Graham, Ron (1989). "Steiner trees on a checkerboard" (PDF). Mathematics Magazine. 62 (2): 83–96. doi:10.2307/2690388. JSTOR 2690388. MR 0991536. Zbl 0681.05018. A90. Graham, Ron; Yao, Frances (1990). "A whirlwind tour of computational geometry" (PDF). American Mathematical Monthly. 97 (8): 687–701. doi:10.2307/2324575. JSTOR 2324575. MR 1072812. Zbl 0712.68097. A15. Butler, Steve; Erdős, Paul; Graham, Ron (2015). "Egyptian fractions with each denominator having three distinct prime divisors" (PDF). Integers. 15: A51. MR 3437526. Zbl 1393.11030. References 1. O'Connor, John J.; Robertson, Edmund F. "Ronald Graham". MacTutor History of Mathematics Archive. University of St Andrews. 2. "2003 Steele Prizes" (PDF). Notices of the American Mathematical Society. Vol. 50, no. 4. April 2003. pp. 462–467. Archived from the original (PDF) on February 6, 2011. Retrieved July 2, 2014. 3. Horgan, John (March 1997). "Profile: Ronald L. Graham – Juggling Act". Scientific American. 276 (3): 28–30. doi:10.1038/scientificamerican0397-28. 4. "Ron Graham Obituary". International Jugglers' Association. July 9, 2020. Retrieved July 13, 2020. 5. "Juggling Numbers: UC San Diego Professor Honored for Work in Applied Mathematics and Computational Science". California Institute for Telecommunications and Information Technology. May 4, 2009. Retrieved July 9, 2020. 6. "Ronald Lewis Graham, 2003–2004 MAA President". Mathematical Association of America. July 7, 2020. Retrieved July 7, 2020. 7. Albers, Donald J. (November 1996). "A Nice Genius". Math Horizons. 4 (2): 18–23. doi:10.1080/10724117.1996.11974993. JSTOR 25678089. 8. Bigelow, Bruce V. (March 18, 2003). "You can count on him: Math expert coolly juggles scientific puzzles and six or seven balls" (PDF). The San Diego Union-Tribune. 9. Ronald Graham at the Mathematics Genealogy Project 10. Hoffman, Paul (1998). The man who loved only numbers: the story of Paul Erdős and the search for mathematical truth. Hyperion. pp. 109–110. ISBN 978-0-7868-6362-4. 11. Rabiner, Larry (February 4, 2000). "Ron Graham – A Biographical Retrospective" (PDF). 12. Chang, Kenneth (July 23, 2020). "Ronald L. Graham, Who Unlocked the Magic of Numbers, Dies at 84". The New York Times. Retrieved January 28, 2021. 13. "The Latest: Ronald Graham, 1935–2020". American Mathematical Society. July 7, 2020. Retrieved July 7, 2020. 14. Ron Graham obituary by Colm Mulcahy, The Guardian, August 3, 2020 15. "Erdos1: coauthors of Paul Erdős, together with their coauthors listed beneath them". Erdős Number Project. Retrieved July 12, 2020. 16. Peck, G. W. (2002). "Kleitman and combinatorics: a celebration". Discrete Mathematics. 257 (2–3): 193–224. doi:10.1016/S0012-365X(02)00595-2. MR 1935723. See in particular Section 4, "The mysterious G. W. Peck", pp. 216–219. 17. Croot, Ernest S., III (2003). "On a coloring conjecture about unit fractions". Annals of Mathematics. 157 (2): 545–556. arXiv:math.NT/0311421. Bibcode:2003math.....11421C. doi:10.4007/annals.2003.157.545. MR 1973054. S2CID 13514070.{{cite journal}}: CS1 maint: multiple names: authors list (link) 18. Roberts, Siobhan (December 10, 2015). "New Erdős Paper Solves Egyptian Fraction Problem". Simons Foundation. 19. Knuth, Donald E. (1990). "A Fibonacci-like sequence of composite numbers". Mathematics Magazine. 63 (1): 21–25. doi:10.2307/2691504. JSTOR 2691504. MR 1042933. 20. Guinness Book of World Records (Rev. American ed.). Sterling Publishing. 1980. p. 193. ISBN 0806901683. 21. Bennett, Jay (October 20, 2017). "The Enormity of the Number TREE(3) Is Beyond Comprehension". Popular Mechanics. Retrieved July 9, 2020. 22. Lamb, Evelyn (May 26, 2016). "Two-hundred-terabyte maths proof is largest ever". Nature. 534 (7605): 17–18. Bibcode:2016Natur.534...17L. doi:10.1038/nature.2016.19990. PMID 27251254. 23. Aigner, Martin; Ziegler, Günter M. (2018). Proofs from THE BOOK (6th ed.). Springer. pp. 79–80. doi:10.1007/978-3-662-57265-8_15. ISBN 978-3-662-57265-8. 24. Shapira, Asaf (2008). "Quasi-randomness and the distribution of copies of a fixed graph". Combinatorica. 28 (6): 735–745. doi:10.1007/s00493-008-2375-0. MR 2488748. S2CID 3212684. 25. Chung, Fan R. K. (1989). "Pebbling in hypercubes". SIAM Journal on Discrete Mathematics. 2 (4): 467–472. doi:10.1137/0402041. 26. Pleanmani, Nopparat (2019). "Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph". Discrete Mathematics, Algorithms and Applications. 11 (6): 1950068, 7. doi:10.1142/s179383091950068x. MR 4044549. S2CID 204207428. 27. Albers, Susanne (2012). Grötschel, Martin (ed.). Ronald Graham: laying the foundations of online optimization. Documenta Mathematica. pp. 239–245. MR 2991486. 28. Garey, M. R.; Johnson, D. S. (1981). "Approximation Algorithms for Bin Packing Problems: A Survey". In Ausiello, G.; Lucertini, M. (eds.). Analysis and Design of Algorithms in Combinatorial Optimization. Courses and Lectures of the International Centre for Mechanical Sciences. Vol. 266. Vienna: Springer. pp. 147–172. doi:10.1007/978-3-7091-2748-3_8. 29. Bastert, Oliver; Matuszewski, Christian (2001). "Layered drawings of digraphs". In Kaufmann, Michael; Wagner, Dorothea (eds.). Drawing Graphs: Methods and Models. Lecture Notes in Computer Science. Vol. 2025. Springer-Verlag. pp. 87–120. doi:10.1007/3-540-44969-8_5. 30. For a recent example, see e.g. Cygan, Marek; Pilipczuk, Marcin; Pilipczuk, Michał; Wojtaszczyk, Jakub Onufry (2014). "Scheduling partially ordered jobs faster than $2^{n}$". Algorithmica. 68 (3): 692–714. doi:10.1007/s00453-012-9694-7. MR 3160651. 31. De Berg, Mark; Cheong, Otfried; Van Kreveld, Marc; Overmars, Mark (2008). Computational Geometry: Algorithms and Applications. Berlin: Springer. pp. 2–14. doi:10.1007/978-3-540-77974-2. ISBN 978-3-540-77973-5. 32. Foster, Jim; Szabo, Tamas (2007). "Diameter graphs of polygons and the proof of a conjecture of Graham". Journal of Combinatorial Theory. Series A. 114 (8): 1515–1525. doi:10.1016/j.jcta.2007.02.006. MR 2360684.. 33. Brass, Peter; Moser, William; Pach, János (2005). Research Problems in Discrete Geometry. New York: Springer. p. 45. ISBN 978-0387-23815-9. MR 2163782. 34. Hadjicostas, Petros; Monico, Chris (2015). "A new inequality related to the Diaconis-Graham inequalities and a new characterisation of the dihedral group". The Australasian Journal of Combinatorics. 63: 226–245. MR 3403376. 35. Hildebrand, Martin (2019). "On a lower bound for the Chung-Diaconis-Graham random process". Statistics & Probability Letters. 152: 121–125. doi:10.1016/j.spl.2019.04.020. MR 3953053. 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JSTOR 2653387.{{cite journal}}: CS1 maint: untitled periodical (link) • Tutte, W. T. (September 2000). SIAM Review. 42 (3): 548–549. JSTOR 2653326.{{cite journal}}: CS1 maint: untitled periodical (link) • Hobbs, Arthur M. (April 2001). American Mathematical Monthly. 108 (4): 379–381. doi:10.2307/2695262. JSTOR 2695262.{{cite journal}}: CS1 maint: untitled periodical (link) • Crilly, Tony (July 2001). The Mathematical Gazette. 85 (503): 375–377. doi:10.2307/3622075. JSTOR 3622075. S2CID 171483616.{{cite journal}}: CS1 maint: untitled periodical (link) 52. Reviews of Magical Mathematics: • Rogovchenko, Yuri V. zbMATH. Zbl 1230.00009.{{cite journal}}: CS1 maint: untitled periodical (link) • Young, Jeffrey R. (October 16, 2011). "The magical mind of Persi Diaconis". The Chronicle of Higher Education. • Cook, John D. (November 2011). "Review". MAA Reviews. Mathematical Association of America. • Howls, C. J. (November 23, 2011). "To create illusions, Fibonacci and algorithms are as important as sleight of hand". Times Higher Education. • Stone, Alex (December 10, 2011). "Pick a card, any card". The Wall Street Journal. • Benjamin, Arthur (2012). "Featured review" (PDF). SIAM Review. 54 (3): 609–612. doi:10.1137/120973238. JSTOR 41642632. MR 2985718. • Wiseman, Richard (February 2012). "Just like that". Nature Physics. 8 (2): 104–105. Bibcode:2012NatPh...8..104W. doi:10.1038/nphys2225. S2CID 120357097. • Davis, Philip J. (March 18, 2012). "Tricky mathematics". SIAM News. • Ó Cairbre, Fiacre (Summer 2012). "Review" (PDF). Irish Mathematical Society Bulletin. 69: 60–62. • Castrillón López, Marco (July 2012). "Review". EMS Reviews. European Mathematical Society. • Van Osdol, Donovan H. (August 2012). Notices of the American Mathematical Society. 59 (7): 960–961. doi:10.1090/noti875.{{cite journal}}: CS1 maint: untitled periodical (link) • Bledsoe, Christie (April 2013). The Mathematics Teacher. 106 (8): 637. doi:10.5951/mathteacher.106.8.0637. JSTOR 10.5951/mathteacher.106.8.0637.{{cite journal}}: CS1 maint: untitled periodical (link) • Robert, Christian (April 2013). Chance. 26 (2): 50–51. doi:10.1080/09332480.2013.794620. S2CID 60760932.{{cite journal}}: CS1 maint: untitled periodical (link) • Scarrabelotti, Jack (2014). "Review". Australian Mathematics Teacher. 70 (1): 29. • Brown, Jill (2015). "Review". Australian Senior Mathematics Journal. 29 (2): 62. 53. Reviews of Handbook of Combinatorics: • Wilf, Herbert S. (March 1997). The Mathematical Intelligencer. 19 (2): 68–69. doi:10.1007/bf03024438.{{cite journal}}: CS1 maint: untitled periodical (link) • Gasarch, William (June 1999). "Review" (PDF). ACM SIGACT News. 30 (2): 7. doi:10.1145/568547.568551. S2CID 3200815. 54. Reviews of The Mathematics of Paul Erdős: • Soifer, A. zbMATH. Zbl 0916.01022.{{cite journal}}: CS1 maint: untitled periodical (link) • Bauer, Craig P. (December 2013). "Review". MAA Reviews. Mathematical Association of America. External links • Graham's UCSD Faculty Research Profile • Papers of Ron Graham – a comprehensive archive of the papers written by Ron Graham • About Ron Graham – a page summarizing some aspects of Graham's life and mathematics – part of Fan Chung's website • "Simons Foundation: Ronald Graham (1935–2020)". Simons Foundation. January 11, 2016. – Extended video interview. • "Three Mathematicians We Lost in 2020: John Conway, Ronald Graham, and Freeman Dyson all explored the world with their minds" Rockmore, Dan. (December 31, 2020) The New Yorker. • Buhler, Joe; Butler, Steve; Spencer, Joel (December 2021). "Ronald Lewis Graham (1935–2020)" (PDF). Notices of the American Mathematical Society. 68 (11): 1931–1950. doi:10.1090/noti2382. • Ronald Graham publications indexed by Google Scholar Juggling and object manipulation Patterns and forms • Box • Cascade • Claw • Columns • Flash • Fountain • Havana • Joggling • Jollyball • Mills' Mess • Multiplex • Notation • Siteswap: list • Passing • Rubenstein's Revenge • Shower • Toss Props • Ball • Bean bag • Bouncy ball • Hacky sack • Cigar box • Club • Indian club • Cup-and-ball • Kendama • Fan dance • Fingerboard • Flagging dance • Flag throwing • Hat • Jianzi • Knife • Plate spinning • Ring • Torch • Trick roping Groups and events • Juggling Conventions • BACAF • BJC • EJC • IJC • Competitions • Combat juggling • IJA • JIS • JISCON • Renegade show • WJF Balance and twirling • Astrojax • Alaskan yo-yo • Baton twirling • Chinese yo-yo • Chinese top • Coin manipulation • Contact juggling • Devil sticks • Fire fan • Fire staff • Flair bartending • Hooping • Hoop rolling • Kemari • Meteor • Keepie uppie • Pen spinning • Poi • tricks • Top • Rattleback • Yo-yo Other • Culture • History • Ancient China • Jugglers • Rhythmic gymnastics • Robot • Terms • World records Presidents of the American Mathematical Society 1888–1900 • John Howard Van Amringe (1888–1890) • Emory McClintock (1891–1894) • George William Hill (1895–1896) • Simon Newcomb (1897–1898) • Robert Simpson Woodward (1899–1900) 1901–1924 • E. H. Moore (1901–1902) • Thomas Fiske (1903–1904) • William Fogg Osgood (1905–1906) • Henry Seely White (1907–1908) • Maxime Bôcher (1909–1910) • Henry Burchard Fine (1911–1912) • Edward Burr Van Vleck (1913–1914) • Ernest William Brown (1915–1916) • Leonard Eugene Dickson (1917–1918) • Frank Morley (1919–1920) • Gilbert Ames Bliss (1921–1922) • Oswald Veblen (1923–1924) 1925–1950 • George David Birkhoff (1925–1926) • Virgil Snyder (1927–1928) • Earle Raymond Hedrick (1929–1930) • Luther P. Eisenhart (1931–1932) • Arthur Byron Coble (1933–1934) • Solomon Lefschetz (1935–1936) • Robert Lee Moore (1937–1938) • Griffith C. Evans (1939–1940) • Marston Morse (1941–1942) • Marshall H. Stone (1943–1944) • Theophil Henry Hildebrandt (1945–1946) • Einar Hille (1947–1948) • Joseph L. Walsh (1949–1950) 1951–1974 • John von Neumann (1951–1952) • Gordon Thomas Whyburn (1953–1954) • Raymond Louis Wilder (1955–1956) • Richard Brauer (1957–1958) • Edward J. 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Ronald Gould (mathematician) Ronald James Gould (born April 15, 1950) is an American mathematician specializing in combinatorics and graph theory. He is a Goodrich C. White professor emeritus in the Emory University Department of Mathematics. Ronald Gould Ronald Gould in 2019 at the Mississippi Discrete Mathematics Workshop Born Ronald James Gould (1950-04-15) April 15, 1950 Other namesRon Gould Alma mater • Western Michigan University (Ph.D.) • State University of New York at Fredonia (B.S.) Scientific career FieldsMathematics InstitutionsEmory University ThesisTraceability in Graphs (1979) Doctoral advisorGary Chartrand Websitehttps://www.math.emory.edu/~rg/ Education and career After attending SUNY Fredonia for his undergraduate degree, Gould received his Ph.D. in 1979 from Western Michigan University. His thesis was titled Traceability in Graphs, and was completed under the supervision of Gary Chartrand.[1][2] He spent a short period as a lecturer at San Jose State University in 1978 and 1979, then moved to Emory University in 1979. He was named to the Goodrich C. White professorship in 2001, and retired in 2016.[3] Gould is most noted for his work in the area of Hamiltonian graph theory.[4] His book Mathematics in Games, Sports, and Gambling: The Games People Play[5] won the American Library Association award for Outstanding Academic Titles, 2010. Gould has over 180 mathematical publications,[6] and has advised 28 Ph.D. students.[1] References 1. Ronald J. Gould at the Mathematics Genealogy Project 2. "Traceability in Graphs (Ph.D. thesis)" (Document). ProQuest 302945071. {{cite document}}: Cite document requires \|publisher= (help) 3. "Curriculum Vita" (PDF). Ron Gould's homepage. Retrieved November 7, 2019. 4. Gould, Ronald J. (2013). "Recent Advances on the Hamiltonian Problem: Survey III". Graphs and Combinatorics. 30 (1): 1–46. doi:10.1007/s00373-013-1377-x. ISSN 0911-0119. MR 3143857. S2CID 33743372. 5. Gould, Ronald (2010). Mathematics in Games, Sports, and Gambling : the Games People Play. Boca Raton: CRC Press. ISBN 978-1-4398-0163-5. OCLC 255899527. (a second edition is also available, OCLC 1003704485) 6. "Ronald J. Gould author profile at MathSciNet". Retrieved November 7, 2019. External links • Ron Gould's homepage at Emory University Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • United States • Latvia • Czech Republic • Netherlands • Poland Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Ronald L. Iman Ronald L. Iman is an American statistician.[1] He was one of the developers of the statistical technique known as Latin hypercube sampling. Books • A Data-Based Approach to Statistics (1994) • A Data-Based Approach to Statistics: Concise Version (1995) • Modern Business Statistics with W. J. Conover, 2e (1989) • Modern Business Statistics with W. J. Conover, (1983) • Introduction to Modern Business Statistics with W. J. Conover, (1983) • A Modern Approach to Statistics with W. J. Conover, (1983) References 1. "[ISI Highly Cited Researchers Version 1.5]". hcr3.isiknowledge.com. Retrieved 2010-12-02. Authority control International • ISNI • VIAF National • Norway • Israel • United States • Czech Republic • Netherlands	Wikipedia
Ron Larson Roland "Ron" Edwin Larson (born October 31, 1941) is a professor of mathematics at Penn State Erie, The Behrend College, Pennsylvania.[1] He is best known for being the author of a series of widely used mathematics textbooks ranging from middle school through the second year of college. Ron Larson Ron Larson, June, 2002 BornRoland Edwin Larson (1941-10-31) October 31, 1941 Fort Lewis, Washington, U.S. OccupationProfessor, author, mathematician CitizenshipAmerican EducationA.A., B.S., M.A., Ph.D. Alma materClark College Lewis & Clark College University of Colorado Spouse Deanna Sue Gilbert (m. 1960) Children2 Website www.ronlarson.com Personal life Ron Larson was born in Fort Lewis near Tacoma, Washington, the second of four children of Mederith[lower-alpha 1] John Larson and Harriet Eleanor Larson. Mederith Larson was an officer in the 321st Engineer Battalion of the United States Army. Mederith Larson served in active duty during World War II, where he was awarded a Bronze Star Medal and a Purple Heart, and the Korean War, where he was awarded an Oak Leaf Cluster and a Silver Star.[2] During the years that Ron was growing up, his father was stationed in several military bases, including Chitose, Hokkaido, Japan and Schofield Barracks, Hawaii. While in Chitose, Ron attended a small DoDDS school, where he was one of only three students in the sixth grade. When Mederith Larson retired from the Army in 1957, he moved with his family to Vancouver, Washington, where he lived until he died (at the age of 89) in 2005.[3] Harriet Larson died (at the age of 95) in the fall of 2009. Larson spent his first two years of high school at Leilehua High School in Wahiawa, Hawaii. In 1957, when his family moved to Vancouver, Washington, Larson enrolled in Battle Ground High School, where he graduated in 1959. On October 29, 1960, at the age of 18, he married Deanna Sue Gilbert, also of Vancouver, Washington. Deanna Gilbert was the second child of Herbert and Dorothy Gilbert. Ron and Deanna Larson have two children, Timothy Roland Larson [4] and Jill Deanna Larson Im, five living grandchildren, and two great-grandchildren. Their first grandchild, Timothy Roland Larson II, died at birth on summer solstice, June 21, 1983. Larson is the third generation of Norwegian and Swedish immigrants who left Scandinavia to homestead in Minnesota in the late 1800s.[5] The surnames and immigration dates of his great-grandparents are Bangen (1866, Norway), Berg (1867, Norway), Larson (1868, Norway), and Watterburg (1879, Sweden). Larson has contributed several thousand dollars to Republican politicians, including Rand Paul, Marco Rubio, Mitt Romney, and Scott Brown.[6] Education From 1959 until 1962, Ron and Deanna Larson started and operated a small business, called Larson's Custom Quilting. In 1962, they sold the business and Ron began attending Clark College in Vancouver, Washington. In 1964, he obtained his associate degree from Clark. Upon graduation from Clark College, Larson was awarded a scholarship from the Alcoa Foundation, which he used to attend Lewis & Clark College in Portland, Oregon. He graduated, with honors, from Lewis & Clark in 1966. During the four years from 1962 through 1966, Ron worked full-time, first at a restaurant and then at a grocery store, in Vancouver and Deanna worked full-time as the secretary to the president of Roberts Motor Company in Portland, Oregon.[7] From 1966 to 1970, Larson attended graduate school at the University of Colorado at Boulder. He received his master's degree in 1968 and his Ph.D. in mathematics in 1970. While at the University of Colorado, Larson was the recipient of an NDEA scholarship and an NSF fellowship. He also served as a teaching assistant. His Ph.D. dissertation "On the Lattice of Topologies" was written under Wolfgang J. Thron.[8] Larson's Ph.D. lineage, as listed by the North Dakota State University, traces back through George David Birkhoff, Joseph Louis Lagrange, Leonhard Euler, and Gottfried Wilhelm Leibniz, the co-developer of calculus.[9] Academic career In 1970, Larson accepted a position of assistant professor at Penn State Erie, The Behrend College in Erie, Pennsylvania. At the time, Behrend College was a 2-year branch campus of the university. In 1971, the Board of Trustees of the University met with the Behrend Advisory Board to announce Behrend College would become the first location outside University Park with the authority to develop baccalaureate program and confer degrees locally.[10] During his first several years at the college, Larson was instrumental in developing a mathematics major at the college. He served as a member of the University Faculty Senate and also as Behrend College's representative on the University Faculty Council. Larson was promoted to associate professor in 1976 and professor in 1983. Early in his career at Penn State, Larson started writing manuscripts for textbooks. He completed and submitted three manuscripts for calculus texts in 1973, 1974, and 1975 ... only to be rejected by several publishers. Larson relates his determination to continue writing in an essay titled A Single Dream.[11] "When Marilyn Monroe was asked if she had been lucky in her career, she said 'When you have a single dream it is more than likely to come true---because you keep working toward it without getting mixed up.' Anyone who has been in my office knows that I am a Marilyn fan. But not just a Marilyn fan---I am a fan of the American Dream." Finally, in 1976 he and his co-author, Robert P. Hostetler, obtained a contract from D. C. Heath and Company. The first edition of their calculus book was published in December, 1978. In 1995, Hostetler left the authorship team and was replaced by Bruce A. Edwards of the University of Florida. Calculus by Larson and Edwards is now in its eleventh edition. It is used worldwide and has been translated in several languages.[12] During the academic year of 1983–84, Larson served as the acting division head for the Division of Science at Penn State Erie. In 1998 Larson was given the Distinguished Alumnus Award from Lewis and Clark College, Portland, Oregon.[13] Books Counting different editions, Larson has written over 400 titles.[14][15] They are used by several million students each year in the United States, as well as by students in other countries. Larson's books have received many awards – for pedagogy, innovation, and design.[16][17] One of these awards was for developing the first completely interactive calculus textbook online. The work on this text was spearheaded by Larson's son, Timothy Larson. The online text, titled Interactive Calculus was posted in 1995. Another award was for innovation in page design. Beginning in 1990, Larson has written all of his mathematics texts to design, so that concepts and examples never break from page to page. The eighth edition of Calculus won the 2005 Benny Award for the best cover in all categories of printing.[18] The middle school series, Big Ideas Math, won the TAA Textbook Excellence Award ("Texty") in 2010 for excellence in secondary mathematics textbook publishing.[19] Larson's textbooks have won awards from the Textbook Authors' Association (TAA) multiple times, including the McGuffey Longevity Award, the TAA Textbook Excellence Award, and the Most Promising New Textbook Excellence Award. Up until 1995, most of Larson's books were published by D. C. Heath, which was owned by Raytheon. In 1995, Raytheon sold D. C. Heath to Houghton Miffin. By 1999, Larson's titles had become a major component of Houghton Mifflin's publications. In that year, he was listed in the company's annual report as one of Houghton Mifflin's major authors. In 2008, the College Division of Houghton Mifflin was sold to Cengage Learning. Larson's textbooks have been translated into Spanish, Portuguese, Chinese, and French.[20][21][22] Company founder In 1984, Larson formed a small company that he called Larson Texts,[4] starting with four employees in an old cottage on the campus of Behrend College. The cottage had been part of the original estate of Ernst Behrend, founder of Hamermill Paper Company.[23] This company grew through a sequence of larger offices. In 1992, Larson gave up his sole proprietorship of the company to form a corporation called Larson Texts, Inc.[24] In the same year the company purchased Typographics, a small typesetting firm in Erie, Pennsylvania. Typographics came with a group of employees who were experienced in design, graphic arts, and composition. In 2000, the company bought and renovated the former Belle Valley School into a 14,000-square-foot (1,300 m2) office building.[25] It has over 70 employees, who work in design, composition, and research ... all connected with the development and production of Larson's textbooks. In 2000, it was listed in the Top Ten Best Places to Work in Pennsylvania for medium-sized companies.[26] In 2008, the company formed a publishing division called Big Ideas Learning which publishes textbooks at the middle and high school levels. These textbooks follow the new curriculum and practices specified by the Common Core State Standards Initiative. In addition to specifying a common curriculum in mathematics, the Common Core State Standards Initiative specifies eight Mathematical Practices that students should be taught. In keeping with this curriculum and practices, Larson has written a book titled Mathematical Practices, with examples of how teachers can implement the curriculum and practices in Grades K-8.[27] Beginning in 2008, Larson Texts began a policy of having the content of its copyrights available free on the Internet. Currently, this policy applies to all of its 7 middle school titles, its 3 high school titles, and its title Math & You. Instructional software In 1992, Larson Texts formed a software division called Meridian Creative Group, later renamed as Larson Learning. The division developed and sold tutorial mathematics software for grades K through 8.[28] In 2005, Larson Learning was sold to Houghton Mifflin for $7 million.[29] 1. Larson, Ron; Robyn Silbey (1998). "Larson's Middle School Math, Grades 6, 7, and 8". Larson Learning 2. Larson, Ron (2000). "Larson's Intermediate Math, Grades 3, 4, 5, and 6". Larson Learning 3. Larson, Ron (2002). "Larson's Elementary Math, Grades Kindergarten, 1, and 2". Larson Learning In 2013, Larson devised and designed an educational game called My Dear Aunt Sally.[30] It was programmed by 3G Studios. The game is intended for Grades 2-8 and teaches operations involving whole numbers, integers, fractions, decimals, and rational numbers. Research During his first few years as an assistant professor at Penn State Erie, Larson continued to do research in the area of his dissertation. His research resulted in the publication of several articles, mostly dealing with the lattice of topologies.[31] By the mid-1970s, however, he switched his writing efforts to textbooks. 1. Larson, R. E., R. P. Hostetler and B. A. Edwards (June 1994, July 1994). "CD-ROM Textbook and Calculus". FOCUS: Mathematics Association of America. Continued involvement with education Larson is an active member of the three American mathematics teaching organizations: the National Council of Teachers of Mathematics, the American Mathematical Association of Two-Year Colleges, and the Mathematical Association of America. He is a frequent speaker at each of these organizations' state and national conferences.[32] Ron and Deanna Larson have been active in philanthropy at Penn State University. They are members of the Mount Nittany Society, which recognizes individuals who have given over $250,000 to the university.[33] Until 2008, all of Larson's textbooks were published by D. C. Heath, McGraw Hill, Houghton Mifflin, Prentice Hall, and McDougal Littell. In 2008, Larson was unable to find a publisher for a new series for middle school to follow the 2006 "Focal Point" recommendations of the National Council of Teachers of Mathematics.[34] He then started a new company to publish the books, Big Ideas Learning, LLC.[35][36] According to his acceptance speech for the Distinguished Alumnus Award in 1998, Ron's interest in writing mathematics textbooks started the summer after his sophomore year in college. "In my sophomore year I decided to switch to math. I wasn't prepared for it. I had forgotten my high school algebra and trig, and I had to spend my sophomore year taking those courses over again. After I was accepted to Lewis & Clark, I made an appointment to talk with the math department chair, Elvy Fredrickson. That was in June 1964. I asked Elvy if she would let me squeeze four years of math into my junior and senior years at Lewis & Clark. To imagine her thoughts, you have to remember that I had not even taken a course in freshman calculus. I didn't then know what Elvy was thinking. I only knew what she said and what she did. She went to a bookshelf in her office in the old math building, scanned the titles, took down a calculus text, handed it to me, and said, 'Study this book during the summer. The week before classes start in the fall, I will give you a test. If you pass, I will let you take your sophomore and junior mathematics courses concurrently. By the time you reach your senior year, you will be on track.' Years later, Elvy told me that she had no idea I would actually do it. But, I had no idea that she had no idea—and so I took her up on her offer. I read the calculus book, passed the test, and started taking third-semester calculus and linear algebra in the fall of 1964."[37] Awards 1. Roland E. Larson, Text and Academic Authors Association McGuffey Longevity Award, 1996, Calculus, 7th Edition (Houghton Mifflin) 2. Roland E. Larson, Text and Academic Authors Association Textbook Excellence Award, 1996, Interactive Calculus: Early Transcendental Functions, (Houghton Mifflin) 3. Roland E. Larson, Text and Academic Authors Association Textbook Excellence Award, 1997, Interactive College Algebra, (Houghton Mifflin) 4. Roland E. Larson, Text and Academic Authors Association Textbook Excellence Award, 1997, Larson's Leapfrog Math, (Meridian Creative Group) 5. Ron Larson, Text and Academic Authors Association McGuffey Longevity Award, 1998, Larson's Leapfrog Math, (Meridian Creative Group) 6. Ron Larson, Lewis and Clark College Distinguished Alumnus Award, 1998 7. Ron Larson, Text and Academic Authors Association McGuffey Longevity Award, 2004, Calculus, 7th Edition, (Houghton Mifflin) 8. Ron Larson, Text and Academic Authors Association Textbook Excellence Award, 2004, Precalculus, 6th Edition, (Houghton Mifflin) 9. Ron Larson, Text and Academic Authors Association McGuffey Longevity Award, 2006, Calculus, 8th Edition, (Houghton Mifflin) 10. Ron Larson, Text and Academic Authors Association Textbook Excellence Award, 2010, Big Ideas Math, 1st Edition, (Big Ideas Learning) 11. Ron Larson, Text and Academic Authors Association McGuffey Longevity Award, 2011, Precalculus: Real Math, Real People, 6th Edition, (Cengage Learning) 12. Ron Larson, Text and Academic Authors Association McGuffey Longevity Award, 2012, Calculus: An Applied Approach, 9th Edition, (Cengage Learning) 13. Ron Larson, Text and Academic Authors Association Textbook Excellence Award, 2012, Big Ideas Math: A Common Core Curriculum, 1st Edition, (Big Ideas Learning) 14. Ron Larson, Text and Academic Authors Association Textbook Excellence Award, 2013, Calculus, 10th Edition, (Cengage Learning) 15. Ron Larson, Text and Academic Authors Association Most Promising New Textbook Award, 2013, Math & You: The Power & Use of Mathematics, 1st Edition, (Larson Texts) 16. Ron Larson, Text and Academic Authors Association Most Promising New Textbook Award, 2013, Big Ideas Math: A Common Core Curriculum Algebra 1, 1st Edition, (Big Ideas Learning) 17. Ron Larson, Text and Academic Authors Association Textbook Excellence Award, 2014, Precalculus, 9th Edition, (Cengage Learning) 18. Ron Larson, Text and Academic Authors Association Textbook Excellence Award, 2014, Big Ideas Math: A Common Core Curriculum, 7 Book Series, 2nd Edition, (Big Ideas Learning Learning) 19. Ron Larson, Text and Academic Authors Association McGuffey Longevity Award, 2014, Calculus: Early Transcendental Functions, 6th Edition, (Cengage Learning) Published books 1. Larson, Roland E.; Robert P. Hostetler (1979), Calculus with Analytic Geometry, D. C. Heath 2. Larson, Roland E.; Robert P. Hostetler (1982), Mathematics for Everyday Living, Saunders 3. Larson, Roland E.; Robert P. Hostetler (1983), Calculus An Applied Approach, D. C. Heath 4. Larson, Roland E.; Robert P. Hostetler (1985), College Algebra, D. C. Heath 5. Larson, Roland E.; Robert P. Hostetler (1985), Algebra and Trigonometry, D. C. Heath 6. Larson, Roland E.; Robert P. Hostetler (1985), Trigonometry, D. C. Heath 7. Larson, Roland E.; Robert P. Hostetler (1985), Precalculus, D. C. Heath 8. Larson, Roland E.; Bruce H. Edwards (1988), Elementary Linear Algebra, D. C. Heath 9. Larson, Roland E.; Bruce H. Edwards (1991), Finite Mathematics, D. C. Heath 10. Larson, Roland E.; Bruce H. Edwards (1991), Finite Mathematics with Calculus, D. C. Heath 11. Larson, Roland E.; Robert P. Hostetler (1992), Elementary Algebra, D. C. Heath 12. Larson, Roland E.; Robert P. Hostetler (1992), Intermediate Algebra, D. C. Heath 13. Larson, Roland E.; Robert P. Hostetler; Anne V. Munn (1992), College Algebra Concepts and Models, D. C. Heath 14. Larson, Roland E.; Robert P. Hostetler, Bruce H. Edwards (1993), College Algebra A Graphing Approach, D. C. Heath 15. Larson, Roland E.; Robert P. Hostetler, Bruce H. Edwards (1993), Algebra and Trigonometry A Graphing Approach, D. C. Heath 16. Larson, Roland E.; Robert P. Hostetler (1993), Precalculus A Graphing Approach, D. C. Heath 17. Larson, Roland E.; Timothy D. Kanold, Lee Stiff (1993), Algebra 1, D. C. Heath 18. Larson, Roland E.; Timothy D. Kanold, Lee Stiff (1993), Algebra 2, D. C. Heath 19. Larson, Roland E.; Laurie Boswell, Lee Stiff (1994), Geometry, D. C. Heath 20. Larson, Roland E.; Robert P. Hostetler, Carolyn F. Neptune (1994), Intermediate Algebra Graphs and Functions, D. C. Heath 21. Larson, Roland E.; Robert P. Hostetler, Carolyn F. Neptune (1994), Algebra for College Students: Graphs and Functions, D. C. Heath 22. Larson, Roland E.; Robert P. Hostetler, Bruce H. Edwards (1995), Precalculus with Limits: A Graphing Approach, D. C. Heath 23. Larson, Roland E.; Robert P. Hostetler, Bruce H. Edwards (1995), Calculus Early Transcendental Functions, D. C. Heath 24. Larson, Roland E.; Robert P. Hostetler, Bruce H. Edwards (1995), Trigonometry A Graphing Approach, D. C. Heath 25. Larson, Roland E.; Laurie Boswell, Timothy D. Kanold, Lee Stiff (1996), Passport to Algebra and Geometry, D. C. Heath, McDougal Littell 26. Larson, Roland E.; Laurie Boswell, Timothy D. Kanold, Lee Stiff (1996), Windows to Algebra and Geometry, D. C. Heath 27. Larson, Roland E.; Laurie Boswell, Lee Stiff (1997), Passport to Mathematics Book 1, D. C. Heath, McDougal Littell 28. Larson, Roland E.; Laurie Boswell, Lee Stiff (1997), Passport to Mathematics Book 2, D. C. Heath, McDougal Littell 29. Larson, Ron; Betsy Farber (2000), Elementary Statistics Picturing the World, Prentice Hall 30. Larson, Ron; Robert P. Hostetler, Bruce H. Edwards (2000), College Algebra: An Internet Approach, Houghton Mifflin 31. Larson, Ron; Robert P. Hostetler, Bruce H. Edwards (2000), Precalculus: An Internet Approach, Houghton Mifflin 32. Larson, Roland E.; Laurie Boswell, Timothy D. Kanold, Lee Stiff (2001), Mathematics Concepts and Skills Course 1, McDougal Littell 33. Larson, Roland E.; Laurie Boswell, Timothy D. Kanold, Lee Stiff (2001), Mathematics Concepts and Skills Course 2, McDougal Littell 34. Larson, Ron; Laurie Boswell, Timothy D. Kanold, Lee Stiff (2001), Algebra 1 Concepts and Skills, McDougal Littell 35. Larson, Ron; Robert P. Hostetler, Bruce H. Edwards (2002), Calculus 1 with Precalculus, Houghton Mifflin 36. Larson, Ron; Laurie Boswell, Lee Stiff (2003), Geometry Concepts and Skills, McDougal Littell 37. Larson, Ron; Laurie Boswell, Timothy Kanold, Lee Stiff (2004), Math Course 1, McDougal Littell 38. Larson, Ron; Laurie Boswell, Timothy Kanold, Lee Stiff (2004), Math Course 2, McDougal Littell 39. Larson, Ron; Laurie Boswell, Timothy Kanold, Lee Stiff (2004), Math Course 3, McDougal Littell 40. Larson, Ron; Laurie Boswell, Timothy Kanold, Lee Stiff (2005), Prealgebra, McDougal Littell 41. Larson, Ron; Robert P. Hostetler (2005), Algebra for College Students, Houghton Mifflin 42. Larson, Ron; Robert Hostetler, Anne V. Hodgkins (2006), College Algebra: A Concise Course, Houghton Mifflin 43. Larson, Ron; Robert Hostetler (2007), Precalculus: A Concise Course, Houghton Mifflin 44. Larson, Ron; Robert Hostetler, Bruce H. Edwards (2008), Essential Calculus: Early Transcendental Functions, Houghton Mifflin 45. Larson, Ron; Laurie Boswell, Timothy D. Kanold, Lee Stiff (2008), Algebra 2 Concepts and Skills, McDougal Littell 46. Larson, Ron (2009), Applied Calculus for the Life and Social Sciences, Houghton Mifflin 47. Larson, Ron (2009), Calculus An Applied Approach, Houghton Mifflin 48. Larson, Ron; Anne V. Hodgkins (2009) College Algebra with Applications for Business and the Life Sciences, Houghton Mifflin 49. Larson, Ron; Bruce H. Edwards (2010), Calculus, Cengage Learning 50. Larson, Ron (2010) Elementary Algebra, Cengage Learning 51. Larson, Ron (2010) Intermediate Algebra, Cengage Learning 52. Larson, Ron; Anne V. Hodgkins (2010), College Algebra and Calclulus: An Applied Approach, Cengage Learning 53. Larson, Ron; Laurie Boswell (2010), Big Ideas Math 1, Big Ideas Learning 54. Larson, Ron; Laurie Boswell (2010), Big Ideas Math 2, Big Ideas Learning 55. Larson, Ron; Laurie Boswell (2010), Big Ideas Math 3, Big Ideas Learning 56. Larson, Ron (2011), Precalculus, Cengage Learning 57. Larson, Ron (2011), Precalculus with Limits, Cengage Learning 58. Larson, Ron (2011), Precalculus: A Concise Course, Cengage Learning 59. Larson, Ron (2011), College Algebra, Cengage Learning 60. Larson, Ron (2011), Trigonometry, Cengage Learning 61. Larson, Ron (2011), Algebra and Trigonometry, Cengage Learning 62. Larson, Ron (2012), Algebra and Trigonometry: Real Mathematics, Real People, Cengage Learning 63. Larson, Ron (2012), Pecalculus: Real Mathematics, Real People, Cengage Learning 64. Larson, Ron (2012), College Algebra: Real Mathematics, Real People, Cengage Learning 65. Larson, Ron (2012), Precalculus with Limits: A Graphing Approach, Cengage Learning 66. Larson, Ron; Bruce Edwards (2012), Calculus 1 with Precalculus, Cengage Learning 67. Larson, Ron(2013), Calculus: An Applied Approach, Cengage Learning 68. Larson, Ron (2013), Brief Calculus: An Applied Approach, Cengage Learning 69. Larson, Ron; Anne Hodgkins (2013), College Algebra with Applications for Business and Life Sciences, Cengage Learning 70. Larson, Ron; Anne Hodgkins (2013), College Algebra and Calculus: An Applied Approach, Cengage Learning 71. Larson, Ron (2013), Elementary Linear Algebra, Cengage Learning 72. Larson, Ron (2013), Math & YOU: The Power & Use of Mathematics, andYOU.com 73. Larson, Ron; Bruce Edwards(2014), Calculus, Cengage Learning 74. Larson, Ron (2014), Precalculus, Cengage Learning 75. Larson, Ron (2014), Precalculus with Limits, Cengage Learning 76. Larson, Ron (2014), Precalculus: A Concise Course, Cengage Learning 77. Larson, Ron (2014), College Algebra, Cengage Learning 78. Larson, Ron (2014), Trigonometry, Cengage Learning 79. Larson, Ron (2014), Algebra and Trigonometry, Cengage Learning 80. Larson, Ron (2014), Elementary Algebra within Reach, Cengage Learning 81. Larson, Ron (2014), Intermediate Algebra within Reach, Cengage Learning 82. Larson, Ron (2014), Elementary and Intermediate Algebra within Reach, Cengage Learning 83. Larson, Ron (2014), College Prep Algebra, Cengage Learning 84. Larson, Ron; Laurie Boswell (2014), Big Ideas Math 1, Big Ideas Learning 85. Larson, Ron; Laurie Boswell (2014), Big Ideas Math 2, Big Ideas Learning 86. Larson, Ron; Laurie Boswell (2014), Big Ideas Math 3, Big Ideas Learning 87. Larson, Ron; Laurie Boswell (2014), Big Ideas Math Accelerated, Big Ideas Learning 88. Larson, Ron; Laurie Boswell (2014), Big Ideas Algebra 1, Big Ideas Learning 89. Larson, Ron; Laurie Boswell (2014), Big Ideas Advanced 1, Big Ideas Learning 90. Larson, Ron; Laurie Boswell (2014), Big Ideas Advanced 2, Big Ideas Learning 91. Larson, Ron; Bruce Edwards (2015), Calculus: Early Transcendental Functions, Cengage Learning 92. Larson, Ron; Besty Farber (2015), Elementary Statistics: Picturing the World, Pearson 93. Larson, Ron; Robyn Silbey (2015), Mathematical Practices: Mathematics for Teachers 94. Larson, Ron; Laurie Boswell (2015), Big Ideas Math Algebra 1, Big Ideas Learning 95. Larson, Ron; Laurie Boswell (2015), Big Ideas Math Geometry, Big Ideas Learning 96. Larson, Ron; Laurie Boswell (2015), Big Ideas Math Algebra 2, Big Ideas Learning Translations 1. Larson, Roland E.; Robert P. Hostetler, Bruce H. Edwards (1995), Cálculo y Geometria Analitica, Vol I, McGraw Hill, ISBN 84-481-1768-9 (Spanish) 2. Larson, Roland E.; Robert P. Hostetler, Bruce H. Edwards (1995), Cálculo y Geometria Analitica, Vol II, McGraw Hill, ISBN 84-481-1769-7 (Spanish) 3. Larson, Roland E.; Robert P. Hostetler, Bruce H. Edwards (1998), Cálculo com Applicações, LTC-Livros Técnicos e Ciêntificos, ISBN 85-216-1144-7 (Portuguese) 4. Larson, Ron; Robert P. Hostetler, Bruce H. Edwards (2002), Cálculo y Geometria Analitica, Vol I, McGraw Hill, ISBN 84-481-1768-9 (Spanish) 5. Larson, Ron; Robert P. Hostetler, Bruce H. Edwards (2002), Cálculo y Geometria Analitica, Vol II, McGraw Hill, ISBN 84-481-1769-7 (Spanish) 6. Larson, Ron; Robert P. Hostetler, Bruce H. Edwards (2002), Calculus 微积分, Houghton Mifflin, ISBN 957-29080-4-9 (Chinese) 7. Larson, Roland E; Bruce H. Edwards, David C. Falvo; (2004), Álgebra Lineal, Pirámide, ISBN 84-368-1878-4 (Spanish) 8. Larson, Ron; Betsy Farber (2004), Estatísticas Applicada, Prentice Hall, ISBN 85-87918-59-1 (Portuguese) 9. Larson, Ron; Robert P. Hostetler, Bruce H. Edwards (2005), Cálculo, Vol I, Pirámide, ISBN 84-368-1707-9 (Spanish) 10. Larson, Ron; Robert P. Hostetler, Bruce H. Edwards (2005), Cálculo, Vol II, Pirámide, ISBN 84-368-1756-7 (Spanish) 11. Larson, Ron; Bruce H. Edwards (2006), Calculus An Applied Approach 微积分一种应用的方法, Houghton Mifflin, ISBN 986-82003-1-8 (Chinese) 12. Larson, Ron; Bruce H. Edwards (2006), Brief Calculus An Applied Approach 微积分一种应用的方法, Houghton Mifflin, ISBN 986-82003-2-6 (Chinese) 13. Larson, Ron; Robert P. Hostetler, Bruce H. Edwards (2006), Calculus 微积分, Houghton Mifflin, ISBN 986-82003-3-4 (Chinese) 14. Larson, Roland E.; Robert P. Hostetler (2008), Precalculo, Reverte, ISBN 84-291-5168-0 (Spanish) 15. Larson, Ron; Bruce H. Edwards (2009), Matematicas I Cálculo Differencial, McGraw Hill, ISBN 970-10-7289-8 (Spanish) 16. Larson, Ron; Bruce H. Edwards (2009), Matematicas II Cálculo Integral, McGraw Hill, ISBN 970-10-7290-1 (Spanish) 17. Larson, Ron; Bruce H. Edwards (2009), Matematicas III Cálculo de Varias Variables, McGraw Hill, ISBN 970-10-7291-X (Spanish) Notes 1. Some sources give the spelling as "Meredith". References 1. Mathematics Faculty at Penn State Erie 2. "U.S. Army Awards of the Silver Star for Conspicuous Gallantry in Action During the Korean War". Archived from the original on 25 January 2013. 3. "OBITS: MEDERITH LARSON". Archived from the original on 9 May 2006. 4. Timothy R. Larson 5. Norwegian Immigration, Minnesota State University 6. "Browse Individual contributions". FEC.gov. Retrieved 27 January 2021. 7. Ron Larson Professor 8. Jones, William B., Ellen E. Reed, and Fred W. Stevenson (Volume 33, Number 2, 2003, 395-403. "Biography of Wolfgang J. Thron". Rocky Mountain Journal of Mathematics. 9. Mathematics Genealogy Project 10. "Behrend College Timeline". Archived from the original on 2016-10-14. Retrieved 2010-08-26. 11. A Single Dream 12. Cengage Learning 13. Lewis & Clark College Distinguished Alumnus Award 14. Library of Congress 15. Library of Congress 16. Text and Academic Authors Association 17. Text and Academic Authors Association 18. Benny Awards, 2005 (Calculus Houghton Mifflin) 19. Texty Awards, 2010 20. Spanish Translation of Calculus 21. Chinese Translation of Calculus An Applied Approach 22. Portuguese Translation of Statistics 23. (July 10, 2008) "Penn State Behrend Unveils Archives at Open House". Penn State Behrend Archives. 24. Pennsylvania Corporations 25. Larson Texts, Inc 26. Best Places to Work in PA 27. MathematicalPractices.com 28. Parent's Choice Awards. "Larson's Elementary Math Activities", Spring 2003. 29. (September 21, 2005). "Houghton Mifflin Acquires Many Larson Learning, Inc. K–12 Products, Including Award-Winning Mathematics Materials". Red Orbit 30. AuntSally.com 31. (Volume 5 Number 2 (Spring 1975). "Rocky Mountain Journal of Mathematics" 32. Association of Mathematics Teachers of New York, 2008 33. Penn State Behrend Magazine, Summer, 2006 34. National Council of Teachers of Mathematics Focal Points 35. Big Ideas Learning, LLC 36. Pennsylvania Corporations 37. Acceptance Speech at Lewis & Clark College Sources • (November 7, 1976, Page 18-A). "2 Behrend Professors Author Text on Calculus". Erie Times News • (March 1, 1981). "Book on Handling Money Penned by Behrend Profs". Erie Times News • Craig, Cindy (February 1, 1982, Volume 37, Number 9, Page 7). "Behrend Profs Achieve Publishing Success". The Collegian, Penn State Erie's Weekly Newspaper • Dile, Robin (May 20, 1984). "Professor Invents the 'Perfect' Dice". Erie Times News • Ross, Michael (March 1, 1985, Volume 33, Number 10, Page 1). "Larson and Hostetler Offer Precalculus Series". The Collegian, Penn State Erie's Weekly Newspaper • Center Spread (March, 1985, Volume 6, Number 1, Page 17). "Dice Game". Physical Science and Engineering, Research Penn State. • Howard, Pat (July 25, 1985, Page 1B). "Textbook Authors Still Get Thrill of Writing". Erie Daily Times • (March 22, 1985). "Penn State Behrend Honors Faculty Authors at Book Day". Erie Times News • (December 13, 1987). "Behrend Honors Philanthropists". Erie Times News • Pellegrini, Mike (August 5, 1991, Page 5). "He Writes Best Sellers for Math Students". Pittsburgh Post Gazette • (Fall, 1991, pages 4–5). "Read Any Good Math Books Lately?". The Behrend Quarterly • (August 15, 1995, Page 12C). "Interactive Computer Companion Developed for Behrend Prof's Text". Erie Times News • (August 20, 1995). "Behrend Professor Offers High Tech Instruction in Math". Erie Times-News • Pawlak, Kim (Notable Author Series, 1997). "Ron Larson: Author and Publisher", Text and Academic Authors, Online Information for TAA Members • Pawlak, Kim (Notable Author Series, 1997). "Instilling a Love for Math", Text and Academic Authors, Online Information for TAA Members • Pawlak, Kim (Notable Author Series, 1997). "Obsessed with Writing", Text and Academic Authors, Online Information for TAA Members • Alumni News (Fall, 1998, Volume 8, Number 1, Page 8). "Three Outstanding Alumni Honored". The Lewis & Clark Chronicle • (April 23, 1998, Page 12C). "Profile Ron Larson, Meridian Creative Group". Morning News, Erie • McQuaid, Deborah (February 10, 1999). "Firm May Move into Former School". Erie Times-News • New Ways to Know (1999), Houghton Mifflin Annual Report • Martin, Jim (March 9, 2003). "At Larson Texts, Success is Academic". Erie Times News • Savory, Jon (November 26, 2003, Part 1). "Numbing the Mind with Numbers: Inside the Head of Mathematics Author Ron Larson". County College of Morris Student Newspaper, The Youngstown Edition • Savory, Jon (December 10, 2003, Part 1). "Does Two Plus Two Really Equal Four". County College of Morris Student Newspaper, The Youngstown Edition • Editorial Review • Cengage Learning Biography of Ron Larson Authority control International • ISNI • VIAF National • Spain • Catalonia • Germany • Israel • United States • Czech Republic Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Rook's graph In graph theory, a rook's graph is an undirected graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's graph represents a square on a chessboard, and there is an edge between any two squares sharing a row (rank) or column (file), the squares that a rook can move between. These graphs can be constructed for chessboards of any rectangular shape. Although rook's graphs have only minor significance in chess lore, they are more important in the abstract mathematics of graphs through their alternative constructions: rook's graphs are the Cartesian product of two complete graphs, and are the line graphs of complete bipartite graphs. The square rook's graphs constitute the two-dimensional Hamming graphs. Rook's graph 8x8 Rook's graph Verticesnm Edgesnm(n + m)/2 − nm Diameter2 Girth3 (if max(n, m) ≥ 3) Chromatic numbermax(n, m) Spectrum{m + n − 2, m − 2, n − 2, −2} Properties • integral • perfect • regular • vertex-transitive • well-covered Table of graphs and parameters Rook's graphs are highly symmetric, having symmetries taking every vertex to every other vertex. In rook's graphs defined from square chessboards, more strongly, every two edges are symmetric, and every pair of vertices is symmetric to every other pair at the same distance in moves (making the graph distance-transitive). For rectangular chessboards whose width and height are relatively prime, the rook's graphs are circulant graphs. With one exception, the rook's graphs can be distinguished from all other graphs using only two properties: the numbers of triangles each edge belongs to, and the existence of a unique 4-cycle connecting each nonadjacent pair of vertices. Rook's graphs are perfect graphs. In other words, every subset of chessboard squares can be colored so that no two squares in a row or column have the same color, using a number of colors equal to the maximum number of squares from the subset in any single row or column (the clique number of the induced subgraph). This class of induced subgraphs are a key component of a decomposition of perfect graphs used to prove the strong perfect graph theorem, which characterizes all perfect graphs. The independence number and domination number of a rook's graph both equal the smaller of the chessboard's width and height. In terms of chess, the independence number is the maximum number of rooks that can be placed without attacking each other; the domination number is the minimum number needed to attack all unoccupied board squares. Rook's graphs are well-covered graphs, meaning that placing non-attacking rooks one at a time can never get stuck until a set of maximum size is reached. Definition and mathematical constructions An n × m rook's graph represents the moves of a rook on an n × m chessboard.[1] Its vertices represent the squares of the chessboard, and may be given coordinates (x, y), where 1 ≤ x ≤ n and 1 ≤ y ≤ m. Two vertices with coordinates (x1, y1) and (x2, y2) are adjacent if and only if either x1 = x2 or y1 = y2. (If x1 = x2, the vertices share a file and are connected by a vertical rook move; if y1 = y2, they share a rank and are connected by a horizontal rook move.)[1] The squares of a single rank or file are all directly connected to each other, so each rank and file forms a clique—a subset of vertices forming a complete graph. The whole rook's graph for an n × m chessboard can be formed from these two kinds of cliques, as the Cartesian product of graphs Kn ◻ Km.[2] Because the rook's graph for a square chessboard is the Cartesian product of equal-size cliques, it is an example of a Hamming graph. Its dimension as a Hamming graph is two, and every two-dimensional Hamming graph is a rook's graph for a square chessboard.[3] Square rook's graphs are also called "Latin square graphs"; applied to a Latin square, its edges describe pairs of squares that cannot contain the same value.[4] The Sudoku graphs are rook's graphs with some additional edges, connecting squares of a Sodoku puzzle that should have unequal values.[5] Geometrically, the rook's graphs can be formed by sets of the vertices and edges (the skeletons) of a family of convex polytopes, the Cartesian products of pairs of neighborly polytopes.[6] For instance, the 3-3 duoprism is a four-dimensional shape formed as the Cartesian product of two triangles, and has a 3 × 3 rook's graph as its skeleton.[7] Regularity and symmetry Strong regularity Moon (1963) and Hoffman (1964) observe that the $m\times n$ rook's graph (or equivalently, as they describe it, the line graph of the complete bipartite graph $K_{m,n}$) has all of the following properties: • It has $mn$ vertices, one for each square of the $m\times n$ chessboard. Each vertex is adjacent to $m+n-2$ edges, connecting it to the $m-1$ squares on the same rank and the $n-1$ squares on the same file. • The triangles within the rook's graph are formed by triples of squares within a single rank or file. When $m\neq n$, exactly $n{\tbinom {m}{2}}$ edges (the ones connecting squares on the same rank) belong to $m-2$ triangles; the remaining $m{\tbinom {n}{2}}$ edges (the ones connecting squares on the same file) belong to $n-2$ triangles. When $m=n$, each edge belongs to $m-2=n-2$ triangles. • Every two nonadjacent vertices belong to a unique $4$-vertex cycle, namely the only rectangle using the two vertices as corners. They show that except in the case $m=n=4$, these properties uniquely characterize the rook's graph. That is, the rook's graphs are the only graphs with these numbers of vertices, edges, triangles per edge, and with a unique 4-cycle through each two non-adjacent vertices.[8][9] When $m=n$, these conditions may be abbreviated by stating that an $n\times n$ rook's graph is a strongly regular graph with parameters $\operatorname {srg} (n^{2},2n-2,n-2,2)$. These parameters describe the number of vertices, the number of edges per vertex, the number of triangles per edge, and the number of shared neighbors for two non-adjacent vertices, respectively.[1] Conversely, every strongly regular graph with these parameters must be an $n\times n$ rook's graph, unless $n=4$.[8][9] When $n=4$, there is another strongly regular graph, the Shrikhande graph, with the same parameters as the $4\times 4$ rook's graph.[10] The Shrikhande graph obeys the same properties listed by Moon and Moser. It can be distinguished from the $4\times 4$ rook's graph in that the neighborhood of each vertex in the Shrikhande graph is connected to form a $6$-cycle. In contrast, in the $4\times 4$ rook's graph, the neighborhood of each vertex forms two triangles, one for its rank and another for its file, without any edges from one part of the neighborhood to the other.[11] Another way of distinguishing the $4\times 4$ rook's graph from the Shrikhande graph uses clique cover numbers: the $n=4$ rook's graph can be covered by four cliques (the four ranks or the four files of the chessboard) whereas six cliques are needed to cover the Shrikhande graph.[10] Symmetry Rook's graphs are vertex-transitive, meaning that they have symmetries taking every vertex to every other vertex. This implies that every vertex has an equal number of edges: they are $(m+n-2)$-regular. The rook's graphs are the only regular graphs formed from the moves of standard chess pieces in this way.[12] When $m\neq n$, the symmetries of the rook's graph are formed by independently permuting the rows and columns of the graph, so the automorphism group of the graph has $m!n!$ elements. When $m=n$, the graph has additional symmetries that swap the rows and columns, so the number of automorphisms is $2n!^{2}$.[13] Any two vertices in a rook's graph are either at distance one or two from each other, according to whether they are adjacent or nonadjacent respectively. Any two nonadjacent vertices may be transformed into any other two nonadjacent vertices by a symmetry of the graph. When the rook's graph is not square, the pairs of adjacent vertices fall into two orbits of the symmetry group according to whether they are adjacent horizontally or vertically, but when the graph is square any two adjacent vertices may also be mapped into each other by a symmetry and the graph is therefore distance-transitive.[14] When $m$ and $n$ are relatively prime, the symmetry group $S_{m}\times S_{n}$ of the rook's graph contains as a subgroup the cyclic group $C_{mn}=C_{m}\times C_{n}$ that acts by cyclically permuting the $mn$ vertices. Therefore, in this case, the rook's graph is a circulant graph.[15] Square rook's graphs are connected-homogeneous, meaning that every isomorphism between two connected induced subgraphs can be extended to an automorphism of the whole graph.[16] Other properties Perfection A rook's graph can also be viewed as the line graph of a complete bipartite graph Kn,m — that is, it has one vertex for each edge of Kn,m, and two vertices of the rook's graph are adjacent if and only if the corresponding edges of the complete bipartite graph share a common endpoint.[2][17] In this view, an edge in the complete bipartite graph from the ith vertex on one side of the bipartition to the jth vertex on the other side corresponds to a chessboard square with coordinates (i, j).[1] Any bipartite graph is a subgraph of a complete bipartite graph, and correspondingly any line graph of a bipartite graph is an induced subgraph of a rook's graph.[18] The line graphs of bipartite graphs are perfect: in them, and in any of their induced subgraphs, the number of colors needed in any vertex coloring is the same as the number of vertices in the largest complete subgraph. Line graphs of bipartite graphs form an important family of perfect graphs: they are one of a small number of families used by Chudnovsky et al. (2006) to characterize the perfect graphs and to show that every graph with no odd hole and no odd antihole is perfect.[19] In particular, rook's graphs are themselves perfect. Because a rook's graph is perfect, the number of colors needed in any coloring of the graph is just the size of its largest clique. The cliques of a rook's graph are the subsets of a single row or a single column, and the largest of these have size max(m, n), so this is also the chromatic number of the graph. An n-coloring of an n × n rook's graph may be interpreted as a Latin square: it describes a way of filling the rows and columns of an n × n grid with n different values in such a way that the same value does not appear twice in any row or column.[20] In the same way, a coloring of a rectangular rook's graph corresponds to a Latin rectangle.[21] Although finding an optimal coloring of a rook's graph is straightforward, it is NP-complete to determine whether a partial coloring can be extended to a coloring of the whole graph (this problem is called precoloring extension). Equivalently, it is NP-complete to determine whether a partial Latin square can be completed to a full Latin square.[22] Independence abcdefgh 8 8 77 66 55 44 33 22 11 abcdefgh A non-attacking placement of eight rooks on a chessboard, forming a maximum independent set in the corresponding rook's graph An independent set in a rook's graph is a set of vertices, no two of which belong to the same row or column of the graph; in chess terms, it corresponds to a placement of rooks no two of which attack each other. Perfect graphs may also be described as the graphs in which, in every induced subgraph, the size of the largest independent set is equal to the number of cliques in a partition of the graph's vertices into a minimum number of cliques. In a rook's graph, the sets of rows or the sets of columns (whichever has fewer sets) form such an optimal partition. The size of the largest independent set in the graph is therefore min(m, n).[1] Rook's graphs are well-covered graphs: every independent set in a rook's graph can be extended to a maximum independent set, and every maximal independent set in a rook's graph has the same size, min(m, n).[23] Domination The domination number of a graph is the minimum cardinality among all dominating sets. On the rook's graph a set of vertices is a dominating set if and only if their corresponding squares either occupy, or are a rook's move away from, all squares on the m × n board. For the m × n board the domination number is min(m, n).[24] On the rook's graph a k-dominating set is a set of vertices whose corresponding squares attack all other squares (via a rook's move) at least k times. A k-tuple dominating set on the rook's graph is a set of vertices whose corresponding squares attack all other squares at least k times and are themselves attacked at least k − 1 times. The minimum cardinality among all k-dominating and k-tuple dominating sets are the k-domination number and the k-tuple domination number, respectively. On the square board, and for even k, the k-domination number is nk/2 when n ≥ (k2 − 2k)/4 and k < 2n. In a similar fashion, the k-tuple domination number is n(k + 1)/2 when k is odd and less than 2n.[25] Hamiltonicity Every rook's graph contains a Hamiltonian cycle.[26] However, these cycles may involve moves between squares that are far apart within a single row or column of the chessboard. Instead, the study of "rook's tours", in the mathematics of chess, has generally concentrated on a special case of these Hamiltonian cycles where the rook is restricted to move only to adjacent squares. These single-step rook's tours only exist on boards with an even number of squares. They play a central role in the proof of Gomory's theorem that, if two squares of opposite colors are removed from a standard chessboard, the remaining squares can always be covered by dominoes.[27] They are featured alongside knight's tours in the first work to discuss chess-piece tours, the 9th century Sanskrit Kavyalankara of Rudrata.[28] Spectrum The spectrum of a rook's graph (the eigenvalues of its adjacency matrix) consists of the four eigenvalues $m+n-2$, $m-2$, $n-2$, and $-2$. Because these are all integers, rook's graphs are integral graphs. There are only three classes of graphs (and finitely many exceptional graphs) that can have four eigenvalues with one of the four being $-2$; one of the three classes is the class of rook's graphs. For most combinations of $m$ and $n$, the $m\times n$ rook's graph is spectrally unique: no other graph has the same spectrum. In particular this is true when $n=2$ or $n=m-1$, or when the two numbers $m$ and $n$ sum to at least 18 and do not have the form $2t^{2}\pm t$.[29] In other graphs The graphs for which the neighbors of each vertex induce a rook's graph have been called locally grid. Examples include the Johnson graphs $J(n,k)$, for which the neighbors of each vertex form a $k\times (n-k)$ rook's graph. Other examples are known, and for some rook's graphs, a complete classification is known. For instance, there are two graphs whose neighborhoods are all $3\times 3$ rook's graphs: they are the Johnson graph $J(6,3)$, and the complement graph of a $4\times 4$ rook's graph.[30] See also • Chessboard complex, the independence complex of the rook's graph • King's graph, knight's graph, queen's graph, and bishop's graph, four other graphs defined from the moves of chess pieces • Lattice graph, the graph of horizontal and vertical adjacencies of squares on a chessboard References 1. Laskar, Renu; Wallis, Charles (1999), "Chessboard graphs, related designs, and domination parameters", Journal of Statistical Planning and Inference, 76 (1–2): 285–294, doi:10.1016/S0378-3758(98)00132-3, MR 1673351. 2. Stones, Douglas S. (2010), "The many formulae for the number of Latin rectangles", Electronic Journal of Combinatorics, 17 (1): Article 1, 46, doi:10.37236/487, MR 2661404 3. Azizoğlu, M. Cemil; Eğecioğlu, Ömer (2003), "Extremal sets minimizing dimension-normalized boundary in Hamming graphs", SIAM Journal on Discrete Mathematics, 17 (2): 219–236, doi:10.1137/S0895480100375053, MR 2032290. 4. Goethals, J.-M.; Seidel, J. J. (1970), "Strongly regular graphs derived from combinatorial designs", Canadian Journal of Mathematics, 22 (3): 597–614, doi:10.4153/CJM-1970-067-9, MR 0282872, S2CID 199082328. 5. Herzberg, Agnes M.; Murty, M. Ram (2007), "Sudoku squares and chromatic polynomials" (PDF), Notices of the American Mathematical Society, 54 (6): 708–717, MR 2327972 6. Matschke, Benjamin; Pfeifle, Julian; Pilaud, Vincent (2011), "Prodsimplicial-neighborly polytopes", Discrete & Computational Geometry, 46 (1): 100–131, arXiv:0908.4177, doi:10.1007/s00454-010-9311-y, MR 2794360, S2CID 2070310 7. Moore, Doug (1992), "Understanding simploids", in Kirk, David (ed.), Graphics Gems III, Academic Press, pp. 250–255, doi:10.1016/b978-0-08-050755-2.50057-9 8. Moon, J. W. (1963), "On the line-graph of the complete bigraph", Annals of Mathematical Statistics, 34 (2): 664–667, doi:10.1214/aoms/1177704179. 9. Hoffman, A. J. (1964), "On the line graph of the complete bipartite graph", Annals of Mathematical Statistics, 35 (2): 883–885, doi:10.1214/aoms/1177703593, MR 0161328. 10. Fiala, Nick C.; Haemers, Willem H. (2006), "5-chromatic strongly regular graphs", Discrete Mathematics, 306 (23): 3083–3096, doi:10.1016/j.disc.2004.03.023, MR 2273138. 11. Burichenko, V. P.; Makhnev, A. A. (2011), "Об автоморфизмах сильно регулярных локально циклических графов" [On automorphisms of strongly regular locally cyclic graphs], Doklady Akademii Nauk (in Russian), 441 (2): 151–155, MR 2953786. Translated in Doklady Mathematics 84 (3): 778–782, 2011, doi:10.1134/S1064562411070076. From the first page of the translation: "The Shrikhande graph is the only strongly regular locally hexagonal graph with parameters (16, 6, 2, 2)." 12. Elkies, Noam (Fall 2004), "Graph theory glossary", Freshman Seminar 23j: Chess and Mathematics, Harvard University Mathematics Department, retrieved 2023-05-03. 13. Harary, Frank (1958), "On the number of bi-colored graphs", Pacific Journal of Mathematics, 8 (4): 743–755, doi:10.2140/pjm.1958.8.743, MR 0103834. See in particular equation (10), p. 748 for the automorphism group of the $n\times n$ rook's graph, and the discussion above the equation for the order of this group. 14. Biggs, Norman (1974), "The symmetry of line graphs", Utilitas Mathematica, 5: 113–121, MR 0347684. 15. This follows from the definition of the rook's graph as a Cartesian product graph, together with Proposition 4 of Broere, Izak; Hattingh, Johannes H. (1990), "Products of circulant graphs", Quaestiones Mathematicae, 13 (2): 191–216, doi:10.1080/16073606.1990.9631612, MR 1068710. 16. Gray, R.; Macpherson, D. (2010), "Countable connected-homogeneous graphs", Journal of Combinatorial Theory, Series B, 100 (2): 97–118, doi:10.1016/j.jctb.2009.04.002, MR 2595694. See in particular Theorem 1, which identifies these graphs as line graphs of complete bipartite graphs. 17. For the equivalence between Cartesian products of complete graphs and line graphs of complete bipartite graphs, see de Werra, D.; Hertz, A. (1999), "On perfectness of sums of graphs" (PDF), Discrete Mathematics, 195 (1–3): 93–101, doi:10.1016/S0012-365X(98)00168-X, MR 1663807. 18. de Werra & Hertz (1999). 19. Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2006), "The strong perfect graph theorem" (PDF), Annals of Mathematics, 164 (1): 51–229, arXiv:math/0212070, doi:10.4007/annals.2006.164.51, JSTOR 20159988, S2CID 119151552. 20. For the equivalence between edge-coloring complete bipartite graphs and Latin squares, see e.g. LeSaulnier, Timothy D.; Stocker, Christopher; Wenger, Paul S.; West, Douglas B. (2010), "Rainbow matching in edge-colored graphs", Electronic Journal of Combinatorics, 17 (1): Note 26, 5, doi:10.37236/475, MR 2651735. 21. Stones, Douglas S. (2010), "The many formulae for the number of Latin rectangles", Electronic Journal of Combinatorics, 17 (1): A1:1–A1:46, doi:10.37236/487, MR 2661404 22. Colbourn, Charles J. (1984), "The complexity of completing partial Latin squares", Discrete Applied Mathematics, 8 (1): 25–30, doi:10.1016/0166-218X(84)90075-1, MR 0739595. 23. For an equivalent statement to the well-covered property of rook's graphs, in terms of matchings in complete bipartite graphs, see Lesk, M.; Plummer, M. D.; Pulleyblank, W. R. (1984), "Equi-matchable graphs", in Bollobás, Béla (ed.), Graph Theory and Combinatorics: Proceedings of the Cambridge Combinatorial Conference, in Honour of Paul Erdös, London: Academic Press, pp. 239–254, MR 0777180. 24. Yaglom, A. M.; Yaglom, I. M. (1987), "Solution to problem 34b", Challenging Mathematical Problems with Elementary Solutions, Dover, p. 77, ISBN 9780486318578. 25. Burchett, Paul; Lane, David; Lachniet, Jason (2009), "K-domination and k-tuple domination on the rook's graph and other results", Congressus Numerantium, 199: 187–204. 26. Hurley, C. B.; Oldford, R. W. (February 2011), "Graphs as navigational infrastructure for high dimensional data spaces", Computational Statistics, 26 (4): 585–612, doi:10.1007/s00180-011-0228-6, S2CID 54220980 27. Watkins, John J. (2004), Across the Board: The Mathematics of Chessboard Problems, Princeton University Press, p. 12, ISBN 9780691130620 28. Murray, H. J. R. (January 1902), "The knight's tour: ancient and oriental", The British Chess Magazine, vol. 22, no. 1, pp. 1–7 29. Doob, Michael (1970), "On characterizing certain graphs with four eigenvalues by their spectra", Linear Algebra and Its Applications, 3 (4): 461–482, doi:10.1016/0024-3795(70)90037-6, MR 0285432 30. Cohen, Arjeh M. (1990), "Local recognition of graphs, buildings, and related geometries" (PDF), in Kantor, William M.; Liebler, Robert A.; Payne, Stanley E.; Shult, Ernest E. (eds.), Finite Geometries, Buildings, and Related Topics: Papers from the Conference on Buildings and Related Geometries held in Pingree Park, Colorado, July 17–23, 1988, Oxford Science Publications, Oxford University Press, pp. 85–94, MR 1072157; see in particular pp. 89–90 External links • Weisstein, Eric W., "Rook Graph", MathWorld	Wikipedia
Rook polynomial In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in the same row or column. The board is any subset of the squares of a rectangular board with m rows and n columns; we think of it as the squares in which one is allowed to put a rook. The board is the ordinary chessboard if all squares are allowed and m = n = 8 and a chessboard of any size if all squares are allowed and m = n. The coefficient of x k in the rook polynomial RB(x) is the number of ways k rooks, none of which attacks another, can be arranged in the squares of B. The rooks are arranged in such a way that there is no pair of rooks in the same row or column. In this sense, an arrangement is the positioning of rooks on a static, immovable board; the arrangement will not be different if the board is rotated or reflected while keeping the squares stationary. The polynomial also remains the same if rows are interchanged or columns are interchanged. abcdefgh 8 8 77 66 55 44 33 22 11 abcdefgh One possible arrangement of rooks on an 8 × 8 chessboard, where no two pieces share a row or column. The term "rook polynomial" was coined by John Riordan.[1] Despite the name's derivation from chess, the impetus for studying rook polynomials is their connection with counting permutations (or partial permutations) with restricted positions. A board B that is a subset of the n × n chessboard corresponds to permutations of n objects, which we may take to be the numbers 1, 2, ..., n, such that the number aj in the j-th position in the permutation must be the column number of an allowed square in row j of B. Famous examples include the number of ways to place n non-attacking rooks on: • an entire n × n chessboard, which is an elementary combinatorial problem; • the same board with its diagonal squares forbidden; this is the derangement or "hat-check" problem (this is a particular case of the problème des rencontres); • the same board without the squares on its diagonal and immediately above its diagonal (and without the bottom left square), which is essential in the solution of the problème des ménages. Interest in rook placements arises in pure and applied combinatorics, group theory, number theory, and statistical physics. The particular value of rook polynomials comes from the utility of the generating function approach, and also from the fact that the zeroes of the rook polynomial of a board provide valuable information about its coefficients, i.e., the number of non-attacking placements of k rooks. Definition The rook polynomial RB(x) of a board B is the generating function for the numbers of arrangements of non-attacking rooks: $R_{B}(x)=\sum _{k=0}^{\min {(m,n)}}r_{k}(B)x^{k},$ where $r_{k}(B)$ is the number of ways to place k non-attacking rooks on the board B. There is a maximum number of non-attacking rooks the board can hold; indeed, there cannot be more rooks than the number of rows or number of columns in the board (hence the limit $\min(m,n)$).[2] Complete boards For rectangular m × n boards Bm,n, we write Rm,n := RBm,n, and if m=n, Rn := Rm,n. The first few rook polynomials on square n × n boards are: ${\begin{aligned}R_{1}(x)&=x+1\\R_{2}(x)&=2x^{2}+4x+1\\R_{3}(x)&=6x^{3}+18x^{2}+9x+1\\R_{4}(x)&=24x^{4}+96x^{3}+72x^{2}+16x+1.\end{aligned}}$ In words, this means that on a 1 × 1 board, 1 rook can be arranged in 1 way, and zero rooks can also be arranged in 1 way (empty board); on a complete 2 × 2 board, 2 rooks can be arranged in 2 ways (on the diagonals), 1 rook can be arranged in 4 ways, and zero rooks can be arranged in 1 way; and so forth for larger boards. The rook polynomial of a rectangular chessboard is closely related to the generalized Laguerre polynomial Lnα(x) by the identity $R_{m,n}(x)=n!x^{n}L_{n}^{(m-n)}(-x^{-1}).$ Matching polynomials A rook polynomial is a special case of one kind of matching polynomial, which is the generating function of the number of k-edge matchings in a graph. The rook polynomial Rm,n(x) corresponds to the complete bipartite graph Km,n . The rook polynomial of a general board B ⊆ Bm,n corresponds to the bipartite graph with left vertices v1, v2, ..., vm and right vertices w1, w2, ..., wn and an edge viwj whenever the square (i, j) is allowed, i.e., belongs to B. Thus, the theory of rook polynomials is, in a sense, contained in that of matching polynomials. We deduce an important fact about the coefficients rk, which we recall given the number of non-attacking placements of k rooks in B: these numbers are unimodal, i.e., they increase to a maximum and then decrease. This follows (by a standard argument) from the theorem of Heilmann and Lieb[3] about the zeroes of a matching polynomial (a different one from that which corresponds to a rook polynomial, but equivalent to it under a change of variables), which implies that all the zeroes of a rook polynomial are negative real numbers. Connection to matrix permanents For incomplete square n × n boards, (i.e. rooks are not allowed to be played on some arbitrary subset of the board's squares) computing the number of ways to place n rooks on the board is equivalent to computing the permanent of a 0–1 matrix. Complete rectangular boards Rooks problems abcdefgh 8 8 77 66 55 44 33 22 11 abcdefgh Fig. 1. The maximal number of non-attacking rooks on an 8 × 8 chessboard is 8. Rook + dots mark the number of squares on a rank, available to each rook, after placing the rooks on the lower ranks. A precursor to the rook polynomial is the classic "Eight rooks problem" by H. E. Dudeney[4] in which he shows that the maximum number of non-attacking rooks on a chessboard is eight by placing them on one of the main diagonals (Fig. 1). The question asked is: "In how many ways can eight rooks be placed on an 8 × 8 chessboard so that neither of them attacks the other?" The answer is: "Obviously there must be a rook in every row and every column. Starting with the bottom row, it is clear that the first rook can be put on any one of eight different squares (Fig. 1). Wherever it is placed, there is the option of seven squares for the second rook in the second row. Then there are six squares from which to select the third row, five in the fourth, and so on. Therefore the number of different ways must be 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 40,320" (that is, 8!, where "!" is the factorial).[5] The same result can be obtained in a slightly different way. Let us endow each rook with a positional number, corresponding to the number of its rank, and assign it a name that corresponds to the name of its file. Thus, rook a1 has position 1 and name "a", rook b2 has position 2 and name "b", etc. Then let us order the rooks into an ordered list (sequence) by their positions. The diagram on Fig. 1 will then transform in the sequence (a,b,c,d,e,f,g,h). Placing any rook on another file would involve moving the rook that hitherto occupied the second file to the file, vacated by the first rook. For instance, if rook a1 is moved to "b" file, rook b2 must be moved to "a" file, and now they will become rook b1 and rook a2. The new sequence will become (b,a,c,d,e,f,g,h). In combinatorics, this operation is termed permutation, and the sequences, obtained as a result of the permutation, are permutations of the given sequence. The total number of permutations, containing 8 elements from a sequence of 8 elements is 8! (factorial of 8). To assess the effect of the imposed limitation "rooks must not attack each other", consider the problem without such limitation. In how many ways can eight rooks be placed on an 8 × 8 chessboard? This will be the total number of combinations of 8 rooks on 64 squares: ${64 \choose 8}={\frac {64!}{8!(64-8)!}}=4,426,165,368.$ Thus, the limitation "rooks must not attack each other" reduces the total number of allowable positions from combinations to permutations which is a factor of about 109,776. A number of problems from different spheres of human activity can be reduced to the rook problem by giving them a "rook formulation". As an example: A company must employ n workers on n different jobs and each job must be carried out only by one worker. In how many ways can this appointment be done? Let us put the workers on the ranks of the n × n chessboard, and the jobs − on the files. If worker i is appointed to job j, a rook is placed on the square where rank i crosses file j. Since each job is carried out only by one worker and each worker is appointed to only one job, all files and ranks will contain only one rook as a result of the arrangement of n rooks on the board, that is, the rooks do not attack each other. The rook polynomial as a generalization of the rooks problem The classical rooks problem immediately gives the value of r8, the coefficient in front of the highest order term of the rook polynomial. Indeed, its result is that 8 non-attacking rooks can be arranged on an 8 × 8 chessboard in r8 = 8! = 40320 ways. Let us generalize this problem by considering an m × n board, that is, a board with m ranks (rows) and n files (columns). The problem becomes: In how many ways can one arrange k rooks on an m × n board in such a way that they do not attack each other? It is clear that for the problem to be solvable, k must be less or equal to the smaller of the numbers m and n; otherwise one cannot avoid placing a pair of rooks on a rank or on a file. Let this condition be fulfilled. Then the arrangement of rooks can be carried out in two steps. First, choose the set of k ranks on which to place the rooks. Since the number of ranks is m, of which k must be chosen, this choice can be done in ${\binom {m}{k}}$ ways. Similarly, the set of k files on which to place the rooks can be chosen in ${\binom {n}{k}}$ ways. Because the choice of files does not depend on the choice of ranks, according to the products rule there are ${\binom {m}{k}}{\binom {n}{k}}$ ways to choose the square on which to place the rook. However, the task is not yet finished because k ranks and k files intersect in k2 squares. By deleting unused ranks and files and compacting the remaining ranks and files together, one obtains a new board of k ranks and k files. It was already shown that on such board k rooks can be arranged in k! ways (so that they do not attack each other). Therefore, the total number of possible non-attacking rooks arrangements is:[6] $r_{k}={\binom {m}{k}}{\binom {n}{k}}k!={\frac {n!m!}{k!(n-k)!(m-k)!}}.$ For instance, 3 rooks can be placed on a conventional chessboard (8 × 8) in $\textstyle {\frac {8!8!}{3!5!5!}}=18,816$ ways. For k = m = n, the above formula gives rk = n! that corresponds to the result obtained for the classical rooks problem. The rook polynomial with explicit coefficients is now: $R_{m,n}(x)=\sum _{k=0}^{\min(m,n)}{\binom {m}{k}}{\binom {n}{k}}k!x^{k}=\sum _{k=0}^{\min(m,n)}{\frac {n!m!}{k!(n-k)!(m-k)!}}x^{k}.$ If the limitation "rooks must not attack each other" is removed, one must choose any k squares from m × n squares. This can be done in: ${\binom {mn}{k}}={\frac {(mn)!}{k!(mn-k)!}}$ ways. If the k rooks differ in some way from each other, e.g., they are labelled or numbered, all the results obtained so far must be multiplied by k!, the number of permutations of k rooks. Symmetric arrangements As a further complication to the rooks problem, let us require that rooks not only be non-attacking but also symmetrically arranged on the board. Depending on the type of symmetry, this is equivalent to rotating or reflecting the board. Symmetric arrangements lead to many problems, depending on the symmetry condition.[7][8][9][10] abcdefgh 8 8 77 66 55 44 33 22 11 abcdefgh Fig. 2. A symmetric arrangement of non-attacking rooks about the centre of an 8 × 8 chessboard. Dots mark the 4 central squares that surround the centre of symmetry. The simplest of those arrangements is when rooks are symmetric about the centre of the board. Let us designate with Gn the number of arrangements in which n rooks are placed on a board with n ranks and n files. Now let us make the board to contain 2n ranks and 2n files. A rook on the first file can be placed on any of the 2n squares of that file. According to the symmetry condition, placement of this rook defines the placement of the rook that stands on the last file − it must be arranged symmetrically to the first rook about the board centre. Let us remove the first and the last files and the ranks that are occupied by rooks (since the number of ranks is even, the removed rooks cannot stand on the same rank). This will give a board of 2n − 2 files and 2n − 2 ranks. It is clear that to each symmetric arrangement of rooks on the new board corresponds a symmetric arrangement of rooks on the original board. Therefore, G2n = 2nG2n − 2 (the factor 2n in this expression comes from the possibility for the first rook to occupy any of the 2n squares on the first file). By iterating the above formula one reaches to the case of a 2 × 2 board, on which there are 2 symmetric arrangements (on the diagonals). As a result of this iteration, the final expression is G2n = 2nn! For the usual chessboard (8 × 8), G8 = 24 × 4! = 16 × 24 = 384 centrally symmetric arrangements of 8 rooks. One such arrangement is shown in Fig. 2. For odd-sized boards (containing 2n + 1 ranks and 2n + 1 files) there is always a square that does not have its symmetric double − this is the central square of the board. There must always be a rook placed on this square. Removing the central file and rank, one obtains a symmetric arrangement of 2n rooks on a 2n × 2n board. Therefore, for such board, once again G2n + 1 = G2n = 2nn!. A little more complicated problem is to find the number of non-attacking arrangements that do not change upon 90° rotation of the board. Let the board have 4n files and 4n ranks, and the number of rooks is also 4n. In this case, the rook that is on the first file can occupy any square on this file, except the corner squares (a rook cannot be on a corner square because after a 90° rotation there would 2 rooks that attack each other). There are another 3 rooks that correspond to that rook and they stand, respectively, on the last rank, the last file, and the first rank (they are obtained from the first rook by 90°, 180°, and 270° rotations). Removing the files and ranks of those rooks, one obtains the rook arrangements for a (4n − 4) × (4n − 4) board with the required symmetry. Thus, the following recurrence relation is obtained: R4n = (4n − 2)R4n − 4, where Rn is the number of arrangements for a n × n board. Iterating, it follows that R4n = 2n(2n − 1)(2n − 3)...1. The number of arrangements for a (4n + 1) × (4n + 1) board is the same as that for a 4n × 4n board; this is because on a (4n + 1) × (4n + 1) board, one rook must necessarily stand in the centre and thus the central rank and file can be removed. Therefore R4n + 1 = R4n. For the traditional chessboard (n = 2), R8 = 4 × 3 × 1 = 12 possible arrangements with rotational symmetry. For (4n + 2) × (4n + 2) and (4n + 3) × (4n + 3) boards, the number of solutions is zero. Two cases are possible for each rook: either it stands in the centre or it doesn't stand in the centre. In the second case, this rook is included in the rook quartet that exchanges squares on turning the board at 90°. Therefore, the total number of rooks must be either 4n (when there is no central square on the board) or 4n + 1. This proves that R4n + 2 = R4n + 3 = 0. The number of arrangements of n non-attacking rooks symmetric to one of the diagonals (for determinacy, the diagonal corresponding to a1–h8 on the chessboard) on a n × n board is given by the telephone numbers defined by the recurrence Qn = Qn − 1 + (n − 1)Qn − 2. This recurrence is derived in the following way. Note that the rook on the first file either stands on the bottom corner square or it stands on another square. In the first case, removal of the first file and the first rank leads to the symmetric arrangement n − 1 rooks on a (n − 1) × (n − 1) board. The number of such arrangements is Qn − 1. In the second case, for the original rook there is another rook, symmetric to the first one about the chosen diagonal. Removing the files and ranks of those rooks leads to a symmetric arrangement n − 2 rooks on a (n − 2) × (n − 2) board. Since the number of such arrangements is Qn − 2 and the rook can be put on the n − 1 square of the first file, there are (n − 1)Qn − 2 ways for doing this, which immediately gives the above recurrence. The number of diagonal-symmetric arrangements is then given by the expression: $Q_{n}=1+{\binom {n}{2}}+{\frac {1}{1\times 2}}{\binom {n}{2}}{\binom {n-2}{2}}+{\frac {1}{1\times 2\times 3}}{\binom {n}{2}}{\binom {n-2}{2}}{\binom {n-4}{2}}+\cdots .$ This expression is derived by partitioning all rook arrangements in classes; in class s are those arrangements in which s pairs of rooks do not stand on the diagonal. In exactly the same way, it can be shown that the number of n-rook arrangements on a n × n board, such that they do not attack each other and are symmetric to both diagonals is given by the recurrence equations B2n = 2B2n − 2 + (2n − 2)B2n − 4 and B2n + 1 = B2n. Arrangements counted by symmetry classes A different type of generalization is that in which rook arrangements that are obtained from each other by symmetries of the board are counted as one. For instance, if rotating the board by 90 degrees is allowed as a symmetry, then any arrangement obtained by a rotation of 90, 180, or 270 degrees is considered to be "the same" as the original pattern, even though these arrangements are counted separately in the original problem where the board is fixed. For such problems, Dudeney[11] observes: "How many ways there are if mere reversals and reflections are not counted as different has not yet been determined; it is a difficult problem." The problem reduces to that of counting symmetric arrangements via Burnside's lemma. References 1. John Riordan, Introduction to Combinatorial Analysis, Princeton University Press, 1980 (originally published by John Wiley and Sons, New York; Chapman and Hall, London, 1958) ISBN 978-0-691-02365-6 (reprinted again in 2002, by Dover Publications). See chapters 7 & 8. 2. Weisstein, Eric W. "Rook Polynomial." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RookPolynomial.html 3. Ole J. Heilmann and Elliott H. Lieb, Theory of monomer-dimer systems. Communications in Mathematical Physics, Vol. 25 (1972), pp. 190–232. 4. Dudeney, Henry E. Amusements In Mathematics. 1917. Nelson. (republished by Plain Label Books: ISBN 1-60303-152-9, also as a collection of newspaper clippings, Dover Publications, 1958; Kessinger Publishing, 2006). The book can be freely downloaded from Project Gutenberg site 5. Dudeney, Problem 295 6. Vilenkin, Naum Ya. Combinatorics (Kombinatorika). 1969. Nauka Publishers, Moscow (In Russian). 7. Vilenkin, Naum Ya. Popular Combinatorics (Populyarnaya kombinatorika). 1975. Nauka Publishers, Moscow (In Russian). 8. Gik, Evgeny Ya. Mathematics on the Chessboard (Matematika na shakhmatnoy doske). 1976. Nauka Publishers, Moscow (In Russian). 9. Gik, Evgeny Ya. Chess and Mathematics (Shakhmaty i matematika). 1983. Nauka Publishers, Moscow (In Russian). ISBN 3-87144-987-3 (GVK-Gemeinsamer Verbundkatalog) 10. Kokhas', Konstantin P. Rook Numbers and Polynomials (Ladeynye chisla i mnogochleny). MCNMO, Moscow, 2003 (in Russian). ISBN 5-94057-114-X www.mccme.ru/free-books/mmmf-lectures/book.26.pdf (GVK-Gemeinsamer Verbundkatalog) 11. Dudeney, Answer to Problem 295	Wikipedia
Room square A Room square, named after Thomas Gerald Room, is an n × n array filled with n + 1 different symbols in such a way that: 1. Each cell of the array is either empty or contains an unordered pair from the set of symbols 2. Each symbol occurs exactly once in each row and column of the array 3. Every unordered pair of symbols occurs in exactly one cell of the array. An example, a Room square of order seven, if the set of symbols is integers from 0 to 7: 0,71,54,62,3 3,41,72,60,5 1,64,52,70,3 0,25,63,71,4 2,51,30,64,7 3,62,40,15,7 0,43,51,26,7 It is known that a Room square (or squares) exist if and only if n is odd but not 3 or 5. History The order-7 Room square was used by Robert Richard Anstice to provide additional solutions to Kirkman's schoolgirl problem in the mid-19th century, and Anstice also constructed an infinite family of Room squares, but his constructions did not attract attention.[1] Thomas Gerald Room reinvented Room squares in a note published in 1955,[2] and they came to be named after him. In his original paper on the subject, Room observed that n must be odd and unequal to 3 or 5, but it was not shown that these conditions are both necessary and sufficient until the work of W. D. Wallis in 1973.[3] Applications Pre-dating Room's paper, Room squares had been used by the directors of duplicate bridge tournaments in the construction of the tournaments. In this application they are known as Howell rotations. The columns of the square represent tables, each of which holds a deal of the cards that is played by each pair of teams that meet at that table. The rows of the square represent rounds of the tournament, and the numbers within the cells of the square represent the teams that are scheduled to play each other at the table and round represented by that cell. Archbold and Johnson used Room squares to construct experimental designs.[4] There are connections between Room squares and other mathematical objects including quasigroups, Latin squares, graph factorizations, and Steiner triple systems.[5] See also • Combinatorial design • Magic square • Square matrices References 1. O'Connor, John J.; Robertson, Edmund F., "Robert Anstice", MacTutor History of Mathematics Archive, University of St Andrews. 2. Room, T. G. (1955), "A new type of magic square", The Mathematical Gazette, 39: 307, doi:10.2307/3608578, JSTOR 3608578, S2CID 125711658 3. Hirschfeld, J. W. P.; Wall, G. E. (1987), "Thomas Gerald Room. 10 November 1902–2 April 1986", Biographical Memoirs of Fellows of the Royal Society, 33: 575–601, doi:10.1098/rsbm.1987.0020, JSTOR 769963, S2CID 73328766; also published in Historical Records of Australian Science 7 (1): 109–122, doi:10.1071/HR9870710109; an abridged version is online at the web site of the Australian Academy of Science 4. Archbold, J. W.; Johnson, N. L. (1958), "A construction for Room's squares and an application in experimental design", Annals of Mathematical Statistics, 29: 219–225, doi:10.1214/aoms/1177706719, MR 0102156 5. Wallis, W. D. (1972), "Part 2: Room squares", in Wallis, W. D.; Street, Anne Penfold; Wallis, Jennifer Seberry (eds.), Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Lecture Notes in Mathematics, vol. 292, New York: Springer-Verlag, pp. 30–121, doi:10.1007/BFb0069909, ISBN 0-387-06035-9; see in particular p. 33 Further reading • Dinitz, J. H.; Stinson, D. R. (1992), "Room squares and related designs", in Dinitz, J. H.; Stinson, D. R. (eds.), Contemporary Design Theory: A Collection of Surveys, Wiley–Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, pp. 137–204, ISBN 0-471-53141-3 • Weisstein, Eric W., "Room Square", MathWorld	Wikipedia
Root-finding algorithms In mathematics and computing, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f, from the real numbers to real numbers or from the complex numbers to the complex numbers, is a number x such that f(x) = 0. As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form, root-finding algorithms provide approximations to zeros, expressed either as floating-point numbers or as small isolating intervals, or disks for complex roots (an interval or disk output being equivalent to an approximate output together with an error bound).[1] Solving an equation f(x) = g(x) is the same as finding the roots of the function h(x) = f(x) – g(x). Thus root-finding algorithms allow solving any equation defined by continuous functions. However, most root-finding algorithms do not guarantee that they will find all the roots; in particular, if such an algorithm does not find any root, that does not mean that no root exists. Most numerical root-finding methods use iteration, producing a sequence of numbers that hopefully converges towards the root as its limit. They require one or more initial guesses of the root as starting values, then each iteration of the algorithm produces a successively more accurate approximation to the root. Since the iteration must be stopped at some point, these methods produce an approximation to the root, not an exact solution. Many methods compute subsequent values by evaluating an auxiliary function on the preceding values. The limit is thus a fixed point of the auxiliary function, which is chosen for having the roots of the original equation as fixed points, and for converging rapidly to these fixed points. The behavior of general root-finding algorithms is studied in numerical analysis. However, for polynomials, root-finding study belongs generally to computer algebra, since algebraic properties of polynomials are fundamental for the most efficient algorithms. The efficiency of an algorithm may depend dramatically on the characteristics of the given functions. For example, many algorithms use the derivative of the input function, while others work on every continuous function. In general, numerical algorithms are not guaranteed to find all the roots of a function, so failing to find a root does not prove that there is no root. However, for polynomials, there are specific algorithms that use algebraic properties for certifying that no root is missed, and locating the roots in separate intervals (or disks for complex roots) that are small enough to ensure the convergence of numerical methods (typically Newton's method) to the unique root so located. Bracketing methods Bracketing methods determine successively smaller intervals (brackets) that contain a root. When the interval is small enough, then a root has been found. They generally use the intermediate value theorem, which asserts that if a continuous function has values of opposite signs at the end points of an interval, then the function has at least one root in the interval. Therefore, they require to start with an interval such that the function takes opposite signs at the end points of the interval. However, in the case of polynomials there are other methods (Descartes' rule of signs, Budan's theorem and Sturm's theorem) for getting information on the number of roots in an interval. They lead to efficient algorithms for real-root isolation of polynomials, which ensure finding all real roots with a guaranteed accuracy. Bisection method The simplest root-finding algorithm is the bisection method. Let f be a continuous function, for which one knows an interval [a, b] such that f(a) and f(b) have opposite signs (a bracket). Let c = (a +b)/2 be the middle of the interval (the midpoint or the point that bisects the interval). Then either f(a) and f(c), or f(c) and f(b) have opposite signs, and one has divided by two the size of the interval. Although the bisection method is robust, it gains one and only one bit of accuracy with each iteration. Therefore, the number of function evaluations required for finding an ε-approximate root is $\log _{2}{\frac {b-a}{\varepsilon }}$. Other methods, under appropriate conditions, can gain accuracy faster. False position (regula falsi) The false position method, also called the regula falsi method, is similar to the bisection method, but instead of using bisection search's middle of the interval it uses the x-intercept of the line that connects the plotted function values at the endpoints of the interval, that is $c={\frac {af(b)-bf(a)}{f(b)-f(a)}}.$ False position is similar to the secant method, except that, instead of retaining the last two points, it makes sure to keep one point on either side of the root. The false position method can be faster than the bisection method and will never diverge like the secant method; however, it may fail to converge in some naive implementations due to roundoff errors that may lead to a wrong sign for f(c); typically, this may occur if the rate of variation of f is large in the neighborhood of the root. ITP method The ITP method is the only known method to bracket the root with the same worst case guarantees of the bisection method while guaranteeing a superlinear convergence to the root of smooth functions as the secant method. It is also the only known method guaranteed to outperform the bisection method on the average for any continuous distribution on the location of the root (see ITP Method#Analysis). It does so by keeping track of both the bracketing interval as well as the minmax interval in which any point therein converges as fast as the bisection method. The construction of the queried point c follows three steps: interpolation (similar to the regula falsi), truncation (adjusting the regula falsi similar to Regula falsi § Improvements in regula falsi) and then projection onto the minmax interval. The combination of these steps produces a simultaneously minmax optimal method with guarantees similar to interpolation based methods for smooth functions, and, in practice will outperform both the bisection method and interpolation based methods under both smooth and non-smooth functions. Interpolation Many root-finding processes work by interpolation. This consists in using the last computed approximate values of the root for approximating the function by a polynomial of low degree, which takes the same values at these approximate roots. Then the root of the polynomial is computed and used as a new approximate value of the root of the function, and the process is iterated. Two values allow interpolating a function by a polynomial of degree one (that is approximating the graph of the function by a line). This is the basis of the secant method. Three values define a quadratic function, which approximates the graph of the function by a parabola. This is Muller's method. Regula falsi is also an interpolation method, which differs from the secant method by using, for interpolating by a line, two points that are not necessarily the last two computed points. Iterative methods Although all root-finding algorithms proceed by iteration, an iterative root-finding method generally uses a specific type of iteration, consisting of defining an auxiliary function, which is applied to the last computed approximations of a root for getting a new approximation. The iteration stops when a fixed point (up to the desired precision) of the auxiliary function is reached, that is when the new computed value is sufficiently close to the preceding ones. Newton's method (and similar derivative-based methods) Newton's method assumes the function f to have a continuous derivative. Newton's method may not converge if started too far away from a root. However, when it does converge, it is faster than the bisection method, and is usually quadratic. Newton's method is also important because it readily generalizes to higher-dimensional problems. Newton-like methods with higher orders of convergence are the Householder's methods. The first one after Newton's method is Halley's method with cubic order of convergence. Secant method Replacing the derivative in Newton's method with a finite difference, we get the secant method. This method does not require the computation (nor the existence) of a derivative, but the price is slower convergence (the order is approximately 1.6 (golden ratio)). A generalization of the secant method in higher dimensions is Broyden's method. Steffensen's method If we use a polynomial fit to remove the quadratic part of the finite difference used in the Secant method, so that it better approximates the derivative, we obtain Steffensen's method, which has quadratic convergence, and whose behavior (both good and bad) is essentially the same as Newton's method but does not require a derivative. Fixed point iteration method We can use the fixed-point iteration to find the root of a function. Given a function $f(x)$ which we have set to zero to find the root ($f(x)=0$), we rewrite the equation in terms of $x$ so that $f(x)=0$ becomes $x=g(x)$ (note, there are often many $g(x)$ functions for each $f(x)=0$ function). Next, we relabel each side of the equation as $x_{n+1}=g(x_{n})$ so that we can perform the iteration. Next, we pick a value for $x_{1}$ and perform the iteration until it converges towards a root of the function. If the iteration converges, it will converge to a root. The iteration will only converge if $\|g'(root)\|<1$. As an example of converting $f(x)=0$ to $x=g(x)$, if given the function $f(x)=x^{2}+x-1$, we will rewrite it as one of the following equations. $x_{n+1}=(1/x_{n})-1$, $x_{n+1}=1/(x_{n}+1)$, $x_{n+1}=1-x_{n}^{2}$, $x_{n+1}=x_{n}^{2}+2x_{n}-1$, or $x_{n+1}=\pm {\sqrt {1-x_{n}}}$. Inverse interpolation The appearance of complex values in interpolation methods can be avoided by interpolating the inverse of f, resulting in the inverse quadratic interpolation method. Again, convergence is asymptotically faster than the secant method, but inverse quadratic interpolation often behaves poorly when the iterates are not close to the root. Combinations of methods Brent's method Brent's method is a combination of the bisection method, the secant method and inverse quadratic interpolation. At every iteration, Brent's method decides which method out of these three is likely to do best, and proceeds by doing a step according to that method. This gives a robust and fast method, which therefore enjoys considerable popularity. Ridders' method Ridders' method is a hybrid method that uses the value of function at the midpoint of the interval to perform an exponential interpolation to the root. This gives a fast convergence with a guaranteed convergence of at most twice the number of iterations as the bisection method. Roots of polynomials This section is an excerpt from Polynomial root-finding algorithms.[edit] Finding polynomial roots is a long-standing problem that has been the object of much research throughout history. A testament to this is that up until the 19th century, algebra meant essentially theory of polynomial equations. Finding roots in higher dimensions The bisection method has been generalized to higher dimensions; these methods are called generalized bisection methods.[2][3] At each iteration, the domain is partitioned into two parts, and the algorithm decides - based on a small number of function evaluations - which of these two parts must contain a root. In one dimension, the criterion for decision is that the function has opposite signs. The main challenge in extending the method to multiple dimensions is to find a criterion that can be computed easily, and guarantees the existence of a root. The Poincaré–Miranda theorem gives a criterion for the existence of a root in a rectangle, but it is hard to verify, since it requires to evaluate the function on the entire boundary of the triangle. Another criterion is given by a theorem of Kronecker.[4] It says that, if the topological degree of a function f on a rectangle is non-zero, then the rectangle must contain at least one root of f. This criterion is the basis for several root-finding methods, such as by Stenger[5] and Kearfott.[6] However, computing the topological degree can be time-consuming. A third criterion is based on a characteristic polyhedron. This criterion is used by a method called Characteristic Bisection.[2]: 19-- It does not require to compute the topological degree - it only requires to compute the signs of function values. The number of required evaluations is at least $\log _{2}(D/\epsilon )$, where D is the length of the longest edge of the characteristic polyhedron.[7]: 11, Lemma.4.7 Note that [7] prove a lower bound on the number of evaluations, and not an upper bound. A fourth method uses an intermediate-value theorem on simplices.[8] Again, no upper bound on the number of queries is given. See also • List of root finding algorithms • Fixed-point computation Broyden's method – Quasi-Newton root-finding method for the multivariable case • Cryptographically secure pseudorandom number generator – Type of functions designed for being unsolvable by root-finding algorithms • GNU Scientific Library • Graeffe's method – Algorithm for finding polynomial roots • Lill's method – Graphical method for the real roots of a polynomial • MPSolve – Software for approximating the roots of a polynomial with arbitrarily high precision • Multiplicity (mathematics) – Number of times an object must be counted for making true a general formula • nth root algorithm • System of polynomial equations – Roots of multiple multivariate polynomials • Kantorovich theorem – About the convergence of Newton's method References 1. Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007). "Chapter 9. Root Finding and Nonlinear Sets of Equations". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8. 2. Mourrain, B.; Vrahatis, M. N.; Yakoubsohn, J. C. (2002-06-01). "On the Complexity of Isolating Real Roots and Computing with Certainty the Topological Degree". Journal of Complexity. 18 (2): 612–640. doi:10.1006/jcom.2001.0636. ISSN 0885-064X. 3. Vrahatis, Michael N. (2020). Sergeyev, Yaroslav D.; Kvasov, Dmitri E. (eds.). "Generalizations of the Intermediate Value Theorem for Approximating Fixed Points and Zeros of Continuous Functions". Numerical Computations: Theory and Algorithms. Lecture Notes in Computer Science. Cham: Springer International Publishing. 11974: 223–238. doi:10.1007/978-3-030-40616-5_17. ISBN 978-3-030-40616-5. S2CID 211160947. 4. "Iterative solution of nonlinear equations in several variables". Guide books. Retrieved 2023-04-16. 5. Stenger, Frank (1975-03-01). "Computing the topological degree of a mapping inRn". Numerische Mathematik. 25 (1): 23–38. doi:10.1007/BF01419526. ISSN 0945-3245. S2CID 122196773. 6. Kearfott, Baker (1979-06-01). "An efficient degree-computation method for a generalized method of bisection". Numerische Mathematik. 32 (2): 109–127. doi:10.1007/BF01404868. ISSN 0029-599X. S2CID 122058552. 7. Vrahatis, M. N.; Iordanidis, K. I. (1986-03-01). "A rapid Generalized Method of Bisection for solving Systems of Non-linear Equations". Numerische Mathematik. 49 (2): 123–138. doi:10.1007/BF01389620. ISSN 0945-3245. 8. Vrahatis, Michael N. (2020-04-15). "Intermediate value theorem for simplices for simplicial approximation of fixed points and zeros". Topology and its Applications. 275: 107036. doi:10.1016/j.topol.2019.107036. ISSN 0166-8641. Further reading • J.M. McNamee: "Numerical Methods for Roots of Polynomials - Part I", Elsevier (2007). • J.M. McNamee and Victor Pan: "Numerical Methods for Roots of Polynomials - Part II", Elsevier (2013). Root-finding algorithms Bracketing (no derivative) • Bisection method • Regula falsi • ITP Method Newton • Newton's method Quasi-Newton • Muller's method • Secant method Hybrid methods • Brent's method • Ridders' method Polynomial methods • Bairstow's method • Jenkins–Traub method • Laguerre's method	Wikipedia
Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution, or Maxwell(ian) distribution, is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. Maxwell–Boltzmann Probability density function Cumulative distribution function Parameters $a>0$ Support $x\in (0;\infty )$ PDF ${\sqrt {\frac {2}{\pi }}}\,{\frac {x^{2}}{a^{3}}}\,\exp \left({\frac {-x^{2}}{2a^{2}}}\right)$ (where exp is the exponential function) CDF $\operatorname {erf} \left({\frac {x}{{\sqrt {2}}a}}\right)-{\sqrt {\frac {2}{\pi }}}\,{\frac {x}{a}}\,\exp \left({\frac {-x^{2}}{2a^{2}}}\right)$ (where erf is the error function) Mean $\mu =2a{\sqrt {\frac {2}{\pi }}}$ Mode ${\sqrt {2}}a$ Variance $\sigma ^{2}={\frac {a^{2}(3\pi -8)}{\pi }}$ Skewness $\gamma _{1}={\frac {2{\sqrt {2}}(16-5\pi )}{(3\pi -8)^{3/2}}}$ Ex. kurtosis $\gamma _{2}={\frac {4(-96+40\pi -3\pi ^{2})}{(3\pi -8)^{2}}}$ Entropy $\ln \left(a{\sqrt {2\pi }}\right)+\gamma -{\frac {1}{2}}$ This article is about particle energy levels and velocities. For system energy states, see Boltzmann distribution. It was first defined and used for describing particle speeds in idealized gases, where the particles move freely inside a stationary container without interacting with one another, except for very brief collisions in which they exchange energy and momentum with each other or with their thermal environment. The term "particle" in this context refers to gaseous particles only (atoms or molecules), and the system of particles is assumed to have reached thermodynamic equilibrium.[1] The energies of such particles follow what is known as Maxwell–Boltzmann statistics, and the statistical distribution of speeds is derived by equating particle energies with kinetic energy. Mathematically, the Maxwell–Boltzmann distribution is the chi distribution with three degrees of freedom (the components of the velocity vector in Euclidean space), with a scale parameter measuring speeds in units proportional to the square root of $T/m$ (the ratio of temperature and particle mass).[2] The Maxwell–Boltzmann distribution is a result of the kinetic theory of gases, which provides a simplified explanation of many fundamental gaseous properties, including pressure and diffusion.[3] The Maxwell–Boltzmann distribution applies fundamentally to particle velocities in three dimensions, but turns out to depend only on the speed (the magnitude of the velocity) of the particles. A particle speed probability distribution indicates which speeds are more likely: a randomly chosen particle will have a speed selected randomly from the distribution, and is more likely to be within one range of speeds than another. The kinetic theory of gases applies to the classical ideal gas, which is an idealization of real gases. In real gases, there are various effects (e.g., van der Waals interactions, vortical flow, relativistic speed limits, and quantum exchange interactions) that can make their speed distribution different from the Maxwell–Boltzmann form. However, rarefied gases at ordinary temperatures behave very nearly like an ideal gas and the Maxwell speed distribution is an excellent approximation for such gases. This is also true for ideal plasmas, which are ionized gases of sufficiently low density.[4] The distribution was first derived by Maxwell in 1860 on heuristic grounds.[5] Boltzmann later, in the 1870s, carried out significant investigations into the physical origins of this distribution. The distribution can be derived on the ground that it maximizes the entropy of the system. A list of derivations are: 1. Maximum entropy probability distribution in the phase space, with the constraint of conservation of average energy $\langle H\rangle =E;$ 2. Canonical ensemble. Distribution function For a system containing a large number of identical non-interacting, non-relativistic classical particles in thermodynamic equilibrium, the fraction of the particles within an infinitesimal element of the three-dimensional velocity space d 3v, centered on a velocity vector of magnitude v, is given by $f(v)~d^{3}v={\biggl [}{\frac {m}{2\pi kT}}{\biggr ]}^{\frac {3}{2}}\,\exp \left(-{\frac {mv^{2}}{2kT}}\right)~d^{3}v,$ where: • m is the particle mass; • k is the Boltzmann constant; • T is thermodynamic temperature; • f (v) is a probability distribution function, properly normalized so that $ \int f(v)\,d^{3}v$ over all velocities is unity. One can write the element of velocity space as $d^{3}v=dv_{x}\,dv_{y}\,dv_{z}$, for velocities in a standard Cartesian coordinate system, or as $d^{3}v=v^{2}\,dv\,d\Omega $ in a standard spherical coordinate system, where $d\Omega $ is an element of solid angle. The Maxwellian distribution function for particles moving in only one direction, if this direction is x, is $f(v_{x})~dv_{x}={\sqrt {\frac {m}{2\pi kT}}}\,\exp \left(-{\frac {mv_{x}^{2}}{2kT}}\right)~dv_{x},$ which can be obtained by integrating the three-dimensional form given above over vy and vz. Recognizing the symmetry of $f(v)$, one can integrate over solid angle and write a probability distribution of speeds as the function[6] $f(v)={\biggl [}{\frac {m}{2\pi kT}}{\biggr ]}^{\frac {3}{2}}\,4\pi v^{2}\exp \left(-{\frac {mv^{2}}{2kT}}\right).$ This probability density function gives the probability, per unit speed, of finding the particle with a speed near v. This equation is simply the Maxwell–Boltzmann distribution (given in the infobox) with distribution parameter $ a={\sqrt {kT/m}}\,.$ The Maxwell–Boltzmann distribution is equivalent to the chi distribution with three degrees of freedom and scale parameter $ a={\sqrt {kT/m}}\,.$ The simplest ordinary differential equation satisfied by the distribution is: ${\begin{aligned}&0=kTvf'(v)+f(v)(mv^{2}-2kT),\\[4pt]&f(1)={\sqrt {\frac {2}{\pi }}}\,{\biggl [}{\frac {m}{kT}}{\biggr ]}^{\frac {3}{2}}\exp \left(-{\frac {m}{2kT}}\right);\end{aligned}}$ or in unitless presentation: ${\begin{aligned}&0=a^{2}xf'(x)+\left(x^{2}-2a^{2}\right)f(x),\\[4pt]&f(1)={\frac {1}{a^{3}}}{\sqrt {\frac {2}{\pi }}}\exp \left(-{\frac {1}{2a^{2}}}\right).\end{aligned}}$ With the Darwin–Fowler method of mean values, the Maxwell–Boltzmann distribution is obtained as an exact result. Relation to the 2D Maxwell–Boltzmann distribution For particles confined to move in a plane, the speed distribution is given by $P(s<\|{\vec {v}}\|<s+ds)={\frac {ms}{kT}}\exp \left(-{\frac {ms^{2}}{2kT}}\right)ds$ This distribution is used for describing systems in equilibrium. However, most systems do not start out in their equilibrium state. The evolution of a system towards its equilibrium state is governed by the Boltzmann equation. The equation predicts that for short range interactions, the equilibrium velocity distribution will follow a Maxwell–Boltzmann distribution. To the right is a molecular dynamics (MD) simulation in which 900 hard sphere particles are constrained to move in a rectangle. They interact via perfectly elastic collisions. The system is initialized out of equilibrium, but the velocity distribution (in blue) quickly converges to the 2D Maxwell–Boltzmann distribution (in orange). Typical speeds The mean speed $\langle v\rangle $, most probable speed (mode) vp, and root-mean-square speed $ {\sqrt {\langle v^{2}\rangle }}$ can be obtained from properties of the Maxwell distribution. This works well for nearly ideal, monatomic gases like helium, but also for molecular gases like diatomic oxygen. This is because despite the larger heat capacity (larger internal energy at the same temperature) due to their larger number of degrees of freedom, their translational kinetic energy (and thus their speed) is unchanged.[7] • The most probable speed, vp, is the speed most likely to be possessed by any molecule (of the same mass m) in the system and corresponds to the maximum value or the mode of f(v). To find it, we calculate the derivative ${\tfrac {df}{dv}},$ set it to zero and solve for v: ${\frac {df(v)}{dv}}=-8\pi {\biggl [}{\frac {m}{2\pi kT}}{\biggr ]}^{\frac {3}{2}}\,v\,\left[{\frac {mv^{2}}{2kT}}-1\right]\exp \left(-{\frac {mv^{2}}{2kT}}\right)=0$ with the solution: ${\frac {mv_{\text{p}}^{2}}{2kT}}=1;\quad v_{\text{p}}={\sqrt {\frac {2kT}{m}}}={\sqrt {\frac {2RT}{M}}}$ where: • R is the gas constant; • M is molar mass of the substance, and thus may be calculated as a product of particle mass, m, and Avogadro constant, NA: $M=mN_{\mathrm {A} }.$ For diatomic nitrogen (N2, the primary component of air)[8] at room temperature (300 K), this gives $v_{\text{p}}\approx {\sqrt {\frac {2\cdot 8.31\ {\text{J}}\cdot {\text{mol}}^{-1}{\text{K}}^{-1}\ 300\ {\text{K}}}{0.028\ {\text{kg}}\cdot {\text{mol}}^{-1}}}}\approx 422\ {\text{m/s}}.$ • The mean speed is the expected value of the speed distribution, setting $ b={\frac {1}{2a^{2}}}={\frac {m}{2kT}}$: ${\begin{aligned}\langle v\rangle &=\int _{0}^{\infty }v\,f(v)\,dv\\[2pt]&=4\pi \left[{\frac {b}{\pi }}\right]^{\frac {3}{2}}\int _{0}^{\infty }v^{3}e^{-bv^{2}}dv\\[2pt]&=4\pi \left[{\frac {b}{\pi }}\right]^{\frac {3}{2}}{\frac {1}{2b^{2}}}={\sqrt {\frac {4}{\pi b}}}\\[2pt]&={\sqrt {\frac {8kT}{\pi m}}}={\sqrt {\frac {8RT}{\pi M}}}={\frac {2}{\sqrt {\pi }}}v_{\text{p}}\end{aligned}}$ • The mean square speed $\langle v^{2}\rangle $ is the second-order raw moment of the speed distribution. The "root mean square speed" $v_{\mathrm {rms} }$ is the square root of the mean square speed, corresponding to the speed of a particle with average kinetic energy, setting $ b={\frac {1}{2a^{2}}}={\frac {m}{2kT}}$: ${\begin{aligned}v_{\mathrm {rms} }&={\sqrt {\langle v^{2}\rangle }}=\left[\int _{0}^{\infty }v^{2}\,f(v)\,dv\right]^{\frac {1}{2}}\\[2pt]&=\left[4\pi \left({\frac {b}{\pi }}\right)^{\frac {3}{2}}\int _{0}^{\infty }v^{4}e^{-bv^{2}}dv\right]^{\frac {1}{2}}\\[2pt]&=\left[4\pi \left({\frac {b}{\pi }}\right)^{\frac {3}{2}}{\frac {3}{8}}\left({\frac {\pi }{b^{5}}}\right)^{\frac {1}{2}}\right]^{\frac {1}{2}}={\sqrt {\frac {3}{2b}}}\\[2pt]&={\sqrt {\frac {3kT}{m}}}={\sqrt {\frac {3RT}{M}}}={\sqrt {\frac {3}{2}}}v_{\text{p}}\end{aligned}}$ In summary, the typical speeds are related as follows: $v_{\text{p}}\approx 88.6\%\ \langle v\rangle <\langle v\rangle <108.5\%\ \langle v\rangle \approx v_{\mathrm {rms} }.$ The root mean square speed is directly related to the speed of sound c in the gas, by $c={\sqrt {\frac {\gamma }{3}}}\ v_{\mathrm {rms} }={\sqrt {\frac {f+2}{3f}}}\ v_{\mathrm {rms} }={\sqrt {\frac {f+2}{2f}}}\ v_{\text{p}},$ where $ \gamma =1+{\frac {2}{f}}$ is the adiabatic index, f is the number of degrees of freedom of the individual gas molecule. For the example above, diatomic nitrogen (approximating air) at 300 K, $f=5$[9] and $c={\sqrt {\frac {7}{15}}}v_{\mathrm {rms} }\approx 68\%\ v_{\mathrm {rms} }\approx 84\%\ v_{\text{p}}\approx 353\ \mathrm {m/s} ,$ the true value for air can be approximated by using the average molar weight of air (29 g/mol), yielding 347 m/s at 300 K (corrections for variable humidity are of the order of 0.1% to 0.6%). The average relative velocity ${\begin{aligned}v_{\rm {rel}}\equiv \langle \|{\vec {v}}_{1}-{\vec {v}}_{2}\|\rangle &=\int \!d^{3}v_{1}\,d^{3}v_{2}\left\|{\vec {v}}_{1}-{\vec {v}}_{2}\right\|f({\vec {v}}_{1})f({\vec {v}}_{2})\\[2pt]&={\frac {4}{\sqrt {\pi }}}{\sqrt {\frac {kT}{m}}}={\sqrt {2}}\langle v\rangle \end{aligned}}$ where the three-dimensional velocity distribution is $f({\vec {v}})\equiv \left[{\frac {2\pi kT}{m}}\right]^{-{\frac {3}{2}}}\exp \left(-{\frac {1}{2}}{\frac {m{\vec {v}}^{2}}{kT}}\right).$ The integral can easily be done by changing to coordinates ${\vec {u}}={\vec {v}}_{1}-{\vec {v}}_{2}$ and ${\vec {U}}={\tfrac {{\vec {v}}_{1}\,+\,{\vec {v}}_{2}}{2}}.$ Derivation and related distributions Maxwell–Boltzmann statistics Main articles: Maxwell–Boltzmann statistics § Derivations, and Boltzmann distribution The original derivation in 1860 by James Clerk Maxwell was an argument based on molecular collisions of the Kinetic theory of gases as well as certain symmetries in the speed distribution function; Maxwell also gave an early argument that these molecular collisions entail a tendency towards equilibrium.[5][10] After Maxwell, Ludwig Boltzmann in 1872[11] also derived the distribution on mechanical grounds and argued that gases should over time tend toward this distribution, due to collisions (see H-theorem). He later (1877)[12] derived the distribution again under the framework of statistical thermodynamics. The derivations in this section are along the lines of Boltzmann's 1877 derivation, starting with result known as Maxwell–Boltzmann statistics (from statistical thermodynamics). Maxwell–Boltzmann statistics gives the average number of particles found in a given single-particle microstate. Under certain assumptions, the logarithm of the fraction of particles in a given microstate is proportional to the ratio of the energy of that state to the temperature of the system: $-\log \left({\frac {N_{i}}{N}}\right)\propto {\frac {E_{i}}{T}}.$ The assumptions of this equation are that the particles do not interact, and that they are classical; this means that each particle's state can be considered independently from the other particles' states. Additionally, the particles are assumed to be in thermal equilibrium.[1][13] This relation can be written as an equation by introducing a normalizing factor: ${\frac {N_{i}}{N}}={\frac {\exp \left(-{\frac {E_{i}}{kT}}\right)}\sum _{j}\exp \left(-{\tfrac {E_{j}}{kT}}\right)}}$ (1) where: • Ni is the expected number of particles in the single-particle microstate i, • N is the total number of particles in the system, • Ei is the energy of microstate i, • the sum over index j takes into account all microstates, • T is the equilibrium temperature of the system, • k is the Boltzmann constant. The denominator in Equation (1) is a normalizing factor so that the ratios $N_{i}:N$ add up to unity — in other words it is a kind of partition function (for the single-particle system, not the usual partition function of the entire system). Because velocity and speed are related to energy, Equation (1) can be used to derive relationships between temperature and the speeds of gas particles. All that is needed is to discover the density of microstates in energy, which is determined by dividing up momentum space into equal sized regions. Distribution for the momentum vector The potential energy is taken to be zero, so that all energy is in the form of kinetic energy. The relationship between kinetic energy and momentum for massive non-relativistic particles is $E={\frac {p^{2}}{2m}}$ (2) where p2 is the square of the momentum vector p = [px, py, pz]. We may therefore rewrite Equation (1) as: ${\frac {N_{i}}{N}}={\frac {1}{Z}}\exp \left(-{\frac {p_{i,x}^{2}+p_{i,y}^{2}+p_{i,z}^{2}}{2mkT}}\right)$ (3) where: • Z is the partition function, corresponding to the denominator in Equation (1); • m is the molecular mass of the gas; • T is the thermodynamic temperature; • k is the Boltzmann constant. This distribution of Ni : N is proportional to the probability density function fp for finding a molecule with these values of momentum components, so: $f_{\mathbf {p} }(p_{x},p_{y},p_{z})\propto \exp \left(-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mkT}}\right)$ (4) The normalizing constant can be determined by recognizing that the probability of a molecule having some momentum must be 1. Integrating the exponential in (4) over all px, py, and pz yields a factor of $\iiint _{-\infty }^{+\infty }\exp \left(-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mkT}}\right)dp_{x}\,dp_{y}\,dp_{z}={\Bigl [}{\sqrt {\pi }}{\sqrt {2mkT}}{\Bigr ]}^{3}$ So that the normalized distribution function is: $f_{\mathbf {p} }(p_{x},p_{y},p_{z})=\left[{\frac {1}{2\pi mkT}}\right]^{\frac {3}{2}}\exp \left(-{\frac {p_{x}^{2}+p_{y}^{2}+p_{z}^{2}}{2mkT}}\right)$ (6) The distribution is seen to be the product of three independent normally distributed variables $p_{x}$, $p_{y}$, and $p_{z}$, with variance $mkT$. Additionally, it can be seen that the magnitude of momentum will be distributed as a Maxwell–Boltzmann distribution, with $a={\sqrt {mkT}}$. The Maxwell–Boltzmann distribution for the momentum (or equally for the velocities) can be obtained more fundamentally using the H-theorem at equilibrium within the Kinetic theory of gases framework. Distribution for the energy The energy distribution is found imposing $f_{E}(E)\,dE=f_{p}({\textbf {p}})\,d^{3}{\textbf {p}},$ (7) where $d^{3}{\textbf {p}}$ is the infinitesimal phase-space volume of momenta corresponding to the energy interval dE. Making use of the spherical symmetry of the energy-momentum dispersion relation $E={\tfrac {\|{\textbf {p}}\|^{2}}{2m}},$ this can be expressed in terms of dE as $d^{3}{\textbf {p}}=4\pi \|{\textbf {p}}\|^{2}d\|{\textbf {p}}\|=4\pi m{\sqrt {2mE}}\ dE.$ (8) Using then (8) in (7), and expressing everything in terms of the energy E, we get ${\begin{aligned}f_{E}(E)dE&=\left[{\frac {1}{2\pi mkT}}\right]^{\frac {3}{2}}\exp \left(-{\frac {E}{kT}}\right)4\pi m{\sqrt {2mE}}\ dE\\[2pt]&=2{\sqrt {\frac {E}{\pi }}}\,\left[{\frac {1}{kT}}\right]^{\frac {3}{2}}\exp \left(-{\frac {E}{kT}}\right)\,dE\end{aligned}}$ and finally $f_{E}(E)=2{\sqrt {\frac {E}{\pi }}}\,\left[{\frac {1}{kT}}\right]^{\frac {3}{2}}\exp \left(-{\frac {E}{kT}}\right)$ (9) Since the energy is proportional to the sum of the squares of the three normally distributed momentum components, this energy distribution can be written equivalently as a gamma distribution, using a shape parameter, $k_{\text{shape}}=3/2$ and a scale parameter, $\theta _{\text{scale}}=kT.$ Using the equipartition theorem, given that the energy is evenly distributed among all three degrees of freedom in equilibrium, we can also split $f_{E}(E)dE$ into a set of chi-squared distributions, where the energy per degree of freedom, ε is distributed as a chi-squared distribution with one degree of freedom,[14] $f_{\varepsilon }(\varepsilon )\,d\varepsilon ={\sqrt {\frac {1}{\pi \varepsilon kT}}}~\exp \left(-{\frac {\varepsilon }{kT}}\right)\,d\varepsilon $ At equilibrium, this distribution will hold true for any number of degrees of freedom. For example, if the particles are rigid mass dipoles of fixed dipole moment, they will have three translational degrees of freedom and two additional rotational degrees of freedom. The energy in each degree of freedom will be described according to the above chi-squared distribution with one degree of freedom, and the total energy will be distributed according to a chi-squared distribution with five degrees of freedom. This has implications in the theory of the specific heat of a gas. Distribution for the velocity vector Recognizing that the velocity probability density fv is proportional to the momentum probability density function by $f_{\mathbf {v} }d^{3}v=f_{\mathbf {p} }\left({\frac {dp}{dv}}\right)^{3}d^{3}v$ and using p = mv we get $f_{\mathbf {v} }(v_{x},v_{y},v_{z})={\biggl [}{\frac {m}{2\pi kT}}{\biggr ]}^{\frac {3}{2}}\exp \left(-{\frac {m(v_{x}^{2}+v_{y}^{2}+v_{z}^{2})}{2kT}}\right)$ which is the Maxwell–Boltzmann velocity distribution. The probability of finding a particle with velocity in the infinitesimal element [dvx, dvy, dvz] about velocity v = [vx, vy, vz] is $f_{\mathbf {v} }\left(v_{x},v_{y},v_{z}\right)\,dv_{x}\,dv_{y}\,dv_{z}.$ Like the momentum, this distribution is seen to be the product of three independent normally distributed variables $v_{x}$, $v_{y}$, and $v_{z}$, but with variance $ {\frac {kT}{m}}$. It can also be seen that the Maxwell–Boltzmann velocity distribution for the vector velocity [vx, vy, vz] is the product of the distributions for each of the three directions: $f_{\mathbf {v} }\left(v_{x},v_{y},v_{z}\right)=f_{v}(v_{x})f_{v}(v_{y})f_{v}(v_{z})$ where the distribution for a single direction is $f_{v}(v_{i})={\sqrt {\frac {m}{2\pi kT}}}\exp \left(-{\frac {mv_{i}^{2}}{2kT}}\right).$ Each component of the velocity vector has a normal distribution with mean $\mu _{v_{x}}=\mu _{v_{y}}=\mu _{v_{z}}=0$ and standard deviation $ \sigma _{v_{x}}=\sigma _{v_{y}}=\sigma _{v_{z}}={\sqrt {\frac {kT}{m}}}$, so the vector has a 3-dimensional normal distribution, a particular kind of multivariate normal distribution, with mean $\mu _{\mathbf {v} }=\mathbf {0} $ and covariance $ \Sigma _{\mathbf {v} }=\left({\frac {kT}{m}}\right)I$, where $I$ is the 3 × 3 identity matrix. Distribution for the speed The Maxwell–Boltzmann distribution for the speed follows immediately from the distribution of the velocity vector, above. Note that the speed is $v={\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}$ and the volume element in spherical coordinates $dv_{x}\,dv_{y}\,dv_{z}=v^{2}\sin \theta \,dv\,d\theta \,d\phi =v^{2}\,dv\,d\Omega $ where $\phi $ and $\theta $ are the spherical coordinate angles of the velocity vector. Integration of the probability density function of the velocity over the solid angles $d\Omega $ yields an additional factor of $4\pi $. The speed distribution with substitution of the speed for the sum of the squares of the vector components: $f(v)={\sqrt {\frac {2}{\pi }}}\,{\biggl [}{\frac {m}{kT}}{\biggr ]}^{\frac {3}{2}}v^{2}\exp \left(-{\frac {mv^{2}}{2kT}}\right).$ In n-dimensional space In n-dimensional space, Maxwell–Boltzmann distribution becomes: $f(v)~d^{n}v={\biggl [}{\frac {m}{2\pi kT}}{\biggr ]}^{\frac {n}{2}}\,\exp \left(-{\frac {m\|v\|^{2}}{2kT}}\right)~d^{n}v$ Speed distribution becomes: $f(v)~dv={\text{const.}}\times \exp \left(-{\frac {mv^{2}}{2kT}}\right)\times v^{n-1}~dv$ The following integral result is useful: ${\begin{aligned}\int _{0}^{+\infty }v^{a}\exp \left(-{\frac {mv^{2}}{2kT}}\right)dv&=\left[{\frac {2kT}{m}}\right]^{\frac {a+1}{2}}\int _{0}^{+\infty }e^{-x}x^{a/2}\,dx^{1/2}\\[2pt]&=\left[{\frac {2kT}{m}}\right]^{\frac {a+1}{2}}\int _{0}^{+\infty }e^{-x}x^{a/2}{\frac {x^{-1/2}}{2}}\,dx\\[2pt]&=\left[{\frac {2kT}{m}}\right]^{\frac {a+1}{2}}{\frac {\Gamma \left({\frac {a+1}{2}}\right)}{2}}\end{aligned}}$ where $\Gamma (z)$ is the Gamma function. This result can be used to calculate the moments of speed distribution function: ${\begin{aligned}\langle v\rangle &={\frac \int _{0}^{+\infty }v\cdot v^{n-1}\exp \left(-{\tfrac {mv^{2}}{2kT}}\right)\,dv}\int _{0}^{+\infty }v^{n-1}\exp \left(-{\tfrac {mv^{2}}{2kT}}\right)\,dv}}\\[4pt]&={\sqrt {\frac {2kT}{m}}}\ {\frac {\Gamma \left({\frac {n+1}{2}}\right)}{\Gamma \left({\frac {n}{2}}\right)}}\end{aligned}}$ which is the mean speed itself $ v_{\mathrm {avg} }=\langle v\rangle ={\sqrt {\frac {2kT}{m}}}\ {\frac {\Gamma \left({\frac {n+1}{2}}\right)}{\Gamma \left({\frac {n}{2}}\right)}}.$ ${\begin{aligned}\langle v^{2}\rangle &={\frac \int _{0}^{+\infty }v^{2}\cdot v^{n-1}\exp \left(-{\tfrac {mv^{2}}{2kT}}\right)\,dv}\int _{0}^{+\infty }v^{n-1}\exp \left(-{\tfrac {mv^{2}}{2kT}}\right)\,dv}}\\[2pt]&=\left[{\frac {2kT}{m}}\right]{\frac {\Gamma ({\frac {n+2}{2}})}{\Gamma ({\frac {n}{2}})}}\\[2pt]&=\left[{\frac {2kT}{m}}\right]{\frac {n}{2}}={\frac {nkT}{m}}\end{aligned}}$ which gives root-mean-square speed $ v_{\rm {rms}}={\sqrt {\langle v^{2}\rangle }}={\sqrt {\frac {nkT}{m}}}.$ The derivative of speed distribution function: ${\frac {df(v)}{dv}}={\text{const.}}\times \exp \left(-{\frac {mv^{2}}{2kT}}\right){\biggl [}-{\frac {mv}{kT}}v^{n-1}+(n-1)v^{n-2}{\biggr ]}=0$ This yields the most probable speed (mode) $ v_{\rm {p}}={\sqrt {\frac {(n-1)kT}{m}}}.$ See also • Quantum Boltzmann equation • Maxwell–Boltzmann statistics • Maxwell–Jüttner distribution • Boltzmann distribution • Rayleigh distribution • Kinetic theory of gases References 1. Statistical Physics (2nd Edition), F. Mandl, Manchester Physics, John Wiley & Sons, 2008, ISBN 9780471915331 2. University Physics – With Modern Physics (12th Edition), H.D. Young, R.A. Freedman (Original edition), Addison-Wesley (Pearson International), 1st Edition: 1949, 12th Edition: 2008, ISBN 978-0-321-50130-1 3. Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN 3-527-26954-1 (Verlagsgesellschaft), ISBN 0-89573-752-3 (VHC Inc.) 4. N.A. Krall and A.W. Trivelpiece, Principles of Plasma Physics, San Francisco Press, Inc., 1986, among many other texts on basic plasma physics 5. See: • Maxwell, J.C. (1860 A): Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 4th Series, vol.19, pp.19-32. • Maxwell, J.C. (1860 B): Illustrations of the dynamical theory of gases. Part II. On the process of diffusion of two or more kinds of moving particles among one another. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 4th Ser., vol.20, pp.21-37. 6. H.J.W. Müller-Kirsten (2013), Basics of Statistical Physics, 2nd ed., World Scientific, ISBN 978-981-4449-53-3, Chapter 2. 7. Raymond A. Serway; Jerry S. Faughn & Chris Vuille (2011). College Physics, Volume 1 (9th ed.). p. 352. ISBN 9780840068484. 8. The calculation is unaffected by the nitrogen being diatomic. Despite the larger heat capacity (larger internal energy at the same temperature) of diatomic gases relative to monatomic gases, due to their larger number of degrees of freedom, ${{3RT} \over {M_{\text{m}}}}$ is still the mean translational kinetic energy. Nitrogen being diatomic only affects the value of the molar mass M = 28 g/mol. See e.g. K. Prakashan, Engineering Physics (2001), 2.278. 9. Nitrogen at room temperature is considered a "rigid" diatomic gas, with two rotational degrees of freedom additional to the three translational ones, and the vibrational degree of freedom not accessible. 10. Gyenis, Balazs (2017). "Maxwell and the normal distribution: A colored story of probability, independence, and tendency towards equilibrium". Studies in History and Philosophy of Modern Physics. 57: 53–65. arXiv:1702.01411. Bibcode:2017SHPMP..57...53G. doi:10.1016/j.shpsb.2017.01.001. S2CID 38272381. 11. Boltzmann, L., "Weitere studien über das Wärmegleichgewicht unter Gasmolekülen." Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, mathematisch-naturwissenschaftliche Classe, 66, 1872, pp. 275–370. 12. Boltzmann, L., "Über die Beziehung zwischen dem zweiten Hauptsatz der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht." Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften in Wien, Mathematisch-Naturwissenschaftliche Classe. Abt. II, 76, 1877, pp. 373–435. Reprinted in Wissenschaftliche Abhandlungen, Vol. II, pp. 164–223, Leipzig: Barth, 1909. Translation available at: http://crystal.med.upenn.edu/sharp-lab-pdfs/2015SharpMatschinsky_Boltz1877_Entropy17.pdf 13. McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3 14. Laurendeau, Normand M. (2005). Statistical thermodynamics: fundamentals and applications. Cambridge University Press. p. 434. ISBN 0-521-84635-8., Appendix N, page 434 Further reading • Physics for Scientists and Engineers – with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0-7167-8964-7 • Thermodynamics, From Concepts to Applications (2nd Edition), A. Shavit, C. Gutfinger, CRC Press (Taylor and Francis Group, USA), 2009, ISBN 978-1-4200-7368-3 • Chemical Thermodynamics, D.J.G. Ives, University Chemistry, Macdonald Technical and Scientific, 1971, ISBN 0-356-03736-3 • Elements of Statistical Thermodynamics (2nd Edition), L.K. Nash, Principles of Chemistry, Addison-Wesley, 1974, ISBN 0-201-05229-6 • Ward, CA & Fang, G 1999, 'Expression for predicting liquid evaporation flux: Statistical rate theory approach', Physical Review E, vol. 59, no. 1, pp. 429–40. • Rahimi, P & Ward, CA 2005, 'Kinetics of Evaporation: Statistical Rate Theory Approach', International Journal of Thermodynamics, vol. 8, no. 9, pp. 1–14. External links Wikimedia Commons has media related to Maxwell–Boltzmann distributions. • "The Maxwell Speed Distribution" from The Wolfram Demonstrations Project at Mathworld Probability distributions (list) Discrete univariate with finite support • Benford • Bernoulli • beta-binomial • binomial • categorical • hypergeometric • negative • Poisson binomial • Rademacher • soliton • discrete uniform • Zipf • Zipf–Mandelbrot with infinite support • beta negative binomial • Borel • Conway–Maxwell–Poisson • discrete phase-type • Delaporte • extended negative binomial • Flory–Schulz • Gauss–Kuzmin • geometric • logarithmic • mixed Poisson • negative binomial • Panjer • parabolic fractal • Poisson • Skellam • Yule–Simon • zeta Continuous univariate supported on a bounded interval • arcsine • ARGUS • Balding–Nichols • Bates • beta • beta rectangular • continuous Bernoulli • Irwin–Hall • Kumaraswamy • logit-normal • noncentral beta • PERT • raised cosine • reciprocal • triangular • U-quadratic • uniform • Wigner semicircle supported on a semi-infinite interval • Benini • Benktander 1st kind • Benktander 2nd kind • beta prime • Burr • chi • chi-squared • noncentral • inverse • scaled • Dagum • Davis • Erlang • hyper • exponential • hyperexponential • hypoexponential • logarithmic • F • noncentral • folded normal • Fréchet • gamma • generalized • inverse • gamma/Gompertz • Gompertz • shifted • half-logistic • half-normal • Hotelling's T-squared • inverse Gaussian • generalized • Kolmogorov • Lévy • log-Cauchy • log-Laplace • log-logistic • log-normal • log-t • Lomax • matrix-exponential • Maxwell–Boltzmann • Maxwell–Jüttner • Mittag-Leffler • Nakagami • Pareto • phase-type • Poly-Weibull • Rayleigh • relativistic Breit–Wigner • Rice • truncated normal • type-2 Gumbel • Weibull • discrete • Wilks's lambda supported on the whole real line • Cauchy • exponential power • Fisher's z • Kaniadakis κ-Gaussian • Gaussian q • generalized normal • generalized hyperbolic • geometric stable • Gumbel • Holtsmark • hyperbolic secant • Johnson's SU • Landau • Laplace • asymmetric • logistic • noncentral t • normal (Gaussian) • normal-inverse Gaussian • skew normal • slash • stable • Student's t • Tracy–Widom • variance-gamma • Voigt with support whose type varies • generalized chi-squared • generalized extreme value • generalized Pareto • Marchenko–Pastur • Kaniadakis κ-exponential • Kaniadakis κ-Gamma • Kaniadakis κ-Weibull • Kaniadakis κ-Logistic • Kaniadakis κ-Erlang • q-exponential • q-Gaussian • q-Weibull • shifted log-logistic • Tukey lambda Mixed univariate continuous- discrete • Rectified Gaussian Multivariate (joint) • Discrete: • Ewens • multinomial • Dirichlet • negative • Continuous: • Dirichlet • generalized • multivariate Laplace • multivariate normal • multivariate stable • multivariate t • normal-gamma • inverse • Matrix-valued: • LKJ • matrix normal • matrix t • matrix gamma • inverse • Wishart • normal • inverse • normal-inverse • complex Directional Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham Degenerate and singular Degenerate Dirac delta function Singular Cantor Families • Circular • compound Poisson • elliptical • exponential • natural exponential • location–scale • maximum entropy • mixture • Pearson • Tweedie • wrapped • Category • Commons	Wikipedia
Root datum In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in SGA III, published in 1970. Definition A root datum consists of a quadruple $(X^{\ast },\Phi ,X_{\ast },\Phi ^{\vee })$, where • $X^{\ast }$ and $X_{\ast }$ are free abelian groups of finite rank together with a perfect pairing between them with values in $\mathbb {Z} $ which we denote by ( , ) (in other words, each is identified with the dual of the other). • $\Phi $ is a finite subset of $X^{\ast }$ and $\Phi ^{\vee }$ is a finite subset of $X_{\ast }$ and there is a bijection from $\Phi $ onto $\Phi ^{\vee }$, denoted by $\alpha \mapsto \alpha ^{\vee }$. • For each $\alpha $, $(\alpha ,\alpha ^{\vee })=2$. • For each $\alpha $, the map $x\mapsto x-(x,\alpha ^{\vee })\alpha $ induces an automorphism of the root datum (in other words it maps $\Phi $ to $\Phi $ and the induced action on $X_{\ast }$ maps $\Phi ^{\vee }$ to $\Phi ^{\vee }$) The elements of $\Phi $ are called the roots of the root datum, and the elements of $\Phi ^{\vee }$ are called the coroots. If $\Phi $ does not contain $2\alpha $ for any $\alpha \in \Phi $, then the root datum is called reduced. The root datum of an algebraic group If $G$ is a reductive algebraic group over an algebraically closed field $K$ with a split maximal torus $T$ then its root datum is a quadruple $(X^{},\Phi ,X_{},\Phi ^{\vee })$, where • $X^{}$ is the lattice of characters of the maximal torus, • $X_{}$ is the dual lattice (given by the 1-parameter subgroups), • $\Phi $ is a set of roots, • $\Phi ^{\vee }$ is the corresponding set of coroots. A connected split reductive algebraic group over $K$ is uniquely determined (up to isomorphism) by its root datum, which is always reduced. Conversely for any root datum there is a reductive algebraic group. A root datum contains slightly more information than the Dynkin diagram, because it also determines the center of the group. For any root datum $(X^{},\Phi ,X_{},\Phi ^{\vee })$, we can define a dual root datum $(X_{},\Phi ^{\vee },X^{},\Phi )$ by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots. If $G$ is a connected reductive algebraic group over the algebraically closed field $K$, then its Langlands dual group ${}^{L}G$ is the complex connected reductive group whose root datum is dual to that of $G$. References • Michel Demazure, Exp. XXI in SGA 3 vol 3 • T. A. Springer, Reductive groups, in Automorphic forms, representations, and L-functions vol 1 ISBN 0-8218-3347-2	Wikipedia
Semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Lie groups and Lie algebras Classical groups • General linear GL(n) • Special linear SL(n) • Orthogonal O(n) • Special orthogonal SO(n) • Unitary U(n) • Special unitary SU(n) • Symplectic Sp(n) Simple Lie groups Classical • An • Bn • Cn • Dn Exceptional • G2 • F4 • E6 • E7 • E8 Other Lie groups • Circle • Lorentz • Poincaré • Conformal group • Diffeomorphism • Loop • Euclidean Lie algebras • Lie group–Lie algebra correspondence • Exponential map • Adjoint representation • Killing form • Index • Simple Lie algebra • Loop algebra • Affine Lie algebra Semisimple Lie algebra • Dynkin diagrams • Cartan subalgebra • Root system • Weyl group • Real form • Complexification • Split Lie algebra • Compact Lie algebra Representation theory • Lie group representation • Lie algebra representation • Representation theory of semisimple Lie algebras • Representations of classical Lie groups • Theorem of the highest weight • Borel–Weil–Bott theorem Lie groups in physics • Particle physics and representation theory • Lorentz group representations • Poincaré group representations • Galilean group representations Scientists • Sophus Lie • Henri Poincaré • Wilhelm Killing • Élie Cartan • Hermann Weyl • Claude Chevalley • Harish-Chandra • Armand Borel • Glossary • Table of Lie groups Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra ${\mathfrak {g}}$, if nonzero, the following conditions are equivalent: • ${\mathfrak {g}}$ is semisimple; • the Killing form, κ(x,y) = tr(ad(x)ad(y)), is non-degenerate; • ${\mathfrak {g}}$ has no non-zero abelian ideals; • ${\mathfrak {g}}$ has no non-zero solvable ideals; • the radical (maximal solvable ideal) of ${\mathfrak {g}}$ is zero. Significance The significance of semisimplicity comes firstly from the Levi decomposition, which states that every finite dimensional Lie algebra is the semidirect product of a solvable ideal (its radical) and a semisimple algebra. In particular, there is no nonzero Lie algebra that is both solvable and semisimple. Semisimple Lie algebras have a very elegant classification, in stark contrast to solvable Lie algebras. Semisimple Lie algebras over an algebraically closed field of characteristic zero are completely classified by their root system, which are in turn classified by Dynkin diagrams. Semisimple algebras over non-algebraically closed fields can be understood in terms of those over the algebraic closure, though the classification is somewhat more intricate; see real form for the case of real semisimple Lie algebras, which were classified by Élie Cartan. Further, the representation theory of semisimple Lie algebras is much cleaner than that for general Lie algebras. For example, the Jordan decomposition in a semisimple Lie algebra coincides with the Jordan decomposition in its representation; this is not the case for Lie algebras in general. If ${\mathfrak {g}}$ is semisimple, then ${\mathfrak {g}}=[{\mathfrak {g}},{\mathfrak {g}}]$. In particular, every linear semisimple Lie algebra is a subalgebra of ${\mathfrak {sl}}$, the special linear Lie algebra. The study of the structure of ${\mathfrak {sl}}$ constitutes an important part of the representation theory for semisimple Lie algebras. History The semisimple Lie algebras over the complex numbers were first classified by Wilhelm Killing (1888–90), though his proof lacked rigor. His proof was made rigorous by Élie Cartan (1894) in his Ph.D. thesis, who also classified semisimple real Lie algebras. This was subsequently refined, and the present classification by Dynkin diagrams was given by then 22-year-old Eugene Dynkin in 1947. Some minor modifications have been made (notably by J. P. Serre), but the proof is unchanged in its essentials and can be found in any standard reference, such as (Humphreys 1972). Basic properties • Every ideal, quotient and product of semisimple Lie algebras is again semisimple.[1] • The center of a semisimple Lie algebra ${\mathfrak {g}}$ is trivial (since the center is an abelian ideal). In other words, the adjoint representation $\operatorname {ad} $ is injective. Moreover, the image turns out[2] to be $\operatorname {Der} ({\mathfrak {g}})$ of derivations on ${\mathfrak {g}}$. Hence, $\operatorname {ad} :{\mathfrak {g}}{\overset {\sim }{\to }}\operatorname {Der} ({\mathfrak {g}})$ :{\mathfrak {g}}{\overset {\sim }{\to }}\operatorname {Der} ({\mathfrak {g}})} is an isomorphism.[3] (This is a special case of Whitehead's lemma.) • As the adjoint representation is injective, a semisimple Lie algebra is a linear Lie algebra under the adjoint representation. This may lead to some ambiguity, as every Lie algebra is already linear with respect to some other vector space (Ado's theorem), although not necessarily via the adjoint representation. But in practice, such ambiguity rarely occurs. • If ${\mathfrak {g}}$ is a semisimple Lie algebra, then ${\mathfrak {g}}=[{\mathfrak {g}},{\mathfrak {g}}]$ (because ${\mathfrak {g}}/[{\mathfrak {g}},{\mathfrak {g}}]$ is semisimple and abelian).[4] • A finite-dimensional Lie algebra ${\mathfrak {g}}$ over a field k of characteristic zero is semisimple if and only if the base extension ${\mathfrak {g}}\otimes _{k}F$ is semisimple for each field extension $F\supset k$.[5] Thus, for example, a finite-dimensional real Lie algebra is semisimple if and only if its complexification is semisimple. Jordan decomposition Each endomorphism x of a finite-dimensional vector space over a field of characteristic zero can be decomposed uniquely into a semisimple (i.e., diagonalizable over the algebraic closure) and nilpotent part $x=s+n\ $ such that s and n commute with each other. Moreover, each of s and n is a polynomial in x. This is the Jordan decomposition of x. The above applies to the adjoint representation $\operatorname {ad} $ of a semisimple Lie algebra ${\mathfrak {g}}$. An element x of ${\mathfrak {g}}$ is said to be semisimple (resp. nilpotent) if $\operatorname {ad} (x)$ is a semisimple (resp. nilpotent) operator.[6] If $x\in {\mathfrak {g}}$, then the abstract Jordan decomposition states that x can be written uniquely as: $x=s+n$ where $s$ is semisimple, $n$ is nilpotent and $[s,n]=0$.[7] Moreover, if $y\in {\mathfrak {g}}$ commutes with x, then it commutes with both $s,n$ as well. The abstract Jordan decomposition factors through any representation of ${\mathfrak {g}}$ in the sense that given any representation ρ, $\rho (x)=\rho (s)+\rho (n)\,$ is the Jordan decomposition of ρ(x) in the endomorphism algebra of the representation space.[8] (This is proved as a consequence of Weyl's complete reducibility theorem; see Weyl's theorem on complete reducibility#Application: preservation of Jordan decomposition.) Structure Let ${\mathfrak {g}}$ be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. The structure of ${\mathfrak {g}}$ can be described by an adjoint action of a certain distinguished subalgebra on it, a Cartan subalgebra. By definition,[9] a Cartan subalgebra (also called a maximal toral subalgebra) ${\mathfrak {h}}$ of ${\mathfrak {g}}$ is a maximal subalgebra such that, for each $h\in {\mathfrak {h}}$, $\operatorname {ad} (h)$ is diagonalizable. As it turns out, ${\mathfrak {h}}$ is abelian and so all the operators in $\operatorname {ad} ({\mathfrak {h}})$ are simultaneously diagonalizable. For each linear functional $\alpha $ of ${\mathfrak {h}}$, let ${\mathfrak {g}}_{\alpha }=\{x\in {\mathfrak {g}}\|\operatorname {ad} (h)x:=[h,x]=\alpha (h)x\,{\text{ for all }}h\in {\mathfrak {h}}\}$. (Note that ${\mathfrak {g}}_{0}$ is the centralizer of ${\mathfrak {h}}$.) Then Root space decomposition — [10] Given a Cartan subalgebra ${\mathfrak {h}}$, it holds that ${\mathfrak {g}}_{0}={\mathfrak {h}}$ and there is a decomposition (as an ${\mathfrak {h}}$-module): ${\mathfrak {g}}={\mathfrak {h}}\oplus \bigoplus _{\alpha \in \Phi }{\mathfrak {g}}_{\alpha }$ where $\Phi $ is the set of all nonzero linear functionals $\alpha $ of ${\mathfrak {h}}$ such that ${\mathfrak {g}}_{\alpha }\neq \{0\}$. Moreover, for each $\alpha ,\beta \in \Phi $, • $[{\mathfrak {g}}_{\alpha },{\mathfrak {g}}_{\beta }]\subseteq {\mathfrak {g}}_{\alpha +\beta }$, which is the equality if $\alpha +\beta \neq 0$. • $[{\mathfrak {g}}_{\alpha },{\mathfrak {g}}_{-\alpha }]\oplus {\mathfrak {g}}_{-\alpha }\oplus {\mathfrak {g}}_{\alpha }\simeq {\mathfrak {sl}}_{2}$ as a Lie algebra. • $\dim {\mathfrak {g}}_{\alpha }=1$; in particular, $\dim {\mathfrak {g}}=\dim {\mathfrak {h}}+\#\Phi $. • ${\mathfrak {g}}_{2\alpha }=\{0\}$; in other words, $2\alpha \not \in \Phi $. • With respect to the Killing form B, ${\mathfrak {g}}_{\alpha },{\mathfrak {g}}_{\beta }$ are orthogonal to each other if $\alpha +\beta \neq 0$; the restriction of B to ${\mathfrak {h}}$ is nondegenerate. (The most difficult item to show is $\dim {\mathfrak {g}}_{\alpha }=1$. The standard proofs all use some facts in the representation theory of ${\mathfrak {sl}}_{2}$; e.g., Serre uses the fact that an ${\mathfrak {sl}}_{2}$-module with a primitive element of negative weight is infinite-dimensional, contradicting $\dim {\mathfrak {g}}<\infty $.) Let $h_{\alpha }\in {\mathfrak {h}},e_{\alpha }\in {\mathfrak {g}}_{\alpha },f_{\alpha }\in {\mathfrak {g}}_{-\alpha }$ with the commutation relations $[e_{\alpha },f_{\alpha }]=h_{\alpha },[h_{\alpha },e_{\alpha }]=2e_{\alpha },[h_{\alpha },f_{\alpha }]=-2f_{\alpha }$; i.e., the $h_{\alpha },e_{\alpha },f_{\alpha }$ correspond to the standard basis of ${\mathfrak {sl}}_{2}$. The linear functionals in $\Phi $ are called the roots of ${\mathfrak {g}}$ relative to ${\mathfrak {h}}$. The roots span ${\mathfrak {h}}^{}$ (since if $\alpha (h)=0,\alpha \in \Phi $, then $\operatorname {ad} (h)$ is the zero operator; i.e., $h$ is in the center, which is zero.) Moreover, from the representation theory of ${\mathfrak {sl}}_{2}$, one deduces the following symmetry and integral properties of $\Phi $: for each $\alpha ,\beta \in \Phi $, • The endomorphism $s_{\alpha }:{\mathfrak {h}}^{}\to {\mathfrak {h}}^{},\,\gamma \mapsto \gamma -\gamma (h_{\alpha })\alpha $ leaves $\Phi $ invariant (i.e., $s_{\alpha }(\Phi )\subset \Phi $). • $\beta (h_{\alpha })$ is an integer. Note that $s_{\alpha }$ has the properties (1) $s_{\alpha }(\alpha )=-\alpha $ and (2) the fixed-point set is $\{\gamma \in {\mathfrak {h}}^{}\|\gamma (h_{\alpha })=0\}$, which means that $s_{\alpha }$ is the reflection with respect to the hyperplane corresponding to $\alpha $. The above then says that $\Phi $ is a root system. It follows from the general theory of a root system that $\Phi $ contains a basis $\alpha _{1},\dots ,\alpha _{l}$ of ${\mathfrak {h}}^{}$ such that each root is a linear combination of $\alpha _{1},\dots ,\alpha _{l}$ with integer coefficients of the same sign; the roots $\alpha _{i}$ are called simple roots. Let $e_{i}=e_{\alpha _{i}}$, etc. Then the $3l$ elements $e_{i},f_{i},h_{i}$ (called Chevalley generators) generate ${\mathfrak {g}}$ as a Lie algebra. Moreover, they satisfy the relations (called Serre relations): with $a_{ij}=\alpha _{j}(h_{i})$, $[h_{i},h_{j}]=0,$ $[e_{i},f_{i}]=h_{i},[e_{i},f_{j}]=0,i\neq j,$ $[h_{i},e_{j}]=a_{ij}e_{j},[h_{i},f_{j}]=-a_{ij}f_{j},$ $\operatorname {ad} (e_{i})^{-a_{ij}+1}(e_{j})=\operatorname {ad} (f_{i})^{-a_{ij}+1}(f_{j})=0,i\neq j$. The converse of this is also true: i.e., the Lie algebra generated by the generators and the relations like the above is a (finite-dimensional) semisimple Lie algebra that has the root space decomposition as above (provided the $[a_{ij}]_{1\leq i,j\leq l}$ is a Cartan matrix). This is a theorem of Serre. In particular, two semisimple Lie algebras are isomorphic if they have the same root system. The implication of the axiomatic nature of a root system and Serre's theorem is that one can enumerate all possible root systems; hence, "all possible" semisimple Lie algebras (finite-dimensional over an algebraically closed field of characteristic zero). The Weyl group is the group of linear transformations of ${\mathfrak {h}}^{}\simeq {\mathfrak {h}}$ generated by the $s_{\alpha }$'s. The Weyl group is an important symmetry of the problem; for example, the weights of any finite-dimensional representation of ${\mathfrak {g}}$ are invariant under the Weyl group.[11] Example root space decomposition in sln(C) For ${\mathfrak {g}}={\mathfrak {sl}}_{n}(\mathbb {C} )$ and the Cartan subalgebra ${\mathfrak {h}}$ of diagonal matrices, define $\lambda _{i}\in {\mathfrak {h}}^{}$ by $\lambda _{i}(d(a_{1},\ldots ,a_{n}))=a_{i}$, where $d(a_{1},\ldots ,a_{n})$ denotes the diagonal matrix with $a_{1},\ldots ,a_{n}$ on the diagonal. Then the decomposition is given by ${\mathfrak {g}}={\mathfrak {h}}\oplus \left(\bigoplus _{i\neq j}{\mathfrak {g}}_{\lambda _{i}-\lambda _{j}}\right)$ where ${\mathfrak {g}}_{\lambda _{i}-\lambda _{j}}={\text{Span}}_{\mathbb {C} }(e_{ij})$ for the vector $e_{ij}$ in ${\mathfrak {sl}}_{n}(\mathbb {C} )$ with the standard (matrix) basis, meaning $e_{ij}$ represents the basis vector in the $i$-th row and $j$-th column. This decomposition of ${\mathfrak {g}}$ has an associated root system: $\Phi =\{\lambda _{i}-\lambda _{j}:i\neq j\}$ sl2(C) For example, in ${\mathfrak {sl}}_{2}(\mathbb {C} )$ the decomposition is ${\mathfrak {sl}}_{2}={\mathfrak {h}}\oplus {\mathfrak {g}}_{\lambda _{1}-\lambda _{2}}\oplus {\mathfrak {g}}_{\lambda _{2}-\lambda _{1}}$ and the associated root system is $\Phi =\{\lambda _{1}-\lambda _{2},\lambda _{2}-\lambda _{1}\}$ sl3(C) In ${\mathfrak {sl}}_{3}(\mathbb {C} )$ the decomposition is ${\mathfrak {sl}}_{3}={\mathfrak {h}}\oplus {\mathfrak {g}}_{\lambda _{1}-\lambda _{2}}\oplus {\mathfrak {g}}_{\lambda _{1}-\lambda _{3}}\oplus {\mathfrak {g}}_{\lambda _{2}-\lambda _{3}}\oplus {\mathfrak {g}}_{\lambda _{2}-\lambda _{1}}\oplus {\mathfrak {g}}_{\lambda _{3}-\lambda _{1}}\oplus {\mathfrak {g}}_{\lambda _{3}-\lambda _{2}}$ and the associated root system is given by $\Phi =\{\pm (\lambda _{1}-\lambda _{2}),\pm (\lambda _{1}-\lambda _{3}),\pm (\lambda _{2}-\lambda _{3})\}$ Examples As noted in #Structure, semisimple Lie algebras over $\mathbb {C} $ (or more generally an algebraically closed field of characteristic zero) are classified by the root system associated to their Cartan subalgebras, and the root systems, in turn, are classified by their Dynkin diagrams. Examples of semisimple Lie algebras, the classical Lie algebras, with notation coming from their Dynkin diagrams, are: • $A_{n}:$ ${\mathfrak {sl}}_{n+1}$, the special linear Lie algebra. • $B_{n}:$ ${\mathfrak {so}}_{2n+1}$, the odd-dimensional special orthogonal Lie algebra. • $C_{n}:$ ${\mathfrak {sp}}_{2n}$, the symplectic Lie algebra. • $D_{n}:$ ${\mathfrak {so}}_{2n}$, the even-dimensional special orthogonal Lie algebra ($n>1$). The restriction $n>1$ in the $D_{n}$ family is needed because ${\mathfrak {so}}_{2}$ is one-dimensional and commutative and therefore not semisimple. These Lie algebras are numbered so that n is the rank. Almost all of these semisimple Lie algebras are actually simple and the members of these families are almost all distinct, except for some collisions in small rank. For example ${\mathfrak {so}}_{4}\cong {\mathfrak {so}}_{3}\oplus {\mathfrak {so}}_{3}$ and ${\mathfrak {sp}}_{2}\cong {\mathfrak {so}}_{5}$. These four families, together with five exceptions (E6, E7, E8, F4, and G2), are in fact the only simple Lie algebras over the complex numbers. Classification See also: Root system Every semisimple Lie algebra over an algebraically closed field of characteristic 0 is a direct sum of simple Lie algebras (by definition), and the finite-dimensional simple Lie algebras fall in four families – An, Bn, Cn, and Dn – with five exceptions E6, E7, E8, F4, and G2. Simple Lie algebras are classified by the connected Dynkin diagrams, shown on the right, while semisimple Lie algebras correspond to not necessarily connected Dynkin diagrams, where each component of the diagram corresponds to a summand of the decomposition of the semisimple Lie algebra into simple Lie algebras. The classification proceeds by considering a Cartan subalgebra (see below) and its adjoint action on the Lie algebra. The root system of the action then both determines the original Lie algebra and must have a very constrained form, which can be classified by the Dynkin diagrams. See the section below describing Cartan subalgebras and root systems for more details. The classification is widely considered one of the most elegant results in mathematics – a brief list of axioms yields, via a relatively short proof, a complete but non-trivial classification with surprising structure. This should be compared to the classification of finite simple groups, which is significantly more complicated. The enumeration of the four families is non-redundant and consists only of simple algebras if $n\geq 1$ for An, $n\geq 2$ for Bn, $n\geq 3$ for Cn, and $n\geq 4$ for Dn. If one starts numbering lower, the enumeration is redundant, and one has exceptional isomorphisms between simple Lie algebras, which are reflected in isomorphisms of Dynkin diagrams; the En can also be extended down, but below E6 are isomorphic to other, non-exceptional algebras. Over a non-algebraically closed field, the classification is more complicated – one classifies simple Lie algebras over the algebraic closure, then for each of these, one classifies simple Lie algebras over the original field which have this form (over the closure). For example, to classify simple real Lie algebras, one classifies real Lie algebras with a given complexification, which are known as real forms of the complex Lie algebra; this can be done by Satake diagrams, which are Dynkin diagrams with additional data ("decorations").[12] Representation theory of semisimple Lie algebras Main article: Representation theory of semisimple Lie algebras Let ${\mathfrak {g}}$ be a (finite-dimensional) semisimple Lie algebra over an algebraically closed field of characteristic zero. Then, as in #Structure, $ {\mathfrak {g}}={\mathfrak {h}}\oplus \bigoplus _{\alpha \in \Phi }{\mathfrak {g}}_{\alpha }$ where $\Phi $ is the root system. Choose the simple roots in $\Phi $; a root $\alpha $ of $\Phi $ is then called positive and is denoted by $\alpha >0$ if it is a linear combination of the simple roots with non-negative integer coefficients. Let $ {\mathfrak {b}}={\mathfrak {h}}\oplus \bigoplus _{\alpha >0}{\mathfrak {g}}_{\alpha }$, which is a maximal solvable subalgebra of ${\mathfrak {g}}$, the Borel subalgebra. Let V be a (possibly-infinite-dimensional) simple ${\mathfrak {g}}$-module. If V happens to admit a ${\mathfrak {b}}$-weight vector $v_{0}$,[13] then it is unique up to scaling and is called the highest weight vector of V. It is also an ${\mathfrak {h}}$-weight vector and the ${\mathfrak {h}}$-weight of $v_{0}$, a linear functional of ${\mathfrak {h}}$, is called the highest weight of V. The basic yet nontrivial facts[14] then are (1) to each linear functional $\mu \in {\mathfrak {h}}^{}$, there exists a simple ${\mathfrak {g}}$-module $V^{\mu }$ having $\mu $ as its highest weight and (2) two simple modules having the same highest weight are equivalent. In short, there exists a bijection between ${\mathfrak {h}}^{}$ and the set of the equivalence classes of simple ${\mathfrak {g}}$-modules admitting a Borel-weight vector. For applications, one is often interested in a finite-dimensional simple ${\mathfrak {g}}$-module (a finite-dimensional irreducible representation). This is especially the case when ${\mathfrak {g}}$ is the Lie algebra of a Lie group (or complexification of such), since, via the Lie correspondence, a Lie algebra representation can be integrated to a Lie group representation when the obstructions are overcome. The next criterion then addresses this need: by the positive Weyl chamber $C\subset {\mathfrak {h}}^{}$, we mean the convex cone $C=\{\mu \in {\mathfrak {h}}^{*}\|\mu (h_{\alpha })\geq 0,\alpha \in \Phi >0\}$ where $h_{\alpha }\in [{\mathfrak {g}}_{\alpha },{\mathfrak {g}}_{-\alpha }]$ is a unique vector such that $\alpha (h_{\alpha })=2$. The criterion then reads:[15] • $\dim V^{\mu }<\infty $ if and only if, for each positive root $\alpha >0$, (1) $\mu (h_{\alpha })$ is an integer and (2) $\mu $ lies in $C$. A linear functional $\mu $ satisfying the above equivalent condition is called a dominant integral weight. Hence, in summary, there exists a bijection between the dominant integral weights and the equivalence classes of finite-dimensional simple ${\mathfrak {g}}$-modules, the result known as the theorem of the highest weight. The character of a finite-dimensional simple module in turns is computed by the Weyl character formula. The theorem due to Weyl says that, over a field of characteristic zero, every finite-dimensional module of a semisimple Lie algebra ${\mathfrak {g}}$ is completely reducible; i.e., it is a direct sum of simple ${\mathfrak {g}}$-modules. Hence, the above results then apply to finite-dimensional representations of a semisimple Lie algebra. Real semisimple Lie algebra For a semisimple Lie algebra over a field that has characteristic zero but is not algebraically closed, there is no general structure theory like the one for those over an algebraically closed field of characteristic zero. But over the field of real numbers, there are still the structure results. Let ${\mathfrak {g}}$ be a finite-dimensional real semisimple Lie algebra and ${\mathfrak {g}}^{\mathbb {C} }={\mathfrak {g}}\otimes _{\mathbb {R} }\mathbb {C} $ the complexification of it (which is again semisimple). The real Lie algebra ${\mathfrak {g}}$ is called a real form of ${\mathfrak {g}}^{\mathbb {C} }$. A real form is called a compact form if the Killing form on it is negative-definite; it is necessarily the Lie algebra of a compact Lie group (hence, the name). Compact case Suppose ${\mathfrak {g}}$ is a compact form and ${\mathfrak {h}}\subset {\mathfrak {g}}$ a maximal abelian subspace. One can show (for example, from the fact ${\mathfrak {g}}$ is the Lie algebra of a compact Lie group) that $\operatorname {ad} ({\mathfrak {h}})$ consists of skew-Hermitian matrices, diagonalizable over $\mathbb {C} $ with imaginary eigenvalues. Hence, ${\mathfrak {h}}^{\mathbb {C} }$ is a Cartan subalgebra of ${\mathfrak {g}}^{\mathbb {C} }$ and there results in the root space decomposition (cf. #Structure) ${\mathfrak {g}}^{\mathbb {C} }={\mathfrak {h}}^{\mathbb {C} }\oplus \bigoplus _{\alpha \in \Phi }{\mathfrak {g}}_{\alpha }$ where each $\alpha \in \Phi $ is real-valued on $i{\mathfrak {h}}$; thus, can be identified with a real-linear functional on the real vector space $i{\mathfrak {h}}$. For example, let ${\mathfrak {g}}={\mathfrak {su}}(n)$ and take ${\mathfrak {h}}\subset {\mathfrak {g}}$ the subspace of all diagonal matrices. Note ${\mathfrak {g}}^{\mathbb {C} }={\mathfrak {sl}}_{n}\mathbb {C} $. Let $e_{i}$ be the linear functional on ${\mathfrak {h}}^{\mathbb {C} }$ given by $e_{i}(H)=h_{i}$ for $H=\operatorname {diag} (h_{1},\dots ,h_{n})$. Then for each $H\in {\mathfrak {h}}^{\mathbb {C} }$, $[H,E_{ij}]=(e_{i}(H)-e_{j}(H))E_{ij}$ where $E_{ij}$ is the matrix that has 1 on the $(i,j)$-th spot and zero elsewhere. Hence, each root $\alpha $ is of the form $\alpha =e_{i}-e_{j},i\neq j$ and the root space decomposition is the decomposition of matrices:[16] ${\mathfrak {g}}^{\mathbb {C} }={\mathfrak {h}}^{\mathbb {C} }\oplus \bigoplus _{i\neq j}\mathbb {C} E_{ij}.$ Noncompact case Suppose ${\mathfrak {g}}$ is not necessarily a compact form (i.e., the signature of the Killing form is not all negative). Suppose, moreover, it has a Cartan involution $\theta $ and let ${\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}$ be the eigenspace decomposition of $\theta $, where ${\mathfrak {k}},{\mathfrak {p}}$ are the eigenspaces for 1 and -1, respectively. For example, if ${\mathfrak {g}}={\mathfrak {sl}}_{n}\mathbb {R} $ and $\theta $ the negative transpose, then ${\mathfrak {k}}={\mathfrak {so}}(n)$. Let ${\mathfrak {a}}\subset {\mathfrak {p}}$ be a maximal abelian subspace. Now, $\operatorname {ad} ({\mathfrak {p}})$ consists of symmetric matrices (with respect to a suitable inner product) and thus the operators in $\operatorname {ad} ({\mathfrak {a}})$ are simultaneously diagonalizable, with real eigenvalues. By repeating the arguments for the algebraically closed base field, one obtains the decomposition (called the restricted root space decomposition):[17] ${\mathfrak {g}}={\mathfrak {g}}_{0}\oplus \bigoplus _{\alpha \in \Phi }{\mathfrak {g}}_{\alpha }$ where • the elements in $\Phi $ are called the restricted roots, • $\theta ({\mathfrak {g}}_{\alpha })={\mathfrak {g}}_{-\alpha }$ for any linear functional $\alpha $; in particular, $-\Phi \subset \Phi $, • ${\mathfrak {g}}_{0}={\mathfrak {a}}\oplus Z_{\mathfrak {k}}({\mathfrak {a}})$. Moreover, $\Phi $ is a root system but not necessarily reduced one (i.e., it can happen $\alpha ,2\alpha $ are both roots). The case of sl(n,C) If ${\mathfrak {g}}=\mathrm {sl} (n,\mathbb {C} )$, then ${\mathfrak {h}}$ may be taken to be the diagonal subalgebra of ${\mathfrak {g}}$, consisting of diagonal matrices whose diagonal entries sum to zero. Since ${\mathfrak {h}}$ has dimension $n-1$, we see that $\mathrm {sl} (n;\mathbb {C} )$ has rank $n-1$. The root vectors $X$ in this case may be taken to be the matrices $E_{i,j}$ with $i\neq j$, where $E_{i,j}$ is the matrix with a 1 in the $(i,j)$ spot and zeros elsewhere.[18] If $H$ is a diagonal matrix with diagonal entries $\lambda _{1},\ldots ,\lambda _{n}$, then we have $[H,E_{i,j}]=(\lambda _{i}-\lambda _{j})E_{i,j}$. Thus, the roots for $\mathrm {sl} (n,\mathbb {C} )$ are the linear functionals $\alpha _{i,j}$ given by $\alpha _{i,j}(H)=\lambda _{i}-\lambda _{j}$. After identifying ${\mathfrak {h}}$ with its dual, the roots become the vectors $\alpha _{i,j}:=e_{i}-e_{j}$ in the space of $n$-tuples that sum to zero. This is the root system known as $A_{n-1}$ in the conventional labeling. The reflection associated to the root $\alpha _{i,j}$ acts on ${\mathfrak {h}}$ by transposing the $i$ and $j$ diagonal entries. The Weyl group is then just the permutation group on $n$ elements, acting by permuting the diagonal entries of matrices in ${\mathfrak {h}}$. Generalizations Main articles: Reductive Lie algebra and Split Lie algebra Semisimple Lie algebras admit certain generalizations. Firstly, many statements that are true for semisimple Lie algebras are true more generally for reductive Lie algebras. Abstractly, a reductive Lie algebra is one whose adjoint representation is completely reducible, while concretely, a reductive Lie algebra is a direct sum of a semisimple Lie algebra and an abelian Lie algebra; for example, ${\mathfrak {sl}}_{n}$ is semisimple, and ${\mathfrak {gl}}_{n}$ is reductive. Many properties of semisimple Lie algebras depend only on reducibility. Many properties of complex semisimple/reductive Lie algebras are true not only for semisimple/reductive Lie algebras over algebraically closed fields, but more generally for split semisimple/reductive Lie algebras over other fields: semisimple/reductive Lie algebras over algebraically closed fields are always split, but over other fields this is not always the case. Split Lie algebras have essentially the same representation theory as semisimple Lie algebras over algebraically closed fields, for instance, the splitting Cartan subalgebra playing the same role as the Cartan subalgebra plays over algebraically closed fields. This is the approach followed in (Bourbaki 2005), for instance, which classifies representations of split semisimple/reductive Lie algebras. Semisimple and reductive groups A connected Lie group is called semisimple if its Lie algebra is a semisimple Lie algebra, i.e. a direct sum of simple Lie algebras. It is called reductive if its Lie algebra is a direct sum of simple and trivial (one-dimensional) Lie algebras. Reductive groups occur naturally as symmetries of a number of mathematical objects in algebra, geometry, and physics. For example, the group $GL_{n}(\mathbb {R} )$ of symmetries of an n-dimensional real vector space (equivalently, the group of invertible matrices) is reductive. See also • Lie algebra • Root system • Lie algebra representation • Compact group • Simple Lie group • Borel subalgebra • Jacobson–Morozov theorem References 1. Serre 2000, Ch. II, § 2, Corollary to Theorem 3. 2. Since the Killing form B is non-degenerate, given a derivation D, there is an x such that $\operatorname {tr} (D\operatorname {ad} y)=B(x,y)$ for all y and then, by an easy computation, $D=\operatorname {ad} (x)$. 3. Serre 2000, Ch. II, § 4, Theorem 5. 4. Serre 2000, Ch. II, § 3, Corollary to Theorem 4. 5. Jacobson 1979, Corollary at the end of Ch. III, § 4. 6. Serre 2000, Ch. II, § 5. Definition 3. 7. Serre 2000, Ch. II, § 5. Theorem 6. 8. Serre 2000, Ch. II, § 5. Theorem 7. 9. This is a definition of a Cartan subalgebra of a semisimple Lie algebra and coincides with the general one. 10. Serre 2000, Ch. VI, § 1. 11. Hall 2015 Theorem 9.3 12. Knapp 2002 Section VI.10 13. A ${\mathfrak {b}}$-weight vector is also called a primitive element, especially in older textbooks. 14. In textbooks, these facts is usually established by the theory of Verma modules. 15. Serre 2000, Ch. VII, § 4, Theorem 3. 16. Knapp 2002, Ch. IV, § 1, Example 1. 17. Knapp 2002, Ch. V, § 2, Proposition 5.9. 18. Hall 2015 Section 7.7.1 • Bourbaki, Nicolas (2005), "VIII: Split Semi-simple Lie Algebras", Elements of Mathematics: Lie Groups and Lie Algebras: Chapters 7–9, Springer, ISBN 9783540434054 • Erdmann, Karin; Wildon, Mark (2006), Introduction to Lie Algebras (1st ed.), Springer, ISBN 1-84628-040-0. • Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666 • Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90053-7. • Jacobson, Nathan (1979) [1962]. Lie algebras. New York: Dover Publications, Inc. ISBN 0-486-63832-4. • Knapp, Anthony W. (2002), Lie groups beyond an introduction (2nd ed.), Birkhäuser • Serre, Jean-Pierre (2000), Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras], translated by Jones, G. A., Springer, ISBN 978-3-540-67827-4. • Varadarajan, V. S. (2004), Lie Groups, Lie Algebras, and Their Representations (1st ed.), Springer, ISBN 0-387-90969-9. Authority control: National • Germany	Wikipedia
Equation solving In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality. A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set. "Solution (mathematics)" redirects here. For solutions of constraint satisfaction problems, see Constraint satisfaction problem § Resolution. For solutions of mathematical optimization problems, see Feasible solution. ${\overset {}{\underset {}{x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}}}$ The quadratic formula, the symbolic solution of the quadratic equation ax2 + bx + c = 0 An equation may be solved either numerically or symbolically. Solving an equation numerically means that only numbers are admitted as solutions. Solving an equation symbolically means that expressions can be used for representing the solutions. For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1. Or x and y can both be treated as unknowns, and then there are many solutions to the equation; a symbolic solution is (x, y) = (a + 1, a), where the variable a may take any value. Instantiating a symbolic solution with specific numbers gives a numerical solution; for example, a = 0 gives (x, y) = (1, 0) (that is, x = 1, y = 0), and a = 1 gives (x, y) = (2, 1). The distinction between known variables and unknown variables is generally made in the statement of the problem, by phrases such as "an equation in x and y", or "solve for x and y", which indicate the unknowns, here x and y. However, it is common to reserve x, y, z, ... to denote the unknowns, and to use a, b, c, ... to denote the known variables, which are often called parameters. This is typically the case when considering polynomial equations, such as quadratic equations. However, for some problems, all variables may assume either role. Depending on the context, solving an equation may consist to find either any solution (finding a single solution is enough), all solutions, or a solution that satisfies further properties, such as belonging to a given interval. When the task is to find the solution that is the best under some criterion, this is an optimization problem. Solving an optimization problem is generally not referred to as "equation solving", as, generally, solving methods start from a particular solution for finding a better solution, and repeating the process until finding eventually the best solution. Overview One general form of an equation is $f\left(x_{1},\dots ,x_{n}\right)=c,$ where f is a function, x1, ..., xn are the unknowns, and c is a constant. Its solutions are the elements of the inverse image $f^{-1}(c)={\bigl \{}(a_{1},\dots ,a_{n})\in D\mid f\left(a_{1},\dots ,a_{n}\right)=c{\bigr \}},$ where D is the domain of the function f. The set of solutions can be the empty set (there are no solutions), a singleton (there is exactly one solution), finite, or infinite (there are infinitely many solutions). For example, an equation such as $3x+2y=21z,$ with unknowns x, y and z, can be put in the above form by subtracting 21z from both sides of the equation, to obtain $3x+2y-21z=0$ In this particular case there is not just one solution, but an infinite set of solutions, which can be written using set builder notation as ${\bigl \{}(x,y,z)\mid 3x+2y-21z=0{\bigr \}}.$ One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation. Solution sets Main article: Solution set The solution set of a given set of equations or inequalities is the set of all its solutions, a solution being a tuple of values, one for each unknown, that satisfies all the equations or inequalities. If the solution set is empty, then there are no values of the unknowns that satisfy simultaneously all equations and inequalities. For a simple example, consider the equation $x^{2}=2.$ This equation can be viewed as a Diophantine equation, that is, an equation for which only integer solutions are sought. In this case, the solution set is the empty set, since 2 is not the square of an integer. However, if one searches for real solutions, there are two solutions, √2 and –√2; in other words, the solution set is {√2, −√2}. When an equation contains several unknowns, and when one has several equations with more unknowns than equations, the solution set is often infinite. In this case, the solutions cannot be listed. For representing them, a parametrization is often useful, which consists of expressing the solutions in terms of some of the unknowns or auxiliary variables. This is always possible when all the equations are linear. Such infinite solution sets can naturally be interpreted as geometric shapes such as lines, curves (see picture), planes, and more generally algebraic varieties or manifolds. In particular, algebraic geometry may be viewed as the study of solution sets of algebraic equations. Methods of solution The methods for solving equations generally depend on the type of equation, both the kind of expressions in the equation and the kind of values that may be assumed by the unknowns. The variety in types of equations is large, and so are the corresponding methods. Only a few specific types are mentioned below. In general, given a class of equations, there may be no known systematic method (algorithm) that is guaranteed to work. This may be due to a lack of mathematical knowledge; some problems were only solved after centuries of effort. But this also reflects that, in general, no such method can exist: some problems are known to be unsolvable by an algorithm, such as Hilbert's tenth problem, which was proved unsolvable in 1970. For several classes of equations, algorithms have been found for solving them, some of which have been implemented and incorporated in computer algebra systems, but often require no more sophisticated technology than pencil and paper. In some other cases, heuristic methods are known that are often successful but that are not guaranteed to lead to success. Brute force, trial and error, inspired guess If the solution set of an equation is restricted to a finite set (as is the case for equations in modular arithmetic, for example), or can be limited to a finite number of possibilities (as is the case with some Diophantine equations), the solution set can be found by brute force, that is, by testing each of the possible values (candidate solutions). It may be the case, though, that the number of possibilities to be considered, although finite, is so huge that an exhaustive search is not practically feasible; this is, in fact, a requirement for strong encryption methods. As with all kinds of problem solving, trial and error may sometimes yield a solution, in particular where the form of the equation, or its similarity to another equation with a known solution, may lead to an "inspired guess" at the solution. If a guess, when tested, fails to be a solution, consideration of the way in which it fails may lead to a modified guess. Elementary algebra Equations involving linear or simple rational functions of a single real-valued unknown, say x, such as $8x+7=4x+35\quad {\text{or}}\quad {\frac {4x+9}{3x+4}}=2\,,$ can be solved using the methods of elementary algebra. Systems of linear equations Smaller systems of linear equations can be solved likewise by methods of elementary algebra. For solving larger systems, algorithms are used that are based on linear algebra. See Gaussian elimination Polynomial equations Main article: Solving polynomial equations See also: System of polynomial equations Polynomial equations of degree up to four can be solved exactly using algebraic methods, of which the quadratic formula is the simplest example. Polynomial equations with a degree of five or higher require in general numerical methods (see below) or special functions such as Bring radicals, although some specific cases may be solvable algebraically, for example $4x^{5}-x^{3}-3=0$ (by using the rational root theorem), and $x^{6}-5x^{3}+6=0\,,$ (by using the substitution x = z1⁄3, which simplifies this to a quadratic equation in z). Diophantine equations In Diophantine equations the solutions are required to be integers. In some cases a brute force approach can be used, as mentioned above. In some other cases, in particular if the equation is in one unknown, it is possible to solve the equation for rational-valued unknowns (see Rational root theorem), and then find solutions to the Diophantine equation by restricting the solution set to integer-valued solutions. For example, the polynomial equation $2x^{5}-5x^{4}-x^{3}-7x^{2}+2x+3=0\,$ has as rational solutions x = −1/2 and x = 3, and so, viewed as a Diophantine equation, it has the unique solution x = 3. In general, however, Diophantine equations are among the most difficult equations to solve. Inverse functions In the simple case of a function of one variable, say, h(x), we can solve an equation of the form h(x) = c for some constant c by considering what is known as the inverse function of h. Given a function h : A → B, the inverse function, denoted h−1 and defined as h−1 : B → A, is a function such that $h^{-1}{\bigl (}h(x){\bigr )}=h{\bigl (}h^{-1}(x){\bigr )}=x\,.$ Now, if we apply the inverse function to both sides of h(x) = c, where c is a constant value in B, we obtain ${\begin{aligned}h^{-1}{\bigl (}h(x){\bigr )}&=h^{-1}(c)\\x&=h^{-1}(c)\\\end{aligned}}$ and we have found the solution to the equation. However, depending on the function, the inverse may be difficult to be defined, or may not be a function on all of the set B (only on some subset), and have many values at some point. If just one solution will do, instead of the full solution set, it is actually sufficient if only the functional identity $h\left(h^{-1}(x)\right)=x$ holds. For example, the projection π1 : R2 → R defined by π1(x, y) = x has no post-inverse, but it has a pre-inverse π−1 1 defined by π−1 1 (x) = (x, 0) . Indeed, the equation π1(x, y) = c is solved by $(x,y)=\pi _{1}^{-1}(c)=(c,0).$ Examples of inverse functions include the nth root (inverse of xn); the logarithm (inverse of ax); the inverse trigonometric functions; and Lambert's W function (inverse of xex). Factorization If the left-hand side expression of an equation P = 0 can be factorized as P = QR, the solution set of the original solution consists of the union of the solution sets of the two equations Q = 0 and R = 0. For example, the equation $\tan x+\cot x=2$ can be rewritten, using the identity tan x cot x = 1 as ${\frac {\tan ^{2}x-2\tan x+1}{\tan x}}=0,$ which can be factorized into ${\frac {\left(\tan x-1\right)^{2}}{\tan x}}=0.$ The solutions are thus the solutions of the equation tan x = 1, and are thus the set $x={\tfrac {\pi }{4}}+k\pi ,\quad k=0,\pm 1,\pm 2,\ldots .$ Numerical methods With more complicated equations in real or complex numbers, simple methods to solve equations can fail. Often, root-finding algorithms like the Newton–Raphson method can be used to find a numerical solution to an equation, which, for some applications, can be entirely sufficient to solve some problem. Matrix equations Equations involving matrices and vectors of real numbers can often be solved by using methods from linear algebra. Differential equations There is a vast body of methods for solving various kinds of differential equations, both numerically and analytically. A particular class of problem that can be considered to belong here is integration, and the analytic methods for solving this kind of problems are now called symbolic integration. Solutions of differential equations can be implicit or explicit.[1] See also • Extraneous and missing solutions • Simultaneous equations • Equating coefficients • Solving the geodesic equations • Unification (computer science) — solving equations involving symbolic expressions References 1. Dennis G. Zill (15 March 2012). A First Course in Differential Equations with Modeling Applications. Cengage Learning. ISBN 978-1-285-40110-2.	Wikipedia
Root of unity modulo n In number theory, a kth root of unity modulo n for positive integers k, n ≥ 2, is a root of unity in the ring of integers modulo n; that is, a solution x to the equation (or congruence) $x^{k}\equiv 1{\pmod {n}}$. If k is the smallest such exponent for x, then x is called a primitive kth root of unity modulo n.[1] See modular arithmetic for notation and terminology. The roots of unity modulo n are exactly the integers that are coprime with n. In fact, these integers are roots of unity modulo n by Euler's theorem, and the other integers cannot be roots of unity modulo n, because they are zero divisors modulo n. A primitive root modulo n, is a generator of the group of units of the ring of integers modulo n. There exist primitive roots modulo n if and only if $\lambda (n)=\varphi (n),$ where $\lambda $ and $\varphi $ are respectively the Carmichael function and Euler's totient function. A root of unity modulo n is a primitive kth root of unity modulo n for some divisor k of $\lambda (n),$ and, conversely, there are primitive kth roots of unity modulo n if and only if k is a divisor of $\lambda (n).$ Roots of unity Properties • If x is a kth root of unity modulo n, then x is a unit (invertible) whose inverse is $x^{k-1}$. That is, x and n are coprime. • If x is a unit, then it is a (primitive) kth root of unity modulo n, where k is the multiplicative order of x modulo n. • If x is a kth root of unity and $x-1$ is not a zero divisor, then $\sum _{j=0}^{k-1}x^{j}\equiv 0{\pmod {n}}$, because $(x-1)\cdot \sum _{j=0}^{k-1}x^{j}\equiv x^{k}-1\equiv 0{\pmod {n}}.$ Number of kth roots For the lack of a widely accepted symbol, we denote the number of kth roots of unity modulo n by $f(n,k)$. It satisfies a number of properties: • $f(n,1)=1$ for $n\geq 2$ • $f(n,\lambda (n))=\varphi (n)$ where λ denotes the Carmichael function and $\varphi $ denotes Euler's totient function • $n\mapsto f(n,k)$ is a multiplicative function • $k\mid \ell \implies f(n,k)\mid f(n,\ell )$ where the bar denotes divisibility • $f(n,\operatorname {lcm} (a,b))=\operatorname {lcm} (f(n,a),f(n,b))$ where $\operatorname {lcm} $ denotes the least common multiple • For prime $p$, $\forall i\in \mathbb {N} \ \exists j\in \mathbb {N} \ f(n,p^{i})=p^{j}$. The precise mapping from $i$ to $j$ is not known. If it were known, then together with the previous law it would yield a way to evaluate $f$ quickly. Examples Let $n=7$ and $k=3$. In this case, there are three cube roots of unity (1, 2, and 4). When $n=11$ however, there is only one cube root of unity, the unit 1 itself. This behavior is quite different from the field of complex numbers where every nonzero number has k kth roots. Primitive roots of unity Properties • The maximum possible radix exponent for primitive roots modulo $n$ is $\lambda (n)$, where λ denotes the Carmichael function. • A radix exponent for a primitive root of unity is a divisor of $\lambda (n)$. • Every divisor $k$ of $\lambda (n)$ yields a primitive $k$th root of unity. One can obtain such a root by choosing a $\lambda (n)$th primitive root of unity (that must exist by definition of λ), named $x$ and compute the power $x^{\lambda (n)/k}$. • If x is a primitive kth root of unity and also a (not necessarily primitive) ℓth root of unity, then k is a divisor of ℓ. This is true, because Bézout's identity yields an integer linear combination of k and ℓ equal to $\gcd(k,\ell )$. Since k is minimal, it must be $k=\gcd(k,\ell )$ and $\gcd(k,\ell )$ is a divisor of ℓ. Number of primitive kth roots For the lack of a widely accepted symbol, we denote the number of primitive kth roots of unity modulo n by $g(n,k)$. It satisfies the following properties: • $g(n,k)={\begin{cases}>0&{\text{if }}k\mid \lambda (n),\\0&{\text{otherwise}}.\end{cases}}$ • Consequently the function $k\mapsto g(n,k)$ has $d(\lambda (n))$ values different from zero, where $d$ computes the number of divisors. • $g(n,1)=1$ • $g(4,2)=1$ • $g(2^{n},2)=3$ for $n\geq 3$, since -1 is always a square root of 1. • $g(2^{n},2^{k})=2^{k}$ for $k\in [2,n-1)$ • $g(n,2)=1$ for $n\geq 3$ and $n$ in (sequence A033948 in the OEIS) • $\sum _{k\in \mathbb {N} }g(n,k)=f(n,\lambda (n))=\varphi (n)$ with $\varphi $ being Euler's totient function • The connection between $f$ and $g$ can be written in an elegant way using a Dirichlet convolution: $f=\mathbf {1} *g$, i.e. $f(n,k)=\sum _{d\mid k}g(n,d)$ One can compute values of $g$ recursively from $f$ using this formula, which is equivalent to the Möbius inversion formula. Testing whether x is a primitive kth root of unity modulo n By fast exponentiation, one can check that $x^{k}\equiv 1{\pmod {n}}$. If this is true, x is a kth root of unity modulo n but not necessarily primitive. If it is not a primitive root, then there would be some divisor ℓ of k, with $x^{\ell }\equiv 1{\pmod {n}}$. In order to exclude this possibility, one has only to check for a few ℓ's equal k divided by a prime. That is, what needs to be checked is: $\forall p{\text{ prime dividing}}\ k,\quad x^{k/p}\not \equiv 1{\pmod {n}}.$ Finding a primitive kth root of unity modulo n Among the primitive kth roots of unity, the primitive $\lambda (n)$th roots are most frequent. It is thus recommended to try some integers for being a primitive $\lambda (n)$th root, what will succeed quickly. For a primitive $\lambda (n)$th root x, the number $x^{\lambda (n)/k}$ is a primitive $k$th root of unity. If k does not divide $\lambda (n)$, then there will be no kth roots of unity, at all. Finding multiple primitive kth roots modulo n Once a primitive kth root of unity x is obtained, every power $x^{\ell }$ is a $k$th root of unity, but not necessarily a primitive one. The power $x^{\ell }$ is a primitive $k$th root of unity if and only if $k$ and $\ell $ are coprime. The proof is as follows: If $x^{\ell }$ is not primitive, then there exists a divisor $m$ of $k$ with $(x^{\ell })^{m}\equiv 1{\pmod {n}}$, and since $k$ and $\ell $ are coprime, there exists an inverse $\ell ^{-1}$ of $\ell $ modulo $k$. This yields $1\equiv ((x^{\ell })^{m})^{\ell ^{-\ell }}\equiv x^{m}{\pmod {n}}$, which means that $x$ is not a primitive $k$th root of unity because there is the smaller exponent $m$. That is, by exponentiating x one can obtain $\varphi (k)$ different primitive kth roots of unity, but these may not be all such roots. However, finding all of them is not so easy. Finding an n with a primitive kth root of unity modulo n In what integer residue class rings does a primitive kth root of unity exist? It can be used to compute a discrete Fourier transform (more precisely a number theoretic transform) of a $k$-dimensional integer vector. In order to perform the inverse transform, divide by $k$; that is, k is also a unit modulo $n.$ A simple way to find such an n is to check for primitive kth roots with respect to the moduli in the arithmetic progression $k+1,2k+1,3k+1,\dots $ All of these moduli are coprime to k and thus k is a unit. According to Dirichlet's theorem on arithmetic progressions there are infinitely many primes in the progression, and for a prime $p$, it holds $\lambda (p)=p-1$. Thus if $mk+1$ is prime, then $\lambda (mk+1)=mk$, and thus there are primitive kth roots of unity. But the test for primes is too strong, and there may be other appropriate moduli. Finding an n with multiple primitive roots of unity modulo n To find a modulus $n$ such that there are primitive $k_{1}{\text{th}},k_{2}{\text{th}},\ldots ,k_{m}{\text{th}}$ roots of unity modulo $n$, the following theorem reduces the problem to a simpler one: For given $n$ there are primitive $k_{1}{\text{th}},\ldots ,k_{m}{\text{th}}$ roots of unity modulo $n$ if and only if there is a primitive $\operatorname {lcm} (k_{1},\ldots ,k_{m})$th root of unity modulo n. Proof Backward direction: If there is a primitive $\operatorname {lcm} (k_{1},\ldots ,k_{m})$th root of unity modulo $n$ called $x$, then $x^{\operatorname {lcm} (k_{1},\ldots ,k_{m})/k_{l}}$ is a $k_{l}$th root of unity modulo $n$. Forward direction: If there are primitive $k_{1}{\text{th}},\ldots ,k_{m}{\text{th}}$ roots of unity modulo $n$, then all exponents $k_{1},\dots ,k_{m}$ are divisors of $\lambda (n)$. This implies $\operatorname {lcm} (k_{1},\dots ,k_{m})\mid \lambda (n)$ and this in turn means there is a primitive $\operatorname {lcm} (k_{1},\ldots ,k_{m})$th root of unity modulo $n$. References 1. Finch, Stephen; Martin, Greg; Sebah, Pascal (2010). "Roots of unity and nullity modulo n" (PDF). Proceedings of the American Mathematical Society. 138 (8): 2729–2743. doi:10.1090/s0002-9939-10-10341-4. Retrieved 2011-02-20.	Wikipedia
Rationalisation (mathematics) In elementary algebra, root rationalisation is a process by which radicals in the denominator of an algebraic fraction are eliminated. If the denominator is a monomial in some radical, say $a{\sqrt[{n}]{x}}^{k},$ with k < n, rationalisation consists of multiplying the numerator and the denominator by ${\sqrt[{n}]{x}}^{n-k},$ and replacing ${\sqrt[{n}]{x}}^{n}$ by x (this is allowed, as, by definition, a nth root of x is a number that has x as its nth power). If k ≥ n, one writes k = qn + r with 0 ≤ r < n (Euclidean division), and ${\sqrt[{n}]{x}}^{k}=x^{q}{\sqrt[{n}]{x}}^{r};$ then one proceeds as above by multiplying by ${\sqrt[{n}]{x}}^{n-r}.$ If the denominator is linear in some square root, say $a+b{\sqrt {x}},$ rationalisation consists of multiplying the numerator and the denominator by $a-b{\sqrt {x}},$ and expanding the product in the denominator. This technique may be extended to any algebraic denominator, by multiplying the numerator and the denominator by all algebraic conjugates of the denominator, and expanding the new denominator into the norm of the old denominator. However, except in special cases, the resulting fractions may have huge numerators and denominators, and, therefore, the technique is generally used only in the above elementary cases. Rationalisation of a monomial square root and cube root For the fundamental technique, the numerator and denominator must be multiplied by the same factor. Example 1: ${\frac {10}{\sqrt {5}}}$ To rationalise this kind of expression, bring in the factor ${\sqrt {5}}$: ${\frac {10}{\sqrt {5}}}={\frac {10}{\sqrt {5}}}\cdot {\frac {\sqrt {5}}{\sqrt {5}}}={\frac {10{\sqrt {5}}}{\left({\sqrt {5}}\right)^{2}}}$ The square root disappears from the denominator, because $\left({\sqrt {5}}\right)^{2}=5$ by definition of a square root: ${\frac {10{\sqrt {5}}}{\left({\sqrt {5}}\right)^{2}}}={\frac {10{\sqrt {5}}}{5}},$ which is the result of the rationalisation. Example 2: ${\frac {10}{\sqrt[{3}]{a}}}$ To rationalise this radical, bring in the factor ${\sqrt[{3}]{a}}^{2}$: ${\frac {10}{\sqrt[{3}]{a}}}={\frac {10}{\sqrt[{3}]{a}}}\cdot {\frac {{\sqrt[{3}]{a}}^{2}}{{\sqrt[{3}]{a}}^{2}}}={\frac {10{\sqrt[{3}]{a}}^{2}}{{\sqrt[{3}]{a}}^{3}}}$ The cube root disappears from the denominator, because it is cubed; so ${\frac {10{\sqrt[{3}]{a}}^{2}}{{\sqrt[{3}]{a}}^{3}}}={\frac {10{\sqrt[{3}]{a}}^{2}}{a}},$ which is the result of the rationalisation. Dealing with more square roots For a denominator that is: ${\sqrt {2}}\pm {\sqrt {3}}\,$ Rationalisation can be achieved by multiplying by the conjugate: ${\sqrt {2}}\mp {\sqrt {3}}\,$ and applying the difference of two squares identity, which here will yield −1. To get this result, the entire fraction should be multiplied by ${\frac {{\sqrt {2}}-{\sqrt {3}}}{{\sqrt {2}}-{\sqrt {3}}}}=1.$ This technique works much more generally. It can easily be adapted to remove one square root at a time, i.e. to rationalise $x\pm {\sqrt {y}}\,$ by multiplication by $x\mp {\sqrt {y}}$ Example: ${\frac {3}{{\sqrt {3}}\pm {\sqrt {5}}}}$ The fraction must be multiplied by a quotient containing ${{\sqrt {3}}\mp {\sqrt {5}}}$. ${\frac {3}{{\sqrt {3}}+{\sqrt {5}}}}\cdot {\frac {{\sqrt {3}}-{\sqrt {5}}}{{\sqrt {3}}-{\sqrt {5}}}}={\frac {3({\sqrt {3}}-{\sqrt {5}})}{{\sqrt {3}}^{2}-{\sqrt {5}}^{2}}}$ Now, we can proceed to remove the square roots in the denominator: ${\frac {3({\sqrt {3}}-{\sqrt {5}})}{{\sqrt {3}}^{2}-{\sqrt {5}}^{2}}}={\frac {3({\sqrt {3}}-{\sqrt {5}})}{3-5}}={\frac {3({\sqrt {3}}-{\sqrt {5}})}{-2}}$ Example 2: This process also works with complex numbers with $i={\sqrt {-1}}$ ${\frac {7}{1\pm {\sqrt {-5}}}}$ The fraction must be multiplied by a quotient containing ${1\mp {\sqrt {-5}}}$. ${\frac {7}{1+{\sqrt {-5}}}}\cdot {\frac {1-{\sqrt {-5}}}{1-{\sqrt {-5}}}}={\frac {7(1-{\sqrt {-5}})}{1^{2}-{\sqrt {-5}}^{2}}}={\frac {7(1-{\sqrt {-5}})}{1-(-5)}}={\frac {7-7{\sqrt {5}}i}{6}}$ Generalizations Rationalisation can be extended to all algebraic numbers and algebraic functions (as an application of norm forms). For example, to rationalise a cube root, two linear factors involving cube roots of unity should be used, or equivalently a quadratic factor. References This material is carried in classic algebra texts. For example: • George Chrystal, Introduction to Algebra: For the Use of Secondary Schools and Technical Colleges is a nineteenth-century text, first edition 1889, in print (ISBN 1402159072); a trinomial example with square roots is on p. 256, while a general theory of rationalising factors for surds is on pp. 189–199.	Wikipedia
Rooted graph In mathematics, and, in particular, in graph theory, a rooted graph is a graph in which one vertex has been distinguished as the root.[1][2] Both directed and undirected versions of rooted graphs have been studied, and there are also variant definitions that allow multiple roots. Rooted graphs may also be known (depending on their application) as pointed graphs or flow graphs. In some of the applications of these graphs, there is an additional requirement that the whole graph be reachable from the root vertex. Variations In topological graph theory, the notion of a rooted graph may be extended to consider multiple vertices or multiple edges as roots. The former are sometimes called vertex-rooted graphs in order to distinguish them from edge-rooted graphs in this context.[3] Graphs with multiple nodes designated as roots are also of some interest in combinatorics, in the area of random graphs.[4] These graphs are also called multiply rooted graphs.[5] The terms rooted directed graph or rooted digraph also see variation in definitions. The obvious transplant is to consider a digraph rooted by identifying a particular node as root.[6][7] However, in computer science, these terms commonly refer to a narrower notion; namely, a rooted directed graph is a digraph with a distinguished node r, such that there is a directed path from r to any node other than r.[8][9][10][11] Authors who give the more general definition may refer to these as connected rooted digraphs[6] or accessible rooted graphs (see § Set theory). The Art of Computer Programming defines rooted digraphs slightly more broadly, namely, a directed graph is called rooted if it has at least one node that can reach all the other nodes. Knuth notes that the notion thus defined is a sort of intermediate between the notions of strongly connected and connected digraph.[12] Applications Flow graphs In computer science, rooted graphs in which the root vertex can reach all other vertices are called flow graphs or flowgraphs.[13] Sometimes an additional restriction is added specifying that a flow graph must have a single exit (sink) vertex.[14] Flow graphs may be viewed as abstractions of flow charts, with the non-structural elements (node contents and types) removed.[15][16] Perhaps the best known sub-class of flow graphs are control-flow graphs, used in compilers and program analysis. An arbitrary flow graph may be converted to a control-flow graph by performing an edge contraction on every edge that is the only outgoing edge from its source and the only incoming edge into its target.[17] Another type of flow graph commonly used is the call graph, in which nodes correspond to entire subroutines.[18] The general notion of flow graph has been called program graph,[19] but the same term has also been used to denote only control-flow graphs.[20] Flow graphs have also been called unlabeled flowgraphs[21] and proper flowgraphs.[15] These graphs are sometimes used in software testing.[15][18] When required to have a single exit, flow graphs have two properties not shared with directed graphs in general: flow graphs can be nested, which is the equivalent of a subroutine call (although there is no notion of passing parameters), and flow graphs can also be sequenced, which is the equivalent of sequential execution of two pieces of code.[22] Prime flow graphs are defined as flow graphs that cannot be decomposed via nesting or sequencing using a chosen pattern of subgraphs, for example the primitives of structured programming.[23] Theoretical research has been done on determining, for example, the proportion of prime flow graphs given a chosen set of graphs.[24] Set theory Peter Aczel has used rooted directed graphs such that every node is reachable from the root (which he calls accessible pointed graphs) to formulate Aczel's anti-foundation axiom in non-well-founded set theory. In this context, each vertex of an accessible pointed graph models a (non-well-founded) set within Aczel's (non-well-founded) set theory, and an arc from a vertex v to a vertex w models that v is an element of w. Aczel's anti-foundation axiom states that every accessible pointed graph models a family of (non-well-founded) sets in this way.[25] Combinatorial game theory Given a purely combinatorial game, there is an associated rooted directed graph whose vertices are game positions and whose edges are moves, and graph traversal starting from the root is used to create a game tree. If the graph contains directed cycles, then a position in the game could repeat infinitely many times, and rules are usually needed to prevent the game from continuing indefinitely. Otherwise, the graph is a directed acyclic graph, and if it isn't a rooted tree, then the game has transpositions. This graph and its topology are important in the study of game complexity, where the state-space complexity is the number of vertices in the graph, the average game length is the average number of vertices traversed from the root to a vertex with no direct successors, and the average branching factor of a game tree is the average outdegree of the graph. Combinatorial enumeration The number of rooted undirected graphs for 1, 2, ... nodes is 1, 2, 6, 20, 90, 544, ... (sequence A000666 in the OEIS). Related concepts A special case of interest are rooted trees, the trees with a distinguished root vertex. If the directed paths from the root in the rooted digraph are additionally restricted to be unique, then the notion obtained is that of (rooted) arborescence—the directed-graph equivalent of a rooted tree.[7] A rooted graph contains an arborescence with the same root if and only if the whole graph can be reached from the root, and computer scientists have studied algorithmic problems of finding optimal arborescences.[26] Rooted graphs may be combined using the rooted product of graphs.[27] See also • k-vertex-connected graph • pointed set References 1. Zwillinger, Daniel (2011), CRC Standard Mathematical Tables and Formulae, 32nd Edition, CRC Press, p. 150, ISBN 978-1-4398-3550-0 2. Harary, Frank (1955), "The number of linear, directed, rooted, and connected graphs", Transactions of the American Mathematical Society, 78 (2): 445–463, doi:10.1090/S0002-9947-1955-0068198-2, MR 0068198. See p. 454. 3. Gross, Jonathan L.; Yellen, Jay; Zhang, Ping (2013), Handbook of Graph Theory (2nd ed.), CRC Press, pp. 764–765, ISBN 978-1-4398-8018-0 4. Spencer, Joel (2001), The Strange Logic of Random Graphs, Springer Science & Business Media, chapter 4, ISBN 978-3-540-41654-8 5. Harary (1955, p. 455). 6. Björner, Anders; Ziegler, Günter M. (1992), "8. Introduction to greedoids" (PDF), in White, Neil (ed.), Matroid Applications, Encyclopedia of Mathematics and its Applications, vol. 40, Cambridge: Cambridge University Press, pp. 284–357, doi:10.1017/CBO9780511662041.009, ISBN 0-521-38165-7, MR 1165537, Zbl 0772.05026, In this context a rooted digraph Δ = (V,E,r) is called connected (or 1-connected) if there is a directed path from the root to every vertex. See in particular p. 307. 7. Gordon, Gary; McMahon, Elizabeth (February 1989), "A greedoid polynomial which distinguishes rooted arborescences" (PDF), Proceedings of the American Mathematical Society, 107 (2): 287, CiteSeerX 10.1.1.308.2526, doi:10.1090/s0002-9939-1989-0967486-0, A rooted subdigraph F is a rooted arborescence if the root vertex ∗ is in F and, for every vertex v in F, there is a unique directed path in F from ∗ to v. Thus, rooted arborescences in digraphs correspond to rooted trees in undirected graphs. 8. Ramachandran, Vijaya (1988), "Fast Parallel Algorithms for Reducible Flow Graphs", Concurrent Computations: 117–138, doi:10.1007/978-1-4684-5511-3_8, ISBN 978-1-4684-5513-7, A rooted directed graph or a flow graph G = (V, A, r) is a directed graph with a distinguished vertex r such that there is a directed path in G from r to every vertex v in V − r.. See in particular p. 122. 9. Okamoto, Yoshio; Nakamura, Masataka (2003), "The forbidden minor characterization of line-search antimatroids of rooted digraphs" (PDF), Discrete Applied Mathematics, 131 (2): 523–533, doi:10.1016/S0166-218X(02)00471-7, A rooted digraph is a triple G=(V,E,r) where (V ∪ {r}, E) is a digraph and r is a specified vertex called the root such that there exists a path from r to every vertex of V.. See in particular p. 524. 10. Jain, Abhinandan (2010), Robot and Multibody Dynamics: Analysis and Algorithms, Springer Science & Business Media, p. 136, ISBN 978-1-4419-7267-5, A rooted digraph is a connected digraph with a single root node that is the ancestor of every other node in the digraph. 11. Chen, Xujin; Zang, Wenan (2006), "An efficient algorithm for finding maximum cycle packings in reducible flow graphs", Algorithmica, 44 (3): 195–211, doi:10.1007/s00453-005-1174-x, hdl:10722/48600, MR 2199991, S2CID 5235131 12. Knuth, Donald (1997), "2.3.4.2. Oriented trees", The Art of Computer Programming, vol. 1 (3rd ed.), Pearson Education, p. 372, ISBN 0-201-89683-4, It is said to be rooted if there is at least one root, that is, at least one vertex R such that there is an oriented path from V to R for all V ≠ R. 13. Gross, Yellen & Zhang (2013, p. 1372). 14. Fenton, Norman Elliott; Hill, Gillian A. (1993), Systems Construction and Analysis: A Mathematical and Logical Framework, McGraw-Hill, p. 319, ISBN 978-0-07-707431-9. 15. Zuse, Horst (1998), A Framework of Software Measurement, Walter de Gruyter, pp. 32–33, ISBN 978-3-11-080730-1 16. Samaroo, Angelina; Thompson, Geoff; Williams, Peter (2010), Software Testing: An ISTQB-ISEB Foundation Guide, BCS, The Chartered Institute, p. 108, ISBN 978-1-906124-76-2 17. Tarr, Peri L.; Wolf, Alexander L. (2011), Engineering of Software: The Continuing Contributions of Leon J. Osterweil, Springer Science & Business Media, p. 58, ISBN 978-3-642-19823-6 18. Jalote, Pankaj (1997), An Integrated Approach to Software Engineering, Springer Science & Business Media, p. 372, ISBN 978-0-387-94899-7 19. Thulasiraman, K.; Swamy, M. N. S. (1992), Graphs: Theory and Algorithms, John Wiley & Sons, p. 361, ISBN 978-0-471-51356-8 20. Cechich, Alejandra; Piattini, Mario; Vallecillo, Antonio (2003), Component-Based Software Quality: Methods and Techniques, Springer Science & Business Media, p. 105, ISBN 978-3-540-40503-0 21. Beineke, Lowell W.; Wilson, Robin J. (1997), Graph Connections: Relationships Between Graph Theory and Other Areas of Mathematics, Clarendon Press, p. 237, ISBN 978-0-19-851497-8 22. Fenton & Hill (1993, p. 323). 23. Fenton & Hill (1993, p. 339). 24. Cooper, C. (2008), "Asymptotic Enumeration of Predicate-Junction Flowgraphs", Combinatorics, Probability and Computing, 5 (3): 215–226, doi:10.1017/S0963548300001991, S2CID 10313545 25. Aczel, Peter (1988), Non-well-founded sets (PDF), CSLI Lecture Notes, vol. 14, Stanford, CA: Stanford University, Center for the Study of Language and Information, ISBN 0-937073-22-9, LCCN 87-17857, MR 0940014, archived from the original (PDF) on 2015-03-26 26. Drescher, Matthew; Vetta, Adrian (2010), "An Approximation Algorithm for the Maximum Leaf Spanning Arborescence Problem", ACM Trans. Algorithms, 6 (3): 46:1–46:18, doi:10.1145/1798596.1798599, S2CID 13987985. 27. Godsil, C. D.; McKay, B. D. (1978), "A new graph product and its spectrum" (PDF), Bull. Austral. Math. Soc., 18 (1): 21–28, doi:10.1017/S0004972700007760, MR 0494910 Further reading • McMahon, Elizabeth W. (1993), "On the greedoid polynomial for rooted graphs and rooted digraphs", Journal of Graph Theory, 17 (3): 433–442, doi:10.1002/jgt.3190170316 • Gordon, Gary (2001), "A characteristic polynomial for rooted graphs and rooted digraphs", Discrete Mathematics, 232 (1–3): 19–33, doi:10.1016/S0012-365X(00)00186-2 External links • Weisstein, Eric W., "Rooted Graph", MathWorld	Wikipedia
Rooted product of graphs In mathematical graph theory, the rooted product of a graph G and a rooted graph H is defined as follows: take \|V(G)\| copies of H, and for every vertex vi of G, identify vi with the root node of the i-th copy of H. More formally, assuming that ${\begin{aligned}V(G)&=\{g_{1},\ldots ,g_{n}\},\\V(H)&=\{h_{1},\ldots ,h_{m}\},\end{aligned}}$ and that the root node of H is h1, define $G\circ H:=(V,E)$, where $V=\left\{(g_{i},h_{j}):1\leq i\leq n,1\leq j\leq m\right\}$ and $E={\Bigl \{}{\bigl (}(g_{i},h_{1}),(g_{k},h_{1}){\bigr )}:(g_{i},g_{k})\in E(G){\Bigr \}}\cup \bigcup _{i=1}^{n}{\Bigl \{}{\bigl (}(g_{i},h_{j}),(g_{i},h_{k}){\bigr )}:(h_{j},h_{k})\in E(H){\Bigr \}}$. If G is also rooted at g1, one can view the product itself as rooted, at (g1, h1). The rooted product is a subgraph of the cartesian product of the same two graphs. Applications The rooted product is especially relevant for trees, as the rooted product of two trees is another tree. For instance, Koh et al. (1980) used rooted products to find graceful numberings for a wide family of trees. If H is a two-vertex complete graph K2, then for any graph G, the rooted product of G and H has domination number exactly half of its number of vertices. Every connected graph in which the domination number is half the number of vertices arises in this way, with the exception of the four-vertex cycle graph. These graphs can be used to generate examples in which the bound of Vizing's conjecture, an unproven inequality between the domination number of the graphs in a different graph product, the cartesian product of graphs, is exactly met (Fink et al. 1985). They are also well-covered graphs. References • Godsil, C. D.; McKay, B. D. (1978), "A new graph product and its spectrum" (PDF), Bull. Austral. Math. Soc., 18 (1): 21–28, doi:10.1017/S0004972700007760, MR 0494910. • Fink, J. F.; Jacobson, M. S.; Kinch, L. F.; Roberts, J. (1985), "On graphs having domination number half their order", Period. Math. Hungar., 16 (4): 287–293, doi:10.1007/BF01848079, MR 0833264. • Koh, K. M.; Rogers, D. G.; Tan, T. (1980), "Products of graceful trees", Discrete Mathematics, 31 (3): 279–292, doi:10.1016/0012-365X(80)90139-9, MR 0584121.	Wikipedia
Rope-burning puzzle In recreational mathematics, rope-burning puzzles are a class of mathematical puzzle in which one is given lengths of rope, fuse cord, or shoelace that each burn for a given amount of time, and matches to set them on fire, and must use them to measure a non-unit amount of time. The fusible numbers are defined as the amounts of time that can be measured in this way. As well as being of recreational interest, these puzzles are sometimes posed at job interviews as a test of candidates' problem-solving ability,[1] and have been suggested as an activity for middle school mathematics students.[2] Example A common and simple version of this problem asks to measure a time of 45 seconds using only two fuses that each burn for a minute. The assumptions of the problem are usually specified in a way that prevents measuring out 3/4 of the length of one fuse and burning it end-to-end, for instance by stating that the fuses burn unevenly along their length.[1][2][3][4] One solution to this problem is to perform the following steps:[3] • Light one end of the first fuse, and both ends of the second fuse. • Once the second fuse has burned out, 30 seconds have elapsed, and there are 30 seconds of burn time left on the first fuse. Light the other end of the first fuse. • Once the first fuse burns out, 45 seconds have elapsed. Many other variations are possible, in some cases using fuses that burn for different amounts of time from each other.[5] Fusible numbers In common versions of the problem, each fuse lasts for a unit length of time, and the only operations used or allowed in the solution are to light one or both ends of a fuse at known times, determined either as the start of the solution or as the time that another fuse burns out. If only one end of a fuse is lit at time $x$, it will burn out at time $x+1$. If both ends of a fuse are lit at times $x$ and $y$, it will burn out at time $(x+y+1)/2$, because a portion of $y-x$ is burnt at the original rate, and the remaining portion of $1-(y-x)$ is burnt at twice the original rate, hence the fuse burns out at $x+(y-x)+[1-(y-x)]/2=(x+y+1)/2$. A number $x$ is a fusible number if it is possible to use unit-time fuses to measure out $x$ units of time using only these operations. For instance, by the solution to the example problem, ${\tfrac {3}{4}}$ is a fusible number.[7] One may assume without loss of generality that every fuse is lit at both ends, by replacing a fuse that is lit only at one end at time $x$ by two fuses, the first one lit at both ends at time $x$ and the second one lit at both ends at time $x+1/2$ when the first fuse burns out. In this way, the fusible numbers can be defined as the set of numbers that can be obtained from the number $0$ by repeated application of the operation $x,y\mapsto (x+y+1)/2$, applied to pairs $x,y$ that have already been obtained and for which $\|x-y\|<1$.[7] The fusible numbers include all of the non-negative integers, and are a well-ordered subset of the dyadic rational numbers, the fractions whose denominators are powers of two. Being well-ordered means that, if one chooses a decreasing sequence of fusible numbers, the sequence must always be finite. Among the well-ordered sets, their ordering can be classified as $\varepsilon _{0}$, an epsilon number (a special case of the infinite ordinal numbers). Because they are well-ordered, for each integer $n$ there is a unique smallest fusible number among the fusible numbers larger than $n$; it has the form $n+1/2^{k}$ for some $k$.[7] This number $k$ grows very rapidly as a function of $n$, so rapidly that for $n=3$ it is (in Knuth's up-arrow notation for large numbers) already larger than $2\uparrow ^{9}16$.[8] The existence of this number $k$, for each $n$, cannot be proven in Peano arithmetic.[7] Lighting more than two points of a fuse If the rules of the fuse-burning puzzles are interpreted to allow fuses to be lit at more points than their ends, a larger set of amounts of time can be measured. For instance, if a fuse is lit in such a way that, while it burns, it always has three ends burning (for instance, by lighting one point in the middle and one end, and then lighting another end or another point in the middle whenever one or two of the current lit points burn out) then it will burn for 1/3 of a unit of time rather than a whole unit. By representing a given amount of time as a sum of unit fractions, and successively burning fuses with multiple lit points so that they last for each unit fraction amount of time, it is possible to measure any rational number of units of time. However, keeping the desired number of flames lit, even on a single fuse, may require an infinite number of re-lighting steps.[4] The problem of representing a given rational number as a sum of unit fractions is closely related to the construction of Egyptian fractions, sums of distinct unit fractions; however, for fuse-burning problems there is no need for the fractions to be different from each other. Using known methods for Egyptian fractions one can prove that measuring a fractional amount of time $x/y$, with $x<y$, needs only $O({\sqrt {\log y}})$ fuses (expressed in big O notation).[9] An unproven conjecture of Paul Erdős on Egyptian fractions suggests that fewer fuses, $O(\log \log y)$, may always be enough.[10] History In a booklet on these puzzles titled Shoelace Clock Puzzles, created by Dick Hess for a 1998 Gathering 4 Gardner conference, Hess credits Harvard statistician Carl Morris as his original source for these puzzles.[4] See also • Water pouring puzzle, another class of puzzles involving the combination of measurements References 1. Mongan, John; Kindler, Noah Suojanen; Giguère, Eric (2012), "Burning fuses", Programming Interviews Exposed: Secrets to Landing Your Next Job (3rd ed.), John Wiley & Sons, p. 234, ISBN 978-1-118-28720-0 2. Brumbaugh, Douglas K. (2013), Teaching Middle School Mathematics, Routledge, pp. 191, 309, ISBN 978-1-136-75622-1 3. Haselbauer, Nathan (2020), 60-Second Brain Teasers Pencil-Free Puzzles: Short Head-Scratchers from the Easy to Near Impossible, Fair Winds Press, pp. 77, 121, ISBN 978-1-63159-927-9 4. Winkler, Peter (2004), "Uses of fuses", Mathematical Puzzles: A Connoisseur's Collection, A K Peters, pp. 2, 6, ISBN 1-56881-201-9 5. Hess, Dick (2009), "Shoelace clocks", All-Star Mathlete Puzzles, Official Mensa Puzzle Books, p. 9, ISBN 978-1-4027-5528-6 6. Jeff Erickson, Fusible Numbers 7. Erickson, Jeff; Nivasch, Gabriel; Xu, Junyan (June 2021), "Fusible numbers and Peano arithmetic", Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2021), IEEE, pp. 1–13, arXiv:2003.14342, doi:10.1109/lics52264.2021.9470703 8. Sloane, N. J. A. (ed.), "Sequence A188545", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation 9. Vose, M. (1985), "Egyptian fractions", Bulletin of the London Mathematical Society, 17: 21, doi:10.1112/blms/17.1.21, MR 0766441 10. Erdős, Pál (1950), "Az $\textstyle {\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+\cdots +{\frac {1}{x_{n}}}={\frac {a}{b}}$ egyenlet egész számú megoldásairól" [On a Diophantine equation] (PDF), Matematikai Lapok (in Hungarian), 1: 192–210, MR 0043117	Wikipedia
Peter Roquette Peter Jaques Roquette (8 October 1927 – 24 February 2023) was a German mathematician working in algebraic geometry, algebra, and number theory. Biography Roquette was born in Königsberg on 8 October 1927. He studied in Erlangen, Berlin, and Hamburg. In 1951 he defended a dissertation at the University of Hamburg under Helmut Hasse, providing a new proof of the Riemann hypothesis for algebraic function fields over a finite field (the first proof was given by André Weil in 1940). In 1951/1952 he was an assistant at the Mathematical Research Institute at Oberwolfach and from 1952 to 1954 at the University of Munich. From 1954 to 1956 he worked at the Institute for Advanced Study in Princeton. In 1954 he was Privatdozent at Munich, and from 1956 to 1959 he worked in the same position at Hamburg. In 1959 he became an associate professor at the University of Saarbrucken and in the same year at the University of Tübingen. From 1967 he was professor at the Ruprecht-Karls-University of Heidelberg, where he retired in 1996. Roquette worked on number and function fields and especially local p-adic fields. He applied the methods of model theory (nonstandard arithmetic) in number theory, joint with Abraham Robinson, with whom he worked on Mahler's theorem (on the finiteness of integral points on a curve of genus g > 0) using non-standard methods. He authored a number of works on the history of mathematics, in particular on the schools of Helmut Hasse and Emmy Noether. In 1975 Roquette was co-editor of the collected essays by Helmut Hasse. From 1978, Roquette was a member of the Heidelberg Academy of Sciences and from 1985, the German Academy of Sciences Leopoldina. He has an honorary doctorate from the University of Duisburg-Essen and was an honorary member of the Mathematical Society of Hamburg. In 1958 he was an invited speaker at the International Congress of Mathematicians in Edinburgh (on the topic of "Some fundamental theorems on abelian function fields"). His doctoral students include Gerhard Frey and Volker Weispfenning. Roquette died in Heidelberg on 24 February 2023, at the age of 95.[1] Selected publications • Analytic theory of elliptic functions over local fields. Vandenhoeck and Ruprecht 1970. • With Franz Lemmermeyer (Editor): The Correspondence of Helmut Hasse and Emmy Noether 1925-1935 Göttingen State and University Library, 2006.. • with Günther Frei (Editor): Emil Artin and Helmut Hasse - correspondence 1923-1934, University of Göttingen Publisher 2008 • The Brauer-Hasse-Noether Theorem in Historical Perspective. Mathem. the-Naturwiss writings. Class of the Heidelberg Academy of Sciences, Springer-Verlag, 2005. • Anthony V. Geramita, Paulo Ribenboim (ed.): Collected Papers of Peter Roquette 3 volumes. Queens Papers in Pure and Applied Mathematics Bd.118, Kingston, Ontario, Queen's University, 2002. • With Alexander Prestel: Formally p-adic Fields. Lecture Notes in Mathematics, Springer-Verlag 1984. • Robinson, A.; Roquette, P. On the finiteness theorem of Siegel and Mahler concerning Diophantine equations. J. Number Theory 7 (1975), 121–176. References 1. "Peter Jaques Roquette". trauer.rnz.de. Retrieved 26 March 2023. External links • Media related to Peter Roquette at Wikimedia Commons • homepage • Peter Roquette at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • Belgium • United States • Czech Republic • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef	Wikipedia
Rosa Donat Rosa María Donat Beneito (born 1960) is a Spanish applied mathematician whose research involves numerical methods for partial differential equations, particularly multiresolution methods for problems modeling fluid dynamics with shock waves or with high Mach number. She is a professor of applied mathematics and vice rector for innovation and transfer at the University of Valencia,[1] and former president of the Spanish Society of Applied Mathematics.[2] Education and career Donat was born in 1960 in La Font de la Figuera.[1] After earning a degree in mathematal sciences from the University of Valencia in 1984, she traveled to the University of California, Los Angeles in 1985 as a Fulbright Scholar, earning a master's degree there in 1987 and a Ph.D. in mathematics in 1990.[2] Her doctoral dissertation, Studies on Error Propagation Into Regions of Smoothness for Certain Nonlinear Approximations to Hyperbolic Equations, was supervised by Stanley Osher.[3] She has held a tenured position at the University of Valencia since 1993, and was given a professorial chair there in 2008.[4] She was elected as president of the Spanish Society of Applied Mathematics in 2016,[4] becoming the society's first woman president[5] and holding office from 2016 to 2020.[2] References 1. "Vice-Principal for Innovation and Transfer", Office of the Principal, University of Valencia, retrieved 2023-04-10 2. Donat Beneito, Rosa Maria (in Spanish), Fundación Gadea Ciencia, retrieved 2023-04-10 3. Rosa Donat at the Mathematics Genealogy Project 4. "La investigadora Rosa Donat, nueva presidenta de la Sociedad Española de Matemática Aplicada (SEMA)", Red de Universidades Valencianas para el fomento de la Investigación, el Desarrollo y la Innovación (in Spanish), 20 June 2016, retrieved 2023-04-10 5. "The Spanish Society of Applied Mathematics", MacTutor History of Mathematics Archive, February 2018, retrieved 2023-04-10 Authority control International • VIAF Academics • DBLP • MathSciNet • Mathematics Genealogy Project • ORCID • Scopus • zbMATH • 2 Other • IdRef	Wikipedia
Rosa M. Miró-Roig Rosa M. Miró-Roig (born August 6, 1960)[1] is a professor of mathematics at the University of Barcelona,[2] specializing in algebraic geometry and commutative algebra. She did her graduate studies at the University of Barcelona, earning a Ph.D. in 1985 under the supervision of Sebastià Xambó-Descamps with a thesis entitled Haces reflexivos sobre espacios proyectivos.[3] Books and editing Miró-Roig has authored and co-authored three mathematics research volumes. Most recently, she co-authored On the Shape of a Pure O-sequence (American Mathematical Society 2012) with Mats Boij, Juan C. Migliore, Uwe Nagel, and Fabrizio Zanello.[4][5] Previously, she authored the research text Determinantal Ideals (Birkhäuser 2007)[6][7] and co-authored the monograph Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness (American Mathematical Society 2001) with Jan O. Kleppe, Juan C. Migliore, Uwe Nagel, and Chris Peterson.[8][9] Miró-Roig is the Chief Editor of the mathematics research journal Collectanea Mathematica (Springer).[10][11] Also, she is on the editorial boards of the mathematics research journals Beiträge zur Algebra und Geometrie (Springer)[12][13] and Journal of Commutative Algebra (Rocky Mountain Mathematics Consortium).[14] Furthermore, she co-edited the mathematics research volumes Projective Varieties with Unexpected Properties (De Gruyter 2008) with Ciro Ciliberto, Antony V. Geramita, Brian Harbourne, and Kristian Ranestad,[15][16] European Congress of Mathematics: Barcelona, July 10–14, 2000 (Birkhäuser 2001) two volumes with Carles Casacuberta, Joan Verdera, and Sebastià Xambó-Descamps,[17][18][19] Six Lectures on Commutative Algebra (Birkhäuser 1998) with J. Elias, J. M. Giral, and S. Zarzuela,[20][21] and Complex Analysis and Geometry (Chapman and Hall/CRC 1997) with V. Ancona, E. Ballico, and A. Silva[22][23] Recognition In 2007, Miró-Roig was awarded the Ferran Sunyer i Balaguer Prize for her work "Determinantal Ideals (Birkhäuser, 2007)".[24] References 1. Birth date from ISNI authority control file, retrieved 2018-11-29. 2. "Rosa Maria Miró-Roig". www.imub.ub.edu. Retrieved 2018-11-22. 3. "Rosa Miró-Roig - The Mathematics Genealogy Project". www.genealogy.math.ndsu.nodak.edu. Retrieved 2018-11-21. 4. "MathSciNet review by Adam L. Van Tuyl". MR 2931681. {{cite journal}}: Cite journal requires \|journal= (help) 5. On the shape of a pure O-sequence. Boij, Mats, 1969-. Providence, R.I.: American Mathematical Society. 2011. ISBN 9780821869109. OCLC 782251800.{{cite book}}: CS1 maint: others (link) 6. "MathSciNet review by Juan C. Migliore". MR 2375719. {{cite journal}}: Cite journal requires \|journal= (help) 7. M., Miró-Roig, Rosa (2008). Determinantal ideals. Basel: Birkhäuser. ISBN 9783764385354. OCLC 233974116.{{cite book}}: CS1 maint: multiple names: authors list (link) 8. "MathSciNet review by Scott R. Nollet". MR 1848976. {{cite journal}}: Cite journal requires \|journal= (help) 9. Gorenstein liaison, complete intersection liaison invariants, and unobstructedness. Kleppe, Jan O. (Jan Oddvar), 1947-. Providence, R.I.: American Mathematical Society. 2001. ISBN 978-0821827383. OCLC 47002207.{{cite book}}: CS1 maint: others (link) 10. "Collectanea Mathematica - incl. option to publish open access (Editorial Board)". springer.com. Retrieved 2018-11-22. 11. "Collectanea Mathematica". Collectanea Mathematica. 2013. doi:10.1007/13348.2038-4815. 12. "Beiträge zur Algebra und Geometrie - incl. option to publish open access (Editorial Board)". springer.com. Retrieved 2018-11-22. 13. "Beiträge zur Algebra und Geometrie". Beiträge zur Algebra und Geometrie. 2013. CiteSeerX 10.1.1.614.5637. doi:10.1007/13366.2191-0383. 14. "Journal of Commutative Algebra Editorial Board". 15. "MathSciNet review". MR 2201829. {{cite journal}}: Cite journal requires \|journal= (help) 16. Projective varieties with unexpected properties : a volume in memory of Giuseppe Veronese : proceedings of the international conference "Varieties with Unexpected Properties," Siena, Italy, June 8-13, 2004. Veronese, Giuseppe, 1854-1917., Ciliberto, C. (Ciro), 1950-. Berlin: Walter de Gruyter. 2005. ISBN 9783110199703. OCLC 228144658.{{cite book}}: CS1 maint: others (link) 17. "MathSciNet review". MR 1905309. {{cite journal}}: Cite journal requires \|journal= (help) 18. European Congress of Mathematics : Barcelona, July 10-14, 2000. Casacuberta, Carlos, 1964-. Basel: Birkhäuser. 2001. ISBN 978-3764364199. OCLC 48053801.{{cite book}}: CS1 maint: others (link) 19. "MathSciNet review". MR 1905345. {{cite journal}}: Cite journal requires \|journal= (help) 20. "MathSciNet review". MR 1648663. {{cite journal}}: Cite journal requires \|journal= (help) 21. Six lectures on commutative algebra. Elias, J. (Juan). Basel: Birkhäuser Verlag. 1998. ISBN 978-3764359515. OCLC 39051720.{{cite book}}: CS1 maint: others (link) 22. "MathSciNet review". MR 1477435. {{cite journal}}: Cite journal requires \|journal= (help) 23. Complex analysis and geometry. Ancona, Vincenzo. Essex, England: Longman. 1997. ISBN 978-0582292765. OCLC 35243565.{{cite book}}: CS1 maint: others (link) 24. "The Ferran Sunyer i Balaguer Prize \| Fundació Ferran Sunyer i Balaguer". ffsb.espais.iec.cat. Retrieved 2018-11-21. Authority control International • ISNI • VIAF National • Norway • Spain • Catalonia • Germany • Israel • United States Academics • MathSciNet • Mathematics Genealogy Project • ORCID • ResearcherID • zbMATH Other • IdRef	Wikipedia
Rosa M. Morris Rosa Margaret Morris (16 July 1914 – 15 October 2011)[1] was a Welsh applied mathematician, working in potential theory and aerodynamics. When she was 23, her research and examination results made national news. In her later career, she taught at the University College of South Wales and Monmouthshire (now Cardiff University), where she co-authored a successful textbook on Mathematical Methods of Physics and became one of the first female Heads of School of Mathematics in the United Kingdom. Rosa Margaret Morris Rosa Morris in her garden, 1938 Born(1914-07-16)16 July 1914 Rogerstone, Monmouthshire, Wales, United Kingdom Died15 October 2011(2011-10-15) (aged 97) EducationUniversity College of South Wales and Monmouthshire Alma materUniversity of Cambridge AwardsUniversity of Wales Fellowship Department of Scientific and Industrial Research Senior Research Award Scientific career FieldsApplied mathematics InstitutionsUniversity College of South Wales and Monmouthshire, Cardiff ThesisTwo-dimensional potential theory, with special reference to aerodynamic problems (1940) Doctoral advisorsGeorge Henry Livens Geoffrey Ingram Taylor Doctoral studentsDavid Edmunds Early life and education Morris was born in Rogerstone, Monmouthshire,[1] the youngest child of John and Mary Aline Morris, née Roberts. Her father, who died when she was young, was the headmaster of the elementary school in Rogerstone,[2][3] and her mother was a schoolmistress.[4] Morris first attended Rogerstone School, then Pontywaun County School, Pontymister, Risca, from 1926 to 1932, with Distinctions in Pure and Applied Mathematics.[2] She studied mathematics at the University College of South Wales and Monmouthshire in Cardiff, graduating with a first class degree in 1936 and continuing as a research student until 1938,[3] working under the supervision of George Henry Livens.[5][6][7] During this time, Morris, aged 23, published her first articles, on potential theory[8] and aerodynamics.[9] Her approach showed "the advantages of using the complex variable in [...] boundary problems of mathematical physics"[10] and made national news, including an interview,[6] with human interest stories focusing on her as a "mathematical genius", having "found a method of solving problems in aerodynamics which have hitherto defied all mathematicians". She was described as a "keen hockey player and accomplished dressmaker", and University College of South Wales and Monmouthshire Principal Frederick Rees stated that at her examination, she would have been entitled to 130 percent compared to the next best student, and a special case had to be made for her to avoid handicapping other students.[4][11][12] Short reports on her achievements were also printed in American local newspapers.[13][14] She won scholarships worth £600 for the first year at Girton College, Cambridge,[4][11] where she was a M. T. Meyer research student.[15] Her fellowships and awards included a University of Wales Fellowship (1938–1940) and a Department of Scientific and Industrial Research Senior Research Award (1939–1941).[3] After three years as a research student, one of them in Cardiff, two in Cambridge,[5] she obtained her PhD from Cambridge University in 1940, with the thesis Two-dimensional potential theory, with special reference to aerodynamic problems.[16] Her Cambridge advisor was Geoffrey Ingram Taylor.[5][17][18] Professional career Morris graduated with a PhD in 1940 and became a faculty member in Cardiff in 1941,[19] where she stayed for the rest of her career. She supervised the 1955 PhD thesis of David Edmunds, who later won the Pólya Prize.[20] Together with Roy Chisholm, Morris wrote a textbook on Mathematical Methods in Physics.[21] Although it was lacking in rigour,[22][23] it was reprinted several times, and, according to Chisholm, "in the late 1960's, North-Holland told us that we had broken their publication record for technical books. We even made a little money."[24] In 1972–1973, while she was a Reader in Fluid Dynamics, she served as one of the first female heads of a mathematics department in the UK, possibly the first at a university.[25][26] Morris was a member of the London Mathematical Society (since 1945)[27] of the Mathematical Association, where she was President of the Cardiff Branch 1955–1956,[28][29] and a Fellow of the Cambridge Philosophical Society until 1983.[30] She was a prolific contributor to Mathematical Reviews, with 188 contributions credited to her name.[31] Publications Morris published research articles on potential theory, fluid dynamics (especially moving aerofoils), and mathematical elasticity theory. • Morris, Rosa M. (1 February 1937). "XXIV. Notes on two-dimensional potential theory.—I. The force and couple in electrostatic problems". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 23 (153): 246–251. doi:10.1080/14786443708561795. ISSN 1941-5982. • Morris, Rosa M. (1 April 1937). "LXXVI. Notes on two-dimensional potential theory.—II. Hydrodynamical problems on the motion of cylinders". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 23 (156): 757–762. doi:10.1080/14786443708561850. ISSN 1941-5982. • Morris, Rosa M. (1 June 1937). "CII. Notes on two-dimensional potential theory.—III. The line source influence problems". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 23 (158): 1082–1096. doi:10.1080/14786443708561878. ISSN 1941-5982. • Morris, Rosa M. (1 July 1937). "IV. Notes on two-dimensional potential theory.—IV. The expression for the fluid energy and its application". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 24 (159): 47–52. doi:10.1080/14786443708561888. ISSN 1941-5982. • Morris, Rosa M. (3 August 1937). "The two-dimensional hydrodynamical theory of moving aerofoils—I". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 161 (906): 406–419. Bibcode:1937RSPSA.161..406M. doi:10.1098/rspa.1937.0152. • Morris, Rosa M. (1 September 1937). "XLIII. Notes on two-dimensional potential theory.— V. The generalized formulæ for the forces and couple on a moving cylinder". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 24 (161): 445–453. doi:10.1080/14786443708561928. ISSN 1941-5982. • Morris, Rosa M. (October 1937). "Two-Dimensional Potential Problems". Mathematical Proceedings of the Cambridge Philosophical Society. 33 (4): 474–484. Bibcode:1937PCPS...33..474M. doi:10.1017/S0305004100077616. ISSN 0305-0041. S2CID 120558044. • Morris, Rosa M. (4 February 1938). "The two-dimensional hydrodynamical theory of moving aerofoils-II". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 164 (918): 346–368. Bibcode:1938RSPSA.164..346M. doi:10.1098/rspa.1938.0022. • Morris, Rosa M. (3 August 1939). "The two-dimensional hydrodynamical theory of moving aerofoils. III". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 172 (949): 213–230. Bibcode:1939RSPSA.172..213M. doi:10.1098/rspa.1939.0100. • Morris, Rosa M. (1 December 1939). "The internal problems of two-dimensional potential theory". Mathematische Annalen. 116 (1): 374–400. doi:10.1007/BF01597363. ISSN 1432-1807. S2CID 121215755. • Morris, Rosa M. (1 December 1940). "The internal problems of two dimensional potential theory". Mathematische Annalen. 117 (1): 31–38. doi:10.1007/BF01450006. ISSN 1432-1807. S2CID 179178042. • Morris, Rosa M. (1940). "Some General Solutions of St. Venant's Flexure and Torsion Problem. I". Proceedings of the London Mathematical Society. s2-46 (1): 81–98. doi:10.1112/plms/s2-46.1.81. ISSN 1460-244X. • Morris, Rosa M. (1946). "Some General Solutions of St. Venant's Torsion and Flexure Problem (II)". Proceedings of the London Mathematical Society. s2-49 (1): 1–18. doi:10.1112/plms/s2-49.1.1. ISSN 1460-244X. • Morris, Rosa M. (25 February 1947). "The two-dimensional hydrodynamical theory of moving aerofoils. IV". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 188 (1015): 439–463. Bibcode:1947RSPSA.188..439M. doi:10.1098/rspa.1947.0019. • Livens, G. H.; Morris, Rosa M. (1 March 1947). "XIX. The boundary-value problems of plane stress". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 38 (278): 153–179. doi:10.1080/14786444708561097. ISSN 1941-5982. • Morris, Rosa M. (1949). "Some General Solutions of St. Venant's Torsion and Flexure Problem (III)". Proceedings of the London Mathematical Society. s2-51 (1): 424–439. doi:10.1112/plms/s2-51.6.424. ISSN 1460-244X. • Morris, Rosa M. (1 January 1951). "The boundary-value problems of plane stress". The Quarterly Journal of Mechanics and Applied Mathematics. 4 (2): 248–256. doi:10.1093/qjmam/4.2.248. ISSN 0033-5614. Archived from the original on 19 April 2021. Retrieved 5 April 2021. • Morris, Rosa M. (1951). "George Henry Livens". Journal of the London Mathematical Society. s1-26 (2): 156–160. doi:10.1112/jlms/s1-26.2.156. ISSN 1469-7750. • Morris, Rosa M.; Hawley, F. J. (1 January 1958). "Torsion and flexure of solid cylinders with cross-section transformable to a ring space". The Quarterly Journal of Mechanics and Applied Mathematics. 11 (4): 462–477. doi:10.1093/qjmam/11.4.462. ISSN 0033-5614. Archived from the original on 19 April 2021. Retrieved 5 April 2021. • Chisholm, John Stephen Roy; Morris, Rosa M (1964). Mathematical methods in physics. Amsterdam: North-Holland. OCLC 610944321. Archived from the original on 19 April 2021. Retrieved 5 April 2021. • Morris, Rosa M.; Price, W. G. (September 1972). "Extension of the Curie principle and constitutive relations for fluids with antisymmetric stress". Mathematical Proceedings of the Cambridge Philosophical Society. 72 (2): 243–251. Bibcode:1972PCPS...72..243M. doi:10.1017/S0305004100047071. ISSN 1469-8064. S2CID 123668313. References 1. "R Morris". South Wales Argus. 27 October 2011. Retrieved 6 March 2021. 2. Osment, Bernard (May–June 2019). "Renowned Mathematician". Risca Directory. p. 12. Archived from the original on 19 April 2021. Retrieved 5 April 2021. 3. Girton College (University of Cambridge) (1948). Girton College Register: 1869–1946. Privately printed for Girton College. p. 549. Archived from the original on 19 April 2021. Retrieved 10 March 2021. 4. "A Mathematical Genius". The Times. 28 July 1938. p. 16. ISSN 0140-0460. Archived from the original on 19 April 2021. Retrieved 9 April 2021. 5. Morris, Rosa Margaret (1940). Two-dimensional potential theory, with special reference to aerodynamic problems (PhD thesis). Cambridge University. 6. Daily Mail Correspondent (29 July 1938). "She Beat Science — By Mistake". Daily Mail. p. 9. ISSN 0307-7578. Retrieved 6 April 2021. 7. Daily Mail Reporter (30 July 1938). "R. A. F. Works on Girl's Formula". Daily Mail. p. 7. ISSN 0307-7578. Retrieved 7 April 2021. 8. Morris, Rosa M. (October 1937). "Two-Dimensional Potential Problems". Mathematical Proceedings of the Cambridge Philosophical Society. 33 (4): 474–484. Bibcode:1937PCPS...33..474M. doi:10.1017/S0305004100077616. ISSN 0305-0041. S2CID 120558044. 9. Morris, Rosa M. (3 August 1937). "The two-dimensional hydrodynamical theory of moving aerofoils—I". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 161 (906): 406–419. Bibcode:1937RSPSA.161..406M. doi:10.1098/rspa.1937.0152. 10. Stevenson, A. C. (1938). "Flexure with Shear and Associated Torsion in Prisms of Uni-Axial and Asymmetric Cross-Sections". Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences. 237 (776): 181. ISSN 0080-4614. JSTOR 91357. 11. "A Mathematical Genius" (PDF). The British Journal of Nursing. 86: 221. August 1938. Archived (PDF) from the original on 19 April 2021. Retrieved 6 March 2021. 12. (NUWT), National Union of Women Teachers (30 September 1938). "A Woman's Triumph in Aero-Dynamics". p. 383. Archived from the original on 19 April 2021. Retrieved 6 April 2021. {{cite magazine}}: Cite magazine requires \|magazine= (help) 13. "The Paris News from Paris, Texas". The Paris News. Paris, Texas. 4 September 1938. p. 28. Retrieved 6 April 2021. 14. "Lead Daily Call from Lead, South Dakota". 30 August 1938. p. 4. Archived from the original on 19 April 2021. Retrieved 6 April 2021. 15. "University News". The Daily Telegraph. 2 July 1938. pp. [18]. ISSN 0307-1235. Archived from the original on 19 April 2021. Retrieved 6 April 2021. 16. Cambridge, University of (1941). Abstracts of Dissertations Approved for the Ph.D., M.Sc., and M.Litt. Degrees in the University of Cambridge. Cambridge University Press. Archived from the original on 19 April 2021. Retrieved 7 April 2021. 17. Board of Research Studies (12 March 1940). "Annual Report of the Board of Research Studies". Cambridge University Reporter. Cambridge University. LXX No.26 (3239): 631. 18. "Rosa M. Morris". Mathematics Genealogy Project. Archived from the original on 3 July 2020. Retrieved 5 April 2021. 19. "Morris, Rosa". mathshistory.st-andrews.ac.uk. Archived from the original on 19 April 2021. Retrieved 5 April 2021. 20. Malcolm Brown, B.; Lang, Jan; Wood, Ian G. (2012). "David Edmunds' Mathematical Work". Spectral theory, function spaces and inequalities : new techniques and recent trends. Basel. pp. ix–x. ISBN 978-3-0348-0263-5. OCLC 761201127.{{cite book}}: CS1 maint: location missing publisher (link) 21. Chisholm, John Stephen Roy; Morris, Rosa M (1964). Mathematical methods in physics. Amsterdam: North-Holland. OCLC 610944321. 22. Mullin, A. A. (1 January 1966). "Mathematical Methods in Physics. J. S. R. Chisholm and R. M. Morris". American Journal of Physics. 34 (1): 79. doi:10.1119/1.1972796. ISSN 0002-9505. Archived from the original on 1 January 2018. Retrieved 5 April 2021. 23. S., D. (1966). "Review of Mathematical Methods in Physics". Mathematics of Computation. 20 (93): 188–189. doi:10.2307/2004316. ISSN 0025-5718. JSTOR 2004316. S2CID 123322185. 24. Chisholm, Roy. "Cardiff 1954 – 62". Roy Chisholm. Archived from the original on 27 January 2019. Retrieved 5 April 2021. 25. Hobbs, Cathy (July 2015). "The first female head of department of mathematical sciences?" (PDF). Newsletter of the London Mathematical Society (449): 21–22. Archived (PDF) from the original on 24 January 2021. Retrieved 9 March 2021. 26. "History of Mathematics at Cardiff". Cardiff University. Retrieved 6 July 2022. 27. Society, London Mathematical (1976). List of Members. p. 32. 28. Williams, W. H. (1956). "Report of the Cardiff Branch, 1955–56". The Mathematical Gazette. 40 (332): iii–iv. doi:10.1017/S0025557200222171. ISSN 0025-5572. JSTOR 3609719. S2CID 185054236. 29. Williams, W. H. (1957). "Cardiff Branch: Report for the Session 1956-1957". The Mathematical Gazette. 41 (337): xix. doi:10.1017/S0025557200037128. ISSN 0025-5572. JSTOR 3609240. S2CID 184617261. 30. "Proceedings of the meetings held during the session 1983–1984". Mathematical Proceedings of the Cambridge Philosophical Society. 96 (3): 555–563. November 1984. Bibcode:1984MPCPS..96..555.. doi:10.1017/S0305004100062496. ISSN 1469-8064. S2CID 251092322. 31. "Items reviewed by Morris, Rosa M.". mathscinet.ams.org. Retrieved 6 April 2021. External links • Rosa M. Morris at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Norway • France • BnF data • Israel • United States • Netherlands Academics • CiNii • Mathematics Genealogy Project Other • IdRef	Wikipedia
Rosana Rodríguez-López Rosana Rodríguez-López is a Spanish mathematician known for her well-cited research publications applying fixed-point theorems to differential equations. She is a professor in the Department of Statistics, Mathematical Analysis and Optimisation at the University of Santiago de Compostela,[1] where she obtained her Ph.D. in 2005 under the supervision of Juan José Nieto Roig.[2] References 1. "Rosana Rodriguez Lopez", Directory, Department of Statistics, Mathematical Analysis and Optimisation, University of Santiago de Compostela, retrieved 2021-09-10 2. Rosana Rodríguez-López at the Mathematics Genealogy Project External links • Rosana Rodríguez-López publications indexed by Google Scholar Authority control International • VIAF National • Germany • Poland Academics • DBLP • Google Scholar • MathSciNet • Mathematics Genealogy Project • ORCID • Scopus • zbMATH	Wikipedia
Rosati involution In mathematics, a Rosati involution, named after Carlo Rosati, is an involution of the rational endomorphism ring of an abelian variety induced by a polarization. Let $A$ be an abelian variety, let ${\hat {A}}=\mathrm {Pic} ^{0}(A)$ be the dual abelian variety, and for $a\in A$, let $T_{a}:A\to A$ be the translation-by-$a$ map, $T_{a}(x)=x+a$. Then each divisor $D$ on $A$ defines a map $\phi _{D}:A\to {\hat {A}}$ via $\phi _{D}(a)=[T_{a}^{}D-D]$. The map $\phi _{D}$ is a polarization if $D$ is ample. The Rosati involution of $\mathrm {End} (A)\otimes \mathbb {Q} $ relative to the polarization $\phi _{D}$ sends a map $\psi \in \mathrm {End} (A)\otimes \mathbb {Q} $ to the map $\psi '=\phi _{D}^{-1}\circ {\hat {\psi }}\circ \phi _{D}$, where ${\hat {\psi }}:{\hat {A}}\to {\hat {A}}$ is the dual map induced by the action of $\psi ^{}$ on $\mathrm {Pic} (A)$. Let $\mathrm {NS} (A)$ denote the Néron–Severi group of $A$. The polarization $\phi _{D}$ also induces an inclusion $\Phi :\mathrm {NS} (A)\otimes \mathbb {Q} \to \mathrm {End} (A)\otimes \mathbb {Q} $ :\mathrm {NS} (A)\otimes \mathbb {Q} \to \mathrm {End} (A)\otimes \mathbb {Q} } via $\Phi _{E}=\phi _{D}^{-1}\circ \phi _{E}$. The image of $\Phi $ is equal to $\{\psi \in \mathrm {End} (A)\otimes \mathbb {Q} :\psi '=\psi \}$ :\psi '=\psi \}} , i.e., the set of endomorphisms fixed by the Rosati involution. The operation $E\star F={\frac {1}{2}}\Phi ^{-1}(\Phi _{E}\circ \Phi _{F}+\Phi _{F}\circ \Phi _{E})$ then gives $\mathrm {NS} (A)\otimes \mathbb {Q} $ the structure of a formally real Jordan algebra. References • Mumford, David (2008) [1970], Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.: American Mathematical Society, ISBN 978-81-85931-86-9, MR 0282985, OCLC 138290 • Rosati, Carlo (1918), "Sulle corrispondenze algebriche fra i punti di due curve algebriche.", Annali di Matematica Pura ed Applicata (in Italian), 3 (28): 35–60, doi:10.1007/BF02419717, S2CID 121620469	Wikipedia
Rose (mathematics) In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied them, Guido Grandi, between the years 1723 and 1728.[1] For the topological usage, see Rose (topology). General overview Specification A rose is the set of points in polar coordinates specified by the polar equation $r=a\cos(k\theta )$[2] or in Cartesian coordinates using the parametric equations $x=r\cos(\theta )=a\cos(k\theta )\cos(\theta )$ $y=r\sin(\theta )=a\cos(k\theta )\sin(\theta )$. Roses can also be specified using the sine function.[3] Since $\sin(k\theta )=\cos \left(k\theta -{\frac {\pi }{2}}\right)=\cos \left(k\left(\theta -{\frac {\pi }{2k}}\right)\right)$. Thus, the rose specified by $\,r=a\sin(k\theta )$ is identical to that specified by $\,r=a\cos(k\theta )$ rotated counter-clockwise by $\pi /2k$ radians, which is one-quarter the period of either sinusoid. Since they are specified using the cosine or sine function, roses are usually expressed as polar coordinate (rather than Cartesian coordinate) graphs of sinusoids that have angular frequency of $k$ and an amplitude of $a$ that determine the radial coordinate $(r)$ given the polar angle $(\theta )$ (though when $k$ is a rational number, a rose curve can be expressed in Cartesian coordinates since those can be specified as algebraic curves[4]). General properties Roses are directly related to the properties of the sinusoids that specify them. Petals • Graphs of roses are composed of petals. A petal is the shape formed by the graph of a half-cycle of the sinusoid that specifies the rose. (A cycle is a portion of a sinusoid that is one period $T=2\pi /k$ long and consists of a positive half-cycle, the continuous set of points where $r\geq 0$ and is $T/2=\pi /k$ long, and a negative half-cycle is the other half where $r\leq 0$.) • The shape of each petal is same because the graphs of half-cycles have the same shape. The shape is given by the positive half-cycle with crest at $(a,0)$ specified by $r=a\cos(k\theta )$ (that is bounded by the angle interval $-T/4\leq \theta \leq T/4$). The petal is symmetric about the polar axis. All other petals are rotations of this petal about the pole, including those for roses specified by the sine function with same values for $a$ and $k$.[5] • Consistent with the rules for plotting points in polar coordinates, a point in a negative half-cycle cannot be plotted at its polar angle because its radial coordinate $r$ is negative. The point is plotted by adding $\pi $ radians to the polar angle with a radial coordinate $\|r\|$. Thus, positive and negative half-cycles can be coincident in the graph of a rose. In addition, roses are inscribed in the circle $r=a$. • When the period $T$of the sinusoid is less than or equal to $4\pi $, the petal's shape is a single closed loop. A single loop is formed because the angle interval for a polar plot is $2\pi $ and the angular width of the half-cycle is less than or equal to $2\pi $. When $T>4\pi $ (or $\|k\|<1/2$) the plot of a half-cycle can be seen as spiraling out from the pole in more than one circuit around the pole until plotting reaches the inscribed circle where it spirals back to the pole, intersecting itself and forming one or more loops along the way. Consequently, each petal forms 2 loops when $4\pi <T\leq 8\pi $ (or $1/4\leq \|k\|<1/2$), 3 loops when $8\pi <T\leq 12\pi $ (or $1/6\leq \|k\|<1/4$), etc. Roses with only one petal with multiple loops are observed for $k=1/3,k=1/5,k=1/7,etc.$ (See the figure in the introduction section.) • A rose's petals will not intersect each other when the angular frequency $k$ is a non-zero integer; otherwise, petals intersect one another. Symmetry All roses display one or more forms of symmetry due to the underlying symmetric and periodic properties of sinusoids. • A rose specified as $r=a\cos(k\theta )$ is symmetric about the polar axis (the line $\theta =0$) because of the identity $a\cos(k\theta )=a\cos(-k\theta )$ that makes the roses specified by the two polar equations coincident. • A rose specified as $r=a\sin(k\theta )$ is symmetric about the vertical line $\theta =\pi /2$ because of the identity $a\sin(k\theta )=a\sin(\pi -k\theta )$ that makes the roses specified by the two polar equations coincident. • Only certain roses are symmetric about the pole. • Individual petals are symmetric about the line through the pole and the petal's peak, which reflects the symmetry of the half-cycle of the underlying sinusoid. Roses composed of a finite number of petals are, by definition, rotationally symmetric since each petal is the same shape with successive petals rotated about the same angle about the pole. Roses with non-zero integer values of k When $k$ is a non-zero integer, the curve will be rose-shaped with $2k$ petals if $k$ is even, and $k$ petals when $k$ is odd.[6] The properties of these roses are a special case of roses with angular frequencies $(k)$ that are rational numbers discussed in the next section of this article. • The rose is inscribed in the circle $r=a$, corresponding to the radial coordinate of all of its peaks. • Because a polar coordinate plot is limited to polar angles between $0$ and $2\pi $, there are $2\pi /T=k$ cycles displayed in the graph. No additional points need be plotted because the radial coordinate at $\theta =0$ is the same value at $\theta =2\pi $ (which are crests for two different positive half-cycles for roses specified by the cosine function). • When $k$ is even (and non-zero), the rose is composed of $2k$ petals, one for each peak in the $2\pi $ interval of polar angles displayed. Each peak corresponds to a point lying on the circle $r=a$. Line segments connecting successive peaks will form a regular polygon with an even number of vertices that has its center at the pole and a radius through each peak, and likewise: • The roses are symmetric about the pole. • The roses are symmetric about each line through the pole and a peak (through the "middle" a petal) with the polar angle between the peaks of successive petals being $2\pi /2k=\pi /k$ radians. Thus, these roses have rotational symmetry of order $2k$. • The roses are symmetric about each line that bisects the angle between successive peaks, which corresponds to half-cycle boundaries and the apothem of the corresponding polygon. • When $k$ is odd, the rose is composed of the $k$ petals, one for each crest (or trough) in the $2\pi $ interval of polar angles displayed. Each peak corresponds to a point lying on the circle $r=a$. These rose's positive and negative half-cycles are coincident, which means that in graphing them, only the positive half-cycles or only the negative half-cycles need to plotted in order to form the full curve. (Equivalently, a complete curve will be graphed by plotting any continuous interval of polar angles that is $\pi $ radians long such as $\theta =0$ to $\theta =\pi $.[7]) Line segments connecting successive peaks will form a regular polygon with an odd number of vertices, and likewise: • The roses are symmetric about each line through the pole and a peak (through the "middle" a petal) with the polar angle between the peaks of successive petals being $2\pi /k$ radians. Thus, these roses have rotational symmetry of order $k$. • The rose’s petals do not overlap. • The roses can be specified by algebraic curves of order $k+1$ when k is odd, and $2(k+1)$ when k is even.[8] The circle A rose with $k=1$ is a circle that lies on the pole with a diameter that lies on the polar axis when $r=a\cos(\theta )$. The circle is the curve's single petal. (See the circle being formed at the end of the next section.) In Cartesian coordinates, the equivalent cosine and sine specifications are $(x-a/2)^{2}+y^{2}=(a/2)^{2}$ and $x^{2}+(y-a/2)^{2}=(a/2)^{2}$, respectively. The quadrifolium A rose with $k=2$ is called a quadrifolium because it has 4 petals. In Cartesian Coordinates the cosine and sine specifications are $(x^{2}+y^{2})^{3}=a^{2}(x^{2}-y^{2})^{2}$ and $(x^{2}+y^{2})^{3}=4(axy)^{2}$, respectively. The trifolium A rose with $k=3$ is called a trifolium[9] because it has 3 petals. The curve is also called the Paquerette de Mélibée. In Cartesian Coordinates the cosine and sine specifications are $(x^{2}+y^{2})^{2}=a(x^{3}-3xy^{2})$ and $(x^{2}+y^{2})^{2}=-a(x^{3}-3xy^{2})$, respectively.[10] (See the trifolium being formed at the end of the next section.) Total and petal areas The total area of a rose with polar equation of the form $r=a\cos(k\theta )$ or $r=a\sin(k\theta )\,$, where $k$ is a non-zero integer, is ${\frac {1}{2}}\int _{0}^{2\pi }(a\cos(k\theta ))^{2}\,d\theta ={\frac {a^{2}}{2}}\left(\pi +{\frac {\sin(4k\pi )}{4k}}\right)={\frac {\pi a^{2}}{2}}$, when $k$ is even; and ${\frac {1}{2}}\int _{0}^{\pi }(a\cos(k\theta ))^{2}\,d\theta ={\frac {a^{2}}{2}}\left({\frac {\pi }{2}}+{\frac {\sin(2k\pi )}{4k}}\right)={\frac {\pi a^{2}}{4}}$, when $k$ is odd.[11] When $k$ is even, there are $2k$ petals; and when $k$ is odd, there are $k$ petals, so the area of each petal is ${\frac {\pi a^{2}}{4k}}$. Roses with rational number values for k In general, when $k$ is a rational number in the irreducible fraction form $k=n/d$, where $n$ and $d$ are non-zero integers, the number of petals is the denominator of the expression $1/2-1/(2k)=(n-d)/2n$.[12] This means that the number of petals is $n$ if both $n$ and $d$ are odd, and $2n$ otherwise.[13] • In the case when both $n$ and $d$ are odd, the positive and negative half-cycles of the sinusoid are coincident. The graph of these roses are completed in any continuous interval of polar angles that is $d\pi $ long.[14] • When $n$ is even and $d$ is odd, or visa versa, the rose will be completely graphed in a continuous polar angle interval $2d\pi $ long.[15] Furthermore, the roses are symmetric about the pole for both cosine and sine specifications.[16] • In addition, when $n$ is odd and $d$ is even, roses specified by the cosine and sine polar equations with the same values of $a$ and $k$ are coincident. For such a pair of roses, the rose with the sine function specification is coincident with the crest of the rose with the cosine specification at on the polar axis either at $\theta =d\pi /2$ or at $\theta =3d\pi /2$. (This means that roses $r=a\cos(k\theta )$ and $r=a\sin(k\theta )$ with non-zero integer values of $k$ are never coincident.) • The rose is inscribed in the circle $r=a$, corresponding to the radial coordinate of all of its peaks. The Dürer folium A rose with $k=1/2$ is called the Dürer folium, named after the German painter and engraver Albrecht Dürer. The roses specified by $r=a\cos(\theta /2)$ and $r=a\sin(\theta /2)$ are coincident even though $a\cos(\theta /2)\neq a\sin(\theta /2)$. In Cartesian Coordinates the rose is specified as $(x^{2}+y^{2})[2(x^{2}+y^{2})-a^{2}]^{2}=a^{4}x^{2}$.[17] The Dürer folium is also a trisectrix, a curve that can be used to trisect angles. The limaçon trisectrix A rose with $k=1/3$ is a limaçon trisectrix that has the property of trisectrix curves that can be used to trisect angles. The rose has a single petal with two loops. (See the animation below.) Examples of roses $r=\cos(k\theta )$ created using gears with different ratios. The rays displayed are the polar axis and $\theta =\pi /2$. Graphing starts at $\theta =2\pi $ when $k$ is an integer, $\theta =2d\pi $ otherwise, and proceeds clock-wise to $\theta =0$. The circle, k=1 (n=1, d=1). The rose is complete when $\theta =\pi $ is reached (one-half revolution of the lighter gear). The limaçon trisectrix, k=1/3 (n=1, d=3), has one petal with two loops. The rose is complete when $\theta =3\pi $ is reached (one and one-half revolution of the lighter gear). The trifolium, k=3 (n=3, d=1). The rose is complete when $\theta =\pi $ is reached (one-half revolution of the lighter gear). The 8 petals of the rose with k=4/5 (n=4, d=5) is each, a single loop that intersect other petals. The rose is symmetric about the pole. The rose is complete at $\theta =0$ (five revolutions of the lighter gear). Roses with irrational number values for k A rose curve specified with an irrational number for $k$ has an infinite number of petals[18] and will never complete. For example, the sinusoid $r=a\cos(\pi \theta )$ has a period $T=2$, so, it has a petal in the polar angle interval $-1/2\leq \theta \leq 1/2$ with a crest on the polar axis; however there is no other polar angle in the domain of the polar equation that will plot at the coordinates $(a,0)$. Overall, roses specified by sinusoids with angular frequencies that are irrational constants form a dense set (i.e., they come arbitrarily close to specifying every point in the disk $r\leq a$). See also • Limaçon trisectrix - has the same shape as the rose with k = 1/3. • Quadrifolium – a rose curve where k = 2. • Maurer rose • Rose (topology) • Sectrix of Maclaurin • Spirograph Notes 1. O'Connor, John J.; Robertson, Edmund F., "Rhodonea", MacTutor History of Mathematics Archive, University of St Andrews 2. Mathematical Models by H. Martyn Cundy and A.P. Rollett, second edition, 1961 (Oxford University Press), p. 73. 3. "Rose (Mathematics)". Retrieved 2021-02-02. 4. Robert Ferreol. "Rose". Retrieved 2021-02-03. 5. Xah Lee. "Rose Curve". Retrieved 2021-02-12. 6. Eric W. Weisstein. "Rose (Mathematics)". Wolfram MathWorld. Retrieved 2021-02-05. 7. "Number of Petals of Odd Index Rhodonea Curve". ProofWiki.org. Retrieved 2021-02-03. 8. Robert Ferreol. "Rose". Retrieved 2021-02-03. 9. "Trifolium". Retrieved 2021-02-02. 10. Eric W. Weisstein. "Paquerette de Mélibée". Wolfram MathWorld. Retrieved 2021-02-05. 11. Robert Ferreol. "Rose". Retrieved 2021-02-03. 12. Jan Wassenaar. "Rhodonea". Retrieved 2021-02-02. 13. Robert Ferreol. "Rose". Retrieved 2021-02-05. 14. Xah Lee. "Rose Curve". Retrieved 2021-02-12. 15. Xah Lee. "Rose Curve". Retrieved 2021-02-12. 16. Jan Wassenaar. "Rhodonea". Retrieved 2021-02-02. 17. Robert Ferreol. "Dürer Folium". Retrieved 2021-02-03. 18. Eric W. Weisstein. "Rose (Mathematics)". Wolfram MathWorld. Retrieved 2021-02-05. External links Applet to create rose with k parameter • Visual Dictionary of Special Plane Curves Xah Lee • Interactive example with JSXGraph • Interactive example with p5	Wikipedia

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