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Wikipedia:Yuri Petunin#0
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Yuri Ivanovich Petunin (Russian: Юрий Иванович Петунин) was a Soviet and Ukrainian mathematician. Petunin was born in the city of Michurinsk (USSR) on September 30, 1937. After graduating from the Tambov State Pedagogical Institute he began his studies at Voronezh State University under the supervision of S.G Krein. He completed his postgraduate studies in 1962, and in 1968 he received his Doctor of Science Degree, the highest scientific degree awarded in the Soviet Union. In 1970 he joined the faculty of the computational mathematics department at Kyiv State University. Yuri Petunin is highly regarded for his results in functional analysis. He developed the theory of Scales in Banach spaces, the theory of characteristics of linear manifolds in conjugate Banach spaces, and with S.G. Krein and E.M. Semenov contributed to the theory of interpolation of linear operators. He solved Banach's problem of norming subspaces in conjugate Banach spaces as well as a problem posted by Calderón and Lions concerning interpolation in factor spaces. In addition to his work in functional analysis, Professor Petunin made significant contributions to pattern recognition and mathematical statistics. He also worked on developing differential diagnostics for oncological disease. The Vysochanskij–Petunin inequality that bears his name formally justifies the so-called 3-sigma rule for unimodal distributions, a rule that has been broadly used in statistics since the time of Gauss. In the area of pattern recognition he developed a theory of linear discriminant rules where he investigated the problems of linear separability of any number of sets in n-dimensional space. In his later years Yuri Petunin returned to the area of functional analysis where he had begun his scientific research. Together with his colleagues at the department of computational mathematics, he successfully worked toward a solution of Hilbert's 20th problem. == See also == Vysochanskii–Petunin inequality == References ==
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Wikipedia:Yurii Egorov#0
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Yurii (or Yuri) Vladimirovich Egorov (Юрий Владимирович Егоров, 11 July 1938 – 6 October 2018) was a Russian-Soviet mathematician who specialised in differential equations. == Life and career == In 1960 he completed his undergraduate studies at the Mechanics and Mathematics Faculty of Moscow State University (MSU). In 1963 from MSU he received his Ph.D. with the thesis "Некоторые задачи теории оптимального управления в бесконечномерных пространствах" ("Some Problems of Optimal Control Theory in Infinite-Dimensional Spaces"). In 1970 from MSU he received his Russian doctorate of sciences (Doctor Nauk) with thesis: "О локальных свойствах псевдодифференциальных операторов главного типа" ("Local Properties of Pseudodifferential Operators of Principal Type"). He was employed at MSU from 1961 to 1992, and he was a full professor in the Department of Differential Equations of the Mechanics and Mathematics Faculty there from 1973 to 1992. Since 1992 he has been a professor of mathematics at Paul Sabatier University (Toulouse III). Egorov's research deals with differential equations and applications in mathematical physics, spectral theory, and optimal control theory. In 1970 he was an Invited Speaker of the ICM in Nice. Egorov died in Toulouse, France on 6 October 2018, at the age of 80. == Awards == 1981 — Lomonosov Memorial Prize (established in 1944) — for his series of publications on "Субэллиптические операторы и их применения к исследованию краевых задач" (Subelliptic operators and their applications to the study of boundary value problems) 1988 — USSR State Prize (with several co-authors) — for their series of publications (1958–1985) on "Исследования краевых задач для дифференциальных операторов и их приложения в математической физике" (Research on boundary value problems and their applications in mathematical physics) 1998 — Petrovsky Award (jointly with V. A. Kondratiev) for their series of publications on "Исследование спектра эллиптических операторов" (The study of the spectra of elliptic operators) == Selected publications == === Articles === "The canonical transformations of pseudodifferential operators." Uspekhi Matematicheskikh Nauk 24, no. 5 (1969): 235–236. "On the solubility of differential equations with simple characteristics." Russian Mathematical Surveys 26, no. 2 (1971): 113. with Mikhail Aleksandrovich Shubin: "Linear partial differential equations. Foundations of the classical theory." Itogi Nauki i Tekhniki. Seriya" Sovremennye Problemy Matematiki. Fundamental'nye Napravleniya" 30 (1988): 5–255. "A contribution to the theory of generalized functions." Russian Mathematical Surveys 45, no. 5 (1990): 1. with Vladimir Aleksandrovich Kondrat'ev and Olga Arsen'evna Oleynik: "Asymptotic behaviour of the solutions of non-linear elliptic and parabolic systems in tube domains." Sbornik: Mathematics 189, no. 3 (1998): 359–382. Victor A. Galaktionov, Vladimir A. Kondratiev, and Stanislav I. Pohozaev: "On the necessary conditions of global existence to a quasilinear inequality in the half-space." Comptes Rendus de l'Académie des Sciences-Series I-Mathematics 330, no. 2 (2000): 93–98. === Books === with Vladimir A. Kondratiev: On spectral theory of elliptic operators. Operator theory, advances and applications; vol. 89. Basel; Boston: Birkhäuser Verlag. 1996. ISBN 9783764353902; x+328 pages{{cite book}}: CS1 maint: postscript (link) with Bert-Wolfgang Schulze: Pseudo-differential operators, singularities, applications. Operator theory, advances and applications; vol. 93. Basel; Boston: Birkhäuser Verlag. 1997. ISBN 9783764354848; xiii+349 pages{{cite book}}: CS1 maint: postscript (link) == References ==
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Wikipedia:Yurii Nesterov#0
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Yurii Nesterov is a Russian mathematician, an internationally recognized expert in convex optimization, especially in the development of efficient algorithms and numerical optimization analysis. He is currently a professor at the University of Louvain (UCLouvain). == Biography == In 1977, Yurii Nesterov graduated in applied mathematics at Moscow State University. From 1977 to 1992 he was a researcher at the Central Economic Mathematical Institute of the Russian Academy of Sciences. Since 1993, he has been working at UCLouvain, specifically in the Department of Mathematical Engineering from the Louvain School of Engineering, Center for Operations Research and Econometrics. In 2000, Nesterov received the Dantzig Prize. In 2009, Nesterov won the John von Neumann Theory Prize. In 2016, Nesterov received the EURO Gold Medal. In 2023, Yurii Nesterov and Arkadi Nemirovski received the WLA Prize in Computer Science or Mathematics, "for their seminal work in convex optimization theory". == Academic work == Nesterov is most famous for his work in convex optimization, including his 2004 book, considered a canonical reference on the subject. His main novel contribution is an accelerated version of gradient descent that converges considerably faster than ordinary gradient descent (commonly referred as Nesterov momentum, Nesterov Acceleration or Nesterov accelerated gradient, in short — NAG). This method, sometimes called "FISTA", was further developed by Beck & Teboulle in their 2009 paper "A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems". His work with Arkadi Nemirovski in their 1994 book is the first to point out that the interior point method can solve convex optimization problems, and the first to make a systematic study of semidefinite programming (SDP). Also in this book, they introduced the self-concordant functions which are useful in the analysis of Newton's method. == References == == External links == Official website
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Wikipedia:Yuriy Drozd#0
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Yuriy Drozd (Ukrainian: Юрій Анатолійович Дрозд; born October 15, 1944) is a Ukrainian mathematician working primarily in algebra. He is a Corresponding Member of the National Academy of Sciences of Ukraine and head of the Department of Algebra and Topology at the Institute of Mathematics of the National Academy of Sciences of Ukraine. == Biography == Drozd graduated from Kyiv University in 1966, pursuing a postgraduate degree at the Institute of Mathematics of the National Academy of Sciences of Ukraine in 1969. His PhD dissertation On Some Questions of the Theory of Integral Representations (1970) was supervised by Igor Shafarevich. From 1969 to 2006 Drozd worked at the Faculty of Mechanics and Mathematics at Kyiv University (at first as lecturer, then as associate professor and full professor). From 1980 to 1998 he headed the Department of Algebra and Mathematical Logic. Since 2006 he has been the head of the Department of Algebra and Topology (until 2014 - the Department of Algebra) of the Institute of Mathematics of the National Academy of Sciences of Ukraine. His doctoral students include Volodymyr Mazorchuk. Since 2022, Drozd has taught at Harvard University. == References == == External links == Personal site. Oberwolfach Photo Collection. Yuriy Drozd, Introduction to Algebraic Geometry (course lecture notes, University of Kaiserslautern). Yuriy Drozd, Vector Bundles over Projective Curves. Yuriy Drozd, General Properties of Surface Singularities. Drozd, Yuriy; Kirichenko, Vladimir (1994). Finite-Dimensional Algebras. Springer. ISBN 978-3-642-76244-4.
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Wikipedia:Yury Drobyshev#0
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Yury Aleksandrovich Drobyshev (Russian: Юрий Александрович Дробышев; 24 June 1955 – 22 April 2024) was a Soviet and Russian scientist, Doctor of Education, professor and rector of Kaluga State University from 2004 to 2010. == Biography == Drobyshev was born on 24 June 1955, in Abakan into a family of employees. In 1972, after graduating from school, he entered the Khakassian State University, which he graduated in 1976 with a degree in mathematics. After graduation, he was retained for teaching work and sent for a one-year internship at Leningrad State University. Starting in 1977, he worked as an assistant and senior lecturer at the Department of Higher Mathematics of the Abakan branch of the Krasnoyarsk State Technical University. In 1984, he went to work at the Abakan State Pedagogical Institute. From 1986 to 1989, he studied in graduate school at the Department of Mathematical Analysis of the Moscow State Correspondence Pedagogical Institute. After graduating from graduate school, from 1990 to 2011, he worked at Kaluga State University and rose through various academic ranking grades. Subsequently, Drobyshev earned the position of Professor of the Department of Higher Mathematics and Statistics of the Financial University under the Government of the Russian Federation. Drobyshev wrote over 110 scientific and educational works, including school textbooks and teaching aids for universities, certified by the Ministry of Education and Science of the Russian Federation. Drobyshev died on 22 April 2024, at the age of 68. == References == == External links == Официальный сайт университета
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Wikipedia:Yury Yershov#0
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Yury Leonidovich Yershov (Russian: Ю́рий Леони́дович Ершо́в, born 1 May 1940 [1]) is a Soviet and Russian mathematician. Yury Yershov was born in 1940 in Novosibirsk. In 1958 he entered the Tomsk State University and in 1963 graduated from the Mathematical Department of the Novosibirsk State University. In 1964 he successfully defended his PhD thesis "Decidable and Undecidable Theories" (advisor Anatoly Maltsev). In 1966 he successfully defended his DrSc thesis "Elementary Theory of Fields" (Элементарные теория полей). Apart from being a mathematician, Yershov was a member of the Communist Party and had different distinguished administrative duties in Novosibirsk State University. Yershov has been accused of antisemitic practices, and his visit to the U.S. in 1980 drew public protests by a number of U.S. mathematicians. Yershov himself denied the validity of these accusations. Yury Yershov is a member of the Russian Academy of Sciences, professor emeritus of Novosibirsk State University and a former Rector of the Novosibirsk State University. He has been working at the Sobolev Institute of Mathematics since 1963. Currently he is Director of this Institute (since 2003). In 1968 he received the title of Full Professor. In 1970 he was elected to be a Correspondent Member of the Academy of Sciences of the USSR, in 1990 he became Full Member (Academician) of the Russian Academy of Sciences. In 1964–2002 he worked at the Novosibirsk State University (as second job): in 1968–2002 as a professor, in addition, in 1973–1976 he was dean of the Mathematical Department of the Novosibirsk State University and rector of this university in 1985–1993. He is editor in chief of the Siberian Mathematical Journal and editor in chief of the journal Algebra i Logika (Algebra and Logic). [2][3] His basic scientific interests are: algebra, field theory, mathematical logic, algorithm theory, model theory, constructive models, computer science and philosophical aspects of mathematics. He proved decidability of the elementary theory of the field of p-adic numbers (independently proven by J. Ax and S. Kochen), undecidability of the elementary theory of finite symmetric groups, decidability of the elementary theory of relatively complemented distributive lattices. Yury Yershov is a laureate of Maltsev's Award of the Russian Academy of Sciences (1992), Russian State Award in the area of Science and Technics (2002, 2003), Lavrentyev's Foundation Award (2007), and is decorated with several Russian State Orders.[4] In 2013 he won the Demidov Prize. In 2023 he was awarded the Lobachevsky Prize. He was an invited speaker at the International Congress of Mathematicians in 1966 at Moscow with the talk Elementary theories of fields and in 1970 at Nice with the talk La theorie des enumerations. == Selected works == Yershov, Yury (1996). Definability and Computability. Springer. ISBN 9780306110399. Yershov, Yury (2001). Multi-Valued Fields. Springer. ISBN 9780306110689. Yershov, Yury (2004). "Local class field theory". St. Petersburg Math. J. 15 (6): 837–846. doi:10.1090/S1061-0022-04-00834-9. MR 2044630. Yershov, Yury (2008). "Tame and purely wild extensions of valued fields". St. Petersburg Math. J. 19 (5): 765–773. doi:10.1090/S1061-0022-08-01019-4. MR 2381943. == References == Yuri L. Ershov at the Mathematics Genealogy Project
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Wikipedia:Yusu Wang#0
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Yusu Wang is a Chinese computer scientist and mathematician who works as a professor at the Halıcıoğlu Data Science Institute at the University of California, San Diego . Her research concerns computational geometry and computational topology, including results on discrete Laplace operators, curve simplification, and Fréchet distance. == Education and career == Wang graduated from Tsinghua University in 1998. She completed her Ph.D. in computer science at Duke University in 2004. Her dissertation, Geometric and Topological Methods in Protein Structure Analysis, was jointly supervised by Pankaj K. Agarwal and Herbert Edelsbrunner. After postdoctoral research with Leonidas J. Guibas at Stanford University, Wang joined the faculty of the Ohio State University in 2005, and she was promoted to the rank of full professor there in 2017. She moved to her current position at the University of California, San Diego in 2020. == Service == Wang is on the editorial boards of the SIAM Journal on Computing and Journal of Computational Geometry. With Gill Barequet, Wang was program co-chair of the 2019 Symposium on Computational Geometry. == References == == External links == Home page Yusu Wang publications indexed by Google Scholar
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Wikipedia:Yutaka Taniyama#0
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Yutaka Taniyama (谷山 豊, Taniyama Yutaka, 12 November 1927 – 17 November 1958) was a Japanese mathematician known for the Taniyama–Shimura conjecture. == Life == Taniyama was born on 22 November 1927 in Kisai, a town in Saitama. He was the sixth of eight children born to a doctor's family. He studied at Urawa High School (present-day Saitama University) after graduating from Fudouoka Middle School. He suspended his college for two years due to his medical condition, but finally graduated in 1950. During Taniyama's college years, he aspired to be a mathematician after reading Teiji Takagi's work. In 1958, Taniyama worked as an Associate Professor after years of assistant at the University of Tokyo. He also obtained his doctorate from the University in May. In October, Taniyama was engaged to be married to Misako Suzuki (鈴木美佐子, Suzuki Misako), while the Institute for Advanced Study in Princeton, New Jersey offered him a position. On 17 November 1958, Taniyama committed suicide by poisoning himself with gas. He left a note explaining how far he had progressed with his teaching duties, and apologizing to his colleagues for the trouble he was causing them. The first paragraph of his suicide note read (quoted in Shimura, 1989): Until yesterday I had no definite intention of killing myself. But more than a few must have noticed that lately I have been tired both physically and mentally. As to the cause of my suicide, I don't quite understand it myself, but it is not the result of a particular incident, nor of a specific matter. Merely may I say, I am in the frame of mind that I lost confidence in my future. There may be someone to whom my suicide will be troubling or a blow to a certain degree. I sincerely hope that this incident will cast no dark shadow over the future of that person. At any rate, I cannot deny that this is a kind of betrayal, but please excuse it as my last act in my own way, as I have been doing my own way all my life. Although his note is mostly enigmatic it does mention tiredness and a loss of confidence in his future. Taniyama's ideas had been criticized as unsubstantiated and his behavior had occasionally been deemed peculiar. Goro Shimura mentioned that he suffered from depression. About a month later, Suzuki also committed suicide by gas, leaving a note reading: "We promised each other that no matter where we went, we would never be separated. Now that he is gone, I must go too in order to join him." After Taniyama's death, Goro Shimura stated that: He was always kind to his colleagues, especially to his juniors, and he genuinely cared about their welfare. He was the moral support of many of those who came into mathematical contact with him, including of course myself. Probably he was never conscious of this role he was playing. But I feel his noble generosity in this respect even more strongly now than when he was alive. And yet nobody was able to give him any support when he desperately needed it. Reflecting on this, I am overwhelmed by the bitterest grief. == Contribution == Taniyama was best known for conjecturing, in modern language, automorphic properties of L-functions of elliptic curves over any number field. A partial and refined case of this conjecture for elliptic curves over rationals is called the Taniyama–Shimura conjecture or the modularity theorem whose statement he subsequently refined in collaboration with Goro Shimura. The names Taniyama, Shimura and Weil have all been attached to this conjecture, but the idea is essentially due to Taniyama. Taniyama's interests were in algebraic number theory. His work has been influenced by André Weil, who had met Taniyama during the symposiums on algebraic number theory in 1955, in which he became famous after proposing his problems at it. Taniyama's problems proposed in 1955 form the basis of a Taniyama–Shimura conjecture, that "every elliptic curve defined over the rational field is a factor of the Jacobian of a modular function field". In 1986, Ken Ribet proved that if the Taniyama–Shimura conjecture held, then so would Fermat's Last Theorem, which inspired Andrew Wiles to work for a number of years in secrecy on it, and to prove enough of it to prove Fermat's Last Theorem. Owing to the pioneering contribution of Wiles and the efforts of a number of mathematicians, the Taniyama–Shimura conjecture was finally proven in 1999. The original Taniyama conjecture for elliptic curves over arbitrary number fields remains open. Goro Shimura stated: Taniyama was not a very careful person as a mathematician. He made a lot of mistakes. But he made mistakes in a good direction and so eventually he got right answers. I tried to imitate him, but I found out that it is very difficult to make good mistakes. == See also == Taniyama group Taniyama's problems == Notes == == Publications == Shimura, Goro; Taniyama, Yutaka (1961), Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, Tokyo: The Mathematical Society of Japan, MR 0125113 This book is hard to find, but an expanded version was later published as Shimura, Goro (1997). Abelian Varieties with Complex Multiplication and Modular Functions (Hardcover ed.). Princeton University Press. ISBN 978-0-691-01656-6. == References == Shimura, Goro (1989), "Yutaka Taniyama and his time. Very personal recollections", The Bulletin of the London Mathematical Society, 21 (2): 186–196, doi:10.1112/blms/21.2.186, ISSN 0024-6093, MR 0976064 Singh, Simon (hardcover, 1998). Fermat's Enigma. Bantam Books. ISBN 0-8027-1331-9 (previously published under the title Fermat's Last Theorem). Weil, André, "Y. Taniyama", Sugaku-no Ayumi, 6 (4): 21–22, Reprinted in Weil's collected works, volume II == External links == O'Connor, John J.; Robertson, Edmund F., "Yutaka Taniyama", MacTutor History of Mathematics Archive, University of St Andrews
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Wikipedia:Yuval Peres#0
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Yuval Peres (Hebrew: יובל פרס; born 5 October 1963) is an Israeli mathematician best known for his research in probability theory, ergodic theory, mathematical analysis, theoretical computer science, and in particular for topics such as fractals and Hausdorff measure, random walks, Brownian motion, percolation and Markov chain mixing times. Peres has been accused of sexual harassment by several female scientists. == Education and career == Peres was born in Israel and obtained his Ph.D. at the Hebrew University of Jerusalem in 1990 under the supervision of Hillel Furstenberg. After his Ph.D. Peres had postdoctoral positions at Stanford and Yale. In 1993 Peres joined the statistics department at UC Berkeley. He later became a professor in both the mathematics and statistics departments. He was also a professor at the Hebrew University. In 2006 Peres joined the Theory Group of Microsoft Research. By 2011 he was principal researcher at Microsoft Research and manager of the Microsoft Research Theory Group, an affiliate professor of mathematics at the University of Washington and an adjunct professor at the University of California, Berkeley. == Recognition == Peres was awarded the Rollo Davidson Prize in 1995 and the Loève Prize in 2001. The work that led to the Loève Prize was surveyed in the Notices of the American Mathematical Society: "A key breakthrough was the observation that certain (hard to prove) intersection properties for Brownian motion and random walks are in fact equivalent to (easier to prove) survival properties of branching processes. This led ultimately to deep work on sample path properties of Brownian motion; for instance, on the fractal dimension of the frontier of two-dimensional Brownian motion and precise study of its thick and thin points and cover times." Peres was an invited speaker at the International Congress of Mathematicians in 2002. In 2011, he was a co-recipient of the David P. Robbins Prize for work on the maximum overhang problem. That year he also delivered the Paul Turán Memorial Lecture. In 2012 he became a fellow of the American Mathematical Society. In 2016, he was elected a foreign associate of the National Academy of Sciences. In July 2017, he was a plenary lecturer at the Mathematical Congress of the Americas. == Allegations of sexual harassment == Peres has been accused of sexual harassment by several female scientists, including Dana Moshkovitz, Anima Anandkumar and Lisha Li. Moshkovitz said she was harassed by Peres on an informal job interview and she reported this to the Microsoft Theory Group. She also said that Peres was promoted shortly after her report. Peres resigned from an affiliate position at the University of Washington in 2012. The university said he resigned "after receiving notice that the university would be investigating allegations of sexual harassment". In November 2018 three Israeli computer scientists Irit Dinur, Oded Goldreich and Ehud Friedgut wrote a letter to the community mentioning some allegations of sexual harassment against Peres and proposed a guideline of not making invitations to junior researchers that may be viewed as intimate. In response Peres wrote a letter to the community and said "I regret all cases in the past where I have not followed this principle. I had no intention to harass anyone but must have been tone deaf not to recognize that I was making some people very uncomfortable. As I wrote above, I promise to adhere to this principle in the future." == Books == Levin, David A.; Peres, Yuval; Wilmer, Elizabeth L. (2009). Markov Chains and Mixing Times. Providence, Rhode Island: American Mathematical Society. ISBN 978-0-8218-4739-8. 2nd ed., 2017. Hough, J. Ben; Krishnapur, Manjunath; Peres, Yuval; Virág, Bálint (2009). Zeros of Gaussian Analytic Functions and Determinantal Processes. Providence, Rhode Island: American Mathematical Society. Mörters, Peter; Peres, Yuval (2010). Brownian Motion. Cambridge University Press. Lyons, Russell; Peres, Yuval (2016). Probability on Trees and Networks. Cambridge University Press. Bishop, Christopher J.; Peres, Yuval (2017). Fractals in Probability and Analysis. Cambridge University Press. Karlin, Anna; Peres, Yuval (2017). Game Theory, Alive. Providence, Rhode Island: American Mathematical Society. == External links == Official Website Yuval Peres at the Mathematics Genealogy Project David P. Robbins Prize for the paper Maximum Overhang, by Mike Paterson, Yuval Peres, Mikkel Thorup, Peter Winkler, and Uri Zwick. Plenary talk by Yuval Peres at the 2011 Joint Mathematics Meetings in New Orleans, LA A New Model For Proof-Checking, by Brown University computer scientist Claire Mathieu, describes Yuval Peres's influence on rigorous proofs in computer science. == References ==
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Wikipedia:Yves Le Jan#0
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Yves Le Jan (born 15 April 1952 in Grenoble) is a French mathematician working in probability theory and stochastic processes. Le Jan studied from 1971 to 1974 at the École normale supérieure, finishing with an Agrégation. 1975 he became a researcher (Attaché de Recherche) at the CNRS (from 1987 Directeur de Recherche) and in 1979 obtained his PhD (Doctorat d´Etat). Since 1993 he is Professor at the University of Paris-Sud. From 2001 to 2004 he was leading its group on probability theory and statistics. In 2006 he was invited speaker at the International Congress of Mathematicians in Madrid (New developments in stochastic dynamics). In 2008 he became Senior Member of the Institut Universitaire de France. In 2011 he was Doob Lecturer at the 8th World Congress in Probability and Statistics in Istanbul. In 2011 he was awarded the Sophie Germain Prize and in 1995 the Poncelet Prize of the French Academy of Sciences. From 2000 to 2006 he was Editor of Annales Henri Poincaré. == Books == with Jacques Franchi Hyperbolic dynamics and Brownian motion : an introduction, Oxford University Press 2012 with K. David Elworthy, Xue-Mei Li The Geometry of Filtering, Birkhäuser 2010 with K. David Elworthy, Xue-Mei Li On the geometry of diffusion operators and stochastic flows, Springer Verlag 1999 Markov paths, loops and fields, École d’Été de Probabilités de Saint-Flour XXXVIII-2008, Springer Verlag 2011 == References == == External links == Homepage Yves Le Jan at the Mathematics Genealogy Project http://www.idref.fr/076172937 http://www.math.u-psud.fr/~lejan/CVanglais.pdf
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Wikipedia:Yves Meyer#0
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Yves F. Meyer (French: [mɛjɛʁ]; born 19 July 1939) is a French mathematician. He is among the progenitors of wavelet theory, having proposed the Meyer wavelet. Meyer was awarded the Abel Prize in 2017. == Biography == Born in Paris, Yves Meyer studied at the Lycée Carnot in Tunis; he won the French General Student Competition (Concours Général) in Mathematics, and was placed first in the entrance examination for the École Normale Supérieure in 1957. He obtained his Ph.D. in 1966, under the supervision of Jean-Pierre Kahane. The Mexican historian Jean Meyer is his cousin. Yves Meyer taught at the Prytanée national militaire during his military service (1960–1963), then was a teaching assistant at the Université de Strasbourg (1963–1966), a professor at Université Paris-Sud (1966–1980), a professor at École Polytechnique (1980–1986), a professor at Université Paris-Dauphine (1985–1995), a senior researcher at the Centre national de la recherche scientifique (CNRS) (1995–1999), an invited professor at the Conservatoire National des Arts et Métiers (2000), a professor at École Normale Supérieure de Cachan (1999–2003), and has been a professor emeritus at Ecole Normale Supérieure de Cachan since 2004. He was awarded the 2010 Gauss Prize for fundamental contributions to number theory, operator theory and harmonic analysis, and his pivotal role in the development of wavelets and multiresolution analysis. He also received the 2017 Abel Prize "for his pivotal role in the development of the mathematical theory of wavelets." == Publications == Meyer, Yves (1970). Nombres de Pisot, nombres de Salem, et analyse harmonique (in French). Berlin New York: Springer-Verlag. ISBN 978-3-540-36243-2. OCLC 295014081. Algebraic numbers and harmonic analysis. Burlington: Elsevier Science. 1972. ISBN 978-0-08-095412-7. OCLC 761646828. Meyer, Yves (1990). Ondelettes et opérateurs (in French). Paris: Hermann. ISBN 978-2-7056-6125-0. OCLC 945745937. Meyer, Yves (22 April 1993). Wavelets and Operators. D. H. Salinger. Cambridge University Press. doi:10.1017/cbo9780511623820. ISBN 978-0-521-42000-6. == Awards and recognitions == He is a member of the Académie des Sciences since 1993. Meyer was an Invited Speaker at the ICM in 1970 in Nice, in 1983 in Warsaw, and in 1990 in Kyoto. In 2010, Yves Meyer was awarded the Carl Friedrich Gauss Prize. In 2012 he became a fellow of the American Mathematical Society. In 2017 he was awarded the Abel Prize for his pivotal role in developing the mathematical theory of wavelets. In 2020 he received the Princess of Asturias Award for Technical and Scientific Research. == See also == == References == == External links == Société Mathématiques de France : Lecture by Yves Meyer (2009) Yves Meyer at the Mathematics Genealogy Project Gauss prize 2010
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Wikipedia:Yves Pomeau#0
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Yves Pomeau, born in 1942, is a French mathematician and physicist, emeritus research director at the CNRS and corresponding member of the French Academy of sciences. He was one of the founders of the Laboratoire de Physique Statistique, École Normale Supérieure, Paris. He is the son of literature professor René Pomeau. == Career == Yves Pomeau did his state thesis in plasma physics, almost without any adviser, at the University of Orsay-France in 1970. After his thesis, he spent a year as a postdoc with Ilya Prigogine in Brussels. He was a researcher at the CNRS from 1965 to 2006, ending his career as DR0 in the Physics Department of the Ecole Normale Supérieure (ENS) (Statistical Physics Laboratory) in 2006. He was a lecturer in physics at the École Polytechnique for two years (1982–1984), then a scientific expert with the Direction générale de l'armement until January 2007. He was Professor, with tenure, part-time at the Department of Mathematics, University of Arizona, from 1990 to 2008. He was visiting scientist at Schlumberger–Doll Laboratories (Connecticut, USA) from 1983 to 1984. He was a visiting professor at MIT in Applied Mathematics in 1986 and in Physics at UC San Diego in 1993. He was Ulam Scholar at CNLS, Los Alamos National Lab, in 2007–2008. He has written 3 books, and published around 400 scientific articles. "Yves Pomeau occupies a central and unique place in modern statistical physics. His work has had a profound influence in several areas of physics, and in particular on the mechanics of continuous media. His work, nourished by the history of scientific laws, is imaginative and profound. Yves Pomeau combines a deep understanding of physical phenomena with varied and elegant mathematical descriptions. Yves Pomeau is one of the most recognized French theorists at the interface of physics and mechanics, and his pioneering work has opened up many avenues of research and has been a continuous source of inspiration for several generations of young experimental physicists and theorists worldwide." == Education == École normale supérieure, 1961–1965. Licence (1962). DEA in Plasma Physics, 1964. Aggregation of Physics 1965. State thesis in plasma physics, University of Orsay, 1970. == Research == In his thesis he showed that in a dense fluid the interactions are different from what they are at equilibrium and propagate through hydrodynamic modes, which leads to the divergence of transport coefficients in 2 spatial dimensions. This aroused his interest in fluid mechanics, and in the transition to turbulence. Together with Paul Manneville they discovered a new mode of transition to turbulence, the transition by temporal Intermittency, which was confirmed by numerous experimental observations and CFD simulations. This is the so-called Pomeau–Manneville scenario, associated with the Pomeau-Manneville maps In papers published in 1973 and 1976, Jean Hardy, Pomeau and Olivier de Pazzis introduced the first lattice Boltzmann model, which is called the HPP model after the authors. Generalizing ideas from his thesis, together with Uriel Frisch and Brosl Hasslacher, they found a very simplified microscopic fluid model (FHP model) which allows simulating very efficiently the complex movements of a real fluid. He was a pioneer of lattice Boltzmann methods and played a historical role in the timeline of computational physics. Reflecting on the situation of the transition to turbulence in parallel flows, he showed that turbulence is caused by a contagion mechanism, and not by local instability. Front can be static or mobile depending on the conditions of the system, and the causes of the motion can be the variation of a free energy, where the most energetically favorable state invades the less favorable one. The consequence is that this transition belongs to the class of directed percolation phenomena in statistical physics, which has also been amply confirmed by experimental and numerical studies. In dynamical systems theory, the structure and length of the attractors of a network corresponds to the dynamic phase of the network. The stability of Boolean network depends on the connections of their nodes. A Boolean network can exhibit stable, critical or chaotic behavior. This phenomenon is governed by a critical value of the average number of connections of nodes ( K c {\displaystyle K_{c}} ), and can be characterized by the Hamming distance as distance measure. If p i = p = c o n s t . {\displaystyle p_{i}=p=const.} for every node, the transition between the stable and chaotic range depends on p {\displaystyle p} . Bernard Derrida and Yves Pomeau proved that, the critical value of the average number of connections is K c = 1 / [ 2 p ( 1 − p ) ] {\displaystyle K_{c}=1/[2p(1-p)]} . A droplet of nonwetting viscous liquid moves on an inclined plane by rolling along it. Together with Lakshminarayanan Mahadevan, he gave a scaling law for the uniform speed of such a droplet. With Christiane Normand, and Manuel García Velarde, he studied convective instability. Apart from simple situations, capillarity remains an area where fundamental questions remain. He showed that the discrepancies appearing in the hydrodynamics of the moving contact line on a solid surface could only be eliminated by taking into account the evaporation/condensation near this line. Capillary forces are almost always insignificant in solid mechanics. Nevertheless, with Serge Mora and collaborators they have shown theoretically and experimentally that soft gel filaments are subject to Rayleigh-Plateau instability, an instability never observed before for a solid. In collaboration with his former PhD student Basile Audoly and Henri Berestycki, he studied the speed of the propagation of a reaction front in a fast steady flow with a given structure in space. With Basile Audoly and Martine Ben Amar, Pomeau developed a theory of large deformations of elastic plates which led them to introduce the concept of "d-cone", that is, a geometrical cone preserving the overall developability of the surface, an idea now taken up by the solid mechanics community. The theory of superconductivity is based on the idea of the formation of pairs of electrons that become more or less bosons undergoing Bose-Einstein condensation. This pair formation would explain the halving of the flux quantum in a superconducting loop. Together with Len Pismen and Sergio Rica they have shown that, going back to Onsager's idea explaining the quantification of the circulation in fundamental quantum states, it is not necessary to use the notion of electron pairs to understand this halving of the circulation quantum. He also analyzed the onset of BEC from the point of view of kinetic theory. Whereas the kinetic equation for a dilute Bose gas had been known for many years, the way it can describe what happens when the gas is cooled down to reach temperature below the temperature of transition. At this temperature the gas gets a macroscopic component in the quantum ground state, as had been predicted by Einstein long ago. Pomeau and collaborators showed that the solution of the kinetic equation becomes singular at zero energies and we did also find how the density of the condensate grows with time after the transition. They also derived the kinetic equation for the Bogoliubov excitations of Bose-Einstein condensates, where they found three collisional processes. Before the surge of interest in super-solids started by Moses Chan experiments, they had shown in an early simulation that a slightly modified NLS equation yields a fair representation of super-solids. With Alan C. Newell, he studied turbulent crystals in macroscopic systems. From his more recent work we must distinguish those concerning a phenomenon typically out of equilibrium, that of the emission of photons by an atom maintained in an excited state by an intense field that creates Rabi oscillations. The theory of this phenomenon requires a precise consideration of the statistical concepts of quantum mechanics in a theory satisfying the fundamental constraints of such a theory. With Martine Le Berre and Jean Ginibre they showed that the good theory was that of a Kolmogorov equation based on the existence of a small parameter, the ratio of the photon emission rate to the atomic frequency itself. == Known for == Timeline of computational physics Lattice Boltzmann Models Intermittency Pomeau–Manneville scenario Pomeau-Manneville maps The stability of Boolean network Front Lattice gas automaton Logistic map Stellar pulsation Hardy-Pomeau-Pazzis (HPP) model The physics of ice skating Lorenz system Saffman-Taylor Fingers Hénon-Pomeau attractor == Prizes and awards == FPS Paul Langevin Award in 1981. FPS Jean Ricard Award in 1985. Perronnet–Bettancourt Prize (1993) awarded by the Spanish government for collaborative research between France and Spain. Chevalier of the Légion d'Honneur since 1991. Elected corresponding member of the French Academy of sciences in 1987 (Mechanical and Computer Sciences). Boltzmann Medal (2016) Three Physicists Prize (2024) == References ==
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Wikipedia:Yvette Kosmann-Schwarzbach#0
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Yvette Kosmann-Schwarzbach (born 30 April 1941) is a French mathematician and professor. == Education and career == Kosmann-Schwarzbach obtained her doctoral degree in 1970 at the University of Paris under supervision of André Lichnerowicz on a dissertation titled Dérivées de Lie des spineurs (Lie derivatives of spinors). She worked at Lille University of Science and Technology, and since 1993 at the École polytechnique. == Research == Kosmann-Schwarzbach is the author of over fifty articles on differential geometry, algebra and mathematical physics, of two books on Lie groups and on the Noether theorem, as well as the co-editor of several books concerning the theory of integrable systems. The Kosmann lift in differential geometry is named after her. == Works == Groups and Symmetries: From Finite Groups to Lie Groups. Translated by Stephanie Frank Singer. Springer 2010, ISBN 978-0387788654. The Noether Theorems: Invariance and Conservation Laws in the Twentieth Century. Translated by Bertram Schwarzbach. Springer 2011, ISBN 978-0387878676. == References ==
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Wikipedia:Yvonne Dold-Samplonius#0
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Yvonne Dold-Samplonius (20 May 1937 – 16 June 2014) was a Dutch mathematician and historian who specialised in the history of Islamic mathematics during the Middle Ages. She was particularly interested in the mathematical methods used by Islamic architects and builders of the Middle Ages for measurements of volumes and measurements of religious buildings or in the design of muqarnas. == Biography == Born on 20 May 1937 in Haarlem, Yvonne Samplonius obtained her degree in mathematics and Arabic from the University of Amsterdam (Doktoratsexamen) in 1966. In 1965 Yvonne Dold-Samplonius married the German mathematician Albrecht Dold. During 1966 and 1967, she studied at Harvard University under the direction of Professor John E. Murdoch. In 1977 she was awarded a PhD for her analysis of the treatise Kitāb al-mafrādāt li Aqāţun (Book of Assumptions of Aqātun) under the supervision of Prof. Evert Marie Bruins and Prof. Juan Vernet. She came into contact with the work of the Persian mathematician, physicist and astronomer Abū Sahl al-Qūhī, who worked in Baghdad in the 10th century and worked on the geometrical forms of buildings. Through his work, she became interested in the geometrical calculations that helped with the building of many of the domes of palaces and mosques, called muqarnas, in the Arab world and Persia. She wrote articles on the Islamic mathematicians Jamshīd al-Kāshī and Abu-Abdullah Muhammad ibn Īsa Māhānī in the Dictionary of the Middle Ages and in the Dictionary of Scientific Biography. In her later years her interest shifted to the use of mathematics in Islamic architecture from an historic point of view. From 1995, she was an associate member of the Interdisciplinary Centre for Scientific Computing (IWR) of the University of Heidelberg, with whom she has published several videos on Islamic geometrical art. In 1985, she was a visiting professor at the University of Siena. In 2000, she organised with Joseph Dauben the conference "2000 Years of Transmission of Mathematical Ideas". In 2002, she became a Corresponding Member of the International Academy of the History of Sciences and was elected an effective member in 2007. She was made honorary citizen of Kashan in Iran in 2000. == Publications == Yvonne Dold-Samplonius, Dissertation: Book of Assumptions by Aqatun (Kitab al-Mafrudat li-Aqatun), Amsterdam 1977. Yvonne Dold-Samplonius : Practical Arabic Mathematics: Measuring the Muqarnas by al-Kashi, Centaurus 35, 193–242, (1992/3). Yvonne Dold-Samplonius : How al-Kashi Measures the Muqarnas: A Second Look, M. Folkerts (Ed.), Mathematische Probleme im Mittelalter: Der lateinische und arabische Sprachbereich, Wolfenbütteler Mittelalter-Studien Vol. 10, 56 – 90, Wiesbaden, (1996). Yvonne Dold-Samplonius : Calculation of Arches and Domes in 15th Century Samarkand, Nexus Network Journal, Vol. 2(3), (2000). Yvonne Dold-Samplonius : Calculating Surface Areas and Volumes in Islamic Architecture, The Enterprise of Science in Islam, New Perspectives, Eds. Jan P. Hogendijk et Abdelhamid I. Sabra, MIT Press, Cambridge Mass. pp. 235–265, (2003). Yvonne Dold-Samplonius, Silvia L. Harmsen : The Muqarnas Plate Found at Takht-i Sulaiman, A New Interpretation, Muqarnas Vol. 22, Leiden, pp. 85–94, (2005). == Videos == Yvonne Dold-Samplonius, Christoph Kindl, Norbert Quien : Qubba for al-Kashi, Video, Interdisciplinary Centre for Scientific Computing (IWR), Heidelberg University, American Mathematical Society, (1996). Qubba for al-Kashi on YouTube Yvonne Dold-Samplonius, Silvia L. Harmsen, Susanne Krömker, Michael Winckler : Magic of Muqarnas, Video, Interdisciplinary Centre for Scientific Computing (IWR), Heidelberg University, (2005). == References ==
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Wikipedia:Yvonne Pothier#0
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Sister Yvonne Marie Pothier (born 1937) is a Canadian mathematics educator and educational psychologist known for her work in the development of numerical concepts in children, and an activist for refugees. She is a professor emerita of education at Mount Saint Vincent University in Halifax, Nova Scotia, and a Sister of Charity of Saint Vincent de Paul in the Roman Catholic Archdiocese of Halifax-Yarmouth. == Mathematics == Pothier graduated from Mount Saint Vincent University in 1966 with a Bachelor of Science, and earned a bachelor of education in 1977 from the University of New Brunswick. She earned a master's degree and Ph.D. at the University of Alberta; her dissertation, Partitioning: Construction of Rational Number in Young Children, was supervised by Daiyo Sawada. She published a condensed version of the same work as an influential journal paper with Sawada. She also coauthored the book Learning Mathematics In Elementary And Middle School: A Learner-Centered Approach (with Nadine Bezuk, W. George Cathcart, and James H. Vance, Pearson, 2003; 6th ed., 2015). == Refugee work == In later life, Pothier became active in work with refugees, coordinating the Refugee Sponsorship Program of the Catholic Archdiocese of Halifax, visiting Sudan in this connection, and assisting in the sponsorship of many refugees in Halifax. For this work she won the Elizabeth Ann Seton Award of the Sisters of Charity, and was commended by the Nova Scotia House of Assembly. == References ==
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Wikipedia:Z-order curve#0
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In mathematical analysis and computer science, functions which are Z-order, Lebesgue curve, Morton space-filling curve, Morton order or Morton code map multidimensional data to one dimension while preserving locality of the data points (two points close together in multidimensions with high probability lie also close together in Morton order). It is named in France after Henri Lebesgue, who studied it in 1904, and named in the United States after Guy Macdonald Morton, who first applied the order to file sequencing in 1966. The z-value of a point in multidimensions is simply calculated by bit interleaving the binary representations of its coordinate values. However, when querying a multidimensional search range in these data, using binary search is not really efficient: It is necessary for calculating, from a point encountered in the data structure, the next possible Z-value which is in the multidimensional search range, called BIGMIN. The BIGMIN problem has first been stated and its solution shown by Tropf and Herzog in 1981. Once the data are sorted by bit interleaving, any one-dimensional data structure can be used, such as simple one dimensional arrays, binary search trees, B-trees, skip lists or (with low significant bits truncated) hash tables. The resulting ordering can equivalently be described as the order one would get from a depth-first traversal of a quadtree or octree. == Coordinate values == The figure below shows the Z-values for the two dimensional case with integer coordinates 0 ≤ x ≤ 7, 0 ≤ y ≤ 7 (shown both in decimal and binary). Interleaving the binary coordinate values (starting to the right with the x-bit (in blue) and alternating to the left with the y-bit (in red)) yields the binary z-values (tilted by 45° as shown). Connecting the z-values in their numerical order produces the recursively Z-shaped curve. Two-dimensional Z-values are also known as quadkey values. The Z-values of the x coordinates are described as binary numbers from the Moser–de Bruijn sequence, having nonzero bits only in their even positions: x[] = {0b000000, 0b000001, 0b000100, 0b000101, 0b010000, 0b010001, 0b010100, 0b010101} The sum and difference of two x values are calculated by using bitwise operations: x[i+j] = ((x[i] | 0b10101010) + x[j]) & 0b01010101 x[i−j] = ((x[i] & 0b01010101) − x[j]) & 0b01010101 if i ≥ j This property can be used to offset a Z-value, for example in two dimensions the coordinates to the top (decreasing y), bottom (increasing y), left (decreasing x) and right (increasing x) from the current Z-value z are: top = (((z & 0b10101010) − 1) & 0b10101010) | (z & 0b01010101) bottom = (((z | 0b01010101) + 1) & 0b10101010) | (z & 0b01010101) left = (((z & 0b01010101) − 1) & 0b01010101) | (z & 0b10101010) right = (((z | 0b10101010) + 1) & 0b01010101) | (z & 0b10101010) And in general to add two two-dimensional Z-values w and z: sum = ((z | 0b10101010) + (w & 0b01010101) & 0b01010101) | ((z | 0b01010101) + (w & 0b10101010) & 0b10101010) == Efficiently building quadtrees and octrees == The Z-ordering can be used to efficiently build a quadtree (2D) or octree (3D) for a set of points. The basic idea is to sort the input set according to Z-order. Once sorted, the points can either be stored in a binary search tree and used directly, which is called a linear quadtree, or they can be used to build a pointer based quadtree. The input points are usually scaled in each dimension to be positive integers, either as a fixed point representation over the unit range [0, 1] or corresponding to the machine word size. Both representations are equivalent and allow for the highest order non-zero bit to be found in constant time. Each square in the quadtree has a side length which is a power of two, and corner coordinates which are multiples of the side length. Given any two points, the derived square for the two points is the smallest square covering both points. The interleaving of bits from the x and y components of each point is called the shuffle of x and y, and can be extended to higher dimensions. Points can be sorted according to their shuffle without explicitly interleaving the bits. To do this, for each dimension, the most significant bit of the exclusive or of the coordinates of the two points for that dimension is examined. The dimension for which the most significant bit is largest is then used to compare the two points to determine their shuffle order. The exclusive or operation masks off the higher order bits for which the two coordinates are identical. Since the shuffle interleaves bits from higher order to lower order, identifying the coordinate with the largest most significant bit, identifies the first bit in the shuffle order which differs, and that coordinate can be used to compare the two points. This is shown in the following Python code: One way to determine whether the most significant bit is smaller is to compare the floor of the base-2 logarithm of each point. It turns out the following operation is equivalent, and only requires exclusive or operations: It is also possible to compare floating point numbers using the same technique. The less_msb function is modified to first compare the exponents. Only when they are equal is the standard less_msb function used on the mantissas. Once the points are in sorted order, two properties make it easy to build a quadtree: The first is that the points contained in a square of the quadtree form a contiguous interval in the sorted order. The second is that if more than one child of a square contains an input point, the square is the derived square for two adjacent points in the sorted order. For each adjacent pair of points, the derived square is computed and its side length determined. For each derived square, the interval containing it is bounded by the first larger square to the right and to the left in sorted order. Each such interval corresponds to a square in the quadtree. The result of this is a compressed quadtree, where only nodes containing input points or two or more children are present. A non-compressed quadtree can be built by restoring the missing nodes, if desired. Rather than building a pointer based quadtree, the points can be maintained in sorted order in a data structure such as a binary search tree. This allows points to be added and deleted in O(log n) time. Two quadtrees can be merged by merging the two sorted sets of points, and removing duplicates. Point location can be done by searching for the points preceding and following the query point in the sorted order. If the quadtree is compressed, the predecessor node found may be an arbitrary leaf inside the compressed node of interest. In this case, it is necessary to find the predecessor of the least common ancestor of the query point and the leaf found. == Use with one-dimensional data structures for range searching == By bit interleaving, the database records are converted to a (possibly very long) sequence of bits. The bit sequences are interpreted as binary numbers and the data are sorted or indexed by the binary values, using any one dimensional data structure, as mentioned in the introduction. However, when querying a multidimensional search range in these data, using binary search is not really efficient. Although Z-order is preserving locality well, for efficient range searches an algorithm is necessary for calculating, from a point encountered in the data structure, the next possible Z-value which is in the multidimensional search range: In this example, the range being queried (x = 2, ..., 3, y = 2, ..., 6) is indicated by the dotted rectangle. Its highest Z-value (MAX) is 45. In this example, the value F = 19 is encountered when searching a data structure in increasing Z-value direction, so we would have to search in the interval between F and MAX (hatched area). To speed up the search, one would calculate the next Z-value which is in the search range, called BIGMIN (36 in the example) and only search in the interval between BIGMIN and MAX (bold values), thus skipping most of the hatched area. Searching in decreasing direction is analogous with LITMAX which is the highest Z-value in the query range lower than F. The BIGMIN problem has first been stated and its solution shown in Tropf and Herzog. For the history after the puplication see. An extensive explanation of the LITMAX/BIGMIN calculation algorithm, together with Pascal Source Code (3D, easy to adapt to nD) and hints on how to handle floating point data and possibly negative data, is provided 2021 by Tropf: Here, bit interleaving is not done explicitly; the data structure has just pointers to the original (unsorted) database records. With a general record comparison function (greater-less-equal, in the sense of z-value), complications with bit sequences length exceeding the computer word length are avoided, and the code can easily be adapted to any number of dimensions and any record key word length. As the approach does not depend on the one dimensional data structure chosen, there is still free choice of structuring the data, so well known methods such as balanced trees can be used to cope with dynamic data, and keeping the tree balance when inserting or deleting takes O(log n) time. The method is also used in UB-trees (balanced). The Free choice makes it easier to incorporate the method into existing databases. This is in contrast for example to R-trees where special considerations are necessary. Applying the method hierarchically (according to the data structure at hand), optionally in both increasing and decreasing direction, yields highly efficient multidimensional range search which is important in both commercial and technical applications, e.g. as a procedure underlying nearest neighbour searches. Z-order is one of the few multidimensional access methods that has found its way into commercial database systems. The method is used in various technical applications of different fields. and in commercial database systems. As long ago as 1966, G.M.Morton proposed Z-order for file sequencing of a static two dimensional geographical database. Areal data units are contained in one or a few quadratic frames represented by their sizes and lower right corner Z-values, the sizes complying with the Z-order hierarchy at the corner position. With high probability, changing to an adjacent frame is done with one or a few relatively small scanning steps. == Related structures == As an alternative, the Hilbert curve has been suggested as it has a better order-preserving behaviour, and, in fact, was used in an optimized index, the S2-geometry. == Applications == === Linear algebra === The Strassen algorithm for matrix multiplication is based on splitting the matrices in four blocks, and then recursively splitting each of these blocks in four smaller blocks, until the blocks are single elements (or more practically: until reaching matrices so small that the Moser–de Bruijn sequence trivial algorithm is faster). Arranging the matrix elements in Z-order then improves locality, and has the additional advantage (compared to row- or column-major ordering) that the subroutine for multiplying two blocks does not need to know the total size of the matrix, but only the size of the blocks and their location in memory. Effective use of Strassen multiplication with Z-order has been demonstrated, see Valsalam and Skjellum's 2002 paper. Buluç et al. present a sparse matrix data structure that Z-orders its non-zero elements to enable parallel matrix-vector multiplication. Matrices in linear algebra can also be traversed using a space-filling curve. Conventional loops traverse a matrix row by row. Traversing with the Z-curve allows efficient access to the memory hierarchy. === Texture mapping === Some GPUs store texture maps in Z-order to increase spatial locality of reference during texture mapped rasterization. This allows cache lines to represent rectangular tiles, increasing the probability that nearby accesses are in the cache. At a larger scale, it also decreases the probability of costly, so called, "page breaks" (i.e., the cost of changing rows) in SDRAM/DDRAM. This is important because 3D rendering involves arbitrary transformations (rotations, scaling, perspective, and distortion by animated surfaces). These formats are often referred to as swizzled textures or twiddled textures. Other tiled formats may also be used. === n-body problem === The Barnes–Hut algorithm requires construction of an octree. Storing the data as a pointer-based tree requires many sequential pointer dereferences to iterate over the octree in depth-first order (expensive on a distributed-memory machine). Instead, if one stores the data in a hashtable, using octree hashing, the Z-order curve naturally iterates the octree in depth-first order. == See also == Geohash Hilbert R-tree Linear algebra Locality preserving hashing Matrix representation Netto's theorem PH-tree Spatial index == References == == External links == STANN: A library for approximate nearest neighbor search, using Z-order curve Methods for programming bit interleaving, Sean Eron Anderson, Stanford University
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Wikipedia:Zafar Usmanov#0
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Zafar Juraevich Usmanov (Tajik: Усмонов Зафар Ҷӯраевич; 26 August 1937 – 13 October 2021) was a Soviet and Tajik mathematician, doctor of physical and mathematical sciences (1974), professor (1983), full member of the Academy of Sciences of the Republic of Tajikistan (1981), Honored scientist of the Republic of Tajikistan (1997), laureate of the State Prize of Tajikistan in the field of science and technology named after Abu Ali ibn Sino (2013). == Biography == Zafar Juraevich Usmanov was born on 26 August 1937 in Dushanbe. Father — Jura Usmanov, historian, journalist, Mother — Hamro (Asrorova) Usmanova, party and state worker. Sibling of the academician of the Academy of Sciences of the Republic of Tajikistan, Pulat Usmanov. 1954—1959 — studies at the Faculty of Mechanics and Mathematics of MSU. 1959—1962 — graduate student of the Department of Mechanics of Moscow State University, 1962—1970 — researcher of the mathematical team of the Academy of Sciences of the Republic of Tatarstan, 1970—1973 — head of the Computing Center, (deputy head) of the Department of Mathematics with the Computing Center of the Academy of Sciences of the Republic of Tajikistan, 1973—1976 — Deputy Director for Science of the Mathematical Institute with the Computing Center of the Academy of Sciences of the Republic of Tajikistan, 1976—1984 — Head of the Computing Center of the Academy of Sciences of the Republic of Tajikistan, 1984—1988 — academician-secretary of the Department of Physical, Mathematical, Chemical and Geological Sciences of the Academy of Sciences of the Republic of Tajikistan, 1988—1999 — Director of the Institute of Mathematics of the Academy of Sciences of the Republic of Tajikistan, From 1999 to the present — Head. Department of Mathematical Modeling, Institute of Mathematics, Academy of Sciences of the Republic of Tajikistan, 1976 — Corresponding Member of the Academy of Sciences of the Republic of Tajikistan, 1981 — Full member of the Academy of Sciences of the Republic of Tajikistan with a degree in mathematics, 1985—1990 — Deputy of the Supreme Council of the Tajik SSR, 1986—1991 — member of the Revision Commission of the Central Committee of the Communist Party of Tajikistan, 1997—2011 — Professor of the Moscow Power Engineering Institute (Technical University), Volzhsky branch of Volzhsky, 1999 — Professor, Department of Informatics, Technological University of Tajikistan, Head of the Department of «Natural Process Metrics» of the Virtual Institute of Interdisciplinary Time Studies, Moscow State University. 2018 — Chairman of the dissertation council 6D.KOA-032 at the Tajik Technical University named after academician M.S. Osimi == Scientific training == The scientific organizer of systemic training at the Institute of Mathematics is about 30 candidates of physical and mathematical sciences on modern problems of computer science. He prepared 18 candidates of sciences in the specialties of differential equations, geometry, computer science, hydromechanics, hydraulics and the history of mathematics, and 1 doctor of sciences in water problems. == Teaching == 1959—1961 — Faculty of Mechanics and Mathematics, Moscow State University M.V. Lomonosova, Moscow, 1965—1994, 2007—2009 — Faculty of Mechanics, Mathematics and Physics, Tajik National University, Dushanbe, 1966—1968 — Faculty of Mathematics, Tajik State Pedagogical University, Dushanbe, 1997—2011 — Department of Power Engineering, Moscow Power Engineering Institute (Technical University), Volzhsky Branch, Volzhsky, 1999 — IV Department of Mathematics, Graz University of Technology, (spring semester), Austria, Faculty of Information Technologies, Technological University of Tajikistan, Dushanbe. == Scientific activity == Usmanov successfully defended his Ph.D. thesis: «Some boundary value problems for systems of differential equations with singular coefficients and their applications to bendings of surfaces with a singular point» / Tajik State National University, (1966) / and doctoral dissertation: "Study of the equations of the theory of infinitesimal bendings of surfaces with positive curvature with a flattening point ", Faculty of Mechanics and Mathematics, / Moscow State University M.V. Lomonosova, (1973) /. The scientific interests of the scientist: Generalized Cauchy-Riemann systems with singularities at isolated points and on a closed line; Deformation of surfaces with an isolated flattening point, a conical point, and a parabolic boundary; Modeling the proper time of an arbitrary process; Modeling of environmental, economic, industrial and technological processes; Automation of information processing in Tajik. === In the field of theoretical mathematics === The theory of generalized Cauchy-Riemann systems with a singular point of the 1st and above 1st order in the coefficients, as well as with the 1st order singularity in the coefficients on the boundary circle, which was a natural generalization of the classical analytical apparatus of I. N. Vekua, developed to study generalized analytic functions. Based on the fundamental achievements in the development of the theory of generalized Cauchy-Riemann systems with singularities, in-depth studies have been carried out on the effect of an isolated flattening point on infinitesimal and exact bends of surfaces of positive curvature. Some progress has been made in solving the generalized Christoffel problem of determining convex surfaces from a predetermined sum of conditional radii of curvature defined on a convex surface with an isolated flattening point (together with A. Khakimov). For a wide class of natural processes described by ordinary differential equations and partial differential equations, natural metrics such as spatio-temporal Minkowski metrics are constructed, based on which a definition of the concept of the intrinsic time of a process and constructive methods for measuring it are proposed. Computational experiments have established the promise of using the new concept to increase the effectiveness of the prognostic properties of mathematical models. === In the field of applied mathematics === A mathematical model has been developed for the evolution of collection material of an arbitrary nature (together with T. I. Khaitov); a mathematical model for describing the evolution of spiral shells by the example of gastropods (together with M.R. Dzhalilov and O.P. Sapov); a mathematical model for determining the gradations of liver failure (together with H. Kh. Mansurov and others); mathematical model of the dynamics of the desert community of the Tigrovaya Balka nature reserve (together with G. N. Sapozhnikov et al.) Some of these results were noted in the reports of the Chief Scientific Secretary of the Presidium of the USSR Academy of Sciences among the most important achievements of the USSR Academy of Sciences in the field of theoretical mathematics in the 70s years and twice in the field of computer science in the 80s. === In the field of informatization of the Tajik language === Created a scientific school in computer linguistics in Tajikistan. He prepared 5 candidates of sciences in mathematical and statistical linguistics. As a leader and direct executor of works, together with his students, he carried out extensive research on the automation of information processing in the Tajik language. He has published over 280 scientific papers on theoretical and applied mathematics in scientific journals of the countries of near and far abroad and has registered 16 intellectual products in the National Patent Information Center of the Ministry of Economic Development and Trade of the Republic of Tatarstan == Major monographs == ZD Usmanov, Generalized Cauchy-Riemann systems with a singular point, Pitman Monographs and Surveys in Pure and Applied Mathematics, 85, Longman, Harlow, 1997, 222 p., ISSN 0269-3666, ISBN 0-582-29280-8 [1 ] Programming the states of a collection, Moscow: Nauka, 1983, −124 p. (Programmer’s library, co-author T.I. Khaitov) Periods, rhythms and cycles in nature, Handbook, Dushanbe, Donish, 1990. −151 p. (co-authors Yu. I. Gorelov, l. I. Sapova) Modeling time, Moscow: Knowledge, 1991, −48 s. (New in life, science, technology. Series Mathematics, Cybernetics). Generalized Cauchy-Riemann systems with a singular point, Math. Institute with the Computing Center of the Academy of Sciences of the Republic of Tajikistan, Dushanbe, 1993, −244 p. Generalized Cauchy-Riemann systems with a singular point, Addison Wesley Longman Ltd., Harlow, England, 1997, −222 p. (Pitman Monographs and Survey in Pure and Applied Mathematics). The experience of computer synthesis of Tajik speech in the text, Technological University of Tajikistan, — Dushanbe, Irfon, 2010. — 146 p. (co-author H.A. Khudoiberdiev) The problem of the layout of characters on a computer keyboard. Technological University of Tajikistan, — Dushanbe: «Irfon», 2010. — 104 p. (co-author O. M. Soliev) Formation of the base of the morphs of the Tajik language. Dushanbe, 2014 .-- 110 s. (co-author G. M. Dovudov) Morphological analysis of Tajik word forms. Dushanbe: Donish, 2015. — 132 p. (co-author G. M. Dovudov) == Implement results == supervised the development and implementation of an automated system for the distribution of paired cocoons in cocoon winding machines for the Dushanbe silk-winding factory; He supervised and directly participated in the development of the mathematical foundations for optimizing the extractant enrichment process in the countercurrent extraction technological chain with the implementation of the results for the practical extraction of sea buckthorn oil from pulp; developed the mathematical basis for the automatic design of slotted grooves of winding drums for the Tajiktekstilmash plant; led the development of a temporary standard for Tajik graphics for use in network technology; development was sent to the Moscow representative office of MICROSOFT for inclusion in the WINDOWS editor (approved by Decree of the Government of the Republic of Tajikistan on 2 August 2004, No. 330); Together with his graduate student O. Soliev, he implemented through the Ministry of Communications of the Republic of Tajikistan the driver we developed for the layout of Tajik letters on a computer keyboard and instructions for installing it for use in everyday work; Together with his graduate student Kh. Khudoiberdiev, he developed and created a software and hardware complex for the automatic unstressed sounding of Tajik texts; together with his students O. Soliev, Kh. Khudoiberdiev developed and created the Tajik computer text editor (Tajik Word); together with S. D. Kholmatova, O. Soliev and H. Khudoyberdiev, developed and created: — Tajik-Russian computer dictionary , — Russian-Tajik computer dictionary; — universal Russian-Tajik-Russian computer dictionary (MultiGanj); together with L. A. Grashchenko and A. Yu. Fomin, he created a computer Tajik-Persian converter of graphic writing systems; Together with students O. Soliev, H. Khudoiberdiev and G. Dovudov, he developed and created the Tajik language pack (spell checker) for OpenOffice.Org and Windows. Usmanov Z.J. is: Member of the Scientific Board of Advisers, American Biographical Society, Head of the Department of «Natural Process Metrics» of the Virtual Institute of Interdisciplinary Study of Time, Moscow State University, Moscow , Member of the Editorial Board of the Central Asian Journal of Mathematics (Central Asian J Math) Wilmington, USA Central Asian Mathematical Journal, Washington, USA. (2005), intravital member of ISAAC (2005), Member of the International Editorial Board of the journal "Bulletin of the Samara State Technical University. Series «Physics and Mathematics» "(2014) Reviewer of articles submitted to the journal «Complex variables & Elliptic equations», University of Delaware, Newark, USA (regular reviewer of CV & EE) (2011), as well as at international conferences: International Multi-Conference on Society, Cybernetics and Informatics: IMSCI; International Conference on Complexity, Cybernetics, and Informing Science and Engineering: CCISE; International Conference on Social and Organizational Informatics and Cybernetics: SOIC). == Rewards == anniversary medal "For Valiant Labor. In commemoration of the centenary of the birth of V. I. Lenin ", (1970), the honorary badge of the All-Union Central Council of Trade Unions «Winner of social competition», (1973), Honorary Badge of the All-Union Society Knowledge «For active work» Certificate of Honor of the Supreme Council of Tajikistan, (1987), Komsomol honorary badge in honor of the 70th anniversary of Komsomol, (1988), honorary badge of the Committee of Physical Culture and Sports under the Council of Ministers of the Tajik SSR "Veteran of Physical Education and Sports Taj. SSR ", (1990), Honored Scientist of the Republic of Tajikistan, (1997), Veteran of Labor of the Russian Federation, (1998), Order of the Lomonosov Committee of Public Awards of the Russian Federation, (2008), Laureate of the State Prize of Tajikistan in the field of science and technology. Abu Ali ibn Sino. (2013). == See also == Tajik Academy of Sciences == References == == External links == Official website
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Wikipedia:Zahid Khalilov#0
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Zahid Ismayil oghlu Khalilov (Azerbaijani: Zahid İsmayıl oğlu Xəlilov, 14 January 1911, Sarachly – 4 February 1974, Baku) was an Azerbaijani mathematician (Professor since 1946) and engineer. Being the founder of Azerbaijani functional analysis school, he was elected President of the Azerbaijan Mathematical Society. Khalilov solved the boundary value problem for polyharmonic equations, proposed abstract generalizations of singular integral operators and made some other contributions. In 1955, Khalilov became a member of Azerbaijan National Academy of Sciences. In 1957—1959 he was its Vice-Chairman and Chairman in 1961–1967. A street in Baku is named after him. == Researches == Khalilov was first to consider the abstract equation with an operator satisfying the condition Q2 = I, within the framework of normed rings. The associated theory was a direct treatment of the singular integral equations theory with continuous coefficients within the subject of an abstract normed ring. Khalilov had also translated the Noether's theorem to the case of abstract singular equations in a normed ring R ux + vS(x) + T(x) = y and gave a general theory of regularizers. Khalilov also investigated the problems of subterranean hydromechanics applied in development of oil and gas deposits. == References == == External links == Profile at the Baku State University site
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Wikipedia:Zariski's finiteness theorem#0
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In algebra, Zariski's finiteness theorem gives a positive answer to Hilbert's 14th problem for the polynomial ring in two variables, as a special case. Precisely, it states: Given a normal domain A, finitely generated as an algebra over a field k, if L is a subfield of the field of fractions of A containing k such that the transcendence degree t r . d e g k ( L ) ≤ 2 {\displaystyle \operatorname {tr.deg} _{k}(L)\leq 2} , then the k-subalgebra L ∩ A {\displaystyle L\cap A} is finitely generated. == References == Zariski, O. (1954). "Interprétations algébrico-géométriques du quatorzième problème de Hilbert". Bull. Sci. Math. (2). 78: 155–168.
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Wikipedia:Zassenhaus algorithm#0
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In computational algebra, the Cantor–Zassenhaus algorithm is a method for factoring polynomials over finite fields (also called Galois fields). The algorithm consists mainly of exponentiation and polynomial GCD computations. It was invented by David G. Cantor and Hans Zassenhaus in 1981. It is arguably the dominant algorithm for solving the problem, having replaced the earlier Berlekamp's algorithm of 1967. It is currently implemented in many computer algebra systems like PARI/GP. == Overview == === Background === The Cantor–Zassenhaus algorithm takes as input a square-free polynomial f ( x ) {\displaystyle f(x)} (i.e. one with no repeated factors) of degree n with coefficients in a finite field F q {\displaystyle \mathbb {F} _{q}} whose irreducible polynomial factors are all of equal degree (algorithms exist for efficiently factoring arbitrary polynomials into a product of polynomials satisfying these conditions, for instance, f ( x ) / gcd ( f ( x ) , f ′ ( x ) ) {\displaystyle f(x)/\gcd(f(x),f'(x))} is a squarefree polynomial with the same factors as f ( x ) {\displaystyle f(x)} , so that the Cantor–Zassenhaus algorithm can be used to factor arbitrary polynomials). It gives as output a polynomial g ( x ) {\displaystyle g(x)} with coefficients in the same field such that g ( x ) {\displaystyle g(x)} divides f ( x ) {\displaystyle f(x)} . The algorithm may then be applied recursively to these and subsequent divisors, until we find the decomposition of f ( x ) {\displaystyle f(x)} into powers of irreducible polynomials (recalling that the ring of polynomials over any field is a unique factorisation domain). All possible factors of f ( x ) {\displaystyle f(x)} are contained within the factor ring R = F q [ x ] ⟨ f ( x ) ⟩ {\displaystyle R={\frac {\mathbb {F} _{q}[x]}{\langle f(x)\rangle }}} . If we suppose that f ( x ) {\displaystyle f(x)} has irreducible factors p 1 ( x ) , p 2 ( x ) , … , p s ( x ) {\displaystyle p_{1}(x),p_{2}(x),\ldots ,p_{s}(x)} , all of degree d, then this factor ring is isomorphic to the direct product of factor rings S = ∏ i = 1 s F q [ x ] ⟨ p i ( x ) ⟩ {\displaystyle S=\prod _{i=1}^{s}{\frac {\mathbb {F} _{q}[x]}{\langle p_{i}(x)\rangle }}} . The isomorphism from R to S, say ϕ {\displaystyle \phi } , maps a polynomial g ( x ) ∈ R {\displaystyle g(x)\in R} to the s-tuple of its reductions modulo each of the p i ( x ) {\displaystyle p_{i}(x)} , i.e. if: g ( x ) ≡ g 1 ( x ) ( mod p 1 ( x ) ) , g ( x ) ≡ g 2 ( x ) ( mod p 2 ( x ) ) , ⋮ g ( x ) ≡ g s ( x ) ( mod p s ( x ) ) , {\displaystyle {\begin{aligned}g(x)&{}\equiv g_{1}(x){\pmod {p_{1}(x)}},\\g(x)&{}\equiv g_{2}(x){\pmod {p_{2}(x)}},\\&{}\ \ \vdots \\g(x)&{}\equiv g_{s}(x){\pmod {p_{s}(x)}},\end{aligned}}} then ϕ ( g ( x ) + ⟨ f ( x ) ⟩ ) = ( g 1 ( x ) + ⟨ p 1 ( x ) ⟩ , … , g s ( x ) + ⟨ p s ( x ) ⟩ ) {\displaystyle \phi (g(x)+\langle f(x)\rangle )=(g_{1}(x)+\langle p_{1}(x)\rangle ,\ldots ,g_{s}(x)+\langle p_{s}(x)\rangle )} . It is important to note the following at this point, as it shall be of critical importance later in the algorithm: Since the p i ( x ) {\displaystyle p_{i}(x)} are each irreducible, each of the factor rings in this direct sum is in fact a field. These fields each have degree q d {\displaystyle q^{d}} . === Core result === The core result underlying the Cantor–Zassenhaus algorithm is the following: If a ( x ) ∈ R {\displaystyle a(x)\in R} is a polynomial satisfying: a ( x ) ≠ 0 , ± 1 {\displaystyle a(x)\neq 0,\pm 1} a i ( x ) ∈ { 0 , − 1 , 1 } for i = 1 , 2 , … , s , {\displaystyle a_{i}(x)\in \{0,-1,1\}{\text{ for }}i=1,2,\ldots ,s,} where a i ( x ) {\displaystyle a_{i}(x)} is the reduction of a ( x ) {\displaystyle a(x)} modulo p i ( x ) {\displaystyle p_{i}(x)} as before, and if any two of the following three sets is non-empty: A = { i ∣ a i ( x ) = 0 } , {\displaystyle A=\{i\mid a_{i}(x)=0\},} B = { i ∣ a i ( x ) = − 1 } , {\displaystyle B=\{i\mid a_{i}(x)=-1\},} C = { i ∣ a i ( x ) = 1 } , {\displaystyle C=\{i\mid a_{i}(x)=1\},} then there exist the following non-trivial factors of f ( x ) {\displaystyle f(x)} : gcd ( f ( x ) , a ( x ) ) = ∏ i ∈ A p i ( x ) , {\displaystyle \gcd(f(x),a(x))=\prod _{i\in A}p_{i}(x),} gcd ( f ( x ) , a ( x ) + 1 ) = ∏ i ∈ B p i ( x ) , {\displaystyle \gcd(f(x),a(x)+1)=\prod _{i\in B}p_{i}(x),} gcd ( f ( x ) , a ( x ) − 1 ) = ∏ i ∈ C p i ( x ) . {\displaystyle \gcd(f(x),a(x)-1)=\prod _{i\in C}p_{i}(x).} === Algorithm === The Cantor–Zassenhaus algorithm computes polynomials of the same type as a ( x ) {\displaystyle a(x)} above using the isomorphism discussed in the Background section. It proceeds as follows, in the case where the field F q {\displaystyle \mathbb {F} _{q}} is of odd-characteristic (the process can be generalised to characteristic 2 fields in a fairly straightforward way. Select a random polynomial b ( x ) ∈ R {\displaystyle b(x)\in R} such that b ( x ) ≠ 0 , ± 1 {\displaystyle b(x)\neq 0,\pm 1} . Set m = ( q d − 1 ) / 2 {\displaystyle m=(q^{d}-1)/2} and compute b ( x ) m {\displaystyle b(x)^{m}} . Since ϕ {\displaystyle \phi } is an isomorphism, we have (using our now-established notation): ϕ ( b ( x ) m ) = ( b 1 m ( x ) + ⟨ p 1 ( x ) ⟩ , … , b s m ( x ) + ⟨ p s ( x ) ⟩ ) . {\displaystyle \phi (b(x)^{m})=(b_{1}^{m}(x)+\langle p_{1}(x)\rangle ,\ldots ,b_{s}^{m}(x)+\langle p_{s}(x)\rangle ).} Now, each b i ( x ) + ⟨ p i ( x ) ⟩ {\displaystyle b_{i}(x)+\langle p_{i}(x)\rangle } is an element of a field of order q d {\displaystyle q^{d}} , as noted earlier. The multiplicative subgroup of this field has order q d − 1 {\displaystyle q^{d}-1} and so, unless b i ( x ) = 0 {\displaystyle b_{i}(x)=0} , we have b i ( x ) q d − 1 = 1 {\displaystyle b_{i}(x)^{q^{d}-1}=1} for each i and hence b i ( x ) m = ± 1 {\displaystyle b_{i}(x)^{m}=\pm 1} for each i. If b i ( x ) = 0 {\displaystyle b_{i}(x)=0} , then of course b i ( x ) m = 0 {\displaystyle b_{i}(x)^{m}=0} . Hence b ( x ) m {\displaystyle b(x)^{m}} is a polynomial of the same type as a ( x ) {\displaystyle a(x)} above. Further, since b ( x ) ≠ 0 , ± 1 {\displaystyle b(x)\neq 0,\pm 1} , at least two of the sets A , B {\displaystyle A,B} and C are non-empty and by computing the above GCDs we may obtain non-trivial factors. Since the ring of polynomials over a field is a Euclidean domain, we may compute these GCDs using the Euclidean algorithm. == Applications == One important application of the Cantor–Zassenhaus algorithm is in computing discrete logarithms over finite fields of prime-power order. Computing discrete logarithms is an important problem in public key cryptography. For a field of prime-power order, the fastest known method is the index calculus method, which involves the factorisation of field elements. If we represent the prime-power order field in the usual way – that is, as polynomials over the prime order base field, reduced modulo an irreducible polynomial of appropriate degree – then this is simply polynomial factorisation, as provided by the Cantor–Zassenhaus algorithm. == Implementation in computer algebra systems == The Cantor–Zassenhaus algorithm is implemented in the PARI/GP computer algebra system as the factormod() function (formerly factorcantor()). == See also == Polynomial factorization Factorization of polynomials over finite fields == References == == External links == https://web.archive.org/web/20200301213349/http://blog.fkraiem.org/2013/12/01/polynomial-factorisation-over-finite-fields-part-3-final-splitting-cantor-zassenhaus-in-odd-characteristic/
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Wikipedia:Zdeněk Dvořák#0
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Zdeněk Dvořák (born April 26, 1981) is a Czech mathematician specializing in graph theory. Dvořák was born in Nové Město na Moravě. He competed on the Czech national team in the 1999 International Mathematical Olympiad, and in the same year in the International Olympiad in Informatics, where he won a gold medal. He earned his Ph.D. in 2007 from Charles University in Prague, under the supervision of Jaroslav Nešetřil. He remained as a research fellow at Charles University until 2010, and then did postdoctoral studies at the Georgia Institute of Technology and Simon Fraser University. He then returned to the Computer Science Institute (IUUK) of Charles University, obtained his habilitation in 2012, and has been a full professor there since 2022. He was one of three winners of the 2015 European Prize in Combinatorics, "for his fundamental contributions to graph theory, in particular for his work on structural aspects of graph theory, including solutions to Havel's 1969 problem and the Heckman–Thomas 14/5 problem on fractional colourings of cubic triangle-free graphs. This refers to two different results of Dvořák: Havel's conjecture is a strengthening of Grötzsch's theorem. It states that there exists a constant d such that, if a planar graph has no two triangles within distance d of each other, then it can be colored with three colors. A proof of this conjecture of Havel was announced by Dvořák and his co-authors in 2009. C. C. Heckman and Robin Thomas conjectured in 2001 that triangle-free graphs of maximum degree three have fractional chromatic number at most 14/5. A proof was announced by Dvořák and his co-authors in 2013 and published by them in 2014. == References == == External links == Home page
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Wikipedia:Zdeněk Frolík#0
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Zdeněk Frolík (March 10, 1933 – May 3, 1989) was a Czech mathematician. His research interests included topology and functional analysis. In particular, his work concerned covering properties of topological spaces, ultrafilters, homogeneity, measures, uniform spaces. He was one of the founders of modern descriptive theory of sets and spaces. Two classes of topological spaces are given Frolík's name: the class P of all spaces X {\displaystyle X} such that X × Y {\displaystyle X\times Y} is pseudocompact for every pseudocompact space Y {\displaystyle Y} , and the class C of all spaces X {\displaystyle X} such that X × Y {\displaystyle X\times Y} is countably compact for every countably compact space Y {\displaystyle Y} . Frolík prepared his Ph.D. thesis under the supervision of Miroslav Katetov and Eduard Čech. == Selected publications == Generalizations of compact and Lindelöf spaces - Czechoslovak Math. J., 9 (1959), pp. 172–217 (in Russian, English summary) The topological product of countably compact spaces - Czechoslovak Math. J., 10 (1960), pp. 329–338 The topological product of two pseudocompact spaces - Czechoslovak Math. J., 10 (1960), pp. 339–349 Generalizations of the Gδ-property of complete metric spaces - Czechoslovak Math. J., 10 (1960), pp. 359–379 On the topological product of paracompact spaces - Bull. Acad. Polon., 8 (1960), pp. 747–750 Locally complete topological spaces - Dokl. Akad. Nauk SSSR, 137 (1961), pp. 790–792 (in Russian) Applications of complete families of continuous functions to the theory of Q-spaces - Czechoslovak Math. J., 11 (1961), pp. 115–133 Invariance of Gδ-spaces under mappings - Czechoslovak Math. J., 11 (1961), pp. 258–260 On almost real compact spaces - Bull. Acad. Polon., 9 (1961), pp. 247–250 On two problems of W.W. Comfort - Comment. Math. Univ. Carolin., 7 (1966), pp. 139–144 Non-homogeneity of βP- P - Comment. Math. Univ. Carolin., 7 (1966), pp. 705–710 Sums of ultrafilters - Bull. Amer. Math. Soc., 73 (1967), pp. 87–91 Homogeneity problems for extremally disconnected spaces - Comment. Math. Univ. Carolin., 8 (1967), pp. 757–763 Baire sets that are Borelian subspaces - Proc. Roy. Soc. A, 299 (1967), pp. 287–290 On the Suslin-graph theorem - Comment Math. Univ. Carolin., 9 (1968), pp. 243–249 A survey of separable descriptive theory of sets and spaces - Czechoslovak Math. J., 20 (1970), pp. 406–467 A measurable map with analytic domain and metrizable range is quotient - Bull. Amer. Math. Soc., 76 (1970), pp. 1112–1117 Luzin sets are additive - Comment Math. Univ. Carolin., 21 (1980), pp. 527–534 Refinements of perfect maps onto metrizable spaces and an application to Čech-analytic spaces - Topology Appl., 33 (1989), pp. 77–84 Decomposability of completely Suslin additive families - Proc. Amer. Math. Soc., 82 (1981), pp. 359–365 Applications of Luzinian separation principles (non-separable case) - Fund. Math., 117 (1983), pp. 165–185 Analytic and Luzin spaces (non-separable case) - Topology Appl., 19 (1985), pp. 129–156 == See also == Wijsman convergence == References ==
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Wikipedia:Zdeněk Hedrlín#0
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Zdeněk Hedrlín (1933 – April 22, 2018) was a Czech mathematician, specializing in universal algebra and combinatorial theory, both in pure and applied mathematics. Zdeněk Hedrlín received his PhD from Prague's Charles University in 1963. His thesis on commutative semigroups was supervised by Miroslav Katětov. Hedrlín held the title of Docent (associated professor) at Charles University, working at the Faculty of Mathematics and Physics for over 60 years until he died at age 85. He was among the first Czech mathematicians to do research on category theory. Already in the mid-1960s, the Prague school of Zdeněk Hedrlín, Aleš Pultr and Věra Trnková had devised a particularly nice notion of concrete categories over Set, the so-called functor-structured categories ... In 1970 Hedrlín was an Invited Speaker at the International Congress of Mathematicians in Nice. In the later part of his career, he focused on applications of relational structures and led very successful special and interdisciplinary seminars. Applications to biological cell behavior earned him and his students a European grant. (He and his students worked on computational cell models of cancer.) Hedrlín was a member of the editorial board of the Journal of Pure and Applied Algebra. His Erdős number is 1. His doctoral students include Vojtěch Rödl. == Selected publication == Hedrlín, Z. (1961). "On common fixed points of commutative mappings" (PDF). Commentationes Mathematicae Universitatis Carolinae. 2 (4): 25–28. Hedrlín, Zdeněk (1962). "On number of commutative mappings from finite set into itself (Preliminary communication)" (PDF). Commentationes Mathematicae Universitatis Carolinae. 3 (1): 32. Hedrlín, Z.; Pultr, A. (1963). "Remark on topological spaces with given semigroups" (PDF). Commentationes Mathematicae Universitatis Carolinae. 4 (4): 161–163. Hedrlín, Z.; Pultr, A. (1964). "Relations (graphs) with given finitely generated semigroups". Monatshefte für Mathematik. 68 (3): 213–217. doi:10.1007/BF01298508. S2CID 120856684. Pultr, A.; Hedrlín, Z. (1964). "Relations (graphs) with given infinite semigroups". Monatshefte für Mathematik. 68 (5): 421–425. doi:10.1007/BF01304185. S2CID 122610862. Baayen, P. C.; Hedrlin, Z. (1964). "On the existence of well distributed sequences in compact spaces" (PDF). Stichting Mathematisch Centrum. Zuivere Wiskunde. Hedrlín, Z.; Pultr, A. (1965). "Symmetric relations (undirected graphs) with given semigroups". Monatshefte für Mathematik. 69 (4): 318–322. doi:10.1007/BF01297617. S2CID 120384797. Vopěnka, P.; Pultr, A.; Hedrlín, Z. (1965). "A rigid relation exists on any set" (PDF). Commentationes Mathematicae Universitatis Carolinae. 6 (2): 149–155. Hedrlín, Zdeněk; Pultr, Aleš (1966). "On full embeddings of categories of algebras". Illinois Journal of Mathematics. 10 (3): 392–406. doi:10.1215/ijm/1256054991. (over 160 citations) Hedrlín, Z.; Pultr, A. (1966). "On Rigid Undirected Graphs". Canadian Journal of Mathematics. 18: 1237–1242. doi:10.4153/CJM-1966-121-7. S2CID 124453196. Hedrlín, Z.; Vopěnka, P. (1966). "An undecidable theorem concerning full embeddings into categories of algebras" (PDF). Commentationes Mathematicae Universitatis Carolinae. 7 (3): 401–409. Hedrlín, Z.; Pultr, A.; Trnková, V. (1967). "Concerning a categorial approach to topological and algebraic theories" (PDF). In: (ed.): General Topology and its Relations to Modern Analysis and Algebra, Proceedings of the second Prague topological symposium, 1966. Academia Publishing House of the Czechoslovak Academy of Sciences, Praha. pp. 176–181. Hedrlín, Zdeněk; Lambek, Joachim (1969). "How comprehensive is the category of semigroups?". Journal of Algebra. 11 (2): 195–212. doi:10.1016/0021-8693(69)90054-4. Hedrlín, Zdeněk (1969). "On universal partly ordered sets and classes" (PDF). Journal of Algebra. 11 (4): 503–509. doi:10.1016/0021-8693(69)90089-1. Hedrlín, Z.; Mendelsohn, E. (1969). "The Category of Graphs with a Given Subgraph-with Applications to Topology and Algebra". Canadian Journal of Mathematics. 21: 1506–1517. doi:10.4153/CJM-1969-165-5. S2CID 124324655. Goralčík, Pavel; Hedrlín, Zdeněk (1971). "On reconstruction of monoids from their table fragments". Mathematische Zeitschrift. 122: 82–92. doi:10.1007/BF01113568. S2CID 120230682. Chvatal, V.; Erdös, P.; Hedrlín, Z. (1972). "Ramsey's theorem and self-complementary graphs". Discrete Mathematics. 3 (4): 301–304. doi:10.1016/0012-365X(72)90087-8. Goralčík, P.; Hedrlín, Z.; Koubek, V.; Ryšunková, J. (1982). "A game of composing binary relations" (PDF). R.A.I.R.O.: Informatique Théorique. 16 (4): 365–369. doi:10.1051/ita/1982160403651. Hedrlín, Z.; Hell, P.; Ko, C.S. (1982). "Homomorphism Interpolation and Approximation". Algebraic and Geometric Combinatorics. North-Holland Mathematics Studies. Vol. 65. pp. 213–227. doi:10.1016/S0304-0208(08)73267-5. ISBN 9780444863652. == References ==
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Wikipedia:Zdzisław Józef Porosiński#0
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Zdzisław Józef Porosiński (19 March 1955 in Kłodzko, Poland – 19 March 2016 in Wrocław, Poland) was a Polish mathematician and statistician. == Biography == In 1979, he graduated in mathematics from Faculty of Fundamental Problems of Technology, Wrocław University of Technology with a master's degree. After graduation, he started working at his alma mater at the Institute of Mathematics (later renamed Institute of Mathematics and Informatics of the Wrocław University of Technology, and from 2015 transformed into the Faculty of Pure and Applied Mathematics). On 17 March 1987, he defended his doctoral dissertation entitled Selected problems of optimal stopping . The research supervisor and thesis supervisor was professor Stanisław Trybuła. On 20 November 2003, the habilitation colloquium was held. On 31 May 2004, he obtained a postdoctoral degree in mathematical sciences. In the years 2001–2008 he worked at the Institute of Basic Sciences State Higher Vocational School in Nysa. In the years 2006–2012 he was the deputy director for didactics at Institute of Mathematics and Computer Science of the Wrocław University of Technology. From 2009 he was a professor Wrocław University of Technology. == Contributions == His research contributions include over 40 papers. His work in probability theory included work on optimal stopping of the sequences of random variables and the statistics of stochastic processes. The most cited results concern the optimal strategies for the optimal stopping problem when the decision horizon is random. He was the doctoral advisor of one student. == References ==
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Wikipedia:Zdzisław Pawlak#0
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Zdzislaw I. Pawlak (10 November 1926 – 7 April 2006) was a Polish mathematician and computer scientist. He was affiliated with several organization, including the Polish Academy of Sciences and the Warsaw School of Information Technology. He served as the director of the Institute of Computer Science at the Warsaw University of Technology (1989–96). Pawlak was known for his contribution to many branches of theoretical computer science. He is credited with introducing the rough set theory and also known for his fundamental works on it. He also introduced the Pawlak flow graphs, a graphical framework for reasoning from data. He was conferred with Order of Polonia Restituta in 1999. He was also a full member of Polish Academy of Sciences. == Education and career == Zdzislaw Pawlak was born on 10 November 1926 in Łódź, Poland. He graduated from a public elementary school in 1939. In 1946 he passed his Baccalaureate Diploma examination, and in 1947 he began studies at the Faculty of Electrical Engineering of the Łódź University of Technology. Two years later, he moved to the Faculty of Telecommunications at the Warsaw University of Technology. He received his M.Sc. degree in Telecommunications in 1951, after which he worked in the Institute of Mathematics of the Polish Academy of Sciences until 1957. == References == == External links == Zdzisław Pawlak at the Mathematics Genealogy Project The International Rough Set Society
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Wikipedia:Zdzisław Skupień#0
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Zdzisław Skupień (27 November 1938 – 1 January 2025) was a Polish mathematician, expert in optimization, discrete mathematics, and graph theory, academic, and dr. hab. (1982). Skupień was born in Świlcza, Poland. In 1964, Skupień introduced the concept of "locally Hamiltonian graphs". In 1976, Skupień introduced the concept of "homogeneously traceable graphs". Skupień authored over 140 publications. He died on 1 January 2025, at the age of 86. == Awards == 1998: Medal of the Commission for National Education (Medal Komisji Edukacji Narodowej) 1988: Order of Polonia Restituta (Krzyż Kawalerski Orderu Odrodzenia Polski) 1983: Cross of Merit (Złoty Krzyż Zasługi) == References ==
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Wikipedia:Zech's logarithm#0
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Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator α {\displaystyle \alpha } . Zech logarithms are named after Julius Zech, and are also called Jacobi logarithms, after Carl G. J. Jacobi who used them for number theoretic investigations. == Definition == Given a primitive element α {\displaystyle \alpha } of a finite field, the Zech logarithm relative to the base α {\displaystyle \alpha } is defined by the equation α Z α ( n ) = 1 + α n , {\displaystyle \alpha ^{Z_{\alpha }(n)}=1+\alpha ^{n},} which is often rewritten as Z α ( n ) = log α ( 1 + α n ) . {\displaystyle Z_{\alpha }(n)=\log _{\alpha }(1+\alpha ^{n}).} The choice of base α {\displaystyle \alpha } is usually dropped from the notation when it is clear from the context. To be more precise, Z α {\displaystyle Z_{\alpha }} is a function on the integers modulo the multiplicative order of α {\displaystyle \alpha } , and takes values in the same set. In order to describe every element, it is convenient to formally add a new symbol − ∞ {\displaystyle -\infty } , along with the definitions α − ∞ = 0 {\displaystyle \alpha ^{-\infty }=0} n + ( − ∞ ) = − ∞ {\displaystyle n+(-\infty )=-\infty } Z α ( − ∞ ) = 0 {\displaystyle Z_{\alpha }(-\infty )=0} Z α ( e ) = − ∞ {\displaystyle Z_{\alpha }(e)=-\infty } where e {\displaystyle e} is an integer satisfying α e = − 1 {\displaystyle \alpha ^{e}=-1} , that is e = 0 {\displaystyle e=0} for a field of characteristic 2, and e = q − 1 2 {\displaystyle e={\frac {q-1}{2}}} for a field of odd characteristic with q {\displaystyle q} elements. Using the Zech logarithm, finite field arithmetic can be done in the exponential representation: α m + α n = α m ⋅ ( 1 + α n − m ) = α m ⋅ α Z ( n − m ) = α m + Z ( n − m ) {\displaystyle \alpha ^{m}+\alpha ^{n}=\alpha ^{m}\cdot (1+\alpha ^{n-m})=\alpha ^{m}\cdot \alpha ^{Z(n-m)}=\alpha ^{m+Z(n-m)}} − α n = ( − 1 ) ⋅ α n = α e ⋅ α n = α e + n {\displaystyle -\alpha ^{n}=(-1)\cdot \alpha ^{n}=\alpha ^{e}\cdot \alpha ^{n}=\alpha ^{e+n}} α m − α n = α m + ( − α n ) = α m + Z ( e + n − m ) {\displaystyle \alpha ^{m}-\alpha ^{n}=\alpha ^{m}+(-\alpha ^{n})=\alpha ^{m+Z(e+n-m)}} α m ⋅ α n = α m + n {\displaystyle \alpha ^{m}\cdot \alpha ^{n}=\alpha ^{m+n}} ( α m ) − 1 = α − m {\displaystyle \left(\alpha ^{m}\right)^{-1}=\alpha ^{-m}} α m / α n = α m ⋅ ( α n ) − 1 = α m − n {\displaystyle \alpha ^{m}/\alpha ^{n}=\alpha ^{m}\cdot \left(\alpha ^{n}\right)^{-1}=\alpha ^{m-n}} These formulas remain true with our conventions with the symbol − ∞ {\displaystyle -\infty } , with the caveat that subtraction of − ∞ {\displaystyle -\infty } is undefined. In particular, the addition and subtraction formulas need to treat m = − ∞ {\displaystyle m=-\infty } as a special case. This can be extended to arithmetic of the projective line by introducing another symbol + ∞ {\displaystyle +\infty } satisfying α + ∞ = ∞ {\displaystyle \alpha ^{+\infty }=\infty } and other rules as appropriate. For fields of characteristic 2, Z α ( n ) = m ⟺ Z α ( m ) = n {\displaystyle Z_{\alpha }(n)=m\iff Z_{\alpha }(m)=n} . == Uses == For sufficiently small finite fields, a table of Zech logarithms allows an especially efficient implementation of all finite field arithmetic in terms of a small number of integer addition/subtractions and table look-ups. The utility of this method diminishes for large fields where one cannot efficiently store the table. This method is also inefficient when doing very few operations in the finite field, because one spends more time computing the table than one does in actual calculation. == Examples == Let α ∈ GF(23) be a root of the primitive polynomial x3 + x2 + 1. The traditional representation of elements of this field is as polynomials in α of degree 2 or less. A table of Zech logarithms for this field are Z(−∞) = 0, Z(0) = −∞, Z(1) = 5, Z(2) = 3, Z(3) = 2, Z(4) = 6, Z(5) = 1, and Z(6) = 4. The multiplicative order of α is 7, so the exponential representation works with integers modulo 7. Since α is a root of x3 + x2 + 1 then that means α3 + α2 + 1 = 0, or if we recall that since all coefficients are in GF(2), subtraction is the same as addition, we obtain α3 = α2 + 1. The conversion from exponential to polynomial representations is given by α 3 = α 2 + 1 {\displaystyle \alpha ^{3}=\alpha ^{2}+1} (as shown above) α 4 = α 3 α = ( α 2 + 1 ) α = α 3 + α = α 2 + α + 1 {\displaystyle \alpha ^{4}=\alpha ^{3}\alpha =(\alpha ^{2}+1)\alpha =\alpha ^{3}+\alpha =\alpha ^{2}+\alpha +1} α 5 = α 4 α = ( α 2 + α + 1 ) α = α 3 + α 2 + α = α 2 + 1 + α 2 + α = α + 1 {\displaystyle \alpha ^{5}=\alpha ^{4}\alpha =(\alpha ^{2}+\alpha +1)\alpha =\alpha ^{3}+\alpha ^{2}+\alpha =\alpha ^{2}+1+\alpha ^{2}+\alpha =\alpha +1} α 6 = α 5 α = ( α + 1 ) α = α 2 + α {\displaystyle \alpha ^{6}=\alpha ^{5}\alpha =(\alpha +1)\alpha =\alpha ^{2}+\alpha } Using Zech logarithms to compute α 6 + α 3: α 6 + α 3 = α 6 + Z ( − 3 ) = α 6 + Z ( 4 ) = α 6 + 6 = α 12 = α 5 , {\displaystyle \alpha ^{6}+\alpha ^{3}=\alpha ^{6+Z(-3)}=\alpha ^{6+Z(4)}=\alpha ^{6+6}=\alpha ^{12}=\alpha ^{5},} or, more efficiently, α 6 + α 3 = α 3 + Z ( 3 ) = α 3 + 2 = α 5 , {\displaystyle \alpha ^{6}+\alpha ^{3}=\alpha ^{3+Z(3)}=\alpha ^{3+2}=\alpha ^{5},} and verifying it in the polynomial representation: α 6 + α 3 = ( α 2 + α ) + ( α 2 + 1 ) = α + 1 = α 5 . {\displaystyle \alpha ^{6}+\alpha ^{3}=(\alpha ^{2}+\alpha )+(\alpha ^{2}+1)=\alpha +1=\alpha ^{5}.} == See also == Gaussian logarithm Irish logarithm, a similar technique derived empirically by Percy Ludgate Finite field arithmetic Logarithm table == References == == Further reading == Fletcher, Alan; Miller, Jeffrey Charles Percy; Rosenhead, Louis (1946) [1943]. An Index of Mathematical Tables (1 ed.). Blackwell Scientific Publications Ltd., Oxford / McGraw-Hill, New York. Conway, John Horton (1968). Churchhouse, Robert F.; Herz, J.-C. (eds.). "A tabulation of some information concerning finite fields". Computers in Mathematical Research. Amsterdam: North-Holland Publishing Company: 37–50. MR 0237467. Lam, Clement Wing Hong; McKay, John K. S. (1973-11-01). "Algorithm 469: Arithmetic over a finite field [A1]". Communications of the ACM. Collected Algorithms of the ACM (CALGO). 16 (11). Association for Computing Machinery (ACM): 699. doi:10.1145/355611.362544. ISSN 0001-0782. S2CID 62794130. toms/469. [1] [2] [3] Kühn, Klaus (2008). "C. F. Gauß und die Logarithmen" (PDF) (in German). Alling-Biburg, Germany. Archived (PDF) from the original on 2018-07-14. Retrieved 2018-07-14.
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Wikipedia:Zeev Rudnick#0
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Zeev Rudnick or Ze'ev Rudnick (Hebrew: זאב רודניק; born 1961 in Haifa, Israel) is a mathematician, specializing in number theory and in mathematical physics, notably quantum chaos. Rudnick is a professor at the School of Mathematical Sciences and the Cissie and Aaron Beare Chair in Number Theory at Tel Aviv University. == Education == Rudnick received his PhD from Yale University in 1990 under the supervision of Ilya Piatetski-Shapiro and Roger Evans Howe. == Career == Rudnick joined Tel Aviv University in 1995, after working as an assistant professor at Princeton and Stanford. In 2003–4 Rudnick was a Leverhulme visiting professor at the University of Bristol and in 2008–2010 and 2015–2016 he was a member of the Institute for Advanced Study at Princeton. In 2012, Rudnick was inducted as a fellow of the American Mathematical Society. == Research == Rudnick has been studying different aspects of quantum chaos and number theory. He has contributed to one of the discoveries concerning the Riemann zeta function, namely, that the Riemann zeros appear to display the same statistics as those which are believed to be present in energy levels of quantum chaotic systems and described by random matrix theory. Together with Peter Sarnak, he has formulated the Quantum Unique Ergodicity conjectures for eigenfunctions on negatively curved manifolds, and has investigated the question arising from Quantum Chaos in other arithmetic models such as the Quantum Cat map (with Par Kurlberg) and the flat torus (with CP Hughes and with Jean Bourgain). Another interest is the interface between function field arithmetic and corresponding problems in number fields. == Education == Ph.D., 1990, Yale University. M.Sc., 1985, The Hebrew University, Jerusalem. B.Sc., 1984, Bar-Ilan University, Ramat Gan. == Awards and fellowships == ERC Advanced Grants, 1.7 million euro, 2013–2018., 2019–2024. Fellow of the American Mathematical Society, 2012–. Annales Henri Poincaré Distinguished Paper Award for the year, 2011. Erdős Prize of the Israel Mathematical Union, 2001. Alon Fellow, 1995. Sloan Foundation Doctoral Dissertation Fellowship, 1989–1990. == Selected works == Duke, William; Rudnick, Zeev; Sarnak, Peter (1993). "Density of integer points on affine homogeneous varieties". Duke Mathematical Journal. 71: 143–179. CiteSeerX 10.1.1.218.5083. doi:10.1215/S0012-7094-93-07107-4. MR 1230289. Z. Rudnick, P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. in Math. Physics 161, 195–213 (1994). W. Luo, Z. Rudnick and P. Sarnak, On Selberg's eigenvalue conjecture, Geom. and Func. Analysis 5 (1995), 387–401. Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory, Duke Mathematical Journal 81 (1996), 269–322 (special volume in honor of J. Nash). P. Kurlberg and Z. Rudnick, Hecke theory and equidistribution for the quantization of linear maps of the torus, Duke Mathematical Journal 103 (2000), 47–78. Z. Rudnick and K. Soundararajan, Lower bounds for moments of L-functions, Proc. of the National Academy of Sciences of the USA, 102 (19), (May 10, 2005), 6837–6838. Z. Rudnick, What is Quantum Chaos?, Notices of the AMS, 55 number 1 (2008), 32–34. J. Bourgain and Z. Rudnick, Restriction of toral eigenfunctions to hypersurfaces, C.R. Math. Acad. Sci. Paris 347 (2009), no 21–22, 1249–1253. Jonathan P. Keating and Zeev Rudnick, The variance of the number of prime polynomials in short intervals and in residue classes. International Mathematics Research Notices 2012; doi: 10.1093/imrn/rns220. Alexei Entin, Edva Roditty-Gershon and Zeev Rudnick, Low-lying zeros of quadratic Dirichlet L-functions, hyper-elliptic curves and Random Matrix Theory, Geom. Funct. Anal. 23 (2013), no. 4, 1230–1261. doi:10.1007/s00039-013-0241-8 == References == == External links == Zeev Rudnick's homepage Articles by Zeev Rudnick on the Arxiv Reviews of publications by Zeev Rudnick by the American Mathematical Society
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Wikipedia:Zero divisor#0
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In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x in R such that ax = 0, or equivalently if the map from R to R that sends x to ax is not injective. Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y in R such that ya = 0. This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element a that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero x such that ax = 0 may be different from the nonzero y such that ya = 0). If the ring is commutative, then the left and right zero divisors are the same. An element of a ring that is not a left zero divisor (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, right regular or right cancellable). An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellable, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A non-zero ring with no nontrivial zero divisors is called a domain. == Examples == In the ring Z / 4 Z {\displaystyle \mathbb {Z} /4\mathbb {Z} } , the residue class 2 ¯ {\displaystyle {\overline {2}}} is a zero divisor since 2 ¯ × 2 ¯ = 4 ¯ = 0 ¯ {\displaystyle {\overline {2}}\times {\overline {2}}={\overline {4}}={\overline {0}}} . The only zero divisor of the ring Z {\displaystyle \mathbb {Z} } of integers is 0 {\displaystyle 0} . A nilpotent element of a nonzero ring is always a two-sided zero divisor. An idempotent element e ≠ 1 {\displaystyle e\neq 1} of a ring is always a two-sided zero divisor, since e ( 1 − e ) = 0 = ( 1 − e ) e {\displaystyle e(1-e)=0=(1-e)e} . The ring of n × n matrices over a field has nonzero zero divisors if n ≥ 2. Examples of zero divisors in the ring of 2 × 2 matrices (over any nonzero ring) are shown here: ( 1 1 2 2 ) ( 1 1 − 1 − 1 ) = ( − 2 1 − 2 1 ) ( 1 1 2 2 ) = ( 0 0 0 0 ) , {\displaystyle {\begin{pmatrix}1&1\\2&2\end{pmatrix}}{\begin{pmatrix}1&1\\-1&-1\end{pmatrix}}={\begin{pmatrix}-2&1\\-2&1\end{pmatrix}}{\begin{pmatrix}1&1\\2&2\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}},} ( 1 0 0 0 ) ( 0 0 0 1 ) = ( 0 0 0 1 ) ( 1 0 0 0 ) = ( 0 0 0 0 ) . {\displaystyle {\begin{pmatrix}1&0\\0&0\end{pmatrix}}{\begin{pmatrix}0&0\\0&1\end{pmatrix}}={\begin{pmatrix}0&0\\0&1\end{pmatrix}}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}.} A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in R 1 × R 2 {\displaystyle R_{1}\times R_{2}} with each R i {\displaystyle R_{i}} nonzero, ( 1 , 0 ) ( 0 , 1 ) = ( 0 , 0 ) {\displaystyle (1,0)(0,1)=(0,0)} , so ( 1 , 0 ) {\displaystyle (1,0)} is a zero divisor. Let K {\displaystyle K} be a field and G {\displaystyle G} be a group. Suppose that G {\displaystyle G} has an element g {\displaystyle g} of finite order n > 1 {\displaystyle n>1} . Then in the group ring K [ G ] {\displaystyle K[G]} one has ( 1 − g ) ( 1 + g + ⋯ + g n − 1 ) = 1 − g n = 0 {\displaystyle (1-g)(1+g+\cdots +g^{n-1})=1-g^{n}=0} , with neither factor being zero, so 1 − g {\displaystyle 1-g} is a nonzero zero divisor in K [ G ] {\displaystyle K[G]} . === One-sided zero-divisor === Consider the ring of (formal) matrices ( x y 0 z ) {\displaystyle {\begin{pmatrix}x&y\\0&z\end{pmatrix}}} with x , z ∈ Z {\displaystyle x,z\in \mathbb {Z} } and y ∈ Z / 2 Z {\displaystyle y\in \mathbb {Z} /2\mathbb {Z} } . Then ( x y 0 z ) ( a b 0 c ) = ( x a x b + y c 0 z c ) {\displaystyle {\begin{pmatrix}x&y\\0&z\end{pmatrix}}{\begin{pmatrix}a&b\\0&c\end{pmatrix}}={\begin{pmatrix}xa&xb+yc\\0&zc\end{pmatrix}}} and ( a b 0 c ) ( x y 0 z ) = ( x a y a + z b 0 z c ) {\displaystyle {\begin{pmatrix}a&b\\0&c\end{pmatrix}}{\begin{pmatrix}x&y\\0&z\end{pmatrix}}={\begin{pmatrix}xa&ya+zb\\0&zc\end{pmatrix}}} . If x ≠ 0 ≠ z {\displaystyle x\neq 0\neq z} , then ( x y 0 z ) {\displaystyle {\begin{pmatrix}x&y\\0&z\end{pmatrix}}} is a left zero divisor if and only if x {\displaystyle x} is even, since ( x y 0 z ) ( 0 1 0 0 ) = ( 0 x 0 0 ) {\displaystyle {\begin{pmatrix}x&y\\0&z\end{pmatrix}}{\begin{pmatrix}0&1\\0&0\end{pmatrix}}={\begin{pmatrix}0&x\\0&0\end{pmatrix}}} , and it is a right zero divisor if and only if z {\displaystyle z} is even for similar reasons. If either of x , z {\displaystyle x,z} is 0 {\displaystyle 0} , then it is a two-sided zero-divisor. Here is another example of a ring with an element that is a zero divisor on one side only. Let S {\displaystyle S} be the set of all sequences of integers ( a 1 , a 2 , a 3 , . . . ) {\displaystyle (a_{1},a_{2},a_{3},...)} . Take for the ring all additive maps from S {\displaystyle S} to S {\displaystyle S} , with pointwise addition and composition as the ring operations. (That is, our ring is E n d ( S ) {\displaystyle \mathrm {End} (S)} , the endomorphism ring of the additive group S {\displaystyle S} .) Three examples of elements of this ring are the right shift R ( a 1 , a 2 , a 3 , . . . ) = ( 0 , a 1 , a 2 , . . . ) {\displaystyle R(a_{1},a_{2},a_{3},...)=(0,a_{1},a_{2},...)} , the left shift L ( a 1 , a 2 , a 3 , . . . ) = ( a 2 , a 3 , a 4 , . . . ) {\displaystyle L(a_{1},a_{2},a_{3},...)=(a_{2},a_{3},a_{4},...)} , and the projection map onto the first factor P ( a 1 , a 2 , a 3 , . . . ) = ( a 1 , 0 , 0 , . . . ) {\displaystyle P(a_{1},a_{2},a_{3},...)=(a_{1},0,0,...)} . All three of these additive maps are not zero, and the composites L P {\displaystyle LP} and P R {\displaystyle PR} are both zero, so L {\displaystyle L} is a left zero divisor and R {\displaystyle R} is a right zero divisor in the ring of additive maps from S {\displaystyle S} to S {\displaystyle S} . However, L {\displaystyle L} is not a right zero divisor and R {\displaystyle R} is not a left zero divisor: the composite L R {\displaystyle LR} is the identity. R L {\displaystyle RL} is a two-sided zero-divisor since R L P = 0 = P R L {\displaystyle RLP=0=PRL} , while L R = 1 {\displaystyle LR=1} is not in any direction. == Non-examples == The ring of integers modulo a prime number has no nonzero zero divisors. Since every nonzero element is a unit, this ring is a finite field. More generally, a division ring has no nonzero zero divisors. A non-zero commutative ring whose only zero divisor is 0 is called an integral domain. == Properties == In the ring of n × n matrices over a field, the left and right zero divisors coincide; they are precisely the singular matrices. In the ring of n × n matrices over an integral domain, the zero divisors are precisely the matrices with determinant zero. Left or right zero divisors can never be units, because if a is invertible and ax = 0 for some nonzero x, then 0 = a−10 = a−1ax = x, a contradiction. An element is cancellable on the side on which it is regular. That is, if a is a left regular, ax = ay implies that x = y, and similarly for right regular. == Zero as a zero divisor == There is no need for a separate convention for the case a = 0, because the definition applies also in this case: If R is a ring other than the zero ring, then 0 is a (two-sided) zero divisor, because any nonzero element x satisfies 0x = 0 = x 0. If R is the zero ring, in which 0 = 1, then 0 is not a zero divisor, because there is no nonzero element that when multiplied by 0 yields 0. Some references include or exclude 0 as a zero divisor in all rings by convention, but they then suffer from having to introduce exceptions in statements such as the following: In a commutative ring R, the set of non-zero-divisors is a multiplicative set in R. (This, in turn, is important for the definition of the total quotient ring.) The same is true of the set of non-left-zero-divisors and the set of non-right-zero-divisors in an arbitrary ring, commutative or not. In a commutative noetherian ring R, the set of zero divisors is the union of the associated prime ideals of R. == Zero divisor on a module == Let R be a commutative ring, let M be an R-module, and let a be an element of R. One says that a is M-regular if the "multiplication by a" map M → a M {\displaystyle M\,{\stackrel {a}{\to }}\,M} is injective, and that a is a zero divisor on M otherwise. The set of M-regular elements is a multiplicative set in R. Specializing the definitions of "M-regular" and "zero divisor on M" to the case M = R recovers the definitions of "regular" and "zero divisor" given earlier in this article. == See also == Zero-product property Glossary of commutative algebra (Exact zero divisor) Zero-divisor graph Sedenions, which have zero divisors == Notes == == References == == Further reading == "Zero divisor", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Michiel Hazewinkel; Nadiya Gubareni; Nadezhda Mikhaĭlovna Gubareni; Vladimir V. Kirichenko. (2004), Algebras, rings and modules, vol. 1, Springer, ISBN 1-4020-2690-0 Weisstein, Eric W. "Zero Divisor". MathWorld.
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Wikipedia:Zero mode#0
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In physics, a zero mode is an eigenvector with a vanishing eigenvalue. In various subfields of physics zero modes appear whenever a physical system possesses a certain symmetry. For example, normal modes of multidimensional harmonic oscillator (e.g. a system of beads arranged around the circle, connected with springs) corresponds to elementary vibrational modes of the system. In such a system zero modes typically occur and are related with a rigid rotation around the circle. The kernel of an operator consists of left zero modes, and the cokernel consists of the right zero modes. == References ==
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Wikipedia:Zero object (algebra)#0
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In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The aforementioned abelian group structure is usually identified as addition, and the only element is called zero, so the object itself is typically denoted as {0}. One often refers to the trivial object (of a specified category) since every trivial object is isomorphic to any other (under a unique isomorphism). Instances of the zero object include, but are not limited to the following: As a group, the zero group or trivial group. As a ring, the zero ring or trivial ring. As an algebra over a field or algebra over a ring, the trivial algebra. As a module (over a ring R), the zero module. The term trivial module is also used, although it may be ambiguous, as a trivial G-module is a G-module with a trivial action. As a vector space (over a field R), the zero vector space, zero-dimensional vector space or just zero space. These objects are described jointly not only based on the common singleton and trivial group structure, but also because of shared category-theoretical properties. In the last three cases the scalar multiplication by an element of the base ring (or field) is defined as: κ0 = 0 , where κ ∈ R. The most general of them, the zero module, is a finitely-generated module with an empty generating set. For structures requiring the multiplication structure inside the zero object, such as the trivial ring, there is only one possible, 0 × 0 = 0, because there are no non-zero elements. This structure is associative and commutative. A ring R which has both an additive and multiplicative identity is trivial if and only if 1 = 0, since this equality implies that for all r within R, r = r × 1 = r × 0 = 0. {\displaystyle r=r\times 1=r\times 0=0.} In this case it is possible to define division by zero, since the single element is its own multiplicative inverse. Some properties of {0} depend on exact definition of the multiplicative identity; see § Unital structures below. Any trivial algebra is also a trivial ring. A trivial algebra over a field is simultaneously a zero vector space considered below. Over a commutative ring, a trivial algebra is simultaneously a zero module. The trivial ring is an example of a rng of square zero. A trivial algebra is an example of a zero algebra. The zero-dimensional vector space is an especially ubiquitous example of a zero object, a vector space over a field with an empty basis. It therefore has dimension zero. It is also a trivial group over addition, and a trivial module mentioned above. == Properties == The zero ring, zero module and zero vector space are the zero objects of, respectively, the category of pseudo-rings, the category of modules and the category of vector spaces. However, the zero ring is not a zero object in the category of rings, since there is no ring homomorphism of the zero ring in any other ring. The zero object, by definition, must be a terminal object, which means that a morphism A → {0} must exist and be unique for an arbitrary object A. This morphism maps any element of A to 0. The zero object, also by definition, must be an initial object, which means that a morphism {0} → A must exist and be unique for an arbitrary object A. This morphism maps 0, the only element of {0}, to the zero element 0 ∈ A, called the zero vector in vector spaces. This map is a monomorphism, and hence its image is isomorphic to {0}. For modules and vector spaces, this subset {0} ⊂ A is the only empty-generated submodule (or 0-dimensional linear subspace) in each module (or vector space) A. === Unital structures === The {0} object is a terminal object of any algebraic structure where it exists, like it was described for examples above. But its existence and, if it exists, the property to be an initial object (and hence, a zero object in the category-theoretical sense) depend on exact definition of the multiplicative identity 1 in a specified structure. If the definition of 1 requires that 1 ≠ 0, then the {0} object cannot exist because it may contain only one element. In particular, the zero ring is not a field. If mathematicians sometimes talk about a field with one element, this abstract and somewhat mysterious mathematical object is not a field. In categories where the multiplicative identity must be preserved by morphisms, but can equal to zero, the {0} object can exist. But not as initial object because identity-preserving morphisms from {0} to any object where 1 ≠ 0 do not exist. For example, in the category of rings Ring the ring of integers Z is the initial object, not {0}. If an algebraic structure requires the multiplicative identity, but neither its preservation by morphisms nor 1 ≠ 0, then zero morphisms exist and the situation is not different from non-unital structures considered in the previous section. == Notation == Zero vector spaces and zero modules are usually denoted by 0 (instead of {0}). This is always the case when they occur in an exact sequence. == See also == Nildimensional space Triviality (mathematics) Examples of vector spaces Field with one element Empty semigroup Zero element List of zero terms == External links == David Sharpe (1987). Rings and factorization. Cambridge University Press. p. 10 : trivial ring. ISBN 0-521-33718-6. Barile, Margherita. "Trivial Module". MathWorld. Barile, Margherita. "Zero Module". MathWorld.
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Wikipedia:Zero of a function#0
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In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f {\displaystyle f} , is a member x {\displaystyle x} of the domain of f {\displaystyle f} such that f ( x ) {\displaystyle f(x)} vanishes at x {\displaystyle x} ; that is, the function f {\displaystyle f} attains the value of 0 at x {\displaystyle x} , or equivalently, x {\displaystyle x} is a solution to the equation f ( x ) = 0 {\displaystyle f(x)=0} . A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f {\displaystyle f} of degree two, defined by f ( x ) = x 2 − 5 x + 6 = ( x − 2 ) ( x − 3 ) {\displaystyle f(x)=x^{2}-5x+6=(x-2)(x-3)} has the two roots (or zeros) that are 2 and 3. f ( 2 ) = 2 2 − 5 × 2 + 6 = 0 and f ( 3 ) = 3 2 − 5 × 3 + 6 = 0. {\displaystyle f(2)=2^{2}-5\times 2+6=0{\text{ and }}f(3)=3^{2}-5\times 3+6=0.} If the function maps real numbers to real numbers, then its zeros are the x {\displaystyle x} -coordinates of the points where its graph meets the x-axis. An alternative name for such a point ( x , 0 ) {\displaystyle (x,0)} in this context is an x {\displaystyle x} -intercept. == Solution of an equation == Every equation in the unknown x {\displaystyle x} may be rewritten as f ( x ) = 0 {\displaystyle f(x)=0} by regrouping all the terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function f {\displaystyle f} . In other words, a "zero of a function" is precisely a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations. == Polynomial roots == Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions). === Fundamental theorem of algebra === The fundamental theorem of algebra states that every polynomial of degree n {\displaystyle n} has n {\displaystyle n} complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs. Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. == Computing roots == There are many methods for computing accurate approximations of roots of functions, the best being Newton's method, see Root-finding algorithm. For polynomials, there are specialized algorithms that are more efficient and may provide all roots or all real roots; see Polynomial root-finding and Real-root isolation. Some polynomial, including all those of degree no greater than 4, can have all their roots expressed algebraically in terms of their coefficients; see Solution in radicals. == Zero set == In various areas of mathematics, the zero set of a function is the set of all its zeros. More precisely, if f : X → R {\displaystyle f:X\to \mathbb {R} } is a real-valued function (or, more generally, a function taking values in some additive group), its zero set is f − 1 ( 0 ) {\displaystyle f^{-1}(0)} , the inverse image of { 0 } {\displaystyle \{0\}} in X {\displaystyle X} . Under the same hypothesis on the codomain of the function, a level set of a function f {\displaystyle f} is the zero set of the function f − c {\displaystyle f-c} for some c {\displaystyle c} in the codomain of f . {\displaystyle f.} The zero set of a linear map is also known as its kernel. The cozero set of the function f : X → R {\displaystyle f:X\to \mathbb {R} } is the complement of the zero set of f {\displaystyle f} (i.e., the subset of X {\displaystyle X} on which f {\displaystyle f} is nonzero). === Applications === In algebraic geometry, the first definition of an algebraic variety is through zero sets. Specifically, an affine algebraic set is the intersection of the zero sets of several polynomials, in a polynomial ring k [ x 1 , … , x n ] {\displaystyle k\left[x_{1},\ldots ,x_{n}\right]} over a field. In this context, a zero set is sometimes called a zero locus. In analysis and geometry, any closed subset of R n {\displaystyle \mathbb {R} ^{n}} is the zero set of a smooth function defined on all of R n {\displaystyle \mathbb {R} ^{n}} . This extends to any smooth manifold as a corollary of paracompactness. In differential geometry, zero sets are frequently used to define manifolds. An important special case is the case that f {\displaystyle f} is a smooth function from R p {\displaystyle \mathbb {R} ^{p}} to R n {\displaystyle \mathbb {R} ^{n}} . If zero is a regular value of f {\displaystyle f} , then the zero set of f {\displaystyle f} is a smooth manifold of dimension m = p − n {\displaystyle m=p-n} by the regular value theorem. For example, the unit m {\displaystyle m} -sphere in R m + 1 {\displaystyle \mathbb {R} ^{m+1}} is the zero set of the real-valued function f ( x ) = ‖ x ‖ 2 − 1 {\displaystyle f(x)=\Vert x\Vert ^{2}-1} . == See also == Root-finding algorithm Bolzano's theorem, a continuous function that takes opposite signs at the end points of an interval has at least a zero in the interval. Gauss–Lucas theorem, the complex zeros of the derivative of a polynomial lie inside the convex hull of the roots of the polynomial. Marden's theorem, a refinement of Gauss–Lucas theorem for polynomials of degree three Sendov's conjecture, a conjectured refinement of Gauss-Lucas theorem zero at infinity Zero crossing, property of the graph of a function near a zero Zeros and poles of holomorphic functions == References == == Further reading == Weisstein, Eric W. "Root". MathWorld.
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Wikipedia:Zero to the power of zero#0
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Zero to the power of zero, denoted as 00, is a mathematical expression with different interpretations depending on the context. In certain areas of mathematics, such as combinatorics and algebra, 00 is conventionally defined as 1 because this assignment simplifies many formulas and ensures consistency in operations involving exponents. For instance, in combinatorics, defining 00 = 1 aligns with the interpretation of choosing 0 elements from a set and simplifies polynomial and binomial expansions. However, in other contexts, particularly in mathematical analysis, 00 is often considered an indeterminate form. This is because the value of xy as both x and y approach zero can lead to different results based on the limiting process. The expression arises in limit problems and may result in a range of values or diverge to infinity, making it difficult to assign a single consistent value in these cases. The treatment of 00 also varies across different computer programming languages and software. While many follow the convention of assigning 00 = 1 for practical reasons, others leave it undefined or return errors depending on the context of use, reflecting the ambiguity of the expression in mathematical analysis. == Discrete exponents == Many widely used formulas involving natural-number exponents require 00 to be defined as 1. For example, the following three interpretations of b0 make just as much sense for b = 0 as they do for positive integers b: The interpretation of b0 as an empty product assigns it the value 1. The combinatorial interpretation of b0 is the number of 0-tuples of elements from a b-element set; there is exactly one 0-tuple. The set-theoretic interpretation of b0 is the number of functions from the empty set to a b-element set; there is exactly one such function, namely, the empty function. All three of these specialize to give 00 = 1. == Polynomials and power series == When evaluating polynomials, it is convenient to define 00 as 1. A (real) polynomial is an expression of the form a0x0 + ⋅⋅⋅ + anxn, where x is an indeterminate, and the coefficients ai are real numbers. Polynomials are added termwise, and multiplied by applying the distributive law and the usual rules for exponents. With these operations, polynomials form a ring R[x]. The multiplicative identity of R[x] is the polynomial x0; that is, x0 times any polynomial p(x) is just p(x). Also, polynomials can be evaluated by specializing x to a real number. More precisely, for any given real number r, there is a unique unital R-algebra homomorphism evr : R[x] → R such that evr(x) = r. Because evr is unital, evr(x0) = 1. That is, r0 = 1 for each real number r, including 0. The same argument applies with R replaced by any ring. Defining 00 = 1 is necessary for many polynomial identities. For example, the binomial theorem ( 1 + x ) n = ∑ k = 0 n ( n k ) x k {\textstyle (1+x)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}x^{k}} holds for x = 0 only if 00 = 1. Similarly, rings of power series require x0 to be defined as 1 for all specializations of x. For example, identities like 1 1 − x = ∑ n = 0 ∞ x n {\textstyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}} and e x = ∑ n = 0 ∞ x n n ! {\textstyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}} hold for x = 0 only if 00 = 1. In order for the polynomial x0 to define a continuous function R → R, one must define 00 = 1. In calculus, the power rule d d x x n = n x n − 1 {\textstyle {\frac {d}{dx}}x^{n}=nx^{n-1}} is valid for n = 1 at x = 0 only if 00 = 1. == Continuous exponents == Limits involving algebraic operations can often be evaluated by replacing subexpressions with their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form. The expression 00 is an indeterminate form: Given real-valued functions f(t) and g(t) approaching 0 (as t approaches a real number or ±∞) with f(t) > 0, the limit of f(t)g(t) can be any non-negative real number or +∞, or it can diverge, depending on f and g. For example, each limit below involves a function f(t)g(t) with f(t), g(t) → 0 as t → 0+ (a one-sided limit), but their values are different: lim t → 0 + t t = 1 , {\displaystyle \lim _{t\to 0^{+}}{t}^{t}=1,} lim t → 0 + ( e − 1 / t 2 ) t = 0 , {\displaystyle \lim _{t\to 0^{+}}\left(e^{-1/t^{2}}\right)^{t}=0,} lim t → 0 + ( e − 1 / t 2 ) − t = + ∞ , {\displaystyle \lim _{t\to 0^{+}}\left(e^{-1/t^{2}}\right)^{-t}=+\infty ,} lim t → 0 + ( a − 1 / t ) − t = a . {\displaystyle \lim _{t\to 0^{+}}\left(a^{-1/t}\right)^{-t}=a.} Thus, the two-variable function xy, though continuous on the set {(x, y) : x > 0}, cannot be extended to a continuous function on {(x, y) : x > 0} ∪ {(0, 0)}, no matter how one chooses to define 00. On the other hand, if f and g are analytic functions on an open neighborhood of a number c, then f(t)g(t) → 1 as t approaches c from any side on which f is positive. This and more general results can be obtained by studying the limiting behavior of the function log ( f ( t ) g ( t ) ) = g ( t ) log f ( t ) {\textstyle \log(f(t)^{g(t)})=g(t)\log f(t)} . == Complex exponents == In the complex domain, the function zw may be defined for nonzero z by choosing a branch of log z and defining zw as ew log z. This does not define 0w since there is no branch of log z defined at z = 0, let alone in a neighborhood of 0. == History == === As a value === In 1752, Euler in Introductio in analysin infinitorum wrote that a0 = 1 and explicitly mentioned that 00 = 1. An annotation attributed to Mascheroni in a 1787 edition of Euler's book Institutiones calculi differentialis offered the "justification" 0 0 = ( a − a ) n − n = ( a − a ) n ( a − a ) n = 1 {\displaystyle 0^{0}=(a-a)^{n-n}={\frac {(a-a)^{n}}{(a-a)^{n}}}=1} as well as another more involved justification. In the 1830s, Libri published several further arguments attempting to justify the claim 00 = 1, though these were far from convincing, even by standards of rigor at the time. === As a limiting form === Euler, when setting 00 = 1, mentioned that consequently the values of the function 0x take a "huge jump", from ∞ for x < 0, to 1 at x = 0, to 0 for x > 0. In 1814, Pfaff used a squeeze theorem argument to prove that xx → 1 as x → 0+. On the other hand, in 1821 Cauchy explained why the limit of xy as positive numbers x and y approach 0 while being constrained by some fixed relation could be made to assume any value between 0 and ∞ by choosing the relation appropriately. He deduced that the limit of the full two-variable function xy without a specified constraint is "indeterminate". With this justification, he listed 00 along with expressions like 0/0 in a table of indeterminate forms. Apparently unaware of Cauchy's work, Möbius in 1834, building on Pfaff's argument, claimed incorrectly that f(x)g(x) → 1 whenever f(x),g(x) → 0 as x approaches a number c (presumably f is assumed positive away from c). Möbius reduced to the case c = 0, but then made the mistake of assuming that each of f and g could be expressed in the form Pxn for some continuous function P not vanishing at 0 and some nonnegative integer n, which is true for analytic functions, but not in general. An anonymous commentator pointed out the unjustified step; then another commentator who signed his name simply as "S" provided the explicit counterexamples (e−1/x)x → e−1 and (e−1/x)2x → e−2 as x → 0+ and expressed the situation by writing that "00 can have many different values". === Current situation === Some authors define 00 as 1 because it simplifies many theorem statements. According to Benson (1999), "The choice whether to define 00 is based on convenience, not on correctness. If we refrain from defining 00, then certain assertions become unnecessarily awkward. ... The consensus is to use the definition 00 = 1, although there are textbooks that refrain from defining 00." Knuth (1992) contends more strongly that 00 "has to be 1"; he draws a distinction between the value 00, which should equal 1, and the limiting form 00 (an abbreviation for a limit of f(t)g(t) where f(t), g(t) → 0), which is an indeterminate form: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side." Other authors leave 00 undefined because 00 is an indeterminate form: f(t), g(t) → 0 does not imply f(t)g(t) → 1. There do not seem to be any authors assigning 00 a specific value other than 1. == Treatment on computers == === IEEE floating-point standard === The IEEE 754-2008 floating-point standard is used in the design of most floating-point libraries. It recommends a number of operations for computing a power: pown (whose exponent is an integer) treats 00 as 1; see § Discrete exponents. pow (whose intent is to return a non-NaN result when the exponent is an integer, like pown) treats 00 as 1. powr treats 00 as NaN (Not-a-Number) due to the indeterminate form; see § Continuous exponents. The pow variant is inspired by the pow function from C99, mainly for compatibility. It is useful mostly for languages with a single power function. The pown and powr variants have been introduced due to conflicting usage of the power functions and the different points of view (as stated above). === Programming languages === The C and C++ standards do not specify the result of 00 (a domain error may occur). But for C, as of C99, if the normative annex F is supported, the result for real floating-point types is required to be 1 because there are significant applications for which this value is more useful than NaN (for instance, with discrete exponents); the result on complex types is not specified, even if the informative annex G is supported. The Java standard, the .NET Framework method System.Math.Pow, Julia, and Python also treat 00 as 1. Some languages document that their exponentiation operation corresponds to the pow function from the C mathematical library; this is the case with Lua's ^ operator and Perl's ** operator (where it is explicitly mentioned that the result of 0**0 is platform-dependent). === Mathematical and scientific software === R, SageMath, and PARI/GP evaluate x0 to 1. Mathematica simplifies x0 to 1 even if no constraints are placed on x; however, if 00 is entered directly, it is treated as an error or indeterminate. Mathematica and PARI/GP further distinguish between integer and floating-point values: If the exponent is a zero of integer type, they return a 1 of the type of the base; exponentiation with a floating-point exponent of value zero is treated as undefined, indeterminate or error. == See also == 0/0 == References == == External links == sci.math FAQ: What is 00? What does 00 (zero to the zeroth power) equal? on AskAMathematician.com
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Wikipedia:Zero-product property#0
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In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, if a b = 0 , then a = 0 or b = 0. {\displaystyle {\text{if }}ab=0,{\text{ then }}a=0{\text{ or }}b=0.} This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. All of the number systems studied in elementary mathematics — the integers Z {\displaystyle \mathbb {Z} } , the rational numbers Q {\displaystyle \mathbb {Q} } , the real numbers R {\displaystyle \mathbb {R} } , and the complex numbers C {\displaystyle \mathbb {C} } — satisfy the zero-product property. In general, a ring which satisfies the zero-product property is called a domain. == Algebraic context == Suppose A {\displaystyle A} is an algebraic structure. We might ask, does A {\displaystyle A} have the zero-product property? In order for this question to have meaning, A {\displaystyle A} must have both additive structure and multiplicative structure. Usually one assumes that A {\displaystyle A} is a ring, though it could be something else, e.g. the set of nonnegative integers { 0 , 1 , 2 , … } {\displaystyle \{0,1,2,\ldots \}} with ordinary addition and multiplication, which is only a (commutative) semiring. Note that if A {\displaystyle A} satisfies the zero-product property, and if B {\displaystyle B} is a subset of A {\displaystyle A} , then B {\displaystyle B} also satisfies the zero product property: if a {\displaystyle a} and b {\displaystyle b} are elements of B {\displaystyle B} such that a b = 0 {\displaystyle ab=0} , then either a = 0 {\displaystyle a=0} or b = 0 {\displaystyle b=0} because a {\displaystyle a} and b {\displaystyle b} can also be considered as elements of A {\displaystyle A} . == Examples == A ring in which the zero-product property holds is called a domain. A commutative domain with a multiplicative identity element is called an integral domain. Any field is an integral domain; in fact, any subring of a field is an integral domain (as long as it contains 1). Similarly, any subring of a skew field is a domain. Thus, the zero-product property holds for any subring of a skew field. If p {\displaystyle p} is a prime number, then the ring of integers modulo p {\displaystyle p} has the zero-product property (in fact, it is a field). The Gaussian integers are an integral domain because they are a subring of the complex numbers. In the strictly skew field of quaternions, the zero-product property holds. This ring is not an integral domain, because the multiplication is not commutative. The set of nonnegative integers { 0 , 1 , 2 , … } {\displaystyle \{0,1,2,\ldots \}} is not a ring (being instead a semiring), but it does satisfy the zero-product property. == Non-examples == Let Z n {\displaystyle \mathbb {Z} _{n}} denote the ring of integers modulo n {\displaystyle n} . Then Z 6 {\displaystyle \mathbb {Z} _{6}} does not satisfy the zero product property: 2 and 3 are nonzero elements, yet 2 ⋅ 3 ≡ 0 ( mod 6 ) {\displaystyle 2\cdot 3\equiv 0{\pmod {6}}} . In general, if n {\displaystyle n} is a composite number, then Z n {\displaystyle \mathbb {Z} _{n}} does not satisfy the zero-product property. Namely, if n = q m {\displaystyle n=qm} where 0 < q , m < n {\displaystyle 0<q,m<n} , then m {\displaystyle m} and q {\displaystyle q} are nonzero modulo n {\displaystyle n} , yet q m ≡ 0 ( mod n ) {\displaystyle qm\equiv 0{\pmod {n}}} . The ring Z 2 × 2 {\displaystyle \mathbb {Z} ^{2\times 2}} of 2×2 matrices with integer entries does not satisfy the zero-product property: if M = ( 1 − 1 0 0 ) {\displaystyle M={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}} and N = ( 0 1 0 1 ) , {\displaystyle N={\begin{pmatrix}0&1\\0&1\end{pmatrix}},} then M N = ( 1 − 1 0 0 ) ( 0 1 0 1 ) = ( 0 0 0 0 ) = 0 , {\displaystyle MN={\begin{pmatrix}1&-1\\0&0\end{pmatrix}}{\begin{pmatrix}0&1\\0&1\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}=0,} yet neither M {\displaystyle M} nor N {\displaystyle N} is zero. The ring of all functions f : [ 0 , 1 ] → R {\displaystyle f:[0,1]\to \mathbb {R} } , from the unit interval to the real numbers, has nontrivial zero divisors: there are pairs of functions which are not identically equal to zero yet whose product is the zero function. In fact, it is not hard to construct, for any n ≥ 2, functions f 1 , … , f n {\displaystyle f_{1},\ldots ,f_{n}} , none of which is identically zero, such that f i f j {\displaystyle f_{i}\,f_{j}} is identically zero whenever i ≠ j {\displaystyle i\neq j} . The same is true even if we consider only continuous functions, or only even infinitely smooth functions. On the other hand, analytic functions have the zero-product property. == Application to finding roots of polynomials == Suppose P {\displaystyle P} and Q {\displaystyle Q} are univariate polynomials with real coefficients, and x {\displaystyle x} is a real number such that P ( x ) Q ( x ) = 0 {\displaystyle P(x)Q(x)=0} . (Actually, we may allow the coefficients and x {\displaystyle x} to come from any integral domain.) By the zero-product property, it follows that either P ( x ) = 0 {\displaystyle P(x)=0} or Q ( x ) = 0 {\displaystyle Q(x)=0} . In other words, the roots of P Q {\displaystyle PQ} are precisely the roots of P {\displaystyle P} together with the roots of Q {\displaystyle Q} . Thus, one can use factorization to find the roots of a polynomial. For example, the polynomial x 3 − 2 x 2 − 5 x + 6 {\displaystyle x^{3}-2x^{2}-5x+6} factorizes as ( x − 3 ) ( x − 1 ) ( x + 2 ) {\displaystyle (x-3)(x-1)(x+2)} ; hence, its roots are precisely 3, 1, and −2. In general, suppose R {\displaystyle R} is an integral domain and f {\displaystyle f} is a monic univariate polynomial of degree d ≥ 1 {\displaystyle d\geq 1} with coefficients in R {\displaystyle R} . Suppose also that f {\displaystyle f} has d {\displaystyle d} distinct roots r 1 , … , r d ∈ R {\displaystyle r_{1},\ldots ,r_{d}\in R} . It follows (but we do not prove here) that f {\displaystyle f} factorizes as f ( x ) = ( x − r 1 ) ⋯ ( x − r d ) {\displaystyle f(x)=(x-r_{1})\cdots (x-r_{d})} . By the zero-product property, it follows that r 1 , … , r d {\displaystyle r_{1},\ldots ,r_{d}} are the only roots of f {\displaystyle f} : any root of f {\displaystyle f} must be a root of ( x − r i ) {\displaystyle (x-r_{i})} for some i {\displaystyle i} . In particular, f {\displaystyle f} has at most d {\displaystyle d} distinct roots. If however R {\displaystyle R} is not an integral domain, then the conclusion need not hold. For example, the cubic polynomial x 3 + 3 x 2 + 2 x {\displaystyle x^{3}+3x^{2}+2x} has six roots in Z 6 {\displaystyle \mathbb {Z} _{6}} (though it has only three roots in Z {\displaystyle \mathbb {Z} } ). == See also == Fundamental theorem of algebra Integral domain and domain Prime ideal Zero divisor == Notes == == References == David S. Dummit and Richard M. Foote, Abstract Algebra (3d ed.), Wiley, 2003, ISBN 0-471-43334-9. == External links == PlanetMath: Zero rule of product
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Wikipedia:Zero-symmetric graph#0
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In the mathematical field of graph theory, a zero-symmetric graph is a connected graph in which each vertex has exactly three incident edges and, for each two vertices, there is a unique symmetry taking one vertex to the other. Such a graph is a vertex-transitive graph but cannot be an edge-transitive graph: the number of symmetries equals the number of vertices, too few to take every edge to every other edge. The name for this class of graphs was coined by R. M. Foster in a 1966 letter to H. S. M. Coxeter. In the context of group theory, zero-symmetric graphs are also called graphical regular representations of their symmetry groups. == Examples == The smallest zero-symmetric graph is a nonplanar graph with 18 vertices. Its LCF notation is [5,−5]9. Among planar graphs, the truncated cuboctahedral and truncated icosidodecahedral graphs are also zero-symmetric. These examples are all bipartite graphs. However, there exist larger examples of zero-symmetric graphs that are not bipartite. These examples also have three different symmetry classes (orbits) of edges. However, there exist zero-symmetric graphs with only two orbits of edges. The smallest such graph has 20 vertices, with LCF notation [6,6,-6,-6]5. == Properties == Every finite zero-symmetric graph is a Cayley graph, a property that does not always hold for cubic vertex-transitive graphs more generally and that helps in the solution of combinatorial enumeration tasks concerning zero-symmetric graphs. There are 97687 zero-symmetric graphs on up to 1280 vertices. These graphs form 89% of the cubic Cayley graphs and 88% of all connected vertex-transitive cubic graphs on the same number of vertices. All known finite connected zero-symmetric graphs contain a Hamiltonian cycle, but it is unknown whether every finite connected zero-symmetric graph is necessarily Hamiltonian. This is a special case of the Lovász conjecture that (with five known exceptions, none of which is zero-symmetric) every finite connected vertex-transitive graph and every finite Cayley graph is Hamiltonian. == See also == Semi-symmetric graph, graphs that have symmetries between every two edges but not between every two vertices (reversing the roles of edges and vertices in the definition of zero-symmetric graphs) == References ==
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Wikipedia:Zeta function regularization#0
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In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to problems in physics, but has its origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory. == Definition == There are several different summation methods called zeta function regularization for defining the sum of a possibly divergent series a1 + a2 + .... One method is to define its zeta regularized sum to be ζA(−1) if this is defined, where the zeta function is defined for large Re(s) by ζ A ( s ) = 1 a 1 s + 1 a 2 s + ⋯ {\displaystyle \zeta _{A}(s)={\frac {1}{a_{1}^{s}}}+{\frac {1}{a_{2}^{s}}}+\cdots } if this sum converges, and by analytic continuation elsewhere. In the case when an = n, the zeta function is the ordinary Riemann zeta function. This method was used by Ramanujan to "sum" the series 1 + 2 + 3 + 4 + ⋯ to ζ(−1) = −1/12. Hawking (1977) showed that in flat space, in which the eigenvalues of Laplacians are known, the zeta function corresponding to the partition function can be computed explicitly. Consider a scalar field φ contained in a large box of volume V in flat spacetime at the temperature T = β−1. The partition function is defined by a path integral over all fields φ on the Euclidean space obtained by putting τ = it which are zero on the walls of the box and which are periodic in τ with period β. In this situation from the partition function he computes energy, entropy and pressure of the radiation of the field φ. In case of flat spaces the eigenvalues appearing in the physical quantities are generally known, while in case of curved space they are not known: in this case asymptotic methods are needed. Another method defines the possibly divergent infinite product a1a2.... to be exp(−ζ′A(0)). Ray & Singer (1971) used this to define the determinant of a positive self-adjoint operator A (the Laplacian of a Riemannian manifold in their application) with eigenvalues a1, a2, ...., and in this case the zeta function is formally the trace of A−s. Minakshisundaram & Pleijel (1949) showed that if A is the Laplacian of a compact Riemannian manifold then the Minakshisundaram–Pleijel zeta function converges and has an analytic continuation as a meromorphic function to all complex numbers, and Seeley (1967) extended this to elliptic pseudo-differential operators A on compact Riemannian manifolds. So for such operators one can define the determinant using zeta function regularization. See "analytic torsion." Hawking (1977) suggested using this idea to evaluate path integrals in curved spacetimes. He studied zeta function regularization in order to calculate the partition functions for thermal graviton and matter's quanta in curved background such as on the horizon of black holes and on de Sitter background using the relation by the inverse Mellin transformation to the trace of the kernel of heat equations. == Example == The first example in which zeta function regularization is available appears in the Casimir effect, which is in a flat space with the bulk contributions of the quantum field in three space dimensions. In this case we must calculate the value of Riemann zeta function at –3, which diverges explicitly. However, it can be analytically continued to s = –3 where hopefully there is no pole, thus giving a finite value to the expression. A detailed example of this regularization at work is given in the article on the detail example of the Casimir effect, where the resulting sum is very explicitly the Riemann zeta-function (and where the seemingly legerdemain analytic continuation removes an additive infinity, leaving a physically significant finite number). An example of zeta-function regularization is the calculation of the vacuum expectation value of the energy of a particle field in quantum field theory. More generally, the zeta-function approach can be used to regularize the whole energy–momentum tensor both in flat and in curved spacetime. [1] [2] [3] The unregulated value of the energy is given by a summation over the zero-point energy of all of the excitation modes of the vacuum: ⟨ 0 | T 00 | 0 ⟩ = ∑ n ℏ | ω n | 2 {\displaystyle \langle 0|T_{00}|0\rangle =\sum _{n}{\frac {\hbar |\omega _{n}|}{2}}} Here, T 00 {\displaystyle T_{00}} is the zeroth component of the energy–momentum tensor and the sum (which may be an integral) is understood to extend over all (positive and negative) energy modes ω n {\displaystyle \omega _{n}} ; the absolute value reminding us that the energy is taken to be positive. This sum, as written, is usually infinite ( ω n {\displaystyle \omega _{n}} is typically linear in n). The sum may be regularized by writing it as ⟨ 0 | T 00 ( s ) | 0 ⟩ = ∑ n ℏ | ω n | 2 | ω n | − s {\displaystyle \langle 0|T_{00}(s)|0\rangle =\sum _{n}{\frac {\hbar |\omega _{n}|}{2}}|\omega _{n}|^{-s}} where s is some parameter, taken to be a complex number. For large, real s greater than 4 (for three-dimensional space), the sum is manifestly finite, and thus may often be evaluated theoretically. The zeta-regularization is useful as it can often be used in a way such that the various symmetries of the physical system are preserved. Zeta-function regularization is used in conformal field theory, renormalization and in fixing the critical spacetime dimension of string theory. == Relation to other regularizations == Zeta function regularization is equivalent to dimensional regularization, see[4]. However, the main advantage of the zeta regularization is that it can be used whenever the dimensional regularization fails, for example if there are matrices or tensors inside the calculations ϵ i , j , k {\displaystyle \epsilon _{i,j,k}} == Relation to Dirichlet series == Zeta-function regularization gives an analytic structure to any sums over an arithmetic function f(n). Such sums are known as Dirichlet series. The regularized form f ~ ( s ) = ∑ n = 1 ∞ f ( n ) n − s {\displaystyle {\tilde {f}}(s)=\sum _{n=1}^{\infty }f(n)n^{-s}} converts divergences of the sum into simple poles on the complex s-plane. In numerical calculations, the zeta-function regularization is inappropriate, as it is extremely slow to converge. For numerical purposes, a more rapidly converging sum is the exponential regularization, given by F ( t ) = ∑ n = 1 ∞ f ( n ) e − t n . {\displaystyle F(t)=\sum _{n=1}^{\infty }f(n)e^{-tn}.} This is sometimes called the Z-transform of f, where z = exp(−t). The analytic structure of the exponential and zeta-regularizations are related. By expanding the exponential sum as a Laurent series F ( t ) = a N t N + a N − 1 t N − 1 + ⋯ {\displaystyle F(t)={\frac {a_{N}}{t^{N}}}+{\frac {a_{N-1}}{t^{N-1}}}+\cdots } one finds that the zeta-series has the structure f ~ ( s ) = a N s − N + ⋯ . {\displaystyle {\tilde {f}}(s)={\frac {a_{N}}{s-N}}+\cdots .} The structure of the exponential and zeta-regulators are related by means of the Mellin transform. The one may be converted to the other by making use of the integral representation of the Gamma function: Γ ( s ) = ∫ 0 ∞ t s − 1 e − t d t {\displaystyle \Gamma (s)=\int _{0}^{\infty }t^{s-1}e^{-t}\,dt} which leads to the identity Γ ( s ) f ~ ( s ) = ∫ 0 ∞ t s − 1 F ( t ) d t {\displaystyle \Gamma (s){\tilde {f}}(s)=\int _{0}^{\infty }t^{s-1}F(t)\,dt} relating the exponential and zeta-regulators, and converting poles in the s-plane to divergent terms in the Laurent series. == Heat kernel regularization == The sum f ( s ) = ∑ n a n e − s | ω n | {\displaystyle f(s)=\sum _{n}a_{n}e^{-s|\omega _{n}|}} is sometimes called a heat kernel or a heat-kernel regularized sum; this name stems from the idea that the ω n {\displaystyle \omega _{n}} can sometimes be understood as eigenvalues of the heat kernel. In mathematics, such a sum is known as a generalized Dirichlet series; its use for averaging is known as an Abelian mean. It is closely related to the Laplace–Stieltjes transform, in that f ( s ) = ∫ 0 ∞ e − s t d α ( t ) {\displaystyle f(s)=\int _{0}^{\infty }e^{-st}\,d\alpha (t)} where α ( t ) {\displaystyle \alpha (t)} is a step function, with steps of a n {\displaystyle a_{n}} at t = | ω n | {\displaystyle t=|\omega _{n}|} . A number of theorems for the convergence of such a series exist. For example, by the Hardy-Littlewood Tauberian theorem, if [5] L = lim sup n → ∞ log | ∑ k = 1 n a k | | ω n | {\displaystyle L=\limsup _{n\to \infty }{\frac {\log \vert \sum _{k=1}^{n}a_{k}\vert }{|\omega _{n}|}}} then the series for f ( s ) {\displaystyle f(s)} converges in the half-plane ℜ ( s ) > L {\displaystyle \Re (s)>L} and is uniformly convergent on every compact subset of the half-plane ℜ ( s ) > L {\displaystyle \Re (s)>L} . In almost all applications to physics, one has L = 0 {\displaystyle L=0} == History == Much of the early work establishing the convergence and equivalence of series regularized with the heat kernel and zeta function regularization methods was done by G. H. Hardy and J. E. Littlewood in 1916[6] and is based on the application of the Cahen–Mellin integral. The effort was made in order to obtain values for various ill-defined, conditionally convergent sums appearing in number theory. In terms of application as the regulator in physical problems, before Hawking (1977), J. Stuart Dowker and Raymond Critchley in 1976 proposed a zeta-function regularization method for quantum physical problems.[7] Emilio Elizalde and others have also proposed a method based on the zeta regularization for the integrals ∫ a ∞ x m − s d x {\displaystyle \int _{a}^{\infty }x^{m-s}dx} , here x − s {\displaystyle x^{-s}} is a regulator and the divergent integral depends on the numbers ζ ( s − m ) {\displaystyle \zeta (s-m)} in the limit s → 0 {\displaystyle s\to 0} see renormalization. Also unlike other regularizations such as dimensional regularization and analytic regularization, zeta regularization has no counterterms and gives only finite results. == See also == Generating function – Formal power series; coefficients encode information about a sequence indexed by natural numbers Perron's formula – Formula to calculate the sum of an arithmetic function in analytic number theory Renormalization – Method in physics used to deal with infinities 1 + 1 + 1 + 1 + ⋯ – Divergent series 1 + 2 + 3 + 4 + ⋯ – Divergent series Analytic torsion – Topological invariant of manifolds that can distinguish homotopy-equivalent manifolds Ramanujan summation – Mathematical techniques for summing divergent infinite series Minakshisundaram–Pleijel zeta function Zeta function (operator) == References == ^ Tom M. Apostol, "Modular Functions and Dirichlet Series in Number Theory", "Springer-Verlag New York. (See Chapter 8.)" ^ A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti and S. Zerbini, "Analytic Aspects of Quantum Fields", World Scientific Publishing, 2003, ISBN 981-238-364-6 ^ G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", Acta Mathematica, 41(1916) pp. 119–196. (See, for example, theorem 2.12) Hawking, S. W. (1977), "Zeta function regularization of path integrals in curved spacetime", Communications in Mathematical Physics, 55 (2): 133–148, Bibcode:1977CMaPh..55..133H, doi:10.1007/BF01626516, ISSN 0010-3616, MR 0524257, S2CID 121650064 ^ V. Moretti, "Direct z-function approach and renormalization of one-loop stress tensor in curved spacetimes, Phys. Rev.D 56, 7797 (1997). Minakshisundaram, S.; Pleijel, Å. (1949), "Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds", Canadian Journal of Mathematics, 1 (3): 242–256, doi:10.4153/CJM-1949-021-5, ISSN 0008-414X, MR 0031145 Ray, D. B.; Singer, I. M. (1971), "R-torsion and the Laplacian on Riemannian manifolds", Advances in Mathematics, 7 (2): 145–210, doi:10.1016/0001-8708(71)90045-4, MR 0295381 "Zeta-function method for regularization", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Seeley, R. T. (1967), "Complex powers of an elliptic operator", in Calderón, Alberto P. (ed.), Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Proceedings of Symposia in Pure Mathematics, vol. 10, Providence, R.I.: Amer. Math. Soc., pp. 288–307, ISBN 978-0-8218-1410-9, MR 0237943 ^ Dowker, J. S.; Critchley, R. (1976), "Effective Lagrangian and energy–momentum tensor in de Sitter space", Physical Review D, 13 (12): 3224–3232, Bibcode:1976PhRvD..13.3224D, doi:10.1103/PhysRevD.13.3224 ^ D. Fermi, L. Pizzocchero, "Local zeta regularization and the scalar Casimir effect. A general approach based on integral kernels", World Scientific Publishing, ISBN 978-981-3224-99-5 (hardcover), ISBN 978-981-3225-01-5 (ebook). doi:10.1142/10570 (2017).
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Wikipedia:Zhang Pingwen#0
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Zhang Pingwen (Chinese: 张平文; born July 1966) is a Chinese mathematician and university administrator, currently serving as president of Wuhan University. He is an academician of the Chinese Academy of Sciences. == Biography == Zhang was born in Changsha County, Hunan, in July 1966. He attended Changsha County No. 1 High School. In 1984, he was accepted to Peking University, where he majored in the Department of Mathematics. After graduating in 1992, Zhang stayed and worked at Peking University, where he was promoted to associate professor in 1994 and to full professor in 1996. In 2015, he was appointed director of the Discipline Construction Office of Peking University, a post he kept until 2019, when he became director of the Peking University Big Data Science Research Center. He moved up the ranks to become assistant president in August 2019 and vice president in December of that same year. On 28 December 2022, the Organization Department of the Chinese Communist Party appointed Zhang as president of Wuhan University, a position at vice-ministerial level. == Honours and awards == 1999 Feng Kang Prize 2014 State Natural Science Award (Second Class) 2015 Member of the Chinese Academy of Sciences (CAS) November 2016 Fellow of The World Academy of Sciences (TWAS) 2020 Fellow of the Society for Industrial and Applied Mathematics (SIAM) 2021 Science and Technology Progress Award of the Ho Leung Ho Lee Foundation == References ==
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Wikipedia:Zhang Qiujian Suanjing#0
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Zhang Qiujian Suanjing (張邱建算經; The Mathematical Classic of Zhang Qiujian) is the only known work of the fifth century Chinese mathematician, Zhang Qiujian. It is one of ten mathematical books known collectively as Suanjing shishu (The Ten Computational Canons). In 656 CE, when mathematics was included in the imperial examinations, these ten outstanding works were selected as textbooks. Jiuzhang suanshu (The Nine Chapters on the Mathematical Art) and Sunzi Suanjing (The Mathematical Classic of Sunzi) are two of these texts that precede Zhang Qiujian suanjing. All three works share a large number of common topics. In Zhang Qiujian suanjing one can find the continuation of the development of mathematics from the earlier two classics. Internal evidences suggest that book was compiled sometime between 466 and 485 CE. "Zhang Qiujian suanjing has an important place in the world history of mathematics: it is one of those rare books before AD 500 that manifests the upward development of mathematics fundamentally due to the notations of the numeral system and the common fraction. The numeral system has a place value notation with ten as base, and the concise notation of the common fraction is the one we still use today." Almost nothing is known about the author Zhang Qiujian, sometimes written as Chang Ch'iu-Chin or Chang Ch'iu-chien. It is estimated that he lived from 430 to 490 CE, but there is no consensus. == Contents == In its surviving form, the book has a preface and three chapters. There are two missing bits, one at the end of Chapter 1 and one at the beginning of Chapter 3. Chapter 1 consists of 32 problems, Chapter 2 of 22 problems and Chapter 3 of 38 problems. In the preface, the author has set forth his objectives in writing the book clearly. There are three objectives: The first is to explain how to handle arithmetical operations involving fractions; the second objective is to put forth new improved methods for solving old problems; and, the third objective is to present computational methods in a precise and comprehensible form. Here is a typical problem of Chapter 1: "Divide 6587 2/3 and 3/4 by 58 ı/2. How much is it?" The answer is given as 112 437/702 with a detailed description of the process by which the answer is obtained. This description makes use of the Chinese rod numerals. The chapter considers several real world problems where computations with fractions appear naturally. In Chapter 2, among others, there are a few problem requiring application of the rule of three. Here is a typical problem: "Now there was a person who stole a horse and rode off with it. After he has traveled 73 li, the owner realized [the theft] and gave chase for 145 li when [the thief] was 23 li ahead before turning back. If he had not turned back but continued to chase, find the distance in li before he reached [the thief]." Answer is given as 238 3/14 li. In Chapter 3, there are several problems connected with volumes of solids which are granaries. Here is an example: "Now there is a pit [in the shape of the frustum of a pyramid] with a rectangular base. The width of the upper [rectangle] is 4 chi and the width of the lower [rectangle] is 7 chi. The length of the upper [rectangle] is 5 chi and the length of the lower [rectangle] is 8 chi. The depth is 1 zhang. Find the amount of millet that it can hold." However, the answer is given in a different set of units. The 37th problem is the "Washing Bowls Problem": "Now there was a woman washing cups by the river. An officer asked, "Why are there so many cups?" The woman replied, "There were guests in the house, but I do not know how many there were. However, every 2 persons had [a cup of] thick sauce, every 3 persons had [a cup of] soup and every 4 persons had [a cup of] rice; 65 cups were used altogether." Find the number of persons." The answer is given as 60 persons. The last problem in the book is the famous Hundred Fowls Problem which is often considered as one of the earliest examples involving equations with indeterminate solutions. "Now one cock is worth 5 qian, one hen 3 qian and 3 chicks 1 qian. It is required to buy 100 fowls with 100 qian. In each case, find the number of cocks, hens and chicks bought." == English translation == In 1969, Ang Tian Se, a student of University of Malaya, prepared an English translation of Zhang Qiujian Suanjing as part of the MA Dissertation. But the translation has not been published. == References ==
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Wikipedia:Zhao Youqin's π algorithm#0
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Zhao Youqin's π algorithm is an algorithm devised by Yuan dynasty Chinese astronomer and mathematician Zhao Youqin (赵友钦, ? – 1330) to calculate the value of π in his book Ge Xiang Xin Shu (革象新书). == Algorithm == Zhao Youqin started with an inscribed square in a circle with radius r. If ℓ {\displaystyle \ell } denotes the length of a side of the square, draw a perpendicular line d from the center of the circle to side l. Let e denotes r − d. Then from the diagram: d = r 2 − ( ℓ 2 ) 2 {\displaystyle d={\sqrt {r^{2}-\left({\frac {\ell }{2}}\right)^{2}}}} e = r − d = r − r 2 − ( ℓ 2 ) 2 . {\displaystyle e=r-d=r-{\sqrt {r^{2}-\left({\frac {\ell }{2}}\right)^{2}}}.} Extend the perpendicular line d to dissect the circle into an octagon; ℓ 2 {\displaystyle \ell _{2}} denotes the length of one side of octagon. ℓ 2 = ( ℓ 2 ) 2 + e 2 {\displaystyle \ell _{2}={\sqrt {\left({\frac {\ell }{2}}\right)^{2}+e^{2}}}} ℓ 2 = 1 2 ℓ 2 + 4 ( r − 1 2 4 r 2 − ℓ 2 ) 2 {\displaystyle \ell _{2}={\frac {1}{2}}{\sqrt {\ell ^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell ^{2}}}\right)^{2}}}} Let l 3 {\displaystyle l_{3}} denotes the length of a side of hexadecagon ℓ 3 = 1 2 ℓ 2 2 + 4 ( r − 1 2 4 r 2 − ℓ 2 2 ) 2 {\displaystyle \ell _{3}={\frac {1}{2}}{\sqrt {\ell _{2}^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell _{2}^{2}}}\right)^{2}}}} similarly ℓ n + 1 = 1 2 ℓ n 2 + 4 ( r − 1 2 4 r 2 − ℓ n 2 ) 2 {\displaystyle \ell _{n+1}={\frac {1}{2}}{\sqrt {\ell _{n}^{2}+4\left(r-{\frac {1}{2}}{\sqrt {4r^{2}-\ell _{n}^{2}}}\right)^{2}}}} Proceeding in this way, he at last calculated the side of a 16384-gon, multiplying it by 16384 to obtain 3141.592 for a circle with diameter = 1000 units, or π = 3.141592. {\displaystyle \pi =3.141592.\,} He multiplied this number by 113 and obtained 355. From this he deduced that of the traditional values of π, that is 3, 3.14, 22/7 and 355/113, the last is the most exact. == See also == Liu Hui's π algorithm == References ==
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Wikipedia:Zhilan Feng#0
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Zhilan Julie Feng (born 1959) is a Chinese-American applied mathematician whose research topics include mathematical biology, population dynamics, and epidemiology. She is a professor of mathematics at Purdue University, and a program director in the Division of Mathematical Sciences at the National Science Foundation. == Education and career == Feng studied mathematics at Jilin University in China, earning a bachelor's degree in 1982 and a master's degree in 1985. She came to Arizona State University for graduate study, completing her Ph.D. in 1994. Her dissertation, A Mathematical Model for the Dynamics of Childhood Diseases Under the Impact of Isolation, was supervised by Horst R. Thieme. After her postdoctoral study at Cornell University, she joined Purdue University as an assistant professor in 1996. She was promoted to full professor in 2005, and became a program director at the National Science Foundation in 2019. == Recognition == Feng was named a Fellow of the American Mathematical Society, in the 2022 class of fellows, "for contributions to applied mathematics, particularly in biology, ecology, and epidemiology". == Books == Feng's books include: Disease Evolution: Models, Concepts, and Data Analyses (American Mathematical Society, 2006, edited with Ulf Dieckmann and Simon A. Levin) Applications of Epidemiological Models to Public Health Policymaking: The role of heterogeneity in model predictions (World Scientific, 2014) Mathematical Models of Plant-Herbivore Interactions (Chapman & Hall / CRC, 2018, with Donald DeAngelis) Mathematical Models in Epidemiology (Springer, 2019, with Fred Brauer and Carlos Castillo-Chavez) == References == == External links == Home page Zhilan Feng publications indexed by Google Scholar
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Wikipedia:Zhongwei Shen#0
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Zhongwei Shen (Chinese: 申仲伟; born c. 1964) is a Chinese-American mathematician, currently a Distinguished Professor at University of Kentucky and a Fellow of the American Mathematical Society. Shen received his B.S. in mathematics from Peking University in 1982 at the age of 18. He has been a visiting scholar at Lanzhou University's School of Mathematics and Statistics on various occasions since 2007. He was named a Changjiang Scholar of Lanzhou University in 2015. == References ==
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Wikipedia:Zhoubi Suanjing#0
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The Zhoubi Suanjing, also known by many other names, is an ancient Chinese astronomical and mathematical work. The Zhoubi is most famous for its presentation of Chinese cosmology and a form of the Pythagorean theorem. It claims to present 246 problems worked out by the Duke of Zhou as well as members of his court, placing its composition during the 11th century BC. However, the present form of the book does not seem to be earlier than the Eastern Han (25–220 AD), with some additions and commentaries continuing to be added for several more centuries. The book was included as part of the Ten Computational Canons. == Names == The work's original title was simply the Zhoubi: the character 髀 is a literary term for the femur or thighbone but in context only refers to one or more gnomons, large sticks whose shadows were used for Chinese calendrical and astronomical calculations. Because of the ambiguous nature of the character 周, it has been alternately understood and translated as 'On the gnomon and the circular paths of Heaven', the 'Zhou shadow gauge manual', the 'Gnomon of the Zhou sundial', and 'Gnomon of the Zhou dynasty'. The honorific Suanjing—'Arithmetical classic', 'Sacred book of arithmetic', 'Mathematical canon', 'Classic of computations',—was added later. == Dating == Examples of the gnomon described in the work have been found from as early as 2300 BC and the Duke of Zhou, was an 11th-century BC regent and noble during the first generation of the Zhou dynasty. The Zhoubi was traditionally dated to the Duke of Zhou's own life and considered to be the oldest Chinese mathematical treatise. However, although some passages seem to come from the Warring States period or earlier, the current text of the work mentions Lü Buwei and is believed to have received its current form no earlier than the Eastern Han, during the 1st or 2nd century. The earliest known mention of the text is from a memorial dedicated to the astronomer Cai Yong in 178 AD. It does not appear at all in the Book of Han's account of calendrical, astronomical, and mathematical works, although Joseph Needham allows that this may have been from its current contents having previously been provided in several different works listed in the Han history which are otherwise unknown. == Contents == The Zhoubi is an anonymous collection of 246 problems encountered by the Duke of Zhou and figures in his court, including the astrologer Shang Gao. Each problem includes an answer and a corresponding arithmetic algorithm. It is an important source on early Chinese cosmology, glossing the ancient idea of a round heaven over a square earth (天圆地方, tiānyuán dìfāng) as similar to the round parasol suspended over some ancient Chinese chariots or a Chinese chessboard. All things measurable were considered variants of the square, while the expansion of a polygon to infinite sides approaches the immeasurable circle. This concept of a 'canopy heaven' (蓋天, gàitiān) had earlier produced the jade bi (璧) and cong objects and myths about Gonggong, Mount Buzhou, Nüwa, and repairing the sky. Although this eventually developed into an idea of a 'spherical heaven' (渾天, hùntiān), the Zhoubi offers numerous explorations of the geometric relationships of simple circles circumscribed by squares and squares circumscribed by circles. A large part of this involves analysis of solar declination in the Northern Hemisphere at various points throughout the year. At one point during its discussion of the shadows cast by gnomons, the work presents a form of the Pythagorean theorem known as the gougu theorem (勾股定理) from the Chinese names—lit. 'hook' and 'thigh'—of the two sides of the carpenter or try square. In the 3rd century, Zhao Shuang's commentary on the Zhoubi included a diagram effectively proving the theorem for the case of a 3-4-5 triangle, whence it can be generalized to all right triangles. The original text being ambiguous on its own, there is disagreement as to whether this proof was established by Zhao or merely represented an illustration of a previously understood concept earlier than Pythagoras. Shang Gao concludes the gougu problem saying "He who understands the earth is a wise man, and he who understands the heavens is a sage. Knowledge is derived from the shadow [straight line], and the shadow is derived from the gnomon [right angle]. The combination of the gnomon with numbers is what guides and rules the ten thousand things." == Commentaries == The Zhoubi has had a prominent place in Chinese mathematics and was the subject of specific commentaries by Zhao Shuang in the 3rd century, Liu Hui in 263, by Zu Gengzhi in the early 6th century, Li Chunfeng in the 7th century, and Yang Hui in 1270. == Translation == A translation to English was published in 1996 by Christopher Cullen, through the Cambridge University Press, entitled Astronomy and mathematics in ancient China: the Zhou bi suan jing. The work includes a preface attributed to Zhao Shuang, as well as his discussions and diagrams for the gougu theorem, the height of the sun, the seven heng and his gnomon shadow table, restored. == See also == Xuan tu Tsinghua Bamboo Slips Dunhuang Star Chart == References == === Citations === === Works cited === "Chinese", Encyclopaedia Britannica, vol. II (1st ed.), Edinburgh: Colin Macfarquhar, 1771, pp. 184–192. Chemla, Karine (2005), Geometrical Figures and Generality in Ancient China and Beyond, Science in Context, ISBN 0-521-55089-0. Cullen, Christopher (1996), Astronomy and Mathematics in Ancient China, Cambridge University Press, ISBN 0-521-55089-0. Cullen, Christopher (2018), "Chinese Astronomy in the Early Imperial Age", The Cambridge History of Science, Vol. I: Ancient Science, Cambridge University Press, ISBN 978-110868262-6. Davis, Philip J.; et al., eds. (1995), "Brief Chronological Table to 1910", The Mathematical Experience, Modern Birkhäuser Classics, Boston: Birkhäuser, pp. 26–29, ISBN 978-081768294-1. Ding, D.X. Daniel (2020), The Historical Roots of Technical Communication in the Chinese Tradition, Newcastle-upon-Tyne: Cambridge Scholars, ISBN 978-152755989-9. Elman, Benjamin (2015), "Early Modern or Late Imperial? The Crisis of Classical Philology in Eighteenth-Century China", World Philology, Cambridge: Harvard University Press, pp. 225–244. Gamwell, Lynn (2016), Mathematics + Art: A Cultural History, Princeton University Press, ISBN 978-069116528-8. Needham, Joseph; et al. (1959), Science & Civilisation in China, Vol. III: Mathematics and the Sciences of the Heavens and the Earth, Cambridge University Press, ISBN 978-052105801-8 {{citation}}: ISBN / Date incompatibility (help). Pang-White, A. Ann (2018), The Confucian Four Books for Women, Oxford University Press, ISBN 978-0-19-046091-4. Tseng, L.Y. Lillian (2011), Picturing Heaven in Early China, East Asian Monographs, Cambridge: Harvard University Asia Center, ISBN 978-0-674-06069-2. Zou Hui (2011), A Jesuit Garden in Beijing and Early Modern Chinese Culture, West Lafayette: Purdue University Press, ISBN 978-155753583-2. == Further reading == 周髀算經 (in Chinese), Chinese Text Project. 周髀算經 (in Chinese), Project Gutenberg. Boyer, Carl B. (1991), A History of Mathematics, John Wiley & Sons, ISBN 0-471-54397-7.
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Wikipedia:Zhusuan#0
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The suanpan (simplified Chinese: 算盘; traditional Chinese: 算盤; pinyin: suànpán), also spelled suan pan or souanpan) is an abacus of Chinese origin, earliest first known written documentation of the Chinese abacus dates to the 2nd century BCE during the Han dynasty, and later, described in a 190 CE book of the Eastern Han dynasty, namely Supplementary Notes on the Art of Figures written by Xu Yue. However, the exact design of this suanpan is not known. Usually, a suanpan is about 20 cm (8 in) tall and it comes in various widths depending on the application. It usually has more than seven rods. There are two beads on each rod in the upper deck and five beads on each rod in the bottom deck. The beads are usually rounded and made of a hardwood. The beads are counted by moving them up or down towards the beam. The suanpan can be reset to the starting position instantly by a quick jerk around the horizontal axis to spin all the beads away from the horizontal beam at the center. Suanpans can be used for functions other than counting. Unlike the simple counting board used in elementary schools, very efficient suanpan techniques have been developed to do multiplication, division, addition, subtraction, square root and cube root operations at high speed. The modern suanpan has 4+1 beads, colored beads to indicate position and a clear-all button. When the clear-all button is pressed, two mechanical levers push the top row beads to the top position and the bottom row beads to the bottom position, thus clearing all numbers to zero. This replaces clearing the beads by hand, or quickly rotating the suanpan around its horizontal center line to clear the beads by centrifugal force. == History == The word "abacus" was first mentioned by Xu Yue (160–220) in his book suanshu jiyi (算数记遗), or Notes on Traditions of Arithmetic Methods, in the Han dynasty. As it described, the original abacus had five beads (suan zhu) bunched by a stick in each column, separated by a transverse rod, and arrayed in a wooden rectangle box. One in the upper part represents five and each of four in the lower part represents one. People move the beads to do the calculation. The long scroll Along the River During Qing Ming Festival painted by Zhang Zeduan (1085–1145) during the Song dynasty (960–1279) might contain a suanpan beside an account book and doctor's prescriptions on the counter of an apothecary. However, the identification of the object as an abacus is a matter of some debate. Zhusuan was an abacus invented in China at the end of the 2nd century CE and reached its peak during the period from the 13th to the 16th century CE. In the 13th century, Guo Shoujing (郭守敬) used Zhusuan to calculate the length of each orbital year and found it to be 365.2425 days. In the 16th century, Zhu Zaiyu (朱載堉) calculated the musical Twelve-interval Equal Temperament using Zhusuan. And again in the 16th century, Wang Wensu (王文素) and Cheng Dawei (程大位) wrote respectively Principles of Algorithms and General Rules of Calculation, summarizing and refining the mathematical algorithms of Zhusuan, thus further boosting the popularity and promotion of Zhusuan. At the end of the 16th century, Zhusuan was introduced to neighboring countries and regions. A 5+1 suanpan appeared in the Ming dynasty, an illustration in a 1573 book on suanpan showed a suanpan with one bead on top and five beads at the bottom. The evident similarity of the Roman abacus to the Chinese one suggests that one may have inspired the other, as there is strong evidence of a trade relationship between the Roman Empire and China. However, no direct connection can be demonstrated, and the similarity of the abaci could be coincidental, both ultimately arising from counting with five fingers per hand. Where the Roman model and Chinese model (like most modern Japanese) has 4 plus 1 bead per decimal place, the old version of the Chinese suanpan has 5 plus 2, allowing less challenging arithmetic algorithms. Instead of running on wires as in the Chinese and Japanese models, the beads of Roman model run in grooves, presumably more reliable since the wires could be bent. Another possible source of the suanpan is Chinese counting rods, which operated with a place value decimal system with empty spot as zero. Although sinologist Nathan Sivin claimed that the abacus, with its limited flexibility, "was useless for the most advanced algebra", and suggested that "the convenience of the abacus" may have paradoxically stymied mathematical innovation from the 14th to 17th centuries, Roger Hart counters that the abacus in fact facilitated new developments during that time, such as Zhu Zaiyu's treatises on musical equal temperament, for which he used nine abacuses to calculate √200 to twenty-five digits. == Beads == There are two types of beads on the suanpan, those in the lower deck, below the separator beam, and those in the upper deck above it. The ones in the lower deck are sometimes called earth beads or water beads, and carry a value of 1 in their column. The ones in the upper deck are sometimes called heaven beads and carry a value of 5 in their column. The columns are much like the places in Indian numerals: one of the columns, usually the rightmost, represents the ones place; to the left of it are the tens, hundreds, thousands place, and so on, and if there are any columns to the right of it, they are the tenths place, hundredths place, and so on. The suanpan is a 2:5 abacus: two heaven beads and five earth beads. If one compares the suanpan to the soroban which is a 1:4 abacus, one might think there are two "extra" beads in each column. In fact, to represent decimal numbers and add or subtract such numbers, one strictly needs only one upper bead and four lower beads on each column. Some "old" methods to multiply or divide decimal numbers use those extra beads like the "Extra Bead technique" or "Suspended Bead technique". The most mysterious and seemingly superfluous fifth lower bead, likely inherited from counting rods as suggested by the image above, was used to simplify and speed up addition and subtraction somewhat, as well as to decrease the chances of error. Its use was demonstrated, for example, in the first book devoted entirely to suanpan: Computational Methods with the Beads in a Tray (Pánzhū Suànfǎ 盤珠算法) by Xú Xīnlǔ 徐心魯 (1573, Late Ming dynasty). A simplify explanation on how to use the 5th beads is shown here. The following two animations show the details of this particular usage: The beads and rods are often lubricated to ensure quick, smooth motion. == Calculating on a suanpan == At the end of a decimal calculation on a suanpan, it is never the case that all five beads in the lower deck are move. Compared with the Chinese abacus, Japanese Soroban can accommodate up to 9 in each digit rod and when it becomes 10, the digit rod changes that will visualise the Decimal system bead in the top deck takes their place. Similarly, if two beads in the top deck are pushed down, they are pushed back up, and one carry bead in the lower deck of the next column to the left is moved up. The result of the computation is read off from the beads clustered near the separator beam between the upper and lower deck. In the past, the chinese used the traditional system of measurements called the Shì yòng zhì (市用制) for its suanpan. In Shì yòng zhì (市用制), the unit of weight the jīn (斤), was defined as 16 liǎng (兩), which made it necessary to perform calculations in hexadecimal. The Suanpan can accommodate up to 15 in each digit rod and when it becomes 16, the digit rod changes that will visualise the Hexadecimal system. That is the reason why the Japanese Soroban's 4 earth beads(when value is 0) is one bead apart from the beam while the suanpan's 5 earth beads(when value is 0) are 2 beads apart from its beam. === Division === There exist different methods to perform division on the suanpan. Some of them require the use of the so-called "Chinese division table". The two most extreme beads, the bottommost earth bead and the topmost heaven bead, are usually not used in addition and subtraction. They are essential (compulsory) in some of the multiplication methods (two of three methods require them) and division method (special division table, Qiuchu 九歸, one amongst three methods). When the intermediate result (in multiplication and division) is larger than 15 (fifteen), the second (extra) upper bead is moved halfway to represent ten (xuanchu, suspended). Thus the same rod can represent up to 20 (compulsory as intermediate steps in traditional suanpan multiplication and division). The mnemonics/readings of the Chinese division method [Qiuchu] has its origin in the use of bamboo sticks [Chousuan], which is one of the reasons that many believe the evolution of suanpan is independent of the Roman abacus. This Chinese division method (i.e. with division table) was not in use when the Japanese changed their abacus to one upper bead and four lower beads in about the 1920s. == Decimal system == This 4+1 abacus works as a bi-quinary based number system (the 5+2 abacus is similar but not identical to bi-quinary) in which carries and shifting are similar to the decimal number system. Since each rod represents a digit in a decimal number, the computation capacity of the suanpan is only limited by the number of rods on the suanpan. When a mathematician runs out of rods, another suanpan can be added to the left of the first. In theory, the suanpan can be expanded indefinitely in this way. The suanpan's extra beads can be used for representing decimal numbers, adding or subtracting decimal numbers, caching carry operations, and base sixteen (hexadecimal) fractions. == Zhusuan == Zhusuan (Chinese: 珠算; literally: "bead calculation") is the knowledge and practices of arithmetic calculation through the suanpan. In the year 2013, it has been inscribed on the UNESCO Representative List of the Intangible Cultural Heritage of Humanity. Zhusuan is named after the Chinese name of abacus, which has been recognised as one of the Fifth Great Innovation in China While deciding on the inscription, the Intergovernmental Committee noted that "Zhusuan is considered by Chinese people as a cultural symbol of their identity as well as a practical tool; transmitted from generation to generation, it is a calculating technique adapted to multiple aspects of daily life, serving multiform socio-cultural functions and offering the world an alternative knowledge system." The movement to get Chinese Zhusuan inscribed in the list was spearheaded by Chinese Abacus and Mental Arithmetic Association. Zhusuan is an important part of the traditional Chinese culture. Zhusuan has a far-reaching effect on various fields of Chinese society, like Chinese folk custom, language, literature, sculpture, architecture, etc., creating a Zhusuan-related cultural phenomenon. For example, ‘Iron Abacus’ (鐵算盤) refers to someone good at calculating; ‘Plus three equals plus five and minus two’ (三下五除二; +3 = +5 − 2) means quick and decisive; ‘3 times 7 equals 21’ indicates quick and rash; and in some places of China, there is a custom of telling children's fortune by placing various daily necessities before them on their first birthday and letting them choose one to predict their future lives. Among the items is an abacus, which symbolizes wisdom and wealth. == Modern usage == Suanpan arithmetic was still being taught in school in Hong Kong as recently as the late 1960s, and in China into the 1990s. In some less-developed industry, the suanpan (abacus) is still in use as a primary counting device and back-up calculating method. However, when handheld calculators became readily available, school children's willingness to learn the use of the suanpan decreased dramatically. In the early days of handheld calculators, news of suanpan operators beating electronic calculators in arithmetic competitions in both speed and accuracy often appeared in the media. Early electronic calculators could only handle 8 to 10 significant digits, whereas suanpans can be built to virtually limitless precision. But when the functionality of calculators improved beyond simple arithmetic operations, most people realized that the suanpan could never compute higher functions – such as those in trigonometry – faster than a calculator. As digitalised calculators seemed to be more efficient and user-friendly, their functional capacities attract more technological-related and large scale industries in application. Nowadays, even though calculators have become more affordable and convenient, suanpans are still commonly used in China. Many parents still tend to send their children to private tutors or school- and government-sponsored after school activities to learn bead arithmetic as a learning aid and a stepping stone to faster and more accurate mental arithmetic, or as a matter of cultural preservation. Speed competitions are still held. Suanpans are still widely used elsewhere in China and Japan, as well as in a few places in Canada and the United States. With its historical value, it has symbolized the traditional cultural identity. It contributes to the advancement of calculating techniques and intellectual development, which closely relate to the cultural-related industry like architecture and folk customs. With their operational simplicity and traditional habit, Suanpans are still generally in use in small-scale shops. In mainland China, formerly accountants and financial personnel had to pass certain graded examinations in bead arithmetic before they were qualified. Starting from about 2002 or 2004, this requirement has been entirely replaced by computer accounting. == Notes == == See also == Abacus Counting rods Soroban == References == Ifrah, Georges (2001). The Universal History of Computing: From the Abacus to the Quantum Computer. New York, NY: John Wiley & Sons, Inc. ISBN 978-0-471-39671-0. Peng Yoke Ho (2000). Li, Qi and Shu: An Introduction to Science and Civilization in China. Courier Dover Publications. ISBN 0-486-41445-0. Martzloff (2006). A History of Chinese Mathematics. Springer-Verlag. ISBN 3-540-33782-2. == External links == Suanpan Tutor - See the steps in addition and subtraction A Traditional Suan Pan Technique for Multiplication Hex to Suanpan UNESCO video on Chinese Zhusuan on YouTube (Published on Dec 4, 2013): Zhusuan
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Wikipedia:Zinovy Reichstein#0
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Zinovy Reichstein (born 1961) is a Russian-born American mathematician. He is a professor at the University of British Columbia in Vancouver. He studies mainly algebra, algebraic geometry and algebraic groups. He introduced (with Joe P. Buhler) the concept of essential dimension. == Early life and education == In high school, Reichstein participated in the national mathematics olympiad in Russia and was the third highest scorer in 1977 and second highest scorer in 1978. Because of the Antisemitism in the Soviet Union at the time, Reichstein was not accepted to Moscow University, even though he had passed the special math entrance exams. He attended a semester of college at Russian University of Transport instead. His family then decided to emigrate, arriving in Vienna, Austria, in August 1979 and New York, United States in the fall of 1980. Reichstein worked as a delivery boy for a short period of time in New York. He was then accepted to and attended California Institute of Technology for his undergraduate studies. Reichstein received his PhD degree in 1988 from Harvard University under the supervision of Michael Artin. Parts of his thesis entitled "The Behavior of Stability under Equivariant Maps" were published in the journal Inventiones Mathematicae. == Career == As of 2011, he is on the editorial board of the mathematics journal Transformation groups. == Awards == Winner of the 2013 Jeffery-Williams Prize awarded by the Canadian Mathematical Society Fellow of the American Mathematical Society, 2012 Invited Speaker to the International Congress of Mathematicians (Hyderabad, India 2010) == References == == External links == Official website Zinovy Reichstein at the Mathematics Genealogy Project
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Wikipedia:Znám's problem#0
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In number theory, Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972, although other mathematicians had considered similar problems around the same time. The initial terms of Sylvester's sequence almost solve this problem, except that the last chosen term equals one plus the product of the others, rather than being a proper divisor. Sun (1983) showed that there is at least one solution to the (proper) Znám problem for each k ≥ 5 {\displaystyle k\geq 5} . Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values. The Znám problem is closely related to Egyptian fractions. It is known that there are only finitely many solutions for any fixed k {\displaystyle k} . It is unknown whether there are any solutions to Znám's problem using only odd numbers, and there remain several other open questions. == The problem == Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the set, plus 1. That is, given k {\displaystyle k} , what sets of integers { n 1 , … , n k } {\displaystyle \{n_{1},\ldots ,n_{k}\}} are there such that, for each i {\displaystyle i} , n i {\displaystyle n_{i}} divides but is not equal to ( ∏ j ≠ i n n j ) + 1 ? {\displaystyle {\Bigl (}\prod _{j\neq i}^{n}n_{j}{\Bigr )}+1?} A closely related problem concerns sets of integers in which each integer in the set is a divisor, but not necessarily a proper divisor, of one plus the product of the other integers in the set. This problem does not seem to have been named in the literature, and will be referred to as the improper Znám problem. Any solution to Znám's problem is also a solution to the improper Znám problem, but not necessarily vice versa. == History == Znám's problem is named after the Slovak mathematician Štefan Znám, who suggested it in 1972. Barbeau (1971) had posed the improper Znám problem for k = 3 {\displaystyle k=3} , and Mordell (1973), independently of Znám, found all solutions to the improper problem for k ≤ 5 {\displaystyle k\leq 5} . Skula (1975) showed that Znám's problem is unsolvable for k < 5 {\displaystyle k<5} , and credited J. Janák with finding the solution { 2 , 3 , 11 , 23 , 31 } {\displaystyle \{2,3,11,23,31\}} for k = 5 {\displaystyle k=5} . == Examples == Sylvester's sequence is an integer sequence in which each term is one plus the product of the previous terms. The first few terms of the sequence are Stopping the sequence early produces a set like { 2 , 3 , 7 , 43 } {\displaystyle \{2,3,7,43\}} that almost meets the conditions of Znám's problem, except that the largest value equals one plus the product of the other terms, rather than being a proper divisor. Thus, it is a solution to the improper Znám problem, but not a solution to Znám's problem as it is usually defined. One solution to the proper Znám problem, for k = 5 {\displaystyle k=5} , is { 2 , 3 , 7 , 47 , 395 } {\displaystyle \{2,3,7,47,395\}} . A few calculations will show that == Connection to Egyptian fractions == Any solution to the improper Znám problem is equivalent (via division by the product of the values x i {\displaystyle x_{i}} ) to a solution to the equation ∑ 1 x i + ∏ 1 x i = y , {\displaystyle \sum {\frac {1}{x_{i}}}+\prod {\frac {1}{x_{i}}}=y,} where y {\displaystyle y} as well as each x i {\displaystyle x_{i}} must be an integer, and conversely any such solution corresponds to a solution to the improper Znám problem. However, all known solutions have y = 1 {\displaystyle y=1} , so they satisfy the equation ∑ 1 x i + ∏ 1 x i = 1. {\displaystyle \sum {\frac {1}{x_{i}}}+\prod {\frac {1}{x_{i}}}=1.} That is, they lead to an Egyptian fraction representation of the number one as a sum of unit fractions. Several of the cited papers on Znám's problem study also the solutions to this equation. Brenton & Hill (1988) describe an application of the equation in topology, to the classification of singularities on surfaces, and Domaratzki et al. (2005) describe an application to the theory of nondeterministic finite automata. == Number of solutions == The number of solutions to Znám's problem for any k {\displaystyle k} is finite, so it makes sense to count the total number of solutions for each k {\displaystyle k} . Sun (1983) showed that there is at least one solution to the (proper) Znám problem for each k ≥ 5 {\displaystyle k\geq 5} . Sun's solution is based on a recurrence similar to that for Sylvester's sequence, but with a different set of initial values. The number of solutions for small values of k {\displaystyle k} , starting with k = 5 {\displaystyle k=5} , forms the sequence 2, 5, 18, 96 (sequence A075441 in the OEIS). Presently, a few solutions are known for k = 9 {\displaystyle k=9} and k = 10 {\displaystyle k=10} , but it is unclear how many solutions remain undiscovered for those values of k {\displaystyle k} . However, there are infinitely many solutions if k {\displaystyle k} is not fixed: Cao & Jing (1998) showed that there are at least 39 solutions for each k ≥ 12 {\displaystyle k\geq 12} , improving earlier results proving the existence of fewer solutions; Sun & Cao (1988) conjecture that the number of solutions for each value of k {\displaystyle k} grows monotonically with k {\displaystyle k} . It is unknown whether there are any solutions to Znám's problem using only odd numbers. With one exception, all known solutions start with 2. If all numbers in a solution to Znám's problem or the improper Znám problem are prime, their product is a primary pseudoperfect number; it is unknown whether infinitely many solutions of this type exist. == See also == Giuga number Primary pseudoperfect number == References == === Notes === === Sources === == External links == Primefan, Solutions to Znám's Problem Weisstein, Eric W., "Znám's Problem", MathWorld
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Wikipedia:Zoghman Mebkhout#0
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Zoghman Mebkhout (born 1949 ) (زغمان مبخوت) is a French-Algerian mathematician. He is known for his work in algebraic analysis, geometry and representation theory, more precisely on the theory of D-modules. == Career == Mebkhout is currently a research director at the French National Centre for Scientific Research and in 2002 Zoghman received the Servant Medal from the CNRS a prize given every two years with an amount of €10,000. == Notable works == In September 1979 Mebkhout presented the Riemann–Hilbert correspondence, which is a generalization of Hilbert's twenty-first problem to higher dimensions. The original setting was for Riemann surfaces, where it was about the existence of regular differential equations with prescribed monodromy groups. In higher dimensions, Riemann surfaces are replaced by complex manifolds of dimension > 1. Certain systems of partial differential equations (linear and having very special properties for their solutions) and possible monodromies of their solutions correspond. An independent proof of this result was presented by Masaki Kashiwara in April 1980. Zoghman is now largely known as a specialist in D-modules theory. == Recognition == Zoghman is one of the first modern international-caliber North-African mathematicians. A symposium in Spain was held on his sixtieth birthday. He was invited to the Institute for Advanced Study and gave a recent talk at Institut Fourier. In his quasi-autobiographical text Récoltes et semailles Alexander Grothendieck wrote extensively about what he for a time thought of as gross mistreatment of Mebkhout, in particular in the context of attribution of credit for the formulation and proof of the Riemann-Hilbert correspondence. However, in May 1986, after being contacted by a number of mathematicians involved in the matter, Grothendieck retracted his former viewpoints (that had been based on direct testimony of Mebkhout) in a number of additions to the manuscript. They were not included in the first edition of the book by Éditions Gallimard, but were added in the later "augmented" edition. == References ==
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Wikipedia:Zoia Ceaușescu#0
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Zoia Ceaușescu (Romanian pronunciation: [ˈzoja tʃe̯a.uˈʃesku]; 28 February 1949 – 20 November 2006) was a Romanian mathematician, the daughter of Communist leader Nicolae Ceaușescu and his wife, Elena. She was also known as Tovarășa Zoia (comrade Zoia). == Biography == Zoia Ceaușescu studied at High School nr. 24 (now Jean Monnet High School) in Bucharest and graduated in 1966. She then continued her studies at the Faculty of Mathematics, University of Bucharest. She received her Ph.D. in 1977 with thesis On Intertwining Dilations written under the direction of Ciprian Foias. Ceaușescu worked as a researcher at the Institute of Mathematics of the Romanian Academy in Bucharest starting in 1974. Her field of specialization was functional analysis. Allegedly, her parents were unhappy with their daughter's choice of doing research in mathematics, so the Institute was disbanded in 1975. She moved on to work for Institutul pentru Creație Științifică și Tehnică (INCREST, Institute for Scientific and Technical Creativity), where she eventually started and headed a new department of mathematics. In 1976, Ceaușescu received the Simion Stoilow Prize for her outstanding contributions to the mathematical sciences. She was married in 1980 to Mircea Oprean, an engineer and professor at the Polytechnic University of Bucharest. During the Romanian Revolution, on 24 December 1989, she was arrested for "undermining the Romanian economy", and released eight months later, on 18 August 1990. After she was freed, she tried unsuccessfully to return to her former job at INCREST, then gave up and retired. After the revolution, some newspapers reported that she had lived a wild life, having numerous lovers and often being drunk. After her parents were executed, the new government confiscated the house where she and her husband lived (the house was used as proof of allegedly stolen wealth), so she and her husband had to live with friends and relatives. After the revolution that ousted her parents, Zoia reported that during her parents' time in power her mother had asked the Securitate to keep an eye on the Ceaușescu children, perhaps she felt, out of a "sense of love". The Securitate "could not touch" the children she said, but the information they provided created a lot of problems for the children. She also remarked that power had a "destructive effect" on her father and that he "lost his sense of judgement". Zoia Ceaușescu believed that her parents were not buried in Ghencea Cemetery; she attempted to have their remains exhumed, but a military court refused her request. The bodies were exhumed for identification and confirmed to be of her parents in 2010, after her death. Zoia was a chain smoker. She died of lung cancer in 2006, at the age of 57, and her remains were cremated at the Cenușa Crematorium. == Selected publications == Zoia Ceaușescu published 22 scientific papers between 1976 and 1988. Some of those are: Ceaușescu, Zoia; Vasilescu, Florian-Horia (1978). "Tensor products and the joint spectrum in Hilbert spaces". Proceedings of the American Mathematical Society. 72 (3). American Mathematical Society: 505–508. doi:10.1090/S0002-9939-1978-0509243-8. JSTOR 2042460. MR 0509243. Ceaușescu, Zoia (1979). "Lifting of a contraction intertwining two isometries". Michigan Mathematical Journal. 26 (2): 231–241. doi:10.1307/mmj/1029002216. MR 0532324. Arsene, Grigore; Ceaușescu, Zoia; Constantinescu, Tiberiu (1988). "Schur analysis of some completion problems". Linear Algebra and Its Applications. 109: 1–35. doi:10.1016/0024-3795(88)90195-4. MR 0961563. == References == == External links == Media related to Zoia Ceaușescu at Wikimedia Commons
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Wikipedia:Zonal polynomial#0
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In mathematics, a zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials. Zonal polynomials appear in special functions with matrix argument which on the other hand appear in matrixvariate distributions such as the Wishart distribution when integrating over compact Lie groups. The theory was started in multivariate statistics in the 1960s and 1970s in a series of papers by Alan Treleven James and his doctoral student Alan Graham Constantine. They appear as zonal spherical functions of the Gelfand pairs ( S 2 n , H n ) {\displaystyle (S_{2n},H_{n})} (here, H n {\displaystyle H_{n}} is the hyperoctahedral group) and ( G l n ( R ) , O n ) {\displaystyle (Gl_{n}(\mathbb {R} ),O_{n})} , which means that they describe canonical basis of the double class algebras C [ H n ∖ S 2 n / H n ] {\displaystyle \mathbb {C} [H_{n}\backslash S_{2n}/H_{n}]} and C [ O d ( R ) ∖ M d ( R ) / O d ( R ) ] {\displaystyle \mathbb {C} [O_{d}(\mathbb {R} )\backslash M_{d}(\mathbb {R} )/O_{d}(\mathbb {R} )]} . The zonal polynomials are the α = 2 {\displaystyle \alpha =2} case of the C normalization of the Jack function. == References == == Literature == Robb Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, Inc., New York, 1984.
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Wikipedia:Zoran Vondraček#0
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Zoran Vondraček (born July 28, 1959) is a Croatian mathematician specializing in Lévy processes, transformed Brownian motions, and probabilistic potential theory. He is a professor of mathematics at the University of Zagreb. == Education and career == Vondraček graduated from the University of Zagreb in 1982, and obtained a master's degree in mathematics there in 1986. He completed a Ph.D. at the University of Florida in 1990 under the supervision of Murali Rao. In 1992 Vondraček was made assistant professor of mathematics at the University of Zagreb. He became an associate professor in 1997, and a full professor in 2002. Vondraček is a member of the Croatian Academy of Sciences and Arts, the American Mathematical Society, and the Institute of Mathematical Statistics. == Recognition == In 2006 Vondraček received the Award for Natural Sciences and Mathematics of the Croatian Academy of Sciences and Arts, alongside oceanographer Nada Krstulović and marine biologist Mladen Šolić. He won the 2010 Croatian National Science Award in the category of natural sciences, alongside Nevenko Bilić and Kristian Vlahoviček. In 2019 he received the University of Florida's Preeminence Award alongside mathematician Hrvoje Šikić. In 2020 he won the Andrija Mohorovičić Prize of the University of Zagreb. == References == == External links == Home page Zoran Vondraček publications indexed by Google Scholar
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Wikipedia:Zuhair Nashed#0
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M. Zuhair Nashed (born May 14, 1936, in Aleppo, Syria) is an American mathematician, working on integral and operator equations, inverse and ill-posed problems, numerical and nonlinear functional analysis, optimization and approximation theory, operator theory, optimal control theory, signal analysis, and signal processing. == Career == Zuhair Nashed received his Master of Science degree in electrical engineering from Massachusetts Institute of Technology in 1958, and his Doctor of Philosophy degree in mathematics, from the University of Michigan Ann Arbor in 1963. He started his academic career in 1963 as an assistant professor at Georgia Institute of Technology, Atlanta, and was promoted to associate professor in 1965 and to full professor in 1969. He moved to the University of Delaware in 1977 to hold the position of Professor of mathematics and electrical engineering. He moved to the University Central Florida, Orlando, in 2002, where he held the position of Professor and Chair 2002 – 2006. Since 2007 he is a professor at the University Central Florida, Orlando. In 2012 he was inducted Fellow of the American Mathematical Society. == References ==
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Wikipedia:Zvi Arad#0
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Zvi Arad (Hebrew: צבי ארד; 16 April 1942, in Petah Tikva, Mandatory Palestine – 4 February 2018, in Petah Tikva, Israel) was an Israeli mathematician, acting president of Bar-Ilan University, and president of Netanya Academic College. == Biography == Zvi Arad began his academic studies in the Mathematics Department of Bar-Ilan University. He received his first degree in 1964 and after army service went on to complete a second and third degree in the Mathematics Department of Tel Aviv University. == Academic career == In 1968 Arad joined the academic staff at Bar-Ilan University as an assistant and in 1983 was appointed a full professor. During the years 1978/9 he held the position of visiting scientist at the University of Chicago, and from 1982 to 1983 held the position of visiting professor at the University of Toronto. Arad held a variety of senior academic posts at Bar-Ilan University. He served as chairman of the Mathematics and Computer Science Department, dean of the Faculty of Natural Sciences and Mathematics, rector and president of the university (succeeding Ernest Krausz, and followed by Shlomo Eckstein). Together with Professor Bernard Pinchuk he founded Gelbart Institute, an international research institute named after Abe Gelbart, and the Emmy Noether Institute (Minerva Center). Together with colleagues he established a journal, the Israel Mathematics Conference Proceedings, distributed by the American Mathematical Society (AMS). From 1984–1985 he served as a member of the Council for Higher Education of the State of Israel. In 1982 he was elected a member of Russia's Academy of Natural Sciences. From 1994 he served on the editorial board of the Algebra Colloquium, a journal of the Chinese Academy of Sciences published by Springer-Verlag. He also serves on the editorial board of various international publications: South East Asian Bulletin of Mathematics of the Asian Mathematical Society, the IMCP of Contemporary Mathematics published by the American Mathematical Society, and the publication Cubo Matemática Educacional, Temuco, Chile. He initiated numerous agreements of cooperation with universities and institutions throughout the world including academic institutes in the former Soviet Union, universities and research centers in America, Canada, Germany, the United Kingdom, Italy, Russia, China, South Africa, etc. He was a member of Israel's first official delegation to the former USSR, under the leadership of President Ezer Weizman. In an official address, President Mikhail Gorbachev mentioned Professor Arad's contributions towards the establishment of scientific communications between Israel and the former USSR. In an effort to advance cooperation in research he has headed delegations of scientists to Russia, China, and East Germany. Haaretz newspaper (January 21, 1998) described him as one of the pioneers of higher education reform in Israel. The Encyclopaedia Hebraica lists Zvi Arad as "fulfilling a key role in the development and advancement of Bar-Ilan University and in the establishment of the University's regional colleges in Safed, Ashkelon and the Jordan Valley)." For this achievement he was awarded a certificate of honor by the mayor of each city. The establishment of these colleges began in 1985 and went on to affect the whole of Israel. These colleges advanced the Galilee and Southern Israel and brought higher education to the peripheries of Israel. == Netanya Academic College == In 1994, at the request of the mayors of the city of Netanya, Yoel Elroi and Zvi Poleg, Arad established the Netanya Academic College. He served as president of the college for 24 years. A partner in the initiation and establishment of the college was Miriam Feirberg, who at that time served as head of the Education Department of the City of Netanya. Today the college is an accredited institute of higher education that grant first and second academic degrees in a variety of fields. == Published works == Together with his colleague Professor Marcel Herzog, Arad wrote Products of Conjugacy Classes, published by Springer-Verlag. The book facilitated the basis of the establishment of mathematical theory and today forms part of the branch of abstract algebra known as Table Algebras, and is attached to central branches in mathematics: Graph theory, algebra combinations, and theory presentation. Arad coauthored two other books on the subject of table algebra. In 2000 his book was published in the series American Mathematical Society Memoirs and in January 2002 another book on table algebras was published in the international publication, Springer. Arad was the editor of Contemporary Mathematics, Volume 402. == See also == Education in Israel == References == == External links == Profile Netanya Academic College Zvi Arad Dun's 100 (in Hebrew)
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Wikipedia:Zvi Galil#0
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Zvi Galil (Hebrew: צבי גליל; born June 26, 1947) is an Israeli-American computer scientist. He has served as the dean of the Columbia University School of Engineering and as president of Tel Aviv University from 2007 through 2009. From 2010 to 2019, he was the dean of the Georgia Institute of Technology College of Computing. His research interests include the design and analysis of algorithms, computational complexity and cryptography. He has been credited with coining the terms stringology and sparsification. He has published over 200 scientific papers and is listed as an ISI highly cited researcher. == Early life and education == Galil was born in Tel Aviv in Mandatory Palestine in 1947. He completed both his B.Sc. (1970) and his M.Sc. (1971) in applied mathematics, both summa cum laude, at Tel Aviv University. In 1975, he earned his Ph.D. in computer science at Cornell University under the supervision of John Hopcroft. He then spent a year working as a post-doctorate researcher at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York. == Career == From 1976 until 1995, he worked in the computer science department at Tel Aviv University, serving as its chair from 1979 to 1982. In 1982, he joined the faculty of Columbia University, serving as the chair of the computer science department from 1989 to 1994. From 1995-2007, he served as the dean of the Columbia University Fu Foundation School of Engineering & Applied Science. In this position, he oversaw the naming of the school in honor of Chinese businessman Z. Y. Fu after a large donation was given in his name. At Columbia, he was appointed the Julian Clarence Levi Professor of Mathematical Methods and Computer Science in 1987, and the Morris and Alma A. Schapiro Dean of Engineering in 1995. In 2007, Galil succeeded Itamar Rabinovich as president of Tel Aviv University. In 2009, he resigned and returned to the faculty, and was succeeded by Joseph Klafter. He was named as the dean of Georgia Tech's college of computing on April 9, 2010. At Georgia Tech, together with Udacity founder Sebastian Thrun, Galil conceived of the college of computing's online Master of Science in computer science (OMSCS) program, and he led the faculty creation of the program. OMSCS went on to become the largest online master’s program in computer science in the United States. OMSCS has been featured in hundreds of articles, including a 2013 front-page article in The New York Times and 2021 interviews in The Wall Street Journal and Forbes. Inside Higher Education noted that OMSCS "suggests that institutions can successfully deliver high-quality, low-cost degrees to students at scale". The Chronicle of Higher Education noted that OMSCS "may have the best chance of changing how much students pay for a traditional degree". A 2023 Forbes article, titled "The Greatest Degree Program Ever", stated that OMSCS "is - by nearly any measure - the most successful degree program in history". Galil stepped down as dean and returned to a regular faculty position in June 2019. He now serves as the Frederick G. Storey Chair in Computing and Executive Advisor to Online Programs at Georgia Tech. === Professional service === In 1982, Galil founded the Columbia University Theory Day and organized the event for the first 15 years. It still exists as the New York Area Theory Day. From 1983 to 1987, Galil served as the chairman of ACM SIGACT, an organization that promotes research in theoretical computer science. He served as managing editor of SIAM Journal on Computing from 1991 to 1997 and editor in chief of Journal of Algorithms from 1988 to 2003. === Research === Galil's research is in the areas of algorithms, particularly string and graph algorithms. complexity, and cryptography. He has also conducted research in experimental design with Jack Kiefer. Galil's real-time algorithms are the fastest possible for string matching and palindrome recognition, and they work even on the most basic computer model, the multi-tape Turing machine. More generally, he formulated a "predictability" condition that allows any complying online algorithm to be converted to a real-time algorithm. With Joel Seiferas, Galil improved the time-optimal algorithms to be space optimal (logarithmic space) as well. Galil worked with Dany Breslauer to design a linear-work, O(loglogn) parallel algorithm for string matching, and they later proved it to have the best possible time complexity among linear work algorithms. With other computer scientists, he designed a constant-time linear-work randomized search algorithm to be used when the pattern preprocessing is given. With his students, Galil designed more than a dozen currently-fastest algorithms for exact or approximate, sequential or parallel, and one- or multi-dimensional string matching. Galil worked with other computer scientists to develop several currently-fastest graph algorithms. Examples include trivalent graph isomorphism and minimum weight spanning trees. With his students, Galil devised a technique he called "sparsification" and a method he called "sparse dynamic programming". The first was used to speed up dynamic graph algorithms. The second was used to speed up the computations of various edit distances between strings. In 1979, together with Ofer Gabber, Galil solved the previously open problem of constructing a family of expander graphs with an explicit expansion ratio, useful in the design of fast graph algorithms. == Awards and honors == In 1995, Galil was inducted as a fellow at the Association for Computing Machinery for "fundamental contributions to the design and analysis of algorithms and outstanding service to the theoretical computer science community," and in 2004, he was elected to the National Academy of Engineering for "contributions to the design and analysis of algorithms and for leadership in computer science and engineering." In 2005, he was selected as a Fellow of the American Academy of Arts and Sciences. In 2008, Columbia University established the Zvi Galil award for student life. In 2009, the Columbia Society of Graduates awarded him the Great Teacher Award. In 2012, The University of Waterloo awarded Galil with an honorary Doctor of Mathematics degree for his "fundamental contributions in the areas of graph algorithms and string matching." In 2020, Academic Influence included Galil in the list of the 10 most influential computer scientists of the last decade, and the advisory board of the College of Computing at Georgia Tech raised over $2 million from over 130 donors to establish an endowed chair named after Galil. In 2024, Columbia University awarded Galil an honorary doctorate. == References == == External links == Home page at Georgia Tech
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Wikipedia:Zvika Brakerski#0
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Zvika Brakerski (Hebrew: צביקה ברקרסקי) is an Israeli mathematician, known for his work on homomorphic encryption, particularly in developing the foundations of the second generation FHE schema, for which he was awarded the 2022 Gödel Prize. Brakerski is an associate professor in the Department of Computer Science and Applied Mathematics at the Weizmann Institute of Science. == Research == In 2011 Brakerski and Vaikuntanathan based Fully Homomorphic Encryption (FHE) on LWE. Together with Gentry, they constructed the Brakerski-Gentry-Vaikuntanathan (BGV) scheme, which can be instantiated in leveled mode without bootstrapping. For these works, they were jointly awarded the Gödel Prize in 2022. In 2012 Brakerski published a paper at the Annual Cryptology Conference "Fully homomorphic encryption without modulus switching from classical GapSVP". This paper formed the basis of the BFV scheme which is - next to BGV - one of the dominant second-generation FHE schema. == References ==
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Wikipedia:Ágnes Szendrei#0
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Ágnes Szendrei is a Hungarian-American mathematician whose research concerns clones, the congruence lattice problem, and other topics in universal algebra. She is a professor of mathematics at the University of Colorado Boulder, and the author of the well-cited book Clones in Universal Algebra (1986). In May 2022, Dr. Szendrei was elected as an external member of the Hungarian Academy of Sciences; such external memberships are for Hungarian scientists who live outside of Hungary and who have made exceptional contributions to scientific research. Szendrei earned a doctorate from the Hungarian Academy of Sciences in 1982, and a habilitation in 1993. Her 1982 dissertation was Clones of Linear Operations and Semi-Affine Algebras, supervised by Béla Csákány. She was on the faculty of the University of Szeged from 1982 until 2003, when she moved to the University of Colorado. Szendrei is a Humboldt Fellow. She won the Kató Rényi Award for undergraduate research in 1975, the Géza Grünwald Commemorative Prize for young researchers of the János Bolyai Mathematical Society in 1978, and the Golden Ring of the Republic in 1979. She was the 1992 winner of the Paul Erdős Prize of the Hungarian Academy of Sciences, and the 2000 winner of the Academy's Farkas Bolyai Award. == References == == External links == Home page Ágnes Szendrei publications indexed by Google Scholar
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Wikipedia:Édouard Le Roy#0
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Édouard Louis Emmanuel Julien Le Roy (French: [edwaʁ ləʁwa] ; 18 June 1870 in Paris – 10 November 1954 in Paris) was a French philosopher and mathematician. == Life == Le Roy entered the École Normale Supérieure in 1892, and received the agrégation in mathematics in 1895. He became Doctor in Sciences in 1898, taught in several high schools, and in 1909 became professor of mathematics at the Lycée Saint-Louis in Paris. From then on, Le Roy took a major interest in philosophy and metaphysics. A friend of Teilhard de Chardin and Henri Bergson's closer disciple, he succeeded Bergson at the College of France (1922) and, in 1945, at the Académie française. In 1919, Le Roy was also elected a member of the Académie des Sciences morales et politiques. Le Roy was especially interested in the relations between science and morality. Along with Henri Poincaré and Pierre Duhem, he supported a conventionalist thesis on the foundation of mathematics. Although a fervent Catholic, he extended this conventionalist theory to revealed truths, which did not, according to him, withdraw any of their strength. In the domain of religious dogmas, he rejected abstract reasoning and speculative theology in favour of instinctive faith, heart and sentiment. He was one of those close to Bergson who encouraged him to turn to the study of mysticism, explored in his later works. His conventionalism led his works, accused of modernism, to be placed on the Index by the Holy See. == Works == Théorie du potentiel newtonien : leçons professées à la Sorbonne pendant le premier semestre (1894-1895) (1896) Sur l'intégration des équations de chaleur (1898) Sur les séries divergentes et les fonctions définies par un développement de Taylor (1899) Science et Philosophie (1899) Dogme et Critique (1907) A New Philosophy: Henri Bergson (Une philosophie nouvelle : Henri Bergson, 1912) What Is a Dogma? (1918) Qu'est-ce-que la Science ?: réponse à André Metz (1926) L'Exigence idéaliste et le fait de l'évolution (1927) Les Origines humaines et l'évolution de l'intelligence (1928) La Pensée Intuitive. Le problème de Dieu (1929) Introduction à l'étude du problème religieux (1944) Discours de réception (1946) Essai d'une philosophie première (1956) Bergson et Bergsonisme (1947) Essai d'une philosophie première : l'exigence idéaliste et l'exigence morale, 2 vol., posthumous (1956-1958) == See also == Noosphere Pragmatism == References == == External links == Works by Edouard Le Roy at Project Gutenberg Works by or about Édouard Le Roy at the Internet Archive A New Philosophy: Henri Bergson
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Wikipedia:Élisabeth Gassiat#0
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Élisabeth Gassiat (née Granier, born 1961) is a French mathematical statistician whose research interests include maximum likelihood estimation for mixture models, latent variables, high-dimensional structured data, the relation between statistics and coding theory, and the statistics of sequence data over finite alphabets. She is a professor at Paris-Sud University. == Education and career == Gassiat was born in 1961 in Paris. She was a student at the École polytechnique from 1980 to 1983. In 1987 she completed a Ph.D. through Paris-Sud University with the dissertation Blind deconvolution supervised by Didier Dacunha-Castelle. After serving as an assistant at the Institut national agronomique Paris Grignon from 1987 to 1988, she became an assistant professor at Paris-Sud University from 1988 to 1993. From 1993 to 1998 she was a professor at the University of Évry Val d'Essonne. She returned to Paris-Sud University in 1998, taking her present position as a professor there. == Recognition == Gassiat became a senior member of the Institut Universitaire de France in 2020. She was named a knight of the Legion of Honour in 2013, and an officer of the Legion of Honour in 2023. A three-day conference in honor of Gassiat's 62nd birthday was held at the Institut de Mathématique d'Orsay in 2023. == Book == Gassiat is the author of the book Universal Coding and Order Identification by Model Selection Methods (Springer Monographs in Mathematics, 2018, translated by Anna Ben-Hamou from a 2014 French edition). == References == == External links == Home page
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Wikipedia:Émile Cotton#0
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Émile Clément Cotton (5 February 1872 – 14 March 1950) was a professor of mathematics at the University of Grenoble. His PhD thesis studied differential geometry in three dimensions, with the introduction of the Cotton tensor. He held the professorship from 1904 until his 1942 retirement. He was awarded the Legion of Honor. His brother was Aimé Cotton; their father was a mathematics professor. == Publications == Cours de mécanique générale. Introduction à l'étude de la mécanique industrielle (1920) == References ==
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Wikipedia:Éric Moulines#0
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Éric Moulines (born in Bordeaux on 24 January 1963) is a French researcher in statistical learning and signal processing. He received the silver medal from the CNRS in 2010, the France Télécom prize awarded in collaboration with the French Academy of Sciences in 2011. He was appointed a Fellow of the European Association for Signal Processing in 2012 and of the Institute of Mathematical Statistics in 2016. He is General Engineer of the Corps des Mines (X81). == Biography == Éric Moulines entered the École polytechnique in 1981, then went to study at Télécom ParisTech. He began his career at the Centre national d'études des télécommunications where he worked on speech synthesis from text. He is involved in the development of new waveform synthesis methods called PSOLA (pitch synchronous overlap and add). After defending his thesis in 1990, he joined the École Nationale Supérieure des Télécommunications as a lecturer. He then became interested in different problems of statistical signal processing. In particular, it contributes to the development of subspaces methods for the identification of multivariate linear systems and source separation and develops new algorithms for adaptive system estimation. He received the authorization to direct research in 2006 and became a professor at Télécom Paris. He then devoted himself mainly to the application of Bayesian methods with applications in signal processing and statistics. Éric Moulines directed 21 theses, was president of the jury for 9 theses, was rapporteur for 10 theses, was member of the jury for 6 theses. == Scientific work == He is interested in the inference of latent variable models and in particular hidden Markov chains, and non-linear state models (non-linear filtering) In particular, it contributes to filtering methods using interacting particle systems. He was more generally interested in the inference of partially observed Markovian models, coupling estimation and simulation problems with Monte Carlo Markov Chain Methods (MCMC). He has also developed numerous theoretical tools for the convergence analysis of MCMC algorithms, obtaining fundamental results on the long time behaviour of Markov chains. Since 2005, he has been working on statistical learning problems, including the analysis of stochastic optimization algorithms. He joined the Centre de mathématiques appliquées de l'École polytechnique as a professor in 2015. He is interested in Bayesian inference from large scale models, with applications in uncertainty quantification in statistical learning. == Honours and awards == Elected member of the French Academy of Sciences in 2017 CNRS Silver Medal in 2010 Officier of the Palmes Académiques. == References ==
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Wikipedia:Étienne Bézout#0
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Étienne Bézout (French: [bezu]; 31 March 1730 – 27 September 1783) was a French mathematician who was born in Nemours, Seine-et-Marne, France, and died in Avon (near Fontainebleau), France. == Work == In 1758 Bézout was elected an adjoint in mechanics of the French Academy of Sciences. Besides numerous minor works, he wrote a Théorie générale des équations algébriques, published at Paris in 1779, which in particular contained much new and valuable matter on the theory of elimination and symmetrical functions of the roots of an equation: he used determinants in a paper in the Histoire de l'académie royale, 1764, but did not treat the general theory. === Publications === Cours de mathématiques, a l'usage du corps de l'artillerie (in French). Vol. 3. Paris: Tilliard. 1798. == Legacy == After his death, a statue was erected in his birth town, Nemours, to commemorate his achievements. In 2000, the minor planet 17285 Bezout was named after him. == See also == Little Bézout's theorem Bézout's theorem Bézout's identity Bézout matrix Bézout domain == References == The original version of this article was taken from the public domain Rouse History of Mathematics Grabiner, Judith (1970–1980). "Bezout, Etienne". Dictionary of Scientific Biography. Vol. 2. New York: Charles Scribner's Sons. pp. 111–114. ISBN 978-0-684-10114-9. == External links == Media related to Étienne Bézout at Wikimedia Commons Étienne Bézout at the Mathematics Genealogy Project
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Wikipedia:Étienne Fouvry#0
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Étienne Fouvry (French pronunciation: [etjɛn fuvʁi], born 1953) is a French mathematician working primarily in analytic number theory. Fouvry defended his dissertation in 1981 at the University of Bordeaux under the joint direction of Henryk Iwaniec and Jean-Marc Deshouillers. He is an emeritus professor at Paris-Saclay University and the 2021 recipient of the Sophie Germain Prize. In 1985, Fouvry showed that the first case of Fermat's Last Theorem is true for infinitely many primes. == References == == External links == Videos of Étienne Fouvry in the AV-Portal of the German National Library of Science and Technology
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Wikipedia:Étienne Pardoux#0
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Étienne Pardoux (born 1947) is a French mathematician working in the field of Stochastic analysis, in particular Stochastic partial differential equations. He is currently Professor at Aix-Marseille University. He obtained his PhD in 1975 at University of Paris-Sud under the supervision of Alain Bensoussan and Roger Meyer Temam. Together with Peng Shige, he developed the Theory of nonlinear Backward Stochastic Differential Equations (BSDE). == References == == External links == Aix-Marseille University faculty page
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Wikipedia:Étienne-Louis Malus#0
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Étienne-Louis Malus (; French: [e.tjɛn.lwi ma.lys]; 23 July 1775 – 23 February 1812) was a French officer, engineer, physicist, and mathematician. Malus was born in Paris, France and studied at the military engineering school at Mezires where he was taught by Gaspard Monge. He participated in Napoleon's expedition into Egypt (1798 to 1801). He was also a member of the mathematics section of the Institut d'Égypte. Malus became a member of the Académie des Sciences in 1810. In 1810 the Royal Society of London awarded him the Rumford Medal. His mathematical work was almost entirely concerned with the study of light. He studied geometric systems called ray systems, closely connected to Julius Plücker's line geometry. He conducted experiments to verify Christiaan Huygens's theories of light and rewrote the theory in analytical form. His discovery of the polarization of light by reflection was published in 1809 and his theory of double refraction of light in crystals, in 1810. Malus attempted to identify the relationship between the polarising angle of reflection that he had discovered, and the refractive index of the reflecting material. While he deduced the correct relation for water, he was unable to do so for glasses due to the low quality of materials available to him (the refractive index of most glasses available at that time varied between the surface and the interior of the glass). It was not until 1815 that Sir David Brewster was able to experiment with higher quality glasses and correctly formulate what is known as Brewster's law. This law was later explained theoretically by Augustin Fresnel, as a special case of his Fresnel equations. Malus is probably best remembered for Malus's law, giving the resultant intensity, when a polariser is placed in the path of an incident beam. A follower of Laplace, both his statement of the Malus's law and his earlier works on polarisation and birefringence were formulated using the corpuscular theory of light. His name is one of the 72 names inscribed on the Eiffel tower. == "Discovery" of polarization == In 1810, Malus, while engaged on the theory of double refraction, casually examined through a doubly refracting prism of quartz the sunlight reflected from the windows of the Luxembourg palace. He was surprised to find that the two rays alternately disappeared as the prism was rotated through successive right angles, in other words, that the reflected light had acquired properties exactly corresponding to those of the rays transmitted through Iceland spar. He named this phenomenon polarization, and thought it could not be explained by wave theory of light. Instead, he explained it by stating that light-corpuscules have polarity (like magnetic poles). == Selected works == Mémoire sur la mesure du pouvoir réfringent des corps opaques. in Nouveau bulletin des sciences de la Société philomathique de Paris, 1 (1807), 77–81 Mémoire sur de nouveaux phénomènes d’optique. ibid., 2 (1811), 291–295 Traité d’optique. in Mémoires présentés à l’Institut des sciences par divers savants, 2 (1811), 214–302 Théorie de la double réfraction de la lumière dans les substances cristallines. ibid., 303–508 == Work == Malus mathematically analyzed the properties of a system of continuous light rays in three dimensions. He found the equation of caustic surfaces and the Malus theorem: Rays of light that are emitted from a point source, after which they have been reflected on a surface, are all normal to a common surface, but after the second refraction they no longer have this property. If the perpendicular surface is identified with a wave front, it is obvious that this result is false, which Malus did not realize because he adhered to Newton's theory of light emission. Malus's theorem was not proved until 1824 by W. R. Hamilton, with Adolphe Quetelet and Joseph Diez Gergonne giving a separate proof in 1825. == See also == Polarimeter Total internal reflection == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Étienne-Louis Malus", MacTutor History of Mathematics Archive, University of St Andrews English translation of his paper "Optique"
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Wikipedia:Ólafur Daníelsson#0
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Ólafur Dan Daníelsson (31 October 1877 – 10 December 1957) was an Icelandic mathematician. He was the first Icelandic mathematician to complete a doctoral degree. He was also the founder of the Icelandic Mathematical Society. == Life == === Early life and education === Danielsson was born in Viðvík in Viðvíkursveit in Skagafjördur. In 1897, he finished his secondary education in Reykjavík, and in the same year, went to study mathematics in the University of Copenhagen. Hieronymus Georg Zeuthen and Julius Petersen were his university tutors. In 1900, his first scientific paper was published in the Danish journal Nyt Tidsskrift for Matematik B. In 1901, he was awarded a gold medal for his mathematical treatise at the University of Copenhagen. In 1904, he was awarded a master's degree, which enabled him to teach in Danish high schools. Returning to Iceland, he applied to be a mathematics teacher at Reykjavik Junior College, where he had studied a few years previously. However, he did not get the job. The successful applicant was an engineer, Sigurður Thoroddsen. He started undertaking PhD research. His thesis built upon the earlier works of Zeuthen and other scientists, such as Rudolph Clebsch, Guido Castelnuevo and Luigi Cremona. In 1909, he submitted his thesis and graduated from the University of Copenhagen. He was the first Icelandic mathematician to be awarded a doctorate. === Career === He became a private tutor and began writing textbooks. In 1906, his first textbook, Reikningsbók/Arithmetic, was published. In 1908, he became the first mathematics teacher in the Iceland Teacher College when it was first established. The students were experienced teachers, but had been lacking formal education themselves. In 1914, his textbook Arithmetic (Reikningsbók) was republished for the students' needs. In 1919, a mathematics stream at Reykjavík Junior College was founded in response to Danielsson's and his friends' initiative. He was tasked with its development, with the goal of enabling students to attend the Polytechnic College in Copenhagen and to pursue university studies in sciences. Prior to that, students needed to spend a preparatory year abroad. At the same time, Danielsson started writing high school mathematics textbooks. In 1920s, his 4 textbooks were republished, including a rewritten version of the Arithmetic book. Additionally, three new subjects were introduced in Icelandic: Um flatarmyndir/On plane geometry, Kenslubók í hornafræði/Trigonometry, and Kenslubók í algebru/A textbook in algebra. These three textbooks were groundbreaking, being the first of their kind in Icelandic. They were adopted for use at Reykjavík High School, along with the advanced Danish textbooks. Later, when Akureyri High School was established in 1930, these textbooks were also incorporated into its curriculum. The mathematician Sigurdur Helgason commented that, "The geometry textbooks by the remarkable mathematician Ólafur Daníelsson, the pioneering founder of mathematics education in Iceland, were written by a man with a real mission". In 1941, Daníelsson concluded his teaching career and retired. His remarkable influence extended over almost seven decades, starting in 1906 when he published his initial textbook and continuing in 1908 when he commenced teaching at Iceland's Teacher College. His significant impact on mathematics education persisted until 1976 when his textbooks were excluded from the reading list of the national entrance examination. There is no doubt about his enduring legacy as a devoted mathematician, as his visionary approach helped shape mathematics education in Iceland. == Research == In the 1920s, Daníelsson dedicated himself to advancing the field of algebraic geometry through his research. He actively participated in the Scandinavian Congress of Mathematicians held in 1925 and 1927. His contributions were instrumental in fostering the development of mathematics in Iceland, which ultimately led to Iceland becoming a full member of the Nordic Congress of Mathematicians in the 1980s. He published several papers in the Danish Matematisk Tidsskrift, with notable contributions in the years 1926, 1940, 1945, and 1948. His research work also appeared in esteemed journals such as Mathematische Annalen, specifically in volumes 102 (1930), 109 (1934), 113 (1937), and 114 (1937). In 1925, Daníelsson participated in the Sixth Scandinavian Congress of Mathematicians held in Copenhagen. Two years later, in 1927, he also attended the seventh congress held in Oslo. He delivered presentations at both congresses, accompanied by the publication of his papers. His first paper, titled "En Lösning af Malfattis problem" [A solution of Malfatti's Problem], was published in Matematisk Tidsskrift. Subsequently, he contributed to Matematische Annalen with a paper entitled "Überkorrespondierende Punkte der Steinerschen Fläche vierter Ordnung und die Hauptpunkte derselben" (Corresponding Points of Steiner's Surface of Fourth Order and their Principal Points). This journal featured the works of renowned mathematicians such as Einstein, van der Waerden, von Neumann, Landau, Ore, and Kolmogorov, among others, and Daníelsson's paper was among the 44 articles published. It is worth noting that Danielsson was the only mathematician from Iceland contributing to Scandinavian Mathematicial journals before the second world war. Daníelsson's fascination with elementary geometry was evident, as he remarked that "it is difficult to find tasks simpler and more elegant than skillful mathematical problems." His final paper was published in both the Journal of the Icelandic Society of Engineers in 1946 and Matematisk Tidsskrift in 1948. == The Icelandic Mathematical Society == On 31 October 1947, the Icelandic Mathematical Society was founded in Reykjavik when Daníelsson was 70. The society records: “On Friday, 31 October 1947, which was the seventieth birthday of Ólafur Daníelsson, he gathered in his home several men and set up a Society. The purpose of the Society is to promote co-operation and promotion of people in Iceland who have completed a university degree in a mathematical subject. The Society holds meetings at which individual members explain their mathematical topics and, if desired, discussions on the topic will be conducted.” The first lecture was delivered by Ólafur Daníelsson himself. He spoke "about the circle transcribed by the outer circumference of the triangle" and calculated its length relative to the radius of the inscribed circle and the circumference of the triangle. This result has been published in the Matematisk Tidsskrift. However, this had been a longstanding interest of him, as the initial foundations of this subject could be traced back to an article he wrote in 1900, published in the same journal. In this regard, the topic itself carried a sense of antiquity, yet it had recently witnessed a fresh comprehension shortly before his presentation. == References ==
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Wikipedia:Ülo Lumiste#0
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Ülo Lumiste (30 June 1929 Vändra – 20 November 2017) was an Estonian mathematician. In 1952 he graduated from the University of Tartu in mathematics. In 1968 he defended his doctoral thesis at Kazan University. Since 1959 he taught at the University of Tartu. Since 1993 he was a member of Estonian Academy of Sciences. His main field of research was differential geometry. In 1960s he established the school of Estonian differential geometry. == Awards == 1999 and 2012: Estonian National Research Award 1999: Order of the White Star, III class. == References ==
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Wikipedia:Āryabhaṭa numeration#0
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Āryabhaṭa numeration is an alphasyllabic numeral system based on Sanskrit phonemes. It was introduced in the early 6th century in India by Āryabhaṭa, in the first chapter titled Gītika Padam of his Aryabhatiya. It attributes a numerical value to each syllable of the form consonant+vowel possible in Sanskrit phonology, from ka = 1 up to hau = 1018. == History == The basis of this number system is mentioned in the second stanza of the first chapter of Aryabhatiya. The Varga (Group/Class) letters ka to ma are to be placed in the varga (square) places (1st, 100th, 10000th, etc.) and Avarga letters like ya, ra, la .. have to be placed in Avarga places (10th, 1000th, 100000th, etc.). The Varga letters ka to ma have values from 1, 2, 3 .. up to 25 and Avarga letters ya to ha have values 30, 40, 50 .. up to 100. In the Varga and Avarga letters, beyond the ninth vowel (place), new symbols can be used. The values for vowels are as follows: a = 1; i = 100; u = 10000; ṛ = 1000000 and so on. Aryabhata used this number system for representing both small and large numbers in his mathematical and astronomical calculations. This system can even be used to represent fractions and mixed fractions. For example, nga is 1⁄5, nja is 1⁄10 and jhardam (jha=9; its half) = 4+1⁄2. == Example == The traditional Indian digit order is reversed compared to the modern way. By consequence, Āryabhaṭa began with the ones before the tens; then the hundreds and the thousands; then the myriad and the lakh (105) and so on. (cf. Indian numbering system) Another example might be ङिशिबुणॢष्खृ ṅiśibuṇḷṣkhṛ, 1582237500. Note that in this case, 106(ṛ) and 108(ḷ) parts are swapped, and 106(ṛ) part is ligature. Another example from Aryabhatiya is a verse for table of sines. makhi bhakhi phakhi dhakhi ṇakhi ñakhi ṅakhi hasjha skaki kiṣga śghakhi kighva ghlaki kigra hakya dhaki kica sga jhaśa ṅva kla pta pha cha kala-ardha-jyāḥ == Numeral table == In citing the values of Āryabhaṭa numbers, the short vowels अ, इ, उ, ऋ, ऌ, ए, and ओ are invariably used. However, the Āryabhaṭa system did not distinguish between long and short vowels. This table only cites the full slate of क-derived (1 x 10x) values, but these are valid throughout the list of numeric syllables. == See also == Aksharapalli Bhutasamkhya system Katapayadi system IAST == References == Kurt Elfering: Die Mathematik des Aryabhata I. Text, Übersetzung aus dem Sanskrit und Kommentar. Wilhelm Fink Verlag, München, 1975, ISBN 3-7705-1326-6 Georges Ifrah: The Universal History of Numbers. From Prehistory to the Invention of the Computer. John Wiley & Sons, New York, 2000, ISBN 0-471-39340-1. B. L. van der Waerden: Erwachende Wissenschaft. Ägyptische, babylonische und griechische Mathematik. Birkhäuser-Verlag, Basel Stuttgart, 1966, ISBN 3-7643-0399-9 Fleet, J. F. (January 1911). "Aryabhata's System of Expressing Numbers". Journal of the Royal Asiatic Society of Great Britain and Ireland. 43: 109–126. doi:10.1017/S0035869X00040995. ISSN 0035-869X. JSTOR 25189823. S2CID 163070211. Fleet, J. F. (1911). "Aryabhata's System of Expressing Numbers". The Journal of the Royal Asiatic Society of Great Britain and Ireland. 43. Royal Asiatic Society of Great Britain and Ireland: 109–126. doi:10.1017/S0035869X00040995. JSTOR 25189823. S2CID 163070211.
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Wikipedia:Āryabhaṭa's sine table#0
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Āryabhata's sine table is a set of twenty-four numbers given in the astronomical treatise Āryabhatiya composed by the fifth century Indian mathematician and astronomer Āryabhata (476–550 CE), for the computation of the half-chords of a certain set of arcs of a circle. The set of numbers appears in verse 12 in Chapter 1 Dasagitika of Aryabhatiya and is the first table of sines. It is not a table in the modern sense of a mathematical table; that is, it is not a set of numbers arranged into rows and columns. Āryabhaṭa's table is also not a set of values of the trigonometric sine function in a conventional sense; it is a table of the first differences of the values of trigonometric sines expressed in arcminutes, and because of this the table is also referred to as Āryabhaṭa's table of sine-differences. Āryabhaṭa's table was the first sine table ever constructed in the history of mathematics. The now lost tables of Hipparchus (c. 190 BC – c. 120 BC) and Menelaus (c. 70–140 CE) and those of Ptolemy (c. AD 90 – c. 168) were all tables of chords and not of half-chords. Āryabhaṭa's table remained as the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table. These endeavors culminated in the eventual discovery of the power series expansions of the sine and cosine functions by Madhava of Sangamagrama (c. 1350 – c. 1425), the founder of the Kerala school of astronomy and mathematics, and the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places. Some historians of mathematics have argued that the sine table given in Āryabhaṭiya was an adaptation of earlier such tables constructed by mathematicians and astronomers of ancient Greece. David Pingree, one of America's foremost historians of the exact sciences in antiquity, was an exponent of such a view. Assuming this hypothesis, G. J. Toomer writes, "Hardly any documentation exists for the earliest arrival of Greek astronomical models in India, or for that matter what those models would have looked like. So it is very difficult to ascertain the extent to which what has come down to us represents transmitted knowledge, and what is original with Indian scientists. ... The truth is probably a tangled mixture of both." == The table == === In modern notations === The values encoded in Āryabhaṭa's Sanskrit verse can be decoded using the numerical scheme explained in Āryabhaṭīya, and the decoded numbers are listed in the table below. In the table, the angle measures relevant to Āryabhaṭa's sine table are listed in the second column. The third column contains the list the numbers contained in the Sanskrit verse given above in Devanagari script. For the convenience of users unable to read Devanagari, these word-numerals are reproduced in the fourth column in ISO 15919 transliteration. The next column contains these numbers in the Hindu-Arabic numerals. Āryabhaṭa's numbers are the first differences in the values of sines. The corresponding value of sine (or more precisely, of jya) can be obtained by summing up the differences up to that difference. Thus the value of jya corresponding to 18° 45′ is the sum 225 + 224 + 222 + 219 + 215 = 1105. For assessing the accuracy of Āryabhaṭa's computations, the modern values of jyas are given in the last column of the table. In the Indian mathematical tradition, the sine ( or jya) of an angle is not a ratio of numbers. It is the length of a certain line segment, a certain half-chord. The radius of the base circle is basic parameter for the construction of such tables. Historically, several tables have been constructed using different values for this parameter. Āryabhaṭa has chosen the number 3438 as the value of radius of the base circle for the computation of his sine table. The rationale of the choice of this parameter is the idea of measuring the circumference of a circle in angle measures. In astronomical computations distances are measured in degrees, minutes, seconds, etc. In this measure, the circumference of a circle is 360° = (60 × 360) minutes = 21600 minutes. The radius of the circle, the measure of whose circumference is 21600 minutes, is 21600 / 2π minutes. Computing this using the value π = 3.1416 known to Aryabhata one gets the radius of the circle as 3438 minutes approximately. Āryabhaṭa's sine table is based on this value for the radius of the base circle. It has not yet been established who is the first ever to use this value for the base radius. But Aryabhatiya is the earliest surviving text containing a reference to this basic constant. == Āryabhaṭa's computational method == The second section of Āryabhaṭiya, titled Ganitapādd, a contains a stanza indicating a method for the computation of the sine table. There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse given by Katz wherein the words in square brackets are insertions of the translator and not translations of texts in the verse. "When the second half-[chord] partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord." This may be referring to the fact that the second derivative of the sine function is equal to the negative of the sine function. == See also == Madhava's sine table Bhaskara I's sine approximation formula == References ==
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Wikipedia:Đuro Kurepa#0
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Đuro Kurepa (Serbian Cyrillic: Ђуро Курепа, pronounced [dʑǔːro kǔrepa]; 16 August 1907 – 2 November 1993) was a Yugoslav and Serbian mathematician, university professor and academic. Throughout his life, Kurepa published over 700 articles, books, papers, and reviews and over 1,000 scientific reviews. He lectured at universities across Europe, as well as those in Canada, Cuba, Iraq, Israel, and the United States, and was quoted saying "I lectured at almost each of [the] nineteen universities of [the former] Yugoslavia..." == Early life == Born as Đurađ Kurepa in Majske Poljane, Kingdom of Croatia-Slavonia, Austria-Hungary to a Serb family. In English, his name was transliterated as Djuro Kurepa while in French he is often attributed as Georges Kurepa. Kurepa was the youngest of Rade and Anđelija Kurepa's fourteen children. His nephew was the mathematician Svetozar Kurepa. He began his schooling in Majske Poljane, continued his education in Glina, and graduated from high school in Križevci. He received a diploma in theoretical mathematics and physics from the University of Zagreb in 1931, and began work as an assistant in the teaching of mathematics the same year. Kurepa then went to the Collège de France and the University of Paris, where he received his doctoral diploma in 1935; his advisor was French mathematician Maurice René Fréchet, and his thesis was titled Ensembles ordonnés et ramifiés. == Career == Kurepa continued to receive post-doctoral education at Warsaw University in Poland and the University of Paris. He became an assistant professor at the University of Zagreb in 1937, associate professor the next year, and assumed the position of full professor in 1948. After the end of World War II and the formation of the Socialist Federal Republic of Yugoslavia, he traveled to five universities in the United States: Harvard University in Cambridge, Massachusetts, the University of Chicago in Chicago, Illinois, the branch of the University of California at Berkeley and the branch at Los Angeles, California the Institute for Advanced Study in Princeton, New Jersey and Columbia University in New York City, New York. Kurepa was an International Congress of Mathematicians Plenary Speaker in 1954 and 1958. In 1965, Kurepa shifted to the University of Belgrade, where he focused on the mathematical fields of logic and set theory. Kurepa was a member of several organizations, including the Serbian Academy of Sciences and Arts and the Yugoslav Academy of Sciences and Arts. He was also the founder and president of the Society of Mathematicians and Physicists of Croatia, the founder and chief editor of the journal Mathematica Balkanica, and the president of the Yugoslav National Committee for Mathematics, the Balkan Mathematical Society, and the Union of Yugoslav Societies of Mathematicians, Physicisists and Astronomers. He received the AVNOJ Award in 1976, and retired from the University of Belgrade in 1977. He published his scientific works in the most notable European and world scientific journals, including: Mathematische Annalen, Izvestiya Akademii nauk SSSR, Fundamenta Mathematicae, Acta Mathematica, Comptes rendus de l'Académie des Sciences, Bulletin de la Société Mathématique de France, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, Journal of Symbolic Logic, Pacific Journal of Mathematics. == Death and legacy == On 1 November 1993, Kurepa was robbed and beaten after retrieving his pension from a bank. He was then hidden from view under a set of stairs. He succumbed to his injuries on 2 November 1993 in a Belgrade emergency ward. He is interred in the Alley of Distinguished Citizens in the Belgrade New Cemetery. As a mathematician, he is known especially for his works on set theory and general topology. Kurepa influenced set theory in several ways, including lending his name to the Kurepa tree. According to Kajetan Šeper, Kurepa's colleague from the University of Zagreb: Professor Kurepa was not only the professional mathematician and teacher, but he was a scientist, philosopher, and humanist as well, in the true sense of these words. He was the founder and pioneer in mathematical logic and the foundations of mathematics in Croatia, and modern mathematical theories in Croatia and Yugoslavia. Generally speaking, he was the catalyzer, the initiator, and the bearer of mathematical science. According to the Mathematics Genealogy Project, Kurepa supervised 27 students, including analytic number theorist Aleksandar Ivić, set theorist Stevo Todorčević and topologist Ljubisa D.R. Kocinac. == Awards == AVNOJ award, 1976 Bernard Bolzano Charter, Prague, 1981 Order of Labour, 1965 Order of Merits for the People with a golden star, 1979 == Selected works == Books Teorija skupova, Belgrade, 1951 ŠTO SU SKUPOVI I KAKVA IM JE ULOGA, Zagreb, 1960 Viša algebra - knjiga prva, Belgrade, 1969 Viša algebra - knjiga druga, Belgrade, 1969 Selected Papers of Đuro Kurepa, Belgrade, 1996–12 Scientific works Kurepa, Đuro (1935), "Ensembles ordonnés et ramifiés", Publ. Math. Univ. Belgrade, 4: 1–138 SUR LES ENSEMBLES ORDONNES DENOMBRABLES, Zagreb, 1948 O REALNIM FUNKCIJAMA U OBITELjI SKUPOVA RACIONALNIH BROJEVA, Zagreb, 1953 On universal ramified sets, Zagreb, 1963 Monotone mappings between some kinds of ordered sets, Zagreb, 1964 Publikovani naučni radovi Đura Kurepe 1961 - 1976, Belgrade, 2012 == References == == External links == Đuro Kurepa at the Mathematics Genealogy Project (thesis in Paris) Biography of Kurepa (in Serbian) Dimitrić, Radoslav. Academician, Professor Ðuro Kurepa (1907--1993). The Review of Modern Logic 1994. 4:401-403
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Wikipedia:Łojasiewicz inequality#0
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In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : U → R be a real analytic function on an open set U in Rn, and let Z be the zero locus of ƒ. Assume that Z is not empty. Then for any compact set K in U, there exist positive constants α and C such that, for all x in K dist ( x , Z ) α ≤ C | f ( x ) | . {\displaystyle \operatorname {dist} (x,Z)^{\alpha }\leq C|f(x)|.} Here, α {\displaystyle \alpha } can be large. The following form of this inequality is often seen in more analytic contexts: with the same assumptions on f, for every p ∈ U there is a possibly smaller open neighborhood W of p and constants θ ∈ (0,1) and c > 0 such that | f ( x ) − f ( p ) | θ ≤ c | ∇ f ( x ) | . {\displaystyle |f(x)-f(p)|^{\theta }\leq c|\nabla f(x)|.} == Polyak inequality == A special case of the Łojasiewicz inequality, due to Polyak, is commonly used to prove linear convergence of gradient descent algorithms. This section is based on Karimi, Nutini & Schmidt (2016) and Liu, Zhu & Belkin (2022). === Definitions === f {\textstyle f} is a function of type R d → R {\textstyle \mathbb {R} ^{d}\to \mathbb {R} } , and has a continuous derivative ∇ f {\displaystyle \nabla f} . X ∗ {\displaystyle X^{*}} is the subset of R d {\displaystyle \mathbb {R} ^{d}} on which f {\displaystyle f} achieves its global minimum (if one exists). Throughout this section we assume such a global minimum value f ∗ {\displaystyle f^{*}} exists, unless otherwise stated. The optimization objective is to find some point x {\displaystyle x} in X ∗ {\displaystyle X^{*}} . μ , L > 0 {\textstyle \mu ,L>0} are constants. ∇ f {\textstyle \nabla f} is L {\displaystyle L} -Lipschitz continuous iff ‖ ∇ f ( x ) − ∇ f ( y ) ‖ ≤ L ‖ x − y ‖ , ∀ x , y {\displaystyle \|\nabla f(x)-\nabla f(y)\|\leq L\|x-y\|,\quad \forall x,y} f {\textstyle f} is μ {\textstyle \mu } -strongly convex iff f ( y ) ≥ f ( x ) + ∇ f ( x ) T ( y − x ) + μ 2 ‖ y − x ‖ 2 ∀ x , y {\displaystyle f(y)\geq f(x)+\nabla f(x)^{T}(y-x)+{\frac {\mu }{2}}\lVert y-x\rVert ^{2}\quad \forall x,y} f {\textstyle f} is μ {\textstyle \mu } -PL (where "PL" means "Polyak-Łojasiewicz") iff 1 2 ‖ ∇ f ( x ) ‖ 2 ≥ μ ( f ( x ) − f ( x ∗ ) ) , ∀ x {\displaystyle {\frac {1}{2}}\|\nabla f(x)\|^{2}\geq \mu \left(f(x)-f(x^{*})\right),\quad \forall x} === Basic properties === === Gradient descent === === Coordinate descent === The coordinate descent algorithm first samples a random coordinate i k {\textstyle i_{k}} uniformly, then perform gradient descent by x k + 1 = x k − η ∂ i k f ( x k ) e i k {\displaystyle x_{k+1}=x_{k}-\eta \partial _{i_{k}}f(x_{k})e_{i_{k}}} === Stochastic gradient descent === In stochastic gradient descent, we have a function to minimize f ( x ) {\textstyle f(x)} , but we cannot sample its gradient directly. Instead, we sample a random gradient ∇ f i ( x ) {\textstyle \nabla f_{i}(x)} , where f i {\textstyle f_{i}} are such that f ( x ) = E i [ f i ( x ) ] {\displaystyle f(x)=\mathbb {E} _{i}[f_{i}(x)]} For example, in typical machine learning, x {\textstyle x} are the parameters of the neural network, and f i ( x ) {\textstyle f_{i}(x)} is the loss incurred on the i {\textstyle i} -th training data point, while f ( x ) {\textstyle f(x)} is the average loss over all training data points. The gradient update step is x k + 1 = x k − η k ∇ f i k ( x k ) {\displaystyle x_{k+1}=x_{k}-\eta _{k}\nabla f_{i_{k}}(x_{k})} where η k > 0 {\textstyle \eta _{k}>0} are a sequence of learning rates (the learning rate schedule). As it is, the proposition is difficult to use. We can make it easier to use by some further assumptions. The second-moment on the right can be removed by assuming a uniform upper bound. That is, if there exists some C > 0 {\textstyle C>0} such that during the SG process, we have E i [ ‖ ∇ f i ( x k ) ‖ 2 ] ≤ C {\displaystyle \mathbb {E} _{i}[\|\nabla f_{i}(x_{k})\|^{2}]\leq C} for all k = 0 , 1 , … {\textstyle k=0,1,\dots } , then E [ f ( x k + 1 ) − f ∗ ] ≤ ( 1 − 2 η k μ ) [ f ( x k ) − f ∗ ] + L C η k 2 2 {\displaystyle \mathbb {E} \left[f\left(x_{k+1}\right)-f^{*}\right]\leq \left(1-2\eta _{k}\mu \right)\left[f\left(x_{k}\right)-f^{*}\right]+{\frac {LC\eta _{k}^{2}}{2}}} Similarly, if ∀ k , E i [ ‖ ∇ f i ( x k ) − ∇ f ( x k ) ‖ 2 ] ≤ C {\displaystyle \forall k,\quad \mathbb {E} _{i}[\|\nabla f_{i}(x_{k})-\nabla f(x_{k})\|^{2}]\leq C} then E [ f ( x k + 1 ) − f ∗ ] ≤ ( 1 − μ ( 2 η k − L η k 2 ) ) [ f ( x k ) − f ∗ ] + L C η k 2 2 {\displaystyle \mathbb {E} \left[f\left(x_{k+1}\right)-f^{*}\right]\leq \left(1-\mu (2\eta _{k}-L\eta _{k}^{2})\right)\left[f\left(x_{k}\right)-f^{*}\right]+{\frac {LC\eta _{k}^{2}}{2}}} ==== Learning rate schedules ==== For constant learning rate schedule, with η k = η = 1 / L {\textstyle \eta _{k}=\eta =1/L} , we have E [ f ( x k + 1 ) − f ∗ ] ≤ ( 1 − μ / L ) [ f ( x k ) − f ∗ ] + C 2 L {\displaystyle \mathbb {E} \left[f\left(x_{k+1}\right)-f^{*}\right]\leq \left(1-\mu /L\right)\left[f\left(x_{k}\right)-f^{*}\right]+{\frac {C}{2L}}} By induction, we have E [ f ( x k ) − f ∗ ] ≤ ( 1 − μ / L ) k [ f ( x 0 ) − f ∗ ] + C 2 μ {\displaystyle \mathbb {E} \left[f\left(x_{k}\right)-f^{*}\right]\leq \left(1-\mu /L\right)^{k}\left[f\left(x_{0}\right)-f^{*}\right]+{\frac {C}{2\mu }}} We see that the loss decreases in expectation first exponentially, but then stops decreasing, which is caused by the C / ( 2 L ) {\textstyle C/(2L)} term. In short, because the gradient descent steps are too large, the variance in the stochastic gradient starts to dominate, and x k {\textstyle x_{k}} starts doing a random walk in the vicinity of X ∗ {\textstyle X^{*}} . For decreasing learning rate schedule with η k = O ( 1 / k ) {\textstyle \eta _{k}=O(1/k)} , we have E [ f ( x k ) − f ∗ ] = O ( 1 / k ) {\displaystyle \mathbb {E} \left[f\left(x_{k}\right)-f^{*}\right]=O(1/k)} . == References == Bierstone, Edward; Milman, Pierre D. (1988), "Semianalytic and subanalytic sets", Publications Mathématiques de l'IHÉS, 67 (67): 5–42, doi:10.1007/BF02699126, ISSN 1618-1913, MR 0972342, S2CID 56006439 Ji, Shanyu; Kollár, János; Shiffman, Bernard (1992), "A global Łojasiewicz inequality for algebraic varieties", Transactions of the American Mathematical Society, 329 (2): 813–818, doi:10.2307/2153965, ISSN 0002-9947, JSTOR 2153965, MR 1046016 Karimi, Hamed; Nutini, Julie; Schmidt, Mark (2016). "Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak–Łojasiewicz Condition". arXiv:1608.04636 [cs.LG]. Liu, Chaoyue; Zhu, Libin; Belkin, Mikhail (2022-07-01). "Loss landscapes and optimization in over-parameterized non-linear systems and neural networks". Applied and Computational Harmonic Analysis. Special Issue on Harmonic Analysis and Machine Learning. 59: 85–116. arXiv:2003.00307. doi:10.1016/j.acha.2021.12.009. ISSN 1063-5203. == External links == "Lojasiewicz inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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Wikipedia:Śaṅkaranārāyaṇa#0
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Sankaranarayana (c. 840 – c. 900 AD) was an Indian astronomer-mathematician in the court of Sthanu Ravi Kulasekhara (c. 844 – c. 870 AD) of the early medieval Chera kingdom in Kerala. He is celebrated as the author of Laghubhaskariyavivarana or Laghubhaskariyavyakha, a detailed commentary on astronomical treatise Laghubhaskariya by 7th century mathematician Bhaskara I (which in turn was based on the works of the 5th century polymath Aryabhata). Sankaranarayana is known to have established an astronomical observatory at the port of Mahodayapuram, present-day Kodungallur, in central Kerala. Laghubhaskariyavivarana (Chapter VII) explicitly states that it was composed in Saka Year 791 (corresponding to 869/70 AD). In the second verse of the commentary Sankaranarayana remembers five major predecessors in the field of mathematics (Aryabhata, Varahamihira, Bhaskara I, Govinda and Haridatta), including his possible master Govinda (c. 800 – c. 860 AD). The commentary notably mentions an expert astronomer who had travelled to the mleccha country. == Scientific contributions == === Astronomical Observatory === According to the commentary, Sankaranarayana installed an astronomical observatory at the Chera capital Mahodayapuram (on the Malabar Coast). There are references to an instrument called "Rasichakra" marked by a "Yantravalaya" in the vivarana. This instrument might be the same as the Golayantra/Chakrayantra mentioned by polymath Aryabhata. The Chakrayantra was developed further and called Phalakayantra by Bhaskara I. "Oh [king] Ravivarmadeva, now deign to tell us quickly, reading off from the armillary sphere installed [at the observatory] in Mahodayapura, duly fitted with all the relevant circles and with the sign (degree-minute) markings, the time of the rising point of the ecliptic (lagna) when the Sun is at 10° in the sign of Capricorn, and also when the Sun is at the end of the sign Libra, which I have noted." "Oh, Ravi, deign to tell us immediately, reading off from the armillary sphere, by means of the reverse vilagna method, the time for offering the daily oblations, when the Sun, shrouded under thick clouds, is 10° in the Sign Leo and also when it is the middle (i.e. 15°) in the Sign Sagittarius. At the expiry of every ghatika (= 24 minutes), drums were sounded by the soldiers (at different corners of the city of Mahodayapuram, such as certain "Balakridesvara") to announce time. === Mathematical contributions === Laghubhaskariyavivarana covers several the standard mathematical methods of Aryabhata I such as the solution of the indeterminate equation [by = ax ± c] (where a, b, and c are integers) in integers (which is then applied to astronomical problems by the author). The standard Indian method involves the use of Euclidean algorithm called kuttakara ("pulveriser"). The most unusual features of the Laghubhaskariyavivarana are the uses of katapayadi system of numeration and the place-value Sanskrit numerals. Sankaranarayana is the earliest author known to use katapayadi numeration with the specific name. == Historiographical significance == Laghubhaskariyavivarana describes "great mansions" in the city of the Mahodayapuram. He marks out the city of Mahodayapuram as a "senamukha". King Ravi Varma had planned to construct an assembly hall in his capital (he had asked the astronomers to fix the purvapararekha and enjoined craftsmen for the construction). A number of specific locations in the capital were also mentioned (such as "Gotramallesvara" — where the royal residence was located — and the "Balakridesvara Ganapati Temple" near to it). Gotramallesvara is identifiable with present-day Lokamallesvaram in Kodungallur. === Identification of king Ravi Kulasekhara with Sthanu === Sankaranarayana says that he was patronized by king Ravi, who had the title Kulasekhara (and thus helps in the identification of Chera ruler Sthanu Ravi with Kulasekhara). The opening verse of the commentary gives an indirect invocation or praise to the lord called "Sthanu" (carefully composed to be applicable to god Siva and the ruling king)."Sa Sthanurjayati trirupasahito lingepi lokarcitah." Laghubhaskariyavivarana, according to the commentary itself, was composed in the 25th regnal year of king Ravi Kulasekhara. === Date of Laghubhaskariyavivarana === Laghubhaskariyavivarana is dated by the author in three methods. ==== As a Kali Date (when the ruler made enquiries regarding solar eclipse) ==== "Angartvambara nanda devamanubhir yate dinanam ganeGraste tigma mayukhamalinitamobhute parahne diviPrsta praggrahanad dvitiyaghatika grasa pramanam raverBharta sri Kulasekharena vilasad velavrtaya bhuva." "Angartvambara nanda devamanubhir yate dinanam gane" Anga = 6, Rtu = 6, Ambara = 0, Nanda = 9, Veda = 4, and Manu = 14 Order - 6609414 Reverse Order - 1449066 Kali Date - 3967 years and 86 days = 25 Mithuna, Kollam Era 41 = 866 AD ==== In the Saka Era ==== "Evam Sakabdah punariha candra randhramuni sankhyaya asambhiravagatah." "Sakabdah punariha candra randhramuni sankhyaya" Candra = 1, Randhra = 9, and Muni = 7 Order - 197 Reverse Order - 791 (Saka Year) = 870 AD ==== In regnal years ==== "Capapravista guru sauri samatva kalamYamyottaram gamanamantaratah pramanamAcaksvya sarvamavagamya bhatoktamargadItyuktavan ravirasena nrpabhivandya.""Tada pancavimsati Varsanyatitani devasya." Meeting of Guru (=Jupiter) and Sauri (=Saturn) in Capa (Dhanu) = 25th regnal year of the king. In the 9th century, these two planets came to Dhanu Rasi simultaneously only in 869 AD. == See also == Indian mathematics History of mathematics List of astronomers and mathematicians of the Kerala school == References ==
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Wikipedia:Štefan Znám#0
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Štefan Znám (9 February 1936, Veľký Blh – 17 July 1993, Bratislava) was a Slovak- Hungarian mathematician, believed to be the first to ponder Znám's problem in modern times. Znám worked in the field of number theory and graph theory. He also co-founded journal Matematické obzory. == External links == Article on Štefan Znám in Matematický ústav SAV "Štefan Znám". PlanetMath.
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