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Wikipedia:Karen Yeats#0
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Karen Amanda Yeats (born 1980) is a Canadian mathematician and mathematical physicist whose research connects combinatorics to quantum field theory. They hold the Canada Research Chair in Combinatorics in Quantum Field Theory at the University of Waterloo. == Biography == Yeats is from Halifax, Nova Scotia. As an undergraduate at the University of Waterloo, they won an honourable mention for the 2003 Morgan Prize for their research in number theory, the theory of Lie groups, and non-standard models of arithmetic. They graduated in 2003, and went to Boston University for graduate school, where they completed their Ph.D. in 2008. Their dissertation, Growth Estimates for Dyson-Schwinger Equations, was supervised by Dirk Kreimer. In 2016, Yeats was awarded a Humboldt Fellowship to visit Kreimer at the Humboldt University of Berlin. Yeats is the author of the books Rearranging Dyson–Schwinger Equations (Memoirs of the American Mathematical Society, 2011) and A Combinatorial Perspective on Quantum Field Theory (Springer, 2017). == References == == External links == Karen Yeats publications indexed by Google Scholar Home page
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Wikipedia:Kari Hag#0
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Kari Jorun Blakkisrud Hag (born April 4, 1941) is a Norwegian mathematician known for her research in complex analysis on quasicircles and quasiconformal mappings, and for her efforts for gender equality in mathematics. She is a professor emerita of mathematics at the Norwegian University of Science and Technology (NTNU). With Frederick Gehring she is the author of the book The Ubiquitous Quasidisk (American Mathematical Society, 2012). == Education and career == Hag is originally from Eidsvoll. She studied at the Norwegian School of Education in Trondheim, completing a cand.mag. in 1963, and then at the University of Oslo, completing a cand.real. in 1967. Following this, she earned her doctorate in 1972 from the University of Michigan. Her dissertation, Quasiconformal Boundary Correspondences and Extremal Mappings, was supervised by Gehring. After completing her doctorate, she joined the Norwegian Institute of Technology (NTH), which later became part of NTNU. She became a full professor at NTNU in 2001, and retired as a professor emerita in 2011. == Awards and honors == NTNU gave Hag their gender equality award in 2000, for her efforts to increase the interest of girls in science and mathematics. In 2018 she was elected as a knight in the Order of St. Olav. == References ==
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Wikipedia:Kari Vilonen#0
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Kari Kaleva Vilonen (born 1955) is a Finnish mathematician, specializing in geometric representation theory. He is currently a professor at the University of Melbourne. == Education == He received in 1983 his Ph.D from Brown University under Robert MacPherson with thesis The Intersection Homology D-module on Hypersurfaces with Isolated Singularities. == Career == From 1983 to 1986 was a C. L. E. Moore instructor at the Massachusetts Institute of Technology, on leave in 1984–1985 at the Mathematical Sciences Research Institute in Berkeley, California. Afterward, Vilonen was a Benjamin Pierce Assistant Professor at Harvard University from 1986 to 1989. From 1989 to 2000 he was a faculty member at Brandeis University, rising to the rank of Professor in 1996. After that, he was a professor at Northwestern University, and then a professor at the University of Helsinki from 2010 to 2015. Starting in 2015, Vilonen has been a professor at the University of Melbourne in Australia. In 2002, with Dennis Gaitsgory and Edward Frenkel, he proved the geometrical Langlands conjecture for curves over finite fields. In 2004, Vilonen, Mark Goresky, Dennis Gaitsgory and Edward Frenkel were awarded a multimillion dollar grant from the Defense Advanced Research Projects Agency (DARPA) to work on a project aimed at establishing links between the Langlands program and dualities in quantum field theory. Later, Frenkel wrote, "We felt like we were in uncharted territory: no mathematicians we knew had ever received grants of this magnitude before." The funds were used to coordinate the work of dozens of mathematicians with the goal of making a concerted effort in a significant area of research. In 2007, with Ivan Mirković, he published "Geometric Langlands duality and representations of algebraic groups over commutative rings", which proved the geometric Satake equivalence, a geometric version of the Satake isomorphism. In 2013, Vilonen received a Humboldt Prize. In 2014, he was awarded a Simons Fellowship from the Simons Foundation. In 2020, the Australian Research Council awarded Vilonen an Australian Laureate Fellowship, their highest award to an individual. This five year grant will allow him to address deep longstanding questions about real groups, algebraic objects which describe the basic symmetries occurring in nature. === Awards and keynote addresses === Vilonen was a Guggenheim Fellow for the academic year 1997/98. In 1998 he was an Invited Speaker with talk Topological methods in representation theory at the International Congress of Mathematicians in Berlin. In 2004 he was elected a member of the Finnish Academy of Science and Letters. == Selected publications == MacPherson, Robert; Vilonen, Kari (1986). "Elementary construction of perverse sheaves". Inventiones Mathematicae. 84 (2). Springer Science and Business Media LLC: 403–435. Bibcode:1986InMat..84..403M. doi:10.1007/bf01388812. ISSN 0020-9910. S2CID 120183452. Mirković, I; Uzawa, T; Vilonen, K (1992). "Matsuki correspondence for sheaves". Inventiones Mathematicae. 109 (1): 231–245. Bibcode:1992InMat.109..231M. doi:10.1007/BF01232026. S2CID 120058836. Vilonen, K (1994). "Perverse sheaves and finite-dimensional algebras". Trans. Amer. Math. Soc. 341 (2): 665–676. doi:10.1090/S0002-9947-1994-1135104-3. Frenkel, E; Gaitsgory, D; Kazhdan, D; Vilonen, K (1998). "Geometric realization of Whittaker functions and the Langlands conjecture". J. Amer. Math. Soc. 11 (2): 451–484. arXiv:alg-geom/9703022. doi:10.1090/S0894-0347-98-00260-4. Schmid, Wilfried; Vilonen, Kari (1998). "Two geometric character formulas for reductive Lie groups". J. Amer. Math. Soc. 11 (4): 799–867. arXiv:math/9801081. Bibcode:1998math......1081S. doi:10.1090/S0894-0347-98-00275-6. Mirković, I; Vilonen, K (1999). "Perverse sheaves on affine Grassmannians and Langlands duality". arXiv:math/9911050. Vilonen, Kari (2000). "Geometric methods in representation theory by K. Vilonen". In J. Adams; D. Vogan (eds.). Representation theory of Lie groups. IAS/Park City Mathematics Series 8. American Mathematical Society. pp. 241–290. arXiv:math/0410032. Bibcode:2004math.....10032V. Schmid, Wilfried; Vilonen, Kari (2000). "Characteristic Cycles and Wave Front Cycles of Representations of Reductive Lie Groups". The Annals of Mathematics. 151 (3). JSTOR: 1071. arXiv:math/0005305. Bibcode:2000math......5305S. doi:10.2307/121129. ISSN 0003-486X. JSTOR 121129. S2CID 3002170. Frenkel, E.; Gaitsgory, D.; Vilonen, K. (31 December 2001). "On the geometric Langlands conjecture". Journal of the American Mathematical Society. 15 (2). American Mathematical Society (AMS): 367–417. doi:10.1090/s0894-0347-01-00388-5. ISSN 0894-0347. Frenkel, E; Gaitsgory, D; Vilonen, K (2002). "On the geometric Langlands conjecture". J. Amer. Math. Soc. 15 (2): 367–417. doi:10.1090/S0894-0347-01-00388-5. Mirković, Ivan; Vilonen, Kari (1 July 2007). "Geometric Langlands duality and representations of algebraic groups over commutative rings". Annals of Mathematics. 166 (1). Annals of Mathematics, Princeton U: 95–143. arXiv:math/0401222. doi:10.4007/annals.2007.166.95. ISSN 0003-486X. Schmid, Wilfried; Vilonen, Kari (2011). "Hodge theory and the unitary representations of reductive Lie groups". Frontiers in Mathematical Sciences. International Press. pp. 397–420. arXiv:1206.5547. Bibcode:2012arXiv1206.5547S. Kashiwara, Masaki; Vilonen, Kari (1 September 2014). "Microdifferential systems and the codimension-three conjecture". Annals of Mathematics. 180 (2): 573–620. arXiv:1209.5124. doi:10.4007/annals.2014.180.2.4. ISSN 0003-486X. S2CID 56382698. == References ==
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Wikipedia:Karin Baur#0
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Karin Baur is a Swiss mathematician who is working in the mathematical fields algebra, representation theory, cluster algebras, cluster categories, combinatorics, Lie algebras. Currently she is a professor at University of Leeds and she also a full professor at University of Graz. From 2007–2012 she has been an assistant professor (SNSF professor) at ETH Zurich. Moreover, she is one of the protagonists of the project Women of Mathematics throughout Europe. == Recognition == In 2018 Baur was awarded a Royal Society Wolfson Fellowship for her work on Surface categories and mutation. For her project Orbit Structures in Representation Spaces, she won an SNSF Professorship in 2007. == Publications == "List of arXiv preprints by Karin Baur". == References == == External links == Karin Baur's webpage at the University of Leeds Karin Baur's webpage at the University of Graz Karin Baur at the Mathematics Genealogy Project
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Wikipedia:Karin Erdmann#0
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Karin Erdmann (born 1948) is a German mathematician specializing in the areas of algebra known as representation theory (especially modular representation theory) and homological algebra (especially Hochschild cohomology). She is notable for her work in modular representation theory which has been cited over 1500 times according to the Mathematical Reviews. Her nephew Martin Erdmann is professor for experimental particle physics at the RWTH Aachen University. == Education == She attended the Justus-Liebig-Universität Gießen and wrote her Ph.D. thesis on "2-Hauptblöcke von Gruppen mit Dieder-Gruppen als 2-Sylow-Gruppen" (Principal 2-blocks of groups with dihedral Sylow 2-subgroups) in 1976 under the direction of Gerhard O. Michler. == Professional career == Erdmann was a Fellow of Somerville College, Oxford. Erdmann is a university lecturer emeritus at the Mathematical Institute at the University of Oxford where she has had 25 doctoral students and 45 descendants. She has published over 115 papers and her work has been cited over 2000 times. She has contributed to the understanding of the representation theory of the symmetric group. == Honors == Erdmann was the inaugural Emmy Noether Lecturer of the German Mathematical Society in 2008. == Selected bibliography == Erdmann, Karin; Wildon, Mark J. (2006), Introduction to Lie algebras, Springer Undergraduate Mathematics Series, London: Springer-Verlag London Ltd., doi:10.1007/1-84628-490-2, ISBN 978-1-84628-040-5, MR 2218355 Erdmann, Karin (1990), Blocks of Tame Representation Type and Related Algebras, Lecture Notes in Mathematics, vol. 1428, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0084003, ISBN 978-3-540-52709-1, MR 1064107 == References == == External links == Mathematical Reviews author profile Home page at Oxford
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Wikipedia:Karin Schnass#0
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Karin Schnass (born 1980) is an Austrian mathematician and computer scientist known for her research on sparse dictionary learning. She is a professor of mathematics at the University of Innsbruck. == Education and career == Schnass was born in Klosterneuburg. She earned a master's degree in mathematics at the University of Vienna in 2004, with a thesis surveying Gabor multipliers supervised by Hans Georg Feichtinger. She completed her Ph.D. in communication and information sciences at the École Polytechnique Fédérale de Lausanne in 2009. Her dissertation was Sparsity & Dictionaries – Algorithms & Design, and her doctoral advisor was Pierre Vandergheynst. After postdoctoral research at the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences in Linz (chosen over Stanford University to stay close to her family) and as an Erwin Schrödinger Research Fellow at the University of Sassari and University of Innsbruck, she joined the Innsbruck Department of Mathematics as an assistant professor in 2016. == Recognition == Schnass was a winner of the Start-Preis of the Austrian Science Fund in 2014. She was a keynote speaker at iTWIST 2016. == References == == External links == Home page Karin Schnass publications indexed by Google Scholar
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Wikipedia:Karl Egil Aubert#0
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Karl Egil Aubert (19 August 1924 – 21 October 1990) was a Norwegian mathematician. Karl Aubert was born in Christiania (now Oslo), Norway. He was the brother of sociologist Vilhelm Aubert. He studied at the University of Oslo and took his Doctor of Science degree at the University of Paris in 1957. He stayed at the Institute for Advanced Study in Princeton from 1958 to 1960. From 1962 to 1990 he was a professor at the University of Oslo. He was also a visiting professor at the University of Washington in Seattle and Tufts University. He chaired the Norwegian Mathematics Society from 1960 to 1967. == References ==
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Wikipedia:Karl F. Sundman#0
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Karl Frithiof Sundman (28 October 1873, in Kaskinen – 28 September 1949, in Helsinki) was a Finnish mathematician who used analytic methods to prove the existence of a convergent infinite series solution to the three-body problem in two papers published in 1907 and 1909. His results gained fame when they were reproduced in Acta Mathematica in 1912. He also published a paper on regularization methods in mechanics in 1912. == Awards, recognition == Sundman was awarded the Pontécoulant prize by the French Academy of Sciences in 1913 for his work on the 3-body problem. In 1908 Sundman was elected member of the Finnish Society of Sciences and Letters and in 1947 foreign member of the Royal Swedish Academy of Sciences. The crater Sundman on the Moon is named after him, as is the asteroid 1424 Sundmania. == See also == Qiudong Wang generalized Sundman's solution to the case of more than three bodies In the 1990s. == References == == Sources == Järnefelt, G.: "Karl Fridhiof Sundman." Soc. Sci. Fenn. Arsbok 30 (2) (1953), 1–13. Järnefelt, G.: "Karl F. Sundman in Memoriam." Acta Mathematica 83 (1950), i–vi. Barrow-Green, June (2010). "The dramatic episode of Sundman. Historia Mathematica, 37 (2): 164-203.
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Wikipedia:Karl Menger#0
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Karl Menger (German: [ˈmɛŋɐ]; January 13, 1902 – October 5, 1985) was an Austrian-born American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebras and the dimension theory of low-regularity ("rough") curves and regions; in graph theory, he is credited with Menger's theorem. Outside of mathematics, Menger has substantial contributions to game theory and social sciences. == Biography == Karl Menger was a student of Hans Hahn and received his PhD from the University of Vienna in 1924. L. E. J. Brouwer invited Menger in 1925 to teach at the University of Amsterdam. In 1927, he returned to Vienna to accept a professorship there. In 1930 and 1931 he was visiting lecturer at Harvard University and the Rice Institute. From 1937 to 1946 he was a professor at the University of Notre Dame. From 1946 to 1971, he was a professor at Illinois Institute of Technology (IIT) in Chicago. In 1983, IIT awarded Menger a Doctor of Humane Letters and Sciences degree. == Contributions to mathematics == His most famous popular contribution was the Menger sponge (mistakenly known as Sierpinski's sponge), a three-dimensional version of the Sierpiński carpet. It is also related to the Cantor set. With Arthur Cayley, Menger is considered one of the founders of distance geometry; especially by having formalized definitions of the notions of angle and of curvature in terms of directly measurable physical quantities, namely ratios of distance values. The characteristic mathematical expressions appearing in those definitions are Cayley–Menger determinants. He was an active participant of the Vienna Circle, which had discussions in the 1920s on social science and philosophy. During that time, he published an influential result on the St. Petersburg paradox with applications to the utility theory in economics; this result has since been criticised as fundamentally misleading. Later he contributed to the development of game theory with Oskar Morgenstern. Meneger's work on topology without points followed Arthur Eddington's approach to geometry without points. Menger was a founding member of the Econometric Society. == Legacy == Menger's longest and last academic post was at the Illinois Institute of Technology, which hosts an annual IIT Karl Menger Lecture and offers the IIT Karl Menger Student Award to an exceptional student for scholarship each year. Menger's memoirs inspired his granddaughter Kirsten Menger-Anderson to write the 2025 novel The Expert of Subtle Revisions, which featured a fictionalized Vienna Circle. == See also == Distance geometry Kuratowski's theorem Selection principle Travelling salesman problem == Notes == == Further reading == Crilly, Tony, 2005, "Paul Urysohn and Karl Menger: papers on dimension theory" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 844–55. Golland, Louise and Sigmund, Karl "Exact Thought in a Demented Time: Karl Menger and his Viennese Mathematical Colloquium" The Mathematical Intelligencer 2000, Vol 22,1, 34-45 == External links == O'Connor, John J.; Robertson, Edmund F., "Karl Menger", MacTutor History of Mathematics Archive, University of St Andrews Karl Menger at the Mathematics Genealogy Project Works by or about Karl Menger at the Internet Archive
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Wikipedia:Karl Sigmund#0
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Karl Sigmund (born July 26, 1945) is a Professor of Mathematics at the University of Vienna and one of the pioneers of evolutionary game theory. == Career == Sigmund was schooled in the Lycée Francais de Vienne. From 1963 to 1968 he studied at the Institute of Mathematics at the University of Vienna, and obtained his Ph.D. under the supervision of Leopold Schmetterer. He spent his postdoctorate years (1968 to 1973) at Manchester ('68-'69), the Institut des hautes études scientifiques in Bures-sur-Yvette near Paris ('69-'70), the Hebrew University in Jerusalem (1970-'71), the University of Vienna (1971-'72) and the Austrian Academy of Sciences (1972-'73). In 1972 he received habilitation. In 1973, Sigmund was appointed C3-professor at the University of Göttingen, and in 1974 became a full professor at the Institute of Mathematics in Vienna. His main scientific interest during these years was in ergodic theory and dynamical systems. From 1977 on, Sigmund became increasingly interested in different fields of biomathematics, and collaborated with Peter Schuster and Josef Hofbauer on mathematical ecology, chemical kinetics and population genetics, but especially on the new field of evolutionary game dynamics and replicator equations. Together with Martin Nowak, Christoph Hauert and Hannelore Brandt, he worked on game dynamical approaches to questions related with the evolution of cooperation in biological and human populations. Since 1984, Sigmund has also worked as a part-time scientist at the International Institute for Applied Systems Analysis (IIASA) in Laxenburg, Lower Austria. == Honours and recognition == Sigmund was head of the Institute of Mathematics at the University of Vienna from 1983 to 1985, managing editor of the scientific journal Monatshefte für Mathematik from 1991 to 2001, vice-president (1995 to 1997) and president (1997 to 2001) of the Austrian Mathematical Society, corresponding member (1996) and full member (1999) of the Austrian Academy of Sciences, and member of the Leopoldina (2003). He has also given many plenary lectures, for instance at the International Congress of Mathematicians in 1998. He was awarded the Gauss Lectureship in 2003. In 2010 he received an honorary doctorate (Doctor Philosophiae Honoris Causa) from the University of Helsinki. In 2012 he received the Isaacs Award. == Other details == During the last decade, Sigmund became increasingly interested in the history of mathematics and in particular, the Vienna Circle. He co-edited the mathematical works of Hans Hahn and Karl Menger and organised in 2001 an exhibition on the exodus of Austrian mathematicians fleeing the Nazis and in 2006 an exhibition on Kurt Gödel. From 2003 to 2005 he was vice-president of the Austrian Science Fund (FWF). Because of his intimate knowledge of the Vienna Circle, Sigmund was invited to the Illinois Institute of Technology to speak at the inaugural Remembering Menger event on April 9, 2007. == Publications == Sigmund's publications include 133 scientific papers, including 18 in Nature; 11 edited volumes; 25 essays; and 5 co-authored books. === Books === Ergodic Theory on Compact Spaces with Manfred Denker and Christian Grillenberger, Springer, 1976. Games of Life: Explorations in Ecology, Evolution, and Behaviour, Oxford University Press, 1993. ISBN 0-19-854665-3 Kurt Gödel: Das Album - The Album with John W. Dawson Jr. and Kurt Mühlberger, Vieweg+Teubner Verlag, 2006. ISBN 978-3-8348-0173-9 The Calculus of Selfishness, Princeton University Press, 2010. ISBN 9780691142753 2016 pbk reprint Games of Life: Explorations in Ecology, Evolution and Behavior, Dover Publications, 2012, updated in 2017. Exact Thinking in Demented Times: The Vienna Circle and the Epic Quest for the Foundations of Science, Basic Books, 2017. With a preface from Douglas Hofstadter who also helped with the translation. ISBN 9781541697829 The Waltz of Reason: The Entanglement of Mathematics and Philosophy, Basic Books, 2023. ISBN 1541602692 == References == == External links == Karl Sigmund's homepage Karl Sigmund at the Mathematics Genealogy Project
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Wikipedia:Karl W. Gruenberg#0
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Karl W. Gruenberg (3 June 1928 – 10 October 2007) was a British mathematician who specialised in group theory, in particular with the cohomology theory of groups. == Education and career == At the age of eleven, Gruenberg was one of the many Jewish children sent from Austria to Great Britain as part of the Kindertransport in 1939. Most of the Kindertransport children never saw their parents again but Karl was lucky and his mother soon joined him, and they moved to London in 1943 where he entered Kilburn Grammar School. In 1946 he won a scholarship to study mathematics at Magdalene College, Cambridge, where he received a BA degree in 1950 (duly upgraded to MA (Cantab.) in 1954. He was appointed as an Assistant Lecturer in Mathematics at Queen Mary College, London University from 1953 to 1955. He got his PhD in 1954 under Philip Hall at Cambridge with his treatise "A Contribution to the Theory of Commutators in Groups and Associative Rings". He was awarded a Commonwealth Fund Fellowship which made it possible for him to spend 1955–56 at Harvard and then 1956–57 at the Institute for Advanced Study in Princeton, New Jersey. In 1948 he became a British citizen. In 1967 he moved back to Queen Mary College where he became a leading figure in the algebra research community and where he remained for the rest of his career. He became a professor in the Department of Pure Mathematics where he worked with Bertram Huppert and Wolfgang Gaschütz organising the group theory conferences at the Mathematical Research Institute of Oberwolfach in Germany. He had a son Mark and a daughter Anne by his first wife Katherine. For thirty years he was married to his second wife Margaret. == Works == papers The Universal Coefficient Theorem in the Cohomology of Groups Journal of the London Mathematical Society 27 May 1966 Some cohomological notes in group theory, Queen Mary College Math. Notes, 1968 Relation modules of finite groups, CBMS Regional Conf. Series Math., American Mathematical Society 1976 books 1970 Cohomological topics in group theory ISBN 978-3-540-36303-3 1977 Linear geometry (with A. J. Weir), Springer-Verlag 1984 Group theory : essays for Philip Hall, J E Roseblade & Philip Hall, London : Academic Press, ISBN 012304880X 1988 The collected works of Philip Hall (with J. E. Roseblade), Clarendon Press, 1988, ISBN 0198532547 2002 Una introduzione all'algebra omologica == References == == External links == Karl W. Gruenberg at the Mathematics Genealogy Project On the occasion of the 65th birthday of Professor K.W. Gruenberg by Alan Camina, Ted C. Hurley, Peter H Kropholler, Publisher: Amsterdam, 1993.
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Wikipedia:Karl Weissenberg#0
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Karl Weissenberg (11 June 1893, Vienna – 6 April 1976, The Hague) was an Austrian physicist, notable for his contributions to rheology and crystallography. == Biography == The Weissenberg effect was named after him, as was the Weissenberg number. He invented a Goniometer to study X-ray diffraction of crystals for which he received the Duddell Medal of the Institute of Physics in 1946, The European Society of Rheology offers a Weissenberg award in his honour. and the Weissenberg rheogoniometer, a type of rheometer. He was born on 11 June 1893 in Vienna, Austria and died in 1976 in the Netherlands. He studied at the Universities of Vienna, Berlin and Jena with Mathematics as his main subject. He published on the theories of Symmetry groups and Tensor and Matrix algebra, then applied mathematics and experimentation to crystallography, rheology and medical science. == References == == Further reading == Publications of Karl Weissenberg (K.W.) and Collaborators Churchill Archives Centre, The Papers of Karl Weissenberg (with brief biography)
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Wikipedia:Karma Dajani#0
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Karma Dajani is a Lebanese-Dutch mathematician whose research interests include ergodic theory, probability theory, and their applications in number theory. She is an associate professor of mathematics at Utrecht University. == Education and career == Dajani was born in Lebanon, and did her undergraduate studies at the American University of Beirut, initially in medicine but switching after a year to mathematics. Because of the Lebanese Civil War, she and her family moved to the US, where she earned her Ph.D. in 1989 from George Washington University. Again, she switched topics, beginning in functional analysis and trying graph theory but ending in ergodic theory. Her dissertation, Simultaneous Recurrence of Weighted Cocycles, was supervised by E. Arthur Robinson Jr., after a previous advisor, Daniel Ullman, shifted his own interests away from ergodic theory. As a student at George Washington University, Dajani was a two-time winner of the university's Taylor Prize in Mathematics. After completing her doctorate, she was a postdoctoral researcher at the University of Maryland, College Park and the University of North Carolina at Chapel Hill. She took a faculty position at the University of Alabama. After marrying a Dutch mathematician, Cor Kraaikamp, she obtained a visiting position at Delft University of Technology and then joined Utrecht University. She spent 25 years as the only female mathematics professor at Utrecht. == Book == With her husband Cor Kraaikamp, Dajani is the author of the book Ergodic Theory of Numbers, published in 2002 by the Mathematical Association of America as volume 29 of their Carus Mathematical Monographs. The book grew out of a course given by Dajani in a 1996 summer program for women in mathematics. == References == == External links == Home page Karma Dajani publications indexed by Google Scholar
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Wikipedia:Karsten Grove#0
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Karsten Grove is a Danish-American mathematician working in metric and differential geometry, differential topology and global analysis, mainly in topics related to global Riemannian geometry, Alexandrov geometry, isometric group actions and manifolds with positive or nonnegative sectional curvature. == Biography == Grove studied mathematics at Aarhus University, where he obtained a Cand. Scient. (equivalent to a M.A.) in 1971 and Lic. Scient. (equivalent to a Ph.D.) in 1974. Between 1971 and 1972 he also acted as an instructor at Aarhus University. From 1972 to 1974 he had a postdoctoral position at the University of Bonn under the supervision of Wilhelm Klingenberg, despite not having yet formally concluded his doctoral degree. In 1974, Grove became an assistant professor at the University of Copenhagen and was promoted to associate professor in 1976, a position he held until 1987. He became a professor at the University of Maryland in 1984, retiring from this position in 2009. Since 2007 he has held the endowed chair of "Rev. Howard J. Kenna, C.S.C. Professor" at the University of Notre Dame. Throughout his career, Grove has had 20 doctoral students, and 51 academic descendants. Grove was an invited speaker at the International Congress of Mathematicians in 1990 in Kyoto (Metric and Topological Measurements on manifolds). He is a fellow of the American Mathematical Society. == Mathematical work == One of Grove's most recognized mathematical contributions to Riemannian Geometry is the Diameter Sphere Theorem, proved jointly with Katsuhiro Shiohama in 1977, which states that a smooth closed Riemannian manifold ( M , g ) {\displaystyle (M,g)} with sec ≥ 1 {\displaystyle \sec \geq 1} and diam > π / 2 {\displaystyle \operatorname {diam} >\pi /2} is homeomorphic to a sphere. Subsequently, the critical point theory for distance functions developed as part of the proof of this result led to several important advances in the area. Another result obtained by Grove, in collaboration with Peter Petersen, is the finiteness of homotopy types of manifolds of a fixed dimension with lower sectional curvature bounds, upper diameter bound, and lower volume bound. == References ==
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Wikipedia:Kasia Rejzner#0
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Katarzyna (Kasia) Anna Rejzner (born 1985) is a Polish mathematical physicist specializing in algebraic quantum field theory and the theory of renormalization, including the Batalin–Vilkovisky formalism. She works as a professor in mathematics at the University of York. == Education and career == Rejzner was born in 1985 in Kraków, the daughter of two architects. She earned a master's degree in physics in 2009 from Jagiellonian University, and completed her Ph.D. in 2011 at the University of Hamburg under the supervision of Klaus Fredenhagen, with a dissertation on the Batalin–Vilkovisky formalism. After postdoctoral studies at the University of Rome Tor Vergata she joined the University of York in 2013, and was promoted to senior lecturer there in 2017. In 2016 and 2017, she visited the Perimeter Institute as an Emmy Noether Visiting Fellow. She was promoted to the position of full professor in Mathematics, University of York in 2024. During the term 2024-2026, Rejzner serves as the President of the International Association of Mathematical Physics (IAMP), being the first female in this position since the foundation of the IAMP. == Book == Rejzner is the author of the book Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians (Mathematical Physics Studies, Springer, 2016). == References == == External links == Home page Kasia Rejzner publications indexed by Google Scholar Media related to Katarzyna Rejzner at Wikimedia Commons
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Wikipedia:Katapayadi system#0
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Kaṭapayādi system (Devanagari: कटपयादि, also known as Paralppēru, Malayalam: പരല്പ്പേര്) of numerical notation is an ancient Indian alphasyllabic numeral system to depict letters to numerals for easy remembrance of numbers as words or verses. Assigning more than one letter to one numeral and nullifying certain other letters as valueless, this system provides the flexibility in forming meaningful words out of numbers which can be easily remembered. == History == The oldest available evidence of the use of Kaṭapayādi (Sanskrit: कटपयादि) system is from Grahacāraṇibandhana by Haridatta in 683 CE. It has been used in Laghu·bhāskarīya·vivaraṇa written by Śaṅkara·nārāyaṇa in 869 CE. In some astronomical texts popular in Kerala planetary positions were encoded in the Kaṭapayādi system. The first such work is considered to be the Chandra-vakyani of Vararuci, who is traditionally assigned to the fourth century CE. Therefore, sometime in the early first millennium is a reasonable estimate for the origin of the Kaṭapayādi system. Aryabhata, in his treatise Ārya·bhaṭīya, is known to have used a similar, more complex system to represent astronomical numbers. There is no definitive evidence whether the Ka-ṭa-pa-yā-di system originated from Āryabhaṭa numeration. == Geographical spread of the use == Almost all evidences of the use of Ka-ṭa-pa-yā-di system is from South India, especially Kerala. Not much is known about its use in North India. However, on a Sanskrit astrolabe discovered in North India, the degrees of the altitude are marked in the Kaṭapayādi system. It is preserved in the Sarasvati Bhavan Library of Sampurnanand Sanskrit University, Varanasi. The Ka-ṭa-pa-yā-di system is not confined to India. Some Pali chronograms based on the Ka-ṭa-pa-yā-di system have been discovered in Burma. == Rules and practices == Following verse found in Śaṅkaravarman's Sadratnamāla explains the mechanism of the system. नञावचश्च शून्यानि संख्या: कटपयादय:। मिश्रे तूपान्त्यहल् संख्या न च चिन्त्यो हलस्वर:॥ Transliteration: nanyāvachaścha śūnyāni sankhyāḥ kaṭapayādayaḥ miśre tūpāntyahal sankhyā na cha chintyo halasvaraḥ Translation: na (न), ña (ञ) and a (अ)-s, i.e., vowels represent zero. The nine integers are represented by consonant group beginning with ka, ṭa, pa, ya. In a conjunct consonant, the last of the consonants alone will count. A consonant without a vowel is to be ignored. Explanation: The assignment of letters to the numerals are as per the following arrangement (In Devanagari, Kannada, Telugu & Malayalam scripts respectively) Consonants have numerals assigned as per the above table. For example, ba (ब) is always 3 whereas 5 can be represented by either nga (ङ) or ṇa (ण) or ma (म) or śha (श). All stand-alone vowels like a (अ) and ṛ (ऋ) are assigned to zero. In case of a conjunct, consonants attached to a non-vowel will be valueless. For example, kya (क्य) is formed by, k (क्) + y (य्) + a (अ). The only consonant standing with a vowel is ya (य). So the corresponding numeral for kya (क्य) will be 1. There is no way of representing the decimal separator in the system. Indians used the Hindu–Arabic numeral system for numbering, traditionally written in increasing place values from left to right. This is as per the rule "अङ्कानां वामतो गतिः" which means numbers go from right to left. === Variations === The consonant, ḷ (Malayālam: ള, Devanāgarī: ळ, Kannada: ಳ) is employed in works using the Kaṭapayādi system, like Mādhava's sine table. Late medieval practitioners do not map the stand-alone vowels to zero. But, it is sometimes considered valueless. == Usage == === Mathematics and astronomy === Mādhava's sine table constructed by 14th century Kerala mathematician-astronomer Mādhava of Saṅgama·grāma employs the Kaṭapayādi system to list the trigonometric sines of angles. Karaṇa·paddhati, written in the 15th century, has the following śloka for the value of pi (π) അനൂനനൂന്നാനനനുന്നനിത്യൈ- സ്സമാഹതാശ്ചക്രകലാവിഭക്താഃ ചണ്ഡാംശുചന്ദ്രാധമകുംഭിപാലൈര്- വ്യാസസ്തദര്ദ്ധം ത്രിഭമൗര്വിക സ്യാത് Transliteration anūnanūnnānananunnanityai ssmāhatāścakra kalāvibhaktoḥ caṇḍāṃśucandrādhamakuṃbhipālair vyāsastadarddhaṃ tribhamaurvika syāt It gives the circumference of a circle of diameter, anūnanūnnānananunnanityai (10,000,000,000) as caṇḍāṃśucandrādhamakuṃbhipālair (31415926536). Śaṅkara·varman's Sad·ratna·mālā uses the Kaṭapayādi system. The first verse of Chapter 4 of the Sad·ratna·mālā ends with the line: (स्याद्) भद्राम्बुधिसिद्धजन्मगणितश्रद्धा स्म यद् भूपगी: Transliteration (syād) bhadrāmbudhisiddhajanmagaṇitaśraddhā sma yad bhūpagīḥ Splitting the consonants in the relevant phrase gives, Reversing the digits to modern-day usage of descending order of decimal places, we get 314159265358979324 which is the value of pi (π) to 17 decimal places, except the last digit might be rounded off to 4. This verse encrypts the value of pi (π) up to 31 decimal places. गोपीभाग्यमधुव्रात-शृङ्गिशोदधिसन्धिग॥ खलजीवितखाताव गलहालारसंधर॥ ಗೋಪೀಭಾಗ್ಯಮಧುವ್ರಾತ-ಶೃಂಗಿಶೋದಧಿಸಂಧಿಗ || ಖಲಜೀವಿತಖಾತಾವ ಗಲಹಾಲಾರಸಂಧರ || This verse directly yields the decimal equivalent of pi divided by 10: pi/10 = 0.31415926535897932384626433832792 గోపీభాగ్యమధువ్రాత-శృంగిశోదధిసంధిగ | ఖలజీవితఖాతావ గలహాలారసంధర || Traditionally, the order of digits are reversed to form the number, in katapayadi system. This rule is violated in this sloka. === Carnatic music === The melakarta ragas of the Carnatic music are named so that the first two syllables of the name will give its number. This system is sometimes called the Ka-ta-pa-ya-di sankhya. The Swaras 'Sa' and 'Pa' are fixed, and here is how to get the other swaras from the melakarta number. Melakartas 1 through 36 have Ma1 and those from 37 through 72 have Ma2. The other notes are derived by noting the (integral part of the) quotient and remainder when one less than the melakarta number is divided by 6. If the melakarta number is greater than 36, subtract 36 from the melakarta number before performing this step. 'Ri' and 'Ga' positions: the raga will have: Ri1 and Ga1 if the quotient is 0 Ri1 and Ga2 if the quotient is 1 Ri1 and Ga3 if the quotient is 2 Ri2 and Ga2 if the quotient is 3 Ri2 and Ga3 if the quotient is 4 Ri3 and Ga3 if the quotient is 5 'Da' and 'Ni' positions: the raga will have: Da1 and Ni1 if remainder is 0 Da1 and Ni2 if remainder is 1 Da1 and Ni3 if remainder is 2 Da2 and Ni2 if remainder is 3 Da2 and Ni3 if remainder is 4 Da3 and Ni3 if remainder is 5 See swaras in Carnatic music for details on above notation. ==== Raga Dheerasankarabharanam ==== The katapayadi scheme associates dha ↔ {\displaystyle \leftrightarrow } 9 and ra ↔ {\displaystyle \leftrightarrow } 2, hence the raga's melakarta number is 29 (92 reversed). 29 less than 36, hence Dheerasankarabharanam has Ma1. Divide 28 (1 less than 29) by 6, the quotient is 4 and the remainder 4. Therefore, this raga has Ri2, Ga3 (quotient is 4) and Da2, Ni3 (remainder is 4). Therefore, this raga's scale is Sa Ri2 Ga3 Ma1 Pa Da2 Ni3 SA. ==== Raga MechaKalyani ==== From the coding scheme Ma ↔ {\displaystyle \leftrightarrow } 5, Cha ↔ {\displaystyle \leftrightarrow } 6. Hence the raga's melakarta number is 65 (56 reversed). 65 is greater than 36. So MechaKalyani has Ma2. Since the raga's number is greater than 36 subtract 36 from it. 65–36=29. 28 (1 less than 29) divided by 6: quotient=4, remainder=4. Ri2 Ga3 occurs. Da2 Ni3 occurs. So MechaKalyani has the notes Sa Ri2 Ga3 Ma2 Pa Da2 Ni3 SA. ==== Exception for Simhendramadhyamam ==== As per the above calculation, we should get Sa ↔ {\displaystyle \leftrightarrow } 7, Ha ↔ {\displaystyle \leftrightarrow } 8 giving the number 87 instead of 57 for Simhendramadhyamam. This should be ideally Sa ↔ {\displaystyle \leftrightarrow } 7, Ma ↔ {\displaystyle \leftrightarrow } 5 giving the number 57. So it is believed that the name should be written as Sihmendramadhyamam (as in the case of Brahmana in Sanskrit). === Representation of dates === Important dates were remembered by converting them using Kaṭapayādi system. These dates are generally represented as number of days since the start of Kali Yuga. It is sometimes called kalidina sankhya. The Malayalam calendar known as kollavarsham (Malayalam: കൊല്ലവര്ഷം) was adopted in Kerala beginning from 825 CE, revamping some calendars. This date is remembered as āchārya vāgbhadā, converted using Kaṭapayādi into 1434160 days since the start of Kali Yuga. Narayaniyam, written by Melpathur Narayana Bhattathiri, ends with the line, āyurārogyasaukhyam (ആയുരാരോഗ്യസൌഖ്യം) which means long-life, health and happiness. This number is the time at which the work was completed represented as number of days since the start of Kali Yuga as per the Malayalam calendar. === Others === Some people use the Kaṭapayādi system in naming newborns. The following verse compiled in Malayalam by Koduṅṅallur Kuññikkuṭṭan Taṃpurān using Kaṭapayādi is the number of days in the months of Gregorian Calendar. പലഹാരേ പാലു നല്ലൂ, പുലര്ന്നാലോ കലക്കിലാം ഇല്ലാ പാലെന്നു ഗോപാലന് – ആംഗ്ലമാസദിനം ക്രമാല് Transliteration palahāre pālu nallū, pularnnālo kalakkilāṃ illā pālennu gopālan – āṃgḷamāsadinaṃ kramāl Translation: Milk is best for breakfast, when it is morning, it should be stirred. But Gopālan says there is no milk – the number of days of English months in order. Converting pairs of letters using Kaṭapayādi yields – pala (പല) is 31, hāre (ഹാരേ) is 28, pālu പാലു = 31, nallū (നല്ലൂ) is 30, pular (പുലര്) is 31, nnālo (ന്നാലോ) is 30, kala (കല) is 31, kkilāṃ (ക്കിലാം) is 31, illā (ഇല്ലാ) is 30, pāle (പാലെ) is 31, nnu go (ന്നു ഗോ) is 30, pālan (പാലന്) is 31. == See also == == References == == External links == Kaṭapayādi Saṅkhyā, a Kaṭapayādi encoding-decoding system. == Further reading == A.A. Hattangadi, Explorations in Mathematics, Universities Press (India) Pvt. Ltd., Hyderabad (2001) ISBN 81-7371-387-1 [3]
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Wikipedia:Katherine Heinrich#0
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Katherine A. Heinrich (born 21 February 1954) is a mathematician and mathematics teacher who was the first female president of the Canadian Mathematical Society. Her research interests include graph theory and the theory of combinatorial designs. Originally from Australia, she moved to Canada where she worked as a professor at Simon Fraser University and as an academic administrator at the University of Regina. == Education and career == Heinrich was born in Murwillumbah, New South Wales. As an undergraduate at the University of Newcastle in Australia, she graduated as a University Medalist in 1976. She continued at Newcastle as a graduate student and completed her doctorate there in 1979. Her dissertation, "Some problems on combinatorial arrays", was supervised by Walter D. Wallis. Heinrich joined the mathematics faculty at Simon Fraser University in 1981, and married another graph theorist there, Brian Alspach. She became a full professor in 1987 and chaired the department from 1991 to 1996. While working at Simon Fraser, she co-ordinated several outreach activities including a conference for pre-teen girls called "Women Do Math" and later "Discover the Possibilities", a shopping-center exhibit called "Math in the Malls", and a series of national conferences on mathematics education. From 1996 to 1998, she was the president of the Canadian Mathematical Society, its first female president. In 1999, she moved to the University of Regina as academic vice president and, in 2003, she was confirmed for a second five-year term as vice president. At Regina, she helped to establish an institute for French-language education and built stronger connections between Regina and the First Nations University of Canada. She retired in 2007 and returned to Newcastle, New South Wales, where she is active in textile arts. == Research == MathSciNet lists 73 publications for Heinrich, dated from 1976 to 2012. Several of her research publications concern orthogonal Latin squares,[A] analogous concepts in graph theory[D] and applications of these concepts in parallel computing.[E] She has also published works on finding spanning subgraphs with constraints on the degree of each vertex[C] and on Alspach's conjecture on disjoint cycle covers of complete graphs,[D] among other topics. == Selected publications == == Recognition == The University of Newcastle gave Heinrich a Gold Medal for Professional Excellence in 1995. In 2005, she won the Adrien Pouliot Award of the Canadian Mathematical Society for her work in mathematics education. == References ==
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Wikipedia:Kathleen Booth#0
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Kathleen Hylda Valerie Booth (née Britten, 9 July 1922 – 29 September 2022) was a British computer scientist and mathematician who wrote the first assembly language and designed the assembler and autocode for the first computer systems at Birkbeck College, University of London. She helped design three different machines including the ARC (Automatic Relay Calculator), SEC (Simple Electronic Computer), and APE(X)C. == Early life and education == Kathleen Britten was born in Stourbridge, Worcestershire, England, on 9 July 1922. She obtained a BSc in mathematics from Royal Holloway, University of London, in 1944 and went on to get a PhD in Applied Mathematics in 1950 from King's College London. She married her colleague Andrew Donald Booth in 1950 and had two children. == Career == Kathleen Booth worked at Birkbeck College, 1946–62. She travelled to the United States as Andrew Booth's research assistant in 1947, visiting with John von Neumann at Princeton University. While at Princeton, she co-authored "General Considerations in the Design of an All Purpose Electronic Digital Computer", describing modifications to the original ARC redesign to the ARC2 using a von Neumann architecture. Part of her contribution was the ARC assembly language. She also built and maintained ARC components. Kathleen and Andrew Booth's team at Birkbeck were considered the smallest of the early British computer groups. From 1947 to 1953, they produced three machines: ARC (Automatic Relay Computer) built with fellow research assistant Xenia Sweeting, an entirely electronic version of the ARC2 called SEC (Simple Electronic Computer), and APE(X)C (All-purpose Electronic (Rayon) Computer). She and Mr. Booth worked on the same team. This was considered a remarkable achievement due to the size of the group and the limited funds at its disposal. Although APE(X)C eventually led to the HEC series manufactured by the British Tabulating Machine Company, the small scale of the Birkbeck group did not place it in the front rank of British computer activity. Booth regularly published papers concerning her work on the ARC and APE(X)C systems and co-wrote "Automatic Digital Calculators" (1953) which illustrated the 'Planning and Coding' programming style. In 1957, Kathleen Booth, her husband Andrew, and J.C. Jennings co-founded Birkbeck College's Department of Numerical Automation, now the School of Computing and Mathematical Sciences. In 1958, she taught a programming course. In 1958, Booth wrote one of the first books describing how to program APE(X)C computers. From 1944 she was a Junior Scientific Officer at the Royal Aircraft Establishment in Farnborough. From 1946 to 1962, Booth was a Research Scientist at British Rubber Producers' Research Association and for ten years from 1952 to 1962 she was Research Fellow and Lecturer at Birkbeck College, University of London. Booth's research on neural networks led to successful programs simulating ways in which animals recognize patterns and characters. She and her husband resigned suddenly from Birkbeck College in 1962 after a chair was not conferred on her husband despite his massive contributions; an ICT 1400 computer was donated to the Department of Numerical Automation but was in fact installed in the London School of Hygiene and Tropical Medicine. On November 11th, 1955, Booth and the research group publicly demoed a machine translation prototype that translated the phrase "This is an example of a translation made by the machine for calculation installed at the laboratory of computation of Birkbeck College, London." from French to English. Booth continued her research into automated translation, becoming the director of a Canadian national project on machine translation in 1965. In 1962, after leaving Birkbeck College the Booth family moved to Canada to where she became a Research Fellow, Lecturer and Associate Professor at the University of Saskatchewan until 1972. At Lakehead University in Canada she became the Professor of Mathematics from 1972 to 1978. Kathleen Booth retired from Lakehead in 1978. Her last current paper was published in 1993 at the age of 71. Titled "Using neural nets to identify marine mammals", it was co-authored by Dr. Ian J. M. Booth, her son. == Personal life and death == She died on 29 September 2022, at the age of 100. == Bibliography == Booth, Andrew D; Britten, Kathleen HV (September 1947), "Principles and Progress in the Construction of High-Speed Digital Computers", Quart. Journ. Mech. And Applied Math., 2 (2): 182–197, doi:10.1093/qjmam/2.2.182. Coding system for the APE(X)C, AU: Murdoch, archived from the original on 7 June 2011, retrieved 22 June 2010. Booth A.D. and Britten K.H.V. (1947) Coding for A.R.C., Institute for Advanced Study, Princeton Booth A.D. and Britten K.H.V. (August 1947, 2nd Edition) General considerations in the design of an all-purpose electronic digital computer, Institute for Advance Study, Princeton Booth A.D. and Britten K.H.V. (1948) "The accuracy of atomic co-ordinates derived from Fourier series in X-ray crystallography Part V", Proc. Roy. Soc. Vol A 193 pp 305–310 Booth A.D. and Booth K.H.V. (1953) Automatic Digital Calculators, Butterworth-Heinmann (Academic Press) London K.H.V Booth, (1958) Programming for an Automatic Digital Calculator, Butterworths, London == References == == External links == The APEXC driver page Principles and Progress in the Construction of High-Speed Digital Computers Andrew Booth Collection, University of Manchester Library. [1], Obituary in The Register.
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Wikipedia:Kathryn E. Hare#0
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Kathryn Elizabeth Hare (born 1959) is a Canadian mathematician specializing in harmonic analysis and fractal geometry. She was the Chair of the Pure Mathematics Department at the University of Waterloo from 2014 to 2018. She retired from the University of Waterloo in 2021. == Education and career == Hare did her undergraduate studies at the University of Waterloo, graduating in 1981. She earned a Ph.D. from the University of British Columbia in 1986. Her dissertation, under the supervision of John J. F. Fournier, was Thin Sets and Strict-Two-Associatedness, and concerned group representation theory. She was an assistant professor at the University of Alberta from 1986 to 1988, before she moved back to Waterloo. == Awards and recognition == In 2011, the Chalmers University of Technology awarded her an Honorary Doctorate for her "prominent research, both in extent and depth, within classical and abstract harmonic analysis". In 2020 she was named as a Fellow of the Canadian Mathematical Society. == Selected publications == Hare, Kathryn E.; Klemes, Ivo (1995), "On permutations of lacunary intervals", Transactions of the American Mathematical Society, 347 (10): 4105–4127, doi:10.2307/2155216, JSTOR 2155216. Graham, Colin C.; Hare, Kathryn E. (2013), Interpolation and Sidon Sets for Compact Groups, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, doi:10.1007/978-1-4614-5392-5, ISBN 978-1-4614-5391-8. Hare, Kathryn E.; He, Jimmy. (2017), "The absolute continuity of convolution products of orbital measures in exceptional symmetric spaces", Monatshefte für Mathematik, 182 (3): 619–635, arXiv:1511.05799, doi:10.1007/s00605-016-0999-5. == References ==
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Wikipedia:Kathy Driver#0
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Kathleen Ann Driver (née Owen) is a retired South African mathematician who lists her research interests as "special functions, orthogonal polynomials and approximation theory". She is an emeritus professor at the University of Cape Town. == Education and career == Driver earned a bachelor's degree in mathematics at the University of the Witwatersrand, in 1966, where she was awarded the William Cullen Medal for the year's best student in the faculty of science, and the South African Association of Women Graduates Award for the best female student at the university. Next, she traveled to Stanford University for a master's degree in 1971. Returning to graduate study in later life, she completed a PhD at the University of the Witwatersrand in 1991, under the supervision of Doron S. Lubinsky. She became a professor of mathematics and head of mathematics at the University of the Witwatersrand from 1999 through 2005, during which time the Department of Mathematics transitioned into a School of Mathematics. She moved to the University of Cape Town as professor of mathematics and dean of the Faculty of Science in 2006. She stepped down as dean in 2011, and retired as an emeritus professor in 2013. == Recognition == Driver is a member of the Academy of Science of South Africa, elected in 2007. == References == == External links == Kathy Driver publications indexed by Google Scholar
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Wikipedia:Kathy Horadam#0
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Kathryn Jennifer Horadam (born 1951) is an Australian mathematician known for her work on Hadamard matrices and related topics in mathematics and information security. She is an emeritus professor at the Royal Melbourne Institute of Technology (RMIT). == Life == Horadam is one of the three children of mathematicians Alwyn Horadam and Eleanor Mollie Horadam, and was born in 1951 in Armidale, New South Wales. She studied mathematics at Australian National University, earning a bachelor's degree in 1972 and completing her PhD in 1977. Her dissertation, The Homology of Groupnets, was supervised by Neville Smythe. She worked for over 30 years at RMIT, becoming a professor of mathematics there in 1995. Additionally, she worked for three years in the Defence Science and Technology Group. == Book == Horadam is the author of the book Hadamard Matrices and Their Applications (Princeton University Press, 2007). == Recognition == Horadam became a fellow of the Institute of Combinatorics and its Applications in 1991 and of the Australian Mathematical Society in 2001. An international workshop on Hadamard matrices was held at RMIT in 2011 in honour of her 60th birthday, and papers from the workshop were published in 2013 as a special issue of the Australasian Journal of Combinatorics. == References == == External links == Kathy Horadam publications indexed by Google Scholar
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Wikipedia:Kato's conjecture#0
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Kato's conjecture is a mathematical problem named after mathematician Tosio Kato, of the University of California, Berkeley. Kato initially posed the problem in 1953. Kato asked whether the square roots of certain elliptic operators, defined via functional calculus, are analytic. The full statement of the conjecture as given by Auscher et al. is: "the domain of the square root of a uniformly complex elliptic operator L = − d i v ( A ∇ ) {\displaystyle L=-\mathrm {div} (A\nabla )} with bounded measurable coefficients in Rn is the Sobolev space H1(Rn) in any dimension with the estimate | | L f | | 2 ∼ | | ∇ f | | 2 {\displaystyle ||{\sqrt {L}}f||_{2}\sim ||\nabla f||_{2}} ". The problem remained unresolved for nearly a half-century, until in 2001 it was jointly solved in the affirmative by Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Philippe Tchamitchian. == References ==
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Wikipedia:Kato's inequality#0
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In functional analysis, a subfield of mathematics, Kato's inequality is a distributional inequality for the Laplace operator or certain elliptic operators. It was proven in 1972 by the Japanese mathematician Tosio Kato. The original inequality is for some degenerate elliptic operators. This article treats the special (but important) case for the Laplace operator. == Inequality for the Laplace operator == Let Ω ⊂ R d {\displaystyle \Omega \subset \mathbb {R} ^{d}} be a bounded and open set, and f ∈ L loc 1 ( Ω ) {\displaystyle f\in L_{\operatorname {loc} }^{1}(\Omega )} such that Δ f ∈ L loc 1 ( Ω ) {\displaystyle \Delta f\in L_{\operatorname {loc} }^{1}(\Omega )} . Then the following holds Δ | f | ≥ Re ( ( sgn f ¯ ) Δ f ) {\displaystyle \Delta |f|\geq \operatorname {Re} \left((\operatorname {sgn} {\overline {f}})\Delta f\right)\quad } in D ′ ( Ω ) {\displaystyle \;{\mathcal {D}}'(\Omega )} , where sgn f ¯ = { f ( x ) ¯ | f ( x ) | if f ≠ 0 0 if f = 0. {\displaystyle \operatorname {sgn} {\overline {f}}={\begin{cases}{\frac {\overline {f(x)}}{|f(x)|}}&{\text{if }}f\neq 0\\0&{\text{if }}f=0.\end{cases}}} L loc 1 {\displaystyle L_{\operatorname {loc} }^{1}} is the space of locally integrable functions – i.e., functions that are integrable on every compact subset of their domains of definition. === Remarks === Sometimes the inequality is stated in the form Δ f + ≥ Re ( 1 [ f ≥ 0 ] Δ f ) {\displaystyle \Delta f^{+}\geq \operatorname {Re} \left(1_{[f\geq 0]}\Delta f\right)\quad } in D ′ ( Ω ) {\displaystyle \;{\mathcal {D}}'(\Omega )} where f + = max ( f , 0 ) {\displaystyle f^{+}=\operatorname {max} (f,0)} and 1 [ f ≥ 0 ] {\displaystyle 1_{[f\geq 0]}} is the indicator function. If f {\displaystyle f} is continuous in Ω {\displaystyle \Omega } then Δ | f | ≥ Re ( ( sgn f ¯ ) Δ f ) {\displaystyle \Delta |f|\geq \operatorname {Re} \left((\operatorname {sgn} {\overline {f}})\Delta f\right)\quad } in D ′ ( { f ≠ 0 } ) {\displaystyle \;{\mathcal {D}}'(\{f\neq 0\})} . == Literature == Brezis, Haı̈m; Ponce, Augusto (2004). "Kato's inequality when Δu is a measure". Comptes Rendus Mathematique. 338 (8): 599–604. doi:10.1016/j.crma.2003.12.032. Arendt, Wolfgang; ter Elst, Antonious F.M. (2019). "Kato's Inequality". Analysis and Operator Theory. Springer Optimization and Its Applications. Springer Optimization and Its Applications. Vol. 146. Cham: Springer. pp. 47–60. doi:10.1007/978-3-030-12661-2_3. ISBN 978-3-030-12660-5. S2CID 191796248. == References ==
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Wikipedia:Katya Scheinberg#0
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Katya Scheinberg is a Russian-American applied mathematician known for her research in continuous optimization and particularly in derivative-free optimization. She is a professor in the School of Industrial and Systems Engineering at Georgia Institute of Technology. == Education and career == Scheinberg was born in Moscow. She completed a bachelor's and master's degree in computational mathematics and cybernetics at Moscow State University in 1992, and earned a Ph.D. in operations research at Columbia University in 1997. Her dissertation, Issues Related to Interior Point Methods for Linear and Semidefinite Programming, was supervised by Donald Goldfarb. Scheinberg worked for IBM Research at the Thomas J. Watson Research Center from 1997 until 2009. After working as a research scientist at Columbia University and as an adjunct faculty member at New York University, she joined the Lehigh faculty in 2010. Scheinberg became Wagner Professor at Lehigh in 2014. In 2019 she moved to Cornell University where she was a professor in the School of Operations Research and Information Engineering. In July 2024 she moved to Georgia Tech. Scheinberg has been editor-in-chief of the SIAM-MOS Book Series on Optimization since 2014, and was the editor of Optima, the newsletter of the Mathematical Programming Society, from 2011 to 2013. == Research == Scheinberg works on the intersection of optimization and machine learning, in particular on kernel support vector machines. With Andrew R. Conn and Luís Nunes Vicente, Scheinberg authored the book Introduction to Derivative Free Optimization (SIAM Press, 2008). == Recognition == In 2015, with Conn and Vicente, she won the Lagrange Prize in Continuous Optimization of the Mathematical Optimization Society and Society for Industrial and Applied Mathematics for their book. The Prize citation wrote that "A small sampling of the direct impact of their work is seen in aerospace engineering, urban transport systems, adaptive meshing for partial differential equations, and groundwater remediation." In 2019, Professor Scheinberg was awarded the Farkas Prize by the Optimization Society in the Institute for Operations Research and the Management Sciences (INFORMS). In 2022 she was named a Fellow of INFORMS, "for outstanding research contributions to continuous optimization, particularly derivative-free optimization and the interface of optimization and machine learning, as well as outstanding service and leadership". == References == == External links == Katya Scheinberg publications indexed by Google Scholar
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Wikipedia:Katz centrality#0
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In graph theory, the Katz centrality or alpha centrality of a node is a measure of centrality in a network. It was introduced by Leo Katz in 1953 and is used to measure the relative degree of influence of an actor (or node) within a social network. Unlike typical centrality measures which consider only the shortest path (the geodesic) between a pair of actors, Katz centrality measures influence by taking into account the total number of walks between a pair of actors. It is similar to Google's PageRank and to the eigenvector centrality. == Measurement == Katz centrality computes the relative influence of a node within a network by measuring the number of the immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under consideration through these immediate neighbors. Connections made with distant neighbors are, however, penalized by an attenuation factor α {\displaystyle \alpha } . Each path or connection between a pair of nodes is assigned a weight determined by α {\displaystyle \alpha } and the distance between nodes as α d {\displaystyle \alpha ^{d}} . For example, in the figure on the right, assume that John's centrality is being measured and that α = 0.5 {\displaystyle \alpha =0.5} . The weight assigned to each link that connects John with his immediate neighbors Jane and Bob will be ( 0.5 ) 1 = 0.5 {\displaystyle (0.5)^{1}=0.5} . Since Jose connects to John indirectly through Bob, the weight assigned to this connection (composed of two links) will be ( 0.5 ) 2 = 0.25 {\displaystyle (0.5)^{2}=0.25} . Similarly, the weight assigned to the connection between Agneta and John through Aziz and Jane will be ( 0.5 ) 3 = 0.125 {\displaystyle (0.5)^{3}=0.125} and the weight assigned to the connection between Agneta and John through Diego, Jose and Bob will be ( 0.5 ) 4 = 0.0625 {\displaystyle (0.5)^{4}=0.0625} . == Mathematical formulation == Let A be the adjacency matrix of a network under consideration. Elements ( a i j ) {\displaystyle (a_{ij})} of A are variables that take a value 1 if a node i is connected to node j and 0 otherwise. The powers of A indicate the presence (or absence) of links between two nodes through intermediaries. For instance, in matrix A 3 {\displaystyle A^{3}} , if element ( a 2 , 12 ) = 1 {\displaystyle (a_{2,12})=1} , it indicates that node 2 and node 12 are connected through some walk of length 3. If C K a t z ( i ) {\displaystyle C_{\mathrm {Katz} }(i)} denotes Katz centrality of a node i, then, given a value α ∈ ( 0 , 1 ) {\displaystyle \alpha \in (0,1)} , mathematically: C K a t z ( i ) = ∑ k = 1 ∞ ∑ j = 1 n α k ( A k ) j i {\displaystyle C_{\mathrm {Katz} }(i)=\sum _{k=1}^{\infty }\sum _{j=1}^{n}\alpha ^{k}(A^{k})_{ji}} Note that the above definition uses the fact that the element at location ( i , j ) {\displaystyle (i,j)} of A k {\displaystyle A^{k}} reflects the total number of k {\displaystyle k} degree connections between nodes i {\displaystyle i} and j {\displaystyle j} . The value of the attenuation factor α {\displaystyle \alpha } has to be chosen such that it is smaller than the reciprocal of the absolute value of the largest eigenvalue of A. In this case the following expression can be used to calculate Katz centrality: C → K a t z = ( ( I − α A T ) − 1 − I ) I → {\displaystyle {\overrightarrow {C}}_{\mathrm {Katz} }=((I-\alpha A^{T})^{-1}-I){\overrightarrow {I}}} Here I {\displaystyle I} is the identity matrix, I → {\displaystyle {\overrightarrow {I}}} is a vector of size n (n is the number of nodes) consisting of ones. A T {\displaystyle A^{T}} denotes the transposed matrix of A and ( I − α A T ) − 1 {\displaystyle (I-\alpha A^{T})^{-1}} denotes matrix inversion of the term ( I − α A T ) {\displaystyle (I-\alpha A^{T})} . An extension of this framework allows for the walks to be computed in a dynamical setting. By taking a time dependent series of network adjacency snapshots of the transient edges, the dependency for walks to contribute towards a cumulative effect is presented. The arrow of time is preserved so that the contribution of activity is asymmetric in the direction of information propagation. Network producing data of the form: { A [ k ] ∈ R N × N } for k = 0 , 1 , 2 , … , M , {\displaystyle \left\{A^{[k]}\in \mathbb {R} ^{N\times N}\right\}\qquad {\text{for}}\quad k=0,1,2,\ldots ,M,} representing the adjacency matrix at each time t k {\displaystyle t_{k}} . Hence: ( A [ k ] ) i j = { 1 there is an edge from node i to node j at time t k 0 otherwise {\displaystyle \left(A^{[k]}\right)_{ij}={\begin{cases}1&{\text{there is an edge from node }}i{\text{ to node }}j{\text{ at time }}t_{k}\\0&{\text{otherwise}}\end{cases}}} The time points t 0 < t 1 < ⋯ < t M {\displaystyle t_{0}<t_{1}<\cdots <t_{M}} are ordered but not necessarily equally spaced. Q ∈ R N × N {\displaystyle Q\in \mathbb {R} ^{N\times N}} for which ( Q ) i j {\displaystyle (Q)_{ij}} is a weighted count of the number of dynamic walks of length w {\displaystyle w} from node i {\displaystyle i} to node j {\displaystyle j} . The form for the dynamic communicability between participating nodes is: Q = ( I − α A [ 0 ] ) − 1 ⋯ ( I − α A [ M ] ) − 1 . {\displaystyle {\mathcal {Q}}=\left(I-\alpha A^{[0]}\right)^{-1}\cdots \left(I-\alpha A^{[M]}\right)^{-1}.} This can be normalized via: Q ^ [ k ] = Q ^ [ k − 1 ] ( I − α A [ k ] ) − 1 ‖ Q ^ [ k − 1 ] ( I − α A [ k ] ) − 1 ‖ . {\displaystyle {\hat {\mathcal {Q}}}^{[k]}={\frac {{\hat {\mathcal {Q}}}^{[k-1]}\left(I-\alpha A^{[k]}\right)^{-1}}{\left\|{\hat {\mathcal {Q}}}^{[k-1]}\left(I-\alpha A^{[k]}\right)^{-1}\right\|}}.} Therefore, centrality measures that quantify how effectively node n {\displaystyle n} can 'broadcast' and 'receive' dynamic messages across the network: C n b r o a d c a s t := ∑ k = 1 N Q n k a n d C n r e c e i v e := ∑ k = 1 N Q k n {\displaystyle C_{n}^{\mathrm {broadcast} }:=\sum _{k=1}^{N}{\mathcal {Q}}_{nk}\quad \mathrm {and} \quad C_{n}^{\mathrm {receive} }:=\sum _{k=1}^{N}{\mathcal {Q}}_{kn}} . === Alpha centrality === Given a graph with adjacency matrix A i , j {\displaystyle A_{i,j}} , Katz centrality is defined as follows: x → = ( I − α A T ) − 1 e → − e → {\displaystyle {\vec {x}}=(I-\alpha A^{T})^{-1}{\vec {e}}-{\vec {e}}\,} where e j {\displaystyle e_{j}} is the external importance given to node j {\displaystyle j} , and α {\displaystyle \alpha } is a nonnegative attenuation factor which must be smaller than the inverse of the spectral radius of A {\displaystyle A} . The original definition by Katz used a constant vector e → {\displaystyle {\vec {e}}} . Hubbell introduced the usage of a general e → {\displaystyle {\vec {e}}} . Half a century later, Bonacich and Lloyd defined alpha centrality as: x → = ( I − α A T ) − 1 e → {\displaystyle {\vec {x}}=(I-\alpha A^{T})^{-1}{\vec {e}}\,} which is essentially identical to Katz centrality. More precisely, the score of a node j {\displaystyle j} differs exactly by e j {\displaystyle e_{j}} , so if e → {\displaystyle {\vec {e}}} is constant the order induced on the nodes is identical. == Applications == Katz centrality can be used to compute centrality in directed networks such as citation networks and the World Wide Web. Katz centrality is more suitable in the analysis of directed acyclic graphs where traditionally used measures like eigenvector centrality are rendered useless. Katz centrality can also be used in estimating the relative status or influence of actors in a social network. The work presented in shows the case study of applying a dynamic version of the Katz centrality to data from Twitter and focuses on particular brands which have stable discussion leaders. The application allows for a comparison of the methodology with that of human experts in the field and how the results are in agreement with a panel of social media experts. In neuroscience, it is found that Katz centrality correlates with the relative firing rate of neurons in a neural network. The temporal extension of the Katz centrality is applied to fMRI data obtained from a musical learning experiment in where data is collected from the subjects before and after the learning process. The results show that the changes to the network structure over the musical exposure created in each session a quantification of the cross communicability that produced clusters in line with the success of learning. A generalized form of Katz centrality can be used as an intuitive ranking system for sports teams, such as in college football. Alpha centrality is implemented in igraph library for network analysis and visualization. == References ==
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Wikipedia:Kaye Stacey#0
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Kaye C. Vale Stacey (born 1948) is an Australian mathematics educator who held the Foundation Chair of Mathematics Education in the Graduate School of Education at the University of Melbourne for 20 years, from 1992 until her retirement in 2012. She is the editor-in-chief of Educational Designer, the journal of the International Society for Design and Development in Education. Stacey has a bachelor's degree from the University of New South Wales. She earned a doctorate (D.Phil.) at the University of Oxford in 1974. Her dissertation The Enumeration of Perfect Quadratic Forms in Seven Variables concerned number theory and was supervised by Bryan John Birch. She also has a Diploma of Education from Monash University. With Leone Burton and John Mason, Stacey is the author of the book Thinking Mathematically (Addison-Wesley, 1982; 2nd ed., Pearson, 2010). In 2003, the Australian Government gave Stacey a Centenary Medal for outstanding services to mathematics education. == References == == External links == Kaye Stacey publications indexed by Google Scholar
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Wikipedia:Kazimierz Bartel#0
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Kazimierz Władysław Bartel (Polish pronunciation: [kaˈʑimjɛʐ vwaˈdɨswav ˈbartɛl]; English: Casimir Bartel; 3 March 1882 – 26 July 1941) was a Polish mathematician, freemason, scholar, diplomat and politician who served as 15th, 17th and 19th Prime Minister of Poland three times between 1926 and 1930 and a Senator of Poland from 1937 until the outbreak of World War II. Bartel was appointed Minister of Railways between 1919 and 1920, in 1922–1930 he was a member of Poland's Sejm. After Józef Piłsudski's May Coup d'état in 1926, he became prime minister and held this post during three broken tenures: 1926, 1928–29, 1929–1930. Bartel was the Deputy Prime Minister between 1926–1928 and Minister of Religious Beliefs and Public Enlightenment, when Piłsudski himself assumed the premiership, however, Bartel was in fact "de facto" prime minister during this period as Piłsudski did not concern himself with the day-to-day functions of the cabinet and the government. In 1930 upon giving up politics, he returned to the university as professor of mathematics. In 1930 he became rector of the Lwów Polytechnic and was soon awarded an honorary doctorate and membership in the Polish Mathematical Association. In 1937 he was appointed a Senator of Poland and held this post until World War II. After the Soviet invasion and occupation of eastern Poland, he was allowed to continue lecturing at the Technical Institute. In 1940 he was summoned to Moscow and offered a seat in the Soviet parliament. On 30 June 1941, in the course of Operation Barbarossa, the German Wehrmacht entered Lwów and began persecuting the local intelligentsia. Bartel was imprisoned two days later by the Gestapo and offered the top post in a Polish puppet government. His ultimate refusal of the German terms was taken as an act of treason by the Germans. By order of Heinrich Himmler, Bartel was murdered on 26 July 1941, shortly after the Massacre of Lwów professors had ended. == Early life and studies == Kazimierz Władysław Bartel was born on 3 March 1882 in Lemberg, Austria-Hungary (later Lwów, Poland, now Lviv in Ukraine) as the son of Michał Bartel and Amalia Chadaczek. Growing up in a working-class family, he graduated from elementary school in Stryj. His railwayman father arranged Bartel to be an apprentice to fitter who taught in craft school. This allowed Bartel to continue his formal education while working as an apprentice. After completing secondary school in 1901, Bartel studied mechanics at the Lwów Polytechnic in the Mechanical Engineering Department. He graduated summa cum laude in 1907 and soon started working for his alma mater as an assistant in descriptive geometry to Placyd Zdzisław Dziwinski. From 1908 to 1909, he also studied mathematics and philosophy at the Franciscan University in Lviv and at the Ludwig Maximilian University of Munich. The travel grant to Munich allowed him to attend the lectures on art history by Karl Dochlemann and mathematics by Aurel Voss and Alfred Pringsheim. He returned to the Polytechnic and earned his doctor of technical sciences in 1909. His dissertation "O utworach szeregów i pęków inwolucyjnych" (Compositions series and involution pencils) allowed him to become one of the first title holders of such doctoral within Austria-Hungary. Bartel gave his habilitation thesis "O płaskich utworach inwolucji stopnia czwartego szeregu zerowego" (On planar products of involution of the fourth series of the zero degree) in 1912, then received the title of associate professor. Bartel became the chair of descriptive geometry after the retirement of Mieczysław Łazarski in 1911 due to blindness. Bartel attained the title of professor of mathematics at the Lwów Polytechnic in 1917. Conscripted into the Austro-Hungarian Army during World War I, after the collapse of Austria-Hungary in 1918 he returned to Lwów, which became part of the newly-established Second Polish Republic. In 1919, as commander of railway troops, he fought in the defence of the city against the Ukrainian siege. Meanwhile, Bartel wrote his first textbook on descriptive geometry and befriended and later supported Poland's future leader, marshal and commander-in-chief, Józef Piłsudski. Since May 1919 he served as the manager of the Armoured Trains Construction Management and Association. His numerous successes in this field led to Prime Minister Leopold Skulski appointing him the Minister of the Railway system of the Republic of Poland. Bartel met other significant and influential politicians and diplomats, most notably Prime Minister Wincenty Witos and Prime Minister Władysław Grabski. After the Polish–Soviet War of 1920, Bartel was nominated as a lieutenant colonel and was left in charge of the railway reserve officers and the Lwów militia. He was awarded a Virtuti Militari cross, a Polish distinction for valor, after the armed conflict. In 1921, Bartel spent six months travelling to museums and galleries in France, Italy, Switzerland and Austria to research on art. Most of his holidays were spent likewise because of his interest stemming from Dochlemann's lectures. He accumulated a good personal archive of notes and photographs for this interest of his. == Political and diplomatic career == In 1922, Bartel was elected a member of Poland's Sejm (parliament) and held that position until 1929. Initially, he was a member of the party PSL "Liberation", but he was not satisfied with the radicalization of the group. In March 1925 at the Congress of the Polish People's Party, he decided to adopt, among others, a reform without compensation. Bartel eventually left the party and the organisation in April 1925, along with Marian Zyndram-Kościałkowski and Bolesław Wysłouch and later founded the parliamentary "Labour Club". This organization quickly came under the direct influence of commander-in-chief Józef Piłsudski. Just before the May Coup of 1926, Bartel received an order from Marshal Piłsudski to prepare for a takeover as prime minister after the expected collapse of President Stanisław Wojciechowski and his government. === First term in office, first government (1926) === On 15 May 1926, after the resignation of the government led by Wincenty Witos and President Wojciechowski after the May Coup, Bartel was appointed by Marshal of the Sejm and the acting head of state Maciej Rataj as the prime minister of the Second Polish Republic, but Bartel later stated in his inauguration speech that he would be the head of government only until the election of a new president. His decision was possibly influenced by the fact that he suffered from kidney and stomach problems and was constantly in pain. One member of the parliament stated, "He was a cheerful and ambitious man, but always in pain. Even his opponents in the Sejm admitted that in personal relationships, it is extremely hard not to be in favour of a man like Bartel. As prime minister, he tried to aid every man possible, even the men and women that opposed his policies and the government, but he was not able to help himself, which led to his early decline in politics and diplomacy of the Polish Republic. He was of weak stature and of weak health and would hardly make a good impression on the public, especially the socialists or communists in the east and therefore, this would not make him an influential Prime Minister nor a diplomat supporting democracy." Bartel's new government consisted mostly of people not connected with any political parties (four of those politicians already were occupied ministerial positions). Bartel was described as ideologically centrist: Prime Minister's newly established office was occupied by both the right-wing and left-wing leaders. Bartel himself took over the Ministry in turn and Piłsudski the Minister of War. Such a system churned mainly the Polish Socialist Party, which supported the May Coup. On 16 May 1926, Prime Minister Bartel made a statement in which he highlighted the principles of his policies. Bartel stated that the cabinet took power in accordance with the law, without any prejudice to the constitutional order. He also called for peace, hard work and dedication to the Polish nation. At the same time, he promised the immediate removal of incompetent and corrupt politicians from any high posts that could negatively influence the future economic growth of the Second Polish Republic. Bartel's closest personal advisor in politics and diplomacy was Marshal Józef Piłsudski, who was in favour of the new minister. Bartel suggested that Ignacy Mościcki should become a candidate for the post of head of state (president), who was also a professor at the Lwów Polytechnic. Bartel's first government was one of the most active in the history of Poland; the politicians and members of parliament gathered every second day and on occasions everyday to discuss political matters. On 4 June 1926, Ignacy Mościcki was elected the president of the Second Polish Republic, and Bartel resigned along with the entire cabinet, but soon after being appointed, President Mościcki designated him again to become prime minister. === Second and third government (1926) === On 8 June 1926, three days after the Mościcki's designation, Bartel formed his second cabinet. On the same day, Józef Piłsudski sent a letter to the head office in which he outlined the conditions of re-entry to the parliament. After his second election, Bartel primarily focused on the restoration of the decree based on the organization of the highest military authorities from 7 January 1921, which enables the free management of the Ministry of War without the vote of the government and the parliament. On 9 June 1926, the decree was officially restored, however, another decree was adopted, which increased the power of the president or head of state over the ministry. Bartel met with representatives of the parliamentary clubs and highlighted in a conversation with them his commitment towards the parliamentary system but also pointed out a more concerning issue:the economic development of the country. At a private meeting with senators, he highlighted his determination and involvement in fighting against bureaucracy, the introduction of an apolitical army and the elimination of the Ministry of Public Works. He vividly stated that before the May Coupn there was no democracy and that Poland was ruled by an oligarchy, nobles and influential leaders of wealthy privately-owned clubs and parties. The supporters of Bartel and his government emphasized his efficiency when they managed the state. His opponents, however, saw it as a tool to limit the role of the Polish Parliament and accused him of deliberate dictatorship and control over the ministers in his "private parliament sittings" - the so-called Sejm Bartlowy (Bartel's Parliament). Bartel was appointed prime minister when Marshal Piłsudski undertook an attempt to communicate with the rebellious senators and members of the Sejm. Bartel himself was considered to be representative of the liberal tendencies in the party and a spokesman of the Sanacja movement. Otherwise the post of the head of government (Prime Minister) was taken by Kazimierz Świtalski or Walery Sławek, both of whom were considered to be uncompromising supporters of the conflict with the parliament. Bartel's government contributed to a marked improvement in administration, which was primarily caused by the Prime Minister's organizational skills and knowledge. He created an efficient system of government action in connection with the Sejm and officials of lower rank: "The ministers of the previous governments generally considered themselves as autonomous rulers, which influenced the private interests of the members of different parties in charge. The government of Kazimierz Bartel was never focused or concentrated on any political ties and friendships. The officials of the Prime Minister were to validate the efficiency of each ministry. Each minister was responsible for the operation of his office and ministers could not engage in any political activities. Before his speech in the parliament on any topic, he had to submit the text to the Prime Minister himself for approval. Bartel demanded such procedures from every minister and senator of his cabinet and personally prepared the agenda for each meeting of the government and disallowed to discuss any topic without his permission or consent." Bartel also tried to improve the situation of the Polish Jews and the Jewish minority around the country. He was determined to eliminate the remnants of regulations dating back to the times of Tsar Nicholas II of Russia and Congress Poland, focused on the persecution of religious minorities, especially the Jews and the Gypsies. Bartel's cabinet announced that it is against such inhumane procedures and actions, and in 1927 the Prime Minister gave permission to adopt a law officially recognizing and granting rights to the Jewish communities. Bartel was also against enforcing certain laws to the nature of the economic sanctions imposed on the Jews. On 2 August 1926, the Parliament adopted an amendment to the Constitution (the so-called "August Novella"), significantly strengthening the role of the president. On 20 September 1926 the Christian Democratic Party raised a vote against two ministers in the government of Bartel: Antoni Sujkowski and Kazimierz Młodzianowski. The party accused them of carrying out political purges in the state administration. Eventually, the vote was passed by the government, which forced Bartel and his cabinet to resign, but Marshal Piłsudski ordered President Mościcki to appoint Bartel as prime minister. Once more, that was not in violation of the Constitution, but the anti-parliamentarian speakers and the socialist politicians, confused with the frequent changes in the administration and the government, threatened the Sejm and even suggested a rebellion or another coup. The conflict made Bartel's third new cabinet last only four days. On 30 September at the Belvedere palace in Warsaw, the council was holding a meeting in the study room, during which it was decided to dissolve the third government. Therefore, Bartel received the document on this subject, which for its validity required President Mościcki's signature. Meanwhile, the Senate immediately demanded that the parliament passes the budget cut policy proposed by the Upper House. Bartel told the Speaker of the Sejm Maciej Rataj, that in such a situation he will personally take the decree to Mościcki and ask for his signature. After the Sejm passed the budget cut policy, Bartel arrived at Mościcki's private residence, but to his surprise, Mościcki refused to sign the document allowing the dissolution. Instead, he ordered Bartel to terminate his employment. Bartel was once again forced to resign, but this time his cabinet would stay intact and his place would be taken by the marshal himself. The former prime minister was very bitter about this turn of events, despite the fact he went along with Piłsudski's and Rataj's plans. In its course, the Marshal warned that, in contrast to the previous government, he will not be "competing" with the ministers and if necessary he will use force if the members would not agree to his radical policies. === Collaboration with Piłsudski's Council === Following his resignation, Bartel was to become the Deputy Prime Minister and Minister of Religious Denominations and Public Enlightenment in Piłsudski's own, private council that operated in case of any unexpected conflict with the current operating government. The Marshal did not devote much attention towards his cabinet, focusing primarily on military and foreign policies. It was Kazimierz Bartel that was to replace the Marshal and take over his duties if absent and become the Speaker of the Sejm. He often spoke, as a representative of the government, on matters related to the budget and finances. These topics were possibly the main subject of a dispute between the "Piłsudskites" and the parliamentary opposition. After the elections in March 1928, Piłsudski decided that Kazimierz Bartel should be appointed to the position of Speaker of the Sejm. On 27 March the "Nonpartisan Bloc for Cooperation with the Government" (BBWR), an ostensibly non-political organization that existed from 1928 to 1935, closely affiliated with Józef Piłsudski and his Sanacja movement, declared Bartel's candidacy. However Piłsudski's plan to place Bartel in charge of the Sejm and nominate him as Marshal Speaker failed, because the senators and members of parliament decided to choose Ignacy Daszyński of the Polish Socialist Party as the Marshal of the Sejm instead. In protest, following the results of the vote, the members and supporters of the parliamentary BBWR party left the room. The year 1928 also marked the release of his first book "Perspektywa Malarska". It dealt with the basic theory of perspective and its extension to architecture and art. It was published by Ksiaznica-Atlas, a publisher in Lwów who provided the negatives for the German translation published by B.G. Teubner in 1934. === Second term in office (1928–1929) === As soon as the new government was formed without Kazimierz Bartel as its head, Józef Piłsudski, temporarily serving as Prime Minister of the country, resigned. He decided, however, that his position will be taken over by Kazimierz Bartel, considered his most trusted and most loyal friend and supporter among the members of the party, although this change was only formal – Bartel was already responsible for leading the ongoing work of the Council of Ministers, even if he was not the Head of Cabinet. Piłsudski's decision greatly dissatisfied the senators of parliament, who would simply demonstrate their anger by not participating in the sessions and sittings of the Sejm. Some politicians dared to even throw rotten food at the ministers that were leaving the voting chamber. The situation worsened in the upcoming months and some ministers raised concerns about their safety, as some demonstrators, often made up of ordinary citizens working on the behalf of the party, tended to physically abuse officials travelling from their homes to the newly constructed government building located on Wiejska Street in Warsaw. Similar events occurred during the inauguration of the first President of the Second Polish Republic, Gabriel Narutowicz, in December 1922. The politicians and ministers were advised to travel with guards, police or at least a weapon that they could defend themselves with, however, the use of weapons may have strengthened the unity of the opposition and of the demonstrators that could use this as an act of violence against the common people and a violation of social democracy. After the beginning of the so-called "Czechowicz affair" in which the opposition discovered that the Chancellor of the Exchequer Gabriel Czechowicz, a strong admirer of Piłsudski, passed 8 million Polish złoty from the state budget for the BBWR campaign between 1927 and 1928, on 12 February 1929 the members of an anti-Sanacja movement have requested to place both Czechowicz and Bartel before the State Tribunal (Court). In protest against this decision, Kazimierz Bartel informed the press of his intention to resign. He also stated that in his opinion the Czechowicz affair was caused by the Parliament and its senators rather than by the doings of one politician. On 13 April 1929 Bartel ordered his government to resign. He was replaced by Kazimierz Świtalski, a stubborn and self-centred man considered to be the cause of relentless struggle with the parliamentary opposition. The following months were marked by disputes between the newly formed government and the Sejm. Bartel's new cabinet began operating on 5 November 1929, however, its first sitting occurred in December on the orders of President Mościcki. After this, the Parliament adopted a motion of no confidence against Świtalski's Cabinet. Kazimierz Bartel became the prime minister once again. === Third term in office (1929–1930) === On 29 December 1929, Bartel was chosen for the third time to be prime minister and formed his fifth government and cabinet, however, he performed his duties with large uncertainties, mainly due to poor health. He had kidney illness and had a ureterolithotomy with help from Tadeusz Pisarski, a urologist he befriended during their conscription in the army. He also suffered from depression and anxiety probably due to the constant disputes with the Sejm and its senators. On 10 January he appeared at a meeting with members of parliament, declaring his willingness to cooperate with the senators and the Sejm, saying "I come with good will and determination gentlemen!" Bartel initially managed to establish cooperation with the Sejm, which resulted in the stabilization of the entire situation and conflict. Later, however, the relations between the cabinet and the parliament deteriorated again. The apogee of another dispute was a request for the adoption of no-confidence motion against the Minister of Labour and Social Welfare Aleksander Prystor. That was done primarily through the initiative of the Polish Socialist Party headed by Ignacy Daszyński and his supporters like Bolesław Limanowski, a Polish socialist politician, historian journalist and advocate of agrarianism who was the oldest member of the Polish Senate until his death in 1935 at the age of 99. On 12 March Bartel gave a speech in the Senate sharply attacking the senators, which were "not able to fulfill the tasks set in order to control the state and the country and that their stubbornness and pride in themselves was an astonishing blow to both the economy and the policies of Poland." He also stated that "being a member of parliament is a profession. It does not require the members to acquire any skills and create new damaging campaigns, only to obey the ruling party. A man focused only on work and career often becomes a man in conflict with others, which entails long political consequences." Bartel believed that the motion of no confidence towards one member of the Senate was the lack of support of the entire government. On 15 March 1930, he decided to leave the office and his resignation was accepted by the president the next day. Soon, he also resigned from his parliamentary seat and left politics. Walery Sławek was appointed the new prime minister of Poland. == Post-candidacy and return to university == After retiring from political life, he returned to the Technical University of Lwów (Polytechnic). In the same year he was elected rector of the university and held that office in the academic year of 1930/1931. He was also awarded an honorary doctorate and membership of the Polish Academy of Sciences: in the years 1930-1932 he was president of the Polish Mathematical Society. During this time he published his most important works, including a series of lectures on the perspective of European painting. It was the first such publication in the world. During his work at the Technical University of Lwów, he expressed strong opposition to plans focused on introducing the so-called "ghetto benches" for students of Jewish origin and ethnicity to separate them from Polish and Christian peers. His opinions, as well as other actions against anti-Semitic students, made Bartel the subject of numerous attacks including throwing eggs and rotten food at the professor or bringing a pig with the sign "Bartel" by Polish nationalists to the university grounds. In 1932, he testified as a witness in the Brest trials, lasting from 26 October 1931 to 13 January 1932, held at the Warsaw Regional Court where leaders of the Centrolew, a "centre-Left" anti-Sanacja political opposition movement, were tried. In 1937, Bartel was appointed Senator of Poland by the President to replace the deceased Emil Bobrowski, and served until the outbreak of World War II. In the autumn of 1938, he was one of the signatories of a document addressed to President Mościcki, which called for the inclusion of representatives of the opposition to the government in connection with the threat of the country's independence. The document also postulated amnesty for politicians of the opposition, who were forced into exile or were imprisoned after the Brest trials. Bartel handed over a memorandum to Mościcki, however, Mościcki did not respond to the proposals. In February 1939 Bartel delivered a speech in the Senate, which has gained wide publicity in the country. In it, he sharply criticized the situation in universities and colleges around Poland; mentioned the widespread anti-Semitism there; and the failed organization of studies, subjects and courses. == World War II == In September 1939, during the defence of Lwów just before the attack of the German troops, Kazimierz Bartel served as the head of the Civic Committee. When Lwów became occupied by the Soviet Union, he was allowed to continue his lectures at the Technical University. In July 1940 he was, along with several other politicians and professors, summoned to Moscow, where he took part in an All-Committee meeting of Universities of the Soviet Union. Conversations and topics mentioned mostly related scientific issues, and Bartel signed a contract with a publishing house to write a textbook of Science and Geometry for the schools of the Soviet Union. He also visited the scientific and cultural institutions like the Tretyakov Gallery and the Institute of Architecture in Moscow. There are some conflicting reports about whether during his stay in Moscow, the Soviets offered him political cooperation. According to some of his closest friends, Stalin issued a proposal for the creation of a new Polish government, but Bartel rejected it. As he wrote to his wife on 16 July 1941: "By listening to private conversations of the officers, I conclude that my position as Prime Minister may be resurrected, but what great duty this will be to control a split, communist country. In Moscow with Joseph Stalin, I had the pleasure of finding out new information from the West - Winston Churchill's speech addressed to Władysław Sikorski about Poland's supposed future." One of the editions of "Paris' Historical Notebooks" described the content of the letter sent to the German Ministry of Foreign Affairs. It stated that Müller, the Deputy Head of the Security Police and Security Service (Reinhard Heydrich) believed that Bartel negotiated in early 1941 with the Soviet authorities about the establishment of a new nation that together with the Soviet Union was to declare war on Nazi Germany. Similar information can be found, among others, in a telegram sent by the Polish Chargé d'Affaires in Switzerland to the Ministry of Foreign Affairs in London dating from 26 September 1940: "It's believed that Moscow professor Bartel has intention to create the Red Government of Poland." This information, however, was never proved to be true. Meanwhile, Maria Bartlowa, the wife of former prime minister, stated that her husband was talking only with the Soviets on the release of his new lecture book. It is also widely believed that Kazimierz Bartel never met Stalin in person. The Prime Minister-in-exile stationing in London, General Władysław Sikorski, had plans to co-operate with Bartel and appoint him an ambassador. Sikorski recognized him as one of the few people from the former political circles who would agree to cooperate on the terms and conditions of the British government. On 19 June 1941 Bartel's candidacy was officially reported by Sikorski during a meeting of the Council of Ministers. The decision was motivated by the political loyalty of the former prime minister, as well as his successful efforts to preserve the Polish character of the Lwów Polytechnic under Soviet occupation. Sikorski, however, failed to find Bartel in the Soviet Union, and Stanisław Kot was appointed ambassador instead. === Arrest and death === On 30 June 1941, soon after the German invasion of the Soviet Union began, the Wehrmacht entered Lwów. Bartel was arrested on 2 July at a meeting with co-workers at the University. Thirty-six other colleagues in the faculty were arrested the next night. Bartel was taken initially to a Gestapo prison on Pelczyńska Street. There, as mentioned by inmate Antoni Stefanowicz, he was treated properly. The former prime minister was allowed to receive, send letters and mathematical books and papers to his wife, and bring food from home. At the time, Bartel was not questioned, because there were some issues regarding the accusations made by the Gestapo. On 21 July, however, he was transferred to a prison at Łąckiego Street, where he was treated poorly. The guards called him a Commie-Jew, as reported by Stefanowicz, and the Nazi officials ordered Bartel to clean the boots of a Ukrainian Hilfsgestapo soldier. Stefanowicz reported that Bartel was mentally devastated and could not understand the essence of the tragedy. According to some sources, the Nazi officials proposed the establishment of a Polish puppet government dependent on the Reich. Such information was given by General Sikorski during a press conference in Cairo in November 1941 (on the way to Moscow). According to his version, Bartel refused and on the orders of Heinrich Himmler was executed on the 26 July 1941 at dawn. He was shot probably near Piaski Janowskie in the context of the Massacre of Lwów professors. Being barred from her daily delivery of food to her husband on Saturday, June 26, Bartel's wife learned of his death the following Monday. According to one account, during the night of October 1943 the Sonderkommando composed of Jewish prisoners unearthed the bodies of the murdered Polish professors that were filed in a mass grave. It was carried out to remove the traces of the murder in connection with the approaching Soviet troops. On 9 October 1943, the corpses were piled. The prisoners were forced to take any personal belongings and clothes, including documents of Kazimierz Bartel and Professor Tadeusz Ostrowski. Later the pile of corpses was set on fire and in the following days the Sonderkommando scattered the ashes on the surrounding fields. In 1966, on the 25th anniversary of the execution of Lwów professors, a plaque with the names of the victims of Nazism was placed on the church of St. Francis of Assisi in Kraków. Next to the memorial there is also a separate epitaph in honour of Kazimierz Bartel. == After death == Knowing the importance that Bartel gave to his work on perspective, his wife saved his manuscript after his death by begging from the Nazi officials. Bartel's library of books were either shipped with some pieces of furniture to Germany or burned with his personal papers. His second book was supposed to be published first in German by B. G. Teubner, who would provide Ksiaznica-Atlas the negatives for the Polish edition. However, war delayed the printing and ultimately caused the destruction of all materials. In the 1950s, the second book was reconstructed in the 1950s by Professor F. Otto of the University of Gdansk using the surviving manuscript and the printer's proofs which Teubner had sent for Bartel's approval. It dealt with analyzing pictures geometrically, artistic reconstruction of geometry exhibited in pictures, and tracing art history using tenets of his theory of perspective. A uniform series was released by Polskie Wydawnictwo Naukowe, the second book in 1958 and the first volume in 1960. == Honours and awards == He was decorated with, among others, the Order of the White Eagle (1932) for outstanding achievements, the French Legion of Honour (class I), the Cross of Valour, the Cross of Independence and the Silver Cross of the Virtuti Militari (1922). == References == == External links == Rzuty cechowane Nürnberg. Crimes against humanity (Volume 5) (Russian) Нюрнбергский процесс. Преступления против человечности (том 5) Москва "Юридическая литература" 1991 ISBN 5-7260-0625-9 (part related to murder of Kazimierz Bartel and his colleagues) Author profile in the database zbMATH
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Wikipedia:Kaṇakkatikāram#0
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Kaṇakkatikāram is a Tamil mathematics book believed to have been written by Kari Nayanar hailing from Korakaiyur in Cholanad. Considering the internal evidences, the work has been dated to 15th century CE. "It is significant that the mathematical methods found in these delve into the material life of the people and approach the dimensions of daily labor enumeratively." According to the author, the contents of the book are based on material available in standard Sanskrit treatises on mathematics like 'Līlāvatī". == Kaṇakkatikāram in Malayalam == Kaṇakkatikāram was popular in Kerala also and it had been in wide use as a textbook for teaching arithmetic to children in the elementary schools in Kerala during the pre-Independence era. A team of researchers at ETH Zurich unearthed as many as nineteen Malayalam manuscripts of Kaṇakkatikāram from different manuscript depositories in Kerala. According these researchers, there is not much common content in these manuscripts except one or two introductory verses and some lists of units of measurements. None of the manuscripts contain any author attribution or any clue to the date of their composition. Because of the variety and diversity of contents in the manuscripts, the researchers suggest that the purported title Kaṇakkatikāram of the works may not be the actual title of a work but might be the name of a genre of work in Malayalam. Regarding the date of composition of the work, the researchers observe thus: "the evidence in this verse and the linguistic evidence detailed above may lead to the conjecture that the Malayalam Kaṇakkatikāram originated around the fifteenth century, somewhere in the Palakkad region, perhaps in the context of merchants that connected the Malabar Coast with Tamil Nadu via the Pallakad Gap. However, this may be a hasty conclusion. The distribution and variety of the manuscripts and the geographical spread of the educational and commercial networks that carried them do not compel us to assume a unified source or origin. Instead, we may surmise that different verses came from different times and places, and were constantly recombined, re-edited and renewed by local teachers and practitioners who created new Kaṇakkatikāram compilations." A Malayalam adaptation of work with some additional mathematical material and explanations, authored by Mavanan Mappila Seyd Muhammed Asan, was printed and published by CMS Press Kottayam in 1863. The title of the Malayalam version of the book is Kaṇakkadhikāram. == Contents == Kaṇakkatikāram deals with the arithmetical calculations that are likely to be required in the daily life of ordinary people. For example, it deals with computations involving quantitative measurements of paddy, wood, other agricultural produce, land, gold, etc. and also the computations of the daily, monthly wages of field workers, etc. The Malayalam version has a 54-page preamble devoted to explaining the notations for whole numbers, fractions and multiplication tables for whole numbers and fractions. == Full texts == A version of the Malayalam Kaṇakkatikāram based on available manuscripts with English translation and detailed explanatory notes: Arun Ashokan, Naresh Keerthi, P. M. Vrinda and Roy Wagner (May 2024). Draft of an edition of Kaṇakkatikāram (PDF). ETH Zurich. Retrieved 6 August 2024.{{cite book}}: CS1 maint: multiple names: authors list (link) A palm leaf manuscript of Kaṇakkatikāram is available at the British Library. A scanned copy of the folios of the manuscript can be viewed at: "கணக்கதிகாரம் (Kanakkatikaram)". eap.bl.uk. The British Library. Retrieved 25 December 2023. (made available as part of the Endangered Archives Programme of the British Library) An old print of Kaṇakkatikāram: Kari Nayanar (1859). Kaṇakkatikāram. Kalvikkaṭal Accukkūṭam. Retrieved 25 December 2023. A modern printing of Kaṇakkatikāram: Kaṇakkatikāram (Internet Archive) Kaṇakkatikāram (Tamil Digital Library) For the Malayalam version of Kaṇakkatikāram: Mavanan Mappila Seyd Muhammed Asan (1863). Kanakkadhikaram. Kottayam: C. M. S. Press. Retrieved 26 December 2023. For a modern book with the same title Kaṇakkatikāram authored by K. Sathyabhama and P. Subrahmanian:Kaṇakkatikāram == See also == Asthana Kolahalam Kaṇita Tīpikai Kanakkusaram == References ==
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Wikipedia:Kaṇita Tīpikai#0
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Kaṇita Tīpikai (Gaṇita Dīpika) is a Tamil book authored by Paṇṭala Rāmasvāmi Nāykkar and published in 1825 dealing with arithmetic. It is the first Tamil book on mathematics ever to be printed and it is the first Tamil book ever to introduce the symbol for zero and also to discuss the decimal place value notation or positional notation using Tamil numerals. Iravatham Mahadevan, a well known Indian epigraphist, and M. V. Bhaskar noted thus in an article published in The Hindu in December 2011: "The convention of using symbols for 10, 100, and 1000 in expressing the higher numerals was current in Tamil Nadu until the advent of printing and the adoption of the international form of Indian numerals with place-value system." Senthi Babu D, a historian and researcher attached to the French Institute of Pondicherry, observes: "Pantulu Ramasamy Naicker authored the first modern arithmetic textbook in Tamil, called the “Kanita Teepikai”, sponsored by the company state, in the year 1825. Interestingly, in his ‘payiram’ to the textbook, he justifies the use of both ‘Aryan language’ and “a bit of kotuntamil” for the sake of “understanding of all”. Ramasamy, one can see, had taken great pains to introduce the notation for zero in Tamil as “0”, which directly brought him to the issue of explaining the concept of place value in numbers. When he had to introduce the concept that the value of a number increases by factors of ten on the left, but decreases when it proceeds to the right, his plea to the reader is to follow the rule as it is a ‘decision of God’." == Full text == The full text of the book is available for free download in Internet Archive: Kanita Dipikai (1825). Pantala Ramasvami Naykkar. Madras: Vepery Mission Press. Retrieved 27 June 2024. == See also == Asthana Kolahalam Kaṇakkatikāram Kanakkusaram == References ==
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Wikipedia:Kees Vuik#0
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Cornelis (Kees) Vuik (Capelle aan den IJssel, Jan. 25, 1959) is a Dutch mathematician and professor. In 1982 he received his master's degree in applied mathematics from Delft University of Technology in Netherlands. He worked at Philips Natuurkundig Laboratorium for six months. He completed his Ph.D. at Utrecht University in 1988. His research focused on moving-boundary problems (Stefan problems) and he was supervised by Prof. dr. E.M.J. Bertin and Prof. dr. A. van der Sluis. Vuik then worked at TU Delft, successively as assistant professor, associate professor, and since 2007 as full professor of Numerical Analysis in the Faculty of Electrical Engineering, Mathematics and Computer Science. Since 2022, he has been department chair of the Delft Institute of Applied Mathematics (DIAM) department. == Research == Vuik's primary research interests are in numerical linear algebra. He also works on simulators for energy networks and high-performance computing. Several dozen Ph.D. students have been supervised by Vuik. His research has led to many publications in international journals. He was the founder of the TU Delft Institute for Computational Science and Engineering (DCSE), of which he has been the director since 2007. The institute connects TU Delft researchers from various disciplines that make use of computational methods. Thanks to his efforts, a High-Performance Computing facility was created within TU Delft, to which he has been a scientific director from 2020 to 2023. From 2012 to 2019, Vuik also served as scientific director of the 4TU.Applied Mathematics Institute (4TU.AMI) of TU Delft, TU Eindhoven, the University of Twente and Wageningen University & Research. Under his leadership, the institute has grown into an important player within the Dutch mathematical community, with connections to similar institutes such as Matheon in Berlin. 4TU.AMI has also contributed to the innovation of mathematics education. In 2013, Vuik participated in a Flemish-Dutch economic mission to Texas as a representative of (at the time) the 3TU federation. == Teaching == Vuik is active in developing mathematical courses, BSc. and MSc. curricula, and educational innovations. He founded the minor Computational Science and Engineering. He is the coordinator of the international program Computer Simulation for Science and Engineering (COSSE), established in cooperation with TU Berlin and KTH Royal Institute of Technology Sweden. Vuik was also closely involved in the Massive Open Online Course (MOOC) Mathematical Modeling Basics. Tens of thousands of students have already participated in this online course. Vuik is a faculty advisor of the SIAM Student Chapter Delft of the Society for Industrial and Applied Mathematics (SIAM). He is further involved in the development and application of Multi-Media Math Education (MUMIE), an open-source e-learning platform in mathematics. == Awards == Vuik has won several awards throughout his career, including: - 2022: Professor of Excellence Award of the Delft University Fund; - 2021: Officer in the Order of Orange Nassau; - 2017: Bronze medal of the TU Delft. == Ancillary positions == Some of Vuik's ancillary positions are: - Jury member of the ASML graduation prize for mathematics at the Royal Holland Society of Sciences and Humanities (KHMW); - Participant in the mathematics council of the Dutch Platform for Mathematics (PWN); - President-director of the Batavian Society of Experimental Philosophy; - Treasurer at EU-Maths-IN; - Chairman of the local organizing committee of ICIAM 2027; - Member of the organizing committee of the SIAM Conference on Computational Science and Engineering in 2021 and 2023. == References ==
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Wikipedia:Keith Geddes#0
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Keith Oliver Geddes (born 1947) is a professor emeritus in the David R. Cheriton School of Computer Science within the Faculty of Mathematics at the University of Waterloo in Waterloo, Ontario. He is a former director of the Symbolic Computation Group in the School of Computer Science. He received a BA in Mathematics at the University of Saskatchewan in 1968; he completed both his MSc and PhD in Computer Science at the University of Toronto. Geddes is probably best known for co-founding the Maple computer algebra system, now in widespread academic use around the world. He is also the Scientific Director at the Ontario Research Centre for Computer Algebra, and is a member of the Association for Computing Machinery, as well as the American and Canadian Mathematical Societies. == Research == Geddes' primary research interest is to develop algorithms for the mechanization of mathematics. More specifically, he is interested in the computational aspects of algebra and analysis. Currently, he is focusing on designing hybrid symbolic-numeric algorithms to perform definite integration and solve ordinary and partial differential equations. Much of his work currently revolves around Maple. == Teaching == Geddes retired from teaching in December 2008. Geddes taught a mixture of both senior-level symbolic computation courses, at both the undergraduate and graduate level, as well as introductory courses on the principles of computer science. == See also == Maple computer algebra system Waterloo Maple Gaston Gonnet — the co-founder of Waterloo Maple Risch algorithm Symbolic integration Derivatives of the incomplete gamma function List of University of Waterloo people == External links == Keith Geddes' home page The Symbolic Computation Group Keith Geddes at the Mathematics Genealogy Project
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Wikipedia:Keith William Morton#0
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Keith William Morton (born 28 May 1930, Ipswich, Suffolk, England) is a British mathematician working on partial differential equations, and their numerical analysis. He obtained his Ph.D. in 1964 under the supervision of Harold Grad at the Courant Institute of Mathematical Sciences of New York University. In 2010, he was awarded the De Morgan Medal. == References ==
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Wikipedia:Kellogg's theorem#0
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Kellogg's theorem is a pair of related results in the mathematical study of the regularity of harmonic functions on sufficiently smooth domains by Oliver Dimon Kellogg. In the first version, it states that, for k ≥ 2 {\displaystyle k\geq 2} , if the domain's boundary is of class C k {\displaystyle C^{k}} and the k-th derivatives of the boundary are Dini continuous, then the harmonic functions are uniformly C k {\displaystyle C^{k}} as well. The second, more common version of the theorem states that for domains which are C k , α {\displaystyle C^{k,\alpha }} , if the boundary data is of class C k , α {\displaystyle C^{k,\alpha }} , then so is the harmonic function itself. Kellogg's method of proof analyzes the representation of harmonic functions provided by the Poisson kernel, applied to an interior tangent sphere. In modern presentations, Kellogg's theorem is usually covered as a specific case of the boundary Schauder estimates for elliptic partial differential equations. == See also == Schauder estimates == Sources == Kellogg, Oliver Dimon (1931), "On the derivatives of harmonic functions on the boundary", Transactions of the American Mathematical Society, vol. 33, no. 2, pp. 486–510, doi:10.2307/1989419, JSTOR 1989419 Gilbarg, David; Trudinger, Neil (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, ISBN 3-540-41160-7
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Wikipedia:Ken-ichi Kawarabayashi#0
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Ken-ichi Kawarabayashi (Japanese: 河原林 健一, born 1975) is a Japanese graph theorist who works as a professor at the National Institute of Informatics and is known for his research on graph theory (particularly the theory of graph minors) and graph algorithms. Kawarabayashi was born on May 22, 1975, in Tokyo. He earned a bachelor's degree in mathematics from Keio University in 1998, a master's degree from Keio in 2000, and a PhD from Keio in 2001, researching the Lovász–Woodall conjecture under the supervision of Katsuhiro Ota. After temporary positions at Vanderbilt University and under the supervision of Paul Seymour at Princeton University, he became an assistant professor at Tohoku University in 2003, and moved to the National Institute of Informatics in 2006. In 2003, Kawarabayashi was one of three winners of the Kirkman Medal of the Institute of Combinatorics and its Applications, an award given by them annually to researchers within four years of their PhD. In 2015, he won the Spring Prize of the Mathematical Society of Japan, its highest honor. He was a keynote speaker at the 2015 International Colloquium on Automata, Languages and Programming. == References == == External links == Google scholar profile
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Wikipedia:Kengo Hirachi#0
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Kengo Hirachi (平地 健吾 Hirachi Kengo, born 30 November 1964) is a Japanese mathematician, specializing in CR geometry and mathematical analysis. Hirachi received from Osaka University his B.S. in 1987, his M.S. in 1989, and his Dr.Sci., advised by Gen Komatsu, in 1994 with dissertation The second variation of the Bergman kernel for ellipsoids. He was a research assistant from 1989 to 1996 and a lecturer from 1996 to 2000 at Osaka University. He was an associate professor from 2000 to 2010 and a full professor from 2010 to the present at the University of Tokyo. He was a visiting professor at the Mathematical Sciences Research Institute from October 1995 to September 1996, at the Erwin Schrödinger Institute for Mathematical Physics from March 2004 to April 2004, at Princeton University from October 2004 to July 2005, and at the Institute for Advanced Study from January 2009 to April 2009. Hirachi’s work employs a wide range of tools in geometry and analysis, including several complex variables, the complex Monge-Ampère equation, microlocal analysis, parabolic invariant theory, explicit computations, and computer algebra packages. In a paper in the Annals of Mathematics (2000) Hirachi constructed CR invariants of strongly pseudoconvex boundaries via a deep study of the logarithmic singularity of the Bergman kernel. He has proved various results linking the Bergman and Szegő kernels, and he has made significant progress to a program in which the Bergman kernel function plays a role analogous to the heat kernel of Riemannian geometry. == Awards and honors == Takebe Senior Prize (1999) of the Mathematical Society of Japan Geometry Prize (2003) of the Mathematical Society of Japan Stefan Bergman Prize (2006) Inoue Prize for Science (2012) Invited lecture at ICM, Seoul 2014 == References == == External links == Kengo Hirachi -- Bibliography, U. of Tokyo website ICM2014 VideoSeries IL8.3 : Kengo Hirachi on Aug14Thu - YouTube
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Wikipedia:Kenneth Pennycuick#0
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Dr. Kenneth Pennycuick (28 May 1911 – 16 January 1995) was a British philatelist who was added to the Roll of Distinguished Philatelists in 1980. He was president and later chairman of the Society of Postal Historians. Pennycuick was a specialist in the philately of East Africa. == References ==
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Wikipedia:Kenneth Walters#0
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Kenneth Walters (14 September 1934 – 28 March 2022) was a British mathematician and rheologist. He was a Distinguished Research Professor at the Institute Of Mathematics, Physics and Computer Science of the Aberystwyth University. == Education == Walters earned his PhD from the University of Swansea in 1959 under the supervision of James G. Oldroyd. His thesis was entitled Some Elastico-Viscous Liquids with Continuous and Discrete Relaxation Spectra. == Work == Walters made contributions to rheology and the development of rheological science in the United Kingdom, and has conducted extensive studies of the behaviour of non-Newtonian fluids, particularly elastic liquids. He made advances in two major areas: the measurement of rheological properties, and the numerical solution of complex flows. In the first area, he extended the theory of viscometric flows, carried out a searching analysis of sources of error in the principal instruments in current use, and was involved in industrial applications arising in the manufacture of lubricants, detergents and paints. His book, Rheometry, is a standard work of reference and the book Numerical Simulation of Non-Newtonian Flow, of which he is joint author, is an influential text in this field of research. == Awards and honours == Prof. Walters was recognized extensively for his contributions to the rheological community by being awarded the Gold Medal from the British Society of Rheology in 1984 and the Weissenberg Award from the European Society of Rheology in 2002. Walters was elected a Fellow of the Royal Society (FRS) in 1991. In 1995 he was made a Foreign Associate of the National Academy of Engineering. He was an active member of the rheological community for many years. He served as President of the British Society of Rheology from 1974 to 1976, President of the European Society of Rheology from 1996 to 2000, and the Chairman of the International Committee on Rheology from 2000 to 2004. He was a founding fellow of the Learned Society of Wales. == Personal life == In 1961, Kenneth began dating Mary Eccles, an Aberystwyth student who had been Student ‘Rag Queen’ before they met. They were both committed Christians and married in 1963. Their mutual faith led them into lay leadership at St Michael's Anglican church in Aberystwyth. They had three children and seven grandchildren. == Death == Walters' death was announced on 30 March 2022. == References ==
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Wikipedia:Kent Mathematics Project#0
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The Kent Mathematics Project (K.M.P.) was an educational system for teaching mathematics to 9-16 year olds. The system comprised task worksheets, booklets, audio compact cassettes and tests. Through the 1970s and 1980s, it was widely adopted in Kent schools, as well as being exported internationally. The system was based on the idea that: mathematical learning should be in terms of levels of concept development and that chronological age should not be the basis of course design. K.M.P. providing materials and structure for non-specialist teachers to teach math classes while sharing the effort required in producing suitable teaching material. == History == Started in 1966, the project originated at Ridgewaye School, Southborough, Tunbridge Wells which closed its doors in 1991. K.M.P went through several titles including "An Auto-Instructional Course in Mathematics" and "The Ridgewaye Individualized Course". The system was inspired by the self-directed learning of the Dalton Plan while attempting to avoid its pitfalls. K.M.P was gradually extended over time, involving trials at a number of schools before being more widely distributed. K.M.P. was adopted by the education authority in 1970 and used in over seventy schools around Kent. In 1988, the project's director objected to the National Curriculum which emphasised goals to be achieved by particular ages. == Usage == To teach a specific concept, the teacher selected a set of twelve tasks called a "matrix" from a material bank for the pupil to complete. The teacher was expected to be available to mentor pupils if they encountered difficulty. The tasks were completed in any order, then self-corrected by the pupil and checked by the teacher. When the entire task matrix is completed, the pupil performed a test to check their understanding. The tests results were then used by the teacher to construct the next task matrix. This allowed children to progress at their own rate, either ahead or behind the rest of the class, while allowing the teacher to customize the course to the pupil's needs. Tasks were assigned a level of 1 to 9, depending on their difficulty. The difficulty of tasks assigned was based on the child's ability, rather than their age. KMP materials were published by Ward Lock & Co. == Impact == Ideas from K.M.P. were adopted in later teaching tools, including Graded Assessment in Mathematics (GAIM) and Secondary Mathematics Individualised Learning Experiment (SMILE). == See also == Programmed learning Mastery learning Educational technology == References ==
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Wikipedia:Kepler–Bouwkamp constant#0
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In plane geometry, the Kepler–Bouwkamp constant (or polygon inscribing constant) is obtained as a limit of the following sequence. Take a circle of radius 1. Inscribe a regular triangle in this circle. Inscribe a circle in this triangle. Inscribe a square in it. Inscribe a circle, regular pentagon, circle, regular hexagon and so forth. The radius of the limiting circle is called the Kepler–Bouwkamp constant. It is named after Johannes Kepler and Christoffel Bouwkamp, and is the inverse of the polygon circumscribing constant. == Numerical value == The decimal expansion of the Kepler–Bouwkamp constant is (sequence A085365 in the OEIS) ∏ k = 3 ∞ cos ( π k ) = 0.1149420448 … . {\displaystyle \prod _{k=3}^{\infty }\cos \left({\frac {\pi }{k}}\right)=0.1149420448\dots .} The natural logarithm of the Kepler-Bouwkamp constant is given by − 2 ∑ k = 1 ∞ 2 2 k − 1 2 k ζ ( 2 k ) ( ζ ( 2 k ) − 1 − 1 2 2 k ) {\displaystyle -2\sum _{k=1}^{\infty }{\frac {2^{2k}-1}{2k}}\zeta (2k)\left(\zeta (2k)-1-{\frac {1}{2^{2k}}}\right)} where ζ ( s ) = ∑ n = 1 ∞ 1 n s {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}} is the Riemann zeta function. If the product is taken over the odd primes, the constant ∏ k = 3 , 5 , 7 , 11 , 13 , 17 , … cos ( π k ) = 0.312832 … {\displaystyle \prod _{k=3,5,7,11,13,17,\ldots }\cos \left({\frac {\pi }{k}}\right)=0.312832\ldots } is obtained (sequence A131671 in the OEIS). == References == == Further reading == Kitson, Adrian R. (2006). "The prime analog of the Kepler–Bouwkamp constant". arXiv:math/0608186. Kitson, Adrian R. (2008). "The prime analogue of the Kepler-Bouwkamp constant". The Mathematical Gazette. 92: 293. doi:10.1017/S0025557200183214. S2CID 117950145. Doslic, Tomislav (2014). "Kepler-Bouwkamp radius of combinatorial sequences". Journal of Integer Sequence. 17: 14.11.3. == External links == Weisstein, Eric W. "Polygon Inscribing". MathWorld.
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Wikipedia:Kerala Mathematical Association#0
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Kerala Mathematical Association is an organisation established in 1962 to serve the mathematical community comprising students, teachers and researchers inside Kerala and outside. It has a membership of around 1000 of which nearly half are life members and about 300 are from outside Kerala and outside India. == Overview == The Association regularly organises, on an average ten to twelve a year, national/international workshops, seminars in different parts of Kerala. The Association also regularly publishes the proceedings of these seminars and workshops. Focusing on teachers and students the Association conducts regional orientation programmes on new developments in mathematics. Since 2004 the association is publishing an international journal Bulletin of Kerala Mathematical Association which contains original research papers on Mathematics and its applications, with two issues in a year. M.S. Samuel (MACFAST, Tiruvalla) is the Executive Editor of the Bulletin. The Kerala Mathematical Association started a regular Prof.T.A. Sarasvati Amma Memorial Lecture in its annual conference in 2002. The Lecture was endowed by R.C. Gupta a historian of Indian mathematics. T. Thrivikraman (formerly Professor, Cochin University of Science and Technology) is currently the President of the Association and Sunny Kuriakose, Professor and Dean, Federal Institute of Science and Technology, Angamaly (formerly, Principal, B.P.C. College, Piravom,) is the Academic Secretary. == See also == Indian Mathematical Society Calcutta Mathematical Society == External links == aim India == References ==
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Wikipedia:Kerala School of Mathematics, Kozhikode#0
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The Kerala School of Mathematics (KSoM) in Kozhikode, India is a research institute in Theoretical sciences with a focus on Mathematics. The institute is a joint venture of the Department of Atomic Energy (DAE) and the Kerala State Council for Science, Technology and Environment (KSCSTE). Kerala School of Mathematics is a center of advanced research and learning in Mathematics and is a meeting ground for leading Mathematicians from around the world. Kerala School of Mathematics has a doctoral program to which students are admitted on an yearly basis. The institute also has an Integrated MSc-PhD program with an option for students to exit the program with an MSc degree at the end of two years. == History == Mathematics in Kerala, during the times of Madhava of Sangamagrama, majorly flourished in the Muziris region of Thrikkandiyur, Thirur, Alattiyur, and Tirunavaya in the Malabar region of Kerala. Kerala school of astronomy and mathematics flourished between the 14th and 16th centuries. Commemorating the rich heritage of Mathematics in the region, Kerala School of Mathematics was hence chosen to be set up in the scenic mountains of the Western Ghats in the city of Kozhikode. The nascent plan to set up Kerala School of Mathematics started forming shape in around 2004. The then DAE chairman Anil Kakodkar and the then executive vice president of KSCSTE, M. S. Valiathan were instrumental in setting up the institute with the guidance of M. S. Raghunathan, Rajeeva Karandikar and Alladi Sitaram. The foundation stone of KSoM was laid by the then Chief Minister A.K. Antony in 2004. The institute was later inaugurated in 2008 by the then Chief Minister V. S. Achuthanandan and finally set up in 2009 with Parameswaran A. J. as the founding director. == External links == Official Website == References ==
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Wikipedia:Kerala school of astronomy and mathematics#0
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The Kerala school of astronomy and mathematics or the Kerala school was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Tirur, Malappuram, Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and its original discoveries seem to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently discovered a number of important mathematical concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha (around 1500), and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (c. 1530), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha. Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series). == Background == Islamic scholars nearly developed a general formula for finding integrals of polynomials by 1000 AD —and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has been found to date. Indian scholars, on the other hand, were by the year 1600 able to use formula similar to Ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Isaac Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed. == Contributions == === Infinite series and calculus === The Kerala school has made a number of contributions to the fields of infinite series and calculus. These include the following infinite geometric series: The Kerala school made intuitive use of mathematical induction, though the inductive hypothesis was not yet formulated or employed in proofs. They used this to discover a semi-rigorous proof of the result: for large n. They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor–Maclaurin) infinite series for sin x {\displaystyle \sin x} , cos x {\displaystyle \cos x} , and arctan x {\displaystyle \arctan x} . The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as: where, for r = 1 , {\displaystyle r=1,} the series reduce to the standard power series for these trigonometric functions, for example: (The Kerala school did not use the "factorial" symbolism.) The Kerala school made use of the rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature (i.e. computation of area under the arc of the circle), was not yet developed.) They also made use of the series expansion of arctan x {\displaystyle \arctan x} to obtain an infinite series expression (later known as Gregory series) for π {\displaystyle \pi } : Their rational approximation of the error for the finite sum of their series are of particular interest. For example, the error, f i ( n + 1 ) {\displaystyle f_{i}(n+1)} , (for n odd, and i = 1, 2, 3) for the series: They manipulated the terms, using the partial fraction expansion of : 1 n 3 − n {\displaystyle {\frac {1}{n^{3}-n}}} to obtain a more rapidly converging series for π {\displaystyle \pi } : They used the improved series to derive a rational expression, 104348 / 33215 {\displaystyle 104348/33215} for π {\displaystyle \pi } correct up to nine decimal places, i.e. 3.141592653 {\displaystyle 3.141592653} . They made use of an intuitive notion of a limit to compute these results. The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions, though the notion of a function, or of exponential or logarithmic functions, was not yet formulated. === Recognition === In 1825 John Warren published a memoir on the division of time in southern India, called the Kala Sankalita, which briefly mentions the discovery of infinite series by Kerala astronomers. The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835. According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries". However, Whish's results were almost completely neglected until over a century later, when the discoveries of the Kerala school were investigated again by C. T. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in Yuktibhasa given in two papers, a commentary on the Yuktibhasa's proof of the sine and cosine series and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary). In 1972 K. V. Sarma published his A History of the Kerala School of Hindu Astronomy which described features of the School such as the continuity of knowledge transmission from the 13th to the 17th century: Govinda Bhattathiri to Parameshvara to Damodara to Nilakantha Somayaji to Jyesthadeva to Acyuta Pisarati. Transmission from teacher to pupil conserved knowledge in "a practical, demonstrative discipline like astronomy at a time when there was not a proliferation of printed books and public schools." In 1994 it was argued that the heliocentric model had been adopted about 1500 A.D. in Kerala. == Possible transmission of Kerala school results to Europe == A. K. Bag suggested in 1979 that knowledge of these results might have been transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries. Kerala was in continuous contact with China and Arabia, and Europe. The suggestion of some communication routes and a chronology by some scholars could make such a transmission a possibility; however, there is no direct evidence by way of relevant manuscripts that such a transmission took place. According to David Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century". V. J. Katz notes some of the ideas of the Kerala school have similarities to the work of 11th-century Iraqi scholar Ibn al-Haytham, suggesting a possible transmission of ideas from Islamic mathematics to Kerala. Both Indian and Arab scholars made discoveries before the 17th century that are now considered a part of calculus. According to Katz, they were yet to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today", like Newton and Leibniz. The intellectual careers of both Newton and Leibniz are well documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate predecessors of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources of which we are not now aware". This is an active area of current research, especially in the manuscript collections of Spain and Maghreb, research that is now being pursued, among other places, at the Centre national de la recherche scientifique in Paris. == See also == Indian astronomy Indian mathematics Indian mathematicians History of mathematics A History of the Kerala School of Hindu Astronomy List of astronomers and mathematicians of the Kerala school == Notes == == References == Bressoud, David (2002), "Was Calculus Invented in India?", The College Mathematics Journal, 33 (1): 2–13, doi:10.2307/1558972, JSTOR 1558972. Gupta, R. C. (1969) "Second Order of Interpolation of Indian Mathematics", Indian Journal of History of Science 4: 92-94 Hayashi, Takao (2003), "Indian Mathematics", in Grattan-Guinness, Ivor (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, vol. 1, pp. 118–130, Baltimore, MD: The Johns Hopkins University Press, 976 pages, ISBN 0-8018-7396-7. Joseph, G. G. (2000), The Crest of the Peacock: The Non-European Roots of Mathematics, Princeton, NJ: Princeton University Press, ISBN 0-691-00659-8. Katz, Victor J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine, 68 (3): 163–174, doi:10.2307/2691411, JSTOR 2691411. Parameswaran, S. (1992) "Whish's showroom revisited", Mathematical Gazette 76, no. 475 pages 28–36 Pingree, David (1992), "Hellenophilia versus the History of Science", Isis, 83 (4): 554–563, Bibcode:1992Isis...83..554P, doi:10.1086/356288, JSTOR 234257, S2CID 68570164 Plofker, Kim (1996), "An Example of the Secant Method of Iterative Approximation in a Fifteenth-Century Sanskrit Text", Historia Mathematica, 23 (3): 246–256, doi:10.1006/hmat.1996.0026. Plofker, Kim (2001), "The "Error" in the Indian "Taylor Series Approximation" to the Sine", Historia Mathematica, 28 (4): 283–295, doi:10.1006/hmat.2001.2331. Plofker, K. (20 July 2007), "Mathematics of India", in Katz, Victor J. (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, 685 pages (published 2007), pp. 385–514, ISBN 978-0-691-11485-9. C. K. Raju. 'Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhâsâ', Philosophy East and West 51, University of Hawaii Press, 2001. Roy, Ranjan (1990), "Discovery of the Series Formula for π {\displaystyle \pi } by Leibniz, Gregory, and Nilakantha", Mathematics Magazine, 63 (5): 291–306, doi:10.2307/2690896, JSTOR 2690896. Sarma, K. V.; Hariharan, S. (1991). "Yuktibhasa of Jyesthadeva : a book of rationales in Indian mathematics and astronomy – an analytical appraisal". Indian J. Hist. Sci. 26 (2): 185–207. Singh, A. N. (1936), "On the Use of Series in Hindu Mathematics", Osiris, 1: 606–628, doi:10.1086/368443, JSTOR 301627, S2CID 144760421 Stillwell, John (2004), Mathematics and its History (2 ed.), Berlin and New York: Springer, 568 pages, ISBN 0-387-95336-1. Tacchi Venturi. 'Letter by Matteo Ricci to Petri Maffei on 1 Dec 1581', Matteo Ricci S.I., Le Lettre Dalla Cina 1580–1610, vol. 2, Macerata, 1613. == External links == An overview of Indian mathematics, MacTutor History of Mathematics archive, 2002. Indian Mathematics: Redressing the balance, MacTutor History of Mathematics archive, 2002. Keralese mathematics, MacTutor History of Mathematics archive, 2002. Possible transmission of Keralese mathematics to Europe, MacTutor History of Mathematics archive, 2002. "Indians predated Newton 'discovery' by 250 years" phys.org, 2007
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Wikipedia:Kernel (algebra)#0
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In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map. The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix. The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective. For some types of structure, such as abelian groups and vector spaces, the possible kernels are exactly the substructures of the same type. This is not always the case, and, sometimes, the possible kernels have received a special name, such as normal subgroup for groups and two-sided ideals for rings. Kernels allow defining quotient objects (also called quotient algebras in universal algebra, and cokernels in category theory). For many types of algebraic structure, the fundamental theorem on homomorphisms (or first isomorphism theorem) states that image of a homomorphism is isomorphic to the quotient by the kernel. The concept of a kernel has been extended to structures such that the inverse image of a single element is not sufficient for deciding whether a homomorphism is injective. In these cases, the kernel is a congruence relation. This article is a survey for some important types of kernels in algebraic structures. == History == The mathematician Pontryagin is credited with using the word "kernel" in 1931 to describe the elements of a group that were sent to the identity element in another group. == Definition == === Group homomorphisms === Let G and H be groups and let f be a group homomorphism from G to H. If eH is the identity element of H, then the kernel of f is the preimage of the singleton set {eH}; that is, the subset of G consisting of all those elements of G that are mapped by f to the element eH. The kernel is usually denoted ker f (or a variation). In symbols: ker f = { g ∈ G : f ( g ) = e H } . {\displaystyle \ker f=\{g\in G:f(g)=e_{H}\}.} Since a group homomorphism preserves identity elements, the identity element eG of G must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set {eG}. If f were not injective, then the non-injective elements can form a distinct element of its kernel: there would exist a, b ∈ G such that a ≠ b and f(a) = f(b). Thus f(a)f(b)−1 = eH. f is a group homomorphism, so inverses and group operations are preserved, giving f(ab−1) = eH; in other words, ab−1 ∈ ker f, and ker f would not be the singleton. Conversely, distinct elements of the kernel violate injectivity directly: if there would exist an element g ≠ eG ∈ ker f, then f(g) = f(eG) = eH, thus f would not be injective. ker f is a subgroup of G and further it is a normal subgroup. Thus, there is a corresponding quotient group G / (ker f). This is isomorphic to f(G), the image of G under f (which is a subgroup of H also), by the first isomorphism theorem for groups. === Ring homomorphisms === Let R and S be rings (assumed unital) and let f be a ring homomorphism from R to S. If 0S is the zero element of S, then the kernel of f is its kernel as additive groups. It is the preimage of the zero ideal {0S}, which is, the subset of R consisting of all those elements of R that are mapped by f to the element 0S. The kernel is usually denoted ker f (or a variation). In symbols: ker f = { r ∈ R : f ( r ) = 0 S } . {\displaystyle \operatorname {ker} f=\{r\in R:f(r)=0_{S}\}.} Since a ring homomorphism preserves zero elements, the zero element 0R of R must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set {0R}. This is always the case if R is a field, and S is not the zero ring. Since ker f contains the multiplicative identity only when S is the zero ring, it turns out that the kernel is generally not a subring of R. The kernel is a subrng, and, more precisely, a two-sided ideal of R. Thus, it makes sense to speak of the quotient ring R / (ker f). The first isomorphism theorem for rings states that this quotient ring is naturally isomorphic to the image of f (which is a subring of S). (Note that rings need not be unital for the kernel definition). === Linear maps === Let V and W be vector spaces over a field (or more generally, modules over a ring) and let T be a linear map from V to W. If 0W is the zero vector of W, then the kernel of T (or null space) is the preimage of the zero subspace {0W}; that is, the subset of V consisting of all those elements of V that are mapped by T to the element 0W. The kernel is usually denoted as ker T, or some variation thereof: ker T = { v ∈ V : T ( v ) = 0 W } . {\displaystyle \ker T=\{\mathbf {v} \in V:T(\mathbf {v} )=\mathbf {0} _{W}\}.} Since a linear map preserves zero vectors, the zero vector 0V of V must belong to the kernel. The transformation T is injective if and only if its kernel is reduced to the zero subspace. The kernel ker T is always a linear subspace of V. Thus, it makes sense to speak of the quotient space V / (ker T). The first isomorphism theorem for vector spaces states that this quotient space is naturally isomorphic to the image of T (which is a subspace of W). As a consequence, the dimension of V equals the dimension of the kernel plus the dimension of the image. One can define kernels for homomorphisms between modules over a ring in an analogous manner. This includes kernels for homomorphisms between abelian groups as a special case. This example captures the essence of kernels in general abelian categories; see Kernel (category theory). === Module homomorphisms === Let R {\displaystyle R} be a ring, and let M {\displaystyle M} and N {\displaystyle N} be R {\displaystyle R} -modules. If φ : M → N {\displaystyle \varphi :M\to N} is a module homomorphism, then the kernel is defined to be: ker φ = { m ∈ M | φ ( m ) = 0 } {\displaystyle \ker \varphi =\{m\in M\ |\ \varphi (m)=0\}} Every kernel is a submodule of the domain module. === Monoid homomorphisms === Let M and N be monoids and let f be a monoid homomorphism from M to N. Then the kernel of f is the subset of the direct product M × M consisting of all those ordered pairs of elements of M whose components are both mapped by f to the same element in N. The kernel is usually denoted ker f (or a variation thereof). In symbols: ker f = { ( m , m ′ ) ∈ M × M : f ( m ) = f ( m ′ ) } . {\displaystyle \operatorname {ker} f=\left\{\left(m,m'\right)\in M\times M:f(m)=f\left(m'\right)\right\}.} Since f is a function, the elements of the form (m, m) must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the diagonal set {(m, m) : m in M}. It turns out that ker f is an equivalence relation on M, and in fact a congruence relation. Thus, it makes sense to speak of the quotient monoid M / (ker f). The first isomorphism theorem for monoids states that this quotient monoid is naturally isomorphic to the image of f (which is a submonoid of N; for the congruence relation). This is very different in flavor from the above examples. In particular, the preimage of the identity element of N is not enough to determine the kernel of f. == Survey of examples == === Group homomorphisms === Let G be the cyclic group on 6 elements {0, 1, 2, 3, 4, 5} with modular addition, H be the cyclic on 2 elements {0, 1} with modular addition, and f the homomorphism that maps each element g in G to the element g modulo 2 in H. Then ker f = {0, 2, 4} , since all these elements are mapped to 0H. The quotient group G / (ker f) has two elements: {0, 2, 4} and {1, 3, 5}. It is indeed isomorphic to H. Given a isomorphism φ : G → H {\displaystyle \varphi :G\to H} , one has ker φ = 1 {\displaystyle \ker \varphi =1} . On the other hand, if this mapping is merely a homomorphism where H is the trivial group, then φ ( g ) = 1 {\displaystyle \varphi (g)=1} for all g ∈ G {\displaystyle g\in G} , so thus ker φ = G {\displaystyle \ker \varphi =G} . Let φ : R 2 → R {\displaystyle \varphi :\mathbb {R} ^{2}\to \mathbb {R} } be the map defined as φ ( ( x , y ) ) = x {\displaystyle \varphi ((x,y))=x} . Then this is a homomorphism with the kernel consisting precisely the points of the form ( 0 , y ) {\displaystyle (0,y)} . This mapping is considered the "projection onto the x-axis." A similar phenomenon occurs with the mapping f : ( R × ) 2 → R × {\displaystyle f:(\mathbb {R} ^{\times })^{2}\to \mathbb {R} ^{\times }} defined as f ( a , b ) = b {\displaystyle f(a,b)=b} , where the kernel is the points of the form ( a , 1 ) {\displaystyle (a,1)} For a non-abelian example, let Q 8 {\displaystyle Q_{8}} denote the Quaternion group, and V 4 {\displaystyle V_{4}} the Klein 4-group. Define a mapping φ : Q 8 → V 4 {\displaystyle \varphi :Q_{8}\to V_{4}} to be: φ ( ± 1 ) = 1 {\displaystyle \varphi (\pm 1)=1} φ ( ± i ) = a {\displaystyle \varphi (\pm i)=a} φ ( ± j ) = b {\displaystyle \varphi (\pm j)=b} φ ( ± k ) = c {\displaystyle \varphi (\pm k)=c} Then this mapping is a homomorphism where ker φ = { ± 1 } {\displaystyle \ker \varphi =\{\pm 1\}} . === Ring homomorphisms === Consider the mapping φ : Z → Z / 2 Z {\displaystyle \varphi :\mathbb {Z} \to \mathbb {Z} /2\mathbb {Z} } where the later ring is the integers modulo 2 and the map sends each number to its parity; 0 for even numbers, and 1 for odd numbers. This mapping turns out to be a homomorphism, and since the additive identity of the later ring is 0, the kernel is precisely the even numbers. Let φ : Q [ x ] → Q {\displaystyle \varphi :\mathbb {Q} [x]\to \mathbb {Q} } be defined as φ ( p ( x ) ) = p ( 0 ) {\displaystyle \varphi (p(x))=p(0)} . This mapping , which happens to be a homomorphism, sends each polynomial to its constant term. It maps a polynomial to zero if and only if said polynomial's constant term is 0. If we instead work with polynomials with real coefficients, then we again receive a homomorphism with its kernel being the polynomials with constant term 0. === Linear maps === If V and W are finite-dimensional and bases have been chosen, then T can be described by a matrix M, and the kernel can be computed by solving the homogeneous system of linear equations Mv = 0. In this case, the kernel of T may be identified to the kernel of the matrix M, also called "null space" of M. The dimension of the null space, called the nullity of M, is given by the number of columns of M minus the rank of M, as a consequence of the rank–nullity theorem. Solving homogeneous differential equations often amounts to computing the kernel of certain differential operators. For instance, in order to find all twice-differentiable functions f from the real line to itself such that x f ″ ( x ) + 3 f ′ ( x ) = f ( x ) , {\displaystyle xf''(x)+3f'(x)=f(x),} let V be the space of all twice differentiable functions, let W be the space of all functions, and define a linear operator T from V to W by ( T f ) ( x ) = x f ″ ( x ) + 3 f ′ ( x ) − f ( x ) {\displaystyle (Tf)(x)=xf''(x)+3f'(x)-f(x)} for f in V and x an arbitrary real number. Then all solutions to the differential equation are in ker T. == Quotient algebras == The kernel of a homomorphism can be used to define a quotient algebra. For instance, if φ : G → H {\displaystyle \varphi :G\to H} denotes a group homomorphism, and we set K = ker φ {\displaystyle K=\ker \varphi } , we can consider G / K {\displaystyle G/K} to be the set of fibers of the homomorphism φ {\displaystyle \varphi } , where a fiber is merely the set of points of the domain mapping to a single chosen point in the range. If X a ∈ G / K {\displaystyle X_{a}\in G/K} denotes the fiber of the element a ∈ H {\displaystyle a\in H} , then we can give a group operation on the set of fibers by X a X b = X a b {\displaystyle X_{a}X_{b}=X_{ab}} , and we call G / K {\displaystyle G/K} the quotient group (or factor group), to be read as "G modulo K" or "G mod K". The terminology arises from the fact that the kernel represents the fiber of the identity element of the range, H {\displaystyle H} , and that the remaining elements are simply "translates" of the kernel, so the quotient group is obtained by "dividing out" by the kernel. The fibers can also be described by looking at the domain relative to the kernel; given X ∈ G / K {\displaystyle X\in G/K} and any element u ∈ X {\displaystyle u\in X} , then X = u K = K u {\displaystyle X=uK=Ku} where: u K = { u k | k ∈ K } {\displaystyle uK=\{uk\ |\ k\in K\}} K u = { k u | k ∈ K } {\displaystyle Ku=\{ku\ |\ k\in K\}} these sets are called the left and right cosets respectively, and can be defined in general for any arbitrary subgroup aside from the kernel. The group operation can then be defined as u K ∘ v K = ( u k ) K {\displaystyle uK\circ vK=(uk)K} , which is well-defined regardless of the choice of representatives of the fibers. According to the first isomorphism theorem, we have an isomorphism μ : G / K → φ ( G ) {\displaystyle \mu :G/K\to \varphi (G)} , where the later group is the image of the homomorphism φ {\displaystyle \varphi } , and the isomorphism is defined as μ ( u K ) = φ ( u ) {\displaystyle \mu (uK)=\varphi (u)} , and such map is also well-defined. For rings, modules, and vector spaces, one can define the respective quotient algebras via the underlying additive group structure, with cosets represented as x + K {\displaystyle x+K} . Ring multiplication can be defined on the quotient algebra the same way as in the group (and be well-defined). For a ring R {\displaystyle R} (possibly a field when describing vector spaces) and a module homomorphism φ : M → N {\displaystyle \varphi :M\to N} with kernel K = ker φ {\displaystyle K=\ker \varphi } , one can define scalar multiplication on G / K {\displaystyle G/K} by r ( x + K ) = r x + K {\displaystyle r(x+K)=rx+K} for r ∈ R {\displaystyle r\in R} and x ∈ M {\displaystyle x\in M} , which will also be well-defined. == Kernel structures == The structure of kernels allows for the building of quotient algebras from structures satisfying the properties of kernels. Any subgroup N {\displaystyle N} of a group G {\displaystyle G} can construct a quotient G / N {\displaystyle G/N} by the set of all cosets of N {\displaystyle N} in G {\displaystyle G} . The natural way to turn this into a group, similar to the treatment for the quotient by a kernel, is to define an operation on (left) cosets by u N ⋅ v N = ( u v ) N {\displaystyle uN\cdot vN=(uv)N} , however this operation is well defined if and only if the subgroup N {\displaystyle N} is closed under conjugation under G {\displaystyle G} , that is, if g ∈ G {\displaystyle g\in G} and n ∈ N {\displaystyle n\in N} , then g n g − 1 ∈ N {\displaystyle gng^{-1}\in N} . Furthermore, the operation being well defined is sufficient for the quotient to be a group. Subgroups satisfying this property are called normal subgroups. Every kernel of a group is a normal subgroup, and for a given normal subgroup N {\displaystyle N} of a group G {\displaystyle G} , the natural projection π ( g ) = g N {\displaystyle \pi (g)=gN} is a homomorphism with ker π = N {\displaystyle \ker \pi =N} , so the normal subgroups are precisely the subgroups which are kernels. The closure under conjugation, however, gives an "internal" criterion for when a subgroup is a kernel for some homomorphism. For a ring R {\displaystyle R} , treating it as a group, one can take a quotient group via an arbitrary subgroup I {\displaystyle I} of the ring, which will be normal due to the ring's additive group being abelian. To define multiplication on R / I {\displaystyle R/I} , the multiplication of cosets, defined as ( r + I ) ( s + I ) = r s + I {\displaystyle (r+I)(s+I)=rs+I} needs to be well-defined. Taking representative r + α {\displaystyle r+\alpha } and s + β {\displaystyle s+\beta } of r + I {\displaystyle r+I} and s + I {\displaystyle s+I} respectively, for r , s ∈ R {\displaystyle r,s\in R} and α , β ∈ I {\displaystyle \alpha ,\beta \in I} , yields: ( r + α ) ( s + β ) + I = r s + I {\displaystyle (r+\alpha )(s+\beta )+I=rs+I} Setting r = s = 0 {\displaystyle r=s=0} implies that I {\displaystyle I} is closed under multiplication, while setting α = s = 0 {\displaystyle \alpha =s=0} shows that r β ∈ I {\displaystyle r\beta \in I} , that is, I {\displaystyle I} is closed under arbitrary multiplication by elements on the left. Similarly, taking r = β = 0 {\displaystyle r=\beta =0} implies that I {\displaystyle I} is also closed under multiplication by arbitrary elements on the right. Any subgroup of R {\displaystyle R} that is closed under multiplication by any element of the ring is called an ideal. Analogously to normal subgroups, the ideals of a ring are precisely the kernels of homomorphisms. == Exact sequence == Kernels are used to define exact sequences of homomorphisms for groups and modules. If A, B, and C are modules, then a pair of homomorphisms ψ : A → B , φ : B → C {\displaystyle \psi :A\to B,\varphi :B\to C} is said to be exact if image ψ = ker φ {\displaystyle {\text{image }}\psi =\ker \varphi } . An exact sequence is then a sequence of modules and homomorphism ⋯ → X n − 1 → X n → X n + 1 → ⋯ {\displaystyle \cdots \to X_{n-1}\to X_{n}\to X_{n+1}\to \cdots } where each adjacent pair of homomorphisms is exact. == Universal algebra == All the above cases may be unified and generalized in universal algebra. Let A and B be algebraic structures of a given type and let f be a homomorphism of that type from A to B. Then the kernel of f is the subset of the direct product A × A consisting of all those ordered pairs of elements of A whose components are both mapped by f to the same element in B. The kernel is usually denoted ker f (or a variation). In symbols: ker f = { ( a , b ) ∈ A × A : f ( a ) = f ( b ) } . {\displaystyle \operatorname {ker} f=\left\{\left(a,b\right)\in A\times A:f(a)=f\left(b\right)\right\}{\mbox{.}}} The homomorphism f is injective if and only if its kernel is exactly the diagonal set {(a, a) : a ∈ A}, which is always at least contained inside the kernel. It is easy to see that ker f is an equivalence relation on A, and in fact a congruence relation. Thus, it makes sense to speak of the quotient algebra A / (ker f). The first isomorphism theorem in general universal algebra states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B). Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely set-theoretic concept. For more on this general concept, outside of abstract algebra, see kernel of a function. == Algebras with nonalgebraic structure == Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations. For example, one may consider topological groups or topological vector spaces, which are equipped with a topology. In this case, we would expect the homomorphism f to preserve this additional structure; in the topological examples, we would want f to be a continuous map. The process may run into a snag with the quotient algebras, which may not be well-behaved. In the topological examples, we can avoid problems by requiring that topological algebraic structures be Hausdorff (as is usually done); then the kernel (however it is constructed) will be a closed set and the quotient space will work fine (and also be Hausdorff). == Kernels in category theory == The notion of kernel in category theory is a generalization of the kernels of abelian algebras; see Kernel (category theory). The categorical generalization of the kernel as a congruence relation is the kernel pair. (There is also the notion of difference kernel, or binary equalizer.) == See also == Kernel (linear algebra) Zero set == References ==
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Wikipedia:Kernel (linear algebra)#0
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In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear map L : V → W between two vector spaces V and W, the kernel of L is the vector space of all elements v of V such that L(v) = 0, where 0 denotes the zero vector in W, or more symbolically: ker ( L ) = { v ∈ V ∣ L ( v ) = 0 } = L − 1 ( 0 ) . {\displaystyle \ker(L)=\left\{\mathbf {v} \in V\mid L(\mathbf {v} )=\mathbf {0} \right\}=L^{-1}(\mathbf {0} ).} == Properties == The kernel of L is a linear subspace of the domain V. In the linear map L : V → W , {\displaystyle L:V\to W,} two elements of V have the same image in W if and only if their difference lies in the kernel of L, that is, L ( v 1 ) = L ( v 2 ) if and only if L ( v 1 − v 2 ) = 0 . {\displaystyle L\left(\mathbf {v} _{1}\right)=L\left(\mathbf {v} _{2}\right)\quad {\text{ if and only if }}\quad L\left(\mathbf {v} _{1}-\mathbf {v} _{2}\right)=\mathbf {0} .} From this, it follows by the first isomorphism theorem that the image of L is isomorphic to the quotient of V by the kernel: im ( L ) ≅ V / ker ( L ) . {\displaystyle \operatorname {im} (L)\cong V/\ker(L).} In the case where V is finite-dimensional, this implies the rank–nullity theorem: dim ( ker L ) + dim ( im L ) = dim ( V ) . {\displaystyle \dim(\ker L)+\dim(\operatorname {im} L)=\dim(V).} where the term rank refers to the dimension of the image of L, dim ( im L ) , {\displaystyle \dim(\operatorname {im} L),} while nullity refers to the dimension of the kernel of L, dim ( ker L ) . {\displaystyle \dim(\ker L).} That is, Rank ( L ) = dim ( im L ) and Nullity ( L ) = dim ( ker L ) , {\displaystyle \operatorname {Rank} (L)=\dim(\operatorname {im} L)\qquad {\text{ and }}\qquad \operatorname {Nullity} (L)=\dim(\ker L),} so that the rank–nullity theorem can be restated as Rank ( L ) + Nullity ( L ) = dim ( domain L ) . {\displaystyle \operatorname {Rank} (L)+\operatorname {Nullity} (L)=\dim \left(\operatorname {domain} L\right).} When V is an inner product space, the quotient V / ker ( L ) {\displaystyle V/\ker(L)} can be identified with the orthogonal complement in V of ker ( L ) {\displaystyle \ker(L)} . This is the generalization to linear operators of the row space, or coimage, of a matrix. == Generalization to modules == The notion of kernel also makes sense for homomorphisms of modules, which are generalizations of vector spaces where the scalars are elements of a ring, rather than a field. The domain of the mapping is a module, with the kernel constituting a submodule. Here, the concepts of rank and nullity do not necessarily apply. == In functional analysis == If V and W are topological vector spaces such that W is finite-dimensional, then a linear operator L: V → W is continuous if and only if the kernel of L is a closed subspace of V. == Representation as matrix multiplication == Consider a linear map represented as a m × n matrix A with coefficients in a field K (typically R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } ), that is operating on column vectors x with n components over K. The kernel of this linear map is the set of solutions to the equation Ax = 0, where 0 is understood as the zero vector. The dimension of the kernel of A is called the nullity of A. In set-builder notation, N ( A ) = Null ( A ) = ker ( A ) = { x ∈ K n ∣ A x = 0 } . {\displaystyle \operatorname {N} (A)=\operatorname {Null} (A)=\operatorname {ker} (A)=\left\{\mathbf {x} \in K^{n}\mid A\mathbf {x} =\mathbf {0} \right\}.} The matrix equation is equivalent to a homogeneous system of linear equations: A x = 0 ⇔ a 11 x 1 + a 12 x 2 + ⋯ + a 1 n x n = 0 a 21 x 1 + a 22 x 2 + ⋯ + a 2 n x n = 0 ⋮ a m 1 x 1 + a m 2 x 2 + ⋯ + a m n x n = 0 . {\displaystyle A\mathbf {x} =\mathbf {0} \;\;\Leftrightarrow \;\;{\begin{alignedat}{7}a_{11}x_{1}&&\;+\;&&a_{12}x_{2}&&\;+\;\cdots \;+\;&&a_{1n}x_{n}&&\;=\;&&&0\\a_{21}x_{1}&&\;+\;&&a_{22}x_{2}&&\;+\;\cdots \;+\;&&a_{2n}x_{n}&&\;=\;&&&0\\&&&&&&&&&&\vdots \ \;&&&\\a_{m1}x_{1}&&\;+\;&&a_{m2}x_{2}&&\;+\;\cdots \;+\;&&a_{mn}x_{n}&&\;=\;&&&0{\text{.}}\\\end{alignedat}}} Thus the kernel of A is the same as the solution set to the above homogeneous equations. === Subspace properties === The kernel of a m × n matrix A over a field K is a linear subspace of Kn. That is, the kernel of A, the set Null(A), has the following three properties: Null(A) always contains the zero vector, since A0 = 0. If x ∈ Null(A) and y ∈ Null(A), then x + y ∈ Null(A). This follows from the distributivity of matrix multiplication over addition. If x ∈ Null(A) and c is a scalar c ∈ K, then cx ∈ Null(A), since A(cx) = c(Ax) = c0 = 0. === The row space of a matrix === The product Ax can be written in terms of the dot product of vectors as follows: A x = [ a 1 ⋅ x a 2 ⋅ x ⋮ a m ⋅ x ] . {\displaystyle A\mathbf {x} ={\begin{bmatrix}\mathbf {a} _{1}\cdot \mathbf {x} \\\mathbf {a} _{2}\cdot \mathbf {x} \\\vdots \\\mathbf {a} _{m}\cdot \mathbf {x} \end{bmatrix}}.} Here, a1, ... , am denote the rows of the matrix A. It follows that x is in the kernel of A, if and only if x is orthogonal (or perpendicular) to each of the row vectors of A (since orthogonality is defined as having a dot product of 0). The row space, or coimage, of a matrix A is the span of the row vectors of A. By the above reasoning, the kernel of A is the orthogonal complement to the row space. That is, a vector x lies in the kernel of A, if and only if it is perpendicular to every vector in the row space of A. The dimension of the row space of A is called the rank of A, and the dimension of the kernel of A is called the nullity of A. These quantities are related by the rank–nullity theorem rank ( A ) + nullity ( A ) = n . {\displaystyle \operatorname {rank} (A)+\operatorname {nullity} (A)=n.} === Left null space === The left null space, or cokernel, of a matrix A consists of all column vectors x such that xTA = 0T, where T denotes the transpose of a matrix. The left null space of A is the same as the kernel of AT. The left null space of A is the orthogonal complement to the column space of A, and is dual to the cokernel of the associated linear transformation. The kernel, the row space, the column space, and the left null space of A are the four fundamental subspaces associated with the matrix A. === Nonhomogeneous systems of linear equations === The kernel also plays a role in the solution to a nonhomogeneous system of linear equations: A x = b or a 11 x 1 + a 12 x 2 + ⋯ + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + ⋯ + a 2 n x n = b 2 ⋮ a m 1 x 1 + a m 2 x 2 + ⋯ + a m n x n = b m {\displaystyle A\mathbf {x} =\mathbf {b} \quad {\text{or}}\quad {\begin{alignedat}{7}a_{11}x_{1}&&\;+\;&&a_{12}x_{2}&&\;+\;\cdots \;+\;&&a_{1n}x_{n}&&\;=\;&&&b_{1}\\a_{21}x_{1}&&\;+\;&&a_{22}x_{2}&&\;+\;\cdots \;+\;&&a_{2n}x_{n}&&\;=\;&&&b_{2}\\&&&&&&&&&&\vdots \ \;&&&\\a_{m1}x_{1}&&\;+\;&&a_{m2}x_{2}&&\;+\;\cdots \;+\;&&a_{mn}x_{n}&&\;=\;&&&b_{m}\\\end{alignedat}}} If u and v are two possible solutions to the above equation, then A ( u − v ) = A u − A v = b − b = 0 {\displaystyle A(\mathbf {u} -\mathbf {v} )=A\mathbf {u} -A\mathbf {v} =\mathbf {b} -\mathbf {b} =\mathbf {0} } Thus, the difference of any two solutions to the equation Ax = b lies in the kernel of A. It follows that any solution to the equation Ax = b can be expressed as the sum of a fixed solution v and an arbitrary element of the kernel. That is, the solution set to the equation Ax = b is { v + x ∣ A v = b ∧ x ∈ Null ( A ) } , {\displaystyle \left\{\mathbf {v} +\mathbf {x} \mid A\mathbf {v} =\mathbf {b} \land \mathbf {x} \in \operatorname {Null} (A)\right\},} Geometrically, this says that the solution set to Ax = b is the translation of the kernel of A by the vector v. See also Fredholm alternative and flat (geometry). == Illustration == The following is a simple illustration of the computation of the kernel of a matrix (see § Computation by Gaussian elimination, below for methods better suited to more complex calculations). The illustration also touches on the row space and its relation to the kernel. Consider the matrix A = [ 2 3 5 − 4 2 3 ] . {\displaystyle A={\begin{bmatrix}2&3&5\\-4&2&3\end{bmatrix}}.} The kernel of this matrix consists of all vectors (x, y, z) ∈ R3 for which [ 2 3 5 − 4 2 3 ] [ x y z ] = [ 0 0 ] , {\displaystyle {\begin{bmatrix}2&3&5\\-4&2&3\end{bmatrix}}{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}},} which can be expressed as a homogeneous system of linear equations involving x, y, and z: 2 x + 3 y + 5 z = 0 , − 4 x + 2 y + 3 z = 0. {\displaystyle {\begin{aligned}2x+3y+5z&=0,\\-4x+2y+3z&=0.\end{aligned}}} The same linear equations can also be written in matrix form as: [ 2 3 5 0 − 4 2 3 0 ] . {\displaystyle \left[{\begin{array}{ccc|c}2&3&5&0\\-4&2&3&0\end{array}}\right].} Through Gauss–Jordan elimination, the matrix can be reduced to: [ 1 0 1 / 16 0 0 1 13 / 8 0 ] . {\displaystyle \left[{\begin{array}{ccc|c}1&0&1/16&0\\0&1&13/8&0\end{array}}\right].} Rewriting the matrix in equation form yields: x = − 1 16 z y = − 13 8 z . {\displaystyle {\begin{aligned}x&=-{\frac {1}{16}}z\\y&=-{\frac {13}{8}}z.\end{aligned}}} The elements of the kernel can be further expressed in parametric vector form, as follows: [ x y z ] = c [ − 1 / 16 − 13 / 8 1 ] ( where c ∈ R ) {\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}=c{\begin{bmatrix}-1/16\\-13/8\\1\end{bmatrix}}\quad ({\text{where }}c\in \mathbb {R} )} Since c is a free variable ranging over all real numbers, this can be expressed equally well as: [ x y z ] = c [ − 1 − 26 16 ] . {\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}=c{\begin{bmatrix}-1\\-26\\16\end{bmatrix}}.} The kernel of A is precisely the solution set to these equations (in this case, a line through the origin in R3). Here, the vector (−1,−26,16)T constitutes a basis of the kernel of A. The nullity of A is therefore 1, as it is spanned by a single vector. The following dot products are zero: [ 2 3 5 ] [ − 1 − 26 16 ] = 0 a n d [ − 4 2 3 ] [ − 1 − 26 16 ] = 0 , {\displaystyle {\begin{bmatrix}2&3&5\end{bmatrix}}{\begin{bmatrix}-1\\-26\\16\end{bmatrix}}=0\quad \mathrm {and} \quad {\begin{bmatrix}-4&2&3\end{bmatrix}}{\begin{bmatrix}-1\\-26\\16\end{bmatrix}}=0,} which illustrates that vectors in the kernel of A are orthogonal to each of the row vectors of A. These two (linearly independent) row vectors span the row space of A—a plane orthogonal to the vector (−1,−26,16)T. With the rank 2 of A, the nullity 1 of A, and the dimension 3 of A, we have an illustration of the rank-nullity theorem. == Examples == If L: Rm → Rn, then the kernel of L is the solution set to a homogeneous system of linear equations. As in the above illustration, if L is the operator: L ( x 1 , x 2 , x 3 ) = ( 2 x 1 + 3 x 2 + 5 x 3 , − 4 x 1 + 2 x 2 + 3 x 3 ) {\displaystyle L(x_{1},x_{2},x_{3})=(2x_{1}+3x_{2}+5x_{3},\;-4x_{1}+2x_{2}+3x_{3})} then the kernel of L is the set of solutions to the equations 2 x 1 + 3 x 2 + 5 x 3 = 0 − 4 x 1 + 2 x 2 + 3 x 3 = 0 {\displaystyle {\begin{alignedat}{7}2x_{1}&\;+\;&3x_{2}&\;+\;&5x_{3}&\;=\;&0\\-4x_{1}&\;+\;&2x_{2}&\;+\;&3x_{3}&\;=\;&0\end{alignedat}}} Let C[0,1] denote the vector space of all continuous real-valued functions on the interval [0,1], and define L: C[0,1] → R by the rule L ( f ) = f ( 0.3 ) . {\displaystyle L(f)=f(0.3).} Then the kernel of L consists of all functions f ∈ C[0,1] for which f(0.3) = 0. Let C∞(R) be the vector space of all infinitely differentiable functions R → R, and let D: C∞(R) → C∞(R) be the differentiation operator: D ( f ) = d f d x . {\displaystyle D(f)={\frac {df}{dx}}.} Then the kernel of D consists of all functions in C∞(R) whose derivatives are zero, i.e. the set of all constant functions. Let R∞ be the direct product of infinitely many copies of R, and let s: R∞ → R∞ be the shift operator s ( x 1 , x 2 , x 3 , x 4 , … ) = ( x 2 , x 3 , x 4 , … ) . {\displaystyle s(x_{1},x_{2},x_{3},x_{4},\ldots )=(x_{2},x_{3},x_{4},\ldots ).} Then the kernel of s is the one-dimensional subspace consisting of all vectors (x1, 0, 0, 0, ...). If V is an inner product space and W is a subspace, the kernel of the orthogonal projection V → W is the orthogonal complement to W in V. == Computation by Gaussian elimination == A basis of the kernel of a matrix may be computed by Gaussian elimination. For this purpose, given an m × n matrix A, we construct first the row augmented matrix [ A I ] , {\displaystyle {\begin{bmatrix}A\\\hline I\end{bmatrix}},} where I is the n × n identity matrix. Computing its column echelon form by Gaussian elimination (or any other suitable method), we get a matrix [ B C ] . {\displaystyle {\begin{bmatrix}B\\\hline C\end{bmatrix}}.} A basis of the kernel of A consists in the non-zero columns of C such that the corresponding column of B is a zero column. In fact, the computation may be stopped as soon as the upper matrix is in column echelon form: the remainder of the computation consists in changing the basis of the vector space generated by the columns whose upper part is zero. For example, suppose that A = [ 1 0 − 3 0 2 − 8 0 1 5 0 − 1 4 0 0 0 1 7 − 9 0 0 0 0 0 0 ] . {\displaystyle A={\begin{bmatrix}1&0&-3&0&2&-8\\0&1&5&0&-1&4\\0&0&0&1&7&-9\\0&0&0&0&0&0\end{bmatrix}}.} Then [ A I ] = [ 1 0 − 3 0 2 − 8 0 1 5 0 − 1 4 0 0 0 1 7 − 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ] . {\displaystyle {\begin{bmatrix}A\\\hline I\end{bmatrix}}={\begin{bmatrix}1&0&-3&0&2&-8\\0&1&5&0&-1&4\\0&0&0&1&7&-9\\0&0&0&0&0&0\\\hline 1&0&0&0&0&0\\0&1&0&0&0&0\\0&0&1&0&0&0\\0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1\end{bmatrix}}.} Putting the upper part in column echelon form by column operations on the whole matrix gives [ B C ] = [ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 3 − 2 8 0 1 0 − 5 1 − 4 0 0 0 1 0 0 0 0 1 0 − 7 9 0 0 0 0 1 0 0 0 0 0 0 1 ] . {\displaystyle {\begin{bmatrix}B\\\hline C\end{bmatrix}}={\begin{bmatrix}1&0&0&0&0&0\\0&1&0&0&0&0\\0&0&1&0&0&0\\0&0&0&0&0&0\\\hline 1&0&0&3&-2&8\\0&1&0&-5&1&-4\\0&0&0&1&0&0\\0&0&1&0&-7&9\\0&0&0&0&1&0\\0&0&0&0&0&1\end{bmatrix}}.} The last three columns of B are zero columns. Therefore, the three last vectors of C, [ 3 − 5 1 0 0 0 ] , [ − 2 1 0 − 7 1 0 ] , [ 8 − 4 0 9 0 1 ] {\displaystyle \left[\!\!{\begin{array}{r}3\\-5\\1\\0\\0\\0\end{array}}\right],\;\left[\!\!{\begin{array}{r}-2\\1\\0\\-7\\1\\0\end{array}}\right],\;\left[\!\!{\begin{array}{r}8\\-4\\0\\9\\0\\1\end{array}}\right]} are a basis of the kernel of A. Proof that the method computes the kernel: Since column operations correspond to post-multiplication by invertible matrices, the fact that [ A I ] {\displaystyle {\begin{bmatrix}A\\\hline I\end{bmatrix}}} reduces to [ B C ] {\displaystyle {\begin{bmatrix}B\\\hline C\end{bmatrix}}} means that there exists an invertible matrix P {\displaystyle P} such that [ A I ] P = [ B C ] , {\displaystyle {\begin{bmatrix}A\\\hline I\end{bmatrix}}P={\begin{bmatrix}B\\\hline C\end{bmatrix}},} with B {\displaystyle B} in column echelon form. Thus A P = B {\displaystyle AP=B} , I P = C {\displaystyle IP=C} , and A C = B {\displaystyle AC=B} . A column vector v {\displaystyle \mathbf {v} } belongs to the kernel of A {\displaystyle A} (that is A v = 0 {\displaystyle A\mathbf {v} =\mathbf {0} } ) if and only if B w = 0 , {\displaystyle B\mathbf {w} =\mathbf {0} ,} where w = P − 1 v = C − 1 v {\displaystyle \mathbf {w} =P^{-1}\mathbf {v} =C^{-1}\mathbf {v} } . As B {\displaystyle B} is in column echelon form, B w = 0 {\displaystyle B\mathbf {w} =\mathbf {0} } , if and only if the nonzero entries of w {\displaystyle \mathbf {w} } correspond to the zero columns of B {\displaystyle B} . By multiplying by C {\displaystyle C} , one may deduce that this is the case if and only if v = C w {\displaystyle \mathbf {v} =C\mathbf {w} } is a linear combination of the corresponding columns of C {\displaystyle C} . == Numerical computation == The problem of computing the kernel on a computer depends on the nature of the coefficients. === Exact coefficients === If the coefficients of the matrix are exactly given numbers, the column echelon form of the matrix may be computed with Bareiss algorithm more efficiently than with Gaussian elimination. It is even more efficient to use modular arithmetic and Chinese remainder theorem, which reduces the problem to several similar ones over finite fields (this avoids the overhead induced by the non-linearity of the computational complexity of integer multiplication). For coefficients in a finite field, Gaussian elimination works well, but for the large matrices that occur in cryptography and Gröbner basis computation, better algorithms are known, which have roughly the same computational complexity, but are faster and behave better with modern computer hardware. === Floating point computation === For matrices whose entries are floating-point numbers, the problem of computing the kernel makes sense only for matrices such that the number of rows is equal to their rank: because of the rounding errors, a floating-point matrix has almost always a full rank, even when it is an approximation of a matrix of a much smaller rank. Even for a full-rank matrix, it is possible to compute its kernel only if it is well conditioned, i.e. it has a low condition number. Even for a well conditioned full rank matrix, Gaussian elimination does not behave correctly: it introduces rounding errors that are too large for getting a significant result. As the computation of the kernel of a matrix is a special instance of solving a homogeneous system of linear equations, the kernel may be computed with any of the various algorithms designed to solve homogeneous systems. A state of the art software for this purpose is the Lapack library. == See also == == Notes and references == == Bibliography == == External links == "Kernel of a matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Khan Academy, Introduction to the Null Space of a Matrix
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Wikipedia:Kernel (set theory)#0
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In set theory, the kernel of a function f {\displaystyle f} (or equivalence kernel) may be taken to be either the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function f {\displaystyle f} can tell", or the corresponding partition of the domain. An unrelated notion is that of the kernel of a non-empty family of sets B , {\displaystyle {\mathcal {B}},} which by definition is the intersection of all its elements: ker B = ⋂ B ∈ B B . {\displaystyle \ker {\mathcal {B}}~=~\bigcap _{B\in {\mathcal {B}}}\,B.} This definition is used in the theory of filters to classify them as being free or principal. == Definition == Kernel of a function For the formal definition, let f : X → Y {\displaystyle f:X\to Y} be a function between two sets. Elements x 1 , x 2 ∈ X {\displaystyle x_{1},x_{2}\in X} are equivalent if f ( x 1 ) {\displaystyle f\left(x_{1}\right)} and f ( x 2 ) {\displaystyle f\left(x_{2}\right)} are equal, that is, are the same element of Y . {\displaystyle Y.} The kernel of f {\displaystyle f} is the equivalence relation thus defined. Kernel of a family of sets The kernel of a family B ≠ ∅ {\displaystyle {\mathcal {B}}\neq \varnothing } of sets is ker B := ⋂ B ∈ B B . {\displaystyle \ker {\mathcal {B}}~:=~\bigcap _{B\in {\mathcal {B}}}B.} The kernel of B {\displaystyle {\mathcal {B}}} is also sometimes denoted by ∩ B . {\displaystyle \cap {\mathcal {B}}.} The kernel of the empty set, ker ∅ , {\displaystyle \ker \varnothing ,} is typically left undefined. A family is called fixed and is said to have non-empty intersection if its kernel is not empty. A family is said to be free if it is not fixed; that is, if its kernel is the empty set. == Quotients == Like any equivalence relation, the kernel can be modded out to form a quotient set, and the quotient set is the partition: { { w ∈ X : f ( x ) = f ( w ) } : x ∈ X } = { f − 1 ( y ) : y ∈ f ( X ) } . {\displaystyle \left\{\,\{w\in X:f(x)=f(w)\}~:~x\in X\,\right\}~=~\left\{f^{-1}(y)~:~y\in f(X)\right\}.} This quotient set X / = f {\displaystyle X/=_{f}} is called the coimage of the function f , {\displaystyle f,} and denoted coim f {\displaystyle \operatorname {coim} f} (or a variation). The coimage is naturally isomorphic (in the set-theoretic sense of a bijection) to the image, im f ; {\displaystyle \operatorname {im} f;} specifically, the equivalence class of x {\displaystyle x} in X {\displaystyle X} (which is an element of coim f {\displaystyle \operatorname {coim} f} ) corresponds to f ( x ) {\displaystyle f(x)} in Y {\displaystyle Y} (which is an element of im f {\displaystyle \operatorname {im} f} ). == As a subset of the Cartesian product == Like any binary relation, the kernel of a function may be thought of as a subset of the Cartesian product X × X . {\displaystyle X\times X.} In this guise, the kernel may be denoted ker f {\displaystyle \ker f} (or a variation) and may be defined symbolically as ker f := { ( x , x ′ ) : f ( x ) = f ( x ′ ) } . {\displaystyle \ker f:=\{(x,x'):f(x)=f(x')\}.} The study of the properties of this subset can shed light on f . {\displaystyle f.} == Algebraic structures == If X {\displaystyle X} and Y {\displaystyle Y} are algebraic structures of some fixed type (such as groups, rings, or vector spaces), and if the function f : X → Y {\displaystyle f:X\to Y} is a homomorphism, then ker f {\displaystyle \ker f} is a congruence relation (that is an equivalence relation that is compatible with the algebraic structure), and the coimage of f {\displaystyle f} is a quotient of X . {\displaystyle X.} The bijection between the coimage and the image of f {\displaystyle f} is an isomorphism in the algebraic sense; this is the most general form of the first isomorphism theorem. == In topology == If f : X → Y {\displaystyle f:X\to Y} is a continuous function between two topological spaces then the topological properties of ker f {\displaystyle \ker f} can shed light on the spaces X {\displaystyle X} and Y . {\displaystyle Y.} For example, if Y {\displaystyle Y} is a Hausdorff space then ker f {\displaystyle \ker f} must be a closed set. Conversely, if X {\displaystyle X} is a Hausdorff space and ker f {\displaystyle \ker f} is a closed set, then the coimage of f , {\displaystyle f,} if given the quotient space topology, must also be a Hausdorff space. A space is compact if and only if the kernel of every family of closed subsets having the finite intersection property (FIP) is non-empty; said differently, a space is compact if and only if every family of closed subsets with F.I.P. is fixed. == See also == Filter (set theory) – Family of sets representing "large" sets == References == == Bibliography == Awodey, Steve (2010) [2006]. Category Theory. Oxford Logic Guides. Vol. 49 (2nd ed.). Oxford University Press. ISBN 978-0-19-923718-0. Dolecki, Szymon; Mynard, Frédéric (2016). Convergence Foundations Of Topology. New Jersey: World Scientific Publishing Company. ISBN 978-981-4571-52-4. OCLC 945169917.
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Wikipedia:Kerstin Jordaan#0
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Kerstin Heidrun Jordaan is a South African mathematician whose research interests include special functions and orthogonal polynomials. She is a professor in the Department of Decision Sciences at the University of South Africa, the executive director of the South African Mathematics Foundation, and the former president of the South African Mathematical Society. == Education and career == Going into her university studies, Jordaan considered medicine and psychology before ending up in mathematics. After earning a bachelor's degree in mathematics and geography at the University of Pretoria, Jordaan earned a second bachelor's degree in mathematics education at the University of the Witwatersrand, and became a secondary school mathematics teacher in Pretoria. She earned a master's degree from the University of Pretoria, before taking a break from academia to raise a family. After doing so, she returned to graduate school for a PhD from the University of the Witwatersrand. Her 2001 dissertation, Zeros of general hypergeometric polynomials, was supervised by Kathy Driver. She was a member of the academic staff at the University of Pretoria until moving to the University of South Africa as a full professor in 2017. She served as president of the South African Mathematical Society from 2016 to 2019. == Recognition == Jordaan is a member of the Academy of Science of South Africa. == References == == External links == Kerstin Jordaan publications indexed by Google Scholar
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Wikipedia:Keti Tenenblat#0
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Keti Tenenblat (born 27 November 1944) is a Turkish-Brazilian mathematician working on Riemannian geometry, the applications of differential geometry to partial differential equations, and Finsler geometry. Together with Chuu-Lian Terng, she generalized Backlund theorem to higher dimensions. == Education == Tenenblat was born on 27 November 1944 in İzmir, Turkey, where she attended elementary and junior high school at an Italian school. In 1957, her family emigrated to Brazil. In Rio de Janeiro, she graduated from high school at Bennett College and joined the National Faculty of Philosophy at the University of Brazil (today UFRJ), in the Mathematics Degree. From 1964 to 1968, she taught mathematics at a secondary school in Rio. She completed her university course in 1967 and began her higher education activities at the Institute of Mathematics of UFRJ in 1968. Between 08/1968 to 07/1969, she attended a master's degree in mathematics at the University of Michigan, USA, while accompanying her husband who was study abroad. Upon returning to Brazil, she returned to teaching at UFRJ and began a doctoral program at IMPA. She defended her doctoral dissertation entitled "An estimate for the length of closed geodesics in Riemannian varieties" in 1972, under the direction of Manfredo P. do Carmo. == Career == From 1973 she joined the faculty of the University of Brasilia (UnB) where she became a Full Professor in 1989. From 1975 to 1978 she pursued a postdoctoral position at the Department of Mathematics at the University of California, Berkeley. During this period, she developed her research under the influence of S. S. Chern and became interested in studying the interaction between differential geometry and differential equations. After 1978, her visits abroad were short-lived. She was a visiting professor at Yale University, MSRI Berkeley, Institute of Theoretical Physics, Santa Barbara, IMA Minnesota, University of Montreal, McGill University, CRM Montreal, Nankai Institute and Fudan Univ. China. She is a recipient of Brazil's National Order of Scientific Merit in Mathematics, Emeritus Professor at the University of Brasília, and was President of the Brazilian Mathematical Society in 1989–1991. She has been a member of the Brazilian Academy of Sciences since 1991. She is also the author of the books Introdução à geometria diferencial (1988), and Transformações de superfícies e aplicações (1981). == Personal life == In 1965, she married Moyses Tenenblat, an engineer graduated from the National School of Engineering. Children Dany (1970), Nitza (1973), Leo (1975) and grandchildren Gabriel (1995), Yuri (1998), Luisa (2000), Clara (2005), Milla (2007), Aylou (2009), and Luca (2014) were born of this marriage. == Selected publications == Tenenblat, Keti; Terng, Chuu Lian (1980), "Bäcklund's theorem for n-dimensional submanifolds of R2n − 1", Annals of Mathematics, Second Series, 111 (3): 477–490, doi:10.2307/1971105, JSTOR 1971105, MR 0577133 Chern, S. S.; Tenenblat, K. (1986), "Pseudospherical surfaces and evolution equations", Studies in Applied Mathematics, 74 (1): 55–83, doi:10.1002/sapm198674155, MR 0827492 Kamran, Niky; Tenenblat, Keti (1996), "Laplace transformation in higher dimensions", Duke Mathematical Journal, 84 (1): 237–266, doi:10.1215/S0012-7094-96-08409-4, MR 1394755 == References ==
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Wikipedia:Kevin M. Short#0
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Kevin M. Short (born June 23, 1963) is an American mathematician and entrepreneur. He is a professor of Applied Mathematics at the University of New Hampshire. He is also co-founder and Chief Technology Officer (CTO) at Setem Technologies, in Newbury, Massachusetts. Since 1994, when he began at UNH, Short's academic research and work has continually focused on tying together nonlinear chaos theory and signal processing so that nonlinearity can play a major role in the future of technology development. == Education == Short grew up and attended high school in Suffern, New York. He completed his undergraduate work at the University of Rochester in 1985, receiving both a B.S. in Physics and a B.A. in geology. He then attended the Imperial College of London on a Marshall Scholarship, where he earned his PhD in Theoretical Physics. In 1994, Short joined the University of New Hampshire's Department of Mathematics as an assistant professor. At UNH, Short presently holds the position of University Professor. == Research == In 1996, Short developed and patented a technology called CCT, or Chaotic Compression Technology. Claimed to be "fundamentally different" from existing technology, CCT used nonlinear mathematical equations to produce complex waveforms. These waveforms were then transmitted through the Internet or any communications device, requiring far less bandwidth to transmit the same amount of data than the existing technology. CCT was widely used whenever music or ringtones were downloaded to a cell phone device. == Business ventures == === Chaoticom === In 2001, Short founded Chaoticom (later renamed Grove Mobile), where he served as the Director and Chief Technology Officer. Chaoticom was the first ever University spin-off company at UNH, and it sought to commercialize Short's research at the university and his patented Chaotic Compression Technology (CCT). Chaoticom applied CCT towards a direct to cell phone mobile music download service, and many innovations within the company led to patenting. The company was acquired by LiveWire Mobile Inc. in March 2008. === Setem Technologies === In 2012, Short co-founded Setem Technologies, where he continues to serve as Chief Technical Officer. Another UNH spinoff company, Setem seeks to use Short's mathematical theorems and signal separation technology to enhance the voice clarity and audio signals in today's voice and speech recognition products (i.e.-cell phones, headsets, hearing aids, voice-activated electronics). == Grammy Award == Short was instrumental in using his Chaotic Compression Technology to restore a bootleg wire recording of a Woody Guthrie concert that is the only known recording of the folk singer performing before a live audience. His work with the project helped earn him and a small team of producers and engineers the 2008 Grammy Award for Best Historical Album: The Live Wire - Woody Guthrie In Performance 1949. Singer-songwriter Nora Guthrie and Jorge Arévalo Mateus were the compilation producers, while Jamie Howarth, Steve Rosenthal, Warren Russell-Smith and Dr. Kevin Short were mastering engineers. == Awards and honors == National Academy of Inventors Fellow (2015) Innovator of the Year (2012) Entrepreneurial Venture Creation Person of the Year (2008) Grammy Award (2008) == Selected publications == K. M. Short; K. Zarringhalam (2008), "An adaptive multiresolution image analysis using compact cupolets", Nonlinear Dynamics, 1–2 (52): 51–70 K. M. Short; R. A. Garcia (May 2006), Signal analysis using complex spectral phase evolution method, Audio Engineering Society 120th Convention K. M. Short (1997), "Detection of teleseismic events in seismic sensor data using nonlinear dynamic forecasting", International Journal of Bifurcation and Chaos, 7 (8): 1833–1845, Bibcode:1997IJBC....7.1833S, doi:10.1142/S0218127497001400 == References ==
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Wikipedia:Khachatur Khachatryan#0
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Khachatur Khachatryan (Armenian: Խաչատուր Աղավարդի Խաչատրյան), (1 July 1982, Armenia) is an Armenian scientist and mathematician. == Early life and education == Khachatur Khachatryan was born on July 1, 1982, in Yerevan, Armenia. In 1998 graduated from Phys. math. school No. 1 named after A. Shahinyan with honors. He graduated from the Faculty of Mathematics of Yerevan State University in 2004 with honors (Department of Differential Equations and Functional Analysis). In 2004, he entered graduate school (aspirant) at Yerevan State University. In 2006, in the specialized council 050 of Yerevan State University, he defended his PhD thesis on the topic “On factorization methods for solving a certain class of integral and integro-differential equations on the semi-axis.” (specialty 01.01.02 - “Differential equations”), scientific supervisor, Doctor of Phys-Math. sciences, prof. N. B. Yengibaryan, official opponents: Doctor of Phys-Math. sciences, prof. Sergeev, Armen Glebovich, candidate of Phys-Math. sciences, Associate Professor A. G. Kamalyan, Leading organization: Armenian State Pedagogical University. In 2011, in the specialized council 050 of Yerevan State University, he defended his doctoral dissertation (= habilitation) on the topic “Questions of the solvability of some nonlinear integral and integro-differential equations with non-compact operators in the critical case” (specialty 01.01.02 - “Differential equations”), official opponents: Doctor of Phys-Math. sciences, prof. Zabreiko, Pyotr Petrovich, Doctor of Phys-Math. sciences, prof. A. S. Krivosheev, Academician of the National Academy of Sciences of Armenia, Doctor of Phys-Math. sciences, prof. A. B. Nersesyan, Leading organization: Steklov Mathematical Institute of the Russian Academy of Sciences. == Career == Since 2018 Khachatryan is a full professor of Mathematics, and the scientific supervisor of seven candidate dissertations (PhD theses). He teaches courses in functional analysis, differential equations, equations of mathematical physics, calculus of variations, mathematical analysis, nonlinear operator equations, convolution type equations. Since 2005, he has been working in the Department of Methods of Mathematical Physics of the Institute of Mathematics of the National Academy of Sciences of Armenia. From 2004 to 2006 and since 2012 he teaches at Yerevan State University (YSU). In 2015-2017 taught at the Yerevan branch of Moscow State University. M. V. Lomonosov (part-time). Since 2019 he has been teaching at the Russian-Armenian Slavic University. Since 2019, the main executor of the Russian Science Foundation grant (project No. 19-11-00223) Moscow State University . From 2020 Head of the Department of Differential Equations, and from 2021, Head of the Department of Theory of Functions and Differential Equations, Yerevan State University. == Scientific interests == Differential equations Equations of mathematical physics Nonlinear analysis Integral and integro-differential equations with Hammerstein and Urysohn type operators Integral equations of convolution type Factorization of integral and integro-differential operators Boltzmann equation p-adic string theory Nonlinear pseudodifferential equations Mathematical modeling of the geographic spread of the epidemic Nonlinear parabolic equations == State awards and honorary titles == 2014 Winner of the "Efficient Researcher" competition (Natural Sciences) (Top 100) 2013 1st-class prize of the RA NAS "Best scientific work of 2013" competition, 2013 Winner of the first prize after Sergey Mergelyan for young scientists (Mathematics and Informatics) 2021 Winner of the "Efficient Researcher" competition (Natural Sciences) (Top 100) 2020 Winner of the "Efficient Researcher" competition (Natural Sciences) (Top 100) 2018 Winner of the "Efficient Researcher" competition (Natural Sciences) (Top 100) 2016 Winner of the "Efficient Researcher" competition (Natural Sciences) (Top 100) 2019 Prize of the President of the Republic of Armenia == Memberships == Deputy Chairman of Yerevan State University 050 Professional Council Member of the editorial board of the Journal of Contemporary Mathematical Analysis Member of Professional Council 058 of Yerevan State University Member of the Scientific Council of the Institute of Mathematics of the National Academy of Sciences of the Republic of Armenia Member of the Scientific Council of the Faculty of Mathematics and Mechanics of Yerevan State University Member of the Scientific Council of the Institute of Mathematics and Informatics of the Armenian-Russian University Member of the Union of Mathematicians of Armenia == Publications == He has more than 150 scientific articles in various scientific international journals. Kh.A. Khachatryan, H.S. Petrosyan, “On a class of nonlinear integral equations of the Hammerstein–Volterra type on a semiaxis”, Russian Math. (Iz. VUZ), 67:1 (2023), 64–73 H.S. Petrosyan, Kh.A. Khachatryan, “Uniqueness of the Solution of a Class of Integral Equations with Sum-Difference. Kernel and with Convex Nonlinearity on the Positive Half-Line”, Math. Notes, 113:4 (2023), 512–524 Kh.A. Khachatryan, H.S. Petrosyan, “On the nontrivial solvability of a system nonlinear integral equations on the whole line”, Izv. RAN. Ser. Mat., 87:5 (2023), 215–231 A.Kh. Khachatryan, Kh.A. Khachatryan, H.S. Petrosyan, “On nonlinear convolution-type integral equations in the theory of p-adic strings”, Theoret. and Math. Phys., 216:1 (2023), 1068–1081 Kh.A. Khachatryan, H.S. Petrosyan, S.M. Andriyan, “On the solubility of a class of two-dimensional integral equations on a quarter plane with monotone nonlinearity”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2022, 2, 19–38 Kh.A. Khachatryan, H.S. Petrosyan, “On summable solutions of a class of nonlinear integral equations on the whole line”, Izv. Math., 86:5 (2022), 980–991 A.Kh. Khachatryan, Kh.A. Khachatryan, A.Zh. Narimanyan, “Existence and uniqueness result for reaction-diffusion model of diffusive population dynamics”, Тр. ММО, 83:2 (2022), 219–239 Kh.A. Khachatryan, H.S. Petrosyan, “On a class of nonlinear integro-differential equations”, Mat. Tr., 25:1 (2022), 192–220 A.G. Sergeev, A.Kh. Khachatryan, Kh.A. Khachatryan, “Математическая модель распространения пандемии типа COVID-19”, Tr. Mosk. Mat. Obs., 83, no. 1, MCCME, M., 2022, 63–75 A.Kh. Khachatryan, Kh.A. Khachatryan, H.S. Petrosyan, “Solvability of Two-Dimensional Integral Equations with Monotone Nonlinearity”, Journal of Mathematical Sciences, New York, 257:3, (2021), 720-731 A.Kh. Khachatryan, Kh.A. Khachatryan, H.S. Petrosyan, “On Positive Bounded Solutions of One Class of Nonlinear Integral Equations with the Hammerstein–Nemytskii Operator”, Differential Equations, vol. 57, iss. 6, (2021) 768–779 A. Kh. Khachatryan, Kh. A. Khachatryan, “On solvability of one infinite system of nonlinear functional equations in the theory of epidemics”, Eurasian Math. J., 11:2 (2020), 52–64 Kh.A.Khachatryan, A.Zh.Narimanyan, A.Kh.Khachatryan, “On Mathematical Modelling of Temporial Spatial Spread of Epidemics” (France), Mathematical Modelling of Natural Phenomena, 15:6 (2020), 1-14 Kh.A. Khachatryan, H.S. Petrosyan. Solvability of a Nonlinear Problem in Open-Closed p-Adic String Theory. Differential Equations 56, 1371–1378 (2020) Kh. A. Khachatryan, “Solvability of some nonlinear boundary value problems for singular integral equations of convolution type”, Trans. Moscow Math. Soc., 81:1 (2020), 1–31 A. G. Sergeev, Kh. A. Khachatryan, “On the solvability of a class of nonlinear integral equations in the problem of a spread of an epidemic”, Trans. Moscow Math. Soc., 80 (2019), 95–11 Kh. A. Khachatryan, “On the solubility of certain classes of non-linear integral equations in p-adic string theory”, Izv. Math., 82:2 (2018), 407–427 Kh. A. Khachatryan, H. S. Petrosyan, “One initial boundary-value problem for integro-differential equation of the second order with power nonlinearity”, Russian Math. (Iz. VUZ), 62:6 (2018), 43–55 A. Kh. Khachatryan, Kh. A. Khachatryan, “Solvability of a nonlinear integral equation in dynamical string theory”, Theoret. and Math. Phys., 195:1 (2018), 529–537 A.Kh.Khachatryan, Kh. A. Khachatryan, “Solvability of a class of nonlinear pseudo-differential equations in R n”, p-adic Numbers, Ultrametric Analysis and Applications, 10:2 (2018), 90-99 A. Kh. Khachatryan, Kh. A. Khachatryan, “A one-parameter family of positive solutions of the non-linear stationary Boltzmann equation (in the framework of a modified model)”, Russian Math. Surveys, 72:3 (2017), 571–573 Kh.A.Khachatryan, “On solvability of one class nonlinear integral equations on whole line with a weak singularity at zero”, p-Adic Numbers, Ultrametric Analysis and Applications, 9:4 (2017), 292-305 A. Kh. Khachatryan, Kh. A. Khachatryan, “Some problems concerning the solvability of the nonlinear stationary Boltzmann equation in the framework of the BGK model”, Trans. Moscow Math. Soc., 77 (2016), 87–106 A. Kh. Khachatryan, Kh. A. Khachatryan, “Solvability of a nonlinear model Boltzmann equation in the problem of a plane shock wave”, Theoret. and Math. Phys., 189:2 (2016), 1609–1623 Kh. A. Khachatryan, “Positive solubility of some classes of non-linear integral equations of Hammerstein type on the semi-axis and on the whole line”, Izv, Math., 79:2 (2015), 411-430. K. A. Khachatryan, Ts. E. Terdzhyan, “On the solvability of one class of nonlinear integral equations in L1(0,+∞)L1(0,+∞)”, Siberian Adv. Math., 25:4 (2015), 268–275 A.Kh.Khachatryan, Kh.A.Khachatryan, Ts. E. Terdjyan, “On solvability of one integral equation on half line with Chebyshev polynomial nonlinearity”, P-Adic Numbers, Ultrametric Analysis, and Applications, 7:3 (2015), 228-237 A.Kh.Khachatryan, Kh.A.Khachatryan, T.H. Sardaryan, “On One Nonlinear Boundary-Value Problem in Kinetic Theory of Gases”, Journal of Mathematical Physics, Analysis, Geometry, 10:3 (2014), 320-327 A.Kh. Khachatryan, Kh. A. Khachatryan, “Qualitative difference between solutions of stationary model Boltzmann equations in the linear and nonlinear cases”, Theoret. and Math. Phys., 180:2 (2014), 990–1004 Khachatur A. Khachatryan and Mikael G. Kostanyan, “On the solvability of a nonlinear second-order integro-differential equation with sum-difference kernel on a semiaxis”, Journal of Mathematical Sciences, 181:1 (2012), 65-77 Aghavard Kh.Khachatryan, Khachatur A.Khachatryan, “Hammerstein -Nemitski type Nonlinear integral equations on half-line in space L1(R+) and Linfinity(R+)”, Acta Univ. Palacki. Olomuc., Fac.rer. nat., Mathematica, 52:1 (2013), 89-100 Kh. A. Khachatryan, “On a class of integral equations of Urysohn type with strong non-linearity”, Izv. Math., 76:1 (2012), 163–189 Kh. A. Khachatryan, “On solvability one Hammerstein–Nemitski type nonlinear integral differential equation with noncompact operator in W11(R+)W11(R+)”, St. Petersburg Math. J., 24:1 (2013), 167–183 A.Kh.Khachatryan, Kh.A.Khachatryan, “Qualitative difference between solutions for a model Boltzmann equation in the linear and nonlinear cases”, Theoretical and Mathematical Physics, 172:3 (2012), 1315-1320; A. Kh. Khachatryan, Kh. A. Khachatryan, “Qualitative difference between solutions for a model of the Boltzmann equation in the linear and nonlinear cases”, Theoret. and Math. Phys., 172:3 (2012), 1315–1320 A. Kh. Khachatryan, Kh. A. Khachatryan, “Existence and Uniqueness Theorem For a Hammerstein Nonlinear Integral Equation”, Opuscula Mathematica (POLAND), 31:3 (2011), 393–398 Kh. A. Khachatryan, “On a Class of Nonlinear Integral Equations With a Noncompact Operator”, Journal of Contemporary Mathematical Analysis, 46:2 (2011), 89–100 A. Khachatryan, Kh. Khachatryan, “On solvability of a nonlinear problem in theory of income distribution”, Eurasian Math. J., 2:2 (2011), 75–88 A. Kh. Khachatryan, Kh. A. Khachatryan, “A nonlinear integral equation of Hammerstein type with a noncompact operator”, Sb. Math., 201:4 (2010), 595–606 Kh. A. Khachatryan, “Solubility of a class of second-order integro-differential equations with monotone non-linearity on a semi-axis”, Izv. Math., 74:5 (2010), 1069–1082 A. Kh. Khachatryan and Kh. A. Khachatryan, “On an Integral Equation With Monotonic Nonlinearity”, Memoirs on Differential Equations and Mathematical Physics, 51:3 (2010), 59–72 == References == == External links == Khachatryan Khachatur. MathNet.ru Khachatur Khachatryan. Yerevan State University Khachatryan Khachatur. Zbmath Khachatryan, Khachatur A. Mathscinet Khachatur Khachatryan: Mathematics is the queen of science and its servant
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Wikipedia:Khairulla Murtazin#0
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Murtazin Khairulla Khabibullovich (Russian: Муртазин Хайрулла Хабибуллович; 4 January 1941 – 17 November 2016) was a Russian mathematician. Since 1978 he has been the Head of the Chair of Mathematical analysis Bashkir State University. == Biography == Murtazin was born in the village Aznash in Uchalinsky District, now in Bashkortostan. He graduated from the Department of Mathematics of Bashkir State University and defended his doctoral thesis in 1994. Since 1978 until the present day he is the head of the Mathematical Analysis chair of the department Scientific activity is devoted to problems of quantum mechanics. Murtazin investigated the asymptotic behavior of the discrete spectrum of the Schrödinger operator, the spectrum of perturbations of partial differential operators, results on the two-particle operators in the class of integrable potentials, conditions for the existence of virtual particles 4. The results of the studies were used in the work on quantum mechanics, nuclear physics and acoustics, for geological and seismic work research and design institute of well logging VNIIGIS (Oktyabrsky, Republic of Bashkortostan). Author of more than 70 scientific papers. == Awards and honours == 2003 – Honored worker of science of the Republic of Bashkortostan 2008 – Honorary Worker of Higher specialized Education of the Russian Federation 2011 – laureate of the State Prize of the Republic of Bashkortostan in the field of science and technology == References == [1]
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Wikipedia:Khinchin's constant#0
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In number theory, Khinchin's constant is a mathematical constant related to the simple continued fraction expansions of many real numbers. In particular Aleksandr Yakovlevich Khinchin proved that for almost all real numbers x, the coefficients ai of the continued fraction expansion of x have a finite geometric mean that is independent of the value of x. It is known as Khinchin's constant and denoted by K0. That is, for x = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 ⋱ {\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots }}}}}}}}\;} it is almost always true that lim n → ∞ ( a 1 a 2 . . . a n ) 1 / n = K 0 . {\displaystyle \lim _{n\rightarrow \infty }\left(a_{1}a_{2}...a_{n}\right)^{1/n}=K_{0}.} The decimal value of Khinchin's constant is given by: K 0 = 2.68545 20010 65306 44530 … {\displaystyle K_{0}=2.68545\,20010\,65306\,44530\dots } (sequence A002210 in the OEIS) Although almost all numbers satisfy this property, it has not been proven for any real number not specifically constructed for the purpose. The following numbers whose continued fraction expansions apparently do have this property (based on empirical data) are: π Roots of equations with a degree > 2, e.g. cubic roots and quartic roots Natural logarithms, e.g. ln(2) and ln(3) The Euler-Mascheroni constant γ Apéry's constant ζ(3) The Feigenbaum constants δ and α Khinchin's constant Among the numbers x whose continued fraction expansions are known not to have this property are: Rational numbers Roots of quadratic equations, e.g. the square roots of integers and the golden ratio φ {\displaystyle \varphi } (however, the geometric mean of all coefficients for square roots of nonsquare integers from 2 to 24 is about 2.708, suggesting that quadratic roots collectively may give the Khinchin constant as a geometric mean); The base of the natural logarithm e. Khinchin is sometimes spelled Khintchine (the French transliteration of Russian Хинчин) in older mathematical literature. == Series expressions == Khinchin's constant can be given by the following infinite product: K 0 = ∏ r = 1 ∞ ( 1 + 1 r ( r + 2 ) ) log 2 r {\displaystyle K_{0}=\prod _{r=1}^{\infty }{\left(1+{1 \over r(r+2)}\right)}^{\log _{2}r}} This implies: ln K 0 = ∑ r = 1 ∞ ln ( 1 + 1 r ( r + 2 ) ) log 2 r {\displaystyle \ln K_{0}=\sum _{r=1}^{\infty }\ln {\left(1+{1 \over r(r+2)}\right)}{\log _{2}r}} Khinchin's constant may also be expressed as a rational zeta series in the form ln K 0 = 1 ln 2 ∑ n = 1 ∞ ζ ( 2 n ) − 1 n ∑ k = 1 2 n − 1 ( − 1 ) k + 1 k {\displaystyle \ln K_{0}={\frac {1}{\ln 2}}\sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}\sum _{k=1}^{2n-1}{\frac {(-1)^{k+1}}{k}}} or, by peeling off terms in the series, ln K 0 = 1 ln 2 [ − ∑ k = 2 N ln ( k − 1 k ) ln ( k + 1 k ) + ∑ n = 1 ∞ ζ ( 2 n , N + 1 ) n ∑ k = 1 2 n − 1 ( − 1 ) k + 1 k ] {\displaystyle \ln K_{0}={\frac {1}{\ln 2}}\left[-\sum _{k=2}^{N}\ln \left({\frac {k-1}{k}}\right)\ln \left({\frac {k+1}{k}}\right)+\sum _{n=1}^{\infty }{\frac {\zeta (2n,N+1)}{n}}\sum _{k=1}^{2n-1}{\frac {(-1)^{k+1}}{k}}\right]} where N is an integer, held fixed, and ζ(s, n) is the complex Hurwitz zeta function. Both series are strongly convergent, as ζ(n) − 1 approaches zero quickly for large n. An expansion may also be given in terms of the dilogarithm: ln K 0 2 = 1 ln 2 [ Li 2 ( − 1 2 ) + 1 2 ∑ k = 2 ∞ ( − 1 ) k Li 2 ( 4 k 2 ) ] . {\displaystyle \ln {\frac {K_{0}}{2}}={\frac {1}{\ln 2}}\left[{\mbox{Li}}_{2}\left({\frac {-1}{2}}\right)+{\frac {1}{2}}\sum _{k=2}^{\infty }(-1)^{k}{\mbox{Li}}_{2}\left({\frac {4}{k^{2}}}\right)\right].} == Integrals == There exist a number of integrals related to Khinchin's constant: ∫ 0 1 log 2 ⌊ x − 1 ⌋ x + 1 d x = ln K 0 {\displaystyle \int _{0}^{1}{\frac {\log _{2}\lfloor x^{-1}\rfloor }{x+1}}\mathrm {d} x=\ln {K_{0}}} ∫ 0 1 log 2 ( Γ ( 2 + x ) Γ ( 2 − x ) ) x ( x + 1 ) d x = ln K 0 − ln 2 {\displaystyle \int _{0}^{1}{\frac {\log _{2}(\Gamma (2+x)\Gamma (2-x))}{x(x+1)}}\mathrm {d} x=\ln K_{0}-\ln 2} ∫ 0 1 1 x ( x + 1 ) log 2 ( π x ( 1 − x 2 ) sin π x ) d x = ln K 0 − ln 2 {\displaystyle \int _{0}^{1}{\frac {1}{x(x+1)}}\log _{2}\left({\frac {\pi x(1-x^{2})}{\sin \pi x}}\right)\mathrm {d} x=\ln K_{0}-\ln 2} ∫ 0 π log 2 ( x | cot x | ) x d x = ln K 0 − 1 2 ln 2 − π 2 12 ln 2 {\displaystyle \int _{0}^{\pi }{\frac {\log _{2}(x|\cot x|)}{x}}\mathrm {d} x=\ln K_{0}-{\frac {1}{2}}\ln 2-{\frac {\pi ^{2}}{12\ln 2}}} == Sketch of proof == The proof presented here was arranged by Czesław Ryll-Nardzewski and is much simpler than Khinchin's original proof which did not use ergodic theory. Since the first coefficient a0 of the continued fraction of x plays no role in Khinchin's theorem and since the rational numbers have Lebesgue measure zero, we are reduced to the study of irrational numbers in the unit interval, i.e., those in I = [ 0 , 1 ] ∖ Q {\displaystyle I=[0,1]\setminus \mathbb {Q} } . These numbers are in bijection with infinite continued fractions of the form [0; a1, a2, ...], which we simply write [a1, a2, ...], where a1, a2, ... are positive integers. Define a transformation T:I → I by T ( [ a 1 , a 2 , … ] ) = [ a 2 , a 3 , … ] . {\displaystyle T([a_{1},a_{2},\dots ])=[a_{2},a_{3},\dots ].\,} The transformation T is called the Gauss–Kuzmin–Wirsing operator. For every Borel subset E of I, we also define the Gauss–Kuzmin measure of E μ ( E ) = 1 ln 2 ∫ E d x 1 + x . {\displaystyle \mu (E)={\frac {1}{\ln 2}}\int _{E}{\frac {dx}{1+x}}.} Then μ is a probability measure on the σ-algebra of Borel subsets of I. The measure μ is equivalent to the Lebesgue measure on I, but it has the additional property that the transformation T preserves the measure μ. Moreover, it can be proved that T is an ergodic transformation of the measurable space I endowed with the probability measure μ (this is the hard part of the proof). The ergodic theorem then says that for any μ-integrable function f on I, the average value of f ( T k x ) {\displaystyle f\left(T^{k}x\right)} is the same for almost all x {\displaystyle x} : lim n → ∞ 1 n ∑ k = 0 n − 1 ( f ∘ T k ) ( x ) = ∫ I f d μ for μ -almost all x ∈ I . {\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=0}^{n-1}(f\circ T^{k})(x)=\int _{I}fd\mu \quad {\text{for }}\mu {\text{-almost all }}x\in I.} Applying this to the function defined by f([a1, a2, ...]) = ln(a1), we obtain that lim n → ∞ 1 n ∑ k = 1 n ln a k = ∫ I f d μ = ∑ r = 1 ∞ ln [ 1 + 1 r ( r + 2 ) ] log 2 r {\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}\ln a_{k}=\int _{I}f\,d\mu =\sum _{r=1}^{\infty }\ln \left[1+{\frac {1}{r(r+2)}}\right]\log _{2}r} for almost all [a1, a2, ...] in I as n → ∞. Taking the exponential on both sides, we obtain to the left the geometric mean of the first n coefficients of the continued fraction, and to the right Khinchin's constant. == Generalizations == The Khinchin constant can be viewed as the first in a series of the Hölder means of the terms of continued fractions. Given an arbitrary series {an}, the Hölder mean of order p of the series is given by K p = lim n → ∞ [ 1 n ∑ k = 1 n a k p ] 1 / p . {\displaystyle K_{p}=\lim _{n\to \infty }\left[{\frac {1}{n}}\sum _{k=1}^{n}a_{k}^{p}\right]^{1/p}.} When the {an} are the terms of a continued fraction expansion, the constants are given by K p = [ ∑ k = 1 ∞ − k p log 2 ( 1 − 1 ( k + 1 ) 2 ) ] 1 / p . {\displaystyle K_{p}=\left[\sum _{k=1}^{\infty }-k^{p}\log _{2}\left(1-{\frac {1}{(k+1)^{2}}}\right)\right]^{1/p}.} This is obtained by taking the p-th mean in conjunction with the Gauss–Kuzmin distribution. This is finite when p < 1 {\displaystyle p<1} . The arithmetic average diverges: lim n → ∞ 1 n ∑ k = 1 n a k = K 1 = + ∞ {\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=1}^{n}a_{k}=K_{1}=+\infty } , and so the coefficients grow arbitrarily large: lim sup n a n = + ∞ {\displaystyle \limsup _{n}a_{n}=+\infty } . The value for K0 is obtained in the limit of p → 0. The harmonic mean (p = −1) is K − 1 = 1.74540566240 … {\displaystyle K_{-1}=1.74540566240\dots } (sequence A087491 in the OEIS). == Open problems == Many well known numbers, such as π, the Euler–Mascheroni constant γ, and Khinchin's constant itself, based on numerical evidence, are thought to be among the numbers for which the limit lim n → ∞ ( a 1 a 2 . . . a n ) 1 / n {\displaystyle \lim _{n\rightarrow \infty }\left(a_{1}a_{2}...a_{n}\right)^{1/n}} converges to Khinchin's constant. However, none of these limits have been rigorously established. In fact, it has not been proven for any real number, which was not specifically constructed for that exact purpose. The algebraic properties of Khinchin's constant itself, e. g. whether it is a rational, algebraic irrational, or transcendental number, are also not known. == See also == Lochs' theorem Lévy's constant Somos' constant List of mathematical constants == References == David H. Bailey; Jonathan M. Borwein; Richard E. Crandall (1995). "On the Khinchine constant" (PDF). Mathematics of Computation. 66 (217): 417–432. doi:10.1090/s0025-5718-97-00800-4. Jonathan M. Borwein; David M. Bradley; Richard E. Crandall (2000). "Computational Strategies for the Riemann Zeta Function" (PDF). J. Comput. Appl. Math. 121 (1–2): 11. Bibcode:2000JCoAM.121..247B. doi:10.1016/s0377-0427(00)00336-8. Thomas Wieting (2007). "A Khinchin Sequence". Proceedings of the American Mathematical Society. 136 (3): 815–824. doi:10.1090/S0002-9939-07-09202-7. Aleksandr Ya. Khinchin (1997). Continued Fractions. New York: Dover Publications. == External links == 110,000 digits of Khinchin's constant 10,000 digits of Khinchin's constant
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Wikipedia:Khintchine inequality#0
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The Khintchine inequality, is a result in probability also frequently used in analysis bounding the expectation a weighted sum of Rademacher random variables with square-summable weights. It is named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet. It states that for each p ∈ ( 0 , ∞ ) {\displaystyle p\in (0,\infty )} there exist constants A p , B p > 0 {\displaystyle A_{p},B_{p}>0} depending only on p {\displaystyle p} such that for every sequence x = ( x 1 , x 2 , … ) ∈ ℓ 2 {\displaystyle x=(x_{1},x_{2},\dots )\in \ell ^{2}} , and i.i.d. Rademacher random variables ϵ 1 , ϵ 2 , … {\displaystyle \epsilon _{1},\epsilon _{2},\dots } , A p ≤ E [ | ∑ n = 1 ∞ ϵ n x n | p ] 1 / p ‖ x ‖ 2 ≤ B p . {\displaystyle A_{p}\leq {\frac {\mathbb {E} \left[\left|\sum _{n=1}^{\infty }\epsilon _{n}x_{n}\right|^{p}\right]^{1/p}}{\|x\|_{2}}}\leq B_{p}.} As a particular case, consider N {\displaystyle N} complex numbers x 1 , … , x N ∈ C {\displaystyle x_{1},\dots ,x_{N}\in \mathbb {C} } , which can be pictured as vectors in a plane. Now sample N {\displaystyle N} random signs ϵ 1 , … , ϵ N ∈ { − 1 , + 1 } {\displaystyle \epsilon _{1},\dots ,\epsilon _{N}\in \{-1,+1\}} , with equal independent probability. The inequality states that | ∑ i ϵ i x i | ≈ | x 1 | 2 + ⋯ + | x N | 2 {\displaystyle {\Big |}\sum _{i}\epsilon _{i}x_{i}{\Big |}\approx {\sqrt {|x_{1}|^{2}+\cdots +|x_{N}|^{2}}}} with a bounded error. == Statement == Let { ε n } n = 1 N {\displaystyle \{\varepsilon _{n}\}_{n=1}^{N}} be i.i.d. random variables with P ( ε n = ± 1 ) = 1 2 {\displaystyle P(\varepsilon _{n}=\pm 1)={\frac {1}{2}}} for n = 1 , … , N {\displaystyle n=1,\ldots ,N} , i.e., a sequence with Rademacher distribution. Let 0 < p < ∞ {\displaystyle 0<p<\infty } and let x 1 , … , x N ∈ C {\displaystyle x_{1},\ldots ,x_{N}\in \mathbb {C} } . Then A p ( ∑ n = 1 N | x n | 2 ) 1 / 2 ≤ ( E | ∑ n = 1 N ε n x n | p ) 1 / p ≤ B p ( ∑ n = 1 N | x n | 2 ) 1 / 2 {\displaystyle A_{p}\left(\sum _{n=1}^{N}|x_{n}|^{2}\right)^{1/2}\leq \left(\operatorname {E} \left|\sum _{n=1}^{N}\varepsilon _{n}x_{n}\right|^{p}\right)^{1/p}\leq B_{p}\left(\sum _{n=1}^{N}|x_{n}|^{2}\right)^{1/2}} for some constants A p , B p > 0 {\displaystyle A_{p},B_{p}>0} depending only on p {\displaystyle p} (see Expected value for notation). More succinctly, ( E | ∑ n = 1 N ε n x n | p ) 1 / p ∈ [ A p , B p ] {\displaystyle \left(\operatorname {E} \left|\sum _{n=1}^{N}\varepsilon _{n}x_{n}\right|^{p}\right)^{1/p}\in [A_{p},B_{p}]} for any sequence x {\displaystyle x} with unit ℓ 2 {\displaystyle \ell ^{2}} norm. The sharp values of the constants A p , B p {\displaystyle A_{p},B_{p}} were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof). It is a simple matter to see that A p = 1 {\displaystyle A_{p}=1} when p ≥ 2 {\displaystyle p\geq 2} , and B p = 1 {\displaystyle B_{p}=1} when 0 < p ≤ 2 {\displaystyle 0<p\leq 2} . Haagerup found that A p = { 2 1 / 2 − 1 / p 0 < p ≤ p 0 , 2 1 / 2 ( Γ ( ( p + 1 ) / 2 ) / π ) 1 / p p 0 < p < 2 1 2 ≤ p < ∞ and B p = { 1 0 < p ≤ 2 2 1 / 2 ( Γ ( ( p + 1 ) / 2 ) / π ) 1 / p 2 < p < ∞ , {\displaystyle {\begin{aligned}A_{p}&={\begin{cases}2^{1/2-1/p}&0<p\leq p_{0},\\2^{1/2}(\Gamma ((p+1)/2)/{\sqrt {\pi }})^{1/p}&p_{0}<p<2\\1&2\leq p<\infty \end{cases}}\\&{\text{and}}\\B_{p}&={\begin{cases}1&0<p\leq 2\\2^{1/2}(\Gamma ((p+1)/2)/{\sqrt {\pi }})^{1/p}&2<p<\infty \end{cases}},\end{aligned}}} where p 0 ≈ 1.847 {\displaystyle p_{0}\approx 1.847} and Γ {\displaystyle \Gamma } is the Gamma function. One may note in particular that B p {\displaystyle B_{p}} matches exactly the moments of a normal distribution. == Uses in analysis == The uses of this inequality are not limited to applications in probability theory. One example of its use in analysis is the following: if we let T {\displaystyle T} be a linear operator between two Lp spaces L p ( X , μ ) {\displaystyle L^{p}(X,\mu )} and L p ( Y , ν ) {\displaystyle L^{p}(Y,\nu )} , 1 < p < ∞ {\displaystyle 1<p<\infty } , with bounded norm ‖ T ‖ < ∞ {\displaystyle \|T\|<\infty } , then one can use Khintchine's inequality to show that ‖ ( ∑ n = 1 N | T f n | 2 ) 1 / 2 ‖ L p ( Y , ν ) ≤ C p ‖ ( ∑ n = 1 N | f n | 2 ) 1 / 2 ‖ L p ( X , μ ) {\displaystyle \left\|\left(\sum _{n=1}^{N}|Tf_{n}|^{2}\right)^{1/2}\right\|_{L^{p}(Y,\nu )}\leq C_{p}\left\|\left(\sum _{n=1}^{N}|f_{n}|^{2}\right)^{1/2}\right\|_{L^{p}(X,\mu )}} for some constant C p > 0 {\displaystyle C_{p}>0} depending only on p {\displaystyle p} and ‖ T ‖ {\displaystyle \|T\|} . == Generalizations == For the case of Rademacher random variables, Pawel Hitczenko showed that the sharpest version is: A ( p ( ∑ n = b + 1 N x n 2 ) 1 / 2 + ∑ n = 1 b x n ) ≤ ( E | ∑ n = 1 N ε n x n | p ) 1 / p ≤ B ( p ( ∑ n = b + 1 N x n 2 ) 1 / 2 + ∑ n = 1 b x n ) {\displaystyle A\left({\sqrt {p}}\left(\sum _{n=b+1}^{N}x_{n}^{2}\right)^{1/2}+\sum _{n=1}^{b}x_{n}\right)\leq \left(\operatorname {E} \left|\sum _{n=1}^{N}\varepsilon _{n}x_{n}\right|^{p}\right)^{1/p}\leq B\left({\sqrt {p}}\left(\sum _{n=b+1}^{N}x_{n}^{2}\right)^{1/2}+\sum _{n=1}^{b}x_{n}\right)} where b = ⌊ p ⌋ {\displaystyle b=\lfloor p\rfloor } , and A {\displaystyle A} and B {\displaystyle B} are universal constants independent of p {\displaystyle p} . Here we assume that the x i {\displaystyle x_{i}} are non-negative and non-increasing. == See also == Marcinkiewicz–Zygmund inequality Burkholder-Davis-Gundy inequality == References == Thomas H. Wolff, "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003. ISBN 0-8218-3449-5 Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math. 70 (1981), no. 3, 231–283 (1982). Fedor Nazarov and Anatoliy Podkorytov, "Ball, Haagerup, and distribution functions", Complex analysis, operators, and related topics, 247–267, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000.
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Wikipedia:Ki-Hang Kim#0
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Ki-Hang Kim (Korean: 김기항; 5 August 1936 – 15 January 2009), also known as Kim Ki-Hang Butler, Hang Kim, Keyhany Keem, or Kim Ki-Hang was a Korean-American Mathematician and Alabama State University professor known for his contributions in semigroups, Boolean matrices, and Social Sciences. He frequently co-wrote with Fred Roush. == Personal life == Kim was born in Anju, Korea, Empire of Japan (now in North Korea) the eldest son of independent farmers Jin Gyong Kim and Mayhryn Hong. A bright child, by 12 years old Kim was capable of speaking some Japanese, Chinese, English and Russian, and had skipped some grades of school; by 14, he was acting as an interpreter for US troops in Korea during the Korean War. In 1950, Kim's region was held by the South Korean and American army. When the North Koreans and Chinese returned, Kim was given six hours to decide whether to take an empty U.S. Air Force seat, and go South with the US Army, which he accepted upon the urging of his father. He didn't see his family again for 30 years. He went to Taegu Airbase, and in 1952 passed the qualifying English exams, securing him the job of interpreter for Colonel Decatur Poindexter Butler. At the war's end, Butler took Kim to the US, for a better education. On 25 November 1954, Kim began the paperwork to immigrate to the US permanently, in order to join the US air force. In 1955, Kim enlisted in the US Army to reduce stress on the Butler family by utilising the G.I. Bill to pay for his further education. He was discharged in 1956. He received US citizenship in 1960. Kim married Myong Ja Hwang on 31 July 1963. They had two children together one of whom is the actress Linda Kim. In 1981, Kim returned to North Korea, reuniting with his family for the first time in 30 years. Kim was also active in the Korean-American community, acting as the first president of the Montgomery Korean-American Association and was a member of the Korean-American Methodist Church. In 1996, Kim was awarded an Honorary Alabama Colonel for his outstanding leadership in the field of mathematics. He was also recognised in the 1972 edition of "Personalities of the South." According to long-time collaborator Fred Roush, there was a period of time where Kim was a bodybuilder. His Erdős number is 2. == Education and career == Kim graduated from the University of Southern Mississippi in 1960 with a B.S. in mathematics. He received a M.S. a year later, in 1961. Unable to fund a Ph.D., Kim taught briefly at University of Hartford. He then obtained a Ph.D. in mathematics from George Washington University in 1970, for On (0,1)-Matrix Semigroups. Kim began teaching at St. Mary's College in 1968, moving to Pembroke State University in 1971. Finally, he accepted the position of professor of mathematics and Director of the Mathematics Research Group at Alabama State University. Kim additionally taught at institutions abroad, in Portugal and India, as well as attending many international conferences, including those in China and Hungary, particularly the conference on Algebraic Semigroup Theory in Szeged, Hungary, where he was the only American invited. He was also active in many conferences within the US, including the American Mathematical Society meeting at Auburn University in 1971, Southeastern Conference on Combinatorics, Graph Theory, Computing in Boca Raton, Florida in 1974. Kim spent 35 years teaching at Alabama State University, ending his tenure in 2007. From 1971 to 1976, Kim published 25 papers on semigroups and Boolean matrices (under the name Kim Butler). Following meeting fellow mathematician Fred Roush, Kim published over 150 more papers over a variety of subjects. He is remembered for bridging the gap between social sciences, particularly economics, psychology, and political sciences. In 1980, he launched and became editor of Mathematical Social Sciences, focusing on Game Theory and Social Choice Theory. Kim also disproved an established theorem dictating the way computer coding was written. Kim wrote seven books, most co-authored by Roush. == Publications and books == Listed below are some early works by Kim, published under Kim Butler: Butler, K.KH. "On Kim's conjecture." Semigroup Forum 2, 281 (1971). Butler, K.KH. (1971). "Binary relations." In: Capobianco, M., Frechen, J.B., Krolik, M. (eds) Recent Trends in Graph Theory. vol 186. Springer, Berlin, Heidelberg. Butler, K.KH. "On a miller and clifford theorem." Semigroup Forum 3, 92–94 (1971). Butler, K.KH. "On (0,1)-matrix semigroups." Semigroup Forum 3, 74–79 (1971). Butler, K.KH. (1972). "The number of partial order graphs." In: Alavi, Y., Lick, D.R., White, A.T. (eds) Graph Theory and Applications. Lecture Notes in Mathematics, vol 303. Springer, Berlin, Heidelberg. Butler, K.KH. "Straddles and splits on semigroups." Acta Mathematica Academiae Scientiarum Hungaricae 24, 113–114 (1973). Butler, K.KH. "Canonical bijection betweenD of (0, 1)-matrix semigroupsof (0, 1)-matrix semigroups." Period. Math. Hung. 4, 303–305 (1973). Butler, K.KH. (1974). "Subgroups of Binary Relations." In: Newman, M.F. (eds) Proceedings of the Second International Conference on the Theory of Groups. Lecture Notes in Mathematics, vol 372. Springer, Berlin, Heidelberg. Butler, K.KH. (1974). "A moore-penrose inverse for boolean relation matrices." In: Holton, D.A. (eds) Combinatorial Mathematics. Lecture Notes in Mathematics, vol 403. Springer, Berlin, Heidelberg. Butler, K.KH. (1974). "Subgroups of binary relations." In: Newman, M.F. (eds) Proceedings of the Second International Conference on The Theory of Groups. Lecture Notes in Mathematics, vol 372. Springer, Berlin, Heidelberg. Ki-Hang Butler, K. "Combinatorial properties of binary semigroups." Period. Math. Hung. 5, 3–46 (1974). Butler, K.KH. "The semigroup of hall relations." Semigroup Forum 9, 253–260 (1974). Over 150 later works can be found here, published under Ki Hang Kim. Additionally, Kim wrote seven books, reviews of which can be found here. They are titled as follows: Lecture notes on (0,1)-matrices (1973) Mathematics for social scientists (1980) with F W Roush. Introduction to mathematical consensus theory (1980) with F W Roush. Boolean matrix theory and applications (1982) Applied abstract algebra (1983) with F W Roush. Competitive economics. Equilibrium and arbitration (1983) with F W Roush. Incline algebra and applications (1984) with Z Q Cao and F W Roush. Team theory (1987) with F W Roush == References == == External links == Kim ki han korean american history museum
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Wikipedia:Kieka Mynhardt#0
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Christina Magdalena (Kieka) Mynhardt (née Steyn; born 1953) is a South African born Canadian mathematician known for her work on dominating sets in graph theory, including domination versions of the eight queens puzzle. She is a professor of mathematics and statistics at the University of Victoria in Canada. == Education and career == Mynhardt was born in Cape Town, and was a student at the Hoërskool Lichtenburg. She completed her Ph.D. at Rand Afrikaans University (now incorporated into the University of Johannesburg) in 1979, supervised by Izak Broere. Her dissertation, The G {\displaystyle {\mathcal {G}}} -constructability of graphs, gave a conjectured construction for the planar graphs by repeatedly adding vertices with prescribed neighborhoods. She became a faculty member at the University of Pretoria and then the University of South Africa before moving to the University of Victoria. == Recognition == In 1995, Mynhardt was selected as one of the founding members of the Academy of Science of South Africa. She was a 2005 recipient of the Dignitas Award of the University of Johannesburg Alumni. == References == == External links == Home page Faces of UVic Research: Kieka Mynhardt (video) Kieka Mynhardt publications indexed by Google Scholar
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Wikipedia:Kiiti Morita#0
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Kiiti Morita (森田 紀一, Morita Kiichi, February 11, 1915 – August 4, 1995) was a Japanese mathematician working in algebra and topology. Morita was born in 1915 in Hamamatsu, Shizuoka Prefecture and graduated from the Tokyo Higher Normal School in 1936. Three years later he was appointed assistant at the Tokyo University of Science. He received his Ph.D. from Osaka University in 1950, with a thesis in topology. After teaching at the Tokyo Higher Normal School, he became professor at the University of Tsukuba in 1951. He held this position until 1978, after which he taught at Sophia University. Morita died of heart failure in 1995 at the Sakakibara Heart Institute in Tokyo; he was survived by his wife, Tomiko, his son, Yasuhiro, and a grandson. He introduced the concepts now known as Morita equivalence and Morita duality which were given wide circulation in the 1960s by Hyman Bass in a series of lectures. The Morita conjectures on normal topological spaces are also named after him. == Publications == Morita, Kiiti (1958). "Duality for modules and its applications to the theory of rings with minimum condition". Science Reports of the Tokyo Kyoiku Daigaku. Section A. 6 (150): 83–142. MR 0096700. Zbl 0080.25702. Morita, Kiiti (1962). "Paracompactness and product spaces". Fundamenta Mathematicae. 50 (3): 223–236. doi:10.4064/fm-50-3-223-236. MR 0132525. Morita, Kiiti (1964). "Products of normal spaces with metric spaces". Mathematische Annalen. 154 (4): 365–382. doi:10.1007/BF01362570. MR 0165491. S2CID 120516038. Morita, Kiiti (1977). "Some problems on normality of products of spaces". In Novák, Josef (ed.). General topology and its relations to modern analysis and algebra, IV (Proc. Fourth Prague Topological Sympos., Prague, 1976), Part B. Prague: Soc. Czechoslovak Mathematicians and Physicists. pp. 296–297. MR 0482657. == References ==
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Wikipedia:Kim-Chuan Toh#0
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Kim-Chuan Toh is a Singaporean mathematician, and Leo Tan Professor in Science at the National University of Singapore (NUS). He is known for his contributions to the theory, practice, and application of convex optimization, especially semidefinite programming and conic programming. == Education == Toh received BSc (Hon.) in 1990 and MSc in 1992, from NUS, and PhD in 1996 from Cornell University. == Awards and honours == Toh received the 2017 INFORMS Optimization Society Farkas Prize, and 2019 President's Science Award (Singapore). He is a fellow of the Society of Industrial and Applied Mathematics (Class of 2018). In 2020, Toh became a laureate of the Asian Scientist 100 by the Asian Scientist. == Selected works == Toh, K. C.; Todd, M. J.; Tütüncü, R. H., "SDPT3—a MATLAB software package for semidefinite programming, version 1.3. Interior point methods". Optim. Methods Softw. 11/12 (1999), no. 1–4, 545–581. Tütüncü, R. H.; Toh, K. C.; Todd, M. J., "Solving semidefinite-quadratic-linear programs using SDPT3. Computational semidefinite and second order cone programming: the state of the art". Math. Program. 95 (2003), no. 2, Ser. B, 189–217. Toh, K. C.; Yun, S., "An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems". Pac. J. Optim. 6 (2010), no. 3, 615–640. Todd, M. J.; Toh, K. C.; Tütüncü, R. H., "On the Nesterov-Todd direction in semidefinite programming". SIAM J. Optim. 8 (1998), no. 3, 769–796. Zhao, X.-Y.; Sun, D.; Toh, K. C., "A Newton-CG augmented Lagrangian method for semidefinite programming". SIAM J. Optim. 20 (2010), no. 4, 1737–1765. Driscoll, T. A.; Toh, K. C.; Trefethen, L. N., "From potential theory to matrix iterations in six steps". SIAM Rev. 40 (1998), no. 3, 547–578. == References == == External links == Homepage of Kim Chuan Toh The Mathematics Genealogy Project – Kim Chuan Toh
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Wikipedia:King's College London Mathematics School#0
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King's College London Mathematics School, also known as King's Maths School or KCLMS, is a maths school located in the Lambeth area of London, England. King's College London Mathematics School is run in partnership with King's College London. The school was inspired by the Kolmogorov Physics and Mathematics School in Moscow, established in 1965 by mathematician Andrey Kolmogorov. The school aims to widen participation in the mathematical sciences by supporting young people from backgrounds currently under-represented in these fields. The school opened in 2014 and specialises in mathematics. It has an approximate 14% acceptance rate. In 2018, the school received nearly 500 applications for 70 places. All prospective students are invited to take a written mathematics aptitude test. Those with a high score on the test are invited to an interview that consists of a mathematics interview and a personal interview. Prospective students are required to obtain GCSE qualifications at grade 8 or 9 (or previous grade A*) in Mathematics and either grade 7 or above (or previous grade A or A*) in Physics or grade 7-7 or above in Combined Science. In addition, prospective students are required to obtain a grade 5 or above (or previous grade C) in a total of at least seven GCSEs, including in English Language. The course structure of King's College London Mathematics School requires all students to study A-levels in mathematics, further mathematics and physics. In their first year, students also choose between an AS-level in either computer science or economics, and complete a substantive, collaborative research project ("King's Certificate") with briefs set by academics and industry professionals. In their second year, students can engage with a unique programme of extension courses ("Curriculum X") and also have the option to complete an Extended Project Qualification (EPQ). In 2019, 60% of all A-level entries were graded A* and 91% of all A-level entries were A*/A. Furthermore, over 25% of leavers received Oxbridge places. These results placed King's College London Mathematics School as the top performing school in the country for A Level attainment. The Sunday Times 2018 School Guide, selected King's College London Mathematics School as the State Sixth Form College of the Year. The Sunday Times also selected it as the Best State Sixth Form college of the Decade in 2021. In December 2024, King’s College London Mathematics School was awarded the titles of Sixth Form College of the Year 2025 and Sixth Form College of the Year for Academic Excellence 2025 by The Sunday Times in the Parent Power schools guide. == References == == External links == King's College London Mathematics School official website
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Wikipedia:Kirchhoff's theorem#0
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In the mathematical field of graph theory, Kirchhoff's theorem or Kirchhoff's matrix tree theorem named after Gustav Kirchhoff is a theorem about the number of spanning trees in a graph, showing that this number can be computed in polynomial time from the determinant of a submatrix of the graph's Laplacian matrix; specifically, the number is equal to any cofactor of the Laplacian matrix. Kirchhoff's theorem is a generalization of Cayley's formula which provides the number of spanning trees in a complete graph. Kirchhoff's theorem relies on the notion of the Laplacian matrix of a graph, which is equal to the difference between the graph's degree matrix (the diagonal matrix of vertex degrees) and its adjacency matrix (a (0,1)-matrix with 1's at places corresponding to entries where the vertices are adjacent and 0's otherwise). For a given connected graph G with n labeled vertices, let λ1, λ2, ..., λn−1 be the non-zero eigenvalues of its Laplacian matrix. Then the number of spanning trees of G is t ( G ) = 1 n λ 1 λ 2 ⋯ λ n − 1 . {\displaystyle t(G)={\frac {1}{n}}\lambda _{1}\lambda _{2}\cdots \lambda _{n-1}\,.} An English translation of Kirchhoff's original 1847 paper was made by J. B. O'Toole and published in 1958. == An example using the matrix-tree theorem == First, construct the Laplacian matrix Q for the example diamond graph G (see image on the right): Q = [ 2 − 1 − 1 0 − 1 3 − 1 − 1 − 1 − 1 3 − 1 0 − 1 − 1 2 ] . {\displaystyle Q=\left[{\begin{array}{rrrr}2&-1&-1&0\\-1&3&-1&-1\\-1&-1&3&-1\\0&-1&-1&2\end{array}}\right].} Next, construct a matrix Q* by deleting any row and any column from Q. For example, deleting row 1 and column 1 yields Q ∗ = [ 3 − 1 − 1 − 1 3 − 1 − 1 − 1 2 ] . {\displaystyle Q^{\ast }=\left[{\begin{array}{rrr}3&-1&-1\\-1&3&-1\\-1&-1&2\end{array}}\right].} Finally, take the determinant of Q* to obtain t(G), which is 8 for the diamond graph. (Notice t(G) is the (1,1)-cofactor of Q in this example.) == Proof outline == (The proof below is based on the Cauchy–Binet formula. An elementary induction argument for Kirchhoff's theorem can be found on page 654 of Moore (2011).) First notice that the Laplacian matrix has the property that the sum of its entries across any row and any column is 0. Thus we can transform any minor into any other minor by adding rows and columns, switching them, and multiplying a row or a column by −1. Thus the cofactors are the same up to sign, and it can be verified that, in fact, they have the same sign. We proceed to show that the determinant of the minor M11 is the number of spanning trees. Let n be the number of vertices of the graph, and m the number of its edges. The incidence matrix E is an n-by-m matrix, which may be defined as follows: suppose that (i, j) is the kth edge of the graph, and that i < j. Then Eik = 1, Ejk = −1, and all other entries in column k are 0 (see oriented incidence matrix for understanding this modified incidence matrix E). For the preceding example (with n = 4 and m = 5): E = [ 1 1 0 0 0 − 1 0 1 1 0 0 − 1 − 1 0 1 0 0 0 − 1 − 1 ] . {\displaystyle E={\begin{bmatrix}1&1&0&0&0\\-1&0&1&1&0\\0&-1&-1&0&1\\0&0&0&-1&-1\\\end{bmatrix}}.} Recall that the Laplacian L can be factored into the product of the incidence matrix and its transpose, i.e., L = EET. Furthermore, let F be the matrix E with its first row deleted, so that FFT = M11. Now the Cauchy–Binet formula allows us to write det ( M 11 ) = ∑ S det ( F S ) det ( F S T ) = ∑ S det ( F S ) 2 {\displaystyle \det \left(M_{11}\right)=\sum _{S}\det \left(F_{S}\right)\det \left(F_{S}^{\mathrm {T} }\right)=\sum _{S}\det \left(F_{S}\right)^{2}} where S ranges across subsets of [m] of size n − 1, and FS denotes the (n − 1)-by-(n − 1) matrix whose columns are those of F with index in S. Then every S specifies n − 1 edges of the original graph, and it can be shown that those edges induce a spanning tree if and only if the determinant of FS is +1 or −1, and that they do not induce a spanning tree if and only if the determinant is 0. This completes the proof. == Particular cases and generalizations == === Cayley's formula === Cayley's formula follows from Kirchhoff's theorem as a special case, since every vector with 1 in one place, −1 in another place, and 0 elsewhere is an eigenvector of the Laplacian matrix of the complete graph, with the corresponding eigenvalue being n. These vectors together span a space of dimension n − 1, so there are no other non-zero eigenvalues. Alternatively, note that as Cayley's formula gives the number of distinct labeled trees of a complete graph Kn we need to compute any cofactor of the Laplacian matrix of Kn. The Laplacian matrix in this case is [ n − 1 − 1 ⋯ − 1 − 1 n − 1 ⋯ − 1 ⋮ ⋮ ⋱ ⋮ − 1 − 1 ⋯ n − 1 ] . {\displaystyle {\begin{bmatrix}n-1&-1&\cdots &-1\\-1&n-1&\cdots &-1\\\vdots &\vdots &\ddots &\vdots \\-1&-1&\cdots &n-1\\\end{bmatrix}}.} Any cofactor of the above matrix is nn−2, which is Cayley's formula. === Kirchhoff's theorem for multigraphs === Kirchhoff's theorem holds for multigraphs as well; the matrix Q is modified as follows: The entry qi,j equals −m, where m is the number of edges between i and j; when counting the degree of a vertex, all loops are excluded. Cayley's formula for a complete multigraph is mn−1(nn−1−(n−1)nn−2) by same methods produced above, since a simple graph is a multigraph with m = 1. === Explicit enumeration of spanning trees === Kirchhoff's theorem can be strengthened by altering the definition of the Laplacian matrix. Rather than merely counting edges emanating from each vertex or connecting a pair of vertices, label each edge with an indeterminate and let the (i, j)-th entry of the modified Laplacian matrix be the sum over the indeterminates corresponding to edges between the i-th and j-th vertices when i does not equal j, and the negative sum over all indeterminates corresponding to edges emanating from the i-th vertex when i equals j. The determinant of the modified Laplacian matrix by deleting any row and column (similar to finding the number of spanning trees from the original Laplacian matrix), above is then a homogeneous polynomial (the Kirchhoff polynomial) in the indeterminates corresponding to the edges of the graph. After collecting terms and performing all possible cancellations, each monomial in the resulting expression represents a spanning tree consisting of the edges corresponding to the indeterminates appearing in that monomial. In this way, one can obtain explicit enumeration of all the spanning trees of the graph simply by computing the determinant. For a proof of this version of the theorem see Bollobás (1998). === Matroids === The spanning trees of a graph form the bases of a graphic matroid, so Kirchhoff's theorem provides a formula for the number of bases in a graphic matroid. The same method may also be used to determine the number of bases in regular matroids, a generalization of the graphic matroids (Maurer 1976). === Kirchhoff's theorem for directed multigraphs === Kirchhoff's theorem can be modified to give the number of oriented spanning trees in directed multigraphs. The matrix Q is constructed as follows: The entry qi,j for distinct i and j equals −m, where m is the number of edges from i to j; The entry qi,i equals the indegree of i minus the number of loops at i. The number of oriented spanning trees rooted at a vertex i is the determinant of the matrix gotten by removing the ith row and column of Q === Counting spanning k-component forests === Kirchhoff's theorem can be generalized to count k-component spanning forests in an unweighted graph. A k-component spanning forest is a subgraph with k connected components that contains all vertices and is cycle-free, i.e., there is at most one path between each pair of vertices. Given such a forest F with connected components F 1 , … , F k {\textstyle F_{1},\dots ,F_{k}} , define its weight w ( F ) = | V ( F 1 ) | ⋅ ⋯ ⋅ | V ( F k ) | {\textstyle w(F)=|V(F_{1})|\cdot \dots \cdot |V(F_{k})|} to be the product of the number of vertices in each component. Then ∑ F w ( F ) = q k , {\displaystyle \sum _{F}w(F)=q_{k},} where the sum is over all k-component spanning forests and q k {\textstyle q_{k}} is the coefficient of x k {\textstyle x^{k}} of the polynomial ( x + λ 1 ) … ( x + λ n − 1 ) x . {\displaystyle (x+\lambda _{1})\dots (x+\lambda _{n-1})x.} The last factor in the polynomial is due to the zero eigenvalue λ n = 0 {\textstyle \lambda _{n}=0} . More explicitly, the number q k {\textstyle q_{k}} can be computed as q k = ∑ { i 1 , … , i n − k } ⊂ { 1 … n − 1 } λ i 1 … λ i n − k . {\displaystyle q_{k}=\sum _{\{i_{1},\dots ,i_{n-k}\}\subset \{1\dots n-1\}}\lambda _{i_{1}}\dots \lambda _{i_{n-k}}.} where the sum is over all n−k-element subsets of { 1 , … , n } {\textstyle \{1,\dots ,n\}} . For example q n − 1 = λ 1 + ⋯ + λ n − 1 = t r Q = 2 | E | q n − 2 = λ 1 λ 2 + λ 1 λ 3 + ⋯ + λ n − 2 λ n − 1 q 2 = λ 1 … λ n − 2 + λ 1 … λ n − 3 λ n − 1 + ⋯ + λ 2 … λ n − 1 q 1 = λ 1 … λ n − 1 {\displaystyle {\begin{aligned}q_{n-1}&=\lambda _{1}+\dots +\lambda _{n-1}=\mathrm {tr} Q=2|E|\\q_{n-2}&=\lambda _{1}\lambda _{2}+\lambda _{1}\lambda _{3}+\dots +\lambda _{n-2}\lambda _{n-1}\\q_{2}&=\lambda _{1}\dots \lambda _{n-2}+\lambda _{1}\dots \lambda _{n-3}\lambda _{n-1}+\dots +\lambda _{2}\dots \lambda _{n-1}\\q_{1}&=\lambda _{1}\dots \lambda _{n-1}\\\end{aligned}}} Since a spanning forest with n−1 components corresponds to a single edge, the k = n−1 case states that the sum of the eigenvalues of Q is twice the number of edges. The k = 1 case corresponds to the original Kirchhoff theorem since the weight of every spanning tree is n. The proof can be done analogously to the proof of Kirchhoff's theorem. An invertible ( n − k ) × ( n − k ) {\displaystyle (n-k)\times (n-k)} submatrix of the incidence matrix corresponds bijectively to a k-component spanning forest with a choice of vertex for each component. The coefficients q k {\textstyle q_{k}} are up to sign the coefficients of the characteristic polynomial of Q. == See also == List of topics related to trees BEST theorem Markov chain tree theorem Minimum spanning tree Prüfer sequence == References ==
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Wikipedia:Kirilo Bojović#0
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Kirilo Bojović (Serbian Cyrillic: Кирило Бојовић; 4 February 1969) is the ruling bishop of the Serbian Orthodox Church abroad with parishes and missions throughout South and Central America, known as the Serbian Orthodox Eparchy of Buenos Aires and South America. == Biography == He was born in Podgorica in Montenegro on 4 February 1969. He completed his mathematical studies in Danilograd, Podgorica and Cetinje, served in the military, and in 2005 traveled to Russia to study philosophy and theology at the prestigious Moscow Spiritual Academy. There he defended a doctoral dissertation on the theme: "Metropolitan Petar II Petrović-Njegoš as a Christian Philosopher". Upon his return to Montenegro, Metropolitan Amfilohije assigned Father Kiril various tasks: professor of the New Testament in the seminary, editor-in-chief of Svetigora, a monthly of the Serbian Orthodox Diocese of Montenegro, with worship in various monasteries. He also pursued post-graduate studies at the Faculty of Theology of the University of Belgrade. Kirilo, in addition to his native Serbian, speaks Russian, Spanish, English, Greek, and Latin. In December 2014, Archimandrite Kirilo arrived in Argentina amid the pastoral visit of Metropolitan Amfilohije to Buenos Aires and South America, to fulfill the responsibility of replacing Vladika Amfilohije upon his return to Montenegro. Then, in 2016, Archimandrite Kirilo was consecrated as a vicar bishop for the Diocese. And in 2018, he was ordained a diocesan bishop and became the new administrator of the diocese. With the blessings of the late Patriarch Irinej and the late Metropolitan Amfilohije, Patriarch Porfirije installed Bishop Kirilo to the Serbian Orthodox Eparchy of Buenos Aires and South America. In May 2023, Bishop Kirilo was named as new administrator of the Metropolitanate of Zagreb and Ljubljana. == References ==
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Wikipedia:Kirsi Peltonen#0
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Kirsi Peltonen is a Finnish mathematician whose research interests include differential geometry and the connections between mathematics and art. She is a Senior University Lecturer in the Department of Mathematics and Systems Analysis at Aalto University, and a docent at the University of Helsinki. Her work has included the design of an innovative interdisciplinary course on mathematics, art, and architecture, the creation of a major exhibit at the Heureka science center near Helsinki, and presentations on mathematics at Finnish schools. Peltonen earned her Ph.D. in 1992, at the University of Helsinki. Her dissertation, On the Existence of Quasiregular Mappings, was supervised by Seppo Rickman. In 2015, Peltonen was the inaugural lecturer for a series of lectures titled Women in Mathematics in Finland and sponsored by European Women in Mathematics. In 2018 the Finnish Mathematics Society awarded their annual mathematics prize to Peltonen, for her work on mathematics and art. == References == == External links == Home page
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Wikipedia:Kirsten Morris#0
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Kirsten Anna Morris (born 1960) is a Canadian applied mathematician specializing in control theory, including work on flexible structures, smart materials, hysteresis, and infinite-dimensional optimization. She is a professor at the University of Waterloo, the former chair of the Society for Industrial and Applied Mathematics Activity Group on Control and Systems, the author of two books on control theory, and an IEEE Fellow. == Education and career == Morris was motivated to study mathematical economics at Queen's University at Kingston by a job doing econometrics at a bank, but lost interest in the economic applications of mathematics after a year, instead switching into a program in mathematics and engineering, which she finished in 1982. She became interested in control theory while studying for a master's degree at the University of Waterloo. After completing the degree in 1984, she continued at Waterloo as a doctoral student, and earned her Ph.D. there in 1989. Her dissertation, Finite-Dimensional Control of Infinite-Dimensional Systems, was supervised by Mathukumalli Vidyasagar. After a year as a staff scientist at the NASA Langley Research Center, she returned to Waterloo as an assistant professor in the Department of Applied Mathematics in 1990. She became a full professor there in 2003, and also holds a cross-appointment in the Department of Mechanical and Mechatronics Engineering. She chaired the Society for Industrial and Applied Mathematics Activity Group on Control and Systems from 2018 to 2019, and has held leadership positions in the IEEE Control Systems Society and the International Federation of Automatic Control. == Books == Morris is the author of the books Introduction to Feedback Control (Harcourt-Brace, 2001) and Controller Design for Distributed Parameter Systems (Springer, 2020). She is the editor of Control of Flexible Structures: Papers from the Workshop on Problems in Sensing, Identification and Control of Flexible Structures held in Waterloo, Ontario, June 1992 (American Mathematical Society, 1993). == Recognition == In 2020, Morris was named an IEEE Fellow, affiliated with the IEEE Control Systems Society, "for contributions to control and estimator design for infinite-dimensional systems". She was named a SIAM Fellow in the 2021 class of fellows, "for contributions to modeling, approximation, and control design for distributed parameter systems". She is also a Fellow of the International Federation of Automatic Control. == References == == External links == Home page
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Wikipedia:Kirsten Wickelgren#0
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Kirsten Graham Wickelgren is an American mathematician whose research interests range over multiple areas including algebraic geometry, algebraic topology, arithmetic geometry, and anabelian geometry. She is a professor of mathematics at Duke University. == Education and career == Wickelgren was one of the finalists in the 1999 Intel Science Talent Search. She majored in mathematics at Harvard University, graduating magna cum laude in 2003. After a year at the École normale supérieure (Paris), she went to Stanford University for doctoral study in mathematics, completing her Ph.D. in 2009. Her dissertation, Lower Central Series Obstructions To Homotopy Sections of Curves Over Number Fields, was supervised by Gunnar Carlsson. She returned to Harvard as a five-year postdoctoral research fellow, funded by the American Institute of Mathematics, and in 2013 became an assistant professor at Georgia Tech. In 2018 she was tenured as associate professor there, and in 2019 she moved to Duke University as a full professor. == Recognition == Wickelgren was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to algebraic topology, algebraic geometry, and number theory". == Family == Wickelgren is daughter of psychologists Norma Graham and Wayne Wickelgren, sister of physicist Peter W. Graham, and half-sister of lawyer Abraham Wickelgren. She is granddaughter of psychologist Frances K. Graham and great-granddaughter of surgeon Evarts Ambrose Graham. == References == == External links == Home page
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Wikipedia:Kiyoshi Oka#0
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Kiyoshi Oka (岡 潔, Oka Kiyoshi, April 19, 1901 – March 1, 1978) was a Japanese mathematician who did fundamental work in the theory of several complex variables. == Biography == Oka was born in Osaka. He went to Kyoto Imperial University in 1919, turning to mathematics in 1923 and graduating in 1924. He was in Paris for three years from 1929, returning to Hiroshima University. He published solutions to the first and second Cousin problems, and work on domains of holomorphy, in the period 1936–1940. He received his Doctor of Science degree from Kyoto Imperial University in 1940. These were later taken up by Henri Cartan and his school, playing a basic role in the development of sheaf theory. The Oka–Weil theorem is due to a work of André Weil in 1935 and Oka's work in 1937. Oka continued to work in the field, and proved Oka's coherence theorem in 1950. Oka's lemma is also named after him. He was a professor at Nara Women's University from 1949 to retirement at 1964. He received many honours in Japan. == Honors == 1951 Japan Academy Prize 1954 Asahi Prize 1960 Order of Culture 1973 Order of the Sacred Treasure, 1st class == Bibliography == KIYOSHI OKA COLLECTED PAPERS Oka, Kiyoshi (1961). Sur les fonctions analytiques de plusieurs variables (in French). Tokyo, Japan: Iwanami Shoten. p. 234. - Includes bibliographical references. Oka, Kiyoshi (1983). Sur les fonctions analytiques de plusieurs variables (in French) (Nouv. ed. augmentee. ed.). Tokyo, Japan: Iwanami. p. 246. Oka, Kiyoshi (1984). Reinhold Remmert (ed.). Kiyoshi Oka Collected Papers. Translated by Raghavan Narasimhan. Commentary: Henri Cartan. Springer-Verlag. p. 223. ISBN 0-387-13240-6. === Selected papers (Sur les fonctions analytiques de plusieurs variables) === Oka, Kiyoshi (1936). "Domaines convexes par rapport aux fonctions rationnelles". Journal of Science of the Hiroshima University. 6: 245–255. doi:10.32917/hmj/1558749869. PDF TeX Oka, Kiyoshi (1937). "Domaines d'holomorphie". Journal of Science of the Hiroshima University. 7: 115–130. doi:10.32917/hmj/1558576819. PDF TeX. Oka, Kiyoshi (1939). "Deuxième problème de Cousin". Journal of Science of the Hiroshima University. 9: 7–19. doi:10.32917/hmj/1558490525. PDF TeX. Oka, Kiyoshi (1941). "Domaines d'holomorphie et domaines rationnellement convexes". Japanese Journal of Mathematics. 17: 517–521. doi:10.4099/jjm1924.17.0_517. PDF TeX. Oka, Kiyoshi (1941). "L'intégrale de Cauchy". Japanese Journal of Mathematics. 17: 523–531. PDF TeX. Oka, Kiyoshi (1942). "Domaines pseudoconvexes". Tôhoku Mathematical Journal. 49: 15–52. PDF TeX. Oka, Kiyoshi (1950). "Sur quelques notions arithmétiques". Bulletin de la Société Mathématique de France. 78: 1–27. doi:10.24033/bsmf.1408. PDF TeX. Oka, Kiyoshi (1951). "Sur les Fonctions Analytiques de Plusieurs Variables, VIII--Lemme Fondamental". Journal of the Mathematical Society of Japan. 3: 204–214, pp. 259–278. doi:10.2969/jmsj/00310204. PDF TeX. Oka, Kiyoshi (1953). "Domaines finis sans point critique intérieur". Japanese Journal of Mathematics. 27: 97–155. doi:10.4099/jjm1924.23.0_97. PDF TeX. Oka, Kiyoshi (1962). "Une mode nouvelle engendrant les domaines pseudoconvexes". Japanese Journal of Mathematics. 32: 1–12. doi:10.4099/jjm1924.32.0_1. PDF TeX. Oka, Kiyoshi (1934). "Note sur les familles de fonctions analytiques multiformes etc". Journal of Science of the Hiroshima University. Ser.A 4: 93–98. doi:10.32917/hmj/1558749763. PDF TeX. Oka, Kiyoshi (1941). "Sur les domaines pseudoconvexes". Proc. Imp. Acad. Tokyo. 17 (1): 7–10. doi:10.3792/pia/1195578912. PDF TeX. Oka, Kiyoshi (1949). "Note sur les fonctions analytiques de plusieurs variables". Kodai Math. Sem. Rep. 1 (5–6): 15–18. doi:10.2996/kmj/1138833536. PDF TeX. == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Kiyoshi Oka", MacTutor History of Mathematics Archive, University of St Andrews Oka library at NWU Photos of Prof. Kiyoshi Oka Related to Works of Dr. Kiyoshi OKA Oka Mathematical Institute
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Wikipedia:Knaster–Kuratowski–Mazurkiewicz lemma#0
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Kazimierz Kuratowski (Polish pronunciation: [kaˈʑimjɛʂ kuraˈtɔfskʲi]; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. He worked as a professor at the University of Warsaw and at the Mathematical Institute of the Polish Academy of Sciences (IM PAN). Between 1946 and 1953, he served as President of the Polish Mathematical Society. He is primarily known for his contributions to set theory, topology, measure theory and graph theory. Some of the notable mathematical concepts bearing Kuratowski's name include Kuratowski's theorem, Kuratowski closure axioms, Kuratowski-Zorn lemma and Kuratowski's intersection theorem. == Life and career == === Early life === Kazimierz Kuratowski was born in Warsaw, (then part of Congress Poland controlled by the Russian Empire), on 2 February 1896. He was a son of Marek Kuratow, a barrister, and Róża Karzewska. He completed a Warsaw secondary school, which was named after general Paweł Chrzanowski. In 1913, he enrolled in an engineering course at the University of Glasgow in Scotland, in part because he did not wish to study in Russian; instruction in Polish was prohibited. He completed only one year of study when the outbreak of World War I precluded any further enrolment. In 1915, Russian forces withdrew from Warsaw and Warsaw University was reopened with Polish as the language of instruction. Kuratowski restarted his university education there the same year, this time in mathematics. He obtained his Ph.D. in 1921, in the newly established Second Polish Republic. === Doctoral thesis === In autumn 1921 Kuratowski was awarded the Ph.D. degree for his groundbreaking work. His thesis statement consisted of two parts. One was devoted to an axiomatic construction of topology via the closure axioms. This first part (republished in a slightly modified form in 1922) has been cited in hundreds of scientific articles. The second part of Kuratowski's thesis was devoted to continua irreducible between two points. This was the subject of a French doctoral thesis written by Zygmunt Janiszewski. Since Janiszewski was deceased, Kuratowski's supervisor was Stefan Mazurkiewicz. Kuratowski's thesis solved certain problems in set theory raised by a Belgian mathematician, Charles-Jean Étienne Gustave Nicolas, Baron de la Vallée Poussin. === Academic career until World War II === Two years later, in 1923, Kuratowski was appointed deputy professor of mathematics at Warsaw University. He was then appointed a full professor of mathematics at Lwów Polytechnic in Lwów, in 1927. He was the head of the Mathematics department there until 1933. Kuratowski was also dean of the department twice. In 1929, Kuratowski became a member of the Warsaw Scientific Society While Kuratowski associated with many of the scholars of the Lwów School of Mathematics, such as Stefan Banach and Stanislaw Ulam, and the circle of mathematicians based around the Scottish Café he kept close connections with Warsaw. Kuratowski left Lwów for Warsaw in 1934, before the famous Scottish Book was begun (in 1935), hence did not contribute any problems to it. He did however, collaborate closely with Banach in solving important problems in measure theory. In 1934 he was appointed the professor at Warsaw University. A year later Kuratowski was nominated as the head of the Mathematics department there. From 1936 to 1939 he was secretary of the Mathematics Committee in The Council of Science and Applied Sciences. === During and after the war === During World War II, he gave lectures at the underground university in Warsaw, since higher education for Poles was forbidden under German occupation. In February 1945, Kuratowski started to lecture at the reopened Warsaw University. In 1945, he became a member of the Polish Academy of Learning, in 1946 he was appointed vice-president of the Mathematics department at Warsaw University, and from 1949 he was chosen to be the vice-president of Warsaw Scientific Society. In 1952 he became a member of the Polish Academy of Sciences, of which he was the vice-president from 1957 to 1968. After World War II, Kuratowski was actively involved in the rebuilding of scientific life in Poland. He helped to establish the State Institute of Mathematics, which was incorporated into the Polish Academy of Sciences in 1952. From 1948 until 1967 Kuratowski was director of the Institute of Mathematics of the Polish Academy of Sciences, and was also a long-time chairman of the Polish and International Mathematics Societies. He served as vice-president of the International Mathematical Union (1963–1966) as well as president of the Scientific Council of the State Institute of Mathematics (1968–1980). From 1948 to 1980 he was the head of the topology section. One of his students was Andrzej Mostowski. == Legacy == Kazimierz Kuratowski was one of a celebrated group of Polish mathematicians who would meet at Lwów's Scottish Café. He was a president of the Polish Mathematical Society (PTM) and a member of the Warsaw Scientific Society (TNW). What is more, he was chief editor in "Fundamenta Mathematicae", a series of publications in "Polish Mathematical Society Annals". Furthermore, Kuratowski worked as an editor in the Polish Academy of Sciences Bulletin. He was also one of the writers of the Mathematical monographs, which were created in cooperation with the Institute of Mathematics of the Polish Academy of Sciences (IMPAN). High quality research monographs of the representatives of Warsaw's and Lwów's School of Mathematics, which concerned all areas of pure and applied mathematics, were published in these volumes. Kazimierz Kuratowski was an active member of many scientific societies and foreign scientific academies, including the Royal Society of Edinburgh, Austria, Germany, Hungary, Italy and the Union of Soviet Socialist Republics (USSR). === Kazimierz Kuratowski Prize === In 1981, IMPAN, the Polish Mathematical Society, and Kuratowski's daughter Zofia Kuratowska established a prize in his name, the Kuratowski Prize, for achievements in mathematics to people under the age of 30 years. The prize is considered the most prestigious of awards for young Polish mathematicians; past recipients have included Józef H. Przytycki, Mariusz Lemańczyk, Tomasz Łuczak, Mikołaj Bojańczyk, and Wojciech Samotij. == Research == Kuratowski's research mainly focused on abstract topological and metric structures. He implemented the closure axioms (known in mathematical circles as the Kuratowski closure axioms). This was fundamental for the development of topological space theory and irreducible continuum theory between two points. The most valuable results, which were obtained by Kuratowski after the war are those that concern the relationship between topology and analytic functions (theory), and also research in the field of cutting Euclidean spaces. Together with Ulam, who was Kuratowski's most talented student during the Lwów Period, he introduced the concept of so-called quasi homeomorphism that opened up a new field in topological studies. Kuratowski's research in the field of measure theory, including research with Banach and Tarski, was continued by many students. Moreover, with Alfred Tarski and Wacław Sierpiński he provided most of the theory concerning Polish spaces (that are indeed named after these mathematicians and their legacy). Knaster and Kuratowski brought a comprehensive and precise study to connected components theory. It was applied to issues such as cutting-plane, with the paradoxical examples of connected components. Kuratowski proved the Kuratowski-Zorn lemma (often called just Zorn's lemma) in 1922. This result has important connections to many basic theorems. Zorn gave its application in 1935. Kuratowski implemented many concepts in set theory and topology. In many cases, Kuratowski established new terminologies and symbolisms. His contributions to mathematics include: a characterization of topological spaces which are now called the Kuratowski closure axioms; proof of the Kuratowski–Zorn lemma; in graph theory, the characterization of planar graphs now known as Kuratowski's theorem; identification of the ordered pair ( x , y ) {\displaystyle (x,y)} with the set { { x } , { x , y } } ; {\displaystyle \{\{x\},\{x,y\}\};} the Kuratowski finite set definition, see Kuratowski-finite; introduction of the Tarski–Kuratowski algorithm; Kuratowski's closure-complement problem; Kuratowski's free set theorem; Kuratowski's intersection theorem; Knaster-Kuratowski fan; Kuratowski-Ulam theorem; Kuratowski convergence of subsets of metric spaces; the Kuratowski and Ryll-Nardzewski measurable selection theorem; Kuratowski's post-war works were mainly focused on three strands: The development of homotopy in continuous functions. The construction of connected space theory in higher dimensions. The uniform depiction of cutting Euclidean spaces by any of its subsets, based on the properties of continuous transformations of these sets. == Publications == Among over 170 published works are valuable monographs and books including Topologie (Vol. I, 1933, translated into English and Russian, and Vol. II, 1950) and Introduction to Set Theory and Topology (Vol. I, 1952, translated into English, French, Spanish, and Bulgarian). He authored "A Half Century of Polish Mathematics 1920-1970: Remembrances and Reflections" (1973) and "Notes to his autobiography" (1981). The latter was published posthumously thanks to Kuratowski's daughter Zofia Kuratowska, who prepared his notes for printing. Kazimierz Kuratowski represented Polish mathematics in the International Mathematical Union where he was vice president from 1963 to 1966. What is more, he participated in numerous international congresses and lectured at dozens of universities around the world. He was an honorary causa doctor at the Universities in Glasgow, Prague, Wroclaw, and Paris. He received the highest national awards, as well as a gold medal of the Czechoslovak Academy of Sciences, and the Polish Academy of Sciences. Kuratowski died on 18 June 1980 in Warsaw. Kuratowski, Kazimierz; Mostowski, Andrzej (1976) [1968], Set theory. With an introduction to descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 86 (Second ed.), Amsterdam-New York-Oxford: North-Holland Publishing Co., MR 0485384 == See also == List of Polish mathematicians Scottish Café List of things named after Kazimierz Kuratowski Timeline of Polish science and technology == Notes == == References == == External links == TOPOLOGIE I, Espaces Métrisables, Espaces Complets Monografie Matematyczne series, vol. 20, Polish Mathematical Society, Warszawa-Lwów, 1948. TOPOLOGIE II, Espaces Compacts, Espaces Connexes, Plan Euclidien Monografie Matematyczne series, vol. 21, Polish Mathematical Society, Warszawa-Lwów, 1950. O'Connor, John J.; Robertson, Edmund F., "Kazimierz Kuratowski", MacTutor History of Mathematics Archive, University of St Andrews Kazimierz Kuratowski at the Mathematics Genealogy Project
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Wikipedia:Knaster–Tarski theorem#0
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In the mathematical areas of order and lattice theory, the Knaster–Tarski theorem, named after Bronisław Knaster and Alfred Tarski, states the following: Let (L, ≤) be a complete lattice and let f : L → L be an order-preserving (monotonic) function w.r.t. ≤. Then the set of fixed points of f in L forms a complete lattice under ≤. It was Tarski who stated the result in its most general form, and so the theorem is often known as Tarski's fixed-point theorem. Some time earlier, Knaster and Tarski established the result for the special case where L is the lattice of subsets of a set, the power set lattice. The theorem has important applications in formal semantics of programming languages and abstract interpretation, as well as in game theory. A kind of converse of this theorem was proved by Anne C. Davis: If every order-preserving function f : L → L on a lattice L has a fixed point, then L is a complete lattice. == Consequences: least and greatest fixed points == Since complete lattices cannot be empty (they must contain a supremum and infimum of the empty set), the theorem in particular guarantees the existence of at least one fixed point of f, and even the existence of a least fixed point (or greatest fixed point). In many practical cases, this is the most important implication of the theorem. The least fixpoint of f is the least element x such that f(x) = x, or, equivalently, such that f(x) ≤ x; the dual holds for the greatest fixpoint, the greatest element x such that f(x) = x. If f(lim xn) = lim f(xn) for all ascending sequences xn, then the least fixpoint of f is lim f n(0) where 0 is the least element of L, thus giving a more "constructive" version of the theorem. (See: Kleene fixed-point theorem.) More generally, if f is monotonic, then the least fixpoint of f is the stationary limit of f α(0), taking α over the ordinals, where f α is defined by transfinite induction: f α+1 = f (f α) and f γ for a limit ordinal γ is the least upper bound of the f β for all β ordinals less than γ. The dual theorem holds for the greatest fixpoint. For example, in theoretical computer science, least fixed points of monotonic functions are used to define program semantics, see Least fixed point § Denotational semantics for an example. Often a more specialized version of the theorem is used, where L is assumed to be the lattice of all subsets of a certain set ordered by subset inclusion. This reflects the fact that in many applications only such lattices are considered. One then usually is looking for the smallest set that has the property of being a fixed point of the function f. Abstract interpretation makes ample use of the Knaster–Tarski theorem and the formulas giving the least and greatest fixpoints. The Knaster–Tarski theorem can be used to give a simple proof of the Cantor–Bernstein–Schroeder theorem and it is also used in establishing the Banach–Tarski paradox. == Weaker versions of the theorem == Weaker versions of the Knaster–Tarski theorem can be formulated for ordered sets, but involve more complicated assumptions. For example: Let L be a partially ordered set with a least element (bottom) and let f : L → L be an monotonic function. Further, suppose there exists u in L such that f(u) ≤ u and that any chain in the subset { x ∈ L ∣ x ≤ f ( x ) , x ≤ u } {\displaystyle \{x\in L\mid x\leq f(x),x\leq u\}} has a supremum. Then f admits a least fixed point. This can be applied to obtain various theorems on invariant sets, e.g. the Ok's theorem: For the monotone map F : P(X ) → P(X ) on the family of (closed) nonempty subsets of X, the following are equivalent: (o) F admits A in P(X ) s.t. A ⊆ F ( A ) {\displaystyle A\subseteq F(A)} , (i) F admits invariant set A in P(X ) i.e. A = F ( A ) {\displaystyle A=F(A)} , (ii) F admits maximal invariant set A, (iii) F admits the greatest invariant set A. In particular, using the Knaster-Tarski principle one can develop the theory of global attractors for noncontractive discontinuous (multivalued) iterated function systems. For weakly contractive iterated function systems the Kantorovich theorem (known also as Tarski-Kantorovich fixpoint principle) suffices. Other applications of fixed-point principles for ordered sets come from the theory of differential, integral and operator equations. == Proof == Let us restate the theorem. For a complete lattice ⟨ L , ≤ ⟩ {\displaystyle \langle L,\leq \rangle } and a monotone function f : L → L {\displaystyle f\colon L\rightarrow L} on L, the set of all fixpoints of f is also a complete lattice ⟨ P , ≤ ⟩ {\displaystyle \langle P,\leq \rangle } , with: ⋁ P = ⋁ { x ∈ L ∣ x ≤ f ( x ) } {\displaystyle \bigvee P=\bigvee \{x\in L\mid x\leq f(x)\}} as the greatest fixpoint of f ⋀ P = ⋀ { x ∈ L ∣ x ≥ f ( x ) } {\displaystyle \bigwedge P=\bigwedge \{x\in L\mid x\geq f(x)\}} as the least fixpoint of f. Proof. We begin by showing that P has both a least element and a greatest element. Let D = {x | x ≤ f(x)} and x ∈ D (we know that at least 0L belongs to D). Then because f is monotone we have f(x) ≤ f(f(x)), that is f(x) ∈ D. Now let u = ⋁ D {\displaystyle u=\bigvee D} (u exists because D ⊆ L and L is a complete lattice). Then for all x ∈ D it is true that x ≤ u and f(x) ≤ f(u), so x ≤ f(x) ≤ f(u). Therefore, f(u) is an upper bound of D, but u is the least upper bound, so u ≤ f(u), i.e. u ∈ D. Then f(u) ∈ D (because f(u) ≤ f(f(u))) and so f(u) ≤ u from which follows f(u) = u. Because every fixpoint is in D we have that u is the greatest fixpoint of f. The function f is monotone on the dual (complete) lattice ⟨ L o p , ≥ ⟩ {\displaystyle \langle L^{op},\geq \rangle } . As we have just proved, its greatest fixpoint exists. It is the least fixpoint of L, so P has least and greatest elements, that is more generally, every monotone function on a complete lattice has a least fixpoint and a greatest fixpoint. For a, b in L we write [a, b] for the closed interval with bounds a and b: {x ∈ L | a ≤ x ≤ b}. If a ≤ b, then ⟨[a, b], ≤⟩ is a complete lattice. It remains to be proven that P is a complete lattice. Let 1 L = ⋁ L {\displaystyle 1_{L}=\bigvee L} , W ⊆ P and w = ⋁ W {\displaystyle w=\bigvee W} . We show that f([w, 1L]) ⊆ [w, 1L]. Indeed, for every x ∈ W we have x = f(x) and since w is the least upper bound of W, x ≤ f(w). In particular w ≤ f(w). Then from y ∈ [w, 1L] follows that w ≤ f(w) ≤ f(y), giving f(y) ∈ [w, 1L] or simply f([w, 1L]) ⊆ [w, 1L]. This allows us to look at f as a function on the complete lattice [w, 1L]. Then it has a least fixpoint there, giving us the least upper bound of W. We've shown that an arbitrary subset of P has a supremum, that is, P is a complete lattice. == Computing a Tarski fixed-point == Chang, Lyuu and Ti present an algorithm for finding a Tarski fixed-point in a totally-ordered lattice, when the order-preserving function is given by a value oracle. Their algorithm requires O ( log L ) {\displaystyle O(\log L)} queries, where L is the number of elements in the lattice. In contrast, for a general lattice (given as an oracle), they prove a lower bound of Ω ( L ) {\displaystyle \Omega (L)} queries. Deng, Qi and Ye present several algorithms for finding a Tarski fixed-point. They consider two kinds of lattices: componentwise ordering and lexicographic ordering. They consider two kinds of input for the function f: value oracle, or a polynomial function. Their algorithms have the following runtime complexity (where d is the number of dimensions, and Ni is the number of elements in dimension i): The algorithms are based on binary search. On the other hand, determining whether a given fixed point is unique is computationally hard: For d=2, for componentwise lattice and a value-oracle, the complexity of O ( log 2 L ) {\displaystyle O(\log ^{2}L)} is optimal. But for d>2, there are faster algorithms: Fearnley, Palvolgyi and Savani presented an algorithm using only O ( log 2 ⌈ d / 3 ⌉ L ) {\displaystyle O(\log ^{2\lceil d/3\rceil }L)} queries. In particular, for d=3, only O ( log 2 L ) {\displaystyle O(\log ^{2}L)} queries are needed. Chen and Li presented an algorithm using only O ( log ⌈ ( d + 1 ) / 2 ⌉ L ) {\displaystyle O(\log ^{\lceil (d+1)/2\rceil }L)} queries. == Application in game theory == Tarski's fixed-point theorem has applications to supermodular games. A supermodular game (also called a game of strategic complements) is a game in which the utility function of each player has increasing differences, so the best response of a player is a weakly-increasing function of other players' strategies. For example, consider a game of competition between two firms. Each firm has to decide how much money to spend on research. In general, if one firm spends more on research, the other firm's best response is to spend more on research too. Some common games can be modeled as supermodular games, for example Cournot competition, Bertrand competition and Investment Games. Because the best-response functions are monotone, Tarski's fixed-point theorem can be used to prove the existence of a pure-strategy Nash equilibrium (PNE) in a supermodular game. Moreover, Topkis showed that the set of PNE of a supermodular game is a complete lattice, so the game has a "smallest" PNE and a "largest" PNE. Echenique presents an algorithm for finding all PNE in a supermodular game. His algorithm first uses best-response sequences to find the smallest and largest PNE; then, he removes some strategies and repeats, until all PNE are found. His algorithm is exponential in the worst case, but runs fast in practice. Deng, Qi and Ye show that a PNE can be computed efficiently by finding a Tarski fixed-point of an order-preserving mapping associated with the game. == See also == Modal μ-calculus == Notes == == References == Andrzej Granas and James Dugundji (2003). Fixed Point Theory. Springer-Verlag, New York. ISBN 978-0-387-00173-9. Forster, T. (2003-07-21). Logic, Induction and Sets. Cambridge University Press. ISBN 978-0-521-53361-4. == Further reading == S. Hayashi (1985). "Self-similar sets as Tarski's fixed points". Publications of the Research Institute for Mathematical Sciences. 21 (5): 1059–1066. doi:10.2977/prims/1195178796. J. Jachymski; L. Gajek; K. Pokarowski (2000). "The Tarski-Kantorovitch principle and the theory of iterated function systems". Bull. Austral. Math. Soc. 61 (2): 247–261. doi:10.1017/S0004972700022243. E.A. Ok (2004). "Fixed set theory for closed correspondences with applications to self-similarity and games". Nonlinear Anal. 56 (3): 309–330. CiteSeerX 10.1.1.561.4581. doi:10.1016/j.na.2003.08.001. B.S.W. Schröder (1999). "Algorithms for the fixed point property". Theoret. Comput. Sci. 217 (2): 301–358. doi:10.1016/S0304-3975(98)00273-4. S. Heikkilä (1990). "On fixed points through a generalized iteration method with applications to differential and integral equations involving discontinuities". Nonlinear Anal. 14 (5): 413–426. doi:10.1016/0362-546X(90)90082-R. R. Uhl (2003). "Smallest and greatest fixed points of quasimonotone increasing mappings". Mathematische Nachrichten. 248–249: 204–210. doi:10.1002/mana.200310016. S2CID 120679842. == External links == J. B. Nation, Notes on lattice theory. An application to an elementary combinatorics problem: Given a book with 100 pages and 100 lemmas, prove that there is some lemma written on the same page as its index
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Wikipedia:Kneser's theorem (differential equations)#0
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In mathematics, the Kneser theorem can refer to two distinct theorems in the field of ordinary differential equations: the first one, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not; the other one, named after Hellmuth Kneser, is about the topology of the set of all solutions of an initial value problem with continuous right hand side. == Statement of the theorem due to A. Kneser == Consider an ordinary linear homogeneous differential equation of the form y ″ + q ( x ) y = 0 {\displaystyle y''+q(x)y=0} with q : [ 0 , + ∞ ) → R {\displaystyle q:[0,+\infty )\to \mathbb {R} } continuous. We say this equation is oscillating if it has a solution y with infinitely many zeros, and non-oscillating otherwise. The theorem states that the equation is non-oscillating if lim sup x → + ∞ x 2 q ( x ) < 1 4 {\displaystyle \limsup _{x\to +\infty }x^{2}q(x)<{\tfrac {1}{4}}} and oscillating if lim inf x → + ∞ x 2 q ( x ) > 1 4 . {\displaystyle \liminf _{x\to +\infty }x^{2}q(x)>{\tfrac {1}{4}}.} === Example === To illustrate the theorem consider q ( x ) = ( 1 4 − a ) x − 2 for x > 0 {\displaystyle q(x)=\left({\frac {1}{4}}-a\right)x^{-2}\quad {\text{for}}\quad x>0} where a {\displaystyle a} is real and non-zero. According to the theorem, solutions will be oscillating or not depending on whether a {\displaystyle a} is positive (non-oscillating) or negative (oscillating) because lim sup x → + ∞ x 2 q ( x ) = lim inf x → + ∞ x 2 q ( x ) = 1 4 − a {\displaystyle \limsup _{x\to +\infty }x^{2}q(x)=\liminf _{x\to +\infty }x^{2}q(x)={\frac {1}{4}}-a} To find the solutions for this choice of q ( x ) {\displaystyle q(x)} , and verify the theorem for this example, substitute the 'Ansatz' y ( x ) = x n {\displaystyle y(x)=x^{n}} which gives n ( n − 1 ) + 1 4 − a = ( n − 1 2 ) 2 − a = 0 {\displaystyle n(n-1)+{\frac {1}{4}}-a=\left(n-{\frac {1}{2}}\right)^{2}-a=0} This means that (for non-zero a {\displaystyle a} ) the general solution is y ( x ) = A x 1 2 + a + B x 1 2 − a {\displaystyle y(x)=Ax^{{\frac {1}{2}}+{\sqrt {a}}}+Bx^{{\frac {1}{2}}-{\sqrt {a}}}} where A {\displaystyle A} and B {\displaystyle B} are arbitrary constants. It is not hard to see that for positive a {\displaystyle a} the solutions do not oscillate while for negative a = − ω 2 {\displaystyle a=-\omega ^{2}} the identity x 1 2 ± i ω = x e ± ( i ω ) ln x = x ( cos ( ω ln x ) ± i sin ( ω ln x ) ) {\displaystyle x^{{\frac {1}{2}}\pm i\omega }={\sqrt {x}}\ e^{\pm (i\omega )\ln {x}}={\sqrt {x}}\ (\cos {(\omega \ln x)}\pm i\sin {(\omega \ln x)})} shows that they do. The general result follows from this example by the Sturm–Picone comparison theorem. === Extensions === There are many extensions to this result, such as the Gesztesy–Ünal criterion. == Statement of the theorem due to H. Kneser == While Peano's existence theorem guarantees the existence of solutions of certain initial values problems with continuous right hand side, H. Kneser's theorem deals with the topology of the set of those solutions. Precisely, H. Kneser's theorem states the following: Let f : R × R n → R n {\displaystyle f\colon \mathbb {R} \times \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{n}} be a continuous function on the region R := [ t 0 , t 0 + a ] × { x ∈ R n : ‖ x − x 0 ‖ ≤ b } {\displaystyle {\mathcal {R}}:=[t_{0},t_{0}+a]\times \{x\in \mathbb {R} ^{n}:\Vert x-x_{0}\Vert \leq b\}} , and such that | f ( t , x ) | ≤ M {\displaystyle |f(t,x)|\leq M} for all ( t , x ) ∈ R {\displaystyle (t,x)\in {\mathcal {R}}} . Given a real number c {\displaystyle c} satisfying t 0 < c ≤ t 0 + min ( a , b / M ) {\displaystyle t_{0}<c\leq t_{0}+\min(a,b/M)} , define the set S c {\displaystyle S_{c}} as the set of points x c {\displaystyle x_{c}} for which there is a solution x = x ( t ) {\displaystyle x=x(t)} of x ˙ = f ( t , x ) {\displaystyle {\dot {x}}=f(t,x)} such that x ( t 0 ) = x 0 {\displaystyle x(t_{0})=x_{0}} and x ( c ) = x c {\displaystyle x(c)=x_{c}} . Then S c {\displaystyle S_{c}} is a closed and connected set. == References ==
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Wikipedia:Knut Sydsæter#0
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Knut Sydsæter (5 October 1937 – 29 September 2012) was a Norwegian mathematician. Professor of Mathematics at the University of Oslo. He is known for having written several books in mathematics for economic analysis, mainly in Norwegian and English. However, his books have been released in several other languages such as Swedish, German, Italian, Chinese, Japanese, Portuguese, Spanish, Russian and Hungarian among others. == References == == External links == Knut Sydsæter's Home Page (archived at web.archive.org)
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Wikipedia:Kochukrishnan Asan#0
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Koccukṛṣṇan Āśān (Kṛṣṇadāsa) (1756 - 1812) was an astronomer/astrologer from Kerala, India. He was born in the Neṭuṃpayil family in Thiruvalla in Kerala as the son of an erudite astrologer Rāman Āśān. Kṛṣṇadāsa studied astronomy and astrology initially under his father and later from his teacher Śūlapāṇi Vāriyar of Kozhikode. His works on astronomy and astrology were all written in the local language Malayalam and they were all addressed to the novice. One of them is of special interest. It is a commentary on Āryabhaṭīya in Malayalam prose. Apart from the fact that the work is in prose, it is also important because it quotes several authorities including Bhāāskara I, Saṅgamagrāma Mādhava and Vaṭaśreṇi Parameśvara. The other works include: Pañcabodha Bhāṣājātakapaddhati (a free rendering-cum-commentary of the Jātakapaddhati of Vaṭaśreṇi Parameśvara incorporating several topics not dealt with in the original Kaṇkkuśāstraṃ Bhāṣāgolayukti == References ==
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Wikipedia:Koecher–Vinberg theorem#0
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Ernest Borisovich Vinberg (Russian: Эрне́ст Бори́сович Ви́нберг; 26 July 1937 – 12 May 2020) was a Soviet and Russian mathematician, who worked on Lie groups and algebraic groups, discrete subgroups of Lie groups, invariant theory, and representation theory. He introduced Vinberg's algorithm and the Koecher–Vinberg theorem. He was a recipient of the 1997 Humboldt Prize. He was on the executive committee of the Moscow Mathematical Society. In 1983, he was an Invited Speaker with a talk on Discrete reflection groups in Lobachevsky spaces at the International Congress of Mathematicians in Warsaw. In 2010, he was elected an International Honorary Member of the American Academy of Arts and Sciences. Ernest Vinberg died from pneumonia caused by COVID-19 on 12 May 2020. == Selected publications == Linear Representations of Groups. Basler Lehrbücher. Vol. 2. Translated by A. Iacob. Birkhäuser. 1989. ISBN 3-7643-2288-8. A Course in Algebra. Graduate Studies in Mathematics. Vol. 56. Translated by Alexander Retakh. American Mathematical Society. 2003. ISBN 0-8218-3413-4. editor and co-author: Lie Groups and Invariant Theory. Advances in the Mathematical Sciences. Vol. 56. American Mathematical Society. 2005. ISBN 0-8218-3733-8. (contains Construction of the exceptional simple Lie algebras) with A. L. Onishchik: Lie Groups and Algebraic Groups. Springer Series in Soviet Mathematics. Springer-Verlag. 1990. ISBN 978-0-387-50614-2. 2012 pbk edition with V. V. Gorbatsevich, A. L. Onishchik: Foundations of Lie Theory and Lie Transformation Groups. Translated by A. Kozlowski. Springer. 1997. ISBN 978-3-642-57999-8. Vinberg, E. B. (28 February 1985). "Hyperbolic reflection groups". Russian Mathematical Surveys. 40 (1): 31–75. Bibcode:1985RuMaS..40...31V. doi:10.1070/RM1985v040n01ABEH003527. S2CID 250912767. (ed.) Geometry II: Spaces of Constant Curvature. Encyclopedia of Mathematical Sciences. Vol. 29. Springer. 1993. doi:10.1007/978-3-662-02901-5. ISBN 978-3-642-08086-9. (contains: Vinberg et alia: Geometry of spaces of constant curvature, Discrete groups of motions of spaces of constant curvature) == References == == External links == Ernest Vinberg at the Mathematics Genealogy Project Humboldt Research Award Ernest Borisovich Vinberg, Moscow Mathematical Journal
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Wikipedia:Kokichi Sugihara#0
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Kōkichi Sugihara (Japanese: 杉原厚吉, born June 29, 1948, in Gifu Prefecture) is a Japanese mathematician and artist known for his three-dimensional optical illusions that appear to make marbles roll uphill, pull objects to the highest point of a building's roof, and make circular pipes look rectangular. His illusions, which often involve videos of three-dimensional objects shown from carefully chosen perspectives, won first place at the Best Illusion of the Year Contest in 2010, 2013, 2018,and 2020 and second place in 2015 and 2016. == Education and career == Sugihara earned bachelor's, master's, and doctoral degrees in mathematical engineering from the University of Tokyo in 1971, 1973, and 1980 respectively. From 1973 to 1981 he worked as a researcher at the Ministry of International Trade and Industry. He then became an associate professor in the Department of Information and Computer Engineering at Nagoya University in 1981, and moved back to the Department of Mathematical Engineering and Information Physics at the University of Tokyo in 1986. Since 2009 he has been a professor at Meiji University. == Illusions == Five of Sugihara's illusions have won awards at the annual Best Illusion of the Year Contest: In 2010, his illusion "Impossible Motion: Magnet Slopes" won first place in the contest. This illusion uses forced perspective to show marbles seemingly rolling up ramps. In 2013 he won first place again (with Jun Ono and Akiyasu Tomoeda) for "Rotation Generated by Translation", an illusion that uses Moiré patterns to create the appearance of rotation from objects moving only by translation. In 2015 his "Ambiguous Garage Roof" won second place. The illusion appears to show a convex roof surface that is reflected in a mirror to a corrugated zig-zag shape. Neither of these appearances accurately describes the true shape of the roof. In 2016 he won second place again for "Ambiguous Cylinder Illusion", which shows a stack of cylinders that from one point of view appear to have a circular cross-section, and from another point of view appear rectangular. In 2018 he won his first place for his "Triply Ambiguous Object", which shows a rectangular object with three different interpretations when seen from three special viewpoints. In 2020 he won his first place for his "3D Schroeder Staircase". His interest in illusions stems from his research in the 1980s on automating the analysis of perspective drawings, which he published in the 1986 MIT Press book Machine Interpretation of Line Drawings. When he asked his computer system to interpret impossible objects such as the ones in the art of M. C. Escher, he discovered that they could be interpreted as drawings of real objects with unexpected shapes. == Other research == Sugihara's research also includes the study of Voronoi diagrams. With three co-authors, he wrote Spatial Tessellations: Concepts and Applications of Voronoi Diagrams (Wiley, 1994; 2nd ed., 2009). == References == == External links == Home page Anomalous Objects – a selection of still images published by Sugihara Optical Illusionist - a video produced by Meiji University featuring an interview with Sugihara showcasing some of his works
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Wikipedia:Kolakoski sequence#0
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In mathematics, the Kolakoski sequence, sometimes also known as the Oldenburger–Kolakoski sequence, is an infinite sequence of symbols {1,2} that is the sequence of run lengths in its own run-length encoding. It is named after the recreational mathematician William Kolakoski (1944–97) who described it in 1965, but it was previously discussed by Rufus Oldenburger in 1939. == Definition == The initial terms of the Kolakoski sequence are: 1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,... (sequence A000002 in the OEIS) Each symbol occurs in a "run" (a sequence of equal elements) of either one or two consecutive terms, and writing down the lengths of these runs gives exactly the same sequence: 1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,2,2,... 1, 2 , 2 ,1,1, 2 ,1, 2 , 2 ,1, 2 , 2 ,1,1, 2 ,1,1, 2 , 2 ,1, 2 ,1,1, 2 ,1, 2 , 2 ,1,1, 2 ,... The description of the Kolakoski sequence is therefore reversible. If K stands for "the Kolakoski sequence", description #1 logically implies description #2 (and vice versa): 1. The terms of K are generated by the runs (i.e., run-lengths) of K 2. The runs of K are generated by the terms of K Accordingly, one can say that each term of the Kolakoski sequence generates a run of one or two future terms. The first 1 of the sequence generates a run of "1", i.e. itself; the first 2 generates a run of "22", which includes itself; the second 2 generates a run of "11"; and so on. Each number in the sequence is the length of the next run to be generated, and the element to be generated alternates between 1 and 2: 1,2 (length of sequence l = 2; sum of terms s = 3) 1,2,2 (l = 3, s = 5) 1,2,2,1,1 (l = 5, s = 7) 1,2,2,1,1,2,1 (l = 7, s = 10) 1,2,2,1,1,2,1,2,2,1 (l = 10, s = 15) 1,2,2,1,1,2,1,2,2,1,2,2,1,1,2 (l = 15, s = 23) As can be seen, the length of the sequence at each stage is equal to the sum of terms in the previous stage. This animation illustrates the process: These self-generating properties, which remain if the sequence is written without the initial 1, mean that the Kolakoski sequence can be described as a fractal, or mathematical object that encodes its own representation on other scales. Bertran Steinsky has created a recursive formula for the i-th term of the sequence. == Research == === Aperiodicity and cube-freeness === The sequence is not eventually periodic, that is, its terms do not have a general repeating pattern (cf. irrational numbers like π and √2). More generally, the sequence is cube-free, i.e., has no substring of the form w w w {\displaystyle www} with w {\displaystyle w} some nonempty finite string. === Density === It seems plausible that the density of 1s in the Kolakoski {1,2}-sequence is 1/2, but this conjecture remains unproved. Václav Chvátal has proved that the upper density of 1s is less than 0.50084. Nilsson has used the same method with far greater computational power to obtain the bound 0.500080. Although calculations of the first 3×108 values of the sequence appeared to show its density converging to a value slightly different from 1/2, later calculations that extended the sequence to its first 1013 values show the deviation from a density of 1/2 growing smaller, as one would expect if the limiting density actually is 1/2. === Connection with tag systems === The Kolakoski sequence can also be described as the result of a simple cyclic tag system. However, as this system is a 2-tag system rather than a 1-tag system (that is, it replaces pairs of symbols by other sequences of symbols, rather than operating on a single symbol at a time) it lies in the region of parameters for which tag systems are Turing complete, making it difficult to use this representation to reason about the sequence. == Algorithms == The Kolakoski sequence may be generated by an algorithm that, in the i-th iteration, reads the value xi that has already been output as the i-th value of the sequence (or, if no such value has been output yet, sets xi = i). Then, if i is odd, it outputs xi copies of the number 1, while if i is even, it outputs xi copies of the number 2. Thus, the first few steps of the algorithm are: The first value has not yet been output, so set x1 = 1, and output 1 copy of the number 1 The second value has not yet been output, so set x2 = 2, and output 2 copies of the number 2 The third value x3 was output as 2 in the second step, so output 2 copies of the number 1. The fourth value x4 was output as 1 in the third step, so output 1 copy of the number 2. Etc. This algorithm takes linear time, but because it needs to refer back to earlier positions in the sequence it needs to store the whole sequence, taking linear space. An alternative algorithm that generates multiple copies of the sequence at different speeds, with each copy of the sequence using the output of the previous copy to determine what to do at each step, can be used to generate the sequence in linear time and only logarithmic space. == See also == Golomb sequence — another self-generating sequence based on run-length Gijswijt's sequence Look-and-say sequence == Notes == == Further reading == Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. p. 337. ISBN 978-0-521-82332-6. Zbl 1086.11015. Dekking, F. M. (1997). "What Is the Long Range Order in the Kolakoski Sequence?". In Moody, R. V. (ed.). Proceedings of the NATO Advanced Study Institute, Waterloo, ON, August 21-September 1, 1995. Dordrecht, Netherlands: Kluwer. pp. 115–125. Fedou, J. M.; Fici, G. (2010). "Some remarks on differentiable sequences and recursivity" (PDF). Journal of Integer Sequences. 13 (3). Article 10.3.2. Keane, M. S. (1991). "Ergodic Theory and Subshifts of Finite Type". In Bedford, T.; Keane, M. (eds.). Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces. Oxford, England: Oxford University Press. pp. 35–70. Lagarias, J. C. (1992). "Number Theory and Dynamical Systems". In Burr, S. A. (ed.). The Unreasonable Effectiveness of Number Theory. Providence, RI: American Mathematical Society. pp. 35–72. ISBN 9780821855010. Păun, Gheorghe; Salomaa, Arto (1996). "Self-Reading Sequences". American Mathematical Monthly. 103 (2): 166–168. doi:10.2307/2975113. JSTOR 2975113. Zbl 0854.68082. Shallit, Jeffrey (1999). "Number theory and formal languages". In Hejhal, Dennis A.; Friedman, Joel; Gutzwiller, Martin C.; Odlyzko, Andrew M. (eds.). Emerging applications of number theory. Based on the proceedings of the IMA summer program, Minneapolis, MN, USA, July 15--26, 1996. The IMA volumes in mathematics and its applications. Vol. 109. Springer-Verlag. pp. 547–570. ISBN 0-387-98824-6. Zbl 0919.00047. == External links == Weisstein, Eric W. "Kolakoski Sequence". MathWorld. Kolakoski Constant to 25000 digits as computed by Olivier Gerard in April 1998 Bellos, Alex (24 July 2017). "The Kolakoski Sequence" (video). Brady Haran. Archived from the original on 2021-12-21. Retrieved 24 July 2017.
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Wikipedia:Kolmogorov–Arnold representation theorem#0
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In real analysis and approximation theory, the Kolmogorov–Arnold representation theorem (or superposition theorem) states that every multivariate continuous function f : [ 0 , 1 ] n → R {\displaystyle f\colon [0,1]^{n}\to \mathbb {R} } can be represented as a superposition of continuous single-variable functions. The works of Vladimir Arnold and Andrey Kolmogorov established that if f is a multivariate continuous function, then f can be written as a finite composition of continuous functions of a single variable and the binary operation of addition. More specifically, f ( x ) = f ( x 1 , … , x n ) = ∑ q = 0 2 n Φ q ( ∑ p = 1 n ϕ q , p ( x p ) ) , {\displaystyle f(\mathbf {x} )=f(x_{1},\ldots ,x_{n})=\sum _{q=0}^{2n}\Phi _{q}\!\left(\sum _{p=1}^{n}\phi _{q,p}(x_{p})\right),} where ϕ q , p : [ 0 , 1 ] → R {\displaystyle \phi _{q,p}\colon [0,1]\to \mathbb {R} } and Φ q : R → R {\displaystyle \Phi _{q}\colon \mathbb {R} \to \mathbb {R} } . There are proofs with specific constructions. It solved a more constrained form of Hilbert's thirteenth problem, so the original Hilbert's thirteenth problem is a corollary. In a sense, they showed that the only true continuous multivariate function is the sum, since every other continuous function can be written using univariate continuous functions and summing.: 180 == History == The Kolmogorov–Arnold representation theorem is closely related to Hilbert's 13th problem. In his Paris lecture at the International Congress of Mathematicians in 1900, David Hilbert formulated 23 problems which in his opinion were important for the further development of mathematics. The 13th of these problems dealt with the solution of general equations of higher degrees. It is known that for algebraic equations of degree 4 the solution can be computed by formulae that only contain radicals and arithmetic operations. For higher orders, Galois theory shows us that the solutions of algebraic equations cannot be expressed in terms of basic algebraic operations. It follows from the so called Tschirnhaus transformation that the general algebraic equation x n + a n − 1 x n − 1 + ⋯ + a 0 = 0 {\displaystyle x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0}=0} can be translated to the form y n + b n − 4 y n − 4 + ⋯ + b 1 y + 1 = 0 {\displaystyle y^{n}+b_{n-4}y^{n-4}+\cdots +b_{1}y+1=0} . The Tschirnhaus transformation is given by a formula containing only radicals and arithmetic operations and transforms. Therefore, the solution of an algebraic equation of degree n {\displaystyle n} can be represented as a superposition of functions of two variables if n < 7 {\displaystyle n<7} and as a superposition of functions of n − 4 {\displaystyle n-4} variables if n ≥ 7 {\displaystyle n\geq 7} . For n = 7 {\displaystyle n=7} the solution is a superposition of arithmetic operations, radicals, and the solution of the equation y 7 + b 3 y 3 + b 2 y 2 + b 1 y + 1 = 0 {\displaystyle y^{7}+b_{3}y^{3}+b_{2}y^{2}+b_{1}y+1=0} . A further simplification with algebraic transformations seems to be impossible which led to Hilbert's conjecture that "A solution of the general equation of degree 7 cannot be represented as a superposition of continuous functions of two variables". This explains the relation of Hilbert's thirteenth problem to the representation of a higher-dimensional function as superposition of lower-dimensional functions. In this context, it has stimulated many studies in the theory of functions and other related problems by different authors. == Variants == A variant of Kolmogorov's theorem that reduces the number of outer functions Φ q {\displaystyle \Phi _{q}} is due to George Lorentz. He showed in 1962 that the outer functions Φ q {\displaystyle \Phi _{q}} can be replaced by a single function Φ {\displaystyle \Phi } . More precisely, Lorentz proved the existence of functions ϕ q , p {\displaystyle \phi _{q,p}} , q = 0 , 1 , … , 2 n {\displaystyle q=0,1,\ldots ,2n} , p = 1 , … , n , {\displaystyle p=1,\ldots ,n,} such that f ( x ) = ∑ q = 0 2 n Φ ( ∑ p = 1 n ϕ q , p ( x p ) ) . {\displaystyle f(\mathbf {x} )=\sum _{q=0}^{2n}\Phi \!\left(\sum _{p=1}^{n}\phi _{q,p}(x_{p})\right).} David Sprecher replaced the inner functions ϕ q , p {\displaystyle \phi _{q,p}} by one single inner function with an appropriate shift in its argument. He proved that there exist real values η , λ 1 , … , λ n {\displaystyle \eta ,\lambda _{1},\ldots ,\lambda _{n}} , a continuous function Φ : R → R {\displaystyle \Phi \colon \mathbb {R} \rightarrow \mathbb {R} } , and a real increasing continuous function ϕ : [ 0 , 1 ] → [ 0 , 1 ] {\displaystyle \phi \colon [0,1]\rightarrow [0,1]} with ϕ ∈ Lip ( ln 2 / ln ( 2 N + 2 ) ) {\displaystyle \phi \in \operatorname {Lip} (\ln 2/\ln(2N+2))} , for N ≥ n ≥ 2 {\displaystyle N\geq n\geq 2} , such that f ( x ) = ∑ q = 0 2 n Φ ( ∑ p = 1 n λ p ϕ ( x p + η q ) + q ) . {\displaystyle f(\mathbf {x} )=\sum _{q=0}^{2n}\Phi \!\left(\sum _{p=1}^{n}\lambda _{p}\phi (x_{p}+\eta q)+q\right).} Phillip A. Ostrand generalized the Kolmogorov superposition theorem to compact metric spaces. For p = 1 , … , m {\displaystyle p=1,\ldots ,m} let X p {\displaystyle X_{p}} be compact metric spaces of finite dimension n p {\displaystyle n_{p}} and let n = ∑ p = 1 m n p {\displaystyle n=\sum _{p=1}^{m}n_{p}} . Then there exists continuous functions ϕ q , p : X p → [ 0 , 1 ] , q = 0 , … , 2 n , p = 1 , … , m {\displaystyle \phi _{q,p}\colon X_{p}\rightarrow [0,1],q=0,\ldots ,2n,p=1,\ldots ,m} and continuous functions G q : [ 0 , 1 ] → R , q = 0 , … , 2 n {\displaystyle G_{q}\colon [0,1]\rightarrow \mathbb {R} ,q=0,\ldots ,2n} such that any continuous function f : X 1 × ⋯ × X m → R {\displaystyle f\colon X_{1}\times \dots \times X_{m}\rightarrow \mathbb {R} } is representable in the form f ( x 1 , … , x m ) = ∑ q = 0 2 n G q ( ∑ p = 1 m ϕ q , p ( x p ) ) . {\displaystyle f(x_{1},\ldots ,x_{m})=\sum _{q=0}^{2n}G_{q}\!\left(\sum _{p=1}^{m}\phi _{q,p}(x_{p})\right).} Kolmogorov-Arnold representation theorem and its aforementioned variants also hold for discontinuous multivariate functions. == Continuous form == In its classic form Kolmogorov-Arnold representation has two layers, where the first, called inner layer, is vector to vector mapping s q = ∑ p = 1 n ϕ q , p ( x p ) , q = 0 , 1 , . . , 2 n {\displaystyle s_{q}=\sum _{p=1}^{n}\phi _{q,p}(x_{p}),\quad q=0,1,..,2n} and the second, outer layer, is vector to scalar mapping f ( x 1 , . . . , x m ) = ∑ q = 0 2 n Φ q ( s q ) . {\displaystyle f(x_{1},...,x_{m})=\sum _{q=0}^{2n}\Phi _{q}\left(s_{q}\right).} The transition from discrete to continuous form for inner layer gives equation of Urysohn with 3D kernel s ( q ) = ∫ p 1 p 2 F [ x ( p ) , p , q ] d p , q ∈ [ q 1 , q 2 ] , {\displaystyle s(q)=\int _{p_{1}}^{p_{2}}F[x(p),p,q]dp,\quad q\in [q_{1},q_{2}],} same transition for the outer layer gives its particular case f = ∫ q 1 q 2 G [ s ( q ) , q ] d q . {\displaystyle f=\int _{q_{1}}^{q_{2}}G[s(q),q]dq.} The generalization of Kolmogorov-Arnold representation known as Kolmogorov-Arnold network in continuous form is a chain of Urysohn equations, where outer equation also may return function or a vector as multiple related targets. Urysohn equation was introduced in 1924 for a different purpose, as function to function mapping with the problem of finding function x ( p ) {\displaystyle x(p)} , provided s ( q ) {\displaystyle s(q)} and F [ x ( p ) , p , q ] {\displaystyle F[x(p),p,q]} . == Limitations == The theorem does not hold in general for complex multi-variate functions, as discussed here. Furthermore, the non-smoothness of the inner functions and their "wild behavior" has limited the practical use of the representation, although there is some debate on this. == Applications == In the field of machine learning, there have been various attempts to use neural networks modeled on the Kolmogorov–Arnold representation. In these works, the Kolmogorov–Arnold theorem plays a role analogous to that of the universal approximation theorem in the study of multilayer perceptrons. == Proof == Here one example is proved. This proof closely follows. A proof for the case of functions depending on two variables is given, as the generalization is immediate. === Setup === Let I {\textstyle I} be the unit interval [ 0 , 1 ] {\textstyle [0,1]} . Let C [ I ] {\textstyle C[I]} be the set of continuous functions of type [ 0 , 1 ] → R {\textstyle [0,1]\to \mathbb {R} } . It is a function space with supremum norm (it is a Banach space). Let f {\textstyle f} be a continuous function of type [ 0 , 1 ] 2 → R {\textstyle [0,1]^{2}\to \mathbb {R} } , and let ‖ f ‖ {\textstyle \|f\|} be the supremum of it on [ 0 , 1 ] 2 {\textstyle [0,1]^{2}} . Let t {\textstyle t} be a positive irrational number. Its exact value is irrelevant. We say that a 5-tuple ( ϕ 1 , … , ϕ 5 ) ∈ C [ I ] 5 {\textstyle (\phi _{1},\dots ,\phi _{5})\in C[I]^{5}} is a Kolmogorov-Arnold tuple if and only if for any f ∈ C [ I 2 ] {\textstyle f\in C[I^{2}]} there exists a continuous function g : R → R {\textstyle g:\mathbb {R} \to \mathbb {R} } , such that f ( x , y ) = ∑ i = 1 5 g ( ϕ i ( x ) + t ϕ i ( y ) ) {\displaystyle f(x,y)=\sum _{i=1}^{5}g(\phi _{i}(x)+t\phi _{i}(y))} In the notation, we have the following: === Proof === Fix a f ∈ C [ I 2 ] {\textstyle f\in C[I^{2}]} . We show that a certain subset U f ⊂ C [ I ] 5 {\textstyle U_{f}\subset C[I]^{5}} is open and dense: There exists continuous g {\textstyle g} such that ‖ g ‖ < 1.01 7 ‖ f ‖ {\textstyle \|g\|<{\frac {1.01}{7}}\|f\|} , and ‖ f ( x , y ) − ∑ i = 1 5 g ( ϕ i ( x ) + t ϕ i ( y ) ) ‖ < 6.01 7 ‖ f ‖ {\displaystyle {\Big \|}f(x,y)-\sum _{i=1}^{5}g(\phi _{i}(x)+t\phi _{i}(y)){\Big \|}<{\frac {6.01}{7}}\|f\|} We can assume that ‖ f ‖ = 1 {\textstyle \|f\|=1} with no loss of generality. By continuity, the set of such 5-tuples is open in C [ I ] 5 {\textstyle C[I]^{5}} . It remains to prove that they are dense. The key idea is to divide [ 0 , 1 ] 2 {\textstyle [0,1]^{2}} into an overlapping system of small squares, each with a unique address, and define g {\textstyle g} to have the appropriate value at each address. ==== Grid system ==== Let ψ 1 ∈ C [ I ] {\textstyle \psi _{1}\in C[I]} . For any ϵ > 0 {\textstyle \epsilon >0} , for all large N {\textstyle N} , we can discretize ψ 1 {\textstyle \psi _{1}} into a continuous function ϕ 1 {\textstyle \phi _{1}} satisfying the following properties: ϕ 1 {\textstyle \phi _{1}} is constant on each of the intervals [ 0 / 5 N , 4 / 5 N ] , [ 5 / 5 N , 9 / 5 N ] , … , [ 1 − 5 / 5 N , 1 − 1 / 5 N ] {\textstyle [0/5N,4/5N],[5/5N,9/5N],\dots ,[1-5/5N,1-1/5N]} . These values are different rational numbers. ‖ ψ 1 − ϕ 1 ‖ < ϵ {\textstyle \|\psi _{1}-\phi _{1}\|<\epsilon } . This function ϕ 1 {\textstyle \phi _{1}} creates a grid address system on [ 0 , 1 ] 2 {\textstyle [0,1]^{2}} , divided into streets and blocks. The blocks are of form [ 0 / 5 N , 4 / 5 N ] × [ 0 / 5 N , 4 / 5 N ] , [ 0 / 5 N , 4 / 5 N ] × [ 5 / 5 N , 9 / 5 N ] , … {\textstyle [0/5N,4/5N]\times [0/5N,4/5N],[0/5N,4/5N]\times [5/5N,9/5N],\dots } . Since f {\textstyle f} is continuous on [ 0 , 1 ] 2 {\textstyle [0,1]^{2}} , it is uniformly continuous. Thus, we can take N {\textstyle N} large enough, so that f {\textstyle f} varies by less than 1 / 7 {\textstyle 1/7} on any block. On each block, ϕ 1 ( x ) + t ϕ 1 ( y ) {\textstyle \phi _{1}(x)+t\phi _{1}(y)} has a constant value. The key property is that, because t {\textstyle t} is irrational, and ϕ 1 {\textstyle \phi _{1}} is rational on the blocks, each block has a different value of ϕ 1 ( x ) + t ϕ 1 ( y ) {\textstyle \phi _{1}(x)+t\phi _{1}(y)} . So, given any 5-tuple ( ψ 1 , … , ψ 5 ) {\textstyle (\psi _{1},\dots ,\psi _{5})} , we construct such a 5-tuple ( ϕ 1 , … , ϕ 5 ) {\textstyle (\phi _{1},\dots ,\phi _{5})} . These create 5 overlapping grid systems. Enumerate the blocks as R i , r {\textstyle R_{i,r}} , where R i , r {\textstyle R_{i,r}} is the r {\textstyle r} -th block of the grid system created by ϕ i {\textstyle \phi _{i}} . The address of this block is a i , r := ϕ i ( x ) + t ϕ i ( y ) {\textstyle a_{i,r}:=\phi _{i}(x)+t\phi _{i}(y)} , for any ( x , y ) ∈ R i , r {\textstyle (x,y)\in R_{i,r}} . By adding a small and linearly independent irrational number (the construction is similar to that of the Hamel basis) to each of ( ϕ 1 , … , ϕ 5 ) {\textstyle (\phi _{1},\dots ,\phi _{5})} , we can ensure that every block has a unique address. By plotting out the entire grid system, one can see that every point in [ 0 , 1 ] 2 {\textstyle [0,1]^{2}} is contained in 3 to 5 blocks, and 2 to 0 streets. ==== Construction of g ==== For each block R i , r {\textstyle R_{i,r}} , if f > 0 {\textstyle f>0} on all of R i , r {\textstyle R_{i,r}} then define g ( a i , r ) = + 1 / 7 {\textstyle g(a_{i,r})=+1/7} ; if f < 0 {\textstyle f<0} on all of R i , r {\textstyle R_{i,r}} then define g ( a i , r ) = − 1 / 7 {\textstyle g(a_{i,r})=-1/7} . Now, linearly interpolate g {\textstyle g} between these defined values. It remains to show this construction has the desired properties. For any ( x , y ) ∈ I 2 {\textstyle (x,y)\in I^{2}} , we consider three cases. If f ( x , y ) ∈ [ 1 / 7 , 7 / 7 ] {\textstyle f(x,y)\in [1/7,7/7]} , then by uniform continuity, f > 0 {\textstyle f>0} on every block R i , r {\textstyle R_{i,r}} that contains the point ( x , y ) {\textstyle (x,y)} . This means that g = 1 / 7 {\textstyle g=1/7} on 3 to 5 of the blocks, and have an unknown value on 2 to 0 of the streets. Thus, we have ∑ i = 1 5 g ( ϕ i ( x ) + t ϕ i ( y ) ) ∈ [ 1 / 7 , 5 / 7 ] {\displaystyle \sum _{i=1}^{5}g(\phi _{i}(x)+t\phi _{i}(y))\in [1/7,5/7]} giving | f ( x , y ) − ∑ i = 1 5 g ( ϕ i ( x ) + t ϕ i ( y ) ) | ∈ [ 0 , 6 / 7 ] {\displaystyle {\Big |}f(x,y)-\sum _{i=1}^{5}g(\phi _{i}(x)+t\phi _{i}(y)){\Big |}\in [0,6/7]} Similarly for f ( x , y ) ∈ [ − 7 / 7 , − 1 / 7 ] {\textstyle f(x,y)\in [-7/7,-1/7]} . If f ( x , y ) ∈ [ − 1 / 7 , 1 / 7 ] {\textstyle f(x,y)\in [-1/7,1/7]} , then since ‖ g ‖ ≤ 1 / 7 {\textstyle \|g\|\leq 1/7} , we still have | f ( x , y ) − ∑ i = 1 5 g ( ϕ i ( x ) + t ϕ i ( y ) ) | ∈ [ 0 , 6 / 7 ] {\displaystyle {\Big |}f(x,y)-\sum _{i=1}^{5}g(\phi _{i}(x)+t\phi _{i}(y)){\Big |}\in [0,6/7]} ==== Baire category theorem ==== Iterating the above construction, then applying the Baire category theorem, we find that the following kind of 5-tuples are open and dense in C [ I ] 5 {\displaystyle C[I]^{5}} : There exists a sequence of g 1 , g 2 , … {\textstyle g_{1},g_{2},\dots } such that ‖ g 1 ‖ < 1.01 7 ‖ f ‖ {\textstyle \|g_{1}\|<{\frac {1.01}{7}}\|f\|} , ‖ g 2 ‖ < 1.01 7 6.01 7 ‖ f ‖ {\textstyle \|g_{2}\|<{\frac {1.01}{7}}{\frac {6.01}{7}}\|f\|} , etc. This allows their sum to be defined: g := ∑ n g n {\textstyle g:=\sum _{n}g_{n}} , which is still continuous and bounded, and it satisfies f ( x , y ) = ∑ i = 1 5 g ( ϕ i ( x ) + t ϕ i ( y ) ) {\displaystyle f(x,y)=\sum _{i=1}^{5}g(\phi _{i}(x)+t\phi _{i}(y))} Since C [ I 2 ] {\textstyle C[I^{2}]} has a countable dense subset, we can apply the Baire category theorem again to obtain the full theorem. === Extensions === The above proof generalizes for n {\textstyle n} -dimensions: Divide the cube [ 0 , 1 ] n {\textstyle [0,1]^{n}} into ( 2 n + 1 ) {\textstyle (2n+1)} interlocking grid systems, such that each point in the cube is on ( n + 1 ) {\textstyle (n+1)} to ( 2 n + 1 ) {\textstyle (2n+1)} blocks, and 0 {\textstyle 0} to n {\textstyle n} streets. Now, since ( n + 1 ) > n {\textstyle (n+1)>n} , the above construction works. Indeed, this is the best possible value. A relatively short proof is given in via dimension theory. In another direction of generality, more conditions can be imposed on the Kolmogorov–Arnold tuples. The proof is given in. (Vituškin, 1954) showed that the theorem is false if we require all functions f , g , ϕ i {\displaystyle f,g,\phi _{i}} to be continuously differentiable. The theorem remains true if we require all ϕ i {\displaystyle \phi _{i}} to be 1-Lipschitz continuous. == References == == Sources == Andrey Kolmogorov, "On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables", Proceedings of the USSR Academy of Sciences, 108 (1956), pp. 179–182; English translation: Amer. Math. Soc. Transl., "17: Twelve Papers on Algebra and Real Functions" (1961), pp. 369–373. Vladimir Arnold, "On functions of three variables", Proceedings of the USSR Academy of Sciences, 114 (1957), pp. 679–681; English translation: Amer. Math. Soc. Transl., "28: Sixteen Papers on Analysis" (1963), pp. 51–54. SpringerLink Vladimir Arnold, "On the representation of continuous functions of three variables as superpositions of continuous functions of two variables", Dokl. Akad. Nauk. SSSR 114:4 (1957), pp. 679–681 (in Russian) SpringerLink Andrey Kolmogorov, "On the representation of continuous functions of several variables as superpositions of continuous functions of one variable and addition", (1957); English translation: Amer. Math. Soc. Transl., "28: Sixteen Papers on Analysis" (1963), PDF == Further reading == S. Ya. Khavinson, Best Approximation by Linear Superpositions (Approximate Nomography), AMS Translations of Mathematical Monographs (1997)
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Wikipedia:Komlós' theorem#0
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Komlós' theorem is a theorem from probability theory and mathematical analysis about the Cesàro convergence of a subsequence of random variables (or functions) and their subsequences to an integrable random variable (or function). It's also an existence theorem for an integrable random variable (or function). There exist a probabilistic and an analytical version for finite measure spaces. The theorem was proven in 1967 by János Komlós. There exists also a generalization from 1970 by Srishti D. Chatterji. == Komlós' theorem == === Probabilistic version === Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space and ξ 1 , ξ 2 , … {\displaystyle \xi _{1},\xi _{2},\dots } be a sequence of real-valued random variables defined on this space with sup n E [ | ξ n | ] < ∞ . {\displaystyle \sup \limits _{n}\mathbb {E} [|\xi _{n}|]<\infty .} Then there exists a random variable ψ ∈ L 1 ( P ) {\displaystyle \psi \in L^{1}(P)} and a subsequence ( η k ) = ( ξ n k ) {\displaystyle (\eta _{k})=(\xi _{n_{k}})} , such that for every arbitrary subsequence ( η ~ n ) = ( η k n ) {\displaystyle ({\tilde {\eta }}_{n})=(\eta _{k_{n}})} when n → ∞ {\displaystyle n\to \infty } then ( η ~ 1 + ⋯ + η ~ n ) n → ψ {\displaystyle {\frac {({\tilde {\eta }}_{1}+\cdots +{\tilde {\eta }}_{n})}{n}}\to \psi } P {\displaystyle P} -almost surely. === Analytic version === Let ( E , A , μ ) {\displaystyle (E,{\mathcal {A}},\mu )} be a finite measure space and f 1 , f 2 , … {\displaystyle f_{1},f_{2},\dots } be a sequence of real-valued functions in L 1 ( μ ) {\displaystyle L^{1}(\mu )} and sup n ∫ E | f n | d μ < ∞ {\displaystyle \sup \limits _{n}\int _{E}|f_{n}|\mathrm {d} \mu <\infty } . Then there exists a function υ ∈ L 1 ( μ ) {\displaystyle \upsilon \in L^{1}(\mu )} and a subsequence ( g k ) = ( f n k ) {\displaystyle (g_{k})=(f_{n_{k}})} such that for every arbitrary subsequence ( g ~ n ) = ( g k n ) {\displaystyle ({\tilde {g}}_{n})=(g_{k_{n}})} if n → ∞ {\displaystyle n\to \infty } then ( g ~ 1 + ⋯ + g ~ n ) n → υ {\displaystyle {\frac {({\tilde {g}}_{1}+\cdots +{\tilde {g}}_{n})}{n}}\to \upsilon } μ {\displaystyle \mu } -almost everywhere. === Explanations === So the theorem says, that the sequence ( η k ) {\displaystyle (\eta _{k})} and all its subsequences converge in Césaro. == Literature == Kabanov, Yuri & Pergamenshchikov, Sergei. (2003). Two-scale stochastic systems. Asymptotic analysis and control. 10.1007/978-3-662-13242-5. Page 250. == References ==
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Wikipedia:Konrad Osterwalder#0
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In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to ordered n-tuples in R d {\displaystyle \mathbb {R} ^{d}} that are pairwise distinct. These functions are called the Schwinger functions (named after Julian Schwinger) and they are real-analytic, symmetric under the permutation of arguments (antisymmetric for fermionic fields), Euclidean covariant and satisfy a property known as reflection positivity. Properties of Schwinger functions are known as Osterwalder–Schrader axioms (named after Konrad Osterwalder and Robert Schrader). Schwinger functions are also referred to as Euclidean correlation functions. == Osterwalder–Schrader axioms == Here we describe Osterwalder–Schrader (OS) axioms for a Euclidean quantum field theory of a Hermitian scalar field ϕ ( x ) {\displaystyle \phi (x)} , x ∈ R d {\displaystyle x\in \mathbb {R} ^{d}} . Note that a typical quantum field theory will contain infinitely many local operators, including also composite operators, and their correlators should also satisfy OS axioms similar to the ones described below. The Schwinger functions of ϕ {\displaystyle \phi } are denoted as S n ( x 1 , … , x n ) ≡ ⟨ ϕ ( x 1 ) ϕ ( x 2 ) … ϕ ( x n ) ⟩ , x k ∈ R d . {\displaystyle S_{n}(x_{1},\ldots ,x_{n})\equiv \langle \phi (x_{1})\phi (x_{2})\ldots \phi (x_{n})\rangle ,\quad x_{k}\in \mathbb {R} ^{d}.} OS axioms from are numbered (E0)-(E4) and have the following meaning: (E0) Temperedness (E1) Euclidean covariance (E2) Positivity (E3) Symmetry (E4) Cluster property === Temperedness === Temperedness axiom (E0) says that Schwinger functions are tempered distributions away from coincident points. This means that they can be integrated against Schwartz test functions which vanish with all their derivatives at configurations where two or more points coincide. It can be shown from this axiom and other OS axioms (but not the linear growth condition) that Schwinger functions are in fact real-analytic away from coincident points. === Euclidean covariance === Euclidean covariance axiom (E1) says that Schwinger functions transform covariantly under rotations and translations, namely: S n ( x 1 , … , x n ) = S n ( R x 1 + b , … , R x n + b ) {\displaystyle S_{n}(x_{1},\ldots ,x_{n})=S_{n}(Rx_{1}+b,\ldots ,Rx_{n}+b)} for an arbitrary rotation matrix R ∈ S O ( d ) {\displaystyle R\in SO(d)} and an arbitrary translation vector b ∈ R d {\displaystyle b\in \mathbb {R} ^{d}} . OS axioms can be formulated for Schwinger functions of fields transforming in arbitrary representations of the rotation group. === Symmetry === Symmetry axiom (E3) says that Schwinger functions are invariant under permutations of points: S n ( x 1 , … , x n ) = S n ( x π ( 1 ) , … , x π ( n ) ) {\displaystyle S_{n}(x_{1},\ldots ,x_{n})=S_{n}(x_{\pi (1)},\ldots ,x_{\pi (n)})} , where π {\displaystyle \pi } is an arbitrary permutation of { 1 , … , n } {\displaystyle \{1,\ldots ,n\}} . Schwinger functions of fermionic fields are instead antisymmetric; for them this equation would have a ± sign equal to the signature of the permutation. === Cluster property === Cluster property (E4) says that Schwinger function S p + q {\displaystyle S_{p+q}} reduces to the product S p S q {\displaystyle S_{p}S_{q}} if two groups of points are separated from each other by a large constant translation: lim b → ∞ S p + q ( x 1 , … , x p , x p + 1 + b , … , x p + q + b ) = S p ( x 1 , … , x p ) S q ( x p + 1 , … , x p + q ) {\displaystyle \lim _{b\to \infty }S_{p+q}(x_{1},\ldots ,x_{p},x_{p+1}+b,\ldots ,x_{p+q}+b)=S_{p}(x_{1},\ldots ,x_{p})S_{q}(x_{p+1},\ldots ,x_{p+q})} . The limit is understood in the sense of distributions. There is also a technical assumption that the two groups of points lie on two sides of the x 0 = 0 {\displaystyle x^{0}=0} hyperplane, while the vector b {\displaystyle b} is parallel to it: x 1 0 , … , x p 0 > 0 , x p + 1 0 , … , x p + q 0 < 0 , b 0 = 0. {\displaystyle x_{1}^{0},\ldots ,x_{p}^{0}>0,\quad x_{p+1}^{0},\ldots ,x_{p+q}^{0}<0,\quad b^{0}=0.} === Reflection positivity === Positivity axioms (E2) asserts the following property called (Osterwalder–Schrader) reflection positivity. Pick any arbitrary coordinate τ and pick a test function fN with N points as its arguments. Assume fN has its support in the "time-ordered" subset of N points with 0 < τ1 < ... < τN. Choose one such fN for each positive N, with the f's being zero for all N larger than some integer M. Given a point x {\displaystyle x} , let x θ {\displaystyle x^{\theta }} be the reflected point about the τ = 0 hyperplane. Then, ∑ m , n ∫ d d x 1 ⋯ d d x m d d y 1 ⋯ d d y n S m + n ( x 1 , … , x m , y 1 , … , y n ) f m ( x 1 θ , … , x m θ ) ∗ f n ( y 1 , … , y n ) ≥ 0 {\displaystyle \sum _{m,n}\int d^{d}x_{1}\cdots d^{d}x_{m}\,d^{d}y_{1}\cdots d^{d}y_{n}S_{m+n}(x_{1},\dots ,x_{m},y_{1},\dots ,y_{n})f_{m}(x_{1}^{\theta },\dots ,x_{m}^{\theta })^{*}f_{n}(y_{1},\dots ,y_{n})\geq 0} where * represents complex conjugation. Sometimes in theoretical physics literature reflection positivity is stated as the requirement that the Schwinger function of arbitrary even order should be non-negative if points are inserted symmetrically with respect to the τ = 0 {\displaystyle \tau =0} hyperplane: S 2 n ( x 1 , … , x n , x n θ , … , x 1 θ ) ≥ 0 {\displaystyle S_{2n}(x_{1},\dots ,x_{n},x_{n}^{\theta },\dots ,x_{1}^{\theta })\geq 0} . This property indeed follows from the reflection positivity but it is weaker than full reflection positivity. ==== Intuitive understanding ==== One way of (formally) constructing Schwinger functions which satisfy the above properties is through the Euclidean path integral. In particular, Euclidean path integrals (formally) satisfy reflection positivity. Let F be any polynomial functional of the field φ which only depends upon the value of φ(x) for those points x whose τ coordinates are nonnegative. Then ∫ D ϕ F [ ϕ ( x ) ] F [ ϕ ( x θ ) ] ∗ e − S [ ϕ ] = ∫ D ϕ 0 ∫ ϕ + ( τ = 0 ) = ϕ 0 D ϕ + F [ ϕ + ] e − S + [ ϕ + ] ∫ ϕ − ( τ = 0 ) = ϕ 0 D ϕ − F [ ( ϕ − ) θ ] ∗ e − S − [ ϕ − ] . {\displaystyle \int {\mathcal {D}}\phi F[\phi (x)]F[\phi (x^{\theta })]^{*}e^{-S[\phi ]}=\int {\mathcal {D}}\phi _{0}\int _{\phi _{+}(\tau =0)=\phi _{0}}{\mathcal {D}}\phi _{+}F[\phi _{+}]e^{-S_{+}[\phi _{+}]}\int _{\phi _{-}(\tau =0)=\phi _{0}}{\mathcal {D}}\phi _{-}F[(\phi _{-})^{\theta }]^{*}e^{-S_{-}[\phi _{-}]}.} Since the action S is real and can be split into S + {\displaystyle S_{+}} , which only depends on φ on the positive half-space ( ϕ + {\displaystyle \phi _{+}} ), and S − {\displaystyle S_{-}} which only depends upon φ on the negative half-space ( ϕ − {\displaystyle \phi _{-}} ), and if S also happens to be invariant under the combined action of taking a reflection and complex conjugating all the fields, then the previous quantity has to be nonnegative. == Osterwalder–Schrader theorem == The Osterwalder–Schrader theorem states that Euclidean Schwinger functions which satisfy the above axioms (E0)-(E4) and an additional property (E0') called linear growth condition can be analytically continued to Lorentzian Wightman distributions which satisfy Wightman axioms and thus define a quantum field theory. === Linear growth condition === This condition, called (E0') in, asserts that when the Schwinger function of order n {\displaystyle n} is paired with an arbitrary Schwartz test function f {\displaystyle f} which vanishes at coincident points, we have the following bound: | S n ( f ) | ≤ σ n | f | C ⋅ n , {\displaystyle |S_{n}(f)|\leq \sigma _{n}|f|_{C\cdot n},} where C ∈ N {\displaystyle C\in \mathbb {N} } is an integer constant, | f | C ⋅ n {\displaystyle |f|_{C\cdot n}} is the Schwartz-space seminorm of order N = C ⋅ n {\displaystyle N=C\cdot n} , i.e. | f | N = sup | α | ≤ N , x ∈ R d | ( 1 + | x | ) N D α f ( x ) | , {\displaystyle |f|_{N}=\sup _{|\alpha |\leq N,x\in \mathbb {R} ^{d}}|(1+|x|)^{N}D^{\alpha }f(x)|,} and σ n {\displaystyle \sigma _{n}} a sequence of constants of factorial growth, i.e. σ n ≤ A ( n ! ) B {\displaystyle \sigma _{n}\leq A(n!)^{B}} with some constants A , B {\displaystyle A,B} . Linear growth condition is subtle as it has to be satisfied for all Schwinger functions simultaneously. It also has not been derived from the Wightman axioms, so that the system of OS axioms (E0)-(E4) plus the linear growth condition (E0') appears to be stronger than the Wightman axioms. === History === At first, Osterwalder and Schrader claimed a stronger theorem that the axioms (E0)-(E4) by themselves imply the Wightman axioms, however their proof contained an error which could not be corrected without adding extra assumptions. Two years later they published a new theorem, with the linear growth condition added as an assumption, and a correct proof. The new proof is based on a complicated inductive argument (proposed also by Vladimir Glaser), by which the region of analyticity of Schwinger functions is gradually extended towards the Minkowski space, and Wightman distributions are recovered as a limit. The linear growth condition (E0') is crucially used to show that the limit exists and is a tempered distribution. Osterwalder's and Schrader's paper also contains another theorem replacing (E0') by yet another assumption called (E0) ˇ {\displaystyle {\check {\text{(E0)}}}} . This other theorem is rarely used, since (E0) ˇ {\displaystyle {\check {\text{(E0)}}}} is hard to check in practice. == Other axioms for Schwinger functions == === Axioms by Glimm and Jaffe === An alternative approach to axiomatization of Euclidean correlators is described by Glimm and Jaffe in their book. In this approach one assumes that one is given a measure d μ {\displaystyle d\mu } on the space of distributions ϕ ∈ D ′ ( R d ) {\displaystyle \phi \in D'(\mathbb {R} ^{d})} . One then considers a generating functional S ( f ) = ∫ e ϕ ( f ) d μ , f ∈ D ( R d ) {\displaystyle S(f)=\int e^{\phi (f)}d\mu ,\quad f\in D(\mathbb {R} ^{d})} which is assumed to satisfy properties OS0-OS4: (OS0) Analyticity. This asserts that z = ( z 1 , … , z n ) ↦ S ( ∑ i = 1 n z i f i ) {\displaystyle z=(z_{1},\ldots ,z_{n})\mapsto S\left(\sum _{i=1}^{n}z_{i}f_{i}\right)} is an entire-analytic function of z ∈ R n {\displaystyle z\in \mathbb {R} ^{n}} for any collection of n {\displaystyle n} compactly supported test functions f i ∈ D ( R d ) {\displaystyle f_{i}\in D(\mathbb {R} ^{d})} . Intuitively, this means that the measure d μ {\displaystyle d\mu } decays faster than any exponential. (OS1) Regularity. This demands a growth bound for S ( f ) {\displaystyle S(f)} in terms of f {\displaystyle f} , such as | S ( f ) | ≤ exp ( C ∫ d d x | f ( x ) | ) {\displaystyle |S(f)|\leq \exp \left(C\int d^{d}x|f(x)|\right)} . See for the precise condition. (OS2) Euclidean invariance. This says that the functional S ( f ) {\displaystyle S(f)} is invariant under Euclidean transformations f ( x ) ↦ f ( R x + b ) {\displaystyle f(x)\mapsto f(Rx+b)} . (OS3) Reflection positivity. Take a finite sequence of test functions f i ∈ D ( R d ) {\displaystyle f_{i}\in D(\mathbb {R} ^{d})} which are all supported in the upper half-space i.e. at x 0 > 0 {\displaystyle x^{0}>0} . Denote by θ f i ( x ) = f i ( θ x ) {\displaystyle \theta f_{i}(x)=f_{i}(\theta x)} where θ {\displaystyle \theta } is a reflection operation defined above. This axioms says that the matrix M i j = S ( f i + θ f j ) {\displaystyle M_{ij}=S(f_{i}+\theta f_{j})} has to be positive semidefinite. (OS4) Ergodicity. The time translation semigroup acts ergodically on the measure space ( D ′ ( R d ) , d μ ) {\displaystyle (D'(\mathbb {R} ^{d}),d\mu )} . See for the precise condition. ==== Relation to Osterwalder–Schrader axioms ==== Although the above axioms were named by Glimm and Jaffe (OS0)-(OS4) in honor of Osterwalder and Schrader, they are not equivalent to the Osterwalder–Schrader axioms. Given (OS0)-(OS4), one can define Schwinger functions of ϕ {\displaystyle \phi } as moments of the measure d μ {\displaystyle d\mu } , and show that these moments satisfy Osterwalder–Schrader axioms (E0)-(E4) and also the linear growth conditions (E0'). Then one can appeal to the Osterwalder–Schrader theorem to show that Wightman functions are tempered distributions. Alternatively, and much easier, one can derive Wightman axioms directly from (OS0)-(OS4). Note however that the full quantum field theory will contain infinitely many other local operators apart from ϕ {\displaystyle \phi } , such as ϕ 2 {\displaystyle \phi ^{2}} , ϕ 4 {\displaystyle \phi ^{4}} and other composite operators built from ϕ {\displaystyle \phi } and its derivatives. It's not easy to extract these Schwinger functions from the measure d μ {\displaystyle d\mu } and show that they satisfy OS axioms, as it should be the case. To summarize, the axioms called (OS0)-(OS4) by Glimm and Jaffe are stronger than the OS axioms as far as the correlators of the field ϕ {\displaystyle \phi } are concerned, but weaker than then the full set of OS axioms since they don't say much about correlators of composite operators. === Nelson's axioms === These axioms were proposed by Edward Nelson. See also their description in the book of Barry Simon. Like in the above axioms by Glimm and Jaffe, one assumes that the field ϕ ∈ D ′ ( R d ) {\displaystyle \phi \in D'(\mathbb {R} ^{d})} is a random distribution with a measure d μ {\displaystyle d\mu } . This measure is sufficiently regular so that the field ϕ {\displaystyle \phi } has regularity of a Sobolev space of negative derivative order. The crucial feature of these axioms is to consider the field restricted to a surface. One of the axioms is Markov property, which formalizes the intuitive notion that the state of the field inside a closed surface depends only on the state of the field on the surface. == See also == Wick rotation Axiomatic quantum field theory Wightman axioms == References ==
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Wikipedia:Konstantin Ardakov#0
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Konstantin Ardakov (born 1979) is professor of pure mathematics at the Mathematical Institute, University of Oxford and fellow and tutor in mathematics at Brasenose College, Oxford. After education at the University of Oxford and the University of Cambridge, he held positions at Cambridge, the University of Sheffield, the University of Nottingham, and Queen Mary University of London, before returning to Oxford. He was awarded the 2019–20 Adams Prize for making "substantial contributions to noncommutative Iwasawa theory, and to the p-adic representation theory of p-adic Lie groups". == References ==
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Wikipedia:Konstantin Khanin#0
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Konstantin "Kostya" Mikhailovich Khanin (Russian: Константин Михайлович Ханин) is a Russian mathematician and physicist. He served as the chair of the Department of Mathematical and Computational Sciences at the University of Toronto Mississauga. == Background == Khanin received his PhD from the Landau Institute of Theoretical Physics in Moscow and continued working there as a Research Associate until 1994. Afterwards, he taught at Princeton University, at the Isaac Newton Institute in Cambridge, and at Heriot-Watt University before joining the faculty at the University of Toronto. Khanin was an invited speaker at the European Congress of Mathematics in Barcelona in 2000. He was a 2013 Simons Foundation Fellow. He held the Jean-Morlet Chair at the Centre International de Rencontres Mathématiques in 2017, and he was an Invited Speaker at the International Congress of Mathematicians in 2018 in Rio de Janeiro. In 2021 he was awarded The Humboldt Prize, also known as the Humboldt Research Award, in recognition of his lifetime's research achievements. == References ==
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Wikipedia:Konstantin Malkov#0
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Malkov (masculine, Russian: Малков) or Malkova (feminine, Russian: Малкова) is a Russian surname. Notable people with the surname include: Anatoli Malkov (born 1981), Russian soccer player Igor Malkov (born 1965), Russian speedskater Konstantin Malkov, American mathematician and businessman Mia Malkova (born 1992), American pornographic actress and livestreamer Vladimir Malkov (disambiguation), multiple people Yevgeni Malkov (born 1988), Russian soccer player == See also == Málkov (disambiguation)
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Wikipedia:Konstantin Posse#0
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Konstantin Alexandrovich Posse (Russian: Константин Александрович Поссе; September 29, 1847 – August 24, 1928) was a Russian mathematician known for contributions to analysis and in particular approximation theory. Veniamin Kagan and D. D. Morduhai-Boltovskoi were among his students. == Selected publications == Possé, C. (1885). "Quelques remarques sur une certaine question de minimum". Mathematische Annalen. 26 (4): 593–596. doi:10.1007/BF01444256. ISSN 0025-5831. JFM 18.0325.01. S2CID 122505974. Possé, C. (1886). Sur quelques applications des fractions continues algébriques. St. Petersburg. JFM 18.0161.02.{{cite book}}: CS1 maint: location missing publisher (link) == Notes == == Further reading == Сергеев, А.А. (1997). Константин Александрович Поссе, 1847–1928. Научно-биографическая литература (in Russian). Moscow: Nauka. ISBN 5-02-003692-7. == External links == Konstantin Posse at the Mathematics Genealogy Project
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Wikipedia:Konstantin Semendyayev#0
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Konstantin Adolfovich Semendyayev (Russian: Константин Адольфович Семендяев; 9 December 1908, Simferopol – 15 November 1988, Moscow) was a Soviet engineer and applied mathematician. He worked in the department of applied mathematics of the Steklov Institute in Moscow. He carried out pioneering work in the area of numerical weather forecasting in the Soviet Union. == Biography == Semendyayev studied at the Lomonosov University with the degree in 1929 and was then at various higher schools. From 1931 to 1936 he was in the Faculty of Mathematics and Mechanics at Lomonosov University. He habilitated in 1940 (Russian doctorate). From 1936 he headed the Department of Mathematical Instruments of the Academy of Sciences of the Soviet Union. He was evacuated to Kazan with the institute during World War II. After World War II, he headed a department for numerical calculations at the Steklov Institute in Moscow and, when the Institute for Applied Mathematics at the Steklov Institute was founded in 1953, his group became the Department of Gas Dynamics. In 1961, he became deputy head of the Institute for Applied Mathematics. In 1963, he went to the Hydrometeorological Center of the USSR, where he led the programming work. He also supported the teaching of applied mathematics at various Moscow educational institutions. Semendyayev is known as the co-author of a handbook of mathematics for engineers and students of technical universities, which he wrote together with Ilya Nikolaevich Bronshtein around the 1939/1940 timeframe. Hot lead typesetting for the work had already started when the Siege of Leningrad prohibited further development and the print matrices were relocated. After the war, they were first considered lost, but could be found again years later, so that the first edition of Справочник по математике для инженеров и учащихся втузов could finally be published in 1945. This was a major success and went through eleven editions in Russia and was translated into various languages, including German and English, until the publisher Nauka planned to replace it with a translation of the American Mathematical Handbook for Scientists and Engineers by Granino and Theresa M. Korn in 1968. However, in a parallel development starting in 1970, the so called "Bronshtein and Semendyayev" (BS), which had been translated into German in 1958, underwent a major overhaul by a team of East German authors around Günter Grosche, Viktor & Dorothea Ziegler (of University of Leipzig), to which Semendyayev contributed as well (a section on computer systems and numerical harmonic analysis). This was published in 1979 and spawned translations into many other languages as well, including a retranslation into Russian and an English edition. In 1986, the 13th Russian edition was published. The German 'Wende' and the later reunification led to considerable changes in the publishing environment in Germany between 1989 and 1991, which eventually resulted in two independent German publishing branches by Eberhard Zeidler (published 1995–2013) and by Gerhard Musiol & Heiner Mühlig (published 1992–2020) to expand and maintain the work up to the present, again with translations into many other languages including English. Semendyayev has been on the editorial board of the Russian journal Journal of Numerical Mathematics and Mathematical Physics (Журнал вычислительной математики и математической физики) since its inception. == Awards and honors == Three Stalin Prizes (1949, 1951, 1953) Three Orders of Lenin (1949, 1955, 1956) Order of the Red Banner of Labour (1953) Jubilee Medal "In Commemoration of the 100th Anniversary of the Birth of Vladimir Ilyich Lenin" Lenin Prize == Publications == With Bronshtein: "Handbook of Mathematics for Engineers and Students of Technical Universities" (Справочник по математике для инженеров и учащихся втузов), Moscow, 1945 == See also == Bronshtein and Semendyayev (BS) Ilya Nikolaevich Bronshtein == References == == Further reading == "Кафедра «Высшая математика» История кафедры" [Department of Higher Mathematics - Department history] (in Russian). Moscow State University of Mechanical Engineering (MAMI). 2016 [2010]. Archived from the original on 2016-04-03. Retrieved 2022-01-23. Volume 29, 1989, pp. 474–475, http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=zvmmf&paperid=3490&option_lang=rus https://web.archive.org/web/20200705112735/http://www.mathnet.ru/links/feea4ab25995d6344cb609e4dcfc8c88/zvmmf3490.pdf https://web.archive.org/web/20211019135410/https://keldysh.ru/memory/index1.htm Keldysh Institute of Applied Mathematics of the Russian Academy of Sciences, 2010 == External links == "Семендяев Константин Адольфович" [Semendyaev, Konstantin Adol'fovich]. Math-Net.Ru (in Russian and English). 2021 [2016]. Archived from the original on 2022-01-23. Retrieved 2022-01-23.
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Wikipedia:Konstantina Trivisa#0
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Konstantina Trivisa is a Greek-American applied mathematician whose research involves nonlinear partial differential equations, fluid dynamics, and the mathematical modeling of flocking. She is a professor of mathematics at the University of Maryland, College Park, where she directs the Institute for Physical Science & Technology. == Education and career == Trivisa graduated from the University of Patras in 1990, and completed her Ph.D. at Brown University in 1996. Her dissertation, A priori estimates in hyperbolic systems of conservation laws via generalized characteristics, was supervised by Constantine Dafermos. After postdoctoral research at Carnegie Mellon University in the Center for Nonlinear Analysis, and a term at Northwestern University as Ralph Boas Assistant Professor of Mathematics, she joined the University of Maryland in 2000. She was promoted to associate professor in 2004 and full professor in 2007. She was director of the Applied Mathematics & Statistics, and Scientific Computation Program from 2007 to 2018. She added affiliations with the Institute for Physical Science & Technology in 2008 and with the Center for Scientific Computation & Mathematical Modeling in 2017. In 2020, she became director of the Institute for Physical Science & Technology. == Recognition == Trivisa won a Sloan Research Fellowship in 2001. She was a recipient of the 2003 Presidential Early Career Award for Scientists and Engineers, "for applying her expertise in applied partial differential equations to the increased understanding of a wide variety of important physical systems modeled by conservation laws". She was elected to the 2023 Class of SIAM Fellows. == References == == External links == Home page Konstantina Trivisa publications indexed by Google Scholar
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Wikipedia:Kos (unit)#0
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The kos (Hindi: कोस), also spelled coss, koss, kosh, koh(in Punjabi), krosh, and krosha, is a unit of measurement which is derived from a Sanskrit term, क्रोश krośa, which means a 'call', as the unit was supposed to represent the distance at which another human could be heard. It is an ancient Indian subcontinental standard unit of distance, in use since at least 4 BCE. According to the Arthashastra, a krośa or kos is about 3,000 metres (9,800 ft). Another conversion is based on the Mughal emperor Akbar, who standardized the unit to 5000 guz in the Ain-i-Akbari. The British in India standardized Akbar's guz to 33 inches (840 mm), making the kos approximately 4,191 metres (13,750 ft). Another conversion suggested a kos to be approximately 2 English miles. == Arthashastra Standard units == The "Arthashastra: Chapter XX. "Measurement of space and time", authored in 4th century BC by Chanakya (Vishnugupta Kauṭilya), sets this standard breakup of Indian units of length: 1 angul (approximate width of a finger) = approx. 3⁄4 of an inch (19 mm) 4 angul = 1 dhanurgrah (bow grip) = 3 inches (76 mm) 8 angul = 1 dhanurmushti (fist with thumb raised) = 6 inches (150 mm) 12 angul = 1 vitastaa (span-distance of stretched out palm between the tips of a person's thumb and the little finger) = 9 inches (230 mm) 2 vitastaa (from the tip of the elbow to the tip of the middle finger) = 1 aratni or hast (cubit or haath) = 18 inches (460 mm) 4 aratni (haath) = 1 dand or dhanush (bow) = 6 feet (1.8 m); 10 dand = 1 rajju = 60 feet (18 m) 2 rajju = 1 paridesh = 120 feet (37 m) 10 rajju = 1 goruta = 219 yards (1/8 mi; 200 m) 10 goruta= 1 krosha/kos = nearly 3,350 yards (3,060 metres; 1.90 miles) == Conversion to SI units and imperial units == Kos may also refer to roughly 1.8 kilometres (1.1 mi) Arthashastra standard unit of kos or krosha is equal to 3075 metres in SI units and 1.91 miles in imperial units. == Usage of kos == Evidence of official usage exists from the Vedic period to the Mughal era. Elderly people in many rural areas of the Indian subcontinent still refer to distances from nearby areas in kos. Most Hindu religious Parikrama circuits are measured in kos, such as 48 kos parikrama of Kurukshetra. Along India's old highways, particularly the Grand Trunk Road, one still finds 16th to early 18th century Kos Minars, or mile markers, erected at distances of a little over two miles. == See also == == References ==
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Wikipedia:Kosaburo Hashiguchi#0
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Kosaburo Hashiguchi (橋口 攻三郎, Hashiguchi Kōsaburō) is a Japanese mathematician and computer scientist at the Toyohashi University of Technology and Okayama University, known for his research in formal language theory. In 1988, he found the first algorithm to determine the star height of a regular language, a problem that had been open since 1963 when Lawrence Eggan solved the related star height problem, showing that there is no finite bound on the star height. Hashiguchi's algorithm for star height is extremely complex, and impractical on all but the smallest examples.[H88] A simpler method, showing also that the problem is PSPACE-complete, was provided in 2005 by Kirsten. Earlier, in 1979, Hashiguchi had also solved another open problem on regular languages, of deciding whether, for a given language A {\displaystyle A} , there exists a finite number n {\displaystyle n} such that A n = A ∗ {\displaystyle A^{n}=A^{*}} .[H79] Hashiguchi is the uncle of Japanese-born American pianist Grace Nikae. == Selected publications == == References == == External links == Home page
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Wikipedia:Kostka number#0
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In mathematics, the Kostka number K λ μ {\displaystyle K_{\lambda \mu }} (depending on two integer partitions λ {\displaystyle \lambda } and μ {\displaystyle \mu } ) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape λ {\displaystyle \lambda } and weight μ {\displaystyle \mu } . They were introduced by the mathematician Carl Kostka in his study of symmetric functions (Kostka (1882)). For example, if λ = ( 3 , 2 ) {\displaystyle \lambda =(3,2)} and μ = ( 1 , 1 , 2 , 1 ) {\displaystyle \mu =(1,1,2,1)} , the Kostka number K λ μ {\displaystyle K_{\lambda \mu }} counts the number of ways to fill a left-aligned collection of boxes with 3 in the first row and 2 in the second row with 1 copy of the number 1, 1 copy of the number 2, 2 copies of the number 3 and 1 copy of the number 4 such that the entries increase along columns and do not decrease along rows. The three such tableaux are shown at right, and K ( 3 , 2 ) ( 1 , 1 , 2 , 1 ) = 3 {\displaystyle K_{(3,2)(1,1,2,1)}=3} . == Examples and special cases == For any partition λ {\displaystyle \lambda } , the Kostka number K λ λ {\displaystyle K_{\lambda \lambda }} is equal to 1: the unique way to fill the Young diagram of shape λ = ( λ 1 , … , λ m ) {\displaystyle \lambda =(\lambda _{1},\dotsc ,\lambda _{m})} with λ 1 {\displaystyle \lambda _{1}} copies of 1, λ 2 {\displaystyle \lambda _{2}} copies of 2, and so on, so that the resulting tableau is weakly increasing along rows and strictly increasing along columns is if all the 1s are placed in the first row, all the 2s are placed in the second row, and so on. (This tableau is sometimes called the Yamanouchi tableau of shape λ {\displaystyle \lambda } .) The Kostka number K λ μ {\displaystyle K_{\lambda \mu }} is positive (i.e., there exist semistandard Young tableaux of shape λ {\displaystyle \lambda } and weight μ {\displaystyle \mu } ) if and only if λ {\displaystyle \lambda } and μ {\displaystyle \mu } are both partitions of the same integer n {\displaystyle n} and λ {\displaystyle \lambda } is larger than μ {\displaystyle \mu } in dominance order. In general, there are no nice formulas known for the Kostka numbers. However, some special cases are known. For example, if μ = ( 1 , 1 , … , 1 ) {\displaystyle \mu =(1,1,\dotsc ,1)} is the partition whose parts are all 1 then a semistandard Young tableau of weight μ {\displaystyle \mu } is a standard Young tableau; the number of standard Young tableaux of a given shape λ {\displaystyle \lambda } is given by the hook-length formula. == Properties == An important simple property of Kostka numbers is that K λ μ {\displaystyle K_{\lambda \mu }} does not depend on the order of entries of μ {\displaystyle \mu } . For example, K ( 3 , 2 ) ( 1 , 1 , 2 , 1 ) = K ( 3 , 2 ) ( 1 , 1 , 1 , 2 ) {\displaystyle K_{(3,2)(1,1,2,1)}=K_{(3,2)(1,1,1,2)}} . This is not immediately obvious from the definition but can be shown by establishing a bijection between the sets of semistandard Young tableaux of shape λ {\displaystyle \lambda } and weights μ {\displaystyle \mu } and μ ′ {\displaystyle \mu ^{\prime }} , where μ {\displaystyle \mu } and μ ′ {\displaystyle \mu ^{\prime }} differ only by swapping two entries. == Kostka numbers, symmetric functions and representation theory == In addition to the purely combinatorial definition above, they can also be defined as the coefficients that arise when one expresses the Schur polynomial s λ {\displaystyle s_{\lambda }} as a linear combination of monomial symmetric functions m μ {\displaystyle m_{\mu }} : s λ = ∑ μ K λ μ m μ , {\displaystyle s_{\lambda }=\sum _{\mu }K_{\lambda \mu }m_{\mu },} where λ {\displaystyle \lambda } and μ {\displaystyle \mu } are both partitions of n {\displaystyle n} . Alternatively, Schur polynomials can also be expressed as s λ = ∑ α K λ α x α , {\displaystyle s_{\lambda }=\sum _{\alpha }K_{\lambda \alpha }x^{\alpha },} where the sum is over all weak compositions α {\displaystyle \alpha } of n {\displaystyle n} and x α {\displaystyle x^{\alpha }} denotes the monomial x 1 α 1 x 2 α 2 … x n α n {\displaystyle x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\dotsc x_{n}^{\alpha _{n}}} . On the level of representations of the symmetric group S n {\displaystyle S_{n}} , Kostka numbers express the decomposition of the permutation module M μ {\displaystyle M_{\mu }} in terms of the irreducible representations V λ {\displaystyle V_{\lambda }} where λ {\displaystyle \lambda } is a partition of n {\displaystyle n} , i.e., M μ = ⨁ λ K λ μ V λ . {\displaystyle M_{\mu }=\bigoplus _{\lambda }K_{\lambda \mu }V_{\lambda }.} On the level of representations of the general linear group G L d ( C ) {\displaystyle \mathrm {GL} _{d}(\mathbb {C} )} , the Kostka number K λ μ {\displaystyle K_{\lambda \mu }} also counts the dimension of the weight space corresponding to μ {\displaystyle \mu } in the unitary irreducible representation U λ {\displaystyle U_{\lambda }} (where we require μ {\displaystyle \mu } and λ {\displaystyle \lambda } to have at most d {\displaystyle d} parts). == Examples == The Kostka numbers for partitions of size at most 3 are as follows: K ∅ ∅ = 1 , {\displaystyle K_{\varnothing \varnothing }=1,} K ( 1 ) ( 1 ) = 1 , {\displaystyle K_{(1)(1)}=1,} K ( 2 ) ( 2 ) = K ( 2 ) ( 1 , 1 ) = 1 , {\displaystyle K_{(2)(2)}=K_{(2)(1,1)}=1,} K ( 1 , 1 ) ( 2 ) = 0 , K ( 1 , 1 ) ( 1 , 1 ) = 1 , {\displaystyle K_{(1,1)(2)}=0,\,K_{(1,1)(1,1)}=1,} K ( 3 ) ( 3 ) = K ( 3 ) ( 2 , 1 ) = K ( 3 ) ( 1 , 1 , 1 ) = 1 , {\displaystyle K_{(3)(3)}=K_{(3)(2,1)}=K_{(3)(1,1,1)}=1,} K ( 2 , 1 ) ( 3 ) = 0 , K ( 2 , 1 ) ( 2 , 1 ) = 1 , K ( 2 , 1 ) ( 1 , 1 , 1 ) = 2 , {\displaystyle K_{(2,1)(3)}=0,\,K_{(2,1)(2,1)}=1,\,K_{(2,1)(1,1,1)}=2,} K ( 1 , 1 , 1 ) ( 3 ) = K ( 1 , 1 , 1 ) ( 2 , 1 ) = 0 , K ( 1 , 1 , 1 ) ( 1 , 1 , 1 ) = 1. {\displaystyle K_{(1,1,1)(3)}=K_{(1,1,1)(2,1)}=0,\,K_{(1,1,1)(1,1,1)}=1.} These values are exactly the coefficients in the expansions of Schur functions in terms of monomial symmetric functions: s ∅ = m ∅ = 1 {\displaystyle s_{\varnothing }=m_{\varnothing }=1} s ( 1 ) = m ( 1 ) {\displaystyle s_{(1)}=m_{(1)}} s ( 2 ) = m ( 2 ) + m ( 1 , 1 ) {\displaystyle s_{(2)}=m_{(2)}+m_{(1,1)}} s ( 1 , 1 ) = m ( 1 , 1 ) {\displaystyle s_{(1,1)}=m_{(1,1)}} s ( 3 ) = m ( 3 ) + m ( 2 , 1 ) + m ( 1 , 1 , 1 ) {\displaystyle s_{(3)}=m_{(3)}+m_{(2,1)}+m_{(1,1,1)}} s ( 2 , 1 ) = m ( 2 , 1 ) + 2 m ( 1 , 1 , 1 ) {\displaystyle s_{(2,1)}=m_{(2,1)}+2m_{(1,1,1)}} s ( 1 , 1 , 1 ) = m ( 1 , 1 , 1 ) {\displaystyle s_{(1,1,1)}=m_{(1,1,1)}} Kostka (1882, pages 118-120) gave tables of these numbers for partitions of numbers up to 8. == Generalizations == Kostka numbers are special values of the 1 or 2 variable Kostka polynomials: K λ μ = K λ μ ( 1 ) = K λ μ ( 0 , 1 ) . {\displaystyle K_{\lambda \mu }=K_{\lambda \mu }(1)=K_{\lambda \mu }(0,1).} == Notes == == References == Stanley, Richard (1999), Enumerative combinatorics, volume 2, Cambridge University Press Kostka, C. (1882), "Über den Zusammenhang zwischen einigen Formen von symmetrischen Funktionen", Crelle's Journal, 93: 89–123, doi:10.1515/crll.1882.93.89 Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853489-1, MR 1354144, archived from the original on 2012-12-11 Sagan, Bruce E. (2001) [1994], "Schur functions in algebraic combinatorics", Encyclopedia of Mathematics, EMS Press
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Wikipedia:Kostka polynomial#0
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In mathematics, Kostka polynomials, named after the mathematician Carl Kostka, are families of polynomials that generalize the Kostka numbers. They are studied primarily in algebraic combinatorics and representation theory. The two-variable Kostka polynomials Kλμ(q, t) are known by several names including Kostka–Foulkes polynomials, Macdonald–Kostka polynomials or q,t-Kostka polynomials. Here the indices λ and μ are integer partitions and Kλμ(q, t) is polynomial in the variables q and t. Sometimes one considers single-variable versions of these polynomials that arise by setting q = 0, i.e., by considering the polynomial Kλμ(t) = Kλμ(0, t). There are two slightly different versions of them, one called transformed Kostka polynomials. The one-variable specializations of the Kostka polynomials can be used to relate Hall-Littlewood polynomials Pμ to Schur polynomials sλ: s λ ( x 1 , … , x n ) = ∑ μ K λ μ ( t ) P μ ( x 1 , … , x n ; t ) . {\displaystyle s_{\lambda }(x_{1},\ldots ,x_{n})=\sum _{\mu }K_{\lambda \mu }(t)P_{\mu }(x_{1},\ldots ,x_{n};t).\ } These polynomials were conjectured to have non-negative integer coefficients by Foulkes, and this was later proved in 1978 by Alain Lascoux and Marcel-Paul Schützenberger. In fact, they show that K λ μ ( t ) = ∑ T ∈ S S Y T ( λ , μ ) t c h a r g e ( T ) {\displaystyle K_{\lambda \mu }(t)=\sum _{T\in SSYT(\lambda ,\mu )}t^{charge(T)}} where the sum is taken over all semi-standard Young tableaux with shape λ and weight μ. Here, charge is a certain combinatorial statistic on semi-standard Young tableaux. The Macdonald–Kostka polynomials can be used to relate Macdonald polynomials (also denoted by Pμ) to Schur polynomials sλ: s λ ( x 1 , … , x n ) = ∑ μ K λ μ ( q , t ) J μ ( x 1 , … , x n ; q , t ) {\displaystyle s_{\lambda }(x_{1},\ldots ,x_{n})=\sum _{\mu }K_{\lambda \mu }(q,t)J_{\mu }(x_{1},\ldots ,x_{n};q,t)\ } where J μ ( x 1 , … , x n ; q , t ) = P μ ( x 1 , … , x n ; q , t ) ∏ s ∈ μ ( 1 − q a r m ( s ) t l e g ( s ) + 1 ) . {\displaystyle J_{\mu }(x_{1},\ldots ,x_{n};q,t)=P_{\mu }(x_{1},\ldots ,x_{n};q,t)\prod _{s\in \mu }(1-q^{arm(s)}t^{leg(s)+1}).\ } Kostka numbers are special values of the one- or two-variable Kostka polynomials: K λ μ = K λ μ ( 1 ) = K λ μ ( 0 , 1 ) . {\displaystyle K_{\lambda \mu }=K_{\lambda \mu }(1)=K_{\lambda \mu }(0,1).\ } == Examples == == References == Macdonald, I. G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853489-1, MR 1354144, archived from the original on December 11, 2012 Nelsen, Kendra; Ram, Arun (2003), "Kostka-Foulkes polynomials and Macdonald spherical functions", Surveys in combinatorics, 2003 (Bangor), London Math. Soc. Lecture Note Ser., vol. 307, Cambridge: Cambridge Univ. Press, pp. 325–370, arXiv:math/0401298, Bibcode:2004math......1298N, MR 2011741 Stembridge, J. R. (2005), Kostka-Foulkes Polynomials of General Type, lecture notes from AIM workshop on Generalized Kostka polynomials == External links == Short tables of Kostka polynomials Long tables of Kostka polynomials
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Wikipedia:Koszul algebra#0
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In abstract algebra, a Koszul algebra R {\displaystyle R} is a graded k {\displaystyle k} -algebra over which the ground field k {\displaystyle k} has a linear minimal graded free resolution, i.e., there exists an exact sequence: ⋯ → ( R ( − i ) ) b i → ⋯ → ( R ( − 2 ) ) b 2 → ( R ( − 1 ) ) b 1 → R → k → 0. {\displaystyle \cdots \rightarrow (R(-i))^{b_{i}}\rightarrow \cdots \rightarrow (R(-2))^{b_{2}}\rightarrow (R(-1))^{b_{1}}\rightarrow R\rightarrow k\rightarrow 0.} for some nonnegative integers b i {\displaystyle b_{i}} . Here R ( − j ) {\displaystyle R(-j)} is the graded algebra R {\displaystyle R} with grading shifted up by j {\displaystyle j} , i.e. R ( − j ) i = R i − j {\displaystyle R(-j)_{i}=R_{i-j}} , and the exponent b i {\displaystyle b_{i}} refers to the b i {\displaystyle b_{i}} -fold direct sum. Choosing bases for the free modules in the resolution, the chain maps are given by matrices, and the definition requires the matrix entries to be zero or linear forms. An example of a Koszul algebra is a polynomial ring over a field, for which the Koszul complex is the minimal graded free resolution of the ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, e.g, R = k [ x , y ] / ( x y ) {\displaystyle R=k[x,y]/(xy)} . The concept is named after the French mathematician Jean-Louis Koszul. == See also == Koszul duality Complete intersection ring == References == Fröberg, R. (1999), "Koszul algebras", Advances in commutative ring theory (Fez, 1997), Lecture Notes in Pure and Applied Mathematics, vol. 205, New York: Marcel Dekker, pp. 337–350, MR 1767430. Loday, Jean-Louis; Vallette, Bruno (2012), Algebraic operads (PDF), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346, Heidelberg: Springer, doi:10.1007/978-3-642-30362-3, ISBN 978-3-642-30361-6, MR 2954392. Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang (1996), "Koszul duality patterns in representation theory", Journal of the American Mathematical Society, 9 (2): 473–527, doi:10.1090/S0894-0347-96-00192-0, MR 1322847. Mazorchuk, Volodymyr; Ovsienko, Serge; Stroppel, Catharina (2009), "Quadratic duals, Koszul dual functors, and applications", Transactions of the American Mathematical Society, 361 (3): 1129–1172, arXiv:math/0603475, doi:10.1090/S0002-9947-08-04539-X, MR 2457393.
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Wikipedia:Kraków School of Mathematics#0
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The Kraków School of Mathematics (Polish: krakowska szkoła matematyczna) was a subgroup of the Polish School of Mathematics represented by mathematicians from the Kraków universities—Jagiellonian University, and the AGH University of Science and Technology–active during the interwar period (1918–1939). Their areas of study were primarily classical analysis, differential equations, and analytic functions. The Kraków School of Differential Equations was founded by Tadeusz Ważewski, a student of Stanisław Zaremba, and was internationally appreciated after World War II. The Kraków School of Analytic Functions was founded by Franciszek Leja. Other notable members included Kazimierz Żorawski, Władysław Ślebodziński, Stanisław Gołąb, and Czesław Olech. == See also == Polish School of Mathematics Lwów School of Mathematics Warsaw School of Mathematics Polish Mathematical Society Kraków School of Mathematics and Astrology == References == Waltoś, Stanisław. "Tradycja i współczesność". Uniwersytet Jagielloński (in Polish). Archived from the original on 2008-02-01. Retrieved 2007-11-26.
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Wikipedia:Kraków School of Mathematics and Astrology#0
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The Kraków School of Mathematics and Astrology (Polish: krakowska szkoła matematyczna i astrologiczna) was an influential mid-to-late-15th-century group of mathematicians and astrologers at the University of Kraków (later Jagiellonian University). == Notable members == Jan of Głogów (1445–1507), author of widely recognized mathematical and astrological tracts Marcin Biem (1470–1540), contributor to the Gregorian calendar Marcin Bylica of Olkusz (1433–93), later court astrologer to King Matthias Corvinus of Hungary Albert Brudzewski (1446–1495), teacher to notable scholars active at European universities Marcin Król of Żurawica (1422–1460) Nicolaus Copernicus (1473–1543), student at Kraków in 1491–95 == See also == Kraków School of Mathematics Polish School of Mathematics == References == Waltoś, Stanisław. "Tradycja i współczesność". Uniwersytet Jagielloński (in Polish). Archived from the original on 2008-02-01. Retrieved 2007-11-26.
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Wikipedia:Krein's condition#0
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In mathematical analysis, Krein's condition provides a necessary and sufficient condition for exponential sums { ∑ k = 1 n a k exp ( i λ k x ) , a k ∈ C , λ k ≥ 0 } , {\displaystyle \left\{\sum _{k=1}^{n}a_{k}\exp(i\lambda _{k}x),\quad a_{k}\in \mathbb {C} ,\,\lambda _{k}\geq 0\right\},} to be dense in a weighted L2 space on the real line. It was discovered by Mark Krein in the 1940s. A corollary, also called Krein's condition, provides a sufficient condition for the indeterminacy of the moment problem. == Statement == Let μ be an absolutely continuous measure on the real line, dμ(x) = f(x) dx. The exponential sums ∑ k = 1 n a k exp ( i λ k x ) , a k ∈ C , λ k ≥ 0 {\displaystyle \sum _{k=1}^{n}a_{k}\exp(i\lambda _{k}x),\quad a_{k}\in \mathbb {C} ,\,\lambda _{k}\geq 0} are dense in L2(μ) if and only if ∫ − ∞ ∞ − ln f ( x ) 1 + x 2 d x = ∞ . {\displaystyle \int _{-\infty }^{\infty }{\frac {-\ln f(x)}{1+x^{2}}}\,dx=\infty .} == Indeterminacy of the moment problem == Let μ be as above; assume that all the moments m n = ∫ − ∞ ∞ x n d μ ( x ) , n = 0 , 1 , 2 , … {\displaystyle m_{n}=\int _{-\infty }^{\infty }x^{n}d\mu (x),\quad n=0,1,2,\ldots } of μ are finite. If ∫ − ∞ ∞ − ln f ( x ) 1 + x 2 d x < ∞ {\displaystyle \int _{-\infty }^{\infty }{\frac {-\ln f(x)}{1+x^{2}}}\,dx<\infty } holds, then the Hamburger moment problem for μ is indeterminate; that is, there exists another measure ν ≠ μ on R such that m n = ∫ − ∞ ∞ x n d ν ( x ) , n = 0 , 1 , 2 , … {\displaystyle m_{n}=\int _{-\infty }^{\infty }x^{n}\,d\nu (x),\quad n=0,1,2,\ldots } This can be derived from the "only if" part of Krein's theorem above. === Example === Let f ( x ) = 1 π exp { − ln 2 x } ; {\displaystyle f(x)={\frac {1}{\sqrt {\pi }}}\exp \left\{-\ln ^{2}x\right\};} the measure dμ(x) = f(x) dx is called the Stieltjes–Wigert measure. Since ∫ − ∞ ∞ − ln f ( x ) 1 + x 2 d x = ∫ − ∞ ∞ ln 2 x + ln π 1 + x 2 d x < ∞ , {\displaystyle \int _{-\infty }^{\infty }{\frac {-\ln f(x)}{1+x^{2}}}dx=\int _{-\infty }^{\infty }{\frac {\ln ^{2}x+\ln {\sqrt {\pi }}}{1+x^{2}}}\,dx<\infty ,} the Hamburger moment problem for μ is indeterminate. == References ==
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Wikipedia:Kripa Shankar Shukla#0
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Kripa Shankar Shukla (10 July 1918 - 22 September 2007) was a historian of Indian mathematics. He was awarded the DLitt degree by Lucknow University in 1955 for his thesis on “Astronomy in the Seventh Century India: Bhāskara I and His Works” which was completed under the guidance of A. N. Singh. He retired in 1979. Shukla published several important source works in Indian mathematics and astronomy with translations, notes, explanations. He also authored a large number of research papers bringing out many previously unknown facts about the historical development of mathematics in India. He continued his active research even after retiring from his official position in Lucknow University. Shukla supervised the research work of several research scholars including that of Yukio Ohashi (1955 - 2019) from Japan whose dissertation was titled "A History of Astronomical Instruments in India". == Publications == === Source works brought out by K. S. Shula === The important source works brought out by Shukla include the following: Sūrya-siddhānta with the commentary of Parameśvara (1957) [1] Pāṭīgaṇita of Śrīdharācārya (1959) [2] Mahābhāskarīya of Bhāskara I (1960) [3] Laghubhāskarīya of Bhāskara I (1963) [4] Dhīkoṭida-karaṇa of Śrīpati (1969) Bījagaṇitāvataṃsa of Nārāyaṇa Paṇḍita (1970) Āryabhaṭīya of Āryabhaṭa (1976) [5] Āryabhaṭīya of Āryabhaṭa with the commentary of Bhāskara I and Someśvara (1976) Karaṇaratna of Devācārya (1979) [6] Vaṭeśvarasiddhānta and Gola of Vaṭeśvara (2 Vols) (1985–86) [7] Laghumānasa of Mañjula (1990) [8] Gaṇitapañcaviṃśī (published posthumously) (2017) === Research papers === Aditya Kolachana, K. Mahesh and K. Ramasubramanian have brought out a 754-page volume containing a collection of selected articles by Shukla under the title "Studies in Indian Mathematics and Astronomy" in 2019. == Awards and accolades == The awards and prizes conferred on Shukla include the following: Awarded Banerji Research Prize of the Lucknow University. Elected Fellow of the National Academy of Sciences, India in 1984 Selected as a Corresponding Member of the International Academy of History of Science, Paris, in 1988. == References ==
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Wikipedia:Kristen Nygaard#0
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Kristen Nygaard (27 August 1926 – 10 August 2002) was a Norwegian computer scientist, programming language pioneer, and politician. Internationally, Nygaard is acknowledged as the co-inventor of object-oriented programming and the programming language Simula with Ole-Johan Dahl in the 1960s. Nygaard and Dahl received the 2001 A. M. Turing Award for their contribution to computer science. == Early life and career == Nygaard was born in Oslo and received his master's degree in mathematics at the University of Oslo in 1956. His thesis on abstract probability theory was entitled "Theoretical Aspects of Monte Carlo methods". Nygaard worked full-time at the Norwegian Defense Research Establishment from 1948 to 1960, in computing and programming (1948–1954) and operational research (1952–1960). From 1957 to 1960, he was head of the first operations research groups in the Norwegian defense establishment. He was cofounder and first chairman of the Norwegian Operational Research Society (1959–1964). In 1960, he was hired by the Norwegian Computing Center (NCC), responsible for building up the NCC as a research institute in the 1960s, becoming its Director of Research in 1962. == Object-oriented programming == With Ole-Johan Dahl, he developed the initial ideas for object-oriented programming (OOP) in the 1960s at the Norwegian Computing Center (Norsk Regnesentral (NR)) as part of the Simula I (1961–1965) and Simula 67 (1965–1968) simulation programming languages, which began as an extended variant and superset of ALGOL 60. The languages introduced the core concepts of object-oriented programming: objects, classes, inheritance, virtual quantities, and multi-threaded (quasi-parallel) program execution. In 2004, the Association Internationale pour les Technologies Objets (AITO) established an annual prize in the name of Ole-Johan Dahl and Kristen Nygaard to honor their pioneering work on object-orientation. This Dahl–Nygaard Prize is awarded annually to two individuals that have made significant technical contributions to the field of object-orientation. The work should be in the spirit of the pioneer conceptual and/or implementation work of Dahl and Nygaard which shaped the present view of object-oriented programming. The prize is presented each year at the ECOOP conference. The prize consists of two awards given to a senior and a junior professional. He conducted research for Norwegian trade unions on planning, control, and data processing, all evaluated in light of the objectives of organised labour (1971–1973), working together with Olav Terje Bergo. His other research and development work included the social impact of computer technology, and the general system description language DELTA (1973–1975), working with Erik Holbaek-Hanssen and Petter Haandlykken. Nygaard was a professor at Aarhus University, Denmark (1975–1976) and then became professor emeritus at the University of Oslo (part-time from 1977, full-time 1984–1996). His work in Aarhus and Oslo included research and education in system development and the social impact of computer technology, and became the foundation of the Scandinavian School in System Development, which is closely linked to the field of participatory design. Starting in 1976, he was engaged in developing and (since 1986) implementing the general object-oriented programming language BETA, together with Bent Bruun Kristensen, Ole Lehrmann Madsen, and Birger Møller-Pedersen. The language is now available on a wide range of computers. == Later career == In the first half of the 1980s, Nygaard was chairman of the steering committee of the Scandinavian research program System Development and Profession Oriented Languages (SYDPOL), coordinating research and supporting working groups in system development, language research, and artificial intelligence. Also in the 1980s, he was chairman of the steering committee for the Cost-13 (European Common Market Commission)-financed research project on the extensions of profession-oriented languages necessary when artificial intelligence and information technology are becoming part of professional work. Nygaard's research from 1995 to 1999 was related to distributed systems. He was the leader of General Object-Oriented Distributed Systems (GOODS), a three-year Norwegian Research Council-supported project starting in 1997, aiming at enriching object-oriented languages and system development methods by new basic concepts that make it possible to describe the relation between layered and/or distributed programs and the computer hardware and people carrying out these computer programs. The GOODS team also included Haakon Bryhni, Dag Sjøberg, and Ole Smørdal. Nygaard's final research interests were studies of the introductory teaching of programming, and creating a process-oriented conceptual platform for informatics. These subjects are to be developed in a new research project named Comprehensive Object-Oriented Learning (COOL), together with several international test sites. He was giving lectures and courses on these subjects in Norway and elsewhere. In November 1999, he became chair of an advisory committee on Broadband Communication for the Norwegian Department for Municipal and Regional Affairs. He held a part-time position at Simula Research Laboratory from 2001, when the research institute was opened. == Recognition == In June 1990, he received an honorary doctorate from Lund University, Sweden. In June 1991, he became the first individual to be given an honorary doctorate by Aalborg University, Denmark. He became a member of the Norwegian Academy of Sciences. In October 1990, Computer Professionals for Social Responsibility awarded him its Norbert Wiener Award for Social and Professional Responsibility. In 1999, he and Dahl became the first people to receive the then new Rosing Prize, awarded by the Norwegian Data Association for exceptional professional achievements. In June 2000, he was awarded an Honorary Fellowship for "his originating of object technology concepts" by the Object Management Group, a technical standards group for object-orientation, which maintains several International Organization for Standardization (ISO) standards. In November 2001, the Institute of Electrical and Electronics Engineers (IEEE) awarded Nygaard and Dahl the IEEE John von Neumann Medal "For the introduction of the concepts underlying object-oriented programming through the design and implementation of Simula 67". In February 2002, he was given, once more with Ole-Johan Dahl, the 2001 A. M. Turing Award by the Association for Computing Machinery (ACM), with the citation: "For ideas fundamental to the emergence of object-oriented programming, through their design of the programming languages Simula I and Simula 67." In August 2000, he was made Commander of the Royal Norwegian Order of St. Olav by then King Harald V of Norway. == Other activities == In 1984 and 1985, Nygaard was chairman of the Informatics Committee of the University of Oslo, and active in the design of the university's plan for developing research, education and computing and communication facilities at all faculties of the university. He was the first chairman of the Environment Protection Committee of the Norwegian Association for the Protection of Nature. He was for 10 years (in the 1970s) Norwegian representative in the Organisation for Economic Co-operation and Development (OECD) activities on information technology. He has been a member of the Research Committee of the Norwegian Federation of Trade Unions, and cooperated with unions in many countries. For several years, he was engaged in running an experimental social institution trying new ways to create humane living conditions for socially outcast alcoholics. Nygaard was active in Norwegian politics. In the mid and late 1960s, he was a member of the National Executive Committee of the Norwegian Liberal Party, and chair of that party's Strategy Committee. He was a minor ballot candidate in the 1949 parliamentary election. During the intense political fight before the 1972 referendum on whether Norway should become a member of the European Common Market (later the European Union), he worked as coordinator for the many youth organisations that worked against membership. From 1971 to 2001, Nygaard was a member of the Labour Party, and a member of their committees on research policies. In November 1988, he became chair of the Information Committee on Norway and the EEC, in August 1990 reorganized as Nei til EF an organization disseminating information about Norway's relation to the Common Market, and coordinating the efforts to keep Norway outside. (No to European Union membership for Norway, literally "No to the EU"). In 1993, when the EEC ratified the Maastricht Treaty and became the European Union the organization changed its name to reflect this. Nei til EF became the largest political organization in Norway (145,000 members in 1994, from a population of 4 million). Nygaard worked with Anne Enger Lahnstein, leader of the anti-EU Centre Party, in this campaign. In the referendum on 28 November 1994, "Nei til EU" succeeded: 52.2% of the electorate voted "No", and the voter participation was the highest ever in Norway's history: 88.8%. The strategy of the campaign, insisted by Nygaard, was that it had to be for something as well as against, i.e., the Scandinavian welfare state Nygaard considered threatened by the Maastricht Agreement. He resigned as chair in 1995, and was later the chair of the organization's strategy committee and a member of its council. In 1996 and 1997, Nygaard was the coordinator of the efforts to establish The European Anti-Maastricht Movement (TEAM), a cooperative network between national organizations opposing the Economic and Monetary Union of the European Union (EMU) and the Maastricht Treaty in European countries within and outside the EU. The European Alliance of EU-critical Movements (TEAM) was successfully started 3 March 1997. == Personal life == Kristen Nygaard married Johanna Nygaard in 1951. She worked at the Norwegian Agency for Aid to Developing Countries. She specialized for a number of years in recruiting and giving administrative support to specialists working in East Africa. Johanna and Kristen Nygaard had three children and seven grandchildren. Nygaard died of a heart attack in 2002. == See also == List of pioneers in computer science == References == Curriculum Vitae for Kristen Nygaard at the Wayback Machine (archived 16 October 2002) (15 February 2002, Long Version) == External links == Curriculum Vitae for Kristen Nygaard Kristen Nygaard bibliography Resources on Ole-Johan Dahl, Kristen Nygaard, and Simula [1] Berntsen D., Elgsaas K., Hegna H. (2010) The Many Dimensions of Kristen Nygaard, Creator of Object-Oriented Programming and the Scandinavian School of System Development. In: Tatnall A. (eds) History of Computing. Learning from the Past. IFIP Advances in Information and Communication Technology, vol 325. Springer, Berlin, Heidelberg. [2] MacTutor History of Mathematics Archive: Kristen Nygaard. [3] Marius Nygaard. Notes on Kristen Nygaard's early years and his political work. Chapter in ”People behind informatics” by Lazlo Bözörményi and Stefan Podlipnig. Institute of Information Technology, University of Klagenfurt 2003
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Wikipedia:Kristian B. Dysthe#0
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Kristian Barstad Dysthe (16 September 1937 – 30 July 2023) was a Norwegian mathematician. == Biography == Dysthe took the cand.real. degree at the University of Bergen in 1962, and the dr.philos. degree in 1972. He became professor in applied mathematics at the University of Tromsø in 1972, and at the University of Bergen from 1992 to retirement in 2007. He has been a visiting scholar at the University of Cambridge, the Scripps Institution of Oceanography and the Stanford University. He was a member of the Norwegian Academy of Science and Letters. Dysthe died on 30 July 2023, at the age of 85. == References ==
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Wikipedia:Kristian Seip#0
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Hans Kristian Seip (6 November 1881 - 25 March 1945) was a Norwegian road engineer and politician for the Liberal Party. He spent most of his professional career in the Norwegian Public Roads Administration. As a politician he was Mayor of Bergen and County Governor of Sogn og Fjordane, and served two terms in the Norwegian Parliament. He is also known as the father of political scientist Jens Arup Seip. == Personal life == He was born in Røyken as the son of priest Jens Laurits Arup Seip (1852–1913) and Marie Fredrikke Aubert (1853–1931), He was the brother of academic Didrik Arup Seip, nephew of educator and politician Karl Seip, and great-grandson of military officer and politician Andreas Martin Seip. Hans Kristian was the father of political scientist Jens Arup Seip and thus father-in-law of historian Anne-Lise Seip. He was also the uncle of forester Hans Kristian Seip. == Career == Hans Kristian Seip graduated from the technical school in Kristiania in 1900, and after working for the Norwegian Public Roads Administration for two years he studied one year at Zürich Polytechnikum. He then returned to Norway to work for the Norwegian Public Roads Administration in Møre from 1903 to 1909, Hordaland from 1909 to 1915 and as engineer in Bergen from 1915 to 1921. From 1921 to 1929 he was the director of roads in Bergen county. Seip was elected to Bergen city council in 1913. He was re-elected several times, and served until 1928; from 1922 to 1924 he served as mayor. He represented the small Prohibition Party. He was also a deputy representative to the Norwegian Parliament during the terms 1925-1927 and 1928–1930, but represented the Liberal Party on the national level. On 2 November 1929 Seip was appointed County Governor of Sogn og Fjordane. While stationed here he was elected to the Norwegian Parliament twice; in 1934 and 1937. The scheduled election in 1940, however, was not held due to the German invasion of Norway in April 1940 and subsequent occupation. In 1941 Seip was removed from the position as County Governor. He retired to Fjærland, and worked on the history of Sogn og Fjordane County Municipality. He died in March 1945, a month and a half before the liberation of Norway. == References ==
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Wikipedia:Kristina Vušković#0
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Kristina L. Vušković (Serbian: Кристина Л. Вушковић, born 6 May 1967) is a Serbian mathematician and theoretical computer scientist working in graph theory. She is Professor in Algorithms and Combinatorics in the School of Computing at the University of Leeds, and a professor of computer science at Union University (Serbia). == Education and career == Vušković was born on 6 May 1967 in Belgrade. She graduated summa cum laude from the Courant Institute of Mathematical Sciences of New York University in 1989, majoring in mathematics and computer science, and completed her PhD in Algorithms, Combinatorics and Optimization at Carnegie Mellon University in 1994. Her dissertation, supervised by Gérard Cornuéjols, was Holes in Bipartite Graphs. After postdoctoral research as an NSERC Canada International Fellow at the University of Waterloo, she became an assistant professor of mathematics at the University of Kentucky, in 1996. She moved to Leeds in 2000, and was given the chair of algorithms and combinatorics at Leeds in 2011. Since 2007 she has also been a professor of computer science at Union University (Serbia). == Research == Vušković's research in graph theory concerns the structure and algorithms of hereditary classes of graphs. Her results include the recognition of perfect graphs in polynomial time; she has also worked in combinatorial algorithms for graph coloring of perfect graphs. == References ==
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Wikipedia:Kriyakramakari#0
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Kriyakramakari (Kriyā-kramakarī) is an elaborate commentary in Sanskrit written by Sankara Variar and Narayana, two astronomer-mathematicians belonging to the Kerala school of astronomy and mathematics, on Bhaskara II's well-known textbook on mathematics Lilavati. Kriyakramakari ('Operational Techniques'), along with Yuktibhasa of Jyeshthadeva, is one of the main sources of information about the work and contributions of Sangamagrama Madhava, the founder of Kerala school of astronomy and mathematics. Also the quotations given in this treatise throw much light on the contributions of several mathematicians and astronomers who had flourished in an earlier era. There are several quotations ascribed to Govindasvami a 9th-century astronomer from Kerala. Sankara Variar (c. 1500 - 1560), the first author of Kriyakramakari, was a pupil of Nilakantha Somayaji and a temple-assistant by profession. He was a prominent member of the Kerala school of astronomy and mathematics. His works include Yukti-dipika an extensive commentary on Tantrasangraha by Nilakantha Somayaji. Narayana (c. 1540–1610), the second author, was a Namputiri Brahmin belonging to the Mahishamangalam family in Puruvanagrama (Peruvanam in modern-day Thrissur District in Kerala). Sankara Variar wrote his commentary of Lilavati up to stanza 199. Variar completed this by about 1540 when he stopped writing due to other preoccupations. Sometimes after his death, Narayana completed the commentary on the remaining stanzas in Lilavati. == On the computation of π == As per K.V. Sarma's critical edition of Lilavati based on Kriyakramakari, stanza 199 of Lilavati reads as follows (Harvard-Kyoto convention is used for the transcription of the Indian characters): vyAse bha-nanda-agni-hate vibhakte kha-bANa-sUryais paridhis sas sUkSmas/ dvAviMzati-ghne vihRte atha zailais sthUlas atha-vA syAt vyavahAra-yogyas// This could be translated as follows; "Multiply the diameter by 3927 and divide the product by 1250; this gives the more precise circumference. Or, multiply the diameter by 22 and divide the product by 7; this gives the approximate circumference which answers for common operations." Taking this verse as a starting point and commenting on it, Sanakara Variar in his Kriyakrakari explicated the full details of the contributions of Sangamagrama Madhava towards obtaining accurate values of π. Sankara Variar commented like this: "The teacher Madhava also mentioned a value of the circumference closer [to the true value] than that: "Gods [thirty-three], eyes [two], elephants [eight], serpents [eight], fires [three], three, qualities [three], Vedas [four], naksatras [twentyseven], elephants [eight], arms [two] (2,827,433,388,233)—the wise said that this is the measure of the circumference when the diameter of a circle is nine nikharva [10^11]." Sankara Variar says here that Madhava's value 2,827,433,388,233 / 900,000,000,000 is more accurate than "that", that is, more accurate than the traditional value for π." Sankara Variar then cites a set of four verses by Madhava that prescribe a geometric method for computing the value of the circumference of a circle. This technique involves calculating the perimeters of successive regular circumscribed polygons, beginning with a square. === An infinite series for π === Sankara Variar then describes an easier method due to Madhava to compute the value of π. "An easier way to get the circumference is mentioned by him (Madhava). That is to say: Add or subtract alternately the diameter multiplied by four and divided in order by the odd numbers like three, five, etc., to or from the diameter multiplied by four and divided by one. Assuming that division is completed by dividing by an odd number, whatever is the even number above [next to] that [odd number], half of that is the multiplier of the last [term]. The square of that [even number] increased by 1 is the divisor of the diameter multiplied by 4 as before. The result from these two (the multiplier and the divisor) is added when [the previous term is] negative, when positive subtracted. The result is an accurate circumference. If division is repeated many times, it will become very accurate." To translate these verses into modern mathematical notations, let C be the circumference and D the diameter of a circle. Then Madhava's easier method to find C reduces to the following expression for C: C = 4D/1 - 4D/3 + 4D/5 - 4D/7 + ... This is essentially the series known as the Gregory-Leibniz series for π. After stating this series, Sankara Variar follows it up with a description of an elaborate geometrical rationale for the derivation of the series. === An infinite series for arctangent === The theory is further developed in Kriyakramakari. It takes up the problem of deriving a similar series for the computation of an arbitrary arc of a circle. This yields the infinite series expansion of the arctangent function. This result is also ascribed to Madhava. "Now, by just the same argument, the determination of the arc of a desired Sine can be [made]. That is as [follows]: The first result is the product of the desired Sine and the radius divided by the Cosine. When one has made the square of the Sine the multiplier and the square of the Cosine the divisor, now a group of results is to be determined from the [previous] results beginning with the first. When these are divided in order by the odd numbers 1, 3, and so forth, and when one has subtracted the sum of the even[-numbered results] from the sum of the odd ones, [that] should be the arc. Here, the smaller of the Sine and Cosine is required to be considered as the desired [Sine]. Otherwise there would be no termination of the results even if repeatedly [computed]." The above formulas state that if for an arbitrary arc θ of a circle of radius R the sine and cosine are known and if we assume that sin θ < cos θ, then we have: θ = (R sin θ)/(1 cos θ) − (R sin3 θ)/(3 cos3 θ) + (R sin5 θ)/(5 cos5 θ) − (R sin7 θ)/(7 cos7 θ)+ . . . == See also == Kerala school of astronomy and mathematics Lilavati Sankara Variar == References ==
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Wikipedia:Kronecker coefficient#0
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In mathematics, Kronecker coefficients gλμν describe the decomposition of the tensor product (= Kronecker product) of two irreducible representations of a symmetric group into irreducible representations. They play an important role in algebraic combinatorics and geometric complexity theory. They were introduced by Murnaghan in 1938. == Definition == Given a partition λ of n, write Vλ for the Specht module associated to λ. Then the Kronecker coefficients gλμν are given by the rule V μ ⊗ V ν = ⨁ λ g μ ν λ V λ . {\displaystyle V_{\mu }\otimes V_{\nu }=\bigoplus _{\lambda }g_{\mu \nu }^{\lambda }V_{\lambda }.} One can interpret this on the level of symmetric functions, giving a formula for the Kronecker product of two Schur polynomials: s μ ⋆ s ν = ∑ λ g μ ν λ s λ . {\displaystyle s_{\mu }\star s_{\nu }=\sum _{\lambda }g_{\mu \nu }^{\lambda }s_{\lambda }.} This is to be compared with Littlewood–Richardson coefficients, where one instead considers the induced representation ↑ S | μ | × S | ν | S | λ | ( V μ ⊗ V ν ) = ⨁ λ c μ ν λ V λ , {\displaystyle \uparrow _{S_{|\mu |}\times S_{|\nu |}}^{S_{|\lambda |}}\left(V_{\mu }\otimes V_{\nu }\right)=\bigoplus _{\lambda }c_{\mu \nu }^{\lambda }V_{\lambda },} and the corresponding operation of symmetric functions is the usual product. Also note that the Littlewood–Richardson coefficients are the analogue of the Kronecker coefficients for representations of GLn, i.e. if we write Wλ for the irreducible representation corresponding to λ (where λ has at most n parts), one gets that W μ ⊗ W ν = ⨁ λ c μ ν λ W λ . {\displaystyle W_{\mu }\otimes W_{\nu }=\bigoplus _{\lambda }c_{\mu \nu }^{\lambda }W_{\lambda }.} == Properties == Bürgisser & Ikenmeyer (2008) showed that computing Kronecker coefficients is #P-hard and contained in GapP. A recent work by Ikenmeyer, Mulmuley & Walter (2017) shows that deciding whether a given Kronecker coefficient is non-zero is NP-hard. This recent interest in computational complexity of these coefficients arises from its relevance in the Geometric Complexity Theory program. A major unsolved problem in representation theory and combinatorics is to give a combinatorial description of the Kronecker coefficients. It has been open since 1938, when Murnaghan asked for such a combinatorial description. A combinatorial description would also imply that the problem is # P-complete in light of the above result. The Kronecker coefficients can be computed as g ( λ , μ , ν ) = 1 n ! ∑ σ ∈ S n χ λ ( σ ) χ μ ( σ ) χ ν ( σ ) , {\displaystyle g(\lambda ,\mu ,\nu )={\frac {1}{n!}}\sum _{\sigma \in S_{n}}\chi ^{\lambda }(\sigma )\chi ^{\mu }(\sigma )\chi ^{\nu }(\sigma ),} where χ λ ( σ ) {\displaystyle \chi ^{\lambda }(\sigma )} is the character value of the irreducible representation corresponding to integer partition λ {\displaystyle \lambda } on a permutation σ ∈ S n {\displaystyle \sigma \in S_{n}} . The Kronecker coefficients also appear in the generalized Cauchy identity ∑ λ , μ , ν g ( λ , μ , ν ) s λ ( x ) s μ ( y ) s ν ( z ) = ∏ i , j , k 1 1 − x i y j z k . {\displaystyle \sum _{\lambda ,\mu ,\nu }g(\lambda ,\mu ,\nu )s_{\lambda }(x)s_{\mu }(y)s_{\nu }(z)=\prod _{i,j,k}{\frac {1}{1-x_{i}y_{j}z_{k}}}.} == See also == Littlewood–Richardson coefficient == References == Bürgisser, Peter; Ikenmeyer, Christian (2008), "The complexity of computing Kronecker coefficients", 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), Discrete Math. Theor. Comput. Sci. Proc., AJ, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, pp. 357–368, MR 2721467
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Wikipedia:Kronecker substitution#0
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Kronecker substitution is a technique named after Leopold Kronecker for determining the coefficients of an unknown polynomial by evaluating it at a single value. If p(x) is a polynomial with integer coefficients, and x is chosen to be both a power of two and larger in magnitude than any of the coefficients of p, then the coefficients of each term of can be read directly out of the binary representation of p(x). One application of this method is to reduce the computational problem of multiplying polynomials to the (potentially simpler) problem of multiplying integers. If p(x) and q(x) are polynomials with known coefficients, then one can use these coefficients to determine a value of x that is a large enough power of two for the coefficients of the product pq(x) to be able to be read off from the binary representation of the number p(x)q(x). Since p(x) and q(x) are themselves straightforward to determine from the coefficients of p and q, this result shows that polynomial multiplication may be performed in the time of a single binary multiplication. == See also == Kronecker product == References ==
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Wikipedia:Kronecker sum of discrete Laplacians#0
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In mathematics, the Kronecker sum of discrete Laplacians, named after Leopold Kronecker, is a discrete version of the separation of variables for the continuous Laplacian in a rectangular cuboid domain. == General form of the Kronecker sum of discrete Laplacians == In a general situation of the separation of variables in the discrete case, the multidimensional discrete Laplacian is a Kronecker sum of 1D discrete Laplacians. === Example: 2D discrete Laplacian on a regular grid with the homogeneous Dirichlet boundary condition === Mathematically, using the Kronecker sum: L = D x x ⊗ I + I ⊗ D y y , {\displaystyle L=\mathbf {D_{xx}} \otimes \mathbf {I} +\mathbf {I} \otimes \mathbf {D_{yy}} ,\,} where D x x {\displaystyle \mathbf {D_{xx}} } and D y y {\displaystyle \mathbf {D_{yy}} } are 1D discrete Laplacians in the x- and y-directions, correspondingly, and I {\displaystyle \mathbf {I} } are the identities of appropriate sizes. Both D x x {\displaystyle \mathbf {D_{xx}} } and D y y {\displaystyle \mathbf {D_{yy}} } must correspond to the case of the homogeneous Dirichlet boundary condition at end points of the x- and y-intervals, in order to generate the 2D discrete Laplacian L corresponding to the homogeneous Dirichlet boundary condition everywhere on the boundary of the rectangular domain. Here is a sample OCTAVE/MATLAB code to compute L on the regular 10×15 2D grid: == Eigenvalues and eigenvectors of multidimensional discrete Laplacian on a regular grid == Knowing all eigenvalues and eigenvectors of the factors, all eigenvalues and eigenvectors of the Kronecker product can be explicitly calculated. Based on this, eigenvalues and eigenvectors of the Kronecker sum can also be explicitly calculated. The eigenvalues and eigenvectors of the standard central difference approximation of the second derivative on an interval for traditional combinations of boundary conditions at the interval end points are well known. Combining these expressions with the formulas of eigenvalues and eigenvectors for the Kronecker sum, one can easily obtain the required answer. === Example: 3D discrete Laplacian on a regular grid with the homogeneous Dirichlet boundary condition === L = D x x ⊗ I ⊗ I + I ⊗ D y y ⊗ I + I ⊗ I ⊗ D z z , {\displaystyle L=\mathbf {D_{xx}} \otimes \mathbf {I} \otimes \mathbf {I} +\mathbf {I} \otimes \mathbf {D_{yy}} \otimes \mathbf {I} +\mathbf {I} \otimes \mathbf {I} \otimes \mathbf {D_{zz}} ,\,} where D x x , D y y {\displaystyle \mathbf {D_{xx}} ,\,\mathbf {D_{yy}} } and D z z {\displaystyle \mathbf {D_{zz}} } are 1D discrete Laplacians in every of the 3 directions, and I {\displaystyle \mathbf {I} } are the identities of appropriate sizes. Each 1D discrete Laplacian must correspond to the case of the homogeneous Dirichlet boundary condition, in order to generate the 3D discrete Laplacian L corresponding to the homogeneous Dirichlet boundary condition everywhere on the boundary. The eigenvalues are λ j x , j y , j z = − 4 h x 2 sin ( π j x 2 ( n x + 1 ) ) 2 − 4 h y 2 sin ( π j y 2 ( n y + 1 ) ) 2 − 4 h z 2 sin ( π j z 2 ( n z + 1 ) ) 2 {\displaystyle \lambda _{jx,jy,jz}=-{\frac {4}{h_{x}^{2}}}\sin \left({\frac {\pi j_{x}}{2(n_{x}+1)}}\right)^{2}-{\frac {4}{h_{y}^{2}}}\sin \left({\frac {\pi j_{y}}{2(n_{y}+1)}}\right)^{2}-{\frac {4}{h_{z}^{2}}}\sin \left({\frac {\pi j_{z}}{2(n_{z}+1)}}\right)^{2}} where j x = 1 , … , n x , j y = 1 , … , n y , j z = 1 , … , n z , {\displaystyle j_{x}=1,\ldots ,n_{x},\,j_{y}=1,\ldots ,n_{y},\,j_{z}=1,\ldots ,n_{z},\,} , and the corresponding eigenvectors are v i x , i y , i z , j x , j y , j z = 2 n x + 1 sin ( i x j x π n x + 1 ) 2 n y + 1 sin ( i y j y π n y + 1 ) 2 n z + 1 sin ( i z j z π n z + 1 ) {\displaystyle v_{ix,iy,iz,jx,jy,jz}={\sqrt {\frac {2}{n_{x}+1}}}\sin \left({\frac {i_{x}j_{x}\pi }{n_{x}+1}}\right){\sqrt {\frac {2}{n_{y}+1}}}\sin \left({\frac {i_{y}j_{y}\pi }{n_{y}+1}}\right){\sqrt {\frac {2}{n_{z}+1}}}\sin \left({\frac {i_{z}j_{z}\pi }{n_{z}+1}}\right)} where the multi-index j x , j y , j z {\displaystyle {jx,jy,jz}} pairs the eigenvalues and the eigenvectors, while the multi-index i x , i y , i z {\displaystyle {ix,iy,iz}} determines the location of the value of every eigenvector at the regular grid. The boundary points, where the homogeneous Dirichlet boundary condition are imposed, are just outside the grid. == Available software == An OCTAVE/MATLAB code http://www.mathworks.com/matlabcentral/fileexchange/27279-laplacian-in-1d-2d-or-3d is available under a BSD License, which computes the sparse matrix of the 1, 2D, and 3D negative Laplacians on a rectangular grid for combinations of Dirichlet, Neumann, and Periodic boundary conditions using Kronecker sums of discrete 1D Laplacians. The code also provides the exact eigenvalues and eigenvectors using the explicit formulas given above.
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Wikipedia:Krull's separation lemma#0
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In abstract algebra, Krull's separation lemma is a lemma in ring theory. It was proved by Wolfgang Krull in 1928. == Statement of the lemma == Let I {\displaystyle I} be an ideal and let M {\displaystyle M} be a multiplicative system (i.e. M {\displaystyle M} is closed under multiplication) in a ring R {\displaystyle R} , and suppose I ∩ M = ∅ {\displaystyle I\cap M=\varnothing } . Then there exists a prime ideal P {\displaystyle P} satisfying I ⊆ P {\displaystyle I\subseteq P} and P ∩ M = ∅ {\displaystyle P\cap M=\varnothing } . == References ==
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Wikipedia:Krull's theorem#0
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In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn's lemma, and in fact is equivalent to Zorn's lemma, which in turn is equivalent to the axiom of choice. == Variants == For noncommutative rings, the analogues for maximal left ideals and maximal right ideals also hold. For pseudo-rings, the theorem holds for regular ideals. An apparently slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows: Let R be a ring, and let I be a proper ideal of R. Then there is a maximal ideal of R containing I. The statement of the original theorem can be obtained by taking I to be the zero ideal (0). Conversely, applying the original theorem to R/I leads to this result. To prove the "stronger" result directly, consider the set S of all proper ideals of R containing I. The set S is nonempty since I ∈ S. Furthermore, for any chain T of S, the union of the ideals in T is an ideal J, and a union of ideals not containing 1 does not contain 1, so J ∈ S. By Zorn's lemma, S has a maximal element M. This M is a maximal ideal containing I. == Notes == == References == Krull, W. (1929). "Idealtheorie in Ringen ohne Endlichkeitsbedingungen". Mathematische Annalen. 101 (1): 729–744. doi:10.1007/BF01454872. S2CID 119883473. Hodges, W. (1979). "Krull implies Zorn". Journal of the London Mathematical Society. s2-19 (2): 285–287. doi:10.1112/jlms/s2-19.2.285.
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Wikipedia:Krull–Akizuki theorem#0
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In commutative algebra, the Krull–Akizuki theorem states the following: Let A be a one-dimensional reduced noetherian ring, K its total ring of fractions. Suppose L is a finite extension of K. If A ⊂ B ⊂ L {\displaystyle A\subset B\subset L} and B is reduced, then B is a noetherian ring of dimension at most one. Furthermore, for every nonzero ideal I {\displaystyle I} of B, B / I {\displaystyle B/I} is finite over A. Note that the theorem does not say that B is finite over A. The theorem does not extend to higher dimension. One important consequence of the theorem is that the integral closure of a Dedekind domain A in a finite extension of the field of fractions of A is again a Dedekind domain. This consequence does generalize to a higher dimension: the Mori–Nagata theorem states that the integral closure of a noetherian domain is a Krull domain. == Proof == First observe that A ⊂ B ⊂ K B {\displaystyle A\subset B\subset KB} and KB is a finite extension of K, so we may assume without loss of generality that L = K B {\displaystyle L=KB} . Then L = K x 1 + ⋯ + K x n {\displaystyle L=Kx_{1}+\cdots +Kx_{n}} for some x 1 , … , x n ∈ B {\displaystyle x_{1},\dots ,x_{n}\in B} . Since each x i {\displaystyle x_{i}} is integral over K, there exists a i ∈ A {\displaystyle a_{i}\in A} such that a i x i {\displaystyle a_{i}x_{i}} is integral over A. Let C = A [ a 1 x 1 , … , a n x n ] {\displaystyle C=A[a_{1}x_{1},\dots ,a_{n}x_{n}]} . Then C is a one-dimensional noetherian ring, and C ⊂ B ⊂ Q ( C ) {\displaystyle C\subset B\subset Q(C)} , where Q ( C ) {\displaystyle Q(C)} denotes the total ring of fractions of C. Thus we can substitute C for A and reduce to the case L = K {\displaystyle L=K} . Let p i {\displaystyle {\mathfrak {p}}_{i}} be minimal prime ideals of A; there are finitely many of them. Let K i {\displaystyle K_{i}} be the field of fractions of A / p i {\displaystyle A/{{\mathfrak {p}}_{i}}} and I i {\displaystyle I_{i}} the kernel of the natural map B → K → K i {\displaystyle B\to K\to K_{i}} . Then we have: A / p i ⊂ B / I i ⊂ K i {\displaystyle A/{{\mathfrak {p}}_{i}}\subset B/{I_{i}}\subset K_{i}} and K ≃ ∏ K i {\displaystyle K\simeq \prod K_{i}} . Now, if the theorem holds when A is a domain, then this implies that B is a one-dimensional noetherian domain since each B / I i {\displaystyle B/{I_{i}}} is and since B ≃ ∏ B / I i {\displaystyle B\simeq \prod B/{I_{i}}} . Hence, we reduced the proof to the case A is a domain. Let 0 ≠ I ⊂ B {\displaystyle 0\neq I\subset B} be an ideal and let a be a nonzero element in the nonzero ideal I ∩ A {\displaystyle I\cap A} . Set I n = a n B ∩ A + a A {\displaystyle I_{n}=a^{n}B\cap A+aA} . Since A / a A {\displaystyle A/aA} is a zero-dim noetherian ring; thus, artinian, there is an l {\displaystyle l} such that I n = I l {\displaystyle I_{n}=I_{l}} for all n ≥ l {\displaystyle n\geq l} . We claim a l B ⊂ a l + 1 B + A . {\displaystyle a^{l}B\subset a^{l+1}B+A.} Since it suffices to establish the inclusion locally, we may assume A is a local ring with the maximal ideal m {\displaystyle {\mathfrak {m}}} . Let x be a nonzero element in B. Then, since A is noetherian, there is an n such that m n + 1 ⊂ x − 1 A {\displaystyle {\mathfrak {m}}^{n+1}\subset x^{-1}A} and so a n + 1 x ∈ a n + 1 B ∩ A ⊂ I n + 2 {\displaystyle a^{n+1}x\in a^{n+1}B\cap A\subset I_{n+2}} . Thus, a n x ∈ a n + 1 B ∩ A + A . {\displaystyle a^{n}x\in a^{n+1}B\cap A+A.} Now, assume n is a minimum integer such that n ≥ l {\displaystyle n\geq l} and the last inclusion holds. If n > l {\displaystyle n>l} , then we easily see that a n x ∈ I n + 1 {\displaystyle a^{n}x\in I_{n+1}} . But then the above inclusion holds for n − 1 {\displaystyle n-1} , contradiction. Hence, we have n = l {\displaystyle n=l} and this establishes the claim. It now follows: B / a B ≃ a l B / a l + 1 B ⊂ ( a l + 1 B + A ) / a l + 1 B ≃ A / ( a l + 1 B ∩ A ) . {\displaystyle B/{aB}\simeq a^{l}B/a^{l+1}B\subset (a^{l+1}B+A)/a^{l+1}B\simeq A/(a^{l+1}B\cap A).} Hence, B / a B {\displaystyle B/{aB}} has finite length as A-module. In particular, the image of I {\displaystyle I} there is finitely generated and so I {\displaystyle I} is finitely generated. The above shows that B / a B {\displaystyle B/{aB}} has dimension at most zero and so B has dimension at most one. Finally, the exact sequence B / a B → B / I → ( 0 ) {\displaystyle B/aB\to B/I\to (0)} of A-modules shows that B / I {\displaystyle B/I} is finite over A. ◻ {\displaystyle \square } == References == Bourbaki, Nicolas (1989). Commutative algebra. Berlin Heidelberg: Springer. ISBN 978-3-540-64239-8.
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