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https://en.wikipedia.org/wiki/SAMPL | SAMPL, which stands for "Stochastic AMPL", is an algebraic modeling language resulting by expanding the well-known language AMPL with extended syntax and keywords. It is designed specifically for representing stochastic programming problems and, through recent extensions, problems with chance constraints, integrated chance constraints and robust optimization problems.
It can generate the deterministic equivalent version of the instances, using all the solvers AMPL connects to, or generate an SMPS representation and use specialized decomposition based solvers, like FortSP.
Language Features
SAMPL shares all language features with AMPL, and adds some constructs specifically designed for expressing scenario based stochastic programming and robust optimization.
Stochastic programming features and constructs
To express scenario-based SP problems, additional constructs describe the tree structure and group the decision variable into stages. Moreover, it is possible to specify which parameter stores the probabilities for each branch of the tree and which set represents the scenario set. Other constructs to easily define chance constraints and integrated chance constraint in an SP problem are available as well.
Using these language constructs allows to retain the structure of the problem, hence making it available to the solvers, which might exploit it using specialized decomposition methods like Benders' decomposition to speed-up the solution.
Robust optimization constructs
SAMPL supports constructs to describe three types of robust optimization formulations:
Soyster
Bertsimas and Sim
Ben-Tal and Nemirovski
Availability
SAMPL is currently available as a part of the software AMPLDev (distributed by www.optirisk-systems.com). It supports many popular 32- and 64-bit platforms including Windows, Linux and Mac OS X. A free evaluation version with limited functionality is available.
A stochastic programming sample model
The following is the SAMPL version of a simple problem (Dakota), to show the SP related constructs. It does not include the data file, which follows the normal AMPL syntax (see the example provided in the AMPL Wikipedia page for further reference).
scenarioset Scen;
tree Tree := twostage;
random param Demand{Prod, Scen};
probability P{Scen};
suffix stage 1;
suffix stage 2;
suffix stage 2;
Solvers connectivity
SAMPL instance level format for SP problems is SMPS, and therefore the problem can be solved by any solver which supports that standard. One of such solvers (FortSP) is included in the standard SAMPL distribution. Regarding robust optimization problems, the needed solver depend on the specific formulation used, as Ben-Tal and Nemirovski formulation need a second-order cone capable solver.
See also
Algebraic modeling language
AIMMS
AMPL
FortSP
GAMS – General Algebraic Modeling System
GLPK – free open source system based on a subset of AMPL
|
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Alexander%20Grothendieck | The mathematician Alexander Grothendieck (1928–2014) is the eponym of many things.
Mathematics
Grothendieck |
https://en.wikipedia.org/wiki/J-structure | In mathematics, a J-structure is an algebraic structure over a field related to a Jordan algebra. The concept was introduced by to develop a theory of Jordan algebras using linear algebraic groups and axioms taking the Jordan inversion as basic operation and Hua's identity as a basic relation. There is a classification of simple structures deriving from the classification of semisimple algebraic groups. Over fields of characteristic not equal to 2, the theory of J-structures is essentially the same as that of Jordan algebras.
Definition
Let V be a finite-dimensional vector space over a field K and j a rational map from V to itself, expressible in the form n/N with n a polynomial map from V to itself and N a polynomial in K[V]. Let H be the subset of GL(V) × GL(V) containing the pairs (g,h) such that g∘j = j∘h: it is a closed subgroup of the product and the projection onto the first factor, the set of g which occur, is the structure group of j, denoted G'''(j).
A J-structure is a triple (V,j,e) where V is a vector space over K, j is a birational map from V to itself and e is a non-zero element of V satisfying the following conditions.
j is a homogeneous birational involution of degree −1
j is regular at e and j(e) = e if j is regular at x, e + x and e + j(x) then
the orbit G e of e under the structure group G = G(j) is a Zariski open subset of V.
The norm associated to a J-structure (V,j,e) is the numerator N of j, normalised so that N(e) = 1. The degree of the J-structure is the degree of N as a homogeneous polynomial map.
The quadratic map of the structure is a map P from V to End(V) defined in terms of the differential dj at an invertible x. We put
The quadratic map turns out to be a quadratic polynomial map on V.
The subgroup of the structure group G generated by the invertible quadratic maps is the inner structure group of the J-structure. It is a closed connected normal subgroup.
J-structures from quadratic forms
Let K have characteristic not equal to 2. Let Q be a quadratic form on the vector space V over K with associated bilinear form Q(x,y) = Q(x+y) − Q(x) − Q(y) and distinguished element e such that Q(e,.) is not trivial. We define a reflection map x* by
and an inversion map j by
Then (V,j,e) is a J-structure.
Example
Let Q be the usual sum of squares quadratic function on Kr for fixed integer r, equipped with the standard basis e1,...,er. Then (Kr, Q, er) is a J-structure of degree 2. It is denoted O2.
Link with Jordan algebras
In characteristic not equal to 2, which we assume in this section, the theory of J-structures is essentially the same as that of Jordan algebras.
Let A be a finite-dimensional commutative non-associative algebra over K with identity e. Let L(x) denote multiplication on the left by x. There is a unique birational map i on A such that i(x).x = e if i is regular on x: it is homogeneous of degree −1 and an involution with i(e) = e. It may be defined by i(x) = L(x)−1.e. We call i the i |
https://en.wikipedia.org/wiki/Hanner%20polytope | In geometry, a Hanner polytope is a convex polytope constructed recursively by Cartesian product and polar dual operations. Hanner polytopes are named after Olof Hanner, who introduced them in 1956.
Construction
The Hanner polytopes are constructed recursively by the following rules:
A line segment is a one-dimensional Hanner polytope.
The Cartesian product of every two Hanner polytopes is another Hanner polytope, whose dimension is the sum of the dimensions of the two given polytopes.
The dual of a Hanner polytope is another Hanner polytope of the same dimension.
They are exactly the polytopes that can be constructed using only these rules: that is, every Hanner polytope can be formed from line segments by a sequence of product and dual operations.
Alternatively and equivalently to the polar dual operation, the Hanner polytopes may be constructed by Cartesian products and direct sums, the dual of the Cartesian products. This direct sum operation combines two polytopes by placing them in two linearly independent subspaces of a larger space and then constructing the convex hull of their union.
Examples
A cube is a Hanner polytope, and can be constructed as a Cartesian product of three line segments. Its dual, the octahedron, is also a Hanner polytope, the direct sum of three line segments. In three dimensions all Hanner polytopes are combinatorially equivalent to one of these two types of polytopes. In higher dimensions the hypercubes and cross polytopes, analogues of the cube and octahedron, are again Hanner polytopes. However, more examples are possible. For instance, the octahedral prism, a four-dimensional prism with an octahedron as its base is also a Hanner polytope, as is its dual, the cubical bipyramid.
Properties
Coordinate representation
Every Hanner polytope can be given vertex coordinates that are 0, 1, or −1. More explicitly, if P and Q are Hanner polytopes with coordinates in this form, then the coordinates of the vertices of the Cartesian product of P and Q are formed by concatenating the coordinates of a vertex in P with the coordinates of a vertex in Q. The coordinates of the vertices of the direct sum of P and Q are formed either by concatenating the coordinates of a vertex in P with a vector of zeros, or by concatenating a vector of zeros with the coordinates of a vertex in Q.
Because the polar dual of a Hanner polytope is another Hanner polytope, the Hanner polytopes have the property that both they and their duals have coordinates in {0,1,−1}.
Number of faces
Every Hanner polytope is centrally symmetric, and has exactly 3d nonempty faces (including the polytope itself as a face but not including the empty set). For instance, the cube has 8 vertices, 12 edges, 6 squares, and 1 cube (itself) as faces; 8 + 12 + 6 + 1 = 27 = 33. The Hanner polytopes form an important class of examples for Kalai's 3d conjecture that all centrally symmetric polytopes have at least 3d nonempty faces.
Pairs of opposite facets and vertices
In |
https://en.wikipedia.org/wiki/Kalai%27s%203%5Ed%20conjecture | {{DISPLAYTITLE:Kalai's 3^d conjecture}}
In geometry, Kalai's 3d conjecture is a conjecture on the polyhedral combinatorics of centrally symmetric polytopes, made by Gil Kalai in 1989. It states that every d-dimensional centrally symmetric polytope has at least 3d nonempty faces (including the polytope itself as a face but not including the empty set).
Examples
In two dimensions, the simplest centrally symmetric convex polygons are the parallelograms, which have four vertices, four edges, and one polygon: . A cube is centrally symmetric, and has 8 vertices, 12 edges, 6 square sides, and 1 solid: . Another three-dimensional convex polyhedron, the regular octahedron, is also centrally symmetric, and has 6 vertices, 12 edges, 8 triangular sides, and 1 solid: .
In higher dimensions, the hypercube [0, 1]d has exactly 3d faces, each of which can be determined by specifying, for each of the d coordinate axes, whether the face projects onto that axis onto the point 0, the point 1, or the interval [0, 1]. More generally, every Hanner polytope has exactly 3d faces. If Kalai's conjecture is true, these polytopes would be among the centrally symmetric polytopes with the fewest possible faces.
Generalizations
In the same work as the one in which the 3d conjecture appears, Kalai conjectured more strongly which the f-vector of every convex centrally symmetric polytope P dominates the f-vector of at least one Hanner polytope H of the same dimension. This means that, for every number i from 0 to the dimension of P, the number of i-dimensional faces of P is greater than or equal to the number of i-dimensional faces of H. If it were true, this would imply the truth of the 3d conjecture; however, the stronger conjecture was later disproven.
Status
The conjecture is known to be true for . It is also known to be true for simplicial polytopes: it follows in this case from a conjecture of that every centrally symmetric simplicial polytope has at least as many faces of each dimension as the cross polytope, proven by . Indeed, these two previous papers were cited by Kalai as part of the basis for making his conjecture. Another special class of polytopes that the conjecture has been proven for are the Hansen polytopes of split graphs, which had been used by to disprove the stronger conjectures of Kalai.
The 3d conjecture remains open for arbitrary polytopes in higher dimensions.
References
Polyhedral combinatorics
Conjectures
Unsolved problems in geometry |
https://en.wikipedia.org/wiki/Peter%20Scholze | Peter Scholze (; born 11 December 1987) is a German mathematician known for his work in arithmetic geometry. He has been a professor at the University of Bonn since 2012 and director at the Max Planck Institute for Mathematics since 2018. He has been called one of the leading mathematicians in the world. He won the Fields Medal in 2018, which is regarded as the highest professional honor in mathematics.
Early life and education
Scholze was born in Dresden and grew up in Berlin. His father is a physicist, his mother a computer scientist, and his sister studied chemistry. He attended the in Berlin-Friedrichshain, a gymnasium devoted to mathematics and science. As a student, Scholze participated in the International Mathematical Olympiad, winning three gold medals and one silver medal.
He studied at the University of Bonn and completed his bachelor's degree in three semesters and his master's degree in two further semesters. He obtained his Ph.D. in 2012 under the supervision of Michael Rapoport.
Career
From July 2011 until 2016, Scholze was a Research Fellow of the Clay Mathematics Institute in New Hampshire. In 2012, shortly after completing his PhD, he was made full professor at the University of Bonn, becoming at the age of 24 the youngest full professor in Germany.
In fall 2014, Scholze was appointed the Chancellor's Professor at University of California, Berkeley, where he taught a course on p-adic geometry.
In 2018, Scholze was appointed as a director of the Max Planck Institute for Mathematics in Bonn.
Work
Scholze's work has concentrated on purely local aspects of arithmetic geometry such as p-adic geometry and its applications. He presented in a more compact form some of the previous fundamental theories pioneered by Gerd Faltings, Jean-Marc Fontaine and later by Kiran Kedlaya. His PhD thesis on perfectoid spaces yields the solution to a special case of the weight-monodromy conjecture.
Scholze and Bhargav Bhatt have developed a theory of prismatic cohomology, which has been described as progress towards motivic cohomology by unifying singular cohomology, de Rham cohomology, ℓ-adic cohomology, and crystalline cohomology.
Scholze and Dustin Clausen proposed a program for condensed mathematics—a project to unify various mathematical subfields, including topology, geometry, functional analysis and number theory.
Awards
In 2012, he was awarded the Prix and Cours Peccot. He was awarded the 2013 SASTRA Ramanujan Prize. In 2014, he received the Clay Research Award. In 2015, he was awarded the Frank Nelson Cole Prize in Algebra, and the Ostrowski Prize.
He received the Fermat Prize 2015 from the Institut de Mathématiques de Toulouse. In 2016, he was awarded the Leibniz Prize 2016 by the German Research Foundation. He declined the $100,000 "New Horizons in Mathematics Prize" of the 2016 Breakthrough Prizes. His turning down of the prize received little media attention.
In 2017 he became a member of the German Academy of Sciences |
https://en.wikipedia.org/wiki/Carlo%20Brancaccio | Carlo Brancaccio (Naples, March 6, 1861 – 1920) was an Italian painter, active mainly in an Impressionist style.
Biography
While he initially had studied mathematics, he abandoned this to study painting by age 22 years. He was mentored by Eduardo Dalbono. His main subjects were city streets, sea- and landscapes, mostly vedute of Naples. At the 1887 Promotrice of Naples he displayed: Passe-partout, and many sketches of the city including the interiors of churches. In 1888, he displayed a large Seascape of Capri; in 1889, Toledo in the Rain; and in same year at the Brera Exposition in Milan, he exhibited the Piazza of the Carmine of Naples.
He won a gold medal at the Exhibition in Rome in 1893. He also painted Neapolitan genre subjects, including: Ore tristi (1898); Impressioni di Napoli (Berlin 1890); and Strada di Almalfi (1897).
Gallery
References
1861 births
1920 deaths
19th-century Italian painters
20th-century Italian painters
Italian vedutisti
Italian male painters
Painters from Naples
Orientalist painters
19th-century Italian male artists
20th-century Italian male artists |
https://en.wikipedia.org/wiki/Steffen%20Lauritzen | Steffen Lilholt Lauritzen FRS (born 22 April 1947) is former Head of the Department of Statistics at the University of Oxford and Fellow of Jesus College, Oxford, and currently Emeritus Professor of Statistics at the University of Copenhagen. He is a leading proponent of mathematical statistics and graphical models.
Education and career
Lauritzen studied statistics at the University of Copenhagen, Denmark, completing the degree of Candidatus statisticae (M.Sc. level) in 1972 and Licentiatus statisticae (PhD level) in 1975. He was appointed there as Lecturer of Statistics and remained until 1981. He continued as Professor of Mathematics and Statistics at Aalborg University, Denmark, from 1981 to 2004. From 2004-2014 he was Professor of Statistics at the University of Oxford, from 2014 to 2021, he was Professor of Statistics at the University of Copenhagen, and since 2021, he is Emeritus Professor of Statistics there.
Honors and awards
Lauritzen was elected a member of the International Statistical Institute in 1984, and a Fellow of the Royal Society in 2011.
Lauritzen was awarded the 1996 Guy Medal in Silver by the Royal Statistical Society. He served, among others, as Editor-in-Chief of the Scandinavian Journal of Statistics from 1998 to 2000.
Lauritzen's book Probabilistic Networks and Expert Systems (1999, Springer-Verlag), written jointly with Robert G. Cowell, Philip Dawid, and David Spiegelhalter, received the 2001 DeGroot Prize from the International Society for Bayesian Analysis.
Selected publications
Spiegelhalter, David J., A. Philip Dawid, Steffen L. Lauritzen and Robert G. Cowell "Bayesian analysis in expert systems" in Statistical Science, 8(3), 1993.
A. Philip Dawid, Uffe Kjærulff, Steffen L. Lauritzen, "Hybrid Propagation in Junction Trees." IPMU 1994
References
1947 births
University of Copenhagen alumni
Elected Members of the International Statistical Institute
20th-century Danish mathematicians
Bayesian statisticians
Danish statisticians
Fellows of Jesus College, Oxford
Living people
Fellows of the Royal Society
21st-century Danish mathematicians
Mathematical statisticians
Academic staff of Aalborg University
Academic staff of the University of Copenhagen |
https://en.wikipedia.org/wiki/Projective%20group%20%28disambiguation%29 | In mathematics, projective group may refer to:
Projective linear group or one of the related linear groups
Projective orthogonal group
Projective unitary group
Projective symplectic group
Projective semilinear group
Projective profinite group, a profinite group with the embedding property |
https://en.wikipedia.org/wiki/Wiley%20Interdisciplinary%20Reviews%3A%20Computational%20Statistics | Wiley Interdisciplinary Reviews: Computational Statistics (WIREs Comp Stats) is a review journal for computational and statistical techniques in the sciences, from the perspectives of both computation and statistics. It contain both tutorial reviews and advanced reviews, as well as opinion pieces and commentaries.
The journal was published by John Wiley & Sons both in print () and online () through 2011. Beginning in 2012, it is published online only. It was started in 2009.
Editors in chief
The initial editors in chief were Edward J. Wegman and Yasmin H. Said of George Mason University, and David W. Scott, of Rice University. As of 2013, the current editors in chief are James E. Gentle, university professor of computational statistics at George Mason University, Karen Kafadar, Rudy Professor of Statistics at Indiana University and David W. Scott, Noah Harding Professor of Statistics at Rice University.
Indexing
The journal is indexed in Compendex and Scopus.
References
External links
Journal home page
Computational statistics journals
Wiley-Blackwell academic journals
Review journals |
https://en.wikipedia.org/wiki/Local%20symbol | In mathematics, local symbol may refer to:
The local Artin symbol in Artin reciprocity
The local symbol used to formulate Weil reciprocity
A Steinberg symbol on a local field |
https://en.wikipedia.org/wiki/Pavel%20Gevorgyan | Pavel Georgyan (born 8 April 1963 in Azokh, Hadrut region, Nagorno-Karabakh, USSR) is a professor, doctor of science in physics and mathematics, corresponding member of Russian Academy of Natural Sciences, Head of Department of Mathematical Analysis of Moscow State Pedagogical University.
Gevorgyan was awarded with the Russian Federation Government Prize in Education (2014).
He is an Honorary Worker of higher professional education of Russian Federation.
Family
Gevorgyan is married and has two children.
Education
In 1980 entered Yerevan State University, Faculty of Mathematics and Mechanics
In 1984 became the winner of Mathematical Olympiad for high school students (Armenia)
In 1984 he transferred to MSU Moscow State University, Department of higher geometry and topology
1985–1989: Postgraduate in Moscow State University, Department of higher geometry and topology
1989: Candidate of Science (equivalent of Ph.D.) in Physics and Mathematics.
Dissertation: “Equivariant movability” (under the supervision of professor Yu.M.Smirnov).
2001: Doctor of Science in Physics and Mathematics
Dissertation: “Generalized shape theory and movability of continuous transformation groups”.
Research area
Topological transformation groups. Equivariant topology. Shape theory.
Career and present positions
1993–1996: Dean of Faculty of Natural Sciences, 1994–1996: Head of Department of Higher Mathematics, 1996–2000: Rector of Artsakh State University (in Nagorno-Karabakh)
2000-2015: Professor of Department of Higher Mathematics of Moscow Power Engineering Institute
2008-2016: Head of Department of Higher and Applied Mathematics of Academy of Labour and Social Relations
2015-2016: Vice-rector of Moscow State Pedagogical University
Since 2015: Head of Department of Mathematical Analysis of Moscow State Pedagogical University
Since 2005: Member of Scientific-Methodological Council on mathematics of Ministry of Education and Science of Russian Federation
2008: Corresponding member of Russian Academy of Natural Sciences
2012: Honorary Worker of Higher Professional Education of Russian Federation
Publications
Gevorgyan P.S., Pop I., Movable morphisms in strong shape category. Topology and its Applications, Elsevier BV (Netherlands), 2019, p. 107001.
Геворкян П.С., Хименес Р., Об эквивариантных расслоениях G-CW-комплексов. Математический сборник, 2019, том 210, № 10, с. 91-98.
Геворкян П.С., Теория шейпов. Фундаментальная и прикладная математика, 2019, том 22, № 6, с. 19-84.
Gevorgyan P.S., Iliadis S.D., Groups of generalized isotopies and generalized G-spaces. Matematicki Vesnik, Drustvo Matematicara SR Srbije (Serbia), 2018, 70, № 2, pp. 110–119.
Gevorgyan P.S., Pop I., Shape dimension of maps. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, Vladimir Andrunachievici Institute of Mathematics and Computer Science (Moldova), 2018, 86, № 1, pp. 3–11.
Геворкян П.С., Группы обратимых бинарных операций топологического пространства. Известия НАН Армении: Матем |
https://en.wikipedia.org/wiki/Simplicial%20group | In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial group is a Kan complex (in particular, its homotopy groups make sense). The Dold–Kan correspondence says that a simplicial abelian group may be identified with a chain complex. In fact it can be shown that
any simplicial abelian group is non-canonically homotopy equivalent to a product of Eilenberg–MacLane spaces,
A commutative monoid in the category of simplicial abelian groups is a simplicial commutative ring.
discusses a simplicial analogue of the fact that a cohomology class on a Kähler manifold has a unique harmonic representative and deduces Kirchhoff's circuit laws from these observations.
References
Charles Weibel, An introduction to homological algebra
External links
What is a simplicial commutative ring from the point of view of homotopy theory?
Simplicial sets |
https://en.wikipedia.org/wiki/Istv%C3%A1n%20Vincze%20%28mathematician%29 | István Vincze ( – ) was a Hungarian mathematician, known for his contributions to number theory, non-parametric statistics, empirical distribution, Cramér–Rao inequality, and information theory. Considered by many, as an expert in theoretical and applied statistics, he was the founder of the Mathematical Institute of the Hungarian Academy, and was the Head of the Statistics Department. He also held the post of professor at Faculty of Science of the Eötvös Loránd University. He wrote over 100 academic papers, authored 10 books, and was a speaker at several conferences, including the Berkeley Symposiums in 1960, 1965, and 1970. He received honors and awards like the Hungarian State Prize and Grauss Ehrenplakette in 1966 and 1978 respectively.
Life
Born in Szeged, Hungary, he graduated from the University of Szeged in 1935.
Around 1950, he founded the Mathematical Institute of the Hungarian Academy, whose director was Alfréd Rényi.
Early in his career, he wrote papers with Paul Erdős, including On the approximation of convex, closed plane curves by multifocal ellipses.
Some of his books that were translated into English are Progress in statistics (1972), and Mathematical methods of statistical quality control (1974).
He participated in conferences and gave seminar talks in the United States, Canada, Argentina, Germany, and China.
He retired from academic teaching in 1980, and died in 1999.
Academic publications
References
1912 births
1999 deaths
Hungarian statisticians
Academic staff of Eötvös Loránd University
20th-century Hungarian mathematicians
Mathematicians from Austria-Hungary |
https://en.wikipedia.org/wiki/Multivariate%20Behrens%E2%80%93Fisher%20problem | In statistics, the multivariate Behrens–Fisher problem is the problem of testing for the equality of means from two multivariate normal distributions when the covariance matrices are unknown and possibly not equal. Since this is a generalization of the univariate Behrens-Fisher problem, it inherits all of the difficulties that arise in the univariate problem.
Notation and problem formulation
Let be independent random samples from two -variate normal distributions with unknown mean vectors and unknown dispersion matrices . The index refers to the first or second population, and the th observation from the th population is .
The multivariate Behrens–Fisher problem is to test the null hypothesis that the means are equal versus the alternative of non-equality:
Define some statistics, which are used in the various attempts to solve the multivariate Behrens–Fisher problem, by
The sample means and sum-of-squares matrices are sufficient for the multivariate normal parameters , so it suffices to perform inference be based on just these statistics. The distributions of and are independent and are, respectively, multivariate normal and Wishart:
Background
In the case where the dispersion matrices are equal, the distribution of the statistic is known to be an F distribution under the null and a noncentral F-distribution under the alternative.
The main problem is that when the true values of the dispersion matrix are unknown, then under the null hypothesis the probability of rejecting via a test depends on the unknown dispersion matrices. In practice, this dependency harms inference when the dispersion matrices are far from each other or when the sample size is not large enough to estimate them accurately.
Now, the mean vectors are independently and normally distributed,
but the sum does not follow the Wishart distribution, which makes inference more difficult.
Proposed solutions
Proposed solutions are based on a few main strategies:
Compute statistics which mimick the statistic and which have an approximate distribution with estimated degrees of freedom (df).
Use generalized p-values based on generalized test variables.
Use Roy's union-intersection principle
Approaches using the T2 with approximate degrees of freedom
Below, indicates the trace operator.
Yao (1965)
(as cited by )
where
Johansen (1980)
(as cited by )
where
and
Nel and Van der Merwe's (1986)
(as cited by )
where
Comments on performance
Kim (1992) proposed a solution that is based on a variant of . Although its power is high, the fact that it is not invariant makes it less attractive. Simulation studies by Subramaniam and Subramaniam (1973) show that the size of Yao's test is closer to the nominal level than that of James's. Christensen and Rencher (1997) performed numerical studies comparing several of these testing procedures and concluded that Kim and Nel and Van der Merwe's tests had the highest power. However, these two procedures are |
https://en.wikipedia.org/wiki/2009%20Kelantan%20FA%20season | The 2009 season was Kelantan FA debut season in the Malaysia Super League. This article shows statistics of the club's players in the season, and also lists all matches that the club played in the season.
Competitions
Super League
Results by match
League table
FA Cup
The 2009 Malaysia FA Cup, known as the TM FA Cup due to the competition's sponsorship by Telekom Malaysia, is the 20th season of the Malaysia FA Cup, a knockout competition for Malaysia's state football association and clubs.
The FA Cup competition has reverted to the old format of play with no more open draws. It will comprise 29 teams 15 Super League and 14 Premier League sides with defending champions Kedah FA, Selangor FA and Terengganu FA receiving byes in the first round.
Malaysia Cup
The 2009 edition of Malaysia Cup started on 26 September 2009. Twenty teams took part in this prestigious competition. The teams were divided into five groups of four. The group leaders and the three best second-placed teams in the groups after six matches qualified to the quarterfinals.
Group E
Player statistics
Squad, appearances and goals
Source: Competitions
Goalscorers
Source: Competitions
Transfers
In
Out
See also
List of Kelantan FA seasons
References
Kelantan F.C.
2009
Kelantan |
https://en.wikipedia.org/wiki/Chad%20Barson | Chad Barson (born February 25, 1991) is an American retired professional soccer player. He is currently the Ohio State Buckeyes men's soccer director of operations.
Career statistics
Sources:
References
External links
1991 births
Living people
American men's soccer players
Akron Zips men's soccer players
Flint City Bucks players
Columbus Crew players
Pittsburgh Riverhounds SC players
Men's association football defenders
Soccer players from Columbus, Ohio
USL League Two players
Major League Soccer players
USL Championship players
United States men's youth international soccer players
United States men's under-20 international soccer players
All-American college men's soccer players
FC Linköping City players
Expatriate men's footballers in Sweden
American expatriate sportspeople in Sweden
American expatriate men's soccer players
Homegrown Players (MLS) |
https://en.wikipedia.org/wiki/R%C3%B3bert%20K%C5%91v%C3%A1ri | Róbert Kővári (born 23 November 1995) is a Hungarian football player who plays for Szeged-Csanád.
Club statistics
Updated to games played as of 1 September 2019.
External links
HLSZ
MLSZ
1995 births
Living people
Footballers from Szekszárd
Hungarian men's footballers
Hungary men's youth international footballers
Men's association football midfielders
Pécsi MFC players
Paksi FC players
Dorogi FC footballers
Soproni VSE players
BFC Siófok players
Szeged-Csanád Grosics Akadémia footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/Fabian%20Holthaus | Fabian Holthaus (born 17 January 1995) is a German professional football left back who plays for Rot-Weiß Oberhausen.
Career statistics
References
External links
1995 births
Living people
Footballers from Hamm
German men's footballers
Men's association football defenders
Germany men's youth international footballers
Hammer SpVg players
VfL Bochum players
VfL Bochum II players
Fortuna Düsseldorf players
Dynamo Dresden players
FC Hansa Rostock players
FC Energie Cottbus players
FC Viktoria Köln players
Rot-Weiß Oberhausen players
Regionalliga players
3. Liga players
2. Bundesliga players |
https://en.wikipedia.org/wiki/Clopper | Clopper may refer to:
Clop (subculture), erotic fan art of the TV show My Little Pony
Hoof, the toe of ungulates
Clopper-Pearson interval, in statistics
Clopper Lake, Seneca Creek State Park, Maryland, US
Clopper Road, Maryland, US
See also
Clop (disambiguation)
Iris Clops, character in the TV show Monster High
Ol' Clip-Clop, a book by Patricia McKissack
Spy Clops, a Lego theme |
https://en.wikipedia.org/wiki/Theory%20of%20Visualization | Theory is becoming an important topic in visualization, expanding from its traditional origins in low-level perception and statistics to an ever-broader array of fields and subfields. It includes color theory, visual cognition, visual grammars, interaction theory, visual analytics and information theory.
References
Visualization (graphics) |
https://en.wikipedia.org/wiki/Charles%20Weibel | Charles Alexander Weibel (born October 28, 1950, in Terre Haute, Indiana) is an American mathematician working on algebraic K-theory, algebraic geometry and homological algebra.
Weibel studied physics and mathematics at the University of Michigan, earning bachelor's degrees in both subjects in 1972. He was awarded a master's degree by the University of Chicago in 1973 and achieved his doctorate in 1977 under the supervision of Richard Swan (Homotopy in Algebraic K-Theory). From 1970 to 1976 he was an "Operations Research Analyst" at Standard Oil of Indiana, and from 1977 to 1978 was at the Institute for Advanced Study. In 1978 he became an assistant professor at the University of Pennsylvania. In 1980 he became an assistant professor at Rutgers University, where he was promoted to professor in 1989.
He joined Vladimir Voevodsky and Markus Rost in proving the (motivic) Bloch–Kato conjecture (2009). It is a generalization of the Milnor conjecture of algebraic K-theory, which was proved by Voevodsky in the 1990s. He was a visiting professor in 1992 at the University of Paris and 1993 at the University of Strasbourg. Since 1983 he has been an editor of the Journal of Pure and Applied Algebra.
He helped found the K-theory Foundation in 2010, and has been a managing editor of the Annals of K-theory since 2014.
In 2014, he became a Fellow of the American Mathematical Society.
Writings
With Eric Friedlander, An overview over algebraic K-theory, in Algebraic K-theory and its applications, World Scientific 1999, pp. 1–119 (1997 Trieste Lecture Notes)
With Carlo Mazza, Vladimir Voevodsky Lectures on Motivic Cohomology, Clay Monographs in Mathematics, American Mathematical Society 2006
An introduction to homological algebra, Cambridge University Press 1994
The proof of the Bloch-Kato conjecture, Trieste Lectures 2007, ICTP Lecture Notes Series 23 (2008), 277–305
Notes
References
The original article was a Google translation of the corresponding article in German Wikipedia.
External links
Homepage
20th-century American mathematicians
21st-century American mathematicians
1950 births
Living people
Rutgers University faculty
Algebraic geometers
University of Michigan alumni
Fellows of the American Mathematical Society
Mathematicians from Indiana
People from Terre Haute, Indiana |
https://en.wikipedia.org/wiki/Dold%E2%80%93Kan%20correspondence | In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the category of simplicial abelian groups. Moreover, under the equivalence, the th homology group of a chain complex is the th homotopy group of the corresponding simplicial abelian group, and a chain homotopy corresponds to a simplicial homotopy. (In fact, the correspondence preserves the respective standard model structures.)
Example: Let C be a chain complex that has an abelian group A in degree n and zero in all other degrees. Then the corresponding simplicial group is the Eilenberg–MacLane space .
There is also an ∞-category-version of the Dold–Kan correspondence.
The book "Nonabelian Algebraic Topology" cited below has a Section 14.8 on cubical versions of the Dold–Kan theorem, and relates them to a previous equivalence of categories between cubical omega-groupoids and crossed complexes, which is fundamental to the work of that book.
Detailed construction
The Dold-Kan correspondence between the category sAb of simplicial abelian groups and the category Ch≥0(Ab) of nonnegatively graded chain complexes can be constructed explicitly through a pair of functorspg 149 so that the compositions of these functors are naturally isomorphic to the respective identity functors. The first functor is the normalized chain complex functorand the second functor is the "simplicialization" functorconstructing a simplicial abelian group from a chain complex.
Normalized chain complex
Given a simplicial abelian group there is a chain complex called the normalized chain complex with termsand differentials given byThese differentials are well defined because of the simplicial identityshowing the image of is in the kernel of each . This is because the definition of gives .
Now, composing these differentials gives a commutative diagramand the composition map . This composition is the zero map because of the simplicial identityand the inclusion , hence the normalized chain complex is a chain complex in . Because a simplicial abelian group is a functorand morphisms are given by natural transformations, meaning the maps of the simplicial identities still hold, the normalized chain complex construction is functorial.
References
Further reading
Jacob Lurie, DAG-I
External links
Simplicial sets
Theorems in abstract algebra |
https://en.wikipedia.org/wiki/Luis%20Davidson%20San%20Juan | Luis J. Davidson San Juan ( – ) was a Cuban mathematician, professor and Doctor in Mathematics, known for his contributions to the fields of mathematics and pedagogy. He is renowned for receiving the Paul Erdős Award in 1992 by the International Commission on Mathematical Instruction, and for being awarded the distinction of Maestro Founder of Mathematics in Ibero-America by the Organization of Ibero-American States for Education, Science and Culture. He was the Vice President of the International Mathematical Olympiad Site Committee in 1988. And he is the author of many books, including Equations and mathematicians (2008), and Problems of elementary mathematics (1987).
Life
Born in Havana, Cuba on September 10, 1921, he earned the Doctorate in Physics and Mathematical Sciences from the [[Un|iversity of Havana]] in 1944 with the thesis entitled Developments on the series of analytic functions.
From 1945 to 1961 he taught at the Instituto de Segunda Enseñanza in Matanzas, and in 1950 he was part of the delegation that represented Cuba at the International Congress of Mathematicians at Harvard University.
In 1960, he held the position of National Inspector of Mathematics. And starting in 1963, he organized competitions in mathematics for pre-collegiate students throughout Cuba.
In 1971, he represented Cuba, as Head of the national delegation in the International Mathematical Olympiad, and in 1988, while a member of the organization, he became the Vice President for the IMO Site Committee.
In 1990 he participated in the First Congress of The World Federation of National Mathematics Competitions at Waterloo, and in 1992 he was awarded the Paul Erdős Award from the International Commission of Mathematical Instruction that was held in Québec, Canada.
Some of his books that were published in Spanish are Contests in mathematics (1974), Problems in elementary mathematics (1987), and Equations and mathematicians (1988). The latter was part of a series that he planned to write about: a story about mathematicians and the problems they have solved.
He died on November 10, 2011.
Publications
References
20th-century Cuban mathematicians
1921 births
2011 deaths
University of Havana alumni |
https://en.wikipedia.org/wiki/Signed%20area | In mathematics, the signed area or oriented area of a region of an affine plane is its area with orientation specified by the positive or negative sign, that is "plus" or "minus" . More generally, the signed area of an arbitrary surface region is its surface area with specified orientation. When the boundary of the region is a simple curve, the signed area also indicates the orientation of the boundary.
Planar area
Polygons
The mathematics of ancient Mesopotamia, Egypt, and Greece had no explicit concept of negative numbers or signed areas, but had notions of shapes contained by some boundary lines or curves, whose areas could be computed or compared by pasting shapes together or cutting portions away, amounting to addition or subtraction of areas. This was formalized in Book I of Euclid's Elements, which leads with several common notions including "if equals are added to equals, then the wholes are equal" and "if equals are subtracted from equals, then the remainders are equal" (among planar shapes, those of the same area were called "equal"). The propositions in Book I concern the properties of triangles and parallelograms, including for example that parallelograms with the same base and in the same parallels are equal and that any triangle with the same base and in the same parallels has half the area of these parallelograms, and a construction for a parallelogram of the same area as any "rectilinear figure" (simple polygon) by splitting it into triangles. Greek geometers often compared planar areas by quadrature (constructing a square of the same area as the shape), and Book II of the Elements shows how to construct a square of the same area as any given polygon.
Just as negative numbers simplify the solution of algebraic equations by eliminating the need to flip signs in separately considered cases when a quantity might be negative, a concept of signed area analogously simplifies geometric computations and proofs. Instead of subtracting one area from another, two signed areas of opposite orientation can be added together, and the resulting area can be meaningfully interpreted regardless of its sign. For example, propositions II.12–13 of the Elements contain a geometric precursor of the law of cosines which is split into separate cases depending on whether the angle of a triangle under consideration is obtuse or acute, because a particular rectangle should either be added or subtracted, respectively (the cosine of the angle is either negative or positive). If the rectangle is allowed to have signed area, both cases can be collapsed into one, with a single proof (additionally covering the right-angled case where the rectangle vanishes).
As with the unoriented area of simple polygons in the Elements, the oriented area of polygons in the affine plane (including those with holes or self-intersections) can be conveniently reduced to sums of oriented areas of triangles, each of which in turn is half of the oriented area of a parallelogram. T |
https://en.wikipedia.org/wiki/David%20Hilbert%20Award | The David Hilbert Award, named after David Hilbert, was established by the World Federation of National Mathematics Competitions to acknowledge mathematicians who have contributed to the development of mathematics worldwide.
Each awardee is selected by the Executive and Advisory Committee of the World Federation of National Mathematics Competitions on the recommendation of the WFNMC Awards Subcommittee.
Past recipients
1991
Edward Barbeau, Canada
Arthur Engel, Germany
Graham Pollard, Australia
1992
Martin Gardner, United States of America
Murray Klamkin, United States of America
Marcin E Kuczma, Poland
1994
María Falk de Losada, United States of America
Peter J. O'Halloran, Australia
1996
Andy Liu, Canada
See also
List of mathematics awards
Notes
References
Sources
Homepage of the award.
Mathematics awards |
https://en.wikipedia.org/wiki/AW%2A-algebra | In mathematics, an AW*-algebra is an algebraic generalization of a W*-algebra. They were introduced by Irving Kaplansky in 1951. As operator algebras, von Neumann algebras, among all C*-algebras, are typically handled using one of two means: they are the dual space of some Banach space, and they are determined to a large extent by their projections. The idea behind AW*-algebras is to forgo the former, topological, condition, and use only the latter, algebraic, condition.
Definition
Recall that a projection of a C*-algebra is a self-adjoint idempotent element. A C*-algebra A is an AW*-algebra if for every subset S of A, the left annihilator
is generated as a left ideal by some projection p of A, and similarly the right annihilator is generated as a right ideal by some projection q:
.
Hence an AW*-algebra is a C*-algebras that is at the same time a Baer *-ring.
The original definition of Kaplansky states that an AW*-algebra is a C*-algebra such that (1) any set of orthogonal projections has a least upper bound, and (2) that each maximal commutative C*-subalgebra is generated by its projections. The first condition states that the projections have an interesting structure, while the second condition ensures that there are enough projections for it to be interesting. Note that the second condition is equivalent to the condition that each maximal commutative C*-subalgebra is monotone complete.
Structure theory
Many results concerning von Neumann algebras carry over to AW*-algebras. For example, AW*-algebras can be classified according to the behavior of their projections, and decompose into types. For another example, normal matrices with entries in an AW*-algebra can always be diagonalized. AW*-algebras also always have polar decomposition.
However, there are also ways in which AW*-algebras behave differently from von Neumann algebras. For example, AW*-algebras of type I can exhibit pathological properties, even though Kaplansky already showed that such algebras with trivial center are automatically von Neumann algebras.
The commutative case
A commutative C*-algebra is an AW*-algebra if and only if its spectrum is a Stonean space. Via Stone duality, commutative AW*-algebras therefore correspond to complete Boolean algebras. The projections of a commutative AW*-algebra form a complete Boolean algebra, and conversely, any complete Boolean algebra is isomorphic to the projections of some commutative AW*-algebra.
References
C*-algebras
Operator theory |
https://en.wikipedia.org/wiki/1974%E2%80%9375%20VfL%20Bochum%20season | The 1974–75 VfL Bochum season was the 37th season in club history.
Matches
Legend
Bundesliga
DFB-Pokal
Squad
Squad and statistics
Squad, appearances and goals scored
Transfers
Summer
In:
Out:
Sources
External links
1974–75 VfL Bochum season at Weltfussball.de
1974–75 VfL Bochum season at kicker.de
1974–75 VfL Bochum season at Fussballdaten.de
Bochum
VfL Bochum seasons |
https://en.wikipedia.org/wiki/Yitang%20Zhang | Yitang Zhang (; born February 5, 1955) is a Chinese-American mathematician primarily working on number theory and a professor of mathematics at the University of California, Santa Barbara since 2015.
Previously working at the University of New Hampshire as a lecturer, Zhang submitted a paper to the Annals of Mathematics in 2013 which established the first finite bound on the least gap between consecutive primes that is attained infinitely often. This work led to a 2013 Ostrowski Prize, a 2014 Cole Prize, a 2014 Rolf Schock Prize, and a 2014 MacArthur Fellowship. Zhang became a professor of mathematics at the University of California, Santa Barbara in fall 2015.
Early life and education
Zhang was born in Shanghai, China, with his ancestral home in Pinghu, Zhejiang. He lived in Shanghai with his grandmother until he went to Peking University. At around the age of nine, he found a proof of the Pythagorean theorem. He first learned about Fermat's Last Theorem and the Goldbach conjecture when he was 10. During the Cultural Revolution, he and his mother were sent to the countryside to work in the fields. He worked as a laborer for 10 years and was unable to attend high school. After the Cultural Revolution ended, Zhang entered Peking University in 1978 as an undergraduate student and received a Bachelor of Science in mathematics in 1982. He became a graduate student of Professor Pan Chengbiao, a number theorist at Peking University, and obtained a Master of Science in mathematics in 1984.
After receiving his master's degree in mathematics, with recommendations from Professor Ding Shisun, the President of Peking University, and Professor Deng Donggao, chair of the university's Math Department, Zhang was granted a full scholarship at Purdue University. Zhang arrived at Purdue in January 1985, studied there for six and a half years, and obtained his PhD in mathematics in December 1991.
Career
Zhang's PhD work was on the Jacobian conjecture. After graduation, Zhang had trouble finding an academic position. In a 2013 interview with Nautilus magazine, Zhang said he did not get a job after graduation. "During that period it was difficult to find a job in academics. That was a job market problem. Also, my advisor [Tzuong-Tsieng Moh] did not write me letters of recommendation." Zhang made this claim again in George Csicsery's documentary film "Counting from Infinity: Yitang Zhang and the Twin Prime Conjecture" while discussing his difficulties at Purdue and in the years that followed. Moh claimed that Zhang never came back to him requesting recommendation letters. In a detailed profile published in The New Yorker magazine in February 2015, Alec Wilkinson wrote Zhang "parted unhappily" with Moh, and that Zhang "left Purdue without Moh's support, and, having published no papers, was unable to find an academic job". In 2018, responding to reports of his treatment of Zhang, Moh posted an update on his website. Moh wrote that Zhang "failed miserably" in provin |
https://en.wikipedia.org/wiki/Klaus%20Schmidt%20%28mathematician%29 | Klaus D. Schmidt (born 25 September 1943) is an Austrian mathematician and retired professor at the Faculty of Mathematics, University of Vienna.
After studying mathematics at the University of Vienna he received his doctorate in 1968 under Edmund Hlawka. He held visiting professorships in Technical University of Vienna, University of Manchester in 1969, Bedford College (1969–1974) and the University of Warwick from 1974 to 1994 after which he came back to the University of Vienna. He retired in 2009. In 1975/76
K. R. Parthasarathy invited Klaus Schmidt to spend 7 months at the new Delhi Centre of Indian Statistical Institute (Parthasarathy was then working at the Indian Institute of Technology, Delhi).
In 1994 he was awarded the Ferran Sunyer i Balaguer Prize for the monograph Dynamical systems of algebraic origin. He is member of the Austrian Academy of Sciences.
He has researched among other things, ergodic theory and its connections with arithmetic, commutative algebra, harmonic analysis, operator algebras and probability theory.
Publications
Positive definite kernels, continuous tensor products, and central limit theorems in probability theory (with KR Parthasarathy). Lecture Notes in Mathematics, Vol 272, Springer Verlag 1972.
Cocycles of ergodic transformation groups. Lecture Notes in Mathematics, Vol 1, MacMillan (India) 1977.
Algebraic Ideas in Ergodic Theory. CBMS Lecture Notes, Vol 76, Amer. Math Soc. 1990.
Dynamical systems of algebraic origin. Progress in Mathematics, Vol 128, Birkhauser Verlag 1995.
References
External links
Klaus Schmidt's Home Page
1943 births
Living people
Austrian mathematicians
Academic staff of the University of Vienna
University of Vienna alumni
Members of the Austrian Academy of Sciences |
https://en.wikipedia.org/wiki/Pascal%20Pellowski | Pascal Pellowski (born 18 December 1988) is a German footballer who most recently played for Astoria Walldorf.
Career
Statistics
References
External links
1988 births
Living people
German men's footballers
3. Liga players
Regionalliga players
SV Darmstadt 98 players
VfL Bochum players
VfL Bochum II players
SV Elversberg players
1. FC Saarbrücken players
FC Astoria Walldorf players
Men's association football defenders
Footballers from Darmstadt
21st-century German people |
https://en.wikipedia.org/wiki/Alan%20Baker%20%28philosopher%29 | Alan R. Baker is a professor of philosophy in Swarthmore College (Pennsylvania, United States), specializing in the philosophy of mathematics and the philosophy of science. He is also a former U.S. shogi champion.
Academic career
Baker did his undergraduate studies at the University of Cambridge, earning a bachelor's degree in philosophy with first class honours in 1991. He then moved to the U.S. for graduate school, earning a master's degree in 1995 and a Ph.D. in 1999, both in philosophy from Princeton University. His doctoral supervisors were Paul Benacerraf and Gideon Rosen. After working as an assistant professor at Xavier University, he moved to Swarthmore in 2003.
Philosophically, Baker is a mathematical realist who has used examples from evolutionary biology to show the necessity of mathematics in scientific reasoning.
In 2005 The New York Times published an excerpt from the exam from his "Introduction to Metaphysics and Epistemology" course in its "pop quiz" column.
Shogi
In 2005, Baker founded a shogi club at Swarthmore College, outside Philadelphia, which is one of only two college-based shogi clubs in the United States. The other club is Cornell University Shogi Club, which was founded in August 2017.
Baker is also a former U.S. shogi champion, having won the 13th U.S. Shogi Championship in 2008. his ELO rating of 2107 placed him in 20th place on the Federation of European Shogi Associations (FESA) bi-annual rating list.
Tournament results:
2008: Winner, 13th U.S. Shogi Championship.
2008: 3rd place, Individual Tournament, 4th International Shogi Forum (Tendō).
2009: 2nd place, British Open Shogi Championship.
2014: Winner, Group B Individual Tournament, 6th International Shogi Forum (Shizuoka)
References
21st-century American philosophers
Philosophers from Pennsylvania
American shogi players
Living people
Year of birth missing (living people)
Philosophers of mathematics
Philosophers of science
Philosophical realism
Swarthmore College faculty |
https://en.wikipedia.org/wiki/Gianni%20Dal%20Maso | Gianni Dal Maso (born 1954) is an Italian mathematician who is active in the fields of partial differential equations, calculus of variations and applied mathematics.
Scientific activity
Dal Maso studied at Scuola Normale Superiore under the guidance of Ennio De Giorgi and is professor of mathematics at the International School for Advanced Studies at Trieste, where he also serves as deputy director. Dal Maso has dealt with a number of questions related to partial differential equations and calculus of variations, covering a range of topics going from lower semicontinuity problems for multiple integrals to existence theorem for so called free discontinuity problems, from the study of asymptotic behaviour of variational problems via so called Γ-convergence methods to fine properties of solutions to obstacle problems. In the last years he has been considerably involved in the study of problems arising from applied mathematics, developing methods aimed at describing the evolution of fractures in plasticity problems.
Recognition
Dal Maso has been awarded the Caccioppoli prize in 1990, the "Medaglia dei XL per la Matematica" by the Accademia Nazionale delle Scienze, the Prize of the Minister for the Cultural Heritage for Mathematics and Mechanics by the Accademia Nazionale dei Lincei in 2003 and the Amerio prize by the Istituto Lombardo Accademia di Scienze e Lettere in 2005. He has been invited speaker and the third European Congress of Mathematics in 2000 and EMS lecturer in 2002. In 2012 he has been awarded an ERC grant.
References
External links
Website at SISSA
3d European Congress of Mathematics
Site of Caccioppoli Prize
Site of European Research Council
1954 births
Living people
21st-century Italian mathematicians
PDE theorists
European Research Council grantees |
https://en.wikipedia.org/wiki/Minimum%20chi-square%20estimation | In statistics, minimum chi-square estimation is a method of estimation of unobserved quantities based on observed data.
In certain chi-square tests, one rejects a null hypothesis about a population distribution if a specified test statistic is too large, when that statistic would have approximately a chi-square distribution if the null hypothesis is true. In minimum chi-square estimation, one finds the values of parameters that make that test statistic as small as possible.
Among the consequences of its use is that the test statistic actually does have approximately a chi-square distribution when the sample size is large. Generally, one reduces by 1 the number of degrees of freedom for each parameter estimated by this method.
Illustration via an example
Suppose a certain random variable takes values in the set of non-negative integers 1, 2, 3, . . . . A simple random sample of size 20 is taken, yielding the following data set. It is desired to test the null hypothesis that the population from which this sample was taken follows a Poisson distribution.
The maximum likelihood estimate of the population average is 3.3. One could apply Pearson's chi-square test of whether the population distribution is a Poisson distribution with expected value 3.3. However, the null hypothesis did not specify that it was that particular Poisson distribution, but only that it is some Poisson distribution, and the number 3.3 came from the data, not from the null hypothesis. A rule of thumb says that when a parameter is estimated, one reduces the number of degrees of freedom by 1, in this case from 9 (since there are 10 cells) to 8. One might hope that the resulting test statistic would have approximately a chi-square distribution when the null hypothesis is true. However, that is not in general the case when maximum-likelihood estimation is used. It is however true asymptotically when minimum chi-square estimation is used.
Finding the minimum chi-square estimate
The minimum chi-square estimate of the population mean λ is the number that minimizes the chi-square statistic
where a is the estimated expected number in the "> 8" cell, and "20" appears because it is the sample size. The value of a is 20 times the probability that a Poisson-distributed random variable exceeds 8, and it is easily calculated as 1 minus the sum of the probabilities corresponding to 0 through 8. By trivial algebra, the last term reduces simply to a. Numerical computation shows that the value of λ that minimizes the chi-square statistic is about 3.5242. That is the minimum chi-square estimate of λ. For that value of λ, the chi-square statistic is about 3.062764. There are 10 cells. If the null hypothesis had specified a single distribution, rather than requiring λ to be estimated, then the null distribution of the test statistic would be a chi-square distribution with 10 − 1 = 9 degrees of freedom. Since λ had to be estimated, one additional degree of freedom is |
https://en.wikipedia.org/wiki/Carlyle%20circle | In mathematics, a Carlyle circle is a certain circle in a coordinate plane associated with a quadratic equation; it is named after Thomas Carlyle. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.
Definition
Given the quadratic equation
x2 − sx + p = 0
the circle in the coordinate plane having the line segment joining the points A(0, 1) and B(s, p) as a diameter is called the Carlyle circle of the quadratic equation.
Defining property
The defining property of the Carlyle circle can be established thus: the equation of the circle having the line segment AB as diameter is
x(x − s) + (y − 1)(y − p) = 0.
The abscissas of the points where the circle intersects the x-axis are the roots of the equation (obtained by setting y = 0 in the equation of the circle)
x2 − sx + p = 0.
Construction of regular polygons
Regular pentagon
The problem of constructing a regular pentagon is equivalent to the problem of constructing the roots of the equation
z5 − 1 = 0.
One root of this equation is z0 = 1 which corresponds to the point P0(1, 0). Removing the factor corresponding to this root, the other roots turn out to be roots of the equation
z4 + z3 + z2 + z + 1 = 0.
These roots can be represented in the form ω, ω2, ω3, ω4 where ω = exp (2i/5). Let these correspond to the points P1, P2, P3, P4. Letting
p1 = ω + ω4, p2 = ω2 + ω3
we have
p1 + p2 = −1, p1p2 = −1. (These can be quickly shown to be true by direct substitution into the quartic above and noting that ω6 = ω, and ω7 = ω2.)
So p1 and p2 are the roots of the quadratic equation
x2 + x − 1 = 0.
The Carlyle circle associated with this quadratic has a diameter with endpoints at (0, 1) and (−1, −1) and center at (−1/2, 0). Carlyle circles are used to construct p1 and p2. From the definitions of p1 and p2 it also follows that
p1 = 2 cos(2/5), p2 = 2 cos(4/5).
These are then used to construct the points P1, P2, P3, P4.
This detailed procedure involving Carlyle circles for the construction of regular pentagons is given below.
Draw a circle in which to inscribe the pentagon and mark the center point O.
Draw a horizontal line through the center of the circle. Mark one intersection with the circle as point B.
Construct a vertical line through the center. Mark one intersection with the circle as point A.
Construct the point M as the midpoint of O and B.
Draw a circle centered at M through the point A. This is the Carlyle circle for x2 + x − 1 = 0. Mark its intersection with the horizontal line (inside the original circle) as the point W and its intersection outside the circle as the point V. These are the points p1 and p2 mentioned above.
Draw a circle of radius OA and center W. It intersects the original circle at two of the vertices of the pentagon.
Draw a circle of radius OA and center V. It intersects t |
https://en.wikipedia.org/wiki/List%20of%20C%2B%2B%20multiple%20precision%20arithmetic%20libraries | The following is an incomplete list of some arbitrary-precision arithmetic libraries for C++.
GMP
MPFR
MPIR
TTMath
Arbitrary Precision Math C++ Package
Class Library for Numbers
Number Theory Library
Apfloat
C++ Big Integer Library
MAPM
ARPREC
InfInt
Universal Numbers
mp++
Footnotes
References
C++ libraries
Numerical software |
https://en.wikipedia.org/wiki/Fontaine%27s%20period%20rings | In mathematics, Fontaine's period rings are a collection of commutative rings first defined by Jean-Marc Fontaine that are used to classify p-adic Galois representations.
The ring BdR
The ring is defined as follows. Let denote the completion of . Let
So an element of is a sequence of elements
such that . There is a natural projection map given by . There is also a multiplicative (but not additive) map defined by , where the are arbitrary lifts of the to . The composite of with the projection is just . The general theory of Witt vectors yields a unique ring homomorphism such that for all , where denotes the Teichmüller representative of . The ring is defined to be completion of with respect to the ideal . The field is just the field of fractions of .
References
Secondary sources
Algebraic number theory
Galois theory
Representation theory of groups
Hodge theory |
https://en.wikipedia.org/wiki/Vector%20logic | Vector logic is an algebraic model of elementary logic based on matrix algebra. Vector logic assumes that the truth values map on vectors, and that the monadic and dyadic operations are executed by matrix operators. "Vector logic" has also been used to refer to the representation of classical propositional logic as a vector space, in which the unit vectors are propositional variables. Predicate logic can be represented as a vector space of the same type in which the axes represent the predicate letters and . In the vector space for propositional logic the origin represents the false, F, and the infinite periphery represents the true, T, whereas in the space for predicate logic the origin represents "nothing" and the periphery represents the flight from nothing, or "something".
Overview
Classic binary logic is represented by a small set of mathematical functions depending on one (monadic) or two (dyadic) variables. In the binary set, the value 1 corresponds to true and the value 0 to false. A two-valued vector logic requires a correspondence between the truth-values true (t) and false (f), and two q-dimensional normalized real-valued column vectors s and n, hence:
and
(where is an arbitrary natural number, and "normalized" means that the length of the vector is 1; usually s and n are orthogonal vectors). This correspondence generates a space of vector truth-values: V2 = {s,n}. The basic logical operations defined using this set of vectors lead to matrix operators.
The operations of vector logic are based on the scalar product between q-dimensional column vectors: : the orthonormality between vectors s and n implies that if , and if , where .
Monadic operators
The monadic operators result from the application , and the associated matrices have q rows and q columns. The two basic monadic operators for this two-valued vector logic are the identity and the negation:
Identity: A logical identity ID(p) is represented by matrix . This matrix operates as follows: Ip = p, p ∈ V2; due to the orthogonality of s with respect to n, we have , and similarly . It is important to note that this vector logic identity matrix is not generally an identity matrix in the sense of matrix algebra.
Negation: A logical negation ¬p is represented by matrix Consequently, Ns = n and Nn = s. The involutory behavior of the logical negation, namely that ¬(¬p) equals p, corresponds with the fact that N2 = I.
Dyadic operators
The 16 two-valued dyadic operators correspond to functions of the type ; the dyadic matrices have q2 rows and q columns.
The matrices that execute these dyadic operations are based on the properties of the Kronecker product. Two properties of this product are essential for the formalism of vector logic:
Using these properties, expressions for dyadic logic functions can be obtained:
Conjunction. The conjunction (p∧q) is executed by a matrix that acts on two vector truth-values: .This matrix reproduces the features of the classical c |
https://en.wikipedia.org/wiki/Maekawa%27s%20theorem | Maekawa's theorem is a theorem in the mathematics of paper folding named after Jun Maekawa. It relates to flat-foldable origami crease patterns and states that at every vertex, the numbers of valley and mountain folds always differ by two in either direction. The same result was also discovered by Jacques Justin and, even earlier, by S. Murata.
Parity and coloring
One consequence of Maekawa's theorem is that the total number of folds at each vertex must be an even number. This implies (via a form of planar graph duality between Eulerian graphs and bipartite graphs) that, for any flat-foldable crease pattern, it is always possible to color the regions between the creases with two colors, such that each crease separates regions of differing colors. The same result can also be seen by considering which side of the sheet of paper is uppermost in each region of the folded shape.
Related results
Maekawa's theorem does not completely characterize the flat-foldable vertices, because it only takes into account the numbers of folds of each type, and not their angles.
Kawasaki's theorem gives a complementary condition on the angles between the folds at a vertex (regardless of which folds are mountain folds and which are valley folds) that is also necessary for a vertex to be flat-foldable.
References
External links
Paper folding |
https://en.wikipedia.org/wiki/Michael%20McQuillan%20%28mathematician%29 | Michael Liam McQuillan is a Scottish mathematician studying algebraic geometry. As of 2019 he is Professor at the University of Rome Tor Vergata.
Career
Michael McQuillan received the doctorate in 1992 at Harvard University under Barry Mazur ("Division points on semi-Abelian varieties").
In 1995, McQuillan proved the Mordell–Lang conjecture. In 1996, MacQuillan gave a new proof of a conjecture of André Bloch (1926) about holomorphic curves in closed subvarieties of Abelian varieties, proved a conjecture of Shoshichi Kobayashi (about the Kobayashi-hyperbolicity of generic hypersurfaces of high degree in projective n-dimensional space) in the three-dimensional case and achieved partial results on a conjecture of Mark Green and Phillip Griffiths (which states that a holomorphic curve on an algebraic surface of general type with cannot be Zariski-dense).
From 1996 to 2001 he was a post-doctoral Research Fellow at All Souls College of the University of Oxford and in 2009 was Professor at the University of Glasgow as well as Advanced Research Fellow of the British Engineering and Physical Sciences Research Council. As of 2019 he is Professor at the University of Rome Tor Vergata and an editor of the European Journal of Mathematics.
Awards
In 2000 McQuillan received the EMS Prize, which was announced from the European Congress of Mathematics in July 2000, for his work:
In 2001 he was awarded the Whitehead Prize of the London Mathematical Society. In 2002 he was invited speaker at the International Congress of Mathematicians in Beijing (Integrating ). In 2001 he received the Whittaker Prize.
References
20th-century Scottish mathematicians
21st-century Scottish mathematicians
Whitehead Prize winners
Living people
Academic staff of the University of Rome Tor Vergata
Harvard University alumni
Year of birth missing (living people)
Algebraic geometers
Sir Edmund Whittaker Memorial Prize winners |
https://en.wikipedia.org/wiki/Plate%20appearances%20per%20strikeout | In baseball statistics, plate appearances per strikeout (PA/SO) represents a ratio of the number of times a batter strikes out to their plate appearance.
This statistic allows a defensive team to examine the opposing team's lineup for hitters who are more prone to strikeout. Such players, when batting, are typically more aggressive than the average hitter. This knowledge permits the pitcher to approach the batter with more pitching options, often throwing more balls out of the strike zone in the hope that the batter will swing and miss.
The number of this statistic can be calculated by dividing a player's total number of plate appearances by their total number of strikeouts. For example, Reggie Jackson collected 2,597 strikeouts and 11,418 plate appearances in his 21-year baseball career, recording a 4.39 PA/SO, which suggests that for every 4.39 plate appearance Jackson had one strikeout.
Sources
MLB.com – Baseball basics abbreviations
FanGraphs Sabermetrics Library
External links
Baseball Almanac – All-time career leaders in Major League strikeouts
The all-time leaders in plate appearances per strikeout – Article by Joe Pawlikowski
Baseball statistics
Batting (baseball)
Baseball terminology |
https://en.wikipedia.org/wiki/Leo%20Paraspondylos | Leo Paraspondylos () was a high-ranking 11th-century Byzantine official, who served as chief minister to Empress Theodora and Emperor Michael VI.
Biography
Leo's surname is in all probability a sobriquet; he seems to have belonged to the Spondylos family, and was possibly a relative of the general Michael Spondyles. He is attested as an official under Michael IV the Paphlagonian (r. 1034–41), and rose in office to become the Empire's chief minister (paradynasteuon) under Theodora (r. 1042–56) and Michael VI (r. 1056–57), holding the posts of synkellos and protosynkellos.
Leo's contemporary, historian and fellow public servant Michael Attaleiates, considered him an excellent administrator, but it was Paraspondylos' refusal to meet the demands of the Empire's leading generals in 1057 that led to their rebellion and the overthrow of Michael VI and the installation of one of their own number, Isaac I Komnenos (r. 1057–59), on the throne. After the rebel victory, Paraspondylos was dismissed from office and exiled from Constantinople, possibly being forcibly tonsured. Although occasionally critical of his rough manners, Michael Psellos intervened on his behalf with Isaac I Komnenos, to little avail. Nothing further is known of Leo thereafter.
References
11th-century Byzantine people
Byzantine officials
Byzantine prisoners and detainees |
https://en.wikipedia.org/wiki/Multipartite%20graph | In graph theory, a part of mathematics, a -partite graph is a graph whose vertices are (or can be) partitioned into different independent sets. Equivalently, it is a graph that can be colored with colors, so that no two endpoints of an edge have the same color. When these are the bipartite graphs, and when they are called the tripartite graphs.
Bipartite graphs may be recognized in polynomial time but, for any it is NP-complete, given an uncolored graph, to test whether it is -partite.
However, in some applications of graph theory, a -partite graph may be given as input to a computation with its coloring already determined; this can happen when the sets of vertices in the graph represent different types of objects. For instance, folksonomies have been modeled mathematically by tripartite graphs in which the three sets of vertices in the graph represent users of a system, resources that the users are tagging, and tags that the users have applied to the resources.
A complete -partite graph is a -partite graph in which there is an edge between every pair of vertices from different independent sets. These graphs are described by notation with a capital letter subscripted by a sequence of the sizes of each set in the partition. For instance, is the complete tripartite graph of a regular octahedron, which can be partitioned into three independent sets each consisting of two opposite vertices. A complete multipartite graph is a graph that is complete -partite for some .
The Turán graphs are the special case of complete multipartite graphs in which each two independent sets differ in size by at most one vertex.
Complete -partite graphs, complete multipartite graphs, and their complement graphs, the cluster graphs, are special cases of cographs, and can be recognized in polynomial time even when the partition is not supplied as part of the input.
References
Graph families |
https://en.wikipedia.org/wiki/Graham%20Hilford%20Pollard | Graham Hilford Pollard is an Australian mathematician, professor, statistician, author, lecturer, and Doctor in Mathematics, recognised for being the recipient of the David Hilbert Award in 1991.
Career
In 1976, he received his PhD from the Australian National University with the thesis entitled A Stochastic Analysis of Scoring Systems.
He is a lecturer in statistics at the Canberra College of Advance Education since 1982, and currently serves as Chairman of the editorial committee of the Australian Mathematics Trust publishing house.
In 1991, he received the David Hilbert Award from the World Federation of National Mathematics Competitions.
Most of his papers have been published by the Australian & New Zealand Journal of Statistics, Journal of Mathematics Competitions, the Journal of Educational Studies in Mathematics, the Journal of the Australian Mathematical Society, and the International Journal of Mathematical Education in Science and Technology.
Academic papers
References
Australian statisticians
Year of birth missing (living people)
Living people
Australian National University alumni |
https://en.wikipedia.org/wiki/Dodecahedral%20pyramid | In 4-dimensional geometry, the dodecahedral pyramid is bounded by one dodecahedron on the base and 12 pentagonal pyramid cells which meet at the apex. Since a dodecahedron's circumradius is greater than its edge length, the pentagonal pyramids require tall isosceles triangle faces.
The dual to the dodecahedral pyramid is an icosahedral pyramid, seen as an icosahedral base, and 20 regular tetrahedra meeting at an apex.
References
External links
Richard Klitzing, Axial-Symmetrical Edge Facetings of Uniform Polyhedra
4-polytopes |
https://en.wikipedia.org/wiki/Hard%E2%80%93easy%20effect | The hard–easy effect is a cognitive bias that manifests itself as a tendency to overestimate the probability of one's success at a task perceived as hard, and to underestimate the likelihood of one's success at a task perceived as easy. The hard-easy effect takes place, for example, when individuals exhibit a degree of underconfidence in answering relatively easy questions and a degree of overconfidence in answering relatively difficult questions. "Hard tasks tend to produce overconfidence but worse-than-average perceptions," reported Katherine A. Burson, Richard P. Larrick, and Jack B. Soll in a 2005 study, "whereas easy tasks tend to produce underconfidence and better-than-average effects."
The hard-easy effect falls under the umbrella of "social comparison theory", which was originally formulated by Leon Festinger in 1954. Festinger argued that individuals are driven to evaluate their own opinions and abilities accurately, and social comparison theory explains how individuals carry out those evaluations by comparing themselves to others.
In 1980, Ferrell and McGoey called it the "discriminability effect"; in 1992, Griffin and Tversky called it the "difficulty effect".
Experiments
In a range of studies, participants have been requested to answer general knowledge questions, each of which had two possible answers, and also to estimate their chances of answering each question correctly. If the participants had a sufficient degree of self-knowledge, their level of confidence in regard to each answer they gave would be high for the questions they answered correctly and lower for the ones they answered wrong. However, this generally is not the case. Many people are overconfident; indeed, studies show that most people systematically overestimate their own abilities. Moreover, people are overconfident about their ability to answer questions that are deemed to be hard but underconfident on questions that are considered easy.
In a study reported in 1997, William M. Goldstein and Robin M. Hogarth gave an experimental group a questionnaire containing general-knowledge questions such as "Who was born first, Aristotle or Buddha?" or "Was the zipper invented before or after 1920?". The subjects filled in the answers they believed to be correct and rated how sure they were of them. The results showed subjects tend to be under-confident of their answers to questions designated by the experimenters as to be easy, and overconfident of their answers to questions designated as hard.
Prevalence
A 2009 study concluded "that all types of judges exhibit the hard-easy effect in almost all realistic situations", and that the presence of the effect "cannot be used to distinguish between judges or to draw support for specific models of confidence elicitation".
The hard-easy effect manifests itself regardless of personality differences. Many researchers agree that it is "a robust and pervasive phenomenon".
A 1999 study suggested that the difference between the dat |
https://en.wikipedia.org/wiki/List%20of%20career%20achievements%20by%20Kevin%20Durant | This page details the records, statistics and career achievements of American basketball player Kevin Durant. Durant is an American basketball small forward who plays for the Phoenix Suns of the National Basketball Association.
NBA career statistics
Statistics are correct through the end of the 2017–18 season.
Regular season
Playoffs
Career-highs
Awards and accomplishments
NBA
2× NBA champion: ,
2× NBA Finals Most Valuable Player: ,
NBA Most Valuable Player:
4× NBA scoring champion: , , ,
NBA Rookie of the Year:
10× All-NBA selection:
6× First team: , , , , ,
4× Second team: , , ,
13× NBA All-Star: , , , , , , , , , , 2021, 2022, 2023
NBA All-Star Game MVP: , 2019
NBA All-Star Game Captain: 2021
2× NBA All-Star Weekend H-O-R-S-E Competition champion: ,
NBA Rookie Challenge MVP:
NBA All-Rookie First Team:
14× NBA Western Conference Player of the Month
1× NBA Eastern Conference Player of the Month
26× NBA Western Conference Player of the Week
3× NBA Eastern Conference Player of the Week
5× NBA Western Conference Rookie of the Month
Best Male Athlete ESPY Award winner: 2014
Best NBA Player ESPY Award winner: 2014
Best Championship Performance ESPY Award winner: 2017
Texas
Consensus National Player of the Year: 2007
Consensus first team All-American: 2007
Adolph Rupp Trophy (first freshman to win this award): 2007
John R. Wooden Award (first freshman to win this award): 2007
National Association of Basketball Coaches Player of the Year (only freshman to ever win this award): 2007
AP Player of the Year (first freshman to win this award): 2007
Naismith Men's College Player of the Year (first freshman to win this award): 2007
USBWA National Freshman of the Year: 2007
Big 12 Player of the Year: 2007
Big 12 Freshman of the Year: 2007
All-Big 12 First Team (unanimous): 2007
Big 12 All-Defensive team: 2007
Big 12 All-Rookie team (unanimous): 2007
Big 12 Tournament Most Valuable Player Award: 2007
4× Big 12 Player of the Week
6× Big 12 Rookie of the Week
NBA achievements
Regular season
Career
Youngest player in NBA history to win an NBA scoring title. (21 years, 197 days)
Youngest player in NBA history to join the 50–40–90 club.
2nd youngest player in NBA History to record 10,000 career points. (24 years, 34 days)
The youngest is LeBron James.
One of five players in NBA history to win the NBA scoring title in 4 or more seasons.
Includes Wilt Chamberlain, Michael Jordan, George Gervin, and Allen Iverson.
One of eight players in NBA history to win the NBA scoring title in three consecutive seasons.
Includes Michael Jordan (twice), George Mikan, Neil Johnston, Wilt Chamberlain, Bob McAdoo, George Gervin, and James Harden.
Season
One of three players in NBA history to score 25 points or more for 40 straight games in one season. (41 games)
Includes Wilt Chamberlain (80 games), Michael Jordan (40 games)
One of four players in NBA history to average 20+ points per game for an entire season as a teenager.
Includes LeBron James, Carmel |
https://en.wikipedia.org/wiki/Kazuya%20Okamura | is a Japanese professional footballer who plays forward for FC Gifu in the J3 League.
Career statistics
Updated to 23 February 2018.
1Includes J2/J3 relegation play-offs.
References
External links
Profile at V-Varen Nagasaki
Profile at Kamatamare Sanuki
1987 births
Living people
Osaka Gakuin University alumni
Association football people from Okayama Prefecture
Japanese men's footballers
J2 League players
J3 League players
Japan Football League players
Roasso Kumamoto players
V-Varen Nagasaki players
Kamatamare Sanuki players
Giravanz Kitakyushu players
FC Gifu players
Men's association football midfielders |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Henri%20Poincar%C3%A9 | In physics and mathematics, a number of ideas are named after Henri Poincaré:
Euler–Poincaré characteristic
Hilbert–Poincaré series
Poincaré–Bendixson theorem
Poincaré–Birkhoff theorem
Poincaré–Birkhoff–Witt theorem, usually known as the PBW theorem
Poincaré algebra
K-Poincaré algebra
Super-Poincaré algebra
Poincaré–Bjerknes circulation theorem
Poincaré complex
Poincaré conjecture, one of the Millennium Prize Problems
Generalized Poincaré conjecture
Poincaré disk model, a model of hyperbolic geometry
Poincaré duality
Twisted Poincaré duality
Poincaré–Einstein synchronization
Poincaré expansion
Poincaré group, the group of isometries of Minkowski spacetime, named in honour of Henri Poincaré
K-Poincaré group
Poincaré half-plane model, a model of two-dimensional hyperbolic geometry
Poincaré homology sphere
Poincaré–Hopf theorem
Poincaré inequality
Poincaré–Wirtinger inequality
Poincaré–Lelong equation
Poincaré lemma
Poincaré-Lefschetz duality
Poincaré–Lindstedt method
Poincaré line bundle
Poincaré map
Poincaré metric
Poincaré–Miranda theorem
Poincaré–Neumann operator
Poincaré plot
Poincaré polynomial
Poincaré recurrence
Poincaré residue
Poincaré separation theorem
Poincaré series (modular form)
Poincaré space
Poincaré sphere (optics)
Poincaré–Steklov operator
Poincaré symmetry
Poincaré wave
Other
Annales Henri Poincaré
Institut Henri Poincaré
Henri Poincaré Prize
Henri Poincaré University
Poincaré and the Three-Body Problem
Poincaré (crater)
Poincaré Medal
Poincaré Seminars
Books with the title "Henri Poincaré"
References
Poincare
Henri Poincaré |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20John%20Milnor | Things named after an American mathematician John Milnor:
Barratt–Milnor sphere
Fáry–Milnor theorem
Milnor conjecture in algebraic K-theory
Milnor conjecture in knot theory
Milnor conjecture concerning manifolds with nonnegative Ricci curvature
Milnor construction
Milnor K-theory
Milnor fibration
Milnor invariants
Milnor manifold
Milnor map
Milnor–Moore theorem
Milnor number
Milnor ring
Milnor sphere
Milnor theorem
Milnor–Thurston kneading theory
Milnor Frame concerning left invariant metrics on three-dimensional Lie groups
Milnor–Wood inequality
Švarc–Milnor lemma
Milnor |
https://en.wikipedia.org/wiki/Isaac%20Argyros | Isaac Argyros (Greek: Ισαάκιος Αργυρός) was a Byzantine mathematician and monk, born about 1312, who wrote a treatise named Easter Rule, along with books on arithmetic, geometry and astronomy.
Works
An Easter Rule, a treatise on Easter
New Tables: An Astronomical treatise, based on Ptolemaic astronomy
Bibliography
Science and Civilisation in China, Volume 3: Mathematics and the Sciences of the Heavens and the Earth, Joseph Needham, Cambridge University Press 1959,
References
1312 births
Isaac
Greek mathematicians
Year of death unknown
14th-century Byzantine writers
14th-century astronomers
14th-century Byzantine scientists
Byzantine astronomers
14th-century Greek people
14th-century Greek scientists
14th-century Greek educators
14th-century Greek mathematicians
14th-century Greek astronomers |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Ferdinand%20Georg%20Frobenius | These are things named after Ferdinand Georg Frobenius, a German mathematician.
Arithmetic and geometric Frobenius
Cauchy–Frobenius lemma
Frobenioid
Frobenius algebra
Frobenius category
Frobenius coin problem
Frobenius number
Frobenius companion matrix
Frobenius covariant
Frobenius element
Frobenius endomorphism (also known as Frobenius morphism, Frobenius map)
Frobenius determinant theorem
Frobenius formula
Frobenius group
Frobenius complement
Frobenius kernel
Frobenius inner product
Frobenius norm
Frobenius manifold
Frobenius matrix
Frobenius method
Frobenius normal form
Frobenius polynomial
Frobenius pseudoprime
Frobenius reciprocity
Frobenius solution to the hypergeometric equation
Frobenius splitting
Frobenius theorem (differential topology)
Frobenius theorem (real division algebras)
Frobenius's theorem (group theory)
Frobenius conjecture
Frobenius–Schur indicator
Perron–Frobenius theorem
Quadratic Frobenius test
Rouché–Frobenius theorem
Quasi-Frobenius Lie algebra
Quasi-Frobenius ring
Frobenius |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Emil%20Artin | These are things named after Emil Artin, a mathematician.
Ankeny–Artin–Chowla congruence
Artin algebra
Artin billiards
Artin braid group
Artin character
Artin conductor
Artin's conjecture for conjectures by Artin. These include
Artin's conjecture on primitive roots
Artin conjecture on L-functions
Artin group
Artin–Hasse exponential
Artin L-function
Artin reciprocity
Artin–Rees lemma
Artin representation
Artin–Schreier theorem
Artin–Schreier theory
Artin's theorem on induced characters
Artin–Zorn theorem
Artinian ideal
Artinian module
Artinian ring
Artin–Tate lemma
Artin–Tits group
Fox–Artin arc
Wedderburn–Artin theorem
Emil Artin Junior Prize in Mathematics
See also
Artinian
Artin |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Niels%20Henrik%20Abel | This is the list of things named after Niels Henrik Abel (1802–1829), a Norwegian mathematician.
Mathematics
Abel's binomial theorem
Abel elliptic functions
Abel equation
Abel equation of the first kind
Abel–Goncharov interpolation
Abel–Plana formula
Abel function
Abel's integral equation
Abel's identity
Abel's inequality
Abel's irreducibility theorem
Abel–Jacobi map
Abel–Jacobi theorem
Abel polynomials
Abel's summation formula
Abelian means
Abel's test
Abel's theorem
Abelian theorem
Abel–Ruffini theorem
Abel transform
Abel transformation
Abelian category
Pre-abelian category
Quasi-abelian category
Abelian group
Abelianization
Metabelian group
Non-abelian group
Abelian extension
Abelian integral
Abelian surface
Abelian variety
Abelian variety of CM-type
Dual abelian variety
Abelian von Neumann algebra
Geography
Abeltoppen, a mountain in Dickson Land at Spitsbergen, Svalbard.
, a street in the 12th arrondissement of Paris in front of the Gare de Lyon mainline station.
Other
Abel, a lunar crater.
Abel Prize
References
Abel
Niels Henrik Abel |
https://en.wikipedia.org/wiki/2012%E2%80%9313%20First%20Vienna%20FC%20season | The 2012–13 First Vienna FC season was the fourth consecutive season in the second highest professional division in Austria after the promotion in 2009.
Squad
Squad and statistics
|-
! colspan="12" style="background:#dcdcdc; text-align:center;"| Goalkeepers
|-
! colspan="12" style="background:#dcdcdc; text-align:center;"| Defenders
|-
! colspan="12" style="background:#dcdcdc; text-align:center;"| Midfielders
|-
! colspan="12" style="background:#dcdcdc; text-align:center;"| Forwards
|}
References
Vienna
First Vienna FC seasons |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20%C3%89lie%20Cartan | These are things named after Élie Cartan (9 April 1869 – 6 May 1951), a French mathematician.
Mathematics and physics
Cartan calculus
Cartan connection, Cartan connection applications
Cartan's criterion
Cartan decomposition
Cartan's equivalence method
Cartan formalism (physics)
Cartan involution
Cartan's magic formula
Cartan relations
Cartan map
Cartan matrix
Cartan pair
Cartan subalgebra
Cartan subgroup
Cartan's method of moving frames
Cartan's theorem, a name for the closed-subgroup theorem
Cartan's theorem, a name for the theorem on highest weights
Cartan's theorem, a name for Lie's third theorem
Einstein–Cartan theory
Einstein–Cartan–Evans theory
Cartan–Ambrose–Hicks theorem
Cartan–Brauer–Hua theorem
Cartan–Dieudonné theorem
Cartan–Hadamard manifold
Cartan–Hadamard theorem
Cartan–Iwahori decomposition
Cartan-Iwasawa-Malcev theorem
Cartan–Kähler theorem
Cartan–Karlhede algorithm
Cartan–Weyl theory
Cartan–Weyl basis
Cartan–Killing form
Cartan–Kuranishi prolongation theorem
CAT(k) space
Maurer–Cartan form
Newton–Cartan theory
Stokes–Cartan's theorem, the generalized fundamental theorem of calculus, proven by Cartan (in its general form), also known as Stokes' theorem although Stokes neither formulated nor proved it.
Other
Cartan (crater)
Élie Cartan Prize
Note some are after Henri Cartan, a son of É. Cartan; e.g.,
Cartan's lemma (potential theory)
Cartan seminar
Cartan's theorems A and B
Cartan–Eilenberg resolution
Cartan |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Erich%20Hecke | These are things named after Erich Hecke, a German mathematician.
Hecke algebra
Hecke algebra of a locally compact group
Hecke algebra of a finite group
Hecke algebra of a pair
Hecke polynomial
Iwahori–Hecke algebra
Affine Hecke algebra
Double affine Hecke algebra
Hecke algebra (disambiguation)
Hecke character
Hecke congruence subgroup
Hecke correspondence
Hecke eigenform
Hecke group
Hecke L-function (disambiguation)
Hecke operator
Hecke ring
Hecke |
https://en.wikipedia.org/wiki/Ruth%20Kellerhals | Ruth Kellerhals (born 17 July 1957) is a Swiss mathematician at the University of Fribourg, whose field of study is hyperbolic geometry, geometric group theory and polylogarithm identities.
Biography
As a child, she went to a gymnasium in Basel and then studied at the University of Basel, graduating in 1982 with a diploma directed by Heinz Huber "On finiteness of the isometry group of a compact negatively curved Riemannian manifold". She received her PhD in 1988, from the same university, with a thesis entitled "On the volumes of hyperbolic polytopes in dimensions three and four". Her advisor was Hans-Christoph Im Hof. During the year 1983–84 she also studied at the University of Grenoble (Fourier Institute).
In 1995 she received her habilitation from the University of Bonn, where she worked at the Max Planck Institute for Mathematics since 1989 until 1995. There, she was an assistant with Professor Friedrich Hirzebruch. Since 1995 she has been an assistant professor at the University of Göttingen, and since 1999 a distinguished professor at the University of Bordeaux 1. In 2000 she became a professor at the University of Fribourg, Switzerland, where she was in 1998 to 1999 as a visiting professor.
Research
Her main research fields include hyperbolic geometry, geometric group theory, geometry of discrete groups (especially reflection groups, Coxeter groups), convex and polyhedral geometry, volumes of hyperbolic polytopes, manifolds and polylogarithms. She does historical research into the works and life of Ludwig Schläfli, a Swiss geometer.
She has been a guest researcher at MSRI, IHES, Mittag-Leffler Institute, the State University of New York at Stony Brook, RIMS in Kyoto, Osaka City University, ETH Zürich, the University of Bern and the University of Auckland. Also she visited numerous research institutes and universities in Helsinki, Berlin and Budapest.
Selected works
References
External links
Homepage
1957 births
Living people
Swiss mathematicians
Swiss women mathematicians
University of Basel alumni
Academic staff of the University of Fribourg |
https://en.wikipedia.org/wiki/Wall%27s%20conjecture | In mathematics, Wall's conjecture denotes one of the following:
a conjecture of C. T. C. Wall in group theory stating that every finitely generated group is accessible
a hypothesis of Donald Dines Wall in number theory on the non-existence of Fibonacci-Wieferich primes |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20W.%20V.%20D.%20Hodge | These are things named after W. V. D. Hodge, a Scottish mathematician.
Hodge algebra
Hodge–Arakelov theory
Hodge bundle
Hodge conjecture
Hodge cycle
Hodge–de Rham spectral sequence
Hodge diamond
Hodge duality
Hodge filtration
Hodge index theorem
Hodge group
Hodge star operator
Hodge structure
Mixed Hodge structure
Hodge–Tate module
Hodge theory
Mixed Hodge module
Hodge–Arakelov theory
p-adic Hodge theory
Hodge |
https://en.wikipedia.org/wiki/Arason%20invariant | In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial discriminant and Clifford invariant over a field k of characteristic not 2, taking values in H3(k,Z/2Z). It was introduced by .
The Rost invariant is a generalization of the Arason invariant to other algebraic groups.
Definition
Suppose that W(k) is the Witt ring of quadratic forms over a field k and I is the ideal of forms of even dimension. The Arason invariant is a group homomorphism from I3 to the Galois cohomology group H3(k,Z/2Z). It is determined by the property that on the 8-dimensional diagonal form with entries 1, –a, –b, ab, -c, ac, bc, -abc (the 3-fold Pfister form«a,b,c») it is given by the cup product of the classes of a, b, c in H1(k,Z/2Z) = k*/k*2. The Arason invariant vanishes on I4, and it follows from the Milnor conjecture proved by Voevodsky that it is an isomorphism from I3/I4 to H3(k,Z/2Z).
References
Algebraic groups |
https://en.wikipedia.org/wiki/Super-Poissonian%20distribution | In mathematics, a super-Poissonian distribution is a probability distribution that has a larger variance than a Poisson distribution with the same mean. Conversely, a sub-Poissonian distribution has a smaller variance.
An example of super-Poissonian distribution is negative binomial distribution.
The Poisson distribution is a result of a process where the time (or an equivalent measure) between events has an exponential distribution, representing a memoryless process.
Mathematical definition
In probability theory it is common to say a distribution, D, is a sub-distribution of another distribution E if D 's moment-generating function, is bounded by E 's up to a constant.
In other words
for some C > 0.
This implies that if and are both from a sub-E distribution, then so is .
A distribution is strictly sub- if C ≤ 1.
From this definition a distribution, D, is sub-Poissonian if
for all t > 0.
An example of a sub-Poissonian distribution is the Bernoulli distribution, since
Because sub-Poissonianism is preserved by sums, we get that the binomial distribution is also sub-Poissonian.
References
Poisson point processes
Types of probability distributions |
https://en.wikipedia.org/wiki/Rost%20invariant | In mathematics, the Rost invariant is a cohomological invariant of an absolutely simple simply connected algebraic group G over a field k, which associates an element of the Galois cohomology group H3(k, Q/Z(2)) to a principal homogeneous space for G. Here the coefficient group Q/Z(2) is the tensor product of the group of roots of unity of an algebraic closure of k with itself. first introduced the invariant for groups of type F4 and later extended it to more general groups in unpublished work that was summarized by .
The Rost invariant is a generalization of the Arason invariant.
Definition
Suppose that G is an absolutely almost simple simply connected algebraic group over a field k. The Rost invariant associates an element a(P) of the Galois cohomology group H3(k,Q/Z(2)) to a G-torsor P.
The element a(P) is constructed as follows. For any extension K of k there is an exact sequence
where the middle group is the étale cohomology group and Q/Z is the geometric part of the cohomology.
Choose a finite extension K of k such that G splits over K and P has a rational point over K. Then the exact sequence splits canonically as a direct sum so the étale cohomology group contains Q/Z canonically. The invariant a(P) is the image of the element 1/[K:k] of Q/Z under the trace map from H(PK,Q/Z(2)) to H(P,Q/Z(2)), which lies in the subgroup H3(k,Q/Z(2)).
These invariants a(P) are functorial in field extensions K of k; in other words the fit together to form an element of the cyclic group Inv3(G,Q/Z(2)) of cohomological invariants of the group G, which consists of morphisms of the functor K→H1(K,G) to the functor K→H3(K,Q/Z(2)). This element of Inv3(G,Q/Z(2)) is a generator of the group and is called the Rost invariant of G.
References
Algebraic groups |
https://en.wikipedia.org/wiki/Outermorphism | In geometric algebra, the outermorphism of a linear function between vector spaces is a natural extension of the map to arbitrary multivectors. It is the unique unital algebra homomorphism of exterior algebras whose restriction to the vector spaces is the original function.
Definition
Let be an -linear map from to . The extension of to an outermorphism is the unique map satisfying
for all vectors and all multivectors and , where denotes the exterior algebra over . That is, an outermorphism is a unital algebra homomorphism between exterior algebras.
The outermorphism inherits linearity properties of the original linear map. For example, we see that for scalars , and vectors , , , the outermorphism is linear over bivectors:
which extends through the axiom of distributivity over addition above to linearity over all multivectors.
Adjoint
Let be an outermorphism. We define the adjoint of to be the outermorphism that satisfies the property
for all vectors and , where is the nondegenerate symmetric bilinear form (scalar product of vectors).
This results in the property that
for all multivectors and , where is the scalar product of multivectors.
If geometric calculus is available, then the adjoint may be extracted more directly:
The above definition of adjoint is like the definition of the transpose in matrix theory. When the context is clear, the underline below the function is often omitted.
Properties
It follows from the definition at the beginning that the outermorphism of a multivector is grade-preserving:
where the notation indicates the -vector part of .
Since any vector may be written as , it follows that scalars are unaffected with . Similarly, since there is only one pseudoscalar up to a scalar multiplier, we must have . The determinant is defined to be the proportionality factor:
The underline is not necessary in this context because the determinant of a function is the same as the determinant of its adjoint. The determinant of the composition of functions is the product of the determinants:
If the determinant of a function is nonzero, then the function has an inverse given by
and so does its adjoint, with
The concepts of eigenvalues and eigenvectors may be generalized to outermorphisms. Let be a real number and let be a (nonzero) blade of grade . We say that a is an eigenblade of the function with eigenvalue if
It may seem strange to consider only real eigenvalues, since in linear algebra the eigenvalues of a matrix with all real entries can have complex eigenvalues. In geometric algebra, however, the blades of different grades can exhibit a complex structure. Since both vectors and pseudovectors can act as eigenblades, they may each have a set of eigenvalues matching the degrees of freedom of the complex eigenvalues that would be found in ordinary linear algebra.
Examples
Simple maps
The identity map and the scalar projection operator are outermorphisms.
Versors
A rotation of a vect |
https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Felix%20Klein | These are things named after Felix Klein (1849 – 1925), a German mathematician.
Mathematics
Klein bottle
Solid Klein bottle
Klein configuration
Klein cubic threefold
Klein four-group
Klein geometry
Klein graphs
Klein's inequality
Klein model
Klein polyhedron
Klein surface
Klein quadric
Klein quartic
Kleinian group
Kleinian integer
Kleinian model
Kleinian ring
Kleinian singularity
Klein's icosahedral cubic surface
Klein's j-invariant
Beltrami–Klein model
Cayley–Klein metric
Clifford–Klein form
Schottky–Klein prime form
Other
Klein's Encyclopedia of Mathematical Sciences
The Felix Klein Protocols
Felix Klein medal, named after the first president of the ICMI (1908–1920), honours a lifetime achievement in mathematics education research.
The Klein project of the IMU and ICMI aims to produce a book for upper secondary teachers that communicates the breadth and vitality of the research discipline of mathematics and connects it to the senior secondary school curriculum.
Klein |
https://en.wikipedia.org/wiki/Gerhard%20Huisken | Gerhard Huisken (born 20 May 1958) is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huisken's monotonicity formula, which is named after him. With Tom Ilmanen, he proved a version of the Riemannian Penrose inequality, which is a special case of the more general Penrose conjecture in general relativity.
Education and career
After finishing high school in 1977, Huisken took up studies in mathematics at Heidelberg University. In 1982, one year after his diploma graduation, he completed his PhD at the same university under the direction of Claus Gerhardt. The topic of his dissertation were non-linear partial differential equations (Reguläre Kapillarflächen in negativen Gravitationsfeldern).
From 1983 to 1984, Huisken was a researcher at the Centre for Mathematical Analysis at the Australian National University (ANU) in Canberra. There, he turned to differential geometry, in particular problems of mean curvature flows and applications in general relativity. In 1985, he returned to the University of Heidelberg, earning his habilitation in 1986. After some time as a visiting professor at the University of California, San Diego, he returned to ANU from 1986 to 1992, first as a Lecturer, then as a Reader. In 1991, he was a visiting professor at Stanford University. From 1992 to 2002, Huisken was a full professor at the University of Tübingen, serving as dean of the faculty of mathematics from 1996 to 1998. From 1999 to 2000, he was a visiting professor at Princeton University.
In 2002, Huisken became a director at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) in Potsdam and, at the same time, an honorary professor at the Free University of Berlin. In April 2013, he took up the post of director at the Mathematical Research Institute of Oberwolfach, together with a professorship at Tübingen University. He remains an external scientific member of the Max Planck Institute for Gravitational Physics.
Huisken's PhD students include Ben Andrews and Simon Brendle, among over twenty-five others.
Work
Huisken's work deals with partial differential equations, differential geometry, and their applications in physics. Numerous phenomena in mathematical physics and geometry are related to surfaces and submanifolds. A dominant theme of Huisken's work has been the study of the deformation of such surfaces, in situations where the rules of deformation are determined by the geometry of those surfaces themselves. Such processes are governed by partial differential equations.
Huisken's contributions to mean curvature flow are particularly fundamental. Through his work, the mean curvature flow of hypersurfaces in various convex settings is largely understood. His discovery of Huisken's monotonicity formula, valid for general mean curvature flows, is a particularly important tool.
In t |
https://en.wikipedia.org/wiki/Julius%20Borcea | Julius Bogdan Borcea (8 June 1968 – 8 April 2009) was a Romanian Swedish mathematician. His scientific work included vertex operator algebra and zero distribution of polynomials and entire functions, via correlation inequalities and statistical mechanics.
Biography
Born in Bacău, Romania, by a math teacher who instilled in her son's intellect the beauty of mathematics, he studied in 1982-1984 at the Lycée Descartes in Rabat, Morocco, and he completed his Baccalaureat at the Lycée Français Prins Henrik of Copenhagen. In 1987–1989 he attended the Lycée Louis-le-Grand in Paris. He obtained his PhD in Mathematics in 1998, at Lund University, under the direction of Arne Meurman. After defending his PhD thesis in 1998, he embarked in postdoctoral studies at the Mittag-Leffler Institute for six months and at the University of Strasbourg for two years. He was appointed Associate Professor in 2001, and Lecturer in 2005 at Stockholm University. A year later he was granted the Swedish Mathematical Society's Wallenberg Prize. Promoted to Full Professor in 2008, he was awarded a Royal Swedish Academy of Sciences Fellowship, in 2009, and the Crafoord prize research grant diploma.
Professional profile
Borcea's scientific work ranged from vertex operator theory to zero distribution of polynomials and entire functions, via correlation inequalities and statistical mechanics. His thesis consists of two seemingly independent parts: one in vertex operator theory and the other devoted to the geometry of zeros of complex polynomials in one variable.
In vertex operator theory, Julius generalized results of Mirko Primc and Arne Meurman and gave a classification of annihilated fields. As concerns complex polynomials, he tackled Sendov’s conjecture on zeros and critical points of complex polynomials in one variable. Using
novel techniques, he proved the conjecture for polynomials of degree not exceeding 7. Earlier (1969) the conjecture had been proven for polynomials of degree not exceeding 5. At Stockholm University, Julius had a steady collaboration with Rikard Bøgvad and Boris Shapiro. They worked on rational approximations of algebraic equations, piecewise harmonic functions and positive Cauchy transforms, and the geometry of zeros of polynomials in one variable. Borcea and Petter Brändén collaborated on a project on the geometry of zeros of polynomials and entire functions. They characterized all linear operators on polynomials
preserving the property of having only real zeros, a problem that goes back to Edmond Laguerre and to George Pólya and Issai Schur. These results were subsequently extended to several variables, and a connection to the Lee–Yang theorem on phase-transitions in statistical physics was made. Together with Tom Liggett (UCLA) they applied their methods to problems in probability theory and were able to prove an important conjecture about the preservation of negative dependence properties in the symmetric exclusion process.
Borcea had a compreh |
https://en.wikipedia.org/wiki/2013%E2%80%9314%20Wycombe%20Wanderers%20F.C.%20season | The 2013–14 Football League Two was Wycombe Wanderers' 126th season in existence and their twentieth season in the Football League. This page shows the statistics of the club's players in the season, and also lists all matches that the club played during the season.
The season also marked the end of Ivor Beeks' 28-year career as Wycombe's chairman, on 21 August 2013. During the 28 years he spent as chairman, he oversaw the club's rise from the Isthmian League into the Football League. He continues his affiliation with the club in the role of vice-president.
Wycombe ended the season in dramatic style, by achieving League Two survival on the final day of the season. On 3 May 2014, Wycombe began the day three points adrift of safety in the relegation zone. However, after a 3–0 victory away at Torquay, coupled with Bristol Rovers' 1–0 defeat to Mansfield, Wycombe stayed up on goal difference, whilst Bristol dropped out of the Football League.
League data
League table
Match results
Legend
Friendlies
League Two
FA Cup
League Cup
League Trophy
Squad statistics
Appearances and goals
|-
|colspan="14"|Players left the club before the end of the season:
|}
Goalscorers
*Hause and Knott left the club before the end of the season
Disciplinary record
Transfers
Charles Dunne agreed a transfer to Blackpool on 24 August 2013, although he immediately returned to Wycombe on a season-long loan.
See also
2013–14 in English football
2013–14 Football League Two
Wycombe Wanderers F.C.
Gareth Ainsworth
Adams Park
References
Wycombe Wanderers F.C. seasons
Wycombe Wanderers |
https://en.wikipedia.org/wiki/Attila%20L%C5%91rinczy | Attila Lőrinczy (born 8 April 1994) is a Hungarian football player who plays for Budapest Honvéd.
Career
On 16 December 2022, Lőrinczy signed a contract with Diósgyőr.
Club statistics
Updated to games played as of 15 May 2021.
References
Sources
MLSZ
HLSZ
1994 births
Living people
Footballers from Budapest
Hungarian men's footballers
Hungary men's youth international footballers
Hungary men's under-21 international footballers
Men's association football midfielders
Budapest Honvéd FC players
Békéscsaba 1912 Előre footballers
Szolnoki MÁV FC footballers
FC Ajka players
Mosonmagyaróvári TE footballers
Soroksár SC players
Budafoki MTE footballers
Diósgyőri VTK players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/Antti%20H%C3%B6lli | Antti Hölli (born July 16, 1987) is a Finnish ice hockey player.
Holli made his SM-liiga debut playing with Tappara during the 2006–07 SM-liiga season.
Career statistics
References
External links
1987 births
Living people
Finnish ice hockey right wingers
Herlev Eagles players
Herning Blue Fox players
Ilves players
Lempäälän Kisa players
MHC Martin players
Milton Keynes Lightning players
Peliitat Heinola players
SønderjyskE Ishockey players
Tappara players
Vaasan Sport players
Finnish expatriate ice hockey players in Denmark
Finnish expatriate ice hockey players in England
Finnish expatriate ice hockey players in Slovakia |
https://en.wikipedia.org/wiki/Aivaras%20Bend%C5%BEius | Aivaras Bendžius (born January 26, 1993) is a Lithuanian ice hockey player.
Bendzius made his SM-liiga debut playing with Ilves during the 2012–13 SM-liiga season.
Career statistics
Regular season and playoffs
International
Junior – U18
Junior – U20
Senior
References
External links
1993 births
Living people
Ilves players
Lithuanian ice hockey forwards
People from Elektrėnai
HDD Jesenice players
Lithuanian expatriate ice hockey people
Lithuanian expatriate sportspeople in Slovenia
Lithuanian expatriate sportspeople in Finland
Lithuanian expatriate sportspeople in Belgium
Lithuanian expatriate sportspeople in France
Lithuanian expatriate sportspeople in the Netherlands
Expatriate ice hockey players in Slovenia
Expatriate ice hockey players in Finland
Expatriate ice hockey players in Belgium
Expatriate ice hockey players in France
Expatriate ice hockey players in the Netherlands
Nijmegen Devils players |
https://en.wikipedia.org/wiki/Grey%20box%20model | In mathematics, statistics, and computational modelling, a grey box model combines a partial theoretical structure with data to complete the model. The theoretical structure may vary from information on the smoothness of results, to models that need only parameter values from data or existing literature. Thus, almost all models are grey box models as opposed to black box where no model form is assumed or white box models that are purely theoretical. Some models assume a special form such as a linear regression or neural network. These have special analysis methods. In particular linear regression techniques are much more efficient than most non-linear techniques. The model can be deterministic or stochastic (i.e. containing random components) depending on its planned use.
Model form
The general case is a non-linear model with a partial theoretical structure and some unknown parts derived from data. Models with unlike theoretical structures need to be evaluated individually, possibly using simulated annealing or genetic algorithms.
Within a particular model structure, parameters or variable parameter relations may need to be found. For a particular structure it is arbitrarily assumed that the data consists of sets of feed vectors f, product vectors p, and operating condition vectors c. Typically c will contain values extracted from f, as well as other values. In many cases a model can be converted to a function of the form:
m(f,p,q)
where the vector function m gives the errors between the data p, and the model predictions. The vector q gives some variable parameters that are the model's unknown parts.
The parameters q vary with the operating conditions c in a manner to be determined. This relation can be specified as q = Ac where A is a matrix of unknown coefficients, and c as in linear regression includes a constant term and possibly transformed values of the original operating conditions to obtain non-linear relations between the original operating conditions and q. It is then a matter of selecting which terms in A are non-zero and assigning their values. The model completion becomes an optimization problem to determine the non-zero values in A that minimizes the error terms m(f,p,Ac) over the data.
Model completion
Once a selection of non-zero values is made, the remaining coefficients in A can be determined by minimizing m(f,p,Ac) over the data with respect to the nonzero values in A, typically by non-linear least squares. Selection of the nonzero terms can be done by optimization methods such as simulated annealing and evolutionary algorithms. Also the non-linear least squares can provide accuracy estimates for the elements of A that can be used to determine if they are significantly different from zero, thus providing a method of term selection.
It is sometimes possible to calculate values of q for each data set, directly or by non-linear least squares. Then the more efficient linear regression can be used to predict q using c th |
https://en.wikipedia.org/wiki/Giovanni%20Alberti%20%28mathematician%29 | Giovanni Alberti (born March 21, 1965) is an Italian mathematician who is active in the fields of calculus of variations, real analysis and geometric measure theory.
Scientific activity
Alberti has studied at Scuola Normale Superiore under the guide of Giuseppe Buttazzo and Ennio De Giorgi; he is professor of mathematics at the University of Pisa. Alberti is mostly known for two remarkable theorems he proved at the beginning of his career, that eventually found applications in various branches of modern mathematical analysis. The first is a very general Lusin type theorem for gradients asserting that every Borel vector field can be realized as the gradient of a continuously differentiable function outside a closed subset of a priori prescribed (small) measure. The second asserts the rank-one property of the distributional derivatives of functions with bounded variation, thereby verifying a conjecture of De Giorgi. This theorem has found several applications, as for instance in the Ambrosio's proof of an open problem posed by Di Perna and Lions concerning the well-posedness of the continuity equation involving BV vector fields. This result is nowadays commonly known as Alberti's rank-one theorem and its proof rests of a very delicate use of sophisticated tools from geometric measure theory; in particular, it makes use of the concept of tangent measure to another measure. Subsequently, Alberti has given contributions to the study of various aspects of Ginzburg-Landau vortices and of the continuity equation.
Recognition
Alberti has been awarded the Caccioppoli prize in 2002 and has been an invited speaker at the fourth European Congress of Mathematics.
References
External links
Website at the University of Pisa
Site of Caccioppoli Prize
Fourth European Congress of Mathematics
1965 births
Living people
20th-century Italian mathematicians
21st-century Italian mathematicians
Academic staff of the University of Pisa
Scuola Normale Superiore di Pisa alumni |
https://en.wikipedia.org/wiki/Vadim%20Cem%C3%AErtan | Vadim Cemîrtan (born 21 July 1987) is a Moldovan footballer who plays as a striker for Florești in the Moldovan Super Liga.
Career statistics
International
References
External links
Profile at Divizia Nationala
1987 births
People from Bender, Moldova
Moldovan men's footballers
Moldovan expatriate men's footballers
Moldova men's international footballers
Living people
Men's association football forwards
FC Tighina players
FC Academia Chișinău players
FC Iskra-Stal players
FC Nistru Otaci players
FC Costuleni players
FC Dinamo-Auto Tiraspol players
FC Dacia Chișinău players
Than Quang Ninh FC players
PFK Nurafshon players
FC Bunyodkor players
FC AGMK players
FC Sfîntul Gheorghe players
FC Zimbru Chișinău players
Ma'an SC players
Uzbekistan Super League players
Moldovan Super Liga players
Jordanian Pro League players
Moldovan expatriate sportspeople in Vietnam
Moldovan expatriate sportspeople in Uzbekistan
Moldovan expatriate sportspeople in Jordan
Expatriate men's footballers in Vietnam
Expatriate men's footballers in Uzbekistan
Expatriate men's footballers in Jordan
Footballers from Transnistria |
https://en.wikipedia.org/wiki/Oleg%20Molla | Oleg Molla (born 22 February 1986, Chișinău, Moldavian SSR) is a Moldavian football striker.
Club statistics
Total matches played in Moldavian First League: 123 matches – 28 goals
References
External links
Profile at Divizia Nationala
Profile at FC Dacia Chișinău
1986 births
Footballers from Chișinău
Moldovan men's footballers
Living people
Men's association football forwards
FC Iskra-Stal players
FC Dacia Chișinău players
FC Sfîntul Gheorghe players
FC Zimbru Chișinău players
FC Tiraspol players
FC Saxan players
FC Spicul Chișcăreni players
Moldovan Super Liga players |
https://en.wikipedia.org/wiki/Ustin%20Cerga | Ustin Cerga (born 25 November 1988, Chișinău, Moldavian SSR) is a Moldavian football goalkeeper who plays for FC Dacia Chișinău.
Club statistics
Total matches played in Moldavian First League: 19 matches – 4 cleansheets
References
External links
Profile at Divizia Nationala
Profile at FC Dacia Chișinău
1988 births
Footballers from Chișinău
Moldovan men's footballers
Living people
Men's association football goalkeepers
FC Sfîntul Gheorghe players |
https://en.wikipedia.org/wiki/Marian%20Stoleru | Marian Stoleru (born 20 November 1988, Chișinău, Moldavian SSR) is a Moldavian football midfielder who plays for German club SV Pars Neu-Isenburg.
Club statistics
Total matches played in Moldavian First League: 63 matches – 8 goals
References
External links
Profile at Divizia Nationala
Marian Stoleru at FuPa
1988 births
Footballers from Chișinău
Moldovan men's footballers
Moldovan expatriate men's footballers
Living people
Men's association football midfielders
FC Dacia Chișinău players
FC Milsami Orhei players
Expatriate men's footballers in Germany
Moldovan expatriate sportspeople in Germany |
https://en.wikipedia.org/wiki/Nicolae%20Nemerenco | Nicolae Nemerenco (born 26 October 1992), is a Moldovan footballer who plays as a forward for Dacia Buiucani.
Club statistics
Total matches played in Moldovan First League: 7 matches – 0 goals
References
External links
1992 births
Living people
Footballers from Chișinău
Moldovan men's footballers
Men's association football forwards
Moldovan Super Liga players
Oman Professional League players
FC Dacia Chișinău players
FC Codru Lozova players
FC Sfîntul Gheorghe players
ACS Foresta Suceava players
Moldovan expatriate men's footballers
Expatriate men's footballers in Oman
Moldovan expatriate sportspeople in Romania
Expatriate men's footballers in Romania
Dacia Buiucani players
Moldovan expatriate sportspeople in Oman |
https://en.wikipedia.org/wiki/Kiril%20Erokhin | Kiril Erokhin (born 22 March 1993, Moscow, Russia) is a Russian football defender who plays for FC Dacia Chișinău.
Club statistics
Total matches played in Moldavian First League: 30 matches – 1 goal
References
External links
Profile at Divizia Nationala
Profile at FC Dacia Chișinău
1993 births
Footballers from Moscow
Moldovan men's footballers
Living people
Men's association football defenders
FC Dacia Chișinău players
FC Zimbru Chișinău players
FC Sfîntul Gheorghe players |
https://en.wikipedia.org/wiki/Elliptic%20pseudoprime | In number theory, a pseudoprime is called an elliptic pseudoprime for (E, P), where E is an elliptic curve defined over the field of rational numbers with complex multiplication by an order in , having equation y2 = x3 + ax + b with a, b integers, P being a point on E and n a natural number such that the Jacobi symbol (−d | n) = −1, if .
The number of elliptic pseudoprimes less than X is bounded above, for large X, by
References
External links
Pseudoprimes |
https://en.wikipedia.org/wiki/Bruce%20Sagan | Bruce E. Sagan (born 1954) is an American Professor of Mathematics at Michigan State University. He specializes in enumerative, algebraic, and topological combinatorics. He is also known as a musician, playing music from Scandinavia and the Balkans.
Early life
Bruce Eli Sagan is the son of Eugene Benjamin Sagan and Arlene Kaufmann Sagan. He grew up in Berkeley, California. He started playing classical violin at a young age under the influence of his mother who was a music teacher and conductor. He received his B.S. in mathematics (1974) from California State University, East Bay (then called California State University, Hayward). He received his Ph.D. in mathematics (1979) from the Massachusetts Institute of Technology. His doctoral thesis "Partially Ordered Sets with Hooklengths – an Algorithmic Approach" was supervised by Richard P. Stanley. He was Stanley's third doctoral student. During his graduate school years he also joined and became music director of the Mandala Folkdance Ensemble.
Mathematical career
Sagan held postdoctoral positions at Université Louis Pasteur (1979–1980), the University of Michigan (1980–1983), University College of Wales, Aberystwyth, Middlebury College (1984–1985), the University of Pennsylvania, and Université du Québec à Montréal (Fall, 1985), before becoming a faculty member at MSU in the Spring of 1986. He has held visiting positions at the Institute for Mathematics and its Applications (Spring, 1988), UCSD (Spring, 1991), the Royal Institute of Technology (1993–1994), MSRI (Winter, 1997), the Isaac Newton Institute (Winter, 2001), Mittag-Leffler Institute (Spring, 2005), and DIMACS (2005–2006). He was also a rotating Program Officer at the National Science Foundation (2007–2010).
Sagan has published over 100 research papers. He has given over 300 talks in North America, Europe, Asia, and Australia. These have included keynote addresses at the Conference on Formal Power Series and Algebraic Combinatorics (2006) and the British Combinatorial Conference (2011). He has graduated 15 Ph.D. students. During his time at Michigan State University, he has won two awards for teaching excellence.
Sagan has been an Editor-in-Chief for the Electronic Journal of Combinatorics since 2004.
Books
Mathematical Essays in Honor of Gian-Carlo Rota (co-edited with Richard P. Stanley), Birkhäuser, Cambridge, 1998, .
The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd edition, Springer-Verlag, New York, 2001, .
Festschrift in Honor of Richard Stanley (special editor), Electronic Journal of Combinatorics, 2004–2006.
Selected papers
.
Musical career
Sagan plays music from the Scandinavian countries and the Balkans on fiddle and native instruments. These include the Swedish nyckelharpa, the Norwegian hardingfele, and the Bulgarian gadulka. In 1985 he and his then wife, Judy Barlas, founded the music and dance camp Scandinavian Week at Buffalo Gap (now known as Nordic F |
https://en.wikipedia.org/wiki/Richard%20Arratia | Richard Alejandro Arratia is a mathematician noted for his work in combinatorics and probability theory.
Contributions
Arratia developed the ideas of interlace polynomials with Béla Bollobás and Gregory Sorkin, found an equivalent formulation of the Stanley–Wilf conjecture as the convergence of a limit, and was the first to investigate the lengths of superpatterns of permutations.
He has also written highly cited papers on the Chen–Stein method on distances between probability distributions, on random walks with exclusion, and on sequence alignment.
He is a coauthor of the book Logarithmic Combinatorial Structures: A Probabilistic Approach.
Education and employment
Arratia earned his Ph.D. in 1979 from the University of Wisconsin–Madison under the supervision of David Griffeath. He is currently a professor of mathematics at the University of Southern California.
Selected publications
Research papers
Books
References
External links
USC faculty page
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
Combinatorialists
Probability theorists
Stuyvesant High School alumni
University of Wisconsin–Madison alumni
University of Southern California faculty |
https://en.wikipedia.org/wiki/Hans%20G%C3%BCnther%20Aach | Hans Günther Aach (2 October 1919 – 4 December 1999) was a German botanist.
Life
Aach was born in Oldenburg. He gained his doctorate in March 1952 in the Faculty of Mathematics and Natural Sciences of the University of Göttingen. In July 1961 he presented his professorial thesis at the University of Cologne. He spent several months as visiting faculty at University of California, Berkeley and Stanford University. On 31 December 1962 he was appointed Extraordinary Professor of Botany at the RWTH Aachen University. From 12 January 1965 he was appointed to the Chair of Botany in the same place and made director of the Botanical Institute. He retired on 1 March 1984.
The emphasis of his scientific work was on proteins in viruses. He collaborated on the Handbuch der Biologie.
Aach died in Aachen in December 1999 at the age of 80.
Works
Abriss der Botanik für Studenten der Medizin und der Naturwissenschaften. Berlin 1948
Über Wachstum und Zusammensetzung von Chlorella pyrenoidosa bei unterschiedlichen Lichtstärken und Nitratmengen. Göttingen 1952
Die Viren. Akademische Verlags-Gesellschaft Athenaion, Konstanz 1960
Zur Konstanz der Aminosäurenzusammensetzung im Eiweißanteil des Tabakmosaikvirus. Köln 1961
Zum Problem der Viruseiweisssynthese in zellfreien Chlorellasystemen. Westdeutscher Verlag, Köln 1968
Literature
Kürschners deutscher Gelehrten-Kalender. Vol. 1, 1966.
Notes and references
External links
1919 births
1999 deaths
20th-century German botanists
People from Oldenburg (city)
Academic staff of RWTH Aachen University
Academic staff of the University of Cologne
University of Göttingen alumni |
https://en.wikipedia.org/wiki/Transfermarkt | Transfermarkt is a German-based website owned by Axel Springer SE that has footballing information, such as scores, results, statistics, transfer news, and fixtures. According to the IVW, it is in the top 25 most visited German websites, and one of the largest sport websites after kicker.de.
The website has scores, results, transfer news, fixtures, and player values. Despite the player values, along with some other facts, being estimates, researchers from the Centre for Economic Performance have found that the "rumours" of player transfers are largely accurate.
These estimated player values are usually updated every few months, considering how the association football market works, these estimates might be slightly lower or higher than what a players' current form and therefore current value might suggest.
History
The website was founded in May 2000, by Matthias Seidel to track players and transfer targets for SV Werder Bremen before expanding to other teams. The website initially focused on players, but gradually expanded to include managers, agents, and other staff. In 2008, Axel Springer publishing house acquired a 51% share in the website. Seidel kept the other 49% of the shares. The English-language version started in 2009.
On 19 May 2014, a relaunch took place for the so-called update to 'version 4'. In the course of this update there were both server-technical as well as data-legal issues, as private data was visible to other users for an indefinite period of time. For 48 hours the site had only a very limited availability, resulting in multiple complaints on Facebook. The biggest criticisms from users was the confusing new design. As a result, Transfermarkt.de publicly apologized for the incidents and issues that were caused during the relaunch.
, the website has 39 million unique monthly visitors and 680,000 registered users.
References
External links
Association football websites
German sport websites
Internet properties established in 2000
2000 establishments in Germany |
https://en.wikipedia.org/wiki/Geometric%20quotient | In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties such that
(i) For each y in Y, the fiber is an orbit of G.
(ii) The topology of Y is the quotient topology: a subset is open if and only if is open.
(iii) For any open subset , is an isomorphism. (Here, k is the base field.)
The notion appears in geometric invariant theory. (i), (ii) say that Y is an orbit space of X in topology. (iii) may also be phrased as an isomorphism of sheaves . In particular, if X is irreducible, then so is Y and : rational functions on Y may be viewed as invariant rational functions on X (i.e., rational-invariants of X).
For example, if H is a closed subgroup of G, then is a geometric quotient. A GIT quotient may or may not be a geometric quotient: but both are categorical quotients, which is unique; in other words, one cannot have both types of quotients (without them being the same).
Relation to other quotients
A geometric quotient is a categorical quotient. This is proved in Mumford's geometric invariant theory.
A geometric quotient is precisely a good quotient whose fibers are orbits of the group.
Examples
The canonical map is a geometric quotient.
If L is a linearized line bundle on an algebraic G-variety X, then, writing for the set of stable points with respect to L, the quotient
is a geometric quotient.
References
Algebraic geometry |
https://en.wikipedia.org/wiki/2016%20Canadian%20census | The 2016 Canadian census was an enumeration of Canadian residents, which counted a population of 35,151,728, a change from its 2011 population of 33,476,688. The census, conducted by Statistics Canada, was Canada's seventh quinquennial census. The official census day was May 10, 2016. Census web access codes began arriving in the mail on May 2, 2016. The 2016 census marked the reinstatement of the mandatory long-form census, which had been dropped in favour of the voluntary National Household Survey for the 2011 census. With a response rate of 98.4%, this census is said to be the best one ever recorded since the 1666 census of New France. This census was succeeded by Canada's 2021 census.
Planning
Consultation with census data users, clients, stakeholders and other interested parties closed in November 2012. Qualitative content testing, which involved soliciting feedback regarding the questionnaire and tests responses to its questions, was scheduled for the fall of 2013, with more extensive testing occurring in May 2014. Statistics Canada was scheduled to submit its census content recommendations for review by the Parliament of Canada in December 2014 for subsequent final approval by the Cabinet of Canada.
On November 5, 2015, during the first Liberal caucus meeting after forming a majority government, the party announced that it would reinstate the mandatory long-form census, starting in 2016. By early January 2016, Statistics Canada had announced a need for 35,000 people to complete this survey to commence in May.
Data release schedule
The release dates for geography products from the 2016 census were:
November 16, 2016, for boundary files (first edition), road network files, hydrography files, reference maps (first edition), attribute information products (correspondence files), and reference guides and documents (first edition); and
February 8, 2017, for boundary files (second edition), reference maps (second edition), attribute information products (GeoSuite and geographic attribute file), and reference guides and documents (second edition).
The release dates for data by release topic from the 2016 census are:
February 8, 2017, for population and dwelling counts;
May 3, 2017, for age and sex, type of dwelling;
May 10, 2017, for Census of Agriculture;
August 2, 2017, for language and families, households and marital status;
September 13, 2017, for income;
October 25, 2017, for immigration and ethnocultural diversity, housing and Aboriginal peoples; and
November 29, 2017, for education, labour, journey to work, language of work and mobility and migration.
Enumeration
Portions of Canada's three territories and remote areas within Alberta, Labrador, Manitoba, Quebec and Saskatchewan were subject to early enumeration between February 1, 2016, and March 31, 2016. Enumeration of the balance of Canada began on May 2, 2016, with the unveiling of the online census questionnaire, eight days prior to the official census day of May 10, 2016. Be |
https://en.wikipedia.org/wiki/Kv%C4%9Bta%20Peschke%20career%20statistics | This is a list of the main career statistics of professional Czech tennis player Květa Peschke.
Performance timelines
Singles
Doubles
Mixed doubles
Notes
At the 2008 Australian Open, Peschke and Martin Damm withdrew before their quarterfinal match, this is not counted as a loss.
At the 2008 Wimbledon Championships, Peschke and Pavel Vízner received a third round walkover, this is not counted as a win.
At the 2013 Australian Open, Peschke and Marcin Matkowski received a second round walkover, this is not counted as a win.
At the 2021 Wimbledon Championships, Peschke and Kevin Krawietz received second and third round walkovers, these are not counted as wins.
Grand Slam tournament finals
Doubles: 3 (1 title, 2 runner-ups)
Mixed doubles: 3 (3 runner-ups)
Other significant finals
Year-end championships finals
Doubles: 3 runner-ups
WTA career finals
Singles: 2 (1 title, 1 runner-up)
Doubles: 78 (36 titles, 42 runner-ups)
ITF Circuit finals
Singles: 16 (10 titles, 6 runner–ups)
Doubles: 12 (8–4)
External links
Peschke |
https://en.wikipedia.org/wiki/Prestack | In algebraic geometry, a prestack F over a category C equipped with some Grothendieck topology is a category together with a functor p: F → C satisfying a certain lifting condition and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object.
Prestacks that appear in nature are typically stacks but some naively constructed prestacks (e.g., groupoid scheme or the prestack of projectivized vector bundles) may not be stacks. Prestacks may be studied on their own or passed to stacks.
Since a stack is a prestack, all the results on prestacks are valid for stacks as well. Throughout the article, we work with a fixed base category C; for example, C can be the category of all schemes over some fixed scheme equipped with some Grothendieck topology.
Informal definition
Let F be a category and suppose it is fibered over C through the functor ; this means that one can construct pullbacks along morphisms in C, up to canonical isomorphisms.
Given an object U in C and objects x, y in , for each morphism in C, after fixing pullbacks , we let
be the set of all morphisms from to ; here, the bracket means we canonically identify different Hom sets resulting from different choices of pullbacks. For each over U, define the restriction map from f to g:
to be the composition
where a canonical isomorphism is used to get the = on the right. Then is a presheaf on the slice category , the category of all morphisms in C with target U.
By definition, F is a prestack if, for each pair x, y, is a sheaf of sets with respect to the induced Grothendieck topology on .
This definition can be equivalently phrased as follows. First, for each covering family , we "define" the category as a category where: writing , etc.,
an object is a set of pairs consisting of objects in and isomorphisms that satisfy the cocycle condition:
a morphism consists of in such that
An object of this category is called a descent datum. This category is not well-defined; the issue is that the pullbacks are determined only up to canonical isomorphisms; similarly fiber products are defined only up to canonical isomorphisms, despite the notational practice to the contrary. In practice, one simply makes some canonical identifications of pullbacks, their compositions, fiber products, etc.; up to such identifications, the above category is well-defined (in other words, it is defined up to a canonical equivalence of categories.)
There is an obvious functor that sends an object to the descent datum that it defines. One can then say: F is a prestack if and only if, for each covering family , the functor is fully faithful. A statement like this is independent of choices of canonical identifications mentioned early.
The essential image of consists precisely of effective descent data (just the definition of "effective"). Thus, F is a stack if a |
https://en.wikipedia.org/wiki/Hasse%20derivative | In mathematics, the Hasse derivative is a generalisation of the derivative which allows the formulation of Taylor's theorem in coordinate rings of algebraic varieties.
Definition
Let k[X] be a polynomial ring over a field k. The r-th Hasse derivative of Xn is
if n ≥ r and zero otherwise. In characteristic zero we have
Properties
The Hasse derivative is a generalized derivation on k[X] and extends to a generalized derivation on the function field k(X), satisfying an analogue of the product rule
and an analogue of the chain rule. Note that the are not themselves derivations in general, but are closely related.
A form of Taylor's theorem holds for a function f defined in terms of a local parameter t on an algebraic variety:
References
Differential algebra |
https://en.wikipedia.org/wiki/Barban%E2%80%93Davenport%E2%80%93Halberstam%20theorem | In mathematics, the Barban–Davenport–Halberstam theorem is a statement about the distribution of prime numbers in an arithmetic progression. It is known that in the long run primes are distributed equally across possible progressions with the same difference. Theorems of the Barban–Davenport–Halberstam type give estimates for the error term, determining how close to uniform the distributions are.
Statement
Let a be coprime to q and
be a weighted count of primes in the arithmetic progression a mod q. We have
where φ is Euler's totient function and the error term E is small compared to x. We take a sum of squares of error terms
Then we have
for and every positive A, where O is Landau's Big O notation.
This form of the theorem is due to Gallagher. The result of Barban is valid only for for some B depending on A, and the result of Davenport–Halberstam has B = A + 5.
See also
Bombieri–Vinogradov theorem
Elliott–Halberstam conjecture
References
Theorems in analytic number theory |
https://en.wikipedia.org/wiki/Concept%20image%20and%20concept%20definition | In mathematics education, concept image and concept definition are two ways of understanding a mathematical concept.
The terms were introduced by . They define a concept image as such:
"We shall use the term concept image to describe the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes. It is built up over the years through experiences of all kinds, changing as the individual meets new stimuli and matures."
A concept definition is similar to the usual notion of a definition in mathematics, with the distinction that it is personal to an individual:
"a personal concept definition can differ from a formal concept definition, the latter being a concept definition which is accepted by the mathematical community at large."
Bibliography
References
External links
David Tall - CONCEPT IMAGE AND CONCEPT DEFINITION
Mathematics education
Definition |
https://en.wikipedia.org/wiki/Component%20analysis%20%28statistics%29 | Component analysis is the analysis of two or more independent variables which comprise a treatment modality. It is also known as a dismantling study.
The chief purpose of the component analysis is to identify the component which is efficacious in changing behavior, if a singular component exists.
Eliminating ineffective or less effective components may help with improving social validity, reducing aversive elements, improving generalization and maintenance, as well as administrative efficacy.
It is also a required skill for the BCBA.
References
Regression analysis
Research |
https://en.wikipedia.org/wiki/Keel%E2%80%93Mori%20theorem | In algebraic geometry, the Keel–Mori theorem gives conditions for the existence of the quotient of an algebraic space by a group. The theorem was proved by .
A consequence of the Keel–Mori theorem is the existence of a coarse moduli space of a separated algebraic stack, which is roughly a "best possible" approximation to the stack by a separated algebraic space.
Statement
All algebraic spaces are assumed of finite type over a locally Noetherian base. Suppose that j:R→X×X is a flat groupoid whose stabilizer j−1Δ is finite over X (where Δ is the diagonal of X×X). The Keel–Mori theorem states that there is an algebraic space that is a geometric and uniform categorical quotient of X by j, which is separated if j is finite.
A corollary is that for any flat group scheme G acting properly on an algebraic space X with finite stabilizers there is a uniform geometric and uniform categorical quotient X/G which is a separated algebraic space. proved a slightly weaker version of this and described several applications.
References
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Six%20operations | In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism. It originally sprang from the relations in étale cohomology that arise from a morphism of schemes . The basic insight was that many of the elementary facts relating cohomology on X and Y were formal consequences of a small number of axioms. These axioms hold in many cases completely unrelated to the original context, and therefore the formal consequences also hold. The six operations formalism has since been shown to apply to contexts such as D-modules on algebraic varieties, sheaves on locally compact topological spaces, and motives.
The operations
The operations are six functors. Usually these are functors between derived categories and so are actually left and right derived functors.
the direct image
the inverse image
the proper (or extraordinary) direct image
the proper (or extraordinary) inverse image
internal tensor product
internal Hom
The functors and form an adjoint functor pair, as do and . Similarly, internal tensor product is left adjoint to internal Hom.
Six operations in étale cohomology
Let be a morphism of schemes. The morphism f induces several functors. Specifically, it gives adjoint functors f* and f* between the categories of sheaves on X and Y, and it gives the functor f! of direct image with proper support. In the derived category, Rf! admits a right adjoint f!. Finally, when working with abelian sheaves, there is a tensor product functor ⊗ and an internal Hom functor, and these are adjoint. The six operations are the corresponding functors on the derived category: , , , , , and .
Suppose that we restrict ourselves to a category of -adic torsion sheaves, where is coprime to the characteristic of X and of Y. In SGA 4 III, Grothendieck and Artin proved that if f is smooth of relative dimension d, then Lf* is isomorphic to , where denote the dth inverse Tate twist and denotes a shift in degree by . Furthermore, suppose that f is separated and of finite type. If is another morphism of schemes, if denotes the base change of X by g, and if f′ and g′ denote the base changes of f and g by g and f, respectively, then there exist natural isomorphisms:
Again assuming that f is separated and of finite type, for any objects M in the derived category of X and N in the derived category of Y, there exist natural isomorphisms:
If i is a closed immersion of Z into S with complementary open immersion j, then there is a distinguished triangle in the derived category:
where the first two maps are the counit and unit, respectively of the adjunctions. If Z and S are regular, then there is an isomorphism:
where and are the units of the tensor product operations (which vary depending on which category of -adic torsion sheaves is under consideration).
If S is regular and , and if K is an invertible object in the derived category on S with resp |
https://en.wikipedia.org/wiki/Categorical%20quotient | In algebraic geometry, given a category C, a categorical quotient of an object X with action of a group G is a morphism that
(i) is invariant; i.e., where is the given group action and p2 is the projection.
(ii) satisfies the universal property: any morphism satisfying (i) uniquely factors through .
One of the main motivations for the development of geometric invariant theory was the construction of a categorical quotient for varieties or schemes.
Note need not be surjective. Also, if it exists, a categorical quotient is unique up to a canonical isomorphism. In practice, one takes C to be the category of varieties or the category of schemes over a fixed scheme. A categorical quotient is a universal categorical quotient if it is stable under base change: for any , is a categorical quotient.
A basic result is that geometric quotients (e.g., ) and GIT quotients (e.g., ) are categorical quotients.
References
Mumford, David; Fogarty, J.; Kirwan, F. Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp.
See also
Quotient by an equivalence relation
Quotient stack
Algebraic geometry |
https://en.wikipedia.org/wiki/Antonio%20Ambrosetti | Antonio Ambrosetti (25 November 1944 – 20 November 2020) was an Italian mathematician who worked in the fields of partial differential equations and calculus of variations.
Scientific activity
Ambrosetti studied at the University of Padua and was professor of mathematics at the International School for Advanced Studies. He is known for his basic work on topological methods in the calculus of variations. These provide tools aimed at establishing the existence of solutions to variational problems when classical direct methods of the calculus of variations cannot be applied. In particular, the so-called mountain pass theorem he established with Paul Rabinowitz is nowadays a classical tool in the context of nonlinear analysis problems.
Recognition
Ambrosetti has been awarded the Caccioppoli prize in 1982, and the Amerio Prize by the Istituto Lombardo Accademia di Scienze e Lettere in 2008. Jointly with Andrea Malchiodi, Ambrosetti has been awarded the 2005 edition of the Ferran Sunyer i Balaguer prize. In 1983 he has been invited speaker at the International Congress of Mathematicians and he was fellow of the Accedemia Nazionale dei Lincei.
References
External links
Website at the International School for Advanced Studies
Site of Caccioppoli Prize
1944 births
2020 deaths
21st-century Italian mathematicians
University of Padua alumni
People from Bari
Variational analysts
Academic staff of the Scuola Normale Superiore di Pisa |
https://en.wikipedia.org/wiki/2013%E2%80%9314%20FK%20Partizan%20season | The 2013–14 season is FK Partizan's 8th season in Serbian SuperLiga. This article shows player statistics and all matches (official and friendly) that the club have and will play during the 2013–14 season.
Players
Squad statistics
Top scorers
Includes all competitive matches. The list is sorted by shirt number when total goals are equal.
Starting 11
Transfers
In
Out
For recent transfers, see List of Serbian football transfers summer 2013 and List of Serbian football transfers winter 2013-14.
Competitions
Overview
Serbian SuperLiga
League table
Results and positions by round
Matches
Serbian Cup
UEFA Champions League
Qualifying phase
UEFA Europa League
Play-off round
Friendlies
Generali Deyna Cup 2013
Sponsors
See also
List of FK Partizan seasons
References
External links
Official website
Partizanopedia 2013-14 (in Serbian)
FK Partizan seasons
Partizan
Partizan
Partizan |
https://en.wikipedia.org/wiki/List%20of%20open-source%20software%20for%20mathematics | This is a list of open-source software to be used for high-order mathematical calculations. This software has played an important role in the field of mathematics. Open-source software in mathematics has become pivotal in education because of the high cost of textbooks.
Computer algebra systems
A computer algebra system is a type of software set that is used in manipulation of mathematical formulae. The principal objective of a computer algebra system is to systematize monotonous and sometimes problematic algebraic manipulation tasks. The primary difference between a computer algebra system and a traditional calculator is the ability to deal with equations symbolically rather than numerically. The precise uses and capabilities of these systems differ greatly from one system to another, yet their purpose remains the same: manipulation of symbolic equations. Computer algebra system often include facilities for graphing equations and provide a programming language for the users' own procedures.
Axiom
Axiom is a general-purpose computer algebra system. It has been in development since 1971 by IBM, and was originally named scratchpad. Richard Jenks originally headed it but over the years Barry Trager who then shaped the direction of the scratchpad project took over the project. It was eventually sold to the Numerical Algorithms Group (NAG) and was renamed Axiom. After a failure to launch as a product, NAG decided to release it as a free software in 2001 with more than 300 man-years worth of research involved. Axiom is licensed under a Modified BSD license.
Cadabra
A Computer Algebra System designed for the solution of problems in field theory. An unpublished computational program written in Pascal called Abra inspired this open-source software. Abra was originally designed for physicists to compute problems present in quantum mechanics. Kespers Peeters then decided to write a similar program in C computing language rather than Pascal, which he renamed Cadabra. However, Cadabra has been expanded for a wider range of uses, it is no longer restricted to physicists.
CoCoA
CoCoA (COmputations in COmmutative Algebra) is open-source software used for computing multivariate polynomials and initiated in 1987. Originally written in Pascal, CoCoA was later translated into C.
GAP
GAP was initiated by RWTH Aachen University in 1986. This was the case until in 1997 when they decided to co-develop GAP further with CIRCA (Centre for Research in Computational Algebra). Unlike MAXIMA and Axiom, GAP is a system for computational discrete algebra with particular emphasis on computational group theory. In March 2005 the GAP Council and the GAP developers have agreed that status and responsibilities of "GAP Headquarters" should be passed to an equal collaboration of a number of "GAP Centres", where there is permanent staff involvement and an element of collective or organizational commitment, while fully recognizing the vital contributions of many individuals outs |
https://en.wikipedia.org/wiki/N-transform | In mathematics, the Natural transform is an integral transform similar to the Laplace transform and Sumudu transform, introduced by Zafar Hayat Khan in 2008. It converges to both Laplace and Sumudu transform just by changing variables. Given the convergence to the Laplace and Sumudu transforms, the N-transform inherits all the applied aspects of the both transforms. Most recently, F. B. M. Belgacem has renamed it the natural transform and has proposed a detail theory and applications.
Formal definition
The natural transform of a function f(t), defined for all real numbers t ≥ 0, is the function R(u, s), defined by:
Khan showed that the above integral converges to Laplace transform when u = 1, and into Sumudu transform for s = 1.
See also
References
Integral transforms |
https://en.wikipedia.org/wiki/D%C3%A1vid%20Pal%C3%A1sthy | Dávid Palásthy (born 10 May 1990) is a professional Hungarian footballer who currently plays for Dunaharaszti MTK.
Club statistics
Updated to games played as of 2 June 2013.
References
External links
HLSZ
1990 births
Living people
Sportspeople from Vác
Hungarian men's footballers
Men's association football goalkeepers
Vác FC players
Ceglédi VSE footballers
Zalaegerszegi TE players
Egri FC players
Soroksár SC players
III. Kerületi TVE footballers
BKV Előre SC footballers
Dunaharaszti MTK players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Footballers from Pest County |
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