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Multiplicative Combinatorics
In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis. Scope Arithmetic combinatorics is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive combinatorics is the special case when only the operations of addition and subtraction are involved. Ben Green explains arithmetic combinatorics in his review of "Additive Combinatorics" by Tao and Vu. Important results Szemerédi's theorem Szemerédi's theorem is a result in arithmetic combinatorics concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured. that every set of integers ''A'' with positive natural density contains a ''k'' term arithmetic progression for every ''k''. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem. Green–Tao theorem and extensions The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green–Tao Theorem
In number theory, the Green–Tao theorem, proven by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k, there exist arithmetic progressions of primes with k terms. The proof is an extension of Szemerédi's theorem. The problem can be traced back to investigations of Lagrange and Waring from around 1770.. Statement Let \pi(N) denote the number of primes less than or equal to N. If A is a subset of the prime numbers such that : \limsup_ \frac>0, then for all positive integers k, the set A contains infinitely many arithmetic progressions of length k. In particular, the entire set of prime numbers contains arbitrarily long arithmetic progressions. In their later work on the generalized Hardy–Littlewood conjecture, Green and Tao stated and conditionally proved the asymptotic formula : (\mathfrak_k + o(1))\frac for the number of ''k'' tuples of primes p_1 < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Product Set
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of sets, also known as an -fold Cartesian product, which can be represented by an -dimensional array, where each element is an -tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Set-theoretic definition A rigorous definition of the Cartesian product requir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Set
In combinatorics, a (v,k,\lambda) difference set is a subset D of cardinality, size k of a group (mathematics), group G of order of a group, order v such that every non-identity element, identity element of G can be expressed as a product d_1d_2^ of elements of D in exactly \lambda ways. A difference set D is said to be ''cyclic'', ''abelian'', ''non-abelian'', etc., if the group G has the corresponding property. A difference set with \lambda = 1 is sometimes called ''planar'' or ''simple''. If G is an abelian group written in additive notation, the defining condition is that every non-zero element of G can be written as a ''difference'' of elements of D in exactly \lambda ways. The term "difference set" arises in this way. Basic facts * A simple counting argument shows that there are exactly k^2-k pairs of elements from D that will yield nonidentity elements, so every difference set must satisfy the equation k^2-k=(v-1)\lambda. * If D is a difference set and g \in G, then gD=\ is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sumset
In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets A and B of an abelian group G (written additively) is defined to be the set of all sums of an element from A with an element from B. That is, :A + B = \. The n-fold iterated sumset of A is :nA = A + \cdots + A, where there are n summands. Many of the questions and results of additive combinatorics and additive number theory can be phrased in terms of sumsets. For example, Lagrange's four-square theorem can be written succinctly in the form :4\,\Box = \mathbb, where \Box is the set of square numbers. A subject that has received a fair amount of study is that of sets with ''small doubling'', where the size of the set A+A is small (compared to the size of A); see for example Freiman's theorem. See also *Restricted sumset * Sidon set *Sum-free set * Schnirelmann density *Shapley–Folkman lemma *X + Y sorting X, or x, is the twenty-fourth Letter (alphabet), letter of the Latin alphab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov's Theorem On Groups Of Polynomial Growth
In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of ''polynomial'' growth, as those groups which have nilpotent subgroups of finite index. Statement The growth rate of a group is a well-defined notion from asymptotic analysis. To say that a finitely generated group has polynomial growth means the number of elements of length at most ''n'' (relative to a symmetric generating set) is bounded above by a polynomial function ''p''(''n''). The ''order of growth'' is then the least degree of any such polynomial function ''p''. A nilpotent group ''G'' is a group with a lower central series terminating in the identity subgroup. Gromov's theorem states that a finitely generated group has polynomial growth if and only if it has a nilpotent subgroup that is of finite index. Growth rates of nilpotent groups There is a vast literature on growth rates, leading up to Gromov's theorem. An earli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freiman's Theorem
In additive combinatorics, a discipline within mathematics, Freiman's theorem is a central result which indicates the approximate structure of sets whose sumset is small. It roughly states that if , A+A, /, A, is small, then A can be contained in a small generalized arithmetic progression. Statement If A is a finite subset of \mathbb with , A+A, \le K, A, , then A is contained in a generalized arithmetic progression of dimension at most d(K) and size at most f(K), A, , where d(K) and f(K) are constants depending only on K. Examples For a finite set A of integers, it is always true that :, A + A, \ge 2, A, -1, with equality precisely when A is an arithmetic progression. More generally, suppose A is a subset of a finite proper generalized arithmetic progression P of dimension d such that , P, \le C, A, for some real C \ge 1. Then , P+P, \le 2^d , P, , so that :, A+A, \le , P+P, \le 2^d , P, \le C2^d, A, . History of Freiman's theorem This result is due to Gregory Freim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximate Group
In mathematics, an approximate group is a subset of a group which behaves like a subgroup "up to a constant error", in a precise quantitative sense (so the term approximate subgroup may be more correct). For example, it is required that the set of products of elements in the subset be not much bigger than the subset itself (while for a subgroup it is required that they be equal). The notion was introduced in the 2010s but can be traced to older sources in additive combinatorics. Formal definition Let G be a group and K \ge 1; for two subsets X, Y \subset G we denote by X\cdot Y the set of all products xy, x\in X, y\in Y. A non-empty subset A \subset G is a ''K-approximate subgroup'' of G if: # It is symmetric, that is if g \in A then g^ \in A; # There exists a subset X \subset G of cardinality , X, \le K such that A \cdot A \subset X\cdot A. It is immediately verified that a finite 1-approximate subgroup is the same thing as a genuine subgroup. Of course this definition is on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Publications Mathématiques De L'IHÉS
''Publications Mathématiques de l'IHÉS'' is a peer-reviewed mathematical journal. It is published by Springer Science+Business Media on behalf of the Institut des Hautes Études Scientifiques, with the help of the Centre National de la Recherche Scientifique. The journal was established in 1959 and was published at irregular intervals, from one to five volumes a year. It is now biannual. The editor-in-chief is Sébastien Boucksom (CNRS, Institut de Mathématique de Jussieu). See also *''Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...'' *'' Journal of the American Mathematical Society'' *'' Inventiones Mathematicae'' External links * Back issues from 1959 to 2010 Mathematics journals Academic journals established in 1959 Springer Science+Business Me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emmanuel Breuillard
Emmanuel Breuillard (born 25 June 1977) is a French mathematician. He was the Sadleirian Professor of Pure Mathematics in the Department of Pure Mathematics and Mathematical Statistics (DPMMS) at the University of Cambridge, and is now Professor of Pure Mathematics at the Mathematical Institute, University of Oxford as of January 1, 2022. He had previously been professor at Paris-Sud 11 University. In 2012, he won an EMS Prize for his contributions to combinatorics and other fields. His area of research has been in group theoretic aspects of geometry, number theory and combinatorics. In 2014 he was an invited speaker at the International Congress of Mathematicians in Seoul. He was elected a member of the Academia Europaea in 2021 and a Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the Fellows of the Royal Society of London to individuals who have made a "substantial contribution to the improvement of natural sc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer-valued Polynomial
In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P(t) is a polynomial whose value P(n) is an integer for every integer ''n''. Every polynomial with integer coefficients is integer-valued, but the converse is not true. For example, the polynomial : P(t) = \frac t^2 + \frac t=\fract(t+1) takes on integer values whenever ''t'' is an integer. That is because one of ''t'' and t + 1 must be an even number. (The values this polynomial takes are the triangular numbers.) Integer-valued polynomials are objects of study in their own right in algebra, and frequently appear in algebraic topology.. See in particular pp. 213–214. Classification The class of integer-valued polynomials was described fully by . Inside the polynomial ring \Q /math> of polynomials with rational number coefficients, the subring of integer-valued polynomials is a free abelian group. It has as basis the polynomials :P_k(t) = t(t-1)\cdots (t-k+1)/k! for k = 0,1,2, \dots, i.e. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Mathematica
''Acta Mathematica'' is a peer-reviewed open-access scientific journal covering research in all fields of mathematics. According to Cédric Villani, this journal is "considered by many to be the most prestigious of all mathematical research journals".. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 4.273, ranking it 5th out of 330 journals in the category "Mathematics". Publication history The journal was established by Gösta Mittag-Leffler in 1882 and is published by Institut Mittag-Leffler, a research institute for mathematics belonging to the Royal Swedish Academy of Sciences. The journal was printed and distributed by Springer from 2006 to 2016. Since 2017, Acta Mathematica has been published electronically and in print by International Press. Its electronic version is open access without publishing fees. Poincaré episode The journal's "most famous episode" (according to Villani) concerns Henri Poincaré, who won a prize offered in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
