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Ergodic Ramsey Theory
Ergodic Ramsey theory is a branch of mathematics where problems motivated by additive combinatorics are proven using ergodic theory. History Ergodic Ramsey theory arose shortly after Endre Szemerédi's proof that a set of positive upper density contains arbitrarily long arithmetic progressions, when Hillel Furstenberg gave a new proof of this theorem using ergodic theory. It has since produced combinatorial results, some of which have yet to be obtained by other means, and has also given a deeper understanding of the structure of measure-preserving dynamical systems. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thick Set
In mathematics, a thick set is a set of integers that contains arbitrarily long intervals. That is, given a thick set T, for every p \in \mathbb, there is some n \in \mathbb such that \ \subset T. Examples Trivially \mathbb is a thick set. Other well-known sets that are thick include non-primes and non-squares. Thick sets can also be sparse, for example: \bigcup_ \. Generalisations The notion of a thick set can also be defined more generally for a semigroup, as follows. Given a semigroup (S, \cdot) and A \subseteq S, A is said to be ''thick'' if for any finite subset F \subseteq S, there exists x \in S such that F \cdot x = \ \subseteq A. It can be verified that when the semigroup under consideration is the natural numbers \mathbb{N} with the addition operation +, this definition is equivalent to the one given above. See also * Cofinal (mathematics) * Cofiniteness * Ergodic Ramsey theory * Piecewise syndetic set * Syndetic set References * J. McLeod,Some Notions of Siz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Syndetic Set
In mathematics, a syndetic set is a subset of the natural numbers having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded. Definition A set S \sub \mathbb is called syndetic if for some finite subset F of \mathbb :\bigcup_ (S-n) = \mathbb where S-n = \. Thus syndetic sets have "bounded gaps"; for a syndetic set S, there is an integer p=p(S) such that , a+1, a+2, ... , a+p\bigcap S \neq \emptyset for any a \in \mathbb. See also * Ergodic Ramsey theory * Piecewise syndetic set * Thick set References * * * {{cite journal , last1=Bergelson , first1=Vitaly , authorlink1=Vitaly Bergelson , last2=Hindman , first2=Neil , title=Partition regular structures contained in large sets are abundant , journal=Journal of Combinatorial Theory The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramsey Theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a question of the form: "how big must some structure be to guarantee that a particular property holds?" More specifically, Ron Graham described Ramsey theory as a "branch of combinatorics". Examples A typical result in Ramsey theory starts with some mathematical structure that is then cut into pieces. How big must the original structure be in order to ensure that at least one of the pieces has a given interesting property? This idea can be defined as partition regularity. For example, consider a complete graph of order ''n''; that is, there are ''n'' vertices and each vertex is connected to every other vertex by an edge. A complete graph of order 3 is called a triangle. Now colour each edge either red or blue. How large must ''n'' be i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Syndetic Set
In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers. A set S \sub \mathbb is called ''piecewise syndetic'' if there exists a finite subset ''G'' of \mathbb such that for every finite subset ''F'' of \mathbb there exists an x \in \mathbb such that :x+F \subset \bigcup_ (S-n) where S-n = \. Equivalently, ''S'' is piecewise syndetic if there is a constant ''b'' such that there are arbitrarily long intervals of \mathbb where the gaps in ''S'' are bounded by ''b''. Properties * A set is piecewise syndetic if and only if it is the intersection of a syndetic set and a thick set. * If ''S'' is piecewise syndetic then ''S'' contains arbitrarily long arithmetic progressions. * A set ''S'' is piecewise syndetic if and only if there exists some ultrafilter ''U'' which contains ''S'' and ''U'' is in the smallest two-sided ideal of \beta \mathbb, the Stone–Čech compactification of the natural numbers. * Partition regularity: if S is piecewise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IP Set
In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set. The finite sums of a set ''D'' of natural numbers are all those numbers that can be obtained by adding up the elements of some finite nonempty subset of ''D''. The set of all finite sums over ''D'' is often denoted as FS(''D''). Slightly more generally, for a sequence of natural numbers (''n''i), one can consider the set of finite sums FS((''n''i)), consisting of the sums of all finite length subsequences of (''n''i). A set ''A'' of natural numbers is an IP set if there exists an infinite set ''D'' such that FS(''D'') is a subset of ''A''. Equivalently, one may require that ''A'' contains all finite sums FS((''n''i)) of a sequence (''n''i). Some authors give a slightly different definition of IP sets: They require that FS(''D'') equal ''A'' instead of just being a subset. The term IP set was coined by Hillel Furstenberg and Benjamin Weiss to abbreviate "infinite-dimensiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal D'Analyse Mathématique
The ''Journal d'Analyse Mathématique'' is a triannual peer-reviewed scientific journal published by Magnes Press ( Hebrew University of Jerusalem). It was established in 1951 by Binyamin Amirà. It covers research in mathematics, especially classical analysis and related areas such as complex function theory, ergodic theory, functional analysis, harmonic analysis, partial differential equations, and quasiconformal mapping. Abstracting and indexing The journal is abstracted and indexed in: *MathSciNet *Science Citation Index Expanded *Scopus * ZbMATH Open According to the ''Journal Citation Reports'', the journal has a 2021 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 1.132. References External links *{{Official website, 1=https://www.springer.com/mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van Der Waerden's Theorem
Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory. Van der Waerden's theorem states that for any given positive integers ''r'' and ''k'', there is some number ''N'' such that if the integers are colored, each with one of ''r'' different colors, then there are at least ''k'' integers in arithmetic progression whose elements are of the same color. The least such ''N'' is the Van der Waerden number ''W''(''r'', ''k''), named after the Dutch mathematician B. L. van der Waerden. Example For example, when ''r'' = 2, you have two colors, say and . ''W''(2, 3) is bigger than 8, because you can color the integers from like this: and no three integers of the same color form an arithmetic progression. But you can't add a ninth integer to the end without creating such a progression. If you add a , then the , , and are in arithmetic progression. Alternatively, if you add a , then the , , and are in arithmetic progression. In fact, there i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Density
In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the desired subset when combing through the interval as ''n '' grows large. Intuitively, it is thought that there are more positive integers than perfect squares, since every perfect square is already positive, and many other positive integers exist besides. However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless if one goes through the natural numbers, the squares become increasingly scarce. The notion of natural density makes this intuition precise for many, but not all, subsets of the naturals (see Schnirelmann density, which is similar to natural density but defined for all subsets of \mathbb). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pál Turán
Pál Turán (; 18 August 1910 – 26 September 1976) also known as Paul Turán, was a Hungarian mathematician who worked primarily in extremal combinatorics. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting 46 years and resulting in 28 joint papers. Life and education Turán was born into a Jewish family in Budapest on 18 August 1910.At the same period of time, Turán and Erdős were famous answerers in the journal '' KöMaL''. He received a teaching degree at the University of Budapest in 1933 and the PhD degree under Lipót Fejér in 1935 at Eötvös Loránd University. As a Jew, he fell victim to numerus clausus, and could not get a university job for several years. He was sent to labour service at various times from 1940-44. He is said to have been recognized and perhaps protected by a fascist guard, who, as a mathematics student, had admired Turán's work. Turán became associate professor at the University of Budapest in 1945 and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Combinatorics
Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are ''inverse problems'': given the size of the sumset ''A'' + ''B'' is small, what can we say about the structures of A and B? In the case of the integers, the classical Freiman's theorem provides a partial answer to this question in terms of multi-dimensional arithmetic progressions. Another typical problem is to find a lower bound for , A + B, in terms of , A, and , B, . This can be viewed as an inverse problem with the given information that , A+B, is sufficiently small and the structural conclusion is then of the form that either A or B is the empty set; however, in literature, such problems are sometimes considered to be direct problems as well. Examples of this type include the Erdős–Heilbronn Conjecture (for a restricted sumset) and the Cauchy–Davenport Theorem. The methods used for tackling such questions often come from many different f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
