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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 Size i ... [...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 points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, ''Menge'', was coined by Bernard Bolzano in his work ''Paradoxes of the Infinite''. Georg Cantor, one of the founders of set theory, gave the following defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other examples of intervals are the set of numbers such that , the set of all real numbers \R, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "length" (or "measure" or "size") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the '' cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an ''infinite set''. For example, the set of all positive integers is infinite: :\. Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set is called finite if there exists a bijection :f\colon S\to\ for some natural number . The number is the set's cardinality, denoted as . The empty set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cofinal (mathematics)
In mathematics, a subset B \subseteq A of a preordered set (A, \leq) is said to be cofinal or frequent in A if for every a \in A, it is possible to find an element b in B that is "larger than a" (explicitly, "larger than a" means a \leq b). Cofinal subsets are very important in the theory of directed sets and nets, where “ cofinal subnet” is the appropriate generalization of "subsequence". They are also important in order theory, including the theory of cardinal numbers, where the minimum possible cardinality of a cofinal subset of A is referred to as the cofinality of A. Definitions Let \,\leq\, be a homogeneous binary relation on a set A. A subset B \subseteq A is said to be or with respect to \,\leq\, if it satisfies the following condition: :For every a \in A, there exists some b \in B that a \leq b. A subset that is not frequent is called . This definition is most commonly applied when (A, \leq) is a directed set, which is a preordered set with additiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cofiniteness
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocountable. These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product topology or direct sum. This use of the prefix "" to describe a property possessed by a set's mplement is consistent with its use in other terms such as " meagre set". Boolean algebras The set of all subsets of X that are either finite or cofinite forms a Boolean algebra, which means that it is closed under the operations of union, intersection, and complementation. This Boolean algebra is the on X. A Boolean algebra A has a unique non-principal ultrafilter (that is, a maximal filter not generated by a single element of the algebra) if and only if there exists an infinite set X such that A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 sy ... [...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'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vitaly Bergelson
Vitaly Bergelson (born 1950 in Kiev) is a mathematical researcher and professor at Ohio State University in Columbus, Ohio. His research focuses on ergodic theory and combinatorics. Bergelson received his Ph.D in 1984 under Hillel Furstenberg at the Hebrew University of Jerusalem. He gave an invited address at the International Congress of Mathematicians in 2006 in Madrid. Among Bergelson's best known results is a polynomial generalization of Szemerédi's theorem. The latter provided a positive solution to the famous Erdős–Turán conjecture from 1936 stating that any set of integers of positive upper density contains arbitrarily long arithmetic progressions. In a 1996 paper Bergelson and Leibman obtained an analogous statement for "polynomial progressions". The Bergelson-Leibman theoremAlexander Soifer, Branko Grünbaum, and Cecil RousseauMathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators. Springer-Verlag, New York, 2008, ; p. 358 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
