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List Of Exceptional Set Concepts
{{Short description, none This is a list of exceptional set concepts. In mathematics, and in particular in mathematical analysis, it is very useful to be able to characterise subsets of a given set ''X'' as 'small', in some definite sense, or 'large' if their Complement (set theory), complement in ''X'' is small. There are numerous concepts that have been introduced to study 'small' or 'exceptional' subsets. In the case of sets of natural numbers, it is possible to define more than one concept of 'density', for example. See also list of properties of sets of reals. *Almost all *Almost always *Almost everywhere *Almost never *Almost surely *Analytic capacity *club set, Closed unbounded set *Cofinal (mathematics) *Cofinite *Dense set *IP set *2-large *Large set (Ramsey theory) *Meagre set *Null set, Measure zero *Natural density *Negligible set *Nowhere dense set *Null set, conull set *Partition regular *Piecewise syndetic set *Schnirelmann density *Small set (combinatorics) *Stationar ... [...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] |
2-large
In Ramsey theory, a set ''S'' of natural numbers is considered to be a large set if and only if Van der Waerden's theorem can be generalized to assert the existence of arithmetic progressions with common difference in ''S''. That is, ''S'' is large if and only if every finite partition of the natural numbers has a cell containing arbitrarily long arithmetic progressions having common differences in ''S''. Examples *The natural numbers are large. This is precisely the assertion of Van der Waerden's theorem. *The even numbers are large. Properties Necessary conditions for largeness include: *If ''S'' is large, for any natural number ''n'', ''S'' must contain at least one multiple (equivalently, infinitely many multiples) of ''n''. *If S=\ is large, it is not the case that ''s''k≥3''s''k-1 for ''k''≥ 2. Two sufficient conditions are: *If ''S'' contains n-cubes for arbitrarily large n, then ''S'' is large. *If S =p(\mathbb) \cap \mathbb where p is a polynomial with p(0)=0 and ... [...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] |
Stationary Set
In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets, and is analogous to a set of non-zero measure in measure theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset. Classical notion If \kappa is a cardinal of uncountable cofinality, S \subseteq \kappa, and S intersects every club set in \kappa, then S is called a stationary set.Jech (2003) p.91 If a set is not stationary, then it is called a thin set. This notion should not be confused with the notion of a thin set in number theory. If S is a stationary set and C is a club set, then their intersection S \cap C is also stationary. This is because if D is any club set, then C \cap D is a club set, thus (S \cap C) \cap D = S \cap (C \cap D) is non empty. Therefore, (S \cap C) must be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Set (combinatorics)
In combinatorial mathematics, a large set of positive integers :S = \ is one such that the infinite sum of the reciprocals :\frac+\frac+\frac+\frac+\cdots diverges. A small set is any subset of the positive integers that is not large; that is, one whose sum of reciprocals converges. Large sets appear in the Müntz–Szász theorem and in the Erdős conjecture on arithmetic progressions. Examples * Every finite subset of the positive integers is small. * The set \ of all positive integers is known to be a large set; this statement is equivalent to the divergence of the harmonic series. More generally, any arithmetic progression (i.e., a set of all integers of the form ''an'' + ''b'' with ''a'' ≥ 1, ''b'' ≥ 1 and ''n'' = 0, 1, 2, 3, ...) is a large set. * The set of square numbers is small (see Basel problem). So is the set of cube numbers, the set of 4th powers, and so on. More generally, the set of positive i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schnirelmann Density
In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it.Schnirelmann, L.G. (1930).On the additive properties of numbers, first published in "Proceedings of the Don Polytechnic Institute in Novocherkassk" (in Russian), vol XIV (1930), pp. 3-27, and reprinted in "Uspekhi Matematicheskikh Nauk" (in Russian), 1939, no. 6, 9–25.Schnirelmann, L.G. (1933). First published asÜber additive Eigenschaften von Zahlen in "Mathematische Annalen" (in German), vol 107 (1933), 649-690, and reprinted asOn the additive properties of numbers in "Uspekhin. Matematicheskikh Nauk" (in Russian), 1940, no. 7, 7–46. Definition The Schnirelmann density of a set of natural numbers ''A'' is defined as :\sigma A = \inf_n \frac, where ''A''(''n'') denotes the number of elements of ''A'' not exceeding ''n'' and inf is infimum.Nathanson (1996) pp.191–19 ... [...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] |
Partition Regular
In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets. Given a set X, a collection of subsets \mathbb \subset \mathcal(X) is called ''partition regular'' if every set ''A'' in the collection has the property that, no matter how ''A'' is partitioned into finitely many subsets, at least one of the subsets will also belong to the collection. That is, for any A \in \mathbb, and any finite partition A = C_1 \cup C_2 \cup \cdots \cup C_n, there exists an ''i'' ≤ ''n'', such that C_i belongs to \mathbb. Ramsey theory is sometimes characterized as the study of which collections \mathbb are partition regular. Examples * the collection of all infinite subsets of an infinite set ''X'' is a prototypical example. In this case partition regularity asserts that every finite partition of an infinite set has an infinite cell (i.e. the infinite pigeonhole principle.) * sets with positive upper density in \mathbb: the ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conull Set
In measure theory, a conull set is a set whose complement is null, i.e., the measure of the complement is zero. For example, the set of irrational numbers is a conull subset of the real line with Lebesgue measure. A property that is true of the elements of a conull set is said to be true almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion t ..... Sep. 62for an example of this usage. References Measure theory {{Settheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nowhere Dense Set
In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. For example, the integers are nowhere dense among the reals, whereas an open ball is not. A countable union of nowhere dense sets is called a meagre set. Meagre sets play an important role in the formulation of the Baire category theorem, which is used in the proof of several fundamental result of functional analysis. Definition Density nowhere can be characterized in different (but equivalent) ways. The simplest definition is the one from density: A subset S of a topological space X is said to be ''dense'' in another set U if the intersection S \cap U is a dense subset of U. S is or in X if S is not dense in any nonempty open subset U of X. Expanding out the negation of density, it is equivalent to require that each nonempty o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negligible Set
In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose. As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function. Negligible sets define several useful concepts that can be applied in various situations, such as truth almost everywhere. In order for these to work, it is generally only necessary that the negligible sets form an ideal; that is, that the empty set be negligible, the union of two negligible sets be negligible, and any subset of a negligible set be negligible. For some purposes, we also need this ideal to be a sigma-ideal, so that countable unions of negligible sets are also negligible. If and are both ideals of subsets of the same set , then one may speak of ''-negligible'' and ''-negligible'' subsets. The opposite of a negligible set is a generic property, which has various forms. Examples Let ... [...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] |
