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Knaster–Kuratowski–Mazurkiewicz Lemma
The Knaster–Kuratowski–Mazurkiewicz lemma is a basic result in mathematical fixed-point theory published in 1929 by Knaster, Kuratowski and Mazurkiewicz. The KKM lemma can be proved from Sperner's lemma and can be used to prove the Brouwer fixed-point theorem. Statement Let \Delta_ be an (n-1)-dimensional simplex with ''n'' vertices labeled as 1,\ldots,n. A KKM covering is defined as a set C_1,\ldots,C_n of closed sets such that for any I \subseteq \, the convex hull of the vertices corresponding to I is covered by \bigcup_C_i. The KKM lemma says that in every KKM covering, the common intersection of all ''n'' sets is nonempty, i.e: :\bigcap_^n C_i \neq \emptyset. Example When n=3, the KKM lemma considers the simplex \Delta_2 which is a triangle, whose vertices can be labeled 1, 2 and 3. We are given three closed sets C_1,C_2,C_3 such that: * C_1 covers vertex 1, C_2 covers vertex 2, C_3 covers vertex 3. * The edge 12 (from vertex 1 to vertex 2) is covered by the sets C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed-point Theorem
In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors claim that results of this kind are amongst the most generally useful in mathematics. In mathematical analysis The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. By contrast, the Brouwer fixed-point theorem (1911) is a non- constructive result: it says that any continuous function from the closed unit ball in ''n''-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma). For example, the cosine function is continuous in 1,1and maps it into 1, 1 and thus must have a fixed point. This is clear when examining a sketched graph of the cos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ravindra Bapat
Ravindra B. Bapat is an Indian mathematician known for the Bapat–Beg theorem. Education He obtained B.Sc. from University of Mumbai, M.Stat. from the Indian Statistical Institute, New Delhi and Ph.D. from the University of Illinois at Chicago in 1981. His research interests include major contributions to matrix inequalities, matrices in graph theory, generalized inverses, and matrix analysis. Career In addition to the numerous research papers in the reputed journals, Bapat has written books on linear algebra published by Hindustan Book Agency, Springer, and Cambridge University Press. He has served on the editorial boards of ''Electronic Journal of Linear Algebra'', ''Indian Journal of Pure and Applied Mathematics'', ''Kerala Mathematical Association Bulletin'', and ''Linear and Multilinear Algebra''. He is national coordinator for the Mathematics Olympiad and was head of the Indian Statistical Institute, Delhi Centre from 2007 to 2011. He is a Visiting Professor of Mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorica
''Combinatorica'' is an international journal of mathematics, publishing papers in the fields of combinatorics and computer science. It started in 1981, with László Babai and László Lovász as the editors-in-chief with Paul Erdős as honorary editor-in-chief. The current editors-in-chief are Imre Bárány and József Solymosi. The advisory board consists of Ronald Graham, Gyula O. H. Katona, Miklós Simonovits, Vera Sós, and Endre Szemerédi. It is published by the János Bolyai Mathematical Society and Springer Verlag. The following members of the '' Hungarian School of Combinatorics'' have strongly contributed to the journal as authors, or have served as editors: Miklós Ajtai, László Babai, József Beck, András Frank, Péter Frankl, Zoltán Füredi, András Hajnal, Gyula Katona, László Lovász, László Pyber, Alexander Schrijver, Miklós Simonovits, Vera Sós, Endre Szemerédi, Tamás Szőnyi, Éva Tardos, Gábor Tardos.{{cite web, url=https://www.springer.com/ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory's Theorem (convex Hull)
Carathéodory's theorem is a theorem in convex geometry. It states that if a point x lies in the convex hull \mathrm(P) of a set P\subset \R^d, then x can be written as the convex combination of at most d+1 points in P. More sharply, x can be written as the convex combination of at most d+1 ''extremal'' points in P, as non-extremal points can be removed from P without changing the membership of ''x'' in the convex hull. Its equivalent theorem for conical combinations states that if a point x lies in the conical hull \mathrm(P) of a set P\subset \R^d, then x can be written as the conical combination of at most d points in P. The similar theorems of Helly and Radon are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice versa. The result is named for Constantin Carathéodory, who proved the theorem in 1911 for the case when P is compact. In 1914 Ernst Steinitz expanded Carathéodory's theorem for arbitrary set. Exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barycenter
In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important concept in fields such as astronomy and astrophysics. The distance from a body's center of mass to the barycenter can be calculated as a two-body problem. If one of the two orbiting bodies is much more massive than the other and the bodies are relatively close to one another, the barycenter will typically be located within the more massive object. In this case, rather than the two bodies appearing to orbit a point between them, the less massive body will appear to orbit about the more massive body, while the more massive body might be observed to wobble slightly. This is the case for the Earth–Moon system, whose barycenter is located on average from Earth's center, which is 75% of Earth's radius of . When the two bodies are of similar mass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Combination
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other words, the operation is equivalent to a standard weighted average, but whose weights are expressed as a percent of the total weight, instead of as a fraction of the ''count'' of the weights as in a standard weighted average. More formally, given a finite number of points x_1, x_2, \dots, x_n in a real vector space, a convex combination of these points is a point of the form :\alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n where the real numbers \alpha_i satisfy \alpha_i\ge 0 and \alpha_1+\alpha_2+\cdots+\alpha_n=1. As a particular example, every convex combination of two points lies on the line segment between the points. A set is convex if it contains all convex combinations of its points. The convex hull of a given set of points is identical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hidetoshi Komiya
Hidetoshi (written: 英寿, 英俊, 英敏, 英利, 秀俊, 秀敏, 秀利 or 秀稔) is a masculine Japanese given name. Notable people with the name include: *, Japanese sumo wrestler *, Japanese sumo wrestler *, Japanese racing driver *, Japanese footballer *, Japanese sculptor and architect *, Japanese voice actor *, Japanese footballer *, Japanese judoka *, Japanese voice actor *, Japanese politician *, Japanese luger *, Japanese sumo wrestler *, Japanese footballer {{given name Japanese masculine given names ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the discrete topology), or no set can be open except the space itself and the empty set (the indiscrete topology). In practice, however, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matching In Hypergraphs
In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph. Definition Recall that a hypergraph is a pair , where is a set of vertices and is a set of subsets of called ''hyperedges''. Each hyperedge may contain one or more vertices. A matching in is a subset of , such that every two hyperedges and in have an empty intersection (have no vertex in common). The matching number of a hypergraph is the largest size of a matching in . It is often denoted by . As an example, let be the set Consider a 3-uniform hypergraph on (a hypergraph in which each hyperedge contains exactly 3 vertices). Let be a 3-uniform hypergraph with 4 hyperedges: : Then admits several matchings of size 2, for example: : : However, in any subset of 3 hyperedges, at least two of them intersect, so there is no matching of size 3. Hence, the matching number of is 2. Intersec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold. Equivalent definitions By definition, a subset A of a topological space (X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its limit points. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
