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Markov–Kakutani Fixed-point Theorem
In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space has a common fixed point. This theorem is a key tool in one of the quickest proofs of amenability of abelian groups. Statement Let ''E'' be a locally convex topological vector space. Let ''C'' be a compact convex subset of ''E''. Let ''S'' be a commuting family of self-mappings ''T'' of ''C'' which are continuous and affine, i.e. ''T''(''tx'' +(1 – ''t'')''y'') = ''tT''(''x'') + (1 – ''t'')''T''(''y'') for ''t'' in ,1and ''x'', ''y'' in ''C''. Then the mappings have a common fixed point in ''C''. Proof for a single affine self-mapping Let ''T'' be a continuous affine self-mapping of ''C''. For ''x'' in ''C'' define other elements of ''C'' by : x(N)=\sum_^N T^n(x). Since ''C'' is compa ... [...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] |
Andrey Markov, Jr
Andrey, Andrej or Andrei (in Cyrillic script: Андрей, Андреј or Андрэй) is a form of Andreas/Ἀνδρέας in Slavic languages and Romanian. People with the name include: * Andrei of Polotsk ( – 1399), Lithuanian nobleman *Andrei Alexandrescu, Romanian computer programmer * Andrey Amador, Costa Rican cyclist * Andrei Arlovski, Belarusian mixed martial artist *Andrey Arshavin, Russian football player *Andrej Babiš, Czech prime minister * Andrey Belousov (born 1959), Russian politician * Andrey Bolotov, Russian agriculturalist and memoirist * Andrey Borodin, Russian financial expert and businessman * Andrei Chikatilo, prolific and cannibalistic Russian serial killer and rapist * Andrei Denisov (weightlifter) (born 1963), Israeli Olympic weightlifter * Andrey Ershov, Russian computer scientist * Andrey Esionov, Russian painter * Andrei Glavina, Istro-Romanian writer and politician * Andrei Gromyko (1909–1989), Belarusian Soviet politician and diplomat * An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shizuo Kakutani
was a Japanese-American mathematician, best known for his eponymous fixed-point theorem. Biography Kakutani attended Tohoku University in Sendai, where his advisor was Tatsujirō Shimizu. At one point he spent two years at the Institute for Advanced Study in Princeton at the invitation of the mathematician Hermann Weyl. While there, he also met John von Neumann. Kakutani received his Ph.D. in 1941 from Osaka University and taught there through World War II. He returned to the Institute for Advanced Study in 1948, and was given a professorship by Yale in 1949, where he won a students' choice award for excellence in teaching. Kakutani received two awards of the Japan Academy, the Imperial Prize and the Academy Prize in 1982, for his scholarly achievements in general and his work on functional analysis in particular. He was a Plenary Speaker of the ICM in 1950 in Cambridge, Massachusetts. Kakutani was married to Keiko ("Kay") Uchida, who was a sister to author Yoshiko U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles. More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios of the lengths of parallel line segments. Consequently, sets of parallel affine subspaces remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. If is the point set of an affine space, then every affine transformation on can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Convex Topological Vector Space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals. Fréchet spaces are locally convex spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm. History Metrizable topol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Intersection Property
In general topology, a branch of mathematics, a non-empty family ''A'' of subsets of a set X is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of A is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of A is infinite. Sets with the finite intersection property are also called centered systems and filter subbases. The finite intersection property can be used to reformulate topological compactness in terms of closed sets; this is its most prominent application. Other applications include proving that certain perfect sets are uncountable, and the construction of ultrafilters. Definition Let X be a set and \mathcal a nonempty family of subsets of that is, \mathcal is a subset of the power set of Then \mathcal is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
