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Ryll-Nardzewski Fixed Point Theorem
In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if E is a normed vector space and K is a nonempty convex subset of E that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of K has at least one fixed point. (Here, a ''fixed point'' of a set of maps is a point that is fixed by each map in the set.) This theorem was announced by Czesław Ryll-Nardzewski. Later Namioka and Asplund gave a proof based on a different approach. Ryll-Nardzewski himself gave a complete proof in the original spirit. Applications The Ryll-Nardzewski theorem yields the existence of a Haar measure on compact groups. See also * 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. In mathematica ...s * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics)
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation (mathematics), transformation. Specifically, for function (mathematics), functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain of a function, domain and the codomain of , and . In particular, cannot have any fixed point if its domain is disjoint from its codomain. If is defined on the real numbers, it corresponds, in graphical terms, to a curve in the Euclidean plane, and each fixed-point corresponds to an intersection of the curve with the line , cf. picture. For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Dugundji
James Dugundji (August 30, 1919 – January 8, 1985) was an American mathematician, a professor of mathematics at the University of Southern California.. See in particulap. 244for a brief biography of Dugundji.Note about the life and work of Dugundji by Andrzej Granas in their book ''Fixed Point Theory'', Springer, 2005. Reprinted in , p. 9. Dugundji's parents emigrated from to , where Dugundji was born in 1919. He studied at |
Fixed-point Theorems In Infinite-dimensional Spaces
In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations. The first result in the field was the Schauder fixed-point theorem, proved in 1930 by Juliusz Schauder (a previous result in a different vein, the Banach fixed-point theorem for contraction mappings in complete metric spaces was proved in 1922). Quite a number of further results followed. One way in which fixed-point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of algebraic topology, first proved for finite simplicial complexes, to spaces of infinite dimension. For example, the research of Jean Leray who founded sheaf theory came out of efforts to extend Schauder's work. Schauder fixed-point theorem: Let ''C'' be a nonempty closed convex subset of a Banach ... [...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. 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 cosine function; the fixed point occurs where the cosine curve ''y'' = cos(''x'') intersect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haar Measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of mathematical analysis, analysis, number theory, group theory, representation theory, mathematical statistics, statistics, probability theory, and ergodic theory. Preliminaries Let (G, \cdot) be a locally compact space, locally compact Hausdorff space, Hausdorff topological group. The Sigma-algebra, \sigma-algebra generated by all open subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g is an element of G and S is a subset of G, then we define the left and right Coset, translates of S by ''g'' as follows: * Left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Czesław Ryll-Nardzewski
Czesław Ryll-Nardzewski (; 7 October 1926 – 18 September 2015) was a Polish mathematician. Life and career Born in Wilno, Second Polish Republic (now Vilnius, Lithuania), he was a student of Hugo Steinhaus. At the age of 26 he became professor at Warsaw University. In 1959, he became a professor at the Wrocław University of Technology. He was the advisor of 18 PhD theses. His main research areas were measure theory, functional analysis, foundations of mathematics and probability theory. Several theorems bear his name: the Ryll-Nardzewski fixed point theorem, “9. Theorem of Ryll-Nardzewski” (p. 171), “(9.6) Theorem (Ryll-Nardzewski)” (p. 174) the Ryll-Nardzewski theorem See Theorem 7.3.1 Cf. (2.10) in model theory, and the Kuratowski and Ryll-Nardzewski measurable selection theorem. See Theorem 6.9.3 on p. 36 and the historical comment on p. 441 He became a member of the Polish Academy of Sciences in 1967. He died in 2015 at the age of 88 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' meaning "equal", and μέτρον ''metron'' meaning "measure". If the transformation is from a metric space to itself, it is a kind of geometric transformation known as a motion. Introduction Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Vector Space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war period. It was badly damaged during World War II (1939–45). In the first thirty years after the war the shipyard again experienced a boom and employed up to 3,000 workers making oil tankers, and then liquid natural gas tankers. Demand dropped off in the 1970s and 1980s. In 1972 the shipyard became Chantiers de France-Dunkerque, and in 1983 merged with others yards to become part of Chantiers du Nord et de la Mediterranee, or Normed. The shipyard closed in 1987. Foundation (1898–99) The Ateliers et Chantiers de France (ACF) company was officially founded on 6 July 1898 by a consortium of six shipping brokers, the Dunkirk chamber of commerce and the state. The state asked that the shipyard be able to build steamships and also four-maste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Map
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 be re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
