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Mazur–Ulam Theorem
In mathematics, the Mazur–Ulam theorem states that if V and W are normed spaces over R and the mapping :f\colon V\to W is a surjective isometry, then f is affine. It was proved by Stanisław Mazur and Stanisław Ulam in response to a question raised by Stefan Banach. For strictly convex spaces the result is true, and easy, even for isometries which are not necessarily surjective. In this case, for any u and v in V, and for any t in ,1/math>, write r=\, u-v\, _V=\, f(u)-f(v)\, _W and denote the closed ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ... of radius around by \bar B(v,R). Then tu+(1-t)v is the unique element of \bar B(v,tr)\cap \bar B(u,(1-t)r), so, since f is injective, f(tu+(1-t)v) is the unique element of f\bigl(\bar B(v,tr)\cap \bar B(u,(1-t)r\bigr) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed 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-masted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map (mathematics)
In mathematics, a map or mapping is a function (mathematics), function in its general sense. These terms may have originated as from the process of making a map, geographical map: ''mapping'' the Earth surface to a sheet of paper. The term ''map'' may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. In category theory, a map may refer to a morphism. The term ''transformation'' can be used interchangeably, but ''transformation (function), transformation'' often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory. Maps as functions In many branches of mathematics, the term ''map'' is used to mean a Function (mathematics), function, sometimes with a specific property of particular importance to that branch. For instance, a "map" is a "continuous f ... [...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] |
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] |
Stanisław Mazur
Stanisław Mieczysław Mazur (; 1 January 1905 – 5 November 1981) was a Polish mathematician and a member of the Polish Academy of Sciences. Mazur made important contributions to geometrical methods in linear and nonlinear functional analysis and to the study of Banach algebras. He was also interested in summability theory, infinite games and computable functions. Lwów and Warsaw Mazur was a student of Stefan Banach at University of Lwów. His doctorate, under Banach's supervision, was awarded in 1935. Mazur, with Juliusz Schauder, was an Invited Speaker of the ICM in 1936 in Oslo. Mazur was a close collaborator with Banach at Lwów and was a member of the Lwów School of Mathematics, where he participated in the mathematical activities at the Scottish Café. On 6 November 1936, he posed the " basis problem" of determining whether every Banach space has a Schauder basis, with Mazur promising a "live goose" as a reward: 37 years later and in a ceremony that was broadcast ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanisław Ulam
Stanisław Marcin Ulam ( ; 13 April 1909 – 13 May 1984) was a Polish and American mathematician, nuclear physicist and computer scientist. He participated in the Manhattan Project, originated the History of the Teller–Ulam design, Teller–Ulam design of thermonuclear weapons, discovered the concept of the cellular automaton, invented the Monte Carlo method, Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he proved a number of theorems and proposed several conjectures. Born into a wealthy Polish Jewish family in Lemberg, Austria-Hungary; Ulam studied mathematics at the Lviv Polytechnic#Second Polish Republic, Lwów Polytechnic Institute, where he earned his PhD in 1933 under the supervision of Kazimierz Kuratowski and Włodzimierz Stożek. In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months. From 1936 to 1939, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an original member of the Lwów School of Mathematics. His major work was the 1932 book, ''Théorie des opérations linéaires'' (Theory of Linear Operations), the first monograph on the general theory of functional analysis. Born in Kraków to a family of Gorals, Goral descent, Banach showed a keen interest in mathematics and engaged in solving mathematical problems during school Recess (break), recess. After completing his secondary education, he befriended Hugo Steinhaus, with whom he established the Polish Mathematical Society in 1919 and later published the scientific journal ''Studia Mathematica''. In 1920, he received an assistantship at the Lwów Polytechnic, subsequently becoming a professor in 1922 and a member of the Polish Academy of Lear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strictly Convex Space
In mathematics, a strictly convex space is a normed vector space (''X'', , , , , ) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is one for which, given any two distinct points ''x'' and ''y'' on the unit sphere ∂''B'' (i.e. the boundary of the unit ball ''B'' of ''X''), the segment joining ''x'' and ''y'' meets ∂''B'' ''only'' at ''x'' and ''y''. Strict convexity is somewhere between an inner product space (all inner product spaces being strictly convex) and a general normed space in terms of structure. It also guarantees the uniqueness of a best approximation to an element in ''X'' (strictly convex) out of a convex subspace ''Y'', provided that such an approximation exists. If the normed space ''X'' is complete and satisfies the slightly stronger property of being uniformly convex (which implies strict convexity), then it is also reflexive by Milman–Pettis theorem. Properties The following properties are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Ball
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the planar region bounded by a circle. In Euclidean 3-space, a ball is taken to be the region of space bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of topology the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandrov–Rassias Problem
The theory of isometries in the framework of Banach spaces has its beginning in a paper by Stanisław Mazur and Stanisław M. Ulam in 1932. They proved the Mazur–Ulam theorem stating that every isometry of a normed real linear space onto a normed real linear space is a linear mapping up to translation. In 1970, Aleksandr Danilovich Aleksandrov asked whether the existence of a single distance that is preserved by a mapping implies that it is an isometry, as it does for Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...s by the Beckman–Quarles theorem. Themistocles M. Rassias posed the following problem: Aleksandrov–Rassias Problem. If and are normed linear spaces and if is a continuous and/or surjective mapping such that whenever vectors and in satisfy \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CRC Press
The CRC Press, LLC is an American publishing group that specializes in producing technical books. Many of their books relate to engineering, science and mathematics. Their scope also includes books on business, forensics and information technology. CRC Press is now a division of Taylor & Francis, itself a subsidiary of Informa. History The CRC Press was founded as the Chemical Rubber Company (CRC) in 1903 by brothers Arthur, Leo and Emanuel Friedman in Cleveland, Ohio, based on an earlier enterprise by Arthur, who had begun selling rubber laboratory aprons in 1900. The company gradually expanded to include sales of laboratory equipment to chemist A chemist (from Greek ''chēm(ía)'' alchemy; replacing ''chymist'' from Medieval Latin ''alchemist'') is a graduated scientist trained in the study of chemistry, or an officially enrolled student in the field. Chemists study the composition of ...s. In 1913 the CRC offered a short (116-page) manual called the ''Rubber Handboo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
