|
Busemann G-space
In mathematics, a Busemann ''G''-space is a type of metric space first described by Herbert Busemann in 1942. If (X,d) is a metric space such that # for every two distinct x, y \in X there exists z \in X-\ such that d(x,z)+d(y,z)=d(x,z) (Menger convexity) # every d-bounded set of infinite cardinality possesses accumulation points # for every w \in X there exists \rho_w such that for any distinct points x,y \in B(w,\rho_w) there exists z \in ( b(w,\rho_w)-\ )^\circ such that d(x,z)+d(y,z)=d(x,z) (geodesics are locally extendable) # for any distinct points x,y \in X, if u,v \in X such that d(x,u)+d(y,u)=d(x,u), d(x,v)+d(y,v)=d(x,v) and d(y,u)=d(y,v) (geodesic extensions are unique). then ''X'' is said to be a ''Busemann'' ''G''-''space''. Every Busemann ''G''-space is a homogenous space. The Busemann conjecture states that every Busemann ''G''-space is a topological manifold. It is a special case of the Bing–Borsuk conjecture In mathematics, the Bing–Borsuk conjecture sta ... [...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] |
Metric Space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Busemann
Herbert Busemann (12 May 1905 – 3 February 1994) was a German-American mathematician specializing in convex and differential geometry. He is the author of Busemann's theorem in Euclidean geometry and geometric tomography. He was a member of the Royal Danish Academy and a winner of the Lobachevsky Medal (1985), the first American mathematician to receive it. He was also a Fulbright scholar in New Zealand in 1952. Biography Herbert Busemann was born in Berlin to a well-to-do family. His father, Alfred Busemann, was a director of Krupp, where Busemann also worked for several years. He studied at University of Munich, Paris, and Rome. He defended his dissertation in University of Göttingen in 1931, where his advisor was Richard Courant. He remained in Göttingen as an assistant until 1933, when he escaped Nazi Germany to Copenhagen (he had a Jewish grandfather). He worked at the University of Copenhagen until 1936, when he left to the United States. There ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Menger Convexity
Menger is a surname. Notable people with the surname include: * Andreas Menger (born 1972), former German football player * Anton Menger (1841–1906), Austrian economist and author; brother of Carl Menger * Carl Menger (1840–1921), Austrian economist and author; founder of the Austrian School of economics * Howard Menger (1922–2009), American who claimed to have met extraterrestrials * Karl Menger (1902–1985), Austrian-born mathematician and son of economist Carl Menger * Kirsten Menger-Anderson (born 1969), American fiction writer See also * Menger Hotel, San Antonio Texas * Menger sponge, a fractal curve * Menger's theorem * Menger–Urysohn dimension; see Inductive dimension * Cayley–Menger determinant; see Distance geometry * * Manger __NOTOC__ A manger or trough is a rack for fodder, or a structure or feeder used to hold food for animals. The word comes from the Old French ''mangier'' (meaning "to eat"), from Latin ''mandere'' (meaning "to chew"). Mangers are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Accumulation Point
In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. A limit point of a set S does not itself have to be an element of S. There is also a closely related concept for sequences. A cluster point or accumulation point of a sequence (x_n)_ in a topological space X is a point x such that, for every neighbourhood V of x, there are infinitely many natural numbers n such that x_n \in V. This definition of a cluster or accumulation point of a sequence generalizes to nets and filters. The similarly named notion of a (respectively, a limit point of a filter, a limit point of a net) by definition refers to a point that the sequence converges to (respectively, the filter converges to, the net converges to). Importantly, although "limit point of a set" is syn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line". The noun '' geodesic'' and the adjective '' geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogenous Space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ''G'' are called the symmetries of ''X''. A special case of this is when the group ''G'' in question is the automorphism group of the space ''X'' – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, ''X'' is homogeneous if intuitively ''X'' looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (differential geometry), or homeomorphism (topology). Some authors insist that the action of ''G'' be faithful (non-identity elements act non-trivially), although the present article does not. Thus there is a group action of ''G'' on ''X'' which can be thought of as preserving some "geometric structure" on ''X'', and making ''X'' into a single ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Busemann Conjecture
Busemann is a German surname. Notable people with the surname include: * Adolf Busemann (1901–1986), German-American aerospace engineer, inventor of Busemann's Biplane * Frank Busemann (born 1975), a German decathlete * Herbert Busemann (1905–1994), a German-American mathematician, the author of Busemann's theorem See also * Busemann biplane The Busemann biplane is a theoretical aircraft configuration invented by Adolf Busemann, which avoids the formation of N-type shock waves and thus does not create a sonic boom or the associated wave drag. However in its original form it does not ... * Busemann's theorem * Busemann function {{surname, Busemann German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Manifold
In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable manifolds are topological manifolds equipped with a differential structure). Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure. Formal definition A topological space ''X'' is called locally Euclidean if there is a non-negative integer ''n'' such that every point in ''X'' has a neighborhood which is homeomo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bing–Borsuk Conjecture
In mathematics, the Bing–Borsuk conjecture states that every n-dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional version of the conjecture implies the Poincaré conjecture. Definitions A topological space is ''homogeneous'' if, for any two points m_1, m_2 \in M, there is a homeomorphism of M which takes m_1 to m_2. A metric space M is an absolute neighborhood retract (ANR) if, for every closed embedding f: M \rightarrow N (where N is a metric space), there exists an open neighbourhood U of the image f(M) which retracts to f(M). There is an alternate statement of the Bing–Borsuk conjecture: suppose M is embedded in \mathbb^ for some m \geq 3 and this embedding can be extended to an embedding of M \times (-\varepsilon, \varepsilon). If M has a mapping cylinder neighbourhood N=C_\varphi of some map \varphi: \partial N \rightarrow M with mapping cyl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Spaces
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
