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Ky Fan Lemma
In mathematics, Ky Fan's lemma (KFL) is a combinatorial lemma about labellings of triangulations. It is a generalization of Tucker's lemma. It was proved by Ky Fan in 1952. Definitions KFL uses the following concepts. * B_n: the closed ''n''-dimensional ball. ** S_: its boundary sphere. * ''T'': a triangulation of B_n. ** ''T'' is called ''boundary antipodally symmetric'' if the subset of simplices of ''T'' which are in S_ provides a triangulation of S_ where if σ is a simplex then so is −σ. * ''L'': a ''labeling'' of the vertices of ''T'', which assigns to each vertex a non-zero integer: L: V(T) \to \mathbb\setminus\. ** ''L'' is called '' boundary odd'' if for every vertex v\in S_, L(-v) = -L(v). * An edge of ''T'' is called a ''complementary edge'' of ''L'' if the labels of its two endpoints have the same size and opposite signs, e.g. . * An ''n''-dimensional simplex of ''T'' is called an ''alternating simplex'' of ''T'' if its labels have different sizes with alternat ... [...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 points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tucker's Lemma
In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker. Let T be a triangulation of the closed ''n''-dimensional ball B_n. Assume T is antipodally symmetric on the boundary sphere S_. That means that the subset of simplices of T which are in S_ provides a triangulation of S_ where if σ is a simplex then so is −σ. Let L:V(T)\to\ be a labeling of the vertices of T which is an odd function on S_, i.e, L(-v) = -L(v) for every vertex v\in S_. Then Tucker's lemma states that T contains a ''complementary edge'' - an edge (a 1-simplex) whose vertices are labelled by the same number but with opposite signs. Proofs The first proofs were non-constructive, by way of contradiction. Later, constructive proofs were found, which also supplied algorithms for finding the complementary edge. Basically, the algorithms are path-based: they start at a certain point or edge of the triangulation, then go from simplex to simplex acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ky Fan
Ky Fan (樊𰋀, , September 19, 1914 – March 22, 2010) was a Chinese-born American mathematician. He was a professor of mathematics at the University of California, Santa Barbara. Biography Fan was born in Hangzhou, the capital of Zhejiang Province, China. His father, named Fan Qi (樊琦, 1879—1947), served in the district courts of Jinhua and Wenzhou. Ky Fan went to Jinhua with his father when he was eight years old and studied at several middle schools in Zhejiang, including the Jinhua High School (currently Jinhua No.1 Middle School), Hangzhou Zongwen High School (currently Hangzhou No.10 Middle School), and Wenzhou High School. Fan obtained his secondary diploma from the Jinhua High School. Fan enrolled into Peking University Department of Mathematics in 1932, and received his B.S. degree from Peking University in 1936. Initially Fan wanted to study engineering, but eventually shifted to mathematics, largely because of the influence of his uncle Feng Zuxun ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ball (mathematics)
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 area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume 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 \operatorname B^n or \o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangulation (topology)
In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, for instance in algebraic topology, in complex analysis or in modeling. Motivation On the one hand, it is sometimes useful to forget about superfluous information of topological spaces: The replacement of the original spaces with simplicial complexes may help to recognize crucial properties and to gain a better understanding of the considered object. On the other hand, simplicial complexes are objects of combinatorial character and therefore one can assign them quantities rising from their combinatorial pattern, for instance, the Euler characteristic. Triangulation allows now to assign such quantities to topological spaces. Investigations concerning the exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a ''k''-simplex is a ''k''-dimensional polytope which is the convex hull of its ''k'' + 1 vertices. More formally, suppose the ''k'' + 1 points u_0, \dots, u_k \in \mathbb^ are affinely independent, which means u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points : C = \left\ This representation in terms of weighted vertices is known as the barycentric coordinate system. A regular simplex is a simplex that is also a regular poly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd Function
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function f(x) = x^n is an even function if ''n'' is an even integer, and it is an odd function if ''n'' is an odd integer. Definition and examples Evenness and oddness are generally considered for real functions, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse. This includes abelian groups, all rings, all fields, and all vector spaces. Thus, for example, a real function could be odd or even (or neither), as could a complex-valued function of a vector variable, and so on. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lemmas
Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a part of a grass plant * Lemma (mathematics), a type of proposition Other uses * ''Lemma'' (album), by John Zorn (2013) * Lemma (logic), an informal contention See also *Analemma, a diagram showing the variation of the position of the Sun in the sky * Dilemma *Lema (other) * Lemmatisation Lemmatisation ( or lemmatization) in linguistics is the process of grouping together the inflected forms of a word so they can be analysed as a single item, identified by the word's lemma, or dictionary form. In computational linguistics, lemmati ... * Neurolemma, part of a neuron {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
