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Topological Combinatorics
The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics. History The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology. In 1978 the situation was reversed—methods from algebraic topology were used to solve a problem in combinatorics—when László Lovász proved the Kneser conjecture, thus beginning the new field of topological combinatorics. Lovász's proof used the Borsuk–Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has been used in the study of fair division problems. In another application of homological methods to graph theory, Lovász proved both the undirected and directed versions of a conjecture of András Frank: Given a ''k''-connected graph ''G'', ''k'' points v_1,\ldo ... [...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] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of compu ... [...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] |
Berlin Institute Of Technology
The Technical University of Berlin (official name both in English and german: link=no, Technische Universität Berlin, also known as TU Berlin and Berlin Institute of Technology) is a public research university located in Berlin, Germany. It was the first German university to adopt the name "Technische Universität" (Technical University). The university alumni and professor list includes several US National Academies members, two National Medal of Science laureates and ten Nobel Prize laureates. TU Berlin is a member of TU9, an incorporated society of the largest and most notable German institutes of technology and of the Top International Managers in Engineering network, which allows for student exchanges between leading engineering schools. It belongs to the Conference of European Schools for Advanced Engineering Education and Research. The TU Berlin is home of two innovation centers designated by the European Institute of Innovation and Technology. The university is labe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Topological Space
In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can lend good insight to a variety of questions". Topologies on a finite set Let X be a finite set. A topology on X is a subset \tau of P(X) (the power set of X ) such that # \varnothing \in \tau and X\in \tau . # if U, V \in \tau then U \cup V \in \tau . # if U, V \in \tau then U \cap V \in \tau . In other words, a subset \tau of P(X) is a topology if \tau contains both \varnothing and X and is closed under finite intersections and arbitrary unions. Elements of \tau are called open sets. Since the power set of a finite s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Topology
In mathematics, combinatorial topology was an older name for algebraic topology, dating from the time when topological invariants of spaces (for example the Betti numbers) were regarded as derived from combinatorial decompositions of spaces, such as decomposition into simplicial complexes. After the proof of the simplicial approximation theorem this approach provided rigour. The change of name reflected the move to organise topological classes such as cycles-modulo-boundaries explicitly into abelian groups. This point of view is often attributed to Emmy Noether, and so the change of title may reflect her influence. The transition is also attributed to the work of Heinz Hopf, who was influenced by Noether, and to Leopold Vietoris and Walther Mayer, who independently defined homology. A fairly precise date can be supplied in the internal notes of the Bourbaki group. While topology was still ''combinatorial'' in 1942, it had become ''algebraic'' by 1944. This corresponds also to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Graph Theory
In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. It also studies immersions of graphs. Embedding a graph in a surface means that we want to draw the graph on a surface, a sphere for example, without two edges intersecting. A basic embedding problem often presented as a mathematical puzzle is the three utilities problem. Other applications can be found in printing electronic circuits where the aim is to print (embed) a circuit (the graph) on a circuit board (the surface) without two connections crossing each other and resulting in a short circuit. Graphs as topological spaces To an undirected graph we may associate an abstract simplicial complex ''C'' with a single-element set per vertex and a two-element set per edge. The geometric realization , ''C'', of the complex consists of a copy of the unit interval ,1per edge, with the endpoints ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Exterior Calculus
In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs and finite element meshes. DEC methods have proved to be very powerful in improving and analyzing finite element methods: for instance, DEC-based methods allow the use of highly non-uniform meshes to obtain accurate results. Non-uniform meshes are advantageous because they allow the use of large elements where the process to be simulated is relatively simple, as opposed to a fine resolution where the process may be complicated (e.g., near an obstruction to a fluid flow), while using less computational power than if a uniformly fine mesh were used. The discrete exterior derivative Stokes' theorem relates the integral of a differential (''n'' − 1)-form ''ω'' over the boundary ∂''M'' of an ''n''-dimensional manifold ''M'' to the integral of d''ω'' (the exterior derivative of ''ω'', and a differential ''n''-form on ''M'') over ''M' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sperner's Lemma
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an simplex contains a cell whose vertices all have different colors. The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain. Sperner colorings have been used for effective computation of fixed points and in root-finding algorithms, and are applied in fair division (cake cutting) algorithms. Finding a Sperner coloring or equivalently a Brouwer fixed point is now believed to be an intractable computational problem, even in the plane, in the general case. The problem is PPAD-complete, a complexity class invented by Christos Papadimitriou. According to the Soviet ''Mathematical Encyclopaedia'' (ed. I.M. Vinogradov), a related 1929 theorem (of Knaster, Borsuk and Mazurkiewicz) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Morse Theory
Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology computation, denoising, mesh compression, and topological data analysis. Notation regarding CW complexes Let X be a CW complex and denote by \mathcal its set of cells. Define the ''incidence function'' \kappa\colon\mathcal \times \mathcal \to \mathbb in the following way: given two cells \sigma and \tau in \mathcal, let \kappa(\sigma,~\tau) be the degree of the attaching map from the boundary of \sigma to \tau. The boundary operator is the endomorphism \partial of the free abelian group generated by \mathcal defined by :\partial(\sigma) = \sum_\kappa(\sigma,\tau)\tau. It is a defining property of boundary operators that \partial\circ\partial \equiv 0. In more axiomatic definitions one can find the requirement that \forall \sigma,\tau^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the ''geometric'' properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology. The central goal of the field of differential topology is the classification of all smooth manifolds up to diffeomorphism. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the ( connected) manifolds in each dimension separately: * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruhat Order
In mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion order on Schubert varieties. History The Bruhat order on the Schubert varieties of a flag manifold or a Grassmannian was first studied by , and the analogue for more general semisimple algebraic groups was studied by . started the combinatorial study of the Bruhat order on the Weyl group, and introduced the name "Bruhat order" because of the relation to the Bruhat decomposition introduced by François Bruhat. The left and right weak Bruhat orderings were studied by . Definition If (''W'', ''S'') is a Coxeter system with generators ''S'', then the Bruhat order is a partial order on the group ''W''. Recall that a reduced word for an element ''w'' of ''W'' is a minimal length expression of ''w'' as a product of elements of ''S'', and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
