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Matroid Polytope
In mathematics, a matroid polytope, also called a matroid basis polytope (or basis matroid polytope) to distinguish it from other polytopes derived from a matroid, is a polytope constructed via the bases of a matroid. Given a matroid M, the matroid polytope P_M is the convex hull of the indicator vectors of the bases of M. Definition Let M be a matroid on n elements. Given a basis B \subseteq \ of M, the indicator vector of B is :\mathbf e_B := \sum_ \mathbf e_i, where \mathbf e_i is the standard ith unit vector in \mathbb^n. The matroid polytope P_M is the convex hull of the set :\ \subseteq \mathbb^n. Examples * Let M be the rank 2 matroid on 4 elements with bases :: \mathcal(M) = \. :That is, all 2-element subsets of \ except \ . The corresponding indicator vectors of \mathcal(M) are :: \. :The matroid polytope of M is : P_M = \text\. :These points form four equilateral triangles at point \, therefore its convex hull is the square pyramid by definition. * Let N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Polymatroid
In mathematics, a polymatroid is a polytope associated with a submodular function. The notion was introduced by Jack Edmonds in 1970. It is also described as the multiset analogue of the matroid. Definition Let E be a finite set and f: 2^E\rightarrow \mathbb_+ a non-decreasing submodular function, that is, for each A\subseteq B \subseteq E we have f(A)\leq f(B) , and for each A, B \subseteq E we have f(A)+f(B) \geq f(A\cup B) + f(A\cap B) . We define the polymatroid associated to f to be the following polytope: P_f= \Big\. When we allow the entries of \textbf to be negative we denote this polytope by EP_f, and call it the extended polymatroid associated to f. An equivalent definition Let E be a finite set and \textbf, \textbf \in \mathbb^E. We call the ''modulus'' of \textbf to be the sum of all of its entries, denoted , \textbf, , and denote \textbf \leq \textbf whenever v_i-u_i\geq 0 for every i \in E (notice that this gives an order to \mathbb_+^E). A polymat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terminology of both linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory. Definition There are many equivalent ( cryptomorphic) ways to define a (finite) matroid.A standard source for basic definitions and results about matroids is Oxley (1992). An older standard source is Welsh (1976). See Brylawsk ... [...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 polytope. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Addition
In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'', i.e., the set : A + B = \. Analogously, the Minkowski difference (or geometric difference) is defined using the complement operation as : A - B = \left(A^c + (-B)\right)^c In general A - B \ne A + (-B). For instance, in a one-dimensional case A = 2, 2/math> and B = 1, 1/math> the Minkowski difference A - B = 1, 1/math>, whereas A + (-B) = A + B = 3, 3 In a two-dimensional case, Minkowski difference is closely related to erosion (morphology) in image processing. The concept is named for Hermann Minkowski. Example For example, if we have two sets ''A'' and ''B'', each consisting of three position vectors (informally, three points), representing the vertices of two triangles in \mathbb^2, with coordinates :A = \ and :B = \ then their Minkowski sum is :A + B = \ which comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete & Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * ''Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents''/Engineering, Computing and Technology Notable articles The articles by Gil Kalai with a proof of a subexponential upper bound on the diameter of a polyhedron and by Samuel Ferguson on the Kepler conjecture, both published in Discrete & Computational geometry, earned their author the Fulkerson Prize The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at e .... References External link ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutohedron
In mathematics, the permutohedron of order ''n'' is an (''n'' − 1)-dimensional polytope embedded in an ''n''-dimensional space. Its vertex coordinates (labels) are the permutations of the first ''n'' natural numbers. The edges identify the shortest possible paths (sets of transpositions) that connect two vertices (permutations). Two permutations connected by an edge differ in only two places (one transposition), and the numbers on these places are neighbors (differ in value by 1). The image on the right shows the permutohedron of order 4, which is the truncated octahedron. Its vertices are the 24 permutations of (1, 2, 3, 4). Parallel edges have the same edge color. The 6 edge colors correspond to the 6 possible transpositions of 4 elements, i.e. they indicate in which two places the connected permutations differ. (E.g. red edges connect permutations that differ in the last two places.) History According to , permutohedra were first studied by . The name ''permut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delta-matroid
In mathematics, a delta-matroid or Δ-matroid is a family of sets obeying an exchange axiom generalizing an axiom of matroids. A non-empty family of sets is a delta-matroid if, for every two sets E and F in the family, and for every element e in their symmetric difference E\triangle F, there exists an f\in E\triangle F such that E\triangle\ is in the family. For the basis sets of a matroid, the corresponding exchange axiom requires in addition that e\in E and f\in F, ensuring that E and F have the same cardinality. For a delta-matroid, either of the two elements may belong to either of the two sets, and it is also allowed for the two elements to be equal. An alternative and equivalent definition is that a family of sets forms a delta-matroid when the convex hull of its indicator vectors (the analogue of a matroid polytope) has the property that every edge length is either one or the square root of two. Delta-matroids were defined by André Bouchet in 1987. Algorithms for matroid inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root Of Two
The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically, the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. The fraction (≈ 1.4142857) is sometimes used as a good rational approximation with a reasonably small denominator. Sequence in the On-Line Encyclopedia of Integer Sequences consists of the digits in the decimal expansion of the square root of 2, here truncated to 65 decimal places: : History The Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) gives an approximation of in four sexagesimal figures, , which is accurate to about six ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersimplex
In polyhedral combinatorics, the hypersimplex \Delta_ is a convex polytope that generalizes the simplex. It is determined by two integers d and k, and is defined as the convex hull of the d-dimensional vectors whose coefficients consist of k ones and d-k zeros. Equivalently, \Delta_ can be obtained by slicing the d-dimensional unit hypercube ,1d with the hyperplane of equation x_1+\cdots+x_d=k and, for this reason, it is a (d-1)-dimensional polytope when 0 Properties The number of vertices of is . The graph formed by the vertices and edges of the hypersimplex is the .Alternative constructions An alternative construction (for ) is to take the convex hull ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terminology of both linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory. Definition There are many equivalent ( cryptomorphic) ways to define a (finite) matroid.A standard source for basic definitions and results about matroids is Oxley (1992). An older standard source is Welsh (1976). See Brylawsk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
