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Distributive Polytope
In the geometry of convex polytopes, a distributive polytope is a convex polytope for which coordinatewise minima and maxima of pairs of points remain within the polytope. For example, this property is true of the unit cube, so the unit cube is a distributive polytope. It is called a distributive polytope because the coordinatewise minimum and coordinatewise maximum operations form the meet and join operations of a continuous distributive lattice on the points of the polytope. Every face of a distributive polytope is itself a distributive polytope. The distributive polytopes all of whose vertex coordinates are 0 or 1 are exactly the order polytopes. See also *Stable matching polytope, a convex polytope that defines a distributive lattice on its points in a different way References {{reflist, refs= {{citation , last1 = Felsner , first1 = Stefan , last2 = Knauer , first2 = Kolja , doi = 10.1016/j.ejc.2010.07.011 , issue = 1 , journal = European Journal of Combinatoric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Cube
A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long.. See in particulap. 671. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.. Unit hypercube The term ''unit cube'' or unit hypercube is also used for hypercubes, or "cubes" in ''n''-dimensional spaces, for values of ''n'' other than 3 and edge length 1. Sometimes the term "unit cube" refers in specific to the set , 1sup>''n'' of all ''n''-tuples of numbers in the interval , 1 The length of the longest diagonal of a unit hypercube of ''n'' dimensions is \sqrt n, the square root of ''n'' and the (Euclidean) length of the vector (1,1,1,....1,1) in ''n''-dimensional space. See also * Doubling the cube * K-cell * Robbins constant, the average distance between two random points in a unit cube * Tychonoff cube, an infinite-dimensional analogue of the unit cube *Unit square *Unit sphere In mathematics, a unit sphere is simply a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Lattice
In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection. Indeed, these lattices of sets describe the scenery completely: every distributive lattice is—up to isomorphism—given as such a lattice of sets. Definition As in the case of arbitrary lattices, one can choose to consider a distributive lattice ''L'' either as a structure of order theory or of universal algebra. Both views and their mutual correspondence are discussed in the article on lattices. In the present situation, the algebraic description appears to be more convenient. A lattice (''L'',∨,∧) is distributive if the following additional identity holds for all ''x'', ''y'', and ''z'' in ''L'': : ''x'' ∧ (''y'' ∨ ''z'') = (''x'' ∧ ''y'') ∨ (''x'' ∧ ''z''). Viewing lattices as part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Polytope
In mathematics, the order polytope of a finite partially ordered set is a convex polytope defined from the set. The points of the order polytope are the monotonic functions from the given set to the unit interval, its vertices correspond to the upper sets of the partial order, and its dimension is the number of elements in the partial order. The order polytope is a distributive polytope, meaning that coordinatewise minima and maxima of pairs of its points remain within the polytope. The order polytope of a partial order should be distinguished from the ''linear ordering polytope'', a polytope defined from a number n as the convex hull of indicator vectors of the sets of edges of n-vertex transitive tournaments. Definition and example A partially ordered set is a pair (S,\le) where S is an arbitrary set and \le is a binary relation on pairs of elements of S that is reflexive (for all x\in S, x\le x), antisymmetric (for all x,y\in S with x\ne y at most one of x\le y and y\le x can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Matching Polytope
In mathematics, economics, and computer science, the stable matching polytope or stable marriage polytope is a convex polytope derived from the solutions to an instance of the stable matching problem. Description The stable matching polytope is the convex hull of the indicator vectors of the stable matchings of the given problem. It has a dimension for each pair of elements that can be matched, and a vertex for each stable matchings. For each vertex, the Cartesian coordinates are one for pairs that are matched in the corresponding matching, and zero for pairs that are not matched. The stable matching polytope has a polynomial number of facets. These include the conventional inequalities describing matchings without the requirement of stability (each coordinate must be between 0 and 1, and for each element to be matched the sum of coordinates for the pairs involving that element must be exactly one), together with inequalities constraining the resulting matching to be stable (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
European Journal Of Combinatorics
European, or Europeans, or Europeneans, may refer to: In general * ''European'', an adjective referring to something of, from, or related to Europe ** Ethnic groups in Europe ** Demographics of Europe ** European cuisine European cuisine comprises the cuisines of Europe "European Cuisine." [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual differe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
