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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] |
Partially Ordered Set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of Comparability, comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indicator Function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\in A, and \mathbf_(x)=0 otherwise, where \mathbf_A is a common notation for the indicator function. Other common notations are I_A, and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, :\mathbf_(x)= \in A For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition The indicator function of a subset of a set is a function \mathbf_A \colon X \to \ defined as \mathbf_A(x) := \begin 1 ~&\text~ x \in A~, \\ 0 ~&\text~ x \notin A~. \end The Iverson bracket provides the equivalent notation, \in A/math> or to be used instead of \mathbf_(x)\,. The function \mathbf_A is sometimes denoted , , , or even just . Nota ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Polynomial
The order polynomial is a polynomial studied in mathematics, in particular in algebraic graph theory and algebraic combinatorics. The order polynomial counts the number of order-preserving maps from a poset to a chain of length n. These order-preserving maps were first introduced by Richard P. Stanley while studying ordered structures and partitions as a Ph.D. student at Harvard University in 1971 under the guidance of Gian-Carlo Rota. Definition Let P be a finite poset with p elements denoted x,y \in P, and let \ be a chain n elements. A map \phi: P \to is order-preserving if x \leq y implies \phi(x) \leq \phi(y). The number of such maps grows polynomially with n, and the function that counts their number is the ''order polynomial'' \Omega(n)=\Omega(P,n). Similarly, we can define an order polynomial that counts the number of strictly order-preserving maps \phi: P \to /math>, meaning x is (strictly) order-preserving, there are p! linear extensions, and both polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehrhart Polynomial
In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane. These polynomials are named after Eugène Ehrhart who studied them in the 1960s. Definition Informally, if is a polytope, and is the polytope formed by expanding by a factor of in each dimension, then is the number of integer lattice points in . More formally, consider a lattice \mathcal in Euclidean space \R^n and a -dimensional polytope in \R^n with the property that all vertices of the polytope are points of the lattice. (A common example is \mathcal = \Z^n and a polytope for which all vertices have integer coordinates.) For any positive integer , let be the -fold dilation of (the polytope formed by multiplying each vertex coordinate, in a basis for the la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time Approximation Scheme
In computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems). A PTAS is an algorithm which takes an instance of an optimization problem and a parameter and produces a solution that is within a factor of being optimal (or for maximization problems). For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most , with being the length of the shortest tour. The running time of a PTAS is required to be polynomial in the problem size for every fixed ε, but can be different for different ε. Thus an algorithm running in time or even counts as a PTAS. Variants Deterministic A practical problem with PTAS algorithms is that the exponent of the polynomial could increase dramatically as ε shrinks, for example if the runtime is . One way of addressing this is to define the efficient polynomial-time a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
♯P-complete
The #P-complete problems (pronounced "sharp P complete" or "number P complete") form a complexity class in computational complexity theory. The problems in this complexity class are defined by having the following two properties: *The problem is in #P, the class of problems that can be defined as counting the number of accepting paths of a polynomial-time non-deterministic Turing machine. *The problem is #P-hard, meaning that every other problem in #P has a Turing reduction or polynomial-time counting reduction to it. A counting reduction is a pair of polynomial-time transformations from inputs of the other problem to inputs of the given problem and from outputs of the given problem to outputs of the other problem, allowing the other problem to be solved using any subroutine for the given problem. A Turing reduction is an algorithm for the other problem that makes a polynomial number of calls to a subroutine for the given problem and, outside of those calls, uses polynomial tim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Extension
In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. Definitions Given any partial orders \,\leq\, and \,\leq^*\, on a set X, \,\leq^*\, is a linear extension of \,\leq\, exactly when (1) \,\leq^*\, is a total order and (2) for every x, y \in X, if x \leq y, then x \leq^* y. It is that second property that leads mathematicians to describe \,\leq^*\, as extending \,\leq. Alternatively, a linear extension may be viewed as an order-preserving bijection from a partially ordered set P to a chain C on the same ground set. Order-extension principle The statement that every partial order can be extended to a total order is known as the order-extension principle. A proof using the axiom of choice was first published by Edward Marczewski in 1930. Marczewski write ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of length (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. In ancient times, volume is measured using similar-shaped natural containers and later on, standardized containers. Some simple three-dimensional shapes can have its volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape's boundary. Zero-, one- and two-dimensional objects have no volume; in fourth and higher dimensions, an analogous concept to the normal vo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. Therefore two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched up completely. Turning the paper over is permitted. In elementary geometry the word ''congruent'' is often used as follows. The word ''equal'' is often used in place of ''congruent'' for these objects. *Two line segments are congruent if they have the same length. *Two angles are congruent if they have the same measure. *Two circles are congruent if they have the ... [...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 sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schläfli Orthoscheme
In geometry, a Schläfli orthoscheme is a type of simplex. The orthoscheme is the generalization of the right triangle to simplex figures of any number of dimensions. Orthoschemes are defined by a sequence of edges (v_0v_1), (v_1v_2), \dots, (v_v_d) \, that are mutually orthogonal. They were introduced by Ludwig Schläfli, who called them ''orthoschemes'' and studied their volume in Euclidean, hyperbolic, and spherical geometries. H. S. M. Coxeter later named them after Schläfli. As right triangles provide the basis for trigonometry, orthoschemes form the basis of a trigonometry of ''n'' dimensions, as developed by Schoute who called it polygonometry. J.-P. Sydler and Børge Jessen studied orthoschemes extensively in connection with Hilbert's third problem. Orthoschemes, also called path-simplices in the applied mathematics literature, are a special case of a more general class of simplices studied by Fiedler, and later rediscovered by Coxeter. These simplices are the co ... [...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] |
