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Criss-cross Algorithm
In optimization (mathematics), mathematical optimization, the criss-cross algorithm is any of a family of algorithms for linear programming. Variants of the criss-cross algorithm also solve more general problems with linear programming, linear inequality constraints and nonlinear programming, nonlinear optimization (mathematics), objective functions; there are criss-cross algorithms for linear-fractional programming problems, quadratic programming, quadratic-programming problems, and linear complementarity problems. Like the simplex algorithm of George Dantzig, George B. Dantzig, the criss-cross algorithm is not a time complexity, polynomial-time algorithm for linear programming. Both algorithms visit all 2''D'' corners of a (perturbed) unit cube, cube in dimension ''D'', the Klee–Minty cube (after Victor Klee and George J. Minty), in the worst-case complexity, worst case. However, when it is started at a random corner, the criss-cross algorithm Average-case com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitcube
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 space, ''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 [0, 1]''n'' of all ''n''-tuples of numbers in the interval [0, 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 (mathematics), ''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 R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tamas Terlaky
Tamas may refer to: * ''Tamas'' (philosophy), a concept of darkness and death in Hindu philosophy * Tamás (name), a given name in Hungarian (Thomas) * ''Tamas'' (novel), a 1975 novel about the partition of India by Bhisham Sahni ** ''Tamas'' (film), a 1987 TV series and film adaptation by Govind Nihalani * Christian Tămaș, Romanian writer * Gabriel Tamaș (born 1983), Romanian footballer * Vladimir Tămaș, Romanian footballer See also * Tama (other) Tama may mean: Languages * Tama language, the language of the Sudanese Tama people * Tama languages, a language family of northern Papua New Guinea Music * Tama Drums, a Japanese brand manufactured by Hoshino Gakki * Tama (percussion), a type o ... {{Disambiguation, surname hu:Tamás ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Operation
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. Arithmetic systems can be distinguished based on the type of numbers they operate on. Integer arithmetic is about calculations with positive and negative integers. Rational number arithmetic involves operations on fractions of integers. Real number arithmetic is about calculations with real numbers, which include both rational and irrational numbers. Another distinction is based on the numeral system employed to perform calculations. Decimal arithmetic is the most common. It uses the basic numerals from 0 to 9 and their combinations to express numbers. Binary arithmetic, by contrast, is used by most computers and represents numbers as combinations of the basic numerals 0 and 1. Computer arithmetic deals with the specificities of the implem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extreme Point
In mathematics, an extreme point of a convex set S in a Real number, real or Complex number, complex vector space is a point in S that does not lie in any open line segment joining two points of S. The extreme points of a line segment are called its ''endpoint (geometry), endpoints''. In linear programming problems, an extreme point is also called ''vertex (geometry), vertex'' or ''corner point'' of S. Definition Throughout, it is assumed that X is a Real number, real or Complex number, complex vector space. For any p, x, y \in X, say that p x and y if x \neq y and there exists a 0 < t < 1 such that If is a subset of and then is called an of if it does not lie between any two points of That is, if there does exist and such that and The s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krein–Milman Theorem
In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). This theorem generalizes to infinite-dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex (i.e. "filled") triangle, including its perimeter and the area "inside of it", is equal to the convex hull of its three vertices, where these vertices are exactly the extreme points of this shape. This observation also holds for any other convex polygon in the plane \R^2. Statement and definitions Preliminaries and definitions Throughout, X will be a real or complex vector space. For any elements x and y in a vector space, the set , y:= \ is called the or closed interval between x and y. The or open interval between x and y is (x, y) := \varnothing when x = y while it is (x, y) := \ when x \neq y; it satisfies (x, y) = , y\setminus \ and , y= (x, y) \cup \. The points ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halfspace
Half-space may refer to: * Half-space (geometry), either of the two parts into which a plane divides Euclidean space ** (Poincaré) Half-space model, a model of hyperbolic geometry using a Euclidean half-space ** Siegel upper half-space In mathematics, the Siegel upper half-space of degree ''g'' (or genus ''g'') (also called the Siegel upper half-plane) is the set of ''g'' × ''g'' symmetric matrices over the complex numbers whose imaginary part is positive definite. It ..., a set of complex matrices with positive definite imaginary part * Half-space (punctuation), a spacing character half the width of a regular space * Half-space model (oceanography), an estimate for seabed height in areas without significant subduction {{disambig ... [...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 avoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl's Theorem
In mathematics, Weyl's theorem or Weyl's lemma might refer to one of a number of results of Hermann Weyl. These include * the Peter–Weyl theorem * Weyl's theorem on complete reducibility, results originally derived from the unitarian trick on representation theory of semisimple groups and semisimple Lie algebras * Weyl's theorem on eigenvalues * Weyl's criterion for equidistribution (Weyl's criterion) * Weyl's lemma on the hypoellipticity of the Laplace equation * results estimating Weyl sums in the theory of exponential sums * Weyl's inequality * Weyl's criterion for a number to be in the essential spectrum of an operator * Weyl's law that describes the asymptotic behavior of large eigenvalues of the Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named aft .. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Farkas Lemma
In mathematics, Farkas' lemma is a solvability theorem for a finite system of linear inequalities. It was originally proven by the Hungarian mathematician Gyula Farkas. Farkas' lemma is the key result underpinning the linear programming duality and has played a central role in the development of mathematical optimization (alternatively, mathematical programming). It is used amongst other things in the proof of the Karush–Kuhn–Tucker theorem in nonlinear programming. Remarkably, in the area of the foundations of quantum theory, the lemma also underlies the complete set of Bell inequalities in the form of necessary and sufficient conditions for the existence of a local hidden-variable theory, given data from any specific set of measurements. Generalizations of the Farkas' lemma are about the solvability theorem for convex inequalities, i.e., infinite system of linear inequalities. Farkas' lemma belongs to a class of statements called "theorems of the alternative": a theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steinitz's Theorem
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedron, convex polyhedra: they are exactly the vertex connectivity, 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. This result provides a classification theorem for the three-dimensional convex polyhedra, something that is not known in higher dimensions. It provides a complete and purely combinatorial description of the graphs of these polyhedra, allowing other results on them, such as Eberhard's theorem on the realization of polyhedra with given types of faces, to be proven more easily, without reference to the geometry of these shapes. Additionally, it has been applied in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathematics), matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
