|
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] |
Jack Edmonds
Jack R. Edmonds (born April 5, 1934) is an American-born and educated computer scientist and mathematician who lived and worked in Canada for much of his life. He has made fundamental contributions to the fields of combinatorial optimization, polyhedral combinatorics, discrete mathematics and the theory of computing. He was the recipient of the 1985 John von Neumann Theory Prize. Early career Edmonds attended Duke University before completing his undergraduate degree at George Washington University in 1957. He thereafter received a master's degree in 1960 at the University of Maryland under Bruce L. Reinhart with a thesis on the problem of embedding graphs into surfaces. From 1959 to 1969 he worked at the National Institute of Standards and Technology (then the National Bureau of Standards), and was a founding member of Alan Goldman’s newly created Operations Research Section in 1961. Goldman proved to be a crucial influence by enabling Edmonds to work in a RAND Corporation-sp ... [...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 mathematician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supermodular
In mathematics, a function :f\colon \mathbb^k \to \mathbb is supermodular if : f(x \uparrow y) + f(x \downarrow y) \geq f(x) + f(y) for all x, y \isin \mathbb^, where x \uparrow y denotes the componentwise maximum and x \downarrow y the componentwise minimum of x and y. If −''f'' is supermodular then ''f'' is called submodular, and if the inequality is changed to an equality the function is modular. If ''f'' is twice continuously differentiable, then supermodularity is equivalent to the condition : \frac \geq 0 \mbox i \neq j. Supermodularity in economics and game theory The concept of supermodularity is used in the social sciences to analyze how one agent's decision affects the incentives of others. Consider a symmetric game with a smooth payoff function \,f defined over actions \,z_i of two or more players i \in . Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval: z_i \in ,b/math>. In this context, supermodularity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Publishing
Springer Publishing Company is an American publishing company of academic journals and books, focusing on the fields of nursing, gerontology, psychology, social work, counseling, public health, and rehabilitation (neuropsychology). It was established in 1951 by Bernhard Springer, a great-grandson of Julius Springer, and is based in Midtown Manhattan, New York City. History Springer Publishing Company was founded in 1950 by Bernhard Springer, the Berlin-born great-grandson of Julius Springer, who founded Springer-Verlag (now Springer Science+Business Media). Springer Publishing's first landmark publications included ''Livestock Health Encyclopedia'' by R. Seiden and the 1952 ''Handbook of Cardiology for Nurses''. The company's books soon branched into other fields, including medicine and psychology. Nursing publications grew rapidly in number, as Modell's ''Drugs in Current Use'', a small annual paperback, sold over 150,000 copies over several editions. Solomon Garb's ''Labo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomial Ideal
In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. A toric ideal is an ideal generated by differences of monomials (provided the ideal is a prime ideal). An affine or projective algebraic variety defined by a toric ideal or a homogeneous toric ideal is an affine or projective toric variety, possibly non-normal. Definitions and Properties Let \mathbb be a field and R = \mathbb /math> be the polynomial ring over \mathbb with ''n'' variables x = x_1, x_2, \dotsc, x_n. A monomial in R is a product x^ = x_1^ x_2^ \cdots x_n^ for an ''n''-tuple \alpha = (\alpha_1, \alpha_2, \dotsc, \alpha_n) \in \mathbb^n of nonnegative integers. The following three conditions are equivalent for an ideal I \subseteq R: # I is generated by monomials, # If f = \sum_ c_ x^ \in I , then x^ \in I , provided that c_ is nonzero. # I is torus fixed, i.e, given (c_1, c_2, \dotsc, c_n) \in (\mathbb^*)^ , then I is fixed under t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics". The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Points
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension n which spans the vector space \mathbb^n. For any basis of \mathbb^n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a regul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spanning Set
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized either as the intersection of all linear subspaces that contain , or as the smallest subspace containing . The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to matroids and modules. To express that a vector space is a linear span of a subset , one commonly uses the following phrases—either: spans , is a spanning set of , is spanned/generated by , or is a generator or generator set of . Definition Given a vector space over a field , the span of a set of vectors (not necessarily infinite) is defined to be the intersection of all subspaces of that contain . is referred to as the subspace ''spanned by'' , or by the vectors in . Conversely, is called a ''spanning set'' of , and we ... [...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] |
Submodular Function
In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an input set decreases as the size of the input set increases. Submodular functions have a natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical networks. Recently, submodular functions have also found immense utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, Active learning (machine learning), active learning, sensor placement, image collection summarization and many other domains. Definition If \Omega is a finite set (mathematics), set, a submodular function is a set function f:2^\righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact (mathematics)
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
