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Sperner Family
In combinatorics, a Sperner family (or Sperner system; named in honor of Emanuel Sperner), or clutter, is a family ''F'' of subsets of a finite set ''E'' in which none of the sets contains another. Equivalently, a Sperner family is an antichain in the inclusion lattice over the power set of ''E''. A Sperner family is also sometimes called an independent system or irredundant set. Sperner families are counted by the Dedekind numbers, and their size is bounded by Sperner's theorem and the Lubell–Yamamoto–Meshalkin inequality. They may also be described in the language of hypergraphs rather than set families, where they are called clutters. Dedekind numbers The number of different Sperner families on a set of ''n'' elements is counted by the Dedekind numbers, the first few of which are :2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 . Although accurate asymptotic estimates are known for larger values of ''n'', it is unknown whether there exists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sperner Family N=2
Emanuel Sperner (9 December 1905 – 31 January 1980) was a German mathematician, best known for two theorems. He was born in Waltdorf (near Neiße, Upper Silesia, now Nysa, Poland), and died in Sulzburg-Laufen, West Germany. He was a student at Carolinum in Nysa and then Hamburg University where his advisor was Wilhelm Blaschke. He was appointed Professor in Königsberg in 1934, and subsequently held posts in a number of universities until 1974. Sperner's theorem, from 1928, says that the size of an antichain in the power set of an ''n''-set (a Sperner family) is at most the middle binomial coefficient(s). It has several proofs and numerous generalizations, including the Sperner property of a partially ordered set. Sperner's lemma, from 1928, states that every Sperner coloring of a triangulation of an ''n''-dimensional simplex contains a cell colored with a complete set of colors. It was proven by Sperner to provide an alternate proof of a theorem of Lebesgue characterizing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free 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 partiall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Art Of Computer Programming
''The Art of Computer Programming'' (''TAOCP'') is a comprehensive multi-volume monograph written by the computer scientist Donald Knuth presenting programming algorithms and their analysis. it consists of published volumes 1, 2, 3, 4A, and 4B, with more expected to be released in the future. The Volumes 1–5 are intended to represent the central core of computer programming for sequential machines; the subjects of Volumes 6 and 7 are important but more specialized. When Knuth began the project in 1962, he originally conceived of it as a single book with twelve chapters. The first three volumes of what was then expected to be a seven-volume set were published in 1968, 1969, and 1973. Work began in earnest on Volume 4 in 1973, but was suspended in 1977 for work on typesetting prompted by the second edition of Volume 2. Writing of the final copy of Volume 4A began in longhand in 2001, and the first online pre-fascicle, 2A, appeared later in 2001. The first published installment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. In 2020, most of the editorial board of ''JCTA'' resigned to form a new, |
Minor (graph Theory)
Minor may refer to: Common meanings * Minor (law), a person not under the age of certain legal activities. * Academic minor, a secondary field of study in undergraduate education Mathematics * Minor (graph theory), a relation of one graph to another * Minor (matroid theory), a relation of one matroid to another * Minor (linear algebra), the determinant of a square submatrix Music * Minor chord * Minor interval * Minor key * Minor scale People * Minor (given name), a masculine given name * Minor (surname), a surname Places in the United States * Minor, Alabama, a census-designated place * Minor, Virginia, an unincorporated community * Minor Creek (California) * Minor Creek (Missouri) * Minor Glacier, Wyoming Sports * Minor, a grade in Gaelic games; also, a person who qualifies to play in that grade * Minor league, a sports league not regarded as a premier league ** Minor League Baseball Minor League Baseball (MiLB) is a professional baseball organization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Menger's Theorem
In the mathematical discipline of graph theory, Menger's theorem says that in a finite graph, the size of a minimum cut set is equal to the maximum number of disjoint paths that can be found between any pair of vertices. Proved by Karl Menger in 1927, it characterizes the connectivity of a graph. It is generalized by the max-flow min-cut theorem, which is a weighted, edge version, and which in turn is a special case of the strong duality theorem for linear programs. Edge connectivity The edge-connectivity version of Menger's theorem is as follows: :Let ''G'' be a finite undirected graph and ''x'' and ''y'' two distinct vertices. Then the size of the minimum edge cut for ''x'' and ''y'' (the minimum number of edges whose removal disconnects ''x'' and ''y'') is equal to the maximum number of pairwise edge-disjoint paths from ''x'' to ''y''. The implication for the graph ''G'' is the following version: :A graph is ''k''-edge-connected (it remains connected after removing fe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kőnig's Theorem (graph Theory)
In the mathematics, mathematical area of graph theory, Kőnig's theorem, proved by , describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs. It was discovered independently, also in 1931, by Jenő Egerváry in the more general case of weighted graphs. Setting A vertex cover in a graph is a set of vertices that includes at least one endpoint of every edge, and a vertex cover is ''minimum'' if no other vertex cover has fewer vertices. A matching (graph theory), matching in a graph is a set of edges no two of which share an endpoint, and a matching is ''maximum'' if no other matching has more edges. It is obvious from the definition that any vertex-cover set must be at least as large as any matching set (since for every edge in the matching, at least one vertex is needed in the cover). In particular, the minimum vertex cover set is at least as large as the maximum matching set. Kőnig's theorem states that, in any bip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Cover
In graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. In computer science, the problem of finding a minimum vertex cover is a classical optimization problem. It is NP-hard, so it cannot be solved by a polynomial-time algorithm if P ≠ NP. Moreover, it is hard to approximate – it cannot be approximated up to a factor smaller than 2 if the unique games conjecture is true. On the other hand, it has several simple 2-factor approximations. It is a typical example of an NP-hard optimization problem that has an approximation algorithm. Its decision version, the vertex cover problem, was one of Karp's 21 NP-complete problems and is therefore a classical NP-complete problem in computational complexity theory. Furthermore, the vertex cover problem is fixed-parameter tractable and a central problem in parameterized complexity theory. The minimum vertex cover problem can be formulated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Ideal
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory. Definitions A subset of a partially ordered set (P, \leq) is an ideal, if the following conditions hold: # is non-empty, # for every ''x'' in and ''y'' in ''P'', implies that ''y'' is in ( is a lower set), # for every ''x'', ''y'' in , there is some element ''z'' in , such that and ( is a directed set). While this is the most general way to define an ideal for arbitrary posets, it was originally defined for lattices only. In this case, the following equivalent definition can be given: a subset of a lattice (P, \leq) is an ideal if and only if it is a lower set that is closed under finite joins ( suprema); that is, it is none ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Simplicial Complex
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. Lee, John M., Introduction to Topological Manifolds, Springer 2011, , p153 For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1). In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems. An abstract simplex can be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra. Definitions A collection of non-empty finite subsets of a set ''S'' is called a set-family. A set-family is called an abstract simplicial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Expansion
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function. The theory of asymptotic series was created by Poincaré (and independently by Stieltjes) in 1886. The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. Since a '' convergent'' Taylor s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
