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Gammoid
In matroid theory, a field within mathematics, a gammoid is a certain kind of matroid, describing sets of vertices that can be reached by vertex-disjoint paths in a directed graph. The concept of a gammoid was introduced and shown to be a matroid by , based on considerations related to Menger's theorem characterizing the obstacles to the existence of systems of disjoint paths. Gammoids were given their name by . and studied in more detail by .. Definition Let G be a directed graph, S be a set of starting vertices, and T be a set of destination vertices (not necessarily disjoint from S). The gammoid \Gamma derived from this data has T as its set of elements. A subset I of T is independent in \Gamma if there exists a set of vertex-disjoint paths whose starting points all belong to S and whose ending points are exactly I.. A strict gammoid is a gammoid in which the set T of destination vertices consists of every vertex in G. Thus, a gammoid is a restriction of a strict gammoid, to ... [...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-independent paths from ''x'' to ''y''. :Extended to all pairs: a graph is ''k''-edge-connected (it remains connected after removing fewer than ''k'' edges) if and only if e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid Theory
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terminology of both linear algebra and graph theory, largely because it is the abstraction of various notions of central importance in these fields. Matroids have found applications in geometry, topology, combinatorial optimization, network theory and coding theory. Definition There are many equivalent ( cryptomorphic) ways to define a (finite) matroid.A standard source for basic definitions and results about matroids is Oxley (1992). An older standard source is Welsh (1976). See Brylawski' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aubrey Ingleton
Aubrey is traditionally a male English given name. The name is from the French derivation Aubry of the Germanic given name Alberic / Old High German given name Alberich, which consists of the elements ALF "elf" and RIK "king", from Proto-Germanic ''*albiz'' "elf", "supernatural being" and ''*rīkaz'' "chieftain", "ruler". Before the Norman conquest, the Anglo-Saxons used the corresponding variant ''Ælf-rīc'' (see Ælfric). The feminine form Aubrey is sometimes from Old French Aubree with a different etymology: Albereda,François de Beaurepaire, ''Les noms des communes et anciennes paroisses de l'Eure'', éditions Picard, 1981, p. 123 sometimes a feminine used of the masculine name Aubrey. However, Aubrey is commonly used as a feminine name in the United States. It was the 15th most popular girl's name in the United States in 2012. People Surname * Andrew Aubrey, Lord Mayor of London in 1339, 1340, and 1351 * Anne Aubrey (born 1935), English actress * Brandon Aubrey (born 19 ... [...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. An electronic, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid Representation
In the mathematical theory of matroids, a matroid representation is a family of vectors whose linear independence relation is the same as that of a given matroid. Matroid representations are analogous to group representations; both types of representation provide abstract algebraic structures (matroids and groups respectively) with concrete descriptions in terms of linear algebra. A linear matroid is a matroid that has a representation, and an ''F''-linear matroid (for a field ''F'') is a matroid that has a representation using a vector space over ''F''. Matroid representation theory studies the existence of representations and the properties of linear matroids. Definitions A (finite) matroid (E,\mathcal) is defined by a finite set E (the elements of the matroid) and a non-empty family \mathcal of the subsets of E, called the independent sets of the matroid. It is required to satisfy the properties that every subset of an independent set is itself independent, and that if one ind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Matroid
In mathematics, a regular matroid is a matroid that can be represented over all fields. Definition A matroid is defined to be a family of subsets of a finite set, satisfying certain axioms. The sets in the family are called "independent sets". One of the ways of constructing a matroid is to select a finite set of vectors in a vector space, and to define a subset of the vectors to be independent in the matroid when it is linearly independent in the vector space. Every family of sets constructed in this way is a matroid, but not every matroid can be constructed in this way, and the vector spaces over different fields lead to different sets of matroids that can be constructed from them. A matroid M is regular when, for every field F, M can be represented by a system of vectors over F.. Properties If a matroid is regular, so is its dual matroid, and so is every one of its minors. Every direct sum of regular matroids remains regular. Every graphic matroid (and every co-graphic matro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid Minor
In the mathematical theory of matroids, a minor of a matroid ''M'' is another matroid ''N'' that is obtained from ''M'' by a sequence of restriction and contraction operations. Matroid minors are closely related to graph minors, and the restriction and contraction operations by which they are formed correspond to edge deletion and edge contraction operations in graphs. The theory of matroid minors leads to structural decompositions of matroids, and characterizations of matroid families by forbidden minors, analogous to the corresponding theory in graphs. Definitions If ''M'' is a matroid on the set ''E'' and ''S'' is a subset of ''E'', then the restriction of ''M'' to ''S'', written ''M'' , ''S'', is the matroid on the set ''S'' whose independent sets are the independent sets of ''M'' that are contained in ''S''. Its circuits are the circuits of ''M'' that are contained in ''S'' and its rank function is that of ''M'' restricted to subsets of ''S''. If ''T'' is an independent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mike Piff
Mike may refer to: Animals * Mike (cat), cat and guardian of the British Museum * Mike the Headless Chicken, chicken that lived for 18 months after his head had been cut off * Mike (chimpanzee), a chimpanzee featured in several books and documentaries Arts * Mike (miniseries), a 2022 Hulu limited series based on the life of American boxer Mike Tyson * Mike (2022 film), a Malayalam film produced by John Abraham * ''Mike'' (album), an album by Mike Mohede * ''Mike'' (1926 film), an American film * MIKE (musician), American rapper, songwriter and record * ''Mike'' (novel), a 1909 novel by P. G. Wodehouse * "Mike" (song), by Elvana Gjata and Ledri Vula featuring John Shahu * Mike (''Twin Peaks''), a character from ''Twin Peaks'' * "Mike", a song by Xiu Xiu from their 2004 album ''Fabulous Muscles'' Businesses * Mike (cellular network), a defunct Canadian cellular network * Mike and Ike, a candies brand Military * MIKE Force, a unit in the Vietnam War * Ivy Mike, the first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a 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 cycles. The two sets U and V may be thought of as a 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 triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Matroid
In matroid theory, the dual of a matroid M is another matroid M^\ast that has the same elements as M, and in which a set is independent if and only if M has a basis set disjoint from it... Matroid duals go back to the original paper by Hassler Whitney defining matroids.. Reprinted in , pp. 55–79. See in particular section 11, "Dual matroids", pp. 521–524. They generalize to matroids the notions of plane graph duality. Basic properties Duality is an involution: for all M, (M^\ast)^\ast=M. An alternative definition of the dual matroid is that its basis sets are the complements of the basis sets of M. The basis exchange axiom, used to define matroids from their bases, is self-complementary, so the dual of a matroid is necessarily a matroid. The flats of M are complementary to the cyclic sets (unions of circuits) of M^\ast, and vice versa. If r is the rank function of a matroid M on ground set E, then the rank function of the dual matroid is r^\ast(S)=r(E \setminus S)+, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
