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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] |
Matroid
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 Bryla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vámos Matroid
In mathematics, the Vámos matroid or Vámos cube is a matroid over a set of eight elements that cannot be represented as a matrix over any field. It is named after English mathematician Peter Vámos, who first described it in an unpublished manuscript in 1968. Definition The Vámos matroid has eight elements, which may be thought of as the eight vertices of a cube or cuboid. The matroid has rank 4: all sets of three or fewer elements are independent, and 65 of the 70 possible sets of four elements are also independent. The five exceptions are four-element circuits in the matroid. Four of these five circuits are formed by faces of the cuboid (omitting two opposite faces). The fifth circuit connects two opposite edges of the cuboid, each of which is shared by two of the chosen four faces. Another way of describing the same structure is that it has two elements for each vertex of the diamond graph, and a four-element circuit for each edge of the diamond graph. Properties *The V� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rota's Conjecture
Rota's excluded minors conjecture is one of a number of conjectures made by mathematician Gian-Carlo Rota. It is considered to be an important problem by some members of the structural combinatorics community. Rota conjectured in 1971 that, for every finite field, the family of matroids that can be represented over that field has only finitely many excluded minors. A proof of the conjecture has been announced by Geelen, Gerards, and Whittle. Statement of the conjecture If S is a set of points in a vector space defined over a field F, then the linearly independent subsets of S form the independent sets of a matroid M; S is said to be a representation of any matroid isomorphic to M. Not every matroid has a representation over every field, for instance, the Fano plane is representable only over fields of characteristic two. Other matroids are representable over no fields at all. The matroids that are representable over a particular field form a proper subclass of all matroids. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Unimodular Matrix
In mathematics, a unimodular matrix ''M'' is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over the integers: there is an integer matrix ''N'' that is its inverse (these are equivalent under Cramer's rule). Thus every equation , where ''M'' and ''b'' both have integer components and ''M'' is unimodular, has an integer solution. The ''n'' × ''n'' unimodular matrices form a group called the ''n'' × ''n'' general linear group over \mathbb, which is denoted \operatorname_n(\mathbb). Examples of unimodular matrices Unimodular matrices form a subgroup of the general linear group under matrix multiplication, i.e. the following matrices are unimodular: * Identity matrix * The inverse of a unimodular matrix * The product of two unimodular matrices Other examples include: * Pascal matrices * Permutation matrices * the three transformation matrices in the ternary tree of primitive Pythagorea ... [...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] |
Fano Plane In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite field with two elements. The standard notation for this plane, as a member of a family of projective spaces, is . Here stands for "projective geometry", the first parameter is the geometric dimension (it is a plane, of dimension 2) and the second parameter is the order (the number of points per line, minus one). The Fano plane is an example of a fin |
