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Geometric Lattice
In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumption of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of flats of finite and infinite matroids, and every geometric or matroid lattice comes from a matroid in this way. Definition A lattice is a poset in which any two elements x and y have both a least upper bound, called the join or supremum, denoted by x\vee y, and a greatest lower bound, called the meet or infimum, denoted by x\wedge y. : The following definitions apply to posets in general, not just lattices, except where otherwise stated. * For a minimal element x, there is no element y such that y y or y y and there is no element z distinct from both x and y so that x > z > y. * A cover of a minimal element is called an atom. * A lattice is atomistic if every element is the supremum of some set of ato ... [...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 Brylawsk ... [...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] |
Robert P
The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honour, praise, renown" and ''berht'' "bright, light, shining"). It is the second most frequently used given name of ancient Germanic origin. It is also in use as a surname. Another commonly used form of the name is Rupert. After becoming widely used in Continental Europe it entered England in its Old French form ''Robert'', where an Old English cognate form (''Hrēodbēorht'', ''Hrodberht'', ''Hrēodbēorð'', ''Hrœdbœrð'', ''Hrœdberð'', ''Hrōðberχtŕ'') had existed before the Norman Conquest. The feminine version is Roberta. The Italian, Portuguese, and Spanish form is Roberto. Robert is also a common name in many Germanic languages, including English, German, Dutch, Norwegian, Swedish, Scots, Danish, and Icelandic. It can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complemented Lattice
In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element ''a'' has a complement, i.e. an element ''b'' satisfying ''a'' ∨ ''b'' = 1 and ''a'' ∧ ''b'' = 0. Complements need not be unique. A relatively complemented lattice is a lattice such that every interval 'c'', ''d'' viewed as a bounded lattice in its own right, is a complemented lattice. An orthocomplementation on a complemented lattice is an involution that is order-reversing and maps each element to a complement. An orthocomplemented lattice satisfying a weak form of the modular law is called an orthomodular lattice. In distributive lattices, complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra. Definition and basic properties A complemented lattice is a bounded lattice (with least eleme ... [...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] |
Canadian Mathematical Bulletin
The ''Canadian Mathematical Bulletin'' (french: Bulletin Canadien de Mathématiques) is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antonio Lei and Javad Mashreghi. The journal publishes short articles in all areas of mathematics that are of sufficient interest to the general mathematical public. Abstracting and indexing The journal is abstracted in: for the Canadian Mathematical Bulletin. * '''' * '' [...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] |
Order Embedding
In order theory, a branch of mathematics, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism. Both of these weakenings may be understood in terms of category theory. Formal definition Formally, given two partially ordered sets (posets) (S, \leq) and (T, \preceq), a function f: S \to T is an ''order embedding'' if f is both order-preserving and order-reflecting, i.e. for all x and y in S, one has : x\leq y \text f(x)\preceq f(y).. Such a function is necessarily injective, since f(x) = f(y) implies x \leq y and y \leq x. If an order embedding between two posets S and T exists, one says that S can be embedded into T. Properties An order isomorphism can be characterized as a surjective order embedding. As a consequence, any order embedding ''f'' restricts to an isomorph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoint or adjunction may mean: * Adjoint of a linear map, also called its transpose * Hermitian adjoint (adjoint of a linear operator) in functional analysis * Adjoint endomorphism of a Lie algebra * Adjoint representation of a Lie group * Adjoint functors in category theory * Adjunction (field theory) * Adjunction formula (algebraic geometry) * Adjunction space in topology * Conjugate transpose of a matrix in linear algebra * Adjugate matrix, related to its inverse * Adjoint equation * The upper and lower adjoints of a Galois connection in order theory * The adjoint of a differential operator with general polynomial coefficients * Kleisli adjunction * Monoidal adjunction * Quillen adjunction * Axiom of adjunction In mathematical set theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (order Theory)
In the mathematical area of order theory, every partially ordered set ''P'' gives rise to a dual (or opposite) partially ordered set which is often denoted by ''P''op or ''P''''d''. This dual order ''P''op is defined to be the same set, but with the inverse order, i.e. ''x'' ≤ ''y'' holds in ''P''op if and only if ''y'' ≤ ''x'' holds in ''P''. It is easy to see that this construction, which can be depicted by flipping the Hasse diagram for ''P'' upside down, will indeed yield a partially ordered set. In a broader sense, two partially ordered sets are also said to be duals if they are dually isomorphic, i.e. if one poset is order isomorphic to the dual of the other. The importance of this simple definition stems from the fact that every definition and theorem of order theory can readily be transferred to the dual order. Formally, this is captured by the Duality Principle for ordered sets: : If a given statement is valid for all partially ordered sets, then its dual statement, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complement (set Theory)
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the set of elements in that are not in . The relative complement of with respect to a set , also termed the set difference of and , written B \setminus A, is the set of elements in that are not in . Absolute complement Definition If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in : A^\complement = U \setminus A. Or formally: A^\complement = \. The absolute complement of is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Element
In mathematics, especially in order theory, a maximal element of a subset ''S'' of some preordered set is an element of ''S'' that is not smaller than any other element in ''S''. A minimal element of a subset ''S'' of some preordered set is defined dually as an element of ''S'' that is not greater than any other element in ''S''. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset S of a preordered set is an element of S which is greater than or equal to any other element of S, and the minimum of S is again defined dually. In the particular case of a partially ordered set, while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements. Specializing further to totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide. As an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
