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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Matroid ： ウィキペディア英語版 In combinatorics, a branch of mathematics, a matroid is a structure that captures and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid, the most significant being in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions.Matroid theory borrows extensively from the terminology of 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==Hereditary property (matroid) redirects to this section title-->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 Bryzlawski's appendix in White (1986) pp.298–302 for a list of equivalent axiom systems.===Independent sets===In terms of independence, a finite matroid M is a pair (E,\mathcal), where E is a finite set (called the ground set) and \mathcal is a family of subsets of E (called the independent sets) with the following properties:, Section 1.2, "Axiom Systems for a Matroid", pp. 7–9. In combinatorics, a branch of mathematics, a matroid is a structure that captures and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid, the most significant being in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions. Matroid theory borrows extensively from the terminology of 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 Bryzlawski's appendix in White (1986) pp.298–302 for a list of equivalent axiom systems.〕 ===Independent sets=== In terms of independence, a finite matroid $M$ is a pair $(E,\backslash mathcal)$, where $E$ is a finite set (called the ground set) and $\backslash mathcal$ is a family of subsets of $E$ (called the independent sets) with the following properties:〔, Section 1.2, "Axiom Systems for a Matroid", pp. 7–9.〕 # The empty set is independent, i.e., $\backslash emptyset\backslash in\backslash mathcal$. Alternatively, at least one subset of $E$ is independent, i.e., $\backslash mathcal\backslash neq\backslash emptyset$. # Every subset of an independent set is independent, i.e., for each $A\text{'}\backslash subset\; A\backslash subset\; E$, if $A\backslash in\backslash mathcal$ then $A\text{'}\backslash in\backslash mathcal$. This is sometimes called the hereditary property. # If $A$ and $B$ are two independent sets of $\backslash mathcal$ and $A$ has more elements than $B$, then there exists an element in $A$ that when added to $B$ gives a larger independent set than $B$. This is sometimes called the augmentation property or the independent set exchange property. The first two properties define a combinatorial structure known as an independence system. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「In combinatorics, a branch of mathematics, a matroid is a structure that captures and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid, the most significant being in terms of independent sets, bases, circuits, closed sets or flats, closure operators, and rank functions.Matroid theory borrows extensively from the terminology of 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==Hereditary property (matroid) redirects to this section title-->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 Bryzlawski's appendix in White (1986) pp.298–302 for a list of equivalent axiom systems.===Independent sets===In terms of independence, a finite matroid M is a pair (E,\mathcal), where E is a finite set (called the ground set) and \mathcal is a family of subsets of E (called the independent sets) with the following properties:, Section 1.2, "Axiom Systems for a Matroid", pp. 7–9.」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.