en>Mark viking: /* Further properties */ Disambig Archimedean

2013-06-25T22:22:28Z

Further properties: Disambig Archimedean

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	~~Hi there~~. ~~Allow me start by introducing~~ the ~~author~~, ~~her name~~ is ~~Sophia Boon~~ but ~~she never really favored~~ that name. ~~For many years he~~'s ~~been living in Alaska and he doesn~~'~~t strategy~~ on ~~changing~~ it. ~~Invoicing~~ is ~~my profession~~. ~~Playing badminton~~ is ~~a factor~~ that he is ~~completely addicted to~~.<br><br>~~My page~~; [~~http:~~//~~chungmuroresidence~~.~~com~~/xe/~~reservation_branch2~~/~~152663 authentic psychic readings~~]		In mathematics, a '''regular matroid''' is a [[matroid]] that can be [[Matroid representation\|represented]] over all [[field (mathematics)\|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 [[Field (mathematics)\|fields]] lead to different sets of matroids that can be constructed from them.

			A matroid <math>M</math> is regular when, for every field <math>F</math>, <math>M</math> can be represented by a system of vectors over <math>F</math>.<ref name="sfo">{{citation
			\| last = Fujishige \| first = Satoru
			\| isbn = 9780444520869
			\| page = 24
			\| publisher = Elsevier
			\| series = Annals of Discrete Mathematics
			\| title = Submodular Functions and Optimization
			\| year = 2005}}.</ref><ref>{{citation
			\| last = Oxley \| first = James G. \| authorlink = James Oxley
			\| isbn = 9780199202508
			\| page = 209
			\| publisher = Oxford University Press
			\| series = Oxford Graduate Texts in Mathematics
			\| title = Matroid Theory
			\| volume = 3
			\| year = 2006}}.</ref>

			==Properties==
			If a matroid is regular, so is its [[dual matroid]],<ref name="sfo"/> and so is every one of its [[matroid minor\|minors]].<ref>{{harvtxt\|Oxley\|2006}}, p. 112.</ref> Every direct sum of regular matroids remains regular.<ref>{{harvtxt\|Oxley\|2006}}, p. 131.</ref>

			Every [[graphic matroid]] (and every co-graphic matroid) is regular.<ref>{{citation
			\| last = Tutte \| first = W. T.
			\| journal = Journal of Research of the National Bureau of Standards
			\| mr = 0179781
			\| pages = 1–47
			\| title = Lectures on matroids
			\| url = http://cdm16009.contentdm.oclc.org/cdm/ref/collection/p13011coll6/id/66650
			\| volume = 69B
			\| year = 1965}}.</ref> Conversely, every regular matroid may be constructed by combining graphic matroids, co-graphic matroids, and a certain ten-element matroid that is neither graphic nor co-graphic, using an operation for combining matroids that generalizes the [[clique-sum]] operation on graphs.<ref>{{citation
			\| last = Seymour \| first = P. D. \| authorlink = Paul Seymour (mathematician)
			\| doi = 10.1016/0095-8956(80)90075-1
			\| issue = 3
			\| journal = [[Journal of Combinatorial Theory]]
			\| mr = 579077
			\| pages = 305–359
			\| series = Series B
			\| title = Decomposition of regular matroids
			\| volume = 28
			\| year = 1980}}.</ref>

			The number of bases in a regular matroid may be computed as the [[determinant]] of an associated matrix, generalizing [[Kirchhoff's theorem\|Kirchhoff's matrix-tree theorem]] for [[graphic matroid]]s.<ref>{{citation
			\| last = Maurer \| first = Stephen B.
			\| issue = 1
			\| journal = SIAM Journal on Applied Mathematics
			\| mr = 0392635
			\| pages = 143–148
			\| title = Matrix generalizations of some theorems on trees, cycles and cocycles in graphs
			\| volume = 30
			\| year = 1976}}.</ref>

			==Characterizations==
			The [[uniform matroid]] <math>U{}^2_4</math> (the four-point line) is not regular: it cannot be realized over the two-element [[finite field]] [[GF(2)]], so it is not a [[binary matroid]], although it can be realized over all other fields. The matroid of the [[Fano plane]] (a rank-three matroid in which seven of the triples of points are dependent) and its dual are also not regular: they can be realized over GF(2), and over all fields of characteristic two, but not over any other fields than those. As {{harvtxt\|Tutte\|1958}} showed, these three examples are fundamental to the theory of regular matroids: every non-regular matroid has at least one of these three as a [[matroid minor\|minor]]. Thus, the regular matroids are exactly the matroids that do not have one of the three forbidden minors <math>U{}^2_4</math>, the Fano plane, or its dual.<ref name="t58">{{citation
			\| last = Tutte \| first = W. T. \| authorlink = W. T. Tutte
			\| journal = Transactions of the American Mathematical Society
			\| mr = 0101526
			\| pages = 144–174
			\| title = A homotopy theorem for matroids. I, II
			\| volume = 88
			\| year = 1958}}.</ref>

			If a matroid is regular, it must clearly be realizable over the two fields GF(2) and GF(3). The converse is true: every matroid that is realizable over both of these two fields is regular. The result follows from a forbidden minor characterization of the matroids realizable over these fields, part of a family of results codified by [[Rota's conjecture]].<ref>{{citation
			\| last = Seymour \| first = P. D. \| authorlink = Paul Seymour (mathematician)
			\| doi = 10.1016/0095-8956(79)90055-8
			\| issue = 2
			\| journal = [[Journal of Combinatorial Theory]]
			\| mr = 532586
			\| pages = 159–173
			\| series = Series B
			\| title = Matroid representation over GF(3)
			\| volume = 26
			\| year = 1979}}.</ref>

			The regular matroids are the matroids that can be defined from a [[unimodular matrix\|totally unimodular matrix]], a matrix in which every square submatrix has determinant 0, 1, or −1. The vectors realizing the matroid may be taken as the rows of the matrix. For this reason, regular matroids are sometimes also called '''unimodular matroids'''.<ref>{{harvtxt\|Oxley\|2006}}, p. 20.</ref> The equivalence of regular matroids and unimodular matrices, and their characterization by forbidden minors, are deep results of [[W. T. Tutte]], originally proved by him using the [[Tutte homotopy theorem]].<ref name="t58"/> {{harvtxt\|Gerards\|1989}} later published an alternative and simpler proof of the characterization of unimodular matrices by forbidden minors.<ref>{{citation\|last=Gerards\|first=A. M. H.\|year=1989\|title=A short proof of Tutte's characterization of totally unimodular matrices\|journal=Linear Algebra and its Applications\|volume=114/115\|pages=207–212\|doi=10.1016/0024-3795(89)90461-8}}.</ref>

			==Algorithms==
			There is a [[polynomial time]] algorithm for testing whether a matroid is regular, given access to the matroid through an [[matroid oracle\|independence oracle]].<ref>{{citation
			\| last = Truemper \| first = K.
			\| issue = 3
			\| journal = European Journal of Combinatorics
			\| mr = 679212
			\| pages = 275–291
			\| title = On the efficiency of representability tests for matroids
			\| volume = 3
			\| year = 1982}}.</ref>

			==References==
			{{reflist\|colwidth=30em}}

			[[Category:Matroid theory]]

en>Amkilpatrick: Help needed: Archimedean

2011-12-25T09:52:14Z

Help needed: Archimedean

New page

Hi there. Allow me start by introducing the author, her name is Sophia Boon but she never really favored that name. For many years he's been living in Alaska and he doesn't strategy on changing it. Invoicing is my profession. Playing badminton is a factor that he is completely addicted to.<br><br>My page; [http://chungmuroresidence.com/xe/reservation_branch2/152663 authentic psychic readings]

Generalised metric - Revision history

en>Mark viking: /* Further properties */ Disambig Archimedean

en>Amkilpatrick: Help needed: Archimedean