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Paving Matroid
In the mathematical theory of matroids, a paving matroid is a matroid in which every circuit has size at least as large as the matroid's rank. In a matroid of rank r every circuit has size at most r+1, so it is equivalent to define paving matroids as the matroids in which the size of every circuit belongs to the set \.. It has been conjectured that almost all matroids are paving matroids. Examples Every simple matroid of rank three is a paving matroid; for instance this is true of the Fano matroid. The Vámos matroid provides another example, of rank four. Uniform matroids of rank r have the property that every circuit is of length exactly r+1 and hence are all paving matroids; the converse does not hold, for example, the cycle matroid of the complete graph K_4 is paving but not uniform. A Steiner system S(t,k,v) is a pair (S,\mathcal) where S is a finite set of size v and \mathcal is a family of k-element subsets of S with the property that every t-element subset of S is also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vamos Matroid
Vamos (Greek language, Greek: Βάμος) is a small town and former municipality in the Chania (regional unit), Chania regional unit, Crete, Greece. Since the 2011 local government reform "Kallikratis" it is a municipal unit, part of the municipality of Apokoronas, serving as its historical capital. It is situated on a small hill at an altitude of above sea level, about from Chania. In Vamos, one can find several restaurants, snack bars and shops in the village, as well as many public services, such as a fully equipped health center, schools, police station and the regional court for the regions of Apokoronas and Sfakia. The village is said to have been founded by Arab invaders during the 8th century, which sought refuge in Crete after being expelled from Andalusia. The first official record of the name (Vamo/Vamu) is found on a map of 1577 made by Francesco Barozzi, and according to the Venetian census of 1583 it had 271 inhabitants. The village followed the fate of the rest of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steiner System
250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and ''t'' = 2 or (recently) ''t'' ≥ 2. A Steiner system with parameters ''t'', ''k'', ''n'', written S(''t'',''k'',''n''), is an ''n''-element set ''S'' together with a set of ''k''-element subsets of ''S'' (called blocks) with the property that each ''t''-element subset of ''S'' is contained in exactly one block. In an alternate notation for block designs, an S(''t'',''k'',''n'') would be a ''t''-(''n'',''k'',1) design. This definition is relatively new. The classical definition of Steiner systems also required that ''k'' = ''t'' + 1. An S(2,3,''n'') was (and still is) called a ''Steiner triple'' (or ''triad'') ''system'', while an S(3,4,''n'') is called a ''Steiner quadr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rota's Basis Conjecture
In linear algebra and matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases, named after Gian-Carlo Rota. It states that, if ''X'' is either a vector space of dimension ''n'' or more generally a matroid of rank ''n'', with ''n'' disjoint bases ''Bi'', then it is possible to arrange the elements of these bases into an ''n'' × ''n'' matrix in such a way that the rows of the matrix are exactly the given bases and the columns of the matrix are also bases. That is, it should be possible to find a second set of ''n'' disjoint bases ''Ci'', each of which consists of one element from each of the bases ''Bi''. Examples Rota's basis conjecture has a simple formulation for points in the Euclidean plane: it states that, given three triangles with distinct vertices, with each triangle colored with one of three colors, it must be possible to regroup the nine triangle vertices into three "rainbow" triangles having one vertex of each col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gian-Carlo Rota
Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, probability theory, and phenomenology. Early life and education Rota was born in Vigevano, Italy. His father, Giovanni, an architect and prominent antifascist, was the brother of the mathematician Rosetta, who was the wife of the writer Ennio Flaiano. Gian-Carlo's family left Italy when he was 13 years old, initially going to Switzerland. Rota attended the Colegio Americano de Quito in Ecuador, and graduated with an A.B. in mathematics from Princeton University in 1953 after completing a senior thesis, titled "On the solubility of linear equations in topological vector spaces", under the supervision of William Feller. He then pursued graduate studies at Yale University, where he received a Ph.D. in mathematics in 1956 after completing a d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henry Crapo (mathematician)
Henry Howland Crapo (; August 12, 1932 – September 3, 2019) was an American-Canadian mathematician who worked in algebraic combinatorics. Over the course of his career, he held positions at several universities and research institutes in Canada and France. He is noted for his work in matroid theory and lattice theory. Education and career Crapo was born in Detroit, Michigan, in 1932. He received his Ph.D. in 1964 under the supervision of Gian-Carlo Rota and Kenneth Hoffman. He held academic positions at the University of Waterloo, Université de Montréal, INRIA Rocquencourt, and École des Hautes Études en Sciences Sociales. During his time in Waterloo, Crapo became a Canadian citizen. Crapo is known for his early work in matroid theory, and for related work in lattice theory. He introduced the beta invariant of a matroid, and published the first paper on the Tutte polynomial (though Tutte had already defined an equivalent polynomial in his thesis). Together with Gia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Enumeration
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets ''S''''i'' indexed by the natural numbers, enumerative combinatorics seeks to describe a ''counting function'' which counts the number of objects in ''S''''n'' for each ''n''. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. The twelvefold way provides a unified framework for counting permutations, combinations and partitions. The simplest such functions are ''closed formulas'', which can be expressed as a composition of elementary functions such as factorials, powers, and so on. For instance, as shown below, the number of different possible orderings of a deck of ''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Family Of Sets
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. The term "collection" is used here because, in some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class rather than a set. A finite family of subsets of a finite set S is also called a '' hypergraph''. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions. Examples The set of all subsets of a given set S is called the power set of S and is denoted by \wp(S). The power set \wp(S) of a given set S is a family of sets over S. A subset of S having k elements is called a k-subset of S. The k-subsets S^ of a set S form a family of sets. Let S = \. An e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: :\. Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set is called finite if there exists a bijection :f\colon S\to\ for some natural number . The number is the set's cardinality, denoted as . The empty set o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has edges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Matroid
In the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the forests in a given finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids. A matroid that is both graphic and co-graphic is sometimes called a planar matroid (but this should not be confused with matroids of rank 3, which generalize planar point configurations); these are exactly the graphic matroids formed from planar graphs. Definition A matroid may be defined as a family of finite sets (called the "independent sets" of the matroid) that is closed under subsets and that satisfies the "exchange property": if sets A and B are both independent, and A is larger than B, then there is an element x\in A\setminus B such that B\cup\ remains independent. If G is an undirected graph, and F is the family of sets of edges that form forests in G, then F is clearly closed under subsets (re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Matroid
In mathematics, a uniform matroid is a matroid in which the independent sets are exactly the sets containing at most ''r'' elements, for some fixed integer ''r''. An alternative definition is that every permutation of the elements is a symmetry. Definition The uniform matroid U^r_n is defined over a set of n elements. A subset of the elements is independent if and only if it contains at most r elements. A subset is a basis if it has exactly r elements, and it is a circuit if it has exactly r+1 elements. The rank of a subset S is \min(, S, ,r) and the rank of the matroid is r. A matroid of rank r is uniform if and only if all of its circuits have exactly r+1 elements. The matroid U^2_n is called the n-point line. Duality and minors The dual matroid of the uniform matroid U^r_n is another uniform matroid U^_n. A uniform matroid is self-dual if and only if r=n/2. Every minor of a uniform matroid is uniform. Restricting a uniform matroid U^r_n by one element (as long as r 0) p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
