|
Biased Graph
{{Short description, Graph with a list of distinguished cycles In mathematics, a biased graph is a graph with a list of distinguished circles (edge sets of simple cycles), such that if two circles in the list are contained in a theta graph, then the third circle of the theta graph is also in the list. A biased graph is a generalization of the combinatorial essentials of a gain graph and in particular of a signed graph. Formally, a biased graph Ω is a pair (''G'', ''B'') where ''B'' is a linear class of circles; this by definition is a class of circles that satisfies the theta-graph property mentioned above. A subgraph or edge set whose circles are all in ''B'' (and which contains no half-edges) is called balanced. For instance, a circle belonging to ''B'' is ''balanced'' and one that does not belong to ''B'' is ''unbalanced''. Biased graphs are interesting mostly because of their matroids, but also because of their connection with multiary quasigroups. See below. Technical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. The symmetric difference of the sets ''A'' and ''B'' is commonly denoted by A \operatorname\Delta B (alternatively, A \operatorname\vartriangle B), A \oplus B, or A \ominus B. It can be viewed as a form of addition modulo 2. The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring. Properties The symmetric difference is equivalent to the union of both relative complements, that is: :A\, \Delta\,B = \left(A \setminus B\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aequationes Mathematicae
''Aequationes Mathematicae'' is a mathematical journal. It is primarily devoted to functional equations, but also publishes papers in dynamical systems, combinatorics, and geometry. As well as publishing regular journal submissions on these topics, it also regularly reports on international symposia on functional equations and produces bibliographies on the subject. János Aczél founded the journal in 1968 at the University of Waterloo, in part because of the long publication delays of up to four years in other journals at the time of its founding. It is currently published by Springer Science+Business Media, with Zsolt Páles of the University of Debrecen as its editor in chief. János Aczél remains its honorary editor in chief. It is frequently listed as a second-quartile mathematics journal by SCImago Journal Rank The SCImago Journal Rank (SJR) indicator is a measure of the prestige of scholarly journals that accounts for both the number of citations received by a journ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element properties are optional. In fact, a nonempty associative quasigroup is a group. A quasigroup that has an identity element is called a loop. Definitions There are at least two structurally equivalent formal definitions of quasigroup: * One defines a quasigroup as a set with one binary operation. * The other, from universal algebra, defines a quasigroup as having three primitive operations. The homomorphic image of a quasigroup that is defined with a single binary operation, however, need not be a quasigroup, in contrast to a quasigroup as having three primitive operations. We begin with the first definition. Algebra A quasigroup is a non-empty set with a binary operation (that is, a magma, indicating that a quasigroup has to sat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dowling Geometry
In combinatorial mathematics, a Dowling geometry, named after Thomas A. Dowling, is a matroid associated with a group. There is a Dowling geometry of each rank for each group. If the rank is at least 3, the Dowling geometry uniquely determines the group. Dowling geometries have a role in matroid theory as universal objects (Kahn and Kung, 1982); in that respect they are analogous to projective geometries, but based on groups instead of fields. A Dowling lattice is the geometric lattice of flats associated with a Dowling geometry. The lattice and the geometry are mathematically equivalent: knowing either one determines the other. Dowling lattices, and by implication Dowling geometries, were introduced by Dowling (1973a,b). A Dowling lattice or geometry of rank ''n'' of a group ''G'' is often denoted by ''Qn''(''G''). The original definitions In his first paper (1973a) Dowling defined the rank-''n'' Dowling lattice of the multiplicative group of a finite field ''F''. It is the se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matroid
In combinatorics, 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 Axiomatic system, 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 simple matroid is equivalent to a geometric lattice. Matroid theory borrows extensively from the terms used in 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 Cryptomorphism, equivalent ways to define a (finite) matroid. Independent sets In terms of independence, a finite matroid M is a pair (E, \mathcal), where E is a finite set (called the ''gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minor (graph Theory)
Minor may refer to: Common meanings * Minor (law), a person not under the age of certain legal activities. * Academic minor, a secondary field of study in undergraduate education Mathematics * Minor (graph theory), a relation of one graph to another * Minor (matroid theory), a relation of one matroid to another * Minor (linear algebra), the determinant of a square submatrix Music * Minor chord * Minor interval * Minor key * Minor scale People * Minor (given name), a masculine given name * Minor (surname), a surname Places in the United States * Minor, Alabama, a census-designated place * Minor, Virginia, an unincorporated community * Minor Creek (California) * Minor Creek (Missouri) * Minor Glacier, Wyoming Sports * Minor, a grade in Gaelic games; also, a person who qualifies to play in that grade * Minor league, a sports league not regarded as a premier league ** Minor League Baseball Minor League Baseball (MiLB) is a professional baseball organization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digon
In geometry, a bigon, digon, or a ''2''-gon, is a polygon with two sides (edge (geometry), edges) and two Vertex (geometry), vertices. Its construction is Degeneracy (mathematics), degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in elliptic space. It may also be viewed as a representation of a graph theory, graph with two vertices, see "Generalized polygon". A regular digon has both angles equal and both sides equal and is represented by Schläfli symbol . It may be constructed on a spherical geometry, sphere as a pair of 180 degree arcs connecting antipodal points, when it forms a spherical lune, lune. The digon is the simplest abstract polytope of rank 2. A truncation (geometry), truncated ''digon'', t is a square, . An Alternation (geometry), alternated digon, h is a monogon, . In different fields In Euclidean geometry The digon can have one of two visual representat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicircular Matroid
In the mathematical subject of matroid theory, the bicircular matroid of a graph ''G'' is the matroid ''B''(''G'') whose points are the edges of ''G'' and whose independent sets are the edge sets of pseudoforests of ''G'', that is, the edge sets in which each connected component contains at most one cycle. The bicircular matroid was introduced by and explored further by and others. It is a special case of the frame matroid of a biased graph. Circuits The circuits, or minimal dependent sets, of this matroid are the bicircular graphs (or bicycles, but that term has other meanings in graph theory); these are connected graphs whose circuit rank is exactly two. There are three distinct types of bicircular graph: *The theta graph consists of three paths joining the same two vertices but not intersecting each other. *The figure eight graph (or tight handcuff) consists of two cycles having just one common vertex. *The loose handcuff (or barbell) consists of two disjoint cycles a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasigroups
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element properties are optional. In fact, a nonempty associative quasigroup is a group. A quasigroup that has an identity element is called a loop. Definitions There are at least two structurally equivalent formal definitions of quasigroup: * One defines a quasigroup as a set with one binary operation. * The other, from universal algebra, defines a quasigroup as having three primitive operations. The homomorphic image of a quasigroup that is defined with a single binary operation, however, need not be a quasigroup, in contrast to a quasigroup as having three primitive operations. We begin with the first definition. Algebra A quasigroup is a non-empty set with a binary operation (that is, a magma, indicating that a quasigroup has to satisf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
