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Generalized Quadrangle
In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles yet containing many quadrangles. A generalized quadrangle is by definition a polar space of rank two. They are the with ''n'' = 4 and near polygon, near 2n-gons with ''n'' = 2. They are also precisely the Partial geometry, partial geometries pg(''s'',''t'',α) with α = 1. Definition A generalized quadrangle is an incidence structure (''P'',''B'',I), with I ⊆ ''P'' × ''B'' an incidence relation, satisfying certain axioms. Elements of ''P'' are by definition the ''points'' of the generalized quadrangle, elements of ''B'' the ''lines''. The axioms are the following: * There is an ''s'' (''s'' ≥ 1) such that on every line there are exactly ''s'' + 1 points. There is at most one point on two distinct lines. * There is a ''t'' (''t'' ≥ 1) such that through every point there are exactly ''t'' + 1 lines. There is at most one line through two distinct points. * F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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GQ(2,2)
''GQ'' (short for ''Gentlemen's Quarterly'' and previously known as ''Apparel Arts'') is an international monthly men's magazine based in New York City and founded in 1931. The publication focuses on fashion, style, and culture for men, though articles on food, movies, fitness, sex, music, travel, celebrities' sports, technology, and books are also featured. History The magazine ''Apparel Arts'' was launched in 1931 in the United States as a men's fashion magazine for the clothing trade, aimed primarily at wholesale buyers and retail sellers. Initially it had a very limited print run and was aimed solely at industry insiders to enable them to advise their customers. The popularity of the magazine among retail customers, who often took the magazine from the retailers, spurred the creation of ''Esquire'' magazine in 1933. ''Apparel Arts'' continued until 1957 when it was transformed into a quarterly magazine for men, which was published for many years by Esquire Inc. Apparel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Girth (graph Theory)
In graph theory, the girth of an undirected graph is the length of a shortest Cycle (graph theory), cycle contained in the graph. If the graph does not contain any cycles (that is, it is a forest (graph theory), forest), its girth is defined to be infinity. For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3. A graph with girth four or more is triangle-free graph, triangle-free. Cages A cubic graph (all vertices have degree three) of girth that is as small as possible is known as a -cage (graph theory), cage (or as a -cage). The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage. There may exist multiple cages for a given girth. For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries–Wong graph. Im ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Joseph A
Joseph is a common male name, derived from the Hebrew (). "Joseph" is used, along with "Josef (given name), Josef", mostly in English, French and partially German languages. This spelling is also found as a variant in the languages of the modern-day Nordic countries. In Portuguese language, Portuguese and Spanish language, Spanish, the name is "José". In Arabic, including in the Quran, the name is spelled , . In Kurdish language, Kurdish (''Kurdî''), the name is , Persian language, Persian, the name is , and in Turkish language, Turkish it is . In Pashto the name is spelled ''Esaf'' (ايسپ) and in Malayalam it is spelled ''Ousep'' (ഔസേപ്പ്). In Tamil language, Tamil, it is spelled as ''Yosepu'' (யோசேப்பு). The name has enjoyed significant popularity in its many forms in numerous countries, and ''Joseph'' was one of the two names, along with ''Robert'', to have remained in the top 10 boys' names list in the US from 1925 to 1972. It is especiall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Prime Power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, … . The prime powers are those positive integers that are divisible by exactly one prime number; in particular, the number 1 is not a prime power. Prime powers are also called primary numbers, as in the primary decomposition. Properties Algebraic properties Prime powers are powers of prime numbers. Every prime power (except powers of 2 greater than 4) has a primitive root; thus the multiplicative group of integers modulo ''p''''n'' (that is, the group of units of the ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Hyperoval
In projective geometry an oval is a point set in a plane that is defined by incidence properties. The standard examples are the nondegenerate conics. However, a conic is only defined in a pappian plane, whereas an oval may exist in any type of projective plane. In the literature, there are many criteria which imply that an oval is a conic, but there are many examples, both infinite and finite, of ovals in pappian planes which are not conics. As mentioned, in projective geometry an oval is defined by incidence properties, but in other areas, ovals may be defined to satisfy other criteria, for instance, in differential geometry by differentiability conditions in the real plane. The higher dimensional analog of an oval is an ovoid in a projective space. A generalization of the oval concept is an abstract oval, which is a structure that is not necessarily embedded in a projective plane. Indeed, there exist abstract ovals which can not lie in any projective plane. Definition of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Hermitian Variety
In geometry, a unital is a set of ''n''3 + 1 points arranged into subsets of size ''n'' + 1 so that every pair of distinct points of the set are contained in exactly one subset. This is equivalent to saying that a unital is a 2-(''n''3 + 1, ''n'' + 1, 1) block design. Some unitals may be embedded in a projective plane of order ''n''2 (the subsets of the design become sets of collinear points in the projective plane). In this case of ''embedded unitals'', every line of the plane intersects the unital in either 1 or ''n'' + 1 points. In the Desarguesian planes, PG(2,''q''2), the classical examples of unitals are given by nondegenerate Hermitian curves. There are also many non-classical examples. The first and the only known unital with non prime power parameters, ''n''=''6'', was constructed by Bhaskar Bagchi and Sunanda Bagchi. It is still unknown if this unital can be embedded in a projective plane of order ''36'', if such a plane exists. Unitals Classical We review some termino ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Quadric
In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids. More generally, a quadric hypersurface (of dimension ''D'') embedded in a higher dimensional space (of dimension ) is defined as the zero set of an irreducible polynomial of degree two in variables; for example, ''D''1 is the case of conic sections ( plane curves). When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a ''degenerate quadric'' or a ''reducible quadric''. A quadric is an affine algebraic variety, or, if it is reducible, an affine algebraic set. Quadrics may also be defined in projective spaces; see , below. Formulation In coordinates , the general quadric is thus defined by the algebraic equationSilvio LevQuadricsin "Geometry Formulas and Facts", excerpted from 3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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ADE Classification
In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, rather than a posteriori verification of a parallelism, was posed in . The complete list of simply laced Dynkin diagrams comprises :A_n, \, D_n, \, E_6, \, E_7, \, E_8. Here "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of \pi/2 = 90^\circ (no edge between the vertices) or 2\pi/3 = 120^\circ (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting B_n and C_n), and three of the five exceptional Dynkin diagrams (omitting F_4 and G_2). This list is non-redundant if one takes n \geq 4 for D_n. If one extends the families to include redundant terms, one obtains the exceptional isomorphisms :D_3 \cong A_3, E_4 \cong A_4, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Root System
In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory. Definitions and examples As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Windmill Graph
In the mathematical field of graph theory, the windmill graph is an undirected graph constructed for and by joining copies of the complete graph at a shared universal vertex. That is, it is a 1-clique-sum of these complete graphs. Properties It has vertices and edges, girth 3 (if ), radius 1 and diameter 2. It has vertex connectivity 1 because its central vertex is an articulation point; however, like the complete graphs from which it is formed, it is -edge-connected. It is trivially perfect and a block graph. Special cases By construction, the windmill graph is the friendship graph , the windmill graph is the star graph and the windmill graph is the butterfly graph. Labeling and colouring The windmill graph has chromatic number and chromatic index . Its chromatic polynomial The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Levi Graph
In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particulap. 181 From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. They are named for Friedrich Wilhelm Levi, who wrote about them in 1942. The Levi graph of a system of points and lines usually has girth at least six: Any 4-cycles would correspond to two lines through the same two points. Conversely any bipartite graph with girth at least six can be viewed as the Levi graph of an abstract incidence structure. Levi graphs of configurations are biregular, and every biregular graph with girth at least six can be viewed as the Levi graph of an abstract configuration.. Levi graphs may also be defined for other types of incidence structure, such as the incidences between points and planes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |