|
Near Polygon
In mathematics, a near polygon is a concept in incidence geometry introduced by Ernest E. Shult and Arthur Yanushka in 1980. Shult and Yanushka showed the connection between the so-called tetrahedrally closed line-systems in Euclidean spaces and a class of point-line geometries which they called near polygons. These structures generalise the notion of generalized polygon as every generalized 2''n''-gon is a near 2''n''-gon of a particular kind. Near polygons were extensively studied and connection between them and dual polar spaces was shown in 1980s and early 1990s. Some sporadic simple groups, for example the Hall-Janko group and the Mathieu groups, act as automorphism groups of near polygons. Definition A near 2''d''-gon is an incidence structure (P,L,I), where P is the set of points, L is the set of lines and I\subseteq P\times L is the incidence relation, such that: * The maximum distance between two points (the so-called diameter) is ''d''. * For every point x and ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
GQ(2,2), The Doily
''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] |
Conjugacy Class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under b = gag^ for all elements g in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element ( singleton set). Functions that are constant for members of the same conjugacy class are called class functions. Definition Let G be a group. Two elements a, b \in G are conjugate if there exists an element g \in G such that gag^ = b, in which case b is called of a and a is called a conjugate of b. In the case of the general linear group \op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Geometriae Dedicata
''Geometriae Dedicata'' is a mathematical journal, founded in 1972, concentrating on geometry and its relationship to topology, group theory and the theory of dynamical systems. It was created on the initiative of Hans Freudenthal in Utrecht, the Netherlands.. It is published by Springer Netherlands. The Editor-in-Chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The editor-in-chief heads all departments of the organization and is held accoun ... is Richard Alan Wentworth. References External links Springer site Geometry journals Springer Science+Business Media academic journals Algebra journals Dynamical systems journals Topology journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Footnotes
In publishing, a note is a brief text in which the author comments on the subject and themes of the book and names supporting citations. In the editorial production of books and documents, typographically, a note is usually several lines of text at the bottom of the page, at the end of a chapter, at the end of a volume, or a house-style typographic usage throughout the text. Notes are usually identified with superscript numbers or a symbol.''The Oxford Companion to the English Language'' (1992) p. 709. Footnotes are informational notes located at the foot of the thematically relevant page, whilst endnotes are informational notes published at the end of a chapter, the end of a volume, or the conclusion of a multi-volume book. Unlike footnotes, which require manipulating the page design (text-block and page layouts) to accommodate the additional text, endnotes are advantageous to editorial production because the textual inclusion does not alter the design of the publication. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hall–Janko Graph
In the mathematical field of graph theory, the Hall–Janko graph, also known as the Hall-Janko-Wales graph, is a 36-regular undirected graph with 100 vertices and 1800 edges. It is a rank 3 strongly regular graph with parameters (100,36,14,12) and a maximum coclique of size 10. This parameter set is not unique, it is however uniquely determined by its parameters as a rank 3 graph. The Hall–Janko graph was originally constructed by D. Wales to establish the existence of the Hall-Janko group as an index 2 subgroup of its automorphism group. The Hall–Janko graph can be constructed out of objects in U3(3), the simple group of order 6048: * In U3(3) there are 36 simple maximal subgroups of order 168. These are the vertices of a subgraph, the U3(3) graph. A 168-subgroup has 14 maximal subgroups of order 24, isomorphic to S4. Two 168-subgroups are called adjacent when they intersect in a 24-subgroup. The U3(3) graph is strongly regular, with parameters (36,14,4,6) * There are 63 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Association Scheme
The theory of association schemes arose in statistics, in the theory of design of experiments, experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory, the theory of error-correcting codes. In algebra, the theory of association schemes generalizes the group character, character theory of group representation, linear representations of groups. Definition An ''n''-class association scheme consists of a Set (mathematics), set ''X'' together with a partition of a set, partition ''S'' of ''X'' × ''X'' into ''n'' + 1 binary relations, ''R''0, ''R''1, ..., ''R''''n'' which satisfy: *R_ = \; it is called the identity relation. *Defining R^* := \, if ''R'' in ''S'', then ''R*'' in ''S''. *If (x,y) \in R_, the number of z \in X such that (x,z) \in R_ and (z,y) \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Partial Linear Space
A partial linear space (also semilinear or near-linear space) is a basic incidence structure in the field of incidence geometry, that carries slightly less structure than a linear space. The notion is equivalent to that of a linear hypergraph. Definition Let S=(,, \textbf) an incidence structure, for which the elements of are called ''points'' and the elements of are called ''lines''. ''S'' is a partial linear space, if the following axioms hold: * any line is incident with at least two points * any pair of distinct points is incident with at most one line If there is a unique line incident with every pair of distinct points, then we get a linear space. Properties The De Bruijn–Erdős theorem shows that in any finite linear space S=(,, \textbf) which is not a single point or a single line, we have , \mathcal, \leq , \mathcal, . Examples * Projective space * Affine space * Polar space * Generalized quadrangle * Generalized polygon * Near polygon References * . * Lyn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Polar Space
In mathematics, in the field of geometry, a polar space of rank ''n'' (), or ''projective index'' , consists of a set ''P'', conventionally called the set of points, together with certain subsets of ''P'', called ''subspaces'', that satisfy these axioms: * Every subspace is isomorphic to a projective space with and ''K'' a division ring. (That is, it is a Desarguesian projective geometry.) For each subspace the corresponding ''d'' is called its dimension. * The intersection of two subspaces is always a subspace. * For each subspace ''A'' of dimension and each point ''p'' not in ''A'', there is a unique subspace ''B'' of dimension containing ''p'' and such that is -dimensional. The points in are exactly the points of ''A'' that are in a common subspace of dimension 1 with ''p''. * There are at least two disjoint subspaces of dimension . It is possible to define and study a slightly bigger class of objects using only the relationship between points and lines: a polar space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Finite Geometry
A finite geometry is any geometry, geometric system that has only a finite set, finite number of point (geometry), points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective space, projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius plane, Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometry, inversive geometries. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine and projective planes so const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Distance-regular Graph
In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices and , the number of vertices at distance from and at distance from depends only upon , , and the distance between and . Some authors exclude the complete graphs and disconnected graphs from this definition. Every distance-transitive graph is distance regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group. Intersection arrays The intersection array of a distance-regular graph is the array ( b_0, b_1, \ldots, b_; c_1, \ldots, c_d ) in which d is the diameter of the graph and for each 1 \leq j \leq d , b_j gives the number of neighbours of u at distance j+1 from v and c_j gives the number of neighbours of u at distance j - 1 from v for any pair of vertices u and v at dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Collinearity
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row". Points on a line In any geometry, the set of points on a line are said to be collinear. In Euclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries (including Euclidean) a line is typically a primitive (undefined) object type, so such visualizations will not necessarily be appropriate. A model for the geometry offers an interpretation of how the points, lines and other object types relate to one another and a notion such as collinearity must be interpreted within the context of that model. For instance, in spherical geometry, where lines are represented in the standard model by great circles of a spher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
