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Truncated Projective Plane
In geometry, a truncated projective plane (TPP), also known as a dual affine plane, is a special kind of a hypergraph or geometric configuration that is constructed in the following way. * Take a finite projective plane. * Remove one of the points (vertices) in the plane. * Remove all lines (edges) containing that point. These objects have been studied in many different settings, often independent of one another, and so, many terminologies have been developed. Also, different areas tend to ask different types of questions about these objects and are interested in different aspects of the same objects. Example: the Pasch hypergraph Consider the Fano plane, which is the projective plane of order 2. It has 7 vertices and 7 edges . It can be truncated e.g. by removing the vertex 7 and the edges containing it. The remaining hypergraph is the TPP of order 2. It has 6 vertices and 4 edges . It is a tripartite hypergraph with sides ,, (which are exactly the neighbors of the removed v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hypergraph
In mathematics, a hypergraph is a generalization of a Graph (discrete mathematics), graph in which an graph theory, edge can join any number of vertex (graph theory), vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a directed hypergraph is a pair (X,E), where X is a set of elements called ''nodes'', ''vertices'', ''points'', or ''elements'' and E is a set of pairs of subsets of X. Each of these pairs (D,C)\in E is called an ''edge'' or ''hyperedge''; the vertex subset D is known as its ''tail'' or ''domain'', and C as its ''head'' or ''codomain''. The order of a hypergraph (X,E) is the number of vertices in X. The size of the hypergraph is the number of edges in E. The order of an edge e=(D,C) in a directed hypergraph is , e, = (, D, ,, C, ): that is, the number of vertices in its tail followed by the number of vertices in its head. The definition above generalizes from a directed graph to a directed hypergraph by defining the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Non-Desarguesian Plane
In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective spaces of dimension not 2; in other words, the only projective spaces of dimension not equal to 2 are the classical projective geometries over a field (or division ring). However, David Hilbert found that some projective planes do not satisfy it. The current state of knowledge of these examples is not complete. Examples There are many examples of both finite and infinite non-Desarguesian planes. Some of the known examples of infinite non-Desarguesian planes include: * The Moulton plane. * Moufang planes over alternative division algebras that are not associative, such as the projective plane over the octonions. Since all finite alternative division rings are fields ( Artin–Zorn theorem), the only non-Desarguesian Moufang planes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...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] |
Fractional Matching
In graph theory, a fractional matching is a generalization of a matching in which, intuitively, each vertex may be broken into fractions that are matched to different neighbor vertices. Definition Given a graph G=(V,E), a fractional matching in G is a function that assigns, to each edge e\in E, a fraction f(e)\in ,1/math>, such that for every vertex v\in V, the sum of fractions of edges adjacent to v is at most one: \forall v\in V: \sum_f(e)\leq 1 A matching in the traditional sense is a special case of a fractional matching, in which the fraction of every edge is either zero or one: f(e)=1 if e is in the matching, and f(e)=0 if it is not. For this reason, in the context of fractional matchings, usual matchings are sometimes called ''integral matchings''. Size The size of an integral matching is the number of edges in the matching, and the matching number \nu(G) of a graph G is the largest size of a matching in G. Analogously, the ''size'' of a fractional matching is the sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fractional Matching In Hypergraphs
In graph theory, a matching in a hypergraph is a set of hyperedges, in which every two hyperedges are disjoint. It is an extension of the notion of matching in a graph. Definition Recall that a hypergraph is a pair , where is a set of vertices and is a set of subsets of called ''hyperedges''. Each hyperedge may contain one or more vertices. A matching in is a subset of , such that every two hyperedges and in have an empty intersection (have no vertex in common). The matching number of a hypergraph is the largest size of a matching in . It is often denoted by . As an example, let be the set Consider a 3-uniform hypergraph on (a hypergraph in which each hyperedge contains exactly 3 vertices). Let be a 3-uniform hypergraph with 4 hyperedges: : Then admits several matchings of size 2, for example: : : However, in any subset of 3 hyperedges, at least two of them intersect, so there is no matching of size 3. Hence, the matching number of is 2. Intersec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ryser's Conjecture
In graph theory, Ryser's conjecture is a conjecture relating the maximum matching size and the minimum transversal size in hypergraphs. This conjecture first appeared in 1971 in the Ph.D. thesis of J. R. Henderson, whose advisor was H. J. Ryser, Herbert John Ryser. Preliminaries A Matching in hypergraphs, matching in a hypergraph is a set of hyperedges such that each vertex appears in at most one of them. The largest size of a matching in a hypergraph ''H'' is denoted by \nu(H). A Vertex cover in hypergraphs, transversal (or vertex cover) in a hypergraph is a set of vertices such that each hyperedge contains at least one of them. The smallest size of a transversal in a hypergraph ''H'' is denoted by \tau(H). For every ''H'', \nu(H)\leq \tau(H), since every cover must contain at least one point from each edge in any matching. If H is ''r''-uniform (each hyperedge has exactly ''r'' vertices), then \tau(H) \leq r\cdot \nu(H), since the union of the edges from any maximal mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Duality (projective Geometry)
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by Point (geometry), points and Line (geometry), lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special Map (mathematics), mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a ''duality''. Such a map can be constructed in many ways. The concept of plane duality readily extends to ''space duality'' and beyond that to duality in any finite-dimensional projective geometry. Principle of duality A projective plane may be defined axiomatically as an incidence structure, in terms of a set of ''points'', a set of ''lines'', and an incidence relation that de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Isomorphic (mathematics)
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Configuration (geometry)
In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points. Although certain specific configurations had been studied earlier (for instance by Thomas Kirkman in 1849), the formal study of configurations was first introduced by Theodor Reye in 1876, in the second edition of his book ''Geometrie der Lage'', in the context of a discussion of Desargues' theorem. Ernst Steinitz wrote his dissertation on the subject in 1894, and they were popularized by Hilbert and Cohn-Vossen's 1932 book ''Anschauliche Geometrie'', reprinted in English as . Configurations may be studied either as concrete sets of points and lines in a specific geometry, such as the Euclidean or projective planes (these are said to be ''realizable'' in that geometry), or as a type of abstract incidence geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
