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Cyclic Polytope
In mathematics, a cyclic polytope, denoted ''C''(''n'',''d''), is a convex polytope formed as a convex hull of ''n'' distinct points on a rational normal curve in R''d'', where ''n'' is greater than ''d''. These polytopes were studied by Constantin Carathéodory, David Gale, Theodore Motzkin, Victor Klee, and others. They play an important role in polyhedral combinatorics: according to the upper bound theorem, proved by Peter McMullen and Richard P. Stanley, Richard Stanley, the boundary ''Δ''(''n'',''d'') of the cyclic polytope ''C''(''n'',''d'') maximizes the number ''f''''i'' of ''i''-dimensional faces among all simplicial spheres of dimension ''d'' − 1 with ''n'' vertices. Definition The moment curve in \mathbb^d is defined by :\mathbf:\mathbb \rightarrow \mathbb^d, \mathbf(t) := \begint,t^2,\ldots,t^d\end^T. The d-dimensional cyclic polytope with n vertices is the convex hull : C(n,d) := \mathbf\ of n > d \ge 2 distinct points \mathbf(t_i) with t_1 < t_2 < \ldots ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanley–Reisner Ring
In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within algebraic combinatorics and combinatorial commutative algebra.Miller & Sturmfels (2005) p.19 Its properties were investigated by Richard Stanley, Melvin Hochster, and Gerald Reisner in the early 1970s. Definition and properties Given an abstract simplicial complex Δ on the vertex set and a field k, the corresponding Stanley–Reisner ring, or face ring, denoted k ” is obtained from the polynomial ring k 'x''1,...,''x''''n''by quotienting out the ideal ''I''Δ generated by the square-free monomials corresponding to the non-faces of Δ: : I_\Delta=(x_\ldots x_: \\notin\Delta), \quad kDeltak _1,\ldots,x_nI_\Delta. The ideal ''I''Δ is called the Stanley–Reisner ideal or the face ideal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter McMullen
Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at the University of Birmingham, where he received his doctorate in 1968. and taught at Western Washington University from 1968 to 1969. In 1978 he earned his Doctor of Science at University College London where he still works as a professor emeritus. In 2006 he was accepted as a corresponding member of the Austrian Academy of Sciences. Contributions McMullen is known for his work in polyhedral combinatorics and discrete geometry, and in particular for proving what was then called the upper bound conjecture and now is the upper bound theorem. This result states that cyclic polytopes have the maximum possible number of faces among all polytopes with a given dimension and number of vertices. McMullen also formulated the g-conjecture, later t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dehn–Sommerville Equations
In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a simplicial polytope. For polytopes of dimension 4 and 5, they were found by Max Dehn in 1905. Their general form was established by Duncan Sommerville in 1927. The Dehn–Sommerville equations can be restated as a symmetry condition for the ''h''-vector'' of the simplicial polytope and this has become the standard formulation in recent combinatorics literature. By duality, analogous equations hold for simple polytopes. Statement Let ''P'' be a ''d''-dimensional simplicial polytope. For ''i'' = 0, 1, ..., ''d'' − 1, let ''f''''i'' denote the number of ''i''-dimensional faces of ''P''. The sequence : f(P)=(f_0,f_1,\ldots,f_) is called the ''f''-vector of the polytope ''P''. Additionally, set : f_=1, f_d=1. Then for any ''k'' = −1, 0, ..., ''d'' − 2, the following Dehn–Sommerville equation ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... was 0.754. External links * Mathematics journals Publications established in 1963 English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighborly Polytope
In geometry and polyhedral combinatorics, a -neighborly polytope is a convex polytope in which every set of or fewer vertices forms a face. For instance, a 2-neighborly polytope is a polytope in which every pair of vertices is connected by an edge, forming a complete graph. 2-neighborly polytopes with more than four vertices may exist only in spaces of four or more dimensions, and in general a -neighborly polytope (other than a simplex) requires a dimension of or more. A -simplex is -neighborly. A polytope is said to be neighborly, without specifying , if it is -neighborly for . If we exclude simplices, this is the maximum possible : in fact, every polytope that is -neighborly for some is a simplex. In a -neighborly polytope with , every 2-face must be a triangle, and in a -neighborly polytope with , every 3-face must be a tetrahedron. More generally, in any -neighborly polytope, all faces of dimension less than are simplices. The cyclic polytopes formed as the convex hul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Polytope
In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only triangular facesPolyhedra, Peter R. Cromwell, 1997. (p.341) and corresponds via Steinitz's theorem to a maximal planar graph. They are topologically dual to simple polytopes. Polytopes which are both simple and simplicial are either simplices or two-dimensional polygons. Examples Simplicial polyhedra include: * Bipyramids * Gyroelongated dipyramids *Deltahedra (equilateral triangles) ** Platonic *** tetrahedron, octahedron, icosahedron ** Johnson solids: ***triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism, gyroelongated square dipyramid * Catalan solids: ** triakis tetrahedron, triakis octahedron, tetrakis hexahedron, disdyakis dodecahedron, triakis icosahedron, pentakis dodecahedron, disdyakis triacontahedron Simplicial tilings: * Regular: ** triangular tiling *Laves tilings: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment Curve
In geometry, the moment curve is an algebraic curve in ''d''-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form :\left( x, x^2, x^3, \dots, x^d \right). In the Euclidean plane, the moment curve is a parabola, and in three-dimensional space it is a twisted cubic. Its closure in projective space is the rational normal curve. Moment curves have been used for several applications in discrete geometry including cyclic polytopes, the no-three-in-line problem, and a geometric proof of the chromatic number of Kneser graphs. Properties Every hyperplane intersects the moment curve in a finite set of at most ''d'' points. If a hyperplane intersects the curve in exactly ''d'' points, then the curve crosses the hyperplane at each intersection point. Thus, every finite point set on the moment curve is in general linear position. Applications The convex hull of any finite set of points on the moment curve is a cyclic polytope. Cyclic polytopes have the l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Normal Curve
In mathematics, the rational normal curve is a smooth, rational curve of degree in projective n-space . It is a simple example of a projective variety; formally, it is the Veronese variety when the domain is the projective line. For it is the plane conic and for it is the twisted cubic. The term "normal" refers to projective normality, not normal schemes. The intersection of the rational normal curve with an affine space is called the moment curve. Definition The rational normal curve may be given parametrically as the image of the map :\nu:\mathbf^1\to\mathbf^n which assigns to the homogeneous coordinates the value :\nu: :T\mapsto \left ^n:S^T:S^T^2:\cdots:T^n \right In the affine coordinates of the chart the map is simply :\nu:x \mapsto \left (x, x^2, \ldots, x^n \right ). That is, the rational normal curve is the closure by a single point at infinity of the affine curve :\left (x, x^2, \ldots, x^n \right ). Equivalently, rational normal curve may be understood ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
