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Cyclohedron
In geometry, the cyclohedron is a -dimensional polytope where can be any non-negative integer. It was first introduced as a combinatorial object by Raoul Bott and Clifford Taubes and, for this reason, it is also sometimes called the Bott–Taubes polytope. It was later constructed as a polytope by Martin Markl and by Rodica Simion. Rodica Simion describes this polytope as an associahedron of type B. The cyclohedron appears in the study of knot invariants. Construction Cyclohedra belong to several larger families of polytopes, each providing a general construction. For instance, the cyclohedron belongs to the generalized associahedra that arise from cluster algebra, and to the graph-associahedra, a family of polytopes each corresponding to a Graph (discrete mathematics), graph. In the latter family, the graph corresponding to the d-dimensional cyclohedron is a cycle on d+1 vertices. In topological terms, the Configuration space (mathematics), configuration space of d+1 distinct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Flip Graph
In mathematics, a flip graph is a graph whose vertices are combinatorial or geometric objects, and whose edges link two of these objects when they can be obtained from one another by an elementary operation called a flip. Flip graphs are special cases of geometric graphs. Among notable flip graphs, one finds the 1-skeleton of polytopes such as associahedra or cyclohedra. Examples A prototypical flip graph is that of a convex n-gon \pi. The vertices of this graph are the triangulations of \pi, and two triangulations are adjacent in it whenever they differ by a single interior edge. In this case, the flip operation consists in exchanging the diagonals of a convex quadrilateral. These diagonals are the interior edges by which two triangulations adjacent in the flip graph differ. The resulting flip graph is both the Hasse diagram of the Tamari lattice and the 1-skeleton of the (n-3)-dimensional associahedron. This basic construction can be generalized in a number of ways. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rodica Simion
Rodica Eugenia Simion (January 18, 1955 – January 7, 2000) was a Romanian-American mathematician. She was the Columbian School Professor of Mathematics at George Washington University. Her research concerned combinatorics: she was a pioneer in the study of permutation patterns, and an expert on noncrossing partitions. Biography Simion was one of the top competitors in the Romanian national mathematical olympiads. She graduated from the University of Bucharest in 1974, and immigrated to the United States in 1976.. She did her graduate studies at the University of Pennsylvania, earning a Ph.D. in 1981 under the supervision of Herbert Wilf. After teaching at Southern Illinois University and Bryn Mawr College, she moved to George Washington University in 1987, and became Columbian School Professor in 1997. Recognition She is included in a deck of playing cards featuring notable women mathematicians published by the Association of Women in Mathematics. Research contributions Simio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Permutohedron
In mathematics, the permutohedron (also spelled permutahedron) of order is an -dimensional polytope embedded in an -dimensional space. Its vertex (geometry), vertex coordinates (labels) are the permutations of the first natural numbers. The edges identify the shortest possible paths (sets of Transposition (mathematics), transpositions) that connect two vertices (permutations). Two permutations connected by an edge differ in only two places (one Transposition (mathematics), transposition), and the numbers on these places are neighbors (differ in value by 1). The image on the right shows the permutohedron of order 4, which is the truncated octahedron. Its vertices are the 24 permutations of . Parallel edges have the same edge color. The 6 edge colors correspond to the 6 possible Transposition (mathematics), transpositions of 4 elements, i.e. they indicate in which two places the connected permutations differ. (E.g. red edges connect permutations that differ in the last two places. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Associahedron
In mathematics, an associahedron is an -dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of letters, and the edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the triangulations of a regular polygon with sides and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal. Associahedra are also called Stasheff polytopes after the work of Jim Stasheff, who rediscovered them in the early 1960s after earlier work on them by Dov Tamari. Examples The one-dimensional associahedron ''K''3 represents the two parenthesizations ((''xy'')''z'') and (''x''(''yz'')) of three symbols, or the two triangulations of a square. It is itself a line segment. The two-dimensional associahedron ''K''4 represents the five parenthesizations of four symbols, or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Permutohedron
In mathematics, the permutohedron (also spelled permutahedron) of order is an -dimensional polytope embedded in an -dimensional space. Its vertex (geometry), vertex coordinates (labels) are the permutations of the first natural numbers. The edges identify the shortest possible paths (sets of Transposition (mathematics), transpositions) that connect two vertices (permutations). Two permutations connected by an edge differ in only two places (one Transposition (mathematics), transposition), and the numbers on these places are neighbors (differ in value by 1). The image on the right shows the permutohedron of order 4, which is the truncated octahedron. Its vertices are the 24 permutations of . Parallel edges have the same edge color. The 6 edge colors correspond to the 6 possible Transposition (mathematics), transpositions of 4 elements, i.e. they indicate in which two places the connected permutations differ. (E.g. red edges connect permutations that differ in the last two places. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Associahedron
In mathematics, an associahedron is an -dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of letters, and the edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the triangulations of a regular polygon with sides and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal. Associahedra are also called Stasheff polytopes after the work of Jim Stasheff, who rediscovered them in the early 1960s after earlier work on them by Dov Tamari. Examples The one-dimensional associahedron ''K''3 represents the two parenthesizations ((''xy'')''z'') and (''x''(''yz'')) of three symbols, or the two triangulations of a square. It is itself a line segment. The two-dimensional associahedron ''K''4 represents the five parenthesizations of four symbols, or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Permutoassociahedron
In mathematics, the permutoassociahedron is an n-dimensional polytope whose vertices correspond to the bracketings of the permutations of n+1 terms and whose edges connect two bracketings that can be obtained from one another either by moving a pair of brackets using associativity or by transposing two consecutive terms that are not separated by a bracket. The permutoassociahedron was first defined as a CW complex by Mikhail Kapranov who noted that this structure appears implicitly in Mac Lane's coherence theorem for symmetric and braided categories as well as in Vladimir Drinfeld's work on the Knizhnik–Zamolodchikov equations. It was constructed as a convex polytope by Victor Reiner and Günter M. Ziegler. Examples When n = 2, the vertices of the permutoassociahedron can be represented by bracketing all the permutations of three terms a, b, and c. There are six such permutations, abc, acb, bac, bca, cab, and cba, and each of them admits two bracketings (obtained from on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Clifford Taubes
Clifford Henry Taubes (born February 21, 1954) is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology. His brother is the journalist Gary Taubes. Early career Taubes received his B.A. from Cornell University in 1975 and his Ph.D. in physics in 1980 from Harvard University under the direction of Arthur Jaffe, having proven results collected in about the existence of solutions to the Landau–Ginzburg vortex equations and the Bogomol'nyi monopole equations. Soon, he began applying his gauge-theoretic expertise to pure mathematics. His work on the boundary of the moduli space of solutions to the Yang-Mills equations was used by Simon Donaldson in his proof of Donaldson's theorem on diagonizability of intersection forms. He proved in that R4 has an uncountable number of smooth structures (see also exotic R4), and (with Raoul Bott in ) proved Witten's rigidity theorem on th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Journal Of Mathematical Physics
The ''Journal of Mathematical Physics'' is a peer-reviewed journal published monthly by the American Institute of Physics devoted to the publication of papers in mathematical physics. The journal was first published bimonthly beginning in January 1960; it became a monthly publication in 1963. The current editor is Jan Philip Solovej from University of Copenhagen The University of Copenhagen (, KU) is a public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in Scandinavia, after Uppsala University. .... Its 2018 Impact Factor is 1.355 Abstracting and indexing This journal is indexed by the following services: 2013. References External ...
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Facet (geometry)
In geometry, a facet is a feature of a polyhedron, polytope, or related geometric structure, generally of dimension one less than the structure itself. More specifically: * In three-dimensional geometry, some authors call a facet of a polyhedron any polygon whose corners are vertices of the polyhedron, including polygons that are not ''Face (geometry), faces''. To ''facetting, facet'' a polyhedron is to find and join such facets to form the faces of a new polyhedron; this is the reciprocal process to ''stellation'' and may also be applied to higher-dimensional polytopes. * In polyhedral combinatorics and in the general theory of polytopes, a Face (geometry), face that has dimension ''n'' − 1 (an (''n'' − 1)-face or hyperface) is called a Face (geometry)#Facet, facet. In this terminology, every facet is a face. * A facet of a simplicial complex is a maximal simplex, that is a simplex that is not a face of another simplex of the complex.. For (boundary complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''Polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
