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Simplicial Sphere
In geometry and combinatorics, a simplicial (or combinatorial) ''d''-sphere is a simplicial complex homeomorphic to the ''d''-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions most simplicial spheres cannot be obtained in this way. One important open problem in the field was the g-conjecture, formulated by Peter McMullen, which asks about possible numbers of faces of different dimensions of a simplicial sphere. In December 2018, the g-conjecture was proven by Karim Adiprasito in the more general context of rational homology spheres. Examples * For any ''n'' ≥ 3, the simple ''n''-cycle ''C''''n'' is a simplicial circle, i.e. a simplicial sphere of dimension 1. This construction produces all simplicial circles. * The boundary of a convex polyhedron in R3 with triangular faces, such as an octahedron or icosahedron, is a simplicial 2-sphere. * More generally, the boundary of any (''d''+1)-dimensional compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Set
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary. In mathematical analysis and related areas of mathematics, a set is called bounded if it is, in a certain sense, of finite measure. Conversely, a set which is not bounded is called unbounded. The word 'bounded' makes no sense in a general topological space without a corresponding metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem .... A bounded set is not necessarily a closed set and vise versa. For example, a subset ''S'' of a 2-dimensional real space R''2'' constrained by two parabolic curves ''x''2 + 1 and ''x''2 - 1 defined in a Cartesian coordinate system is a closed but is not b ... [...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] |
Richard P
Richard is a male given name. It originates, via Old French, from Frankish language, Old Frankish and is a Compound (linguistics), compound of the words descending from Proto-Germanic language, Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", "Dick (nickname), Dick", "Dickon", "Dickie (name), Dickie", "Rich (given name), Rich", "Rick (given name), Rick", "Rico (name), Rico", "Ricky (given name), Ricky", and more. Richard is a common English, German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Catalan "Ricard" and the Italian "Riccardo", among others (see comprehensive variant list below). People ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Bound Theorem
In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of the central results of polyhedral combinatorics. Originally known as the upper bound conjecture, this statement was formulated by Theodore Motzkin, proved in 1970 by Peter McMullen, and strengthened from polytopes to subdivisions of a sphere in 1975 by Richard P. Stanley. Cyclic polytopes The cyclic polytope \Delta(n,d) may be defined as the convex hull of n vertices on the moment curve, the set of d-dimensional points with coordinates (t,t^2,t^3,\dots). The precise choice of which n points on this curve are selected is irrelevant for the combinatorial structure of this polytope. The number of i-dimensional faces of \Delta(n,d) is given by the formula f_i(\Delta(n,d)) = \binom \quad \textrm \quad 0 \leq i < \left\lfloor\frac\right\rfloor and com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gil Kalai
Gil Kalai (born 1955) is the Henry and Manya Noskwith Professor Emeritus of Mathematics at the Hebrew University of Jerusalem, Israel, Professor of Computer Science at the Interdisciplinary Center, Herzliya, and adjunct Professor of mathematics and of computer science at Yale University, United States. Biography Kalai received his PhD from Hebrew University in 1983, under the supervision of Micha Perles, and joined the Hebrew University faculty in 1985 after a postdoctoral fellowship at the Massachusetts Institute of Technology.Profile at the Technical University of Eindhoven as an instructor of a minicourse on polyhedral combinatorics. He was the recipient of the Pólya Prize ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branko Grünbaum
Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentBranko Grünbaum Hrvatska enciklopedija LZMK. and a professor at the in . He received his Ph.D. in 1957 from Hebrew University of Jerusalem< ...
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Steinitz's Theorem
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. This result provides a classification theorem for the three-dimensional convex polyhedra, something that is not known in higher dimensions. It provides a complete and purely combinatorial description of the graphs of these polyhedra, allowing other results on them, such as Eberhard's theorem on the realization of polyhedra with given types of faces, to be proven more easily, without reference to the geometry of these shapes. Additionally, it has been applied in graph drawing, as a way to construct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Steinitz
Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician. Biography Steinitz was born in Laurahütte (Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and his wife Auguste Cohen; he had two brothers. He studied at the University of Breslau and the University of Berlin, receiving his Ph.D. from Breslau in 1894. Subsequently, he took positions at Charlottenburg (now the Technical University of Berlin), Breslau, and the University of Kiel, Germany, where he died in 1928. Steinitz married Martha Steinitz and had one son. Mathematical works Steinitz's 1894 thesis was on the subject of projective configurations; it contained the result that any abstract description of an incidence structure of three lines per point and three points per line could be realized as a configuration of straight lines in the Euclidean plane with the possible exception of one of the lines. His thesis also contains the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barycentric Subdivision
In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension on simplicial complexes is a canonical method to refine them. Therefore, the barycentric subdivision is an important tool in algebraic topology. Motivation The barycentric subdivision is an operation on simplicial complexes. In algebraic topology it is sometimes useful to replace the original spaces with simplicial complexes via triangulations: The substitution allows to assign combinatorial invariants as the Euler characteristic to the spaces. One can ask if there is an analogous way to replace the continuous functions defined on the topological spaces by functions that are linear on the simplices and which are homotopic to the original maps (see also simplicial approximation). In general, such an assignment requires a refinement of the given complex, meaning, one replaces bigger simplices by a union of smaller simplices. A standard way to effectuate such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
