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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] |
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] |
Intersection Cohomology
In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years. Intersection cohomology was used to prove the Kazhdan–Lusztig conjectures and the Riemann–Hilbert correspondence. It is closely related to ''L''2 cohomology. Goresky–MacPherson approach The homology groups of a compact, oriented, connected, ''n''-dimensional manifold ''X'' have a fundamental property called Poincaré duality: there is a perfect pairing : H_i(X,\Q) \times H_(X,\Q) \to H_0(X,\Q) \cong \Q. Classically—going back, for instance, to Henri Poincaré—this duality was understood in terms of intersection theory. An element of :H_j(X) is represented by a ''j''-dimensional cycle. If an ''i''-dimensional and an (n-i)-dimensional cycle are in general position, then their intersection is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Günter M
Gunter or Günter may refer to: * Gunter rig, a type of rig used in sailing, especially in small boats * Gunter Annex, Alabama, a United States Air Force installation * Gunter, Texas, city in the United States People Surname * Chris Gunter (born 1989), Welsh footballer with Cardiff City, Tottenham Hotspur, Nottingham Forest and Reading * Cornell Gunter (1936–1990), American R&B singer, brother of Shirley Gunter * David Gunter (1933–2005), English footballer with Southampton, brother of Phil Gunter * Edmund Gunter (1581–1626), British mathematician and inventor, known for: ** Gunter's chain ** Gunter's rule * James Gunter (1745–1819), English confectioner, fruit grower and scientific gardener * Jen Gunter (born 1966), Canadian-American gynecologist & author * Gordon Gunter (1909–1998), American marine biologist and fisheries scientist * Matthew Alan Gunter (born 1957), United States Episcopal bishop * Phil Gunter (1932–2007), English footballer with Portsmout ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
JSTOR
JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of journals in the humanities and social sciences. It provides full-text searches of almost 2,000 journals. , more than 8,000 institutions in more than 160 countries had access to JSTOR. Most access is by subscription but some of the site is public domain, and open access content is available free of charge. JSTOR's revenue was $86 million in 2015. History William G. Bowen, president of Princeton University from 1972 to 1988, founded JSTOR in 1994. JSTOR was originally conceived as a solution to one of the problems faced by libraries, especially research and university libraries, due to the increasing number of academic journals in existence. Most libraries found it prohibitively expensive in terms of cost and space to maintain a comprehen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The Royal Society
''Proceedings of the Royal Society'' is the main research journal of the Royal Society. The journal began in 1831 and was split into two series in 1905: * Series A: for papers in physical sciences and mathematics. * Series B: for papers in life sciences. Many landmark scientific discoveries are published in the Proceedings, making it one of the most historically significant science journals. The journal contains several articles written by the most celebrated names in science, such as Paul Dirac, Werner Heisenberg, Ernest Rutherford, Erwin Schrödinger, William Lawrence Bragg, Lord Kelvin, J.J. Thomson, James Clerk Maxwell, Dorothy Hodgkin and Stephen Hawking. In 2004, the Royal Society began ''The Journal of the Royal Society Interface'' for papers at the interface of physical sciences and life sciences. History The journal began in 1831 as a compilation of abstracts of papers in the ''Philosophical Transactions of the Royal Society'', the older Royal Society publication ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duncan Sommerville
Duncan MacLaren Young Sommerville (1879–1934) was a Scottish mathematician and astronomer. He compiled a bibliography on non-Euclidean geometry and also wrote a leading textbook in that field. He also wrote ''Introduction to the Geometry of N Dimensions'', advancing the study of polytopes. He was a co-founder and the first secretary of the New Zealand Astronomical Society. Sommerville was also an accomplished watercolourist, producing a series New Zealand landscapes. The middle name 'MacLaren' is spelt using the old orthography M'Laren in some sources, for example the records of the Royal Society of Edinburgh. Early life Sommerville was born on 24 November 1879 in Beawar in India, where his father the Rev Dr James Sommerville, was employed as a missionary by the United Presbyterian Church of Scotland. His father had been responsible for establishing the hospital at Jodhpur, Rajputana. The family returned home to Perth, Scotland, where Duncan spent 4 years at a private sc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytopes
''Convex Polytopes'' is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perles, and G. C. Shephard, and published in 1967 by John Wiley & Sons. It went out of print in 1970. A second edition, prepared with the assistance of Volker Kaibel, Victor Klee, and Günter M. Ziegler, was published by Springer-Verlag in 2003, as volume 221 of their book series Graduate Texts in Mathematics. ''Convex Polytopes'' was the winner of the 2005 Leroy P. Steele Prize for mathematical exposition, given by the American Mathematical Society. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries. Topics The book has 19 chapters. After two chapters introducing background material in linear algebra, topology, and convex geometry, two more chapters ... [...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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Poincaré Duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact and without boundary), then the ''k''th cohomology group of ''M'' is isomorphic to the (n-k)th homology group of ''M'', for all integers ''k'' :H^k(M) \cong H_(M). Poincaré duality holds for any coefficient ring, so long as one has taken an orientation with respect to that coefficient ring; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation. History A form of Poincaré duality was first stated, without proof, by Henri Poincaré in 1893. It was stated in terms of Betti numbers: The ''k''th and (n-k)th Betti numbers of a closed (i.e., compact and without boundary) orientable ''n''-manifold are equal. The ''cohomology'' concept was at that time about 40 y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toric Variety
In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special, but still quite general class of toric varieties, this information is also encoded in a polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space. Toric varieties from tori The original motivation to study toric varieties was to study torus embeddings. Given the algebraic t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Max Dehn
Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. However, he was forced to retire in 1935 and eventually fled Germany in 1939 and emigrated to the United States. Dehn was a student of David Hilbert, and in his habilitation in 1900 Dehn resolved Hilbert's third problem, making him the first to resolve one of Hilbert's well-known 23 problems. Dehn's students include Ott-Heinrich Keller, Ruth Moufang, Wilhelm Magnus, and the artists Dorothea Rockburne and Ruth Asawa. Biography Dehn was born to a family of Jewish origin in Hamburg, Imperial Germany. He studied the foundations of geometry with Hilbert at Göttingen in 1899, and obtained a proof of the Jordan curve theorem for polygons. In 1900 he wrote his dissertation on the role of the Legendre angle sum theorem in axiomatic geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables with coefficients in ''k'', that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of \mathbb^n. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the quotient ring :k _0, \ldots, x_nI is called the homogeneous coordinate ring of ''X''. Basic invariants of ''X'' such as the degree and the dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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