Vinberg's Algorithm

	Vinberg's Algorithm In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group. used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II25,1 in terms of the Leech lattice. Description of the algorithm Let \Gamma < \mathrm(\mathbb^n) be a hyperbolic reflection group. Choose any point $v_0 \in \mathbb^n$ ; we shall call it the basic (or initial) point. The fundamental domain $P_0$ of its stabilizer $\Gamma_$ is a polyhedral cone in $\mathbb^n$ . Let $H_1,...,H_m$ be the faces of this cone, and let $a_1,...,a_m$ be outer normal vectors to it. Consider the half-spaces $H_k^- = \.$ There exists a unique fundamental polyhedron $P$ of $\Gamma$ contained in $P_0$ and containing the point $v_0$ . Its faces containing [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ernest Borisovich Vinberg Ernest Borisovich Vinberg (; 26 July 1937 – 12 May 2020) was a Soviet and Russian mathematician, who worked on Lie groups and algebraic groups, discrete subgroups of Lie groups, invariant theory, and representation theory. He introduced Vinberg's algorithm and the Koecher–Vinberg theorem. He was a recipient of the 1997 Humboldt Prize. He was on the executive committee of the Moscow Mathematical Society. In 1983, he was an Invited Speaker with a talk on ''Discrete reflection groups in Lobachevsky spaces'' at the International Congress of Mathematicians in Warsaw. In 2010, he was elected an International Honorary Member of the American Academy of Arts and Sciences. Ernest Vinberg died from pneumonia caused by COVID-19 Coronavirus disease 2019 (COVID-19) is a contagious disease caused by the coronavirus SARS-CoV-2. In January 2020, the disease spread worldwide, resulting in the COVID-19 pandemic. The symptoms of COVID‑19 can vary but often include fever ... on 12 May ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Fundamental Domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits. There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells. Hints at a general definition Given an action of a group ''G'' on a topological space ''X'' by homeomorphisms, a fundamental domain for this action is a set ''D'' of representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hyperbolic Reflection Group Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics Hyperbolic geometry, a non-Euclidean geometry Hyperbolic functions, analogues of ordinary trigonometric functions, defined using the hyperbola * of or pertaining to hyperbole, the use of exaggeration as a rhetorical device or figure of speech * ''Hyperbolic'' (album), by Pnau, 2024 See also * Exaggeration * Hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ... {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	II25,1 In mathematics, II25,1 is the even 26-dimensional Lorentzian unimodular lattice. It has several unusual properties, arising from Conway's discovery that it has a norm zero Weyl vector. In particular it is closely related to the Leech lattice Λ, and has the Conway group Co1 at the top of its automorphism group. Construction Write ''R''''m'',''n'' for the ''m''+''n''-dimensional vector space ''R''''m''+''n'' with the inner product of (''a''1,...,''a''''m''+''n'') and (''b''1,...,''b''''m''+''n'') given by :''a''1''b''1+...+''a''''m''''b''''m'' − ''a''''m''+1''b''''m''+1 − ... − ''a''''m''+''n''''b''''m''+''n''. The lattice II25,1 is given by all vectors (''a''1,...,''a''26) in ''R''25,1 such that either all the ''ai'' are integers or they are all integers plus 1/2, and their sum is even. Reflection group The lattice II25,1 is isomorphic to Λ⊕H where: Λ is the Leech lattice, H is the 2-dimensional even Lorentzian lattice, generated by 2 norm 0 vectors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Leech Lattice In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by Ernst Witt in 1940. Characterization The Leech lattice Λ24 is the unique lattice in 24-dimensional Euclidean space, E24, with the following list of properties: It is unimodular lattice, unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix (mathematics), matrix with determinant 1. It is even; i.e., the square of the length of each vector in Λ24 is an even integer. The length of every non-zero vector in Λ24 is at least 2. The last condition is equivalent to the condition that unit balls centered at the points of Λ24 do not overlap. Each is tangent to 196,560 neighbors, and this is known to be the largest number of non-overlapping 24-dimensional unit balls that can kissing number, simultaneously t ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Journal Of Algebra ''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier Elsevier ( ) is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell (journal), Cell'', the ScienceDirect collection of electronic journals, .... ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1984. From 1985 until 2000, Walter Feit served as its editor-in-chief. In 2004, ''Journal of Algebra'' announced (vol. 276, no. 1 and 2) the creation of a new section on computational algebra, with a separate editorial board. The first issue completely devoted to computational algebra was vol. 292, no. 1 (October 2005). The Editor-in-Chief of the ''Journal of Algebra'' is Michel Broué, Université Paris Diderot, and Gerhard Hiß, Rheinisch-Westfälische Technische Hochschule Aachen ( R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Oxford University Press Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the Vice Chancellor, vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, Walton Street, Oxford, opposite Somerville College, Oxford, Somerville College, in the inner suburb of Jericho, Oxford, Jericho. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Hyperbolic Geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane (mathematics), plane where every point is a saddle point. Hyperbolic plane geometry is also the geometry of pseudosphere, pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they local property, locally resemble the hyperbolic plane. The hyperboloid model of hyperbolic geometry provides a representation of event (relativity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]