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ADE Classification
In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, rather than a posteriori verification of a parallelism, was posed in . The complete list of simply laced Dynkin diagrams comprises :A_n, \, D_n, \, E_6, \, E_7, \, E_8. Here "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of \pi/2 = 90^\circ (no edge between the vertices) or 2\pi/3 = 120^\circ (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting B_n and C_n), and three of the five exceptional Dynkin diagrams (omitting F_4 and G_2). This list is non-redundant if one takes n \geq 4 for D_n. If one extends the families to include redundant terms, one obtains the exceptional isomorphisms :D_3 \cong A_3, E_4 \cong A_4, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Laced Dynkin Diagrams
Simply may refer to: * ''Simply'' (Blossom Dearie album), 1982 * ''Simply'' (K. T. Oslin album), 2015 * "Simply", a song by De La Soul from the 2001 album '' AOI: Bionix'' * Simply Market, a French supermarket chain * Simply Beverages, an American fruit juice company See also * Simple (other) {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Platonic Solid
In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (identical in shape and size) regular polygons (all angles congruent and all edge (geometry), edges congruent), and the same number of faces meet at each Vertex (geometry), vertex. There are only five such polyhedra: Geometers have studied the Platonic solids for thousands of years. They are named for the ancient Greek philosopher Plato, who hypothesized in one of his dialogues, the ''Timaeus (dialogue), Timaeus'', that the classical elements were made of these regular solids. History The Platonic solids have been known since antiquity. It has been suggested that certain carved stone balls created by the late Neolithic people of Scotland represent these shapes; however, these balls have rounded knobs rather than being polyhedral, the num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Matrices
In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan. Lie algebras A (symmetrizable) generalized Cartan matrix is a square matrix A = (a_) with integer entries such that # For diagonal entries, a_ = 2 . # For non-diagonal entries, a_ \leq 0 . # a_ = 0 if and only if a_ = 0 # A can be written as DS, where D is a diagonal matrix, and S is a symmetric matrix. For example, the Cartan matrix for ''G''2 can be decomposed as such: : \begin 2 & -3 \\ -1 & 2 \end = \begin 3&0\\ 0&1 \end\begin \frac & -1 \\ -1 & 2 \end. The third condition is not independent but is really a consequence of the first and fourth conditions. We can always choose a ''D'' with positive diagonal entries. In that case, if ''S'' in the above d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Laplace Operator
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a Graph (discrete mathematics), graph or a lattice (group), discrete grid. For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete Laplace operator is more commonly called the Laplacian matrix. The discrete Laplace operator occurs in physics problems such as the Ising model and loop quantum gravity, as well as in the study of discrete dynamical systems. It is also used in numerical analysis as a stand-in for the continuous Laplace operator. Common applications include image processing, where it is known as the Laplace filter, and in machine learning for cluster analysis, clustering and semi-supervised learning on neighborhood graphs. Definitions Graph Laplacians There are various definitions of the ''discrete Laplacian'' for Graph (discrete mathematics), graphs, differing by sign and scale factor (sometim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Slodowy
Peter Slodowy (12 October 1948, in Leverkusen – 19 November 2002, in Bonn) was a German mathematician who worked on singularity theory and algebraic geometry. He completed his Ph.D. thesis at the University of Regensburg The University of Regensburg () is a public research university located in the city of Regensburg, Germany. The university was founded on 18 July 1962 by the Landtag of Bavaria as the fourth full-fledged university in Bavaria. Following groundbr ... in 1978 under the direction of Theodor Bröcker and Egbert Brieskorn. The Slodowy correspondence is named after him. Publications * References * * 20th-century German mathematicians 1948 births 2002 deaths University of Regensburg alumni {{Germany-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Du Val Singularity
In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on a double branched cover of the plane, with minimal resolution obtained by replacing the singular point with a tree of smooth rational curves, with intersection pattern dual to a Dynkin diagram of A-D-E singularity type. They are the canonical singularities (or, equivalently, rational Gorenstein singularities) in dimension 2. They were studied by Patrick du Val and Felix Klein. The Du Val singularities also appear as quotients of \Complex^2 by a finite subgroup of SL2(\Complex); equivalently, a finite subgroup of SU(2), which are known as binary polyhedral groups. The rings of invariant polynomials of these finite group actions were computed by Klein, and are essentially the coordinate rings of the singularities; this is a classic result in invariant theory. Classification The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. Definitions of orbifold have been given several times: by Ichirō Satake in the context of automorphic forms in the 1950s under the name ''V-manifold''; by William Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name ''orbifold'', after a vote by his students; and by André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name ''orbihedron''. Historically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group \mathrm(2,\Z) on the upper half-plane: a version of the Riemann–Roch theorem holds after the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coxeter Group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; for example, the symmetry group of each regular polyhedron is a finite Coxeter group. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups, and finite Coxeter groups were classified in 1935. Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical than others. The best known is the ( convex, non- stellated) regular icosahedron—one of the Platonic solids—whose faces are 20 equilateral triangles. Regular icosahedra There are two objects, one convex and one nonconvex, that can both be called regular icosahedra. Each has 30 edges and 20 equilateral triangle faces with five meeting at each of its twelve vertices. Both have icosahedral symmetry. The term "regular icosahedron" generally refers to the convex variety, while the nonconvex form is called a ''great icosahedron''. Convex regular icosahedron The convex regular icosahedron is usually referred to simply as the ''regular icosahedron'', one of the five regular Platonic solids, and is represented by its Schläfli symbol , contai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedron
In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Poinsot polyhedron, regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The #Pyritohedron, pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the #Tetartoid, tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling polyhedra, space-filling. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of irregular octahedra also exist, including both convex set, convex and non-convex shapes. Combinatorially equivalent to the regular octahedron The following polyhedra are combinatorially equivalent to the regular octahedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of it: * Triangular antiprisms: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. The regular octahedron is a special case in which the six lateral triangles are also equilateral. * Tetragonal bipyramids, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all thr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It is a type of parallelepiped, with pairs of parallel opposite faces, and more specifically a rhombohedron, with congruent edges, and a rectangular cuboid, with right angles between pairs of intersecting faces and pairs of intersecting edges. It is an example of many classes of polyhedra: Platonic solid, regular polyhedron, parallelohedron, zonohedron, and plesiohedron. The dual polyhedron of a cube is the regular octahedron. The cube can be represented in many ways, one of which is the graph known as the cubical graph. It can be constructed by using the Cartesian product of graphs. The cube is the three-dimensional hypercube, a family of polytopes also including the two-dimensional square and four-dimensional tesseract. A cube with 1, unit s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
