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Integer Lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice. Automorphism group The automorphism group (or group of congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order 2''n'' ''n''!. As a matrix group it is given by the set of all ''n''×''n'' signed permutation matrices. This group is isomorphic to the semidirect product :(\mathbb Z_2)^n \rtimes S_n where the symmetric group ''S''''n'' acts on (Z2)''n'' by permutation (this is a classic example of a wreath product). For the square lattice, this is the group of the square, or the dihedral group of order 8; for the three-dimensional cubic lattice, we get the group of the cube, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Pentagon Approximation Integer Vertex Coordinates
The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger song), "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses in arts, entertainment, and media * Regular character, a main character who appears more frequently and/or prominently than a recurring character * Regular division of the plane, a series of drawings by the Dutch artist M. C. Escher which began in 1936 * ''Regular Show'', an animated television sitcom * ''The Regular Guys'', a radio morning show Language * Regular inflection, the formation of derived forms such as plurals in ways that are typical for the language ** Regular verb * Regular script, the newest of the Chinese script styles Mathematics There are an extremely large number of unrelated notions of "regular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signed Permutation Matrices
In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value. An example of a generalized permutation matrix is :\begin 0 & 0 & 3 & 0\\ 0 & -7 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & \sqrt2\end. Structure An invertible matrix ''A'' is a generalized permutation matrix if and only if it can be written as a product of an invertible diagonal matrix ''D'' and an (implicitly invertible) permutation matrix ''P'': i.e., :A = DP. Group structure The set of ''n'' × ''n'' generalized permutation matrices with entries in a field ''F'' forms a subgroup of the general linear group GL(''n'', ''F''), in which the group of nonsingular diagonal matrices Δ(''n'', ''F'') forms a normal subgroup. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logic, logical system in which each result is ''mathematical proof, proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Grid
A regular grid is a tessellation of ''n''-dimensional Euclidean space by congruent parallelotopes (e.g. bricks). Its opposite is irregular grid. Grids of this type appear on graph paper and may be used in finite element analysis, finite volume methods, finite difference methods, and in general for discretization of parameter spaces. Since the derivatives of field variables can be conveniently expressed as finite differences, structured grids mainly appear in finite difference methods. Unstructured grids offer more flexibility than structured grids and hence are very useful in finite element and finite volume methods. Each cell in the grid can be addressed by index (i, j) in two dimensions or (i, j, k) in three dimensions, and each vertex has coordinates (i\cdot dx, j\cdot dy) in 2D or (i\cdot dx, j\cdot dy, k\cdot dz) in 3D for some real numbers ''dx'', ''dy'', and ''dz'' representing the grid spacing. Related grids A Cartesian grid is a special case where the elements are uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coarse Structure
In the mathematical fields of geometry and topology, a coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topological spaces to be defined. The concern of traditional geometry and topology is with the small-scale structure of the space: properties such as the continuity of a function depend on whether the inverse images of small open sets, or neighborhoods, are themselves open. Large-scale properties of a space—such as boundedness, or the degrees of freedom of the space—do not depend on such features. ''Coarse geometry'' and ''coarse topology'' provide tools for measuring the large-scale properties of a space, and just as a metric or a topology contains information on the small-scale structure of a space, a coarse structure contains information on its large-scale properties. Properly, a coarse structure is not the large-scale analog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Diophantine Graph
An Erdős–Diophantine graph is an object in the mathematical subject of Diophantine equations consisting of a set of integer points at integer distances in the plane that cannot be extended by any additional points. Equivalently, in geometric graph theory, it can be described as a complete graph with vertices located on the integer square grid \mathbb^2 such that all mutual distances between the vertices are integers, while all other grid points have a non-integer distance to at least one vertex. Erdős–Diophantine graphs are named after Paul Erdős and Diophantus of Alexandria. They form a subset of the set of Diophantine figures, which are defined as complete graphs in the Diophantine plane for which the length of all edges are integers (unit distance graphs). Thus, Erdős–Diophantine graphs are exactly the Diophantine figures that cannot be extended. The existence of Erdős–Diophantine graphs follows from the Erdős–Anning theorem The Erdős–Anning theorem states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\times B = \. A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product is taken, the cells of the table contain ordered pairs of the form . One can similarly define the Cartesian product of ''n'' sets, also known as an ''n''-fold Cartesian product, which can be represented by an ''n''-dimensional array, where each element is an ''n''-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product. Examples A deck of cards An ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Geometry
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. Four theorems in Diophantine geometry which are of fundamental importance include: * Mordell–Weil Theorem * Roth's Theorem * Siegel's Theorem * Faltings's Theorem Background Serge Lang published a book ''Diophantine Geometry'' in the area in 1962, and by this book he coined the term "Diophantine Geometry". The traditional arrangement of material on Diophantine equations was by degree and number of variables, as in Mordell's ''Diophantine Equations'' (1969). Mordell's book starts with a remark on homogeneous equations ''f'' = 0 over the rational field, attributed to C. F. Gauss, that non-zero solutions in integers (even primitive lattice points) exist if non-zero rational solutions do, and notes a caveat of L. E. D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedral Group
A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedron that is dual to an octahedron. The group of orientation-preserving symmetries is ''S''4, the symmetric group or the group of permutations of four objects, since there is exactly one such symmetry for each permutation of the four diagonals of the cube. Details Chiral and full (or achiral) octahedral symmetry are the discrete point symmetries (or equivalently, symmetries on the sphere) with the largest symmetry groups compatible with translational symmetry. They are among the crystallographic point groups of the cubic crystal system. As the hyperoctahedral group of dimension 3 the full octahedral group is the wreath product S_2 \wr S_3 \simeq S_2^3 \rtimes S_3,and a natural way to identify its elements is as pairs (m, n) with m \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dihedral Group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, or refers to the symmetries of the -gon, a group of order . In abstract algebra, refers to this same dihedral group. This article uses the geometric convention, . Definition Elements A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. Usually, we take n \ge 3 here. The associated rotations and reflections make up the dihedral group \mathrm_n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides and n/2 axes of symmetry connecting oppo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wreath Product
In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used in the classification of permutation groups and also provide a way of constructing interesting examples of groups. Given two groups A and H (sometimes known as the ''bottom'' and ''top''), there exist two variations of the wreath product: the unrestricted wreath product A \text H and the restricted wreath product A \text H. The general form, denoted by A \text_ H or A \text_ H respectively, requires that H acts on some set \Omega; when unspecified, usually \Omega = H (a regular wreath product), though a different \Omega is sometimes implied. The two variations coincide when A, H, and \Omega are all finite. Either variation is also denoted as A \wr H (with \wr for the LaTeX symbol) or ''A'' ≀ ''H'' (Unicode U+2240). The notion ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
