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Thin Group (algebraic Group Theory)
In algebraic group theory, a thin group is a discrete Zariski topology, Zariski-dense subgroup of ''G''(R) that has infinite covolume, where ''G'' is a semisimple algebraic group over the real number, reals. This is in contrast to a lattice (group), lattice, which is a discrete subgroup of finite covolume. The theory of "group expansion" (expander graph properties of related Cayley graphs) for particular thin groups has been applied to arithmetic properties of Apollonian circles and in Zaremba's conjecture. References * Linear algebraic groups {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski Topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semisimple Algebraic Group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group ''GL''(''n'') of invertible matrices, the special orthogonal group ''SO''(''n''), and the symplectic group ''Sp''(2''n''). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive. Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limit (mathematics), limits, continuous function, continuity and derivatives. The set of real numbers is mathematical notation, denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers subset, include t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (group)
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension n which spans the vector space \mathbb^n. For any basis of \mathbb^n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expander Graph
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes. Definitions Intuitively, an expander graph is a finite, undirected multigraph in which every subset of the vertices that is not "too large" has a "large" boundary. Different formalisations of these notions give rise to different notions of expanders: ''edge expanders'', ''vertex expanders'', and ''spectral expanders'', as defined below. A disconnected graph is not an expander, since the boundary of a connected component is empty. Every connected graph is an expander; however, different connected graphs have different expansion parameters. The complete graph has the best expansion property, but it has largest possible degree. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing families of expander graphs. Definition Let G be a group and S be a generating set of G. The Cayley graph \Gamma = \Gamma(G,S) is an edge-colored directed graph constructed as follows: In his Collected Mathematical Papers 10: 403–405. * Each element g of G is assigned a vertex: the vertex set of \Gamma is identified with G. * Each element s of S is assigned a color c_s. * For every g \in G and s \in S, there is a directed edge of color c_s from the vertex corresponding to g to the one corresponding to gs. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apollonian Circles
In geometry, Apollonian circles are two families ( pencils) of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates. They were discovered by Apollonius of Perga, a renowned Greek geometer. Definition The Apollonian circles are defined in two different ways by a line segment denoted ''CD''. Each circle in the first family (the blue circles in the figure) is associated with a positive real number ''r'', and is defined as the locus of points ''X'' such that the ratio of distances from ''X'' to ''C'' and to ''D'' equals ''r'', :\left\. For values of ''r'' close to zero, the corresponding circle is close to ''C'', while for values of ''r'' close to ∞, the corresponding circle is close to ''D''; for the intermediate value ''r'' = 1, the circle degenerates to a line, the perpendicular bisector of ''CD''. The equation defining these circles as a loc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zaremba's Conjecture
In mathematics, and more particularly in the analytic theory of regular continued fractions, an infinite regular continued fraction ''x'' is said to be ''restricted'', or composed of restricted partial quotients, if the sequence of denominators of its partial quotients is bounded; that is :x = _0;a_1,a_2,\dots= a_0 + \cfrac = a_0 + \underset \frac,\, and there is some positive integer ''M'' such that all the (integral) partial denominators ''ai'' are less than or equal to ''M''. Periodic continued fractions A regular periodic continued fraction consists of a finite initial block of partial denominators followed by a repeating block; if : \zeta = _0;a_1,a_2,\dots,a_k,\overline\, then ζ is a quadratic irrational number, and its representation as a regular continued fraction is periodic. Clearly any regular periodic continued fraction consists of restricted partial quotients, since none of the partial denominators can be greater than the largest of ''a''0 through ''a''''k''+'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
