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Superstrong Approximation
Superstrong approximation is a generalisation of strong approximation in algebraic groups ''G'', to provide spectral gap results. The spectrum in question is that of the Laplacian matrix associated to a family of quotients of a discrete group Γ; and the gap is that between the first and second eigenvalues (normalisation so that the first eigenvalue corresponds to constant functions as eigenvectors). Here Γ is a subgroup of the rational points of ''G'', but need not be a lattice: it may be a so-called thin group. The "gap" in question is a lower bound (absolute constant) for the difference of those eigenvalues. A consequence and equivalent of this property, potentially holding for Zariski dense subgroups Γ of the special linear group over the integers, and in more general classes of algebraic groups ''G'', is that the sequence of Cayley graphs for reductions Γ''p'' modulo prime numbers ''p'', with respect to any fixed set ''S'' in Γ that is a symmetric set and generating set, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Approximation In Algebraic Groups
In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ... to algebraic groups ''G'' over global fields ''k''. History proved strong approximation for some classical groups. Strong approximation was established in the 1960s and 1970s, for semisimple simply-connected algebraic groups over global fields. The results for number fields are due to and ; the global field, function field case, over finite fields, is due to and . In the number field case Platonov also proved a related result over local fields called the Kneser–Tits conjecture. Formal definitions and properties Let ''G'' be a linear algebraic group over a global field ''k'', and ''A'' the adele ring of ''k''. If ''S'' is a non-emp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Set
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set. The larger set is then said to be generated by the smaller set. It is commonly the case that the generating set has a simpler set of properties than the generated set, thus making it easier to discuss and examine. It is usually the case that properties of the generating set are in some way preserved by the act of generation; likewise, the properties of the generated set are often reflected in the generating set. List of generators A list of examples of generating sets follow. * Generating set or spanning set of a vector space: a set that spans the vector space * Generating set of a group: A subset of a group that is not contained in any subgro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Groups
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 theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Growth Rate (group Theory)
In the mathematical subject of geometric group theory, the growth rate of a group with respect to a symmetric generating set describes how fast a group grows. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length ''n''. Definition Suppose ''G'' is a finitely generated group; and ''T'' is a finite ''symmetric'' set of generators (symmetric means that if x \in T then x^ \in T ). Any element x \in G can be expressed as a word in the ''T''-alphabet : x = a_1 \cdot a_2 \cdots a_k \text a_i\in T. Consider the subset of all elements of ''G'' that can be expressed by such a word of length ≤ ''n'' :B_n(G,T) = \. This set is just the closed ball of radius ''n'' in the word metric ''d'' on ''G'' with respect to the generating set ''T'': :B_n(G,T) = \. More geometrically, B_n(G,T) is the set of vertices in the Cayley graph with respect to ''T'' that are within distan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximate Subgroup
An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ''ad-'' (''ad-'' before ''p'' becomes ap- by assimilation) meaning ''to''. Words like ''approximate'', ''approximately'' and ''approximation'' are used especially in technical or scientific contexts. In everyday English, words such as ''roughly'' or ''around'' are used with a similar meaning. It is often found abbreviated as ''approx.'' The term can be applied to various properties (e.g., value, quantity, image, description) that are nearly, but not exactly correct; similar, but not exactly the same (e.g., the approximate time was 10 o'clock). Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws. In science, approximation can refer to us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Lubotzky
Alexander Lubotzky ( he, אלכסנדר לובוצקי; born 28 June 1956) is an Israeli mathematician and former politician who is currently a professor at the Weizmann Institute of Science and an adjunct professor at Yale University. He served as a member of the Knesset for Third Way (Israel), The Third Way party between 1996 and 1999. In 2018 he won the Israel Prize for his accomplishments in mathematics and computer science. Biography Alexander (Alex) Lubotzky was born in Tel Aviv to Holocaust survivors. His father, Iser Lubotzky was a Partisan (military), Partisan, Irgun officer and the legal advisor of Herut. After school, Lubotzky did his Israel Defence Forces, IDF national service as a captain officer in a special intelligence and communication unit. He studied mathematics at Bar-Ilan University during highschool, gaining a Bachelor of Arts, BA (summa cum laude) and continuing directly with studying for his PhD under the supervision of Hillel Furstenberg. Lubotzky married ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kazhdan's Property (T)
In mathematics, a locally compact topological group ''G'' has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Spectrum of a C*-algebra, Fell topology. Informally, this means that if ''G'' acts unitary representation, unitarily on a Hilbert space and has "almost invariant vectors", then it has a nonzero invariant vector. The formal definition, introduced by David Kazhdan (#CITEREFKazhdan1967, 1967), gives this a precise, quantitative meaning. Although originally defined in terms of irreducible representations, property (T) can often be checked even when there is little or no explicit knowledge of the unitary dual. Property (T) has important applications to group representation theory, Grigory Margulis, lattices in algebraic groups over local fields, ergodic theory, geometric group theory, Expander graph, expanders, operator algebras and the expanding graph, theory of networks. Definitions Let ''G'' be a σ-compact, locally compact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expander Family
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. Informal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Set
In mathematics, a nonempty subset of a group is said to be symmetric if it contains the inverses of all of its elements. Definition In set notation a subset S of a group G is called if whenever s \in S then the inverse of s also belongs to S. So if G is written multiplicatively then S is symmetric if and only if S = S^ where S^ := \left\. If G is written additively then S is symmetric if and only if S = - S where - S := \. If S is a subset of a vector space then S is said to be a if it is symmetric with respect to the additive group structure of the vector space; that is, if S = - S, which happens if and only if - S \subseteq S. The of a subset S is the smallest symmetric set containing S, and it is equal to S \cup - S. The largest symmetric set contained in S is S \cap - S. Sufficient conditions Arbitrary unions and intersections of symmetric sets are symmetric. Any vector subspace in a vector space is a symmetric set. Examples In \R, examples of symmetric s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Gap
In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to other properties of the system. See also * Cheeger constant (graph theory) * Cheeger constant (Riemannian geometry) * Eigengap * Spectral gap (physics) * Spectral radius In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectru ... References External links * {{Mathanalysis-stub Spectral theory ... [...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. Not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Linear Group
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant :\det\colon \operatorname(n, F) \to F^\times. where ''F''× is the multiplicative group of ''F'' (that is, ''F'' excluding 0). These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When ''F'' is a finite field of order ''q'', the notation is sometimes used. Geometric interpretation The special linear group can be characterized as the group of ''volume and orientation preserving'' linear transformations of R''n''; this corresponds to the interpretation of the determinant as measuring change in volume and orientation. Lie subgroup When ''F'' is R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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