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Bers Area Inequality
In the mathematical theory of Kleinian groups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem was proved by , apart from a gap that was filled by . The Ahlfors finiteness theorem states that if Γ is a finitely-generated Kleinian group with region of discontinuity Ω, then Ω/Γ has a finite number of components, each of which is a compact Riemann surface with a finite number of points removed. Bers area inequality The Bers area inequality is a quantitative refinement of the Ahlfors finiteness theorem proved by . It states that if Γ is a non-elementary finitely-generated Kleinian group with ''N'' generators and with region of discontinuity Ω, then :Area(Ω/Γ) ≤ with equality only for Schottky groups. (The area is given by the Poincaré metric in each component.) Moreover, if Ω1 is an invariant component then :Area(Ω/Γ) ≤ 2Area(Ω1/Γ) with equality only for Fuchsian group In math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleinian Group
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with , is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by . has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball in . The group of Möbius transformations is also related as the non-orientation-preserving isometry group of , . So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces. History The theory of general Kleinian groups was founded by and , who named them after Felix Klein. The special case of Schottky groups had been studied a few years earlier, in 1877, by Schottky. Definitions One modern definition of Kleinian group is as a group which acts on the 3-ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schottky Group
In mathematics, a Schottky group is a special sort of Kleinian group, first studied by . Definition Fix some point ''p'' on the Riemann sphere. Each Jordan curve not passing through ''p'' divides the Riemann sphere into two pieces, and we call the piece containing ''p'' the "exterior" of the curve, and the other piece its "interior". Suppose there are 2''g'' disjoint Jordan curves ''A''1, ''B''1,..., ''A''''g'', ''B''''g'' in the Riemann sphere with disjoint interiors. If there are Möbius transformations ''T''''i'' taking the outside of ''A''''i'' onto the inside of ''B''''i'', then the group generated by these transformations is a Kleinian group. A Schottky group is any Kleinian group that can be constructed like this. Properties By work of , a finitely generated Kleinian group is Schottky if and only if it is finitely generated, free, has nonempty domain of discontinuity, and all non-trivial elements are loxodromic. A fundamental domain for the action of a Schottky gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuchsian Group
In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup of PSL(2,R). Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In this case, the group may be called the Fuchsian group of the surface. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry. Some Escher graphics are based o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen Smale, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal D'Analyse Mathématique
The ''Journal d'Analyse Mathématique'' is a triannual peer-reviewed scientific journal published by Magnes Press (Hebrew University of Jerusalem). It was established in 1951 by Binyamin Amirà. It covers research in mathematics, especially classical analysis and related areas such as complex function theory, ergodic theory, functional analysis, harmonic analysis, partial differential equations, and quasiconformal mapping. Abstracting and indexing The journal is abstracted and indexed in: *MathSciNet *Science Citation Index Expanded *Scopus *ZbMATH Open According to the ''Journal Citation Reports'', the journal has a 2021 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 1.132. References External links *{{Official website, 1=https://www.springer.com/mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Groups
Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a group with the discrete topology *Discrete category, category whose only arrows are identity arrows *Discrete mathematics, the study of structures without continuity *Discrete optimization, a branch of optimization in applied mathematics and computer science *Discrete probability distribution, a random variable that can be counted *Discrete space, a simple example of a topological space *Discrete spline interpolation, the discrete analog of ordinary spline interpolation *Discrete time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ..., non-c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Groups
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kleinian Groups
In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space . The latter, identifiable with , is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by . has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball in . The group of Möbius transformations is also related as the non-orientation-preserving isometry group of , . So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces. History The theory of general Kleinian groups was founded by and , who named them after Felix Klein. The special case of Schottky groups had been studied a few years earlier, in 1877, by Schottky. Definitions One modern definition of Kleinian group is as a group which acts on the 3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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