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Orbital Integral
In mathematics, an orbital integral is an integral transform that generalizes the spherical mean operator to homogeneous spaces. Instead of integrating over spheres, one integrates over generalized spheres: for a homogeneous space ''X'' = ''G''/''H'', a generalized sphere centered at a point ''x''0 is an orbit of the isotropy group of ''x''0. Definition The model case for orbital integrals is a Riemannian symmetric space ''G''/''K'', where ''G'' is a Lie group and ''K'' is a symmetric compact subgroup. Generalized spheres are then actual geodesic spheres and the spherical averaging operator is defined as :M^rf(x) = \int_K f(gk\cdot y)\,dk, where * the dot denotes the action of the group ''G'' on the homogeneous space ''X'' * ''g'' ∈ ''G'' is a group element such that ''x'' = ''g''·''o'' * ''y'' ∈ ''X'' is an arbitrary element of the geodesic sphere of radius ''r'' centered at ''x'': ''d''(''x'',''y'') = ''r'' * the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗. More precisely, is a subgroup of if the Restriction (mathematics), restriction of ∗ to is a group operation on . This is often denoted , read as " is a subgroup of ". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group is a subgroup which is a subset, proper subset of (that is, ). This is often represented notationally by , read as " is a proper subgroup of ". Some authors also exclude the trivial group from being proper (that is, ). If is a subgroup of , then is sometimes called an overgroup of . The same definitions apply more generally when is an arbitrary se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Formula (other)
Trace formula may refer to: * Arthur–Selberg trace formula, also known as invariant trace formula, Jacquet's relative trace formula, simple trace formula, stable trace formula * Grothendieck trace formula, an analogue in algebraic geometry of the Lefschetz fixed-point theorem in algebraic topology, used to express the Hasse–Weil zeta function. * Gutzwiller trace formula: See Quantum chaos * Kuznetsov trace formula, an extension of the Petersson trace formula. * Local trace formula, an analog of Arthur–Selberg trace formula * Petersson trace formula *Selberg trace formula * Behrend's trace formula, or Behrend's fixed point formula a generalization of the Grothendieck–Lefschetz trace formula, that may be interpreted as a Selberg trace formula. See also * List of zeta functions * List of fixed point theorems In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), unde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphic Forms
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the group action (mathematics), action of a discrete subgroup \Gamma \subset G of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are holomorphic automorphic forms defined over the groups SL2(R), SL(2, R) or PSL2(R), PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the Adele ring, adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group ''G''(A''F''), for an algebraic group ''G'' and an algebraic number field ''F'', is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Funk Transform
In the mathematical field of integral geometry, the Funk transform (also known as Minkowski–Funk transform, Funk–Radon transform or spherical Radon transform) is an integral transform defined by integrating a function on great circles of the sphere. It was introduced by Paul Funk in 1911, based on the work of . It is closely related to the Radon transform. The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere. Definition The Funk transform is defined as follows. Let ''ƒ'' be a continuous function on the 2-sphere S2 in R3. Then, for a unit vector x, let :Ff(\mathbf) = \int_ f(\mathbf)\,ds(\mathbf) where the integral is carried out with respect to the arclength ''ds'' of the great circle ''C''(x) consisting of all unit vectors perpendicular to x: :C(\mathbf) = \. Inversion The Funk transform annihilates all odd functions, and so it is natural to confine attention to the case when ''ƒ'' is even. In that case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations. Classical context Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory. The early work of Luis Santaló and Wilhelm Blaschke was in this connection. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations. There is a sample space of lines, one on which the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unimodular Group
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory. Preliminaries Let (G, \cdot) be a locally compact Hausdorff topological group. The \sigma-algebra generated by all open subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g is an element of G and S is a subset of G, then we define the left and right translates of S by ''g'' as follows: * Left translate: g S = \. * Right translate: S g = \. Left and right translates map Borel sets onto Borel sets. A measure \mu on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haar Measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This Measure (mathematics), measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of mathematical analysis, analysis, number theory, group theory, representation theory, mathematical statistics, statistics, probability theory, and ergodic theory. Preliminaries Let (G, \cdot) be a locally compact space, locally compact Hausdorff space, Hausdorff topological group. The Sigma-algebra, \sigma-algebra generated by all open subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g is an element of G and S is a subset of G, then we define the left and right Coset, translates of S by ''g'' as follows: * Left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a " straight line". The noun '' geodesic'' and the adjective '' geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of havi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory. In the following we will assume all groups are Hausdorff spaces. Compact Lie groups Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include * the circle group T and the torus groups T''n'', * the orthogonal group O(''n''), the special orthogonal group SO(''n'') and its covering spin group Spin(''n''), * the unitary group U(''n'') and the special unitary group SU(''n''), * the compact forms of the exceptional Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Transform
In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, that is called the kernel or nucleus of the transform. Some kernels have an associated ''inverse kernel'' K^( u,t ) which (roughly speaking) yields an inverse transform: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
