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Clifford Analysis
Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but are not limited to, the Hodge–Dirac operator, d+d on a Riemannian manifold, the Dirac operator in euclidean space and its inverse on C_^(\mathbf^) and their conformal equivalents on the sphere, the Laplacian in euclidean ''n''-space and the Atiyah–Singer–Dirac operator on a spin manifold, Rarita–Schwinger/Stein–Weiss type operators, conformal Laplacians, spinorial Laplacians and Dirac operators on SpinC manifolds, systems of Dirac operators, the Paneitz operator, Dirac operators on hyperbolic space, the hyperbolic Laplacian and Weinstein equations. Euclidean space In Euclidean space the Dirac operator has the form :D=\sum_^e_\frac where ''e''1, ..., ''e''''n'' is an orthonormal basis for R''n'', and R''n'' is considered to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford Algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford. The most familiar Clifford algebras, the orthogonal Clifford algebras, are also referred to as (''pseudo-'')''Riemannian Clifford algebras'', as distinct from ''symplectic Clifford algebras''.see for ex. Introduction and basic properties A Clifford algebra is a unital associative algebra that contains and is generated by a vector space over a field , where is equipped with a qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beurling–Ahlfors Transform
In complex analysis and geometric function theory, the Grunsky matrices, or Grunsky operators, are infinite matrices introduced in 1939 by Helmut Grunsky. The matrices correspond to either a single holomorphic function on the unit disk or a pair of holomorphic functions on the unit disk and its complement. The Grunsky inequalities express boundedness properties of these matrices, which in general are contraction operators or in important special cases unitary operators. As Grunsky showed, these inequalities hold if and only if the holomorphic function is univalent. The inequalities are equivalent to the inequalities of Goluzin, discovered in 1947. Roughly speaking, the Grunsky inequalities give information on the coefficients of the logarithm of a univalent function; later generalizations by Milin, starting from the Lebedev–Milin inequality, succeeded in exponentiating the inequalities to obtain inequalities for the coefficients of the univalent function itself. The Grunsky m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy Spaces
In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real analysis Hardy spaces are certain spaces of distributions on the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the ''Lp'' spaces of functional analysis. For 1 ≤ ''p'' < ∞ these real Hardy spaces ''Hp'' are certain subsets of ''Lp'', while for ''p'' < 1 the ''Lp'' spaces have some undesirable properties, and the Hardy spaces are much better behaved. There are also higher-dimensional generalizations, consisting of certain holomorphic functions on |
Plemelj Operator
Josip Plemelj (December 11, 1873 – May 22, 1967) was a Slovene mathematician, whose main contributions were to the theory of analytic functions and the application of integral equations to potential theory. He was the first chancellor of the University of Ljubljana. Life Plemelj was born in the village of Bled near Bled Castle in Austria-Hungary (now Slovenia); he died in Ljubljana, Yugoslavia (now Slovenia). His father, Urban, a carpenter and crofter, died when Josip was only a year old. His mother Marija, née , found bringing up the family alone very hard, but she was able to send her son to school in Ljubljana, where Plemelj studied from 1886 to 1894. Due to a bench thrown into Tivoli Pond by him or his friends, he could not attend the school after he finished the fourth class and had to pass the final exam privately. After leaving and obtaining the necessary examination results he went to the University of Vienna in 1894 where he had applied to Faculty of Arts to stu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Szegő Kernel
In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions. It is named for its discoverer, the Hungarian mathematician Gábor Szegő. Let Ω be a bounded domain in C''n'' with ''C''2 boundary, and let ''A''(Ω) denote the space of all holomorphic functions in Ω that are continuous on \overline. Define the Hardy space ''H''2(∂Ω) to be the closure in ''L''2(∂Ω) of the restrictions of elements of ''A''(Ω) to the boundary. The Poisson integral implies that each element ''ƒ'' of ''H''2(∂Ω) extends to a holomorphic function ''Pƒ'' in Ω. Furthermore, for each ''z'' ∈ Ω, the map :f\mapsto Pf(z) defines a continuous linear functional on ''H''2(∂Ω). By the Riesz representation theorem, this linear functional is represented by a kernel ''k''''z'', which is to say :Pf(z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bergman Kernel
In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is the reproducing kernel for the Hilbert space (RKHS) of all square integrable holomorphic functions on a domain ''D'' in C''n''. In detail, let L2(''D'') be the Hilbert space of square integrable functions on ''D'', and let ''L''2,''h''(''D'') denote the subspace consisting of holomorphic functions in L2(''D''): that is, :L^(D) = L^2(D)\cap H(D) where ''H''(''D'') is the space of holomorphic functions in ''D''. Then ''L''2,''h''(''D'') is a Hilbert space: it is a closed linear subspace of ''L''2(''D''), and therefore complete in its own right. This follows from the fundamental estimate, that for a holomorphic square-integrable function ''ƒ'' in ''D'' for every compact subset ''K'' of ''D''. Thus convergence of a sequence of holomorphic functions in ''L''2(''D'') implies also compact convergence, and so the limit function is also holomorphic. Another consequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Transform
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics. Biogra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two '' directed lines'' in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative. Quaternions are generally represented in the form :a + b\ \mathbf i + c\ \mathbf j +d\ \mathbf k where , and are real numbers; and , and are the ''basic quaternions''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Spinor
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the .... The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors. It was first published in 1928. Formal definition In general, let ''D'' be a first-order differential operator acting on a vector bundle ''V'' over a Riemannian manifold ''M''. If :D^2=\Delta, \, where ∆ is the Laplacian of ''V'', then ''D'' is called a Dirac operator. In high-energy physics, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kelvin Inverse
The Kelvin transform is a device used in classical potential theory to extend the concept of a harmonic function, by allowing the definition of a function which is 'harmonic at infinity'. This technique is also used in the study of subharmonic and superharmonic functions. In order to define the Kelvin transform ''f''* of a function ''f'', it is necessary to first consider the concept of inversion in a sphere in R''n'' as follows. It is possible to use inversion in any sphere, but the ideas are clearest when considering a sphere with centre at the origin. Given a fixed sphere ''S''(0,''R'') with centre 0 and radius ''R'', the inversion of a point ''x'' in R''n'' is defined to be x^* = \frac x. A useful effect of this inversion is that the origin 0 is the image of \infty, and \infty is the image of 0. Under this inversion, spheres are transformed into spheres, and the exterior of a sphere is transformed to the interior, and vice versa. The Kelvin transform of a function is then def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
