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Singular Integral Operators On Closed Curves
In mathematics, singular integral operators on closed curves arise in problems in analysis, in particular complex analysis and harmonic analysis. The two main singular integral operators, the Hilbert transform and the Cauchy transform, can be defined for any smooth Jordan curve in the complex plane and are related by a simple algebraic formula. In the special case of Fourier series for the unit circle, the operators become the classical Cauchy transform, the orthogonal projection onto Hardy space, and the Hilbert transform a real orthogonal linear complex structure. In general the Cauchy transform is a non-self-adjoint idempotent and the Hilbert transform a non-orthogonal complex structure. The range of the Cauchy transform is the Hardy space of the bounded region enclosed by the Jordan curve. The theory for the original curve can be deduced from that of the unit circle, where, because of rotational symmetry, both operators are classical singular integral operators of convolution typ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Schmidt Operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm \, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ^2_H, where \ is an orthonormal basis. The index set I need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This definition is independent of the choice of the orthonormal basis. In finite-dimensional Euclidean space, the Hilbert–Schmidt norm \, \cdot\, _\text is identical to the Frobenius norm. , , ·, , is well defined The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if \_ and \_ are such bases, then \sum_i \, Ae_i\, ^2 = \sum_ \left, \langle Ae_i, f_j\rangle \^2 = \sum_ \left, \langle e_i, A^*f_j\rangle \^2 = \sum_j\, A^* f_j\, ^2. If e_i = f_i, then \sum_i \, Ae_i\, ^2 = \sum_i\, A^* e_i\, ^2. As for any bounded operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Studies In Mathematics
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published ihardcoverane-bookformats. List of books *1 ''The General Topology of Dynamical Systems'', Ethan Akin (1993, ) *2 ''Combinatorial Rigidity'', Jack Graver, Brigitte Servatius, Herman Servatius (1993, ) *3 ''An Introduction to Gröbner Bases'', William W. Adams, Philippe Loustaunau (1994, ) *4 ''The Integrals of Lebesgue, Denjoy, Perron, and Henstock'', Russell A. Gordon (1994, ) *5 ''Algebraic Curves and Riemann Surfaces'', Rick Miranda (1995, ) *6 ''Lectures on Quantum Groups'', Jens Carsten Jantzen (1996, ) *7 ''Algebraic Number Fields'', Gerald J. Janusz (1996, 2nd ed., ) *8 ''Discovering Modern Set Theory. I: The Basics'', Winfried Just, Martin Weese (1996, ) *9 ''An Invitation to Arithmetic Geometry'', Dino Lorenzini (1996, ) *10 ''Representations of Finite and Compact Groups'', Barry Simon (199 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hölder Continuous
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Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natura ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory Kernel Theorem
In mathematics, the Carathéodory kernel theorem is a result in complex analysis and geometric function theory established by the Greek mathematician Constantin Carathéodory in 1912. The uniform convergence on compact sets of a sequence of holomorphic univalent functions, defined on the unit disk in the complex plane and fixing 0, can be formulated purely geometrically in terms of the limiting behaviour of the images of the functions. The kernel theorem has wide application in the theory of univalent functions and in particular provides the geometric basis for the Loewner differential equation. Kernel of a sequence of open sets Let ''U''''n'' be a sequence of open sets in C containing 0. Let ''V''''n'' be the connected component of the interior of containing 0. The kernel of the sequence is defined to be the union of the ''V''''n'''s, provided it is non-empty; otherwise it is defined to be \. Thus the kernel is either a connected open set containing 0 or the one point set \. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue Calculus
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. Contour integration methods include: * direct integration of a complex-valued function along a curve in the complex plane (a ''contour''); * application of the Cauchy integral formula; and * application of the residue theorem. One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums. Curves in the complex plane In complex analysis a contour is a type of curve in the complex plane. In contour integration, contours provide a precise definition of the curves on which an integral may be suitably defined. A curve in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy's Integral Formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis. Theorem Let be an open subset of the complex plane , and suppose the closed disk defined as :D = \bigl\ is completely contained in . Let be a holomorphic function, and let be the circle, oriented counterclockwise, forming the boundary of . Then for every in the interior of , :f(a) = \frac \oint_\gamma \frac\,dz.\, The proof of this statement uses the Cauchy integral theorem and like that theorem, it only requires t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Operator Topology
In functional analysis, a branch of mathematics, the strong operator topology, often abbreviated SOT, is the locally convex topology on the set of bounded operators on a Hilbert space ''H'' induced by the seminorms of the form T\mapsto\, Tx\, , as ''x'' varies in ''H''. Equivalently, it is the coarsest topology such that, for each fixed ''x'' in ''H'', the evaluation map T\mapsto Tx (taking values in ''H'') is continuous in T. The equivalence of these two definitions can be seen by observing that a subbase for both topologies is given by the sets U(T_0,x,\epsilon) = \ (where ''T0'' is any bounded operator on ''H'', ''x'' is any vector and ε is any positive real number). In concrete terms, this means that T_i\to T in the strong operator topology if and only if \, T_ix-Tx\, \to 0 for each ''x'' in ''H''. The SOT is stronger than the weak operator topology and weaker than the norm topology. The SOT lacks some of the nicer properties that the weak operator topology has, but being ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bergman Space
In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain ''D'' of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for , the Bergman space is the space of all holomorphic functions f in ''D'' for which the p-norm is finite: :\, f\, _ := \left(\int_D , f(x+iy), ^p\,\mathrm dx\,\mathrm dy\right)^ < \infty. The quantity is called the ''norm'' of the function ; it is a true if . Thus is the subspace of holomorphic functions that are in the space L''p''(''D''). The Bergman spaces are |
Green's Theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively oriented, piecewise smooth, simple closed curve in a plane, and let be the region bounded by . If and are functions of defined on an open region containing and have continuous partial derivatives there, then \oint_C (L\, dx + M\, dy) = \iint_ \left(\frac - \frac\right) dx\, dy where the path of integration along is anticlockwise. In physics, Green's theorem finds many applications. One is solving two-dimensional flow integrals, stating that the sum of fluid outflowing from a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter. Proof when ''D'' is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Frédéric Frenet
Jean Frédéric Frenet (; 7 February 1816 – 12 June 1900) was a French mathematician, astronomer, and meteorologist. He was born and died in Périgueux, France. He is best known for being an independent co-discoverer of the Frenet–Serret formulas. He wrote six out of the nine formulas, which at that time were not expressed in vector notation. These formulas are important in the theory of space curves (differential geometry), and they were presented in his doctoral thesis at Toulouse in 1847. That year he became a professor at Toulouse, and one year later, 1848, he became professor of mathematics at Lyon. He also was director of an astronomical observatory at Lyon. Four years later, in 1852, he published the Frenet formulas in the ''Journal de Mathématiques Pures et Appliquées''. In 1856 his calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature ''at a point'' of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
