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Conformal Welding
In mathematics, conformal welding (sewing or gluing) is a process in geometric function theory for producing a Riemann surface by joining together two Riemann surfaces, each with a disk removed, along their boundary circles. This problem can be reduced to that of finding univalent holomorphic maps ''f'', ''g'' of the unit disk and its complement into the extended complex plane, both admitting continuous extensions to the closure of their domains, such that the images are complementary Jordan domains and such that on the unit circle they differ by a given quasisymmetric homeomorphism. Several proofs are known using a variety of techniques, including the Beltrami equation In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation : = \mu . for ''w'' a complex distribution of the complex variable ''z'' in some open set ''U'', with derivatives that are locally ''L''2 ..., the Hilbert transform on the circle and elementary approximati ... [...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] |
Geometric Function Theory
Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem. Topics in geometric function theory The following are some of the most important topics in geometric function theory: Conformal maps A conformal map is a function which preserves angles locally. In the most common case the function has a domain and range in the complex plane. More formally, a map, : f: U \rightarrow V\qquad with U,V \subset \mathbb^n is called conformal (or angle-preserving) at a point u_0 if it preserves oriented angles between curves through u_0 with respect to their orientation (i.e., not just the magnitude of the angle). Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. Quasiconformal maps In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm. Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasisymmetric Map
In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the relative sizes of sets: if two sets ''A'' and ''B'' have diameters ''t'' and are no more than distance ''t'' apart, then the ratio of their sizes changes by no more than a multiplicative constant. These maps are also related to quasiconformal maps, since in many circumstances they are in fact equivalent. Definition Let (''X'', ''d''''X'') and (''Y'', ''d''''Y'') be two metric spaces. A homeomorphism ''f'':''X'' → ''Y'' is said to be η-quasisymmetric if there is an increasing function ''η'' : 0_if_for_all_triples_of_distinct_points_x,y,z_in_X,_then :_.html" ;"title=", ∞) → [0, ∞) such that for any triple ''x'', ''y'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beltrami Equation
In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation : = \mu . for ''w'' a complex distribution of the complex variable ''z'' in some open set ''U'', with derivatives that are locally ''L''2, and where ''μ'' is a given complex function in ''L''∞(''U'') of norm less than 1, called the Beltrami coefficient, and where \partial / \partial z and \partial / \partial \overline are Wirtinger derivatives. Classically this differential equation was used by Gauss to prove the existence locally of isothermal coordinates on a surface with analytic Riemannian metric. Various techniques have been developed for solving the equation. The most powerful, developed in the 1950s, provides global solutions of the equation on C and relies on the L''p'' theory of the Beurling transform, a singular integral operator defined on L''p''(C) for all 1 0, ''EG'' − ''F''2 > 0) that varies smoothly with ''x'' and ''y''. The Beltrami coefficient of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Integral Operators Of Convolution Type
In mathematics, singular integral operators of convolution type are the singular integral operators that arise on R''n'' and T''n'' through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on ''L''2 is evident because the Fourier transform converts them into multiplication operators. Continuity on ''Lp'' spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general cri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Trick
Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander. Statement Two homeomorphisms of the ''n''-dimensional ball D^n which agree on the boundary sphere S^ are isotopic. More generally, two homeomorphisms of ''D''''n'' that are isotopic on the boundary are isotopic. Proof Base case: every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary. If f\colon D^n \to D^n satisfies f(x) = x \text x \in S^, then an isotopy connecting ''f'' to the identity is given by : J(x,t) = \begin tf(x/t), & \text 0 \leq \, x\, 0 the transformation J_t replicates f at a different scale, on the disk of radius t, thus as t\rightarrow 0 it is reasonable to expect that J_t merges to the identity. The subtlety is that at t=0, f "disappears": the germ at the origin "jumps" from an infinitely stretched version of f to the identity. Each of the steps in the homotopy could be smoothed (smoot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy Space
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 s 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 |
Fredholm Operator
In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : ''X'' → ''Y'' between two Banach spaces with finite-dimensional kernel \ker T and finite-dimensional (algebraic) cokernel \mathrm\,T = Y/\mathrm\,T, and with closed range \mathrm\,T. The last condition is actually redundant. The '' index'' of a Fredholm operator is the integer : \mathrm\,T := \dim \ker T - \mathrm\,\mathrm\,T or in other words, : \mathrm\,T := \dim \ker T - \mathrm\,\mathrm\,T. Properties Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored." The formally correct statement follows. A bounded operator ''T'' : ''X'' → ''Y'' between Banach spaces ''X'' and ''Y'' is Fredholm if and only if it is invertible modulo compact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Argument Principle
In complex analysis, the argument principle (or Cauchy's argument principle) relates the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative. Specifically, if ''f''(''z'') is a meromorphic function inside and on some closed contour ''C'', and ''f'' has no zeros or poles on ''C'', then : \frac\oint_ \, dz=Z-P where ''Z'' and ''P'' denote respectively the number of zeros and poles of ''f''(''z'') inside the contour ''C'', with each zero and pole counted as many times as its multiplicity and order, respectively, indicate. This statement of the theorem assumes that the contour ''C'' is simple, that is, without self-intersections, and that it is oriented counter-clockwise. More generally, suppose that ''f''(''z'') is a meromorphic function on an open set Ω in the complex plane and that ''C'' is a closed curve in Ω which avoids all zeros and poles of ''f'' and is contractible to a point inside Ω. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Analysis
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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