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Grunsky Matrix
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 ... [...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 functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, '' holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which dea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Field Theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points. Scale invariance vs conformal invariance In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions to argue that it should appear in nature. The basic idea behind its plausibility is that ''local'' scale invariant theories have t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Louis De Branges
Louis may refer to: People * Louis (given name), origin and several individuals with this name * Louis (surname) * Louis (singer), Serbian singer Other uses * Louis (coin), a French coin * HMS ''Louis'', two ships of the Royal Navy See also * Derived terms * King Louis (other) * Saint Louis (other) * Louis Cruise Lines * Louis dressing, for salad * Louis Quinze, design style Associated terms * Lewis (other) * Louie (other) * Luis (other) * Louise (other) * Louisville (other) Associated names * * Chlodwig, the origin of the name Ludwig, which is translated to English as "Louis" * Ladislav and László - names sometimes erroneously associated with "Louis" * Ludovic, Ludwig, Ludwick, Ludwik Ludwik () is a Polish given name. Notable people with the name include: * Ludwik Czyżewski, Polish WWII general * Ludwik Fleck (1896–1961), Polish medical doctor and biologist * Ludwik Gintel (1899–1973), Polish- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a constant called the ''center'' of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center ''c'' is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gronwall's Area Theorem
In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following: Koebe Quarter Theorem. The image of an injective analytic function f:\mathbf\to\mathbb from the unit disk \mathbf onto a subset of the complex plane contains the disk whose center is f(0) and whose radius is , f'(0), /4. The theorem is named after Paul Koebe, who conjectured the result in 1907. The theorem was proven by Ludwig Bieberbach in 1916. The example of the Koebe function shows that the constant 1/4 in the theorem cannot be improved (increased). A related result is the Schwarz lemma, and a notion related to both is conformal radius. Grönwall's area theorem Suppose that :g(z) = z +b_1z^ + b_2 z^ + \cdots is univalent in , z, >1. Then :\sum_^\infty n, b_n, ^2 \le 1. In fact, if r > 1, the complement of the image of the disk , z, >r is a bounded domain X(r). Its area is given by : \int_ dx\,dy = \int_\overline\,dz = \int_\overline\,dg=\pi r^2 - \pi\sum_^\infty n, b_n, ^2 r^. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reproducing Kernel
In functional analysis, a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H of functions from a set X (to \mathbb or \mathbb) is an RKHS if the point-evaluation functional L_x:H\to\mathbb, L_x(f)=f(x), is continuous for every x\in X. Equivalently, H is an RKHS if there exists a function K_x \in H such that, for all f \in H,\langle f, K_x \rangle = f(x).The function K_x is then called the ''reproducing kernel'', and it reproduces the value of f at x via the inner product. An immediate consequence of this property is that convergence in norm implies uniform convergence on any subset of X on which \, K_x\, is bounded. However, the converse does not necessarily hold. Often the set X carries a topology, and \, K_x\, depends continuously on x\in X, in which case: convergence in norm implies uniform convergence on compact subsets of X. It is not entirely straightforward ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, Complete metric space, completeness means that there are enough limit (mathematics), limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, mathematical formulation of quantum mechanics, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy–Schwarz Inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics. Inner products of vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors in Hilbert spaces). The inequality for sums was published by . The corresponding inequality for integrals was published by and . Schwarz gave the modern proof of the integral version. Statement of the inequality The Cauchy–Schwarz inequality states that for all vectors \mathbf and \mathbf of an inner product space where \langle \cdot, \cdot \rangle is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coincides with the standard measure of length, area, or volume. In general, it is also called '-dimensional volume, '-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by \lambda(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential 1-form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f(x) \, dx is an example of a -form, and can be integrated over an interval ,b/math> contained in the domain of f: \int_a^b f(x)\,dx. Similarly, the expression f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz is a -form that can be integrated over a surface S: \int_S \left(f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz\right). The symbol \wedge denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form f(x,y,z) \, dx \wedge dy \wedge dz represents a volume element that can be integrated over a region of space. In general, a -form is an object th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
