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Fekete Constant
In mathematics, the conformal radius is a way to measure the size of a simply connected planar domain ''D'' viewed from a point ''z'' in it. As opposed to notions using Euclidean distance (say, the radius of the largest inscribed disk with center ''z''), this notion is well-suited to use in complex analysis, in particular in conformal maps and conformal geometry. A closely related notion is the transfinite diameter or (logarithmic) capacity of a compact simply connected set ''D'', which can be considered as the inverse of the conformal radius of the complement ''E'' = ''Dc'' viewed from infinity. Definition Given a simply connected domain ''D'' ⊂ C, and a point ''z'' ∈ ''D'', by the Riemann mapping theorem there exists a unique conformal map ''f'' : ''D'' → D onto the unit disk (usually referred to as the uniformizing map) with ''f''(''z'') = 0 ∈ D and ''f''′(''z'') ∈ R+. The conformal radius of ''D'' from ''z'' is then defined as : \mathrm(z,D) := \frac\,. The simples ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whenev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gábor Szegő
Gábor Szegő () (January 20, 1895 – August 7, 1985) was a Hungarian-American mathematician. He was one of the foremost mathematical analysts of his generation and made fundamental contributions to the theory of orthogonal polynomials and Toeplitz matrices building on the work of his contemporary Otto Toeplitz. Life Szegő was born in Kunhegyes, Austria-Hungary (today Hungary), into a Jewish family as the son of Adolf Szegő and Hermina Neuman.Biography on the homepage of Kunhegyes (in Hungarian) He married the chemist Anna Elisabeth Neményi in 1919, with whom he had two children. In 1912 he started studies in at the |
Michael Fekete
Michael (Mihály) Fekete ( he, מיכאל פקטה; 19 July 1886 – 13 May 1957) was a Hungarian-Israeli mathematician. Biography Fekete was born in 1886 in Zenta, Austria-Hungary (today Senta, Serbia). He received his PhD in 1909 from the University of Budapest (later renamed to Eötvös Loránd University), under the stewardship of Lipót Fejér, among whose students were other mathematicians such as Paul Erdős, John von Neumann, Pál Turán and George Pólya. After completing his PhD he left to University of Göttingen, which in those days was considered a mathematics hub, and subsequently returned to the University of Budapest, where he attained the title of Privatdozent. In addition, Fekete engaged in private mathematics tutoring. Among his students was János Neumann, who was later known in the United States as John von Neumann. In 1922, Fekete published a paper together with von Neumann in the subject of extremal polynomials. This was von Neumann's first scienti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Capacity Of A Set
In mathematics, the capacity of a set in Euclidean space is a measure of the "size" of that set. Unlike, say, Lebesgue measure, which measures a set's volume or physical extent, capacity is a mathematical analogue of a set's ability to hold electrical charge. More precisely, it is the capacitance of the set: the total charge a set can hold while maintaining a given potential energy. The potential energy is computed with respect to an idealized ground at infinity for the harmonic or Newtonian capacity, and with respect to a surface for the condenser capacity. Historical note The notion of capacity of a set and of "capacitable" set was introduced by Gustave Choquet in 1950: for a detailed account, see reference . Definitions Condenser capacity Let Σ be a closed, smooth, (''n'' − 1)-dimensional hypersurface in ''n''-dimensional Euclidean space ℝ''n'', ''n'' ≥ 3; ''K'' will denote the ''n''-dimensional compact (i.e., closed and bounded) set of which Σ is the b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary (topology)
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include \operatorname(S), \operatorname(S), and \partial S. Some authors (for example Willard, in ''General Topology'') use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, ''Metric Spaces'' by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Koebe 1/4 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_ 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 n, b_n, ^2 r^. Since the area i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz Lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions. Statement Let \mathbf = \ be the open unit disk in the complex plane \mathbb centered at the origin, and let f : \mathbf\rightarrow \mathbb be a holomorphic map such that f(0) = 0 and , f(z), \leq 1 on \mathbf. Then , f(z), \leq , z, for all z \in \mathbf, and , f'(0), \leq 1. Moreover, if , f(z), = , z, for some non-zero z or , f'(0), = 1, then f(z) = az for some a \in \mathbb with , a, = 1.Theorem 5.34 in Proof The proof is a straightforward application of the maximum modulus principle on the function :g(z) = \begin \frac\, & \mbox z \neq 0 \\ f'(0) & \mbox z = 0, \end which is holomorphic on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0. Complex plane Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to the set of complex numbers with positive imaginary part: :\mathcal \equiv \ ~. The term arises from a common visualization of the complex number as the point in the plane endowed with Cartesian coordinates. When the axis is oriented vertically, the "upper half-plane" corresponds to the region above the axis and thus complex numbers for which > 0. It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by 0. Proposition: Let ''A'' and ''B'' be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes ''A'' to ''B''. :Proof: First shift the center of ''A'' to (0,0). Then take λ = (diame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century. The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. In some applications in statistic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
