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Conformal Radius
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 : \operatorname(z,D) := \frac\,. The ... [...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 (topology), path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such a hole cannot be continuously transformed into each other. 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 (topology), 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schramm–Loewner Evolution
In probability theory, the Schramm–Loewner evolution with parameter ''κ'', also known as stochastic Loewner evolution (SLE''κ''), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Given a parameter ''κ'' and a domain ''U'' in the complex plane, it gives a family of random curves in ''U'', with ''κ'' controlling how much the curve turns. There are two main variants of SLE, ''chordal SLE'' which gives a family of random curves from two fixed boundary points, and ''radial SLE'', which gives a family of random curves from a fixed boundary point to a fixed interior point. These curves are defined to satisfy conformal invariance and a domain Markov property. It was discovered by as a conjectured scaling limit of the planar uniform spanning tree (UST) and the planar loop-erased random walk (LERW) probabilistic processes, and developed by him together with Greg Lawler and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign. History Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled ''CRC Concise Encyclopedia of Mathematics''. The free online version became only partially accessible to the public. In 1999 Weisstein we ... [...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, ... [...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 [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Fekete
Michael (Mihály) Fekete (; 19 July 1886 – 13 May 1957) was a Hungarian-Israelis, Israeli mathematician. Biography Michael Fekete was born in Zenta, Austria-Hungary (today Senta, Serbia). He received his PhD in 1909 from the University of Budapest (later renamed Eötvös Loránd University). He studied under Lipót Fejér. After completing his PhD he switched to University of Göttingen, which was considered a mathematics hub. In 1914, he returned to the University of Budapest, where he attained the title of Privatdozent. Fekete also worked as a private math tutor. Among his students was János Neumann. In 1922, Fekete published a paper together with Neumann on extremal polynomials, which was Neumann's first scientific paper. Fekete dedicated the majority of his scientific work to the transfinite diameter. In 1928 Fekete immigrated to Mandate Palestine and was among the first instructors of the Institute of Mathematics at the Hebrew University of Jerusalem. In 1929 he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull, 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 set, 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 projective duality, dual problem of intersecting Half-space (geome ... [...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 (topology), closure of not belonging to the Interior (topology), 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 Manifold#Manifold with boundary, 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 Felix Hausdorff, Hausdorff's border, which is defined as the intersection ... [...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_^\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] |
Schwarz Lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex differential geometry that estimates the (squared) pointwise norm , \partial f , ^2 of a holomorphic map f:(X,g_X) \to (Y,g_Y) between Hermitian manifolds under curvature assumptions on g_X and g_Y. The classical Schwarz lemma is a result in complex analysis typically viewed to be about holomorphic functions from the open set, open unit disk \mathbb := \to itself. The Schwarz lemma has opened several branches of complex geometry, and become an essential tool in the use of geometric PDE methods in complex geometry. Statement of the classical Schwarz Lemma Let \mathbf = \ be the open unit disk in the complex number, complex plane \mathbb centered at the origin (mathematics), 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), = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants. Affine geometry The affine transformations of the upper half-plane include # shifts (x,y)\mapsto (x+c,y), c\in\mathbb, and # dilations (x,y)\mapsto (\lambda x,\lambda y), \lambda > 0. Proposition: Let and 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 to Then take \lambda=(\text\ B)/(\text\ A) and dilate. Then shift to the center of Inversive geometry Definition: \mathcal := \left\ . can be recognized as the circle of radius centered at and as the polar plot of \rho(\theta) = \cos \theta. Proposition: in and are collinear points. In ... [...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 the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras. In the Greek deductive geometry exemplified by Euclid's ''Elements'', distances were not represented as numbers but line segments of the same length, which were considered "equal". The notion of distance is inherent in the compass tool used to draw a circle, whose points all have the same distance from a common center point. 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 b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
