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Analytic Capacity
In the mathematical discipline of complex analysis, the analytic capacity of a compact subset ''K'' of the complex plane is a number that denotes "how big" a bounded analytic function on C \ ''K'' can become. Roughly speaking, ''γ''(''K'') measures the size of the unit ball of the space of bounded analytic functions outside ''K''. It was first introduced by Lars Ahlfors in the 1940s while studying the removability of singularities of bounded analytic functions. Definition Let ''K'' ⊂ C be compact. Then its analytic capacity is defined to be :\gamma(K) = \sup \ Here, \mathcal^\infty (U) denotes the set of bounded analytic functions ''U'' → C, whenever ''U'' is an open subset of the complex plane. Further, : f'(\infty):= \lim_z\left(f(z)-f(\infty)\right) : f(\infty):= \lim_f(z) Note that f'(\infty) = g'(0), where g(z) = f(1/z). However, usually f'(\infty)\neq \lim_ f'(z). If ''A'' ⊂ C is an arbitrary set, then we define :\gamma(A) = \sup \. Removable ... [...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] |
Removable Singularity
In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point. For instance, the (unnormalized) sinc function : \text(z) = \frac has a singularity at . This singularity can be removed by defining \text(0) := 1, which is the limit of as tends to 0. The resulting function is holomorphic. In this case the problem was caused by being given an indeterminate form. Taking a power series expansion for \frac around the singular point shows that : \text(z) = \frac\left(\sum_^ \frac \right) = \sum_^ \frac = 1 - \frac + \frac - \frac + \cdots. Formally, if U \subset \mathbb C is an open subset of the complex plane \mathbb C, a \in U a point of U, and f: U\setminus \ \rightarrow \mathbb C is a holomorphic function, then a is called a removable singularity for f if there exists a holomorp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Xavier Tolsa
Xavier Tolsa (born 1966) is a Catalan mathematician, specializing in analysis. Tolsa is a professor at the Autonomous University of Barcelona and at the ''Institució Catalana de Recerca i Estudis Avançats'' (ICREA), the Catalan Institute for Advanced Scientific Studies. Tolsa does research on harmonic analysis (Calderón-Zygmund theory), complex analysis, geometric measure theory, and potential theory. Specifically, he is known for his research on analytic capacity and removable sets. He solved the problem of A. G. Vitushkin about the semi-additivity of analytic capacity. This enabled him to solve an even older problem of Paul Painlevé on the geometric characterization of removable sets. Tolsa succeeded in solving the Painlevé problem by using the concept of so-called curvatures of measures introduced by Mark Melnikov in 1995. Tolsa's proof involves estimates of Cauchy transforms. He has also done research on the so-called David- Semmes problem involving Riesz transforms an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Guy David (mathematician)
Guy David (born 1957) is a French mathematician, specializing in analysis. Biography David studied from 1976 to 1981 at the École normale supérieure, graduating with ''Agrégation'' and ''Diplôme d'études approfondies'' (DEA). At the University of Paris-Sud (Paris XI) he received in 1981 his doctoral degree (''Thèse du 3ème cycle'') and in 1986 his higher doctorate (''Thèse d'État'') with thesis ''Noyau de Cauchy et opérateurs de Caldéron-Zygmund'' supervised by Yves Meyer. David was from 1982 to 1989 an ''attaché de recherches'' (research associate) at the ''Centre de mathématiques Laurent Schwartz'' of the CNRS. At the University of Paris-Sud he was from 1989 to 1991 a professor and from 1991 to 2001 a professor first class, and is since 1991 a professor of the ''Classe exceptionelle''. (with CV) David is known for his research on Hardy spaces and on singular integral equations using the methods of Alberto Calderón. In 1998 David solved a special case of a problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John B
John Bryn Williams (born 1977), known as John B, is an English disc jockey and electronic music producer. He is widely recognised for his eccentric clothing and wild hair and his production of several cutting edge drum and bass tracks. John B ranked number 76 in ''DJ Magazine''s 2010 Top 100 DJs annual poll, announced on 27 October 2010. Career Williams was born on 12 July 1977 in Maidenhead, Berkshire. He started producing music around the age of 14, and now is the head of drum and bass record label Beta Recordings, together with its more specialist drum and bass sub-labels Nu Electro, Tangent, and Chihuahua. He also has releases on Formation Records, Metalheadz and Planet Mu. Williams was ranked 92nd drum and bass DJ on the 2009 ''DJ Magazine'' top 100. Style While his trademark sound has evolved through the years, it generally involves female vocals and trance-like synths (a style which has been dubbed "trance and bass", "trancestep" and "futurestep" by listeners). His m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anatoli Georgievich Vitushkin
Anatoli Georgievich Vitushkin (russian: Анато́лий Гео́ргиевич Виту́шкин) (June 25, 1931 – May 9, 2004) was a Soviet mathematician noted for his work on analytic capacity and other parts of mathematical analysis. Early life Anatoli Georgievich Vitushkin was born on 25 June 1931 in Moscow. He was blind. Career He entered Moscow State University in 1949 after graduating from the Tula Suvorov Military School where mathematics was taught as part of a broader education for potential officers. He graduated in 1954. He studied under Andrey Kolmogorov and benefited from participation in Alexander Kronrod's circle. He joined the Steklov Institute of Mathematics staff in 1965. For many years he was a member of the Editorial board of the Russian journal; ''Mathematical Notes ''Mathematical Notes'' is a peer-reviewed mathematical journal published by Springer Science+Business Media on behalf of the Russian Academy of Sciences that covers all aspects of ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in ,∞to each set in \R^n or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. Likewise, the one-dimensional Hausdorff measure of a simple curve in \R^n is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue-measurable subset of \R^2 is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of counting, length, and area. It also generalizes volume. In fact, there are ''d''-dimensional Hausdorff measures for any ''d'' ≥ 0, which is not necessarily an integer. These measures are fundamenta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Montel's Theorem
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel, and give conditions under which a family of holomorphic functions is normal. Locally uniformly bounded families are normal The first, and simpler, version of the theorem states that a family of holomorphic functions defined on an open subset of the complex numbers is normal if and only if it is locally uniformly bounded. This theorem has the following formally stronger corollary. Suppose that \mathcal is a family of meromorphic functions on an open set D. If z_0\in D is such that \mathcal is not normal at z_0, and U\subset D is a neighborhood of z_0, then \bigcup_f(U) is dense in the complex plane. Functions omitting two values The stronger version of Montel's Theorem (occasionally referred to as the Fundamental Normality Test) states that a family of holomorphic functions, all of which omit t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singleton (mathematics)
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains, thus 1 and are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as \ is a singleton as it contains a single element (which itself is a set, however, not a singleton). A set is a singleton if and only if its cardinality is . In von Neumann's set-theoretic construction of the natural numbers, the number 1 is ''defined'' as the singleton \. In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set ''A'', the axiom applied to ''A'' and ''A'' asserts the existence of \, which is the same a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the discrete topology), or no set can be open except the space itself and the empty set (the indiscrete topology). In practice, however, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Subset
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
