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Carathéodory's Theorem (other)
In mathematics, Carathéodory's theorem may refer to one of a number of results of Constantin Carathéodory: *Carathéodory's theorem (conformal mapping), about the extension of conformal mappings to the boundary *Carathéodory's theorem (convex hull), about the convex hulls of sets in Euclidean space *Carathéodory's existence theorem, about the existence of solutions to ordinary differential equations *Carathéodory's extension theorem, about the extension of a measure *Carathéodory–Jacobi–Lie theorem, a generalization of Darboux's theorem in symplectic topology *Carathéodory's criterion, a necessary and sufficient condition for a measurable set *Carathéodory kernel theorem, a geometric criterion for local uniform convergence of univalent functions *Borel–Carathéodory theorem, about the boundedness of a complex analytic function *Vitali–Carathéodory theorem In mathematics, the Vitali–Carathéodory theorem is a result in real analysis that shows that, under the co ... [...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] |
Constantin Carathéodory
Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant contributions to real and complex analysis, the calculus of variations, and measure theory. He also created an axiomatic formulation of thermodynamics. Carathéodory is considered one of the greatest mathematicians of his era and the most renowned Greek mathematician since antiquity. Origins Constantin Carathéodory was born in 1873 in Berlin to Greek parents and grew up in Brussels. His father Stephanos, a lawyer, served as the Ottoman ambassador to Belgium, St. Petersburg and Berlin. His mother, Despina, née Petrokokkinos, was from the island of Chios. The Carathéodory family, originally from Bosnochori or Vyssa, was well established and respected in Constantinople, and its members held many important governmental positions. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory's Theorem (conformal Mapping)
In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, first proved in 1913, states that the conformal mapping sending the unit disk to the region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve. The result is one of Carathéodory's results on prime ends and the boundary behaviour of univalent holomorphic functions. Proofs of Carathéodory's theorem The first proof of Carathéodory's theorem presented here is a summary of the short self-contained account in ; there are related proofs in and . Clearly if ''f'' admits an extension to a homeomorphism, then ∂''U'' must be a Jordan curve. Conversely if ∂''U'' is a Jordan curve, the first step is to prove ''f'' extends continuously to the closure of ''D''. In fact this will hold if and only if ''f'' is uniformly continuous on ''D'': for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory's Theorem (convex Hull)
Carathéodory's theorem is a theorem in convex geometry. It states that if a point x lies in the convex hull \mathrm(P) of a set P\subset \R^d, then x can be written as the convex combination of at most d+1 points in P. More sharply, x can be written as the convex combination of at most d+1 ''extremal'' points in P, as non-extremal points can be removed from P without changing the membership of ''x'' in the convex hull. Its equivalent theorem for conical combinations states that if a point x lies in the conical hull \mathrm(P) of a set P\subset \R^d, then x can be written as the conical combination of at most d points in P. The similar theorems of Helly and Radon are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice versa. The result is named for Constantin Carathéodory, who proved the theorem in 1911 for the case when P is compact. In 1914 Ernst Steinitz expanded Carathéodory's theorem for arbitrary set. Exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory's Existence Theorem
In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations. The theorem is named after Constantin Carathéodory. Introduction Consider the differential equation : y'(t) = f(t,y(t)) with initial condition : y(t_0) = y_0, where the function ƒ is defined on a rectangular domain of the form : R = \. Peano's existence theorem states that if ƒ is continuous, then the differential equation has at least one solution in a neighbourhood of the initial condition. However, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation : y'(t) = H(t), \quad y(0) = 0, where ''H'' denotes the He ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory's Extension Theorem
In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets ''R'' of a given set ''Ω'' can be extended to a measure on the σ-algebra generated by ''R'', and this extension is unique if the pre-measure is σ-finite. Consequently, any pre-measure on a ring containing all intervals of real numbers can be extended to the Borel algebra of the set of real numbers. This is an extremely powerful result of measure theory, and leads, for example, to the Lebesgue measure. The theorem is also sometimes known as the Carathéodory– Fréchet extension theorem, the Carathéodory– Hopf extension theorem, the Hopf extension theorem and the Hahn–Kolmogorov extension theorem. Introductory statement Several very similar statements of the theorem can be given. A slightly more involved one, based on semi-rings of sets, is given further down below. A shorter, simpler statem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory–Jacobi–Lie Theorem
The Carathéodory– Jacobi– Lie theorem is a theorem in symplectic geometry which generalizes Darboux's theorem. Statement Let ''M'' be a 2''n''-dimensional symplectic manifold with symplectic form ω. For ''p'' ∈ ''M'' and ''r'' ≤ ''n'', let ''f''1, ''f''2, ..., ''f''r be smooth functions defined on an open neighborhood ''V'' of ''p'' whose differentials are linearly independent at each point, or equivalently :df_1(p) \wedge \ldots \wedge df_r(p) \neq 0, where = 0. (In other words, they are pairwise in involution.) Here is the Poisson bracket. Then there are functions ''f''r+1, ..., ''f''n, ''g''1, ''g''2, ..., ''g''n defined on an open neighborhood ''U'' ⊂ ''V'' of ''p'' such that (fi, gi) is a symplectic chart of ''M'', i.e., ω is expressed on ''U'' as :\omega = \sum_^n df_i \wedge dg_i. Applications As a direct application we have the following. Given a Hamiltonian system A Hamiltonian system is a dynamical syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory's Criterion
Carathéodory's criterion is a result in measure theory that was formulated by Greek mathematician Constantin Carathéodory that characterizes when a set is Lebesgue measurable. Statement Carathéodory's criterion: Let \lambda^* : \wp(\R^n) \to , \infty/math> denote the Lebesgue outer measure on \R^n, where \wp(\R^n) denotes the power set of \R^n, and let M \subseteq \R^n. Then M is Lebesgue measurable if and only if \lambda^*(S) = \lambda^* (S \cap M) + \lambda^* (S \cap M^\mathrm) for every S \subseteq \R^n, where M^\mathrm denotes the complement of M. Notice that S is not required to be a measurable set. Generalization The Carathéodory criterion is of considerable importance because, in contrast to Lebesgue's original formulation of measurability, which relies on certain topological properties of \R, this criterion readily generalizes to a characterization of measurability in abstract spaces. Indeed, in the generalization to abstract measures, this theorem is sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory Kernel Theorem
In mathematics, the Carathéodory kernel theorem is a result in complex analysis and geometric function theory established by the Greek mathematician Constantin Carathéodory in 1912. The uniform convergence on compact sets of a sequence of holomorphic univalent functions, defined on the unit disk in the complex plane and fixing 0, can be formulated purely geometrically in terms of the limiting behaviour of the images of the functions. The kernel theorem has wide application in the theory of univalent functions and in particular provides the geometric basis for the Loewner differential equation. Kernel of a sequence of open sets Let ''U''''n'' be a sequence of open sets in C containing 0. Let ''V''''n'' be the connected component of the interior of containing 0. The kernel of the sequence is defined to be the union of the ''V''''n'''s, provided it is non-empty; otherwise it is defined to be \. Thus the kernel is either a connected open set containing 0 or the one point set \. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel–Carathéodory Theorem
In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named for Émile Borel and Constantin Carathéodory. Statement of the theorem Let a function f be analytic on a closed disc of radius ''R'' centered at the origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu .... Suppose that ''r'' 1, so we may assume ''f'' is nonconstant. First let ''f''(0) = 0. Since Re ''f'' is harmonic, Re ''f''(0) is equal to the average of its values around any circle centered at 0. That is, : \operatorname f(0) = \frac \int_ \operatorname f(z) dz. Since ''f'' is regular and nonconstant, we have that Re ''f'' is also nonconstant. Since Re ''f''(0) = 0, we must ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vitali–Carathéodory Theorem
In mathematics, the Vitali–Carathéodory theorem is a result in real analysis that shows that, under the conditions stated below, integrable functions can be approximated in L1 from above and below by lower- and upper-semicontinuous functions, respectively. It is named after Giuseppe Vitali and Constantin Carathéodory. Statement of the theorem Let ''X'' be a locally compact Hausdorff space equipped with a Borel measure, µ, that is finite on every compact set, outer regular, and tight Tight may refer to: Clothing * Skin-tight garment, a garment that is held to the skin by elastic tension * Tights, a type of leg coverings fabric extending from the waist to feet * Tightlacing, the practice of wearing a tightly-laced corset ... when restricted to any Borel set that is open or of finite mass. If ''f'' is an element of L1(µ) then, for every ''ε'' > 0, there are functions ''u'' and ''v'' on ''X'' such that ''u'' ≤ ''f'' ≤ ''v'', ''u'' is upper-semicontinuous and bou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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