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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] |
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constituent states, Berlin is surrounded by the State of Brandenburg and contiguous with Potsdam, Brandenburg's capital. Berlin's urban area, which has a population of around 4.5 million, is the second most populous urban area in Germany after the Ruhr. The Berlin-Brandenburg capital region has around 6.2 million inhabitants and is Germany's third-largest metropolitan region after the Rhine-Ruhr and Rhine-Main regions. Berlin straddles the banks of the Spree, which flows into the Havel (a tributary of the Elbe) in the western borough of Spandau. Among the city's main topographical features are the many lakes in the western and southeastern boroughs formed by the Spree, Havel and Dahme, the largest of which is Lake Müggelsee. Due to its l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Minkowski
Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number theory, mathematical physics, and the theory of relativity. Minkowski is perhaps best known for his foundational work describing space and time as a four-dimensional space, now known as "Minkowski spacetime", which facilitated geometric interpretations of Albert Einstein's special theory of relativity (1905). Personal life and family Hermann Minkowski was born in the town of Aleksota, the Suwałki Governorate, the Kingdom of Poland, part of the Russian Empire, to Lewin Boruch Minkowski, a merchant who subsidized the building of the choral synagogue in Kovno, and Rachel Taubmann, both of Jewish descent. Hermann was a younger brother of the medical researcher Oskar (born 1858). In different sources Minkowski's nationality is variously giv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory's Lemma
In mathematics, Nevanlinna's criterion in complex analysis, proved in 1920 by the Finnish mathematician Rolf Nevanlinna, characterizes holomorphic univalent functions on the unit disk which are starlike. Nevanlinna used this criterion to prove the Bieberbach conjecture for starlike univalent functions. Statement of criterion A univalent function ''h'' on the unit disk satisfying ''h''(0) = 0 and ''h(0) = 1 is starlike, i.e. has image invariant under multiplication by real numbers in ,1 if and only if z h^\prime(z)/h(z) has positive real part for , ''z'', < 1 and takes the value 1 at 0. Note that, by applying the result to ''a''•''h''(''rz''), the criterion applies on any disc , ''z'', < r with only the requirement that ''f''(0) = 0 and ''f(0) ≠ 0. Proof of criterion Let ''h''(''z'') be a starlike univalent function on , ''z'', < 1 with ''h''(0) = 0 and ''h(0) = 1. For ''t'' < 0, define : |
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] |
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] |
Carathéodory Metric
In mathematics, the Carathéodory metric is a metric defined on the open unit ball of a complex Banach space that has many similar properties to the Poincaré metric of hyperbolic geometry. It is named after the Greek mathematician Constantin Carathéodory. Definition Let (''X'', , , , , ) be a complex Banach space and let ''B'' be the open unit ball in ''X''. Let Δ denote the open unit disc in the complex plane C, thought of as the Poincaré disc model for 2-dimensional real/1-dimensional complex hyperbolic geometry. Let the Poincaré metric ''ρ'' on Δ be given by :\rho (a, b) = \tanh^ \frac (thus fixing the curvature to be −4). Then the Carathéodory metric ''d'' on ''B'' is defined by :d (x, y) = \sup \. What it means for a function on a Banach space to be holomorphic is defined in the article on Infinite dimensional holomorphy. Properties * For any point ''x'' in ''B'', ::d(0, x) = \rho(0, \, x \, ). * ''d'' can also be given by the following formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory Function
In mathematical analysis, a Carathéodory function (or Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measurable functions does not have to be Lebesgue-measurable as well. Nevertheless, a composition of a measurable function with a continuous function is indeed Lebesgue-measurable, but in many situations, continuity is a too restrictive assumption. Carathéodory functions are more general than continuous functions, but still allow a composition with Lebesgue-measurable function to be measurable. Carathéodory functions play a significant role in calculus of variation, and it is named after the Greek mathematician Constantin Carathéodory. Definition W:\Omega\times\mathbb^\rightarrow\mathbb\cup\left\ , for \Omega\subseteq\mathbb^ endowed with the Lebesgue measure, is a Carathéodory function if: 1. The mapping x\mapsto W\left(x,\xi\right) is Lesbegue-measurable for every \xi\in\m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory Conjecture
In differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ..., the Carathéodory conjecture is a mathematical conjecture attributed to Constantin Carathéodory by Hans Ludwig Hamburger in a session of the Berlin Mathematical Society in 1924.''Sitzungsberichte der Berliner Mathematischen Gesellschaft'', 210. Sitzung am 26. März 1924, Dieterichsche Universitätsbuchdruckerei, Göttingen 1924 Carathéodory did publish a paper on a related subject, but never committed the conjecture into writing. In, John Edensor Littlewood mentions the conjecture and Hamburger's contributionHans Ludwig Hamburger, H. Hamburger, ''Beweis einer Caratheodoryschen Vermutung. I'', Annals of Mathematics, Ann. Math. (2) 41, 63—86 (1940); ''Beweis einer Caratheodoryschen Vermutung. II'', Acta Math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nazim Terzioglu
Subahdar, also known as Nazim or in English as a "Subah", was one of the designations of a governor of a Subah (province) during the Khalji dynasty of Bengal, Mamluk dynasty (Delhi), Khalji dynasty, Tughlaq dynasty, Mughal era ( of India who was alternately designated as Sahib-i-Subah or Nazim. The word, ''Subahdar'' is of Persian origin. According to sources, Subahdar Awlia Khan was a famous and trusted Subahdar of the Khalji dynasty of Bengal (1204-1231) whose title was Saheb-i-Subah could not be ascertained.He belonged to the Oghuz Turks Kayı (tribe) and his ancestors came to the region during the expansion of The Great Seljuk Empire to establish good governance and justice in Islam. Subahdar Awlia Khan was a friend of Muhammad bin Bakhtiyar Khalji Later, during the conquest of Bengal, Awlia Khan was his fellow warrior. Today the descendants of the great Subahdar Awlia Khan have been living in Fuldi village of Gazipur district of Bangladesh for almost 900 years and Mesbah ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wladimir Seidel
Wladimir P. Seidel (December 21, 1907 – January 12, 1981) was a Russian-born German-American mathematician, and Doctor of Mathematics. He held a fellowship as a Benjamin Peirce Professor in Harvard University. During World War II, he was with the ''Montreal Theory'' group for the National Research Council of Canada. Life He was born in Odessa, Russian Empire on December 21, 1907. Career He earned his Ph.D. from Ludwig-Maximilians-Universität in München (February 26, 1930) on a dissertation entitled ''Über die Ränderzuordnung bei konformen Abbildungen'', advised by Constantin Carathéodory. He joined the faculty of Mathematics at Harvard University (as Benjamin Peirce Instructor, 1932–33), at University of Rochester (1941–55), at The Institute for Advanced Study in Princeton (1952–53), at University of Notre Dame (1955–63), and at Wayne State University in Detroit (since 1963). During World War II, he was with the ''Montreal Theory'' group for the National Res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Peschl
Ernst Ferdinand Peschl (1 September 1906 – 9 June 1986) was a German mathematician. Early life Ernst Peschl came from a family of brewery owners. He was born to Eduard Ferdinand Peschl and his wife, Ulla (née Adler) in 1906. Education and academic appointments After finishing secondary school in 1925 in Passau, Peschl started studying mathematics, physics, and astronomy in Munich. He received his doctorate in 1931 from the University of Munich under the supervision of Constantin Carathéodory with a dissertation titled ''Über die Krümmung von Niveaukurven bei der konformen Abbildung einfachzusammenhängender Gebiete auf das Innere eines Kreises; eine Verallgemeinerung eines Satzes von E. Study'' ("On the curvature of level curves of the conformal mapping of simply connected domains to the interior of a circle: A generalization of a theorem of Eduard Study"). This was followed by some years spent working as an assistant with Robert König in Jena and Heinrich Behnke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Boerner
Hermann Boerner, also written "Börner" at library service center (11 July 1906 – 3 June 1982) was a German mathematician who worked on variation calculus, , and theory. Publications [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
