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Werner Fenchel
Moritz Werner Fenchel (; 3 May 1905 – 24 January 1988) was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory which would, in time, serve as the foundation for nonlinear programming. A German-born Jew and early refugee from Nazi suppression of intellectuals, Fenchel lived most of his life in Denmark. Fenchel's monographs and lecture notes are considered influential. Biography Early life and education Fenchel was born on 3 May 1905 in Berlin, Germany, his younger brother was the Israeli film director and architect Heinz Fenchel. Fenchel studied mathematics and physics at the University of Berlin between 1923 and 1928. He wrote his doctorate thesis in geometry (''Über Krümmung und Windung geschlossener Raumkurven'') under Ludwig Bieberbach. Professorship in Germany From 1928 to 1933, Fenchel was Professor E. Landau's Assistant at the Univ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
German Empire
The German Empire (),Herbert Tuttle wrote in September 1881 that the term "Reich" does not literally connote an empire as has been commonly assumed by English-speaking people. The term literally denotes an empire – particularly a hereditary empire led by an emperor, although has been used in German to denote the Roman Empire because it had a weak hereditary tradition. In the case of the German Empire, the official name was , which is properly translated as "German Empire" because the official position of head of state in the constitution of the German Empire was officially a "presidency" of a confederation of German states led by the King of Prussia who would assume "the title of German Emperor" as referring to the German people, but was not emperor of Germany as in an emperor of a state. –The German Empire" ''Harper's New Monthly Magazine''. vol. 63, issue 376, pp. 591–603; here p. 593. also referred to as Imperial Germany, the Second Reich, as well as simply Germany, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fenchel–Moreau Theorem
In convex analysis, the Fenchel–Moreau theorem (named after Werner Fenchel and Jean Jacques Moreau) or Fenchel biconjugation theorem (or just biconjugation theorem) is a theorem which gives necessary and sufficient conditions for a function to be equal to its biconjugate. This is in contrast to the general property that for any function f^ \leq f. This can be seen as a generalization of the bipolar theorem. It is used in duality theory to prove strong duality (via the perturbation function). Statement Let (X,\tau) be a Hausdorff locally convex space, for any extended real valued function f: X \to \mathbb \cup \ it follows that f = f^ if and only if one of the following is true # f is a proper, lower semi-continuous In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, ro ..., and c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Berlin
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative of Wilhelm von Humboldt, Johann Gottlieb Fichte and Friedrich Ernst Daniel Schleiermacher as the University of Berlin () in 1809, and opened in 1810, making it the oldest of Berlin's four universities. From 1828 until its closure in 1945, it was named Friedrich Wilhelm University (german: Friedrich-Wilhelms-Universität). During the Cold War, the university found itself in East Berlin and was ''de facto'' split in two when the Free University of Berlin opened in West Berlin. The university received its current name in honour of Alexander and Wilhelm von Humboldt in 1949. The university is divided into nine faculties including its medical school shared with the Freie Universität Berlin. The university has a student enrollment of aroun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heinz Fenchel
Heinz Fenchel (1906–1988) was a German art director, who designed the sets for a number of films in Weimar Germany and Czechoslovakia. He worked on several films with the director Pál Fejös.Büttner p.231-33 The Jewish Fenchel later emigrated to Israel where he worked as an architect. Selected filmography * '' Ship in Distress'' (1929) * ''The Model from Montparnasse'' (1929) * ''Phantoms of Happiness'' (1929) * '' The Black Forest Girl'' (1929) * '' The Convict from Istanbul'' (1929) * ''Troika'' (1930) * '' The Squeaker'' (1931) * '' The Forester's Daughter'' (1931) * '' Mamsell Nitouche'' (1932) * '' Kiki'' (1932) * '' The Ringer'' (1932) * ''Ways to a Good Marriage'' (1933) * ''The Golden Smile ''The Golden Smile'' (Danish:''Det gyldne smil'') is a 1935 Danish drama film directed by Pál Fejös and starring Bodil Ipsen, Helen von Münchofen and John Price.Sadoul & Morris p.80 The film's sets were designed by the art director Heinz F ...'' (1935) References Biblio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berlin
Berlin ( , ) is the capital and List of cities in Germany by population, largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's List of cities in the European Union by population within city limits, most populous city, according to population within city limits. One of Germany's States of Germany, sixteen constituent states, Berlin is surrounded by the Brandenburg, 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 Metropolitan Region, Berlin-Brandenburg capital region has around 6.2 million inhabitants and is Metropolitan regions in Germany, Germany's third-largest metropolitan region after the Rhine-Ruhr and Frankfurt Rhine-Main, Rhine-Main regions. Berlin straddles the banks of the Spree (river), Spree, which flows into the Havel (a tributary of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Programming
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or stationary points) of an objective function over a set of unknown real variables and conditional to the satisfaction of a system of equalities and inequalities, collectively termed constraints. It is the sub-field of mathematical optimization that deals with problems that are not linear. Applicability A typical non-convex problem is that of optimizing transportation costs by selection from a set of transportation methods, one or more of which exhibit economies of scale, with various connectivities and capacity constraints. An example would be petroleum product transport given a selection or combination of pipeline, rail tanker, road tanker, river barge, or coastal tankship. Owing to economic batch size the cost functions may have disco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of some vector space X is if it satisfies any of the following equivalent conditions: #If 0 \leq r \leq 1 is real and x, y \in C then r x + (1 - r) y \in C. #If 0 is a if holds for any real 0 is called if \operatorname f \neq \varnothing and f(x) > -\infty for x \in \operatorname f. Alternatively, this means that there exists some x in the domain of f at which f(x) \in \mathbb and f is also equal to -\infty. In words, a function is if its domain is not empty, it never takes on the value -\infty, and it also is not identically equal to +\infty. If f : \mathbb^n \to \infty, \infty/math> is a proper convex function then there exist some vector b \in \mathbb^n and some r \in \mathbb such that :f(x) \geq x \cdot b - r for every x where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Royal Danish Academy Of Sciences And Letters
{{Infobox organization , name = The Royal Danish Academy of Sciences and Letters , full_name = , native_name = Det Kongelige Danske Videnskabernes Selskab , native_name_lang = , logo = Royal Danish Academy of Sciences and Letters seal.svg , logo_size = 150 , logo_alt = , logo_caption = , image = Carlsbergfondet.JPG , image_size = , alt = , caption = The building on H.C. Andersens Boulevard. , map = , map_size = , map_alt = , map_caption = , map2 = , map2_size = , map2_alt = , map2_caption = , abbreviation = , nickname = , pronounce = , pronounce ref = , pronounce comment = , pronounce 2 = , named_after = , motto = , predecessor = , merged ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rockefeller Fellowship
The Rockefeller Foundation is an American private foundation and philanthropic medical research and arts funding organization based at 420 Fifth Avenue, New York City. The second-oldest major philanthropic institution in America, after the Carnegie Corporation, the foundation was ranked as the 39th largest U.S. foundation by total giving as of 2015. By the end of 2016, assets were tallied at $4.1 billion (unchanged from 2015), with annual grants of $173 million. According to the OECD, the foundation provided US$103.8 million for development in 2019. The foundation has given more than $14 billion in current dollars. The foundation was started by Standard Oil magnate John D. Rockefeller ("Senior") and son "Junior", and their primary business advisor, Frederick Taylor Gates, on May 14, 1913, when its charter was granted by New York. The foundation has had an international reach since the 1930s and major influence on global non-governmental organizations. The World Health Org ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre–Fenchel Transformation
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel). It allows in particular for a far reaching generalization of Lagrangian duality. Definition Let X be a real topological vector space and let X^ be the dual space to X. Denote by :\langle \cdot , \cdot \rangle : X^ \times X \to \mathbb the canonical dual pairing, which is defined by \left( x^*, x \right) \mapsto x^* (x). For a function f : X \to \mathbb \cup \ taking values on the extended real number line, its is the function :f^ : X^ \to \mathbb \cup \ whose value at x^* \in X^ is defined to be the supremum: :f^ \left( x^ \right) := \sup \left\, or, equivalently, in terms of the infimum: :f^ \left( x^ \right) := - \inf \left\. This definition can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
