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Herbert Grötzsch
Camillo Herbert Grötzsch (21 May 1902 – 15 May 1993) was a German mathematician. He was born in Döbeln and died in Halle. Grötzsch worked in graph theory. He was the discoverer and eponym of the Grötzsch graph, a triangle-free graph that requires four colors in any graph coloring, and Grötzsch's theorem, the result that every triangle-free planar graph requires at most three colors. A student of Paul Koebe, he made important contributions to the theory of conformal mappings and univalent functions: he was the first to introduce the concept of a quasiconformal mapping In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let ''f'' : ''D' .... Publications *Herbert Grötzsch, Über die Verzerrung bei schlichten nicht-konformen Abbildungen und über eine damit zusammenhängende Erweiterung de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Univalent Function
In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective. Examples The function f \colon z \mapsto 2z + z^2 is univalent in the open unit disc, as f(z) = f(w) implies that f(z) - f(w) = (z-w)(z+w+2) = 0. As the second factor is non-zero in the open unit disc, f must be injective. Basic properties One can prove that if G and \Omega are two open connected sets in the complex plane, and :f: G \to \Omega is a univalent function such that f(G) = \Omega (that is, f is surjective), then the derivative of f is never zero, f is invertible, and its inverse f^ is also holomorphic. More, one has by the chain rule :(f^)'(f(z)) = \frac for all z in G. Comparison with real functions For real analytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function :f: (-1, 1) \to (-1, 1) \, given by ''ƒ''(''x'')& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theorists
Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discrete mathematics * Graph of a function *Graph of a relation *Graph paper *Chart, a means of representing data (also called a graph) Computing * Graph (abstract data type), an abstract data type representing relations or connections *graph (Unix), Unix command-line utility *Conceptual graph, a model for knowledge representation and reasoning Other uses * HMS ''Graph'', a submarine of the UK Royal Navy See also *Complex network *Graf *Graff (other) *Graph database *Grapheme, in linguistics *Graphemics *Graphic (other) *-graphy (suffix from the Greek for "describe," "write" or "draw") *List of information graphics software This is a list of software to create any kind of information graphics: * either includes the ability t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1993 Deaths
File:1993 Events Collage.png, From left, clockwise: The Oslo I Accord is signed in an attempt to resolve the Israeli–Palestinian conflict; The Russian White House is shelled during the 1993 Russian constitutional crisis; Czechoslovakia is peacefully dissolved into the Czech Republic and Slovakia; In the United States, the ATF besieges a compound belonging to David Koresh and the Branch Davidians in a search for illegal weapons, which ends in the building being set alight and killing most inside; Eritrea gains independence; A major snow storm passes over the United States and Canada, leading to over 300 fatalities; Drug lord and narcoterrorist Pablo Escobar is killed by Colombian special forces; Ramzi Yousef and other Islamic terrorists detonate a truck bomb in the subterranean garage of the North Tower of the World Trade Center in the United States., 300x300px, thumb rect 0 0 200 200 Oslo I Accord rect 200 0 400 200 1993 Russian constitutional crisis rect 400 0 600 200 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1902 Births
Nineteen or 19 may refer to: * 19 (number), the natural number following 18 and preceding 20 * one of the years 19 BC, AD 19, 1919, 2019 Films * ''19'' (film), a 2001 Japanese film * ''Nineteen'' (film), a 1987 science fiction film Music * 19 (band), a Japanese pop music duo Albums * ''19'' (Adele album), 2008 * ''19'', a 2003 album by Alsou * ''19'', a 2006 album by Evan Yo * ''19'', a 2018 album by MHD * ''19'', one half of the double album ''63/19'' by Kool A.D. * ''Number Nineteen'', a 1971 album by American jazz pianist Mal Waldron * ''XIX'' (EP), a 2019 EP by 1the9 Songs * "19" (song), a 1985 song by British musician Paul Hardcastle. * "Nineteen", a song by Bad4Good from the 1992 album '' Refugee'' * "Nineteen", a song by Karma to Burn from the 2001 album ''Almost Heathen''. * "Nineteen" (song), a 2007 song by American singer Billy Ray Cyrus. * "Nineteen", a song by Tegan and Sara from the 2007 album '' The Con''. * "XIX" (song), a 2014 song by Slipkn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horst Tietz
Horst may refer to: Science * Horst (geology), a raised fault block bounded by normal faults or graben People * Horst (given name) * Horst (surname) * ter Horst, Dutch surname * van der Horst, Dutch surname Places Settlements Germany * Horst, Steinburg, a municipality in the district of Steinburg in Schleswig-Holstein * Horst, Lauenburg, a municipality in the district of Lauenburg in Schleswig-Holstein * Horst, Mecklenburg-Vorpommern, a village and district in the municipality of Sundhagen, Mecklenburg-Vorpommern * , a district in the city of Gelsenkirchen, North Rhine-Westphalia * , a town in the municipality of Seevetal, Lower Saxony Netherlands * Horst aan de Maas, a municipality in the province of Limburg ** Horst, Limburg, the municipal seat of Horst aan de Maas * , a hamlet in the municipality of Ermelo, Gelderland * , a village in the municipality of Gilze en Rijen, North Brabant * Schothorst, , and , districts in the city and municipality of Amersfoort, Utrecht Po ... [...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, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasiconformal Mapping
In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Intuitively, let ''f'' : ''D'' → ''D''′ be an orientation-preserving homeomorphism between open sets in the plane. If ''f'' is continuously differentiable, then it is ''K''-quasiconformal if the derivative of ''f'' at every point maps circles to ellipses with eccentricity bounded by ''K''. Definition Suppose ''f'' : ''D'' → ''D''′ where ''D'' and ''D''′ are two domains in C. There are a variety of equivalent definitions, depending on the required smoothness of ''f''. If ''f'' is assumed to have continuous partial derivatives, then ''f'' is quasiconformal provided it satisfies the Beltrami equation for some complex valued Lebesgue measurable μ satisfying sup , μ, 0. Then ''f'' satisfies () precisely when it is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Mapping
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in U if it preserves angles between directed curves through u_0, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix ( orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix. For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locall ... [...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 Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
