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Émile Picard
Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924. Life He was born in Paris on 24 July 1856 and educated there at the Lycée Henri-IV. He then studied mathematics at the École Normale Supérieure. Picard's mathematical papers, textbooks, and many popular writings exhibit an extraordinary range of interests, as well as an impressive mastery of the mathematics of his time. Picard's little theorem states that every nonconstant entire function takes every value in the complex plane, with perhaps one exception. Picard's great theorem states that an analytic function with an essential singularity takes every value infinitely often, with perhaps one exception, in any neighborhood of the singularity. He made important contributions in the theory of differential equations, including work on Picard–Vessiot theory, Painlevé transcendents and his introduction o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Royal Society Of London
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, recognising excellence in science, supporting outstanding science, providing scientific advice for policy, education and public engagement and fostering international and global co-operation. Founded on 28 November 1660, it was granted a royal charter by King Charles II as The Royal Society and is the oldest continuously existing scientific academy in the world. The society is governed by its Council, which is chaired by the Society's President, according to a set of statutes and standing orders. The members of Council and the President are elected from and by its Fellows, the basic members of the society, who are themselves elected by existing Fellows. , there are about 1,700 fellows, allowed to use the postnominal title FRS (Fellow of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simion Stoilow
Simion Stoilow or Stoilov ( – 4 April 1961) was a Romanian mathematician, creator of the Romanian school of complex analysis, and author of over 100 publications. Biography He was born in Bucharest, and grew up in Craiova. His father, Colonel Simion Stoilow, fought at the in the Romanian War of Independence. After studying at the Obedeanu elementary school and the Carol I High School, Stoilow went in 1907 to the University of Paris, where he earned a B.S. degree in 1910 and a Ph.D. in Mathematics in 1916. His doctoral dissertation was written under the direction of Émile Picard. He returned to Romania in 1916 to fight in the Romanian Campaign of World War I, first in Dobrudja, then in Moldavia. After the war, he became professor of mathematics at the University of Iași (1919–1921) and the University of Cernăuți (1921–1939). He was an Invited Speaker of the International Congress of Mathematicians in 1920 at Strasbourg, in 1928 at Bologna, and in 1936 at Oslo. In 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard–Lindelöf Theorem
In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy. Theorem Let D \subseteq \R \times \R^nbe a closed rectangle with (t_0, y_0) \in D. Let f: D \to \R^n be a function that is continuous in t and Lipschitz continuous in y. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval _0-\varepsilon, t_0+\varepsilon/math>. Note that D is often instead required to be open but even under such an assumption, the proof only uses a closed rectangle within D. Proof sketch The proof relies on transforming the differential equation, and applying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard Variety
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group :H^1 (X, \mathcal_X^).\, For integral schemes the Picard group is isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group. The name is in honour of Émile Picard's theories, in particular of divisors on algebraic surfaces. Examples * The Picard group of the spectrum of a Dedekind domain is its ''ideal class group''. * The invertible sheaves on projective space P''n''(''k'') for ''k'' a field, are the twisting sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard Theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. The theorems Little Picard Theorem: If a function f: \mathbb \to\mathbb is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. Sketch of Proof: Picard's original proof was based on properties of the modular lambda function, usually denoted by λ, and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc. This function is explicitly constructed in the theory of elliptic functions. If ''f'' omits two values, then the composition of ''f'' with the inverse of the modular function maps the plane into the unit disc which implies that ''f'' is constant by Liouville's theorem. This theorem is a significant strengthening of Liouville's theorem which states that the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard Modular Surface
In mathematics, a Picard modular surface, studied by , is a complex surface constructed as a quotient of the unit ball in C2 by a Picard modular group. Picard modular surfaces are some of the simplest examples of Shimura varieties and are sometimes used as a test case for the general theory of Shimura varieties. See also *Hilbert modular surface *Siegel modular variety In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally pola ... References * *{{Citation , last1=Picard , first1=Émile , authorlink=Émile Picard, title= Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques , url= http://www.numdam.org/item?id=ASENS_1881_2_10__305_0 , year=1881 , journal=Annales Scientifiques de l'École Normale Supérieure , series=Série 2 , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard Modular Group
In mathematics, a Picard modular group, studied by , is a group of the form SU(''J'',''L''), where ''L'' is a 3-dimensional lattice over the ring of integers of an imaginary quadratic field and ''J'' is a hermitian form on ''L'' of signature (2, 1). Picard modular groups act on the unit sphere in C2 and the quotient is called a Picard modular surface. See also *Fuchsian group *Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ... References * *{{Citation , last1=Picard , first1=Émile , authorlink=Émile Picard, title= Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques , url= http://www.numdam.org/item?id=ASENS_1881_2_10__305_0 , year=1881 , journal=Annales Scientifiques de l'École Norm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard Horn
A Picard horn, also called the Picard topology or Picard model, is one of the oldest known hyperbolic 3-manifolds, first described by Émile Picard in 1884. The manifold is the quotient of the upper half-plane model of hyperbolic 3-space by the projective special linear group, \operatorname_2(\mathbf . It was proposed as a model for the shape of the universe in 2004. The term "horn" is due to pseudosphere models of hyperbolic space. Geometry and topology A modern description, in terms of fundamental domain and identifications, can be found in section 3.2, page 63 of Grunewald and Huntebrinker, along with the first 80 eigenvalues of the Laplacian, tabulated on page 72, where \Upsilon_1 is a fundamental domain of the Picard space.Fritz Grunewald and Wolfgang Huntebrinker, A numerical study of eigenvalues of the hyperbolic Laplacian for polyhedra with one cusp', Experiment. Math. Volume 5, Issue 1 (1996), 57-80 Cosmology The term was coined in 2004 by Ralf Aurich, Sven Lustig, Frank ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard Group
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group :H^1 (X, \mathcal_X^).\, For integral schemes the Picard group is isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group. The name is in honour of Émile Picard's theories, in particular of divisors on algebraic surfaces. Examples * The Picard group of the spectrum of a Dedekind domain is its '' ideal class group''. * The invertible sheaves on projective space P''n''(''k'') for ''k'' a field, are the twisting shea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard Functor
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group :H^1 (X, \mathcal_X^).\, For integral schemes the Picard group is isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group. The name is in honour of Émile Picard's theories, in particular of divisors on algebraic surfaces. Examples * The Picard group of the spectrum of a Dedekind domain is its ''ideal class group''. * The invertible sheaves on projective space P''n''(''k'') for ''k'' a field, are the twisting sheaves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanisław Zaremba (mathematician)
Stanisław Zaremba (3 October 1863 – 23 November 1942) was a Polish mathematician and engineer.. His research in partial differential equations, applied mathematics and classical analysis, particularly on harmonic functions, gained him a wide recognition. He was one of the mathematicians who contributed to the success of the Polish School of Mathematics through his teaching and organizational skills as well as through his research. Apart from his research works, Zaremba wrote many university textbooks and monographies. He was a professor of the Jagiellonian University (since 1900), member of Academy of Learning (since 1903), co-founder and president of the Polish Mathematical Society (1919). He should not be confused with his son Stanisław Krystyn Zaremba, also a mathematician. Biography Zaremba was born on 3 October 1863 in Romanówka, present-day Ukraine. The son of an engineer, he was educated at a grammar school in Saint Petersburg and studied at the Institute of Techn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
