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Marc Krasner
Marc Krasner (1912 – 13 May 1985, in Paris) was a Russian Empire-born French mathematician, who worked on algebraic number theory. Krasner emigrated from the Soviet Union to France and received in 1935 his PhD from the University of Paris under Jacques Hadamard with thesis ''Sur la théorie de la ramification des idéaux de corps non-galoisiens de nombres algébriques''. From 1937 to 1960 he was a scientist at CNRS and from 1960 professor at the University of Clermont-Ferrand. From 1965 he was a professor at the University of Paris VI (Pierre et Marie Curie), where he retired in 1980 as professor emeritus. Krasner did research on p-adic analysis. In 1944 he introduced the concept of ultrametric spaces,''Nombres semi-réels et espaces ultramétriques'', Comptes Rendus de l'Académie des Sciences, Tome II, vol. 219, p. 433 to which p-adic numbers belong. In 1951, alongside Lev Kaluznin, he proved the Krasner-Kaloujnine universal embedding theorem, which states that every exten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. Since the 17th century, Paris has been one of the world's major centres of finance, diplomacy, commerce, fashion, gastronomy, and science. For its leading role in the arts and sciences, as well as its very early system of street lighting, in the 19th century it became known as "the City of Light". Like London, prior to the Second World War, it was also sometimes called the capital of the world. The City of Paris is the centre of the Île-de-France region, or Paris Region, with an estimated population of 12,262,544 in 2019, or about 19% of the population of France, making the region France's primate city. The Paris Region had a GDP of €739 billion ($743 billion) in 2019, which is the highest in Europe. According to the Economist Intelli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Embedding Theorem
The universal embedding theorem, or Krasner–Kaloujnine universal embedding theorem, is a theorem from the mathematical discipline of group theory first published in 1951 by Marc Krasner and Lev Kaluznin. The theorem states that any group extension of a group by a group is isomorphic to a subgroup of the regular wreath product The theorem is named for the fact that the group is said to be ''universal'' with respect to all extensions of by Statement Let and be groups, let be the set of all functions from to and consider the action of on itself by right multiplication. This action extends naturally to an action of on defined by \phi(g).h=\phi(gh^), where \phi\in K, and and are both in This is an automorphism of so we can define the semidirect product called the ''regular wreath product'', and denoted or A\wr H. The group (which is isomorphic to \) is called the ''base group'' of the wreath product. The Krasner–Kaloujnine universal embedding theorem states tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1985 Deaths
The year 1985 was designated as the International Youth Year by the United Nations. Events January * January 1 ** The Internet's Domain Name System is created. ** Greenland withdraws from the European Economic Community as a result of a new agreement on fishing rights. * January 7 – Japan Aerospace Exploration Agency launches ''Sakigake'', Japan's first interplanetary spacecraft and the first deep space probe to be launched by any country other than the United States or the Soviet Union. * January 15 – Tancredo Neves is elected president of Brazil by the Congress, ending the 21-year military rule. * January 20 – Ronald Reagan is privately sworn in for a second term as President of the United States. * January 27 – The Economic Cooperation Organization (ECO) is formed, in Tehran. * January 28 – The charity single record "We Are the World" is recorded by USA for Africa. February * February 4 – The border between Gibraltar and Spai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1912 Births
Year 191 ( CXCI) was a common year starting on Friday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Apronianus and Bradua (or, less frequently, year 944 ''Ab urbe condita''). The denomination 191 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Parthia * King Vologases IV of Parthia dies after a 44-year reign, and is succeeded by his son Vologases V. China * A coalition of Chinese warlords from the east of Hangu Pass launches a punitive campaign against the warlord Dong Zhuo, who seized control of the central government in 189, and held the figurehead Emperor Xian hostage. After suffering some defeats against the coalition forces, Dong Zhuo forcefully relocates the imperial capital from Luoyang to Chang'an. Before leaving, Dong Zhuo orders his troops to loot the tombs of the H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Académie Des Sciences
The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at the forefront of scientific developments in Europe in the 17th and 18th centuries, and is one of the earliest Academies of Sciences. Currently headed by Patrick Flandrin (President of the Academy), it is one of the five Academies of the Institut de France. History The Academy of Sciences traces its origin to Colbert's plan to create a general academy. He chose a small group of scholars who met on 22 December 1666 in the King's library, near the present-day Bibliothèque Nationals, and thereafter held twice-weekly working meetings there in the two rooms assigned to the group. The first 30 years of the Academy's existence were relatively informal, since no statutes had as yet been laid down for the institution. In contrast to its British ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prix Paul Doistau–Émile Blutet
The Prix Paul Doistau–Émile Blutet is a biennial prize awarded by the French Academy of Sciences in the fields of mathematics and physical sciences since 1954. Each recipient receives 3000 euros. The prize is also awarded quadrennially in biology. The award is also occasionally awarded in other disciplines. List of laureates Mathematics * 1958 Marc Krasner * 1980 Jean-Michel Bony * 1982 Jean-Pierre Ramis * 1982 Gérard Maugin * 1985 Dominique Foata * 1986 Pierre-Louis Lions * 1987 Pierre Bérard * 1987 Lucien Szpiro * 1999 Wendelin Werner * 2001 Hélène Esnault * 2004 Laurent Stolovitch * 2006 Alice Guionnet * 2008 Isabelle Gallagher * 2010 Yves André * 2012 Serge Cantat * 2014 Sébastien Boucksom * 2016 Hajer Bahouri * 2018 Physical sciences * 2002 * 2005 Mustapha Besbes * 2007 * 2009 Hasnaa Chennaoui-Aoudjehane * 2011 Henri-Claude Nataf * 2013 * 2015 Philippe André * 2019 Integrative biology * 2000 Jérôme Giraudat * 2004 Marie-Claire Verdus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krasner's Lemma
In number theory, more specifically in ''p''-adic analysis, Krasner's lemma is a basic result relating the topology of a complete non-archimedean field to its algebraic extensions. Statement Let ''K'' be a complete non-archimedean field and let be a separable closure of ''K''. Given an element α in , denote its Galois conjugates by ''α''2, ..., ''α''''n''. Krasner's lemma states: :if an element ''β'' of is such that ::\left, \alpha-\beta\ 1 with coefficients in a Henselian field (''K'', ''v'') and roots in the algebraic closure . Let ''I'' and ''J'' be two disjoint, non-empty sets with union . Moreover, consider a polynomial ::g=\prod_(X-\alpha_i) with coefficients and roots in . Assume ::\forall i\in I\forall j\in J: v(\alpha_i-\alpha_i^*)>v(\alpha_i^*-\alpha_j^*). Then the coefficients of the polynomials ::g^*:=\prod_(X-\alpha_i^*),\ h^*:=\prod_(X-\alpha_j^*) are contained in the field extension of ''K'' generated by the coefficients of ''g''. (The original Kras ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lev Kaluznin
Lev Arkad'evich Kaluznin ( rus, Лев Аркадьевич Калужнин) (31 January 1914 – 6 December 1990) was a Russian mathematician. Other transliterations of his name used by himself include ''Kalužnin'' and ''Kaluzhnin'', while he used the transliteration ''Léo Kaloujnine'' in publications while he lived in France. Biography and education Kaluznin was born in Moscow. His parents divorced not long after his birth, and his father, Arkadii Rubin, moved to England and was not part of Kaluznin's life. His mother, Maria Pavlovna Kaluznina, moved with the young Kaluznin to Petrograd (present-day Saint Petersburg), where she brought him up. She shared her love for Russian culture, including music and literature, with her son, and she would remain an important part of his life. In 1923, Kaluznin and his mother moved to Germany. She worked as a governess, while Kaluznin was enrolled at a ''Realgymnasium'' (secondary school), graduating in 1933. His school offered a thoro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and Algebraic function field, function fields. These properties, such as whether a ring (mathematics), ring admits unique factorization, the behavior of ideal (ring theory), ideals, and the Galois groups of field (mathematics), fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History of algebraic number theory Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrametric Space
In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems for ultrametric spaces may seem strange at a first glance, they appear naturally in many applications. Formal definition An ultrametric on a set is a real-valued function :d\colon M \times M \rightarrow \mathbb (where denote the real numbers), such that for all : # ; # (''symmetry''); # ; # if then ; # (strong triangle inequality or ultrametric inequality). An ultrametric space is a pair consisting of a set together with an ultrametric on , which is called the space's associated distance function (also called a metric). If satisfies all of the conditions except possibly condition 4 then is called an ultrapseudometric on . An ultrapseudometric space is a pair consisting of a set and an ultrapseudometric on . In the case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Analysis
In mathematics, ''p''-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of ''p''-adic numbers. The theory of complex-valued numerical functions on the ''p''-adic numbers is part of the theory of locally compact groups. The usual meaning taken for ''p''-adic analysis is the theory of ''p''-adic-valued functions on spaces of interest. Applications of ''p''-adic analysis have mainly been in number theory, where it has a significant role in diophantine geometry and diophantine approximation. Some applications have required the development of ''p''-adic functional analysis and spectral theory. In many ways ''p''-adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of ''p''-adic numbers is much simpler. Topological vector spaces over ''p''-adic fields show distinctive features; for example aspects relating to convexity and the Hahn–Banach theorem a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
