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Irregularity Of A Surface
In mathematics, the irregularity of a complex surface ''X'' is the Hodge number h^= \dim H^1(\mathcal_X), usually denoted by ''q.'' The irregularity of an algebraic surface is sometimes defined to be this Hodge number, and sometimes defined to be the dimension of the Picard variety, which is the same in characteristic 0 but can be smaller in positive characteristic. The name "irregularity" comes from the fact that for the first surfaces investigated in detail, the smooth complex surfaces in P3, the irregularity happens to vanish. The irregularity then appeared as a new "correction" term measuring the difference p_g - p_a of the geometric genus and the arithmetic genus of more complicated surfaces. Surfaces are sometimes called regular or irregular depending on whether or not the irregularity vanishes. For a complex analytic manifold ''X'' of general dimension, the Hodge number h^= \dim H^1(\mathcal_X) is called the irregularity of X, and is denoted by ''q''. Complex surfaces Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Surface
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * Complex (album), ''Complex'' (album), by Montaigne, 2019, and its title track * Complex (EP), ''Complex'' (EP), by Rifle Sport, 1985 * Complex (song), "Complex" (song), by Gary Numan, 1979 * Complex Networks, publisher of magazine ''Complex'', now online Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-protein intracellular complex * Protein complex, a group of two or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Characteristic
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, is the smallest positive number such that: :\underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: :\underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their requirements for a ring (see Multiplicative identity and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bockstein Operation
In homological algebra, the Bockstein homomorphism, introduced by , is a connecting homomorphism associated with a short exact sequence :0 \to P \to Q \to R \to 0 of abelian groups, when they are introduced as coefficients into a chain complex ''C'', and which appears in the homology groups as a homomorphism reducing degree by one, :\beta\colon H_i(C, R) \to H_(C,P). To be more precise, ''C'' should be a complex of free, or at least torsion-free, abelian groups, and the homology is of the complexes formed by tensor product with ''C'' (some flat module condition should enter). The construction of β is by the usual argument (snake lemma). A similar construction applies to cohomology groups, this time increasing degree by one. Thus we have :\beta\colon H^i(C, R) \to H^(C,P). The Bockstein homomorphism \beta associated to the coefficient sequence :0 \to \Z/p\Z\to \Z/p^2\Z\to \Z/p\Z\to 0 is used as one of the generators of the Steenrod algebra. This Bockstein homomorphism ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Cartier (mathematician)
Pierre Émile Cartier (born 10 June 1932) is a French mathematician. An associate of the Bourbaki group and at one time a colleague of Alexander Grothendieck, his interests have ranged over algebraic geometry, representation theory, mathematical physics, and category theory. He studied at the École Normale Supérieure in Paris under Henri Cartan and André Weil. Since his 1958 thesis on algebraic geometry he has worked in a number of fields. He is known for the introduction of the Cartier operator in algebraic geometry in characteristic ''p'', and for work on duality of abelian varieties and on formal groups. He is the eponym of Cartier divisors and Cartier duality. From 1961 to 1971 he was a professor at the University of Strasbourg. In 1970 he was an Invited Speaker at the International Congress of Mathematicians in Nice. He was awarded the 1978 Prize Ampère of the French Academy of Sciences. In 2012 he became a fellow of the American Mathematical Society. Publications ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen Smale, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enriques Surface
In mathematics, Enriques surfaces are algebraic surfaces such that the irregularity ''q'' = 0 and the canonical line bundle ''K'' is non-trivial but has trivial square. Enriques surfaces are all projective (and therefore Kähler over the complex numbers) and are elliptic surfaces of genus 0. Over fields of characteristic not 2 they are quotients of K3 surfaces by a group of order 2 acting without fixed points and their theory is similar to that of algebraic K3 surfaces. Enriques surfaces were first studied in detail by as an answer to a question discussed by about whether a surface with ''q'' = ''p''''g'' = 0 is necessarily rational, though some of the Reye congruences introduced earlier by are also examples of Enriques surfaces. Enriques surfaces can also be defined over other fields. Over fields of characteristic other than 2, showed that the theory is similar to that over the complex numbers. Over fields of characteristic 2 the definition is modified, and there are two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jun-Ichi Igusa
was a Japanese mathematician who for over three decades was on the faculty at Johns Hopkins University. He is known for his contributions to algebraic geometry and number theory. The Igusa zeta-function, the Igusa quartic, Igusa subgroups, Igusa curves, and Igusa varieties are named after him. He was an invited speaker for the 1962 International Congress of Mathematicians in Stockholm. He was awarded Japan's Order of the Sacred Treasure. In 2012 he became a fellow of the American Mathematical Society. Life and career Igusa was born in Kiyosato village, Gunma Prefecture, Japan, on 30 January 1924. He graduated from the University of Tokyo in 1945 and received his Ph.D. from Kyoto University in 1953, after which he became professor of mathematics at the University of Tsukuba. After a brief period spent at Harvard University, he took up a permanent position at Johns Hopkins University, in Baltimore. Igusa taught at Johns Hopkins from 1955 to 1993. He joined the staff of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The National Academy Of Sciences
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Sciences, published since 1915, and publishes original research, scientific reviews, commentaries, and letters. According to ''Journal Citation Reports'', the journal has a 2021 impact factor of 12.779. ''PNAS'' is the second most cited scientific journal, with more than 1.9 million cumulative citations from 2008 to 2018. In the mass media, ''PNAS'' has been described variously as "prestigious", "sedate", "renowned" and "high impact". ''PNAS'' is a delayed open access journal, with an embargo period of six months that can be bypassed for an author fee ( hybrid open access). Since September 2017, open access articles are published under a Creative Commons license. Since January 2019, ''PNAS'' has been online-only, although print issues are a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kähler Surface
Kähler may refer to: ;People * Alexander Kähler (born 1960), German television journalist * Birgit Kähler (born 1970), German high jumper *Erich Kähler (1906–2000), German mathematician *Heinz Kähler (1905–1974), German art historian and archaeologist *Luise Kähler (1869–1955), German trade union leader and politician *Martin Kähler (1835–1912), German theologian *Otto Kähler (1894–1967), German admiral *Wilhelmine Kähler (1864–1941), German politician ;Other * Kähler Keramik, a Danish ceramics manufacturer *Kähler manifold, an important geometric complex manifold See also *Kahler (other) Kahler may refer to: Places *Kahler, Luxembourg, a small town in the commune of Garnich *Kahler Asten, a German mountain range Other uses *Kahler (surname) *Kahler's disease, a cancer otherwise known as ''multiple myeloma'' *Kahler Tremolo System, ... {{disambiguation, surname Occupational surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Number
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic. The theory was developed by Hodge in the 1930s to study algebraic geometry, and it built on the work of Georges de Rham on de Rham cohomology. It has major applications in two settings: Riemannian manifolds and Kähler manifolds. Hodge's primary motivation, the study of complex projective varieties, is encompassed by the latter case. Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic cycles. While Hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annales Scientifiques De L'École Normale Supérieure
''Annales Scientifiques de l'École Normale Supérieure'' is a French scientific journal of mathematics published by the Société Mathématique de France. It was established in 1864 by the French chemist Louis Pasteur and published articles in mathematics, physics, chemistry, biology, and geology. In 1900, it became a purely mathematical journal. It is published with help of the Centre national de la recherche scientifique. Its web site is hosted by the mathematics department of the École Normale Supérieure École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, S .... External links * Archive(1864–2013) Mathematics journals Publications established in 1864 Multilingual journals Multidisciplinary scientific journals Société Mathématique de France academic journals {{mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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