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Eisenstein–Kronecker Number
In mathematics, Eisenstein–Kronecker numbers are an analogue for imaginary quadratic fields of generalized Bernoulli numbers. They are defined in terms of classical Eisenstein–Kronecker series, which were studied by Kenichi Bannai and Shinichi Kobayashi using the Poincaré bundle. Eisenstein–Kronecker numbers are algebraic and satisfy congruences that can be used in the construction of two-variable ''p''-adic ''L''-functions. They are related to critical ''L''-values of Hecke characters. Definition When is the area of the fundamental domain of \Gamma divided by \pi, where \Gamma is a lattice in \mathbb: e_^(z_0,w_0):=\sum_\frac\langle\gamma,w_0\rangle_\Gamma, when \mathbb_0:=\mathbb\cup\, \,\,\,z_0,w_0\in\mathbb, where \langle z,w\rangle_\Gamma:=e^\frac and \overline is the complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imaginary Quadratic Field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 and 1. If d>0, the corresponding quadratic field is called a real quadratic field, and, if d<0, it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a subfield of the field of the s. Quadratic fields have been studied in great depth, initially as part of the theory of s. There remain some unsolve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Bernoulli Numbers
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set theory, set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of falsifiability, verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Research Institute For Mathematical Sciences
The is a research institute attached to Kyoto University, hosting researchers in the mathematical sciences from all over Japan. RIMS was founded in April 1963. List of directors * Masuo Fukuhara (1963.5.1 – 1969.3.31) * Kōsaku Yosida (1969.4.1 – 1972.3.31) * Hisaaki Yoshizawa (1972.4.1 – 1976.3.31) * Kiyoshi Itō (1976.4.1 – 1979.4.1) * Nobuo Shimada (1979.4.2 – 1983.4.1) * Heisuke Hironaka (1983.4.2 – 1985.1.30) * Nobuo Shimada (1985.1.31 – 1987.1.30) * Mikio Sato (1987.1.31 – 1991.1.30) * Satoru Takasu (1991.1.31 – 1993.1.30) * Huzihiro Araki (1993.1.31 – 1996.3.31) * Kyoji Saito, Kyōji Saitō (1996.4.1 – 1998.3.31) * Masatake Mori (1998.4.1 – 2001.3.31) * Masaki Kashiwara (2001.4.1 – 2003.3.31) * Yōichirō Takahashi (2003.4.1 – 2007.3.31) * Masaki Kashiwara (2007.4.1 – 2009.3.31) * Shigeru Morishige (2009.4.1 – 2011.3.31) * Shigefumi Mori (2011.4.1 – 2014.3.31) * Shigeru Mukai (2014.4.1 – 2017.3.31) * Michio Yamada (2017.4.1 – present ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
European Mathematical Society
The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current president is Jan Philip Solovej, professor at the Department of Mathematics at the University of Copenhagen. Goals The Society seeks to serve all kinds of mathematicians in universities, research institutes and other forms of higher education. Its aims are to #Promote mathematical research, both pure and applied, #Assist and advise on problems of mathematical education, #Concern itself with the broader relations of mathematics to society, #Foster interaction between mathematicians of different countries, #Establish a sense of identity amongst European mathematicians, #Represent the mathematical community in supra-national institutions. The EMS is itself an Affiliate Member of the International Mathematical Union and an Associate Member ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Bundle
In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field ''k''. A 1-dimensional abelian variety is an elliptic curve, and every elliptic curve is isomorphic to its dual, but this fails for higher-dimensional abelian varieties, so the concept of dual becomes more interesting in higher dimensions. Definition Let ''A'' be an abelian variety over a field ''k''. We define \operatorname^0 (A) \subset \operatorname (A) to be the subgroup of the Picard group consisting of line bundles ''L'' such that m^*L \cong p^*L \otimes q^*L, where m, p, q are the multiplication and projection maps A \times_k A \to A respectively. An element of \operatorname^0(A) is called a degree 0 line bundle on ''A''. To ''A'' one then associates a dual abelian variety ''A''v (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrized by a ''k''-variety ''T'' is defined to be a line bundle ''L'' on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forum Of Mathematics, Sigma
Forum or The Forum may refer to: Common uses *Forum (legal), designated space for public expression in the United States *Forum (Roman), open public space within a Roman city **Roman Forum, most famous example * Internet forum, discussion board on the Internet Arts and entertainment * Forum & Forum Expanded, a section of the Berlin International Film Festival * ''Forum'' (album), a 2001 pop/soft rock album by Invertigo *The Forum (vocal group), organized by American musician Les Baxter *Forum theatre, a type of theatrical technique created by Brazilian theatre director Augusto Boal * Forum Theatre (Washington, D.C.), a former theatre group Buildings Shopping centres * Forum (shopping centre), Helsinki, Finland *The Forum (shopping mall), Bangalore, India * Forum Mall (Kolkata), Kolkata, India *Forum The Shopping Mall, Singapore * The Forum on Peachtree Parkway, Peachtree Corners, Georgia, United States *The Forum Shops at Caesars, Las Vegas, Nevada, United States Sports and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number
In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is an algebraic number, because it is a root of the polynomial X^2 - X - 1, i.e., a solution of the equation x^2 - x - 1 = 0, and the complex number 1 + i is algebraic as a root of X^4 + 4. Algebraic numbers include all integers, rational numbers, and nth root, ''n''-th roots of integers. Algebraic complex numbers are closed under addition, subtraction, multiplication and division, and hence form a field (mathematics), field, denoted \overline. The set of algebraic real numbers \overline \cap \R is also a field. Numbers which are not algebraic are called transcendental number, transcendental and include pi, and . There are countable set, countably many algebraic numbers, hence almost all real (or complex) numbers (in the sense of Lebesgue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic L-function
In mathematics, a ''p''-adic zeta function, or more generally a ''p''-adic ''L''-function, is a function analogous to the Riemann zeta function, or more general L-function, ''L''-functions, but whose domain of a function, domain and codomain, target are ''p-adic'' (where ''p'' is a prime number). For example, the domain could be the p-adic integer, ''p''-adic integers Z''p'', a profinite group, profinite ''p''-group, or a ''p''-adic family of Galois representations, and the image could be the p-adic number, ''p''-adic numbers Q''p'' or its algebraic closure. The source of a ''p''-adic ''L''-function tends to be one of two types. The first source—from which Tomio Kubota and Heinrich-Wolfgang Leopoldt gave the first construction of a ''p''-adic ''L''-function —is via the ''p''-adic interpolation of special values of L-functions, special values of ''L''-functions. For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a ''p''-adic ''L''- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duke Mathematical Journal
''Duke Mathematical Journal'' is a peer-reviewed mathematics journal published by Duke University Press. It was established in 1935. The founding editors-in-chief were David Widder, Arthur Coble, and Joseph Miller Thomas. The first issue included a paper by Solomon Lefschetz. Leonard Carlitz served on the editorial board for 35 years, from 1938 to 1973. The current managing editor is Richard Hain (Duke University). Impact According to the journal homepage, the journal has a 2018 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 2.194, ranking it in the top ten mathematics journals in the world. References External links * Mathematics journals Mathematical Journal Academic journals established in 1935 Multilingual journals English-language journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Character
In number theory, a Hecke character is a generalisation of a Dirichlet character, introduced by Erich Hecke to construct a class of ''L''-functions larger than Dirichlet ''L''-functions, and a natural setting for the Dedekind zeta-functions and certain others which have functional equations analogous to that of the Riemann zeta-function. Definition A Hecke character is a character of the idele class group of a number field or global function field. It corresponds uniquely to a character of the idele group which is trivial on principal ideles, via composition with the projection map. This definition depends on the definition of a character, which varies slightly between authors: It may be defined as a homomorphism to the non-zero complex numbers (also called a "quasicharacter"), or as a homomorphism to the unit circle in \mathbb ("unitary"). Any quasicharacter (of the idele class group) can be written uniquely as a unitary character times a real power of the norm, so there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
