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Harmonic Maass Form
In mathematics, a weak Maass form is a smooth function f on the upper half plane, transforming like a modular form under the action of the modular group, being an eigenfunction of the corresponding hyperbolic Laplace operator, and having at most linear exponential growth at the cusps. If the eigenvalue of f under the Laplacian is zero, then f is called a harmonic weak Maass form, or briefly a harmonic Maass form. A weak Maass form which has actually moderate growth at the cusps is a classical Maass wave form. The Fourier expansions of harmonic Maass forms often encode interesting combinatorial, arithmetic, or geometric generating functions. Regularized theta lifts of harmonic Maass forms can be used to construct Arakelov Green functions for special divisors on orthogonal Shimura varieties. Definition A complex-valued smooth function f on the upper half-plane is called a weak Maass form of integral weight (for the group ) if it satisfies the following three conditions: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans Petersson
Hans Petersson (24 September 1902 in Bentschen – 9 November 1984 in Münster) was a German mathematician. He introduced the Petersson inner product and is also known for the Ramanujan–Petersson conjecture In mathematics, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients of the cusp form of weight :\Delta(z)= \sum_\tau(n)q^n=q\prod_\left (1-q^n \right)^ = q-24q^2+252q^3- 1472q^4 + 4830q^5-\c .... See also * Weil–Petersson metric References * * {{DEFAULTSORT:Petersson, Hans 1902 births 1984 deaths People from Zbąszyń 20th-century German mathematicians People from the Province of Posen Academic staff of Charles University Nazi Party members Academic staff of the University of Münster ... [...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 scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as ... of 2.194, ranking it in the top ten mathematics journals in the world. References External links * Mathematics journals Mathematical Journal Publications established in 1935 Multilingual journals English-language journals French- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nick Katz
Nicholas Michael Katz (born December 7, 1943) is an American mathematician, working in arithmetic geometry, particularly on ''p''-adic methods, monodromy and moduli problems, and number theory. He is currently a professor of Mathematics at Princeton University and an editor of the journal ''Annals of Mathematics''. Life and work Katz graduated from Johns Hopkins University (BA 1964) and from Princeton University, where in 1965 he received his master's degree and in 1966 he received his doctorate under supervision of Bernard Dwork with thesis ''On the Differential Equations Satisfied by Period Matrices''. After that, at Princeton, he was an instructor, an assistant professor in 1968, associate professor in 1971 and professor in 1974. From 2002 to 2005 he was the chairman of faculty there. He was also a visiting scholar at the University of Minnesota, the University of Kyoto, Paris VI, Orsay Faculty of Sciences, the Institute for Advanced Study and the IHES. While in Fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zwegers
Sander Pieter Zwegers (born April 16, 1975) is a Dutch mathematician who made a connection between Maass forms and Srinivasa Ramanujan's mock theta functions in 2002. He was born in Oosterhout. After a period at the Max-Planck Institute in Bonn, he became an assistant professor at the University College Dublin in 2008. Since 2011, he has been is professor of number theory at the University of Cologne. Research In 1976, the American mathematician George Andrews found what is nowadays known as the "Lost Notebook Ramanujan's lost notebook is the manuscript in which the Indian mathematician Srinivasa Ramanujan recorded the mathematical discoveries of the last year (1919–1920) of his life. Its whereabouts were unknown to all but a few mathematicians until i ..." of Ramanujan. It contains many remarkable results, including the mysterious mock theta functions. This notebook contains what many specialists regard as Ramanujan’s deepest work. It was Sander Zwegers who, as a PdD stude ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory. His 1872 Erlangen program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time. Life Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province. His mother was Sophie Elise Klein (1819–1890, née Kayser). He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ... in .) It is always understood that the curve is really sitting in the projective plane, with the point being the uniqu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Eichler
Martin Maximilian Emil Eichler (29 March 1912 – 7 October 1992) was a German number theorist. Eichler received his Ph.D. from the Martin Luther University of Halle-Wittenberg in 1936. Eichler and Goro Shimura developed a method to construct elliptic curves from certain modular forms. The converse notion that every elliptic curve has a corresponding modular form would later be the key to the proof of Fermat's Last Theorem. Selected publications * ''Quadratische Formen und orthogonale Gruppen'', Springer 1952, 1974 * * ''Einführung in die Theorie der algebraischen Zahlen und Funktionen'', Birkhäuser 1963; Eng. trans. 1966''Introduction to the theory of algebraic numbers and functions'' in which a section on modular forms is added; pbk 2014 reprint of 1963 German original * ''Projective varieties and modular forms'' 1971 ( Riemann–Roch theorem); * with Don Zagier: ''The Theory of Jacobi forms'', Birkhäuser 1985; ''Über die Einheiten der Divisionsalgebren'', Ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Zeta Function
In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named for Karl Weierstrass. The relation between the sigma, zeta, and \wp functions is analogous to that between the sine, cotangent, and squared cosecant functions: the logarithmic derivative of the sine is the cotangent, whose derivative is negative the squared cosecant. Weierstrass sigma function The Weierstrass sigma function associated to a two-dimensional lattice \Lambda\subset\Complex is defined to be the product : \begin \operatorname&=z\prod_ \left(1-\frac\right) e^ \\ &=z\prod_^\infty \left(1-\frac\right) e^ \end where \Lambda^ denotes \Lambda-\ or \ are a '' fundamental pair of periods''. Through careful manipulation of the Weierstrass factorization theorem as it relates also to the sine function, another potentially more manageable infinite product definition is : \operatorname=\frace^\sin\prod_^\infty\left(1-\frac\r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whittaker Function
In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric. More generally, introduced Whittaker functions of reductive groups over local fields, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL2(R). Whittaker's equation is :\frac+\left(-\frac+\frac+\frac\right)w=0. It has a regular singular point at 0 and an irregular singular point at ∞. Two solutions are given by the Whittaker functions ''M''κ,μ(''z''), ''W''κ,μ(''z''), defined in terms of Kummer's confluent hypergeometric functions ''M'' and ''U'' by :M_\left(z\right) = \exp\left(-z/2\right)z^M\left(\mu-\kappa+\tfrac, 1+2\mu, z\right) :W_\left(z\right) = \exp\left(-z/2\right)z^U\left(\mu-\kappa+\tfrac, 1+2\mu, z\right). The Whittaker function W_(z) is the same as those with opposite v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology. Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré disc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mock Modular Form
In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight . The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his lost notebook. Sander Zwegers discovered that adding certain non-holomorphic functions to them turns them into harmonic weak Maass forms. History Ramanujan's 12 January 1920 letter to Hardy listed 17 examples of functions that he called mock theta functions, and his lost notebook contained several more examples. (Ramanujan used the term "theta function" for what today would be called a modular form.) Ramanujan pointed out that they have an asymptotic expansion at the cusps, similar to that of modular forms of weight , possibly with poles at cusps, but cannot be expressed in terms of "ordinary" theta functions. He called functions with similar properties "mock theta functions". Zwegers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
