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Mock Theta Function
In mathematics, a mock modular form is the Holomorphic function, holomorphic part of a harmonic weak Maass wave form, 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 Ramanujan's lost notebook, 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 funct ... [...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] |
Ken Ono
Ken Ono (born March 20, 1968) is an American mathematician with fields of study in number theory. He is the STEM Advisor to the Provost and the Marvin Rosenblum Professor of Mathematics at the University of Virginia. Early life and education Ono was born on March 20, 1968, in Philadelphia, Pennsylvania. He is the son of mathematician Takashi Ono (mathematician), Takashi Ono, who emigrated from Japan to the United States after World War II. Ono was born in the United States as his father returned to the United States from the University of British Columbia in Canada for a position at the University of Pennsylvania. In the 1980s, Ono attended Towson High School, but dropped out. He later enrolled at the University of Chicago without a high school diploma. There he raced bicycles, and he was a member of the Pepsi–Miyata (bicycle), Miyata Cycling Team. He received his BA from the University of Chicago in 1989, where he was a member of the Psi Upsilon fraternity. He earned his PhD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Émile Appell
:''M. P. Appell is the same person: it stands for Monsieur Paul Appell''. Paul Émile Appell (27 September 1855 in Strasbourg – 24 October 1930 in Paris) was a French mathematician and Rector of the University of Paris. Appell polynomials and Appell's equations of motion are named after him, as is rue Paul Appell in the 14th arrondissement of Paris and the minor planet 988 Appella. Life Paul Appell entered the École Normale Supérieure in 1873. He was elected to the French Academy of Sciences in 1892. In 1895, he became a Professor at the École Centrale Paris. Between 1903 and 1920 he was Dean of the Faculty of Science of the University of Paris, then Rector of the University of Paris from 1920 to 1925. Appell was the President of the Société astronomique de France (SAF), the French astronomical society, from 1919 to 1921. His daughter Marguerite Appell (1883–1969), who married the mathematician Émile Borel, is known as a novelist under her pen-name Camille Mar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambert Series
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form :S(q)=\sum_^\infty a_n \frac . It can be resummed formally by expanding the denominator: :S(q)=\sum_^\infty a_n \sum_^\infty q^ = \sum_^\infty b_m q^m where the coefficients of the new series are given by the Dirichlet convolution of ''a''''n'' with the constant function 1(''n'') = 1: :b_m = (a*1)(m) = \sum_ a_n. \, This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform. Examples Since this last sum is a typical number-theoretic sum, almost any natural multiplicative function will be exactly summable when used in a Lambert series. Thus, for example, one has :\sum_^\infty q^n \sigma_0(n) = \sum_^\infty \frac where \sigma_0(n)=d(n) is the number of positive divisors of the number ''n''. For the higher order sum-of-divisor functions, one has :\sum_^\infty q^n \sigma_\alpha(n) = \sum_^\infty \frac = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Don Zagier
Don Bernard Zagier (born 29 June 1951) is an American-German mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany. He was a professor at the ''Collège de France'' in Paris from 2006 to 2014. Since October 2014, he is also a Distinguished Staff Associate at the International Centre for Theoretical Physics ( ICTP). Among his doctoral students are Fields medalists Maxim Kontsevich and Maryna Viazovska. Background Zagier was born in Heidelberg, West Germany. His mother was a psychiatrist, and his father was the dean of instruction at the American College of Switzerland. His father held five different citizenships, and he spent his youth living in many different countries. After finishing high school (at age 13) and attending Winchester College for a year, he studied for three years at MIT, completing his bachelor's and master's degrees and being named a Putnam Fellow in 1967 at t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where the infinite series representation which initially defined the function becomes divergent. The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology. Initial discussion Suppose ''f'' is an analytic function defined on a non-empty open subset ''U'' of the complex plane If ''V'' is a larger open subset of containing ''U'', and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incomplete Gamma Function
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which are defined similarly to the gamma function but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit. Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity. Definition The upper incomplete gamma function is defined as: \Gamma(s,x) = \int_x^ t^\,e^\, dt , whereas the lower incomplete gamma function is defined as: \gamma(s,x) = \int_0^x t^\,e^\, dt . In both cases is a complex parameter, such that the real part of is positive. Properties By integration by parts we find the recurrence relati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maass Form
In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup \Gamma of \mathrm_(\R) as modular forms. They are eigenforms of the hyperbolic Laplace operator \Delta defined on \mathcal and satisfy certain growth conditions at the cusps of a fundamental domain of \Gamma. In contrast to modular forms, Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949. General remarks The group : G := \mathrm_(\R) = \left\ operates on the upper half plane :\mathcal = \ by fractional linear transformations: :\begin a & b \\ c & d \\ \end \cdot z := \frac. It can be extended to an operation on \mathcal \cup \ \cup \mathbb by defining: :\begin a & b \\ c & d \\ \end\cdot z :=\begin \frac & \text cz+d \neq 0, \\ \infty & \text cz+d=0,\end :\begin a & b \\ c & d \\ \end \cdot \inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metaplectic Group
In mathematics, the metaplectic group Mp2''n'' is a double cover of the symplectic group Sp2''n''. It can be defined over either real or ''p''-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles. The metaplectic group has a particularly significant infinite-dimensional linear representation, the Weil representation. It was used by André Weil to give a representation-theoretic interpretation of theta functions, and is important in the theory of modular forms of half-integral weight and the theta correspondence. Definition The fundamental group of the symplectic Lie group Sp2n(R) is infinite cyclic, so it has a unique connected double cover, which is denoted Mp2''n''(R) and called the metaplectic group. The metaplectic group Mp2(R) is ''not'' a matrix group: it has no faithful finite-dimensional representations. Therefore, the question of its explicit realization is nontrivial. It has faithful ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Linear Group
In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel (algebra), kernel of the determinant :\det\colon \operatorname(n, R) \to R^\times. where R^\times is the multiplicative group of R (that is, R excluding 0 when R is a field). These elements are "special" in that they form an Algebraic variety, algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant is polynomial in the entries). When R is the finite field of order q, the notation \operatorname(n,q) is sometimes used. Geometric interpretation The special linear group \operatorname(n,\R) can be characterized as the group of ''volume and orientation (mathematics), orientation preserving'' linear tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
