Hypergeometric Functions, My Love
Hypergeometric Functions, My Love

Author: Masaaki Yoshida

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 301

ISBN-13: 3322901661

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The classical story - of the hypergeometric functions, the configuration space of 4 points on the projective line, elliptic curves, elliptic modular functions and the theta functions - now evolves, in this book, to the story of hypergeometric funktions in 4 variables, the configuration space of 6 points in the projective plane, K3 surfaces, theta functions in 4 variables. This modern theory has been established by the author and his collaborators in the 1990's; further development to different aspects is expected. It leads the reader to a fascinating 4-dimensional world. The author tells the story casually and visually in a plain language, starting form elementary level such as equivalence relations, the exponential function, ... Undergraduate students should be able to enjoy the text.

Theory of Hypergeometric Functions
Theory of Hypergeometric Functions

Author: Kazuhiko Aomoto

Publisher: Springer Science & Business Media

Published: 2011-05-21

Total Pages: 327

ISBN-13: 4431539387

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This book presents a geometric theory of complex analytic integrals representing hypergeometric functions of several variables. Starting from an integrand which is a product of powers of polynomials, integrals are explained, in an open affine space, as a pair of twisted de Rham cohomology and its dual over the coefficients of local system. It is shown that hypergeometric integrals generally satisfy a holonomic system of linear differential equations with respect to the coefficients of polynomials and also satisfy a holonomic system of linear difference equations with respect to the exponents. These are deduced from Grothendieck-Deligne’s rational de Rham cohomology on the one hand, and by multidimensional extension of Birkhoff’s classical theory on analytic difference equations on the other.

Arithmetic and Geometry Around Hypergeometric Functions
Arithmetic and Geometry Around Hypergeometric Functions

Author: Rolf-Peter Holzapfel

Publisher: Springer Science & Business Media

Published: 2007-06-28

Total Pages: 441

ISBN-13: 3764382848

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This volume comprises lecture notes, survey and research articles originating from the CIMPA Summer School Arithmetic and Geometry around Hypergeometric Functions held at Galatasaray University, Istanbul, June 13-25, 2005. It covers a wide range of topics related to hypergeometric functions, thus giving a broad perspective of the state of the art in the field.

Hessian Polyhedra, Invariant Theory And Appell Hypergeometric Functions
Hessian Polyhedra, Invariant Theory And Appell Hypergeometric Functions

Author: Lei Yang

Publisher: World Scientific

Published: 2018-03-13

Total Pages: 317

ISBN-13: 9813209496

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Our book gives the complex counterpart of Klein's classic book on the icosahedron. We show that the following four apparently disjoint theories: the symmetries of the Hessian polyhedra (geometry), the resolution of some system of algebraic equations (algebra), the system of partial differential equations of Appell hypergeometric functions (analysis) and the modular equation of Picard modular functions (arithmetic) are in fact dominated by the structure of a single object, the Hessian group $mathfrak{G}’_{216}$. It provides another beautiful example on the fundamental unity of mathematics.

Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions
Encyclopedia of Special Functions: The Askey-Bateman Project: Volume 2, Multivariable Special Functions

Author: Tom H. Koornwinder

Publisher: Cambridge University Press

Published: 2020-10-15

Total Pages: 442

ISBN-13: 1108916554

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This is the second of three volumes that form the Encyclopedia of Special Functions, an extensive update of the Bateman Manuscript Project. Volume 2 covers multivariable special functions. When the Bateman project appeared, study of these was in an early stage, but revolutionary developments began to be made in the 1980s and have continued ever since. World-renowned experts survey these over the course of 12 chapters, each containing an extensive bibliography. The reader encounters different perspectives on a wide range of topics, from Dunkl theory, to Macdonald theory, to the various deep generalizations of classical hypergeometric functions to the several variables case, including the elliptic level. Particular attention is paid to the close relation of the subject with Lie theory, geometry, mathematical physics and combinatorics.

Multivariate Normal Distribution, The: Theory And Applications
Multivariate Normal Distribution, The: Theory And Applications

Author: Thu Pham-gia

Publisher: World Scientific

Published: 2021-05-05

Total Pages: 494

ISBN-13: 9811235309

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This book provides the reader with user-friendly applications of normal distribution. In several variables it is called the multinormal distribution which is often handled using matrices for convenience. The author seeks to make the arguments less abstract and hence, starts with the univariate case and moves progressively toward the vector and matrix cases. The approach used in the book is a gradual one, going from one scalar variable to a vector variable and to a matrix variable. The author presents the unified aspect of normal distribution, as well as addresses several other issues, including random matrix theory in physics. Other well-known applications, such as Herrnstein and Murray's argument that human intelligence is substantially influenced by both inherited and environmental factors, will be discussed in this book. It is a better predictor of many personal dynamics — including financial income, job performance, birth out of wedlock, and involvement in crime — than are an individual's parental socioeconomic status, or education level, and deserve to be mentioned and discussed.

Gröbner Deformations of Hypergeometric Differential Equations
Gröbner Deformations of Hypergeometric Differential Equations

Author: Mutsumi Saito

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 261

ISBN-13: 366204112X

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The theory of Gröbner bases is a main tool for dealing with rings of differential operators. This book reexamines the concept of Gröbner bases from the point of view of geometric deformations. The algorithmic methods introduced in this book are particularly useful for studying the systems of multidimensional hypergeometric PDE's introduced by Gelfand, Kapranov, and Zelevinsky. A number of original research results are contained in the book, and many open problems are raised for future research in this rapidly growing area of computational mathematics.

Hypergeometric Functions and Their Applications
Hypergeometric Functions and Their Applications

Author: James B. Seaborn

Publisher: Springer Science & Business Media

Published: 2013-04-09

Total Pages: 261

ISBN-13: 1475754434

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Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mathe matical Sciences (AMS) series, which will focus on advanced textbooks and research level monographs. Preface A wide range of problems exists in classical and quantum physics, engi neering, and applied mathematics in which special functions arise. The procedure followed in most texts on these topics (e. g. , quantum mechanics, electrodynamics, modern physics, classical mechanics, etc. ) is to formu late the problem as a differential equation that is related to one of several special differential equations (Hermite's, Bessel's, Laguerre's, Legendre's, etc. ).