Hypergeometric Functions, My Love

Hypergeometric Functions, My Love PDF

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 PDF

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 PDF

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 PDF

Author: Yang Lei

Publisher: World Scientific

Published: 2018-03-13

Total Pages: 316

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 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 PDF

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 PDF

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 PDF

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.

Algebra, Geometry and Software Systems

Algebra, Geometry and Software Systems PDF

Author: Michael Joswig

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 332

ISBN-13: 3662051486

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A collection of surveys and research papers on mathematical software and algorithms. The common thread is that the field of mathematical applications lies on the border between algebra and geometry. Topics include polyhedral geometry, elimination theory, algebraic surfaces, Gröbner bases, triangulations of point sets and the mutual relationship. This diversity is accompanied by the abundance of available software systems which often handle only special mathematical aspects. This is why the volume also focuses on solutions to the integration of mathematical software systems. This includes low-level and XML based high-level communication channels as well as general frameworks for modular systems.

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