Exponential Sums And Differential Equations
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| Author | : Nicholas M. Katz |
| Publisher | : Princeton University Press |
| Total Pages | : 445 |
| Release | : 2016-03-02 |
| Genre | : Mathematics |
| ISBN | : 1400882435 |
This book is concerned with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. These areas are the theory of linear differential equations in one complex variable with polynomial coefficients, and the theory of one parameter families of exponential sums over finite fields. After reviewing some results from representation theory, the book discusses results about differential equations and their differential galois groups (G) and one-parameter families of exponential sums and their geometric monodromy groups (G). The final part of the book is devoted to comparison theorems relating G and G of suitably "corresponding" situations, which provide a systematic explanation of the remarkable "coincidences" found "by hand" in the hypergeometric case.
| Author | : Nicholas M. Katz |
| Publisher | : Princeton University Press |
| Total Pages | : 444 |
| Release | : 1990-09-21 |
| Genre | : Mathematics |
| ISBN | : 0691085994 |
This book is concerned with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. These areas are the theory of linear differential equations in one complex variable with polynomial coefficients, and the theory of one parameter families of exponential sums over finite fields. After reviewing some results from representation theory, the book discusses results about differential equations and their differential galois groups (G) and one-parameter families of exponential sums and their geometric monodromy groups (G). The final part of the book is devoted to comparison theorems relating G and G of suitably "corresponding" situations, which provide a systematic explanation of the remarkable "coincidences" found "by hand" in the hypergeometric case.
| Author | : S. W. Graham |
| Publisher | : Cambridge University Press |
| Total Pages | : 133 |
| Release | : 1991-01-25 |
| Genre | : Mathematics |
| ISBN | : 0521339278 |
This book is a self-contained account of the one- and two-dimensional van der Corput method and its use in estimating exponential sums. These arise in many problems in analytic number theory. It is the first cohesive account of much of this material and will be welcomed by graduates and professionals in analytic number theory. The authors show how the method can be applied to problems such as upper bounds for the Riemann-Zeta function. the Dirichlet divisor problem, the distribution of square free numbers, and the Piatetski-Shapiro prime number theorem.
| Author | : G. R. H. Greaves |
| Publisher | : Cambridge University Press |
| Total Pages | : 360 |
| Release | : 1997-01-30 |
| Genre | : Mathematics |
| ISBN | : 0521589576 |
State-of-the-art analytic number theory proceedings.
| Author | : Zhi-Zhong Sun |
| Publisher | : Walter de Gruyter GmbH & Co KG |
| Total Pages | : 226 |
| Release | : 2020-08-24 |
| Genre | : Mathematics |
| ISBN | : 3110615304 |
Starting with an introduction to fractional derivatives and numerical approximations, this book presents finite difference methods for fractional differential equations, including time-fractional sub-diffusion equations, time-fractional wave equations, and space-fractional differential equations, among others. Approximation methods for fractional derivatives are developed and approximate accuracies are analyzed in detail.
| Author | : Igor Shparlinski |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 253 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 940111806X |
This volume presents an exhaustive treatment of computation and algorithms for finite fields. Topics covered include polynomial factorization, finding irreducible and primitive polynomials, distribution of these primitive polynomials and of primitive points on elliptic curves, constructing bases of various types, and new applications of finite fields to other araes of mathematics. For completeness, also included are two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number generators, modular arithmetic etc.), and computational number theory (primality testing, factoring integers, computing in algebraic number theory, etc.) The problems considered here have many applications in computer science, coding theory, cryptography, number theory and discrete mathematics. The level of discussion presuppose only a knowledge of the basic facts on finite fields, and the book can be recommended as supplementary graduate text. For researchers and students interested in computational and algorithmic problems in finite fields.
| Author | : Harry Dankowicz |
| Publisher | : SIAM |
| Total Pages | : 585 |
| Release | : 2013-01-01 |
| Genre | : Science |
| ISBN | : 1611972574 |
This book provides a comprehensive introduction to the mathematical methodology of parameter continuation, the computational analysis of families of solutions to nonlinear mathematical equations. It develops a systematic formalism for constructing abstract representations of continuation problems and for implementing these in an existing computational platform. Recipes for Continuation lends equal importance to theoretical rigor, algorithm development, and software engineering; demonstrates the use of fully developed toolbox templates for single- and multisegment boundary-value problems to the analysis of periodic orbits in smooth and hybrid dynamical systems, quasi-periodic invariant tori, and homoclinic and heteroclinic connecting orbits between equilibria and/or periodic orbits; shows the use of vectorization for optimal computational efficiency, an object-oriented paradigm for the modular construction of continuation problems, and adaptive discretization algorithms for guaranteed bounds on estimated errors; and contains extensive and fully worked examples that illustrate the application of the MATLAB®-based Computational Continuation Core (COCO) to problems from recent research literature that are relevant to dynamical system models from mechanics, electronics, biology, economics, and neuroscience.
| Author | : Percy Deift |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 284 |
| Release | : 1996 |
| Genre | : Mathematics |
| ISBN | : 9780821897003 |
This volume contains some of the lectures presented in June 1994 during the AMS-SIAM Summer Seminar at the Mathematical Sciences Research Institute in Berkeley. The goal of the seminar was to introduce participants to as many interesting and active applications of dynamical systems and probabilistic methods to problems in applied mathematics as possible. As a result, this book covers a great deal of ground. Nevertheless, the pedagogical orientation of the lectures has been retained, and therefore the book will serve as an ideal introduction to these varied and interesting topics.
| Author | : B L J Braaksma |
| Publisher | : World Scientific |
| Total Pages | : 343 |
| Release | : 2002-12-10 |
| Genre | : Mathematics |
| ISBN | : 9814487430 |
This volume is the record of a workshop on differential equations and the Stokes phenomenon, held in May 2001 at the University of Groningen. It contains expanded versions of most of the lectures given at the workshop. To a large extent, both the workshop and the book may be regarded as a sequel to a conference held in Groningen in 1995 which resulted in the book The Stokes Phenomenon and Hilbert's 16th Problem (B L J Braaksma, G K Immink and M van der Put, editors), also published by World Scientific (1996).Both books offer a snapshot concerning the state of the art in the areas of differential, difference and q-difference equations. Apart from the asymptotics of solutions, Painlevé properties and the algebraic theory, new topics addressed in the second book include arithmetic theory of linear equations, and Galois theory and Lie symmetries of nonlinear differential equations.
| Author | : Boele Lieuwe Jan Braaksma |
| Publisher | : World Scientific |
| Total Pages | : 352 |
| Release | : 2002 |
| Genre | : Mathematics |
| ISBN | : 9789812381729 |
Offers a snapshot concerning the state of the art in the areas of differential, difference and q-difference equations.
