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Number Theory, Fourier Analysis and Geometric Discrepancy

Author: Giancarlo Travaglini

Publisher: Cambridge University Press

Published: 2014-06-12

Total Pages: 251

ISBN-13: 1139992821

The study of geometric discrepancy, which provides a framework for quantifying the quality of a distribution of a finite set of points, has experienced significant growth in recent decades. This book provides a self-contained course in number theory, Fourier analysis and geometric discrepancy theory, and the relations between them, at the advanced undergraduate or beginning graduate level. It starts as a traditional course in elementary number theory, and introduces the reader to subsequent material on uniform distribution of infinite sequences, and discrepancy of finite sequences. Both modern and classical aspects of the theory are discussed, such as Weyl's criterion, Benford's law, the Koksma–Hlawka inequality, lattice point problems, and irregularities of distribution for convex bodies. Fourier analysis also features prominently, for which the theory is developed in parallel, including topics such as convergence of Fourier series, one-sided trigonometric approximation, the Poisson summation formula, exponential sums, decay of Fourier transforms, and Bessel functions.

A Panorama of Discrepancy Theory

Author: William Chen

Publisher: Springer

Published: 2014-10-07

Total Pages: 695

ISBN-13: 3319046969

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This is the first work on Discrepancy Theory to show the present variety of points of view and applications covering the areas Classical and Geometric Discrepancy Theory, Combinatorial Discrepancy Theory and Applications and Constructions. It consists of several chapters, written by experts in their respective fields and focusing on the different aspects of the theory. Discrepancy theory concerns the problem of replacing a continuous object with a discrete sampling and is currently located at the crossroads of number theory, combinatorics, Fourier analysis, algorithms and complexity, probability theory and numerical analysis. This book presents an invitation to researchers and students to explore the different methods and is meant to motivate interdisciplinary research.

Geometric Discrepancy

Author: Jiri Matousek

Publisher: Springer Science & Business Media

Published: 2009-12-02

Total Pages: 293

ISBN-13: 3642039421

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What is the "most uniform" way of distributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? This book is an accessible and lively introduction to the area of geometric discrepancy theory, with numerous exercises and illustrations. In separate, more specialized parts, it also provides a comprehensive guide to recent research.

Number Theory, Fourier Analysis and Geometric Discrepancy

Author: Giancarlo Travaglini

Publisher: Cambridge University Press

Published: 2014-06-12

Total Pages: 251

ISBN-13: 1107044030

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Classical number theory is developed from scratch leading to geometric discrepancy theory, with Fourier analysis introduced along the way.

Discrepancy Theory

Author: Dmitriy Bilyk

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2020-01-20

Total Pages: 303

ISBN-13: 3110651203

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The contributions in this book focus on a variety of topics related to discrepancy theory, comprising Fourier techniques to analyze discrepancy, low discrepancy point sets for quasi-Monte Carlo integration, probabilistic discrepancy bounds, dispersion of point sets, pair correlation of sequences, integer points in convex bodies, discrepancy with respect to geometric shapes other than rectangular boxes, and also open problems in discrepany theory

Fourier Analysis and Convexity

Author: Luca Brandolini

Publisher: Springer Science & Business Media

Published: 2011-04-27

Total Pages: 268

ISBN-13: 0817681728

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Explores relationship between Fourier Analysis, convex geometry, and related areas; in the past, study of this relationship has led to important mathematical advances Presents new results and applications to diverse fields such as geometry, number theory, and analysis Contributors are leading experts in their respective fields Will be of interest to both pure and applied mathematicians

Fourier Analysis: Volume 1, Theory

Author: Adrian Constantin

Publisher: Cambridge University Press

Published: 2016-05-31

Total Pages: 368

ISBN-13: 1316670805

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Fourier analysis aims to decompose functions into a superposition of simple trigonometric functions, whose special features can be exploited to isolate specific components into manageable clusters before reassembling the pieces. This two-volume text presents a largely self-contained treatment, comprising not just the major theoretical aspects (Part I) but also exploring links to other areas of mathematics and applications to science and technology (Part II). Following the historical and conceptual genesis, this book (Part I) provides overviews of basic measure theory and functional analysis, with added insight into complex analysis and the theory of distributions. The material is intended for both beginning and advanced graduate students with a thorough knowledge of advanced calculus and linear algebra. Historical notes are provided and topics are illustrated at every stage by examples and exercises, with separate hints and solutions, thus making the exposition useful both as a course textbook and for individual study.

Fourier Analysis on Polytopes and the Geometry of Numbers

Author: Sinai Robins

Publisher: American Mathematical Society

Published: 2024-04-24

Total Pages: 352

ISBN-13: 1470470330

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This book offers a gentle introduction to the geometry of numbers from a modern Fourier-analytic point of view. One of the main themes is the transfer of geometric knowledge of a polytope to analytic knowledge of its Fourier transform. The Fourier transform preserves all of the information of a polytope, and turns its geometry into analysis. The approach is unique, and streamlines this emerging field by presenting new simple proofs of some basic results of the field. In addition, each chapter is fitted with many exercises, some of which have solutions and hints in an appendix. Thus, an individual learner will have an easier time absorbing the material on their own, or as part of a class. Overall, this book provides an introduction appropriate for an advanced undergraduate, a beginning graduate student, or researcher interested in exploring this important expanding field.

Random Graphs, Geometry and Asymptotic Structure

Author: Michael Krivelevich

Publisher: Cambridge University Press

Published: 2016-04-25

Total Pages: 129

ISBN-13: 1316552942

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The theory of random graphs is a vital part of the education of any researcher entering the fascinating world of combinatorics. However, due to their diverse nature, the geometric and structural aspects of the theory often remain an obscure part of the formative study of young combinatorialists and probabilists. Moreover, the theory itself, even in its most basic forms, is often considered too advanced to be part of undergraduate curricula, and those who are interested usually learn it mostly through self-study, covering a lot of its fundamentals but little of the more recent developments. This book provides a self-contained and concise introduction to recent developments and techniques for classical problems in the theory of random graphs. Moreover, it covers geometric and topological aspects of the theory and introduces the reader to the diversity and depth of the methods that have been devised in this context.

The Geometry of Celestial Mechanics

Author: Hansjörg Geiges

Publisher: Cambridge University Press

Published: 2016-03-24

Total Pages: 241

ISBN-13: 1107125405

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A first course in celestial mechanics emphasising the variety of geometric ideas that have shaped the subject.