Many Variations Of Mahler Measures

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Many Variations of Mahler Measures

Author : François Brunault
Publisher : Cambridge University Press
ISBN 13 : 1108889190
Total Pages : 185 pages
Book Rating : 4.1/5 (88 download)

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Book Synopsis Many Variations of Mahler Measures by : François Brunault

Download or read book Many Variations of Mahler Measures written by François Brunault and published by Cambridge University Press. This book was released on 2020-05-14 with total page 185 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Mahler measure is a fascinating notion and an exciting topic in contemporary mathematics, interconnecting with subjects as diverse as number theory, analysis, arithmetic geometry, special functions and random walks. This friendly and concise introduction to the Mahler measure is a valuable resource for both graduate courses and self-study. It provides the reader with the necessary background material, before presenting the recent achievements and the remaining challenges in the field. The first part introduces the univariate Mahler measure and addresses Lehmer's question, and then discusses techniques of reducing multivariate measures to hypergeometric functions. The second part touches on the novelties of the subject, especially the relation with elliptic curves, modular forms and special values of L-functions. Finally, the Appendix presents the modern definition of motivic cohomology and regulator maps, as well as Deligne–Beilinson cohomology. The text includes many exercises to test comprehension and challenge readers of all abilities.

Around the Unit Circle

Author : James McKee
Publisher : Springer Nature
ISBN 13 : 3030800318
Total Pages : 444 pages
Book Rating : 4.0/5 (38 download)

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Book Synopsis Around the Unit Circle by : James McKee

Download or read book Around the Unit Circle written by James McKee and published by Springer Nature. This book was released on 2021-12-08 with total page 444 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mahler measure, a height function for polynomials, is the central theme of this book. It has many interesting properties, obtained by algebraic, analytic and combinatorial methods. It is the subject of several longstanding unsolved questions, such as Lehmer’s Problem (1933) and Boyd’s Conjecture (1981). This book contains a wide range of results on Mahler measure. Some of the results are very recent, such as Dimitrov’s proof of the Schinzel–Zassenhaus Conjecture. Other known results are included with new, streamlined proofs. Robinson’s Conjectures (1965) for cyclotomic integers, and their associated Cassels height function, are also discussed, for the first time in a book. One way to study algebraic integers is to associate them with combinatorial objects, such as integer matrices. In some of these combinatorial settings the analogues of several notorious open problems have been solved, and the book sets out this recent work. Many Mahler measure results are proved for restricted sets of polynomials, such as for totally real polynomials, and reciprocal polynomials of integer symmetric as well as symmetrizable matrices. For reference, the book includes appendices providing necessary background from algebraic number theory, graph theory, and other prerequisites, along with tables of one- and two-variable integer polynomials with small Mahler measure. All theorems are well motivated and presented in an accessible way. Numerous exercises at various levels are given, including some for computer programming. A wide range of stimulating open problems is also included. At the end of each chapter there is a glossary of newly introduced concepts and definitions. Around the Unit Circle is written in a friendly, lucid, enjoyable style, without sacrificing mathematical rigour. It is intended for lecture courses at the graduate level, and will also be a valuable reference for researchers interested in Mahler measure. Essentially self-contained, this textbook should also be accessible to well-prepared upper-level undergraduates.

Heights of Polynomials and Entropy in Algebraic Dynamics

Author : Graham Everest
Publisher : Springer Science & Business Media
ISBN 13 : 1447138988
Total Pages : 217 pages
Book Rating : 4.4/5 (471 download)

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Book Synopsis Heights of Polynomials and Entropy in Algebraic Dynamics by : Graham Everest

Download or read book Heights of Polynomials and Entropy in Algebraic Dynamics written by Graham Everest and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main theme of this book is the theory of heights as they appear in various guises. This includes a large body of results on Mahlers measure of the height of a polynomial. The authors'approach is very down to earth as they cover the rationals, assuming no prior knowledge of elliptic curves. The chapters include examples and particular computations, with all special calculation included so as to be self-contained. The authors devote space to discussing Mahlers measure and to giving some convincing and original examples to explain this phenomenon. XXXXXXX NEUER TEXT The main theme of this book is the theory of heights as it appears in various guises. To this §End.txt.Int.:, it examines the results of Mahlers measure of the height of a polynomial, which have never before appeared in book form. The authors take a down-to-earth approach that includes convincing and original examples. The book uncovers new and interesting connections between number theory and dynamics and will be interesting to researchers in both number theory and nonlinear dynamics.

Polynomials with Special Regard to Reducibility

Author : A. Schinzel
Publisher : Cambridge University Press
ISBN 13 : 9781139426718
Total Pages : 590 pages
Book Rating : 4.4/5 (267 download)

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Book Synopsis Polynomials with Special Regard to Reducibility by : A. Schinzel

Download or read book Polynomials with Special Regard to Reducibility written by A. Schinzel and published by Cambridge University Press. This book was released on 2000-04-27 with total page 590 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers most of the known results on reducibility of polynomials over arbitrary fields, algebraically closed fields and finitely generated fields. Results valid only over finite fields, local fields or the rational field are not covered here, but several theorems on reducibility of polynomials over number fields that are either totally real or complex multiplication fields are included. Some of these results are based on recent work of E. Bombieri and U. Zannier (presented here by Zannier in an appendix). The book also treats other subjects like Ritt's theory of composition of polynomials, and properties of the Mahler measure, and it concludes with a bibliography of over 300 items. This unique work will be a necessary resource for all number theorists and researchers in related fields.

Topics in Number Theory

Author : Scott D. Ahlgren
Publisher : Springer
ISBN 13 : 9781461379881
Total Pages : 0 pages
Book Rating : 4.3/5 (798 download)

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Book Synopsis Topics in Number Theory by : Scott D. Ahlgren

Download or read book Topics in Number Theory written by Scott D. Ahlgren and published by Springer. This book was released on 2011-09-17 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: From July 31 through August 3,1997, the Pennsylvania State University hosted the Topics in Number Theory Conference. The conference was organized by Ken Ono and myself. By writing the preface, I am afforded the opportunity to express my gratitude to Ken for beng the inspiring and driving force behind the whole conference. Without his energy, enthusiasm and skill the entire event would never have occurred. We are extremely grateful to the sponsors of the conference: The National Sci ence Foundation, The Penn State Conference Center and the Penn State Depart ment of Mathematics. The object in this conference was to provide a variety of presentations giving a current picture of recent, significant work in number theory. There were eight plenary lectures: H. Darmon (McGill University), "Non-vanishing of L-functions and their derivatives modulo p. " A. Granville (University of Georgia), "Mean values of multiplicative functions. " C. Pomerance (University of Georgia), "Recent results in primality testing. " C. Skinner (Princeton University), "Deformations of Galois representations. " R. Stanley (Massachusetts Institute of Technology), "Some interesting hyperplane arrangements. " F. Rodriguez Villegas (Princeton University), "Modular Mahler measures. " T. Wooley (University of Michigan), "Diophantine problems in many variables: The role of additive number theory. " D. Zeilberger (Temple University), "Reverse engineering in combinatorics and number theory. " The papers in this volume provide an accurate picture of many of the topics presented at the conference including contributions from four of the plenary lectures.

Research Directions in Number Theory

Author : Alina Bucur
Publisher : Springer Nature
ISBN 13 : 303151677X
Total Pages : 325 pages
Book Rating : 4.0/5 (315 download)

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Book Synopsis Research Directions in Number Theory by : Alina Bucur

Download or read book Research Directions in Number Theory written by Alina Bucur and published by Springer Nature. This book was released on with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Number Theory and Polynomials

Author : James Fraser McKee
Publisher : Cambridge University Press
ISBN 13 : 0521714672
Total Pages : 350 pages
Book Rating : 4.5/5 (217 download)

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Book Synopsis Number Theory and Polynomials by : James Fraser McKee

Download or read book Number Theory and Polynomials written by James Fraser McKee and published by Cambridge University Press. This book was released on 2008-05-08 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contributions by leading experts in the field provide a snapshot of current progress in polynomials and number theory.

Approximation by Algebraic Numbers

Author : Yann Bugeaud
Publisher : Cambridge University Press
ISBN 13 : 1139455672
Total Pages : 292 pages
Book Rating : 4.1/5 (394 download)

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Book Synopsis Approximation by Algebraic Numbers by : Yann Bugeaud

Download or read book Approximation by Algebraic Numbers written by Yann Bugeaud and published by Cambridge University Press. This book was released on 2004-11-08 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible and broad account of the approximation and classification of real numbers suited for graduate courses on Diophantine approximation (some 40 exercises are supplied), or as an introduction for non-experts. Specialists will appreciate the collection of over 50 open problems and the comprehensive list of more than 600 references.

Zeta and L-Functions of Varieties and Motives

Author : Bruno Kahn
Publisher : Cambridge University Press
ISBN 13 : 1108574912
Total Pages : 217 pages
Book Rating : 4.1/5 (85 download)

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Book Synopsis Zeta and L-Functions of Varieties and Motives by : Bruno Kahn

Download or read book Zeta and L-Functions of Varieties and Motives written by Bruno Kahn and published by Cambridge University Press. This book was released on 2020-05-07 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: The amount of mathematics invented for number-theoretic reasons is impressive. It includes much of complex analysis, the re-foundation of algebraic geometry on commutative algebra, group cohomology, homological algebra, and the theory of motives. Zeta and L-functions sit at the meeting point of all these theories and have played a profound role in shaping the evolution of number theory. This book presents a big picture of zeta and L-functions and the complex theories surrounding them, combining standard material with results and perspectives that are not made explicit elsewhere in the literature. Particular attention is paid to the development of the ideas surrounding zeta and L-functions, using quotes from original sources and comments throughout the book, pointing the reader towards the relevant history. Based on an advanced course given at Jussieu in 2013, it is an ideal introduction for graduate students and researchers to this fascinating story.

Analytic Methods In Number Theory: When Complex Numbers Count

Author : Wadim Zudilin
Publisher : World Scientific
ISBN 13 : 9811279330
Total Pages : 192 pages
Book Rating : 4.8/5 (112 download)

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Book Synopsis Analytic Methods In Number Theory: When Complex Numbers Count by : Wadim Zudilin

Download or read book Analytic Methods In Number Theory: When Complex Numbers Count written by Wadim Zudilin and published by World Scientific. This book was released on 2023-08-22 with total page 192 pages. Available in PDF, EPUB and Kindle. Book excerpt: There is no surprise that arithmetic properties of integral ('whole') numbers are controlled by analytic functions of complex variable. At the same time, the values of analytic functions themselves happen to be interesting numbers, for which we often seek explicit expressions in terms of other 'better known' numbers or try to prove that no such exist. This natural symbiosis of number theory and analysis is centuries old but keeps enjoying new results, ideas and methods.The present book takes a semi-systematic review of analytic achievements in number theory ranging from classical themes about primes, continued fractions, transcendence of π and resolution of Hilbert's seventh problem to some recent developments on the irrationality of the values of Riemann's zeta function, sizes of non-cyclotomic algebraic integers and applications of hypergeometric functions to integer congruences.Our principal goal is to present a variety of different analytic techniques that are used in number theory, at a reasonably accessible — almost popular — level, so that the materials from this book can suit for teaching a graduate course on the topic or for a self-study. Exercises included are of varying difficulty and of varying distribution within the book (some chapters get more than other); they not only help the reader to consolidate their understanding of the material but also suggest directions for further study and investigation. Furthermore, the end of each chapter features brief notes about relevant developments of the themes discussed.

People, Places, and Mathematics

Author : Thomas Ward
Publisher : Springer Nature
ISBN 13 : 3031390741
Total Pages : 353 pages
Book Rating : 4.0/5 (313 download)

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Book Synopsis People, Places, and Mathematics by : Thomas Ward

Download or read book People, Places, and Mathematics written by Thomas Ward and published by Springer Nature. This book was released on 2023-10-31 with total page 353 pages. Available in PDF, EPUB and Kindle. Book excerpt: This memoir chronicles the journey of an academic, tracing a path from primary school in Zambia to a career in higher education as a mathematician and educational leader. Set against the backdrop of the 20th century, the book explores how early influences and historical events shape an individual's life and professional trajectory. The author shares childhood experiences across three parts of Africa, providing an original perspective as a witness to the post-colonial period. Through personal reflections, the memoir delves into the emergence of ideas and collaborations in mathematics and how these shape career choices. It also offers candid observations on the major changes in British higher education since the 1980s. Intended for a general audience, this book provides a compelling read for anyone interested in the experience of becoming a mathematician, and higher education in general.

Neverending Fractions

Author : Jonathan Borwein
Publisher : Cambridge University Press
ISBN 13 : 0521186498
Total Pages : 223 pages
Book Rating : 4.5/5 (211 download)

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Book Synopsis Neverending Fractions by : Jonathan Borwein

Download or read book Neverending Fractions written by Jonathan Borwein and published by Cambridge University Press. This book was released on 2014-07-03 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introductory text covers a variety of applications to interest every reader, from researchers to amateur mathematicians.

Fractal Geometry and Stochastics II

Author : Christoph Bandt
Publisher : Springer Science & Business Media
ISBN 13 : 9783764362157
Total Pages : 308 pages
Book Rating : 4.3/5 (621 download)

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Book Synopsis Fractal Geometry and Stochastics II by : Christoph Bandt

Download or read book Fractal Geometry and Stochastics II written by Christoph Bandt and published by Springer Science & Business Media. This book was released on 2000-02-14 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: A collection of contributions by outstanding mathematicians, highlighting the principal directions of research on the combination of fractal geometry and stochastic methods. Clear expositions introduce the most recent results and problems on these subjects and give an overview of their historical development.

Pisot and Salem Numbers

Author : Marie J. Bertin
Publisher : Birkhäuser
ISBN 13 : 3034886322
Total Pages : 297 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Pisot and Salem Numbers by : Marie J. Bertin

Download or read book Pisot and Salem Numbers written by Marie J. Bertin and published by Birkhäuser. This book was released on 2012-12-06 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: the attention of The publication of Charles Pisot's thesis in 1938 brought to the mathematical community those marvelous numbers now known as the Pisot numbers (or the Pisot-Vijayaraghavan numbers). Although these numbers had been discovered earlier by A. Thue and then by G. H. Hardy, it was Pisot's result in that paper of 1938 that provided the link to harmonic analysis, as discovered by Raphael Salem and described in a series of papers in the 1940s. In one of these papers, Salem introduced the related class of numbers, now universally known as the Salem numbers. These two sets of algebraic numbers are distinguished by some striking arith metic properties that account for their appearance in many diverse areas of mathematics: harmonic analysis, ergodic theory, dynamical systems and alge braic groups. Until now, the best known and most accessible introduction to these num bers has been the beautiful little monograph of Salem, Algebraic Numbers and Fourier Analysis, first published in 1963. Since the publication of Salem's book, however, there has been much progress in the study of these numbers. Pisot had long expressed the desire to publish an up-to-date account of this work, but his death in 1984 left this task unfulfilled.

The Wonderful World of Relativity

Author : Andrew Steane
Publisher : Oxford University Press
ISBN 13 : 0199694613
Total Pages : 249 pages
Book Rating : 4.1/5 (996 download)

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Book Synopsis The Wonderful World of Relativity by : Andrew Steane

Download or read book The Wonderful World of Relativity written by Andrew Steane and published by Oxford University Press. This book was released on 2011-10-06 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a lively and visual introduction to Einstein's theory of relativity. It begins by introducing spacetime, in the familiar context of low velocities. It then shows how Einstein's theory forces us to understand time in a new way. Paradoxes and puzzles are introduced and resolved, and the book culminates in a thorough unfolding of the relation between mass and energy.

Rational Points on Varieties

Author : Bjorn Poonen
Publisher : American Mathematical Soc.
ISBN 13 : 1470437732
Total Pages : 358 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Rational Points on Varieties by : Bjorn Poonen

Download or read book Rational Points on Varieties written by Bjorn Poonen and published by American Mathematical Soc.. This book was released on 2017-12-13 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces. The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere.

Gustav Mahler's Symphonic Landscapes

Author : Thomas Peattie
Publisher : Cambridge University Press
ISBN 13 : 9781108456548
Total Pages : 0 pages
Book Rating : 4.4/5 (565 download)

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Book Synopsis Gustav Mahler's Symphonic Landscapes by : Thomas Peattie

Download or read book Gustav Mahler's Symphonic Landscapes written by Thomas Peattie and published by Cambridge University Press. This book was released on 2018-04-26 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The relationship between Gustav Mahler's career as conductor and his symphonic writing has remained largely unexplored territory with respect to his provocative re-invention of the Austro-German symphony at the turn of the twentieth century. This study offers a new account of these works by allowing Mahler's decisive contribution to the genre to emerge in light of his sustained engagement with the musical, theatrical, and aesthetic traditions of the Austrian fin de siècle. Appealing to ideas of landscape, mobility, and theatricality, Thomas Peattie elaborates a richly interdisciplinary framework that draws attention to the composer's unique symphonic idiom in terms of its radical attitude toward the presentation and ordering of musical events. The identification of a fundamental tension between the music's episodic nature and its often-noted narrative impulse in turn suggests a highly original symphonic dramaturgy, one that is ultimately characterized by an abstract theatricality.