Number theory Books

Book Synopsis Diese Einführung besticht durch zwei ungewöhnliche Aspekte: Sie gibt einen Einblick in die Mathematik als Bestandteil unserer Kultur, und sie vermittelt die Hintergründe der Mathematik vom Schulstoff ausgehend bis zum Niveau von Mathematikvorlesungen im ersten Studienjahr. Die Stoffdarstellung geht vom Aufbau der natürlichen Zahlen aus; der Schwerpunkt liegt aber in den exakten Begründungen der Zahlenbegriffe, der Geometrie der Ebene und der Funktionen einer Veränderlichen. Dabei werden alle Sätze bis hin zum Hauptsatz der Algebra vollständig bewiesen. Der klare Aufbau des Buches mit Stichwortregister wichtiger Begriffe erleichtert das systematische Lernen und Nachschlagen. Die zweite Auflage enthält teilweise ausführliche Darstellungen für die Lösungen der zahlreichen Übungsaufgaben.Da viele Aspekte zur Sprache kommen, die so weder im Unterricht noch im Studium behandelt werden, ergänzt die Einführung ideal den Vorlesungsstoff für Lehramtskandidaten und Diplomstudenten.Trade Review"...dies ist eine Art "Brückenkurs"', der Aspekte der Schulmathematik von höherer Warte aus diskutiert... Der Autor steckt sich im Vorwort selbst das ehrgeizige Ziel, einen ‚Einblick in die Mathematik als einen Bestandteil unserer Kultur‘ zu geben, indem er sich ‚am Schulstoff (zwar) orientiert, aber über diesen hinausgeht und ihn hinterfragt.‘ Die Erreichbarkeit dieses Zieles stellt er mit diesem schönen Buch sehr überzeugend unter Beweis. Dabei wird beileibe nicht der Schulstoff ‚formalisiert‘, und noch weniger der Universitätsstoff ‚trivialisiert‘, sondern es kommen Aspekte zur Sprache, die im Mathematikunterricht wegen ihrer Schwierigkeit und im Mathematikstudium aus Zeitgründen kaum zur Sprache kommen. Dies ist ebenso verdienstvoll wie ungewöhnlich; als Ergebnis ist ein Buch herausgekommen, welches im ausufernden Markt tatsächlich eine Lücke füllt. Man kann grob drei Stoffgebiete unterscheiden, die behandelt werden, nämlich Zahlen (Kapitel 1-4 und 9), Geometrie (Kapitel 5 und 10) und Reelle Analysis (Kapitel 6-8). Wie ernst der Autor seine Aufgabe genommen hat, zeigt die sehr lesenswerte Einleitung, die auch den formalen Aufbau und inhaltliche Einzelheiten erklärt. Man kann allen Erstsemesterstudenten der Mathematik und Physik wärmstens empfehlen, dieses Buch als Ergänzung zu der von ihrem Dozenten empfohlenen Literatur zu kaufen und regelmäßig zu konsultieren." Jürgen Appell, Würzburg, in Zentralblatt MATH Table of ContentsNatürliche Zahlen.- Die 0 und die ganzen Zahlen.- Rationale Zahlen.- Reelle Zahlen.- Euklidische Geometrie der Ebene.- Reelle Funktionen einer Veränderlichen.- Maß und Integral.- Trigonometrie.- Die komplexen Zahlen.- Nicht-euklidische Geometrie.- Lösungen der Aufgaben.

Read more less

Book SynopsisThis work provides an introduction to four central problems in analytic number theory. These are (1) the problems of estimating the number of integer points in planar domains, (2) the problem of the distribution of prime numbers in the sequence of all natural numbers and in arithmetic progressions, (3) Goldbach's problems on sums of primes, and (4) Waring's problem on sums of k-th powers. The following fundamental methods of analytic number theory are used to solve these problems: complex integration, I.M. Vinogradov's method of trigonometric sums, and the circle method of G.H. Hardy, J.E. Littlewood, and S. Ramanujan. There are numerous exercises at the end of each chapter. These exercises either refine the theorems proved in the text, or lead to new ideas in number theory. The author also includes a section of hints for the solution of the exercises.

Read more less

Book SynopsisThis volume contains three long lecture series by J.L. Colliot-Thelene, Kazuya Kato and P. Vojta. Their topics are respectively the connection between algebraic K-theory and the torsion algebraic cycles on an algebraic variety, a new approach to Iwasawa theory for Hasse-Weil L-function, and the applications of arithemetic geometry to Diophantine approximation. They contain many new results at a very advanced level, but also surveys of the state of the art on the subject with complete, detailed profs and a lot of background. Hence they can be useful to readers with very different background and experience. CONTENTS: J.L. Colliot-Thelene: Cycles algebriques de torsion et K-theorie algebrique.- K. Kato: Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions.- P. Vojta: Applications of arithmetic algebraic geometry to diophantine approximations.Table of ContentsCycles algébriques de torsion et K-théorie algébrique Cours au C.I.M.E., juin 1991.- Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR. Part I.- Applications of arithmetic algebraic geometry to diophantine approximations.- Arithmetic algebraic geometry, Trento, Italy 1991.

Read more less

Book SynopsisFrom the reviews: "A well-written, very thorough account ... Among the topics are lattices, reduction, Minkowskis Theorem, distance functions, packings, and automorphs; some applications to number theory; excellent bibliographical references." The American Mathematical MonthlyTrade ReviewFrom the reviews:"The work is carefully written. It is well motivated, and interesting to read, even if it is not always easy... historical material is included... the author has written excellent account of an interesting subject." -Mathematical Gazette"A well-written, very thorough account ... Among the topi are lattices, reduction, Minkowskis Theorem, distance functions, packings, and automorphs; some applications to number theory; excellent bibliographical references." -The American Mathematical Monthly“It is very clearly written, and assumes little in the way of prerequisites. In particular, it is accessible to an undergraduate who is willing to work a bit, and I speak from experience as I first read the book the summer before I started graduate school. At the same time, it is a serious work giving an exhaustive (and not at all watered down) account of Minkowski’s theory. … This book certainly earns its place in a series on the ‘Classics in Mathematics.’” (Darren Glass, The Mathematical Association of America, January, 2011)Table of ContentsNotation Prologue Chapter I. Lattices 1. Introduction 2. Bases and sublattices 3. Lattices under linear transformation 4. Forms and lattices 5. The polar lattice Chapter II. Reduction 1. Introduction 2. The basic process 3. Definite quadratic forms 4. Indefinite quadratic forms 5. Binary cubic forms 6. Other forms Chapter III. Theorems of Blichfeldt and Minkowski 1. Introduction 2. Blichfeldt's and Mnowski's theorems 3. Generalisations to non-negative functions 4. Characterisation of lattices 5. Lattice constants 6. A method of Mordell 7. Representation of integers by quadratic forms Chapter IV. Distance functions 1. Introduction 2. General distance-functions 3. Convex sets 4. Distance functions and lattices Chapter V. Mahler's compactness theorem 1. Introduction 2. Linear transformations 3. Convergence of lattices 4. Compactness for lattices 5. Critical lattices 6. Bounded star-bodies 7. Reducibility 8. Convex bodies 9. Speres 10. Applications to diophantine approximation Chapter VI. The theorem of Minkowski-Hlawka 1. Introduction 2. Sublattices of prime index 3. The Minkowski-Hlawka theorem 4. Schmidt's theorems 5. A conjecture of Rogers 6. Unbounded star-bodies Chapter VII. The quotient space 1. Introduction 2. General properties 3. The sum theorem Chapter VIII. Successive minima 1. Introduction 2. Spheres 3. General distance-functions Chapter IX. Packings 1. Introduction 2. Sets with V(/varphi) =n^2/Delta(/varphi) 3. Voronoi's results 4. Preparatory lemmas 5. Fejes Tóth's theorem 6. Cylinders 7. Packing of spheres 8. The proudctio of n linear forms Chapter X. Automorphs 1. Introduction 2. Special forms 3. A method of Mordell 4. Existence of automorphs 5. Isolation theorems 6. Applications of isolation 7. An infinity of solutions 8. Local methods Chapter XI. Ihomogeneous problems 1. Introduction 2. Convex sets 3. Transference theorems for convex sets 4. The producti of n linear forms Appendix References Index quotient space. successive minima. Packings. Automorphs. Inhomogeneous problems.

Read more less

Book SynopsisIt is gratifying that this textbook is still sufficiently popular to warrant a third edition. I have used the opportunity to improve and enlarge the book. When the second edition was prepared, only two pages on algebraic geometry codes were added. These have now been removed and replaced by a relatively long chapter on this subject. Although it is still only an introduction, the chapter requires more mathematical background of the reader than the remainder of this book. One of the very interesting recent developments concerns binary codes defined by using codes over the alphabet 7l.4• There is so much interest in this area that a chapter on the essentials was added. Knowledge of this chapter will allow the reader to study recent literature on 7l. -codes. 4 Furthermore, some material has been added that appeared in my Springer Lec­ ture Notes 201, but was not included in earlier editions of this book, e. g. Generalized Reed-Solomon Codes and Generalized Reed-Muller Codes. In Chapter 2, a section on "Coding Gain" ( the engineer's justification for using error-correcting codes) was added. For the author, preparing this third edition was a most welcome return to mathematics after seven years of administration. For valuable discussions on the new material, I thank C.P.l.M.Baggen, I. M.Duursma, H.D.L.Hollmann, H. C. A. van Tilborg, and R. M. Wilson. A special word of thanks to R. A. Pellikaan for his assistance with Chapter 10.Table of Contents1 Mathematical Background.- 1.1. Algebra.- 1.2. Krawtchouk Polynomials.- 1.3. Combinatorial Theory.- 1.4. Probability Theory.- 2 Shannon’s Theorem.- 2.1. Introduction.- 2.2. Shannon’s Theorem.- 2.3. On Coding Gain.- 2.4. Comments.- 2.5. Problems.- 3 Linear Codes.- 3.1. Block Codes.- 3.2. Linear Codes.- 3.3. Hamming Codes.- 3.4. Majority Logic Decoding.- 3.5. Weight Enumerators.- 3.6. The Lee Metric.- 3.7. Comments.- 3.8. Problems.- 4 Some Good Codes.- 4.1. Hadamard Codes and Generalizations.- 4.2. The Binary Golay Code.- 4.3. The Ternary Golay Code.- 4.4. Constructing Codes from Other Codes.- 4.5. Reed—Muller Codes.- 4.6. Kerdock Codes.- 4.7. Comments.- 4.8. Problems.- 5 Bounds on Codes.- 5.1. Introduction: The Gilbert Bound.- 5.2. Upper Bounds.- 5.3. The Linear Programming Bound.- 5.4. Comments.- 5.5. Problems.- 6 Cyclic Codes.- 6.1. Definitions.- 6.2. Generator Matrix and Check Polynomial.- 6.3. Zeros of a Cyclic Code.- 6.4. The Idempotent of a Cyclic Code.- 6.5. Other Representations of Cyclic Codes.- 6.6. BCH Codes.- 6.7. Decoding BCH Codes.- 6.8. Reed—Solomon Codes.- 6.9. Quadratic Residue Codes.- 6.10. Binary Cyclic Codes of Length 2n(n odd).- 6.11. Generalized Reed—Muller Codes.- 6.12. Comments.- 6.13. Problems.- 7 Perfect Codes and Uniformly Packed Codes.- 7.1. Lloyd’s Theorem.- 7.2. The Characteristic Polynomial of a Code.- 7.3. Uniformly Packed Codes.- 7.4. Examples of Uniformly Packed Codes.- 7.5. Nonexistence Theorems.- 7.6. Comments.- 7.7. Problems.- 8 Codes over ?4.- 8.1. Quaternary Codes.- 8.2. Binary Codes Derived from Codes over ?4.- 8.3. Galois Rings over ?4.- 8.4. Cyclic Codes over ?4.- 8.5. Problems.- 9 Goppa Codes.- 9.1. Motivation.- 9.2. Goppa Codes.- 9.3. The Minimum Distance of Goppa Codes.- 9.4. Asymptotic Behaviour of Goppa Codes.- 9.5. Decoding Goppa Codes.- 9.6. Generalized BCH Codes.- 9.7. Comments.- 9.8. Problems.- 10 Algebraic Geometry Codes.- 10.1. Introduction.- 10.2. Algebraic Curves.- 10.3. Divisors.- 10.4. Differentials on a Curve.- 10.5. The Riemann—Roch Theorem.- 10.6. Codes from Algebraic Curves.- 10.7. Some Geometric Codes.- 10.8. Improvement of the Gilbert—Varshamov Bound.- 10.9. Comments.- 10.10.Problems.- 11 Asymptotically Good Algebraic Codes.- 11.1. A Simple Nonconstructive Example.- 11.2. Justesen Codes.- 11.3. Comments.- 11.4. Problems.- 12 Arithmetic Codes.- 12.1. AN Codes.- 12.2. The Arithmetic and Modular Weight.- 12.3. Mandelbaum—Barrows Codes.- 12.4. Comments.- 12.5. Problems.- 13 Convolutional Codes.- 13.1. Introduction.- 13.2. Decoding of Convolutional Codes.- 13.3. An Analog of the Gilbert Bound for Some Convolutional Codes.- 13.4. Construction of Convolutional Codes from Cyclic Block Codes.- 13.5. Automorphisms of Convolutional Codes.- 13.6. Comments.- 13.7. Problems.- Hints and Solutions to Problems.- References.

Read more less

Book SynopsisThis edition has been called ‘startlingly up-to-date’, and in this corrected second printing you can be sure that it’s even more contemporaneous. It surveys from a unified point of view both the modern state and the trends of continuing development in various branches of number theory. Illuminated by elementary problems, the central ideas of modern theories are laid bare. Some topics covered include non-Abelian generalizations of class field theory, recursive computability and Diophantine equations, zeta- and L-functions. This substantially revised and expanded new edition contains several new sections, such as Wiles' proof of Fermat's Last Theorem, and relevant techniques coming from a synthesis of various theories.Trade ReviewFrom the reviews of the second edition: "Here is a welcome update to Number theory I. Introduction to number theory by the same authors … . the book now brings the reader up to date with some of the latest results in the field. … The book is generally well-written and should be of interest to both the general, non-specialist reader of Number Theory as well as established researchers who are seeking an overview of some of the latest developments in the field." Philip Maynard, The Mathematical Gazette, Vol. 90 (519), 2006 [...] the first edition was a very good book; this edition is even better. [...] Embedded in the text are a lot of interesting ideas, insights, and clues to how the authors think about the subject. [...] Things get more interesting in Part II (by far the largest of the tree parts)[...] This part of the book covers such things as approaches through logic, algebraic number theory, arithmetic of algebraic varieties, zeta functions, and modular forms, followed by an extensive (50+ pages ) account of Wiles' proof of Fermat's Last Theorem. This is a valuable addition, new in this edition, and serves as a vivid example of the power of the "ideas and theories" that dominate this part of the book. Also new and very interesting is Part III, entitled "Analogies and Visions," [...] The best surveys of mathematics are those written by deeply insightful mathematicians who are not afraid to infuse their ideas and insights into their outline of subject. This is what we have here, and the result is an essential book. I only wish the price were lower so that I could encourage my students buy themselves a copy. Maybe I'll do that anyway. Fernado Q. Gouvêa, on 09/10/2005 "This book is a revised and updated version of the first English translation. … Overall, the book is very well written, and has an impressive reference list. It is an excellent resource for those who are looking for both deep and wide understanding of number theory." (Alexander A. Borisov, Mathematical Reviews, Issue 2006 j) "This edition feels altogether different from the earlier one … . There is much new and more in this edition than in the 1995 edition: namely, one hundred and fifty extra pages. … For my part, I come to praise this fine volume. This book is a highly instructive read with the usual reminder that there lots of facts one does not know … . the quality, knowledge, and expertise of the authors shines through. … The present volume is almost startlingly up-to-date … ." (Alf van der Poorten, Gazette of the Australian Mathematical Society, Vol. 34 (1), 2007)Table of ContentsProblems and Tricks.- Number Theory.- Some Applications of Elementary Number Theory.- Ideas and Theories.- Induction and Recursion.- Arithmetic of algebraic numbers.- Arithmetic of algebraic varieties.- Zeta Functions and Modular Forms.- Fermat’s Last Theorem and Families of Modular Forms.- Analogies and Visions.- Introductory survey to part III: motivations and description.- Arakelov Geometry and Noncommutative Geometry (d’après C. Consani and M. Marcolli, [CM]).

Read more less

Book SynopsisThe Local Langlands Conjecture for GL(2) contributes an unprecedented text to the so-called Langlands theory. It is an ambitious research program of already 40 years and gives a complete and self-contained proof of the Langlands conjecture in the case n=2. It is aimed at graduate students and at researchers in related fields. It presupposes no special knowledge beyond the beginnings of the representation theory of finite groups and the structure theory of local fields.Trade ReviewFrom the reviews:"In this book the authors present a complete proof of the Langlands conjecture for GL (2) over a non-archimedean local field, which uses local methods and is accessible to students. … The book is very well written and easy to read." (J. G. M. Mars, Zentralblatt MATH, Vol. 1100 (2), 2007)"The book under review gives a complete and self-contained insight into the theory of representations of G. … We highly recommend this book to Ph.D. students as well as to specialists. The book contains a huge amount of information, definition and facts … . The book has a Bibliography containing 91 references … ." (Alexandru Ioan Badulescu, Mathematical Reviews, Issue 2007 m)“The aim of this monograph is to present a complete and self-contained proof of the Langlands conjecture for GL(2) over a non-archimedean local field. … This volume presents a large amount of difficult material in a clear and readable manner. It can be recommended to anyone interested in representations of linear algebraic groups.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 154 (4), August, 2008)Table of ContentsSmooth Representations.- Finite Fields.- Induced Representations of Linear Groups.- Cuspidal Representations.- Parametrization of Tame Cuspidals.- Functional Equation.- Representations of Weil Groups.- The Langlands Correspondence.- The Weil Representation.- Arithmetic of Dyadic Fields.- Ordinary Representations.- The Dyadic Langlands Correspondence.- The Jacquet-Langlands Correspondence.

Read more less

Book SynopsisThis book surveys the current state of the "small" sieve methods developed by Brun, Selberg and later workers. The book is suitable for university graduates making their first acquaintance with the subject, leading them towards the frontiers of modern research and unsolved problems in the subject area.Trade ReviewFrom the reviews of the first edition: "The author presents a self-contained account of the small sieve. … This well-written book will become my primary source for the small sieve … . I recommend it to everybody who is interested in the technically complicated theory on sieve methods." (R. Tijdeman, Nieuw Archief voor Wiskunde, Vol. 4 (3), 2003) "The author’s choice of subjects provides a good background in the basic ideas of the sieve … . This text also supplies excellent background for some of the important unsolved problems of the subject. … In conclusion, the reviewer recommends this book strongly to students of sieve methods in the opening years of the twenty-first century. It will likely become one of the standard references on the subject." (Sidney W. Graham, Zentralblatt MATH, Vol. 1003 (03), 2003) "The book being reviewed is an excellent survey on sieve methods. … The book is well written indeed, and most of the material can be described as self-contained. It can therefore be read by university graduates making their first acquaintance with the subject … ." (P. Shiu, The Mathematical Gazette, Vol. 86 (507), 2002)Table of Contents1. The Structure of Sifting Arguments.- 2. Selberg’s Upper Bound Method.- 3. Combinatorial Methods.- 4. Rosser’s Sieve.- 5. The Sieve with Weights.- 6. The Remainder Term in the Linear Sieve.- 7. Lower Bound Sieves when ? > 1.- References.

Read more less

Book SynopsisThis book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory.Trade Review"...In the breadth, depth and inevitability of treatment of this beautiful material, the author has made a contribution to the mathematical community consistent with the distinction of his career. That he has succeeded in compressing this treatment into a succinct monograph of fewer than 190 pages is a testament to his taste, discipline and powers of exposition."-- MATHEMATICAL REVIEWSTable of ContentsI. Periods of meromorphic functions.- § 1. Meromorphic functions.- § 2. Periodic meromorphic functions.- § 3. Jacobi’s lemma.- § 4. Elliptic functions.- § 5. The modular group and modular functions.- Notes on Chapter I.- II. General properties of elliptic functions.- §1. The period parallelogram.- § 2. Elementary properties of elliptic functions.- Notes on Chapter II.- III. Weierstrass’s elliptic function ?(z).- §1. The convergence of a double series.- § 2. The elliptic function ?(z).- § 3. The differential equation associated with ?(z).- § 4. The addition-theorem.- § 5. The generation of elliptic functions.- Appendix I. The cubic equation.- Appendix II. The biquadratic equation.- Notes on Chapter III.- IV. The zeta-function and the sigma-function of Weierstrass.- § 1. The function ?(z).- §2. The function ?(z).- § 3. An expression for elliptic functions.- Notes on Chapter IV.- V. The theta-functions.- §1. The function ?(?, ?).- § 2. The four sigma-functions.- § 3. The four theta-functions.- § 4. The differential equation.- § 5. Jacobi’s formula for ?’ (0, ?).- § 6. The infinite products for the theta-functions.- § 7. Theta-functions as solutions of functional equations.- § 8. The transformation formula connecting ?3(v, ?) and ?3(?, ?1/?) ..- Notes on Chapter V.- VI. The modular function J(?).- § 1. Definition of J(?).- § 2. The functions g2(?) and g3(?).- § 3. Expansion of the function J(?) and the connexion with theta-functions.- § 4. The function J(?) in a fundamental domain of the modular group ..- § 5. Relations between the periods and the invariants of ?(u).- § 6. Elliptic integrals of the first kind.- Notes on Chapter VI.- VII. The Jacobian elliptic functions and the modular function ?(?).- § 1. The functions sn u, en u, dn u of Jacobi.- § 2. Definition by theta-functions.- § 3. Connexion with the sigma-functions.- § 4. The differential equation.- § 5. Infinite products for the Jacobian elliptic functions.- § 6. Addition-theorems for sn u, cn u, dn u.- § 7. The modular function ?(?).- §8. Mapping properties of ?(?) and Picard’s theorem.- Notes on Chapter VII.- VIII. Dedekind’s ?-function and Euler’s theorem on pentagonal numbers.- § 1. Connexion with the invariants of the ?-function and with the theta-functions.- § 2. Euler’s theorem and Jacobi’s proof.- § 3. The transformation formula connecting ?(z) and ?(?½).- §4. Siegel’s proof of Theorem 1.- §5. Connexion between ?(z) and the modular functions J(z), ?(z).- Notes on Chapter VIII.- IX. The law of quadratic reciprocity.- § 1. Reciprocity of generalized Gaussian sums.- § 2. Quadratic residues.- §3. The law of quadratic reciprocity.- Notes on Chapter IX.- X. The representation of a number as a sum of four squares ..- §1. The theorems of Lagrange and of Jacobi.- § 2. Proof of Jacobi’s theorem by means of theta-functions.- §3. Siegel’s proof of Jacobi’s theorem.- Notes on Chapter X.- XI. The representation of a number by a quadratic form.- §1. Positive-definite quadratic forms.- § 2. Multiple theta-series and quadratic forms.- § 3. Theta-functions associated to positive-definite forms.- § 4. Representation of an even integer by a positive-definite form.- Notes on Chapter XI.- Chronological table.

Read more less

Book Synopsis

Read more less

Book SynopsisTable of Contents1. Matrices 2. Simultaneous linear equations 3. The inverse 4. An introduction to optimization 5. Determinants 6. Eigenvalues and eigenvectors 7. Matrix calculus 8. Linear differential equations 9. Probability and Markov chains 10. Real inner products and least square 11. Sabermetrics e An introduction 12. Sabermetrics e A module Appendix: A word on technology Answers and hints to selected problems

Read more less

Book SynopsisNumbers and politics are inter-related at almost every level--be it the abstract geometry of understandings of territory, the explosion of population statistics and measures of economic standards, the popularity of Utilitarianism, Rawlsian notions of justice, the notion of value, or simply the very idea of political science. Time and space are reduced to co-ordinates, illustrating a very real take on the political: a way of measuring and controlling it.This book engages with the relation between politics and number through a reading, exegesis and critique of the work of Martin Heidegger. The importance of mathematics and the role played by the understandings of calculation is a recurrent concern in his writing and is regularly contrasted with understandings of speech and language. This book provides the most detailed analysis of the relation between language, politics and mathematics in Heidegger''s work. It insists that questions of language and calculation in Heidegger are inherently political, and that a far broader range of his work is concerned with politics than is usually admitted.Trade ReviewElden should be applauded for writing with such sharp focus, while simultaneously never reducing the genuine complexity of Heidegger's thought. Contemporary Political Theory Elden is a careful scholar, who writes in a clear, accessible prose. He has identified all the important texts germane to his argument and provides a good rationale to the volume as proposed. -- Dr Laurence Hemming, Heythrop College, University of London I wholeheartedly recommend this book with its rich lode of expositions of Heidegger's texts on the political in its ancient, modern and postmodern manifestations. -- Professor Theodore Kisiel, Northern Illinois University Stuart Elden's Speaking Against Number takes full advantage of the most recent volumes of Heidegger's previously unpublished lectures and manuscripts to develop a rich new approach to his political thought. The resulting book should be widely read, especially by everyone who thinks they already know all there is to know about this topic. -- Professor Robert Bernasconi, University of Memphis This volume shows wide-ranging and sound scholarship. Elden has done a superior job of weaving together many important strands of Heidegger's thought. -- Richard Polt Continental Philosophy Review Elden's book manages to reinvigorate a seemingly tired debate regarding Heidegger's political engagement. This is a unique achievement in that he succeeds in re-opening a question that continues to haunt readers of Heidegger: to what extent can we separate the man from his thought? -- Paul Ennis, UCD Borderlands e-journal An importantly original contribution to the question of Heidegger and the political. -- Babette E. Babich, Fordham University, New York Political Theory Elden should be applauded for writing with such sharp focus, while simultaneously never reducing the genuine complexity of Heidegger's thought. Elden is a careful scholar, who writes in a clear, accessible prose. He has identified all the important texts germane to his argument and provides a good rationale to the volume as proposed. I wholeheartedly recommend this book with its rich lode of expositions of Heidegger's texts on the political in its ancient, modern and postmodern manifestations. Stuart Elden's Speaking Against Number takes full advantage of the most recent volumes of Heidegger's previously unpublished lectures and manuscripts to develop a rich new approach to his political thought. The resulting book should be widely read, especially by everyone who thinks they already know all there is to know about this topic. This volume shows wide-ranging and sound scholarship. Elden has done a superior job of weaving together many important strands of Heidegger's thought. Elden's book manages to reinvigorate a seemingly tired debate regarding Heidegger's political engagement. This is a unique achievement in that he succeeds in re-opening a question that continues to haunt readers of Heidegger: to what extent can we separate the man from his thought? An importantly original contribution to the question of Heidegger and the political.Table of ContentsIntroduction; 1. Speaking: Rhetorical Politics; 2. Against: Polemical Politics; 3. Number: Calculative Politics; Conclusion: Taking the Measure of the Political.

Read more less

Book SynopsisMotivation.- Ergodicity, Recurrence and Mixing.- Continued Fractions.- Invariant Measures for Continuous Maps.- Conditional Measures and Algebras.- Factors and Joinings.- Furstenberg's Proof of Szemeredi's Theorem.- Actions of Locally Compact Groups.- Geodesic Flow on Quotients of the Hyperbolic Plane.- Nilrotation.- More Dynamics on Quotients of the Hyperbolic Plane.- Appendix A: Measure Theory.- Appendix B: Functional Analysis.- Appendix C: Topological GroupsTrade ReviewFrom the reviews:“The book is an introduction to ergodic theory and dynamical systems. … The book is intended for graduate students and researchers with some background in measure theory and functional analysis. Definitely, it is a book of great interest for researchers in ergodic theory, homogeneous dynamics or number theory.” (Antonio Díaz-Cano Ocaña, The European Mathematical Society, January, 2014)“A book with a wider perspective on ergodic theory, and yet with a focus on the interaction with number theory, remained a glaring need in the overall context of the development of the subject. … The book under review goes a long way in fulfilling this need. … it covers a good deal of conventional ground in ergodic theory … . a very welcome addition and would no doubt inspire interest in the area among researchers as well as students, and cater to it successfully.” (S. G. Dani, Ergodic Theory and Dynamical Systems, Vol. 32 (3), June, 2012)“The book under review is an introductory textbook on ergodic theory, written with applications to number theory in mind. … it aims both to provide the reader with a solid comprehensive background in the main results of ergodic theory, and of reaching nontrivial applications to number theory. … The book should also be very appealing to more advanced readers already conducting research in representation theory or number theory, who are interested in understanding the basis of the recent interaction with ergodic theory.” (Barak Weiss, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 114, 2012)“This introductory book, which goes beyond the standard texts and allows the reader to get a glimpse of modern developments, is a timely and welcome addition to the existing and ever-growing ergodic literature. … This book is highly recommended to graduate students and indeed to anyone who is interested in acquiring a better understanding of contemporary developments in mathematics.” (Vitaly Bergelson, Mathematical Reviews, Issue 2012 d)“The book contains a presentation of the ergodic theory field, focusing mainly on results applicable to number theory. … of interest for researchers, specialists, professors and students that work within some other areas than precisely the ergodic theory. … ‘Ergodic Theory. With a view toward number theory’ is now an indispensable reference in the domain and offers important instruments of research for other theoretical fields.” (Adrian Atanasiu, Zentralblatt MATH, Vol. 1206, 2011)Table of ContentsMotivation.- Ergodicity, Recurrence and Mixing.- Continued Fractions.- Invariant Measures for Continuous Maps.- Conditional Measures and Algebras.- Factors and Joinings.- Furstenberg’s Proof of Szemeredi’s Theorem.- Actions of Locally Compact Groups.- Geodesic Flow on Quotients of the Hyperbolic Plane.- Nilrotation.- More Dynamics on Quotients of the Hyperbolic Plane.- Appendix A: Measure Theory.- Appendix B: Functional Analysis.- Appendix C: Topological Groups

Read more less

Book SynopsisFrom Fontaine-Mazur Conjecture to Analytic Pro-p-Groups: A Survey (Abdellatif).- Orientations and Cycles in Supersingular Isogeny Graphs (Stange).- Generalized Ramanujan-Sato Series Arising from Modular Forms (Swisher).- Mock Theta Functions and Related Combinatorics (Ballantine).- Transcendental Lattices of Certain Singular K3 Surfaces(Bertin).- Power-Saving Error Terms for the Number of D4-Quartic Extensions over a Number Field Ordered by Discriminant (Lopez).- Dynamical Mahler Measure: a Survey and Some Recent Results (Lalin).- Geometric Decomposition of Abelian Varieties of Order 1 (Kedlaya).- On Marko  Type Surfaces over Number Fields and the Arithmetic of Marko  Numbers (Sivaraman).- p-Adic Measures for Reciprocals of L-Functions of Totally Real Number Fields (Taha).

Read more less

Book Synopsis1 Integers.- 2 Congruences and Residue Class Rings.- 3 Encryption.- 4 Probability and Perfect Secrecy.- 5 DES.- 6 AES.- 7 Prime Number Generation.- 8 Public-Key Encryption.- 9 Factoring.- 10 Discrete Logarithms.- 11 Cryptographic Hash Functions.- 12 Digital Signatures.- 13 Other Systems.- 14 Identification.- 15 Secret Sharing.- 16 Public-Key Infrastructures.- Solutions of the exercises.- References.Trade ReviewFrom the reviews: Zentralblatt Math "[......] Of the three books under review, Buchmann's is by far the most sophisticated, complete and up-to-date. It was written for computer-science majors - German ones at that - and might be rough going for all but the best American undergraduates. It is amazing how much Buchmann is able to do in under 300 pages: self-contained explanations of the relevant mathematics (with proofs); a systematic introduction to symmetric cryptosystems, including a detailed description and discussion of DES; a good treatment of primality testing, integer factorization, and algorithms for discrete logarithms, clearly written sections describing most of the major types of cryptosystems, and explanations of basic concepts of practical cryptography such as hash functions, message authentication codes, signatures, passwords, certification authorities, and certificate chains. This book is an excellent reference, and I believe that it would also be a good textbook for a course for mathematics or computer science majors, provided that the instructor is prepared to supplement it with more leisurely treatments of some of the topics." N. Koblitz (Seattle, WA) - American Math. Society Monthly. J.A. Buchmann Introduction to Cryptography "It gives a clear and systematic introduction into the subject whose popularity is ever increasing, and can be recommended to all who would like to learn about cryptography. The book contains many exercises and examples. It can be used as a textbook and is likely to become popular among students. The necessary definitions and concepts from algebra, number theory and probability theory are formulated, illustrated by examples and applied to cryptography." —ZENTRALBLATT MATH "For those of use who wish to learn more about cryptography and/or to teach it, Johannes Buchmann has written this book. … The book is mathematically complete and a satisfying read. There are plenty of homework exercises … . This is a good book for upperclassmen, graduate students, and faculty. … This book makes a superior reference and a fine textbook." (Robert W. Vallin, MathDL, January, 2001) "Buchmann’s book is a text on cryptography intended to be used at the undergraduate level. … the intended audiences of this book are ‘readers who want to learn about modern cryptographic algorithms and their mathematical foundations … . I enjoy reading this book. … Readers will find a good exposition of the techniques used in developing and analyzing these algorithms. … These make Buchmann’s text an excellent choice for self study or as a text for students … in elementary number theory and algebra." (Andrew C. Lee, SIGACT News, Vol. 34 (4), 2003) From the reviews of the second edition: "This is the english translation of the second edition of the author’s prominent german textbook ‘Einführung in die Kryptographie’. The original text grew out of several courses on cryptography given by the author at the Technical University Darmstadt; it is aimed at readers who want to learn about modern cryptographic techniques and its mathematical foundations … . As compared with the first edition the number of exercises has almost been doubled and some material … has been added." (R. Steinbauer, Monatshefte für Mathematik, Vol. 150 (4), 2007)Table of ContentsIntegers.- Congruences and Residue Class Rings.- Encryption.- Probability and Perfect Secrecy.- DES.- AES.- Prime Number Generation.- Public-Key Encryption.- Factoring.- Discrete Logarithms.- Cryptographic Hash Functions.- Digital Signatures.- Other Systems.- Identification.- Public-Key Infrastructures.- Solutions of the Odd Exercises.- Subject Index.- Bibliography.

Read more less

Book Synopsis1. Divisibility, the Fundamental Theorem of Number Theory.- 2. Congruences.- 3. Rational and Irrational Numbers. Approximation of Numbers by Rational Numbers (Diophantine Approximation).- 4. Geometric Methods in Number Theory.- 5. Properties of Prime Numbers.- 6. Sequences of Integers.- 7. Diophantine Problems.- 8. Arithmetic Functions.- Hints to the More Difficult Exercises.Trade ReviewFrom the reviews: "Read this book just for Erdös’s (Erdos’s) characteristic turn of thought, or for results hard to find elsewhere, such as a finiteness theorem concerning odd perfect numbers with a fixed number of factors. Summing Up: Recommended. Lower-division undergraduates through professionals." (D.V. Feldman, CHOICE, December, 2003) "This is an English translation of the second edition of a book originally published over 40 years ago … . The contents should be accessible to, and inspire and challenge, keen pre-university students as well as giving the experienced mathematician food for thought. The proofs are elementary and largely self-contained, and the problems and results well motivated. … This translation makes a very clearly and nicely written book available to many more readers who should benefit and gain much pleasure from studying it." (Eira J. Scourfield, Zentralblatt MATH, Issue 1018, 2003) "This rather unique book is a guided tour through number theory. While most introductions to number theory provide a systematic and exhaustive treatment of the subject, the authors have chosen instead to illustrate the many varied subjects by associating recent discoveries, interesting methods, and unsolved problems. … János Surányi’s vast teaching experience successfully complements Paul Erdös’s ability to initiate new directions of research by suggesting new problems and approaches." (L’Enseignement Mathematique, Vol. 49 (1-2), 2003) "This is a somewhat enlarged translation of the Hungarian book … . It goes without saying that the text is masterly written. It contains on comparatively few lines the fundamental ideas of not only elementary Number Theory: it contains also irrationality proofs ... . The book is hence by far not an n-th version of always the same matter. The style reminds me on the celebrated book of Pólya … . It is desirable that the book under discussion should have a similar success." (J. Schoissengeier, Monatshefte für Mathematik, Vol. 143 (2), 2004) "This an introduction to elementary number theory in which the authors present the main notions of that theory and ‘try to give glimpses into the deeper related mathematics’, as they write in the preface. There are 8 chapters … . Each of them brings not only the notions and theorems (sometimes with unconventional proofs) which usually appear in introductory texts, but discusses also topics found rarely … . One also finds several interesting historical comments." (W. Narkiewicz, Mathematical Reviews, 2003j)Table of Contents* Preface * Facts Used Without Proof in the Book * Divisibility, the Fundamental Theorem of Number Theory * Congruences * Rational and irrational numbers. Approximation of numbers by rational numbers. (Diophantine approximation.) * Geometric methods in number theory * Properties of prime numbers * Sequences of integers * Diophantine Problems * Arithmetic Functions * Hints to the more difficult exercises * Bibliography * Index

Read more less

Book SynopsisA Course in Computational Number Theory uses the computer as a tool for motivation and explanation. The book is designed for the reader to quickly access a computer and begin doing personal experiments with the patterns of the integers. It presents and explains many of the fastest algorithms for working with integers. Traditional topics are covered, but the text also explores factoring algorithms, primality testing, the RSA public-key cryptosystem, and unusual applications such as check digit schemes and a computation of the energy that holds a salt crystal together. Advanced topics include continued fractions, Pell's equation, and the Gaussian primes.Table of ContentsPreface. Notation. Chapter 1 Fundamentals. 1.0 Introduction. 1.1 A Famous Sequence of Numbers. 1.2 The Euclidean ALgorithm. The Oldest Algorithm. Reversing the Euclidean Algorithm. The Extended GCD Algorithm. The Fundamental Theorem of Arithmetic. Two Applications. 1.3 Modular Arithmetic. 1.4 Fast Powers. A Fast Alforithm for ExponentiationPowers of Matrices, Big-O Notation. Chapter 2 Congruences, Equations, and Powers. 2.0 Introduction. 2.1 Solving Linear Congruences. Linear Diophantine Equations in Two Variables. The Conductor. An Importatnt Quadratic Congruence. 2.2 The Chinese Remainder Theorem. 2.3 PowerMod Patterns. Fermat's Little Theorem. More Patterns in Powers. 2.4 Pseudoprimes. Using the Pseudoprime Test. Chapter 3 Euler's Function. 3.0 Introduction. 3.1 Euler's Function. 3.2 Perfect Numbers and Their Relatives. The Sum of Divisors Function. Perfect Numbers. Amicalbe, Abundant, and Deficient Numbers. 3.3 Euler's Theorem. 3.4 Primitive Roots for Primes. The order of an Integer. Primes Have PRimitive roots. Repeating Decimals. 3.5 Primitive Roots for COmposites. 3.6 The Universal Exponent. Universal Exponents. Power Towers. The Form of Carmichael Numbers. Chapter 4 Prime Numbers. 4.0 Introduction. 4.1 The Number of Primes. We'll Never Run Out of Primes. The Sieve of Eratosthenes. Chebyshev's Theorem and Bertrand's Postulate. 4.2 Prime Testing and Certification. Strong Pseudoprimes. Industrial-Grade Primes. Prime Certification Via Primitive Roots. An Improvement. Pratt Certificates. 4.3 Refinements and Other Directions. Other PRimality Tests. Strong Liars are Scarce. Finding the nth Prime. 4.4 A Doszen Prime Mysteries. Chapter 5 Some Applications. 5.0 Introduction. 5.1 Coding Secrets. Tossing a Coin into a Well. The RSA Cryptosystem. Digital Signatures. 5.2 The Yao Millionaire Problem. 5.3 Check Digits. Basic Check Digit Schemes. A Perfect Check Digit Method. Beyond Perfection: Correcting Errors. 5.4 Factoring Algorithms. Trial Division. Fermat's Algorithm. Pollard Rho. Pollard p-1. The Current Scene. Chapter 6 Quadratic Residues. 6.0 Introduction. 6.1 Pepin's Test. Quadratic Residues. Pepin's Test. Primes Congruent to 1 (Mod. 6.2 Proof of Quadratic Reciprocity. Gauss's Lemma. Proof of Quadratic Recipocity. Jacobi's Extension. An Application to Factoring. 6.3 Quadratic Equations. Chapter 7 Continuec Faction. 7.0 Introduction. 7.1 FInite COntinued Fractions. 7.2 Infinite Continued Fractions. 7.3 Periodic Continued Fractions. 7.4 Pell's Equation. 7.5 Archimedes and the Sun God's Cattle. Wurm's Version: Using Rectangular Bulls. The Real Cattle Problem. 7.6 Factoring via Continued Fractions. Chapter 8 Prime Testing with Lucas Sequences. 8.0 Introduction. 8.1 Divisibility Properties of Lucas Sequencese. 8.2 Prime Tests Using Lucas Sequencesse. Lucas Certification. The Lucas-Lehmer Algorithm Explained. Luca Pseudoprimes. Strong Quadratic Pseudoprimes. Primality Testing's Holy Grail. Chapter 9 Prime Imaginaries and Imaginary Primes. 9.0 Introduction. 9.1 Sums of Two Squares. 9.2 The Gaussian Intergers. Complex Number Theory. Gaussian Primes. The Moat Problem. The Gaussian Zoo. 9.3 Higher Reciprocity 325. Appendix A. Maathematica Basics. 1.0 Introduction. A.1 Plotting. A.2 Typesetting. Sending Files By E-Mail. A.3 Types of Functions. A.4 Lists. A.5 Programs. A.6 Solving Equations. A.7 Symbolic Algebra. Appendix B Lucas Certificates Exist. References. Index of Mathematica Objects. Subject Index.

Read more less

Book SynopsisNumber Theory: A Lively Introduction with Proofs, Applications, and Stories, is a new book that provides a rigorous yet accessible introduction to elementary number theory along with relevant applications. Readable discussions motivate new concepts and theorems before their formal definitions and statements are presented. Many theorems are preceded by Numerical Proof Previews, which are numerical examples that will help give students a concrete understanding of both the statements of the theorems and the ideas behind their proofs, before the statement and proof are formalized in more abstract terms. In addition, many applications of number theory are explained in detail throughout the text, including some that have rarely (if ever) appeared in textbooks. A unique feature of the book is that every chapter includes a math myth, a fictional story that introduces an important number theory topic in a friendly, inviting manner. Many of the exerciTable of ContentsPreface. To the Student. To the Instructor. Acknowledgements. 0. Prologue. 1. Numbers, Rational and Irrational. (Historical figures: Pythagoras and Hypatia). 1.1 Numbers and the Greeks. 1.2 Numbers you know. 1.3 A First Look at Proofs. 1.4 Irrationality of he square root of 2. 1.5 Using Quantifiers. 2. Mathematical Induction. (Historical figure: Noether). 2.1.The Principle of Mathematical Induction. 2.2 Strong Induction and the Well Ordering Principle. 2.3 The Fibonacci Sequence and the Golden Ratio. 2.4 The Legend of the Golden Ratio. 3. Divisibility and Primes. (Historical figure: Eratosthenes). 3.1 Basic Properties of Divisibility. 3.2 Prime and Composite Numbers. 3.3 Patterns in the Primes. 3.4 Common Divisors and Common Multiples. 3.5 The Division Theorem. 3.6 Applications of gcd and lcm. 4.The Euclidean Algorithm. (Historical figure: Euclid). 4.1 The Euclidean Algorithm. 4.2 Finding the Greatest Common Divisor. 4.3 A Greeker Argument that the square root of 2 is Irrational. 5. Linear Diophantine Equations. (Historical figure: Diophantus). 5.1 The Equation aX + bY = 1. 5.2 Using the Euclidean Algorithm to Find a Solution. 5.3 The Diophantine Equation aX + bY = n. 5.4 Finding All Solutions to a Linear Diophantine Equation. 6. The Fundamental Theorem of Arithmetic. (Historical figure: Mersenne). 6.1 The Fundamental Theorem. 6.2 Consequences of the Fundamental Theorem. 7. Modular Arithmetic. (Historical figure: Gauss). 7.1 Congruence modulo n. 7.2 Arithmetic with Congruences. 7.3 Check Digit Schemes. 7.4 The Chinese Remainder Theorem. 7.5 The Gregorian Calendar. 7.6 The Mayan Calendar. 8. Modular Number Systems. (Historical figure: Turing). 8.1 The Number System Zn: an Informal View. 8.2 The Number System Zn: Definition and Basic Properties. 8.3 Multiplicative Inverses in Zn. 8.4 Elementary Cryptography. 8.5 Encryption Using Modular Multiplication. 9. Exponents Modulo n. (Historical figure: Fermat). 9.1 Fermat's Little Theorem. 9.2 Reduced Residues and the Euler \phi-function. 9.3 Euler's Theorem. 9.4 Exponentiation Ciphers with a Prime modulus. 9.5 The RSA Encryption Algorithm. 10. Primitive Roots. (Historical figure: Lagrange). 10.1 Zn. 10.2 Solving Polynomial Equations in Zn. 10.3 Primitive Roots. 10.4 Applications of Primitive Roots. 11. Quadratic Residues. (Historical figure: Eisenstein) 11.1 Squares Modulo n 11.2 Euler's Identity and the Quadratic Character of -1 11.3 The Law of Quadratic Reciprocity 11.4 Gauss's Lemma 11.5 Quadratic Residues and Lattice Points. 11.6 The Proof of Quadratic Reciprocity. 12. Primality Testing. (Historical figure: Erdös). 12.1 Primality testing. 12.2 Continued Consideration of Charmichael Numbers. 12.3 The Miller-Rabin Primality test. 12.4 Two Special Polynomial Equations in Zp. 12.5 Proof that Millar-Rabin is Effective. 12.6 Prime Certificates. 12.7 The AKS Deterministic Primality Test. 13. Gaussian Integers. (Historical figure: Euler). 13.1 Definition of Gaussian Integers 13.2 Divisibility and Primes in Z[i]. 13.3 The Division Theorem for the Gaussian Integers. 13.4 Unique Factorization in Z[i]. 13.5 Gaussian Primes. 13.6 Fermat's Two Squares Theorem. 14. Continued Fractions. (Historical figure: Ramanujan). 14.1 Expressing Rational Numbers as Continued Fractions. 14.2 Expressing Irrational Numbers as Continued Fractions. 14.3 Approximating Irrational Numbers Using Continued Fractions. 14.4 Proving that Convergents are Fantastic Approximations. 15. Some Nonlinear Diophantine Equations. (Historical figure: Germain). 15.1 Pell's Equation 15.2 Fermat's Last Theorem 15.3 Proof of Fermat's Last Theorem for n = 4. 15.4 Germain's Contributions to Fermat's Last Theorem 15.5 A Geometric look at the Equation x4 + y4 = z2. Appendix: Axioms of Number Theory. A.1 What is a Number System? A.2 Order Properties of the Integers. A.3 Building Results From Our Axioms. A.4 The Principle of Mathematical Induction.

Read more less

Book SynopsisDevised in the 19th century, Gauss and Jacobi Sums are classical formulas that form the basis for contemporary research in many of today's sciences. This book offers readers a solid grounding on the origin of these abstract, general theories.Table of ContentsGauss Sums. Jacobi Sums and Cyclotomic Numbers. Evaluation of Jacobi Sums Over Fp. Determination of Gauss Sums Over Fp. Difference Sets. Jacobsthal Sums Over Fp. Residuacity. Reciprocity Laws. Congruences for Binomial Coefficients. Diagonal Equations over Finite Fields. Gauss Sums over Fq. Eisenstein Sums. Brewer Sums. A General Eisenstein Reciprocity Law. Research Problems. Bibliography. Notation. Indexes.

Read more less

Book SynopsisA complete, self-contained introduction to a powerful and resurging mathematical discipline. Combinatorial Geometry presents and explains with complete proofs some of the most important results and methods of this relatively young mathematical discipline, started by Minkowski, Fejes Toth, Rogers, and Erd???s.Table of ContentsARRANGEMENTS OF CONVEX SETS. Geometry of Numbers. Approximation of a Convex Set by Polygons. Packing and Covering with Congruent Convex Discs. Lattice Packing and Lattice Covering. The Method of Cell Decomposition. Methods of Blichfeldt and Rogers. Efficient Random Arrangements. Circle Packings and Planar Graphs. ARRANGEMENTS OF POINTS AND LINES. Extremal Graph Theory. Repeated Distances in Space. Arrangement of Lines. Applications of the Bounds on Incidences. More on Repeated Distances. Geometric Graphs. Epsilon Nets and Transversals of Hypergraphs. Geometric Discrepancy. Hints to Exercises. Bibliography. Indexes.

Read more less

Book SynopsisExplores fundamental concepts in arithmetic. This book begins with the definitions and properties of algebraic fields. It then discusses the theory of divisibility from an axiomatic viewpoint, rather than by the use of ideals. It also gives an introduction to p-adic numbers and their uses, which are important in modern number theory.Table of ContentsCh. I Algebraic Fields 1 Ch. II Theory of Divisibility (Kronecker, Dedekind) 33 Ch. III Local Primadic Analysis (Kummer, Hensel) 71 Ch. IV Algebraic Number Fields 141 Amendments 223

Read more less

Book SynopsisDevelops a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. This book formulates simple general conditions on the spectral theory of the group and the regularity of the averaging sets, which suffice to guarantee convergence to the ergodic mean.Table of ContentsPreface vii 0.1 Main objectives vii 0.2 Ergodic theory and amenable groups viii 0.3 Ergodic theory and nonamenable groups x Chapter 1. Main results: Semisimple Lie groups case 1 1.1 Admissible sets 1 1.2 Ergodic theorems on semisimple Lie groups 2 1.3 The lattice point-counting problem in admissible domains 4 1.4 Ergodic theorems for lattice subgroups 6 1.5 Scope of the method 8 Chapter 2. Examples and applications 11 2.1 Hyperbolic lattice points problem 11 2.2 Counting integral unimodular matrices 12 2.3 Integral equivalence of general forms 13 2.4 Lattice points in S-algebraic groups 15 2.5 Examples of ergodic theorems for lattice actions 16 Chapter 3. Definitions, preliminaries, and basic tools 19 3.1 Maximal and exponential-maximal inequalities 19 3.2 S-algebraic groups and upper local dimension 21 3.3 Admissible and coarsely admissible sets 21 3.4 Absolute continuity and examples of admissible averages 23 3.5 Balanced and well-balanced families on product groups 26 3.6 Roughly radial and quasi-uniform sets 27 3.7 Spectral gap and strong spectral gap 29 3.8 Finite-dimensional subrepresentations 30 Chapter 4. Main results and an overview of the proofs 33 4.1 Statement of ergodic theorems for S-algebraic groups 33 4.2 Ergodic theorems in the absence of a spectral gap: overview 35 4.3 Ergodic theorems in the presence of a spectral gap: overview 38 4.4 Statement of ergodic theorems for lattice subgroups 40 4.5 Ergodic theorems for lattice subgroups: overview 42 4.6 Volume regularity and volume asymptotics: overview 44 Chapter 5. Proof of ergodic theorems for S-algebraic groups 47 5.1 Iwasawa groups and spectral estimates 47 5.2 Ergodic theorems in the presence of a spectral gap 50 5.3 Ergodic theorems in the absence of a spectral gap, I 56 5.4 Ergodic theorems in the absence of a spectral gap, II 57 5.5 Ergodic theorems in the absence of a spectral gap, III 60 5.6 The invariance principle and stability of admissible averages 67 Chapter 6. Proof of ergodic theorems for lattice subgroups 71 6.1 Induced action 71 6.2 Reduction theorems 74 6.3 Strong maximal inequality 75 6.4 Mean ergodic theorem 78 6.5 Pointwise ergodic theorem 83 6.6 Exponential mean ergodic theorem 84 6.7 Exponential strong maximal inequality 87 6.8 Completion of the proofs 90 6.9 Equidistribution in isometric actions 91 Chapter 7. Volume estimates and volume regularity 93 7.1 Admissibility of standard averages 93 7.2 Convolution arguments 98 7.3 Admissible, well-balanced, and boundary-regular families 101 7.4 Admissible sets on principal homogeneous spaces 105 7.5 Tauberian arguments and Holder continuity 107 Chapter 8. Comments and complements 113 8.1 Lattice point-counting with explicit error term 113 8.2 Exponentially fast convergence versus equidistribution 115 8.3 Remark about balanced sets 116 Bibliography 117 Index 121

Read more less

Book SynopsisStudies the intersection cohomology of the Shimura varieties associated to unitary groups of any rank over Q. The author also uses the method developed by Langlands, Ihara, and Kottwitz, which is to compare the Grothendieck-Lefschetz fixed point formula and the Arthur-Selberg trace formula.Trade Review"This book is a research monograph, yet the author takes care in recalling in detail the relevant notation and previous results instead of just referring to the literature. Also, explicit calculations are given, making the book readable not only for experts but also for interested advanced students."--Eva Viehmann, Mathematical ReviewsTable of ContentsPreface vii Chapter 1: The fixed point formula 1 Chapter 2: The groups 31 Chapter 3: Discrete series 47 Chapter 4: Orbital integrals at p 63 Chapter 5: The geometric side of the stable trace formula 79 Chapter 6: Stabilization of the fixed point formula 85 Chapter 7: Applications 99 Chapter 8: The twisted trace formula 119 Chapter 9: The twisted fundamental lemma 157 Appendix: Comparison of two versions of twisted transfer factors 189 Bibliography 207 Index 215

Read more less

Book SynopsisWeyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. This book proves foundational results about these series and develops their combinatorics.Table of Contents*FrontMatter, pg. i*Contents, pg. v*Preface, pg. vii*Chapter One. Type A Weyl Group Multiple Dirichlet Series, pg. 1*Chapter Two. Crystals and Gelfand-Tsetlin Patterns, pg. 10*Chapter Three. Duality, pg. 22*Chapter Four. Whittaker Functions, pg. 26*Chapter Five. Tokuyama's Theorem, pg. 31*Chapter Six. Outline of the Proof, pg. 36*Chapter Seven. Statement B Implies Statement A, pg. 51*Chapter Eight. Cartoons, pg. 54*Chapter Nine. Snakes, pg. 58*Chapter Ten. Noncritical Resonances, pg. 64*Chapter Eleven. Types, pg. 67*Chapter Twelve. Knowability, pg. 74*Chapter Thirteen. The Reduction to Statement D, pg. 77*Chapter Fourteen. Statement E Implies Statement D, pg. 87*Chapter Fifteen. Evaluation of LAMBDAGAMMA and LAMBDADELTA, and Statement G, pg. 89*Chapter Sixteen. Concurrence, pg. 96*Chapter Seventeen. Conclusion of the Proof, pg. 104*Chapter Eighteen. Statement B and Crystal Graphs, pg. 108*Chapter Nineteen. Statement B and the Yang-Baxter Equation, pg. 115*Chapter Twenty. Crystals and p-adic Integration, pg. 132*Bibliography, pg. 143*Notation, pg. 149*Index, pg. 155

Read more less

Book SynopsisIn the earlier monograph Pseudo-reductive Groups, Brian Conrad, Ofer Gabber, and Gopal Prasad explored the general structure of pseudo-reductive groups. In this new book, Classification of Pseudo-reductive Groups, Conrad and Prasad go further to study the classification over an arbitrary field. An isomorphism theorem proved here determines the autoTrade Review"This book is beautiful and will be at the origin of many advances in the general theory of arbitrary algebraic groups."--Bertrand Remy, MathSciNetTable of Contents*Frontmatter, pg. i*Contents, pg. v*1. Introduction, pg. 1*2. Preliminary notions, pg. 15*3. Field-theoretic and linear-algebraic invariants, pg. 28*4. Central extensions and groups locally of minimal type, pg. 57*5. Universal smooth k-tame central extension, pg. 66*6. Automorphisms, isomorphisms, and Tits classification, pg. 79*7. Constructions with regular degenerate quadratic forms, pg. 108*8. Constructions when PHI has a double bond, pg. 138*9. Generalization of the standard construction, pg. 171*A. Pseudo-isogenies, pg. 181*B. Clifford constructions, pg. 187*C. Pseudo-split and quasi-split forms, pg. 206*D. Basic exotic groups of type F4 of relative rank 2, pg. 230*Bibliography, pg. 239*Index, pg. 241

Read more less

Book SynopsisTrade Review"Fascinating... Chamberland offers enticing explanations that will leave readers hungry to know more. This wonderful book never loses its focus or momentum."--Publishers Weekly "[B]oth amateur and professional mathematicians alike will find new items of interest here... [A] welcome, splendid, fruitful addition to my math bookshelf."--Math Tango blog "The collection is outright delightful. It will agitate the minds of students and shake the sense of know-all off many a professional and most of the amateurs."--Alexander Bogomolny, Cut the Knot blog "Boring deep into the innocuous-looking number one, Chamberland opens an unexpected entry point into a dizzying maze of infinities... A bracing mathematical adventure."--Booklist "The exotics like pi and e have gotten their share of attention in the world of popular mathematical writing. Now it's time to give proper attention to the integers 1 through 9... [Single Digits] is consistently entertaining and well-written."--MAA Reviews "Chamberland takes readers on a fascinating exploration of small numbers, from one to nine, looking at their history, applications, and connections to various areas of mathematics, including number theory, geometry, chaos theory, numerical analysis, and mathematical physics... Appealing to high-school and college students, professional mathematicians, and those mesmerized by patterns, this book shows that single digits offer a plethora of possibilities that readers can count on."--DVD, Lunar and Planetary Information Bulletin "Chamberland makes this an entertaining and historical exposition, using wit and humor throughout."--Math Horizons "To put it simply, this book is a delight. Chamberland has assembled a fascinating collection of vignettes, each tied to a digit from one to nine, that inform, entertain, and intrigue... This wide spectrum of ideas is consistently interesting, and the author's skill in mining each nugget is worthy of great respect."--Choice "The range of topics included virtually guarantees that any reader will find new and unfamiliar material to enjoy... [Single Digits] is a very enjoyable book which, at many points, makes some very deep mathematics quite accessible. Highly recommended."--Keith Johnson, CMS Notes "For instructors of math courses of all levels, the vignettes in Single Digits can provide a very readable introduction or jumping-off point for discussions and projects... In an introductory group theory course, it would be a good exercise for students to consider perfect riffle shuffles in decks of size other than 52. Finally, a statistics class collecting and analyzing real-world data sets could consider whether Benford's Law applies in their situation."--Matthew Welz, MAA Focus "I highly recommend Single Digits: In Praise of Small Numbers. It would be a fine addition to any high school or math department library. As a carefully curated set of interesting topics, it would serve as a good place to start exploring the ocean of ideas in mathematics."--Bruce Cohen, NCTMTable of Contents*Frontmatter, pg. i*Contents, pg. v*Preface, pg. xi*Chapter 1. The Number One, pg. 1*Chapter 2. The Number Two, pg. 24*Chapter 3. The Number Three, pg. 69*Chapter 4. The Number Four, pg. 111*Chapter 5. The Number Five, pg. 132*Chapter 6. The Number Six, pg. 156*Chapter 7. The Number Seven, pg. 170*Chapter 8. The Number Eight, pg. 191*Chapter 9. The Number Nine, pg. 205*Chapter 10. Solutions, pg. 216*Further reading, pg. 219*Credits for illustrations, pg. 223*Index, pg. 225

Read more less

Book SynopsisTrade Review"[Berkeley lectures on p-adic] represents a new beginning advancing p-adic geometry and its relation to these other paramount areas. It should be treated now as a ‘must have’ in any aspiring p-adic arithmetic geometer’s library and a critical resource for all researchers in the field."---Lance Edward Miller, MathSciNet

Read more less

a huge range and FREE tracked UK delivery on ALL orders.

Read more less

Book SynopsisContains papers presented at a conference held in April 2007 at the CRM in Montreal honouring the remarkable contributions of John McKay. This title features the papers that cover a wide range of topics, including group theory, symmetries, modular functions, and geometry, with focus on 2 areas: 'Monstrous Moonshine' and the 'McKay Correspondence'.

Read more less

Book SynopsisIn these volumes, a reader will find all of John Tate's published mathematical papers-spanning more than six decades-enriched by new comments made by the author. Included also is a selection of his letters. His letters give us a close view of how he works and of his ideas in process of formation.Table of Contents Part I: Fourier analysis in number fields and Hecke's zeta-functions by J. T. Tate A note on finite ring extensions by E. Artin and J. T. Tate On the relation between extremal points of convex sets and homomorphisms of algebras by J. Tate Genus change in inseparable extensions of function fields by J. Tate On Chevalley's proof of Luroth's theorem by S. Lang and J. Tate The higher dimensional cohomology groups of class field theory by J. Tate The cohomology groups of algebraic number fields by J. T. Tate On the Galois cohomology of unramified extensions of function fields in one variable by Y. Kawada and J. Tate On the characters of finite groups by R. Brauer and J. Tate Homology of Noetherian rings and local rings by J. Tate WC-groups over $p$-adic fields by J. Tate On the inequality of Castelnuovo-Severi by E. Artin and J. Tate On the inequality of Castelnuovo-Severi, and Hodge's theorem by J. Tate Principal homogeneous spaces over abelian varieties by S. Lang and J. Tate Principal homogeneous spaces for abelian varieties by J. Tate A different with an odd class by A. Frohlich, J.-P. Serre, and J. Tate Nilpotent quotient groups by J. Tate Duality theorems in Galois cohomology over number fields by J. Tate Ramification groups of local fields by S. Sen and J. Tate Formal complex multiplication in local fields by J. Lubin and J. Tate Algebraic cycles and poles of zeta functions by J. T. Tate Elliptic curves and formal groups by J. Lubin, J. Serre, and J. Tate On the conjectures of Birch and Swinnerton-Dyer and a geometric analog by J. Tate Formal moduli for one-parameter formal Lie groups by J. Lubin and J. Tate The cohomology groups of tori in finite Galois extensions of number fields by J. Tate Global class field theory by J. T. Tate Endomorphisms of abelian varieties over finite fields by J. Tate The rank of elliptic curves by J. T. Tate and I. R. Safarevic Residues of differentials on curves by J. Tate $p$-divisible groups by J. T. Tate The work of David Mumford by J. Tate Classes d'isogenie des varietes abeliennes sur un corps fini (d'apres T. Honda) by J. Tate Good reduction of abelian varieties by J.-P. Serre and J. Tate Group schemes of prime order by J. Tate and F. Oort Symbols in arithmetic by J. Tate Rigid analytic spaces by J. Tate The Milnor ring of a global field by H. Bass and J. Tate Appendix by H. Bass and J. Tate Letter from Tate to Iwasawa on a relation between $K_2$ and Galois cohomology by J. Tate Points of order 13 on elliptic curves by B. Mazur and J. Tate The arithmetic of elliptic curves by J. T. Tate The 1974 Fields Medals (I): An algebraic geometer by J. Tate Algorithm for determining the type of a singular fiber in an elliptic pencil by J. Tate Letters by J. Tate

Read more less

Book SynopsisAn exciting approach to the history and mathematics of number theory . . . the author's style is totally lucid and very easy to read . . .the result is indeed a wonderful story. Mathematical Reviews Written in a unique and accessible style for readers of varied mathematical backgrounds, the Second Edition of Primes of the Form p = x2+ ny2 details the history behind how Pierre de Fermat's work ultimately gave birth to quadratic reciprocity and the genus theory of quadratic forms. The book also illustrates how results of Euler and Gauss can be fully understood only in the context of class field theory, and in addition, explores a selection of the magnificent formulas of complex multiplication. Primes of the Form p = x2 + ny2, Second Edition focuses on addressing the question of when a prime p is of the form x2<Table of ContentsPreface to the First Edition ixPreface to the Second Edition xiNotation xiiiIntroduction 1Chapter One: From Fermat to GaussChapter Two: Class Field TheoryChapter Three: Complex MultiplicationChapter Four: Additional TopicsRefrencesAdditional ReferencesIndex

Read more less

Book SynopsisVolume II provides an advanced approach to the extended gibonacci family, which includes Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas, Vieta, Vieta-Lucas, and Chebyshevpolynomials of both kinds. This volume offers a uniquely unified, extensive, and historical approach that will appeal to both students and professional mathematicians. As in Volume I, Volume II focuses on problem-solving techniques such as pattern recognition;conjecturing; proof-techniques, and applications. It offers a wealth of delightful opportunities toexplore and experiment, as well as plentiful material for group discussions, seminars, presentations, and collaboration. In addition, the material covered in this book promotes intellectual curiosity, creativity, and ingenuity. Volume II features: A wealth of examples, applications, and exercises of varying degrees of difficulty and sophistication. Numerous combinatorial and graph-theoretic proofs and techniques. A uniquely thorough discussTable of ContentsList of Symbols xiii Preface xv 31. Fibonacci and Lucas Polynomials I 1 31.1. Fibonacci and Lucas Polynomials 3 31.2. Pascal’s Triangle 18 31.3. Additional Explicit Formulas 22 31.4. Ends of the Numbers ln 25 31.5. Generating Functions 26 31.6. Pell and Pell–Lucas Polynomials 27 31.7. Composition of Lucas Polynomials 33 31.8. De Moivre-like Formulas 35 31.9. Fibonacci–Lucas Bridges 36 31.10. Applications of Identity (31.51) 37 31.11. Infinite Products 48 31.12. Putnam Delight Revisited 51 31.13. Infinite Simple Continued Fraction 54 32. Fibonacci and Lucas Polynomials II 65 32.1. Q-Matrix 65 32.2. Summation Formulas 67 32.3. Addition Formulas 71 32.4. A Recurrence for n2 76 32.5. Divisibility Properties 82 33. Combinatorial Models II 87 33.1. A Model for Fibonacci Polynomials 87 33.2. Breakability 99 33.3. A Ladder Model 101 33.4. A Model for Pell–Lucas Polynomials: Linear Boards 102 33.5. Colored Tilings 103 33.6. A New Tiling Scheme 104 33.7. A Model for Pell–Lucas Polynomials: Circular Boards 107 33.8. A Domino Model for Fibonacci Polynomials 114 33.9. Another Model for Fibonacci Polynomials 118 34. Graph-Theoretic Models II 125 34.1. Q-Matrix and Connected Graph 125 34.2. Weighted Paths 126 34.3. Q-Matrix Revisited 127 34.4. Byproducts of the Model 128 34.5. A Bijection Algorithm 136 34.6. Fibonacci and Lucas Sums 137 34.7. Fibonacci Walks 140 35. Gibonacci Polynomials 145 35.1. Gibonacci Polynomials 145 35.2. Differences of Gibonacci Products 159 35.3. Generalized Lucas and Ginsburg Identities 174 35.4. Gibonacci and Geometry 181 35.5. Additional Recurrences 184 35.6. Pythagorean Triples 188 36. Gibonacci Sums 195 36.1. Gibonacci Sums 195 36.2. Weighted Sums 206 36.3. Exponential Generating Functions 209 36.4. Infinite Gibonacci Sums 215 37. Additional Gibonacci Delights 233 37.1. Some Fundamental Identities Revisited 233 37.2. Lucas and Ginsburg Identities Revisited 238 37.3. Fibonomial Coefficients 247 37.4. Gibonomial Coefficients 250 37.5. Additional Identities 260 37.6. Strazdins’ Identity 264 38. Fibonacci and Lucas Polynomials III 269 38.1. Seiffert’s Formulas 270 38.2. Additional Formulas 294 38.3. Legendre Polynomials 314 39. Gibonacci Determinants 321 39.1. A Circulant Determinant 321 39.2. A Hybrid Determinant 323 39.3. Basin’s Determinant 333 39.4. Lower Hessenberg Matrices 339 39.5. Determinant with a Prescribed First Row 343 40. Fibonometry II 347 40.1. Fibonometric Results 347 40.2. Hyperbolic Functions 356 40.3. Inverse Hyperbolic Summation Formulas 361 41. Chebyshev Polynomials 371 41.1. Chebyshev Polynomials Tn(x) 372 41.2. Tn(x) and Trigonometry 384 41.3. Hidden Treasures in Table 41.1 386 41.4. Chebyshev Polynomials Un(x) 396 41.5. Pell’s Equation 398 41.6. Un(x) and Trigonometry 399 41.7. Addition and Cassini-like Formulas 401 41.8. Hidden Treasures in Table 41.8 402 41.9. A Chebyshev Bridge 404 41.10. Tn and Un as Products 405 41.11. Generating Functions 410 42. Chebyshev Tilings 415 42.1. Combinatorial Models for Un 415 42.2. Combinatorial Models for Tn 420 42.3. Circular Tilings 425 43. Bivariate Gibonacci Family I 429 43.1. Bivariate Gibonacci Polynomials 429 43.2. Bivariate Fibonacci and Lucas Identities 430 43.3. Candido’s Identity Revisited 439 44. Jacobsthal Family 443 44.1. Jacobsthal Family 444 44.2. Jacobsthal Occurrences 450 44.3. Jacobsthal Compositions 452 44.4. Triangular Numbers in the Family 459 44.5. Formal Languages 468 44.6. A USA Olympiad Delight 480 44.7. A Story of 1, 2, 7, 42, 429,…483 44.8. Convolutions 490 45. Jacobsthal Tilings and Graphs 499 45.1. 1 × n Tilings 499 45.2. 2 × n Tilings 505 45.3. 2 × n Tubular Tilings 510 45.4. 3 × n Tilings 514 45.5. Graph-Theoretic Models 518 45.6. Digraph Models 522 46. Bivariate Tiling Models 537 46.1. A Model for 𝑓n(x, y) 537 46.2. Breakability 539 46.3. Colored Tilings 542 46.4. A Model for ln(x, y) 543 46.5. Colored Tilings Revisited 545 46.6. Circular Tilings Again 547 47. Vieta Polynomials 553 47.1. Vieta Polynomials 554 47.2. Aurifeuille’s Identity 567 47.3. Vieta–Chebyshev Bridges 572 47.4. Jacobsthal–Chebyshev Links 573 47.5. Two Charming Vieta Identities 574 47.6. Tiling Models for Vn 576 47.7. Tiling Models for 𝑣n(x) 582 48. Bivariate Gibonacci Family II 591 48.1. Bivariate Identities 591 48.2. Additional Bivariate Identities 594 48.3. A Bivariate Lucas Counterpart 599 48.4. A Summation Formula for 𝑓2n(x, y) 600 48.5. A Summation Formula for l2n(x, y) 602 48.6. Bivariate Fibonacci Links 603 48.7. Bivariate Lucas Links 606 49. Tribonacci Polynomials 611 49.1. Tribonacci Numbers 611 49.2. Compositions with Summands 1, 2, and 3 613 49.3. Tribonacci Polynomials 616 49.4. A Combinatorial Model 618 49.5. Tribonacci Polynomials and the Q-Matrix 624 49.6. Tribonacci Walks 625 49.7. A Bijection between the Two Models 627 Appendix 631 A.1. The First 100 Fibonacci and Lucas Numbers 631 A.2. The First 100 Pell and Pell–Lucas Numbers 634 A.3. The First 100 Jacobsthal and Jacobsthal–Lucas Numbers 638 A.4. The First 100 Tribonacci Numbers 642 Abbreviations 644 Bibliography 645 Solutions to Odd-Numbered Exercises 661 Index 725

Read more less

Book SynopsisA. Prime Numbers.- B. Divisibility.- C. Additive Number Theory.- D. Diophantine Equations.- E. Sequences of Integers.- F. None of the Above.- Index of Authors Cited.- General Index.Trade ReviewFrom the reviews of the third edition: "This is the third edition of Richard Guy’s well-known problem book on number theory … . The earlier editions have served well in providing beginners as well as seasoned researchers in number theory with a good supply of problems. … many of the problems from earlier editions have been expanded with more up-to-date comments and remarks. … There is little doubt that a new generation of talented young mathematicians will make very good use of this book … ." (P. Shiu, The Mathematical Gazette, Vol. 89 (516), 2005)"The earlier editions of this book are among the most-opened books on the shelves of many practicing number theorists. The descriptions of state-of-the-art results on every topic and the extensive bibliographies in each section provide valuable ports of entry to the vast literature. A new and promising addition to this third edition is the inclusion of frequent references to entries in the Online encyclopedia of integer sequences at the end of each topic." (Greg Martin, Mathematical Reviews, Issue 2005 h)Table of ContentsPreface to the First Edition Preface to the Second Edition Preface to the Third Edition Glossary of Symbols A. Prime Numbers. A1. Prime values of quadratic functions. A2. Primes connected with factorials. A3. Mersenne primes. Repunits. Fermat numbers. Primes of shape k · 2n + 1. A4. The prime number race. A5. Arithmetic progressions of primes. A6. Consecutive primes in A.P. A7. Cunningham chains. A8. Gaps between primes. Twin primes. A9. Patterns of primes. A10. Gilbreath's conjecture. A11. Increasing and decreasing gaps. A12. Pseudoprimes. Euler pseudoprimes. Strong pseudoprimes. A13. Carmichael numbers. A14. 'Good' primes and the prime number graph. A15. Congruent products of consecutive numbers. A16. Gaussian primes. Eisenstein-Jacobi primes. A17. Formulas for primes. A18. The Erd½os-Selfridge classi.cation of primes. A19. Values of n making n - 2k prime. Odd numbers not of the form ±pa ± 2b. A20. Symmetric and asymmetric primes. B. Divisibility B1. Perfect numbers. B2. Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers. B3. Unitary perfect numbers. B4. Amicable numbers. B5. Quasi-amicable or betrothed numbers. B6. Aliquot sequences. B7. Aliquot cycles. Sociable numbers. B8. Unitary aliquot sequences. B9. Superperfect numbers. B10. Untouchable numbers. B11. Solutions of mó(m) = nó(n). B12. Analogs with d(n), ók(n). B13. Solutions of ó(n) = ó(n + 1). B14. Some irrational series. B15. Solutions of ó(q) + ó(r) = ó(q + r). B16. Powerful numbers. Squarefree numbers. B17. Exponential-perfect numbers B18. Solutions of d(n) = d(n + 1). B19. (m, n + 1) and (m+1, n) with same set of prime factors. The abc-conjecture. B20. Cullen and Woodallnumbers. B21. k · 2n + 1 composite for all n. B22. Factorial n as the product of n large factors. B23. Equal products of factorials. B24. The largest set with no member dividing two others. B25. Equal sums of geometric progressions with prime ratios. B26. Densest set with no l pairwise coprime. B27. The number of prime factors of n + k which don't divide n + i, 0 ¡Ü i < k. B28. Consecutive numbers with distinct prime factors. B29. Is x determined by the prime divisors of x + 1, x + 2,. . ., x + k? B30. A small set whose product is square. B31. Binomial coeffcients. B32. Grimm's conjecture. B33. Largest divisor of a binomial coeffcient. B34. If there's an i such that n - i divides _nk_. B35. Products of consecutive numbers with the same prime factors. B36. Euler's totient function. B37. Does ö(n) properly divide n - 1? B38. Solutions of ö(m) = ó(n). B39. Carmichael's conjecture. B40. Gaps between totatives. B41. Iterations of ö and ó. B42. Behavior of ö(ó(n)) and ó(ö(n)). B43. Alternating sums of factorials. B44. Sums of factorials. B45. Euler numbers. B46. The largest prime factor of n. B47. When does 2a -2b divide na - nb? B48. Products taken over primes. B49. Smith numbers. C. Additive Number Theory C1. Goldbach's conjecture. C2. Sums of consecutive primes. C3. Lucky numbers. C4. Ulam numbers. C5. Sums determining members of a set. C6. Addition chains. Brauer chains. Hansen chains. C7. The money-changing problem. C8. Sets with distinct sums of subsets. C9. Packing sums of pairs. C10. Modular di.erence sets and error correcting codes. C11. Three-subsets with distinct sums. C12. The postage stamp problem. C13. The corresponding modular covering problem. Harmonious labelling of graphs. C14.

Read more less

Book SynopsisIntended for advanced level students in computer science and mathematics, this key text, now in a brand new edition, provides a survey of recent progress in primality testing and integer factorization, with implications for factoring based public key cryptography.Trade ReviewFrom the reviews of the second edition:"The well-written and self-contained second edition ‘is designed for a professional audience composed of researchers practitioners in industry.’ In addition, ‘this book is also suitable as a secondary text for graduate-level students in computer science, mathematics, and engineering,’ as it contains about 300 problems. … Overall … ‘this monograph provides a survey of recent progress in Primality Testing and Integer Factorization, with implications in factoring-based Public Key Cryptography.’" (Hao Wang, ACM Computing Reviews, April, 2009)“This is the second edition of a book originally published in 2004. … I used it as a reference in preparing lectures for an advanced cryptography course for undergraduates, and it proved to be a wonderful source for a general description of the algorithms. … the book will be a valuable addition to any good reference library on cryptography and number theory … . It contains descriptions of all the main algorithms, together with explanations of the key ideas behind them.” (S. C. Coutinho, SIGACT News, April, 2012)Table of ContentsPreface to the Second Edition.- Preface to the First Edition.- Number-Theoretic Preliminaries.- Problems in Number Theory. Divisibility Properties. Euclid's Algorithm and Continued Fractions. Arithmetic Functions. Linear Congruences. Quadratic Congruences. Primitive Roots and Power Residues. Arithmetic of Elliptic Curves. Chapter Notes and Further Reading.- Primality Testing and Prime Generation.- Computing with Numbers and Curves. Riemann Zeta and Dirichlet L Functions. Rigorous Primality Tests. Compositeness and Pseudoprimality Tests. Lucas Pseudoprimality Test. Elliptic Curve Primality Tests. Superpolynomial-Time Tests. Polynomial-Time Tests. Primality Tests for Special Numbers. Prime Number Generation. Chapter Notes and Further Reading.- Integer Factorization and Discrete Logarithms.- Introduction. Simple Factoring Methods. Elliptic Curve Method (ECM). General Factoring Congruence. Continued FRACtion Method (CFRAC). Quadratic Sieve (QS). Number Field Sieve (NFS). Quantum Factoring Algorithm. Discrete Logarithms. kth Roots. Elliptic Curve Discrete Logarithms. Chapter Notes and Further Reading.- Number-Theoretic Cryptography.- Public-Key Cryptography. RSA Cryptosystem. Rabin Cryptography. Quadratic Residuosity Cryptography. Discrete Logarithm Cryptography. Elliptic Curve Cryptography. Zero-Knowledge Techniques. Deniable Authentication. Non-Factoring Based Cryptography. Chapter Notes and Further Reading.- Bibliography.- Index.- About the Author.

Read more less

Book Synopsis​This book is a history of complex function theory from its origins to 1914, when the essential features of the modern theory were in place.Trade Review“There is much in this book that will educate, be appreciated by, and no doubt provoke mathematicians as well as historians of mathematics and of science. … It stands its ground as a scholarly treatise that fills many lacunae in the extant historical literature. It will surely provoke further debate and research. As a bonus, it comes filled with treasures for both the specialist and the novice.” (Tushar Das, MAA Reviews, July, 2015)“The book is devoted to the history of complex (analytic) function theory from its origins to 1914. … The book is highly recommended for historians of mathematics, mathematicians with historical interests, and everyone who is interested in complex function theory and its history. It offers a wealth of information that is well documented.” (Karl-Heinz Schlote, Mathematical Reviews, October, 2014)“This comprehensive, massively researched volume … is a detailed historical account of the development of analytic function theory in the 19th century, tracing its rise and ramification through that period up until about 1910. … It is a very dense and scholarly work, suitable for specialists. Summing Up: Recommended. Graduate students, researchers/faculty, and professionals/practitioners.” (D. Robbins, Choice, Vol. 51 (9), May, 2014)“This book is the first one devoted to the history of complex function theory. The authors present the rise of analytic function theory from its origins to 1914. … This book is of great interest and help, not only for mathematicians interested in complex function theory, but also for everyone who likes the history of mathematics.” (Agnieszka Wisniowska-Wajnryb, zbMATH, Vol. 1276, 2014)Table of ContentsList of Figures.- Introduction.- 1. Elliptic Functions.- 2. From real to complex.- 3. Cauch.- 4. Elliptic integrals.- 5. Riemann.- 6. Weierstrass.- 7. Differential equations.- 8. Advanced topics.- 9. Several variables.- 10. Textbooks.

Read more less

Book SynopsisOffers a hands-on transcendental approach to differential Galois theory, based on the Riemann-Hilbert correspondence. Along the way, it provides a smooth, down-to-earth introduction to algebraic geometry, category theory and tannakian duality. A large variety of examples, exercises, and theoretical constructions offers an accessible entry into this exciting area.Trade ReviewJacques Sauloy's book is an introduction to differential Galois theory, an important area of mathematics having different powerful applications (for example, to the classical problem of integrability of dynamical systems in mechanics and physics)...Sauloy offers an alternative approach to the subject which is based on the monodromy representation...Enriching the understanding of differential Galois theory, this point of view also brings new solutions, which makes the book especially valuable...There are a lot of nice exercises, both inside and at the end of each chapter." — Renat R. Gontsov, Mathematical Reviews"The book is an elementary introduction to the differential Galois theory and is intended for undergraduate students of mathematical departments. It is not overloaded with redundant definitions, constructs and results. Everything that is minimally necessary for understanding the whole presentation is given in full. The reader can find the rest [of the] details from a well-designed references system. And at the same time, the book contains quite a lot of carefully selected examples and exercises." — Mykola Grygorenko, Zentralblatt MATH"It's an excellent book about a beautiful and deep subject...There are loads of exercises, and I think the book is very well-paced, as well as very clearly written. It's a fabulous entry in the AMS GSM series." — Michael Berg, MAA ReviewsTable of Contents Part 1. A quick introduction to complex analytic functions: The complex exponential function Power series Analytic functions The complex logarithm From the local to the global Part 2. Complex linear differential equations and their monodromy: Two basic equations and their monodromy Linear complex analytic differential equations A functorial point of view on analytic continuation: Local systems Part 3. The Riemann-Hilbert correspondence: Regular singular points and the local Riemann-Hilbert correspondence Local Riemann-Hilbert correspondence as an equivalence of categories Hypergeometric series and equations The global Riemann-Hilbert correspondence Part 4. Differential Galois theory: Local differential Galois theory The local Schlesinger density theorem The universal (Fuchsian local) Galois group The universal group as proalgebraic hull of the fundamental group Beyond local Fuchsian differential Galois theory Appendix A. Another proof of the surjectivity of $\mathrm{exp}:\mathrm{Mat}_n(\mathbf{C})\rightarrow \mathrm{GL}_n(\mathbf{C})$ Appendix B. Another construction of the logarithm of a matrix Appendix C. Jordan decomposition in a linear algebraic group Appendix D. Tannaka duality without schemes Appendix E. Duality for diagonalizable algebraic groups Appendix F. Revision problems Bibliography Index.

Read more less

Book SynopsisContains the proceedings of the Arizona Winter School 2016, held in March 2016 at The University of Arizona. The School provided a unique opportunity to introduce graduate students to analytic methods in arithmetic geometry.Table of Contents A. C. Cojocaru, Primes, elliptic curves and cyclic groups H. A. Helfgott, Growth and expansion in algebraic groups over finite fields E. Fouvry, E. Kowalski, P. Michel, and W. Sawin, Lectures on applied $\ell$-adic cohomology A. V. Sutherland, Sato-Tate distributions.

Read more less

Book SynopsisPresents original research articles covering a large range of topics, including weight enumerators for codes, function field analogs of the Brauer-Siegel theorem, the computation of cohomological invariants of curves, the trace distributions of algebraic groups, and applications of the computation of zeta functions of curves.Table of Contents J. D. Achter and E. W. Howe, Hasse-Witt and Cartier-Manin matrices: A warning and a request M. Hindry, Analogues of Brauer-Siegel theorem in arithmetic geometry J. Javanpeykar and J. Voight, The Belyi degree of a curve is computable N. Kaplan, Weight enumerators of Reed-Muller codes from cubic curves and their duals G. Lachaud, The distribution of the trace in the compact group of type $G_2$ B. Malmskog, R. Pries, and C. Weir, The de Rham cohomology of the Suzuki curves F. Pazuki, Decompositions en hauteurs locales B. Poonen, Using zeta functions to factor polynomials over finite fields J. Sijsling, Canonical models of arithmetic $(1;\infty)$-curves A. V. Sutherland and J. F. Voloch, Maps between curves and arithmetic obstructions.

Read more less

Book SynopsisContains the proceedings of the conference Dynamics: Topology and Numbers, held in July 2018. The papers cover diverse fields of mathematics with a unifying theme of relation to dynamical systems. These include arithmetic geometry, flat geometry, complex dynamics, graph theory, relations to number theory, and topological dynamics.Table of Contents L. Snoha, The life and mathematics of Sergii Kolyada I. Kolyada, A. Blokh, and L. Snoha, Recollections about Sergii Kolyada P. Moree, Sergiy and the MPIM Y. I. Manin and M. Marcolli, Homotopy types and geometries below ${\rm Spec}(\mathbb{Z})$ A. Fel'shtyn and M. Zietek, Dynamical zeta functions of Riedemeister type and representations spaces O. Jenkinson and M. Pollicott, Rigorous dimension estimates for Cantor sets arising in Zaremba theory P. Colognese and M. Pollicott, Volume growth for infinite graphs and translation surfaces J. Byszewski, G. Cornelissen, M. Houben, and L. Van Der Meijden, Dynamically affine maps in positive characteristic S. Kolyada, M. Misiurewicz, and L. Snoha, Special $\alpha$-limit sets E. Shi and X. Ye, Equicontinuity of minimal sets for amenable group actions on dendrites E. Akin, E. Glasner, and B. Weiss, On weak rigidity and weakly mixing enveloping semigroups A. Ganguly and A. Ghosh, The inhomogenous Sprindzhuk conjecture over a local field of positive characteristic A. Blokh, L. Oversteegen, and V. Timorin, Dynamical generation of parameter laminations P. Oprocha, T. Yu, and Guohua Zhang, Multi-sensitivity, multi-transitivity and $\delta$-transitivity R. Sharp, Convergence of zeta functions for amenable group extensions of shifts S. Bezuglyi and O. Karpel, Invariant measures for Cantor dynamical systems M. Kapovich, Periods of abelian differentials and dynamics J. Riedl and D. Schleicher, Crossed renormalization of quadratic polynomials.

Read more less

Book SynopsisProvides an introduction to number theory at the undergraduate level, emphasizing geometric aspects of the subject. The geometric approach is exploited to explore in depth the classical topic of quadratic forms with integer coefficients, a central topic of the book.Table of Contents A preview The Farey diagram Continued fractions Symmetries of the Farey diagram Quadratic forms Classification of quadratic forms Representations by quadratic forms The class group for quadratic forms Quadratic fields Tables Glossary of nonstandard terminology Bibliography Index

Read more less

Book SynopsisThe polynomials studied in this book take their origin in number theory. The authors show how, by drawing simple pictures, one can prove some long-standing conjectures and formulate new ones. The theory presented here touches upon many different fields of mathematics.Table of Contents Introduction. Dessins d'enfants: From polynomials through Belyi functions to weighted trees. Existence theorem. Recapitulation and perspective. Classification of unitrees. Computation of Davenport-Zannier pairs for unitrees. Primitive monodromy groups of weighted trees. Trees with primitive monodromy groups. A zoo of examples and constructions. Diophantine invariants. Enumeration. What remains to be done. Bibliography. Index.

Read more less

Book SynopsisPresents a dialogue between a professor and eight students in a summer problem solving camp and allows for a conversational approach to the problems as well as some mathematical humour and a few non-mathematical digressions. The problems have been selected for their entertainment value, elegance, trickiness, and unexpectedness.Table of Contents The first day Polynomials Base mathematics A mysterious visitor Set theory Triangles Independence day Independence aftermath Amanda An aesthetical error Miraculous cancellation Probability theory Geometry Hodegepodge Self-referential mathematics All good things must come to an end Bibliography Index.

Read more less